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Theoretical and Experimental Estimation of the Hysteretic Component of Friction for a Visco-Elastic Material Sliding on a Rigid Rough Surface

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According to the most recent approaches, the friction coefficient arising between tyre rubber and road can be seen as constituted by three components: adhesion, deforming hysteresis and wear. This paper deals with the estimation of the hysteretic component of friction. This component is due to indentation phenomena regarding contact mechanics of deformable bodies sliding on the asperities of a rough substrate, which exert oscillating forces at the interface, leading to cyclic deformations of the rubber and then to energy "dissipation" via its internal damping. To evaluate this component, a physical model able to calculate energy dissipation of a rubber block indented by a rigid asperity in sliding contact has been built. To this aim the visco-elastic characteristics of the material have been taken into account. The road profile has been modelled as the sum of two sinusoidal waves characterizing the macro and the micro roughness scales. The results coming out from the physical model have been validated by means both of FEM models results and of experimental tests carried on a pin on disk tribometer.
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International Review on Modelling and Simulations (I.RE.MO.S.), Vol. 6, N. 3
ISSN 1974-9821 June 2013
Manuscript received and revised May 2013, accepted June 2013 Copyright © 2013 Praise Worthy Prize S.r.l. - All rights reserved
1023
Theoretical and Experimental Estimation of the
Hysteretic Component of Friction for a Visco-Elastic Material
Sliding on a Rigid Rough Surface
Flavio Farroni, Riccardo Russo, Francesco Timpone
Abstract According to the most recent approaches, the friction coefficient arising between tyre
rubber and road can be seen as constituted by three components: adhesion, deforming hysteresis
and wear. This paper deals with the estimation of the hysteretic component of friction. This
component is due to indentation phenomena regarding contact mechanics of deformable bodies
sliding on the asperities of a rough substrate, which exert oscillating forces at the interface,
leading to cyclic deformations of the rubber and then to energy "dissipation" via its internal
damping. To evaluate this component, a physical model able to calculate energy dissipation of a
rubber block indented by a rigid asperity in sliding contact has been built. To this aim the visco-
elastic characteristics of the material have been taken into account. The road profile has been
modelled as the sum of two sinusoidal waves characterizing the macro and the micro roughness
scales. The results coming out from the physical model have been validated by means both of FEM
models results and of experimental tests carried on a pin on disk tribometer. Copyright © 2013
Praise Worthy Prize S.r.l. - All rights reserved.
Keywords: Visco-Elastic Material, Sliding Friction, Hysteresis Model
Nomenclature
µ Friction coefficient [-]
λ
Roughness wavelength [m]
Ra Roughness index [m]
ε Strain [-]
σ Stress [Pa]
E'
Rubber storage modulus [Pa]
E''
Rubber loss modulus [Pa]
E
Rubber complex dynamic modulus [Pa]
δ Phase angle between storage and loss modulus
[rad]
f
Frequency [Hz]
T Temperature [K]
t Time [s]
F
T
Tyre friction force [N]
F
N
Normal load [N]
Vs Sliding velocity [m/s]
p
Average contact pressure [Pa]
0 Nominal area
W
DISS
Dissipated power [W]
VTOT Total tread volume in contact with
an asperity [m3]
N
Contact half length [m]
R Curvature radius of asperity apex [m]
T0 Period of stress/strain cycle [s]
K
Kuznetsov adhesion parameter
ν Poisson's ratio
I. Introduction
Knowledge about phenomena concerning with
adherence is a key factor in the automotive field, in
particular as concerns the braking/traction and stability
control systems design [1], [2], [3], [4], [5] and also in
other fields, as for example the friction based
transmissions [6], [7], [8].
The continuous drivers’ seeking of the optimal grip
conditions, makes the development of a physical friction
model an essential instrument for the investigation of the
factors acting on indentation and adhesion mechanisms
on which tyre/road interaction is based. Rubber/asphalt
friction, in fact, is influenced by a great number of
variables and parameters, often hard to be controlled and
measured: macro and micro roughness of the bodies in
contact [9], [10], pressure arising at their interface [11],
materials stiffness characteristics [12] and their
frequency and temperature dependence [13], relative
motion direction and speed [14].
The possibility offered by a physical model to provide
a better comprehension of the cited factors allows to act
on them with a wide range of aims: studying soil textures
structured in order to increase drivers' safety both in dry
and in wet conditions, producing more performing rubber
compounds, able to optimize frictional behaviour under
certain temperatures or frequencies and, in particular in
race applications - for which the presented studies have
been originally carried out - in order to configure optimal
vehicle setup and driving strategies [15], [16].
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F. Farroni, R. Russo, F. Timpone
Copyright © 2013 Praise Worthy Prize S.r.l. - All rights reserved International Review on Modelling and Simulations, Vol. 6, N. 3
1024
A deep knowledge of the mechanisms involved with
tyre/road friction is a key factor in the design of the
suspension system: an optimal setting of tyre working
angles, operated in order to optimize temperature, contact
pressure and sliding velocity distributions, can be
efficiently provided by a physical grip model able to
indicate the best wheel configuration at the boundary
conditions changes.
II. Basic Hypotheses
Asphalt and common road profiles are widely
considered as well described by the sum of two
sinusoidal waves characterizing the macro and the micro
roughness scales; for a sort of superposition principle it
can be stated that macro roughness (modelled as a
sinusoid characterized by wavelength value in the range
0.5 50 mm) is responsible of the indentation
phenomena concerning with hysteretic component of
rubber friction, while micro roughness (modelled as a
sinusoid characterized by wavelength value in the range
0 – 0.5 mm) is linked with the attitude that bodies in
sliding contact have to link each other by means of
intermolecular adhesive bonds [17].
As concerns the modelling of global tyre system
hysteretic dissipation due to interaction forces with road
[11], an analytical model has already been developed
[18], [19]. In the presented study the analyses will be
focused on the only hysteretic friction component due to
asphalt macro roughness and consequently a single tread
element will be considered. In order to model the
interactions between a tyre tread element and an asphalt
asperity, it has been necessary to focus on the behavior of
an elementary parallelepiped of rubber in sliding contact
with a 3D sinusoidal wave. Tread elementary volume has
been defined as a square-based parallelepiped; its height
is equal to tyre tread thickness and the base side to road
macro-roughness wavelength λMACRO (Fig. 1).
Wavelength λ and roughness index Ra [20]
characterizing soil profile have been estimated by means
of proper algorithms employed to analyze data acquired
experimentally by laser scan on different dry tracks and
to reproduce the best-fitting sinusoidal wave
corresponding to the road profile (Fig. 2).
Fig. 1. Elementary tread volume and coordinates system
Fig. 2. Acquired road profile and plot of a 2D section
The chosen Cartesian reference system, as showed in
Fig. 1, has its origin in the centre of the upper
parallelepiped face; x-axis is in the tread surface plane,
oriented in the sliding direction of the indenter, z-axis is
oriented in the direction of tread width, and y-axis is
oriented in order to obtain a right handed coordinates
system.
Rubber and road are considered as isotropic and
homogeneous materials; moreover, road is modelled as
perfectly rigid.
III. Visco-Elastic Phenomena
Characterization
Rubbery state of a polymer is determined by the so-
called glass transition temperature Tg. If the working
temperature is above Tg the polymer shows a rubbery
behavior, below Tg a glassy one.
Polymers, in particular in the neighborhood of the
thermal transition zone, do not follow reversible stress-
strain behaviour, but the strain lags behind the stress with
a delay of δ/ω, where ω is the pulsation (rad/s).
This visco-elastic effect cannot be described by the
classical perfectly elastic dynamic modulus E: in order to
model this hysteresis it is necessary to introduce a
dynamic storage modulus E' (Fig. 3) and a dynamic loss
modulus E''. An index frequently used to estimate
hysteretic attitude of polymers is the loss angle, defined
as tan(δ) = E''/E' (Fig. 4).
When both the frequency and the temperature vary, it
is possible to make use of the property whereby an
appropriate shift operation is capable of combining the
effect of them: the main element on which the
temperature - frequency equivalence principle is based is
that the values of the complex modulus components at
any reference frequency and temperature (f1, T1) are
identical to the ones observable at any other frequency f2
at a properly shifted value of temperature α(T1):
1 1 2 1
E f ,T E f , T
 
(1)
The most widely used relationship able to describe the
equivalence principle is the Williams-Landel-Ferry
(WLF) transform [21].
For passenger tyre rubbers it can be employed in a
simplified way in order to determine the unknown
equivalent temperature T*= α(T1):
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F. Farroni, R. Russo, F. Timpone
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1025
2 1
1
*
f T T
log f T
 
 
  (2)
in which a common ΔT value, identifiable, once known
rubber thermodynamics characteristics [22], by means of
DMA and DSC tests [23] at different frequencies, is
about 8°C.
The physical meaning of the law is that rubber
stressed at high frequency behaves like if the stress is
applied at lower frequency but at the same time, at a
colder working temperature.
High frequency acts reducing the time between two
consecutive stresses, not allowing the complete relax of
the rubber, in the same way as a low working
temperature would do.
Rubber dynamic modulus is modelled by means of
temperature and frequency DMA characterizations,
whose outputs have been showed in Fig. 3 and 4.
Fig. 3. Temperature/Frequency equivalence principle –
Storage Modulus
Fig. 4. Temperature/Frequency equivalence principle – tan (δ)
Defining stress frequency f of sliding tyre tread over
rough surface as the ratio between sliding velocity Vs and
macro-scale wavelength λ, visco-elastic material model -
and consequently the whole hysteretic friction model - is
able to take into account of the effects that temperature,
sliding velocity and road roughness have on indentation
phenomena.
IV. Hysteretic Power Dissipation Model
The modelling of the hysteresis starts from the
expression of the power dissipated by a rubber block
sliding with speed Vs under a vertical load FZ over a
generically rough surface.
In general, considering the volume VTOT of the
elementary tread element, it is possible to express the
dissipated power Wdiss at time t as [24]:
 
   
TOT
TOT
diss
V
V
W t w x ,y,z,t dV
d
x,y,z,t x,y,z,t dV
dt
 
 
 
 
(3)
in which W (x, y, z, t) represents the dissipated power in
each point of the deformed volume at time t.
Hypothesizing the Vs constant in the sliding over a
single asperity, each stress/strain cycle can be considered
as performed in a period T0 equal to the inverse ratio of
the frequency f at which rubber is stressed if sliding on
macro roughness.
The average value of dissipated power must be thus
evaluated over such this time period:
 
0
00
1T
w x,y,z w x,y,z,t dt
T
(4)
In the same time period, considering tangential force
FT as constant, the global dissipated power can be
expressed as:
0diss T s N s s
W F V F V pA V
 
  (5)
in which p represents the average contact pressure in the
nominal area A0, equal to λ2. In this way, expressing the
stress as σand the strain as ε, the balance between global
and local dissipated powers is:
   
0
0
00
1
TOT
T
s
V
d
pA V x,y,z,t x,y,z,t dt dV
T dt
 
 
 
 
 
  (6)
allowing to formulate the final hysteretic friction
expression:
   
0
00
0
1
TOT
T
V
s
d
x,y,z,t x,y,z,t dt dV
T dt
pA V
 
 
 
 
 
(7)
In order to estimate this friction coefficient, knowing
polymer characteristics, road wavelength and input
variables, it is necessary to provide the stress and the
strain derivative in each point of the discretized
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F. Farroni, R. Russo, F. Timpone
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1026
elementary tread volume, and, in particular, of their
components along x, y and z directions.
Thanks to the studies of Y. A. Kuznetsov [25], [26],
[27], it is possible to calculate the stress state induced in
a rubber elastic body by a periodic sinusoidal perfectly
rigid indenter in sliding contact with it.
Once determined the radius of curvature R of the road
sinusoidal indenter at the apex:
 
2
1
2
λa
R
R
 
 
 
 
 
 
 
(8)
it is possible to estimate by means of Kuznetsov formula,
the half-length N of the contact area (Fig. 5) as a function
of the radius R, of the average contact pressure p in the
nominal area A0, of the rubber dynamic modulus E
calculated taking into account of the working conditions
acting on the examined elementary volume:
 
2
14 1pR
l
N sin l E
 
 
 
 
 
 
(9)
Kuznetsov's method for planar stress calculation can
be used to determine the three-dimensional field starting
from the vertical planes x-z (with sliding velocity) and y-
z (without sliding velocity) showed in Figs. 6 and 7,
localized under asperity apex.
The model does not need the direction of the sliding
velocity on each asperity as an input because the
orientation of the elementary volume adapts
automatically to it, considering x axis parallel to Vs. In
this way the only cinematic input is the modulus of the
sliding velocity and it is taken into account by means of
the frequency effect that it has on rubber for WLF law.
Fig. 5. Detail of the contact between sinusoidal asperity apex
and tyre tread
Fig. 6. σx distribution in the x-z plane under a pressure of 125 kPa
Fig. 7. σy distribution in the y-z plane under a pressure of 125 kPa
The essential relationship between adhesive and
hysteretic friction is taken into account by means of
Kuznetsov's parameter K, that is supposed to be equal to
the adhesive friction coefficient.
Thanks to a wide experimental campaign, performed
by the authors of the present paper with the aim to
investigate tyre/road adhesive component of friction
arising in the sliding contact with micro-rough profiles
[28], [29], it has been possible to identify the optimal
value of K for each kind of analysis.
Applying Kuznetsov equations in the x-z plane, it is
possible to calculate the stress components σx and σz
generated by the sliding indenter in the direction of
sliding velocity. Because of the self-orientation of the
elementary volume, it is possible to state that in the plane
y-z sliding velocity components are absent: it means that
Kuznetsov’s equations must be applied in this plane
imposing K equal to zero, obtaining perfectly symmetric
tangential stress that can be employed in order to provide
an estimation of the stress component σy. With the aim to
extend the bi-dimensional results, reported in Fig. 8, to
the whole three-dimensional elementary volume, the
planar components have been scaled basing on the
information obtained by the stress estimation in the
perpendicular planes. As an example, the extension of
the σx component has been carried out as follows: a
scaling function SF, equal to:
 
 
 
max abs
y
y
y
SF y
(10)
has been defined with the aim to scale stress components
along y direction. Thanks to SF, it is possible to provide
an estimation of σx in the whole elementary volume:
x x
x, y, z x, z SF y
 
(11)
showed in Fig. 9.
Fig. 8. σx in the xz plane (left) and σy in the yz plane (right),
depth = 2mm, temperature = 40°C, pressure = 150kPa
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F. Farroni, R. Russo, F. Timpone
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1027
Fig. 9. σx scaling procedure (left) and final σx extension in
the 3D space (right)
The same results can be applied extending σz along y
direction and σx along x direction. Stress components τxy,
τxz and τyz are usually neglected in power dissipation
analyses because their contribution is about one order of
magnitude lower than the one relative to components σ.
The knowledge of the described stress components
allows to calculate the numerical derivative of the strain
/dt of Eq. (7) extended to the elementary tread volume.
At this aim, the volume has been discretized in 200 nodes
along x and y and 50 along z; this number represents an
optimal trade off between the stability of the results and
the computation performances.
Fig. 10. Employment of a single volume strain for the calculation
of strain derivative over multiple asperities
In the hypothesis of Vs constant for the sliding over
two consecutive identical asperities, rubber deformations,
calculated as σ/E, can be supposed to be equal under each
different sinusoidal indenter.
For this reason, instead of considering ε values at
different times in the numerical derivative of strain:
0 0
t t t
t
 

(12)
it will be enough to localize in the same elementary
volume the value assumed by ε after a time Δt (Fig. 10)
substituting Δx = Vs Δt in the derivative:
0 0
x x x
x
 

(13)
with Δt < 0.1 ms in order to avoid aliasing problems.
V. Results and Validation
The proposed model has been validated by means of
multiple solutions: FEM analyses performed thanks to
the commercial software ANSYS have highlighted a
good correspondence with stress and strain distributions
obtained with the showed methodologies (Fig. 11);
contact area extension, moreover, calculated by means of
the cited FEM software for varying loads, compounds
and road profiles, confirmed the results provided by Eq.
(9) (Fig. 12). An experimental activity is actually
providing comparisons between model simulations
results and friction tests carried out with a pin on disk
tribometer.
Fig. 11. Comparison between proposed estimation and FEM calculation
of σx in xz plane under a pressure of 125 kPa
0 10 20 30 40 50 60
0
0.5
1
1.5 x 10-3
Half lengt h of the c ontact area N [ m]
Force [N]
FEM
Model estimation
Fig. 12. Comparison between proposed estimation
and FEM calculation of static contact area extension
This kind of tester is often employed to measure
friction and sliding wear properties of dry or lubricated
surfaces of a variety of bulk materials and coatings.
The elements of the machine are:
an electric motor, driven by an inverter;
a metal disk, moved by the motor through a belt, that
can be covered with other different materials;
an arm on which a tyre rubber specimen is housed;
a load cell, interposed between the specimen and the
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1028
arm, that allows the tangential force measurement;
an incremental encoder, installed on the disk axis in
order to measure its angular position and velocity;
an optical pyrometer pointed on the disk surface in
proximity of the contact exit edge, that provides an
estimation of the temperature at the interface;
a thermocouple located in the neighborhood of the
specimen, used to measure ambient temperature.
During the test the arm is vertically approached to the
rotating disk surface and through the application of
calibrated weights, the normal force between specimen
and disk can be varied.
Fig. 13. The Pin on Disk tribometer
Fig. 14. Passenger tyre E’ thermal characterization data
(frequency=1Hz) and its detail in the tyre working range
In the experimental campaign, aimed to investigate
hysteretic contribution to friction, the disk has been
covered with 3M anti-slip tape (Ra = 27 µm) and its
surface has been kept wet during the contact by means of
a thin water film. The role played by water is to avoid the
adhesive interaction between rubber and 3M tape,
depurating global friction from this contribution.
Fig. 15. Passenger tyre tan(δ) thermal characterization data
(frequency=1Hz)
Fig. 16. Estimated Hysteretic Friction compared
with experimental points for 3M tape at 40°C
In this way road asperities, able to break the water
film, reach the specimens surface and indent it, isolating
the pure hysteretic contribution.
The specimens employed in the tests have been
extracted from two slabs made up of two different
compounds (C1 and C2) and characterized by means of
DMA procedures, obtaining E' (Fig. 14) and tan (δ) (Fig.
15) curves as a function of temperature. Hysteresis model
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F. Farroni, R. Russo, F. Timpone
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1029
outputs have been compared with the experimental points
highlighting a good correspondance for both compounds
(Fig. 16).
Hysteretic friction trend plotted as a function of the
main variables involved in the phenomenon (Vs, T, p)
showed to be in good accordance with the expected one,
deductible from the experimental results and from data
available in literature [30]; differences between dry and
wet results can be attributed to the substantial reduction
of adhesive component of friction in wet conditions.
In Fig. 17 a map of hysteretic friction at a temperature
of 40°C is reported for compound C1; it is possible to
observe that the shape of the map along pressure axis is
strongly influenced by the indentation mechanisms
linked with road roughness and rubber deformability, and
that the decreasing trend along Vs axis is mainly due to
WLF law and polymer characterization.
Fig. 17. Map of hysteretic friction at a temperature of 40°C
IV. Conclusion
In this paper a physical model able to estimate the
hysteretic component of the friction coefficient arising
between a visco-elastic material in sliding contact with a
rigid rough surface has been presented.
It allows also to assess the actual contact area between
them. The model is based on an energetic balance which
takes into account the material visco-elastic
characteristics and 3D stress- strain distributions inside
it.
The results provided are in good agreement both with
FEM simulations carried on an analogous setup and with
experimental results coming out from a pin on disk
tribometer.
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REPRINT
F. Farroni, R. Russo, F. Timpone
Copyright © 2013 Praise Worthy Prize S.r.l. - All rights reserved International Review on Modelling and Simulations, Vol. 6, N. 3
1030
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[27] Kuznetsov, Y. A., Gorokhovsky, G. A. - "Effect of tangential
load on the stressed state of rubbing rough bodies" - Wear, 73,
1981.
[28] Farroni, F., Russo, M., Russo, R., Timpone, F. - "Tyre - Road
Interaction: Experimental Investigations about the Friction
Coefficient Dependence on Contact Pressure, Road Roughness,
Slide Velocity and Temperature" - Proceedings of the ASME
2012 11th Biennial Conference On Engineering Systems Design
And Analysis, 2012.
[29] Farroni, F., Rocca, E., Russo, R., Savino, S., Timpone, F. -
"Experimental investigations about adhesion component of
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Authors’ information
Department of Industrial Engineering, University of Naples Federico II,
Via Claudio 21, 80125 Naples, Italy.
F. Farroni (Naples, 7th of March 1985)
received his M.Sc. degree in Mechanical
Engineering in 2010 at University of Naples
“Federico II” where is actually Ph.D. bursary
holder in Mechanical System Engineering. He is
technical consultant for vehicle dynamics at
Ferrari S.p.A. and tyre modeler at GES racing
department. His recent work has focused on the
development of interaction models accounting friction and
thermodynamics phenomena and on experimental activities in the field
of contact mechanics for the optimization of dry and wet grip
performances.
Tel: +39 333 3742646
Fax: +39 081 2394165
E-mail: flavio.farroni@unina.it
R Russo Sarno, Salerno (Italy). Graduated in
Mechanical Engineering at the University
Federico II of Naples Italy in 1984. He is
Associate Professor in Mechanical and Thermal
Measurements. He is the author or co-author of
more than seventy publications in both the
applied mechanics and the mechanical
measurements fields. Prof. Russo is a member
of the Italian Mechanical and Thermal Measurements group
F. Timpone was born in Naples (Italy) the 5th
of September 1974. He received the M.Sc.
degree in Mechanical Engineering in 1999 and
the Ph.D. degree in Thermomechanical System
Engineering in 2004 both from the University of
Naples “Federico II”. He is Assistant Professor
at the University of Naples “Federico II” and his
research interests include the dynamics and the
control of mechanical systems.
REPRINT
... In a deeper analysis of tire/road interaction in motorsport sector, the focus moves necessarily to the optimization of the contact conditions in reference to the behavior of materials constituting tire tread and inner layers [2,3]. In particular, the adhesive bonding [4] and the power dissipated in the local indentation of road asperities [5], are highly influenced by the viscoelastic properties of tire polymers; such properties mainly vary with stress frequency [6], local displacement [7], and temperature [6,8]. ...
... Sci. 2020, 10, x FOR PEER REVIEW 2 of 13 local indentation of road asperities [5], are highly influenced by the viscoelastic properties of tire polymers; such properties mainly vary with stress frequency [6], local displacement [7], and temperature [6,8]. ...
Article
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While in the automotive field the relationship between road adherence and tire temperature is mainly investigated with the aim to enhance the vehicle performance in motorsport, the motorcycle sector is highly sensitive to such theme also from less extreme applications. The small extension of the footprint, along with the need to guarantee driver stability and safety in the widest possible range of riding conditions, requires that tires work as most as possible at a temperature able to let the viscoelastic compounds-constituting the tread and the composite materials of the whole carcass structure-provide the highest interaction force with road. Moreover, both for tire manufacturing companies and for single track vehicles designers and racing teams, a deep knowledge of the thermodynamic phenomena involved at the ground level is a key factor for the development of optimal solutions and setup. This paper proposes a physical model based on the application of the Fourier thermodynamic equations to a three-dimensional domain, accounting for all the sources of heating like friction power at the road interface and the cyclic generation of heat because of rolling and to asphalt indentation, and for the cooling effects because of the air forced convection, to road conduction and to turbulences in the inflation chamber. The complex heat exchanges in the system are fully described and modeled, with particular reference to the management of contact patch position, correlated to camber angle and requiring the adoption of an innovative multi-ribbed and multi-layered tire structure. The completely physical approach induces the need of a proper parameterization of the model, whose main stages are described, both from the experimental and identification points of view, with particular reference to non-destructive procedures for thermal parameters definition. One of the most peculiar and challenging features of the model is linked with its topological and analytical structure, allowing to run in real-time, usefully for the application in co-simulation vehicle dynamics platforms, for performance prediction and setup optimization applications.
... In a deeper analysis of tire/road interaction in motorsport sector, the focus moves necessarily to the optimization of the contact conditions in reference to the behaviour of materials constituting tire tread and inner layers [2] [3]. In particular, the adhesive bonding [4] and the power dissipated in the local indentation of road asperities [5], are highly influenced by the viscoelastic properties of tire polymers; such properties mainly vary with stress frequency [6], local displacement [7] and temperature [6] [8]. ...
Conference Paper
While in the automotive field the relationship between road adherence and tire temperature is mainly investigated with the aim to enhance the vehicle performance in motorsport, the motorcycle sector is highly sensitive to such theme also from less extreme applications. The small extension of the footprint, along with the need to guarantee driver stability and safety in the widest possible range of riding conditions, require that tires work as most as possible at a temperature able to let the viscoelastic compounds - constituting the tread and the composite materials of the whole carcass structure - provide the highest interaction force with soil. Moreover, both for tire manufacturing companies and for single track vehicles designers and racing teams, a deep knowledge of the thermodynamic phenomena involved at the ground level is a key factor for the development of optimal solutions and setup. This paper proposes a physical model based on the application of the Fourier thermodynamic equations to a three-dimensional domain, accounting for all the sources of heating like friction power at the road interface and the cyclic generation of heat due to rolling and to asphalt indentation, and for the cooling effects due to air forced convection, to road conduction and to turbulences in the inflation chamber. The complex heat exchanges in the system are fully described and modelled, with particular reference to the management of contact patch position, correlated to camber angle and requiring the adoption of an innovative multi-ribbed and multi-layered tire structure. The completely physical approach induces the need of a proper parameterization of the model, whose main stages are described, both from the experimental and identification points of view, with particular reference to non-destructive procedures for thermal parameters definition.
... Also in this case a "mean function" has been adopted, different from the previous one for a preliminary selection criteria that allows to arrange points inside ortho-slip ranges, providing the results highlighted in figure 6 by different colours. Points selected for G function identification show a friction and generalized stiffness trend ( Fig. 8 and 9) in good agreement with the expected theoretical ones [15] [16]. ...
Chapter
One of the most diffused tire/road interaction models, widely employed in simulation applications, is the Pacejka’s Magic Formula (MF) [1, 2]. It is a semi-empirical model able to fit full scale test data, characterized by a large number of coefficients, often called micro-parameters, grouped basing on physical considerations in order to create specific functions, called macro-parameters. MF model coefficients provided by tire manufacturers are generally not fully representative of the behaviour of tires in contact with road. This is due to the testing conditions employed to identify model coefficients: tests are usually performed on a specific rolling bench or on a flat-trac (Tire testing system, commercialised by MTS Systems Corporation. It applies forces and motions to a tire running on a continuous flat belt.), that keep the tire in contact with a steel or an abrasive paper covered belt. The impossibility to test the tires under real working conditions causes unavoidable approximation errors, mainly due to differences in thermal exchanges and wear phenomena [3] between tire/belt and real tire/road contact. Therefore it is commonly necessary to modify the MF coefficients in order to improve the bench data correlation and to be able to validate vehicle models with data coming from experimental tests. The aim of the developed tool, called TRIP-ID (Tire/Road Interaction Parameters IDentification), is to provide an innovative procedure to identify the Pacejka coefficients basing on the experimental tests carried out measuring global vehicle data during outdoor track sessions. In the presented application, the procedure collects and processes the data provided by TRICK tool [4], allowing to eliminate the outlier points, to discriminate wear and thermal phenomena, taking into account the combined slip condition and the effects of vertical load and camber angle on the global grip. The innovative approach proposed can be useful to reproduce in real time simulation applications the feedback that high performances tires give to sport vehicle drivers, whose interest and skills are focused on keeping them in the optimal thermal range. The coupling of a properly modified MF model with a thermal and with a friction model can provide a reliable simulation and analysis instrument for drivers, carmakers and tire producers.
... The structure has a fixed-base natural period of 0.4 s and a damping ratio of 2%. The base isolation frequency has been obtained by defining the structure mass and taking into account an equivalent horizontal stiffness of the seismic isolator K h,eq =800 kN/m, even if the reaction force of the isolator is nonlinear due hysteresis phenomenon [22,23]. The system is divided into two parts: (1) the physical substructure, consisting of the seismic isolator anchored between the sliding table and the reaction structure (see Fig. 3 for reference), and (2) the numerical substructure, consisting of an elastic SDOF structure. ...
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In this paper, the implementation and validation of hybrid tests for base-isolated structures are presented. Hybrid testing is a technique that combines physical testing and numerical simulation. In this study, two different types of seismic isolators have been considered: fiber reinforced bearings and high damping rubber bearings. The seismic bearings constitute the physical models and the structure is simulated by a numerical model. The experimental results demonstrate the feasibility of the proposed procedure in order to evaluate the performance of the considered seismic bearings.
... Moreover, vehicle dynamics control systems [12], [13], [14], [15], [16], [17] are usually based on the hypothesis that the tyre is always in contact with the road and this not always verified assumption can cause their improper intervention. Also tyre friction [18], [19], [20], [21], [22], [23], [24] and thermal modelling [25] are influenced by the effects of this phenomenon. ...
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This paper deals with tyre enveloping behaviour on uneven road surface. To build realistic and reliable vehicle dynamics models it is of fundamental importance to study the influence of road obstacles and irregularities on forces and displacements involving a rolling tyre. After a brief review about early studies concerning this phenomenon, the tandem model with elliptical cams is introduced and described, highlighting its hypothesis and the parameters on which it is based, the most important of which are the cams profiles and the distance between them. The main aim of this paper is the executive project of a test rig aimed to carry out an experimental campaign for the identification of the parameters of the tandem cam model and for its validation. The idea is to experimentally acquire the path of the patch centre of a tyre rolling over an obstacle, to define the parameters of the curves employed for the cam profiles and the distance between them in the tandem model. It is important to highlight that these parameters are strictly connected to tyre properties and need tests to be investigated and identified. The design started from a test bench for motorcycles already available at DII’s Tyre Lab, introducing proper changes without compromising original test bench destination. © 2016, International Association of Engineers. All rights reserved.
... Tyre dissipates energy according to various mechanisms: rolling resistance, viscosity, hysteresis, friction energy, etc. This dissipation of energy contributes to influence tyre temperature[13], contact conditions[14],[15],[16]and the resulting friction coefficient[17],[18],[19]. In this paper the structural dissipations are taken into account as structural dampings. ...
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A Numerical–physical tyre model was developed . The whole model allows to obtain the road–tyre interactions so it can be used in vehicle dynamic simulations. In this article are presented its capabilities in normal interaction analysis. The normal interaction, i.e. the relationship between the normal load and the normal deflection, influences the tangential (longitudinal plus lateral) one, which determines the vehicle handling behaviour. The parameters used in this model depend on the structure of the tyre and they can be measured on the real tyre. The tyre has been schematized as composed by a flexible belt , the sidewalls and a rigid ring (Rim). The flexible belt is composed by a number of lumped masses linked by extensional and bending stiffnesses and dampers. The tyre model has been developed using the finite segment method. Using these method could be possible to include in the tyre simulations various non-linear structural effects due to large displacements and rotations. The model allows to simulate both steady state and transient conditions.
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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.
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