The Karhunen-Loève expansion is a powerful spectral technique for the analysis and synthesis of dynamical systems. It consists in decomposing a spatial correlation matrix, which can be obtained through numerical or physical experiments. The decomposition produces orthogonal eigenfunctions or proper orthogonal modes, and eigenvalues that provide a measurement of how much energy is contained in each mode. The relation between KL modes and mode shapes of linear vibrating systems has already been derived and demonstrated for two and three dofs mass-spring-damper systems. The purpose of this paper is to extend this investigation to more complex distributed-parameter linear systems. A plane truss and a simply supported plate subjected to impulsive forces, commonly used in modal analysis are studied. The resulting KL modes are compared to the analytical mode shapes. Damping and random noise effects in the procedure performance are evaluated. Two methods for indirectly obtaining natural frequencies are also presented.
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).
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