On the flexural in-plane vibrations of a thin circular ring made from an elastic solid with cubic crystalline symmetry

We present a rigorous analysis of the in-plane flexural vibrations of a thin circular ring, made from anisotropic material with cubic symmetry (e.g. single-crystalline silicon). We do not make any assumptions on the mode shapes of vibration, but rather derive them as part of the solution. Rings in which the cross-section has a finite radius of gyration (i.e. with a finite radial thickness) are considered. In our analysis we present an analytic proof of the degeneracy of odd-ordered modes. The results are compared with previous studies and numerical simulations.