Complexity of the response of linear systems with a random coefficient and propagation of uncertainties

In the modelling of dynamical systems, uncertainties are present and 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 is random. The function connecting the response of the system to the stiffness, one of the coefficients of the linear equation, is highly nonlinear. The linearity exists only if one considers input, the forcing term, and the output, the response. The novelty of the paper is that the computations are done analytically whenever possible. The propagation of uncertainties is then characterised. 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 present 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 behaviour of the mean system can be quite different from the mean of the realisations. More complex systems, discrete and continuous, are also discussed and show similar behaviour.