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 closedform 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. haracterization of the natural frequencies and the mode-shapes play a fundamental role in the analysis and design of engineering dynamic systems. The determination of natural frequency and mode shapes require the solution of an eigenvalue problem. Eigenvalue problems also arise in the context of the stability analysis of structures. This problem could either be a difierential eigenvalue problem or a matrix eigenvalue problem, depending on whether a continuous model or a discrete model is used to describe the given vibrating system. Description of real-life engineering structural systems is inevitably associated with some amount of uncertainty in specifying material properties, geometric parameters, boundary conditions and applied loads. When we take account of these uncertainties, it is necessary to consider random eigenvalue problems. Several studies have been conducted on this topic since the mid-sixties. The study of probabilistic characterization of the eigensolutions of random matrix and difierential operators is now an important research topic in the fleld of stochastic structural mechanics. The paper by Boyce 1 and the book by Scheidt and Purkert 2 are useful sources of information on early work in this area of research and also provide a systematic account of difierent approaches to random eigenvalue problems. Several review papers, for example, by Ibrahim, 3 Benaroya and Rehak, 4 Benaroya, 5 Manohar and Ibrahim, 6 and Manohar and Gupta 7 have appeared in this fleld which summarize the current as well as the earlier works. In this paper we obtain a closed-form expression of arbitrary order joint moments of the natural frequencies of discrete linear systems or discretized continuous systems. The random eigenvalue problem of undamped or proportionally damped systems can be expressed by K(x)`j = ! 2 jM(x)`j