Statistical Linearization of Nonlinear Structural Systems with Singular Matrices
Publication: Journal of Engineering Mechanics
Volume 142, Issue 9
Abstract
A generalized statistical linearization technique is developed for determining approximately the stochastic response of nonlinear dynamic systems with singular matrices. This system modeling can arise when a greater than the minimum number of coordinates is utilized, and can be advantageous, for instance, in cases of complex multibody systems where the explicit formulation of the equations of motion can be a nontrivial task. In such cases, the introduction of additional/redundant degrees of freedom can facilitate the formulation of the equations of motion in a less labor-intensive manner. Specifically, relying on the generalized matrix inverse theory and on the Moore-Penrose (M-P) matrix inverse, a family of optimal and response-dependent equivalent linear matrices is derived. This set of equations in conjunction with a generalized excitation-response relationship for linear systems leads to an iterative determination of the system response mean vector and covariance matrix. Further, it is proved that setting the arbitrary element in the M-P solution for the equivalent linear matrices equal to zero yields a mean square error at least as low as the error corresponding to any nonzero value of the arbitrary element. This proof greatly facilitates the practical implementation of the technique because it promotes the utilization of the intuitively simplest solution among a family of possible solutions. A pertinent numerical example demonstrates the validity of the generalized technique.
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Acknowledgments
The first and the third authors would like to gratefully acknowledge the support of this work through the EPSRC and ESRC Centre for Doctoral Training on Quantification and Management of Risk & Uncertainty in Complex Systems & Environments (EP/L015927/1). Further, the authors gratefully acknowledge the fruitful interaction with Professor Antonina Pirrotta (University of Palermo and University of Liverpool).
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Published In
Journal of Engineering Mechanics
Volume 142 • Issue 9 • September 2016
Copyright
© 2016 American Society of Civil Engineers.
History
Received: Dec 16, 2015
Accepted: Mar 25, 2016
Published online: May 18, 2016
Published in print: Sep 1, 2016
Discussion open until: Oct 18, 2016
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