The eigenvalue problem for structural systems with statistical properties.

Collins, Jon D.; Thomson, WlLLIAM T.

doi:10.2514/3.5180

Cited by 188 publications

(36 citation statements)

References 3 publications

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“…which is identical to equation (10). A sufficient condition for convergence of the algorithm derived by either approach is that any of the norms of the matrix &A,-lAl should be less than unity (see Reference 10, p. 59).…”

Section: (A+ Cai) And+ Sx) = Bo+ Eb1mentioning

confidence: 98%

Design parameter variation and structural response

Romstad

Hutchinson

Runge

1973

Numerical Meth Engineering

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SUMMARYA general power series method is developed to aid in obtaining changes in the response variables due to changes in design parameters. The approach is considered for eigenvalue problems and systems of simultaneous equations which may represent static, dynamic and stability response problems. Particular emphasis is placed upon the eigenvalue problem and techniques for improving the calculations from the power series are suggested utilizing modified Rayleigh quotient expressions. Each of the methods proposed is treated cornputationally by the formulation of a finite element solution of column elastic stability. Investigations of solution accuracy versus percentage variation in element stiffness, whether the method underestimates or overestimates the changes, and the required number of eigenvalue solutions are presented.

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Section: (A+ Cai) And+ Sx) = Bo+ Eb1mentioning

confidence: 98%

Design parameter variation and structural response

Romstad

Hutchinson

Runge

1973

Numerical Meth Engineering

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“…The SFEM is based on the classical deterministic Finite Element Method and consider the elements properties as random [6]. It is mainly divided into three main approaches: statistical approaches, also referred to as Monte-Carlo Simulations (MCS) [7,8], perturbation methods [9][10][11] and the Spectral Stochastic Finite Element Method (SSFEM) [6,[12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

“…Finally, perturbation approaches provide a framework both straightforward and easy to implement to reduce the computational cost of MCS. It consists in the approximation of the random quantities using Taylor (or Neumann) expansion [9][10][11]21]. They are usually limited to first or second order and have been applied to a large field of applications [22].…”

Section: Introductionmentioning

confidence: 99%

“…They are usually limited to first or second order and have been applied to a large field of applications [22]. The first order Taylor expansion allows to obtain a closed-form expression of the joint probability density function of the random eigenvalues when input random variables are assumed to be Gaussian [10]. If random eigenvalues cannot be modeled as Gaussian random variables, perturbation methods (if necessary using higher order expansion) allows to approximate statistical moments of the random eigenvalue in order to identify their probability density function (pdf) [9].…”

Section: Introductionmentioning

confidence: 99%

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Beyond the limitations of perturbation methods for real random eigenvalue problems using Exceptional Points and analytic continuation

Ghienne

Nennig

2020

Journal of Sound and Vibration

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A numerical method is proposed to approximate the solution of parametric eigenvalue problem when the variability of the parameters exceed the radius of convergence of low order perturbation methods. The radius of convergence of eigenvalue perturbation methods, based on Taylor series, is known to decrease when eigenvalues are getting closer to each other. This phenomenon, knwon as veering in structural dynamics, is a direct consequence of the existence of branch point singularity in the complex plane of the varying parameters where some eigenvalues are defective. When this degeneracy, referred to as Exceptional Point (EP), is close to the real axis, the veering becomes stronger.The main idea of the proposed approach is to combined a pair of eigenvalues to remove this singularity. To do so, two analytic auxiliary functions are introduced and are computed through high order derivatives of the eigenvalue pair with respect to the parameter. This yields a new robust eigenvalue reconstruction scheme which is compared to Taylor and Puiseux series. In all cases, theoretical bounds are established and all approximations are compared numerically on a three degrees of freedom toy model. This system illustrate the ability of the method to handle the vibrations of a structure with a randomly varying parameter. Computationally efficient, the proposed algorithm could also be relevant for actual numerical models of large size, arising from other applications involving parametric eigenvalue problems, e.g., waveguides, rotating machinery or instability problems such as squeal or flutter.

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“…In this category, the main focus is perturbation methods and spectral methods . In 1969, Collins and Thompson presented a first‐order perturbation method for dynamic analysis of structures with parameter uncertainties. A detailed overview of the perturbation method for algebraic random eigenvalue problems can be found in the monograph of Kleiber and Hien .…”

Section: Introductionmentioning

confidence: 99%

Homotopy approach for random eigenvalue problem

Huang

Zhang

Phoon

2017

Numerical Meth Engineering

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A novel approach, referred to as the homotopy stochastic finite element method, is proposed to solve the eigenvalue problem of a structure associated with some amount of uncertainty based on the homotopy analysis method.For this approach, an infinite multivariate series of the involved random variables is proposed to express the random eigenvalue or even a random eigenvector. The coefficients of the multivariate series are determined using the homotopy analysis method. The convergence domain of the derived series is greatly expanded compared with the Taylor series due to the use of an approach function of the parameter h. Therefore, the proposed method is not limited to random parameters with small fluctuation. However, in practice, only singlevariable and double-variable approximations are employed to simplify the calculation. The numerical examples show that with a suitable choice of the auxiliary parameter h, the suggested approximations can produce very accurate results and require reduced or similar computational efforts compared with the existing methods.KEYWORDS homotopy analysis method, perturbation method, random eigenvalue problem, stochastic finite element method, Taylor series 1 | INTRODUCTION Algebraic eigenvalue problems are a class of basic and significant problems in various fields, such as structural dynamics and structural stability. Currently, the computation of eigenvalues and eigenvectors is well comprehended for deterministic problems. 1,2 In many practical cases, however, the physical properties of the structural systems are not deterministic. For instance, the stiffness of a beam can be affected by material imperfections such that the stiffness distribution along the beam is irregular and difficult to measure and the boundary constraints of beam structures are usually uncertain. 3,4 Therefore, it is extremely necessary to use random variables to more realistically describe the uncertain characteristics that exist in eigenvalue problems in engineering.Due to the randomness of the input parameters, such as the modulus of elasticity, of a physical problem, the desired output or eigenvalues will also be random. The methods for computing these random outputs are generally composed of 2 categories. The first category includes simulation-based methods. All orders of statistical moments and all probability density functions of the eigenvalues can be determined by repeatedly carrying out computations of the deterministic problem in the simulation-based methods. Direct Monte-Carlo (DMC) simulation is the most important and fundamental simulation-based method, 5-9 but it requires considerable computational effort, especially for large systems. Even so, DMC simulation can be conducted and is considered a closed-form solution for evaluating approximation methods. The second category for random analysis, stochastic finite element methods (SFEM), 10-13

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The eigenvalue problem for structural systems with statistical properties.

Cited by 188 publications

References 3 publications

Design parameter variation and structural response

Design parameter variation and structural response

Beyond the limitations of perturbation methods for real random eigenvalue problems using Exceptional Points and analytic continuation

Homotopy approach for random eigenvalue problem

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