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TOE, ACKERMANN, CORNERING FORCE, TURN-IN AND RELATED PUZZLEMENTS
Following on from your latest chassis newsletter regarding the advantages of static toe out to aid turn in, what are your views on using anti-Ackermann geometry to put more steering angle onto the tyre with the greater slip angle?
Last month I mentioned that a car gets quicker initial turn-in when it has some static toe-out at the front wheels. Iíll stand by that, but I think I need to clarify a bit what quicker initial turn-in actually means in this context. It doesnít necessarily mean that the car negotiates that segment at a higher speed, or that the lap time will be lower. It means that it takes a smaller handwheel movement to produce the desired initial yaw acceleration to make the car follow a desired line, and the car feels more responsive.
This may improve lap time, when a very rapid increase in yaw velocity is desired, as in a chicane or a sharp turn in autocross. But in such cases it will matter at least as much what the toe condition is at large handwheel displacements.
In most oval track and road course turns, it is easy for the driver to make too abrupt a yaw input, with any front end settings. Getting the car to respond quickly isnít really the key to good segment time, as much as being smooth is. The beginning driver usually needs to work on having slower hands.
Letís review a bit of terminology. Anti-Ackermann, or negative Ackermann, is steering geometry that makes the front wheels toe in with respect to each other when steered away from center. Zero Ackermann is parallel steer: front wheels maintain static wheel-to-wheel toe setting as they steer. Positive Ackermann is steering geometry that creates wheel-to-wheel toe-out with steer. 100% Ackermann is steering geometry that gives an amount of toe-out that produces scuff-free operation in low-speed turning.
The turn center is the point about which the carís c.g. or origin is instantaneously moving when the car is traveling in a curved path Ė in other words, the instantaneous center of curvature of the carís path.
The carís origin is a point chosen as its center for modeling purposes. It may be at the carís c.g., or it may be at the midpoint of its wheelbase and tracks. I tend to prefer the latter, at least for the purposes at hand.
Slip angle is the angular difference between the direction a wheel or other object is aimed (its heading) and the direction itís actually traveling (its bearing).
A tire thatís cornering hard runs at a slip angle. On a given surface, at any given load (Fz), pressure, temperature, camber, and so on, the tire will have a characteristic relationship between slip angle and lateral or y-axis force or Fy. When graphed with slip angle on the horizontal axis and Fy on the vertical axis, the function displays as a concave-down curve that rises, peaks, and then descends. The peak occurs at the slip angle where the tire generates the greatest Fy under those conditions.
That peak occurs at a greater slip angle when the load on the tire is greater. Consequently, a pair of unequally loaded tires will generate the greatest total Fy when the more heavily loaded one is running at a somewhat greater slip angle than the less heavily loaded one. This leads some to conclude that the wheels need to be toed in a bit when cornering, to achieve that slip angle relationship, and therefore a race car should have negative Ackermann.
Unfortunately, the relationships involved are far from simple. In particular, there is not a simple relationship between toe and slip angle, contrary to what one might suppose.
At low speed, with all steering done by the front wheels, and little or no slip angle on any tire, as seen from above the car, the turn center lies on the rear axle line. For the front tires to be at zero slip, as seen from above their axes need to pass through the turn center. This means that the outside front wheel must steer less than the inside front wheel, and the axis of the outer front wheel must lie rearward of the inner front wheelís axis. We may then say that the inside front wheel is leading the outside front wheel through the turn (or the outer is trailing the inner), and neither rear wheel is leading the other.
In this case, the front wheels must have toe-out to have equal slip angles. If they are parallel, the outside one will have a positive slip angle and the inside one will have a negative slip angle. Both will be scuffing, in opposite directions. Interestingly, the car as a whole, at its origin, has a negative slip angle: the body points out of the turn, and the nose tracks outside of the tail. The rear wheels track inside the fronts.
If we start adding speed, the car as a whole will assume a drift or attitude angle, and all the tires will start to assume slip angles, if they had none at very low speed. If the car has neutral handling, or an
understeer gradient of zero, the driver will not need to move the handwheel to keep the car on course, but the car will change attitude relative to its direction of travel. The tail will run further out,
and the nose will run further in. The turn center will no longer be on the rear axle line, but will be forward of it.
As we keep increasing speed, and the turn center keeps moving forward with respect to the car, we will reach a point where the turn center lies on a line perpendicular to the carís centerline, and passing through the origin. At this point, the body has a slip angle of zero. The outside rear wheel leads the inside rear wheel. For the rear wheels to have equal slip angles, they need to have toe-in. The inside front wheel still leads the outside front, but by less than at very low speed. For the front wheels to have equal slip angles, they need to have toe-out, but less than at very low speed.
The rear wheels will now be making tracks that lie approximately on top of the front tire tracks, assuming front and rear track widths are close to equal.
Since the car is now making some lateral acceleration, it will now have some lateral load transfer. This will make the outside tires more heavily loaded than the inside ones, and there will be a case for having less toe-out at the front, and more toe-in at the rear, because of that. However, the exact amount of this that we want will depend on the exact nature of the tire properties weíre dealing with. It wonít be the same for all cars, or all tires.
As we add more speed, the slip angles will increase further, and the turn center will continue to move forward with respect to the car. The car as a whole will now have a positive slip angle, and the rear tires will be tracking outside the fronts. At some point, we will reach a condition where the turn center lies on the front axle line. In this condition, we finally have the relationship we might have imagined between front toe and front slip angle: toe-in definitely implies greater slip angle on the outside tire; toe-out definitely implies greater slip angle on the inside one; parallel front wheels actually do have equal slip angles.
The rear tires are now tracking well outside the fronts. The outside rear is leading the inside rear by more than before. The rear wheels now need more toe-in than before, to have equal slip angles.
Finally, as we add more speed, the turn center can move forward of the front axle line. In this situation, the outside front wheel leads the inside front, and the inside front has the greater slip angle if the front wheels are parallel. The front wheels now need toe-in to have equal slip angles.
We can thus define three turn center zones:
∑ Zone 1: Turn center between rear axle and origin. Rear wheels track inside fronts. Fronts need toe-out to have equal slip angles.
∑ Zone 2: Turn center between origin and front axle. Rear wheels track outside fronts. Fronts need toe-out to have equal slip angles, but not as much as for Zone 1.
∑ Zone 3: Turn center ahead of front axle. Rear wheels track outside fronts, more than in Zone 2. Front wheels need toe-in to have equal slip angles.
Note that because we have assumed a zero understeer gradient, the handwheel and the front wheels (with respect to the car) are at the same angle through all of this. In the real world, the understeer gradient is usually positive, or in some cases may be negative Ė it is seldom truly zero. If the car has the front wheels steered, say, an average of four degrees to the left, it could be understeering through a left-hand sweeper with the rear wheels at a very small slip angle (turn center in Zone 1), or it could be on a smaller radius left turn with the rears sliding considerably (turn center in Zone 2), or it could be powersliding around a left hairpin (turn center in Zone 3). The car could even be turning right, in a state of pronounced oversteer, with the driver countersteering (turn center in Zone 3, and possibly to the right of the carís centerline).
The car could be cornering, and have the front wheels pointed straight (zero handwheel displacement). This implies some amount of oversteer, and implies that the rear tires are tracking outside the fronts and the turn center is in Zone 3. The turn could be a sweeper, with the rear wheels sliding just a little, or it could be a tighter turn, with the rear wheels sliding a lot. In any such case, if the handwheel displacement is zero, Ackermann will have no effect at all, and the toe setting will be about the same as static, altered only slightly by compliances and roll steer.
Each of these situations requires a different front toe condition for equal slip angles, or for a given amount of slip angle inequality. Yet any conventional steering system can only provide one fixed relationship between toe and average steer. Therefore, there is no way to create steering geometry that is optimal for all conditions.
However, we can make some useful generalizations, at least for cars running on pavement and having small understeer gradients:
∑ Very tight turns will be taken with the front wheels steered hard into the turn, and the rear wheels will track inside the fronts, unless the driver is severely horsing the car with the throttle. The inside front will lead, and the turn center will be in Zone 1. Even if we want the outside front tire to operate at somewhat greater slip angle than the inside front, we will need some toe-out. Since the front wheels are steered a lot, Ackermann will have a lot of influence on toe.
∑ Sweepers will be taken with the front wheels steered very little, and the rear wheels will track well outside the fronts. The outside front will lead, and the turn center will be in Zone 3. We will need slight toe-in, even to have equal slip angles on the front wheels. To get greater slip angle on the outside front, we will need some added amount of toe-in. Unfortunately, it will be hard to get this using Ackermann, because we have very little handwheel displacement. It will have to come mainly from our static setting.
∑ There will be some intermediate range of turn radii where the handwheel displacement is moderate, but enough so that Ackermann has noticeable influence, and the turn center is in Zone 2 Ė meaning we need a little toe-out for equal front slip angles, or roughly zero toe or a little toe-in if we want more slip angle on the outside front.
How do we get a compromise that at least roughly matches these requirements? By using the combination that is standard practice for road cars: a bit of static toe-in, and some positive
Ackermann. To tailor the same car for autocross or hillclimb use, we might use some static toe-out instead.
But it isnít really crucial to get this perfect. Lots of races have been won with negative Ackermann, and static front toe-out. Negative Ackermann is advantageous when catching oversteer slides. In many front-steer cars, packaging constraints confine us to negative Ackermann, and we have no choice but to make the best of it.
Finally, there is some reason not to want the inside and outside tires to reach peak cornering force exactly together. That gives us the highest peak cornering force, but also may make breakaway a bit more sudden. If the inside and outside tires optimize at slightly different lateral accelerations, that may spread the cornering power curve and make the car more forgiving. This is somewhat analogous to tuning an engineís intake and exhaust systems for somewhat different rpm, to spread the engineís power curve.