Free Vibration of Four Sided Simply Supported Trapezoidal Orthotropic Plate with Varying Thickness and Thermal Effect

Abstract

newline Visco-elastic plates are widely used in various mechanical structures, aircrafts and industries. These plates are developed with an aim that scientists and design engineers can make a use of it with a practical approach. Visco-elastic plates may be regarded as an approximation to the wings, blades and variation in thickness may also lead to reduction in weight of structure of aircraft, space shuttle or satellite. The plates of variable thickness are being mostly used in aerospace engineering, marine engineering, mechanical engineering, civil engineering, and nuclear power plants etc. An interest towards the effect of high temperatures on non-homogeneous plates of variable thickness is developed due to application in various fields. The reason for these is that during heating up periods, structures are exposed to high intensity heat fluxes and material properties undergo significant changes: in particular the thermal effect cannot be taken as negligible. The information about first two modes of vibration is essential for construction engineer before finalizing a design. The vibration of plates of various shapes homogeneous and non homogeneous, orthotropic or isotropic with or without variation in thickness has been studied by lot of researchers with or without considering the effect of temperature. In present study five problems have been solved on the basis of classical plate theory of elasticity. newlineIn first problem, the effect of non-homogeneity on vibration of orthotropic trapezoidal plate with bi-linear thickness variation has been studied. Thermal induced vibration of these plates has been assumed as bi-linear temperature circulation. For non-homogeneity of the plate material density is taken to be constant. The governing differential equation has been solved with the help of variables separation method. The approximated frequency equation is derived by using Rayleigh-Ritz method by two term deflection function. Frequency values for the first two modes of vibration have been calculated for a simply supported trapezoidal plate for various values of thermal gradient, taper constants and aspect ratio. newlineIn second problem, free vibration of orthotropic trapezoidal plate under the effect of bi-parabolic thickness variation and bi-linear temperature. Density (due to non-homogeneity) of the plate is taken to be constant. The governing differential equation has been solved with the help of variables separation method. The newlineABSTRACT newlineviii newlineapproximated frequency equation is derived by using Rayleigh-Ritz method by two term deflection function. Frequency values for the first two modes of vibration have been calculated for a simply supported trapezoidal plate for various values of thermal gradient, aspect ratio and taper constants with the help of MAPLE (latest computational software). newlineIn third problem, free vibration of orthotropic trapezoidal plate under the effect of bi-linear thickness variation and bi-parabolic temperature has been studied. Density (due to non-homogeneity) of the plate is taken to be constant. The governing differential equation has been solved with the help of variables separation method. The approximated frequency equation is derived by using Rayleigh-Ritz method by two term deflection function. Frequency values for the first two modes of vibration have been calculated for a simply supported trapezoidal plate for various values of thermal gradient, taper constants and aspect ratio. newlineIn fourth problem, free vibration of orthotropic trapezoidal plate under the effect of bi-parabolic thickness and bi-parabolic temperature has been studied. Density (due to non-homogeneity) of the plate is taken to be constant. The governing differential equation has been solved with the help of variables separation method. The approximated frequency equation is derived by using Rayleigh-Ritz method by two term deflection function. Frequency values for the first two modes of vibration have been calculated for a simply supported trapezoidal plate for various values of thermal gradient, taper constants and aspect ratio. newlineIn fifth problem, free vibration of orthotropic trapezoidal plate under the effect of bi-exponential thickness and bi-exponential temperature has been studied. For non-homogeneity of material, density of the plate is taken to be constant. By use of separation of variables method, the governing differential equation has been solved for orthotropic trapezoidal plate. The approximated frequency equation is derived by using Rayleigh-Ritz method by two term deflection function. Latest computational software MAPLE has been used to calculate Frequency values for the first two modes of vibration at various values of plate s parameters (thermal gradient, taper constants, aspect ratio). Results are shown in form of tables and graphs.