On the natural frequencies of vibrating systems

Abstract

The basic problem of Vibration Theory is to calculate for a given system the modes and associated frequencies of its “normal” free oscilla­tions. These are components into which the whole motion can be resolved when the system vibrates freely, and through small distances, about its position of equilibrium. Each one is wholly independent of every other, and has its own (in general) distinct phase and frequency, which are common to all parts of the system. Relatively to one another the amplitudes of different parts are invariant, but the phase and magnitude of a normal oscillation are not (in theory) restricted. Exact calculation is difficult even when attention is confined to the gravest (i. e. lowest) natural frequency, and on that account great value attaches to a theorem of Lord Rayleigh whereby a close estimate of this frequency can be based on a comparatively rough assumption in regard to the corresponding mode. It is known that the result will err, if at all, in the direction of over-estimation : if then by equally simple calculations it were possible to obtain a second figure close to it and known to be an under-estimate , such knowledge would for practical purposes be very nearly as useful as an exact result.