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278 Multibody Systems Approach to Vehicle Dynamics
camber angle or a combination of the two. The resulting offset tyre load
introduces an additional component of moment M x . Attention to the sign
convention associated with the tyre reference frame is again needed if the
moment is to be included in a tyre model. In Figure 5.31, to assist understanding,
F z is represented as the tyre load acting on the tyre rather than the
negative normal force computed in the Z SAE direction.
A consideration of the overturning moment is generally more important
where relatively large displacements in the tyre occur, as with aircraft tyres
(Smiley, 1957; Smiley and Horne, 1960). Overturning effects are also of
major importance for motorcycle tyres, particularly in terms of matching
the behaviour of front and rear tyres. The lateral offset, y, also applies to
the longitudinal forces and is responsible for the ‘stand up under light braking’
that all motorcycles display.
5.4.11 Combined traction and cornering (comprehensive slip)
The treatment of longitudinal braking or driving forces and lateral cornering
forces has so far dealt with the two components of force in isolation. The
simulation of vehicle behaviour involving tyre forces acting in this manner
leads to what is termed pure cornering or pure tractive (i.e. driving or braking)
behaviour. In reality longitudinal and lateral forces often occur simultaneously
during vehicle manoeuvres. A typical situation would be to
initiate braking before entering a bend and continue braking into the corner.
It is also typical, once the driver feels sufficient confidence, to begin
applying throttle, and driving forces, during cornering before exiting the
bend. For such situations a tyre model must be able to deal with combined
tractive and cornering forces, a situation referred to as comprehensive slip.
The basic law of friction relating frictional force to normal force can be of
assistance when considering combinations of longitudinal driving or braking
forces with lateral cornering forces. The treatment here concentrates on
lateral forces due to slip angle with camber angle set to zero. Figure 5.32 initially
shows a tyre subject to pure braking or cornering force where in each
case the slip in the ground plane is such that the tyre force produced is a peak
value this being F z , the peak coefficient of friction multiplied by tyre load.
For pure cornering the peak force will occur at a relatively large slip angle
where in Figure 5.32 some lateral distortion of the contact patch is indicated
together with a small amount of pneumatic trail.
For a tyre running at a large slip angle with additional braking force the resultant
ground plane force is still equal to F z but the resultant force direction
opposes the direction of sliding. The longitudinal and lateral forces F x and
F y are now components of the resultant force. Thus it can be seen that the
simultaneous action of longitudinal and lateral slip reduces the amount of
cornering or braking/driving force that may be obtained independently.
Figure 5.33 shows a plot of lateral force against longitudinal force for a
range of slip angles at a given tyre load and with the camber angle set to
zero. The x-axis represents the tyre at zero slip angle running from a maximum
braking force value equal to F z at point A to a maximum driving
force value equal to F z at point B, these points being consistent with the
slip ratios that would produce peak force for a straight running tyre.