Shock and Vibration

Volume 2016 (2016), Article ID 3586230, 22 pages

http://dx.doi.org/10.1155/2016/3586230

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

^{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

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.

#### Abstract

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 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 in particular for functions that are strongly nonlinear in the random space. In this last case, surrogate models based on the multielement GPC can be useful [20]. The response surface methodology (RMS) is also proposed in [21] to deal with the stability, reliability, and sensitivity analysis of brake systems submitted to interval and random uncertainties. The proposed surrogate model consists of using basis functions (defined by monomials in the uncertain parameters) to express the system eigenvalues. The same surrogate model is proposed in [22] for the optimization design of brake system under interval and random uncertainties. Another approach consists of constructing surrogate models based on the perturbation principle [23]. In this specific case, the main principle consists in expressing system’s eigenvalues by means of Taylor expansions near the mean value of uncertain parameters. For example, the first-order perturbation method has been proposed by Butlin and Woodhouse [24] to quantify sensitivity of friction-induced instabilities to the design parameters. Moreover the recent study of Nobari et al. [25] proposes a second-order Taylor expansion to estimate statistical properties of eigenvalues characterizing mode coupling instabilities. Despite its efficiency, this approach has limitations, especially when standard deviations of parameters are important.

Another type of surrogate models can be constructed based on the kriging method [26, 27]. This approach exploits spatial correlations between a small number of function values at some samples generated from a suitable experience plan to predict unknown values of the function within its design space of interest. The kriging based model consists in two main parts; the regression model roughly represents the global tendency of the analyzed function while the second part is defined by a stochastic process representing the spatial correlations in the design space of interest. Extensive reviews of kriging metamodeling in simulation and other applications as in the sensitivity analysis and optimization in design process can be found in [28, 29].

In the recent past, the prediction of squeal instability in brake systems via surrogate modeling has been introduced by Nobari et al. [30] and Nechak et al. [13] in order to construct a predictor of squeal instability. However, these two interesting studies were limited to the estimation of unstable frequencies without considering the prediction of all the stable modes or the behavior of the system before the emergence of squeal instability. Extension of the prediction of both stable and unstable behaviors (i.e., estimation of all the real parts and imaginary parts of eigenvalues) for mechanical systems subjected to friction-induced vibrations via surrogate modeling is proposed in the present study. Furthermore, the construction of a surrogate model for each separated mode is carried out with a careful selection of output data, due to the evolution of the complex eigenvalues and the order of the modes when some specific parameters change. If the surrogate model is not constructed by paying attention to this point, errors due to an improper surrogate model will appear. To overcome this difficulty, a tracking process based on the Modal Assurance Criterion (MAC) is proposed in conjunction with the generation of kriging surrogate models characterized by their tuning parameters, namely, the order of the regression model, the spatial correlation model, and the size of the experience plan. The two previous studies [13, 30] have not analyzed exhaustively the effects of these tuning parameters on the accuracy of kriging for instability predictions of mechanical system subjected to friction-induced vibrations.

So the main objective and originality of the present paper lies in the analysis of performances (in terms of accuracy and cost) of kriging surrogate models with respect to their tuning parameters, when dealing with the prediction of not only mode coupling instabilities but also the complete approximation of the real and imaginary parts of both stable and unstable modes. To undertake such a study and to validate the methodology of constructing kriging surrogate models with a tracking process, numerical simulations will be performed on a minimal four-degree-of-freedom model. Two specific numerical cases will be investigated: the first one will be a classical baseline with “a simple mode coupling mechanism” (i.e., coalescence of two modes, one being unstable and the other unstable). The second case deals with more complex modes coupling mechanisms with the successive appearances and disappearances of instabilities, and the crossing phenomenon between modes.

This study is organized as follows. First, the mechanical system under study is presented and the classical stability analysis (i.e., CEA) is briefly discussed. Then, the methodology for constructing kriging surrogate models is developed. The proposed procedure allows not only the prediction of mode coupling instabilities but also the prediction of all the complex eigenvalues of the mechanical system (i.e., the real parts and imaginary parts for both stable and unstable modes) by performing a selection and arrangement of the modes via a suitable MAC criterion. The last part of the present study is devoted to the presentation and discussion of the numerical results.

#### 2. Mechanical System and Stability Analysis

##### 2.1. Description of the Phenomenological Model

Figure 1 shows the minimal four-degree-of-freedom model to be used in the present study. This phenomenological model has its origins in the previous two-degree-of-freedom model proposed by Hultén [31, 32] and was used to point out the role of the damping and the destabilization paradox [33] and for the prediction of mode coupling instabilities submitted to parameter uncertainties [15–17, 34]. In the present study, an extension of this minimal Hultén model is proposed in order to investigate the case of multi-instabilities. The model consists of two masses and held against moving bands disposed as shown in Figure 1. Contacts between masses and bands are modeled by plates supported by springs and damping. The masses and are linearly coupled by a spring (i.e., stiffness ) and the associated damping . For the sake of simplicity, it is assumed that the two masses and the three band surfaces are always in contact due to a preload applied to the mechanical system. Considering the friction forces between the four plates and the three bands, the coefficient of friction is assumed to be constant. Its value is the same for all contacts and the classical Coulomb law is applied. The velocity of the moving bands is considered as constant. Moreover, the relative velocities between the band speed and the displacements of masses are assumed to be positive so that the direction of the tangential friction force does not change. According to the Coulomb law, the tangential friction force is assumed to be proportional to the normal force (i.e., , where is the friction coefficient).