A non–Poisson model for the vibration analysis of uncertain dynamic systems

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Abstract
The response of a dynamic system to medium- or high-frequency excitation can be very sensitive to small changes in the physical properties of the system. Previous work has investigated this issue by adopting a Poisson model of the system natural frequencies; other studies have performed numerical simulations that reveal the effect of parameters that cannot be included in the Poisson model. The present work considers a point-process model of the natural frequencies that encompasses the Poisson model as a limiting case but also allows the influence of key parameters such as the statistical-overlap factor to be considered. A number of general results are derived and these are then applied to the special case of Gaussian natural frequencies. A closed-form solution for the variance of a frequency-response function is derived that depends upon a single parameter: this parameter is a combination of the statistical-overlap factor, the modal overlap factor, and the degree of correlation between the natural frequencies. Attention is also directed at the statistical distribution of the natural frequency spacing, and various results are derived in terms of the cumulant functions of the point-process model. The analytical results obtained are confirmed by comparison with simulation results.

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