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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...

The natural vibrations of anisotropic rectangular plates of varying thickness with complex boundary conditions are studied using the spline-collocation and discrete-orthogonalization methods. The basic principles of the approach are outlined. The natural vibrations of orthotropic plates with parabolically varying thickness are calculated. The results (natural frequencies and modes) obtained with different boundary conditions are analyzed Keywords: anisotropic rectangular plates, spline-collocation method, discrete-orthogonalization method, natural frequencies and modesIntroduction. Plates of varying thickness are widely used in structures of various designations. To design them, it is necessary to determine the natural frequencies and modes with high accuracy, which are needed to describe the response of plates to the operating conditions. For plates with constant thickness and hinged opposite edges, the solution can be constructed in a closed form [5,13]. With boundary conditions of other types, however, it is impossible to obtain a similar solution for natural vibrations of elastic plates. The natural vibrations of orthotropic plates with such boundary conditions were studied quite actively, which was reflected in a number of publications. The solutions for forced and natural vibrations of orthotropic plates were obtained in [21] in the form of double trigonometric series. Lagrangian multipliers were used in [20] to solve a similar problem with allowance for shear strains in several first modes. The superposition method was used in [14] to table natural frequencies for a certain range of stiffness ratios. In [23], the superposition method and affine transformation were used to determine the natural frequencies of orthotropic plates partially clamped and partially simply supported. The Kantorovich method was used in [7] to study the natural vibrations of clamped plates. The natural vibrations of complex anisotropic plates were studied in [2,9] using variational methods and the R-function method. The natural vibrations of rectangular plates of varying thickness were addressed by many authors. For example, the paper [12] is concerned with the general natural-vibration problem for plates of varying thickness. The transverse vibrations of plates with exponentially varying thickness are studied in [10] and inhomogeneous rectangular plates with parabolically varying thickness in [22]. The natural vibrations of simply supported plates with linearly varying thickness were investigated in [6,9,11,19].Thus, we may conclude that there is a variety of approximate approaches to natural-vibration problems for anisotropic rectangular plates with boundary conditions that do not allow closed-form solutions. Recently, computational mathematics, mathematical physics, and mechanics have widely employed spline functions to solve such problems. This is due to the following advantages of the spline-approximation method over the other ones: stability of splines against local perturbations, i.e., the behavior of a spline in the neighbor...

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The problem of free vibrations of a solid cylinder with different boundary conditions is solved using the three-dimensional theory of elasticity and a numerical analytic approach. The spline-approximation and collocation methods are used to reduce the partial differential equations of elasticity to systems of ordinary differential equations of high order with respect to the radial coordinate. These equations are solved by stable numerical discrete orthogonalization and incremental search. Calculated results are presented for transversely isotropic and inhomogeneous materials of the cylinder and for several types of boundary conditions at its ends Keywords: free vibrations, three-dimensional theory of elasticity, solid cylinder, finite length, spline-collocationIntroduction. Solving three-dimensional dynamic problems for elastic solids involves significant difficulties because of the complexity of the associated system of partial differential equations and the necessity to satisfy boundary conditions. There are only few publications [1-6, 12-22] that use the three-dimensional theory of elasticity to study the vibrations of finite-length cylinders.To solve such problems in computational mathematics, mathematical physics, and mechanics, spline functions are widely used. For example, the spline approximation method was used in [7][8][9][10][11] to study the mechanical behavior of various plates and shells.The present paper proposes an efficient numerical procedure to determine the natural frequencies and modes of axisymmetric vibrations of transversely isotropic solid finite-length cylinders with different boundary conditions. The procedure uses spline-approximation in one of the coordinate directions followed by solution of an eigenvalue boundary-value problem for systems of ordinary differential equations of high order with variable (in the general case) coefficients by stable numerical discrete orthogonalization in combination with incremental search. The approach makes it possible to study the free vibrations of finite-length cylinders made of inhomogeneous materials.

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