Acta acustica – ACUSTICA
Vol. 00 (2007) 1-13
____________________________________________________________________________________________________________
Calculation of tyre noise radiation with a mixed approach
P.Jean, N. Noe, F. Gaudaire
CSTB, 24 rue Joseph Fourier,38400 Saint Martin d’Hères, France, [email protected]
_________________________________________________________________________________________________________________
Summary
The radiation of tyre noise is computed using a hybrid integral/ beam tracing approach called GRIM. It is simply an integral over
the vibrating tyre, of the product of the known velocity by the full Green function which takes into account all the multiple
reflections and diffraction between tyre, ground and car body. This Green function can be calculated by means of a beam tracing
program named ICARE -which can deal with curved surfaces and multiple diffraction. This approach makes the link between tyre
models and reception points well outside the car. In order to deal with small scale phenomena, close to contact, a combined use of
BEM and beam tracing is showed to bypass the limitations of both geometrical and BEM approaches.
PACS no.43.28.Js
circumference. One should also mention analytical descriptions
such as the beam-like tyre model proposed by R. Pennington
1. Introduction
[16] which provides physical insight at very low computational
cost.
The prediction of tyre noise has recently gained much attention
The second component in the GRIM approach is the Green
and many research projects have been dedicated to this subject.
function between the blocked tyre surface and the receiver. This
The reasons for such a focus are well known. Firstly, traffic
acoustical term of the GRIM integral can be computed by any
noise is a well established and increasing nuisance. Secondly,
means such as BEM [5], infinite elements [17] or geometrical
the reduction of other noise sources in cars such as engine noise
approach such as beam-tracing [18]. The BEM approach is very
has led to the present concern for tyre noise which is
precise and apt to model detailed small-scale geometries close to
acknowledged to be dominant above 50 km/h. An important
contact but will become time-consuming when dealing with
literature both experimental and numerical [1,2] exists on this
extended geometries such as the whole car body. On the other
subject. Several noise mechanisms have been observed and
hand, geometrical models can handle large geometries with
addressed. The basic mechanisms are well described in [2]. The
reduced cost but are limited to wavelengths smaller than
main problem involves tyre interaction with the ground [3]
geometrical dimensions. A combination of both BEM and beam
which will lead to the vibration of the tyre and its radiation into
tracing is proposed in this article where BEM is employed close
the surrounding medium. Geometrical aspects will then come
to the tyre up to a fictitious surface, called a skirt, positioned
into play: the shape of the tyre/road profile often called ‘horn
close to contact. Beyond the skirt beam tracing is employed.
enhancement’ will strongly amplify the noise radiated [4,5].
This mixed BEM-beam tracing will be necessary if one wants to
Also important [6] but less studied, is the presence of the car
introduce detailed tyre shapes.
body. Including the car into numerical schemes is often beyond
Other mechanisms such as small scale geometrical effects
the possibilities of the numerical approaches employed. In the
involving the small cavities in the road –often called air
present paper an attempt in this direction has been made by
pumping- or Helmholtz and quarter-wavelength effects in the
employing a mixed integral/ ray approach called GRIM (Green
grooves are not considered in the work here reported. It is
Ray Integral Approach) [7-10]. It relies on a Rayleigh-integral
nevertheless the belief of the authors that these aspects, not fully
which combines a known surface velocity of the tyre and a
understood at present and also little reported in the literature [20-
particular Green function between tyre surface and receiver.
21] deserve full attention. Work carried out at CSTB on this
The velocity can be estimated using many available tyre models.
subject is the object of separate publications [22].
A Finite Element description can be employed [12,13] but FEM
is usually time consuming and limited to low frequencies, the
Finally, the road itself also acts in several ways. First, its outer
high frequency limit depending on the mesh refinement. A
surface acts in the building up of contact forces [3] acting on the
complementary well-used model is that of W. Kropp [11,14]
tyre, but also as part of air-pumping mechanisms. Its inner
which relies on a “two-layer plate on springs” description and
structure may also play an important role. In the present work,
neglects the curvature of the tyre. A good compromise to the
the ground is either rigid or locally reacting whereas, in reality, it
previous approaches is the spectral finite element method
acts as a propagating medium. The part of the tyre in contact
proposed by S. Finnveden [15] which employs a finite element
with the ground vibrates and generates waves which will travel,
description along the tyre width and a wave description along the
reradiate in the air and thus contribute to the overall perceived
noise. Also, a wave reflection on the ground surface is not
Received 5 May 2005
exactly a local action since waves enter the ground and reradiate
Accepted ** *** ****
1
Acta acustica – ACUSTICA
Vol. 00 (2007) 1-13
____________________________________________________________________________________________________________
further away. This aspect is being studied at CSTB and will also
( ∆ + k 2 )GV ( M , Q ) = −δ Q
be the object of separate publications [23].
is recalled; in part 3 numerical validations are reported; in part 4,
∂GV ( M , Q )
=0
∀Q ∈ SV
∂n
∂GV ( M , Q )
= − jωρσ GV ( M , Q ) ∀Q ∈ S A
∂n
car effects are simulated.
Sommerfeld conditions
This paper is structured as follows: in part 2, the GRIM method
∀M , Q ∈ Ω
(2)
This more complex Green function GV is, again, the pressure at
2. A decoupled approach
M due to a unit source placed at Q, but it now includes all
As mentioned in the introduction, many physical aspects will
contribute to the overall noise radiated by a rolling tyre. Some
authors have concentrated [4,5] on geometrical amplification of
radiation from the vibrating tyre surface through the so-called
reflections and diffractions on S and it assumes SV to be rigid.
Importing GV into (1) leads to a much simpler expression, since
the second integral cancels out and the derivative of GV on SV is
identically equal to zero, so that
“horn effect” due to the particular shape formed by a tyre resting
P ( M ) = − jωρ ∫ V (Q )GV ( M , Q )dS (Q )
on a road. However, actual tyre velocities can easily be
incorporated in the calculation of radiated noise if one assumes
that the tyre velocity is known independently of its surroundings,
meaning the rest of the car.
(3)
SV
Equation (3) is an exact expression where V and P are coupled. It
could be used in a classical FEM/BEM scheme with the main
advantage of reducing the surface of integration to SV, leading to
a matrix expression to be solved. Fortunately, in most airloading situations the assumption that the velocity can be
2.1 The GRIM approach
estimated a priori is possible so that one simply ends up with a
The method employed -called GRIM: Green Ray Integral
Method [7]- has been previously applied to several vibroacoustical problems [8-10].
simple integration of the product of two known quantities. GV
can be computed by any means such as the Boundary Element
Methods (BEM) or by geometrical approaches. BEM is a precise
approach well suited when only the tyre is to be modelled.
Geometrical approaches are rather high frequency oriented and
more suited for the modelling of the full car. Consequently, both
methods have been employed. BEM has been first used to give a
precise reference solution and, as will be latter showed, as a
complementary solution to beam tracing in the contact zone. The
beam tracing program ICARE [18,24,50] has been employed.
From (3), the expression for acoustic velocity can be expressed
as
Figure 1. The GRIM approach for tyre radiation. SV is a
vibrating boundary (vibrating tyre: belt and side walls), SA is an
acoustic boundary complementary to SV.
Figure 1 shows the general problem to be solved. The time
dependence ejωt is implicit. The air domain Ω is bounded by
surface S=SV ∪ SA, where SV is a vibrating surface having a
velocity V and SA is an ‘acoustic’ surface with local reaction
described by an acoustic admittance σ.
The acoustic pressure P at any point M ⊂ Ω can be expressed as
P( M ) =
⎡
∂G ( M , Q ) ⎤
∫ ⎢⎢ jωρV (Q)G (M , Q) − P(Q) ∂nQ ⎥⎥ dS (Q)
SV ⎣
⎦
V ( M ) = ∫ V (Q )
SV
∂GV ( M , Q )
dS (Q )
∂n M
(4)
2.2 Beam tracing computation with ICARE
In order to apply beam tracing to the computation of Green
functions between a rigid tyre and any receiving point, one needs
to use somewhat more evolved than common algorithms since
reflections on curved surfaces, multiple reflections and
diffraction must be taken into account. The computer software
ICARE has been developed to account for these aspects in an
(1)
⎡
∂G ( M , Q ) ⎤
− ∫ P (Q ) ⎢ jωρσ (Q ).G ( M , Q ) +
⎥ dS (Q )
∂nQ
⎢
⎥⎦
SA
⎣
efficient way. ICARE is a commercial software. A more detailed
description of the underlying theory and algorithms can be found
in [18,50]. Only the main principles are summarized below.
Different geometrical approaches can be found in the literature
where G is the free field Green function between M and any
[25-32]. They are all based upon an analogy between optics and
point Q on S. Next, we define a Green function GV solution of
acoustics where the propagation of sound is analysed by means
of acoustical rays. Such approaches encounter low frequency
limitations where the validity of the geometrical simplifications
requires that acoustical wavelengths must be smaller than
characteristic dimensions of the problem analysed. Several
published methods deal with diffraction [32-40] but none of
2
Acta acustica – ACUSTICA
Vol. 00 (2007) 1-13
____________________________________________________________________________________________________________
these beam tracing algorithms can precisely deal with curved
(B1 and B2) are followed as they start propagating. Beam B1
surfaces since they rely on pyramidal beams (where rays
reflects on S1 and then encounters the wedge S2. Since part of B1
converge into a point, i.e the image-source). With curved
strikes the wedge while the upper part goes over its top, B1 must
surfaces, there is no image-source and consequently pyramidal
be slit into two beams, the separation being marked by a dashed
beams can not be used.
line. Beam B2 encounters a spherical obstacle S3 and must also
be split.
The ray tracing approach consists in emitting a large number of
rays from the source and in following their propagation. It is
S1
very straightforward to implement. One of the problems with
this method is continuity of the solution. An artificial width is
usually added to each beam in the form of a disk-like weighing
function with pseudo-Gaussian properties across its width [25].
However, aliasing problems still remain. An alternative is to
employ beam tracing where emitted rays are replaced by beams.
Reflection on plane surfaces or elements (planar discretisation of
boundaries such as obtained from finite elements descriptions)
can then be derived analytically. The approach employed in
ICARE is a combination of ray tracing and beam tracing where
each emitted beam is defined by a set of three rays, but not
assuming pyramidal beams. Figure 2 represents an example of
beam-splitting of sound radiated by a point-like source.
Figure 3. Splitting of sound beams. Beams B1 and B2 must be
further separated into sub beams due to obstacles (dashed
lines).
A second important feature of ICARE is the treatment of
diffraction which is done by means of the Geometrical Theory of
Diffraction (GTD) as described in [33]. A ray reaching a
diffracting edge, at a point D, with an angle β relative to the
edge’s tangent will reradiate within a cone of opening β as
shown in Figure 4.
Figure 2. A point source decomposed into beams formed by the
emitting point and three rays.
D
The follow-up of the beam history as it travels in the medium is
simply the follow-up of the set of three rays. This allows
introducing any type of boundary since the reflection of a ray on
a curved surface is entirely defined by the knowledge of the
normal and the tangents to the surface at the impact point.
Therefore, ICARE deals exactly with curved surfaces when they
are mathematically or parametrically defined (cylinders, spheres,
nurbs,..); the alternative, not so precise, would consist in
Figure 4. Diffracting cone according to the GTD. For an incident
ray on the edge, the potentially diffracted rays have the same angle
with the edge than the incident ray.
approximating the curvature by a set of polygonal shapes which
would also be a much slower process.
One key aspect of the algorithms in ICARE is its automatic
In ICARE, diffraction edges must be chosen by the user.
adapting to object encountered as the beams travel. If necessary,
Successive diffractions can be defined. In practice, more than
the beam will be split into sub-beams. This occurs for instance
two successive diffractions are not recommended due to a very
when a beam impacts two different surfaces or an edge. Also,
significant computation time and probable lack of precision. A
angular widening as the beam travels will necessitate sub-
beam impinging on a diffraction edge will intersect the edge
dividing so that the supporting rays remain compact up to chosen
along a diffracting segment D1D2. The union of the two
limits (i.e. the interior of the beam defined by its rays can be
diffracting cones at points D1 and D2 will define a diffracting
overestimated with a bounding volume that does not cover all
source. This source is then decomposed into source beams, as
space). Figure 3 gives, for the sake of clarity, a 2D
represented in Figure 5, now defined by four supporting rays
representation of a source emitting several beams, two of which
which are, as for source beams, propagated, reflected and
3
Acta acustica – ACUSTICA
Vol. 00 (2007) 1-13
____________________________________________________________________________________________________________
eventually sub divided as they travel and reach boundaries. It
is deterministic, since it combines the contribution of a finite
must mentioned that the direct use of GTD neglects the finite
series of elementary terms. Convergence of the computed
length of diffracting edges This is coherent with the basic
pressure can only be controlled at the acoustic stage of the
assumption of geometrical acoustics where surfaces are also
calculations since it depends on attenuations in air or absorption
supposed to be very large compared to acoustic wavelengths.
at the boundaries (the first geometric part of the computations is
The introduction of corrective terms for the finite dimensions of
independent of material and fluid characteristics); the number of
edges is a subject of on-going research.
terms (eq. (5)) to be summed to obtain convergence may become
important for multiple reflections in a reverberant environment.
Inside a volume with many absorbent boundaries –inside a car
for instance- convergence may be obtained rapidly when
individual rays (or beams) have been reflected only a few times
(between five and eight times each). In order to deal with the
more reverberant situations, where a large number of successive
reflections is necessary for convergence, a statistical model has
been implemented [25]. The solution is then the sum of low
order reflection contributions and complementary terms based on
energy considerations. A model based on radiosity [41] has also
been introduced. These models have not been employed in this
work since they have been introduced in ICARE only recently.
An important feature of ICARE is its possibility to consider
surface sources SV having a known velocity – the surface of a
Figure 5. A secondary source due to diffraction of an incoming
beam on a wedge is decomposed into beams defined by four
rays.
For given receivers and propagating beams, the exact
propagations paths are computed by sequential numerical
refinements until rays are found passing close enough to the
receivers. In ICARE, two very different computing steps are
considered. First, the geometrical part is done independently of
physical properties and frequency; it only depends on
geometrical data. As an output, a binary file containing all
propagation information is created. At a second stage (the
acoustic stage), the combination of the physical data and
propagation data will give the acoustic pressures at receiver
positions M as a combination process of different paths where
each path is expressed as:
P(M) =
A .Đ .
Đ
directly used by using reciprocity and shooting beams from the
receiver position onto SV. As the propagating beams intersect
with SV, within triangular portions Ti of SV, the elementary
contributions of integral (3) on Ti’s are piled up into P(M). An
alternative is to compute separately the values of GV(M,Q)
between receivers M and SV at pre-selected positions of Q on SV.
The latter approach allows carrying out the integration for
different velocity profiles independently of the geometric
computation.
Figure 6 shows a comparison, at 2000 Hz,, of computations
made both with BEM (MICADO software, [42-43]) and ICARE
in the case of a monopole placed over a convex (top graphs) or a
concave surface (lower graphs), both straight and infinite in the y
direction perpendicular to the graphic representation (xz).
∏ Ri . ∏ Dj
i
The
tyre in the present study. The GRIM approach can then be
(5)
j
term refers to geometrical divergence and is equal
to ∆Ω / S ( R ) where ∆Ω represents the solid angle of the beam
originating from the source point and S(R) is the surface area of
the wavefront at the receiver point; these values are computed
during the geometric calculation from the characteristics of the
associated beam; Đ =1/R for a point source.
A
represents air
absorption; this term may become important for long distances,
in open air or for reverberant situations. Ri’s and Dj’s represent
reflection and diffraction coefficients. This second acoustic stage
is much faster than the geometrical stage and consequently can
be iterated for acoustic optimisation.
MICADO is based on a variational approach either in 2D or in
3D and can deal, through a Fourier-like transform [43] of 2D
results, with 2.5D problems such as the previous one. BEM
computations will be employed in the following computations
either to obtain reference solutions or as part of combined BEM
and beam tracing computations (paragraph 3.5). The surfaces
have a radius of curvature of 3 m. The monopole source is
placed at (0.65,0,0.65) and is defined such that the free field
acoustic pressure at 1m is 1/4π. The acoustic pressure is
computed in the xz plane at y=0. Beam tracing computations
with ICARE include both reflections on the curved surfaces and
end diffractions. The agreement between both computations is
quite satisfying.
One must keep in mind that, like all geometrical approaches,
derivations in ICARE are in theory only valid at medium and
high frequency when acoustical wavelengths become smaller
than geometrical details.
Reflection is computed with the classical plane wave or spherical
wave reflection coefficient [26] based upon the knowledge of the
incident angle and the local impedance. The previous expression
4
Acta acustica – ACUSTICA
Vol. 00 (2007) 1-13
____________________________________________________________________________________________________________
Figure 7. Beam/ray tracing by ICARE on a baffled tyre. The
strip in lighter grey shows the tyre of width w.
Figure 6. Validation of beam tracing computation for curved
surfaces. The surface is infinite and straight in the y
(unrepresented) dimension. It is finite in the represented xz
plane. A unit monopole is placed at (0.65,0,0.65). Pressure
represented in the y=0 plane.
This problem has been solved with ICARE and also with the
BEM program MICADO [42,43]. The case of a cylinder of
radius 32 cm is considered. It has either point contact at the
origin O (unloaded tyre, four top graphs) or a flat contact over 32
degrees (loaded tyre, four lower graphs). A receiver M is placed
at (2, 0, 1). Figure 8 compares the Green functions G(Q,M) for
3.
Numerical results
In all examples, the same coordinate frame is employed. The
origin, placed on the ground, corresponds to the centre of the
tyre. The x axis is on the ground in the rolling direction, the y
axis corresponds to the tyre width in 3D cases and the z axis is
vertical (see Figure 7). ℜ and w denote respectively the radius
and the width of the tyre.
different positions Q on the tyre facing M, computed either with
MICADO and ICARE. At Q1 (20 degrees from O) and for the
unloaded tyre, the comparison of both computations shows
comparable orders of magnitude but different minima locations.
When the tyre is loaded, the angle between tyre and road, near
contact, is less sharp and a very good agreement at Q1 can be
observed above 300 Hz. At other Q locations, the agreement
between both models is very satisfactory. The results for both
loading cases are very similar at Q3 and Q4, therefore not
influenced by the geometry of the tyre. Differences between the
3.1 Horn amplification in 2.5D: ICARE validation
loaded and unloaded cases can still be seen at Q2 (50 degrees
from the vertical).
Much focus has been placed on the study of geometrical
amplification through the horn-like shape close to the contact
point of an unloaded tyre resting on the ground. This problem is
directly related to the computation of the Green function in
equation (3) and can be used in the validation process.
Therefore, the first problem herein addressed is that of an infinite
cylinder with point-like sources and receivers. The sources are
disposed on a thin circumferential strip (figure 7). This
configuration can therefore be viewed as the case of a baffled
tyre. This problem is named a 2.5 D problem since the geometry
is infinite along one direction and sources and receivers are
points. Figure 7 illustrates the propagation of rays computed by
ICARE between points Q on the tyre surface and an external
point M. For a given pair (Q,M) the sum of all rays propagating
between Q and M will give the Green function G(M,Q). Note
that due to reciprocity either point can be source or receiver. One
can see that for points closer to contact the number of incoming
rays rapidly increases.
Figure 8. Comparison of Green functions: ICARE versus BEM
computations. The cylinder is rigid and has a radius of 32 cm. It
has point contact with the rigid ground at the coordinate origin
5
Acta acustica – ACUSTICA
Vol. 00 (2007) 1-13
____________________________________________________________________________________________________________
O, or flat contact over 32 degrees. Sources Q1 to Q4 are at
small air gap h between tyre and ground may have a significant
20,50,80 or 110° from O on the tyre. The receiver M is at
effect. Figure 10 compares the horn amplification for h=0, 0.1,
(2,0,1). _____ MICADO, o
0.5, 1 and 3 mm computed with MICADO in 2D.
------- o ICARE
3.2 Horn amplification in 2D
In order to gain confidence with the BEM computations made
with MICADO, comparison of horn amplification calculations in
2D is made against results found in the literature. In [4], a
commercial BEM software (SYSNOISE [48]) has been used as a
reference for 2D computations made by a multipole synthesis
approach [4] (hereafter named MSA). The MSA approach
consists in replacing the tyre resting on the ground by two
multipoles on either side of the surface representing the road.
One multipole is situated inside the tyre contour. The second
multipole is situated symmetrically below the road surface. The
Figure 10. Effect of space h between tyre (ℜ=31 cm) and
ground. Source placed at (d,0,0) and receiver at (1,0). ----- h =
1 mm;
0; *-–––* h=0.1 mm; o-–––o h=0.5 mm; -–––
∆-–––∆ 3 mm
key point is that both multipoles together have to fulfil the
boundary conditions given on both the tyre surface (prescribed
Even between situations with direct contact (h=0) or with a
velocity) and on the road (given impedance). In Figure 9,
slight uplift (h=1 mm), different amplification values can be
SYSNOISE and MSA results are compared with MICADO, for a
obtained, the closer the source to the tyre, the stronger the
tyre with radius ℜ=31 cm. Two sources are placed at (d, 0) for
influence of the shift h. This effect emphasizes the importance of
d = 4 or 8 cm, and the receiver M is placed at (1, 0). The
local details close to contact where the actual contact patch will
quantity represented is the horn amplification which is defined
be unsmooth and where longitudinal or transversal grooves will
as the increase of noise level when a tyre is added over an
create uneven contact. Such effects are not included in the 3D
existing road surface. A very good agreement can be seen
computations latter reported.
between MSA and MICADO whereas SYSNOISE results
slightly overestimate the horn amplification.
3.3 Horn amplification in 3D
Comparisons of 3D BEM computations with MICADO have
also been done against results obtained by R. GRAF [5]
measured and computed using 3D BEM. The case considered is
that of an unloaded cylindrical tyre with a radius of ℜ=32 cm
and a width w=20 cm. Both tyre and ground are assumed to be
rigid. A source is placed on the ground close to contact which is
a line contact (point contact in 2D) at position (d, 0, 0). The
receiver is placed at (3, 0, 1). Figure 11 shows the horn
amplification for two values of d and good agreement between
Figure 9. Horn effect on a rigid cylinder with radius ℜ=31 cm.
Source and receiver lay on the rigid ground. The source is either
4 cm or 8 cm away from the lower part of the cylinder (contact
point). The receiver is 1 m away. Three computations: -–––
MSA; *-–––* SYSNOISE (BEM); o-–––o MICADO (BEM)
MICADO and GRAF’s results.
It must be noted that the calculations were all made for a tyre
lifted by one millimetre above ground since computations with
MSA were facilitated by using a slightly lifted tyre. Using BEM
for a tyre directly resting on the ground must also be handled
with care by using two pressure values on both sides of the
contact point and an increased number of Gauss points in the
numerical integration scheme. The slight disagreement between
MSA and SYSNOISE probably comes from the use of an
insufficient number of Gauss points close to contact in the
SYSNOISE model. The present BEM computations with
Figure 11. 3D Horn amplification. Validation of BEM computation
against [5]. Rigid cylindrical tyre with radius ℜ=32 cm and width
w= 20 cm. Source on the ground at distance d from line contact.
Receiver at (3, 0, 1). . ■ measured (Graf), ---- 3D BEM(Graf);
___ 3D BEM (MICADO).
MICADO employ a variable number of Gauss points depending
on the singularity. As already showed in [4] the presence of a
6
Acta acustica – ACUSTICA
Vol. 00 (2007) 1-13
____________________________________________________________________________________________________________
3.4 Integrated results (GRIM computations)
3.5 The skirt technique
The computation of sound pressure levels radiated from a known
An alternative to the previous full geometrical computations has
tyre velocity profile, by means of equation (3), has first been
been found by extending the use of the GRIM approach [49].
tested in 2.5D. We assume a uniform unit velocity on the belt so
Figure 13 represents, in 2D, what has been called the ‘skirt
that acoustic pressures computed using either BEM (MICADO)
approach’. The tyre belt S is split into a lower portion SL facing
or ICARE estimates of GV can be compared. A baffled tyre
the receiver and a complementary upper surface SU. Since
having a radius of 32 cm and a width w=22 cm (see Figure 7) is
equation (3) is valid for any limiting surface S, it can be applied
considered. The tyre has point contact with the ground. The
on S*=SS ∪ SU rather than on S=SL∪ SU where SL is replaced by
computation of the Green functions is made for sources and
a skirt-like surface SS which surrounds the lower portion of the
receivers in the same y plane so that only one source point is
tyre (see Figure 13) and is defined by its circumferential width
taken across the tyre width. Figure 12 compares the sound
2α. In order to apply equation (4) on S*, the acoustical velocity
pressure levels at M (2,0,1) for both computations of GV. The left
must be estimated on SS. The whole process can therefore be
graph assumes unit velocity on the whole circumference
separated into three steps:
showing poor agreement between both computations. In the right
graph, however, a zero velocity is assumed between 0 and 12.5
a)
A single tyre model is made using BEM. The tyre is
degrees –i.e. for a circumferential portion close to the point
supposed to have zero velocity on its upper part SU.
contact. Good agreement can then be observed.
Therefore, only the contribution from the lower part (SL)
is considered in this first stage. The GRIM approach
(equation 4) is employed to compute the acoustical
velocity at chosen positions M(x,y,z) on the skirt SS–the
fictitious surface surrounding SL. The whole tyre is
meshed but complementary boundaries such as the car
body are not included in this computation. Therefore the
resultant velocity is an approximation.
b)
In a second stage, the actual full situation (one tyre, four
tyres, four tyres+car body,… etc..) is modified by adding
a rigid skirt SS which entirely surrounds the lower part of
the tyre. The knowledge of V on the skirt SS and of the
original velocity on the upper part of the tyre (SU) assures
that this problem is similar to the original one [8]. The
Figure 12. Unloaded tyre with radius 32 cm. 2.5D GRIM
integration: on total periphery (left graph) or omitting the lower
12.5 degrees (right graph). Unit uniform radial velocity.
Receiver at (2,0,1). ____ GV(BEM-MICADO);
- - - - GV(ICARE)
Green function G*(M,Q) between the receiver M and the
new boundary S*=SS+SU can be computed using beam
tracing with the major advantage that the horn zone is
now replaced by a vertical boundary more suitable to
geometric acoustics.
The BEM computation being very precise, the discrepancy
between both computations in the left graph of Figure 12 has
c)
The GRIM integration is carried out on S*. The resultant
been attributed to a poor estimation of the Green functions
pressure is a close approximation of the real value as will
between the receiver at M and points Q on the tyre close to
be showed.
contact (see Figure 8). The horn-shaped geometry in the case of
point contact is indeed not well suited for beam tracing
If the velocity on SL can be assumed to be slow varying (it can be
computations since as Q gets closer to the ground the number of
approximated by a mean value, say) pre-computed velocity
reflected rays increases rapidly and numerical convergence is
transfer functions H(SL,SS) can be computed independently at a
very difficult to obtain. Note, that omitting only 12.5 degrees,
separate pre-processing stage, only once for a given tyre and
changes significantly the integrated results; this confirms the
ground configuration. Different velocity profiles can latter be
paramount role of the horn zone in the radiation process. This
considered by simply combining the values of V on SL with the
effect would even be more important for real velocity fields
pre-computed H’s.
which have relatively higher amplitudes close to contact.
The skirt-technique has been applied both in 2D and in 3D.
Last, we mention that R. GRAF [5] proposes an analytical
algorithm for geometrical computations in 2D for a straight
wedge. But this solution can not easily be adapted to 3D
problems.
7
Acta acustica – ACUSTICA
Vol. 00 (2007) 1-13
____________________________________________________________________________________________________________
of a cosine velocity distribution (right graph) is also considered.
This more realistic cosine profile corresponds to a tyre with a
unit velocity at contact and a null velocity on its upper part. In
both cases the lower part gives the major contribution to the
overall level, even more so for the cosine profile (similar
observations have been made with the computed velocity profile
of Figure 23).
Figure 13. Geometry of the tyre with skirt. The skirt SS is a
fictitious boundary enveloping the lower part of the tyre within
an angular aperture 2α. The 3 stage process consists in 1)
computing the actual velocity V on SS, 2) computing the Green
functions G* between the receiver at M and SS assumed rigid,
3) making the GRIM integration on SS ∪ SU
instead of SL ∪ SU.
3.5.1 The skirt technique in 2D
Figure 15. Tyre with a unit or cosine velocity distribution on the
belt. Separate contributions to the total sound pressure level at M
(2,1). 2D computation. *---* lower part of the belt (SL);
o
o upper part of the belt (SU);
total
Figure 14 reports a 2D application where the SS surface (skirt) is
simply a straight vertical line of length H=7.3 mm. Again, a unit
velocity is assumed on the whole belt. Two skirt computations
are presented either using one or three integration points on SS.
3.5.2 The skirt technique in 3D
The receiver is placed at (2, 1).The reference curves in full line
correspond to a reference GRIM computation where the Green
Figure 16 shows results in case of a 3D unloaded tyre of
function is entirely computed by means of BEM in order to have
cylindrical shape (radius ℜ=31 cm and width w=22 cm) lifted
a precise reference curve. The dashed line is obtained by means
1 mm above ground. A unit velocity is assumed on the lower
of the skirt technique; as previously explained: BEM is used to
part of the belt along twice 20 degrees (surface SL ) and zero
compute the normal velocity V on the fictitious skirt; the final
elsewhere. The sound pressure level obtained by applying the
GRIM integration (3) is applied on S* with G* functions -
GRIM integration on SL is compared with the integration on a
between the receiver and the now rigid skirt- computed by
skirt (SS) which envelops the lower part of the tyre as shown in
means of the beam tracing technique. Very good agreement is
Figure 13. In this second computation the acoustic velocity on
obtained even with only one integration point. Using three points
the skirt is computed by means of 3D BEM (3D version of
improves the results only above 3500 Hz which corresponds to a
MICADO). A good agreement, within 3 dB, can be found
meshing requirement on SS of more than one point every twelve
between both computations. One could argue that since
wavelengths.
integration on both surfaces give the similar results, then why
use the skirt technique? The answer lies in the fact that the skirt
technique can be employed for more complex geometries such as
tyres with complex tread profile where the direct use of beam
tracing in small air gaps around contact would not be precise
enough. The decomposition of calculations into precise but
expensive computation (BEM or other) close to contact and
faster beam tracing further away can be seen as an efficient
hybrid approach suited for complex problems where ground, tyre
and car geometries must be considered.
Figure 14. Unloaded 32 cm tyre. 2D GRIM computations.
Receiver at M (2,1). Left graph: one integration point; right
graph: 3 integration points on the vertical skirt SS.; ____
integration on the full tyre; … integration on tyre + skirt surface
.
Figure 15 shows the separate contributions from lower and upper
parts of the tyre to the total sound pressure level. In addition to
the previous case of uniform unit velocity (left graph), the case
8
Acta acustica – ACUSTICA
Vol. 00 (2007) 1-13
____________________________________________________________________________________________________________
Figure 16. Skirt approach. 3D case. Simple cylindrical tyre of
radius 31 cm and width 22 cm lifted 1 mm above ground. Skirt
placed at α=20 degrees from centre line. Unit velocity on the
lower part of the belt (surface SL,), null elsewhere. Pressure level
at position (2,0,1) – 2 m away and 1 m above ground. Solid line
for exact integration on the whole tyre, dotted line for results
with the skirt technique.
Figure 17. Loaded tyre with radius 32 cm. 2.5D GRIM
integration on total periphery. Unit velocity on the belt. ____
G(BEM), - - - - G(ICARE). Receiver at (2,0,1).
Figure 18 compares the radiated sound pressure levels for
unloaded and loaded 32 cm-tyres (contact along 2α=32 degrees)
3.6 Surface contact
at two different receiver locations, showing that a significant
reduction of tyre noise is achieved for the loaded tyre.
In Figure 12, the direct application of the GRIM technique has
been made for an unloaded tyre which results in point contact in
the 2D plane. Many results in the literature are related to such an
academic contacts. With regard to beam tracing, point contact
has the drawback of leading to sharp horn regions, whereas in
reality a tyre is loaded by the car and the contact between tyre
and ground occurs on a surface patch. The case of a tyre of
radius 32 cm with flat contact on 2α = 32 degrees is considered.
Figure 17, corresponds to the left graph of Figure 12 for this
more realistic situation, where a unit tyre velocity is assumed on
the whole belt. Contrary to the case of point contact, the direct
use of equation (3) with a beam tracing computation of G on the
whole belt, shows a very good agreement with the reference
computations made with BEM. The difference in a third-octave
Figure 18. Effect of loaded tyre shape: ____unloaded (point
contact in 2D) ,- - - - loaded (flat line contact in 2D). Sound
pressure level at two receiver positions: (2,0,1) left graph,
(7, 0.1) right graph.
band representation would show differences mostly smaller to 3
dB. As already mentioned, the reduced sharpness of the horn
region is more favourable for beam tracing when the tyre is
3.7 Comparison of 2.5D and 3D GRIM computations.
loaded and the skirt-technique is not required if perfect contact
can be assumed. Nevertheless, this technique will be required for
Last, before studying more complex situations, a comparison of
unsmooth tyres with small-scale tread profile not suitable for
2.5D and 3D GRIM computations has been made. The 2.5D
beam tracing computation near contact. The integration of
radiation corresponds to a baffled tyre. The 2.5D Green function
complex effects such as “air-pumping” can also be considered,
is obtained in two steps. First, a 2D-BEM computation made for
by using the previous skirt approach and by separating the air
a rigid geometry (ground and tyre), provides a 2D solution
domain into several regions where different computation
which is integrated in a Fourier-like manner [43] to provide the
techniques can be employed [22].
elementary solution for an infinite cylinder -but point-like source
and receiver pairs- suitable for integration on a baffled strip-like
surface (2.5 D).
The case of a cylindrical tyre of radius 32 cm and width 22 cm is
considered, either in 3D or baffled. The tyre is lifted by 1 mm
and a unit velocity is applied on the whole belt. Figure 19
compares the Sound Pressure Levels computed by equation (3)
both for 2.5D and 3D BEM estimated Green functions. The
receiver at (2, 0, 1) is in the rolling plane of symmetry (y=0),
2 m away from the centre of the tyre (x=2) and 1 m above
ground (z=1). It must be noted that the 2.5D computation would
be the optimal result obtainable using Green functions computed
by means of beam tracing without diffraction on the “corner”
between belt and side walls. 2.5D and 3D results show the same
trends, but local differences can be seen. As the receiver point
9
Acta acustica – ACUSTICA
Vol. 00 (2007) 1-13
____________________________________________________________________________________________________________
moves away from the plane of symmetry the quality of
agreement has been found to degrade significantly. This
emphasizes the necessity of carrying 3D computations since, in
practice, receiver points are not in the rolling plane. Angular
dependency for 3D unloaded tyres can be found in [5]. The
authors want to stress that the main objective of 2.5D
computations was for validation and parametric analysis. In
chapter 4, real 3D situations are considered.
Figure 20. Tyre + chassis. Green functions G(Q,M). Q1 at
α1=23°, Q1 at α2=30.3° , M at (1.1,0.2) ___BEM, …ICARE.
Figure 19. GRIM generated sound pressure levels for a unit
uniform velocity on the belt of a cylindrical tyre, at position
(2,0,1). Comparison of results obtained using 3D (_____) and
2.5D (- - - -) Green functions.
Figure 21 shows, for the same configuration, the comparison of
the sound pressure level at M, obtained after using the GRIM
integration of equation (3), where the Green functions are
computed either with MICADO (BEM) or ICARE (beam
tracing) as shown in Figure 20. A unit uniform velocity on the
4. Influence of the car body
tyre is again considered. Good agreement between both
computations can be observed and consequently more complex
4.1 Tyre and plate configuration in 2.5 D
3D situations can be considered as shown in the next section.
Adding the car loading on the tyre has been seen to significantly
reduce the noise radiated since the horn region was less sharp. In
the same manner the study of an isolated tyre can be seen as
quite academic. In practice, the proximity of the car full
geometry is far from simple since multiple reflections will occur
between the ground, the chassis and the four wheels. Also, the
tyre casings will have a confining action. An intermediate step
towards the full problem is first considered in 2.5D. A loaded
(α=16 degrees) and baffled tyre (ℜ=32 cm) is considered.
Again, a uniform unit velocity is assumed on the belt. A 1.5 mlong, and 1-cm thick plate is placed between points (0.35,0.31)
and (1.85,0.31) (see Figure 20). The plate is added in order to
Figure 21. Tyre + reflecting surface. Validation of the GRIM
approach with ICARE. SPL at M at (1.1,0.2) ___ BEM,
…ICARE.
represent the chassis Since the tyre is loaded, ICARE is
employed for the computation of the Green functions without the
skirt technique. Figure 20 compares the Green functions
4.2 Simplified car
computed by BEM and beam tracing at M (1.1 0.2) for two
source points Q1 and Q2 on the tyre, respectively at α1= 16+10 °
A very simplified car is considered. Figure 22 shows the
and α2=16+17.3°. Good agreement can be seen. The peak at
geometry of the car. The four tyres have dimensions ℜ=32 cm
2800 Hz in the BEM result is not recovered with ICARE which
and w=20 cm. The chassis is 0.31 m above ground. The tyre
has been used without diffraction.
rectangular casings have dimensions (0.70 m x 0.25m x 0.32 m).
The separation between the front and rear axles is 2.2 m. The
overall horizontal dimensions are Lx=3.3 m and Ly=1.4 m. The
vertical dimensions above the chassis have little effect on the
results. The purpose of this situation is to show the influence of a
car body on the radiated sound pressure levels. The car loading
creates a 2α=26 degrees contact patch. Computation of the
Green functions has been made with ICARE. All lower edges
including the edges around the tyres have been included as
diffracting edges.
10
Acta acustica – ACUSTICA
Vol. 00 (2007) 1-13
____________________________________________________________________________________________________________
impedances -with a porosity of 20 %, a flow resistance of 5000
Ns/m4 and a tortuosity of 5.
Figures 24 (rigid ground) and 25 (asphalt ground) show the
effect of the car body as an insertion loss Hcar=SPL(car) –
SPL(four tyres). Each Figure has 3 graphs corresponding to a
constant velocity, a cosine velocity and the velocity distribution
obtained from Kropp’s model. Each graph shows three plots
corresponding to the 3 points represented as M1, M2, M3 in
Figure 22, respectively at angles 30, 60 and 90 degrees from the
front of the car. A third-octave band representation has been
used in order to smooth the strong frequency fluctuations.
Figure 22. Beam tracing with ICARE. Rays between points on the
left front tyre and a receiver M1, 7.5 m away and 1.2 m above
ground are displayed.
Three different velocity profiles on the belt have been
considered. The first profile is a constant velocity. The second
profile is a |cos(θ/2)| velocity distribution around the tyre, giving
a maximal value close to the ground and a zero velocity at the
top of the tyre. The third velocity distribution has been obtained
by means of Kropp’s tyre model [11,14]. This well known model
is based on a “two layer plate on springs” description of the tyre
and provides the velocity on the belt and on the side walls as a
function of frequency. Figure 23 represents the variation of the
Figure 24. Effect of car body for three different velocity profiles
on tyre belts. Rigid ground.*-–––* M1; o-–––o M2; -–––
M3
normal velocity V here employed. V is averaged over the tyre
width, so that it only depends on frequency and angular position.
These computations have been made for an NCT5 Goodyear tyre
placed on an ISO road [45]. One can see that the velocity is not
symmetric between leading and trailing edges.
Figure 25. Effect of car body for three different velocity profiles
on tyre belts. Asphalt surface.
M3
*-–––* M1; o-–––o M2; -–––
Figure 23. Velocity computed by Kropp’s plate model. RMS
average over the tyre width.
The presence of the car body tends to increase or decrease the
noise levels by several dB down to -7 dB and up to +15 dB. This
is mainly due to the multiple reflections that can be seen in
For each profile the velocity is assumed to be identical on the
Figure 22, principally between car and road. Reflections between
four tyres. The velocity on the side walls has been omitted. The
tyre and casing have been found to be less important particularly
GRIM approach is employed using ICARE for the computation
when realistic velocity profiles -with maximum levels close to
of the Green functions. The contribution from the four tyres is
tyre-ground contact- are considered. The increase or decrease of
added incoherently.
noise level around the car is strongly affected by the position of
the receiver, the type of road and by the type of velocity profile
Receivers are placed on a horizontal circle 7.5 m away from the
on the tyre. The use of velocities computed by Kropp’s model
medium point between the four tyres, 1.2 m above ground. For
show values of Hcar of positive amplitudes at all frequencies
each velocity profile, two cases have been considered: (i) four
while the constant and cosine velocity show the strongest
wheels without car body and (ii) four wheels with the car body.
amplifications around 1200-1600 Hz. These results indicate that
The car is assumed to be perfectly rigid. Two types of ground are
the modelling of noise radiation from tyres must include the car
considered, either rigid or made of 4 cm of asphalt over a rigid
body.
underlayer where Hamet’s model [44] is used to compute surface
11
Acta acustica – ACUSTICA
Vol. 00 (2007) 1-13
____________________________________________________________________________________________________________
Effect of Road absorption
medium including the car body. A combination of precise BEM
computations close to contact and ray tracing by means of the
One would expect the type of road to influence the results in a
ICARE software has been presented.
significant manner when the car body is considered since
Other physical aspects should also be considered. Comparisons
multiple reflections between car and road will be strongly
with measured sound pressure levels have shown that high
attenuated when absorption effects come into play.
frequency phenomena such as air-pumping should also be
Figure 26 shows the insertion loss for ground effects, defined as
considered. Many local small-scale phenomena should therefore
the sound pressure level for an asphalt road minus the sound
be added to the present model. The proposed so-called “skirt-
pressure level for a rigid road. The solid line delimits regions
approach” here presented can be used for detailed modelling
with positive and negative values. The left graph corresponds to
close to contact in order to introduce tyre and road unsmooth
the academic cosine velocity profile whereas the right graph has
profiles [22]. Another aspect currently studied at CSTB is the
been obtained with the velocity represented in Figure 23. Both
wave transfer through the ground [23] which is also important
graphs show noise reduction –up to 10 dB- mainly around
since a few centimetres-thick layer of asphalt will act as a wave
1000 Hz and 2500 Hz which correspond to the maximum
guide and convey tyre vibrations and add to global noise
absorption obtained by Hamet’s model. The use of computed
radiation at distant locations. These complementary aspects have
velocities leads to non symmetric results since, as already
been studied in the sequel European project ITARI (2004-2007)
pointed out, the velocity profile is asymmetric. Also the angular
[51] and will be the object of further publications.
distribution of maximum absorption is different according to the
One could conclude by insisting on the complexity of tyre road
velocity profile employed. To summarize these results, it can be
interaction and its associated radiation of sound. This paper does
said that the estimation of noise radiated by tyres is affected by
not intend to present a full model but rather to introduce some
many interacting parameters and that the study of isolated tyres
innovative aspects such as combining tyre surface velocities and
only provides part of the information.
a mixed propagation scheme based on both BEM and beam
tracing which is in our opinion an efficient way of introducing
the whole car plus tyres problem. Finally, one should not forget
that urban noise is a combination of cars and buildings. Ongoing
research is currently being made towards this global goal by
adding building facades beyond the car [47].
Acknowledgement
The authors are in great debt to Europe funding through the
Figure 26. Effect of type of road. SPL(asphalt)-SPL(rigid ground).
The black contour line delimits the regions of positive and negative
values. Left graph: cosine velocity distribution, right graph:
velocities computed by Kropp’s plate model (only up to 2800 Hz).
European RATIN project which has offered a means to several
European research Centres to develop their ideas in a joined
effort and to meet fellow researchers in the field of tyre noise.
We also thank W. Kropp and his colleagues for providing the
velocity profile employed in Figure 23.
5. Conclusion
Tyre noise computation is often made for single unloaded tyres.
It has been confirmed that the true geometry of the tyre is
compulsory, meaning that the angle formed between tyre and
road has a strong influence on radiated sound pressure levels.
The consideration of full problems including the whole car body
and the four tyres can best be obtained by mixing approaches.
First, we assume that the tyre velocity on the belt is known
independently of its surroundings. The present work has been
developed within the European research project RATIN
[45,46,50] -running between 2000 and 2003- where velocities
were computed using different tyre models [13-16] for single
tyres. Next, noise radiation is carried out using an integral
formalism named GRIM which is based on a simple integration
on the tyre of the product GV where V is the tyre velocity and G
is the Green function between tyres and receivers. The Green
References
[1] U. Sandberg, J. A. Ejsmont: Tyre/road noise reference book.
Published by Informex, Printed by MODENA, Poland (2002)
[2] P. Andersson: High frequency tyre vibrations. Report F 0204. University of Chalmers, Gothenburg (2002)
[3] P. Andersson: Modelling interfacial details in tyre/road
contact- Adhesion forces and non-linear contact stiffness. Ph. D
thesis, University of Chalmers, Gothenburg (2005)
[4] W. Kropp, F.-X. Becot, S. Barrelet: On the sound radiation
by tyres. Acustica 86 (2000) 769-779.
[5] R.A.G. Graf, C.-Y. Kuo, A. P. Dowling, W. R. Graham: On
the horn effect of a tyre/road interface, part. I: experimental and
computation. Journal of Sound and Vibration. 256 (2002) 417431.
functions will integrate the complexity of the surrounding
12
Acta acustica – ACUSTICA
Vol. 00 (2007) 1-13
____________________________________________________________________________________________________________
[6] P. Jean, J.-F. Rondeau, F. Gaudaire : Emploi d’une méthode
[25] D. Van Maercke, J. Martin: The prediction of echograms
mixte
and impulse responses within the Epidaure software. Applied
pour
le
calcul
du
rayonnement
acoustique
des
pneumatiques. Proceedings of Confort automobile et ferroviaire.
Acoustics 38 (1993) 93-114.
Le Mans 2002.
[26] M. Gensane and F. Santon: Prediction of sound fields in
[7] P. Jean: Coupling integral and geometrical representations
rooms of arbitrary shape: validity of the image sources method.
for vibro-acoustical problems. Journal of Sound and Vibration
Journal of Sound and Vibration 63 (1979) 97-108
224 (1999) 475-487.
[27] L. T. Santos, M. Tygel: Impedance-type approximations of
[8] P. Jean: Coupling geometrical and integral methods for
the P-P elastic reflection coefficient : modeling and AVO
indoor and outdoor sound propagation - validation examples.
inversion. Geophysics 69 (2003) 592-598.
Acta Acustica 87 (2001) 236-246.
[28] T. Lewers: A combined beam tracing and radiant exchange
[9] P. Jean, F. Gaudaire, J. Martin: Applications de la méthode
computer model of room acoustics. Applied Acoustics 38 (1993)
GRIM (Green Ray Integral Method). Proceedings of Confort
161-178.
automobile et ferroviaire. Le Mans 15-16 Nov. 2000.
[29] I. A. Drumm, Y. W. Lam: The adaptive beam-tracing
[10] P. Jean, J. Roland: Application of the Green Ray Integral
algorithm. Journal of the Acoustical Society of America 107
Method (GRIM) to sound transmission problems. Building
(2000) 1405-1412.
Acoustics 8, (2001)139-156.
[30] N. Campo, P. Rissone, M. Toderi: Adaptive pyramid
[11] W. Kropp: Ein modell zur Beschreibung des Rollgeräusches
tracing: a new technique for room acoustics. Applied Acoustics
eines unprofilierten Gürtelreifens auf rauher Strassenoberfläche.
61, (2000) 199-221.
PhD Thesis, VDI-Fortschrittberichte Reihe 11, Vol. 166, VDI
[31] T. Funkhouser, N. Tsingos, I. Carlbom, G. Elko, M. Sondhi,
Verlag, Düsseldorf, Germany.1992.
James, E. West, G. Pingali, Patrick Min, A. Ngan: A beam
[12] A. Fadavi, D. Duhamel D., H. Yin: Tyre/road noise : Finite
tracing method for interactive architectural acoustics. Journal of
element modelling of tyre vibrations. Proceedings of Internoise,
the Acoustical Society of America 115 (2004) 739-756.
La Haye, August 26-31, 2001.
[32] J. Keller: Geometrical theory of diffraction. Journal of the
[13] A. Pietrzyk: Prediction of the dynamic response of a tire.
Optical Society of America 52 (1962) 116-130
Proceedings of Internoise, Nice, 2001.
[33] R. B. Kouyoumjian, P.H. Pathak: A Uniform Geometrical
[14] W. Kropp, K. Larsson, F. Wullens, P. Andersson, X.-F.
Theory of Diffraction for an Edge in a Perfectly Conducting
Becot: The modelling of tyre/road noise - a quasi three-
Surface. Proceedings of the IEEE, 62, (1974) 1448-1461.
dimensional model. Proceedings of Internoise, The Hague. 2001.
[34] T. Kawai: Sound diffraction by a many sided barrier or
[15] S. Finnveden: Tyre vibration analysis with conical
pillar. Journal of Sound and Vibration. 79, (1981) 229-242.
waveguide finite elements. Proceedings of Internoise 2002,
[35] H. Medwin: Shadowing by finite noise barriers, Journal of
Dearborn US.
the Acoustical Society of America 69 (1981) 1060-1064.
[16] R. J. Pinnington: A wave model of a circular tyre.
[36] H. Medwin: Emily Childs, Gary M. Jebsen: Impulse studies
Proceedings of Forum Acusticum Sevilla. 2002
of double diffraction: A discrete Huygens interpretation. Journal
[17] J.-L. Migeot, J.-P. Coyette, T. Leclercq, J.-D. Thiébaut,
of the Acoustical Society of America 72 (1982) 1005-1013.
L. Hazard, W. Gnörich, A. Ossipov: Acoustic radiation from
[37] J. Vanderkooy: A simple theory of cabinet edge diffraction.
tyres using finite and infinite elements: coupled and uncoupled
Journal of the Auditory Engineering Society 39 (1991) 923-933.
approaches. Proceedings of Euronoise, Naples, 2003.
[38] U. Stephenson: Quantized pyramidal beam tracing - a new
[18] ICARE software. Users Manuel. CSTB France. 2000-2006.
[19] J. F. Hamet, C. DEFRAYET, C. PALLAS: Air pumping
algorithm for room acoustics and noise emission prognosis:
Acustica united with Acta Acustica 82 (1996) 517-525.
phenomena in road cavities. Proceedings of the International
[39] R. R. Torres, U. P. Svensson, M. Kleiner: Computation of
Tire/Road Noise Conference, Gothenburg, 8-10 August, 1990.
edge diffraction for more accurate room acoustics auralization.
[20] R. E. Hayden: Roadside noise from the interaction of a
Journal of the Acoustical Society of America 109 (2001) 600-
rolling tire with the road surface. Proceedings of the Purdue
610.
Noise Control Conference, Purdue University, 1971.
[40] F. Antonacci, M. Foco, A. Sarti, S. Tubaro: Fast modeling
[21] M. J. Gagen: Novel acoustic sources from squeezed cavities
of acoustic reflections and diffraction in complex environments
in car tires. Journal of the Acoustical Society of America 106
using visibility diagrams. Proceedings of Eusipco 2004 1773-
(1999) 794-801.
1776.
[22] F. Conte, P. Jean: CFD modelling of air compression and
[41] G. Rougeron, F. Gaudaire, Y. Gabillet, K. Bouatouch:
release in road cavities during tyre/road interaction. Proceedings
Simulation
of Euronoise, Tampere, Finland, 2006.
electromagnetic
[23] P. Jean: A multi domain BEM approach. Application to
algorithm, Computer & Graphics 26,(2002) 125-141.
outdoor sound propagation and tyre noise. Proceedings of
[42] P. Jean: A variational approach for the study of outdoor
Euroise, Tampere, Finland, 2006.
sound propagation and application to railway noise. Journal of
[24] F. Gaudaire, N. Noe, J. Martin, P. Jean, D.Van
sound and vibration 212 (1998) 275-294.
Maercke: Une méthode de tirs de rayon pour caractériser la
[43] P. Jean, G. Defrance, Y. Gabillet: The importance of source
propagation sonore dans les volumes complexes. Proceedings of
type on the assessment of noise barriers. Journal of Sound and
Confort automobile et ferroviaire, Le Mans, 2000.
Vibration 226 (1996) 201-216.
of
the
indoor
wave
with
propagation
a
of
a
time-dependence
60
GHz
radiosity
13
Acta acustica – ACUSTICA
Vol. 00 (2007) 1-13
____________________________________________________________________________________________________________
[44] J.-F. Hamet, M. Berengier: Acoustical characteristics of
porous
pavements:
a
new
phenomenological
approach.
Internoise, Leuven ,1993.
[45] RATIN . Final Technical report. EEC contract GRD1-9910583. January 2004.
[46] P. Jean, J.F. Rondeau, F. Gaudaire: Calculation of tyre noise
with a mixed approach. Proceedings of Forum Acusticum,
Sevilla, 2002.
[47]
SONVERT
project.
Financed
by
« Agence
De
l’Environnement et de la Maîtrise de l’Energie » (ADEME).
2005-2007.
[48] SYSNOISE 5.4. Users Manual, LMS International, Leuven,
Belgium. (1993-1999).
[49] P. Jean and J.-F. Rondeau A model for the calculation of
noise transmission inside dwellings. Application to aircraft
noise. Applied Acoustics 65 (2004) 861-882
[50] P. Jean, F. Gaudaire, N. Noe An hybrid method for the
calculation of sound radiation from tyres: an integral approach
combined with a ray tracing computation of Green function.
EEC project RATIN, Final report from CSTB. January 2004.
[51] ITARI: Integrated Tyre And Road Interaction. Project no.
FP6-PL-0506437. Activity report – period 2. March 2006.
14