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

摘要

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 IH B 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 theor y 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 IH B 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 a l l frequency components and their corresponding amplitudes even at critical points. Results of the Van der Pol- Mathieu equation obtained from the IH B 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.