Abstract:The periodic responses and quasi-periodic motions of a van der Pol-Mathieu equation subjected to three excitations, i.e., self-excited, parametric excitation, and external excitation, are studied in this paper. A new characteristic is observed that the spectra of the quasi-periodic motions contain uniformly spaced sideband frequencies. Firstly, the traditional incremental harmonic balance (IHB) method is used to obtain periodic responses of the van der Pol-Mathieu equation and to trace their nonlinear frequency response curves automaically. Then the Floquet theory is used to analyze stability of the periodic responses and their bifurcations. Based on the characteristic that the spectra of quasi-periodic motions contain two incommensurate basic frequencies, i.e., the excitation frequency and a priori unknown frequency related to uniformly spaced sideband frequencies. Then the IHB method with two time-scales basing on the two basic frequencies is formulated to accurately calculate all frequency components and their corresponding amplitudes even at critical points. All the results obtained from the IHB method with two time-scales are in excellent agreement with those from numerical integration using the fourth-order Runge-Kutta method. Finally, this investigation reveals rich dynamic characteristics of the van der Pol-Mathieu equation in a range of excitation frequencies.