Free vibration analysis of a fluid-filled functionally graded spherical shell subjected to internal pressure

An analytical solution is developed to study the free vibration of a thin functionally graded (FG) spherical shell under initial internal static pressure based on Love's first approximation theory. A coupled vibro-acoustic analytical model is presented for spherical shells filled with compressible nonviscous fluid. The non-homogenous material properties are assumed to be graded according to a power-law distribution of the constituents through the shell thickness. By introducing a stress function, the reformulated coupled equations of motion of FG spherical shells under the influence of initial stresses are obtained. The wave equation is used to model the internal acoustic domain. The boundary conditions of continuity of fluid and shell velocities, as well as the normal pressure acting on the internal surface of the shell from the fluid are imposed. The frequency equation of the coupled system is obtained utilizing modal expansion along with the orthogonality properties of the mode shapes. Exact solutions for the free vibration of pressurized empty and fluid-filled shells are obtained in terms of products of trigonometric and Legendre functions in a spherical coordinate system. Numerical results are validated with the results of simple cases available in the literature as well as finite element modeling. Effects of different parameters including material constants, geometry, initial pressure and vibro-acoustic coupling on natural frequencies are studied. The presented analytical solution is an attempt to describe the vibrational behavior of FG pressurized fluid-filled spherical shells.