You have seen in lecture and exercises how to model a spring-mass system with damping and external forcing. One application for the model is an automobile suspension system, where the forcing term is the road (with or without frost heaves) and shock absorbers provide the damping.
In this project, you will study the behavior of the solution with different types of damping terms. Recognize that once you allow the damping coefficient to depend on the solution, the equation is no longer linear and most analytic solution methods (such as characteristic equations and undetermined coefficients) are useless. You have no choice but to study the system through numerical methods.
One of the best ways to picture the solution curves for this model is in the (x,v) = (position, velocity) plane. Derive a system of first-order differential equations for x(t) and and plot solution curves in the (x,v) plane. Pay particular attention to the limiting behavior of the solutions. How does the limit depend on the initial data? How does it depend on other parameters in the model?
Do some numerical experiments with the Renault shock absorber. Explain in detail how you program the damping coefficient in your equation. Take for the forcing function and describe the limiting behavior of the solution for a range of values. Does the new shock absorber give a ``smoother'' ride than the original shock absorber? Justify your recommendations.
Final Version Due: Thursday, March 6
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