Steady-state response of a random dynamical system described with Padé approximants and random eigenmodes

Designing a random dynamical system requires the prediction of the statistics of the response, knowing the random model of the uncertain parameters. Direct Monte Carlo simulation (MCS) is the reference method for propagating uncertainties but its main drawback is the high numerical cost. A surrogate model based on a polynomial chaos expansion (PCE) can be built as an alternative to MCS. However, some previous studies have shown poor convergence properties around the deterministic eigenfrequencies. In this study, an extended Pade approximant approach is proposed not only to accelerate the convergence of the PCE but also to have a better representation of the exact frequency response, which is a rational function of the uncertain parameters. A second approach is based on the random mode expansion of the response, which is widely used for deterministic dynamical systems. A PCE approach is used to calculate the random modes. Both approaches are tested on an example to check their efficiency

Double synthèse modale : bilan et perspectives

Depuis une quinzaine d'années, 1'Equipe D2S (Dynamique des Structures et des Systèmes) a mis en œuvre des méthodes de synthèse modale spécifiques basées sur une formulation hybride des problèmes de dynamique. Leur particularité réside dans l'utilisation de modes d'interfaces reliés à la notion de modes de branche et dans l'analyse des effets de troncature. L'utilisation conjointe de modes internes, libres ou encastrés suivant le choix de la formulation, et des modes d'interfaces permet d'obtenir des modèles dynamiques à très faible nombre de degrés de liberté. En rattachant ces modèles aux problèmes intermédiaires de Weinstein, il est possible d'analyser leur convergence dans un cadre mathématique rigoureux

Stochastic study of a non-linear self-excited system with friction

ThispaperproposestwomethodsbasedonthePolynomialChaostocarryout the stochastic study of a self-excited non-linear system with friction which is commonly used to represent brake-squeal phenomenon. These methods are illustrated using three uncertain configurations and validated using comparison with Monte Carlo simulation results. First, the stability of the static equilibrium point is examined by computing stochastic eigenvalues. Then, for unstable ranges of the equilibrium point, a constrained harmonic balance method is developed to determine subsequent limit cycles in the deterministic case; it is then adapted to the stochastic case. This demonstrates the effectiveness of the methods to fit complex eigenmodes as well as limit cycles dispersion with a good accuracy

Polynomial Chaos Expansion With Fuzzy and Random Uncertainties in Dynamical Systems

This paper proposes a surrogate model which is able to deal with mixed uncertain dynamical systems: some uncertain parameters are modelled by random variables whereas others are represented by fuzzy variables. Polynomial chaos expansions (PCE) were developed for uncertainty propagation through random systems. However, fuzzy variables may also be described with polynomial chaos: in particular, Legendre polynomials are well adapted to fuzzy variables. Hence we propose a polynomial chaos expansion, which is able to describe an uncertain dynamical system with both random and fuzzy variables. The method is applied to simulate an uncertain bar and the results are compared to MCS results. Thus the PCE is successfully applied to dynamical systems with both fuzzy and random uncertainties. This study also highlights the issue of the description of the outputs: which quantities should be calculated to represent the behaviour of the uncertain outputs? In the case studies, the fuzzy mean and the fuzzy standard deviation are used to describe the output properties

Polynomial chaos expansion with random and fuzzy variables

A dynamical uncertain system is studied in this paper. Two kinds of uncertainties are addressed, where the uncertain parameters are described through random variables and/or fuzzy variables. A general framework is proposed to deal with both kinds of uncertainty using a polynomial chaos expansion (PCE). It is shown that fuzzy variables may be expanded in terms of polynomial chaos when Legendre polynomials are used. The components of the PCE are a solution of an equation that does not depend on the nature of uncertainty. Once this equation is solved, the post-processing of the data gives the moments of the random response when the uncertainties are random or gives the response interval when the variables are fuzzy. With the PCE approach, it is also possible to deal with mixed uncertainty, when some parameters are random and others are fuzzy. The results provide a fuzzy description of the response statistical moments