Analysis and Control of Mathematical Models of Fluttering Plates

气动弹性领域的一个基本问题是气动弹性颤振失稳的预报和控制。当气动载荷激发结构的自然振动模式时，在气流中的柔性结构中会发生颤振；结果是结构的位移和气流中的摄动之间可能会发生剧烈的相互作用。这种现象可能会在许多应用中发生，包括：大风中的建筑物和桥梁、国旗状结构、人体呼吸系统以及空中和陆地车辆上的面板和襟翼结构。在飞机方面，颤振是一个关键问题。如果颤振引起的结构位移足够大，则可能发生结构破坏。长时间持续的微小振动也可能导致结构中代价高昂的和/或危险的疲劳。最近，有人提出利用颤振(自然发生的或诱导的)的想法，通过压电体的“收获”来提供另一种能源。由于这些原因，人们对建立描述颤振现象的数学模型非常感兴趣，以便深入了解流-结构耦合的机制并基于其物理参数预测系统的动力学。该项目包括偏微分方程组(PDE)的控制，对与壁板和襟翼颤振相关的主要模型的分析。从偏微分方程组分析得到的结果是有价值的，原因有很多。它们：(I)指导和简化实验和数值颤振阈值的确定，(Ii)可以提高实验的成本效益和缩短原型的设计时间，(Iii)指出对于给定的流板构型，什么类型和位置的阻尼将是有效的。要获得流动-结构系统的结果是很困难的，因为在界面上动力学的失配规律性和耦合中定义不明确或无界的迹项出现了问题。双曲道正则性理论、抽象耦合模型、时滞偏微分方程组和几何约束阻尼的最新进展使流板模型的现代偏微分方程组分析变得容易进行。本文研究了一类非线性流板模型在反馈控制下的适定性和稳定性，该模型包括部分自由板边界条件和板边缘附近的动态流边界条件。考虑了结构中平面内和平面外运动的完全非线性模型，以及非线性流体。此外，最近的稳定性分析将扩展到中间的“跨音速”流型和活塞理论的高超声速流型。除了这些模型的适定性外，还将考虑动力学的时间收敛性质(即吸引子)，以确定系统的非暂态行为对板的边界条件和外部载荷的敏感性。目前的建议可以被看作是对气动弹性模型的分析，通过比较适定偏微分方程组的定性性质与实验观察到的和/或数值近似的行为。

One of the fundamental problems in the field of aeroelasticity is the prediction and control of the instability known as aeroelastic flutter. Flutter occurs in a flexible structure immersed in a gas flow when aerodynamic loading excites the natural oscillatory modes of the structure; the result is a potentially violent interaction between the displacements of the structure and perturbations in the gas flow field. This phenomenon may occur in a multitude of applications including: buildings and bridges in strong winds, flag-like structures, the human respiratory system, and panel and flap structures on air and land vehicles. In the context of aircraft, flutter is a key concern. If the magnitude of the structural displacements due to flutter is sufficiently large, structural failure can occur. Small oscillations sustained over long periods of time may also bring about costly and/or hazardous fatigue in the structure. Very recently, the idea of harnessing flutter (naturally occurring, or induced) has been suggested to provide an alternative source of energy via piezoelectric "harvesting". For these reasons there is great interest in producing mathematical models that describe the flutter phenomenon in order to gain insight into the mechanisms of flow-structure coupling and predict the dynamics of the system based on its physical parameters. This project comprises a control of partial differential equations (PDEs) analysis of the principal model associated to panel and flap flutter. Results derived from PDE analyses are valuable for a variety of reasons. They: (i) guide and streamline experimental and numerical flutter threshold determination, (ii) can improve cost-effectiveness of experimentation and cut-down on design time of prototypes, (iii) indicate what types and locations of damping will be effective for a given flow-plate configuration. The proposed investigations are based upon very recent progress in aeroelasticity that has permitted extensions of a classical flow-plate model used over the last 50 years. Obtaining results for flow-structure systems is demanding, as problems arise in the mismatching regularity of dynamics at the interface and ill defined or unbounded trace terms in the coupling. Recent advances in hyperbolic trace regularity theory, abstract coupled models, PDEs with delay, and geometrically constrained damping make modern PDE analysis of flow-plate models tractable. This proposal centers on well-posedness and stability in the presence of feedback controls for a class of nonlinear flow-plate models which include partially free plate boundary conditions and dynamic flow boundary conditions near plate edges. Fully nonlinear models accounting for both in-plane and out-of-plane motion in the structure, as well as nonlinear fluids, are considered. Moreover, recent stability analyses will be extended to the intermediary "transonic" flow regime and the piston-theoretic, hypersonic regime. Beyond well-posedness of these models, time convergence properties (i.e., attractors) of the dynamics will be considered to determine the sensitivity of non-transient behavior of the system to the plate's boundary conditions and external loading. The current proposal can be viewed as an analysis of models arising in aeroelasticity by providing a comparison between qualitative properties of well-posed PDEs to experimentally observed and/or numerically approximated behaviors.