Analysis and Control of Mathematical Models of Fluttering Plates

在气动弹性领域中的基本问题之一是被称为气动弹性颤振的不稳定性的预测和控制。当气动载荷激发结构的固有振动模式时，颤振发生在浸没在气流中的柔性结构中;结果是结构的位移与气流场中的扰动之间潜在的剧烈相互作用。这种现象可能发生在许多应用中，包括：强风中的建筑物和桥梁，旗帜状结构，人类呼吸系统以及空中和陆地车辆上的面板和襟翼结构。在飞机的背景下，颤振是一个关键的问题。如果由于颤振引起的结构位移的幅度足够大，则可能发生结构失效。长时间持续的小振荡也可能在结构中带来代价高昂和/或危险的疲劳。最近，已经提出利用颤振（自然发生的或诱导的）的想法，以通过压电“收获”提供替代的能量源。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，并根据其物理参数预测系统的动力学。该项目包括与壁板和襟翼颤振相关的主模型的偏微分方程（PDE）控制分析。由于各种原因，PDE分析得出的结果是有价值的。他们：（i）指导和简化颤振阈值的实验和数值确定，（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配置，什么类型和位置的阻尼将是有效的。拟议的调查是基于最近的进展，气动弹性，已允许扩展的经典流板模型在过去的50年中使用。获得流结构系统的结果是苛刻的，因为在界面处的动力学的不匹配规律性和耦合中的不好定义或无界迹项中出现问题。在双曲迹正则性理论、抽象耦合模型、时滞偏微分方程和几何约束阻尼等方面的最新进展使得流板模型的现代偏微分方程分析变得容易处理。本文主要研究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动态流动边界条件。完全非线性模型占在平面内和平面外的运动结构，以及非线性流体，被认为是。此外，最近的稳定性分析将扩展到中间的“跨音速”流态和活塞理论的高超音速流态。除了这些模型的适定性之外，时间收敛特性（即，吸引子）的动力学将被认为是确定灵敏度的非瞬态行为的系统板的边界条件和外部负载。目前的建议可以被看作是一个分析模型所产生的气动弹性，通过提供一个定性性质之间的比较适定偏微分方程的实验观察到的和/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.