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 should point, 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-wheeldrive 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). Camberangle is also zero. It is only changes to the "steer-angles" that are being considered. The analysis uses specific 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 steerangle 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. For example, 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 forces are 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 It follows that the wheel with the larger steerangle will have the larger total longitudinal force. the whole car can be considered to have been yawed slightly to the left. the forces will be balanced. then both wheels will have a leftwards steer-angle. then the Toe-In car will still have the greater longitudinal force at its right side. while the same forces in lateral and longitudinal "car-coordinates" are shown as solid arrows. and an associated leftwards axial-force. Now one wheel has zero steer-angle. and zero slip-angle. That is. 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. Before making adjustments to roll-centre heights or spring-rates. Alternately. hence it has an increased axial-force. sinA ~ A. In each case the "wheel-coordinate" forces are shown as hollow arrows.degree change in steer-angle of the same wheel carrying a vertical load of 8kN. putting more toe-in on the car will generate differential-longitudinal-forces that act to yaw the car away from the turn. fy. when the steering-wheel is in the straight-ahead position. and thus only a rearwards rolling-drag force (or some rearwards braking-force). plus the same rolling-drag or braking-force as the first wheel. The longitudinal component of the drag or braking-force. For small steer-angles the size of the longitudinal component of the axial-force. will be approximately equal to the steer-angle x fy. while putting more toe-out on the car will generate differential-longitudinal-forces that act to yaw the car into the turn. The difference in these two longitudinal forces acts to yaw the car to the right. typically less than 1 degree. which in turn will change the sizes of the lateral and longitudinal components of the forces. But in general. can produce a change in axial-force of 3kN. and the "differential-longitudinal-force" –Delta Fx -acts to yaw the car to the left. and Figure 2b depicts a car with static-toe-out. and the Toe-Out car will have the greater longitudinal force at its left side. it remains unchanged. the wheel with the greater steer-angle (which is its effective slip-angle) will generate the greater axial-force. while the car is still travelling straight-ahead. For a small angle A (less than about 10 degrees). Figure 2a depicts a car with static-toe-in. and cosA ~ 1 (A measured in radians). However. the longitudinal component of force is directly proportional to the steer-angle. Variations in wheel loading will change the size of the axial-force for a given slip-angle. in Figure 2a -Toe-In -there is a greater longitudinal force at the rightwheel. If the steering is turned further to the left. Conversely. The other wheel has an increased steerangle and slip-angle. The slip-angles (equal to the static-toeangles) and the forces acting on the wheels are shown. then at the left-wheel. will be approximately equal to fx -that is. Static-toe angles are relatively small. while the car is still travelling straight-ahead. the racecar engineer should make sure that the wheels are pointing in the right direction. in Figure2b -Toe-Out -the greater longitudinal force is at the left-wheel. The lower parts of Figure 2 show what happens when a small steering movement is made to the left. The upper parts of Figure 2 depict the cars when travelling along a straight road. But even if we assume that the left and right axial-forces are equal. It can be seen that while everything is symmetric. fx. . This difference in the left and right longitudinal forces arises from the difference of the steer-angles of the two wheels. STATIC-TOE -"Static-toe" refers to the steer-angles of the front wheels. relative to the car's centreline. For equal left and right wheel loads. then the steering geometry is said to have "parallel-steer". the differential-longitudinal-forces are stabilizing with toe-in (giving stable high speed cruising. The term "Ackermann" doesn't seem to have a universally accepted definition. than that of the centre of the rear-axle. This rear-slip-angle is depicted in the graphic. If both rear-wheels are aligned with the centreline of the car. or pro-Ackermann". then it is possible for the car to . It refers to the direction of travel of the centre of the rear-axle. then the steering is said to have "dynamic-toe-in".and destabilizing with toe-out (giving "nervous" straight line driving. At full-lock. relative to the outer-wheelprint's direction of travel. which is the angle between the centreline of the car. and "sharp" turn-in). Dynamic-toe-in is often referred to as "negative-. DYNAMIC-TOE -"Dynamic-toe" refers to the change in steer-angle of one front-wheel. This is typical of low speed travel when the horizontal forces on the rear tyres are low. The vertical axis indicates Alpha(Inner)-Alpha(Outer). and thus their slip-angles are minimal. As a generalization. relative to the centreline of the car. such as that available on some production cars. if the car has some form of rear-steer. If the steer-angles of the front-wheels remain equal to each other as they move from straight-ahead to full-lock. relative to the centreline of the car. With toe-out the wheel with the greater steer-angle has a longer moment-arm for its axial-force. If the front-wheels toe-out relative to each other as they move towards full-lock. for any specific motion of the car. The other two curves indicate the KSA of the front-wheels when the centre of the rear-axle has a slipangle of 10 degrees and 30 degrees. 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.Another way to look at this. The rightmost curve indicates the KSA of the front-wheels when the centre of the rear-axle has zero slip-angle –that is. then they should be set-up with some static-toe-in. and the inner-rear-wheel will have a slightly larger slipangle. and the direction of travel of the outer-wheelprint. or anti-Ackermann". so that the axles of the wheels will be pointing directly at the instantaneous centre of the car's motion. Three curves are shown. Dynamic-toe is a function of the steering geometry. then the steering is said to have "dynamic-toe-out". relative to the other front-wheel. and if there is a significant amount of rear-slip. Rather. when the Instant Centre lies on an extension of the rear-axleline. as the steering is turned away from straight-ahead. They can be interpreted as the steer-angles that are required of the front-wheels. The horizontal axis indicates Alpha(Outer). while dynamic-toe-out is sometimes called "positive-. On the other hand. This would suggest that if the rear-wheels are to run at equal slip-angles. which is the dynamic-toe-out of the inner-wheelprint's direction of travel. Figure 3 indicates the "Kinematic Steer-Angles" (KSA) of the front-wheels of a car. then the outerrear-wheel will have a slightly smaller slip-angle. If the front-wheels toe-in relative to each other as they move towards full-lock. dynamic-toe can result in a difference of the two front-wheel steerangles of 10 degrees or more. as the car rotates around various "Instant Centres". Note that these angles don't refer to the actual steer-angles of the front-wheels. they indicate the direction that the centres of the wheelprints are travelling. and "sluggish" turn-in). or anti-Ackermann). This argument is based on the observation of Figure 1.3 degrees.3(KSA) + 9(slip-angle) = +3. then the car needs progressively less front-wheel steer-angle to negotiate corners. and Inner-Wheel Steer-Angle = 5. 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). Toe-Out of Inner-Wheel = -0." curve of Figure 3.of each curve indicates when the car is travelling in a straight line -infinite radius. when the frontwheels are steered about 5 degrees away from straight-ahead. Also indicated on each curve are the radii of cornering (prefixed with "R" and taken to the centre of the car) for different KSAs. then the greater the slip-angle that it must run at. and the more lightly loaded inner-wheel develop sits peak-axial-force at 9 degrees. They are partially presented here to show that the curves will eventually return to zero dynamic-toe-out. or anti-Ackermann. opposite-lock. The leftmost point of each curve indicates when the car is travelling in a straight line . Outer-Wheel Steer-Angle = -5(KSA) + 10(slip-angle) = +5 degrees.7 degrees as indicated at the left of Figure 4. stiff rear tyres. while the rear-slip is kept to a small value (say 0degrees). consider Figure 4. For example. If the rearslip-angle is large enough. KSA of Outer-Wheel = -5 degrees. or rear-wheel-steer -and the car is turning a reasonably tight corner.3) = -5.3 degrees If the heavily loaded outer-wheel develops its peak-axial-force at a slip-angle of 10 degrees. The curves end at a rightmost point which can be considered to be "full-lock". That is. . Since the outer-wheel of a cornering car will carry a greater load than the inner-wheel. (Steer-angles beyond this point are only applicable to high-maneuverability vehicles. 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.run with large. For a 5m radius corner. when the outerwheel has rotated a full 180 degrees. but still equal. and the "Rear-Slip = 10 deg. it follows that the outer-wheel must run at a greater slip-angle than the inner-wheel. then the front-wheel Kinematic Steer-Angles are negative -that is. about 8 degrees of dynamic-toe-out is required. in order to develop its maximum cornering force. then quite large values of dynamic-toe-out are required if both front-wheels are to operate at similar slip-angles. so KSA of Inner-Wheel = -5 + (-0. that the greater the load on a tyre. rear-wheel-slip-angles. The dynamic-toe-out required in this situation is also negative -that is. racecars will generally corner faster with some dynamic-toe-in (negative-. The curves are asymmetric in this format because the outer-wheel becomes the inner-wheel part-way through its travel.With 10 degrees of rear-slip. and if the corner radius is also large enough. if both wheels areto develop their maximum cornering forces. there should be about 1. together with the cross-ply tyre curves at the right of Figure 1. dynamic-toe-in. For a full-lock turn more than 16 degrees of dynamic-toe-out may be required. and a corner radius of about 30m. As the rear-slip-angle increases.) If a car has a small rear-slip-angle -due to either slow speeds.3 degrees of actual dynamic-toe-in. then for maximum cornering force. due to excessive toe-in. If the outer-wheel reaches peak-axialforce before the inner-wheel. . if the inner-wheel reaches peak-force first. or dynamic-toe-in from the steering may be helpful. Likewise. then any increase in slip-angles will decrease the outer-wheel's axial-force. It follows that it is desirable to fit a car with tyres that develop their peak-force at small slipangles -for example. In this situation. then any increase in slipangles will result in a stabilizing understeer yaw moment acting on the car. and its drag component of force would be increased. and the inner-wheel slip-angle would be greater than 9 degrees. Its "centripetal" component of force would be reduced. The race-tracks of that period also had fewer tight-radius corners such as "chicanes". If the car in Figure 4 had a steering linkage that generated dynamic-toe-out. The slip-angle thus generates a "drag" component of the axial-force that is rearwards to the direction of wheel print travel. large slip-angles are not desirable because they generate drag. Conversely. but. 2. it is not necessarily the quickest way through a corner. While sideways might look fast. As explained previously. Much of the racing involved long straights followed by high-speed large-radius "power-limited" bends.7degrees. it points 9 degrees behind this radial line. They are: 1. some anti-Ackermann. low-profile radial-ply tyres-and to minimise the load variations of the wheels. then the inner-wheel steerangle would be greater than 3. and thus engine power and fuel. One of the main benefits of radial-ply tyres is that they corner at lower slip-angles. The tyre would thus be running at a point past its peak-axial-force. The maximum amount of dynamic-toe-in needed is about 3 degrees for dirt racers. The above philosophy was developed quite a few years ago. to overcome it. This "slip-angle-drag" is an undesirable force as it requires forward thrust.The upper-right section of Figure 4 gives a more detailed view of the situation at the inner-front-wheel. Large slip-angles. the need for dynamic-toe-in is considerably reduced. With low slip-angle. The car's total cornering force would thus be reduced. evenly loaded wheels. and for dirt-track cars with very high rearslip-angles. The axial-force Fy doesn't point directly at the Instant Centre. which is destabilizing. This is the essence of the anti-Ackermann argument. as this reduces the maximum available horizontal wheel forces via the tyre-load-sensitivity effect. There are several details that should be kept in mind when considering this argument. and it would require more engine power for the car to maintain the same speed. when racecars had the high slipangle tyres used in the above example. than do cross-ply tyres. by the definition of its 9 degree slipangle. and increase the inner-wheel's axial-force. it is not desirable to have widely different loads on the wheels. 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. 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. and less than 2 degrees for road racers. and thus with less drag. due to insufficient toe-in. An anti-Ackermann steering geometry will produce a dynamic-toe curve that is initially horizontal. Effective-toe-angles. Since we often don't know the individual front-wheel slip-angles. . 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. 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. and Figure 2). and the car is still travelling straight-ahead (see the "Static-Toe" section. This large rearwards movement of the inner-wheel's axial-force vector will effectively drag that side of the car backwards. or. exerting a large yaw moment on the car. the included angle between the two front-wheels). We will use the term "effective-toe" to refer to the discrepancy between the KSAs and the actual steerangles. A steering geometry that generates large dynamic-toe-out angles will cause the inner-wheel of the car to develop a larger steer-angle. than the outer-wheel. If such a racecar has a steering geometry that gives parallel-steer. There are many categories of racecar that are required to turn sharp corners -trials. 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.These effects may be small. It is only when the car has a yaw angle to its direction of travel that its rear-wheels can develop a lateral force. Transient manoeuvres. Adjust the static-toe of your car to its maximum safe extent (leave enough thread in the adjustments to hold the track-rods together). then its wheels will be "effectively" toed-in even more. 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. then it will have a dynamic-toe curve that is a horizontal line in Figure 3 (zero toe-change). if measured at each wheel. for the fence-sitters. a driver will typically turn the steering wheel through a large angle -perhaps half-lock or more. 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). 4. and so on. or for accident avoidance. It might be useful for the reader to conduct a small experiment to clarify the above situation. rallycross. hillclimb. whenever a large steering movement is made. than a car with"peak-force-toe-in". but then drops below the horizontal axis of Figure 3. parallel-steer. try to get at least 5 degrees per wheel -that is. and thus push the rear of the car sideways. autocross. Now drive the car slowly along a quiet road. but they would suggest that a racecar with some "peak-force-toeout" (in the sense that the inner-wheel reaches peak-force first) would be more stable "at the limit". During a fast lane-change. and slip-angle. Many racing textbooks advocate steering geometries that are anti-Ackermann. 3. are equal to the tyre's slip-angles. thus improving the car's transient responsetimes. Sharp corners. Regardless of whether it is toe-in or toe-out. If this racecar has anti-Ackermann steering. at least 10 degrees of toe difference between the two wheels. Formula SAE/Student. as may be necessary during overtaking. minus the actual total-toe-out of the front-wheels (that is. 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. be trying to push the nose of the car out of the corner. but while both front-wheels are on the ground the anti-Ackermann car won't like sharp corners. 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 slipangle peak. In each case the wheel-hub has a rearward mounted "steer-arm". and RS = 10 degrees. 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. Thirdly. the front-wheels must run at a non-zero slip-angle. Figure 5 shows the KSA curves of Figure 3 for rear-slip-angles of RS = 0degrees. or beyond its slip-angle peak. This curve is also an approximate average of the curves that have practical front and rear-slipangles. has the steer-arm centrelines intersecting at the wheelbase mid-point. and they will protest loudly. the tyres won't like it. 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 experiment is an inexpensive version of the tyre-testing machines that produce the curves of Figure 1. or anti-Ackermann. where the centrelines of the two steer-arms intersect at the centre of the rear-axle. does to the front-wheels of a car whenever it turns a sharp corner. STEERING GEOMETRY -The previous sections discussed the vehicle dynamic responses that we might expect from different dynamic-toe behaviours. Secondly. so that all four wheels can run at an appropriate slip-angle. and RS = 10 degrees. the nose of the car will behave like an overeager puppy trying to sniff every tree either side of the road. If the car has stiff front springs. The four curves give an indication of the dynamic-toe angles that are required of a steering geometry. To generate a cornering force. and it will be reluctant to respond to any steering inputs from the driver. Speed and noise will make this fight less obvious. The accompanying curves show the relationship . This section shows how the different dynamic-toe behaviours are generated by different steering geometries. the steering will be "light". 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 experiment also shows what parallel-steer. The second steering geometry. and the ends of the two steer-arms are connected by a track-rod. despite the driver's steering efforts. and stop it from fighting the outer-wheel. on sealed road surfaces. thus changing the direction of the car. and there are rear-slip-angles of RS = 0 degrees. The traditional "Ackermann geometry" is shown as "A". Figure 5 also shows two steering systems typical of beam-axle suspensions. Both tyres will initially be operating close to their slip-angle-peaks where there is a lot of real "slip". Firstly.Several things should become apparent. and if it can corner fast enough. These curves give the "ideal-steer-angles" for the front-wheels. then it may be able to lift the innerfront-wheel off the ground. It gives a slow-motion view of the tyres as they are operating close to their slip-angle peaks. At least one wheel(typically the outer-wheel) will be close to. Most of each tyre print will be sliding so there will be little self-aligning torque or steering feel. if they were to run at a "front-slip-angle" of FS = 0degrees. The inner-wheel may. in fact. "B". The wheels are forced to run at a large effective-total-toe-in. and some effective-toe-out at full-lock. Four curves are shown for different angles of the centreline of the steer-arm. An important observation to be made of Figures 5. as it is mounted in the chassis. and the dynamic-toe-out of the inner-wheel. Only when the steer-arm-to-track-rod joint is moved past 20 degrees (that is. A compromise geometry somewhere between the two shown. 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. The second geometry. the inner-wheel turns more. but this time the steer-arm angle is kept at straightahead. The curves end when the R&P runs out of travel. the wheels maintain effectivetoe-in throughout the range of steering travel. A practical fulllock limit would stop the inner-wheel at least 10 degrees before the track-rod goes straight. The curves show that Ackermann geometry. As Alpha(T) becomes more acute. for the 150mm long steer-arm)do the wheels start to develop effective-toe-out. The choice of layout depends on packaging issues such as R&P placement in the chassis. If the steering geometry has rearward facing steer-arms (as in Figure 5). which in this case is +/100 mm from centre. and 7 is the size of the angle Alpha(T) between the steer-arm and the track-rod. 6. A disadvantage of geometry "B" is that it has a lower maximum outer-wheel steer-angle. has a similar dynamic-toe curve to a R&P mounted 50mm behind the axle with a steer-arm angle of 0 degrees. Figure 7 shows a similar R&P system to Figure 6. and the steering geometry approximates parallel-steer. Even with the "30 degree Steer-Arm" curve (steer-arm-endjoint about 75mm outboard of the king-pin)this toe-out only becomes significant close to full-lock. When Alpha(T) is around 90 degrees there is minimal dynamic-toe change. the track-rod cannot rotate the inner-wheel-hub any further. and the dynamic-toe-out increases. While the steer-arm angle is less than 20 degrees from straight-ahead. A shorter steer-arm would require less R&P travel. is varied. for these two geometries. is only an approximation to the KSA curve.and a rear mounted R&P. and thus a larger minimum turning circle than geometry "A". more than 52mm outboard of the king-pin. has the wheels running with an effective-total-toe-out throughout the steering range. "A". to help the car around hairpins. The curves "A" and "B" end (at their top-right) when the track-rod becomes "straight" with the innerwheel steer-arm -that is. For most of the steering range the wheels will be running with an effective-total-toe-in of about 3 to 5 degrees.between the outer-wheel steer-angles. and the longitudinal location of the R&P. changes to . A R&P mounted100mm in front of the axle with a steer-arm angle of about 25 degrees. plus some static-toe-in. That is. and at about full-lock (for the given dimensions). would give stable toe-in for high-speed large-radius corners. so that the steering can't jam at full-lock. and its effect on the dynamic-toe curve. and in fact it only gives Kinematic Steer-Angles at zero steer-angle. the dynamic-toe curves will still behave in a similar manner to that described above. implying a lot of wheel-scrub during low-speed parking manoeuvres. "B". for a given linear movement of the track-rod. At full-lock the wheels have a lot of toe-out. versus steer-arm placement within the wheel assembly. and the outer-wheel turns less. The curves of Figure 7 are similar to those of Figure 6. It differs in that its left-steer-arm is slightly shorter than the right-steer-arm. that has the R&P100mm in front of the frontaxle line. The two acute angles in the "Z-Bar" linkage (idlerrear-arm to outer-track-rod. while the right-steer-arm points straight-ahead. It also pulls the rightmost end of the curve to the right (more left-steer-arm-angle). both idler-front-arms are always at the same angle. the steering-wheel is turned to the left -and Right Steer. with the steering straight-ahead. Figure 9 shows two steering linkages that generate asymmetric dynamic-toe curves. Again. Likewise. Geometry "B" is similar to the example in Figure 6 with a 10 degree steer-arm-angle (10 degrees on both sides. 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. and the central-track-rod. However. Either of the idlers could be driven by a steering-box. or more toe-in). with responsive turn-in to tighter corners. and negligible wheel scrub during low-speed fulllock manoeuvres. because of intrusion into the footwell or engine space. Often it is not possible to use large steer-arm angles. suitably driven. such as those that race on ovals and spend most of the time either going straight. or turning left. Another advantage is that the increased number of angles between the space. and down (less toe-out. as in Figure 6. then the two idlers can be merged into a single central idler.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. Figure 8 shows a steering geometry that may appear more complicated than the previous systems.because the brake-disc is in the way. it may not be possible to move the R&P behind the front-axle line. They are connected via their front-arms to a central-track-rod. or the centraltrack-rod could be replaced by a R&P and two short track-rods. the asymmetry has the effect of rotating the previously symmetric curve clockwise. One advantage of this layout is that it can improve packaging convenience. or R&P. hence an included angle of 20degrees between the two steer-arms). and via their rear-arms to outer-track-rods which then connect to the steer-arms. With this layout two "idlers" are mounted either side of the chassis. as in Figure 7. but it has several advantages. These asymmetric curves are suitable for asymmetric racecars. For example. as the idler-rear-arm-to-track-rod angle straightens (for the inner-wheel) the dynamic-toe curve is pulled back down. can be mounted well forward of the front axle line for increased footwell space. this layout would give stable high-speed cruising and cornering. It differs in that. With some initial static-toe-in. makes it easier to tailor the shape of the dynamic-toe curve. If the suspension wishbones are long enough. The steer-arms can be straight-ahead. it takes only a little experimentation to get the dynamic-toe curve to track the KSA curve to within half a degree. This change has the effect of rotating the previously symmetric curve clockwise. . Geometry "A" is similar to the example in Figures 6 and 7. The specific layout shown in Figure 8 has the idlers and central-track-rod acting as a parallelogram -that is. Another advantage is that the increased number of angles between the track-rods and the rotating idlers and steer-arms. the 20 degree angle is at the left-steer-arm. The dynamic-toe curves are shown for both Left Steer -that is. and has zero steer-arm angles. although feasible. Extensions of this model include the reduction in cornering power that is due to lateral-load-transfer. This car could even negotiate Formula 1 style racetracks. as may happen if the driver has to counter-steer through the left turns due to a large rearslip-angle. according to the different conditions during a race. and is beyond the scope of this article. But the yawing power that is available from such differential-longitudinal-forces isn't specifically banned. 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. Each wheel is assigned a representative lateral stiffness -the initial slope of the curves in Figure 1. FURTHER CONSIDERATIONS -About 50 years ago. anyone who has used fiddle-brakes will respect the yawing power that can be generated by a small rearwards force on one side of the car. camber change. However. That might require some kind of an "active-steering-linkage". and so on. generate differentiallongitudinal-forces that act to yaw the car away from the corner. Parallel-steer. This would imply sharp turn-in when turning into the left-hand corners. and especially in example "B". There can be no differential-longitudinal-forces. Overall the car would be slower. the steering has considerable dynamictoe-out when turning left. 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 their resulting dynamic-toe curves. or "adapt". one rear-wheel. then they will appreciate how powerful the differential-longitudinal-forces can be when it comes to turning a vehicle. When the dynamic effects of driving this model around a particular corner at a particular speed are calculated. . Dynamic-toe-out of the front-wheels generates just the right sort of differential-longitudinal-forces that help yaw the car into a corner. The simplest version of this model is a 2-D plan-view of a vehicle. presented to various learned societies around the world. although its tyre wear would be increased through the tighter corners. But one thing that we can't expect of these mechanical steering linkages. ABS and similar systems are banned from most forms of racing. If the reader has driven a tank. the bicycle-model takes little account of the vehicle's width. For many Vehicle Dynamicists. example "B" regains some dynamic-toe-out. is that their dynamic-toe curves will change. and anti-Ackermann steering. One of the easiest ways to take advantage of this yawing power is to use dynamic-toe changes. bulldozer. with one front-wheel. isn't particularly efficient. if not thousands of papers based on this model. Many more variations of steering linkages.. are possible. Besides which. the bicycle-model is their bedrock. Likewise. With larger rightwards steer-angles.. or other skid-steer vehicle. When turning right. 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 or oversteer behaviour. then the steering has some dynamic-toe-in –as required by the anti-Ackermann argument. and thus to control understeer/oversteer. Quite complicated versions of the model have been developed. and uses only a joy-stick controlled ABS system for steering.In both of the above examples. the theoretical "Bicycle-Model" was developed to aid in the understanding of vehicle dynamics. The vehicle is symmetric about its centreline. and a centre-of-mass located somewhere between. Steering a car by using the brakes. Over the years there have been hundreds. fine-point biro. It can count. The top-left-corner of this rectangle corresponds to the front-left-wheel. Use the protractor to mark out a line from the centre of the rear-axle. If the reader prefers to use one of the more expensive. towards the right and down. and the surface of the sphere traced out by the track-rod-end-joint -there can be 0. ~~~oOo~~~ THE $2 SUPER COMPUTER===================== There is a children's song "The Super Computer". 20. straightedge (ruler). The maths are straightforward.No new hardware has to be developed to exploit these yawing forces. (If many different steering geometries are to be investigated. if available). it can smile. 3. it can talk. mark out lines from the front-left-wheel at angles of 10. It doesn't use buttons. There's a super computer inside your head. the only additional "coprocessors" needed to produce the curves shown in this article are A4 paper (5mm graph. The example in Figure 3 was drawn at a scale of 1:50. The method is as follows: 1. and a protractor. 1 or 2 intersection points. has nerves instead. by Don Spencer. might be appropriate.. until these lines intersect the rear-slip line. then a dedicated computer program executing this same method. " Since we all come thus equipped. that has the lyrics: "There's a super computer that can do anything. and various rear-slip-angles. then the method of solution would be much the same as outlined here..) The first curves to be produced are the Kinematic Steer-Angle curves. for the specific wheelbase and track dimensions. at the required rear-slip-angle -a horizontal line for 0 degrees rear-slip. it can cry. it can jump. then they should be available at the local newsagent for less than $2 total (the author bought a nice protractor. 2. and an adequate compass. at the required rear-slip-angle -a horizontal towards the right and up. Similarly. . although undoubtedly more fashionable boxes of electronic mischief as their coprocessor. For the generalised 3-D case the program must find the correct intersection point of the circular arctraced out by the steerarm-end-joint. it can sing. and also to . for 50 cents each). For example. These intersections are the Instant Centres (ICs) of the turn. It can run. compass. that is 52mm x 32mm. it can laugh. a computer aided drafting program can be used in a similar manner to that described below. as shown in Figure 3. Draw a rectangle proportional to the car's wheelbase and track in the lower left corner of the page. towards the right and up. 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. If the reader doesn't have any of these. These lines correspond to changes of angle of the front-left-wheel's axle.30. With its own compact and portable power supply. and 40 degrees. only the . Provided that the "rackmarks" are symmetric about the rack "centremark". the leftward angles represent the inner-wheel steer-angles. Use the protractor to measure the angles of these lines. Join the points with straight lines. Align the "zero" of the protractor to the original steer-arm centreline. If a bigger sheet of paper is available. 4. If there is a lot of rear-slip. Draw an arc of radius equal to the steerarm length. If drawn as in Figure 10. say 20mm. draw a curve of outer-wheel steer-angles versus dynamic-toe change of the inner-wheel. Use a scale that is as large as possible -30% to 50% for A4 paper (see below for comments regarding the scale). positions to the left and the right of the rack-end-joint. 50%. then measure the angles to each of the newly drawn steer-arm-angle lines.this wheel's steer-angle from straight-ahead. to the left and right of the already drawn "centred" position. Draw lines from each of these intersections to the kingpin. Draw up a table of corresponding steer-arm-angles to the left. draw the basic steering geometry across the bottom of the page. The method is as follows: 1. Use the compass to measure the track-rod length. For a quick feeling of the shape of the dynamic-toe curve. 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. and so on. Use the straightedge to draw lines from the front-right-wheel to the ICs. Only draw one half of the steering system. then some lines at +/-5 degrees can be added. 5. Mark off positions along the rack centreline corresponding to the position of the end-joint at 25%. of the original steerarm line. Draw in the centreline of the steer-arm. If the steering is via a rotating idler instead of a rack. right(inner-wheel) angle minus left (outer-wheel) angle. With the compass point on each of the above marked rack positions. or with a "Frenchcurve" or a "spline" if available. Draw the rack centreline and the rack end-joint (with the rack "centred") in the correct position relative to the kingpin. if the reader is not drawing his curves at the kitchen table while his children are doing their homework. and also the difference of the angles -that is. and the rightward angles represent the outer-wheel steer-angles.Draw up a table of corresponding front-left and front-right-wheel angles. mark positions at regular. Draw the track-rod from the rack-end-joint to the steer-arm-end-joint. then each inner-wheel steer-angle will have a corresponding outer-wheel steer-angle. 3. draw an arc that intersects the steer-arm arc. although at these angles the wheels are almost parallel. 2. and "Kinematic Toe-Out" values (inner minus outer-wheel angles) along the vertical axis. and to the right. then mark of symmetric positions to the left and the right of the centred idler position. then lines at -10 degrees. 75% and 100% full-lock.Transfer the above values to a graph. will be required. with the outer-wheel angles along the horizontal axis. Also note on the table the differences of inner minus outer-wheel steer-angle.On a similar graph to the KSA. 4. As with Figure 10. Done! Figure 10 shows how to produce the Dynamic-Toe curves for a specific R&P steering geometry. Position the kingpin first. 5. for that particular displacement of the rack. Alternatively. then it is probable that a mistake was made. This could be due to wrong compass point positioning. then very quick and accurate curves can be expected. If a finished curve has a "kink" in it. then the real linkage will also be "compliant" at this point in its travel.half and full-lock angles have to be plotted. then the accuracy will probably exceed that on the real car. If an "A0" size drawing board with built-in protractor is available. regarding the simplified2-D analysis. . In this case. If the arc of the track-rod is almost parallel with the arc of the steer-arm. But a 90% accurate answer today. making the intersection point difficult to determine. should be kept in mind when interpreting the curves produced by this method.9999% accurate answer "sometime tomorrow". given the manufacturing tolerances and compliance of the linkage and suspension under load. if full-scale or larger drawings are used. If these look good then more points can be plotted for a more accurate curve. or misreading of the protractor. The caveats at the beginning of the main article. poorly drawn steer-arm-angle lines. flex of a cheap compass. may well be better than a 99.