tire force and moment properties for … force and... · tire force and moment properties for...

TIRE FORCE AND MOMENT PROPERTIES FOR COMBINED SLIP CONDITIONS

CONSIDERING CAMBER EFFECT

Nan Xu, Konghui Guo

ASCL State Key Lab, Jilin University

Changchun Jilin, China

April 20-21, 2015

Introduction

Tire axis system and slip ratios

Key factors for developing the analytical tire model

Analytical tire forces and moments model

Simulation analysis and experiment validation

Conclusions

Outline

2

Tire mechanics characteristics under combined cornering/braking and camber situations have significant effects on the vehicle directional control.

The analytical models, which usually include carcass model with elastic tread elements, establish the relationship between tire structure parameters and tire behaviors.

Some key factors are considered in the model of this paper: arbitrary pressure distribution; translational, bending and twisting compliance of the carcass; effective carcass camber, dynamic friction coefficient; anisotropic stiffness properties and tire width.

Combined longitudinal slip, lateral slip and camber.

The analytical model in this paper is valuable for understanding tire properties and developing the semi-empirical models.

Introduction

3

Tire axis system and slip ratios

Fx

Zt

Yt

Xt

Tire revolution

axis Wheel traveling

direction

Fy

My

Mz

Fz

Mx

Tire revolution

direction

V

Ot

cossx ex

r e

V V RS

V R

Unified definition for slip ratios:

Sliding speed /updating speed

sinsy

y

r e

V VS

V R

4

Arbitrary pressure distribution

Unified 3 parameter expression of CPD: for diff

tire structures, loads, inflations & resistances

Key factors for developing the

analytical tire model

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

n=1

n=2

n=3

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

=0.5

=1

=2

η(u

)

Relative longitudinal coordinate u

=1

n=2

Relative longitudinal coordinate u

η(u

)

2 1 4 1

2 4 1

n nA

n n

3 2 3 4 3 4 1

2 1 4 1 4 3 3

n n nB

n n n a

uBuuAu nn 111)( 22

5

Carcass structure parameters

lateral translating deformation

lateral bending deformation

twisting deformation

longitudinal translating deformation

effective carcass camber


0 0c y cyy F K

( ) ( )cb y cby x F K u

( ) zy x M N x

0 0c x cxx F K

6

0cy

x

y

a a0

)(xycb

x

y

a a0

x

y

a

a0

)(xy

zF

xM

xM

zF

zq

e

yF

e c c y x

y x

F MF M

Dynamic friction coefficient


2 2exp log exps sd s m s h

sm sm

V VN

V V

vs

d

0

s

vsm

m h : describe the variation tendency

where, N(usually N=0.8) is a factor to make the friction coefficient increase slightly around the origin.

7

Anisotropic stiffness properties


F ad

F ad 180 200 220 240 260 280

180

200

220

240

260

280

Slip direction [deg]

Sh

ea

r fo

rce

dire

ctio

n

[de

g]

=14

=1

=4

measured force direction

slip direction

adhesion direction

.50-

0

180 200 220 240 260 280180

200

220

240

260

280


Sh

ea

r fo

rce

dire

ctio

n

[de

g]

=14

=1

=4

measured force direction

slip direction

adhesion direction

.50-

0

In sliding region:

In adhesion region:

X axis: Slip direction arctans y xS S

Y axis: Force direction arctanF y xF F

•Generally, contact area includes both adhesion & sliding region

•Resultant direction varied from pure adhesion to totally slip

arctanF ad y y x xK S K S

arctanF s y xS S

Adhesion direction

Slip direction

||↑

||↑

8

Deformation of carcass and tread element under combined slip condition


9

O

X

Y

V

o

x

y

xc0

yc0 － ycr

α

θa-aA

D

Pt

Vtcosα

VrtΔx

Δy

x

Pc

deflected carcass due to camber

C

B

wheel plane

belt

contact line wheel spin axis

tread element

Tire forces and moments without sliding


10

22

sin

sin

x t x

y y y y

z m y m

F a k S

F K S K

M K S K

2 1yK A A3 1yK A A

013

m b y y

aK K K

1 013

m b y y

aK K b K

1 1 1 3 3b x b x xA a S a S cF

2 0 01 3x y mA a S K K

3 1 0 1 0 21 3 3x y y x x tA a S b K ab K S F b r

2

0 2y tK a k3

0

2

3m tK a k

3

0

2

3y t

e

K a kR

0 0

1 1

cx cy

cK K

2b t cbak K 32

3ta k N

Tire forces and moments in general case with sliding region


11

1 9 9 4

2 2 1 2 6

4 2

3 1 10 11 3 5 1 2

4 1 5 7

11

1

2 2

31 ,

x z sx

x x z sx x

cb

tx z sx x t

e e

P B B S B F aN

P B B S B S a B F cFK

a k aP b B B S B B F b F b r

R R

P B P Ba

1 0 2 1 3 0

4 1 5 2 6

2

3 2 3

7 8 9

22 4 3 2

10 11

1 1 1, ,

2 2 2

1 1 1, ,

2 2 2

1 1 1, 2 3 , 1 3 2

4 4 4

1 1 1 1, 1

4 4 3 2

b c b c c

c c D c

c

c c c c

c c c cc

B D u B D u B m u

B m u B m u B m u

uB B u u B u u

u u u uB B u

7 3

1 7 0 5 9 0 1 8 1 0 5 3 1 3 5 4

2 5 1 4

2 7 0 4 9 0 2 8 1 0 4 3 2 3 4 4

2 5 1 4

sin

sin

x x x z sx

y m y y z sy

y

y m y y z sy

z

F B K S B F

PB K P B K S PB b K P P PB P B a FF

P P PP

P B K P B K S P B b K P P P B P B a FM

P P PP

Tire width effect


12

y

z

eR y

y a y

a y

e

vertical load, contact patch length and effective rolling radii have opposite variations in width.

e x exw

e e

R S yS y

R y

ew e eR y R y

mw m ez y z y

2 2w ew mwa y R y z y

2

2

w

zw z

a yF y F

a

0 0.2 0.4 0.6 0.8 13

4

5

6

7

8x 10

4

Corn

ering s

tiff

ness K

y [

N/r

ad]

Bending characteristic ratio b [-]

=0

cornering stiffness

Effect of carcass compliance on tire cornering stiffness


1 1 1 3 3b x b x xA a S a S cF

2 0 01 3x y mA a S K K

2 1yK A A

0

1

1 1y y

b

K K

0xS

bending characteristic ratio εb

twisting characteristic ratio εθ

The cornering stiffness will decrease

with the increase of εb or εθ

13

Cornering stiffness under combined slip condition


1 1 1

1 1 3 3

y x b

ypure b x b x x

K S

K S S a cF

-0.05 0 0.050.9

0.95

1

1.05

1.1

1.15

No

rma

lize

d c

orn

ering

stiff

ne

ss [

-]

Longitudinal slip ratio [-]

cornering stiffness will increase when tire has a slight braking. 14

Effect of carcass compliance on aligning moment under combined slip condition


0 0

1 1

cx cy

cK K

translating compliance coefficient:

-3000 -2000 -1000 0 1000 2000 3000-50

0

50

Alig

nin

g m

om

en

t M

z [

Nm

]

Longitudinal force Fx [N]

=2

Fz=3000N

c = 0

c = -5×10-6

c = -10×10-6

c = -15×10-6

15

180 200 220 240 260 280 300 320 340 360180

200

220

240

260

280

300

320

340

360R

esultant

forc

e d

irection [

deg]


=1

2

=1

2

4

4

braking

driving

adhesion direction

slip direction

resultant force direction

Simulation results of tire forces and moments under combined slip condition


Anisotropic stiffness properties. The anisotropy of tire slip stiffness will cause the variation of resultant force direction under different combined slip conditions.

16

Side slip with camber


The shift of lateral force The variation of curvature near the peak side force The severe asymmetry of aligning moment

17

-20 -10 0 10 20-3000

-2000

-1000

0

1000

2000

3000

Late

ral F

orc

e [

N]

Slip Angle [deg]

=-10

=10

0

Fz = 3000N

-20 -10 0 10 20-60

-40

-20

0

20

40

60

Alig

nin

g M

om

ent

[Nm

]

Slip Angle [deg]

=-10

=0

=10

Fz = 3000N

Longitudinal slip with camber


Side force has an extreme variation when applying braking/driving force, and Fy even changes its sign in the driving half of the diagram. The aligning torque, aroused by longitudinal force and shifted point of action due to camber, will generate an additional distortion of the carcass which results in an effective slip angle. 18

-1 -0.5 0 0.5 1-1000

-500

0

500

1000

Late

ral F

orc

e [

N]

Longitudinal Slip Ratio [-]

-2

-5

=-10

=10

Fz = 3000N

-1 -0.5 0 0.5 1-150

-100

-50

0

50

100

150

Alig

nin

g M

om

ent

[Nm

]


=-2

=-5

=-10

=10

Fz = 3000N

Combined slip with camber


With the increase of slip angle a , the longitudinal force will decrease because of the limitation of friction coefficient the sliding velocity dependent friction coefficient can be observed the carcass compliance and width effect are reflected by the asymmetry of Fy and Mz 19

-1 -0.5 0 0.5 1-3000

-2000

-1000

0

1000

2000

3000

Lo

ng

itu

din

al F

orc

e [

N]


-2

=-10

=8

Fz = 3000N

-1 -0.5 0 0.5 1-3000

-2000

-1000

0

1000

2000

3000

La

tera

l F

orc

e [

N]


-8

-4

-2

2

4

=-10

=8Fz = 3000N

-1 -0.5 0 0.5 1-200

-150

-100

-50

0

50

100

150

200

Alig

nin

g M

om

en

t [N

m]


=-10

=8

-8-2

2

Fz = 3000N

Experiment validation--side slip with camber


20

-30 -20 -10 0 10 20 30-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

Slip Angle [deg]

La

tera

l F

orc

e [

N]

= 10

Fz = 7200N

= -10

Test Data

Analytical Tire Model

-30 -20 -10 0 10 20 30-300

-200

-100

0

100

200

300

Slip Angle [deg]

Alig

nin

g M

om

ent

[Nm

]

= -10

= 10 Fz = 7200N

Test Data


Experiment validation--longitudinal slip with camber


21

-1 -0.5 0 0.5 1-600

-400

-200

0

200

400

600

800

Longitudinal slip ratio [-]

La

tera

l F

orc

e [

N]

Fz = 4000N

= 0, = -10

Test Data


-1 -0.5 0 0.5 1-300

-200

-100

0

100

200

300


Alig

nin

g M

om

ent

[Nm

] Fz = 4000N

= 0, = -10

Test Data


Experiment validation--combined slip with camber


22

-4000 -2000 0 2000 4000-3500

-3000

-2500

-2000

-1500

-1000

-500

0

Longitudinal Force [N]

La

tera

l F

orc

e [

N]

Fz = 4000N

= 4, = -5

Test Data


-1 -0.5 0 0.5 1-200

-150

-100

-50

0

50

100

150

200


Alig

nin

g M

om

ent

[Nm

]

Fz = 4000N

= 4, = -5

Test Data


Firstly, arbitrary pressure distribution, translational, bending andtwisting compliance of the carcass, effective carcass camber,dynamic friction coefficient anisotropic stiffness properties and tirewidth are the key factors for developing the analytical tire model.

Secondly, the considerable and interesting effects on tire force andmoment due to camber can be reflected well by the analyticalmodel. It will be very helpful for researchers to understand themechanism of tire force generation.

Thirdly, for variety of cases with camber, the severe asymmetryand dramatic variations of lateral force and aligning moment aremainly due to the carcass compliance and tire width.

Finally, considering all key factors, the analytical tire model iscapable of describing all kinds of tire properties reasonably andaccurately. The model parameters can also be identified from tiremeasurements and the computational results using the analyticalmodel show good agreement with test data.

Conclusions

23

The authors would like to thank the previous joint project between the Research and Development Center of General motors and the State Key Laboratory of Automotive Simulation and Control at Jilin University, from which the test data presented in this paper is produced.

Special thanks are due to the National Natural Science Foundation of China (51405185) and the National Basic Research Program of China (973 Program) (2011CB711201) for supporting authors’ research.

Acknowledgments

24

Any Questions?

Thanks for your time and attention!

Affiliation: ASCL State Key Lab, Jilin UniversityMailing address: No.5988 Renmin Avenue, Changchun, Jilin, 130025, P.R.ChinaEmail: [email protected], [email protected]: 0431-85095090-6108

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