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Brake squeal reduction of vehicle disc brake system with interval parameters by uncertain optimization
Journal of Sound and Vibration 333 (2014) 7313–7325
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Brake squeal reduction of vehicle disc brake system with interval parameters by uncertain optimization Hui Lü, Dejie Yu n State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan 410082, People's Republic of China
a r t i c l e i n f o
abstract
Article history: Received 11 December 2013 Received in revised form 18 July 2014 Accepted 24 August 2014 Handling Editor: H. Ouyang Available online 16 September 2014
An uncertain optimization method for brake squeal reduction of vehicle disc brake system with interval parameters is presented in this paper. In the proposed method, the parameters of frictional coefficient, material properties and the thicknesses of wearing components are treated as uncertain parameters, which are described as interval variables. Attention is focused on the stability analysis of a brake system in squeal, and the stability of brake system is investigated via the complex eigenvalue analysis (CEA) method. The dominant unstable mode is extracted by performing CEA based on a linear finite element (FE) model, and the negative damping ratio corresponding to the dominant unstable mode is selected as the indicator of instability. The response surface method (RSM) is applied to approximate the implicit relationship between the unstable mode and the system parameters. A reliability-based optimization model for improving the stability of the vehicle disc brake system with interval parameters is constructed based on RSM, interval analysis and reliability analysis. The Genetic Algorithm is used to get the optimal values of design parameters from the optimization model. The stability analysis and optimization of a disc brake system are carried out, and the results show that brake squeal propensity can be reduced by using stiffer back plates. The proposed approach can be used to improve the stability of the vehicle disc brake system with uncertain parameters effectively. & 2014 Elsevier Ltd. All rights reserved.
1. Introduction Due to the consistent customer complaints and high warranty costs, brake squeal noise has become one of the important concerns associated with the automotive brake systems [1–3]. If the brake system comes into an unstable state in the working process, the strong vibration and a harsh noise may be caused. The brake squeal noise, especially in the frequency range between 1 and 16 kHz, is most annoying to passengers' hearing [4,5]. Several classic theories have been formulated to explain the mechanism of the disc brake squeal phenomenon. However, there is neither a comprehensive understanding of the problem nor a generalized theory of the brake squeal mechanism yet [5,6]. For the last several decades, researches on brake vibration and noise have been conducted by using the theoretical, experimental, and numerical approaches [3,7]. The theoretical approaches provide a good insight into the mechanism of squeal, but the complicated brake system has to be considerably simplified and the accurate analysis results cannot be n
Corresponding author. Tel.: þ86 73188821915; fax: þ 86 73188823946. E-mail address: [email protected] (D. Yu).
http://dx.doi.org/10.1016/j.jsv.2014.08.027 0022-460X/& 2014 Elsevier Ltd. All rights reserved.
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provided. The experimental approaches can effectively investigate the effects of different parameters and operating conditions on the squeal of a brake system, however, the experiments are mostly expensive and time-consuming. Instead of a simple schematic model, the numerical methods can simulate different structures, material compositions and operating conditions of a disc brake or of different brakes or of even different vehicles, when used rightly. CEA method is a widely used approach for brake squeal investigation. Since a good correlation between the CEA numerical analysis results and the available experimental data is already verified experimentally, the CEA approach has already become an effective approach to the brake squeal investigation [8,9]. Many numerical approaches have been presented to explore the squeal phenomenon with the CEA approach. In the 1980s, Liles [10] presented a method for determining the geometric stability characteristics of a disc brake assembly by the use of finite element method (FEM). In this research, a CEA method was performed to determine which modes were unstable and therefore likely to produce squeal. In the 1990s, Chargin et al. [11] developed a unique nonlinear method accounted for both superelement modes and surface friction data for the analysis of brake squeal, both transient responses and complex eigenvalues were provided for the analysis of brake systems. In the 21st century, a front disc brake system was used as an example for the investigation of the low frequency squeal by Kung et al. [12]. Many different modifications to the brake system were proposed. Ouyang et al. [13] developed a moving-load model for the disc-brake stability analysis, and solved the dynamic instability as a nonlinear eigenvalue problem. Compared with the non-moving-load model, a more realistic disc-brake model was constructed by their method; more unstable frequencies were predicted and correlated very well with the experimental squeal frequencies. As the complex eigenvalue analysis and the dynamic transient analysis were typically two different methodologies that could be used to predict squeal in a disc brake [11], Abubakar and Ouyang [14] explored a proper way of conducting both types of analyses and investigating the correlation between them for a large degree-of-freedom disc brake model. Guan et al. [15] explored the sensitivity analysis methods to determine the dominant modal parameters of substructures of a brake system for the brake squeal suppression analysis, the related formulas of sensitivities of the positive real part of the squeal mode to substructures' modal parameters were derived. Fritz et al. [8,16] computed the brake system eigenvalues by using a technique based on FEM, and the effects of damping on the coalescence patterns of system eigenvalues were investigated. Liu et al. [17] and Junior et al. [18] investigated the effects of system parameters on the disc brake squeal, and the insulation and damping materials were applied to suppress the brake squeal. Dai and Lim [19] applied an enhanced dynamic FE model with friction coupling to optimize the design of the disc brake pad structure for squeal noise reduction and the analysis showed that the eigenvalues possessing positive real parts tended to produce unstable modes with the propensity towards the generation of squeal noise. In order to improve the FE model accuracy, Nishizawa et al. [20] and Nonaka et al. [21] conducted researches aiming to incorporate the dynamic stiffness of the pads into FE modal analysis, and the change in the squeal frequency resulting from the change in the thicknesses of the pads was reproduced. In recent years, Nouby et al. [22] proposed a mathematical approach to investigating the influencing factors of the brake pad on the disc brake squeal by integrating finite element simulations with statistical regression techniques. Their predicted results showed that the brake squeal propensity could be reduced by increasing Young's modulus of back plate and modifying the shape of friction material by adding chamfers and slots. Lakkam and Koetniyom [23] proposed an optimization method for brake squeal by using FEM, assumed-coupling mode method and experiment operations. According to this research, the position/geometry of the constrained layer damping patch could be optimized by minimizing the strain energy of vibrating pads with constrained layer. Sarrouy et al. [24] proposed numerical approaches to dealing with the brake stability problem with uncertainties. The work of Sarrouy et al. was based on polynomial chaos expansions and took place in the context of uncertain systems. Through their methods, the stochastic eigenvalue problems of a disc brake system could be processed efficiently and accurately compared with Monte Carlo simulations. Although so many researches on brake squeal have been conducted through CEA numerical approaches, however, only a few investigate the brake squeal problem with uncertainties. In engineering practices, many different factors appear to affect brake squeal, such as geometrical dimensions, material properties and loading conditions. Variabilities always exist in these factors in reality [6,24]. When taking those variabilities into account, parameter uncertainties have to be introduced into the analysis model of brake systems for obtaining more reliable results. To deal with the uncertain problem with limited information, the interval model, in which the fluctuations of uncertain parameters are assumed to fall into a hyperrectangle, has been developed. It just needs to obtain the upper and lower bounds of an uncertain parameter, if the uncertain parameter is treated as an interval variable. The lower and upper bounds of interval variables can be easily obtained in engineering practice. Therefore, the analysis methods based on the interval analysis have been well developed and widely applied [25]. In addition, carrying out reliability analysis while considering the uncertainties can ensure the system with uncertain parameters is always in a reliable state [26]. In order to improve the performance of the system with uncertainties, the optimization under uncertainty should be carried out during the design process. For the uncertain problems with limited information, several theories for the optimization with non-probabilistic parameters have been proposed for the designs of engineering structures in recent years [27–30]. Thus it is necessary to conduct uncertain optimization for reducing the brake squeal of a real brake system. In this paper, attention is focused on the stability analysis and the optimization of the vehicle disc brake system with uncertainties. The stability of a disc brake system is investigated via the CEA method. Based on the FE model, the dominant unstable mode is extracted. And the damping ratio corresponding to the dominant unstable mode is chosen as the indicator of instability. Uncertain parameters are employed to deal with the uncertainties existing in the geometrical, material and loading properties of brake system components. For simplicity, only the frictional coefficient, the elastic properties of
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materials and the thicknesses of wearing components are treated as the uncertain parameters and they are described as interval variables. The RSM method is used to build the models to approximate the functional relationship between the dominant unstable mode and system parameters. Based on the RSM, the interval analysis and the reliability analysis, a reliability-based optimization model for improving the stability of the vehicle disc brake system with interval parameters is constructed. Genetic Algorithm is used to get the optimal value of design parameter from the optimization model. The stability analysis and optimization of a disc brake system are carried out, and the results show that the proposed approach can be used to improve the stability of the vehicle disc brake system with uncertainties effectively. The remainder of this paper is organized as follows. In Section 2, the concept of non-probabilistic reliability is introduced. In Section 3, the stability analysis of vehicle disc brake systems considering uncertainties is discussed. In Section 4, the method of reliability-based optimization of the brake system with interval parameters is presented. In Section 5, a numerical example of the reliability-based optimization of a vehicle disc brake system is provided to verify the effectiveness of the proposed method. The main conclusions of the work are summarized in Section 6. 2. Non-probabilistic reliability 2.1. Interval parameter The interval analysis approach, based on the work of Moore [31] and Alefeld [32], expresses uncertain system parameters as interval variables with lower and upper bounds. For any uncertain system parameter x, if its lower and upper bounds can be obtained, then it can be written as the following form: x A ½xL ; xU L
(1)
U
where x is called as interval or interval variable, x and x are the lower and upper bounds of x, respectively. The midpoint and the radius of the interval x are defined as xC ¼
xL þ xU ; 2
xR ¼
xU xL 2
(2)
where xC is the midpoint of the interval, xR is the radius of the interval [33], respectively. 2.2. Non-probabilistic reliability In practical engineering problems, uncertainties always exist in material properties, geometrical dimensions and loading conditions. Those uncertainties are usually modeled by uncertain parameters. Traditionally, uncertain parameters are treated as the stochastic variables with certain probability distributions. Unfortunately, for many practical engineering problems, no sufficient information can be obtained to construct the precise probability distributions of uncertain parameters. Under this case, it is desirable to describe the uncertain parameters as the interval variables. The lower and upper bounds of the interval variables are comparatively easy to be obtained in engineering practices. The reliability analysis based on the model which only has interval variables is called as the non-probabilistic reliability analysis [34]. It is shown that the probabilistic reliability may be very sensitive to small inaccuracy in the probabilistic model. Consequently, the nonprobabilistic reliability is useful when sufficient information is not available for verifying a probabilistic model. Defining X as the m-dimensional vector of the interval variables representing uncertainties X ¼ fx1 ; x2 ; …; xm gT ;
xi A ½xLi ; xUi ;
i ¼ 1; 2; …; m
(3)
where the superscripts L and U represent the lower and upper bounds of an interval, respectively. m is the total number of the interval variables. Denoting the limit-state function as [35] M ¼ GðXÞ ¼ Gfx1 ; x2 ; …; xm g
(4)
where GðXÞ is the limit-state function. If GðXÞ 4 0, the system is in the safe state; if GðXÞ o 0, the system is in the failed state; and if GðXÞ ¼ 0, the system is in the critical state. Apparently, M is not a deterministic value but an interval variable. It can be expressed as M A ½M L ; MU L
(5)
U
where M and M are the lower and upper bound of the interval variable M, they can be determined by M L ¼ min GðXÞ ¼ min Gfx1 ; x2 ; …; xm g
(6)
MU ¼ max GðXÞ ¼ max Gfx1 ; x2 ; …; xm g
(7)
The non-probabilistic reliability index is defined as [34,36]
η¼
MC M
R
¼
ML þM U MU ML
(8)
where η is the non-probabilistic reliability index, MC is the midpoint of the interval M, and MR is the radius of the interval M.
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Fig. 1. A simplified model of disc brake system.
It can be known from Eqs. (4) and (8) that, if η 41, then G(X)40, and the system is in the safe state; if η o 1, then G(X) o0, and the system is in the failed state; if 1 o η o 1, then G(X) may be greater or less than zero, and the system may or may not be in the safe state. Therefore, in order to ensure that the system is completely reliable, η 41 must be ensured. The greater the η value is, the higher the reliability of the system is [34,36].
3. Stability analysis of vehicle disc brake systems considering uncertainties 3.1. Stability analysis of brake systems using FEM An automotive disc brake system generally consists of brake disc, stationary pads, carrier bracket, calliper and guide pins [37]. The disc is rigidly mounted on the axle hub and therefore rotates with the wheel. The pair of brake pads, which consist of friction materials and back plates, is pressed against the disc in order to generate a frictional torque to slow the rotation of the disc and the wheel. For the purpose of simulating the vibration characteristics of a brake system reasonably with an acceptable workload of computation, a simplified model of the vehicle disc brake system is used according to the works of Abubakar and Ouyang [14], Junior et al. [18] and Sarrouy et al. [24], which is shown in Fig. 1. Eigenvalue analysis is an available technique for evaluating the stability of brake systems, and the complex eigenvalues can be used as the measurement of the stability of brake systems. In recent years, the FEM has become an indispensable tool for modeling the disc brake systems and providing new insights into the problem of brake squeal. Based on the FEM model of a brake assembly system, the equation of motion of the brake system can be written as [18] € þ Cu _ þ ðK Kf Þu ¼ 0 Mu
(9)
where M, C and K are the mass, damping and stiffness matrices of the brake system, respectively. u is the generalized displacement vector and the dot denotes the derivative with respect to time. Kf is the friction stiffness matrix which is determined by the pad-rotor interface properties of the brake. A pseudo forcing function in the stiffness term is introduced in Eq. (9). The stiffness matrix ðK Kf Þ may be asymmetric caused by the introduction of friction, meaning that the system characteristic roots and eigenvectors are complex numbers under certain conditions. The complementary solution to the homogenous, second order, matrix differential Eq. (9) is in the form of u ¼ φest
(10)
Substituting Eq. (10) into Eq. (9), we can get the complex eigenvalue problem ðs2 M þ sC þK Kf Þφ ¼ 0
(11)
where φ is the eigenvector, s is the eigenvalue. Certain values of s which are called as complex eigenvalues may occur in the form of complex numbers, and in the form of complex modes in the FE analysis. The complex eigenvalue corresponding to the kth order complex mode can be expressed as sk ¼ σ k 7 jωk
(12)
where σk is the real part and ωk is the imaginary part of the complex eigenvalue. Whether the brake system is stable or not can be determined by the real parts of the complex eigenvalues. It is widely known that a brake system is not stable when the real part of one complex eigenvalue of the brake system is positive, so the modes that are unstable and likely to produce brake noise can be revealed by examining the real part of the brake system eigenvalues. An extra term, the damping ratio, is defined and calculated by using the following equation [17]:
ζ¼
σ π jωj
(13)
where ζ is the damping ratio of a mode. Apparently, if the damping ratio is negative (σ 40), the brake system is unstable, and vice versa. The main aim of this analysis is to increase the damping ratio value of the dominant unstable mode.
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3.2. Response surface model of the dominant unstable mode of the disc brake system with uncertainties Due to the complex working environment and operating conditions, the brake squeal is generated with great randomness, and it is difficult to be captured and reproduced artificially. This randomness is closely related to the uncertainties of system parameters. The brake squeal is the stochastic outcomes of the combined effect of many uncertainties. Therefore, the generation mechanism of the brake noise has not yet been comprehensively understood until now. The uncertainties in parameters of a disc brake system are mainly summarized and described as follows. (1) The friction coefficient is variable, but not constant, with respect to the sliding velocity between the brake pad and the disc [3]. (2) Wear is one of the most important factors that influent brake squeal [38,39], and the thicknesses of the friction material and the disc decrease continuously due to the wear in the course of brake working. (3) The material properties of brake components are very uncertain, by the influence of different operations or working environments. If the probability density functions of the uncertain parameters are defined unambiguously, the probabilistic method is the popular approach for the analysis of the structures with uncertain parameters [40]. Based on the probability density functions of the input parameters, the probability density functions of the output quantities can be achieved. Unfortunately, the statistical information to establish the probability density functions is not sufficient for the brake system. Under this case, it is advisable to employ the interval model to represent the uncertain parameters. In the FE analysis of a brake system, these parameters such as the friction coefficient, the material properties and the geometric dimensions of brake components may all influence the brake system modes. The behaviors of brake system modes can be improved by the optimization of some parameters. In the conventional parameter design for a brake using the finite element method, such as the works of Liu et al. [17] and Junior et al. [18], the optimization is usually carried out by varying a single parameter while keeping all the other parameters fixed at a specific set of conditions. Apparently, this method is time-consuming and does not consider the effects of the combination of parameters. RSM, which is a collection of mathematical and statistical techniques, is beneficial for the modeling and analysis of problems in which a response is influenced by several variables. The computation time and cost of the optimization of complex systems can be greatly reduced by RSM. Recently, RSM has been widely employed to optimize and understand the performance of complex systems [41,42]. By the application of RSM, the implicit functional relationships between the vibration modes and the system parameters of a brake system can be approximated with a limited number of finite element analysis experiments. Referring to the research of Nouby et al. [22], the complex eigenvalue corresponding to an unstable mode of a brake system can be fitted by a second-order model in the form of quadratic polynomial equation given below: n
n
n
i¼1
i¼1
1 ¼ ioj
n
n
n
i¼1
i¼1
1 ¼ ioj
σ ¼ a0 þ ∑ ai xi þ ∑ aii x2i þ ∑ aij xi xj ω ¼ b0 þ ∑ bi xi þ ∑ bii x2i þ ∑ bij xi xj
(14)
(15)
where σ and ω are the real and imaginary parts of the complex eigenvalue corresponding to the unstable mode of a brake system, respectively. a0, b0, ai, bi, aii, bii, aij and bij are the unknown coefficients that can be determined by the experimental design and the least square method. xi (i ¼ 1; 2; …; n) is the ith system parameter, including the uncertain parameters and the design parameters. If any values of the uncertain parameters in the interval are all considered in the experimental design space, then the stability state of the brake system corresponding to any values of the uncertain parameters in intervals can be described by Eqs. (14) and (15). In particular, if all the changes of the thicknesses of friction materials and disc resulted from the wear throughout the lifetime cycle are considered in the experimental design space for a brake system, then the stability state of the brake system corresponding to any wear state can be predicted by Eqs. (14) and (15). For establishing a response surface model, it is necessary to conduct the analysis of variance (ANOVA) to test the adequacy and significance of the model, so as to ensure the fitting accuracy of the model [42]. 4. Reliability-based optimization of the brake system with interval parameters 4.1. Reliability-based optimization model of the brake system with interval parameters In engineering practices, the traditional deterministic design optimization model has been successfully applied to reduce the cost and improve the quality of structures. However, the existence of uncertainties in either engineering simulations or manufacturing processes calls for a reliability-based design optimization model for robust and cost-effective designs. The solution from the reliability-based design optimization provides not only an improved design but also a higher level of confidence in the design. While the damping ratio of a vibration mode of a brake system is greater than the constant value ζ c ¼ 0:01, the brake system can be considered to be stable [43], then the limit-state function describing the stability of a brake system can be obtained by the following expression M ¼ GðXÞ ¼ ζ ðXÞ ζ c ¼ ζ ðXÞ þ0:01
(16)
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According to Eqs. (6)–(8) and (16), the reliability index of a brake system stability can be obtained as follows:
ηðXÞ ¼ ¼
maxðζ ðXÞ þ0:01Þ þ minðζ ðXÞ þ 0:01Þ maxðζ ðXÞ þ 0:01Þ minðζ ðXÞ þ 0:01Þ
maxðζ ðXÞ þ 0:01Þ þ maxð ζ ðXÞ 0:01Þ maxðζ ðXÞ þ 0:01Þ maxð ζ ðXÞ 0:01Þ
(17)
In order to ensure the stability of a brake system, it is required to ensure the reliability index η 41. The reliability-based optimization model of the stability of the vehicle disc brake system with interval parameters can then be expressed as min f ðXÞ s:t: ηðXÞ 4 1 X ¼ fXT1 ; XT2 gT X1 A ½XL1 ; XU1 ;
X2l rX2 r X2u
(18)
where f ðXÞ is the objective function, its mathematical expression and physical meaning can be determined based on the actual situation. X is the vector of variables of the brake system, X1 is the vector of non-design variables and X2 is the vector of design variables. XL1 and XU1 represent the lower and upper bounds of X1, respectively. X2l and X2u represent the lower and upper values of the variation range of X2, respectively. It can be easily seen from Eqs. (17) and (18) that the maximum and minimum of the limit-state function ζ ðXÞ þ0:01 need to be evaluated in this optimization model, at the same time, the minimum of the objective function f ðXÞ needs to be searched. So the optimization problem described by Eq. (18) is a typical two-layer nesting optimization problem. In the outer layer optimization, design parameters are used to build up the design variable space. In the inner layer optimization, uncertain parameters are used to build up the non-design variable space. The best optimal results can be found by the inner layer optimization and the outer layer optimization together.
4.2. The procedure for reliability-based optimization of brake systems The procedure for reliability-based optimization of the vehicle disc brake system with interval parameters can be summarized as (1) Define the system parameters, and divide the parameters into design variables and non-design variables. (2) Create the FE model of the brake system (shown in Eq. (11)), carry out experimental design, and perform CEA based on the FE model to obtain the unstable modes. (3) Create the response surface models of complex eigenvalues corresponding to unstable modes (shown in Eqs. (14) and (15)). (4) Build up the reliability-based optimization model of the brake system with interval parameters (shown in Eqs. (17) and (18)). (5) Solve the reliability-based optimization model, and get the optimal design results. The flow chart of the reliability-based optimization of the brake system with interval parameters is shown in Fig. 2.
5. Numerical example 5.1. Complex eigenvalue analysis of the brake system with interval parameters A simplified three-dimensional FE model of a disc brake assembly is developed in Altair HyperMesh software and illustrated in Fig. 3. This brake assembly consists of only a brake disc and a pair of brake pads which consist of friction materials and back plates. The disc is made of gray cast iron which is wear resistant and relatively inexpensive. The friction material which is made of an organic friction material is firmly mounted to the rigid brake plate which is made of steel. The FE meshes are generated by using the three-dimensional continuum elements, including 26,125 elements and 37,043 nodes, and the fine meshes are used in the contact regions. The friction contact interactions are defined between both sides of the disc and the contact plates of the pads. The CEA of the brake system is carried out by using ABQUS/Standard code. Fig. 4 presents the constraints and loadings of the brake assembly. The disc is completely fixed at the five counter-bolt holes and the ears of the back plates are constrained to allow only axial movements. The caliper–piston assembly is not defined in the simplified model of the disc brake system. Hence, the hydraulic pressure is directly applied to the back plates at the contact regions between the inner pad and the piston and between the outer pad and the caliper, and it is assumed that an equal magnitude of force acts on each back plate. For this brake, the friction materials are made of organic materials and have the anisotropic properties. Referring to the works of Junior et al. [18] and Cao et al. [37], here the friction materials are treated as isotropic materials but with varied Young's modulus. This assumption is reasonable to a certain degree, especially in an uncertainty analysis. Based on this
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Start Define system parameters
Creat FE model of brake system (shown in Eq.(13))
Uncertain parameters and design parameters space
Performe CEA base on FE model
Carry out experimental design
Create response surface models of complex eigenvalues corresponding to unstable modes (shown in Eqs. (16), (17) and (18))
Build up the reliability-based optimization model for the stability of the brake system with interval parameters (shown in Eqs. (20) and (21)).
Outer layer optimization Design variables space X2 Inner layer optimization Non-design variables space X1 Objective function value
No
Stopping criterion Yes Output optimal design results
Fig. 2. The flow chart of the reliability-based optimization of the brake system with interval parameters.
assumption and the unpredictability of wear, material properties, contact conditions between brake pads and disc, the uncertain variables and design variables are selected and described as follows: (1) The friction coefficient μ, disc thickness h1, friction material thickness h2, Young's modulus of disc E1, Young's modulus of back plate E2 and Young's modulus of friction material E3 are selected as uncertain variables and assumed as interval variables. The interval values of μ, E1, E2 and E3 are determined by the potential uncertainties, the interval values of h1 and h2 are determined according to the brake wear. The nominal and interval values of these variables are listed in Table 1. (2) Generally, in a real brake, the thickness of the whole brake pad cannot be allowed to change as any such a change would lead to the change of calliper and perhaps other brake components. In order to provide guidance for brake designers at the concept stage, here the back plate thickness h3 is taken as a design variable. Its initial value is 6.0 mm, and the design range is taken from 5.5 mm to 8.5 mm.
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Fig. 3. FE model of a disc brake system.
Fig. 4. Constraints and loading of the simplified brake system.
In a brake system, the units of the independent variables differ from one another. Even if some of the parameters have the same unit, not all of these parameters will be tested over the same range. Since parameters have different units and/or ranges in the experimental domain, the regression analysis should not be performed. Instead, one must firstly normalize the parameters before performing a regression analysis. Each of the coded variables is forced to range from 1 to 1, so all the parameters will affect the response evenly and the units of the parameters are irrelevant. Commonly used equations for coding are shown below:
μ 0:35
h1 19:0 h2 8:0 ; x2 ¼ ; x3 ¼ ; 0:05 1:0 3:0 h3 7:0 E1 125 E2 210 x4 ¼ ; x5 ¼ ; x6 ¼ ; 1:5 6:25 10:5 E3 5:94 x7 ¼ 0:594
x1 ¼
(19)
where xi (i ¼ 1; 2; …; 7) are the normalized parameters. A total number of 70 group samples in the space of normalized parameters are obtained by Latin Hypercube Sampling [44], which is a widely-used sampling method for constructing the response surface model. CEA approach is performed based upon the FE model of the disc brake system for each group samples from 0 to 16 kHz, which is the frequency range where the brake squeal commonly occurs. The analysis results show that, the unstable modes appear mainly at the frequency of around 1.9 kHz in this research, and the damping ratios of these unstable modes are almost higher than that of the other unstable modes. Based on the earlier studies of Abubakar and Ouyang [14] and Nouby et al. [22], here we take the unstable modes at the certain frequency for studying. Therefore, the unstable mode at 1.9 kHz for each group sample is regarded as the dominant unstable mode whose damping ratio is negative and is needed to be reduced to improve the system stability. As an example, the complex eigenvalues of this brake system corresponding to some group samples are shown as Fig. 5. These samples are selected randomly. It can be seen clearly that the unstable modes appear mostly at the frequency of around 1.9 kHz. 5.2. Response surface models of the complex eigenvalue corresponding to the dominant unstable mode Based on the CEA results of the disc brake system for each group samples and the method of the least squares, the quadratic polynomial response surface approximation models of the real part and the imaginary part of the complex eigenvalue corresponding to the dominant unstable mode are established as
σ d ðXÞ ¼ 62:75 þ13:25x1 þ 5:06x2 þ 30:91x3 27:40x4 þ9:13x5 3:77x6 þ 1:78x7
0:94x1 x2 þ0:54x1 x3 2:68x1 x4 5:69x1 x5 þ1:48x1 x6 4:29x1 x7 2:19x2 x3
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Table 1 The nominal values, lower and upper bounds of interval variables. Interval variables
Nominal values
Lower bounds
Upper bounds
Friction coefficient m Disc thickness h1 (mm) Friction material thickness h2 (mm) Young's modulus of disc E1 (GPa) Young's modulus of back plate E2 (GPa) Young's modulus of friction material E3 (GPa)
0.35 20.0 11.0 125 210 5.94
0.30 18.0 5.00 118.75 199.5 5.346
0.40 20.0 11.0 131.25 220.5 6.534
Fig. 5. Complex eigenvalues of the brake corresponding to some group samples.
0:63x2 x4 þ1:77x2 x5 þ 5:88x2 x6 þ 7:32x2 x7 þ2:06x3 x4 þ 3:46x3 x5 1:55x3 x6 þ 1:26x3 x7 þ2:04x4 x5 þ 1:97x4 x6 þ 4:55x4 x7 þ0:01x5 x6 4:06x5 x7 þ 4:65x6 x7 3:52x21 þ 2:17x22 0:49x23 26:67x24 3:92x25 2:60x26 þ 0:06x27
(20)
ωd ðXÞ ¼ 1911:29 0:19x1 þ 57:43x2 11:30x3 þ6:72x4 þ 45:16x5 þ1:95x6 þ 1:20x7
1:16x1 x2 þ 0:13x1 x3 1:84x1 x4 þ 0:26x1 x5 0:09x1 x6 0:05x1 x7 0:34x2 x3 2:55x2 x4 þ 1:49x2 x5 0:33x2 x6 þ 0:20x2 x7 3:90x3 x4 þ1:02x3 x5 0:73x3 x6 þ 0:27x3 x7 2:00x4 x5 þ2:15x4 x6 0:49x4 x7 þ 0:51x5 x6 0:19x5 x7 þ 0:69x6 x7 0:48x21 þ 0:30x22 þ 0:85x23 þ 4:46x24 0:64x25 þ 0:10x26 þ0:05x27
(21)
where X ¼ ½x1 ; x2 ; x3 ; x4 ; x5 ; x6 ; x7 T , σ d ðXÞ and ωd ðXÞ are the real and imaginary parts of the complex eigenvalue corresponding to the dominant unstable mode, respectively. Since any values of the uncertain parameters in intervals are all considered in the experimental design space, then the state of the brake system corresponding to any values of the uncertain parameters in intervals can be described by Eqs. (20) and (21). For example, the thickness parameters of the friction materials and disc, resulted from the wear, are all considered in the experimental design space, thus the stability state of the brake system corresponding to any wear state can be described by Eqs. (20) and (21). ANOVA is required to test the significance and adequacy of the response surface model. The ANOVA results for the quadratic response surface model of σ d and ωd are shown in Tables 2 and 3, respectively. The coefficient of determination (R2) is defined as the ratio of the explained variable to the total variation and a measurement of the degree of fitness. When R2 is close to 1, the model fits the actual experimental data better. The smaller the value of R2, the less relevant the model fits the actual data. The suggested R2 value should be at least 0.80 for a good fit of a model [45,46]. It can be seen from Tables 2 and 3 that the R2 value of the regression models of σ d and ωd are all higher than 0.80, which indicates that the regression models of σ d and ωd can explain the observed response well. Thus, the regression models of σ d and ωd are considered to be quite satisfactory. While the p value is lower than 0.01, it indicates that the model is considered to be statistical significant [44,47]. It can be seen from Table 2 that the second-order polynomial model of σ d is highly significant as the p value of the model is less than 0.0001. The F value for the model is 60.37. There is only a 0.01 percent chance that the “model F value” could occur because of the noise. As for the second-order polynomial model of ωd , the same conclusions can be drawn from Table 3. Hence, the response surface models developed in this section for σ d and ωd are successful in describing the correlation between the system parameters and the dominant unstable mode.
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Table 2 ANOVA results for the quadratic response surface model of σ d . Source
Sum of squares
Degree of freedom
Mean squares
F value
p value
Regression Residuals Total
52,234.01 840.51 53,074.52 R¼ 0.9921
35 34 69
1492.40 24.72
60.37
o 0.0001
R2 ¼ 0.9842
Table 3 ANOVA results for the quadratic response surface model of ωd . Source
Sum of squares
Degree of freedom
Mean squares
F value
p value
Regression Residuals Total
137,100 50 137,150 R¼ 0.9998
35 34 69
3916.68 1.47
2662.95
o 0.0001
R2 ¼0.9996
5.3. Reliability-based optimization of the brake system According to the actual situation, and in order to provide guidance for brake designers at the concept stage, the thickness parameter of back plate is selected as design variable, and the rest of the system parameters, namely, the friction coefficient, the thickness of disc, the thickness of friction material, Young's modulus of disc, Young's modulus of back plate and Young's modulus of friction material are selected as non-design variables. In order to guarantee the stiffness and strength of the back plate, the thickness of back plate should not be less than the initial value, thus the constraint for the thickness of back plate can be taken as 6:0 r h3 r8:5
(22)
By combining Eq. (20), it can be written with normalized variable as 0:6667 rx4 r1
(23)
For the purpose of lightweight design, the mass of back plate should be minimized, thus the thickness of back plate can be taken as the optimization objective. From Eq. (18), the reliability-based optimization model of this brake system stability with uncertain parameters can be expressed as follows when the dominant unstable mode is considered min h3 s:t: ηðμ; h1 ; h2 ; h3 ; E1 ; E2 ; E3 Þ 4 1 0:3 r μ r 0:4 18 rh1 r 20 5 rh2 r 11 6:0 r h3 r 8:5 118:75 r E1 r 131:25 199:5 r E2 r 220:5 5:346 r E3 r 6:534
(24)
By combining Eq. (19), it can be expressed with normalized variable as min x4 s:t: ηðXÞ 4 1; X ¼ fXT1 ; XT2 gT X1 ¼ fx1 ; x2 ; x3 ; x5 ; x6 ; x7 gT ; X2 ¼ fx4 g;
xi A ½ 1; 1;
i ¼ 1; 2; 3; 5; 6; 7
0:6667 r x4 r 1
(25)
where
ηðXÞ ¼
maxðð σ d ðXÞ=π jωd ðXÞjÞ þ 0:01Þ þ maxððσ d ðXÞ=π jωd ðXÞjÞ 0:01Þ maxðð σ d ðXÞ=π jωd ðXÞjÞ þ 0:01Þ maxððσ d ðXÞ=π jωd ðXÞjÞ 0:01Þ
(26)
The Genetic Algorithm [48] is selected to optimize the model described by Eq. (25). Here, the interval variables x1 ; x2 ; x3 ; x5 ; x6 ; x7 are selected as non-design variables and build up the non-design space X1. Whereas x4 is selected as design variable and makes up the design space X2. In the outer layer optimization, the optimal value of x4 is searched in the design space. In the inner layer optimization, the upper and lower bounds of the limit-state function described by Eq. (16)
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Fig. 6. System complex eigenvalues for a group parameter values after optimization.
Fig. 7. The reliability index of the brake stability for different values of h3.
and the reliability index of the brake system stability described by Eq. (26) are searched at different values of x4, in the nondesign space. The best optimal results are found by the inner layer optimization and the outer layer optimization together. The final optimization results are x4 ¼ 0:8667 and ηðXÞ ¼ 1:0184. The results mean that when h3 ¼8.3 mm, this brake system can always stay in a stable state, even if considering the influence of uncertain factors. Thus, the robustness of the brake system stability is improved greatly. The prediction accuracies of the fitted models expressed by Eqs. (20) and (21) are checked by the optimization results. By substituting the group normalized value (x1 ¼ 0, x2 ¼ 0, x3 ¼ 0, x4 ¼ 0:8667, x5 ¼ 0, x6 ¼ 0, x7 ¼ 0), in which x4 takes the optimized value and the other normalized variables take the values of zero, into Eqs. (20) and (21). The calculation results are σ d;RSM ¼ 18:97 and ωd;RSM ¼ 1920:5. Complex eigenvalue analysis is performed based upon the FE model of this brake for this group value, the corresponding results are σ d;FEM ¼ 19:03 and ωd;FEM ¼ 1921:2, which show that the prediction accuracy of the fitted models is very well. Fig. 6 shows the system complex eigenvalues for the above CEA results, it can be seen that all the asterisks representing system complex eigenvalues are far away from the line corresponding to the damping ratio ¼ 0.01, that is, the system is in a quite stable state under this group of parameters. To a certain extent, the purpose of suppressing brake squeal is achieved. If the reliability index η 41, the brake system will be in an absolutely stable state even though the influence of uncertain factors is considered. In order to show the influence of design variable h3 on this commercial brake system stability more clearly, the reliability index of the brake system stability for different values of h3 is given in Fig. 7. It can be seen from Fig. 7 that the stability and robustness of this commercial brake system are increased as the thickness of back plate h3 increases. The reason is that a larger thickness will induce larger supporting stiffness of the back plate. It can be known from the works of Liles [10] and Nouby et al. [22] that the stiffer back plates can reduce the propensity of brake squeal. This is quite coincided with the results of this research. Therefore, in engineering practices, it may improve the stability and robustness of a real brake and reduce brake squeal through increasing the stiffness of back plates, even though the influence of uncertain conditions are considered. In order to achieve the purpose of improving the supporting stiffness of back plate, one can make the back plate with larger thickness in the conceptual design stage, or change the material of back plate into the material with larger Young's modulus, or consider the material properties and geometrical dimensions together during the design process. 6. Conclusions In this paper, a reliability-based optimization method for the brake squeal reduction of the vehicle disc brake system with interval parameters is presented. In the proposed method, the dominant unstable mode is extracted by performing CEA
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based on the linear FE model of the brake system, and the negative damping ratio corresponding to the dominant unstable mode is chosen as an indicator of instability. RSM is used to build up the models to express the functional relationship between the dominant unstable mode and the system parameters. ANOVA is applied to test the adequacy and significance of the response surface models to ensure the fitting accuracy. The efficiency of the stability analysis of the brake systems is greatly improved by the application of RSM. Uncertain parameters are employed to deal with the uncertainties existing in geometrical dimensions, material and loading properties of the brake system. The frictional coefficient, the parameters of material properties, and the thicknesses of wearing components are treated as the uncertain parameters. Usually, there is no sufficient information to construct the precise probability distributions of uncertain parameters. Thus, these uncertain parameters are described as interval variables in the proposed method. A reliability-based stability optimization model for the vehicle disc brake system considering uncertainties is constructed based on the RSM, the interval analysis and the reliability analysis. The stability analysis and optimization of a disc brake system with interval parameters are carried out. The results show that the proposed approach can improve the stability of a brake system effectively under uncertain conditions.
Acknowledgments The paper is supported by Hunan Provincial Innovation Foundation for Postgraduate (Grant no. CX2013B143), Independent Research Project of State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body in Hunan University (Grant no. 51375002). The authors would also like to thank reviewers for their valuable suggestions. References [1] M.R. Nishiwaki, Review of study on brake squeal, Japan Society of Automobile Engineering Review 11 (4) (1990) 48–54. [2] S. Yang, R.F. Gibson, Brake vibration and noise: reviews, comments, and proposals, International Journal of Materials and Product Technology 12 (4–6) (1997) 496–513. [3] N.M. Kinkaid, O.M. O'Reilly, P. Papadopoulos, Automotive disc brake squeal: a review, Journal of Sound and Vibration 267 (1) (2003) 105–166. [4] M. Nishiwaki, Generalized theory of brake noise, Proceedings of the Institution of Mechanical Engineers 207 (3) (1993) 195–202. [5] A. Papinniemi, J.C.S. Lai, J. Zhao, L. Loader, Brake squeal: a literature review, Applied Acoustics 63 (4) (2002) 391–400. [6] S. Oberst, J.C.S. Lai, Statistical analysis of brake squeal noise, Journal of Sound and Vibration 330 (12) (2011) 2978–2994. [7] H. Ouyang, W. Nack, Y. Yuan, F. Chen, Numerical analysis of automotive disc brake squeal: a review, International Journal of Vehicle Noise and Vibration 1 (3–4) (2005) 207–231. [8] G. Fritz, J.J. Sinou, J.M. Duffal, L. Jézéquel, Effects of damping on brake squeal coalescence patterns application on a finite element model, Mechanics Research Communications 34 (2) (2007) 181–190. [9] W.V. Nack, Brake squeal analysis by finite elements, International Journal of Vehicle Design 23 (3–4) (2000) 263–275. [10] G.D. Liles, Analysis of disc brake squeal using finite element methods, SAE Paper No. 891150, 1989. [11] M.L. Chargin, L.W. Dunne, D.N. Herting, Nonlinear dynamics of brake squeal, Finite Elements in Analysis and Design 28 (1) (1997) 69–82. [12] S.W. Kung, K.B. Dunlap, R.S. Ballinger, Complex eigenvalue analysis for reducing low frequency brake squeal, SAE Paper No. 2000-01-4444, 2000. [13] H. Ouyang, W. Li, J.E. Mottershead, A moving-load model for disc-brake stability analysis, Journal of Vibration and Acoustics 125 (1) (2003) 53–58. [14] A.R. AbuBakar, H. Ouyang, Complex eigenvalue analysis and dynamic transient analysis in predicting disc brake squeal, International Journal of Vehicle Noise and Vibration 2 (2) (2006) 143–155. [15] D. Guan, X. Su, F. Zhang, Sensitivity analysis of brake squeal tendency to substructures' modal parameters, Journal of Sound and Vibration 291 (1–2) (2006) 72–80. [16] G. Fritz, J.J. Sinou, J.M. Duffal, L. Jézéquel, Investigation of the relationship between damping and mode-coupling patterns in case of brake squeal, Journal of Applied Mechanics 307 (3–5) (2007) 591–609. [17] P. Liu, H. Zheng, C. Cai, Y.Y. Wang, C. Lu, K.H. Ang, G.R. Liu, Analysis of disc brake squeal using the complex eigenvalue method, Applied Acoustics 68 (6) (2007) 603–615. [18] M.T. Junior, S.N.Y. Gerges, R. Jordan, Analysis of brake squeal noise using the finite element method: a parametric study, Applied Acoustics 69 (2) (2008) 147–162. [19] Y. Dai, T.C. Lim, Suppression of brake squeal noise applying finite element brake and pad model enhanced by spectral-based assurance criteria, Applied Acoustics 69 (3) (2008) 196–214. [20] Y. Nishizawa, S. Wakamatsu, H. Yanagida, Y. Takahara, Study of dynamic pad stiffness influencing brake squeal, SAE Paper No. 2007-01-3956, 2007. [21] H. Nonaka, Y. Nishizawa, Y. Kurita, Y. Oura, Considering the dynamic pad stiffness in FEM analysis of disk brake squeal, SAE Paper No. 2010-01-1716, 2010. [22] M. Nouby, D. Mathivanan, K. Srinivasan, A combined approach of complex eigenvalue analysis and design of experiments (DOE) to study disc brake squeal, International Journal of Engineering, Science and Technology 1 (1) (2009) 254–271. [23] S. Lakkam, S. Koetniyom, Optimization of constrained layer damping for strain energy minimization of vibrating pads, Songklanakarin Journal of Science and Technology 34 (2) (2012) 179–187. [24] E. Sarrouy, O. Dessombz, J.J. Sinou, Piecewise polynomial chaos expansion with an application to brake squeal of a linear brake system, Journal of Sound and Vibration 332 (3–4) (2013) 577–594. [25] R. Moore, W. Lodwick, Interval analysis and fuzzy set theory, Fuzzy Sets and Systems 135 (1) (2003) 5–9. [26] R. Rackwitz, Reliability analysis—a review and some perspectives, Structural Safety 23 (4) (2001) 365–395. [27] F. Massa, A. Leroux, B. Lallemand, T. Tison, S. Mary, F. Buffe, Fuzzy vibration analysis and optimization of engineering structures: application to Demeter satellite, IUTAM Bookseries 27 (1) (2011) 63–76. [28] F. Massa, B. Lallemand, T. Tison, Fuzzy multiobjective optimization of mechanical structures, Computer Methods in Applied Mechanics and Engineering 198 (5) (2009) 631–643. [29] O. Kelesoglu, Fuzzy multiobjective optimization of truss-structures using genetic algorithm, Advances in Engineering Software 38 (2007) 717–721. [30] J. Ramík, Optimal solutions in optimization problem with objective function depending on fuzzy parameters, Fuzzy Sets and Systems 158 (2007) 1873–1881. [31] R. Moore, Interval Analysis, Prentice Hall, Englewood Cliff, 1966. [32] G. Alefeld, J. Herzberber, Introductions to Interval Computations, Academic Press, New York, 1983.
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[33] D. Moens, D. Vandepitte, A survey of non-probabilistic uncertainty treatment in finite element analysis, Computer Methods in Applied Mechanics and Engineering 194 (12–16) (2005) 1527–1555. [34] Y.B. Haim, A non-probabilistic concept of reliability, Structural Safety 14 (4) (1994) 227–245. [35] Z.P. Qiu, J. Wang, The interval estimation of reliability for probabilistic and non-probabilistic hybrid structural system, Engineering Failure Analysis 17 (5) (2010) 1142–1154. [36] J. Tao, C.J. Jun, X.Y. Lan, A semi-analytic method for calculating non-probabilistic reliability index based on interval models, Applied Mathematical Modelling 31 (7) (2007) 1362–1370. [37] Q. Cao, H. Ouyang, M.I. Friswell, J.E. Mottershead, Linear eigenvalue analysis of the disc-brake squeal problem, International Journal for Numerical Methods in Engineering 61 (9) (2004) 1546–1563. [38] A.R.A. Bakar, H. Ouyang, S. James, L. Li, Finite element analysis of wear and its effect on squeal generation, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 222 (7) (2008) 1153–1165. [39] A.R. AbuBakar, H. Ouyang, Wear prediction of friction material and brake squeal using the finite element method, Wear 264 (11–12) (2008) 1069–1076. [40] G. Stefanou, The stochastic finite element method: past, present and future, Computer Methods in Applied Mechanics and Engineering 198 (9–12) (2009) 1031–1051. [41] D.C. Montgomery, Design and Analysis of Experiments, Wiley Publications, New York, 1991. [42] R.H. Mayers, D.C. Montgomery, Response Surface Methodology: Process and Product Optimization Using Designed Experiments, John Wiley & Sons, New Jersey, 2002. [43] J. Hou, X.X. Guo, G.F. Tan, Complex mode analysis on disc brake squeal and design improvement, SAE Paper No. 2009-01-2101, 2009. [44] M. Papila, Accuracy of Response Surface Approximations for Weight Equations Based on Structural Optimization, PhD thesis, University of Florida, 2001. [45] J. Fu, Y. Zhao, Q. Wu, Optimising photoelectrocatalytic oxidation of fulvic acid using response surface methodology, Journal of Hazardous Materials 144 (1–2) (2007) 499–505. [46] A.M. Joglekar, A.T. May, Product excellence through design of experiments, Cereal Foods World 32 (1987) 857–868. [47] H.K. Kim, J.G. Kim, J.D. Cho, J.W. Hong, Optimization and characterization of UV-curable adhesives for optical communications by response surface methodology, Polymer Testing 22 (8) (2003) 899–906. [48] D.E. Goldberg, Genetic Algorithms in Search, Optimization & Machine Learning, Addison–Wesley, New York, 1989.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Brake squeal reduction of vehicle disc brake system with interval parameters by uncertain optimization Hui Lü, Dejie Yu n State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan 410082, People's Republic of China
a r t i c l e i n f o
abstract
Article history: Received 11 December 2013 Received in revised form 18 July 2014 Accepted 24 August 2014 Handling Editor: H. Ouyang Available online 16 September 2014
An uncertain optimization method for brake squeal reduction of vehicle disc brake system with interval parameters is presented in this paper. In the proposed method, the parameters of frictional coefficient, material properties and the thicknesses of wearing components are treated as uncertain parameters, which are described as interval variables. Attention is focused on the stability analysis of a brake system in squeal, and the stability of brake system is investigated via the complex eigenvalue analysis (CEA) method. The dominant unstable mode is extracted by performing CEA based on a linear finite element (FE) model, and the negative damping ratio corresponding to the dominant unstable mode is selected as the indicator of instability. The response surface method (RSM) is applied to approximate the implicit relationship between the unstable mode and the system parameters. A reliability-based optimization model for improving the stability of the vehicle disc brake system with interval parameters is constructed based on RSM, interval analysis and reliability analysis. The Genetic Algorithm is used to get the optimal values of design parameters from the optimization model. The stability analysis and optimization of a disc brake system are carried out, and the results show that brake squeal propensity can be reduced by using stiffer back plates. The proposed approach can be used to improve the stability of the vehicle disc brake system with uncertain parameters effectively. & 2014 Elsevier Ltd. All rights reserved.
1. Introduction Due to the consistent customer complaints and high warranty costs, brake squeal noise has become one of the important concerns associated with the automotive brake systems [1–3]. If the brake system comes into an unstable state in the working process, the strong vibration and a harsh noise may be caused. The brake squeal noise, especially in the frequency range between 1 and 16 kHz, is most annoying to passengers' hearing [4,5]. Several classic theories have been formulated to explain the mechanism of the disc brake squeal phenomenon. However, there is neither a comprehensive understanding of the problem nor a generalized theory of the brake squeal mechanism yet [5,6]. For the last several decades, researches on brake vibration and noise have been conducted by using the theoretical, experimental, and numerical approaches [3,7]. The theoretical approaches provide a good insight into the mechanism of squeal, but the complicated brake system has to be considerably simplified and the accurate analysis results cannot be n
Corresponding author. Tel.: þ86 73188821915; fax: þ 86 73188823946. E-mail address: [email protected] (D. Yu).
http://dx.doi.org/10.1016/j.jsv.2014.08.027 0022-460X/& 2014 Elsevier Ltd. All rights reserved.
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provided. The experimental approaches can effectively investigate the effects of different parameters and operating conditions on the squeal of a brake system, however, the experiments are mostly expensive and time-consuming. Instead of a simple schematic model, the numerical methods can simulate different structures, material compositions and operating conditions of a disc brake or of different brakes or of even different vehicles, when used rightly. CEA method is a widely used approach for brake squeal investigation. Since a good correlation between the CEA numerical analysis results and the available experimental data is already verified experimentally, the CEA approach has already become an effective approach to the brake squeal investigation [8,9]. Many numerical approaches have been presented to explore the squeal phenomenon with the CEA approach. In the 1980s, Liles [10] presented a method for determining the geometric stability characteristics of a disc brake assembly by the use of finite element method (FEM). In this research, a CEA method was performed to determine which modes were unstable and therefore likely to produce squeal. In the 1990s, Chargin et al. [11] developed a unique nonlinear method accounted for both superelement modes and surface friction data for the analysis of brake squeal, both transient responses and complex eigenvalues were provided for the analysis of brake systems. In the 21st century, a front disc brake system was used as an example for the investigation of the low frequency squeal by Kung et al. [12]. Many different modifications to the brake system were proposed. Ouyang et al. [13] developed a moving-load model for the disc-brake stability analysis, and solved the dynamic instability as a nonlinear eigenvalue problem. Compared with the non-moving-load model, a more realistic disc-brake model was constructed by their method; more unstable frequencies were predicted and correlated very well with the experimental squeal frequencies. As the complex eigenvalue analysis and the dynamic transient analysis were typically two different methodologies that could be used to predict squeal in a disc brake [11], Abubakar and Ouyang [14] explored a proper way of conducting both types of analyses and investigating the correlation between them for a large degree-of-freedom disc brake model. Guan et al. [15] explored the sensitivity analysis methods to determine the dominant modal parameters of substructures of a brake system for the brake squeal suppression analysis, the related formulas of sensitivities of the positive real part of the squeal mode to substructures' modal parameters were derived. Fritz et al. [8,16] computed the brake system eigenvalues by using a technique based on FEM, and the effects of damping on the coalescence patterns of system eigenvalues were investigated. Liu et al. [17] and Junior et al. [18] investigated the effects of system parameters on the disc brake squeal, and the insulation and damping materials were applied to suppress the brake squeal. Dai and Lim [19] applied an enhanced dynamic FE model with friction coupling to optimize the design of the disc brake pad structure for squeal noise reduction and the analysis showed that the eigenvalues possessing positive real parts tended to produce unstable modes with the propensity towards the generation of squeal noise. In order to improve the FE model accuracy, Nishizawa et al. [20] and Nonaka et al. [21] conducted researches aiming to incorporate the dynamic stiffness of the pads into FE modal analysis, and the change in the squeal frequency resulting from the change in the thicknesses of the pads was reproduced. In recent years, Nouby et al. [22] proposed a mathematical approach to investigating the influencing factors of the brake pad on the disc brake squeal by integrating finite element simulations with statistical regression techniques. Their predicted results showed that the brake squeal propensity could be reduced by increasing Young's modulus of back plate and modifying the shape of friction material by adding chamfers and slots. Lakkam and Koetniyom [23] proposed an optimization method for brake squeal by using FEM, assumed-coupling mode method and experiment operations. According to this research, the position/geometry of the constrained layer damping patch could be optimized by minimizing the strain energy of vibrating pads with constrained layer. Sarrouy et al. [24] proposed numerical approaches to dealing with the brake stability problem with uncertainties. The work of Sarrouy et al. was based on polynomial chaos expansions and took place in the context of uncertain systems. Through their methods, the stochastic eigenvalue problems of a disc brake system could be processed efficiently and accurately compared with Monte Carlo simulations. Although so many researches on brake squeal have been conducted through CEA numerical approaches, however, only a few investigate the brake squeal problem with uncertainties. In engineering practices, many different factors appear to affect brake squeal, such as geometrical dimensions, material properties and loading conditions. Variabilities always exist in these factors in reality [6,24]. When taking those variabilities into account, parameter uncertainties have to be introduced into the analysis model of brake systems for obtaining more reliable results. To deal with the uncertain problem with limited information, the interval model, in which the fluctuations of uncertain parameters are assumed to fall into a hyperrectangle, has been developed. It just needs to obtain the upper and lower bounds of an uncertain parameter, if the uncertain parameter is treated as an interval variable. The lower and upper bounds of interval variables can be easily obtained in engineering practice. Therefore, the analysis methods based on the interval analysis have been well developed and widely applied [25]. In addition, carrying out reliability analysis while considering the uncertainties can ensure the system with uncertain parameters is always in a reliable state [26]. In order to improve the performance of the system with uncertainties, the optimization under uncertainty should be carried out during the design process. For the uncertain problems with limited information, several theories for the optimization with non-probabilistic parameters have been proposed for the designs of engineering structures in recent years [27–30]. Thus it is necessary to conduct uncertain optimization for reducing the brake squeal of a real brake system. In this paper, attention is focused on the stability analysis and the optimization of the vehicle disc brake system with uncertainties. The stability of a disc brake system is investigated via the CEA method. Based on the FE model, the dominant unstable mode is extracted. And the damping ratio corresponding to the dominant unstable mode is chosen as the indicator of instability. Uncertain parameters are employed to deal with the uncertainties existing in the geometrical, material and loading properties of brake system components. For simplicity, only the frictional coefficient, the elastic properties of
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materials and the thicknesses of wearing components are treated as the uncertain parameters and they are described as interval variables. The RSM method is used to build the models to approximate the functional relationship between the dominant unstable mode and system parameters. Based on the RSM, the interval analysis and the reliability analysis, a reliability-based optimization model for improving the stability of the vehicle disc brake system with interval parameters is constructed. Genetic Algorithm is used to get the optimal value of design parameter from the optimization model. The stability analysis and optimization of a disc brake system are carried out, and the results show that the proposed approach can be used to improve the stability of the vehicle disc brake system with uncertainties effectively. The remainder of this paper is organized as follows. In Section 2, the concept of non-probabilistic reliability is introduced. In Section 3, the stability analysis of vehicle disc brake systems considering uncertainties is discussed. In Section 4, the method of reliability-based optimization of the brake system with interval parameters is presented. In Section 5, a numerical example of the reliability-based optimization of a vehicle disc brake system is provided to verify the effectiveness of the proposed method. The main conclusions of the work are summarized in Section 6. 2. Non-probabilistic reliability 2.1. Interval parameter The interval analysis approach, based on the work of Moore [31] and Alefeld [32], expresses uncertain system parameters as interval variables with lower and upper bounds. For any uncertain system parameter x, if its lower and upper bounds can be obtained, then it can be written as the following form: x A ½xL ; xU L
(1)
U
where x is called as interval or interval variable, x and x are the lower and upper bounds of x, respectively. The midpoint and the radius of the interval x are defined as xC ¼
xL þ xU ; 2
xR ¼
xU xL 2
(2)
where xC is the midpoint of the interval, xR is the radius of the interval [33], respectively. 2.2. Non-probabilistic reliability In practical engineering problems, uncertainties always exist in material properties, geometrical dimensions and loading conditions. Those uncertainties are usually modeled by uncertain parameters. Traditionally, uncertain parameters are treated as the stochastic variables with certain probability distributions. Unfortunately, for many practical engineering problems, no sufficient information can be obtained to construct the precise probability distributions of uncertain parameters. Under this case, it is desirable to describe the uncertain parameters as the interval variables. The lower and upper bounds of the interval variables are comparatively easy to be obtained in engineering practices. The reliability analysis based on the model which only has interval variables is called as the non-probabilistic reliability analysis [34]. It is shown that the probabilistic reliability may be very sensitive to small inaccuracy in the probabilistic model. Consequently, the nonprobabilistic reliability is useful when sufficient information is not available for verifying a probabilistic model. Defining X as the m-dimensional vector of the interval variables representing uncertainties X ¼ fx1 ; x2 ; …; xm gT ;
xi A ½xLi ; xUi ;
i ¼ 1; 2; …; m
(3)
where the superscripts L and U represent the lower and upper bounds of an interval, respectively. m is the total number of the interval variables. Denoting the limit-state function as [35] M ¼ GðXÞ ¼ Gfx1 ; x2 ; …; xm g
(4)
where GðXÞ is the limit-state function. If GðXÞ 4 0, the system is in the safe state; if GðXÞ o 0, the system is in the failed state; and if GðXÞ ¼ 0, the system is in the critical state. Apparently, M is not a deterministic value but an interval variable. It can be expressed as M A ½M L ; MU L
(5)
U
where M and M are the lower and upper bound of the interval variable M, they can be determined by M L ¼ min GðXÞ ¼ min Gfx1 ; x2 ; …; xm g
(6)
MU ¼ max GðXÞ ¼ max Gfx1 ; x2 ; …; xm g
(7)
The non-probabilistic reliability index is defined as [34,36]
η¼
MC M
R
¼
ML þM U MU ML
(8)
where η is the non-probabilistic reliability index, MC is the midpoint of the interval M, and MR is the radius of the interval M.
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Fig. 1. A simplified model of disc brake system.
It can be known from Eqs. (4) and (8) that, if η 41, then G(X)40, and the system is in the safe state; if η o 1, then G(X) o0, and the system is in the failed state; if 1 o η o 1, then G(X) may be greater or less than zero, and the system may or may not be in the safe state. Therefore, in order to ensure that the system is completely reliable, η 41 must be ensured. The greater the η value is, the higher the reliability of the system is [34,36].
3. Stability analysis of vehicle disc brake systems considering uncertainties 3.1. Stability analysis of brake systems using FEM An automotive disc brake system generally consists of brake disc, stationary pads, carrier bracket, calliper and guide pins [37]. The disc is rigidly mounted on the axle hub and therefore rotates with the wheel. The pair of brake pads, which consist of friction materials and back plates, is pressed against the disc in order to generate a frictional torque to slow the rotation of the disc and the wheel. For the purpose of simulating the vibration characteristics of a brake system reasonably with an acceptable workload of computation, a simplified model of the vehicle disc brake system is used according to the works of Abubakar and Ouyang [14], Junior et al. [18] and Sarrouy et al. [24], which is shown in Fig. 1. Eigenvalue analysis is an available technique for evaluating the stability of brake systems, and the complex eigenvalues can be used as the measurement of the stability of brake systems. In recent years, the FEM has become an indispensable tool for modeling the disc brake systems and providing new insights into the problem of brake squeal. Based on the FEM model of a brake assembly system, the equation of motion of the brake system can be written as [18] € þ Cu _ þ ðK Kf Þu ¼ 0 Mu
(9)
where M, C and K are the mass, damping and stiffness matrices of the brake system, respectively. u is the generalized displacement vector and the dot denotes the derivative with respect to time. Kf is the friction stiffness matrix which is determined by the pad-rotor interface properties of the brake. A pseudo forcing function in the stiffness term is introduced in Eq. (9). The stiffness matrix ðK Kf Þ may be asymmetric caused by the introduction of friction, meaning that the system characteristic roots and eigenvectors are complex numbers under certain conditions. The complementary solution to the homogenous, second order, matrix differential Eq. (9) is in the form of u ¼ φest
(10)
Substituting Eq. (10) into Eq. (9), we can get the complex eigenvalue problem ðs2 M þ sC þK Kf Þφ ¼ 0
(11)
where φ is the eigenvector, s is the eigenvalue. Certain values of s which are called as complex eigenvalues may occur in the form of complex numbers, and in the form of complex modes in the FE analysis. The complex eigenvalue corresponding to the kth order complex mode can be expressed as sk ¼ σ k 7 jωk
(12)
where σk is the real part and ωk is the imaginary part of the complex eigenvalue. Whether the brake system is stable or not can be determined by the real parts of the complex eigenvalues. It is widely known that a brake system is not stable when the real part of one complex eigenvalue of the brake system is positive, so the modes that are unstable and likely to produce brake noise can be revealed by examining the real part of the brake system eigenvalues. An extra term, the damping ratio, is defined and calculated by using the following equation [17]:
ζ¼
σ π jωj
(13)
where ζ is the damping ratio of a mode. Apparently, if the damping ratio is negative (σ 40), the brake system is unstable, and vice versa. The main aim of this analysis is to increase the damping ratio value of the dominant unstable mode.
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3.2. Response surface model of the dominant unstable mode of the disc brake system with uncertainties Due to the complex working environment and operating conditions, the brake squeal is generated with great randomness, and it is difficult to be captured and reproduced artificially. This randomness is closely related to the uncertainties of system parameters. The brake squeal is the stochastic outcomes of the combined effect of many uncertainties. Therefore, the generation mechanism of the brake noise has not yet been comprehensively understood until now. The uncertainties in parameters of a disc brake system are mainly summarized and described as follows. (1) The friction coefficient is variable, but not constant, with respect to the sliding velocity between the brake pad and the disc [3]. (2) Wear is one of the most important factors that influent brake squeal [38,39], and the thicknesses of the friction material and the disc decrease continuously due to the wear in the course of brake working. (3) The material properties of brake components are very uncertain, by the influence of different operations or working environments. If the probability density functions of the uncertain parameters are defined unambiguously, the probabilistic method is the popular approach for the analysis of the structures with uncertain parameters [40]. Based on the probability density functions of the input parameters, the probability density functions of the output quantities can be achieved. Unfortunately, the statistical information to establish the probability density functions is not sufficient for the brake system. Under this case, it is advisable to employ the interval model to represent the uncertain parameters. In the FE analysis of a brake system, these parameters such as the friction coefficient, the material properties and the geometric dimensions of brake components may all influence the brake system modes. The behaviors of brake system modes can be improved by the optimization of some parameters. In the conventional parameter design for a brake using the finite element method, such as the works of Liu et al. [17] and Junior et al. [18], the optimization is usually carried out by varying a single parameter while keeping all the other parameters fixed at a specific set of conditions. Apparently, this method is time-consuming and does not consider the effects of the combination of parameters. RSM, which is a collection of mathematical and statistical techniques, is beneficial for the modeling and analysis of problems in which a response is influenced by several variables. The computation time and cost of the optimization of complex systems can be greatly reduced by RSM. Recently, RSM has been widely employed to optimize and understand the performance of complex systems [41,42]. By the application of RSM, the implicit functional relationships between the vibration modes and the system parameters of a brake system can be approximated with a limited number of finite element analysis experiments. Referring to the research of Nouby et al. [22], the complex eigenvalue corresponding to an unstable mode of a brake system can be fitted by a second-order model in the form of quadratic polynomial equation given below: n
n
n
i¼1
i¼1
1 ¼ ioj
n
n
n
i¼1
i¼1
1 ¼ ioj
σ ¼ a0 þ ∑ ai xi þ ∑ aii x2i þ ∑ aij xi xj ω ¼ b0 þ ∑ bi xi þ ∑ bii x2i þ ∑ bij xi xj
(14)
(15)
where σ and ω are the real and imaginary parts of the complex eigenvalue corresponding to the unstable mode of a brake system, respectively. a0, b0, ai, bi, aii, bii, aij and bij are the unknown coefficients that can be determined by the experimental design and the least square method. xi (i ¼ 1; 2; …; n) is the ith system parameter, including the uncertain parameters and the design parameters. If any values of the uncertain parameters in the interval are all considered in the experimental design space, then the stability state of the brake system corresponding to any values of the uncertain parameters in intervals can be described by Eqs. (14) and (15). In particular, if all the changes of the thicknesses of friction materials and disc resulted from the wear throughout the lifetime cycle are considered in the experimental design space for a brake system, then the stability state of the brake system corresponding to any wear state can be predicted by Eqs. (14) and (15). For establishing a response surface model, it is necessary to conduct the analysis of variance (ANOVA) to test the adequacy and significance of the model, so as to ensure the fitting accuracy of the model [42]. 4. Reliability-based optimization of the brake system with interval parameters 4.1. Reliability-based optimization model of the brake system with interval parameters In engineering practices, the traditional deterministic design optimization model has been successfully applied to reduce the cost and improve the quality of structures. However, the existence of uncertainties in either engineering simulations or manufacturing processes calls for a reliability-based design optimization model for robust and cost-effective designs. The solution from the reliability-based design optimization provides not only an improved design but also a higher level of confidence in the design. While the damping ratio of a vibration mode of a brake system is greater than the constant value ζ c ¼ 0:01, the brake system can be considered to be stable [43], then the limit-state function describing the stability of a brake system can be obtained by the following expression M ¼ GðXÞ ¼ ζ ðXÞ ζ c ¼ ζ ðXÞ þ0:01
(16)
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According to Eqs. (6)–(8) and (16), the reliability index of a brake system stability can be obtained as follows:
ηðXÞ ¼ ¼
maxðζ ðXÞ þ0:01Þ þ minðζ ðXÞ þ 0:01Þ maxðζ ðXÞ þ 0:01Þ minðζ ðXÞ þ 0:01Þ
maxðζ ðXÞ þ 0:01Þ þ maxð ζ ðXÞ 0:01Þ maxðζ ðXÞ þ 0:01Þ maxð ζ ðXÞ 0:01Þ
(17)
In order to ensure the stability of a brake system, it is required to ensure the reliability index η 41. The reliability-based optimization model of the stability of the vehicle disc brake system with interval parameters can then be expressed as min f ðXÞ s:t: ηðXÞ 4 1 X ¼ fXT1 ; XT2 gT X1 A ½XL1 ; XU1 ;
X2l rX2 r X2u
(18)
where f ðXÞ is the objective function, its mathematical expression and physical meaning can be determined based on the actual situation. X is the vector of variables of the brake system, X1 is the vector of non-design variables and X2 is the vector of design variables. XL1 and XU1 represent the lower and upper bounds of X1, respectively. X2l and X2u represent the lower and upper values of the variation range of X2, respectively. It can be easily seen from Eqs. (17) and (18) that the maximum and minimum of the limit-state function ζ ðXÞ þ0:01 need to be evaluated in this optimization model, at the same time, the minimum of the objective function f ðXÞ needs to be searched. So the optimization problem described by Eq. (18) is a typical two-layer nesting optimization problem. In the outer layer optimization, design parameters are used to build up the design variable space. In the inner layer optimization, uncertain parameters are used to build up the non-design variable space. The best optimal results can be found by the inner layer optimization and the outer layer optimization together.
4.2. The procedure for reliability-based optimization of brake systems The procedure for reliability-based optimization of the vehicle disc brake system with interval parameters can be summarized as (1) Define the system parameters, and divide the parameters into design variables and non-design variables. (2) Create the FE model of the brake system (shown in Eq. (11)), carry out experimental design, and perform CEA based on the FE model to obtain the unstable modes. (3) Create the response surface models of complex eigenvalues corresponding to unstable modes (shown in Eqs. (14) and (15)). (4) Build up the reliability-based optimization model of the brake system with interval parameters (shown in Eqs. (17) and (18)). (5) Solve the reliability-based optimization model, and get the optimal design results. The flow chart of the reliability-based optimization of the brake system with interval parameters is shown in Fig. 2.
5. Numerical example 5.1. Complex eigenvalue analysis of the brake system with interval parameters A simplified three-dimensional FE model of a disc brake assembly is developed in Altair HyperMesh software and illustrated in Fig. 3. This brake assembly consists of only a brake disc and a pair of brake pads which consist of friction materials and back plates. The disc is made of gray cast iron which is wear resistant and relatively inexpensive. The friction material which is made of an organic friction material is firmly mounted to the rigid brake plate which is made of steel. The FE meshes are generated by using the three-dimensional continuum elements, including 26,125 elements and 37,043 nodes, and the fine meshes are used in the contact regions. The friction contact interactions are defined between both sides of the disc and the contact plates of the pads. The CEA of the brake system is carried out by using ABQUS/Standard code. Fig. 4 presents the constraints and loadings of the brake assembly. The disc is completely fixed at the five counter-bolt holes and the ears of the back plates are constrained to allow only axial movements. The caliper–piston assembly is not defined in the simplified model of the disc brake system. Hence, the hydraulic pressure is directly applied to the back plates at the contact regions between the inner pad and the piston and between the outer pad and the caliper, and it is assumed that an equal magnitude of force acts on each back plate. For this brake, the friction materials are made of organic materials and have the anisotropic properties. Referring to the works of Junior et al. [18] and Cao et al. [37], here the friction materials are treated as isotropic materials but with varied Young's modulus. This assumption is reasonable to a certain degree, especially in an uncertainty analysis. Based on this
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Start Define system parameters
Creat FE model of brake system (shown in Eq.(13))
Uncertain parameters and design parameters space
Performe CEA base on FE model
Carry out experimental design
Create response surface models of complex eigenvalues corresponding to unstable modes (shown in Eqs. (16), (17) and (18))
Build up the reliability-based optimization model for the stability of the brake system with interval parameters (shown in Eqs. (20) and (21)).
Outer layer optimization Design variables space X2 Inner layer optimization Non-design variables space X1 Objective function value
No
Stopping criterion Yes Output optimal design results
Fig. 2. The flow chart of the reliability-based optimization of the brake system with interval parameters.
assumption and the unpredictability of wear, material properties, contact conditions between brake pads and disc, the uncertain variables and design variables are selected and described as follows: (1) The friction coefficient μ, disc thickness h1, friction material thickness h2, Young's modulus of disc E1, Young's modulus of back plate E2 and Young's modulus of friction material E3 are selected as uncertain variables and assumed as interval variables. The interval values of μ, E1, E2 and E3 are determined by the potential uncertainties, the interval values of h1 and h2 are determined according to the brake wear. The nominal and interval values of these variables are listed in Table 1. (2) Generally, in a real brake, the thickness of the whole brake pad cannot be allowed to change as any such a change would lead to the change of calliper and perhaps other brake components. In order to provide guidance for brake designers at the concept stage, here the back plate thickness h3 is taken as a design variable. Its initial value is 6.0 mm, and the design range is taken from 5.5 mm to 8.5 mm.
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Fig. 3. FE model of a disc brake system.
Fig. 4. Constraints and loading of the simplified brake system.
In a brake system, the units of the independent variables differ from one another. Even if some of the parameters have the same unit, not all of these parameters will be tested over the same range. Since parameters have different units and/or ranges in the experimental domain, the regression analysis should not be performed. Instead, one must firstly normalize the parameters before performing a regression analysis. Each of the coded variables is forced to range from 1 to 1, so all the parameters will affect the response evenly and the units of the parameters are irrelevant. Commonly used equations for coding are shown below:
μ 0:35
h1 19:0 h2 8:0 ; x2 ¼ ; x3 ¼ ; 0:05 1:0 3:0 h3 7:0 E1 125 E2 210 x4 ¼ ; x5 ¼ ; x6 ¼ ; 1:5 6:25 10:5 E3 5:94 x7 ¼ 0:594
x1 ¼
(19)
where xi (i ¼ 1; 2; …; 7) are the normalized parameters. A total number of 70 group samples in the space of normalized parameters are obtained by Latin Hypercube Sampling [44], which is a widely-used sampling method for constructing the response surface model. CEA approach is performed based upon the FE model of the disc brake system for each group samples from 0 to 16 kHz, which is the frequency range where the brake squeal commonly occurs. The analysis results show that, the unstable modes appear mainly at the frequency of around 1.9 kHz in this research, and the damping ratios of these unstable modes are almost higher than that of the other unstable modes. Based on the earlier studies of Abubakar and Ouyang [14] and Nouby et al. [22], here we take the unstable modes at the certain frequency for studying. Therefore, the unstable mode at 1.9 kHz for each group sample is regarded as the dominant unstable mode whose damping ratio is negative and is needed to be reduced to improve the system stability. As an example, the complex eigenvalues of this brake system corresponding to some group samples are shown as Fig. 5. These samples are selected randomly. It can be seen clearly that the unstable modes appear mostly at the frequency of around 1.9 kHz. 5.2. Response surface models of the complex eigenvalue corresponding to the dominant unstable mode Based on the CEA results of the disc brake system for each group samples and the method of the least squares, the quadratic polynomial response surface approximation models of the real part and the imaginary part of the complex eigenvalue corresponding to the dominant unstable mode are established as
σ d ðXÞ ¼ 62:75 þ13:25x1 þ 5:06x2 þ 30:91x3 27:40x4 þ9:13x5 3:77x6 þ 1:78x7
0:94x1 x2 þ0:54x1 x3 2:68x1 x4 5:69x1 x5 þ1:48x1 x6 4:29x1 x7 2:19x2 x3
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Table 1 The nominal values, lower and upper bounds of interval variables. Interval variables
Nominal values
Lower bounds
Upper bounds
Friction coefficient m Disc thickness h1 (mm) Friction material thickness h2 (mm) Young's modulus of disc E1 (GPa) Young's modulus of back plate E2 (GPa) Young's modulus of friction material E3 (GPa)
0.35 20.0 11.0 125 210 5.94
0.30 18.0 5.00 118.75 199.5 5.346
0.40 20.0 11.0 131.25 220.5 6.534
Fig. 5. Complex eigenvalues of the brake corresponding to some group samples.
0:63x2 x4 þ1:77x2 x5 þ 5:88x2 x6 þ 7:32x2 x7 þ2:06x3 x4 þ 3:46x3 x5 1:55x3 x6 þ 1:26x3 x7 þ2:04x4 x5 þ 1:97x4 x6 þ 4:55x4 x7 þ0:01x5 x6 4:06x5 x7 þ 4:65x6 x7 3:52x21 þ 2:17x22 0:49x23 26:67x24 3:92x25 2:60x26 þ 0:06x27
(20)
ωd ðXÞ ¼ 1911:29 0:19x1 þ 57:43x2 11:30x3 þ6:72x4 þ 45:16x5 þ1:95x6 þ 1:20x7
1:16x1 x2 þ 0:13x1 x3 1:84x1 x4 þ 0:26x1 x5 0:09x1 x6 0:05x1 x7 0:34x2 x3 2:55x2 x4 þ 1:49x2 x5 0:33x2 x6 þ 0:20x2 x7 3:90x3 x4 þ1:02x3 x5 0:73x3 x6 þ 0:27x3 x7 2:00x4 x5 þ2:15x4 x6 0:49x4 x7 þ 0:51x5 x6 0:19x5 x7 þ 0:69x6 x7 0:48x21 þ 0:30x22 þ 0:85x23 þ 4:46x24 0:64x25 þ 0:10x26 þ0:05x27
(21)
where X ¼ ½x1 ; x2 ; x3 ; x4 ; x5 ; x6 ; x7 T , σ d ðXÞ and ωd ðXÞ are the real and imaginary parts of the complex eigenvalue corresponding to the dominant unstable mode, respectively. Since any values of the uncertain parameters in intervals are all considered in the experimental design space, then the state of the brake system corresponding to any values of the uncertain parameters in intervals can be described by Eqs. (20) and (21). For example, the thickness parameters of the friction materials and disc, resulted from the wear, are all considered in the experimental design space, thus the stability state of the brake system corresponding to any wear state can be described by Eqs. (20) and (21). ANOVA is required to test the significance and adequacy of the response surface model. The ANOVA results for the quadratic response surface model of σ d and ωd are shown in Tables 2 and 3, respectively. The coefficient of determination (R2) is defined as the ratio of the explained variable to the total variation and a measurement of the degree of fitness. When R2 is close to 1, the model fits the actual experimental data better. The smaller the value of R2, the less relevant the model fits the actual data. The suggested R2 value should be at least 0.80 for a good fit of a model [45,46]. It can be seen from Tables 2 and 3 that the R2 value of the regression models of σ d and ωd are all higher than 0.80, which indicates that the regression models of σ d and ωd can explain the observed response well. Thus, the regression models of σ d and ωd are considered to be quite satisfactory. While the p value is lower than 0.01, it indicates that the model is considered to be statistical significant [44,47]. It can be seen from Table 2 that the second-order polynomial model of σ d is highly significant as the p value of the model is less than 0.0001. The F value for the model is 60.37. There is only a 0.01 percent chance that the “model F value” could occur because of the noise. As for the second-order polynomial model of ωd , the same conclusions can be drawn from Table 3. Hence, the response surface models developed in this section for σ d and ωd are successful in describing the correlation between the system parameters and the dominant unstable mode.
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Table 2 ANOVA results for the quadratic response surface model of σ d . Source
Sum of squares
Degree of freedom
Mean squares
F value
p value
Regression Residuals Total
52,234.01 840.51 53,074.52 R¼ 0.9921
35 34 69
1492.40 24.72
60.37
o 0.0001
R2 ¼ 0.9842
Table 3 ANOVA results for the quadratic response surface model of ωd . Source
Sum of squares
Degree of freedom
Mean squares
F value
p value
Regression Residuals Total
137,100 50 137,150 R¼ 0.9998
35 34 69
3916.68 1.47
2662.95
o 0.0001
R2 ¼0.9996
5.3. Reliability-based optimization of the brake system According to the actual situation, and in order to provide guidance for brake designers at the concept stage, the thickness parameter of back plate is selected as design variable, and the rest of the system parameters, namely, the friction coefficient, the thickness of disc, the thickness of friction material, Young's modulus of disc, Young's modulus of back plate and Young's modulus of friction material are selected as non-design variables. In order to guarantee the stiffness and strength of the back plate, the thickness of back plate should not be less than the initial value, thus the constraint for the thickness of back plate can be taken as 6:0 r h3 r8:5
(22)
By combining Eq. (20), it can be written with normalized variable as 0:6667 rx4 r1
(23)
For the purpose of lightweight design, the mass of back plate should be minimized, thus the thickness of back plate can be taken as the optimization objective. From Eq. (18), the reliability-based optimization model of this brake system stability with uncertain parameters can be expressed as follows when the dominant unstable mode is considered min h3 s:t: ηðμ; h1 ; h2 ; h3 ; E1 ; E2 ; E3 Þ 4 1 0:3 r μ r 0:4 18 rh1 r 20 5 rh2 r 11 6:0 r h3 r 8:5 118:75 r E1 r 131:25 199:5 r E2 r 220:5 5:346 r E3 r 6:534
(24)
By combining Eq. (19), it can be expressed with normalized variable as min x4 s:t: ηðXÞ 4 1; X ¼ fXT1 ; XT2 gT X1 ¼ fx1 ; x2 ; x3 ; x5 ; x6 ; x7 gT ; X2 ¼ fx4 g;
xi A ½ 1; 1;
i ¼ 1; 2; 3; 5; 6; 7
0:6667 r x4 r 1
(25)
where
ηðXÞ ¼
maxðð σ d ðXÞ=π jωd ðXÞjÞ þ 0:01Þ þ maxððσ d ðXÞ=π jωd ðXÞjÞ 0:01Þ maxðð σ d ðXÞ=π jωd ðXÞjÞ þ 0:01Þ maxððσ d ðXÞ=π jωd ðXÞjÞ 0:01Þ
(26)
The Genetic Algorithm [48] is selected to optimize the model described by Eq. (25). Here, the interval variables x1 ; x2 ; x3 ; x5 ; x6 ; x7 are selected as non-design variables and build up the non-design space X1. Whereas x4 is selected as design variable and makes up the design space X2. In the outer layer optimization, the optimal value of x4 is searched in the design space. In the inner layer optimization, the upper and lower bounds of the limit-state function described by Eq. (16)
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Fig. 6. System complex eigenvalues for a group parameter values after optimization.
Fig. 7. The reliability index of the brake stability for different values of h3.
and the reliability index of the brake system stability described by Eq. (26) are searched at different values of x4, in the nondesign space. The best optimal results are found by the inner layer optimization and the outer layer optimization together. The final optimization results are x4 ¼ 0:8667 and ηðXÞ ¼ 1:0184. The results mean that when h3 ¼8.3 mm, this brake system can always stay in a stable state, even if considering the influence of uncertain factors. Thus, the robustness of the brake system stability is improved greatly. The prediction accuracies of the fitted models expressed by Eqs. (20) and (21) are checked by the optimization results. By substituting the group normalized value (x1 ¼ 0, x2 ¼ 0, x3 ¼ 0, x4 ¼ 0:8667, x5 ¼ 0, x6 ¼ 0, x7 ¼ 0), in which x4 takes the optimized value and the other normalized variables take the values of zero, into Eqs. (20) and (21). The calculation results are σ d;RSM ¼ 18:97 and ωd;RSM ¼ 1920:5. Complex eigenvalue analysis is performed based upon the FE model of this brake for this group value, the corresponding results are σ d;FEM ¼ 19:03 and ωd;FEM ¼ 1921:2, which show that the prediction accuracy of the fitted models is very well. Fig. 6 shows the system complex eigenvalues for the above CEA results, it can be seen that all the asterisks representing system complex eigenvalues are far away from the line corresponding to the damping ratio ¼ 0.01, that is, the system is in a quite stable state under this group of parameters. To a certain extent, the purpose of suppressing brake squeal is achieved. If the reliability index η 41, the brake system will be in an absolutely stable state even though the influence of uncertain factors is considered. In order to show the influence of design variable h3 on this commercial brake system stability more clearly, the reliability index of the brake system stability for different values of h3 is given in Fig. 7. It can be seen from Fig. 7 that the stability and robustness of this commercial brake system are increased as the thickness of back plate h3 increases. The reason is that a larger thickness will induce larger supporting stiffness of the back plate. It can be known from the works of Liles [10] and Nouby et al. [22] that the stiffer back plates can reduce the propensity of brake squeal. This is quite coincided with the results of this research. Therefore, in engineering practices, it may improve the stability and robustness of a real brake and reduce brake squeal through increasing the stiffness of back plates, even though the influence of uncertain conditions are considered. In order to achieve the purpose of improving the supporting stiffness of back plate, one can make the back plate with larger thickness in the conceptual design stage, or change the material of back plate into the material with larger Young's modulus, or consider the material properties and geometrical dimensions together during the design process. 6. Conclusions In this paper, a reliability-based optimization method for the brake squeal reduction of the vehicle disc brake system with interval parameters is presented. In the proposed method, the dominant unstable mode is extracted by performing CEA
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based on the linear FE model of the brake system, and the negative damping ratio corresponding to the dominant unstable mode is chosen as an indicator of instability. RSM is used to build up the models to express the functional relationship between the dominant unstable mode and the system parameters. ANOVA is applied to test the adequacy and significance of the response surface models to ensure the fitting accuracy. The efficiency of the stability analysis of the brake systems is greatly improved by the application of RSM. Uncertain parameters are employed to deal with the uncertainties existing in geometrical dimensions, material and loading properties of the brake system. The frictional coefficient, the parameters of material properties, and the thicknesses of wearing components are treated as the uncertain parameters. Usually, there is no sufficient information to construct the precise probability distributions of uncertain parameters. Thus, these uncertain parameters are described as interval variables in the proposed method. A reliability-based stability optimization model for the vehicle disc brake system considering uncertainties is constructed based on the RSM, the interval analysis and the reliability analysis. The stability analysis and optimization of a disc brake system with interval parameters are carried out. The results show that the proposed approach can improve the stability of a brake system effectively under uncertain conditions.
Acknowledgments The paper is supported by Hunan Provincial Innovation Foundation for Postgraduate (Grant no. CX2013B143), Independent Research Project of State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body in Hunan University (Grant no. 51375002). The authors would also like to thank reviewers for their valuable suggestions. References [1] M.R. Nishiwaki, Review of study on brake squeal, Japan Society of Automobile Engineering Review 11 (4) (1990) 48–54. [2] S. Yang, R.F. Gibson, Brake vibration and noise: reviews, comments, and proposals, International Journal of Materials and Product Technology 12 (4–6) (1997) 496–513. [3] N.M. Kinkaid, O.M. O'Reilly, P. Papadopoulos, Automotive disc brake squeal: a review, Journal of Sound and Vibration 267 (1) (2003) 105–166. [4] M. Nishiwaki, Generalized theory of brake noise, Proceedings of the Institution of Mechanical Engineers 207 (3) (1993) 195–202. [5] A. Papinniemi, J.C.S. Lai, J. Zhao, L. Loader, Brake squeal: a literature review, Applied Acoustics 63 (4) (2002) 391–400. [6] S. Oberst, J.C.S. Lai, Statistical analysis of brake squeal noise, Journal of Sound and Vibration 330 (12) (2011) 2978–2994. [7] H. Ouyang, W. Nack, Y. Yuan, F. Chen, Numerical analysis of automotive disc brake squeal: a review, International Journal of Vehicle Noise and Vibration 1 (3–4) (2005) 207–231. [8] G. Fritz, J.J. Sinou, J.M. Duffal, L. Jézéquel, Effects of damping on brake squeal coalescence patterns application on a finite element model, Mechanics Research Communications 34 (2) (2007) 181–190. [9] W.V. Nack, Brake squeal analysis by finite elements, International Journal of Vehicle Design 23 (3–4) (2000) 263–275. [10] G.D. Liles, Analysis of disc brake squeal using finite element methods, SAE Paper No. 891150, 1989. [11] M.L. Chargin, L.W. Dunne, D.N. Herting, Nonlinear dynamics of brake squeal, Finite Elements in Analysis and Design 28 (1) (1997) 69–82. [12] S.W. Kung, K.B. Dunlap, R.S. Ballinger, Complex eigenvalue analysis for reducing low frequency brake squeal, SAE Paper No. 2000-01-4444, 2000. [13] H. Ouyang, W. Li, J.E. Mottershead, A moving-load model for disc-brake stability analysis, Journal of Vibration and Acoustics 125 (1) (2003) 53–58. [14] A.R. AbuBakar, H. Ouyang, Complex eigenvalue analysis and dynamic transient analysis in predicting disc brake squeal, International Journal of Vehicle Noise and Vibration 2 (2) (2006) 143–155. [15] D. Guan, X. Su, F. Zhang, Sensitivity analysis of brake squeal tendency to substructures' modal parameters, Journal of Sound and Vibration 291 (1–2) (2006) 72–80. [16] G. Fritz, J.J. Sinou, J.M. Duffal, L. Jézéquel, Investigation of the relationship between damping and mode-coupling patterns in case of brake squeal, Journal of Applied Mechanics 307 (3–5) (2007) 591–609. [17] P. Liu, H. Zheng, C. Cai, Y.Y. Wang, C. Lu, K.H. Ang, G.R. Liu, Analysis of disc brake squeal using the complex eigenvalue method, Applied Acoustics 68 (6) (2007) 603–615. [18] M.T. Junior, S.N.Y. Gerges, R. Jordan, Analysis of brake squeal noise using the finite element method: a parametric study, Applied Acoustics 69 (2) (2008) 147–162. [19] Y. Dai, T.C. Lim, Suppression of brake squeal noise applying finite element brake and pad model enhanced by spectral-based assurance criteria, Applied Acoustics 69 (3) (2008) 196–214. [20] Y. Nishizawa, S. Wakamatsu, H. Yanagida, Y. Takahara, Study of dynamic pad stiffness influencing brake squeal, SAE Paper No. 2007-01-3956, 2007. [21] H. Nonaka, Y. Nishizawa, Y. Kurita, Y. Oura, Considering the dynamic pad stiffness in FEM analysis of disk brake squeal, SAE Paper No. 2010-01-1716, 2010. [22] M. Nouby, D. Mathivanan, K. Srinivasan, A combined approach of complex eigenvalue analysis and design of experiments (DOE) to study disc brake squeal, International Journal of Engineering, Science and Technology 1 (1) (2009) 254–271. [23] S. Lakkam, S. Koetniyom, Optimization of constrained layer damping for strain energy minimization of vibrating pads, Songklanakarin Journal of Science and Technology 34 (2) (2012) 179–187. [24] E. Sarrouy, O. Dessombz, J.J. Sinou, Piecewise polynomial chaos expansion with an application to brake squeal of a linear brake system, Journal of Sound and Vibration 332 (3–4) (2013) 577–594. [25] R. Moore, W. Lodwick, Interval analysis and fuzzy set theory, Fuzzy Sets and Systems 135 (1) (2003) 5–9. [26] R. Rackwitz, Reliability analysis—a review and some perspectives, Structural Safety 23 (4) (2001) 365–395. [27] F. Massa, A. Leroux, B. Lallemand, T. Tison, S. Mary, F. Buffe, Fuzzy vibration analysis and optimization of engineering structures: application to Demeter satellite, IUTAM Bookseries 27 (1) (2011) 63–76. [28] F. Massa, B. Lallemand, T. Tison, Fuzzy multiobjective optimization of mechanical structures, Computer Methods in Applied Mechanics and Engineering 198 (5) (2009) 631–643. [29] O. Kelesoglu, Fuzzy multiobjective optimization of truss-structures using genetic algorithm, Advances in Engineering Software 38 (2007) 717–721. [30] J. Ramík, Optimal solutions in optimization problem with objective function depending on fuzzy parameters, Fuzzy Sets and Systems 158 (2007) 1873–1881. [31] R. Moore, Interval Analysis, Prentice Hall, Englewood Cliff, 1966. [32] G. Alefeld, J. Herzberber, Introductions to Interval Computations, Academic Press, New York, 1983.
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[33] D. Moens, D. Vandepitte, A survey of non-probabilistic uncertainty treatment in finite element analysis, Computer Methods in Applied Mechanics and Engineering 194 (12–16) (2005) 1527–1555. [34] Y.B. Haim, A non-probabilistic concept of reliability, Structural Safety 14 (4) (1994) 227–245. [35] Z.P. Qiu, J. Wang, The interval estimation of reliability for probabilistic and non-probabilistic hybrid structural system, Engineering Failure Analysis 17 (5) (2010) 1142–1154. [36] J. Tao, C.J. Jun, X.Y. Lan, A semi-analytic method for calculating non-probabilistic reliability index based on interval models, Applied Mathematical Modelling 31 (7) (2007) 1362–1370. [37] Q. Cao, H. Ouyang, M.I. Friswell, J.E. Mottershead, Linear eigenvalue analysis of the disc-brake squeal problem, International Journal for Numerical Methods in Engineering 61 (9) (2004) 1546–1563. [38] A.R.A. Bakar, H. Ouyang, S. James, L. Li, Finite element analysis of wear and its effect on squeal generation, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 222 (7) (2008) 1153–1165. [39] A.R. AbuBakar, H. Ouyang, Wear prediction of friction material and brake squeal using the finite element method, Wear 264 (11–12) (2008) 1069–1076. [40] G. Stefanou, The stochastic finite element method: past, present and future, Computer Methods in Applied Mechanics and Engineering 198 (9–12) (2009) 1031–1051. [41] D.C. Montgomery, Design and Analysis of Experiments, Wiley Publications, New York, 1991. [42] R.H. Mayers, D.C. Montgomery, Response Surface Methodology: Process and Product Optimization Using Designed Experiments, John Wiley & Sons, New Jersey, 2002. [43] J. Hou, X.X. Guo, G.F. Tan, Complex mode analysis on disc brake squeal and design improvement, SAE Paper No. 2009-01-2101, 2009. [44] M. Papila, Accuracy of Response Surface Approximations for Weight Equations Based on Structural Optimization, PhD thesis, University of Florida, 2001. [45] J. Fu, Y. Zhao, Q. Wu, Optimising photoelectrocatalytic oxidation of fulvic acid using response surface methodology, Journal of Hazardous Materials 144 (1–2) (2007) 499–505. [46] A.M. Joglekar, A.T. May, Product excellence through design of experiments, Cereal Foods World 32 (1987) 857–868. [47] H.K. Kim, J.G. Kim, J.D. Cho, J.W. Hong, Optimization and characterization of UV-curable adhesives for optical communications by response surface methodology, Polymer Testing 22 (8) (2003) 899–906. [48] D.E. Goldberg, Genetic Algorithms in Search, Optimization & Machine Learning, Addison–Wesley, New York, 1989.