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302 Multibody Systems Approach to Vehicle Dynamics
F x
M z
Slip angle
Slip ratio
F y
Fig. 5.57
Typical form of tyre force and moment curves from steady state testing
It is important to note that the data used to generate the tyre model is obtained
from steady state testing. The lateral force F y and the aligning moment
M z are measured during pure cornering, i.e. cornering without braking, and
the longitudinal braking force during pure braking, i.e. braking without
cornering.
The basis of this model is that tyre force and moment curves obtained
under pure slip conditions and shown in Figure 5.57 look like sine functions
that have been modified by introducing an arctangent function to
‘stretch’ the slip values on the x-axis.
The general form of the model as presented in Bakker et al. (1986) is:
y(x) D sin[C arctan{Bx E(Bx arctan(Bx))}] (5.49)
where
Y(X) y(x) S v (5.50)
x X S h (5.51)
S h horizontal shift
S v vertical shift
In this case Y is either the side force F y , the aligning moment M z or the
longitudinal force F x and X is either the slip angle or the longitudinal
slip, for which Pacejka uses . The physical significance of the coefficients
in the formula become more meaningful when considering Figure 5.58.
For lateral force or aligning moment the offsets S v and S h arise due to
adding camber or physical features in the tyre such as conicity and ply
steer. For the longitudinal braking force this is due to rolling resistance.
Working from the offset XY-axis system the main coefficients are:
D – is the peak value.
C – is a shape factor that controls the ‘stretching’ in the x direction. The
value is determined by whether the curve represents lateral force, aligning
moment, or longitudinal braking force. These values can be modified to