Quasi-Periodic Solutions of a Damped Nonlinear Quasi-Periodic Mathieu Equation by the Incremental Harmonic Balance Method with Two Time Scales

Abstract

Quasi-periodic (QP) solutions of a damped nonlinear QP Mathieu's equation with cubic nonlinearity are investigated by using the incremental harmonic balance (IHB) method with two time scales. The damped nonlinear QP Mathieu's equation contains two incommensurable harmonic excitation frequencies, one is a small frequency while the other nearly equals to twice the linear natural frequency. It is found that Fourier spectra of QP solutions of the equation consist of uniformly spaced sidebands due to cubic nonlinearity. The IHB method with two time scales, which relates to the two excitation frequencies, is adopted to trace solution curves of the equation in an automatical way and all frequencies of solutions and their corresponding amplitudes. Effects of parametric excitation are studied in detail. The stability of a QP solution is evaluated from the Floquet theory via examining the perturbation superposing on the QP solution. Three types of QP solutions can be obtained from the IHB method, which agree very well with the results from numerical integration. However, the perturbation method using the double-step method of multiple scales (MMS) obtains only one type of QP solutions since the ratio of the small frequency to the linear natural frequency of the first reduced-modulation equation is nearly 1 in the second perturbation procedure, while the other two types of QP solutions from the IHB method do not need the ratio. Furthermore, the results from the double-step MMS are different from those from numerical integration and the IHB method.