Abstract: In this paper, the method of relaxed boundary conditions is applied to rectangular plates with edges which are a sort of the mixture of simply supported portions and clamped portions, so that the lower limit of fundamental frequency of such plates is evaluated. A kind of polynomial satisfying the displacement boundary conditions is designed, wich makes it possoble to evaluate the upper limit of fundamental frequency by Ritz method. The practical calculation examples solved by these methods have given satisfactory results. At the end of this paper, it is pointed out that the so-called exact solution of such plates usually evaluated by the force superposition method is essentially a kind of lower limit of solution, if the truncated error of series which occurs in actual calculation is considered.