Nonlinear and stochastic analysis of dynamical instabilities based on Chebyshev polynomial properties and applied to a mechanical system with friction

Abstract

In this paper, mechanical systems presenting instabilities induced by friction such as brakes and clutches are studied using a phenomenological model. For these systems, the stability and vibration levels are strongly linked to the values of certain parameters such as the friction coefficient, contact stiffness, etc. Therefore, it is particularly important to take into account the uncertainties of these parameters to obtain robust and predictive models. In this paper, the modelling of uncertainties is done using probabilistic theory with a parametric view. Each quantity that is a function of the random variables is decomposed in a basis using polynomial chaos. For the model studied, the choice of a classical intrusive approach to solve the mechano-stochastic problem induces particularly long computation times. This is due to the need to perform probabilistic numerical integrations to determine integral terms associated with generalized stochastic forces. As the integration steps in these numerical quadratures must be very small, the calculation times are very long. This is why, in this paper, a modified intrusive approach is presented with the main objective of solving a mechano-stochastic problem in reasonable times and thus avoiding approximation of stochastic generalized forces with numerical integrations. This modified intrusive approach is composed of two parts. In the first part, an approximation of the mechanical action torsor (forces and moments) at the contact and friction interface, is performed. To achieve this, behaviour laws (between torsors) are constructed using potentials and dissipation functions based on physical considerations and reinforced by an influence study. In the second part, the properties of Chebyshev polynomials of the second kind are exploited to avoid numerical integration, several times per time step, of the terms associated with generalized stochastic forces. This approach is initially used in a linear framework to perform a stability study of fixed points. It is then used in a nonlinear framework to carry out temporal integrations. Finally, the results of temporal integration and simulation times from the modified intrusive approach are compared to those of the non-intrusive approach. For the latter, Chebyshev polynomials of the second kind are also used to interpolate the results of deterministic temporal integrations.