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44 4. STABILITY OF TRIPPED VEHICLE ROLLOVER
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
LTR
New RI
6420 8 10
Time (s)
(c) Under a combined rollover situation
Rollover Index
Figure 4.1: (Continued.) Rollover indices under diﬀerent situations.
Case III
In this case a combined untripped and tripped rollover due to an unpredictable road bump under
the right wheel while driving in a step steering is studied. e ﬁnal value of the steering angle
of the front wheel is ı D 2
ı
, the maximum height of the road bump is 0.15 m, and the vehicle
speed is 100 km/h. e rollover indices of the vehicle in this case is shown in Figure 4.1c.
e same conclusions can be drawn as the above two cases. So, the improved rollover
index can predict the risk for both tripped and untripped rollover of a vehicle by measuring the
accelerations and roll angle in real time.
4.1.2 ROLLOVER INDEX ON BANKED ROADS
Much progress has been made for the bus rollover warning in the past decades. But much of this
research has not taken account for dynamic road bank. To ﬁll in the gap, some researchers present
real-time rollover trend prediction to indicate bus rollover risk with road bank estimation [4, 5,
36, 40, 41]. Dahmani et al. used the estimated roll angle and roll rate to compute the rollover
index which is based on the prediction of the lateral load transfer [31]. Lateral load transfer is
the change in the normal force acting on the tyres due to both the acceleration of the CG, and
the shifting of the position of the CG in the y direction due to the movement of the suspension.
Figure 4.2 illustrates lateral load transfer in the vertical plane.
4.1. ROLL INDEX OF TRIPPED VEHICLE ROLLOVER 45
k
ϕ
,
c
ϕ
ϕB
F
zy
F
zy
ma
y
z
y
m
s
g
ϕ
B
ϕ
s
Figure 4.2: Vehicle roll model.
e estimation of the LTR is very diﬃcult since normal force sensors are expensive. An
expression for LTR which depends on the roll states and vehicle parameters can be obtained.
is is denoted by LTR
d
. In order to derive LTR
d
, we resolve weight (m
s
g) and pseudo-force
(m
s
a
y
) into components in the vehicle-ﬁxed y and z directions. e following dynamics are
obtained:
m
s
a
y
h C m
s
gh
.
s
C
B
/
c
P
s
C k
s
D 0; (4.9)
where m
s
is the sprung mass of vehicle and a
y
is the lateral acceleration. k
and c
are the
combined roll stiﬀness coeﬃcient and combined roll damping coeﬃcient, respectively. h is the
CG height from roll axis.
e torque balance for the sprung and unsprung masses about the left tyre roll axis is
written as follows:
F
z2
T
w
C m
s
a
y
h C m
s
gh
B
m
s
g
T
w
2
h
s
m
u
g
T
w
2
D 0; (4.10)
where F
z2
are the vertical right tyre forces, m
u
is the unsprung mass of vehicle.
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