Calculation of tyre noise radiation with a mixed approach

Transcription

1 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, Jean@cstb.fr 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 js 1. Introduction The prediction of tyre noise has recently gained much attention and many research projects have been dedicated to this subject. The reasons for such a focus are well known. Firstly, traffic noise is a well established and increasing nuisance. Secondly, the reduction of other noise sources in cars such as engine noise has led to the present concern for tyre noise which is acknowledged to be dominant above 50 km/h. An important literature both experimental and numerical [1,2] exists on this subject. Several noise mechanisms have been observed and addressed. The basic mechanisms are well described in [2]. The main problem involves tyre interaction with the ground [3] which will lead to the vibration of the tyre and its radiation into the surrounding medium. Geometrical aspects will then come into play: the shape of the tyre/road profile often called horn enhancement will strongly amplify the noise radiated [4,5]. Also important [6] but less studied, is the presence of the car body. Including the car into numerical schemes is often beyond the possibilities of the numerical approaches employed. In the present paper an attempt in this direction has been made by employing a mixed integral/ ray approach called GRIM (Green Ray Integral Approach) [7-10]. It relies on a Rayleigh-integral which combines a known surface velocity of the tyre and a particular Green function between tyre surface and receiver. The velocity can be estimated using many available tyre models. A Finite Element description can be employed [12,13] but FEM is usually time consuming and limited to low frequencies, the high frequency limit depending on the mesh refinement. A complementary well-used model is that of W. Kropp [11,14] which relies on a two-layer plate on springs description and neglects the curvature of the tyre. A good compromise to the previous approaches is the spectral finite element method proposed by S. Finnveden [15] which employs a finite element description along the tyre width and a wave description along the Received 5 May 2005 Accepted ** *** **** circumference. One should also mention analytical descriptions such as the beam-like tyre model proposed by R. Pennington [16] which provides physical insight at very low computational cost. The second component in the GRIM approach is the Green function between the blocked tyre surface and the receiver. This acoustical term of the GRIM integral can be computed by any means such as BEM [5], infinite elements [17] or geometrical approach such as beam-tracing [18]. The BEM approach is very precise and apt to model detailed small-scale geometries close to contact but will become time-consuming when dealing with extended geometries such as the whole car body. On the other hand, geometrical models can handle large geometries with reduced cost but are limited to wavelengths smaller than geometrical dimensions. A combination of both BEM and beam tracing is proposed in this article where BEM is employed close to the tyre up to a fictitious surface, called a skirt, positioned close to contact. Beyond the skirt beam tracing is employed. This mixed BEM-beam tracing will be necessary if one wants to introduce detailed tyre shapes. Other mechanisms such as small scale geometrical effects involving the small cavities in the road often called air pumping- or Helmholtz and quarter-wavelength effects in the grooves are not considered in the work here reported. It is nevertheless the belief of the authors that these aspects, not fully understood at present and also little reported in the literature [20-21] deserve full attention. Work carried out at CSTB on this subject is the object of separate publications [22]. Finally, the road itself also acts in several ways. First, its outer surface acts in the building up of contact forces [3] acting on the tyre, but also as part of air-pumping mechanisms. Its inner structure may also play an important role. In the present work, the ground is either rigid or locally reacting whereas, in reality, it acts as a propagating medium. The part of the tyre in contact with the ground vibrates and generates waves which will travel, reradiate in the air and thus contribute to the overall perceived noise. Also, a wave reflection on the ground surface is not exactly a local action since waves enter the ground and reradiate 1

2 further away. This aspect is being studied at CSTB and will also be the object of separate publications [23]. This paper is structured as follows: in part 2, the GRIM method is recalled; in part 3 numerical validations are reported; in part 4, car effects are simulated. 2 ( + k ) GV ( M, Q) = δ Q M, Q Ω GV ( M, Q) = 0 Q S (2) V n GV ( M, Q) = jωρσ GV ( M, Q) Q S A n Sommerfeld conditions 2. A decoupled approach 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 horn effect due to the particular shape formed by a tyre resting 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. 2.1 The GRIM approach The method employed -called GRIM: Green Ray Integral Method [7]- has been previously applied to several vibroacoustical problems [8-10]. Figure 1. The GRIM approach for tyre radiation. S V is a vibrating boundary (vibrating tyre: belt and side walls), S A is an acoustic boundary complementary to S V. Figure 1 shows the general problem to be solved. The time dependence e jωt is implicit. The air domain Ω is bounded by surface S=S V S A, where S V is a vibrating surface having a velocity V and S A 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 G( M, Q) P( M ) = jωρ V ( Q) G( M, Q) P( Q) ds( Q) (1) n S V Q G( M, Q) P( Q) jωρσ ( Q). G( M, Q) + ds( Q) n S A Q where G is the free field Green function between M and any point Q on S. Next, we define a Green function G V solution of This more complex Green function G V is, again, the pressure at M due to a unit source placed at Q, but it now includes all reflections and diffractions on S and it assumes S V to be rigid. Importing G V into (1) leads to a much simpler expression, since the second integral cancels out and the derivative of G V on S V is identically equal to zero, so that P( M ) = jωρ V ( Q) G ( M, Q) ds( Q) S V V 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 S V, leading to a matrix expression to be solved. Fortunately, in most airloading situations the assumption that the velocity can be estimated a priori is possible so that one simply ends up with a simple integration of the product of two known quantities. G V 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 GV ( M, Q) V ( M ) = V ( Q) ds( Q) n SV 2.2 Beam tracing computation with ICARE M 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 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 [25-32]. They are all based upon an analogy between optics and 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 (3) (4) 2

3 these beam tracing algorithms can precisely deal with curved surfaces since they rely on pyramidal beams (where rays converge into a point, i.e the image-source). With curved surfaces, there is no image-source and consequently pyramidal beams can not be used. The ray tracing approach consists in emitting a large number of rays from the source and in following their propagation. It is 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. (B 1 and B 2 ) are followed as they start propagating. Beam B 1 reflects on S 1 and then encounters the wedge S 2. Since part of B 1 strikes the wedge while the upper part goes over its top, B 1 must be slit into two beams, the separation being marked by a dashed line. Beam B 2 encounters a spherical obstacle S 3 and must also be split. S 1 Figure 3. Splitting of sound beams. Beams B 1 and B 2 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 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 adapting to object encountered as the beams travel. If necessary, the beam will be split into sub-beams. This occurs for instance when a beam impacts two different surfaces or an edge. Also, angular widening as the beam travels will necessitate subdividing so that the supporting rays remain compact up to chosen limits (i.e. the interior of the beam defined by its rays can be overestimated with a bounding volume that does not cover all space). Figure 3 gives, for the sake of clarity, a 2D representation of a source emitting several beams, two of which 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. In ICARE, diffraction edges must be chosen by the user. Successive diffractions can be defined. In practice, more than two successive diffractions are not recommended due to a very significant computation time and probable lack of precision. A beam impinging on a diffraction edge will intersect the edge along a diffracting segment D 1 D 2. The union of the two diffracting cones at points D 1 and D 2 will define a diffracting source. This source is then decomposed into source beams, as represented in Figure 5, now defined by four supporting rays which are, as for source beams, propagated, reflected and 3

4 eventually sub divided as they travel and reach boundaries. It must mentioned that the direct use of GTD neglects the finite length of diffracting edges This is coherent with the basic assumption of geometrical acoustics where surfaces are also supposed to be very large compared to acoustic wavelengths. The introduction of corrective terms for the finite dimensions of edges is a subject of on-going research. 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. Ri. Dj (5) i j The Đ 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. 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 is deterministic, since it combines the contribution of a finite series of elementary terms. Convergence of the computed pressure can only be controlled at the acoustic stage of the calculations since it depends on attenuations in air or absorption at the boundaries (the first geometric part of the computations is independent of material and fluid characteristics); the number of 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 S V having a known velocity the surface of a tyre in the present study. The GRIM approach can then be directly used by using reciprocity and shooting beams from the receiver position onto S V. As the propagating beams intersect with S V, within triangular portions Ti of S V, the elementary contributions of integral (3) on Ti s are piled up into P(M). An alternative is to compute separately the values of G V (M,Q) between receivers M and S V at pre-selected positions of Q on S V. 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). 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. 4

5 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. 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. 3.1 Horn amplification in 2.5D: ICARE validation 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 different positions Q on the tyre facing M, computed either with MICADO and ICARE. At Q 1 (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 Q 1 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 Q 3 and Q 4, therefore not influenced by the geometry of the tyre. Differences between the loaded and unloaded cases can still be seen at Q 2 (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

6 O, or flat contact over 32 degrees. Sources Q 1 to Q 4 are at 20,50,80 or 110 from O on the tyre. The receiver M is at (2,0,1). MICADO, o o ICARE small air gap h between tyre and ground may have a significant effect. Figure 10 compares the horn amplification for h=0, 0.1, 0.5, 1 and 3 mm computed with MICADO in 2D. 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 key point is that both multipoles together have to fulfil the boundary conditions given on both the tyre surface (prescribed velocity) and on the road (given impedance). In Figure 9, SYSNOISE and MSA results are compared with MICADO, for a tyre with radius R=31 cm. Two sources are placed at (d, 0) for d = 4 or 8 cm, and the receiver M is placed at (1, 0). The quantity represented is the horn amplification which is defined as the increase of noise level when a tyre is added over an existing road surface. A very good agreement can be seen between MSA and MICADO whereas SYSNOISE results slightly overestimate the horn amplification. 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 = 0; * - * h=0.1 mm; o- o h=0.5 mm; - 1 mm; - 3 mm Even between situations with direct contact (h=0) or with a slight uplift (h=1 mm), different amplification values can be obtained, the closer the source to the tyre, the stronger the influence of the shift h. This effect emphasizes the importance of local details close to contact where the actual contact patch will be unsmooth and where longitudinal or transversal grooves will create uneven contact. Such effects are not included in the 3D computations latter reported. 3.3 Horn amplification in 3D 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) 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 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 MICADO employ a variable number of Gauss points depending on the singularity. As already showed in [4] the presence of a 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), D BEM(Graf); 3D BEM (MICADO). 6

7 3.4 Integrated results (GRIM computations) 3.5 The skirt technique The computation of sound pressure levels radiated from a known tyre velocity profile, by means of equation (3), has first been tested in 2.5D. We assume a uniform unit velocity on the belt so that acoustic pressures computed using either BEM (MICADO) or ICARE estimates of G V can be compared. A baffled tyre having a radius of 32 cm and a width w=22 cm (see Figure 7) is considered. The tyre has point contact with the ground. The computation of the Green functions is made for sources and receivers in the same y plane so that only one source point is taken across the tyre width. Figure 12 compares the sound pressure levels at M (2,0,1) for both computations of G V. The left graph assumes unit velocity on the whole circumference showing poor agreement between both computations. In the right graph, however, a zero velocity is assumed between 0 and 12.5 degrees i.e. for a circumferential portion close to the point contact. Good agreement can then be observed. 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). G V (BEM-MICADO); G V (ICARE) The BEM computation being very precise, the discrepancy between both computations in the left graph of Figure 12 has been attributed to a poor estimation of the Green functions between the receiver at M and points Q on the tyre close to contact (see Figure 8). The horn-shaped geometry in the case of point contact is indeed not well suited for beam tracing computations since as Q gets closer to the ground the number of reflected rays increases rapidly and numerical convergence is very difficult to obtain. Note, that omitting only 12.5 degrees, changes significantly the integrated results; this confirms the paramount role of the horn zone in the radiation process. This effect would even be more important for real velocity fields which have relatively higher amplitudes close to contact. 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. An alternative to the previous full geometrical computations has been found by extending the use of the GRIM approach [49]. Figure 13 represents, in 2D, what has been called the skirt approach. The tyre belt S is split into a lower portion S L facing the receiver and a complementary upper surface S U. Since equation (3) is valid for any limiting surface S, it can be applied on S*=S S S U rather than on S=S L S U where S L is replaced by a skirt-like surface S S which surrounds the lower portion of the tyre (see Figure 13) and is defined by its circumferential width 2α. In order to apply equation (4) on S*, the acoustical velocity must be estimated on S S. The whole process can therefore be separated into three steps: a) A single tyre model is made using BEM. The tyre is supposed to have zero velocity on its upper part S U. Therefore, only the contribution from the lower part (S L ) 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 S S the fictitious surface surrounding S L. 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 S S which entirely surrounds the lower part of the tyre. The knowledge of V on the skirt S S and of the original velocity on the upper part of the tyre (S U ) assures that this problem is similar to the original one [8]. The Green function G*(M,Q) between the receiver M and the new boundary S*=S S +S U 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. c) The GRIM integration is carried out on S*. The resultant pressure is a close approximation of the real value as will be showed. If the velocity on S L can be assumed to be slow varying (it can be approximated by a mean value, say) pre-computed velocity transfer functions H(S L,S S ) can be computed independently at a separate pre-processing stage, only once for a given tyre and ground configuration. Different velocity profiles can latter be considered by simply combining the values of V on S L with the pre-computed H s. The skirt-technique has been applied both in 2D and in 3D. 7

8 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 S S 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 S S, 2) computing the Green functions G* between the receiver at M and S S assumed rigid, 3) making the GRIM integration on S S S U instead of S L S U The skirt technique in 2D Figure 14 reports a 2D application where the S S 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 S S. The receiver is placed at (2, 1).The reference curves in full line correspond to a reference GRIM computation where the Green function is entirely computed by means of BEM in order to have a precise reference curve. The dashed line is obtained by means of the skirt technique; as previously explained: BEM is used to compute the normal velocity V on the fictitious skirt; the final GRIM integration (3) is applied on S* with G* functions - between the receiver and the now rigid skirt- computed by means of the beam tracing technique. Very good agreement is obtained even with only one integration point. Using three points improves the results only above 3500 Hz which corresponds to a meshing requirement on S S of more than one point every twelve wavelengths. 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 (S L ); o o upper part of the belt (S U ); total The skirt technique in 3D Figure 16 shows results in case of a 3D unloaded tyre of cylindrical shape (radius R=31 cm and width w=22 cm) lifted 1 mm above ground. A unit velocity is assumed on the lower part of the belt along twice 20 degrees (surface S L ) and zero elsewhere. The sound pressure level obtained by applying the GRIM integration on S L is compared with the integration on a skirt (S S ) which envelops the lower part of the tyre as shown in Figure 13. In this second computation the acoustic velocity on the skirt is computed by means of 3D BEM (3D version of MICADO). A good agreement, within 3 db, can be found between both computations. One could argue that since 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 S S.; 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

9 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 S L, ), 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. 3.6 Surface contact 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 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 loaded and the skirt-technique is not required if perfect contact can be assumed. Nevertheless, this technique will be required for unsmooth tyres with small-scale tread profile not suitable for beam tracing computation near contact. The integration of complex effects such as air-pumping can also be considered, by using the previous skirt approach and by separating the air domain into several regions where different computation techniques can be employed [22]. 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) at two different receiver locations, showing that a significant reduction of tyre noise is achieved for the loaded tyre. 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. 3.7 Comparison of 2.5D and 3D GRIM computations. Last, before studying more complex situations, a comparison of 2.5D and 3D GRIM computations has been made. The 2.5D radiation corresponds to a baffled tyre. The 2.5D Green function is obtained in two steps. First, a 2D-BEM computation made for a rigid geometry (ground and tyre), provides a 2D solution which is integrated in a Fourier-like manner [43] to provide the 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

10 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). Q 1 at α 1 =23, Q 1 at α 2 =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. 4. Influence of the car body 4.1 Tyre and plate configuration in 2.5 D 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 tyre is again considered. Good agreement between both computations can be observed and consequently more complex 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 m- long, 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 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 computed by BEM and beam tracing at M ( ) for two source points Q1 and Q2 on the tyre, respectively at α 1 = and α 2 = Good agreement can be seen. The peak at 2800 Hz in the BEM result is not recovered with ICARE which has been used without diffraction. Figure 21. Tyre + reflecting surface. Validation of the GRIM approach with ICARE. SPL at M at (1.1,0.2) BEM, ICARE. 4.2 Simplified car A very simplified car is considered. Figure 22 shows the geometry of the car. The four tyres have dimensions R=32 cm and w=20 cm. The chassis is 0.31 m above ground. The tyre 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

11 impedances -with a porosity of 20 %, a flow resistance of 5000 Ns/m 4 and a tortuosity of 5. Figure 22. Beam tracing with ICARE. Rays between points on the left front tyre and a receiver M 1, 7.5 m away and 1.2 m above ground are displayed. 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 M 1, M 2, M 3 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. 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 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 24. Effect of car body for three different velocity profiles on tyre belts. Rigid ground. * - * M1; o- o M2; - M3 Figure 25. Effect of car body for three different velocity profiles on tyre belts. Asphalt surface. *- * M1; o- o M2; - M3 Figure 23. Velocity computed by Kropp s plate model. RMS average over the tyre width. For each profile the velocity is assumed to be identical on the four tyres. The velocity on the side walls has been omitted. The GRIM approach is employed using ICARE for the computation of the Green functions. The contribution from the four tyres is added incoherently. Receivers are placed on a horizontal circle 7.5 m away from the medium point between the four tyres, 1.2 m above ground. For each velocity profile, two cases have been considered: (i) four wheels without car body and (ii) four wheels with the car body. The car is assumed to be perfectly rigid. Two types of ground are considered, either rigid or made of 4 cm of asphalt over a rigid underlayer where Hamet s model [44] is used to compute surface 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 Figure 22, principally between car and road. Reflections between tyre and casing have been found to be less important particularly when realistic velocity profiles -with maximum levels close to tyre-ground contact- are considered. The increase or decrease of 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 on the tyre. The use of velocities computed by Kropp s model show values of Hcar of positive amplitudes at all frequencies while the constant and cosine velocity show the strongest amplifications around Hz. These results indicate that the modelling of noise radiation from tyres must include the car body. 11

12 Effect of Road absorption One would expect the type of road to influence the results in a significant manner when the car body is considered since multiple reflections between car and road will be strongly attenuated when absorption effects come into play. Figure 26 shows the insertion loss for ground effects, defined as the sound pressure level for an asphalt road minus the sound pressure level for a rigid road. The solid line delimits regions with positive and negative values. The left graph corresponds to the academic cosine velocity profile whereas the right graph has been obtained with the velocity represented in Figure 23. Both graphs show noise reduction up to 10 db- mainly around 1000 Hz and 2500 Hz which correspond to the maximum absorption obtained by Hamet s model. The use of computed velocities leads to non symmetric results since, as already pointed out, the velocity profile is asymmetric. Also the angular distribution of maximum absorption is different according to the velocity profile employed. To summarize these results, it can be said that the estimation of noise radiated by tyres is affected by many interacting parameters and that the study of isolated tyres only provides part of the information. medium including the car body. A combination of precise BEM computations close to contact and ray tracing by means of the ICARE software has been presented. Other physical aspects should also be considered. Comparisons with measured sound pressure levels have shown that high frequency phenomena such as air-pumping should also be considered. Many local small-scale phenomena should therefore be added to the present model. The proposed so-called skirtapproach here presented can be used for detailed modelling close to contact in order to introduce tyre and road unsmooth profiles [22]. Another aspect currently studied at CSTB is the wave transfer through the ground [23] which is also important since a few centimetres-thick layer of asphalt will act as a wave guide and convey tyre vibrations and add to global noise radiation at distant locations. These complementary aspects have been studied in the sequel European project ITARI ( ) [51] and will be the object of further publications. One could conclude by insisting on the complexity of tyre road interaction and its associated radiation of sound. This paper does not intend to present a full model but rather to introduce some innovative aspects such as combining tyre surface velocities and 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 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). 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 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 functions will integrate the complexity of the surrounding The authors are in great debt to Europe funding through the 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. 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 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) [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)