You guys are in luck. 1) because Mark Ortiz is back with us. And 2) Because his Chassis newsletter that he just sent out is on Ackermann.
For those of you who don't know, Mark was rammed from behind by a hit-and-run driver while bicycling, and nearly died last November. He truly is one of the good guys.
Here's the News Letter: I'm going to cut/paste, I hope this works.
Kevin
The Mark Ortiz Automotive
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Mark Ortiz Automotive is a chassis consulting service primarily serving oval track and road racers.
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INS AND OUTS OF TOE AND ACKERMANN
Your comments regarding Ackermann, or anti-Ackermann, or perhaps why bother with Ackermann,
would be appreciated. It occurs to me that the more heavily loaded outside front tire must operate at
a larger slip angle than is possible for the opposite unloaded front while cornering. The result is that
the lightly loaded tire is pulled sliding across the pavement not doing much work. I assume it is
reasonable to distinguish between the slip angle of a tire which is being deflected by cornering forces
and one which is sliding. Would it be more efficient to use front roll stiffness to completely unload the
inside front during hard cornering? Does Ackermann aid turn in? Would anti-Ackermann provide an
advantage? Is it possible that toe out in combination with anti-Ackermann might be effective? Is the
turning angle of the front wheels in most hard cornering racing situations so small that Ackermann is
not a factor? What are your thoughts?
For the benefit of newbies, Ackermann effect is a property of steering geometry that causes the front
wheels to toe out as steering angle increases. If the front wheels toe in as steering angle increases,
that is called negative Ackermann or anti-Ackermann. If the toe angle does not vary with steering
input, that is zero Ackermann, or parallel steer.
Ackermann effect must not be considered in isolation. The tires do not know what kind of Ackermann
properties the steering system has. They only know how much they are toed in or out, at a particular
instant. How much the wheels are toed in or out at a particular instant depends on a combination of
Ackermann effect and static toe setting.
Additionally, there can be toe changes due to bump steer or compliance steer. For simplicity, we will
disregard these effects here.
The static setting provides a starting point when the steering is centered, and Ackermann effect adds
toe-out from there, in a fixed relationship to handwheel (steering wheel) angle.
Trouble is, the optimal toe angle in terms of tire performance is not a constant, nor does it have a
fixed relationship to handwheel input. Unless we are prepared to engineer some sort of elaborately
programmed steer-by-wire system that controls the right and left front wheels independently, we
cannot obtain optimal geometry for all possible situations. We are stuck with striking a compromise
for a particular set of conditions.
The nature of that compromise depends in part on how much extra slip angle we wish to give the
outside front wheel. One can reasonably argue that the more heavily loaded wheel reaches peak
cornering force at a greater slip angle than a more lightly loaded one, so the front tires achieve the
greatest peak cornering force when the outside tire has a greater slip angle than the inner one.
The questioner asks whether there is a difference between slip angle of a tire that is sliding and slip
angle of a tire that is not sliding. More precisely, we might talk about a tire that is only partially
sliding, in the rear portion of the contact patch, and one where sliding is occurring in the entire
contact patch. There is no difference in the definition of slip angle; it is simply the angular difference
between the tire's instantaneous direction of travel and its instantaneous direction of aim: the
difference between its bearing and its heading. However, there is a difference in the effect of adding
slip angle in the two cases. If the tire is below the slip angle where its lateral force peaks, adding slip
angle adds lateral force and also adds drag. If the tire is above the slip angle where its lateral force
peaks, adding slip angle does not add cornering force and indeed probably reduces it. However, up to
slip angles associated with total loss of control, drag continues to increase as we add slip angle.
One thing that makes all this a bit complex is that when a tire is near its peak cornering force, lateral
force greatly exceeds drag force, yet moderate slip angle changes have a fairly small effect on lateral
force, but a relatively large effect on drag force. This makes it difficult to evaluate the effects of toe
changes on cornering capability, purely by observing changes in car balance or amount of understeer.
Drag forces at the tires do not turn the car in the sense of accelerating it laterally, or centripetally
(toward the center of the turn), but they do tend to steer the car: accelerate it in yaw, or rotate it.
This means that it is possible to have a case where we are adding moderate amounts of cornering
force at the front by increasing outside tire slip angle, yet the steering trace and the driver feedback
may show increased understeer, and the car may be slower! In such a case, we can, at least in
theory, dial a bit of oversteer in by juggling tire load distribution, and then we may have a slightly
faster car than we started with. The only way to know is to try this rather than immediately backing
up on the toe or Ackermann change.
We have so far been assuming that what we're after is the highest peak cornering force. We get that if
both tires are at their optimum slip angle for lateral force at the same time. However, one could also
make a case for having the tires not peak together, to make breakaway gentler and make the car
more driver-friendly. This is somewhat analogous to the question of whether to tune the engine's
exhaust and induction systems for the same rpm, to get the highest peak power, or tune them for
different speeds, to spread the power band.
Is this complicated enough yet? We're just getting started.
Suppose we have sufficient information to decide what slip angles we want on the two front tires, or
what difference we want in their slip angles. Does that allow us to say what our toe-out or toe-in
should be at a particular instant? Nope. Without knowing the geometry of the car and the turn, and
without knowing what the rear wheels are doing, we can't even get close.
For simplicity, let's suppose we don't want any difference in the inside and outside tires' slip angles.
Let's take a look at what it would take to get that, in various situations.
Some readers will be familiar with the concept of a turn center. This is the point about which the car's
center of mass or c.g. (sometimes approximated as the midpoint of the car's centerline in plan view) is
instantaneously revolving, as it negotiates the turn. If the car has a constant attitude angle – that is, if
it is drifting or sliding a steady amount – all points on the car are moving about the turn center.
At any given instant, the car has an instantaneous direction of travel, which is always a tangent to its
path of motion. If the car is traveling in a curved path, that path has an instantaneous radius r. This
radius is equal to square of the car's instantaneous speed along its instantaneous direction of travel,
divided by its centripetal acceleration: r = v2/a. In a totally unbanked turn, with the tires sliding only a
little, these two quantities are approximately equal to the car's speed as read by a speedometer or
wheel speed sensor, and the car's lateral acceleration as measured by an on-board accelerometer. (If
the turn is banked, or the car is sliding dramatically, these approximations become much poorer.)
If, in plan or top view, we construct at the c.g. a perpendicular to the car's instantaneous direction of
travel, and define a point on that line a distance r from the car's center of mass in the direction of the
turn, that point is the turn center.
In the car's frame of reference, the turn center can be anywhere from the rear axle line to well ahead
of the front axle line. (It could even be behind the rear axle line, if the rear wheels have a negative slip
angle. This could occur when negotiating a banked turn at low speed. Normally we can ignore this
case when studying behavior at racing speeds.)
The simplest situation is a small-radius turn, taken so gently that tire slip angles are negligible. In this
case, the turn center is on, or very nearly on, the rear axle line in plan view. For zero slip angle at both
front wheels, the front wheel axes should ideally meet in plan view at the turn center. That implies
that the front wheels will have substantial toe-out.
The larger the turn radius, and the larger the rear wheel slip angle, the further forward the turn center
moves. At some point, the turn center will lie on the front axle line in plan view. In this situation, we
have equal slip angles on the two front wheels when the toe-out is zero.
In high-speed, large-radius turns, the turn center is usually ahead of the front axle line. We now have a
condition where the front wheels need to be toed in for the slip angles to be equal.
As a broad generalization, the front wheels are steered more in small-radius turns than in large ones,
and the turn center is further rearward in the car's frame of reference. This argues for having at least
some Ackermann in the geometry, but it is harder to come up with a general rule to calculate exactly
how much.
In high-speed turns, steering inputs are generally very small, and consequently Ackermann effect has
far less influence than static toe setting. Ackermann has greatest influence in autocross and hillclimb
cars.
Does Ackermann aid turn-in? Basically, yes, and so does static toe-out, at least up to a point. Really
excessive toe-out, whether from static setting or Ackermann, will hurt front grip and the effect will
reverse, but within a sane range, turn-in will at least feel quicker with some toe-out. This is partly due
to the early yaw moment from inside front tire drag as the handwheel is first turned.
Does using anti-Ackermann along with static toe-out make sense? It's certainly done successfully, on
winged single-seaters on high-speed ovals. Logically, however, it makes more sense to use static
toe-in with positive Ackermann. This is conventional practice in passenger cars.
What about completely unloading the inside front wheel using front roll resistance? Well, it does make
Ackermann academic, at least during the time that the wheel actually is airborne, and it eliminates
any concerns about tire drag due to the front tires fighting each other. Unfortunately, in many cases
using that much front roll resistance will create excessive understeer.