The resonances of forced dynamical systems occur when either the amplitude of the frequency response undergoes a local maximum (amplitude resonance) or phase lag quadrature takes places (phase resonance). This study focuses on the phase resonance of nonlinear mechanical systems subjected to single-point, single-harmonic excitation. In this context, the main contribution of this paper is to develop a computational framework which can predict the mode shapes and oscillation frequencies at phase resonance. The resulting nonlinear modes are termed phase resonance nonlinear modes (PRNMs). A key property of PRNMs is that, besides primary resonances, they can accurately characterize superharmonic, subharmonic and ultra-subharmonic resonances for which, as shall be shown in this paper, phase lags at resonance may be different from $\pi ∕2$. The proposed developments are demonstrated using one- and two-degree-of-freedom systems featuring a cubic nonlinearity.

当频率响应的振幅经历局部最大值（振幅共振）或发生相位滞后正交（相位共振）时，就会发生受迫动力系统的共振。本研究侧重于受单点、单谐波激励的非线性机械系统的相位共振。在这种情况下，本文的主要贡献是开发一个计算框架，可以预测相位共振时的模式形状和振荡频率。产生的非线性模式称为相位共振非线性模式 (PRNM)。PRNM 的一个关键特性是，除了初级谐振之外，它们还可以准确地表征超谐波、次谐波和超次谐波谐振，如本文所述，谐振时的相位滞后可能不同于$\pi ∕2$. 使用具有三次非线性特征的一自由度和二自由度系统演示了所提出的发展。