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258 Multibody Systems Approach to Vehicle Dynamics
SPIN
AXIS
OVERTURNING
MOMENT
(M x )
TRACTIVE FORCE
(F x )
WC
{X SAE } 1
α
P
ROLLING RESISTANCE
MOMENT (M y )
SELF-ALIGNING
MOMENT
(M y )
{Z SAE } 1
{Y SAE } 1
LATERAL FORCE
(F y )
NORMAL FORCE
(F z )
Fig. 5.12
Tyre forces and moments shown acting in the SAE tyre axis system
The use of the term stiffness can also add confusion to newcomers to the
subject area. A traditional static force/displacement approach is used by
Moore (1975) to define longitudinal, lateral and torsional stiffness of a
tyre. In each case a non-rolling tyre is mounted on a plate and incrementally
loaded as indicated in Figure 5.13 until complete sliding occurs.
Plotting graphs of force or moment against displacement or rotation allows
the stiffness parameters to be obtained from the slopes at the origin.
We will see later that terms such as cornering stiffness and aligning moment
stiffness are associated with a rolling tyre and should not be confused with
the lateral and torsional stiffness defined here. The term longitudinal stiffness
can be particularly misleading as another definition is commonly used
when longitudinal tractive forces due to driving and braking are discussed.
The explanations that follow will initially deal with each force mechanism
in isolation, for example lateral forces arising due to slip angle and camber
angle are considered separately with no simultaneous longitudinal tractive
force. Following this, a more complex treatment involving combinations of
the various force components will be addressed.
5.4.2 Normal (vertical) force calculations
The calculation of normal force in the tyre is relatively straightforward compared
with the calculation of longitudinal or lateral forces. The normal force
will, however, always be negative when computed using the SAE tyre axis
system. This is not particularly elegant when presenting the dependencies of
other force components on the normal force. To overcome this a positive
value of this force component is often referred to as vertical force or tyre
load. In SAE J670e (SAE Publication, 1976) vertical load is taken as the
negative of normal force.