Vibrations of tori with hollow elliptical cross-section from a three-dimensional theory

Natural frequencies of a toroidal shells of revolution with hollow elliptical cross-section are determined by the Ritz method from a three-dimensional (3-D) theory while traditional shell theories are mathematically two-dimensional (2-D). The Legendre polynomials, which are mathematically orthonomal, are used instead of ordinary algebraic polynomials as admissible functions. The present analysis is based upon the circular cylindrical coordinates while the toroidal coordinates have been used in general. Potential and kinetic energies of the torus are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the torus. Comparisons are made between the frequencies from the present 3-D method, a 2-D thin shell theory, and thin and thick ring theories. The present method is applicable to very thick toroidal shells as well as thin ones.