10 - Parametric excitation

Summary

It has long been known that a resonant system can be set into oscillation by periodically varying the parameters. A very extensive analysis was given by Rayleigh, who drew attention to analogies with rather different physical processes – the perturbation of a planetary orbit by another planet having a quite different period in some near-integral relationship, and the propagation of waves in periodically stratified media. This latter example relates to a phenomenon of much greater physical interest now than in Rayleigh's day since it is basic to Bragg reflection of X-rays and to the motion of electrons in solids. Nevertheless, these are analogies only in the sense that the same type of differential equation is involved in all of them, as well as in the theory of vibrating elliptical membranes. It can hardly be claimed that the solution of one of these problems adds anything to the intuitive understanding of another. A mathematical framework, however, is well worth having, for these are not easy problems. The examples presented here earn a place in their own right as showing yet another aspect of the variety of oscillatory phenomena in nature. At the same time, there are important technological applications in the form of parametric amplifiers, as will be discussed in outline at the end of the chapter.

Probably the easiest way to demonstrate parametric excitation is to set up a simple pendulum whose length can be slightly varied at twice the natural frequency.