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Vehicle System Dynamics

Vol. 45, Supplement, 2007, 79–99

UniTire: uniﬁed tire model for vehicle dynamic simulation

K. GUO and D. LU*

Automobile Dynamic Simulation State Key Lab, Jilin University, Changchun, Jilin, 130025,

People’s Republic of China

UniTire is a uniﬁed non-linear and non-steady tire model for vehicle dynamic simulation and control

under complex wheel motion inputs, involving large lateral slip, longitudinal slip, turn-slip, and camber.

The model is now installed in an ADSL driving simulator at Jilin University for studying vehicle

dynamics and their control systems.

In this paper, ﬁrst, a brief history of UniTire development is introduced; then the application scope

of UniTire and available interfaces to MBS software are presented; thirdly, a more detailed description

of UniTire is given; fourthly, a tool aiming at parameterization of UniTire is also demonstrated; and

ﬁnally, some comments on TMPT are made.

Keywords: UniTire; Tire model; Nonlinear and non-steady; TMPT; Simulation

1. History of UniTire

UniTire is a uniﬁed nonlinear and non-steady tire model for vehicle dynamic simulation and

control under complex wheel motion inputs. Under pure or combined slip conditions of the

tire, and using the velocity of the wheel center V , slip angle α, longitudinal slip ratio S

x

,

camber γ , turn-slip ϕ, and vertical load F

z

as input variables, UniTire model calculates the

lateral force F

y

, longitudinal force F

x

, overturning moment M

x

, rolling resistance moment

M

y

, and aligning moment M

z

, where the road acts on the tire.

In 1973, to improve the handling stability of vehicles, Professor Konghui Guo began

research in tire mechanics theory and experiment, by designing the ﬂat plank tire test machine

QY7329 in Changchun Automobile Research Institute, which was the ﬁrst tire test equipment

in China [1].

In 1986, based on various tests and theoretical analysis, a pure lateral slip UniTire model was

proposed:

F

y0

= 1 − exp(−φ

y

− E

y

φ

3

y

), where F

y0

is the dimensionless lateral force, φ

y

the

normalized lateral slip ratio, and E

y

the curvature factor of lateral force. Later, a combined slip

UniTire model was developed as:

F = 1 −exp(−φ − Eφ

3

), where F is the dimensionless

resultant force (combined longitudinal and lateral), φ the normalized combined slip ratio, and

E the curvature factor of combined slip resultant force.

In 1995, the combined slip UniTire model was improved as:

F = 1 −exp(−φ − Eφ

2

−

(E

2

+ 1/12)φ

3

). Compared with F = 1 − exp(−φ − Eφ

3

), the new one satisﬁes up to

*Corresponding author. Email: lu_dang@vip.sohu.com

Vehicle System Dynamics

ISSN 0042-3114 print/ISSN 1744-5159 online © 2007 Taylor & Francis

http://www.informaworld.com

DOI: 10.1080/00423110701816742

80 K. Guo and D. Lu

the third-order derivatives of initial (φ → 0) and ﬁnal (φ →∞) boundary conditions for

simpliﬁed physical tire model, and as a result, an accurate UniTire with fewer parameters was

achieved [2].

In 1998, to satisfy the needs in complex and extreme operating conditions, such as starting

and braking with low speeds, steering with sharp angles, and drastic combined slip scenarios,

a non-steady-state UniTire model for a low-frequency range (<1c/m) was proposed, which

has a semi-physical form obtained from physical non-steady-state model through the concepts

of slip propagation E-functions and quasi-steady state [3]. Later developments have been

integrated into the latest version UniTire 2.0.

2. Application scope of UniTire and available interfaces to MBS software

At present, the main application scope of UniTire is for handling dynamics with frequency

range up to 8 Hz. UniTire is now installed in an ADSL driving simulator at Jilin University

(shown as ﬁgure 1) for studying vehicle dynamics and their control systems, which has proven

to be a real-time tire model of nice adaptability and high accuracy.

Figure 1. ADSL driving simulator at Jilin University.

Figure 2. Interface between UniTire and CarSim.

Uniﬁed tire model for vehicle dynamic simulation 81

The Standard Tire Interface (STI) presented at the Second International Colloquium on Tire

Models for Vehicle Dynamics Analysis is employed as an interface between UniTire and MBS

software, such as MSC.ADAMS, and also a user-deﬁned interface is used between UniTire

and other multibody codes, such as CarSim. Figure 2 demonstrates the interface between

UniTire and CarSim.

3. Features of UniTire model

In this section, ﬁrst, the deﬁnitions of a tire coordinate system and slip ratios are introduced,

which are very important for understanding the UniTire model, though this information is

elementary, and then, a detailed description of UniTire model will be given.

3.1 Tire coordinate system

The right-hand orthogonal axis system is employed (ﬁgure 3), and because the updating leading

point of the contact patch is determined by the tire revolution direction, the positive directions

of X

t

and Y

t

-axes are coincident with the tire revolution direction (not wheel center traveling

direction), which leads to uniﬁed deﬁnitions of slip ratios and very simple uniﬁed expressions

for longitudinal, lateral and resultant forces, and moments (c.f. below). The traveling velocity

of the wheel center is denoted as V , the direction of which gives a slip angle α, with respect

to the central wheel plane.

3.2 Slip ratios

The longitudinal and lateral slip ratios can all be deﬁned in the uniﬁed form as the sliding

speed over the rolling speed (the updating speed of the contact patch) of the tire with the

coordinate system:

S

x

=

−V

sx

R

e

,S

y

=

−V

sy

R

e

,S

x

∈ (−∞, +∞), S

y

∈ (−∞, +∞), (1)

where is the angular velocity, R

e

is the effective rolling radius, and V

sx

and V

sy

are the

relative sliding speeds in the contact patch with respect to the road surface. Notice that S

x

and

S

y

are both symmetrically deﬁned in the ranges of (−∞, +∞).

Figure 3. Tire coordinate system for UniTire.

82 K. Guo and D. Lu

The normalized longitudinal, lateral, and combined slip ratios are deﬁned as:

φ

x

=

K

x

S

x

F

xm

,φ

y

=

K

y

S

y

F

ym

,φ=

φ

2

x

+ φ

2

y

, (2)

with K

x

andK

y

, the longitudinal slip stiffness and cornering stiffness of the tire, respectively,

and F

xm

and F

ym

the potential extreme values of longitudinal and lateral forces, respectively,

which are deﬁned as:

F

xm

= μ

x

F

z

,F

ym

= μ

y

F

z

, (3)

where F

z

is tire vertical load, and μ

x

and μ

y

are, respectively, the longitudinal and lateral

friction coefﬁcients between tires and road surface.

3.3 Normalized pressure distribution of the contact patch

The contact pressure distribution over the contact patch strongly inﬂuences tire behaviors.

The distributions of contact pressure vary with different structures of tire, load, and internal

pressure, so it is necessary to simplify the contact pressure distribution for a tire analytical

modeling. To simulate different kinds of contact pressure distribution, the contact pressure

distribution over the contact length 2a is expressed as [4–6].

q

z

(x

t

) =

F

z

2a

η

x

t

a

, (4)

where F

z

is the tire vertical load and η(x

t

/a) the normalized pressure distribution function.

With σ = x

t

/a, η(σ ) should satisfy the following conditions:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

η(1) = η(−1) = 0,

η(σ ) ≥ 0,σ∈[−1, 1],

η(σ ) = 0,σ/∈[−1, 1],

1

−1

η(σ )dσ = 2,

1

−1

η(σ )σdσ = 2

a

,

(5)

where is the front shift of the gravity center of contact pressure, shown in ﬁgure 4. η(σ) is

recommended to be the below expression:

η(σ ) = c

1

(1 − σ

2n

)(1 + λσ

2n

)(1 − c

2

σ). (6)

According to equation (5), coefﬁcients c

1

and c

2

can be expressed as:

⎧

⎪

⎪

⎨

⎪

⎪

⎩

c

1

=

(2n + 1)(4n + 1)

2n(4n + 1 + λ)

,

c

2

=−

3(2n + 3)(4n + 3)(4n + 1 + λ)

(2n + 1)(4n + 1)(4n + 3 + 3λ)

a

.

(7)

Parameters n, λ, and are functions of tire vertical load and can be obtained through the ﬁtting

of measurement data of contact pressure distribution. With these three parameters, equation (6)

can be employed to express an arbitrary pressure distribution over the contact patch, shown

in ﬁgure 5.

Uniﬁed tire model for vehicle dynamic simulation 83

Figure 4. Illustration of tire contact pressure distribution.

Figure 5. Curves of normalized pressure distribution function.

3.4 Simpliﬁed physical tire model

The physical model of tire is simpliﬁed as shown in ﬁgure 6, assuming that the carcass of the

tire can merely be deformed along the directions of X

t

and Y

t

axes translationally, neglecting

any bending and twisting deformations.

Under the condition of combined cornering and braking/driving, the deformations of carcass

and tread are shown in ﬁgure 7. In this ﬁgure, the origin O

t

denotes the contact center, X

t

O

t

Y

t

is the tire coordinate system for describing the deformations of both carcass and tread, and

x

t

o

t

y

t

is a relative coordinate system for describing the tread deformation with respect to

Figure 6. The physical model of a tire.

84 K. Guo and D. Lu

Figure 7. Deformation of carcass and contact patch.

carcass. The origin o

t

coincides with O

t

before it is deformed. The central line of the contact

patch, which coincides with the wheel central line O

t

X

t

, is now taking a new position ABC,

due to the forces and moment in the contact patch. X

c

and Y

c

are the deformations of carcass

along axes X

t

and Y

t

caused by the longitudinal and lateral forces, which are expressed as:

⎧

⎪

⎪

⎨

⎪

⎪

⎩

X

c

=

F

x

K

cx

,

Y

c

=

F

y

K

cy

,

(8)

where K

cx

and K

cy

are the longitudinal and lateral stiffnesses of the carcass.

From ﬁgure 7, a point in adhesion region, which begins to contact at the point A, is now

reaching the position P

t

, after rolling for a period of time t. Meanwhile, the corresponding

point on carcass moves from point A to P

c

. The deformations of the tread along x and y axes

are expressed as:

x = S

x

(a − x

t

),

y = S

y

(a − x

t

),

(9)

where a denotes half of the contact length, if the stiffness of tread material in x

t

and y

t

directions

are k

tx

and k

ty

, then the shear stresses of point P

t

, in the adhesion region, in both directions,

are as follows:

⎧

⎪

⎪

⎨

⎪

⎪

⎩

q

x

= k

tx

x = k

tx

S

x

(a − x

t

) = φ

x

μ

x

F

z

2a

(1 − σ),

q

y

= k

ty

y = k

ty

S

y

(a − x

t

) = φ

y

μ

y

F

z

2a

(1 − σ),

(10)

where K

x

= 2a

2

k

tx

, K

y

= 2a

2

k

ty

, and σ = x

t

/a is the relative longitudinal coordinate. The

magnitude of resultant shear stress becomes:

q =

q

2

x

+ q

2

y

= φ

μ

x

φ

x

φ

2

+

μ

y

φ

y

φ

2

F

z

2a

(1 − σ). (11)

Uniﬁed tire model for vehicle dynamic simulation 85

With μ serving as the friction coefﬁcient along the resultant shear stress direction, the

maximum shear stress q

max

in this direction can be expressed as:

q

max

= μq

z

, (12)

where q

z

is the contact pressure along the contact patch length, which can be expressed in an

uniﬁed form as:

q

z

=

F

z

2a

η(σ ). (13)

According to the friction ellipse concept shown in ﬁgure 8, we have:

q

x max

μ

x

q

z

2

+

q

y max

μ

y

q

z

2

= 1, (14)

and with equation (12), it yields:

q

x max

μ

x

q

z

2

+

q

y max

μ

y

q

z

2

=

q

max

μq

z

2

. (15)

In the adhesion region,

q

x

q

x max

=

q

y

q

y max

=

q

q

max

. (16)

So,

q

x

μ

x

q

z

2

+

q

y

μ

y

q

z

2

=

q

μq

z

2

. (17)

Substituted with equations (2), (10), and (11), the directional friction coefﬁcient for calculating

the extreme value of the resultant force can be derived from equation (17) as:

μ =

μ

x

φ

x

φ

2

+

μ

y

φ

y

φ

2

. (18)

With equations (11)–(13) and (18), the relative coordinate of initial sliding point σ

c

(φ) can be

solved by the uniﬁed initial sliding condition function:

η(σ

c

)

1 − σ

c

= φ (19)

Figure 8. Friction ellipse concept.

86 K. Guo and D. Lu

Deﬁned the normalized longitudinal force, the lateral and resultant forces, respectively, are

as follows:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

F

x

=

F

x

μ

x

F

z

,

F

y

=

F

y

μ

y

F

z

,

F =

F

μF

z

,

(20)

and conditionally assumed that the direction of the resultant shear stress in the sliding region

will be the same as that in the adhesion region (in section 3.6.3, the error caused by this

assumption will be discussed and the means of error-correction will be given), and with

equation (10), we have:

F

x

F

y

=

q

x

q

y

=

φ

x

μ

x

φ

y

μ

y

. (21)

Because the resultant force is related to the longitudinal and lateral forces as:

F

2

= F

2

x

+ F

2

y

, (22)

and substituted with equations (19), (21), and (22), it yields:

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

F

x

= F

φ

x

φ

,

F

y

= F

φ

y

φ

,

F

2

= F

2

x

+ F

2

y

.

(23)

The resultant force can be obtained as follows:

F =

σ

c

−1

μ

F

z

2a

η(σ )a dσ +

1

σ

c

qa dσ (24)

or

F(φ) =

φ

4

(1 − σ

c

)

2

+

m

0

(σ

c

)

2

, (25)

where m

0

(σ

c

) is the zero-order moment of η(σ ) in the sliding region and can be expressed as:

m

0

(σ

c

) =

σ

c

−1

η(σ )dσ. (26)

The aligning moment can similarly be calculated as follows:

M

z

=

1

−1

q

y

(X

c

+ aσ)a dσ +

1

−1

q

x

Y

c

a dσ (27)

or

M

z

= F

y

(D

x

+ X

c

) − F

x

Y

c

. (28)

The pneumatic trail D

x

is calculated as follows,

D

x

(φ)

a

=

φ(1 − σ

c

)

3

/6 − φ(1 −σ

c

)

2

/4 + m

1

(σ

c

)/2

F

, (29)

where m

1

(σ

c

) is the ﬁrst-order moment of η(σ) in the sliding region and can be expressed as:

m

1

(σ

c

) =

σ

c

−1

σ η(σ )dσ. (30)

Uniﬁed tire model for vehicle dynamic simulation 87

3.5 Boundary conditions of simpliﬁed physical tire model

From the simpliﬁed physical model, the boundary conditions of the resultant force and

pneumatic trail can be derived as [4–6]:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

lim

φ→0

F = 0,

lim

φ→0

dF

dφ

= 1,

lim

φ→0

d

2

F

dφ

2

=

2

D

,

lim

φ→0

d

3

F

dφ

3

=

2

D

2

3 −

2D

D

,

lim

φ→∞

F = 1,

lim

φ→∞

dF

dφ

= 0,

(31)

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

lim

φ→0

D

x

= D

x0

,

lim

φ→0

dD

x

dφ

=

2D

x0

D

,

lim

φ→∞

D

x

= D

e

lim

φ→∞

dD

x

dφ

= 0,

(32)

where

D =

dη(σ )

dσ

σ =−1

,D

=

d

2

η(σ )

dσ

2

σ =−1

. (33)

3.6 Uniﬁed semi-physical tire model for steady state

3.6.1 Recommended steady-state semi-physical model. Satisfactory with the above

boundary condition of simpliﬁed physical tire model, the semi-physical expression of the

resultant force and pneumatic trail are written as [2, 4, 6]:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

F = 1 −exp

−φ − E

1

φ

2

−

E

2

1

+

1

12

φ

3

,

D

x

= (D

x0

+ D

e

) exp(−D

1

φ − D

2

φ

2

) − D

e

F

x

= F

φ

x

φ

μ

x

F

z

F

y

= F

φ

y

φ

μ

y

F

z

M

z

= F

y

(D

x

+ X

c

) − F

x

Y

c

(34)

3.6.2 Expression of dynamic friction coefﬁcient. In the semi-physical steady-state tire

model, the longitudinal or lateral friction coefﬁcient can be expressed separately. In general,

88 K. Guo and D. Lu

the slip velocity of the contact patch has a very signiﬁcant effect on the tire friction coefﬁcient;

here the following friction model (modiﬁed from Savkoor’s formula to have a ﬂat range at

origin [7]) is employed to describe the relationship between the friction coefﬁcients and slip

velocity:

μ

d

= μ

s

+ (μ

0

− μ

s

) exp

−h

2

log

2

V

s

v

m

+ exp

−

V

s

v

m

, (35)

where μ

d

denotes μ

x

or μ

y

; μ

0

,μ

s

,h, and v

m

are the friction characteristic parameters for

μ

x

or μ

y

separately; and V

s

is the slip velocity of the contact patch in longitudinal or lateral

direction, which can be expressed as:

⎧

⎪

⎪

⎨

⎪

⎪

⎩

V

sx

=

S

x

S

x

− 1

V cos α,

V

sy

=

S

y

S

x

− 1

V cos α.

(36)

3.6.3 Modiﬁcation of the direction of resultant force. In the previous simpliﬁed physical

tire model, there is an assumption: ‘the direction of resultant shear stress in sliding region

will be the same as that in adhesion region’, and it is true for the condition of equality of

longitudinal slip and cornering stiffnesses. However, in most conditions, the longitudinal slip

stiffness is not the same as the cornering stiffness. Thus the semi-physical model will have

some errors, especially under large combined slip conditions, and needs a slight modiﬁcation

for the model. By the introduction of a factor λ, the normalized longitudinal and lateral forces

can be expressed as:

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

F

x

= F

λφ

x

(λφ

x

)

2

+ φ

2

y

,

F

y

= F

φ

y

(λφ

x

)

2

+ φ

2

y

,

(37)

where the modiﬁcation factor λ is deﬁned as:

λ = 1 +

K

y

K

x

− 1

F, (38)

which approaches to 1 for a small slip condition where

F → 0, then

F

x

F

y

=

K

x

S

x

K

y

S

y

, (39)

and for a large slip condition where

F → 1, then

F

x

F

y

=

S

x

S

y

. (40)

3.7 Non-steady-state tire model

3.7.1 Analytical non-steady state tire model. On the basis of the tire cornering property

in non-steady state, an analytical model with small transient lateral inputs (yaw and lateral

Uniﬁed tire model for vehicle dynamic simulation 89

Figure 9. Illustration of a dynamic tire system.

motions) is established (ﬁgure 9) [3, 8–11]:

F

y

(s) = ψ(s)G

fψ

(s) + Y(s)G

fy

(s),

M

z

(s) = ψ(s)G

mψ

(s) + Y(s)G

my

(s),

(41)

where G

fψ

(s), G

fy

(s), G

mψ

(s), and G

my

(s) can be expressed as follows if neglecting the

bending and twisting deformations of carcass:

⎧

⎪

⎪

⎨

⎪

⎪

⎩

G

fy

(s) =−

K

y

a

E(s)

1 + ε

0

E(s)

,

G

my

(s) =

K

y

D

x0

a

E

t

(s)

1 + ε

0

E(s)

,

(42)

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

G

fψ

(s) = K

y

1 − E(s)

1 + ε

0

E(s)

,

G

mψ

(s) =−K

y

D

x0

1 − E

t

(s) + ε

0

(E(s) −E

t

(s))

1 + ε

0

E(s)

+ L

tw

E(s)

,

(43)

where s denotes the operator of Laplace transformation in the spatial domain.

Functions E(s) and E

t

(s), which are called E-functions, are deﬁned as:

E(s) =

1

2

2

0

(1 − e

−aσs

)dσ,

E

t

(s) =

3

2

2

0

(σ − 1)(1 − e

−aσs

)dσ,

(44)

and, the characteristic ratio ε

0

and L

tw

reads:

ε

0

=

K

y

aK

cy

,L

tw

=

b

2

K

x

a

2

K

y

. (45)

The analytical non-steady-state tire model not only describes the transient tire property

with small lateral motions (without any sliding), but also provides the basis for studying the

dynamic tire property with large lateral slip inputs (the shear stresses of sliding zone of the

contact patch are determined by the friction coefﬁcient and contact pressure).

90 K. Guo and D. Lu

3.7.2 Simpliﬁed analytical non-steady tire model with ﬁrst-order approximation.

Expanding the E-function expressions to Taylor’s series, and neglecting the higher-order

terms of s, equation (44) can be simpliﬁed as [9–11]:

E(s) ≈ as, E

t

(s) ≈ as. (46)

Substituting equation (46) into equation (41) and considering equations (42) and (43), it yields

⎧

⎪

⎪

⎨

⎪

⎪

⎩

F

y

(s) = K

y

(1 − as)ψ(s) − sY (s)

1 + l

y

s

,

M

z

(s) =−K

y

D

x0

(1 − as)ψ(s) − sY (s)

1 + l

y

s

− K

m

sψ(s),

(47)

where

l

y

= aε

0

=

K

y

K

cy

(48)

is deﬁned as the lateral relaxation length, and

K

m

= K

y

D

x0

L

tw

a =

b

2

a

K

x

D

x0

(49)

represents the additional moment against turn-slip dψ/dX developed by the tire width.

Considering the additional part K

m

sψ(s) as an extra damping moment and transforming

equation (47) into spatial domain, we obtain:

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

F

y

+ l

y

dF

y

dX

= K

y

ψ −

dY

dX

− a

dψ

dX

M

z

+ l

y

dM

z

dX

=−K

y

D

xn

ψ −

dY

dX

− a

dψ

dX

(50)

By introducing the quasi-steady-state concept, the transient lateral force and aligning moment

at large lateral slip inputs can be calculated on the basis of the semi-physical steady-state tire

model, in which the effective slip ratio accounts for the tire slip conditions (ﬁgure 10).

With notice that ψ − dY/dX = tan α is the nominal lateral slip ratio, and dψ/dX = ϕ is

the turn-slip ratio, the generation of dynamic F

y

and M

z

= F

y

D

x

+ M

z

under transient tire

inputs of lateral motion dY/dX and yaw angle ψ, which are equivalent to the inputs of tan α

and ϕ, are shown in ﬁgure 10.

Figure 10. Block diagram of the ﬁrst-order model.

Uniﬁed tire model for vehicle dynamic simulation 91

3.7.3 High-order non-steady tire model. The E-functions determine the dynamic tire

property of side force and aligning moment to a great extent. The neglect of higher-order terms

of Taylor’s series leads to dissatisfaction with the measurement data for aligning moment

response (the ﬁrst-order model). However, if expanding the E-functions with the Taylor’s

series to second-order terms, the approximate expressions will result in an unstable system.

Thus, a pair of fractional expressions of higher order is employed and the theoretical boundary

conditions of E-functions provide a key clue to determine the expressions.

At very low frequency, where s approaches to zero, we have:

lim

s→0

E(s) = 0, lim

s→0

E

t

(s) = 0, (51)

the ﬁrst-order derivatives of E-functions become:

lim

s→0

E

(s) = a, lim

s→0

E

t

(s) = a, (52)

and the second-order derivatives have:

lim

s→0

E

(s) =−

4

3

a

2

, lim

s→0

E

t

(s) =−2a

2

. (53)

Now E-functions are approximated with fractional expressions as follows:

E(s) =

as

1 + (2/3)as

E

t

(s) =

as

(1 + (1/3)as)(1 + (2/3)as)

(54)

It is apparent from equation (54) that the approximate expression, which is called the high-

order approximation, comply with all the boundary conditions of E-functions determined by

equations (52) and (53).

Substituting equation (54) into equation (41) and considering equations (42) and (43) for a

small slip input, it yields:

F

y

(s) = K

y

tan α − (1/3)aϕ

1 + (2/3 + ε

0

)as

, (55)

and the aligning moment response consists of a normal part and a additional damping part:

M

z

(s) =−K

y

D

xn

(1 + (1/3)as)

tan α + (1/3)aϕ(2/3 + ε

0

)as

(1 + (2/3 + ε

0

)as)

, (56)

M

z

(s) =−

K

m

ϕ

1 + (2/3)as

. (57)

With the quasi-steady-state concept, the effective slip ratio with respect to the side force can

be derived from equation (55):

S

y

=

S

yn

1 + (2/3 + ε

0

)as

, (58)

where the nominal slip ratio S

yn

reads:

S

yn

= tan α −

1

3

aϕ (59)

According to equation (56), deﬁne the effective slip ratio with respect to the pneumatic trail

S

yd

as follows:

S

yd

= S

y

+

1

3

aϕ, (60)

which can be regarded as the effective slip ratio with respect to the side force coupling with

the turn-slip effect.

92 K. Guo and D. Lu

Figure 11. Block diagram of the high-order model.

The factor 1/(1 +as/3) in equation (56) is considered as a relaxation effect on the quasi-

steady pneumatic trailD

xn

, where the length constant is a/3. Whereas the factor 1/(1 +2as/3)

in equation (57) can be regarded as the relaxation effect on the normalized slip ratio of damping

moment, with a relaxation length 2a/3. The simulation diagram of the high-order model is

shown in ﬁgure 11.

3.8 Overturning moment

Tire overturning moment is caused by the shift of the application point of vertical force, which

was inﬂuenced by two factors: tire carcass lateral translation deformation yielded by lateral

force and the effective carcass camber related with tire camber and lateral force. UniTire

describes the tire overturning moment as [12]:

M

x

= M

x1

+ M

x2

+ M

xR

, (61)

where M

x1

can be expressed as

M

x1

= F

z

F

y

K

cy

. (62)

M

x2

is induced by the effective carcass camber and can be written as:

M

x2

=−K

1

γ

e

− (K

2

γ

e

)

3

, (63)

where the effective carcass camber γ

e

can be described as

γ

e

= arctan

F

y

/K

cy

+ R

l

sin γ

R

l

cos γ

. (64)

K

1

and K

2

are the stiffness parameters, which can be expressed as a function of vertical load as:

K

1

= K

11

+ K

12

F

zn

+ K

13

F

2

zn

,

K

2

= K

21

+ K

22

F

zn

+ K

23

F

2

zn

,

(65)

where F

zn

= F

z

/F

z_rated

, F

z_rated

means the tire-rated load. M

xR

is the residual overturning

moment, which can be written as a function of vertical load:

M

xR

= M

xR1

+ M

xR2

F

zn

+ M

xR3

F

2

zn

. (66)

Uniﬁed tire model for vehicle dynamic simulation 93

3.9 Rolling resistance moment

Steady-state rolling resistance moment M

ys

can be calculated with

M

ys

=−F

z

fR

l

1 + h tan

π

2

cr

, (67)

where f is the coefﬁcient of rolling resistance, h is the coefﬁcient of rolling resistance depend-

ing on , the angular velocity of wheel, and

cr

is the critical angular velocity of the wheel

when standing wave occurs and the rolling resistance moment becomes inﬁnite.

The formula for a non-steady state rolling resistance moment M

y

reads,

˙

M

y

=

θ

r

(M

ys

− M

y

), (68)

where θ

r

is the relaxation angle of rolling resistance.

3.10 Loaded radius

The vertical load F

z

, lateral force F

y

, and camber angle γ will cause a tire loaded radius

change, the relation among R

l

and F

z

, F

y

, γ can be expressed as [12]:

R

l

= R

l_F z

+ R

l_γ

+ R

l_Fy

, (69)

where R

l_Fz

denotes the relation between loaded radius and vertical load, which can be

expressed as

R

l_Fz

= R

1

+ R

2

F

zn

+ R

3

F

2

zn

. (70)

R

l_γ

describes the relation between the increment of tire loaded radius and camber angle

and reads:

R

l_γ

=

R

1

+ R

2

F

zn

+ R

3

F

2

zn

γ

2

. (71)

R

l_Fy

is the increment of tire loaded radius caused by the variation of lateral force, which

can be written as:

R

l_Fy

= K

Rl

(F

y

− F

y_shift

)

2

, (72)

and K

Rl

is a function of vertical load; F

y_shift

can also be expressed as a function of vertical

load and camber angle.

3.11 Simulation diagram of UniTire model

For the longitudinal non-steady state force, with the similar but simpler processing, we obtain

the effective longitudinal slip ratio from the following equation:

S

x

=

S

xn

1 + l

x

s

, (73)

where S

xn

=−V

sx

/R

e

and l

x

= K

x

/K

cx

. Combining equation (73) with the UniTire steady-

state formulas, the tire longitudinal force can be obtained under small or large slip conditions.

94 K. Guo and D. Lu

Figure 12. The simulation diagram of the UniTire model.

Further, considering the relationship between the camber and turn-slip, the nominal effective

slip ratios with respect to the side force and the effective slip ratio with respect to the aligning

moment in the high-order model become:

⎧

⎪

⎪

⎨

⎪

⎪

⎩

S

yn

= tan α −

a

3

ϕ − λ

c

sin γ

R

e

,

S

y

=

S

yn

1 + l

y

s

,

(74)

S

yd

= S

y

+

a

3

ϕ − λ

d

sin γ

R

e

. (75)

Combining equations (74) and (75) with ﬁgure 11, with λ

s

=

√

(1 − F

2

) accounting for the

saturation effect of existing shear forces, the simulation diagram of UniTire model associated

with lateral slip, longitudinal slip, turn-slip, and camber is shown in ﬁgure 12. (Notice that K

x

and K

y

are functions of F

z

, l

x

and l

y

are functions ofF

z

, and S

x

, S

y

and μ

x

, μ

y

are functions

ofF

z

, V

sx

, V

sy

).

4. UniTire tool

To improve the efﬁciency of parameterization for an UniTire model, the UniTire-Tool is devel-

oped. The UniTire-Tool has four relatively independent parts: Data Preprocessing module,

Fitting module, Drawing module, and Analysing module.

Data Preprocessing module aims at preprocessing original tire test data with different

forms to a standard format and saves the preprocessed data into a

∗

.dat ﬁle. The GUI of

Data Preprocessing module is shown in ﬁgure 13.

Uniﬁed tire model for vehicle dynamic simulation 95

Figure 13. Data Preprocessing module of UniTire-Tool.

Figure 14. Fitting module of UniTire-Tool.

With Fitting module, the calculation of parameters from the preprocessed data can be eas-

ily performed by employing regression techniques. The GUI of Fitting module is shown in

ﬁgure 14.

Drawing module is designed for special comparison between test data and calculation results

with UniTire, and generating ﬁgures for reports. The GUI of Drawing module is shown in

ﬁgure 15.

Analysing module is useful and convenient for a developer to make an intensive study of

interesting parameters in UniTire, such as cornering stiffness, lateral friction coefﬁcient, and

so on. The GUI of Analysing module is shown in ﬁgure 16.

5. UniTire experimental validation

Figures 17–20 show the comparisons between UniTire Model and tire test data for a

P245/75R16 tire under steady-state conditions with 60 Km/h tire traveling velocity: including

pure lateral slip, pure longitudinal slip, and combined slips.

96 K. Guo and D. Lu

Figure 15. Drawing module of UniTire-Tool.

Figure 16. Analysing module of UniTire-Tool.

Figure 17. Lateral force comparison under pure lateral slip.

Uniﬁed tire model for vehicle dynamic simulation 97

Figure 18. Aligning moment comparison under pure lateral slip.

Figure 19. Braking force comparison under pure longitudinal slip.

In addition, with the formula (76), the error of UniTire simulation results for TMPT is

calculated, as shown in table 1 [13]:

ε =

n

i=1

(y

i,sim

− y

i,test

)

2

n

i=1

y

2

i,test

× 100%. (76)

98 K. Guo and D. Lu

Figure 20. Lateral and longitudinal forces comparison under combined slips.

Table 1. UniTire modeling error for TMPT validation

tests.

2.5 Bar (%) 2.0 Bar (%)

Lateral force Fy 1.1239 1.4438

Self aligning moment Mz 5.4103 6.9301

Longitudinal force Fx 1.4719 2.0799

According to TMPT handling tests deﬁnition, all simulations were done in ADAMS with

UniTire model. The report of ‘UniTire model for TMPT Validation & Capability Tests’ can

be downloaded from http://www.unitire.com.cn.

6. Conclusion

UniTire Model is presented in this paper, and the features of the model include:

• semi-physical model based on the uniﬁed analytical model;

• better accuracy with fewer parameters via realization of analytical boundary conditions;

• uniﬁed deﬁnitions of slip ratios yielding uniﬁed analytical normalized force functions,

leading to predictability from pure slips force and moment to those of combined slips and

thus the test work needed for identiﬁcation of tire model parameters being signiﬁcantly

reduced

• ‘plug-in’ arbitrary road surface with dynamic friction properties;

• potential of prediction for force and moment under different speeds;

• uniﬁcation of steady and non-steady via higher-order approximation of transfer matrix and

effective slip ratio concept;

• turn-slip included.

Uniﬁed tire model for vehicle dynamic simulation 99

Acknowledgements

The authors would like to thank Professors Peter Lugner and Mandred Plöchl for their effort

for organizing TMPT and thank Mr Van Oosten for his kind help in understanding STI in

ADAMS.

References

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