Gaussian orthogonal ensemble spacing statistics and the statistical overlap factor applied to dynamic systems

This paper examines the extent to which Gaussian orthogonal ensemble (GOE) statistics is applicable to the natural frequencies of a dynamic system. The natural frequencies of a simply supported plate or a rectangular room tend to have an exponential spacing distribution. However, any disruption of the system symmetries has been shown to promote GOE statistics, for which the modal spacing distribution is Rayleigh. In this paper, the effect of a range of uncertainties on the modal statistics of structures is numerically characterised. This is achieved by examining the modal statistics of mass and/or spring loaded plates and plates coupled by springs. The natural frequencies of the aforementioned structures have been derived using the Lagrange–Rayleigh–Ritz technique. The degree of uncertainty required to effect the transition from an exponential to a Rayleigh distribution and to achieve universality of the statistical properties is investigated. A further measure of the randomness required to produce GOE statistics can be obtained by examining the amount of mixing and veering between the modes of a dynamic system. The statistical overlap factor is a non-dimensional parameter related to the random variation in an individual natural frequency from its mean value, and is useful to quantify the frequency beyond which the resonant behaviour of individual modes no longer dominates the response statistics. Using a first-order perturbation analysis, an approximate expression for the statistical overlap factor has been developed for the randomised plates, to estimate the modal range for the occurrence of GOE statistics.