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A spline-collocation approach is proposed for studying the natural vibrations of orthotropic rectangular plates of variable thickness. The approach is based on the spline-approximation method and the method of discrete orthogonalization coupled with the step-by-step search method. The study is carried out within the framework of the classical and refined theories of plates. The dynamic response of the plates is studied depending on the variation of the plate thickness, the mechanical parameters, and the type of boundary conditions.

For determining the dynamic characteristics of free vibrations of circular unclosed cylindrical shells of variable thickness in two coordinate directions, we have used the spline-collocation method together with the method of discrete orthogonalization. The problem has been solved within the framework of the refined Timoshenko-Mindlin theory. We have also investigated the influence of different laws of change in the shell thickness on the character of its natural vibrations. Our calculations have been carried out for different geometrical and elastic parameters of the shell under study and different boundary conditions.

The problem of the free axisymmetric vibrations of longitudinally polarized piezoceramic hollow cylinders is solved by a numerical analytic method. The spline-collocation method with respect to the longitudinal coordinate is used to reduce the original problem of electroelasticity to an eigenvalue boundary-value problem for ordinary differential equations 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 for a wide range of their geometric characteristics Keywords: free vibrations, hollow piezoceramic cylinder, spline-approximation, boundary conditions Introduction. Piezoceramic materials have significant advantages over natural piezoelectrics (quartz, tourmaline, Rochelle salt, etc.) such as good moldability, low cost, high sensitivity, and high thermal stability. Modern piezoceramic elements and devices are solid-state and can be made in three-dimensional, two-dimensional, and integrated forms. They show high noise immunity, low intrinsic noise, and high radiation resistance. This is why piezoceramic materials are widely used in various fields of science and engineering, subject to intensive development, and their properties are of huge and ever-increasing interest.The widespread use of piezoceramic elements and devices is due to the tendency to better describe real processes in piezoceramic structural materials and to reveal and study three-dimensional effects occurring in thick-walled elements. 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 [1-3, 9, 10].Recent trends in computational mathematics, mathematical physics, and mechanics are toward the use of spline-functions. This is due to the following advantages of spline-approximations: (i) stability of splines against local perturbations (the local behavior of a spline at a point does not affect its overall behavior, unlike, for example, polynomial approximation); (ii) good convergence of spline-interpolation (unlike polynomial interpolation); and (iii) simplicity and convenience of numerical implementation of spline algorithms. When used in various variational, projective, and other discrete-continuous methods, spline-functions yield much better results than classical polynomials do, simplify the numerical implementation of these methods, and improve the accuracy of the solution. Noteworthy are the publications [5][6][7][8], which use spline-approximation to study the mechanical behavior of various plates and shells.In the present paper, we study the free vibrations of a longitudinally polarized hollow piezoceramic cylinder. The lateral surfaces of the cylinder are free from external loads. The cylinders are considered either hinged or clamped.1. Governing Equations. The vibrations of piezoelectric bodies are described by the equations of elasti...