Method of averaging

From Wikipedia, the free encyclopedia

In the study of dynamical systems, the method of averaging is used to study certain time-varying systems by analyzing easier, time-invariant systems obtained by averaging the original system.

[edit] Definition

Consider a general, nonlinear dynamical system

$\dot{x} = \epsilon f( t, x , \epsilon )$

where $f (t, x)$ is periodic in $t$ with period $T$ . The corresponding averaged system is

$\dot{x}^{a} = \epsilon \frac{1}{T}\int_{0}^{T}f(\tau,x,0) d\tau$

The primary benefit of averaging is that it is usually easier to analyze equilibria (and their stability) of time-invariant (autonomous) systems.

[edit] Example

Consider a simple pendulum whose point of suspension is vibrated vertically by a small amplitude, high frequency signal (this is usually known as dithering). The equation of motion for such a pendulum is given by

$m(l\ddot{\theta} - a\omega^2 \sin \omega t \sin \theta) = -mg \sin \theta - k(l\dot{\theta} + a\omega \cos \omega t \sin \theta)$