Project number: FP6-PL-0506437

ITARI

Integrated Tyre And Road Interaction

SPECIFIC TARGETED RESEARCH OR INNOVATION PROJECT

PRIORITY 6 Sustainable development, global change &ecosystems

D4.4: Tyre models for tyre deformation and prediction model for rolling resistance Final validated version

Due date of deliverable: month 36 Actual submission date: month 40

Start date of project:1 February 2003 Duration: 36 month Organisation name of lead contractor for this deliverable: KTH Final version

Project co-funded by the European Commission within the Sixth Framework Programme (2002-2006) Dissemination Level

PU Public X PP Restricted to other programme participants (including the Commission Services) RE Restricted to a group specified by the consortium (including the Commission Services) CO Confidential, only for members of the consortium (including the Commission Services)

D4.4: Tyre models for tyre deformation and

prediction model for rolling resistance - Final

validated version.

Martin Fraggstedt and Svante Finnveden

The Marcus Wallenberg Laboratory for Sound and Vibration Research (MWL),Royal Institute of Technology, 100 44 Stockholm, Sweden

E-mail: mfragg@kth.se

1 Introduction

This the follow-up to the Deliverable D4.3 ”Tyre models for tyre deformationand prediction model for rolling resistance”. This version comes with the nameextension Final validated version.

The main thing that has changed since D4.3 was written is that the dampinghas been changed slightly in the tyre model. Before the damping was mainlyin the tread and in the shell elements just below the tread. The damping of theshell elements has been changed so that it is now evenly distributed over allshell elements. This has lead to a more even distribution of dissipated powerover the tyres sub structures. This is more in line with litterature.

Another difference is that the KTH flexibility matrix was used in the contactforce calculation done by CTH. In D4.3 Chalmers flexibility matrix was used.

This document also adresses the validation in a deeper way. D4.3 showedthat the rolling resistance results were in the right range but D4.4 contains acomparison between measurements on real existing roads and the calculatedrolling resistance based on scanned road profiles from these roads.

The results does not correlate in an absolute sense but a ranking of the roadsis possible if the road textures are different enough.

Rolling resistance is mainly low frequency (below 100 Hz). The damping atlow frequencies is therefore important.

The intension was to introduce frequency dependant damping to adress this is-sue. However, to accurately describe the damping at low frequency has proven

to be difficult and has therefore not yet been successful. This is still work inprogress and will have to be reported elsewhere.

2 Introduction

Traffic has a negative effect on the environment, for over fifty years it has beenan irritating noise polluter. For higher speeds tyres have been found to be themajor contributor for traffic noise. Also the interior noise in the vehicle due tothe tyres are becoming more important as other noise sources such as engines,exhaust systems and gear boxes are better managed.

Traffic is also a major source of green house gases. In the United States thecontribution from the road transport sector to CO2 emissions from fossil fuelconsumption was 24 % in 2002 [1]. The energy consumed by a car, travelingat constant speed, is due to engine ineffiency, internal friction, and the en-ergy needed to overcome resisting forces such as aerodynamic drag and rollingresistance. Tyre rolling resistance accounts for approximately 20 % of CO2emissions from cars and 30 to 40 % from trucks. This means that approx-imately 5 % of the total CO2 emissions from fossil fuel is generated in thetyres. Here we are studying the effect of rolling resistance, by looking at thepower dissipated, caused by visco-elastic forces as a tyre is rolling on a roughroad.

2.1 Car tyres

Car tyres are made of several different materials including steel, fabric andof course numerous rubber compounds, see Figure 1. The three major subregions of the tyre are: the lower side wall, the upper side wall and the centralarea. The ply is a layer of fabric that is embedded in rubber.At the lower sidewalls the ply encloses a volume filled with both steel wires and hard rubbermaterials, this makes the lower side wall areas relatively stiff. The upper sidewall areas are on the other hand quite flexible, since the ply layer there isplain and there is less steel in there. The central area is made up of the beltand the tread. The belt consists of a rubber embedded steel lining (breakers)in the circumferential direction to assure support and rigidity. The tread is anabout 13 mm thick rubber layer which provides the grip.

The objective of this work is to estimate the hysteretic losses due to the de-formation of the tyre as it is rolling on a rough road. To do that a waveguideFinite Elements (FE) model for the tyre, described in reference [2], is used.The model includes: the curvature, the geometry of the cross-section, the pre-

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Fig. 1. The tyre consists of three major sub regions. Upper side wall, lower side walland the central area. The central area is in turn divided into the belt and the tread.

stress due to inflation pressure, the anisotropic material properties and therigid body properties of the rim. The model is based on design data. The mo-tion of the tyre belt and side wall is described with quadratic anisotropic, deepshell elements that include pre-stress and the motion of the tread on top ofthe tyre by quadratic, Lagrange type, homogenous, isotropic two dimensionalelements. The non proportional damping used in the model is based on mea-sured mobilities. The external forces acting on the tyre as it is rolling comesfrom a non-linear contact model [3]. Only normal forces are considered.

The tyre in the present study is a Goodyear, radial, passenger car tyre, withthe dimensions 205/55ZR16, mounted on an Argos rim. The tyre is ’slick’, i.e.it does not have a tread pattern or groves, but in all other aspects has prop-erties typical of a production tyre. The tread pattern has an effect on the tyrevibration [4]. When the tread pattern is cut out of the tread, mass is removedand the bending stiffness is reduced. This generally leads to higher eigenfre-quencies and higher levels for the point mobility. Circumferential groves canbe included in the model since this would not alter the rotational symmetry.Commercial tread patterns however would have to be modeled as an equiva-lent tread pattern. It should me noted though, that the tread pattern is ac-counted for in the contact force calculation, so, as long as the model is slightlyadapted to fit mobility measurements the methodology presented here shouldwork. Local deformation in the tread and tread pattern becomes increasingly

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important above roughly 800 Hz [5] which is well above the frequencies wherepower dissipation occurs for a rolling tyre.

The remaining of this section discusses rolling resistance. In the upcomingsection the waveguide FE model is described. After that a brief description ofthe contact forces follows. The contact forces are then in Section 5 inserted intothe tyre model and the displacement response is calculated. The calculationof the power consumed by the tyre is described and is followed by results andconcluding remarks.

2.2 Influence of tyre parameters on rolling resistance

The rolling resistance Fr is defined as the energy consumed per unit of distancetravelled [6]. The unit is Nm/m = N which is equivalent to a drag force inNewtons. Rubber is a visco elastic material, as it deforms a part of the energyis stored elastically but the remainder is dissipated as heat. These hystereticlosses, as well as aerodynamic drag and friction in the contact patch and withthe rim are losses that contribute to the total drag force on a moving vehicle.Rolling resistance plays a rather large role when it comes to fuel economy. Animprovement in rolling resistance of 10 % can yield fuel consumption improve-ments ranging from 0.5 to 1.5 % for passenger cars and light trucks and 1.5to 3 % for heavy trucks [7].

The rolling resistance is usually given as a dimensionless constant times thegravity force,

Fr = Cr m g, (1)

where m is the mass, g is the constant of gravity and Cr is the rolling resistancecoefficient. Cr is normally in the range 0.01-0.02, with a typical value of 0.012for a passenger car tyre on dry asphalt [8].

The power consumed by this force is

P = V Fr = V Cr m g (2)

where V is the speed of the vehicle. In this uncomplicated semi empiricalmodel for the power dissipation the only explicit parameters are the load andthe speed. Any non-linear dependence on these parameters and all the otherparameters are hidden in Cr. This model is usually good enough for someapplications but studies have shown that the rolling resistance coefficient isinfluenced by a number of parameters which will be discussed in the remainingof this section.

There are two different kinds of passenger car tyres, radial and diagonal, whichdiffers in construction. The radial tyre has been completely dominant the last

4

decade. The rolling resistance of a radial tyre is allways lower than for diagonaltyres. Although radial tyres have a larger deflection, the internal deformationof a diagonal tyre is much bigger because of movement within the differentlayers, which does not occur for radial tyres [9].

The rubber compound has a large effect on the rolling resistance. By changingto a tread compound with a smaller loss factor, for example silica, the rollingresistance can be reduced with as much as 20 % [9]. Traditionally, loweringthe rolling resistance has led to a reduction of the wet grip performance, butthis problem seems to be manageable today.

The relation of the rolling resistance of passenger car tyres to tyre load isalmost linear. Over the practical load range, the rolling resistance coefficientis nearly constant, only slightly decreasing with increasing load [9].

Another important parameter is the inflation pressure. When the inflationpressure is increased the deflection and deformation of the tyre will be smallerleading to less hysteretic losses and a lower rolling resistance. The effect ismore pronounced for higher loads [9].

Rolling resistance will increase with deflection. Deflection is defined as thedifference between the loaded radius and the unloaded radius. However, usingdeflection in rolling loss studies is often avoided because of the difficulty ofmeasuring it accurately. For increasing vertical load, rolling resistance willincrease at constant deflection [10]

The rolling resistance can be considered to be constant until a certain speed.When exceeding this speed, standing waves will occur in the periphery of thetyre, which will increase the rolling resistance. The speed where this effect willoccur depends on the construction of the tyre [9].

The rolling loss will increase with increasing braking and driving torque. Yet,almost all rolling loss tests are done under free-rolling conditions. Tests withbraking and driving torques are more difficult to execute. An increase of 500Nm, can double the rolling loss [10].

The material characteristics of rubber show a strong temperature dependence[11], see Figure 2. In the later end of the transition region, and in the rubberregion a temperature increase leads to a softer rubber with smaller losses. Thisis the background to the discussion that will now follow.

With the increase of the internal tyre temperature, the hysteresis will de-crease and thus the rolling resistance will decrease. If the ambient tempera-ture and speed are constant, the internal tyre temperature will change untilit reaches the equilibrium condition. When a cold tyre starts to roll, the in-ternal temperature is increasing (approximately 40C for passenger car tyres

5

Fig. 2. Schematic characteristics of rubber material temperature dependence [11].

until equilibrium temperature is reached) and the rolling resistance drops by30 % since the hysteretic losses are smaller at higher temperature for rubbercompounds. This means that driving short distances, during which the tyredoes not reach its equilibrium temperature, has a negative impact on rollingresistance. Tyres optimised for rolling resistance, will not develop their fullpotential during short urban trips [9].

The vehicle speed and the tyre temperature are directly related. In the non-steady state temperature condition, the speed change will have a large in-fluence on the rolling resistance. At time zero, the temperature of the tyreis independent of speed and is equal to the ambient temperature. As timeprogresses, rolling loss values decreases and temperature rises until no furtherchanges occur, and the equilibrium temperature has been reached. A suddenspeed increase of 20 km/h, momentarily increases the rolling loss by 9 %.In practice the tyre will warm up much faster, than in a laboratory environ-ment, because of the fact that driving a car on the road includes acceleration,deceleration and cornering [12].

Increasing the ambient temperature will increase the equilibrium tyre temper-ature by the same amount. Experimental data suggests that for each degreeincrease in ambient temperature, rolling loss decreases by 0.4 % for heavyvehicle tyres to 0.8 % for passenger car tyres [12].

By reducing tread thickness e.g. because of tyre wear, the rolling resistancewill decrease. The difference in rolling resistance between a new and worn tyrecan reach 25 % [9].

The relation between tyre dimensions and rolling resistance is inconsistent.The reason is that it is very difficult to compare different sizes of tyres, becauseby increasing the tyre size, it’s construction (e.g. tread, belt, and cords) willalso change [9].

Another central parameter is the road and its shape. Depending on the wave-

6

length of the unevenness, the unevenness is divided into road texture and roadroughness [12]. The road texture concerns wavelengths up to the length of thecontact patch (typically 100 up to 150 mm) and is in turn divided into: i)micro texture (wavelength smaller than 1 mm), ii) macro texture (wavelengthbetween 1 mm and 10 mm), and iii) mega texture (wavelength between 10mm and 100 mm) [9]. The full texture range from micro to mega texture hasa significant influence on the rolling resistance [9]. An increase of the macrotexture will increase the rolling resistance [9].

For wavelengths in the same range as the contact path or larger, the un-evenness is called road roughness. The road roughness causes a deflection ofthe contact path as a whole and also excites the suspension system [12] and istherefore best studied in outdoor tests. Since the suspension is involved in thispower loss mechanism it is sometimes called driving resistance to separate itfrom the rolling resistance [9].

The road condition has a significant influence on the rolling resistance. Therolling resistance is almost doubled at a velocity of 120 km/h when comparinga dry and a wet road with 0.5 mm water depth [10].

Hall [7] claims, based on temperature distribution in the tyre, that roughly 40% of the rolling resistance is generated in the shoulder (the part of the centralarea which is closest to the upper side wall, see Figure 1), 30 % in the crown(the middle part of the central area), 15 % in the upper side wall and 15 % inthe lower side wall.

2.3 Measuring rolling resistance

The rolling resistance of tyres can be measured in different ways: 1) In a lab-oratory environment such as a test drum, a twin drum or a flat-belt, or 2) Bymeasurements on the road such as traction measurements, coast-down tests,tow force measurements of a trailer and fue1 consumption measurements.

Rolling resistance is linked directly to energy consumption and because ofthat another way to determine rolling resistance is through measuring thetyre temperature in relation to road and ambient temperature.

In the ISO standard [6] the test drum method is proclaimed. The drum shouldbe at least 1.5 m in diameter. There are four alternative methods to do therolling resistance measurement: a) the force method where the reaction forceat the spindle is measured. b) the tourque method where the tourque inputis measured at the test drum. c) the power method where the input power ismeasured at the test drum. d) the deceleration method where the decelerationof the test drum and tyre assembly is measured.

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Each way of measuring the rolling resistance has advantages and/or disadvan-tages such as accuracy, correlation to real use, necessary hardware and timespan. For that reason it is difficult to compare the results from different testsand sources. The knowledge of the environment, such as ambient tempera-ture, wind and road type is also important. The rolling resistance of a tyre istherefore not an absolute quantity or property.

A round robin test was performed, in the autumn of 2004, to investigate theeffect of different roads for the rolling resistance and the energy consumption[13]. For the rolling resistance measurements two different measurement sys-tems based on trailers were used. Two kinds of tyres, slicks and profiled, wereused on 8 roads for the slick tyre and 11 roads for the profiled tyre. For the pro-filed tyre the measurement systems have a correlation coefficient of 85 %. Forthe slick tyre the correlation coefficient is only 34 %. The conclusion drawn in[13] is that for profiled tyres differences in rolling resistance for different roadscan be quantified even though the absolute values cannot.

For the energy consumption an electric car with the ability to measure theconsumed energy and a diesel car where the fuel consumption can be closelymonitored were used. The correlation coefficient is in this case 28 % (46 % ifone measurement is excluded). The conclusion is that external circumstancesinfluence the measurements so much that differences in energy consumptionfor different roads are poorly estimated.

2.4 Modelling of rolling resistance

The rolling resistance models can be categorised as empirical, thermal, viscoelastic or thermo-visco elastic [12]. The only models discussed here will be thevisco elastic models.

Stutts and Soedel [14] modeled the pneumatic tyre as a tension band on avisco elastic foundation. The ground contact region of the rolling tyre in steadystate was found by solving a two-point boundary value problem with specifiedradial deflections, treating the ground contact end points as unknowns. Therolling resistance was determined by integrating the forces acting throughthe elastic foundation on the wheel axle due to the steady state deformationof the tread band. Bending stiffness of the tension band is neglected anddissipative forces are introduced through one lumped viscous force term for thefoundation. They revealed the presence of a critical angular velocity beyondwhich the response changes from a damped exponential to a damped harmonicform. The physical parameters, of the tyre were calculated from experimentaldata. Without damping the contact region is symmetric and the net rollingresistance force is zero. When damping is introduced there is a forward shift

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of the contact region and the rolling resistance force appears.

Kim and Savkoor [15] used an elastic ring supported on a visco elastic foun-dation. Additional elastic spring elements on the outer surface of the ring areincluded to model the compliance of the tread rubber. The contact problemof a free-rolling tyre is formulated for prescribed normal deflection and sub-jected to constraints of both normal contact and friction. Coulomb friction isassumed to apply locally, Coriolis effects are accounted for and bending of thetreadband is allowed according to the Bernoulli-Euler assumption. The equa-tions of motion are solved with a modal expansion method. Three differentdamping models are applied, viscous, structural and a Maxwell four-elementmodel. The structural damping leads to a speed independent rolling resistancein accordance with litterature.

Yam et al [16] based their calculation on experimental modal parameters andthe equilibrium and kinematics of a rolling tyre. The tyre carcass deformationis calculated using the transfer matrix resulting from an experimental modalanalysis. The calculation of tyre rolling properties starts from the static statusof the tyre, and the results after the tyre rotates through one discretized angleare calculated successively, until the rolling of the tyre is steady. The distri-butions of vertical reaction and horizontal tractional forces on the footprintof a rolling tyre are calculated. The rolling resistance, under different verticalloads, inflation pressures and rolling speeds are investigated numerically andis verified against experiments and litterature.

Popov et al [17] modelled a truck tyre, based on the model developed by Kimand Savkoor [15]. The stiffness and damping parameters needed, came from anexperimental modal analyis. Structural damping is used. Sophisticated rollingresistance measurements using a large test drum were also performed. Themodel is used to obtain the rolling resistance due to tread compression, whichamounts to 56 % of the measured rolling resistance force. All contributionsare however not calculated.

The models above are all based on rather simple equivalent structures, suchas elastic rings, whereas the model used in the present study has the correctgeometry and stiffness parameters as it is based on design data. None of themodels above are treating a rough road even though the road texture androughness have a significant effect on the rolling resistance.

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3 Waveguide finite element model

3.1 Variational statement

Without external forces and with dissipation neglected, Hamilton’s principlestates that the true motion of the system is the one that minimises the differ-ence between the time integrals of the strain and kinetic energy.

∫ t2

t1

δ(U − T )dt =∫ t2

t1

δ(L)dt =0 (3)

where t is the time and U , T are the strain and kinetic energy. The functionalU −T will be referred to as the Lagrangian L. By considering a harmonic timedependence of the form e−iωt, where ω is the angular frequency, one can lettime stretch from minus infinity to plus infinity and apply Parseval’s identity.For linear vibration the different frequency components do not couple, it istherefore possible to consider one frequency at a time. The resulting bilinearfunctional is,

L =1

2

1

2π

∫

V(ε∗)T C ε − ρω2(u∗)T udV (4)

where ∗ denotes complex conjugate, T denotes transpose, C is the rigiditymatrix, which is real valued for a conservative system and ρ is the density.The vector ε contains the engineering components of strain, which are linearfunctionals of the displacement u and it’s spatial derivatives.

When dissipation is included the above functional (equation (4)) does not havea stationary minimum for the true displacements. The losses may, however, behandled by using a variational principle similar to that of Hamilton [18],[19].Dissipative forces are introduced through the use of a complex, possibly fre-quency dependent, rigidity matrix. By replacing the complex conjugates ofthe strain and displacement with the complex conjugate of strain and dis-placement in an adjoint negatively damped system the new functional willbe stationary for true motion. This is conceptually more complex but doesnot add to the calculation burden. The work done by external forces f is alsoadded as in reference [2]. The resulting functional is the new lagrangian L,

L =∫

V

(

εaT C ε − ω2ρuaT u − fH u − uaT f)

dV, (5)

where H denotes complex conjugate and transpose and a denotes complexconjugate of the response in the adjoint system.

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3.2 Tyre model

A waveguide FE model is a good tool when the structure has constant cross-sectional material and geometrical properties along one coordinate direction,e.g. smooth car tyres. The idea is to discretise the cross-section using standardFE technique but to keep the analytical approach along the direction of thewaveguide. For these structures the response can be described by

u(x, r, φ) = Ψ(x, r)Tv(φ), (6)

where u is the response variable, x, r are the coordinates of the cross-section,φ measures the angle around the structure, Ψ is a vector of polynomial FEshape functions, each non zero within one finite element only, and the entriesof the vector v are the ’nodal’ displacement.

The tyre cross-section mesh with 42 elements, 113 nodes and 516 degrees offreedom (DOF) is seen in Figure 3. The motion of the tyre belt is describedwith quadratic anisotropic, deep shell elements that include pre stress and themotion of the tread on top of the tyre by quadratic, Lagrange type, isotropictwo dimensional elements.

3.3 Input data definition

The geometry and most of the elastic data that defines the tyre model arebased on an Abaqus finite element model given by Goodyear. The big dis-advantage with this procedure is the need to rely on a tyre manufacturer forinput data. The advantage is that the model is based on design data, thereforecommunication with the design engineers is possible. Also, it is of very goodquality.

The input data file for the Abaqus model is converted to fit the syntax usedby the waveguide FE program (written in Matlab). In the Abaqus model theelastic data for the tread is simply given by a single value for the shear modulusG = 5.81MPa. Since rubber is nearly incompressible the value of Poisson’sratio was set to ν = 0.4995. The data for the deep shell elements given byGoodyear was adapted slightly. A discussion of these adaption follows in thesubsequent sections.

3.3.1 Pre-load

The amount of pre-load force (per unit length) attached to each element isgiven as 3 membrane forces and 3 bending forces per element. The waveguide

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model only considers the membrane forces [20], hence the flexural forces areignored, which could be one of the reasons that slight adaptation is needed.When studying the pre-load forces it is seen that the shear forces and thecircumferential forces at the side wall are very small. In fact, they have arandom appearance. For stability reasons these elements in the static forcevectors are set to zero. It is also noted that the force in the direction alongthe cross-section (ξ -direction) is zero at the rim. This cannot be correct sincestatic equilibrium for a strip going around the tyre requires the force in the ξ-direction to be inversely proportional to the radius. The pre-load used hereis defined by

Nξ(n) = Nξ,m

Rm

R(n), (7)

where n is the element number, R(n) is the radius at the middle of the element,Nξ(n) is the pre load at element n, Rm is the radius at the middle of the tyreand Nξ,m is the pre-load at the middle of the tyre. The pre-load defined Eq.(7) gives static equilibrium and is just a slight adjustement of the pre-loaddefined by the Abaqus model.

The peaks were shifted a bit in frequency between the simulation and themeasurements. Since the pre-load is in a first approximation proportional tothe inflation pressure, which is measured with a rather simple pressure gauge,it is multiplied with a factor 0.95 to have better agreement with measurements.

3.3.2 Transverse shear rigidity

The Abaqus model defines the 6 x 6 matrix in the upper left corner of therigidity matrix D in [20] Eq. (76). The transverse shear rigidity is not explicitlydefined, therefore the transverse shear rigidities are set to be equal to theinplane shear rigidity, i.e.

D[7, 7] = D[8, 8] = D[3, 3] (8)

3.3.3 Visco-elastic data

The elastic data, as mentioned above, is defined by Goodyear’s Abaqus model.The dissipative properties on the other hand are defined by an ad hoc curvefitting procedure based on measured mobilities.

Solid rubber is highly damped and based on numerical experiments a lossfactor η = 0.3 was used for the tread. To find a baseline for the damping ofthe deep shell elements, the 3-dB band widths of the resonances in the lowfrequency region (100-200 Hz) were estimated. At these frequencies the tyrebehaves like a pre-stressed curved beam on an elastic foundation. When thebase line damping was introduced the damping of the higher order beam modes

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became too high and therefore some of the baseline damping was distributed todamping proportional to the pre-load. Following this idea the rigidity matrixof the deep shell elements D is multiplied with a factor 1− iηb, where ηb = 0.01and the pre-load by a factor 1 − iηp, where ηp = 0.01.

To get the right frequency trend concerning damping, extra damping wasadded to the deep shell elements. Thus, the diagonal elements of D which de-scribes the rigidity of inplane cross-sectional membrane strain, bending acrossthe belt and bending in the circumferential direction is multiplied by factors1 − iη1 , 1 − iη4 and 1 − iη5 respectively, where η1 = 0.03, η4 = 0.03 andη5 = 0.05. The increase of the apparent damping at even higher frequencies ismodelled through a factor 1 + iηv multiplying the mass matrix, where

ηv (f) =

0, f < 260 Hz

0.05 + 0.05 · f/3000, f ≥ 260 Hz(9)

This is the only frequency dependent damping even though it is well knownthat the visco-elastic properties of rubber vary largely with frequency. For theconsidered tyre, however, the large increase of apparent damping at the ’cut-on’ of the higher order cross-sectional modes, around 350 Hz, is most probablydue to that a large portion of the strain energy is in the rubber, and less inthe steel wires within the belt.

The damping model is simplistic but gives quite good results. In Figure (4) acomparison between the measured and simulated point mobility can be seen.The point mobility was measured with excitation perpendicular to the cross-section at the middle point of the tread. The level of damping looks good.

Work in progress considers a frequency dependant material description forboth the tread and the belt elements.

3.4 Equations of motion

In this investigation the rigid rim is assumed to be blocked. The response atthe connection to the rim is therfore zero,

vc1 = vc2 = 0, (10)

where vc1 and vc2 contains the DOFs associated with the two nodes connectedto the rim.

The modified version of Hamilton’s principle mentioned above is employed,displacements of the form (6) are assumed, standard FE procedures for ele-ment formulation and assembling are used, and finally variation of the adjoint

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−0.1 −0.05 0 0.05 0.1

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

x [m]

r [m

]

Fig. 3. Tyre mesh. The belt is modelled with quadratic anisotropic, deep shell el-ements that includes prestress and the tread on top of the tyre by quadratic ,Lagrange type, isotropic two dimensional elements.

systems displacements, leads to the following Euler-Lagrange equation [20].

[

−A11∂2

∂φ2+ (A01 − A10)

∂

∂φ+ A00 − ω2M

]

v(φ, ω) = f(φ, ω), (11)

where f is the corresponding generalised consistent force vector, the matricesAij and M describe the elastic and inertia forces of the structure. Dissipativeforces are introduced through the imaginary parts of the matrices Aij and M.

The solutions to equation (11) will be periodic with respect to the circum-ferential angle φ, since the tyre is a circular structure. This means that thesolution can be expressed as an exponential Fourier series

v(φ, ω) =∑

n

vn(ω)einφ, (12)

and similarly for the force vector. In equation (12) n is equivalent to thenumber of wavelengths going around the tyre in the circumferential directionand will here be referred to as the wave order.

To arrive at the desired equation the assumed solution is inserted into equa-tion (11), which is then multiplied by e−imφ, finally the reulting expression isintegrated around the tyre. This procedure filters out the coeficients due tothe orthogonality of the complex exponential basis function over the intervall

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[0 2π]. The result is

[

n2A11 + in(A01 − A10) + A00 − ω2M]

vn(ω) = fn(ω), (13)

which also can be written

Dn(ω)vn(ω) = fn(ω). (14)

4 Contact Forces

The contact forces are calculated by Chalmers University of Technology (CTH),as described in reference [3]. The contact force prediction is based on a non-linear contact model in which the tyre structure is described by its flexibilitymatrix. Topographies of the surface are scanned, the tread pattern is accountedfor, and then the tyre is ’rolled’ over it. The procedure is done in the timedomain and uses a Lagrange multipliers approach. The nonlinear conditionsused are: i) the tyre cannot indent into the road, ii) if a point is not in contactthe force is zero and iii) that the force cannot be negative (road pulling tyredown) ([21] chapter 6). Only forces acting normal to the road is considered.

The tyre in this investigation has, as mentioned earlier, no tread pattern.The tread pattern has an effect on the tyre vibration [4]. The tread pattern is,however, taken into account in the contact force computation, so as long as thetyre model is somewhat tailored to fit mobility measurements the procedurepresented here should work. The power dissipation is mainly at low frequencies(below 100 Hz), so in a first approximation the tread does not have to bemodelled in great detail, but further investigation on the influence of the treadis needed.

4.1 Evaluation Of Generalised Contact Forces

The contact forces were determined for a speed of 80 km/h and consist ofan integer number of full revolutions. The demonstration below is for a casewhere two full revolutions were used. The total load on the tyre varies withtime but the mean load is within 10 % of 300 kg for all roads. The pressure,p (force per unit length and unit circumferential angle) resulting from thecontact model, is described by a three-dimensional force matrix, F, specifying

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the values of the contact pressure [3].

p(x, φ, t) = F(p, q, k)∆t

∆φ∆xδ(t − tk)

· (H(φ − (φq − ∆φ/2)) − H(φ − (φq + ∆φ/2)))

· (H(x − (xp − ∆x/2)) − H(x − (xp + ∆x/2))) ,

(15)

φq = 2π(q − 1)

Q; q = 1, 2, ..., Q; Q = 512

xp = [−0.07 0.06 ... 0.07] ; p = 1, 2, ..., 15

tk = (k − 1)∆t; k = 1, 2, ... , K; K = 1024

∆x = 0.01 m, ∆φ =2π

Qrad, ∆t = 0.1734ms

where H is the Heaviside function, p is the pressure and ∆x ∆φ is the areaupon which the force acts. The coordinate φ is around the tyre, x is in thetransverse direction along the tyre contour and t is the time.

The nodal force vector, f = f(φ, t) (force per unit angle), is produced byweighting of the pressure with the FE shape functions and integrating over x.

f(φ, t) =∫

Ψ(x, r)p(x, φ, t)dx (16)

The part of p which depends on x is defined as

F(s, q, k) =∫

Ψ(x, r)F(p, q, k)1

∆x· (H(x − (xp − ∆x/2)) − H(x − (xp + ∆x/2))) dx

(17)

where the new index s represents the nodes in the tyre road contact zone.With this formulation the nodal force vector can be written:

f(φ, t) = F(s, q, k)∆t

∆φδ(t − tk)

· (H(φ − (φq − ∆φ/2)) − H(φ − (φq + ∆φ/2)))(18)

The next task is to make a Fourier series expansion in time and in the cir-cumferential direction.In the circumferential direction the series is given by

f(φ, t) =N

∑

n=−N

fn(t)einφ. (19)

The Fourier coefficients in (19) are given by

fn(t) =1

2π

∫ π

−πf(φ, t)e−inφdφ =

1

2π

∫ 2π

0f(φ, t)e−inφdφ. (20)

16

Using equation (18) and that f (φ, t) is a periodic function in φ :

fn(t) =∆t

2πδ(t − tk)

∫ 2π

0

F(s, q, k)

∆φe−inφ

· (H(φ − (φq − ∆φ/2)) − H(φ − (φq + ∆φ/2))) dφ

=∆t

2πδ(t − tk)

sin(n∆φ/2)

n∆φ/2

Q∑

q=1

F(s, q, k)e−inφq

(21)

The contact forces are mostly zero but when the investigated location is incontact, there is a rather wild outburst. It would be unfortunate if such anoutburst occurred at the ends of the summation in (21) and if this is the case,the entries are shifted. Thus, the entries to f are shifted so that the first entryis for φ = φs:

fn(t) =1

2π

∫ 2π

0f(φ, t)e−inφdφ =

e−inφs

2π

∫ 2π

0f(θ + φs, t)e

−inθdθ (22)

which follows from that f is a periodic function of φ. This procedure determinesfn for n = 0, 1, ... , Q−1 ,, which equally gives fn for n = −Q/2, ... , Q/2−1 ,since f−n = fQ−n.

Finally, the generalised nodal force vector is transformed to the frequencydomain. Thus, it is assumed that the forces are periodic, described by thefollowing Fourier series:

fn(t) =N

∑

n=−N

fn(ωm)eiωmt (23)

where

fn(ωm) =1

T

∫ T

0fn(t)e−iωmtdt (24)

ωm = 2πm/T, T = K∆t, M = K/2 (25)

fn(ωm) =1

T

∫ T

0fn(t)e−iωmtdt

=1

T

∫ T

0

K∑

k=1

fn(tk) ∆tδ(t − tk)e−iωmtdt

=∆t

T

K∑

k=1

fn(tk) e−iωmtk

=1

K

K∑

k=1

fn(tk) e−i2πm(k−1)

K

(26)

17

The summation in (26) is made with the Matlab procedure fft. This producesthe Fourier coefficients for ωm, m = 0, . . . , (K − 1) , which equally deter-mines, after reordering, the coefficients for m = −M, . . . , (M − 1), sincefn (ω−m) = fn (ωK−m). The contact forces fn (t) are not truly periodic and, toavoid Gibb’s phenomena, a Hanning window is applied before the FFT.The resulting relation is:

f (φ, t) =N

∑

n=−N

M−1∑

m=−(M−1)

fn (ωm) ei n φ+iωmt (27)

4.1.1 On Negative Frequencies

The nodal force vector f (φ, t) is a real valued quantity and it follows that:

fn (t) = f∗−n (t) (28)

The spatially Fourier transformed force fn (t), on the other hand, is not real anda similar result for the generalised force vector fn (ω) is perhaps not obvious.However, by analysing the real and imaginary parts of fn (t), which both arereal by definition, and using the relation (28), it follows that

fn (ω) = f∗−n (−ω) (29)

Moreover, the system matrix in equation (14) has this property and it followsthat the response vector vn (ω) is also such that

vn (ω) = v∗

−n (−ω) (30)

The response at positive frequencies thus gives the one at negative frequen-cies. Consequently, the calculation burden and the memory requirements arereduced if the response for all wave orders and positive frequencies are con-sidered only.

5 Vibration Response Of A Rolling Tyre

The tyre is rotating with a fixed angular speed, Ω , which is given by

Ω = Qπ/T , (31)

where Q is the number of full revolutions.

Now, Newton’s law applies for a fixed piece of matter: a particle. The equationsof motion are therefore solved in a Lagrangian coordinate system that is fixed

18

to the rotating tyre. In doing so, the Corioli forces are neglected. Also, thecentrifugal force is neglected and it is assumed that the increased static tensionand radial expansion that it induces are already included in the definitionof the tyre’s steady state. Upon this basis, the tyre vibration is predicted.Thus, the generalised force vectors fn (ωm) are inserted into equation (13).The solution of these equations produces the generalised tyre displacementvectors un (ωm). The tyre displacement as a function of location and time isthen given by

v (φ, t) =N

∑

n=−N

M−1∑

m=−(M−1)

vn (ωm) ei n φ+iωmt (32)

6 Rotating accelerometer

To go from the displacement given in equation(32) to acceleration the dis-placement is multiplied with −ω2

a (φ, t) =N

∑

n=−N

M−1∑

m=−(M−1)

(−ω2m)vn (ωm) ei n φ+iωmt (33)

A comparision between a measured acceleration signal from an accelerometerplaced in the groove of a tyre rotating on a test drum and simulated accel-eration signals are in Figure 5. The results are presented as power spectraldensity of the acceleration signal. An average over all angles around the tyreis used in the simulation seing as we want the tyre to ’see’ more of the roadsurface.

At the big low frequency peak the response is underestimated by around 2dB. At higher frequencies the discrepencies are larger.

The simulation for a rough road seems to agree better than that for the ISOroad even though the test drum has an ISO texture

At frequencies above 1000 Hz the tyre model is not believed to be valid sothose dicrepencies are expected.

7 Dissipated power

Energy is always conserved, it is just transformed from one form to another.The energy going into the system must therefore equal the dissipated energy

19

in the system. The same argument is valid for the power, consequently, theinjected power must equal the dissipated power.

7.1 Dissipated power based on external forces

The direct approach to calculate the time average of the input power is basedon the following relation:

P =1

T

∫ 2π

0

∫ T

0fT(φ, t)

∂v

∂tdtdφ, (34)

where P is the time averaged input power, f is the nodal force vector and∂v(φ, t)/∂t the nodal velocity vector.

The displacement and the force are given by (32) and (27) respectively. Thevelocity is easily derived from the displacement.

∂v(φ, t)

∂t=

N∑

n=−N

M−1∑

m=−(M−1)

iωmvn(ωm)einφ+iωmt (35)

Since only the coefficients for positive frequencies has been calculated this isre-written using the relations (29) and (30),

∂v(φ, t)

∂t=

N∑

n=−N

einφ

0 · ivn(0) +M−1∑

m=1

iωmvn(ωm)eiωmt

− iωmvn(−ωm)e - iωmt

(36)

Which can be simplified to:

∂v(φ, t)

∂t=

N∑

n=−N

einφ

[

M−1∑

m=0

iωmvn(ωm)eiωmt − iωmv∗

−n(ωm)e−iωmt.

]

(37)

f(φ, t) =N

∑

n=−N

einφ

[

fn(0) +M−1∑

m=1

fn(ωm)eiωmt+f∗−n(ωm)e−iωmt

]

(38)

These formulas are now inserted in equation (34)

P =1

T

∫ 2π

0

∫ T

0

N∑

q=−N

eiqφ

fTq (0) +

M−1∑

p=1

fTq

(ωp)eiωpt + fH

−q(ωp)e

- iωpt

·N

∑

n=−N

[

M−1∑

m=0

iωmvn(ωm)eiωmt − iωmv∗

−n(ωm)e - iωmt

]

einφdtdφ

(39)

20

Since,∫ 2π

0einφeiqφdφ =

0 if n 6= −q,

2π if n = −q,

many terms cancel leading to

P =2π

T

N∑

n=−N

∫ T

0

fT−n

(0) +M−1∑

p=1

fT−n

(ωp)eiωpt + fH

n (ωp)e- iωpt

·

[

M−1∑

m=0

iωmvn(ωm)eiωmt − iωmv∗

−n(ωm)e - iωmt

]

dt

(40)

Once again the integral vanishes for many terms since

∫ T

0eiωmteiωptdt =

0 if p 6= −m,

T if p = −m.

Finally the injected power is given by

P = 2πN

∑

n=−N

M−1∑

m=1

iωmfHn (ωm)vn(ωm) − iωmfT

−n(ωm)v∗

−n(ωm) . (41)

7.2 Dissipated Power Using Internal Energy Considerations

Another way to form the time average of the input power is through the powerconsumed by dissipation. Starting from the power expression (equation 41),the nodal force can be expressed in terms of the nodal displacement via theequation of motion

Dn(ω)vn(ω) = fn(ω). (42)

This formal expression can be expanded to(

A00 + inA01 − inA10 + n2A11 − ω2M)

vn = fn, (43)

where the matrices Aij describe the stiffness of the structure and M is themass matrix. Inserting this expression for the force into the power expression(equation 41) and making use of symmetry of the stiffness matrices and themass matrix, leads to the following expression for the dissipated power

P = 4πN

∑

n=−N

M−1∑

m=1

ωm(vHn (ωm) Im(A00)vn(ωm) + invH

n (ωm) Im(A01)vn(ωm)

− invHn (ωm) Im(A10)vn(ωm) + n2vH

n (ωm) Im(A11)vn(ωm)

− ω2mvH

n (ωm) Im(M)vn(ωm))

(44)

21

Observe that since losses proportional to the mass matrix are included in themodel the mass matrix is also present in the expression.

The advantage with this procedure is that it can be done element vise, whichmeans that the elements where a substantial part of the power is consumedcan be identified.

8 Results

The above mentioned procedures to calculate the dissipated power has beenemployed for two different surfaces: a test road replica with a surface accordingto ISO standards for pass by noise testing which can be classified as smooth,and a road replica with a rough surface. The ”roads” are mounted on a testdrum managed by Goodyear. The tyre referred to as airplane tyre differs onlyin tread pattern to the tyre modeled in this investigation (slick tyre). It hasthree longitudinal grooves. The calculations take a stiff two hours on a modern(2006) laptop for 512 frequencies, 200 wave order and 516 degrees of freedom.The speed was 80 km/h and the load was close to 300 kg. The results aregiven in Table 1.

Case Dissipated Power [W]

Airplane tyre measured by TUG on safety walk (sandpaper) 601.68

Airplane tyre measured by TUG on rough road () 866.55

Airplane tyre on ISO road (mean load 293 kg) 712.66

Airplane tyre on rough road (mean load 324 kg) 852.41

Slick tyre on ISO road (mean load 280 kg) 758.92

Table 1Measured and calculated dissipated power for different roads.

For comparison Table 1 also presents values coming from a measurement bythe Gdansk University of Technology (TUG). These were made on the airplanetyre with three longitudinal grooves mentioned above. The results from thesemeasurements were given as a rolling resistance coefficient which has beenscaled in accordance with the semi-empirical model (load 300kg) to producevalues of the dissipated power, see equation (2). As can be seen in Table 1, thecalculated power gives reasonable values. The rough drum replica gives largerlosses than the smooth drum, which also is in accordance with literature.

The dissipated power for the smooth tyre on the ISO road as a functionof frequency and wave order can be seen in Figure 6 and 7 respectively. Asubstantial part of the dissipation occurs below 100 Hz and at a wave orderaround 3.

22

Most of the dissipated power occurs in and below the tread. This is illustrated,for the slick tyre on the ISO road in Figure 8 and 9. Around 15 % of the lossesoccurs in the side wall which is low compared to the 30 % reported by [7].The total damping level in the model is estimated quite accurately (see Figure(4), but the distribution of the damping, in the different parts of the tyre, isprobably wrong. which possibly explains the discrepancy. No results at otherspeeds loads or inflation pressure are available.

9 Validation study

Some of the roads from the round robin test [13] have recently been scanned.That is the road texture have been measured using laser equipment.

The nomenclature of the presented roads are the same as in [13]. A briefdesription of the roads is in Table 2.

Road Description

B1 Cement concrete with Burlap (CC-smooth)

C1 Gussasphalt 0/11

D1 Cement concrete with Burlap (CC-smooth)

NL3 Dense asphalt concrete (DAC 0/16),

NL4 Double layer porous asphalt with a grading size of 2/6

NL5 Double layer porous asphalt with a grading size of 4/8

Table 2Description of investigated roads

The results from the simulations together with measuered values is in Table3 and 4. The simulated values are presented as an average over four laps. Theairplane tyre described in the previous section is used in the simulation. Thesimulated dissipated power is once again in the right range

As can be seen in Table 3 the ranking between B1 and D1 is ok but the relativedifference is smaller in the simulation.

The simulation is not able to predict the difference between B1 and D1 cor-rectly. The simulation predicts that D1 leads to a slightly larger power dis-sipation whereas the measurements indicates the complete opposite. This isprobably because B1 and D1 are very similar in terms of texture propertiesand the relative difference between them is in the accuracy range of the mea-surement procedure.

23

Case Bast Profiled Bast Slick TUG Profiled TUG Slick Simulation Airplane

B1 660.0000 840.0000 526.6667 693.3333 821.3400

C1 826.6667 973.3333 740.0000 886.6667 941.0375

D1 606.6667 733.3333 486.6667 - 869.9550

C1/B1 1.2525 1.1587 1.4051 1.2788 1.1456

D1/B1 0.9192 0.8730 0.9241 - 1.0501

Table 3Measured and calculated dissipated power for different roads. The values givenrepresent power dissipation [W]

Case Bast Profiled Bast Slick TUG Profiled TUG Slick Simulation Airplane

NL3 773.3333 886.6667 680.0000 853.3333 838.5550

NL4 700.0000 840.0000 586.6667 740.0000 821.0350

NL5 733.3333 846.6667 620.0000 766.6667 844.4050

NL4/NL3 0.9052 0.9474 0.8627 0.8672 0.9791

NL5/NL3 0.9483 0.9549 0.9118 0.8984 1.0070

Table 4Measured and calculated dissipated power for different roads. The values givenrepresent power dissipation [W]

Table 4 shows that the ranking between NL3 and NL4 is ok but once againthe relative difference is smaller in the simulation.

All meaurements show the same trend. NL3 has the highest rolling resistancefollowed by NL5. NL4 has the lowest rolling resistance. In the simulations NL5has the highest rolling resistance followed by NL3 and NL4

The reason that the correlation isn’t better could be that the road profilesused in the contact force calculations are too short. The rolling resistance isdominated by the low frequency response and only 2 meters of road profilehas been used in the calculations.

Another reason could be that the contact stiffness used in the contact forcecalculation is to low, This is indicated in Figure 5 where the simulated powerspectral density of the acceleration signal for the ISO texture is much lowerthan the measured signal above the low frequency peak.

24

10 Conclusions

A car tyre is modeled with waveguide finite elements. The model is based ondesign data and accounts for: the curvature, the geometry of the cross-section,the pre-stress due to inflation pressure, the anisotropic material properties andthe rigid body properties of the rim. The motion of the tyre belt and side wallis described with quadratic anisotropic, deep shell elements that include pre-stress and the motion of the tread on top of the tyre by quadratic, Lagrangetype, isotropic two dimensional elements.

The non proportional damping used in the model is based on measured mo-bilities.

External forces resulting from a non-linear contact model, for three differentroads are inserted and the responses are calculated. The dissipated power isthen calculated through the injected power and the power dissipated withinthe elements.

The results obtained with the two methods agrees perfectly with each other.Even if the power calculation should be the same for the different methods itis still nice to see that they agree since there is an ocean of FE juggling andadministration between them. The calculation result compares favorably withthose from literature [8] and measurements. The rough road dissipates morepower then the smooth road. Around 15 % of the losses occurrs in the side wallwhich is low compared to the 30 % reported by [7]. The total damping level inthe model is fine, but the distribution of the damping, in the different parts ofthe tyre, is probably wrong, which possibly explains the discrepancies. Sincethe visco elastic data is extremely important for a rolling resistance estimation,the damping should be established in a more scientific way, concluding thatthe results for for rolling resistance are promising but that further work isneeded.

When it comes to the validation of real roads it can be concluded that if theroads are different enough a ranking of the surfaces is possible. The relativedifference between the different roads is smaller in the calculation than in themeasurements.

In D4.3 it was speculated that by increasing the number of laps available fromthe contact force calculation the results would be more accurate due to thefiner frequency resolution. The finer frequency resolution is needed to resolveall the energy of the eigenmodes of the tyre. As it turns out this is not agreat idea. The laps are very similar. If the laps were identical the fourierseries would only have non zero components at frequencies f = mf0 = m/Twhere T is the period time of one lap and m = 1, 2, 3, ..... The results whenone or many laps are used are similar. Using more than one lap increases the

25

calculation burden witout improving the results.

It should be possible to spread out the energy over more frequency componentsif one just makes sure that the input energy is the same. Some attempts havebeen made but they have not been successful.

11 Future work

Future work consists in fine tuning the tyre model with regards to damping.Based on measurements of the dynamic shear modulus a frequency dependanttread will be introduced. The damping of the belt and side wall will also needfurther work.

An investigation of the influence of certain tyre parameters would also beinteresting. It would be possible to change the speed, the external load, andperhaps the wear of the tyre (by reducing the tread height).

References

[1] Environmental protection agency, EPA 430-R-02-003, Inventoryof U.S. Greenhouse gas emissions and sinks 2002.

[2] Fraggstedt M. and Finnveden S. A Waveguide Finite ElementModel of a Pneumatic Tyre. Paper A in this thesis, 2006.

[3] Wullens F. Excitation of tyre vibrations due to tyre/roadinteraction, PhD thesis, Applied Acoustics, Chalmers Universityof Technology, 2004.

[4] Andersson P., Larsson K., Wullens F. and Kropp W. HighFrequency Dynamic Behaviour of Smooth and PatternedPassenger Car Tyres, Acta Acustica United With Acustica Vol.90 (2004) 445 -456.

[5] Larsson K. and Kropp W. A high-frequency three-dimensionaltyre model based on two coupled elastic layers. Journal of Soundand Vibration, 253(4):889-908, 2002.

[6] ISO 18164 Passenger car, truck, bus and motorcycle tyres -Methods of measuring rolling resistance, 2005.

[7] Hall D.E. and Moreland J.C. Fundamentals of rolling resistance,Rubber Chemistry and Technology 74 (3): 525-539 JUL-AUG,2001.

26

[8] Wennerstrm E. Fordonsteknik, 8th edition, In swedish, KTH,2004.

[9] Hoogvelt R.B.J., Hogt R.M.M., Meyer M.T.M. and Kuiper E.Rolling resistance of passenger car and heavy vehicle tyres aliterature survey, TNO report 01.OR.VD.036.1/RH, December11th 2001.

[10] Schuring D. (Firestone Tire & Rubber Compagny) Rolling loss ofpneumatic tires, Rubber chemistry and technology, volume 53 p.600-727, 1980.

[11] Sjoberg M. On Dynamic properties of rubber isolators, PhDthesis, Department of Vehicle Engineering, KTH 2002, ISSN 1103-470x.

[12] Schuring D. and Futamura S. (Central Research, BridgestoneFirestone Inc.) Rolling loss of pneumatic highway tires in theeighties, Rubber chemistry and technology, Vol. 63, pp. 3 15-367,1990.

[13] Round Robin Test Rolling Resistance / Energy ConsumptionDWW-2005-046, 2005.

[14] Stutts D.S. and Soedel W. A Simplified Dynamic Model of theEffect of Internal Damping on the rolling resistance in pneumatictires, Journal of Sound and Vibrarion 155 (1), 153-164, 1992.

[15] Kim S.-J., and Savkoor A.R. The Contact Problem of In-PlaneRolling of Tires on a Flat Road, Vehicle System DynamicsSupplement 27, pp. 189-206, 1997.

[16] Yam L.H., Guan D.H., Shang J. and Zhang A.Q. Study ontyre rolling resistance using experimental modal analysis, Int. J.Vehicle Design, Vol. 30, No. 3, pp. 251-262, 2002.

[17] Popov A.A., Cole D.J., Cebon D. and Winkler C.B. EnergyLoss in Truck Tyres and Suspensions. Vehicle System DynamicsSupplement 33 , pp. 516-527, 1999.

[18] Morse P.M. and Feshbach H. Methods of theoretical physics,Chapter 3. 1953.

[19] Finnveden S. Exact spectral finite element analysis of a railwaycar structure, Acta Acustica, 2, 461-482, 1994.

[20] Finnveden S. and Fraggstedt M. Waveguide finite elements forcurved structures, TRITA-AVE 2006:38.

[21] Andersson P. Modelling interfacial details in tyre/road contact-Adhesion forces and non-linear contact stiffness, PhD thesis,Applied Acoustics, Chalmers University of Technology, 2005.

27

101

102

103

−70

−65

−60

−55

−50

−45

−40

−35

Frequency [Hz]

Mag

nitu

de o

f poi

nt m

obili

ty d

B r

el 1

(m

/Ns)

2

Fig. 4. Magnitude of point mobility for excitation in the middle position. Measured(solid) and calculated (dashed).

28

0 500 1000 1500 2000 2500 3000−15

−10

−5

0

5

10

15

20

25

30

35

Frequency (Hz)

Acc

eler

atio

n P

SD

(dB

rel

1 (

m2 /s

4 / H

z)

Fig. 5. Power spectral density of a rotating accelerometer signal. Measured ISO roadaveraged over 40 laps (Solid), simulated rough road (Dashed) and simulated ISOroad (Dotted). The simulated curves are taken as an average over all angles aroundthe tyre.

0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

6

7

8

9

10

11

Frequency [Hz]

Pow

er s

pect

ral d

ensi

ty [W

/Hz]

Fig. 6. Power spectral density of the dissipated power. Most of the dissipation occursbelow 100 Hz.

29

0 5 10 15 20 25 300

20

40

60

80

100

120

Waveorder

Pow

er [W

]

Fig. 7. Dissipated power as a function of wave order. A substantial part of thedissipated power occur at a wave order of around 3.

0 5 10 15 20 25 300

5

10

15

20

25

30

35

40

45

50

Element number

Pow

er [W

]

Fig. 8. Power dissipation in the different elements. Belt elements (solid), Treadelements (dashed)

30

−0.1 −0.05 0 0.05 0.1

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

x [m]

r [m

]

Fig. 9. The arrow indicates elements where a lot of power is consumed.

31