Extremely strong convergence of eigenvalue-density of linear stochastic dynamical systems
Eigenvalue problems play a crucial role in the stability and dynamics of engineering systems modeled using the linear mechanical theory. When uncertainties, either in the parameters or in the modelling, are considered, the eigenvalue problem becomes a random eigenvalue problem. Over the past half a century, random eigenvalue problems have received extensive attentions from the physicists, applied mathematicians and engineers. Within the context of civil, mechanical and aerospace engineering, significant work has been done on perturbation method based approaches in conjunction with the stochastic finite element method. The perturbation based methods work very well in the low frequency region which is often sufficient for many engineering applications. In the high frequency region however, which is necessary for some practical applications, these methods often fail to capture crucial physics, such as the veering and modal overlap. In this region one needs to consider the complete spectrum of the eigenvalues as opposed to the individual eigenvalues often considered in the low frequency applications. In this paper we consider the density of the eigenvalues of a discrete or discretised continuous system with uncertainty. It has been rigorously proved that the density of eigenvalues of random dynamical systems reaches a non-random limit for large systems. This fact has been demonstrated by numerical examples. The implications of this result for the response calculation of large stochastic structural dynamical systems have been highlighted.
KeywordsStrong Convergence Random Matrix Random Matrix Theory Random Dynamical System Random Matrix Model
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