Electroelastic Vibrations of Heterogeneous Piezoceramic Hollow Spheres

Abstract

The axisymmetric and nonaxisymmetric problems of natural and forced vibrations of a hollow sphere made of a functionally gradient piezoelectric material theory are considered based on 3D electroelasticity. The properties of the material vary along the radial coordinate. The external surface of the sphere is free of tractions and is either insulated or short-circuited by electrodes for the analysis of the natural vibrations of the system. Two cases of forced vibrations are investigated: an electric excitation—when an electrostatic potential with an alternating sign is applied to the external surface of the spheres; and mechanical excitation—when pressure with an alternating sign is applied to its external surface. Separation of variables and series of the components of the mechanical and electric displacements were used. The electric potential and of the stress tensor is expressed in terms of spherical functions. As a result, the initially three-dimensional problem described by the partial differential equations with variable coefficients is reduced to a boundary-value problem for the systems of the ordinary differential equations. A boundary-value eigenvalue problem for the case of natural vibrations is arrived at in the process. It is solved by discrete-orthogonalization methods combined with a step-by-step search method. The influence of the geometric, mechanical, and electric parameters on the frequency spectrum in the case of nonaxisymmetric natural vibrations of an inhomogeneous piezoceramic thick-walled sphere was analyzed. An inhomogeneous boundary-value problem is obtained for the case of forced vibrations. This problem is solved by a stable discrete-orthogonalization method. The influence of the geometric and electric parameters on the kinematic (mechanical displacement and electrostatic potential) and dynamic (mechanical stress and electric displacement) characteristics was analyzed. Different variants of polarized piezoceramic materials are considered. Significant attention is paid to the validation of the reliability of the results obtained by numerical calculations.