On Pressure Distributions of Drum Brakes

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Yuan Mao HuangProfessor

e-mail: [email protected]

J. S. ShyrResearch Assistant

Department of Mechanical Engineering,National Taiwan University,

Taipei, Taiwan,Republic of China

On Pressure Distributions of DrumBrakesBased on the assumptions that there is perfect contact at the interface between thedrum and the lining plates, the friction coefficient is constant, the thermal effect isglected, and the brake drum is a rigid body, the pressure distributions of drum brwere studied by using the boundary element method. The constant element is usedtwo-dimensional model of the drum brake for simplicity and economy. The friction fversus the effective lift at the actuation edge and the location of the maximum pressucompared and indicate a good correlation with existing data. The effects of the Yomodulus of elasticity of the metal shoe, the arc lengths of the metal shoe and liningthe location, the thickness, the friction coefficient, the Young’s modulus of elasticity olining plate, and the angle of actuation force on the pressure distributions werestudied. By selecting proper values of these parameters, a drum brake can be desighave a more uniform pressure distribution and a longer life.@DOI: 10.1115/1.1427694#

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Introduction

Drum brakes, which are an important device and are installevehicles and motor cycles, are used to reduce the speed aassure the safety of the drivers and passengers on the roadschematic drawing of a drum brake is shown in Fig. 1. It primarconsists of a brake drum and brake shoes with lining plates. Quties it should have include: good braking performance; unifobraking on all wheels; good endurance without failure; and eoperation, inspection, adjustment and maintenance. Advantoffered by the drum brake are its capability from the seenergizing effect to reduce the required actuating force andrelatively lesser cost.

Day et al.@1# used the finite element method to analyze a twdimensional model of the drum brake. Thereafter, the thermalpansion, imperfect contact, the decay of braking capability andkinds of transient phenomena were analyzed by using the fielement method@2–6#. Watson and Newcomb@7# argued that it ismore expensive to use the finite element program in the mframe and that the pressure distribution on the lining plate dnot vary significantly along the axis of the brake. The measument of the torque or the wear can be used to calculate the psure distribution@8#. Nevertheless, these methods are indirmethods. It is difficult to obtain directly the pressure distributifrom an experiment@2,5#. Recently, the applications of the bounary element method~BEM! have been increased and showed tthe BEM can reduce the time required to prepare numericaland can provide good results in the linear problems@9–14#.

The purpose of this study is to analyze the effect of the effeclift at the actuation end on the friction force, effects of the Younmodulus of elasticity for metal shoes, the angles of the arc lenof metal shoes and lining plates, respectively, the leading locaof the lining plates, the thickness of the lining plate, the fricticoefficient, the Young’s modulus of elasticity for lining plates, athe location of the actuating force of the drum brake on the psure distributions. A two-dimensional model of the drum brakedeveloped by using the BEM, and independent and less expensoftware is generated for the personal computer to reduce thefor designing drum brakes. By selecting proper values of thparameters, a drum brake can be designed to have a more unpressure distribution and a longer life.

Contributed by the Design Automation Committee for publication in the JOUR-NAL OF MECHANICAL DESIGN. Manuscript received Nov. 1999. Associate EditoH. Lankarani.

Copyright © 2Journal of Mechanical Design

aded 03 Nov 2008 to 140.112.113.225. Redistribution subject to ASME

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Method of ApproachDuring the derivation of the boundary integration equations,

solution of an infinite domain subjected to a concentrated loaany internal location is required. If a unit force acts on an interpoint M, determination of the effect of this force on any pointQmust satisfy two requirements. The first one is that all of tstresses must be zero when the distance between the pointsM andQ becomes infinite. The second one is that the stress becoinfinite, or the pointM is a singular point when the distance btween the pointsM andQ becomes zero. Combining the differential equation of the stress, the relationship of the stress andstrain and the relationship of the strain and the displacemyields

]2ux

]x21

]2ux

]y2 S 1

122n D S ]2ux

]x21

]2ux

]x]yD 52 f x

Gs

]2uy

]x21

]2uy

]y21S 1

122n D S ]2uy

]x21

]2uy

]x]yD 52 f y

Gs6 (1)

which is the Navier two-dimensional differential equations acan be solved by using the complementary function and thecial integration. If the displacements are expressed in terms ofGalerkin vector,

ux5]2Gx

]x21

]2Gx

]y22

1

2~12n! S ]2Gx

]x21

]2Gy

]x]yDuy5

]2Gy

]x21

]2Gy

]y22

1

2~12n! S ]2Gy

]y21

]2Gx

]x]yD 6 (2)

the Galerkin vector can be determined as

Gx5Gy51

8pGsr 2 lnF1

r G (3)

and the displacement is

ui5Ui j ej (4)

Substituting Eq.~4! into the relationship of the stress and thdisplacement yields the traction force

t i5Ti j ej (5)r:

002 by ASME MARCH 2002, Vol. 124 Õ 115

license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

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To minimize the error of numerical approximation, the basolution of the displacement as a weighting factor is multipliedevery term and integrated to yield the Somigliana equation

uli1E

GTlkukdG5E

GUlktkdG1E

VUlk f kdV (6)

The displacement of any internal point can be determined ifdisplacements of the boundary points and the traction forcesdetermined. The Somigliana equation can also be applied toboundary points and yields

clki uk

i 1EGTlkukdG5E

GUlktkdG1E

VUlk f kdV (7)

Four variablesu1 , u2 , t1 and t2 exist for each nodal point in atwo-dimensional problem; therefore, there will be 4N variablesfor N nodal points. If there is only one solution for an elasproblem, only two values of these four variables are knowneach nodal point and 2N equations are required to solve the uknown variables.

The geometric configuration, the displacements and the tracforces must be described. If there is a unit force acting onpoint i and the effect of body force can be neglected, the tintegrated terms related to the nodal pointsi and j form the influ-ence coefficients,Gi j andH i j . If

Hi j 5H i j iÞ j

Hi j 5H i j 1ci i 5 jJ (8)

Eq. ~7! becomes

(j 5 l

N

Hi j uj5(j 5 l

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Applying to all boundary points and using local coordinates asbasis of the boundary value yields

HT8U5GT8P (10)

where the matricesH andG are 2N32N matrices. The values othe displacement matrixU and the traction force matrixP are allrelative to global coordinates. The boundary condition shouldgiven after calculation of all coefficients for the matricesH andGin order to obtain the unique solution. Since it is not a symmematrix, a direct method such as the Gaussian elimination mecan be used.

Model and Numerical AnalysisIt is assumed that the variations of the pressure and the de

mation in the axial direction are negligible for simplicity, that ndeformation occurs during braking, that there is the perfect inicontact between the contact surfaces of the lining plate andcircular profile of brake drum, that the friction coefficient of thlining plate and the brake drum is constant, that the wear is nligible, that the thermal effect is neglected and that the brake d

Fig. 1 Schematic drawing of drum brake

116 Õ Vol. 124, MARCH 2002


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is a rigid body. The two-dimensional model of the brake shoe tis used in this analysis can be divided into two parts basedmaterials, the metal shoe and the lining plate. They are diviinto 28 elements, respectively, due to the limited size of the coputer memory. The displacement and the traction force at theter point in the constant element are used for the element toplify the integration. Applying Eq.~10! to the metal shoe and thlining plate, respectively, combining the points on the contact sface of the lining plate and the metal shoe and the indepenportion of the lining surface, and combining the points on tcontact surface of the metal shoe and the lining plate andindependent portion of the metal shoe with the traction equirium condition

PLB5PS

C (11)

and the displacement compatibility condition

ULB52US

C (12)

yields

FHLA HL

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0 2HSC 2GS

C HSDGFUL

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ULB

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A 0 0 0

0 0 0 GSDGF PL

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where the matricesH andG have been multiplied by the transfematrix, the superscriptB stands for the common portion of thlining plate and the metal shoe and the superscriptA stands for theindependent portion of the lining plate, the superscript C stafor the common part of the metal shoe and the lining plate andsuperscriptD stands for the independent portion of the metal sh

In the local coordinates, four variables for each nodal pointthe displacementsUn andUt in the normal direction and the tangential direction, respectively, and the traction forcesPn andPt inthe normal direction and the tangential direction, respectively. Tboundary conditions along the friction surface are

Un50

Pt52mPnJ (14)

Two variablesUt and Pn are unknown. Equation~13! can bewritten

@ .. 2~Gn2mGt! Ht ..#3.

.

Pn

Ut

.

.

4 5@ .. 2Hn 0 ..#3.

.

Un

0

.

.

4 (15)

The boundary conditions for the fixed anchor are

Un50

Ut50 J (16)

The supporting portion for the sliding abutment can be rewritt

@ ..2Gni Ht

i2 Gni 11 Ht

i 11..#@ ..Pni Ut

i Pni 11 Ut

i 11..#T

5@ ..2Hni Gt

i 2Hni 11 Gt

i 11..#@ ..Uni Pt

i Uni 11 Pn

i 11..#T (17)

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where the superscript stands for the number of the nodal pointthe subscript stands for the direction. Similarly, the supportportion for the fixed anchor can be rewritten

@ ..2Gni 2 Gt

i2 Gni 112 Gt

i 11..#@ ..Pni Pt

i Pni 11 Pt

i 11..#T

5@ ..2Hni 2Ht

i 2Hni 11 2Ht

i 11..#@ ..Uni Ut

i Uni 11 Ut

i 11..#T

(18)

After considering the boundary condition, the matrix can bewritten

H8U85G8P8 (19)

where the matricesH8 andG8 are the matrices after arrangemenand the matrixU8 is obtained from the unknown displacemeand the traction force. The matrixP8 is obtained from all theboundary conditions. After obtaining the traction forces anddisplacements of nodes, the torque can be obtained from

Torque5(n51

m

mPDA (20)

whereDA is the area of the element.

ResultThe data of the first drum brake with the supporting end fix

by a pin as shown in Fig. 1 are listed in Table 1. The anglea ismeasured from the line of the pin center and the brake dcenter to the line of the brake drum center to near edge oflining plate. The anglef is measured from the line of the actution force to the normal vector of the surface at the load point. Tfriction force versus the effective lift at the actuation edge thanormal to the acting surface is shown in Fig. 2. The data obtaifrom Day and Harding@15# by using the finite element method aalso shown for comparison.

The data of the second drum brake are shown in Table 2. If

Fig. 2 Friction force versus effective lift at actuation end

Table 1 Data of first drum brake.

Journal of Mechanical Design


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Table 2 Data of second drum brake.

Fig. 3 Pressure distribution with high Young’s modulus ofelasticity for rigid metal shoe

Fig. 4 Pressure distribution versus Young’s modulus of elas-ticity of metal shoe

Fig. 5 Pressure distribution versus lining arc for various arclengths of metal shoe

MARCH 2002, Vol. 124 Õ 117


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Young’s modulus of elasticity increases, the material becomstiffer. At first, the Young’s modulus of elasticity of the brake shin Table 2 is replaced by a value four times larger, 3.313106

N/mm2, and the result of the pressure distribution is compawith the sinusoidal curve of the rigid brake obtained from Offn@16# as shown in Fig. 3. Based on the data shown in Table 1the displacement of the surface along the normal direction ofsurface to be 0.1 mm, the pressure distributions versus the liarc for various Young’s moduli of elasticity of metal shoes ashown in Fig. 4. The effects of the arc lengths of the metal sand the lining plate with the same center location on the pres

Fig. 6 Pressure distribution versus lining arc for various arclengths of lining plate

Fig. 7 Pressure distribution versus lining arc for various lead-ing locations of lining plate

Fig. 8 Pressure distribution versus lining arc for various lin-ing plate thicknesses

118 Õ Vol. 124, MARCH 2002


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distribution are shown in Figs. 5 and 6, respectively. Shiftinglining plate, the pressure distribution is shown in Fig. 7. The prsure distributions versus the lining arc for the thickness oflining plate to be 3, 4 and 5 mm are shown in Fig. 8. It is wknown that the friction coefficient of material can affect the braing performance. The effect of the friction coefficient of the lininplate on the pressure distribution is shown in Fig. 9. ChangingYoung’s modulus of elasticity of the lining plate, the pressudistribution is shown in Fig. 10. If the ratio of the Young’s modlus of elasticity of the lining plate to the Young’s moduluselasticity of the metal shoe remains constant and both Younmoduli of elasticity of the lining plate and the metal shoe increa

Fig. 9 Pressure distribution versus lining arc for various fric-tion coefficients of lining plate

Fig. 10 Pressure distribution versus lining arc for variousYoung’s moduli of elasticity of lining plate

Fig. 11 Pressure distribution versus lining arc for variousYoung’s moduli of elasticity of metal shoe and lining plate



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20% and decrease 20%, respectively, the pressure distributionshown in Fig. 11. The effect of the angle measured from theof the actuating force to the normal vector of the surface atload point with the traction pressure 50 N/mm2 on the pressuredistribution is shown in Fig. 12.

DiscussionWatson and Newcomb@7# argued that it is more expensive t

use the finite element program in the main frame. This study uthe boundary element method in the personal computer. Neveless, the computer running time is not available for comparisUsing the linear elements or the second order elements crediscontinuous phenomena along the irregular boundary. The cmon nodal point has the different normal vector and boundconditions. It is necessary to have an extra equation to providunique solution for the final linear equation@12#. Therefore, theseelements are inappropriate to treat the corner problem. Usingconstant element can get rid of this problem. In addition, sithere is no shape function and no need of the Jacobin coordtransform, integrating the equation is easier and writing a coputer code is simpler for the constant element than the secorder element. Watson and Newcomb@7# also argued that thepressure distribution on the lining plate does not vary significanalong the axis of the brake. Therefore, two-dimensional modeused in this analysis to reduce the computer time.

It is difficult to obtain directly the pressure distribution from aexperiment@2,5#. Day @17# studied two distinct brake operatinconditions experimentally to investigate the effect of vehispeed on the pressure distribution with time. However, the dimsions of the brake are not specified completely, and the resultsnot appropriate for comparison with this study. The friction foris proportional to the effective lift at the actuating edge as shoin Fig. 2. The deviation of the calculated friction force from thobtained by Day and Harding@15# increases when the effectivlift increases. It may come from the insufficient number of telements and the accumulative error from the numerical calction, and it can be reduced if the number of the elements iscreased.

The maximum pressure on the lining plate that occurs at 60Fig. 3 is located at 85° from the center of the pin. The convtional angular location of the maximum pressure from the sporting point for the rigid drum brake is at 90°@16#. Figure 4shows that the angular location of the maximum pressure shfurther from the supporting point when the Young’s moduluselasticity of metal shoe increases. The location of the maximpressure should be close to 90° from the supporting point ifYoung’s modulus of elasticity increases to simulate the ribrake. Therefore, the mathematical model is feasible for the sof the pressure distribution of lining plates. When the Younmodulus of elasticity of the metal shoe decreases, the pressu

Fig. 12 Pressure distribution versus lining arc for variousangles of actuation force

Journal of Mechanical Design


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the center of the metal shoe decreases, but the pressure incrtoward the supporting point. This result may come from the buling effect that the pressure at the center of the brake sholowered.

The pressure close to the supporting point decreases andpressure distribution is more uniform if the arc length of the meshoe increases as shown in Fig. 5. The similar result of a muniform pressure can be obtained if the arc length of the linplate increases as shown in Fig. 6. If the arc length of lining pldecreases, a local high pressure may occur. Figure 7 showsthe maximum pressure decreases if the lining plate is shiftedward the supporting point and the pressure distribution becomore uniform. If the thickness of the lining plate increases,maximum pressure decreases and a more uniform pressure dbution can be obtained as shown in Fig. 8. Figure 9 shows thamaximum pressure decreases and the pressure distributioncomes more uniform if the friction coefficient decreases. If tYoung’s modulus of elasticity of the lining plate decreases,maximum pressure decreases and a more uniform pressure dbution can be achieved as shown in Fig. 10. Figure 11 showsthe maximum pressure and the pressure distribution decreaboth Young’s moduli of elasticity of the metal shoe and the liniplate decrease while their ratio remains the same. The shapthe pressure distribution remain the same. This result identthat obtained by Wintle@4#. The maximum pressure decreases athe pressure distribution becomes more uniform if the angle ofactuation force increases as shown in Fig. 12. Reducing the aof the actuation force decreases the pressure at the actuationand may result in the separation of the lining plate at the actuaend from the brake drum.

Frictional work in a brake is primarily converted into heat thincreases the temperature of shoes and lining plates of the band changes the properties, particularly in the thermal expansThe circumferential distribution of pressure along the brake linarc, that plays an important role in determining the torque genated, depends upon the thermal expansion, flexural deflectioncontact at the interface. Heat generated at the sliding interbetween the friction material and the mating surface of the bris not uniformly distributed and leads to pronounced variationhigh surface temperature and the high interface pressure@2#. Fric-tion interface temperature in the brake is affected by the interfpressure distribution, so that regions of local high interface prsure will also be regions of high surface temperature@6#. Theeffects in the areas of frictional heat generation and dissipawith thermal consideration on the distribution of pressure, thavery important in minimizing brake thermal problems, is requirin order to incorporate the non-linear characteristics of frictimaterials to obtain the uniform friction interface pressure.

The distribution of pressure is significantly affected by druflexibility, which relates to the Young’s modulus of elasticity. Thpressure generated for the flexible drum is reduced because aactuation displacement is absorbed by drum deflection therreducing both effective cam lift and actuation force.

ConclusionBased on the similarity between the calculated results and

available data, the results suggest that the software developedbe used to predict the pressure distribution of the lining platethe drum brake. If the deflections of the metal shoe and the linplate and the thermal effect are neglected, the pressure distribuis a sinusoidal curve. If these deflections are considered inanalysis, the pressure distribution will be changed. In orderdesign a drum brake with a more uniform pressure acting onlining plate and to prolong the life of the friction material the alengths of the brake shoe and the lining plate should be longer

MARCH 2002, Vol. 124 Õ 119


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location of the lining plate should be close to the supporting pothe lining plate should be thicker; the friction coefficient shouldsmaller; the Young’s modulus of elasticity for the material usedthe brake shoe and the lining plate should be smaller; andangle of the actuation force should be larger. However, it shobe noted that the material with the small Young’s moduluselasticity might wear easily and faster. Although the material wa small friction coefficient can provide more uniform pressuacting on the lining plate, it should not be too small. Otherwisewill lower the braking effect. The friction coefficient of materiaabout 0.2 is a good selection for application. In addition, a laractuation force should be applied with the actuation angle ofto compensate and increase the braking effort.

AcknowledgmentThe authors would like to express their sincere appreciation

the grant no. NSC88-2212-E-002-036 from the National ScieCouncil of the Republic of China for this study and to GrantHuang for comments and revisions made on this manuscript.

Nomenclature

A 5 areae 5 unit vector inj directionF 5 actuation forcef 5 body force in thei direction

G 5 Galerkin vectorGi j ,Hi j 5 influence coefficient matrix regarding pointsi and j

GS 5 shear modulusP 5 traction force matrixr 5 distance between pointsM andQT 5 traction basis solution matrix or transverse matrix

T8 5 coordinate transform matrixt 5 traction force basic solution

U 5 displacement matrixu 5 displacement basic solutionn 5 Poisson’s ratio

x,y 5 coordinateV 5 integration domainm 5 friction coefficienta 5 angle measured from line of centers of drum brake

and pin to line of center of drum brake and near edof lining plate

f 5 angle measured from the line of actuation force tothe normal vector of the surface at the load point

f 5 angle of arc length of lining plate

120 Õ Vol. 124, MARCH 2002


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force.

ge

G 5 boundary

Subscripts

ij 5 ith row andjth column in matrixn 5 normal componentl 5 in l directiont 5 tangential component

L 5 lining plateS 5 metal shoe

x,y 5 direction

Superscripts

i 5 point iij 5 displacement vector ofQ in j direction with a unit

force acted on pointM in i direction

References@1# Day, A. J., Harding, P. R. J., and Newcomb, T. P., 1979, ‘‘A Finite Eleme

Approach to Drum Brake Analysis,’’ Proc. ImechE,193, pp. 401–406.@2# Day, A. J., Harding, P. R. J., and Newcomb, T. P., 1984, ‘‘Combined Ther

and Mechanical Analysis of Drum Brakes,’’ Proc. ImechE, No. 15,198D, pp.287–294.

@3# Mashinostroeniya, V., 1986, ‘‘Calculation of the Shoe of Drum Brakes,’’ Sviet Engineering Research,6, No. 7, pp. 23–25.

@4# Wintle, J. B., 1978, ‘‘Torque Variations of Drum Brakes,’’ MSc. ThesiLoughborough University of Technology.

@5# Day, A. J., 1991, ‘‘Drum Brake Interface Pressure Distribution,’’ Proc. Imech205, pp. 127–136.

@6# Day, A. J., Tirovic, M., and Newcomb, T. P., 1991, ‘‘Thermal Effects anPressure Distribution in Brakes,’’ Proc. ImechE,205D, pp. 199–205.

@7# Watson, C., and Newcomb, T. P., 1990, ‘‘A Three-dimensional Finite Approto Drum Brake Analysis,’’ Proc. ImechE,204, pp. 93–102.

@8# Scieszka, S. F., and Barecki, Z., 1984, ‘‘Geometry of Contact Between BrShoes and Drums,’’ The South Africa Mech., Engr., pp. 324–329.

@9# Becker, A. A., 1992,The Boundary Element Method in Engineering, McGraw-Hill, Inc., pp. 62-90, 161-179.

@10# Rizzo, F. J., 1967, ‘‘An Integral Equation Approach to Boundary Value Prolems of Classical Elastostatics,’’ Q. Appl. Math.,25, pp. 83–95.

@11# Cruse, T. A., 1968, ‘‘A Direct Formulation and Numerical Solution of thGeneral Transient Elastodynamic Problems-II,’’ J. Math. Anal. Appl.,22, pp.341–355.

@12# Brebbia, C. A., 1980,Boundary Element Techniques in Engineering, Butter-worth & Co. Ltd., pp. 120-150.

@13# Rizzo, F. J., and Shippy, D. J., 1968, ‘‘A Formulation and Solution Procedfor the General Non-homogeneous Elastic Inclusion Problem,’’ Int. J. SoStruct.,4, pp. 1161–1179.

@14# Swedlow, J. L., and Cruse, T. A., 1971, ‘‘Formulation of Boundary IntegEquations for 3-D Elastoplastic Flow,’’ Int. J. Solids Struct.,7, pp. 1673–1683.

@15# Day, A. J., and Harding, P. R. J., 1983, ‘‘Performance Variation of Cam Oerated Drum Brake,’’ Proc. R. Soc. London, Ser. A, pp. 60–77.

@16# Offner, D., 1969, ‘‘Generalizing the Analysis of Shoe-type Brake-Clutch Stems,’’ ASME J. Eng. Ind., pp. 694–701.

@17# Day, A. J., 1988, ‘‘An Analysis of Speed, Temperature, and Performance Cacteristics of Automotive Drum Brakes,’’ ASME J. Tribol.,110, p. 298–305.



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On Pressure Distributions of Drum Brakes