Nonaxisymmetric vibrations of radially polarized hollow cylinders made of functionally gradient piezoelectric materials

摘要

The nonaxisymmetric problem of natural vibrations of radially polarized hollow cylinders made of functionally gradient piezoelectric materials is solved. The properties of the material change continuously along a radial coordinate according to an exponential law. The lateral surfaces of the cylinder are free of external tractions and short circuited by electrodes. After separation of variables and representation of the components of the displacement vector in the form of standing circumferential waves, the initially three-dimensional problem is reduced to a two-dimensional partial differential equation problem. By using the method of spline-collocations with respect to a longitudinal coordinate, this two-dimensional problem is reduced to a boundary-value problem for the eigenvalues expressed in terms of ordinary differential equations. This problem is solved by the stable discrete-orthogonalization technique in combination with a step-by-step search method with respect to the radial coordinate. Results were obtained numerically and subsequently analyzed in this paper.