Collaborative Research: Experiment, Theory, and Simulation of Aeroelastic Limit Cycle Oscillations for Energy Harvesting Applications

在强调无碳能源、电力系统最小化和分散发电的背景下，基于振动的能量收集技术的发展在过去的20年里已经成为人们非常感兴趣的话题。典型的气动弹性能量收集方案是将一个薄的弹性结构浸入流体（水或空气）流动中。在一定条件下，流动的存在会引起结构自激振动，即颤振，从而导致周期振荡。最近有研究表明，通过粘贴压电材料，可以从这些结构振荡中收集电能。这种能源有望成为一种替代能源，但也带来了数学建模和实验实施的挑战。细长悬臂板在轴向流动中尤其容易发生颤振失稳，即使在低流速下也是如此。为了有效地从这个系统中获取能量，必须了解所产生的极限环振荡的定性特征。这是通过建立适当的偏微分方程模型，分析其数学性质，通过科学计算模拟动力学，并将这些结果与实验数据进行比较来实现的。该项目将为本科生和研究生的培训提供机会和支持。从技术角度来看，上述分析需要理解由流动驱动的大悬臂挠度。所研究的模型必须捕捉到与非线性悬臂动力耦合的流动的失稳效应。与传统的大挠度弹性（基于结构的拉伸能力）不同，悬臂梁的不可扩展性会引起主要的非线性效应：非局部惯性和非线性刚度。本项目推导并分析了悬臂结构在轴向流动中颤振后行为的PDE模型。具体来说，它：(i)改进悬臂建模并做出预测，这将导致更好的能量收集设备；（ii）为一种新的弹性模型建立了严格的PDE解理论；（iii）解决了数学文献中关于不稳定非线性悬臂梁的空白；（iv）改进了非线性悬臂梁的谱和有限元计算方法；(v)建立模型、理论、计算和风洞实验的协同作用；（vi）提供了一个清晰的表述，具有挑战性的数学问题，与工程应用有直接和可实现的联系。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Given the emphasis on carbon-free energy, power system minimization, and decentralized power generation, the development of vibration-based energy harvesting technologies has become a topic of great interest over the past 20 years. A typical aeroelastic energy harvesting scheme involves immersing a thin, elastic structure in a fluid (water or air) flow. Under certain conditions, the presence of the flow brings about a structural self-excited vibration, known as flutter, resulting in periodic oscillations. It has been recently shown that, via affixed piezo-electric materials, electrical energy can be harvested from these structural oscillations. This sort of energy holds promise as an alternative energy source, but also brings about mathematical modeling and experimental implementation challenges. Slender cantilever plates in an axial flow are particularly prone to the flutter instability, even at low flow speeds. To effectively and efficiently harvest energy from this system, one must understand the qualitative features of the resulting limit cycle oscillation. This is done by formulating appropriate partial differential equation models, analyzing their mathematical properties, simulating dynamics through scientific computation, and comparing these results against experimental data. This project will provide opportunities and support for the training of undergraduate and graduate students. From a technical point of view, the analysis described above requires understanding large cantilever deflections driven by a flow. The models studied must capture the de-stabilizing effects of the flow as coupled to nonlinear cantilever dynamics. Unlike traditional large deflection elasticity (based on the structure's ability to stretch), a cantilever's inextensibility gives rise to the primary nonlinear effects of interest: nonlocal inertia and nonlinear stiffness. This project derives and analyzes PDE models for the post-flutter behavior of cantilevered structures in an axial flow of fluid. Specifically, it: (i) improves cantilever modeling and makes predictions that will lead to better energy harvesting devices; (ii) develops a rigorous theory of PDE solutions for a novel elasticity model; (iii) addresses a gap in the mathematical literature concerning unstable nonlinear cantilevers; (iv) refines spectral and finite element computational methods for nonlinear cantilevers; (v) creates a synergy of modeling, theory, computation, and wind-tunnel experimentation; (vi) provides a clearly formulated, challenging mathematical problem with a direct and realizable connection to engineering applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.