Stochastic sensitivity analysis of eigenvalues and eigenvectors

Song, Datong; Chen, Suhuan; Qiu, Zhipin

doi:10.1016/0045-7949(94)00386-h

Cited by 31 publications

(10 citation statements)

References 7 publications

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“…Several authors [40][41][42][43][44][45][46][47][48][49][50] proposed mean-centered perturbation methods. Recently Adhikari [51,52] and Adhikari and Friswell [53] developed an asymptotic approach and an optimal series-expansion methods to obtain second and higherorder joint statistics of eigenvalues.…”

Section: Article In Pressmentioning

confidence: 99%

Distributed parameter model updating using the Karhunen–Loève expansion

Adhikari

Friswell

2010

Mechanical Systems and Signal Processing

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Section: Article In Pressmentioning

confidence: 99%

Distributed parameter model updating using the Karhunen–Loève expansion

Adhikari

Friswell

2010

Mechanical Systems and Signal Processing

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“…Over the past three decades, several authors [14,15,26,[31][32][33][34][35]40] have applied first-order perturbation method in various problems of practical interest. Adhikari [1] used the first-order perturbation method in complex random eigenvalue problems arising in non-proportionally damped systems.…”

Section: Joint Natural Frequency Statistics Using the Perturbation Mementioning

confidence: 99%

“…The current literature on random eigenvalue problems arising in engineering systems is dominated by the meancentered perturbation methods [7,14,15,26,[31][32][33][34][35]40]. These methods work well if the uncertainties are small and the parameters follow a Gaussian distribution.…”

Section: Introductionmentioning

confidence: 99%

Joint statistics of natural frequencies of stochastic dynamic systems

Adhikari

2006

Comput Mech

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The description of real-life engineering structural systems is usually associated with some amount of uncertainty in specifying material properties, geometric parameters and boundary conditions. In the context of structural dynamics it is necessary to consider joint probability distribution of the natural frequencies in order to account for these uncertainties. Current methods to deal with such problems are dominated by approximate perturbation methods. In this paper a new approach based on an asymptotic approximation of multidimensional integrals is proposed. A closed-form expression for general order joint moments of arbitrary number of natural frequencies of linear stochastic systems is derived. The proposed method does not employ the small-randomness and Gaussian random variable assumption usually used in the perturbation based methods. Joint distributions of the natural frequencies are investigated using numerical examples and the results are compared with Monte Carlo simulation.Keywords Random eigenvalue problem · Asymptotic method · Stochastic dynamical systems · Probabilistic structural mechanics Nomenclature κ (r 1 ,r 2 ) jk (r 1 , r 2 )th order cumulant of jth and kth natural frequencies D (•) (x) Hessian matrix of (•) at x d (•) (x) gradient vector of (•) at x

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“…Historically, many studies have focused on evaluating the statistics of natural frequencies considering variations in elements of mass and stiffness. The existing studies can be broadly classified into Perturbation methods Possibilistic approaches Maximum entropy approaches Random matrix theory …”

Section: Motivationmentioning

confidence: 99%

Sampling of closely‐spaced ordered set of uniformly distributed random variables

Tadinada

Gupta

2011

Numerical Meth Engineering

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SUMMARYWhen the individual PDFs of closely-spaced random variables such as natural frequencies of a structure overlap, generation of sample sets by assuming the frequencies to be independent random variables can lead to incorrect sets of frequencies in the sense that the frequencies do not remain as ordered sets. Rejection of such disordered sample sets results in individual density functions that are significantly different from the distributions initially assumed for sampling each random variable. One way to overcome this constraint in the simulation of an ordered set of random variables is to consider them in an implicit manner using a joint PDF. In this paper, we present a formulation for a joint density function that is developed using fundamental probability approaches. The formulation ensures that the sampled random variables always remain as ordered sets and maintain the individual density functions for each variable. Application of the proposed formulation is illustrated for cases with not just two closely-spaced variables but also for a case with multiple closely-spaced variables such that the PDFs of more than two random variables overlap with each other. An expression is presented to determine the exact number of terms needed in the formulation. However, it is also illustrated that only two terms are sufficient in most applications even when the exact number of terms needed is very high.

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Stochastic sensitivity analysis of eigenvalues and eigenvectors

Cited by 31 publications

References 7 publications

Distributed parameter model updating using the Karhunen–Loève expansion

Distributed parameter model updating using the Karhunen–Loève expansion

Joint statistics of natural frequencies of stochastic dynamic systems

Sampling of closely‐spaced ordered set of uniformly distributed random variables

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