TRANSVERSE VIBRATIONS OF SOME NON-HOMOGENEOUS RECTANGULAR PLATES

Abstract

In the thesis, a theoretical study on the vibration characteristics of isotropic/ orthotropic nonhomogeneous rectangular plates of uniform/ non-uniform thickness together with and without the effect of elastic foundation and in-plane loading has been presented on the basis of Kirchhoff plate theory. The new models for the unidirectional/ bidirectional non-homogeneity of the plate material along the edges have been proposed with their usage and significance in various technological situations. It consists of nine chapters. Chapter I present an up-to-date survey of the literature on the vibration of plates: particularly, on rectangular plates with other complicating effects such as non-homogeneity, elastic foundation, orthotropy, thickness variation and in-plane force etc. and their wide industrial applications which inspire the work of the thesis. In chapters II-VIII, their analysis with different considerations have presented applying two different numerical techniques, namely, generalized differential quadratue (GDQ) method and Ritz- generalized differential quadratue (GDQ) method. The conclusions and future scope have been presented in Chapter IX. The numerical results have been obtained using the software MATLAB. A chapter-wise summary is given as follows: Chapter II provides an analysis for the buckling and vibrational behaviour of nonhomogeneous rectangular plates of uniform thickness when the two opposite edges are simply supported and these are subjected to linearly varying in-plane force. For non-homogeneity of the plate material, the Young’s modulus and density of the plate material are assumed to vary exponentially along one of the axial directions of the plate. The governing differential equation of motion for such a plate has been derived by Hamilton’s energy principle. This partial differential equation has been reduced to an ordinary differential equation using the sine function for mode shapes between the simply supported edges. The frequency equations from the resulting equation are then obtained by using the generalized differential quadrature (GDQ) method for three different combinations of clamped, simply supported and free boundary conditions at the other two edges. These frequency equations have been solved numerically using MATLAB. The effect of various parameters such as non-homogeneity parameter, density parameter, aspect ratio, in-plane force parameter and loading parameter has been studied on the natural frequencies for the first three modes of vibration. Critical buckling loads have been computed for various parameters. Comparison has been made with the known results. ii In chapter III, the analysis of chapter II has been extended to include the effect of Pasternak foundation on the vibrations of non-homogeneous rectangular plates when the two opposite edges are simply supported and these are subjected to linearly varying in-plane force. The governing differential equation of motion for such a model of the plate has been obtained using Hamilton’s energy principle. The frequency equations for two different combinations of clamped and simply supported boundary conditions at the other two edges have been obtained. The lowest three roots of these equations have been reported as first three natural frequencies. The effect of foundation parameters together with varying values of other parameters such as non-homogeneity parameter, density parameter, aspect ratio, in-plane force parameter and loading parameter has been investigated on the frequencies for the first three modes of vibration. By allowing the frequency to approach zero, the critical buckling loads have been computed. Chapter IV is concerned with the free transverse vibrations of two-dimensional nonhomogeneous isotropic rectangular plates of uniform thickness. The non-homogeneity of the plate material is assumed to vary along both the axial directions of the plate and arise due to the arbitrary variations in the Young’s modulus and density of the plate material with the inplane coordinates. Hamilton’s energy principle has been used to obtain the governing differential equation of motion for such a plate model. The resulting equation has been reduced to an eigenvalue problem using two-dimensional GDQ method for four different combinations of boundary conditions at the edges namely, (i) all the edges are clamped i.e. fully clamped (ii) two opposite edges are clamped and other two are simply supported (iii) two opposite edges are clamped and other two are free (iv) two opposite edges are simply supported and other two are free. The lowest three eigenvalues have been reported as the first three natural frequencies for the first three modes of vibration. Numerical results have been computed for linear as well as parabolic variations in non-homogeneity for various values of non-homogeneity parameters, density parameters and aspect ratio. A comparison of results with those available in literature has been presented which shows a very good agreement. In Chapter V, the analysis of chapter IV has been extended for the plates of variable thickness. The thickness of the plate is varying bidirectionally and taken as the Cartesian product of linear variations along the two concurrent edges of the plate. An approximate solution for the governing differential equation of motion of such a plate has been obtained by employing two-dimensional GDQ method. The effect of various parameters such as thickness parameters, non-homogeneity parameters, density parameters, and aspect ratio on the frequencies for the iii first three modes of vibration has been investigated for all the four plates. Comparison of results with those obtained by other methods has been presented. In Chapter VI, the effect of two-dimensional non-homogeneity on the free transverse vibrations of orthotropic rectangular plates has been studied. The governing differential equation of motion for such a model of the plate has been derived using Hamilton’s principle. For all the boundary conditions the corresponding eigenvalue problem has been obtained by using two-dimensional GDQ method. The mechanical properties of the plate material i.e. Young’s moduli, shear modulus and density of the plate are assumed to vary exponentially with the in-plane coordinates in the directions of orthotropy. The influence of various plate parameters such as non-homogeneity parameter, density parameter and aspect ratio on natural frequencies has been studied for the first three modes of vibration. In order to verify the accuracy of present method, numerical results have been compared by the results available in the literature. Boron-epoxy has been taken as an example of an orthotropic material. In Chapter VII, the effect of bilinear thickness variation on the transverse vibrations of nonhomogeneous orthotropic rectangular plates has been investigated by extending the analysis given in Chapter VI. The thickness of the plate is taken as the Cartesian product of linear variations along the two concurrent edges of the plate. For non-homogeneity of the plate material the Young’s moduli, shear modulus and density of the plate material are assumed to vary exponentially with the in-plane coordinates along the directions of orthotropy in distinct manner. For such plate model, the governing differential equation of motion with the incorporation of thickness variation has been obtained using Hamilton’s energy principle. The effect of thickness variation along with other plate parameters such as non-homogeneity parameters, density parameters and the aspect ratio on the natural frequencies has been analysed for the first three modes of vibration. The comparison of results has been carried out for homogeneous isotropic and orthotropic square plates with those results available in the literature. In Chapter VIII, the analysis of chapter VII has been extended to study the free transverse vibrations of non-homogeneous orthotropic rectangular plates of variable thickness resting on a two-parameter elastic foundation. The governing differential equation of the motion for such a plate model has been derived using Hamilton’s energy principle. Mixed Ritz-GDQ method has been applied to obtain the eigenvalue problems for four different combinations of clamped and simply supported edges namely (i) all edges are clamped (ii) three edges are clamped and one are simply supported (iii) two opposite edges are clamped and other two are simply iv supported (iv) all edges are simply supported, respectively. The lowest three eigenvalues have been reported as the first three natural frequencies. The influence of Winkler as well as shear stiffness parameters together with non-homogeneity parameters, density parameters, thickness parameter and aspect ratio on the frequencies has been investigated for first three modes of vibration for all the four boundary conditions. A comparison of results has been presented.