-
ib
ka S
Received 6 September 2010
Keywords:BrakingPad/disc systemFrictional heatingMoving heat
sourceHeat conductionFinite element method
andmembers of the disc brake system basing on two- and
three-dimensional FE modelling techniques and
bers o
hazardous environments such as coal mines [1e4].Complexity of
the friction and wear processes state major dif-
culty of formulating universal physical model to determine
criticaloperation conditions for specied case of braking action.
Exactanalytical solutions of temperature of friction pair may be
obtainedwith restriction to semi-spaces, plane parallel strip or
semi-planes.
heating systems should be noticed.Rotating systems such as disc
brakes in which pads cover solely
the segment of rubbing path of a disc, are intrinsically
submitted tonon-axisymmetric thermal load. Simplications of a real
three-dimensional modelling techniques into
two-dimensionalityrelating to the heat rate uniformly distributed
in circumferentialdirection were so far accomplished [7e11]. In
point of fact theyenter simplications of three-dimensional process
of heating,which is omitted in systems where the friction surface
of a body
* Corresponding author. Tel.: 48 85 746 93 12; fax: 48 85 746 92
10.
Contents lists availab
a
sev
Applied Thermal Engineering 31 (2011) 1003e1012E-mail address:
[email protected] (P. Grzes).kinetic energy conversion
into heat at the pad/disc interface. Theincrease of friction moment
is a limited quantity and depends onthe coefcient of friction,
radius of rubbing path, and forces that acton the pads. The process
of slipping leads the increase of temper-ature, whereas its peak
value is one of the most crucial factor in thecourse of action to
occur. The temperature on the contact surfacesof the tribosystem
during emergency braking intensied bysignicant thermal load due to
frictional forces as well as the highvelocity of the process is, in
particular, important to predict in
gular prismand circular source on a rotating
cylinderwereproposedin article [5]. The temperature and the thermal
constriction resis-tance as a function of geometric characters and
velocity weredetermined. The temperature and the thermal stresses
of the pad(the strip) sliding with the constant retardation on a
surface of thedisc (the semi-space) both during heating and after
the moment ofstandstill were studied [6]. However these geometric
congurationsmay correlate with actual engineering applications,
absence of theexact solutions, primarily application of nite areas
of frictional1. Introduction
The sliding contact of the mem1359-4311/$ e see front matter
2010 Elsevier Ltd.doi:10.1016/j.applthermaleng.2010.12.016second,
the three-dimensional rotor is subjected to the non-axisymmetric
thermal load to simulaterealistic thermal behaviour of the brake
action. Operation conditions, thermo-physical properties
ofmaterials and dimensions of the brake system were adopted from
the real representation of the brakingprocess of the passenger
vehicle. Arbitrarily selected four values of the velocities at the
moment of brakeengagement were applied to the models so as to
investigate theirs inuence on the obtained solutions ofthe
temperature evolutions on the contact surface of the disc volume
referring to two separated niteelement analysis. The large amount
of heat generated at the pad/disc interface during emergency
brakingindisputably evokes non-uniform temperature distributions in
the domain of the rotor, whereas the padelement is constantly
heated during mutual sliding. The obtained results of the original
code of three-dimensional modelling technique implemented to the
conventional FE software revel high agreementwith the solution of
simplied process of friction heating.
2010 Elsevier Ltd. All rights reserved.
f disc brake results in
Typically the heat ux condition is applied at the region of
contact.The three-dimensional temperature distributions of a moving
heatsource problem with a rectangular and elliptic source on a
rectan-where the intensity of heat ux was assumed to be uniformly
distributed on the friction surface of discduring braking process,
and the heat is transferred exclusively in axial direction, whereas
during theAccepted 6 December 2010Available online 15 December
2010
complexity of the phenomenon. First step of the analysis based
on the previously developed modelAnalysis of disc brake temperature
distrunder non-axisymmetric load
Adam Adamowicz, Piotr Grzes*
Faculty of Mechanical Engineering, Bialystok University of
Technology (BUT), 45C Wiejs
a r t i c l e i n f o
Article history:
a b s t r a c t
This paper aims to study
Applied Therm
journal homepage: www.elAll rights reserved.ution during single
braking
treet, Bialystok 15-351, Poland
compare the temperature distributions caused by mutual sliding
of two
le at ScienceDirect
l Engineering
ier .com/locate/apthermeng
-
T0 initial temperature, C{T} temperature vectorV velocity of the
vehicle, km/hV0 initial velocity of the vehicle, km/hDx the mesh
size (smallest element dimension), mz axial coordinate, m
Greek symbolsg heat partition ratiod thickness3 coefcient of
thermal activityq circumferential coordinate, degr density,
kg/m3
f0 cover angle of pad, degu angular velocity, 1/su0 initial
angular velocity, 1/s
rmal Engineering 31 (2011) 1003e1012and counterbody is equal
aircraft brakes and clutch systems [12].Two models of heat
dissipation utilizing axisymmetric arrange-ment of a disc brake:
namely macroscopic and microscopic modelwere implemented in
articles [7,8]. In the macroscopic model rstlaw of thermodynamics
has been taken into account and formicroscopic model various
characteristics such as braking time,velocity of the vehicle,
thermo-physical properties of materials,contact pressure, and
dimensions of a real disc brake assembly havebeen studied. Greens
functions were used to determine tempera-ture distributions in the
disc and pad volume [8].
Formulation of the heat ux activity during frictional
heatingindependent of circumferential coordinate q may cause
unrealisticcontact conditions and falsify actual, elastic
distortions. In order tosimulate reasonable emergency braking
process, the three-dimensional FE model assuming nonlinear pressure
distributionand angular velocity variability was proposed in
article [13]. The
Nomenclature
c specic heat, J/kg K[C] heat capacity matrixf coefcient of
frictionh heat transfer coefcient, W/(m2 K)k thermal diffusivity,
m2/sK thermal conductivity, W/(m K)[K] conductivity matrixp
pressure, MPap0 contact pressure, MPaq intensity of heat ux,
W/m2
r inner radius, mR outer radius, m{R} heat source vectort time,
sts braking time, st0s time of braking with constant deceleration,
sDt time step, sT temperature, CTN ambient temperature, C
A. Adamowicz, P. Grzes / Applied The1004thermo-physical
properties of materials independent on tempera-ture have been
used.
Operation of disc brake above the certain range of the
velocitymay lead to thermoelastic distortions and in consequence to
non-uniform pressure distribution due the interchanged moments
ofcontact and its absence during rotation, known as
thermoelasticinstability (TEI) [14]. The upwind scheme in nite
elementformulation to prevent possible perturbations owing high
Pecletnumber was developed [15].
The conventional nite element method is well adopted
instationaryproblems, however three-dimensionalmodellingof
partsbeing inmotion imposes very nemesh due to high values of
Pecletnumber, which determines the range of the velocity, above
whichoscillations may occur. The hybrid method combining the
niteelement method and the fast Fourier transform (FFT) technique,
asan alternative approach in order to reduce computational
timewithout loss of the temperaturealterationsowing the
circumferenceof a disc brake was used [3,4,9,10,15,16]. The
temperature distribu-tions during different operation conditions
were presented. Thereview of FEM-solutions of thermal problems of
friction duringbraking is given in the article of Yevtushenko and
Grzes [17].
In order to predict the temperature on the contact surfaces
ofelements of disc brake, experimental examinations
includinginfrared techniques such as two colour pyrometry [18],
infraredmapping [3,4] as well as thermocouples [1,19,20] were
developed.In this paper three-dimensional nite element analysis
regardingmovable behaviour of the disc brake system was developed
andcompared with the two-dimensional modelling of frictional
heatingproblemderived from previous authors study [11]. In order to
assureaccuracy of the solution several nite element meshes of the
twospecied models of the real disc brake was tested.
Investigationcomprises dissimilar evolutions of the external
loadduringmountaindescent with constant velocity and application of
single, emergencybraking to standstill. For the purpose of
comparison of obtainedresults, dimensions of the disc brake,
operation conditions andthermo-physical material properties were
adopted from the studydeveloped previously [8]. Special concern is
focused on the descrip-tionof theFEmodelling techniqueof
themovingheat sourceproblemcorresponding to axial conguration of
the same phenomenon.
Subscriptsd indicates discp indicates padw indicates wheel2.
Statement of the problem
The disc brake system comprises in the majority two
elements:rotating axisymmetric disc and immovable non-axisymmetric
pad(Fig. 1). When the braking process occurs, the hydraulic
pressureforces the piston and therefore pads and disc brake are in
sliding
Fig. 1. A schematic diagram with three-dimensional nite element
mesh of a pad/discbrake system.
-
ermapad in operation is equal to the apparent surface in the
slidingmotion. The contact pressure is uniformly distributed over
allfriction surfaces hence the heat generation of the midplane
isconsidered as symmetric;
(3) The average of the intensity of heat ux into disc on the
contactarea equals [21]:
qdr; q; tjzdd 1 g fprut; rp r Rp;0 q 2p; 0 t ts; 1
qpr;q;tjzdp gfprut; rp rRp; 0 qf0; 0 t ts; (2)
(4) The heat partitioning factor representing the fraction of
fric-tional heat ux entering the pad has the following form
[22]
g 11 3; (3)
where
3 Kdkp
pKp
kd
p ; (4)is the thermal activity coefcient [23]
(5) The frictional heat due to Newtons law has been dissipated
toatmosphere on the other surfaces. The heat transfer coefcienth is
constant during simulation of braking process;
(6) Radiation is neglected by virtue of short braking time
andhence relatively low temperature;
(7) The wear on the contact surface is negligible.
In the three-dimensional model of solid disc, single surface of
itssymmetry in axial direction is insulated owing nature of
consideredphenomenon of heating. On both, the external, internal
surface offactory outcomes. Thus primarily relevant in the present
study wasto examine proposed technique of moving heat source
modellingproblem. The solid disc brake was analyzed, where the
dimensions,operating parameters and properties of materials were
adoptedfrom the study of Talati and Jalalifar [8].
For both types of disc models it has been assumed as
follows:
(1) Material properties are isotropic and independent of
thetemperature;
(2) The nominal surface of contact between the disc brake and
thecontact. The friction at pad/disc interface resists the movement
andthe vehicle slows down, remains at the same level of the
velocityduring mountain descents or eventually, stops. The
frictionbetween disc and pads always opposes motion and the heat
isgenerated due to conversion of the kinetic energy, whose portion
isdissipated by convection to the atmosphere in accordance
toNewtons law. However radiation as a third type of heat
exchangealways takes place, owing its negligibly amount is omitted
in themodelling of the presented phenomenon.
In this paper non-axisymmetric thermal load due to the
fric-tional heat generated during the single braking process
imple-mented in the three-dimensional model is investigated to
compareobtained solution of the temperature evolution on the disc
frictionsurface with the two-dimensional representation of the
constantheating studied previously [19] and to answer if there is
theaccurate range of the velocity under which the uniform heat
uxratio upon the circumference of the disc may results in the
satis-
A. Adamowicz, P. Grzes / Applied Ththe disc and contact surface
free from friction, the convectionconditions are prescribed due to
the Newtons law of cooling. In thezone of temporary contact of the
pad and disc, the thermal ux isassigned, which differs in the area
of disc at any instant of brakingtime corresponding to the
components of the intensity of heat uxproduct Eq. (2). The contact
pressure p0 is constant during theanalysis, whereas the velocity
for the rst case of the analysisdecreases linearly with time
ut u0 1 t
t0s
!; 0 t t0s ; (5)
and during the second constant value of the velocity is
assumed.
3. Mathematical model
In order to determine the temperature distributions,
bothanalytical and numerical techniques have been employed.
Thestarting point of the analysis of the temperature elds in the
discvolume, is the parabolic heat conduction equation given in
thecylindrical coordinate system (r, q, z) [24]
v2Tvr2
1rvTvr
1r2
v2T
vq2 v
2Tvz2
1kd
vTvt
uvTvq
; rd r Rd;
0 q 2p; 0 < z < dd; t > 0 6
The boundary and initial conditions of non-stationary problemare
established as follows (Fig. 1)
KdvTvz
z0 qdr; q; t; rp r Rp; 0 q 2p;0 t ts; G 7
KdvTvz
z0 hTN Tr; q; t; rd r rp; 0 q 2p;t 0; U1 8
KdvTvr
rRd hTN Tq; z; t; 0 q 2p; 0 z dd;t 0; U2 9
KdvTvr
rrd hTN Tq; z; t; 0 q 2p; 0 z dd;t 0; U3 10
vTvz
zdd 0; rd r Rd; 0 q 2p; t 0; U4 (11)Tr; q; z;0 T0; rd r Rd; 0 q
2p; 0 z dd (12)
4. FE formulation
The object of this section is to develop approximate
time-step-ping procedures for axisymmetric transient governing
equations.Using Galerkins approach the following matrix form of the
Eq. (6)is formulated [25]
CdfTgdt
KfTg fRg (13)
In order to solve the ordinary differential equation (13)
the
l Engineering 31 (2011) 1003e1012 1005Crank-Nicolson method was
used. Based on the assumption that
-
were partially coated, the elements for the pad were
simulta-neously uncovered. The process was repeated and the time of
padimaginary contact area with the constant number of
elementsduring computations was successively longer compatibly to
therate of deceleration until standstill. In the case of braking
withconstant velocity, the time of heating phase of
three-dimensionalmodel equals f0=2p of time of one rotation,
whereas cooling phaselasts longer due to angular dimension of pad
element and equals1 f0=2p of time of one rotation of the wheel.
The thermal ux entering the disc acted in the shape of
theintensity of heat ux applied to three-dimensional nite
elementsin the area of pad operation during braking. Instead of
automaticmesh generation capabilities, the original programming of
thebuilt-in commends of nite element software covering the
algo-rithm of moving heat source described above, to assure
correctnessof the boundary conditions prescribed to specic
elements, in
rmal Engineering 31 (2011) 1003e1012temperature {T}t and {T}tDt
at time t and t Dt respectively, thefollowing relation is
specied
1Dt
fTgtDtfTgt 1 bdTdt
tbdTdt
tDt
(14)
Substituting Eq. (14) to Eq. (13) we obtain the following
implicitalgebraic equation
C bDtKfTgtDt C 1 bKDtfTgt 1 bDtfRgtbDtfRgtDt 15
where b is the factor which ranges from 0.5 to 1 and is given
todetermine an integration accuracy and stable scheme.
The transientnite element analysiswas developedusing
theMDPatran/MD Nastran software package [26,27]. The nite
elementmesh of the disc model chosen for the analysis is
illustrated in Fig. 1.The accuracyof the solutionwasobtainedby
testingdifferent grids ofnite elements, all of which had its own
global number of elementsdue to specic division in circumferential,
radial, and axial direction.The investigated, individual grids at
the initial phase of the compu-tations consisted of the 180, 240,
360, 450, 540 elements in thecircumference,10,15, 20, 25, 30 in the
radial direction of the rubbingpath, and 3, 3, 4, 4, 5 elements in
the axial direction, respectively. Thecalculationsof transient
temperatureof the rotorwere carriedout forthe braking process with
constant deceleration from the initialvelocity of 25km/h. Themeshof
thenite elementswas selecteddueto the difference of the obtained
peak values of temperature relatingto the nest mesh (model with the
540 elements in the circumfer-ence). The FE model of disc employed
for the transient analysisconsisted of 43,200 eight-node hexagonal
elements e HEX8 (360elements in the circumference and 4 in axial
direction) and 33,693nodes was used in the thermal analysis. As the
mesh should becapable to reproduce the rapid temperature variations
in theimmediate vicinity of the contact surface, the size of the
niteelement increased with the distance from the region of
generatedsurface of friction. To avoid inaccurate or unstable
results, a properxed time step associated with spatial mesh size is
essential [26].
Dt Dx2 rdcd10Kd
(16)
In order to simulate moving heat source problem in the processof
emergency braking, avoiding inaccuracies and oscillations tooccur
due to Peclet number which in presented case markedlyexceeds the
critical value of Pe 2, time-stepping procedure cor-responding to
the relative pad/disc location was developed. Theknown amount of
the intensity of heat ux entering the disc atsucceeding instants of
time, determined from the product of radialdistance from the axis
of disc on the friction surface, the contactpressure, velocity of
the vehicle with the rate of deceleration Eq. (5)and friction
coefcient was implemented to the FE models and theproblem was
solved based on the programming technique imple-mented to the
commercial FEM programme [26,27]. In conse-quence spatial scheme of
heating issue was accomplished and timedependent boundary
conditions due to rotating pad activity wereestablished. At the
beginning of the process, after the brakeengagement the amount of
heat (Eq. (1)) was applied to theselected nite elements of pad area
of the three-dimensionalmodel. At the next time step, smaller than
computed from the Eq.(16), the corresponding motion of heat source
(brake pad) wascalculated and displaced to the adjacent elements
according tomutual sliding direction of the members of braking
system. Thisprocess was modelled by the function, which imitated
the processof covering of elements of the model during relative
motion of
A. Adamowicz, P. Grzes / Applied The1006rotating disc and xed
pad. While elements near by the front of padparticular, on the
contact surface of disc was developed.
5. Results and discussion
In this paper temperature distributions of the disc brake
withoutpad have been investigated. The disc rotor is subjected to
high non-axisymmetric thermal loadwhichmay lead to
non-uniformpressureandtemperaturedistributions. Therefore
three-dimensional analysisfacilitates to examine temperature
alterations in the circumferenceand theirs inuence on the area
inside the disc. Both convection andconduction have been analyzed.
Particularly conduction wasconsidered to be the most important mode
of heat transfer.
In order to validate proposed transient numerical analysis
twodifferent types of the FE models were investigated, namely
two-and three-dimensional conguration [8]. The part of
presentedtemperature evolutions for two-dimensional model (braking
withthe constant retardation from the velocity of V0 100
km/h)originates from previous authors study [11]. The transient
solutionwas performed for four selected initial velocities and
relateddurations of braking process with constant deceleration. For
thecase of braking with constant velocity, exclusively the action
ofV 100 km/h (ts 3.96 s) was tested. Material properties
andoperation conditions adopted in the analysis were the same
forboth types of FE models and are given in Table 1.
In Fig. 2 numerical solutions of three-dimensional
(continuousline) transient analysis of the disc contact surface
temperatureevolutions for specied radii ofdisc andposition in the
circumferenceq 0 confronted with the results obtained from the
two-dimen-sional analysis (dashed line) are shown. In order to
illustrate effect offrictionallyexcitedheatingover the
frictionsurface, eachofdescribedgures covers four characteristic
points of the position along the
Table 1Thermo-physical properties of materials, dimensions and
operation conditions forthe transient analysis (from Talati and
Jalalifar [8]).
Items Disc Pad
Thermal conductivity, K [W/(m K)] 43 12Heat capacity, c [J/(kg
K)] 445 900Density, r [kg/m3] 7850 2500Inner radius, r [mm] 66
76.5Outer radius, R [mm] 113.5Cover angle of pad, 40 64.5Thickness,
d [mm] 5.5 10Radius of the wheel, Rw [mm] 314Initial velocity of
the vehicle, V0 [km/h] 100 75 50 25Time of braking, ts [s] 3.96
2.97 1.98 0.99Pressure, p0 [MPa] 3.17Coefcient of friction, f
0.5Heat transfer coefcient, h [W/(m2 K)] 60Initial temperature, T0
[C] 20
Ambient temperature, TN [C] 20
-
Fig. 3. Evolutions of temperature on the contact surface of the
disc brake duringbraking from the initial velocity V0 75 km/h at
selected radial locations for three-
ermal Engineering 31 (2011) 1003e1012 1007radius, namely the
external radius of the disc Rd, the mean radius ofrubbing path rm,
theminimal radius of pad rp, and the internal radiusof the disc rd.
The temperature curves directly correspond with thedivers
representations of the disc brake model congurations. It
isnoticeable, that the value of temperature in each case of
axisym-metric heating of disc rapidly rises at the beginning of
brakingprocess, reaches its maximal value, then decreases to the
lower leveland eventually stops, which is coherent with the studies
[7,8,11,13].However uctuations of temperatures have a presence in
the solu-tion of three-dimensional model of frictional heating, the
approxi-
Fig. 2. Evolutions of temperature on the contact surface of the
disc brake duringbraking from the initial velocity V0 100 km/h at
selected radial locations for three-(solid curves) and
two-dimensional (dashed curves) models.
A. Adamowicz, P. Grzes / Applied Thmated values remain the
approximate conrming the stability of theFE modelling in the
two-dimensionality. The temperature curvesexpose saw-shaped
character, which stems from the mutual rota-tional motion of the
disc over the xed pad [13,20]. The presentedtemperature evolution
is obtained for certain, xed spot on thecircumference of a disc,
therefore periods of heating and coolingphases may be
distinguished. When the specic nite element oftemperature
calculations on the contact surface of disc is covered bypad
(heating phase) the increase of temperature is noticeablebecause of
accumulation of the frictional heat. On the contrarywhenthe pad is
out of considered spot on the rubbing path, the coolingconditions
according to Newtons law are established and thetemperature
decreases. Each revolution of the wheel strictly corre-sponds to
one cycle of heating and cooling state. It is evident, that
thetemperaturedistributioncorrelates intermediately to the
intensityofheat ux entering the disc, whose value in the plane
model linearlydecreases with time until the standstill, whereas
spatial represen-tation accessorily complies non-continuous heating
of disc over thecircumference. In the solutionof
two-dimensionalmodel the highesttemperature T 227.94 C is reached
at the radial positionr113.5mm, after time t3.022 s,whereas
thehighest temperatureT 259.34 C of fully three-dimensional disc,
occurs at the sameradius r 113.5 mm after time t 2.688 s. The
discrepancy oftemperatures is lowerat theendof
theprocessandequalsT3.52 C.At the radial locationof76.5mmthehighest
temperatureof two- andthree-dimensional FE model equals T 98.98 C
and T 108.31 C,respectively. Exclusively at the internal surface of
disc r 66 mm inboth FE models the highest value of temperatures (T
47.32 C andT 47.43 C) is attained at the end of the braking
process. Themaximal temperature at the radii of 76.5, 95
and113.5mmof the 3-Dmodel occurs at the same time t 2.688 s,
whereas identical radialpositions of axisymmetric case gives the
solutions of time equalledt 3.36 s, t 3.098 s and t 3.022 s
respectively.
The temperature evolutions on the contact surface of
discconditioned by the obtained results of two types of braking
processsimulations from the initial velocity V0 75 km/h are shown
inFig. 3. In the two-dimensional model the temperature curves
ofsurface of friction continuously alter with time analogously as
wasduring braking from V0 100 km/h (Fig. 2). In the spatial model,
the
(solid curves) and two-dimensional (dashed curves)
models.increase of temperature is noticeable after the moment
without
Fig. 4. Evolutions of temperature on the contact surface of the
disc brake duringbraking from the initial velocity V0 50 km/h at
selected radial locations for three-(solid curves) and
two-dimensional (dashed curves) models.
-
sliding contact at the specied location in the circumference q
0,then the maximal temperature is attained, and succeeding periodof
pad absence effects with its rapid descend. The nature of
therepeated heating and cooling states (Figs. 2e5) indicates two
typesof temperature curves, the rst, describing period of heating
is theconcave curve, the second part of one rotation of disc is
describedby the convex curve, which is caused by the extortion of
frictionallygenerated heat impulse and its absence after the pad
transitionwith the convective cooling. The time of these periods
differs due tothe velocity of braking and is constant at its
specied value at eachradial location on the friction surface. The
temperature curves atthe radii of 66, 76.5 mm almost coincide near
the time of full stopwith both solutions owing complexity of the
model, whereas thediscrepancy of temperatures during the action and
after standstilloverlaps less at radii r 95 and r 113.5 mm.
In Fig. 4 temperature evolutions on the contact surface
duringbraking from the initial velocity of 50 km/h are presented.
Thesimilar pattern of temperature progress with regard to Figs. 2
and 3may be observed. The averages of temperatures curves of
3D-modelagree highlywith the results obtained from themodel drawn
on theintensity of heat ux uniformly distributed in the
circumference ofthe disc. The maximal temperature reached during
barking from50 km/h equals T 81.86 C and T 112.54 C for two- and
three-dimensional model respectively. The seventh rotation ends at
theposition of pad covering the tested location in the
circumferenceq 0 (twenty eighth rotation in Fig. 2). Therefore the
temperatureafter the full stop is closer to the value obtained in
the axisymmetricproblem of frictional heating. Relating to the
radial location of thepresented temperaturecurves, proportionof
thedistance from zaxis
Fig. 5. Evolutions of temperature on the contact surface of the
disc brake duringbraking from the initial velocity V0 25 km/h at
selected radial locations for three-(solid curves) and
two-dimensional (dashed curves) models.
Fig. 6. Temperature distributions on the contact surface at the
moment of standstill of brakV0 25 km/h for three- (solid curves)
and two-dimensional (dashed curves) models.
A. Adamowicz, P. Grzes / Applied Thermal Engineering 31 (2011)
1003e10121008ing from the initial velocity: (a) V0 100 km/h, (b) V0
75 km/h, (c) V0 50 km/h, (d)
-
is not equalled to the corresponding values of temperature.
Thisphenomenonmay be attributed to the contact surface of disc
whichis situated on itsmargin, thus the area beneath rubbing path
absorbsmore heat during action and temperature is adequately
lower.
The evolutions of temperatures during braking from the
lowesttested velocity V0 25 km/h are shown in Fig. 5. It may be
observedthat only one rotationwas accomplished within sliding
process. Theplotted curves at each radius reveal signicant
disagreement of thetwopresented solutions. Themaximal value of
temperature of three-dimensional model attained in the action, at
radius of 113.5 mmequals T 64.73 C, whereas in opposite approach of
modelling,temperature equals T 41.21 C. Such a spread of results,
relating tothe simplied process of heating may mislead the actual
effecttemperature variations. Themoment,when the highest
temperatureoccurs evidently depends on the investigated location in
thecircumference, and inparticular casemay be identical to the
solutionof two-dimensional model. At the radius of 66 mm the
temperatureremains unchanged in both cases.
Fig. 6 depicts the temperature elds on contact surface in
thecircumference at the moment of standstill for selected radial
loca-tions and different initial velocities: a) V 100 km/h, b) V 75
km/h,c) V 50 km/h, d) V 25 km/h. The temperature curves of
three-dimensional model are plotted with regard to the constant
temper-ature of two-dimensional FEmodel. In fact temperature
distributionof axialmodel in Fig. 6 shouldbe illustrated as a
point, but to facilitateclarity straight line (dashed) is used. It
may be observed that thetemperature rises when the pad passes
specied position on thefriction surface of disc and decreases to
the level beneath the distri-bution of two-dimensional event. The
highest calculated range of
amplitude of temperature occurs on the external edge of disc in
eachcase of braking. For r 76.5 mm the temperature is more
smoothunder pad transition, whereas on the edge of external surface
isalmost constant in the circumference because of the distance
fromthe rubbing path. The presented plots of temperature drawn
alongthe circumferential direction which correspond to the articles
ofFloquet and Dubourg [9] and Cho and Ahn [16].
The average temperature of spatial problem during brakingfrom V
25 km/h (Fig. 6d) at the radius r 113.5, 95, 76.5 and66 mm equals T
35.59, 33.47, 25.74, 20.09 C respectively,whereas temperature at
the end of braking of 2D model equalsT 35.65, 33.52, 25.65, 20.13
C, therefore when the initial angularvelocity equals u0 22.116 s1
(V0 25 km/h), the mean temper-ature coincides in each case of the
solution with the relative errorlower than 0.5%, whereas for the
initial velocity u0 44.232,66.348, 88.464 s1 (V0 50, 75, 100 km/h)
equals 1, 2, and 3%(Fig. 6aec), respectively. However this
arithmetic mean of spatialdistribution of temperature is not able
to include realistic responseof material heating of disc during
process of braking. The level oftemperature in each case of brake
engagement owing differentinitial velocities corresponds to
temperatures at the moment ofstandstill presented in Figs. 2e5. The
temperature distributionsexpose importance of place under
examination in the circumfer-ence of spatial model and its parallel
time.
Fig. 7 shows the temperature distributions that evolved on
theexternal radius of disc (r 113.5mm) at different locations in
depthduring braking from the previously selected initial velocities
of thevehicle: a) V0 100 km/h, b) V0 75 km/h, c) V0 50 km/h, d)V0
25 km/h. The solutions of spatial model are confronted with
A. Adamowicz, P. Grzes / Applied Thermal Engineering 31 (2011)
1003e1012 1009Fig. 7. Evolutions of the disc temperature at
different axial distances from the contact surfac(b) V0 75 km/h,
(c) V0 50 km/h, (d) V0 25 km/h for three- (solid curves) and
two-die at the radius of 113.5 mm during braking from the initial
velocity: (a) V0 100 km/h,mensional (dashed curves) models.
-
the axisymmetric representation. The permanent rise of
tempera-ture until attainment of its maximal value and slightly
descendafter, near the moment of full stop on the contact surface(z
0 mm) and for z 1, 2 mm is noticeable. The bigger distance(z 3.5,
5.5 mm) results in constant increase of temperature untilthe end of
the process. The character of evolution of temperature onthe axial
position z 1, 2 mm slightly differs from the trace at theposition
of 0 mm, whereas temperature on the depth of 2, 3.5,5.5 mm evolves
almost identically to the curves generated inaxisymmetric model.
The presented evolutions of temperature areplotted for the specied
location in the circumference, thusagreement of the results owing
type of the frictionally excitedheating process strongly depends on
the position of testing as well.Nevertheless chosen point q 0 in
the circumference of the three-dimensional model during braking
from V 100 km/h (Fig. 7a) andV 50 km/h (Fig. 7c) almost overlapped
selected axial distancesfrom contact surface of two-dimensional
model. It stems from thefact that during the immediate moment of
standstill, pad covers thespot of disc under examination and
therefore causes slight rise oftemperature sufcient to improve
agreement of plane and spatialsolution of heating. It has to be
noticed that in the two-dimensionalmodel convective terms on the
friction surface have been neglectedand constant heating with the
same value of thermal ux in thecircumference during braking was
established. All of the temper-ature curves in Fig. 7 which
represents two-dimensional model onthe depth of z 2, 3.5, 5.5 mm
exceed values of the related
time ts obtained during braking from the initial velocityV0 100
km/h is shown. The circumferential location of presentedtemperature
curves was chosen due to the disc/pad related positionnext to
sliding pad. The temperature for the particular distancesfrom the
axis of the rotor correlates with presented curves plottedversus
braking time for four characteristic radii (Fig. 2). The value
oftemperature on the internal surface of disc coincides within
bothconguration of the model of frictional heating at any step of
time.At the end of braking process, the temperature on the
frictionsurface along the radius is approximate. Nevertheless for
the timet 25%ts, 50%ts and 75%ts the temperature in the contact
zoneobtained from spatial model highly exceeds the
correspondingvalues of the two-dimensional phenomenon, which stems
from thefact that during braking with linear decrease of time the
padinuences directly the level of temperature until themoment of
fullstop.
In Fig. 9 the temperature evolutions on the contact surface
ofdisc versus time during braking with the constant velocity of100
km/h are illustrated. The temperature of the
two-dimensionalsolution at the position r 76.5, 95, and 113.5 mm
rapidly rises atthe beginning of action, then linear increase is
noticeable untilstandstill, whereas in case of spatial model the
delay of tempera-ture variations is may be seen at the beginning of
braking, afterwhich impulse nature of heating takes place. The
process of heatingrelating to the average of temperature highly
agrees withindifferent FE model of the same phenomenon. The maximal
value of
A. Adamowicz, P. Grzes / Applied Thermal Engineering 31 (2011)
1003e10121010temperature evolutions of spatial model during braking
process.However temporary peak values of temperature during pad
passingare higher than smoothed evolution of temperature of
uniformheating in the circumference, owing the average amount
oftemperature of spatial model its inuence on heating is
lowered.The temperature evolutions of braking from the angular
velocityu0 22.116 s1 (Fig. 7d) conrm that for that case the
solutions ofheat transfer in disc brake are obviously dissimilar
when considertwo- and three-dimensional model.
In Fig. 8 the temperature distribution on the contact
surfacealong the radius of disc at four different moments of time
braking
Fig. 8. Temperature distributions on the friction surface versus
radial direction during
braking from the initial velocity V0 100 km/h for three- (solid
curves) and two-dimensional (dashed curves) models.temperature on
the contact surface is reached at the external radiusof disc at the
end of braking for two-dimensional modelT 494.35 C and for the last
pad rotation at the time of t 3.906 s,T 526.63 C. The similar
approach of simulation of brakingprocess with constant velocity
using three-dimensional model hasbeen investigated [2]. Assumptions
regardless circumferentialconductive ux were made with the
assessment of the enterederror.
The temperature variations at different locations in depth
fromthe disc/pad interface z 0 mm to the surface of symmetry of
discz 5.5 mm are shown in Fig. 10. With a distinction to the case
of
Fig. 9. Evolutions of temperature on the contact surface of the
disc brake duringbraking with constant velocity V 100 km/h at
selected radial locations for the two-
and three-dimensional problem for three- (solid curves) and
two-dimensional (dashedcurves) models.
-
Fig. 10. Evolutions of the disc temperature at different axial
distances from the contactsurface at the radius of 113.5 mm during
braking with constant velocity V 100 km/hfor three- (solid curves)
and two-dimensional (dashed curves) models.
A. Adamowicz, P. Grzes / Applied Thermabraking with linearly
decreased velocity of the vehicle (Fig. 7a)braking with the
constant velocity results in the increase oftemperature after the
initial moment of time nearly linear until fullstop. The
temperatures of two-dimensional model at the positionz1,2,3.5,5.5
mm are higher at any instant of braking time.
The temperature distributions on the friction surface along
thecircumference of disc for the case of braking with constant
velocityof 100 km/h are shown in Fig. 11. It is clearly noticeable
for r 76.5,
95, 113.5 mm that the temperature on the contact surface
corre-sponds to pad transition over the rotational disc.
Fig. 11. Temperature distributions on the contact surface at the
moment of standstill,process of braking with the constant velocity
V 100 km/h for three- (solid curves)and two-dimensional (dashed
curves) models.Fig. 12 depicts the corresponding temperature
distributions onthe contact surface of disc brake for selected
moments of time. Thetemperature prole for the time equalled 0.25 of
ts has the longestlinear section in the middle of braking path. At
the subsequentmoments of time this spot is more rounded. In the
contrary to Fig. 8,this case of braking with constant velocity V
100 km/h lasting3.96 s results in aligned respective plots of
temperature elds dueto the two- and three-dimensional description
of the analyzed
Fig. 12. Temperature distributions on the friction surface
versus radial direction duringbraking with the constant velocity V
100 km/h for three- (solid curves) and two-dimensional (dashed
curves) models.
l Engineering 31 (2011) 1003e1012 1011phenomenon.
6. Conclusions
In this paper three-dimensional nite element analysis wascarried
out for temperature distributions assessment in disc brakesystem
during single braking. The disc rotor was examined withoutpad
presence. The heat conductivity problem was divided into twocases
of different congurations of the disc brake FE models
owingcomplexity of the problem.
From the obtained results we can conclude, that the tempera-ture
of disc on the contact surface of two-dimensional model andaveraged
solution of spatial solution during braking with theconstant
deceleration sharply rises at the beginning of the process,reached
its maximal value and eventually stops on the lower level,whereas
if the velocity of the vehicle is constant the temperatureafter the
initial moment of time increases approximately linearly.
The character of temperature evolution on the contact surface
ofdisc and its inuence in depth reveals high coincidence with
regardto the three-dimensional model and simplied
two-dimensionalrepresentation of the considered problem. Therefore
validation ofthe outcomes of previously conducted study of
frictional heating ofdisc with uniformly distributed heat ux has
been made.
Fully three-dimensional analysis under non-axisymmetricthermal
load provides information of realistic behaviour oftemperature
alterations distinguishing period of heating (concavecurve) and
cooling (convex curve) in the selected spot on the fric-tion
surface during both single braking to full stop and brakingduring
mountain descent with the constant velocity.
-
Based on the investigated individual cases of single brakingfrom
the different initial velocities, it may be observed that
thecompatibility of two- and three-dimensional model lowers withthe
decrease of the velocity of the vehicle. The above
axisymmetricsolution of the temperature elds of disc indicates that
the solu-tion is reliable if the angular velocity of disc exceeds
u0 44.232 s1.
The developed nite element analysis of friction heating of
discduring emergency braking has conrmed the solution in the
two-dimensionality feasible further to carry the fully transient
simula-tionwith the time dependent material properties and
coefcient offriction due to adequately low computer storage
requirements.
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Analysis of disc brake temperature distribution during single
braking under non-axisymmetric loadIntroductionStatement of the
problemMathematical modelFE formulationResults and
discussionConclusionsReferences