Joint Distribution of Eigenvalues of Linear Stochastic Systems

Abstract

Description of real-life engineering structural systems is associated with some amount of uncertainty in specifying 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. Current methods to deal with such problems are dominated by approximate perturbation methods. Some exact methods to obtain joint distribution of the natural frequencies are reviewed and their applicability in the context of real-life engineering problems are discussed. A new approach based on an asymptotic approximation of multidimensional integrals is proposed. A closed-form expression for general order joint moments of arbitrary number of natural frequencies of linear stochastic systems is derived. The proposed method does not employ the ‘small randomness ’ assumption unusually used in perturbation based methods. Joint distributions of the natural frequencies are investigated using numerical examples and the results are compared with Monte Carlo Simulation. I.

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