Abstract: The Rayleigh-Ritz method is utilized to determine the minimum stiffness and optimal position of elastic point (simple) supports, so that the fundamental natural frequency of a rectangular plate can be raised to the original second frequency. The plate has only one boundary edge restrained (clamped or simply supported), and the additional supports are placed along the free edge opposite to the restrained boundary. Lagrange multipliers are applied to enforce the optimality conditions for the position design of a support. The minimum stiffness of the support can be found numerically by solving a characteristic eigenvalue problem. Illustrative examples are employed to verify the solution procedure and show that the present method is feasible and effective for the optimal design of a flexible support of a finite stiffness.