Lateral Vibrations and Whirling Speeds

Abstract

In the types of vibration considered in previous chapters it has always been possible to assume, without introducing undue errors, that the system consisted of one or more rigid bodies connected by massless elastic elements. Following from this it became relatively straightforward to formulate the equations of motion in terms of the chosen displacement co-ordinates, and to solve for the natural frequencies according to the number of degrees of freedom. For various reasons this approach cannot be applied directly to the vibrations of beams and shafts. Both mass and elasticity are generally ‘distributed’ along the length and mathematically there are then infinite degrees of freedom. Even when ‘point’ masses predominate and the mass of the beam is neglected it is not possible (except in the single-mass case) to write down the restoring force on any mass in terms of its displacement at any instant. In consequence, ‘approximate’ methods have been developed for general use, ‘exact’ analysis being applied only to a few standard cases. These approximate methods are usually limited to the determination of the fundamental (that is, first mode) frequency which, as in most multi-degree-of-freedom vibrations, is of prime practical importance.