Three-dimensional vibrations of solid cones with and without an axial circular cylindrical hole

Abstract

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of solid cones with and without an axial circular cylindrical hole, having arbitrary constraints on their boundaries. The method is based upon the 3-D dynamic equations of elasticity. Displacement components u(r), u(0), and u(z) in the radial, circumferential, and axial directiom, respectively, are taken to be sinusoidal in time, periodic in theta, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the cones are formulated, the Ritz method is used to solve the eigenvalue problem, 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. Novel numerical results are presented for solid cones with and without an axial circular cylindrical hole. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the cones. (C) 2004 Published by Elsevier Ltd.