Page 349 - Automotive Engineering Powertrain Chassis System and Vehicle Body

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

intercepts under 45 (da 2 /da 1 ¼ 1): consists of only the main branch through the origin. If the
rear axle characteristic (at least in the end) is higher than
F y1 ða 1 Þ F y2 ða Þ the front axle characteristic, the vehicle will show (at
2
¼ (11.1.97) least in the limit) an understeer nature and unstable
F z1 F z2
singular points cannot occur. This at least if for the case
Fig. 11.1-22 illustrates the way these isoclines are of initial oversteer the speed remains under the critical
constructed. The system of Fig. 11.1-17 with k ¼ a ¼ b, speed. In such cases, the domain of attraction is theo-
d ¼ 0.04 rad and V¼ 50 km/h has been considered. Note retically unbounded so that for all initial conditions ul-
that the normalised tyre characteristics appear in the timately the stable equilibrium is attained. The domain
left-hand diagram for the construction of the isoclines. of Fig. 11.1-22 appears to be open on two sides which
The three points of intersection of the isoclines are the means that initial conditions, in a certain range of (r/v)
singular points. They correspond to the points I, II and III values, do not require to be limited in order to reach the
of Fig. 11.1-17. The stable point is a focus (spiral) point stable point. Obviously, disturbance impulses acting in
with a complex pair of solutions of the characteristic front of the centre of gravity may give rise to such
equation with a negative real part. The two unstable combinations of initial conditions.
points are of the saddle type corresponding to a real pair In Figs. 11.1-23 and 11.1-24 the influence of an in-
of solutions, one of which is positive. The direction in crease in steer angle d on the stability margin (distance
which the motion follows the trajectories is still a ques- between stable point and separatrix) has been shown for
tion to be examined. Also for this purpose the alternative the two vehicles considered in Fig. 11.1-20. The system of
set of axes with r and v as coordinates (multiplied with Fig. 11.1-23 is clearly much more sensitive. An increase in
a factor) has been introduced in the diagram after using d (but also an increase in speed V) reduces the stability
the relations (11.1.44). margin until it is totally vanished as soon as the two sin-
From the original equations (11.1.42) it can be found gular points merge (also the corresponding points I and II
that the isocline (11.1.97) forms the boundary between on the handling curve of Fig. 11.1-17) and the domain
areas with _ r > 0 and _ r < 0 (indicated in Fig. 11.1-22). breaks open. As a result, all trajectories starting above the
Now it is easy to ascertain the direction along the tra- lower separatrix tend to leave the area. This can only be
jectories. We note that the system exhibits a bounded stopped by either quickly reducing the steer angle or en-
domain of attraction. The boundaries are called separa- larging d to around 0.2 rad or more. The latter situation
trices. Once outside the domain, the motion finds itself appears to be stable again (focus) as has been stated
in an unstable situation. Remains the disturbance limited before. For the understeered vehicle of Fig. 11.1-24 sta-
so that resulting initial conditions of the state variables bility is practically always ensured.
stay within the boundaries, then ultimately the steady- For a further appreciation of the phase diagram it is
state condition is reached again. of interest to determine the new initial state (r o , v o )
For systems with normalised characteristics showing after the action of a lateral impulse to the vehicle
everywhere a positive slope, a handling curve arises that (cf. Fig. 11.1-25). For an impulse S acting at a distance x

Fig. 11.1-22 Isoclines for the construction of trajectories in the phase-plane. Also shown: the three singular points I, II and III
(cf. Fig. 11.1-17) and the separatrices constituting the boundary of the domain of attraction. Point I represents the stable cornering motion
at steer angle d.

350