Natural frequencies of prolate spheroidal shells of constant thickness

The general displacement-equilibrium equations, which include the effects of transverse shear and rotary inertia, have been derived for a prolate spheroidal shell of constant thickness subject to an harmonically time-varying, arbitrary spatially distributed force normal to the shell surface. The approximate solutions for the two non-torsional displacements of the shell middle surface and the non-torsional rotation of the shell cross-section are obtained by using Galerkin's variational method. Numerical results are presented for the seven lowest axisymmetric natural frequencies of the shell. When 15 term solutions are used for both thick and thin shells, which have eccentricities that vary from 0·13 to 0·89, the approximate natural frequencies for the first seven flexural modes are all found to converage to within less than 8% of the final values given, with most converging to within less than 2%. Good agreement with other published results is obtained for the approximate natural frequencies of a thin prolate spheroidal shell and for the exact natural frequencies of a thick spherical shell. Additional results are presented for the natural frequencies of moderately thick shells as a function of shell eccentricity, mode number and shell thickness.