Saturation of Almost Periodic and Chaotic Aeroelastic Oscillations of Plates Under a Resonant Multimode Force

Abstract

A new approach based on the solution of a singular integral equation to analyzing the aeroelastic oscillations of thin plates is proposed. The velocity circulation is expanded into a series in the generalized coordinates of oscillations of the plate. This allows using a system with a finite number of degrees of freedom for the generalized coordinates to describe the aeroelastic oscillations. The following scenario is analyzed: the motion of the plate undergoes Neimark bifurcations and transform into quasiperiodic oscillations, which, in turn, transform into chaotic oscillations Here we propose a new method for analyzing the aeroelastic oscillations of plates undergoing geometrically nonlinear deformation. To determine the circulation density, we will solve a hypersingular integral equation. The velocity circulation will be expanded into a series in the generalized coordinates of flexural oscillations of the plate. The set of circulation density functions does not depend on time and is determined only once during the solution of the problem. The effectiveness of this method is in using a nonlinear system with a finite number of degrees of freedom for the generalized coordinates of flexural oscillations of the plate. This dynamic system of low dimension can be analyzed in detail. 2. Problem Formulation and Basic Equations. Consider a cantilevered thin rectangular plate (Fig. 1). Since it is thin, its shear and rotary inertia can be neglected. The stresses and strains of the plate are related by Hooke's law. When a gas flow is past a thin plate on both its sides (Fig. 1), flutter (self-sustained oscillations) occurs. These unstable oscillations of the plate are constrained by the forces due to geometrically nonlinear deformation. Then the displacements of the plate are commensurable with its thickness. Such deformation of the plate is described by the von Karman equations