Frequency Response Functions of Random Linear Mechanical Systems and Propagation of Uncertainties

Abstract

In the modeling of dynamical systems, uncertainties are present and they must be taken into account to improve the prediction of the models. It is very important to understand how they propagate and how random systems behave. The aim of this work is to discuss the probability distribution function (PDF) of the amplitude and phase of the response of random linear mechanical systems when the stiffness are random. The novelty of the paper is that the computations are done analytically whenever possible. The propagation of uncertainties is then characterized. The PDF of the response of a system with random stiffness near the resonant frequency of the mean system has a complex structure and can presents multimodality in certain conditions. In Statistics a mode is a maximum of the PDF, and the modes describe the most probable values of the random variable. This multimodality makes approximations of the statistics, the mean for example, very difficult and sometimes meaningless since the behavior of the mean system can be quite different of the mean of the realizations. More complex systems, discrete and continuous, are also discussed and they show similar behaviour.