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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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