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282 Multibody Systems Approach to Vehicle Dynamics
Lateral force versus time
Lateral force F y (N)
F y max
0.632F y max
Steady state
t 0
t 1
τ
t 2
Time (s)
Fig. 5.35
Development of lateral force following step steering input
history plot, similar to that shown in Figure 5.35, indicating the build-up in
lateral force.
The results obtained (Loeb et al., 1990) for the lateral force response
appear exponential indicating a first order dynamic system where the time
constant , equal to t 2 t 1 in Figure 5.35, is the time required to achieve
63.2% of the final steady state response.
Incorporation of a lag effect for tyre lateral force within an MBS program
requires an understanding of the mathematical integration process used to
solve the equations of motion as discussed in Chapter 3 of this text. For the
MSC.ADAMS program the approach taken is to compute a theoretical
value of slip angle, l , that includes a lag effect and to input this to the
appropriate tyre model algorithm for lateral force due to slip angle. As a
starting point the tyre relaxation length L R is taken as an input parameter
from which, for a forward speed V x , the time constant can be found using
L R /V x (5.24)
Thus by this definition the relaxation length L R is the distance through
which the tyre must roll in order to develop 63.2% of the required lateral
force. This leads to an initial expression:
dl
c l
(5.25)
dt
where
c is the computed value of slip angle (instantaneous) at the current time
l is the value of slip angle corrected to account for lag
An estimate of the term d l /dt in equation (5.25) can be obtained from
equation (5.26). Additional understanding of the terms can be obtained by
reference to Figure 5.36 where for clarity the integration time step, t t last ,
is shown with exaggerated magnitude:
dl
l
last
≈
(5.26)
d t t
t
last