The asymptotic method by Krylov-Bogoljubov-Mitropolsky is directly applied in this paper for finding the solutions and differential equations of the first approximation for amplitude and phase of the one-frequency regime vibrations of the plate, expressed by a partial differential equation of the plate, so that it can be applied to any form and boundary conditions of thin plate leaning, if its proper functions and proper circular frequencies of the unperturbed form of vibrations are known. After that energy interpretation of the first approximation is given for the amplitude and phase of vibrations in the one-frequency regime and, at the same time, the expression for the mean value of the virtual work of the perturbing force acting upon the plate. By means of derived equations the proper nonlinear vibrations of the thin plate on the nonlinear elastic substratum are studied, especially in relation to the nonlinear law of the elastic substratum and the coefficient of the effect of the nonlinearity law for the elasticity of substratum material to the change of the proper circular frequency. In the case of a rectangular free leaned plate to the elastic substratum there is calculated the value of that coefficient.