Three-dimensional vibration analysis of thick shells of revolution

A 3D method of analysis is presented for determining the foe vibration freq uencies and mode shapes of hollow bodies of revolution (i.e., thick shells) , not limited to straight-line generators or constant thickness. The middle surface of the shell may have arbitrary curvatures, and the wall thickness may vary arbitrarily. Displacement components u(phi), u(z), and u(theta) i n the meridional, normal, and circumferential directions. respectively, are taken to be sinusoidal in time, periodic in theta, and algebraic polynomia ls in the phi and z directions. Potential (strain) and kinetic energies of the entire body are formulated, and upper-bound values of the frequencies a re obtained by minimizing the frequencies. As the degree of the polynomials are increased, frequencies converge to the exact values. Novel numerical r esults are presented for two types of thick conical shells and thick spheri cal shell segments having linear thickness variations and completely free b oundaries. Convergence to four-digit exactitude is demonstrated for the fir st five frequencies of both types of shells. The method is applicable to th in shells, as well as thick and very thick ones.