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 x²_max = 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.