Resonance of a nonlinear forced system with two-frequency parametric and self-excitations

The resonance of a single-degree-of-freedom forced nonlinear system with two-frequency parametric and self-excitations is investigated. In particular, the case in which the parametric excitation terms and the excitation force with close frequencies is examined. The method of multiple scales is used to determine the equations of modulation of amplitude and phase. The behaviour of steady-state responses and their stability are studied. The effects of damping, and magnitudes of self-excitation, parametric excitation and the excitation force are analyzed. The theoretical results are verified by numerical integration results of the governing equation and the modulation equations.