In-plane vibrations of rectangular plates with rectangular cutouts

This work presents highly accurate numerical calculations of the natural frequencies and modes for the in plane vibrations of rectangular plates with rectangular cutouts. The solutions are obtained using the multi term extended Kantorovich method. The solution is sought as the sum of multiplications of two one dimensional functions. In this method a solution is assumed in one direction of the plate, and this enables to transform the coupled partial differential equations of the plate equilibrium into an a set of ordinary differential equation. These equation are solved exactly by the exact element method [1], and an approximate natural frequency is obtained. In the second step, the derived solution is now taken as the assumed solution in one direction, and the process is repeated to find an improved approximation of the natural frequency. This process converges with a small number of solution cycles. For plates with cutouts this process yields very accurate values even with 1 term expansion, and in some cases one can improve these values by adding additional functions in the expansion. As an example the natural frequencies of a square plate fully restrained along all four edges, with centrally located square cutout are given in the Table. The size of the cutout is 0.4 of the edge size. The first 5 frequencies of vibration are given together with the results from a finite element analysis using ANSYS (with 8520 DOF). Many more results will be given.