# Dependence of the contact area on the velocity of a rolling tire.

**ABSTRACT** It is known that the eigenfrequencies of a rolling tire depend on the velocity of rotation. We distinguish two causes for the stiffness increase: the frequency dependence of the complex modulus of the materials and the geometrical stiffness. The real part of the Young's modulus is monotonic according to the frequency. It contributes for an important part to the stiffening. The geometrical stiffness also increases with the rotational velocity. A consequence of these effects is the modification of the size of the contact area for different velocities of a rolling tire. Here we first present experimental results estimating the size of the contact area for a tire in statics and for different rolling velocities. Differences of 20% can be observed. Then the viscoelastic behaviours of the tire materials are presented and experimental results showing the frequency dependence of the complex modulus of the tire constitutive materials are given. Then finite element computations are presented with a real distribution of materials in the tire section and the size of the contact area is estimated and compared to experimental measurements. These results could improve the modelling of tire road interaction for tire noise predictions.

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Dependence of the contact area on the velocity of a

rolling tire

H.H. Nguyena, J. Cesbronb, F. Anfosso-Ledeec, H.P. Yina, S. Erlicherdand D.

Duhamela

aENPC, UR Navier, 6 et 8 Avenue Blaise Pascal, Cit´ e Descartes, Champs sur Marne, 77455

Marne la Vall´ ee, France

bUniversit´ e d’Evry - Val d’Essonne, Laboratoire de M´ ecanique d’Evry, EA3332, 40, rue du

Pelvoux, 91 020 Evry Cedex, France

cLaboratoire Central des Ponts et Chauss´ ees, BP 4129, 44341 Bouguenais Cedex, France

dUniversit´ e Paris-Est, 6 et 8 avenue Blaise Pascal, Cit´ e Descartes - Champs sur Marne, 77455

Marne la Vall´ ee Cedex 2, France

nguyen.hong-hai@lami.enpc.fr

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Abstract :

Tires are made of viscoelastic materials with stiffness quite dependent on the frequency. Generally,

two causes of the stiffness increase are distinguished: a frequency dependence complex modulus and a

geometrical stiffness. In this paper, an experimental and theoretical study on the relaxation and frequency

dependence of complex moduli of the tire constitutive materials are presented and validated. Expressions

of the viscoelastic behavior are presented in time and frequency domains. The results show that the real

part of the Young’s modulus is monotonic according to the frequency. It contributes to an important

part of the stiffening.

A numerical approach simulating the experimental results of the contact area of Cesbron is also

presented. The tire is modeled with a real material distribution in the tire section. The geometrical

stiffness also increases with the rotational velocity and it varies with the vibration frequencies. Static

and dynamic computations for different rolling velocities are done. The results show that the contact

area depends on the velocity of the rolling tire.

computations show a good agrement and a decrease of about 20 % in the contact areas when the tire

rolls compared to a static tire. This difference can be explained by the viscoelastic properties of the

materials.

Comparisons between the measurements and the

Keywords: Complex modulus, rolling tire, contact area, velocity

1Introduction

The understanding of the tire behaviour during the rolling

and of the tire noise production mechanisms are still ma-

jor challenges in the field of car design. A correct mod-

elling of these mechanisms requires a deep knowledge of

parameters affecting the tire-road contact, in particular

the contact area. In detail, it has been proven exper-

imentally [1] that the contact area depends on several

factors, like the tire geometry, the asperities of the road,

the rolling velocity and the material distribution in the

tire section.

In this paper, two factors affecting the tire-road con-

tact area are analyzed: (i) the frequency dependence of

the material constitutive properties and (ii) the rolling

velocity.

The composite material constituting a tire is thought

as a homogenized material, having a visco-elastic be-

haviour. It is supposed here that the corresponding re-

laxation function has the form of a standard Prony’s se-

ries. In the first part of this article, the relaxation tests

are presented and discussed. In particular, the com-

plex modulus is derived from the relaxation function

accounting for the influence of the initial fast loading

phase preceding the true relaxation phase with constant

strain. By this procedure, it is possible to get a correct

expression of the complex modulus, removing the errors

associated to the usual formulas based on the assump-

tion that the constant strain of the relaxation test is

applied instantaneously. The analysis of the relaxation

test data obtained at the laboratory shows that the stor-

age modulus (i.e. the real part of the complex modu-

lus) at zero frequency is sensibly less than at frequencies

of some tenth of Hertz. The dynamic tests performed

with a hammer-sensor excitation-measurement system

confirm this behaviour. This increment of the storage

modulus for increasing frequencies shows that there is a

stiffening effect associated with the material under dy-

namic conditions.

An experimental campaign has been realized at the LCPC

(Laboratoire Central des Ponts et Chausses France), in

order to evaluate the influence of rolling velocity on

the tire-road contact area. For both static and rolling

conditions, the contact area has been measured for a

smooth/non-smooth tire on a smooth/non-smooth road.

In the static load case, the tire supports only the car

weight. In the other case, the contact area is calculated

from pressure measurements [1]. The results show that

the contact area has a tendency to converge toward an

asymptotic value for increasing rolling velocity (up to

50 km/h in the available test data).

To simulate this, the tire-road contact is modeled with

the finite element code ANSYS. The real geometry and

material distribution of the tire are implemented. Only

the case of smooth road is considered. First, a static

analysis is performed with the static material proper-

ties obtained from experimental measures and the corre-

sponding contact area is computed. Then, the rolling ef-

fect is taken into account into the tire finite element sim-

ulation. The constitutive material properties obtained

from the relaxation tests are implemented and the con-

tact area is evaluated also in this case and it proves to

be smaller than in the static case. We can conclude

that there is a good agreement between experimental

and numerical results.

2Complex moduli identification

In principle, the viscoelastic behavior can be identified

by the relaxation tension test. The viscoelasticity is pre-

sented by the characteristic times. These times have an

important role in the dynamical domain. The Fourier

transform shows that the complex modulus at high fre-

quencies is mainly influenced by the smallest times. In

this section, it is assumed that the relaxation modulus

is expressed in a Prony’s series form in which the char-

acteristic times appear in the exponential term. Since

the initial time is not zero, an identification technique

with the non-zero time is presented. The real stress is

analytically expressed via the Prony’s series terms. The

coefficients respective to the exponential form are deter-

minated by Curve Fitting Toolbox in Matlab. Hence,

the relaxation modulus is specified.

The rubber takes a major part in the material distri-

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bution of the tire section. Though, the tread belt layer

supports the most of the static stresses occurring in the

tire. The Fig. 1 presents the constitutive materials in

the tire section and the tread belt macro layer which

is composed by two elementary layers with the parallel

wires and a rubber layer. In particular the mechanical

properties this layer are studied in the next part of this

paper.

Figure 1: Composition of a tire and of the tread belt

macro layer

2.1Relaxation modulus and relaxation

stress

The relaxation stress can be expressed by a Stieltjes

integral [3, 2],:

?t

?t

or by the tensor form:

σij(t) =

0

Rijkl(t − τ)∂εkl(τ)

∂τ

dτ + Rijkl(t)ε(0)

= Rijkl(0)ε(t) +

0

∂Rijkl(τ)

∂τ

εkl(t − τ)dτ

(1)

σ(t) = R(t) ∗Dε

Dτ

(2)

where R(t) is the relaxation modulus tensor of degree 4,

Dε

Dτis the distribution derivation (includes the disconti-

nuity in time). The relaxation function is presented by

the Prony’s series:

R(t) = A0+

N

?

i=1

Aie−t

τi

(3)

In practice, an instantaneous strain can be not carried

out ε(t) = H(t)ε0 at the instant t = 0 (H(t) is the

Heaviside function).Thus, the strain is supposed to

reach the maximal value ε0 in the interval [0,t0] by a

linear variation of ε(t).This quantity is analytically

expressed (Fig. 2):

?

ε(t) = ε0

H(t − t0) +t

t0

(H(t) − H(t − t0))

?

(4)

Figure 2: Real time relaxation strain and stress

The viscoelastic behaviour is established in two cases:

• 0 ≤ t < t0:

Assuming that in the interval [0,t0], the strain varies

linearly, ε(t) = tε0

t0, we obtain:

?t

σ(t) =ε0

t0

0

R(t − τ)dτ =ε0

t0

?t

0

R(τ)dτ

(5)

• t0≤ t:

The derivative of the strain in the interval [t0,t] is can-

celled.

?t

Hence, the relaxation stress is deduced in term of the

characteristic times and the associated magnitudes :

?

i=1

?N

i=1

Knowing that the exponential function is a convex base,

the stress relaxation can be identified only where the

convexity is ensured. In this case, the studied time is

divided in two intervals: before and after the instant

t0. In each interval, the stress is convex but it is not

convex in the whole time. The free coefficient A0is the

elastic modulus and the interval [t0,Tmax] is chosen for

the identification. The stress is depicted as below:

?

i=1

σ(t) =

0

R(t − τ)∂ε(τ)

∂τ

dτ =ε0

t0

?t0

0

R(t − τ)dτ

(6)

σ(t) =ε0

t0

H(t − t0)

?N

?

Aiτi

?

e

t0−t

τi − 1

?

− A0(t − t0)

?

−H(t)

?

Aiτi

?

e

−t

τi − 1

?

− A0t

??

(7)

σ(t) =ε0

t0

A0t0+

N

?

Aiτie−t

τi

?

e

t0

τi − 1

??

(8)

The Curve Fitting Toolbox in Matlab is used to identify

the parameters. The fitting consists in minimizing the

error function, which is the difference between the mea-

sured and estimated values. The number of exponential

terms is chosen as small as possible. The value N is

increased until the best solution is found. Its is noted

that a bigger N does signify a better solution because

the minimization algorithm can conduct to a local min-

imum. Another control is to limit the elastic modulus

A0to reach a good fit at the end of the curve.

The elementary layer is supposed to be composed of the

parallel wires and the rubber. The macro layer of the

tread belt is composed of two elementary layers and a

rubber layer at the middle (Fig. 1). The thickness of

each layer is 0.6mm. This macro layer supports the high-

est circumferential tension stress in the tire. However,

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it takes the role to keep a good contact road-tire and to

protect the tire. This layer is mainly studied.

The result of the modulus identification by the relax-

ation stress is presented in the Fig. 3 and in the Tab. 1

and 2.

Figure 3: Relaxation stress identification for the

moduli in the two main tire directions

A0

A1

A2

A3

Ex

356.5974 34.150659.7337139.8615

Ey

9.92381.24886.0636 3.1896

Table 1: Term coefficients of tests in the two main

directions

t0

t1

t2

t3

Ex

1.2000212.2070 7.3106 0.5625

Ey

1.200046.78114.29290.5652

Table 2: Initial and characteristic times of tests in the

two main directions

2.2 Complex modulus

The complex modulus depends on the frequency be-

cause of the viscoelasticity. In the same way, the re-

laxation modulus of the elementary layer is measured.

All the moduli tested can be expressed analytically in

the Prony’s series form. With a homogenization pro-

cess for a multi-layer composite, the apparent modulus

of the macro layer can be calculated from the data of

the elementary layers.

In the other way, the Fourier transform from the time

domain into the frequency domain gives us the analyt-

ical expression of the macro layer in term of the fre-

quency. Finally, the direct measurement of the complex

modulus by the beam vibration formula is used. The

FRF (Frequency Response Function) is measured by the

Bruel & Kjaer sensors. These signals are treated by the

software PulseLabShop. Analytically, the eigenfrequen-

cies are expressed as [4]:

fn=(βnl)2

2πl2

?

EI

ρA

(9)

where βnl is the wave number on the beam length. In

the clamped-free case, it satisfies the equation:

cos(βnl)cosh(βnl) + 1=0(10)

The measured eigenfrequencies give the real part of the

complex modulus. The damping ratio detected by Pulse-

LabShop gives the imaginary one. The calculated and

measured apparent moduli are showed in the Figs. 4

and 5.

Figure 4: Circumferential apparent modulus

comparison of the macro layer exploited from the

relaxation data, calculated by the homogenization

process and in several beam lengths

Figure 5: Transverse apparent modulus comparison of

the macro layer exploited from the relaxation data and

calculated by the homogenization process

3Tire-road contact area test

The contact test has been realized by Cesbron. This

author did an experimental and numerical study on the

multi-asperity contact. However, in this paper, only the

contact test results of the smooth tire with a plane foun-

dation is used to compare with the numerical one.

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3.1Contact measurement method

The area and the contact pressure can be measured with

the help of the sensors disposed at the interface of two

solids. There are two sensors categories: the film which

is sensible to the contact pressure post-processed by an

image treatment and the real time pressure acquisition

cells. With the rolling tire case, the pressure is measured

by the Tekscan 3150 cells . The description of this equip-

ment is shown in the Fig. 6. The measurement is real-

ized on a real vehicle (Renault Scenic, 2.0L/16V). This

vehicle is equipped of two standard front tires and two

smooth back tires. These two smooth tires are competi-

tion tires. The contact pressure on the right back tire is

measured. The other tires are placed at the same level

to ensure the identical forces on four tires. The vehicle

body is elevated on the contact level and positioned with

precision to apply the tire load on the contact plate.

Figure 6: Experimental disposition of the contact test:

1.Test vehicle 2.Lifting bridge 3.Studied contact zone

4.Smooth tire 5.Tekscan sensor 6.Reference plate

7.Tekscan box [1]

According to the constructor, the stiffness of the used

competition tire is of the same order as the standard

ones. Finally, the pumping pressure is 0.22 MPa (2.2

bars) and this value is controlled before each test.

3.2Test result

At a moment, each cell gives a contact pressure in term

of time. The real time pressure reconstruction of all the

cells gives us the contact pressure distribution of the

studied zone. The contact area is measured in several

cases of road asperity (Fig. 7). It is calculated by the

formula:

A = V

t1

where V is the vehicle velocity and L(t) is the contact

length measured at each instant, t1,t2are the beginning

and final contact times.

The result is presented in the Fig. 8 and the Tab. 3.

?t2

L(t)dt

(11)

4Tire modeling and comparison

with the experiments

4.1Geometry and load applied

In this paper, the tire modeling is realized with the

smooth tire. The section geometry is measured from a

Figure 7: Tested road surface [1]

Figure 8: Contact area in the static case and for the 3

different velocities considered [1]

real section. A revolution process generates the whole of

the model. The wheel is modeled as the fixed boundary

condition. The road is supposed smooth and modeled

by a plate. The contact is therefore between two smooth

surfaces. The model is presented in the Fig. 9.

Figure 9: Modeling of the road-tire contact in ANSYS

The applied force on the tire is replaced by the pressure

in the plate. The tested force is F = 3300N giving a

pressure

p =F

a2=3300

0.42= 20625N/m2

(12)

where a is the length size of the square plate. The tire is

fixed on the wheel. To stabilize the model, we fix some

nodes of the plate in the horizontal displacement. The

inner pressure is taken equal to 2.2 bars.

The finite element software ANSYS is used to solve this

problem. The rolling velocity is introduced by the ro-

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Road surfaceStatic30km/h40km/h50km/h

A’156123123122

C 155130129 123

E1 164128 124125

E2151133137131

M2151 134131136

L1 148121126122

L2160122122121

Table 3: Evolution of contact area in term of the

velocity [1]

tating angular velocity Ω =

the static moduli into the model to calculate the contact

area in the static case. The contact area includes the

zones where the pressure contact or the stress intensity

is considered as large enough.

V

Rtire. First, we introduce

4.2Result and comparison

The result is saved in graphic form (Fig.

contact area is determinated by the connected elements

on the two contact sides (Fig. 12). The results show a

good agrement with the measurement.

10).The

Figure 10: Vertical static stress in the road-tire contact

problem

Figure 11: Stress intensity and calculated contact area

Figure 12: Road-tire contact areas calculated by

ANSYS

5Conclusion

An identification technique of the relaxation stress is

presented in this paper. The analytical expressions of

the relaxation modulus are deduced. The Fourier trans-

form gives the dependence of complex moduli with the

frequency. An homogenization process is done to cal-

culate the apparent moduli of the macro layer. The vi-

bration test gives the equivalent dynamical moduli and

confirms the stiffening of the materials when the fre-

quency is increased.

A road-tire contact test is realized in the frame of the

Cesbron’s thesis [1]. The test shows that when the ve-

locity varies, the contact area varies too. However, there

is a coupling between the velocity dependence and the

dependence with the frequency of the complex modulus

and it is the cause of the variation of the contact area.

When the vehicle moves,the tire rotates and the mate-

rials work at different frequencies. As the rigidity of the

materials is quite different in rotation and in dynamics,

this explains the reduction in contact area as the tire

rolls.

References

[1] J. Cesbron, ”Influence de la texture de chausse sur

le bruit de contact pneumatique/chausses”, PhD

Thesis-Ecole Central de Nantes, (2007)

[2] J. Lemaitre and J.-L. Chaboche, ”Mcanique des ma-

triaux solides”, Dunod, (2004)

[3] Y. Chevalier, ”Comportements lastique et visco-

lastique des composites”, Techniques de l’Ingnieur,

trait Plastiques et Composites - Vol ARCH1, (1988)

[4] J. Courbon, ”Vibrations des poutres”, Techniques de

l’Ingnieur - trait Construction, (1984)

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