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28 3. STABILITY OF UNTRIPPED VEHICLE ROLLOVER
14.5
14
13.5
13
12.5
12
11.5
11
10.5
1 2 3 4 5 6 7 8
Roll Stiﬀness of Suspension (N.m/rad)
Critical Forward Speed (m/s)
stable
× 10
4
Figure 3.7: Stability region on (k
, U
c
) plane.
ing function to variable
1
. Hence, DSF increases with the decreasing of
1
, as shown
in Figure 3.8. DSF varies from the steer coeﬃcient c
f
and c
r
induced by roll, as seen
from Equation (3.3). DSF will be increased by decreasing c
f
or increasing c
r
, as shown in
Figures 3.93.10.
3.1.3 LATERAL LOAD TRANSFER RATIO
At present, the Lateral Load Transfer Ratio (LTR) is the most widely used evaluation index
to predict rollover risk [13, 2528]. When vehicle is running, the vertical load of the vehicle
will gradually shift from the inside to the outside. So, the LTR evaluates the rollover stability
according to the vertical load of the wheels during the driving. It is believed that the vehicle
will roll over, when the vertical load on one side of the wheel is reduced to 0. e fundamental
deﬁnition of the LTR is described as follows:
LTR D
F
z1
F
z2
F
z1
C F
z2
; (3.4)
where F
z1
and F
z2
are the total vertical loads of the left and right wheels of vehicle, respectively.
A vehicle is considered to roll over when LTR is more than 1 or less than 1. It should be
noted that, F
z1
equals to F
z2
and LTR D 0 when a vehicle is traveling straight. If F
z1
D 0, then
3.1. ROLL INDEX OF UNTRIPPED VEHICLE ROLLOVER 29
28
26
24
22
20
18
16
14
12
0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5
Ratio of the Cornering Stiﬀness of a Front Tire to the Rear
Critical Forward Speed (m/s)
stable
Figure 3.8: Stability region on (
1
, U
c
) plane.
14.9
14.8
14.7
14.6
14.5
14.4
14.3
14.2
14.1
14.0
0.0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
e Steering Coeﬃcient Induced by Roll at the Front Axle
Critical Forward Speed (m/s)
stable
Figure 3.9: Stability region on (c
f
, U
c
) plane.
30 3. STABILITY OF UNTRIPPED VEHICLE ROLLOVER
14.7
14.6
14.5
14.4
14.3
14.2
14.1
14.0
13.9
13.8
0.0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
e Steering Coeﬃcient Induced by Roll at the Rear Axle
Critical Forward Speed (m/s)
stable
Figure 3.10: Stability region on (c
r
, U
c
) plane.
LTR D 1 and the left wheel just lifts oﬀ the ground. Also, if F
z2
D 0, then LTR D 1 and the
right wheel just lifts oﬀ the ground.
Figure 3.11 shows the rollover stability at diﬀerent conditions ( J-turn, Fishhook, and
Double Lane Change maneuvers) and diﬀerent vehicle speed (60 km/h, 80 km/h, and 100 km/h)
based on the LTR. It is observed that LTR can eﬀectively predict the vehicle rollover. It also
shows that the higher the speed, and the more likely it is to be rollover.
LTR is considered as a very useful index to study the dynamics and simulation of vehicle
rollover, while the vertical load of each wheel of the vehicle is diﬃcult to be measured or esti-
mated in real time. So, the LTR cannot be used to predict the risk of vehicle rollover directly,
especially in the case of emergency. Many researchers derived new rollover indexes based on the
vehicle rollover dynamics model and the deﬁnition of the LTR [15, 2936]. For example, ignore
the vertical motion of the vehicle, so
(
F
z1
F
z2
D F
s2
F
s1
F
z1
C F
z2
D mg:
(3.5)
F
s1
and F
s2
are the left and right supporting force of suspension. According to moment
equilibrium equations, the diﬀerence value between the left and right suspension force can be
described as follows:
T
w
2
.
F
s2
F
s1
/
D k
.
s
u
/
c
P
s
P
u
; (3.6)
3.1. ROLL INDEX OF UNTRIPPED VEHICLE ROLLOVER 31
1
0.8
0.6
0.4
0.2
0.0
0 2 4 6 8 10
t (s)
(a) J-turn condition
LTR
100 km/h
80 km/h
60 km/h
1
0.5
0.0
-0.5
-1.0
0 5 10
t (s)
(b) Fishhook condition
LTR
100 km/h
80 km/h
60 km/h
Figure 3.11: Stability analysis at typical conditions. (Continues.)
32 3. STABILITY OF UNTRIPPED VEHICLE ROLLOVER
1
0.5
0.0
-0.5
-1.0
0 100 15050 200 250
Mileage (m)
(c) Double lane change condition
LTR
100 km/h
80 km/h
60 km/h
Figure 3.11: (Continued.) Stability analysis at typical conditions.
where k
is equivalent roll damping coeﬃcient of suspension, c
is equivalent roll damping
coeﬃcient of suspension, and
s
/
u
represents roll angle of sprung/unsprung mass.
e torque balance of the vehicle under roll motion of sprung mass is given as below:
I
x
R
s
D m
s
ha
y
C m
s
hg
s
k
.
s
u
/
c
P
s
P
u
; (3.7)
where I
x
is the roll inertia of sprung mass, m
s
is sprung mass, h is the height between the center
of sprung mass and the roll center, and a
y
is the lateral acceleration of the vehicle.
us,
F
s2
F
s1
D
2
T
w
I
x
R
s
m
s
ha
y
m
s
hg
s
: (3.8)
So, the rollover index (RI) can be represented as
RI D
2
T
w
I
x
R
s
m
s
ha
y
m
s
hg
s
mg
: (3.9)
Figure 3.12 shows the comparison between fundamental LTR and new rollover index RI
in J-turn and Fishhook condition. In Figure 3.12a for J-turn maneuver, the maximum error
between fundamental LTR and RI is about 5%. And the peak value of RI is a little bigger
than LTR. at means RI can be more sensitive. In Figure 3.12b, there is a good ﬁt between
RI and LTR for Fishhook maneuver. In general, the new rollover index RI agrees with the
fundamental deﬁnition of LTR when the vehicle rolls with all wheels keeping on road due to
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