ABSTRACTThe brake torque variation (BTV) generated due to geometricirregularities in the disc surface is generally accepted as thefundamental source of brake judder; geometric imperfectionsor waviness in a disc brake caliper system is often quantifiedas the disc thickness variation (DTV). Prior research hasmainly focused on the vibration path(s) and receiver(s),though such approaches grossly simplify the source(frictional contact) dynamics and often ignore caliperdynamics. Reduction of the effective interfacial contactstiffness could theoretically reduce the friction-inducedtorque given a specific DTV, although this method wouldseverely increase static compliance and fluid volumedisplacement. An experiment is designed to quantify theeffect of disc-pad contact modifications within a floatingcaliper design on BTV as well as on static compliance. Themajor objective of this experiment is to determine if changesin the disc-pad contact geometry can also reduce BTVwithout limiting the static compliance of the caliper system.A conceptual half-caliper model is proposed to explain theobserved effects of pad modifications. This simplified elasto-kinematic model uses the elastic center concept on a padsubject to spatially phased periodic displacement inputs(DTV) at the disc-pad interface. It is utilized to determine theeffective variation in normal load. The model is finallyemployed to determine the sensitivity of key physicalparameters and to identify trends that might reduce BTV.

INTRODUCTIONBrake judder is defined as the vibration at frequenciesproportional to wheel speed (10-20 Hz) experienced by adriver during a high-speed braking event. The fundamentalsource of this vibration is believed to be the geometricirregularities on the disc surface (caused by a variety ofmechanisms, such as manufacturing process effects, thermalexpansion, wear, and misalignments). Disc surface distortions(say quantified by ξ(t) where t is time) cause variations in thenormal load at the disc-pad interface and thereby producevariations in brake torque. Some well known causes,observed effects, and analysis methods are summarized byJacobsson [1] in an extensive review of literature as of 2003.Much of the available literature focuses on the on-vehiclepath modifications and on an attenuation of the observedeffects at the driver's location. No prior study has explicitlyexamined the role of caliper dynamics.

Using a conventional brake dynamometer, judder is

traditionally quantified in terms of BTV ( ) produced by adisc and caliper system during a braking event. AlthoughDTV (ζ(t)) is typically required to produce judder, certainattributes of a disc and caliper system can significantly

contribute to . Most mathematical or computationalmodels for brake judder assume a uniform brake disc-padcontact surface [2,3,4]. However, a disc which possesses awavy braking surface will create a non-uniform contactinterface between it and the brake pad. This non-uniformityon the interfacial braking surface can affect heat distributionpatterns on the disc surface as well as create a time-varyingeffective center of pressure [5]. Changes in the center of

Effect of Disc-Pad Contact Modification on theBrake Judder Source Using a Simplified Elasto-Kinematic Model

2013-01-1907Published

05/13/2013

Jason Dreyer, John Drabison, Jared Liette and Rajendra SinghThe Ohio State University

Osman Taha SenIstanbul Technical University

Copyright © 2013 SAE International

doi:10.4271/2013-01-1907

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pressure can affect several parameters which can ultimately

affect . Previous literature has focused on the sensitivity to stiffness distribution within the caliper systemand some coupling with the hydraulic system [2,3,4]. Theseworks suggest that judder sensitivity at the caliper can bereduced by decreasing the effective stiffness of the calipersystem, a change which can adversely affect pedal feel. Theprior work does not take into account the stiffnessdistribution from the leading to trailing edge of the contactinterface in the disc-pad or pad-caliper interfaces during abraking event, which may uncover a means to reduce juddersensitivity at the caliper without augmenting calipercompliance. Effects of such leading-to-trailing stiffnessdistributions within the disc-pad and pad-caliper interfacesare well-known for high frequency brake squeal [6, 7], but, tothe best of our knowledge, it has not yet been viewed as asignificant contributor to brake judder.

PROBLEM FORMULATIONThe scope of this paper will be to examine the effect of disc-

pad contact modifications on with respect to leading andtrailing dimensions. The objectives of the study include thefollowing: (i) design an experiment (different pad geometriesexamined on the same brake caliper and disc) to quantify the

effect of disc-pad contact modification on ; (ii) propose aconceptual model to explain the effects of pad modifications

in a floating caliper braking system on and validate themodel with an experiment; and (iii) determine sensitivity tosome elasto-kinematic parameters and identify trends that

might reduce the source of .

In order to investigate the sensitivity of caliper designsrelative to leading and trailing variations in disc-pad contactgeometries, multiple cases of pads with different disc-padcontact interfaces (Aab) are measured in a floating caliperbraking system. The study is limited to symmetric leading-to-trailing pad contact modifications to the disc-pad interface.Often these types of modifications are made to reduce brakesqueal or mitigate pad wear. The pad-caliper interface itself isnot modified. The different pad modification casesconsidered for this study are shown in Figure 1 and quantifiedin terms of an index r. For each pad set with the same pad

material composition, is measured under the samerepresentative on-brake conditions, with no resonant growth.The same disc with ζ(t) is employed. The measured ζ(t) isfound to be approximately 20 μm peak-to-peak. The staticcompliance of the system is experimentally quantified (usingload-deflection measurements) to understand variation instiffness parameters due to the change loading interfaceswithin the caliper.

Figure 1. Pad modifications (r = 2 and 3) considered forthis study with the unmodified pad set (r = 1) considered

as benchmark; shaded regions are disc-pad contactinterfaces.

SIMPLIFIED ELASTO-KINEMATICMODELEach disc and caliper system is comprised of a disc-padcontact interface, as illustrated in Figure 2 where Aabrepresents the interface between the (rotating) disc and pad,and through which a point on the disc enters (leading edge)and then leaves (trailing edge) the caliper. Both inboard andoutboard pads are assumed to have equivalent disc-padcontact properties, with a center of contact defined at aneffective point R on the disc. The rotational speed of the discis defined as Ω in revolutions per min (rpm).

Figure 2. Illustration of the disc-pad interface (Aab) withleading and trailing edges.

The brake torque T developed at Aab subjected to theboundary conditions of the caliper is defined by both a meantorque and an alternating peak-to-peak torque component, given by

(1)

For the sake of simplicity, is assumed not to be varyingwith time, which is appropriate for the constant pressurebraking events considered in this study.

The mean load (N) on the pad for the specific caliper underbraking conditions with the trailing abutment of the padloaded is represented in Figure 3, where is the mean

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normal load at the disc-pad interface, is the mean hydraulicpressure applied by the caliper, Ah is the effective hydraulicarea of the caliper, µ is the coefficient of friction at Aab, µ2 isthe coefficient of friction at the contact point between the padbacking plate and caliper abutment, u is the distance from theeffective mean hydraulic force on the pad to the loadedabutment, w is the distance from at the disc-pad interfaceto the loaded abutment, and z is the thickness of the frictionmaterial. For both pads under this loading, is given by

(2)

with constant µ and R. From this relationship, the mean loadat the disc-pad interface is given by

(3)

Figure 3. Mean load distribution on the pad underbraking conditions.

For the alternating case, the following simplified model isproposed in Figure 4 for a half-caliper. Here, the peak-to-peak alternating normal load at the disc-pad interface isassumed to be alternating around a sufficiently high . Thecorresponding condition is given by

(4)

such that is not overcome, preventing the pad to slide atthe pad-abutment interface. In this case, the pad-abutmentinterface may be assumed as a “pinned” support at point O,with a pad rotation about this support defined as θ. Thedistance from the leading and trailing edge of Aab to point Ois given by a and b, respectively; and the distance from theleading and trailing edge of the pad-caliper interface (Acd) topoint O is given by c and d, respectively. For thisformulation, a>b and c>d.

Figure 4. Simplified half-caliper elasto-kinematic model.

The model assumes a phased displacement excitation

amplitude, given by at the leading edge and at thetrailing edge, and a compressive pad stiffness component kp,distributed equally between the leading and trailing edges ofAab. Based on an arc length along a circle, the phase shift of αradians is denoted as , where

(5)

The pad is assumed to be supported by the compressivecaliper stiffness kc, distributed equally between the leadingand trailing edges of Acd. According to equation (5), thisformulation assumes that the contact length of the Aab on thedisc surface (a-b) with respect to R provides a reasonableapproximation of phase between leading and trailing edges.

Based on observations from prior dynamic models of thecaliper [2,3,4], the common judder frequency input falls inthe stiffness controlled region of caliper normal loaddynamics. This suggests that elastic contact elements andtheir kinematic relationships are sufficient for estimating thedynamic behavior of the caliper system. Our modelincorporates the leading and trailing edge effects of thesystem geometry as well as the center of pressure using theconcept of an elastic center. The angular motion of the padsubjected to a general disc surface input ξ is given by

(6)

The normal load variation at Aab is given by

(7)

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If the input ξ is approximated by a finite term (say M) Fourierseries as

(8)where Ξm is the Fourier amplitude and ϕm is the phase for the

mth order term, then measured at Aab can be expressed as

(9a)where

(9b)

Here, ke can be interpreted as the effective dynamic stiffnessof the caliper system.

From , can be calculated as follows, similar to equation(2),

(10)

The condition in equation (4) can now be verified for aspecific set of conditions. Under experimental conditions, and cannot easily be measured; however, and are easyto measure. In this case, the condition of equation (4) can bealtered and then verified by

(11)

It is important to note the difference between the mean andalternating calculations of torque. Often, is reported as afunction of mean pressure, such that substituting equation (3)into equation (2) yields

(12)where µe is defined as the effectiveness of the caliper system,given by

(13)

Equation (13) clearly shows that the observed effectiveness isnot necessarily the same as the coefficient of friction µ at Aab,

which is used in the calculation of .

In addition, it is important to note that our model can producedifferent levels of at the same values of kc and kp. Themodel assumes no interactions among parameters; however,in actual caliper systems, changes in geometry at the loadinginterfaces can feasibly change the interfacial loadingdistributions within the caliper and the effective kc.

EXPERIMENTAL STUDY ANDMODEL VALIDATIONThe dimensions of Aab and Acd critical for our model areshown in Figure 5. The dimensions of c and d for inboard andoutboard pads are slightly different. An average valuebetween the inboard and outboard pads for each case is usedin this study. If c and d vary greatly between inboard andoutboard pads, the model could be modified as thesummation of two different half-caliper models. Values for aand b are shown in Figure 5 for the unmodified pad set (r =1). Values of a and b for the modified pad sets (r = 2 and 3)are taken at the edges of the leading and trailing contactsurfaces, as indicated by the shaded regions in Figure 1.

Figure 5. Inboard and outboard disc-pad (Aab) and pad-caliper (Acd) contact dimensions for unmodified pads

(r = 1).

The stiffness values of the model are obtained using load-deflection measurements, representative of an off-brakecondition (no pad-abutment load path) at comparable meanloads as seen in the experiment described in Figure 6. For kp,a static compression test is performed on the pad between twoflat surfaces without squeal noise insulators (shims). Thisstiffness is essentially the compressive stiffness of the padmaterial. For kc, the caliper is loaded under static conditionsand effective displacements of the caliper structure aremeasured at different locations over its surface; the effectivemeasured stiffness is corrected for the contribution of kp. Thevalue of kc is measured with pads and shims; therefore, it

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incorporates the contact stiffness effects of the shims andpad-caliper supports as well as the stiffness contributions ofthe hydraulic system and caliper structure. In general, kc≪kp.

A brake-dynamometer-like laboratory experiment is used to

measure changes in for the different pad contactmodification cases. A schematic of the setup is given Figure6. The experiment is designed to provide a consistentmechanically governed braking condition for the differentpad sets. The flywheel (approximately 15 kg-m2) isaccelerated to a constant rotational speed by an electric motor(not pictured), which is then mechanically decoupled fromthe system. The braking system is then applied at a constanthydraulic pressure (around 350 kPa) beginning at a rotationalspeed of 1050 rpm. The T(t) is measured for the same appliedbrake pressure for each setup and reported over the sameangular speed range, between 1000 and 900 rpm (outside ofthe resonant growth regime of the torsional system). Thetemperature and surface profile of the disc are measured foreach pad set to ensure the same conditions for each test. Themodified pads are created by removing material from thesurface of new pads to change Aab and thus avoidsignificantly affecting the flexural stiffness of the pad. Thevariation among the pad materials prior to modification isalso measured but is found to be of the same order as thevariation within each pad set test. Each pad set is run sixtimes on the experimental setup. The condition given inequation (11) is confirmed for each test case.

Figure 6. Laboratory judder experiment.

Results from the model predictions and experimentalmeasurements for the different pad cases (r = 1, 2, 3) arecompared in Table 1. The model results assume the same kpand kc for each of the pad sets and the same µ and R. Themodel also uses two terms in the Fourier series

approximation of ξ. Both experimental and model results are

normalized to the average of (baseline pad set r = 1), givenby

(14)where the subscript r denotes the pad set.

Table 1. Experimental and model results in terms ofnormalized torque variations T* given by equation (14).

The 95% confidence intervals calculated for pad setmeasurements are reported to quantify the variation of thisexperiment including an inherent variability associated withbraking systems. Measurements for the pad sets agree wellwith the model predictions, each falling within the 95%confidence interval for each pad set. In general, the results

suggest that a reduction in Aab can reduce . Themeasured µe is also dependent on the pad set (with r = 1 asthe highest and r = 3 as the lowest). Nevertheless, each ofthese values of µe falls within a reasonable range given byequation (13).

SENSITIVITY OF KEYPARAMETERS AND LIMITATIONSThe effective kc for the different pad sets used in this study ismeasured, as well as the effective volume displacement ofhydraulic fluid under off-brake conditions. A maximumreduction in the effective stiffness corresponding to anincrease in hydraulic fluid displacement with respect to anominal state (pad set r = 1) is observed and quantified asabout 20%. This reduction suggests an interaction of thecontact geometry and kc, which can be attributed to theredistribution of interfacial forces within the caliper.However, the caliper load-deflection test repeatability for agiven pad set is of the order of 20% as well. This variabilitycan be attributed to a variation in the seating of the pistonseals depending on installation of pads within system.Sensitivity to such variation in kc can be investigated usingthe model.

The proposed model is exercised by varying the stiffnessparameters by +/− 5% around the nominal stiffness used forthe model correlation in Table 1. The results of thissensitivity analysis are reported in terms of the percent

change in due to an increase in a certain stiffness

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parameter. The results in Table 2 suggest that a larger Aab ismore sensitive to changes in kp, while a smaller Aab is moresensitive to changes in kc. This sensitivity also is important tounderstand potential effects at different braking conditions,specifically if kc or kp are dependent on .

Table 2. Sensitivity of model to changes in caliper andpad stiffness for different pad sets.

The model may be utilized to determine the sensitivity ofadditional variations in Aab and Acd. For the nominal stiffnessvalues for this caliper system, the general trend shows that anequal reduction in Aab and Acd together is the most effective

means to reduce , while a reduction in Acd yields only

minor reductions in .

Furthermore, as can be seen from equation (10), decreases in

µ or R will also proportionally reduce . Radial disc-padcontact modifications are not explored in this study, but canbe represented as modifications to the value of R used in themodel. A change in R may also change the interfacial loadingwithin the brake caliper system; and therefore, the effect onkc should also be investigated. This model assumes constantvalues for R and µ; however, in the case of corrosion orsevere thermal conditions, µ and R may also become afunction of time or angular position similar to ξ.

Although not reported in this paper, this model can also beused to investigate asymmetric variation of dimensionswithin the caliper; however, there are many differentcombinations possible that can yield either increases or

decreases in . Caution should be exercised in arbitrarilyvarying parameters in such a model, as they may often lead todesigns that have detrimental effects on other aspects ofbrake performance, such as wear, thermal management, andbrake noise.

CONCLUSIONThe chief contribution is a proposed simplified half-caliperelasto-kinematic model that incorporates leading and trailingstiffness elements on the disc and caliper sides of the pad topredict the generation of BTV in the presence of DTV. Thismodel and the associated study demonstrate how differentcaliper designs with the same compressive stiffness attributescan produce different levels of BTV. The cases in this studysuggest that a more compact leading-to-trailing disc-padcontact interface can reduce BTV with minimal impact oncaliper compliance.

Although the proposed model qualitatively agrees with theexperimental findings, extension to a predictive model forother brake designs will require significant effort. Additionalelastic elements at the contact interfaces may improve themodel as the complexity in disc surface profile increases dueto the presence of higher orders, such as those observedduring hot judder. The proposed formulation can be extendedto a full caliper with inclusion of separate inboard andoutboard contact dimensions and stiffness elements. Thermaland wear effects on the model parameters should also beinvestigated. Influence of the caliper bracket supports, suchas guide pins or bushings, displacements due to disc orbearing deflection, different pad abutment designs, andfunctional spring components should also be considered in anextension to this formulation. This simplified model does notaddress brake pressure variation, and in order to adequatelyaddress this, a model with a mechanically coupled hydraulicsystem is required.

It should be noted that the proposed elasto-kinematic theoryhas been validated on a simple laboratory test stand, whichremoves many of the inherent complexities of a vehiclesystem, such as phasing among the dynamics of the brakecorners. The proposed disc-pad modifications must still beexperimentally validated in vehicle tests. Nevertheless,concepts in this paper should lead to more refinedmeasurements in both laboratory and vehicle tests.

REFERENCES1. Jacobsson, H., “Aspects of Disc Brake Judder,” Proc.IMechE Part D: J. Automob. Eng. 217: 419-430, 2003.

2. Kang, J. and Choi S., “Brake Dynamometer ModelPredicting Brake Torque Variation to Disc ThicknessVariation,” Proc. IMechE Part D: J. Automob. Eng.,221:49-55, 2007

3. Leslie, A., “Mathematical Model of Brake Caliper toDetermine Brake Torque Variation Associated with DiscThickness Variation (DTV) Input,” SAE Technical Paper2004-01-2777, 2004, doi:10.4271/2004-01-2777.

4. Kim, S.-H., Han E.-J., Kang S.-W., and Cho S.-S.,“Investigation of Influential Factors of a Brake CornerSystem to Reduce Brake Torque Variation,” Int. J.of Auto.Tech. 9(2): 233-247, 2008.

5. Tirovic, M. and Day A. J., “Disc Brake Interface PressureDistributions,” Proc. IMechE Part D: J. Automob. Eng.205(2): 137-146, 1991.

6. Dunlap, K., Riehle, M., and Longhouse, R., “AnInvestigative Overview of Automotive Disc Brake Noise,”SAE Technical Paper 1999-01-0142, 1999, doi:10.4271/1999-01-0142.

7. Fieldhouse, J., Ashraf, N., and Talbot, C., “TheMeasurement and Analysis of the Disc/Pad InterfaceDynamic Centre of Pressure and Its Influence on Brake

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Noise,” SAE Int. J. Passeng. Cars - Mech. Syst. 1(1):736-745,2009, doi:10.4271/2008-01-0826.

CONTACT INFORMATIONProfessor Rajendra SinghAcoustics and Dynamics LaboratoryNSF I/UCRC Smart Vehicle Concepts CenterDepartment of Mechanical and Aerospace EngineeringThe Ohio State Universitysingh.3@osu.eduPhone: 614-292-9044www.autonvh.org

ACKNOWLEDGMENTSThe authors gratefully acknowledge Honda R&D Americas,Inc., OSU-Honda Partnership Program, and OSU-TREPEndowment for supporting this research.

LIST OF SYMBOLSAab - Disc-pad contact interface [mm]

Acd - Pad-caliper contact interface [mm]

Ah - Effective hydraulic area [mm2]

a - Distance from loaded pad abutment to leading contactstiffness on disc-pad interface [mm]b - Distance from loaded pad abutment to trailing contactstiffness on disc-pad interface [mm]c - Distance from loaded pad abutment to leading contactstiffness on pad-caliper interface [mm]d - Distance from loaded pad abutment to trailing contactstiffness on pad-caliper interface [mm]kc - Caliper compressive stiffness [N/mm]

ke - Effective dynamic system stiffness [N/mm]

kp - Pad compressive stiffness [N/mm]

M - Total terms in Fourier seriesm - Harmonic index in Fourier seriesN - Normal load [N]

- Mean normal load [N]

- Peak-to-peak normal load variation [N]O - Point of rotation of pad at pad-abutment interface

- Mean hydraulic pressure [MPa]R - Effective contact radius [m]r - Index denoting pad sett - Time [s]T - Brake torque [Nm]

- Mean brake torque [Nm]

- Peak-to-peak brake torque variation [Nm]u - Distance from loaded pad abutment to center of pressureon disc-pad interface [mm]w - Distance from loaded pad abutment to center of pressureon pad-caliper interface [mm]z - Pad thickness [mm]α - Phase angle [rad]θ - Rotational angle of pad motion [rad]µ - Coefficient of friction at disc-pad interface [-]µ2 - Coefficient of friction at pad abutment [-]

µe - Observed brake effectiveness [-]

ξ - Periodic disc thickness variation [μm]

Ξm - mth order disc thickness variation amplitude [μm]

ϕm - mth order disc thickness variation phase [rad]

Ω - Rotational speed of disc [rpm] - Phase shift

AbbreviationsBTV - Brake torque variationDTV - Disc thickness variation

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