TY - JOUR

T1 - Linear random vibration of structural systems with singular matrices

AU - Fragkoulis, Vasileios C.

AU - Kougioumtzoglou, Ioannis A.

AU - Pantelous, Athanasios A.

PY - 2016/2/1

Y1 - 2016/2/1

N2 - 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.

AB - 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.

KW - Moore-Penrose inverse

KW - Random vibration

KW - Singular matrix

KW - Structural dynamics

UR - http://www.scopus.com/inward/record.url?scp=84978484591&partnerID=8YFLogxK

U2 - 10.1061/(ASCE)EM.1943-7889.0001000

DO - 10.1061/(ASCE)EM.1943-7889.0001000

M3 - Article

AN - SCOPUS:84978484591

VL - 142

JO - Journal of Engineering Mechanics - ASCE

JF - Journal of Engineering Mechanics - ASCE

SN - 0733-9399

IS - 2

M1 - 040150811

ER -