DYNAMICTIREFORCESWITHSMOOTHTRANSITIONTO STAND-STILL

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7th EUROMECH Solid Mechanics Conference

J. Ambrosio et.al. (eds.)

Lisbon, Portugal, 711 September 2009

DYNAMIC TIRE FORCES WITH SMOOTH TRANSITION TO

STAND-STILL

G. Rill1 and W. Hirschberg2

1Regensburg University of Applied Sciences, GermanyFakultat Maschinenbau, Galgenbergstr. 30, 93053 Regensburg

e-mail: [email protected]

2Graz University of Technology, AustriaInstitut fur Fahrzeugtechnik, Inffeldgasse 11/I, 8010 Graz

e-mail: [email protected]

Keywords: Tire Modeling, Dynamic Tire Forces and Torques, Vehicle Dynamics.

Abstract. Sophisticated tests of electronic devices controlling the wheel slip include braking

to stand-still, drive-away and starting on a slope maneuvers. Today, costly and time-consuming

field tests are more and more supplemented or completely substituted by hardware in the loop

(HIL) simulation techniques. In vehicle dynamics the reliability and accuracy of the simulationresults strongly depend on the tire model. The handling tire model TMeasy [1] is characterized

by a useful compromise between user-friendliness, model-complexity and efficiency in compu-

tation time on the one hand, and precision in representation on the other hand. The slip based

description of the steady state tire forces and torques which is common to all handling tire

models usually causes numerical problems when approaching stand still [8]. By taking the

compliance of the tire into account the steady state approach can easily be extended to a first

order dynamic tire model. By small modifications in the slip definition the TMeasy approach

can handle not even hard braking maneuvers but also grants a smooth transition to stand-still.

The simulation results of a vehicle breaking to a full stop and moving downhill will document

this particular TMeasy model capacity.

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G. Rill and W. Hirschberg

1 INTRODUCTION

For the dynamic simulation of on-road vehicles, the model-element tire/road is of special

importance, according to its influence on the achievable results. It can be said that the suffi-

cient description of the interactions between tire and road is one of the most important tasks

of vehicle modeling, because all the other components of the chassis influence the vehicle dy-

namic properties via the tire contact forces and torques. Therefore, in the interest of balanced

modeling, the precision of the complete vehicle model should stand in reasonable relation to

the performance of the applied tire model. At present, two groups of models can be identified,

handling models and structural or high frequency models [4].

Complex tire models are computer time consuming and they need a lot of data. Usually,

they are used for stochastic vehicle vibrations occurring during rough road rides and causing

strength-relevant component loads [5].

Comparatively lean tire models are suitable for vehicle dynamics simulations, while, with

the exception of some elastic partial structures such as twist-beam axles in cars or the vehicle

frame in trucks, the elements of the vehicle structure can be seen as rigid. In contrast to purelyphysical tire models semi-physical tire models, will rely also on measured and observed force-

slip characteristics. This class of tire models is characterized by an useful compromise between

user-friendliness, model-complexity and efficiency in computation time on the one hand, and

precision in representation on the other hand, [2].

Measurements show that the dynamic reaction of the tire forces and torques to disturbances

can be approximated quite well by first order systems [3]. The TMeasy model approach auto-

matically generates a first order dynamic approximation to tire forces and torques with a smooth

transition to stand-still.

2 STEADY STATE TIRE FORCES AND TORQUES

2.1 Contact Geometry

Within TMeasy it is assumed that the contact patch is sufficiently flat. Four road points Q1toQ4located in the front, in the rear, to the left and to the right of the tire patch are used to definethe normal vector eNof a local track plane and to calculate the geometric contact point P onrough roads, Fig. 1. As in reality, sharp bends and discontinuities, which will occur at step- or

ramp-sized obstacles, are smoothed by this approach.

x

Q1Q2

P

en

M

+x

unevenroad

undeflectedtire contour

longitudinalinclination

unevenroad

y

undeflectedtire contour

Q4Q3 P

en

M

+y

lateralinclination

Figure 1: Track normal and geometric contact point on rough roads

The direction of the longitudinal and lateral force as well as the tire camber angle are then

derived from the direction of the wheel rotation axis and the track normal.

The tire deflectionzwhich normally is the difference between the undeflected tire radius

r0and the static radiusrSis calculated via equivalent deflection areas on a cambered tire, Fig. 2.

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rS

r0

eyR

en

P=Q

z

rSL

r0

eyR

en

P

rSR

r0

eyR

en

P

rSR

full contact partial contact

= 0 = 0 /

rS rS

Fz Fz

Q

Fz

Q

Figure 2: Tire deflection and static contact point

In consequence the geometric contact point P is shifted to the static contact pointQ wherethe resulting wheel loadFz will be applied.By taking into account that the tire deformation consists of the belt and flank deformation

a realistic approximation of the length L of the contact patch is possible. The dynamic rollingradius rDof the tire which is needed for average transport velocity of tread particles is calculatedby a weighted sum of the undeflected and the static tire radius.

2.2 Normal Force

The normal force or wheel load is separated into a static and a dynamic part

Fz = FSz +F

Dz . (1)

The static partFSz is described as a nonlinear function of the tire deflectionzand the dynamicpartFDz is roughly approximated by a damping force proportional to the time derivativezofthe tire deflection, [1]. Because the tire can only apply pressure forces to the road the normal

force is restricted toFz 0.

2.3 Longitudinal and Lateral Forces

The longitudinal force as a function of the longitudinal slipFx = Fx(sx)and the lateral forcedepending on the lateral slipFy = Fy(sy) are defined by characteristic parameters: the initialinclinationdF0x ,dF

0

y , the locationsMx ,s

My and the magnitude of the maximumF

Mx ,F

My as well

as the sliding limitsSx ,sSy and the sliding forceFSx,FSy , Fig. 3.A simple brush model delivers the longitudinal and lateral slip as

sx =(vx rD)

rD || and sy =

vyrD ||

(2)

wheredenotes the angular velocity of the wheel, rD describes the dynamic rolling radius andvx, vy are the components of the contact point velocity in the longitudinal and lateral direc-tion. In TMeasy both slips which in general driving situations will appear simultaneously are

vectorially added to the generalized slip

s = sxsx

2 + sysy

2 = sNx 2 + sNy 2 (3)3


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Fy

sx

ssy

S

FS

M

FM

dF0

F(s)

dF

S

y

FyFyM

SsyMsy

0

Fy

sy

dFx0

FxM Fx

SFx

sxM

sxS

sx

Fx

s

s

Figure 3: Simple and generalized tire characteristics

The longitudinal and lateral slip were normalized, sx sNx andsy s

Ny , in order to achieve

a nearly equally weighted contribution to the generalized slip. The normalizing factors sx andsy take characteristic properties of the longitudinal and lateral tire force characteristics intoaccount. If the longitudinal and the lateral tire characteristics do not differ too much, the nor-

malizing factors will be approximately equal to one.

If the wheel locks, the average transport velocity will vanish, rD || = 0. Hence, longitu-dinal, lateral, and generalized slip will tend to infinity, s . To avoid this problem, the

normalized slipssNx andsNy are modified to

sNx = sxsx

= (vx rD)

rD || sx sNx =

(vx rD)

rD || sx+vN(4)

and

sNy = sysy

= vyrD || sy

sNy = vy

rD || sy+ vN. (5)

In normal driving situations, whererD || vNholds, the difference between the original slipsand the modified slips are hardly noticeable. However, the fictitious velocity vN>0 avoids thesingularities at rD || = 0 and will produce in this particular case a generalized slip whichpoints exactly into the direction of the sliding velocity of a locked wheel.

By combining the longitudinal and lateral slip to a generalized slip s the combined forcecharacteristicF = F(s)can be automatically generated by the characteristic tire parameter inlongitudinal and lateral direction, [1]. In reverse, the longitudinal and lateral tire forces are then

given by the projection of the generalized force characteristic into the longitudinal and lateral

direction

Fx=Fx cos= FsNxs

=F

ssNx and Fy =F sinF

sNys

=F

ssNy (6)

whereF/srepresents the global derivative of the combined force characteristic.

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Then, Eqs. (7) and (8) will read as

FDx F

ssNx +

F

s

1

rD||sx+vNxe (13)

FDy FssNy + Fs

1rD||sy+vN

ye (14)

where according to Eq. (6) the steady state tire forces FSx andFSy were replaced by the terms

FssNx and

FssNy . On the other hand, the dynamic tire forces can be derived from

FDx = cxxe + dx xe (15)

FDy = cyye + dy ye (16)

wherecx,cy anddx,dy denote stiffness and damping properties of the tire in longitudinal andlateral direction. Inserting the normalized longitudinal slips defined by Eqs. (4) and (5) into

Eqs. (13) and (14) and combining them with Eqs. (15) and (16) yields two first order differential

equations for the longitudinal and lateral tire deflectiondx+

F

s

1

rD||sx+vN

xe =

F

s

(vx rD)

rD||sx+vN cxxe (17)

dy+F

s

1

rD||sy+vN

ye =

F

s

vyrD||sy+vN

cyye (18)

Multiplying these differential equations with the modified transport velocities

v

Tx = rD || sx+vN and v

Ty = rD || sy+vN (19)

finally results in

vTxdx +

F

s

xe =

F

s (vx rD) v

Txcxxe (20)vTydy +

F

s

ye =

F

svy v

Tycyye (21)

where Eqs. (15) and (16) will then provide the tire forces Fx = FDx andFy = F

Dy . A corre-

sponding dynamic model of the bore torque which is needed for simulating the parking effort is

described in Ref. [6].This first order dynamic tire force model is completely characterized by the combined force

characteristicF = F(s) as well as the stiffness cx, cy and damping dx, dy properties of thetire. Via the steady state tire characteristics the dynamics of the tire deflections and hence the

dynamics of the tire forces automatically depends on the wheel loadFzand the longitudinal andlateral slip.

3.2 Transition to Stand-Still

At stand still the contact point velocities vx,vy and the angular velocity of the wheel willvanish. At = 0 the fictitious velocityvN replaces the modified transport velocitiesv

Tx and

v

Ty defined in Eq. (19) and avoids the singularities in the normalized slips defined by Eqs. (4)

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forceFMx =3200Nat slipsMx = 0.1and the sliding forceF

Sx =3100Nats

Sx= 0.4. The fictitious

velocity was set tovN= 0.05m/s.The vehicle starts from stand still on a hill with the inclination angle = 15. At first it

rolls backwards (v r andv 0). Then, the vehicle is accelerated by a driving torque

TD > 0. After a short period a braking torqueTB < 0 is applied which causes the wheel tolock in an instant (r = 0). As described in [7] the braking torque is generated here by anenhanced dry friction model. When the vehicle comes to a stand still att 6s the enhanceddry friction model automatically provides a positive braking torque TB > 0 which preventsthe vehicle from moving downhill again. As the brake is not released yet the vehicle oscillates

some time in longitudinal direction. During this period where the wheel is locked the system

vehicle and tire represents a damped oscillator. The stiffness and damping properties of the

tire in longitudinal direction cx = 180000N/m anddx = 1500N/(m/s) together with thecorresponding overall vehicle mass ofm= 400 kg result here in a frequency off= 3.4Hz.

Att 8s the vibrations have completely ceased and the vehicle is in steady state. The tire

force ofFx = 1.0165kNwhich is needed to compensate the downhill force m g sin slowlydecays to the value ofFx = 1.0156kN in a time interval of80 s. This causes the vehicle tocreep from the stopping position atx= 32.615mdown tox= 32.572mwith a sliding velocityofv 0.6mm/sin same time interval. Even in this considerably large stopping interval thecreepage of the vehicle is hardly noticeable, Fig. 5.

Finally, att = 98s the brake is fully releasedTB = 0and the vehicle starts to roll downhillagain.

5 CONCLUSION

This simple but effective extension to first order dynamic tire forces and torques allows a

smooth transition from normal driving situations to stand still and keeps the dynamics of the

system finite. The simulation results show that it will serve as a good approximation to a

discontinuous stick slip model.

REFERENCES

[1] W. Hirschberg, G. Rill, H. Weinfurter, Tyre Model TMeasy. Vehicle System Dynamics,

Volume 45, Issue S1 2007, pages 101-119.

[2] W. Hirschberg, F. Paleak, G. Rill and J. otnk, Reliable Vehicle Dynamics Simulation

in Spite of Uncertain Input Data. In: Proceedings of 12th EAEC European Automotive

Congress, Bratislava, 2009.

[3] P. van der Jagt,The Road to Virtual Vehicle Prototyping; new CAE-models for accelerated

vehicle dynamics development, Tech. Univ. Eindhoven 2000, ISBN 90-386-2552-9 NUGI

834.

[4] P. Lugner and H. Pacejka and M. Plochl, Recent advances in tyre models and testing

procedures. Vehicle System Dynamics, 2005, Vol. 43, No. 67, pp. 413436.

[5] A. Riepl, W. Reinalter and G. Fruhmann,Rough Road Simulation with tire model RMOD-

K and FTire. Proc. of the 18th IAVSD Symposium on the Dynamics of vehicles on Roads

and on Tracks. Kanagawa, Japan, Taylor & Francis, London 2003.

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[6] G. Rill, First Order Tire Dynamics. In: Proceedings of the III European Conference on

Computational Mechanics Solids, Structures and Coupled Problems in Engineering. Lis-

bon, Portugal, 2006.

[7] G. Rill. A Modified Implicit Euler Algorithm for Solving Vehicle Dynamic Equations.

Multibody System Dynamics, Vol. 15, Issue 1, pp. 1-24, 2006.

[8] G. Rill, Wheel Dynamics. In: P.S. Varoto and M.A.Trindade (editors), Proceedings of

the XII International Symposium on Dynamic Problems of Mechanics (DINAME 2007),

ABCM, 2007.

[9] G. Rill, C. Chucholowski, Real Time Simulation of Large Vehicle Systems. ECCOMAS

Multibody Dynamics, Mailand, Italien 2007.

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