Details
| Original language | English |
|---|---|
| Article number | 04016063 |
| Journal | Journal of engineering mechanics |
| Volume | 142 |
| Issue number | 9 |
| Publication status | Published - 1 Sept 2016 |
| Externally published | Yes |
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.
Keywords
- Moore-Penrose inverse, Random vibration, Singular matrices, Statistical linearization, Structural dynamics
ASJC Scopus subject areas
- Engineering(all)
- Mechanics of Materials
- Engineering(all)
- Mechanical Engineering
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In: Journal of engineering mechanics, Vol. 142, No. 9, 04016063, 01.09.2016.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Statistical linearization of nonlinear structural systems with singular matrices
AU - Fragkoulis, Vasileios C.
AU - Kougioumtzoglou, Ioannis A.
AU - Pantelous, Athanasios A.
N1 - Publisher Copyright: © 2016 American Society of Civil Engineers. Copyright: Copyright 2017 Elsevier B.V., All rights reserved.
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
JF - Journal of engineering mechanics
SN - 0733-9399
IS - 9
M1 - 04016063
ER -