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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)
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... The dynamic behaviour of the vehicle can be deeply affected by the tyre operating conditions, including thermodynamic and wear effects. One of the biggest factors is tyre temperature itself, playing a fundamental role in high-performance motor sport applications (Farroni et al., 2014). Figure 1.1 shows the effect of tyre temperature on pure longitudinal grip. ...
... The physical model (1D-heatflow) presented by Rosa et al. (2008) set the foundations for the 3D physical model later presented by Farroni et al. (2014;2015) for motorsport applications "to estimate the temperature distribution even of the deepest tyre layers". As stated by the authors, these models have shown great real-time applicability for DiL simulations. ...
... Lastly, the conduction between different tyre layers due to temperature difference is also considered. For more modelling details, the reader can refer to (Farroni et al., 2014;Farroni, Sakhnevych, and Timpone, 2015). ...
... Later, the original model, designed for vehicle handling applications, has been reformulated to include the internal pressure effect [21,22] and to extend the applicability in dynamic scenarios with higher frequency [23]. The MF model has been further enhanced in [24], where the authors have proposed an advanced multiphysical MF-based (MF-evo) realtime tyre model with the aim to extend the Pacejka's Magic Formula tyre model in the whole range of the tyre operating conditions, taking into account its internal temperature distribution [25,26], inflation pressure [22], tread wear [27,28], compound viscoelastic characteristics and road roughness [29,30]. The potential risks, related with the employment of empirical models, are linked with their parametrisation and the quality of data, since the adoption of numeric data-based techniques makes it possible to completely misinterpret the tyre behaviour even in case of a good fitting towards experimental results. ...
... To demonstrate the augmented reliability of the methodology, both standard MF and MF-evo have been calibrated and compared towards the experimental dataset acquired in dedicated handling track session, characterised by a typical measurement noise. In the case under study, the additional physical variables, consisting in temperature, pressure and compound wear level, have been co-simulated thanks to specific tyre thermal model [25,26] and a wear model [30], respectively. ...
... Indeed, once properly calibrated and validated towards experimental data, the MF-evo-based real-time co-simulation tyre system can be employed within offline vehicle setup optimisation routines, advanced data analysis algorithms, hard real-time simulation environments for driver-in-the-loop, software-inthe-loop and hardware-in-the-loop, and, finally, embedded onboard model-based control logics. In [25,26], the authors describe the real-time tyre thermal model, able to calculate the temperatures governing the grip and stiffness properties, whereas the procedure to take into account of tread wear and compound degradation thanks to physical grip model is presented in [28]. ...
Article
To cite this article: Aleksandr Sakhnevych (2021): Multiphysical MF-based tyre modelling and parametrisation for vehicle setup and control strategies optimisation, Vehicle System Dynamics, ABSTRACT Starting from the earliest phases of design of the vehicle and its control systems, the understanding of tyres is of fundamental importance to govern the overall vehicle dynamics. A properly charac-terised tyre-road interaction model is essential to achieve a reliable vehicle dynamics model on which more design variations can be studied directly in simulation environment optimising both cost and time. The possibility to count on computationally efficient and reliable formulations represents nowadays a great advantage, and the multiphysical Pacejka's Magic Formula (MF-evo) tyre model presented is one of the best trade-off solutions to meet the strict real-time requirements and to reproduce multiphysical variations of the tyre dynamic behaviour towards temperature, pressure and wear effects. A specific methodology has been developed to characterise and to identify the MF-evo parameters with a high grade of accuracy and reliability directly from experimental data. The proposed technique is based on a pre-processing procedure to remove non-physical outliers and to cluster the data, which allows to optimise the multidimensional parameterisation process. To the purpose of validation of the parametrisation routine, data from a motorsport case, exceptionally difficult to reproduce in simulation due particularly significant variations of the tyre dynamics during a single test, have been employed demonstrating the MF-evo model potential and robustness. ARTICLE HISTORY
... Therefore, in order to avoid accidents, it is necessary to monitor the tyre state. During the motion, due to the multi-material interaction and the viscoelastic rubber matrix compositions, the dynamic characteristics of such component may vary considerably during their life cycle as a function of road, inner pressure, ageing and temperature that affects the tyre friction and stiffness [2]. The existing widely accepted vehicle dynamics motion controllers are still sub-optimal in the sense that they don't consider the various multi-physical phenomena of the tyre such as tire temperature, tire wear, etc [5]. ...
Conference Paper
Tire operating conditions, such as thermodynamic or wear status can profoundly affect vehicle dynamic behavior by playing a key role especially in motorsport applications as they change the tire's potential stiffness and grip. To this purpose, the presented work aims to enrich an ABS controller based on nonlinear model predictive control (NMPC) with the knowledge of tire temperature and to evaluate the improvements obtained.
... Tyre models typically relate the tyre load, lateral and longitudinal slips, and camber angle to the tyre forces and moments. More advanced models can include further sensitivity to vehicle speed, tread wear, inflation pressure [41] and temperature [42,43], amongst other parameters. ...
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Electric vehicles and low carbon technology are currently at the forefront of research due to the need to rapidly reduce global carbon emissions. Significant effort has been invested into the improvement of electric cars but comparatively little for electric motorcycles, especially high-performance electric motorcycles. To achieve high-performance it is important to capture relevant design trade-offs and plan for vehicle optimisation prior to starting detailed design. These design trade-offs typically involve optimal sizing of the vehicle battery, electric motor, and motor drive, as well as the determination of the optimum lift-to-drag ratio. A full vehicle analysis including pertinent mechanical and electrical elements is required to perform this properly, as the system is highly interdependent. Existing models are shown to be lacking in key areas, notably the integration of an appropriate battery model, a realistic electric motor model (reflecting modern high-performance electric motorcycle design practices), and an appropriate tyre model, amongst other issues. The work in this thesis builds and validates a full vehicle model of a modern high-performance electric motorcycle. This is accomplished by first developing a rigid body dynamics motorcycle model that includes a full tyre model, the effects of downforce, differing front and rear tyres, and front-wheel drive. Further work is then undertaken to increase the depth and suitability of the electric powertrain modelling for high-performance electric motorcycles. Here, the battery thermal and electrical responses are modelled as well as the powertrain torque response, including saturation and loss modelling of the motor, motor drive and final drive. To validate these models both motor dynamometer testing and battery cycle testing is performed. An accelerated battery testing procedure is also developed to reduce the time required to properly evaluate and characterise test cells for performance evaluation. Having developed the vehicle model, a lap simulation procedure is then developed, implemented, and validated. Validation uses lap data acquired at multiple events including the Isle of Man TT Zero, Pikes Peak International Hillclimb (PPHIC) and Elvington Airfield Land speed record attempts. The lap simulation is then extended to include the effects of energy deployment strategy on lap time. This includes a different methodology for designs that are limited by the battery thermal performance and those that are not. This deployment strategy implementation is shown to significantly affect lap time. The work continues with lap time simulations of the Isle of Man TT Zero and PPHIC, investigating the respective influence of energy management on battery sizing. This shows that it is important to include the energy management strategy into the design evaluation and that the energy management trade-offs are specific to each race event. Additionally, analysis shows that situations, where battery temperature management strategies dominate energy management strategies, should be avoided by the proper design of a battery cooling system. This is because the penalty associated with reducing battery temperature through power and velocity limitations is higher than that of including sufficient cooling. The lap time sensitivity to mass, motor inertia, winglet lift-to-drag ratios and other design variables are explored with recommendations made for the Isle of Man TT Zero race and PPHIC. It is shown that by properly including representations of the underlying physics using a holistic modelling approach, and utilising a quantifiable objective, the relative contribution of individual elements can be quantified and directly compared. The significance of this from a full vehicle design standpoint is large as now vehicle development can be accurately targeted into areas that provide significant benefit. This can greatly improve the efficiency of the development process and the ultimate performance of the motorcycle.
... The physical model (1D-heatflow) presented by Rosa et al. [24] set the foundations for the 3D physical model later presented by Farroni et al. [1,25] for motorsport applications 'to estimate the temperature distribution even of the deepest tyre layers'. As stated by the authors, these models have shown great real-time applicability for DiL simulations. ...
Article
Full-text available
Vehicle dynamics can be deeply affected by various tyre operating conditions, including thermodynamic and wear effects. Indeed, tyre temperature plays a fundamental role in high performance applications due to the dependencies of the cornering stiffness and potential grip in such conditions. This work is focused on the evaluation of a potentially improved control strategy’s performance when the control model is fed by instantaneously varying tyre parameters, taking into account the continuously evolving external surface temperature and the vehicle boundary conditions. To this end, a simplified tyre thermal model has been integrated into a model predictive control strategy in order to exploit the thermal dynamics’ dependents within a proposed advanced ABS control system. We evaluate its performance in terms of the resulting braking distance. In particular, a non-linear model predictive control (NMPC) based ABS controller with tyre thermal knowledge has been integrated. The chosen topic can possibly lay a foundation for future research into autonomous control where the detailing of decision-making of the controllers will reach the level of multi-physical phenomena concerning the tyre–road interaction.
... The aim of the study is not limited to a quantitative analysis on the effects of the variables variation on some specific KPIs or metrics, for which the DOE would have been a perfect solution, but one of the main intents is the analysis of the behavioral dynamics of the whole vehicle, due to the under/over-steering effects linked to tire state, as for example in terms of its trajectory. Another aspect to highlight is that varying the initial temperature on front tire means varying the initial state of all the layers in which the tire is radially discretized [21,35,37]; the same can be said for rear axle. Track and external air temperatures have been both set to 30 • C, defined as an arbitrary reference; the intention of the study, independently from the quantitative aspects related to the reference temperature values, aims to extrapolate relative deductions, useful to understand the global trends at decoupled boundary and starting conditions. ...
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The handling behavior of a vehicle is one of its most important properties because of its relation to performance and safety and to its deep link with concepts such as “over-steer” or “under-steer”. Tire-road interaction models play a fundamental role in the vehicle system modelling, since tires are responsible for the generation of forces arising within contact patches, fundamental for both handling and ride/comfort. Among the models used to reproduce such forces, Pacejka’s Magic Formula (MF) is undoubtedly one of the most used ones in real-time automotive simulation environments because of its ability to fit quite easily a large amount of experimental data, but its original formulation did not take into account of the tire thermodynamics and wear conditions, which clearly affect tire and vehicle dynamics and are not negligible, especially for high level applications, such as motorsport competitions. Exploiting a multiphysical tire model, which consists in an evolved version of the standard MF model (MF-evo), and a vehicle model properly validated throughout experimental data acquired in outdoor testing sessions carried out with an industrial partner, the current work presents a study on vehicle behavior variation induced by thermodynamic and wear parameters, defining a series of metrics to analyze and show results. One of the elements of interest on which the focus is placed is the possibility to highlight how under-over-steering behavior of a car changes according to different thermodynamic states of tires; to do this, a commercial software VI CarRealTime has been used to perform a series of objective steady-state maneuvers and long runs, exploiting the logic of a lap time optimizer.
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Chapter
In steady-state conditions, explicit expressions for tyre characteristics may be derived using the theoretical framework provided by the brush theory. This chapter is, thus, dedicated to addressing the stationary problem from both the local and global perspectives. The fundamental concepts of critical slip and spin are introduced with respect to an isotropic tyre, and the deformation of the bristles inside the contact patch is investigated for different operating conditions of the tyre. Analytical functions describing the tyre forces and moment acting inside the contact patch are obtained for the particular case of a rectangular contact patch. The analysis is qualitative in nature.
Chapter
At a high level of description, the tyre may be thought of as a nonlinear dynamical system, which produces certain outputs, often referred to as tyre characteristics, when subjected to opportune inputs. This interpretation allows defining some fundamental quantities that contribute to determining both the steady-state and the transient response of the tyre. Amongst these, the slip variables play the most important role. In steady-state conditions, the tyre characteristics may be described as real analytic functions of the slips, which may be defined in three main different ways. The Jacobians of the steady-state characteristics with respect to the slip variables are often called matrices of generalised slip stiffnesses. The local properties of the steady-state tyre characteristics may be deduced from the entries of these matrices.
Chapter
This chapter deals with tyre mechanics and it has a particular focus on thermal effects on its dynamical behaviour. In the first part the typical tyre structure is introduced together with the tyre mechanical/dynamical behaviour according to a classical approach, so recalling the main kinematic and dynamic quantities involved in tyre pure and combined interactions. The core of this chapter is the description of a physical-analytical tyre thermal model able to determine the thermal status in each part of the tyre useful for vehicle dynamics modelling and driving simulations in order to take into account thermal effects on tyre interactions and consequently on vehicle dynamical behaviour. Successively also the tyre wear modelling is faced, after a brief introduction to the different models available in literature some considerations are reported concerning the thermal effects on wear.
Conference Paper
Full-text available
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.
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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.
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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.
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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.