Eigenvalue density of linear stochastic dynamical systems: A random matrix approach

Adhikari, S. , Pastur, L., Lytova, A. and Du Bois, J. (2012) Eigenvalue density of linear stochastic dynamical systems: A random matrix approach. Journal of Sound and Vibration, 331(5), pp. 1042-1058. (doi: 10.1016/j.jsv.2011.10.027)

Abstract

Eigenvalue problems play an important role in the dynamic analysis of engineering systems modeled using the theory of linear structural mechanics. When uncertainties are considered, the eigenvalue problem becomes a random eigenvalue problem. In this paper the density of the eigenvalues of a discretized continuous system with uncertainty is discussed by considering the model where the system matrices are the Wishart random matrices. An analytical expression involving the Stieltjes transform is derived for the density of the eigenvalues when the dimension of the corresponding random matrix becomes asymptotically large. The mean matrices and the dispersion parameters associated with the mass and stiffness matrices are necessary to obtain the density of the eigenvalues in the frameworks of the proposed approach. The applicability of a simple eigenvalue density function, known as the Marenko–Pastur (MP) density, is investigated. The analytical results are demonstrated by numerical examples involving a plate and the tail boom of a helicopter with uncertain properties. The new results are validated using an experiment on a vibrating plate with randomly attached spring–mass oscillators where 100 nominally identical samples are physically created and individually tested within a laboratory framework.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Adhikari, Professor Sondipon
• Authors: Adhikari, S., Pastur, L., Lytova, A., and Du Bois, J.
• College/School: College of Science and Engineering > School of Engineering > Infrastructure and Environment
• Journal Name: Journal of Sound and Vibration
• Publisher: Elsevier
• ISSN: 0022-460X
• ISSN (Online): 1095-8568