MA2051 - Ordinary Differential Equations
D96 Project #2A - Nonlinear Shocks 




   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.





    

  1.   To start, describe the limiting behavior of the standard model

    
How does the amplitude of the limiting solution depend on ? 
How does it depend on the initial data?

    
Compare your results with the graphs on page 280 and the 
formula given in Exercise 12 on page  282 in the text.

    

    

  2. 
One of the best ways to analyze the solution curves for 
a second-order equation is to 
graph them in the  (position, velocity) plane.
Derive a system of first-order differential equations for
 and  and  plot solution curves in the
 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?

    

    

  3.  
Mathematicians and engineers at the Renault Automobile Company 
have studied a new type of shock absorber which controls the  
damping coefficient.  Their new shock absorber can increase the 
damping coefficient whenever the mass (i.e., the car) is returning 
toward equilibrium.  In the mathematical 
model, the damping coefficient is a discontinuous function of 
both  and .

    
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?  Do 
you think that Renault should pursue its development of this new shock 
absorber?

    

    

  4. 
Design your own shock absorber and analyze the resulting
system of differential equations.   Can yo do better then
Renault?

    






  

    

      © 1996 by Will Brother.
      All rights Reserved. File last modified on April 25, 1996.