Self-excitation, Limit Cycle Oscillations, and Control of Large Deflection Plate Models in Engineering Applications

这项工作的重点是预测，抑制和优化的弹性不稳定性在三个工程设置：（i）桥面，（ii）压电能量采集器，和（iii）飞机和弹丸镶板。在这些设置中，气流可能导致颤振不稳定性的发生，从而引起结构中的普遍振荡行为。我们进行了全面的分析，这种现象，通过检查特定的偏微分方程模型的柔性板。虽然相关的模型是基于应用，这些板方程是最近。研究工作将确定薄结构是否、何时以及如何被环境气流破坏。因此，可以在工程中实现和理解大挠度建模方面进行改进，并将提高预测能力，以开发技术，如下一代射弹镶板和压电弹性能量采集器。理解这些非线性动力学转化为有效设计新设备的能力。这项研究可能会影响其他应用领域，包括人体组织的生物控制。该奖项还为本科生和研究生以及博士后学者提供研究机会。我们解决三个板块配置在一个统一的方式，通过专注于非线性板块建模和相关的演化偏微分方程的稳定性。我们确定并执行相关的计算方法和数值研究。物理配置需要混合和更高的边界条件的四阶偏微分方程的空间域上的角。在每种配置中，我们分析了PDE解决方案的定性性质，这需要了解系统参数和几何形状的函数的动态。 通过自失稳框架，我们研究了由这些板块系统产生的动力系统的极限环振荡（和全局吸引子）。定性研究使我们能够反思我们的建模选择，这项工作：（i）改进板建模，以产生更好的预测;（ii）开发一个严格的理论，新的模型;（iii）解决了文献中的差距，不稳定的非线性板与混合边界条件;（iv）代表建模，理论，该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的评估支持影响审查标准。

This work focuses on the prediction, suppression, and optimization of elastic instability in three engineering settings: (i) bridge decks, (ii) piezo-electric energy harvesters, and (iii) aircraft and projectile paneling. In these settings, airflow may lead to the onset of flutter instability, giving rise to pervasive oscillatory behaviors in the structure. We perform a comprehensive analysis of this phenomenon through the examination of particular partial differential equation models of flexible plates. Although the relevant models are based on applications, these plate equations are quite recent. The research work will determine if, when, and how thin structures are destabilized by ambient airflows. Accordingly, improvements can be made in the implementation and understanding of large deflection modeling in engineering and will improve predictive capabilities for the purposes of developing technologies, such as next-generation projectile paneling and piezo-elastic energy harvesters. Understanding these nonlinear dynamics translates into the ability to design new devices efficiently and effectively. This research may impact additional application areas, including biological control of human tissue. This award also provides research opportunities for undergraduate and graduate students, and postdoctoral scholars. We address three plate configurations in a unified manner through a focus on nonlinear plate modeling and the (in)stability of associated evolutionary partial differential equations. We determine and execute related computational methods and numerical studies. The physical configurations require mixed and higher boundary conditions for 4th-order PDEs on spatial domains with corners. In each configuration, we analyze the qualitative properties of PDE solutions, which requires an understanding of the dynamics as a function of system parameters and geometry. Via the self-destabilization framework, we investigate limit cycle oscillations (and global attractors) for the dynamical systems generated by these plate systems. Qualitative studies allow us to reflect on our modeling choices, and this work: (i) improves plate modeling to yield better predictions; (ii) develops a rigorous theory for novel models; (iii) addresses a gap in the literature concerning unstable nonlinear plates with mixed boundary conditions; (iv) represents a synergy of modeling, theory, and computation.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.