From 5586d14a2f2294fee20bd84d20b2efa05bede2d6 Mon Sep 17 00:00:00 2001 From: rbds Date: Wed, 25 Nov 2015 20:33:42 -0500 Subject: [PATCH 01/61] start project --- Double_Pendulum_Problem.ipynb | 275 ++++++++++++++++++++++++++++++++++ figures/diagram.png | Bin 0 -> 9380 bytes numericalmoocstyle.css | 142 ++++++++++++++++++ 3 files changed, 417 insertions(+) create mode 100644 Double_Pendulum_Problem.ipynb create mode 100644 figures/diagram.png create mode 100644 numericalmoocstyle.css diff --git a/Double_Pendulum_Problem.ipynb b/Double_Pendulum_Problem.ipynb new file mode 100644 index 0000000..8185832 --- /dev/null +++ b/Double_Pendulum_Problem.ipynb @@ -0,0 +1,275 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Dynamics of a Double Pendulum" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In this module, we will study the dynamics of a double pendulum. We first derive the equations of motion from the Lagrangian. Next, we take a look at three different numerical integration schemes: Euler's method, Runge-Kutta, and a symplectic integrator. We will investigate how the initial conditions affect the behavior of the system, as well as some of the sources of error in these simulations. Finally, we will look at the properties of the system in phase space using phase plots and Poincare sections." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Background\n", + "\n", + "The double pendulum problem is a simple system which can produce surprisingly complex movements. It is a chaotic system, meaning it is unpredictable, and there is no closed form solution to the motion of the two masses. This is a frequent example problem in the study of dynamic systems, nonlinear controls, and mechanics.\n", + "\n", + "In this notebook we will be treating the two rods as massless and ignoring the effects of friction. For a treatment of this problem without these assumptions, see: \n", + "\n", + "See the image below for the definition of constants." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "![Image](figures/diagram.png)\n", + "#### Figure 1. Diagram of Double Pendulum System" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Equations of Motion\n", + "\n", + "To derive the equations of motion, we will start with the Lagrangian. This is defined by:\n", + "\n", + "$$\\begin{equation}\n", + "L = T- U\n", + "\\end{equation}$$\n", + "\n", + "where T is the total Kinetic Energy in the system, and U is the total Potential Energy. The equations of motion are then found using the Lagrange Equation:\n", + "\n", + "$$\\begin{equation}\n", + "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x_i}}\\right) - \\frac{\\partial L}{\\partial x_i} = 0\n", + "\\end{equation}$$\n", + "\n", + "\n", + "So the equations of motion for the system become:\n", + "$$\\begin{eqnarray}\n", + "L &=& C_L S \\times \\frac{1}{2} \\rho v^2 \\\\\n", + "D &=& C_D S \\times \\frac{1}{2} \\rho v^2\n", + "\\end{eqnarray}$$\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# This cell loads the style of the notebook, which is taken from the \n", + "# Numerical Methods in Python Course: http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about\n", + "\n", + "from IPython.core.display import HTML\n", + "css_file = 'numericalmoocstyle.css'\n", + "HTML(open(css_file, \"r\").read())" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.4.3" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +} diff --git a/figures/diagram.png b/figures/diagram.png new file mode 100644 index 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zzxi^WzoGUamj;lhic|4UctrLi{{}jkvBH|tnxlfN-90?}sDA3!rhw4vI5kVNij%R- z5`w7on;^`9d|(ZlqEd|n%VF^K9P)c;B!m5Fj`0Q?b8rbl9f}}m!3KSSrvqO)556k{ zPXqVh4t#Ed{(P92IdlLD<+@)j^9xm1~F?$%Y!@$2QD1EJbP3z$Q0|(OV0RR91 literal 0 HcmV?d00001 diff --git a/numericalmoocstyle.css b/numericalmoocstyle.css new file mode 100644 index 0000000..16a3c5c --- /dev/null +++ b/numericalmoocstyle.css @@ -0,0 +1,142 @@ + + + + + + + From 19910bea0e07bb1506ba5b05c92053935163092c Mon Sep 17 00:00:00 2001 From: rbds Date: Sat, 28 Nov 2015 21:01:30 +0000 Subject: [PATCH 02/61] update derivation of EoM --- Double_Pendulum_Problem.ipynb | 21 +++++++++++++++------ 1 file changed, 15 insertions(+), 6 deletions(-) diff --git a/Double_Pendulum_Problem.ipynb b/Double_Pendulum_Problem.ipynb index 8185832..9632af2 100644 --- a/Double_Pendulum_Problem.ipynb +++ b/Double_Pendulum_Problem.ipynb @@ -47,7 +47,19 @@ "L = T- U\n", "\\end{equation}$$\n", "\n", - "where T is the total Kinetic Energy in the system, and U is the total Potential Energy. The equations of motion are then found using the Lagrange Equation:\n", + "where $T$ is the total Kinetic Energy in the system, and $U$ is the total Potential Energy. To calculate potential energy:\n", + "\n", + "$$\\begin{equation}\n", + "U = m_1g(-y_1) + m_2g(-y_2)\n", + "\\end{equation}$$\n", + "\n", + "$$\\begin{equation}\n", + "U = m_1g(l_1cos(\\theta_1) + m_2g(l_1cos\\theta_1 + l_2cos\\theta_2)\n", + "\\end{equation}$$\n", + "\n", + "\n", + "\n", + "The equations of motion are then found using the Lagrange Equation:\n", "\n", "$$\\begin{equation}\n", "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x_i}}\\right) - \\frac{\\partial L}{\\partial x_i} = 0\n", @@ -55,10 +67,7 @@ "\n", "\n", "So the equations of motion for the system become:\n", - "$$\\begin{eqnarray}\n", - "L &=& C_L S \\times \\frac{1}{2} \\rho v^2 \\\\\n", - "D &=& C_D S \\times \\frac{1}{2} \\rho v^2\n", - "\\end{eqnarray}$$\n" + "|" ] }, { @@ -267,7 +276,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.4.3" + "version": "3.5.0" } }, "nbformat": 4, From 3a1d4c84dc778be2b0037e17fe6a86c9fea5967d Mon Sep 17 00:00:00 2001 From: rbds Date: Sun, 29 Nov 2015 19:18:20 -0500 Subject: [PATCH 03/61] derive EoM --- Double_Pendulum_Problem.ipynb | 126 +++++++++++++++++++++++++++++----- 1 file changed, 110 insertions(+), 16 deletions(-) diff --git a/Double_Pendulum_Problem.ipynb b/Double_Pendulum_Problem.ipynb index 9632af2..f7a2e9f 100644 --- a/Double_Pendulum_Problem.ipynb +++ b/Double_Pendulum_Problem.ipynb @@ -41,52 +41,146 @@ "source": [ "## Equations of Motion\n", "\n", - "To derive the equations of motion, we will start with the Lagrangian. This is defined by:\n", + "You are encouraged to follow this derivation ON PAPER on your own. The following is a method that will work in nearly any situation to find equations of motion, often working well where a force analysis becomes difficult. This is good practice for anyone studying mechanics. To derive the equations of motion, we will start with the Lagrangian. This is defined by:\n", "\n", "$$\\begin{equation}\n", "L = T- U\n", "\\end{equation}$$\n", "\n", - "where $T$ is the total Kinetic Energy in the system, and $U$ is the total Potential Energy. To calculate potential energy:\n", + "where $T$ is the total Kinetic Energy in the system, and $U$ is the total Potential Energy. Potential energy is calculated as:\n", "\n", "$$\\begin{equation}\n", "U = m_1g(-y_1) + m_2g(-y_2)\n", "\\end{equation}$$\n", "\n", "$$\\begin{equation}\n", - "U = m_1g(l_1cos(\\theta_1) + m_2g(l_1cos\\theta_1 + l_2cos\\theta_2)\n", + "U = -m_1g(l_1cos(\\theta_1) - m_2g(l_1cos\\theta_1 + l_2cos\\theta_2)\n", "\\end{equation}$$\n", "\n", + "Kinetic Energy is calculated as:\n", "\n", + "$$\\begin{equation}\n", + "T = \\frac{1}{2}mv_1^2+\\frac{1}{2}mv_2^2\n", + "\\end{equation}$$\n", "\n", - "The equations of motion are then found using the Lagrange Equation:\n", + "$$\\begin{equation}\n", + "T = \\frac{1}{2}m_1(l_1\\dot\\theta_1)^2 + \\frac{1}{2}m_2(l_1\\dot\\theta_1 + l_2\\dot\\theta_2)^2\n", + "\\end{equation}$$\n", "\n", "$$\\begin{equation}\n", - "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x_i}}\\right) - \\frac{\\partial L}{\\partial x_i} = 0\n", + "T = \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2+\\frac{1}{2}m_2\\left(2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+l_2^2\\dot\\theta_2^2\\right)\n", "\\end{equation}$$\n", "\n", "\n", - "So the equations of motion for the system become:\n", - "|" + "So the Lagrangian quantity becomes:\n", + "$$\\begin{equation}\n", + "L = \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2+m_2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+(m_1+m_2)l_1gcos\\theta_1 + m_2l_2gcos\\theta_2\n", + "\\end{equation}$$\n" ] }, { - "cell_type": "code", - "execution_count": null, + "cell_type": "markdown", "metadata": { "collapsed": true }, - "outputs": [], - "source": [] + "source": [ + "The equations of motion are then found using the Lagrange Equation:\n", + "\n", + "$$\\begin{equation}\n", + "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x_i}}\\right) - \\frac{\\partial L}{\\partial x_i} = 0\n", + "\\end{equation}$$\n", + "\n", + "\n", + "So the equation of motion for $\\theta_1$ is calculated by the following steps:\n", + "\n", + "$$\\begin{equation}\n", + "\\frac{\\partial L}{\\partial \\dot{\\theta_1}} = (m_1+m_2)l_1^2\\dot\\theta_1+m_2l_1l_2\\dot\\theta_2cos(\\theta_2-\\theta_1)\n", + "\\end{equation}$$\n", + "\n", + "$$\\begin{equation}\n", + "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{\\theta_1}}\\right) = (m_1+m_2)l_1^2\\ddot\\theta_1+m_2l_1l_2\\ddot\\theta_2cos(\\theta_2-\\theta_1)-m_2l_1l_2\\dot\\theta_2^2sin(\\theta_2-\\theta_1)+m_2l_1l_2\\dot\\theta_1\\dot\\theta_2sin(\\theta_2-\\theta_1)\n", + "\\end{equation}$$\n", + "\n", + "$$\\begin{equation}\n", + "\\frac{\\partial L}{\\partial \\theta_1} = m_2l_1l_2\\dot\\theta_1\\dot\\theta_2sin(\\theta_2-\\theta_1)+(m_1+m_2)gl_1sin\\theta_1\n", + "\\end{equation}$$\n", + "\n", + "Some of these terms will cancel out, and we are left with a final equation of motion:\n", + "\n", + "$$\\begin{equation}\n", + "0 = (m_1+m_2)l_1^2\\ddot\\theta_1+m_2l_1l_2\\ddot\\theta_2cos(\\theta_2-\\theta_1)-m_2l_1l_2\\dot\\theta_2^2sin(\\theta_2-\\theta_1)+(m_1+m_2)gl_1sin\\theta_1\n", + "\\end{equation}$$\n", + "\n", + "We will also need the equation of motion for $\\theta_2$. This can be derived using the same process. It should work out to the following:\n", + "\n", + "$$\\begin{equation}\n", + "0 = m_2l_2^2\\ddot\\theta_2+m_2l_1l_2\\ddot\\theta_1cos(\\theta_2-\\theta_1) - m_2l_2l_1\\dot\\theta_1^2sin(\\theta_2-\\theta_1)+ l_2m_2gsin\\theta_2\n", + "\\end{equation}$$" + ] }, { - "cell_type": "code", - "execution_count": null, + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Problem setup\n", + "\n", + "Notice that these equations of motion are implicit equations for $\\ddot\\theta_1$ and $\\ddot\\theta_2$. We have talked about implicit equations in this class, and we solve them by forming a system of equations. In this case, we will need to solve the equations of motion at each time step. We will characterize our system in state space using the angle and angular velocity of each pendulum. So our state becomes:\n", + "\n", + "$$\\begin{eqnarray}\n", + "\\vec{x}(t) = \\begin{pmatrix} \\theta_1\\\\ \\theta_2\\\\ \\dot\\theta_1 \\\\ \\dot\\theta_2 \\end{pmatrix}\n", + "\\end{eqnarray}$$\n", + "\n", + "The equations of motion can be rewritten as a system of equations:\n", + "\n", + "$$ A\\dot x=f(x)$$\n", + "\n", + "where \n", + "\n", + "\\begin{equation}\n", + "A = \n", + "\\left[\\begin{array}{c}\n", + "1 &0 &0 &0 \\\\\n", + "0 &1 &0 &0 \\\\\n", + "0 &0 &(m_1+m_2)l_1^2 &m_2l_1l_2cos(x_2-x_1) \\\\\n", + "0 &0 &m_2l_1l_2cos(x_2-x_1) &m_2l_2^2\n", + "\\end{array}\\right]\n", + "\\end{equation}\n", + "\n", + "and\n", + "\n", + "$$\\begin{eqnarray}\n", + "f(\\vec{x}(t)) = \\begin{pmatrix} x_3\\\\ x_4\\\\ m_2l_1l_2x_4^2sin(x_2-x_1) -(m_1+m_2)gl_1sin(x_1)\\\\ m_2l_1l_2x_3^2sin(x_2-x_1)-l_2m_2gsinx_2 \\end{pmatrix}\n", + "\\end{eqnarray}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Integration\n", + "\n", + "#### Euler's Method\n", + "\n", + "The first method we will use is Euler integration, covered in lesson 1 of the Numerical Methods course. This is the simplest form of numerical integration, but it can still be effective. Using our equations for $\\dot x$ and $f(x)$:\n", + "\n", + "$$x_i^{n+1} = x_i^n + h\\dot{x}_i^n$$\n", + "\n", + "where $h$ is the timestep.\n", + "\n", + "#### Runge-Kutta Method (RK4)\n", + "\n" + ] + }, + { + "cell_type": "markdown", "metadata": { "collapsed": true }, - "outputs": [], - "source": [] + "source": [ + "## Solve!\n", + "\n", + "Here we define functions for each of the integration methods, set our parameters and initial conditions, and then numerically integrate the motion of the pendulums. The plots below compare the different integration techniques." + ] }, { "cell_type": "code", @@ -276,7 +370,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.5.0" + "version": "3.4.3" } }, "nbformat": 4, From 0bd568b2913385ec0c7379f42e7822d443d21d6a Mon Sep 17 00:00:00 2001 From: rbds Date: Mon, 30 Nov 2015 22:20:52 +0000 Subject: [PATCH 04/61] start simple pendulum section --- Double_Pendulum_Problem.ipynb | 125 +++++++++++++++++++++++++++++++++- 1 file changed, 123 insertions(+), 2 deletions(-) diff --git a/Double_Pendulum_Problem.ipynb b/Double_Pendulum_Problem.ipynb index f7a2e9f..3927841 100644 --- a/Double_Pendulum_Problem.ipynb +++ b/Double_Pendulum_Problem.ipynb @@ -4,7 +4,128 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "# Dynamics of a Double Pendulum" + "# Integration of Mechanical Systems\n", + "\n", + "In this module, we will be exploring numerical integrators and how they perform for different systems." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Simple Harmonic Oscillator\n", + "\n", + "The simple harmonic oscillator is a simple physical system described by the second-order differential equation below. Despite its simplicity, this is a system that shows up in a similar form in many different fields of engineering. The same equation describes the behavior of a spring-mass-damper system, RLC circuit, or the motion of a (small-angle) pendulum.\n", + "\n", + "(Insert diagram for spring mass damper)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The differential equation describing the motion of a simple harmonic oscillator is:\n", + "\n", + "$$\\begin{equation}\n", + "0 = m\\ddot{x} + kx\n", + "\\end{equation}$$\n", + "\n", + "We can describe the state of the system by its position $x$ and its velocity $\\dot{x}$. We will call this state vector $q$, as:\n", + "\n", + "$$\\begin{eqnarray}\n", + "\\vec{q} = \\begin{bmatrix} x\\\\ \\dot{x} \\end{bmatrix}\n", + "\\end{eqnarray}$$\n", + "\n", + "This system can then be broken into two first order differential equations:\n", + "\n", + "$$\\begin{eqnarray}\n", + "\\vec{\\dot{q}} = \\begin{bmatrix} q_2\\\\ -\\frac{k}{m}q_1 \\end{bmatrix}\n", + "\\end{eqnarray}$$\n", + "\n", + "To integrate this system numerically, we can start with some initial conditions for $q$, then use a method such as Euler's method to integrate forward in time. This \n", + "\n", + "This system is a nice example for numerical integration, because it has an analytical form which we can compare to. The analytical solution is\n", + "\n", + "$$\\begin{equation}\n", + "x(t) = Acos(\\omega t + \\phi)\n", + "\n", + "\\end{equation}$$\n", + "\n", + "where $\\omega$ is the natural frequency of the system given by $\\sqrt{\\frac{k}{m}}$ . The amplitude, $A$, and the phase, $\\phi$, of oscillation are determined from the initial conditions. " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "### Adding a damping term" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "### Adding a " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Dynamics of a Double Pendulum" ] }, { @@ -370,7 +491,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.4.3" + "version": "3.5.0" } }, "nbformat": 4, From 2ff46c0b39f7aabdb6864b9bd13774a8f4894da9 Mon Sep 17 00:00:00 2001 From: rbds Date: Mon, 30 Nov 2015 22:55:37 +0000 Subject: [PATCH 05/61] equations for Euler, RK4 --- Double_Pendulum_Problem.ipynb | 59 +++++++++++++++++++++++++++++++++-- 1 file changed, 56 insertions(+), 3 deletions(-) diff --git a/Double_Pendulum_Problem.ipynb b/Double_Pendulum_Problem.ipynb index 3927841..46eed7c 100644 --- a/Double_Pendulum_Problem.ipynb +++ b/Double_Pendulum_Problem.ipynb @@ -42,13 +42,66 @@ "\\vec{\\dot{q}} = \\begin{bmatrix} q_2\\\\ -\\frac{k}{m}q_1 \\end{bmatrix}\n", "\\end{eqnarray}$$\n", "\n", - "To integrate this system numerically, we can start with some initial conditions for $q$, then use a method such as Euler's method to integrate forward in time. This \n", + "To look at the behavior of this system, we can start with some initial conditions for $q$, then use a numerical integration method such as Euler's method to integrate forward in time. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Euler's method\n", + "\n", + "Euler's method is discussed in lesson 1 of the Numerical Methods course. The method goes as follows:\n", + "$$x_i^{n+1} = x_i^n + h\\dot{x}_i^n$$\n", + "where $h$ is the timestep." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Runge-Kutta Integration (RK4)\n", "\n", + "Depending on the application, Euler's method may provide enough accuracy with a small timestep, especially a simple harmonic oscillator. However, Euler is a first-order method and we can do better. One option is a Runge-Kutta scheme. This is a popular numerical integration method which is easily extended to higher orders. Here we will use a fourth order Runge-Kutta, also known as RK4:\n", + "\n", + "$$\\begin{equation}\n", + "q_{n+1} = q_n + \\frac{h}{6}\\left(k_1+2k_2+2k_3+k_4\\right)\n", + "\\end{equation}$$\n", + "where\n", + "$$\\begin{eqnarray}\n", + "k_1 = f(t_n, x_n)\\\\\n", + "k_2 = f(t_n+\\frac{h}{2}, x_n+\\frac{h}{2}k_1)\\\\\n", + "k_3 = f(t_n+\\frac{h}{2}, x_n+\\frac{h}{2}k_2)\\\\\n", + "k_4 = f(t_n+h, y_n+hk_3)\\\\\n", + "t_{n+1} = t_n + h\n", + "\\end{eqnarray}$$\n", + "\n", + "For more information on the Runge-Kutta method, including the derivation and explanation of coefficients, see (cite)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ "This system is a nice example for numerical integration, because it has an analytical form which we can compare to. The analytical solution is\n", "\n", "$$\\begin{equation}\n", "x(t) = Acos(\\omega t + \\phi)\n", - "\n", "\\end{equation}$$\n", "\n", "where $\\omega$ is the natural frequency of the system given by $\\sqrt{\\frac{k}{m}}$ . The amplitude, $A$, and the phase, $\\phi$, of oscillation are determined from the initial conditions. " @@ -56,7 +109,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 3, "metadata": { "collapsed": true }, From 0d5761245f3e51cdb52467b8a5117ecd33b4a9ef Mon Sep 17 00:00:00 2001 From: rbds Date: Tue, 1 Dec 2015 00:52:51 +0000 Subject: [PATCH 06/61] update text --- Double_Pendulum_Problem.ipynb | 18 +++++++++++------- 1 file changed, 11 insertions(+), 7 deletions(-) diff --git a/Double_Pendulum_Problem.ipynb b/Double_Pendulum_Problem.ipynb index 46eed7c..f1ad2ff 100644 --- a/Double_Pendulum_Problem.ipynb +++ b/Double_Pendulum_Problem.ipynb @@ -39,7 +39,7 @@ "This system can then be broken into two first order differential equations:\n", "\n", "$$\\begin{eqnarray}\n", - "\\vec{\\dot{q}} = \\begin{bmatrix} q_2\\\\ -\\frac{k}{m}q_1 \\end{bmatrix}\n", + "\\vec{\\dot{q}} = f(\\vec{q}) = \\begin{bmatrix} q_2\\\\ -\\frac{k}{m}q_1 \\end{bmatrix}\n", "\\end{eqnarray}$$\n", "\n", "To look at the behavior of this system, we can start with some initial conditions for $q$, then use a numerical integration method such as Euler's method to integrate forward in time. " @@ -69,11 +69,11 @@ "\\end{equation}$$\n", "where\n", "$$\\begin{eqnarray}\n", - "k_1 = f(t_n, x_n)\\\\\n", - "k_2 = f(t_n+\\frac{h}{2}, x_n+\\frac{h}{2}k_1)\\\\\n", - "k_3 = f(t_n+\\frac{h}{2}, x_n+\\frac{h}{2}k_2)\\\\\n", - "k_4 = f(t_n+h, y_n+hk_3)\\\\\n", - "t_{n+1} = t_n + h\n", + "k_1 &=& f(t_n, x_n)\\\\\n", + "k_2 &=& f(t_n+\\frac{h}{2}, x_n+\\frac{h}{2}k_1)\\\\\n", + "k_3 &=& f(t_n+\\frac{h}{2}, x_n+\\frac{h}{2}k_2)\\\\\n", + "k_4 &=& f(t_n+h, y_n+hk_3)\\\\\n", + "t_{n+1} &=& t_n + h\n", "\\end{eqnarray}$$\n", "\n", "For more information on the Runge-Kutta method, including the derivation and explanation of coefficients, see (cite)\n" @@ -82,7 +82,11 @@ { "cell_type": "markdown", "metadata": {}, - "source": [] + "source": [ + "### Verlet method\n", + "\n", + "Symplectic integrator" + ] }, { "cell_type": "markdown", From cad05767b2074deb1daac1b2b715ddb80d02d226 Mon Sep 17 00:00:00 2001 From: rbds Date: Mon, 30 Nov 2015 22:15:56 -0500 Subject: [PATCH 07/61] add symplectic integrators --- Double_Pendulum_Problem.ipynb | 156 ++++++++++++++++++++++++++-------- 1 file changed, 120 insertions(+), 36 deletions(-) diff --git a/Double_Pendulum_Problem.ipynb b/Double_Pendulum_Problem.ipynb index f1ad2ff..a10ae5f 100644 --- a/Double_Pendulum_Problem.ipynb +++ b/Double_Pendulum_Problem.ipynb @@ -6,7 +6,7 @@ "source": [ "# Integration of Mechanical Systems\n", "\n", - "In this module, we will be exploring numerical integrators and how they perform for different systems." + "In this module, we will be exploring numerical integrators and how they perform for different mechanical systems." ] }, { @@ -45,6 +45,11 @@ "To look at the behavior of this system, we can start with some initial conditions for $q$, then use a numerical integration method such as Euler's method to integrate forward in time. " ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [] + }, { "cell_type": "markdown", "metadata": {}, @@ -53,7 +58,8 @@ "\n", "Euler's method is discussed in lesson 1 of the Numerical Methods course. The method goes as follows:\n", "$$x_i^{n+1} = x_i^n + h\\dot{x}_i^n$$\n", - "where $h$ is the timestep." + "\n", + "where $h$ is the timestep. This isn't a complicated scheme, so for details see the code below." ] }, { @@ -83,15 +89,48 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "### Verlet method\n", + "### Symplectic Integrators\n", + "\n", + "Symplectic integrators are similar to the methods discussed above, but they use equations of motion derived from Hamiltonian mechanics. According to [Berkeley source], syplectic integrators preserve the conserved Hamiltonian quantities. In practical terms, this works out to mean the methods reflect conservation of energy, down to a truncation error.\n", "\n", - "Symplectic integrator" + "The Hamiltonian equations of motion can be derived from the total energy in the system, much like the Lagrangian. The coordinates we use are position ($q$) and momentum ($p = m\\dot x$). The Hamiltonian is:\n", + "$$ H = T+V$$\n", + "where $T$ is the kinetic energy in the system, and $V$ is the potential energy in the system. So, we get:\n", + "$$T = \\frac{1}{2}m\\dot{x}^2 = \\frac{p^2}{2m}$$\n", + "and\n", + "$$V = mgq$$\n", + "\n", + "The Hamilton's equations are:\n", + "$$\\begin{eqnarray*}\n", + "\\frac{dp}{dt} &=& -\\frac{\\partial H}{\\partial q}\\\\\n", + "\\frac{dq}{dt} &=& \\frac{\\partial H}{\\partial p}\n", + "\\end{eqnarray*}$$\n", + "\n", + "So our equations of motion become:\n", + "$$\\begin{equation*}\n", + "\\begin{bmatrix} \\dot q \\\\ \\dot p \\end{bmatrix} = \\begin{bmatrix} -\\frac{\\partial H}{\\partial q}\\\\ \\frac{\\partial H}{\\partial p} \\end{bmatrix} = \\begin{bmatrix} -mg \\\\ \\frac{p}{m} \\end{bmatrix}\n", + "\\end{equation*}$$" ] }, { "cell_type": "markdown", "metadata": {}, - "source": [] + "source": [ + "Let's now take a look at symplectic integration methods. We'll start with a first order method - symplectic Euler's method. This uses the equations:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "p_{n+1} = p_n - h \\frac{\\partial H}{\\partial q} \\Bigr|_{p_{n+1},q_n}\\\\\n", + "q_{n+1} = q_n + \\frac{h}{2} \\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1},q_n}\n", + "\\end{eqnarray*}$$\n", + "\n", + "We can then plug in the partial derivatives and integrate the system. This system will do a better job of conserving energy than Euler's method, $\\textit{but only to a truncation error.}$ Again, we can do better than first-order. Let's try a second-order symplectic scheme, also called Verlet integration. We'll have the same equations of motion, but this time a different set of integration equations:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "p_{n+1/2} &=& p_n - \\frac{h}{2} \\frac{\\partial H}{\\partial q} \\Bigr|_{p_{n+1/2},q_n}\\\\\n", + "q_{n+1} &=& q_n + \\frac{h}{2} \\left(\\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1/2},q_n} + \\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1/2},q_{n+1/2}} \\right) \\\\\n", + "p_{n+1} &=& p_{n+1/2} - \\frac{h}{2} \\frac{\\partial H}{\\partial q}\\Bigr|_{p_{n+1/2},q_{n+1/2}}\n", + "\\end{eqnarray*}$$" + ] }, { "cell_type": "markdown", @@ -102,24 +141,44 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "This system is a nice example for numerical integration, because it has an analytical form which we can compare to. The analytical solution is\n", + "### Analytical Solution\n", + "The simple harmonic oscillator is a nice example for numerical integration, because it has an analytical form which we can compare to. The analytical solution is\n", "\n", "$$\\begin{equation}\n", - "x(t) = Acos(\\omega t + \\phi)\n", + "x(t) = Acos(\\omega_n t + \\phi)\n", "\\end{equation}$$\n", "\n", - "where $\\omega$ is the natural frequency of the system given by $\\sqrt{\\frac{k}{m}}$ . The amplitude, $A$, and the phase, $\\phi$, of oscillation are determined from the initial conditions. " + "where $\\omega_n$ is the natural frequency of the system given by $\\sqrt{\\frac{k}{m}}$ . The amplitude, $A$, and the phase, $\\phi$, of oscillation are determined from the initial conditions. " ] }, { - "cell_type": "code", - "execution_count": 3, + "cell_type": "markdown", "metadata": { "collapsed": true }, - "outputs": [], "source": [ - "### Adding a damping term" + "### Adding a damping term\n", + "\n", + "A more realistic oscillator also has a damping term. This might correspond to friction, or to some other non-conservative force. For now, we will assume that the damping is linear, and is proportional to the velocity (this is called the -- model of friction). We get the following equation of motion:\n", + "\n", + "$$\\begin{equation*}\n", + "0 = m\\ddot{x} + c\\dot{x} + kx\n", + "\\end{equation*}$$\n", + "\n", + "This gives us the differential equations:\n", + "$$\\begin{eqnarray*}\n", + "\\vec{\\dot{q}} = f(\\vec{q}) = \\begin{bmatrix} q_2\\\\ -\\frac{c}{m}\\dot{q_1} -\\frac{k}{m}q_1 \\end{bmatrix}\n", + "\\end{eqnarray*}$$\n", + "\n", + "We also have a new analytical solution. Assuming the system is underdamped (add explanation of damping ratio), the solution becomes:\n", + "\n", + "\n", + "$$\\begin{equation}\n", + "x(t) = Acos(\\omega_d t + \\phi)\n", + "\\end{equation}$$\n", + "\n", + "where $\\omega_d = \\omega_n\\sqrt{1 - \\zeta^2}$ is called the damped natural frequency.\n", + "\n" ] }, { @@ -227,27 +286,18 @@ "\n", "where $T$ is the total Kinetic Energy in the system, and $U$ is the total Potential Energy. Potential energy is calculated as:\n", "\n", - "$$\\begin{equation}\n", - "U = m_1g(-y_1) + m_2g(-y_2)\n", - "\\end{equation}$$\n", - "\n", - "$$\\begin{equation}\n", - "U = -m_1g(l_1cos(\\theta_1) - m_2g(l_1cos\\theta_1 + l_2cos\\theta_2)\n", - "\\end{equation}$$\n", + "$$\\begin{eqnarray}\n", + "U &=& m_1g(-y_1) + m_2g(-y_2) \\\\\n", + " &=& -m_1g(l_1cos(\\theta_1) - m_2g(l_1cos\\theta_1 + l_2cos\\theta_2)\n", + "\\end{eqnarray}$$\n", "\n", "Kinetic Energy is calculated as:\n", "\n", - "$$\\begin{equation}\n", - "T = \\frac{1}{2}mv_1^2+\\frac{1}{2}mv_2^2\n", - "\\end{equation}$$\n", - "\n", - "$$\\begin{equation}\n", - "T = \\frac{1}{2}m_1(l_1\\dot\\theta_1)^2 + \\frac{1}{2}m_2(l_1\\dot\\theta_1 + l_2\\dot\\theta_2)^2\n", - "\\end{equation}$$\n", - "\n", - "$$\\begin{equation}\n", - "T = \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2+\\frac{1}{2}m_2\\left(2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+l_2^2\\dot\\theta_2^2\\right)\n", - "\\end{equation}$$\n", + "$$\\begin{eqnarray*}\n", + "T &=& \\frac{1}{2}mv_1^2+\\frac{1}{2}mv_2^2 \\\\\n", + "&=& \\frac{1}{2}m_1(l_1\\dot\\theta_1)^2 + \\frac{1}{2}m_2(l_1\\dot\\theta_1 + l_2\\dot\\theta_2)^2 \\\\\n", + "&=& \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2+\\frac{1}{2}m_2\\left(2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+l_2^2\\dot\\theta_2^2\\right)\n", + "\\end{eqnarray*}$$\n", "\n", "\n", "So the Lagrangian quantity becomes:\n", @@ -328,7 +378,13 @@ "\n", "$$\\begin{eqnarray}\n", "f(\\vec{x}(t)) = \\begin{pmatrix} x_3\\\\ x_4\\\\ m_2l_1l_2x_4^2sin(x_2-x_1) -(m_1+m_2)gl_1sin(x_1)\\\\ m_2l_1l_2x_3^2sin(x_2-x_1)-l_2m_2gsinx_2 \\end{pmatrix}\n", - "\\end{eqnarray}$$" + "\\end{eqnarray}$$\n", + "\n", + "In order to solve this, we can evaluate matrix A and vector $f(x)$ at each time step, then invert A and premultiply the result to each side of the equation. This gives us an explicit system of equations of motion:\n", + "\n", + "$$ \\dot x=A^{-1}f(x)$$\n", + "\n", + "You may notice that these operations require a significant computational load at each time step. In order to reduce this, we can analytically invert the matrix A, then simply fill in the necessary values at each time step and multiply by $f(x)$. This will significantly speed up running time, as inverting a matrix is an expensive operation.\n" ] }, { @@ -337,7 +393,7 @@ "source": [ "## Integration\n", "\n", - "#### Euler's Method\n", + "### Euler's Method\n", "\n", "The first method we will use is Euler integration, covered in lesson 1 of the Numerical Methods course. This is the simplest form of numerical integration, but it can still be effective. Using our equations for $\\dot x$ and $f(x)$:\n", "\n", @@ -345,7 +401,7 @@ "\n", "where $h$ is the timestep.\n", "\n", - "#### Runge-Kutta Method (RK4)\n", + "### Runge-Kutta Method (RK4)\n", "\n" ] }, @@ -360,9 +416,28 @@ "Here we define functions for each of the integration methods, set our parameters and initial conditions, and then numerically integrate the motion of the pendulums. The plots below compare the different integration techniques." ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Sources: \n", + "\n", + "http://scienceworld.wolfram.com/physics/DoublePendulum.html\n", + "\n", + "http://www.phy.uct.ac.za/courses/opencontent/phylab2/worksheet9_09.pdf\n", + "\n", + "http://www.phys.lsu.edu/faculty/gonzalez/Teaching/Phys7221/DoublePendulum.pdf\n", + "\n", + "http://www.iontrap.wabash.edu/adlab/papers/F2011_foster_groninger_tang_chaos.pdf\n", + "\n", + "https://math.berkeley.edu/~alanw/242papers99/markiewicz.pdf\n", + "\n", + "http://www.unige.ch/~hairer/poly_geoint/week2.pdf" + ] + }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 2, "metadata": { "collapsed": false }, @@ -517,7 +592,7 @@ "" ] }, - "execution_count": 4, + "execution_count": 2, "metadata": {}, "output_type": "execute_result" } @@ -530,6 +605,15 @@ "css_file = 'numericalmoocstyle.css'\n", "HTML(open(css_file, \"r\").read())" ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] } ], "metadata": { @@ -548,7 +632,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.5.0" + "version": "3.4.3" } }, "nbformat": 4, From 7d1581e259aad843e6e5fe82ed3ad18615dd240f Mon Sep 17 00:00:00 2001 From: rbds Date: Tue, 1 Dec 2015 18:38:19 -0500 Subject: [PATCH 08/61] move files around --- .../Double_Pendulum_Problem.ipynb | 25 ++++++++---------- {figures => randy_schur/figures}/diagram.png | Bin .../numericalmoocstyle.css | 0 3 files changed, 11 insertions(+), 14 deletions(-) rename Double_Pendulum_Problem.ipynb => randy_schur/Double_Pendulum_Problem.ipynb (95%) rename {figures => randy_schur/figures}/diagram.png (100%) rename numericalmoocstyle.css => randy_schur/numericalmoocstyle.css (100%) diff --git a/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb similarity index 95% rename from Double_Pendulum_Problem.ipynb rename to randy_schur/Double_Pendulum_Problem.ipynb index a10ae5f..3fcaa39 100644 --- a/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -321,25 +321,19 @@ "\n", "So the equation of motion for $\\theta_1$ is calculated by the following steps:\n", "\n", - "$$\\begin{equation}\n", - "\\frac{\\partial L}{\\partial \\dot{\\theta_1}} = (m_1+m_2)l_1^2\\dot\\theta_1+m_2l_1l_2\\dot\\theta_2cos(\\theta_2-\\theta_1)\n", - "\\end{equation}$$\n", - "\n", - "$$\\begin{equation}\n", - "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{\\theta_1}}\\right) = (m_1+m_2)l_1^2\\ddot\\theta_1+m_2l_1l_2\\ddot\\theta_2cos(\\theta_2-\\theta_1)-m_2l_1l_2\\dot\\theta_2^2sin(\\theta_2-\\theta_1)+m_2l_1l_2\\dot\\theta_1\\dot\\theta_2sin(\\theta_2-\\theta_1)\n", - "\\end{equation}$$\n", - "\n", - "$$\\begin{equation}\n", - "\\frac{\\partial L}{\\partial \\theta_1} = m_2l_1l_2\\dot\\theta_1\\dot\\theta_2sin(\\theta_2-\\theta_1)+(m_1+m_2)gl_1sin\\theta_1\n", - "\\end{equation}$$\n", + "$$\\begin{eqnarray*}\n", + "\\frac{\\partial L}{\\partial \\dot{\\theta_1}} &=& (m_1+m_2)l_1^2\\dot\\theta_1+m_2l_1l_2\\dot\\theta_2cos(\\theta_2-\\theta_1) \\\\\n", + "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{\\theta_1}}\\right) &=& (m_1+m_2)l_1^2\\ddot\\theta_1+m_2l_1l_2\\ddot\\theta_2cos(\\theta_2-\\theta_1)-m_2l_1l_2\\dot\\theta_2^2sin(\\theta_2-\\theta_1)+m_2l_1l_2\\dot\\theta_1\\dot\\theta_2sin(\\theta_2-\\theta_1) \\\\\n", + "-\\frac{\\partial L}{\\partial \\theta_1} &=& -m_2l_1l_2\\dot\\theta_1\\dot\\theta_2sin(\\theta_2-\\theta_1)-(m_1+m_2)gl_1sin\\theta_1\n", + "\\end{eqnarray*}$$\n", "\n", - "Some of these terms will cancel out, and we are left with a final equation of motion:\n", + "We can add together the second and third equations to find the first equation of motion. Some of these terms will cancel out, and we are left with a final equation of motion:\n", "\n", "$$\\begin{equation}\n", "0 = (m_1+m_2)l_1^2\\ddot\\theta_1+m_2l_1l_2\\ddot\\theta_2cos(\\theta_2-\\theta_1)-m_2l_1l_2\\dot\\theta_2^2sin(\\theta_2-\\theta_1)+(m_1+m_2)gl_1sin\\theta_1\n", "\\end{equation}$$\n", "\n", - "We will also need the equation of motion for $\\theta_2$. This can be derived using the same process. It should work out to the following:\n", + "We will also need the equation of motion for $\\theta_2$. This can be derived using the same process. It should work out to the following (check this by yourself!):\n", "\n", "$$\\begin{equation}\n", "0 = m_2l_2^2\\ddot\\theta_2+m_2l_1l_2\\ddot\\theta_1cos(\\theta_2-\\theta_1) - m_2l_2l_1\\dot\\theta_1^2sin(\\theta_2-\\theta_1)+ l_2m_2gsin\\theta_2\n", @@ -432,7 +426,10 @@ "\n", "https://math.berkeley.edu/~alanw/242papers99/markiewicz.pdf\n", "\n", - "http://www.unige.ch/~hairer/poly_geoint/week2.pdf" + "http://www.unige.ch/~hairer/poly_geoint/week2.pdf\n", + "\n", + "http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1994CeMDA..60..409T&defaultprint=YES&filetype=.pdf\n", + "\n" ] }, { diff --git a/figures/diagram.png b/randy_schur/figures/diagram.png similarity index 100% rename from figures/diagram.png rename to randy_schur/figures/diagram.png diff --git a/numericalmoocstyle.css b/randy_schur/numericalmoocstyle.css similarity index 100% rename from numericalmoocstyle.css rename to randy_schur/numericalmoocstyle.css From 398cf3cf5e475ff41947fa4c2c52bb5c384f0fea Mon Sep 17 00:00:00 2001 From: rbds Date: Tue, 1 Dec 2015 19:34:00 -0500 Subject: [PATCH 09/61] update damped Hamiltonian derivation --- Double_Pendulum_Problem.ipynb | 706 ++++++++++++++++++++++++++++++++++ 1 file changed, 706 insertions(+) create mode 100644 Double_Pendulum_Problem.ipynb diff --git a/Double_Pendulum_Problem.ipynb b/Double_Pendulum_Problem.ipynb new file mode 100644 index 0000000..59df853 --- /dev/null +++ b/Double_Pendulum_Problem.ipynb @@ -0,0 +1,706 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Integration of Mechanical Systems\n", + "\n", + "In this module, we will be exploring numerical integrators and how they perform for different mechanical systems." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Simple Harmonic Oscillator\n", + "\n", + "The simple harmonic oscillator is a simple physical system described by the second-order differential equation below. Despite its simplicity, this is a system that shows up in a similar form in many different fields of engineering. The same equation describes the behavior of a spring-mass-damper system, RLC circuit, or the motion of a (small-angle) pendulum.\n", + "\n", + "(Insert diagram for spring mass damper)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The differential equation describing the motion of a simple harmonic oscillator is:\n", + "\n", + "$$\\begin{equation}\n", + "0 = m\\ddot{x} + kx\n", + "\\end{equation}$$\n", + "\n", + "We can describe the state of the system by its position $x$ and its velocity $\\dot{x}$. We will call this state vector $q$, as:\n", + "\n", + "$$\\begin{eqnarray}\n", + "\\vec{q} = \\begin{bmatrix} x\\\\ \\dot{x} \\end{bmatrix}\n", + "\\end{eqnarray}$$\n", + "\n", + "This system can then be broken into two first order differential equations:\n", + "\n", + "$$\\begin{eqnarray}\n", + "\\vec{\\dot{q}} = f(\\vec{q}) = \\begin{bmatrix} q_2\\\\ -\\frac{k}{m}q_1 \\end{bmatrix}\n", + "\\end{eqnarray}$$\n", + "\n", + "To look at the behavior of this system, we can start with some initial conditions for $q$, then use a numerical integration method such as Euler's method to integrate forward in time. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Equations of Motion from Lagrangian\n", + "\n", + "You are encouraged to follow this derivation ON PAPER on your own. We gave you the equations of motion for a simple harmonic oscillator, but what if we don't have this equation? The following is a method that will work in nearly any situation to find equations of motion, often working well where a force analysis becomes difficult. This is good practice for anyone studying mechanics. To derive the equations of motion, we will start with the Lagrangian. This is defined by:\n", + "\n", + "$$\\begin{equation}\n", + "L = T- U\n", + "\\end{equation}$$\n", + "\n", + "where $T$ is the total Kinetic Energy in the system, and $U$ is the total Potential Energy. Potential energy is calculated as:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "U &=& \\frac{1}{2}kx^2\\\\\n", + "\\end{eqnarray*}$$\n", + "\n", + "Kinetic Energy is calculated as:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "T &=& \\frac{1}{2}mv^2 \\\\\n", + "&=& \\frac{1}{2}m(\\dot x)^2 \n", + "\\end{eqnarray*}$$\n", + "\n", + "\n", + "So the Lagrangian quantity becomes:\n", + "$$\\begin{equation}\n", + "L = \\frac{1}{2}m(\\dot x)^2 - \\frac{1}{2}kx^2\\\\\n", + "\\end{equation}$$\n", + "\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The equations of motion are then found using the Lagrange Equation:\n", + "\n", + "$$\\begin{equation}\n", + "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x_i}}\\right) - \\frac{\\partial L}{\\partial x_i} = 0\n", + "\\end{equation}$$\n", + "\n", + "Since we have only one state (position), we need only one equation of motion. We'll calculate this in steps:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "\\frac{\\partial L}{\\partial \\dot{x}} &=& m\\dot{x} \\\\\n", + "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x}}\\right) &=& m\\ddot{x}\\\\\n", + "-\\frac{\\partial L}{\\partial x} &=& + kx\n", + "\\end{eqnarray*}$$\n", + "\n", + "We can add together the second and third equations to find the first equation of motion. Some of these terms will cancel out, and we are left with a final equation of motion:\n", + "\n", + "$$\\begin{equation}\n", + "0 = m\\ddot{x} + kx\n", + "\\end{equation}$$\n", + "\n", + "This is the same equation we have before! So, we can be relatively confident that this derivation is correct." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Integration Methods\n", + "\n", + "### Euler's method\n", + "\n", + "Euler's method is discussed in lesson 1 of the Numerical Methods course. The method goes as follows:\n", + "$$x_i^{n+1} = x_i^n + h\\dot{x}_i^n$$\n", + "\n", + "where $h$ is the timestep. This isn't a complicated scheme, so for details see the code below." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Runge-Kutta Integration (RK4)\n", + "\n", + "Depending on the application, Euler's method may provide enough accuracy with a small timestep, especially a simple harmonic oscillator. However, Euler is a first-order method and we can do better. One option is a Runge-Kutta scheme. This is a popular numerical integration method which is easily extended to higher orders. Here we will use a fourth order Runge-Kutta, also known as RK4:\n", + "\n", + "$$\\begin{equation}\n", + "q_{n+1} = q_n + \\frac{h}{6}\\left(k_1+2k_2+2k_3+k_4\\right)\n", + "\\end{equation}$$\n", + "where\n", + "$$\\begin{eqnarray}\n", + "k_1 &=& f(t_n, x_n)\\\\\n", + "k_2 &=& f(t_n+\\frac{h}{2}, x_n+\\frac{h}{2}k_1)\\\\\n", + "k_3 &=& f(t_n+\\frac{h}{2}, x_n+\\frac{h}{2}k_2)\\\\\n", + "k_4 &=& f(t_n+h, y_n+hk_3)\\\\\n", + "t_{n+1} &=& t_n + h\n", + "\\end{eqnarray}$$\n", + "\n", + "For more information on the Runge-Kutta method, including the derivation and explanation of coefficients, see (cite)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Symplectic Integrators\n", + "\n", + "Symplectic integrators are similar to the methods discussed above, but they use equations of motion derived from Hamiltonian mechanics. According to [Berkeley source], syplectic integrators preserve the conserved Hamiltonian quantities. In practical terms, this works out to mean the methods reflect conservation of energy, down to a truncation error.\n", + "\n", + "The Hamiltonian equations of motion can be derived from the total energy in the system, much like the Lagrangian. The coordinates we use are position ($q$) and momentum ($p = m\\dot x$). The Hamiltonian is:\n", + "$$ H = T+V$$\n", + "where $T$ is the kinetic energy in the system, and $V$ is the potential energy in the system. So, we get:\n", + "$$\\begin{eqnarray*}\n", + "T &=& \\frac{1}{2}m\\dot{x}^2 = \\frac{p^2}{2m}\\\\\n", + "V &=& mgq\\\\\n", + "H &=& \\frac{p^2}{2m} + mgq\n", + "\\end{eqnarray*}$$\n", + "\n", + "\n", + "The Hamilton's equations are:\n", + "$$\\begin{eqnarray*}\n", + "\\frac{dp}{dt} &=& -\\frac{\\partial H}{\\partial q}\\\\\n", + "\\frac{dq}{dt} &=& \\frac{\\partial H}{\\partial p}\n", + "\\end{eqnarray*}$$\n", + "\n", + "So our equations of motion become:\n", + "$$\\begin{equation*}\n", + "\\begin{bmatrix} \\dot p \\\\ \\dot q \\end{bmatrix} = \\begin{bmatrix} -\\frac{\\partial H}{\\partial q}\\\\ \\frac{\\partial H}{\\partial p} \\end{bmatrix} = \\begin{bmatrix} -mg \\\\ \\frac{p}{m} \\end{bmatrix}\n", + "\\end{equation*}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's now take a look at symplectic integration methods. We'll start with a first order method - symplectic Euler's method. This uses the equations:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "p_{n+1} = p_n - h \\frac{\\partial H}{\\partial q} \\Bigr|_{p_{n+1},q_n}\\\\\n", + "q_{n+1} = q_n + \\frac{h}{2} \\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1},q_n}\n", + "\\end{eqnarray*}$$\n", + "\n", + "We can then plug in the partial derivatives and integrate the system. This system will do a better job of conserving energy than Euler's method, $\\textit{but only to a truncation error.}$ Again, we can do better than first-order. Let's try a second-order symplectic scheme, also called Verlet integration. We'll have the same equations of motion, but this time a different set of integration equations:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "p_{n+1/2} &=& p_n - \\frac{h}{2} \\frac{\\partial H}{\\partial q} \\Bigr|_{p_{n+1/2},q_n}\\\\\n", + "q_{n+1} &=& q_n + \\frac{h}{2} \\left(\\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1/2},q_n} + \\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1/2},q_{n+1/2}} \\right) \\\\\n", + "p_{n+1} &=& p_{n+1/2} - \\frac{h}{2} \\frac{\\partial H}{\\partial q}\\Bigr|_{p_{n+1/2},q_{n+1/2}}\n", + "\\end{eqnarray*}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Analytical Solution\n", + "The simple harmonic oscillator is a nice example for numerical integration, because it has an analytical form which we can compare to. The analytical solution is\n", + "\n", + "$$\\begin{equation}\n", + "x(t) = Acos(\\omega_n t + \\phi)\n", + "\\end{equation}$$\n", + "\n", + "where $\\omega_n$ is the natural frequency of the system given by $\\omega_n = \\sqrt{\\frac{k}{m}}$ . The amplitude, $A$, and the phase, $\\phi$, of oscillation are determined from the initial conditions. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Code section?" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "### Adding a damping term\n", + "\n", + "A more realistic oscillator also has a damping term. This might correspond to friction, or to some other non-conservative force. For now, we will assume that the damping is linear, and is proportional to the velocity (this is called the -- model of friction). We get the following equation of motion:\n", + "\n", + "$$\\begin{equation*}\n", + "0 = m\\ddot{x} + c\\dot{x} + kx\n", + "\\end{equation*}$$\n", + "\n", + "This gives us the differential equations:\n", + "$$\\begin{eqnarray*}\n", + "\\vec{\\dot{q}} = f(\\vec{q}) = \\begin{bmatrix} q_2\\\\ -\\frac{c}{m}\\dot{q_1} -\\frac{k}{m}q_1 \\end{bmatrix}\n", + "\\end{eqnarray*}$$\n", + "\n", + "The Euler's method and Runge-Kutta integration work the same way with this new equation. We also have a new analytical solution. Assuming the system is underdamped (add explanation of damping ratio), the solution becomes:\n", + "\n", + "\n", + "$$\\begin{equation}\n", + "x(t) = Acos(\\omega_d t + \\phi)\n", + "\\end{equation}$$\n", + "\n", + "where $\\omega_d = \\omega_n\\sqrt{1 - \\zeta^2}$ is called the damped natural frequency.\n", + "\n", + "The big change comes in the Hamiltonian equations. If these are dependent on convservation of energy, how do we deal with a system where energy is not conserved?\n", + "\n", + "Our first step is to re-examine the Lagrangian. Since losses due to friction are neither potential nor kinetic energy, we'll need a new term. The Lagrangian equation of motion with non-conservative forces is:\n", + "\n", + "$$\\begin{equation}\n", + "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x_i}}\\right) - \\frac{\\partial L}{\\partial x_i} = Q\n", + "\\end{equation}$$\n", + "\n", + "where $Q$ is the sum of all non-conservative forces. In this case, \n", + "$$Q = c\\frac{\\partial x}{\\partial t}$$\n", + "\n", + "Now that we have this, we can add a similar term to the Hamiltonian equations of motion to deal with the same problem. For more information on the derivation of the Hamiltonian from the Lagrangian, see Tveter (cite). The Hamiltonian equations of motion become:\n", + "\n", + "$$\\begin{equation*}\n", + "\\begin{bmatrix} \\dot p \\\\ \\dot q \\end{bmatrix} = \\begin{bmatrix} -\\frac{\\partial H}{\\partial q} + Q\\\\ \\frac{\\partial H}{\\partial p} \\end{bmatrix} = \\begin{bmatrix} -mg + \\frac{c}{m}q \\\\ \\frac{p}{m} \\end{bmatrix}\n", + "\\end{equation*}$$\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Dynamics of a Double Pendulum" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In this module, we will study the dynamics of a double pendulum. We first derive the equations of motion from the Lagrangian. Next, we take a look at three different numerical integration schemes: Euler's method, Runge-Kutta, and a symplectic integrator. We will investigate how the initial conditions affect the behavior of the system, as well as some of the sources of error in these simulations. Finally, we will look at the properties of the system in phase space using phase plots and Poincare sections." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Background\n", + "\n", + "The double pendulum problem is a simple system which can produce surprisingly complex movements. It is a chaotic system, meaning it is unpredictable, and there is no closed form solution to the motion of the two masses. This is a frequent example problem in the study of dynamic systems, nonlinear controls, and mechanics.\n", + "\n", + "In this notebook we will be treating the two rods as massless and ignoring the effects of friction. For a treatment of this problem without these assumptions, see: \n", + "\n", + "See the image below for the definition of constants." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "![Image](figures/diagram.png)\n", + "#### Figure 1. Diagram of Double Pendulum System" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Equations of Motion \n", + "\n", + "You are encouraged to follow this derivation ON PAPER on your own. The following is a method that will work in nearly any situation to find equations of motion, often working well where a force analysis becomes difficult. This is good practice for anyone studying mechanics. To derive the equations of motion, we will start with the Lagrangian. This is defined by:\n", + "\n", + "$$\\begin{equation}\n", + "L = T- U\n", + "\\end{equation}$$\n", + "\n", + "where $T$ is the total Kinetic Energy in the system, and $U$ is the total Potential Energy. Potential energy is calculated as:\n", + "\n", + "$$\\begin{eqnarray}\n", + "U &=& m_1g(-y_1) + m_2g(-y_2) \\\\\n", + " &=& -m_1g(l_1cos(\\theta_1) - m_2g(l_1cos\\theta_1 + l_2cos\\theta_2)\n", + "\\end{eqnarray}$$\n", + "\n", + "Kinetic Energy is calculated as:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "T &=& \\frac{1}{2}mv_1^2+\\frac{1}{2}mv_2^2 \\\\\n", + "&=& \\frac{1}{2}m_1(l_1\\dot\\theta_1)^2 + \\frac{1}{2}m_2(l_1\\dot\\theta_1 + l_2\\dot\\theta_2)^2 \\\\\n", + "&=& \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2+\\frac{1}{2}m_2\\left(2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+l_2^2\\dot\\theta_2^2\\right)\n", + "\\end{eqnarray*}$$\n", + "\n", + "\n", + "So the Lagrangian quantity becomes:\n", + "$$\\begin{equation}\n", + "L = \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2+m_2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+(m_1+m_2)l_1gcos\\theta_1 + m_2l_2gcos\\theta_2\n", + "\\end{equation}$$\n" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "The equations of motion are then found using the Lagrange Equation:\n", + "\n", + "$$\\begin{equation}\n", + "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x_i}}\\right) - \\frac{\\partial L}{\\partial x_i} = 0\n", + "\\end{equation}$$\n", + "\n", + "\n", + "So the equation of motion for $\\theta_1$ is calculated by the following steps:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "\\frac{\\partial L}{\\partial \\dot{\\theta_1}} &=& (m_1+m_2)l_1^2\\dot\\theta_1+m_2l_1l_2\\dot\\theta_2cos(\\theta_2-\\theta_1) \\\\\n", + "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{\\theta_1}}\\right) &=& (m_1+m_2)l_1^2\\ddot\\theta_1+m_2l_1l_2\\ddot\\theta_2cos(\\theta_2-\\theta_1)-m_2l_1l_2\\dot\\theta_2^2sin(\\theta_2-\\theta_1)+m_2l_1l_2\\dot\\theta_1\\dot\\theta_2sin(\\theta_2-\\theta_1) \\\\\n", + "-\\frac{\\partial L}{\\partial \\theta_1} &=& -m_2l_1l_2\\dot\\theta_1\\dot\\theta_2sin(\\theta_2-\\theta_1)-(m_1+m_2)gl_1sin\\theta_1\n", + "\\end{eqnarray*}$$\n", + "\n", + "We can add together the second and third equations to find the first equation of motion. Some of these terms will cancel out, and we are left with a final equation of motion:\n", + "\n", + "$$\\begin{equation}\n", + "0 = (m_1+m_2)l_1^2\\ddot\\theta_1+m_2l_1l_2\\ddot\\theta_2cos(\\theta_2-\\theta_1)-m_2l_1l_2\\dot\\theta_2^2sin(\\theta_2-\\theta_1)+(m_1+m_2)gl_1sin\\theta_1\n", + "\\end{equation}$$\n", + "\n", + "We will also need the equation of motion for $\\theta_2$. This can be derived using the same process. It should work out to the following (check this by yourself!):\n", + "\n", + "$$\\begin{equation}\n", + "0 = m_2l_2^2\\ddot\\theta_2+m_2l_1l_2\\ddot\\theta_1cos(\\theta_2-\\theta_1) - m_2l_2l_1\\dot\\theta_1^2sin(\\theta_2-\\theta_1)+ l_2m_2gsin\\theta_2\n", + "\\end{equation}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Problem setup\n", + "\n", + "Notice that these equations of motion are implicit equations for $\\ddot\\theta_1$ and $\\ddot\\theta_2$. We have talked about implicit equations in this class, and we solve them by forming a system of equations. In this case, we will need to solve the equations of motion at each time step. We will characterize our system in state space using the angle and angular velocity of each pendulum. So our state becomes:\n", + "\n", + "$$\\begin{eqnarray}\n", + "\\vec{x}(t) = \\begin{pmatrix} \\theta_1\\\\ \\theta_2\\\\ \\dot\\theta_1 \\\\ \\dot\\theta_2 \\end{pmatrix}\n", + "\\end{eqnarray}$$\n", + "\n", + "The equations of motion can be rewritten as a system of equations:\n", + "\n", + "$$ A\\dot x=f(x)$$\n", + "\n", + "where \n", + "\n", + "\\begin{equation}\n", + "A = \n", + "\\left[\\begin{array}{c}\n", + "1 &0 &0 &0 \\\\\n", + "0 &1 &0 &0 \\\\\n", + "0 &0 &(m_1+m_2)l_1^2 &m_2l_1l_2cos(x_2-x_1) \\\\\n", + "0 &0 &m_2l_1l_2cos(x_2-x_1) &m_2l_2^2\n", + "\\end{array}\\right]\n", + "\\end{equation}\n", + "\n", + "and\n", + "\n", + "$$\\begin{eqnarray}\n", + "f(\\vec{x}(t)) = \\begin{pmatrix} x_3\\\\ x_4\\\\ m_2l_1l_2x_4^2sin(x_2-x_1) -(m_1+m_2)gl_1sin(x_1)\\\\ m_2l_1l_2x_3^2sin(x_2-x_1)-l_2m_2gsinx_2 \\end{pmatrix}\n", + "\\end{eqnarray}$$\n", + "\n", + "In order to solve this, we can evaluate matrix A and vector $f(x)$ at each time step, then invert A and premultiply the result to each side of the equation. This gives us an explicit system of equations of motion:\n", + "\n", + "$$ \\dot x=A^{-1}f(x)$$\n", + "\n", + "You may notice that these operations require a significant computational load at each time step. In order to reduce this, we can analytically invert the matrix A, then simply fill in the necessary values at each time step and multiply by $f(x)$. This will significantly speed up running time, as inverting a matrix is an expensive operation.\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Integration\n", + "\n", + "### Euler's Method\n", + "\n", + "The first method we will use is Euler integration, covered in lesson 1 of the Numerical Methods course. This is the simplest form of numerical integration, but it can still be effective. Using our equations for $\\dot x$ and $f(x)$:\n", + "\n", + "$$x_i^{n+1} = x_i^n + h\\dot{x}_i^n$$\n", + "\n", + "where $h$ is the timestep.\n", + "\n", + "### Runge-Kutta Method (RK4)\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "## Solve!\n", + "\n", + "Here we define functions for each of the integration methods, set our parameters and initial conditions, and then numerically integrate the motion of the pendulums. The plots below compare the different integration techniques." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Sources: \n", + "\n", + "http://scienceworld.wolfram.com/physics/DoublePendulum.html\n", + "\n", + "http://www.phy.uct.ac.za/courses/opencontent/phylab2/worksheet9_09.pdf\n", + "\n", + "http://www.phys.lsu.edu/faculty/gonzalez/Teaching/Phys7221/DoublePendulum.pdf\n", + "\n", + "http://www.iontrap.wabash.edu/adlab/papers/F2011_foster_groninger_tang_chaos.pdf\n", + "\n", + "https://math.berkeley.edu/~alanw/242papers99/markiewicz.pdf\n", + "\n", + "http://www.unige.ch/~hairer/poly_geoint/week2.pdf\n", + "\n", + "http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1994CeMDA..60..409T&defaultprint=YES&filetype=.pdf\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "execution_count": 2, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# This cell loads the style of the notebook, which is taken from the \n", + "# Numerical Methods in Python Course: http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about\n", + "\n", + "from IPython.core.display import HTML\n", + "css_file = './randy_schur/numericalmoocstyle.css'\n", + "HTML(open(css_file, \"r\").read())" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.4.3" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +} From 94457d64208b7d9d41e4693b153869cad265ea25 Mon Sep 17 00:00:00 2001 From: rbds Date: Tue, 1 Dec 2015 20:26:57 -0500 Subject: [PATCH 10/61] in class changes 12/01 --- Double_Pendulum_Problem.ipynb | 706 --------------------- randy_schur/Double_Pendulum_Problem.ipynb | 204 ++++-- randy_schur/figures/spring-mass-damper.png | Bin 0 -> 11939 bytes 3 files changed, 137 insertions(+), 773 deletions(-) delete mode 100644 Double_Pendulum_Problem.ipynb create mode 100644 randy_schur/figures/spring-mass-damper.png diff --git a/Double_Pendulum_Problem.ipynb b/Double_Pendulum_Problem.ipynb deleted file mode 100644 index 59df853..0000000 --- a/Double_Pendulum_Problem.ipynb +++ /dev/null @@ -1,706 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Integration of Mechanical Systems\n", - "\n", - "In this module, we will be exploring numerical integrators and how they perform for different mechanical systems." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Simple Harmonic Oscillator\n", - "\n", - "The simple harmonic oscillator is a simple physical system described by the second-order differential equation below. Despite its simplicity, this is a system that shows up in a similar form in many different fields of engineering. The same equation describes the behavior of a spring-mass-damper system, RLC circuit, or the motion of a (small-angle) pendulum.\n", - "\n", - "(Insert diagram for spring mass damper)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The differential equation describing the motion of a simple harmonic oscillator is:\n", - "\n", - "$$\\begin{equation}\n", - "0 = m\\ddot{x} + kx\n", - "\\end{equation}$$\n", - "\n", - "We can describe the state of the system by its position $x$ and its velocity $\\dot{x}$. We will call this state vector $q$, as:\n", - "\n", - "$$\\begin{eqnarray}\n", - "\\vec{q} = \\begin{bmatrix} x\\\\ \\dot{x} \\end{bmatrix}\n", - "\\end{eqnarray}$$\n", - "\n", - "This system can then be broken into two first order differential equations:\n", - "\n", - "$$\\begin{eqnarray}\n", - "\\vec{\\dot{q}} = f(\\vec{q}) = \\begin{bmatrix} q_2\\\\ -\\frac{k}{m}q_1 \\end{bmatrix}\n", - "\\end{eqnarray}$$\n", - "\n", - "To look at the behavior of this system, we can start with some initial conditions for $q$, then use a numerical integration method such as Euler's method to integrate forward in time. " - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Equations of Motion from Lagrangian\n", - "\n", - "You are encouraged to follow this derivation ON PAPER on your own. We gave you the equations of motion for a simple harmonic oscillator, but what if we don't have this equation? The following is a method that will work in nearly any situation to find equations of motion, often working well where a force analysis becomes difficult. This is good practice for anyone studying mechanics. To derive the equations of motion, we will start with the Lagrangian. This is defined by:\n", - "\n", - "$$\\begin{equation}\n", - "L = T- U\n", - "\\end{equation}$$\n", - "\n", - "where $T$ is the total Kinetic Energy in the system, and $U$ is the total Potential Energy. Potential energy is calculated as:\n", - "\n", - "$$\\begin{eqnarray*}\n", - "U &=& \\frac{1}{2}kx^2\\\\\n", - "\\end{eqnarray*}$$\n", - "\n", - "Kinetic Energy is calculated as:\n", - "\n", - "$$\\begin{eqnarray*}\n", - "T &=& \\frac{1}{2}mv^2 \\\\\n", - "&=& \\frac{1}{2}m(\\dot x)^2 \n", - "\\end{eqnarray*}$$\n", - "\n", - "\n", - "So the Lagrangian quantity becomes:\n", - "$$\\begin{equation}\n", - "L = \\frac{1}{2}m(\\dot x)^2 - \\frac{1}{2}kx^2\\\\\n", - "\\end{equation}$$\n", - "\n", - "\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The equations of motion are then found using the Lagrange Equation:\n", - "\n", - "$$\\begin{equation}\n", - "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x_i}}\\right) - \\frac{\\partial L}{\\partial x_i} = 0\n", - "\\end{equation}$$\n", - "\n", - "Since we have only one state (position), we need only one equation of motion. We'll calculate this in steps:\n", - "\n", - "$$\\begin{eqnarray*}\n", - "\\frac{\\partial L}{\\partial \\dot{x}} &=& m\\dot{x} \\\\\n", - "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x}}\\right) &=& m\\ddot{x}\\\\\n", - "-\\frac{\\partial L}{\\partial x} &=& + kx\n", - "\\end{eqnarray*}$$\n", - "\n", - "We can add together the second and third equations to find the first equation of motion. Some of these terms will cancel out, and we are left with a final equation of motion:\n", - "\n", - "$$\\begin{equation}\n", - "0 = m\\ddot{x} + kx\n", - "\\end{equation}$$\n", - "\n", - "This is the same equation we have before! So, we can be relatively confident that this derivation is correct." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Integration Methods\n", - "\n", - "### Euler's method\n", - "\n", - "Euler's method is discussed in lesson 1 of the Numerical Methods course. The method goes as follows:\n", - "$$x_i^{n+1} = x_i^n + h\\dot{x}_i^n$$\n", - "\n", - "where $h$ is the timestep. This isn't a complicated scheme, so for details see the code below." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Runge-Kutta Integration (RK4)\n", - "\n", - "Depending on the application, Euler's method may provide enough accuracy with a small timestep, especially a simple harmonic oscillator. However, Euler is a first-order method and we can do better. One option is a Runge-Kutta scheme. This is a popular numerical integration method which is easily extended to higher orders. Here we will use a fourth order Runge-Kutta, also known as RK4:\n", - "\n", - "$$\\begin{equation}\n", - "q_{n+1} = q_n + \\frac{h}{6}\\left(k_1+2k_2+2k_3+k_4\\right)\n", - "\\end{equation}$$\n", - "where\n", - "$$\\begin{eqnarray}\n", - "k_1 &=& f(t_n, x_n)\\\\\n", - "k_2 &=& f(t_n+\\frac{h}{2}, x_n+\\frac{h}{2}k_1)\\\\\n", - "k_3 &=& f(t_n+\\frac{h}{2}, x_n+\\frac{h}{2}k_2)\\\\\n", - "k_4 &=& f(t_n+h, y_n+hk_3)\\\\\n", - "t_{n+1} &=& t_n + h\n", - "\\end{eqnarray}$$\n", - "\n", - "For more information on the Runge-Kutta method, including the derivation and explanation of coefficients, see (cite)\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Symplectic Integrators\n", - "\n", - "Symplectic integrators are similar to the methods discussed above, but they use equations of motion derived from Hamiltonian mechanics. According to [Berkeley source], syplectic integrators preserve the conserved Hamiltonian quantities. In practical terms, this works out to mean the methods reflect conservation of energy, down to a truncation error.\n", - "\n", - "The Hamiltonian equations of motion can be derived from the total energy in the system, much like the Lagrangian. The coordinates we use are position ($q$) and momentum ($p = m\\dot x$). The Hamiltonian is:\n", - "$$ H = T+V$$\n", - "where $T$ is the kinetic energy in the system, and $V$ is the potential energy in the system. So, we get:\n", - "$$\\begin{eqnarray*}\n", - "T &=& \\frac{1}{2}m\\dot{x}^2 = \\frac{p^2}{2m}\\\\\n", - "V &=& mgq\\\\\n", - "H &=& \\frac{p^2}{2m} + mgq\n", - "\\end{eqnarray*}$$\n", - "\n", - "\n", - "The Hamilton's equations are:\n", - "$$\\begin{eqnarray*}\n", - "\\frac{dp}{dt} &=& -\\frac{\\partial H}{\\partial q}\\\\\n", - "\\frac{dq}{dt} &=& \\frac{\\partial H}{\\partial p}\n", - "\\end{eqnarray*}$$\n", - "\n", - "So our equations of motion become:\n", - "$$\\begin{equation*}\n", - "\\begin{bmatrix} \\dot p \\\\ \\dot q \\end{bmatrix} = \\begin{bmatrix} -\\frac{\\partial H}{\\partial q}\\\\ \\frac{\\partial H}{\\partial p} \\end{bmatrix} = \\begin{bmatrix} -mg \\\\ \\frac{p}{m} \\end{bmatrix}\n", - "\\end{equation*}$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let's now take a look at symplectic integration methods. We'll start with a first order method - symplectic Euler's method. This uses the equations:\n", - "\n", - "$$\\begin{eqnarray*}\n", - "p_{n+1} = p_n - h \\frac{\\partial H}{\\partial q} \\Bigr|_{p_{n+1},q_n}\\\\\n", - "q_{n+1} = q_n + \\frac{h}{2} \\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1},q_n}\n", - "\\end{eqnarray*}$$\n", - "\n", - "We can then plug in the partial derivatives and integrate the system. This system will do a better job of conserving energy than Euler's method, $\\textit{but only to a truncation error.}$ Again, we can do better than first-order. Let's try a second-order symplectic scheme, also called Verlet integration. We'll have the same equations of motion, but this time a different set of integration equations:\n", - "\n", - "$$\\begin{eqnarray*}\n", - "p_{n+1/2} &=& p_n - \\frac{h}{2} \\frac{\\partial H}{\\partial q} \\Bigr|_{p_{n+1/2},q_n}\\\\\n", - "q_{n+1} &=& q_n + \\frac{h}{2} \\left(\\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1/2},q_n} + \\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1/2},q_{n+1/2}} \\right) \\\\\n", - "p_{n+1} &=& p_{n+1/2} - \\frac{h}{2} \\frac{\\partial H}{\\partial q}\\Bigr|_{p_{n+1/2},q_{n+1/2}}\n", - "\\end{eqnarray*}$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Analytical Solution\n", - "The simple harmonic oscillator is a nice example for numerical integration, because it has an analytical form which we can compare to. The analytical solution is\n", - "\n", - "$$\\begin{equation}\n", - "x(t) = Acos(\\omega_n t + \\phi)\n", - "\\end{equation}$$\n", - "\n", - "where $\\omega_n$ is the natural frequency of the system given by $\\omega_n = \\sqrt{\\frac{k}{m}}$ . The amplitude, $A$, and the phase, $\\phi$, of oscillation are determined from the initial conditions. " - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Code section?" - ] - }, - { - "cell_type": "markdown", - "metadata": { - "collapsed": true - }, - "source": [ - "### Adding a damping term\n", - "\n", - "A more realistic oscillator also has a damping term. This might correspond to friction, or to some other non-conservative force. For now, we will assume that the damping is linear, and is proportional to the velocity (this is called the -- model of friction). We get the following equation of motion:\n", - "\n", - "$$\\begin{equation*}\n", - "0 = m\\ddot{x} + c\\dot{x} + kx\n", - "\\end{equation*}$$\n", - "\n", - "This gives us the differential equations:\n", - "$$\\begin{eqnarray*}\n", - "\\vec{\\dot{q}} = f(\\vec{q}) = \\begin{bmatrix} q_2\\\\ -\\frac{c}{m}\\dot{q_1} -\\frac{k}{m}q_1 \\end{bmatrix}\n", - "\\end{eqnarray*}$$\n", - "\n", - "The Euler's method and Runge-Kutta integration work the same way with this new equation. We also have a new analytical solution. Assuming the system is underdamped (add explanation of damping ratio), the solution becomes:\n", - "\n", - "\n", - "$$\\begin{equation}\n", - "x(t) = Acos(\\omega_d t + \\phi)\n", - "\\end{equation}$$\n", - "\n", - "where $\\omega_d = \\omega_n\\sqrt{1 - \\zeta^2}$ is called the damped natural frequency.\n", - "\n", - "The big change comes in the Hamiltonian equations. If these are dependent on convservation of energy, how do we deal with a system where energy is not conserved?\n", - "\n", - "Our first step is to re-examine the Lagrangian. Since losses due to friction are neither potential nor kinetic energy, we'll need a new term. The Lagrangian equation of motion with non-conservative forces is:\n", - "\n", - "$$\\begin{equation}\n", - "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x_i}}\\right) - \\frac{\\partial L}{\\partial x_i} = Q\n", - "\\end{equation}$$\n", - "\n", - "where $Q$ is the sum of all non-conservative forces. In this case, \n", - "$$Q = c\\frac{\\partial x}{\\partial t}$$\n", - "\n", - "Now that we have this, we can add a similar term to the Hamiltonian equations of motion to deal with the same problem. For more information on the derivation of the Hamiltonian from the Lagrangian, see Tveter (cite). The Hamiltonian equations of motion become:\n", - "\n", - "$$\\begin{equation*}\n", - "\\begin{bmatrix} \\dot p \\\\ \\dot q \\end{bmatrix} = \\begin{bmatrix} -\\frac{\\partial H}{\\partial q} + Q\\\\ \\frac{\\partial H}{\\partial p} \\end{bmatrix} = \\begin{bmatrix} -mg + \\frac{c}{m}q \\\\ \\frac{p}{m} \\end{bmatrix}\n", - "\\end{equation*}$$\n", - "\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Dynamics of a Double Pendulum" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this module, we will study the dynamics of a double pendulum. We first derive the equations of motion from the Lagrangian. Next, we take a look at three different numerical integration schemes: Euler's method, Runge-Kutta, and a symplectic integrator. We will investigate how the initial conditions affect the behavior of the system, as well as some of the sources of error in these simulations. Finally, we will look at the properties of the system in phase space using phase plots and Poincare sections." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Background\n", - "\n", - "The double pendulum problem is a simple system which can produce surprisingly complex movements. It is a chaotic system, meaning it is unpredictable, and there is no closed form solution to the motion of the two masses. This is a frequent example problem in the study of dynamic systems, nonlinear controls, and mechanics.\n", - "\n", - "In this notebook we will be treating the two rods as massless and ignoring the effects of friction. For a treatment of this problem without these assumptions, see: \n", - "\n", - "See the image below for the definition of constants." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "![Image](figures/diagram.png)\n", - "#### Figure 1. Diagram of Double Pendulum System" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Equations of Motion \n", - "\n", - "You are encouraged to follow this derivation ON PAPER on your own. The following is a method that will work in nearly any situation to find equations of motion, often working well where a force analysis becomes difficult. This is good practice for anyone studying mechanics. To derive the equations of motion, we will start with the Lagrangian. This is defined by:\n", - "\n", - "$$\\begin{equation}\n", - "L = T- U\n", - "\\end{equation}$$\n", - "\n", - "where $T$ is the total Kinetic Energy in the system, and $U$ is the total Potential Energy. Potential energy is calculated as:\n", - "\n", - "$$\\begin{eqnarray}\n", - "U &=& m_1g(-y_1) + m_2g(-y_2) \\\\\n", - " &=& -m_1g(l_1cos(\\theta_1) - m_2g(l_1cos\\theta_1 + l_2cos\\theta_2)\n", - "\\end{eqnarray}$$\n", - "\n", - "Kinetic Energy is calculated as:\n", - "\n", - "$$\\begin{eqnarray*}\n", - "T &=& \\frac{1}{2}mv_1^2+\\frac{1}{2}mv_2^2 \\\\\n", - "&=& \\frac{1}{2}m_1(l_1\\dot\\theta_1)^2 + \\frac{1}{2}m_2(l_1\\dot\\theta_1 + l_2\\dot\\theta_2)^2 \\\\\n", - "&=& \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2+\\frac{1}{2}m_2\\left(2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+l_2^2\\dot\\theta_2^2\\right)\n", - "\\end{eqnarray*}$$\n", - "\n", - "\n", - "So the Lagrangian quantity becomes:\n", - "$$\\begin{equation}\n", - "L = \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2+m_2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+(m_1+m_2)l_1gcos\\theta_1 + m_2l_2gcos\\theta_2\n", - "\\end{equation}$$\n" - ] - }, - { - "cell_type": "markdown", - "metadata": { - "collapsed": true - }, - "source": [ - "The equations of motion are then found using the Lagrange Equation:\n", - "\n", - "$$\\begin{equation}\n", - "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x_i}}\\right) - \\frac{\\partial L}{\\partial x_i} = 0\n", - "\\end{equation}$$\n", - "\n", - "\n", - "So the equation of motion for $\\theta_1$ is calculated by the following steps:\n", - "\n", - "$$\\begin{eqnarray*}\n", - "\\frac{\\partial L}{\\partial \\dot{\\theta_1}} &=& (m_1+m_2)l_1^2\\dot\\theta_1+m_2l_1l_2\\dot\\theta_2cos(\\theta_2-\\theta_1) \\\\\n", - "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{\\theta_1}}\\right) &=& (m_1+m_2)l_1^2\\ddot\\theta_1+m_2l_1l_2\\ddot\\theta_2cos(\\theta_2-\\theta_1)-m_2l_1l_2\\dot\\theta_2^2sin(\\theta_2-\\theta_1)+m_2l_1l_2\\dot\\theta_1\\dot\\theta_2sin(\\theta_2-\\theta_1) \\\\\n", - "-\\frac{\\partial L}{\\partial \\theta_1} &=& -m_2l_1l_2\\dot\\theta_1\\dot\\theta_2sin(\\theta_2-\\theta_1)-(m_1+m_2)gl_1sin\\theta_1\n", - "\\end{eqnarray*}$$\n", - "\n", - "We can add together the second and third equations to find the first equation of motion. Some of these terms will cancel out, and we are left with a final equation of motion:\n", - "\n", - "$$\\begin{equation}\n", - "0 = (m_1+m_2)l_1^2\\ddot\\theta_1+m_2l_1l_2\\ddot\\theta_2cos(\\theta_2-\\theta_1)-m_2l_1l_2\\dot\\theta_2^2sin(\\theta_2-\\theta_1)+(m_1+m_2)gl_1sin\\theta_1\n", - "\\end{equation}$$\n", - "\n", - "We will also need the equation of motion for $\\theta_2$. This can be derived using the same process. It should work out to the following (check this by yourself!):\n", - "\n", - "$$\\begin{equation}\n", - "0 = m_2l_2^2\\ddot\\theta_2+m_2l_1l_2\\ddot\\theta_1cos(\\theta_2-\\theta_1) - m_2l_2l_1\\dot\\theta_1^2sin(\\theta_2-\\theta_1)+ l_2m_2gsin\\theta_2\n", - "\\end{equation}$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Problem setup\n", - "\n", - "Notice that these equations of motion are implicit equations for $\\ddot\\theta_1$ and $\\ddot\\theta_2$. We have talked about implicit equations in this class, and we solve them by forming a system of equations. In this case, we will need to solve the equations of motion at each time step. We will characterize our system in state space using the angle and angular velocity of each pendulum. So our state becomes:\n", - "\n", - "$$\\begin{eqnarray}\n", - "\\vec{x}(t) = \\begin{pmatrix} \\theta_1\\\\ \\theta_2\\\\ \\dot\\theta_1 \\\\ \\dot\\theta_2 \\end{pmatrix}\n", - "\\end{eqnarray}$$\n", - "\n", - "The equations of motion can be rewritten as a system of equations:\n", - "\n", - "$$ A\\dot x=f(x)$$\n", - "\n", - "where \n", - "\n", - "\\begin{equation}\n", - "A = \n", - "\\left[\\begin{array}{c}\n", - "1 &0 &0 &0 \\\\\n", - "0 &1 &0 &0 \\\\\n", - "0 &0 &(m_1+m_2)l_1^2 &m_2l_1l_2cos(x_2-x_1) \\\\\n", - "0 &0 &m_2l_1l_2cos(x_2-x_1) &m_2l_2^2\n", - "\\end{array}\\right]\n", - "\\end{equation}\n", - "\n", - "and\n", - "\n", - "$$\\begin{eqnarray}\n", - "f(\\vec{x}(t)) = \\begin{pmatrix} x_3\\\\ x_4\\\\ m_2l_1l_2x_4^2sin(x_2-x_1) -(m_1+m_2)gl_1sin(x_1)\\\\ m_2l_1l_2x_3^2sin(x_2-x_1)-l_2m_2gsinx_2 \\end{pmatrix}\n", - "\\end{eqnarray}$$\n", - "\n", - "In order to solve this, we can evaluate matrix A and vector $f(x)$ at each time step, then invert A and premultiply the result to each side of the equation. This gives us an explicit system of equations of motion:\n", - "\n", - "$$ \\dot x=A^{-1}f(x)$$\n", - "\n", - "You may notice that these operations require a significant computational load at each time step. In order to reduce this, we can analytically invert the matrix A, then simply fill in the necessary values at each time step and multiply by $f(x)$. This will significantly speed up running time, as inverting a matrix is an expensive operation.\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Integration\n", - "\n", - "### Euler's Method\n", - "\n", - "The first method we will use is Euler integration, covered in lesson 1 of the Numerical Methods course. This is the simplest form of numerical integration, but it can still be effective. Using our equations for $\\dot x$ and $f(x)$:\n", - "\n", - "$$x_i^{n+1} = x_i^n + h\\dot{x}_i^n$$\n", - "\n", - "where $h$ is the timestep.\n", - "\n", - "### Runge-Kutta Method (RK4)\n", - "\n" - ] - }, - { - "cell_type": "markdown", - "metadata": { - "collapsed": true - }, - "source": [ - "## Solve!\n", - "\n", - "Here we define functions for each of the integration methods, set our parameters and initial conditions, and then numerically integrate the motion of the pendulums. The plots below compare the different integration techniques." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Sources: \n", - "\n", - "http://scienceworld.wolfram.com/physics/DoublePendulum.html\n", - "\n", - "http://www.phy.uct.ac.za/courses/opencontent/phylab2/worksheet9_09.pdf\n", - "\n", - "http://www.phys.lsu.edu/faculty/gonzalez/Teaching/Phys7221/DoublePendulum.pdf\n", - "\n", - "http://www.iontrap.wabash.edu/adlab/papers/F2011_foster_groninger_tang_chaos.pdf\n", - "\n", - "https://math.berkeley.edu/~alanw/242papers99/markiewicz.pdf\n", - "\n", - "http://www.unige.ch/~hairer/poly_geoint/week2.pdf\n", - "\n", - "http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1994CeMDA..60..409T&defaultprint=YES&filetype=.pdf\n", - "\n" - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "execution_count": 2, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "# This cell loads the style of the notebook, which is taken from the \n", - "# Numerical Methods in Python Course: http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about\n", - "\n", - "from IPython.core.display import HTML\n", - "css_file = './randy_schur/numericalmoocstyle.css'\n", - "HTML(open(css_file, \"r\").read())" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.4.3" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index 3fcaa39..125771d 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -15,9 +15,16 @@ "source": [ "## Simple Harmonic Oscillator\n", "\n", - "The simple harmonic oscillator is a simple physical system described by the second-order differential equation below. Despite its simplicity, this is a system that shows up in a similar form in many different fields of engineering. The same equation describes the behavior of a spring-mass-damper system, RLC circuit, or the motion of a (small-angle) pendulum.\n", + "The simple harmonic oscillator is a simple physical system described by the second-order differential equation below. Despite its simplicity, this is a system that shows up in a similar form in many different fields of engineering. The same equation describes the behavior of a spring-mass-damper system, RLC circuit, or the motion of a (small-angle) pendulum.\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "![Image](figures/spring-mass-damper.png)\n", "\n", - "(Insert diagram for spring mass damper)" + "##### Figure 1. Diagram of Spring-Mass-Damper system (from https://en.wikipedia.org/wiki/Harmonic_oscillator)" ] }, { @@ -48,12 +55,70 @@ { "cell_type": "markdown", "metadata": {}, - "source": [] + "source": [ + "### Equations of Motion from Lagrangian\n", + "\n", + "You are encouraged to follow this derivation ON PAPER on your own. We gave you the equations of motion for a simple harmonic oscillator, but what if we don't have this equation? The following is a method that will work in nearly any situation to find equations of motion, often working well where a force analysis becomes difficult. This is good practice for anyone studying mechanics. To derive the equations of motion, we will start with the Lagrangian. This is defined by:\n", + "\n", + "$$\\begin{equation}\n", + "L = T- U\n", + "\\end{equation}$$\n", + "\n", + "where $T$ is the total Kinetic Energy in the system, and $U$ is the total Potential Energy. Potential energy is calculated as:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "U &=& \\frac{1}{2}kx^2\\\\\n", + "\\end{eqnarray*}$$\n", + "\n", + "Kinetic Energy is calculated as:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "T &=& \\frac{1}{2}mv^2 \\\\\n", + "&=& \\frac{1}{2}m(\\dot x)^2 \n", + "\\end{eqnarray*}$$\n", + "\n", + "\n", + "So the Lagrangian quantity becomes:\n", + "$$\\begin{equation}\n", + "L = \\frac{1}{2}m(\\dot x)^2 - \\frac{1}{2}kx^2\\\\\n", + "\\end{equation}$$\n", + "\n", + "\n" + ] }, { "cell_type": "markdown", "metadata": {}, "source": [ + "The equations of motion are then found using the Lagrange Equation:\n", + "\n", + "$$\\begin{equation}\n", + "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x_i}}\\right) - \\frac{\\partial L}{\\partial x_i} = 0\n", + "\\end{equation}$$\n", + "\n", + "Since we have only one generalized coordinate (position), we need only one equation of motion. We'll calculate this in steps:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "\\frac{\\partial L}{\\partial \\dot{x}} &=& m\\dot{x} \\\\\n", + "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x}}\\right) &=& m\\ddot{x}\\\\\n", + "-\\frac{\\partial L}{\\partial x} &=& + kx\n", + "\\end{eqnarray*}$$\n", + "\n", + "We can add together the second and third equations to find the first equation of motion. Some of these terms will cancel out, and we are left with a final equation of motion:\n", + "\n", + "$$\\begin{equation}\n", + "0 = m\\ddot{x} + kx\n", + "\\end{equation}$$\n", + "\n", + "This is the same equation we have before! So, we can be relatively confident that this derivation is correct." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Integration Methods\n", + "\n", "### Euler's method\n", "\n", "Euler's method is discussed in lesson 1 of the Numerical Methods course. The method goes as follows:\n", @@ -96,9 +161,12 @@ "The Hamiltonian equations of motion can be derived from the total energy in the system, much like the Lagrangian. The coordinates we use are position ($q$) and momentum ($p = m\\dot x$). The Hamiltonian is:\n", "$$ H = T+V$$\n", "where $T$ is the kinetic energy in the system, and $V$ is the potential energy in the system. So, we get:\n", - "$$T = \\frac{1}{2}m\\dot{x}^2 = \\frac{p^2}{2m}$$\n", - "and\n", - "$$V = mgq$$\n", + "$$\\begin{eqnarray*}\n", + "T &=& \\frac{1}{2}m\\dot{x}^2 = \\frac{p^2}{2m}\\\\\n", + "V &=& mgq\\\\\n", + "H &=& \\frac{p^2}{2m} + mgq\n", + "\\end{eqnarray*}$$\n", + "\n", "\n", "The Hamilton's equations are:\n", "$$\\begin{eqnarray*}\n", @@ -108,7 +176,7 @@ "\n", "So our equations of motion become:\n", "$$\\begin{equation*}\n", - "\\begin{bmatrix} \\dot q \\\\ \\dot p \\end{bmatrix} = \\begin{bmatrix} -\\frac{\\partial H}{\\partial q}\\\\ \\frac{\\partial H}{\\partial p} \\end{bmatrix} = \\begin{bmatrix} -mg \\\\ \\frac{p}{m} \\end{bmatrix}\n", + "\\begin{bmatrix} \\dot p \\\\ \\dot q \\end{bmatrix} = \\begin{bmatrix} -\\frac{\\partial H}{\\partial q}\\\\ \\frac{\\partial H}{\\partial p} \\end{bmatrix} = \\begin{bmatrix} -mg \\\\ \\frac{p}{m} \\end{bmatrix}\n", "\\end{equation*}$$" ] }, @@ -123,7 +191,7 @@ "q_{n+1} = q_n + \\frac{h}{2} \\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1},q_n}\n", "\\end{eqnarray*}$$\n", "\n", - "We can then plug in the partial derivatives and integrate the system. This system will do a better job of conserving energy than Euler's method, $\\textit{but only to a truncation error.}$ Again, we can do better than first-order. Let's try a second-order symplectic scheme, also called Verlet integration. We'll have the same equations of motion, but this time a different set of integration equations:\n", + "We can then plug in the partial derivatives and integrate the system. This system will do a better job of conserving energy than Euler's method, $\\textit{but only to a truncation error.}$ Again, we can do better than first-order. Let's try a second-order symplectic scheme, also called Verlet integration. We'll have the same equations of motion, but this time a different set of integration equations:\n", "\n", "$$\\begin{eqnarray*}\n", "p_{n+1/2} &=& p_n - \\frac{h}{2} \\frac{\\partial H}{\\partial q} \\Bigr|_{p_{n+1/2},q_n}\\\\\n", @@ -132,11 +200,6 @@ "\\end{eqnarray*}$$" ] }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [] - }, { "cell_type": "markdown", "metadata": {}, @@ -148,7 +211,14 @@ "x(t) = Acos(\\omega_n t + \\phi)\n", "\\end{equation}$$\n", "\n", - "where $\\omega_n$ is the natural frequency of the system given by $\\sqrt{\\frac{k}{m}}$ . The amplitude, $A$, and the phase, $\\phi$, of oscillation are determined from the initial conditions. " + "where $\\omega_n$ is the natural frequency of the system given by $\\omega_n = \\sqrt{\\frac{k}{m}}$ . The amplitude, $A$, and the phase, $\\phi$, of oscillation are determined from the initial conditions. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Code section?" ] }, { @@ -170,7 +240,7 @@ "\\vec{\\dot{q}} = f(\\vec{q}) = \\begin{bmatrix} q_2\\\\ -\\frac{c}{m}\\dot{q_1} -\\frac{k}{m}q_1 \\end{bmatrix}\n", "\\end{eqnarray*}$$\n", "\n", - "We also have a new analytical solution. Assuming the system is underdamped (add explanation of damping ratio), the solution becomes:\n", + "The Euler's method and Runge-Kutta integration work the same way with this new equation. We also have a new analytical solution. Assuming the system is underdamped (add explanation of damping ratio), the solution becomes:\n", "\n", "\n", "$$\\begin{equation}\n", @@ -178,18 +248,34 @@ "\\end{equation}$$\n", "\n", "where $\\omega_d = \\omega_n\\sqrt{1 - \\zeta^2}$ is called the damped natural frequency.\n", - "\n" + "\n", + "The big change comes in the Hamiltonian equations. If these are dependent on convservation of energy, how do we deal with a system where energy is not conserved?\n", + "\n", + "Our first step is to re-examine the Lagrangian. Since losses due to friction are neither potential nor kinetic energy, we'll need a new term. The Lagrangian equation of motion with non-conservative forces is:\n", + "\n", + "$$\\begin{equation}\n", + "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x_i}}\\right) - \\frac{\\partial L}{\\partial x_i} = Q\n", + "\\end{equation}$$\n", + "\n", + "where $Q$ is the sum of all non-conservative forces. In this case, \n", + "$$Q = c\\frac{\\partial x}{\\partial t}$$\n", + "\n", + "Now that we have this, we can add a similar term to the Hamiltonian equations of motion to deal with the same problem. For more information on the derivation of the Hamiltonian from the Lagrangian, see Tveter (cite). The Hamiltonian equations of motion become:\n", + "\n", + "$$\\begin{equation*}\n", + "\\begin{bmatrix} \\dot p \\\\ \\dot q \\end{bmatrix} = \\begin{bmatrix} -\\frac{\\partial H}{\\partial q} + Q\\\\ \\frac{\\partial H}{\\partial p} \\end{bmatrix} = \\begin{bmatrix} -mg + \\frac{c}{m}q \\\\ \\frac{p}{m} \\end{bmatrix}\n", + "\\end{equation*}$$\n", + "\n", + "From here, the we can use the same Verlet integration method as before." ] }, { - "cell_type": "code", - "execution_count": null, + "cell_type": "markdown", "metadata": { "collapsed": true }, - "outputs": [], "source": [ - "### Adding a " + "### Duffing equation (Ading a non-linear damping term)" ] }, { @@ -228,15 +314,6 @@ "outputs": [], "source": [] }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, { "cell_type": "markdown", "metadata": {}, @@ -248,7 +325,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "In this module, we will study the dynamics of a double pendulum. We first derive the equations of motion from the Lagrangian. Next, we take a look at three different numerical integration schemes: Euler's method, Runge-Kutta, and a symplectic integrator. We will investigate how the initial conditions affect the behavior of the system, as well as some of the sources of error in these simulations. Finally, we will look at the properties of the system in phase space using phase plots and Poincare sections." + "In this module, we will study the dynamics of a double pendulum. Like before, we'll derive the equations of motion from the Lagrangian and the Hamiltonian. We can use the same methods of integration as before, but this time there is no analytical solution to compare to. Finally, we will look at the properties of the system in phase space using phase plots and Poincare sections." ] }, { @@ -259,7 +336,7 @@ "\n", "The double pendulum problem is a simple system which can produce surprisingly complex movements. It is a chaotic system, meaning it is unpredictable, and there is no closed form solution to the motion of the two masses. This is a frequent example problem in the study of dynamic systems, nonlinear controls, and mechanics.\n", "\n", - "In this notebook we will be treating the two rods as massless and ignoring the effects of friction. For a treatment of this problem without these assumptions, see: \n", + "In this notebook we will be treating the two rods as massless and ignoring the effects of friction. For a treatment of this problem without these assumptions, see (cite).\n", "\n", "See the image below for the definition of constants." ] @@ -276,20 +353,18 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "## Equations of Motion\n", - "\n", - "You are encouraged to follow this derivation ON PAPER on your own. The following is a method that will work in nearly any situation to find equations of motion, often working well where a force analysis becomes difficult. This is good practice for anyone studying mechanics. To derive the equations of motion, we will start with the Lagrangian. This is defined by:\n", + "## Equations of Motion \n", "\n", - "$$\\begin{equation}\n", + "As a reminder, the Lagrangian is\n", + "$$\\begin{equation*}\n", "L = T- U\n", - "\\end{equation}$$\n", - "\n", + "\\end{equation*}$$\n", "where $T$ is the total Kinetic Energy in the system, and $U$ is the total Potential Energy. Potential energy is calculated as:\n", "\n", - "$$\\begin{eqnarray}\n", + "$$\\begin{eqnarray*}\n", "U &=& m_1g(-y_1) + m_2g(-y_2) \\\\\n", " &=& -m_1g(l_1cos(\\theta_1) - m_2g(l_1cos\\theta_1 + l_2cos\\theta_2)\n", - "\\end{eqnarray}$$\n", + "\\end{eqnarray*}$$\n", "\n", "Kinetic Energy is calculated as:\n", "\n", @@ -301,9 +376,9 @@ "\n", "\n", "So the Lagrangian quantity becomes:\n", - "$$\\begin{equation}\n", + "$$\\begin{equation*}\n", "L = \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2+m_2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+(m_1+m_2)l_1gcos\\theta_1 + m_2l_2gcos\\theta_2\n", - "\\end{equation}$$\n" + "\\end{equation*}$$\n" ] }, { @@ -314,12 +389,12 @@ "source": [ "The equations of motion are then found using the Lagrange Equation:\n", "\n", - "$$\\begin{equation}\n", + "$$\\begin{equation*}\n", "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x_i}}\\right) - \\frac{\\partial L}{\\partial x_i} = 0\n", - "\\end{equation}$$\n", + "\\end{equation*}$$\n", "\n", "\n", - "So the equation of motion for $\\theta_1$ is calculated by the following steps:\n", + "There are two generalized coordinates ($\\theta_1 and \\theta_2$. So the equation of motion for $\\theta_1$ is calculated by the following steps:\n", "\n", "$$\\begin{eqnarray*}\n", "\\frac{\\partial L}{\\partial \\dot{\\theta_1}} &=& (m_1+m_2)l_1^2\\dot\\theta_1+m_2l_1l_2\\dot\\theta_2cos(\\theta_2-\\theta_1) \\\\\n", @@ -329,15 +404,15 @@ "\n", "We can add together the second and third equations to find the first equation of motion. Some of these terms will cancel out, and we are left with a final equation of motion:\n", "\n", - "$$\\begin{equation}\n", + "$$\\begin{equation*}\n", "0 = (m_1+m_2)l_1^2\\ddot\\theta_1+m_2l_1l_2\\ddot\\theta_2cos(\\theta_2-\\theta_1)-m_2l_1l_2\\dot\\theta_2^2sin(\\theta_2-\\theta_1)+(m_1+m_2)gl_1sin\\theta_1\n", - "\\end{equation}$$\n", + "\\end{equation*}$$\n", "\n", "We will also need the equation of motion for $\\theta_2$. This can be derived using the same process. It should work out to the following (check this by yourself!):\n", "\n", - "$$\\begin{equation}\n", + "$$\\begin{equation*}\n", "0 = m_2l_2^2\\ddot\\theta_2+m_2l_1l_2\\ddot\\theta_1cos(\\theta_2-\\theta_1) - m_2l_2l_1\\dot\\theta_1^2sin(\\theta_2-\\theta_1)+ l_2m_2gsin\\theta_2\n", - "\\end{equation}$$" + "\\end{equation*}$$" ] }, { @@ -349,16 +424,16 @@ "Notice that these equations of motion are implicit equations for $\\ddot\\theta_1$ and $\\ddot\\theta_2$. We have talked about implicit equations in this class, and we solve them by forming a system of equations. In this case, we will need to solve the equations of motion at each time step. We will characterize our system in state space using the angle and angular velocity of each pendulum. So our state becomes:\n", "\n", "$$\\begin{eqnarray}\n", - "\\vec{x}(t) = \\begin{pmatrix} \\theta_1\\\\ \\theta_2\\\\ \\dot\\theta_1 \\\\ \\dot\\theta_2 \\end{pmatrix}\n", + "\\vec{q}(t) = \\begin{pmatrix} \\theta_1\\\\ \\theta_2\\\\ \\dot\\theta_1 \\\\ \\dot\\theta_2 \\end{pmatrix}\n", "\\end{eqnarray}$$\n", "\n", "The equations of motion can be rewritten as a system of equations:\n", "\n", - "$$ A\\dot x=f(x)$$\n", + "$$ A\\dot q=f(q)$$\n", "\n", "where \n", "\n", - "\\begin{equation}\n", + "\\begin{equation*}\n", "A = \n", "\\left[\\begin{array}{c}\n", "1 &0 &0 &0 \\\\\n", @@ -366,19 +441,19 @@ "0 &0 &(m_1+m_2)l_1^2 &m_2l_1l_2cos(x_2-x_1) \\\\\n", "0 &0 &m_2l_1l_2cos(x_2-x_1) &m_2l_2^2\n", "\\end{array}\\right]\n", - "\\end{equation}\n", + "\\end{equation*}\n", "\n", "and\n", "\n", - "$$\\begin{eqnarray}\n", - "f(\\vec{x}(t)) = \\begin{pmatrix} x_3\\\\ x_4\\\\ m_2l_1l_2x_4^2sin(x_2-x_1) -(m_1+m_2)gl_1sin(x_1)\\\\ m_2l_1l_2x_3^2sin(x_2-x_1)-l_2m_2gsinx_2 \\end{pmatrix}\n", - "\\end{eqnarray}$$\n", + "$$\\begin{eqnarray*}\n", + "f(\\vec{q}(t)) = \\begin{pmatrix} q_3\\\\ q_4\\\\ m_2l_1l_2x_4^2sin(q_2-q_1) -(m_1+m_2)gl_1sin(q_1)\\\\ m_2l_1l_2q_3^2sin(q_2-q_1)-l_2m_2gsinq_2 \\end{pmatrix}\n", + "\\end{eqnarray*}$$\n", "\n", "In order to solve this, we can evaluate matrix A and vector $f(x)$ at each time step, then invert A and premultiply the result to each side of the equation. This gives us an explicit system of equations of motion:\n", "\n", - "$$ \\dot x=A^{-1}f(x)$$\n", + "$$ \\dot q=A^{-1}f(q)$$\n", "\n", - "You may notice that these operations require a significant computational load at each time step. In order to reduce this, we can analytically invert the matrix A, then simply fill in the necessary values at each time step and multiply by $f(x)$. This will significantly speed up running time, as inverting a matrix is an expensive operation.\n" + "You may notice that these operations require a significant computational load at each time step. In order to reduce this, we can analytically invert the matrix A, then simply fill in the necessary values at each time step and multiply by $f(q)$. This will significantly speed up running time, as inverting a matrix is an expensive operation.\n" ] }, { @@ -387,15 +462,10 @@ "source": [ "## Integration\n", "\n", - "### Euler's Method\n", - "\n", - "The first method we will use is Euler integration, covered in lesson 1 of the Numerical Methods course. This is the simplest form of numerical integration, but it can still be effective. Using our equations for $\\dot x$ and $f(x)$:\n", - "\n", - "$$x_i^{n+1} = x_i^n + h\\dot{x}_i^n$$\n", + "For Euler and Runge-Kutta, we can use the equation above to integrate our system. The results will be VERY dependent on initial conditions; this is why we call it a chaotic system.\n", "\n", - "where $h$ is the timestep.\n", + "For the symplectic integrator, we'll need the equations of motion from the Hamiltonian. If you want good practice, you can derive these on your own. The Hamiltonian equations become:\n", "\n", - "### Runge-Kutta Method (RK4)\n", "\n" ] }, @@ -434,7 +504,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 1, "metadata": { "collapsed": false }, @@ -589,7 +659,7 @@ "" ] }, - "execution_count": 2, + "execution_count": 1, "metadata": {}, "output_type": "execute_result" } @@ -599,7 +669,7 @@ "# Numerical Methods in Python Course: http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about\n", "\n", "from IPython.core.display import HTML\n", - "css_file = 'numericalmoocstyle.css'\n", + "css_file = './randy_schur/numericalmoocstyle.css'\n", "HTML(open(css_file, \"r\").read())" ] }, diff --git a/randy_schur/figures/spring-mass-damper.png b/randy_schur/figures/spring-mass-damper.png new file mode 100644 index 0000000000000000000000000000000000000000..6d473b7743810a379b1b68dd02bfcfb5bde94bb0 GIT binary patch literal 11939 zcmZvicRbba|NqZ$DB_TntPmJIF<L{!WfQUTu=Trsr8Hj|89AZ8Y%vy~&j(sOt;=pzaGnDn0HvKgWWCICZ^2^E~kR 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zbjp!fy<-(Km2b1!vX=(kXR&%@kAYyDo*g`+Vi+JTqCJ*Gts4*Q%|UfYWJspr$5i6X zj>?;h?%3m=CLPfDrDQXz)KARFB;z=3Xdp#E)Qm7OYZ%|fufzwJT#d8mDo$ZMO)Bm9 z%rd?9C)8>+;g>p~{KSC@(@&e2codqeI(_@0S+V3vXu66r#vkffDJSc})qxRXc8+q8 zKl`)Ap3@U#Gvl^PKhY_bIMmTC8_@LsBV02yoxvX}#Zn`ly|D)Fn1j@mwUtT~tU~?= DQ=vp- literal 0 HcmV?d00001 From 6c7ef12ee3ba553cfd168b314ce11c3f27f8d499 Mon Sep 17 00:00:00 2001 From: rbds Date: Fri, 4 Dec 2015 18:33:39 -0500 Subject: [PATCH 11/61] bad job starting programming --- randy_schur/Double_Pendulum_Problem.ipynb | 118 ++++++++++++++++++++-- 1 file changed, 112 insertions(+), 6 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index 125771d..6696c5c 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -218,9 +218,115 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Code section?" + "### Let's Get to Programming" ] }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy\n", + "from scipy.linalg import solve\n", + "from math import pi\n", + "\n", + "from matplotlib import pyplot\n", + "%matplotlib notebook\n", + "from matplotlib import rcParams, cm\n", + "rcParams['font.family'] = 'serif'\n", + "rcParams['font.size'] = 16" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "#Set up parameters:\n", + "\n", + "m = 3\n", + "k = 17;\n", + "\n", + "T = 2.5; #[seconds]\n", + "dt = .01; #\n", + "N = int(T/dt)+1\n", + "t = numpy.linspace(0.0, T, N)\n", + "\n", + "#Initial Conditions\n", + "x0 = numpy.array([pi/2, 0])\n", + "xdot0 = numpy.zeros_like(x0)\n" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def euler(u, f, dt):\n", + " \"\"\" Euler's method for integrating a system of differential equations.\n", + " \n", + " Parameters:\n", + " IC - \n", + " x -\n", + " dt-\n", + " \n", + " Returns: \n", + " x - array of values at time T.\n", + " \"\"\"\n", + " \n", + " return u + dt*f(u)\n", + " " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def f(u):\n", + " \"\"\"Returns RHS of harmonic oscillator EOM\n", + " \n", + " \n", + " \"\"\"\n", + " pos = u[0]\n", + " vel = u[1]\n", + " \n", + " return numpy.array([vel, k/m*vel])\n", + " \n", + " \n", + " " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, { "cell_type": "markdown", "metadata": { @@ -685,21 +791,21 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3", + "display_name": "Python 2", "language": "python", - "name": "python3" + "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", - "version": 3 + "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.4.3" + "pygments_lexer": "ipython2", + "version": "2.7.10" } }, "nbformat": 4, From bb0a2b05918f50b68a073dc16a66ff8beb501705 Mon Sep 17 00:00:00 2001 From: rbds Date: Sat, 5 Dec 2015 16:25:54 -0500 Subject: [PATCH 12/61] better start to programming --- randy_schur/Double_Pendulum_Problem.ipynb | 855 +++++++++++++++++++++- 1 file changed, 830 insertions(+), 25 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index 6696c5c..d2b7906 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -223,7 +223,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 60, "metadata": { "collapsed": false }, @@ -231,7 +231,7 @@ "source": [ "import numpy\n", "from scipy.linalg import solve\n", - "from math import pi\n", + "from math import pi, cos, sin, sqrt\n", "\n", "from matplotlib import pyplot\n", "%matplotlib notebook\n", @@ -242,30 +242,32 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 61, "metadata": { - "collapsed": true + "collapsed": false }, "outputs": [], "source": [ "#Set up parameters:\n", "\n", "m = 3\n", - "k = 17;\n", + "k = 12;\n", "\n", - "T = 2.5; #[seconds]\n", + "T = 5; #[seconds]\n", "dt = .01; #\n", "N = int(T/dt)+1\n", "t = numpy.linspace(0.0, T, N)\n", "\n", "#Initial Conditions\n", - "x0 = numpy.array([pi/2, 0])\n", - "xdot0 = numpy.zeros_like(x0)\n" + "x0 = 2\n", + "xdot0 = 0\n", + "\n", + "x_init = numpy.array([x0, xdot0])" ] }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 62, "metadata": { "collapsed": true }, @@ -275,45 +277,848 @@ " \"\"\" Euler's method for integrating a system of differential equations.\n", " \n", " Parameters:\n", - " IC - \n", - " x -\n", - " dt-\n", + " u - state at current step \n", + " f - RHS of equation\n", + " dt- time step size\n", " \n", " Returns: \n", " x - array of values at time T.\n", " \"\"\"\n", " \n", - " return u + dt*f(u)\n", + " return u + dt*f(u) " + ] + }, + { + "cell_type": "code", + "execution_count": 63, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def RK4(u, f, dt):\n", + " k1 = f(u)\n", + " k2 = f(u) + 0.5*dt*k1\n", + " k3 = f(u) + 0.5*dt*k2\n", + " k4 = f(u) + dt*k3\n", + " \n", + " return u + dt/6*(k1+2*k2+2*k3+k4)\n", " " ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 64, "metadata": { "collapsed": true }, "outputs": [], "source": [ - "def f(u):\n", + "def f_harmonic_oscillator(q):\n", " \"\"\"Returns RHS of harmonic oscillator EOM\n", " \n", + " Parameters:\n", + " q - initial state\n", + " \n", + " Returns:\n", + " f - RHS of harmonic oscillator eqn.\n", " \n", " \"\"\"\n", - " pos = u[0]\n", - " vel = u[1]\n", + " pos = q[0]\n", + " vel = q[1]\n", " \n", - " return numpy.array([vel, k/m*vel])\n", + " return numpy.array([vel, -k/m*pos])" + ] + }, + { + "cell_type": "code", + "execution_count": 66, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "application/javascript": [ + "/* Put everything inside the global mpl namespace */\n", + "window.mpl = {};\n", + "\n", + "mpl.get_websocket_type = function() {\n", + " if (typeof(WebSocket) !== 'undefined') {\n", + " return WebSocket;\n", + " } else if (typeof(MozWebSocket) !== 'undefined') {\n", + " return MozWebSocket;\n", + " } else {\n", + " alert('Your browser does not have WebSocket support.' +\n", + " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", + " 'Firefox 4 and 5 are also supported but you ' +\n", + " 'have to enable WebSockets in about:config.');\n", + " };\n", + "}\n", + "\n", + "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", + " this.id = figure_id;\n", + "\n", + " this.ws = websocket;\n", + "\n", + " this.supports_binary = (this.ws.binaryType != undefined);\n", + "\n", + " if (!this.supports_binary) {\n", + " var warnings = document.getElementById(\"mpl-warnings\");\n", + " if (warnings) {\n", + " warnings.style.display = 'block';\n", + " warnings.textContent = (\n", + " \"This browser does not support binary websocket messages. \" +\n", + " \"Performance may be slow.\");\n", + " }\n", + " }\n", + "\n", + " this.imageObj = new Image();\n", + "\n", + " this.context = undefined;\n", + " this.message = undefined;\n", + " this.canvas = undefined;\n", + " this.rubberband_canvas = undefined;\n", + " this.rubberband_context = undefined;\n", + " this.format_dropdown = undefined;\n", + "\n", + " this.image_mode = 'full';\n", + "\n", + " this.root = $('
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');\n", + " titlebar.append(titletext)\n", + " this.root.append(titlebar);\n", + " this.header = titletext[0];\n", + "}\n", + "\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._init_canvas = function() {\n", + " var fig = this;\n", + "\n", + " var canvas_div = $('
');\n", + "\n", + " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", + "\n", + " function canvas_keyboard_event(event) {\n", + " return fig.key_event(event, event['data']);\n", + " }\n", + "\n", + " canvas_div.keydown('key_press', canvas_keyboard_event);\n", + " canvas_div.keyup('key_release', canvas_keyboard_event);\n", + " this.canvas_div = canvas_div\n", + " this._canvas_extra_style(canvas_div)\n", + " this.root.append(canvas_div);\n", + "\n", + " var canvas = $('');\n", + " canvas.addClass('mpl-canvas');\n", + " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", + "\n", + " this.canvas = canvas[0];\n", + " this.context = canvas[0].getContext(\"2d\");\n", + "\n", + " var rubberband = $('');\n", + " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", + "\n", + " var pass_mouse_events = true;\n", + "\n", + " canvas_div.resizable({\n", + " start: function(event, ui) {\n", + " pass_mouse_events = false;\n", + " },\n", + " resize: function(event, ui) {\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " stop: function(event, ui) {\n", + " pass_mouse_events = true;\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " });\n", + "\n", + " function mouse_event_fn(event) {\n", + " if (pass_mouse_events)\n", + " return fig.mouse_event(event, event['data']);\n", + " }\n", + "\n", + " rubberband.mousedown('button_press', mouse_event_fn);\n", + " rubberband.mouseup('button_release', mouse_event_fn);\n", + " // Throttle sequential mouse events to 1 every 20ms.\n", + " rubberband.mousemove('motion_notify', mouse_event_fn);\n", + "\n", + " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", + " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", + "\n", + " canvas_div.on(\"wheel\", function (event) {\n", + " event = event.originalEvent;\n", + " event['data'] = 'scroll'\n", + " if (event.deltaY < 0) {\n", + " event.step = 1;\n", + " } else {\n", + " event.step = -1;\n", + " }\n", + " mouse_event_fn(event);\n", + " });\n", + "\n", + " canvas_div.append(canvas);\n", + " canvas_div.append(rubberband);\n", + "\n", + " this.rubberband = rubberband;\n", + " this.rubberband_canvas = rubberband[0];\n", + " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", + " this.rubberband_context.strokeStyle = \"#000000\";\n", + "\n", + " this._resize_canvas = function(width, height) {\n", + " // Keep the size of the canvas, canvas container, and rubber band\n", + " // canvas in synch.\n", + " canvas_div.css('width', width)\n", + " canvas_div.css('height', height)\n", + "\n", + " canvas.attr('width', width);\n", + " canvas.attr('height', height);\n", + "\n", + " rubberband.attr('width', width);\n", + " rubberband.attr('height', height);\n", + " }\n", + "\n", + " // Set the figure to an initial 600x600px, this will subsequently be updated\n", + " // upon first draw.\n", + " this._resize_canvas(600, 600);\n", + "\n", + " // Disable right mouse context menu.\n", + " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", + " return false;\n", + " });\n", + "\n", + " function set_focus () {\n", + " canvas.focus();\n", + " canvas_div.focus();\n", + " }\n", + "\n", + " window.setTimeout(set_focus, 100);\n", + "}\n", + "\n", + "mpl.figure.prototype._init_toolbar = function() {\n", + " var fig = this;\n", + "\n", + " var nav_element = $('
')\n", + " nav_element.attr('style', 'width: 100%');\n", + " this.root.append(nav_element);\n", + "\n", + " // Define a callback function for later on.\n", + " function toolbar_event(event) {\n", + " return fig.toolbar_button_onclick(event['data']);\n", + " }\n", + " function toolbar_mouse_event(event) {\n", + " return fig.toolbar_button_onmouseover(event['data']);\n", + " }\n", + "\n", + " for(var toolbar_ind in mpl.toolbar_items) {\n", + " var name = mpl.toolbar_items[toolbar_ind][0];\n", + " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", + " var image = mpl.toolbar_items[toolbar_ind][2];\n", + " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", + "\n", + " if (!name) {\n", + " // put a spacer in here.\n", + " continue;\n", + " }\n", + " var button = $('');\n", + " button.click(method_name, toolbar_event);\n", + " button.mouseover(tooltip, toolbar_mouse_event);\n", + " nav_element.append(button);\n", + " }\n", + "\n", + " // Add the status bar.\n", + " var status_bar = $('');\n", + " nav_element.append(status_bar);\n", + " this.message = status_bar[0];\n", + "\n", + " // Add the close button to the window.\n", + " var buttongrp = $('
');\n", + " var button = $('');\n", + " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", + " button.mouseover('Close figure', toolbar_mouse_event);\n", + " buttongrp.append(button);\n", + " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", + " titlebar.prepend(buttongrp);\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(el){\n", + " // this is important to make the div 'focusable\n", + " el.attr('tabindex', 0)\n", + " // reach out to IPython and tell the keyboard manager to turn it's self\n", + " // off when our div gets focus\n", + "\n", + " // location in version 3\n", + " if (IPython.notebook.keyboard_manager) {\n", + " IPython.notebook.keyboard_manager.register_events(el);\n", + " }\n", + " else {\n", + " // location in version 2\n", + " IPython.keyboard_manager.register_events(el);\n", + " }\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._key_event_extra = function(event, name) {\n", + " var manager = IPython.notebook.keyboard_manager;\n", + " if (!manager)\n", + " manager = IPython.keyboard_manager;\n", + "\n", + " // Check for shift+enter\n", + " if (event.shiftKey && event.which == 13) {\n", + " this.canvas_div.blur();\n", + " event.shiftKey = false;\n", + " // Send a \"J\" for go to next cell\n", + " event.which = 74;\n", + " event.keyCode = 74;\n", + " manager.command_mode();\n", + " manager.handle_keydown(event);\n", + " }\n", + "}\n", + "\n", + "mpl.figure.prototype.handle_save = function(fig, msg) {\n", + " fig.ondownload(fig, null);\n", + "}\n", + "\n", + "\n", + "mpl.find_output_cell = function(html_output) {\n", + " // Return the cell and output element which can be found *uniquely* in the notebook.\n", + " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", + " // IPython event is triggered only after the cells have been serialised, which for\n", + " // our purposes (turning an active figure into a static one), is too late.\n", + " var cells = IPython.notebook.get_cells();\n", + " var ncells = cells.length;\n", + " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", + " data = data.data;\n", + " }\n", + " if (data['text/html'] == html_output) {\n", + " return [cell, data, j];\n", + " }\n", + " }\n", + " }\n", + " }\n", + "}\n", + "\n", + "// Register the function which deals with the matplotlib target/channel.\n", + "// The kernel may be null if the page has been refreshed.\n", + "if (IPython.notebook.kernel != null) {\n", + " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", + "}\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "#Euler\n", + "q1 = numpy.zeros((N,2)) \n", + "q1[0] = x_init.copy() #set initial conditions\n", + "for n in range(N-1): #integrate with Euler\n", + " q1[n+1] = euler(q1[n], f_damped_SHM, dt)\n", + " #print(q1[n])\n", + " \n", + "#Runge-Kutta \n", + "q2 = numpy.zeros((N,2))\n", + "q2[0] = x_init.copy()\n", + "for n in range(N-1):\n", + " q2[n+1] = RK4(q2[n], f_damped_SHM, dt)\n", + "\n", + "#Symplectic\n", + "q3 = numpy.zeros((N,2))\n", + "q3[0] = x_init_H.copy()\n", + "for n in range(N-1):\n", + " q3[n+1] = verlet_SHM(q3[n], dt)\n", + "\n", + "#Analytical Solution \n", + "A = x0 #with no forcing, the max amplitude is equal to initial amplitude\n", + "phi = 0\n", + "q_analytical = A*numpy.cos(numpy.sqrt(k/m)*t + phi)\n", + " \n", + "pyplot.figure(figsize=(12,8));\n", + "pyplot.grid(True);\n", + "pyplot.xlabel(r't', fontsize=18);\n", + "pyplot.ylabel(r'position (meters)', fontsize=18);\n", + "pyplot.title('Harmonic oscillator position');\n", + "pyplot.plot(t, q1[:,0], lw=2, label='Euler');\n", + "pyplot.plot(t, q2[:,0], 'r-', lw=2, label='RK4');\n", + "pyplot.plot(t, q3[:,0], 'g', lw=2, label='Verlet');\n", + "pyplot.plot(t, q_analytical, 'k--', lw=2, label='analytical');\n", + "pyplot.legend();\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, { "cell_type": "markdown", "metadata": { From 23dacc591ee4d9a1422d63b4fe97407a006fb9e5 Mon Sep 17 00:00:00 2001 From: rbds Date: Sun, 6 Dec 2015 09:08:10 -0500 Subject: [PATCH 17/61] need to calculate analytical velocity for SHM --- randy_schur/Double_Pendulum_Problem.ipynb | 1262 ++++++--------------- 1 file changed, 333 insertions(+), 929 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index c16acba..54e4c6a 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -223,11 +223,20 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 1, "metadata": { "collapsed": false }, - "outputs": [], + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/home/randy/anaconda3/lib/python3.4/site-packages/IPython/kernel/__init__.py:13: ShimWarning: The `IPython.kernel` package has been deprecated. You should import from ipykernel or jupyter_client instead.\n", + " \"You should import from ipykernel or jupyter_client instead.\", ShimWarning)\n" + ] + } + ], "source": [ "import numpy\n", "from scipy.linalg import solve\n", @@ -242,7 +251,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 2, "metadata": { "collapsed": false }, @@ -255,14 +264,14 @@ "#g = 9.8 #[m/s**2]\n", "\n", "T = 25; #[seconds]\n", - "dt = .01; #\n", + "dt = .02; #\n", "N = int(T/dt)+1\n", "t = numpy.linspace(0.0, T, N)" ] }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 3, "metadata": { "collapsed": true }, @@ -285,7 +294,7 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 4, "metadata": { "collapsed": true }, @@ -311,13 +320,13 @@ }, { "cell_type": "code", - "execution_count": 41, + "execution_count": 5, "metadata": { "collapsed": true }, "outputs": [], "source": [ - "def f_harmonic_oscillator(q):\n", + "def f_harmonic_oscillator(u):\n", " \"\"\"Returns RHS of harmonic oscillator EOM\n", " \n", " Parameters:\n", @@ -327,15 +336,15 @@ " f - RHS of harmonic oscillator eqn.\n", " \n", " \"\"\"\n", - " pos = q[0]\n", - " vel = q[1]\n", + " pos = u[0]\n", + " vel = u[1]\n", " \n", " return numpy.array([vel, -k/m*pos])" ] }, { "cell_type": "code", - "execution_count": 42, + "execution_count": 6, "metadata": { "collapsed": true }, @@ -355,909 +364,15 @@ " p_half = pos - dt/2*mom/m\n", " q_half = mom + dt/2*(k*pos)\n", " \n", - " q = mom + dt/2*(2*k*p_half)\n", - " p = p_half - dt/2*q/m\n", - " \n", - " return numpy.array([p, q]) " - ] - }, - { - "cell_type": "code", - "execution_count": 43, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "#Initial Conditions\n", - "x0 = 0.5 #[m]\n", - "xdot0 = 0 #[m/s]\n", - "\n", - "p0 = x0\n", - "q0 = m*xdot0\n", - "\n", - "x_init = numpy.array([x0, xdot0])\n", - "x_init_H = numpy.array([p0, q0])" - ] - }, - { - "cell_type": "code", - "execution_count": 44, - "metadata": { - "collapsed": false, - "scrolled": true - }, - "outputs": [ - { - "data": { - "application/javascript": [ - "/* Put everything inside the global mpl namespace */\n", - "window.mpl = {};\n", - "\n", - "mpl.get_websocket_type = function() {\n", - " if (typeof(WebSocket) !== 'undefined') {\n", - " return WebSocket;\n", - " } else if (typeof(MozWebSocket) !== 'undefined') {\n", - " return MozWebSocket;\n", - " } else {\n", - " alert('Your browser does not have WebSocket support.' +\n", - " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", - " 'Firefox 4 and 5 are also supported but you ' +\n", - " 'have to enable WebSockets in about:config.');\n", - " };\n", - "}\n", - "\n", - "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", - " this.id = figure_id;\n", - "\n", - " this.ws = websocket;\n", - "\n", - " this.supports_binary = (this.ws.binaryType != undefined);\n", - "\n", - " if (!this.supports_binary) {\n", - " var warnings = document.getElementById(\"mpl-warnings\");\n", - " if (warnings) {\n", - " warnings.style.display = 'block';\n", - " warnings.textContent = (\n", - " \"This browser does not support binary websocket messages. \" +\n", - " \"Performance may be slow.\");\n", - " }\n", - " }\n", - "\n", - " this.imageObj = new Image();\n", - "\n", - " this.context = undefined;\n", - " this.message = undefined;\n", - " this.canvas = undefined;\n", - " this.rubberband_canvas = undefined;\n", - " this.rubberband_context = undefined;\n", - " this.format_dropdown = undefined;\n", - "\n", - " this.image_mode = 'full';\n", - "\n", - " this.root = $('
');\n", - " this._root_extra_style(this.root)\n", - " this.root.attr('style', 'display: inline-block');\n", - "\n", - " $(parent_element).append(this.root);\n", - "\n", - " this._init_header(this);\n", - " this._init_canvas(this);\n", - " this._init_toolbar(this);\n", - "\n", - " var fig = this;\n", - "\n", - " this.waiting = false;\n", - "\n", - " this.ws.onopen = function () {\n", - " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", - " fig.send_message(\"send_image_mode\", {});\n", - " fig.send_message(\"refresh\", {});\n", - " }\n", - "\n", - " this.imageObj.onload = function() {\n", - " if (fig.image_mode == 'full') {\n", - " // Full images could contain transparency (where diff images\n", - " // almost always do), so we need to clear the canvas so that\n", - " // there is no ghosting.\n", - " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", - " }\n", - " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", - " };\n", - "\n", - " this.imageObj.onunload = function() {\n", - " this.ws.close();\n", - " }\n", - "\n", - " this.ws.onmessage = this._make_on_message_function(this);\n", - "\n", - " this.ondownload = ondownload;\n", - "}\n", - "\n", - "mpl.figure.prototype._init_header = function() {\n", - " var titlebar = $(\n", - " '
');\n", - " var titletext = $(\n", - " '
');\n", - " titlebar.append(titletext)\n", - " this.root.append(titlebar);\n", - " this.header = titletext[0];\n", - "}\n", - "\n", - "\n", - "\n", - "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", - "\n", - "}\n", - "\n", - "\n", - "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", - "\n", - "}\n", - "\n", - "mpl.figure.prototype._init_canvas = function() {\n", - " var fig = this;\n", - "\n", - " var canvas_div = $('
');\n", - "\n", - " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", - "\n", - " function canvas_keyboard_event(event) {\n", - " return fig.key_event(event, event['data']);\n", - " }\n", - "\n", - " canvas_div.keydown('key_press', canvas_keyboard_event);\n", - " canvas_div.keyup('key_release', canvas_keyboard_event);\n", - " this.canvas_div = canvas_div\n", - " this._canvas_extra_style(canvas_div)\n", - " this.root.append(canvas_div);\n", - "\n", - " var canvas = $('');\n", - " canvas.addClass('mpl-canvas');\n", - " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", - "\n", - " this.canvas = canvas[0];\n", - " this.context = canvas[0].getContext(\"2d\");\n", - "\n", - " var rubberband = $('');\n", - " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", - "\n", - " var pass_mouse_events = true;\n", - "\n", - " canvas_div.resizable({\n", - " start: function(event, ui) {\n", - " pass_mouse_events = false;\n", - " },\n", - " resize: function(event, ui) {\n", - " fig.request_resize(ui.size.width, ui.size.height);\n", - " },\n", - " stop: function(event, ui) {\n", - " pass_mouse_events = true;\n", - " fig.request_resize(ui.size.width, ui.size.height);\n", - " },\n", - " });\n", - "\n", - " function mouse_event_fn(event) {\n", - " if (pass_mouse_events)\n", - " return fig.mouse_event(event, event['data']);\n", - " }\n", - "\n", - " rubberband.mousedown('button_press', mouse_event_fn);\n", - " rubberband.mouseup('button_release', mouse_event_fn);\n", - " // Throttle sequential mouse events to 1 every 20ms.\n", - " rubberband.mousemove('motion_notify', mouse_event_fn);\n", - "\n", - " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", - " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", - "\n", - " canvas_div.on(\"wheel\", function (event) {\n", - " event = event.originalEvent;\n", - " event['data'] = 'scroll'\n", - " if (event.deltaY < 0) {\n", - " event.step = 1;\n", - " } else {\n", - " event.step = -1;\n", - " }\n", - " mouse_event_fn(event);\n", - " });\n", - "\n", - " canvas_div.append(canvas);\n", - " canvas_div.append(rubberband);\n", - "\n", - " this.rubberband = rubberband;\n", - " this.rubberband_canvas = rubberband[0];\n", - " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", - " this.rubberband_context.strokeStyle = \"#000000\";\n", - "\n", - " this._resize_canvas = function(width, height) {\n", - " // Keep the size of the canvas, canvas container, and rubber band\n", - " // canvas in synch.\n", - " canvas_div.css('width', width)\n", - " canvas_div.css('height', height)\n", - "\n", - " canvas.attr('width', width);\n", - " canvas.attr('height', height);\n", - "\n", - " rubberband.attr('width', width);\n", - " rubberband.attr('height', height);\n", - " }\n", - "\n", - " // Set the figure to an initial 600x600px, this will subsequently be updated\n", - " // upon first draw.\n", - " this._resize_canvas(600, 600);\n", - "\n", - " // Disable right mouse context menu.\n", - " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", - " return false;\n", - " });\n", - "\n", - " function set_focus () {\n", - " canvas.focus();\n", - " canvas_div.focus();\n", - " }\n", - "\n", - " window.setTimeout(set_focus, 100);\n", - "}\n", - "\n", - "mpl.figure.prototype._init_toolbar = function() {\n", - " var fig = this;\n", - "\n", - " var nav_element = $('
')\n", - " nav_element.attr('style', 'width: 100%');\n", - " this.root.append(nav_element);\n", - "\n", - " // Define a callback function for later on.\n", - " function toolbar_event(event) {\n", - " return fig.toolbar_button_onclick(event['data']);\n", - " }\n", - " function toolbar_mouse_event(event) {\n", - " return fig.toolbar_button_onmouseover(event['data']);\n", - " }\n", - "\n", - " for(var toolbar_ind in mpl.toolbar_items) {\n", - " var name = mpl.toolbar_items[toolbar_ind][0];\n", - " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", - " var image = mpl.toolbar_items[toolbar_ind][2];\n", - " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", - "\n", - " if (!name) {\n", - " // put a spacer in here.\n", - " continue;\n", - " }\n", - " var button = $('');\n", + " button.click(method_name, toolbar_event);\n", + " button.mouseover(tooltip, toolbar_mouse_event);\n", + " nav_element.append(button);\n", + " }\n", + "\n", + " // Add the status bar.\n", + " var status_bar = $('');\n", + " nav_element.append(status_bar);\n", + " this.message = status_bar[0];\n", + "\n", + " // Add the close button to the window.\n", + " var buttongrp = $('
');\n", + " var button = $('');\n", + " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", + " button.mouseover('Close figure', toolbar_mouse_event);\n", + " buttongrp.append(button);\n", + " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", + " titlebar.prepend(buttongrp);\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(el){\n", + " // this is important to make the div 'focusable\n", + " el.attr('tabindex', 0)\n", + " // reach out to IPython and tell the keyboard manager to turn it's self\n", + " // off when our div gets focus\n", + "\n", + " // location in version 3\n", + " if (IPython.notebook.keyboard_manager) {\n", + " IPython.notebook.keyboard_manager.register_events(el);\n", + " }\n", + " else {\n", + " // location in version 2\n", + " IPython.keyboard_manager.register_events(el);\n", + " }\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._key_event_extra = function(event, name) {\n", + " var manager = IPython.notebook.keyboard_manager;\n", + " if (!manager)\n", + " manager = IPython.keyboard_manager;\n", + "\n", + " // Check for shift+enter\n", + " if (event.shiftKey && event.which == 13) {\n", + " this.canvas_div.blur();\n", + " event.shiftKey = false;\n", + " // Send a \"J\" for go to next cell\n", + " event.which = 74;\n", + " event.keyCode = 74;\n", + " manager.command_mode();\n", + " manager.handle_keydown(event);\n", + " }\n", + "}\n", + "\n", + "mpl.figure.prototype.handle_save = function(fig, msg) {\n", + " fig.ondownload(fig, null);\n", + "}\n", + "\n", + "\n", + "mpl.find_output_cell = function(html_output) {\n", + " // Return the cell and output element which can be found *uniquely* in the notebook.\n", + " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", + " // IPython event is triggered only after the cells have been serialised, which for\n", + " // our purposes (turning an active figure into a static one), is too late.\n", + " var cells = IPython.notebook.get_cells();\n", + " var ncells = cells.length;\n", + " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", + " data = data.data;\n", + " }\n", + " if (data['text/html'] == html_output) {\n", + " return [cell, data, j];\n", + " }\n", + " }\n", + " }\n", + " }\n", + "}\n", + "\n", + "// Register the function which deals with the matplotlib target/channel.\n", + "// The kernel may be null if the page has been refreshed.\n", + "if (IPython.notebook.kernel != null) {\n", + " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", + "}\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "pyplot.figure(figsize=(10,6))\n", + "pyplot.tick_params(axis='both', labelsize=14)\n", + "pyplot.grid(True)\n", + "pyplot.xlabel('$\\Delta t$', fontsize=16)\n", + "pyplot.ylabel('Error', fontsize=16)\n", + "pyplot.loglog(dt_values, Euler_error_values, 'bo-', label='Euler')\n", + "pyplot.loglog(dt_values, RK_error_values, 'ro-', label='RK4')\n", + "pyplot.loglog(dt_values, Verlet_error_values, 'go-', label='Symplectic')\n", + "pyplot.axis('equal')\n", + "pyplot.legend();" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Is this what you would expect to see based on the plots of position above? What are the sources of this error? We have the usual suspects: truncation error, discretization error, possible precision errors. But ordinarily we don't see a second order scheme (Verlet) outperforming a fourth order scheme (RK4) by three orders of magnitude. What is going on?\n", + "\n", + "One thing we notice in the plots of position is that for Euler and Runge-Kutta, the spring is stretched farther and farther each oscillation. This means it has more and more potential energy. With no outside forces adding energy to the system and no friction to remove energy, Conservation of Energy says that value should be constant! Let's take a look at total energy in the system. Since we derived equations of motion from the Lagrangian, this should be easy! All we need to do is calculate kinetic and potential energy at each time step, and we already have the information we need to do this. " + ] + }, + { + "cell_type": "code", + "execution_count": 40, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def get_Energy(u):\n", + " \"\"\" Calculates total energu in the system at given time step for a simple harmonic oscillator\n", + " Parameters:\n", + " u - state of system [pos, vel]\n", + " Returns:\n", + " total energy in system.\n", + " \"\"\"\n", + " pos = u[0]\n", + " vel = u[1]\n", + " T = 0.5*m*vel**2\n", + " V = 0.5*k*pos**2\n", + " \n", + " return T+V" + ] + }, + { + "cell_type": "code", + "execution_count": 41, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "#Euler total energy\n", + "Euler_energy = numpy.zeros_like(t)\n", + "for i in range(N):\n", + " Euler_energy[i] = get_Energy(q1[i,:])\n", + " \n", + "\n", + "#RK4 total energy\n", + "RK4_energy = numpy.zeros_like(t)\n", + "for i in range(N):\n", + " RK4_energy[i] = get_Energy(q2[i,:])\n", + "\n", + "#Verlet total energy\n", + "#Here we will convert from momentum to velocity so we can use the same get_energy function.\n", + "#Momemtum (q) is mass*velocity.\n", + "Verlet_energy = numpy.zeros_like(t)\n", + "Verlet_vel = q3[:,1]/m\n", + "for i in range(N):\n", + " Verlet_energy[i] = get_Energy(numpy.array([q3[i,0], Verlet_vel[i]]))\n", + "\n", + "\n", + "#analytic total energy\n", + "analytical_energy = numpy.zeros_like(t)\n", + "anayltical_state = numpy.array([q_analytical, vel_analytical]).T\n", + "for i in range(N):\n", + " analytical_energy[i] = get_Energy(anayltical_state[i,:])" + ] + }, + { + "cell_type": "code", + "execution_count": 44, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "application/javascript": [ + "/* Put everything inside the global mpl namespace */\n", + "window.mpl = {};\n", + "\n", + "mpl.get_websocket_type = function() {\n", + " if (typeof(WebSocket) !== 'undefined') {\n", + " return WebSocket;\n", + " } else if (typeof(MozWebSocket) !== 'undefined') {\n", + " return MozWebSocket;\n", + " } else {\n", + " alert('Your browser does not have WebSocket support.' +\n", + " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", + " 'Firefox 4 and 5 are also supported but you ' +\n", + " 'have to enable WebSockets in about:config.');\n", + " };\n", + "}\n", + "\n", + "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", + " this.id = figure_id;\n", + "\n", + " this.ws = websocket;\n", + "\n", + " this.supports_binary = (this.ws.binaryType != undefined);\n", + "\n", + " if (!this.supports_binary) {\n", + " var warnings = document.getElementById(\"mpl-warnings\");\n", + " if (warnings) {\n", + " warnings.style.display = 'block';\n", + " warnings.textContent = (\n", + " \"This browser does not support binary websocket messages. \" +\n", + " \"Performance may be slow.\");\n", + " }\n", + " }\n", + "\n", + " this.imageObj = new Image();\n", + "\n", + " this.context = undefined;\n", + " this.message = undefined;\n", + " this.canvas = undefined;\n", + " this.rubberband_canvas = undefined;\n", + " this.rubberband_context = undefined;\n", + " this.format_dropdown = undefined;\n", + "\n", + " this.image_mode = 'full';\n", + "\n", + " this.root = $('
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');\n", + " var titletext = $(\n", + " '
');\n", + " titlebar.append(titletext)\n", + " this.root.append(titlebar);\n", + " this.header = titletext[0];\n", + "}\n", + "\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._init_canvas = function() {\n", + " var fig = this;\n", + "\n", + " var canvas_div = $('
');\n", + "\n", + " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", + "\n", + " function canvas_keyboard_event(event) {\n", + " return fig.key_event(event, event['data']);\n", + " }\n", + "\n", + " canvas_div.keydown('key_press', canvas_keyboard_event);\n", + " canvas_div.keyup('key_release', canvas_keyboard_event);\n", + " this.canvas_div = canvas_div\n", + " this._canvas_extra_style(canvas_div)\n", + " this.root.append(canvas_div);\n", + "\n", + " var canvas = $('');\n", + " canvas.addClass('mpl-canvas');\n", + " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", + "\n", + " this.canvas = canvas[0];\n", + " this.context = canvas[0].getContext(\"2d\");\n", + "\n", + " var rubberband = $('');\n", + " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", + "\n", + " var pass_mouse_events = true;\n", + "\n", + " canvas_div.resizable({\n", + " start: function(event, ui) {\n", + " pass_mouse_events = false;\n", + " },\n", + " resize: function(event, ui) {\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " stop: function(event, ui) {\n", + " pass_mouse_events = true;\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " });\n", + "\n", + " function mouse_event_fn(event) {\n", + " if (pass_mouse_events)\n", + " return fig.mouse_event(event, event['data']);\n", + " }\n", + "\n", + " rubberband.mousedown('button_press', mouse_event_fn);\n", + " rubberband.mouseup('button_release', mouse_event_fn);\n", + " // Throttle sequential mouse events to 1 every 20ms.\n", + " rubberband.mousemove('motion_notify', mouse_event_fn);\n", + "\n", + " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", + " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", + "\n", + " canvas_div.on(\"wheel\", function (event) {\n", + " event = event.originalEvent;\n", + " event['data'] = 'scroll'\n", + " if (event.deltaY < 0) {\n", + " event.step = 1;\n", + " } else {\n", + " event.step = -1;\n", + " }\n", + " mouse_event_fn(event);\n", + " });\n", + "\n", + " canvas_div.append(canvas);\n", + " canvas_div.append(rubberband);\n", + "\n", + " this.rubberband = rubberband;\n", + " this.rubberband_canvas = rubberband[0];\n", + " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", + " this.rubberband_context.strokeStyle = \"#000000\";\n", + "\n", + " this._resize_canvas = function(width, height) {\n", + " // Keep the size of the canvas, canvas container, and rubber band\n", + " // canvas in synch.\n", + " canvas_div.css('width', width)\n", + " canvas_div.css('height', height)\n", + "\n", + " canvas.attr('width', width);\n", + " canvas.attr('height', height);\n", + "\n", + " rubberband.attr('width', width);\n", + " rubberband.attr('height', height);\n", + " }\n", + "\n", + " // Set the figure to an initial 600x600px, this will subsequently be updated\n", + " // upon first draw.\n", + " this._resize_canvas(600, 600);\n", + "\n", + " // Disable right mouse context menu.\n", + " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", + " return false;\n", + " });\n", + "\n", + " function set_focus () {\n", + " canvas.focus();\n", + " canvas_div.focus();\n", + " }\n", + "\n", + " window.setTimeout(set_focus, 100);\n", + "}\n", + "\n", + "mpl.figure.prototype._init_toolbar = function() {\n", + " var fig = this;\n", + "\n", + " var nav_element = $('
')\n", + " nav_element.attr('style', 'width: 100%');\n", + " this.root.append(nav_element);\n", + "\n", + " // Define a callback function for later on.\n", + " function toolbar_event(event) {\n", + " return fig.toolbar_button_onclick(event['data']);\n", + " }\n", + " function toolbar_mouse_event(event) {\n", + " return fig.toolbar_button_onmouseover(event['data']);\n", + " }\n", + "\n", + " for(var toolbar_ind in mpl.toolbar_items) {\n", + " var name = mpl.toolbar_items[toolbar_ind][0];\n", + " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", + " var image = mpl.toolbar_items[toolbar_ind][2];\n", + " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", + "\n", + " if (!name) {\n", + " // put a spacer in here.\n", + " continue;\n", + " }\n", + " var button = $('');\n", + " button.click(method_name, toolbar_event);\n", + " button.mouseover(tooltip, toolbar_mouse_event);\n", + " nav_element.append(button);\n", + " }\n", + "\n", + " // Add the status bar.\n", + " var status_bar = $('');\n", + " nav_element.append(status_bar);\n", + " this.message = status_bar[0];\n", + "\n", + " // Add the close button to the window.\n", + " var buttongrp = $('
');\n", + " var button = $('');\n", + " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", + " button.mouseover('Close figure', toolbar_mouse_event);\n", + " buttongrp.append(button);\n", + " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", + " titlebar.prepend(buttongrp);\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(el){\n", + " // this is important to make the div 'focusable\n", + " el.attr('tabindex', 0)\n", + " // reach out to IPython and tell the keyboard manager to turn it's self\n", + " // off when our div gets focus\n", + "\n", + " // location in version 3\n", + " if (IPython.notebook.keyboard_manager) {\n", + " IPython.notebook.keyboard_manager.register_events(el);\n", + " }\n", + " else {\n", + " // location in version 2\n", + " IPython.keyboard_manager.register_events(el);\n", + " }\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._key_event_extra = function(event, name) {\n", + " var manager = IPython.notebook.keyboard_manager;\n", + " if (!manager)\n", + " manager = IPython.keyboard_manager;\n", + "\n", + " // Check for shift+enter\n", + " if (event.shiftKey && event.which == 13) {\n", + " this.canvas_div.blur();\n", + " event.shiftKey = false;\n", + " // Send a \"J\" for go to next cell\n", + " event.which = 74;\n", + " event.keyCode = 74;\n", + " manager.command_mode();\n", + " manager.handle_keydown(event);\n", + " }\n", + "}\n", + "\n", + "mpl.figure.prototype.handle_save = function(fig, msg) {\n", + " fig.ondownload(fig, null);\n", + "}\n", + "\n", + "\n", + "mpl.find_output_cell = function(html_output) {\n", + " // Return the cell and output element which can be found *uniquely* in the notebook.\n", + " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", + " // IPython event is triggered only after the cells have been serialised, which for\n", + " // our purposes (turning an active figure into a static one), is too late.\n", + " var cells = IPython.notebook.get_cells();\n", + " var ncells = cells.length;\n", + " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", + " data = data.data;\n", + " }\n", + " if (data['text/html'] == html_output) {\n", + " return [cell, data, j];\n", + " }\n", + " }\n", + " }\n", + " }\n", + "}\n", + "\n", + "// Register the function which deals with the matplotlib target/channel.\n", + "// The kernel may be null if the page has been refreshed.\n", + "if (IPython.notebook.kernel != null) {\n", + " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", + "}\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "#Euler\n", + "q1 = numpy.zeros((N,2)) \n", + "q1[0] = x_init.copy() #set initial conditions\n", + "for n in range(N-1): #integrate with Euler\n", + " q1[n+1] = euler(q1[n], f_damped_HM, dt)\n", + " #print(q1[n])\n", + " \n", + "#Runge-Kutta \n", + "q2 = numpy.zeros((N,2))\n", + "q2[0] = x_init.copy()\n", + "for n in range(N-1):\n", + " q2[n+1] = RK4(q2[n], f_damped_HM, dt)\n", + "\n", + "#Symplectic\n", + "q3 = numpy.zeros((N,2))\n", + "q3[0] = x_init_H.copy()\n", + "for n in range(N-1):\n", + " q3[n+1] = Verlet_damped_HM(q3[n], dt)\n", "\n", "#Analytical Solution \n", "A = x0 #with no forcing, the max amplitude is equal to initial amplitude\n", From ee9438f9745b15395a7523c921e7fc7725925834 Mon Sep 17 00:00:00 2001 From: rbds Date: Sun, 6 Dec 2015 13:50:02 -0500 Subject: [PATCH 20/61] damped harmonic oscillations --- randy_schur/Double_Pendulum_Problem.ipynb | 265 ++++++++++++++++++---- randy_schur/numericalmoocstyle.css | 5 +- 2 files changed, 221 insertions(+), 49 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index b1a1efb..fb997f8 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -223,11 +223,20 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 4, "metadata": { "collapsed": false }, - "outputs": [], + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/home/randy/anaconda3/lib/python3.4/site-packages/IPython/kernel/__init__.py:13: ShimWarning: The `IPython.kernel` package has been deprecated. You should import from ipykernel or jupyter_client instead.\n", + " \"You should import from ipykernel or jupyter_client instead.\", ShimWarning)\n" + ] + } + ], "source": [ "import numpy\n", "from scipy.linalg import solve\n", @@ -242,7 +251,7 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 5, "metadata": { "collapsed": false }, @@ -262,7 +271,7 @@ }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 6, "metadata": { "collapsed": true }, @@ -285,7 +294,7 @@ }, { "cell_type": "code", - "execution_count": 30, + "execution_count": 7, "metadata": { "collapsed": true }, @@ -311,7 +320,7 @@ }, { "cell_type": "code", - "execution_count": 31, + "execution_count": 8, "metadata": { "collapsed": true }, @@ -335,7 +344,7 @@ }, { "cell_type": "code", - "execution_count": 32, + "execution_count": 9, "metadata": { "collapsed": true }, @@ -363,7 +372,7 @@ }, { "cell_type": "code", - "execution_count": 33, + "execution_count": 10, "metadata": { "collapsed": true }, @@ -382,7 +391,7 @@ }, { "cell_type": "code", - "execution_count": 34, + "execution_count": 11, "metadata": { "collapsed": false, "scrolled": true @@ -1189,7 +1198,7 @@ }, { "cell_type": "code", - "execution_count": 35, + "execution_count": 12, "metadata": { "collapsed": true }, @@ -1212,7 +1221,7 @@ }, { "cell_type": "code", - "execution_count": 38, + "execution_count": 13, "metadata": { "collapsed": false }, @@ -1233,7 +1242,7 @@ }, { "cell_type": "code", - "execution_count": 39, + "execution_count": 14, "metadata": { "collapsed": false }, @@ -2011,7 +2020,7 @@ }, { "cell_type": "code", - "execution_count": 40, + "execution_count": 15, "metadata": { "collapsed": true }, @@ -2034,7 +2043,7 @@ }, { "cell_type": "code", - "execution_count": 41, + "execution_count": 16, "metadata": { "collapsed": false }, @@ -2069,7 +2078,7 @@ }, { "cell_type": "code", - "execution_count": 44, + "execution_count": 17, "metadata": { "collapsed": false }, @@ -2893,7 +2902,7 @@ }, { "cell_type": "code", - "execution_count": 48, + "execution_count": 18, "metadata": { "collapsed": true }, @@ -2917,7 +2926,7 @@ }, { "cell_type": "code", - "execution_count": 49, + "execution_count": 46, "metadata": { "collapsed": true }, @@ -2934,11 +2943,11 @@ " pos = u[0]\n", " mom = u[1]\n", " \n", - " p_half = pos - dt/2*(mom/m - c/m)\n", - " q_half = mom + dt/2*(k*pos)\n", + " p_half = pos - dt/2*(mom/m)\n", + " #q_half = mom + dt/2*(k*pos)\n", " \n", - " q = mom + dt/2*(2*k*p_half)\n", - " p = p_half - dt/2*(q/m - c/m)\n", + " q = mom + dt*(k*p_half - c/m*mom)\n", + " p = p_half - dt/2*(q/m)\n", " \n", " return numpy.array([p, q]) \n", " " @@ -2946,7 +2955,7 @@ }, { "cell_type": "code", - "execution_count": 56, + "execution_count": 47, "metadata": { "collapsed": true }, @@ -2970,12 +2979,14 @@ "x_init = numpy.array([x0, xdot0])\n", "x_init_H = numpy.array([p0, q0])\n", "zeta = 0.1 #[N/(m/s)]\n", - "c = zeta*2*m*sqrt(k/m)" + "wn = sqrt(k/m)\n", + "wd = wn*sqrt(1-zeta**2)\n", + "c = zeta*2*m*wn" ] }, { "cell_type": "code", - "execution_count": 57, + "execution_count": 48, "metadata": { "collapsed": false }, @@ -3719,7 +3730,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -3731,49 +3742,53 @@ ], "source": [ "#Euler\n", - "q1 = numpy.zeros((N,2)) \n", - "q1[0] = x_init.copy() #set initial conditions\n", + "q1_d = numpy.zeros((N,2)) \n", + "q1_d[0] = x_init.copy() #set initial conditions\n", "for n in range(N-1): #integrate with Euler\n", - " q1[n+1] = euler(q1[n], f_damped_HM, dt)\n", + " q1_d[n+1] = euler(q1_d[n], f_damped_HM, dt)\n", " #print(q1[n])\n", " \n", "#Runge-Kutta \n", - "q2 = numpy.zeros((N,2))\n", - "q2[0] = x_init.copy()\n", + "q2_d = numpy.zeros((N,2))\n", + "q2_d[0] = x_init.copy()\n", "for n in range(N-1):\n", - " q2[n+1] = RK4(q2[n], f_damped_HM, dt)\n", + " q2_d[n+1] = RK4(q2_d[n], f_damped_HM, dt)\n", "\n", "#Symplectic\n", - "q3 = numpy.zeros((N,2))\n", - "q3[0] = x_init_H.copy()\n", + "q3_d = numpy.zeros((N,2))\n", + "q3_d[0] = x_init_H.copy()\n", "for n in range(N-1):\n", - " q3[n+1] = Verlet_damped_HM(q3[n], dt)\n", + " q3_d[n+1] = Verlet_damped_HM(q3_d[n], dt)\n", "\n", "#Analytical Solution \n", "A = x0 #with no forcing, the max amplitude is equal to initial amplitude\n", "phi = 0\n", - "q_analytical = A*numpy.cos(numpy.sqrt(k/m)*t + phi)\n", + "q_d_analytical = A*numpy.exp(-zeta*wn*t)*numpy.cos(wd*t + phi)\n", " \n", - "pyplot.figure(figsize=(12,8));\n", + "pyplot.figure(figsize=(10,8));\n", "pyplot.grid(True);\n", "pyplot.xlabel(r't', fontsize=18);\n", "pyplot.ylabel(r'position (meters)', fontsize=18);\n", "pyplot.title('Harmonic oscillator position');\n", - "pyplot.plot(t, q1[:,0], lw=2, label='Euler');\n", - "pyplot.plot(t, q2[:,0], 'r-', lw=2, label='RK4');\n", - "pyplot.plot(t, q3[:,0], 'g', lw=2, label='Verlet');\n", - "pyplot.plot(t, q_analytical, 'k--', lw=2, label='analytical');\n", + "pyplot.plot(t, q1_d[:,0], lw=2, label='Euler');\n", + "pyplot.plot(t, q2_d[:,0], 'r-', lw=2, label='RK4');\n", + "pyplot.plot(t, q3_d[:,0], 'g', lw=2, label='Verlet');\n", + "pyplot.plot(t, q_d_analytical, 'k--', lw=2, label='analytical');\n", "pyplot.legend();\n" ] }, { - "cell_type": "code", - "execution_count": null, + "cell_type": "markdown", "metadata": { "collapsed": true }, - "outputs": [], - "source": [] + "source": [ + "How does each integrator perform compared to the case of simple harmonic motion? From the eye test, it seems like the error is actually better here. Why is that? \n", + "\n", + "The effect of the damper is to remove energy from the system. So, in this case the damping term is helping to remove some energy from the system that the integration adds. As time goes to infinity, these solutions will converge, and the Euler and Runge-Kutta methods are stable. \n", + "\n", + "Another way to investigate stability is with a phase plot. Instead of position vs. time, we can plot velocity vs. position. This plot is in phase space (a subset of state space). For any given state (position and velocity), we should be able to determine what the next state will be. This gives us an entire direction field, over which we can plot the actual behavior of the system to see if they match up. If the phase plot becomes repetitive, we can call it stable" + ] }, { "cell_type": "markdown", @@ -4010,17 +4025,173 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 23, "metadata": { "collapsed": false }, - "outputs": [], + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "execution_count": 23, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "# This cell loads the style of the notebook, which is taken from the \n", "# Numerical Methods in Python Course: http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about\n", "\n", "from IPython.core.display import HTML\n", - "css_file = './randy_schur/numericalmoocstyle.css'\n", + "css_file = './numericalmoocstyle.css'\n", "HTML(open(css_file, \"r\").read())" ] }, diff --git a/randy_schur/numericalmoocstyle.css b/randy_schur/numericalmoocstyle.css index 16a3c5c..aca3fcb 100644 --- a/randy_schur/numericalmoocstyle.css +++ b/randy_schur/numericalmoocstyle.css @@ -15,12 +15,13 @@ } div.cell { /* set cell width */ - width: 750px; + width: 1000px; } div #notebook { /* centre the content */ background: #fff; /* white background for content */ - width: 1000px; + width: 125 +0px; margin: auto; padding-left: 0em; } From 942444c759646482ca5cbc52dc8a24dd29760f04 Mon Sep 17 00:00:00 2001 From: rbds Date: Sun, 6 Dec 2015 13:55:20 -0500 Subject: [PATCH 21/61] adjust formatting css --- randy_schur/Double_Pendulum_Problem.ipynb | 38 ++++++++++++++++++++++- randy_schur/numericalmoocstyle.css | 7 ++--- 2 files changed, 40 insertions(+), 5 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index fb997f8..45c8aed 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -3787,9 +3787,45 @@ "\n", "The effect of the damper is to remove energy from the system. So, in this case the damping term is helping to remove some energy from the system that the integration adds. As time goes to infinity, these solutions will converge, and the Euler and Runge-Kutta methods are stable. \n", "\n", - "Another way to investigate stability is with a phase plot. Instead of position vs. time, we can plot velocity vs. position. This plot is in phase space (a subset of state space). For any given state (position and velocity), we should be able to determine what the next state will be. This gives us an entire direction field, over which we can plot the actual behavior of the system to see if they match up. If the phase plot becomes repetitive, we can call it stable" + "Another way to investigate stability is with a phase plot. Instead of position vs. time, we can plot velocity vs. position. This plot is in phase space (a subset of state space). For any given state (position and velocity), we should be able to determine what the next state will be. This gives us an entire direction field, over which we can plot the actual behavior of the system to see if they match up. If the phase plot becomes repetitive, we can call it stable, and if the phase plot goes to the origin, we can call it asymptotically stable." ] }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, { "cell_type": "markdown", "metadata": { diff --git a/randy_schur/numericalmoocstyle.css b/randy_schur/numericalmoocstyle.css index aca3fcb..e6a8035 100644 --- a/randy_schur/numericalmoocstyle.css +++ b/randy_schur/numericalmoocstyle.css @@ -20,8 +20,7 @@ div.cell { /* set cell width */ div #notebook { /* centre the content */ background: #fff; /* white background for content */ - width: 125 -0px; + width: 1250px; margin: auto; padding-left: 0em; } @@ -49,7 +48,7 @@ div.text_cell_render{ line-height: 140%; font-size: 125%; font-weight: 400; - width:600px; + width:800px; margin-left:auto; margin-right:auto; } @@ -129,7 +128,7 @@ div.text_cell_render{ MathJax.Hub.Config({ TeX: { extensions: ["AMSmath.js"], - equationNumbers: { autoNumber: "AMS", useLabelIds: true} + equationNumbers: { autoNumber: "AMS", useLabelIds: false} }, tex2jax: { inlineMath: [ ['$','$'], ["\\(","\\)"] ], From c22093e9d85da3ccc71b01f643bc2861c36442ed Mon Sep 17 00:00:00 2001 From: rbds Date: Sun, 6 Dec 2015 14:24:18 -0500 Subject: [PATCH 22/61] phase portrait --- randy_schur/Double_Pendulum_Problem.ipynb | 866 +++++++++++++++++++++- 1 file changed, 837 insertions(+), 29 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index 45c8aed..e377cd8 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -223,7 +223,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 1, "metadata": { "collapsed": false }, @@ -243,6 +243,7 @@ "from math import pi, cos, sin, sqrt\n", "\n", "from matplotlib import pyplot\n", + "from matplotlib.pyplot import quiver\n", "%matplotlib notebook\n", "from matplotlib import rcParams, cm\n", "rcParams['font.family'] = 'serif'\n", @@ -251,7 +252,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 2, "metadata": { "collapsed": false }, @@ -271,7 +272,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 3, "metadata": { "collapsed": true }, @@ -294,7 +295,7 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 4, "metadata": { "collapsed": true }, @@ -320,7 +321,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 5, "metadata": { "collapsed": true }, @@ -344,7 +345,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 6, "metadata": { "collapsed": true }, @@ -372,7 +373,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 7, "metadata": { "collapsed": true }, @@ -391,7 +392,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 8, "metadata": { "collapsed": false, "scrolled": true @@ -1198,7 +1199,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 9, "metadata": { "collapsed": true }, @@ -1221,7 +1222,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 10, "metadata": { "collapsed": false }, @@ -1242,7 +1243,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 11, "metadata": { "collapsed": false }, @@ -2020,7 +2021,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 12, "metadata": { "collapsed": true }, @@ -2043,7 +2044,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 13, "metadata": { "collapsed": false }, @@ -2078,7 +2079,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 14, "metadata": { "collapsed": false }, @@ -2902,7 +2903,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 15, "metadata": { "collapsed": true }, @@ -2926,7 +2927,7 @@ }, { "cell_type": "code", - "execution_count": 46, + "execution_count": 16, "metadata": { "collapsed": true }, @@ -2955,7 +2956,7 @@ }, { "cell_type": "code", - "execution_count": 47, + "execution_count": 17, "metadata": { "collapsed": true }, @@ -2986,7 +2987,7 @@ }, { "cell_type": "code", - "execution_count": 48, + "execution_count": 29, "metadata": { "collapsed": false }, @@ -3764,6 +3765,7 @@ "A = x0 #with no forcing, the max amplitude is equal to initial amplitude\n", "phi = 0\n", "q_d_analytical = A*numpy.exp(-zeta*wn*t)*numpy.cos(wd*t + phi)\n", + "vel_d_analytical = -A*numpy.exp(-zeta*wn*t)*wd*numpy.sin(wd*t+phi)\n", " \n", "pyplot.figure(figsize=(10,8));\n", "pyplot.grid(True);\n", @@ -3792,12 +3794,819 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 32, "metadata": { - "collapsed": true + "collapsed": false }, - "outputs": [], - "source": [] + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "0.5\n", + "-0.366458550519\n", + "0.778366222918\n", + "-1.06721090077\n" + ] + } + ], + "source": [ + "print(max(q_d_analytical))\n", + "print(min(q_d_analytical))\n", + "print(max(vel_d_analytical))\n", + "print(min(vel_d_analytical))" + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "application/javascript": [ + "/* Put everything inside the global mpl namespace */\n", + "window.mpl = {};\n", + "\n", + "mpl.get_websocket_type = function() {\n", + " if (typeof(WebSocket) !== 'undefined') {\n", + " return WebSocket;\n", + " } else if (typeof(MozWebSocket) !== 'undefined') {\n", + " return MozWebSocket;\n", + " } else {\n", + " alert('Your browser does not have WebSocket support.' +\n", + " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", + " 'Firefox 4 and 5 are also supported but you ' +\n", + " 'have to enable WebSockets in about:config.');\n", + " };\n", + "}\n", + "\n", + "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", + " this.id = figure_id;\n", + "\n", + " this.ws = websocket;\n", + "\n", + " this.supports_binary = (this.ws.binaryType != undefined);\n", + "\n", + " if (!this.supports_binary) {\n", + " var warnings = document.getElementById(\"mpl-warnings\");\n", + " if (warnings) {\n", + " warnings.style.display = 'block';\n", + " warnings.textContent = (\n", + " \"This browser does not support binary websocket messages. \" +\n", + " \"Performance may be slow.\");\n", + " }\n", + " }\n", + "\n", + " this.imageObj = new Image();\n", + "\n", + " this.context = undefined;\n", + " this.message = undefined;\n", + " this.canvas = undefined;\n", + " this.rubberband_canvas = undefined;\n", + " this.rubberband_context = undefined;\n", + " this.format_dropdown = undefined;\n", + "\n", + " this.image_mode = 'full';\n", + "\n", + " this.root = $('
');\n", + " this._root_extra_style(this.root)\n", + " this.root.attr('style', 'display: inline-block');\n", + "\n", + " $(parent_element).append(this.root);\n", + "\n", + " this._init_header(this);\n", + " this._init_canvas(this);\n", + " this._init_toolbar(this);\n", + "\n", + " var fig = this;\n", + "\n", + " this.waiting = false;\n", + "\n", + " this.ws.onopen = function () {\n", + " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", + " fig.send_message(\"send_image_mode\", {});\n", + " fig.send_message(\"refresh\", {});\n", + " }\n", + "\n", + " this.imageObj.onload = function() {\n", + " if (fig.image_mode == 'full') {\n", + " // Full images could contain transparency (where diff images\n", + " // almost always do), so we need to clear the canvas so that\n", + " // there is no ghosting.\n", + " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", + " }\n", + " fig.context.drawImage(fig.imageObj, 0, 0);\n", + " fig.waiting = false;\n", + " };\n", + "\n", + " this.imageObj.onunload = function() {\n", + " this.ws.close();\n", + " }\n", + "\n", + " this.ws.onmessage = this._make_on_message_function(this);\n", + "\n", + " this.ondownload = ondownload;\n", + "}\n", + "\n", + "mpl.figure.prototype._init_header = function() {\n", + " var titlebar = $(\n", + " '
');\n", + " var titletext = $(\n", + " '
');\n", + " titlebar.append(titletext)\n", + " this.root.append(titlebar);\n", + " this.header = titletext[0];\n", + "}\n", + "\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._init_canvas = function() {\n", + " var fig = this;\n", + "\n", + " var canvas_div = $('
');\n", + "\n", + " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", + "\n", + " function canvas_keyboard_event(event) {\n", + " return fig.key_event(event, event['data']);\n", + " }\n", + "\n", + " canvas_div.keydown('key_press', canvas_keyboard_event);\n", + " canvas_div.keyup('key_release', canvas_keyboard_event);\n", + " this.canvas_div = canvas_div\n", + " this._canvas_extra_style(canvas_div)\n", + " this.root.append(canvas_div);\n", + "\n", + " var canvas = $('');\n", + " canvas.addClass('mpl-canvas');\n", + " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", + "\n", + " this.canvas = canvas[0];\n", + " this.context = canvas[0].getContext(\"2d\");\n", + "\n", + " var rubberband = $('');\n", + " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", + "\n", + " var pass_mouse_events = true;\n", + "\n", + " canvas_div.resizable({\n", + " start: function(event, ui) {\n", + " pass_mouse_events = false;\n", + " },\n", + " resize: function(event, ui) {\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " stop: function(event, ui) {\n", + " pass_mouse_events = true;\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " });\n", + "\n", + " function mouse_event_fn(event) {\n", + " if (pass_mouse_events)\n", + " return fig.mouse_event(event, event['data']);\n", + " }\n", + "\n", + " rubberband.mousedown('button_press', mouse_event_fn);\n", + " rubberband.mouseup('button_release', mouse_event_fn);\n", + " // Throttle sequential mouse events to 1 every 20ms.\n", + " rubberband.mousemove('motion_notify', mouse_event_fn);\n", + "\n", + " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", + " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", + "\n", + " canvas_div.on(\"wheel\", function (event) {\n", + " event = event.originalEvent;\n", + " event['data'] = 'scroll'\n", + " if (event.deltaY < 0) {\n", + " event.step = 1;\n", + " } else {\n", + " event.step = -1;\n", + " }\n", + " mouse_event_fn(event);\n", + " });\n", + "\n", + " canvas_div.append(canvas);\n", + " canvas_div.append(rubberband);\n", + "\n", + " this.rubberband = rubberband;\n", + " this.rubberband_canvas = rubberband[0];\n", + " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", + " this.rubberband_context.strokeStyle = \"#000000\";\n", + "\n", + " this._resize_canvas = function(width, height) {\n", + " // Keep the size of the canvas, canvas container, and rubber band\n", + " // canvas in synch.\n", + " canvas_div.css('width', width)\n", + " canvas_div.css('height', height)\n", + "\n", + " canvas.attr('width', width);\n", + " canvas.attr('height', height);\n", + "\n", + " rubberband.attr('width', width);\n", + " rubberband.attr('height', height);\n", + " }\n", + "\n", + " // Set the figure to an initial 600x600px, this will subsequently be updated\n", + " // upon first draw.\n", + " this._resize_canvas(600, 600);\n", + "\n", + " // Disable right mouse context menu.\n", + " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", + " return false;\n", + " });\n", + "\n", + " function set_focus () {\n", + " canvas.focus();\n", + " canvas_div.focus();\n", + " }\n", + "\n", + " window.setTimeout(set_focus, 100);\n", + "}\n", + "\n", + "mpl.figure.prototype._init_toolbar = function() {\n", + " var fig = this;\n", + "\n", + " var nav_element = $('
')\n", + " nav_element.attr('style', 'width: 100%');\n", + " this.root.append(nav_element);\n", + "\n", + " // Define a callback function for later on.\n", + " function toolbar_event(event) {\n", + " return fig.toolbar_button_onclick(event['data']);\n", + " }\n", + " function toolbar_mouse_event(event) {\n", + " return fig.toolbar_button_onmouseover(event['data']);\n", + " }\n", + "\n", + " for(var toolbar_ind in mpl.toolbar_items) {\n", + " var name = mpl.toolbar_items[toolbar_ind][0];\n", + " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", + " var image = mpl.toolbar_items[toolbar_ind][2];\n", + " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", + "\n", + " if (!name) {\n", + " // put a spacer in here.\n", + " continue;\n", + " }\n", + " var button = $('');\n", + " button.click(method_name, toolbar_event);\n", + " button.mouseover(tooltip, toolbar_mouse_event);\n", + " nav_element.append(button);\n", + " }\n", + "\n", + " // Add the status bar.\n", + " var status_bar = $('');\n", + " nav_element.append(status_bar);\n", + " this.message = status_bar[0];\n", + "\n", + " // Add the close button to the window.\n", + " var buttongrp = $('
');\n", + " var button = $('');\n", + " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", + " button.mouseover('Close figure', toolbar_mouse_event);\n", + " buttongrp.append(button);\n", + " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", + " titlebar.prepend(buttongrp);\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(el){\n", + " // this is important to make the div 'focusable\n", + " el.attr('tabindex', 0)\n", + " // reach out to IPython and tell the keyboard manager to turn it's self\n", + " // off when our div gets focus\n", + "\n", + " // location in version 3\n", + " if (IPython.notebook.keyboard_manager) {\n", + " IPython.notebook.keyboard_manager.register_events(el);\n", + " }\n", + " else {\n", + " // location in version 2\n", + " IPython.keyboard_manager.register_events(el);\n", + " }\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._key_event_extra = function(event, name) {\n", + " var manager = IPython.notebook.keyboard_manager;\n", + " if (!manager)\n", + " manager = IPython.keyboard_manager;\n", + "\n", + " // Check for shift+enter\n", + " if (event.shiftKey && event.which == 13) {\n", + " this.canvas_div.blur();\n", + " event.shiftKey = false;\n", + " // Send a \"J\" for go to next cell\n", + " event.which = 74;\n", + " event.keyCode = 74;\n", + " manager.command_mode();\n", + " manager.handle_keydown(event);\n", + " }\n", + "}\n", + "\n", + "mpl.figure.prototype.handle_save = function(fig, msg) {\n", + " fig.ondownload(fig, null);\n", + "}\n", + "\n", + "\n", + "mpl.find_output_cell = function(html_output) {\n", + " // Return the cell and output element which can be found *uniquely* in the notebook.\n", + " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", + " // IPython event is triggered only after the cells have been serialised, which for\n", + " // our purposes (turning an active figure into a static one), is too late.\n", + " var cells = IPython.notebook.get_cells();\n", + " var ncells = cells.length;\n", + " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", + " data = data.data;\n", + " }\n", + " if (data['text/html'] == html_output) {\n", + " return [cell, data, j];\n", + " }\n", + " }\n", + " }\n", + " }\n", + "}\n", + "\n", + "// Register the function which deals with the matplotlib target/channel.\n", + "// The kernel may be null if the page has been refreshed.\n", + "if (IPython.notebook.kernel != null) {\n", + " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", + "}\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "#create a phase plot\n", + "NP = 15\n", + "pos = numpy.linspace(-x0, x0, NP)\n", + "vel = numpy.linspace(-2*x0, 2*x0, NP)\n", + "\n", + "X1, X2 = numpy.meshgrid(pos, vel)\n", + "u = numpy.zeros_like(X1)\n", + "v = numpy.zeros_like(X2)\n", + "\n", + "for i in range(NP):\n", + " for j in range(NP):\n", + " x = X1[i, j]\n", + " y = X2[i, j]\n", + " yprime = f_damped_HM([x, y])\n", + " u[i,j] = yprime[0]\n", + " v[i,j] = yprime[1]\n", + "\n", + "\n", + "pyplot.figure(figsize=(10,8));\n", + "pyplot.quiver(X1, X2, u, v, pivot='mid', color='k', label='direction field');\n", + "pyplot.plot(q_d_analytical, vel_d_analytical, 'k-', label='analytical');\n", + "pyplot.plot(q1_d[:,0], q1_d[:,1],'r-', label='Euler');\n", + "pyplot.plot(q2_d[:,0], q2_d[:,1], 'b-', label='RK4');\n", + "pyplot.plot(q3_d[:,0], -Verlet_vel_d, 'g-', label='Verlet');\n", + "pyplot.xlabel('Postion ($x_1$)')\n", + "pyplot.ylabel('Velocity ($x_2$)')\n", + "pyplot.legend();\n", + "pyplot.title('Phase portrait for damped harmonic oscillator');\n" + ] + }, + { + "cell_type": "code", + "execution_count": 46, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "pyplot.close(\"all\") #This line is for cleanup - There are a lot of plots in this notebook. Comment it out \n", + " #if you want to keep all of the plots open." + ] }, { "cell_type": "code", @@ -4691,7 +5479,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "In this module, we will study the dynamics of a double pendulum. Like before, we'll derive the equations of motion from the Lagrangian and the Hamiltonian. We can use the same methods of integration as before, but this time there is no analytical solution to compare to. Finally, we will look at the properties of the system in phase space using phase plots and Poincare sections." + "In this section, we will study the dynamics of a double pendulum. Like before, we'll derive the equations of motion from the Lagrangian and the Hamiltonian. We can use the same methods of integration as before, but this time there is no analytical solution to compare to. Finally, we will look at the properties of the system in phase space using phase plots and Poincare sections." ] }, { @@ -4700,7 +5488,7 @@ "source": [ "## Background\n", "\n", - "The double pendulum problem is a simple system which can produce surprisingly complex movements. It is a chaotic system, meaning it is unpredictable, and there is no closed form solution to the motion of the two masses. This is a frequent example problem in the study of dynamic systems, nonlinear controls, and mechanics.\n", + "The double pendulum problem is a relatively simple system which can produce surprisingly complex movements. It is a chaotic system, meaning it is unpredictable and small changes in initial conditions lead to large changes in motion. There is no closed form solution to the motion of the two masses. This is a frequent example problem in the study of dynamic systems, nonlinear controls, and mechanics.\n", "\n", "In this notebook we will be treating the two rods as massless and ignoring the effects of friction. For a treatment of this problem without these assumptions, see (cite).\n", "\n", @@ -4870,7 +5658,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 21, "metadata": { "collapsed": false }, @@ -5025,7 +5813,7 @@ "" ] }, - "execution_count": 20, + "execution_count": 21, "metadata": {}, "output_type": "execute_result" } From b1173a800901bc3163e9e17591bd6d7ad284652d Mon Sep 17 00:00:00 2001 From: rbds Date: Sun, 6 Dec 2015 22:14:56 -0500 Subject: [PATCH 24/61] derive double pendulum from Hamiltonian --- randy_schur/Double_Pendulum_Problem.ipynb | 323 +++++++++++++++++----- 1 file changed, 260 insertions(+), 63 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index ee97532..54f8b7c 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -223,23 +223,15 @@ }, { "cell_type": "code", - "execution_count": 1, + "execution_count": 26, "metadata": { "collapsed": false }, - "outputs": [ - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/home/randy/anaconda3/lib/python3.4/site-packages/IPython/kernel/__init__.py:13: ShimWarning: The `IPython.kernel` package has been deprecated. You should import from ipykernel or jupyter_client instead.\n", - " \"You should import from ipykernel or jupyter_client instead.\", ShimWarning)\n" - ] - } - ], + "outputs": [], "source": [ "import numpy\n", "from scipy.linalg import solve\n", + "from numpy.linalg import det\n", "from math import pi, cos, sin, sqrt\n", "\n", "from matplotlib import pyplot\n", @@ -252,7 +244,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 13, "metadata": { "collapsed": false }, @@ -272,7 +264,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 14, "metadata": { "collapsed": true }, @@ -295,7 +287,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 15, "metadata": { "collapsed": true }, @@ -321,7 +313,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 16, "metadata": { "collapsed": true }, @@ -345,7 +337,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 17, "metadata": { "collapsed": true }, @@ -373,7 +365,7 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 18, "metadata": { "collapsed": true }, @@ -392,7 +384,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 19, "metadata": { "collapsed": false, "scrolled": true @@ -1193,13 +1185,13 @@ "source": [ "If we look closely at the plot, we can see that Euler and Runge-Kutta integration are close to each other, but begin to diverge as time goes on. This should be the behavior we expect for the integration of a non-linear system. But how close are these methods to the analytical solution? Does Verlet do any better? In this case it is obvious that Euler and Runge-Kutta are not correct (and possible unstable). Let's do an error analysis.\n", "\n", - "#### Error Analysis\n", + "### Error Analysis\n", "The error analysis code below is based on the numerical methods MOOC notebook on phugoid oscillation. " ] }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 20, "metadata": { "collapsed": true }, @@ -1222,7 +1214,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 21, "metadata": { "collapsed": false }, @@ -1243,7 +1235,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 22, "metadata": { "collapsed": false }, @@ -2021,7 +2013,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 23, "metadata": { "collapsed": true }, @@ -2044,7 +2036,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 24, "metadata": { "collapsed": false }, @@ -2079,7 +2071,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 25, "metadata": { "collapsed": false }, @@ -2903,7 +2895,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 27, "metadata": { "collapsed": true }, @@ -2927,7 +2919,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 28, "metadata": { "collapsed": true }, @@ -2956,7 +2948,7 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 29, "metadata": { "collapsed": true }, @@ -2987,7 +2979,7 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 30, "metadata": { "collapsed": false }, @@ -3793,7 +3785,7 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 31, "metadata": { "collapsed": false }, @@ -4600,19 +4592,11 @@ }, { "cell_type": "code", - "execution_count": 42, + "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [ - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/home/randy/anaconda3/lib/python3.4/site-packages/matplotlib/pyplot.py:424: RuntimeWarning: More than 20 figures have been opened. Figures created through the pyplot interface (`matplotlib.pyplot.figure`) are retained until explicitly closed and may consume too much memory. (To control this warning, see the rcParam `figure.max_open_warning`).\n", - " max_open_warning, RuntimeWarning)\n" - ] - }, { "data": { "application/javascript": [ @@ -5395,7 +5379,7 @@ }, { "cell_type": "code", - "execution_count": 46, + "execution_count": 33, "metadata": { "collapsed": false }, @@ -5517,7 +5501,7 @@ "\n", "$$\\begin{eqnarray*}\n", "U &=& m_1g(-y_1) + m_2g(-y_2) \\\\\n", - " &=& -m_1g(l_1cos(\\theta_1) - m_2g(l_1cos\\theta_1 + l_2cos\\theta_2)\n", + " &=& -m_1gl_1cos\\theta_1 - m_2g(l_1cos\\theta_1 + l_2cos\\theta_2)\n", "\\end{eqnarray*}$$\n", "\n", "Kinetic Energy is calculated as:\n", @@ -5577,13 +5561,13 @@ "\n", "Notice that these equations of motion are implicit equations for $\\ddot\\theta_1$ and $\\ddot\\theta_2$. We have talked about implicit equations in this class, and we solve them by forming a system of equations. In this case, we will need to solve the equations of motion at each time step. We will characterize our system in state space using the angle and angular velocity of each pendulum. So our state becomes:\n", "\n", - "$$\\begin{eqnarray}\n", - "\\vec{q}(t) = \\begin{pmatrix} \\theta_1\\\\ \\theta_2\\\\ \\dot\\theta_1 \\\\ \\dot\\theta_2 \\end{pmatrix}\n", - "\\end{eqnarray}$$\n", + "$$\\begin{eqnarray*}\n", + "\\vec{x}(t) = \\begin{pmatrix} \\theta_1\\\\ \\theta_2\\\\ \\dot\\theta_1 \\\\ \\dot\\theta_2 \\end{pmatrix}\n", + "\\end{eqnarray*}$$\n", "\n", "The equations of motion can be rewritten as a system of equations:\n", "\n", - "$$ A\\dot q=f(q)$$\n", + "$$ A\\dot x=f(x)$$\n", "\n", "where \n", "\n", @@ -5600,65 +5584,278 @@ "and\n", "\n", "$$\\begin{eqnarray*}\n", - "f(\\vec{q}(t)) = \\begin{pmatrix} q_3\\\\ q_4\\\\ m_2l_1l_2x_4^2sin(q_2-q_1) -(m_1+m_2)gl_1sin(q_1)\\\\ m_2l_1l_2q_3^2sin(q_2-q_1)-l_2m_2gsinq_2 \\end{pmatrix}\n", + "f(\\vec{x}(t)) = \\begin{pmatrix} x_3\\\\ x_4\\\\ m_2l_1l_2x_4^2sin(x_2-x_1) -(m_1+m_2)gl_1sin(x_1)\\\\ m_2l_1l_2x_3^2sin(x_2-x_1)-l_2m_2gsinx_2 \\end{pmatrix}\n", "\\end{eqnarray*}$$\n", "\n", "In order to solve this, we can evaluate matrix A and vector $f(x)$ at each time step, then invert A and premultiply the result to each side of the equation. This gives us an explicit system of equations of motion:\n", "\n", - "$$ \\dot q=A^{-1}f(q)$$\n", + "$$ \\dot x=A^{-1}f(x)$$\n", "\n", - "You may notice that these operations require a significant computational load at each time step. In order to reduce this, we can analytically invert the matrix A, then simply fill in the necessary values at each time step and multiply by $f(q)$. This will significantly speed up running time, as inverting a matrix is an expensive operation.\n" + "You may notice that these operations require a significant computational load at each time step. In order to reduce this, we can analytically invert the matrix A, then simply fill in the necessary values at each time step and multiply by $f(q)$. This will significantly speed up running time, as inverting a matrix is a computationally expensive operation.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "## Integration\n", + "### Equations of Motion from Hamiltonian\n", "\n", - "For Euler and Runge-Kutta, we can use the equation above to integrate our system. The results will be VERY dependent on initial conditions; this is why we call it a chaotic system.\n", + "Here we derive the equations of motion from the Hamiltonian. If you want good practice, you can derive these on your own. We start with the same potential and kinetic energy, but we'll need to use postion ($\\theta$) and momentum ($I\\omega$) to give us:\n", "\n", - "For the symplectic integrator, we'll need the equations of motion from the Hamiltonian. If you want good practice, you can derive these on your own. The Hamiltonian equations become:\n", + "$$\\begin{eqnarray*}\n", + "U &=& m_1g(-y_1) + m_2g(-y_2) \\\\\n", + " &=& -m_1g(l_1cos\\theta_1 - m_2g(l_1cos\\theta_1 + l_2cos\\theta_2)\n", + "\\end{eqnarray*}$$\n", "\n", - "\n" + "Kinetic Energy is calculated as:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "T &=& \\frac{1}{2}mv_1^2+\\frac{1}{2}mv_2^2 \\\\\n", + "&=& \\frac{q_1^2}{2I_1} + \\frac{q_2^2}{2I_2}\n", + "\\end{eqnarray*}$$\n", + "\n", + "where $I_i = m_il_i^2$. So our Hamiltonian quantity becomes:\n", + "$$\\begin{equation*}\n", + "H = \\frac{1}{2}\\left(\\frac{q_1^2}{I_1} + \\frac{q_2^2}{I_2}\\right) - (m_1+m_2)gl_1cos\\theta_1 - m_2gl_2cos\\theta_2\n", + "\\end{equation*}$$\n", + "\n", + "Since we have two degrees of freedom, we will have four equations of motion. These will be the following:\n", + "$$\\begin{eqnarray*}\n", + "\\dot{p_1} &=& -\\frac{\\partial H}{\\partial q_1} = \\frac{q_1}{I_1} \\\\\n", + "\\dot{q_1} &=& \\frac{\\partial H}{\\partial p_1} = (m_1+m_2)gl_1sinp_1 \\\\\n", + "\\dot{p_2} &=& -\\frac{\\partial H}{\\partial q_2} = \\frac{q_2}{I_2} \\\\\n", + "\\dot{q_2} &=& \\frac{\\partial H}{\\partial p_2} = m_2gl_2sinp_2 \\\\\n", + "\\end{eqnarray*}$$\n" ] }, { "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Integration\n", + "\n", + "For Euler and Runge-Kutta, we can use the matrix equation above to integrate our system. The results will be VERY dependent on initial conditions; this is why we call it a chaotic system. For the symplectic integrator, we'll use the Hamiltonian equations of motion above. " + ] + }, + { + "cell_type": "code", + "execution_count": 64, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def A_inv(A):\n", + " \"\"\"Returns the inverse of matrix A (from EOM analysis)\n", + " \n", + " Parameters: \n", + " A - Matrix of coefficients\n", + " Returns:\n", + " A^-1 - Inverse of matrix A\n", + " \"\"\"\n", + " d = det(A)\n", + " Ai = 1/d*([[A[2,2]*A[3,3]-A[2,3]*A[3,2], 0, 0, 0], [0, A[2,2]*A[3,3]-A[2,3]*A[3,2], 0, 0]\\\n", + " , [0, 0, A[3,3], -A[2,3]], [0, 0, -A[3,2], A[2,2]]])\n", + " \n", + " return Ai " + ] + }, + { + "cell_type": "code", + "execution_count": 101, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def f_double_pendulum(u):\n", + " \"\"\"Returns RHS of double pendulum EOM\n", + " \n", + " Parameters:\n", + " q - initial state\n", + " \n", + " Returns:\n", + " RHS - RHS of harmonic oscillator eqn.\n", + " \n", + " \"\"\"\n", + " x1 = u[0]\n", + " x2 = u[1]\n", + " x3 = u[2]\n", + " x4 = u[3]\n", + " \n", + " f1 = x3\n", + " f2 = x4\n", + " f3 = m2*l1*l2*x4**2*sin(x2 - x1) - (m1+m2)*g*l1*sin(x1)\n", + " f4 = m2*l1*l2*x3**2*sin(x2 - x1) - l2*m2*g*sin(x2)\n", + " \n", + " A = numpy.matrix([[1, 0, 0, 0],[0, 1, 0, 0],[0, 0, (m1+m2)*l1**2, m2*l1*l2*cos(x2-x1)],[0, 0, m2*l1*l2*cos(x2-x1), m2*l1**2]])\n", + " print(A)\n", + " Ai = A_inv(A)\n", + " f = numpy.array([f1, f2, f3, f4])\n", + " print(f)\n", + " RHS = Ai*f.T\n", + " \n", + " return numpy.array([f1, f2, f3, f4])" + ] + }, + { + "cell_type": "code", + "execution_count": 102, "metadata": { "collapsed": true }, + "outputs": [], "source": [ - "## Solve!\n", + "#Set up parameters:\n", + "g = 9.8 #[m/s^2]\n", + "m1 = 2\n", + "m2 = 1.5;\n", + "l1 = 1\n", + "l2 = 1.5\n", + "\n", + "T = 25; #[seconds]\n", + "dt = .02; #\n", + "N = int(T/dt)+1\n", + "t = numpy.linspace(0.0, T, N)\n", + "\n", + "#Initial Conditions\n", + "theta1_0 = 2 #[radians]\n", + "theta2_0 = 1\n", + "theta1_dot_0 = 0 #[rad/s]\n", + "theta2_dot_0 = 0 #\n", "\n", - "Here we define functions for each of the integration methods, set our parameters and initial conditions, and then numerically integrate the motion of the pendulums. The plots below compare the different integration techniques." + "#p0 = x0\n", + "#q0 = m*xdot0\n", + "x_init_dp = numpy.array([theta1_0, theta2_0, theta1_dot_0, theta2_dot_0])\n", + "#x_init_H = numpy.array([p0, q0])\n" + ] + }, + { + "cell_type": "code", + "execution_count": 103, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[[ 1. 0. 0. 0. ]\n", + " [ 0. 1. 0. 0. ]\n", + " [ 0. 0. 3.5 1.21568019]\n", + " [ 0. 0. 1.21568019 1.5 ]]\n", + "[ 0. 0. -31.18890174 -18.55443522]\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/home/randy/anaconda3/lib/python3.4/site-packages/ipykernel/__main__.py:10: DeprecationWarning: using a non-integer number instead of an integer will result in an error in the future\n" + ] + }, + { + "ename": "ValueError", + "evalue": "operands could not be broadcast together with shapes (0,) (4,) ", + "output_type": "error", + "traceback": [ + "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[1;31mValueError\u001b[0m Traceback (most recent call last)", + "\u001b[1;32m\u001b[0m in \u001b[0;36m\u001b[1;34m()\u001b[0m\n\u001b[0;32m 3\u001b[0m \u001b[0mq1_dp\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mx_init_dp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mcopy\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;31m#set initial conditions\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 4\u001b[0m \u001b[1;32mfor\u001b[0m \u001b[0mn\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mN\u001b[0m\u001b[1;33m-\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m \u001b[1;31m#integrate with Euler\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 5\u001b[1;33m \u001b[0mq1_dp\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mn\u001b[0m\u001b[1;33m+\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0meuler\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mq1_dp\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mn\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mf_double_pendulum\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdt\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 6\u001b[0m \u001b[1;31m#print(q1[n])\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 7\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;32m\u001b[0m in \u001b[0;36meuler\u001b[1;34m(u, f, dt)\u001b[0m\n\u001b[0;32m 11\u001b[0m \"\"\"\n\u001b[0;32m 12\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 13\u001b[1;33m \u001b[1;32mreturn\u001b[0m \u001b[0mu\u001b[0m \u001b[1;33m+\u001b[0m \u001b[0mdt\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mu\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m", + "\u001b[1;32m\u001b[0m in \u001b[0;36mf_double_pendulum\u001b[1;34m(u)\u001b[0m\n\u001b[0;32m 24\u001b[0m \u001b[0mf\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnumpy\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0marray\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mf1\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mf2\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mf3\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mf4\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 25\u001b[0m \u001b[0mprint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 26\u001b[1;33m \u001b[0mRHS\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mAi\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mT\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 27\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 28\u001b[0m \u001b[1;32mreturn\u001b[0m \u001b[0mnumpy\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0marray\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mf1\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mf2\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mf3\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mf4\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;31mValueError\u001b[0m: operands could not be broadcast together with shapes (0,) (4,) " + ] + } + ], + "source": [ + "#Euler\n", + "q1_dp = numpy.zeros((N,4)) \n", + "q1_dp[0] = x_init_dp.copy() #set initial conditions\n", + "for n in range(N-1): #integrate with Euler\n", + " q1_dp[n+1,:] = euler(q1_dp[n,:], f_double_pendulum, dt)\n", + " #print(q1[n])\n", + " \n", + "pyplot.figure(figsize=(10,8));\n", + "pyplot.grid(True);\n", + "pyplot.xlabel(r't', fontsize=18);\n", + "pyplot.ylabel(r'position (meters)', fontsize=18);\n", + "pyplot.title('Harmonic oscillator position');\n", + "pyplot.plot(t, q1_d[:,0], lw=2, label='Euler');\n", + "pyplot.legend();" + ] + }, + { + "cell_type": "code", + "execution_count": 97, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "0\n" + ] + } + ], + "source": [ + "a = numpy.matrix([[1,2],[3,4]])\n", + "print(n)" ] }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [] + }, { "cell_type": "markdown", "metadata": {}, "source": [ "Sources: \n", "\n", - "http://scienceworld.wolfram.com/physics/DoublePendulum.html\n", + "[1] http://scienceworld.wolfram.com/physics/DoublePendulum.html\n", "\n", - "http://www.phy.uct.ac.za/courses/opencontent/phylab2/worksheet9_09.pdf\n", + "[2] http://www.phy.uct.ac.za/courses/opencontent/phylab2/worksheet9_09.pdf\n", "\n", - "http://www.phys.lsu.edu/faculty/gonzalez/Teaching/Phys7221/DoublePendulum.pdf\n", + "[3] http://www.phys.lsu.edu/faculty/gonzalez/Teaching/Phys7221/DoublePendulum.pdf\n", "\n", - "http://www.iontrap.wabash.edu/adlab/papers/F2011_foster_groninger_tang_chaos.pdf\n", + "[4] http://www.iontrap.wabash.edu/adlab/papers/F2011_foster_groninger_tang_chaos.pdf\n", "\n", - "https://math.berkeley.edu/~alanw/242papers99/markiewicz.pdf\n", + "[5] https://math.berkeley.edu/~alanw/242papers99/markiewicz.pdf\n", "\n", - "http://www.unige.ch/~hairer/poly_geoint/week2.pdf\n", + "[6] http://www.unige.ch/~hairer/poly_geoint/week2.pdf\n", "\n", - "http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1994CeMDA..60..409T&defaultprint=YES&filetype=.pdf\n", + "[7] http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1994CeMDA..60..409T&defaultprint=YES&filetype=.pdf\n", "\n" ] }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 1, "metadata": { "collapsed": false }, @@ -5813,7 +6010,7 @@ "" ] }, - "execution_count": 21, + "execution_count": 1, "metadata": {}, "output_type": "execute_result" } From b1f00cd9b75f71dfbed721ae78c35880762fe434 Mon Sep 17 00:00:00 2001 From: rbds Date: Sun, 6 Dec 2015 22:41:33 -0500 Subject: [PATCH 25/61] evening commit --- randy_schur/Double_Pendulum_Problem.ipynb | 1220 ++++++++++++++++----- 1 file changed, 918 insertions(+), 302 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index 54f8b7c..421339e 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -223,11 +223,20 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 1, "metadata": { "collapsed": false }, - "outputs": [], + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/home/randy/anaconda3/lib/python3.4/site-packages/IPython/kernel/__init__.py:13: ShimWarning: The `IPython.kernel` package has been deprecated. You should import from ipykernel or jupyter_client instead.\n", + " \"You should import from ipykernel or jupyter_client instead.\", ShimWarning)\n" + ] + } + ], "source": [ "import numpy\n", "from scipy.linalg import solve\n", @@ -244,7 +253,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 2, "metadata": { "collapsed": false }, @@ -264,7 +273,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 3, "metadata": { "collapsed": true }, @@ -287,7 +296,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 4, "metadata": { "collapsed": true }, @@ -313,7 +322,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 5, "metadata": { "collapsed": true }, @@ -337,7 +346,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 6, "metadata": { "collapsed": true }, @@ -365,7 +374,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 7, "metadata": { "collapsed": true }, @@ -384,7 +393,7 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 8, "metadata": { "collapsed": false, "scrolled": true @@ -1129,7 +1138,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1191,7 +1200,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 9, "metadata": { "collapsed": true }, @@ -1214,7 +1223,7 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 10, "metadata": { "collapsed": false }, @@ -1235,7 +1244,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 11, "metadata": { "collapsed": false }, @@ -1979,7 +1988,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2013,7 +2022,7 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 12, "metadata": { "collapsed": true }, @@ -2036,7 +2045,7 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 13, "metadata": { "collapsed": false }, @@ -2071,7 +2080,7 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 14, "metadata": { "collapsed": false }, @@ -2815,7 +2824,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2895,7 +2904,7 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 15, "metadata": { "collapsed": true }, @@ -2919,7 +2928,7 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 16, "metadata": { "collapsed": true }, @@ -2948,7 +2957,7 @@ }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 17, "metadata": { "collapsed": true }, @@ -2979,7 +2988,7 @@ }, { "cell_type": "code", - "execution_count": 30, + "execution_count": 18, "metadata": { "collapsed": false }, @@ -3723,7 +3732,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -3785,7 +3794,7 @@ }, { "cell_type": "code", - "execution_count": 31, + "execution_count": 19, "metadata": { "collapsed": false }, @@ -4529,7 +4538,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -4592,7 +4601,7 @@ }, { "cell_type": "code", - "execution_count": 32, + "execution_count": 20, "metadata": { "collapsed": false }, @@ -5336,7 +5345,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -5379,7 +5388,7 @@ }, { "cell_type": "code", - "execution_count": 33, + "execution_count": 21, "metadata": { "collapsed": false }, @@ -5639,7 +5648,7 @@ }, { "cell_type": "code", - "execution_count": 64, + "execution_count": 90, "metadata": { "collapsed": true }, @@ -5654,7 +5663,7 @@ " A^-1 - Inverse of matrix A\n", " \"\"\"\n", " d = det(A)\n", - " Ai = 1/d*([[A[2,2]*A[3,3]-A[2,3]*A[3,2], 0, 0, 0], [0, A[2,2]*A[3,3]-A[2,3]*A[3,2], 0, 0]\\\n", + " Ai = 1/d*numpy.array([[A[2,2]*A[3,3]-A[2,3]*A[3,2], 0, 0, 0], [0, A[2,2]*A[3,3]-A[2,3]*A[3,2], 0, 0]\\\n", " , [0, 0, A[3,3], -A[2,3]], [0, 0, -A[3,2], A[2,2]]])\n", " \n", " return Ai " @@ -5662,7 +5671,7 @@ }, { "cell_type": "code", - "execution_count": 101, + "execution_count": 91, "metadata": { "collapsed": false }, @@ -5688,21 +5697,23 @@ " f3 = m2*l1*l2*x4**2*sin(x2 - x1) - (m1+m2)*g*l1*sin(x1)\n", " f4 = m2*l1*l2*x3**2*sin(x2 - x1) - l2*m2*g*sin(x2)\n", " \n", - " A = numpy.matrix([[1, 0, 0, 0],[0, 1, 0, 0],[0, 0, (m1+m2)*l1**2, m2*l1*l2*cos(x2-x1)],[0, 0, m2*l1*l2*cos(x2-x1), m2*l1**2]])\n", - " print(A)\n", + " A = numpy.array([[1, 0, 0, 0],[0, 1, 0, 0],[0, 0, (m1+m2)*l1**2, m2*l1*l2*cos(x2-x1)],[0, 0, m2*l1*l2*cos(x2-x1), m2*l1**2]])\n", + " #print(numpy.shape(A))\n", + " #print(type(A))\n", " Ai = A_inv(A)\n", - " f = numpy.array([f1, f2, f3, f4])\n", - " print(f)\n", - " RHS = Ai*f.T\n", + " f = numpy.array([[f1, f2, f3, f4]])\n", + " #print(numpy.shape(f.T))\n", + " #print(type(f))\n", + " RHS = Ai.dot(f.T)\n", " \n", - " return numpy.array([f1, f2, f3, f4])" + " return RHS" ] }, { "cell_type": "code", - "execution_count": 102, + "execution_count": 92, "metadata": { - "collapsed": true + "collapsed": false }, "outputs": [], "source": [ @@ -5727,294 +5738,899 @@ "#p0 = x0\n", "#q0 = m*xdot0\n", "x_init_dp = numpy.array([theta1_0, theta2_0, theta1_dot_0, theta2_dot_0])\n", - "#x_init_H = numpy.array([p0, q0])\n" + "#x_init_H = numpy.array([p0, q0])\n", + "\n" ] }, { "cell_type": "code", - "execution_count": 103, + "execution_count": 93, "metadata": { - "collapsed": false + "collapsed": true }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[[ 1. 0. 0. 0. ]\n", - " [ 0. 1. 0. 0. ]\n", - " [ 0. 0. 3.5 1.21568019]\n", - " [ 0. 0. 1.21568019 1.5 ]]\n", - "[ 0. 0. -31.18890174 -18.55443522]\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/home/randy/anaconda3/lib/python3.4/site-packages/ipykernel/__main__.py:10: DeprecationWarning: using a non-integer number instead of an integer will result in an error in the future\n" - ] - }, - { - "ename": "ValueError", - "evalue": "operands could not be broadcast together with shapes (0,) (4,) ", - "output_type": "error", - "traceback": [ - "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[1;31mValueError\u001b[0m Traceback (most recent call last)", - "\u001b[1;32m\u001b[0m in \u001b[0;36m\u001b[1;34m()\u001b[0m\n\u001b[0;32m 3\u001b[0m \u001b[0mq1_dp\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mx_init_dp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mcopy\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;31m#set initial conditions\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 4\u001b[0m \u001b[1;32mfor\u001b[0m \u001b[0mn\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mN\u001b[0m\u001b[1;33m-\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m \u001b[1;31m#integrate with Euler\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 5\u001b[1;33m \u001b[0mq1_dp\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mn\u001b[0m\u001b[1;33m+\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0meuler\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mq1_dp\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mn\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mf_double_pendulum\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdt\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 6\u001b[0m \u001b[1;31m#print(q1[n])\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 7\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n", - "\u001b[1;32m\u001b[0m in \u001b[0;36meuler\u001b[1;34m(u, f, dt)\u001b[0m\n\u001b[0;32m 11\u001b[0m \"\"\"\n\u001b[0;32m 12\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 13\u001b[1;33m \u001b[1;32mreturn\u001b[0m \u001b[0mu\u001b[0m \u001b[1;33m+\u001b[0m \u001b[0mdt\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mu\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m", - "\u001b[1;32m\u001b[0m in \u001b[0;36mf_double_pendulum\u001b[1;34m(u)\u001b[0m\n\u001b[0;32m 24\u001b[0m \u001b[0mf\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnumpy\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0marray\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mf1\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mf2\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mf3\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mf4\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 25\u001b[0m \u001b[0mprint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 26\u001b[1;33m \u001b[0mRHS\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mAi\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mT\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 27\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 28\u001b[0m \u001b[1;32mreturn\u001b[0m \u001b[0mnumpy\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0marray\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mf1\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mf2\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mf3\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mf4\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", - "\u001b[1;31mValueError\u001b[0m: operands could not be broadcast together with shapes (0,) (4,) " - ] - } - ], + "outputs": [], "source": [ - "#Euler\n", - "q1_dp = numpy.zeros((N,4)) \n", - "q1_dp[0] = x_init_dp.copy() #set initial conditions\n", - "for n in range(N-1): #integrate with Euler\n", - " q1_dp[n+1,:] = euler(q1_dp[n,:], f_double_pendulum, dt)\n", - " #print(q1[n])\n", + "def euler_DP(u, f, dt):\n", + " \"\"\" Euler's method for integrating a system of differential equations.\n", " \n", - "pyplot.figure(figsize=(10,8));\n", - "pyplot.grid(True);\n", - "pyplot.xlabel(r't', fontsize=18);\n", - "pyplot.ylabel(r'position (meters)', fontsize=18);\n", - "pyplot.title('Harmonic oscillator position');\n", - "pyplot.plot(t, q1_d[:,0], lw=2, label='Euler');\n", - "pyplot.legend();" + " Parameters:\n", + " u - state at current step \n", + " f - RHS of equation\n", + " dt- time step size\n", + " \n", + " Returns: \n", + " x - array of values at next time step.\n", + " \"\"\"\n", + " #print(f(u))\n", + " #print(u)\n", + " \n", + " return u + dt*f(u).T " ] }, { "cell_type": "code", - "execution_count": 97, + "execution_count": 101, "metadata": { "collapsed": false }, "outputs": [ { - "name": "stdout", + "name": "stderr", "output_type": "stream", "text": [ - "0\n" + "/home/randy/anaconda3/lib/python3.4/site-packages/ipykernel/__main__.py:18: RuntimeWarning: overflow encountered in double_scalars\n", + "/home/randy/anaconda3/lib/python3.4/site-packages/ipykernel/__main__.py:19: RuntimeWarning: overflow encountered in double_scalars\n", + "/home/randy/anaconda3/lib/python3.4/site-packages/numpy/linalg/linalg.py:1729: RuntimeWarning: invalid value encountered in det\n", + " return _umath_linalg.det(a, signature=signature).astype(result_t)\n", + "/home/randy/anaconda3/lib/python3.4/site-packages/ipykernel/__main__.py:10: RuntimeWarning: divide by zero encountered in double_scalars\n", + "/home/randy/anaconda3/lib/python3.4/site-packages/ipykernel/__main__.py:10: RuntimeWarning: invalid value encountered in multiply\n" ] - } - ], - "source": [ - "a = numpy.matrix([[1,2],[3,4]])\n", - "print(n)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, - { - "cell_type": "markdown", - "metadata": { - "collapsed": true - }, - "source": [] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Sources: \n", - "\n", - "[1] http://scienceworld.wolfram.com/physics/DoublePendulum.html\n", - "\n", - "[2] http://www.phy.uct.ac.za/courses/opencontent/phylab2/worksheet9_09.pdf\n", - "\n", - "[3] http://www.phys.lsu.edu/faculty/gonzalez/Teaching/Phys7221/DoublePendulum.pdf\n", - "\n", - "[4] http://www.iontrap.wabash.edu/adlab/papers/F2011_foster_groninger_tang_chaos.pdf\n", - "\n", - "[5] https://math.berkeley.edu/~alanw/242papers99/markiewicz.pdf\n", - "\n", - "[6] http://www.unige.ch/~hairer/poly_geoint/week2.pdf\n", - "\n", - "[7] http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1994CeMDA..60..409T&defaultprint=YES&filetype=.pdf\n", - "\n" - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false - }, - "outputs": [ + }, { "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "execution_count": 1, - "metadata": {}, - "output_type": "execute_result" - } - ], + "\n", + "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._init_canvas = function() {\n", + " var fig = this;\n", + "\n", + " var canvas_div = $('
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')\n", + " nav_element.attr('style', 'width: 100%');\n", + " this.root.append(nav_element);\n", + "\n", + " // Define a callback function for later on.\n", + " function toolbar_event(event) {\n", + " return fig.toolbar_button_onclick(event['data']);\n", + " }\n", + " function toolbar_mouse_event(event) {\n", + " return fig.toolbar_button_onmouseover(event['data']);\n", + " }\n", + "\n", + " for(var toolbar_ind in mpl.toolbar_items) {\n", + " var name = mpl.toolbar_items[toolbar_ind][0];\n", + " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", + " var image = mpl.toolbar_items[toolbar_ind][2];\n", + " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", + "\n", + " if (!name) {\n", + " // put a spacer in here.\n", + " continue;\n", + " }\n", + " var button = $('');\n", + " button.click(method_name, toolbar_event);\n", + " button.mouseover(tooltip, toolbar_mouse_event);\n", + " nav_element.append(button);\n", + " }\n", + "\n", + " // Add the status bar.\n", + " var status_bar = $('');\n", + " nav_element.append(status_bar);\n", + " this.message = status_bar[0];\n", + "\n", + " // Add the close button to the window.\n", + " var buttongrp = $('
');\n", + " var button = $('');\n", + " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", + " button.mouseover('Close figure', toolbar_mouse_event);\n", + " buttongrp.append(button);\n", + " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", + " titlebar.prepend(buttongrp);\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(el){\n", + " // this is important to make the div 'focusable\n", + " el.attr('tabindex', 0)\n", + " // reach out to IPython and tell the keyboard manager to turn it's self\n", + " // off when our div gets focus\n", + "\n", + " // location in version 3\n", + " if (IPython.notebook.keyboard_manager) {\n", + " IPython.notebook.keyboard_manager.register_events(el);\n", + " }\n", + " else {\n", + " // location in version 2\n", + " IPython.keyboard_manager.register_events(el);\n", + " }\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._key_event_extra = function(event, name) {\n", + " var manager = IPython.notebook.keyboard_manager;\n", + " if (!manager)\n", + " manager = IPython.keyboard_manager;\n", + "\n", + " // Check for shift+enter\n", + " if (event.shiftKey && event.which == 13) {\n", + " this.canvas_div.blur();\n", + " event.shiftKey = false;\n", + " // Send a \"J\" for go to next cell\n", + " event.which = 74;\n", + " event.keyCode = 74;\n", + " manager.command_mode();\n", + " manager.handle_keydown(event);\n", + " }\n", + "}\n", + "\n", + "mpl.figure.prototype.handle_save = function(fig, msg) {\n", + " fig.ondownload(fig, null);\n", + "}\n", + "\n", + "\n", + "mpl.find_output_cell = function(html_output) {\n", + " // Return the cell and output element which can be found *uniquely* in the notebook.\n", + " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", + " // IPython event is triggered only after the cells have been serialised, which for\n", + " // our purposes (turning an active figure into a static one), is too late.\n", + " var cells = IPython.notebook.get_cells();\n", + " var ncells = cells.length;\n", + " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", + " data = data.data;\n", + " }\n", + " if (data['text/html'] == html_output) {\n", + " return [cell, data, j];\n", + " }\n", + " }\n", + " }\n", + " }\n", + "}\n", + "\n", + "// Register the function which deals with the matplotlib target/channel.\n", + "// The kernel may be null if the page has been refreshed.\n", + "if (IPython.notebook.kernel != null) {\n", + " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", + "}\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "#Euler\n", + "q1_dp = numpy.zeros((N,4)) \n", + "q1_dp[0,:] = x_init_dp.copy() #set initial conditions\n", + "for n in range(N-1): #integrate with Euler\n", + " q1_dp[n+1,:] = euler_DP(q1_dp[n,:], f_double_pendulum, dt)\n", + " #print(q1[n])\n", + " \n", + "#Runge-Kutta\n", + "q2_dp = numpy.zeros((N,4)) \n", + "q2_dp[0,:] = x_init_dp.copy() #set initial conditions\n", + "for n in range(N-1): #integrate with Euler\n", + " q2_dp[n+1,:] = RK4_DP(q2_dp[n,:], f_double_pendulum, dt)\n", + " \n", + "pyplot.figure(figsize=(10,8));\n", + "pyplot.grid(True);\n", + "pyplot.xlabel(r't', fontsize=18);\n", + "pyplot.ylabel(r'position (meters)', fontsize=18);\n", + "pyplot.title('Double Pendulum Euler');\n", + "pyplot.plot(t, q1_dp[:,0], lw=2, label='Joint 1');\n", + "pyplot.plot(t, q1_dp[:,1], lw=2, label='Joint 2')\n", + "pyplot.legend();\n", + "\n", + "pyplot.figure(figsize=(10,8));\n", + "pyplot.grid(True);\n", + "pyplot.xlabel(r't', fontsize=18);\n", + "pyplot.ylabel(r'position (meters)', fontsize=18);\n", + "pyplot.title('Double Pendulum RK4');\n", + "pyplot.plot(t, q2_dp[:,0], lw=2, label='Joint 1');\n", + "pyplot.plot(t, q2_dp[:,1], lw=2, label='Joint 2')\n", + "pyplot.legend();" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [] }, { "cell_type": "code", @@ -6652,21 +7428,21 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3", + "display_name": "Python 2", "language": "python", - "name": "python3" + "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", - "version": 3 + "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.4.3" + "pygments_lexer": "ipython2", + "version": "2.7.10" } }, "nbformat": 4, diff --git a/randy_schur/figures/POS_computer.png b/randy_schur/figures/POS_computer.png new file mode 100644 index 0000000000000000000000000000000000000000..131df2d4a15ea42251f9d3ab248c4b39ce98553a GIT binary patch literal 82838 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z{sUGWrvs&d!3HcpfjhaDykTzmw`0n*X{Xc}8k~S{_pGf*w|R$V%O4}<`Te`@r3Eq z+ksmJo`Y2aOJ3vLTP8Pmt_8MCfpfXQUdmnIT&u*-XS4Hxr)CHO&ri6>%1{eB7kKVj<_wM3Fi(Ehl+^n~_Q1tg#sn1c?3%Z~#$Wh?oEY`p}#k-wP zHs!w`aFoiT;zI)Pyh>o(>@IM02e>)$XZ^qD^}rLG{(oqP+gq_&wu&F?pz^noG$`5)i5x4 My85}Sb4q9e0E?&BL;wH) literal 0 HcmV?d00001 From 1ba78db66df5947b147a9529b1d561d82ae8be59 Mon Sep 17 00:00:00 2001 From: rbds Date: Mon, 7 Dec 2015 15:24:57 -0500 Subject: [PATCH 27/61] fix derivation for double pendulum EOM --- randy_schur/Double_Pendulum_Problem.ipynb | 86 +++++++++-------------- 1 file changed, 33 insertions(+), 53 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index 456bdf6..08b4cb1 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -5568,39 +5568,51 @@ "source": [ "## Problem setup\n", "\n", - "Notice that these equations of motion are implicit equations for $\\ddot\\theta_1$ and $\\ddot\\theta_2$. We have talked about implicit equations in this class, and we solve them by forming a system of equations. In this case, we will need to solve the equations of motion at each time step. We will characterize our system in state space using the angle and angular velocity of each pendulum. So our state becomes:\n", + "Notice that these equations of motion are implicit equations for $\\ddot\\theta_1$ and $\\ddot\\theta_2$. We have talked about implicit equations in this class, and we solve them by forming a system of equations. In this case, we will need to solve the equations of motion at each time step. We will characterize our system in state space using the angle and angular velocity of each pendulum. Let's rewrite the equations of motion in a more readable form. We can use the standard form of this equation (which you'll be familiar with if you have studied mechanics) by separating out mass terms (anything multiplied by $\\ddot{x}$), centripetal terms (anything multiplied by $\\dot{x}$), and gravitational terms (anything multiplied by $x$). We get the following equations of state:\n", "\n", "$$\\begin{eqnarray*}\n", - "\\vec{x}(t) = \\begin{pmatrix} \\theta_1\\\\ \\theta_2\\\\ \\dot\\theta_1 \\\\ \\dot\\theta_2 \\end{pmatrix}\n", + "0 &=&\\textbf{M}\\vec{\\ddot{x}}(t) + \\textbf{V}\\dot{x} + \\textbf{G} \n", "\\end{eqnarray*}$$\n", "\n", - "The equations of motion can be rewritten as a system of equations:\n", + "The equations of motion can be solved for $\\ddot{x}$:\n", "\n", - "$$ A\\dot x=f(x)$$\n", + "$$ \\ddot{x} = \\textbf{M}^{-1}\\left[ \\textbf{V}\\dot x + \\textbf{G} \\right]$$\n", "\n", "where \n", "\n", + "\\begin{eqnarray*}\n", + "M = \n", + "\\left[\\begin{array}{c}\n", + "(m_1+m_2)l_1 & m_2l_1l_2cos(\\theta_2 - \\theta_1)\\\\\n", + " m_2l_1l_2cos(\\theta_2 - \\theta_1) &m_2l_2^2\n", + "\\end{array}\\right]\n", + "\\end{eqnarray*}\n", + "\n", + "\n", "\\begin{equation*}\n", - "A = \n", + "V = \n", "\\left[\\begin{array}{c}\n", - "1 &0 &0 &0 \\\\\n", - "0 &1 &0 &0 \\\\\n", - "0 &0 &(m_1+m_2)l_1^2 &m_2l_1l_2cos(x_2-x_1) \\\\\n", - "0 &0 &m_2l_1l_2cos(x_2-x_1) &m_2l_2^2\n", + "0 & -m_2l_1l_2sin(\\theta_2 - \\theta_1)\\\\\n", + " -m_2l_1l_2sin(\\theta_2 - \\theta_1) &0\n", "\\end{array}\\right]\n", "\\end{equation*}\n", "\n", + "\n", + "\\begin{eqnarray*}\n", + "G = \n", + "\\left[\\begin{array}{c}\n", + "(m_1+m_2)l_1gsin\\theta_1\\\\\n", + " m_2l_2gsin\\theta_2\n", + "\\end{array}\\right]\n", + "\\end{eqnarray*}\n", + "\n", "and\n", "\n", "$$\\begin{eqnarray*}\n", - "f(\\vec{x}(t)) = \\begin{pmatrix} x_3\\\\ x_4\\\\ m_2l_1l_2x_4^2sin(x_2-x_1) -(m_1+m_2)gl_1sin(x_1)\\\\ m_2l_1l_2x_3^2sin(x_2-x_1)-l_2m_2gsinx_2 \\end{pmatrix}\n", + "\\vec{x}(t) = \\begin{pmatrix} \\theta_1\\\\ \\theta_2\\\\ \\end{pmatrix}, \\vec{\\dot{x}(t)} = \\begin{pmatrix} \\dot \\theta_1 \\\\ \\dot \\theta_2 \\end{pmatrix}\n", "\\end{eqnarray*}$$\n", "\n", - "In order to solve this, we can evaluate matrix A and vector $f(x)$ at each time step, then invert A and premultiply the result to each side of the equation. This gives us an explicit system of equations of motion:\n", - "\n", - "$$ \\dot x=A^{-1}f(x)$$\n", - "\n", - "You may notice that these operations require a significant computational load at each time step. In order to reduce this, we can analytically invert the matrix A, then simply fill in the necessary values at each time step and multiply by $f(q)$. This will significantly speed up running time, as inverting a matrix is a computationally expensive operation.\n" + "We can now treat this system the same way we handled the simple harmonic motion; break the two second order ODEs into four first-order ODEs, then use Euler or Runge-Kutta on the results." ] }, { @@ -5653,27 +5665,11 @@ "collapsed": true }, "outputs": [], - "source": [ - "def A_inv(A):\n", - " \"\"\"Returns the inverse of matrix A (from EOM analysis)\n", - " \n", - " Parameters: \n", - " A - Matrix of coefficients\n", - " Returns:\n", - " A^-1 - Inverse of matrix A\n", - " \"\"\"\n", - " A2 = A[2:4, 2:4]\n", - " #print(A2)\n", - " d = det(A2)\n", - " Ai = numpy.array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, A[3,3]/d, -A[2,3]/d], [0, 0, -A[3,2]/d, A[2,2]/d]])\n", - "\n", - " \n", - " return Ai " - ] + "source": [] }, { "cell_type": "code", - "execution_count": 173, + "execution_count": 191, "metadata": { "collapsed": false }, @@ -5694,22 +5690,6 @@ " x3 = u[2]\n", " x4 = u[3]\n", " \n", - " #f1 = x3\n", - " #f2 = x4\n", - " #f3 = m2*l1*l2*x4**2*sin(x2 - x1) - (m1+m2)*g*l1*sin(x1)\n", - " #f4 = m2*l1*l2*x3**2*sin(x2 - x1) - l2*m2*g*sin(x2)\n", - " \n", - " #A = numpy.array([[1, 0, 0, 0],[0, 1, 0, 0],[0, 0, (m1+m2)*l1**2, m2*l1*l2*cos(x2-x1)],[0, 0, m2*l1*l2*cos(x2-x1), m2*l1**2]])\n", - " #print(numpy.shape(A))\n", - " \n", - " #i = A_inv(A)\n", - " #print(type(Ai))\n", - " #print(Ai)\n", - " #f = numpy.array([[f1, f2, f3, f4]])\n", - " #print(numpy.shape(f.T))\n", - " #print(type(f))\n", - " #RHS = \n", - " \n", " M = numpy.array([[(m1+m2)*l1**2, m2*l1*l2*cos(x2-x1)],[m2*l1*l2*cos(x2-x1), m2*l2**2]])\n", " V = numpy.array([[0, -m2*l1*l2*sin(x2-x1)],[-m2*l1*l2*sin(x2-x1), 0]])\n", " G = numpy.array([[(m1+m2)*g*l1*sin(x1)],[m2*l2*g*sin(x2)]])\n", @@ -5725,7 +5705,7 @@ }, { "cell_type": "code", - "execution_count": 186, + "execution_count": 192, "metadata": { "collapsed": false }, @@ -5758,7 +5738,7 @@ }, { "cell_type": "code", - "execution_count": 187, + "execution_count": 193, "metadata": { "collapsed": true }, @@ -5783,7 +5763,7 @@ }, { "cell_type": "code", - "execution_count": 188, + "execution_count": 194, "metadata": { "collapsed": false }, @@ -5809,7 +5789,7 @@ }, { "cell_type": "code", - "execution_count": 190, + "execution_count": 195, "metadata": { "collapsed": false }, From c23801eb828c5b077064362acff2153a641e34ff Mon Sep 17 00:00:00 2001 From: rbds Date: Mon, 7 Dec 2015 15:34:07 -0500 Subject: [PATCH 28/61] split notebooks --- ...on Techniques for Mechanical Systems.ipynb | 5419 +++++++++++++++++ 1 file changed, 5419 insertions(+) create mode 100644 randy_schur/Integration Techniques for Mechanical Systems.ipynb diff --git a/randy_schur/Integration Techniques for Mechanical Systems.ipynb b/randy_schur/Integration Techniques for Mechanical Systems.ipynb new file mode 100644 index 0000000..d66d98c --- /dev/null +++ b/randy_schur/Integration Techniques for Mechanical Systems.ipynb @@ -0,0 +1,5419 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Integration of Mechanical Systems\n", + "\n", + "In this module, we will be exploring numerical integrators and how they perform for different mechanical systems." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Simple Harmonic Oscillator\n", + "\n", + "The simple harmonic oscillator is a simple physical system described by the second-order differential equation below. Despite its simplicity, this is a system that shows up in a similar form in many different fields of engineering. The same equation describes the behavior of a spring-mass-damper system, RLC circuit, or the motion of a (small-angle) pendulum.\n", + "\n", + "![Image](figures/spring-mass-damper.png)\n", + "\n", + "##### Figure 1. Diagram of Spring-Mass-Damper system (from https://en.wikipedia.org/wiki/Harmonic_oscillator)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The differential equation describing the motion of a simple harmonic oscillator is:\n", + "\n", + "$$\\begin{equation}\n", + "0 = m\\ddot{x} + kx\n", + "\\end{equation}$$\n", + "\n", + "We can describe the state of the system by its position $x$ and its velocity $\\dot{x}$. We will call this state vector $q$, as:\n", + "\n", + "$$\\begin{eqnarray}\n", + "\\vec{q} = \\begin{bmatrix} x\\\\ \\dot{x} \\end{bmatrix}\n", + "\\end{eqnarray}$$\n", + "\n", + "This system can then be broken into two first order differential equations:\n", + "\n", + "$$\\begin{eqnarray}\n", + "\\vec{\\dot{q}} = f(\\vec{q}) = \\begin{bmatrix} q_2\\\\ -\\frac{k}{m}q_1 \\end{bmatrix}\n", + "\\end{eqnarray}$$\n", + "\n", + "To look at the behavior of this system, we can start with some initial conditions for $q$, then use a numerical integration method such as Euler's method to integrate forward in time. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Equations of Motion from Lagrangian\n", + "\n", + "You are encouraged to follow this derivation ON PAPER on your own. We gave you the equations of motion for a simple harmonic oscillator, but what if we don't have this equation? The following is a method that will work in nearly any situation to find equations of motion, often working well where a force analysis becomes difficult. This is good practice for anyone studying mechanics. To derive the equations of motion, we will start with the Lagrangian. This is defined by:\n", + "\n", + "$$\\begin{equation}\n", + "L = T- U\n", + "\\end{equation}$$\n", + "\n", + "where $T$ is the total Kinetic Energy in the system, and $U$ is the total Potential Energy. Potential energy in this system comes from the stretching or compressing of the spring, and is calculated as:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "U &=& \\frac{1}{2}kx^2\\\\\n", + "\\end{eqnarray*}$$\n", + "\n", + "Kinetic Energy in this system is from the motion of the mass, and is calculated as:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "T &=& \\frac{1}{2}mv^2 \\\\\n", + "&=& \\frac{1}{2}m(\\dot x)^2 \n", + "\\end{eqnarray*}$$\n", + "\n", + "\n", + "So the Lagrangian quantity becomes:\n", + "$$\\begin{equation}\n", + "L = \\frac{1}{2}m(\\dot x)^2 - \\frac{1}{2}kx^2\\\\\n", + "\\end{equation}$$\n", + "\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The equations of motion are then found using the Lagrange Equation:\n", + "\n", + "$$\\begin{equation}\n", + "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x_i}}\\right) - \\frac{\\partial L}{\\partial x_i} = 0\n", + "\\end{equation}$$\n", + "\n", + "Since we have only one generalized coordinate (position), we need only one equation of motion. We'll calculate this in steps:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "\\frac{\\partial L}{\\partial \\dot{x}} &=& m\\dot{x} \\\\\n", + "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x}}\\right) &=& m\\ddot{x}\\\\\n", + "-\\frac{\\partial L}{\\partial x} &=& + kx\n", + "\\end{eqnarray*}$$\n", + "\n", + "We can add together the second and third equations to find the first equation of motion. Some of these terms will cancel out, and we are left with a final equation of motion:\n", + "\n", + "$$\\begin{equation}\n", + "0 = m\\ddot{x} + kx\n", + "\\end{equation}$$\n", + "\n", + "This is the same equation we have before! So, we can be relatively confident that this derivation is correct." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Integration Methods\n", + "\n", + "### Euler's method\n", + "\n", + "Euler's method is discussed in lesson 1 of the Numerical Methods course. The method goes as follows:\n", + "$$x_i^{n+1} = x_i^n + h\\dot{x}_i^n$$\n", + "\n", + "where $h$ is the timestep. This isn't a complicated scheme, so for details see the code below." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Runge-Kutta Integration (RK4)\n", + "\n", + "Depending on the application, Euler's method may provide enough accuracy with a small timestep, especially a simple harmonic oscillator. However, Euler is a first-order method and we can do better. One option is a Runge-Kutta scheme. This is a popular numerical integration method which is easily extended to higher orders. Here we will use a fourth order Runge-Kutta, also known as RK4:\n", + "\n", + "$$\\begin{equation}\n", + "q_{n+1} = q_n + \\frac{h}{6}\\left(k_1+2k_2+2k_3+k_4\\right)\n", + "\\end{equation}$$\n", + "where\n", + "$$\\begin{eqnarray}\n", + "k_1 &=& f(t_n, x_n)\\\\\n", + "k_2 &=& f(t_n+\\frac{h}{2}, x_n+\\frac{h}{2}k_1)\\\\\n", + "k_3 &=& f(t_n+\\frac{h}{2}, x_n+\\frac{h}{2}k_2)\\\\\n", + "k_4 &=& f(t_n+h, y_n+hk_3)\\\\\n", + "t_{n+1} &=& t_n + h\n", + "\\end{eqnarray}$$\n", + "\n", + "For more information on the Runge-Kutta method, including the derivation and explanation of coefficients, see (cite)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Symplectic Integrators\n", + "\n", + "Symplectic integrators are similar to the methods discussed above, but they use equations of motion derived from Hamiltonian mechanics. According to [Berkeley source], syplectic integrators preserve the conserved Hamiltonian quantities. In practical terms, this works out to mean the methods reflect conservation of momentum, down to a truncation error.\n", + "\n", + "The Hamiltonian equations of motion can be derived from the total energy in the system, much like the Lagrangian. The coordinates we use are position ($q$) and momentum ($p = m\\dot x$). The Hamiltonian is:\n", + "$$ H = T+V$$\n", + "where $T$ is the kinetic energy in the system, and $V$ is the potential energy in the system. So, we get:\n", + "$$\\begin{eqnarray*}\n", + "T &=& \\frac{1}{2}m\\dot{x}^2 = \\frac{q^2}{2m}\\\\\n", + "V &=& \\frac{1}{2}k{x}^2 = \\frac{kp^2}{2} \\\\\n", + "H &=& \\frac{q^2}{2m} + \\frac{kp^2}{2}\n", + "\\end{eqnarray*}$$\n", + "\n", + "\n", + "The Hamilton's equations are:\n", + "$$\\begin{eqnarray*}\n", + "\\frac{dp}{dt} &=& -\\frac{\\partial H}{\\partial q}\\\\\n", + "\\frac{dq}{dt} &=& \\frac{\\partial H}{\\partial p}\n", + "\\end{eqnarray*}$$\n", + "\n", + "So our equations of motion become:\n", + "$$\\begin{equation*}\n", + "\\begin{bmatrix} \\dot p \\\\ \\dot q \\end{bmatrix} = \\begin{bmatrix} -\\frac{\\partial H}{\\partial q}\\\\ \\frac{\\partial H}{\\partial p} \\end{bmatrix} = \\begin{bmatrix} -\\frac{q}{m} \\\\ kp\\end{bmatrix}\n", + "\\end{equation*}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's now take a look at symplectic integration methods. We'll start with a first order method - symplectic Euler's method. This uses the equations:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "p_{n+1} = p_n - h \\frac{\\partial H}{\\partial q} \\Bigr|_{p_{n+1},q_n}\\\\\n", + "q_{n+1} = q_n + \\frac{h}{2} \\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1},q_n}\n", + "\\end{eqnarray*}$$\n", + "\n", + "We can then plug in the partial derivatives and integrate the system. This system will do a better job of conserving energy than Euler's method, $\\textit{but only to a truncation error.}$ Again, we can do better than first-order. Let's try a second-order symplectic scheme, also called Verlet integration. We'll have the same equations of motion, but this time a different set of integration equations:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "p_{n+1/2} &=& p_n - \\frac{h}{2} \\frac{\\partial H}{\\partial q} \\Bigr|_{p_{n+1/2},q_n}\\\\\n", + "q_{n+1} &=& q_n + \\frac{h}{2} \\left(\\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1/2},q_n} + \\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1/2},q_{n+1/2}} \\right) \\\\\n", + "p_{n+1} &=& p_{n+1/2} - \\frac{h}{2} \\frac{\\partial H}{\\partial q}\\Bigr|_{p_{n+1/2},q_{n+1/2}}\n", + "\\end{eqnarray*}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Analytical Solution\n", + "The simple harmonic oscillator is a nice example for numerical integration, because it has an analytical form which we can compare to. The analytical solution is\n", + "\n", + "$$\\begin{equation}\n", + "x(t) = Acos(\\omega_n t + \\phi)\n", + "\\end{equation}$$\n", + "\n", + "where $\\omega_n$ is the natural frequency of the system given by $\\omega_n = \\sqrt{\\frac{k}{m}}$ . The amplitude, $A$, and the phase, $\\phi$, of oscillation are determined from the initial conditions. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Let's Get to Programming" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy\n", + "from scipy.linalg import solve\n", + "from numpy.linalg import det\n", + "from math import pi, cos, sin, sqrt\n", + "\n", + "from matplotlib import pyplot\n", + "from matplotlib.pyplot import quiver\n", + "%matplotlib notebook\n", + "from matplotlib import rcParams, cm\n", + "rcParams['font.family'] = 'serif'\n", + "rcParams['font.size'] = 16" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "#Set up parameters:\n", + "\n", + "m = 12\n", + "k = 150;\n", + "#g = 9.8 #[m/s**2]\n", + "\n", + "T = 25; #[seconds]\n", + "dt = .02; #\n", + "N = int(T/dt)+1\n", + "t = numpy.linspace(0.0, T, N)" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def euler(u, f, dt):\n", + " \"\"\" Euler's method for integrating a system of differential equations.\n", + " \n", + " Parameters:\n", + " u - state at current step \n", + " f - RHS of equation\n", + " dt- time step size\n", + " \n", + " Returns: \n", + " x - array of values at next time step.\n", + " \"\"\"\n", + " \n", + " return u + dt*f(u) " + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def RK4(u, f, dt):\n", + " \"\"\"Runge Kutta fourth order integration method\n", + " \n", + " Parameters:\n", + " u - state of the system at time t\n", + " f - function for RHS of state equations\n", + " dt - time step\n", + " \n", + " Returns: array of state values at next time step.\n", + " \"\"\"\n", + " k1 = f(u)\n", + " k2 = f(u) + 0.5*dt*k1\n", + " k3 = f(u) + 0.5*dt*k2\n", + " k4 = f(u) + dt*k3\n", + " \n", + " return u + dt/6*(k1+2*k2+2*k3+k4)" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def f_harmonic_oscillator(u):\n", + " \"\"\"Returns RHS of harmonic oscillator EOM\n", + " \n", + " Parameters:\n", + " q - initial state\n", + " \n", + " Returns:\n", + " f - RHS of harmonic oscillator eqn.\n", + " \n", + " \"\"\"\n", + " pos = u[0]\n", + " vel = u[1]\n", + " \n", + " return numpy.array([vel, -k/m*pos])" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def verlet_SHM(u, dt):\n", + " \"\"\" Verlet integration for integrating from Hamiltonian equations of motion\n", + " for a simple harmonic oscillator\n", + " Parameters:\n", + " u - state at current time step\n", + " dt - time step size\n", + " Returns: state at next time step.\n", + " \"\"\"\n", + " pos = u[0]\n", + " mom = u[1]\n", + " \n", + " p_half = pos - dt/2*mom/m\n", + " q_half = mom + dt/2*(k*pos)\n", + " \n", + " q = mom + dt/2*(2*k*p_half)\n", + " p = p_half - dt/2*q/m\n", + " \n", + " return numpy.array([p, q]) " + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "#Initial Conditions\n", + "x0 = 0.5 #[m]\n", + "xdot0 = 0 #[m/s]\n", + "\n", + "p0 = x0\n", + "q0 = m*xdot0\n", + "\n", + "x_init = numpy.array([x0, xdot0])\n", + "x_init_H = numpy.array([p0, q0])" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "application/javascript": [ + "/* Put everything inside the global mpl namespace */\n", + "window.mpl = {};\n", + "\n", + "mpl.get_websocket_type = function() {\n", + " if (typeof(WebSocket) !== 'undefined') {\n", + " return WebSocket;\n", + " } else if (typeof(MozWebSocket) !== 'undefined') {\n", + " return MozWebSocket;\n", + " } else {\n", + " alert('Your browser does not have WebSocket support.' +\n", + " 'Please try Chrome, Safari or Firefox ≥ 6. 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What are the sources of this error? We have the usual suspects: truncation error, discretization error, possible precision errors. But ordinarily we don't see a second order scheme (Verlet) outperforming a fourth order scheme (RK4) by three orders of magnitude. What is going on?\n", + "\n", + "One thing we notice in the plots of position is that for Euler and Runge-Kutta, the spring is stretched farther and farther each oscillation. This means it has more and more potential energy. With no outside forces adding energy to the system and no friction to remove energy, Conservation of Energy says that value should be constant! Let's take a look at total energy in the system. Since we derived equations of motion from the Lagrangian, this should be easy! All we need to do is calculate kinetic and potential energy at each time step, and we already have the information we need to do this. " + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def get_Energy(u):\n", + " \"\"\" Calculates total energu in the system at given time step for a simple harmonic oscillator\n", + " Parameters:\n", + " u - state of system [pos, vel]\n", + " Returns:\n", + " total energy in system.\n", + " \"\"\"\n", + " pos = u[0]\n", + " vel = u[1]\n", + " T = 0.5*m*vel**2\n", + " V = 0.5*k*pos**2\n", + " \n", + " return T+V" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "#Euler total energy\n", + "Euler_energy = numpy.zeros_like(t)\n", + "for i in range(N):\n", + " Euler_energy[i] = get_Energy(q1[i,:])\n", + " \n", + "\n", + "#RK4 total energy\n", + "RK4_energy = numpy.zeros_like(t)\n", + "for i in range(N):\n", + " RK4_energy[i] = get_Energy(q2[i,:])\n", + "\n", + "#Verlet total energy\n", + "#Here we will convert from momentum to velocity so we can use the same get_energy function.\n", + "#Momemtum (q) is mass*velocity.\n", + "Verlet_energy = numpy.zeros_like(t)\n", + "Verlet_vel = q3[:,1]/m\n", + "for i in range(N):\n", + " Verlet_energy[i] = get_Energy(numpy.array([q3[i,0], Verlet_vel[i]]))\n", + "\n", + "\n", + "#analytic total energy\n", + "analytical_energy = numpy.zeros_like(t)\n", + "anayltical_state = numpy.array([q_analytical, vel_analytical]).T\n", + "for i in range(N):\n", + " analytical_energy[i] = get_Energy(anayltical_state[i,:])" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "application/javascript": [ + "/* Put everything inside the global mpl namespace */\n", + "window.mpl = {};\n", + "\n", + "mpl.get_websocket_type = function() {\n", + " if (typeof(WebSocket) !== 'undefined') {\n", + " return WebSocket;\n", + " } else if (typeof(MozWebSocket) !== 'undefined') {\n", + " return MozWebSocket;\n", + " } else {\n", + " alert('Your browser does not have WebSocket support.' +\n", + " 'Please try Chrome, Safari or Firefox ≥ 6. 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From the eye test, it seems like the error is actually better here. Why is that? \n", + "\n", + "The effect of the damper is to remove energy from the system. So, in this case the damping term is helping to remove some energy from the system that the integration adds. As time goes to infinity, these solutions will converge, and the Euler and Runge-Kutta methods are stable. We can again look at the energy in the system according to each method. \n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "application/javascript": [ + "/* Put everything inside the global mpl namespace */\n", + "window.mpl = {};\n", + "\n", + "mpl.get_websocket_type = function() {\n", + " if (typeof(WebSocket) !== 'undefined') {\n", + " return WebSocket;\n", + " } else if (typeof(MozWebSocket) !== 'undefined') {\n", + " return MozWebSocket;\n", + " } else {\n", + " alert('Your browser does not have WebSocket support.' +\n", + " 'Please try Chrome, Safari or Firefox ≥ 6. 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');\n", + " var button = $('');\n", + " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", + " button.mouseover('Close figure', toolbar_mouse_event);\n", + " buttongrp.append(button);\n", + " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", + " titlebar.prepend(buttongrp);\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(el){\n", + " // this is important to make the div 'focusable\n", + " el.attr('tabindex', 0)\n", + " // reach out to IPython and tell the keyboard manager to turn it's self\n", + " // off when our div gets focus\n", + "\n", + " // location in version 3\n", + " if (IPython.notebook.keyboard_manager) {\n", + " IPython.notebook.keyboard_manager.register_events(el);\n", + " }\n", + " else {\n", + " // location in version 2\n", + " IPython.keyboard_manager.register_events(el);\n", + " }\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._key_event_extra = function(event, name) {\n", + " var manager = IPython.notebook.keyboard_manager;\n", + " if (!manager)\n", + " manager = IPython.keyboard_manager;\n", + "\n", + " // Check for shift+enter\n", + " if (event.shiftKey && event.which == 13) {\n", + " this.canvas_div.blur();\n", + " event.shiftKey = false;\n", + " // Send a \"J\" for go to next cell\n", + " event.which = 74;\n", + " event.keyCode = 74;\n", + " manager.command_mode();\n", + " manager.handle_keydown(event);\n", + " }\n", + "}\n", + "\n", + "mpl.figure.prototype.handle_save = function(fig, msg) {\n", + " fig.ondownload(fig, null);\n", + "}\n", + "\n", + "\n", + "mpl.find_output_cell = function(html_output) {\n", + " // Return the cell and output element which can be found *uniquely* in the notebook.\n", + " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", + " // IPython event is triggered only after the cells have been serialised, which for\n", + " // our purposes (turning an active figure into a static one), is too late.\n", + " var cells = IPython.notebook.get_cells();\n", + " var ncells = cells.length;\n", + " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", + " data = data.data;\n", + " }\n", + " if (data['text/html'] == html_output) {\n", + " return [cell, data, j];\n", + " }\n", + " }\n", + " }\n", + " }\n", + "}\n", + "\n", + "// Register the function which deals with the matplotlib target/channel.\n", + "// The kernel may be null if the page has been refreshed.\n", + "if (IPython.notebook.kernel != null) {\n", + " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", + "}\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "#create a phase plot\n", + "NP = 15\n", + "pos = numpy.linspace(-x0, x0, NP)\n", + "vel = numpy.linspace(-2*x0, 2*x0, NP)\n", + "\n", + "X1, X2 = numpy.meshgrid(pos, vel)\n", + "u = numpy.zeros_like(X1)\n", + "v = numpy.zeros_like(X2)\n", + "\n", + "for i in range(NP):\n", + " for j in range(NP):\n", + " x = X1[i, j]\n", + " y = X2[i, j]\n", + " yprime = f_damped_HM([x, y])\n", + " u[i,j] = yprime[0]\n", + " v[i,j] = yprime[1]\n", + "\n", + "\n", + "pyplot.figure(figsize=(10,8));\n", + "pyplot.quiver(X1, X2, u, v, pivot='mid', color='k', label='direction field');\n", + "pyplot.plot(q_d_analytical, vel_d_analytical, 'k-', label='analytical');\n", + "pyplot.plot(q1_d[:,0], q1_d[:,1],'r-', label='Euler');\n", + "pyplot.plot(q2_d[:,0], q2_d[:,1], 'b-', label='RK4');\n", + "pyplot.plot(q3_d[:,0], -Verlet_vel_d, 'g-', label='Verlet');\n", + "pyplot.xlabel('Postion ($x_1$)')\n", + "pyplot.ylabel('Velocity ($x_2$)')\n", + "pyplot.legend();\n", + "pyplot.title('Phase portrait for damped harmonic oscillator');\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 2", + "language": "python", + "name": "python2" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 2 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython2", + "version": "2.7.10" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +} From 2b151d7e0d676cc58b9ee6ea9ea6e2251f3e6d36 Mon Sep 17 00:00:00 2001 From: rbds Date: Mon, 7 Dec 2015 15:53:04 -0500 Subject: [PATCH 29/61] damping for double pendulum --- randy_schur/Double_Pendulum_Problem.ipynb | 4408 +---------------- ...on Techniques for Mechanical Systems.ipynb | 205 +- 2 files changed, 463 insertions(+), 4150 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index 08b4cb1..ba51bfe 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -4,282 +4,286 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "# Integration of Mechanical Systems\n", - "\n", - "In this module, we will be exploring numerical integrators and how they perform for different mechanical systems." + "# Dynamics of a Double Pendulum" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "## Simple Harmonic Oscillator\n", - "\n", - "The simple harmonic oscillator is a simple physical system described by the second-order differential equation below. Despite its simplicity, this is a system that shows up in a similar form in many different fields of engineering. The same equation describes the behavior of a spring-mass-damper system, RLC circuit, or the motion of a (small-angle) pendulum.\n" + "In this section, we will study the dynamics of a double pendulum. Like before, we'll derive the equations of motion from the Lagrangian and the Hamiltonian. We can use the same methods of integration as before, but this time there is no analytical solution to compare to. Finally, we will look at the properties of the system in phase space using phase plots and Poincare sections." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "![Image](figures/spring-mass-damper.png)\n", + "## Background\n", "\n", - "##### Figure 1. Diagram of Spring-Mass-Damper system (from https://en.wikipedia.org/wiki/Harmonic_oscillator)" + "The double pendulum problem is a relatively simple system which can produce surprisingly complex movements. It is a chaotic system, meaning it is unpredictable and small changes in initial conditions lead to large changes in motion. There is no closed form solution to the motion of the two masses. This is a frequent example problem in the study of dynamic systems, nonlinear controls, and mechanics.\n", + "\n", + "In this notebook we will be treating the two rods as massless and we'll start out ignoring the effects of friction. For a treatment of this problem without these assumptions, see (cite).\n", + "\n", + "See the image below for the definition of constants." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "The differential equation describing the motion of a simple harmonic oscillator is:\n", - "\n", - "$$\\begin{equation}\n", - "0 = m\\ddot{x} + kx\n", - "\\end{equation}$$\n", - "\n", - "We can describe the state of the system by its position $x$ and its velocity $\\dot{x}$. We will call this state vector $q$, as:\n", - "\n", - "$$\\begin{eqnarray}\n", - "\\vec{q} = \\begin{bmatrix} x\\\\ \\dot{x} \\end{bmatrix}\n", - "\\end{eqnarray}$$\n", - "\n", - "This system can then be broken into two first order differential equations:\n", - "\n", - "$$\\begin{eqnarray}\n", - "\\vec{\\dot{q}} = f(\\vec{q}) = \\begin{bmatrix} q_2\\\\ -\\frac{k}{m}q_1 \\end{bmatrix}\n", - "\\end{eqnarray}$$\n", - "\n", - "To look at the behavior of this system, we can start with some initial conditions for $q$, then use a numerical integration method such as Euler's method to integrate forward in time. " + "![Image](figures/diagram.png)\n", + "#### Figure 1. Diagram of Double Pendulum System" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "### Equations of Motion from Lagrangian\n", - "\n", - "You are encouraged to follow this derivation ON PAPER on your own. We gave you the equations of motion for a simple harmonic oscillator, but what if we don't have this equation? The following is a method that will work in nearly any situation to find equations of motion, often working well where a force analysis becomes difficult. This is good practice for anyone studying mechanics. To derive the equations of motion, we will start with the Lagrangian. This is defined by:\n", + "## Equations of Motion \n", "\n", - "$$\\begin{equation}\n", + "As a reminder, the Lagrangian is\n", + "$$\\begin{equation*}\n", "L = T- U\n", - "\\end{equation}$$\n", - "\n", - "where $T$ is the total Kinetic Energy in the system, and $U$ is the total Potential Energy. Potential energy in this system comes from the stretching or compressing of the spring, and is calculated as:\n", + "\\end{equation*}$$\n", + "where $T$ is the total Kinetic Energy in the system, and $U$ is the total Potential Energy. Potential energy is calculated as:\n", "\n", "$$\\begin{eqnarray*}\n", - "U &=& \\frac{1}{2}kx^2\\\\\n", + "U &=& m_1g(-y_1) + m_2g(-y_2) \\\\\n", + " &=& -m_1gl_1cos\\theta_1 - m_2g(l_1cos\\theta_1 + l_2cos\\theta_2)\n", "\\end{eqnarray*}$$\n", "\n", - "Kinetic Energy in this system is from the motion of the mass, and is calculated as:\n", + "Kinetic Energy is calculated as:\n", "\n", "$$\\begin{eqnarray*}\n", - "T &=& \\frac{1}{2}mv^2 \\\\\n", - "&=& \\frac{1}{2}m(\\dot x)^2 \n", + "T &=& \\frac{1}{2}mv_1^2+\\frac{1}{2}mv_2^2 \\\\\n", + "&=& \\frac{1}{2}m_1(l_1\\dot\\theta_1)^2 + \\frac{1}{2}m_2(l_1\\dot\\theta_1 + l_2\\dot\\theta_2)^2 \\\\\n", + "&=& \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2+\\frac{1}{2}m_2\\left(2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+l_2^2\\dot\\theta_2^2\\right)\n", "\\end{eqnarray*}$$\n", "\n", "\n", "So the Lagrangian quantity becomes:\n", - "$$\\begin{equation}\n", - "L = \\frac{1}{2}m(\\dot x)^2 - \\frac{1}{2}kx^2\\\\\n", - "\\end{equation}$$\n", - "\n", - "\n" + "$$\\begin{equation*}\n", + "L = \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2+m_2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+(m_1+m_2)l_1gcos\\theta_1 + m_2l_2gcos\\theta_2\n", + "\\end{equation*}$$\n" ] }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "collapsed": true + }, "source": [ "The equations of motion are then found using the Lagrange Equation:\n", "\n", - "$$\\begin{equation}\n", + "$$\\begin{equation*}\n", "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x_i}}\\right) - \\frac{\\partial L}{\\partial x_i} = 0\n", - "\\end{equation}$$\n", + "\\end{equation*}$$\n", "\n", - "Since we have only one generalized coordinate (position), we need only one equation of motion. We'll calculate this in steps:\n", + "\n", + "There are two generalized coordinates ($\\theta_1 and \\theta_2$. So the equation of motion for $\\theta_1$ is calculated by the following steps:\n", "\n", "$$\\begin{eqnarray*}\n", - "\\frac{\\partial L}{\\partial \\dot{x}} &=& m\\dot{x} \\\\\n", - "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x}}\\right) &=& m\\ddot{x}\\\\\n", - "-\\frac{\\partial L}{\\partial x} &=& + kx\n", + "\\frac{\\partial L}{\\partial \\dot{\\theta_1}} &=& (m_1+m_2)l_1^2\\dot\\theta_1+m_2l_1l_2\\dot\\theta_2cos(\\theta_2-\\theta_1) \\\\\n", + "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{\\theta_1}}\\right) &=& (m_1+m_2)l_1^2\\ddot\\theta_1+m_2l_1l_2\\ddot\\theta_2cos(\\theta_2-\\theta_1)-m_2l_1l_2\\dot\\theta_2^2sin(\\theta_2-\\theta_1)+m_2l_1l_2\\dot\\theta_1\\dot\\theta_2sin(\\theta_2-\\theta_1) \\\\\n", + "-\\frac{\\partial L}{\\partial \\theta_1} &=& -m_2l_1l_2\\dot\\theta_1\\dot\\theta_2sin(\\theta_2-\\theta_1)-(m_1+m_2)gl_1sin\\theta_1\n", "\\end{eqnarray*}$$\n", "\n", "We can add together the second and third equations to find the first equation of motion. Some of these terms will cancel out, and we are left with a final equation of motion:\n", "\n", - "$$\\begin{equation}\n", - "0 = m\\ddot{x} + kx\n", - "\\end{equation}$$\n", + "$$\\begin{equation*}\n", + "0 = (m_1+m_2)l_1^2\\ddot\\theta_1+m_2l_1l_2\\ddot\\theta_2cos(\\theta_2-\\theta_1)-m_2l_1l_2\\dot\\theta_2^2sin(\\theta_2-\\theta_1)+(m_1+m_2)gl_1sin\\theta_1\n", + "\\end{equation*}$$\n", + "\n", + "We will also need the equation of motion for $\\theta_2$. This can be derived using the same process. It should work out to the following (check this by yourself!):\n", "\n", - "This is the same equation we have before! So, we can be relatively confident that this derivation is correct." + "$$\\begin{equation*}\n", + "0 = m_2l_2^2\\ddot\\theta_2+m_2l_1l_2\\ddot\\theta_1cos(\\theta_2-\\theta_1) - m_2l_2l_1\\dot\\theta_1^2sin(\\theta_2-\\theta_1)+ l_2m_2gsin\\theta_2\n", + "\\end{equation*}$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "## Integration Methods\n", + "## Problem setup\n", "\n", - "### Euler's method\n", + "Notice that these equations of motion are implicit equations for $\\ddot\\theta_1$ and $\\ddot\\theta_2$. We have talked about implicit equations in this class, and we solve them by forming a system of equations. In this case, we will need to solve the equations of motion at each time step. We will characterize our system in state space using the angle and angular velocity of each pendulum. Let's rewrite the equations of motion in a more readable form. We can use the standard form of this equation (which you'll be familiar with if you have studied mechanics) by separating out mass terms (anything multiplied by $\\ddot{x}$), centripetal terms (anything multiplied by $\\dot{x}$), and gravitational terms (anything multiplied by $x$). We get the following equations of state:\n", "\n", - "Euler's method is discussed in lesson 1 of the Numerical Methods course. The method goes as follows:\n", - "$$x_i^{n+1} = x_i^n + h\\dot{x}_i^n$$\n", + "$$\\begin{eqnarray*}\n", + "0 &=&\\textbf{M}\\vec{\\ddot{x}}(t) + \\textbf{V}\\dot{x} + \\textbf{G} \n", + "\\end{eqnarray*}$$\n", "\n", - "where $h$ is the timestep. This isn't a complicated scheme, so for details see the code below." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Runge-Kutta Integration (RK4)\n", + "The equations of motion can be solved for $\\ddot{x}$:\n", "\n", - "Depending on the application, Euler's method may provide enough accuracy with a small timestep, especially a simple harmonic oscillator. However, Euler is a first-order method and we can do better. One option is a Runge-Kutta scheme. This is a popular numerical integration method which is easily extended to higher orders. Here we will use a fourth order Runge-Kutta, also known as RK4:\n", + "$$ \\ddot{x} = \\textbf{M}^{-1}\\left[ \\textbf{V}\\dot x + \\textbf{G} \\right]$$\n", "\n", - "$$\\begin{equation}\n", - "q_{n+1} = q_n + \\frac{h}{6}\\left(k_1+2k_2+2k_3+k_4\\right)\n", - "\\end{equation}$$\n", - "where\n", - "$$\\begin{eqnarray}\n", - "k_1 &=& f(t_n, x_n)\\\\\n", - "k_2 &=& f(t_n+\\frac{h}{2}, x_n+\\frac{h}{2}k_1)\\\\\n", - "k_3 &=& f(t_n+\\frac{h}{2}, x_n+\\frac{h}{2}k_2)\\\\\n", - "k_4 &=& f(t_n+h, y_n+hk_3)\\\\\n", - "t_{n+1} &=& t_n + h\n", - "\\end{eqnarray}$$\n", + "where \n", "\n", - "For more information on the Runge-Kutta method, including the derivation and explanation of coefficients, see (cite)\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Symplectic Integrators\n", + "\\begin{eqnarray*}\n", + "M = \n", + "\\left[\\begin{array}{c}\n", + "(m_1+m_2)l_1 & m_2l_1l_2cos(\\theta_2 - \\theta_1)\\\\\n", + " m_2l_1l_2cos(\\theta_2 - \\theta_1) &m_2l_2^2\n", + "\\end{array}\\right]\n", + "\\end{eqnarray*}\n", "\n", - "Symplectic integrators are similar to the methods discussed above, but they use equations of motion derived from Hamiltonian mechanics. According to [Berkeley source], syplectic integrators preserve the conserved Hamiltonian quantities. In practical terms, this works out to mean the methods reflect conservation of momentum, down to a truncation error.\n", "\n", - "The Hamiltonian equations of motion can be derived from the total energy in the system, much like the Lagrangian. The coordinates we use are position ($q$) and momentum ($p = m\\dot x$). The Hamiltonian is:\n", - "$$ H = T+V$$\n", - "where $T$ is the kinetic energy in the system, and $V$ is the potential energy in the system. So, we get:\n", - "$$\\begin{eqnarray*}\n", - "T &=& \\frac{1}{2}m\\dot{x}^2 = \\frac{q^2}{2m}\\\\\n", - "V &=& \\frac{1}{2}k{x}^2 = \\frac{kp^2}{2} \\\\\n", - "H &=& \\frac{q^2}{2m} + \\frac{kp^2}{2}\n", - "\\end{eqnarray*}$$\n", + "\\begin{equation*}\n", + "V = \n", + "\\left[\\begin{array}{c}\n", + "0 & -m_2l_1l_2sin(\\theta_2 - \\theta_1)\\\\\n", + " -m_2l_1l_2sin(\\theta_2 - \\theta_1) &0\n", + "\\end{array}\\right]\n", + "\\end{equation*}\n", + "\n", + "\n", + "\\begin{eqnarray*}\n", + "G = \n", + "\\left[\\begin{array}{c}\n", + "(m_1+m_2)l_1gsin\\theta_1\\\\\n", + " m_2l_2gsin\\theta_2\n", + "\\end{array}\\right]\n", + "\\end{eqnarray*}\n", "\n", + "and\n", "\n", - "The Hamilton's equations are:\n", "$$\\begin{eqnarray*}\n", - "\\frac{dp}{dt} &=& -\\frac{\\partial H}{\\partial q}\\\\\n", - "\\frac{dq}{dt} &=& \\frac{\\partial H}{\\partial p}\n", + "\\vec{x}(t) = \\begin{pmatrix} \\theta_1\\\\ \\theta_2\\\\ \\end{pmatrix}, \\vec{\\dot{x}(t)} = \\begin{pmatrix} \\dot \\theta_1 \\\\ \\dot \\theta_2 \\end{pmatrix}\n", "\\end{eqnarray*}$$\n", "\n", - "So our equations of motion become:\n", - "$$\\begin{equation*}\n", - "\\begin{bmatrix} \\dot p \\\\ \\dot q \\end{bmatrix} = \\begin{bmatrix} -\\frac{\\partial H}{\\partial q}\\\\ \\frac{\\partial H}{\\partial p} \\end{bmatrix} = \\begin{bmatrix} -\\frac{q}{m} \\\\ kp\\end{bmatrix}\n", - "\\end{equation*}$$" + "We can now treat this system the same way we handled the simple harmonic motion; break the two second order ODEs into four first-order ODEs, then use Euler or Runge-Kutta on the results." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Let's now take a look at symplectic integration methods. We'll start with a first order method - symplectic Euler's method. This uses the equations:\n", + "### Equations of Motion from Hamiltonian\n", + "\n", + "Here we derive the equations of motion from the Hamiltonian. If you want good practice, you can derive these on your own. We start with the same potential and kinetic energy, but we'll need to use postion ($\\theta$) and momentum ($I\\omega$) to give us:\n", + "\n", + "$$\\begin{eqnarray*}\n", + "U &=& m_1g(-y_1) + m_2g(-y_2) \\\\\n", + " &=& -m_1g(l_1cos\\theta_1 - m_2g(l_1cos\\theta_1 + l_2cos\\theta_2)\n", + "\\end{eqnarray*}$$\n", + "\n", + "Kinetic Energy is calculated as:\n", "\n", "$$\\begin{eqnarray*}\n", - "p_{n+1} = p_n - h \\frac{\\partial H}{\\partial q} \\Bigr|_{p_{n+1},q_n}\\\\\n", - "q_{n+1} = q_n + \\frac{h}{2} \\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1},q_n}\n", + "T &=& \\frac{1}{2}mv_1^2+\\frac{1}{2}mv_2^2 \\\\\n", + "&=& \\frac{q_1^2}{2I_1} + \\frac{q_2^2}{2I_2}\n", "\\end{eqnarray*}$$\n", "\n", - "We can then plug in the partial derivatives and integrate the system. This system will do a better job of conserving energy than Euler's method, $\\textit{but only to a truncation error.}$ Again, we can do better than first-order. Let's try a second-order symplectic scheme, also called Verlet integration. We'll have the same equations of motion, but this time a different set of integration equations:\n", + "where $I_i = m_il_i^2$. So our Hamiltonian quantity becomes:\n", + "$$\\begin{equation*}\n", + "H = \\frac{1}{2}\\left(\\frac{q_1^2}{I_1} + \\frac{q_2^2}{I_2}\\right) - (m_1+m_2)gl_1cos\\theta_1 - m_2gl_2cos\\theta_2\n", + "\\end{equation*}$$\n", "\n", + "Since we have two degrees of freedom, we will have four equations of motion. These will be the following:\n", "$$\\begin{eqnarray*}\n", - "p_{n+1/2} &=& p_n - \\frac{h}{2} \\frac{\\partial H}{\\partial q} \\Bigr|_{p_{n+1/2},q_n}\\\\\n", - "q_{n+1} &=& q_n + \\frac{h}{2} \\left(\\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1/2},q_n} + \\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1/2},q_{n+1/2}} \\right) \\\\\n", - "p_{n+1} &=& p_{n+1/2} - \\frac{h}{2} \\frac{\\partial H}{\\partial q}\\Bigr|_{p_{n+1/2},q_{n+1/2}}\n", - "\\end{eqnarray*}$$" + "\\dot{p_1} &=& -\\frac{\\partial H}{\\partial q_1} = \\frac{q_1}{I_1} \\\\\n", + "\\dot{q_1} &=& \\frac{\\partial H}{\\partial p_1} = (m_1+m_2)gl_1sinp_1 \\\\\n", + "\\dot{p_2} &=& -\\frac{\\partial H}{\\partial q_2} = \\frac{q_2}{I_2} \\\\\n", + "\\dot{q_2} &=& \\frac{\\partial H}{\\partial p_2} = m_2gl_2sinp_2 \\\\\n", + "\\end{eqnarray*}$$\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "### Analytical Solution\n", - "The simple harmonic oscillator is a nice example for numerical integration, because it has an analytical form which we can compare to. The analytical solution is\n", - "\n", - "$$\\begin{equation}\n", - "x(t) = Acos(\\omega_n t + \\phi)\n", - "\\end{equation}$$\n", + "## Integration\n", "\n", - "where $\\omega_n$ is the natural frequency of the system given by $\\omega_n = \\sqrt{\\frac{k}{m}}$ . The amplitude, $A$, and the phase, $\\phi$, of oscillation are determined from the initial conditions. " + "For Euler and Runge-Kutta, we can use the matrix equation above to integrate our system. The results will be VERY dependent on initial conditions; this is why we call it a chaotic system. For the symplectic integrator, we'll use the Hamiltonian equations of motion above. " ] }, { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Let's Get to Programming" - ] + "cell_type": "code", + "execution_count": 159, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] }, { "cell_type": "code", - "execution_count": 1, + "execution_count": 202, "metadata": { "collapsed": false }, - "outputs": [ - { - "name": "stderr", - "output_type": "stream", - "text": [ - "C:\\Users\\rschur\\Anaconda2\\lib\\site-packages\\IPython\\kernel\\__init__.py:13: ShimWarning: The `IPython.kernel` package has been deprecated. You should import from ipykernel or jupyter_client instead.\n", - " \"You should import from ipykernel or jupyter_client instead.\", ShimWarning)\n" - ] - } - ], + "outputs": [], "source": [ - "import numpy\n", - "from scipy.linalg import solve\n", - "from numpy.linalg import det\n", - "from math import pi, cos, sin, sqrt\n", - "\n", - "from matplotlib import pyplot\n", - "from matplotlib.pyplot import quiver\n", - "%matplotlib notebook\n", - "from matplotlib import rcParams, cm\n", - "rcParams['font.family'] = 'serif'\n", - "rcParams['font.size'] = 16" + "def f_double_pendulum(u):\n", + " \"\"\"Returns RHS of double pendulum EOM\n", + " \n", + " Parameters:\n", + " q - initial state\n", + " \n", + " Returns:\n", + " RHS - RHS of harmonic oscillator eqn.\n", + " \n", + " \"\"\"\n", + " x1 = u[0]\n", + " x2 = u[1]\n", + " x3 = u[2]\n", + " x4 = u[3]\n", + " \n", + " M = numpy.array([[(m1+m2)*l1**2, m2*l1*l2*cos(x2-x1)],[m2*l1*l2*cos(x2-x1), m2*l2**2]])\n", + " V = numpy.array([[0, -m2*l1*l2*sin(x2-x1)],[-m2*l1*l2*sin(x2-x1), 0]])\n", + " G = numpy.array([[(m1+m2)*g*l1*sin(x1)],[m2*l2*g*sin(x2)]])\n", + " qdd = numpy.linalg.inv(M).dot(V.dot(numpy.array([[x3],[x4]])) +G) #- 2*numpy.array([[x3],[x4]])) \n", + " #print(qdd)\n", + " #print(M)\n", + " #print(V.dot(numpy.array([[x3],[x4]])))\n", + " #print(G)\n", + " RHS = numpy.array([[x3, x4, qdd[0], qdd[1]]])\n", + " #print(RHS.T)\n", + " return RHS.T" ] }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 203, "metadata": { "collapsed": false }, "outputs": [], "source": [ "#Set up parameters:\n", - "\n", - "m = 12\n", - "k = 150;\n", - "#g = 9.8 #[m/s**2]\n", + "g = -9.8 #[m/s^2]\n", + "m1 = 2\n", + "m2 = 1.5;\n", + "l1 = 1\n", + "l2 = 1.5\n", "\n", "T = 25; #[seconds]\n", "dt = .02; #\n", "N = int(T/dt)+1\n", - "t = numpy.linspace(0.0, T, N)" + "t = numpy.linspace(0.0, T, N)\n", + "\n", + "#Initial Conditions\n", + "theta1_0 = 1 #[radians]\n", + "theta2_0 = pi\n", + "theta1_dot_0 = 0 #[rad/s]\n", + "theta2_dot_0 = 0 #\n", + "\n", + "#p0 = x0\n", + "#q0 = m*xdot0\n", + "x_init_dp = numpy.array([theta1_0, theta2_0, theta1_dot_0, theta2_dot_0])\n", + "#x_init_H = numpy.array([p0, q0])\n", + "\n" ] }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 204, "metadata": { "collapsed": true }, "outputs": [], "source": [ - "def euler(u, f, dt):\n", + "def euler_DP(u, f, dt):\n", " \"\"\" Euler's method for integrating a system of differential equations.\n", " \n", " Parameters:\n", @@ -290,3511 +294,41 @@ " Returns: \n", " x - array of values at next time step.\n", " \"\"\"\n", + " #print(f(u))\n", + " #print(u)\n", " \n", - " return u + dt*f(u) " - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "def RK4(u, f, dt):\n", - " \"\"\"Runge Kutta fourth order integration method\n", - " \n", - " Parameters:\n", - " u - state of the system at time t\n", - " f - function for RHS of state equations\n", - " dt - time step\n", - " \n", - " Returns: array of state values at next time step.\n", - " \"\"\"\n", - " k1 = f(u)\n", - " k2 = f(u) + 0.5*dt*k1\n", - " k3 = f(u) + 0.5*dt*k2\n", - " k4 = f(u) + dt*k3\n", - " \n", - " return u + dt/6*(k1+2*k2+2*k3+k4)" - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "def f_harmonic_oscillator(u):\n", - " \"\"\"Returns RHS of harmonic oscillator EOM\n", - " \n", - " Parameters:\n", - " q - initial state\n", - " \n", - " Returns:\n", - " f - RHS of harmonic oscillator eqn.\n", - " \n", - " \"\"\"\n", - " pos = u[0]\n", - " vel = u[1]\n", - " \n", - " return numpy.array([vel, -k/m*pos])" - ] - }, - { - "cell_type": "code", - "execution_count": 6, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "def verlet_SHM(u, dt):\n", - " \"\"\" Verlet integration for integrating from Hamiltonian equations of motion\n", - " for a simple harmonic oscillator\n", - " Parameters:\n", - " u - state at current time step\n", - " dt - time step size\n", - " Returns: state at next time step.\n", - " \"\"\"\n", - " pos = u[0]\n", - " mom = u[1]\n", - " \n", - " p_half = pos - dt/2*mom/m\n", - " q_half = mom + dt/2*(k*pos)\n", - " \n", - " q = mom + dt/2*(2*k*p_half)\n", - " p = p_half - dt/2*q/m\n", - " \n", - " return numpy.array([p, q]) " - ] - }, - { - "cell_type": "code", - "execution_count": 7, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "#Initial Conditions\n", - "x0 = 0.5 #[m]\n", - "xdot0 = 0 #[m/s]\n", - "\n", - "p0 = x0\n", - "q0 = m*xdot0\n", - "\n", - "x_init = numpy.array([x0, xdot0])\n", - "x_init_H = numpy.array([p0, q0])" - ] - }, - { - "cell_type": "code", - "execution_count": 8, - "metadata": { - "collapsed": false, - "scrolled": true - }, - "outputs": [ - { - "data": { - "application/javascript": [ - "/* Put everything inside the global mpl namespace */\n", - "window.mpl = {};\n", - "\n", - "mpl.get_websocket_type = function() {\n", - " if (typeof(WebSocket) !== 'undefined') {\n", - " return WebSocket;\n", - " } else if (typeof(MozWebSocket) !== 'undefined') {\n", - " return MozWebSocket;\n", - " } else {\n", - " alert('Your browser does not have WebSocket support.' +\n", - " 'Please try Chrome, Safari or Firefox ≥ 6. 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');\n", - " var button = $('');\n", - " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", - " buttongrp.append(button);\n", - " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", - " titlebar.prepend(buttongrp);\n", - "}\n", - "\n", - "\n", - "mpl.figure.prototype._canvas_extra_style = function(el){\n", - " // this is important to make the div 'focusable\n", - " el.attr('tabindex', 0)\n", - " // reach out to IPython and tell the keyboard manager to turn it's self\n", - " // off when our div gets focus\n", - "\n", - " // location in version 3\n", - " if (IPython.notebook.keyboard_manager) {\n", - " IPython.notebook.keyboard_manager.register_events(el);\n", - " }\n", - " else {\n", - " // location in version 2\n", - " IPython.keyboard_manager.register_events(el);\n", - " }\n", - "\n", - "}\n", - "\n", - "mpl.figure.prototype._key_event_extra = function(event, name) {\n", - " var manager = IPython.notebook.keyboard_manager;\n", - " if (!manager)\n", - " manager = IPython.keyboard_manager;\n", - "\n", - " // Check for shift+enter\n", - " if (event.shiftKey && event.which == 13) {\n", - " this.canvas_div.blur();\n", - " event.shiftKey = false;\n", - " // Send a \"J\" for go to next cell\n", - " event.which = 74;\n", - " event.keyCode = 74;\n", - " manager.command_mode();\n", - " manager.handle_keydown(event);\n", - " }\n", - "}\n", - "\n", - "mpl.figure.prototype.handle_save = function(fig, msg) {\n", - " fig.ondownload(fig, null);\n", - "}\n", - "\n", - "\n", - "mpl.find_output_cell = function(html_output) {\n", - " // Return the cell and output element which can be found *uniquely* in the notebook.\n", - " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", - " // IPython event is triggered only after the cells have been serialised, which for\n", - " // our purposes (turning an active figure into a static one), is too late.\n", - " var cells = IPython.notebook.get_cells();\n", - " var ncells = cells.length;\n", - " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", - " data = data.data;\n", - " }\n", - " if (data['text/html'] == html_output) {\n", - " return [cell, data, j];\n", - " }\n", - " }\n", - " }\n", - " }\n", - "}\n", - "\n", - "// Register the function which deals with the matplotlib target/channel.\n", - "// The kernel may be null if the page has been refreshed.\n", - "if (IPython.notebook.kernel != null) {\n", - " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", - "}\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/html": [ - "" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "pyplot.figure(figsize=(10,6))\n", - "pyplot.tick_params(axis='both', labelsize=14)\n", - "pyplot.grid(True)\n", - "pyplot.xlabel('$\\Delta t$', fontsize=16)\n", - "pyplot.ylabel('Error', fontsize=16)\n", - "pyplot.loglog(dt_values, Euler_error_values, 'bo-', label='Euler')\n", - "pyplot.loglog(dt_values, RK_error_values, 'ro-', label='RK4')\n", - "pyplot.loglog(dt_values, Verlet_error_values, 'go-', label='Symplectic')\n", - "pyplot.axis('equal')\n", - "pyplot.legend();" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Is this what you would expect to see based on the plots of position above? What are the sources of this error? We have the usual suspects: truncation error, discretization error, possible precision errors. But ordinarily we don't see a second order scheme (Verlet) outperforming a fourth order scheme (RK4) by three orders of magnitude. What is going on?\n", - "\n", - "One thing we notice in the plots of position is that for Euler and Runge-Kutta, the spring is stretched farther and farther each oscillation. This means it has more and more potential energy. With no outside forces adding energy to the system and no friction to remove energy, Conservation of Energy says that value should be constant! Let's take a look at total energy in the system. Since we derived equations of motion from the Lagrangian, this should be easy! All we need to do is calculate kinetic and potential energy at each time step, and we already have the information we need to do this. " - ] - }, - { - "cell_type": "code", - "execution_count": 12, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "def get_Energy(u):\n", - " \"\"\" Calculates total energu in the system at given time step for a simple harmonic oscillator\n", - " Parameters:\n", - " u - state of system [pos, vel]\n", - " Returns:\n", - " total energy in system.\n", - " \"\"\"\n", - " pos = u[0]\n", - " vel = u[1]\n", - " T = 0.5*m*vel**2\n", - " V = 0.5*k*pos**2\n", - " \n", - " return T+V" - ] - }, - { - "cell_type": "code", - "execution_count": 13, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "#Euler total energy\n", - "Euler_energy = numpy.zeros_like(t)\n", - "for i in range(N):\n", - " Euler_energy[i] = get_Energy(q1[i,:])\n", - " \n", - "\n", - "#RK4 total energy\n", - "RK4_energy = numpy.zeros_like(t)\n", - "for i in range(N):\n", - " RK4_energy[i] = get_Energy(q2[i,:])\n", - "\n", - "#Verlet total energy\n", - "#Here we will convert from momentum to velocity so we can use the same get_energy function.\n", - "#Momemtum (q) is mass*velocity.\n", - "Verlet_energy = numpy.zeros_like(t)\n", - "Verlet_vel = q3[:,1]/m\n", - "for i in range(N):\n", - " Verlet_energy[i] = get_Energy(numpy.array([q3[i,0], Verlet_vel[i]]))\n", - "\n", - "\n", - "#analytic total energy\n", - "analytical_energy = numpy.zeros_like(t)\n", - "anayltical_state = numpy.array([q_analytical, vel_analytical]).T\n", - "for i in range(N):\n", - " analytical_energy[i] = get_Energy(anayltical_state[i,:])" - ] - }, - { - "cell_type": "code", - "execution_count": 14, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "application/javascript": [ - "/* Put everything inside the global mpl namespace */\n", - "window.mpl = {};\n", - "\n", - "mpl.get_websocket_type = function() {\n", - " if (typeof(WebSocket) !== 'undefined') {\n", - " return WebSocket;\n", - " } else if (typeof(MozWebSocket) !== 'undefined') {\n", - " return MozWebSocket;\n", - " } else {\n", - " alert('Your browser does not have WebSocket support.' +\n", - " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", - " 'Firefox 4 and 5 are also supported but you ' +\n", - " 'have to enable WebSockets in about:config.');\n", - " };\n", - "}\n", - "\n", - "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", - " this.id = figure_id;\n", - "\n", - " this.ws = websocket;\n", - "\n", - " this.supports_binary = (this.ws.binaryType != undefined);\n", - "\n", - " if (!this.supports_binary) {\n", - " var warnings = document.getElementById(\"mpl-warnings\");\n", - " if (warnings) {\n", - " warnings.style.display = 'block';\n", - " warnings.textContent = (\n", - " \"This browser does not support binary websocket messages. \" +\n", - " \"Performance may be slow.\");\n", - " }\n", - " }\n", - "\n", - " this.imageObj = new Image();\n", - "\n", - " this.context = undefined;\n", - " this.message = undefined;\n", - " this.canvas = undefined;\n", - " this.rubberband_canvas = undefined;\n", - " this.rubberband_context = undefined;\n", - " this.format_dropdown = undefined;\n", - "\n", - " this.image_mode = 'full';\n", - "\n", - " this.root = $('
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');\n", - " var button = $('');\n", - " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", - " buttongrp.append(button);\n", - " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", - " titlebar.prepend(buttongrp);\n", - "}\n", - "\n", - "\n", - "mpl.figure.prototype._canvas_extra_style = function(el){\n", - " // this is important to make the div 'focusable\n", - " el.attr('tabindex', 0)\n", - " // reach out to IPython and tell the keyboard manager to turn it's self\n", - " // off when our div gets focus\n", - "\n", - " // location in version 3\n", - " if (IPython.notebook.keyboard_manager) {\n", - " IPython.notebook.keyboard_manager.register_events(el);\n", - " }\n", - " else {\n", - " // location in version 2\n", - " IPython.keyboard_manager.register_events(el);\n", - " }\n", - "\n", - "}\n", - "\n", - "mpl.figure.prototype._key_event_extra = function(event, name) {\n", - " var manager = IPython.notebook.keyboard_manager;\n", - " if (!manager)\n", - " manager = IPython.keyboard_manager;\n", - "\n", - " // Check for shift+enter\n", - " if (event.shiftKey && event.which == 13) {\n", - " this.canvas_div.blur();\n", - " event.shiftKey = false;\n", - " // Send a \"J\" for go to next cell\n", - " event.which = 74;\n", - " event.keyCode = 74;\n", - " manager.command_mode();\n", - " manager.handle_keydown(event);\n", - " }\n", - "}\n", - "\n", - "mpl.figure.prototype.handle_save = function(fig, msg) {\n", - " fig.ondownload(fig, null);\n", - "}\n", - "\n", - "\n", - "mpl.find_output_cell = function(html_output) {\n", - " // Return the cell and output element which can be found *uniquely* in the notebook.\n", - " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", - " // IPython event is triggered only after the cells have been serialised, which for\n", - " // our purposes (turning an active figure into a static one), is too late.\n", - " var cells = IPython.notebook.get_cells();\n", - " var ncells = cells.length;\n", - " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", - " data = data.data;\n", - " }\n", - " if (data['text/html'] == html_output) {\n", - " return [cell, data, j];\n", - " }\n", - " }\n", - " }\n", - " }\n", - "}\n", - "\n", - "// Register the function which deals with the matplotlib target/channel.\n", - "// The kernel may be null if the page has been refreshed.\n", - "if (IPython.notebook.kernel != null) {\n", - " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", - "}\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/html": [ - "" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "#Euler\n", - "q1_d = numpy.zeros((N,2)) \n", - "q1_d[0] = x_init.copy() #set initial conditions\n", - "for n in range(N-1): #integrate with Euler\n", - " q1_d[n+1] = euler(q1_d[n], f_damped_HM, dt)\n", - " #print(q1[n])\n", - " \n", - "#Runge-Kutta \n", - "q2_d = numpy.zeros((N,2))\n", - "q2_d[0] = x_init.copy()\n", - "for n in range(N-1):\n", - " q2_d[n+1] = RK4(q2_d[n], f_damped_HM, dt)\n", - "\n", - "#Symplectic\n", - "q3_d = numpy.zeros((N,2))\n", - "q3_d[0] = x_init_H.copy()\n", - "for n in range(N-1):\n", - " q3_d[n+1] = Verlet_damped_HM(q3_d[n], dt)\n", - "\n", - "#Analytical Solution \n", - "A = x0 #with no forcing, the max amplitude is equal to initial amplitude\n", - "phi = 0\n", - "q_d_analytical = A*numpy.exp(-zeta*wn*t)*numpy.cos(wd*t + phi)\n", - "vel_d_analytical = -A*numpy.exp(-zeta*wn*t)*wd*numpy.sin(wd*t+phi)\n", - " \n", - "pyplot.figure(figsize=(10,8));\n", - "pyplot.grid(True);\n", - "pyplot.xlabel(r't', fontsize=18);\n", - "pyplot.ylabel(r'position (meters)', fontsize=18);\n", - "pyplot.title('Harmonic oscillator position');\n", - "pyplot.plot(t, q1_d[:,0], lw=2, label='Euler');\n", - "pyplot.plot(t, q2_d[:,0], 'r-', lw=2, label='RK4');\n", - "pyplot.plot(t, q3_d[:,0], 'g', lw=2, label='Verlet');\n", - "pyplot.plot(t, q_d_analytical, 'k--', lw=2, label='analytical');\n", - "pyplot.legend();\n" + " return u + dt*f(u).T " ] }, { - "cell_type": "markdown", + "cell_type": "code", + "execution_count": 205, "metadata": { - "collapsed": true + "collapsed": false }, + "outputs": [], "source": [ - "How does each integrator perform compared to the case of simple harmonic motion? From the eye test, it seems like the error is actually better here. Why is that? \n", - "\n", - "The effect of the damper is to remove energy from the system. So, in this case the damping term is helping to remove some energy from the system that the integration adds. As time goes to infinity, these solutions will converge, and the Euler and Runge-Kutta methods are stable. We can again look at the energy in the system according to each method. \n", - "\n" + "def RK4_DP(u,f, dt):\n", + " \"\"\"Runge Kutta fourth order integration method\n", + " \n", + " Parameters:\n", + " u - state of the system at time t\n", + " f - function for RHS of state equations\n", + " dt - time step\n", + " \n", + " Returns: array of state values at next time step.\n", + " \"\"\"\n", + " k1 = f(u).T\n", + " k2 = f(u).T + 0.5*dt*k1\n", + " k3 = f(u).T + 0.5*dt*k2\n", + " k4 = f(u).T + dt*k3 \n", + " \n", + " return u + dt/6*(k1+2*k2+2*k3+k4)" ] }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 206, "metadata": { "collapsed": false }, @@ -4538,7 +1072,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -4546,66 +1080,7 @@ }, "metadata": {}, "output_type": "display_data" - } - ], - "source": [ - "#Euler total energy\n", - "Euler_energy_d = numpy.zeros_like(t)\n", - "for i in range(N):\n", - " Euler_energy_d[i] = get_Energy(q1_d[i,:])\n", - " \n", - "\n", - "#RK4 total energy\n", - "RK4_energy_d = numpy.zeros_like(t)\n", - "for i in range(N):\n", - " RK4_energy_d[i] = get_Energy(q2_d[i,:])\n", - "\n", - "#Verlet total energy\n", - "#Here we will convert from momentum to velocity so we can use the same get_energy function.\n", - "#Momemtum (q) is mass*velocity.\n", - "Verlet_energy_d = numpy.zeros_like(t)\n", - "Verlet_vel_d = q3_d[:,1]/m\n", - "for i in range(N):\n", - " Verlet_energy_d[i] = get_Energy(numpy.array([q3_d[i,0], Verlet_vel_d[i]]))\n", - "\n", - "\n", - "#analytic total energy\n", - "analytical_energy_d = numpy.zeros_like(t)\n", - "anayltical_state_d = numpy.array([q_d_analytical, vel_d_analytical]).T\n", - "for i in range(N):\n", - " analytical_energy_d[i] = get_Energy(anayltical_state_d[i,:])\n", - " \n", - "pyplot.figure(figsize=(8,6));\n", - "pyplot.grid(True);\n", - "pyplot.xlabel(r't', fontsize=18);\n", - "pyplot.ylabel(r'Total energy in system (J)', fontsize=18);\n", - "pyplot.title('Energy in Damped Harmonic Oscillator');\n", - "pyplot.plot(t, Euler_energy_d, lw=2, label='Euler');\n", - "pyplot.plot(t, RK4_energy_d, 'r-', lw=2, label='RK4');\n", - "pyplot.plot(t, Verlet_energy_d, 'g', lw=2, label='Verlet');\n", - "pyplot.plot(t, analytical_energy_d, 'k--', lw=2, label='analytical');\n", - "pyplot.legend();" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can see some oscillations in the total energy, which come from the discretization error. Overall, we see again that Verlet performs better here than Euler or Runge-Kutta (although not perfectly - it does have induced oscillations from the damping term), but just like the analytical solution the energy in the system asymptotically approaches zero.\n", - "\n", - "\n", - "### Phase Portrait\n", - "\n", - "Another way to investigate stability is with a phase plot. Instead of position vs. time, we can plot velocity vs. position. This plot is in phase space (a subset of state space). For any given state (position and velocity), we should be able to determine what the next state will be. This gives us an entire direction field, over which we can plot the actual behavior of the system to see if they match up. If the phase plot becomes repetitive, we can call it stable, and if the phase plot goes to the origin, we can call it asymptotically stable." - ] - }, - { - "cell_type": "code", - "execution_count": 20, - "metadata": { - "collapsed": false - }, - "outputs": [ + }, { "data": { "application/javascript": [ @@ -5340,342 +1815,87 @@ ] }, "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/html": [ - "" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "#create a phase plot\n", - "NP = 15\n", - "pos = numpy.linspace(-x0, x0, NP)\n", - "vel = numpy.linspace(-2*x0, 2*x0, NP)\n", - "\n", - "X1, X2 = numpy.meshgrid(pos, vel)\n", - "u = numpy.zeros_like(X1)\n", - "v = numpy.zeros_like(X2)\n", - "\n", - "for i in range(NP):\n", - " for j in range(NP):\n", - " x = X1[i, j]\n", - " y = X2[i, j]\n", - " yprime = f_damped_HM([x, y])\n", - " u[i,j] = yprime[0]\n", - " v[i,j] = yprime[1]\n", - "\n", - "\n", - "pyplot.figure(figsize=(10,8));\n", - "pyplot.quiver(X1, X2, u, v, pivot='mid', color='k', label='direction field');\n", - "pyplot.plot(q_d_analytical, vel_d_analytical, 'k-', label='analytical');\n", - "pyplot.plot(q1_d[:,0], q1_d[:,1],'r-', label='Euler');\n", - "pyplot.plot(q2_d[:,0], q2_d[:,1], 'b-', label='RK4');\n", - "pyplot.plot(q3_d[:,0], -Verlet_vel_d, 'g-', label='Verlet');\n", - "pyplot.xlabel('Postion ($x_1$)')\n", - "pyplot.ylabel('Velocity ($x_2$)')\n", - "pyplot.legend();\n", - "pyplot.title('Phase portrait for damped harmonic oscillator');\n" - ] - }, - { - "cell_type": "code", - "execution_count": 21, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "pyplot.close(\"all\") #This line is for cleanup - There are a lot of plots in this notebook. Comment it out \n", - " #if you want to keep all of the plots open." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, - { - "cell_type": "markdown", - "metadata": { - "collapsed": true - }, - "source": [ - "### Duffing equation (Adding a non-linear damping term)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Dynamics of a Double Pendulum" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this section, we will study the dynamics of a double pendulum. Like before, we'll derive the equations of motion from the Lagrangian and the Hamiltonian. We can use the same methods of integration as before, but this time there is no analytical solution to compare to. Finally, we will look at the properties of the system in phase space using phase plots and Poincare sections." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Background\n", - "\n", - "The double pendulum problem is a relatively simple system which can produce surprisingly complex movements. It is a chaotic system, meaning it is unpredictable and small changes in initial conditions lead to large changes in motion. There is no closed form solution to the motion of the two masses. This is a frequent example problem in the study of dynamic systems, nonlinear controls, and mechanics.\n", - "\n", - "In this notebook we will be treating the two rods as massless and ignoring the effects of friction. For a treatment of this problem without these assumptions, see (cite).\n", - "\n", - "See the image below for the definition of constants." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "![Image](figures/diagram.png)\n", - "#### Figure 1. Diagram of Double Pendulum System" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Equations of Motion \n", - "\n", - "As a reminder, the Lagrangian is\n", - "$$\\begin{equation*}\n", - "L = T- U\n", - "\\end{equation*}$$\n", - "where $T$ is the total Kinetic Energy in the system, and $U$ is the total Potential Energy. Potential energy is calculated as:\n", - "\n", - "$$\\begin{eqnarray*}\n", - "U &=& m_1g(-y_1) + m_2g(-y_2) \\\\\n", - " &=& -m_1gl_1cos\\theta_1 - m_2g(l_1cos\\theta_1 + l_2cos\\theta_2)\n", - "\\end{eqnarray*}$$\n", - "\n", - "Kinetic Energy is calculated as:\n", - "\n", - "$$\\begin{eqnarray*}\n", - "T &=& \\frac{1}{2}mv_1^2+\\frac{1}{2}mv_2^2 \\\\\n", - "&=& \\frac{1}{2}m_1(l_1\\dot\\theta_1)^2 + \\frac{1}{2}m_2(l_1\\dot\\theta_1 + l_2\\dot\\theta_2)^2 \\\\\n", - "&=& \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2+\\frac{1}{2}m_2\\left(2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+l_2^2\\dot\\theta_2^2\\right)\n", - "\\end{eqnarray*}$$\n", - "\n", - "\n", - "So the Lagrangian quantity becomes:\n", - "$$\\begin{equation*}\n", - "L = \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2+m_2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+(m_1+m_2)l_1gcos\\theta_1 + m_2l_2gcos\\theta_2\n", - "\\end{equation*}$$\n" - ] - }, - { - "cell_type": "markdown", - "metadata": { - "collapsed": true - }, - "source": [ - "The equations of motion are then found using the Lagrange Equation:\n", - "\n", - "$$\\begin{equation*}\n", - "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x_i}}\\right) - \\frac{\\partial L}{\\partial x_i} = 0\n", - "\\end{equation*}$$\n", - "\n", - "\n", - "There are two generalized coordinates ($\\theta_1 and \\theta_2$. So the equation of motion for $\\theta_1$ is calculated by the following steps:\n", - "\n", - "$$\\begin{eqnarray*}\n", - "\\frac{\\partial L}{\\partial \\dot{\\theta_1}} &=& (m_1+m_2)l_1^2\\dot\\theta_1+m_2l_1l_2\\dot\\theta_2cos(\\theta_2-\\theta_1) \\\\\n", - "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{\\theta_1}}\\right) &=& (m_1+m_2)l_1^2\\ddot\\theta_1+m_2l_1l_2\\ddot\\theta_2cos(\\theta_2-\\theta_1)-m_2l_1l_2\\dot\\theta_2^2sin(\\theta_2-\\theta_1)+m_2l_1l_2\\dot\\theta_1\\dot\\theta_2sin(\\theta_2-\\theta_1) \\\\\n", - "-\\frac{\\partial L}{\\partial \\theta_1} &=& -m_2l_1l_2\\dot\\theta_1\\dot\\theta_2sin(\\theta_2-\\theta_1)-(m_1+m_2)gl_1sin\\theta_1\n", - "\\end{eqnarray*}$$\n", - "\n", - "We can add together the second and third equations to find the first equation of motion. Some of these terms will cancel out, and we are left with a final equation of motion:\n", - "\n", - "$$\\begin{equation*}\n", - "0 = (m_1+m_2)l_1^2\\ddot\\theta_1+m_2l_1l_2\\ddot\\theta_2cos(\\theta_2-\\theta_1)-m_2l_1l_2\\dot\\theta_2^2sin(\\theta_2-\\theta_1)+(m_1+m_2)gl_1sin\\theta_1\n", - "\\end{equation*}$$\n", - "\n", - "We will also need the equation of motion for $\\theta_2$. This can be derived using the same process. It should work out to the following (check this by yourself!):\n", - "\n", - "$$\\begin{equation*}\n", - "0 = m_2l_2^2\\ddot\\theta_2+m_2l_1l_2\\ddot\\theta_1cos(\\theta_2-\\theta_1) - m_2l_2l_1\\dot\\theta_1^2sin(\\theta_2-\\theta_1)+ l_2m_2gsin\\theta_2\n", - "\\end{equation*}$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ - "## Problem setup\n", - "\n", - "Notice that these equations of motion are implicit equations for $\\ddot\\theta_1$ and $\\ddot\\theta_2$. We have talked about implicit equations in this class, and we solve them by forming a system of equations. In this case, we will need to solve the equations of motion at each time step. We will characterize our system in state space using the angle and angular velocity of each pendulum. Let's rewrite the equations of motion in a more readable form. We can use the standard form of this equation (which you'll be familiar with if you have studied mechanics) by separating out mass terms (anything multiplied by $\\ddot{x}$), centripetal terms (anything multiplied by $\\dot{x}$), and gravitational terms (anything multiplied by $x$). We get the following equations of state:\n", - "\n", - "$$\\begin{eqnarray*}\n", - "0 &=&\\textbf{M}\\vec{\\ddot{x}}(t) + \\textbf{V}\\dot{x} + \\textbf{G} \n", - "\\end{eqnarray*}$$\n", - "\n", - "The equations of motion can be solved for $\\ddot{x}$:\n", - "\n", - "$$ \\ddot{x} = \\textbf{M}^{-1}\\left[ \\textbf{V}\\dot x + \\textbf{G} \\right]$$\n", - "\n", - "where \n", - "\n", - "\\begin{eqnarray*}\n", - "M = \n", - "\\left[\\begin{array}{c}\n", - "(m_1+m_2)l_1 & m_2l_1l_2cos(\\theta_2 - \\theta_1)\\\\\n", - " m_2l_1l_2cos(\\theta_2 - \\theta_1) &m_2l_2^2\n", - "\\end{array}\\right]\n", - "\\end{eqnarray*}\n", - "\n", - "\n", - "\\begin{equation*}\n", - "V = \n", - "\\left[\\begin{array}{c}\n", - "0 & -m_2l_1l_2sin(\\theta_2 - \\theta_1)\\\\\n", - " -m_2l_1l_2sin(\\theta_2 - \\theta_1) &0\n", - "\\end{array}\\right]\n", - "\\end{equation*}\n", - "\n", - "\n", - "\\begin{eqnarray*}\n", - "G = \n", - "\\left[\\begin{array}{c}\n", - "(m_1+m_2)l_1gsin\\theta_1\\\\\n", - " m_2l_2gsin\\theta_2\n", - "\\end{array}\\right]\n", - "\\end{eqnarray*}\n", - "\n", - "and\n", - "\n", - "$$\\begin{eqnarray*}\n", - "\\vec{x}(t) = \\begin{pmatrix} \\theta_1\\\\ \\theta_2\\\\ \\end{pmatrix}, \\vec{\\dot{x}(t)} = \\begin{pmatrix} \\dot \\theta_1 \\\\ \\dot \\theta_2 \\end{pmatrix}\n", - "\\end{eqnarray*}$$\n", + "#Euler\n", + "q1_dp = numpy.zeros((N,4)) \n", + "q1_dp[0,:] = x_init_dp.copy() #set initial conditions\n", + "for n in range(N-1): #integrate with Euler\n", + " q1_dp[n+1,:] = euler_DP(q1_dp[n,:], f_double_pendulum, dt)\n", + " #print(q1[n])\n", + " \n", + "#Runge-Kutta\n", + "q2_dp = numpy.zeros((N,4)) \n", + "q2_dp[0,:] = x_init_dp.copy() #set initial conditions\n", + "for n in range(N-1): #integrate with Euler\n", + " q2_dp[n+1,:] = RK4_DP(q2_dp[n,:], f_double_pendulum, dt)\n", + " \n", + "pyplot.figure(figsize=(10,8));\n", + "pyplot.grid(True);\n", + "pyplot.xlabel(r't', fontsize=18);\n", + "pyplot.ylabel(r'position (meters)', fontsize=18);\n", + "pyplot.title('Double Pendulum Euler');\n", + "pyplot.plot(t, q1_dp[:,0], lw=2, label='Joint 1');\n", + "pyplot.plot(t, q1_dp[:,1], lw=2, label='Joint 2')\n", + "pyplot.legend();\n", "\n", - "We can now treat this system the same way we handled the simple harmonic motion; break the two second order ODEs into four first-order ODEs, then use Euler or Runge-Kutta on the results." + "pyplot.figure(figsize=(10,8));\n", + "pyplot.grid(True);\n", + "pyplot.xlabel(r't', fontsize=18);\n", + "pyplot.ylabel(r'position (meters)', fontsize=18);\n", + "pyplot.title('Double Pendulum RK4');\n", + "pyplot.plot(t, q2_dp[:,0], lw=2, label='Joint 1');\n", + "pyplot.plot(t, q2_dp[:,1], lw=2, label='Joint 2')\n", + "pyplot.legend();" ] }, { "cell_type": "markdown", - "metadata": {}, + "metadata": { + "collapsed": false + }, "source": [ - "### Equations of Motion from Hamiltonian\n", - "\n", - "Here we derive the equations of motion from the Hamiltonian. If you want good practice, you can derive these on your own. We start with the same potential and kinetic energy, but we'll need to use postion ($\\theta$) and momentum ($I\\omega$) to give us:\n", + "Again, we can see that these integrators cause the system to go unstable. In fact, there is a physical interpretation for what is happening. In the RK4 plot, you can see the position of joint 1 oscillating betwee -pi and pi (corresponding to straight up in the air), until it finally reaches a value less than -pi. So, once joint 1 'flips around,' the system diverges.\n", "\n", - "$$\\begin{eqnarray*}\n", - "U &=& m_1g(-y_1) + m_2g(-y_2) \\\\\n", - " &=& -m_1g(l_1cos\\theta_1 - m_2g(l_1cos\\theta_1 + l_2cos\\theta_2)\n", - "\\end{eqnarray*}$$\n", - "\n", - "Kinetic Energy is calculated as:\n", - "\n", - "$$\\begin{eqnarray*}\n", - "T &=& \\frac{1}{2}mv_1^2+\\frac{1}{2}mv_2^2 \\\\\n", - "&=& \\frac{q_1^2}{2I_1} + \\frac{q_2^2}{2I_2}\n", - "\\end{eqnarray*}$$\n", - "\n", - "where $I_i = m_il_i^2$. So our Hamiltonian quantity becomes:\n", - "$$\\begin{equation*}\n", - "H = \\frac{1}{2}\\left(\\frac{q_1^2}{I_1} + \\frac{q_2^2}{I_2}\\right) - (m_1+m_2)gl_1cos\\theta_1 - m_2gl_2cos\\theta_2\n", - "\\end{equation*}$$\n", + "Just like with simple harmonic motion, we can add in a damping term to help remove some of the energy that the integration scheme adds in. In the problem setup, we actually used an abbreviated form of a standard equation. The full equation is here:\n", "\n", - "Since we have two degrees of freedom, we will have four equations of motion. These will be the following:\n", - "$$\\begin{eqnarray*}\n", - "\\dot{p_1} &=& -\\frac{\\partial H}{\\partial q_1} = \\frac{q_1}{I_1} \\\\\n", - "\\dot{q_1} &=& \\frac{\\partial H}{\\partial p_1} = (m_1+m_2)gl_1sinp_1 \\\\\n", - "\\dot{p_2} &=& -\\frac{\\partial H}{\\partial q_2} = \\frac{q_2}{I_2} \\\\\n", - "\\dot{q_2} &=& \\frac{\\partial H}{\\partial p_2} = m_2gl_2sinp_2 \\\\\n", - "\\end{eqnarray*}$$\n" + "\\begin{equation}\n", + "\\textbf{M}(x)\\vec{\\ddot{x}}(t) + \\textbf{V}(x, \\dot x ) \\dot{x} + \\textbf{G(x)} + f(\\dot x) + \\tau_d(t) = \\tau(t)\n", + "\\end{equation}" ] }, { "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Integration\n", - "\n", - "For Euler and Runge-Kutta, we can use the matrix equation above to integrate our system. The results will be VERY dependent on initial conditions; this is why we call it a chaotic system. For the symplectic integrator, we'll use the Hamiltonian equations of motion above. " - ] - }, - { - "cell_type": "code", - "execution_count": 159, "metadata": { "collapsed": true }, - "outputs": [], - "source": [] + "source": [ + "where \\textbf{M, V}, and \\textbf{G} are defined as above, $f$ is frictional or damping terms, and the $\\tau$ terms represent added torque to the system. Since we aren't adding any torque, the $\\tau$ terms will remain 0. However, we can add in some linear damping by including the frictional term. This requires redefining our RHS function, but no other changes!" + ] }, { "cell_type": "code", - "execution_count": 191, + "execution_count": 207, "metadata": { "collapsed": false }, "outputs": [], "source": [ - "def f_double_pendulum(u):\n", + "def f_damped_double_pendulum(u):\n", " \"\"\"Returns RHS of double pendulum EOM\n", " \n", " Parameters:\n", @@ -5693,7 +1913,8 @@ " M = numpy.array([[(m1+m2)*l1**2, m2*l1*l2*cos(x2-x1)],[m2*l1*l2*cos(x2-x1), m2*l2**2]])\n", " V = numpy.array([[0, -m2*l1*l2*sin(x2-x1)],[-m2*l1*l2*sin(x2-x1), 0]])\n", " G = numpy.array([[(m1+m2)*g*l1*sin(x1)],[m2*l2*g*sin(x2)]])\n", - " qdd = numpy.linalg.inv(M).dot(V.dot(numpy.array([[x3],[x4]])) +G) #- 2*numpy.array([[x3],[x4]])) \n", + " f = c*numpy.array([[x3],[x4]])\n", + " qdd = numpy.linalg.inv(M).dot(V.dot(numpy.array([[x3],[x4]]))+G -f) \n", " #print(qdd)\n", " #print(M)\n", " #print(V.dot(numpy.array([[x3],[x4]])))\n", @@ -5705,91 +1926,7 @@ }, { "cell_type": "code", - "execution_count": 192, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "#Set up parameters:\n", - "g = -9.8 #[m/s^2]\n", - "m1 = 2\n", - "m2 = 1.5;\n", - "l1 = 1\n", - "l2 = 1.5\n", - "\n", - "T = 25; #[seconds]\n", - "dt = .02; #\n", - "N = int(T/dt)+1\n", - "t = numpy.linspace(0.0, T, N)\n", - "\n", - "#Initial Conditions\n", - "theta1_0 = 0 #[radians]\n", - "theta2_0 = 1\n", - "theta1_dot_0 = 0 #[rad/s]\n", - "theta2_dot_0 = 0 #\n", - "\n", - "#p0 = x0\n", - "#q0 = m*xdot0\n", - "x_init_dp = numpy.array([theta1_0, theta2_0, theta1_dot_0, theta2_dot_0])\n", - "#x_init_H = numpy.array([p0, q0])\n", - "\n" - ] - }, - { - "cell_type": "code", - "execution_count": 193, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "def euler_DP(u, f, dt):\n", - " \"\"\" Euler's method for integrating a system of differential equations.\n", - " \n", - " Parameters:\n", - " u - state at current step \n", - " f - RHS of equation\n", - " dt- time step size\n", - " \n", - " Returns: \n", - " x - array of values at next time step.\n", - " \"\"\"\n", - " #print(f(u))\n", - " #print(u)\n", - " \n", - " return u + dt*f(u).T " - ] - }, - { - "cell_type": "code", - "execution_count": 194, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "def RK4_DP(u,f, dt):\n", - " \"\"\"Runge Kutta fourth order integration method\n", - " \n", - " Parameters:\n", - " u - state of the system at time t\n", - " f - function for RHS of state equations\n", - " dt - time step\n", - " \n", - " Returns: array of state values at next time step.\n", - " \"\"\"\n", - " k1 = f(u).T\n", - " k2 = f(u).T + 0.5*dt*k1\n", - " k3 = f(u).T + 0.5*dt*k2\n", - " k4 = f(u).T + dt*k3 \n", - " \n", - " return u + dt/6*(k1+2*k2+2*k3+k4)" - ] - }, - { - "cell_type": "code", - "execution_count": 195, + "execution_count": 208, "metadata": { "collapsed": false }, @@ -6533,7 +2670,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -7281,7 +3418,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -7292,26 +3429,29 @@ } ], "source": [ + "#We use the same initial conditions as the example without damping, so they aren't redefined here.\n", + "c = 2 #[N/(rad/s)] define some damping\n", + "\n", "#Euler\n", - "q1_dp = numpy.zeros((N,4)) \n", - "q1_dp[0,:] = x_init_dp.copy() #set initial conditions\n", + "q1d_dp = numpy.zeros((N,4)) \n", + "q1d_dp[0,:] = x_init_dp.copy() #set initial conditions\n", "for n in range(N-1): #integrate with Euler\n", - " q1_dp[n+1,:] = euler_DP(q1_dp[n,:], f_double_pendulum, dt)\n", + " q1d_dp[n+1,:] = euler_DP(q1d_dp[n,:], f_damped_double_pendulum, dt)\n", " #print(q1[n])\n", " \n", "#Runge-Kutta\n", - "q2_dp = numpy.zeros((N,4)) \n", - "q2_dp[0,:] = x_init_dp.copy() #set initial conditions\n", + "q2d_dp = numpy.zeros((N,4)) \n", + "q2d_dp[0,:] = x_init_dp.copy() #set initial conditions\n", "for n in range(N-1): #integrate with Euler\n", - " q2_dp[n+1,:] = RK4_DP(q2_dp[n,:], f_double_pendulum, dt)\n", + " q2d_dp[n+1,:] = RK4_DP(q2d_dp[n,:], f_damped_double_pendulum, dt)\n", " \n", "pyplot.figure(figsize=(10,8));\n", "pyplot.grid(True);\n", "pyplot.xlabel(r't', fontsize=18);\n", "pyplot.ylabel(r'position (meters)', fontsize=18);\n", "pyplot.title('Double Pendulum Euler');\n", - "pyplot.plot(t, q1_dp[:,0], lw=2, label='Joint 1');\n", - "pyplot.plot(t, q1_dp[:,1], lw=2, label='Joint 2')\n", + "pyplot.plot(t, q1d_dp[:,0], lw=2, label='Joint 1');\n", + "pyplot.plot(t, q1d_dp[:,1], lw=2, label='Joint 2')\n", "pyplot.legend();\n", "\n", "pyplot.figure(figsize=(10,8));\n", @@ -7319,44 +3459,20 @@ "pyplot.xlabel(r't', fontsize=18);\n", "pyplot.ylabel(r'position (meters)', fontsize=18);\n", "pyplot.title('Double Pendulum RK4');\n", - "pyplot.plot(t, q2_dp[:,0], lw=2, label='Joint 1');\n", - "pyplot.plot(t, q2_dp[:,1], lw=2, label='Joint 2')\n", + "pyplot.plot(t, q2d_dp[:,0], lw=2, label='Joint 1');\n", + "pyplot.plot(t, q2d_dp[:,1], lw=2, label='Joint 2')\n", "pyplot.legend();" ] }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, { "cell_type": "markdown", "metadata": { "collapsed": true }, - "source": [] + "source": [ + "### Investigating system behavior\n", + "\n" + ] }, { "cell_type": "markdown", @@ -7388,7 +3504,7 @@ }, "outputs": [], "source": [ - "# This cell loads the style of the notebook, which is taken from the \n", + "# This cell loads the style of the notebook, which is modified from the \n", "# Numerical Methods in Python Course: http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about\n", "\n", "from IPython.core.display import HTML\n", diff --git a/randy_schur/Integration Techniques for Mechanical Systems.ipynb b/randy_schur/Integration Techniques for Mechanical Systems.ipynb index d66d98c..4594f0f 100644 --- a/randy_schur/Integration Techniques for Mechanical Systems.ipynb +++ b/randy_schur/Integration Techniques for Mechanical Systems.ipynb @@ -5369,12 +5369,209 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 26, "metadata": { - "collapsed": true + "collapsed": false }, - "outputs": [], - "source": [] + "outputs": [ + { + "ename": "SyntaxError", + "evalue": "invalid syntax (, line 1)", + "output_type": "error", + "traceback": [ + "\u001b[1;36m File \u001b[1;32m\"\"\u001b[1;36m, line \u001b[1;32m1\u001b[0m\n\u001b[1;33m Sources:\u001b[0m\n\u001b[1;37m ^\u001b[0m\n\u001b[1;31mSyntaxError\u001b[0m\u001b[1;31m:\u001b[0m invalid syntax\n" + ] + } + ], + "source": [ + "Sources: \n", + "\n", + "[1] http://scienceworld.wolfram.com/physics/DoublePendulum.html\n", + "\n", + "[2] http://www.phy.uct.ac.za/courses/opencontent/phylab2/worksheet9_09.pdf\n", + "\n", + "[3] http://www.phys.lsu.edu/faculty/gonzalez/Teaching/Phys7221/DoublePendulum.pdf\n", + "\n", + "[4] http://www.iontrap.wabash.edu/adlab/papers/F2011_foster_groninger_tang_chaos.pdf\n", + "\n", + "[5] https://math.berkeley.edu/~alanw/242papers99/markiewicz.pdf\n", + "\n", + "[6] http://www.unige.ch/~hairer/poly_geoint/week2.pdf\n", + "\n", + "[7] http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1994CeMDA..60..409T&defaultprint=YES&filetype=.pdf\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "execution_count": 27, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# This cell loads the style of the notebook, which is modified from the \n", + "# Numerical Methods in Python Course: http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about\n", + "\n", + "from IPython.core.display import HTML\n", + "css_file = './numericalmoocstyle.css'\n", + "HTML(open(css_file, \"r\").read())" + ] }, { "cell_type": "code", From 52478e3afbc66804e6a6fbf5e4a6daaa933281e0 Mon Sep 17 00:00:00 2001 From: rbds Date: Mon, 7 Dec 2015 16:51:18 -0500 Subject: [PATCH 30/61] Delete 0e1p4ebd.wag.txt --- 0e1p4ebd.wag.txt | 13 ------------- 1 file changed, 13 deletions(-) delete mode 100644 0e1p4ebd.wag.txt diff --git a/0e1p4ebd.wag.txt b/0e1p4ebd.wag.txt deleted file mode 100644 index d5e5a32..0000000 --- a/0e1p4ebd.wag.txt +++ /dev/null @@ -1,13 +0,0 @@ - -# Please enter the commit message for your changes. Lines starting -# with '#' will be ignored, and an empty message aborts the commit. -# On branch master -# Your branch is up-to-date with 'origin/master'. -# -# Changes to be committed: -# new file: figures/POS_computer.png -# -# Changes not staged for commit: -# modified: Double_Pendulum_Problem.ipynb -# - From c58c649ed88c610665ece81969b716319f360630 Mon Sep 17 00:00:00 2001 From: rbds Date: Mon, 7 Dec 2015 19:03:59 -0500 Subject: [PATCH 31/61] plot Poincare section --- randy_schur/Double_Pendulum_Problem.ipynb | 897 +++++++++++++++++++++- 1 file changed, 874 insertions(+), 23 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index ba51bfe..080d2bd 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -197,15 +197,6 @@ "For Euler and Runge-Kutta, we can use the matrix equation above to integrate our system. The results will be VERY dependent on initial conditions; this is why we call it a chaotic system. For the symplectic integrator, we'll use the Hamiltonian equations of motion above. " ] }, - { - "cell_type": "code", - "execution_count": 159, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, { "cell_type": "code", "execution_count": 202, @@ -244,7 +235,7 @@ }, { "cell_type": "code", - "execution_count": 203, + "execution_count": 287, "metadata": { "collapsed": false }, @@ -257,7 +248,7 @@ "l1 = 1\n", "l2 = 1.5\n", "\n", - "T = 25; #[seconds]\n", + "T = 50; #[seconds]\n", "dt = .02; #\n", "N = int(T/dt)+1\n", "t = numpy.linspace(0.0, T, N)\n", @@ -277,7 +268,7 @@ }, { "cell_type": "code", - "execution_count": 204, + "execution_count": 288, "metadata": { "collapsed": true }, @@ -302,7 +293,7 @@ }, { "cell_type": "code", - "execution_count": 205, + "execution_count": 289, "metadata": { "collapsed": false }, @@ -328,7 +319,7 @@ }, { "cell_type": "code", - "execution_count": 206, + "execution_count": 290, "metadata": { "collapsed": false }, @@ -1072,7 +1063,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1820,7 +1811,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1871,10 +1862,44 @@ "source": [ "Again, we can see that these integrators cause the system to go unstable. In fact, there is a physical interpretation for what is happening. In the RK4 plot, you can see the position of joint 1 oscillating betwee -pi and pi (corresponding to straight up in the air), until it finally reaches a value less than -pi. So, once joint 1 'flips around,' the system diverges.\n", "\n", + "With simple harmonic motion, the symplectic integrator was able to accurately represent the state of the system without adding energy. Let's see if it is successful with the double pendulum. " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def euler_DP(u, f, dt):\n", + " \"\"\" Euler's method for integrating a system of differential equations.\n", + " \n", + " Parameters:\n", + " u - state at current step \n", + " f - RHS of equation\n", + " dt- time step size\n", + " \n", + " Returns: \n", + " x - array of values at next time step.\n", + " \"\"\"\n", + " #print(f(u))\n", + " #print(u)\n", + " \n", + " return u + dt*f(u).T " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Damped Double Pendulum\n", + "\n", "Just like with simple harmonic motion, we can add in a damping term to help remove some of the energy that the integration scheme adds in. In the problem setup, we actually used an abbreviated form of a standard equation. The full equation is here:\n", "\n", "\\begin{equation}\n", - "\\textbf{M}(x)\\vec{\\ddot{x}}(t) + \\textbf{V}(x, \\dot x ) \\dot{x} + \\textbf{G(x)} + f(\\dot x) + \\tau_d(t) = \\tau(t)\n", + "\\textbf{M}(x)\\vec{\\ddot{x}}(t) + \\textbf{V}(x, \\dot x ) \\vec{\\dot{x}} + \\textbf{G(x)} + f(\\dot x) + \\tau_d(t) = \\tau(t)\n", "\\end{equation}" ] }, @@ -1884,12 +1909,12 @@ "collapsed": true }, "source": [ - "where \\textbf{M, V}, and \\textbf{G} are defined as above, $f$ is frictional or damping terms, and the $\\tau$ terms represent added torque to the system. Since we aren't adding any torque, the $\\tau$ terms will remain 0. However, we can add in some linear damping by including the frictional term. This requires redefining our RHS function, but no other changes!" + "where $\\textbf{M, V}$, and $\\textbf{G}$ are defined as above, $f$ is frictional or damping terms, and the $\\tau$ terms represent added torque to the system. Since we aren't adding any torque, the $\\tau$ terms will remain 0. However, we can add in some linear damping by including the frictional term. This requires redefining our RHS function, but no other changes!" ] }, { "cell_type": "code", - "execution_count": 207, + "execution_count": 291, "metadata": { "collapsed": false }, @@ -1926,7 +1951,7 @@ }, { "cell_type": "code", - "execution_count": 208, + "execution_count": 292, "metadata": { "collapsed": false }, @@ -2670,7 +2695,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -3418,7 +3443,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -3471,7 +3496,833 @@ }, "source": [ "### Investigating system behavior\n", - "\n" + "\n", + "In order to investigate system behavior for damped harmonic motion, we used a phase portrait where we plotted position vs. velocity. In this system, we have 2 positions and 2 velocities, so a phase plot won't accurately capture the behavior of the system. Instead we are going to use something called a Poincare section (https://en.wikipedia.org/wiki/Poincar%C3%A9_map). The way these work is to capture the system behavior at a specific, repetetive point. For example, each time that $\\theta_1 = 0$ (mass 1 is hanging straight down) we can capture the angular position and velocity of mass 2. This way, we are taking a 'snapshot' of the system, and plotting the state each time. We can now investigate system behavior on a simple 2D plot, similar to the phase portrait." + ] + }, + { + "cell_type": "code", + "execution_count": 294, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def poincare(state, sf, val, s1, s2):\n", + " '''Creates a poincare section, capturing system state each time q[state]=val\n", + " \n", + " Parameters:\n", + " state - state vector\n", + " sf - the row in q to fix\n", + " val - the value to capture state at \n", + " s1 - state to capture 1\n", + " s2 - state to capture 2\n", + " \n", + " Returns:\n", + " v - array with information for poincare section\n", + " '''\n", + " v = numpy.zeros((2,1))\n", + " for i in range(len(state[:,sf])-1):\n", + " if (state[i,sf]>val):\n", + " if (state[i+1,sf] < val):\n", + " # print(v)\n", + " # print([[state[i, s1],[state[i, s2]]]])\n", + " v = numpy.append(v, [[state[i, s1]],[state[i, s2]]], axis=1)\n", + " \n", + " if (state[i,sf] < val):\n", + " if (state[i+1,sf] > val):\n", + " v = numpy.append(v, [[state[i, s1]],[state[i, s2]]], axis=1)\n", + " \n", + " return v\n", + " " + ] + }, + { + "cell_type": "code", + "execution_count": 296, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "application/javascript": [ + "/* Put everything inside the global mpl namespace */\n", + "window.mpl = {};\n", + "\n", + "mpl.get_websocket_type = function() {\n", + " if (typeof(WebSocket) !== 'undefined') {\n", + " return WebSocket;\n", + " } else if (typeof(MozWebSocket) !== 'undefined') {\n", + " return MozWebSocket;\n", + " } else {\n", + " alert('Your browser does not have WebSocket support.' +\n", + " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", + " 'Firefox 4 and 5 are also supported but you ' +\n", + " 'have to enable WebSockets in about:config.');\n", + " };\n", + "}\n", + "\n", + "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", + " this.id = figure_id;\n", + "\n", + " this.ws = websocket;\n", + "\n", + " this.supports_binary = (this.ws.binaryType != undefined);\n", + "\n", + " if (!this.supports_binary) {\n", + " var warnings = document.getElementById(\"mpl-warnings\");\n", + " if (warnings) {\n", + " warnings.style.display = 'block';\n", + " warnings.textContent = (\n", + " \"This browser does not support binary websocket messages. \" +\n", + " \"Performance may be slow.\");\n", + " }\n", + " }\n", + "\n", + " this.imageObj = new Image();\n", + "\n", + " this.context = undefined;\n", + " this.message = undefined;\n", + " this.canvas = undefined;\n", + " this.rubberband_canvas = undefined;\n", + " this.rubberband_context = undefined;\n", + " this.format_dropdown = undefined;\n", + "\n", + " this.image_mode = 'full';\n", + "\n", + " this.root = $('
');\n", + " this._root_extra_style(this.root)\n", + " this.root.attr('style', 'display: inline-block');\n", + "\n", + " $(parent_element).append(this.root);\n", + "\n", + " this._init_header(this);\n", + " this._init_canvas(this);\n", + " this._init_toolbar(this);\n", + "\n", + " var fig = this;\n", + "\n", + " this.waiting = false;\n", + "\n", + " this.ws.onopen = function () {\n", + " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", + " fig.send_message(\"send_image_mode\", {});\n", + " fig.send_message(\"refresh\", {});\n", + " }\n", + "\n", + " this.imageObj.onload = function() {\n", + " if (fig.image_mode == 'full') {\n", + " // Full images could contain transparency (where diff images\n", + " // almost always do), so we need to clear the canvas so that\n", + " // there is no ghosting.\n", + " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", + " }\n", + " fig.context.drawImage(fig.imageObj, 0, 0);\n", + " fig.waiting = false;\n", + " };\n", + "\n", + " this.imageObj.onunload = function() {\n", + " this.ws.close();\n", + " }\n", + "\n", + " this.ws.onmessage = this._make_on_message_function(this);\n", + "\n", + " this.ondownload = ondownload;\n", + "}\n", + "\n", + "mpl.figure.prototype._init_header = function() {\n", + " var titlebar = $(\n", + " '
');\n", + " var titletext = $(\n", + " '
');\n", + " titlebar.append(titletext)\n", + " this.root.append(titlebar);\n", + " this.header = titletext[0];\n", + "}\n", + "\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._init_canvas = function() {\n", + " var fig = this;\n", + "\n", + " var canvas_div = $('
');\n", + "\n", + " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", + "\n", + " function canvas_keyboard_event(event) {\n", + " return fig.key_event(event, event['data']);\n", + " }\n", + "\n", + " canvas_div.keydown('key_press', canvas_keyboard_event);\n", + " canvas_div.keyup('key_release', canvas_keyboard_event);\n", + " this.canvas_div = canvas_div\n", + " this._canvas_extra_style(canvas_div)\n", + " this.root.append(canvas_div);\n", + "\n", + " var canvas = $('');\n", + " canvas.addClass('mpl-canvas');\n", + " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", + "\n", + " this.canvas = canvas[0];\n", + " this.context = canvas[0].getContext(\"2d\");\n", + "\n", + " var rubberband = $('');\n", + " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", + "\n", + " var pass_mouse_events = true;\n", + "\n", + " canvas_div.resizable({\n", + " start: function(event, ui) {\n", + " pass_mouse_events = false;\n", + " },\n", + " resize: function(event, ui) {\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " stop: function(event, ui) {\n", + " pass_mouse_events = true;\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " });\n", + "\n", + " function mouse_event_fn(event) {\n", + " if (pass_mouse_events)\n", + " return fig.mouse_event(event, event['data']);\n", + " }\n", + "\n", + " rubberband.mousedown('button_press', mouse_event_fn);\n", + " rubberband.mouseup('button_release', mouse_event_fn);\n", + " // Throttle sequential mouse events to 1 every 20ms.\n", + " rubberband.mousemove('motion_notify', mouse_event_fn);\n", + "\n", + " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", + " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", + "\n", + " canvas_div.on(\"wheel\", function (event) {\n", + " event = event.originalEvent;\n", + " event['data'] = 'scroll'\n", + " if (event.deltaY < 0) {\n", + " event.step = 1;\n", + " } else {\n", + " event.step = -1;\n", + " }\n", + " mouse_event_fn(event);\n", + " });\n", + "\n", + " canvas_div.append(canvas);\n", + " canvas_div.append(rubberband);\n", + "\n", + " this.rubberband = rubberband;\n", + " this.rubberband_canvas = rubberband[0];\n", + " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", + " this.rubberband_context.strokeStyle = \"#000000\";\n", + "\n", + " this._resize_canvas = function(width, height) {\n", + " // Keep the size of the canvas, canvas container, and rubber band\n", + " // canvas in synch.\n", + " canvas_div.css('width', width)\n", + " canvas_div.css('height', height)\n", + "\n", + " canvas.attr('width', width);\n", + " canvas.attr('height', height);\n", + "\n", + " rubberband.attr('width', width);\n", + " rubberband.attr('height', height);\n", + " }\n", + "\n", + " // Set the figure to an initial 600x600px, this will subsequently be updated\n", + " // upon first draw.\n", + " this._resize_canvas(600, 600);\n", + "\n", + " // Disable right mouse context menu.\n", + " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", + " return false;\n", + " });\n", + "\n", + " function set_focus () {\n", + " canvas.focus();\n", + " canvas_div.focus();\n", + " }\n", + "\n", + " window.setTimeout(set_focus, 100);\n", + "}\n", + "\n", + "mpl.figure.prototype._init_toolbar = function() {\n", + " var fig = this;\n", + "\n", + " var nav_element = $('
')\n", + " nav_element.attr('style', 'width: 100%');\n", + " this.root.append(nav_element);\n", + "\n", + " // Define a callback function for later on.\n", + " function toolbar_event(event) {\n", + " return fig.toolbar_button_onclick(event['data']);\n", + " }\n", + " function toolbar_mouse_event(event) {\n", + " return fig.toolbar_button_onmouseover(event['data']);\n", + " }\n", + "\n", + " for(var toolbar_ind in mpl.toolbar_items) {\n", + " var name = mpl.toolbar_items[toolbar_ind][0];\n", + " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", + " var image = mpl.toolbar_items[toolbar_ind][2];\n", + " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", + "\n", + " if (!name) {\n", + " // put a spacer in here.\n", + " continue;\n", + " }\n", + " var button = $('');\n", + " button.click(method_name, toolbar_event);\n", + " button.mouseover(tooltip, toolbar_mouse_event);\n", + " nav_element.append(button);\n", + " }\n", + "\n", + " // Add the status bar.\n", + " var status_bar = $('');\n", + " nav_element.append(status_bar);\n", + " this.message = status_bar[0];\n", + "\n", + " // Add the close button to the window.\n", + " var buttongrp = $('
');\n", + " var button = $('');\n", + " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", + " button.mouseover('Close figure', toolbar_mouse_event);\n", + " buttongrp.append(button);\n", + " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", + " titlebar.prepend(buttongrp);\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(el){\n", + " // this is important to make the div 'focusable\n", + " el.attr('tabindex', 0)\n", + " // reach out to IPython and tell the keyboard manager to turn it's self\n", + " // off when our div gets focus\n", + "\n", + " // location in version 3\n", + " if (IPython.notebook.keyboard_manager) {\n", + " IPython.notebook.keyboard_manager.register_events(el);\n", + " }\n", + " else {\n", + " // location in version 2\n", + " IPython.keyboard_manager.register_events(el);\n", + " }\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._key_event_extra = function(event, name) {\n", + " var manager = IPython.notebook.keyboard_manager;\n", + " if (!manager)\n", + " manager = IPython.keyboard_manager;\n", + "\n", + " // Check for shift+enter\n", + " if (event.shiftKey && event.which == 13) {\n", + " this.canvas_div.blur();\n", + " event.shiftKey = false;\n", + " // Send a \"J\" for go to next cell\n", + " event.which = 74;\n", + " event.keyCode = 74;\n", + " manager.command_mode();\n", + " manager.handle_keydown(event);\n", + " }\n", + "}\n", + "\n", + "mpl.figure.prototype.handle_save = function(fig, msg) {\n", + " fig.ondownload(fig, null);\n", + "}\n", + "\n", + "\n", + "mpl.find_output_cell = function(html_output) {\n", + " // Return the cell and output element which can be found *uniquely* in the notebook.\n", + " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", + " // IPython event is triggered only after the cells have been serialised, which for\n", + " // our purposes (turning an active figure into a static one), is too late.\n", + " var cells = IPython.notebook.get_cells();\n", + " var ncells = cells.length;\n", + " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", + " data = data.data;\n", + " }\n", + " if (data['text/html'] == html_output) {\n", + " return [cell, data, j];\n", + " }\n", + " }\n", + " }\n", + " }\n", + "}\n", + "\n", + "// Register the function which deals with the matplotlib target/channel.\n", + "// The kernel may be null if the page has been refreshed.\n", + "if (IPython.notebook.kernel != null) {\n", + " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", + "}\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "#Verlet integration\n", + "q3_dp = numpy.zeros((N,4)) \n", + "q3_dp[0,:] = x_init_H_dp.copy() #set initial conditions\n", + "print(q3_dp[0,:])\n", + "for n in range(N-1): #integrate with Euler\n", + " q3_dp[n+1,:] = Verlet_DP(q3_dp[n,:], dt)\n", + " \n", + "pyplot.figure(figsize=(10,8));\n", + "pyplot.grid(True);\n", + "pyplot.xlabel(r't', fontsize=18);\n", + "pyplot.ylabel(r'position (meters)', fontsize=18);\n", + "pyplot.title('Double Pendulum Euler');\n", + "pyplot.plot(t, q3_dp[:,0], lw=2, label='Joint 1');\n", + "pyplot.plot(t, q3_dp[:,2], lw=2, label='Joint 2')\n", + "pyplot.legend();" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, { "cell_type": "markdown", "metadata": {}, diff --git a/randy_schur/Integration Techniques for Mechanical Systems.ipynb b/randy_schur/Integration Techniques for Mechanical Systems.ipynb index 4594f0f..f4e1e02 100644 --- a/randy_schur/Integration Techniques for Mechanical Systems.ipynb +++ b/randy_schur/Integration Techniques for Mechanical Systems.ipynb @@ -379,7 +379,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 28, "metadata": { "collapsed": false }, @@ -1123,7 +1123,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1154,7 +1154,7 @@ " q3[n+1] = verlet_SHM(q3[n], dt)\n", "\n", "#Analytical Solution \n", - "A = x0 #with no forcing, the max amplitude is equal to initial amplitude\n", + "A = x0 #with no forcing and no initial velocity, the max amplitude is equal to initial amplitude\n", "phi = 0\n", "omega = sqrt(k/m)\n", "q_analytical = A*numpy.cos(omega*t + phi)\n", From f069866e87665f2a434c7a806c91c2cc9e5a9add Mon Sep 17 00:00:00 2001 From: rbds Date: Tue, 8 Dec 2015 00:51:23 +0000 Subject: [PATCH 33/61] add screenshots --- randy_schur/Double_Pendulum_Problem.ipynb | 4563 +---------------- ...on Techniques for Mechanical Systems.ipynb | 491 +- randy_schur/figures/Error_64bit.png | Bin 0 -> 24164 bytes randy_schur/figures/response_64bit.png | Bin 0 -> 69838 bytes 4 files changed, 285 insertions(+), 4769 deletions(-) create mode 100644 randy_schur/figures/Error_64bit.png create mode 100644 randy_schur/figures/response_64bit.png diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index 1267ee8..b1c0004 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -199,7 +199,7 @@ }, { "cell_type": "code", - "execution_count": 202, + "execution_count": 1, "metadata": { "collapsed": false }, @@ -235,11 +235,23 @@ }, { "cell_type": "code", - "execution_count": 319, + "execution_count": 2, "metadata": { "collapsed": false }, - "outputs": [], + "outputs": [ + { + "ename": "NameError", + "evalue": "name 'numpy' is not defined", + "output_type": "error", + "traceback": [ + "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[1;31mNameError\u001b[0m Traceback (most recent call last)", + "\u001b[1;32m\u001b[0m in \u001b[0;36m\u001b[1;34m()\u001b[0m\n\u001b[0;32m 11\u001b[0m \u001b[0mdt\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m.02\u001b[0m\u001b[1;33m;\u001b[0m \u001b[1;31m#\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 12\u001b[0m \u001b[0mN\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mT\u001b[0m\u001b[1;33m/\u001b[0m\u001b[0mdt\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m+\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 13\u001b[1;33m \u001b[0mt\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnumpy\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mlinspace\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m0.0\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mT\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mN\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 14\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 15\u001b[0m \u001b[1;31m#Initial Conditions\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;31mNameError\u001b[0m: name 'numpy' is not defined" + ] + } + ], "source": [ "#Set up parameters:\n", "g = -9.8 #[m/s^2]\n", @@ -273,7 +285,7 @@ }, { "cell_type": "code", - "execution_count": 320, + "execution_count": null, "metadata": { "collapsed": true }, @@ -298,7 +310,7 @@ }, { "cell_type": "code", - "execution_count": 321, + "execution_count": null, "metadata": { "collapsed": false }, @@ -324,1508 +336,11 @@ }, { "cell_type": "code", - "execution_count": 322, + "execution_count": null, "metadata": { "collapsed": false }, - "outputs": [ - { - "data": { - "application/javascript": [ - "/* Put everything inside the global mpl namespace */\n", - "window.mpl = {};\n", - "\n", - "mpl.get_websocket_type = function() {\n", - " if (typeof(WebSocket) !== 'undefined') {\n", - " return WebSocket;\n", - " } else if (typeof(MozWebSocket) !== 'undefined') {\n", - " return MozWebSocket;\n", - " } else {\n", - " alert('Your browser does not have WebSocket support.' +\n", - " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", - " 'Firefox 4 and 5 are also supported but you ' +\n", - " 'have to enable WebSockets in about:config.');\n", - " };\n", - "}\n", - "\n", - "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", - " this.id = figure_id;\n", - "\n", - " this.ws = websocket;\n", - "\n", - " this.supports_binary = (this.ws.binaryType != undefined);\n", - "\n", - " if (!this.supports_binary) {\n", - " var warnings = document.getElementById(\"mpl-warnings\");\n", - " if (warnings) {\n", - " warnings.style.display = 'block';\n", - " warnings.textContent = (\n", - " \"This browser does not support binary websocket messages. \" +\n", - " \"Performance may be slow.\");\n", - " }\n", - " }\n", - "\n", - " this.imageObj = new Image();\n", - "\n", - " this.context = undefined;\n", - " this.message = undefined;\n", - " this.canvas = undefined;\n", - " this.rubberband_canvas = undefined;\n", - " this.rubberband_context = undefined;\n", - " this.format_dropdown = undefined;\n", - "\n", - " this.image_mode = 'full';\n", - "\n", - " this.root = $('
');\n", - " this._root_extra_style(this.root)\n", - " this.root.attr('style', 'display: inline-block');\n", - "\n", - " $(parent_element).append(this.root);\n", - "\n", - " this._init_header(this);\n", - " this._init_canvas(this);\n", - " this._init_toolbar(this);\n", - "\n", - " var fig = this;\n", - "\n", - " this.waiting = false;\n", - "\n", - " this.ws.onopen = function () {\n", - " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", - " fig.send_message(\"send_image_mode\", {});\n", - " fig.send_message(\"refresh\", {});\n", - " }\n", - "\n", - " this.imageObj.onload = function() {\n", - " if (fig.image_mode == 'full') {\n", - " // Full images could contain transparency (where diff images\n", - " // almost always do), so we need to clear the canvas so that\n", - " // there is no ghosting.\n", - " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", - " }\n", - " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", - " };\n", - "\n", - " this.imageObj.onunload = function() {\n", - " this.ws.close();\n", - " }\n", - "\n", - " this.ws.onmessage = this._make_on_message_function(this);\n", - "\n", - " this.ondownload = ondownload;\n", - "}\n", - "\n", - "mpl.figure.prototype._init_header = function() {\n", - " var titlebar = $(\n", - " '
');\n", - " var titletext = $(\n", - " '
');\n", - " titlebar.append(titletext)\n", - " this.root.append(titlebar);\n", - " this.header = titletext[0];\n", - "}\n", - "\n", - "\n", - "\n", - "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", - "\n", - "}\n", - "\n", - "\n", - "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", - "\n", - "}\n", - "\n", - "mpl.figure.prototype._init_canvas = function() {\n", - " var fig = this;\n", - "\n", - " var canvas_div = $('
');\n", - "\n", - " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", - "\n", - " function canvas_keyboard_event(event) {\n", - " return fig.key_event(event, event['data']);\n", - " }\n", - "\n", - " canvas_div.keydown('key_press', canvas_keyboard_event);\n", - " canvas_div.keyup('key_release', canvas_keyboard_event);\n", - " this.canvas_div = canvas_div\n", - " this._canvas_extra_style(canvas_div)\n", - " this.root.append(canvas_div);\n", - "\n", - " var canvas = $('');\n", - " canvas.addClass('mpl-canvas');\n", - " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", - "\n", - " this.canvas = canvas[0];\n", - " this.context = canvas[0].getContext(\"2d\");\n", - "\n", - " var rubberband = $('');\n", - " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", - "\n", - " var pass_mouse_events = true;\n", - "\n", - " canvas_div.resizable({\n", - " start: function(event, ui) {\n", - " pass_mouse_events = false;\n", - " },\n", - " resize: function(event, ui) {\n", - " fig.request_resize(ui.size.width, ui.size.height);\n", - " },\n", - " stop: function(event, ui) {\n", - " pass_mouse_events = true;\n", - " fig.request_resize(ui.size.width, ui.size.height);\n", - " },\n", - " });\n", - "\n", - " function mouse_event_fn(event) {\n", - " if (pass_mouse_events)\n", - " return fig.mouse_event(event, event['data']);\n", - " }\n", - "\n", - " rubberband.mousedown('button_press', mouse_event_fn);\n", - " rubberband.mouseup('button_release', mouse_event_fn);\n", - " // Throttle sequential mouse events to 1 every 20ms.\n", - " rubberband.mousemove('motion_notify', mouse_event_fn);\n", - "\n", - " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", - " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", - "\n", - " canvas_div.on(\"wheel\", function (event) {\n", - " event = event.originalEvent;\n", - " event['data'] = 'scroll'\n", - " if (event.deltaY < 0) {\n", - " event.step = 1;\n", - " } else {\n", - " event.step = -1;\n", - " }\n", - " mouse_event_fn(event);\n", - " });\n", - "\n", - " canvas_div.append(canvas);\n", - " canvas_div.append(rubberband);\n", - "\n", - " this.rubberband = rubberband;\n", - " this.rubberband_canvas = rubberband[0];\n", - " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", - " this.rubberband_context.strokeStyle = \"#000000\";\n", - "\n", - " this._resize_canvas = function(width, height) {\n", - " // Keep the size of the canvas, canvas container, and rubber band\n", - " // canvas in synch.\n", - " canvas_div.css('width', width)\n", - " canvas_div.css('height', height)\n", - "\n", - " canvas.attr('width', width);\n", - " canvas.attr('height', height);\n", - "\n", - " rubberband.attr('width', width);\n", - " rubberband.attr('height', height);\n", - " }\n", - "\n", - " // Set the figure to an initial 600x600px, this will subsequently be updated\n", - " // upon first draw.\n", - " this._resize_canvas(600, 600);\n", - "\n", - " // Disable right mouse context menu.\n", - " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", - " return false;\n", - " });\n", - "\n", - " function set_focus () {\n", - " canvas.focus();\n", - " canvas_div.focus();\n", - " }\n", - "\n", - " window.setTimeout(set_focus, 100);\n", - "}\n", - "\n", - "mpl.figure.prototype._init_toolbar = function() {\n", - " var fig = this;\n", - "\n", - " var nav_element = $('
')\n", - " nav_element.attr('style', 'width: 100%');\n", - " this.root.append(nav_element);\n", - "\n", - " // Define a callback function for later on.\n", - " function toolbar_event(event) {\n", - " return fig.toolbar_button_onclick(event['data']);\n", - " }\n", - " function toolbar_mouse_event(event) {\n", - " return fig.toolbar_button_onmouseover(event['data']);\n", - " }\n", - "\n", - " for(var toolbar_ind in mpl.toolbar_items) {\n", - " var name = mpl.toolbar_items[toolbar_ind][0];\n", - " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", - " var image = mpl.toolbar_items[toolbar_ind][2];\n", - " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", - "\n", - " if (!name) {\n", - " // put a spacer in here.\n", - " continue;\n", - " }\n", - " var button = $('');\n", - " button.click(method_name, toolbar_event);\n", - " button.mouseover(tooltip, toolbar_mouse_event);\n", - " nav_element.append(button);\n", - " }\n", - "\n", - " // Add the status bar.\n", - " var status_bar = $('');\n", - " nav_element.append(status_bar);\n", - " this.message = status_bar[0];\n", - "\n", - " // Add the close button to the window.\n", - " var buttongrp = $('
');\n", - " var button = $('');\n", - " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", - " buttongrp.append(button);\n", - " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", - " titlebar.prepend(buttongrp);\n", - "}\n", - "\n", - "\n", - "mpl.figure.prototype._canvas_extra_style = function(el){\n", - " // this is important to make the div 'focusable\n", - " el.attr('tabindex', 0)\n", - " // reach out to IPython and tell the keyboard manager to turn it's self\n", - " // off when our div gets focus\n", - "\n", - " // location in version 3\n", - " if (IPython.notebook.keyboard_manager) {\n", - " IPython.notebook.keyboard_manager.register_events(el);\n", - " }\n", - " else {\n", - " // location in version 2\n", - " IPython.keyboard_manager.register_events(el);\n", - " }\n", - "\n", - "}\n", - "\n", - "mpl.figure.prototype._key_event_extra = function(event, name) {\n", - " var manager = IPython.notebook.keyboard_manager;\n", - " if (!manager)\n", - " manager = IPython.keyboard_manager;\n", - "\n", - " // Check for shift+enter\n", - " if (event.shiftKey && event.which == 13) {\n", - " this.canvas_div.blur();\n", - " event.shiftKey = false;\n", - " // Send a \"J\" for go to next cell\n", - " event.which = 74;\n", - " event.keyCode = 74;\n", - " manager.command_mode();\n", - " manager.handle_keydown(event);\n", - " }\n", - "}\n", - "\n", - "mpl.figure.prototype.handle_save = function(fig, msg) {\n", - " fig.ondownload(fig, null);\n", - "}\n", - "\n", - "\n", - "mpl.find_output_cell = function(html_output) {\n", - " // Return the cell and output element which can be found *uniquely* in the notebook.\n", - " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", - " // IPython event is triggered only after the cells have been serialised, which for\n", - " // our purposes (turning an active figure into a static one), is too late.\n", - " var cells = IPython.notebook.get_cells();\n", - " var ncells = cells.length;\n", - " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", - " data = data.data;\n", - " }\n", - " if (data['text/html'] == html_output) {\n", - " return [cell, data, j];\n", - " }\n", - " }\n", - " }\n", - " }\n", - "}\n", - "\n", - "// Register the function which deals with the matplotlib target/channel.\n", - "// The kernel may be null if the page has been refreshed.\n", - "if (IPython.notebook.kernel != null) {\n", - " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", - "}\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/html": [ - "" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "#Euler\n", "q1_dp = numpy.zeros((N,4)) \n", @@ -1872,7 +387,7 @@ }, { "cell_type": "code", - "execution_count": 327, + "execution_count": null, "metadata": { "collapsed": false }, @@ -1905,767 +420,11 @@ }, { "cell_type": "code", - "execution_count": 328, + "execution_count": null, "metadata": { "collapsed": false }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[ 1. 0. 3.14159265 0. ]\n" - ] - }, - { - "data": { - "application/javascript": [ - "/* Put everything inside the global mpl namespace */\n", - "window.mpl = {};\n", - "\n", - "mpl.get_websocket_type = function() {\n", - " if (typeof(WebSocket) !== 'undefined') {\n", - " return WebSocket;\n", - " } else if (typeof(MozWebSocket) !== 'undefined') {\n", - " return MozWebSocket;\n", - " } else {\n", - " alert('Your browser does not have WebSocket support.' +\n", - " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", - " 'Firefox 4 and 5 are also supported but you ' +\n", - " 'have to enable WebSockets in about:config.');\n", - " };\n", - "}\n", - "\n", - "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", - " this.id = figure_id;\n", - "\n", - " this.ws = websocket;\n", - "\n", - " this.supports_binary = (this.ws.binaryType != undefined);\n", - "\n", - " if (!this.supports_binary) {\n", - " var warnings = document.getElementById(\"mpl-warnings\");\n", - " if (warnings) {\n", - " warnings.style.display = 'block';\n", - " warnings.textContent = (\n", - " \"This browser does not support binary websocket messages. \" +\n", - " \"Performance may be slow.\");\n", - " }\n", - " }\n", - "\n", - " this.imageObj = new Image();\n", - "\n", - " this.context = undefined;\n", - " this.message = undefined;\n", - " this.canvas = undefined;\n", - " this.rubberband_canvas = undefined;\n", - " this.rubberband_context = undefined;\n", - " this.format_dropdown = undefined;\n", - "\n", - " this.image_mode = 'full';\n", - "\n", - " this.root = $('
');\n", - " this._root_extra_style(this.root)\n", - " this.root.attr('style', 'display: inline-block');\n", - "\n", - " $(parent_element).append(this.root);\n", - "\n", - " this._init_header(this);\n", - " this._init_canvas(this);\n", - " this._init_toolbar(this);\n", - "\n", - " var fig = this;\n", - "\n", - " this.waiting = false;\n", - "\n", - " this.ws.onopen = function () {\n", - " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", - " fig.send_message(\"send_image_mode\", {});\n", - " fig.send_message(\"refresh\", {});\n", - " }\n", - "\n", - " this.imageObj.onload = function() {\n", - " if (fig.image_mode == 'full') {\n", - " // Full images could contain transparency (where diff images\n", - " // almost always do), so we need to clear the canvas so that\n", - " // there is no ghosting.\n", - " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", - " }\n", - " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", - " };\n", - "\n", - " this.imageObj.onunload = function() {\n", - " this.ws.close();\n", - " }\n", - "\n", - " this.ws.onmessage = this._make_on_message_function(this);\n", - "\n", - " this.ondownload = ondownload;\n", - "}\n", - "\n", - "mpl.figure.prototype._init_header = function() {\n", - " var titlebar = $(\n", - " '
');\n", - " var titletext = $(\n", - " '
');\n", - " titlebar.append(titletext)\n", - " this.root.append(titlebar);\n", - " this.header = titletext[0];\n", - "}\n", - "\n", - "\n", - "\n", - "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", - "\n", - "}\n", - "\n", - "\n", - "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", - "\n", - "}\n", - "\n", - "mpl.figure.prototype._init_canvas = function() {\n", - " var fig = this;\n", - "\n", - " var canvas_div = $('
');\n", - "\n", - " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", - "\n", - " function canvas_keyboard_event(event) {\n", - " return fig.key_event(event, event['data']);\n", - " }\n", - "\n", - " canvas_div.keydown('key_press', canvas_keyboard_event);\n", - " canvas_div.keyup('key_release', canvas_keyboard_event);\n", - " this.canvas_div = canvas_div\n", - " this._canvas_extra_style(canvas_div)\n", - " this.root.append(canvas_div);\n", - "\n", - " var canvas = $('');\n", - " canvas.addClass('mpl-canvas');\n", - " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", - "\n", - " this.canvas = canvas[0];\n", - " this.context = canvas[0].getContext(\"2d\");\n", - "\n", - " var rubberband = $('');\n", - " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", - "\n", - " var pass_mouse_events = true;\n", - "\n", - " canvas_div.resizable({\n", - " start: function(event, ui) {\n", - " pass_mouse_events = false;\n", - " },\n", - " resize: function(event, ui) {\n", - " fig.request_resize(ui.size.width, ui.size.height);\n", - " },\n", - " stop: function(event, ui) {\n", - " pass_mouse_events = true;\n", - " fig.request_resize(ui.size.width, ui.size.height);\n", - " },\n", - " });\n", - "\n", - " function mouse_event_fn(event) {\n", - " if (pass_mouse_events)\n", - " return fig.mouse_event(event, event['data']);\n", - " }\n", - "\n", - " rubberband.mousedown('button_press', mouse_event_fn);\n", - " rubberband.mouseup('button_release', mouse_event_fn);\n", - " // Throttle sequential mouse events to 1 every 20ms.\n", - " rubberband.mousemove('motion_notify', mouse_event_fn);\n", - "\n", - " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", - " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", - "\n", - " canvas_div.on(\"wheel\", function (event) {\n", - " event = event.originalEvent;\n", - " event['data'] = 'scroll'\n", - " if (event.deltaY < 0) {\n", - " event.step = 1;\n", - " } else {\n", - " event.step = -1;\n", - " }\n", - " mouse_event_fn(event);\n", - " });\n", - "\n", - " canvas_div.append(canvas);\n", - " canvas_div.append(rubberband);\n", - "\n", - " this.rubberband = rubberband;\n", - " this.rubberband_canvas = rubberband[0];\n", - " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", - " this.rubberband_context.strokeStyle = \"#000000\";\n", - "\n", - " this._resize_canvas = function(width, height) {\n", - " // Keep the size of the canvas, canvas container, and rubber band\n", - " // canvas in synch.\n", - " canvas_div.css('width', width)\n", - " canvas_div.css('height', height)\n", - "\n", - " canvas.attr('width', width);\n", - " canvas.attr('height', height);\n", - "\n", - " rubberband.attr('width', width);\n", - " rubberband.attr('height', height);\n", - " }\n", - "\n", - " // Set the figure to an initial 600x600px, this will subsequently be updated\n", - " // upon first draw.\n", - " this._resize_canvas(600, 600);\n", - "\n", - " // Disable right mouse context menu.\n", - " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", - " return false;\n", - " });\n", - "\n", - " function set_focus () {\n", - " canvas.focus();\n", - " canvas_div.focus();\n", - " }\n", - "\n", - " window.setTimeout(set_focus, 100);\n", - "}\n", - "\n", - "mpl.figure.prototype._init_toolbar = function() {\n", - " var fig = this;\n", - "\n", - " var nav_element = $('
')\n", - " nav_element.attr('style', 'width: 100%');\n", - " this.root.append(nav_element);\n", - "\n", - " // Define a callback function for later on.\n", - " function toolbar_event(event) {\n", - " return fig.toolbar_button_onclick(event['data']);\n", - " }\n", - " function toolbar_mouse_event(event) {\n", - " return fig.toolbar_button_onmouseover(event['data']);\n", - " }\n", - "\n", - " for(var toolbar_ind in mpl.toolbar_items) {\n", - " var name = mpl.toolbar_items[toolbar_ind][0];\n", - " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", - " var image = mpl.toolbar_items[toolbar_ind][2];\n", - " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", - "\n", - " if (!name) {\n", - " // put a spacer in here.\n", - " continue;\n", - " }\n", - " var button = $('');\n", - " button.click(method_name, toolbar_event);\n", - " button.mouseover(tooltip, toolbar_mouse_event);\n", - " nav_element.append(button);\n", - " }\n", - "\n", - " // Add the status bar.\n", - " var status_bar = $('');\n", - " nav_element.append(status_bar);\n", - " this.message = status_bar[0];\n", - "\n", - " // Add the close button to the window.\n", - " var buttongrp = $('
');\n", - " var button = $('');\n", - " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", - " buttongrp.append(button);\n", - " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", - " titlebar.prepend(buttongrp);\n", - "}\n", - "\n", - "\n", - "mpl.figure.prototype._canvas_extra_style = function(el){\n", - " // this is important to make the div 'focusable\n", - " el.attr('tabindex', 0)\n", - " // reach out to IPython and tell the keyboard manager to turn it's self\n", - " // off when our div gets focus\n", - "\n", - " // location in version 3\n", - " if (IPython.notebook.keyboard_manager) {\n", - " IPython.notebook.keyboard_manager.register_events(el);\n", - " }\n", - " else {\n", - " // location in version 2\n", - " IPython.keyboard_manager.register_events(el);\n", - " }\n", - "\n", - "}\n", - "\n", - "mpl.figure.prototype._key_event_extra = function(event, name) {\n", - " var manager = IPython.notebook.keyboard_manager;\n", - " if (!manager)\n", - " manager = IPython.keyboard_manager;\n", - "\n", - " // Check for shift+enter\n", - " if (event.shiftKey && event.which == 13) {\n", - " this.canvas_div.blur();\n", - " event.shiftKey = false;\n", - " // Send a \"J\" for go to next cell\n", - " event.which = 74;\n", - " event.keyCode = 74;\n", - " manager.command_mode();\n", - " manager.handle_keydown(event);\n", - " }\n", - "}\n", - "\n", - "mpl.figure.prototype.handle_save = function(fig, msg) {\n", - " fig.ondownload(fig, null);\n", - "}\n", - "\n", - "\n", - "mpl.find_output_cell = function(html_output) {\n", - " // Return the cell and output element which can be found *uniquely* in the notebook.\n", - " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", - " // IPython event is triggered only after the cells have been serialised, which for\n", - " // our purposes (turning an active figure into a static one), is too late.\n", - " var cells = IPython.notebook.get_cells();\n", - " var ncells = cells.length;\n", - " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", - " data = data.data;\n", - " }\n", - " if (data['text/html'] == html_output) {\n", - " return [cell, data, j];\n", - " }\n", - " }\n", - " }\n", - " }\n", - "}\n", - "\n", - "// Register the function which deals with the matplotlib target/channel.\n", - "// The kernel may be null if the page has been refreshed.\n", - "if (IPython.notebook.kernel != null) {\n", - " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", - "}\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/html": [ - "" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/javascript": [ - "/* Put everything inside the global mpl namespace */\n", - "window.mpl = {};\n", - "\n", - "mpl.get_websocket_type = function() {\n", - " if (typeof(WebSocket) !== 'undefined') {\n", - " return WebSocket;\n", - " } else if (typeof(MozWebSocket) !== 'undefined') {\n", - " return MozWebSocket;\n", - " } else {\n", - " alert('Your browser does not have WebSocket support.' +\n", - " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", - " 'Firefox 4 and 5 are also supported but you ' +\n", - " 'have to enable WebSockets in about:config.');\n", - " };\n", - "}\n", - "\n", - "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", - " this.id = figure_id;\n", - "\n", - " this.ws = websocket;\n", - "\n", - " this.supports_binary = (this.ws.binaryType != undefined);\n", - "\n", - " if (!this.supports_binary) {\n", - " var warnings = document.getElementById(\"mpl-warnings\");\n", - " if (warnings) {\n", - " warnings.style.display = 'block';\n", - " warnings.textContent = (\n", - " \"This browser does not support binary websocket messages. \" +\n", - " \"Performance may be slow.\");\n", - " }\n", - " }\n", - "\n", - " this.imageObj = new Image();\n", - "\n", - " this.context = undefined;\n", - " this.message = undefined;\n", - " this.canvas = undefined;\n", - " this.rubberband_canvas = undefined;\n", - " this.rubberband_context = undefined;\n", - " this.format_dropdown = undefined;\n", - "\n", - " this.image_mode = 'full';\n", - "\n", - " this.root = $('
');\n", - " this._root_extra_style(this.root)\n", - " this.root.attr('style', 'display: inline-block');\n", - "\n", - " $(parent_element).append(this.root);\n", - "\n", - " this._init_header(this);\n", - " this._init_canvas(this);\n", - " this._init_toolbar(this);\n", - "\n", - " var fig = this;\n", - "\n", - " this.waiting = false;\n", - "\n", - " this.ws.onopen = function () {\n", - " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", - " fig.send_message(\"send_image_mode\", {});\n", - " fig.send_message(\"refresh\", {});\n", - " }\n", - "\n", - " this.imageObj.onload = function() {\n", - " if (fig.image_mode == 'full') {\n", - " // Full images could contain transparency (where diff images\n", - " // almost always do), so we need to clear the canvas so that\n", - " // there is no ghosting.\n", - " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", - " }\n", - " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", - " };\n", - "\n", - " this.imageObj.onunload = function() {\n", - " this.ws.close();\n", - " }\n", - "\n", - " this.ws.onmessage = this._make_on_message_function(this);\n", - "\n", - " this.ondownload = ondownload;\n", - "}\n", - "\n", - "mpl.figure.prototype._init_header = function() {\n", - " var titlebar = $(\n", - " '
');\n", - " var titletext = $(\n", - " '
');\n", - " titlebar.append(titletext)\n", - " this.root.append(titlebar);\n", - " this.header = titletext[0];\n", - "}\n", - "\n", - "\n", - "\n", - "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", - "\n", - "}\n", - "\n", - "\n", - "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", - "\n", - "}\n", - "\n", - "mpl.figure.prototype._init_canvas = function() {\n", - " var fig = this;\n", - "\n", - " var canvas_div = $('
');\n", - "\n", - " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", - "\n", - " function canvas_keyboard_event(event) {\n", - " return fig.key_event(event, event['data']);\n", - " }\n", - "\n", - " canvas_div.keydown('key_press', canvas_keyboard_event);\n", - " canvas_div.keyup('key_release', canvas_keyboard_event);\n", - " this.canvas_div = canvas_div\n", - " this._canvas_extra_style(canvas_div)\n", - " this.root.append(canvas_div);\n", - "\n", - " var canvas = $('');\n", - " canvas.addClass('mpl-canvas');\n", - " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", - "\n", - " this.canvas = canvas[0];\n", - " this.context = canvas[0].getContext(\"2d\");\n", - "\n", - " var rubberband = $('');\n", - " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", - "\n", - " var pass_mouse_events = true;\n", - "\n", - " canvas_div.resizable({\n", - " start: function(event, ui) {\n", - " pass_mouse_events = false;\n", - " },\n", - " resize: function(event, ui) {\n", - " fig.request_resize(ui.size.width, ui.size.height);\n", - " },\n", - " stop: function(event, ui) {\n", - " pass_mouse_events = true;\n", - " fig.request_resize(ui.size.width, ui.size.height);\n", - " },\n", - " });\n", - "\n", - " function mouse_event_fn(event) {\n", - " if (pass_mouse_events)\n", - " return fig.mouse_event(event, event['data']);\n", - " }\n", - "\n", - " rubberband.mousedown('button_press', mouse_event_fn);\n", - " rubberband.mouseup('button_release', mouse_event_fn);\n", - " // Throttle sequential mouse events to 1 every 20ms.\n", - " rubberband.mousemove('motion_notify', mouse_event_fn);\n", - "\n", - " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", - " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", - "\n", - " canvas_div.on(\"wheel\", function (event) {\n", - " event = event.originalEvent;\n", - " event['data'] = 'scroll'\n", - " if (event.deltaY < 0) {\n", - " event.step = 1;\n", - " } else {\n", - " event.step = -1;\n", - " }\n", - " mouse_event_fn(event);\n", - " });\n", - "\n", - " canvas_div.append(canvas);\n", - " canvas_div.append(rubberband);\n", - "\n", - " this.rubberband = rubberband;\n", - " this.rubberband_canvas = rubberband[0];\n", - " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", - " this.rubberband_context.strokeStyle = \"#000000\";\n", - "\n", - " this._resize_canvas = function(width, height) {\n", - " // Keep the size of the canvas, canvas container, and rubber band\n", - " // canvas in synch.\n", - " canvas_div.css('width', width)\n", - " canvas_div.css('height', height)\n", - "\n", - " canvas.attr('width', width);\n", - " canvas.attr('height', height);\n", - "\n", - " rubberband.attr('width', width);\n", - " rubberband.attr('height', height);\n", - " }\n", - "\n", - " // Set the figure to an initial 600x600px, this will subsequently be updated\n", - " // upon first draw.\n", - " this._resize_canvas(600, 600);\n", - "\n", - " // Disable right mouse context menu.\n", - " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", - " return false;\n", - " });\n", - "\n", - " function set_focus () {\n", - " canvas.focus();\n", - " canvas_div.focus();\n", - " }\n", - "\n", - " window.setTimeout(set_focus, 100);\n", - "}\n", - "\n", - "mpl.figure.prototype._init_toolbar = function() {\n", - " var fig = this;\n", - "\n", - " var nav_element = $('
')\n", - " nav_element.attr('style', 'width: 100%');\n", - " this.root.append(nav_element);\n", - "\n", - " // Define a callback function for later on.\n", - " function toolbar_event(event) {\n", - " return fig.toolbar_button_onclick(event['data']);\n", - " }\n", - " function toolbar_mouse_event(event) {\n", - " return fig.toolbar_button_onmouseover(event['data']);\n", - " }\n", - "\n", - " for(var toolbar_ind in mpl.toolbar_items) {\n", - " var name = mpl.toolbar_items[toolbar_ind][0];\n", - " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", - " var image = mpl.toolbar_items[toolbar_ind][2];\n", - " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", - "\n", - " if (!name) {\n", - " // put a spacer in here.\n", - " continue;\n", - " }\n", - " var button = $('');\n", - " button.click(method_name, toolbar_event);\n", - " button.mouseover(tooltip, toolbar_mouse_event);\n", - " nav_element.append(button);\n", - " }\n", - "\n", - " // Add the status bar.\n", - " var status_bar = $('');\n", - " nav_element.append(status_bar);\n", - " this.message = status_bar[0];\n", - "\n", - " // Add the close button to the window.\n", - " var buttongrp = $('
');\n", - " var button = $('');\n", - " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", - " buttongrp.append(button);\n", - " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", - " titlebar.prepend(buttongrp);\n", - "}\n", - "\n", - "\n", - "mpl.figure.prototype._canvas_extra_style = function(el){\n", - " // this is important to make the div 'focusable\n", - " el.attr('tabindex', 0)\n", - " // reach out to IPython and tell the keyboard manager to turn it's self\n", - " // off when our div gets focus\n", - "\n", - " // location in version 3\n", - " if (IPython.notebook.keyboard_manager) {\n", - " IPython.notebook.keyboard_manager.register_events(el);\n", - " }\n", - " else {\n", - " // location in version 2\n", - " IPython.keyboard_manager.register_events(el);\n", - " }\n", - "\n", - "}\n", - "\n", - "mpl.figure.prototype._key_event_extra = function(event, name) {\n", - " var manager = IPython.notebook.keyboard_manager;\n", - " if (!manager)\n", - " manager = IPython.keyboard_manager;\n", - "\n", - " // Check for shift+enter\n", - " if (event.shiftKey && event.which == 13) {\n", - " this.canvas_div.blur();\n", - " event.shiftKey = false;\n", - " // Send a \"J\" for go to next cell\n", - " event.which = 74;\n", - " event.keyCode = 74;\n", - " manager.command_mode();\n", - " manager.handle_keydown(event);\n", - " }\n", - "}\n", - "\n", - "mpl.figure.prototype.handle_save = function(fig, msg) {\n", - " fig.ondownload(fig, null);\n", - "}\n", - "\n", - "\n", - "mpl.find_output_cell = function(html_output) {\n", - " // Return the cell and output element which can be found *uniquely* in the notebook.\n", - " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", - " // IPython event is triggered only after the cells have been serialised, which for\n", - " // our purposes (turning an active figure into a static one), is too late.\n", - " var cells = IPython.notebook.get_cells();\n", - " var ncells = cells.length;\n", - " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", - " data = data.data;\n", - " }\n", - " if (data['text/html'] == html_output) {\n", - " return [cell, data, j];\n", - " }\n", - " }\n", - " }\n", - " }\n", - "}\n", - "\n", - "// Register the function which deals with the matplotlib target/channel.\n", - "// The kernel may be null if the page has been refreshed.\n", - "if (IPython.notebook.kernel != null) {\n", - " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", - "}\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/html": [ - "" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/plain": [ - "[]" - ] - }, - "execution_count": 296, - "metadata": {}, - "output_type": "execute_result" - } - ], + "outputs": [], "source": [ "v = poincare(q2d_dp, 0, 0, 2, 3) #use states from RK4 integration\n", "\n", @@ -5178,21 +681,21 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 2", + "display_name": "Python 3", "language": "python", - "name": "python2" + "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", - "version": 2 + "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", - "pygments_lexer": "ipython2", - "version": "2.7.10" + "pygments_lexer": "ipython3", + "version": "3.5.0" } }, "nbformat": 4, diff --git a/randy_schur/Integration Techniques for Mechanical Systems.ipynb b/randy_schur/Integration Techniques for Mechanical Systems.ipynb index f4e1e02..5997a83 100644 --- a/randy_schur/Integration Techniques for Mechanical Systems.ipynb +++ b/randy_schur/Integration Techniques for Mechanical Systems.ipynb @@ -218,7 +218,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 22, "metadata": { "collapsed": false }, @@ -239,7 +239,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 23, "metadata": { "collapsed": true }, @@ -259,7 +259,7 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 24, "metadata": { "collapsed": true }, @@ -282,7 +282,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 25, "metadata": { "collapsed": true }, @@ -308,7 +308,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 26, "metadata": { "collapsed": true }, @@ -332,7 +332,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 27, "metadata": { "collapsed": true }, @@ -360,7 +360,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 28, "metadata": { "collapsed": true }, @@ -379,7 +379,7 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 29, "metadata": { "collapsed": false }, @@ -459,7 +459,6 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -790,11 +789,13 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -843,6 +844,19 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", + "/*\n", + " * return a copy of an object with only non-object keys\n", + " * we need this to avoid circular references\n", + " * http://stackoverflow.com/a/24161582/3208463\n", + " */\n", + "function simpleKeys (original) {\n", + " return Object.keys(original).reduce(function (obj, key) {\n", + " if (typeof original[key] !== 'object')\n", + " obj[key] = original[key]\n", + " return obj;\n", + " }, {});\n", + "}\n", + "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -856,7 +870,8 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step});\n", + " step: event.step,\n", + " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -896,7 +911,8 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value});\n", + " this.send_message(name, {key: value,\n", + " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", @@ -913,7 +929,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -972,6 +988,8 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", + " fig.root.unbind('remove')\n", + "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -979,8 +997,12 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.send_message('closing', {});\n", - " fig.ws.close()\n", + " fig.close_ws(fig, msg);\n", + "}\n", + "\n", + "mpl.figure.prototype.close_ws = function(fig, msg){\n", + " fig.send_message('closing', msg);\n", + " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -1035,14 +1057,20 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -1123,7 +1151,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1185,7 +1213,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 30, "metadata": { "collapsed": true }, @@ -1208,7 +1236,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 31, "metadata": { "collapsed": true }, @@ -1229,7 +1257,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 32, "metadata": { "collapsed": false }, @@ -1309,7 +1337,6 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -1640,11 +1667,13 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -1693,6 +1722,19 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", + "/*\n", + " * return a copy of an object with only non-object keys\n", + " * we need this to avoid circular references\n", + " * http://stackoverflow.com/a/24161582/3208463\n", + " */\n", + "function simpleKeys (original) {\n", + " return Object.keys(original).reduce(function (obj, key) {\n", + " if (typeof original[key] !== 'object')\n", + " obj[key] = original[key]\n", + " return obj;\n", + " }, {});\n", + "}\n", + "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -1706,7 +1748,8 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step});\n", + " step: event.step,\n", + " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -1746,7 +1789,8 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value});\n", + " this.send_message(name, {key: value,\n", + " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", @@ -1763,7 +1807,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -1822,6 +1866,8 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", + " fig.root.unbind('remove')\n", + "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -1829,8 +1875,12 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.send_message('closing', {});\n", - " fig.ws.close()\n", + " fig.close_ws(fig, msg);\n", + "}\n", + "\n", + "mpl.figure.prototype.close_ws = function(fig, msg){\n", + " fig.send_message('closing', msg);\n", + " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -1885,14 +1935,20 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -1973,7 +2029,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2007,7 +2063,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 12, "metadata": { "collapsed": true }, @@ -2030,7 +2086,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 13, "metadata": { "collapsed": true }, @@ -2065,7 +2121,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 14, "metadata": { "collapsed": false }, @@ -2145,7 +2201,6 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -2476,11 +2531,13 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -2529,6 +2586,19 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", + "/*\n", + " * return a copy of an object with only non-object keys\n", + " * we need this to avoid circular references\n", + " * http://stackoverflow.com/a/24161582/3208463\n", + " */\n", + "function simpleKeys (original) {\n", + " return Object.keys(original).reduce(function (obj, key) {\n", + " if (typeof original[key] !== 'object')\n", + " obj[key] = original[key]\n", + " return obj;\n", + " }, {});\n", + "}\n", + "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -2542,7 +2612,8 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step});\n", + " step: event.step,\n", + " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -2582,7 +2653,8 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value});\n", + " this.send_message(name, {key: value,\n", + " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", @@ -2599,7 +2671,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -2658,6 +2730,8 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", + " fig.root.unbind('remove')\n", + "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -2665,8 +2739,12 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.send_message('closing', {});\n", - " fig.ws.close()\n", + " fig.close_ws(fig, msg);\n", + "}\n", + "\n", + "mpl.figure.prototype.close_ws = function(fig, msg){\n", + " fig.send_message('closing', msg);\n", + " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -2721,14 +2799,20 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -2809,7 +2893,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2887,7 +2971,7 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 15, "metadata": { "collapsed": true }, @@ -2911,7 +2995,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 16, "metadata": { "collapsed": true }, @@ -2940,7 +3024,7 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 17, "metadata": { "collapsed": true }, @@ -2971,7 +3055,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 18, "metadata": { "collapsed": false }, @@ -3051,7 +3135,6 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -3382,11 +3465,13 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -3435,6 +3520,19 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", + "/*\n", + " * return a copy of an object with only non-object keys\n", + " * we need this to avoid circular references\n", + " * http://stackoverflow.com/a/24161582/3208463\n", + " */\n", + "function simpleKeys (original) {\n", + " return Object.keys(original).reduce(function (obj, key) {\n", + " if (typeof original[key] !== 'object')\n", + " obj[key] = original[key]\n", + " return obj;\n", + " }, {});\n", + "}\n", + "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -3448,7 +3546,8 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step});\n", + " step: event.step,\n", + " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -3488,7 +3587,8 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value});\n", + " this.send_message(name, {key: value,\n", + " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", @@ -3505,7 +3605,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -3564,6 +3664,8 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", + " fig.root.unbind('remove')\n", + "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -3571,8 +3673,12 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.send_message('closing', {});\n", - " fig.ws.close()\n", + " fig.close_ws(fig, msg);\n", + "}\n", + "\n", + "mpl.figure.prototype.close_ws = function(fig, msg){\n", + " fig.send_message('closing', msg);\n", + " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -3627,14 +3733,20 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -3715,7 +3827,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -3775,7 +3887,7 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 19, "metadata": { "collapsed": false }, @@ -3855,7 +3967,6 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -4186,11 +4297,13 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -4239,6 +4352,19 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", + "/*\n", + " * return a copy of an object with only non-object keys\n", + " * we need this to avoid circular references\n", + " * http://stackoverflow.com/a/24161582/3208463\n", + " */\n", + "function simpleKeys (original) {\n", + " return Object.keys(original).reduce(function (obj, key) {\n", + " if (typeof original[key] !== 'object')\n", + " obj[key] = original[key]\n", + " return obj;\n", + " }, {});\n", + "}\n", + "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -4252,7 +4378,8 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step});\n", + " step: event.step,\n", + " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -4292,7 +4419,8 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value});\n", + " this.send_message(name, {key: value,\n", + " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", @@ -4309,7 +4437,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -4368,6 +4496,8 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", + " fig.root.unbind('remove')\n", + "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -4375,8 +4505,12 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.send_message('closing', {});\n", - " fig.ws.close()\n", + " fig.close_ws(fig, msg);\n", + "}\n", + "\n", + "mpl.figure.prototype.close_ws = function(fig, msg){\n", + " fig.send_message('closing', msg);\n", + " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -4431,14 +4565,20 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -4519,7 +4659,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -4582,7 +4722,7 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 20, "metadata": { "collapsed": false }, @@ -4662,7 +4802,6 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -4993,11 +5132,13 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -5046,6 +5187,19 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", + "/*\n", + " * return a copy of an object with only non-object keys\n", + " * we need this to avoid circular references\n", + " * http://stackoverflow.com/a/24161582/3208463\n", + " */\n", + "function simpleKeys (original) {\n", + " return Object.keys(original).reduce(function (obj, key) {\n", + " if (typeof original[key] !== 'object')\n", + " obj[key] = original[key]\n", + " return obj;\n", + " }, {});\n", + "}\n", + "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -5059,7 +5213,8 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step});\n", + " step: event.step,\n", + " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -5099,7 +5254,8 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value});\n", + " this.send_message(name, {key: value,\n", + " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", @@ -5116,7 +5272,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -5175,6 +5331,8 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", + " fig.root.unbind('remove')\n", + "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -5182,8 +5340,12 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.send_message('closing', {});\n", - " fig.ws.close()\n", + " fig.close_ws(fig, msg);\n", + "}\n", + "\n", + "mpl.figure.prototype.close_ws = function(fig, msg){\n", + " fig.send_message('closing', msg);\n", + " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -5238,14 +5400,20 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -5326,7 +5494,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -5369,17 +5537,17 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "ename": "SyntaxError", - "evalue": "invalid syntax (, line 1)", + "evalue": "invalid syntax (, line 1)", "output_type": "error", "traceback": [ - "\u001b[1;36m File \u001b[1;32m\"\"\u001b[1;36m, line \u001b[1;32m1\u001b[0m\n\u001b[1;33m Sources:\u001b[0m\n\u001b[1;37m ^\u001b[0m\n\u001b[1;31mSyntaxError\u001b[0m\u001b[1;31m:\u001b[0m invalid syntax\n" + "\u001b[1;36m File \u001b[1;32m\"\"\u001b[1;36m, line \u001b[1;32m1\u001b[0m\n\u001b[1;33m Sources:\u001b[0m\n\u001b[1;37m ^\u001b[0m\n\u001b[1;31mSyntaxError\u001b[0m\u001b[1;31m:\u001b[0m invalid syntax\n" ] } ], @@ -5404,166 +5572,11 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": null, "metadata": { "collapsed": false }, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "execution_count": 27, - "metadata": {}, - "output_type": "execute_result" - } - ], + "outputs": [], "source": [ "# This cell loads the style of the notebook, which is modified from the \n", "# Numerical Methods in Python Course: http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about\n", @@ -5594,21 +5607,21 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 2", + "display_name": "Python 3", "language": "python", - "name": "python2" + "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", - "version": 2 + "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", - "pygments_lexer": "ipython2", - "version": "2.7.10" + "pygments_lexer": "ipython3", + "version": "3.5.0" } }, "nbformat": 4, diff --git a/randy_schur/figures/Error_64bit.png b/randy_schur/figures/Error_64bit.png new file mode 100644 index 0000000000000000000000000000000000000000..b4cae05a13a6e3f228b6bf89dad15aa2b901e2ea GIT binary patch literal 24164 zcma&O2Rzn&`#ye236YWPl2wvXcFI;}2pQQ4A$!kD*-1h+q3j(>HrZL(D;OM{T1XS@Np?|Q79C?l%%*K3Uy)wg~AxX zJ^_Du6g-Us-!N@NrIfL;v4?)hkHN2mws+KQl`ITx9rUaXP)bkiY;6s!^`C!(w_ZX? 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!McUxr0zngnJs`V$yS`1{V4^7P{!Km9W^;m3dG$p4sU@JqZs7}5{7DyYf`LPXPepC7`8@LtR< z+r{BBe}8{J0?)kvWwj&pWx?$N>4o$1xMb+Tg9nZ!!fb37K<<}6?#XB?!iX3Y155l} zi2h&G@qf?a@Rwl6dYbiF@Od9viqp^&WJe}7EiGhiXYP~ea`n09qxe@?s(@4tza(HK zsi>*~I~DZaaPv_Z Date: Mon, 7 Dec 2015 20:04:00 -0500 Subject: [PATCH 34/61] POS computer screenshots --- ...on Techniques for Mechanical Systems.ipynb | 137 ++++++------------ randy_schur/figures/combined_response.png | Bin 0 -> 212034 bytes randy_schur/figures/error_combined.png | Bin 0 -> 94263 bytes 3 files changed, 46 insertions(+), 91 deletions(-) create mode 100644 randy_schur/figures/combined_response.png create mode 100644 randy_schur/figures/error_combined.png diff --git a/randy_schur/Integration Techniques for Mechanical Systems.ipynb b/randy_schur/Integration Techniques for Mechanical Systems.ipynb index 5997a83..f97971e 100644 --- a/randy_schur/Integration Techniques for Mechanical Systems.ipynb +++ b/randy_schur/Integration Techniques for Mechanical Systems.ipynb @@ -218,7 +218,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 2, "metadata": { "collapsed": false }, @@ -239,7 +239,7 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 3, "metadata": { "collapsed": true }, @@ -259,7 +259,7 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 4, "metadata": { "collapsed": true }, @@ -282,7 +282,7 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 5, "metadata": { "collapsed": true }, @@ -308,7 +308,7 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 6, "metadata": { "collapsed": true }, @@ -332,7 +332,7 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 7, "metadata": { "collapsed": true }, @@ -360,7 +360,7 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 8, "metadata": { "collapsed": true }, @@ -379,7 +379,7 @@ }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 9, "metadata": { "collapsed": false }, @@ -459,6 +459,7 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", + " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -789,13 +790,11 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", - " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", - " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -844,19 +843,6 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", - "/*\n", - " * return a copy of an object with only non-object keys\n", - " * we need this to avoid circular references\n", - " * http://stackoverflow.com/a/24161582/3208463\n", - " */\n", - "function simpleKeys (original) {\n", - " return Object.keys(original).reduce(function (obj, key) {\n", - " if (typeof original[key] !== 'object')\n", - " obj[key] = original[key]\n", - " return obj;\n", - " }, {});\n", - "}\n", - "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -870,8 +856,7 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step,\n", - " guiEvent: simpleKeys(event)});\n", + " step: event.step});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -911,8 +896,7 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value,\n", - " guiEvent: simpleKeys(event)});\n", + " this.send_message(name, {key: value});\n", " return false;\n", "}\n", "\n", @@ -929,7 +913,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -988,8 +972,6 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", - " fig.root.unbind('remove')\n", - "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -997,12 +979,8 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.close_ws(fig, msg);\n", - "}\n", - "\n", - "mpl.figure.prototype.close_ws = function(fig, msg){\n", - " fig.send_message('closing', msg);\n", - " // fig.ws.close()\n", + " fig.send_message('closing', {});\n", + " fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -1057,20 +1035,14 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('

');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", + " button.mouseover('Close figure', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", - "mpl.figure.prototype._root_extra_style = function(el){\n", - " var fig = this\n", - " el.on(\"remove\", function(){\n", - "\tfig.close_ws(fig, {});\n", - " });\n", - "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -1151,7 +1123,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1213,7 +1185,7 @@ }, { "cell_type": "code", - "execution_count": 30, + "execution_count": 10, "metadata": { "collapsed": true }, @@ -1236,7 +1208,7 @@ }, { "cell_type": "code", - "execution_count": 31, + "execution_count": 11, "metadata": { "collapsed": true }, @@ -1257,7 +1229,7 @@ }, { "cell_type": "code", - "execution_count": 32, + "execution_count": 12, "metadata": { "collapsed": false }, @@ -1337,6 +1309,7 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", + " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -1667,13 +1640,11 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", - " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", - " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -1722,19 +1693,6 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", - "/*\n", - " * return a copy of an object with only non-object keys\n", - " * we need this to avoid circular references\n", - " * http://stackoverflow.com/a/24161582/3208463\n", - " */\n", - "function simpleKeys (original) {\n", - " return Object.keys(original).reduce(function (obj, key) {\n", - " if (typeof original[key] !== 'object')\n", - " obj[key] = original[key]\n", - " return obj;\n", - " }, {});\n", - "}\n", - "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -1748,8 +1706,7 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step,\n", - " guiEvent: simpleKeys(event)});\n", + " step: event.step});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -1789,8 +1746,7 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value,\n", - " guiEvent: simpleKeys(event)});\n", + " this.send_message(name, {key: value});\n", " return false;\n", "}\n", "\n", @@ -1807,7 +1763,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -1866,8 +1822,6 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", - " fig.root.unbind('remove')\n", - "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -1875,12 +1829,8 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.close_ws(fig, msg);\n", - "}\n", - "\n", - "mpl.figure.prototype.close_ws = function(fig, msg){\n", - " fig.send_message('closing', msg);\n", - " // fig.ws.close()\n", + " fig.send_message('closing', {});\n", + " fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -1935,20 +1885,14 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", + " button.mouseover('Close figure', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", - "mpl.figure.prototype._root_extra_style = function(el){\n", - " var fig = this\n", - " el.on(\"remove\", function(){\n", - "\tfig.close_ws(fig, {});\n", - " });\n", - "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -2029,7 +1973,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2056,7 +2000,18 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Is this what you would expect to see based on the plots of position above? What are the sources of this error? We have the usual suspects: truncation error, discretization error, possible precision errors. But ordinarily we don't see a second order scheme (Verlet) outperforming a fourth order scheme (RK4) by three orders of magnitude. What is going on?\n", + "Is this what you would expect to see based on the plots of position above? What are the sources of this error? We have the usual suspects: truncation error, discretization error, possible precision errors. Don't believe that these are significant? Take a look at the following screenshot, which shows the plots above, built with identical code on two different computers. The responses on the left are from a 64-bit computer, and the responses on the right are from a 32-bit computer. Look especially at the phase shift in the system response. That's a huge difference! Even the precision of the machine can have a significant impact on numerical integration.\n", + "\n", + "![Image](figures/combined_response.png)\n", + "\n", + "![Image](figures/error_combined.png)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Despite the numerical errors, ordinarily we don't see a second order scheme (Verlet) outperforming a fourth order scheme (RK4) by three orders of magnitude. What is going on?\n", "\n", "One thing we notice in the plots of position is that for Euler and Runge-Kutta, the spring is stretched farther and farther each oscillation. This means it has more and more potential energy. With no outside forces adding energy to the system and no friction to remove energy, Conservation of Energy says that value should be constant! Let's take a look at total energy in the system. Since we derived equations of motion from the Lagrangian, this should be easy! All we need to do is calculate kinetic and potential energy at each time step, and we already have the information we need to do this. 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512 +- 2 files changed, 4884 insertions(+), 280 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index b1c0004..2f8e1d9 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -11,7 +11,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "In this section, we will study the dynamics of a double pendulum. Like before, we'll derive the equations of motion from the Lagrangian and the Hamiltonian. We can use the same methods of integration as before, but this time there is no analytical solution to compare to. Finally, we will look at the properties of the system in phase space using phase plots and Poincare sections." + "In this notebook, we will study the dynamics of a double pendulum. Like before, we'll derive the equations of motion from the Lagrangian and the Hamiltonian. We can use the same methods of integration as before, but this time there is no analytical solution to compare to. Finally, we will look at the properties of the system in phase space using Poincare sections." ] }, { @@ -20,7 +20,7 @@ "source": [ "## Background\n", "\n", - "The double pendulum problem is a relatively simple system which can produce surprisingly complex movements. It is a chaotic system, meaning it is unpredictable and small changes in initial conditions lead to large changes in motion. There is no closed form solution to the motion of the two masses. This is a frequent example problem in the study of dynamic systems, nonlinear controls, and mechanics.\n", + "The double pendulum problem is a relatively simple system which can produce surprisingly complex motion. It is a chaotic system, meaning it is unpredictable and small changes in initial conditions lead to large changes in motion. There is no closed form solution to the motion of the two masses. This is a frequent example problem in the study of dynamic systems, nonlinear controls, and mechanics.\n", "\n", "In this notebook we will be treating the two rods as massless and we'll start out ignoring the effects of friction. For a treatment of this problem without these assumptions, see (cite).\n", "\n", @@ -80,7 +80,7 @@ "\\end{equation*}$$\n", "\n", "\n", - "There are two generalized coordinates ($\\theta_1 and \\theta_2$. So the equation of motion for $\\theta_1$ is calculated by the following steps:\n", + "There are two generalized coordinates ($\\theta_1$ and $\\theta_2$). So the equation of motion for $\\theta_1$ is calculated by the following steps:\n", "\n", "$$\\begin{eqnarray*}\n", "\\frac{\\partial L}{\\partial \\dot{\\theta_1}} &=& (m_1+m_2)l_1^2\\dot\\theta_1+m_2l_1l_2\\dot\\theta_2cos(\\theta_2-\\theta_1) \\\\\n", @@ -107,13 +107,13 @@ "source": [ "## Problem setup\n", "\n", - "Notice that these equations of motion are implicit equations for $\\ddot\\theta_1$ and $\\ddot\\theta_2$. We have talked about implicit equations in this class, and we solve them by forming a system of equations. In this case, we will need to solve the equations of motion at each time step. We will characterize our system in state space using the angle and angular velocity of each pendulum. Let's rewrite the equations of motion in a more readable form. We can use the standard form of this equation (which you'll be familiar with if you have studied mechanics) by separating out mass terms (anything multiplied by $\\ddot{x}$), centripetal terms (anything multiplied by $\\dot{x}$), and gravitational terms (anything multiplied by $x$). We get the following equations of state:\n", + "Notice that these equations of motion are implicit equations for $\\ddot\\theta_1$ and $\\ddot\\theta_2$. Let's rewrite the equations of motion in a more readable form. We can use the standard form of this equation (which you'll be familiar with if you have studied any dynamics) by separating out mass terms (anything multiplied by $\\ddot{x}$), centripetal terms (anything multiplied by $\\dot{x}$), and gravitational terms (anything multiplied by $x$). We get the following equation of motion:\n", "\n", "$$\\begin{eqnarray*}\n", "0 &=&\\textbf{M}\\vec{\\ddot{x}}(t) + \\textbf{V}\\dot{x} + \\textbf{G} \n", "\\end{eqnarray*}$$\n", "\n", - "The equations of motion can be solved for $\\ddot{x}$:\n", + "The equation of motion can be solved for $\\ddot{x}$:\n", "\n", "$$ \\ddot{x} = \\textbf{M}^{-1}\\left[ \\textbf{V}\\dot x + \\textbf{G} \\right]$$\n", "\n", @@ -148,7 +148,7 @@ "and\n", "\n", "$$\\begin{eqnarray*}\n", - "\\vec{x}(t) = \\begin{pmatrix} \\theta_1\\\\ \\theta_2\\\\ \\end{pmatrix}, \\vec{\\dot{x}(t)} = \\begin{pmatrix} \\dot \\theta_1 \\\\ \\dot \\theta_2 \\end{pmatrix}\n", + "\\vec{x}(t) = \\begin{bmatrix} \\theta_1\\\\ \\theta_2\\\\ \\end{bmatrix}, \\vec{\\dot{x}(t)} = \\begin{bmatrix} \\dot \\theta_1 \\\\ \\dot \\theta_2 \\end{bmatrix}\n", "\\end{eqnarray*}$$\n", "\n", "We can now treat this system the same way we handled the simple harmonic motion; break the two second order ODEs into four first-order ODEs, then use Euler or Runge-Kutta on the results." @@ -167,23 +167,22 @@ " &=& -m_1g(l_1cos\\theta_1 - m_2g(l_1cos\\theta_1 + l_2cos\\theta_2)\n", "\\end{eqnarray*}$$\n", "\n", - "Kinetic Energy is calculated as:\n", + "Kinetic Energy for rotational motion calculated as:\n", "\n", "$$\\begin{eqnarray*}\n", - "T &=& \\frac{1}{2}mv_1^2+\\frac{1}{2}mv_2^2 \\\\\n", - "&=& \\frac{q_1^2}{2I_1} + \\frac{q_2^2}{2I_2}\n", + "T &=& \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2+\\frac{1}{2}m_2\\left(2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+l_2^2\\dot\\theta_2^2\\right)\n", "\\end{eqnarray*}$$\n", "\n", - "where $I_i = m_il_i^2$. So our Hamiltonian quantity becomes:\n", + "where $I_i$ is the angular moment of inertia and $\\omega$ is the angular velocity. So our Hamiltonian quantity becomes:\n", "$$\\begin{equation*}\n", "H = \\frac{1}{2}\\left(\\frac{q_1^2}{I_1} + \\frac{q_2^2}{I_2}\\right) - (m_1+m_2)gl_1cos\\theta_1 - m_2gl_2cos\\theta_2\n", "\\end{equation*}$$\n", "\n", "Since we have two degrees of freedom, we will have four equations of motion. These will be the following:\n", "$$\\begin{eqnarray*}\n", - "\\dot{p_1} &=& -\\frac{\\partial H}{\\partial q_1} = \\frac{q_1}{I_1} \\\\\n", + "\\dot{p_1} &=& -\\frac{\\partial H}{\\partial q_1} = -\\frac{q_1}{I_1} \\\\\n", "\\dot{q_1} &=& \\frac{\\partial H}{\\partial p_1} = (m_1+m_2)gl_1sin(p_1) \\\\\n", - "\\dot{p_2} &=& -\\frac{\\partial H}{\\partial q_2} = \\frac{q_2}{I_2} \\\\\n", + "\\dot{p_2} &=& -\\frac{\\partial H}{\\partial q_2} = -\\frac{q_2}{I_2} \\\\\n", "\\dot{q_2} &=& \\frac{\\partial H}{\\partial p_2} = m_2gl_2sin(p_2) \\\\\n", "\\end{eqnarray*}$$\n" ] @@ -203,6 +202,36 @@ "metadata": { "collapsed": false }, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/home/randy/anaconda3/lib/python3.4/site-packages/IPython/kernel/__init__.py:13: ShimWarning: The `IPython.kernel` package has been deprecated. You should import from ipykernel or jupyter_client instead.\n", + " \"You should import from ipykernel or jupyter_client instead.\", ShimWarning)\n" + ] + } + ], + "source": [ + "import numpy\n", + "from scipy.linalg import solve\n", + "from numpy.linalg import det\n", + "from math import pi, cos, sin, sqrt\n", + "\n", + "from matplotlib import pyplot\n", + "from matplotlib.pyplot import quiver\n", + "%matplotlib notebook\n", + "from matplotlib import rcParams, cm\n", + "rcParams['font.family'] = 'serif'\n", + "rcParams['font.size'] = 16" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, "outputs": [], "source": [ "def f_double_pendulum(u):\n", @@ -235,23 +264,11 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 3, "metadata": { "collapsed": false }, - "outputs": [ - { - "ename": "NameError", - "evalue": "name 'numpy' is not defined", - "output_type": "error", - "traceback": [ - "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[1;31mNameError\u001b[0m Traceback (most recent call last)", - "\u001b[1;32m\u001b[0m in \u001b[0;36m\u001b[1;34m()\u001b[0m\n\u001b[0;32m 11\u001b[0m \u001b[0mdt\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m.02\u001b[0m\u001b[1;33m;\u001b[0m \u001b[1;31m#\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 12\u001b[0m \u001b[0mN\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mT\u001b[0m\u001b[1;33m/\u001b[0m\u001b[0mdt\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m+\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 13\u001b[1;33m \u001b[0mt\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnumpy\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mlinspace\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m0.0\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mT\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mN\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 14\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 15\u001b[0m \u001b[1;31m#Initial Conditions\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", - "\u001b[1;31mNameError\u001b[0m: name 'numpy' is not defined" - ] - } - ], + "outputs": [], "source": [ "#Set up parameters:\n", "g = -9.8 #[m/s^2]\n", @@ -285,7 +302,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 4, "metadata": { "collapsed": true }, @@ -310,7 +327,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 5, "metadata": { "collapsed": false }, @@ -336,11 +353,1508 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 6, "metadata": { "collapsed": false }, - "outputs": [], + "outputs": [ + { + "data": { + "application/javascript": [ + "/* Put everything inside the global mpl namespace */\n", + "window.mpl = {};\n", + "\n", + "mpl.get_websocket_type = function() {\n", + " if (typeof(WebSocket) !== 'undefined') {\n", + " return WebSocket;\n", + " } else if (typeof(MozWebSocket) !== 'undefined') {\n", + " return MozWebSocket;\n", + " } else {\n", + " alert('Your browser does not have WebSocket support.' +\n", + " 'Please try Chrome, Safari or Firefox ≥ 6. 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')\n", + " nav_element.attr('style', 'width: 100%');\n", + " this.root.append(nav_element);\n", + "\n", + " // Define a callback function for later on.\n", + " function toolbar_event(event) {\n", + " return fig.toolbar_button_onclick(event['data']);\n", + " }\n", + " function toolbar_mouse_event(event) {\n", + " return fig.toolbar_button_onmouseover(event['data']);\n", + " }\n", + "\n", + " for(var toolbar_ind in mpl.toolbar_items) {\n", + " var name = mpl.toolbar_items[toolbar_ind][0];\n", + " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", + " var image = mpl.toolbar_items[toolbar_ind][2];\n", + " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", + "\n", + " if (!name) {\n", + " // put a spacer in here.\n", + " continue;\n", + " }\n", + " var button = $('');\n", + " button.click(method_name, toolbar_event);\n", + " button.mouseover(tooltip, toolbar_mouse_event);\n", + " nav_element.append(button);\n", + " }\n", + "\n", + " // Add the status bar.\n", + " var status_bar = $('');\n", + " nav_element.append(status_bar);\n", + " this.message = status_bar[0];\n", + "\n", + " // Add the close button to the window.\n", + " var buttongrp = $('
');\n", + " var button = $('');\n", + " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", + " button.mouseover('Close figure', toolbar_mouse_event);\n", + " buttongrp.append(button);\n", + " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", + " titlebar.prepend(buttongrp);\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(el){\n", + " // this is important to make the div 'focusable\n", + " el.attr('tabindex', 0)\n", + " // reach out to IPython and tell the keyboard manager to turn it's self\n", + " // off when our div gets focus\n", + "\n", + " // location in version 3\n", + " if (IPython.notebook.keyboard_manager) {\n", + " IPython.notebook.keyboard_manager.register_events(el);\n", + " }\n", + " else {\n", + " // location in version 2\n", + " IPython.keyboard_manager.register_events(el);\n", + " }\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._key_event_extra = function(event, name) {\n", + " var manager = IPython.notebook.keyboard_manager;\n", + " if (!manager)\n", + " manager = IPython.keyboard_manager;\n", + "\n", + " // Check for shift+enter\n", + " if (event.shiftKey && event.which == 13) {\n", + " this.canvas_div.blur();\n", + " event.shiftKey = false;\n", + " // Send a \"J\" for go to next cell\n", + " event.which = 74;\n", + " event.keyCode = 74;\n", + " manager.command_mode();\n", + " manager.handle_keydown(event);\n", + " }\n", + "}\n", + "\n", + "mpl.figure.prototype.handle_save = function(fig, msg) {\n", + " fig.ondownload(fig, null);\n", + "}\n", + "\n", + "\n", + "mpl.find_output_cell = function(html_output) {\n", + " // Return the cell and output element which can be found *uniquely* in the notebook.\n", + " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", + " // IPython event is triggered only after the cells have been serialised, which for\n", + " // our purposes (turning an active figure into a static one), is too late.\n", + " var cells = IPython.notebook.get_cells();\n", + " var ncells = cells.length;\n", + " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", + " data = data.data;\n", + " }\n", + " if (data['text/html'] == html_output) {\n", + " return [cell, data, j];\n", + " }\n", + " }\n", + " }\n", + " }\n", + "}\n", + "\n", + "// Register the function which deals with the matplotlib target/channel.\n", + "// The kernel may be null if the page has been refreshed.\n", + "if (IPython.notebook.kernel != null) {\n", + " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", + "}\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "
" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "#Euler\n", "q1_dp = numpy.zeros((N,4)) \n", @@ -387,7 +1901,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 7, "metadata": { "collapsed": false }, @@ -407,24 +1921,780 @@ " p2 = u[2]\n", " q2 = u[3]\n", " \n", - " p1_half = p1 - dt/2*(q1/I1)\n", - " q1 = q1 + dt*((m1+m2)*g*l1*sin(p1_half))\n", - " p1 = p1_half - dt/2*(q1/I1)\n", + " p1_half = p1 + dt/2*(q1/I1)\n", + " q1 = q1 - dt*((m1+m2)*g*l1*sin(p1_half))\n", + " p1 = p1_half + dt/2*(q1/I1)\n", " \n", - " p2_half = p2 - dt/2*(q2/I2)\n", - " q2 = q2 + dt*m2*g*l2*sin(p2_half)\n", - " p2 = p2_half - dt/2*(q2/I2)\n", + " p2_half = p2 + dt/2*(q2/I2)\n", + " q2 = q2 - dt*m2*g*l2*sin(p2_half)\n", + " p2 = p2_half + dt/2*(q2/I2)\n", " \n", " return numpy.array([p1, q1, p2, q2]) " ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 8, "metadata": { "collapsed": false }, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[ 1. 0. 3.14159265 0. ]\n" + ] + }, + { + "data": { + "application/javascript": [ + "/* Put everything inside the global mpl namespace */\n", + "window.mpl = {};\n", + "\n", + "mpl.get_websocket_type = function() {\n", + " if (typeof(WebSocket) !== 'undefined') {\n", + " return WebSocket;\n", + " } else if (typeof(MozWebSocket) !== 'undefined') {\n", + " return MozWebSocket;\n", + " } else {\n", + " alert('Your browser does not have WebSocket support.' +\n", + " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", + " 'Firefox 4 and 5 are also supported but you ' +\n", + " 'have to enable WebSockets in about:config.');\n", + " };\n", + "}\n", + "\n", + "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", + " this.id = figure_id;\n", + "\n", + " this.ws = websocket;\n", + "\n", + " this.supports_binary = (this.ws.binaryType != undefined);\n", + "\n", + " if (!this.supports_binary) {\n", + " var warnings = document.getElementById(\"mpl-warnings\");\n", + " if (warnings) {\n", + " warnings.style.display = 'block';\n", + " warnings.textContent = (\n", + " \"This browser does not support binary websocket messages. \" +\n", + " \"Performance may be slow.\");\n", + " }\n", + " }\n", + "\n", + " this.imageObj = new Image();\n", + "\n", + " this.context = undefined;\n", + " this.message = undefined;\n", + " this.canvas = undefined;\n", + " this.rubberband_canvas = undefined;\n", + " this.rubberband_context = undefined;\n", + " this.format_dropdown = undefined;\n", + "\n", + " this.image_mode = 'full';\n", + "\n", + " this.root = $('
');\n", + " this._root_extra_style(this.root)\n", + " this.root.attr('style', 'display: inline-block');\n", + "\n", + " $(parent_element).append(this.root);\n", + "\n", + " this._init_header(this);\n", + " this._init_canvas(this);\n", + " this._init_toolbar(this);\n", + "\n", + " var fig = this;\n", + "\n", + " this.waiting = false;\n", + "\n", + " this.ws.onopen = function () {\n", + " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", + " fig.send_message(\"send_image_mode\", {});\n", + " fig.send_message(\"refresh\", {});\n", + " }\n", + "\n", + " this.imageObj.onload = function() {\n", + " if (fig.image_mode == 'full') {\n", + " // Full images could contain transparency (where diff images\n", + " // almost always do), so we need to clear the canvas so that\n", + " // there is no ghosting.\n", + " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", + " }\n", + " fig.context.drawImage(fig.imageObj, 0, 0);\n", + " fig.waiting = false;\n", + " };\n", + "\n", + " this.imageObj.onunload = function() {\n", + " this.ws.close();\n", + " }\n", + "\n", + " this.ws.onmessage = this._make_on_message_function(this);\n", + "\n", + " this.ondownload = ondownload;\n", + "}\n", + "\n", + "mpl.figure.prototype._init_header = function() {\n", + " var titlebar = $(\n", + " '
');\n", + " var titletext = $(\n", + " '
');\n", + " titlebar.append(titletext)\n", + " this.root.append(titlebar);\n", + " this.header = titletext[0];\n", + "}\n", + "\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._init_canvas = function() {\n", + " var fig = this;\n", + "\n", + " var canvas_div = $('
');\n", + "\n", + " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", + "\n", + " function canvas_keyboard_event(event) {\n", + " return fig.key_event(event, event['data']);\n", + " }\n", + "\n", + " canvas_div.keydown('key_press', canvas_keyboard_event);\n", + " canvas_div.keyup('key_release', canvas_keyboard_event);\n", + " this.canvas_div = canvas_div\n", + " this._canvas_extra_style(canvas_div)\n", + " this.root.append(canvas_div);\n", + "\n", + " var canvas = $('');\n", + " canvas.addClass('mpl-canvas');\n", + " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", + "\n", + " this.canvas = canvas[0];\n", + " this.context = canvas[0].getContext(\"2d\");\n", + "\n", + " var rubberband = $('');\n", + " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", + "\n", + " var pass_mouse_events = true;\n", + "\n", + " canvas_div.resizable({\n", + " start: function(event, ui) {\n", + " pass_mouse_events = false;\n", + " },\n", + " resize: function(event, ui) {\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " stop: function(event, ui) {\n", + " pass_mouse_events = true;\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " });\n", + "\n", + " function mouse_event_fn(event) {\n", + " if (pass_mouse_events)\n", + " return fig.mouse_event(event, event['data']);\n", + " }\n", + "\n", + " rubberband.mousedown('button_press', mouse_event_fn);\n", + " rubberband.mouseup('button_release', mouse_event_fn);\n", + " // Throttle sequential mouse events to 1 every 20ms.\n", + " rubberband.mousemove('motion_notify', mouse_event_fn);\n", + "\n", + " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", + " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", + "\n", + " canvas_div.on(\"wheel\", function (event) {\n", + " event = event.originalEvent;\n", + " event['data'] = 'scroll'\n", + " if (event.deltaY < 0) {\n", + " event.step = 1;\n", + " } else {\n", + " event.step = -1;\n", + " }\n", + " mouse_event_fn(event);\n", + " });\n", + "\n", + " canvas_div.append(canvas);\n", + " canvas_div.append(rubberband);\n", + "\n", + " this.rubberband = rubberband;\n", + " this.rubberband_canvas = rubberband[0];\n", + " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", + " this.rubberband_context.strokeStyle = \"#000000\";\n", + "\n", + " this._resize_canvas = function(width, height) {\n", + " // Keep the size of the canvas, canvas container, and rubber band\n", + " // canvas in synch.\n", + " canvas_div.css('width', width)\n", + " canvas_div.css('height', height)\n", + "\n", + " canvas.attr('width', width);\n", + " canvas.attr('height', height);\n", + "\n", + " rubberband.attr('width', width);\n", + " rubberband.attr('height', height);\n", + " }\n", + "\n", + " // Set the figure to an initial 600x600px, this will subsequently be updated\n", + " // upon first draw.\n", + " this._resize_canvas(600, 600);\n", + "\n", + " // Disable right mouse context menu.\n", + " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", + " return false;\n", + " });\n", + "\n", + " function set_focus () {\n", + " canvas.focus();\n", + " canvas_div.focus();\n", + " }\n", + "\n", + " window.setTimeout(set_focus, 100);\n", + "}\n", + "\n", + "mpl.figure.prototype._init_toolbar = function() {\n", + " var fig = this;\n", + "\n", + " var nav_element = $('
')\n", + " nav_element.attr('style', 'width: 100%');\n", + " this.root.append(nav_element);\n", + "\n", + " // Define a callback function for later on.\n", + " function toolbar_event(event) {\n", + " return fig.toolbar_button_onclick(event['data']);\n", + " }\n", + " function toolbar_mouse_event(event) {\n", + " return fig.toolbar_button_onmouseover(event['data']);\n", + " }\n", + "\n", + " for(var toolbar_ind in mpl.toolbar_items) {\n", + " var name = mpl.toolbar_items[toolbar_ind][0];\n", + " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", + " var image = mpl.toolbar_items[toolbar_ind][2];\n", + " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", + "\n", + " if (!name) {\n", + " // put a spacer in here.\n", + " continue;\n", + " }\n", + " var button = $('');\n", + " button.click(method_name, toolbar_event);\n", + " button.mouseover(tooltip, toolbar_mouse_event);\n", + " nav_element.append(button);\n", + " }\n", + "\n", + " // Add the status bar.\n", + " var status_bar = $('');\n", + " nav_element.append(status_bar);\n", + " this.message = status_bar[0];\n", + "\n", + " // Add the close button to the window.\n", + " var buttongrp = $('
');\n", + " var button = $('');\n", + " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", + " button.mouseover('Close figure', toolbar_mouse_event);\n", + " buttongrp.append(button);\n", + " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", + " titlebar.prepend(buttongrp);\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(el){\n", + " // this is important to make the div 'focusable\n", + " el.attr('tabindex', 0)\n", + " // reach out to IPython and tell the keyboard manager to turn it's self\n", + " // off when our div gets focus\n", + "\n", + " // location in version 3\n", + " if (IPython.notebook.keyboard_manager) {\n", + " IPython.notebook.keyboard_manager.register_events(el);\n", + " }\n", + " else {\n", + " // location in version 2\n", + " IPython.keyboard_manager.register_events(el);\n", + " }\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._key_event_extra = function(event, name) {\n", + " var manager = IPython.notebook.keyboard_manager;\n", + " if (!manager)\n", + " manager = IPython.keyboard_manager;\n", + "\n", + " // Check for shift+enter\n", + " if (event.shiftKey && event.which == 13) {\n", + " this.canvas_div.blur();\n", + " event.shiftKey = false;\n", + " // Send a \"J\" for go to next cell\n", + " event.which = 74;\n", + " event.keyCode = 74;\n", + " manager.command_mode();\n", + " manager.handle_keydown(event);\n", + " }\n", + "}\n", + "\n", + "mpl.figure.prototype.handle_save = function(fig, msg) {\n", + " fig.ondownload(fig, null);\n", + "}\n", + "\n", + "\n", + "mpl.find_output_cell = function(html_output) {\n", + " // Return the cell and output element which can be found *uniquely* in the notebook.\n", + " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", + " // IPython event is triggered only after the cells have been serialised, which for\n", + " // our purposes (turning an active figure into a static one), is too late.\n", + " var cells = IPython.notebook.get_cells();\n", + " var ncells = cells.length;\n", + " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", + " data = data.data;\n", + " }\n", + " if (data['text/html'] == html_output) {\n", + " return [cell, data, j];\n", + " }\n", + " }\n", + " }\n", + " }\n", + "}\n", + "\n", + "// Register the function which deals with the matplotlib target/channel.\n", + "// The kernel may be null if the page has been refreshed.\n", + "if (IPython.notebook.kernel != null) {\n", + " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", + "}\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "
" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "application/javascript": [ + "/* Put everything inside the global mpl namespace */\n", + "window.mpl = {};\n", + "\n", + "mpl.get_websocket_type = function() {\n", + " if (typeof(WebSocket) !== 'undefined') {\n", + " return WebSocket;\n", + " } else if (typeof(MozWebSocket) !== 'undefined') {\n", + " return MozWebSocket;\n", + " } else {\n", + " alert('Your browser does not have WebSocket support.' +\n", + " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", + " 'Firefox 4 and 5 are also supported but you ' +\n", + " 'have to enable WebSockets in about:config.');\n", + " };\n", + "}\n", + "\n", + "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", + " this.id = figure_id;\n", + "\n", + " this.ws = websocket;\n", + "\n", + " this.supports_binary = (this.ws.binaryType != undefined);\n", + "\n", + " if (!this.supports_binary) {\n", + " var warnings = document.getElementById(\"mpl-warnings\");\n", + " if (warnings) {\n", + " warnings.style.display = 'block';\n", + " warnings.textContent = (\n", + " \"This browser does not support binary websocket messages. \" +\n", + " \"Performance may be slow.\");\n", + " }\n", + " }\n", + "\n", + " this.imageObj = new Image();\n", + "\n", + " this.context = undefined;\n", + " this.message = undefined;\n", + " this.canvas = undefined;\n", + " this.rubberband_canvas = undefined;\n", + " this.rubberband_context = undefined;\n", + " this.format_dropdown = undefined;\n", + "\n", + " this.image_mode = 'full';\n", + "\n", + " this.root = $('
');\n", + " this._root_extra_style(this.root)\n", + " this.root.attr('style', 'display: inline-block');\n", + "\n", + " $(parent_element).append(this.root);\n", + "\n", + " this._init_header(this);\n", + " this._init_canvas(this);\n", + " this._init_toolbar(this);\n", + "\n", + " var fig = this;\n", + "\n", + " this.waiting = false;\n", + "\n", + " this.ws.onopen = function () {\n", + " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", + " fig.send_message(\"send_image_mode\", {});\n", + " fig.send_message(\"refresh\", {});\n", + " }\n", + "\n", + " this.imageObj.onload = function() {\n", + " if (fig.image_mode == 'full') {\n", + " // Full images could contain transparency (where diff images\n", + " // almost always do), so we need to clear the canvas so that\n", + " // there is no ghosting.\n", + " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", + " }\n", + " fig.context.drawImage(fig.imageObj, 0, 0);\n", + " fig.waiting = false;\n", + " };\n", + "\n", + " this.imageObj.onunload = function() {\n", + " this.ws.close();\n", + " }\n", + "\n", + " this.ws.onmessage = this._make_on_message_function(this);\n", + "\n", + " this.ondownload = ondownload;\n", + "}\n", + "\n", + "mpl.figure.prototype._init_header = function() {\n", + " var titlebar = $(\n", + " '
');\n", + " var titletext = $(\n", + " '
');\n", + " titlebar.append(titletext)\n", + " this.root.append(titlebar);\n", + " this.header = titletext[0];\n", + "}\n", + "\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._init_canvas = function() {\n", + " var fig = this;\n", + "\n", + " var canvas_div = $('
');\n", + "\n", + " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", + "\n", + " function canvas_keyboard_event(event) {\n", + " return fig.key_event(event, event['data']);\n", + " }\n", + "\n", + " canvas_div.keydown('key_press', canvas_keyboard_event);\n", + " canvas_div.keyup('key_release', canvas_keyboard_event);\n", + " this.canvas_div = canvas_div\n", + " this._canvas_extra_style(canvas_div)\n", + " this.root.append(canvas_div);\n", + "\n", + " var canvas = $('');\n", + " canvas.addClass('mpl-canvas');\n", + " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", + "\n", + " this.canvas = canvas[0];\n", + " this.context = canvas[0].getContext(\"2d\");\n", + "\n", + " var rubberband = $('');\n", + " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", + "\n", + " var pass_mouse_events = true;\n", + "\n", + " canvas_div.resizable({\n", + " start: function(event, ui) {\n", + " pass_mouse_events = false;\n", + " },\n", + " resize: function(event, ui) {\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " stop: function(event, ui) {\n", + " pass_mouse_events = true;\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " });\n", + "\n", + " function mouse_event_fn(event) {\n", + " if (pass_mouse_events)\n", + " return fig.mouse_event(event, event['data']);\n", + " }\n", + "\n", + " rubberband.mousedown('button_press', mouse_event_fn);\n", + " rubberband.mouseup('button_release', mouse_event_fn);\n", + " // Throttle sequential mouse events to 1 every 20ms.\n", + " rubberband.mousemove('motion_notify', mouse_event_fn);\n", + "\n", + " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", + " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", + "\n", + " canvas_div.on(\"wheel\", function (event) {\n", + " event = event.originalEvent;\n", + " event['data'] = 'scroll'\n", + " if (event.deltaY < 0) {\n", + " event.step = 1;\n", + " } else {\n", + " event.step = -1;\n", + " }\n", + " mouse_event_fn(event);\n", + " });\n", + "\n", + " canvas_div.append(canvas);\n", + " canvas_div.append(rubberband);\n", + "\n", + " this.rubberband = rubberband;\n", + " this.rubberband_canvas = rubberband[0];\n", + " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", + " this.rubberband_context.strokeStyle = \"#000000\";\n", + "\n", + " this._resize_canvas = function(width, height) {\n", + " // Keep the size of the canvas, canvas container, and rubber band\n", + " // canvas in synch.\n", + " canvas_div.css('width', width)\n", + " canvas_div.css('height', height)\n", + "\n", + " canvas.attr('width', width);\n", + " canvas.attr('height', height);\n", + "\n", + " rubberband.attr('width', width);\n", + " rubberband.attr('height', height);\n", + " }\n", + "\n", + " // Set the figure to an initial 600x600px, this will subsequently be updated\n", + " // upon first draw.\n", + " this._resize_canvas(600, 600);\n", + "\n", + " // Disable right mouse context menu.\n", + " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", + " return false;\n", + " });\n", + "\n", + " function set_focus () {\n", + " canvas.focus();\n", + " canvas_div.focus();\n", + " }\n", + "\n", + " window.setTimeout(set_focus, 100);\n", + "}\n", + "\n", + "mpl.figure.prototype._init_toolbar = function() {\n", + " var fig = this;\n", + "\n", + " var nav_element = $('
')\n", + " nav_element.attr('style', 'width: 100%');\n", + " this.root.append(nav_element);\n", + "\n", + " // Define a callback function for later on.\n", + " function toolbar_event(event) {\n", + " return fig.toolbar_button_onclick(event['data']);\n", + " }\n", + " function toolbar_mouse_event(event) {\n", + " return fig.toolbar_button_onmouseover(event['data']);\n", + " }\n", + "\n", + " for(var toolbar_ind in mpl.toolbar_items) {\n", + " var name = mpl.toolbar_items[toolbar_ind][0];\n", + " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", + " var image = mpl.toolbar_items[toolbar_ind][2];\n", + " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", + "\n", + " if (!name) {\n", + " // put a spacer in here.\n", + " continue;\n", + " }\n", + " var button = $('');\n", + " button.click(method_name, toolbar_event);\n", + " button.mouseover(tooltip, toolbar_mouse_event);\n", + " nav_element.append(button);\n", + " }\n", + "\n", + " // Add the status bar.\n", + " var status_bar = $('');\n", + " nav_element.append(status_bar);\n", + " this.message = status_bar[0];\n", + "\n", + " // Add the close button to the window.\n", + " var buttongrp = $('
');\n", + " var button = $('');\n", + " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", + " button.mouseover('Close figure', toolbar_mouse_event);\n", + " buttongrp.append(button);\n", + " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", + " titlebar.prepend(buttongrp);\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(el){\n", + " // this is important to make the div 'focusable\n", + " el.attr('tabindex', 0)\n", + " // reach out to IPython and tell the keyboard manager to turn it's self\n", + " // off when our div gets focus\n", + "\n", + " // location in version 3\n", + " if (IPython.notebook.keyboard_manager) {\n", + " IPython.notebook.keyboard_manager.register_events(el);\n", + " }\n", + " else {\n", + " // location in version 2\n", + " IPython.keyboard_manager.register_events(el);\n", + " }\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._key_event_extra = function(event, name) {\n", + " var manager = IPython.notebook.keyboard_manager;\n", + " if (!manager)\n", + " manager = IPython.keyboard_manager;\n", + "\n", + " // Check for shift+enter\n", + " if (event.shiftKey && event.which == 13) {\n", + " this.canvas_div.blur();\n", + " event.shiftKey = false;\n", + " // Send a \"J\" for go to next cell\n", + " event.which = 74;\n", + " event.keyCode = 74;\n", + " manager.command_mode();\n", + " manager.handle_keydown(event);\n", + " }\n", + "}\n", + "\n", + "mpl.figure.prototype.handle_save = function(fig, msg) {\n", + " fig.ondownload(fig, null);\n", + "}\n", + "\n", + "\n", + "mpl.find_output_cell = function(html_output) {\n", + " // Return the cell and output element which can be found *uniquely* in the notebook.\n", + " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", + " // IPython event is triggered only after the cells have been serialised, which for\n", + " // our purposes (turning an active figure into a static one), is too late.\n", + " var cells = IPython.notebook.get_cells();\n", + " var ncells = cells.length;\n", + " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", + " data = data.data;\n", + " }\n", + " if (data['text/html'] == html_output) {\n", + " return [cell, data, j];\n", + " }\n", + " }\n", + " }\n", + " }\n", + "}\n", + "\n", + "// Register the function which deals with the matplotlib target/channel.\n", + "// The kernel may be null if the page has been refreshed.\n", + "if (IPython.notebook.kernel != null) {\n", + " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", + "}\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "v = poincare(q2d_dp, 0, 0, 2, 3) #use states from RK4 integration\n", - "\n", + "print(numpy.shape(v))\n", "pyplot.figure(figsize=(10,8));\n", "pyplot.grid(True);\n", "pyplot.xlabel(r'$\\theta_2$', fontsize=18);\n", - "pyplot.ylabel(r'$\\dot{\\theta_2}$', fontsize=18);\n", - "pyplot.title('Poincare Section');\n", + "pyplot.ylabel(r'$\\dot{\\theta_2}$', fontsize=18);#pyplot.title('Poincare Section');\n", "pyplot.plot(v[0,:], v[1,:], 'bo', lw=2, );\n", - "#pyplot.plot(t, q2d_dp[:,1], lw=2, label='Joint 2')\n", - "#pyplot.legend();\n" + "\n", + "from matplotlib import animation\n", + "from JSAnimation.IPython_display import display_animation\n", + "def animate(data):\n", + " print(numpy.shape(data))\n", + " x = data[0,:]\n", + " y = data[1,:]\n", + " line.set_data(x, y)\n", + " return line," + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# #(try to animate plot)\n", + "#fig = pyplot.figure();\n", + "#ax = pyplot.axes(xlim=(-10, 10), ylim=(-10,10),xlabel=(r'$\\theta_2$'), ylabel=(r'$\\dot{\\theta_2}$'))\n", + "#line, = ax.plot([],[],'bo', lw=2);\n", + "\n", + "#anim = animation.FuncAnimation(fig, animate, frames= v[:,:], interval=10)\n", + "#display_animation(anim, default_mode='once')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "(try to animate plot)\n", - "\n", - "Other than a few outliers, the plot converges to 0,0. So, this system is stable! This would not be the case for the undamped double pendulum, as integrated by RK4. " + "Other than a few outliers, the plot converges to (0,0). So, this system is stable! This would not be the case for the undamped double pendulum, as integrated by RK4. " ] }, { @@ -695,7 +5239,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.5.0" + "version": "3.4.3" } }, "nbformat": 4, diff --git a/randy_schur/Integration Techniques for Mechanical Systems.ipynb b/randy_schur/Integration Techniques for Mechanical Systems.ipynb index f97971e..dec9b79 100644 --- a/randy_schur/Integration Techniques for Mechanical Systems.ipynb +++ b/randy_schur/Integration Techniques for Mechanical Systems.ipynb @@ -6,7 +6,7 @@ "source": [ "# Integration of Mechanical Systems\n", "\n", - "In this module, we will be exploring numerical integrators and how they perform for different mechanical systems." + "In this module, we will be exploring numerical integrators and how they perform on different mechanical systems." ] }, { @@ -32,7 +32,7 @@ "0 = m\\ddot{x} + kx\n", "\\end{equation}$$\n", "\n", - "We can describe the state of the system by its position $x$ and its velocity $\\dot{x}$. We will call this state vector $q$, as:\n", + "Since this is a second order system, we will need two states to accurately describe the system. We can use position $x$ and velocity $\\dot{x}$. This is called the state vector $q$, as:\n", "\n", "$$\\begin{eqnarray}\n", "\\vec{q} = \\begin{bmatrix} x\\\\ \\dot{x} \\end{bmatrix}\n", @@ -44,7 +44,7 @@ "\\vec{\\dot{q}} = f(\\vec{q}) = \\begin{bmatrix} q_2\\\\ -\\frac{k}{m}q_1 \\end{bmatrix}\n", "\\end{eqnarray}$$\n", "\n", - "To look at the behavior of this system, we can start with some initial conditions for $q$, then use a numerical integration method such as Euler's method to integrate forward in time. " + "To look at the behavior of this system, we can start with some initial conditions for $q$, then use a numerical integration method such as Euler's method to integrate forward in time. This is equivalent to pulling (or pushing) on the spring, then letting go at t=0." ] }, { @@ -53,7 +53,7 @@ "source": [ "### Equations of Motion from Lagrangian\n", "\n", - "You are encouraged to follow this derivation ON PAPER on your own. We gave you the equations of motion for a simple harmonic oscillator, but what if we don't have this equation? The following is a method that will work in nearly any situation to find equations of motion, often working well where a force analysis becomes difficult. This is good practice for anyone studying mechanics. To derive the equations of motion, we will start with the Lagrangian. This is defined by:\n", + "You are encouraged to follow this derivation ON PAPER on your own. The equations of motion are given above for a simple harmonic oscillator, but what if we didn't have this equation ahead of time? The following is a method that will work in nearly any situation to find equations of motion, often working well where a force analysis becomes difficult. This is especially good practice for anyone studying mechanics. To derive the equations of motion, we will start with the Lagrangian. This is defined by:\n", "\n", "$$\\begin{equation}\n", "L = T- U\n", @@ -69,13 +69,13 @@ "\n", "$$\\begin{eqnarray*}\n", "T &=& \\frac{1}{2}mv^2 \\\\\n", - "&=& \\frac{1}{2}m(\\dot x)^2 \n", + "&=& \\frac{1}{2}m\\dot{x}^2 \n", "\\end{eqnarray*}$$\n", "\n", "\n", "So the Lagrangian quantity becomes:\n", "$$\\begin{equation}\n", - "L = \\frac{1}{2}m(\\dot x)^2 - \\frac{1}{2}kx^2\\\\\n", + "L = \\frac{1}{2}m\\dot{x}^2 - \\frac{1}{2}kx^2\\\\\n", "\\end{equation}$$\n", "\n", "\n" @@ -128,10 +128,10 @@ "source": [ "### Runge-Kutta Integration (RK4)\n", "\n", - "Depending on the application, Euler's method may provide enough accuracy with a small timestep, especially a simple harmonic oscillator. However, Euler is a first-order method and we can do better. One option is a Runge-Kutta scheme. This is a popular numerical integration method which is easily extended to higher orders. Here we will use a fourth order Runge-Kutta, also known as RK4:\n", + "Depending on the application, Euler's method may provide enough accuracy with a small timestep, especially a simple harmonic oscillator. However, Euler is a first-order method and we can do better. One option is a Runge-Kutta scheme. This is a popular numerical integration method which is easily extended to higher orders. Here we will use a fourth order Runge-Kutta, also called RK4:\n", "\n", "$$\\begin{equation}\n", - "q_{n+1} = q_n + \\frac{h}{6}\\left(k_1+2k_2+2k_3+k_4\\right)\n", + "q_i^{n+1} = q_i^n + \\frac{h}{6}\\left(k_1+2k_2+2k_3+k_4\\right)\n", "\\end{equation}$$\n", "where\n", "$$\\begin{eqnarray}\n", @@ -151,17 +151,21 @@ "source": [ "### Symplectic Integrators\n", "\n", - "Symplectic integrators are similar to the methods discussed above, but they use equations of motion derived from Hamiltonian mechanics. According to [Berkeley source], syplectic integrators preserve the conserved Hamiltonian quantities. In practical terms, this works out to mean the methods reflect conservation of momentum, down to a truncation error.\n", + "Symplectic integrators are similar to the methods discussed above, but they use equations of motion derived from Hamiltonian mechanics. According to [5], syplectic integrators preserve the conserved Hamiltonian quantities. In practical terms, this works out to mean the methods reflect conservation of energy, down to a truncation error.\n", "\n", "The Hamiltonian equations of motion can be derived from the total energy in the system, much like the Lagrangian. The coordinates we use are position ($q$) and momentum ($p = m\\dot x$). The Hamiltonian is:\n", "$$ H = T+V$$\n", "where $T$ is the kinetic energy in the system, and $V$ is the potential energy in the system. So, we get:\n", "$$\\begin{eqnarray*}\n", "T &=& \\frac{1}{2}m\\dot{x}^2 = \\frac{q^2}{2m}\\\\\n", - "V &=& \\frac{1}{2}k{x}^2 = \\frac{kp^2}{2} \\\\\n", - "H &=& \\frac{q^2}{2m} + \\frac{kp^2}{2}\n", + "V &=& \\frac{1}{2}k{x}^2 = \\frac{kp^2}{2} \n", "\\end{eqnarray*}$$\n", "\n", + "and \n", + "\n", + "$$H = \\frac{q^2}{2m} + \\frac{kp^2}{2}$$\n", + "\n", + "\n", "\n", "The Hamilton's equations are:\n", "$$\\begin{eqnarray*}\n", @@ -179,14 +183,14 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Let's now take a look at symplectic integration methods. We'll start with a first order method - symplectic Euler's method. This uses the equations:\n", + "Let's now take a look at symplectic integration methods [7]. We'll start with a first order method - symplectic Euler's method. This uses the equations:\n", "\n", "$$\\begin{eqnarray*}\n", "p_{n+1} = p_n - h \\frac{\\partial H}{\\partial q} \\Bigr|_{p_{n+1},q_n}\\\\\n", "q_{n+1} = q_n + \\frac{h}{2} \\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1},q_n}\n", "\\end{eqnarray*}$$\n", "\n", - "We can then plug in the partial derivatives and integrate the system. This system will do a better job of conserving energy than Euler's method, $\\textit{but only to a truncation error.}$ Again, we can do better than first-order. Let's try a second-order symplectic scheme, also called Verlet integration. We'll have the same equations of motion, but this time a different set of integration equations:\n", + "We can then plug in the partial derivatives and integrate the system. This system will do a better job of conserving energy than Euler's method (we'll investigate this later), $\\textit{but only to a truncation error.}$ Again, we can do better than first-order. Let's try a second-order symplectic scheme, also called Verlet integration. We'll have the same equations of motion, but this time a different set of integration equations [6]:\n", "\n", "$$\\begin{eqnarray*}\n", "p_{n+1/2} &=& p_n - \\frac{h}{2} \\frac{\\partial H}{\\partial q} \\Bigr|_{p_{n+1/2},q_n}\\\\\n", @@ -222,7 +226,16 @@ "metadata": { "collapsed": false }, - "outputs": [], + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/home/randy/anaconda3/lib/python3.4/site-packages/IPython/kernel/__init__.py:13: ShimWarning: The `IPython.kernel` package has been deprecated. You should import from ipykernel or jupyter_client instead.\n", + " \"You should import from ipykernel or jupyter_client instead.\", ShimWarning)\n" + ] + } + ], "source": [ "import numpy\n", "from scipy.linalg import solve\n", @@ -239,27 +252,25 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 8, "metadata": { - "collapsed": true + "collapsed": false }, "outputs": [], "source": [ - "#Set up parameters:\n", - "\n", + "#Set up system parameters:\n", "m = 12\n", - "k = 150;\n", - "#g = 9.8 #[m/s**2]\n", + "k = 150\n", "\n", "T = 25; #[seconds]\n", - "dt = .02; #\n", - "N = int(T/dt)+1\n", - "t = numpy.linspace(0.0, T, N)" + "dt = .02; #time step size\n", + "N = int(T/dt)+1 #number of time steps\n", + "t = numpy.linspace(0.0, T, N) #array of time values" ] }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 9, "metadata": { "collapsed": true }, @@ -282,7 +293,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 10, "metadata": { "collapsed": true }, @@ -296,7 +307,8 @@ " f - function for RHS of state equations\n", " dt - time step\n", " \n", - " Returns: array of state values at next time step.\n", + " Returns: \n", + " state values at next time step.\n", " \"\"\"\n", " k1 = f(u)\n", " k2 = f(u) + 0.5*dt*k1\n", @@ -308,7 +320,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 11, "metadata": { "collapsed": true }, @@ -318,10 +330,10 @@ " \"\"\"Returns RHS of harmonic oscillator EOM\n", " \n", " Parameters:\n", - " q - initial state\n", + " u - initial state\n", " \n", " Returns:\n", - " f - RHS of harmonic oscillator eqn.\n", + " RHS of harmonic oscillator eqn.\n", " \n", " \"\"\"\n", " pos = u[0]\n", @@ -332,7 +344,7 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 12, "metadata": { "collapsed": true }, @@ -344,7 +356,8 @@ " Parameters:\n", " u - state at current time step\n", " dt - time step size\n", - " Returns: state at next time step.\n", + " Returns: \n", + " state at next time step.\n", " \"\"\"\n", " pos = u[0]\n", " mom = u[1]\n", @@ -360,7 +373,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 13, "metadata": { "collapsed": true }, @@ -370,6 +383,7 @@ "x0 = 0.5 #[m]\n", "xdot0 = 0 #[m/s]\n", "\n", + "#Equivalent initial conditions for the Hamiltonian system\n", "p0 = x0\n", "q0 = m*xdot0\n", "\n", @@ -379,7 +393,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 14, "metadata": { "collapsed": false }, @@ -1123,7 +1137,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1134,20 +1148,20 @@ } ], "source": [ - "#Euler\n", - "q1 = numpy.zeros((N,2)) \n", + "#Euler Integration\n", + "q1 = numpy.zeros((N,2)) #initialize array\n", "q1[0] = x_init.copy() #set initial conditions\n", "for n in range(N-1): #integrate with Euler\n", " q1[n+1] = euler(q1[n], f_harmonic_oscillator, dt)\n", " #print(q1[n])\n", " \n", - "#Runge-Kutta \n", + "#Runge-Kutta Integration\n", "q2 = numpy.zeros((N,2))\n", "q2[0] = x_init.copy()\n", "for n in range(N-1):\n", " q2[n+1] = RK4(q2[n], f_harmonic_oscillator, dt)\n", "\n", - "#Symplectic\n", + "#Symplectic Integration\n", "q3 = numpy.zeros((N,2))\n", "q3[0] = x_init_H.copy()\n", "for n in range(N-1):\n", @@ -1177,7 +1191,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "If we look closely at the plot, we can see that Euler and Runge-Kutta integration are close to each other, but begin to diverge as time goes on. This should be the behavior we expect for the integration of a non-linear system. But how close are these methods to the analytical solution? Does Verlet do any better? In this case it is obvious that Euler and Runge-Kutta are not correct (and possible unstable). Let's do an error analysis.\n", + "If we look closely at the plot, we can see that Euler and Runge-Kutta integration are close to each other, but begin to diverge as time goes on. This should be the behavior we expect for the integration of a non-linear system. But how close are these methods to the analytical solution? Does Verlet do any better? In this case it is obvious that Euler and Runge-Kutta are not correct (and possibly unstable). Let's do an error analysis.\n", "\n", "### Error Analysis\n", "The error analysis code below is based on the numerical methods MOOC notebook on phugoid oscillation. " @@ -1185,7 +1199,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 15, "metadata": { "collapsed": true }, @@ -1208,7 +1222,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 16, "metadata": { "collapsed": true }, @@ -1229,7 +1243,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 17, "metadata": { "collapsed": false }, @@ -1973,7 +1987,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2000,7 +2014,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Is this what you would expect to see based on the plots of position above? What are the sources of this error? We have the usual suspects: truncation error, discretization error, possible precision errors. Don't believe that these are significant? Take a look at the following screenshot, which shows the plots above, built with identical code on two different computers. The responses on the left are from a 64-bit computer, and the responses on the right are from a 32-bit computer. Look especially at the phase shift in the system response. That's a huge difference! Even the precision of the machine can have a significant impact on numerical integration.\n", + "Is this what you would expect to see based on the plots of position above? What are the sources of this error? We have the usual suspects: truncation error, discretization error, possible precision errors. Don't believe that these are significant? Take a look at the following screenshot, which shows the plots above, built with identical code on two different computers. The responses on the left are from a 64-bit computer, and the responses on the right are from a 32-bit computer. Look closely at the phase shift in the system response. That's a huge difference! Even the precision of the machine can have a significant impact on numerical integration.\n", "\n", "![Image](figures/combined_response.png)\n", "\n", @@ -2018,7 +2032,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 18, "metadata": { "collapsed": true }, @@ -2041,7 +2055,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 19, "metadata": { "collapsed": true }, @@ -2076,7 +2090,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 20, "metadata": { "collapsed": false }, @@ -2156,6 +2170,7 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", + " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -2486,13 +2501,11 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", - " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", - " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -2541,19 +2554,6 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", - "/*\n", - " * return a copy of an object with only non-object keys\n", - " * we need this to avoid circular references\n", - " * http://stackoverflow.com/a/24161582/3208463\n", - " */\n", - "function simpleKeys (original) {\n", - " return Object.keys(original).reduce(function (obj, key) {\n", - " if (typeof original[key] !== 'object')\n", - " obj[key] = original[key]\n", - " return obj;\n", - " }, {});\n", - "}\n", - "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -2567,8 +2567,7 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step,\n", - " guiEvent: simpleKeys(event)});\n", + " step: event.step});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -2608,8 +2607,7 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value,\n", - " guiEvent: simpleKeys(event)});\n", + " this.send_message(name, {key: value});\n", " return false;\n", "}\n", "\n", @@ -2626,7 +2624,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -2685,8 +2683,6 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", - " fig.root.unbind('remove')\n", - "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -2694,12 +2690,8 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.close_ws(fig, msg);\n", - "}\n", - "\n", - "mpl.figure.prototype.close_ws = function(fig, msg){\n", - " fig.send_message('closing', msg);\n", - " // fig.ws.close()\n", + " fig.send_message('closing', {});\n", + " fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -2754,20 +2746,14 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", + " button.mouseover('Close figure', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", - "mpl.figure.prototype._root_extra_style = function(el){\n", - " var fig = this\n", - " el.on(\"remove\", function(){\n", - "\tfig.close_ws(fig, {});\n", - " });\n", - "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -2848,7 +2834,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2875,7 +2861,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "It looks like Euler and Runge-Kutta integration are adding a significant amount of energy to the system. Our symplectic integrator is more accurate because it seems to preserve energy. Maybe all of that work deriving the equations of motion was worth it! We now know that the Hamiltonian equations do better with preserved quantities, and this can make a big difference even in a simple system." + "It looks like Euler and Runge-Kutta integration are adding a significant amount of energy to the system. Our symplectic integrator is more accurate because it seems to preserve energy. Maybe all of that work deriving the equations of motion was worth it! We now know that the Hamiltonian equations do better with Conservation of Energy, and this can make a big difference even in a simple system." ] }, { @@ -2884,7 +2870,7 @@ "source": [ "## Adding a damping term\n", "\n", - "A more realistic oscillator also has a damping term. This might correspond to friction, or to some other non-conservative force. For now, we will assume that the damping is linear, and is proportional to the velocity (this is called the -- model of friction). We get the following equation of motion:\n", + "A more realistic oscillator also has a damping term. This might correspond to friction, or to some other non-conservative force. For now, we will assume that the damping is linear, and is proportional to the velocity (this is called the viscous damping model of friction). We get the following equation of motion:\n", "\n", "$$\\begin{equation*}\n", "0 = m\\ddot{x} + c\\dot{x} + kx\n", @@ -2895,7 +2881,21 @@ "\\vec{\\dot{q}} = f(\\vec{q}) = \\begin{bmatrix} q_2\\\\ -\\frac{c}{m}\\dot{q_1} -\\frac{k}{m}q_1 \\end{bmatrix}\n", "\\end{eqnarray*}$$\n", "\n", - "The Euler's method and Runge-Kutta integration work the same way with this new equation. We also have a new analytical solution. Assuming the system is underdamped (add explanation of damping ratio), the solution becomes:\n", + "We can define a damping ratio, $\\zeta = \\frac{c}{2m\\omega_n} \\geq 0$, to express how large the damping is relative to the other system parameters. This differential equation has several types of solutions, depending on the damping ratio. These lead to a few different behaviors:\n", + "\n", + "1. Undamped $\\zeta =0$ - This is simple harmonic motion, which we have already examined\n", + "2. Underdamped $\\zeta < 1$ - The system will oscillate, but will lose energy over time. We will assume this case for the rest of this notebook, since it leads to the most interesting results.\n", + "3. Critically damped $\\zeta = 1$ - A special case which will lead to no oscillation (the system may overshoot its equilibrium position one time, but it won't oscillate) \n", + "4. Overdamped $\\zeta>1$ - This system will lead to no oscillation\n", + "\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Euler's method and Runge-Kutta integration work the same way as in simple harmonic motion. We also have a new analytical solution. Assuming the system is underdamped (which we will do from here on out), the solution becomes:\n", "\n", "\n", "$$\\begin{equation}\n", @@ -2921,12 +2921,12 @@ "\\begin{bmatrix} \\dot p \\\\ \\dot q \\end{bmatrix} = \\begin{bmatrix} -\\frac{\\partial H}{\\partial q} + Q\\\\ \\frac{\\partial H}{\\partial p} \\end{bmatrix} = \\begin{bmatrix} -\\frac{q}{m} - \\frac{c}{m}q \\\\ kp \\end{bmatrix}\n", "\\end{equation*}$$\n", "\n", - "From here, the we can use the same Verlet integration method as before." + "From here, the we can use the same Verlet integration method as before. If this explanation has too much hand-waving for you, please see [7]." ] }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 21, "metadata": { "collapsed": true }, @@ -2950,7 +2950,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 22, "metadata": { "collapsed": true }, @@ -2979,7 +2979,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 23, "metadata": { "collapsed": true }, @@ -3010,7 +3010,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 24, "metadata": { "collapsed": false }, @@ -3090,6 +3090,7 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", + " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -3420,13 +3421,11 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", - " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", - " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -3475,19 +3474,6 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", - "/*\n", - " * return a copy of an object with only non-object keys\n", - " * we need this to avoid circular references\n", - " * http://stackoverflow.com/a/24161582/3208463\n", - " */\n", - "function simpleKeys (original) {\n", - " return Object.keys(original).reduce(function (obj, key) {\n", - " if (typeof original[key] !== 'object')\n", - " obj[key] = original[key]\n", - " return obj;\n", - " }, {});\n", - "}\n", - "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -3501,8 +3487,7 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step,\n", - " guiEvent: simpleKeys(event)});\n", + " step: event.step});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -3542,8 +3527,7 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value,\n", - " guiEvent: simpleKeys(event)});\n", + " this.send_message(name, {key: value});\n", " return false;\n", "}\n", "\n", @@ -3560,7 +3544,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -3619,8 +3603,6 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", - " fig.root.unbind('remove')\n", - "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -3628,12 +3610,8 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.close_ws(fig, msg);\n", - "}\n", - "\n", - "mpl.figure.prototype.close_ws = function(fig, msg){\n", - " fig.send_message('closing', msg);\n", - " // fig.ws.close()\n", + " fig.send_message('closing', {});\n", + " fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -3688,20 +3666,14 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", + " button.mouseover('Close figure', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", - "mpl.figure.prototype._root_extra_style = function(el){\n", - " var fig = this\n", - " el.on(\"remove\", function(){\n", - "\tfig.close_ws(fig, {});\n", - " });\n", - "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -3782,7 +3754,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -3834,7 +3806,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "How does each integrator perform compared to the case of simple harmonic motion? From the eye test, it seems like the error is actually better here. Why is that? \n", + "How does each integrator perform compared to the case of simple harmonic motion? Just by looking at the system response, it seems like the error is actually better here. Why is that? \n", "\n", "The effect of the damper is to remove energy from the system. So, in this case the damping term is helping to remove some energy from the system that the integration adds. As time goes to infinity, these solutions will converge, and the Euler and Runge-Kutta methods are stable. We can again look at the energy in the system according to each method. \n", "\n" @@ -3842,7 +3814,7 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 25, "metadata": { "collapsed": false }, @@ -3922,6 +3894,7 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", + " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -4252,13 +4225,11 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", - " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", - " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -4307,19 +4278,6 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", - "/*\n", - " * return a copy of an object with only non-object keys\n", - " * we need this to avoid circular references\n", - " * http://stackoverflow.com/a/24161582/3208463\n", - " */\n", - "function simpleKeys (original) {\n", - " return Object.keys(original).reduce(function (obj, key) {\n", - " if (typeof original[key] !== 'object')\n", - " obj[key] = original[key]\n", - " return obj;\n", - " }, {});\n", - "}\n", - "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -4333,8 +4291,7 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step,\n", - " guiEvent: simpleKeys(event)});\n", + " step: event.step});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -4374,8 +4331,7 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value,\n", - " guiEvent: simpleKeys(event)});\n", + " this.send_message(name, {key: value});\n", " return false;\n", "}\n", "\n", @@ -4392,7 +4348,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -4451,8 +4407,6 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", - " fig.root.unbind('remove')\n", - "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -4460,12 +4414,8 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.close_ws(fig, msg);\n", - "}\n", - "\n", - "mpl.figure.prototype.close_ws = function(fig, msg){\n", - " fig.send_message('closing', msg);\n", - " // fig.ws.close()\n", + " fig.send_message('closing', {});\n", + " fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -4520,20 +4470,14 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", + " button.mouseover('Close figure', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", - "mpl.figure.prototype._root_extra_style = function(el){\n", - " var fig = this\n", - " el.on(\"remove\", function(){\n", - "\tfig.close_ws(fig, {});\n", - " });\n", - "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -4614,7 +4558,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -4677,7 +4621,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 29, "metadata": { "collapsed": false }, @@ -4757,6 +4701,7 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", + " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -5087,13 +5032,11 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", - " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", - " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -5142,19 +5085,6 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", - "/*\n", - " * return a copy of an object with only non-object keys\n", - " * we need this to avoid circular references\n", - " * http://stackoverflow.com/a/24161582/3208463\n", - " */\n", - "function simpleKeys (original) {\n", - " return Object.keys(original).reduce(function (obj, key) {\n", - " if (typeof original[key] !== 'object')\n", - " obj[key] = original[key]\n", - " return obj;\n", - " }, {});\n", - "}\n", - "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -5168,8 +5098,7 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step,\n", - " guiEvent: simpleKeys(event)});\n", + " step: event.step});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -5209,8 +5138,7 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value,\n", - " guiEvent: simpleKeys(event)});\n", + " this.send_message(name, {key: value});\n", " return false;\n", "}\n", "\n", @@ -5227,7 +5155,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -5286,8 +5214,6 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", - " fig.root.unbind('remove')\n", - "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -5295,12 +5221,8 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.close_ws(fig, msg);\n", - "}\n", - "\n", - "mpl.figure.prototype.close_ws = function(fig, msg){\n", - " fig.send_message('closing', msg);\n", - " // fig.ws.close()\n", + " fig.send_message('closing', {});\n", + " fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -5355,20 +5277,14 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", + " button.mouseover('Close figure', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", - "mpl.figure.prototype._root_extra_style = function(el){\n", - " var fig = this\n", - " el.on(\"remove\", function(){\n", - "\tfig.close_ws(fig, {});\n", - " });\n", - "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -5449,7 +5365,7 @@ { "data": { "text/html": [ - "" + "
" ], "text/plain": [ "" @@ -5487,25 +5403,14 @@ "pyplot.xlabel('Postion ($x_1$)')\n", "pyplot.ylabel('Velocity ($x_2$)')\n", "pyplot.legend();\n", - "pyplot.title('Phase portrait for damped harmonic oscillator');\n" + "pyplot.title('Phase portrait for damped harmonic oscillator');" ] }, { - "cell_type": "code", - "execution_count": 21, + "cell_type": "markdown", "metadata": { "collapsed": false }, - "outputs": [ - { - "ename": "SyntaxError", - "evalue": "invalid syntax (, line 1)", - "output_type": "error", - "traceback": [ - "\u001b[1;36m File \u001b[1;32m\"\"\u001b[1;36m, line \u001b[1;32m1\u001b[0m\n\u001b[1;33m Sources:\u001b[0m\n\u001b[1;37m ^\u001b[0m\n\u001b[1;31mSyntaxError\u001b[0m\u001b[1;31m:\u001b[0m invalid syntax\n" - ] - } - ], "source": [ "Sources: \n", "\n", @@ -5527,11 +5432,166 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 28, "metadata": { "collapsed": false }, - "outputs": [], + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "execution_count": 28, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "# This cell loads the style of the notebook, which is modified from the \n", "# Numerical Methods in Python Course: http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about\n", @@ -5562,21 +5622,21 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 2", + "display_name": "Python 3", "language": "python", - "name": "python2" + "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", - "version": 2 + "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", - "pygments_lexer": "ipython2", - "version": "2.7.11" + "pygments_lexer": "ipython3", + "version": "3.4.3" } }, "nbformat": 4, From 43c1db174679ab5616d94409ca7baf5b09f2f075 Mon Sep 17 00:00:00 2001 From: rbds Date: Mon, 7 Dec 2015 23:15:08 -0500 Subject: [PATCH 36/61] fix Hamiltonian EOM --- randy_schur/Double_Pendulum_Problem.ipynb | 73 ++++++++++++----------- 1 file changed, 39 insertions(+), 34 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index 2f8e1d9..4b86a69 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -22,7 +22,7 @@ "\n", "The double pendulum problem is a relatively simple system which can produce surprisingly complex motion. It is a chaotic system, meaning it is unpredictable and small changes in initial conditions lead to large changes in motion. There is no closed form solution to the motion of the two masses. This is a frequent example problem in the study of dynamic systems, nonlinear controls, and mechanics.\n", "\n", - "In this notebook we will be treating the two rods as massless and we'll start out ignoring the effects of friction. For a treatment of this problem without these assumptions, see (cite).\n", + "In this notebook we will be treating the two rods as massless and we'll start out ignoring the effects of friction. For a treatment of this problem without these assumptions, see [4].\n", "\n", "See the image below for the definition of constants." ] @@ -160,31 +160,45 @@ "source": [ "### Equations of Motion from Hamiltonian\n", "\n", - "Here we derive the equations of motion from the Hamiltonian. If you want good practice, you can derive these on your own. We start with the same potential and kinetic energy, but we'll need to use postion ($\\theta$) and momentum ($I\\omega$) to give us:\n", + "Here we derive the equations of motion from the Hamiltonian. If you want good practice, you can derive these on your own. If you want to see another derivation with nice animations, you can see [1]. We start with the same potential and kinetic energy as before, but we'll need to use postion ($\\theta$) and momentum ($I\\omega$) to give us:\n", "\n", "$$\\begin{eqnarray*}\n", "U &=& m_1g(-y_1) + m_2g(-y_2) \\\\\n", - " &=& -m_1g(l_1cos\\theta_1 - m_2g(l_1cos\\theta_1 + l_2cos\\theta_2)\n", + " &=& -m_1gl_1cos\\theta_1 - m_2g(l_1cos\\theta_1 + l_2cos\\theta_2) \\\\\n", + " &=& -m_1gl_1cos(p_1) - m_2g(l_1cos(p_1) + l_2cos(p_2))\n", "\\end{eqnarray*}$$\n", "\n", "Kinetic Energy for rotational motion calculated as:\n", "\n", "$$\\begin{eqnarray*}\n", - "T &=& \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2+\\frac{1}{2}m_2\\left(2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+l_2^2\\dot\\theta_2^2\\right)\n", + "T &=& \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2+\\frac{1}{2}m_2\\left(2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+l_2^2\\dot\\theta_2^2\\right)\\\\\n", + "&=& \\frac{1}{2}(I_1 + m_2l_1^2)\\omega_1^2+\\left[m_2l_1l_2\\omega_1\\omega_2cos(\\theta_2-\\theta_1)+\\frac{I_2\\omega_2^2}{2}\\right] \\\\\n", + "&=& \\frac{q_1^2}{2I_1}+\\frac{m_2l_1^2q_1^2}{2m_1I_1} + \\left[ \\frac{q_2}{l_2} \\frac{q_1}{m_1l_1}cos(p_2-p_1) + \\frac{q_2^2}{2I_2} \\right]\n", "\\end{eqnarray*}$$\n", - "\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ "where $I_i$ is the angular moment of inertia and $\\omega$ is the angular velocity. So our Hamiltonian quantity becomes:\n", "$$\\begin{equation*}\n", - "H = \\frac{1}{2}\\left(\\frac{q_1^2}{I_1} + \\frac{q_2^2}{I_2}\\right) - (m_1+m_2)gl_1cos\\theta_1 - m_2gl_2cos\\theta_2\n", - "\\end{equation*}$$\n", - "\n", + "H = \\frac{q_1^2}{2I_1}+\\frac{m_2l_1^2q_1^2}{2m_1I_1} + \\left[ \\frac{q_2}{l_2} \\frac{q_1}{m_1l_1}cos(p_2-p_1) + \\frac{q_2^2}{2I_2} \\right] -m_1gl_1cos(p_1) - m_2g\\left( l_1cos(p_1) + l_2cos(p_2) \\right)\n", + "\\end{equation*}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ "Since we have two degrees of freedom, we will have four equations of motion. These will be the following:\n", "$$\\begin{eqnarray*}\n", - "\\dot{p_1} &=& -\\frac{\\partial H}{\\partial q_1} = -\\frac{q_1}{I_1} \\\\\n", - "\\dot{q_1} &=& \\frac{\\partial H}{\\partial p_1} = (m_1+m_2)gl_1sin(p_1) \\\\\n", - "\\dot{p_2} &=& -\\frac{\\partial H}{\\partial q_2} = -\\frac{q_2}{I_2} \\\\\n", - "\\dot{q_2} &=& \\frac{\\partial H}{\\partial p_2} = m_2gl_2sin(p_2) \\\\\n", - "\\end{eqnarray*}$$\n" + "\\dot{p_1} &=& -\\frac{\\partial H}{\\partial q_1} = -\\frac{q_1}{I_1} - \\frac{m_2l_1^2q_1}{m_1I_1} - \\frac{q_1}{l_2m_1l_1}cos(p_2-p_1) \\\\\n", + "\\dot{q_1} &=& \\frac{\\partial H}{\\partial p_1} = \\frac{q_1q_2}{m_1l_1l_2}sin(p_2-p_1) + (m_1+m_2)gl_1sin(p_1) \\\\\n", + "\\dot{p_2} &=& -\\frac{\\partial H}{\\partial q_2} = -\\frac{q_1}{m_1l_1l_2}cos(p_2-p_1) - \\frac{q_2}{2I_2} \\\\\n", + "\\dot{q_2} &=& \\frac{\\partial H}{\\partial p_2} = -\\frac{q_1q_2}{m_1l_1l_2}sin(p_2-p_1) + m_2l_2gsin(p_2)\\\\\n", + "\\end{eqnarray*}$$" ] }, { @@ -198,20 +212,11 @@ }, { "cell_type": "code", - "execution_count": 1, + "execution_count": 19, "metadata": { "collapsed": false }, - "outputs": [ - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/home/randy/anaconda3/lib/python3.4/site-packages/IPython/kernel/__init__.py:13: ShimWarning: The `IPython.kernel` package has been deprecated. You should import from ipykernel or jupyter_client instead.\n", - " \"You should import from ipykernel or jupyter_client instead.\", ShimWarning)\n" - ] - } - ], + "outputs": [], "source": [ "import numpy\n", "from scipy.linalg import solve\n", @@ -228,7 +233,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 20, "metadata": { "collapsed": false }, @@ -264,7 +269,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 21, "metadata": { "collapsed": false }, @@ -302,7 +307,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 22, "metadata": { "collapsed": true }, @@ -327,7 +332,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 23, "metadata": { "collapsed": false }, @@ -353,7 +358,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 24, "metadata": { "collapsed": false }, @@ -1097,7 +1102,7 @@ { "data": { "text/html": [ - "
" + "" ], "text/plain": [ "" @@ -1845,7 +1850,7 @@ { "data": { "text/html": [ - "
" + "" ], "text/plain": [ "" @@ -2685,7 +2690,7 @@ { "data": { "text/html": [ - "
" + "" ], "text/plain": [ "" @@ -3527,7 +3532,7 @@ { "data": { "text/html": [ - "
" + "" ], "text/plain": [ "" @@ -4275,7 +4280,7 @@ { "data": { "text/html": [ - "
" + "" ], "text/plain": [ "" From 692463f0e41dc084b168207c9b08a622e0b378c3 Mon Sep 17 00:00:00 2001 From: rbds Date: Mon, 7 Dec 2015 23:33:58 -0500 Subject: [PATCH 37/61] Verlet for double pendulum - still needs checking --- randy_schur/Double_Pendulum_Problem.ipynb | 207 +++++++++++++++++++--- 1 file changed, 181 insertions(+), 26 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index 4b86a69..7d8febf 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -196,7 +196,7 @@ "$$\\begin{eqnarray*}\n", "\\dot{p_1} &=& -\\frac{\\partial H}{\\partial q_1} = -\\frac{q_1}{I_1} - \\frac{m_2l_1^2q_1}{m_1I_1} - \\frac{q_1}{l_2m_1l_1}cos(p_2-p_1) \\\\\n", "\\dot{q_1} &=& \\frac{\\partial H}{\\partial p_1} = \\frac{q_1q_2}{m_1l_1l_2}sin(p_2-p_1) + (m_1+m_2)gl_1sin(p_1) \\\\\n", - "\\dot{p_2} &=& -\\frac{\\partial H}{\\partial q_2} = -\\frac{q_1}{m_1l_1l_2}cos(p_2-p_1) - \\frac{q_2}{2I_2} \\\\\n", + "\\dot{p_2} &=& -\\frac{\\partial H}{\\partial q_2} = -\\frac{q_1}{m_1l_1l_2}cos(p_2-p_1) - \\frac{q_2}{I_2} \\\\\n", "\\dot{q_2} &=& \\frac{\\partial H}{\\partial p_2} = -\\frac{q_1q_2}{m_1l_1l_2}sin(p_2-p_1) + m_2l_2gsin(p_2)\\\\\n", "\\end{eqnarray*}$$" ] @@ -212,7 +212,7 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 25, "metadata": { "collapsed": false }, @@ -233,7 +233,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 31, "metadata": { "collapsed": false }, @@ -269,7 +269,7 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 32, "metadata": { "collapsed": false }, @@ -307,7 +307,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 33, "metadata": { "collapsed": true }, @@ -332,7 +332,7 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 34, "metadata": { "collapsed": false }, @@ -358,7 +358,7 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 35, "metadata": { "collapsed": false }, @@ -1906,7 +1906,7 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 52, "metadata": { "collapsed": false }, @@ -1926,20 +1926,20 @@ " p2 = u[2]\n", " q2 = u[3]\n", " \n", - " p1_half = p1 + dt/2*(q1/I1)\n", - " q1 = q1 - dt*((m1+m2)*g*l1*sin(p1_half))\n", - " p1 = p1_half + dt/2*(q1/I1)\n", + " p1_half = p1 - dt/2*(q1/I1 + m2*l1**2*q1/(m1*I1) + q1/(m1*l1*l2)*cos(p2-p1))\n", + " q1 = q1 + dt*(q1*q2/(m1*l1*l2)*sin(p2-p1_half) + (m1+m2)*g*l1*sin(p1_half))\n", + " p1 = p1_half - dt/2*(q1/I1 + m2*l1**2*q1/(m1*I1) + q1/(m1*l1*l2)*cos(p2-p1))\n", " \n", - " p2_half = p2 + dt/2*(q2/I2)\n", - " q2 = q2 - dt*m2*g*l2*sin(p2_half)\n", - " p2 = p2_half + dt/2*(q2/I2)\n", + " p2_half = p2 - dt/2*(q1/(m1*l1*l2)*cos(p2-p1) - q2/I2)\n", + " q2 = q2 - dt*(-q1*q2/(m1*l1*l2)*sin(p2_half-p1) + m2*l2*g*sin(p2))\n", + " p2 = p2_half - dt/2*(q1/(m1*l1*l2)*cos(p2-p1) - q2/I2)\n", " \n", " return numpy.array([p1, q1, p2, q2]) " ] }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 54, "metadata": { "collapsed": false }, @@ -2690,7 +2690,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2713,7 +2713,7 @@ "pyplot.xlabel(r't', fontsize=18);\n", "pyplot.ylabel(r'position (meters)', fontsize=18);\n", "pyplot.title('Double Pendulum Euler');\n", - "pyplot.plot(t, q3_dp[:,0], lw=2, label='Joint 1');\n", + "#pyplot.plot(t, q3_dp[:,0], lw=2, label='Joint 1');\n", "pyplot.plot(t, q3_dp[:,2], lw=2, label='Joint 2')\n", "pyplot.legend();" ] @@ -2751,7 +2751,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 38, "metadata": { "collapsed": false }, @@ -2788,7 +2788,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 39, "metadata": { "collapsed": false }, @@ -3532,7 +3532,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -4280,7 +4280,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -4339,7 +4339,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 40, "metadata": { "collapsed": false }, @@ -4376,7 +4376,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 41, "metadata": { "collapsed": false }, @@ -5158,7 +5158,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 42, "metadata": { "collapsed": false }, @@ -5204,11 +5204,166 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 43, "metadata": { "collapsed": false }, - "outputs": [], + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "execution_count": 43, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "# This cell loads the style of the notebook, which is modified from the \n", "# Numerical Methods in Python Course: http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about\n", From 987be27ff25b5472fa3068e69d7210e17f117cc3 Mon Sep 17 00:00:00 2001 From: rbds Date: Mon, 7 Dec 2015 23:37:01 -0500 Subject: [PATCH 38/61] evening commit --- randy_schur/Double_Pendulum_Problem.ipynb | 15 +++++++++++---- 1 file changed, 11 insertions(+), 4 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index 7d8febf..a8ec71d 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -1906,7 +1906,7 @@ }, { "cell_type": "code", - "execution_count": 52, + "execution_count": 55, "metadata": { "collapsed": false }, @@ -1939,7 +1939,7 @@ }, { "cell_type": "code", - "execution_count": 54, + "execution_count": 56, "metadata": { "collapsed": false }, @@ -2720,12 +2720,19 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 57, "metadata": { "collapsed": true }, "outputs": [], - "source": [] + "source": [ + "#To Do:\n", + "# plot total energy in system\n", + "# animate motion of double pendulum\n", + "# Poincare section for symplectic DP\n", + "# animate Poincare sections\n", + "# check through sources" + ] }, { "cell_type": "markdown", From 16024076179d1f7fd75b421889c7edf4d7e6c565 Mon Sep 17 00:00:00 2001 From: rbds Date: Tue, 8 Dec 2015 16:48:23 -0500 Subject: [PATCH 39/61] fix naming convention --- randy_schur/Double_Pendulum_Problem.ipynb | 97 +++++---- ...on Techniques for Mechanical Systems.ipynb | 198 +++++++++--------- 2 files changed, 157 insertions(+), 138 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index a8ec71d..e4275e4 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -173,7 +173,7 @@ "$$\\begin{eqnarray*}\n", "T &=& \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2+\\frac{1}{2}m_2\\left(2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+l_2^2\\dot\\theta_2^2\\right)\\\\\n", "&=& \\frac{1}{2}(I_1 + m_2l_1^2)\\omega_1^2+\\left[m_2l_1l_2\\omega_1\\omega_2cos(\\theta_2-\\theta_1)+\\frac{I_2\\omega_2^2}{2}\\right] \\\\\n", - "&=& \\frac{q_1^2}{2I_1}+\\frac{m_2l_1^2q_1^2}{2m_1I_1} + \\left[ \\frac{q_2}{l_2} \\frac{q_1}{m_1l_1}cos(p_2-p_1) + \\frac{q_2^2}{2I_2} \\right]\n", + "&=& \\frac{q_1^2}{2I_1}+\\frac{m_2q_1^2}{2m_1I_1} + \\left[ \\frac{q_2}{l_2} \\frac{q_1}{m_1l_1}cos(p_2-p_1) + \\frac{q_2^2}{2I_2} \\right]\n", "\\end{eqnarray*}$$\n", "\n" ] @@ -184,7 +184,7 @@ "source": [ "where $I_i$ is the angular moment of inertia and $\\omega$ is the angular velocity. So our Hamiltonian quantity becomes:\n", "$$\\begin{equation*}\n", - "H = \\frac{q_1^2}{2I_1}+\\frac{m_2l_1^2q_1^2}{2m_1I_1} + \\left[ \\frac{q_2}{l_2} \\frac{q_1}{m_1l_1}cos(p_2-p_1) + \\frac{q_2^2}{2I_2} \\right] -m_1gl_1cos(p_1) - m_2g\\left( l_1cos(p_1) + l_2cos(p_2) \\right)\n", + "H = \\frac{q_1^2}{2I_1}+\\frac{m_2q_1^2}{2m_1I_1} + \\left[ \\frac{q_2}{l_2} \\frac{q_1}{m_1l_1}cos(p_2-p_1) + \\frac{q_2^2}{2I_2} \\right] -m_1gl_1cos(p_1) - m_2g\\left( l_1cos(p_1) + l_2cos(p_2) \\right)\n", "\\end{equation*}$$" ] }, @@ -194,7 +194,7 @@ "source": [ "Since we have two degrees of freedom, we will have four equations of motion. These will be the following:\n", "$$\\begin{eqnarray*}\n", - "\\dot{p_1} &=& -\\frac{\\partial H}{\\partial q_1} = -\\frac{q_1}{I_1} - \\frac{m_2l_1^2q_1}{m_1I_1} - \\frac{q_1}{l_2m_1l_1}cos(p_2-p_1) \\\\\n", + "\\dot{p_1} &=& -\\frac{\\partial H}{\\partial q_1} = -\\frac{q_1}{I_1} - \\frac{m_2q_1}{m_1I_1} - \\frac{q_1}{l_2m_1l_1}cos(p_2-p_1) \\\\\n", "\\dot{q_1} &=& \\frac{\\partial H}{\\partial p_1} = \\frac{q_1q_2}{m_1l_1l_2}sin(p_2-p_1) + (m_1+m_2)gl_1sin(p_1) \\\\\n", "\\dot{p_2} &=& -\\frac{\\partial H}{\\partial q_2} = -\\frac{q_1}{m_1l_1l_2}cos(p_2-p_1) - \\frac{q_2}{I_2} \\\\\n", "\\dot{q_2} &=& \\frac{\\partial H}{\\partial p_2} = -\\frac{q_1q_2}{m_1l_1l_2}sin(p_2-p_1) + m_2l_2gsin(p_2)\\\\\n", @@ -212,11 +212,20 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 2, "metadata": { "collapsed": false }, - "outputs": [], + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "C:\\Users\\rschur\\Anaconda2\\lib\\site-packages\\IPython\\kernel\\__init__.py:13: ShimWarning: The `IPython.kernel` package has been deprecated. You should import from ipykernel or jupyter_client instead.\n", + " \"You should import from ipykernel or jupyter_client instead.\", ShimWarning)\n" + ] + } + ], "source": [ "import numpy\n", "from scipy.linalg import solve\n", @@ -233,7 +242,7 @@ }, { "cell_type": "code", - "execution_count": 31, + "execution_count": 3, "metadata": { "collapsed": false }, @@ -269,7 +278,7 @@ }, { "cell_type": "code", - "execution_count": 32, + "execution_count": 4, "metadata": { "collapsed": false }, @@ -307,7 +316,7 @@ }, { "cell_type": "code", - "execution_count": 33, + "execution_count": 5, "metadata": { "collapsed": true }, @@ -332,7 +341,7 @@ }, { "cell_type": "code", - "execution_count": 34, + "execution_count": 6, "metadata": { "collapsed": false }, @@ -358,7 +367,7 @@ }, { "cell_type": "code", - "execution_count": 35, + "execution_count": 7, "metadata": { "collapsed": false }, @@ -1102,7 +1111,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1850,7 +1859,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1906,7 +1915,7 @@ }, { "cell_type": "code", - "execution_count": 55, + "execution_count": 8, "metadata": { "collapsed": false }, @@ -1926,9 +1935,9 @@ " p2 = u[2]\n", " q2 = u[3]\n", " \n", - " p1_half = p1 - dt/2*(q1/I1 + m2*l1**2*q1/(m1*I1) + q1/(m1*l1*l2)*cos(p2-p1))\n", + " p1_half = p1 - dt/2*(q1/I1 + m2**q1/(m1*I1) + q1/(m1*l1*l2)*cos(p2-p1))\n", " q1 = q1 + dt*(q1*q2/(m1*l1*l2)*sin(p2-p1_half) + (m1+m2)*g*l1*sin(p1_half))\n", - " p1 = p1_half - dt/2*(q1/I1 + m2*l1**2*q1/(m1*I1) + q1/(m1*l1*l2)*cos(p2-p1))\n", + " p1 = p1_half - dt/2*(q1/I1 + m2**q1/(m1*I1) + q1/(m1*l1*l2)*cos(p2-p1))\n", " \n", " p2_half = p2 - dt/2*(q1/(m1*l1*l2)*cos(p2-p1) - q2/I2)\n", " q2 = q2 - dt*(-q1*q2/(m1*l1*l2)*sin(p2_half-p1) + m2*l2*g*sin(p2))\n", @@ -1939,7 +1948,7 @@ }, { "cell_type": "code", - "execution_count": 56, + "execution_count": 9, "metadata": { "collapsed": false }, @@ -2690,7 +2699,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2720,7 +2729,7 @@ }, { "cell_type": "code", - "execution_count": 57, + "execution_count": 10, "metadata": { "collapsed": true }, @@ -2758,7 +2767,7 @@ }, { "cell_type": "code", - "execution_count": 38, + "execution_count": 11, "metadata": { "collapsed": false }, @@ -2795,7 +2804,7 @@ }, { "cell_type": "code", - "execution_count": 39, + "execution_count": 12, "metadata": { "collapsed": false }, @@ -3539,7 +3548,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -4287,7 +4296,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -4346,7 +4355,7 @@ }, { "cell_type": "code", - "execution_count": 40, + "execution_count": 13, "metadata": { "collapsed": false }, @@ -4383,7 +4392,7 @@ }, { "cell_type": "code", - "execution_count": 41, + "execution_count": 15, "metadata": { "collapsed": false }, @@ -5134,7 +5143,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -5142,6 +5151,16 @@ }, "metadata": {}, "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "[]" + ] + }, + "execution_count": 15, + "metadata": {}, + "output_type": "execute_result" } ], "source": [ @@ -5153,19 +5172,19 @@ "pyplot.ylabel(r'$\\dot{\\theta_2}$', fontsize=18);#pyplot.title('Poincare Section');\n", "pyplot.plot(v[0,:], v[1,:], 'bo', lw=2, );\n", "\n", - "from matplotlib import animation\n", - "from JSAnimation.IPython_display import display_animation\n", - "def animate(data):\n", - " print(numpy.shape(data))\n", - " x = data[0,:]\n", - " y = data[1,:]\n", - " line.set_data(x, y)\n", - " return line," + "#from matplotlib import animation\n", + "#from JSAnimation.IPython_display import display_animation\n", + "#def animate(data):\n", + "# print(numpy.shape(data))\n", + "# x = data[0,:]\n", + "# y = data[1,:]\n", + "# line.set_data(x, y)\n", + "# return line," ] }, { "cell_type": "code", - "execution_count": 42, + "execution_count": 16, "metadata": { "collapsed": false }, @@ -5392,21 +5411,21 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3", + "display_name": "Python 2", "language": "python", - "name": "python3" + "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", - "version": 3 + "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.4.3" + "pygments_lexer": "ipython2", + "version": "2.7.11" } }, "nbformat": 4, diff --git a/randy_schur/Integration Techniques for Mechanical Systems.ipynb b/randy_schur/Integration Techniques for Mechanical Systems.ipynb index dec9b79..5d7175b 100644 --- a/randy_schur/Integration Techniques for Mechanical Systems.ipynb +++ b/randy_schur/Integration Techniques for Mechanical Systems.ipynb @@ -157,25 +157,25 @@ "$$ H = T+V$$\n", "where $T$ is the kinetic energy in the system, and $V$ is the potential energy in the system. So, we get:\n", "$$\\begin{eqnarray*}\n", - "T &=& \\frac{1}{2}m\\dot{x}^2 = \\frac{q^2}{2m}\\\\\n", - "V &=& \\frac{1}{2}k{x}^2 = \\frac{kp^2}{2} \n", + "T &=& \\frac{1}{2}m\\dot{x}^2 = \\frac{p^2}{2m}\\\\\n", + "V &=& \\frac{1}{2}k{x}^2 = \\frac{kq^2}{2} \n", "\\end{eqnarray*}$$\n", "\n", "and \n", "\n", - "$$H = \\frac{q^2}{2m} + \\frac{kp^2}{2}$$\n", + "$$H = \\frac{p^2}{2m} + \\frac{kq^2}{2}$$\n", "\n", "\n", "\n", "The Hamilton's equations are:\n", "$$\\begin{eqnarray*}\n", - "\\frac{dp}{dt} &=& -\\frac{\\partial H}{\\partial q}\\\\\n", - "\\frac{dq}{dt} &=& \\frac{\\partial H}{\\partial p}\n", + "\\frac{dq}{dt} &=& -\\frac{\\partial H}{\\partial p}\\\\\n", + "\\frac{dp}{dt} &=& \\frac{\\partial H}{\\partial q}\n", "\\end{eqnarray*}$$\n", "\n", "So our equations of motion become:\n", "$$\\begin{equation*}\n", - "\\begin{bmatrix} \\dot p \\\\ \\dot q \\end{bmatrix} = \\begin{bmatrix} -\\frac{\\partial H}{\\partial q}\\\\ \\frac{\\partial H}{\\partial p} \\end{bmatrix} = \\begin{bmatrix} -\\frac{q}{m} \\\\ kp\\end{bmatrix}\n", + "\\begin{bmatrix} \\dot q \\\\ \\dot p \\end{bmatrix} = \\begin{bmatrix} -\\frac{\\partial H}{\\partial p}\\\\ \\frac{\\partial H}{\\partial q} \\end{bmatrix} = \\begin{bmatrix} -\\frac{p}{m} \\\\ kq\\end{bmatrix}\n", "\\end{equation*}$$" ] }, @@ -222,7 +222,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 3, "metadata": { "collapsed": false }, @@ -231,7 +231,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "/home/randy/anaconda3/lib/python3.4/site-packages/IPython/kernel/__init__.py:13: ShimWarning: The `IPython.kernel` package has been deprecated. You should import from ipykernel or jupyter_client instead.\n", + "C:\\Users\\rschur\\Anaconda2\\lib\\site-packages\\IPython\\kernel\\__init__.py:13: ShimWarning: The `IPython.kernel` package has been deprecated. You should import from ipykernel or jupyter_client instead.\n", " \"You should import from ipykernel or jupyter_client instead.\", ShimWarning)\n" ] } @@ -252,7 +252,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 4, "metadata": { "collapsed": false }, @@ -270,7 +270,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 5, "metadata": { "collapsed": true }, @@ -293,7 +293,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 6, "metadata": { "collapsed": true }, @@ -320,7 +320,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 7, "metadata": { "collapsed": true }, @@ -344,7 +344,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 8, "metadata": { "collapsed": true }, @@ -362,18 +362,18 @@ " pos = u[0]\n", " mom = u[1]\n", " \n", - " p_half = pos - dt/2*mom/m\n", - " q_half = mom + dt/2*(k*pos)\n", + " q_half = pos - dt/2*mom/m\n", + " #p_half = mom + dt/2*(k*pos)\n", " \n", - " q = mom + dt/2*(2*k*p_half)\n", - " p = p_half - dt/2*q/m\n", + " p = mom + dt/2*(2*k*q_half)\n", + " q = q_half - dt/2*p/m\n", " \n", - " return numpy.array([p, q]) " + " return numpy.array([q, p]) " ] }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 9, "metadata": { "collapsed": true }, @@ -384,16 +384,16 @@ "xdot0 = 0 #[m/s]\n", "\n", "#Equivalent initial conditions for the Hamiltonian system\n", - "p0 = x0\n", - "q0 = m*xdot0\n", + "q0 = x0\n", + "p0 = m*xdot0\n", "\n", "x_init = numpy.array([x0, xdot0])\n", - "x_init_H = numpy.array([p0, q0])" + "x_init_H = numpy.array([q0, p0])" ] }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 10, "metadata": { "collapsed": false }, @@ -1137,7 +1137,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1149,29 +1149,29 @@ ], "source": [ "#Euler Integration\n", - "q1 = numpy.zeros((N,2)) #initialize array\n", - "q1[0] = x_init.copy() #set initial conditions\n", + "x1 = numpy.zeros((N,2)) #initialize array\n", + "x1[0] = x_init.copy() #set initial conditions\n", "for n in range(N-1): #integrate with Euler\n", - " q1[n+1] = euler(q1[n], f_harmonic_oscillator, dt)\n", + " x1[n+1] = euler(x1[n], f_harmonic_oscillator, dt)\n", " #print(q1[n])\n", " \n", "#Runge-Kutta Integration\n", - "q2 = numpy.zeros((N,2))\n", - "q2[0] = x_init.copy()\n", + "x2 = numpy.zeros((N,2))\n", + "x2[0] = x_init.copy()\n", "for n in range(N-1):\n", - " q2[n+1] = RK4(q2[n], f_harmonic_oscillator, dt)\n", + " x2[n+1] = RK4(x2[n], f_harmonic_oscillator, dt)\n", "\n", "#Symplectic Integration\n", - "q3 = numpy.zeros((N,2))\n", - "q3[0] = x_init_H.copy()\n", + "x3 = numpy.zeros((N,2))\n", + "x3[0] = x_init_H.copy()\n", "for n in range(N-1):\n", - " q3[n+1] = verlet_SHM(q3[n], dt)\n", + " x3[n+1] = verlet_SHM(x3[n], dt)\n", "\n", "#Analytical Solution \n", "A = x0 #with no forcing and no initial velocity, the max amplitude is equal to initial amplitude\n", "phi = 0\n", "omega = sqrt(k/m)\n", - "q_analytical = A*numpy.cos(omega*t + phi)\n", + "x_analytical = A*numpy.cos(omega*t + phi)\n", "#analytical velocity:\n", "vel_analytical = -A*omega*numpy.sin(omega*t + phi)\n", " \n", @@ -1180,10 +1180,10 @@ "pyplot.xlabel(r't', fontsize=18);\n", "pyplot.ylabel(r'position (meters)', fontsize=18);\n", "pyplot.title('Harmonic oscillator position');\n", - "pyplot.plot(t, q1[:,0], lw=2, label='Euler');\n", - "pyplot.plot(t, q2[:,0], 'r-', lw=2, label='RK4');\n", - "pyplot.plot(t, q3[:,0], 'g', lw=2, label='Verlet');\n", - "pyplot.plot(t, q_analytical, 'k--', lw=2, label='analytical');\n", + "pyplot.plot(t, x1[:,0], lw=2, label='Euler');\n", + "pyplot.plot(t, x2[:,0], 'r-', lw=2, label='RK4');\n", + "pyplot.plot(t, x3[:,0], 'g', lw=2, label='Verlet');\n", + "pyplot.plot(t, x_analytical, 'k--', lw=2, label='analytical');\n", "pyplot.legend();" ] }, @@ -1199,7 +1199,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 11, "metadata": { "collapsed": true }, @@ -1222,7 +1222,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 12, "metadata": { "collapsed": true }, @@ -1235,15 +1235,15 @@ "Verlet_error_values = Euler_error_values.copy()\n", "for i, dt in enumerate(dt_values):\n", " ###Call error function\n", - " Euler_error_values[i] = get_error(q1[:,0], q_analytical, dt)\n", - " RK_error_values[i] = get_error(q2[:,0], q_analytical, dt)\n", - " Verlet_error_values[i] = get_error(q3[:,0], q_analytical, dt)\n", + " Euler_error_values[i] = get_error(x1[:,0], x_analytical, dt)\n", + " RK_error_values[i] = get_error(x2[:,0], x_analytical, dt)\n", + " Verlet_error_values[i] = get_error(x3[:,0], x_analytical, dt)\n", " \n" ] }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 13, "metadata": { "collapsed": false }, @@ -1987,7 +1987,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2032,7 +2032,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 16, "metadata": { "collapsed": true }, @@ -2055,42 +2055,42 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 17, "metadata": { - "collapsed": true + "collapsed": false }, "outputs": [], "source": [ "#Euler total energy\n", "Euler_energy = numpy.zeros_like(t)\n", "for i in range(N):\n", - " Euler_energy[i] = get_Energy(q1[i,:])\n", + " Euler_energy[i] = get_Energy(x1[i,:])\n", " \n", "\n", "#RK4 total energy\n", "RK4_energy = numpy.zeros_like(t)\n", "for i in range(N):\n", - " RK4_energy[i] = get_Energy(q2[i,:])\n", + " RK4_energy[i] = get_Energy(x2[i,:])\n", "\n", "#Verlet total energy\n", "#Here we will convert from momentum to velocity so we can use the same get_energy function.\n", "#Momemtum (q) is mass*velocity.\n", "Verlet_energy = numpy.zeros_like(t)\n", - "Verlet_vel = q3[:,1]/m\n", + "Verlet_vel = x3[:,1]/m\n", "for i in range(N):\n", - " Verlet_energy[i] = get_Energy(numpy.array([q3[i,0], Verlet_vel[i]]))\n", + " Verlet_energy[i] = get_Energy(numpy.array([x3[i,0], Verlet_vel[i]]))\n", "\n", "\n", "#analytic total energy\n", "analytical_energy = numpy.zeros_like(t)\n", - "anayltical_state = numpy.array([q_analytical, vel_analytical]).T\n", + "anayltical_state = numpy.array([x_analytical, vel_analytical]).T\n", "for i in range(N):\n", " analytical_energy[i] = get_Energy(anayltical_state[i,:])" ] }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 18, "metadata": { "collapsed": false }, @@ -2834,7 +2834,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2918,7 +2918,7 @@ "Now that we have this, we can add a similar term to the Hamiltonian equations of motion to deal with the same problem. For more information on the derivation of the Hamiltonian from the Lagrangian, see Tveter (cite). The Hamiltonian equations of motion become:\n", "\n", "$$\\begin{equation*}\n", - "\\begin{bmatrix} \\dot p \\\\ \\dot q \\end{bmatrix} = \\begin{bmatrix} -\\frac{\\partial H}{\\partial q} + Q\\\\ \\frac{\\partial H}{\\partial p} \\end{bmatrix} = \\begin{bmatrix} -\\frac{q}{m} - \\frac{c}{m}q \\\\ kp \\end{bmatrix}\n", + "\\begin{bmatrix} \\dot q \\\\ \\dot p \\end{bmatrix} = \\begin{bmatrix} -\\frac{\\partial H}{\\partial p} + Q\\\\ \\frac{\\partial H}{\\partial q} \\end{bmatrix} = \\begin{bmatrix} -\\frac{p}{m} - \\frac{c}{m}p \\\\ kq \\end{bmatrix}\n", "\\end{equation*}$$\n", "\n", "From here, the we can use the same Verlet integration method as before. If this explanation has too much hand-waving for you, please see [7]." @@ -2926,7 +2926,7 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 19, "metadata": { "collapsed": true }, @@ -2950,7 +2950,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 21, "metadata": { "collapsed": true }, @@ -2967,19 +2967,19 @@ " pos = u[0]\n", " mom = u[1]\n", " \n", - " p_half = pos - dt/2*(mom/m)\n", + " q_half = pos - dt/2*(mom/m)\n", " #q_half = mom + dt/2*(k*pos)\n", " \n", - " q = mom + dt*(k*p_half - c/m*mom)\n", - " p = p_half - dt/2*(q/m)\n", + " p = mom + dt*(k*q_half - c/m*mom)\n", + " q = q_half - dt/2*(p/m)\n", " \n", - " return numpy.array([p, q]) \n", + " return numpy.array([q, p]) \n", " " ] }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 22, "metadata": { "collapsed": true }, @@ -2997,11 +2997,11 @@ "x0 = 1 #[m]\n", "xdot0 = 0 #[m/s]\n", "\n", - "p0 = x0\n", - "q0 = m*xdot0\n", + "q0 = x0\n", + "p0 = m*xdot0\n", "\n", "x_init = numpy.array([x0, xdot0])\n", - "x_init_H = numpy.array([p0, q0])\n", + "x_init_H = numpy.array([q0, p0])\n", "zeta = 0.1 #[N/(m/s)]\n", "wn = sqrt(k/m)\n", "wd = wn*sqrt(1-zeta**2)\n", @@ -3010,7 +3010,7 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 23, "metadata": { "collapsed": false }, @@ -3754,7 +3754,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -3766,28 +3766,28 @@ ], "source": [ "#Euler\n", - "q1_d = numpy.zeros((N,2)) \n", - "q1_d[0] = x_init.copy() #set initial conditions\n", + "x1_d = numpy.zeros((N,2)) \n", + "x1_d[0] = x_init.copy() #set initial conditions\n", "for n in range(N-1): #integrate with Euler\n", - " q1_d[n+1] = euler(q1_d[n], f_damped_HM, dt)\n", + " x1_d[n+1] = euler(x1_d[n], f_damped_HM, dt)\n", " #print(q1[n])\n", " \n", "#Runge-Kutta \n", - "q2_d = numpy.zeros((N,2))\n", - "q2_d[0] = x_init.copy()\n", + "x2_d = numpy.zeros((N,2))\n", + "x2_d[0] = x_init.copy()\n", "for n in range(N-1):\n", - " q2_d[n+1] = RK4(q2_d[n], f_damped_HM, dt)\n", + " x2_d[n+1] = RK4(x2_d[n], f_damped_HM, dt)\n", "\n", "#Symplectic\n", - "q3_d = numpy.zeros((N,2))\n", - "q3_d[0] = x_init_H.copy()\n", + "x3_d = numpy.zeros((N,2))\n", + "x3_d[0] = x_init_H.copy()\n", "for n in range(N-1):\n", - " q3_d[n+1] = Verlet_damped_HM(q3_d[n], dt)\n", + " x3_d[n+1] = Verlet_damped_HM(x3_d[n], dt)\n", "\n", "#Analytical Solution \n", "A = x0 #with no forcing, the max amplitude is equal to initial amplitude\n", "phi = 0\n", - "q_d_analytical = A*numpy.exp(-zeta*wn*t)*numpy.cos(wd*t + phi)\n", + "x_d_analytical = A*numpy.exp(-zeta*wn*t)*numpy.cos(wd*t + phi)\n", "vel_d_analytical = -A*numpy.exp(-zeta*wn*t)*wd*numpy.sin(wd*t+phi)\n", " \n", "pyplot.figure(figsize=(10,8));\n", @@ -3795,10 +3795,10 @@ "pyplot.xlabel(r't', fontsize=18);\n", "pyplot.ylabel(r'position (meters)', fontsize=18);\n", "pyplot.title('Harmonic oscillator position');\n", - "pyplot.plot(t, q1_d[:,0], lw=2, label='Euler');\n", - "pyplot.plot(t, q2_d[:,0], 'r-', lw=2, label='RK4');\n", - "pyplot.plot(t, q3_d[:,0], 'g', lw=2, label='Verlet');\n", - "pyplot.plot(t, q_d_analytical, 'k--', lw=2, label='analytical');\n", + "pyplot.plot(t, x1_d[:,0], lw=2, label='Euler');\n", + "pyplot.plot(t, x2_d[:,0], 'r-', lw=2, label='RK4');\n", + "pyplot.plot(t, x3_d[:,0], 'g', lw=2, label='Verlet');\n", + "pyplot.plot(t, x_d_analytical, 'k--', lw=2, label='analytical');\n", "pyplot.legend();\n" ] }, @@ -4558,7 +4558,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -4572,26 +4572,26 @@ "#Euler total energy\n", "Euler_energy_d = numpy.zeros_like(t)\n", "for i in range(N):\n", - " Euler_energy_d[i] = get_Energy(q1_d[i,:])\n", + " Euler_energy_d[i] = get_Energy(x1_d[i,:])\n", " \n", "\n", "#RK4 total energy\n", "RK4_energy_d = numpy.zeros_like(t)\n", "for i in range(N):\n", - " RK4_energy_d[i] = get_Energy(q2_d[i,:])\n", + " RK4_energy_d[i] = get_Energy(x2_d[i,:])\n", "\n", "#Verlet total energy\n", "#Here we will convert from momentum to velocity so we can use the same get_energy function.\n", "#Momemtum (q) is mass*velocity.\n", "Verlet_energy_d = numpy.zeros_like(t)\n", - "Verlet_vel_d = q3_d[:,1]/m\n", + "Verlet_vel_d = x3_d[:,1]/m\n", "for i in range(N):\n", - " Verlet_energy_d[i] = get_Energy(numpy.array([q3_d[i,0], Verlet_vel_d[i]]))\n", + " Verlet_energy_d[i] = get_Energy(numpy.array([x3_d[i,0], Verlet_vel_d[i]]))\n", "\n", "\n", "#analytic total energy\n", "analytical_energy_d = numpy.zeros_like(t)\n", - "anayltical_state_d = numpy.array([q_d_analytical, vel_d_analytical]).T\n", + "anayltical_state_d = numpy.array([x_d_analytical, vel_d_analytical]).T\n", "for i in range(N):\n", " analytical_energy_d[i] = get_Energy(anayltical_state_d[i,:])\n", " \n", @@ -4621,7 +4621,7 @@ }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 26, "metadata": { "collapsed": false }, @@ -5365,7 +5365,7 @@ { "data": { "text/html": [ - "
" + "" ], "text/plain": [ "" @@ -5396,10 +5396,10 @@ "\n", "pyplot.figure(figsize=(10,8));\n", "pyplot.quiver(X1, X2, u, v, pivot='mid', color='k', label='direction field');\n", - "pyplot.plot(q_d_analytical, vel_d_analytical, 'k-', label='analytical');\n", - "pyplot.plot(q1_d[:,0], q1_d[:,1],'r-', label='Euler');\n", - "pyplot.plot(q2_d[:,0], q2_d[:,1], 'b-', label='RK4');\n", - "pyplot.plot(q3_d[:,0], -Verlet_vel_d, 'g-', label='Verlet');\n", + "pyplot.plot(x_d_analytical, vel_d_analytical, 'k-', label='analytical');\n", + "pyplot.plot(x1_d[:,0], x1_d[:,1],'r-', label='Euler');\n", + "pyplot.plot(x2_d[:,0], x2_d[:,1], 'b-', label='RK4');\n", + "pyplot.plot(x3_d[:,0], -Verlet_vel_d, 'g-', label='Verlet');\n", "pyplot.xlabel('Postion ($x_1$)')\n", "pyplot.ylabel('Velocity ($x_2$)')\n", "pyplot.legend();\n", @@ -5622,21 +5622,21 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3", + "display_name": "Python 2", "language": "python", - "name": "python3" + "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", - "version": 3 + "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.4.3" + "pygments_lexer": "ipython2", + "version": "2.7.11" } }, "nbformat": 4, From d881ab9df349807778f9c31044dbe9e07c7a0419 Mon Sep 17 00:00:00 2001 From: rbds Date: Tue, 8 Dec 2015 16:53:19 -0500 Subject: [PATCH 40/61] fix naming convention - Double pendulum --- randy_schur/Double_Pendulum_Problem.ipynb | 100 ++++++++---------- ...on Techniques for Mechanical Systems.ipynb | 2 +- 2 files changed, 43 insertions(+), 59 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index e4275e4..53124ab 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -212,20 +212,11 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 17, "metadata": { "collapsed": false }, - "outputs": [ - { - "name": "stderr", - "output_type": "stream", - "text": [ - "C:\\Users\\rschur\\Anaconda2\\lib\\site-packages\\IPython\\kernel\\__init__.py:13: ShimWarning: The `IPython.kernel` package has been deprecated. You should import from ipykernel or jupyter_client instead.\n", - " \"You should import from ipykernel or jupyter_client instead.\", ShimWarning)\n" - ] - } - ], + "outputs": [], "source": [ "import numpy\n", "from scipy.linalg import solve\n", @@ -242,7 +233,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 18, "metadata": { "collapsed": false }, @@ -278,7 +269,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 19, "metadata": { "collapsed": false }, @@ -304,19 +295,19 @@ "theta1_dot_0 = 0 #[rad/s]\n", "theta2_dot_0 = 0 #\n", "\n", - "p1_0 = theta1_0\n", - "q1_0 = m1*theta1_dot_0\n", - "p2_0 = theta2_0\n", - "q2_0 = m2*theta2_dot_0\n", + "q1_0 = theta1_0\n", + "p1_0 = m1*theta1_dot_0\n", + "q2_0 = theta2_0\n", + "p2_0 = m2*theta2_dot_0\n", "\n", "x_init_dp = numpy.array([theta1_0, theta2_0, theta1_dot_0, theta2_dot_0])\n", - "x_init_H_dp = numpy.array([p1_0, q1_0, p2_0, q2_0])\n", + "x_init_H_dp = numpy.array([q1_0, p1_0, q2_0, p2_0])\n", "\n" ] }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 20, "metadata": { "collapsed": true }, @@ -341,7 +332,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 21, "metadata": { "collapsed": false }, @@ -367,7 +358,7 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 22, "metadata": { "collapsed": false }, @@ -1871,25 +1862,25 @@ ], "source": [ "#Euler\n", - "q1_dp = numpy.zeros((N,4)) \n", - "q1_dp[0,:] = x_init_dp.copy() #set initial conditions\n", + "x1_dp = numpy.zeros((N,4)) \n", + "x1_dp[0,:] = x_init_dp.copy() #set initial conditions\n", "for n in range(N-1): #integrate with Euler\n", - " q1_dp[n+1,:] = euler_DP(q1_dp[n,:], f_double_pendulum, dt)\n", + " x1_dp[n+1,:] = euler_DP(x1_dp[n,:], f_double_pendulum, dt)\n", " #print(q1[n])\n", " \n", "#Runge-Kutta\n", - "q2_dp = numpy.zeros((N,4)) \n", - "q2_dp[0,:] = x_init_dp.copy() #set initial conditions\n", + "x2_dp = numpy.zeros((N,4)) \n", + "x2_dp[0,:] = x_init_dp.copy() #set initial conditions\n", "for n in range(N-1): #integrate with Euler\n", - " q2_dp[n+1,:] = RK4_DP(q2_dp[n,:], f_double_pendulum, dt)\n", + " x2_dp[n+1,:] = RK4_DP(x2_dp[n,:], f_double_pendulum, dt)\n", " \n", "pyplot.figure(figsize=(10,8));\n", "pyplot.grid(True);\n", "pyplot.xlabel(r't', fontsize=18);\n", "pyplot.ylabel(r'position (meters)', fontsize=18);\n", "pyplot.title('Double Pendulum Euler');\n", - "pyplot.plot(t, q1_dp[:,0], lw=2, label='Joint 1');\n", - "pyplot.plot(t, q1_dp[:,1], lw=2, label='Joint 2')\n", + "pyplot.plot(t, x1_dp[:,0], lw=2, label='Joint 1');\n", + "pyplot.plot(t, x1_dp[:,1], lw=2, label='Joint 2')\n", "pyplot.legend();\n", "\n", "pyplot.figure(figsize=(10,8));\n", @@ -1897,8 +1888,8 @@ "pyplot.xlabel(r't', fontsize=18);\n", "pyplot.ylabel(r'position (meters)', fontsize=18);\n", "pyplot.title('Double Pendulum RK4');\n", - "pyplot.plot(t, q2_dp[:,0], lw=2, label='Joint 1');\n", - "pyplot.plot(t, q2_dp[:,1], lw=2, label='Joint 2')\n", + "pyplot.plot(t, x2_dp[:,0], lw=2, label='Joint 1');\n", + "pyplot.plot(t, x2_dp[:,1], lw=2, label='Joint 2')\n", "pyplot.legend();" ] }, @@ -1915,7 +1906,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 29, "metadata": { "collapsed": false }, @@ -1930,36 +1921,29 @@ " Returns: \n", " state at next time step.\n", " \"\"\"\n", - " p1 = u[0]\n", - " q1 = u[1]\n", - " p2 = u[2]\n", - " q2 = u[3]\n", + " q1 = u[0]\n", + " p1 = u[1]\n", + " q2 = u[2]\n", + " p2 = u[3]\n", " \n", - " p1_half = p1 - dt/2*(q1/I1 + m2**q1/(m1*I1) + q1/(m1*l1*l2)*cos(p2-p1))\n", - " q1 = q1 + dt*(q1*q2/(m1*l1*l2)*sin(p2-p1_half) + (m1+m2)*g*l1*sin(p1_half))\n", - " p1 = p1_half - dt/2*(q1/I1 + m2**q1/(m1*I1) + q1/(m1*l1*l2)*cos(p2-p1))\n", + " q1_half = q1 - dt/2*(q1/I1 + m2**q1/(m1*I1) + q1/(m1*l1*l2)*cos(p2-p1))\n", + " p1 = p1 + dt*(q1*q2/(m1*l1*l2)*sin(p2-q1_half) + (m1+m2)*g*l1*sin(q1_half))\n", + " q1 = q1_half - dt/2*(q1/I1 + m2**q1/(m1*I1) + q1/(m1*l1*l2)*cos(p2-p1))\n", " \n", - " p2_half = p2 - dt/2*(q1/(m1*l1*l2)*cos(p2-p1) - q2/I2)\n", - " q2 = q2 - dt*(-q1*q2/(m1*l1*l2)*sin(p2_half-p1) + m2*l2*g*sin(p2))\n", - " p2 = p2_half - dt/2*(q1/(m1*l1*l2)*cos(p2-p1) - q2/I2)\n", + " q2_half = q2 - dt/2*(q1/(m1*l1*l2)*cos(p2-p1) - q2/I2)\n", + " p2 = p2 - dt*(-q1*q2/(m1*l1*l2)*sin(q2_half-p1) + m2*l2*g*sin(p2))\n", + " q2 = q2_half - dt/2*(q1/(m1*l1*l2)*cos(p2-p1) - q2/I2)\n", " \n", - " return numpy.array([p1, q1, p2, q2]) " + " return numpy.array([q1, p1, q2, p2]) " ] }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[ 1. 0. 3.14159265 0. ]\n" - ] - }, { "data": { "application/javascript": [ @@ -2699,7 +2683,7 @@ { "data": { "text/html": [ - "" + "
" ], "text/plain": [ "" @@ -2711,19 +2695,19 @@ ], "source": [ "#Verlet integration\n", - "q3_dp = numpy.zeros((N,4)) \n", - "q3_dp[0,:] = x_init_H_dp.copy() #set initial conditions\n", - "print(q3_dp[0,:])\n", + "x3_dp = numpy.zeros((N,4)) \n", + "x3_dp[0,:] = x_init_H_dp.copy() #set initial conditions\n", + "#print(q3_dp[0,:])\n", "for n in range(N-1): #integrate with Euler\n", - " q3_dp[n+1,:] = Verlet_DP(q3_dp[n,:], dt)\n", + " x3_dp[n+1,:] = Verlet_DP(x3_dp[n,:], dt)\n", " \n", "pyplot.figure(figsize=(10,8));\n", "pyplot.grid(True);\n", "pyplot.xlabel(r't', fontsize=18);\n", "pyplot.ylabel(r'position (meters)', fontsize=18);\n", "pyplot.title('Double Pendulum Euler');\n", - "#pyplot.plot(t, q3_dp[:,0], lw=2, label='Joint 1');\n", - "pyplot.plot(t, q3_dp[:,2], lw=2, label='Joint 2')\n", + "pyplot.plot(t, x3_dp[:,0], lw=2, label='Joint 1');\n", + "pyplot.plot(t, x3_dp[:,2], lw=2, label='Joint 2')\n", "pyplot.legend();" ] }, diff --git a/randy_schur/Integration Techniques for Mechanical Systems.ipynb b/randy_schur/Integration Techniques for Mechanical Systems.ipynb index 5d7175b..9b21a8a 100644 --- a/randy_schur/Integration Techniques for Mechanical Systems.ipynb +++ b/randy_schur/Integration Techniques for Mechanical Systems.ipynb @@ -4621,7 +4621,7 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 27, "metadata": { "collapsed": false }, From 314a53d15e746f9756c8ebcbbe55a18e803a0f73 Mon Sep 17 00:00:00 2001 From: rbds Date: Tue, 8 Dec 2015 17:16:35 -0500 Subject: [PATCH 41/61] add figure --- randy_schur/Double_Pendulum_Problem.ipynb | 20 +- ...on Techniques for Mechanical Systems.ipynb | 1339 ++++++++++++----- 2 files changed, 1007 insertions(+), 352 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index 53124ab..3d5ecd5 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -1939,7 +1939,7 @@ }, { "cell_type": "code", - "execution_count": 30, + "execution_count": 38, "metadata": { "collapsed": false }, @@ -2683,7 +2683,7 @@ { "data": { "text/html": [ - "
" + "" ], "text/plain": [ "" @@ -2707,7 +2707,7 @@ "pyplot.ylabel(r'position (meters)', fontsize=18);\n", "pyplot.title('Double Pendulum Euler');\n", "pyplot.plot(t, x3_dp[:,0], lw=2, label='Joint 1');\n", - "pyplot.plot(t, x3_dp[:,2], lw=2, label='Joint 2')\n", + "#pyplot.plot(t, x3_dp[:,2], lw=2, label='Joint 2')\n", "pyplot.legend();" ] }, @@ -2751,7 +2751,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 32, "metadata": { "collapsed": false }, @@ -2788,7 +2788,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 33, "metadata": { "collapsed": false }, @@ -4339,7 +4339,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 34, "metadata": { "collapsed": false }, @@ -4376,7 +4376,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 35, "metadata": { "collapsed": false }, @@ -5139,10 +5139,10 @@ { "data": { "text/plain": [ - "[]" + "[]" ] }, - "execution_count": 15, + "execution_count": 35, "metadata": {}, "output_type": "execute_result" } @@ -5168,7 +5168,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 36, "metadata": { "collapsed": false }, diff --git a/randy_schur/Integration Techniques for Mechanical Systems.ipynb b/randy_schur/Integration Techniques for Mechanical Systems.ipynb index 9b21a8a..9df054b 100644 --- a/randy_schur/Integration Techniques for Mechanical Systems.ipynb +++ b/randy_schur/Integration Techniques for Mechanical Systems.ipynb @@ -220,180 +220,983 @@ "### Let's Get to Programming" ] }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "C:\\Users\\rschur\\Anaconda2\\lib\\site-packages\\IPython\\kernel\\__init__.py:13: ShimWarning: The `IPython.kernel` package has been deprecated. You should import from ipykernel or jupyter_client instead.\n", + " \"You should import from ipykernel or jupyter_client instead.\", ShimWarning)\n" + ] + } + ], + "source": [ + "import numpy\n", + "from scipy.linalg import solve\n", + "from numpy.linalg import det\n", + "from math import pi, cos, sin, sqrt\n", + "\n", + "from matplotlib import pyplot\n", + "from matplotlib.pyplot import quiver\n", + "%matplotlib notebook\n", + "from matplotlib import rcParams, cm\n", + "rcParams['font.family'] = 'serif'\n", + "rcParams['font.size'] = 16" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "#Set up system parameters:\n", + "m = 12\n", + "k = 150\n", + "\n", + "T = 25; #[seconds]\n", + "dt = .02; #time step size\n", + "N = int(T/dt)+1 #number of time steps\n", + "t = numpy.linspace(0.0, T, N) #array of time values" + ] + }, { "cell_type": "code", "execution_count": 3, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def euler(u, f, dt):\n", + " \"\"\" Euler's method for integrating a system of differential equations.\n", + " \n", + " Parameters:\n", + " u - state at current step \n", + " f - RHS of equation\n", + " dt- time step size\n", + " \n", + " Returns: \n", + " x - array of values at next time step.\n", + " \"\"\"\n", + " \n", + " return u + dt*f(u) " + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def RK4(u, f, dt):\n", + " \"\"\"Runge Kutta fourth order integration method\n", + " \n", + " Parameters:\n", + " u - state of the system at time t\n", + " f - function for RHS of state equations\n", + " dt - time step\n", + " \n", + " Returns: \n", + " state values at next time step.\n", + " \"\"\"\n", + " k1 = f(u)\n", + " k2 = f(u) + 0.5*dt*k1\n", + " k3 = f(u) + 0.5*dt*k2\n", + " k4 = f(u) + dt*k3\n", + " \n", + " return u + dt/6*(k1+2*k2+2*k3+k4)" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def f_harmonic_oscillator(u):\n", + " \"\"\"Returns RHS of harmonic oscillator EOM\n", + " \n", + " Parameters:\n", + " u - initial state\n", + " \n", + " Returns:\n", + " RHS of harmonic oscillator eqn.\n", + " \n", + " \"\"\"\n", + " pos = u[0]\n", + " vel = u[1]\n", + " \n", + " return numpy.array([vel, -k/m*pos])" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def verlet_SHM(u, dt):\n", + " \"\"\" Verlet integration for integrating from Hamiltonian equations of motion\n", + " for a simple harmonic oscillator\n", + " Parameters:\n", + " u - state at current time step\n", + " dt - time step size\n", + " Returns: \n", + " state at next time step.\n", + " \"\"\"\n", + " pos = u[0]\n", + " mom = u[1]\n", + " \n", + " q_half = pos - dt/2*mom/m\n", + " #p_half = mom + dt/2*(k*pos)\n", + " \n", + " p = mom + dt/2*(2*k*q_half)\n", + " q = q_half - dt/2*p/m\n", + " \n", + " return numpy.array([q, p]) " + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "#Initial Conditions\n", + "x0 = 0.5 #[m]\n", + "xdot0 = 0 #[m/s]\n", + "\n", + "#Equivalent initial conditions for the Hamiltonian system\n", + "q0 = x0\n", + "p0 = m*xdot0\n", + "\n", + "x_init = numpy.array([x0, xdot0])\n", + "x_init_H = numpy.array([q0, p0])" + ] + }, + { + "cell_type": "code", + "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { - "name": "stderr", - "output_type": "stream", - "text": [ - "C:\\Users\\rschur\\Anaconda2\\lib\\site-packages\\IPython\\kernel\\__init__.py:13: ShimWarning: The `IPython.kernel` package has been deprecated. You should import from ipykernel or jupyter_client instead.\n", - " \"You should import from ipykernel or jupyter_client instead.\", ShimWarning)\n" - ] + "data": { + "application/javascript": [ + "/* Put everything inside the global mpl namespace */\n", + "window.mpl = {};\n", + "\n", + "mpl.get_websocket_type = function() {\n", + " if (typeof(WebSocket) !== 'undefined') {\n", + " return WebSocket;\n", + " } else if (typeof(MozWebSocket) !== 'undefined') {\n", + " return MozWebSocket;\n", + " } else {\n", + " alert('Your browser does not have WebSocket support.' +\n", + " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", + " 'Firefox 4 and 5 are also supported but you ' +\n", + " 'have to enable WebSockets in about:config.');\n", + " };\n", + "}\n", + "\n", + "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", + " this.id = figure_id;\n", + "\n", + " this.ws = websocket;\n", + "\n", + " this.supports_binary = (this.ws.binaryType != undefined);\n", + "\n", + " if (!this.supports_binary) {\n", + " var warnings = document.getElementById(\"mpl-warnings\");\n", + " if (warnings) {\n", + " warnings.style.display = 'block';\n", + " warnings.textContent = (\n", + " \"This browser does not support binary websocket messages. \" +\n", + " \"Performance may be slow.\");\n", + " }\n", + " }\n", + "\n", + " this.imageObj = new Image();\n", + "\n", + " this.context = undefined;\n", + " this.message = undefined;\n", + " this.canvas = undefined;\n", + " this.rubberband_canvas = undefined;\n", + " this.rubberband_context = undefined;\n", + " this.format_dropdown = undefined;\n", + "\n", + " this.image_mode = 'full';\n", + "\n", + " this.root = $('
');\n", + " this._root_extra_style(this.root)\n", + " this.root.attr('style', 'display: inline-block');\n", + "\n", + " $(parent_element).append(this.root);\n", + "\n", + " this._init_header(this);\n", + " this._init_canvas(this);\n", + " this._init_toolbar(this);\n", + "\n", + " var fig = this;\n", + "\n", + " this.waiting = false;\n", + "\n", + " this.ws.onopen = function () {\n", + " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", + " fig.send_message(\"send_image_mode\", {});\n", + " fig.send_message(\"refresh\", {});\n", + " }\n", + "\n", + " this.imageObj.onload = function() {\n", + " if (fig.image_mode == 'full') {\n", + " // Full images could contain transparency (where diff images\n", + " // almost always do), so we need to clear the canvas so that\n", + " // there is no ghosting.\n", + " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", + " }\n", + " fig.context.drawImage(fig.imageObj, 0, 0);\n", + " fig.waiting = false;\n", + " };\n", + "\n", + " this.imageObj.onunload = function() {\n", + " this.ws.close();\n", + " }\n", + "\n", + " this.ws.onmessage = this._make_on_message_function(this);\n", + "\n", + " this.ondownload = ondownload;\n", + "}\n", + "\n", + "mpl.figure.prototype._init_header = function() {\n", + " var titlebar = $(\n", + " '
');\n", + " var titletext = $(\n", + " '
');\n", + " titlebar.append(titletext)\n", + " this.root.append(titlebar);\n", + " this.header = titletext[0];\n", + "}\n", + "\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._init_canvas = function() {\n", + " var fig = this;\n", + "\n", + " var canvas_div = $('
');\n", + "\n", + " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", + "\n", + " function canvas_keyboard_event(event) {\n", + " return fig.key_event(event, event['data']);\n", + " }\n", + "\n", + " canvas_div.keydown('key_press', canvas_keyboard_event);\n", + " canvas_div.keyup('key_release', canvas_keyboard_event);\n", + " this.canvas_div = canvas_div\n", + " this._canvas_extra_style(canvas_div)\n", + " this.root.append(canvas_div);\n", + "\n", + " var canvas = $('');\n", + " canvas.addClass('mpl-canvas');\n", + " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", + "\n", + " this.canvas = canvas[0];\n", + " this.context = canvas[0].getContext(\"2d\");\n", + "\n", + " var rubberband = $('');\n", + " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", + "\n", + " var pass_mouse_events = true;\n", + "\n", + " canvas_div.resizable({\n", + " start: function(event, ui) {\n", + " pass_mouse_events = false;\n", + " },\n", + " resize: function(event, ui) {\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " stop: function(event, ui) {\n", + " pass_mouse_events = true;\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " });\n", + "\n", + " function mouse_event_fn(event) {\n", + " if (pass_mouse_events)\n", + " return fig.mouse_event(event, event['data']);\n", + " }\n", + "\n", + " rubberband.mousedown('button_press', mouse_event_fn);\n", + " rubberband.mouseup('button_release', mouse_event_fn);\n", + " // Throttle sequential mouse events to 1 every 20ms.\n", + " rubberband.mousemove('motion_notify', mouse_event_fn);\n", + "\n", + " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", + " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", + "\n", + " canvas_div.on(\"wheel\", function (event) {\n", + " event = event.originalEvent;\n", + " event['data'] = 'scroll'\n", + " if (event.deltaY < 0) {\n", + " event.step = 1;\n", + " } else {\n", + " event.step = -1;\n", + " }\n", + " mouse_event_fn(event);\n", + " });\n", + "\n", + " canvas_div.append(canvas);\n", + " canvas_div.append(rubberband);\n", + "\n", + " this.rubberband = rubberband;\n", + " this.rubberband_canvas = rubberband[0];\n", + " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", + " this.rubberband_context.strokeStyle = \"#000000\";\n", + "\n", + " this._resize_canvas = function(width, height) {\n", + " // Keep the size of the canvas, canvas container, and rubber band\n", + " // canvas in synch.\n", + " canvas_div.css('width', width)\n", + " canvas_div.css('height', height)\n", + "\n", + " canvas.attr('width', width);\n", + " canvas.attr('height', height);\n", + "\n", + " rubberband.attr('width', width);\n", + " rubberband.attr('height', height);\n", + " }\n", + "\n", + " // Set the figure to an initial 600x600px, this will subsequently be updated\n", + " // upon first draw.\n", + " this._resize_canvas(600, 600);\n", + "\n", + " // Disable right mouse context menu.\n", + " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", + " return false;\n", + " });\n", + "\n", + " function set_focus () {\n", + " canvas.focus();\n", + " canvas_div.focus();\n", + " }\n", + "\n", + " window.setTimeout(set_focus, 100);\n", + "}\n", + "\n", + "mpl.figure.prototype._init_toolbar = function() {\n", + " var fig = this;\n", + "\n", + " var nav_element = $('
')\n", + " nav_element.attr('style', 'width: 100%');\n", + " this.root.append(nav_element);\n", + "\n", + " // Define a callback function for later on.\n", + " function toolbar_event(event) {\n", + " return fig.toolbar_button_onclick(event['data']);\n", + " }\n", + " function toolbar_mouse_event(event) {\n", + " return fig.toolbar_button_onmouseover(event['data']);\n", + " }\n", + "\n", + " for(var toolbar_ind in mpl.toolbar_items) {\n", + " var name = mpl.toolbar_items[toolbar_ind][0];\n", + " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", + " var image = mpl.toolbar_items[toolbar_ind][2];\n", + " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", + "\n", + " if (!name) {\n", + " // put a spacer in here.\n", + " continue;\n", + " }\n", + " var button = $('');\n", - " button.click(method_name, toolbar_event);\n", - " button.mouseover(tooltip, toolbar_mouse_event);\n", - " nav_element.append(button);\n", - " }\n", - "\n", - " // Add the status bar.\n", - " var status_bar = $('');\n", - " nav_element.append(status_bar);\n", - " this.message = status_bar[0];\n", - "\n", - " // Add the close button to the window.\n", - " var buttongrp = $('
');\n", - " var button = $('');\n", - " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", - " buttongrp.append(button);\n", - " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", - " titlebar.prepend(buttongrp);\n", - "}\n", - "\n", - "\n", - "mpl.figure.prototype._canvas_extra_style = function(el){\n", - " // this is important to make the div 'focusable\n", - " el.attr('tabindex', 0)\n", - " // reach out to IPython and tell the keyboard manager to turn it's self\n", - " // off when our div gets focus\n", - "\n", - " // location in version 3\n", - " if (IPython.notebook.keyboard_manager) {\n", - " IPython.notebook.keyboard_manager.register_events(el);\n", - " }\n", - " else {\n", - " // location in version 2\n", - " IPython.keyboard_manager.register_events(el);\n", - " }\n", - "\n", - "}\n", - "\n", - "mpl.figure.prototype._key_event_extra = function(event, name) {\n", - " var manager = IPython.notebook.keyboard_manager;\n", - " if (!manager)\n", - " manager = IPython.keyboard_manager;\n", - "\n", - " // Check for shift+enter\n", - " if (event.shiftKey && event.which == 13) {\n", - " this.canvas_div.blur();\n", - " event.shiftKey = false;\n", - " // Send a \"J\" for go to next cell\n", - " event.which = 74;\n", - " event.keyCode = 74;\n", - " manager.command_mode();\n", - " manager.handle_keydown(event);\n", - " }\n", - "}\n", - "\n", - "mpl.figure.prototype.handle_save = function(fig, msg) {\n", - " fig.ondownload(fig, null);\n", - "}\n", - "\n", - "\n", - "mpl.find_output_cell = function(html_output) {\n", - " // Return the cell and output element which can be found *uniquely* in the notebook.\n", - " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", - " // IPython event is triggered only after the cells have been serialised, which for\n", - " // our purposes (turning an active figure into a static one), is too late.\n", - " var cells = IPython.notebook.get_cells();\n", - " var ncells = cells.length;\n", - " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", - " data = data.data;\n", - " }\n", - " if (data['text/html'] == html_output) {\n", - " return [cell, data, j];\n", - " }\n", - " }\n", - " }\n", - " }\n", - "}\n", - "\n", - "// Register the function which deals with the matplotlib target/channel.\n", - "// The kernel may be null if the page has been refreshed.\n", - "if (IPython.notebook.kernel != null) {\n", - " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", - "}\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/html": [ - "" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "def euler(u, f, dt):\n", + " \"\"\" Euler's method for integrating a system of differential equations.\n", + " \n", + " Parameters:\n", + " u - state at current step \n", + " f - RHS of equation\n", + " dt- time step size\n", + " \n", + " Returns: \n", + " x - array of values at next time step.\n", + " \"\"\"\n", + " \n", + " return u + dt*f(u) " + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": true + }, + "outputs": [], "source": [ - "#Euler Integration\n", - "x1 = numpy.zeros((N,2)) #initialize array\n", - "x1[0] = x_init.copy() #set initial conditions\n", - "for n in range(N-1): #integrate with Euler\n", - " x1[n+1] = euler(x1[n], f_harmonic_oscillator, dt)\n", - " #print(q1[n])\n", + "def RK4(u, f, dt):\n", + " \"\"\"Runge Kutta fourth order integration method\n", " \n", - "#Runge-Kutta Integration\n", - "x2 = numpy.zeros((N,2))\n", - "x2[0] = x_init.copy()\n", - "for n in range(N-1):\n", - " x2[n+1] = RK4(x2[n], f_harmonic_oscillator, dt)\n", - "\n", - "#Symplectic Integration\n", - "x3 = numpy.zeros((N,2))\n", - "x3[0] = x_init_H.copy()\n", - "for n in range(N-1):\n", - " x3[n+1] = verlet_SHM(x3[n], dt)\n", - "\n", - "#Analytical Solution \n", - "A = x0 #with no forcing and no initial velocity, the max amplitude is equal to initial amplitude\n", - "phi = 0\n", - "omega = sqrt(k/m)\n", - "x_analytical = A*numpy.cos(omega*t + phi)\n", - "#analytical velocity:\n", - "vel_analytical = -A*omega*numpy.sin(omega*t + phi)\n", + " Parameters:\n", + " u - state of the system at time t\n", + " f - function for RHS of state equations\n", + " dt - time step\n", + " \n", + " Returns: \n", + " state values at next time step.\n", + " \"\"\"\n", + " k1 = f(u)\n", + " k2 = f(u) + 0.5*dt*k1\n", + " k3 = f(u) + 0.5*dt*k2\n", + " k4 = f(u) + dt*k3\n", " \n", - "pyplot.figure(figsize=(12,8));\n", - "pyplot.grid(True);\n", - "pyplot.xlabel(r't', fontsize=18);\n", - "pyplot.ylabel(r'position (meters)', fontsize=18);\n", - "pyplot.title('Harmonic oscillator position');\n", - "pyplot.plot(t, x1[:,0], lw=2, label='Euler');\n", - "pyplot.plot(t, x2[:,0], 'r-', lw=2, label='RK4');\n", - "pyplot.plot(t, x3[:,0], 'g', lw=2, label='Verlet');\n", - "pyplot.plot(t, x_analytical, 'k--', lw=2, label='analytical');\n", - "pyplot.legend();" + " return u + dt/6*(k1+2*k2+2*k3+k4)" ] }, { - "cell_type": "markdown", - "metadata": {}, + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": true + }, + "outputs": [], "source": [ - "If we look closely at the plot, we can see that Euler and Runge-Kutta integration are close to each other, but begin to diverge as time goes on. This should be the behavior we expect for the integration of a non-linear system. But how close are these methods to the analytical solution? Does Verlet do any better? In this case it is obvious that Euler and Runge-Kutta are not correct (and possibly unstable). One thing to check is whether the behavior changes as we change our time step size. Maybe the timestep is too large to capture some aspect of the system response." + "def f_harmonic_oscillator(u):\n", + " \"\"\"Returns RHS of harmonic oscillator EOM\n", + " \n", + " Parameters:\n", + " u - initial state\n", + " \n", + " Returns:\n", + " RHS of harmonic oscillator eqn.\n", + " \n", + " \"\"\"\n", + " pos = u[0]\n", + " vel = u[1]\n", + " \n", + " return numpy.array([vel, -k/m*pos])" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def verlet_SHM(u, dt):\n", + " \"\"\" Verlet integration for integrating from Hamiltonian equations of motion\n", + " for a simple harmonic oscillator\n", + " Parameters:\n", + " u - state at current time step\n", + " dt - time step size\n", + " Returns: \n", + " state at next time step.\n", + " \"\"\"\n", + " pos = u[0]\n", + " mom = u[1]\n", + " \n", + " q_half = pos - dt/2*mom/m\n", + " #p_half = mom + dt/2*(k*pos)\n", + " \n", + " p = mom + dt/2*(2*k*q_half)\n", + " q = q_half - dt/2*p/m\n", + " \n", + " return numpy.array([q, p]) " + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "#Initial Conditions\n", + "x0 = 0.5 #[m]\n", + "xdot0 = 0 #[m/s]\n", + "\n", + "#Equivalent initial conditions for the Hamiltonian system\n", + "q0 = x0\n", + "p0 = m*xdot0\n", + "\n", + "x_init = numpy.array([x0, xdot0])\n", + "x_init_H = numpy.array([q0, p0])" ] }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 8, "metadata": { "collapsed": false }, @@ -1276,7 +464,6 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -1607,11 +794,13 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -1660,6 +849,19 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", + "/*\n", + " * return a copy of an object with only non-object keys\n", + " * we need this to avoid circular references\n", + " * http://stackoverflow.com/a/24161582/3208463\n", + " */\n", + "function simpleKeys (original) {\n", + " return Object.keys(original).reduce(function (obj, key) {\n", + " if (typeof original[key] !== 'object')\n", + " obj[key] = original[key]\n", + " return obj;\n", + " }, {});\n", + "}\n", + "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -1673,7 +875,8 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step});\n", + " step: event.step,\n", + " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -1713,7 +916,8 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value});\n", + " this.send_message(name, {key: value,\n", + " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", @@ -1730,7 +934,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -1789,6 +993,8 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", + " fig.root.unbind('remove')\n", + "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -1796,8 +1002,12 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.send_message('closing', {});\n", - " fig.ws.close()\n", + " fig.close_ws(fig, msg);\n", + "}\n", + "\n", + "mpl.figure.prototype.close_ws = function(fig, msg){\n", + " fig.send_message('closing', msg);\n", + " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -1852,14 +1062,20 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -1940,7 +1156,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1951,49 +1167,42 @@ } ], "source": [ - "# Use the same initial conditions and parameters, just change dt.\n", - "T = 25; #[seconds]\n", - "dt_2 = .005; #time step size\n", - "N_2 = int(T/dt_2)+1 #number of time steps\n", - "t_2 = numpy.linspace(0.0, T, N_2) #array of time values\n", - "\n", - "\n", "#Euler Integration\n", - "x1_2 = numpy.zeros((N_2,2)) #initialize array\n", - "x1_2[0] = x_init.copy() #set initial conditions\n", - "for n in range(N_2-1): #integrate with Euler\n", - " x1_2[n+1] = euler(x1_2[n], f_harmonic_oscillator, dt)\n", + "x1 = numpy.zeros((N,2)) #initialize array\n", + "x1[0] = x_init.copy() #set initial conditions\n", + "for n in range(N-1): #integrate with Euler\n", + " x1[n+1] = euler(x1[n], f_harmonic_oscillator, dt)\n", " #print(q1[n])\n", " \n", "#Runge-Kutta Integration\n", - "x2_2 = numpy.zeros((N_2,2))\n", - "x2_2[0] = x_init.copy()\n", - "for n in range(N_2-1):\n", - " x2_2[n+1] = RK4(x2_2[n], f_harmonic_oscillator, dt)\n", + "x2 = numpy.zeros((N,2))\n", + "x2[0] = x_init.copy()\n", + "for n in range(N-1):\n", + " x2[n+1] = RK4(x2[n], f_harmonic_oscillator, dt)\n", "\n", "#Symplectic Integration\n", - "x3_2 = numpy.zeros((N_2,2))\n", - "x3_2[0] = x_init_H.copy()\n", - "for n in range(N_2-1):\n", - " x3_2[n+1] = verlet_SHM(x3_2[n], dt)\n", + "x3 = numpy.zeros((N,2))\n", + "x3[0] = x_init_H.copy()\n", + "for n in range(N-1):\n", + " x3[n+1] = verlet_SHM(x3[n], dt)\n", "\n", "#Analytical Solution \n", "A = x0 #with no forcing and no initial velocity, the max amplitude is equal to initial amplitude\n", "phi = 0\n", "omega = sqrt(k/m)\n", - "x_2_analytical = A*numpy.cos(omega*t_2 + phi)\n", + "x_analytical = A*numpy.cos(omega*t + phi)\n", "#analytical velocity:\n", - "vel_2_analytical = -A*omega*numpy.sin(omega*t_2 + phi)\n", + "vel_analytical = -A*omega*numpy.sin(omega*t + phi)\n", " \n", "pyplot.figure(figsize=(12,8));\n", "pyplot.grid(True);\n", "pyplot.xlabel(r't', fontsize=18);\n", "pyplot.ylabel(r'position (meters)', fontsize=18);\n", "pyplot.title('Harmonic oscillator position');\n", - "pyplot.plot(t_2, x1_2[:,0], lw=2, label='Euler');\n", - "pyplot.plot(t_2, x2_2[:,0], 'r-', lw=2, label='RK4');\n", - "pyplot.plot(t_2, x3_2[:,0], 'g', lw=2, label='Verlet');\n", - "pyplot.plot(t_2, x_2_analytical, 'k--', lw=2, label='analytical');\n", + "pyplot.plot(t, x1[:,0], lw=2, label='Euler');\n", + "pyplot.plot(t, x2[:,0], 'r-', lw=2, label='RK4');\n", + "pyplot.plot(t, x3[:,0], 'g', lw=2, label='Verlet');\n", + "pyplot.plot(t, x_analytical, 'k--', lw=2, label='analytical');\n", "pyplot.legend();" ] }, @@ -2001,15 +1210,20 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "So changing the timestep does make a difference. Let's do an error analysis to examine how the error changes with the timestep size.\n", - "\n", + "If we look closely at the plot, we can see that Euler and Runge-Kutta integration are close to each other, but begin to diverge as time goes on. This should be the behavior we expect for the integration of a non-linear system. But how close are these methods to the analytical solution? Does Verlet do any better? In this case it is obvious that Euler and Runge-Kutta are not correct (and possibly unstable). One thing to check is whether the behavior changes as we change our time step size. Maybe the timestep is too large to capture some aspect of the system response." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ "### Error Analysis\n", "The error analysis code below is based on the numerical methods MOOC notebook on phugoid oscillation. " ] }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 11, "metadata": { "collapsed": true }, @@ -2032,7 +1246,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 12, "metadata": { "collapsed": true }, @@ -2053,7 +1267,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 13, "metadata": { "collapsed": false }, @@ -2133,7 +1347,6 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -2464,11 +1677,13 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -2517,6 +1732,19 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", + "/*\n", + " * return a copy of an object with only non-object keys\n", + " * we need this to avoid circular references\n", + " * http://stackoverflow.com/a/24161582/3208463\n", + " */\n", + "function simpleKeys (original) {\n", + " return Object.keys(original).reduce(function (obj, key) {\n", + " if (typeof original[key] !== 'object')\n", + " obj[key] = original[key]\n", + " return obj;\n", + " }, {});\n", + "}\n", + "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -2530,7 +1758,8 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step});\n", + " step: event.step,\n", + " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -2570,7 +1799,8 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value});\n", + " this.send_message(name, {key: value,\n", + " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", @@ -2587,7 +1817,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -2646,6 +1876,8 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", + " fig.root.unbind('remove')\n", + "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -2653,8 +1885,12 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.send_message('closing', {});\n", - " fig.ws.close()\n", + " fig.close_ws(fig, msg);\n", + "}\n", + "\n", + "mpl.figure.prototype.close_ws = function(fig, msg){\n", + " fig.send_message('closing', msg);\n", + " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -2709,14 +1945,20 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -2797,7 +2039,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2842,7 +2084,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 14, "metadata": { "collapsed": true }, @@ -2865,7 +2107,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 15, "metadata": { "collapsed": false }, @@ -2900,7 +2142,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 16, "metadata": { "collapsed": false }, @@ -2980,7 +2222,6 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -3311,11 +2552,13 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -3364,6 +2607,19 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", + "/*\n", + " * return a copy of an object with only non-object keys\n", + " * we need this to avoid circular references\n", + " * http://stackoverflow.com/a/24161582/3208463\n", + " */\n", + "function simpleKeys (original) {\n", + " return Object.keys(original).reduce(function (obj, key) {\n", + " if (typeof original[key] !== 'object')\n", + " obj[key] = original[key]\n", + " return obj;\n", + " }, {});\n", + "}\n", + "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -3377,7 +2633,8 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step});\n", + " step: event.step,\n", + " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -3417,7 +2674,8 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value});\n", + " this.send_message(name, {key: value,\n", + " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", @@ -3434,7 +2692,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -3493,6 +2751,8 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", + " fig.root.unbind('remove')\n", + "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -3500,8 +2760,12 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.send_message('closing', {});\n", - " fig.ws.close()\n", + " fig.close_ws(fig, msg);\n", + "}\n", + "\n", + "mpl.figure.prototype.close_ws = function(fig, msg){\n", + " fig.send_message('closing', msg);\n", + " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -3556,14 +2820,20 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -3644,7 +2914,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -3736,7 +3006,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 17, "metadata": { "collapsed": true }, @@ -3760,7 +3030,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 18, "metadata": { "collapsed": true }, @@ -3789,7 +3059,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 19, "metadata": { "collapsed": true }, @@ -3820,7 +3090,7 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 20, "metadata": { "collapsed": false }, @@ -3900,7 +3170,6 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -4231,11 +3500,13 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -4284,6 +3555,19 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", + "/*\n", + " * return a copy of an object with only non-object keys\n", + " * we need this to avoid circular references\n", + " * http://stackoverflow.com/a/24161582/3208463\n", + " */\n", + "function simpleKeys (original) {\n", + " return Object.keys(original).reduce(function (obj, key) {\n", + " if (typeof original[key] !== 'object')\n", + " obj[key] = original[key]\n", + " return obj;\n", + " }, {});\n", + "}\n", + "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -4297,7 +3581,8 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step});\n", + " step: event.step,\n", + " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -4337,7 +3622,8 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value});\n", + " this.send_message(name, {key: value,\n", + " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", @@ -4354,7 +3640,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -4413,6 +3699,8 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", + " fig.root.unbind('remove')\n", + "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -4420,8 +3708,12 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.send_message('closing', {});\n", - " fig.ws.close()\n", + " fig.close_ws(fig, msg);\n", + "}\n", + "\n", + "mpl.figure.prototype.close_ws = function(fig, msg){\n", + " fig.send_message('closing', msg);\n", + " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -4476,14 +3768,20 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -4564,7 +3862,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -4624,7 +3922,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 21, "metadata": { "collapsed": false }, @@ -4704,7 +4002,6 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -5035,11 +4332,13 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -5088,6 +4387,19 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", + "/*\n", + " * return a copy of an object with only non-object keys\n", + " * we need this to avoid circular references\n", + " * http://stackoverflow.com/a/24161582/3208463\n", + " */\n", + "function simpleKeys (original) {\n", + " return Object.keys(original).reduce(function (obj, key) {\n", + " if (typeof original[key] !== 'object')\n", + " obj[key] = original[key]\n", + " return obj;\n", + " }, {});\n", + "}\n", + "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -5101,7 +4413,8 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step});\n", + " step: event.step,\n", + " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -5141,7 +4454,8 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value});\n", + " this.send_message(name, {key: value,\n", + " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", @@ -5158,7 +4472,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -5217,6 +4531,8 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", + " fig.root.unbind('remove')\n", + "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -5224,8 +4540,12 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.send_message('closing', {});\n", - " fig.ws.close()\n", + " fig.close_ws(fig, msg);\n", + "}\n", + "\n", + "mpl.figure.prototype.close_ws = function(fig, msg){\n", + " fig.send_message('closing', msg);\n", + " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -5280,14 +4600,20 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -5368,7 +4694,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -5431,7 +4757,7 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 22, "metadata": { "collapsed": false }, @@ -5511,7 +4837,6 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -5842,11 +5167,13 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -5895,6 +5222,19 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", + "/*\n", + " * return a copy of an object with only non-object keys\n", + " * we need this to avoid circular references\n", + " * http://stackoverflow.com/a/24161582/3208463\n", + " */\n", + "function simpleKeys (original) {\n", + " return Object.keys(original).reduce(function (obj, key) {\n", + " if (typeof original[key] !== 'object')\n", + " obj[key] = original[key]\n", + " return obj;\n", + " }, {});\n", + "}\n", + "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -5908,7 +5248,8 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step});\n", + " step: event.step,\n", + " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -5948,7 +5289,8 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value});\n", + " this.send_message(name, {key: value,\n", + " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", @@ -5965,7 +5307,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -6024,6 +5366,8 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", + " fig.root.unbind('remove')\n", + "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -6031,8 +5375,12 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.send_message('closing', {});\n", - " fig.ws.close()\n", + " fig.close_ws(fig, msg);\n", + "}\n", + "\n", + "mpl.figure.prototype.close_ws = function(fig, msg){\n", + " fig.send_message('closing', msg);\n", + " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -6087,14 +5435,20 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -6175,7 +5529,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -6423,21 +5777,21 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 2", + "display_name": "Python 3", "language": "python", - "name": "python2" + "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", - "version": 2 + "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", - "pygments_lexer": "ipython2", - "version": "2.7.11" + "pygments_lexer": "ipython3", + "version": "3.5.0" } }, "nbformat": 4, From 0d9581b0059d05054f4cde942428e6e4a466d634 Mon Sep 17 00:00:00 2001 From: rbds Date: Tue, 8 Dec 2015 23:24:56 +0000 Subject: [PATCH 44/61] fix Verlet for DP --- randy_schur/Double_Pendulum_Problem.ipynb | 26 +++++++++++------------ 1 file changed, 13 insertions(+), 13 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index ed1bb56..be1d3bb 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -358,7 +358,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 18, "metadata": { "collapsed": false }, @@ -1962,7 +1962,7 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 19, "metadata": { "collapsed": false }, @@ -1982,20 +1982,20 @@ " q2 = u[2]\n", " p2 = u[3]\n", " \n", - " q1_half = q1 - dt/2*(q1/I1 + m2**q1/(m1*I1) + q1/(m1*l1*l2)*cos(p2-p1))\n", - " p1 = p1 + dt*(q1*q2/(m1*l1*l2)*sin(p2-q1_half) + (m1+m2)*g*l1*sin(q1_half))\n", - " q1 = q1_half - dt/2*(q1/I1 + m2**q1/(m1*I1) + q1/(m1*l1*l2)*cos(p2-p1))\n", + " p1_half = p1 - dt/2*(p1*p2/(m1*l1*l2)*sin(q2-q1)+(m1+m2)*g*l1*sin(q1))\n", + " q1_plus = q1 + dt*(-p1_half/I1 - m2*p1_half/(m1*I1) - p2/(m1*l1*l2)*cos(q2-q1))\n", + " p1_plus = p1_half - dt/2*(p1*p2/(m1*l1*l2)*sin(q2-q1_plus)+(m1+m2)*g*l1*sin(q1_plus))\n", " \n", - " q2_half = q2 - dt/2*(q1/(m1*l1*l2)*cos(p2-p1) - q2/I2)\n", - " p2 = p2 - dt*(-q1*q2/(m1*l1*l2)*sin(q2_half-p1) + m2*l2*g*sin(p2))\n", - " q2 = q2_half - dt/2*(q1/(m1*l1*l2)*cos(p2-p1) - q2/I2)\n", + " p2_half = p2 - dt/2*( -p1*p2/(m1*l1*l2)*sin(q2-q1) + m2*g*l2*sin(q2))\n", + " q2_plus = q2 + dt*(-p1/(m1*l1*l2)*cos(q2-q1) - p2_half/I2)\n", + " p2_plus = p2_half - dt/2*( -p1*p2/(m1*l1*l2)*sin(q2_plus-q1) + m2*g*l2*sin(q2_plus))\n", " \n", - " return numpy.array([q1, p1, q2, p2]) " + " return numpy.array([q1_plus, p1_plus, q2_plus, p2_plus]) " ] }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 22, "metadata": { "collapsed": false }, @@ -2767,7 +2767,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2791,13 +2791,13 @@ "pyplot.ylabel(r'position (meters)', fontsize=18);\n", "pyplot.title('Double Pendulum Euler');\n", "pyplot.plot(t, x3_dp[:,0], lw=2, label='Joint 1');\n", - "#pyplot.plot(t, x3_dp[:,2], lw=2, label='Joint 2')\n", + "pyplot.plot(t, x3_dp[:,2], lw=2, label='Joint 2')\n", "pyplot.legend();" ] }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 21, "metadata": { "collapsed": true }, From 14bb5f858c539aa75907cb68c081bb20cf1dd5e9 Mon Sep 17 00:00:00 2001 From: rbds Date: Tue, 8 Dec 2015 23:48:14 -0500 Subject: [PATCH 45/61] evening commit --- randy_schur/Double_Pendulum_Problem.ipynb | 1895 +++++++++++++++++++-- 1 file changed, 1773 insertions(+), 122 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index be1d3bb..f5587f4 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -269,7 +269,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 77, "metadata": { "collapsed": false }, @@ -285,12 +285,12 @@ "I2 = m2*l2**2\n", "\n", "T = 25; #[seconds]\n", - "dt = .02; #\n", + "dt = .01; #\n", "N = int(T/dt)+1\n", "t = numpy.linspace(0.0, T, N)\n", "\n", "#Initial Conditions\n", - "theta1_0 = 1 #[radians]\n", + "theta1_0 = pi/2 #[radians]\n", "theta2_0 = pi\n", "theta1_dot_0 = 0 #[rad/s]\n", "theta2_dot_0 = 0 #\n", @@ -307,7 +307,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 78, "metadata": { "collapsed": true }, @@ -332,7 +332,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 79, "metadata": { "collapsed": false }, @@ -358,7 +358,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 80, "metadata": { "collapsed": false }, @@ -908,7 +908,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -1130,7 +1130,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1684,7 +1684,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -1906,7 +1906,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1955,47 +1955,1748 @@ "collapsed": false }, "source": [ - "Again, we can see that these integrators cause the system to go unstable. In fact, there is a physical interpretation for what is happening. In the RK4 plot, you can see the position of joint 1 oscillating betwee -pi and pi (corresponding to straight up in the air), until it finally reaches a value less than -pi. So, once joint 1 'flips around,' the system diverges.\n", + "Again, we can see that these integrators cause the system to go unstable. In fact, there is a physical interpretation for what is happening. In the RK4 plot, you can see the position of joint 1 oscillating between almost -pi and almost pi (corresponding to straight up in the air), until it finally reaches a value greater than pi. So, once joint 1 'flips around,' the system diverges.\n", "\n", "With simple harmonic motion, the symplectic integrator was able to accurately represent the state of the system without adding energy. Let's see if it is successful with the double pendulum. " ] }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 82, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def Verlet_DP(u, dt):\n", + " \"\"\" Verlet integration for integrating from Hamiltonian equations of motion\n", + " for a ddouble pendulum\n", + " Parameters:\n", + " u - state at current time step\n", + " dt - time step size\n", + " Returns: \n", + " state at next time step.\n", + " \"\"\"\n", + " q1 = u[0]\n", + " p1 = u[1]\n", + " q2 = u[2]\n", + " p2 = u[3]\n", + " \n", + " p1_half = p1 - dt/2*(p1*p2/(m1*l1*l2)*sin(q2-q1)+(m1+m2)*g*l1*sin(q1))\n", + " q1_plus = q1 + dt*(-p1_half/I1 - m2*p1_half/(m1*I1) - p2/(m1*l1*l2)*cos(q2-q1))\n", + " p1_plus = p1_half - dt/2*(p1*p2/(m1*l1*l2)*sin(q2-q1_plus)+(m1+m2)*g*l1*sin(q1_plus))\n", + " \n", + " p2_half = p2 - dt/2*( -p1*p2/(m1*l1*l2)*sin(q2-q1) + m2*g*l2*sin(q2))\n", + " q2_plus = q2 + dt*(-p1/(m1*l1*l2)*cos(q2-q1) - p2_half/I2)\n", + " p2_plus = p2_half - dt/2*( -p1*p2/(m1*l1*l2)*sin(q2_plus-q1) + m2*g*l2*sin(q2_plus))\n", + " \n", + " return numpy.array([q1_plus, p1_plus, q2_plus, p2_plus]) " + ] + }, + { + "cell_type": "code", + "execution_count": 83, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "application/javascript": [ + "/* Put everything inside the global mpl namespace */\n", + "window.mpl = {};\n", + "\n", + "mpl.get_websocket_type = function() {\n", + " if (typeof(WebSocket) !== 'undefined') {\n", + " return WebSocket;\n", + " } else if (typeof(MozWebSocket) !== 'undefined') {\n", + " return MozWebSocket;\n", + " } else {\n", + " alert('Your browser does not have WebSocket support.' +\n", + " 'Please try Chrome, Safari or Firefox ≥ 6. 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');\n", + " var button = $('');\n", + " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", + " buttongrp.append(button);\n", + " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", + " titlebar.prepend(buttongrp);\n", + "}\n", + "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(el){\n", + " // this is important to make the div 'focusable\n", + " el.attr('tabindex', 0)\n", + " // reach out to IPython and tell the keyboard manager to turn it's self\n", + " // off when our div gets focus\n", + "\n", + " // location in version 3\n", + " if (IPython.notebook.keyboard_manager) {\n", + " IPython.notebook.keyboard_manager.register_events(el);\n", + " }\n", + " else {\n", + " // location in version 2\n", + " IPython.keyboard_manager.register_events(el);\n", + " }\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._key_event_extra = function(event, name) {\n", + " var manager = IPython.notebook.keyboard_manager;\n", + " if (!manager)\n", + " manager = IPython.keyboard_manager;\n", + "\n", + " // Check for shift+enter\n", + " if (event.shiftKey && event.which == 13) {\n", + " this.canvas_div.blur();\n", + " event.shiftKey = false;\n", + " // Send a \"J\" for go to next cell\n", + " event.which = 74;\n", + " event.keyCode = 74;\n", + " manager.command_mode();\n", + " manager.handle_keydown(event);\n", + " }\n", + "}\n", + "\n", + "mpl.figure.prototype.handle_save = function(fig, msg) {\n", + " fig.ondownload(fig, null);\n", + "}\n", + "\n", + "\n", + "mpl.find_output_cell = function(html_output) {\n", + " // Return the cell and output element which can be found *uniquely* in the notebook.\n", + " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", + " // IPython event is triggered only after the cells have been serialised, which for\n", + " // our purposes (turning an active figure into a static one), is too late.\n", + " var cells = IPython.notebook.get_cells();\n", + " var ncells = cells.length;\n", + " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", + " data = data.data;\n", + " }\n", + " if (data['text/html'] == html_output) {\n", + " return [cell, data, j];\n", + " }\n", + " }\n", + " }\n", + " }\n", + "}\n", + "\n", + "// Register the function which deals with the matplotlib target/channel.\n", + "// The kernel may be null if the page has been refreshed.\n", + "if (IPython.notebook.kernel != null) {\n", + " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", + "}\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "Verlet_energy = numpy.zeros_like(t)\n", + "Verlet_vel1 = x3_dp[:,1]/m1\n", + "Verlet_vel2 = x3_dp[:,3]/m2\n", + "\n", + "\n", + "u = numpy.array([x3_dp[:,0], Verlet_vel1, x3_dp[:,2], Verlet_vel2])\n", + "#print(u[0,:])\n", + "for i in range(N):\n", + " Verlet_energy[i] = get_Energy(u[:,i])\n", + "\n", + "pyplot.figure(figsize=(8,6));\n", + "pyplot.grid(True);\n", + "pyplot.xlabel(r't', fontsize=18);\n", + "pyplot.ylabel(r'Total energy in system', fontsize=18);\n", + "pyplot.title('Energy in System');\n", + "pyplot.plot(t, Verlet_energy, lw=2, label='Verlet');" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "So, we are very far from perfect. But, at least the energy in the system is stable, even if it isn't constant! " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "### Investigating system behavior\n", + "\n", + "In order to investigate system behavior for damped harmonic motion, we used a phase portrait where we plotted position vs. velocity. In this system, we have 2 positions and 2 velocities, so a phase plot won't accurately capture the behavior of the system. Instead we are going to use something called a Poincare section (https://en.wikipedia.org/wiki/Poincar%C3%A9_map). The way these work is to capture the system behavior at a specific, repetetive point. For example, each time that $\\theta_2 = 0$ or any multiple of 2*pi (mass 2 is hanging straight down) we can capture the angular position and velocity of mass 2. This way, we are taking a 'snapshot' of the system, and plotting the state each time. We can now investigate system behavior on a simple 2D plot, similar to the phase portrait." + ] + }, + { + "cell_type": "code", + "execution_count": 96, "metadata": { "collapsed": false }, "outputs": [], "source": [ - "def Verlet_DP(u, dt):\n", - " \"\"\" Verlet integration for integrating from Hamiltonian equations of motion\n", - " for a ddouble pendulum\n", - " Parameters:\n", - " u - state at current time step\n", - " dt - time step size\n", - " Returns: \n", - " state at next time step.\n", - " \"\"\"\n", - " q1 = u[0]\n", - " p1 = u[1]\n", - " q2 = u[2]\n", - " p2 = u[3]\n", - " \n", - " p1_half = p1 - dt/2*(p1*p2/(m1*l1*l2)*sin(q2-q1)+(m1+m2)*g*l1*sin(q1))\n", - " q1_plus = q1 + dt*(-p1_half/I1 - m2*p1_half/(m1*I1) - p2/(m1*l1*l2)*cos(q2-q1))\n", - " p1_plus = p1_half - dt/2*(p1*p2/(m1*l1*l2)*sin(q2-q1_plus)+(m1+m2)*g*l1*sin(q1_plus))\n", + "def poincare(state, sf, val, s1, s2):\n", + " '''Creates a poincare section, capturing system state each time q[state]=val\n", " \n", - " p2_half = p2 - dt/2*( -p1*p2/(m1*l1*l2)*sin(q2-q1) + m2*g*l2*sin(q2))\n", - " q2_plus = q2 + dt*(-p1/(m1*l1*l2)*cos(q2-q1) - p2_half/I2)\n", - " p2_plus = p2_half - dt/2*( -p1*p2/(m1*l1*l2)*sin(q2_plus-q1) + m2*g*l2*sin(q2_plus))\n", - " \n", - " return numpy.array([q1_plus, p1_plus, q2_plus, p2_plus]) " + " Parameters:\n", + " state - state vector\n", + " sf - the row in q to fix\n", + " val - the value to capture state at \n", + " s1 - state to capture 1\n", + " s2 - state to capture 2\n", + " \n", + " Returns:\n", + " v - array with information for poincare section\n", + " '''\n", + " v = numpy.zeros((2,1))\n", + " for i in range(len(state[:,sf])-1):\n", + " if (state[i,sf]>val):\n", + " if (state[i+1,sf] < val):\n", + " v = numpy.append(v, [[state[i, s1]],[state[i, s2]]], axis=1)\n", + " \n", + " if (state[i,sf] < val):\n", + " if (state[i+1,sf] > val):\n", + " v = numpy.append(v, [[state[i, s1]],[state[i, s2]]], axis=1)\n", + " \n", + " return v" ] }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 97, "metadata": { "collapsed": false }, @@ -2545,7 +4246,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -2767,7 +4468,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2778,36 +4479,32 @@ } ], "source": [ - "#Verlet integration\n", - "x3_dp = numpy.zeros((N,4)) \n", - "x3_dp[0,:] = x_init_H_dp.copy() #set initial conditions\n", - "#print(q3_dp[0,:])\n", - "for n in range(N-1): #integrate with Euler\n", - " x3_dp[n+1,:] = Verlet_DP(x3_dp[n,:], dt)\n", - " \n", + "v = poincare(x3_dp, 1, 2*pi, 0, 2) #use states from RK4 integration\n", "pyplot.figure(figsize=(10,8));\n", "pyplot.grid(True);\n", - "pyplot.xlabel(r't', fontsize=18);\n", - "pyplot.ylabel(r'position (meters)', fontsize=18);\n", - "pyplot.title('Double Pendulum Euler');\n", - "pyplot.plot(t, x3_dp[:,0], lw=2, label='Joint 1');\n", - "pyplot.plot(t, x3_dp[:,2], lw=2, label='Joint 2')\n", - "pyplot.legend();" + "pyplot.xlabel(r'$\\theta_2$', fontsize=18);\n", + "pyplot.ylabel(r'$\\dot{\\theta_2}$', fontsize=18);\n", + "pyplot.title('Poincare Section');\n", + "pyplot.plot(v[0,:], v[1,:], 'bo', lw=2, );" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "There isn't a clear pattern here. So, we can say that our system is NOT likely to be asymptotically stable. However, this shouldn't be surprising. If we continued simulating, our system would continue rotating since there is no damping. Let's see what happens when there is some damping in the system. We're going to use RK4 for the following analysis." ] }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 98, "metadata": { "collapsed": true }, "outputs": [], "source": [ "#To Do:\n", - "# plot total energy in system\n", "# animate motion of double pendulum\n", - "# Poincare section for symplectic DP\n", - "# animate Poincare sections\n", "# check through sources" ] }, @@ -2835,7 +4532,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 99, "metadata": { "collapsed": false }, @@ -2872,7 +4569,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 100, "metadata": { "collapsed": false }, @@ -3422,7 +5119,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -3644,7 +5341,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -4198,7 +5895,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -4420,7 +6117,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -4468,66 +6165,20 @@ }, { "cell_type": "markdown", - "metadata": { - "collapsed": true - }, + "metadata": {}, "source": [ - "### Investigating system behavior\n", + "It looks like the system is chaotic at first, but becomes periodic as it settles.\n", "\n", - "In order to investigate system behavior for damped harmonic motion, we used a phase portrait where we plotted position vs. velocity. In this system, we have 2 positions and 2 velocities, so a phase plot won't accurately capture the behavior of the system. Instead we are going to use something called a Poincare section (https://en.wikipedia.org/wiki/Poincar%C3%A9_map). The way these work is to capture the system behavior at a specific, repetetive point. For example, each time that $\\theta_1 = 0$ (mass 1 is hanging straight down) we can capture the angular position and velocity of mass 2. This way, we are taking a 'snapshot' of the system, and plotting the state each time. We can now investigate system behavior on a simple 2D plot, similar to the phase portrait." - ] - }, - { - "cell_type": "code", - "execution_count": 13, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "def poincare(state, sf, val, s1, s2):\n", - " '''Creates a poincare section, capturing system state each time q[state]=val\n", - " \n", - " Parameters:\n", - " state - state vector\n", - " sf - the row in q to fix\n", - " val - the value to capture state at \n", - " s1 - state to capture 1\n", - " s2 - state to capture 2\n", - " \n", - " Returns:\n", - " v - array with information for poincare section\n", - " '''\n", - " v = numpy.zeros((2,1))\n", - " for i in range(len(state[:,sf])-1):\n", - " if (state[i,sf]>val):\n", - " if (state[i+1,sf] < val):\n", - " # print(v)\n", - " # print([[state[i, s1],[state[i, s2]]]])\n", - " v = numpy.append(v, [[state[i, s1]],[state[i, s2]]], axis=1)\n", - " \n", - " if (state[i,sf] < val):\n", - " if (state[i+1,sf] > val):\n", - " v = numpy.append(v, [[state[i, s1]],[state[i, s2]]], axis=1)\n", - " \n", - " return v\n", - " " + "Let's do a Poincare section of this plot as well. For brevity, we'll use only the RK4 system response." ] }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 101, "metadata": { "collapsed": false }, "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "(2, 19)\n" - ] - }, { "data": { "application/javascript": [ @@ -5073,7 +6724,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -5295,7 +6946,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -5307,21 +6958,21 @@ { "data": { "text/plain": [ - "[]" + "[]" ] }, - "execution_count": 14, + "execution_count": 101, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "v = poincare(q2d_dp, 0, 0, 2, 3) #use states from RK4 integration\n", - "print(numpy.shape(v))\n", + "v = poincare(q2d_dp, 0, 0, 1, 3) #use states from RK4 integration\n", "pyplot.figure(figsize=(10,8));\n", "pyplot.grid(True);\n", "pyplot.xlabel(r'$\\theta_2$', fontsize=18);\n", - "pyplot.ylabel(r'$\\dot{\\theta_2}$', fontsize=18);#pyplot.title('Poincare Section');\n", + "pyplot.ylabel(r'$\\dot{\\theta_2}$', fontsize=18);\n", + "pyplot.title('Poincare Section');\n", "pyplot.plot(v[0,:], v[1,:], 'bo', lw=2, );\n", "\n", "#from matplotlib import animation\n", @@ -5336,7 +6987,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 92, "metadata": { "collapsed": false }, @@ -5355,7 +7006,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Other than a few outliers, the plot converges to (0,0). So, this system is stable! This would not be the case for the undamped double pendulum, as integrated by RK4. " + "Other than a few outliers, the plot converges seems to converge. So, this system is stable! This would not be the case for the undamped double pendulum, as integrated by RK4. " ] }, { @@ -5382,7 +7033,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 19, "metadata": { "collapsed": false }, @@ -5537,7 +7188,7 @@ "" ] }, - "execution_count": 16, + "execution_count": 19, "metadata": {}, "output_type": "execute_result" } @@ -5577,7 +7228,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.5.0" + "version": "3.4.3" } }, "nbformat": 4, From 68c6243dad992c4e663983079e9a4be19400a634 Mon Sep 17 00:00:00 2001 From: rbds Date: Thu, 10 Dec 2015 12:32:57 +0000 Subject: [PATCH 46/61] fix RK4 --- randy_schur/Double_Pendulum_Problem.ipynb | 16 ++++---- ...on Techniques for Mechanical Systems.ipynb | 39 +++++++++++++------ 2 files changed, 36 insertions(+), 19 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index f5587f4..54eefa1 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -269,7 +269,7 @@ }, { "cell_type": "code", - "execution_count": 77, + "execution_count": 3, "metadata": { "collapsed": false }, @@ -307,7 +307,7 @@ }, { "cell_type": "code", - "execution_count": 78, + "execution_count": 4, "metadata": { "collapsed": true }, @@ -332,7 +332,7 @@ }, { "cell_type": "code", - "execution_count": 79, + "execution_count": 5, "metadata": { "collapsed": false }, @@ -358,7 +358,7 @@ }, { "cell_type": "code", - "execution_count": 80, + "execution_count": 6, "metadata": { "collapsed": false }, @@ -908,7 +908,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -1684,7 +1684,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -1955,7 +1955,7 @@ "collapsed": false }, "source": [ - "Again, we can see that these integrators cause the system to go unstable. In fact, there is a physical interpretation for what is happening. In the RK4 plot, you can see the position of joint 1 oscillating between almost -pi and almost pi (corresponding to straight up in the air), until it finally reaches a value greater than pi. So, once joint 1 'flips around,' the system diverges.\n", + "Again, we can see that these integrators cause the system to go unstable. In fact, there is a physical interpretation for what is happening. In the Euler plot, you can see the position of joint 1 oscillating between almost -pi and almost pi (corresponding to straight up in the air), until it finally reaches a value greater than pi. So, once joint 1 'flips around,' the system diverges.\n", "\n", "With simple harmonic motion, the symplectic integrator was able to accurately represent the state of the system without adding energy. Let's see if it is successful with the double pendulum. " ] @@ -7228,7 +7228,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.4.3" + "version": "3.5.0" } }, "nbformat": 4, diff --git a/randy_schur/Integration Techniques for Mechanical Systems.ipynb b/randy_schur/Integration Techniques for Mechanical Systems.ipynb index 32fdc22..a67690c 100644 --- a/randy_schur/Integration Techniques for Mechanical Systems.ipynb +++ b/randy_schur/Integration Techniques for Mechanical Systems.ipynb @@ -222,7 +222,7 @@ }, { "cell_type": "code", - "execution_count": 1, + "execution_count": 5, "metadata": { "collapsed": false }, @@ -243,7 +243,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 6, "metadata": { "collapsed": false }, @@ -261,7 +261,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 7, "metadata": { "collapsed": true }, @@ -284,7 +284,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 8, "metadata": { "collapsed": true }, @@ -311,7 +311,24 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 24, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def RK4_2(u, f, dt):\n", + " k1 = dt*f(u)\n", + " k2 = dt*f(u+ k1/2)\n", + " k3 = dt*f(u + k2/2)\n", + " k4 = dt*f(u + k3)\n", + " \n", + " return u + 1/6*(k1 + 2*k2 + 2*k3 + k4)" + ] + }, + { + "cell_type": "code", + "execution_count": 25, "metadata": { "collapsed": true }, @@ -335,7 +352,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 26, "metadata": { "collapsed": true }, @@ -364,9 +381,9 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 27, "metadata": { - "collapsed": true + "collapsed": false }, "outputs": [], "source": [ @@ -384,7 +401,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 29, "metadata": { "collapsed": false }, @@ -1156,7 +1173,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1178,7 +1195,7 @@ "x2 = numpy.zeros((N,2))\n", "x2[0] = x_init.copy()\n", "for n in range(N-1):\n", - " x2[n+1] = RK4(x2[n], f_harmonic_oscillator, dt)\n", + " x2[n+1] = RK4_2(x2[n], f_harmonic_oscillator, dt)\n", "\n", "#Symplectic Integration\n", "x3 = numpy.zeros((N,2))\n", From ccadd6271ee5a0a43a7f47fa64b9122e7cc48e6b Mon Sep 17 00:00:00 2001 From: rbds Date: Thu, 10 Dec 2015 13:59:41 +0000 Subject: [PATCH 47/61] adding in RK2 --- ...on Techniques for Mechanical Systems.ipynb | 2702 ++++++++++++++++- 1 file changed, 2590 insertions(+), 112 deletions(-) diff --git a/randy_schur/Integration Techniques for Mechanical Systems.ipynb b/randy_schur/Integration Techniques for Mechanical Systems.ipynb index a67690c..26a16a3 100644 --- a/randy_schur/Integration Techniques for Mechanical Systems.ipynb +++ b/randy_schur/Integration Techniques for Mechanical Systems.ipynb @@ -243,7 +243,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 45, "metadata": { "collapsed": false }, @@ -261,7 +261,7 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 46, "metadata": { "collapsed": true }, @@ -284,13 +284,13 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 47, "metadata": { "collapsed": true }, "outputs": [], "source": [ - "def RK4(u, f, dt):\n", + "def RK2(u, f, dt):\n", " \"\"\"Runge Kutta fourth order integration method\n", " \n", " Parameters:\n", @@ -301,23 +301,30 @@ " Returns: \n", " state values at next time step.\n", " \"\"\"\n", - " k1 = f(u)\n", - " k2 = f(u) + 0.5*dt*k1\n", - " k3 = f(u) + 0.5*dt*k2\n", - " k4 = f(u) + dt*k3\n", - " \n", - " return u + dt/6*(k1+2*k2+2*k3+k4)" + " k1 = u + 0.5*dt*f(u)\n", + " k2 = u + dt*f(k1)\n", + " return k2" ] }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 48, "metadata": { "collapsed": true }, "outputs": [], "source": [ - "def RK4_2(u, f, dt):\n", + "def RK4(u, f, dt):\n", + " \"\"\"Runge Kutta fourth order integration method\n", + " \n", + " Parameters:\n", + " u - state of the system at time t\n", + " f - function for RHS of state equations\n", + " dt - time step\n", + " \n", + " Returns: \n", + " state values at next time step.\n", + " \"\"\"\n", " k1 = dt*f(u)\n", " k2 = dt*f(u+ k1/2)\n", " k3 = dt*f(u + k2/2)\n", @@ -328,7 +335,7 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 49, "metadata": { "collapsed": true }, @@ -352,7 +359,7 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 50, "metadata": { "collapsed": true }, @@ -381,7 +388,7 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 51, "metadata": { "collapsed": false }, @@ -401,7 +408,7 @@ }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 52, "metadata": { "collapsed": false }, @@ -1173,7 +1180,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1191,17 +1198,25 @@ " x1[n+1] = euler(x1[n], f_harmonic_oscillator, dt)\n", " #print(q1[n])\n", " \n", - "#Runge-Kutta Integration\n", + "#Runge-Kutta Integration Second Order\n", "x2 = numpy.zeros((N,2))\n", "x2[0] = x_init.copy()\n", "for n in range(N-1):\n", - " x2[n+1] = RK4_2(x2[n], f_harmonic_oscillator, dt)\n", + " x2[n+1] = RK2(x2[n], f_harmonic_oscillator, dt)\n", "\n", + "\n", + "#Runge-Kutta Integration Fourth Order\n", + "x4 = numpy.zeros((N,2))\n", + "x4[0] = x_init.copy()\n", + "for n in range(N-1):\n", + " x4[n+1] = RK4(x4[n], f_harmonic_oscillator, dt)\n", + " \n", + " \n", "#Symplectic Integration\n", - "x3 = numpy.zeros((N,2))\n", - "x3[0] = x_init_H.copy()\n", + "xv = numpy.zeros((N,2))\n", + "xv[0] = x_init_H.copy()\n", "for n in range(N-1):\n", - " x3[n+1] = verlet_SHM(x3[n], dt)\n", + " xv[n+1] = verlet_SHM(xv[n], dt)\n", "\n", "#Analytical Solution \n", "A = x0 #with no forcing and no initial velocity, the max amplitude is equal to initial amplitude\n", @@ -1217,8 +1232,9 @@ "pyplot.ylabel(r'position (meters)', fontsize=18);\n", "pyplot.title('Harmonic oscillator position');\n", "pyplot.plot(t, x1[:,0], lw=2, label='Euler');\n", - "pyplot.plot(t, x2[:,0], 'r-', lw=2, label='RK4');\n", - "pyplot.plot(t, x3[:,0], 'g', lw=2, label='Verlet');\n", + "pyplot.plot(t, x2[:,0], 'r', lw=2, label='RK2');\n", + "pyplot.plot(t, x4[:,0], 'c', lw=2, label='RK4');\n", + "pyplot.plot(t, xv[:,0], 'g', lw=2, label='Verlet');\n", "pyplot.plot(t, x_analytical, 'k--', lw=2, label='analytical');\n", "pyplot.legend();" ] @@ -1227,7 +1243,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "If we look closely at the plot, we can see that Euler and Runge-Kutta integration are close to each other, but begin to diverge as time goes on. This should be the behavior we expect for the integration of a non-linear system. But how close are these methods to the analytical solution? Does Verlet do any better? In this case it is obvious that Euler and Runge-Kutta are not correct (and possibly unstable). One thing to check is whether the behavior changes as we change our time step size. Maybe the timestep is too large to capture some aspect of the system response." + "If we look closely at the plot, we can see that Verlet and Runge-Kutta integration are close to each other, but something is clearly off with Euler's method. TBut how close are these methods to the analytical solution? Does Verlet do any better than we expect? In this case it is obvious that Euler is not correct (and possibly unstable). One thing to check is whether the behavior changes as we change our time step size. Maybe the timestep is too large to capture some aspect of the system response." ] }, { @@ -1240,7 +1256,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 53, "metadata": { "collapsed": true }, @@ -1263,28 +1279,30 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 55, "metadata": { - "collapsed": true + "collapsed": false }, "outputs": [], "source": [ "dt_values = numpy.array([0.1, 0.05, 0.01, 0.005, 0.001, 0.0001])\n", "\n", "Euler_error_values = numpy.zeros_like(dt_values)\n", - "RK_error_values = Euler_error_values.copy()\n", + "RK2_error_values = Euler_error_values.copy()\n", + "RK4_error_values = Euler_error_values.copy()\n", "Verlet_error_values = Euler_error_values.copy()\n", "for i, dt in enumerate(dt_values):\n", " ###Call error function\n", " Euler_error_values[i] = get_error(x1[:,0], x_analytical, dt)\n", - " RK_error_values[i] = get_error(x2[:,0], x_analytical, dt)\n", - " Verlet_error_values[i] = get_error(x3[:,0], x_analytical, dt)\n", + " RK2_error_values[i] = get_error(x2[:,0], x_analytical, dt)\n", + " RK4_error_values[i] = get_error(x4[:,0], x_analytical, dt)\n", + " Verlet_error_values[i] = get_error(xv[:,0], x_analytical, dt)\n", " \n" ] }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 56, "metadata": { "collapsed": false }, @@ -2056,7 +2074,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2073,7 +2091,8 @@ "pyplot.xlabel('$\\Delta t$', fontsize=16)\n", "pyplot.ylabel('Error', fontsize=16)\n", "pyplot.loglog(dt_values, Euler_error_values, 'bo-', label='Euler')\n", - "pyplot.loglog(dt_values, RK_error_values, 'ro-', label='RK4')\n", + "pyplot.loglog(dt_values, RK2_error_values, 'ro-', label='RK2')\n", + "pyplot.loglog(dt_values, RK4_error_values, 'co-', label='RK4')\n", "pyplot.loglog(dt_values, Verlet_error_values, 'go-', label='Symplectic')\n", "pyplot.axis('equal')\n", "pyplot.legend();" @@ -2087,83 +2106,31 @@ "\n", "![Image](figures/combined_response.png)\n", "\n", - "![Image](figures/error_combined.png)\n" + "![Image](figures/error_combined.png)\n", + " # FIX THESE IMAGES" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Despite the numerical errors, ordinarily we don't see a second order scheme (Verlet) outperforming a fourth order scheme (RK4) by three orders of magnitude. What is going on?\n", - "\n", - "One thing we notice in the plots of position is that for Euler and Runge-Kutta, the spring is stretched farther and farther each oscillation. This means it has more and more potential energy. With no outside forces adding energy to the system and no friction to remove energy, Conservation of Energy says that value should be constant! Let's take a look at total energy in the system. Since we derived equations of motion from the Lagrangian, this should be easy! All we need to do is calculate kinetic and potential energy at each time step, and we already have the information we need to do this. " - ] - }, - { - "cell_type": "code", - "execution_count": 14, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "def get_Energy(u):\n", - " \"\"\" Calculates total energu in the system at given time step for a simple harmonic oscillator\n", - " Parameters:\n", - " u - state of system [pos, vel]\n", - " Returns:\n", - " total energy in system.\n", - " \"\"\"\n", - " pos = u[0]\n", - " vel = u[1]\n", - " T = 0.5*m*vel**2\n", - " V = 0.5*k*pos**2\n", - " \n", - " return T+V" - ] - }, - { - "cell_type": "code", - "execution_count": 15, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "#Euler total energy\n", - "Euler_energy = numpy.zeros_like(t)\n", - "for i in range(N):\n", - " Euler_energy[i] = get_Energy(x1[i,:])\n", - " \n", - "\n", - "#RK4 total energy\n", - "RK4_energy = numpy.zeros_like(t)\n", - "for i in range(N):\n", - " RK4_energy[i] = get_Energy(x2[i,:])\n", - "\n", - "#Verlet total energy\n", - "#Here we will convert from momentum to velocity so we can use the same get_energy function.\n", - "#Momemtum (q) is mass*velocity.\n", - "Verlet_energy = numpy.zeros_like(t)\n", - "Verlet_vel = x3[:,1]/m\n", - "for i in range(N):\n", - " Verlet_energy[i] = get_Energy(numpy.array([x3[i,0], Verlet_vel[i]]))\n", - "\n", - "\n", - "#analytic total energy\n", - "analytical_energy = numpy.zeros_like(t)\n", - "anayltical_state = numpy.array([x_analytical, vel_analytical]).T\n", - "for i in range(N):\n", - " analytical_energy[i] = get_Energy(anayltical_state[i,:])" + "Despite the numerical errors, why is one second order scheme (Verlet) outperforming another (RK2) by so much? What is going on? We can check the order of convergence to make sure that these schemes are actually 2nd order." ] }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 65, "metadata": { "collapsed": false }, "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/home/mint/miniconda3/lib/python3.5/site-packages/ipykernel/__main__.py:24: DeprecationWarning: using a non-integer number instead of an integer will result in an error in the future\n" + ] + }, { "data": { "application/javascript": [ @@ -2931,7 +2898,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2942,23 +2909,2534 @@ } ], "source": [ - "pyplot.figure(figsize=(8,6));\n", - "pyplot.grid(True);\n", - "pyplot.xlabel(r't', fontsize=18);\n", - "pyplot.ylabel(r'Total energy in system (J)', fontsize=18);\n", - "pyplot.title('Energy in Simple Harmonic Oscillator');\n", - "pyplot.plot(t, Euler_energy, lw=2, label='Euler');\n", - "pyplot.plot(t, RK4_energy, 'r-', lw=2, label='RK4');\n", - "pyplot.plot(t, Verlet_energy, 'g', lw=2, label='Verlet');\n", - "pyplot.plot(t, analytical_energy, 'k--', lw=2, label='analytical');\n", - "pyplot.legend();" + "def get_diffgrid(u_current, u_fine, dt):\n", + " \"\"\"Returns the difference between one grid and the fine one using L-1 norm.\n", + " \n", + " Parameters\n", + " ----------\n", + " u_current : array of float\n", + " solution on the current grid.\n", + " u_finest : array of float\n", + " solution on the fine grid.\n", + " dt : float\n", + " time-increment on the current grid.\n", + " \n", + " Returns\n", + " -------\n", + " diffgrid : float\n", + " difference computed in the L-1 norm.\n", + " \"\"\"\n", + " \n", + " N_current = len(u_current[:,0])\n", + " N_fine = len(u_fine[:,0])\n", + " \n", + " grid_size_ratio = numpy.ceil(N_fine/N_current)\n", + " \n", + " diffgrid = dt * numpy.sum( numpy.abs(\\\n", + " u_current[:,0]- u_fine[::grid_size_ratio,0])) \n", + " \n", + " return diffgrid\n", + "\n", + "# use a for-loop to compute the solution on different grids - this code is modified from Numerical MOOC lesson 1 module 4\n", + "dt_values = numpy.array([0.1, 0.05, 0.01, 0.005, 0.001])\n", + "u1_values = numpy.empty_like(dt_values, dtype=numpy.ndarray)\n", + "u2_values = u1_values.copy()\n", + "u4_values = u1_values.copy()\n", + "uv_values = u1_values.copy()\n", + "\n", + "#At each value of dt, integrate the system using 4 different methods from t=0 to T\n", + "for i, dt in enumerate(dt_values):\n", + " N = int(T/dt)+1 \n", + " t = numpy.linspace(0.0, T, N)\n", + " u1 = numpy.empty((N,2))\n", + " u1[0] = numpy.array([x0, xdot0])\n", + " u2 = u1.copy()\n", + " u4 = u1.copy()\n", + " uv = u1.copy()\n", + " for n in range(N-1):\n", + " u1[n+1] = euler(u1[n], f_harmonic_oscillator, dt)\n", + " u2[n+1] = RK2(u2[n], f_harmonic_oscillator, dt)\n", + " u4[n+1] = RK4(u4[n], f_harmonic_oscillator, dt)\n", + " uv[n+1] = verlet_SHM(uv[n], dt)\n", + " # store the value of u related to one grid\n", + " u1_values[i] = u1\n", + " u2_values[i] = u2\n", + " u4_values[i] = u4\n", + " uv_values[i] = uv\n", + " \n", + "# compute diffgrid\n", + "diffgrid1 = numpy.empty_like(dt_values)\n", + "diffgrid2 = diffgrid1.copy()\n", + "diffgrid4 = diffgrid1.copy()\n", + "diffgridv = diffgrid1.copy()\n", + "for i, dt in enumerate(dt_values):\n", + " diffgrid1[i] = get_diffgrid(u1_values[i], u1_values[-1], dt)\n", + " diffgrid2[i] = get_diffgrid(u2_values[i], u2_values[-1], dt)\n", + " diffgrid4[i] = get_diffgrid(u4_values[i], u4_values[-1], dt)\n", + " diffgridv[i] = get_diffgrid(uv_values[i], uv_values[-1], dt)" ] }, { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "It looks like Euler and Runge-Kutta integration are adding a significant amount of energy to the system. Our symplectic integrator is more accurate because it seems to preserve energy. Maybe all of that work deriving the equations of motion was worth it! We now know that the Hamiltonian equations do better with Conservation of Energy, and this can make a big difference even in a simple system." + "cell_type": "code", + "execution_count": 70, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "application/javascript": [ + "/* Put everything inside the global mpl namespace */\n", + "window.mpl = {};\n", + "\n", + "mpl.get_websocket_type = function() {\n", + " if (typeof(WebSocket) !== 'undefined') {\n", + " return WebSocket;\n", + " } else if (typeof(MozWebSocket) !== 'undefined') {\n", + " return MozWebSocket;\n", + " } else {\n", + " alert('Your browser does not have WebSocket support.' +\n", + " 'Please try Chrome, Safari or Firefox ≥ 6. 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');\n", + " var titletext = $(\n", + " '
');\n", + " titlebar.append(titletext)\n", + " this.root.append(titlebar);\n", + " this.header = titletext[0];\n", + "}\n", + "\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._init_canvas = function() {\n", + " var fig = this;\n", + "\n", + " var canvas_div = $('
');\n", + "\n", + " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", + "\n", + " function canvas_keyboard_event(event) {\n", + " return fig.key_event(event, event['data']);\n", + " }\n", + "\n", + " canvas_div.keydown('key_press', canvas_keyboard_event);\n", + " canvas_div.keyup('key_release', canvas_keyboard_event);\n", + " this.canvas_div = canvas_div\n", + " this._canvas_extra_style(canvas_div)\n", + " this.root.append(canvas_div);\n", + "\n", + " var canvas = $('');\n", + " canvas.addClass('mpl-canvas');\n", + " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", + "\n", + " this.canvas = canvas[0];\n", + " this.context = canvas[0].getContext(\"2d\");\n", + "\n", + " var rubberband = $('');\n", + " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", + "\n", + " var pass_mouse_events = true;\n", + "\n", + " canvas_div.resizable({\n", + " start: function(event, ui) {\n", + " pass_mouse_events = false;\n", + " },\n", + " resize: function(event, ui) {\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " stop: function(event, ui) {\n", + " pass_mouse_events = true;\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " });\n", + "\n", + " function mouse_event_fn(event) {\n", + " if (pass_mouse_events)\n", + " return fig.mouse_event(event, event['data']);\n", + " }\n", + "\n", + " rubberband.mousedown('button_press', mouse_event_fn);\n", + " rubberband.mouseup('button_release', mouse_event_fn);\n", + " // Throttle sequential mouse events to 1 every 20ms.\n", + " rubberband.mousemove('motion_notify', mouse_event_fn);\n", + "\n", + " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", + " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", + "\n", + " canvas_div.on(\"wheel\", function (event) {\n", + " event = event.originalEvent;\n", + " event['data'] = 'scroll'\n", + " if (event.deltaY < 0) {\n", + " event.step = 1;\n", + " } else {\n", + " event.step = -1;\n", + " }\n", + " mouse_event_fn(event);\n", + " });\n", + "\n", + " canvas_div.append(canvas);\n", + " canvas_div.append(rubberband);\n", + "\n", + " this.rubberband = rubberband;\n", + " this.rubberband_canvas = rubberband[0];\n", + " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", + " this.rubberband_context.strokeStyle = \"#000000\";\n", + "\n", + " this._resize_canvas = function(width, height) {\n", + " // Keep the size of the canvas, canvas container, and rubber band\n", + " // canvas in synch.\n", + " canvas_div.css('width', width)\n", + " canvas_div.css('height', height)\n", + "\n", + " canvas.attr('width', width);\n", + " canvas.attr('height', height);\n", + "\n", + " rubberband.attr('width', width);\n", + " rubberband.attr('height', height);\n", + " }\n", + "\n", + " // Set the figure to an initial 600x600px, this will subsequently be updated\n", + " // upon first draw.\n", + " this._resize_canvas(600, 600);\n", + "\n", + " // Disable right mouse context menu.\n", + " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", + " return false;\n", + " });\n", + "\n", + " function set_focus () {\n", + " canvas.focus();\n", + " canvas_div.focus();\n", + " }\n", + "\n", + " window.setTimeout(set_focus, 100);\n", + "}\n", + "\n", + "mpl.figure.prototype._init_toolbar = function() {\n", + " var fig = this;\n", + "\n", + " var nav_element = $('
')\n", + " nav_element.attr('style', 'width: 100%');\n", + " this.root.append(nav_element);\n", + "\n", + " // Define a callback function for later on.\n", + " function toolbar_event(event) {\n", + " return fig.toolbar_button_onclick(event['data']);\n", + " }\n", + " function toolbar_mouse_event(event) {\n", + " return fig.toolbar_button_onmouseover(event['data']);\n", + " }\n", + "\n", + " for(var toolbar_ind in mpl.toolbar_items) {\n", + " var name = mpl.toolbar_items[toolbar_ind][0];\n", + " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", + " var image = mpl.toolbar_items[toolbar_ind][2];\n", + " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", + "\n", + " if (!name) {\n", + " // put a spacer in here.\n", + " continue;\n", + " }\n", + " var button = $('');\n", - " button.click(method_name, toolbar_event);\n", - " button.mouseover(tooltip, toolbar_mouse_event);\n", - " nav_element.append(button);\n", - " }\n", - "\n", - " // Add the status bar.\n", - " var status_bar = $('');\n", - " nav_element.append(status_bar);\n", - " this.message = status_bar[0];\n", - "\n", - " // Add the close button to the window.\n", - " var buttongrp = $('
');\n", - " var button = $('');\n", - " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", - " buttongrp.append(button);\n", - " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", - " titlebar.prepend(buttongrp);\n", - "}\n", - "\n", - "mpl.figure.prototype._root_extra_style = function(el){\n", - " var fig = this\n", - " el.on(\"remove\", function(){\n", - "\tfig.close_ws(fig, {});\n", - " });\n", - "}\n", - "\n", - "mpl.figure.prototype._canvas_extra_style = function(el){\n", - " // this is important to make the div 'focusable\n", - " el.attr('tabindex', 0)\n", - " // reach out to IPython and tell the keyboard manager to turn it's self\n", - " // off when our div gets focus\n", - "\n", - " // location in version 3\n", - " if (IPython.notebook.keyboard_manager) {\n", - " IPython.notebook.keyboard_manager.register_events(el);\n", - " }\n", - " else {\n", - " // location in version 2\n", - " IPython.keyboard_manager.register_events(el);\n", - " }\n", - "\n", - "}\n", - "\n", - "mpl.figure.prototype._key_event_extra = function(event, name) {\n", - " var manager = IPython.notebook.keyboard_manager;\n", - " if (!manager)\n", - " manager = IPython.keyboard_manager;\n", - "\n", - " // Check for shift+enter\n", - " if (event.shiftKey && event.which == 13) {\n", - " this.canvas_div.blur();\n", - " event.shiftKey = false;\n", - " // Send a \"J\" for go to next cell\n", - " event.which = 74;\n", - " event.keyCode = 74;\n", - " manager.command_mode();\n", - " manager.handle_keydown(event);\n", - " }\n", - "}\n", - "\n", - "mpl.figure.prototype.handle_save = function(fig, msg) {\n", - " fig.ondownload(fig, null);\n", - "}\n", - "\n", - "\n", - "mpl.find_output_cell = function(html_output) {\n", - " // Return the cell and output element which can be found *uniquely* in the notebook.\n", - " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", - " // IPython event is triggered only after the cells have been serialised, which for\n", - " // our purposes (turning an active figure into a static one), is too late.\n", - " var cells = IPython.notebook.get_cells();\n", - " var ncells = cells.length;\n", - " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", - " data = data.data;\n", - " }\n", - " if (data['text/html'] == html_output) {\n", - " return [cell, data, j];\n", - " }\n", - " }\n", - " }\n", - " }\n", - "}\n", - "\n", - "// Register the function which deals with the matplotlib target/channel.\n", - "// The kernel may be null if the page has been refreshed.\n", - "if (IPython.notebook.kernel != null) {\n", - " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", - "}\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/html": [ - "" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "#Verlet integration\n", - "x3_dp = numpy.zeros((N,4)) \n", - "x3_dp[0,:] = x_init_H_dp.copy() #set initial conditions\n", - "#print(q3_dp[0,:])\n", - "for n in range(N-1): #integrate with Euler\n", - " x3_dp[n+1,:] = Verlet_DP(x3_dp[n,:], dt)\n", - " \n", - "pyplot.figure(figsize=(10,8));\n", - "pyplot.grid(True);\n", - "pyplot.xlabel(r't', fontsize=18);\n", - "pyplot.ylabel(r'position (radians)', fontsize=18);\n", - "pyplot.title('Double Pendulum Verlet Integration');\n", - "pyplot.plot(t, x3_dp[:,0], lw=2, label='Joint 1');\n", - "pyplot.plot(t, x3_dp[:,2], lw=2, label='Joint 2')\n", - "pyplot.legend();" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This certainly looks better! It's an interesting result. There is clearly some periodicity, although the motion isn't exactly periodic. But how accurate is this simulation? The short answer is that since we don't have an analytical solution, it is difficult to know exactly. However, there are still some properties of the system that we can look at. For one thing, the total energy in the system should be constant. Let's see if that is the case.\n", - "\n", - "Recall from the analysis above that the total energy is the combined potential and kinetic energy at each time step.\n", - "\n", - "$$\\begin{eqnarray*}\n", - "U &=& -m_1gl_1cos(q_1) - m_2g(l_1cos(q_1) + l_2cos(q_2))\\\\\n", - "T &=& \\frac{p_1^2}{2I_1}+\\frac{m_2p_1^2}{2m_1I_1} + \\left[ \\frac{p_2}{l_2} \\frac{p_1}{m_1l_1}cos(q_2-q_1) + \\frac{p_2^2}{2I_2} \\right]\n", - "\\end{eqnarray*}$$\n", - "\n" - ] - }, - { - "cell_type": "code", - "execution_count": 84, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "def get_Energy(u):\n", - " \"\"\" Calculates total energu in the system at given time step for a simple harmonic oscillator\n", - " Parameters:\n", - " u - state of system [pos, vel]\n", - " Returns:\n", - " total energy in system.\n", - " \"\"\"\n", - " q1 = u[0]\n", - " p1 = u[1]\n", - " q2 = u[2]\n", - " p2 = u[3]\n", - " \n", - " \n", - " \n", - " V = -m1*g*l1*cos(q1) - m2*g*(l1*cos(q1)+l2*cos(q2))\n", - " T = p1**2/(2*I1)+m2*p1**2/(2*m1*I1) + p1*p2/(m1*l1*l2)*cos(q2-q1)+p2**2/(2*I1)\n", - " #print(T)\n", - " #print(V)\n", - " return T+V" - ] - }, - { - "cell_type": "code", - "execution_count": 85, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "application/javascript": [ - "/* Put everything inside the global mpl namespace */\n", - "window.mpl = {};\n", - "\n", - "mpl.get_websocket_type = function() {\n", - " if (typeof(WebSocket) !== 'undefined') {\n", - " return WebSocket;\n", - " } else if (typeof(MozWebSocket) !== 'undefined') {\n", - " return MozWebSocket;\n", - " } else {\n", - " alert('Your browser does not have WebSocket support.' +\n", - " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", - " 'Firefox 4 and 5 are also supported but you ' +\n", - " 'have to enable WebSockets in about:config.');\n", - " };\n", - "}\n", - "\n", - "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", - " this.id = figure_id;\n", - "\n", - " this.ws = websocket;\n", - "\n", - " this.supports_binary = (this.ws.binaryType != undefined);\n", - "\n", - " if (!this.supports_binary) {\n", - " var warnings = document.getElementById(\"mpl-warnings\");\n", - " if (warnings) {\n", - " warnings.style.display = 'block';\n", - " warnings.textContent = (\n", - " \"This browser does not support binary websocket messages. \" +\n", - " \"Performance may be slow.\");\n", - " }\n", - " }\n", - "\n", - " this.imageObj = new Image();\n", - "\n", - " this.context = undefined;\n", - " this.message = undefined;\n", - " this.canvas = undefined;\n", - " this.rubberband_canvas = undefined;\n", - " this.rubberband_context = undefined;\n", - " this.format_dropdown = undefined;\n", - "\n", - " this.image_mode = 'full';\n", - "\n", - " this.root = $('
');\n", - " this._root_extra_style(this.root)\n", - " this.root.attr('style', 'display: inline-block');\n", - "\n", - " $(parent_element).append(this.root);\n", - "\n", - " this._init_header(this);\n", - " this._init_canvas(this);\n", - " this._init_toolbar(this);\n", - "\n", - " var fig = this;\n", - "\n", - " this.waiting = false;\n", - "\n", - " this.ws.onopen = function () {\n", - " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", - " fig.send_message(\"send_image_mode\", {});\n", - " fig.send_message(\"refresh\", {});\n", - " }\n", - "\n", - " this.imageObj.onload = function() {\n", - " if (fig.image_mode == 'full') {\n", - " // Full images could contain transparency (where diff images\n", - " // almost always do), so we need to clear the canvas so that\n", - " // there is no ghosting.\n", - " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", - " }\n", - " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " };\n", - "\n", - " this.imageObj.onunload = function() {\n", - " this.ws.close();\n", - " }\n", - "\n", - " this.ws.onmessage = this._make_on_message_function(this);\n", - "\n", - " this.ondownload = ondownload;\n", - "}\n", - "\n", - "mpl.figure.prototype._init_header = function() {\n", - " var titlebar = $(\n", - " '
');\n", - " var titletext = $(\n", - " '
');\n", - " titlebar.append(titletext)\n", - " this.root.append(titlebar);\n", - " this.header = titletext[0];\n", - "}\n", - "\n", - "\n", - "\n", - "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", - "\n", - "}\n", - "\n", - "\n", - "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", - "\n", - "}\n", - "\n", - "mpl.figure.prototype._init_canvas = function() {\n", - " var fig = this;\n", - "\n", - " var canvas_div = $('
');\n", - "\n", - " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", - "\n", - " function canvas_keyboard_event(event) {\n", - " return fig.key_event(event, event['data']);\n", - " }\n", - "\n", - " canvas_div.keydown('key_press', canvas_keyboard_event);\n", - " canvas_div.keyup('key_release', canvas_keyboard_event);\n", - " this.canvas_div = canvas_div\n", - " this._canvas_extra_style(canvas_div)\n", - " this.root.append(canvas_div);\n", - "\n", - " var canvas = $('');\n", - " canvas.addClass('mpl-canvas');\n", - " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", - "\n", - " this.canvas = canvas[0];\n", - " this.context = canvas[0].getContext(\"2d\");\n", - "\n", - " var rubberband = $('');\n", - " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", - "\n", - " var pass_mouse_events = true;\n", - "\n", - " canvas_div.resizable({\n", - " start: function(event, ui) {\n", - " pass_mouse_events = false;\n", - " },\n", - " resize: function(event, ui) {\n", - " fig.request_resize(ui.size.width, ui.size.height);\n", - " },\n", - " stop: function(event, ui) {\n", - " pass_mouse_events = true;\n", - " fig.request_resize(ui.size.width, ui.size.height);\n", - " },\n", - " });\n", - "\n", - " function mouse_event_fn(event) {\n", - " if (pass_mouse_events)\n", - " return fig.mouse_event(event, event['data']);\n", - " }\n", - "\n", - " rubberband.mousedown('button_press', mouse_event_fn);\n", - " rubberband.mouseup('button_release', mouse_event_fn);\n", - " // Throttle sequential mouse events to 1 every 20ms.\n", - " rubberband.mousemove('motion_notify', mouse_event_fn);\n", - "\n", - " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", - " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", - "\n", - " canvas_div.on(\"wheel\", function (event) {\n", - " event = event.originalEvent;\n", - " event['data'] = 'scroll'\n", - " if (event.deltaY < 0) {\n", - " event.step = 1;\n", - " } else {\n", - " event.step = -1;\n", - " }\n", - " mouse_event_fn(event);\n", - " });\n", - "\n", - " canvas_div.append(canvas);\n", - " canvas_div.append(rubberband);\n", - "\n", - " this.rubberband = rubberband;\n", - " this.rubberband_canvas = rubberband[0];\n", - " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", - " this.rubberband_context.strokeStyle = \"#000000\";\n", - "\n", - " this._resize_canvas = function(width, height) {\n", - " // Keep the size of the canvas, canvas container, and rubber band\n", - " // canvas in synch.\n", - " canvas_div.css('width', width)\n", - " canvas_div.css('height', height)\n", - "\n", - " canvas.attr('width', width);\n", - " canvas.attr('height', height);\n", - "\n", - " rubberband.attr('width', width);\n", - " rubberband.attr('height', height);\n", - " }\n", - "\n", - " // Set the figure to an initial 600x600px, this will subsequently be updated\n", - " // upon first draw.\n", - " this._resize_canvas(600, 600);\n", - "\n", - " // Disable right mouse context menu.\n", - " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", - " return false;\n", - " });\n", - "\n", - " function set_focus () {\n", - " canvas.focus();\n", - " canvas_div.focus();\n", - " }\n", - "\n", - " window.setTimeout(set_focus, 100);\n", - "}\n", - "\n", - "mpl.figure.prototype._init_toolbar = function() {\n", - " var fig = this;\n", - "\n", - " var nav_element = $('
')\n", - " nav_element.attr('style', 'width: 100%');\n", - " this.root.append(nav_element);\n", - "\n", - " // Define a callback function for later on.\n", - " function toolbar_event(event) {\n", - " return fig.toolbar_button_onclick(event['data']);\n", - " }\n", - " function toolbar_mouse_event(event) {\n", - " return fig.toolbar_button_onmouseover(event['data']);\n", - " }\n", - "\n", - " for(var toolbar_ind in mpl.toolbar_items) {\n", - " var name = mpl.toolbar_items[toolbar_ind][0];\n", - " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", - " var image = mpl.toolbar_items[toolbar_ind][2];\n", - " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", - "\n", - " if (!name) {\n", - " // put a spacer in here.\n", - " continue;\n", - " }\n", - " var button = $('');\n", - " button.click(method_name, toolbar_event);\n", - " button.mouseover(tooltip, toolbar_mouse_event);\n", - " nav_element.append(button);\n", - " }\n", - "\n", - " // Add the status bar.\n", - " var status_bar = $('');\n", - " nav_element.append(status_bar);\n", - " this.message = status_bar[0];\n", - "\n", - " // Add the close button to the window.\n", - " var buttongrp = $('
');\n", - " var button = $('');\n", - " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", - " buttongrp.append(button);\n", - " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", - " titlebar.prepend(buttongrp);\n", - "}\n", - "\n", - "mpl.figure.prototype._root_extra_style = function(el){\n", - " var fig = this\n", - " el.on(\"remove\", function(){\n", - "\tfig.close_ws(fig, {});\n", - " });\n", - "}\n", - "\n", - "mpl.figure.prototype._canvas_extra_style = function(el){\n", - " // this is important to make the div 'focusable\n", - " el.attr('tabindex', 0)\n", - " // reach out to IPython and tell the keyboard manager to turn it's self\n", - " // off when our div gets focus\n", - "\n", - " // location in version 3\n", - " if (IPython.notebook.keyboard_manager) {\n", - " IPython.notebook.keyboard_manager.register_events(el);\n", - " }\n", - " else {\n", - " // location in version 2\n", - " IPython.keyboard_manager.register_events(el);\n", - " }\n", - "\n", - "}\n", - "\n", - "mpl.figure.prototype._key_event_extra = function(event, name) {\n", - " var manager = IPython.notebook.keyboard_manager;\n", - " if (!manager)\n", - " manager = IPython.keyboard_manager;\n", - "\n", - " // Check for shift+enter\n", - " if (event.shiftKey && event.which == 13) {\n", - " this.canvas_div.blur();\n", - " event.shiftKey = false;\n", - " // Send a \"J\" for go to next cell\n", - " event.which = 74;\n", - " event.keyCode = 74;\n", - " manager.command_mode();\n", - " manager.handle_keydown(event);\n", - " }\n", - "}\n", - "\n", - "mpl.figure.prototype.handle_save = function(fig, msg) {\n", - " fig.ondownload(fig, null);\n", - "}\n", - "\n", - "\n", - "mpl.find_output_cell = function(html_output) {\n", - " // Return the cell and output element which can be found *uniquely* in the notebook.\n", - " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", - " // IPython event is triggered only after the cells have been serialised, which for\n", - " // our purposes (turning an active figure into a static one), is too late.\n", - " var cells = IPython.notebook.get_cells();\n", - " var ncells = cells.length;\n", - " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", - " data = data.data;\n", - " }\n", - " if (data['text/html'] == html_output) {\n", - " return [cell, data, j];\n", - " }\n", - " }\n", - " }\n", - " }\n", - "}\n", - "\n", - "// Register the function which deals with the matplotlib target/channel.\n", - "// The kernel may be null if the page has been refreshed.\n", - "if (IPython.notebook.kernel != null) {\n", - " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", - "}\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/html": [ - "" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "v = poincare(x3_dp, 1, 2*pi, 0, 2) #use states from RK4 integration\n", - "pyplot.figure(figsize=(10,8));\n", - "pyplot.grid(True);\n", - "pyplot.xlabel(r'$\\theta_2$', fontsize=18);\n", - "pyplot.ylabel(r'$\\dot{\\theta_2}$', fontsize=18);\n", - "pyplot.title('Poincare Section');\n", - "pyplot.plot(v[0,:], v[1,:], 'bo', lw=2, );" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "There isn't a clear pattern here. So, we can say that our system is NOT likely to be asymptotically stable. However, this shouldn't be surprising. If we continued simulating, our system would continue rotating since there is no damping. Let's see what happens when there is some damping in the system. We're going to use RK4 for the following analysis." - ] - }, - { - "cell_type": "code", - "execution_count": 98, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "#To Do:\n", - "# animate motion of double pendulum\n", - "# check through sources" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Damped Double Pendulum\n", - "\n", - "Just like with simple harmonic motion, we can add in a damping term to help remove some of the energy that the integration scheme adds in. In the problem setup, we actually used an abbreviated form of a standard equation. The full equation is here:\n", - "\n", - "\\begin{equation}\n", - "\\textbf{M}(x)\\vec{\\ddot{x}}(t) + \\textbf{V}(x, \\dot x ) \\vec{\\dot{x}} + \\textbf{G(x)} + f(\\dot x) + \\tau_d(t) = \\tau(t)\n", - "\\end{equation}" - ] - }, - { - "cell_type": "markdown", - "metadata": { - "collapsed": true - }, - "source": [ - "where $\\textbf{M, V}$, and $\\textbf{G}$ are defined as above, $f$ is frictional or damping terms, and the $\\tau$ terms represent added torque to the system. Since we aren't adding any torque, the $\\tau$ terms will remain 0. However, we can add in some linear damping by including the frictional term. This requires redefining our RHS function, but no other changes!" - ] - }, - { - "cell_type": "code", - "execution_count": 99, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "def f_damped_double_pendulum(u):\n", - " \"\"\"Returns RHS of double pendulum EOM\n", - " \n", - " Parameters:\n", - " q - initial state\n", - " \n", - " Returns:\n", - " RHS - RHS of harmonic oscillator eqn.\n", - " \n", - " \"\"\"\n", - " x1 = u[0]\n", - " x2 = u[1]\n", - " x3 = u[2]\n", - " x4 = u[3]\n", - " \n", - " M = numpy.array([[(m1+m2)*l1**2, m2*l1*l2*cos(x2-x1)],[m2*l1*l2*cos(x2-x1), m2*l2**2]])\n", - " V = numpy.array([[0, -m2*l1*l2*sin(x2-x1)],[-m2*l1*l2*sin(x2-x1), 0]])\n", - " G = numpy.array([[(m1+m2)*g*l1*sin(x1)],[m2*l2*g*sin(x2)]])\n", - " f = c*numpy.array([[x3],[x4]])\n", - " qdd = numpy.linalg.inv(M).dot(V.dot(numpy.array([[x3],[x4]]))+G -f) \n", - " #print(qdd)\n", - " #print(M)\n", - " #print(V.dot(numpy.array([[x3],[x4]])))\n", - " #print(G)\n", - " RHS = numpy.array([[x3, x4, qdd[0], qdd[1]]])\n", - " #print(RHS.T)\n", - " return RHS.T" - ] - }, - { - "cell_type": "code", - "execution_count": 100, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "application/javascript": [ - "/* Put everything inside the global mpl namespace */\n", - "window.mpl = {};\n", - "\n", - "mpl.get_websocket_type = function() {\n", - " if (typeof(WebSocket) !== 'undefined') {\n", - " return WebSocket;\n", - " } else if (typeof(MozWebSocket) !== 'undefined') {\n", - " return MozWebSocket;\n", - " } else {\n", - " alert('Your browser does not have WebSocket support.' +\n", - " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", - " 'Firefox 4 and 5 are also supported but you ' +\n", - " 'have to enable WebSockets in about:config.');\n", - " };\n", - "}\n", - "\n", - "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", - " this.id = figure_id;\n", - "\n", - " this.ws = websocket;\n", - "\n", - " this.supports_binary = (this.ws.binaryType != undefined);\n", - "\n", - " if (!this.supports_binary) {\n", - " var warnings = document.getElementById(\"mpl-warnings\");\n", - " if (warnings) {\n", - " warnings.style.display = 'block';\n", - " warnings.textContent = (\n", - " \"This browser does not support binary websocket messages. \" +\n", - " \"Performance may be slow.\");\n", - " }\n", - " }\n", - "\n", - " this.imageObj = new Image();\n", - "\n", - " this.context = undefined;\n", - " this.message = undefined;\n", - " this.canvas = undefined;\n", - " this.rubberband_canvas = undefined;\n", - " this.rubberband_context = undefined;\n", - " this.format_dropdown = undefined;\n", - "\n", - " this.image_mode = 'full';\n", - "\n", - " this.root = $('
');\n", - " this._root_extra_style(this.root)\n", - " this.root.attr('style', 'display: inline-block');\n", - "\n", - " $(parent_element).append(this.root);\n", - "\n", - " this._init_header(this);\n", - " this._init_canvas(this);\n", - " this._init_toolbar(this);\n", - "\n", - " var fig = this;\n", - "\n", - " this.waiting = false;\n", - "\n", - " this.ws.onopen = function () {\n", - " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", - " fig.send_message(\"send_image_mode\", {});\n", - " fig.send_message(\"refresh\", {});\n", - " }\n", - "\n", - " this.imageObj.onload = function() {\n", - " if (fig.image_mode == 'full') {\n", - " // Full images could contain transparency (where diff images\n", - " // almost always do), so we need to clear the canvas so that\n", - " // there is no ghosting.\n", - " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", - " }\n", - " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " };\n", - "\n", - " this.imageObj.onunload = function() {\n", - " this.ws.close();\n", - " }\n", - "\n", - " this.ws.onmessage = this._make_on_message_function(this);\n", - "\n", - " this.ondownload = ondownload;\n", - "}\n", - "\n", - "mpl.figure.prototype._init_header = function() {\n", - " var titlebar = $(\n", - " '
');\n", - " var titletext = $(\n", - " '
');\n", - " titlebar.append(titletext)\n", - " this.root.append(titlebar);\n", - " this.header = titletext[0];\n", - "}\n", - "\n", - "\n", - "\n", - "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", - "\n", - "}\n", - "\n", - "\n", - "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", - "\n", - "}\n", - "\n", - "mpl.figure.prototype._init_canvas = function() {\n", - " var fig = this;\n", - "\n", - " var canvas_div = $('
');\n", - "\n", - " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", - "\n", - " function canvas_keyboard_event(event) {\n", - " return fig.key_event(event, event['data']);\n", - " }\n", - "\n", - " canvas_div.keydown('key_press', canvas_keyboard_event);\n", - " canvas_div.keyup('key_release', canvas_keyboard_event);\n", - " this.canvas_div = canvas_div\n", - " this._canvas_extra_style(canvas_div)\n", - " this.root.append(canvas_div);\n", - "\n", - " var canvas = $('');\n", - " canvas.addClass('mpl-canvas');\n", - " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", - "\n", - " this.canvas = canvas[0];\n", - " this.context = canvas[0].getContext(\"2d\");\n", - "\n", - " var rubberband = $('');\n", - " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", - "\n", - " var pass_mouse_events = true;\n", - "\n", - " canvas_div.resizable({\n", - " start: function(event, ui) {\n", - " pass_mouse_events = false;\n", - " },\n", - " resize: function(event, ui) {\n", - " fig.request_resize(ui.size.width, ui.size.height);\n", - " },\n", - " stop: function(event, ui) {\n", - " pass_mouse_events = true;\n", - " fig.request_resize(ui.size.width, ui.size.height);\n", - " },\n", - " });\n", - "\n", - " function mouse_event_fn(event) {\n", - " if (pass_mouse_events)\n", - " return fig.mouse_event(event, event['data']);\n", - " }\n", - "\n", - " rubberband.mousedown('button_press', mouse_event_fn);\n", - " rubberband.mouseup('button_release', mouse_event_fn);\n", - " // Throttle sequential mouse events to 1 every 20ms.\n", - " rubberband.mousemove('motion_notify', mouse_event_fn);\n", - "\n", - " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", - " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", - "\n", - " canvas_div.on(\"wheel\", function (event) {\n", - " event = event.originalEvent;\n", - " event['data'] = 'scroll'\n", - " if (event.deltaY < 0) {\n", - " event.step = 1;\n", - " } else {\n", - " event.step = -1;\n", - " }\n", - " mouse_event_fn(event);\n", - " });\n", - "\n", - " canvas_div.append(canvas);\n", - " canvas_div.append(rubberband);\n", - "\n", - " this.rubberband = rubberband;\n", - " this.rubberband_canvas = rubberband[0];\n", - " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", - " this.rubberband_context.strokeStyle = \"#000000\";\n", - "\n", - " this._resize_canvas = function(width, height) {\n", - " // Keep the size of the canvas, canvas container, and rubber band\n", - " // canvas in synch.\n", - " canvas_div.css('width', width)\n", - " canvas_div.css('height', height)\n", - "\n", - " canvas.attr('width', width);\n", - " canvas.attr('height', height);\n", - "\n", - " rubberband.attr('width', width);\n", - " rubberband.attr('height', height);\n", - " }\n", - "\n", - " // Set the figure to an initial 600x600px, this will subsequently be updated\n", - " // upon first draw.\n", - " this._resize_canvas(600, 600);\n", - "\n", - " // Disable right mouse context menu.\n", - " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", - " return false;\n", - " });\n", - "\n", - " function set_focus () {\n", - " canvas.focus();\n", - " canvas_div.focus();\n", - " }\n", - "\n", - " window.setTimeout(set_focus, 100);\n", - "}\n", - "\n", - "mpl.figure.prototype._init_toolbar = function() {\n", - " var fig = this;\n", - "\n", - " var nav_element = $('
')\n", - " nav_element.attr('style', 'width: 100%');\n", - " this.root.append(nav_element);\n", - "\n", - " // Define a callback function for later on.\n", - " function toolbar_event(event) {\n", - " return fig.toolbar_button_onclick(event['data']);\n", - " }\n", - " function toolbar_mouse_event(event) {\n", - " return fig.toolbar_button_onmouseover(event['data']);\n", - " }\n", - "\n", - " for(var toolbar_ind in mpl.toolbar_items) {\n", - " var name = mpl.toolbar_items[toolbar_ind][0];\n", - " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", - " var image = mpl.toolbar_items[toolbar_ind][2];\n", - " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", - "\n", - " if (!name) {\n", - " // put a spacer in here.\n", - " continue;\n", - " }\n", - " var button = $('');\n", - " button.click(method_name, toolbar_event);\n", - " button.mouseover(tooltip, toolbar_mouse_event);\n", - " nav_element.append(button);\n", - " }\n", - "\n", - " // Add the status bar.\n", - " var status_bar = $('');\n", - " nav_element.append(status_bar);\n", - " this.message = status_bar[0];\n", - "\n", - " // Add the close button to the window.\n", - " var buttongrp = $('
');\n", - " var button = $('');\n", - " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", - " buttongrp.append(button);\n", - " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", - " titlebar.prepend(buttongrp);\n", - "}\n", - "\n", - "mpl.figure.prototype._root_extra_style = function(el){\n", - " var fig = this\n", - " el.on(\"remove\", function(){\n", - "\tfig.close_ws(fig, {});\n", - " });\n", - "}\n", - "\n", - "mpl.figure.prototype._canvas_extra_style = function(el){\n", - " // this is important to make the div 'focusable\n", - " el.attr('tabindex', 0)\n", - " // reach out to IPython and tell the keyboard manager to turn it's self\n", - " // off when our div gets focus\n", - "\n", - " // location in version 3\n", - " if (IPython.notebook.keyboard_manager) {\n", - " IPython.notebook.keyboard_manager.register_events(el);\n", - " }\n", - " else {\n", - " // location in version 2\n", - " IPython.keyboard_manager.register_events(el);\n", - " }\n", - "\n", - "}\n", - "\n", - "mpl.figure.prototype._key_event_extra = function(event, name) {\n", - " var manager = IPython.notebook.keyboard_manager;\n", - " if (!manager)\n", - " manager = IPython.keyboard_manager;\n", - "\n", - " // Check for shift+enter\n", - " if (event.shiftKey && event.which == 13) {\n", - " this.canvas_div.blur();\n", - " event.shiftKey = false;\n", - " // Send a \"J\" for go to next cell\n", - " event.which = 74;\n", - " event.keyCode = 74;\n", - " manager.command_mode();\n", - " manager.handle_keydown(event);\n", - " }\n", - "}\n", - "\n", - "mpl.figure.prototype.handle_save = function(fig, msg) {\n", - " fig.ondownload(fig, null);\n", - "}\n", - "\n", - "\n", - "mpl.find_output_cell = function(html_output) {\n", - " // Return the cell and output element which can be found *uniquely* in the notebook.\n", - " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", - " // IPython event is triggered only after the cells have been serialised, which for\n", - " // our purposes (turning an active figure into a static one), is too late.\n", - " var cells = IPython.notebook.get_cells();\n", - " var ncells = cells.length;\n", - " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", - " data = data.data;\n", - " }\n", - " if (data['text/html'] == html_output) {\n", - " return [cell, data, j];\n", - " }\n", - " }\n", - " }\n", - " }\n", - "}\n", - "\n", - "// Register the function which deals with the matplotlib target/channel.\n", - "// The kernel may be null if the page has been refreshed.\n", - "if (IPython.notebook.kernel != null) {\n", - " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", - "}\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/html": [ - "" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "#We use the same initial conditions as the example without damping, so they aren't redefined here.\n", - "c = 2 #[N/(rad/s)] define some damping\n", - "\n", - "#Euler\n", - "q1d_dp = numpy.zeros((N,4)) \n", - "q1d_dp[0,:] = x_init_dp.copy() #set initial conditions\n", - "for n in range(N-1): #integrate with Euler\n", - " q1d_dp[n+1,:] = euler_DP(q1d_dp[n,:], f_damped_double_pendulum, dt)\n", - " #print(q1[n])\n", - " \n", - "#Runge-Kutta\n", - "q2d_dp = numpy.zeros((N,4)) \n", - "q2d_dp[0,:] = x_init_dp.copy() #set initial conditions\n", - "for n in range(N-1): #integrate with Euler\n", - " q2d_dp[n+1,:] = RK4_DP(q2d_dp[n,:], f_damped_double_pendulum, dt)\n", - " \n", - "pyplot.figure(figsize=(10,8));\n", - "pyplot.grid(True);\n", - "pyplot.xlabel(r't', fontsize=18);\n", - "pyplot.ylabel(r'position (meters)', fontsize=18);\n", - "pyplot.title('Double Pendulum Euler');\n", - "pyplot.plot(t, q1d_dp[:,0], lw=2, label='Joint 1');\n", - "pyplot.plot(t, q1d_dp[:,1], lw=2, label='Joint 2')\n", - "pyplot.legend();\n", - "\n", - "pyplot.figure(figsize=(10,8));\n", - "pyplot.grid(True);\n", - "pyplot.xlabel(r't', fontsize=18);\n", - "pyplot.ylabel(r'position (meters)', fontsize=18);\n", - "pyplot.title('Double Pendulum RK4');\n", - "pyplot.plot(t, q2d_dp[:,0], lw=2, label='Joint 1');\n", - "pyplot.plot(t, q2d_dp[:,1], lw=2, label='Joint 2')\n", - "pyplot.legend();" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "It looks like the system is chaotic at first, but becomes periodic as it settles.\n", - "\n", - "Let's do a Poincare section of this plot as well. For brevity, we'll use only the RK4 system response." - ] - }, - { - "cell_type": "code", - "execution_count": 101, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "application/javascript": [ - "/* Put everything inside the global mpl namespace */\n", - "window.mpl = {};\n", - "\n", - "mpl.get_websocket_type = function() {\n", - " if (typeof(WebSocket) !== 'undefined') {\n", - " return WebSocket;\n", - " } else if (typeof(MozWebSocket) !== 'undefined') {\n", - " return MozWebSocket;\n", - " } else {\n", - " alert('Your browser does not have WebSocket support.' +\n", - " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", - " 'Firefox 4 and 5 are also supported but you ' +\n", - " 'have to enable WebSockets in about:config.');\n", - " };\n", - "}\n", - "\n", - "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", - " this.id = figure_id;\n", - "\n", - " this.ws = websocket;\n", - "\n", - " this.supports_binary = (this.ws.binaryType != undefined);\n", - "\n", - " if (!this.supports_binary) {\n", - " var warnings = document.getElementById(\"mpl-warnings\");\n", - " if (warnings) {\n", - " warnings.style.display = 'block';\n", - " warnings.textContent = (\n", - " \"This browser does not support binary websocket messages. \" +\n", - " \"Performance may be slow.\");\n", - " }\n", - " }\n", - "\n", - " this.imageObj = new Image();\n", - "\n", - " this.context = undefined;\n", - " this.message = undefined;\n", - " this.canvas = undefined;\n", - " this.rubberband_canvas = undefined;\n", - " this.rubberband_context = undefined;\n", - " this.format_dropdown = undefined;\n", - "\n", - " this.image_mode = 'full';\n", - "\n", - " this.root = $('
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');\n", - " var titletext = $(\n", - " '
');\n", - " titlebar.append(titletext)\n", - " this.root.append(titlebar);\n", - " this.header = titletext[0];\n", - "}\n", - "\n", - "\n", - "\n", - "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", - "\n", - "}\n", - "\n", - "\n", - "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", - "\n", - "}\n", - "\n", - "mpl.figure.prototype._init_canvas = function() {\n", - " var fig = this;\n", - "\n", - " var canvas_div = $('
');\n", - "\n", - " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", - "\n", - " function canvas_keyboard_event(event) {\n", - " return fig.key_event(event, event['data']);\n", - " }\n", - "\n", - " canvas_div.keydown('key_press', canvas_keyboard_event);\n", - " canvas_div.keyup('key_release', canvas_keyboard_event);\n", - " this.canvas_div = canvas_div\n", - " this._canvas_extra_style(canvas_div)\n", - " this.root.append(canvas_div);\n", - "\n", - " var canvas = $('');\n", - " canvas.addClass('mpl-canvas');\n", - " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", - "\n", - " this.canvas = canvas[0];\n", - " this.context = canvas[0].getContext(\"2d\");\n", - "\n", - " var rubberband = $('');\n", - " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", - "\n", - " var pass_mouse_events = true;\n", - "\n", - " canvas_div.resizable({\n", - " start: function(event, ui) {\n", - " pass_mouse_events = false;\n", - " },\n", - " resize: function(event, ui) {\n", - " fig.request_resize(ui.size.width, ui.size.height);\n", - " },\n", - " stop: function(event, ui) {\n", - " pass_mouse_events = true;\n", - " fig.request_resize(ui.size.width, ui.size.height);\n", - " },\n", - " });\n", - "\n", - " function mouse_event_fn(event) {\n", - " if (pass_mouse_events)\n", - " return fig.mouse_event(event, event['data']);\n", - " }\n", - "\n", - " rubberband.mousedown('button_press', mouse_event_fn);\n", - " rubberband.mouseup('button_release', mouse_event_fn);\n", - " // Throttle sequential mouse events to 1 every 20ms.\n", - " rubberband.mousemove('motion_notify', mouse_event_fn);\n", - "\n", - " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", - " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", - "\n", - " canvas_div.on(\"wheel\", function (event) {\n", - " event = event.originalEvent;\n", - " event['data'] = 'scroll'\n", - " if (event.deltaY < 0) {\n", - " event.step = 1;\n", - " } else {\n", - " event.step = -1;\n", - " }\n", - " mouse_event_fn(event);\n", - " });\n", - "\n", - " canvas_div.append(canvas);\n", - " canvas_div.append(rubberband);\n", - "\n", - " this.rubberband = rubberband;\n", - " this.rubberband_canvas = rubberband[0];\n", - " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", - " this.rubberband_context.strokeStyle = \"#000000\";\n", - "\n", - " this._resize_canvas = function(width, height) {\n", - " // Keep the size of the canvas, canvas container, and rubber band\n", - " // canvas in synch.\n", - " canvas_div.css('width', width)\n", - " canvas_div.css('height', height)\n", - "\n", - " canvas.attr('width', width);\n", - " canvas.attr('height', height);\n", - "\n", - " rubberband.attr('width', width);\n", - " rubberband.attr('height', height);\n", - " }\n", - "\n", - " // Set the figure to an initial 600x600px, this will subsequently be updated\n", - " // upon first draw.\n", - " this._resize_canvas(600, 600);\n", - "\n", - " // Disable right mouse context menu.\n", - " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", - " return false;\n", - " });\n", - "\n", - " function set_focus () {\n", - " canvas.focus();\n", - " canvas_div.focus();\n", - " }\n", - "\n", - " window.setTimeout(set_focus, 100);\n", - "}\n", - "\n", - "mpl.figure.prototype._init_toolbar = function() {\n", - " var fig = this;\n", - "\n", - " var nav_element = $('
')\n", - " nav_element.attr('style', 'width: 100%');\n", - " this.root.append(nav_element);\n", - "\n", - " // Define a callback function for later on.\n", - " function toolbar_event(event) {\n", - " return fig.toolbar_button_onclick(event['data']);\n", - " }\n", - " function toolbar_mouse_event(event) {\n", - " return fig.toolbar_button_onmouseover(event['data']);\n", - " }\n", - "\n", - " for(var toolbar_ind in mpl.toolbar_items) {\n", - " var name = mpl.toolbar_items[toolbar_ind][0];\n", - " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", - " var image = mpl.toolbar_items[toolbar_ind][2];\n", - " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", - "\n", - " if (!name) {\n", - " // put a spacer in here.\n", - " continue;\n", - " }\n", - " var button = $('');\n", - " button.click(method_name, toolbar_event);\n", - " button.mouseover(tooltip, toolbar_mouse_event);\n", - " nav_element.append(button);\n", - " }\n", - "\n", - " // Add the status bar.\n", - " var status_bar = $('');\n", - " nav_element.append(status_bar);\n", - " this.message = status_bar[0];\n", - "\n", - " // Add the close button to the window.\n", - " var buttongrp = $('
');\n", - " var button = $('');\n", - " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", - " buttongrp.append(button);\n", - " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", - " titlebar.prepend(buttongrp);\n", - "}\n", - "\n", - "mpl.figure.prototype._root_extra_style = function(el){\n", - " var fig = this\n", - " el.on(\"remove\", function(){\n", - "\tfig.close_ws(fig, {});\n", - " });\n", - "}\n", - "\n", - "mpl.figure.prototype._canvas_extra_style = function(el){\n", - " // this is important to make the div 'focusable\n", - " el.attr('tabindex', 0)\n", - " // reach out to IPython and tell the keyboard manager to turn it's self\n", - " // off when our div gets focus\n", - "\n", - " // location in version 3\n", - " if (IPython.notebook.keyboard_manager) {\n", - " IPython.notebook.keyboard_manager.register_events(el);\n", - " }\n", - " else {\n", - " // location in version 2\n", - " IPython.keyboard_manager.register_events(el);\n", - " }\n", - "\n", - "}\n", - "\n", - "mpl.figure.prototype._key_event_extra = function(event, name) {\n", - " var manager = IPython.notebook.keyboard_manager;\n", - " if (!manager)\n", - " manager = IPython.keyboard_manager;\n", - "\n", - " // Check for shift+enter\n", - " if (event.shiftKey && event.which == 13) {\n", - " this.canvas_div.blur();\n", - " event.shiftKey = false;\n", - " // Send a \"J\" for go to next cell\n", - " event.which = 74;\n", - " event.keyCode = 74;\n", - " manager.command_mode();\n", - " manager.handle_keydown(event);\n", - " }\n", - "}\n", - "\n", - "mpl.figure.prototype.handle_save = function(fig, msg) {\n", - " fig.ondownload(fig, null);\n", - "}\n", - "\n", - "\n", - "mpl.find_output_cell = function(html_output) {\n", - " // Return the cell and output element which can be found *uniquely* in the notebook.\n", - " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", - " // IPython event is triggered only after the cells have been serialised, which for\n", - " // our purposes (turning an active figure into a static one), is too late.\n", - " var cells = IPython.notebook.get_cells();\n", - " var ncells = cells.length;\n", - " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", - " data = data.data;\n", - " }\n", - " if (data['text/html'] == html_output) {\n", - " return [cell, data, j];\n", - " }\n", - " }\n", - " }\n", - " }\n", - "}\n", - "\n", - "// Register the function which deals with the matplotlib target/channel.\n", - "// The kernel may be null if the page has been refreshed.\n", - "if (IPython.notebook.kernel != null) {\n", - " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", - "}\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/html": [ - "" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "#Euler Integration\n", - "x1 = numpy.zeros((N,2)) #initialize array\n", - "x1[0] = x_init.copy() #set initial conditions\n", - "for n in range(N-1): #integrate with Euler\n", - " x1[n+1] = euler(x1[n], f_harmonic_oscillator, dt)\n", - " #print(q1[n])\n", - " \n", - "#Runge-Kutta Integration Second Order\n", - "x2 = numpy.zeros((N,2))\n", - "x2[0] = x_init.copy()\n", - "for n in range(N-1):\n", - " x2[n+1] = RK2(x2[n], f_harmonic_oscillator, dt)\n", - "\n", - "\n", - "#Runge-Kutta Integration Fourth Order\n", - "x4 = numpy.zeros((N,2))\n", - "x4[0] = x_init.copy()\n", - "for n in range(N-1):\n", - " x4[n+1] = RK4(x4[n], f_harmonic_oscillator, dt)\n", - " \n", - " \n", - "#Symplectic Integration\n", - "xv = numpy.zeros((N,2))\n", - "xv[0] = x_init_H.copy()\n", - "for n in range(N-1):\n", - " xv[n+1] = verlet_SHM(xv[n], dt)\n", - "\n", - "#Analytical Solution \n", - "A = x0 #with no forcing and no initial velocity, the max amplitude is equal to initial amplitude\n", - "phi = 0\n", - "omega = sqrt(k/m)\n", - "x_analytical = A*numpy.cos(omega*t + phi)\n", - "#analytical velocity:\n", - "vel_analytical = -A*omega*numpy.sin(omega*t + phi)\n", - " \n", - "pyplot.figure(figsize=(11,8));\n", - "pyplot.grid(True);\n", - "pyplot.xlabel(r't', fontsize=18);\n", - "pyplot.ylabel(r'position (meters)', fontsize=18);\n", - "pyplot.title('Harmonic oscillator position');\n", - "pyplot.plot(t, x1[:,0], lw=2, label='Euler');\n", - "pyplot.plot(t, x2[:,0], 'r', lw=2, label='RK2');\n", - "pyplot.plot(t, x4[:,0], 'c', lw=2, label='RK4');\n", - "pyplot.plot(t, xv[:,0], 'g', lw=2, label='Verlet');\n", - "pyplot.plot(t, x_analytical, 'k--', lw=2, label='analytical');\n", - "pyplot.legend();" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we look closely at the plot, we can see that Verlet and Runge-Kutta integration are close to each other, but something is clearly off with Euler's method. But how close are these methods to the analytical solution? Does Verlet do any better than we expect? In this case it is obvious that Euler is not correct (and possibly unstable). One thing to check is whether the behavior changes as we change our time step size. Maybe the timestep is too large to capture some aspect of the system response." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Error Analysis\n", - "The error analysis code below is based on the numerical methods MOOC notebook on phugoid oscillation. " - ] - }, - { - "cell_type": "code", - "execution_count": 12, + "execution_count": 7, "metadata": { "collapsed": true }, "outputs": [], "source": [ - "def get_error(y, z, dt):\n", - " \"\"\"Returns error relative to analytical solution using L-1 norm.\n", - " \n", + "def verlet_SHM(u, dt):\n", + " \"\"\" Verlet integration for integrating from Hamiltonian equations of motion\n", + " for a simple harmonic oscillator\n", " Parameters:\n", - " y - numerical solution\n", - " z - analytical solution\n", + " u - state at current time step\n", " dt - time step size\n", - " \n", - " Returns:\n", - " error - L_1 norm of the error W.R.T. analytical solution \n", + " Returns: \n", + " state at next time step.\n", " \"\"\"\n", - " err = dt*numpy.sum(numpy.abs(y-z))\n", - " return err" + " pos = u[0]\n", + " mom = u[1]\n", + " \n", + " q_half = pos - dt/2*mom/m\n", + " #p_half = mom + dt/2*(k*pos)\n", + " \n", + " p = mom + dt/2*(2*k*q_half)\n", + " q = q_half - dt/2*p/m\n", + " \n", + " return numpy.array([q, p]) " ] }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [], "source": [ - "dt_values = numpy.array([0.1, 0.05, 0.01, 0.005, 0.001, 0.0001])\n", + "#Initial Conditions\n", + "x0 = 0.5 #[m]\n", + "xdot0 = 0 #[m/s]\n", "\n", - "Euler_error_values = numpy.zeros_like(dt_values)\n", - "RK2_error_values = Euler_error_values.copy()\n", - "RK4_error_values = Euler_error_values.copy()\n", - "Verlet_error_values = Euler_error_values.copy()\n", - "for i, dt in enumerate(dt_values):\n", - " ###Call error function\n", - " Euler_error_values[i] = get_error(x1[:,0], x_analytical, dt)\n", - " RK2_error_values[i] = get_error(x2[:,0], x_analytical, dt)\n", - " RK4_error_values[i] = get_error(x4[:,0], x_analytical, dt)\n", - " Verlet_error_values[i] = get_error(xv[:,0], x_analytical, dt)\n", - " \n" + "#Equivalent initial conditions for the Hamiltonian system\n", + "q0 = x0\n", + "p0 = m*xdot0\n", + "\n", + "x_init = numpy.array([x0, xdot0])\n", + "x_init_H = numpy.array([q0, p0])" ] }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 9, "metadata": { "collapsed": false }, @@ -2074,7 +1180,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2085,16 +1191,51 @@ } ], "source": [ - "pyplot.figure(figsize=(10,6))\n", - "pyplot.tick_params(axis='both', labelsize=14)\n", - "pyplot.grid(True)\n", - "pyplot.xlabel('$\\Delta t$', fontsize=16)\n", - "pyplot.ylabel('Error', fontsize=16)\n", - "pyplot.loglog(dt_values, Euler_error_values, 'bo-', label='Euler')\n", - "pyplot.loglog(dt_values, RK2_error_values, 'ro-', label='RK2')\n", - "pyplot.loglog(dt_values, RK4_error_values, 'co-', label='RK4')\n", - "pyplot.loglog(dt_values, Verlet_error_values, 'go-', label='Symplectic')\n", - "pyplot.axis('equal')\n", + "#Euler Integration\n", + "x1 = numpy.zeros((N,2)) #initialize array\n", + "x1[0] = x_init.copy() #set initial conditions\n", + "for n in range(N-1): #integrate with Euler\n", + " x1[n+1] = euler(x1[n], f_harmonic_oscillator, dt)\n", + " #print(q1[n])\n", + " \n", + "#Runge-Kutta Integration Second Order\n", + "x2 = numpy.zeros((N,2))\n", + "x2[0] = x_init.copy()\n", + "for n in range(N-1):\n", + " x2[n+1] = RK2(x2[n], f_harmonic_oscillator, dt)\n", + "\n", + "\n", + "#Runge-Kutta Integration Fourth Order\n", + "x4 = numpy.zeros((N,2))\n", + "x4[0] = x_init.copy()\n", + "for n in range(N-1):\n", + " x4[n+1] = RK4(x4[n], f_harmonic_oscillator, dt)\n", + " \n", + " \n", + "#Symplectic Integration\n", + "xv = numpy.zeros((N,2))\n", + "xv[0] = x_init_H.copy()\n", + "for n in range(N-1):\n", + " xv[n+1] = verlet_SHM(xv[n], dt)\n", + "\n", + "#Analytical Solution \n", + "A = x0 #with no forcing and no initial velocity, the max amplitude is equal to initial amplitude\n", + "phi = 0\n", + "omega = sqrt(k/m)\n", + "x_analytical = A*numpy.cos(omega*t + phi)\n", + "#analytical velocity:\n", + "vel_analytical = -A*omega*numpy.sin(omega*t + phi)\n", + " \n", + "pyplot.figure(figsize=(11,8));\n", + "pyplot.grid(True);\n", + "pyplot.xlabel(r't', fontsize=18);\n", + "pyplot.ylabel(r'position (meters)', fontsize=18);\n", + "pyplot.title('Harmonic oscillator position');\n", + "pyplot.plot(t, x1[:,0], lw=2, label='Euler');\n", + "pyplot.plot(t, x2[:,0], 'r', lw=2, label='RK2');\n", + "pyplot.plot(t, x4[:,0], 'c', lw=2, label='RK4');\n", + "pyplot.plot(t, xv[:,0], 'g', lw=2, label='Verlet');\n", + "pyplot.plot(t, x_analytical, 'k--', lw=2, label='analytical');\n", "pyplot.legend();" ] }, @@ -2102,35 +1243,70 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Is this what you would expect to see based on the plots of position above? What are the sources of this error? We have the usual suspects: truncation error, discretization error, possible precision errors. Don't believe that these are significant? The discretization is the only difference between the first two plots above, and with a smaller time step the response from Runge-Kutta and Euler are greatly improved. To exame precision errors, take a look at the following screenshot, which shows the plots above, built with identical code on two different computers. The responses on the left are from a 64-bit computer, and the responses on the right are from a 32-bit computer. Look closely at the phase shift in the system response. That's a huge difference! Even the precision of the machine can have a significant impact on numerical integration.\n", - "\n", - "![Image](figures/combined_response.png)\n", - "\n", - "![Image](figures/error_combined.png)\n", - " # FIX THESE IMAGES" + "If we look closely at the plot, we can see that Verlet and Runge-Kutta integration are close to each other, but something is clearly off with Euler's method. But how close are these methods to the analytical solution? Does Verlet do any better than we expect? In this case it is obvious that Euler is not correct (and possibly unstable). One thing to check is whether the behavior changes as we change our time step size. Maybe the timestep is too large to capture some aspect of the system response." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Despite the numerical errors, why is one second order scheme (Verlet) outperforming another (RK2) by so much? What is going on? We can check the order of convergence to make sure that these schemes are actually 2nd order." + "### Error Analysis\n", + "The error analysis code below is based on the numerical methods MOOC notebook on phugoid oscillation. " + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def get_error(y, z, dt):\n", + " \"\"\"Returns error relative to analytical solution using L-1 norm.\n", + " \n", + " Parameters:\n", + " y - numerical solution\n", + " z - analytical solution\n", + " dt - time step size\n", + " \n", + " Returns:\n", + " error - L_1 norm of the error W.R.T. analytical solution \n", + " \"\"\"\n", + " err = dt*numpy.sum(numpy.abs(y-z))\n", + " return err" ] }, { "cell_type": "code", - "execution_count": 65, + "execution_count": 13, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "dt_values = numpy.array([0.1, 0.05, 0.01, 0.005, 0.001, 0.0001])\n", + "\n", + "Euler_error_values = numpy.empty_like(dt_values)\n", + "RK2_error_values = Euler_error_values.copy()\n", + "RK4_error_values = Euler_error_values.copy()\n", + "Verlet_error_values = Euler_error_values.copy()\n", + "for i, dt in enumerate(dt_values):\n", + " ###Call error function\n", + " Euler_error_values[i] = get_error(x1[:,0], x_analytical, dt)\n", + " RK2_error_values[i] = get_error(x2[:,0], x_analytical, dt)\n", + " RK4_error_values[i] = get_error(x4[:,0], x_analytical, dt)\n", + " Verlet_error_values[i] = get_error(xv[:,0], x_analytical, dt)\n", + " \n" + ] + }, + { + "cell_type": "code", + "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/home/mint/miniconda3/lib/python3.5/site-packages/ipykernel/__main__.py:24: DeprecationWarning: using a non-integer number instead of an integer will result in an error in the future\n" - ] - }, { "data": { "application/javascript": [ @@ -2898,7 +2074,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2908,6 +2084,54 @@ "output_type": "display_data" } ], + "source": [ + "pyplot.figure(figsize=(10,6))\n", + "pyplot.tick_params(axis='both', labelsize=14)\n", + "pyplot.grid(True)\n", + "pyplot.xlabel('$\\Delta t$', fontsize=16)\n", + "pyplot.ylabel('Error', fontsize=16)\n", + "pyplot.loglog(dt_values, Euler_error_values, 'bo-', label='Euler')\n", + "pyplot.loglog(dt_values, RK2_error_values, 'ro-', label='RK2')\n", + "pyplot.loglog(dt_values, RK4_error_values, 'co-', label='RK4')\n", + "pyplot.loglog(dt_values, Verlet_error_values, 'go-', label='Symplectic')\n", + "pyplot.axis('equal')\n", + "pyplot.legend();" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Is this what you would expect to see based on the plots of position above? What are the sources of this error? We have the usual suspects: truncation error, discretization error, possible precision errors. Don't believe that these are significant? The discretization is the only difference between the first two plots above, and with a smaller time step the response from Runge-Kutta and Euler are greatly improved. To exame precision errors, take a look at the following screenshot, which shows the plots above, built with identical code on two different computers. The responses on the left are from a 64-bit computer, and the responses on the right are from a 32-bit computer. Look closely at the phase shift in the system response. That's a huge difference! Even the precision of the machine can have a significant impact on numerical integration.\n", + "\n", + "![Image](figures/combined_response.png)\n", + "\n", + "![Image](figures/error_combined.png)\n", + " # FIX THESE IMAGES" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Despite the numerical errors, why is one second order scheme (Verlet) outperforming another (RK2) by so much? What is going on? We can check the order of convergence to make sure that these schemes are actually 2nd order." + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/home/mint/miniconda3/lib/python3.5/site-packages/ipykernel/__main__.py:24: DeprecationWarning: using a non-integer number instead of an integer will result in an error in the future\n" + ] + } + ], "source": [ "def get_diffgrid(u_current, u_fine, dt):\n", " \"\"\"Returns the difference between one grid and the fine one using L-1 norm.\n", @@ -2978,7 +2202,7 @@ }, { "cell_type": "code", - "execution_count": 70, + "execution_count": 21, "metadata": { "collapsed": false }, @@ -3791,7 +3015,7 @@ }, { "cell_type": "code", - "execution_count": 71, + "execution_count": 22, "metadata": { "collapsed": true }, @@ -3814,7 +3038,7 @@ }, { "cell_type": "code", - "execution_count": 78, + "execution_count": 24, "metadata": { "collapsed": false }, @@ -3834,7 +3058,7 @@ "#Verlet total energy\n", "#Here we will convert from momentum to velocity so we can use the same get_energy function.\n", "#Momemtum (q) is mass*velocity.\n", - "Verlet_vel = x3[:,1]/m\n", + "Verlet_vel = xv[:,1]/m\n", "#analytic total energy\n", "analytical_energy = numpy.zeros_like(t)\n", "anayltical_state = numpy.array([x_analytical, vel_analytical]).T\n", @@ -3849,7 +3073,7 @@ }, { "cell_type": "code", - "execution_count": 83, + "execution_count": 25, "metadata": { "collapsed": false }, @@ -5501,7 +4725,7 @@ }, { "cell_type": "code", - "execution_count": 84, + "execution_count": 26, "metadata": { "collapsed": true }, @@ -5525,7 +4749,7 @@ }, { "cell_type": "code", - "execution_count": 85, + "execution_count": 27, "metadata": { "collapsed": true }, @@ -5554,7 +4778,7 @@ }, { "cell_type": "code", - "execution_count": 86, + "execution_count": 28, "metadata": { "collapsed": true }, @@ -5585,7 +4809,7 @@ }, { "cell_type": "code", - "execution_count": 91, + "execution_count": 29, "metadata": { "collapsed": false }, @@ -6415,7 +5639,7 @@ }, { "cell_type": "code", - "execution_count": 95, + "execution_count": 30, "metadata": { "collapsed": false }, @@ -7247,7 +6471,7 @@ }, { "cell_type": "code", - "execution_count": 96, + "execution_count": 31, "metadata": { "collapsed": false }, @@ -8019,7 +7243,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -8094,166 +7318,11 @@ }, { "cell_type": "code", - "execution_count": 90, + "execution_count": null, "metadata": { "collapsed": false }, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "execution_count": 90, - "metadata": {}, - "output_type": "execute_result" - } - ], + "outputs": [], "source": [ "# This cell loads the style of the notebook, which is modified from the \n", "# Numerical Methods in Python Course: http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about\n", From 6f6710397b52ba94ea7ed664b16039d8c9a6092e Mon Sep 17 00:00:00 2001 From: rbds Date: Thu, 10 Dec 2015 16:27:54 +0000 Subject: [PATCH 51/61] update DP notebook - check EOM again --- randy_schur/Double_Pendulum_Problem.ipynb | 5424 ++++++++++++++++- ...on Techniques for Mechanical Systems.ipynb | 12 +- 2 files changed, 5136 insertions(+), 300 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index f26a4a8..840ad08 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -266,7 +266,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 15, "metadata": { "collapsed": false }, @@ -304,7 +304,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 16, "metadata": { "collapsed": true }, @@ -329,7 +329,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 17, "metadata": { "collapsed": false }, @@ -357,7 +357,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 18, "metadata": { "collapsed": false }, @@ -1960,7 +1960,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 19, "metadata": { "collapsed": false }, @@ -1993,287 +1993,4190 @@ }, { "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "#Verlet integration\n", - "x3_dp = numpy.zeros((N,4)) \n", - "x3_dp[0,:] = x_init_H_dp.copy() #set initial conditions\n", - "#print(q3_dp[0,:])\n", - "for n in range(N-1): #integrate with Euler\n", - " x3_dp[n+1,:] = Verlet_DP(x3_dp[n,:], dt)\n", - " \n", - "pyplot.figure(figsize=(10,8));\n", - "pyplot.grid(True);\n", - "pyplot.xlabel(r't', fontsize=18);\n", - "pyplot.ylabel(r'position (radians)', fontsize=18);\n", - "pyplot.title('Double Pendulum Verlet Integration');\n", - "pyplot.plot(t, x3_dp[:,0], lw=2, label='Joint 1');\n", - "pyplot.plot(t, x3_dp[:,2], lw=2, label='Joint 2')\n", - "pyplot.legend();" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This certainly looks better! It's an interesting result. There is clearly some periodicity, although the motion isn't exactly periodic. But how accurate is this simulation? The short answer is that since we don't have an analytical solution, it is difficult to know exactly. However, there are still some properties of the system that we can look at. For one thing, the total energy in the system should be constant. Let's see if that is the case.\n", - "\n", - "Recall from the analysis above that the total energy is the combined potential and kinetic energy at each time step.\n", - "\n", - "$$\\begin{eqnarray*}\n", - "U &=& -m_1gl_1cos(q_1) - m_2g(l_1cos(q_1) + l_2cos(q_2))\\\\\n", - "T &=& \\frac{p_1^2}{2I_1}+\\frac{m_2p_1^2}{2m_1I_1} + \\left[ \\frac{p_2}{l_2} \\frac{p_1}{m_1l_1}cos(q_2-q_1) + \\frac{p_2^2}{2I_2} \\right]\n", - "\\end{eqnarray*}$$\n", - "\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "def get_Energy(u):\n", - " \"\"\" Calculates total energu in the system at given time step for a simple harmonic oscillator\n", - " Parameters:\n", - " u - state of system [pos, vel]\n", - " Returns:\n", - " total energy in system.\n", - " \"\"\"\n", - " q1 = u[0]\n", - " p1 = u[1]\n", - " q2 = u[2]\n", - " p2 = u[3]\n", - " \n", - " \n", - " \n", - " V = -m1*g*l1*cos(q1) - m2*g*(l1*cos(q1)+l2*cos(q2))\n", - " T = p1**2/(2*I1)+m2*p1**2/(2*m1*I1) + p1*p2/(m1*l1*l2)*cos(q2-q1)+p2**2/(2*I1)\n", - " #print(T)\n", - " #print(V)\n", - " return T+V" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "Verlet_energy = numpy.zeros_like(t)\n", - "Verlet_vel1 = x3_dp[:,1]/m1\n", - "Verlet_vel2 = x3_dp[:,3]/m2\n", - "\n", - "\n", - "u = numpy.array([x3_dp[:,0], Verlet_vel1, x3_dp[:,2], Verlet_vel2])\n", - "#print(u[0,:])\n", - "for i in range(N):\n", - " Verlet_energy[i] = get_Energy(u[:,i])\n", - "\n", - "pyplot.figure(figsize=(8,6));\n", - "pyplot.grid(True);\n", - "pyplot.xlabel(r't', fontsize=18);\n", - "pyplot.ylabel(r'Total energy in system', fontsize=18);\n", - "pyplot.title('Energy in System');\n", - "pyplot.plot(t, Verlet_energy, lw=2, label='Verlet');" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "So, we are very far from perfect. But, at least the energy in the system is stable, even if it isn't constant! " - ] - }, - { - "cell_type": "markdown", - "metadata": { - "collapsed": true - }, - "source": [ - "### Investigating system behavior\n", - "\n", - "In order to investigate system behavior for damped harmonic motion, we used a phase portrait where we plotted position vs. velocity. In this system, we have 2 positions and 2 velocities, so a phase plot won't accurately capture the behavior of the system. Instead we are going to use something called a Poincare section (https://en.wikipedia.org/wiki/Poincar%C3%A9_map). The way these work is to capture the system behavior at a specific, repetetive point. For example, each time that $\\theta_2 = 0$ or any multiple of 2*pi (mass 2 is hanging straight down) we can capture the angular position and velocity of mass 2. This way, we are taking a 'snapshot' of the system, and plotting the state each time. We can now investigate system behavior on a simple 2D plot, similar to the phase portrait." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "def poincare(state, sf, val, s1, s2):\n", - " '''Creates a poincare section, capturing system state each time q[state]=val\n", - " \n", - " Parameters:\n", - " state - state vector\n", - " sf - the row in q to fix\n", - " val - the value to capture state at \n", - " s1 - state to capture 1\n", - " s2 - state to capture 2\n", - " \n", - " Returns:\n", - " v - array with information for poincare section\n", - " '''\n", - " v = numpy.zeros((2,1))\n", - " for i in range(len(state[:,sf])-1):\n", - " if (state[i,sf]>val):\n", - " if (state[i+1,sf] < val):\n", - " v = numpy.append(v, [[state[i, s1]],[state[i, s2]]], axis=1)\n", - " \n", - " if (state[i,sf] < val):\n", - " if (state[i+1,sf] > val):\n", - " v = numpy.append(v, [[state[i, s1]],[state[i, s2]]], axis=1)\n", - " \n", - " return v" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "v = poincare(x3_dp, 1, 2*pi, 0, 2) #use states from RK4 integration\n", - "pyplot.figure(figsize=(10,8));\n", - "pyplot.grid(True);\n", - "pyplot.xlabel(r'$\\theta_2$', fontsize=18);\n", - "pyplot.ylabel(r'$\\dot{\\theta_2}$', fontsize=18);\n", - "pyplot.title('Poincare Section');\n", - "pyplot.plot(v[0,:], v[1,:], 'bo', lw=2, );" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "There isn't a clear pattern here. So, we can say that our system is NOT likely to be asymptotically stable. However, this shouldn't be surprising. If we continued simulating, our system would continue rotating since there is no damping. Let's see what happens when there is some damping in the system. We're going to use RK4 for the following analysis." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "#To Do:\n", - "# animate motion of double pendulum\n", - "# check through sources" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Damped Double Pendulum\n", - "\n", - "Just like with simple harmonic motion, we can add in a damping term to help remove some of the energy that the integration scheme adds in. In the problem setup, we actually used an abbreviated form of a standard equation. The full equation is here:\n", - "\n", - "\\begin{equation}\n", - "\\textbf{M}(x)\\vec{\\ddot{x}}(t) + \\textbf{V}(x, \\dot x ) \\vec{\\dot{x}} + \\textbf{G(x)} + f(\\dot x) + \\tau_d(t) = \\tau(t)\n", - "\\end{equation}" - ] - }, - { - "cell_type": "markdown", - "metadata": { - "collapsed": true - }, - "source": [ - "where $\\textbf{M, V}$, and $\\textbf{G}$ are defined as above, $f$ is frictional or damping terms, and the $\\tau$ terms represent added torque to the system. Since we aren't adding any torque, the $\\tau$ terms will remain 0. However, we can add in some linear damping by including the frictional term. This requires redefining our RHS function, but no other changes!" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "def f_damped_double_pendulum(u):\n", - " \"\"\"Returns RHS of double pendulum EOM\n", - " \n", - " Parameters:\n", - " q - initial state\n", - " \n", - " Returns:\n", - " RHS - RHS of harmonic oscillator eqn.\n", - " \n", - " \"\"\"\n", - " x1 = u[0]\n", - " x2 = u[1]\n", - " x3 = u[2]\n", - " x4 = u[3]\n", - " \n", - " M = numpy.array([[(m1+m2)*l1**2, m2*l1*l2*cos(x2-x1)],[m2*l1*l2*cos(x2-x1), m2*l2**2]])\n", - " V = numpy.array([[0, -m2*l1*l2*sin(x2-x1)],[-m2*l1*l2*sin(x2-x1), 0]])\n", - " G = numpy.array([[(m1+m2)*g*l1*sin(x1)],[m2*l2*g*sin(x2)]])\n", - " f = c*numpy.array([[x3],[x4]])\n", - " qdd = numpy.linalg.inv(M).dot(V.dot(numpy.array([[x3],[x4]]))+G -f) \n", - " #print(qdd)\n", - " #print(M)\n", - " #print(V.dot(numpy.array([[x3],[x4]])))\n", - " #print(G)\n", - " RHS = numpy.array([[x3, x4, qdd[0], qdd[1]]])\n", - " #print(RHS.T)\n", - " return RHS.T" - ] - }, - { - "cell_type": "code", - "execution_count": null, + "execution_count": 32, "metadata": { "collapsed": false }, - "outputs": [], - "source": [ - "#We use the same initial conditions as the example without damping, so they aren't redefined here.\n", - "c = 2 #[N/(rad/s)] define some damping\n", - "\n", - "#Euler\n", - "q1d_dp = numpy.zeros((N,4)) \n", - "q1d_dp[0,:] = x_init_dp.copy() #set initial conditions\n", - "for n in range(N-1): #integrate with Euler\n", - " q1d_dp[n+1,:] = euler_DP(q1d_dp[n,:], f_damped_double_pendulum, dt)\n", - " #print(q1[n])\n", - " \n", - "#Runge-Kutta\n", - "q2d_dp = numpy.zeros((N,4)) \n", - "q2d_dp[0,:] = x_init_dp.copy() #set initial conditions\n", - "for n in range(N-1): #integrate with Euler\n", - " q2d_dp[n+1,:] = RK4_DP(q2d_dp[n,:], f_damped_double_pendulum, dt)\n", - " \n", - "pyplot.figure(figsize=(10,8));\n", - "pyplot.grid(True);\n", - "pyplot.xlabel(r't', fontsize=18);\n", - "pyplot.ylabel(r'position (meters)', fontsize=18);\n", - "pyplot.title('Double Pendulum Euler');\n", - "pyplot.plot(t, q1d_dp[:,0], lw=2, label='Joint 1');\n", - "pyplot.plot(t, q1d_dp[:,1], lw=2, label='Joint 2')\n", - "pyplot.legend();\n", - "\n", - "pyplot.figure(figsize=(10,8));\n", + "outputs": [ + { + "data": { + "application/javascript": [ + "/* Put everything inside the global mpl namespace */\n", + "window.mpl = {};\n", + "\n", + "mpl.get_websocket_type = function() {\n", + " if (typeof(WebSocket) !== 'undefined') {\n", + " return WebSocket;\n", + " } else if (typeof(MozWebSocket) !== 'undefined') {\n", + " return MozWebSocket;\n", + " } else {\n", + " alert('Your browser does not have WebSocket support.' +\n", + " 'Please try Chrome, Safari or Firefox ≥ 6. 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');\n", + " var button = $('');\n", + " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", + " buttongrp.append(button);\n", + " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", + " titlebar.prepend(buttongrp);\n", + "}\n", + "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(el){\n", + " // this is important to make the div 'focusable\n", + " el.attr('tabindex', 0)\n", + " // reach out to IPython and tell the keyboard manager to turn it's self\n", + " // off when our div gets focus\n", + "\n", + " // location in version 3\n", + " if (IPython.notebook.keyboard_manager) {\n", + " IPython.notebook.keyboard_manager.register_events(el);\n", + " }\n", + " else {\n", + " // location in version 2\n", + " IPython.keyboard_manager.register_events(el);\n", + " }\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._key_event_extra = function(event, name) {\n", + " var manager = IPython.notebook.keyboard_manager;\n", + " if (!manager)\n", + " manager = IPython.keyboard_manager;\n", + "\n", + " // Check for shift+enter\n", + " if (event.shiftKey && event.which == 13) {\n", + " this.canvas_div.blur();\n", + " event.shiftKey = false;\n", + " // Send a \"J\" for go to next cell\n", + " event.which = 74;\n", + " event.keyCode = 74;\n", + " manager.command_mode();\n", + " manager.handle_keydown(event);\n", + " }\n", + "}\n", + "\n", + "mpl.figure.prototype.handle_save = function(fig, msg) {\n", + " fig.ondownload(fig, null);\n", + "}\n", + "\n", + "\n", + "mpl.find_output_cell = function(html_output) {\n", + " // Return the cell and output element which can be found *uniquely* in the notebook.\n", + " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", + " // IPython event is triggered only after the cells have been serialised, which for\n", + " // our purposes (turning an active figure into a static one), is too late.\n", + " var cells = IPython.notebook.get_cells();\n", + " var ncells = cells.length;\n", + " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", + " data = data.data;\n", + " }\n", + " if (data['text/html'] == html_output) {\n", + " return [cell, data, j];\n", + " }\n", + " }\n", + " }\n", + " }\n", + "}\n", + "\n", + "// Register the function which deals with the matplotlib target/channel.\n", + "// The kernel may be null if the page has been refreshed.\n", + "if (IPython.notebook.kernel != null) {\n", + " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", + "}\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "Verlet_energy = numpy.zeros_like(t)\n", + "Verlet_vel1 = xv_dp[:,1]/m1\n", + "Verlet_vel2 = xv_dp[:,3]/m2\n", + "\n", + "\n", + "u = numpy.array([xv_dp[:,0], Verlet_vel1, xv_dp[:,2], Verlet_vel2])\n", + "#print(u[0,:])\n", + "for i in range(N):\n", + " Verlet_energy[i] = get_Energy(u[:,i])\n", + "\n", + "pyplot.figure(figsize=(8,6));\n", + "pyplot.grid(True);\n", + "pyplot.xlabel(r't', fontsize=18);\n", + "pyplot.ylabel(r'Total energy in system', fontsize=18);\n", + "pyplot.title('Energy in System');\n", + "pyplot.plot(t, Verlet_energy, lw=2, label='Verlet');" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "So, we are very far from perfect. But, at least the energy in the system is stable, even if it isn't constant! " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "### Investigating system behavior\n", + "\n", + "In order to investigate system behavior for damped harmonic motion, we used a phase portrait where we plotted position vs. velocity. In this system, we have 2 positions and 2 velocities, so a phase plot won't accurately capture the behavior of the system. Instead we are going to use something called a Poincare section (https://en.wikipedia.org/wiki/Poincar%C3%A9_map). The way these work is to capture the system behavior at a specific, repetetive point. For example, each time that $\\theta_2 = 0$ or any multiple of 2*pi (mass 2 is hanging straight down) we can capture the angular position and velocity of mass 2. This way, we are taking a 'snapshot' of the system, and plotting the state each time. We can now investigate system behavior on a simple 2D plot, similar to the phase portrait." + ] + }, + { + "cell_type": "code", + "execution_count": 36, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def poincare(state, sf, val, s1, s2):\n", + " '''Creates a poincare section, capturing system state each time q[state]=val\n", + " \n", + " Parameters:\n", + " state - state vector\n", + " sf - the row in q to fix\n", + " val - the value to capture state at \n", + " s1 - state to capture 1\n", + " s2 - state to capture 2\n", + " \n", + " Returns:\n", + " v - array with information for poincare section\n", + " '''\n", + " v = numpy.zeros((2,1))\n", + " for i in range(len(state[:,sf])-1):\n", + " if (state[i,sf]>val):\n", + " if (state[i+1,sf] < val):\n", + " v = numpy.append(v, [[state[i, s1]],[state[i, s2]]], axis=1)\n", + " \n", + " if (state[i,sf] < val):\n", + " if (state[i+1,sf] > val):\n", + " v = numpy.append(v, [[state[i, s1]],[state[i, s2]]], axis=1)\n", + " \n", + " return v" + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "application/javascript": [ + "/* Put everything inside the global mpl namespace */\n", + "window.mpl = {};\n", + "\n", + "mpl.get_websocket_type = function() {\n", + " if (typeof(WebSocket) !== 'undefined') {\n", + " return WebSocket;\n", + " } else if (typeof(MozWebSocket) !== 'undefined') {\n", + " return MozWebSocket;\n", + " } else {\n", + " alert('Your browser does not have WebSocket support.' +\n", + " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", + " 'Firefox 4 and 5 are also supported but you ' +\n", + " 'have to enable WebSockets in about:config.');\n", + " };\n", + "}\n", + "\n", + "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", + " this.id = figure_id;\n", + "\n", + " this.ws = websocket;\n", + "\n", + " this.supports_binary = (this.ws.binaryType != undefined);\n", + "\n", + " if (!this.supports_binary) {\n", + " var warnings = document.getElementById(\"mpl-warnings\");\n", + " if (warnings) {\n", + " warnings.style.display = 'block';\n", + " warnings.textContent = (\n", + " \"This browser does not support binary websocket messages. \" +\n", + " \"Performance may be slow.\");\n", + " }\n", + " }\n", + "\n", + " this.imageObj = new Image();\n", + "\n", + " this.context = undefined;\n", + " this.message = undefined;\n", + " this.canvas = undefined;\n", + " this.rubberband_canvas = undefined;\n", + " this.rubberband_context = undefined;\n", + " this.format_dropdown = undefined;\n", + "\n", + " this.image_mode = 'full';\n", + "\n", + " this.root = $('
');\n", + " this._root_extra_style(this.root)\n", + " this.root.attr('style', 'display: inline-block');\n", + "\n", + " $(parent_element).append(this.root);\n", + "\n", + " this._init_header(this);\n", + " this._init_canvas(this);\n", + " this._init_toolbar(this);\n", + "\n", + " var fig = this;\n", + "\n", + " this.waiting = false;\n", + "\n", + " this.ws.onopen = function () {\n", + " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", + " fig.send_message(\"send_image_mode\", {});\n", + " fig.send_message(\"refresh\", {});\n", + " }\n", + "\n", + " this.imageObj.onload = function() {\n", + " if (fig.image_mode == 'full') {\n", + " // Full images could contain transparency (where diff images\n", + " // almost always do), so we need to clear the canvas so that\n", + " // there is no ghosting.\n", + " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", + " }\n", + " fig.context.drawImage(fig.imageObj, 0, 0);\n", + " };\n", + "\n", + " this.imageObj.onunload = function() {\n", + " this.ws.close();\n", + " }\n", + "\n", + " this.ws.onmessage = this._make_on_message_function(this);\n", + "\n", + " this.ondownload = ondownload;\n", + "}\n", + "\n", + "mpl.figure.prototype._init_header = function() {\n", + " var titlebar = $(\n", + " '
');\n", + " var titletext = $(\n", + " '
');\n", + " titlebar.append(titletext)\n", + " this.root.append(titlebar);\n", + " this.header = titletext[0];\n", + "}\n", + "\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._init_canvas = function() {\n", + " var fig = this;\n", + "\n", + " var canvas_div = $('
');\n", + "\n", + " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", + "\n", + " function canvas_keyboard_event(event) {\n", + " return fig.key_event(event, event['data']);\n", + " }\n", + "\n", + " canvas_div.keydown('key_press', canvas_keyboard_event);\n", + " canvas_div.keyup('key_release', canvas_keyboard_event);\n", + " this.canvas_div = canvas_div\n", + " this._canvas_extra_style(canvas_div)\n", + " this.root.append(canvas_div);\n", + "\n", + " var canvas = $('');\n", + " canvas.addClass('mpl-canvas');\n", + " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", + "\n", + " this.canvas = canvas[0];\n", + " this.context = canvas[0].getContext(\"2d\");\n", + "\n", + " var rubberband = $('');\n", + " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", + "\n", + " var pass_mouse_events = true;\n", + "\n", + " canvas_div.resizable({\n", + " start: function(event, ui) {\n", + " pass_mouse_events = false;\n", + " },\n", + " resize: function(event, ui) {\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " stop: function(event, ui) {\n", + " pass_mouse_events = true;\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " });\n", + "\n", + " function mouse_event_fn(event) {\n", + " if (pass_mouse_events)\n", + " return fig.mouse_event(event, event['data']);\n", + " }\n", + "\n", + " rubberband.mousedown('button_press', mouse_event_fn);\n", + " rubberband.mouseup('button_release', mouse_event_fn);\n", + " // Throttle sequential mouse events to 1 every 20ms.\n", + " rubberband.mousemove('motion_notify', mouse_event_fn);\n", + "\n", + " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", + " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", + "\n", + " canvas_div.on(\"wheel\", function (event) {\n", + " event = event.originalEvent;\n", + " event['data'] = 'scroll'\n", + " if (event.deltaY < 0) {\n", + " event.step = 1;\n", + " } else {\n", + " event.step = -1;\n", + " }\n", + " mouse_event_fn(event);\n", + " });\n", + "\n", + " canvas_div.append(canvas);\n", + " canvas_div.append(rubberband);\n", + "\n", + " this.rubberband = rubberband;\n", + " this.rubberband_canvas = rubberband[0];\n", + " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", + " this.rubberband_context.strokeStyle = \"#000000\";\n", + "\n", + " this._resize_canvas = function(width, height) {\n", + " // Keep the size of the canvas, canvas container, and rubber band\n", + " // canvas in synch.\n", + " canvas_div.css('width', width)\n", + " canvas_div.css('height', height)\n", + "\n", + " canvas.attr('width', width);\n", + " canvas.attr('height', height);\n", + "\n", + " rubberband.attr('width', width);\n", + " rubberband.attr('height', height);\n", + " }\n", + "\n", + " // Set the figure to an initial 600x600px, this will subsequently be updated\n", + " // upon first draw.\n", + " this._resize_canvas(600, 600);\n", + "\n", + " // Disable right mouse context menu.\n", + " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", + " return false;\n", + " });\n", + "\n", + " function set_focus () {\n", + " canvas.focus();\n", + " canvas_div.focus();\n", + " }\n", + "\n", + " window.setTimeout(set_focus, 100);\n", + "}\n", + "\n", + "mpl.figure.prototype._init_toolbar = function() {\n", + " var fig = this;\n", + "\n", + " var nav_element = $('
')\n", + " nav_element.attr('style', 'width: 100%');\n", + " this.root.append(nav_element);\n", + "\n", + " // Define a callback function for later on.\n", + " function toolbar_event(event) {\n", + " return fig.toolbar_button_onclick(event['data']);\n", + " }\n", + " function toolbar_mouse_event(event) {\n", + " return fig.toolbar_button_onmouseover(event['data']);\n", + " }\n", + "\n", + " for(var toolbar_ind in mpl.toolbar_items) {\n", + " var name = mpl.toolbar_items[toolbar_ind][0];\n", + " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", + " var image = mpl.toolbar_items[toolbar_ind][2];\n", + " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", + "\n", + " if (!name) {\n", + " // put a spacer in here.\n", + " continue;\n", + " }\n", + " var button = $('');\n", + " button.click(method_name, toolbar_event);\n", + " button.mouseover(tooltip, toolbar_mouse_event);\n", + " nav_element.append(button);\n", + " }\n", + "\n", + " // Add the status bar.\n", + " var status_bar = $('');\n", + " nav_element.append(status_bar);\n", + " this.message = status_bar[0];\n", + "\n", + " // Add the close button to the window.\n", + " var buttongrp = $('
');\n", + " var button = $('');\n", + " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", + " buttongrp.append(button);\n", + " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", + " titlebar.prepend(buttongrp);\n", + "}\n", + "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(el){\n", + " // this is important to make the div 'focusable\n", + " el.attr('tabindex', 0)\n", + " // reach out to IPython and tell the keyboard manager to turn it's self\n", + " // off when our div gets focus\n", + "\n", + " // location in version 3\n", + " if (IPython.notebook.keyboard_manager) {\n", + " IPython.notebook.keyboard_manager.register_events(el);\n", + " }\n", + " else {\n", + " // location in version 2\n", + " IPython.keyboard_manager.register_events(el);\n", + " }\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._key_event_extra = function(event, name) {\n", + " var manager = IPython.notebook.keyboard_manager;\n", + " if (!manager)\n", + " manager = IPython.keyboard_manager;\n", + "\n", + " // Check for shift+enter\n", + " if (event.shiftKey && event.which == 13) {\n", + " this.canvas_div.blur();\n", + " event.shiftKey = false;\n", + " // Send a \"J\" for go to next cell\n", + " event.which = 74;\n", + " event.keyCode = 74;\n", + " manager.command_mode();\n", + " manager.handle_keydown(event);\n", + " }\n", + "}\n", + "\n", + "mpl.figure.prototype.handle_save = function(fig, msg) {\n", + " fig.ondownload(fig, null);\n", + "}\n", + "\n", + "\n", + "mpl.find_output_cell = function(html_output) {\n", + " // Return the cell and output element which can be found *uniquely* in the notebook.\n", + " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", + " // IPython event is triggered only after the cells have been serialised, which for\n", + " // our purposes (turning an active figure into a static one), is too late.\n", + " var cells = IPython.notebook.get_cells();\n", + " var ncells = cells.length;\n", + " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", + " data = data.data;\n", + " }\n", + " if (data['text/html'] == html_output) {\n", + " return [cell, data, j];\n", + " }\n", + " }\n", + " }\n", + " }\n", + "}\n", + "\n", + "// Register the function which deals with the matplotlib target/channel.\n", + "// The kernel may be null if the page has been refreshed.\n", + "if (IPython.notebook.kernel != null) {\n", + " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", + "}\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "application/javascript": [ + "/* Put everything inside the global mpl namespace */\n", + "window.mpl = {};\n", + "\n", + "mpl.get_websocket_type = function() {\n", + " if (typeof(WebSocket) !== 'undefined') {\n", + " return WebSocket;\n", + " } else if (typeof(MozWebSocket) !== 'undefined') {\n", + " return MozWebSocket;\n", + " } else {\n", + " alert('Your browser does not have WebSocket support.' +\n", + " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", + " 'Firefox 4 and 5 are also supported but you ' +\n", + " 'have to enable WebSockets in about:config.');\n", + " };\n", + "}\n", + "\n", + "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", + " this.id = figure_id;\n", + "\n", + " this.ws = websocket;\n", + "\n", + " this.supports_binary = (this.ws.binaryType != undefined);\n", + "\n", + " if (!this.supports_binary) {\n", + " var warnings = document.getElementById(\"mpl-warnings\");\n", + " if (warnings) {\n", + " warnings.style.display = 'block';\n", + " warnings.textContent = (\n", + " \"This browser does not support binary websocket messages. \" +\n", + " \"Performance may be slow.\");\n", + " }\n", + " }\n", + "\n", + " this.imageObj = new Image();\n", + "\n", + " this.context = undefined;\n", + " this.message = undefined;\n", + " this.canvas = undefined;\n", + " this.rubberband_canvas = undefined;\n", + " this.rubberband_context = undefined;\n", + " this.format_dropdown = undefined;\n", + "\n", + " this.image_mode = 'full';\n", + "\n", + " this.root = $('
');\n", + " this._root_extra_style(this.root)\n", + " this.root.attr('style', 'display: inline-block');\n", + "\n", + " $(parent_element).append(this.root);\n", + "\n", + " this._init_header(this);\n", + " this._init_canvas(this);\n", + " this._init_toolbar(this);\n", + "\n", + " var fig = this;\n", + "\n", + " this.waiting = false;\n", + "\n", + " this.ws.onopen = function () {\n", + " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", + " fig.send_message(\"send_image_mode\", {});\n", + " fig.send_message(\"refresh\", {});\n", + " }\n", + "\n", + " this.imageObj.onload = function() {\n", + " if (fig.image_mode == 'full') {\n", + " // Full images could contain transparency (where diff images\n", + " // almost always do), so we need to clear the canvas so that\n", + " // there is no ghosting.\n", + " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", + " }\n", + " fig.context.drawImage(fig.imageObj, 0, 0);\n", + " };\n", + "\n", + " this.imageObj.onunload = function() {\n", + " this.ws.close();\n", + " }\n", + "\n", + " this.ws.onmessage = this._make_on_message_function(this);\n", + "\n", + " this.ondownload = ondownload;\n", + "}\n", + "\n", + "mpl.figure.prototype._init_header = function() {\n", + " var titlebar = $(\n", + " '
');\n", + " var titletext = $(\n", + " '
');\n", + " titlebar.append(titletext)\n", + " this.root.append(titlebar);\n", + " this.header = titletext[0];\n", + "}\n", + "\n", + "\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "\n", + "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._init_canvas = function() {\n", + " var fig = this;\n", + "\n", + " var canvas_div = $('
');\n", + "\n", + " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", + "\n", + " function canvas_keyboard_event(event) {\n", + " return fig.key_event(event, event['data']);\n", + " }\n", + "\n", + " canvas_div.keydown('key_press', canvas_keyboard_event);\n", + " canvas_div.keyup('key_release', canvas_keyboard_event);\n", + " this.canvas_div = canvas_div\n", + " this._canvas_extra_style(canvas_div)\n", + " this.root.append(canvas_div);\n", + "\n", + " var canvas = $('');\n", + " canvas.addClass('mpl-canvas');\n", + " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", + "\n", + " this.canvas = canvas[0];\n", + " this.context = canvas[0].getContext(\"2d\");\n", + "\n", + " var rubberband = $('');\n", + " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", + "\n", + " var pass_mouse_events = true;\n", + "\n", + " canvas_div.resizable({\n", + " start: function(event, ui) {\n", + " pass_mouse_events = false;\n", + " },\n", + " resize: function(event, ui) {\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " stop: function(event, ui) {\n", + " pass_mouse_events = true;\n", + " fig.request_resize(ui.size.width, ui.size.height);\n", + " },\n", + " });\n", + "\n", + " function mouse_event_fn(event) {\n", + " if (pass_mouse_events)\n", + " return fig.mouse_event(event, event['data']);\n", + " }\n", + "\n", + " rubberband.mousedown('button_press', mouse_event_fn);\n", + " rubberband.mouseup('button_release', mouse_event_fn);\n", + " // Throttle sequential mouse events to 1 every 20ms.\n", + " rubberband.mousemove('motion_notify', mouse_event_fn);\n", + "\n", + " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", + " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", + "\n", + " canvas_div.on(\"wheel\", function (event) {\n", + " event = event.originalEvent;\n", + " event['data'] = 'scroll'\n", + " if (event.deltaY < 0) {\n", + " event.step = 1;\n", + " } else {\n", + " event.step = -1;\n", + " }\n", + " mouse_event_fn(event);\n", + " });\n", + "\n", + " canvas_div.append(canvas);\n", + " canvas_div.append(rubberband);\n", + "\n", + " this.rubberband = rubberband;\n", + " this.rubberband_canvas = rubberband[0];\n", + " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", + " this.rubberband_context.strokeStyle = \"#000000\";\n", + "\n", + " this._resize_canvas = function(width, height) {\n", + " // Keep the size of the canvas, canvas container, and rubber band\n", + " // canvas in synch.\n", + " canvas_div.css('width', width)\n", + " canvas_div.css('height', height)\n", + "\n", + " canvas.attr('width', width);\n", + " canvas.attr('height', height);\n", + "\n", + " rubberband.attr('width', width);\n", + " rubberband.attr('height', height);\n", + " }\n", + "\n", + " // Set the figure to an initial 600x600px, this will subsequently be updated\n", + " // upon first draw.\n", + " this._resize_canvas(600, 600);\n", + "\n", + " // Disable right mouse context menu.\n", + " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", + " return false;\n", + " });\n", + "\n", + " function set_focus () {\n", + " canvas.focus();\n", + " canvas_div.focus();\n", + " }\n", + "\n", + " window.setTimeout(set_focus, 100);\n", + "}\n", + "\n", + "mpl.figure.prototype._init_toolbar = function() {\n", + " var fig = this;\n", + "\n", + " var nav_element = $('
')\n", + " nav_element.attr('style', 'width: 100%');\n", + " this.root.append(nav_element);\n", + "\n", + " // Define a callback function for later on.\n", + " function toolbar_event(event) {\n", + " return fig.toolbar_button_onclick(event['data']);\n", + " }\n", + " function toolbar_mouse_event(event) {\n", + " return fig.toolbar_button_onmouseover(event['data']);\n", + " }\n", + "\n", + " for(var toolbar_ind in mpl.toolbar_items) {\n", + " var name = mpl.toolbar_items[toolbar_ind][0];\n", + " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", + " var image = mpl.toolbar_items[toolbar_ind][2];\n", + " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", + "\n", + " if (!name) {\n", + " // put a spacer in here.\n", + " continue;\n", + " }\n", + " var button = $('');\n", + " button.click(method_name, toolbar_event);\n", + " button.mouseover(tooltip, toolbar_mouse_event);\n", + " nav_element.append(button);\n", + " }\n", + "\n", + " // Add the status bar.\n", + " var status_bar = $('');\n", + " nav_element.append(status_bar);\n", + " this.message = status_bar[0];\n", + "\n", + " // Add the close button to the window.\n", + " var buttongrp = $('
');\n", + " var button = $('');\n", + " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", + " buttongrp.append(button);\n", + " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", + " titlebar.prepend(buttongrp);\n", + "}\n", + "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(el){\n", + " // this is important to make the div 'focusable\n", + " el.attr('tabindex', 0)\n", + " // reach out to IPython and tell the keyboard manager to turn it's self\n", + " // off when our div gets focus\n", + "\n", + " // location in version 3\n", + " if (IPython.notebook.keyboard_manager) {\n", + " IPython.notebook.keyboard_manager.register_events(el);\n", + " }\n", + " else {\n", + " // location in version 2\n", + " IPython.keyboard_manager.register_events(el);\n", + " }\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._key_event_extra = function(event, name) {\n", + " var manager = IPython.notebook.keyboard_manager;\n", + " if (!manager)\n", + " manager = IPython.keyboard_manager;\n", + "\n", + " // Check for shift+enter\n", + " if (event.shiftKey && event.which == 13) {\n", + " this.canvas_div.blur();\n", + " event.shiftKey = false;\n", + " // Send a \"J\" for go to next cell\n", + " event.which = 74;\n", + " event.keyCode = 74;\n", + " manager.command_mode();\n", + " manager.handle_keydown(event);\n", + " }\n", + "}\n", + "\n", + "mpl.figure.prototype.handle_save = function(fig, msg) {\n", + " fig.ondownload(fig, null);\n", + "}\n", + "\n", + "\n", + "mpl.find_output_cell = function(html_output) {\n", + " // Return the cell and output element which can be found *uniquely* in the notebook.\n", + " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", + " // IPython event is triggered only after the cells have been serialised, which for\n", + " // our purposes (turning an active figure into a static one), is too late.\n", + " var cells = IPython.notebook.get_cells();\n", + " var ncells = cells.length;\n", + " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", + " data = data.data;\n", + " }\n", + " if (data['text/html'] == html_output) {\n", + " return [cell, data, j];\n", + " }\n", + " }\n", + " }\n", + " }\n", + "}\n", + "\n", + "// Register the function which deals with the matplotlib target/channel.\n", + "// The kernel may be null if the page has been refreshed.\n", + "if (IPython.notebook.kernel != null) {\n", + " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", + "}\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "[]" + ] + }, + "execution_count": 51, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ - "v = poincare(q2d_dp, 0, 0, 1, 3) #use states from RK4 integration\n", + "v = poincare(q4d_dp, 0, 0, 1, 3) #use states from RK4 integration\n", "pyplot.figure(figsize=(10,8));\n", "pyplot.grid(True);\n", "pyplot.xlabel(r'$\\theta_2$', fontsize=18);\n", @@ -2314,7 +7004,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 52, "metadata": { "collapsed": false }, @@ -2360,11 +7050,166 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 53, "metadata": { "collapsed": false }, - "outputs": [], + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "execution_count": 53, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "# This cell loads the style of the notebook, which is modified from the \n", "# Numerical Methods in Python Course: http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about\n", @@ -2373,15 +7218,6 @@ "css_file = './numericalmoocstyle.css'\n", "HTML(open(css_file, \"r\").read())" ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] } ], "metadata": { diff --git a/randy_schur/Integration Techniques for Mechanical Systems.ipynb b/randy_schur/Integration Techniques for Mechanical Systems.ipynb index 78cd23b..6f7e3b3 100644 --- a/randy_schur/Integration Techniques for Mechanical Systems.ipynb +++ b/randy_schur/Integration Techniques for Mechanical Systems.ipynb @@ -308,7 +308,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 32, "metadata": { "collapsed": true }, @@ -335,7 +335,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 33, "metadata": { "collapsed": true }, @@ -359,7 +359,7 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 34, "metadata": { "collapsed": true }, @@ -388,7 +388,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 35, "metadata": { "collapsed": false }, @@ -408,7 +408,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 36, "metadata": { "collapsed": false }, @@ -1180,7 +1180,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" From 6ab30c3dc2f53dbd5d132e6b28300fac2b71608d Mon Sep 17 00:00:00 2001 From: rbds Date: Thu, 10 Dec 2015 20:22:05 +0000 Subject: [PATCH 52/61] RK4 Double Pendulum adding energy --- randy_schur/Double_Pendulum_Problem.ipynb | 118 +++++----- ...on Techniques for Mechanical Systems.ipynb | 206 ++++++++++++++++-- 2 files changed, 246 insertions(+), 78 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index 840ad08..8698c1f 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -233,7 +233,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 63, "metadata": { "collapsed": false }, @@ -256,8 +256,10 @@ " x4 = u[3]\n", " #print(x1)\n", " \n", - " M = numpy.array([[(m1+m2)*l1**2, m2*l1*l2*cos(x2-x1)],[m2*l1*l2*cos(x2-x1), m2*l2**2]])\n", - " V = numpy.array([[0, -m2*l1*l2*sin(x2-x1)],[-m2*l1*l2*sin(x2-x1), 0]])\n", + " #M = numpy.array([[(m1+m2)*l1**2, m2*l1*l2*cos(x2-x1)],[m2*l1*l2*cos(x2-x1), m2*l2**2]])\n", + " M = numpy.array([[(m1+m2)*l1**2, m2*l1*l2*cos(x1-x2)],[m2*l1*l2*cos(x1-x2), m2*l2**2]])\n", + " #V = numpy.array([[0, -m2*l1*l2*sin(x2-x1)],[-m2*l1*l2*sin(x2-x1), 0]])\n", + " V = numpy.array([[0, -m2*l1*l2*sin(x1-x2)],[-m2*l1*l2*sin(x1-x2), 0]])\n", " G = numpy.array([[(m1+m2)*g*l1*sin(x1)],[m2*l2*g*sin(x2)]])\n", " qdd = numpy.linalg.inv(M).dot(V.dot(numpy.array([[x3],[x4]])) +G) #- 2*numpy.array([[x3],[x4]])) \n", " RHS = numpy.array([x3, x4, qdd[0], qdd[1]])\n", @@ -266,7 +268,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 64, "metadata": { "collapsed": false }, @@ -281,7 +283,7 @@ "I1 = m1*l1**2\n", "I2 = m2*l2**2\n", "\n", - "T = 25; #[seconds]\n", + "T = 50; #[seconds]\n", "dt = .01; #\n", "N = int(T/dt)+1\n", "t = numpy.linspace(0.0, T, N)\n", @@ -304,7 +306,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 65, "metadata": { "collapsed": true }, @@ -329,7 +331,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 66, "metadata": { "collapsed": false }, @@ -357,7 +359,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 67, "metadata": { "collapsed": false }, @@ -1129,7 +1131,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1905,7 +1907,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1960,7 +1962,7 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 47, "metadata": { "collapsed": false }, @@ -1993,7 +1995,7 @@ }, { "cell_type": "code", - "execution_count": 32, + "execution_count": 48, "metadata": { "collapsed": false }, @@ -2765,7 +2767,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2781,7 +2783,7 @@ "xv_dp[0,:] = x_init_H_dp.copy() #set initial conditions\n", "#print(q3_dp[0,:])\n", "for n in range(N-1): #integrate with Euler\n", - " xv_dp[n+1,:] = Verlet_DP(x3_dp[n,:], dt)\n", + " xv_dp[n+1,:] = Verlet_DP(xv_dp[n,:], dt)\n", " \n", "pyplot.figure(figsize=(10,8));\n", "pyplot.grid(True);\n", @@ -2810,7 +2812,7 @@ }, { "cell_type": "code", - "execution_count": 33, + "execution_count": 49, "metadata": { "collapsed": true }, @@ -2839,7 +2841,7 @@ }, { "cell_type": "code", - "execution_count": 35, + "execution_count": 70, "metadata": { "collapsed": false }, @@ -3611,7 +3613,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -3619,25 +3621,40 @@ }, "metadata": {}, "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "[]" + ] + }, + "execution_count": 70, + "metadata": {}, + "output_type": "execute_result" } ], "source": [ "Verlet_energy = numpy.zeros_like(t)\n", "Verlet_vel1 = xv_dp[:,1]/m1\n", "Verlet_vel2 = xv_dp[:,3]/m2\n", + "RK4_energy = numpy.zeros_like(t)\n", "\n", "\n", "u = numpy.array([xv_dp[:,0], Verlet_vel1, xv_dp[:,2], Verlet_vel2])\n", "#print(u[0,:])\n", "for i in range(N):\n", " Verlet_energy[i] = get_Energy(u[:,i])\n", + " RK4_energy[i] = get_Energy(x4_dp[i,:])\n", + "\n", + " \n", "\n", - "pyplot.figure(figsize=(8,6));\n", + "pyplot.figure(figsize=(11,6));\n", "pyplot.grid(True);\n", "pyplot.xlabel(r't', fontsize=18);\n", "pyplot.ylabel(r'Total energy in system', fontsize=18);\n", "pyplot.title('Energy in System');\n", - "pyplot.plot(t, Verlet_energy, lw=2, label='Verlet');" + "pyplot.plot(t, Verlet_energy, lw=2, label='Verlet');\n", + "pyplot.plot(t, RK4_energy, 'r', lw=2, label='RK4')" ] }, { @@ -3660,7 +3677,7 @@ }, { "cell_type": "code", - "execution_count": 36, + "execution_count": 51, "metadata": { "collapsed": false }, @@ -3694,7 +3711,7 @@ }, { "cell_type": "code", - "execution_count": 37, + "execution_count": 52, "metadata": { "collapsed": false }, @@ -4466,7 +4483,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -4490,12 +4507,12 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "There isn't a clear pattern here. So, we can say that our system is NOT likely to be asymptotically stable. However, this shouldn't be surprising. If we continued simulating, our system would continue rotating since there is no damping. Let's see what happens when there is some damping in the system. We're going to use RK4 for the following analysis." + "There isn't a clear pattern here, although there is some clustering. So, we can say that our system is NOT likely to be asymptotically stable. However, this shouldn't be surprising. If we continued simulating, our system would continue rotating since there is no damping. Let's see what happens when there is some damping in the system. We're going to use RK4 for the following analysis." ] }, { "cell_type": "code", - "execution_count": 38, + "execution_count": 53, "metadata": { "collapsed": true }, @@ -4530,7 +4547,7 @@ }, { "cell_type": "code", - "execution_count": 45, + "execution_count": 54, "metadata": { "collapsed": false }, @@ -4564,29 +4581,7 @@ }, { "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - " #print(u)\n", - " x1 = u[0]\n", - " x2 = u[1]\n", - " x3 = u[2]\n", - " x4 = u[3]\n", - " #print(x1)\n", - " \n", - " M = numpy.array([[(m1+m2)*l1**2, m2*l1*l2*cos(x2-x1)],[m2*l1*l2*cos(x2-x1), m2*l2**2]])\n", - " V = numpy.array([[0, -m2*l1*l2*sin(x2-x1)],[-m2*l1*l2*sin(x2-x1), 0]])\n", - " G = numpy.array([[(m1+m2)*g*l1*sin(x1)],[m2*l2*g*sin(x2)]])\n", - " qdd = numpy.linalg.inv(M).dot(V.dot(numpy.array([[x3],[x4]])) +G) #- 2*numpy.array([[x3],[x4]])) \n", - " RHS = numpy.array([x3, x4, qdd[0], qdd[1]])" - ] - }, - { - "cell_type": "code", - "execution_count": 50, + "execution_count": 55, "metadata": { "collapsed": false }, @@ -5358,7 +5353,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -6134,7 +6129,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -6191,7 +6186,7 @@ }, { "cell_type": "code", - "execution_count": 51, + "execution_count": 56, "metadata": { "collapsed": false }, @@ -6963,7 +6958,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -6975,10 +6970,10 @@ { "data": { "text/plain": [ - "[]" + "[]" ] }, - "execution_count": 51, + "execution_count": 56, "metadata": {}, "output_type": "execute_result" } @@ -7004,7 +6999,7 @@ }, { "cell_type": "code", - "execution_count": 52, + "execution_count": 41, "metadata": { "collapsed": false }, @@ -7050,7 +7045,7 @@ }, { "cell_type": "code", - "execution_count": 53, + "execution_count": 42, "metadata": { "collapsed": false }, @@ -7205,7 +7200,7 @@ "" ] }, - "execution_count": 53, + "execution_count": 42, "metadata": {}, "output_type": "execute_result" } @@ -7218,6 +7213,15 @@ "css_file = './numericalmoocstyle.css'\n", "HTML(open(css_file, \"r\").read())" ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] } ], "metadata": { diff --git a/randy_schur/Integration Techniques for Mechanical Systems.ipynb b/randy_schur/Integration Techniques for Mechanical Systems.ipynb index 6f7e3b3..927c75b 100644 --- a/randy_schur/Integration Techniques for Mechanical Systems.ipynb +++ b/randy_schur/Integration Techniques for Mechanical Systems.ipynb @@ -308,7 +308,7 @@ }, { "cell_type": "code", - "execution_count": 32, + "execution_count": 5, "metadata": { "collapsed": true }, @@ -335,7 +335,7 @@ }, { "cell_type": "code", - "execution_count": 33, + "execution_count": 6, "metadata": { "collapsed": true }, @@ -359,7 +359,7 @@ }, { "cell_type": "code", - "execution_count": 34, + "execution_count": 7, "metadata": { "collapsed": true }, @@ -388,7 +388,7 @@ }, { "cell_type": "code", - "execution_count": 35, + "execution_count": 8, "metadata": { "collapsed": false }, @@ -408,7 +408,7 @@ }, { "cell_type": "code", - "execution_count": 36, + "execution_count": 9, "metadata": { "collapsed": false }, @@ -1180,7 +1180,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1279,7 +1279,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 11, "metadata": { "collapsed": false }, @@ -1302,7 +1302,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 12, "metadata": { "collapsed": false }, @@ -2119,7 +2119,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 13, "metadata": { "collapsed": false }, @@ -2202,7 +2202,7 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 14, "metadata": { "collapsed": false }, @@ -3015,7 +3015,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 17, "metadata": { "collapsed": true }, @@ -3038,7 +3038,7 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 18, "metadata": { "collapsed": false }, @@ -3073,7 +3073,7 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 19, "metadata": { "collapsed": false }, @@ -4725,7 +4725,7 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 20, "metadata": { "collapsed": true }, @@ -4749,7 +4749,7 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 21, "metadata": { "collapsed": true }, @@ -4778,7 +4778,7 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 22, "metadata": { "collapsed": true }, @@ -4809,7 +4809,7 @@ }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 23, "metadata": { "collapsed": false }, @@ -5639,7 +5639,7 @@ }, { "cell_type": "code", - "execution_count": 30, + "execution_count": 24, "metadata": { "collapsed": false }, @@ -6471,7 +6471,7 @@ }, { "cell_type": "code", - "execution_count": 31, + "execution_count": 25, "metadata": { "collapsed": false }, @@ -7318,11 +7318,166 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 26, "metadata": { "collapsed": false }, - "outputs": [], + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "execution_count": 26, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "# This cell loads the style of the notebook, which is modified from the \n", "# Numerical Methods in Python Course: http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about\n", @@ -7332,6 +7487,15 @@ "HTML(open(css_file, \"r\").read())" ] }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, { "cell_type": "code", "execution_count": null, From b0d17c9d478cf41afa529f5b91d25b80670fe740 Mon Sep 17 00:00:00 2001 From: rbds Date: Sun, 13 Dec 2015 22:01:50 -0500 Subject: [PATCH 53/61] Update README.md --- README.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/README.md b/README.md index e39f9a5..9cab8a3 100644 --- a/README.md +++ b/README.md @@ -1,2 +1,4 @@ # assignment-bank-2015 For submission of class projects at GW, Fall 2015 + +The code in this repository is under Creative Commons Attribution license CC-BY 4.0, code under MIT license © 2015 Randall Schur From e8e43bccae0475822fbdb17c4cfd6f0af61bf163 Mon Sep 17 00:00:00 2001 From: rbds Date: Sun, 13 Dec 2015 22:02:51 -0500 Subject: [PATCH 54/61] edit wording --- randy_schur/Double_Pendulum_Problem.ipynb | 120 +++++------------- ...n_Techniques_for_Mechanical_Systems.ipynb} | 52 ++++---- 2 files changed, 61 insertions(+), 111 deletions(-) rename randy_schur/{Integration Techniques for Mechanical Systems.ipynb => Integration_Techniques_for_Mechanical_Systems.ipynb} (99%) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index 8698c1f..2c51bba 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -233,7 +233,7 @@ }, { "cell_type": "code", - "execution_count": 63, + "execution_count": 17, "metadata": { "collapsed": false }, @@ -268,7 +268,7 @@ }, { "cell_type": "code", - "execution_count": 64, + "execution_count": 18, "metadata": { "collapsed": false }, @@ -306,7 +306,7 @@ }, { "cell_type": "code", - "execution_count": 65, + "execution_count": 19, "metadata": { "collapsed": true }, @@ -331,7 +331,7 @@ }, { "cell_type": "code", - "execution_count": 66, + "execution_count": 20, "metadata": { "collapsed": false }, @@ -359,7 +359,7 @@ }, { "cell_type": "code", - "execution_count": 67, + "execution_count": 21, "metadata": { "collapsed": false }, @@ -909,7 +909,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -1685,7 +1685,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -1962,7 +1962,7 @@ }, { "cell_type": "code", - "execution_count": 47, + "execution_count": 22, "metadata": { "collapsed": false }, @@ -1995,7 +1995,7 @@ }, { "cell_type": "code", - "execution_count": 48, + "execution_count": 23, "metadata": { "collapsed": false }, @@ -2545,7 +2545,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -2812,7 +2812,7 @@ }, { "cell_type": "code", - "execution_count": 49, + "execution_count": 24, "metadata": { "collapsed": true }, @@ -2841,7 +2841,7 @@ }, { "cell_type": "code", - "execution_count": 70, + "execution_count": 25, "metadata": { "collapsed": false }, @@ -3391,7 +3391,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -3613,7 +3613,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -3621,16 +3621,6 @@ }, "metadata": {}, "output_type": "display_data" - }, - { - "data": { - "text/plain": [ - "[]" - ] - }, - "execution_count": 70, - "metadata": {}, - "output_type": "execute_result" } ], "source": [ @@ -3654,7 +3644,8 @@ "pyplot.ylabel(r'Total energy in system', fontsize=18);\n", "pyplot.title('Energy in System');\n", "pyplot.plot(t, Verlet_energy, lw=2, label='Verlet');\n", - "pyplot.plot(t, RK4_energy, 'r', lw=2, label='RK4')" + "pyplot.plot(t, RK4_energy, 'r', lw=2, label='RK4')\n", + "pyplot.legend();" ] }, { @@ -3677,7 +3668,7 @@ }, { "cell_type": "code", - "execution_count": 51, + "execution_count": 26, "metadata": { "collapsed": false }, @@ -3711,7 +3702,7 @@ }, { "cell_type": "code", - "execution_count": 52, + "execution_count": 27, "metadata": { "collapsed": false }, @@ -4261,7 +4252,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -4510,19 +4501,6 @@ "There isn't a clear pattern here, although there is some clustering. So, we can say that our system is NOT likely to be asymptotically stable. However, this shouldn't be surprising. If we continued simulating, our system would continue rotating since there is no damping. Let's see what happens when there is some damping in the system. We're going to use RK4 for the following analysis." ] }, - { - "cell_type": "code", - "execution_count": 53, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "#To Do:\n", - "# animate motion of double pendulum\n", - "# check through sources" - ] - }, { "cell_type": "markdown", "metadata": {}, @@ -4547,7 +4525,7 @@ }, { "cell_type": "code", - "execution_count": 54, + "execution_count": 29, "metadata": { "collapsed": false }, @@ -4573,15 +4551,13 @@ " G = numpy.array([[(m1+m2)*g*l1*sin(x1)],[m2*l2*g*sin(x2)]])\n", " f = c*numpy.array([[x3],[x4]])\n", " qdd = numpy.linalg.inv(M).dot(V.dot(numpy.array([[x3],[x4]]))+G -f) \n", - " #print(qdd)\n", " RHS = numpy.array([x3, x4, qdd[0], qdd[1]])\n", - " #print(RHS.T)\n", " return RHS.T" ] }, { "cell_type": "code", - "execution_count": 55, + "execution_count": 31, "metadata": { "collapsed": false }, @@ -5131,7 +5107,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -5353,7 +5329,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -5907,7 +5883,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -6129,7 +6105,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -6159,7 +6135,7 @@ "pyplot.figure(figsize=(10,8));\n", "pyplot.grid(True);\n", "pyplot.xlabel(r't', fontsize=18);\n", - "pyplot.ylabel(r'position (meters)', fontsize=18);\n", + "pyplot.ylabel(r'position (radians)', fontsize=18);\n", "pyplot.title('Double Pendulum Euler');\n", "pyplot.plot(t, q1d_dp[:,0], lw=2, label='Joint 1');\n", "pyplot.plot(t, q1d_dp[:,1], lw=2, label='Joint 2')\n", @@ -6168,7 +6144,7 @@ "pyplot.figure(figsize=(10,8));\n", "pyplot.grid(True);\n", "pyplot.xlabel(r't', fontsize=18);\n", - "pyplot.ylabel(r'position (meters)', fontsize=18);\n", + "pyplot.ylabel(r'position (radians)', fontsize=18);\n", "pyplot.title('Double Pendulum RK4');\n", "pyplot.plot(t, q4d_dp[:,0], lw=2, label='Joint 1');\n", "pyplot.plot(t, q4d_dp[:,1], lw=2, label='Joint 2')\n", @@ -6186,7 +6162,7 @@ }, { "cell_type": "code", - "execution_count": 56, + "execution_count": 33, "metadata": { "collapsed": false }, @@ -6736,7 +6712,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -6970,10 +6946,10 @@ { "data": { "text/plain": [ - "[]" + "[]" ] }, - "execution_count": 56, + "execution_count": 33, "metadata": {}, "output_type": "execute_result" } @@ -6985,33 +6961,7 @@ "pyplot.xlabel(r'$\\theta_2$', fontsize=18);\n", "pyplot.ylabel(r'$\\dot{\\theta_2}$', fontsize=18);\n", "pyplot.title('Poincare Section');\n", - "pyplot.plot(v[0,:], v[1,:], 'bo', lw=2, );\n", - "\n", - "#from matplotlib import animation\n", - "#from JSAnimation.IPython_display import display_animation\n", - "#def animate(data):\n", - "# print(numpy.shape(data))\n", - "# x = data[0,:]\n", - "# y = data[1,:]\n", - "# line.set_data(x, y)\n", - "# return line," - ] - }, - { - "cell_type": "code", - "execution_count": 41, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "# #(try to animate plot)\n", - "#fig = pyplot.figure();\n", - "#ax = pyplot.axes(xlim=(-10, 10), ylim=(-10,10),xlabel=(r'$\\theta_2$'), ylabel=(r'$\\dot{\\theta_2}$'))\n", - "#line, = ax.plot([],[],'bo', lw=2);\n", - "\n", - "#anim = animation.FuncAnimation(fig, animate, frames= v[:,:], interval=10)\n", - "#display_animation(anim, default_mode='once')" + "pyplot.plot(v[0,:], v[1,:], 'bo', lw=2);\n" ] }, { @@ -7045,7 +6995,7 @@ }, { "cell_type": "code", - "execution_count": 42, + "execution_count": 34, "metadata": { "collapsed": false }, @@ -7200,7 +7150,7 @@ "" ] }, - "execution_count": 42, + "execution_count": 34, "metadata": {}, "output_type": "execute_result" } @@ -7240,7 +7190,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.5.0" + "version": "3.4.3" } }, "nbformat": 4, diff --git a/randy_schur/Integration Techniques for Mechanical Systems.ipynb b/randy_schur/Integration_Techniques_for_Mechanical_Systems.ipynb similarity index 99% rename from randy_schur/Integration Techniques for Mechanical Systems.ipynb rename to randy_schur/Integration_Techniques_for_Mechanical_Systems.ipynb index 927c75b..371b290 100644 --- a/randy_schur/Integration Techniques for Mechanical Systems.ipynb +++ b/randy_schur/Integration_Techniques_for_Mechanical_Systems.ipynb @@ -958,7 +958,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -1852,7 +1852,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -2128,7 +2128,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "/home/mint/miniconda3/lib/python3.5/site-packages/ipykernel/__main__.py:24: DeprecationWarning: using a non-integer number instead of an integer will result in an error in the future\n" + "/home/randy/anaconda3/lib/python3.4/site-packages/ipykernel/__main__.py:24: DeprecationWarning: using a non-integer number instead of an integer will result in an error in the future\n" ] } ], @@ -2752,7 +2752,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -3003,7 +3003,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "So yes, Verlet and RK2 both exhibit second order convergence as we expected. RK4 integration looks like fourth order. Euler integration looks weird. It appears to get exponentially worse with the time discretization. What is going on? One explanation is that there is some behavior that the first order method of Euler integration fails to capture." + "So yes, Verlet and RK2 both exhibit second order convergence as we expected. RK4 integration looks like fourth order. Euler integration looks weird. It appears to get exponentially worse with the time discretization. What is going on? One explanation is that there is some behavior that the first order method of Euler integration fails to capture until the time step is extremely small." ] }, { @@ -3015,7 +3015,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 15, "metadata": { "collapsed": true }, @@ -3038,7 +3038,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 16, "metadata": { "collapsed": false }, @@ -3073,7 +3073,7 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 17, "metadata": { "collapsed": false }, @@ -3623,7 +3623,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -4399,7 +4399,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -4660,7 +4660,7 @@ "source": [ "It looks like Euler integration is adding a significant amount of energy to the system.\n", "\n", - "The really interesting results come from the other integrators. RK2 continuously adds energy to the system. Our symplectic integrator looks like it may be oscillating, but the energy in the system is stable! Maybe all of that work deriving the equations of motion was worth it! We now know that the Hamiltonian equations do better with Conservation of Energy, and this can make a big difference even in a simple system. " + "The really interesting results come from the other integrators. RK2 continuously adds energy to the system. Our symplectic integrator looks like it may be oscillating, but the energy in the system is stable! Maybe all of that work deriving the equations of motion was worth it! We now know that the Hamiltonian equations do better with Conservation of Energy, and this can make a difference even in a simple system. " ] }, { @@ -4705,7 +4705,7 @@ "\n", "The big change comes in the Hamiltonian equations. If these are dependent on convservation of energy, how do we deal with a system where energy is not conserved?\n", "\n", - "Our first step is to re-examine the Lagrangian. Since losses due to friction are neither potential nor kinetic energy, we'll need a new term. The Lagrangian equation of motion with non-conservative forces is:\n", + "Our first step is to re-examine the Lagrangian. Since losses due to friction are neither potential nor kinetic energy, we'll need a new term. The Lagrangian equation with non-conservative forces is:\n", "\n", "$$\\begin{equation}\n", "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x_i}}\\right) - \\frac{\\partial L}{\\partial x_i} = Q\n", @@ -4725,7 +4725,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 18, "metadata": { "collapsed": true }, @@ -4749,7 +4749,7 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 19, "metadata": { "collapsed": true }, @@ -4778,7 +4778,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 20, "metadata": { "collapsed": true }, @@ -4809,7 +4809,7 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 21, "metadata": { "collapsed": false }, @@ -5359,7 +5359,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -5631,7 +5631,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "How does each integrator perform compared to the case of simple harmonic motion? Just by looking at the system response, it seems like the error is actually better here. Why is that? It is important to note here that the integrators perform differently for different system models. This is part of why we study numerical methods! There isn't a single answer to which scheme to use, it is dependent on the characteristics of both the scheme and the system mode.\n", + "How does each integrator perform compared to the case of simple harmonic motion? Just by looking at the system response, it seems like the error is actually better here. Why is that? It is important to note here that the integrators perform differently for different system models. This is part of why we study numerical methods! There isn't a single answer to which scheme to use, it is dependent on the characteristics of both the scheme and the system.\n", "\n", "The effect of the damper is to remove energy from the system. So, in this case the damping term is helping to remove some energy from the system that the integration adds. As time goes to infinity, these solutions will converge, and the Euler and Runge-Kutta methods are stable. We can again look at the energy in the system according to each method. \n", "\n" @@ -5639,7 +5639,7 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 22, "metadata": { "collapsed": false }, @@ -6189,7 +6189,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -6461,17 +6461,17 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "We can see some oscillations in the total energy, which come from the discretization error. Overall, we see that Verlet and the Runge-Kutta methods perform better here than Euler (although not perfectly - it does have induced oscillations from the damping term), but just like the analytical solution the energy in the system asymptotically approaches zero.\n", + "We can see some oscillations in the total energy, which come from the discretization error. Overall, we see that Verlet and the Runge-Kutta methods perform better here than Euler (although not perfectly - there are induced oscillations from the damping term), but just like the analytical solution the energy in the system asymptotically approaches zero.\n", "\n", "\n", "### Phase Portrait\n", "\n", - "Another way to investigate stability is with a phase plot. Instead of position vs. time, we can plot velocity vs. position. This plot is in phase space (a subset of state space). For any given state (position and velocity), we should be able to determine what the next state will be. This gives us an entire direction field, over which we can plot the actual behavior of the system to see if they match up. If the phase plot becomes repetitive, we can call it stable, and if the phase plot goes to the origin, we can call it asymptotically stable." + "Another way to investigate stability is with a phase plot. Instead of position vs. time, we can plot velocity vs. position. This plot is in phase space (this is similar to state space, if you have a control systems background). For any given state (position and velocity), we should be able to determine what the next state will be. This gives us an entire direction field, over which we can plot the actual behavior of the system to see if they match up. If the phase plot becomes repetitive, we can call it stable, and if the phase plot goes to the origin, we can call it asymptotically stable." ] }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 23, "metadata": { "collapsed": false }, @@ -7021,7 +7021,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", + "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -7289,7 +7289,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "In this case, it appears that Verlet, RK2, and RK4 give almost identical results, which are close but not exactly the same as the analytical solution. So for the damped harmonic oscillator, there is less of an advantage to using a more complex integration scheme." + "In this case, it appears that Verlet, RK2, and RK4 give almost identical results, which are close but not exactly the same as the analytical solution. So for the damped harmonic oscillator, there is less of an advantage to using a more complex integration scheme. Try modifying the code above to make a phase portrait for the simple harmonic oscillator. We should see a circle (or an ellipse) form, which is called a limit cycle. Which integrators create this behavior?" ] }, { @@ -7522,7 +7522,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.5.0" + "version": "3.4.3" } }, "nbformat": 4, From f0658a1a8a4604085311291dfff6f4bd9b2dc894 Mon Sep 17 00:00:00 2001 From: rbds Date: Sun, 13 Dec 2015 22:27:15 -0500 Subject: [PATCH 55/61] evening commit --- randy_schur/Double_Pendulum_Problem.ipynb | 101 +++++++++++++--------- 1 file changed, 60 insertions(+), 41 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index 2c51bba..266c871 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -233,7 +233,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 2, "metadata": { "collapsed": false }, @@ -268,7 +268,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 3, "metadata": { "collapsed": false }, @@ -306,7 +306,7 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 4, "metadata": { "collapsed": true }, @@ -331,7 +331,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 5, "metadata": { "collapsed": false }, @@ -359,7 +359,7 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 6, "metadata": { "collapsed": false }, @@ -1962,7 +1962,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 7, "metadata": { "collapsed": false }, @@ -1995,7 +1995,7 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 8, "metadata": { "collapsed": false }, @@ -2812,7 +2812,7 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 9, "metadata": { "collapsed": true }, @@ -2829,19 +2829,25 @@ " p1 = u[1]\n", " q2 = u[2]\n", " p2 = u[3]\n", + " #make sure q1 and q1 are between 0 and 2pi. \n", + " while q1 > 2*pi:\n", + " q1 -= 2*pi\n", + " while q2 > 2*pi:\n", + " q2 -= 2*pi\n", + " while q1 < 0:\n", + " q1+=2*pi\n", + " while q2 <0:\n", + " q2+=2*pi \n", " \n", " \n", - " \n", - " V = -m1*g*l1*cos(q1) - m2*g*(l1*cos(q1)+l2*cos(q2))\n", - " T = p1**2/(2*I1)+m2*p1**2/(2*m1*I1) + p1*p2/(m1*l1*l2)*cos(q2-q1)+p2**2/(2*I1)\n", - " #print(T)\n", - " #print(V)\n", - " return T+V" + " V = -m1*g*l1*cos(q1) - m2*g*l1*cos(q1)+l2*cos(q2)\n", + " T = p1**2/(2*I1)+m2*p1**2/(2*m1*I1) + p1*p2/(m1*l1*l2)*cos(q2-q1)+p2**2/(2*I2)\n", + " return T+V\n" ] }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 10, "metadata": { "collapsed": false }, @@ -3613,7 +3619,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -3621,6 +3627,16 @@ }, "metadata": {}, "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 10, + "metadata": {}, + "output_type": "execute_result" } ], "source": [ @@ -3634,25 +3650,38 @@ "#print(u[0,:])\n", "for i in range(N):\n", " Verlet_energy[i] = get_Energy(u[:,i])\n", - " RK4_energy[i] = get_Energy(x4_dp[i,:])\n", + " #RK4_energy[i] = get_Energy(x4_dp[i,:])\n", "\n", " \n", "\n", - "pyplot.figure(figsize=(11,6));\n", + "pyplot.figure(figsize=(10,8));\n", "pyplot.grid(True);\n", - "pyplot.xlabel(r't', fontsize=18);\n", - "pyplot.ylabel(r'Total energy in system', fontsize=18);\n", + "\n", "pyplot.title('Energy in System');\n", - "pyplot.plot(t, Verlet_energy, lw=2, label='Verlet');\n", - "pyplot.plot(t, RK4_energy, 'r', lw=2, label='RK4')\n", - "pyplot.legend();" + "ax1 = pyplot.gca()\n", + "pyplot.plot(t, Verlet_energy, lw=2, label='Verlet', color='b');\n", + "pyplot.xlabel('time (s)')\n", + "pyplot.ylabel('Energy')\n", + "\n", + "## use to plot RK4 energy on same graph.\n", + "#for tl in ax1.get_yticklabels():\n", + "# tl.set_color('b')\n", + "\n", + "#ax2 = ax1.twinx();\n", + "#ax2.plot(t, RK4_energy, 'r', lw=2, label='RK4')\n", + "#for tl in ax2.get_yticklabels():\n", + "# tl.set_color('r')\n", + "\n", + "#ax1.set_xlabel('time (s)')\n", + "#ax1.set_ylabel('Verlet', color='b')\n", + "#ax2.set_ylabel('RK4', color='r')\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "So, we are very far from perfect. But, at least the energy in the system is stable, even if it isn't constant! " + "So, we are very far from perfect. But, at least with Verlet the energy in the system is stable, even if it isn't constant! " ] }, { @@ -3668,7 +3697,7 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 11, "metadata": { "collapsed": false }, @@ -3702,7 +3731,7 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 12, "metadata": { "collapsed": false }, @@ -4525,7 +4554,7 @@ }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 13, "metadata": { "collapsed": false }, @@ -4557,7 +4586,7 @@ }, { "cell_type": "code", - "execution_count": 31, + "execution_count": 14, "metadata": { "collapsed": false }, @@ -6162,7 +6191,7 @@ }, { "cell_type": "code", - "execution_count": 33, + "execution_count": 15, "metadata": { "collapsed": false }, @@ -6942,16 +6971,6 @@ }, "metadata": {}, "output_type": "display_data" - }, - { - "data": { - "text/plain": [ - "[]" - ] - }, - "execution_count": 33, - "metadata": {}, - "output_type": "execute_result" } ], "source": [ @@ -6995,7 +7014,7 @@ }, { "cell_type": "code", - "execution_count": 34, + "execution_count": 16, "metadata": { "collapsed": false }, @@ -7150,7 +7169,7 @@ "" ] }, - "execution_count": 34, + "execution_count": 16, "metadata": {}, "output_type": "execute_result" } From 4a9c04921ff850f1860ae0c06b7d6e67b43a3e99 Mon Sep 17 00:00:00 2001 From: rbds Date: Mon, 14 Dec 2015 10:13:58 -0500 Subject: [PATCH 56/61] stupid computer --- ...on_Techniques_for_Mechanical_Systems.ipynb | 1475 +++-------------- randy_schur/figures/POS_computer.png | Bin 82838 -> 64372 bytes randy_schur/figures/combined_response.png | Bin 212034 -> 170544 bytes 3 files changed, 260 insertions(+), 1215 deletions(-) diff --git a/randy_schur/Integration_Techniques_for_Mechanical_Systems.ipynb b/randy_schur/Integration_Techniques_for_Mechanical_Systems.ipynb index 371b290..16abbe7 100644 --- a/randy_schur/Integration_Techniques_for_Mechanical_Systems.ipynb +++ b/randy_schur/Integration_Techniques_for_Mechanical_Systems.ipynb @@ -226,7 +226,16 @@ "metadata": { "collapsed": false }, - "outputs": [], + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "C:\\Users\\rschur\\Anaconda2\\lib\\site-packages\\IPython\\kernel\\__init__.py:13: ShimWarning: The `IPython.kernel` package has been deprecated. You should import from ipykernel or jupyter_client instead.\n", + " \"You should import from ipykernel or jupyter_client instead.\", ShimWarning)\n" + ] + } + ], "source": [ "import numpy\n", "from scipy.linalg import solve\n", @@ -488,6 +497,7 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", + " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -818,13 +828,11 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", - " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", - " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -873,19 +881,6 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", - "/*\n", - " * return a copy of an object with only non-object keys\n", - " * we need this to avoid circular references\n", - " * http://stackoverflow.com/a/24161582/3208463\n", - " */\n", - "function simpleKeys (original) {\n", - " return Object.keys(original).reduce(function (obj, key) {\n", - " if (typeof original[key] !== 'object')\n", - " obj[key] = original[key]\n", - " return obj;\n", - " }, {});\n", - "}\n", - "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -899,8 +894,7 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step,\n", - " guiEvent: simpleKeys(event)});\n", + " step: event.step});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -940,8 +934,7 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value,\n", - " guiEvent: simpleKeys(event)});\n", + " this.send_message(name, {key: value});\n", " return false;\n", "}\n", "\n", @@ -1017,8 +1010,6 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", - " fig.root.unbind('remove')\n", - "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -1026,12 +1017,8 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.close_ws(fig, msg);\n", - "}\n", - "\n", - "mpl.figure.prototype.close_ws = function(fig, msg){\n", - " fig.send_message('closing', msg);\n", - " // fig.ws.close()\n", + " fig.send_message('closing', {});\n", + " fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -1086,20 +1073,14 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", + " button.mouseover('Close figure', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", - "mpl.figure.prototype._root_extra_style = function(el){\n", - " var fig = this\n", - " el.on(\"remove\", function(){\n", - "\tfig.close_ws(fig, {});\n", - " });\n", - "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -1180,7 +1161,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1249,960 +1230,60 @@ { "cell_type": "markdown", "metadata": {}, - "source": [ - "### Error Analysis\n", - "The error analysis code below is based on the numerical methods MOOC notebook on phugoid oscillation. " - ] - }, - { - "cell_type": "code", - "execution_count": 10, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "def get_error(y, z, dt):\n", - " \"\"\"Returns error relative to analytical solution using L-1 norm.\n", - " \n", - " Parameters:\n", - " y - numerical solution\n", - " z - analytical solution\n", - " dt - time step size\n", - " \n", - " Returns:\n", - " error - L_1 norm of the error W.R.T. analytical solution \n", - " \"\"\"\n", - " err = dt*numpy.sum(numpy.abs(y-z))\n", - " return err" - ] - }, - { - "cell_type": "code", - "execution_count": 11, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "dt_values = numpy.array([0.1, 0.05, 0.01, 0.005, 0.001, 0.0001])\n", - "\n", - "Euler_error_values = numpy.empty_like(dt_values)\n", - "RK2_error_values = Euler_error_values.copy()\n", - "RK4_error_values = Euler_error_values.copy()\n", - "Verlet_error_values = Euler_error_values.copy()\n", - "for i, dt in enumerate(dt_values):\n", - " ###Call error function\n", - " Euler_error_values[i] = get_error(x1[:,0], x_analytical, dt)\n", - " RK2_error_values[i] = get_error(x2[:,0], x_analytical, dt)\n", - " RK4_error_values[i] = get_error(x4[:,0], x_analytical, dt)\n", - " Verlet_error_values[i] = get_error(xv[:,0], x_analytical, dt)\n", - " \n" - ] - }, - { - "cell_type": "code", - "execution_count": 12, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "application/javascript": [ - "/* Put everything inside the global mpl namespace */\n", - "window.mpl = {};\n", - "\n", - "mpl.get_websocket_type = function() {\n", - " if (typeof(WebSocket) !== 'undefined') {\n", - " return WebSocket;\n", - " } else if (typeof(MozWebSocket) !== 'undefined') {\n", - " return MozWebSocket;\n", - " } else {\n", - " alert('Your browser does not have WebSocket support.' +\n", - " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", - " 'Firefox 4 and 5 are also supported but you ' +\n", - " 'have to enable WebSockets in about:config.');\n", - " };\n", - "}\n", - "\n", - "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", - " this.id = figure_id;\n", - "\n", - " this.ws = websocket;\n", - "\n", - " this.supports_binary = (this.ws.binaryType != undefined);\n", - "\n", - " if (!this.supports_binary) {\n", - " var warnings = document.getElementById(\"mpl-warnings\");\n", - " if (warnings) {\n", - " warnings.style.display = 'block';\n", - " warnings.textContent = (\n", - " \"This browser does not support binary websocket messages. \" +\n", - " \"Performance may be slow.\");\n", - " }\n", - " }\n", - "\n", - " this.imageObj = new Image();\n", - "\n", - " this.context = undefined;\n", - " this.message = undefined;\n", - " this.canvas = undefined;\n", - " this.rubberband_canvas = undefined;\n", - " this.rubberband_context = undefined;\n", - " this.format_dropdown = undefined;\n", - "\n", - " this.image_mode = 'full';\n", - "\n", - " this.root = $('
');\n", - " this._root_extra_style(this.root)\n", - " this.root.attr('style', 'display: inline-block');\n", - "\n", - " $(parent_element).append(this.root);\n", - "\n", - " this._init_header(this);\n", - " this._init_canvas(this);\n", - " this._init_toolbar(this);\n", - "\n", - " var fig = this;\n", - "\n", - " this.waiting = false;\n", - "\n", - " this.ws.onopen = function () {\n", - " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", - " fig.send_message(\"send_image_mode\", {});\n", - " fig.send_message(\"refresh\", {});\n", - " }\n", - "\n", - " this.imageObj.onload = function() {\n", - " if (fig.image_mode == 'full') {\n", - " // Full images could contain transparency (where diff images\n", - " // almost always do), so we need to clear the canvas so that\n", - " // there is no ghosting.\n", - " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", - " }\n", - " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " };\n", - "\n", - " this.imageObj.onunload = function() {\n", - " this.ws.close();\n", - " }\n", - "\n", - " this.ws.onmessage = this._make_on_message_function(this);\n", - "\n", - " this.ondownload = ondownload;\n", - "}\n", - "\n", - "mpl.figure.prototype._init_header = function() {\n", - " var titlebar = $(\n", - " '
');\n", - " var titletext = $(\n", - " '
');\n", - " titlebar.append(titletext)\n", - " this.root.append(titlebar);\n", - " this.header = titletext[0];\n", - "}\n", - "\n", - "\n", - "\n", - "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", - "\n", - "}\n", - "\n", - "\n", - "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", - "\n", - "}\n", - "\n", - "mpl.figure.prototype._init_canvas = function() {\n", - " var fig = this;\n", - "\n", - " var canvas_div = $('
');\n", - "\n", - " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", - "\n", - " function canvas_keyboard_event(event) {\n", - " return fig.key_event(event, event['data']);\n", - " }\n", - "\n", - " canvas_div.keydown('key_press', canvas_keyboard_event);\n", - " canvas_div.keyup('key_release', canvas_keyboard_event);\n", - " this.canvas_div = canvas_div\n", - " this._canvas_extra_style(canvas_div)\n", - " this.root.append(canvas_div);\n", - "\n", - " var canvas = $('');\n", - " canvas.addClass('mpl-canvas');\n", - " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", - "\n", - " this.canvas = canvas[0];\n", - " this.context = canvas[0].getContext(\"2d\");\n", - "\n", - " var rubberband = $('');\n", - " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", - "\n", - " var pass_mouse_events = true;\n", - "\n", - " canvas_div.resizable({\n", - " start: function(event, ui) {\n", - " pass_mouse_events = false;\n", - " },\n", - " resize: function(event, ui) {\n", - " fig.request_resize(ui.size.width, ui.size.height);\n", - " },\n", - " stop: function(event, ui) {\n", - " pass_mouse_events = true;\n", - " fig.request_resize(ui.size.width, ui.size.height);\n", - " },\n", - " });\n", - "\n", - " function mouse_event_fn(event) {\n", - " if (pass_mouse_events)\n", - " return fig.mouse_event(event, event['data']);\n", - " }\n", - "\n", - " rubberband.mousedown('button_press', mouse_event_fn);\n", - " rubberband.mouseup('button_release', mouse_event_fn);\n", - " // Throttle sequential mouse events to 1 every 20ms.\n", - " rubberband.mousemove('motion_notify', mouse_event_fn);\n", - "\n", - " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", - " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", - "\n", - " canvas_div.on(\"wheel\", function (event) {\n", - " event = event.originalEvent;\n", - " event['data'] = 'scroll'\n", - " if (event.deltaY < 0) {\n", - " event.step = 1;\n", - " } else {\n", - " event.step = -1;\n", - " }\n", - " mouse_event_fn(event);\n", - " });\n", - "\n", - " canvas_div.append(canvas);\n", - " canvas_div.append(rubberband);\n", - "\n", - " this.rubberband = rubberband;\n", - " this.rubberband_canvas = rubberband[0];\n", - " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", - " this.rubberband_context.strokeStyle = \"#000000\";\n", - "\n", - " this._resize_canvas = function(width, height) {\n", - " // Keep the size of the canvas, canvas container, and rubber band\n", - " // canvas in synch.\n", - " canvas_div.css('width', width)\n", - " canvas_div.css('height', height)\n", - "\n", - " canvas.attr('width', width);\n", - " canvas.attr('height', height);\n", - "\n", - " rubberband.attr('width', width);\n", - " rubberband.attr('height', height);\n", - " }\n", - "\n", - " // Set the figure to an initial 600x600px, this will subsequently be updated\n", - " // upon first draw.\n", - " this._resize_canvas(600, 600);\n", - "\n", - " // Disable right mouse context menu.\n", - " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", - " return false;\n", - " });\n", - "\n", - " function set_focus () {\n", - " canvas.focus();\n", - " canvas_div.focus();\n", - " }\n", - "\n", - " window.setTimeout(set_focus, 100);\n", - "}\n", - "\n", - "mpl.figure.prototype._init_toolbar = function() {\n", - " var fig = this;\n", - "\n", - " var nav_element = $('
')\n", - " nav_element.attr('style', 'width: 100%');\n", - " this.root.append(nav_element);\n", - "\n", - " // Define a callback function for later on.\n", - " function toolbar_event(event) {\n", - " return fig.toolbar_button_onclick(event['data']);\n", - " }\n", - " function toolbar_mouse_event(event) {\n", - " return fig.toolbar_button_onmouseover(event['data']);\n", - " }\n", - "\n", - " for(var toolbar_ind in mpl.toolbar_items) {\n", - " var name = mpl.toolbar_items[toolbar_ind][0];\n", - " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", - " var image = mpl.toolbar_items[toolbar_ind][2];\n", - " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", - "\n", - " if (!name) {\n", - " // put a spacer in here.\n", - " continue;\n", - " }\n", - " var button = $('');\n", + " button.click(method_name, toolbar_event);\n", + " button.mouseover(tooltip, toolbar_mouse_event);\n", + " nav_element.append(button);\n", + " }\n", + "\n", + " // Add the status bar.\n", + " var status_bar = $('');\n", + " nav_element.append(status_bar);\n", + " this.message = status_bar[0];\n", + "\n", + " // Add the close button to the window.\n", + " var buttongrp = $('
');\n", + " var button = $('');\n", + " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", + " buttongrp.append(button);\n", + " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", + " titlebar.prepend(buttongrp);\n", + "}\n", + "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", + "\n", + "mpl.figure.prototype._canvas_extra_style = function(el){\n", + " // this is important to make the div 'focusable\n", + " el.attr('tabindex', 0)\n", + " // reach out to IPython and tell the keyboard manager to turn it's self\n", + " // off when our div gets focus\n", + "\n", + " // location in version 3\n", + " if (IPython.notebook.keyboard_manager) {\n", + " IPython.notebook.keyboard_manager.register_events(el);\n", + " }\n", + " else {\n", + " // location in version 2\n", + " IPython.keyboard_manager.register_events(el);\n", + " }\n", + "\n", + "}\n", + "\n", + "mpl.figure.prototype._key_event_extra = function(event, name) {\n", + " var manager = IPython.notebook.keyboard_manager;\n", + " if (!manager)\n", + " manager = IPython.keyboard_manager;\n", + "\n", + " // Check for shift+enter\n", + " if (event.shiftKey && event.which == 13) {\n", + " this.canvas_div.blur();\n", + " event.shiftKey = false;\n", + " // Send a \"J\" for go to next cell\n", + " event.which = 74;\n", + " event.keyCode = 74;\n", + " manager.command_mode();\n", + " manager.handle_keydown(event);\n", + " }\n", + "}\n", + "\n", + "mpl.figure.prototype.handle_save = function(fig, msg) {\n", + " fig.ondownload(fig, null);\n", + "}\n", + "\n", + "\n", + "mpl.find_output_cell = function(html_output) {\n", + " // Return the cell and output element which can be found *uniquely* in the notebook.\n", + " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", + " // IPython event is triggered only after the cells have been serialised, which for\n", + " // our purposes (turning an active figure into a static one), is too late.\n", + " var cells = IPython.notebook.get_cells();\n", + " var ncells = cells.length;\n", + " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", + " data = data.data;\n", + " }\n", + " if (data['text/html'] == html_output) {\n", + " return [cell, data, j];\n", + " }\n", + " }\n", + " }\n", + " }\n", + "}\n", + "\n", + "// Register the function which deals with the matplotlib target/channel.\n", + "// The kernel may be null if the page has been refreshed.\n", + "if (IPython.notebook.kernel != null) {\n", + " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", + "}\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/html": [ + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "pyplot.figure(figsize=(10,6))\n", + "pyplot.tick_params(axis='both', labelsize=14)\n", + "pyplot.grid(True)\n", + "pyplot.xlabel('$\\Delta t$', fontsize=16)\n", + "pyplot.ylabel('Error', fontsize=16)\n", + "pyplot.loglog(dt_values, Euler_error_values, 'bo-', label='Euler')\n", + "pyplot.loglog(dt_values, RK2_error_values, 'ro-', label='RK2')\n", + "pyplot.loglog(dt_values, RK4_error_values, 'co-', label='RK4')\n", + "pyplot.loglog(dt_values, Verlet_error_values, 'go-', label='Symplectic')\n", + "pyplot.axis('equal')\n", + "pyplot.legend();" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Is this what you would expect to see based on the plots of position above? What are the sources of this error? We have the usual suspects: truncation error, discretization error, possible precision errors. Don't believe that these are significant? The discretization is the only difference between the first two plots above, and with a smaller time step the response from Runge-Kutta and Euler are greatly improved. To exame precision errors, take a look at the following screenshot, which shows the plots above, built with identical code on two different computers. The responses on the left are from a 64-bit computer, and the responses on the right are from a 32-bit computer. Look closely at the phase shift in the system response. That's a huge difference! Even the precision of the machine can have a significant impact on numerical integration.\n", + "\n", + "![Image](figures/combined_response.png)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Despite the numerical errors, why is one second order scheme (Verlet) outperforming another (RK2) by so much? What is going on? We can check the order of convergence to make sure that these schemes are actually 2nd order." + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/home/mint/miniconda3/lib/python3.5/site-packages/ipykernel/__main__.py:24: DeprecationWarning: using a non-integer number instead of an integer will result in an error in the future\n" + ] + } + ], + "source": [ + "def get_diffgrid(u_current, u_fine, dt):\n", + " \"\"\"Returns the difference between one grid and the fine one using L-1 norm.\n", + " \n", + " Parameters\n", + " ----------\n", + " u_current : array of float\n", + " solution on the current grid.\n", + " u_finest : array of float\n", + " solution on the fine grid.\n", + " dt : float\n", + " time-increment on the current grid.\n", + " \n", + " Returns\n", + " -------\n", + " diffgrid : float\n", + " difference computed in the L-1 norm.\n", + " \"\"\"\n", + " \n", + " N_current = len(u_current[:,0])\n", + " N_fine = len(u_fine[:,0])\n", + " \n", + " grid_size_ratio = numpy.ceil(N_fine/N_current)\n", + " \n", + " diffgrid = dt * numpy.sum( numpy.abs(\\\n", + " u_current[:,0]- u_fine[::grid_size_ratio,0])) \n", + " \n", + " return diffgrid\n", + "\n", + "# use a for-loop to compute the solution on different grids - this code is modified from Numerical MOOC lesson 1 module 4\n", + "dt_values = numpy.array([0.1, 0.05, 0.01, 0.005, 0.001])\n", + "u1_values = numpy.empty_like(dt_values, dtype=numpy.ndarray)\n", + "u2_values = u1_values.copy()\n", + "u4_values = u1_values.copy()\n", + "uv_values = u1_values.copy()\n", + "\n", + "#At each value of dt, integrate the system using 4 different methods from t=0 to T\n", "for i, dt in enumerate(dt_values):\n", - " ###Call error function\n", - " Euler_error_values[i] = get_error(x1[:,0], x_analytical, dt)\n", - " RK2_error_values[i] = get_error(x2[:,0], x_analytical, dt)\n", - " RK4_error_values[i] = get_error(x4[:,0], x_analytical, dt)\n", - " Verlet_error_values[i] = get_error(xv[:,0], x_analytical, dt)\n", - " \n" + " N = int(T/dt)+1 \n", + " t = numpy.linspace(0.0, T, N)\n", + " u1 = numpy.empty((N,2))\n", + " u1[0] = numpy.array([x0, xdot0])\n", + " u2 = u1.copy()\n", + " u4 = u1.copy()\n", + " uv = u1.copy()\n", + " for n in range(N-1):\n", + " u1[n+1] = euler(u1[n], f_harmonic_oscillator, dt)\n", + " u2[n+1] = RK2(u2[n], f_harmonic_oscillator, dt)\n", + " u4[n+1] = RK4(u4[n], f_harmonic_oscillator, dt)\n", + " uv[n+1] = verlet_SHM(uv[n], dt)\n", + " # store the value of u related to one grid\n", + " u1_values[i] = u1\n", + " u2_values[i] = u2\n", + " u4_values[i] = u4\n", + " uv_values[i] = uv\n", + " \n", + "# compute diffgrid\n", + "diffgrid1 = numpy.empty_like(dt_values)\n", + "diffgrid2 = diffgrid1.copy()\n", + "diffgrid4 = diffgrid1.copy()\n", + "diffgridv = diffgrid1.copy()\n", + "for i, dt in enumerate(dt_values):\n", + " diffgrid1[i] = get_diffgrid(u1_values[i], u1_values[-1], dt)\n", + " diffgrid2[i] = get_diffgrid(u2_values[i], u2_values[-1], dt)\n", + " diffgrid4[i] = get_diffgrid(u4_values[i], u4_values[-1], dt)\n", + " diffgridv[i] = get_diffgrid(uv_values[i], uv_values[-1], dt)" ] }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 14, "metadata": { "collapsed": false }, @@ -1363,7 +2279,6 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -1694,11 +2609,13 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -1747,6 +2664,19 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", + "/*\n", + " * return a copy of an object with only non-object keys\n", + " * we need this to avoid circular references\n", + " * http://stackoverflow.com/a/24161582/3208463\n", + " */\n", + "function simpleKeys (original) {\n", + " return Object.keys(original).reduce(function (obj, key) {\n", + " if (typeof original[key] !== 'object')\n", + " obj[key] = original[key]\n", + " return obj;\n", + " }, {});\n", + "}\n", + "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -1760,7 +2690,8 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step});\n", + " step: event.step,\n", + " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -1800,7 +2731,8 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value});\n", + " this.send_message(name, {key: value,\n", + " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", @@ -1817,7 +2749,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -1876,6 +2808,8 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", + " fig.root.unbind('remove')\n", + "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -1883,8 +2817,12 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.send_message('closing', {});\n", - " fig.ws.close()\n", + " fig.close_ws(fig, msg);\n", + "}\n", + "\n", + "mpl.figure.prototype.close_ws = function(fig, msg){\n", + " fig.send_message('closing', msg);\n", + " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -1939,14 +2877,20 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -2027,7 +2971,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2037,138 +2981,6 @@ "output_type": "display_data" } ], - "source": [ - "pyplot.figure(figsize=(10,6))\n", - "pyplot.tick_params(axis='both', labelsize=14)\n", - "pyplot.grid(True)\n", - "pyplot.xlabel('$\\Delta t$', fontsize=16)\n", - "pyplot.ylabel('Error', fontsize=16)\n", - "pyplot.loglog(dt_values, Euler_error_values, 'bo-', label='Euler')\n", - "pyplot.loglog(dt_values, RK2_error_values, 'ro-', label='RK2')\n", - "pyplot.loglog(dt_values, RK4_error_values, 'co-', label='RK4')\n", - "pyplot.loglog(dt_values, Verlet_error_values, 'go-', label='Symplectic')\n", - "pyplot.axis('equal')\n", - "pyplot.legend();" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Is this what you would expect to see based on the plots of position above? What are the sources of this error? We have the usual suspects: truncation error, discretization error, possible precision errors. Don't believe that these are significant? The discretization is the only difference between the first two plots above, and with a smaller time step the response from Runge-Kutta and Euler are greatly improved. To exame precision errors, take a look at the following screenshot, which shows the plots above, built with identical code on two different computers. The responses on the left are from a 64-bit computer, and the responses on the right are from a 32-bit computer. Look closely at the phase shift in the system response. That's a huge difference! Even the precision of the machine can have a significant impact on numerical integration.\n", - "\n", - "![Image](figures/combined_response.png)\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Despite the numerical errors, why is one second order scheme (Verlet) outperforming another (RK2) by so much? What is going on? We can check the order of convergence to make sure that these schemes are actually 2nd order." - ] - }, - { - "cell_type": "code", - "execution_count": 13, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "name": "stderr", - "output_type": "stream", - "text": [ - "C:\\Users\\rschur\\Anaconda2\\lib\\site-packages\\ipykernel\\__main__.py:24: DeprecationWarning: using a non-integer number instead of an integer will result in an error in the future\n" - ] - }, - { - "ename": "ValueError", - "evalue": "operands could not be broadcast together with shapes (251,) (253,) ", - "output_type": "error", - "traceback": [ - "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[1;31mValueError\u001b[0m Traceback (most recent call last)", - "\u001b[1;32m\u001b[0m in \u001b[0;36m\u001b[1;34m()\u001b[0m\n\u001b[0;32m 59\u001b[0m \u001b[0mdiffgridv\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdiffgrid1\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mcopy\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 60\u001b[0m \u001b[1;32mfor\u001b[0m \u001b[0mi\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdt\u001b[0m \u001b[1;32min\u001b[0m \u001b[0menumerate\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdt_values\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 61\u001b[1;33m \u001b[0mdiffgrid1\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mi\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mget_diffgrid\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mu1_values\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mi\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mu1_values\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;33m-\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdt\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 62\u001b[0m \u001b[0mdiffgrid2\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mi\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mget_diffgrid\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mu2_values\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mi\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mu2_values\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;33m-\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdt\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 63\u001b[0m \u001b[0mdiffgrid4\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mi\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mget_diffgrid\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mu4_values\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mi\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mu4_values\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;33m-\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdt\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", - "\u001b[1;32m\u001b[0m in \u001b[0;36mget_diffgrid\u001b[1;34m(u_current, u_fine, dt)\u001b[0m\n\u001b[0;32m 22\u001b[0m \u001b[0mgrid_size_ratio\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnumpy\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mceil\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mN_fine\u001b[0m\u001b[1;33m/\u001b[0m\u001b[0mN_current\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 23\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 24\u001b[1;33m \u001b[0mdiffgrid\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdt\u001b[0m \u001b[1;33m*\u001b[0m \u001b[0mnumpy\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0msum\u001b[0m\u001b[1;33m(\u001b[0m \u001b[0mnumpy\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mabs\u001b[0m\u001b[1;33m(\u001b[0m \u001b[0mu_current\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m-\u001b[0m \u001b[0mu_fine\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m:\u001b[0m\u001b[0mgrid_size_ratio\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 25\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 26\u001b[0m \u001b[1;32mreturn\u001b[0m \u001b[0mdiffgrid\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", - "\u001b[1;31mValueError\u001b[0m: operands could not be broadcast together with shapes (251,) (253,) " - ] - } - ], - "source": [ - "def get_diffgrid(u_current, u_fine, dt):\n", - " \"\"\"Returns the difference between one grid and the fine one using L-1 norm.\n", - " \n", - " Parameters\n", - " ----------\n", - " u_current : array of float\n", - " solution on the current grid.\n", - " u_finest : array of float\n", - " solution on the fine grid.\n", - " dt : float\n", - " time-increment on the current grid.\n", - " \n", - " Returns\n", - " -------\n", - " diffgrid : float\n", - " difference computed in the L-1 norm.\n", - " \"\"\"\n", - " \n", - " N_current = len(u_current[:,0])\n", - " N_fine = len(u_fine[:,0])\n", - " \n", - " grid_size_ratio = numpy.ceil(N_fine/N_current)\n", - " \n", - " diffgrid = dt * numpy.sum( numpy.abs(\\\n", - " u_current[:,0]- u_fine[::grid_size_ratio,0])) \n", - " \n", - " return diffgrid\n", - "\n", - "# use a for-loop to compute the solution on different grids - this code is modified from Numerical MOOC lesson 1 module 4\n", - "dt_values = numpy.array([0.1, 0.05, 0.01, 0.005, 0.001])\n", - "u1_values = numpy.empty_like(dt_values, dtype=numpy.ndarray)\n", - "u2_values = u1_values.copy()\n", - "u4_values = u1_values.copy()\n", - "uv_values = u1_values.copy()\n", - "\n", - "#At each value of dt, integrate the system using 4 different methods from t=0 to T\n", - "for i, dt in enumerate(dt_values):\n", - " N = int(T/dt)+1 \n", - " t = numpy.linspace(0.0, T, N)\n", - " u1 = numpy.empty((N,2))\n", - " u1[0] = numpy.array([x0, xdot0])\n", - " u2 = u1.copy()\n", - " u4 = u1.copy()\n", - " uv = u1.copy()\n", - " for n in range(N-1):\n", - " u1[n+1] = euler(u1[n], f_harmonic_oscillator, dt)\n", - " u2[n+1] = RK2(u2[n], f_harmonic_oscillator, dt)\n", - " u4[n+1] = RK4(u4[n], f_harmonic_oscillator, dt)\n", - " uv[n+1] = verlet_SHM(uv[n], dt)\n", - " # store the value of u related to one grid\n", - " u1_values[i] = u1\n", - " u2_values[i] = u2\n", - " u4_values[i] = u4\n", - " uv_values[i] = uv\n", - " \n", - "# compute diffgrid\n", - "diffgrid1 = numpy.empty_like(dt_values)\n", - "diffgrid2 = diffgrid1.copy()\n", - "diffgrid4 = diffgrid1.copy()\n", - "diffgridv = diffgrid1.copy()\n", - "for i, dt in enumerate(dt_values):\n", - " diffgrid1[i] = get_diffgrid(u1_values[i], u1_values[-1], dt)\n", - " diffgrid2[i] = get_diffgrid(u2_values[i], u2_values[-1], dt)\n", - " diffgrid4[i] = get_diffgrid(u4_values[i], u4_values[-1], dt)\n", - " diffgridv[i] = get_diffgrid(uv_values[i], uv_values[-1], dt)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], "source": [ "# plot using the matplotlib function loglog()\n", "pyplot.figure(figsize=(7,10))\n", @@ -2200,7 +3012,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 15, "metadata": { "collapsed": true }, @@ -2223,7 +3035,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 16, "metadata": { "collapsed": false }, @@ -2258,7 +3070,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 17, "metadata": { "collapsed": false }, @@ -2338,7 +3150,6 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -2669,11 +3480,13 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -2722,6 +3535,19 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", + "/*\n", + " * return a copy of an object with only non-object keys\n", + " * we need this to avoid circular references\n", + " * http://stackoverflow.com/a/24161582/3208463\n", + " */\n", + "function simpleKeys (original) {\n", + " return Object.keys(original).reduce(function (obj, key) {\n", + " if (typeof original[key] !== 'object')\n", + " obj[key] = original[key]\n", + " return obj;\n", + " }, {});\n", + "}\n", + "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -2735,7 +3561,8 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step});\n", + " step: event.step,\n", + " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -2775,7 +3602,8 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value});\n", + " this.send_message(name, {key: value,\n", + " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", @@ -2792,7 +3620,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -2851,6 +3679,8 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", + " fig.root.unbind('remove')\n", + "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -2858,8 +3688,12 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.send_message('closing', {});\n", - " fig.ws.close()\n", + " fig.close_ws(fig, msg);\n", + "}\n", + "\n", + "mpl.figure.prototype.close_ws = function(fig, msg){\n", + " fig.send_message('closing', msg);\n", + " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -2914,14 +3748,20 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -3002,7 +3842,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -3086,7 +3926,6 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -3417,11 +4256,13 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -3470,6 +4311,19 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", + "/*\n", + " * return a copy of an object with only non-object keys\n", + " * we need this to avoid circular references\n", + " * http://stackoverflow.com/a/24161582/3208463\n", + " */\n", + "function simpleKeys (original) {\n", + " return Object.keys(original).reduce(function (obj, key) {\n", + " if (typeof original[key] !== 'object')\n", + " obj[key] = original[key]\n", + " return obj;\n", + " }, {});\n", + "}\n", + "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -3483,7 +4337,8 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step});\n", + " step: event.step,\n", + " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -3523,7 +4378,8 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value});\n", + " this.send_message(name, {key: value,\n", + " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", @@ -3540,7 +4396,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -3599,6 +4455,8 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", + " fig.root.unbind('remove')\n", + "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -3606,8 +4464,12 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.send_message('closing', {});\n", - " fig.ws.close()\n", + " fig.close_ws(fig, msg);\n", + "}\n", + "\n", + "mpl.figure.prototype.close_ws = function(fig, msg){\n", + " fig.send_message('closing', msg);\n", + " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -3662,14 +4524,20 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -3750,7 +4618,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -3854,7 +4722,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 18, "metadata": { "collapsed": true }, @@ -3878,7 +4746,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 19, "metadata": { "collapsed": true }, @@ -3907,7 +4775,7 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 20, "metadata": { "collapsed": true }, @@ -3938,7 +4806,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 21, "metadata": { "collapsed": false }, @@ -4018,7 +4886,6 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -4349,11 +5216,13 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -4402,6 +5271,19 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", + "/*\n", + " * return a copy of an object with only non-object keys\n", + " * we need this to avoid circular references\n", + " * http://stackoverflow.com/a/24161582/3208463\n", + " */\n", + "function simpleKeys (original) {\n", + " return Object.keys(original).reduce(function (obj, key) {\n", + " if (typeof original[key] !== 'object')\n", + " obj[key] = original[key]\n", + " return obj;\n", + " }, {});\n", + "}\n", + "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -4415,7 +5297,8 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step});\n", + " step: event.step,\n", + " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -4455,7 +5338,8 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value});\n", + " this.send_message(name, {key: value,\n", + " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", @@ -4472,7 +5356,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -4531,6 +5415,8 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", + " fig.root.unbind('remove')\n", + "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -4538,8 +5424,12 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.send_message('closing', {});\n", - " fig.ws.close()\n", + " fig.close_ws(fig, msg);\n", + "}\n", + "\n", + "mpl.figure.prototype.close_ws = function(fig, msg){\n", + " fig.send_message('closing', msg);\n", + " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -4594,14 +5484,20 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -4682,7 +5578,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -4740,7 +5636,7 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 22, "metadata": { "collapsed": false }, @@ -4820,7 +5716,6 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -5151,11 +6046,13 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -5204,6 +6101,19 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", + "/*\n", + " * return a copy of an object with only non-object keys\n", + " * we need this to avoid circular references\n", + " * http://stackoverflow.com/a/24161582/3208463\n", + " */\n", + "function simpleKeys (original) {\n", + " return Object.keys(original).reduce(function (obj, key) {\n", + " if (typeof original[key] !== 'object')\n", + " obj[key] = original[key]\n", + " return obj;\n", + " }, {});\n", + "}\n", + "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -5217,7 +6127,8 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step});\n", + " step: event.step,\n", + " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -5257,7 +6168,8 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value});\n", + " this.send_message(name, {key: value,\n", + " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", @@ -5274,7 +6186,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -5333,6 +6245,8 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", + " fig.root.unbind('remove')\n", + "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -5340,8 +6254,12 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.send_message('closing', {});\n", - " fig.ws.close()\n", + " fig.close_ws(fig, msg);\n", + "}\n", + "\n", + "mpl.figure.prototype.close_ws = function(fig, msg){\n", + " fig.send_message('closing', msg);\n", + " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -5396,14 +6314,20 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -5484,7 +6408,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -5544,7 +6468,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 23, "metadata": { "collapsed": false }, @@ -5624,7 +6548,6 @@ " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", - " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", @@ -5955,11 +6878,13 @@ " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", + " fig.waiting = false;\n", " return;\n", " }\n", "\n", @@ -6008,6 +6933,19 @@ " return {\"x\": x, \"y\": y};\n", "};\n", "\n", + "/*\n", + " * return a copy of an object with only non-object keys\n", + " * we need this to avoid circular references\n", + " * http://stackoverflow.com/a/24161582/3208463\n", + " */\n", + "function simpleKeys (original) {\n", + " return Object.keys(original).reduce(function (obj, key) {\n", + " if (typeof original[key] !== 'object')\n", + " obj[key] = original[key]\n", + " return obj;\n", + " }, {});\n", + "}\n", + "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", @@ -6021,7 +6959,8 @@ " var y = canvas_pos.y;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", - " step: event.step});\n", + " step: event.step,\n", + " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", @@ -6061,7 +7000,8 @@ "\n", " this._key_event_extra(event, name);\n", "\n", - " this.send_message(name, {key: value});\n", + " this.send_message(name, {key: value,\n", + " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", @@ -6078,7 +7018,7 @@ "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", - "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", + "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", @@ -6137,6 +7077,8 @@ "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", + " fig.root.unbind('remove')\n", + "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", @@ -6144,8 +7086,12 @@ " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", - " fig.send_message('closing', {});\n", - " fig.ws.close()\n", + " fig.close_ws(fig, msg);\n", + "}\n", + "\n", + "mpl.figure.prototype.close_ws = function(fig, msg){\n", + " fig.send_message('closing', msg);\n", + " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", @@ -6200,14 +7146,20 @@ "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", - " var button = $('');\n", + " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", - " button.mouseover('Close figure', toolbar_mouse_event);\n", + " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", + "mpl.figure.prototype._root_extra_style = function(el){\n", + " var fig = this\n", + " el.on(\"remove\", function(){\n", + "\tfig.close_ws(fig, {});\n", + " });\n", + "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", @@ -6288,7 +7240,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -6363,7 +7315,7 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 25, "metadata": { "collapsed": false }, @@ -6518,7 +7470,7 @@ "" ] }, - "execution_count": 26, + "execution_count": 25, "metadata": {}, "output_type": "execute_result" } @@ -6531,43 +7483,25 @@ "css_file = './numericalmoocstyle.css'\n", "HTML(open(css_file, \"r\").read())" ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] } ], "metadata": { "kernelspec": { - "display_name": "Python 2", + "display_name": "Python 3", "language": "python", - "name": "python2" + "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", - "version": 2 + "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", - "pygments_lexer": "ipython2", - "version": "2.7.11" + "pygments_lexer": "ipython3", + "version": "3.5.0" } }, "nbformat": 4, From 8f93e8d9ccdeb7021088a63c662cadf57f49d831 Mon Sep 17 00:00:00 2001 From: rbds Date: Mon, 14 Dec 2015 11:48:32 -0500 Subject: [PATCH 58/61] editing --- randy_schur/Double_Pendulum_Problem.ipynb | 26 ++++----- ...on_Techniques_for_Mechanical_Systems.ipynb | 55 ++++++++++--------- 2 files changed, 40 insertions(+), 41 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index b361b53..ca0d848 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -63,7 +63,7 @@ "\n", "So the Lagrangian quantity becomes:\n", "$$\\begin{equation*}\n", - "L = \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2+m_2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+(m_1+m_2)l_1gcos\\theta_1 + m_2l_2gcos\\theta_2\n", + "L = \\frac{1}{2}(m_1+m_2)l_1^2\\dot\\theta_1^2 + \\frac{1}{2}m_2l_2^2\\dot{\\theta_2}^2+m_2l_1l_2\\dot\\theta_1\\dot\\theta_2cos(\\theta_2-\\theta_1)+(m_1+m_2)l_1gcos\\theta_1 + m_2l_2gcos\\theta_2\n", "\\end{equation*}$$\n" ] }, @@ -85,7 +85,7 @@ "$$\\begin{eqnarray*}\n", "\\frac{\\partial L}{\\partial \\dot{\\theta_1}} &=& (m_1+m_2)l_1^2\\dot\\theta_1+m_2l_1l_2\\dot\\theta_2cos(\\theta_2-\\theta_1) \\\\\n", "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{\\theta_1}}\\right) &=& (m_1+m_2)l_1^2\\ddot\\theta_1+m_2l_1l_2\\ddot\\theta_2cos(\\theta_2-\\theta_1)-m_2l_1l_2\\dot\\theta_2^2sin(\\theta_2-\\theta_1)+m_2l_1l_2\\dot\\theta_1\\dot\\theta_2sin(\\theta_2-\\theta_1) \\\\\n", - "-\\frac{\\partial L}{\\partial \\theta_1} &=& -m_2l_1l_2\\dot\\theta_1\\dot\\theta_2sin(\\theta_2-\\theta_1)-(m_1+m_2)gl_1sin\\theta_1\n", + "-\\frac{\\partial L}{\\partial \\theta_1} &=& -m_2l_1l_2\\dot\\theta_1\\dot\\theta_2sin(\\theta_2-\\theta_1)+(m_1+m_2)gl_1sin\\theta_1\n", "\\end{eqnarray*}$$\n", "\n", "We can add together the second and third equations to find the first equation of motion. Some of these terms will cancel out, and we are left with a final equation of motion:\n", @@ -97,7 +97,7 @@ "We will also need the equation of motion for $\\theta_2$. This can be derived using the same process. It should work out to the following (check this by yourself!):\n", "\n", "$$\\begin{equation*}\n", - "0 = m_2l_2^2\\ddot\\theta_2+m_2l_1l_2\\ddot\\theta_1cos(\\theta_2-\\theta_1) - m_2l_2l_1\\dot\\theta_1^2sin(\\theta_2-\\theta_1)+ l_2m_2gsin\\theta_2\n", + "0 = m_2l_2^2\\ddot\\theta_2+m_2l_1l_2\\ddot\\theta_1cos(\\theta_2-\\theta_1) + m_2l_2l_1\\dot\\theta_1^2sin(\\theta_2-\\theta_1)+ l_2m_2gsin\\theta_2\n", "\\end{equation*}$$" ] }, @@ -132,7 +132,7 @@ "V = \n", "\\left[\\begin{array}{c}\n", "0 & -m_2l_1l_2sin(\\theta_2 - \\theta_1)\\\\\n", - " -m_2l_1l_2sin(\\theta_2 - \\theta_1) &0\n", + " m_2l_1l_2sin(\\theta_2 - \\theta_1) &0\n", "\\end{array}\\right]\n", "\\end{equation*}\n", "\n", @@ -233,7 +233,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 58, "metadata": { "collapsed": false }, @@ -255,7 +255,7 @@ " x4 = u[3]\n", " \n", " M = numpy.array([[(m1+m2)*l1**2, m2*l1*l2*cos(x1-x2)],[m2*l1*l2*cos(x1-x2), m2*l2**2]])\n", - " V = numpy.array([[0, -m2*l1*l2*sin(x1-x2)],[-m2*l1*l2*sin(x1-x2), 0]])\n", + " V = numpy.array([[0, m2*l1*l2*sin(x1-x2)],[-m2*l1*l2*sin(x1-x2), 0]])\n", " G = numpy.array([[(m1+m2)*g*l1*sin(x1)],[m2*l2*g*sin(x2)]])\n", " qdd = numpy.linalg.inv(M).dot(V.dot(numpy.array([[x3],[x4]])) +G) #- 2*numpy.array([[x3],[x4]])) \n", " RHS = numpy.array([x3, x4, qdd[0], qdd[1]])\n", @@ -264,7 +264,7 @@ }, { "cell_type": "code", - "execution_count": 39, + "execution_count": 59, "metadata": { "collapsed": false }, @@ -301,7 +301,7 @@ }, { "cell_type": "code", - "execution_count": 40, + "execution_count": 60, "metadata": { "collapsed": true }, @@ -318,15 +318,13 @@ " Returns: \n", " x - array of values at next time step.\n", " \"\"\"\n", - " #print(f(u))\n", - " #print(u)\n", " \n", " return u + dt*f(u)" ] }, { "cell_type": "code", - "execution_count": 41, + "execution_count": 61, "metadata": { "collapsed": false }, @@ -354,7 +352,7 @@ }, { "cell_type": "code", - "execution_count": 42, + "execution_count": 62, "metadata": { "collapsed": false }, @@ -1126,7 +1124,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1902,7 +1900,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" diff --git a/randy_schur/Integration_Techniques_for_Mechanical_Systems.ipynb b/randy_schur/Integration_Techniques_for_Mechanical_Systems.ipynb index aecb47b..56e6b02 100644 --- a/randy_schur/Integration_Techniques_for_Mechanical_Systems.ipynb +++ b/randy_schur/Integration_Techniques_for_Mechanical_Systems.ipynb @@ -138,7 +138,7 @@ "k_1 &=& f(t_n, x_n)\\\\\n", "k_2 &=& f(t_n+\\frac{h}{2}, x_n+\\frac{h}{2}k_1)\\\\\n", "k_3 &=& f(t_n+\\frac{h}{2}, x_n+\\frac{h}{2}k_2)\\\\\n", - "k_4 &=& f(t_n+h, y_n+hk_3)\\\\\n", + "k_4 &=& f(t_n+h, x_n+hk_3)\\\\\n", "t_{n+1} &=& t_n + h\n", "\\end{eqnarray}$$\n", "\n", @@ -183,11 +183,11 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Let's now take a look at symplectic integration methods [7]. We'll start with a first order method - symplectic Euler's method. This uses the equations:\n", + "Let's now take a look at symplectic integration methods [6]. We'll start with a first order method - symplectic Euler's method. This uses the equations:\n", "\n", "$$\\begin{eqnarray*}\n", "p_{n+1} = p_n - h \\frac{\\partial H}{\\partial q} \\Bigr|_{p_{n+1},q_n}\\\\\n", - "q_{n+1} = q_n + \\frac{h}{2} \\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1},q_n}\n", + "q_{n+1} = q_n + h\\frac{\\partial H}{\\partial p} \\Bigr|_{p_{n+1},q_n}\n", "\\end{eqnarray*}$$\n", "\n", "We can then plug in the partial derivatives and integrate the system. This system will do a better job of conserving energy than Euler's method (we'll investigate this later), $\\textit{but only to a truncation error.}$ Again, we can do better than first-order. Let's try a second-order symplectic scheme, also called Verlet integration. We'll have the same equations of motion, but this time a different set of integration equations [6]:\n", @@ -222,7 +222,7 @@ }, { "cell_type": "code", - "execution_count": 1, + "execution_count": 31, "metadata": { "collapsed": false }, @@ -243,7 +243,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 32, "metadata": { "collapsed": false }, @@ -261,7 +261,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 33, "metadata": { "collapsed": true }, @@ -284,7 +284,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 34, "metadata": { "collapsed": true }, @@ -308,7 +308,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 35, "metadata": { "collapsed": true }, @@ -335,7 +335,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 36, "metadata": { "collapsed": true }, @@ -359,7 +359,7 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 37, "metadata": { "collapsed": true }, @@ -388,7 +388,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 38, "metadata": { "collapsed": false }, @@ -408,7 +408,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 39, "metadata": { "collapsed": false }, @@ -1256,7 +1256,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 40, "metadata": { "collapsed": true }, @@ -1279,7 +1279,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 41, "metadata": { "collapsed": false }, @@ -1302,7 +1302,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 42, "metadata": { "collapsed": false }, @@ -2074,7 +2074,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2095,6 +2095,7 @@ "pyplot.loglog(dt_values, RK4_error_values, 'co-', label='RK4')\n", "pyplot.loglog(dt_values, Verlet_error_values, 'go-', label='Symplectic')\n", "pyplot.axis('equal')\n", + "pyplot.title('Error in Simple Harmonic Motion Integration')\n", "pyplot.legend();" ] }, @@ -2102,7 +2103,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Is this what you would expect to see based on the plots of position above? What are the sources of this error? We have the usual suspects: truncation error, discretization error, possible precision errors. Don't believe that these are significant? The discretization is the only difference between the first two plots above, and with a smaller time step the response from Runge-Kutta and Euler are greatly improved. To exame precision errors, take a look at the following screenshot, which shows the plots above, built with identical code on two different computers. The responses on the left are from a 64-bit computer, and the responses on the right are from a 32-bit computer. Look closely at the phase shift in the system response. That's a huge difference! Even the precision of the machine can have a significant impact on numerical integration.\n", + "Is this what you would expect to see based on the plots of position above? What are the sources of this error? We have the usual suspects: truncation error, discretization error, possible precision errors. Don't believe that these are significant? To examine precision errors, take a look at the following screenshot, which shows the plots above, built with identical code on two different computers. The responses on the right are from a 64-bit computer, and the responses on the left are from a 32-bit computer. Look closely at the phase shift in the system response. That's a huge difference! Even the precision of the machine can have a significant impact on numerical integration.\n", "\n", "![Image](figures/combined_response.png)\n" ] @@ -2116,7 +2117,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 43, "metadata": { "collapsed": false }, @@ -2199,7 +2200,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 44, "metadata": { "collapsed": false }, @@ -3012,7 +3013,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 45, "metadata": { "collapsed": true }, @@ -3035,7 +3036,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 46, "metadata": { "collapsed": false }, @@ -3070,7 +3071,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 47, "metadata": { "collapsed": false }, @@ -4722,7 +4723,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 48, "metadata": { "collapsed": true }, @@ -4746,7 +4747,7 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 49, "metadata": { "collapsed": true }, @@ -4775,7 +4776,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 50, "metadata": { "collapsed": true }, @@ -4806,7 +4807,7 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 51, "metadata": { "collapsed": false }, @@ -5636,7 +5637,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 52, "metadata": { "collapsed": false }, From 1ced5e8b1dd564e8efa4884da780f43188def14a Mon Sep 17 00:00:00 2001 From: rbds Date: Mon, 14 Dec 2015 13:58:24 -0500 Subject: [PATCH 59/61] energy plots in DP for RK4 --- randy_schur/Double_Pendulum_Problem.ipynb | 72 ++++++++--------------- 1 file changed, 26 insertions(+), 46 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index ca0d848..62a614e 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -212,7 +212,7 @@ }, { "cell_type": "code", - "execution_count": 1, + "execution_count": 10, "metadata": { "collapsed": false }, @@ -233,7 +233,7 @@ }, { "cell_type": "code", - "execution_count": 58, + "execution_count": 11, "metadata": { "collapsed": false }, @@ -264,7 +264,7 @@ }, { "cell_type": "code", - "execution_count": 59, + "execution_count": 12, "metadata": { "collapsed": false }, @@ -301,7 +301,7 @@ }, { "cell_type": "code", - "execution_count": 60, + "execution_count": 13, "metadata": { "collapsed": true }, @@ -324,7 +324,7 @@ }, { "cell_type": "code", - "execution_count": 61, + "execution_count": 14, "metadata": { "collapsed": false }, @@ -352,7 +352,7 @@ }, { "cell_type": "code", - "execution_count": 62, + "execution_count": 15, "metadata": { "collapsed": false }, @@ -1955,7 +1955,7 @@ }, { "cell_type": "code", - "execution_count": 43, + "execution_count": 17, "metadata": { "collapsed": false }, @@ -1988,7 +1988,7 @@ }, { "cell_type": "code", - "execution_count": 44, + "execution_count": 18, "metadata": { "collapsed": false }, @@ -2805,23 +2805,23 @@ }, { "cell_type": "code", - "execution_count": 45, + "execution_count": 32, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def get_Energy(u):\n", - " \"\"\" Calculates total energy in the system at given time step for a simple harmonic oscillator\n", + " \"\"\" Calculates total energy in the system at given time step for a double pendulum\n", " Parameters:\n", " u - state of system [pos, vel]\n", " Returns:\n", " total energy in system.\n", " \"\"\"\n", " q1 = u[0]\n", - " p1 = u[1]\n", + " w1 = u[1]\n", " q2 = u[2]\n", - " p2 = u[3]\n", + " w2 = u[3]\n", " #make sure q1 and q1 are between 0 and 2pi. \n", " while q1 > 2*pi:\n", " q1 -= 2*pi\n", @@ -2834,13 +2834,13 @@ " \n", " \n", " V = -m1*g*l1*cos(q1) - m2*g*l1*cos(q1)+l2*cos(q2)\n", - " T = p1**2/(2*I1)+m2*p1**2/(2*m1*I1) + p1*p2/(m1*l1*l2)*cos(q2-q1)+p2**2/(2*I2)\n", - " return T+V\n" + " T = 1/2*(m1+m2)*l1**2*w1**2+1/2*m2*2*l1*l2*w1*w2*cos(q1-q2)+l2**2*w2**2\n", + " return T+V" ] }, { "cell_type": "code", - "execution_count": 46, + "execution_count": 33, "metadata": { "collapsed": false }, @@ -3612,7 +3612,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -3620,16 +3620,6 @@ }, "metadata": {}, "output_type": "display_data" - }, - { - "data": { - "text/plain": [ - "" - ] - }, - "execution_count": 46, - "metadata": {}, - "output_type": "execute_result" } ], "source": [ @@ -3643,7 +3633,7 @@ "#print(u[0,:])\n", "for i in range(N):\n", " Verlet_energy[i] = get_Energy(u[:,i])\n", - " #RK4_energy[i] = get_Energy(x4_dp[i,:])\n", + " RK4_energy[i] = get_Energy(x4_dp[i,:])\n", "\n", " \n", "\n", @@ -3655,19 +3645,9 @@ "pyplot.plot(t, Verlet_energy, lw=2, label='Verlet', color='b');\n", "pyplot.xlabel('time (s)')\n", "pyplot.ylabel('Energy')\n", - "\n", - "## use to plot RK4 energy on same graph.\n", - "#for tl in ax1.get_yticklabels():\n", - "# tl.set_color('b')\n", - "\n", - "#ax2 = ax1.twinx();\n", - "#ax2.plot(t, RK4_energy, 'r', lw=2, label='RK4')\n", - "#for tl in ax2.get_yticklabels():\n", - "# tl.set_color('r')\n", - "\n", - "#ax1.set_xlabel('time (s)')\n", - "#ax1.set_ylabel('Verlet', color='b')\n", - "#ax2.set_ylabel('RK4', color='r')\n" + "pyplot.plot(t, RK4_energy, 'r', lw=2, label='RK4')\n", + "pyplot.legend();\n", + "\n" ] }, { @@ -3690,7 +3670,7 @@ }, { "cell_type": "code", - "execution_count": 47, + "execution_count": 24, "metadata": { "collapsed": false }, @@ -3724,7 +3704,7 @@ }, { "cell_type": "code", - "execution_count": 48, + "execution_count": 26, "metadata": { "collapsed": false }, @@ -4507,7 +4487,7 @@ } ], "source": [ - "v = poincare(xv_dp, 1, 2*pi, 0, 2) #use states from RK4 integration\n", + "v = poincare(xv_dp, 1, 2*pi, 0, 2) #use states from Verlet integration\n", "pyplot.figure(figsize=(10,8));\n", "pyplot.grid(True);\n", "pyplot.xlabel(r'$\\theta_2$', fontsize=18);\n", @@ -4547,7 +4527,7 @@ }, { "cell_type": "code", - "execution_count": 49, + "execution_count": 27, "metadata": { "collapsed": false }, @@ -4579,7 +4559,7 @@ }, { "cell_type": "code", - "execution_count": 50, + "execution_count": 28, "metadata": { "collapsed": false }, @@ -6184,7 +6164,7 @@ }, { "cell_type": "code", - "execution_count": 51, + "execution_count": 29, "metadata": { "collapsed": false }, From ed144e237dd346a3f5d7e4a154174b2e39bf6de3 Mon Sep 17 00:00:00 2001 From: rbds Date: Tue, 15 Dec 2015 14:16:10 -0500 Subject: [PATCH 60/61] Update README.md --- README.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/README.md b/README.md index 9cab8a3..00b523b 100644 --- a/README.md +++ b/README.md @@ -1,4 +1,6 @@ # assignment-bank-2015 -For submission of class projects at GW, Fall 2015 +For submission of class project at GW, Fall 2015 + +This assignment examines numerical integration techniques using mechanical systems. In the notebook Integration_Techniques_for_Mechanical_Systems, I start by introducing a simple harmonic oscillator and three different types of integrators (Euler, Runge-Kutta, and Verlet). Next I introduce damping and look at how the system behavior changes for each method of integration. The Double_Pendulum_Problem notebook looks at the double pendulum problem using Runge-Kutta and Verlet integration. The code in this repository is under Creative Commons Attribution license CC-BY 4.0, code under MIT license © 2015 Randall Schur From b41a3aa040e2e17f29fbb335796654036f19f195 Mon Sep 17 00:00:00 2001 From: rbds Date: Mon, 28 Dec 2015 07:33:47 -0500 Subject: [PATCH 61/61] address pull request comments --- randy_schur/Double_Pendulum_Problem.ipynb | 34 +++--- ...on_Techniques_for_Mechanical_Systems.ipynb | 112 ++++++++++++------ 2 files changed, 91 insertions(+), 55 deletions(-) diff --git a/randy_schur/Double_Pendulum_Problem.ipynb b/randy_schur/Double_Pendulum_Problem.ipynb index 62a614e..e9c8071 100644 --- a/randy_schur/Double_Pendulum_Problem.ipynb +++ b/randy_schur/Double_Pendulum_Problem.ipynb @@ -212,7 +212,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 1, "metadata": { "collapsed": false }, @@ -233,7 +233,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 2, "metadata": { "collapsed": false }, @@ -264,7 +264,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 3, "metadata": { "collapsed": false }, @@ -301,7 +301,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 4, "metadata": { "collapsed": true }, @@ -324,7 +324,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 5, "metadata": { "collapsed": false }, @@ -352,7 +352,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 6, "metadata": { "collapsed": false }, @@ -1955,7 +1955,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 7, "metadata": { "collapsed": false }, @@ -1988,7 +1988,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 8, "metadata": { "collapsed": false }, @@ -2805,7 +2805,7 @@ }, { "cell_type": "code", - "execution_count": 32, + "execution_count": 9, "metadata": { "collapsed": true }, @@ -2840,7 +2840,7 @@ }, { "cell_type": "code", - "execution_count": 33, + "execution_count": 10, "metadata": { "collapsed": false }, @@ -3670,7 +3670,7 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 11, "metadata": { "collapsed": false }, @@ -3704,7 +3704,7 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 12, "metadata": { "collapsed": false }, @@ -4527,7 +4527,7 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 13, "metadata": { "collapsed": false }, @@ -4559,7 +4559,7 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 14, "metadata": { "collapsed": false }, @@ -6164,7 +6164,7 @@ }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 15, "metadata": { "collapsed": false }, @@ -6987,7 +6987,7 @@ }, { "cell_type": "code", - "execution_count": 52, + "execution_count": 16, "metadata": { "collapsed": false }, @@ -7142,7 +7142,7 @@ "" ] }, - "execution_count": 52, + "execution_count": 16, "metadata": {}, "output_type": "execute_result" } diff --git a/randy_schur/Integration_Techniques_for_Mechanical_Systems.ipynb b/randy_schur/Integration_Techniques_for_Mechanical_Systems.ipynb index 56e6b02..0ee3e41 100644 --- a/randy_schur/Integration_Techniques_for_Mechanical_Systems.ipynb +++ b/randy_schur/Integration_Techniques_for_Mechanical_Systems.ipynb @@ -142,7 +142,7 @@ "t_{n+1} &=& t_n + h\n", "\\end{eqnarray}$$\n", "\n", - "For more information on the Runge-Kutta method, including the derivation and explanation of coefficients, see (cite)\n" + "For more information on the Runge-Kutta method, including the derivation and explanation of coefficients, see [9].\n" ] }, { @@ -222,7 +222,7 @@ }, { "cell_type": "code", - "execution_count": 31, + "execution_count": 2, "metadata": { "collapsed": false }, @@ -243,7 +243,7 @@ }, { "cell_type": "code", - "execution_count": 32, + "execution_count": 3, "metadata": { "collapsed": false }, @@ -261,7 +261,7 @@ }, { "cell_type": "code", - "execution_count": 33, + "execution_count": 4, "metadata": { "collapsed": true }, @@ -284,14 +284,14 @@ }, { "cell_type": "code", - "execution_count": 34, + "execution_count": 5, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def RK2(u, f, dt):\n", - " \"\"\"Runge Kutta fourth order integration method\n", + " \"\"\"Runge Kutta second order integration method\n", " \n", " Parameters:\n", " u - state of the system at time t\n", @@ -308,7 +308,7 @@ }, { "cell_type": "code", - "execution_count": 35, + "execution_count": 6, "metadata": { "collapsed": true }, @@ -335,7 +335,7 @@ }, { "cell_type": "code", - "execution_count": 36, + "execution_count": 7, "metadata": { "collapsed": true }, @@ -359,7 +359,7 @@ }, { "cell_type": "code", - "execution_count": 37, + "execution_count": 8, "metadata": { "collapsed": true }, @@ -388,7 +388,7 @@ }, { "cell_type": "code", - "execution_count": 38, + "execution_count": 9, "metadata": { "collapsed": false }, @@ -408,7 +408,7 @@ }, { "cell_type": "code", - "execution_count": 39, + "execution_count": 10, "metadata": { "collapsed": false }, @@ -1180,7 +1180,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -1236,7 +1236,7 @@ "pyplot.plot(t, x4[:,0], 'c', lw=2, label='RK4');\n", "pyplot.plot(t, xv[:,0], 'g', lw=2, label='Verlet');\n", "pyplot.plot(t, x_analytical, 'k--', lw=2, label='analytical');\n", - "pyplot.legend();" + "pyplot.legend(loc='upper left');" ] }, { @@ -1256,7 +1256,7 @@ }, { "cell_type": "code", - "execution_count": 40, + "execution_count": 11, "metadata": { "collapsed": true }, @@ -1279,7 +1279,7 @@ }, { "cell_type": "code", - "execution_count": 41, + "execution_count": 12, "metadata": { "collapsed": false }, @@ -1302,7 +1302,7 @@ }, { "cell_type": "code", - "execution_count": 42, + "execution_count": 13, "metadata": { "collapsed": false }, @@ -2117,7 +2117,7 @@ }, { "cell_type": "code", - "execution_count": 43, + "execution_count": 14, "metadata": { "collapsed": false }, @@ -2200,7 +2200,7 @@ }, { "cell_type": "code", - "execution_count": 44, + "execution_count": 17, "metadata": { "collapsed": false }, @@ -2972,7 +2972,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -2980,6 +2980,16 @@ }, "metadata": {}, "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 17, + "metadata": {}, + "output_type": "execute_result" } ], "source": [ @@ -2994,7 +3004,8 @@ "pyplot.loglog(dt_values[:-1], diffgrid1[:-1], label='Euler', color='b', ls='--', lw=2, marker='o');\n", "pyplot.loglog(dt_values[:-1], diffgrid2[:-1], label='RK2', color = 'r', ls='--', lw=2, marker='o');\n", "pyplot.loglog(dt_values[:-1], diffgrid4[:-1], label='RK4', color = 'c', ls='--', lw=2, marker='o');\n", - "pyplot.loglog(dt_values[:-1], diffgridv[:-1], label='Verlet', color = 'g', ls='--', lw=2, marker='o');" + "pyplot.loglog(dt_values[:-1], diffgridv[:-1], label='Verlet', color = 'g', ls='--', lw=2, marker='o');\n", + "pyplot.legend(loc='upper left')" ] }, { @@ -3008,12 +3019,12 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "In the plots of position using Euler integration, we notice that the spring is stretched farther and farther each oscillation. This means it has more and more potential energy. With no outside forces adding energy to the system and no friction to remove energy, Conservation of Energy says that value should be constant! Let's take a look at total energy in the system. Since we derived equations of motion from the Lagrangian, this should be easy! All we need to do is calculate kinetic and potential energy at each time step, and we already have the information we need to do this. " + "In the plots of position using Euler integration, we notice that the spring is stretched farther and farther each oscillation. This means it has more and more potential energy. With no outside forces adding energy to the system and no friction to remove energy, conservation of energy says that value should be constant! Let's take a look at total energy in the system. Since we derived equations of motion from the Lagrangian, this should be easy! All we need to do is calculate kinetic and potential energy at each time step, and we already have the information we need to do this. " ] }, { "cell_type": "code", - "execution_count": 45, + "execution_count": 19, "metadata": { "collapsed": true }, @@ -3036,7 +3047,7 @@ }, { "cell_type": "code", - "execution_count": 46, + "execution_count": 20, "metadata": { "collapsed": false }, @@ -3066,12 +3077,12 @@ " RK2_energy[i] = get_Energy(x2[i,:])\n", " RK4_energy[i] = get_Energy(x4[i,:])\n", " Verlet_energy[i] = get_Energy(numpy.array([xv[i,0], Verlet_vel[i]]))\n", - " analytical_energy[i] = get_Energy(anayltical_state[i,:]) " + " analytical_energy[i] = get_Energy(anayltical_state[i,:])" ] }, { "cell_type": "code", - "execution_count": 47, + "execution_count": 22, "metadata": { "collapsed": false }, @@ -4619,7 +4630,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -4639,7 +4650,7 @@ "pyplot.plot(t, Euler_energy, 'b-', lw=2, label='Euler');\n", "\n", "#Plot other methods on a separate figure, so we can actually see behavior\n", - "pyplot.figure(figsize=(8,6));\n", + "pyplot.figure(figsize=(10,6));\n", "pyplot.grid(True);\n", "pyplot.xlabel(r't', fontsize=18);\n", "pyplot.ylabel(r'Total energy in system (J)', fontsize=18);\n", @@ -4649,7 +4660,7 @@ "pyplot.plot(t, RK4_energy, 'c-', lw=2, label='RK4');\n", "pyplot.plot(t, Verlet_energy, 'g-', lw=2, label='Verlet');\n", "pyplot.plot(t, analytical_energy, 'k--', lw=2, label='analytical');\n", - "pyplot.legend();" + "pyplot.legend(loc='upper left');" ] }, { @@ -4718,12 +4729,12 @@ "\\begin{bmatrix} \\dot q \\\\ \\dot p \\end{bmatrix} = \\begin{bmatrix} -\\frac{\\partial H}{\\partial p} + Q\\\\ \\frac{\\partial H}{\\partial q} \\end{bmatrix} = \\begin{bmatrix} -\\frac{p}{m} - \\frac{c}{m}p \\\\ kq \\end{bmatrix}\n", "\\end{equation*}$$\n", "\n", - "From here, the we can use the same Verlet integration method as before. If this explanation has too much hand-waving for you, please see [7]." + "From here, we can use the same Verlet integration method as before. If this explanation has too much hand-waving for you, please see [7]." ] }, { "cell_type": "code", - "execution_count": 48, + "execution_count": 23, "metadata": { "collapsed": true }, @@ -4747,7 +4758,7 @@ }, { "cell_type": "code", - "execution_count": 49, + "execution_count": 24, "metadata": { "collapsed": true }, @@ -4776,7 +4787,7 @@ }, { "cell_type": "code", - "execution_count": 50, + "execution_count": 25, "metadata": { "collapsed": true }, @@ -4807,7 +4818,7 @@ }, { "cell_type": "code", - "execution_count": 51, + "execution_count": 26, "metadata": { "collapsed": false }, @@ -5629,6 +5640,8 @@ "cell_type": "markdown", "metadata": {}, "source": [ + "*Note: In the figure above, the energy in the system according to Verlet, RK2, and RK4 is so close that the lines are appearing on top of each other. If you zoom in really closely you can see the differences. \n", + "\n", "How does each integrator perform compared to the case of simple harmonic motion? Just by looking at the system response, it seems like the error is actually better here. Why is that? It is important to note here that the integrators perform differently for different system models. This is part of why we study numerical methods! There isn't a single answer to which scheme to use, it is dependent on the characteristics of both the scheme and the system.\n", "\n", "The effect of the damper is to remove energy from the system. So, in this case the damping term is helping to remove some energy from the system that the integration adds. As time goes to infinity, these solutions will converge, and the Euler and Runge-Kutta methods are stable. We can again look at the energy in the system according to each method. \n", @@ -7287,7 +7300,18 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "In this case, it appears that Verlet, RK2, and RK4 give almost identical results, which are close but not exactly the same as the analytical solution. So for the damped harmonic oscillator, there is less of an advantage to using a more complex integration scheme. Try modifying the code above to make a phase portrait for the simple harmonic oscillator. We should see a circle (or an ellipse) form, which is called a limit cycle. Which integrators create this behavior?" + "In this case, it appears that Verlet, RK2, and RK4 give almost identical results, which are close but not exactly the same as the analytical solution. So for the damped harmonic oscillator, there is less of an advantage to using a more complex integration scheme. Try modifying the code above to make a phase portrait for the simple harmonic oscillator. We should see a circle (or an ellipse) form because the system behavior repeats itself. This is called a limit cycle. Which integrators create this behavior?" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Conclusion\n", + "\n", + "In this notebook we discussed four different numerical integrators and applied these to a mechanical system. Even for the relatively straightforward case of harmonic motion, we found that different integrators can have very different behavior. In order to choose the best method (not too complex, but still accurate) we investigated how each method affects conservation of energy. We also found that not every situation is the same - the integrators performed differently for damped and undamped systems.\n", + "\n", + "In the next notebook, we will take these numerical integrators and apply them to the more complex double pendulum system. We'll use some of the same tools to analyze the behavior of each numerical integrator. However, there is no closed form solution for the double pendulum, so we'll use what we learned in this notebook to try and determine the effectiveness of each method on the more complex system." ] }, { @@ -7296,7 +7320,7 @@ "collapsed": false }, "source": [ - "Sources: \n", + "Sources and additional resources: \n", "\n", "[1] http://scienceworld.wolfram.com/physics/DoublePendulum.html\n", "\n", @@ -7311,12 +7335,15 @@ "[6] http://www.unige.ch/~hairer/poly_geoint/week2.pdf\n", "\n", "[7] http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1994CeMDA..60..409T&defaultprint=YES&filetype=.pdf\n", - "\n" + "\n", + "[8] http://www.ujfi.fei.stuba.sk/fyzika-ballo/Runge-Kutta.pdf\n", + "\n", + "[9] http://web.mit.edu/10.001/Web/Course_Notes/Differential_Equations_Notes/node5.html" ] }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 18, "metadata": { "collapsed": false }, @@ -7471,7 +7498,7 @@ "" ] }, - "execution_count": 25, + "execution_count": 18, "metadata": {}, "output_type": "execute_result" } @@ -7484,6 +7511,15 @@ "css_file = './numericalmoocstyle.css'\n", "HTML(open(css_file, \"r\").read())" ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] } ], "metadata": {