A comprehensive overview of a non-parametric probabilistic approach of model uncertainties for predictive models in structural dynamics

In structural dynamics, a predictive model is constructed by developing a mathematical–mechanical model of a designed system in order to predict the response of the real system which is the manufactured system realized from the designed system. The mathematical–mechanical modelling process of the designed system introduces two fundamental types of uncertainties: the data uncertainties and the model uncertainties. Uncertainties have to be taken into account for improving the predictability of the model. Model uncertainties cannot be modelled by using the usual parametric probabilistic approach. Recently, a general non-parametric probabilistic approach of model uncertainties for dynamical systems has been proposed using the random matrix theory. This paper gives a comprehensive overview of this approach in developing its foundations in simple terms and in illustrating all the concepts and the tools introduced in the general theory, by using a simple example. This paper deals with (1) notions of designed systems, real systems, mean models as predictive models, errors and uncertainties; (2) the definition of a simple example in linear elastodynamics; (3) a comprehensive overview of the non-parametric probabilistic approach of model uncertainties for predictive models in structural dynamics; (4) a summary of the random matrix ensembles which are necessary for the non-parametric modelling of random uncertainties; (5) the estimation of the dispersion parameters of the non-parametric probabilistic model using experimental data; (6) the method to solve the stochastic equation of the dynamical system with non-parametric probabilistic model of random uncertainties; (7) a numerical simulation and the validation for the simple example.