Damped Simple Harmonic Motion
Up to now we have discussed systems in which the force is proportional to the displacement, but
pointed in an opposite direction. In these cases, the motion of the system can be described by
simple harmonic motion. However, if we include the friction force, the motion will not be simple
harmonic anymore. The system will still oscillate, but its amplitude will slowly decrease over
time.
Suppose the total force acting on a mass is not only proportional to its displacement, but also to
its velocity. The total force can be represented in the following way
In this formula, b is called the
damping constant
. Substituting the expression for the force in
terms of the acceleration we obtain the following differential equation
The general solution of this differential equation will have the form
Substituting this expression in the differential equation we obtain
This equation can be rewritten as
and the solutions for [omega] are
Substituting this in the expression for x(t) we obtain

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