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

本项目的目标是对弹性固体结构（长而薄的矩形板）与轴向气流(平行于板的气流？S轴)，并在此基础上对结构的颤振控制问题进行了研究。板的动力学是由两个耦合的双曲偏微分方程系统控制的，在气动弹性学中称为戈兰模型。气流是平行于主板块轴线的气流的一个小扰动，假定它是无粘的、势的、等熵的和亚音速的。气流的动力学是由欧拉流体动力学方程控制的。基于上述物理假设，欧拉方程可以简化为一个三维线性双曲方程。该方程通过一组特定的边界条件与结构方程组耦合：(a)流动切线条件，(b)库塔-儒科夫斯基条件，(c)远场条件。该项目的目标包括：(a)模型的渐近分析、谱分析和稳定性分析，(b)可能的颤振控制机制分析，(c)将线性模型推广到涉及非线性结构方程的模型（dowwell - 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)软腭的扑动（腭扑动）导致打鼾和睡眠呼吸暂停，(b)压电能量收集。该项目是PI在飞机机翼模型的渐近、频谱和稳定性分析以及颤振控制方面12年工作的延续。固体结构与空气或流体流动相互作用的例子包括：飞机的机翼和尾部、悬索桥、电力线路、血管壁和支气管气道等。将所有例子结合起来的现象是颤振，即结构的突然混沌振动，当空气或流体流动的速度达到某个称为绝对速度的临界值时发生。该项目涉及最近开发的轴向气流（平行于板的主轴的气流）弹性固体板的数学模型的理论分析。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，并且是设计颤振控制机构所必需的。轴向流动情况比正常流动情况更具挑战性(流动垂直于板？S主轴)，这发生在所有的飞机机翼模型。该模型的两大主要应用之一是医疗，它处理腭瓣：软腭不受控制的振动导致打鼾，甚至睡眠呼吸暂停。目前，治疗方法包括外科手术和设计防打鼾设备。第二个应用是新兴的压电能量收集领域。该研究方向的目标是开发一种新技术，用于提供替代电力来源和/或为电池或电容器等存储设备充电。这个概念在减少化学废物和潜在的金钱收益方面具有生态影响，因为它显著降低了维护成本。

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.