How to retrieve the normal modes using the POD

Rana Elias (1) , Nelly Point (2) , Joanna Bodgi (3) , Pierre Argoul (2)

1 navier umr 8205 - Laboratoire Navier
2 Dynamique des structures et identification
3 USJ - Université Saint-Joseph de Beyrouth

Proper orthogonal decomposition (POD) is an increasingly popular way to analyze data, and to obtain either a low- dimensional approximated description of a high-dimensional process, or useful information for damage assessment. In the case of a dynamic system with n degrees of freedom, the purpose of POD is to retrieve the modal properties from the measured response. Until now, POD has been used for systems with a diagonal mass matrix. The aim of this presentation is double: first, to demonstrate that POD can also be used for a non-diagonal symmetric mass matrix; and second, to present sufficient conditions on the response sampling, in order to retrieve the modal characteristics with a prescribed accuracy. The conditions to obtain this approximation with a given accuracy are first explicitly given for the case without damping. Then the case of proportional damping is addressed and similar conditions are shown. The critical case of two modal frequencies close together is also studied, given that it requires particular conditions. The obtained conditions show that the expected accuracy is explicitly limited by the damping ratio. Some numerical tests illustrate the accuracy evolution of the approximated normal modes obtained by this method, with respect to the variation of the observation time and the damping ratio. This careful analysis can be useful for finding the cause of poor approximation properties in more complex cases, such as analysis of variation in nonlinear normal modes.

Mots clés: dynamic system; POD; normal modes; accuracy of an approximation

Domaines: Vibrations [physics.class-ph]; Vibrations [physics.class-ph]

Rana Elias, Nelly Point, Joanna Bodgi, Pierre Argoul. How to retrieve the normal modes using the POD. XVIII symposium Vibrations chocs et Bruit & ASTELAB, Jul 2012, EDF CLAMART, France. 12 p. ⟨hal-00736330⟩