10 2. DYNAMIC MODEL OF VEHICLE ROLLOVER
Vertical motions of unsprung masses:
m
1
Rz
u1
D F
s1
k
t1
.
z
u1
z
r1
/
(2.21)
m
2
Rz
u2
D F
s2
k
t2
.
z
u2
z
r2
/
: (2.22)
In these equations,
m
1
and m
2
are the left unsprung mass and the right unsprung mass,
respectively; k
t1
and k
t2
are the vertical stiffness of the left tires and right tires, respectively;
z
c
is the vertical displacement of sprung mass; z
u1
and z
u2
are the vertical displacement of the
left unsprung mass and the right unsprung mass, respectively; z
r1
and z
r2
are the road input
of the left tires and the right tires, respectively; F
s1
and F
s2
are the dynamic forces of the left
suspension and the right suspension due to vertical acceleration, respectively; and M
B
is the
anti-yaw torque.
e dynamic forces of the left and right suspensions due to vertical acceleration can be
written as:
F
s1
D k
s1
.
z
s1
z
u1
/
c
s1
.
Pz
s1
Pz
u1
/
(2.23)
F
s2
D k
s2
.
z
s2
z
u2
/
c
s2
.
P
z
s2
Pz
u2
/
: (2.24)
In Equations (2.23) and (2.24), k
s1
and k
s2
are the vertical stiffness of the left suspension and
the right suspension, respectively; c
s1
and c
s2
are the equivalent damping coefficient of the left
suspension and the right suspension; and z
s1
and z
s2
are the vertical displacement of sprung
mass on the left and on the right, respectively.
By taking the coupling relationship between the vertical motion and lateral motion of the
sprung mass into consideration, the equation can be obtained as follows:
z
s1
z
s2
D G
T
z
c
'
; (2.25)
where
G D
1 1
T
w
=2 T
w
=2
:
2.5 MULTI-FREEDOM MODEL
Given the exclusive features of heavy-duty vehicles such as the high center of gravity, the big
wheel tread, the long wheelbase, the large number of passengers’ capacity, and the variable dis-
tribution of passengers having an impact on its rollover property, the above-mentioned rollover
model cannot accurately describe the roll motion. erefore, it is necessary to establish a multi-
freedom rollover dynamics model to represent the motion state of heavy-duty vehicles. In this
section, a six degree of freedom rollover dynamics model is established for a triaxle bus which
has complex structure. For a triaxle bus, the middle axle and the rear axle are on the same side of
the center of mass, and the distance between the middle axle and the rear axle is short such that
2.5. MULTI-FREEDOM MODEL 11
the roll coupling between the middle axle and the rear axle is neglected. erefore, the middle
axle and the rear axle of the triaxle bus is equivalent to a virtual rear axle, as shown in Figure 2.8.
l
e
Y
Y
X
X
a
a
c
b
Figure 2.8: e equivalent model of the triaxle bus.
In addition, a twisted bar with a constant stiffness is assumed to link between the first axle
and the virtual rear axle. For the sake of simplicity, the effects of the lateral wind, the pitching
motion, and the longitudinal motion are neglected since they are of secondary importance in
studying the rollover of such a vehicle, the road profile is regarded as symmetric with respect
to the x axle. us, a 6-DOF vehicle model moving at a constant speed and constant steering
angle is established, as shown in Figure 2.9.
X
Y
u
a
δ
f
r
β
r
β
f
F
Y1
F
Y1
le
T
f
m
uf
m
ur
φuf
φur
m
sf
m
sr
h
sf
h
sr
φ
sf
φ
sf
φ
sr
φ
sr
Figure 2.9: Dynamic model of the triaxle bus rollover.
From D’Alembert’s principle, the equations of the above model are as follows.
Lateral motion:
ma
y
m
sf
h
f
R'
sf
m
sr
h
r
R'
sr
D 2F
f
cos ı C 2F
Yr
: (2.26)
12 2. DYNAMIC MODEL OF VEHICLE ROLLOVER
Yaw motion:
I
Z
Pr D 2aF
f
cos ı C M
r
: (2.27)
Roll motion of the sprung mass of the front axle:
I
Xf
R'
sf
D m
sf
h
f
a
y
C m
sf
gh
f
'
sf
k
f
'
sf
'
uf
l
f
P'
sf
P'
uf
C k
b
'
sf
'
sr
:
(2.28)
Roll motion of the sprung mass of the rear axle:
I
Xr
R'
sr
D m
sr
h
r
a
y
C m
sr
gh
r
'
sr
k
r
.
'
sr
'
ur
/
l
r
.
P'
sr
P'
ur
/
k
b
'
sr
'
sf
:
(2.29)
Roll motion of the unsprung mass of the front axle:
2F
Y 1
h
c
C m
uf
h
uf
h
cf
a
y
D k
uf
'
uf
m
uf
g
h
uf
h
cf
'
uf
k
f
'
sf
'
uf
l
f
P'
sf
P'
uf
:
(2.30)
Roll motion of the unsprung mass of the virtual rear axle:
2
.
F
Y 2
C F
Y 3
/
h
c
C m
ur
.
h
ur
h
cr
/
a
y
D k
ur
'
ur
m
ur
g
.
h
ur
h
cr
/
'
ur
k
r
.
'
sr
'
ur
/
l
r
.
P'
sr
P'
ur
/
;
(2.31)
where
F
Yr
D F
m
C F
r
M
r
D 2b
1
F
m
C 2c
1
F
r
:
(2.32)
In the above-mentioned equations, m
sf
represents the equivalent sprung mass of the front
axle; m
sr
indicates the equivalent sprung mass of the rear axle; m
uf
refers to the unsprung mass
of the front axle; m
ur
is the unsprung mass of the rear two axles; b
1
and c are the longitudinal
distance from the CG to the middle axle and rear axle, respectively; h
f
is the height between
the center of front sprung mass and the roll center; h
r
is the height between the center of rear
sprung mass and the roll center; h
uf
and h
ur
are the height of the center of the front unsprung
mass and the rear unsprung mass, measured upward from the road; h
cf
and h
cr
are the height of
the front roll center and the rear roll center, measured upward from the road, respectively; I
Xf
and I
Xr
are the roll inertia of the front sprung mass and the rear sprung mass, measured about
the roll axle; '
sf
and '
sr
are the roll angle of the front sprung mass and the rear sprung mass; '
uf
and '
ur
are the roll angle of the front unsprung mass and the rear unsprung mass; F
Yr
the lateral
force of the tires at the virtual axle; M
r
is the yaw moment caused by the virtual rear axle; F
f
,
F
m
, and F
r
are the lateral force of the tires at the first axle, the middle axle, and the rear axle,
respectively; k
f
and k
r
are the equivalent roll stiffness coefficient of the front suspension and
the rear suspension; k
uf
and k
ur
are the equivalent roll stiffness coefficient of the front unsprung
mass and the rear unsprung mass; l
f
and l
r
are the equivalent roll damping coefficient of the
2.5. MULTI-FREEDOM MODEL 13
front suspension and the rear suspension; and k
b
is the torsion stiffness coefficient of vehicle
frame.
In addition, the steering angle of front wheels ı is assumed to be sufficiently small that
cos ı 1 in Equations (2.26) and (2.27) holds.
e lateral forces in Equations (2.26) and (2.27) mainly come from the contact between
the road and tires at the front, middle, and rear axle, depending on the physical properties of the
tire and the corresponding side slip angles ˇ
f
, ˇ
m
, and ˇ
r
observed on the front wheels, middle
wheels, and rear wheels, respectively. In addition, the two wheels at the front axle will rotate
around the king bolt, the two middle wheels and rear wheels rotate around the axle vertical to
the road, due to the roll motion. erefore, the slip angle of a tire can be determined from the
simple geometric relations, as follows:
8
ˆ
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
ˆ
:
ˇ
f
D arctan
v C ar
U
ı
ˇ
m
D arctan
v br
U
ˇ
r
D arctan
v cr
U
:
(2.33)
Using a simple tire model with a linear constant cornering stiffness, the lateral forces of
tires can be obtained:
8
ˆ
<
ˆ
:
F
f
D k
f
ˇ
f
F
m
D k
m
ˇ
m
F
r
D k
r
ˇ
r
;
(2.34)
where k
f
, k
m
, and k
r
are the cornering stiffness of the front wheels, the middle wheels, and the
rear wheels, respectively.
Setting U
s
D
'
sf
'
sr
'
uf
'
ur
T
, V D
P'
sf
P'
sr
T
, and substituting the Equa-
tions (2.33) and (2.34) into Equations (2.26)–(2.32), the state space equations of vehicle rollover
system can be obtained in a matrix form as follows:
M
q
2
6
6
6
6
4
Pv
Pr
P
U
s
P
V
3
7
7
7
7
5
D A
q
2
6
6
6
6
4
v
r
U
s
V
3
7
7
7
7
5
C B
q
ı; (2.35)
14 2. DYNAMIC MODEL OF VEHICLE ROLLOVER
where
M
q
D
2
6
4
M
1
0
24
M
2
M
3
M
4
M
5
0
22
M
6
0
22
3
7
5
I A
q
D
2
6
4
A
1
0
24
0
22
A
2
A
3
A
4
0
22
0
24
A
5
3
7
5
I
B
q
D
2K
f
2aK
f
0 0 2h
f
K
f
0 0 0
T
I
M
1
D
"
m 0
0 I
Z
#
I M
2
D
"
h
f
m
sf
h
r
m
sr
0 0
#
I
M
3
D
2
6
6
6
6
4
h
f
m
sf
0
h
r
m
sr
0
m
uf
.h
uf
h
cf
/ 0
m
ur
.h
ur
h
cr
/ 0
3
7
7
7
7
5
I M
4
D
2
6
6
6
6
4
0 0 0 0
0 0 0 0
0 0 l
f
0
0 0 0 l
r
3
7
7
7
7
5
I
M
5
D
2
6
6
6
6
4
I
Xf
0
0 I
Xr
0 0
0 0
3
7
7
7
7
5
I M
6
D
"
1 0 0 0
0 1 0 0
#
I
A
1
D
2
6
6
4
2K
f
C 2K
m
C 2K
r
U
mu
2
C 2aK
f
2b
1
K
m
2c
1
K
r
U
2aK
f
2bK
m
2cK
r
U
2a
2
K
f
C 2b
2
1
K
m
C 2c
2
1
K
r
U
3
7
7
5
I
A
2
D
2
6
6
6
6
6
6
6
4
0 h
f
m
s
U
0 h
r
m
sr
U
2K
f
h
c
U
m
uf
U
2
h
uf
h
cf
2aK
f
h
cf
U
2K
m
C K
r
/h
c
U
m
ur
U
2
.
h
ur
h
cr
/
C 2bK
m
h
cr
C 2cK
r
h
cr
u
3
7
7
7
7
7
7
7
5
I
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