Adhikari, S. and Langley, R. S.
In A Der-Kiureghian, Samer Madanat, and Juan M. Pestana, editors,
Proceedings of the ninth International Conference on
Applications of Statistics and Probability in Civil Engineering
(ICASP 9), San Fransisco, California, USA, volume 1 of
Applications of Statistics and Probability in Civil
Engineering, pages 201-207, Rotterdam, Netherlands, July 6-9,
2003. Millpress.
Dynamic characteristics of linear structural systems are governed by the natural frequencies and the mode-shapes. In this paper the statistical properties of the eigenvalues of linear dynamic systems are considered. It is assumed that the mass and the stiffness matrices are smooth, continuous and at least twice differentiable functions of a random parameter vector. The random parameter vector is assumed to be standard Gaussian or can be transformed to standard Gaussian. Two approaches are proposed to obtain moments, cumulants and probability density functions of the eigenvalues. The first approach is based on a perturbation expansion of the eigenvalues about an optimal point which is ‘best’ is some sense. This optimal point is obtained by using the concepts borrowed from structural reliability analysis. The second approach is based on asymptotic analysis. Moments of the eigenvalues are obtained by asymptotic expansion of a multidimensional integral involving the joint probability density function of the random variables. Based on theses methods, two simple expressions of the probability density functions of the eigenvalues are derived. A numerical example is given to compare the proposed methods with Monte Carlo simulations.
@INPROCEEDINGS{cp8,
AUTHOR={S. Adhikari and R. S. Langley},
TITLE={Distribution of eigenvalues of linear stochastic systems},
BOOKTITLE={Proceedings of the ninth International Conference on Applications of Statistics
and Probability in Civil Engineering (ICASP 9), San Fransisco, California, USA},
YEAR={2003},
Editor={A Der-Kiureghian and Samer Madanat and Juan M. Pestana},
Volume={1},
Series={Applications of Statistics and Probability in Civil Engineering},
Pages={201-207},
Address={Rotterdam, Netherlands},
Month={July},
Publisher={Millpress},
Note={}
}