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TRT: Thermo racing tyre a physical model to predict the tyre temperature distribution

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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 University of Naples Federico II and a top ranking motorsport team. The model is three-dimensional and takes into account all the heat flows and the generative terms occurring 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 friction energy developed in the contact patch, the strain energy loss is evaluated. The model inputs come out from telemetry data, while its thermodynamic parameters come either from literature or from dedicated experimental tests. The model gives in output the temperature circumferential distribution in the different tyre layers (surface, 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.
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1 23
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
1 23
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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]
Tair 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·(TrTi)·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·(TrT)·A0(20)
˙
Q=C2·hforc ·(TT)·Aconv (21)
where:
C1=Aeff
A0(22)
C2=1+(1k1)·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 ·(T6T2)·Y ·Z1
2k1
X ·(T2T58)
·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·T62·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
X22·k1
Y 22·k1
Z2
12·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
X22·k1
Y 22·k1
Z2
12·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
X22·k1
Y 22·k1
Z2
12·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
X22·k1
Y 22·k1
Z2
12·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
X22·k2
Y 2k2
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
X22·k2
Y 2k2
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
X22·k2
Y 22·k2
Z2
22·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
X22·k2
Y 22·k2
Z2
22·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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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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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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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 ·(T6T2)·Y ·Z1
2k1
X ·(T2T58)·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·T62·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 ·(T6T2)·Y ·Z1
2k1
X
·(T2T58)·Y ·Z1
2+k1
Y ·(T1T2)·X
·Z1
2k1
Y ·(T2T3)·X ·Z1
2+k1
Z1
·(T62 T2)·X ·Y +CR ·Fx·vx+Fy·vy
A
·X ·Y +Hc·(TrT2)
·X ·Y =m2·cv1·T2
t (A)
in the case of contact with external air
˙
QSEL +k1
X ·(T6T2)·Y ·Z1
2k1
X
·(T2T58)·Y ·Z1
2+k1
Y ·(T1T2)·X
·Z1
2k1
Y ·(T2T3)·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
X22·k1
Y 22·k1
Z2
12·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
X22·k1
Y 22·k1
Z2
12·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 ·(T5T1)·Y
2·Z1
2k1
X
·(T1T57)·Y
2·Z1
2+k1
Y ·(T2T1)·X
·Z1
2+k1
Z1·(T61 T1)·X ·Y
2+CR
·Fx·vx+Fy·vy
A·X ·Y
2+Hc·(TrT1)
·X ·Y
2=m1·cv1·T1
t (C)
in the case of contact with external air
˙
QSEL +k1
X ·(T5T1)·Y
2·Z1
2k1
X
·(T1T57)·Y
2·Z1
2+k1
Y ·(T2T1)·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
X22·k1
Y 22·k1
Z2
12·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
X22·k1
Y 22·k1
Z2
12·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
X22·k2
Y 2k2
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
2k1
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
X22·k2
Y 2k2
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
2k2
X
·(T122 T158)·Y ·Z2
2+k2
Y
·(T121 T122)·X ·Z2
2k2
Y
·(T122 T123)·X ·Z2
2+k2
Z2
·(T62 T122)·X ·Y +hint ·(Tair_int T122)
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·X ·Y =m122 ·cv2·T122
t (G)
that simplified returns:
T122
t =1
ρ·cv2·2·˙
QSEL
X ·Y ·Z2
+2·k2
X22·k2
Y 22·k2
Z2
22·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
2k2
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
X22·k2
Y 22·k2
Z2
22·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)
References
1. Pacejka HB (2007) Tyre and vehicle dynamics.
Butterworth-Heinemann, Stoneham
2. Capone G, Giordano D, Russo M, Terzo M, Timpone F
(2008) Ph.An.Ty.M.H.A.: a physical analytical tyre model
for handling analysis—the normal interaction. Veh Syst
Dyn 47:15–27
3. Castagnetti D, Dragoni E, Scirè Mammano G (2008) Elas-
tostatic contact model of rubber-coated truck wheels loaded
to the ground. Proc Inst Mech Eng, Part L 222:245–257
4. Farroni F, Rocca E, Russo R, Savino S, Timpone F (2012)
Experimental investigations about adhesion component of
friction coefficient dependence on road roughness, contact
pressure, slide velocity and dry/wet conditions. In: Pro-
ceedings of the 14th mini conference on vehicle system dy-
namics, identification and anomalies, VSDIA
5. Farroni F, Russo M, Russo R, Timpone F (2012) Tyre-road
interaction: experimental investigations about the friction
coefficient dependence on contact pressure, road rough-
ness, slide velocity and temperature. In: Proceedings of the
ASME 11th biennial conference on engineering systems
design and analysis, ESDA2012
6. Park HC, Youn ISK, Song TS, Kim NJ (1997) Analysis of
temperature distribution in a rolling tire due to strain energy
dissipation. Tire Sci Technol 25:214–228
7. Lin YJ, Hwang SJ (2004) Temperature prediction of rolling
tires by computer simulation. Math Comput Simul 67:235–
249
8. De Rosa R, Di Stazio F, Giordano D, Russo M, Terzo
M (2008) Thermo tyre: tyre temperature distribution dur-
ing handling maneuvers. Veh Syst Dyn 46:831–844. ISSN:
0042-3114
9. Allouis C, Amoresano A, Giordano D, Russo M, Timpone
F (2012) Measurement of the thermal diffusivity of a tyre
compound by mean of infrared optical technique. Int Rev
Mech Eng 6:1104–1108. ISSN: 1970-8734
10. Kreith F, Manglik RM, Bohn MS (2010) Principles of heat
transfer, 6th edn. Brooks/Cole, New York
11. Gent AN, Walter JD (2005). The pneumatic tire. NHTSA,
Washington
12. Johnson KL (1985) Contact mechanics. Cambridge Univer-
sity Press, Cambridge
13. Agrawal R, Saxena NS, Mathew G, Thomas S, Sharma
KB (2000) Effective thermal conductivity of three-phase
styrene butadiene composites. J Appl Polym Sci 76:1799–
1803
14. Dashora P (1994) A study of variation of thermal conduc-
tivity of elastomers with temperature. Phys Scr 49:611–614
15. Brancati R, Rocca E, Timpone F (2010) An experimental
test rig for pneumatic tyre mechanical parameters measure-
ment. In: Proceedings of the 12th mini conference on vehi-
cle system dynamics, identification and anomalies, VSDIA
16. Giordano D (2009) Temperature prediction of high perfor-
mance racing tyres. PhD thesis
17. Brancati R, Strano S, Timpone F (2011) An analytical
model of dissipated viscous and hysteretic energy due to
interaction forces in a pneumatic tire: theory and exper-
iments. Mech Syst Signal Process 25:2785–2796. ISSN:
0888-3270
Author's personal copy
Meccanica
18. Chadbourn BA, Luoma JA, Newcomb DE, Voller VR
(1996) Consideration of hot-mix asphalt thermal proper-
ties during compaction—quality management of hot-mix
asphalt. In: ASTM STP 1299
19. Browne AL, Wickliffe LE (1980) Parametric study of con-
vective heat transfer coefficients at the tire surface. Tire Sci
Technol 8:37–67
20. Fluent 6.3 user’s guide (2006) Fluent Inc, Chap 3
21. Karniadakis GEM (1988) Numerical simulation of forced
convection heat transfer from a cylinder in crossflow. Int J
Heat Mass Transf 31:107–118
22. van der Steen R (2007) Tyre/road friction modeling.
PhD thesis, Eindhoven University of Technology, Depart-
ment of Mechanical Engineering
Author's personal copy
... The thermoRIDE model is a physical-analytical tire model designed to comprehensively study and analyze the complex interactions between a tire, its external environment, and the inner wheel chamber as can be seen in Figure 2 This model is capable of simulating the temperature distribution in the tire layers and of relating this distribution to the heat exchange mechanisms. thermoRIDE model considers several critical heat-related phenomena, including heat generation and exchange processes within the tire structure and with the external environment [5]: ...
... Equation 5, divided on both sides by the quantity ρ · dV · c v · dt, defines the Fourier heat equation: ...
... The tire is considered parallelepiped-shaped and it is discretized by means of a grid, whose nodes represent the points in which the temperature will be determined instant by instant thanks to the written Fourier Equation 7 [5]. The discretization of the tire, which caters to its unique attributes such as dimensions, diffusivity, and inertia, can substantially vary. ...
Conference Paper
Full-text available
Optimizing the performance of racing motorcycles is a central goal for competition teams. The necessity to ensure driver stability and a good level of grip in the widest possible range of riding conditions makes it necessary for tires to work in the right temperature window, capable of ensuring the highest interaction force between tire and road. Specifically, the internal temperature of the tire is a parameter that can be difficult to measure and control but has a significant impact on motorcycle performance and, also, on driver stability. Deepening knowledge of internal tire temperature in racing motorcycles can improve performance optimization on the track and finding the right motorcycle setup. In this work, a physical thermal model is adopted for an activity concerning the development of a moto-student vehicle, to predict the racing motorcycle setup allowing the tire to work in a thermal window that optimizes grip and maximizes tire life. More in detail, a focus has been placed on the effects of the motorcycle’s wheelbase and pivot height variations on internal tire temperatures. Indeed, the stability and handling of the vehicle are highly dependent on the geometric properties of the chassis. Several values of such quantities have been tested in a properly implemented vehicle model developed in the “VI-BikeRealTime” environment, validated by outdoor tests, able to provide forces acting on the tires, slip indices, and speeds, needed by the thermal model as inputs. Through the analysis of the internal temperatures calculated by the model, reached by the various layers of the tire, it has been possible to investigate which of the simulated conditions cause a too-fast thermal activation of the tire and which of them can avoid overheating and underheating phenomena. Lately, this research has delved into the correlation between motorcycle riders' paths and temperature fluctuations with the aim of comprehending how minor alterations in routine maneuvers may influence tire energy activation, particularly in the context of racing and qualifying conditions.
... Notably, friction is not solely dependent on road surface conditions. The tire-road friction is known to exhibit significant temperature dependence owing to rubber characteristics and the internal structure [1]. Recently, tire models capable of taking tire temperature dependency and thermodynamics into account have been developed. ...
... where, α tire is the partition coefficient which defines the fraction of the slip loss power added to the tire itself (the rest is conducted to the road surface via the contact patch) [1], ǫ tire is the rolling resistance coefficient of the tire. V sx and V sy are the longitudinal and lateral slip velocities, respectively, defined further below. ...
Preprint
Automated vehicles need to estimate tire-road friction information, as it plays a key role in safe trajectory planning and vehicle dynamics control. Notably, friction is not solely dependent on road surface conditions, but also varies significantly depending on the tire temperature. However, tire parameters such as the friction coefficient have been conventionally treated as constant values in automated vehicle motion planning. This paper develops a simple thermodynamic model that captures tire friction temperature variation. To verify the model, it is implemented into trajectory planning for automated drifting - a challenging application that requires leveraging an unstable, drifting equilibrium at the friction limits. The proposed method which captures the hidden tire dynamics provides a dynamically feasible trajectory, leading to more precise tracking during experiments with an LQR (Linear Quadratic Regulator) controller.
... First, a tool called T.R.I.C.K. (Tire-Road Interaction Characterization & Knowledge) [10] is presented. This tool had been developed with the aim to process data acquired from experimental test sessions, estimating tires interaction forces, slip indices and inclination angle; the output of this tool is a sort of "virtual telemetry" which can be used to feed the thermoRIDE, a thermodynamic model [11][12][13] which provides in output the lateral and circumferential temperature distributions, in all the different tire layers: tread at three different depths, carcass and inner liner. Then, it is shown how this thermodynamic model can work coupled with a specific wear model [14], called weaRIDE, thanks to which it is possible to calculate tire tread thickness variation and to evaluate its effect on tire temperature and pressure. ...
... thermoRIDE model is a physical-analytical tire model, developed to analyze and reproduce the phenomena concerning the tire thermal behavior during its interaction with both the external environment and the inner wheel chamber (inner air, rim, brakes, etc.). The thermodynamic evolution of the tire system is described by the diffusion equation of Fourier applied to a three-dimensional domain [11]. The model takes into account of the following physical phenomena, also shown in Fig. 4: ▪ Heat Generation due to: ▪ Friction Power ▪ Strain Energy Loss (SEL) ▪ Heat Exchange with the External Environment due to: ...
... To include the effect of the tyre surface temperature on friction curves, the variation of the Pacejka's parameters can be studied as well, see [27]. In real-time simulation environments, moreover, a tyre thermal model (see e.g., [17,28,29]) is frequently employed, to account for temperature variations during the operation of the tyre. ...
... However, to consider the tyre temperature variability while driving the vehicle, a subsequent thermal model is necessary. In the current literature, similarly to tyre wear models, thermal models range from lumped models, to complex distributed ones based on finite elements, some examples of the two approaches are in [16,17,[28][29][30]. To maintain the same paradigm of the tyre wear framework discussed in this paper, we propose to use a simpler version of the lumped model shown in [17,29]: ...
... The tire is characterized as having a parallelepiped-like shape, and it is divided into discrete nodes using a grid [7]. Within this grid, nodes are designated as points where temperature calculations are made at each moment. ...
Conference Paper
Full-text available
Bicycle mobility has become increasingly popular as a sustainable and healthy means of transportation. Bicycles are not only a cost-effective transportation mode but also help reduce traffic congestion and air pollution. However, the efficiency and safety of bicycling largely depend on the optimization of bicycle components, such as the tires. The importance of bike tire optimization cannot be underestimated as it can affect both bicycle dynamics and bicycle performance. Due to the lack of multi-physical mathematical models able to analyze and reproduce complex tire/road contact phenomena, useful to predict the wide range of working conditions, this research aims to the development of a bicycle tire thermal model. The main outcome is to provide the full temperature local distribution inside the tire’s inner rubber layers and the inflation chamber. Such kind of information plays a fundamental role in the definition of the optimal adherence conditions, for both safety and performance maximization, and as an indicator of the proper tire design for various applications, each requiring specific heat generation and management. The experimental validation has been carried out thanks to an innovative test-rig developed at Politecnico di Milano. It is known as VetyT (acronym of Velo Tyre Testing), and it complies with the standard ISO 9001-2015. It has been specifically instrumented for the activity, acquiring the external tire temperatures to be compared with the respective simulated ones, under various workingconditions.
... m, and advanced, allowing belt deflection [16][17][18][19], suitable for higher frequencies and shorter wavelengths. Finally, there are also models dedicated to the investigation of specific issues, such as tyre-temperature distribution [20], tyre wear [21], contact pressure [22,23], impulsive loading [24], hydroplaning [25,26]. ...
Article
Full-text available
Several approaches have been developed over the years for the modelling of the tyre behaviour in vehicle-dynamic applications. The so-called ‘rigid-ring’ models are among the classics for the modelling of the belt dynamics. Although there are several works dealing with the vibrating properties of tyres, the problem of the identification of the related rigid-ring model parameters has not been described other than qualitatively or partially. The aim of this work is thus to fill this gap and to devise a procedure for the experimental characterisation of such parameters, namely the frequency and damping of the in-plane and out-of-plane belt vibration modes as well as the associated masses and inertias. An experimental modal analysis (EMA) approach is employed, which involves an instrumented hammer combined with three-axial accelerometers roving on 16 stations equally spaced along the tyre circumference. The method is numerically demonstrated on the finite-element models of a motorcycle tyre and a car tyre. The approach is also experimentally validated on a real tyre. The rigid-ring vibration modes of the motorcycle tyre are in the range 70–220 Hz, while those of the car tyre are in the range 51–85 Hz. The ratios of the mass/inertia of the rigid ring to the mass/inertia of the tyre are in the range 40–87% and 68–74% for the motorcycle and car respectively.
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Featured Application: Advanced thermo-mechanical tyre models, real time vehicle dynamic simulation , heat transfer calcualtion. Abstract: Tyres are one of the most important elements of a vehicle because they are the link to the road and have a huge impact on traffic-related pollution. Knowing their behaviour, thus being able to use them at their best and reducing their wear rate, is one of the means of improving their lifetime, which means decreasing traffic environmental impact. In order to understand how tyres behave and to predict the real-time tyre-road coefficient of friction, which is strongly influenced by the temperature, in the last few years several complex thermo-mechanical models of heat transfer inside the tyre have been developed. However, in the current state of the art of the literature and practice, there is still an important parameter regarding such models that is not deeply studied. This parameter is the heat transfer coefficient between the tyre and the road at the contact patch, which usually is considered as a constant. The current research paper allows understanding that such an approximation is not always valid for all of the speeds and tyre loads of city and race cars; instead, it is developed an equation that, for the first time, calculates the real-time, dynamic tyre-road heat transfer coefficient, taking into account the tyre's travelling speed and the footprint length. The equation results are in good agreement with the empirical values coming from the literature and permit understanding how much such a parameter can vary, depending on the tyre use range. The formulation is simple enough to be easily implemented in existing thermodynamic tyre models without requiring meaningful computational time.
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High friction and large deformation under high-speed sideslip landing conditions cause rapid heating and severe abrasion of aircraft tire, which seriously threatens landing safety. Therefore, theories of energy dissipation and transient heat transfer are discussed. A thermo-mechanical-abrasive (TMA) coupling analysis method is proposed for solving the thermomechanical problem. Laws of temperature and abrasion under different slip angles are revealed experimentally. Results show that sideslip conditions lead to increase of friction energy dissipation and wear as well as an asymmetric distribution of temperature field; Considering abrasion can effectively improve prediction accuracy of thermomechanical analysis (an increase of 27.65%), the predicted temperature and abrasion profile are in good agreement with the experiment data.
Conference Paper
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The results of an experimental activity carried out with the aim to investigate on the adhesive behaviour of visco-elastic materials in sliding contact with road asperities are presented. Experiments are carried out using a prototype of pin on disk machine in which pin is constituted by a specimen of rubber coming from a commercial tire, while different disks are realized in glass, marble, steel and in abrasive paper of different roughness. Tests are performed in both dry and wet conditions. Roughness of the test surfaces is evaluated by a rugosimeter, while pressure is evaluated, off-line, analysing the extension of the contact patch left by the pin on a sheet of graph paper under known applied loads. Slide velocity is imposed by an inverter controlled motor driving the disk. Basing on well known theoretical hypotheses, adhesive component of friction coefficient is estimated making the specimens slide on surfaces characterized by low values of macro-roughness, in order to underline the differences in rubber behaviour respect to micro-roughness surface variations. The results confirmed adhesion dependence on pressure and sliding velocity in both cases of smooth surfaces, where the main friction mechanism is the adhesive one, and of rough surfaces, where the main friction mechanism is the hysteretic one. Analysing various surfaces roughness it is possible to notice a maximized adhesive contribution on flat surfaces; it reduces with increasing roughness, while hysteretic friction comes over instead of it because of asperities penetration into rubber sliding surface. Moreover in the case of rough surfaces the separation between static and dynamic friction coefficient is evident and the static coefficient is greater than the dynamic one. On the other hand in case of smooth surface the absence of indentation phenomena doesn't allow to recognize, in the measured force time history, the "classical" peak usually associated to the static friction coefficient. Dry and wet tests performed on different micro-roughness profiles highlighted that friction coefficient in dry conditions is greater on smoother surfaces, while an opposite tendency is shown in wet condition, when asperities are greater enough to break the thin water layer, providing a certain degree of indentation. A proposal for a methodology able to estimate the only adhesive friction component, developed thanks to wet contact tests, is expressed in the end of the paper.
Conference Paper
Full-text available
In this paper the results of an experimental activity carried out with the aim to investigate on the frictional behaviour of visco-elastic materials in sliding contact with road asperities is presented. Experiments are carried out using a prototype of pin on disk machine whose pin is constituted by a specimen of rubber coming from a commercial tyre while the disk may be in glass, marble or abrasive paper. Tests are performed both in dry and wet conditions. Roughness of the disk materials is evaluated by a tester and by a laser scan device. Temperature in proximity of the contact patch is measured by pyrometer pointed on the disk surface in the pin trailing edge, while room temperature is measured by a thermocouple. Sliding velocity is imposed by an inverter controlled motor driving the disk and measured by an incremental encoder. Vertical load is imposed applying calibrated weights on the pin and friction coefficients are measured acquiring the longitudinal forces signal by means of a load cell. As regards to the road roughness, the experimental results show a marked dependence with road Ra index. Dry and wet tests performed on different micro-roughness profiles (i.e. glass and marble) highlighted that friction coefficient in dry conditions is greater on smoother surfaces, while an opposite tendency is shown in wet conditions. Although affected by uncertainties the results confirm the dependence of friction on temperature, vertical load and track conditions.
Article
A computer program was developed at the University of Minnesota to predict asphalt concrete cooling times for road construction during adverse weather conditions. Cooling models require extensive experimental data on the thermal properties of hot-mix paving materials. A sensitivity analysis was performed to determine which thermal properties affect pavement cooling times significantly. The results indicated that more information on asphalt thermal conductivity and thermal diffusivity is required. Two suitable test methods for determining these properties at typical paving temperatures and densities were developed, and preliminary results for dense-graded and stone-matrix asphalt (SMA) mixes agreed well with values reported in the literature.
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The thirteen chapters of this book are introduced by a preface and followed by five appendices. The main chapter headings are: motion and forces at a point of contact; line loading of an elastic half-space; point loading of an elastic half-space; normal contact of elastic solids - Hertz theory; non-Hertzian normal contact of elastic bodies; normal contact of inelastic solids; tangential loading and sliding contact; rolling contact of elastic bodies; rolling contact of inelastic bodies; calendering and lubrication; dynamic effects and impact; thermoelastic contact; and rough surfaces. (C.J.A.)
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This paper addresses the systematic procedure using sequential approach for the analysis of the coupled thermo-mechanical behavior of a steady rolling tire. Not only the knowledge of mechanical stresses but also of the temperature loading in a rolling tire are very important because material damage and material properties are significantly affected by the temperature. In general, the thermo-mechanical behavior of a pneumatic tire is highly complex transient phenomenon that requires the solution of a dynamic nonlinear coupled themoviscoelasticity problem with heat source resulting from internal dissipation and friction. In this paper, a sequential approach, with effective calculation schemes, to modeling this system is presented in order to predict the temperature distribution with reasonable sccuracies in a steady state rolling tire. This approach has the three major analysis modules-deformation, dissipation, and thermal modules. In the dissipation module, an analytic method for the calculation of the heat source in a rolling tire is established using viscoelastic theory. For the verification of the calculated temperature profiles and rolling resistance at different velocities, they were compared with the measured ones.
Book
PRINCIPLES OF HEAT TRANSFER was first published in 1959, and since then it has grown to be considered a classic within the field, setting the standards for coverage and organization within all other Heat Transfer texts. The book is designed for a one-semester course in heat transfer at the junior or senior level, however, flexibility in pedagogy has been provided. Following several recommendations of the ASME Committee on Heat Transfer Education, Kreith, Manglik, and Bohn present relevant and stimulating content in this fresh and comprehensive approach to heat transfer, acknowledging that in today's world classical mathematical solutions to heat transfer problems are often less influential than computational analysis. This acknowledgement is met with the emphasize that students must still learn to appreciate both the physics and the elegance of simple mathematics in addressing complex phenomena, aiming at presenting the principles of heat transfer both within the framework of classical mathematics and empirical correlations.
Article
Rotary nosing with 'relieved die' is proposed for the shrinking tip diameter of a tube. Press nosing or spinning is conventionally used for shrinking the tip diameter. In press nosing, while the forming limit is low, productivity is high. In spinning, although the forming limit is very high, the productivity is very low. The present proposed rotary nosing is an intermediate method that is expected to achieve a higher forming limit than press nosing and realize higher productivity than spinning. In this present research, the forming limit was first examined experimentally and the mechanism was studied by finite element analysis. Based on the results, rotary nosing with a relieved die was proposed. The relieved die consists of three partial cones so that the die geometry is easily cut out using an ordinary lathe. This type of die can be easily and widely applied for industrial use. The effect of the proposed method was numerically evaluated with regard to the pushing force and hoop stress. Finally, the effect of rotary nosing was verified experimentally and the range was clarified where rotary nosing has a higher forming limit than conventional press nosing.
Article
Analyses have shown that the thermal state of a tire is influenced by both the size of and variation in the value of the convective heat transfer coefficient at the tire surface. In the work reported here, a test facility was constructed to permit the determination of convective heat transfer coefficients under a broad range of operating conditions. Data were obtained to show the effects of air speed, boundary layer thickness and turbulence level, humidity, tire surface contamination, tire surface roughness and unevenness, and tire surface wetness on convective heat transfer coefficients. The significance of these results to tire power loss is discussed.