Principal parametric resonances of non-linear two degree of freedom dynamical systems with quadratic and cuibic non linearities subjected to parametric excitation are studied. The relations between the parametric frequencies of the system are such that excited into parametric resonances in two cases, the first case without internal resonance, second case in the presence of different posibilities of internal resonances (one-to-one, one-to-two, one-to-three, two-to-one and three-to-one). The method of averaging is used to construct a first order non-linear ordinary differential equations governing the modulation of the amplitudes and phases of the two modes. In all cases the steady state solutions and their stability are determined. Numerical results depicting the various resonances are present. |
