Internal resonance isa unique phenomenon in nonlinear structures. It can lead to pronounced energy transfer between vibrating modes, causing unexpected stress concentrations in local areas and posing a risk to structural integrity. This paper presents a general model updating procedure for nonlinear structures with such internal resonances using experimentally measured data. The test involves using multiple amplitudes of sinusoidal excitations to drive the structure to several required vibration amplitudes, enabling the underlying linear dynamics, uncoupled nonlinear dynamics, and internal resonances to be activated individually. The underlying linear system, uncoupled nonlinear terms, and cross-coupling terms in the dynamic equation are updated sequentially, with updating the cross-coupling terms being the most challenging task. Unlike conventional practices that heuristically pre-assume a model for the cross-coupling force terms, this paper minimises their number by using the coupling potential energies and identifies their type via an interaction map. A novel mode shape residual is then proposed to tune the coefficients of the cross-coupling terms, which allows energy transfer between coupled modes to be captured quantitatively. A diesis-like structure featuring 1:1 bending-torsion coupling was experimentally investigated using multi-level stepped-sine excitations to demonstrate the proposed model updating procedure. The vibrations of its first resonance veered from a bending-dominant motion to a bending-torsion coexisting motion as the driving forces increased, indicating strong amplitude-dependent mode shapes on the occurrence of internal resonance. The experimental data also showed that the softening and hardening trend of the torsional mode was dependent on the vibration amplitude of the bending mode, a peculiar vibration phenomenon seldom reported for realistic structures in the literature. Using the proposed model updating procedure, a two-mode coupled nonlinear model was identified from the frequency response functions and validated by comparing the backbone curves and amplitude-dependent mode shapes. Results showed that the updated model could predict the resonant vibration patterns of the structure away from and around its internal resonance with sufficient accuracy. It was also revealed that the softening and hardening behaviours of the torsional mode were attributed to different branches of Nonlinear Normal Modes (NNMs).

中文翻译：

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使用振型残差对具有 1：1 内部谐振的非线性结构进行模型更新


内部共振是非线性结构中的独特现象。它可能导致振动模式之间明显的能量传递，导致局部区域出现意外的应力集中，并对结构完整性构成风险。本文使用实验测量数据，为具有这种内部共振的非线性结构提供了一种通用的模型更新程序。该测试涉及使用多个正弦激励振幅将结构驱动到几个所需的振幅，从而能够单独激活底层线性动力学、解耦非线性动力学和内部共振。动态方程中的基础线性系统、非耦合非线性项和交叉耦合项按顺序更新，其中更新交叉耦合项是最具挑战性的任务。与启发式地预先假设交叉耦合力项模型的传统做法不同，本文通过使用耦合势能来最小化它们的数量，并通过相互作用图确定它们的类型。然后提出了一种新的振型残差来调整交叉耦合项的系数，从而可以定量捕获耦合模之间的能量传递。使用多级阶跃正弦激励对具有 1：1 弯曲-扭转耦合的 diesis 状结构进行了实验研究，以演示所提出的模型更新程序。随着驱动力的增加，其第一次共振的振动从弯曲主导运动转变为弯曲-扭转共存运动，这表明在发生内部共振时具有很强的振幅依赖性振型。 实验数据还表明，扭转模式的软化和硬化趋势取决于弯曲模式的振动幅度，这是文献中很少报道的实际结构的特殊振动现象。使用所提出的模型更新程序，从频率响应函数中确定了一个双模耦合非线性模型，并通过比较主干曲线和幅度相关振型进行了验证。结果表明，更新后的模型可以足够准确地预测结构远离其内部共振和周围的共振振动模式。研究还揭示了扭转模态的软化和硬化行为归因于非线性正则模态 （NNM） 的不同分支。