Reduced-Order Model for an Impacting Beam using the Karhunen-Loµeve Expansion

Abstract

The Karhunen-Loµeve expansion (KLE), also known in the literature as the proper orthogonal decomposition, is a powerful tool for the model reduction of structural systems. Although the method has been used for quite some time in turbulence studies to uncover spatial coherent structures in °ow fields, only recently has it been applied to structural dynamics problems. The KL method is a primarily statistical one where the system dynamics is assumed to be a second-order stochastic process. It consists in obtaining a set of orthogonal eigenfunctions where the dynamics is to be projected. Practically, one constructs a spatial autocorrelation tensor and then performs its spectral decomposition. The resulting eigenfunctions will provide the required proper orthogonal modes (POMs) or empirical eigenmodes and the correspondent empirical eigenvalues (or proper orthogonal values, POVs) represent the mean energy contained in that projection. Finally, one uses a number of the computed modes in Galerkin's method in order to obtain a reduced dimension system (this is sometimes called the KLG method). Although this method can also be applied to linear systems, its main application stems from nonlinear ones. In this present work, such a system is studied, namely a linear clamped beam impacting a °exible barrier. The KLE will be used to look into the dynamics of the system and to generate a reduced-order model (ROM).