SOME PROBLEMS ON FREE TRANSVERSE VIBRATIONS OF NONHOMOGENEOUS ELASTIC PLATES

Abstract

The work reported in this thesis is an attempt to study the vibrational behaviour of isotropic/orthotropic nonhomogeneous plates. It is composed of nine chapters. Chapter I presents an up-to-date survey of the literature on the vibration of plates of various geometries with various complicating effects such as nonhomogeneity, thickness variation, elastic foundation and in-plane force. It reveals that almost no work has been done dealing with nonhomogeneous isotropic/orthotropic rectangular/skew/triangular plates in which nonhomogeneity is a function of two variables. In chapters II to VII, the effect of nonhomogeneity with other complicating effects has been studied on the vibration characteristics of isotropic/orthotropic rectangular plates. Further, chapters VIII and IX deal with skew and triangular orthotropic plates, respectively. The numerical results have been computed employing two different numerical techniques for the first/lowest three modes of vibration on the basis of classical plate theory. Mode shapes for specified plates have been illustrated in each chapter. The results would be of great interest to design engineers. The chapter-wise summary has been given as follows: Chapter II analyses the free transverse vibrations of nonhomogeneous orthotropic rectangular plates of variable thickness. The nonhomogeneity of the plate material is assumed to arise due to the exponential variation in Young's moduli and density along one direction. Following Levy approach i.e. the two parallel edges are simply supported, the fourth order differential equation governing the motion of exponentially varying thickness in one direction has been solved by using differential quadrature method for three different combinations of clamped, simply supported and free boundary conditions at the other two edges. Effect of the nonhomogeneity together with other plate parameters such as aspect ratio and thickness variation has been illustrated on the natural frequencies for the first three modes of vibration. Comparison of results with published literature demonstrates the computational efficiency of the approach. In chapter III, the effect of Winkler-type foundation has been investigated on the free transverse vibrations of nonhomogeneous orthotropic rectangular plates with exponential variation in Young's moduli, density and thickness of the plate along x- direction. The two opposite edges (y=0, y=b) are assumed to be simply supported while the other two edges (x=0, x=a) may have a combination of clamped, simply supported and free edge conditions. A semi-analytical approach has been used for the solution. Assuming the transverse displacement w to vary as sin (pay 1 b), the partial differential equation governing the motion is reduced to an ordinary differential equation in x with variable coefficients. The resulting ordinary differential equation is then solved numerically by the method of differential quadrature for three different combinations of clamped, simply supported and free boundary conditions. The first three frequencies have been computed to study the behavior of foundation parameter together with other plate parameters. A comparison of results with those obtained by other methods has been presented. In chapter IV, an analysis for the free transverse vibrations of nonhomogeneous isotropic rectangular plates of varying thickness has been presented employing two dimensional boundary characteristic orthogonal polynomials in Rayleigh-Ritz method. Gram-Schmidt process has been used to generate these polynomials. The thickness of the plate is varying bidirectionally and is the Cartesian product of linear variations along the two concurrent edges of the plate. The nonhomogeneity of the plate material is assumed to arise due to linear variations in Young's modulus and density with both the in-plane variables. Out of many possible combinations of classical boundary conditions at the edges, the following four different combinations of edge conditions, namely: CCCC- all the four edges are clamped; SCSC- two opposite edges are clamped and the other two are simply supported; FCFC- two opposite edges are clamped and the other two are free; FSFS- two opposite edges are simply supported and the other two are free, have been considered in the present work. The effect of the nonhomogeneity together with thickness variation and aspect ratio has been investigated for the lowest three modes of vibration. Comparison of frequencies with those available in the literature has been presented. In chapter V, Rayleigh-Ritz method with boundary characteristic orthogonal polynomials has been used to study the vibrational behaviour of nonhomogeneous orthotropic rectangular plates of bilinearly varying thickness. The mechanical properties of the plate material i.e. Young's moduli, shear modulus and density of the plate are assumed to vary linearly with the variables x and y. Assuming the variation for thickness of the plate as in chapter IV, the effect of orthotropy together with other plate parameters on the natural frequencies has been illustrated for the lowest three modes of vibration for all the four boundary conditions CCCC, SCSC, FCFC and FSFS. Results have been compared with published one. Chapter VI deals with the free transverse vibrations of nonhomogeneous orthotropic rectangular plates of non-uniform thickness resting on Winkler foundation. It is assumed that Young's moduli, shear modulus and density of the plate material are linear functions of in-plane variables x and y. The variation for thickness of the plate is bilinear. The lowest three natural frequencies of such plates have been obtained using Rayleigh-Ritz method with boundary characteristic orthogonal polynomials for four different combinations of clamped, simply supported and free edge conditions. Comparison of results has been presented. Chapter VII deals with the buckling and vibration behaviour of nonhomogeneous orthotropic rectangular plates of bilinearly varying thickness using boundary characteristic orthogonal polynomials in Rayleigh-Ritz method. The two opposite edges (y=0, y=b) are subjected to uniform in-plane force. Assuming that the mechanical properties and thickness of the plate are varying as in chapter V, the effect of nonhomogeneity, orthotropy together with varying values of aspect ratio and in-plane force parameter has been studied for the lowest three modes of vibration for all the four boundary conditions as considered in chapter V. By allowing the frequency to approach zero, the critical buckling loads in compression for various values of plate parameters have been computed and compared with those available in the literature. Chapter VIII presents the free transverse vibrations of nonhomogeneous orthotropic skew plates with variable thickness. The mechanical properties i.e. Young's moduli, shear modulus and density of the plate material are assumed as linear functions of in-plane variables. Rayleigh-Ritz method with boundary characteristic orthogonal polynomials has been used to obtain lowest three natural frequencies of such plates for four different combinations of clamped, simply supported and free edge conditions. The effect of nonhomogeneity parameters, density parameters and thickness parameters together with aspect ratio for different values of skew angle has been illustrated for all the four boundary conditions CCCC, SCSC, FCFC and FSFS. A comparison of results have been presented, wherever possible Chapter IX is concerned with the free transverse vibrations of nonhomogeneous orthotropic triangular plates of varying thickness. The thickness of the plate and elastic properties of the plate material vary with both the in-plane variables. The approximate solution for the lowest three modes of vibration has been obtained using Rayleigh-Ritz method with boundary characteristic orthogonal polynomials for four different combinations of clamped, simply supported and free edge conditions, namely CCC, CSC, CFC and SSS. The use of these orthogonal polynomials in Rayleigh-Ritz method leads to a standard eigenvalue problem which has been solved numerically using Jacobi method. The lowest three eigenvalues correct to three decimals have been reported for equilateral triangular plates for different values of plate parameters. The effect of nonhomogeneity and thickness variation on the lowest three natural frequencies has been studied. Frequencies have been compared with published work