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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

Author's personal copy

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

Author's personal copy

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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Meccanica

•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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Meccanica

+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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