On the free transverse vibrations of a uniform circular disc clamped at its centre; and on the effects of rotation

Abstract

1. A thin circular disc, entirely unconstrained, can vibrate transversely in modes which may be characterized by any number of nodal concentric circles, crossed by uniformly spaced nodal diameters. The natural frequencies, for a disc of uniform thickness, have been calculated by Kirchhoff, and a recent investigation by Prof. Lamb and the present author has shown the amount by which they are increased when the disc rotates at any given speed about its axis. This extension of Kirchhoff’s analysis was suggested by published accounts of failures in turbine discs, the circumstances of which indicated that they had their origin in forced vibrations, intensified by resonance. It was realised that the results would be only indirectly applicable to design, mainly because in practice discs are constructed with curved profiles (the thickness increasing from the rim to the boss), but also because the calculated frequencies, in some instances, will be seriously affected by the “clamping” which is introduced when the disc is attached at its centre to a shaft and rotor of considerable inertia.