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280 Multibody Systems Approach to Vehicle Dynamics
Lateral force F y
D
Friction circle
= 4°
Resultant force
F R
= 2°
= 1°
A Braking force F x
Driving force F x B
Fig. 5.33 Plotting lateral force against longitudinal force (friction circle)
Figure 5.33 shows, for a typical tyre, the general form of the friction circle
diagram for the full range of driving and braking forces. Note that only lateral
forces due to positive slip angle are presented and a similar diagram
would exist for measurements taken at negative slip angles. Point D represents
an example of a position where the tyre is operating at the friction
limit for combined braking and cornering, as shown in Figure 5.32 where
it is clear that the amount of braking or cornering force that could be produced
independently is reduced and that the magnitude of F R is simply
2
F F F
2 (5.23)
R x x
C
= 10°
= 6°
It can also be noted that the curves are not symmetric in that lateral forces
initially increase slightly as braking force is applied. As discussed earlier
the braking force adds circumferential tension to the tyre material entering
the contact patch. This stress stiffening effect can be seen to raise the lateral
force slightly while the reversal of longitudinal force to driving leads
to a reduction. As the curves approach the friction limit it can be observed
that they turn inwards. For a fixed slip angle the longitudinal slip is
increased moving along the curves, for either braking or driving, until the
point where both lateral force and longitudinal force reduce, hence causing
the curves to bend back.
Plotting curves of aligning moment on the y-axis against longitudinal force
on the x-axis it is possible to show that the opposite can occur (Phillips,
2000) and that adding braking force reduces aligning moment and adding
driving force adds to it. Referring back to Figure 5.32, the bottom diagram
shows a tyre running at moderate slip angle producing a lateral force F y
along a line of action set back from the centre by the pneumatic trail. The
contact patch is shown displaced laterally due to the cornering force so that
for simultaneous braking the braking force produces a moment that would
subtract from the existing aligning moment due to the product of lateral
force and pneumatic trail. From the diagram at the bottom of Figure 5.32 it is
also clear that the simultaneous application of a driving force would produce