Combination Resonances of a Circular Plate With Three-Mode Interaction

A nonlinear analysis is presented for combination resonances in the symmetric responses of a clamped circular plate with the internal resonance, $ω3≈ω1+2ω2$⁠. The combination resonances occur when the frequency of the excitation are near a combination of the natural frequencies, that is, when $Ω≈2ω1+ω2$⁠. By means of the internal resonance condition, the frequency of the excitation is also near another combination of the natural frequencies, that is, $Ω≈ω1−ω2+ω3$⁠. The effect of two near combination resonance frequencies on the response of the plate is examined. The method of multiple scales is used to solve the nonlinear nonautonomous system of equations governing the generalized coordinates in Galerkin’s procedure. For steady-state responses, we determine the equilibrium points of the autonomous system transformed from the nonautonomous system and examine their stability. It has been found that in some cases resonance responses with nonzero-amplitude modes don’t exist, and the amplitudes of the responses decrease with the excitation amplitude. We integrate numerically the nonautonomous system to find the long-term behaviors of the plate and to check the validity of the analytical solution. It is found that there exist multiple stable responses resulting in jumps. In this case the long-term response of the plate depends on the initial condition. In order to visualize total responses depending on the initial conditions, we draw the deflection curves of the plate.