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22 3. STABILITY OF UNTRIPPED VEHICLE ROLLOVER
can be obtained from Equation (3.1) as follows:
SSF D
a
y
g
D
T
w
2H
: (3.2)
It can be seen from Equation (3.2) that the value of SSF only relates to track and height of
center of mass, which is convenient for application. Ronald Huston [24] used SSF and dynamic
stability factor” which is deﬁned in the same way as the SSF but for the dynamic case to predict
rollover. Figures 3.1 and 3.2 show the variation of vehicle body critical title angle
cr
with SSF
and with the tire/road friction [24]. As expected, the greater the SSF value, the larger is the
tilt angle, and thus the less likely the vehicle is to roll over and, conversely, the smaller the SSF,
the greater is the rollover propensity.
50
45
40
35
30
25
20
15
10
5
0
0.9 1.0 1.1 1.2 1.3 1.4
θ
cr
(deg)
μ = 0.10
μ = 0.25
μ = 0.40
μ = 0.55
μ = 0.70
μ = 0.85
S = SSF
Figure 3.1: Critical tilt angle as a function of the SSF for various values of the coeﬃcient of
friction.
is measure of rollover propensity reﬂects only the most fundamental relation and does
not take the eﬀects of suspension and tire compliance into account. It cannot eﬀectively predict
rollover risk during a dynamic case. erefore, it is essential to propose a rollover index which
applies to both steady case and dynamic case.
3.1.2 DYNAMICS STABILITY FACTOR
Because the SSF does not work well in a dynamic condition. To detect wheel liftoﬀ condi-
tions when a vehicle is moving, Jin et al. deﬁned a Dynamics Stability Factor (DSF) based on
3.1. ROLL INDEX OF UNTRIPPED VEHICLE ROLLOVER 23
50
45
40
35
30
25
20
15
10
5
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
θ
cr
(deg)
S = 1.40
S = 1.30
S = 1.20
S = 1.10
S = 1.00
S = 0.90
Coeﬃcient of Friction μ
Figure 3.2: Critical tilt angle as a function of the coeﬃcient of friction for various values of the
SSF.
SSF [13]:
DSF D
T
w
2H
u
2
m
s
h
2
ı
LH
k
m
s
hg
"
1
mu
2
L
2
a
2k
r
b
2k
f
cos ı
u
2
m
s
h
c
f
c
r
L
k
m
s
hg
#
: (3.3)
In the steady state motion, the lateral acceleration at the rollover threshold can be pre-
dicted by the value of DSF. e vehicle with a larger DSF implies a higher threshold of rollover.
Compared with the traditional SSF, the dynamic stability factor has a number of features as fol-
lows.
1. From the conditions in Equation (3.3), the second item of DSF (including minus) is always
negative. Hence, DSF is always smaller than SSF, and can predict the trend of vehicle
rollover more precisely than SSF, as shown in Figure 3.3.
2. DSF includes the eﬀects of the track width T
w
and the height of the center of gravity
H on vehicle dynamic stability. DSF increases with an increase in the track width and
the decrease in the height of the center of gravity. at is, increasing the track width and
decreasing the height of the center of gravity can improve the stability of vehicle rollover.
is fact gets an agreement with the traditional SSF. As shown in Equation (3.3), the
24 3. STABILITY OF UNTRIPPED VEHICLE ROLLOVER
1.4
1.25
1.185
1
0.8
0.6
0.4
0.2
0
0 1.0
Time (s)
Lateral Acceleration (g)
1.5 2.00.5
(a) U = 13 m/s
Roll Motion in Cornering
SSF Boundary
DSF Boundary
1.4
1.25
1.172
1
0.8
0.6
0.4
0.2
0
0 1.0
Time (s)
Lateral Acceleration (g)
1.5 2.00.5
(b) U = 14.5 m/s
Roll Motion in Cornering
SSF Boundary
DSF Boundary
Figure 3.3: Lateral acceleration of roll motion at three forward speeds. (Continues.)
3.1. ROLL INDEX OF UNTRIPPED VEHICLE ROLLOVER 25
1.5
1.25
1.158
1.0
0.5
0
0 1.0
Time (s)
Lateral Acceleration (g)
1.5 2.00.5
(c) U = 16 m/s
Roll Motion in Cornering
SSF Boundary
DSF Boundary
Figure 3.3: (Continued.) Lateral acceleration of roll motion at three forward speeds.
second item of DSF has nothing to do with the track width T
w
, but inversely relates to the
center of gravity H . us, decreasing H receives better improvement on vehicle stability
than increasing T
w
, as shown in Figure 3.4, while SSF does not show this tendency.
3. DSF takes the longitudinal location of the center of gravity into consideration. As section
(b), when wheelbase L is ﬁxed as constant, substituting b D L a into the second item of
DSF yields a function f .a/. is is a decreasing function to variable a. erefore, moving
the center of the sprung mass close to the front axle can improve the stability of vehicle
rollover, as shown in Figure 3.5.
4. DSF takes the forward speed and the steering angle into account. From Equation (3.3),
DSF increases with the decreasing of the forward speed U and the front wheel-steering
angle ı. erefore, this evidence enables one to improve the stability of vehicle rollover by
using either a low speed or a small steering angle, as shown in Figure 3.6.
5. DSF varies from the equivalent roll stiﬀness of the suspension, as seen from Equa-
tion (3.3). It is reasonable to expect that enhancing this fact enables one to improve the
stability of vehicle rollover, as shown in Figure 3.7.
6. Furthermore, DSF includes the eﬀects of the properties of tires on vehicle dynamic sta-
bility. e ratio of the cornering stiﬀness of a front tire and the rear as
1
D k
f
=k
r
and
substituting it into the second item of DSF yields a function f .
1
/, which is a decreas-
26 3. STABILITY OF UNTRIPPED VEHICLE ROLLOVER
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.6
0.7 1.0 1.1 1.2 1.3
Height of Center of Gravity (m)
Critical Lateral Acceleration (g)
1.40.8 0.9
(a) Critical lateral acceleration vs. the height of CG
SSF
DSF
stable
1.5
1.4
1.3
1.2
1.1
1.0
0.9
0.8
1.5 1.6 1.7 2.0 2.1 2.2
Track Width [m]
Critical Lateral Acceleration (g)
1.8 1.9
(b) Critical lateral acceleration vs. the track width T
w
SSF
DSF
stable
Figure 3.4: e critical lateral acceleration with respect to the height of center of gravity and the
track width, respectively, for SSF and DSF.
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