Insight Into Brake Squeal Source Mechanism Considering .

Insight into brake squeal source mechanism considering kinematicnonlinearity and pad-disc separationOsman Taha Sena)Department of Mechanical Engineering, Istanbul Technical University, Istanbul, 34469, TurkeyRajendra Singhb)Acoustics and Dynamics Laboratory, NSF Smart Vehicle Concepts Center, The Ohio StateUniversity, Columbus, Ohio, 43210, USAThe chief goal of this study is to propose an improved mathematical model for the brakesqueal 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 initiallyin contact with the brake disc. In this model, the pad has two translational and onerotational degrees of freedom and unlike prior formulations, its leading and trailing edgesare defined. The governing equations include kinematic nonlinearities (arising from thearrangement of the braking force vectors) and separation effects between the pad and thedisc. These equations are numerically solved for a few braking force vector arrangements,and the resulting dynamic responses are compared for several cases. The importance ofpad-disc separation is better understood with the proposed model. In conclusion, animproved insight for the brake squeal source mechanisms is obtained while overcoming thelimitation of prior models.1INTRODUCTIONBrake squeal is a friction-induced high frequency noise problem that is observed in manyautomotive brake systems1, 2. Minimal order models are often employed since they reveal thephysics of the source mechanisms which include stick-slip vibrations due to negative damping3;sprag-slip phenomenon4; self-excited vibrations with constant friction coefficient5; modecoupling/splitting phenomenon6; and hammering7. In particular, Hamabe et al.8 and Hoffmann etal.9 studied the effect of brake force angular arrangements on brake squeal noise generation witha two degree of freedom model. However, they ignored the kinematic nonlinearity that arises dueto these brake force arrangements in their model. Furthermore, brake pad and disc were assumedto be in contact at all times in such studies. Sen et al.10 extended two prior models8, 9 bya)b)email: senos@itu.edu.tremail: singh.3@osu.edu

overcoming the aforementioned oversimplification and studied the problem with a two degree offreedom formulation. Specifically Sen et al.10 illustrated that the brake pad and disc can losecontact at some brake force angular arrangements; therefore the previously conducted stabilityanalysis in which the contact loss nonlinearity is ignored may not be accurate. The chief goal ofthe current paper is to develop an improved brake squeal source model in which both kinematicnonlinearities and contact loss effects are simultaneously considered. In addition, the leading andtrailing edge dynamics of the brake pad are also taken into account.2 IMPROVED BRAKE SQUEAL SOURCE MODEL WITH LEADING/TRAILINGEDGESThe main objectives of the current study are as follows: 1) Develop an improved nonlinearmodel of the squeal source model by considering kinematic nonlinearities and padleading/trailing edge dynamics; 2) Numerically calculate the steady-state response of thenonlinear model for different brake force angular arrangements; and 3) Compare the results ofnew model with a prior model10 in order to highlight the importance of leading/trailing edgedynamics of the brake pad.The three degree of freedom model for the brake squeal investigation is proposed in Fig. 1.Unlike the prior models8-10, 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 (2hand 2w). As seen in Fig. 1, the brake pad (m, I) is free to translate and rotate about the Ex-Eyplane, and attached to the common ground with two linear springs (k1 and k2) at two arbitraryangles (α1 and α2). Furthermore, the pad is positioned over a brake disc that translates at aconstant speed V. The contact between the pad and the disc are defined with two point contactelements with linear springs at leading (k3) and trailing (k4) edges of the pad.2wk1k2α2yα1xθ2hm, Ik3EyO (0,0)Exk4VFig. 1 –Improved three degree of freedom brake squeal source modelThe governing equations of the system of Fig. 1 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 ( ri ) ends ofeach spring are defined. For the sake of completeness, corresponding position vectors only forthe spring k1 are as follows where (x0, y0, θ0) corresponds to the position of the pad at staticequilibrium, and (x, y, θ) is the motion of the pad with respect to the static equilibrium:

!!!r f 1 x0 d cos ( β θ 0 ) L1 cos (α 1 θ 0 ) Ex y0 d sin ( β θ 0 ) L1 sin (α 1 θ 0 ) E y ,(1a)!!!r1 x1 x d cos ( β θ1 θ ) Ex y1 y d sin ( β θ1 θ ) E y .(1b)()(())()!Furthermore d w2 h2 and β sin 1 ( h d ) . Second, the elastic force vectors ( Fs,i ) for eachspring are obtained from Hooke’s law as:! !!ri r f ,i! !Fs,i ki ri r f ,i Li ! ! ,ri r f ,i(!)!!i 1, 2, 3, 4.(2)!Note that Fs,1 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 beextended beyond theirfree 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. 1 are obtained as:m!!x Fs,1x 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,1y Fs,2 y h3 ( x,θ , L3 ) Fs,3 y h4 ( x,θ , L4 ) Fs,4 y ,(4b)Iθ!! Fs,1x d sin ( β θ 0 θ ) Fs,1y 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 thatdefines the contact loss between the brake pad and disc is given as: 1 h j x,θ , L j 0 ()! !rj r f , j L j,! !rj r f , j L jj 3, 4.(5)Since hj (x, θ, Lj) is a discontinuous function, the governing equations given (by Eqn. (4)) cantake four different forms, i.e. the system of Fig. 1 may assume the following four states: a. Leadingand 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 edgeis in contact.3RESULTS AND DISCUSSIONGoverning equations given by Eqn. (4) form a set of coupled ordinary nonlinear differentialequations. Nonlinearities in these equations arise from: 1) Arrangements of the springs k1 and k2; 2)Surface separation at contact interfaces (hj (x, θ, Lj)); and 3) Dry friction model. However, the frictionterm can be dropped for the sake of simplicity since Vrel 0 at all times. Furthermore, the piecewiselinear function hj (x, θ, Lj) is also smoothened with a continuous function as given below where σ isthe regularizing factor (often a large number).()h j x,θ , L j ((! !1 tanh σ rj r f , j L j2))j 3, 4.(6)To simulate the actuation pressure acting on the brake pad, pretensions are defined onsprings k3 and k4, and the static equilibrium point of the pad is evaluated from the force andmoment balance equations. Finally, a cohesive set of four governing equations, Eqn. (4) in whichhj (x, θ, Lj) is replaced with Eqn. (6), is numerically solved.The solutions of the system of Fig. 1 are obtained for different α1-α2 parameter sets. Theresults are given in terms of normalized leading/trailing edge contact forces Fs,3 y and Fs,4 y wherethe normalization is done by the total actuation (brake) force. First, the case where α1 90 andα2 180 is investigated and corresponding normalized contact forces are given in Fig. 2 forboth proposed and prior models. As seen in the figure, consecutive impact-like contact forces aredeveloped at the leading and trailing edges of the pad (Fig. 2(a) and Fig. 2(b)). Furthermore, thecontact force at the leading edge is almost as large as the constant actuation force (Fig. 2(a));however, the contact force generated at the trailing edge is about four times of the actuation force(Fig. 2(c)). In the second case, α1 180 and α2 90 arrangement is solved, and the results aregiven in Fig. 3. While the trailing edge of the pad is always in contact with the disc (Fig. 3(b)),the leading edge loses contact during the event (Fig. 3(a)). Like the first case, the trailing edgecontact force is always greater than the leading edge force. Since these two cases (α1 90 , α2 180 and α1 180 , α2 90 ) simulate one of the squeal mechanisms called as the hammering7,the proposed model successfully predicts a squeal initiation event. Moreover, the solutions of theprior model10 in which the leading/trailing edge dynamics of the pad is ignored, are also shownin Fig. 2(c) and Fig. 3(c) for the sake of comparison; observe that no contact loss is observedwith the prior model. In addition, the solutions in Fig. 2(c) and Fig. 3(c) are exactly the sameeven though the angular arrangements of the springs k1 and k2 are very different. Therefore, it iscrucial to include the rotational degree of freedom for the pad in order to have a more accuraterepresentation of the squeal problem.

(a)(c)10.81.11Fs,3y0.60.90.40.80.2000.20.3t 0.30.0020.0040.0060.0080.01t [sec]t [sec]Fig. 2 – Time histories for contact forces for α1 90 and α2 180 . (a) Leading contact forceFs,3 y (proposed model); (b) Trailing contact force Fs,4 y (proposed model); (c) Contactforce Fc (prior 040.0060.0080.01t .01t [sec]t [sec]Fig. 3 – Time histories for contact forces for α1 180 and α2 90 . (a) Leading contact forceFs,3 y (proposed model); (b) Trailing contact force Fs,4 y (proposed model); (c) Contactforce Fc (prior model).

0.0080.01t 080.01t [sec]t [sec]Fig. 4 – Time histories for contact forces for α1 α2 135 . (a) Leading contact force Fs,3 y(proposed model); (b) Trailing contact force Fs,4 y (proposed model); (c) Contact forceFc (prior .0060.0080.01t 040.0060.0080.01000.0020.0040.0060.0080.01t [sec]t [sec]Fig. 5 – Time histories for contact forces for α1 α2 150 . (a) Leading contact force Fs,3 y(proposed model); (b) Trailing contact force Fs,4 y (proposed model); (c) Contact forceFc (prior model).