Analysis of Nonlinear Random Vibrations Using Orthogonal Decompositions

Abstract

Orthogonal decompositions provide a powerful tool for random vibrations analysis. The most popular orthogonal decomposition is the Karhunen-Loève Decomposition (KLD). The KLD is a statistical analysis technique for finding the coherent structures in an ensemble of spatially distributed data. The structures (or KL modes) are defined as the eigenvectors of the covariance matrix of the associated random process. Recently, a modified KLD named Smooth Decomposition (SD) has been proposed. The SD can be viewed as a projection of an ensemble of spatially distributed data such that the vector directions of the projection not only keeps the maximum possible variance but also the motions resulting along the vector directions are as smooth as possible in time. The vector directions (or S modes) are defined as the eigenvectors of the generalized eigenproblem defined from the covariance matrix of the random process and the covariance matrix of the associated time derivative random process. It was shown that the SD is an interesting tool to linear random analysis. In this paper, the SD will be used to analyze nonlinear random vibrations. We first focus on the physical interpretation of the S modes. It will be shown that the S modes can be related to the normal modes of the associated linearized system.
Finally the ability of KLD and SD to analyze random vibration problem is demonstrated considering an energy pumping phenomena in a linear chain with nonlinear end-attachment.