TY - JOUR

T1 - Statistical linearization of nonlinear structural systems with singular matrices

AU - Fragkoulis, Vasileios C.

AU - Kougioumtzoglou, Ioannis A.

AU - Pantelous, Athanasios A.

PY - 2016/9/1

Y1 - 2016/9/1

N2 - 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.

AB - 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.

KW - Moore-Penrose inverse

KW - Random vibration

KW - Singular matrices

KW - Statistical linearization

KW - Structural dynamics

UR - http://www.scopus.com/inward/record.url?scp=84982260950&partnerID=8YFLogxK

U2 - 10.1061/(ASCE)EM.1943-7889.0001119

DO - 10.1061/(ASCE)EM.1943-7889.0001119

M3 - Article

AN - SCOPUS:84982260950

VL - 142

JO - Journal of Engineering Mechanics - ASCE

JF - Journal of Engineering Mechanics - ASCE

SN - 0733-9399

IS - 9

M1 - 04016063

ER -