On the nature of random system matrices in structural dynamics

Abstract

This work is concerned with the nature of random mass and stiffness matrices arising in linear dynamic systems due to inherent uncertainties in the system parameters. In an important recent paper, Soize [Prob. Eng. Mech. 15, 277–294 (2000)] has developed an expression for the probability density function of such matrices by using the mean value of the matrices in conjunction with the entropy optimization principle. Although mathematically optimal given knowledge of only the mean values of the matrices, it is not entirely clear how well the results obtained will match the statistical properties of a physical system. This issue is investigated here by considering the structure of random matrices arising from Guyan-reduced dynamic models. It is shown that, under rather general conditions, the random parts of the mass and the stiffness matrices of the reduced system in the modal coordinates resemble the so-called ‘‘Gaussian Orthogonal Ensemble (GOE).’’ A single parameter σ (the standard deviation of the matrix diagonals) characterizes a GOE, and this offers the possibility of a very straightforward Monte Carlo simulation technique for the system matrices and response. A limited number of numerical results have been shown in favor of the proposed result.