Automotive brake squeal simulation and optimization.
This work carries out complex modal analyses and optimizations to resolve an 1800 Hz front brake squeal issue encountered in a vehicle program development phase. The stability theory of complex modes for brake squeal simulation is briefly explained. A brake system finite element model is constructed, and the model is validated by the measurement in accordance with the SAE 2521 procedure. The key parameters for evaluating the stability of the brake system complex modes are determined. The modal contributions of relevant components to unstable modes are analyzed and ranked. Finally, in order to resolve the squeal issue, the design improvements of rotor, caliper and pad are proposed and numerical simulations are carried out. The obtained results demonstrate that the optimized rotor and pad design can alleviate the squeal issue significantly while the optimized clipper design could essentially eliminate the squeal issue. The analytical solutions obtained from the proposed pad design are validated by the experimental measurements.
CITATION: Yang, S., Sun, Z., Liu, Y. , Lu, B. et al., "Automotive Brake Squeal Simulation and Optimization," SAE Int. J. Passeng. Cars - Mech. Syst. 9(1):2016.
Brake system complaints are among the top vehicle quality concerns as demonstrated by market surveys such as J.D. Power IQS (Initial Quality Study). Among those brake related complaints, NVH problems often claim the lion's share. The most notorious brake system NVH issues are squeal, rattle, groan, judder and moan.
The automobile brake squeal problem is usually a single frequency event, characterized by high intensity up to 120 dB(A), poor sound quality, and a frequency that falls into a wide range from 0.9 to 18 kHz. Its mechanisms are characterized by excitation, modal coupling and mode lock-in. It is well accepted from a dynamics perspective that it is caused by brake system instability, represented by a self-sustained energy-feed-in cycle due to the physics of negative damping. Thus, when a brake system possesses intrinsic characteristics prone to instability and a proper excitation source happens to be present, the brake system may generate squeal. The understanding, analysis and prevention of the issue are further complicated by the nonlinear and random nature of the problem, such as contact mechanism, stick-slip process and tribological effects. Therefore, once the problem emerges, it is difficult to resolve, and in fact it is one of the toughest issues during the brake system development effort.
In recent years, considerable efforts have been devoted to the understanding and resolution of the problem as evidenced in many published literatures, e.g. . OEM's and brake system suppliers have carried out extensive studies on the mechanism, simulation methodology, and experimental technology. This work concentrates on a typical brake squeal problem encountered in a prototype vehicle for a vehicle program. During a program development stage, a front wheel brake system repeatedly developed a typical 1800 Hz brake squeal noise. The factors influencing such event are many, such as the pad, caliper, anchor bracket, knuckle, other brake components, and the interactions of them. Many proposed efforts, such as various bench test and vehicle measurement, could not find an effective fix. In this work, based on the theory of the stability of the complex mode for brake systems, a finite element model of the brake system is established first, and then the complex modal analysis is carried out. The evaluation method and factors that influence brake squeal are looked into. And finally, the proposed resolution measure is investigated and validated by testing.
1. ROOT CAUSES OF UNSTABLE MODES
For a linear vibration system with N degrees of freedom and viscous damping, the dynamic mathematical expression can be written as:
M[??] + C[??] + Ku = Q,
where u is the displacement vector, M is the mass matrix, C is the damping matrix which includes the friction effect, K is the stiffness matrix, and Q is the external load. In the case of Q being 0, the above equation becomes
M[??] + C[??] + Ku = 0,
which describes the inherent properties of the system. In general, the damping matrix and the stiffness matrix are asymmetric, thus the above equation can not be solved for real modes. In this case, it necessitates the complex mode analysis.
In general, the above equation has a solution of the form:
where [lambda] and [phi] are the complex eigenvalues and eigenvectors respectively. The complex eigenvalues are of the form [[lambda].sub.j] = [[alpha].sub.j] [+ or -] i[[omega].sub.j], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[omega].sub.j] represent the real part and imaginary part respectively with j = 1,2, ..., N.
From the nature of the solution, it is seen that the real part of the eigenvalue represents the attenuation of the vibration. When [[alpha].sub.j] < 0, the complex mode will cause the system to undergo a damped vibration with decreasing amplitude, and therefore this mode is a stable one. When [[alpha].sub.j] = 0, this complex mode represents an undamped vibration, which could be considered as a critical stable mode. And when [[alpha].sub.j] > 0, the complex mode represents a vibration with increasing amplitude, and thus it diverges as an unstable mode. The imaginary part of the eigenvalue represents the frequency of the vibration mode.
If the contact mechanism and the friction behavior between the brake rotor and pad are not considered, the system would possess a positive-definite M, a symmetric C and a symmetric K. This situation will not lead to the occurrence of any unstable modes. If the contact friction mechanism is included in the formulation, after discretization and linearization, the following equation is derived:
M[??] + (C + [C.sub.f])[??] + (K + [K.sub.f])u = 0
where [C.sub.f] and [K.sub.f] are the friction damping matrix and friction stiffness matrix respectively due to the contact mechanism and friction effects between the rotor and pad. It is this addition of the extra quantities that leads to the likelihood of unstable modes.
Figure 1 shows the modeling of the contact mechanism for the rotor and pad. Suppose the element coordinate system is (U,V), the coefficient of friction is [mu], and the contact stiffness is k. Then the normal force and tangential force between points n,m, can be expressed as:
It is apparent that the element stiffness matrix is asymmetric, and all elements connecting the contact surface have asymmetric stiffness matrix. Therefore, K + [K.sub.f] is asymmetric. This means that the eigenvalues could be of a complex nature, and their real part might be positive.
The establishment of [C.sub.f] requires detailed information regarding the relationship between the friction force and relative speed. Figure 2 displays the familiar Stribeck model for the relation between the friction force and relative speed . It is evident that in the Stribeck region, the friction coefficient is negative. As a result, the damping matrix C + [C.sub.f] might be negative.
2. ESTABLISHMENT OF FINITE ELEMENT COMPLEX MODE ANALYSIS MODEL
The brake squeal is usually caused by unstable high frequency modes between 0.9 to 18 kHz. In order to carry out the analysis, it is necessary to establish an accurate and reliable brake system finite element model that can correctly capture the characteristics of such high frequency modes. To ensure the model accuracy, the friction coefficient, the material properties of the pad and silencing sheet, hydraulic cylinder oil film, bearing, stiffness of the suspension joints, tire and bushings, are all obtained by testing. Based on the measured data, the models of brake system and the suspension system are assembled, and a brake system complex stability analysis model is established as shown in Figure 3.
The model contains 251269 solid and shell elements with 359192 nodes, 12 contact friction pairs including pad, rotor, and caliper, as well as 5 other friction pairs in the suspension system.
The anisotropic material properties of the friction pad are obtained by ultrasonic testing . By comparing the modal results obtained by testing with the calculated modal results using these measured material properties, the material parameters are modified so as to ensure that the modal results from testing and simulation are correlated. In this way, a correct set of material parameters is obtained. The comparison of the first and second modes for the pad from the testing and simulation is shown in Figure 4. The modified material properties are listed in Table 1, where E is the elastic modulus, G the shear modulus, and [nu] the Poisson's ratio, with 1 denoting the vertical direction and 2 the tangential direction.
3. BRAKE SYSTEM COMPLEX MODE STABILITY ANALYSIS
3.1. Simulation Setup
Brake squeals occur in a specific brake pressure, speed, temperature, and friction condition. In theory, it is desirable to analyze the stability of brake system considering all design variables involved in a complicated brake system, along with many operating and environmental conditions such as brake pressure, friction coefficient and rotating speed. In reality, as these parameters are continuously changing, it is impractical and impossible to carry out such a thorough and exhaustive calculation covering all possible combinations of parameters and events. Based on past simulation works for many programs and the specific situation of the current brake system, this work establishes a load case matrix for analyses as shown in Table 2.
3.2. Stability Criterion
Whether an unstable mode could lead to brake squeal or not also depends on the nature of negative damping as well as the modal frequency . This work assesses the stability of the brake system by using the system's negative damping coefficient which is defined by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
where [[alpha].sub.j] and [[omega].sub.j] are the real part and imaginary part of the jth eigenvalue respectively. If the negative damping coefficients are of the same value, the squeal propensity is construed the same. In this case, the real part of the eigenvalue, i.e [[alpha].sub.j], is linearly proportional to the frequency, which is equivalent to saying that a higher frequency mode can afford to have a higher real part of the eigenvalue.
In theory, the mode with its negative damping coefficient less than 0 is an unstable mode. However, a more realistic assessment is that only when the negative damping coefficient reaches a certain value, the brake squeal would be likely to occur. Therefore, in order to assess whether a brake system is imminent to become unstable, it is important to establish a critical value of the negative damping coefficient which signals the inception of instability.
According to past experience and specific requirements of the current project, the critical negative damping coefficient is taken as -0.01. If a negative damping coefficient of a system is less than this value, it is construed as the occurrence of brake squeal is more likely. It needs to point out here that this requirement is not considered very stringent.
3.3. Simulation Result
Using the load condition matrix in table 2 and the analysis procedure outlined in Section 3.1, 35 complex modal analyses are carried out. The computer resource involves a top line desktop, and the computation takes more than 130 hours of runtime. There are many simulation results obtained, and only the main results corresponding to 2 friction coefficient values of 0.5 and 0.6 are selected to be displayed here in Figures 5 and 6 respectively. In each diagram, one friction coefficient is combined with 5 brake pressure values and one rotating speed under the forward driving condition,.
From Figures 5 and 6, the results demonstrate the occurrence of negative damping coefficients in the vicinity of 1800 Hz. It is seen that as the brake pressure decreases, the situation becomes worse. In the case of the friction coefficient 0.6 and the pressure 3 bar, the negative damping coefficient reaches -0.36, which is the worst negative damping coefficient value among all cases displayed.
Similar analyses are carried using the reverse operating mode, and the obtained results are displayed in Figures 7 and 8 for the friction coefficients 0.5 and 0.6 respectively. It can be seen that there are occurrences of unstable modes around 1800 Hz as well, with the worst negative damping coefficient value of -0.32 under 3 bar for both friction coefficient values of 0.5 and 0.6. In addition, there exist unstable modes around 1100 Hz and 3000 Hz as well for the reverse condition, albeit the values of the negative damping coefficients are relatively insignificant compared to those around 1800 Hz.
According to the criterion outlined in Section 3.2, the negative damping coefficients around 1800 Hz are much worse than the critical value of -0.01, which indicates that the brake squeal issues are very likely to occur. This conclusion is consistent with the repeated occurrence of unacceptable brake squeal issues around 1800 Hz for the prototype vehicles.
The analytical result of the structural vibration of the brake system around 1800 Hz is shown in Figure 9. This vibration mode mainly consists of third order out of plane rotor disc mode and the first order bending of the caliper mode.
4. MODEL VALIDATION
A brake system bench test is carried out in accordance with the international standard SAEJ2521. The results of the bench test are shown in Figure 10, which clearly exhibits the 1800 Hz brake squeal issue. Therefore, the numerical simulation has correctly predicted and captured the brake squeal phenomenon. The results of cumulative percentage of noisy stops obtained from the bench test are shown in Figure 11.
5. RESOLUTION MEASURE ANALYSES
5.1. Components and Their Modal Contributions to Unstable Modes
In order to rank the contributions of the main components to the brake squeal issue, component and modal contribution analyses are carried out. Similar methodology in modal alignment perspective is used in many analytical studies, e.g. .
Figure 12 shows the component contributions to the unstable mode around 1800 Hz, which indicates that the most important contributors are the caliper, rotor and lining.
Figures 13 and 14 display the results of modal contributions from the caliper (housing) and rotor. The first order torsional mode of the brake caliper as shown in Figure 13 and the third out of plane brake rotor mode as shown in Figure 14 are the most prominent contributors in this case.
5.2. Disc Optimization
As pointed out in the last section, the main contributors to the brake system unstable mode are the disc rotor and caliper, i.e. the coupling of the first order torsional caliper mode with the third out-of-plane rotor mode. To improve system stability, the most straightforward method would be to decouple the rotor mode with the caliper mode. Since it is more costly to revise the brake caliper design, this work proposes to shift the frequency of the rotor.
In order to raise the modal frequency of the rotor, the rotor disc ventilation pattern is changed from the original straight design to the spiral design as described in Figure 15. The structure modes of the brake rotor are summarized in Figure 16. Obviously, the third order out-of-plane rotor mode is improved by 118Hz, while the first and second modes are not changed as much.
Figures 17 and 18 show the results of the brake system complex mode stability analyses under forwarding condition for the cases of 0.5 and 0.6 friction coefficients respectively. And Figures 19 and 20 display the results under reverse condition.
All these results show a significant improvement in the values of the negative damping coefficient compared to the original results shown in Figures 5 to 8. Figure 21 summarizes the improvements from the comparison of Figures 5, 6, 7, 8 representing the original design with Figures 17, 18, 19, 20 corresponding to the proposed design.
5.3. Lining Optimization
The previous simulation analysis shows that the brake pad contributes significantly to the unstable mode. The geometrical pattern and lining material certainly impact the system stability. Two design changes are proposed. One is the material change and the other the geometry modification.
The original metallic material is replaced by a relatively soft nonmetallic material, with its anisotropic properties shown in Table 3. The involved parameters are measured and modified in the same way as the original material properties outlined in Section 2.
Figure 22 shows the original and proposed pad design. The proposed design integrates a slot in the middle, and adds a tapered angle to the pad, similar to worn-off angles. Figures 23, 24, 25, 26 show the analytical results for the proposed chamfered and slotted pad with non-metallic lining material.
Figure 27 summarizes the improved results displayed in Figures 23, 24, 25, 26. It can be seen that the optimization of the lining material and the shape of the pad lining has some improvement, but it is not as good as that of the proposed brake rotor disc design.
5.4. Caliper Optimization
As pointed out in Figures 12 and 13, the maximum contribution to the 1800 Hz unstable mode of the brake system is the first order torsional mode of the caliper, which indicates that shifting the caliper modal frequency could improve the system stability. In order to accomplish this frequency shifting, a new caliper design is proposed. This design basically incorporates a stiffer structure approach which is shown in Figure 28, where the left picture and right picture display the original design and proposed design respectively.
Figure 29 shows the calculated fst torsional mode results for the two caliper designs. The first torsional modal frequency of the new caliper reaches 2592 Hz which is 630 Hz higher than that of the original caliper.
Figures 30, 31, 32, 33 display the results based on the new caliper design, and Figure 34 summarizes the results from which it can be seen that the proposed new caliper design leads to a significant improvement with values of negative damping coefficient in all cases much better than the critical value of -0.01. This demonstrates that the proposed caliper design can eliminate the 1800 Hz brake squeal issue. Although a 3000 Hz unstable mode shows up in this case, its negative damping coefficient is not considered significant enough to cause a concern nonetheless.
5. PROGRAM IMPLEMENTATION
Due to the program timing pressure and the cost concern associated with the proposed rotor and caliper design change, so far only the proposed pad design was adopted by the program team. According to the proposed pad design, a test sample was fabricated and an experimental measurement was performed according to SAEJ2521. Figure 35 shows the test results. In comparison with the original results displayed in Figure 11, it is seen that the number of occurrences and magnitudes of the brake squeal are greatly reduced, in agreement with the simulation results. Drive evaluations found that the original brake squeal issue was mitigated to an acceptable level for the program.
A brake squeal issue surfaced in a development stage of a passenger vehicle program. This article outlined the basic theory of the complex modal analysis and the root cause of the brake system instability. And then a complex modal analysis FE model was established. The calculation results showed that there exists an 1800 Hz instability mode in the brake system which correlates to the observed outcome. The contribution analysis of components and their modes exhibited that the first mode of caliper, the third mode of rotor and lining are the main contributors to the squeal issue. The optimization results showed that the proposed rotor disc design with spiral ventilation pattern, and the proposed chamfered and slotted pad with nonmetallic lining material could improve the instability mode, although could not completely eliminate the issue. The proposed stiffer caliper design could essentially eliminate this instability mode. Due to program timing and cost constraints, only the proposed pad design change was implemented. A test sample of the proposed pad design was fabricated, and the experimental measurement was performed. The test result showed that the 1800 Hz squeal was greatly reduced, which validated the simulation results. And the program target was finally met.
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Shukai Yang, Zuokui Sun, Yingjie Liu, Bingwu Lu, Tao Liu, and Hangsheng Hou
FAW R&D Center
The authors would like to thank these colleagues at FAW for facilitating various activities of this work: Min Wang, JianHua Wang. Thanks also go to Le Xi and his colleagues from the supplier: Continental Brake Systems (Shanghai) Co., Ltd. This work was supported by the 2012 technology innovation project of FAW coded kg12073.
Table 1. Measured and modified friction pad anisotropic material properties. [E.sub.1] [E.sub.2] [E.sub.3] [v.sub.12] [v.sub.13] Measured 1.0 1.0 2.8 0.19 0.32 Modified 6.5 6.5 1.8 0.19 0.32 [v.sub.23] [G.sub.12] [G.sub.13] [G.sub.23] Measured 0.32 4.2 2.1 2.1 Modified 0.32 2.7 1.3 1.3 Table 2. Operating condition matrix used for analyses. Rotating Speed Friction coefficient Brake Pressure(Bar) 0.4 3 8 16 24 30 Forward 5 r/s 0.5 3 8 16 24 30 0.6 3 8 16 24 30 0.4 3 8 16 24 30 Backward 2 r/s 0.5 3 8 16 24 30 0.6 3 8 16 24 30 0.7 3 8 16 24 30 Table 3. Measured and modified friction pad anisotropic material properties for the proposed pad material. [E.sub.1] [E.sub.2] [E.sub.3] [v.sub.12] [v.sub.13] Measuied 9. 9. 2.3 0.22 0.29 Modified 6.2 6.1 1.5 0.22 0.29 [v.sub.23] [G.sub.12] [G.sub.13] [G.sub.23] Measuied 0.29 3.9 1.7 1.7 Modified 0.29 2.5 1.1 1.1
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|Author:||Yang, Shukai; Sun, Zuokui; Liu, Yingjie; Lu, Bingwu; Liu, Tao; Hou, Hangsheng|
|Publication:||SAE International Journal of Passenger Cars - Mechanical Systems|
|Date:||Apr 1, 2016|
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