Kriging Surrogate Models for Predicting the Complex Eigenvalues of Mechanical Systems Subjected to Friction-Induced Vibration

Hindawi Publishing Corporation Shock and Vibration Volume 2016, Article ID 3586230, 22 pages http://dx.doi.org/10.1155/2016/3586230 Research Article Kriging Surrogate Models for Predicting the Complex Eigenvalues of Mechanical Systems Subjected to Friction-Induced Vibration E. Denimal,1,2 L. Nechak,1 J.-J. Sinou,1,3 and S. Nacivet2 1 Laboratoire de Tribologie et Dynamique des Systèmes, UMR CNRS 5513, École Centrale de Lyon, 36 avenue Guy de Collongue, 69134 Écully Cedex, France 2 PSA Peugeot Citroën, Centre Technique de la Garenne Colombes, 18 rue des Fauvelles, 92250 La Garenne-Colombes, France 3 Institut Universitaire de France, 75005 Paris, France Correspondence should be addressed to J.-J. Sinou; Received 7 June 2016; Revised 2 September 2016; Accepted 18 September 2016 Academic Editor: Matteo Aureli Copyright © 2016 E. Denimal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This study focuses on the kriging based metamodeling for the prediction of parameter-dependent mode coupling instabilities. The high cost of the currently used parameter-dependent Complex Eigenvalue Analysis (CEA) has induced a growing need for alternative methods. Hence, this study investigates capabilities of kriging metamodels to be a suitable alternative. For this aim, kriging metamodels are proposed to predict the stability behavior of a four-degree-of-freedom mechanical system submitted to friction-induced vibrations. This system is considered under two configurations defining two stability behaviors with coalescence patterns of different complexities. Efficiency of kriging is then assessed on both configurations. In this framework, the proposed kriging surrogate approach includes a mode tracking method based on the Modal Assurance Criterion (MAC) in order to follow the physical modes of the mechanical system. Based on the numerical simulations, it is demonstrated by a comparison with the reference parameter-dependent CEA that the proposed kriging surrogate model can provide efficient and reliable predictions of mode coupling instabilities with different complex patterns. 1. Introduction Studies of mechanical systems subjected to friction-induced vibrations benefit from a growing interest due to the large amount of applications in the field of mechanical engineering. The different and complex mechanisms that can be responsible for undesirable dynamic characteristics and appearance of instabilities in many mechanical systems have been extensively studied in the last decades [1–5]. There are typically two different analyses and categories of mechanisms available for defining the origin of friction-induced system instability: the first one is mainly due to tribological properties whereas the second one relies on geometrical conditions. While the variation of the friction coefficient is considered as one of the most important factors for the emergence of instability in the first category (i.e., in the case of a tribological approach), the origin of friction-induced vibrations is rather related to kinematic constraints or sprag-slip phenomenon [6] and modal coupling in the second case (i.e., in the case of a structural dynamics approach based on geometrical conditions). In this last case, the emergence of instability can be detected even with a constant friction coefficient. In the present study, this last approach that is based on structural coupling mechanism will be discussed. Nowadays, two kinds of analysis are classically used to undertake numerical studies of friction-induced vibrations and dynamic instabilities on mechanical systems: the Complex Eigenvalue Analysis (CEA) to detect unstable frequencies [7, 8] and time analysis to determine self-excited vibrations [9, 10]. As explained in previous papers [9, 11, 12], both approaches have their pros and cons. However CEA based methods and the calculations of self-excited vibrations may become too costly when parametric analysis and/or 2 uncertainty propagation are needed for engineering design problems [13]. In these cases, it may be worthwhile to work towards the development of sophisticated methods based on surrogate models in order to perform design optimization or design space approximation (i.e., emulation). The main aim is to substitute any complex model by a suitable surrogate model which offers a convenient compromise between the accuracy of its predictions and the cost related to its implementation. In the present study, one is interested in estimating the occurrence of instability in a predefined design space approximation. In this context, the main purpose of the surrogate modeling is the generation of a surrogate that is as accurate as possible for the prediction of the occurrence of instabilities in the complete design space of interest, using as few simulation evaluations as possible. Such approximation models, known as metamodels or emulators, mimic the behavior of the simulation model (i.e., estimation of all the real and imaginary parts of eigenvalues in our case) as closely as possible while being computationally cheaper to evaluate. It may be noted that the accuracy of the surrogate depends on the number and location of samples in the design space of interest required for its implementation. Moreover, surrogate models are characterized by some tuning parameters that control their accuracy. In the field of friction-induced vibrations, numerous formalisms have been developed to define surrogate models for the prediction of mode coupling instabilities. Surrogate models that are based on the Generalized Polynomial Chaos (GPC) formalism [14] have been proposed this last decade to deal with the stability of mechanical systems subjected to friction-induced vibrations under uncertainties [15–18]. This approach has been proposed for propagating uncertainties described by probability density functions in systems submitted to friction-induced instabilities, a task which is prohibitive when performed by using the Monte Carlo method. The latter was exploited for estimating of the probability of squeal occurrence in [19] and in several other studies as a reference method [15–17]. So the main idea governing the GPC formalism consists of expressing the system’s degrees of freedom or eigenvalues within a functional space built from polynomials that are orthogonal with respect to probabilistic measures associated with the system’s design parameters. The chaos order is the most important tuning parameter which is fixed to a suitable value from a convergence study. This probabilistic surrogate model has shown an interesting efficiency in propagating and quantifying uncertainties on the stability behavior of such systems. However, it may present some limits when the number of uncertain parameters is relatively high and/or when high chaos orders are required i (...truncated)