Asymptotic and Spectral Analysis and Control Problems for Aeroelastic Energy Harvester Models

本项目的目的是对压电气动弹性能量采集器模型进行严格的数学分析。能量收集器是一种将弹性结构（横梁或飞机机翼）的机械振动能量转化为电能的装置。能量收集是一个新兴的领域。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。压电能量收集器由弹性结构（例如，梁）和粘接在其上的压电材料层组成。如果结构在外力（例如，环境气流）的影响下振动，那么每个压电层都会发生变形，从而在层的表面上出现电压。这种电压产生的电流的能量可以被收集。在这个项目中，研究者研究了三种越来越复杂的气动弹性收割机模型，即收割机的底层弹性结构是飞机机翼的模型。该项目是由研究者进行的两个研究项目的结果和模型的概括。第一个是对飞机机翼模型的严格调查。这是研究者在飞机机翼模型数学分析和机翼颤振控制方面15年工作的自然延续。第二个项目是对能量收集器模型进行详细的数学分析，该模型具有最基本的底层弹性结构：欧拉-伯努利梁。对该工程模型进行了数值研究和实验验证。该研究人员在最近的四篇论文中提出了该模型的数学分析。特别对该模型的控制问题进行了分析。压电能量采集器的数学模型由以下几个部分组成：1)底层结构方程；2)覆盖压电陶瓷层上下面的电极与外部电负载形成的电路方程；3)描述直接和反向压电效应的附加术语。本项目的目标是研究三种具有物理上真实的底层弹性结构和相似电路的压电收割机模型。1、结构为不受气流影响的细长飞机机翼的弯扭振动模型（地面振动机翼模型）。该模型是由两个耦合的双曲方程系统给出的，双曲方程表示横向挠度和扭转角。2。弹性结构是在与机翼前缘垂直的不可压缩气流中相同的弯曲-扭转机翼模型。结构方程包含附加的时间卷积积分项，表示气流施加在机翼上的力和力矩。3。弹性结构是轴向（平行于翼展）不可压缩气流中的短刚性结构（打鼾/呼吸暂停的腭扑动模型）。上述模型的目标是：a)推导电气动弹性模态和模态振型的渐近表示；b)模态振型Riesz基性质的证明；C)将结果应用于精确可控性和输出跟踪问题。长期目标是将上述程序扩展到可压缩气流情况。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The objective of this project is a rigorous mathematical analysis of piezoelectric aeroelastic energy harvester models. An energy harvester is a device that transforms the energy of mechanical vibration of an elastic structure (a beam or an aircraft wing) into electric energy. Energy harvesting is a newly emerging area. 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. A piezoelectric energy harvester consists of an elastic structure (e.g., a beam) and layers of a piezoelectric material bonded to it. If the structure vibrates under an influence of external forces (e.g., an ambient airflow) then each of the piezoelectric layers undergoes a deformation which results in the appearance of an electric voltage on the faces of the layers. This voltage produces an electric current whose energy can be harvested. In this project the investigator studies three increasingly more complicated aeroelastic harvester models, i.e., models of a harvester whose underlying elastic structure is an aircraft wing. The project is a generalization of the results and models of two research projects carried out by the investigator. The first one is a rigorous investigation of aircraft wing models. It is a natural continuation of the investigator's 15 years of work on mathematical analysis of aircraft wing models and on wing flutter control. The second project is a detailed mathematical analysis of an energy harvester model with the most basic underlying elastic structure: the Euler-Bernoulli beam. The model, well known in engineering, was studied numerically and validated experimentally. Mathematical analysis of the model is presented in four recent papers by the investigator. In particular, control problems for this model were analyzed.A mathematical model of a piezoelectric energy harvester is composed of the following: 1) the equation(s) of the underlying structure; 2) the equation of the electric circuit formed by the electrodes covering the top and bottom faces of the piezoceramic layer and by the external electric load; 3) additional terms describing the direct and inverse piezoelectric effects. The goal of this project is to investigate three piezoelectric harvester models with physically realistic underlying elastic structures and similar electric circuits. I. The structure is the bending-torsion vibration model describing a long slender aircraft wing, not affected by an airflow (ground vibration wing model). The model is given as a system of two coupled hyperbolic equations for the transverse deflection and the torsion angle. II. The elastic structure is the same bending-torsion wing model in an ambient incompressible air flow that is normal to the leading edge of the wing. The structural equations contain additional time-convolution integral terms representing the forces and moments exerted on the wing by the flow. III. The elastic structure is a short stiff structure in an axial (parallel to wing span) incompressible airflow (a palatal flutter model for snoring/apnea.) The goals for the above models are: a) derivation of asymptotic representation for the electroaeroelastic modes and mode shapes; b) proof of the Riesz basis property of the mode shapes; c) application of results to exact controllability and output tracking problems. The long term goal is to extend the above program to compressible airflow cases.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.