Three-dimensional vibration analysis of thick hyperboloidal shells of revolution

Abstract

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, hyperboloidal shells of revolution. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u r , u θ , and u z in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the hyperboloidal shells are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies 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 hyperboloidal shells of revolution. Numerical results are tabulated for 18 configurations of completely free hyperboloidal shells of revolution having two different shell thickness ratios, three variant axis ratios, and three types of shell height ratios. Poisson's ratio ( ν) is fixed at 0.3. Comparisons are made among the frequencies for these hyperboloidal shells and ones which are cylindrical or nearly cylindrical (small meridional curvature). The method is applicable to thin hyperboloidal shells, as well as thick and very thick ones.