Random matrix theory and non-parametric model of random uncertainties in vibration analysis

Abstract

To cite this version:Christian Soize. Random matrix theory and non-parametric model of random uncertainties in vibration analysis. Journal of Sound and Vibration, Elsevier, 2003, 263 (4) of random matrices constituted of symmetric positive-definite real random matrices. This ensemble differs from the GOE and from the other known ensembles of the random matrix theory. The present paper has three main objectives. The first one is to study the statistics of the random eigenvalues of random matrices belonging to this new ensemble and to compare with the GOE. The second one is to compare this new ensemble of random matrices with the GOE in the context of the non-parametric approach of random uncertainties in structural dynamics for the low-frequency range. The last objective is to give a new validation for the non-parametric model of random uncertainties in structural dynamics in comparing, in the low-frequency range, the dynamical response of a simple system having random uncertainties modelled by the parametric and the non-parametric methods. These three objectives will allow us to conclude about the validity of the different theories.

Stochastic modeling of uncertainties in computational structural dynamics—Recent theoretical advances

“…The random matrix theory [16] (see also [17,18,19] for such a theory in the context of linear acoustics) is used to construct the prior probability distribution of the random matrices modeling the uncertain operators of the mean computational model. This prior probability distribution is constructed by using the Maximum Entropy Principle [20] (from Information Theory [21]) for which the constraints are defined by the available information [13,14,22,15]. Since the paper [13], many works have been published in order to validate the nonparametric probabilistic approach of model uncertainties with experimental results (see for instance [23,24,25,26,27,28,15,29]), to extend the applicability of the theory to other areas [30,31,32,33,34,35,36,37,38,39,40,41], to extend the theory to new ensembles of positive-definite random matrices yielding a more flexible description of the dispersion levels [42], to apply the theory for the analysis of complex dynamical systems in the medium-frequency range, including vibroacoustic systems, [43,44,23,45,25,26,27,28,46,…”

“…The Gaussian Orthogonal Ensemble (GOE), for which the mean value is the unity matrix [I n ] (see [22]), is the set of random matrices [G GOE ]) which can be written as …”

“…The objective of this section is then to summarize the construction given in [13,14,22,15] of the ensemble SG + 0 of random matrices [G 0 ] defined on the probability space (Θ, T , P), with values in the set M + n (R) of all the positive definite symmetric (n × n) real matrices and such that…”

Polynomial chaos representation of a stochastic preconditioner

“…Note that the arguments for L inc and D inc in this context refer to random variables ξ and a measure of bandwidth, ℓ, and are different from arguments in equation (12).…”

“…In many problems of science and engineering the quest for accuracy in predicting the behavior of the associated physical systems has motivated the adoption of stochastic equations as viable representative models [4,11,12]. In many of these models, the governing equations take the form of partial differential equations with coefficients represented as stochastic processes or variables [4,2].…”