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George Mason UniversityZakaria Tarik ZerhouniSource: Home > Project 3 > ConclusionMath 447: Numerical Analysis
To conclude we will quickly summarize the results of our investigation. Steps one and two addressed the construction of a simulation of a forced, damped pendulum by means of a system of differential equations and the behaviour that could be attained by changing the forcing parameter \(A\). Periodic trajectories were discovered for an adequately small \(A\). However, based on the step size of the computation, a different trajectory could be achieved. Clearly one of these is incorrect, and thus the choice of step size if crucial for the accuracy of the results. Once \(A\) was large enough, chaotic motion was produced in the pendulum. We went on in steps three and four to simulate a damped pendulum with an oscillating pivot, and discovered on what intervals the parameter \(A\) could use parametric resonance to create or undo stability. Depending on any choice in time interval and the criteria by which one chooses to define stability, it was possible by interval halving to compute very accurately where an inverted or hanging pendulum can be stabilized or otherwise destabilized by the choice of the parameter \(A\). Finally, step five simulated a double pendulum and provided some animation of its behaviour. Stability was only achieved in this case by relying on the force of gravity to overwhelm and bound the movement of the double pendulum, which can easily enter into completely bewildering behaviour. What was discovered in the analysis of the modeled pendulum was that visual observation of the animation was in fact very difficult and time consuming. In many ways, utilizing non visual methods of checking the change in velocity for our stored computational results were more easily interpretable than the output of video. All are encouraged to learn more about the solutions of differential equations and the pendulum model by investigating these computer simulations themselves. The MatLab programming code provided with the results and animation of the pendulum models may be used to reproduce the results or even to discover new ones. Many thanks to Dr. Timothy Sauer and our classmates for their invaluable guidance and aid.
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