Vibration of Toroidal Shells with Hollow Circular Cross-Sections

This paper presents a three-dimensional (3D) analysis for the natural frequencies of completely free, toroidal shells of revolution with hollow circular cross-sections by the Ritz method. The displacement components [Formula: see text], [Formula: see text], and [Formula: see text] in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in [Formula: see text], and of the ordinary algebraic simple polynomials in the [Formula: see text] and [Formula: see text] directions. The potential (strain) and kinetic energies of the torus are formulated, and the upper bound values of the frequencies are obtained by a minimization procedure. As the degree of the polynomials increases, the frequencies computed converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the torus. Comparisons are made between the frequencies from the present 3D method, a 3D finite element method, experimental methods, and thin and thick ring theories.