Random matrix eigenvalue problems in structural dynamics

Abstract

SUMMARYNatural frequencies and mode shapes play a fundamental role in the dynamic characteristics of linear structural systems. Considering that the system parameters are known only probabilistically, we obtain the moments and the probability density functions of the eigenvalues of discrete linear stochastic dynamic systems. Current methods to deal with such problems are dominated by mean-centred perturbation-based methods. Here two new approaches are proposed. The first approach is based on a perturbation expansion of the eigenvalues about an optimal point which is 'best' in some sense. The second approach is based on an asymptotic approximation of multidimensional integrals. A closed-form expression is derived for a general rth-order moment of the eigenvalues. Two approaches are presented to obtain the probability density functions of the eigenvalues. The first is based on the maximum entropy method and the second is based on a chi-square distribution. Both approaches result in simple closed-form expressions which can be easily calculated. The proposed methods are applied to two problems and the analytical results are compared with Monte Carlo simulations. It is expected that the 'small randomness' assumption usually employed in mean-centred-perturbation-based methods can be relaxed considerably using these methods.

Application of random eigenvalue analysis to assess bridge flutter probability

“…It must be noticed that the developments hereinabove differ from Adhikari and Friswell (2007), since complex-valued eigenvalues are considered. Moreover, this approach requires to solve a set of 2Pðn þ 1Þ equations for each eigenpair and to consider in this set the coupling between the eigenvalues and the eigenvector components, which may become computationally cumbersome.…”

“…The pair λ i and Y i results from the eigenproblem (12) by setting θ ¼ θ. Adhikari and Friswell (2007) propose an iterative improvement of this method, by changing the point θ A Θ. However, the mean value θ remains a convenient guess, especially for slightly dispersed random system, independently of the probability distribution of θ.…”

CFD Based Aeroelastic Stability Predictions Under the Influence of Structural Variability

“…If the statistical moments are known then the probability density function can be derived by the maximum entropy method 16,17 as…”

Stochastic dynamic systems with complex‐valued eigensolutions

“…. ,ẏ 7 . Figures 6(a) and (b) show the mean and standard The univariate and bivariate decomposition methods require 37 and 613 solutions, respectively, of the matrix characteristic equation, whereas 100 000 such solutions (sample size) are involved in the direct Monte Carlo simulation.…”

“…However, these methods have two major limitations: both the uncertainty of random input and the non-linearity of the random eigenvalue or eigenvector with respect to random input must be small. Methods other than the perturbation methods include the iteration method [1], the Ritz method [4], the crossing theory [5], the stochastic reduced basis [6], the asymptotic method [7], the polynomial chaos expansion [8], and the recently developed dimensional decomposition method [9]. However, most if not all, of these past studies focus strictly on undamped or proportionally damped systems.…”