Vibration Analysis of Toroidal Shells with Hollow Circular Cross-Section Having Variable Thickness

A three-dimensional (3D) Ritz method of analysis is presented for determining the free vibration frequencies of completely free, toroidal shells of revolution with hollow circular cross-section having variable thickness. Displacement components ur, u, and uz in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in , and ordinary algebraic polynomials in the r and z directions. Strain and kinetic energies of the torus are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies 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 Ritz method, a 3D finite-element method, experimental methods, and thin and thick ring theories.