Tire ModelingLateral and Longitudinal Tire Forces

Gurkan Erdogan, PhD

April 27, 2009

Why Tires are important

for Vehicle Control Systems?

• Tires generate the forces that drive and maneuver the vehicle.

• The knowledge of magnitude, direction and limit of the tire

forces are essential and valuable for vehicle control systems.forces are essential and valuable for vehicle control systems.

• However, the estimation of these variables in all driving

conditions and in real-time is a very challenging task.

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Horizontal Tire Forces

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How tire force is generated?

• Tire forces are generated inside the contact patch, in

other words between the tire and the ground.

• Tire forces are a combination of two factors: • Tire forces are a combination of two factors:

– Friction/sliding in the contact patch, and

– Elastic deformations/slipping of the tire.

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Background – Friction Forces

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Background – Friction Forces

• Stribeck Effect

– Stribeck (1902) observed that the friction force is decreasing

continuously with increasing velocities for low velocities.

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Background – Elastic Force

xkF ×=( )εσ ×= E

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Background – Friction/Elastic Forces

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Classification of Tire Mathematical Models

• Based on how you attack the problem…

from experimental

data only

using similarity

methods

through simple

physical model

through complex

physical model

empirical theoretical

• Based on which time behavior you can capture…

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Steady State

Models

Dynamic (Transient)

Models

Tire Modelsthrough simple physical models

• There are four main players in tire modeling through simple physical models

1. Tread deflection (with and w/o)

2. Carcass/belt deflection (with and w/o)

3. Distribution of contact pressure (symmetric and asymmetric dist.)

4. Tire-road friction properties (variable friction)

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Tire Modelsthrough simple physical models

• Brush Model– BM (Rigid Carcass)

– BM + Linear Carcass Defection

– BM + Parabolic Carcass Defection

– BM + Asymmetric Carcass Deflection– BM + Asymmetric Carcass Deflection

• String Model– Stretched String (No Tread Element)

– BM + Stretched String

• Beam Model

Ideally:

Some Concepts - Instantaneous Center of Rotation

Reality:

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ωωωωw

Some Concepts – Effective Radius

tr

t

arV

eff

weffeff

φ

ω

=

==

Rstat

Rw

a

φ

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t

φa

reff =

weffstat RrR <<

Slip Ratio: ( )weffx

xweff

xrV

Vr

ω

ωσ

,max

−=

Brush Model – Pure Longitudinal Slipthrough simple physical models

Longitudinal Tire Deformation

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xweffo VrV −= ω

0=gV

Remember…

BM – Pure Longitudinal Slip

Brush Model – Pure Side Slipthrough simple physical models

weffr ω

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Vθδα −=Slip Angle:

Brush Model – Pure Side Slipthrough simple physical models

• Uniform Normal Force Distribution

weffr ω

Fz

( ) ∫−

==a

a

zzz dxqFa

Fxp

2

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weffr ω

• Uniform Normal Force Distribution

Brush Model – Pure Side Slipthrough simple physical models

( )( )

( ) ( )zz

py

zzy

FFM

ac

FFF

µµ

αµ

µtan8

32

'

2

2

−=

−=Lateral Tire Force :

Tire Moment:

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( )( )

( )( )

( )bs

z

py

py

z

py

zz

xF

ac

ac

F

ac

FM

αµ

αµ

αµ

tan2

tan48tan8 23

'

=

−=

Friction Coefficient :

Tire Moment:

( ) 21

tan

1

222

max

−=−=α

µµµµ

ac

F

a

FF

xyFF

py

zzz

bsbzy

• Parabolic Normal Force Distribution

weffr ω

Fz

Brush Model – Pure Side Slipthrough simple physical models

( ) ( )∫−

=

−=

a

a

zz dxxpF

a

xa

a

Fxp

2

22

4

3

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weffr ω

Brush Model – Pure Side Slipthrough simple physical models

• Parabolic Normal Force Distribution

αθδ =− V

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• Parabolic Normal Force Distribution

– Small Slip Angle (as αααα����0, tan(αααα)���� αααα)

( ) ( )( )α

αxa

xay

−=

−= tan

ycq pyy =

Brush Model – Pure Side Slipthrough simple physical models

( )

y

py

a

a

py

a

a

yz

Fa

aac

dxxxac

dxxqM

3

32 2

=

−=

−=

=

∫

∫

−

−

α

α( )

α

α

α

αF

py

a

a

py

a

a

yy

C

ac

dxxac

dxqF

=

=

−=

=

∫

∫

−

−

22

( )αxa −=

Brush Model – Pure Side Slipthrough simple physical models

• Parabolic Normal Force Distribution

– Large Slip Angles

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( )( )

( )( )

( )( )

+−=

slslsl

zy FFαα

αα

αα

µ2

2

tan

tan

3

1

tan

tan1

tan

tan3

( )slz

py

F

acαµ tan

3

2 2

=

Lateral Tire Force :

Friction Coefficient :

( )( )

( )( )

( )( )

( )( )

}|tan

tan|)

tan

tan(3|

tan

tan|31{

tan

tan 32'

slslslsl

zz aFMαα

αα

αα

αα

µ −+−−=Tire Moment (Lateral):

• Theoretical Slips

( )

( )

−=+

−=

=

−=−

=+

−=−

=

=

driveV

V

brakeV

V

driveV

V

V

VV

brakeV

V

V

VV

sy

x

sy

y

sxxr

x

sx

x

xr

x

κα

α

σ

κκ

κ

σ

1

tan

tan

1

Brush Model – Combined Slipthrough simple physical models

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−=+

−==+

driveV

driveVV

rrr κκ 11

weffr rV ω= ( )

( )

22

yx

y

x

xav

xau

σσσ

σ

σ

+=

−=

−=

+−=

2

2

3

113

mmm

zFFσσ

σσ

σσ

µTire Force :

}331{3

3

2

2'

mmmm

zz aFMσσ

σσ

σσ

σσ

µ −+−−=Tire Moment:

Brush Model – Combined Slipthrough simple physical models

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m

z

p

F

acσµ

3

2 2

=Friction Coefficient :

mmmm

Friction Circle:

• Dugoff’s Model

_

_

yliny

xlinx

CF

CF

=

=

σ

σ

α

κ

Brush Model – Combined Slipthrough simple physical models

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( )( )

2

_

2

_

2

_

2

_2

_

2

_

_

_2

_

2

_

_

2

2

2,

2

1

1,

2

linylinx

z

linyy

linxxzlinylinx

linyy

linxxzlinylinx

yliny

FF

F

FF

FFFFFif

FF

FFFFFif

CF

+=

−×=

−×=>+

×=

×=≤+

=

µ

λ

λλλλµ

µ

σα

String Modelthrough simple physical models

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• The steady state tire models are handy when we have constant linear and angular velocities.

• Experimental data used to validate the slip/force curves are obtained using specialized equipment that allows independent linear and angular velocity modulation so as to cover the whole slip range.

Dynamic (Transient) Tire Modelsthrough simple physical models

• This steady-state point of view is rarely true in reality, especially when the vehicle goes through continuous successive phases between acceleration and braking.

• Dynamic models capture the transient behavior of the tire-road contact forces under time-varying velocity conditions.– Bliman

– Kinematic TM

– Dahl TM

– LuGre TM

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Dynamic Models – Relaxation Length

• Relaxation length

– is related to the distance needed by the tire to reach a the steady state situation after a step change in slip.

– is the distance needed to build up the steady state tire forces.

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( )veFz

FF

coulomb

o

F

x

coulombo

frictionndeformatio

sgn1

−=

=

−σ

σ zF

vvz

coulomb

oσ−=&

Dynamic Models – Dahl Modelsthrough simple physical models

The Dahl model is essentially Coulomb friction with a lag.

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( )

( )vv

ze

vvez

dt

dF

dt

dF

coulomb

o

coulomb

o

F

x

F

x

frictionndeformatio

sgn

sgn

&

&

=

=

=

−

−

σ

σ

Derivation

( )vFz

zF

vv

zz

o

coulombss

ss

coulomb

o

ss

sgn

0

0

σ

σ

=

−=

⇒=&

Steady State:

( ) ( )

( ) ( ) ( )veFvfzz

orveFz

FF

vg

x

coulombo

vg

x

coulombo

frictionndeformatio

o

o

sgn1

,sgn1

1

−=++

−=

=

−

−

σ

σ

σσ

σ

&

Dynamic Models –LuGre Modelsthrough simple physical models

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( )z

vg

vvz

oσ−=&

( ) ( )( ) vvf

eFFFvg stribeckvv

coulombstaticcoulomb

2σ

α

=

−+= −

Tire Modelsfrom experimental data only

• Pacejka TM

• Burckhardt TM

• Kiencke and Daiss TM

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TIREBLACK BOX

Create

TIRE

DEFORMATION

Measure

TIRE FORCE

( )SlipfForce =

Pacejka Tire Model (Magic Formula)from experimental data only

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References

• Vehicle Dynamics and Control, 2005, R. Rajamani

• Tire and Vehicle Dynamics, 2005, H.B. Pacejka

• Contact Mechanics, 1987, K.L. Johnson

• Vehicle Dynamics: Theory and Application, 2009, R. N. Jazar

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THANKS …

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