Vibrations of rectangular plates by the extended Kantorovich method

Exact solutions for plate vibrations are available for limited number of boundary condition combinations on the four edges of the plate. One of the approximate methods that were proposed for the solution was the extended Kantorovich method. The single term Kantorovich method proposes to assume the solution as a multiplication of a function of one direction by another function in the second direction. In this paper, the proposed solution is assumed as a sum of single variable function multiplications. Then, the partial differential equation of motion of the plate is converted into a set of coupled ordinary differential equations. Then, the functions in one direction are assumed, and the functions in the second direction are solved for the natural frequencies and vibration modes. In the extended method, after the first solution, the derived functions are used as the known functions, and the solution in the second direction is found. After several cycles the solution converges to highly accurate results. The solution is obtained by finding the dynamic stiffness matrix using the exact element method. Results are given for completely clamped square plates, and compared with FE results.