Incremental Harmonic Balance with Two Time Scales for a Nonlinear Quasi-Periodic Mathieu Equation

Quasi-periodic solution of a damped quasi-periodic (QP) Mathieu’s equation with cubic nonlinearity is obtained by using the incremental harmonic balance (IHB) with two time scales. It is found that the QP equation consists of stable and unstable QP solutions and every solution consists of carrier frequencies and their spaced uniformly sidebands. The IHB with two time scales is applied to trace the response curve of the amplitude of the parameter excitation. Besides, three types of QP solutions are found, which all agree very well with the results from numerical integration. While the perturbation method using the double-step method of multiple scales (MMS) obtains only one type of QP solutions, since the small excitation frequency is nearly the natural frequency of the second perturbation equation. Finally the stability of QP solutions is analyzed by the Floquet theory.