### Damped harmonic motion

No energy is lost during SHM. In reality, energy is dissipated---this is known as damping. Damped harmonic motion arises when energy loss is included.

A natural model for damping is to assume that the resistive force is opposite and proportional to the velocity. Then the equation of motion is:

F = - k x - b v = m a

Writing v and a in terms of time derivatives of the displacement, the equation of motion is:
 d2x dt2
= -
 k m
x -
 b m
 dx dt

The solution to this equation is:   x(t) = A e-bt/2m cos(ωt+φ)
where
ω = (
 k m
-
 b2 4m2
) ½ ( ωo2 -
 b2 4m2
) ½

There are 4 different behaviors that depend on the damping constant b:

1. No damping, b=0:  The motion reduces to SHM.
2. Underdamping, 0 < b < 2mω0 Decaying oscillations. A larger value of b leads to faster decay of oscillations.

In the underdamped regime, the energy decays exponentially in time:

E(t) = ½ k x2max = ½ k A e-bt/m ≡ E0 e-t/τ

where τ is the time constant of the damping.

3. Overdamping, 2mω0< b:  Very slow monotonic decay.
4. Critical damping, 2mω0= b:  Quickest monotonic decay. Critical damping is the principle underlying shock absorbers.