A Biased Presentation of Smooth Decomposition and Some Applications

Abstract

Smooth Decomposition (SD) is a multivariate data or statistical analysis method used to identify normal modes, natural frequencies and energy partition of systems. The method is based on the knowledge of the system response (spatial data field) to a random excitation. It should be noted that only the output data of the system is needed for the identification. The excitation has to satisfy some properties, normally well met by a white noise, but doesn’t need to be measured. This turns the method the ideal way to deal with the identification of systems under ambient excitations, as wind or waves for instance, which can be hard to compute or to describe. The output data of the system response is then projected into a basis and an optimization problem is created. It consists of finding the basis that gives the maximum variance of the displacement-projection and the minimum variance of the velocity- projection. This optimization problem can then be written as an eigenvalue problem with the covariance matrices of the displacement field, and of the corresponding velocity field. Solving this problem the system is identified and no further considerations and approximations are needed. From the eigenvalues, the “energy” participation of each normal mode in the response during the simulation or the experimental test can be evaluated. Since this information is crucial for non-linear systems identification, the Smooth Decomposition method can be used to identify linear and non-linear systems. The objective of the paper is to explain the Smooth Decomposition method and to present an application of it. First we present the method and show how the results of SD can be interpreted. Then, an application of SD on a simulated numerical model of a cantilever beam is performed and discussed to understand how SD can be a nice tool for modal analysis.