Modal interaction of resonantly forced oscillations of two-degree-of-freedom structure

摘要

An analysis is presented for simultaneous primary resonances of two-degree-of-freedom system with quadratic and cubic nonlinearities to external excitations having the frequencies Ω1 and Ω2. The relations between the external frequencies of the system are such that they are excited into external resonances in two cases: the first case without internal resonances and the second case in the presence of different possibilities of internal resonances (one-to-one, two-to-one, three-to-one, one-to-two, and one-to-three). The method of multiple scales (MMS) is used to determine four first-order ordinary differential equations, which describe the modulation of the amplitudes and phases of both modes. In order to use the MMS properly, we introduce new detuning parameters of both modes. It is found analytically that two-mode responses can occur at each primary resonance. In all cases the steady-state solutions and their stability are determined. Numerical results depicting the resonances are presented.