Flexural vibrations of anisotropic thin rotating rings

Abstract

We present a rigorous analysis of the in-plane flexural vibrations of a thin rotating circular ring. The ring is made from an anisotropic material with cubic symmetry, as in (100) single-crystalline silicon. The ring is assumed to be sufficiently thin, such that it can be considered as an inextensible Euler-Bernoulli beam. In this study, both the natural frequencies and their related mode shapes are analytically derived using an asymptotic method, for the modes numbered n = 2, 3 and 4. We show that due to a rate of rotation, these modes exhibit a precession-like response, which was previously shown to occur in isotropic rings. However, for rotating rings that are made from an anisotropic material, we show that the even-ordered modes n = 2 and n = 4, exhibit a ‘breathing’ phenomenon in the precessing mode. In deriving the asymptotic approximation of the frequencies and mode shapes, we adopt a technique used in quantum mechanics, and modify it appropriately for the problem at hand. The theoretical predictions are verified by comparing them to finite element simulations, showing good agreement. This work is relevant to the emerging technology of vibrating ring gyroscopes that are made from (100) single-crystalline silicon.