46 4. STABILITY OF TRIPPED VEHICLE ROLLOVER
By substituting Equations (4.9) and (4.10) into Equation (3.4), the following expression
for LTR
d
can be obtained:
LTR
d
D
2
mgT
w
c
P
s
C k
s
: (4.11)
4.2 ENERGY METHODS
Besides the methods mention above, there have been some efforts to predict vehicle rollover
using the vehicle roll energy and the rollover potential energy. Choi proposed a new rollover
index using energy method [42]. Figure 4.3 shows the diagram of the front view of a vehicle
sprung mass where y and z axes are fixed to the CG of the sprung mass and rotate with the
mass. a
y
is the lateral acceleration measured by the accelerometer attached to the vehicle sprung
mass. e measured acceleration is partly from the vehicle acceleration and partly from gravity.
is the absolute roll angle of the vehicle sprung mass with respect to the earth coordinates due
to the lateral acceleration and/or the super elevation angle of the road surface.
g cosϕ
a
y
z
y
h
d
C.G.
⤶
⤶
Figure 4.3: Diagram of vehicle coordinates.
If the artificial angle is defined by the tangent ratio of a
y
and g cos./ as follows:
tan
a
y
g cos
: (4.12)
en, Figure 4.3 can be reconfigured as Figure 4.4.
In the reconfigured coordinates, z axis is defined as parallel to the direction of the net
force on the vehicle sprung mass. Defining the net acceleration g cos./= cos./ on the vehicle
mass as virtual gravity, the problem can be reduced to a mass on a degree hill with a gravity
constant of g cos./= cos./.
In Figure 4.4, the current height of CG is d sin. / C h cos./, and the critical height of
CG, where the vehicle is at the verge of rollover, is
p
d
2
C h
2
. erefore, defining the height
4.2. ENERGY METHODS 47
g
cosϕ
z’
y
’
σ
C.G.
cosσ
⤶
Figure 4.4: Diagram of a vehicle in virtual gravity coordinates.
change of CG required for rollover as h:
h D
p
d
2
C h
2
.d sin C h cos /
D
p
d
2
C h
2
da
y
C hg cos
q
g
2
cos
2
C a
2
y
:
(4.13)
e minimum amount of potential energy—normalized with the vehicle mass—required
for the rollover is defined as .g cos = cos / h using the concept of virtual gravity constant.
Since the lateral kinetic energy of a vehicle can be converted to potential energy very quickly
through the roll motion, a vehicle has the potential to rollover as long as the lateral energy is
larger or equal to the minimum required potential energy, i.e.,
1
2
v
2
>
g cos
cos
h D
q
g
2
cos
2
C a
2
y
p
d
2
C h
2
.da
y
C hg cos /: (4.14)
e lateral velocity v can be calculated from longitudinal velocity u and vehicle side slip
angle ˇ as
v D Uˇ: (4.15)
Motivated by the above inequality condition, a rollover potentiality index ˆ
0
is defined
as follows:
ˆ
0
D
1
2
j
Uˇ
j
2
q
g
2
C a
2
y
p
d
2
C h
2
C da
y
C hg: (4.16)
Positive ˆ
0
means that the vehicle has the potential to rollover, and the possibility of
rollover increases with ˆ
0
. However, large ˆ
0
alone does not mean that the vehicle will rollover.
e large kinetic energy needs to be converted to roll dynamic energy. It usually happens when
a vehicle hits a high surface or a bump after a large side slip typically on a low surface. If the
vehicle hits a high surface, the lateral acceleration of the vehicle increases very quickly. Simula-
tion results show that the measured lateral acceleration needs to be more than 80% of statically
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