Linear Random Vibration of Structural Systems with Singular Matrices
Publication: Journal of Engineering Mechanics
Volume 142, Issue 2
Abstract
A framework is developed for determining the stochastic response of linear multi-degree-of-freedom (MDOF) structural systems with singular matrices. This system modeling can arise when using more than the minimum number of coordinates, and can be advantageous, for instance, in cases of complex multibody systems whose dynamics satisfy a number of constraints. In such cases the explicit formulation of the equations of motion can be a nontrivial task, whereas the introduction of additional/redundant degrees of freedom can facilitate the formulation of the equations of motion in a less labor-intensive manner. Relying on the generalized matrix inverse theory and on the Moore-Penrose (M-P) matrix inverse, standard concepts, relationships, and equations of the linear random vibration theory are extended and generalized herein to account for systems with singular matrices. Adopting a state-variable formulation, equations governing the system response mean vector and covariance matrix are formed and solved. Further, it is shown that a complex modal analysis treatment, unlike the standard system modeling case, does not lead to decoupling of the equations of motion. However, relying on a singular value decomposition of the system transition matrix significantly facilitates the efficient computation of the system response statistics. A linear structural system with singular matrices is considered as a numerical example for demonstrating the applicability of the methodology and for elucidating certain related numerical aspects.
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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.
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Published In
Journal of Engineering Mechanics
Volume 142 • Issue 2 • February 2016
Copyright
© 2015 American Society of Civil Engineers.
History
Received: Mar 26, 2015
Accepted: Jul 20, 2015
Published online: Sep 1, 2015
Published in print: Feb 1, 2016
Discussion open until: Feb 1, 2016
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