Final Project_Xiaofan Liu #17
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My final project
My final project. A little correction from the previous one.
My final project
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Nice job. One thing that you should definitely take more care with is that you are using images from an un-cited source. It looks like they are figures from a textbook. You must add an image credit to these, with a note to their copyright! You might even add a note about their inclusion under fair use, because this is an educational work. (In general, though, one cannot reuse copyrighted images without permission of the rights holder!) Notice that you are loading the NumPy library twice! Pick a way to load it, and stick with it. We prefer Mechanics of vibrations is not fresh in my memory, so I followed your derivations like your regular reader. In doing so, I was confused at times. The expression x r=gamma theta, for example, does not make sense to me. Something seems off there. If the angle theta is small, then the arc A–A' would be given by r times theta. (By the way, you forgot to define r when you first use it—it only appears on the second figure, but not defined in the text.) And when looking at a thin cross-sectional slice of the shaft, the shear strain would be given by shear angle (gamma) times dx. Maybe you got some variables turned around here? About the stability constraint: where did you get that the CFL condition for this equation discretized with central differences was �sigma<0.5. And then why would you choose a time step that is 10 times smaller than your stability limit?! In fact, the wave equation is subject to the typical CFL condition of sigma<1 for explicit methods. To get your numerical solution started, you used the Taylor series of theta, the known initial value of the angular velocity, and a finite-difference approximation of the second derivative. You missed the opportunity of making a statement here about the order of accuracy of this starting scheme. It's an important detail because in many starting schemes one gets stuck with a lower-order accuracy than the overall solution. Is that the case here? Now, your solution with the explicit method looks strange ... and, as you can see, the solution is different than the one you get with the implicit method. Something is wrong ... I suspect you have a bug. Finally, the title could be more informative about the topic of the notebook if it said that it refers to torsional vibrations. Maybe just "Torsional vibrations of a shaft" would do. A couple of LaTeX typos: Typos and Style: Based on these information—>this information "with what have gotten."—> with what we have? |
My final project