Page 352 - Automotive Engineering Powertrain Chassis System and Vehicle Body

Tyre characteristics and vehicle handling and stability C HAPTER 11.1

At moderate braking with deceleration –a x ¼ F L /m the 11.1.3.5 The moment method
load transfer remains small and we may use the linearised
approximation of the variation of cornering stiffness with Possible steady-state cornering conditions, stable or un-
vertical load: stable, have been portrayed in the handling diagram of
! Fig. 11.1-17.In Fig. 11.1-22 motions tending to or
departing from these steady-state conditions have been
vC i
C i ¼ C io þ z ai DF zi with z ai ¼ depicted. These motions are considered to occur after
vF zi
F zio a sudden change in steer angle. The potential available to
(11.1.103) deviate from the steady turn depends on the margin of the
front and rear side forces to increase in magnitude. For
The understeer gradient (11.1.101) can now be
each point on the handling curve it is possible to assess the
expressed in terms of the longitudinal acceleration a x degree of manoeuvrability in terms of the moment that
(which might be: minus the forward component of the can be generated by the tyre side forces about the vehicle
acceleration due to gravity parallel to the road). We centre of gravity. Note that at the steady-state equilib-
obtain:
rium condition the tyre side forces are balanced with the
centrifugal force and the moment equals zero.
a x
h ¼ h þ l (11.1.104) In general, the handling curve holds for a given speed
o
g
of travel. That is so, when e.g. the aerodynamic down
forces are essential in the analysis. In Fig. 11.1-27 a dia-
with the determining factor l approximately expressed as:
gram has been presented that is designated as the
2 2 Milliken Moment Method (MMM) diagram and is
h F z1o h F z2o computed for a speed of 60 mph. The force-moment
l ¼ z a1 þ z a2 (11.1.105)
b C 1o a C 2o concept was originally proposed by W.F. Milliken in 1952
and thereafter continuously further developed by the
and h o denoting the original value not including the effect Cornell Aeronautical Laboratory staff and by Milliken
of longitudinal forces. Obviously, since z a1,2 is usually Research Associates. A detailed description is given in
positive, negative longitudinal accelerations a x , corre- Milliken’s book.
sponding to braking, will result in a decrease of the The graph shows curves of the resulting tyre moment
degree of understeer. N vs the resulting tyre side force Y in non-dimensional
To illustrate the magnitude of the effect we use the form. The resulting force and moment result from the
parameter values given in Table 11.1-1 (above individual side forces and act from ground to vehicle. For
Eq.(11.1.77)) and add the c.g. height h ¼ 0.6 m and the greater accuracy, one may take the effect of the pneu-
cornering stiffness vs load gradients z ai ¼ 0.5C io /F zio . The matic trails into consideration. Two sets of curves have
resulting factor appears to take the value l ¼ 0.052. This been plotted: one set for constant values of the vehicle
constitutes an increase of h equal to0.052a x /g. Apparently, side slip angle b with the steering wheel angle d stw as
the effect of a x on the understeer gradient is considerable parameter and the other set for contant steer angle and
when regarding the original value h o ¼ 0.0174. varying slip angle. Along the horizontal axis the moment is
As illustrated in Fig. 11.1-9 the peak side force will be zero and we have the steady-state equilibrium cornering
diminished if a longitudinal driving or braking force is situation that corresponds with the handling curve. It is
transmitted by the tyre. This will have an impact on the observed that for the constant speed considered in the
resulting handling diagram in the higher range of lateral diagram, the steer angle increases when the total side
acceleration. The resulting situation may be represented force Y or lateral acceleration a y is chosen larger which
by the second and third diagrams of Fig. 11.1-18 corre- indicates that the motion remains stable. At the limit
sponding to braking (or driving) at the front or rear re- (near number 2) the maximum steady-state lateral ac-
spectively. The problem becomes considerably more celeration is attained. At that point the ability to generate
complex when we realise that at the front wheels the a positive moment is exhausted. Only a negative moment
components of the longitudinal forces perpendicular to may still be developed by the car that tends to straighten
the x axis of the vehicle are to be taken into account. the curve that is being negotiated. As we have seen in
Obviously, we find that at braking of the front wheels Fig. 11.1-18, second diagram, there is still some side force
these components will counteract the cornering effect of margin at the rear tyre which can be used to increase the
the side forces and thus will make the car more un- lateral acceleration in a transient fashion. At the same
dersteer. The opposite occurs when these wheels are time, however, the car yaws outwards because the asso-
driven (more oversteer). ciated moment is negative (cf. diagram near number 8).
At hard braking, possibly up to wheel lock, stability How to get at points below the equilibrium point near the
and steerability may deteriorate severely.

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