Nonlinear oscillations of self-excited systems under multifrequency parametric excitation

A self·excited system with weak nonlinearities and multifrequency parametric excitation is investigated in this study. The method of multiple scales is used to analyze the system under four different resonances relating parametric excitation frequencies with the natural frequencies. In the first case, the parametric excitation frequency is approximately equal to twice the natural frequency, λ≃2ω. In the second case, the parametric excitation frequency is approximately equal to the natural frequency, λ≃ω. The third case treats a system with two parametric excitation frequencies under the condition λ₁±λ₂≃2ω. In the last case, a two-degree-of-freedom system with natural frequencies ω, and ω, is considered and the resonance λ₁+ λ₂≃ω_r- ω_q , is analyzed. Different parameters (the 1' load amplitudes, a detuning parameter, and a system stiffness parameter) are varied in each case and the responses obtained are presented in plots. The stability of the solutions is affected by all the parameters mentioned, especially the load amplitudes and the detuning parameter.