Three-dimensional vibration analysis of joined thick conical — Cylindrical shells of revolution with variable thickness

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies of joined thick conical-cylindrical shells of revolution with variable thickness. 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 ur, uθ, and uz in the radial, circumferential, and axial directions, respectively, are taken to be periodic in θ and in time, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the joined 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. Natural frequencies are presented for different boundary conditions. Comparisons are made between the frequencies from the present 3-D Ritz method and 2-D thin shell theories by previous researchers.