Forced Vibration of Isotropic and Nonhomogeneous Rectangular Plate of Linearly Varying Thickness

Abstract

On the basis of classical plate theory forced vibrations of non-homogeneous rectangular plate of variable thickness analysed. Due to the variation in density the non-homogeneity of the plate material is assumed to arise vary linearly. For estimating the maximum deflection of a rectangular plate approximate formulae are proposed which are subject to a uniformly distributed harmonic lateral load. The fundamental frequency of vibration Maximum deflection for the different values vibration is computed for a simply supported-free-simply supported-free plate non-homogeneity constant and aspect ratios, for various values of taper constant. Graphical presentation of present work is shown in this paper. Two of these take modal solutions of the three-dimensional equations just mentioned; these solutions satisfy the differential equations and boundary conditions on the major surfaces of the plate exactly, and a number of such solutions are summed so as to satisfy the remaining boundary conditions on the minor surfaces approximately, using either the variation formulation or the method of least squares. Analysis and numerical results are presented for free transverse vibrations of isotropic rectangular plates having arbitrarily varying non-homogeneity with the in-plane coordinates along the two concurrent edges on the basis of Kirchhoff plate theory. For the non-homogeneity, a general type of variation for Young’s modulus and density of the plate material has been assumed. Generalized differential quadrature method has been used to obtain the eigenvalue problem for such model of plates for four different combinations of boundary conditions at the edges namely, (i) fully clamped, (ii) two opposite edges are clamped and other two are simply supported. By solving these eigenvalue problems using software MATLAB, the lowest three eigenvalues have been reported as the first three natural frequencies for the first three modes of vibration. The effect of various plate parameters on the vibration characteristics has been analyzed. Three dimensional mode shapes have been plotted. A comparison of results with those available in literature has been presented.