In this paper, we introduce a new nonlocal modal hydrodynamic theory for fluid–structure interactions (FSI) of light, flexible cantilever beams and plates undergoing small amplitude vibrations in Newtonian, incompressible, viscous, heavy fluids otherwise at rest. For low aspect ratio flexible structures and high mode numbers, three dimensional (3D) and nonlocal fluid effects become prominent drivers of the coupled dynamics, to the point that existing local hydrodynamic theories based on two dimensional (2D) fluid approximations become inadequate to predict the system response. On the other hand, our approach is based on a rigorous, yet efficient, 3D treatment of the hydrodynamic loading on cantilevered thin structures. The off-line solution of the FSI problem results in the so-called nonlocal modal hydrodynamic function matrix, that is, the representation of the nonlocal hydrodynamic load operator on a basis formed by the structural modes. Our theory then integrates the nonlocal hydrodynamics within a fully coupled structural modal model in the frequency domain. We compare and discuss our theory predictions in terms of frequency response functions, mode shapes, hydrodynamic loads, quality factors, added mass ratios with the predictions of the classical local approaches, for different actuation scenarios, identifying the limitations of the hypotheses underlying existing treatments. Importantly, we also validate our new model with experiments conducted on flexible square plates. While computationally efficient, our fully coupled theory is exact up to numerical truncation and can bridge knowledge gaps in the design and analysis of FSI systems based on low aspect ratio flexible beams and plates.

中文翻译：

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粘性流体中梁和板振动的非局部模态流体动力函数理论

在本文中，我们介绍了一种新的非局部模态流体动力学理论，用于研究轻质柔性悬臂梁和板的流固相互作用（FSI），这些悬臂梁和板在牛顿、不可压缩、粘性、重流体中经历小振幅振动，否则处于静止状态。对于低纵横比柔性结构和高模态数，三维 (3D) 和非局部流体效应成为耦合动力学的主要驱动因素，以至于基于二维 (2D) 流体近似的现有局部流体动力学理论不足以预测系统响应。另一方面，我们的方法基于对悬臂薄结构上的流体动力载荷进行严格而高效的 3D 处理。 FSI问题的离线求解得到所谓的非局部模态水动力函数矩阵，即基于结构模态形成的非局部水动力载荷算子的表示。然后，我们的理论将非局部流体动力学集成到频域中的完全耦合结构模态模型中。我们针对不同的驱动场景，在频率响应函数、振型、水动力载荷、品质因数、添加质量比方面对我们的理论预测与经典局部方法的预测进行了比较和讨论，确定了现有治疗方法假设的局限性。重要的是，我们还通过在柔性方形板上进行的实验来验证我们的新模型。虽然计算效率很高，但我们的全耦合理论在数值截断方面是精确的，可以弥补基于低纵横比柔性梁和板的 FSI 系统设计和分析方面的知识差距。