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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 47

The Random Eigenvalue Problem for Stochastic Systems

S. Kadry and A. Chateauneuf

LaMI-IFMA Laboratory, Clermont-Ferrand, France

Full Bibliographic Reference for this paper
S. Kadry, A. Chateauneuf, "The Random Eigenvalue Problem for Stochastic Systems", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 47, 2006. doi:10.4203/ccp.83.47
Keywords: random eigenvalue problem, transformation method, Rayleigh method, probability density function, statistical distributions, linear stochastic systems.

Summary
The characterization of the natural frequencies and mode shapes play a fundamental role in the analysis and the design of dynamic systems. The determination of this information requires the solution of an eigenvalue problem. Eigenvalue problems also arise in the context of the stability analysis of structures. This problem could be either a differential eigenvalue problem or a matrix eigenvalue problem, depending on whether a continuous model or a discrete model is used to describe the given vibrating system. Several studies have been conducted on this topic since the mid-sixties.

The study of probabilistic characterization of the eigensolutions of random matrix and differential operators is now an important research topic in the field of stochastic structural mechanics. The paper by Boyce [1] and the book by Scheidt and Purkert [2] are useful sources of information on early work in this area of research and also provide a systematic account of different approaches to random eigenvalue problems. Several review papers, have appeared in this field which summarize the current as well as the earlier works. The current literature on random eigenvalue arising in engineering systems is dominated by the mean-centered perturbation methods. These methods work well when the uncertainties are small and the parameter distribution is Gaussian. Methods which are not based on mean-centered perturbation but still have the generality and computational efficiency to be applicable for engineering dynamic systems are rare.

The description of real-life engineering structures systems is associated with some amount of uncertainty related to material properties, geometric parameters, boundary conditions and applied loads. In the context of structural dynamics, it is necessary to consider random eigenvalue problems in order to account for these uncertainties. A proposed approach based on the combination of the probabilistic transformation methods [3] for a random variable and the Rayleigh method [4] in order to evaluate the probability density function of the eigenvalue of stochastic systems. This approach has the advantage of giving directly the whole density function (closed-form) of the eigenvalues, which is very helpful for probabilistic analysis. To show the accuracy of the proposed method, an example of a beam is analyzed for an uncertainty in the material (Young's modulus) and the geometry (beam length). The results are compared with Monte Carlo Simulations.

References
1
Boyce W.E. Random Eigenvalue Problems. Probabilistic methods in applied mathematics. New York: Academic Press 1968.
2
Scheidt J.V., Purkert W. Random Eigenvalue Problems. New York: North Holland 1983.
3
Soong T., Random Differential Equations in Science and Engineering. Academic Press 1973.
4
Boswell L.F., Mello C. Dynamics of Structural Systems. Blackwell Scientific Publications 1993.

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