Page 337 - Automotive Engineering Powertrain Chassis System and Vehicle Body

CHAP TER 1 1. 1 Tyre characteristics and vehicle handling and stability

By taking the inverse, the expression for the steer s ffiffiffiffiffiffiffi
angle required to negotiate a curve with a given radius R V crit ¼ gl ðh < 0Þ (11.1.57)
is obtained: h

1 2 aC 1 bC 2 As will be shown later, the motion becomes unstable
d ¼ l mV (11.1.53)
R lC 1 C 2 when the critical speed is surpassed. Apparently, this can
only happen when the vehicle shows oversteer.
It is convenient to introduce the so-called understeer For an understeered car a counterpart has been de-
coefficient or gradient h. For our model, this quantity is fined which is the so-called characteristic speed. It is the
defined as: speed where the steer angle required to maintain the
same curvature increases to twice the angle needed at
mg aC 1 bC 2 s mgC
h ¼ ¼ (11.1.54) speeds approaching zero. We may also say that at the
l C 1 C 2 l C 1 C 2 characteristic speed the path curvature response to steer
angle has decreased to half its value at very low speed.
with g denoting the acceleration due to gravity. After Also interesting is the fact that at the characteristic
having defined the lateral acceleration which in the speed the yaw rate response to steer angle r/d reaches
present linear analysis equals the centripetal acceleration:
a maximum (the proof of which is left to the reader). We
have for the characteristic velocity:
V 2
a y ¼ Vr ¼ (11.1.55) s ffiffiffiffi
R
gl
V ¼ ðh > 0Þ (11.1.58)
Eq.(11.1.53) can be written in the more convenient char h
form as:
! Expression (11.1.54) for the understeer gradient h is
l V 2 l a y
d ¼ 1 þ h ¼ þ h (11.1.56) simplified when the following expressions for the front
R gl R g and rear axle loads are used:

The meaning of understeer vs oversteer becomes clear F z1 ¼ b mg and F z2 ¼ a mg (11.1.59)
when the steer angle is plotted against the centripetal l l
acceleration while the radius R is kept constant. In
Fig. 11.1-10 (left-hand diagram) this is done for three We obtain:
types of vehicles showing understeer, neutral steer and
F z1 F z2
oversteer. Apparently, for an understeered vehicle, the h ¼ (11.1.60)
steer angle needs to be increased when the vehicle is going C 1 C 2
to run at a higher speed. At neutral steer the steer angle which says that a vehicle exhibits an understeer nature
can be kept constant while at oversteer a reduction in steer when the relative cornering compliance of the tyres at
angle is needed when the speed of travel is increased and the front is larger than at the rear. It is important to note
at the same time a constant turning radius is maintained. that in (11.1.59) and (11.1.60) the quantities F z1,2
According to Eq.(11.1.56) the steer angle changes sign denote the vertical axle loads that occur at stand-still and
when for an oversteered car the speed increases beyond thus represent the mass distribution of the vehicle.
the critical speed that is expressed by: Changes of these loads due to aerodynamic down forces

Fig. 11.1-10 The steer angle vs lateral acceleration at constant path curvature (left graph). The difference in slip angle versus lateral
acceleration and the required steer angle at a given path curvature (right graph). The understeer gradient h.

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