Flutter Analysis and Control for Elastic Structure in Axial Air Flow: Applications to Palatal Flutter and Energy Harvesting

本项目的目的是对弹性固体结构(长薄板)与轴向气流(平行于板的气流？S轴)相互作用的数学模型进行详细的分析，并在此分析结果的基础上研究结构的颤振控制问题。板的动力学由两个耦合的双曲型偏微分方程组控制，在气动弹性中称为Goland模型。气流是平行于主板轴线的气流的微小扰动，假定为无粘性、位势、等熵和亚音速。气流的动力学受欧拉流体动力学方程的支配。由于上述物理假设，欧拉方程可以简化为一个关于微扰势的三维线性双曲方程。该方程通过一组特定的边界条件与结构方程组相耦合：(A)流-切线条件，(B)库塔-朱可夫斯基条件，和(C)远场条件。该项目的目标包括：(A)模型的渐近、频谱和稳定性分析，(B)分析可能的颤振控制机制，(C)将线性模型推广到包含非线性结构方程的模型(Dowell-Hodges模型)，并将颤振作为极限环振荡进行研究。该模型有两个主要的实际应用：(A)导致打呼和睡眠呼吸暂停的软腭的颤动，(B)压电能量的收集。该项目是PI 12年来对飞机机翼模型进行渐近、频谱和稳定性分析以及颤振控制工作的继续。固体结构与空气或流体相互作用的例子包括：飞机机翼和尾翼、悬索桥、输电线、血管壁和支气管气道等。将所有这些例子结合在一起的现象是颤振，即结构的突然混沌振动，当空气或流体的速度达到称为绝对速度的某个临界值时发生。该项目对最近发展起来的弹性固体平板在轴向气流(平行于平板主轴的气流)中的数学模型进行了理论分析。科学界已经开始对该模型进行实验和计算研究。然而，理论分析的重要性是显而易见的：它可以提供新的见解，并且对于颤振控制机构的设计是必要的。轴向流动情况比正常流动情况(流动垂直于平板？S主轴)更具挑战性，而正常流动情况存在于所有机翼模型中。该模型的两个主要应用之一是医学，它处理的是腭音：软腭的不受控制的振动导致打鼾声，甚至睡眠呼吸暂停。目前，治疗方法包括外科手术和设计抗鼾器。第二个应用是从完全振动中获取压电能的一个新兴领域。这一研究方向的目标是开发一种新技术，用于提供替代电源和/或为电池或电容器等存储设备充电。这一概念在减少化学废物方面具有生态意义，并通过显著降低维护成本获得潜在的经济收益。

The objective of this project is to carry out a detailed analysis of a mathematical model of an elastic solid structure (a long thin rectangular plate) interacting with an axial air flow (a flow parallel to the plate?s axis), and based on the results of this analysis, to investigate the problem of a flutter control for the structure. The dynamics of the plate is governed by a system of two coupled hyperbolic partial differential equations known in aeroelasticity as the Goland model. The airflow, which is a small perturbation of a stream parallel to the main plate's axis, is assumed to be inviscid, potential, isentropic, and subsonic. The dynamics of the airflow is governed by the Euler hydrodynamic equation. Owing to the above physical assumptions, the Euler equation can be reduced to a single three-dimensional linear hyperbolic equation for the perturbation potential. This equation is coupled with the system of structural equations by a set of specific boundary conditions: (a) the flow-tangency condition, (b) the Kutta-Joukowski condition, and (c) the far-field condition. The goals of the project include the following: (a) asymptotic, spectral, and stability analysis of the model, (b) analysis of possible flutter control mechanisms, (c) generalization of linear model to the model involving nonlinear structural equations (the Dowell-Hodges model) and investigation of flutter as limit cycle oscillations. The model has two major practical applications: (a) flutter of a soft palate (the palatal flutter) resulting in snoring and sleep apnoea, (b) piezoelectric power harvesting. The project is a continuation of the PI's 12-year work on asymptotic, spectral, and stability analysis and on flutter control for aircraft wing models.Examples of solid structures interacting with an air or fluid flow include: aircraft wings and tails, suspension bridges, electric power lines, walls of blood vessels and bronchial airways, etc. The phenomenon that unites all the examples is flutter, i.e., sudden chaotic vibrations of the structure, which occur when the speed of the air or fluid flow reaches certain critical value called the utter speed. The project deals with theoretical analysis of a recently developed mathematical model of an elastic solid plate in an axial air flow (an air flow parallel to the plate's main axis). An experimental and computational investigation of the model has already begun in the scientific community. However, an importance of theoretical analysis is obvious: it can provide new insights and is necessary for a design of flutter control mechanisms. The axial flow case is more challenging than a normal flow case (the flow is perpendicular to the plate?s main axis), which occurs in all aircraft wing models. One of the two major applications of the model is medical, which deals with palatal utter: uncontrolled vibrations of a soft palate resulting in snoring and even sleep apnoae. Currently, treatments involve surgical procedures and designing anti-snoring devices. The second application is a newly emerging area of piezoelectric energy harvesting from utter vibrations. The goal of this research direction is to develop a new technology for providing alternative sources of electric power and/or recharging storage devices such as batteries or capacitors. The concept has ecological ramifications in reducing the chemical waste and potential monetary gains by significantly reducing maintenance cost.