Dynamic Response to Moving Concentrated Masses of Highly Prestressed Orthotropic Rectangular Plate Resting on Pasternak Foundation

Abstract

In this paper, the response to moving concentrated masses of highly prestressed orthotropic rectangular plate resting on two parameter elastic sub-grades is investigated. In the equation governing the motion of the plate under the action of the travelling masses, it is observed that a small parameter multiplies the highest order operator which is peculiar of singular perturbation problems. From the literatures, problems in which the small parameter multiplies the highest derivatives in the governing differential equation do not have closed-form solution.  Problems such as this, defile conventional mathematical methods used to obtain approximate solution, since the small parameter affects the problem in such a way that the solution varies rapidly in some regions of the domain of definition of the problem and slowly in the other parts. It is a well-known fact that when the problem's small parameter multiplies its highest order derivatives in the governing differential equation, it is only amenable to the methods of singular perturbation. Since of all the singular perturbation methods, the method of matched asymptotic expansions is more user-friendly, it is adopted for the solution of this plate problem. Analyses of the solution obtained for this plate dynamical problem are shown in plotted curves. These reveal that increase in the value of prestress or shear modulus leads to increase in the critical speed of the orthotropic rectangular plate traversed by moving concentrated mass. Also, as the rotatory inertia or mass ratio increases, the critical speed decreases. As observed, there is more than one resonance condition in this dynamical system which involves plate flexure under moving concentrated masses, and then, the smaller the mass ratio, the better the improvement in critical speed.