4569846498

Modelling and assembly of the full vehicle 391
a y lateral acceleration (m/s 2 )
torque around z-axis (Nm)
steer angle (rad)
side slip angle (rad)
ij wheel slip angles (rad)
. yaw rate (rad/s)
F zij vertical forces on each wheel (N)
ij position: i front(f )/rear(r), j left(l)/right(r)
Note that steer angle and the velocity of the vehicle’s centre of gravity
v cog are specified as model inputs.
The relationship between the dynamic vehicle parameters can be formulated
as differential equations. Most of these can be found in the standard
literature. Using formulas by Wong (2001) and Will and Żak (1997) the
following differential equations for acceleration, torque and yaw rate can
be derived:
1 ( )
v˙ cos sin cos sin ˙
x m F xfl F yfl F xfr F yfr F xrl F xrr v y
1 ( )
(6.28)
v˙ cos sin cos sin ˙
y m F yfl F xfl F yfr F xfr F yrl F yrr v x
(6.29)
t f
t f tr
tr
Fxfl
Fxfr
Fxrl
Fxrr bFyfl bFyfr cFyrl cFyrr
2 2 2 2
M M M M
(6.30)
zfl zfr zrl zrr
˙˙ J z
(6.31)
where the additional parameters are defined as:
F xij longitudinal forces on tyre ij (N)
F yij lateral forces on tyre ij (N)
F xij longitudinal forces on tyre ij in the vehicle’s co-ordinate system (N)
F yij lateral forces on tyre ij in the vehicle’s co-ordinate system (N)
M zij self-aligning moment on tyre ij (Nm)
m mass of vehicle (kg)
J z moment of inertia around vertical axis (Nm 2 )
t f , t r front and rear track width (m)
b, c position of centre of gravity between wheels (m)
Other important states are the wheel slip angles ij and the body slip angle
, defined as follows:
fl/
r
⎛ ⎞
vy
b˙
arctan
⎜ ⎟
1
⎜ vx
tf˙
⎟
⎝ 2 ⎠
(6.32)