12 - Coupled vibrators

Summary

A resonant system acted upon by an oscillatory force presents a straightforward enough problem if it is passive and linear, especially if the force is applied by some prime mover that is uninfluenced by the response it excites. Such problems are the subject matter of chapter 6, while nonlinear passive systems, as discussed in chapter 9, are more complicated to handle. If the prime mover is influenced by the response, additional complexities enter, and this chapter treats of some of these. As is to be expected, linear systems present the least difficulty, and we shall begin with the behaviour of two coupled resonant systems, each of which may be thought of as driving the other and being perturbed by the reaction of the other back on it. Examples have already appeared earlier, as for instance the coupled pendulums discussed in chapter 2, and the coupled resonant lines in chapter 7. In both cases we noted a very general characteristic of such systems, that even if they are tuned to the same frequency before being coupled, they do not vibrate at this frequency when coupled, but have resonances which move progressively further from the original frequency as the coupling is strengthened. It would perhaps be logical, having considered this problem, to proceed to coupled, passive, non-linear vibrators; but these are so difficult that we shall leave them alone. It is not quite so hard to derive useful results for the behaviour of self-maintained oscillators when perturbed either by the injection of a steady signal or by being coupled to a similar oscillator.