Vibrations of a Torus with Hollow Elliptical Cross Section Having Variable Thickness

Abstract

The natural frequencies of toroidal shells of revolution with a hollow elliptical cross section and variable thickness are determined by the Ritz method from a three-dimensional theory, whereas traditional shell theories are mathematically two-dimensional. Instead of ordinary algebraic polynomials, the Legendre polynomials, which are mathematically orthonormal, are used as admissible functions. The present analysis is based upon the circular cylindrical coordinates, whereas toroidal coordinates have been used in general. The potential and kinetic energies of the torus are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. Convergence to a four-digit exactitude is demonstrated for the first five frequencies of the torus. Comparisons are made between the frequencies from the present three-dimensional method, a two-dimensional thin-shell theory, and thin- and thick-ring theories. The present method is applicable to very thick shells as well as thin shells.