Prediction and optimization of the disc brake squeal performance with Altair HyperWorks and ANSYS

Transcription

1 Journal of Advances in Vehicle Engineering 2(2) (2016) Prediction and optimization of the disc brake squeal performance with Altair HyperWorks and ANSYS Xiaofeng Wang *a, Bin Yang a, Qing Li b a State Key Laboratory of Automotive Safety and Energy, Department of Automotive Engineering, Tsinghua University, Beijing , China b Automotive Engineering Institute, Changan Automobile Stock Co., Ltd., Chongqing , China (Manuscript Received: 17 Feb 2016; Revised: 28 May 2016; Accepted: 10 Jun, 2016) Abstract This paper aims at seeking practical methods for predicting and optimizing the brake squeal performance of a passenger car disc brake. The finite element models of the disc brake were built with HyperMesh of Altair HyperWorks 8.0 and the complex modes analysis of each model was performed with Linear Non-prestressed Modal Analysis method of ANSYS The predicted brake squeal frequencies were compared with those measured with a brake dynamometer. In order to improve the complex modes analysis results, the effect of changing the friction coefficient between the disc and pads was studied, which was equivalent to calibrating the finite element model by use of the dynamometer test results and resulted in a more accurate model. The topology optimization of the pad and caliper were performed with Optistruct of Alair HyperWorks to eliminate the critical squeal frequencies. Keywords: Disc brake squeal; finite element model; Complex modes analysis; Friction coefficient; Topology optimization Introduction Disc brake squeal of passenger cars has been a major warranty issue, which is the result of friction-induced vibration or self-excited vibration and whose frequency range is about from 1 to 20 khz [1, 2]. And brake squeal has also been a challenging problem for many engineers and researchers due to its immense complexity and random occurrence [3, 4]. In order to understand squeal phenomenon, predict and eliminate it, a variety of tools have been developed which include both experimental and analytical methods. The most common experimental method is to utilize the brake dynamometer to simulate brake events under different brake conditions [1-9]. Analytical methods mainly utilize finite element analysis methods to predict squeal noise and seek design changes to eliminate it [1, 3-7, 9]. Although many squeal problems have been successfully solved, a comprehensive understanding of the problem is still lacking and the squeal problem has remained. The main analytical methods on disc brake squeal include complex modes analysis, transient analysis, parametrical analysis, and operational simulation, each of which has its own advantages and limitations [1-5]. The key lies in that the analysts can choose right methods to tackle the squeal problems facing them. The complex modes analysis in frequency domain is the analytical method currently used in solving most brake squeal problems in automotive industries, which linearizes the brake * Corresponding author. X Wang (*) State Key Laboratory of Automotive Safety and Energy, Department of Automotive Engineering, Tsinghua University, Beijing , China wangxf60@mail.tsinghua.edu.cn squeal solution at the static steady sliding position [3], in which a modal analysis of the pre-stressed structure is performed. An unsymmetric stiffness matrix is a result of the friction coupling between the brake pad and disc, which may lead to complex eigenfrequencies. If the real part of the complex frequency is positive, then the system is unstable as the vibrations grow exponentially over time and the squeal occurs. ANSYS 12.0 presents three different methods to perform a brake squeal analysis (Refer to Brake Squeal Analysis of Structural Analysis Guide in ANSYS 12.0 Help): (1) Full Nonlinear Prestressed Modal Analysis; (2) Linear Non-prestressed Modal Analysis; (3) Partial Prestressed Modal Analysis. A static contact analysis is first performed to establish initial condition (or static steady sliding position) and Newton- Raphson iterations are usually required in the methods (1) and (3), resulting in very long computation durations. If the stress stiffening effects are not critical, the method (2) can be applied to perform the complex modes analysis, in which the static contact analysis is no longer performed and Newton-Raphson iterations are not required. Thus, the computation duration needed in the method (2) is much shorter than those needed in the methods (1) and (3). However, in theory, the method (2) is less accurate than the other two methods. And some work may need doing for improving the effectiveness of the method (2) in predicting the brake squeal frequencies. In this paper, the finite element models of a passenger car disc brake were built with Altair HyperWorks 8.0 and the complex modes analysis of each model was performed with the ANSYS method (2) above. And the results obtained are 94

2 compared with the test results obtained with a brake dynamometer in accordance with SAE J2521 standard, with the focus on how to improve the complex modes analysis results obtained with the ANSYS method (2). Based on the knowledge obtained above, the optimization of the disc brake pad and caliper was performed to eliminate the critical squeal modes by use of OptiStruct module of Altair HyperWorks Disc Brake and Dynamometer Test Results Fig. 1 shows the 3D numerical model of the disc brake studied, which consists of a ventilated disc, a single-piston floating caliper, two pads, an anchor bracket, two guidance pins, a wheel hub and a knuckle, with the pad having no chamfer and having two parallel slots which is named as Pad 1, as shown in Fig. 2. No damping insulator is applied in the brake. The disc brake was tested on a brake dynamometer in accordance with SAE J2521 standard and Fig. 3 shows the results. It can be seen that there were plenty of squeal frequencies, at which the noise levels were above 70dB (A). The following frequencies were more critical, 4050, 8800, 10400, and 13200Hz, at which both the maximum noise levels and occurrence frequencies are high. In order to improve the squeal performance of the disc brake, its pad design was changed. A chamfer was made at each end of the pad and the two parallel slots (as shown in Figure 2) were eliminated, with the improved pad or Pad 2 shown in Fig. 4. And the damping insulators were applied in the brake. The disc brake with the improved pads above was also tested on the brake dynamometer in accordance with SAE J2521 standard and Fig. 5 shows the results. It can be seen that the noise levels were comparatively high at 2700 and 4000Hz, but the maximum noise levels were significantly lower than those shown in Fig. 3. And the number of the squeal frequencies at which the noise levels exceeded 70dB (A) and their occurrence frequencies were markedly reduced. Thus the above design changes of pad significantly improved the squeal performance of the disc brake. The squeal frequencies of the disc brake with the different pad designs above will be predicted by combined use of Altair HyperWorks 8.0 and ANSYS Figure 1. 3D numerical model of the disc brake studied Figure 2. Finite element model of the pad without chamfers and with two parallel slots or Pad 1 95

3 Figure 3. Dynamometer test results of the disc brake with the pads without chamfers and with two parallel slots or Pad 1 Figure 4. Finite element model of the pad with chamfers and without slots or Pad 2 Figure 5. Dynamometer test results of the disc brake with the pads with chamfers and without slots or Pad 2 Figure 6. Anchor bracket in its test of experimental modes analysis 96

4 Table 1. Properties of the materials of the disc brake components Component Modulus Elasticity (MPa) Poisson Ratio Density (t/mm 3 ) Pad back-plate* 2.0e e-09 Pad friction material 1.15e e-09 Caliper** 1.55e e-09 Anchor bracket** 1.6e e-09 Disc*** 1.1e e-09 Piston* 2.0e e-09 Guidance pin* 2.0e e-09 Wheel hub* 2.0e e-09 *The material of the components is steel. **The material of the components is spheroidal graphite cast iron. ***The material of the component is gray cast iron. Figure 7. Finite element model of Pad 2 in ANSYS Figure 8. Finite element model of the caliper in ANSYS Figure 9. Finite element model of the anchor bracket in ANSYS Figure 10. Finite element model of the disc in ANSYS Table 2. Pad mode frequencies obtained with ANSYS and EMA* Modal frequency(hz) Mode No. ANSYS EMA* Error (%) * EMA is an initial word of Experimental Modes Analysis. 97

5 Table 3. Caliper mode frequencies obtained with ANSYS and EMA Modal frequency(hz) Mode No. ANSYS EMA* Error (%) * EMA is an initial word of Experimental Modes Analysis. Table 4. Anchor bracket mode frequencies obtained with ANSYS and EMA Modal frequency(hz) Mode No. ANSYS EMA* Error (%) * EMA is an initial word of Experimental Modes Analysis. Table 5. Disc mode frequencies obtained with ANSYS and EMA Modal frequency(hz) Mode No. ANSYS EMA* Error (%) * EMA is an initial word of Experimental Modes Analysis. 3. Finite Element Model of Disc Brake The HyperMesh module of Altaire HyperWorks 8.0 was applied to construct the finite element models of the disc, pads, caliper, piston, guidance pins and anchor bracket of the disc brake and the wheel hub all with the solid elements of tetra4 type or SOLID45 type in ANSYS whose dimensions were 98

6 between 2 and 4mm. The models were exported to ANSYS *.cdb format files. The *.cdb format files were then read, the models were reconstructed and finally, the normal modes analysis of each of them in free-free state was performed in ANSYS The normal mode analysis results of the disc, pad, caliper and anchor bracket were compared with the corresponding experimental modes analysis results to determine the appropriate modulus of elasticity of each component material. Fig. 6 shows the anchor bracket in its test of experimental modes analysis. Table 1 shows the properties of the materials of the components of the disc brake. Figs. 7, 8, 9 and 10 show the finite element models of the pad, caliper, anchor bracket and disc, respectively. Tables 2, 3, 4 and 5 show their modal frequencies obtained with ANSYS and experimental modes analysis, respectively, with the frequency error calculated with the following formula: frequency calculated frequency measured Error frequecy measured (1) It can be seen that the frequency errors are between 0.02% and 8.02%. Thus the finite element models of the components are reasonable and can be used to construct the disc brake finite element model for performing complex modes analysis to predict the brake squeal frequencies. 4. Disc brake finite element models for performing complex modes analysis The finite element models of the disc, pads, caliper, piston, guidance pins, anchor bracket and the wheel hub were assembled in HyerMesh to form the disc brake models for performing complex modes analysis to predict the brake squeal frequencies. Fig. 11 shows the disc brake model with the pads with the chamfers and without slots or Pad 2 (as shown in Fig. 4), in which the pads, caliper, piston, guidance pins, anchor bracket were linked together with the common nodes in their corresponding interfaces and all the translational and rotational degrees of freedom of all the anchor bracket nodes in the contact areas with the knuckle were restrained, as shown in Fig. 11(a). The disc and wheel hub were joined together with the common nodes in their interfaces. A virtual bearing made of steel was constructed to support the wheel hub, with X, Y and Z translational degrees of freedom of one node and X, Z translational degrees of freedom of another node on the disc rotation axis restrained, as shown in Fig. 11(a) and (b). Thus the assembly of the disc, wheel hub and the virtual bearing could rotate about the disc rotation axis. 3D contact elements were constructed to simulate the surface-to-surface contact between the disc and pads, with the contact elements CONTA173 and TARGE170 used in the pad-disc contact couples, as shown in Fig. 11. The contact elements constructed had the ANSYS contact element defaults except for the friction coefficient of 0.38 which is a generally accepted nominal value of friction coefficient in calculating brake performance. The disc brake model above is named as Model A. Another disc brake model or Model B was also constructed, which was basically the same as Model A except for the pads, with Pad 2 (as shown in Fig. 4) replaced with Pad 1 (as shown in Fig. 2). Then the disc brake models above were exported to ANSYS *.cdb format files. And the obtained *.cdb format files were read, the models were reconstructed and finally, the complex modes analysis of each of them was performed in ANSYS Figure 11. Disc brake model (Model A) with the pads with the chamfers and without the parallel slots or Pad 2 in HyperMesh 99

7 5. Performing complex modes analysis of the disc brakes with ANSYS Complex modes analysis of Model B Linear Non-prestressed Modal Analysis or method (2) of ANSYS mentioned above was applied to perform complex modes analysis of the disc brake models reconstructed in ANSYS Model B or the disc brake model with Pad 1s (as shown in Fig. 2) was analyzed first. Fig. 12 shows the model in ANSYS. in Fig. 12) and hit the key enter, the complex modes analysis of the brake model began to be performed. Table 6 and Fig. 13 show the obtained complex modes frequencies whose real parts are positive, at which the brake squeal may occur, and the brake dynamometer test results, which indicates that the complex modes analysis results are not fully satisfactory with many squeal frequencies not predicted, particularly the critical frequency Hz. Thus, it is necessary to study how to obtain better complex modes analysis results with ANSYS method (2). Table 8. Model B complex modes frequencies whose real parts are positive (Pad-disc friction coefficient is 0.38) and the dynamometer Figure 12. Disc brake model with Pad 1s or Model B in ANSYS The following are the ANSYS commands: /solu $ antype,static $ nropt,unsym $ ematwrite,yes $ /solu $ cmrotate,disc_v,0,-3,0 $ cmrotate,shaft_v,0,-3,0 $ cmrotate,bearing_v,0,-3,0 $ nsubst, 1,1,1 $ psolve,elform,cndi $ finish $ /solu $ antype,modal $ modopt,qrdamp,200 $ mxpand,200 $ psolve,eigqrda $ psolve,eigexp $ finish $ where the command cmrotate,disc_v,0,-3,0 is for determining the component(disc_v) containing the elements of disc in the disc brake finite element model, the rotation axis and angular speed of the disc with (0,-3,0) indicating that the disc rotation axis is Y axis and the angular speed is -3 which indicates that the wheel rotates forwards. The command cmrotate,shaft_v,0,-3,0 and cmrotate,bearing_v,0,-3,0 are for determining the components (shaft_v and bearing_v) containing the elements of wheel hub and virtual bearing, respectively, the rotation axis and angular speed of them. Thus, the disc, wheel hub and virtual bearing had the same rotation axis and angular speed. Having the commands above pasted to the command input area in the ANSYS GUI (Graphical User Interface, as shown squeal frequencies Complex modes frequency Mode No. Real Part Frequency (Hz) Dyno* squeal frequency (Hz) Error (%) (critical) (critical) (critical) (critical) (critical) *Dynamometer 100

8 Figure 13. Model B complex modes frequencies whose real parts are positive (Pad-disc friction coefficient is 0.38) and the dynamometer test results 5.2. Effect of disc angular speed on the complex modes analysis results The effect of disc angular speed on the complex modes analysis results was studied by using different disc angular speeds about its rotation axis, which were -1, -3, -6, -12 and - 20, respectively, in the corresponding ANSYS command with the wheel hub and virtual bearing always having the same rotation axis and angular speed as the disc. And it was found that such changes in the angular speed had no influence on the complex modes analysis results Effect of the friction coefficient between the disc and pad As introduced in Brake Squeal Analysis of Structural Analysis Guide in ANSYS 12.0 Help, the static contact analysis for establishing the static steady sliding position (or initial condition) is not performed in ANSYS method (2) and the initial condition is completely determined by the original state of the brake model constructed. And the forces applied to the piston and caliper by the brake hydraulic pressure in the caliper cylinder are not taken into account, which press the pads against the disc. However, it is generally accepted that the disc brake squeal is the result of friction-induced vibration or self-excited vibration and the friction forces between the disc and pads play a critical role. And the friction force is determined by the normal pressure force and friction coefficient. The forces applied to the piston and caliper are the pressure forces which are not taken into account and have no effect on the results of complex modes analysis in ANSYS method (2) due to its arithmetic. Thus, it is thought that changing the friction coefficient of the pad-disc interface is the only way to adjust the friction forces between the disc and pads, which may have an effect on the complex modes analysis results. Table 7 through Table 12, Fig. 14 and Fig. 15 show Model B complex modes frequencies whose real parts are positive for different friction coefficients between the disc and pads and the dynamometer test results, indicating that the complex modes frequencies for the pad-disc friction coefficients of 0.50, 0.60, 0.65, 0.70 and 0.80 were all quite close to the corresponding dynamometer squeal frequencies, with most dynamometer squeal frequencies having their corresponding complex frequencies with positive real part and all the critical dynamometer squeal frequencies (4050, 8800, 10400, and Hz) having their corresponding complex frequencies with positive real part. However, it should be pointed out that the friction coefficients above may not be the real ones between the disc and pads but represent the combined effect of the normal pressure forces applied and the real friction coefficient. It can also be thought that the dynamometer test results were used to calibrate the disc brake finite element model by finding an appropriate friction coefficient between the disc and pads for making the complex modes analysis results as close as possible to the dynamometer test squeal frequencies. In such a way, a more accurate model could be obtained, based on which the improvements on the disc brake design could be sought to eliminate the critical squeal frequencies. Thus, according to the results above with Model B, friction coefficient 0.65, the median of the good friction coefficients 0.50, 0.60, 0.65, 0.70 and 0.80, could be taken as the nominal friction coefficient, based on which the improvements on the disc brake design could be sought to eliminate the critical squeal frequencies. Table 7. Model B complex modes frequencies whose real parts are positive (Pad-disc friction coefficient is 0.50) and the dynamometer squeal Mode no. frequencies Complex modes frequency Real part Frequency Dyno* (Hz) frequency squeal Error (%) (Hz) (critical) (critical) (critical) (critical) (critical)

9 Table 8. Model B complex modes frequencies whose real parts are positive (Pad-disc friction coefficient is 0.60) and the dynamometer squeal frequencies Complex modes frequency Mode No. Real part Frequency (Hz) Dyno* squeal frequency (Hz) Error (%) (critical) (critical) (critical) (critical) (critical) Table 9. Model B complex modes frequencies whose real parts are positive (Pad-disc friction coefficient is 0.65) and the dynamometer squeal frequencies Complex modes frequency Mode no. Real part Frequency (Hz) Dyno* squeal frequency (Hz) Error (%) (critical) (critical) (critical) (critical) (critical)

10 Table 10. Model B complex modes frequencies whose real parts are positive (Pad-disc friction coefficient is 0.70) and the dynamometer squeal frequencies Complex modes frequency Mode No. Real part Frequency (Hz) Dyno* squeal frequency (Hz) Error (%) (critical) (critical) (critical) (critical) (critical) Table 11. Model B complex modes frequencies whose real parts are positive (Pad-disc friction coefficient is 0.80) and the dynamometer squeal frequencies Complex modes frequency Mode No. Real part Frequency (Hz) Dyno* squeal frequency (Hz) Error (%) (critical) (critical) (critical) (critical) (critical)

11 Table 12. Model B complex modes frequencies whose real parts are positive (Pad-disc friction coefficients are 0.38, 0.50, 0.60, 0.65, 0.70 and 0.80, respectively) and the dynamometer squeal frequencies Friction coefficient Dyno* squeal frequency (Hz) Complex modes frequency with positive real part (Hz) (critical) (critical) (critical) (critical) (critical) Figure 14. Model B complex modes frequencies whose real parts are positive (Pad-disc friction coefficients are 0.38, 0.50 and 0.60, respectively) and the dynamometer test results Figure 15. Model B complex modes frequencies whose real parts are positive (Pad-disc friction coefficients are 0.65, 0.70 and 0.80, respectively) and the dynamometer test results 104

12 Table 13. Model A complex modes frequencies whose real parts are positive (Pad-disc friction coefficient is 0.87 equivalent to 0.65 in Model B) and the dynamometer squeal frequencies Complex modes frequency Mode No. Real part Frequency (Hz) Dyno* squeal frequency (Hz) Error (%) (critical) (critical) Figure 16. Brake dynamometer squeal frequencies and Model A complex modes frequencies whose real parts are positive with the pad-disc friction coefficient being 0.87 which is equivalent to 0.65 in Model B Figure th order mode shape ( Hz unstable mode) 105

13 Figure th order mode shape ( Hz unstable mode) Table 14. Model A complex modes frequencies whose real parts are positive (Pad-disc friction coefficient is 0.80 equivalent to 0.60 in Model B) and the dynamometer squeal frequencies Complex modes frequency Mode no Real part Frequency (Hz) Dyno* squeal frequency (Hz) Error (%) (critical) (critical) Figure 19. Brake dynamometer squeal frequencies and Model A complex modes frequencies whose real parts are positive with the pad-disc friction coefficients being 0.80, 0.87 and 0.93, respectively. 106

14 Figure 20. Finite element model of Pad 3 with the slots filled Table 15. Model A complex modes frequencies whose real parts are positive (Pad-disc friction coefficient is 0.93 equivalent to 0.70 in Model B) and the dynamometer squeal frequencies Complex modes frequency Mode No. Real part Frequency (Hz) Dyno* squeal frequency (Hz) Error (%) (critical) (critical) Table 17. Normal modes analysis results of Pad 1 and Pad 2 Normal modes frequency(hz) Mode No. Pad 2 Pad 1 Difference in normal modes frequency(hz)

15 Table 18. Normal modes analysis results of different pads* Normal modes frequency(hz) Mode No. Pad 2 Pad 1 Pad 3 Pad 4 Pad 5 Pad Figure 21. Finite element model of Pad 4 or optimized Pad 3 Table 19. Model A, Model C, Model D, Model E and Model F complex modes frequencies whose real parts are positive (Pad-disc friction coefficient is 0.87) and the dynamometer squeal frequencies Complex modes frequency with positive real part (Hz) Model A Model C Model D Model E Model F Model G Dyno* squeal frequency (Hz) (critical) (critical)

16 Figure 22. Redefined design variables of Pad 3 Figure 23. Finite element model of Pad 5, Pad 6 and Pad 2; (a) Pad 5, (b) Simplified Pad 5 or Pad 6, (c) Pad 5 and Pad 6, (d) Pad 2 and Pad Complex modes analysis of Model A Based on the knowledge obtained with Model B, another disc brake model with Pad 2 (as shown in Fig. 4) or Model A was analyzed with ANSYS, in which the friction coefficient 0.65 in the friction interface between the pad and disc is adopted. But the contact area between the disc and one such pad is only about 75% of that for Pad 1. And in order to develop the same brake moment, the same normal pressure forces should be applied for the same real friction coefficient. But, as introduced above, the pressure forces cannot be adjusted and the only way to alter the friction forces is to change the friction coefficient in ANSYS method (2). And it is assumed that the pressure in the disc-pad interface is kept constant in ANSYS method (2). Thus the product of the disc-pad contact area and friction coefficient should be the same to develop the same brake moment when the disc-pad contact areas are different. So the friction coefficient between the disc and pad for Model A should be 0.65/0.75=0.87. Table 13 and Figure 16 show Model A complex modes frequencies whose real parts are positive for the friction coefficient 0.87 between the disc and pad and the dynamometer test results, indicating that the critical dynamometer squeal frequencies(2700 and 4000 Hz) having their corresponding complex frequencies with positive real parts. Thus, the complex modes analysis results obtained with Model A were satisfactory. Fig. 17 and Fig. 18 show the mode shapes of two predicted unstable complex frequencies with Hz (19 th order) and Hz (26 th order) corresponding to the critical dynamometer squeal frequencies 2700Hz and 4000Hz, respectively. In order to check the analysis robustness to the variation in friction coefficient, different friction coefficients were used to perform the complex modes analysis of Model A. Table 13 through Table 16 and Fig. 19 show the Model A complex modes frequencies whose real parts are positive for the friction coefficients 0.87(equivalent to 0.65 in Model B), 0.80 (equivalent to 0.6 in Model B) and 0.93 (equivalent to 0.70 in Model B) between the disc and pad, respectively, and the dynamometer test results, indicating that the critical dynamometer squeal frequencies (2700 and 4000 Hz) all had their corresponding complex frequencies with positive real part. Thus, the complex modes analysis results obtained with Model A for the above friction coefficients were satisfactory and the analysis method was robust. Therefore, the analysis method above is practical and effective. 109

17 6. Optimization of the Pad As indicated above, the squeal performance of the disc brake studied was significantly improved only by modification of the geometry shape of the pad friction material, which may be the easiest, fastest and cheapest way to reach the goal of improvement. Thus the optimization of the disc brake pad was studied. Some researchers reported different ways to modify the pad geometry shapes to restrain the disc brake squeal (Liu et al., 2006; Stanef et al., 2006). Table 17 shows the normal modes analysis results of Pad 2 (as shown in Fig. 4) and Pad 1 (as shown in Figure 2), which were obtained with OptiStruct module of Altair HyperWorks 8.0. As mentioned above, the disc brake with Pad 2 had a significantly better squeal performance than that with Pad 1. And it can be seen in Table 17 that each normal mode frequency of Pad 2 is higher than the corresponding one of Pad 1, with the maximum difference being Hz and mean difference being about 480Hz, which indicates that increasing each mode frequency of the pad may be one direction of performing its optimization. In order to increase each mode frequency of the Pad 1, the slots in its friction material were filled with the same pad friction material, obtaining Pad 3 (as shown in Fig. 20). Table 18 shows its normal modes analysis results. And it can be seen that each mode frequency of Pad 3 is lower than the corresponding one of Pad 2 from normal mode order 7 to order 13. Thus Pad 3 should be optimized to increase its normal modes frequencies of above orders. * Pad 2 is as shown in Fig. 4; Pad 1 is as shown in Fig. 2; Pad 3 is as shown in Fig. 20; Pad 4 is the optimized Pad 3 (as shown in Fig. 21); Pad 5 is the newly optimized Pad 3 (as shown in Fig. 23); Pad 6 is the simplified Pad 5 (as shown in Fig. 24). The topology optimization of Pad 3 was performed with the OptiStruct module of Altair HyperWorks 8.0, with the optimization setup as follows: (1) Design variables and draw direction The design variables included all the elements of the pad friction material. The design variables and draw direction are shown in Fig. 20. (2) Defining constraints The defined constraints included: 7 th order frequency 4000Hz; 8 th order frequency 5730Hz; 9 th order frequency 8514Hz; 10 th order frequency 10630Hz; 11 th order frequency 11073Hz; 12 th order frequency 14206Hz; 13 th order frequency 15714Hz. That is, the lower limit for each modes frequency is the corresponding modes frequency added by about 600Hz. (3) Defining the objective function The defined objective function is minimizing the volume of the design variables. The topology optimization problem above was solved with OptiStruct module. And the optimized Pad 3 or Pad 4 is shown in Figure 21 and its normal modes analysis results shown in Table 18, in which it can be seen that only the modes frequencies of modes 12, 13 and 14 of Pad 4 are lower than the corresponding ones of Pad 2, with the frequency differences being Hz, Hz and 90.89Hz, respectively. It seems that only the 12 th order modes frequency of Pad 4 is less desirable. Thus, generally speaking, the topology optimization reached its goals A new brake finite element model or Model C for complex modes analysis was constructed, which was based on Model B with the Pad 1s (as shown in Fig. 2) replaced with the Pad 4s (as shown in Fig. 21), with the contact elements newly constructed having the friction coefficient being 0.87 because the disc-pad contact area of Pad 4 was close to that of Pad 2(as shown in Fig. 4). Table 19 shows the complex modes frequencies with positive real part of Model A and Model C, which were obtained with ANSYS. And it can be seen that Model C had similar squeal performance to Model A, which indicates that Pad 4 had similar squeal performance to Pad 2. It can be seen in Fig. 21 that the geometric shape of Pad 4 is complicated which is hard to be manufactured. In order to obtaining the optimized pad of comparatively simple shape, the topology optimization setup of Pad 3 above was changed, in which only the design variables were changed which merely included the elements in the end parts and the filled slots of the pad friction material, as shown in Fig. 22. And this topology optimization problem was solved with OptiStruct module, with the newly optimized Pad 3 or Pad 5 shown in Fig. 23(a) and its normal modes analysis results shown in Table 18. Based on Pad 5, Pad 6 was obtained with four planes replacing the complicated curves of Pad 6, as shown in Fig. 23(b) and (c). And Figure 23(d) shows the difference between Pad 2 and Pad 6. As shown in Table 18, all the modes frequencies of Pad 6 are higher than the corresponding ones of Pad 2 with the maximum difference being 80.33Hz. Thus, Pad 6 had quite similar dynamic behaviors to Pad 2 and was desirable. A new brake finite element model or Model D for complex modes analysis was constructed, which was based on Model B with the Pad 1s (as shown in Fig. 2) replaced with the Pad 6s (as shown in Fig. 23(b)), with the contact elements newly constructed having the friction coefficient being 0.87 because the disc-pad contact area of Pad 4 was close to that of Pad 2 (as shown in Fig. 4). Table 19 shows the complex modes frequencies with positive real part of Model D, which were obtained with ANSYS. And it can be seen that Model D had similar squeal performance to Model A, which indicates that Pad 6 had similar squeal performance to Pad Normal Modes Analysis of the Disc Bake As shown in Fig. 5, 2700Hz and 4000Hz were critical squeal frequencies with Pad 2 (as shown in Fig. 4) applied in 110

18 the disc brake, which were dynamometer test results in accordance with SAE J2521 standard. As shown in Table 18, Hz and Hz, which were obtained with Model A and Model C, respectively, were corresponding to 2700Hz; Hz and Hz, which were obtained with Model A and Model C, respectively, were corresponding to 4000Hz. And it was found that it was difficult to eliminate those two critical squeal frequencies only by optimizing the shape of the pad. Thus, it was necessary to find other ways to eliminate them. The normal modes analysis of Model C was performed with OptiStruct module for seeking some clues to eliminate the critical squeal frequencies, in which the contact elements were not recognized and used, having no effect on the results. Thus, there is no interaction between the disc and two pads. Table 20 shows the normal modes frequencies of Model C in the vicinity of 2700 and 4000Hz. Fig. 24 and Fig. 25 show the corresponding modes, respectively. According to Fig. 24, it was thought that the interaction between 22 nd order mode ( Hz) and two disc modes(23 rd and 24 th order modes) might be the main reason of the unstable mode of Model C whose squeal frequency was Hz corresponding to 2700Hz in the dynamometer test. According to Fig. 25, it was thought that the interaction between 32 nd order mode ( Hz) and two disc modes(30 th and 31 st order modes) might be the main reason of the unstable mode of Model C whose squeal frequency was Hz corresponding to 4000Hz in the dynamometer test. As shown in Fig. 24(b) and Fig. 25(d), 22 nd and 32 nd order modes all included the caliper ear part bending. Thus these two critical squeal frequencies might be eliminated by controlling the caliper ear part bending or correcting the modes of the caliper. 8. Optimization of the Caliper In order to control the caliper ear part bending, two blocks made of the same material as the caliper were added to the caliper ear parts in Model C, as shown in Fig. 26, obtaining a new model or Model E. Table 20 also shows the normal modes analysis results of Model E obtained with OptiStruct module. It can be seen that 22 nd order mode whose frequency is Hz and 32 nd order mode whose frequency is Hz of Model C have been eliminated. Table 19 also shows the complex modes analysis results of Model E obtained with ANSYS with the two unstable critical modes corresponding to 2700 and 4000Hz obtained with the dynamometer were eliminated. Thus this modification of the caliper shape was quite effective in eliminating the two critical unstable modes. In order to reduce the weight of the blocks added to the ear parts of the caliper, the topology optimization of them was performed with OptiStruct module of Altair HyperWorks 8.0, with the optimization setup as follows: (1) Design variables and draw direction The design variables included the elements of the blocks added to the caliper ear parts. The design variables and draw direction are shown in Fig. 28. (2) Defining constraints The defined constraints included: 24 th order frequency 3217Hz; 32 nd order frequency 4505Hz; The objective of defining such constraints is to increase the 24 th and 32 nd order frequencies which are all caliper mode frequencies, making them sufficiently higher than the disc mode frequencies nearest to them or Hz and Hz, respectively. (3) Defining the objective function The defined objective function is minimizing the volume of the design variables. The topology optimization problem was solved with OptiStruct module. And Fig. 29 shows Model F or Model E with the optimized blocks added to the caliper. Table 19 also shows its complex modes analysis results obtained with ANSYS, which indicates that the optimization had little effects on the complex modes analysis results and only contributed to the lightweight of the disc brake. It can be seen in Fig. 29 that the geometric shapes of the optimized blocks added to the caliper are complicated which are hard to be manufactured. Based on the above blocks, simplified blocks were obtained with one plane replacing the complicated curves of each optimized block, as shown in Fig. 30. Based on Model F, Model G was constructed with the simplified blocks replacing the optimized blocks in Model F, as shown in Fig. 30(e). Table 19 shows its complex modes analysis results obtained with ANSYS, which indicates that Model G almost had the same unstable complex modes frequencies as Model F except for only one mode frequency(15159hz) and thus the way for simplifying the optimized blocks is reasonable. The simplified blocks were also added to the calipers in Model A with Pad 2 and Model D with Pad 6, respectively, obtaining Model H and Model I. Table 21 shows the complex modes analysis results of Model A, Model H and Model I obtained with ANSYS, which indicates that the unstable mode frequencies corresponding to the critical squeal frequency 2700 and 4000Hz obtained with the dynamometer were eliminated in Model H and Model I, with these two models better than Model A in this sense. But some new unstable modes were introduced in Model H and Model I, whether they cause new squeal problems or not should be verified with appropriate tests. 111

19 Table 20. Normal modes frequencies of Model C and Model E in the vicinities of 2700 and 4000Hz Model C Model E Mode no. Frequency (Hz) Mode no. Frequency (Hz) * * * * * * * * * * * * * * *Disc modes Figure 24. Model C normal modes in the vicinity of the critical squeal frequency 2700Hz; (a) 21 st order mode ( Hz), (b) 22 nd order mode ( Hz), (c) 23 rd order mode ( Hz,disc mode), (d) 24 th order mode ( Hz,disc mode), (e) 25 th order mode ( Hz,disc mode), (f) 26 th order mode ( Hz,disc mode), (g) 27 th order mode ( Hz) Figure 26. Model E with the blocks added to the ear parts of the caliper 112

20 Figure 27. Wheel and brake assembly Figure 28. Design variables and draw direction of Model E Figure 29. Model F 113

21 Figure 30. Optimized and simplified blocks added to the caliper Table 21. Model A, Model H and Model I complex modes frequencies whose real parts are positive (Pad-disc friction coefficient is 0.87) and the dynamometer squeal frequencies Complex modes frequency with positive real part Model A Model H Model I Dyno* squeal frequency (Hz) Real part Frequency(Hz) Real part Frequecy(Hz) Real part Frequency(Hz) (critical) (critical)

22 9. Conclusion (1) The brake squeal frequencies of the disc brake studied can be effectively predicted by combined use of HyperMesh of Altair HyperWorks 8.0 and Linear Non-prestressed Modal Analysis method (method (2)) of ANSYS (2) The complex modes analysis results obtained with Linear Non-prestressed Modal Analysis method of ANSYS 12.0 can be improved by selecting an appropriate friction coefficient between the disc and pads, making the predicted brake squeal frequencies, particularly the critical squeal frequencies, better tally with the measured squeal frequencies with the brake dynamometer. In such a way, the dynamometer test results are used to calibrate the disc brake finite element model and a more accurate model can be obtained, based on which the improvements on the disc brake design can be sought to eliminate the critical squeal frequencies. (3) For the disc brake studied, raising each mode frequency of its pad or Pad 3 was effective in improving the squeal performance. The topology optimizations of Pad 3 performed to raise its modes frequencies resulted in Pad 4, Pad 5 and Pad 6 which were similar to Pad 2 in improving the disc brake squeal performance. (4) For the disc brake, the normal modes analysis of the disc brake finite element model for complex modes analysis was effective in revealing the clue for improving the caliper to eliminate the critical squeal frequencies corresponding to 2700 and 4000Hz, respectively, based on which two blocks made of the same material as the caliper were added to the caliper ear parts. The modification of the caliper geometric shape was proved effective in eliminating the two critical squeal frequencies above by the analysis performed. And the topology optimization of the two blocks only contributed to the lightweight of them and had little effect on the squeal performance. (5) Some new unstable modes were introduced in Model H and Model I, whether they cause new squeal problems or not should be verified with appropriate tests. On Automotive Disc Brake Squeal Part II: Simulation and Analysis. SAE Paper No [4] Stanef, D., Papinniemi, A. and Zhao J. (2006). From Prototype to Production The Practical Nature of Brake Squeal Noise. SAE Paper No [5] Hou, J., Guo, X. and Tan, G. (2009). Complex Mode Analysis on Disc Brake Squeal and Design Improvement. SAE Paper No [6] Jaber, N., Wang, A., Yue, Y. and Ralph Allgaier, R. (2006). Brake Noise Study (Part II) High Frequency Squeal. SAE Paper No [7] Liu, W., Vyletel, G. and Jerry Li, J. (2006). A Rapid Design Tool and Methodology for Reducing High Frequency Brake Squeal. SAE Paper No [8] Yokoyama, T., Matsushima, T., Matsui, N. and Misumi, R. (2012). A Study of Reduction for Brake Squeal in Disc InPlane Mode. SAE Paper No [9] Nelagadde, M. and Smith, E. (2009). Optimization and Sensitivity Analysis of Brake Rotor Frequencies. SAE Paper No Acknowledgements The authors would like to thank Prof. Dihua Guan for her contribution to this study in performing the experimental modes analysis of the disc brake components and providing the results. References [1] Yue, Y., Allgaier, R., Jaber, N. and Wang, A. (2006). Brake Noise Study (Part I) - Low Frequency Squeal. SAE Paper No [2] Chen, F., Quaglia, R. and Tan, C. (2003). On Automotive Disc Brake Squeal Part I: Mechanisms and Causes. SAE Paper No [3] Ouyang, H., Nack, W., Yuan, Y. and Frank Chen, F. (2003). 115