Vibration Analysis of Homogeneous Transradially Isotropic Generalized Thermoelastic Spheres

Abstract

The exact free vibration analysis of stress free or rigidly fixed, thermally insulated/isothermal, transradially (spherically) isotropic thermoelastic solid sphere has been presented in context of nonclassical thermoelasticity. The transradially isotropic is also frequently referred as spherically isotropic in the literature. The basic governing equations of linear generalized thermoelastic, transradially isotropic, sphere have been uncoupled and simplified with the help of Helmholtz decomposition theorem. The formal solution of the coupled system of partial differential equations has been obtained by employing matrix Fröbenius method of extended series. The secular equations for the existence of possible modes of vibrations in the sphere have been derived by employing boundary conditions. The special cases of spheroidal (S-mode) and toroidal (T-mode) vibrations have also been deduced and discussed. It is found that the toroidal motion gets decoupled from the spheroidal one and remains independent of thermal variations and thermal relaxation time. In order to illustrate the analytical development, the numerical solution of secular equations for spheroidal motion (S-mode) is carried out with respect of magnesium and solid helium spheres. The lowest frequency and damping factor of vibrational modes have been computed with the help of MATLAB programming and the results are presented graphically. The study may find applications in aerospace, navigation, geophysics tribology, and other industries where spherical structures are in frequent use.