6.10 Map of Achievable Performance (MAP)—A New Global Approach
6.10.2 MAP: Vehicle Slip Angle β vs Curvature ρ
Other very useful handling maps can be drawn in the plane (ρ, β), that is maps which show the relationship between the curvatureρ and the vehicle slip angleβ.
Again, it is possible at a glance to appreciate the difference between different vehi- cles.
The achievable region for an understeer vehicle is shown in Fig. 6.25. It is bounded by four lines, each with a precise physical meaning:
(1) upper line: zero lateral acceleration and forward speed;
(2) lower line: maximum lateral acceleration;
(3) left line: maximum forward speed;
(4) right line: maximum steer angle.
But more interesting are the MAPs (Map of Achievable Performance) that can be drawn inside the achievable region.
Curves at constant speeduand also lines at constant steer angleδare shown in Fig.6.26for an understeer vehicle (the same of Fig.6.20). As expected, moving top to bottom along the lines at constant steer angles, that is with increasing speed, brings smaller values of the curvatureρ. Also interesting is to observe that at low speed the slip angleβgrows withδ, whereas at high speed it is the other way around.
At intermediate speeds,β initially grows and then decreases.
Fig. 6.26 Understeer vehicle:β–ρMAP with curves at constant speedu and lines at constant steer angleδ
Fig. 6.27 Understeer vehicle:β–ρMAP with curves at constant lateral accelerationa˜yand lines at constant steer angleδ
Lines at constant lateral accelerationa˜y along, again, with lines at constantδ, are shown in Fig.6.27 for the same understeer vehicle. As expected, the vehicle slip angle β grows steadily if the steer angleδ is increased with constant lateral accelerationa˜y.
Combining Figs.6.26and6.27we obtain Fig.6.28: quite an informative picture to grasp the global vehicle behavior. We can appreciate the interplay between a lot of relevant handling quantities. Again, in Fig.6.28, all lines at constant speed inter- sect all lines at constant lateral acceleration. This is typical of all vehicles without significant aerodynamic vertical loads.
The achievable region in the plane(ρ, β), that is theβ–ρMAP, for an oversteer vehicle (the same of Figs.6.24and6.23) is shown in Fig.6.29, along with curves at constant speeduand lines at constant steer angleδ. As expected, moving top to bottom along the lines at constant steer angles, that is with increasing speed, brings bigger values of the curvatureρ.
Very instructive is the comparison between Figs.6.26and6.29, that is between an understeer and an oversteer vehicle. The two achievable regions have different
6.10 Map of Achievable Performance (MAP)—A New Global Approach 167 Fig. 6.28 Understeer
vehicle:β–ρMAP with lines at constantu,a˜yandδ
Fig. 6.29 Oversteer vehicle:
β–ρMAP with curves at constant speeduand lines at constant steer angleδ
Fig. 6.30 Vehicle with too much understeer:β–ρMAP with lines at constantu,a˜y andδ
Fig. 6.31 Effects of rear steering on the achievable region: rear wheels turning opposite of the front wheels (left), rear wheels turning like the front wheels (right)
Fig. 6.32 Achievable region of a vehicle with rear wheels turning opposite of the front wheels at low speed and like the front wheels at high speed
shapes also because an oversteer vehicle becomes unstable for certain combina- tions of speed and steer angle, as already pointed out when discussing Fig.6.24.
These critical combinations form a sort of stability boundary which collects all points where theu-curves andδ-lines are tangent to each other, as shown in both Figs.6.24and6.29.
On the opposite side, a vehicle with too much understeer has an achievable region like in Fig.6.30, which comes with Fig.6.21.
The effects of rear steering (in addition to front steering, of course) are shown in Fig.6.31. The picture on the left is for the case of rear wheels turning opposite of the front wheels, withχˆ= −0.1 in (6.74), whereas the picture on the right is for rear wheels turning like the front wheels, withχˆ=0.1. The vehicle slip angleβis pretty much affected (cf. Fig.6.28). Basically, a negativeχˆ moves the achievable region
6.11 Vehicle in Transient Conditions (Stability and Control Derivatives) 169
Fig. 6.33 Effect of steering onβ: front steering only (top) and also rear steering (bottom). All cases have the sameα1andα2
upwards, and vice versa. On the other hand,χˆ does not impinge on the available region in the plane(δ, ρ).
To have a narrower achievable region we have to move down the upper part and move up the lower part in the plane(ρ, β). This is indeed the effect of a steer- ing system with rear wheels turning opposite of the front wheels at low speed, and turning like the front wheels at high speed. That is a steering system with, e.g.,χ (u)ˆ = − ˆχ0cos(π u/umax). The net result can be appreciated by comparing Fig.6.32with Fig.6.28. The vehicle behaves better ifβspans a smaller range.
From all these figures, it is also clear for which combinations ofδ anda˜y we have positive or negativeβ. The achievable region provides a much better insight into rear steering than by looking at, e.g., Fig.6.33.