Under certain conditions, the dynamics of a nonlinear mechanical system can be represented by a single nonlinear modal oscillator. This holds, in particular, under external excitation near primary resonance or under self-excitation by negative damping of the respective mode. The properties of the modal oscillator can be determined by computational or experimental nonlinear modal analysis. The simplification to a single-nonlinear-mode model facilitates qualitative and global analysis, and substantially reduces the computational effort required for probabilistic methods and design optimization. Important limitations of this theory are that only purely mechanical systems can be analyzed and that the respective nonlinear mode has to be recomputed when the system’s structural properties are varied. With the theoretical extension proposed in this work, it becomes feasible to attach linear subsystems to the primary mechanical system, and to approximate the dynamics of this coupled system using only the nonlinear mode of the primary mechanical system. The attachments must be described by linear ordinary or differential-algebraic equations with time-invariant coefficient matrices. The attachments do not need to be of purely mechanical nature, but may contain, for instance, electric, magnetic, acoustic, thermal or aerodynamic models. This considerably extends the range of utility of nonlinear modes to applications as diverse as model updating or vibration energy harvesting. As long as the attachments do not significantly deteriorate the host system’s modal deflection shape, it is shown that their effect can be reduced to a complex-valued modal impedance and an imposed modal forcing term. In the present work, the proposed approach is computationally assessed for the analysis of exciter-structure interaction. More specifically, the force drop typically encountered in frequency response testing is revisited. A cantilevered beam with cubic spring and an attached electro-dynamical shaker serves as benchmark. The proposed approach shows excellent accuracy. Mainly the already known limitations of single-nonlinear-mode theory reappear. In particular, higher harmonics should not be too pronounced. In the transient case, the time scales of vibration and amplitude-phase modulation should be well separated, and the attachment dynamics should be in quasi-steady state.

在某些条件下，非线性机械系统的动力学可以用单个非线性模态振荡器来表示。这尤其在接近初级共振的外部激励下或在通过相应模式的负阻尼的自激励下成立。模态振荡器的特性可以通过计算或实验非线性模态分析来确定。简化为单非线性模式的模型有助于进行定性和全局分析，并大大减少了概率方法和设计优化所需的计算量。该理论的重要局限性在于，只能分析纯机械系统，并且当系统的结构特性发生变化时，必须重新计算相应的非线性模式。通过这项工作中提出的理论扩展，将线性子系统连接到主要机械系统，并仅使用主要机械系统的非线性模式来逼近此耦合系统的动力学变得可行。附件必须通过具有时不变系数矩阵的线性普通或微分代数方程来描述。附件不必具有纯粹的机械性质，而可以包含例如电，磁，声，热或空气动力学模型。这极大地将非线性模式的应用范围扩展到了模型更新或振动能量收集等各种应用中。只要附件不会显着恶化主机系统的模态偏转形状，就表明它们的作用可以减小为复数值模态阻抗和强加的模态强迫项。在目前的工作中，所提出的方法在计算上被评估用于激振器-结构相互作用的分析。更具体地说，重新讨论了频率响应测试中通常遇到的力下降问题。带有立方弹簧的悬臂梁和附加的电动振动筛作为基准。所提出的方法显示出极好的准确性。主要是出现了单非线性模式理论的已知局限性。特别是，高次谐波不应太明显。在瞬态情况下，应将振动和幅度相位调制的时间尺度很好地分开，并且附件动力学应处于准稳态。再次探讨了频率响应测试中通常遇到的力下降问题。带有立方弹簧的悬臂梁和附加的电动振动筛作为基准。所提出的方法显示出极好的准确性。主要是出现了单非线性模式理论的已知局限性。特别是，高次谐波不应太明显。在瞬态情况下，应将振动和幅度相位调制的时间尺度很好地分开，并且附件动力学应处于准稳态。再次探讨了频率响应测试中通常遇到的力下降问题。带有立方弹簧的悬臂梁和附加的电动振动筛作为基准。所提出的方法显示出极好的准确性。主要是出现了单非线性模式理论的已知局限性。特别是，高次谐波不应太明显。在瞬态情况下，应将振动和幅度相位调制的时间尺度很好地分开，并且附件动力学应处于准稳态。