Three-Dimensional Free Vibration Analysis of a Homogeneous Transradially Isotropic Thermoelastic Sphere

In the present paper, an exact three-dimensional vibration analysis of a transradially isotropic, thermoelastic solid sphere subjected to stress-free, thermally insulated, or isothermal boundary conditions has been carried out. Nondimensional basic governing equations of motion and heat conduction for the considered thermoelastic sphere are uncoupled and simplified by using Helmholtz decomposition theorem. By using a spherical wave solution, a system of governing partial differential equations is further reduced to a coupled system of three ordinary differential equations in radial coordinate in addition to uncoupled equation for toroidal motion. Matrix Fröbenious method of extended power series is used to investigate motion along radial coordinate from the coupled system of equations. Secular equations for the existence of various types of possible modes of vibrations in the sphere are derived in the compact form by employing boundary conditions. Special cases of spheroidal and toroidal modes of vibrations of a solid sphere have also been deduced and discussed. It is observed that the toroidal motion remains independent of thermal variations as expected and spheroidal modes are in general affected by thermal variations. Finally, the numerical solution of the secular equation for spheroidal motion (⁠$S$-modes) is carried out to compute lowest frequency and dissipation factor of different modes with MATLAB programming for zinc and cobalt materials. Computer simulated results have been presented graphically. The analyses may find applications in aerospace, navigation, and other industries where spherical structures are in frequent use.