Free Vibration of Thin Spherical Shells

Abstract

This research introduces a new approach to analytically derive the differential equations of motion of a thin spherical shell. The approach presented is used to obtain an expression for the relationship between the transverse and surface displacements of the shell. This relationship, which is more explicit than the one that can be obtained through use of the Airy stress function, is used to uncouple the surface and normal displacements in the spatial differential equation for transverse motion. The associated Legendre polynomials are utilized to obtain analytical solutions for the resulting spatial differential equation. The spatial solutions are found to exactly satisfy the boundary conditions for the simply supported and the clamped hemispherical shell. The results to the equations of motion indicate that the eigenfrequencies of the thin spherical shell are independent of the azimuthal coordinate. As a result, there are several mode shapes for each eigenfrequency. The results also indicate that the effects of midsurface tensions are more significant than bending at low mode numbers but become negligible as the mode number increases.