A Simple Proper Orthogonal Decomposition Method forvon Karman Plate undergoing Supersonic Flow

Abstract

We apply a simple proper orthogonal decomposition (POD) method to compute the nonlinear oscillations of a degenerate two-dimensional fluttering plate undergoing supersonic flow. First, the von Karman's large deflection theory and quasi-steady aerodynamic theory are employed in constructing the governing equations of the simply supported plate. Then, the governing equations are solved by both the Galerkin method and the POD method. The Galerkin method is ac- curate but sometimes computationally expensive, since the number of degrees of freedom (dofs) required is relatively large provided that nonlinearity is strong. The POD method can be used to capture the complex dynamics of a strongly nonlinear system using very few degrees of freedom, much fewer than the Galerkin approach. The presently proposed POD method has two advantages over the conventional one. i) a simple numerical difference technique is first introduced to the POD method to avoid the complicated mode-to-mode projection between POD modes and Galerkin modes. ii) POD based reduced order models (POD/ROM) are constructed by us- ing a set of general modes which is extracted from chaotic responses. That is to say POD modes extracted from one set of parameters can be applied to various parameter variations for the same dynamic system. Moveover, results for the buck- led, LCO and chaotic responses of the plate are presented and compared with the Galerkin solutions. Numerical examples demonstrate the accuracy and efficiency of the present POD method.