An incremental harmonic balance method with two time-scales for quasi-periodic responses of a Van der Pol–Mathieu equation

Abstract

An incremental harmonic balance (IHB) method with two time-scales is presented and used for calculating accurate quasi-periodic responses of a one-degree-of-freedom Van der Pol–Mathieu equation with coupled self-excited vibration and parametrically excited vibration with 1:2 resonance. For periodic responses of the Van der Pol–Mathieu equation with only one basic frequency, the traditional IHB method is used to automatically trace their nonlinear frequency response curves. Stability and bifurcations of the periodic responses for given parameters are then determined by the Floquet theory using the precise Hsu’s method. It is found that a jump from a periodic response to a quasi-periodic response at a critical point results from a saddle node bifurcation. For quasi-periodic responses of the Van der Pol–Mathieu equation, their spectra contain uniformly spaced sideband frequencies that have not been observed heretofore, which involve two incommensurate basic frequencies, i.e., the parametric excitation frequency and a priori unknown frequency related to uniformly spaced sideband frequencies. The IHB method with two time-scales is formulated to deal with cases where one basic frequency is unknown a priori, in order to automatically trace nonlinear frequency response curves of quasi-periodic responses of the Van der Pol–Mathieu equation with 1:2 resonance and accurately calculate all frequency components and their corresponding amplitudes even at critical points. Results of the Van der Pol–Mathieu equation 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. This investigation reveals rich dynamic characteristics of the Van der Pol–Mathieu equation in a wide range of parametric excitation frequencies.