Nonlinear oscillations under multifrequency parametric excitation
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A second-order system of differential equations containing a multifrequency parametric excitation and weak quadratic and cubic nonlinearities is investigated. The method of multiple scales is used to carry out a general analysis, and three resonance conditions are considered in detail. First, the case in which the sum of two excitation frequencies is near two times a natural frequency, Î»s + Î»t ~2Ï q, is examined. Second, the influence of an internal resonance, Ï q~3Ï r, on the previous case is studied. Finally, the effect of the internal resonance wr~3wq on the resonance Î»s + Î»t ~2Ï q is investigated. Results are presented as plots of response amplitudes as functions of a detuning parameter, excitation amplitude, and, for the first case, a measure of the relative values of Î»s + Î»t.
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