Exercise 5.7

Consider the differential equation of a harmonic oscillator given by a damped mass-spring-damper system (eq. 4, Chapter 3):

Suppose we have a spring that stretches 20 cm in length caused by a weight of 98 Newton. Thus N/m and kg. The damping is chosen such that holds, that is sub-critically damped.

This differential equation is preferably written as set of first order differential equations.
Suppose and , then we get the following equations:

a)
Plot the solution with initial conditions and on a time-interval that is large enough and plot this solution in the phase plane as well.
What happens if you choose other initial conditions?
b)
Plot for the super-critically damped system, , the solution. Use the same initial conditions.
c)
See if you can find initial conditions in the case of critical damping, , for which the solution at least possesses one zero (at least a crossing of the equilibrium position).

Esteur 2010-03-22