Free Vibrations of Thin Isotropic Oblate Spheroidal Shells

Abstract

The free-vibration problem of a thin, isotropic, oblate spheroidal shell was solved by Galerkin's method. Membrane theory and harmonic axisymmetric motion were assumed to derive the differential equations of motion. The equations of motion lead to two ordinary differential equations with variable coefficients. The two differential equations can be reduced to one ordinary second-order differential equation with variable coefficients. This eigenvalue problem is solved by Galerkin's method. It was shown that Galerkin's solution for the oblate spheroid yields the exact solution for the sphere as the eccentricity of the oblate spheroid goes to zero. It is shown that two sets of frequencies exist for the oblate spheroidal shell. Galerkin's method converges rapidly for those of the lower set. After the frequencies are determined, the tangential displacements are obtained as solutions of a homogeneous system of equations. For numerical illustration, an oblate spheroidal shell with eccentricity equal to 0.688 was considered.