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

本项目的目标是对压电式气动弹性能量采集器模型进行严格的数学分析。能量采集器是一种将弹性结构(梁或机翼)的机械振动能量转化为电能的装置。能源采集是一个新兴的领域。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。压电式能量采集器由弹性结构(如梁)和粘结在其上的多层压电材料组成。如果结构在外力(例如，环境气流)的影响下振动，则每个压电层经历变形，这导致在层的表面上出现电压。这种电压产生一种电流，其能量可以被收集。在这个项目中，研究人员研究了三个越来越复杂的气动弹性收割机模型，即其底层弹性结构为飞机机翼的收割机模型。该项目是对研究人员开展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。这是研究人员15年来在飞机机翼模型数学分析和机翼颤振控制方面工作的自然延续。第二个项目是对具有最基本弹性结构的能量收集器模型进行详细的数学分析：欧拉-伯努利梁。对工程上熟知的该模型进行了数值研究和实验验证。这位研究人员在最近的四篇论文中对该模型进行了数学分析。特别是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包括：1)底层结构的方程(S)；2)覆盖在压电层顶面和底面上的电极与外部电载荷形成的电路的方程；3)描述正、逆压电效应的附加项。本项目的目标是研究三种具有物理上真实的基础弹性结构和相似电路的压电收割机模型。结构是描述不受气流影响的细长机翼的弯曲-扭转振动模型(地面振动机翼模型)。该模型由横向挠度和扭转角的两个耦合双曲方程组组成。弹性结构与机翼前缘垂直的环境不可压缩气流中的弯曲-扭转机翼模型相同。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。弹性结构是轴向(平行于翼展)不可压缩气流中的短而僵硬的结构(用于鼾声/呼吸暂停的腭部颤动模型)。上述模型的目的是：a)推导电气弹振型和振型的渐近表示；b)证明振型的Riesz基性质；c)将结果应用于精确的可控性和输出跟踪问题。长期目标是将上述计划扩展到可压缩气流案例。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.