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

Abstract

The Karhunen-Loève 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 flow 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 flexible barrier. The KLE will be used to look into the dynamics of the system and to generate a reduced-order model (ROM).

Proper orthogonal decomposition for model reduction of a vibroimpact system

“…POD-basis is the best basis of projection for the Galerkin method in the sense that it minimizes the average squared distance between the original data and its reduced linear representation. The group of PUC-Rio has been working on this subject for a while ; ; Trindade and Sampaio (2001); Wolter et al (2002)). Recently three works have been published by Sampaio (2006, 2009a,b).…”

Remarks on the efficiency of POD for model reduction in non‐linear dynamics of continuous elastic systems

“…This case is very different from the case of continuous systems because the dimension is finite. The second class of papers deals with the continuous case and the eigenvalue problem that one has to solve to compute the basis is effectively solved using the snapshot method (see, for instance [3][4][5][6][7][8][9][10], for parabolic equations, for fluid dynamics problems, for non-linear heat conduction problems, vibroimpact problems and for reconstruction of travelling waves). In general, in the two classes of papers, there is neither comparison of different bases of reduction nor of their efficiency in the reduction.…”