Non-linear interactions in asymmetric vibrations of a circular plate

In order to investigate the non-linear asymmetric vibrations of a clamped circular plate on an elastic foundation, the primary resonances of a plate with an internal resonance, in which the natural frequencies of two asymmetric modes are commensurable are considered. The response is expressed as an expansion in terms of the linear, free oscillation modes, and its amplitude is considered to be small but finite. The method of multiple scales is used to reduce the non-linear governing equations to a system of autonomous ordinary differential equations for amplitude and phase variables. For a numerical example the case of internal resonance (a commensurable relationship between natural frequencies), ω32≈3ω11, where the first subscript refers to the number of nodal diameters and the second subscript the number of nodal circles including boundary is considered. When the frequency of excitation is near ω11, there exist at most five stable steady-state responses. Four of them are superpositions of traveling wave components and one is a superposition of standing wave components. The result shows the interaction between modes corresponding to ω11 and ω32 by showing non-vanishing amplitudes of the mode not directly excited. When the frequency of excitation is near ω32, similarly the interaction between modes is shown to exist. All of the responses with non-vanishing amplitudes of modes excited indirectly, however, turn out to be unstable, which is a peculiar phenomenon.