Natural frequencies of nonaxisymmetric vibration of prolate spheroidal shells

Abstract

The equations of motion for nonaxisymmetric vibration of prolate spheroidal shells of constant thickness were derived using Hamilton’s principle. The thin shell theory used in this derivation includes three displacements and two changes of curvature. The effects of membrane, bending, shear deformations, and rotatory inertias are included in this theory. The resulting five partial differential equations are self-adjoint and positive definite. The nonaxisymmetric modal solutions are expanded in a doubly infinite series of comparison functions. These include associated Legendre functions in terms of the prolate spheroidal angular coordinate, and circular functions of the circumferential coordinate. The natural frequencies and the mode shapes were obtained by the Galerkin method for each circumferential mode. Numerical results were obtained for several shell thickness-to-length ratios ranging from 0.005 to 0.1, and for various diameter-to-length ratios, including the limiting case of a spherical shell. [Work supported by Office of Naval Research and the Navy/ASEE Summer Faculty Program.]