TY - JOUR

T1 - Eigenvalue density of linear stochastic dynamical systems

T2 - A random matrix approach

AU - Adhikari, Sondipon

AU - Pastur, L

AU - Lytova, A

AU - du Bois, Jonathan

PY - 2012/2/27

Y1 - 2012/2/27

N2 - 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 MarenkoPastur (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 springmass oscillators where 100 nominally identical samples are physically created and individually tested within a laboratory framework.

AB - 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 MarenkoPastur (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 springmass oscillators where 100 nominally identical samples are physically created and individually tested within a laboratory framework.

UR - http://www.scopus.com/inward/record.url?scp=82955193838&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1016/j.jsv.2011.10.027

U2 - 10.1016/j.jsv.2011.10.027

DO - 10.1016/j.jsv.2011.10.027

M3 - Article

SN - 0022-460X

VL - 331

SP - 1042

EP - 1058

JO - Journal of Sound and Vibration

JF - Journal of Sound and Vibration

IS - 5

ER -