Modeling and PDE Theory for The Large Deflections

Abstract

Flutter is defined as a self-excitation of a thin structure where a surrounding flow destabilizes its natural elastic modes. Cantilevers are particularly prone to flutter, and it has been shown that this instability can induce large displacements from which mechanical energy can be captured via piezoelectric laminates. To effectively harvest energy in this manner, one must have viable models that describe the behavior of the cantilever's large deflections after the onset of flutter. The aim of this dissertation is to introduce the modeling and the mathematical analysis that correspond to such systems. The first part of this dissertation focuses on a recent PDE model that derives the equations of motion for an inextensible cantilevered beam via Hamilton's principle. The theoretical results are centered around the existence, uniqueness, and decay of strong solutions. In addition, numerical results are available where a modal approach is used to provide insight into the features and limitations of this model. The next part of this dissertation is centered around two-dimensional cantilevered plates. Firstly, the modeling of large deflections for a cantilevered plate is addressed. Various modeling hypotheses are explored and Hamilton's principle is employed to derive the corresponding equations of motion. Following this, the linear (Kirchhoff-Love) cantilevered plate is used to develop a semigroup argument that addresses the well-posedness of this configuration. In the last part, a system that considers the coupling between the structure of a linear cantilevered beam with a full potential flow is introduced. This flow is given by a perturbed wave equation but taken with mixed boundary conditions of Kutta-Joukowsky type.