Response of two-degree-of-freedom systems to multifrequency parametric excitations

Abstract The method of multiple scales is used to analyze the response of two-degree-of-freedom systems to multifrequency parametric excitations. The equations describing the modulation of the amplitudes with time are derived for the case of four simultaneous resonances: main resonances of the two modes, combination resonance of the difference type, and combination resonance of the summed type. When only two simultaneous resonances exist, the modulation equations can be transformed into a system of four real first-order differential equations with constant coefficients whose solution depends on the eigenvalues of a four by four constant coefficient matrix. When three or more simultaneous resonances exist, the modulation equations cannot be made autononous, in general, and hence one may not be able to solve them except numerically.