Insight into brake squeal source mechanism considering kinematic nonlinearity and pad-disc separation Osman Taha Sen a Department of Mechanical Engineering, Istanbul Technical University, Istanbul, 34469, Turkey Rajendra Singh b Acoustics and Dynamics Laboratory, NSF Smart Vehicle Concepts Center, The Ohio State University, Columbus, Ohio, 4320, USA The chief goal of this study is to propose an improved mathematical model for the brake squeal phenomenon, a dynamic instability problem seen in vehicle brake systems. Accordingly, a three degree of freedom model of the brake pad is developed that is initially in contact with the brake disc. In this model, the pad has two translational and one rotational degrees of freedom and unlike prior formulations, its leading and trailing edges are defined. The governing equations include kinematic nonlinearities arising from the arrangement of the braking force vectors and separation effects between the pad and the disc. These equations are numerically solved for a few braking force vector arrangements, and the resulting dynamic responses are compared for several cases. The importance of pad-disc separation is better understood with the proposed model. In conclusion, an improved insight for the brake squeal source mechanisms is obtained while overcoming the limitation of prior models. INTRODUCTION Brake squeal is a friction-induced high frequency noise problem that is observed in many automotive brake systems, 2. Minimal order models are often employed since they reveal the physics of the source mechanisms which include stick-slip vibrations due to negative damping 3 ; sprag-slip phenomenon 4 ; self-excited vibrations with constant friction coefficient 5 ; mode coupling/splitting phenomenon 6 ; and hammering 7. In particular, Hamabe et al. 8 and Hoffmann et al. 9 studied the effect of brake force angular arrangements on brake squeal noise generation with a two degree of freedom model. However, they ignored the kinematic nonlinearity that arises due to these brake force arrangements in their model. Furthermore, brake pad and disc were assumed to be in contact at all times in such studies. Sen et al. 0 extended two prior models 8, 9 by a email: senos@itu.edu.tr b email: singh.3@osu.edu
overcoming the aforementioned oversimplification and studied the problem with a two degree of freedom formulation. Specifically Sen et al. 0 illustrated that the brake pad and disc can lose contact at some brake force angular arrangements; therefore the previously conducted stability analysis in which the contact loss nonlinearity is ignored may not be accurate. The chief goal of the current paper is to develop an improved brake squeal source model in which both kinematic nonlinearities and contact loss effects are simultaneously considered. In addition, the leading and trailing edge dynamics of the brake pad are also taken into account. 2 IMPROVED BRAKE SQUEAL SOURCE MODEL WITH LEADING/TRAILING EDGES The main objectives of the current study are as follows: Develop an improved nonlinear model of the squeal source model by considering kinematic nonlinearities and pad leading/trailing edge dynamics; 2 Numerically calculate the steady-state response of the nonlinear model for different brake force angular arrangements; and 3 Compare the results of new model with a prior model 0 in order to highlight the importance of leading/trailing edge dynamics of the brake pad. The three degree of freedom model for the brake squeal investigation is proposed in Fig.. Unlike the prior models 8-0, the brake pad has two translations x and y and one rotation θ; hence, the pad is no longer just a particle but an inertial element with geometric dimensions 2h and 2w. As seen in Fig., the brake pad m, I is free to translate and rotate about the E x -E y plane, and attached to the common ground with two linear springs k and k 2 at two arbitrary angles α and α 2. Furthermore, the pad is positioned over a brake disc that translates at a constant speed V. The contact between the pad and the disc are defined with two point contact elements with linear springs at leading k 3 and trailing k 4 edges of the pad. 2w k k 2 α y α 2 2h θ m, I x E y k 4 k 3 O 0,0 E x V Fig. Improved three degree of freedom brake squeal source model The governing equations of the system of Fig. are derived by defining the elastic force vectors that act upon the pad. First, the position vectors of the fixed r f,i and free r i ends of each spring are defined. For the sake of completeness, corresponding position vectors only for the spring k are as follows where x 0, y 0, θ 0 corresponds to the position of the pad at static equilibrium, and x, y, θ is the motion of the pad with respect to the static equilibrium:
r f = x0 d cos β θ 0 + L cos α + θ 0 Ex + y0 + d sin β θ 0 + L sin α + θ 0 E y, a r = x + x d cos β θ θ Ex + y + y + d sin β θ θ E y. b Furthermore d = w2 + h2 and β = sin h d. Second, the elastic force vectors Fs,i for each spring are obtained from Hooke s law as: ri r f,i Fs,i = ki ri r f,i Li, ri r f,i i =, 2, 3, 4. 2 Note that Fs, and Fs,2 have components in both Ex and E y directions; however, the contact forces Fs,3 and Fs,4 have only single components in E y direction. Note that the conditions Fs,3 > 0 and Fs,4 > 0 must always be satisfied, since k3 and k4 represent the contact springs and both cannot be extended beyond their free lengths L3 and L4. Observe that the contact forces also generate friction forces that are in Ex direction. The dry friction model is assumed as follows where Vrel = V y. µ Vrel = µ sign Vrel. 3 Finally, the governing equations of the system given of Fig. are obtained as: m x = Fs,x + Fs,2 x + µ Vrel h3 x,θ, L3 Fs,3 y + µ Vrel h4 x,θ, L4 Fs,4 y, 4a m y = Fs,y + Fs,2 y + h3 x,θ, L3 Fs,3 y + h4 x,θ, L4 Fs,4 y, 4b Iθ = Fs,x d sin β θ 0 θ Fs,y d cos β θ 0 θ Fs,2 x d sin β + θ 0 + θ + Fs,2 y d cos β + θ 0 + θ h3 x,θ, L3 Fs,3 y d cos β + θ 0 + θ + µ Vrel h3 x,θ, L3 Fs,3 y d sin β + θ 0 + θ, 4c +h4 x,θ, L4 Fs,4 y d cos β θ 0 θ + µ Vrel h4 x,θ, L4 Fs,4 y d sin β θ 0 θ where hj x, θ, Lj is a piecewise linear function and j = 3, 4. This piecewise linear function that defines the contact loss between the brake pad and disc is given as: h j x,θ, L j = 0 rj r f, j < L j, rj r f, j L j j = 3, 4. 5 Since hj x, θ, Lj is a discontinuous function, the governing equations given by Eqn. 4 can take four different forms, i.e. the system of Fig. may assume the following four states: a. Leading and trailing edges are both in contact; b. Both leading and trailing edges lose contact; c. Leading edge
is in contact, but the trailing edge is not; and d. Leading edge is not in contact while the trailing edge is in contact. 3 RESULTS AND DISCUSSION Governing equations given by Eqn. 4 form a set of coupled ordinary nonlinear differential equations. Nonlinearities in these equations arise from: Arrangements of the springs k and k 2 ; 2 Surface separation at contact interfaces h j x, θ, L j ; and 3 Dry friction model. However, the friction term can be dropped for the sake of simplicity since V rel > 0 at all times. Furthermore, the piecewise linear function h j x, θ, L j is also smoothened with a continuous function as given below where σ is the regularizing factor often a large number. h j tanh σ r x,θ, L j j r f, j L j = 2 j = 3, 4. 6 To simulate the actuation pressure acting on the brake pad, pretensions are defined on springs k 3 and k 4, and the static equilibrium point of the pad is evaluated from the force and moment balance equations. Finally, a cohesive set of four governing equations, Eqn. 4 in which h j x, θ, L j is replaced with Eqn. 6, is numerically solved. The solutions of the system of Fig. are obtained for different α -α 2 parameter sets. The results are given in terms of normalized leading/trailing edge contact forces and F s,4 y where the normalization is done by the total actuation brake force. First, the case where α = 90 and α 2 = 80 is investigated and corresponding normalized contact forces are given in Fig. 2 for both proposed and prior models. As seen in the figure, consecutive impact-like contact forces are developed at the leading and trailing edges of the pad Fig. 2a and Fig. 2b. Furthermore, the contact force at the leading edge is almost as large as the constant actuation force Fig. 2a; however, the contact force generated at the trailing edge is about four times of the actuation force Fig. 2c. In the second case, α = 80 and α 2 = 90 arrangement is solved, and the results are given in Fig. 3. While the trailing edge of the pad is always in contact with the disc Fig. 3b, the leading edge loses contact during the event Fig. 3a. Like the first case, the trailing edge contact force is always greater than the leading edge force. Since these two cases α = 90, α 2 = 80 and α = 80, α 2 = 90 simulate one of the squeal mechanisms called as the hammering 7, the proposed model successfully predicts a squeal initiation event. Moreover, the solutions of the prior model 0 in which the leading/trailing edge dynamics of the pad is ignored, are also shown in Fig. 2c and Fig. 3c for the sake of comparison; observe that no contact loss is observed with the prior model. In addition, the solutions in Fig. 2c and Fig. 3c are exactly the same even though the angular arrangements of the springs k and k 2 are very different. Therefore, it is crucial to include the rotational degree of freedom for the pad in order to have a more accurate representation of the squeal problem.
a c. 0.9 b 0 0 0. 5 4 F s,4y 3 2 0 0 0. Fig. 2 Time histories for contact forces for α = 90 and α 2 = 80. a Leading contact force proposed model; b Trailing contact force F s,4 y proposed model; c Contact force prior model. a c. 0.9 0. b 0 F s,4y 5 5 Fig. 3 Time histories for contact forces for α = 80 and α 2 = 90. a Leading contact force proposed model; b Trailing contact force F s,4 y proposed model; c Contact force prior model.
a c. 0.9 b F s,4y Fig. 4 Time histories for contact forces for α = α 2 = 35. a Leading contact force proposed model; b Trailing contact force F s,4 y proposed model; c Contact force prior model. a 5 c b 5 5 F s,4y 0. 0 Fig. 5 Time histories for contact forces for α = α 2 = 50. a Leading contact force proposed model; b Trailing contact force F s,4 y proposed model; c Contact force prior model.
For the third case with α = α 2 = 35, the results are displayed in Fig. 4. Both edges of the brake pad remain in contact with the disc at all times, though the leading edge contact force Fig. 4a is higher than the trailing edge Fig. 4b unlike the previous cases. Moreover, the solution of the prior model Fig. 4c is again same with the solution of the first two cases. Finally, the problem is solved for the α = α 2 = 50 case, and the contact forces are shown in Fig. 5. Comparing Fig. 4 and Fig. 5, the leading edge contact force Fig. 5a is found to be much higher than the trailing edge contact force Fig. 5b in the case with α = α 2 = 50. While the solution of the prior model 0 for α = α 2 = 50 Fig. 5c represents surface separation, this is not observed in the proposed model for this case. In order to summarize the results, mean and peak-to-peak values of the numerically calculated contact forces are compared in Table for two models. Note that the mean values for and F s,4 y are not given for Case, since these are impact-like forces. Table Numerically calculated mean and peak-to-peak values of the contact forces for the proposed and prior models. Configuration of Fig. Case α = 90 α 2 = 80 Case 2 α = 80 α 2 = 90 Case 3 α = α 2 = 35 Case 4 α = α 2 = 50 Proposed Model Prior Model 0 F s3,y,mean F s3,y,pp F s4,y,mean F s4,y,pp,mean,pp -.09-4.57 7 7 0.9 0 0 0.3 7 7 4 4 6 4 7 7 0.7 5 0.7 0.00 4 CONCLUSION This paper proposes an improved mathematical minimal order model that describes some inherent nonlinear mechanisms of the brake squeal source. First, the nonlinear governing equations are derived where the nonlinearities arise due to the kinematic arrangements, surface separation effects and dry friction. Second, the model is numerically solved for several α and α 2 values, and the new predictions are compared with the solutions of a prior model 0. The comparative results show that the contribution of the rotational degree of freedom of the pad is important. Essentially, ignoring the leading/trailing edge dynamics of the pad may yield misleading result. Finally, the proposed model successfully simulates the hammering phenomenon, which is believed to be one of the sources mechanisms for brake squeal problem 7. 5 ACKNOWLEDGEMENTS The first author acknowledges the Scientific and Technological Research Council of Turkey 300 Starting R&D Projects Funding Program, Project No: 5M002 for supporting this fundamental study. The second author thanks the Smart Vehicle Concepts Center www.smartvehiclecenter.org and the National Science Foundation Industry/University
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