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6 2. DYNAMIC MODEL OF VEHICLE ROLLOVER
Roll Center
ay
ay
ϕ
s
ϕ
u
ϕ = ϕ
u
+ ϕ
s
Figure 2.4: Roll model.
Alternatively, Equation (2.9) can be shown in the following form:
T
roll
.z/ D k
.
z z
1
/ .
z z
2
/ .
z Nz
2
/
.
z p
1
/ .
z Np
1
/ .
z p
2
/ .
z Np
2
/
; (2.10)
where
k D
.
1 p
1
/ .
1 Np
1
/ .
1 p
2
/ .
1 Np
2
/
.
1 z
1
/ .
1 z
2
/ .
1 Nz
2
/
;
z
1
, z
2
, and Nz
2
are zeros, and p
1
, Np
1
, p
2
, Np
2
are pole of T
roll
.z/.
2.3 LATERAL-YAW-ROLL MODEL
Whether the roll plane model or the yaw-roll model can only partially analyze the rollover
problem. erefore, a mathematical model that can fully describe rollover problem is needed. In
recent years, the 3-DOF vehicle model including lateral, yaw, and roll motion becomes the most
commonly used model [812]. e three degrees of freedom vehicle model can be established
as follows [13].
For the vehicle model shown in Figure 2.5, a frame of coordinates is ﬁxed on the vehicle
body. In studying the rollover dynamics of the vehicle moving at a constant steering angle and
a constant speed, the dynamic equations of lateral, yaw, and roll motions can be established as
follows:
8
ˆ
<
ˆ
:
ma
y
m
s
h
R
D 2F
f
cos ı C 2F
r
I
z
Pr D 2aF
f
cos ı 2bF
r
I
x
R
m
s
ha
y
D m
s
gh sin c
P
k
;
(2.11)
2.3. LATERAL-YAW-ROLL MODEL 7
where the lateral acceleration is given below:
a
y
D Pv C ur: (2.12)
y
a
r
b
x
z
o
u
v
F
f
F
r
f
β
β
δ
r
R
m
s
a
y
m
s
g
F
L
F
g
T
w
h
h
c
m
ϕ
Figure 2.5: Lateral-yaw-roll model.
Equation (2.11) describes the balance relations of the lateral forces on the entire vehicle,
the yaw moments of the entire vehicle, and the roll moments on the sprung mass, respectively.
e lateral forces in Equation (2.11) mainly come from the contact between the tire and
the road surface at each front and rear wheel and is a function of the physical properties of the
tire and the corresponding sideslip angles ˇ
f
or ˇ
r
observed on the front wheel or rear wheel,
respectively. e slip angle of a tire can be determined from the simple geometric relations shown
in Figure 2.6 as follows:
ˇ
f
D arctan
v C ar
U
ı; ˇ
r
D arctan
v br
U
: (2.13)
In this study, a simple tire model with linear constant cornering stiﬀness will be used so
that the lateral forces of tires yield
F
f
D k
f
ˇ
f
; F
r
D k
r
ˇ
r
: (2.14)
As the vehicle is moving in cornering, the lateral velocity and yaw rate do not vanish.
So, the dynamics of vehicle rollover can be described by Equations (2.11), (2.12), (2.13), and
(2.14) in partial unknown state variables v, r, , and
P
. at is, the dynamic equation of vehicle
8 2. DYNAMIC MODEL OF VEHICLE ROLLOVER
δ
β
f
Total Velocity
v+ar
F
f
u
Figure 2.6: Zoom view of a front wheel.
rollover can be recast as
P
X D f .X; /; X; f 2 <
4
; (2.15)
where
X D
.
x
1
; x
2
; x
3
; x
4
/
T
D
v; r; ;
P
T
; f
.
f
1
; f
2
; f
3
; f
4
/
T
; (2.16)
where
f
1
D
2k
f
ˇ
f
cos ı 2k
r
ˇ
r
I
x
C m
2
s
h
2
g sin m
s
hk
m
s
hc
P
mI
x
m
2
s
h
2
Ur
mI
x
m
2
s
h
2
f
2
D
2bk
r
ˇ
r
2ak
f
ˇ
f
cos ı
I
z
; f
3
D x
4
f
4
D
2k
f
ˇ
f
cos ı 2k
r
ˇ
r
m
s
h C m
s
hmg sin mk
mc
P
mI
x
m
2
s
h
2
ı; U; m; m
s
; h; I
x
; I
z
; k
f
; k
r
; k
; c
; a; b
T
:
In these equations, a and b represents longitudinal distance from the center of gravity to the
front and the rear axle; c
'
is equivalent roll damping coeﬃcient of suspension; F
f
=F
r
is lateral
force of a front/rear tire; F
L
=F
R
is vertical load on left/right tires; h is the length of roll arm
measured from the center of gravity to the roll center; I
x
represents roll moment of inertia of
the sprung mass, measured about the roll axis; I
z
represents yaw moment of inertia of the total
mass, measured about the z axis; k
f
=k
r
represents cornering stiﬀness coeﬃcient of a front/rear
tire; k
is the equivalent roll stiﬀness coeﬃcient of suspension; L is the wheelbase of vehicle; m
is the total mass of vehicle; m
s
is sprung mass of vehicle; r is yaw rate of the sprung mass; T
w
is track width of vehicle; U is forward speed of vehicle; v is lateral speed of vehicle; ı represents
steering angle of front wheels; is the roll angle;
P
is the roll rate; and
R
is roll acceleration.
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