Use of Random Excitation to Develop POD Based Reduced Order Models for Nonlinear Structural Dynamics
This paper considers use of Proper Orthogonal Decomposition (POD), also known as the Karhunen-Loeve (K-L) method, to obtain reduced order dynamic models of nonlinear structural systems. The study applies POD to simulated time series data to extract dominant “modes” that describe the system behavior. The “POD modes” are used to formulate reduced order differential equation models (ROM’s) of the structure in which the dependent variables are the POD modal coordinates. Two example systems are considered: 1) a clamped beam whose tip is placed between attracting magnets; POD analysis of this system was done by Feeny and Kerschen  using harmonic excitation to excite chaotic motions, which were analyzed to develop the POD modes for reduced order modeling, and 2) a chain of oscillators having an isolated nonlinear Duffing element. Our approach is to generate the POD modes for model reduction by using strong, band limited random excitation to excite vibratory motions. The richness of this type of excitation is intended to provide responses whose POD-based reduced models can be used with reasonable accuracy for system parameters that differ from those used to generate the reduced order model (this is the central issue in using POD to generate reduced order nonlinear models of structures). Our results indicate that, due to the spectral richness of the random excitation, this type of excitation can be used with reasonable accuracy for conditions that differ from those used to generate the reduced order model. The method works well for the chain of oscillator system, but less well for the magnetic beam system, due to the presence of multiple stable equilibria in this system. A useful result of this work is the finding that the number of POD modes required to achieve accurate reduced order differential equation models may be considerably larger than the number of POD modes needed to accurately project the full model responses onto a subspace defined by dominant POD modes. Also, it is shown that for the clamped beam problem with multiple equilibria, we require more modes to develop a reduced order model than we require for a chain of oscillators.
N. Kumar and T. D. Burton, "Use of Random Excitation to Develop POD Based Reduced Order Models for Nonlinear Structural Dynamics," ASME Proceedings: 21st Biennial Conference on Mechanical Vibration and Noise, American Society of Mechanical Engineers (ASME), Jan 2008.
The definitive version is available at http://dx.doi.org/10.1115/DETC2007-35539
ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
Mechanical and Aerospace Engineering
Keywords and Phrases
Structural Dynamics; Random Excitation
Article - Conference proceedings
© 2008 American Society of Mechanical Engineers (ASME), All rights reserved.