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Meccanica
An International Journal of Theoretical
and Applied Mechanics AIMETA
ISSN 0025-6455
Meccanica
DOI 10.1007/s11012-013-9821-9
TRT: thermo racing tyre a physical model
to predict the tyre temperature distribution
Flavio Farroni, Daniele Giordano,
Michele Russo & Francesco Timpone
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Meccanica
DOI 10.1007/s11012-013-9821-9
TRT: thermo racing tyre a physical model to predict the tyre
temperature distribution
Flavio Farroni ·Daniele Giordano ·
Michele Russo ·Francesco Timpone
Received: 7 February 2013 / Accepted: 9 October 2013
© Springer Science+Business Media Dordrecht 2013
Abstract In the paper a new physical tyre thermal
model is presented. The model, called Thermo Racing
Tyre (TRT) was developed in collaboration between
the Department of Industrial Engineering of the Uni-
versity of Naples Federico II and a top ranking motor-
sport team.
The model is three-dimensional and takes into ac-
count all the heat flows and the generative terms occur-
ring in a tyre. The cooling to the track and to external
air and the heat flows inside the system are modelled.
Regarding the generative terms, in addition to the fric-
tion energy developed in the contact patch, the strain
energy loss is evaluated. The model inputs come out
from telemetry data, while its thermodynamic param-
eters come either from literature or from dedicated ex-
perimental tests.
The model gives in output the temperature circum-
ferential distribution in the different tyre layers (sur-
face, bulk, inner liner), as well as all the heat flows.
These information have been used also in interaction
models in order to estimate local grip value.
Keywords Tyre temperature ·Real time thermal
model ·Strain energy loss ·Friction power ·Tyre heat
flows
F. Farroni (B)·D. Giordano ·M. Russo ·F. Timpone
Dipartimento di Ingegneria Industriale, Università degli
Studi di Napoli Federico II, Via Claudio 21, 80125 Naples,
Italy
e-mail: flavio.farroni@unina.it
Symbols
Ttemperature [K]
Tair air temperature [K]
T∞air temperature at an infinite
distance [K]
Trroad surface temperature [K]
ttime [s]
α=k
ρ·cvthermal diffusivity; αttyre, αr
road [m2
s]
˙qGheat generated per unit of
volume and time [J
s·m3]
ρdensity [kg
m3]
cvspecific heat at constant volume
[J
kg·K]
cpspecific heat at constant pressure
[J
kg·K]
kt,krtyre and road thermal
conductivity [W
m·K]
Hcheat transfer coefficient [W
m2·K]
hexternal air natural convection
coefficient [W
m2·K]
hforc external air forced convection
coefficient [W
m2·K]
hint internal air natural convection
coefficient [W
m2·K]
x,y,zcoordinates
Fx,Fylongitudinal and lateral tyre-road
interaction forces [N]
Fznormal load acting on the single
wheel [N]
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vx,vylongitudinal and lateral slip
velocity [m
s]
Atyre-road contact area [m2]
Atot total area of external surface
[m2]
kair air thermal conductivity [W
m·K]
Vair velocity [m
s]
νair kinematic viscosity [m2
s]
μair dynamic viscosity [ kg
s·m]
L=1
1
De+1
W
characteristic length of the heat
exchange surface [m]
Wtread width [m]
La contact patch length [m]
De tyre external diameter [m]
pi tyre inflating pressure [bar]
ggravity acceleration [m
s2]
βcoefficient of thermal air
expansion [1/T]
Gr =g·β·L3·(T −T∞)
ν2Grashof number [–]
Pr =μ·cp
Kair Prandtl number [–]
1 Introduction
In automobile racing world, where reaching the limit
is the standard and the time advantage in an extremely
short time period is a determining factor for the out-
come, predicting in advance the behaviour of the ve-
hicle system in different conditions is a pressing need.
Moreover, new regulations limits to the track test ses-
sions made the “virtual experimentation” fundamental
in the development of new solutions.
Through the wheels, the vehicle exchanges forces
with the track [1,2] which depend on the structure
of the tyres [3] and on their adherence, strongly in-
fluenced by temperature [4,5].
Theoretical and experimental studies, aimed to
predict temperature distribution in steady state pure
rolling conditions, useful to evaluate its effects on en-
ergetic dissipation phenomena, are quite diffused in
literature [6,7]. Less widespread are analyses con-
ducted in transient conditions involving tyre tempera-
ture effects on vehicle dynamics. A thermal tyre model
for racing vehicles, in addition to predict the temper-
ature with a high degree of accuracy, must be able to
simulate the high-frequency dynamics characterizing
this kind of systems. Furthermore, the model has to be
able to estimate the temperature distribution even of
the deepest tyre layers, usually not easily measurable
on-line; it must predict the effects that fast temperature
variations induce in visco-elastic materials behaviour,
and it must take into account the dissipative phenom-
ena related to the tyre deformations.
With the aim to understand the above phenomena
and to evaluate the influence of the physical variables
on the thermal behaviour of the tyre, an analytical-
physical model has been developed and called Thermo
Racing Tyre (TRT).
At present time there are not physical models avail-
able in literature able to describe the thermal be-
haviour of the tyres in a sufficiently detailed way to
meet the needs of a racing company. The TRT model
may be considered as an evolution of the Thermo-
Tyre model [8] that allows to determine in a sim-
plified way the surface temperature of such system,
neglecting the heat produced by cyclic deformations
and not considering the structure of the different lay-
ers.
The above mentioned limitations of ThermoTyre
have been removed in the implementation of TRT, that
results in an accurate physical model useful for the
thermal analysis of the tyre and characterized by pre-
dictive attitudes since it is based on physical param-
eters known from literature or measurable by specific
tests [9].
2 Tyre modeling and base hypotheses
The tyre is considered as unrolled in the circumferen-
tial direction (and then parallelepiped-shaped), lack-
ing of sidewalls and grooves (so the tyre is modelled
as slick), discretized by means of a grid, whose nodes
represent the points in which the temperature will be
determined instant by instant (Fig. 1).
The parallelepiped is constituted by three layers in
the radial direction z, which will be hereinafter indi-
cated as surface (outer surface of the tyre), bulk (inter-
mediate layer), and inner liner (inner surface).
The number of nodes of the grid is given by the
product (numx ·numy ·numz) where numx represents
the number of nodes along the xdirection, numy the
number of nodes along the ydirection and numz is the
number of nodes along the zdirection. Nodes enumer-
ation has been carried out starting from the first layer
in contact with the road, proceeding transversely. Each
layer is subdivided in 15 elements in the longitudinal
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Fig. 1 Discretization of the tyre
direction and 4 elements in the transversal direction,
so the entire tyre results discretized in 180 elements
(but clearly the discretization can be modified).
The chosen Cartesian reference system has its ori-
gin in the node 1; x-axis is oriented in the circumfer-
ential direction of the tyre enrolled in a plane, y-axis is
oriented in the direction of its width, and finally z-axis
is oriented in the thickness direction; the positive di-
rection is defined in all the three cases by the increas-
ing numbering of the nodes. Along the radial direc-
tion, i.e. along z, two zones of homogeneous material
are identified: the first one (thickness =z1), local-
ized between surface and bulk nodes, correspondent
to the tread; the second one (thickness =z2)tothe
tyre carcass.
Tread is mainly constituted by visco-elastic vulcan-
ized polymers and fillers, while carcass includes also
reinforcements.
Each one of them is characterized by the following
physical parameters, accounting for the material com-
position:
•Density ρ
•Specific heat c
•Thermal conductivity K
for the last two quantities it has been taken into ac-
count their variability with temperature.
To the generic i-th node a parallelepiped volume
was associated, equal to
Vi=x ·y ·Zm,i (1)
in which x and y are respectively the dimensions
along the directions xand y, while the quantity Zm,i
represents the dimension along the z-direction of the
i-th layer defined so that once multiplied the obtained
volume Viby the density, the mass results equal to the
expected one for each single element.
Each node will then have a mass expressed as fol-
lows:
mi=C·Vi·ρ(2)
where the Ccoefficient depends on the position in the
grid. Indeed, from Fig. 1it is easy to notice that the
volumes associated to the external nodes (e.g. node 1)
are characterized to be parallelepiped-shaped, having
sides in the direction yand zrespectively equal to
Y /2 and to Zm,i /2. Therefore, in this case, the co-
efficient Cwill be equal to 1/4. With the aim to char-
acterize the coefficients Cfor each kind of node, the
following list is proposed:
•C=1/4 for a node of the first layer, external in the
transversal direction
•C=1/2 for a node of the first layer, internal in the
transversal direction
•C=1/2 for a node of the bulk layer, external in the
transversal direction
•C=1 for a node of the bulk layer, internal in the
transversal direction
•C=1/4 for a node of the inner liner, external in the
transversal direction
•C=1/2 for a node of the inner liner, internal in the
transversal direction
With the aim of modeling heat flows and tyre lay-
ers temperatures, the following assumptions have been
adopted:
•Road is isotropic and homogeneous in all its charac-
teristics, without irregularities, schematized as a ge-
ometric plane, whose surface temperature is known
and equal to Ts
•It is assumed that the contact area is rectangular in
shape, characterized by width Wequal to the width
of the tread, and length Ladepending on the radial
stiffness of the tyre and on the normal load. The as-
sumption of rectangular shape of the contact area is
realistic in the case of sport tyres, characterized by
high width values.
•Camber angle is assumed equal to zero
•During rolling it is assumed that the tyre keeps the
deformed configuration, and consequently contact
patch extension, reached under the application of
the static load.
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•Thetyreisalsoassumedmotionless,inaLa-
grangian approach, with variable boundary condi-
tions
•The radiation heat transfer mechanism is neglected.
3 Thermodynamic model
The developed thermodynamic tyre model is based on
the use of the diffusion equation of Fourier applied to
a three-dimensional domain.
The complexity of the phenomena under study and
the degree of accuracy required have made that it be-
comes necessary to take into account the dependence
of the thermodynamic quantities and in particular of
the thermal conductivity on the temperature.
Furthermore, the non-homogeneity of the tyre has
made it necessary to consider the variation of the
above parameters also along the thickness.
Therefore, the Fourier equation takes the following
formulation [10]:
∂T
∂t =˙qG
ρ·cv+1
ρ·cv·∂2k(z,T ) ·T
∂x2
+∂2k(z,T ) ·T
∂y2+∂2k(z,T ) ·T
∂z2(3)
Writing the balance equations for each generic node
needs the modeling of heat generation and of heat ex-
changes with the external environment.
For the tyre system, the heat is generated in two
different ways: for friction phenomena arising at
the interface with the asphalt and because of stress-
deformation cycles to which the entire mass is sub-
jected during the exercise.
3.1 Friction power
The first heat generation mechanism is connected with
the thermal power produced at tyre-road interface be-
cause of interaction; in particular, it is due to the tan-
gential stresses that, in the sliding zone of the contact
patch [11], do work dissipated in heat. This power is
called “friction power” and will be indicated in the fol-
lowing with FP. In the balance equations writing, FP
can be associated directly to the nodes involved in the
contact with the ground.
Since the lack in local variables availability, FP is
calculated as referred to global values of force and
sliding velocity, assumed to be equal in the whole con-
tact patch:
FP =Fx·vx+Fy·vy
AW
m2(4)
A part of this thermal power is transferred to the tyre
and the remaining to the asphalt. This is taken into ac-
count by means of a partition coefficient CR.
To determine the partition coefficient, the following
expression can be used [12]:
CR =kt
kr·αr
αt
(5)
in which thermal diffusivity αcan be expressed as α=
Kk
ρ·cv.
Considering the following road properties:
kr=0.55 W
m·K(6)
ρr=2200 kg
m3(7)
cvr=920 J
kg·K(8)
and the properties of the SBR (Styrene and Butadiene
mixture used for the production of passenger tyres),
available in literature [13,14], the resulting calculated
value of CR is about 0.55, which means that the 55 %
of the generated power is directed to the tyre.
Since the model takes into account the variability
of the thermal conductivity of rubber with the temper-
ature, also the CR coefficient will be a function of the
calculated temperature; this results in a variation be-
tween 0.5 and 0.8.
Since Fxand Fyare global forces between tyre and
road, and not known the contribution of each node to
these interaction forces, heat generated by means of
friction power mechanism transferred to the tyre has
been equally distributed to all the nodes in contact with
the ground. The model allows not uniform local heat
distributions as soon as local stresses and velocities
distributions are known.
3.2 Strain energy loss (SEL)
The energy dissipated by the tyre as a result of cyclic
deformations is called Strain Energy Loss (SEL). This
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Fig. 2 Hysteresis cycle for a front tyre
dissipation is due to a superposition of several phe-
nomena: intra-plies friction, friction inside plies, non-
linear visco-elastic behavior of all rubbery compo-
nents.
The cyclic deformations to which the system is sub-
ject occur with a frequency corresponding to the tyre
rotational speed. During the rolling, in fact, portions of
tyre, entering continuously in the contact area, are sub-
mitted to deformations which cause energy loss and
then heat dissipation.
In the model the amount of heat generated by defor-
mation (SEL) is estimated through experimental tests
carried out deforming cyclically the tyre in three di-
rections (radial, longitudinal and lateral) [15]. These
tests are conducted on a proper test bench and a test
plan, based on the range of interaction forces and fre-
quencies at which tyre is usually stressed, has been
developed [16]. For each testing parameters combina-
tion, the acquired and measured area of the hysteresis
cycle is representative of the energy dissipated in the
deformation cycle (Fig. 2).
Estimated energies do not exactly coincide with the
ones dissipated in the actual operative conditions, as
the deformation mechanism is different; it is however
possible to identify a correlation between them on the
basis of coefficients estimated from real data teleme-
try.
Interpolating all the results obtained by means of
the test plan, an analytic function has been identi-
fied [17]; it expresses the SEL as a function of the
parameters (amplitude of the interaction force compo-
nents and applying frequency) on which it depends.
3.3 Heat transfers modelling
As regards the heat exchange between the tyre and the
external environment, it can be classified as follows:
•Heat exchange with the road (called “cooling to the
ground”);
•Heat exchange with the outside air;
•Heat exchange with the inflating gas.
As said, the radiation mechanism of heat exchange is
neglected. The same has to be said about the convec-
tive heat exchange with the external air along the sur-
face of the sidewalls because the air flow is directed al-
most tangentially to them; for this reason the value of
convective heat exchange coefficient is small. More-
over, being the rubber characterized by very low ther-
mal conductivity, belt thermal dynamics do not influ-
ence significantly sidewall dynamics and vice versa
The phenomenon of thermal exchange with the
asphalt has been modeled through Newton’s for-
mula [18], schematizing the whole phenomenon by
means of an appropriate coefficient of heat exchange.
The term for such exchanges, for the generic i-th node
will be equal to:
Hc·(Tr−Ti)·X ·Y (9)
The heat exchange with the outside air is described
by the mechanism of forced convection, when there
is relative motion between the car and the air, and by
natural convection, when such motion is absent.
The determination of the convection coefficient h,
both forced and natural, is based on the classical ap-
proach of the dimensionless analysis [3].
Considering the tyre invested by the air similarly to
a cylinder invested transversely from an air flux, the
forced convection coefficient is provided by the fol-
lowing formulation [10,19]:
hforc =kair
L·0.0239 ·V·L
ν0.805(10)
in which, Kair is evaluated at an average temperature
between the effective air one and outer tyre surface
one. Vis considered to be coincident with the forward
speed of the vehicle (air speed is supposed to be zero);
the values of hforc calculated with the above approach
are close to those obtained by means of CFD simula-
tions [20,21].
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The natural convection coefficient h, also obtained
by the dimensionless analysis, can be expressed as:
h=Nu ·kair
L(11)
in which, for this case:
Nu =0.53 ·Gr0.25 ·Pr0.25 (12)
The last heat exchange, the convection with the inflat-
ing gas, can be expressed my means of a mechanism of
natural convection, as the indoor air is considered sta-
tionary with respect to the tyre during rolling. In this
case, by modeling the system as a horizontal cylinder
coaxial with the inflating gas contained in a cavity, the
heat exchange coefficient is:
hint =kair
δ·0.40 ·g·β·δ3·(T −T∞)
ν20.20
·μ·cp
k0.20(13)
with δequal to the difference between effective rolling
radius and the rim radius.
3.4 Contact area calculation
The size and the shape of the contact area are func-
tion of the vertical load acting on each wheel, of the
inflation pressure and of camber and toe angles.
In the T.R.T. model the contact area is assumed to
be rectangular in shape, as already said, with constant
width W, equal to the tread width, and length La,vari-
able with the above mentioned parameters, except the
toe angle.
The extension of the patch depends on the number
of nodes in contact with the road and it is calculated
as:
A0=NEC ·x ·y (14)
NEC is given by (NECx)·(NECy).
NECxis the number of nodes in contact along xmi-
nus one, calculated as explained in the following and
NECyis the number of nodes in contact with the road
along yminus one, identified by the ratio between the
width Wof the tread and the lateral dimension y of
the single element.
The area is indicated with A0to emphasize that it
is not variable during the simulation after having been
calculated in pre-processing. The real number of nodes
in contact is calculated from the effective area of con-
tact Aeff , which is obtained by means of diagrams as
the ones showed in Figs. 3and 4, taking into account
actual vertical load and inflating pressure:
NECef f =Aef f
W·x ·NECy(15)
in which for the amount Aef f
W·x , representing the num-
ber of nodes in contact with the road in the xdirection
minus one, it is considered the nearest integer.
The effective area of contact has been obtained on
the basis of the results provided by FEM simulations
(Figs. 3and 41), both for front and for rear tyre. The
used tyre FE model was validated on measured static
contact patch and on measured static and dynamic tyre
profiles [22].
Below are shown the extensions of the effective
contact area as a function of the vertical load and of the
camber angle for a value of the inflation pressure equal
to the one employed in usual working conditions.
Effective contact area values have been adimen-
sionalized respect to a reference value for confiden-
tiality reasons.
In Fig. 4it is possible to observe the influence of
inflating pressure variations on the contact area.
The obtained analytical expressions have been op-
timized around the average value of camber angle as-
sumed by each axle in typical working conditions and
they are of the type:
Aeff =f(F
z,γ,p
i)·groove factor (16)
in which groove factor is a coefficient taking into ac-
count the presence or not of grooves on the tread and
represents the ratio between the effective area of a
grooved tyre and a of slick one with the same nomi-
nal dimensions. By definition, then, this coefficient as-
sumes unitary value in the case of a slick tyre.
Then, considering that in steady state conditions the
variations of the inflation pressure are small and that
camber angle does not have a great influence on the
size of the contact area, for simplicity, these dependen-
cies have been neglected. As a result, it is possible to
1In Figs. 3and 4camber values A,B,C, vertical load values
FzA,FzB,FzC and inflating pressure values A,B,Care in-
side typical working ranges of the considered tyres. Their rel-
ative order is specified in figure captions and they are not ex-
plicited for confidentiality reasons.
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Fig. 3 (a) Contact area as a function of the vertical load for
different camber angles—front tyre (Camber C>Camber B>
Camber A).(b) Contact area as a function of the camber an-
gle for different vertical loads—front tyre (FzA,FzB =2FzA,
FzC =3FzA). (c) Contact area as a function of the verti-
cal load for different camber angles—rear tyre (Camber C>
Camber B>Camber A). (d) Contact area as a function of
the camber angle for different vertical loads—rear tyre (FzA,
FzB =2FzA,FzC =3FzA)
Fig. 4 (a) Contact area as a function of the vertical load for dif-
ferent values of the inflation pressure—front tyre (Press.C>
Press.B>Press.A). (b) Contact area as a function of the verti-
cal load for different values of the inflation pressure—rear tyre
(Press.C>Press.B>Press.A)
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consider an expression, optimized on internal pressure
typical values at medium values of speed and camber,
of the type:
Aeff =f(F
z)·groove factor (17)
As said, in order to avoid excessive computational
loads, the number of nodes in contact has been con-
sidered constant during a simulation. So for its deter-
mination the average normal load acting on the sin-
gle wheel has been considered. This average normal
load is determined considering the dynamic behaviour
of the car, taking into account longitudinal and lateral
load transfers and aerodynamics downforces. There-
fore, it results:
NECx=Aef f [f(F
z,aver age )]·groove factor
W·x (18)
To take into account the variation of the contact area
extension as a function of the normal instantaneous
load in the model, the values of the coefficients char-
acterizing the heat exchanges, depending on the varia-
tions of the size of the area (in particular Hc, for what
concerns the conductive exchange with the asphalt and
hforc for the remaining area of the surface) have to be
scaled, having decided not to act directly on NECxand
NECy.
Since heat exchanges are expressed by relations of
the type:
˙
Q=h·T ·A(19)
the effect of the contact patch variations can be trans-
ferred to the heat transfer coefficients by means of fac-
tors which are proportional to the ratio between the ex-
tension of the effective area with respect to the static
one.
The equations of heat exchange become, therefore,
the following:
˙
Q=C1·Hc·(Tr−T)·A0(20)
˙
Q=C2·hforc ·(T∞−T)·Aconv (21)
where:
C1=Aeff
A0(22)
C2=1+(1−k1)·A0
Aconv
(23)
Aconv =Atot −A0(24)
3.5 The constitutive equations
On the basis of the previous considerations it is pos-
sible to write the power balance equations, based on
heat transfers, for each elementary mass associated to
each node. These equations are different for each node,
depending on its position in the grid.
The conductivity between the surface and the bulk
layers is indicated with k1, while with k2is indicated
the conductivity associated to the exchange between
the bulk and the inner liner layers.
Image depicting the control volume associated with
the node 2 (surface layer) are reported in Figs. 5and 6.
The images show the thermal powers exchanged in all
directions respectively for the two cases: road contact
(Fig. 5) and contact with the external air (Fig. 6).
As an example, the only heat balance equation for
node 2 along the xdirection is reported, recalling that,
for the performed discretization, the nodes adjacent to
2 are 6 and 58:
k1
X ·(T6−T2)·Y ·Z1
2−k1
X ·(T2−T58)
·Y ·Z1
2=m2·cv1·T2
t (25)
Substituting the expression of the mass (2) (reminding
that in this case C=1/2) leads to the equation:
T2
t =1
ρ·cv1·k1
X2·T6−2·k1
X2·T2+k1
X2·T58
(26)
Taking into account the exchanges along all directions
and all the possible heat generations, the equation of
node 2 can be written (see Appendix):
•in the case of contact with the road
T2
t =1
ρ·cv1·2·˙
QSEL
X ·Y ·Z1
+−2·k1
X2−2·k1
Y 2−2·k1
Z2
1−2·Hc
Z1
·T2+k1
Y 2·T1+k1
Y 2·T3+k1
X2·T6
+k1
X2·T58 +2·k1
Z2
1·T62 +2·FP
Z1
+2·Hc
Z1·Tr(27)
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Fig. 5 Control volume associated with the node 2, assumed in
contact with the road
Fig. 6 Control volume associated with the node 2, assumed in
contact with the external air
•in the case of contact with external air
T2
t =1
ρ·cv1·2·˙
QSEL
X ·Y ·Z1
+−2·k1
X2−2·k1
Y 2−2·k1
Z2
1−2·hforc
Z1
·T2+k1
Y 2·T1+k1
Y 2·T3+k1
X2·T6
+k1
X2·T58 +2·k1
Z2
1·T62
+2·hforc
Z1·Tair (28)
having denoted by ˙
QSEL the power dissipated by
cyclic deformation.
Note the presence, in Eq. (27), of the generative
term identified by FP and of the term identifying the
cooling with the road (characterized by the presence of
the Hccoefficient). On the other hand, in (28) it is pos-
sible to notice the absence of the generative term (FP)
and the presence of the term identifying the exchange
with the outside air (characterized by the presence of
the hforc coefficient).
In the model the tyre has been considered motion-
less and the boundary conditions rotating around it to
take into account the fact that elements belonging to
the surface layer will be affected alternatively by the
boundary conditions corresponding to the contact with
the road and to the forced convective exchange with
the external air.
The equations showed for node 2 are valid for all
the nodes belonging to the surface layer, localized in-
ternally in lateral direction.
For a node still belonging to the surface layer, but
external in lateral direction (C=1/4), for example
node 1, the equations are (see Appendix):
•in the case of contact with the road
T1
t =1
ρ·cv1·4·˙
QSEL
X ·Y ·Z1
+−2·k1
X2−2·k1
Y 2−2·k1
Z2
1−2·Hc
Z1
·T1+2·k1
Y 2·T2+k1
X2·T5+k1
X2·T57
+2·k1
Z2
1·T61 +2·FP
Z1+2·Hc
Z1·Tr(29)
•in the case of contact with external air
T1
t =1
ρ·cv1·4·˙
QSEL
X ·Y ·Z1
+−2·k1
X2−2·k1
Y 2−2·k1
Z2
1−2·hforz
Z1
·T1+2·k1
Y 2·T2+k1
X2·T5+k1
X2
·T57 +2·k1
Z2
1·T61 +2·hforc
Z1·Tair (30)
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The equation relating to the bulk layer, for an internal
node in the lateral direction (C=1), e.g. node 62, is
(see Appendix):
T62
t =1
ρ·cv2·˙
QSEL
X ·Y ·(Z1
2+Z2
2)
+−2·k2
X2−2·k2
Y 2−k2
Z2·(Z1
2+Z2
2)
−k1
Z1·(Z1
2+Z2
2)·T62 +k2
Y 2
·T61 +k2
Y 2·T63 +k2
X2·T66 +k2
X2
·T118 +k2
Z2·(Z1
2+Z2
2)·T122
+k1
Z1·(Z1
2+Z2
2)·T2(31)
Similarly, relatively to a bulk external node in the
transverse direction (C=1/2), it results (see
Appendix):
T61
t =1
ρ·cv2·2·˙
QSEL
X ·Y ·(Z1
2+Z2
2)
+−2·k2
X2−2·k2
Y 2−k2
Z2·(Z1
2+Z2
2)
−k1
Z1·(Z1
2+Z2
2)·T61 +2·k2
Y 2
·T62 +k2
X2·T65 +k2
X2·T117
+k2
Z2·(Z1
2+Z2
2)·T121
+k1
Z1·(Z1
2+Z2
2)·T1(32)
As concerns the innermost layer, the inner liner, the
equation of exchange for an internal node in the trans-
verse direction (C=1/2), e.g. 122, is (see Appendix):
T122
t =1
ρ·cv2·2·˙
QSEL
X ·Y ·Z2
+−2·k2
X2−2·k2
Y 2−2·k2
Z2
2−2·hint
Z2
·T122 +k2
Y 2·T121 +k2
Y 2·T123
+k2
X2·T126 +k2
X2·T158 +2·k2
Z2
2
·T62 +2·hint
Z2·Tair_int(33)
Finally, for an external node in the transverse direc-
tion belonging to the Inner liner (C=1/4), it is (see
Appendix):
T121
t =1
ρ·cv2·4·˙
QSEL
X ·Y ·Z2
+−2·k2
X2−2·k2
Y 2−2·k2
Z2
2−2·hint
Z2
·T121 +2·k2
Y 2·T122 +k2
X2·T125
+k2
X2·T157 +2·k2
Z2
2·T61
+2·hint
Z2·Tair_int(34)
In conclusion, the matrix equation at the basis of the
model is:
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
∂T1
∂t
∂T2
∂t
∂T3
∂t
···
···
∂Tn
∂t
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
b1
b2
···
···
···
bn
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠+1
ρ·cv
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
a11 ··· a1n
a21 ··· a2n
··· ···
··· ···
··· ···
··· ···
an1ann
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
·⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
T1
T2
···
···
···
Tn
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(35)
in which aij is the generic coefficient, relative to the
energy balance equation of the node i, that multiplies
the jth node temperature, while biis the generic coef-
ficient not multiplying nodes temperatures.
To properly operate in order to provide the tyre tem-
perature distribution, the model requires the following
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Meccanica
Fig. 7 Comparison between measured and simulated surface temperatures both for rear and front tyres
input data: normal, longitudinal and lateral interaction
tyre-road forces, longitudinal and lateral slip speeds,
forward speed at the wheel center, air and road temper-
atures. The structural characteristics and thermal prop-
erties of the tyre and the thermal conductivity of the
track are also required.
Some of these data result from the measures of
telemetry available for different circuits and are pre-
liminary analyzed in order to check their reliability;
others, such as in particular the ones related to struc-
tural and thermal characteristics of the tyre, are es-
timated on the basis of measurements and tests con-
ducted on the tyres [9].
At the end of the model development, sensitivity
analyses have been performed; it resulted that em-
ployed instruments, characterized by high accuracy,
are able to guarantee low uncertainty levels that do not
affect the goodness of model results.
In addition to surface, bulk and inner liner tempera-
ture distributions, the model also provides the thermal
flows affecting the tyre, such as the flow due to the ex-
ternal air cooling, the one due to the cooling with the
road, the one with the inflation air as well as the flows
due to friction, hysteresis and exchanges between the
different layers.
4 Results and discussion
The model needs an initial tuning phase to be carried
out only once for each season, because of changes in
car setup and tyres construction, aimed to the identi-
fication of the values of some scaling factors. This is
done on the basis of the results obtained during pre-
season testing, which commonly take place early in
the year before the season starts. This phase also al-
lows the direct experimental check of tyre thermal pa-
rameters.
Once developed through this operation it can be
used in a predictive manner, known all inputs, with ref-
erence to the various operating conditions of the dif-
ferent circuits. The results obtained are in good agree-
ment with the telemetry data.
This is clearly shown in Figs. 7,8and 9, which il-
lustrate a comparison between the temperatures pro-
vided by the telemetry (measured my means of in-
frared sensors, pointing the middle line of the tyre)
during a race and the results provided by the model
in simulation. The signals show a certain periodicity
because they refer to race laps.
Figure 7in particular shows a comparison between
the temperature of surface measured and simulated for
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Meccanica
Fig. 8 Bulk simulated temperature and comparison between measured and simulated inner liner temperatures both for rear and front
tyres
all the four wheels. As can be seen the agreement be-
tween the model and telemetry is excellent.
With regard to the front wheels, the fragmentary
telemetry data is due to the fact that when the steer-
ing angle exceeds a certain threshold, the temperature
measurement is not trusted because the sensor detects
temperature values corresponding to different zones of
the tyre. Substantially when the steering wheel is over
a certain value the reliability of the temperature signal
is lost.
In Fig. 8, the temperatures of the inner liner mea-
sured and those calculated with the model are reported.
Also in this case, for all four wheels the agreement is
excellent. In the figure are also reported bulk temper-
atures estimated by means of model simulations. For
bulk temperatures no data are available from teleme-
try.
Proper time ranges have been selected to highlight
thermal dynamics characteristic of each layer; in par-
ticular, as concerns bulk and inner liner (Fig. 8), tem-
perature decreasing trend is due to a vehicle slowdown
before a pit stop.
Finally in Fig. 9, with reference to a different cir-
cuit, the comparisons between the measured temper-
atures and those supplied by the model for all four
wheels of the vehicle are reported.
Even in this case, despite the fragmentary telemetry
data of the front tyres surface temperature, the agree-
ment between the telemetry data and those evaluated
with the model is good.
5 Conclusions
The Thermo Racing Tyre model presented in this pa-
per is an indispensable instrument to optimize racing
tyres performances since tyre surface temperatures as
well as bulk ones have great influence on the tyre-track
interaction. The interaction forces reach their maxi-
mum values only within a narrow temperature range,
while decay significantly outside of it. The ability to
predict the temperature distribution on the surface, and
also within the tyre in the different operating situations
during the race, allows to identify the tyre conditions
during the race, so it is possible to ensure the optimum
temperature to maximize the forces exchanged with
the track.
Moreover, having the model the possibility to turn
in real time, it is suitable for applications on a driving
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Meccanica
Fig. 9 Another example of bulk and inner liner simulated temperature and comparison between measured and simulated surface
temperatures both for rear and front tyres
simulator where it is necessary to reproduce the real
operating conditions including the tyre temperatures.
The physical nature of the model, based on ana-
lytic equations containing known or measurable phys-
ical parameters, in addition to give to the model the
predictive ability, also allows an analysis of the influ-
ence of different parameters including the constructive
characteristics and chemical-physical properties of the
rubber. This is extremely useful in the design phase
of the tyres, but also for the choice of the tyres ac-
cording to the various circuits characteristics and to
the methodology of their use.
Naturally, the model needs a preliminary tuning
phase before it can be used and this stage is possible if
a sufficient wide and varied amount of data from mul-
tiple circuits through the telemetry is available.
This phase is typically placed in the activities of
pre-season testing on the track. Once developed the
model, it will provide, on the basis of inputs from
telemetry or from models if used on a driving simu-
lator, the output temperature of the surface, bulk and
inner liner as well as heat flows in input and in output
from the tyre. The knowledge of heat flows and hence
their balance is another important instrument for the
identification of optimum operating conditions in or-
der to maximize tyre performances.
Appendix
As an example, heat balance equation for node 2 along
the xdirection is reported, recalling that, for the per-
formed discretization, the nodes adjacent to 2 are 6
and 58:
k1
X ·(T6−T2)·Y ·Z1
2−k1
X ·(T2−T58)·Y
·Z1
2=m2·cv1·T2
t (25)
Substituting the expression of the mass (2) (reminding
that in this case C=1/2) leads to the equation:
T2
t =1
ρ·cv1·k1
X2·T6−2·k1
X2·T2+k1
X2·T58
(26)
Taking into account the exchanges along all direc-
tions and all the possible heat generations, the equa-
tion of node 2 can be written:
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•in the case of contact with the road
˙
QSEL +k1
X ·(T6−T2)·Y ·Z1
2−k1
X
·(T2−T58)·Y ·Z1
2+k1
Y ·(T1−T2)·X
·Z1
2−k1
Y ·(T2−T3)·X ·Z1
2+k1
Z1
·(T62 −T2)·X ·Y +CR ·Fx·vx+Fy·vy
A
·X ·Y +Hc·(Tr−T2)
·X ·Y =m2·cv1·T2
t (A)
•in the case of contact with external air
˙
QSEL +k1
X ·(T6−T2)·Y ·Z1
2−k1
X
·(T2−T58)·Y ·Z1
2+k1
Y ·(T1−T2)·X
·Z1
2−k1
Y ·(T2−T3)·X ·Z1
2+k1
Z1
·(T62 −T2)·X ·Y +hforc ·(Tair −T2)
·X ·Y =m2·cv1·T2
t (B)
having denoted by ˙
QSEL the power dissipated by
cyclic deformation.
Once developed, the two expressions lead respec-
tively to:
•in the case of contact with the road
T2
t =1
ρ·cv1·2·˙
QSEL
X ·Y ·Z1
+−2·k1
X2−2·k1
Y 2−2·k1
Z2
1−2·Hc
Z1
·T2+k1
Y 2·T1+k1
Y 2·T3+k1
X2·T6
+k1
X2·T58 +2·k1
Z2
1·T62
+2·FP
Z1+2·Hc
Z1·Tr(27)
•in the case of contact with external air
T2
t =1
ρ·cv1·2·˙
QSEL
X ·Y ·Z1
+−2·k1
X2−2·k1
Y 2−2·k1
Z2
1−2·hforc
Z1
·T2+k1
Y 2·T1+k1
Y 2·T3+k1
X2·T6
+k1
X2·T58 +2·k1
Z2
1·T62
+2·hforc
Z1·Tair (28)
The equations showed for node 2 are valid for all the
nodes belonging to the surface layer, localized inter-
nally in lateral direction.
For a node still belonging to the surface layer, but
external in lateral direction (C=1/4), for example
node 1, the complete equations are:
•in the case of contact with the road
˙
QSEL +k1
X ·(T5−T1)·Y
2·Z1
2−k1
X
·(T1−T57)·Y
2·Z1
2+k1
Y ·(T2−T1)·X
·Z1
2+k1
Z1·(T61 −T1)·X ·Y
2+CR
·Fx·vx+Fy·vy
A·X ·Y
2+Hc·(Tr−T1)
·X ·Y
2=m1·cv1·T1
t (C)
•in the case of contact with external air
˙
QSEL +k1
X ·(T5−T1)·Y
2·Z1
2−k1
X
·(T1−T57)·Y
2·Z1
2+k1
Y ·(T2−T1)·X
·Z1
2+k1
Z1·(T61 −T1)·X ·Y
2+hforc
·(Tair −T1)·X ·Y
2=m1·cv1·T1
t (D)
leading, respectively, to:
•for the first case
T1
t =1
ρ·cv1·4·˙
QSEL
X ·Y ·Z1
+−2·k1
X2−2·k1
Y 2−2·k1
Z2
1−2·Hc
Z1·T1
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+2·k1
Y 2·T2+k1
X2·T5+k1
X2·T57
+2·k1
Z2
1·T61 +2·FP
Z1+2·Hc
Z1·Tr(29)
•for the second case
T1
t =1
ρ·cv1·4·˙
QSEL
X ·Y ·Z1
+−2·k1
X2−2·k1
Y 2−2·k1
Z2
1−2·hforz
Z1
·T1+2·k1
Y 2·T2+k1
X2·T5+k1
X2
·T57 +2·k1
Z2
1·T61 +2·hforc
Z1·Tair (30)
The equation relating to the bulk layer, for an internal
node in the lateral direction (C=1), e.g. node 62, is:
˙
QSEL +k2
X ·(T66 −T62)·Y ·Z1
2+Z2
2
−k2
X ·(T62 −T118)·Y ·Z1
2+Z2
2
+k2
Y ·(T61 −T62)·X ·Z1
2+Z2
2
−k2
Y ·(T62 −T63)·X ·Z1
2+Z2
2
+k2
Z2·(T122 −T62)·X ·Y −k1
Z1
·(T62 −T2)·X ·Y =m62 ·cv2·T62
t (E)
Such expression, suitably developed, leads to:
T62
t =1
ρ·cv2·˙
QSEL
X ·Y ·(Z1
2+Z2
2)
+−2·k2
X2−2·k2
Y 2−k2
Z2·(Z1
2+Z2
2)
−k1
Z1·(Z1
2+Z2
2)·T62 +k2
Y 2·T61
+k2
Y 2·T63 +k2
X2·T66 +k2
X2·T118
+k2
Z2·(Z1
2+Z2
2)·T122
+k1
Z1·(Z1
2+Z2
2)·T2(31)
Similarly, relatively to a bulk external node in the
transverse direction (C=1/2), it results:
˙
QSEL +k2
X ·(T65 −T61)·Y
2·Z1
2+Z2
2
−k2
X ·(T61 −T117)·Y
2·Z1
2+Z2
2
+k2
Y ·(T62 −T61)·X ·Z1
2+Z2
2
+k2
Z2·(T121 −T61)·X ·Y
2−k1
Z1
·(T61 −T1)·X ·Y
2=m61 ·cv2·T61
t (F)
that becomes:
T61
t =1
ρ·cv2·2·˙
QSEL
X ·Y ·(Z1
2+Z2
2)
+−2·k2
X2−2·k2
Y 2−k2
Z2·(Z1
2+Z2
2)
−k1
Z1·(Z1
2+Z2
2)·T61 +2·k2
Y 2·T62
+k2
X2·T65 +k2
X2·T117
+k2
Z2·(Z1
2+Z2
2)·T121
+k1
Z1·(Z1
2+Z2
2)·T1(32)
As concerns the innermost layer, the inner liner, the
equation of exchange for an internal node in the trans-
verse direction (C=1/2), e.g. 122, is:
˙
QSEL +k2
X ·(T126 −T122)·Y ·Z2
2−k2
X
·(T122 −T158)·Y ·Z2
2+k2
Y
·(T121 −T122)·X ·Z2
2−k2
Y
·(T122 −T123)·X ·Z2
2+k2
Z2
·(T62 −T122)·X ·Y +hint ·(Tair_int −T122)
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Meccanica
·X ·Y =m122 ·cv2·T122
t (G)
that simplified returns:
T122
t =1
ρ·cv2·2·˙
QSEL
X ·Y ·Z2
+−2·k2
X2−2·k2
Y 2−2·k2
Z2
2−2·hint
Z2
·T122 +k2
Y 2·T121 +k2
Y 2·T123
+k2
X2·T126 +k2
X2·T158 +2·k2
Z2
2
·T62 +2·hint
Z2·Tair_int(33)
Finally, for an external node in the transverse direction
belonging to the Inner liner (C=1/4), it is:
˙
QSEL +k2
X ·(T125 −T121)·Y
2·Z2
2−k2
X
·(T121 −T157)·Y
2·Z2
2+k2
Y
·(T122 −T121)·X ·Z2
2+k2
Z2
·(T61 −T121)·X ·Y
2
+hint ·(Tair_int −T121)·X ·Y
2
=m121 ·cv2·T121
t (H)
which, simplified, provides:
T121
t =1
ρ·cv2·4·˙
QSEL
X ·Y ·Z2
+−2·k2
X2−2·k2
Y 2−2·k2
Z2
2−2·hint
Z2
·T121 +2·k2
Y 2·T122 +k2
X2·T125
+k2
X2·T157 +2·k2
Z2
2·T61
+2·hint
Z2·Tair_int(34)
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