principles of ackerman erik zapletal

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Written by Erik Zapletal, (C) March 2001.This article may be freely disseminated, as long as it is not

altered from its present form. If sections are "cut and paste'd" then the original authorship should be

acknowledged.

~~~~oOo~~~~

TO TOE-IN, OR TO TOE-OUT?

======================THAT IS THE QUESTION!===================

INTRODUCTION -The defining characteristic of a "wheel" is that it is a structure which will roll freely in a

direction perpendicular to its "axle", but it will resist movements in the direction of its axle.

It follows that if we want a racecar to go fast in a straight line, then all four wheels should point in the

same direction -straight-ahead. But if we want the racecar to go fast around corners -that is, we want

the car to accelerate sideways by using the wheels' resistance to axial motions –then in just which

directions should the wheels point?

Specifically, during cornering, should the front-wheels toe-in, remain parallel, or toe-out? How will these

toe changes effect the dynamics of the car? And when we have decided which way the wheels shouldpoint, how should we design the steering geometry so that it actually points the wheels in the right

directions? This article attempts to answer these questions.

It should be noted that this article considers mainly the steering of the front-wheels of a rear-wheel-

drive car being driven on a sealed road. However, many of the principles also apply to rear-steer, or

front-wheel-drive, or dirt, clay or ice road surfaces.

The analysis is based on a simplified two-dimensional plan view of the car. No suspension geometry is

considered. "Zero-point", or "centrepoint", steering is used -that is, a vertical steering axis passing

through the centre of the tyre print (giving zero offset, trail, castor-angle and kingpin-angle). Camber-

angle is also zero. It is only changes to the "steer-angles" that are being considered. The analysis usesspecific dimensions -the car has a wheelbase of 2.6m, and a front track of 1.6m.Different dimensions will

yield different results. The method used for this analysis is outlined in the section "The $2 Super

Computer".

More ackerman and toe

TYRES -Since a car's performance is so dependent on the interaction between the tyres and the road, we

should briefly consider this area.

Figure 1 shows two sets of curves. One set is for a wide, low profile, radial-ply tyre. The other set is for a

narrow, tall, cross-ply tyre. Each set indicates performance at two different vertical loads on the tyre.

The horizontal axis indicates the so called "slip-angle" that exists between the horizontal heading of the

wheel-hub, and the actual horizontal direction in which the wheel is travelling. The vertical axis indicates

the force Fy that acts between the tyre and the road. This force acts at ground level, and is, by

definition, horizontal and parallel to the wheel's axle in plan view. This force is often referred to as the

"tyre lateral force",but to avoid confusion with the lateral forces that act on the chassis, we shall refer to

it as the "axial-force".



The graphic to the side of the curves depicts (with exaggeration) the situation at the lower part of the

curves. Here the tyre print isn't actually slipping on the road. Rather, the cornering force causes the

flexible sidewalls to distort elastically, with the greatest distortion being towards the rear of the tyre

print. This distortion allows the circular "hoop" of the tread to adopt a different camber-angle and steer-

angle to that of the wheel-hub. The change in steer-angle of the tyre tread allows the wheel-hub to

"crab" sideways even though there is minimal slippage between the tyre print and the road.

As the axial-force increases, the rearmost parts of the tyre print start to slip. This increases the angle

between the wheel-hub heading and its actual direction of travel, and the curves bend to the right. As

more of the rear section of the tyre print slides, the "vector" of the distributed tyre print forces moves

forward, causing the self-aligning torque of the tyre to diminish and the steering to feel lighter.

The curves reach their peak-axial-force when most of the rear section of the tyre print is sliding. Beyond

this point the axial-force drops off slightly, and then levels out as all of the tyre print slides. There is now

almost no directional control from the steering -small changes to the steer-angle of the wheels will not

significantly change the axial-force.

As the vertical load, Fz, on the tyre increases, the peak-axial-force increases by a lesser ratio. Forexample, if the vertical load is doubled, then the peak-axial-force is less than doubled. There is a

reduction in the apparent "Coefficient of Friction" (Cf = PeakFy/Fz) of the tyre as the vertical load

increases. Another effect, which we will come back to later, is that as the vertical load increases, the

peak-axial-force is developed at a greater slip-angle.

During accelerating or braking there is a "longitudinal" force (horizontal and perpendicular to the wheel

axle) developed between the tyre and the road. The curves for these longitudinal forces are similar to

Figure 1,with the exception that "slip-angle" is replaced by "slip-ratio" (which can be defined in several

different ways).The creation of the longitudinal forces involves an expenditure of energy.

The creation of the longitudinal forces involves an expenditure of energy. During acceleration the forcesare creating kinetic energy, and thus require fuel to be burnt. During braking, the previously created

kinetic energy, and its fuel cost, are dumped as heat. On the other hand, the tyre axial-forces are almost

free. There is a small slip-angle-drag cost, proportional to slip-angle size, which we will come back to

later. But because the axial-forces are almost orthogonal to the direction of motion, they can accelerate

the car towards the centre of the corner at almost no energy cost. Cornering power not only wins races,

but, thermodynamically speaking, it is almost free.

The abovementioned reduction of Cf of a tyre with load, often called the "tyre-load-sensitivity", can be

used to adjust the understeer/oversteer handling balance of a car. For example, if two of the radial-ply

tyres in Figure 1 are fitted to one axle of a car, and each carries a vertical load of 4kN, then they can

together develop a total axial-force of about 10kN.If during rapid cornering only the outer-wheel is

carrying the combined vertical load of 8kN, then it can only develop an axial-force of about8.8kN -a

difference of 1.2kN. These "lateral-load-transfer" effects can be achieved by changes to the roll-centre

height and spring-rates of the axle, compared with the roll-centre height and spring-rates of the other

axle.

Often the roll-centre and spring-rate changes are considered to be the most important influences on

handling balance. However, it should be noted that a change in steer-angle of only 1 degree, on just one

of the above wheels carrying a vertical load of 4kN, can produce a change in axial-force of2kN. A 1



degree change in steer-angle of the same wheel carrying a vertical load of 8kN, can produce a change in

axial-force of 3kN. Before making adjustments to roll-centre heights or spring-rates, the racecar

engineer should make sure that the wheels are pointing in the right direction.

STATIC-TOE -"Static-toe" refers to the steer-angles of the front wheels, relative to the car's centreline,

when the steering-wheel is in the straight-ahead position. Static-toe angles are relatively small, typically

less than 1 degree.

Figure 2a depicts a car with static-toe-in, and Figure 2b depicts a car with static-toe-out. The upper parts

of Figure 2 depict the cars when travelling along a straight road. The slip-angles (equal to the static-toe-

angles) and the forces acting on the wheels are shown. It can be seen that while everything is

symmetric, the forces will be balanced.

The lower parts of Figure 2 show what happens when a small steering movement is made to the left,

while the car is still travelling straight-ahead. Alternately, the whole car can be considered to have been

yawed slightly to the left. Now one wheel has zero steer-angle, and zero slip-angle, and thus only a

rearwards rolling-drag force (or some rearwards braking-force). The other wheel has an increased steer-

angle and slip-angle, hence it has an increased axial-force, plus the same rolling-drag or braking-force asthe first wheel. In each case the "wheel-coordinate" forces are shown as hollow arrows, while the same

forces in lateral and longitudinal "car-coordinates" are shown as solid arrows.

In both of the lower parts of Figure 2 there is the same total of lateral force and longitudinal force acting

on the front of the car. However, in Figure 2a -Toe-In -there is a greater longitudinal force at the right-

wheel, then at the left-wheel. The difference in these two longitudinal forces acts to yaw the car to the

right. Conversely, in Figure2b -Toe-Out -the greater longitudinal force is at the left-wheel, and the

"differential-longitudinal-force" –Delta Fx -acts to yaw the car to the left.

If the steering is turned further to the left, while the car is still travelling straight-ahead, then both

wheels will have a leftwards steer-angle, and an associated leftwards axial-force. For equal left and rightwheel loads, the wheel with the greater steer-angle (which is its effective slip-angle) will generate the

greater axial-force. But even if we assume that the left and right axial-forces are equal, then the Toe-In

car will still have the greater longitudinal force at its right side, and the Toe-Out car will have the greater

longitudinal force at its left side.

This difference in the left and right longitudinal forces arises from the difference of the steer-angles of

the two wheels. For a small angle A (less than about 10 degrees), sinA ~ A, and cosA ~ 1 (A measured in

radians). For small steer-angles the size of the longitudinal component of the axial-force, fy, will be

approximately equal to the steer-angle x fy. That is, the longitudinal component of force is directly

proportional to the steer-angle. The longitudinal component of the drag or braking-force, fx, will be

approximately equal to fx -that is, it remains unchanged. It follows that the wheel with the larger steer-

angle will have the larger total longitudinal force.

Variations in wheel loading will change the size of the axial-force for a given slip-angle, which in turn will

change the sizes of the lateral and longitudinal components of the forces. But in general, putting more

toe-in on the car will generate differential-longitudinal-forces that act to yaw the car away from the

turn, while putting more toe-out on the car will generate differential-longitudinal-forces that act to yaw

the car into the turn.



Another way to look at this, is that with toe-in the wheel with the greater steer-angle has a short

moment-arm for its axial-force about the car's centre-of-mass. With toe-out the wheel with the greater

steer-angle has a longer moment-arm for its axial-force.

As a generalization, the differential-longitudinal-forces are stabilizing with toe-in (giving stable high

speed cruising, and "sluggish" turn-in),and destabilizing with toe-out (giving "nervous" straight line

driving, and "sharp" turn-in).

DYNAMIC-TOE -"Dynamic-toe" refers to the change in steer-angle of one front-wheel, relative to the

other front-wheel, as the steering is turned away from straight-ahead. Dynamic-toe is a function of the

steering geometry. At full-lock, dynamic-toe can result in a difference of the two front-wheel steer-

angles of 10 degrees or more.

If the steer-angles of the front-wheels remain equal to each other as they move from straight-ahead to

full-lock, then the steering geometry is said to have "parallel-steer". If the front-wheels toe-in relative to

each other as they move towards full-lock, then the steering is said to have "dynamic-toe-in". If the

front-wheels toe-out relative to each other as they move towards full-lock, then the steering is said to

have "dynamic-toe-out".

Dynamic-toe-in is often referred to as "negative-, or anti-Ackermann", while dynamic-toe-out is

sometimes called "positive-, or pro-Ackermann". The term "Ackermann" doesn't seem to have a

universally accepted definition.

Figure 3 indicates the "Kinematic Steer-Angles" (KSA) of the front-wheels of a car, as the car rotates

around various "Instant Centres". Note that these angles don't refer to the actual steer-angles of the

front-wheels. Rather, they indicate the direction that the centres of the wheelprints are travelling,

relative to the centreline of the car, for any specific motion of the car. They can be interpreted as the

steer-angles that are required of the front-wheels, so that the axles of the wheels will be pointing

directly at the instantaneous centre of the car's motion.

The horizontal axis indicates Alpha(Outer), which is the angle between the centreline of the car, and the

direction of travel of the outer-wheelprint. The vertical axis indicates Alpha(Inner)-Alpha(Outer), which

is the dynamic-toe-out of the inner-wheelprint's direction of travel, relative to the outer-wheelprint's

direction of travel.

Three curves are shown. The rightmost curve indicates the KSA of the front-wheels when the centre of

the rear-axle has zero slip-angle –that is, when the Instant Centre lies on an extension of the rear-axle-

line. This is typical of low speed travel when the horizontal forces on the rear tyres are low, and thus

their slip-angles are minimal.

The other two curves indicate the KSA of the front-wheels when the centre of the rear-axle has a slip-

angle of 10 degrees and 30 degrees. This rear-slip-angle is depicted in the graphic. It refers to the

direction of travel of the centre of the rear-axle, relative to the centreline of the car. If both rear-wheels

are aligned with the centreline of the car, and if there is a significant amount of rear-slip, then the outer-

rear-wheel will have a slightly smaller slip-angle, and the inner-rear-wheel will have a slightly larger slip-

angle, than that of the centre of the rear-axle. This would suggest that if the rear-wheels are to run at

equal slip-angles, then they should be set-up with some static-toe-in. On the other hand, if the car has

some form of rear-steer, such as that available on some production cars, then it is possible for the car to



run with large, but still equal, rear-wheel-slip-angles, while the rear-slip is kept to a small value (say

0degrees).

Also indicated on each curve are the radii of cornering (prefixed with "R" and taken to the centre of the

car) for different KSAs. The leftmost point of each curve indicates when the car is travelling in a straight

line - of each curve indicates when the car is travelling in a straight line -infinite radius. The curves end at

a rightmost point which can be considered to be "full-lock".

(Steer-angles beyond this point are only applicable to high-maneuverability vehicles. They are partially

presented here to show that the curves will eventually return to zero dynamic-toe-out, when the outer-

wheel has rotated a full 180 degrees. The curves are asymmetric in this format because the outer-wheel

becomes the inner-wheel part-way through its travel.)

If a car has a small rear-slip-angle -due to either slow speeds, stiff rear tyres, or rear-wheel-steer -and

the car is turning a reasonably tight corner, then quite large values of dynamic-toe-out are required if

both front-wheels are to operate at similar slip-angles. For a 5m radius corner, about 8 degrees of

dynamic-toe-out is required.

For a full-lock turn more than 16 degrees of dynamic-toe-out may be required. As the rear-slip-angle

increases, then the car needs progressively less front-wheel steer-angle to negotiate corners. If the rear-

slip-angle is large enough, and if the corner radius is also large enough, then the front-wheel Kinematic

Steer-Angles are negative -that is, opposite-lock. The dynamic-toe-out required in this situation is also

negative -that is, dynamic-toe-in. This condition of opposite-lock and dynamic-toe-in occurs when the

Instant Centre of the car's motion is in front of the front-axle-line.

THE "ANTI-ACKERMANN" ARGUMENT –There is a school of thought that argues that while production

cars can benefit from dynamic-toe-out during cornering (positive-Ackermann), racecars will generally

corner faster with some dynamic-toe-in (negative-, or anti-Ackermann). This argument is based on the

observation of Figure 1, that the greater the load on a tyre, then the greater the slip-angle that it mustrun at, in order to develop its maximum cornering force.

Since the outer-wheel of a cornering car will carry a greater load than the inner-wheel, it follows that

the outer-wheel must run at a greater slip-angle than the inner-wheel, if both wheels areto develop

their maximum cornering forces. For example, consider Figure 4, together with the cross-ply tyre curves

at the right of Figure 1, and the "Rear-Slip = 10 deg." curve of Figure 3.With 10 degrees of rear-slip, and

a corner radius of about 30m;

KSA of Outer-Wheel = -5 degrees,

Toe-Out of Inner-Wheel = -0.3 degrees, so

KSA of Inner-Wheel = -5 + (-0.3) = -5.3 degrees

If the heavily loaded outer-wheel develops its peak-axial-force at a slip-angle of 10 degrees, and the

more lightly loaded inner-wheel develop sits peak-axial-force at 9 degrees, then for maximum cornering

force; Outer-Wheel Steer-Angle = -5(KSA) + 10(slip-angle) = +5 degrees, and Inner-Wheel Steer-Angle = -

5.3(KSA) + 9(slip-angle) = +3.7 degrees as indicated at the left of Figure 4.

That is, there should be about 1.3 degrees of actual dynamic-toe-in, or anti-Ackermann, when the front-

wheels are steered about 5 degrees away from straight-ahead.



The upper-right section of Figure 4 gives a more detailed view of the situation at the inner-front-wheel.

The axial-force Fy doesn't point directly at the Instant Centre, but, by the definition of its 9 degree slip-

angle, it points 9 degrees behind this radial line. The slip-angle thus generates a "drag" component of

the axial-force that is rearwards to the direction of wheel print travel. This "slip-angle-drag" is an

undesirable force as it requires forward thrust, and thus engine power and fuel, to overcome it. One of

the main benefits of radial-ply tyres is that they corner at lower slip-angles, and thus with less drag, than

do cross-ply tyres.

If the car in Figure 4 had a steering linkage that generated dynamic-toe-out, then the inner-wheel steer-

angle would be greater than 3.7degrees, and the inner-wheel slip-angle would be greater than 9

degrees. The tyre would thus be running at a point past its peak-axial-force. Its "centripetal" component

of force would be reduced, and its drag component of force would be increased. The car's total

cornering force would thus be reduced, and it would require more engine power for the car to maintain

the same speed. This is the essence of the anti-Ackermann argument.

There are several details that should be kept in mind when considering this argument. They are:

1.

Large slip-angles.

The above philosophy was developed quite a few years ago, when racecars had the high slip-

angle tyres used in the above example. The race-tracks of that period also had fewer tight-radius

corners such as "chicanes". Much of the racing involved long straights followed by high-speed

large-radius "power-limited" bends. In this situation, and for dirt-track cars with very high rear-

slip-angles, some anti-Ackermann, or dynamic-toe-in from the steering may be helpful. The

maximum amount of dynamic-toe-in needed is about 3 degrees for dirt racers, and less than 2

degrees for road racers.

As explained previously, large slip-angles are not desirable because they generate drag. While

sideways might look fast, it is not necessarily the quickest way through a corner. Likewise, it isnot desirable to have widely different loads on the wheels, as this reduces the maximum

available horizontal wheel forces via the tyre-load-sensitivity effect.

It follows that it is desirable to fit a car with tyres that develop their peak-force at small slip-

angles -for example, low-profile radial-ply tyres-and to minimise the load variations of the

wheels. With low slip-angle, evenly loaded wheels, the need for dynamic-toe-in is considerably

reduced.

2. Which tyre should peak first? A racecar might not have the ideal steering geometry that allows

its front-wheels to reach peak-axial-force simultaneously. If the outer-wheel reaches peak-axial-

force before the inner-wheel, due to excessive toe-in, then any increase in slip-angles will

decrease the outer-wheel's axial-force, and increase the inner-wheel's axial-force. A

consideration of the car-longitudinal components of these forces would suggest that such a

change will exert an oversteering yaw moment on the car, which is destabilizing. Conversely, if

the inner-wheel reaches peak-force first, due to insufficient toe-in, then any increase in slip-

angles will result in a stabilizing understeer yaw moment acting on the car.



These effects may be small, but they would suggest that a racecar with some "peak-force-toe-

out" (in the sense that the inner-wheel reaches peak-force first) would be more stable "at the

limit", than a car with"peak-force-toe-in".

3. Transient manoeuvres. During a fast lane-change, as may be necessary during overtaking, or for

accident avoidance, a driver will typically turn the steering wheel through a large angle -perhaps

half-lock or more. The purpose of the large steering movement is to make the front-wheels yaw

the front of the car to one side as quickly as possible. It is only when the car has a yaw angle to

its direction of travel that its rear-wheels can develop a lateral force, and thus push the rear of

the car sideways.

A steering geometry that generates large dynamic-toe-out angles will cause the inner-wheel of

the car to develop a larger steer-angle, and slip-angle, than the outer-wheel, whenever a large

steering movement is made, and the car is still travelling straight-ahead (see the "Static-Toe"

section, and Figure 2). This large rearwards movement of the inner-wheel's axial-force vector

will effectively drag that side of the car backwards, exerting a large yaw moment on the car,

thus improving the car's transient responsetimes.

4. Sharp corners. Many racing textbooks advocate steering geometries that are anti-Ackermann,

or, for the fence-sitters, parallel-steer. They do so largely because these geometries can be

useful on high-speed large-radius (small steer-angle)corners -conditions that apply to many of

the top categories of motorsports.

There are many categories of racecar that are required to turn sharp corners -trials, autocross,

rallycross, hillclimb, Formula SAE/Student, and so on. If such a racecar has a steering geometry

that gives parallel-steer, then it will have a dynamic-toe curve that is a horizontal line in Figure 3

(zero toe-change). An anti-Ackermann steering geometry will produce a dynamic-toe curve that

is initially horizontal, but then drops below the horizontal axis of Figure 3.

A racecar that must turn sharp corners may need well over 10 degrees of dynamic-toe-out to enable its

front-wheels to operate at similar slip-angles. If this racecar has parallel-steer then its wheels will be

"effectively" toed-in at over 5 degrees per wheel whenever it is turning a sharp corner. If this racecar

has anti-Ackermann steering, then its wheels will be "effectively" toed-in even more.

We will use the term "effective-toe" to refer to the discrepancy between the KSAs and the actual steer-

angles. Effective-toe-angles, if measured at each wheel, are equal to the tyre's slip-angles. Since we

often don't know the individual front-wheel slip-angles, we can refer to the "effective-total-toe-in"

between the two front-wheels as being equal to the KSA Toe-Out (for the car's specific instantaneous

motion), minus the actual total-toe-out of the front-wheels (that is, the included angle between the two

front-wheels).

It might be useful for the reader to conduct a small experiment to clarify the above situation. Adjust the

static-toe of your car to its maximum safe extent (leave enough thread in the adjustments to hold the

track-rods together). Regardless of whether it is toe-in or toe-out, try to get at least 5 degrees per wheel

-that is, at least 10 degrees of toe difference between the two wheels. Now drive the car slowly along a

quiet road.



Several things should become apparent. Firstly, the tyres won't like it, and they will protest loudly. Both

tyres will initially be operating close to their slip-angle-peaks where there is a lot of real "slip". Secondly,

the steering will be "light". Most of each tyre print will be sliding so there will be little self-aligning

torque or steering feel. Thirdly, despite the driver's steering efforts, the nose of the car will behave like

an overeager puppy trying to sniff every tree either side of the road. One tyre will occasionally get a

better grip (due to a change in road surface or wheel loading) and it will push the other tyre over its slip-

angle peak, thus changing the direction of the car.

This experiment is an inexpensive version of the tyre-testing machines that produce the curves of Figure

1. It gives a slow-motion view of the tyres as they are operating close to their slip-angle peaks. The

experiment also shows what parallel-steer, or anti-Ackermann, does to the front-wheels of a car

whenever it turns a sharp corner. The wheels are forced to run at a large effective-total-toe-in. At least

one wheel(typically the outer-wheel) will be close to, or beyond its slip-angle peak, and it will be

reluctant to respond to any steering inputs from the driver. The inner-wheel may, in fact, be trying to

push the nose of the car out of the corner.

If the car has stiff front springs, and if it can corner fast enough, then it may be able to lift the inner-

front-wheel off the ground, and stop it from fighting the outer-wheel. Speed and noise will make thisfight less obvious, but while both front-wheels are on the ground the anti-Ackermann car won't like

sharp corners.

STEERING GEOMETRY -The previous sections discussed the vehicle dynamic responses that we might

expect from different dynamic-toe behaviours. This section shows how the different dynamic-toe

behaviours are generated by different steering geometries.

Figure 5 shows the KSA curves of Figure 3 for rear-slip-angles of RS = 0degrees, and RS = 10 degrees.

These curves give the "ideal-steer-angles" for the front-wheels, if they were to run at a "front-slip-angle"

of FS = 0degrees.

To generate a cornering force, the front-wheels must run at a non-zero slip-angle. Two additional curves

are shown that indicate the ideal-steer-angles (ISAs) that are required if both front-wheels are to have a

front-slip-angle of FS=5 degrees, and there are rear-slip-angles of RS = 0 degrees, and RS = 10 degrees. It

can be seen that these two ISA curves are simply the previous two KSA curves translated sideways by

the additional front-wheel slip-angle of 5 degrees.

The four curves give an indication of the dynamic-toe angles that are required of a steering geometry, so

that all four wheels can run at an appropriate slip-angle. The KSA curve of FS = RS = 0 degrees is

repeated in Figures 6 to 9 to aid in the comparison of the various steering geometries. This curve is used

because it would be the ideal dynamic-toe curve if the car had "ideal" wheels that cornered with zero

slip-angle. This curve is also an approximate average of the curves that have practical front and rear-slip-

angles, on sealed road surfaces.

Figure 5 also shows two steering systems typical of beam-axle suspensions. In each case the wheel-hub

has a rearward mounted "steer-arm", and the ends of the two steer-arms are connected by a track-rod.

The traditional "Ackermann geometry" is shown as "A", where the centrelines of the two steer-arms

intersect at the centre of the rear-axle. The second steering geometry, "B", has the steer-arm

centrelines intersecting at the wheelbase mid-point. The accompanying curves show the relationship



between the outer-wheel steer-angles, and the dynamic-toe-out of the inner-wheel, for these two

geometries.

The curves "A" and "B" end (at their top-right) when the track-rod becomes "straight" with the inner-

wheel steer-arm -that is, the track-rod cannot rotate the inner-wheel-hub any further. A practical full-

lock limit would stop the inner-wheel at least 10 degrees before the track-rod goes straight, so that the

steering can't jam at full-lock.

The curves show that Ackermann geometry, "A", is only an approximation to the KSA curve, and in fact it

only gives Kinematic Steer-Angles at zero steer-angle, and at about full-lock (for the given dimensions).

For most of the steering range the wheels will be running with an effective-total-toe-in of about 3 to 5

degrees. The second geometry, "B", has the wheels running with an effective-total-toe-out throughout

the steering range.

A disadvantage of geometry "B" is that it has a lower maximum outer-wheel steer-angle, and thus a

larger minimum turning circle than geometry "A". At full-lock the wheels have a lot of toe-out, implying

a lot of wheel-scrub during low-speed parking manoeuvres. A compromise geometry somewhere

between the two shown, plus some static-toe-in, would give stable toe-in for high-speed large-radiuscorners, and some effective-toe-out at full-lock, to help the car around hairpins.

Figure 6 shows a rack-and-pinion (R&P) steering geometry as typically used on racecars. The R&P and

the steer-arms are mounted in front of the front-axle line. Four curves are shown for different angles of

the centreline of the steer-arm. The curves end when the R&P runs out of travel, which in this case is +/-

100 mm from centre. A shorter steer-arm would require less R&P travel.

While the steer-arm angle is less than 20 degrees from straight-ahead, the wheels maintain effective-

toe-in throughout the range of steering travel. Only when the steer-arm-to-track-rod joint is moved past

20 degrees (that is, more than 52mm outboard of the king-pin, for the 150mm long steer-arm)do the

wheels start to develop effective-toe-out. Even with the "30 degree Steer-Arm" curve (steer-arm-end- joint about 75mm outboard of the king-pin)this toe-out only becomes significant close to full-lock.

Figure 7 shows a similar R&P system to Figure 6, but this time the steer-arm angle is kept at straight-

ahead, and the longitudinal location of the R&P, as it is mounted in the chassis, is varied.

The curves of Figure 7 are similar to those of Figure 6. A R&P mounted100mm in front of the axle with a

steer-arm angle of about 25 degrees, has a similar dynamic-toe curve to a R&P mounted 50mm behind

the axle with a steer-arm angle of 0 degrees. The choice of layout depends on packaging issues such as

R&P placement in the chassis, versus steer-arm placement within the wheel assembly.

An important observation to be made of Figures 5, 6, and 7 is the size of the angle Alpha(T) between the

steer-arm and the track-rod, and its effect on the dynamic-toe curve. When Alpha(T) is around 90

degrees there is minimal dynamic-toe change, and the steering geometry approximates parallel-steer.

As Alpha(T) becomes more acute, the inner-wheel turns more, and the outer-wheel turns less, for a

given linear movement of the track-rod, and the dynamic-toe-out increases.

If the steering geometry has rearward facing steer-arms (as in Figure 5),and a rear mounted R&P, the

dynamic-toe curves will still behave in a similar manner to that described above. That is, changes to



steer-arm angle and R&P placement that make Alpha(T) more acute (as it is shown in Figure 5) will tend

to give more dynamic-toe-out.

Often it is not possible to use large steer-arm angles, as in Figure 6,because the brake-disc is in the way.

Likewise, it may not be possible to move the R&P behind the front-axle line, as in Figure 7, because of

intrusion into the footwell or engine space.

Figure 8 shows a steering geometry that may appear more complicated than the previous systems, but it

has several advantages. With this layout two "idlers" are mounted either side of the chassis. They are

connected via their front-arms to a central-track-rod, and via their rear-arms to outer-track-rods which

then connect to the steer-arms. Either of the idlers could be driven by a steering-box, or the central-

track-rod could be replaced by a R&P and two short track-rods. If the suspension wishbones are long

enough, then the two idlers can be merged into a single central idler, suitably driven.

One advantage of this layout is that it can improve packaging convenience. The steer-arms can be

straight-ahead, and the central-track-rod, or R&P, can be mounted well forward of the front axle line for

increased footwell space. Another advantage is that the increased number of angles between the space.

Another advantage is that the increased number of angles between the track-rods and the rotatingidlers and steer-arms, makes it easier to tailor the shape of the dynamic-toe curve. For example, it takes

only a little experimentation to get the dynamic-toe curve to track the KSA curve to within half a degree.

The specific layout shown in Figure 8 has the idlers and central-track-rod acting as a parallelogram -that

is, both idler-front-arms are always at the same angle. The two acute angles in the "Z-Bar" linkage (idler-

rear-arm to outer-track-rod, and outer-track-rod to steer-arm) cause the dynamic-toe curve to rise

rapidly during initial steering movement -more rapidly than in any of the previous R&P curves. However,

as the idler-rear-arm-to-track-rod angle straightens (for the inner-wheel) the dynamic-toe curve is pulled

back down. With some initial static-toe-in, this layout would give stable high-speed cruising and

cornering, with responsive turn-in to tighter corners, and negligible wheel scrub during low-speed full-

lock manoeuvres.

Figure 9 shows two steering linkages that generate asymmetric dynamic-toe curves. These asymmetric

curves are suitable for asymmetric racecars, such as those that race on ovals and spend most of the time

either going straight, or turning left. The dynamic-toe curves are shown for both Left Steer -that is, the

steering-wheel is turned to the left -and Right Steer.

Geometry "A" is similar to the example in Figures 6 and 7, that has the R&P100mm in front of the front-

axle line, and has zero steer-arm angles. It differs in that its left-steer-arm is slightly shorter than the

right-steer-arm. This change has the effect of rotating the previously symmetric curve clockwise. It also

pulls the rightmost end of the curve to the right (more left-steer-arm-angle), and down (less toe-out, or

more toe-in).

Geometry "B" is similar to the example in Figure 6 with a 10 degree steer-arm-angle (10 degrees on both

sides, hence an included angle of 20degrees between the two steer-arms). It differs in that, with the

steering straight-ahead, the 20 degree angle is at the left-steer-arm, while the right-steer-arm points

straight-ahead. Again, the asymmetry has the effect of rotating the previously symmetric curve

clockwise.



In both of the above examples, and especially in example "B", the steering has considerable dynamic-

toe-out when turning left. This would imply sharp turn-in when turning into the left-hand corners. When

turning right, as may happen if the driver has to counter-steer through the left turns due to a large rear-

slip-angle, then the steering has some dynamic-toe-in –as required by the anti-Ackermann argument.

With larger rightwards steer-angles, example "B" regains some dynamic-toe-out.

Many more variations of steering linkages, and their resulting dynamic-toe curves, are possible. But one

thing that we can't expect of these mechanical steering linkages, is that their dynamic-toe curves will

change, or "adapt", according to the different conditions during a race. That might require some kind of

an "active-steering-linkage", and is beyond the scope of this article...

FURTHER CONSIDERATIONS -About 50 years ago, the theoretical "Bicycle-Model" was developed to aid

in the understanding of vehicle dynamics. The simplest version of this model is a 2-D plan-view of a

vehicle, with one front-wheel, one rear-wheel, and a centre-of-mass located somewhere between. Each

wheel is assigned a representative lateral stiffness -the initial slope of the curves in Figure 1. When the

dynamic effects of driving this model around a particular corner at a particular speed are calculated,

then the model gives reasonably accurate predictions of the "differential-lateral" forces -the difference

between the front-wheel and the rear-wheel lateral-forces -that are responsible for understeer oroversteer behaviour.

Extensions of this model include the reduction in cornering power that is due to lateral-load-transfer,

camber change, and so on. Quite complicated versions of the model have been developed. Over the

years there have been hundreds, if not thousands of papers based on this model, presented to various

learned societies around the world. For many Vehicle Dynamicists, the bicycle-model is their bedrock.

However, the bicycle-model takes little account of the vehicle's width. The vehicle is symmetric about its

centreline. There can be no differential-longitudinal-forces. If the reader has driven a tank, bulldozer, or

other skid-steer vehicle, then they will appreciate how powerful the differential-longitudinal-forces can

be when it comes to turning a vehicle. Likewise, anyone who has used fiddle-brakes will respect theyawing power that can be generated by a small rearwards force on one side of the car.

Many production cars are being fitted with Active-Stability-Control systems(or some such acronym) that

use the Anti-lock Brake System (ABS) hardware to brake one side of the car to control yaw motion, and

thus to control understeer/oversteer. It would be entirely feasible to build a high-speed large-radius

Indy-style racecar that has all four wheels fixed rigidly straight-ahead, and uses only a joy-stick

controlled ABS system for steering. This car could even negotiate Formula 1 style racetracks, although its

tyre wear would be increased through the tighter corners.

Steering a car by using the brakes, although feasible, isn't particularly efficient. Overall the car would be

slower. Besides which, ABS and similar systems are banned from most forms of racing. But the yawing

power that is available from such differential-longitudinal-forces isn't specifically banned.

One of the easiest ways to take advantage of this yawing power is to use dynamic-toe changes.

Dynamic-toe-out of the front-wheels generates just the right sort of differential-longitudinal-forces that

help yaw the car into a corner. Parallel-steer, and anti-Ackermann steering, generate differential-

longitudinal-forces that act to yaw the car away from the corner.



No new hardware has to be developed to exploit these yawing forces. All cars have the basic hardware -

it is just a matter of adjusting the cars have the basic hardware -it is just a matter of adjusting the

geometry.

~~~oOo~~~

THE $2 SUPER COMPUTER=====================

There is a children's song "The Super Computer", by Don Spencer, that has the lyrics:

"There's a super computer that can do anything.

It can count, it can smile, it can talk, it can sing.

It can run, it can jump, it can laugh, it can cry,

With its own compact and portable power supply.

...

It doesn't use buttons, has nerves instead.

There's a super computer inside your head. "

Since we all come thus equipped, the only additional "coprocessors" needed to produce the curvesshown in this article are A4 paper (5mm graph, if available), fine-point biro, straightedge (ruler),

compass, and a protractor. If the reader doesn't have any of these, then they should be available at the

local newsagent for less than $2 total (the author bought a nice protractor, and an adequate compass,

for 50 cents each).

If the reader prefers to use one of the more expensive, although undoubtedly more fashionable boxes of

electronic mischief as their coprocessor, then the method of solution would be much the same as

outlined here. For example, a computer aided drafting program can be used in a similar manner to that

described below.

(If many different steering geometries are to be investigated, then a dedicated computer programexecuting this same method, might be appropriate. The maths are straightforward. For the generalised

3-D case the program must find the correct intersection point of the circular arctraced out by the steer-

arm-end-joint, and the surface of the sphere traced out by the track-rod-end-joint -there can be 0, 1 or 2

intersection points.)

The first curves to be produced are the Kinematic Steer-Angle curves, as shown in Figure 3, for the

specific wheelbase and track dimensions, and various rear-slip-angles. The method is as follows:

1. Draw a rectangle proportional to the car's wheelbase and track in the lower left corner of the page.

The example in Figure 3 was drawn at a scale of 1:50, that is 52mm x 32mm. The top-left-corner of this

rectangle corresponds to the front-left-wheel.

2. Use the protractor to mark out a line from the centre of the rear-axle, towards the right and up, at the

required rear-slip-angle -a horizontal towards the right and up, at the required rear-slip-angle -a

horizontal line for 0 degrees rear-slip.

3. Similarly, mark out lines from the front-left-wheel at angles of 10, 20,30, and 40 degrees, towards the

right and down, until these lines intersect the rear-slip line. These intersections are the Instant Centres

(ICs) of the turn. These lines correspond to changes of angle of the front-left-wheel's axle, and also to



this wheel's steer-angle from straight-ahead. If there is a lot of rear-slip, then lines at -10 degrees, and

so on, will be required. If a bigger sheet of paper is available, then some lines at +/-5 degrees can be

added, although at these angles the wheels are almost parallel.

4. Use the straightedge to draw lines from the front-right-wheel to the ICs. Use the protractor to

measure the angles of these lines.

5.Draw up a table of corresponding front-left and front-right-wheel angles, and also the difference of

the angles -that is, right(inner-wheel) angle minus left (outer-wheel) angle.Transfer the above values to

a graph, with the outer-wheel angles along the horizontal axis, and "Kinematic Toe-Out" values (inner

minus outer-wheel angles) along the vertical axis. Join the points with straight lines, or with a "French-

curve" or a "spline" if available. Done!

Figure 10 shows how to produce the Dynamic-Toe curves for a specific R&P steering geometry. The

method is as follows:

1. As with Figure 10, draw the basic steering geometry across the bottom of the page. Only draw one

half of the steering system. Use a scale that is as large as possible -30% to 50% for A4 paper (see belowfor comments regarding the scale). Position the kingpin first. Draw an arc of radius equal to the steer-

arm length. Draw in the centreline of the steer-arm. Draw the rack centreline and the rack end-joint

(with the rack "centred") in the correct position relative to the kingpin. Draw the track-rod from the

rack-end-joint to the steer-arm-end-joint.

2. Mark off positions along the rack centreline corresponding to the position of the end-joint at 25%,

50%, 75% and 100% full-lock, to the left and right of the already drawn "centred" position. Alternatively,

mark positions at regular, say 20mm, positions to the left and the right of the rack-end-joint. If the

steering is via a rotating idler instead of a rack, then mark of symmetric positions to the left and the right

of the centred idler position.

3. Use the compass to measure the track-rod length. With the compass point on each of the above

marked rack positions, draw an arc that intersects the steer-arm arc. Draw lines from each of these

intersections to the kingpin. Align the "zero" of the protractor to the original steer-arm centreline, then

measure the angles to each of the newly drawn steer-arm-angle lines.

4. Draw up a table of corresponding steer-arm-angles to the left, and to the right, of the original steer-

arm line. If drawn as in Figure 10, the leftward angles represent the inner-wheel steer-angles, and the

rightward angles represent the outer-wheel steer-angles. Provided that the "rackmarks" are symmetric

about the rack "centremark", then each inner-wheel steer-angle will have a corresponding outer-wheel

steer-angle, for that particular displacement of the rack. Also note on the table the differences of inner

minus outer-wheel steer-angle.

5.On a similar graph to the KSA, draw a curve of outer-wheel steer-angles versus dynamic-toe change of

the inner-wheel. Done!

Using the above method it should be possible to draw the dynamic-toe curve for a specific steering

layout in about 15 minutes -less, if the reader is not drawing his curves at the kitchen table while his

children are doing their homework. For a quick feeling of the shape of the dynamic-toe curve, only the



half and full-lock angles have to be plotted. If these look good then more points can be plotted for a

more accurate curve.

If an "A0" size drawing board with built-in protractor is available, then very quick and accurate curves

can be expected. In this case, if full-scale or larger drawings are used, then the accuracy will probably

exceed that on the real car, given the manufacturing tolerances and compliance of the linkage and

suspension under load.

If a finished curve has a "kink" in it, then it is probable that a mistake was made. This could be due to

wrong compass point positioning, flex of a cheap compass, poorly drawn steer-arm-angle lines, or

misreading of the protractor. If the arc of the track-rod is almost parallel with the arc of the steer-arm,

making the intersection point difficult to determine, then the real linkage will also be "compliant" at this

point in its travel.

The caveats at the beginning of the main article, regarding the simplified2-D analysis, should be kept in

mind when interpreting the curves produced by this method. But a 90% accurate answer today, may

well be better than a 99.9999% accurate answer "sometime tomorrow".

principles of ackerman erik zapletal

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