NUMERICAL METHODS FOR FREQUENCIES AND MODE SHAPES OF PLATES WITH UNIFORM AND VARIABLE THICKNESS

Abstract

The present work deals with the numerical solution for frequencies and mode shapes of free transverse vibrations of isotropic elastic plates of various shapes under different types of boundary conditions at the edge. Except for a few cases the thickness of the plate has been taken as variable. Extensive numerical work has been done to find the frequencies and the associated mode shapes. The three dimensional mode shapes and contour lines have been drawn in almost all cases. Comparison has been made with all known results. The first chapter is an introduction to the subject matter of the present work. The chapters from the second to the sixth are all having the same common approach based upon the Rayleigh-Ritz method. These deal with triangular and circular plates of variable thickness and different types of boundary conditions. This forms the major part of the thesis. The basis functions are chosen to satisfy the essential boundary conditions. A large number of approximations have been worked out to ensure convergence of the results. Unfortunately, the method is highly unstable as the matrices generated are ill-conditioned. So all the computations are done using double precision arithmetic. The last chapter that is i. e. the seventh, is significantly different from the rest of the thesis. Here a finite difference method has been evolved to solve elliptic problems in general. As a special case the results for the vibration of rectangular plates and membranes have been derived. To solve the resulting eigenvalue problem the well-known generalized Jacobi method has been used. In whole of the study classical plate theory has been considered. Although these geometries have been studied extensively, a lot of information is still missing in the existing literature. The purpose of the present work is to fill this gap. The success of the Rayleigh-Ritz method largely depends upon the choice of the basis functions. Unfortunately, there is no general rule to find out the best set of such functions so as to have a faster convergence. In this work we have chosen the basis functions of the form mi ni ()i(x,y) = B (x,y) x y , where B(x,y) = 0, is the equation of the boundary of the plate and P = 0, 1 or 2 depends upon whether it is free, simply-supported or clamped.Theconstantsm.andn.are nonnegative integers. It is clear that the above choice of basis functions will satisfy the essential boundary conditions. When the boundary consists of s segments having different boundary conditions we can take the factor P1 P2 Ps B1 B ... Bs in place of BP. The next challenging problem is the accurate evaluation of the integrals involved. If the boundary has equation which is a polynomial in x and y and the thickness variation is also approximated by a polynomial, it is possible to evolve formula 'to get all the integrals involved in closed form for almost all common geometries. 27. As far as the plotting of mode shapes is concerned, we have used tools of computer graphics and Turbo C++ language to develop the three -dimensional surfaces for various modes of vibration. Perspective projection has been used in all cases. To economize the computations the displacements are calculated at some selected points and surface is generated by spline interpolation. From this additional points are generated and the curves obtained by joining these points. In all the cases contour lines are also drawn. These are the curves of constant displacements....