Driven harmonic motion

What happens when an oscillating system also experiences a sinusoidal external force? The response depends on the relation between the frequency of the external force and the natural oscillation frequency of the system.

A good example is a kid on a playground swing. You naturally match the frequency of your pushes to the frequency of the swing.

Forcing at the natural frequency is known as resonance. Here the amplitude can be substantial, even when the driving force has a small amplitude.

This resonance is naturally found by plotting the amplitude of the oscillator as a function of the driving frequency (see transparency).

In addition to amplitude versus driving frequency, the phase relation between the driving force and the oscillator has interesting features: