Frequency analysis of a uniform ring perturbed by point masses and springs

Abstract

Perturbation expansions of solutions for a uniform, thin, linear elastic ring perturbed by point masses and radial massless springs are developed. The perturbation locations divide the ring into uniform segments so a variational formulation is used to determine the boundary conditions that must be satisfied between adjoining segments. The motion of each segment can be represented as a weighted sum of the eigenfunctions for the uniform thin ring so when the boundary conditions are enforced, the resulting algebraic relations are expanded as a function of the perturbation parameter (the perturbation mass normalized by the ring mass). A series of algebraic problems are sequentially solved to yield perturbation expansions for the modal frequencies and eigenmodes. Single-mass, dual-mass, and mass-spring case studies are considered. The perturbation results show excellent agreement with finite element analysis of a thin ring for mass perturbations up to 15% of the nominal ring mass. The results are also compared to Rayleigh-Ritz analysis.