MA2051 - Ordinary Differential Equations
Project 2a - Nonlinear Shocks - C97 






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

     displaymath21 
    
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  tex2html_wrap_inline27  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?

  2.  
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 x(t) and v(t).

    
Do some numerical experiments with the Renault shock absorber. 
Explain in detail how you program the damping coefficient in  your equation. 
Take  tex2html_wrap_inline35  for the forcing function and 
describe the limiting behavior of the solution for a range of  tex2html_wrap_inline37  values. 
Does the new shock absorber give a ``smoother'' ride than the 
original shock absorber?  Justify your recommendations.

  3. 
It is not clear that ``limiting behavior'' is the right way to
measure the quality of a shock absorber.  Can you come up with other
criteria better suited to the real application?

    




Final Version Due: Thursday, March 6



 




  

    

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All rights Reserved. File last modified on February 27, 1997