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Tyre characteristics and modelling 283
Steady state
Slip angle
c
l
last
d l
dt
Integration time step
t 0
Fig. 5.36
t last t
Time (s)
Build-up of slip angle in an MBS model to represent tyre lag
where
last is the value of l computed at the last successful integration time step
t is the current simulation time
t last is the time for the last successful integration time step
An estimate of the term ( c l )/ in equation (5.25) can be obtained from
(5.27)
Combining equations (5.26) and (5.27) allows the value of l required to
compute lateral force at the current time step to be obtained from
⎛ c l⎞
l
tt
⎝
(5.28)
Using data for the baseline vehicle used throughout this text it is possible
to carry out a calculation to estimate a time delay, tyre lag, for a vehicle
travelling at 100 kph. The load on the tyre F z is taken as 4500 N and
the radial stiffness of the tyre k z is taken as 160 N/mm. From this it is
possible to calculate the static tyre deflection z :
Fz
z
28.1 mm
(5.29)
kz
Referring back to equation (5.10) the effective rolling radius, R e , can be
calculated using the tyre deflection z from equation (5.29) and an
unloaded tyre radius, R u , of 318.5 mm from
R
c l c last
e
R
309.1 mm
u
z
3
last
⎠ ( )
last
(5.30)
Typically a tyre would roll through between 0.5 and 1 revolution
(Gillespie, 1992) in order to develop the lateral force following a change
in slip angle. If we assume that the tyre must complete 0.5 revolutions then
for a speed of 100 kph the tyre lag on this vehicle is 0.035 s.