1

TIRE FORCE AND MOMENT PROPERTIES FOR COMBINED SLIP CONDITIONS CONSIDERING CAMBER EFFECT

Nan Xu, Konghui Guo State Key Laboratory of Automotive Simulation and Control, Jilin University,

No.5988 Renming Street Changchun Jilin, China xu.nan0612@gmail.com

In this paper, an analytical tire model with flexible carcass for combined longitudinal, side slip and camber is presented, which can describe tire behaviors well and can also be used conveniently for studying vehicle dynamics. Some key factors are considered in the developed model to describe complicated tire characteristics and tire-road friction: arbitrary pressure distribution; translational, bending and twisting compliance of the carcass; effective carcass camber; dynamic friction coefficient; anisotropic stiffness properties and tire width. The complex tire force and moment under combined slip conditions can be described accurately by considering these key modelling factors.

1 INTRODUCTION An accurate description of tire properties for combined longitudinal and lateral slip is essentially necessary for vehicle simulation and directional controls [1-3]. And the camber also has important effect on tire force and moment properties. Variety of tire models has been proposed for use in calculating these properties at the tire-road interface [4-12]. In general, steady state tire models can be divided into two main categories, namely empirical and physical models [2]. Empirical models rely basically on curve-fitted experimental data and can provide a good representation of experimental data for longitudinal force, lateral force and aligning moment, which are often employed for the vehicle simulation and control. The most widely used empirical model is the magic formula initially conceived by Pacejka and Bakker [4]. However, empirical models would become quite complicated and have numerous parameters while considering all kinds of operating variables, such as slip angle, inclination angle, slip ratio, vertical load, inflation pressure, road friction, and rolling speed. On the other hand, analytical models are concerned with the mechanism of tyre force generation, which usually include carcass model (beam, string or rigid) with elastic tread elements, such as Brush Model, Fiala Model and String Model [2]. So it can be employed to derive some useful qualitative conclusions for understanding tire properties, but most of these models are either relatively simple or more complicated, which limit their practical use. In this paper, we don’t see the carcass as an actual beam or string, and the carcass deformation is described with relatively simple and general forms, composed by translational, bending and twisting deformations. Besides, the arbitrary pressure distribution, effective carcass camber, dynamic friction coefficient, anisotropic stiffness properties and tire width are also considered. Consequently, the analytical model would become more suitable for application and also appropriate for analyzing tire properties in detail. In this paper, the key factors for developing the analytical tire model are firstly discussed; then, considering all these factors, the analytical tire model with flexible carcass for combined slips is introduced. By employing the model, effects of carcass compliance on tire properties are discussed, which are valuable for understanding tire properties. Some important and interesting phenomena induced by camber can also be observed clearly through this model; at the end, the analytical model is validated by test data.

2 TIRE AXIS SYSTEM AND SLIP RATIOS The tire axis system employed in this paper is shown in Fig.1. The positive direction of X-axis and Y-axis are coincident with tire revolution direction; the Z axis is perpendicular to road plane and upward. The wheel traveling speed is denoted as V, is slip angle and is inclination angle. The figure shows all the forces and moments associated with a wheel.

2

Fx

Z

Y

X

Tire revolutionaxis Wheel traveling

direction

Fy

My

Mz

Fz

MxTire revolution

direction

V

O

Figure 1: Tire axis system

The longitudinal and lateral slip ratios are defined in the unified form [5]: cos

= ,

sin ,

sx ex x

e e

syy y

e e

V V RS S

R RV VS SR R

where, is the angular speed, Re is the effective rolling radius. Vsx and Vsy are the longitudinal and lateral sliding speeds of tire with respect to road surface.

3 KEY FACTORS FOR ANALYTICAL TIRE MODEL

3.1 Arbitrary pressure distribution Contact pressure distribution over contact patch strongly influences tire behaviors. Here, the contact pressure distribution is assumed to be uniform in lateral direction and of arbitrary form in circumferential direction to represent different kinds of operating conditions. The contact pressure qz along the contact patch length 2a is expressed as an arbitrary form as

2z

zF

q x ua

(2)

where Fz is the tire vertical load; u=x/a is the relative coordinate. (u) is the normalized pressure distribution function, and is expressed as:

2 2( ) 1 1 1

2 1 4 1 3 2 3 4 3 4 1,

2 4 1 2 1 4 1 4 3 3

n nu A u u B u

n n n n nA B

n n n n n a

(3)

where, n, and Δ are the parameters which determine the shape of pressure distribution. With these three parameters, Eq.(3) can be employed to express arbitrary pressure distribution over contact patch, shown in Fig.2.

-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=1n=2n=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

=1n=2

Relative longitudinal coordinate u

η(u)

Figure 2: Pressure distribution

3.2 Carcass structure parameters It is well known that the deformation of carcass has an important effect on tire properties, especially for combined slip conditions. In this paper, three stiffness parameters for tire lateral carcass deformation are introduced, that is, lateral translation stiffness of tire carcass Kcy0, carcass bending stiffness Kcb and

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carcass twisting stiffness N . The corresponding carcass deformations including lateral translating part yc0, bending part ycb, twisting part y , are shown in Fig.3. The carcass longitudinal deformation is assumed to be the longitudinal translating part xc0. The lateral translating deformation of carcass can be calculated as

0 0c y cyy F K (4) where, Kcy0 is the lateral translation stiffness of carcass; Fy is the lateral force. The lateral bending deformation of carcass can be calculated as

ycb

cb

F xy xK a

(5)

where Kcb is the carcass bending stiffness; ξ(x/a) is the general function of carcass bending deformation. The zero-order moment and first-order moment of ξ(u) are expressed as:

1 1

0 1( ) ( ) , ( ) ( )u u

D u u du D u u u du (6)

The twisting deformation of carcass is calculated by y x x (7)

where θ=Mz /N is carcass twisting angle; N is carcass twisting stiffness; Mz is aligning moment. The longitudinal translating deformation of carcass is expressed as

0 0c x cxx F K (8) where, Kcx0 is the longitudinal translation stiffness of carcass; Fx is the longitudinal force.

0cy

x

y

a a0)(xycb

x

y

a a0 x

y

aa0

)(xy

Figure 3: Tire lateral carcass deformation

3.3 Effective carcass camber For a cambered tire, overturning moment is generated due to the contact pressure distribution tends to the direction of the wheel cambered. With the effect of lateral force and overturning moment, an additional camber angle will be added to the tire, forming the effective carcass camber e (shown in figure 4) known as the ‘effect of effective carcass camber’[13]. This will lead to the actual camber smaller than the nominal value. The general shape of the deflected carcass centre line due to camber is similar to a parabola. Therefore, in the contact range the deformation resulting from the camber angle may be represented by [10]

2 2

0 1 sin2 e

a xy bR

(9)

where is the nominal camber; b1 is a correction factor which usually will be smaller than unity.

zFxM

xM

zF

zq

e

yF

Figure 4: Effective camber effect

3.4 Dynamic friction coefficient The dynamic friction coefficient is used in the analytical tire model, which considers the significant influence of slip speed. The expression is as follows [14]:

4

2 2exp log exps sd s m s h

sm sm

V VN

V V (10)

where μ0,μs, μh and Vsm are friction characteristic parameters; N(usually N=0.8)is a factor to make the friction coefficient increase slightly around the origin; Vs is the slip speed between the road and tire.

3.5 Anisotropic stiffness properties The anisotropy of tire slip stiffness will arouse the difference of tire shear stress direction in adhesion region and sliding region, which are expressed as [15]:

: arctan

: arctan

ad y y x x

s y x

adhesion K S K S

sliding S S (11)

where Kx, Ky represent the longitudinal slip and cornering stiffness respectively.

4 ANALYTICAL TIRE FORCES AND MOMENTS MODEL

4.1 Analytical tire model without tire width effect For now, we assume a thin tire model with tire width equals zero, in order to simplify the modelling and reveal most of tire properties clearly. The influence of tire width, primarily the aligning torque when tire cambered, will be discussed in the next part.

Tire forces and moments without sliding In order to obtain the shear force in the contact patch, the deformations of the tread and carcass along X and Y axes must be known firstly. The longitudinal and lateral deformations of the tread and carcass, under combined slip condition, are shown in Fig.5. In this figure, XOY is a coordinate system before the carcass is deformed, and xoy is a relative coordinate system for describing the tread deformation and bending and twisting deformation of carcass. xc0 and yc0 are the longitudinal and lateral translating deformation of carcass. ‘ABC’ is the contact line of the contact patch under combined slip condition. V is the wheel travelling speed; α is the slip angle and θ is the carcass twisting angle. In general case, the whole length of contact patch, 2a, is divided into two parts, the adhesion region ‘AB’, and the sliding region ‘BC’, by the initial sliding point ‘B’.

OX

YV

ox

y

xc0

yc0 ycr

α

θa-aA

D

Pt

Vtcosα

VrtΔx

Δy

x

Pc

deflectedcarcass dueto camber

C

B

wheel plane

belt

contact line wheel spin axis

tread element

Figure 5: Deformation of carcass and tread element under combined slip condition

PcPt in the figure represents the tread element after rolling for a period of time t. The upper point of tread element, Pc, is attached to the belt of tire. Its coordinate could be written as

2 2

1 sin2

pc

pc cbe

x x

a xy y x y x bR

(12)

The lower point of tread element, Pt, is contacted with the ground. Its coordinate could be written as cos

sinpt r x

pt y

x x Vt V t x a x S

y Vt a a a x S (13)

Therefore, the longitudinal and lateral deformations of tread element is

5

2 2

1 sin2

pt pc x

yzpt pc y

cb e

x x x a x S

FM x a xy y y a x S bN K a R

(14)

The main source of anisotropy is due to different tire structural flexibility in lateral and longitudinal direction, whereas tread anisotropy is present but comparatively small. In this paper, the stiffness of tread, denoted as kt, is considered to be isotropic. The shear stresses of tread element in X and Y directions can be expressed as:

2 2

1 sin2

x t t x

yzy t t y

cb e

q k x k a x S

FM x a xq k y k a x S bN K a R

(15)

The shear force in the direction X and Y can be determined as follows:

0 0

a

x xaa

y yaa a

z x pc y pc x c c y ca a

F q dx

F q dx

M q y dx q x dx F y y F x

(16)

where 2 sinc ty b r is the displacement of a point with force Fx due to camber [16]. rt is the radius of curvature of the tread, and b2 is a correction factor. Substituting into the previous equations (4)~(8), (12) and (15), and converting into explicit expression, the forces and moments could be expressed as

sin

sin

x x x

y y y y

z m y m

F K SF K S K

M K S K

(17)

with

2

0 0

1 0 1 0 2

0

1 0

21 3

1 1 3 3

1 3 31 1 3 3

13

13

x t

x y my

b x b x x

x y y x x ty

b x b x x

m b y y

m b y y

K a ka S K K

Ka S a S cF

a S b K ab K S F b rK

a S a S cFaK K K

aK K b K

(18)

where Kx is the longitudinal slip stiffness, Ky is the cornering stiffness, Kyγ is the inclination stiffness, Km is the aligning stiffness and Kmγ is the aligning stiffness contributed by camber. Ky0, Kyγ, and Km0 are the cornering stiffness, inclination stiffness and aligning stiffness respectively when the carcass is assumed to be rigid, which are expressed as:

2 3 30 0 0

2 22 ,3 3y t m t y t

e

K a k K a k K a kR

(19)

The bending characteristic ratio and twisting characteristic ratio have been introduced that are defined by:

322 ,3b t cb tak K a k N (20)

c in the Eq.(18) is the translating compliance coefficient that is defined by:

0 0

1 1

cx cy

cK K

(21)

6

Tire forces and moments in general case with sliding region Considering that the sliding region might exist, the shear force in this area should be expressed as q(x) =μqz(x), based on which the coordinate of initial sliding point, x=xc, or the relative coordinate uc=xc/a, can be solved. The coordinate of initial sliding point will satisfy the following equation:

2 2

2z

x y cF

q q q ua

(22)

Integrating through both adhesion and sliding regions, the tire forces and moments can be derived as,

0 0

2

2

2

2

c

c

c

c

c

c c

c

a x zx t sxx a

a x zy t syx a

a a x zz t pc t pc sx pcx x a

x zsy pc x c c y ca

F xF k xdx dxa aF xF k ydx dxa a

F xM k x y dx k y x dx y dxa a

F x x dx F y y F xa a

(23)

where θsx and θsy are, respectively, the longitudinal and lateral components of shear stress direction θs in sliding region. Substituted with Eq.(4)~(11), (12) and (14), Eq.(23) becomes,

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 syy

y m y y z syz

F B K S B F

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

P P P P

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

P P P P

(24)

with

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

2 231 ,

x z sx

x x z sx xcb

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 rR R

P B P Ba

(25)

and

1 0 2 1 3 0

4 1 5 2 6

23 2 3

7 8 9

22 4 3 2

10 11

1 1 1, ,2 2 21 1 1, ,2 2 21 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

cc 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

(26)

where m0(uc), m1(uc), m2(uc) and mD(uc) are defined by:

0 1= cu

cm u u du , 1 1= cu

cm u u u du , 22 1

= cu

cm u u u du , 1

= cu

D cm u u u du (27)

4.2 Analytical tire model with tire width effect For a cambered tire, the width of the contact patch has a considerable effect on the tire properties. The

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vertical load, contact patch length and effective rolling radii are all different and have opposite variations in width, as shown in Fig.6. The longitudinal slip ratios are different for the case of camber because of a difference in effective rolling radii. Consequently, braking force is generated on one side and driving force on the other side, which lead to a considerable torque and indirectly on the side force.

y

z

eR y

ya y

a y

e

Figure 6: Variation of contact patch length and effective rolling radii due to camber

To include the tire width effect, the effective rolling radii of a cambered tire could be expressed by ew e eR y R y (28)

where e is the effective carcass camber; y is the lateral coordinate in the contact patch. According the Eq.(9), 1 1arcsin sine b b . The radial deflection of tire changes in width,

mw m ez y z y (29) Therefore, the half length of contact patch could be obtained,

2 2w ew mwa y R y z y (30) And the vertical load can be expressed as

2

2w

zw z

a yF y F

a (31)

The longitudinal slip ratios along the width direction for the case of camber are calculated by e e xew x

xwew e e

R y VR y VS y

R y R y (32)

Considering the nominal longitudinal slip ratio x e x eS R V R , which can also be seen as the slip ratio of the wheel center plane, the Eq.(32) can be rewritten as

e x exw

e e

R S yS y

R y (33)

According the definition of slip ratios, we can also get the lateral slip ratios, 1 tanyw xwS y S y (34)

To get the expressions of tire model and calculate the tire forces and moments specifically, we consider a model with a left and a right row of tread elements which have the same distance from the wheel center plane, positioned at a distance yb1 and yb2 (yb1=-yb2). According to Eq.(28)~Eq.(34), the corresponding effective rolling radii(Re1, Re2), half length of contact patch(a1, a2), vertical load(Fz1, Fz2), longitudinal and lateral slip ratio(Sx1, Sx2, Sy1, Sy2) could be obtained. Similar to the modelling process of the analytical model without width effect in section 4.1, we can get the models respectively for the position yb1 and yb2. Then, by adding the longitudinal forces, lateral forces and aligning moments of the two rows, the total tire properties could be obtained. It is noticed that the total aligning moment should be calculated relative to the tire contact center located on the wheel center plane.

5 SIMULATION ANALYSIS AND EXPERIMENT VALIDATION According to the analytical tire model, the tire forces and moments under combined slip conditions can be simulated and the effects of carcass structure parameters can be analyzed. Furthermore, the analytical

8

tire model can also be used in vehicle dynamics simulation by identifying the model parameters with test data.

5.1 Simulation analysis The tire properties under combined slip conditions without camber and the corresponding influence of carcass compliance have been analysed by authors in [17]. In brief, the cornering stiffness will decrease, and the lateral force and aligning moment will become asymmetric, severely for Mz, by carcass compliance under combined braking/driving and cornering conditions. In this paper, all kinds of operating conditions with camber effect will be discussed in detail.

Side slip with camber Tire properties for side slip with camber could be calculated by assuming Sx=0 with the analytical tire model. Simulation results are shown in Fig.7-Fig.8. The curves show good qualitative agreement with measured characteristics. The shift of lateral force, including the variation of curvature near the peak side force, is revealed clearly in Fig.7. The severe asymmetry of aligning moment and the width effect reflected by different Mz at α=0, can also be described satisfactorily in Fig.8.

-20 -10 0 10 20-3000

-2000

-1000

0

1000

2000

3000

Late

ral F

orce

[N

]

Slip Angle [deg]

=-10

=10

0

Fz = 3000N

-20 -10 0 10 20-60

-40

-20

0

20

40

60

Alig

ning

Mom

ent

[Nm

]

Slip Angle [deg]

=-10

=0

=10Fz = 3000N

Figure 7: Lateral force Figure 8: Aligning moment

Longitudinal slip with camber Tire properties for longitudinal slip with camber are calculated by setting α=0. Simulation results are shown in Fig.9-Fig.10. The figure of longitudinal force hasn’t been given because of the tiny influence due to camber. Considerable and interesting effects can be found in side force, which has an extreme variation when applying braking/driving force, and Fy even changes its sign in the driving half of the diagram. The mechanism of this phenomenon can easily be understood by the analytical tire model, in brief, 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. The corresponding aligning moment is shown in Fig.8. These entire phenomenons correspond reasonably well with the experimental results given later in this paper.

-1 -0.5 0 0.5 1-1000

-500

0

500

1000

Late

ral F

orce

[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

ning

Mom

ent

[Nm

]

Longitudinal Slip Ratio [-]

=-2

=-5

=-10

=10

Fz = 3000N

Figure 9: Lateral force Figure 10: Aligning moment

Combined slip with camber In this part, simulated results of combined slip with camber using the analytical tire model have been presented. Fig.11 shows the longitudinal force under combined slip conditions. With the increase of slip

9

angle , the longitudinal force will decrease because of the limitation of friction force between the tire and the road. More important, because we adopt the dynamic friction coefficient as shown in Eq.(10), the sliding velocity dependent friction coefficient can be seen obviously in Fig.11. The lateral force and aligning moment are shown in Fig.12-Fig.13, the carcass compliance and width effect are reflected by the asymmetry of Fy and Mz, as discussed in previous sections. The obvious variation of Fy when slip angle is small in Fig.12 are just due to distortion of the carcass.

-1 -0.5 0 0.5 1-3000

-2000

-1000

0

1000

2000

3000

Long

itudi

nal F

orce

[N

]

Longitudinal Slip Ratio [-]

-2

=-10

=8

Fz = 3000N

-1 -0.5 0 0.5 1-3000

-2000

-1000

0

1000

2000

3000

Late

ral F

orce

[N

]

Longitudinal Slip Ratio [-]

-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

ning

Mom

ent

[Nm

]

Longitudinal Slip Ratio [-]

=-10

=8

-8-2

2

Fz = 3000N

Figure 11: Longitudinal force Figure 12: Lateral force Figure 13: Aligning moment

5.2 Experiment validation Test data for the P245/65R17 specification at different combined slip conditions are used to validate the analytical tire model.Fig.14 and Fig.15 show the tire forces and moments respectively under different operating conditions. These figures show that the analytical tire model can describe the tire characteristics satisfactorily.

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

-6000

-4000

-2000

0

2000

4000

6000

8000

Slip Angle [deg]

Late

ral F

orce

[N

]

= 10

Fz = 7200N = -10

Test DataAnalytical Tire Model

-1 -0.5 0 0.5 1-600

-400

-200

0

200

400

600

800

Longitudinal slip ratio [-]

Late

ral F

orce

[N

]

Fz = 4000N = 0 , = -10

Test DataAnalytical Tire Model

-4000 -2000 0 2000 4000-3500

-3000

-2500

-2000

-1500

-1000

-500

0

Longitudinal Force [N]

Late

ral F

orce

[N

]

Fz = 4000N = 4 , = -5

Test DataAnalytical Tire Model

(a) side slip with camber (b)longitudinal slip with camber (c) combined slip with camber

Figure 14: Tire force characteristics at different conditions

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

-200

-100

0

100

200

300

Slip Angle [deg]

Alig

ning

Mom

ent

[Nm

]

= -10

= 10 Fz = 7200N

Test DataAnalytical Tire Model

-1 -0.5 0 0.5 1-300

-200

-100

0

100

200

300

Longitudinal Slip Ratio [-]

Alig

ning

Mom

ent

[Nm

] Fz = 4000N = 0 , = -10

Test DataAnalytical Tire Model

-1 -0.5 0 0.5 1-200

-150

-100

-50

0

50

100

150

200

Longitudinal Slip Ratio [-]

Alig

ning

Mom

ent

[Nm

]

Fz = 4000N = 4 , = -5

Test DataAnalytical Tire Model

(a) side slip with camber (b)longitudinal slip with camber (c) combined slip with camber

Figure 15: Tire moment characteristics at different conditions

6 CONCLUSIONS This paper presented an analytical tire model for combined slips considering camber effect. Some key factors are considered in the developed model to describe complicated tire characteristics and tire-road friction. With employing the model, the complex and interesting tire properties of side slip, longitudinal slip and combined slip with camber are discussed. The simulation results and experiment validation of these properties are also provided. Some conclusions are summarized as follows: Firstly, arbitrary pressure distribution, translational, bending and twisting compliance of the carcass, effective carcass camber, dynamic friction coefficient anisotropic stiffness properties and tire width are the key factors for developing the analytical tire model.

10

Secondly, the considerable and interesting effects on tire force and moment due to camber can be reflected well by the analytical model. It will be very helpful for researchers to understand the mechanism of tire force generation. For variety of cases with camber, the severe asymmetry and dramatic variations of lateral force and aligning moment are mainly due to the carcass compliance and tire width. Finally, considering all key factors, the analytical tire model is capable of describing all kinds of tire properties reasonably and accurately. The model parameters can also be identified from tire measurements and the computational results using the analytical model show good agreement with test data.

ACKNOWLEDGMENTS 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. The authors also thank the staffs in State Key Laboratory of Automobile Dynamics Simulation for their contribution to this study.

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