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

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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.