Free Axisymmetric and Nonaxisymmetric Vibrations of Hollow Homogeneous and Inhomogeneous Piezoceramic Cylinders of Finite Length with Different Polarization

Abstract

Various approaches to the solution of linear problems of elasticity and electroelasticity of anisotropic inhomogeneous finite-length cylinders based on discrete–continuous methods and three-dimensional formulations are presented. The advantage of these method consists in the reduction of the partial differential equations of the considered problems to the associated one-dimensional problems (the spline-collocation method) and exact satisfaction of the boundary conditions. The approach leads to practically exact solutions of boundary-value problems and eigenvalue problems described by a system of ordinary differential equations with variable coefficients (by the discrete-orthogonalization method). Axisymmetric and nonaxisymmetric problems of natural vibrations of hollow inhomogeneous elastic cylinders and cylinders with piezoelectric properties based on 3D elasticity and electroelasticity are considered. The properties of the material vary along a radial coordinate. We consider two types of inhomogeneous materials: when the properties of the material are piecewise constant (layered structures with metal and dielectric layers) and when they vary continuously (functionally gradient and functionally gradient piezoelectric materials FGM and FGPM). The external surface of the cylinder is free of tractions and either insulated or short-circuited by electrodes. After separation of variables and representation of the components of the mechanical displacement vector and electric potential 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 the longitudinal coordinate, this two-dimensional problem is reduced to a one-dimensional eigenvalue problem (described by 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. A nontraditional approach to solving problems of the above class is proposed. Different variants of polarized piezoceramic materials are considered. The effect of variation in mechanical and electrical parameters through thickness, and the influence of boundary conditions on natural frequencies and vibration modes of the finite-length cylinders with inhomogeneous elastic and piezoelectric properties is analyzed. Significant attention is paid to the validation of the reliability of the results obtained by numerical calculations.