Damping arises when we have dissipative forces which cause an oscillation to die out.
A simple case is when the damping force is proportional to the velocity of the oscillating object. That is, we have the force , where is the damping constant
This corresponds to the following Differential Equation

Which, for small damping, is solved by
An oscillator achieves critical damping when the angular frequency is . That is, when satisfies

or

Here the object no longer oscillates and returns to its equilibrium position when displaced and released
An oscillator is overdamped when . Here there is no oscillation but the system returns to equilibrium more slowly than with critical damping
An oscillator is underdamped when . Here the system oscillates with steadily decreasing amplitude.
In damped oscillation, the damping force is non-conservative. We have

This derives from the expression for energy in simple harmonic motion.

Resonance

A driving force is a force which is applied to a (damped) oscillator to maintain the amplitude.
The force can be applied with an angular frequency . The motion is called a driven oscillation. We distinguish from the natural angular frequency obtained from when we do not apply the driving force.

The force is applied periodically say
- The amplitude of the driven oscillator is given by
- When , has a maximum near . The height of the curve is proportional to .
- When , we have a constant displacement
Resonance - pertains to the peaking of the amplitude at driving frequencies close to the natural frequency of the system.

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