Oscillations of weakly non-linear, self-excited systems under multi-frequency parametric excitation

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A class of self-excited mechanical or structural systems subjected to parametric excitation is considered. The systems have an arbitrary number of generalized co-ordinates. The equations of motion include weak quadratic and cubic non-linearities in the stiffness, small “negative” linear damping terms, and small “positive” cubic damping terms. Special cases are van der Pol's equation and Rayleigh's equation. The parametric excitation includes multiple frequencies λ m . By using the method of multiple scales, the following resonances are analyzed: λ s ≈ 2 ω q (principal parametric resonance), λ s ≈ ω q , λ s ± λ t ≈ 2 ω q , and λ s + λ t ≈ ω r − ω q , where the ω n are natural frequencies of the system. Steady state response amplitudes are determined and plotted as functions of a detuning parameter, excitation amplitudes and, when λ s ± λ t ≈ 2 ω q , a measure of the relative values of λ s and λ t .