2.3. LATERAL-YAW-ROLL MODEL 7
where the lateral acceleration is given below:
a
y
D Pv C ur: (2.12)
y
a
r
b
x
z
o
u
v
F
f
F
r
f
β
β
δ
r
R
m
s
a
y
m
s
g
F
L
F
g
T
w
h
h
c
m
road
⤶
ϕ⤶
Figure 2.5: Lateral-yaw-roll model.
Equation (2.11) describes the balance relations of the lateral forces on the entire vehicle,
the yaw moments of the entire vehicle, and the roll moments on the sprung mass, respectively.
e lateral forces in Equation (2.11) mainly come from the contact between the tire and
the road surface at each front and rear wheel and is a function of the physical properties of the
tire and the corresponding sideslip angles ˇ
f
or ˇ
r
observed on the front wheel or rear wheel,
respectively. e slip angle of a tire can be determined from the simple geometric relations shown
in Figure 2.6 as follows:
ˇ
f
D arctan
v C ar
U
ı; ˇ
r
D arctan
v br
U
: (2.13)
In this study, a simple tire model with linear constant cornering stiffness will be used so
that the lateral forces of tires yield
F
f
D k
f
ˇ
f
; F
r
D k
r
ˇ
r
: (2.14)
As the vehicle is moving in cornering, the lateral velocity and yaw rate do not vanish.
So, the dynamics of vehicle rollover can be described by Equations (2.11), (2.12), (2.13), and
(2.14) in partial unknown state variables v, r, , and
P
. at is, the dynamic equation of vehicle