Research Article Stability Optimization Of A Disc Brake .
Hindawi Publishing CorporationShock and VibrationVolume 2016, Article ID 3497468, 13 h ArticleStability Optimization of a Disc Brake System withHybrid Uncertainties for Squeal ReductionHui Lü and Dejie YuState Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan 410082, ChinaCorrespondence should be addressed to Dejie Yu; djyu@hnu.edu.cnReceived 2 November 2015; Revised 28 January 2016; Accepted 28 January 2016Academic Editor: Tai ThaiCopyright 2016 H. Lü and D. Yu. 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.A hybrid uncertain model is introduced to deal with the uncertainties existing in a disc brake system in this paper. By the hybriduncertain model, the uncertain parameters of the brake with enough sampling data are treated as probabilistic variables, while theuncertain parameters with limited data are treated as interval probabilistic variables whose distribution parameters are expressedas interval variables. Based on the hybrid uncertain model, the reliability-based design optimization (RBDO) of a disc brake withhybrid uncertainties is proposed to explore the optimal design for squeal reduction. In the optimization, the surrogate model ofthe real part of domain unstable eigenvalue of the brake system is established, and the upper bound of its expectation is adoptedas the optimization objective. The lower bounds of the functions related to system stability, the mass, and the stiffness of designcomponent are adopted as the optimization constraints. The combinational algorithm of Genetic Algorithm and Monte-Carlomethod is employed to perform the optimization. The results of a numerical example demonstrate the effectiveness of the proposedoptimization on improving system stability and reducing squeal propensity of a disc brake under hybrid uncertainties.1. IntroductionBrake squeal is a noise problem caused by self-frictioninduced vibrations which can induce dynamic instabilities ofbrake systems [1]. Poor stability performance of a disc brakesystem could lead to an inconvenient squealing noise. Thisfrequently leads to customer complaints and results in enormous warranty costs. Therefore, extensive efforts have beenundertaken by industrial corporations as well as by the scientific communities to resolve the squeal problem. Earlyresearches on brake squeal are mostly dedicated to the rootcauses of brake squeal and several representative mechanismshave been attributed to brake squeal phenomenon, such asstick-slip, sprag-slip, hammering, mode-coupling, and timedelay [2–5]. Moreover, several comprehensive review paperscan be found on the researches of brake squeal [6–10]. Ofthose mechanisms, the mode-coupling has attracted the mostattention in these literatures. However, the root cause of brakesqueal is still not fully understood due to its complexity.The optimization designs of disc brake systems for squealreduction have received many researchers’ attentions recently.For example, Guan et al. [11] explored sensitivity analysismethods to determine the dominant modal parameters of thesubstructures of a brake system for squeal suppression, andthe dominant modal parameters were set as the target of optimization design to seek the modifications of the disc and thebracket to eliminate the squeal mode. Spelsberg-Korspeter[12, 13] performed a structural optimization of brake rotorsto emphasize the potential of designing rotor geometryrobust against self-excited vibrations, and the mathematicaldifficulties of such an optimization were discussed. Lakkamand Koetniyom [14] proposed an optimization study on brakesqueal aimed at minimizing the strain energy of vibratingpads with constrained layer damping; the results of thisresearch could be a guide to specify the position of the constrained layer damping patch under pressure conditions.Shintani and Azegami [15] presented a solution to a nonparametric shape optimization problem of a disc brake modelto suppress squeal noise; the optimum shape of the pad wasfound and the real part of the complex eigenvalue representing the cause of brake squeal was minimized.The abovementioned studies on structural optimizationof disc brake systems are restricted to deterministic optimization, in which all design variables and parameters involved
2are regarded certain. In practical engineering, uncertain factors widely exist in the structures and systems, as well as in thebrake systems. Under uncertain cases, the optimum obtainedfrom a deterministic optimization could easily violate theconstraints and cause unreliable solutions and designs [16–18]. In order to take into account various uncertainties, theRBDO is introduced and has been intensively studied bothin the methodology and in applications [19–21]. Comparedwith the deterministic optimization, the RBDO aims to seek areliable optimum by converting the deterministic constraintsinto probabilistic constraints. Therefore, the RBDO can beconsidered as the potential method to improve the stabilityperformance of the uncertain disc brake systems for squealreduction.Due to the effects of manufacturing errors, aggressiveenvironment factors, wear and unpredictable external excitations, uncertainties associated with material properties,geometric dimensions, and contact conditions are unavoidable. On the investigation of disc brake squeal consideringuncertainties, just only a few research papers have been published at present. Chittepu [22] has carried out a robustnessevaluation of a brake system with a stochastic model, in whichthe material scatter was modeled by random variables andthe geometrical tolerances were modeled by random fields.Based on the polynomial chaos expansions, the stochasticeigenvalue problems and the stability of a simplified discbrake have been investigated by Sarrouy et al. [23], in whichthe friction coefficient and the contact stiffness of the discbrake system were modeled as random parameters. Tison etal. [24] have proposed a complete strategy to improve theprediction of brake squeal by introducing uncertainty androbustness concepts into the simulations. The strategy mainlyrelied on the integration of complex eigenvalue calculations,probabilistic analysis, and a robustness criterion. Lü and Yu[25, 26] have proposed practical approaches for the stabilityanalysis and optimization of disc brakes with probabilisticuncertainties or interval uncertainties for squeal reduction,in which an uncertain parameter is treated as either probabilistic variable or interval variable. It can be seen that theinvestigations on uncertain brake squeal problem are notstill comprehensive currently. It is necessary to develop moredifferent uncertain methods for the uncertain researches ondisc brake squeal reduction.It is well known that probabilistic methods are thetraditional approach to cope with these uncertainties arisingin practical engineering problems, just as we can see in theabovementioned studies [22–24]. In the probabilistic methods, the uncertain parameters are treated as probabilisticvariables whose probability distributions are defined unambiguously [27]. To construct the precise probability distributions of probabilistic variables, a large amount of statisticalinformation or experimental data is required. Unfortunately,in the design stage of a disc brake system, the informationto construct the precise probability distributions of someprobabilistic variables (e.g., the friction coefficient or thethicknesses of wearing components) is not always sufficientdue to the immeasurability or wear effects. In the last twodecades, the interval probabilistic model has been proposedfor overcoming the deficiencies of probabilistic methods. InShock and Vibrationthe interval probabilistic model, the uncertain parameters aretreated as interval probabilistic variables whose distributionparameters with limited information are expressed as intervalvariables instead of precise values. The interval probabilisticmodel was firstly proposed by Elishakoff et al. [28, 29] andsubsequently applied to the structural response analysis [30]and the structural reliability analysis [31]. From the overallperspective, researches on the interval probabilistic model arestill in the preliminary stage and some important issues arestill unsolved. For example, the application of the intervalprobabilistic model in the stability analysis of disc brakes isnot yet explored.The purpose of this paper is to take into account thehybrid uncertainties existing in a disc brake and developan effective optimization for improving its system stabilityand suppressing its squeal propensity. For this purpose, aRBDO approach based on the hybrid uncertain model isproposed. In the hybrid uncertain model, the uncertainties ofthe thickness of nonwear component, the mass densities andelastic modulus of component materials, and the brake pressure are represented by using probabilistic variables, whosedistribution parameters are well defined, whereas the uncertainties of the friction coefficient, the thickness of disc, andfriction material are modeled by interval probabilistic variables, whose distribution parameters are expressed as intervalvariables. To facilitate computation, the response surface (RS)model is used to replace the time-consuming FE simulations.Based on the methods of RSM, CEA, and hybrid uncertainanalysis, the RBDO model for improving the system stabilityof the disc brake is constructed. In the RBDO model, theupper bound of the expectation of the real part of the domainunstable eigenvalue is selected as design objective, while thelower bounds of the functions related to system stability, themass and the stiffness of design component, are adopted asoptimization constraints. The effectiveness of the proposedoptimization is demonstrated by a numerical example.The rest of this paper is organized as follows: Section 2introduces the CEA of disc brakes; Section 3 describes theconstruction of the RBDO of a disc brake with hybrid uncertainties; Section 4 presents the procedure of the proposedRBDO; Section 5 provides a numerical example of the application of the proposed optimization, and finally Section 6draws the major conclusions.2. Complex Eigenvalue Analysis of Disc Brake2.1. A Simplified Disc Brake. For the purpose of simulatingthe vibration characteristics of a brake system reasonably withan acceptable computation, a simplified disc brake system istaken for investigation. The simplified brake consists of a discand a pair of brake pads, as shown in Figure 1. The disc isrigidly mounted on the axle hub and therefore rotates withthe wheel. The pair of brake pad assemblies, which consist ofthe friction materials and the back plates, is pressed againstthe disc in order to generate a frictional torque to slow therotation of the disc. The similar simplified model has beenpreviously considered and successfully used by some studies,such as [23, 32–34].
Shock and Vibration3where 𝜆 is the complex eigenvalue and 𝜙 is the complex eigenmode. The complex eigenvalues of the system can be obtainedby solving the complex eigenvalue problem of (6). The 𝑖thcomplex eigenvalue 𝜆 𝑖 can be expressed asBrake discBrake padFriction material𝜆 𝑖 𝛼𝑖 𝑗𝛽𝑖 ,Back plateFigure 1: A simplified model of a disc brake system.2.2. Complex Eigenvalue Analysis. The CEA can be carriedout by two stages. In the first stage, the steady state of thebrake system is found and in the second stage, the CEA isperformed.In the steady state of the brake system, the vector ofapplied forces F(y,̇ y) is balanced by the vector of nonlinearstiffness forces F𝑘 (y). It can be expressed asF𝑘 (y) F (y,̇ y) ,(1)where y is the steady-state displacement vector and ẏ is thefirst derivative of y with respect to time. The nonlinearity ofstiffness forces F𝑘 (y) is mainly due to the contact formulation,in which the contact stiffness is commonly added at thedisc/pad interface. The disc is about one hundred times stifferthan the frictional material which is a mixture of manycomponents; thus deformation of a pad under loading ismuch greater and nonlinearity behavior occurs. For a morecomplete discussion of system nonlinearities, the readers canrefer to [35].After the steady state of the system is found, in thesecond stage, the stability of system is assessed. The extendednonlinear equation of motion including the mass matrix Mand the damping matrix D is given byMÿ Dẏ F𝑘 (y) F (y,̇ y) ,(2)where ÿ is the second derivative of y with respect to time. Thesystem can be linearized about the steady statey y0 Δy,(3)where y0 is the steady-state displacement vector and Δydenotes small perturbation in the vicinity of the steady state.After linearization, the resulting equation is given byM Δÿ D Δẏ K Δy H1 Δẏ H0 Δy,(4)where matrices H0 and H1 are the partial derivatives ofF(y,̇ y) with respect to the displacements and the velocities,respectively. Equation (4) can be simplified tô Δẏ K̂ Δy 0,M Δÿ D(5)̂ and K̂ are the new damping matrix and stiffnesswhere Dmatrix; then the complex eigenvalue problem can be formulated aŝ K)̂ 𝜙 0,(M𝜆2 D𝜆(6)(7)where 𝛼𝑖 and 𝛽𝑖 are the real and imaginary parts of thecomplex eigenvalue 𝜆 𝑖 , respectively.It is widely known that a vibration system will becomeunstable if the real part of a complex eigenvalue of the systemis positive. Therefore, the positive real parts of the eigenvaluescan be taken as an indicator for system stability and squealpropensity. The main aim of this study is to minimize thepositive real part of the dominant unstable eigenvalue of thebrake system.It is worth mentioning that the damping is not taken intoaccount in the current study. Just as the recent study [36]points out, the damping models of friction materials or othercomponents have not been extensively studied at present andthere is no reliable data on the damping of brake systems.In most cases, the damping can stabilize brake systems andexcluding it will provide more potential unstable eigenvalues.Thus, industrial researchers tend to exclude the damping andset a target value for the real part or the damping ratio ofan unstable eigenvalue in CEA of practical engineering. Ifthe real part or the damping ratio is not larger than thetarget value, the brake system will be considered as stableand acceptable [36]. However, two main effects of dampingon coalescence patterns have been noticed in fact, namely,the shifting effect and the smoothing effect [37]. The first onealways stabilizes the brake, whereas the second one may causeinstability. For more detailed and extensive discussion ofdamping on system stability, the readers can refer to [37].Moreover, the main limitation of CEA is the use of a linearmodel which excludes dynamic contact. Sinou et al. [38, 39]pointed out that CEA may lead to underestimation or overestimation of the unstable modes of a brake. In [38], it wasshown that additional unstable modes appeared during transient oscillations, which were not predicted by CEA. Thus,the nonlinear methods have been proposed and developedto approximate the transient behavior in the time domain[40], or the limit cycle of frictional systems subjected to selfexcited vibration and squeal noise in the frequency domain[41]. Besides, the stochastic studies have been recently carriedout to determine the limit cycles of a self-excited nonlinearsystem with friction which is commonly used to representbrake squeal phenomenon [42]. However, we mainly focusour attention on the stability analysis and optimization of alinear brake system with uncertainties in this work, considering that CEA is a fast and efficient strategy for an industrialfinite element model. Therefore, just the CEA approach isused in this paper.3. RBDO of the Disc Brake System withHybrid Uncertainties3.1. Surrogate Model Based on RMS. In engineering design,the coupling of optimization algorithm with finite element
4Shock and Vibrationanalysis (FEA) may be inefficient, since the iterative analysesduring optimization usually require enormous iterations andhigh computational cost, especially for RBDO problem. Asa result, the surrogate models have been widely adoptedin engineering applications to alleviate the computationalburden. By surrogate models, the explicit mathematical relationship between functional responses and design variablescan be established with a moderate number of FEA runs.RS model is one of the simplest and most popular surrogatemodels, which can be treated as an effective alternative to FEAand has been widely adopted in design optimization [43].Mathematically, by the RS model, the real part of acomplex eigenvalue of FEA can be defined in terms of basisfunction as𝑁𝛼 (x) 𝑎𝑗 𝜑𝑗 (x) ,(8)𝑗 1where 𝛼(x) is the real part of a complex eigenvalue, 𝜑𝑗 (x) arethe basis functions, 𝑁 is the number of basis functions, and 𝑎𝑗represents the regression coefficients. When using a quadraticmodel, the full set of the second-order polynomials of 𝜑𝑗 (x)is given as1, 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 , 𝑥12 , 𝑥1 𝑥2 , . . . , 𝑥1 𝑥𝑛 , . . . , 𝑥𝑛2(9)and the surrogate model of 𝛼(x) could be thus defined as𝑛𝑛𝛼 (x) 𝑎0 𝑎𝑖 𝑥𝑖 𝑎𝑖𝑗 𝑥𝑖 𝑥𝑗 𝑎𝑖𝑖 𝑥𝑖2 ,𝑖 1𝑖𝑗(𝑖 𝑗)𝑖 1(10)where x [𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ]𝑇 denotes the variable vector thatdetermines the response 𝛼(x) and 𝑎0 , 𝑎𝑖 , 𝑎𝑖𝑗 , and 𝑎𝑖𝑖 are theestimated regression coefficients which can be obtained fromthe design of experiment (DOE) and by the least squaremethod (LSM) [44]. The cross product terms 𝑥𝑖 𝑥𝑗 representthe two-variable interactions and the square terms 𝑥𝑖2 represent the second-order nonlinearities. 𝑛 is the number of thevariables.After the RS model is established, its accuracy should beassessed. It is necessary to conduct an analysis of variance(ANOVA) to test the model, so as to ensure its fitting accuracyand significance [45].It is worth mentioning that the surrogate model withkriging predictor has been recently proposed to investigatethe propensity of brake squeal by Nobari et al. [36] andNechak et al. [46]. However, the current study is not focusedon the surrogate models; thus just the most widely used surrogate model (RS model) is applied here. Indeed, it has greatsignificance to carry out further studies on the application ofdifferent surrogate models to squeal problem in the future.3.2. Disc Brake System with Hybrid Uncertainties. Due to thecomplex working environment and operating conditions, theoccurrence of brake squeal is intermittent or perhaps evencasual in general, and it is difficult to be captured and reproduced artificially. This is possibly related to the uncertaintiesexisting in the brake system. For instance, the friction coefficient is changed in the course of braking. This characteristicof friction coefficient makes it possible to provide the energysource for brake squeal [9]. So it will be more reasonable andbe of great significance, if the uncertainties are taken intoaccount for the stability analysis of the disc brake systems.For the disc brake system, uncertainties in material, geometry, and loading properties and manufacturing errors areunavoidable. Usually, the most common approach to modelthe uncertainties is the probabilistic model. By this approach,𝛼(x) in (10) for an uncertain optimization problem can begenerally expressed as𝛼 (x) 𝛼 (x𝐷, x𝑃 ) ,(11)where x𝐷 [𝑥𝐷,1 , 𝑥𝐷,2 , . . . , 𝑥𝐷,𝑘 ]𝑇 is the design variablevector and 𝑘 is the number of design variables; x𝑃 [𝑥𝑃,1 , 𝑥𝑃,2 , . . . , 𝑥𝑃,𝑙 ]𝑇 is the probabilistic variable vector whichis taken as nondesign variables and 𝑙 is the number ofprobabilistic variables.In the real case, the disc brake is a rather complex systemwhich is related to frictional contact and wear [47]. It isdifficult or even impossible to obtain the precise values of thedistribution parameters of
disc brake squeal reduction. It is well known that probabilistic methods are the traditional approach to cope with these uncertainties arising in practical engineering problems, just as we can see in the abovementioned studies [ ]. In the probabilistic meth-
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Since the eld { also referred to as black-box optimization, gradient-free optimization, optimization without derivatives, simulation-based optimization and zeroth-order optimization { is now far too expansive for a single survey, we focus on methods for local optimization of continuous-valued, single-objective problems.
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