Geometrically nonlinear vibrations of circular plate with two cut-outs are simulated by the von Karman equations with respect to displacements. The combination of the Rayleigh–Ritz method and the -function method, which allows satisfying all boundary conditions, is applied to obtain the vibration modes of the plate. The nonlinear vibrations are expanded using these vibrations modes. The dynamical system with three degrees of freedom is derived by Galerkin method. The influence of cut-outs size on linear and nonlinear vibrations of the plate is analyzed. For different parameters of cut-outs, different internal resonances occur in the plate. The nonlinear vibrations of the system for different internal resonances are analyzed by multiple scale method.