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.

提出了一种具有两个时间尺度的增量谐波平衡 (IHB) 方法，用于计算具有耦合自激振动和参数激励振动的单自由度范德波尔-马蒂厄方程的准确准周期响应1:2 共振。对于只有一个基本频率的 Van der Pol-Mathieu 方程的周期响应，使用传统的 IHB 方法来自动跟踪它们的非线性频率响应曲线。然后通过 Floquet 理论使用精确的 Hsu 方法确定给定参数的周期性响应的稳定性和分岔。发现在临界点从周期响应跳跃到准周期响应是由鞍点分叉引起的。对于 Van der Pol-Mathieu 方程的准周期响应，它们的频谱包含迄今未观察到的均匀间隔的边带频率，其涉及两个不公约的基本频率，即参数激励频率和与均匀间隔的边带频率相关的先验未知频率。制定了具有两个时间尺度的 IHB 方法来处理一个基本频率先验未知的情况，以便自动跟踪具有 1:2 共振的 Van der Pol-Mathieu 方程的准周期响应的非线性频率响应曲线甚至在关键点准确计算所有频率分量及其相应的幅度。从具有两个时间尺度的 IHB 方法获得的 Van der Pol-Mathieu 方程的结果与使用四阶 Runge-Kutta 方法的数值积分获得的结果非常一致。