The accuracy of Rayleigh’s method of calculating the natural frequencies of vibrating systems

In this paper a theorem of Kato (1949) which provides upper and lower bounds for the eigenvalues of a Hermitian operator is modified and generalized so as to give upper and lower bounds for the normal frequencies of oscillation of a conservative dynamical system. The method given here is directly applicable to a system specified by generalized co-ordinates with both elastic and inertial couplings. It can be applied to any one of the normal modes of vibration of the system. The bounds obtained are much closer than those given by Rayleigh’s comparison theorems in which the inertia or elasticity of the system is changed, and they are in fact the ‘best possible’ bounds. The principles of the computation of upper and lower bounds is explained in this paper and will be illustrated by some numerical examples in a subsequent paper.