Tyre Model TMeasy

CCG Seminar TV 4.08Tyre Models in Vehicle Dynamics: Theory and Application

Sept 20-21, 2010, Vienna

Wolfgang Hirschberg∗, Georg Rill†, Heinz Weinfurter‡

The present contribution is concerned with the semi-physical tyre model TMeasy for vehicledynamics and handling analyses. Even in the case of more or less weak testing input data,the effort for the application of TMeasy remains limited due to its consequent “easy to use”orientation. One particular feature of TMeasy is the widely physical meaning of its smartparameter set, which allows to sustain the identification process even under uncertain conditions.After a general introduction, the modeling concept of TMeasy is compactly described in thiscontribution. Taking the interface STI to MBS software into account, the way how to applyTMeasy is briefly shown. This includes selected examples of application.

1 Introduction

1.1 Overall concept of TMeasy

TMeasy is a semi-physical tyre model for low frequency applications in vehicle dynamics. It hasconsequently been following an “easy to use” strategy which takes the existing insufficiencies inthe availability of reliable testing data into account. In order to fulfil this aim, the number ofmodel parameters of TMeasy is rather limited according to the limited availability and accuracyof the basic experimental data. On the other hand the more or less direct physical meaningof the TMeasy model parameters enables their identification also in the case of unsure or evenincomplete measurement data sets.

Within TMeasy the contact force characteristics in longitudinal and lateral direction are de-scribed by a handful of physical parameters, which takes the degressive influence of decreasingtyre load into account. The load influence can be easily decsribed by providing these parame-ters for the nominal payload and its double value. The combined force characteristics are thendirectly generated via a generalised slip approach which does not need any additional fitting

∗Graz University of Technology, Austria†Regensburg University of Applied Sciences, Germany‡IBH, St.Ulrich/Steyr, Austria

1

1 Introduction

parameters. TMeasy also computes the tyre load as the third component of the tyre contactforces.

The self aligning torque is approximated by the product of lateral force and the pneumatictrail, which again is described by few physical parameters. TMeasy also includes the calculationof the rolling resistance torque, the tipping torque and the bore (turn) torque around the normalaxis. The approximative calculation of the contact geometry delivers the camber angle whichinfluence on the contact forces and torques is considered within a certain range of application.As long as the contact patch remains as a closed area TMeasy can handle uneven roads too.Typical applications are wave tracks, longitudinal grooves and change of road inclination, whichare offered as special road types additionally to the standard even road including an arbitrarysubarea of deviating friction.

In order to model the time delay at transient force changes TMeasy includes nonlinear firstorder dynamics for the longitudinal and lateral tyre forces as well as for the aligning torque andthe bore torque. The latter allows to consider the steering effort while parking. Again there areno additional parameters spent for the first order filters, whereas the variable relaxation lengthsare directly calculated from known physical tyre properties.

1.2 History

TMeasy was developed under the primary aspect of practical applicability in vehicle dynamics.The first version was published in [8]. It has to be mentioned that from that time some TMeasyderivatives are widely in use which are also named “TMeasy”. Known reproductions are used bydSPACE in the Vehicle Dynamics Simulation Package ASM [17] and in the program for accidentreconstruction PC-Crash [16]. Another implementation of TMeasy into Dymola is known, [18].Of course, these externally created versions are excluded from the official TMeasy softwaremaintenance.

Based on increasing experiences from passenger car and truck applications, an improved ver-sion 2.0 was created. The steady state part is published in [3] and the modeling concept fordynamic tyre forces and torques can be found in [9]. In 2003, enhancements in the discretisationof the contact geometry led to version 3.0. Besides some enhancements with respect to MBSintegration via the Standard Tyre Interface (STI), the version TMeasy 4.0 includes a parkingtorque model. This actual version was applied for the Tyre Performance Test programme TMPTwhich was organised by the Vienna University of Technology in 2007, [4].

A new version 5.0 is currently in preparation which include a sophisticated model for thecomputation of the dynamic tyre forces which provides a smooth transition from stand still tonormal driving situations.

1.3 Applications

For many years MAN Nutzfahrzeuge AG have been using TMeasy within the simulation systemSIMPACK [20] for the investigation of the dynamics and safety of heavy trucks, [5], [14]. In themeantime, TMeasy has been offered as an official, optional tyre model for vehicle dynamics inSIMPACK.

Furthermore, TMeasy is integrated as the standard tyre model into the simulation systemveDyna [21], which is widely applied for off-line and on-line simulation of road vehicles. Recently,the implementation of TMeasy into the MBS system Adams [15] was successfully carried outand validated, [1] [7] [2].

2

2 Modeling Concept

2 Modeling Concept

2.1 Contact Geometry

2.1.1 Local Track Plane and Geometric Contact Point

In any point of contact between the tyre and the road surface normal and friction forces aretransmitted. The effect of the contact forces can be fully described by a resulting force vectorapplied at a specific point of the contact patch and a torque vector.

−∆x

Q1Q2

P

en

M+∆x

unevenroad

undeflectedtire contour

longitudinalinclination

unevenroad

−∆y

undeflectedtire contour

Q4Q3 P

en

M

+∆y

lateralinclination

Figure 1: Inclination of local track plane in longitudinal and lateral direction

To calculate the geometric contact point an uneven road described by a function of twospatial coordinates z = z(x, y) is approximated by a local track plane. In order to get a goodapproximation to the local track inclination in longitudinal and lateral direction four points willbe used to determine the local track normal. The points Q1 to Q4 are placed on the track inthe front, in the rear, to the left, and to the right of the wheel center, Fig. 1. The vector rQ2Q1

pointing from Q1 to Q2 and the vector rQ4Q3 pointing from Q3 to Q4 define the inclination ofthe local track plane in longitudinal and lateral direction. Hence, the local track normal reads

en =rQ2Q1×rQ4Q3

| rQ2Q1×rQ4Q3 |. (1)

As in reality, sharp bends and discontinuities, which will occur at step- or ramp-sized obstacles,are smoothed by this approach.

The rim center plane is defined by the unit vector eyR into the direction of the wheel rotationaxis, Fig. 2. The unit vector ex into the direction of the intersection line of the local track planeand the rim center plane defines the direction of the longitudinal tyre force. The direction of thelateral tyre force is described by the unit vector ey which is mutual perpendicular to the tracknormal en and the unit vector ex into the direction longitudinal tyre force. The tyre camberangle

γ = arcsin(eTyR en

)(2)

describes the inclination of the wheel rotation axis eyR against the local track normal en. Thepoint P on the intersection line with the shortest distance to the wheel center M serves asgeometric contact point. Its location is described by the vector

r0P = r0M + rMP , (3)

where r0M defines the momentary position of the rim center M with respect to the earth-fixedreference frame 0 and the vector from the rim center to the geometric contact track point canbe written as

rMP = −rS ezR , (4)

3

2 Modeling Concept

road: z = z ( x , y )

eyR

M

en

0P

tire

x0

0y0

z0*P

P

x0

0y0

z0

eyR

M

en

ex

γ

ey

rimcentreplane

local road plane

ezR

rMP

wheelcarrier

0P ab

Figure 2: Contact geometry

where rS names the static tyre radius and the unit vector ezR = ex× eyR defines the radialdirection.

2.1.2 Static Contact Point

Assuming that the pressure distribution on a cambered tyre with full road contact correspondswith the trapezoidal shape of the deflected tyre area, the acting point of the resulting verticaltyre force FZ will be shifted from the geometric contact point P to the static contact point Q,Fig. 3.

en

P

γ

wC

rS

Q

Fzr0-rSL r0-rSR

yQ

ey

A

A∆z

en

γ

P

wC

Q

Fzy

ey

b/2

Figure 3: Lateral deviation of contact point at full and partial contact

The center of the trapezoidal area determines the lateral deviation yQ. As long as the tyre isin full contact with road, the relation

yQ = − b2

124ztan γ

cos γ(5)

will hold, where b is the width of the tyre, 4z denotes the tyre deflection and γ names the tyrecamber angle. If the cambered tyre has only a partial contact to the road then, according to

4

2 Modeling Concept

the deflection area a triangular pressure distribution will be assumed. Now, the location of thestatic contact point Q is given by

yQ = ±(

1

3wC −

b

2 cos γ

), (6)

where wC defines the actual width of the contact patch and the term b/(2 cos γ) describes thedistance from the geometric contact point P to the outer corner of the contact patch. The plussign holds for positive and the minus sign for negative camber angles. The static contact pointdescribed by the vector

r0Q = r0P + yQ ey (7)

represents the contact patch much better than the geometric contact point.

2.1.3 Contact Point Velocity

The absolute velocity of the static contact point will be obtained from

v0Q,0 = r0M,0 + rMQ,0 , (8)

where r0M,0 = v0M,0 denotes the absolute velocity of the wheel center and rMQ describes theposition of static contact point Q relative to the wheel center M . The vector rMQ contains thetyre deflection 4z normal to the road and it takes part on all those motions of the wheel carrierwhich do not contain elements of the wheel rotation. Hence, its time derivative can be calculatedfrom

rMQ,0 = ω∗0R,0×rMQ,0 + 4z en,0 , (9)

where ω∗0R is the angular velocity of the wheel rim without any component in the direction ofthe wheel rotation axis, 4z denotes the change of the tyre deflection, and en describes the roadnormal. As the static contact point Q lies on the track, v0Q,0 must not contain any componentnormal to the track

eTn,0 v0Q,0 = 0 or eTn,0(v0M,0 + ω∗0R,0×rMQ,0

)+ 4z eTn,0 en,0 = 0 . (10)

As en,0 is a unit vector, eTn,0 en,0 = 1 will hold, and then, the time derivative of the tyre deformationis simply given by

4z = − eTn,0(v0M,0 + ω∗0R,0×rMQ,0

). (11)

Finally, the components of the contact point velocity in longitudinal and lateral direction areobtained from

vx = eTx,0 v0Q,0 and vy = eTy,0 v0Q,0 . (12)

2.2 Wheel Load and Tipping Torque

The vertical tyre force Fz can be calculated as a function of the normal tyre deflection 4z andthe deflection velocity 4z. In a first approximation it is separated into a static and a dynamicpart

Fz(4z, 4z) = F Sz (4z) + FD

z (4z) . (13)

Because the tyre can only apply pressure forces to the road the normal force is restricted toFz ≥ 0. The static part is described as a nonlinear function of the normal tyre deflection

F Sz = a14z + a2 (4z)2 , (14)

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2 Modeling Concept

where the constants a1 and a2 may be calculated from the radial stiffness at nominal and doublepayload. The parabolic approximation in Eq. (14) fits very well to the measurements, [3]. Thedynamic part is roughly approximated by

FDz = dR4z , (15)

where dR is a constant describing the radial tyre damping, and the derivative of the tyre defor-mation 4z is given by Eq. (11).

The lateral shift of the vertical tyre force Fz from the geometric contact point P to the staticcontact point Q is equivalent to a force applied in P and a tipping torque Tx acting around alongitudinal axis in P , Fig. 4.

en

γ

P Q

Fzy

ey

en

γ

P

Fz

ey

Tx

∼en

γ

P

Q

Fzy

ey

Figure 4: Cambered tyre with full and partial contact

The use of the tipping torque instead of shifting the contact point is limited to those caseswhere the tyre has full or nearly full contact to the road. If the cambered tyre has only partlycontact to the road, the geometric contact point P may even be located outside the contact areawhereas the static contact point Q is still a real contact point.

2.3 Generalised Tyre Force

During general driving situations, e.g. acceleration or deceleration in curves, longitudinal slipand lateral slip defined by

sx =−(vx − rD Ω)

rD |Ω|and sy =

−vyrD |Ω|

(16)

appear simultaneously. Here, vx and vy are the components of the contact point velocity inlongitudinal and lateral direction, Ω describes the angular velocity of the wheel and rD is thedynamic rolling radius. Both slips can vectorially be added to a generalised slip

s =

√(sxsx

)2

+

(sysy

)2

=

√(sNx)2

+(sNy)2, (17)

where a normalization was performed, sx → sNx and sy → sNy , in order to achieve a nearly equallyweighted contribution to the generalised slip. If the wheel locks, the average transport velocitywill vanish, rD |Ω| = 0. Hence, longitudinal, lateral, and generalised slip will tend to infinity,s→∞. To avoid this problem, the normalised slips sNx and sNy are modified to

sNx =sxsx

=−(vx − rD Ω)

rD |Ω| sx⇒ sNx =

−(vx − rD Ω)

rD |Ω| sx + vN(18)

and

sNy =sysy

=−vy

rD |Ω| sy⇒ sNy =

−vyrD |Ω| sy + vN

. (19)

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2 Modeling Concept

When choosing small values vN > 0 the singularity at rD |Ω| = 0 is avoided. In addition, thegeneralised slip points then into the direction of the sliding velocity for a locked wheel. Innormal driving situations, where rD |Ω| vN holds, the differences between the primary andthe modified slips are hardly noticeable.

Fy

sx

ssy

S

ϕ

FS

M

FM

dF0

F(s)

dF

S

y

FyFy

M

SsyMsy

0

Fy

sy

dFx0

FxM Fx

SFx

sxM

sxS

sx

Fx

s

s

Figure 5: Generalised tyre characteristics

The graph F = F (s) of the generalised tyre force can be defined by the characteristic param-eters dF 0, sM , FM , sS and F S, Fig. 5. These parameters are calculated from the correspondingvalues of the longitudinal and lateral force characteristics. An elliptic function grants a smoothtransition from the characteristic curve of longitudinal to the curve of lateral forces in the rangeof ϕ = 0 to ϕ = 90. The longitudinal and the lateral forces follow then from the accordingprojections in longitudinal

Fx = F cosϕ = FsNxs

=F

ssNx = f sNx (20)

and lateral direction

Fy = F sinϕ = FsNys

=F

ssNy = f sNy , (21)

where f = F/s describes the global derivative of the generalised tyre force characteristics.The generalised tyre force characteristics F = F (s) is now approximated in intervals by

appropriate functions, Fig. 6. In the first interval a rational fraction is used which is defined bythe initial inclination dF 0 and the location sM and the magnitude FM of the maximum tyreforce. Then, the generalised tyre force characteristics is smoothly continued by two parabolasuntil it finally reaches the sliding area, were the generalised tyre force is simply approximatedby a straight line.

2.4 Self Aligning Torque

The distribution of the lateral forces over the contact patch length also defines the point ofapplication of the resulting lateral force. At small slip values this point lies behind the center

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2 Modeling Concept

FM

FS

F

dF0

sSsM s*

para-bola

straightline

parabola

rationalfunction

s

Figure 6: Approximation of generalised tyre characteristics

of the contact patch (contact point P). With increasing slip values it moves forward, sometimeseven before the center of the contact patch. At extreme slip values, when practically all particlesare sliding, the resulting force is applied at the center of the contact patch. The resulting lateralforce Fy with the dynamic tyre offset or pneumatic trail n as a lever arm generates the selfaligning torque

TS = −nFy . (22)

The dynamic tyre offset can be normalised by the length of the contact patch L.

n/L

0

sysySsy

0

(n/L)

n/L

0

sysy0

(n/L)

Figure 7: Normalised tyre offset with and without overshoot

The normalised dynamic tyre offset starts at sy = 0 with an initial value (n/L)0 > 0 and, ittends to zero, n/L → 0 at large slip values, sy ≥ sSy . Sometimes the normalised dynamic tyreoffset overshoots to negative values before it reaches zero again. This behavior can be modeledby introducing the slip values s0

y and sSy where the normalised dynamic tyre offset overshootsand reaches zero again as additional model parameter, Fig. 7.

2.5 Bore Torque

In particular during steering motions the angular velocity of the wheel has a component ωn 6= 0 indirection of the track normal which will cause a bore motion. If the wheel moves in longitudinaland lateral direction too then, a very complicated deflection profile of the tread particles in thecontact patch will occur. However, by a simple approach the resulting bore torque (also namedturn torque) can be approximated quite well by the parameters of the generalised tyre forcecharacteristics.

At first, the complex shape of a tyre’s contact patch roughly described by its length L andwidth B is approximated by a circle with the radius RP , Fig. 8.

At large bore motions all particles in the contact patch are sliding. Then, the maximum boretorque is given by

TmaxB =2

3RP F

S , (23)

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2 Modeling Concept

normal shape of contact patchcircularapproximation

ϕdϕ

F

RP

r

dr

ωn

B

L

ex

ey

Figure 8: Bore torque approximation

where F S denotes the maximum sliding force and RB = 23RP can be considered as the bore

radius of the contact patch.For small slip values the force transmitted in the patch element can be approximated by

F ≈ dF 0 s, where s denotes the slip of the patch element, and dF 0 is the initial inclination ofthe generalised tyre force characteristics. Similar to Eq. (16) one can define

s =−r ωnrD |Ω|

(24)

where r ωn describes the sliding velocity in the patch element and the term rD |Ω| represents theaverage transport velocity of the tread particles. By setting r = RB we get the average bore slip

sB =−RB ωn

rD |Ω|+ vN, (25)

where similar to (18) and (19) the artificial velocity vN ≥ 0 was added in the denominatorin order to avoid numerical problems at a locked wheel. Now, the bore torque can simply beapproximated by

TB = RB dF0 sB . (26)

Via the initial inclination dF 0 and the bore radius RB the bore torque TB automatically takesthe actual tyre properties into account. The bore torque is limited by its maximum value,|TB | ≤ TmaxB which is defined by (23).

2.6 Different Influences

2.6.1 Wheel Load

The resistance of a real tyre against deformations has the effect that with increasing wheel loadthe distribution of pressure over the contact patch becomes more and more uneven. The treadparticles are deflected just as they are transported through the contact patch. The pressurepeak in the front of the contact patch cannot be used, for these tread particles are far awayfrom the adhesion limit because of their small deflection. In the rear of the contact patch thepressure drop leads to a reduction of the maximally transmittable friction force. With risingimperfection of the pressure distribution over the contact patch, the ability to transmit forcesof friction between tyre and road lessens. In practice, this leads to a digressive influence of thewheel load on the characteristic curves of longitudinal and lateral forces.

9

2 Modeling Concept

In order to respect this fact in a tyre model, the characteristic data for two nominal wheel loadsFNz and 2FN

z will be provided. From this data the initial inclinations dF 0x , dF 0

y , the maximalforces FM

x , FMy and the sliding forces F S

x , F Sy for arbitrary wheel loads Fz are calculated by

quadratic functions. For the maximum longitudinal force it reads

FMx (Fz) =

FzFNz

[2FM

x (FNz )− 1

2FMx (2FN

z )−(FMx (FN

z )− 12FMx (2FN

z ))FzFNz

]. (27)

The location of the maxima sMx , sMy , and the slip values, sSx , sSy , at which full sliding appears, aredefined as linear functions of the wheel load Fz. For the location of the maximum longitudinalforce this will result in

sMx (Fz) = sMx (FNz ) +

(sMx (2FN

z )− sMx (FNz ))( Fz

FNz

− 1

). (28)

The self-aligning torque is modeled via the lateral force and the dynamic tyre offset. Thecharacteristic curve parameters describing the dynamic tyre offset will be provided for the singleand double pay load too. Similar to Eq. (28) the parameters for arbitrary wheel loads werecalculated by linear inter- or extrapolation.

2.6.2 Coefficient of Friction

The tyre characteristics are valid for one specific tyre road combination only. Hence, differenttyre road combinations will demand for different sets of model parameter. A reduced or changedfriction coefficient mainly influences the maximum force and the sliding force, whereas the initialinclination will remain unchanged. So, by setting

sM → µLµ0

sM , FM → µLµ0

FM , sS → µLµ0

sS , F S → µLµ0

F S , (29)

the essential tyre model parameter which primarily depend on the friction coefficient µ0 areadjusted to the new or a local friction coefficient µL.

If the road model will not only provide the unevenness information z = fR(x, y) but alsothe local friction coefficient [z, µL] = fR(x, y) then, braking on µ-split maneuvers can easily besimulated, [10].

2.6.3 Camber

At a cambered tyre, Fig. 9, the angular velocity of the wheel Ω has a component normal to theroad

Ωn = Ω sin γ , (30)

where γ denotes the camber angle. Now, the tread particles in the contact patch have a lateralvelocity which depends on their momentary position. At the contact point it vanishes whereasat the end of the contact patch it takes on the same value as at the beginning, however, pointinginto the opposite direction. Assuming that the tread particles stick to the track, a parabolicdeflection profile will be generated.

The lateral displacements of the tread particles caused by a cambered tyre are compared nowwith the ones caused by pure lateral slip. For small lateral slips the equivalent camber slip isgiven by

sγy =1

3sγ . (31)

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2 Modeling Concept

eyR

vγ(ξ)

rimcentreplane

Ω

γ

yγ(ξ)

Ωn

ξ

rD |Ω|ex

ey

en

Figure 9: Velocity state of tread particles at cambered tyre

Then, the lateral camber force can be modeled by

F γy =

∂dFy∂sy

∣∣∣∣sy=0

sγy , (32)

where∣∣F γ

y

∣∣ ≤ FM limits the camber force to the maximum tyre force. By replacing the partialderivative of the lateral tyre force at a vanishing lateral slip by the global derivative of thegeneralised tyre force the camber force will be automatically reduced when approaching thesliding area. By introducing a load dependent weighting factor in Eq. (32) the camber force canbe adjusted to measurements.

2.7 First Order Tyre Dynamics

Measurements [6] show that the dynamic reaction of the tyre forces and torques to disturbancescan be approximated quite well by first order systems.

tire

rim

dycy

Fyvyye

rim

tire

vx - rDΩFxxe

cxdx

Figure 10: Tyre deflection in longitudinal and lateral direction

The tyre forces Fx and Fy acting in the contact patch deflect the tyre in longitudinal andlateral direction, Fig. 10. In a first order approximation the dynamic tyre forces in longitudinaland lateral direction are given by

Fx (vx + xe)︸ ︷︷ ︸FDx

≈ Fx (vx)︸ ︷︷ ︸F Sx

+∂Fx∂vx

xe and Fy (vy + ye)︸ ︷︷ ︸FDy

≈ Fy (vy)︸ ︷︷ ︸F Sy

+∂Fy∂vy

ye , (33)

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2 Modeling Concept

where xe and ye name the longitudinal and the lateral tyre deflection. In steady state thelongitudinal tyre forces F S

x and F Sy will be provided by Eqs. (20) and (21) as functions of the

normalised slips sNx and sNy . Their derivatives with respect to the components of the contactpoint velocity are given by

∂F Sx

∂vx=

∂F Sx

∂sNx

∂sNx∂vx

=∂F S

x

∂sNx

−1

rD|Ω|sx + vN(34)

∂F Sy

∂vy=

∂F Sy

∂sNy

∂sNy∂vy

=∂F S

y

∂sNy

−1

rD|Ω|sy + vN(35)

where the definition of the normalised longitudinal slip in Eqs. (18) and (19) were used to generatethe derivatives of the slips with respect to the components of the contact point velocity. Corre-sponding to the first order approximations in Eq. (33) the partial derivatives of the steady statetyre forces with respect to the normalised slips will be approximated by their global derivatives

∂F Sx

∂sNx≈ F S

x

sNx=

f sNxsNx

= f and∂F S

y

∂sNy≈

F Sy

sNy=

f sNysNy

= f , (36)

Then, Eq. (33) will read as

FDx ≈ f sNx + f

−1

rD|Ω|sx + vNxe and FD

y ≈ f sNy + f−1

rD|Ω|sy + vNye , (37)

where according to Eqs. (20) and (21) the steady state tyre forces F Sx and F S

y were replaced bythe terms f sNx and f sNy . On the other hand, the dynamic tyre forces can be derived from

FDx = cx xe + dx xe and FD

y = cy ye + dy ye , (38)

where cx, cy and dx, dy denote stiffness and damping properties of the tyre in longitudinal andlateral direction. Inserting the normalised longitudinal slips defined by Eqs. (18) and (19) intothe Eq. (37) and combining them with Eq. (38) yields two first order differential equations forthe longitudinal and lateral tyre deflection

(v∗Tx dx + f ) xe = − f (vx − rD Ω) − v∗Tx cx xe , (39)(v∗Ty dy + f

)ye = − f vy − v∗Ty cy ye , (40)

where the modified transport velocities

v∗Tx = rD |Ω| sx + vN and v∗Ty = rD |Ω| sy + vN (41)

were introduced to shorten the equations.This first order dynamic tyre force model is completely characterised by the generalised steady

state tyre characteristics f , and the stiffness cx, cy and damping dx, dy properties of the tyre. Viathe steady state tyre characteristics the dynamics of the tyre deflections and hence the dynamicsof the tyre forces automatically depends on the wheel load Fz and the longitudinal and lateralslip. The differential equations (39) and (40) are even valid for locked wheels. But, at standstill the tyre deflections and in consequence the tyre forces too will decay exponentially in time.However, by a small modification [11] the differential equations can be transformed to a stickslip model which means that now tyre forces which are needed to compensate downhill forcesare perfectly maintained as long as the wheel is not rotating.

12

2 Modeling Concept

2.8 Torque Dynamics

Following the calculation of the maximum bore torque the contact patch can be reduced to anequivalent contact ring, Fig. 11. During bore motions the wheel rim rotates with the angle ϕWaround an axis normal to the contact patch. The position of the contact ring relative to thewheel is described by the twist angle ϕ.

dc

F

ϕW

ϕ

contact ring

wheel rim

RP

RB

C

Figure 11: Simple bore torque model

The contact ring with a radius which is equal to the bore radius RB is attached to the rimby a spring damper element with the torsional stiffness cϕ and the torsional damping dϕ Thedynamic bore torque is then given by

TDB = cϕ ϕ + dϕ ϕ . (42)

Similar to Eqs. (39) and (40) this model approach results in a first order differential equation forthe tyre twist angle ϕ(

dF0R2B + rD |Ω| dϕ

)ϕ = − dF0R

2B ϕW − rD |Ω| cϕ ϕ . (43)

At stand still (Ω = 0) the simple differential equation

ϕ = −ϕW (44)

remains here which means that the torsional tyre deflection ϕ is increased or decreased as longas steering motions ϕW 6= 0 are performed. But, the differential equation (44) is only valid aslong as the resulting bore torque does not exceed the maximum value. To take this effect intoaccount at first the steady state torque is limited |cϕ ϕ| ≤ TmaxB . Then, adhesion is assumedwhich is described by

ϕA = −dF0R2B ϕW + rD |Ω|T stB

dF0R2B + rD |Ω| dϕ

. (45)

The resulting dynamic bore torque

TDB = cϕ ϕ+ dϕ ϕA (46)

now allows to check for sliding which finally is done by

ϕ =

ϕA if |TDB | < TmaxB

0 if |TDB | ≥ TmaxB

(47)

This model approach provides a continuous transition from stand still, rD |Ω| = 0, to normaldriving situations, rD |Ω| > 0, [9].

13

3 Interface to MBS Software

3 Interface to MBS Software

The implementation of TMeasy into a multi-body simulation system can easily be done via theStandard Tyre Interface (STI, [12]). This interface (current version 1.4) is supported today bymost of the commercial simulation systems and allows the link of any STI-compatible tyre model,as far as they represent per definition vehicle dynamic models with an idealised contact point,cf. Fig. 12.

STIStandard Tyre

Interface

Simulationhost

Internalstandard

tyre model

USRMODTMeasy

ROADTMroad

F, M ... Force, moment w.r.t. wheel point coordinates

F, M ... Force, moment w.r.t. wheel carrier coordinates

F, M

F, M

Wheelmotion

x, y

Tyre type 1

[Tyre type 2] [optional]

SlipCamberDeflect.

z, µ(x, y)

P1

P1

P2

P2

[Model param.]

Model param.

Road param. Road type R

Figure 12: Implementation of TMeasy into a simulation program

The simulation program delivers the necessary wheel motion values in the sequence of wheelsWi, i = 1, 2 ... nW at every time step to STI, which are here transformed into the internal motionvalues of the applied tyre model and are passed to it. As output, STI delivers the actual vectorsof the tyre forces Fi and torques Mi in the specified form back to the simulation program. Onnecessity, from there they can be passed on to the according post processor as well as additionaltyre variables for any control purpose.

The coefficients of the chosen tyre type(s) and the road parameters, e.g. road geometry andfriction distribution [z, µL] = fR(x, y) are provided via independent model data. There are twodifferent ways to forward the model parameters. One may prefer the direct import of the neededdata from the selcted tyre and road parameter files such as depicted as path P1 in Fig. 12.The other way is to pass the set of parameters from the simulation host’s preprocessor via thetherefore reserved parameter arrays to the tyre model, marked as path P2.

A complete set of parameters for a vehicle model thus consists of at least one road file andone tyre file for each group of identical vehicle tyres, therefore, at least of one tyre file. Thecorrect assignment of the tyre to its related model body “wheel Wi” is defined in the model fileand again this is directed by STI.

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4 How to Apply TMeasy

4 How to Apply TMeasy

The following tasks have to be prepared before the application of TMeasy for any certain vehicledynamics simulation.

• Definition of the tyre type for each of the wheels

• Preparation of the parameter set for each tyre

• Definition of the road type

• Preparation of the road parameters for the selected road type

4.1 Identify the Parameters

First of all, a tyre property file is needed for TMeasy which contains its model parameters. Withmeasurement data from test rigs (Fx/sl, Fy/α, . . . ) the parameters can be easily identified. Asthe lateral force measurements usually correspond to the slip angle α, the longitudinal forcecharacteristics are related to different slip definitions sl. This process is supported by the utility

“TFView”, which is implemented in Matlab [19]. TFView reads all available measurement dataand plots it together with the approximated curves of TMeasy in one plot for each categorylike Fy/α. The measurement data should be provided in matrix-form as ASCII-files, such asstandardised in the Tydex format [12].

Figure 13: User utility TFView

A typical plot can be seen in Fig. 14, where the circle symbols represent the measurementdata, and the lines depict the approximated curves for several vertical loads.

A single curve is determined by a set of five parameters as shown in Fig. 15. Because oftheir physical meaning, these values can directly be obtained from measurement data. ApplyingTFView, the correctness of the identified parameters can then be checked with respect to theavailable measurement data set. The characteristic parameters for a lateral force graph are theinitial inclination (cornering stiffness) dF 0

y , location sMy and magnitude of the maximum FMy ,

begin of full sliding sGy and the sliding force FGy .

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4 How to Apply TMeasy

−20 −15 −10 −5 0 5 10 15 20−6000

−4000

−2000

0

2000

4000

6000

Slip angle AL [deg]

La

tera

l fo

rce

−F

y [

N]

Lateral Tyre Force

Radial 205/50 R15, 6J, p=2.0 bar, v=60.0 km/h, cam=2.0 deg

TMeasy tyre model 4.3.07 + STI V1.4 − Hirschberg, Rill, Weinfurter 14.02.2006TFView 2.2 (c) 2002 W.Hirschberg 29−Aug−2007

Fn=1800 N

Fn=3200 N

Fn=4600 N

Fn=6000 N

Approx 1

Approx 2

Approx 3

Approx 4

Figure 14: Side force characteristics

full slidingadhesion

adhesion/

sliding

Fy

M sy

Fy nomM

Fy nomS

sy nom sy nomS

dFy nomS

Figure 15: Lateral force graph

These five parameters have to be identified for the nominal vertical load Fz nom and for thedoubled vertical load 2Fz nom to take the degressive behaviour of side force capability due toincreasing tyre load into account. Therefore, a set of ten parameters describe the whole sideforce characteristics as shown in Fig. 14. This characteristic is plotted by selecting Fy/α inTFView as shown in Fig. 13. The identification of the parameters for the longitudinal forces Fxis done in the same way.

The self aligning torque is calculated by the product of lateral force and pneumatic trail whichis described by three additional parameters: The normalised pneumatic trail ptr0, ptrs as theside slip value where the pneumatic trail changes the sign, and ptrz where the trail tends to zero.Again those parameters have to be set for Fz nom and 2Fz nom.

The combined tyre force characteristics can be directly generated via the above shown gener-alised slip approach, which does not need any additional fitting parameters.

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4 How to Apply TMeasy

4.2 User Roads

TMeasy offers some basic road models which can be linked via the STI Road interface (currentversion SRI 1.2). As an example, beside the standard even road, the road type “track grooves”representing a pair of water grooves with depth t is shown in Fig. 16. Optionally, other specialroad types provided by the particular MBS application are available by direct MBS access.

3

Rd_spr.dat

x

y

ψ0

x0

C

xR

s

yR

tb

r

r

z0z

zR

yR

y

C

y0

O

O

Ebene Fahrbahn mit 2 Spurrillen in beliebiger Richtung Figure 16: Road type “track grooves”

4.3 Examples of Application

In the following, three selected SIMPACK [20] - TMeasy applications are shown.Firstly, a single wheel test bench is considered. The test rig model is used to evaluate tyre-

characteristics in simulation environments. Multiple models are available to verify all tyre char-acteristics during only one simulation. With the post-processing tool, the simulation data canbe plotted together with the related measurement data.

Overturning a passenger car: This example shows an overturning vehicle on a tilting platform.It demonstrates the possibility to handle high camber angles of the tyres. In the initial state,the car has got a small yaw angle. Therefore, the non-braked car starts slightly rolling whentilting the plate where it stands on. In the shown case, the passenger car overturns because thevehicle’s COG is set to a height of 1.0 m above ground. In the more realistic case of COGz=0.5m the vehicle is drifting downward after a short distance of rolling.

The driving manoeuvre “lane change in a limit situation” is a challenge for simulation solvers.The tyre model has to handle large slip angles, high amounts of camber angles and the possibilityof loosing and re-contacting the ground whilst saving calculation time. The example shows aMAN TGA 460 tractor-semitrailer combination with a gross vehicle weight of 40 tons. The lane

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4 How to Apply TMeasy

v

yC

zC

xC

C

Figure 17: Single wheel test bench

Figure 18: Overturning a car

change manoeuvre which is carried out within the range of 25 m at a speed of 80 km/h servesto investigate driving stability and rollover prevention with ESP-function [14].

Figure 19: Lane change of a tractor-trailer combination

The real-time capability of TMeasy is one condition to support any related application. Incase of using constant step size integrators, the maximum step size is limited by the accuracy andnumerical stability which are immediately influenced by tyre stiffness and the mass distribution

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4 How to Apply TMeasy

between tyre and chassis. Former investigations concerning real-time were published in [13].

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References

References

[1] L. Herak. Verification of semiphysical model of tyre TMeasy (Adams Implementation).Master thesis, Slovak University of Technology in Bratislava, 2008.

[2] W. Hirschberg, F. Palcak, G. Rill and J. Sotnik. Reliable vehicle dynamics simulation in spiteof uncertain input data. EAEC Conf. 2009 Europe in the Second Century of Auto Mobility,Bratislava: (CD) 2009.

[3] W. Hirschberg, G. Rill and H. Weinfurter. User-appropriate tyre-modeling for vehicle dy-namics in standard and limit situations. Vehicle System Dynamics, 38(2): 103–125, 2002.

[4] W. Hirschberg, G. Rill and H. Weinfurter. Tire Model TMeasy. Vehicle System Dynamics,Vol. 45, Supplement 1: 101–119, 2007.

[5] W. Hirschberg, H. Weinfurter and C. Jung. Ermittlung der Potenziale zur LKW-Stabilisierung durch Fahrdynamiksimulation. VDI-Berichte 1559, Dusseldorf: 167–188, 2000.

[6] P. van der Jagt. The Road to Virtual Vehicle Prototyping; new CAE-models for acceleratedvehicle dynamics development. ISBN 90-386-2552-9 NUGI 834, Tech. Univ. Eindhoven, 2000.

[7] P. Kintler. Validierung des Reifenmodells TMeasy mittels eines Vollfahrzeugmodells.Master thesis, Slovak University of Technology in Bratislava, 2009.

[8] G. Rill. Simulation von Kraftfahrzeugen. Vieweg Verlag, ISBN 3-528-08931-8, Braun-schweig/Wiesbaden, Deutschland, 1994.

[9] G. Rill. First order tire dynamics. In Proc. of the III European Conference on ComputationalMechanics Solids, Structures and Coupled Problems in Engineering, Lisbon, Portugal, 2006.

[10] G. Rill and C. Chucholowski. Modeling concepts for modern steering systems. In ECCOMASMultibody Dynamics, Madrid, Spain, 2005.

[11] G. Rill and C. Chucholowski. Real time simulation of large vehicle systems. In ECCOMASMultibody Dynamics, Milano, Italy, 2007.

[12] J.J.M. van Oosten et al. Tydex Workshop: Standardisation of Data Exchange in TyreTesting and Tyre Modelling. Proc. 2nd Int. Colloquium on Tyre Models for Vehicle DynamicAnalysis, Swets&Zeitlinger, Lisse 1997.

[13] S. R. Waser. Applikation des Reifenmodells TMeasy fur den Tyre Model Performance TestTMPT. Diploma thesis, Graz University of Technology, Austria, 2005.

[14] H. Weinfurter, W. Hirschberg and E. Hipp. Entwicklung einer Storgrossenkompensation furNutzfahrzeuge mittels Steer-by-Wire durch Simulation. In VDI-Berichte 1846, Dusseldorf:923–941, 2004.

[15] http://www.mscsoftware.com/, dated Aug 27, 2010.

[16] http://www.dsd.at/, dated Aug 27, 2010.

[17] http://www.dspace.de/ww/de/gmb/home/products/sw/automotive simulation models.cfm/, dated Aug 27, 2010.

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References

[18] http://www.dynasim.com/documents/0709 dysm fly 04vehi cl lo.pdf/,dated Sept 11, 2007.

[19] http://www.mathworks.com/, dated Aug 27, 2010.

[20] http://www.simpack.de/, dated Aug 27, 2010.

[21] http://www.tesis.de/, dated Aug 27, 2010.

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