Free nonaxisymmetric vibrations of radially polarized hollow piezoceramic cylinders of finite length

Abstract

The three-dimensional problem of free nonaxisymmetric vibrations of hollow piezoceramic cylinders with axial polarization is considered. An efficient numerical analytic method to solve boundary-value problems is proposed. The original three-dimensional problem of electroelasticity is reduced to a two-dimensional problem by representing the displacement components as standing circumferential waves. Spline collocation with respect to the axial coordinate is used to reduce this two-dimensional problem to an eigenvalue boundary-value problem with respect to the radial coordinate. This problem is solved by the stable discrete-orthogonalization and incremental-search methods. Numerical results are presented and the natural frequencies of the cylinders are analyzed in a wide range of their geometric characteristics Keywords: free vibrations, three-dimensional problem of electroelasticity, hollow piezoceramic cylinder, spline-collocation Introduction. Cylindrical piezoelectric elements are widely used in acoustoelectronics. Therefore, it is important to study the dynamic processes in piezoceramic cylinders. Solving dynamic problems for thick-walled elements as three-dimensional problems of elasticity encounters considerable difficulties associated with the complexity of the original partial differential equations and the necessity of satisfying the boundary conditions on the faces of the body. These difficulties are even more severe in the case of coupled fields and anisotropic materials. Despite the great number of relevant publications, there are only few studies on the vibrations of piezoceramic cylinders of finite length based on the three-dimensional theory of elasticity [4-3, 11, 12]. As is shown in [7-10], spline approximation can successfully be used to study the mechanical behavior of various plates and shells. In [1-3], this method was employed to study the axisymmetric longitudinal and torsional vibrations of piezoceramic cylinders. In the present paper, we will solve the more general problem of the nonaxisymmetric vibrations of piezoceramic cylinders, the above-mentioned axisymmetric problem being its special case.1. Problem Formulation. We will examine the frequency spectrum of free nonaxisymmetric vibrations of a hollow piezoceramic cylinder with radial polarization. The cylinder is clamped at the ends and its lateral surfaces are free from external loads and are covered by thin short-circuited electrodes.The closed system of equations for this problem in cylindrical coordinates ( , , ) r z q includes