Download - Contact Analysis for Drum Brakes and Disk Brakes Using Adina

Contact analysis for drum brakes and disk brakes usingADINA

C. Hohmann*, K. Schi�ner, K. Oerter, H. Reese

Institute of Engineering Mechanics and Control Engineering, University of Siegen, Paul-Bonatz Str. 9±11, Fachbereich 11,

Mashinentechnik, 57076, Siegen, Germany

Abstract

Brakes in cars and trucks are safety parts. Requirements not only in performance but also in comfort,serviceability and working lifetime are high and rising. Optimal design of today's brake systems is found usingadditional calculations based on ®nite element methods. For both types of brake systems, drum brakes and disk

brakes, the di�erent parts of brakes, i.e. the brake pad with the friction material, the counter body and calliper, canbe modelled. Two examples are given in this paper: a drum brake of a trailer and a typical disk brake used inpassenger cars. The main problem to be solved is the calculation of the distribution of contact forces between brakepad and counter body (drum or disk). The contact problem includes friction and is solved using the ADINA 7.1

sparse solver. After the brake pressure is applied, the turning moment on the axle rises constantly until the drum ordisk respectively changes from sticking to sliding condition. It is shown that the sparse solver is highly e�cient forthis sophisticated nonlinear problem. Results include deformation, stress distribution, contact pressure and showing

which regions of the contact area are in sticking or sliding condition. # 1999 Elsevier Science Ltd. All rightsreserved.

1. Introduction

1.1. Brake construction

Brakes in cars are expected to work properly with a

minimum amount of service. The purpose of brakes isto reduce the velocity or to maintain it when the ve-hicle is driving downhill. Without brakes it would not

be possible to control the speed of the vehicle.Nevertheless the design of brakes is generally underes-timated. In brakes high amounts of energy are trans-formed during short periods. This is underlined by the

fact, that often the braking power is several times

higher than the power of the engine.

In cars and trucks di�erent types of braking systems

are used. In this paper only wheel brakes based on

friction are considered. Generally two design forms are

used: disk and drum brakes. The demands made by

the vehicle industry are strict. The requirements will

also rise with future development of lighter and more

economical cars. This supports the need for e�cient

methods of calculating brake systems. The ®nite el-

ement method is an ideal tool for this purpose. It is

suited for analysis of both stress and temperature. In

this paper the contact problem is described for disk

brakes and drum brakes. For example a drum brake

used in trucks and a disk brake used in small passen-

ger cars are presented.

Computers and Structures 72 (1999) 185±198

0045-7949/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.

PII: S0045-7949(99 )00007-3

* Corresponding author. Fax: +49-271-7402436.

In order to transfer the kinetic energy of the vehicleinto heat, friction brakes are commonly used at each

wheel of the vehicle. The area of contact is the originof the heat sources. Cooling surfaces emit heat into theenvironment. Nevertheless the temperatures can get as

high as 9008C (16508F) at the contact area.For proper operation the following criteria have to

be considered

. high and stable coe�cient of friction;

. good thermal capacity;

. good wear resistance of the tribo system (brake lin-

ings and disk respectively drum);. mechanical resistance of material;

Fig. 1. Frequently used types of friction brakes: (a) disk brake and (b) drum brake.

Fig. 2. Transmission of forces in a brake system [1].

C. Hohmann et al. / Computers and Structures 72 (1999) 185±198186

. weight optimized construction;

. and use of environmental suited materials.

Despite di�erent geometric design (see Fig. 1), both

types of brakes use the same principle to create thebraking force: ®xed brake shoes are pressed against arotating counter body. Due to friction the brake forceis acting contrary to the motion of the counter body

thereby reducing its velocity. The resulting frictionforce is proportional to the normal force N and thecoe�cient of friction m. The brake parameter C � is

de®ned as the relation between friction forces and theapplied force and is used in the comparison of di�erentbrake designs.

A typical brake used in cars consists of the operatingdevice (pedal or hand lever), the transfer unit and thewheel brake (Fig. 2). For disk and drum brakes thenormal force N is applied, using mechanical, pneu-

matic or hydraulic transfer units. A servo unit raisesthe force FBet applied by the driver:

N � iAFBet �1�

The factor iA is the total transmission ratio of the

brake. The friction force R=mN acts on the rubbingsurface. The distance between the friction force andthe centre of revolution is the e�ective radius re�. The

origin of the friction force for disk brakes is approxi-mately the middle of the friction surface, depending onthe shape of the friction area. In the case of drum

brakes the e�ective radius is the inner radius of thedrum.E�cient software and hardware is needed for the

simulation of complete brake systems. ADINA

Version 7.1 simpli®es the construction of the complexthree-dimensional structures together with providingreliable algorithms for the solution of the contact pro-

blem. Using the sparse solver the calculation time isreduced signi®cantly.

2. Design of drum brakes

2.1. Analytical calculation

The analytical calculation of drum brakes is basedon the assumptions of Koessler [2,3]. He describes thedistribution of the normal pressure between drum and

brake linings with cosine functions (see Fig. 3a). Theassumed distribution only ®ts if the curvatures of thebrake lining and of the drum are equal. The radius of

the brake lining is changed during operation due towear. This results in irregular distributions of the con-tact pressure, which are shown in Fig. 3b and c.

At each point of the contact area the coe�cient m ofsliding friction is constant. The distributed frictionforces are caused by the pressure and are proportionalto the coe�cient of sliding friction:

dR � m dN �2�

The braking moment can be calculated with the dis-tance r of the contact point to the centre of revolution.

The summated friction forces make up the peripheralforce R1 for the leading shoe and R2 for the trailingshoe. Both act against the rotation of the drum with

Fig. 3. Normal pressure distribution, assumed by Koessler (a), for a new (b) and an old lining (c).

C. Hohmann et al. / Computers and Structures 72 (1999) 185±198 187

the lever arm r:

R1 ��a2ÿa1

dR, R2 ��a2ÿa1

dR �3�

The brake shoes are spread by a rotation of the S-cam

clamping device. Therefore the forces S1 and S2 actupon the rolls (Fig. 4a). They are in balance with the

normal pressure and peripheral pressure acting on thelining surfaces (Fig. 4b) and the resulting forces F1 and

F2 acting at the bolts. The static conditions for equili-brium can be formulated for each direction (x, y and

a ) and both brake shoes in the form:

Fig. 4. Applied forces and reaction forces for the trailing shoe (a), and normal and tangential pressure for a contact point (b).

Fig. 5. Parts of an S-cam driven drum brake and FEM discretization.


Fig.6.Boundary

conditionsandapplied

displacements.


S1 cos g1 ÿ�ÿa2ÿa1

dN �x 1�

�ÿa2ÿa1

dR �x 1� F1 cos d1 � 0

S2 cos g2 ÿ�ÿa2ÿa1

dN �x 2�

�ÿa2ÿa1

dR �x 2� F2 cos d2 � 0

S1 sin g1 ÿ�ÿa2ÿa1

dN �y 1ÿ

�ÿa2ÿa1

dR �y 1� F1 sin d1 � 0

S2 sin g2 ÿ�ÿa2ÿa1

dN �y 2ÿ

�ÿa2ÿa1

dR �y 2� F2 sin d2 � 0

S1 � cÿ�ÿa2ÿa1

r � dRÿ F1 � a � 0

S2 � cÿ�ÿa2ÿa1

r � dRÿ F2 � a � 0 �4�

These equations have to be solved for both shoes. Thesolution of the integrals results in a relationshipbetween applied forces and the braking moment. With

a0=a1+a2 the brake parameter C � is:

C � � R1 � R2

S1 � S2� 2

f �a0�a0r�

f �a0�a0r

�2

ÿm2h

rm with

f �a0� � a0 � sin�a0�4 sin

�1

2a0

��5�

Type of drum brake: Simplex Duo-duplex Duo-servo

Brake parameter: C �1 2.0 C �13.0 C �1 5.0

The ®nite element model is used, to calculate a more

realistic distribution of the contact pressure, consider-ing the elastic deformations of both the linings and thedrum.

2.2. Finite element calculation

2.2.1. Description of the model

A complete three-dimensional structure of the drumbrake has to be modelled. The ®nite element modelconsists of three independent parts: the drum, the lead-

ing shoe and the trailing shoe (see Fig. 5). All parts,with the exception of the brake linings, are made ofsteel. Normally the brake linings are riveted to thebrake shoe. In the model, however, uniform connec-

tion between lining and shoe is considered. Threedimensional elements with eight nodes are used tomodel the solids. The de®nitions of nodes and elements

are created with the PATRAN-preprocessor. AFORTRAN program translates the node and elementlists for ADINA-IN. Contact areas are de®ned for the

rubbing faces of the drum and the four lining pad sur-faces.The ®xed bolts are modelled with truss elements.

This enables the brake shoes to rotate around the ®xedbolts (see Fig. 6). The complete S-cam clamping deviceis neglected in this model. Since the S-cam providesstraight spreading of the brake shoes, displacements

are applied to the centre points of the rolls. The rollsare also modelled with truss elements. The correlationbetween S-cam rotation and the displacements of the

rolls is given by the manufacturer. The connectionbetween the rolls and the shoes is also modelled usingtruss elements.

2.2.2. Solution techniqueAlthough the mesh is quite coarse, the number of

nodes is as high as 22,000. Although a static analysis is

performed this is a large scale problem. The model isloaded in two intervals, Fig. 7. In the ®rst interval the

Fig. 7. Time functions for the spreading of the shoes (a) and rotation of the drum (b).


brake shoes are spread for an initial contact between

the brake linings and the drum. The value and angleof the displacements are de®ned by the geometry ofthe S-cam.

After the spreading of the shoes, the drum is rotatedin the second interval. Again displacements are used todescribe the rotation. For the nodes at the ¯ange of

the drum, skew systems are de®ned and the displace-ments in the radial and the axial directions are ®xed(Fig. 8). The drum is still a free system with respect to

the rotation. In order to avoid a singular system sti�-ness matrix, for the nodal points of the drum ¯ange

displacements are prescribed in the circumferential

direction.

2.2.3. Results of the drum brake analysis

Deformation of the drum and the brakes shoes iscalculated using ADINA Version 7.0. The results areshown in respect to the rotated drum. Figs. 9 and 10

show the contact forces on the brake linings. The con-tact surfaces are laid out from the bolt end to the endclose to the rolls. It can be seen, that the contact forces

have a non-uniform distribution which di�ers from theassumed distribution of the analytic solution by

Fig. 8. Skew system (a) for the de®nition of prescribed displacements (b) for drum rotation.

Fig. 9. Distribution of contact forces on the trailing shoe.


Fig. 10. Distribution of contact forces on the leading shoe.

Fig. 11. E�ective stresses for the drum in N/mm2.


Koessler [2]. High peaks are apparent at the top end

of the lining, especially on the trailing shoe, close to

the roll. The distribution of the contact pressure tends

to that of the old lining shown in Fig. 3c for the ana-

lytic solution.

Furthermore the reaction forces at the ®xed bolts

and the rolls are compared with the results of the ana-

lytic calculation for the supporting forces F and S.

Although the magnitudes of the supporting forcesmatch, the angles of the forces do not correspond to

the results of the analytical approach given by

Koessler [2].

Since the calculated contact situation is di�erent

from the assumed contact situation in the analytic the-

ory, the supporting forces of the two situations also

di�er. Reaction forces are also gained at the nodes of

the drum, for which displacements are prescribed.

These reaction forces are used to calculate the braking

moment.

Drum deformations and e�ective stresses are shown

in Fig. 11. The stresses are low because of the massive

construction of the drum. Nevertheless, it is not over-

dimensioned because most of the heat generated by

friction is transferred into the drum. Maximum stresses

at the shoes (Figs. 12 and 13) are higher than those ofthe drum. The shown stress distribution can be used in

optimizing the structure of the shoes.

2. Design of disk brakes

3.1. Analytical calculation

Unlike the drum brake, the friction surfaces of disk

brakes are planar. The main advantage of the diskbrake is that the heat can be transferred into the en-vironment directly over the free surfaces of the disk[4]. Further improvements for the heat convection are

gained by ventilation channels or holes in the frictionring. However, in this example a solid disk is con-sidered.

Normal forces are generated by the hydraulic press-ure in order to press the brake pads against the disk.The piston presses the pad against the disk. A reaction

takes place where the calliper transfers the force to thepad on the other side of the disk (Fig. 13). The normalforce for each pad is

Fig. 12. E�ective stresses for both brake shoes in N/mm2.


N � pK � AK �6�

The tangential stress generated on the friction surfaceresults in the friction forces on each side of the disk.

With the coe�cient m of friction, this force is

R � m �N �7�

As the friction forces act on both sides of the disk, thebrake parameter C � for this type of brakes is

C � � 2R

N� 2mN

N� 2m �8�

3.2. FEM calculation

3.2.1. ModelThe ®nite element model consists of the disk, the

calliper and the two brake pads (see also Watson andNewcomb [5]). The geometry is taken directly from thepart and the model is created using the geometric el-

ements provided by ADINA-IN. Mapped meshing isused to create the element mesh of the disk. For thecomplex structures of the pads and the calliper a free

formed mesh with a su�cient density is constructed.

The entire mesh consists out of parabolic three-dimen-

sional-solid elements (21 nodes for one brick element).

Linear elastic material models are used for both the

lining material and the brake parts which are made of

steel. The lining is glued to the supporting plates of

the pads by the pressing process.

The calliper as well as the brake pads are free to

move in the axial direction. Together they are guided

by bolts through the sprocket holes and grooves at the

sides of the supporting plate. By this means they press

against the supporting structure of the steering knuckle

which is not modelled with ®nite elements but simu-

lated by boundary conditions. The grooves and the

sprocket holes are represented by nodes ®xed in the

directions perpendicular to the axial direction. Since

the disk is free to rotate, it is ®xed to a point on the

axle by beam elements with high sti�ness.

For the contact surfaces on the linings and the fric-

tion ring of the disk a friction coe�cient of m=0.4 is

de®ned. Although in experiments di�erent values for

static and sliding friction are found, in the calculation

the value is constant due to program restrictions.

Fig. 13. Finite element model of the disk brake (calliper in sectional drawing).


3.2.2. Solution technique

Again the structure is loaded in two intervals. First

the hydraulic pressure is applied on the brake pad and

the calliper (Time=1.00). In the detail of Fig. 13 one

can see the surfaces on which the pressure is applied.

In the second interval (time=2.00) the disk is rotated

about the axis. The second interval corresponds to the

situation of a vehicle with an automatic transmission

stopping with the gear switched in driving position.

The motor moment is acting against the braking

moment. When the motor moment is higher than the

braking moment the car begins to move. The rotation

is prescribed about the centre point of the beam struc-

ture. The beam structure transfers the rotation to the

disk.

Since the calliper and the pads are free to move in

the axial direction, truss elements with low sti�ness areapplied in order to avoid rigid body movement. Theseelements do not alter the results but make the sti�ness

matrix positive de®nite. They also stabilize the processof ®nding contact in the ®rst load step.For this problem the sparse solver with automatic

time stepping is used. The time steps have to be verysmall, especially at the start and the end of the ®rstinterval (application of brake pressure).

3.2.3. ResultsThe sparse solver is very e�cient. Contact is found

easily for the plain surfaces of the linings and the disk.

The e�ective stress and the contact pressure are com-pared for two states. The ®rst state is after applicationof the brake pressure. Maximum stress occurs at the

Fig. 14. E�ective stresses in N/m2 for an applied brake pressure.


calliper as it is bent (Fig. 14). This causes higher con-tact pressure at the outer radius of the linings (Fig.16). For the second interval the disk is rotated, thereby

pressing the brake pads against the guidance of thesteering knuckle (boundary conditions). This results inhigher stresses in the supporting plate of the brake

pads (Fig. 15). First the disk sticks to the linings. Ananalysis of node displacements of lining and disk sur-faces reveals that they are moved together. This con-dition is called static friction. The static friction forces

H at each contact node are always smaller than thede®ned friction forces R for sliding friction:

HRR � mN �9�

After the disk is rotated through a certain angle, thedisk breaks free. The peripheral forces have then

reached the limit de®ned by the coe�cient of friction(H=R ). Fig. 17 shows that the contact pressurechanges [6]. The exact point of change from sticking to

sliding condition can be found. Fig. 18 marks thedi�erent regions of the contact surface as well as de®n-ing their condition. Depending upon the load step,

regions of the lining surface are shown which are stillin a sticking condition and those which are in a slidingcondition. This is achieved by band plotting of theresultant de®ned in the following way:

condition � friction force

contact forceRm �10�

In the case of sliding, this condition is equal to thecoe�cient of friction input for m in ADINA-IN. This

Fig. 15. E�ective stresses in N/m2 for an applied brake pressure and a rotated disk.


Fig. 16. Contact pressure in N/m2 for an applied brake pressure.

Fig. 17. Contact pressure in N/m2 for an applied brake pressure and a rotated disk.


analysis is the basis for dynamic calculations of fric-

tion-induced vibrations.

4. Conclusion

Only a short introduction into the fundamentals ofbrake constructions is presented in this paper. The

basic formulas for the analytic calculation are givenfor di�erent brake types. Drum and disk brakes aremodelled with ®nite elements using ADINA-IN. It is

possible to import lists of nodal coordinates and el-ement de®nitions, which are created using a preproces-sor such as PATRAN. It is more ¯exible to use the

geometry de®nitions provided by ADINA-IN.The correct calculation of contact is essential for the

design of friction brakes. The sparse solver im-plemented in ADINA Version 7.1 reduces job duration

time of these large models to a few hours (DEC AlphaServer 4000). The reduction of elapsed time makes itpossible to investigate the e�ect of altering design vari-

ables.The presented results are a small overview of what is

possible. In particular the detailed analysis of the

change from stick to shift is a step in understandingthe friction process. Additional friction laws which dis-tinguish between the coe�cient of static friction and ofsliding friction would be a helpful tool for investigating

stick/slip problems. Consideration of dynamic e�ects

will be possible with more powerful hardware in thenear future.Thermal analysis can also be performed using these

models in ADINA-T. The coupling of thermal and

elastomechanical calculations is a great advantage ofthe ADINA system.

References

[1] Wallentowitz H. LaÈ ngsdynamik von Kraftfahrzeugen.

Vorlesungsumdruck Kraftfahrzeugtechnik II. Institut fuÈ r

Kraftfahrwesen, RWTH Aachen: Schriftenreihe

Automobiltechnik, 1997.

[2] Koessler P. Berechnung von Innenbacken-Bremsen fuÈ r

Kraftfahrzeuge. Stuttgart: Franckh'sche, 1957.

[3] Koessler P. Grundlagen der Fahrzeugtechnik,

Originalausgabe. MuÈ nchen: Heine, 1985.

[4] Day AJ, Newcomb TP. The dissipation of frictional

energy from the interface of an annular disk brake. Proc

Inst Mech Engng 1984;198D(11):201±9.

[5] Watson C, Newcomb TT. A three-dimensional ®nite el-

ement approach to drum brake analysis. Proc Inst Mech

Engng, D: J Automobile Engng 1990;204(D2):93±101.

[6] Tirovic M, Day AJ. Disk brake interface pressure distri-

bution. Proc Inst Mech Engng, D: J Automobile Engng

1991;205:137±46.

Fig. 18. Contact areas in sticking or sliding condition for a certain time step.


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