Forced oscillation and resonance

Lesson 7M

An important example of non-homogeneous differential equations are forced oscillations. An example of a forced oscillation equation with damping is \[ mu'' + \gamma u' + ku = R \sin(\omega t).\] The solutions to this differential equation will have two parts: One part involving \(e^{-rt}\), and one part involving sines and cosines with the same frequency as the forcing function (\(\omega\), that is).

In the long run, the only part that is not near zero is part that comes from the forcing function. This is often called the steady state solution, or the response.

The textbook has a good discussion of this in Example 1 of section 3.8 and the page that follows the example.

Resonance

Resonance is the phenomenon that a small forcing function (that is, a forcing function with small amplitude) can cause a relatively large response (again, measured in terms of amplitude) when the forcing frequency is near the natural frequency of an oscillating system.

Here is an animation of how a spring oscillation changes as you adjust the forcing frequency.

And here is one famous example of resonance causing problems.