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392 Multibody Systems Approach to Vehicle Dynamics
rl/
r
⎛ ⎞
vy
c˙
arctan
⎜ ⎟
1
⎜ vx
tr˙
⎟
⎝ 2 ⎠
(6.33)
⎛
arctan v y ⎞
⎜ ⎟
⎝ v ⎠
(6.34)
In this model roll and pitch of the vehicle are neglected but weight transfer
is included to determine the vertical load at each wheel as defined by
Milliken and Milliken (1995):
F mg m ah y c
zfl/
r ⎛ 1
ma
2
⎞
⎜
⎟
⎝ t ⎠
l
F mg m ah y b
zrl/
r ⎛ 1
ma
2
⎞
⎜
⎟
⎝ t ⎠
l
(6.35)
(6.36)
The additional parameters are the height h of the vehicle’s centre of gravity,
the wheelbase and the gravitational acceleration g.
In equations (6.37) and (6.38) it has to be considered that a x v . x and
a y v . y. The yaw motion of the vehicle has to be taken in account (Wong,
2001) giving:
a v˙
v
˙
(6.37)
x x y
a v˙
v
˙
y y x
x
x
x
h
l
h
l
(6.38)
In this work the author (Wenzel et al., 2003) has simulated a range of
vehicle manoeuvres using both the ‘Magic Formula’ and Fiala tyre models
described in Chapter 5. The example shown here is for the lane change
manoeuvre used in this case study with a reduced steer input applied at the
wheels as shown in the bottom of Figure 6.63.
Also shown in Figure 6.63 are the results from the Simulink model and a
simulation run with the MSC.ADAMS linkage model. For this manoeuvre
and vehicle data set the Simulink and MSC.ADAMS models can be seen to
produce similar results.
In completing this case study there are some conclusions that can be drawn.
For vehicle handling simulations it has been shown here that simple models
such as the equivalent roll stiffness model can provide good levels of accuracy.
It is known, however, that roll centres will ‘migrate’ as the vehicle
rolls, particularly as the vehicle approaches limit conditions. The plots for
Case Study 1 in Chapter 4 show the vertical movement of the roll centre
along the centre line of the vehicle as the suspension moves between bump
and rebound. On the complete vehicle the roll centre will also move laterally
off the centre line as the vehicle rolls.
Using a multibody systems approach to develop a simple model may also
throw up some surprises for the unsuspecting analyst. The equivalent roll