Properties of the Damped Oscillator

There are several properties of the damped oscillator that are important to know.

• The amplitude damps exponentially as time advances. After a certain amount of time, the amplitude is halved. After the same amount of time, it is halved again.
• The frequency is shifted.
• The oscillator can be (under)damped, critically damped, or overdamped.

The oscillator is underdamped if $\omega'$ is real, which will be true if $4km > b^2$. It will undergo true oscillations, eventually approaching zero amplitude due to damping.

The oscillator is critically dampled if $\omega'$ is zero, when $4km = b^2$. The oscillator will not oscillate - it will go to zero exponentially in the shortest possible time.

The oscillator is overdamped if $\omega'$ is imaginary, which will be true if $4km < b^2$. In this case $\alpha$ is entirely real and has a component that damps very slowly. The amplitude goes to zero exponentially as before, but over a longer (possibly much longer) time and does not oscillate through zero at all.

A car's shock absorbers should be barely underdamped. If the car "bounces" once and then damps to zero when you push down on a fender and suddenly release it, the shocks are good. If it bounces three of four time the shocks are too underdamped and dangerous as you could lose control after a big bump. If it doesn't bounce up and back down at all at all and instead slowly oozes back up to level from below, it is overdamped and dangerous, as a succession of sharp bumps could leave your shocks still compressed and unable to absorb the impack.

Tall buildings also have dampers to keep them from swaying in a strong wind. Houses are build with lots of dampers in them to keep them quiet. Fully understanding damped (and eventually driven) oscillation is essential to many sciences as well as both mechanical and electrical engineering.