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# ARTICLE IN PRESS

JOURNAL OF
SOUND AND
VIBRATION
Journal of Sound and Vibration 290 (2006) 101–132
www.elsevier.com/locate/jsvi

## A wave model of a circular tyre. Part 1: belt modelling

R.J. Pinnington
Institute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ, UK
Received 7 January 2004; received in revised form 11 February 2005; accepted 16 March 2005
Available online 14 July 2005

Abstract

The equations of motion of a curved tyre belt are derived for one-dimensional waves propagating around
the belt and a standing wave across the belt. The effects of curvature, shear stiffness, rotary inertia, tension,
rotational speed and air pressure are included. These are combined to give a sixth-order wave equation, the
solution of which gives three pairs of wavenumber as a function of frequency. The application of the
boundary conditions at the contact leads to the input and transfer mobilities for both in-plane and out of
plane excitation. Observed are: low-frequency rigid-body modes, belt bending modes and in-plane ring
modes. At high frequencies only travelling waves occur.
r 2005 Elsevier Ltd. All rights reserved.

1. Introduction

When a rotating tyre interacts with the road surface, as seen in Fig. 1, the time varying
deformations are transmitted causing noise interior and exterior to the vehicle. To calculate this
interaction with the road and also to determine the resulting vibration of the tyre surface it is ﬁrst
necessary to make a dynamic model of the tyre. Various tyre models have been developed for use
in different frequency regimes. Finite element models  and the lumped parameter ‘ring model’
 are used for low-frequency vehicle handling and ride comfort, generally below 50 Hz. Vehicle
interior noise is greatest between 50 and 500 Hz and the ﬂexural modes of the belt that contribute
have been modelled using both ﬁnite elements  and modal solutions to a wave equation [3–5].

Corresponding author. Platia Skra 13, Kalamaria 55132, Thessaloniki, Greece. Tel./fax: +30 2310 480 645.
E-mail address: rjpinnington@compuserve.com.

0022-460X/\$ - see front matter r 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jsv.2005.03.023
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Ω radians/sec

Force
tyre speed c
radiated
sound wr (x)
road
profile
x

## Fig. 1. Rotating tyre on a road proﬁle.

The main exterior noise occurs between 500 and 3000 Hz, a region where there is little modal
behaviour of the belt ﬂexural waves. A ﬂat orthotropic plate , and a deep ﬂat two-layered plate
, have been used with modal and travelling wave solutions. Also an inﬁnite ﬂat belt model that
included tension, transverse shear, rotary inertia and bending was made for this region [6,7]. The
dynamic behaviour was described by a fourth-order wave equation and input and transfer
mobilities were obtained for transverse excitation. Below about 1 kHz, only tension and bending
control the single propagating transverse wave; above this frequency a transverse shear wave and
an in-plane rotational wave propagate.
The objective here is to extend this wave model to include: curvature, side-walls, asymmetric
belt and tyre rotation to the other parameters, and thus make a complete circular belt model. A
circular beam in Refs. [8,9], without the complications of side-wall, tension and cross modes has
been investigated previously, for an energy ﬂow application.
The main effect of curvature is to couple radial and circumferential motion thereby including
the effects of both normal and tangential forces in all vibration responses of the tyre. A single
model will therefore be made to embrace the whole frequency range of interest, from zero to
3 kHz, for both normal and tangential excitation. One-dimensional waves are allowed to
propagate around the belt with a standing wave pattern in the transverse direction. The side-wall
is modelled here from the static, pressure dependent, stiffness values taken from Ref. , a
frequency-dependent side-wall is considered in the companion paper . The waves in the air
cavity are also neglected here, as they are only noticed at the ﬁrst cavity resonance. They are,
however, also included in Ref. , but not with full structural–acoustic coupling.
Tyre rotation has three effects that are included here. Firstly, waves in the direction of rotation
have a corresponding speed increase, while waves in the opposite direction are slowed by the same
amount. Secondly, the increase in belt tension to resist centrifugal forces. Finally, radial velocity
couples with the rotation to cause Coriolis forces.
The equations of motion are combined to give a sixth-order wave equation for the
circumferential waves of a particular transverse mode order m. There are three pairs of roots
corresponding to the three waves that exist simultaneously at each frequency. The wavenumbers
of these waves are plotted against frequency in the ‘dispersion curves’. The ratio between radial
and circumferential motion for each of these waves is also generated.
The boundary conditions at what would be the contact with the road are applied, giving the
input and transfer mobilities for both radial and circumferential excitation. The transfer functions
show: low-frequency rigid-body modes, mid-frequency tension/bending modes, and the in-plane
mode group, beginning with the ring frequency around 500 Hz.
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For line excitation only the waves of the m ¼ 0 transverse order are included to give the transfer
function. For point excitation the transfer functions from all the transverse mode orders are
summed.
The results from the belt model developed here are compared with measurements in the
companion paper .

## 2. Basic equations of motion

The overall scheme is shown in Fig. 1, where a tyre rotates in a positive or anti-clockwise sense
at velocity V and angular velocity O. In the ﬁrst instance it is assumed all the excitation arises
from a transverse line of contact. To calculate the response to these forces the equations of motion
for the tyre must be found.
Fig. 2a shows a segment of length dc and width dz, in a tyre belt of radius a and width b,
displaying the sign convention for positive directions, rotations, forces and moments. It is
assumed that the belt rotates about the offset wire reinforced neutral axis seen in Fig. 2b. It is also
assumed that the averaged material properties of the cross-section are known, as reference will
only be made to these single material values for the cross-section. Using the wave approach it is
simple to make these material properties dependent on frequency, if required; but it is not
necessary to state this dependency until the programming stage.

δz
z Nz+∂Nz /∂z.δz
Mz+∂∂Mz /∂z.δz

N+∂N /∂c.δc
δθ Mc+∂Mc /∂c.δc Qz+∂Qz /∂z.δz
θ Qz

Mz Qc+∂Qc /∂c.δc

Nz

Qc
Mc

N
z
Φc
u, c

(a) w

Φc

neutral u, c
axis ∆

(b) w

Fig. 2. (a) Sign convention for belt element and (b) asymmetric belt element.
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The tyre could be described as a curved, tensioned, Mindlin plate on distributed two-directional
stiffness. The belt is subjected to a net static pressure P which causes a static tension/length N c , N z
in the circumferential and transverse directions c; z. Qc , Qz are the shear forces/length. N is the
total static and dynamic circumferential force/width. M c , M z are the bending moments/length.
The accompanying displacements are u; w, in the circumferential and radial directions. y describes
the geometric position and is related to the circumferential coordinate c by ay. The tyre rotates
in the positive y direction with velocity V. The belt is restrained either side by a side-wall of
stiffness/belt length of K c , K r , in the circumferential and radial directions.
In this section three groups of equations are presented; kinematic relationships, equilibrium of
forces and moments, and the linking Hooke’s Law relationships.

## 2.1. Kinematic relationships

The segment motion can be described by three variables; displacements u; w and the kinematic
rotation F. For a Mindlin plate that can deform in both bending and shear, the slope at any
position s in the s and z directions are, respectively,
qw
¼ bc þ gc , (1a)
qc
qw
¼ bz þ gz , (1b)
qz
where bc , bz and gc ; gz are the slopes due to bending and shear, respectively. The kinematic
rotation, in the c direction, of the element Fc , seen in Figs. 2a and b, including that due to the
circumferential displacement u, is therefore
u qw
Fc ¼  . (2a)
a qc
There is no geometric curvature in the z direction and so the rotation is simply
qw
Fz ¼  . (2b)
qz
The total angle j in the c direction is the sum of the geometric rotation y and the kinematic
rotation Fc . The small change in slope over the length dc is therefore
 
1 qFc
dj ¼ þ dc. (3)
a qc
The circumferential strain ec also has contributions from both displacements
qu w
þ .
ec ¼ (4a)
qc a
The transverse strain ez only has a contribution from the transverse displacement uz
quz
ez ¼, (4b)
qz
however, the transverse strain is ignored here and accordingly set to zero.
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## 2.2. Force equilibrium equations

There are four equations of equilibrium: forces in the radial direction, forces in the
circumferential direction (c), and for moments in the s and transverse (z) directions. The
equilibrium of radial forces are taken in line with the circumferential shear force/width Qc .
Substitution of the segment rotation in Eq. (2a) yields the equation for equilibrium. For
convenience, it is separated here into the equations for static and dynamic equilibrium. Eq. (5a)
for static equilibrium is also calculated in Appendix A:

2N z sin f1 N c
P þ mO2 a   ¼ 0,
b a

   
qQc qQz 1 qFc qFc q2 w P 2
pðc; zÞ þ þ  N cd þ  Nc þ N z 2 ¼ mw€  þ mO w. (5a,b)
qc qz a qc qc qz a

P is the net static air pressure. pðc; zÞ is the dynamic pressure due to the side-wall and external
radial force, it is calculated in Section 2.24. N z is the static transverse tensile force/length; N c and
N cd are, respectively, the static and dynamic circumferential tensile forces/width; Qz is the shear
force/width in the radial direction. m is the belt mass/area.
By resolving forces in the c direction in line with the circumferential force/width N, in Fig. 2a,
and substitution of the segment rotation in Eq. (2a), gives
 
tðc; zÞ þ
qN qQzc
þ þ Qc þ
1 qFc
¼ mðu€ þ DF € c Þ þ 2mw_ ðc þ u_ Þ þ 2K c u, (6)
qc qz a qc a b

where K c is the circumferential stiffness/belt length of a single side-wall. The tangential external
stress is t. Qzc is the shear force/length in the circumferential direction. The wire reinforced neutral
axis displaces u, however, the asymmetry of the belt about the neutral axis gives an inertial
centroid which is displaced D, seen in Fig. 2b. The inertial centroid therefore accelerates by an
additional rotation included in Eq. (6). The second term from the right is the Coriolis force, which
gives gyroscopic coupling between the circumferential and radial motion. The nonlinear product
of the radial and axial velocities is ignored in the later analysis.
The net moment, taken about the right-hand end z-axis of the segment is responsible only for
the angular acceleration due to bending b€ c (as the shear component of slope is a distortion that
does not involve overall rotation). The external moment/width M 0 that arises from the offset h
from the neutral axis of the external shear force t is also included
qM 0
¼ th. (7a)
qc
Moment equilibrium about the z-axis yields

qM c qM 0 ðs; zÞ
Qc  ¼ I c b€ c þ , (7b)
qc qc
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where I c is the belt moment of inertia/width. Similarly, the net moment about the right-hand side
c-axis yields
qM z
Qz  ¼ I z b€ z , (7c)
qz
where I z is the belt moment of inertia/length about the c-axis.

## 2.3. Hooke’s Law relationships

The three types of forces in Section 2.2 are assumed to be linearly related to the three
deformation types given in Section 2.1. The constants of proportionality are the various elastic
moduli of the belt section.

## 2.3.1. Circumferential stress–strain

The circumferential force/width N has a static component N c and a dynamic component N cd ¼
Ac ec arising from the circumferential strain
N ¼ N c þ N cd , (8a)
where Ac is the belt circumferential stiffness/width. The tension N c , calculated in Appendix A,
Fig. 15, has two components, one determined from the pressure and tyre geometry, the other from
resisting the datum centrifugal force mO2 a
 
l s sin f1
N c ¼ Pa 1  þ mO2 a2 ,
b fs
where l s is the side-wall proﬁle length and 2fs is the angle subtended by the side-wall proﬁle. The
side-wall attaches to the belt at an angle f1 . The static force/length in the transverse direction is
also given in Appendix A
1 Pl s
Nz ¼ . (8b)
2 fs

## 2.3.2. Shear stress–strain relationships

In the circumferential direction the shear force/width Qc is related to the shear strain gc by the
belt shear stiffness/width Sc :
Qc ¼ Sc gc . (9a)

Likewise in the transverse direction the shear force/width Qz is related to the shear strain gz , by the
belt shear stiffness/length S z :
Qz ¼ Sz gz . (9b)

The shear force in the circumferential direction normal to the z-axis Qzc is controlled by the shear
stiffness S zc :
qu
Qzc ¼ Szc . (9c)
qz
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## 2.3.3. Bending moment–curvature

For a slender pre-curved beam of initial radius a the bending moment M c is related to the total
radius of curvature r and the bending stiffness/width Bc :
 
1 1
M c ¼ Bc  . (10)
r a
Ref.  gives the radius of curvature r as a function of the strain and kinematic rotation as
 
1 1 qFc
¼ þ ð1  ec Þ. (11)
r a qc
Therefore combining Eqs. (10) and (11) with a large initial curvature assumption, gives the
bending moment-to-curvature relationship as
 
qFc ec
M c ¼ Bc  . (12)
qc a
Making substitutions from Eqs. (1a), (2a) and (4a) the bending moment in the circumferential
direction can now be written as
 
qbc w
M c ¼ Bc þ . (13a)
qc a2
The ﬁrst term in the brackets is the curvature for a ﬂat plate. The second term is the moment due
to stretching of the neutral axis, becoming zero for a ﬂat plate as the radius a tends to inﬁnity.
This is a simpler and slightly different expression for this relationship than is given in Ref. .
Likewise substitutions from Eqs. (1b), (2b) and (4b) give the bending moment in the transverse
direction as
 
qbz
M z ¼ Bz . (13b)
qz
This corresponds to the plane bending expression for a ﬂat plate, where Bz is the bending stiffness
in the transverse direction.

## 2.4. The ratio of shear rotation to bending rotation

For the circumferential direction, Eq. (1a) shows that the slope or rotation has two components
gc , bc , from shear and bending. The relative sizes of the bending slope to the total slope can now
be found by eliminating unwanted variables from Eqs. (1a), (7b) and (13a) using a harmonic time
dependency eiot , where o and t are frequency in radians/second, and time.
 
qw Bc q2 bc
þ S c ¼ B c þ ðSc  o2 I c Þbc . (14a)
qc a2 qc2
The corresponding version for the shear slope in the circumferential direction is found by
substitution of Eq. (1a). Likewise in the transverse direction, the ratio of total slope to bending
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## slope is given from Eqs. (1b), (3b), and (13b)

qw q2 b
ðS z Þ ¼ Bz 2z þ ðS z  o2 I z Þbz . (14b)
qz qz

## 2.5. Equivalent modal stiffness

Eq. (5b) for radial equilibrium contains two terms with differentials in z and also a dynamic
pressure term pðc; zÞ, associated with the side-wall stiffness. These three terms describing the belt
transverse properties are all expressed as dynamic stiffness for the mth transverse mode of the belt.
This procedure reduces Eq. (5) and the subsequent analysis to one-dimensional form, with only a
c variation for each belt transverse mode order m.

## 2.5.1. Modal stiffness from the shear force and tension

If a harmonic wave solution of the form bz ¼ bz expðikz zÞ is applied to Eq. (14b) the bending
rotation can be written as
!
Sz qw
bz ¼ 2
. (15)
Bz kz þ Sz  o I z qz
2

Substitution of this equation into Eqs. (1b) and (9b) gives the shear force in the transverse
direction in terms of w:
Bz k2z  o2 I z qw
Qz ¼ S z . (16)
Bz k2z þ Sz  o2 I z qz
This shear force term and the tension term N z in Eq. (5) can now be written as a modal stiffness if
some mode shapes can be selected to describe the radial motion in the z direction. To simplify this
procedure it is assumed here that the belt construction is much heavier than that of the side-wall,
which provides some justiﬁcation for choosing boundary conditions at z ¼ b=2 that are;
unconstrained in the radial direction but with zero slope. Also as the purpose here is not to
accurately predict resonance frequencies but rather to demonstrate the physical behaviour, the
simplest form, namely sine and cosine mode shapes are selected to satisfy these boundary
conditions. These occur for a discrete set of wavenumbers kz ¼ mp=b, m ¼ 0; 1; 2; 3 . . ., i.e.
X mpz X mpz
wðc; zÞ ¼ wm ðcÞ cos þ wm ðcÞ sin . (17)
m¼0;2;4...
b m¼1;3;5...
b
Using Eqs. (16) and (17) the two z-dependant terms in Eq. (5) can now be replaced by a modal
stiffness K mz , i.e.
qQz q2 w
þ N z 2 ¼ K mz w, (18)
qz qz
where
!
mp2 Bz ðmp=bÞ2  o2 I z
K mz ¼ N z þ Sz .
b Bz ðmp=bÞ2 þ S z  o2 I z
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R.J. Pinnington / Journal of Sound and Vibration 290 (2006) 101–132 109

For the beam modes of the belt, denoted m ¼ 0, there is no curvature in the z direction and so this
modal stiffness is zero. For higher-order modes the stiffness increases with the square of the
transverse belt mode order m, which determines the mode cut-on frequency. Above the cut-on
frequency the mode forms in Eq. (17) propagate in the direction c.

## 2.5.2. The modal stiffness from the side-wall

The pressure on the belt pðc; zÞ has two components, the external load p0 and the pressure ps
from the side-wall displacement w at z ¼ b=2. These give rise to a pressure distribution on the
belt
pðc; zÞ ¼ K r ½dðz  b=2Þ þ dðz þ b=2Þwðc; zÞ þ p0 ðc; zÞ (19)
which may be expressed as a sum of modal pressures Pm :
X mpz X mpz
pðc; zÞ ¼ Pm ðcÞ cos þ Pm ðcÞ sin . (20)
m¼0;2;4:::
b m¼1;3;5:::
b

Substitution of Eq. (17) into Eq. (19), and application of the orthogonality relations to Eqs. (19)
and (20) allows the modal pressure terms to be written using the external modal pressure P0m and
a modal stiffness K rm
Pm ¼ K rm wm þ P0m , (21)
where for m ¼ 0, K rm ¼ 2K r =b; for m40 K rm ¼ 4K r =b.
All the terms in z found in Eq. (5) can now be replaced with the modal stiffness terms of
Eqs. (18) and (21). The analysis now proceeds considering only wave motion in the c direction.

## 3. Equations of equilibrium in kinematic parameters

The three groups of equations of the previous sections can be combined to give the two
equations of equilibrium of force only in terms of the displacements u; w.

## 3.1. Equation of radial equilibrium

The equation of radial equilibrium is found with three preliminary steps. These are to ﬁnd the
shear gradient, bending gradient, and the associated shear force. Eq. (5) of radial equilibrium is
ﬁrst expanded using Eqs. (1a), (2a), (4a), (7b), (8a), (9a), (12), (18) and (21), giving the gradients of
the shear stress or shear strain.
The radial displacement for all subsequent analysis becomes wm as each equation refers only to
the displacement for the mth transverse mode, as expressed in Eq. (17), the other kinematic and
force variables likewise have the modal subscript m, i.e. um ; gm ; bm :
qgm qum q2 wm wm
Sc ¼ P0m þ ðN c þ Ac Þ  N c 2
þ ðAc þ ðK zm þ K rm Þa2  Pa  ðo2 þ O2 Þma2 Þ 2 .
qc aqc qc a
(22)
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The gradient of the slope due to bending is the derivative of Eq. (1a)
qbm q2 wm qgm
¼  . (23)
qc qc2 qc
The shear force/width can be written, in the ﬁrst instance, by expanding Eq. (7b) with Eq. (13a)
 2 
q bm qwm qM 0m
Qm ¼ Bc þ  o2 I c b m þ . (24)
qc2 a2 qc qc

By substituting Eq. (22) into Eq. (24), the shear force becomes
8 3
9
>
> 2 q wm qwm 2 >
>a ðSc þ N c Þ  ðAc  Sc  Pa þ ðK zm þ K rm Þa  ðo þ O Þma Þ >
2 2 2
>
Bc < qc3 qc =
Qm ¼  2
a Sc >> q2 um qP0m >
>
>
: a ðN c þ Ac Þ  a2 >
;
qc2 qc
qM 0m
 o2 I c bm þ . ð25Þ
qc
This expression for shear force can now be differentiated with respect to c, then combined with
Eq. (23) to eliminate qbm =qc. The resulting expression for qgm =qc is then substituted into Eq. (22),
giving the equation for vertical equilibrium only in terms of um and wm . It is presented here in a
non-dimensional form which will be employed later in the wave equation
q 4 wm q2 wm wm
0 ¼  a3 ðS̄ c þ N̄ c Þ þ a fC 1  S̄c ð1 þ z2c Þ  N̄ c C 2 g þ C1C2
qc4 qc2 a
 3 
2 q um qum
þ ð1 þ N̄ c Þ a þ C2 þ C2C3 þ f m. ð26Þ
qc3 qc

In Eq. (26) some parameters have been grouped together in brackets ( ) as they share some
physical signiﬁcance
C 1 ¼ 1 þ K̄ m  P̄a  z2Lc  Zce ,

C 2 ¼ z2c  R̄c ,

C 3 ¼ P̄0m ,

q2 P̄0m qt̄
f m ¼ a2 2
þ hR̄s , (27a2c)
qc qc
where
Sc Nc Pa Sc K rm þ K zm mO2
S̄ c ¼ ; N̄ c ¼ ; P̄a ¼ ; R̄c ¼ a2 ; K̄ m ¼ a2 ; Z ce ¼ a2 .
Ac Ac Ac Bc Ac Ac
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The external loading term f m uses Eq. (7a) to write the moment M 0m in terms of shear stress t.
The normalised external pressures and shear stresses are deﬁned as
P0m tm
P̄0m ¼ a ; t̄m ¼ a .
Ac Ac
The normalised non-dimensional circumferential wavenumber zLc is deﬁned by
m
z2Lc ¼ ðaoÞ2 .
Ac
There is a wave that cuts on at high frequencies that involves only belt rotation and no
translation, here it is called the ‘rotational wave’ , it could also be called the ﬁrst asymmetric
Lamb wave . The wavenumber zc is deﬁned by
Ic
z2c ¼ ðaoÞ2 .
Bc
When C 1 ¼ 0 the ring frequency occurs for the m ¼ 0 waves, it increases with the side-wall radial
stiffness K r , but decreases with inﬂation pressure P and rotation speed. An instability occurs when
the tyre speed drives the ring frequency to zero as the radial stiffness becomes zero.
When C 2 ¼ 0 the rotational wave cuts on. This cut-on frequency increases with the section
shear stiffness (proportional to belt depth), but decreases with the section moment of inertia
(proportional to the cube of belt depth). The cut-on frequency is therefore inversely proportional
to belt thickness.
For an inﬁnite cylinder with no side-wall, static equilibrium gives C 3 ¼ 0. For a ﬁnite width tyre
it can be assumed that this term is negative, as the side-wall will take some of the tension.
The external load f m has two components, the ﬁrst is the radial load and the second is a
moment from the tangential force.
For a ﬂat belt with zero: tension, shear stiffness, side-wall stiffness, Eq. (26) reverts to the Euler
beam bending equation.

## 3.2. The equation of circumferential equilibrium

Eq. (6) for circumferential equilibrium can be expanded with Eqs. (1a), (8a) and (9a) to yield
 
q2 um qwm 2 qFm um wm
t̄m þ a 2
þ ð1  zLc D̄Þ þ S̄c gm 1 þ a ¼ ðK̄ cm þ K̄ zcm  z2Lc Þ þ Z co , (28)
qc qc qc a a
where the normalised radial side-wall stiffness derived in Eq. (21) is kz ¼ mp=b. The normalised
cross mode shear stiffness is found from Eqs. (6) and (9c) by setting
Kc Kc mp2 a2
K̄ cm ¼ 2a2 for m ¼ 0; K̄ cm ¼ 4a2 for m40; K̄ zcm ¼ S zc .
Ac b Ac b b Ac
The Coriolis term and centroid offset are
2iomc D
Zco ¼ ; D̄ ¼ .
Ac a
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This equation will now be simpliﬁed using the following justiﬁcation. The non-dimensional
shear stiffness S̄ c , deﬁned in Eq. (26), is the ratio of the average section shear stiffness to
circumferential stiffness. For a tyre belt the shear stiffness is dominated by the rubber, and the
circumferential stiffness by the embedded steel wires; the shear stiffness ratio is likely to be of the
order of 0.01, allowing this ﬁnal term on the left hand to be ignored in the proceeding analysis.
For a homogeneous material it is also probably safe to ignore this term for three reasons. First
the second term in the bracket forms a nonlinear product gm qfm =qc, which is also very small. The
greatest value of the shear ratio for a homogeneous material e.g. for steel, S̄ c  0:3, and as from
Eq. (1) gpqw=qc, the greatest possible error in the shear wavenumber from the omission is 30%.
The equation for circumferential equilibrium can therefore be written in the reduced form below
q2 um qwm um wm
t̄m þ a þ ð1  z2Lc D̄Þ ¼ ðK̄ cm þ K̄ zcm  z2Lc Þ þ Z co . (29)
qc2 qc a a
For a ﬂat beam a ! 1 and Eq. (29) becomes that for longitudinal motion of a rod. The cross-
coupling to radial motion w is purely geometric, and increases with decreasing radius a. The term
in brackets, C
C 4 ¼ K̄ cm þ K̄ zcm  z2Lc (30)
becomes zero at the rotational rigid-body resonance
pﬃﬃﬃ of the belt (n ¼ 0). At this frequency the belt
rotates on the side-wall shear stiffness. At 1= 2 of this frequency the n ¼ 1 translation rigid-body
resonance occurs. Below these frequencies there is mainly rigid-body motion of the belt.

## 3.3. The wave equation for a circular tyre belt

The wave equation for a free belt is given by combining Eqs. (26) and (29), but setting the load
terms f m and tm to zero. A harmonic solution for a wave travelling in the positive c direction, of
the form eikc , is applied. Then by substituting for wm using Eq. (29), the sixth-order wave
equation is obtained in the normalized wavenumber zm ¼ km a
0 ¼ z6m ðS̄ c þ N̄ c Þ
 z4m ½z2Lc ð1  D̄ð1 þ N̄ c ÞÞ þ P̄a þ Zce  K̄ m  N̄ c R̄c þ ðS̄ c þ N̄ c Þð1 þ z2c þ z2Lc  K̄ cm  K̄ zcm Þ
" #
ðK̄ cm  K̄ zcm  z2Lc Þð1 þ K̄ m  P̄a  S̄ c  z2Lc þ Zce þ N̄ c R̄c  z2c ðS̄ c þ N̄ c ÞÞ
2
þ zm
þ ðz2c  R̄c ÞðN̄ c  K̄ m þ P̄a þ Z ce þ z2Lc ð1  D̄ð1 þ N̄ c ÞÞÞ
þ ð1 þ K̄ m  P̄a  Z ce  z2Lc Þðz2c  R̄c Þðz2Lc  K̄ cm  K̄ zcm Þ
 ið1 þ N̄ c Þðz3m  ðz2c  R̄c Þzm ÞZco . ð31Þ
The components within the brackets are functions of frequency or constants. The material
properties can be complex to include hysteretic damping; they can also be written as functions of
frequency (as is the case for polymers).
The ﬁnal term with the odd wavenumber orders is the Coriolis coupling, the  is for the anti-
clockwise and clockwise waves, respectively. This term may be signiﬁcant in the contact patch but
not for a circular tyre, and so is ignored in this initial study.
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The equation in z2m is solved at each frequency to give three pairs of roots p ¼ 1; 2; 3 for each
transverse mode group m. The normalised wavenumbers are of the form zpm i.e. z1m ; z2m ; z3m ,
‘þ’ denotes the positive or anti-clockwise travelling wave, ‘’ denotes the negative or clockwise
travelling wave. The subscript m will be omitted in the following discussion for the sake of
simplicity.
In this analysis the three selected wavenumbers kp will take the sign and form of the wave that
exists in the positive direction. The corresponding wave in the negative direction is always assumed
to take the opposite sign. The roots are complex in general, and all the possibilities are plotted in
Fig. 4. The true roots or waves are those that decay in the anti-clockwise direction and have the
possibilities ðkr  iki Þ, where kr and ki are the real and imaginary wavenumbers. These roots are
those of the lower half of the ﬁgure. The correct anti-clockwise wave therefore takes the form
expðikp cÞ ¼ expðikr cÞ expðki cÞ.
This wave can be of three types

1. kr bki , a propagating or travelling wave, seen just below the real axis. For zero damping this
root lies on the axis, damping causes this root and all the others to rotate in a clockwise
direction.
2. ki bkr , the evanescent bending wave is purely imaginary for zero damping, the damping gives
small negative real part.
3. kr  ki is termed the ‘complex wave’, which always occurs in a pair kr  iki (labelled 1, 2) to
form a rapidly decaying standing wave, which usually describes a local stiffness characteristic.

## 3.4. Wavenumbers modified by tyre rotation

All the displacements are functions of wavenumber, deﬁned in the frame of reference
of the tyre. However, in practice the tyre is rotating in relation to the axle, and wavenumbers
are perceived from this reference to differ from those of the reference frame of the tyre. This
is true only for travelling waves, where energy is stored within the tyre as both kinetic and
strain energy. However, the situation is less clear for the evanescent and complex ‘waves’
which exist only to satisfy the boundary conditions in the locality of the excitation. The complex
wave describes stiffness and will therefore not be affected by the rotation, which acts on
inertia. Alternatively, it could be said that the phase speed for a complex wave is inﬁnite and so
will not be signiﬁcantly perturbed by rotation. The evanescent bending wave represents inertia
and so is expected to be inﬂuenced by rotation, but this effect is ignored here because of the
difﬁculty in isolating the root in the program. The changes for the travelling waves only were
made as follows.
The wave phase speeds for anti-clockwise and clockwise travelling waves are the same, i.e. V p ,
p ¼ 1; 2; 3 for no rotation. If the tyre rotates anti-clockwise at velocity V, the travelling waves
change velocity relative to the wheel axle and line of contact (but the waves still travel at their
original velocities in relation to the belt). The modiﬁed wave speeds relative to the axle are
V pa ¼ V p þ V ,
V pb ¼ V p  V . ð32Þ
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The anti-clockwise wave becomes V pa , faster by V; while the clockwise wave V pb slows by V.
Using the relationship, kp ¼ oV p , the wavenumbers become
kp kp
kpa ¼ ; kpb ¼ . (33)
1 þ Refkp gV =o 1  Refkp gV =o

Note that both wavenumbers are positive here, the appropriate negative sign for the negative
travelling wave is included in the analysis of the next section.
Only the real part of wavenumber is related to energy transfer and is employed in the
denominator. If there is no rotation: kb ¼ kp , kpa ¼ kp :

## 4. Belt transfer functions

Fig. 3 shows the tyre belt subjected to external forces at the contact point y ¼ 0. All the forces,
moments, kinematic quantities, and wavenumbers in the following analysis are assumed to refer to a
particular belt lateral mode of order m, until the final discussion on the transfer functions.
If a normal force F y or a tangential force F x is applied to the tyre it causes responses at every
point c, in both the circumferential u and radial w directions. These responses are quantiﬁed in the
transfer functions. All the transfer functions can be calculated from the summation of the three
pairs of waves that exist at each frequency. The amplitudes of these waves are found from the six
boundary conditions at the contact line with the force input, corresponding to the road contact
with the tyre.
A matrix of radial wave amplitudes is written. These are used to give: slopes f, in-plane
displacements, shear rotation a, bending rotation b, in-plane forces N, shear forces Q and
moments M. The six boundary conditions are then expressed as a matrix in terms of the radial
displacements. The solution of this matrix gives the wave amplitudes and hence the transfer
functions.

θ, Φc

Qc(0)
Mc(0)
Mc(lc) Φc
u, c

N(lc) N(0) w

Qc(lc) Fy, uy

Fx, ux

## Fig. 3. Forces on the belt at the contact line.

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## 4.1. The wave field

The radial displacement w, and circumferential displacement u, at any point c are written as a
sum of the three wave pairs. The gradient of each wave also has two components due to bending
and shear, respectively.

## 4.1.1. Radial displacement

The wavenumbers for the three wave pairs obtained from the wave equation are: kpa ; kpb ,
p ¼ 1; 2; 3; a (anti-clockwise), b (clockwise), denote the direction of the wave propagation. If there
is no rotation, kpa ¼ kpb ¼ kp for all waves. From Eq. (33), rotation only causes a change to the
travelling wave, i.e. 0okpa okp okpb .
The radial amplitudes of each of the three pairs of waves: wpa ; wpb , are given in a column matrix wp .
8 9
>
> w1a >>
>
> >
>
>
> w >
1b >
>
> >
>
>
< w2a >=
wp ¼ . (34)
>
> w2b >>
>
> >
>
>
> w3a >>
>
> >
>
>
:w > ;
3b

At position c each wave becomes phase-lagged and attenuated in accordance with the 66 diagonal
matrix Lpc :
Lpc ¼ diag½eik1a c ; a1b eik1b c ; eik2a c ; a2b eik2b c ; eik3a c ; a3b eik3b c , (35)
where the exponential attenuation and phase-lag over the belt circumference l c ¼ 2pa is
apa ¼ expðikpa l c Þ; apc ¼ expðikpb l c Þ.
The column matrix for weighted wave amplitude and phase is wpc :
wpc ¼ Lpc wp . (36)
The radial displacement wðcÞ at any point c is the sum of the six waves in Eq. (36)
wðcÞ ¼ IT wpc , (37)
T
where the row matrix I ¼ f1 1 1 1 1 1g is just a device to make the sum.
The slope at any point c is obtained using the differential operator Dp :
qw
¼ IT Dp wpc , (38)
qc
where Dp ¼ diag½ik1a ; ik1b ; ik2a ; ik2b ; ik3a ; ik3b .

## 4.1.2. Circumferential displacement

The circumferential displacement upa is the radial displacement wpa weighted by the
displacement ratios Apa and Apb for anti-clockwise and clockwise pth wave.
upa ¼ Apa wpa ; upb ¼ Apb wpb . (39a,b)
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The displacement ratios for the anti-clockwise and clockwise waves are found by substituting the
harmonic travelling wave solutions expðikp cÞ into Eq. (29)

## izpa ð1  z2L D̄Þ  Z co

Apa ; Apb ¼ . (40a,b)
z2pa  z2L þ K̄ cm þ K̄ zcm

The Coriolis coupling Z co is set to zero in this study. The total circumferential displacement uðcÞ is
the sum of the six wave displacements up in Eq. (36)

## The diagonal matrix of displacement ratios is

Ap ¼ diag½A1a ; A1b ; A2a ; A2b ; A3a ; A3b .

## 4.1.3. The slope due to bending

The gradient qw=qc is given in Eq. (1a) as the sum of the bending slope b and the shear g. Slope
due to bending is related to the gradient by Eq. (14a). On substitution of the harmonic solution
for a clockwise or anti-clockwise travelling wave p, and using the normalisation of Eq. (26)
qwp
bp ¼ Ep , (42)
qc
where

1 þ R̄c
E pa ¼ E pb ¼ .
z2p  z2c þ R̄c

## The diagonal matrix of wave weighting functions E pa , E pb is

Ep ¼ diag½E 1a ; E 1b ; E 2a ; E 2b ; E 3a ; E 3b . (43)

## 4.2. The boundary conditions

There are six boundary conditions all at the contact line. Three are geometric continuity
relations linking the belt either side of the contact line. The other three are the resolution of the
two external forces and moment, into the internal forces. All the boundary conditions are
expressed in terms of the six radial wave amplitudes wp .

## 4.2.1. Continuity of radial displacement

There is continuity in the radial displacement at y ¼ 0; y ¼ 2p, or c ¼ 0, c ¼ l c , where l c is the
belt length. From Eqs. (36) and (37)
0 ¼ B1p wp , (44)
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where
B1p ¼ IT ðLpc ð0Þ  Lpc ðl c ÞÞ
or
B1p ¼ ½1  a1a a1b  1 1  a2a a2b  1 1  a3a a3b  1.

## 4.2.2. Continuity of circumferential displacement

There is continuity of circumferential displacement at c ¼ 0; c ¼ l c , so equating these
displacements using Eqs. (36) and (41) gives
0 ¼ B2p wp , (45)
where
B2p ¼ IT Ap ðLpc ð0Þ  Lpc ðl c ÞÞ
or
B2p ¼ ½A1a ð1  a1a Þ A1b ða1b  1Þ A2a ð1  a2a Þ A2b ða2b  1Þ A3a ð1  a3a Þ A3b ða3b  1Þ.

## 4.2.3. Continuity of slope due to bending

Slope due to bending b, is continuous across the input line. However slope due to the shear g, is
discontinuous because of the external shear force. Continuity of slope due to bending can be
expressed as bð0Þ ¼ bðl c Þ, or using Eqs. (36), (38), (42) and (43)
0 ¼ B3p wp , (46)
where
B3p ¼ IT Ep Dp ðLpc ð0Þ  Lpc ðl c ÞÞ.
This can be expanded to give the 61 row matrix
B3p ¼ ½ikpa E pa ð1  apa Þ ikpb E pb ðapb  1Þ::::::::::::::::::; p ¼ 1; 2; 3.

## 4.2.4. Bending moment

An external moment will be applied in any practical situation because the external in-plane
force is applied at the tread surface rather than at the neutral axis. The external moment M 0 is
balanced by the internal moments at c ¼ 0 and l c , i.e.
M 0 ¼ M c ðl c Þ  M c ð0Þ.
If substitution is made from Eq. (13a)
 
qbc w qbc w
M 0 ¼ Bc   Bc  . (47)
qc a c¼0 qc a c¼l c
By differentiating Eq. (42) with respect to c, and using the continuity in w, this becomes
M 0 ¼ B4p wp , (48)
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118 R.J. Pinnington / Journal of Sound and Vibration 290 (2006) 101–132

where
B4p ¼ Bc IT Ep D2p ðLpc ð0Þ  Lpc ðl c ÞÞ.
Expansion of this expression gives a 61 row matrix
B4p ¼ Bc ½k2pa E pa ð1  apa Þ  k2pb E pb ðapb  1Þ::::::::::::::::::; p ¼ 1; 2; 3.

## 4.2.5. Normal force resolution

The normal force/width F y , seen in Fig. 3, can be resolved into the component internal shear
forces/width Qc , and circumferential forces/width N, at c ¼ 0; c ¼ l c
F y ¼ ½Qc  NFc c¼0  ½Qc  NFc c¼l c , (49)

where the kinematic rotation Fc , is given in Eq. (2a). The circumferential force/width N is given
from Eq. (8a) as the dynamic force and N c , the static in-plane force/width. The shear force for the
pth wave can be expanded using Eqs. (1a), (9a) and (43). I0 is the unit matrix
qwp
Qp ¼ S c ðI0  Ep Þ . (50)
qc
If only the static circumferential force is considered, i.e. N ¼ N c , then substitution into Eq. (49) of
Eqs. (2a), (38), (41), (43) and (50), yield
F y ¼ B5p wp , (51)

where
 
T Nc
B5p ¼ I ðS c þ N c ÞDp  S c Ep Dp  Ap ðLpc ð0Þ  Lpc ðl c ÞÞ.
a

On expanding the terms this becomes the following 61 row matrix:
. .
B5p ¼ ½B51a B51b .. B52a B52b .. B53a B53b .

For the pth pair of anti-clockwise and clockwise coefﬁcients B5pa B5pb
2  3
Nc
6 ikpa ðS c ð1  E pa Þ þ N c Þ  a Apa ðapa  1Þ 7
6 7
½B5pa B5pb  ¼ 6   7; p ¼ 1; 2; 3.
4 Nc 5
ikpb ðS c ð1  E pb Þ þ N c Þ  Apb ðapb  1Þ
a

## 4.2.6. Tangential force resolution

Resolution of the tangential forces seen in Fig. 3 yields

## F x ¼ Nðl c Þ  Nð0Þ, (52)

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where F x is the external tangential force/width, N is the circumferential force/width deﬁned in Eq.
(8a). By making substitutions from Eqs. (8a) and (4a), the tangential force becomes
   
qu w qu w
F x ¼ Ac þ  Ac þ (53)
qc a c¼l c qc a c¼0

which is, on using the amplitude ratio of Eq. (41) with Eqs. (36) and (38)

F x ¼ B6p wp , (54)

where
 
T 1
B6p ¼ Ac I Dp Ap þ I0 ðLpc ð0Þ  Lpc ðl c ÞÞ.
a

## Eq. (54) may be further expanded as a row matrix of three pairs

. .
B6p ¼ ½B61a B61b ..B62a B62b ..B63a B63b .

## The pth pair is

    
1 1
½B6pa B6pb  ¼ Ac ðapa  1Þ ikpa Apa þ Ac ð1  apb Þ ikpb Apb þ ; p ¼ 1; 2; 3.
a a

## 4.3. The belt transfer functions

The six boundary conditions may now be assembled into a single matrix. The solution of this
matrix gives the required transfer functions for either normal or tangential excitation. The
boundary conditions that sum to zero are found in Eqs. (44)–(46), (50), those with an external
force are in Eqs. (51), (54). The boundary conditions are written as a stiffness matrix K
F ¼ Kwp , (55)
where the force matrix F includes the two external forces, wp are the wave radial displacement
amplitudes
8 9 2 3 8 9
>
> 0 >
> B 1p >
> w 1a >
>
>
> >
> 6B 7 >
> >
>
>
> 0 >
> 6 7 >
> w >
>
>
> >
> 6
2p
7 >
> 1b >
>
>
< 0 = > 6 B3p 7 >
< w2a >=
6 7
F¼ ; K¼6 7; wp ¼ .
>
> M0 >> 6 B4p 7 >
> w2b >>
>
> >
> 6 7 >
> >
>
>
> > 6 B5p 7 > w3a >
> Fy >
> >
> 4 5 >
>
>
>
>
>
>
:F > ; >
: >
x B6p w3b ;

## Inversion of Eq. (55) gives the radial wave amplitudes

wp ¼ K1 F. (56)
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120 R.J. Pinnington / Journal of Sound and Vibration 290 (2006) 101–132

The circumferential wave amplitudes weight these by the amplitude ratio in Eq. (41)
up ¼ Ap wp . (57)

The displacements wðcÞ, uðcÞ at any point c are found by multiplying the wave amplitudes with the
position weighting 61 row matrix from Eqs. (36) and (37)
IT Lpc ¼ fexpðikpa cÞ apb expðikpb cÞ:::::::::::::::g; p ¼ 1; 2; 3,

## 4.3.1. Radial and circumferential mobility for a normal force

The radial and circumferential mobility for a normal force is obtained by modifying
the force column matrix in Eq. (55); Fx ¼ 0, M0 ¼ 0, Fy ¼ Fym , where Fym refers speciﬁcally
to the force applied to the mth transverse belt mode. The negative sign arises because
the applied force in Fig. 3 is in the negative radial direction. The responses in Eq. (58)
are also replaced by the speciﬁed modal equivalent, i.e. wm ðcÞ ¼ wðcÞ, um ðcÞ ¼ uðcÞ. Then Eqs.
(56)–(58) give
w_ m ðcÞ
¼ Y yy T 1
m ðcÞ ¼ ioI Lpc K F,
F ym

u_ m ðcÞ
¼ Y xy 1
m ðcÞ ¼ ioAp K F. (59)
F ym
Y yy xy
m ðcÞ, Y m ðcÞ, are the modal mobilities at position c in the radial and circumferential directions,
respectively. The modal forces for any line force F ym ðzÞ can be found from the modal
decomposition equation:
X mpz X mpz
F y ðzÞ ¼ F ym cos þ F ym sin . (60)
m¼0;2;4...
b m¼1;3;5...
b

For the particular case of a point force at z, i.e. F y ðzÞ ¼ F 0 dðz  z0 Þ the orthogonality relations
applied to Eq. (60), i.e.
Z z¼b=2  mpz mpz
:::::: cos þ sin dz
z¼b=2 b b
yields
 
F0 2F 0 mpz0 mpz0
F y0 ¼ ; F ym ¼ cos þ sin . (61)
b m¼0 b b m¼2;4... b m¼1;3...
The modal force from a line force of total magnitude F 0 applied over the belt width b is the ﬁrst
term in Eq. (61) i.e. F y0 ¼ F 0 =b.
The velocity response in the radial and circumferential directions, to a unit point radial
force, is a sum of the modal contributions in Eq. (59), with the modal forces available
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X
Y yy ¼ Y yy
m,
m¼0;1;2;3::::
X
xy
Y ¼ Y xy
m. ð62a; bÞ
m¼0;1;2;3:::

## 4.3.2. Radial and circumferential mobilities for a circumferential force

The mobilities for a unit line circumferential force acting on the neutral axis are obtained by
modifying the force column matrix in Eqs. (55), (59); F x ¼ 1=b, F y ¼ 0, M 0 ¼ 0, where b is the
belt width.

4.3.3. Radial and circumferential mobilities for a tangential force on the surface
The mobilities for a unit line circumferential force acting tangentially on the surface are obtained
by modifying the force column matrix in Eqs. (55), (59); F x ¼ 1=b, F y ¼ 0, M 0 ¼ t=b, where b is the
belt width and t the distance from the surface to the neutral axis. The circumferential displacement
on the surface u0ps includes a component from the belt rotation from Eqs. (36)–(38)
u0pc ¼ upc þ tDp wpc . (63)

5. Parameter study

Some parameters seen in Table 1, were selected for the physical properties of a typical
automobile tyre; they are a modiﬁed version of those used in Refs. [6,7]. Using these properties the

Table 1
Belt parameters

## Internal pressure P 2  105 N=m2

Belt thickness t 1:6  102 m
Belt radius a 0.3 m
Belt width b 0.16 m
Side-wall length ls 0.1 m
Side-wall angle ys p=6 rad
Tensile force/width Nc Nz Eq. (A.4) P  l s =2ys N/m
Shear moduli Gc Gz 5  107 5  107 N=m2
Shear stiffness/width Sc Sz Gc  t Gz  t N/m
Bending stiffness/width Bc Bz 107 90 Nm
Axial stiffness/width Ac 1  107 1 N/m
Belt mass/area m 16 kg=m2
Rotational inertia/width Ic Iz m  t2 =3 m  t2 =3 kg
Bending and shear loss factor Z1 Z1 0.15 0.15
Axial and tension loss factor Z2 Z2 0.1 0.1
Side-wall stiffness/length Kr Kc 1:4  106 2:1  105 N=m2
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following related quantities are generated and discussed for the m ¼ 0 wavegroup generated by
line excitation, namely, the wavenumbers, the ratio between radial and longitudinal motion, the
slopes due to bending and shear, and radial and circumferential mobilities. Finally, a modal
summation is made and the point and transfer mobilities are plotted.

## 5.1. Wavenumber as a function of frequency

For Figs. 5–7, a consistent notation is employed to identify the roots of the anticlockwise
wavenumbers: — root 1, - - - - root 2, :::::: root 3. The properties of each root will be discussed with
reference to these ﬁgures. The issues are: the signiﬁcant frequency ranges, and the wave
mechanism in each frequency range.
All the results are plotted between 10 Hz and 10 kHz, covering the practical range of interest.
Fig. 5a shows the three wavenumber magnitudes, or wavenumber radii given in Fig. 4. Fig. 5b
displays the corresponding phase F of each wavenumber, where F ¼ tan1 ðki =kr Þ. The phase of

ki Evanescent
Complex 1 Complex 2

travelling (-ve)
Φ
-kr +kr

travelling (+ve)
rotation of roots for
increasing damping

Complex 2 Complex 1
Evanescent
-ki

## Fig. 4. Types of wavenumber roots.

3
10 0

-0.2
wavenumber (k) (phase,radians)

-0.4
2
10
-0.6
wavenumber (k)

-0.8
1
10 -1

-1.2

-1.4
0
10
-1.6

-1.8
-1
10 -2
1 2 3 4 1 2 3 4
10 10 10 10 10 10 10 10
(a) Frequency (Hz) (b) Frequency (Hz)

Fig. 5. (a) Modulus and (b) phase of wavenumbers: root 1, -  -  -  - root 2,       root 3.
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2
10 2

## Amplitude Ratio u/w (phase, radians)

1.5

1
10 1
Amplitude Ratio u/w

0.5

0
10 0

-0.5

-1
10 -1

-1.5

-2
10 -2
1 2 3 4 1 2 3 4
10 10 10 10 10 10 10 10
(a) Frequency (Hz) (b) Frequency (Hz)

Fig. 6. (a) Modulus and (b) phase of amplitude ratio u=w: root 1, 2  2  2 root 2,       root 3.

all anti-clockwise wavenumbers lie between 0 and p as was indicated in Fig. 4. Figs. 6a and b
give the modulus and phase of the ratio Ap of longitudinal motion to transverse motion for each
root. Figs. 7a and b display, for each wave, the ratio of shear slope to bending slope.

## 5.1.1. Wavenumber root 1

Roots 1 and 2 seen in Fig. 5a are a complex pair of constant equal modulii below 120 Hz. The
roots are symmetric about the negative imaginary axis in Fig. 4; or alternatively in Fig. 5b, the
wavenumber phases are equally greater and less than the imaginary axis value of p=2. In Fig. 6a
it is shown that their dominant motion is radial rather than circumferential. In this frequency
range, below the rigid body translational mode at 120 Hz, these two roots are responsible for the
local stiffness controlled deformation around the contact line (Section 5.2.1).
Between 120 Hz and 3 kHz root 1 is an evanescent bending wave. This is evident from the
modulus, in Fig. 5a, increasing as ðfrequencyÞ1=2 , while the phase is near p=2. The phase is
slightly more negative than this because of the material damping. Fig. 6a also indicates that the
motion is predominately radial. At 3 kHz there is a dip in the wavenumber modulus and a large
step in the phase in Fig. 6b, to almost 0 rad. This behaviour marks the cut-on of the rotational
travelling wave, identiﬁable as a travelling wave from a wavenumber phase near zero. The form of
this wave is suggested from Figs. 6 and 7. From Fig. 6 it is seen to be a radial wave. From Fig. 7a
and b the bending and shear components are in anti-phase, but the shear part is just dominant.
This implies that this wave does not have a large net radial component and may therefore be able
to propagate in the tyre contact patch, where radial motion is not encouraged excitation.
The wave speed at high frequencies is close to that of the speed of sound, and so this could
contribute to sound radiation, particularly for in-plane shear excitation in the contact region.

## 5.1.2. Wavenumber root 2

The second root forms the complex complement of root 1 until the rigid-body translational
resonance at 120 Hz, as discussed above. Between 120 Hz and the ring-frequency at 450 Hz the
phase of near p=2 suggest no propagation. Above the ring-frequency the root cuts on and
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1
10 3

## Shear slope / Bending slope (phase, radians)

2
Shear slope / Bending slope

1
0
10
0

-1
-1
10
-2

-3

-2
10 -4
1 2 3 4 1 2 3 4
10 10 10 10 10 10 10 10
(a) Frequency (Hz) (b) Frequency (Hz)

Fig. 7. (a) Modulus and (b) phase of shear slope/bending slope: root 1, 2  2  2 root 2,       root 3.

propagates as a longitudinal wave. This wave is identiﬁed from three things: the slope
proportional to frequency (in Fig. 5a), a near zero phase (in Fig. 5b), and a dominant longitudinal
motion in Fig. 6a. The wave speed is 800 m/s and is therefore able to radiate sound freely.

## 5.1.3. Wavenumber root 3

At 60 Hz the rigid-body rotation resonance occurs, of the belt mass upon the side-wall shear
stiffness. Below this frequency root 3 is responsible for describing the rigid-body motion. The
wavelength can be calculated, from the wavenumber in Fig. 5a, to be one tyre circumference as for
the n ¼ 1 mode.
At higher frequencies a propagating wave cuts on, seen from the near zero phase in Fig. 5a.
Figs. 6a and 7a show it to be a bending wave below the rotational wave cut-on at 3 kHz. The wave
only propagates above the frequency of the n ¼ 1 mode at 120 Hz, when the belt resonates on the
side-wall. Above this frequency the travelling wave is responsible for any discernible standing
wave below the ring-frequency at 450 Hz. The wave is controlled by belt tension and bending
stiffness, although the slope in Fig. 5a above 200 Hz is proportional to ðfrequencyÞ1=2 suggesting
the bending is dominant here.
The wave becomes shear dominated above 3 kHz as may be deduced from: the predominately
radial motion in Fig. 6a, the greater shear motion in Fig. 7a. The wave speed is 200 m/s at 10 kHz
which is less than the speed of sound in air. Radiation of sound is only expected from the
discontinuity at the excitation line.

5.2. Transfer functions for line excitation in the radial and circumferential directions

The wavenumbers can now be used in the assembly of transfer functions using the
procedures of Section 4. The force is normal or tangential and is applied at the contact line at
0 . The velocity response is calculated at the same position to give input mobility, it is also
calculated at 180 to give the transfer mobility. Radial and longitudinal (circumferential)
mobilities are given in Figs. 8 and 9. The tangential mobility at the tyre surface due to longitudinal
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-3
10

Mobility Yy (m/s/N)
-4
10

-5
10

1 2 3 4
10 10 10 10
Frequency (Hz)

Fig. 8. Radial mobilities: — transfer modulus at 180 , input modulus,       input real part.

-3
10
Mobility Yx (m/s/N)

-4
10

-5
10

1 2 3 4
10 10 10 10
Frequency (Hz)

Fig. 9. Longitudinal mobilities: — transfer modulus at 180 , input modulus,       input real part.

and radial motion is given in Fig. 10. Finally, two parameter variations are presented in Figs. 11
and 12, showing the effects of pressure and tyre speed.

## 5.2.1. Radial mobilities

Fig. 8 gives radial mobilities at the input at 0 , and 180 , from a normal force at the contact line
at 0 . The real part of the input mobility is always positive which implies that the model is
physically possible. Below the rigid body transverse mode n ¼ 1 at 120 Hz, the input mobility
modulus is controlled by the local stiffness, generated from the complex roots 1 and 2. The
transfer mobility is accordingly at least an order of magnitude lower. Between 120 and 450 Hz the
n ¼ 1; 2; 3; . . . bending wave resonance frequencies, from root 3, can be seen. The transfer mobility
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Fig. 10. Input tangential mobilities: moment modulus,       real part for moment, — total tangential modulus,
2  2  2 total tangential real part.

-2
10
Mobility Yy (m/s/N)

-3
10

-4
10
1 2 3 4
10 10 10 10
Frequency (Hz)

Fig. 11. Input radial mobility as a function of inﬂation pressure: — 0.5 bar, 2 bar,       4 bar.

levels are lower because there is no contribution from the evanescent bending wave, and the waves
are attenuated by the damping.
At 450 Hz the ring-frequency occurs, n ¼ 0, when a longitudinal wavelength (root 2), ﬁts
around the circumference. A larger peak is seen at this frequency at the input; but a trough occurs
in the transfer mobility. The ring mode is therefore only local to the excitation as suggested by the
non-zero root 2 wavenumber phase, in Fig. 5b. This may explain why this phenomenon is not
easily observed in measurements. The second longitudinal wave resonance is seen near 1 kHz on
all plots.
The input modulus and real part of mobility have a slope of (frequency)1=2 until 3 kHz as for a
beam in pure bending. At 3 kHz the rotation wave cuts on, but the main response above this
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R.J. Pinnington / Journal of Sound and Vibration 290 (2006) 101–132 127

Fig. 12. Real part of input radial mobility as a function of speed: — 0 m/s, 40 m/s,       80 m/s.

frequency is from the transverse shear wave, identiﬁed as root 3 in Fig. 5. For a different
excitation, for example by a moment, the rotation wave may be larger. Above 1 kHz the transfer
mobility reduces sharply as these waves have a short wavelength giving a high attenuation.

## 5.2.2. Longitudinal mobility on the neutral axis

Fig. 9 displays the real part and modulus of the input longitudinal mobility, and also the
modulus of the transfer mobility at 180 . At 60 Hz the belt has a rigid body n ¼ 0 rotational
resonance. Below this frequency the tyre rotates and translates as a rigid body, as described by
root 3 in Fig. 5; the side-wall shear stiffness controls the response level. At 120 Hz the belt rigid
body translation resonance occurs, denoted as n ¼ 1. Above this frequency the response level
drops until a minimum at the ring frequency at 450 Hz. The coupling with transverse motion
causes a slight excitation to the bending modes in this region. At 600 Hz the lowest excited
longitudinal mode is seen, the remainder of the sequence is clearly visible until 3 kHz. In this
region the transfer mobility is similar to the input mobility, which could mean that these waves are
sufﬁciently large to make a detectable contribution to radiated sound levels. The mean input
mobility takes the value of a longitudinally excited inﬁnite rod.

## 5.2.3. Moment mobility and tangential mobility

In Section 4.3.3, it was shown that the tangential input mobility is the sum of the in-plane
response seen in Fig. 9, and the response to the moment about the neutral axis. The real
component and modulus of this moment component is displayed in Fig. 10. Also shown is the real
part and modulus of the total input tangential mobility. Below about 1 kHz the tangential
mobility is similar to the longitudinal mobility. However, near 3 kHz the cut on of the rotational
wave causes a major peak in the response. This peak dominates the response due to the moment
alone. It seems likely that this local resonance could contribute signiﬁcantly in the tyre squeal
phenomena as it is excited by tangential forces and moves primarily tangentially to the surface.
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## 5.2.4. The effect of tyre pressure on transverse input mobility

Fig. 11 displays the transverse input mobility modulus as a function of tyre pressure. It is seen
that the increase in pressure has two effects. The most signiﬁcant is that it increases the radial side-
wall stiffness thereby raising the frequency of the lowest radial natural frequency and decreasing
the total response level. The second less noticeable effect is the resulting increase of tyre tension.
This causes a stiffening of the belt below the ring frequency and so tends to raise natural
frequencies and decrease the response in this region. These two effects are at low frequencies and
would inﬂuence the vibration and noise within the vehicle.

## 5.2.5. The effect of tyre speed on the input transverse mobility

The tyre speed has two effects on the tyre vibration. The ﬁrst is the change to the clockwise
and anti-clockwise wavenumbers, which was discussed in Section 3.5. The second is the
increasing centrifugal force on the belt which contributes to the tension, as described in Appendix
A. These two inﬂuences are seen in Fig. 12 for the mobility modulus and real part. The overall
response becomes lower from the belt stiffening and the decrease of the anti-clockwise
wavenumbers. These two mechanisms also cause a decrease in wavenumber which leads to
increase in sound radiation. Mode splitting is only noticeable on the ﬁrst mode because of the high
attenuation of the clockwise wave due to the increase in wavenumber. The range in the ﬁgure,
0–40 m/s, covers normal driving operation where the change in mobility due to speed is less than a
factor of two.
When the rotational speed is equal to the phase speed due to tension, a shock wave will develop
at the leading edge of the contact zone. A lot of power is absorbed by the tyre as the impedance
becomes very large, making failure likely. The 80 m/s seen in the ﬁgure is near the limit for this
particular tyre. If a puncture occurs in a tyre, the loss in pressure will cause a decrease in
wavespeed giving the possibility of this failure mechanism at lower driving speeds. Tyres with a
high pressure would be more vulnerable.

Fig. 13. Radial point mobility: input modulus,       input real part, — transfer modulus at 180 .
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R.J. Pinnington / Journal of Sound and Vibration 290 (2006) 101–132 129

-1
10

-2
10

Mobility m/s/N
-3
10

-4
10

-5
10
1 2 3 4
10 10 10 10
Frequency Hz

Fig. 14. Tangential point mobility: input modulus,       input real part, — transfer modulus at 180 .

## 5.2.6. The effect of an offset neutral axis

The effect of the non-alignment of the neutral axis and centroidal axis was investigated, but the
changes were so small that it was considered that a plot was unnecessary.

## 5.3. Tyre mobilities for point excitation

If there is point excitation rather than line excitation, the higher order mode groups, m40, are
summed with the m ¼ 0 response in accordance with Eq. (62a). Figs. 13 and 14 show the modulus
and real part of the input point mobility along with the transfer mobility at 180 . The excitation
and response were displaced from the centre line by a distance z0 ¼ b=10.
Fig. 13 gives the input and transfer mobilities in the radial direction. Below 300 Hz the
frequency structure is similar to the m ¼ 0 version in Fig. 8, except that the input response is
greater on account of the presence of the stiffness component of the higher order modes. At about
0.6, 1.1, 1.7, 2.2 kHz humps in the input response indicate the cut-on of the m ¼ 1; 2; 3; 4; . . .
transverse mode groups. Only modes m ¼ 0–8 were included in the summation with the top cut-
on frequency at about 4 kHz. This explains why the slope is less than proportional to frequency, as
would be expected for a plate controlled by shear stiffness. Limiting the mode count in this way
simulates the effect of an excitation diameter of b=8.
Fig. 14 shows the input and transfer mobilities in the circumferential direction for tangential
excitation. The result is similar to that for m ¼ 0, in Fig. 10, except that the levels above 500 Hz
are greater due to the summation of transverse modes. The four cut-on frequencies of the
transverse modes m ¼ 1; 2; 3; 4 are the same as for normal excitation, as the response is dominated
by the radial wave motion excited by the input moment. The ‘resonance’ from the rotational wave
cut on at 3 kHz is a clear feature, and is therefore expected to have an important role in the
dynamics of tangential motion and high-frequency sound generation.
Similar high frequency, normal and tangential excitation results are seen in Ref. .
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130 R.J. Pinnington / Journal of Sound and Vibration 290 (2006) 101–132

6. Conclusions

A sixth-order differential wave equation was derived to describe the motion of a curved tyre in the
frequency range between 0 Hz and 4 kHz, for point or line excitation, in the radial and tangential
directions. The model included the effects of: longitudinal stiffness, bending stiffness, shear stiffness,
tension, pressure, speed, side-wall stiffness in two directions, mass and rotary inertia.
The solution of this equation is three pairs of waves at each frequency. These waves are
described in terms of frequency dependency, ratio of circumferential to radial motion, ratio of
shear slope to bending slope, and amplitude and phase. The waves are used to assemble radial and
longitudinal transfer functions.
The behaviour of the tyre may be roughly divided into three frequency regions. First is the belt
rigid body region that occurs below the rigid body translational resonance frequency. At this
resonance the belt mass is mainly restrained by the side-wall radial stiffness. There is also local
stiffness deformation at the contact. The belt mass, pressure and tyre geometry are main
inﬂuences. If the pressure increases the rigid body frequency is raised and the tyre becomes stiffer,
lowering the mobility.
The second frequency region lies between the rigid body resonance and the belt ring frequency,
around 500 Hz. Standing waves are observed on the belt in the radial direction. The behaviour is
controlled by belt tension and bending stiffness. The tension is a function of pressure and tyre
rotational speed; increasing these increases the stiffness and raises the resonant frequencies. The
belt moves as a rigid body in the longitudinal direction.
At about 500 Hz the belt ring frequency occurs. Above this frequency longitudinal waves can
propagate within the belt and the coupling due to curvature between longitudinal to transverse
motion decreases. The ring frequency mode seems to be a local resonance around the excitation
line. The longitudinal waves propagate with relatively low attenuation and because of their long
wavelengths may contribute to the sound radiation.
In the lateral direction waves become strongly attenuated and only travelling waves are observed,
the belt behaves as if of inﬁnite extent. Around 3 kHz a new wave cuts on which involves rotation
without much transverse motion. At the same time the bending wave converts to a transverse shear
wave, controlled by the belt shear stiffness. This wave dominates the radial motion.
The effect of tyre speed was investigated. The general trend is to reduce the mobility at all
frequencies due to the combined inﬂuence of centrifugal force, and speed dependent wavenumber
changes. This would increase the power absorbed and radiated by the tyre.

Acknowledgement

The support by the EC ﬁfth framework program of the RATIN project is gratefully
acknowledged.

## Appendix A. Tyre belt and side-wall tension

A segment of length 2ady and belt radius a, of a tyre cross-section is displayed in Fig. 15. The
belt mass/area is m. The transverse and circumferential belt tension/width N z , N c and the side-wall
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R.J. Pinnington / Journal of Sound and Vibration 290 (2006) 101–132 131

δθ δθ

φs radius a s
pressure P
Nc Nc c φs Ns s
φ1
belt width b
Ω radians/sec

## Fig. 15. Belt segment and cross-section.

elements, are only a function of the pressure and geometry. The belt tension/length N c is found
from the equations of equilibrium in the horizontal and radial directions. The tyre rotates at
O rad=s giving a belt velocity c m/s. The side-wall is the arc of a circle of radius as subtending an
angle 2fs . The arc is rotated making an angle f1 with the horizontal.
By considering the horizontal equilibrium of the right-hand side-wall element
N z ¼ P:as , (A.1)
where the side-wall radius as is related to the side-wall arc length l s by
ls
as ¼ . (A.2)
2fs
The radial equilibrium of the belt segment can now be taken along the centre line
0 ¼ ððP þ O2 maÞa:b  N c b  2N z a sin f1 Þ2dy. (A.3)
This includes contributions from the pressure, centrifugal force, belt tension and side-wall tension,
respectively. By making substitutions from Eqs. (A.1) and (A.2), Eq. (A.3) can be re-arranged to
give the belt tension:
 
l s sin f1
N c ¼ Pa 1  þ O2 a2 m. (A.4)
b fs

References

 H. Ishiara, Development of a three dimensional membrane element for the ﬁnite element analysis of tires, Tire
Science and Technology 13 (2) (1985).
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Vehicles on Roads and Tracks, Vehicle System Dynamics, Swets & Zeitlinger, 1994.
 W. Kropp, Structure-sound on a smooth tyre, Applied Acoustics 20 (3) (1989) 181–193.
 K. Larsson, W. Kropp, A high frequency tyre model based on two coupled elastic layers, Journal of Sound and
Vibration 253 (4) (2001) 889–908.
 S. Gong, A Study of In-plane Dynamics of Tires, PhD Thesis, Delft University of Technology, Faculty of
Mechanical Engineering and Marine Technology, 1993.
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Vibration 253 (2002) 961–983.
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Vibration 221 (1999) 887–902.
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(2000) 455–488.
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Sound and Vibration; doi:10.1016/j.jsv.2005.03.024.
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