Solution of the problem of nonaxisymmetric free vibrations of piezoceramic hollow cylinders with axial polarization

Abstract

We consider the nonaxisymmetric problem of free vibrations of hollow piezoceramic cylinders for some types of boundary conditions at their end faces. The piezoceramics is polarized in the axial direction. The cylinder lateral surfaces are free from external loads and covered with thin electrodes that are shortcircuited. The initial three-dimensional problem is reduced to two-dimensional after separation of variables and representation of the components of the displacement vector and electrostatic potential in the form of standing waves in the circumferential direction. The method of solution of the problem is based on the joint use of the spline-collocation method along the longitudinal coordinate and the method of discrete orthogonalization combined with step-by-step search by the radial coordinate. We also present some results of numerical analysis of the behavior of a cylinder made of PZT 4 piezoceramics over a wide range of changes in the geometrical characteristics of the cylinder. Piezoceramic structural elements of cylindrical shape are widely used in radio electronics, various automatic devices, computing machinery, and measuring tools. A high efficiency of the transformation of electron energy into mechanical and the inverse process, together with comparatively small sizes, enable one to use such materials in nanotechnologies. The solution of dynamic problems for thick-walled elements as three-dimensional problems of the theory of elasticity is connected with significant difficulties, which are due to the complexity of the system of initial partial differential equations as well as by the necessity to satisfy boundary conditions on the surfaces confining the body. These difficulties grow substantially in the cases where the fields under study are coupled and the piezoelectric materials are anisotropic [1, 2, 8, 9]. Therefore, it becomes necessary to develop efficient numerical– analytic approaches for the solution of these classes of problems. During recent years, for the solution of such problems of computational mathematics, mathematical physics, and mechanics, spline functions are widely used. This is explained by the advantages of the apparatus of spline approximations as compared with other methods. Among the main advantages of this apparatus, we should mention