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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