A bit of light holiday reading for the mathematically minded member. Reproduced without permission from "Motorcycle Tuning - Chassis" by John Robinson. (ISBN 0 7506 0798 X) I thoroughly recommend this book. It is great reading, and has heaps of cool pictures too! Think of this as just a taster, pp2-8. Who thought steering a motorcycle was simple?

All diagrams and tables omitted. CoG==Centre of Gravity.

Enjoy... Tim Walker ZX7R

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The way in which a bike steers is the focal point of its handling and stability. At very low speeds, a bike steers by turning the front wheel into the corner. With the bike essentially vertical, the trail shifts the front tyres contact patch to the left in a right turn. This means that the CoG is now to the right of a line drawn between the two contact patches - the line on which the bike is supported. It would, therefore, try to fall over to the right.

At the same time it is moving in an arc to the right; the centre of this curve would be where lines drawn through each wheel spindle cross one another. It has acceleration (centripetal acceleration) towards this point, even though its speed, as recorded by a speedometer, is constant. The acceleration is v²/r, where v is the speed and r is the radius of the arc which the bike is following. The force which provides this acceleration is mv²/r, where m is the mass of the bike, and it is generated at ground level. The reaction caused by the inertia of the bike equals this force but is in the opposite direction, away from the centre of the turn (centrifugal force) and is considered to act through the bikes CoG - which is well above ground level.

The centrifugal force sets up a couple which tries to make the bike fall over to the left. The strength of this couple is ymv²/r, where y is the height of the CoG. The couple trying to make the bike topple to the right is mgx, where mg is the total weight and x is the amount the CoG has been displaced from the bikes line of support.

When the bike is turning in a balanced fashion it does not fall over, so these two must be equal:

ymv²/r = mgx, or v² = rgx/y

Now g is constant, and so is y for a given bike, while x depends on the steering geometry and the steer angle, and r also depends on the steer angle. For a given bike and a given steer angle, rgx/y is constant. There can be only one value for the speed v which will satisfy this and keep the bike in balance (there are two actually, +v and -v, because the maths allows for you to be able to ride backwards at the same speed).

If the rider goes slower, then ymv²/r will be too small and the bike will fall to the right, into the turn. If the rider goes faster, ymv²/r will be too big, and the bike will fall to the left, away from the turn.

As an aside, the rider can shift the CoG, especially on a very light bike, by standing up (increase y) and by leaning his body to one side (increase or reduce x). So by making x and y variable, a trials rider can give himself a range of speeds for which the bike is balanced in a given turn.

If v is steadily increased and the height of the CoG stays the same, then the term rx will also have to be increased. To increase x, the steering has to be turned further into the turn, but this action reduces r, the radius of the turn, so we quickly reach a value v for which the steering geometry cannot cope. It happens, on conventional machines, at something in the region of 2 to 4 ft/s, but it is worth remembering that this critical speed exists.

If the speed is too great - as it must be if the bike is to exceed 4 ft/s - then the effect of a right steer effort is to make the machine fall, or roll, to the left. So now we have a bike travelling at very low speed, but more than 4 ft/s, and the result of applying a right steer angle is that it rolls left. The immediate effect of this is that the CoG is now displaced to the left of the line on which the bike is supported, and both wheels are leaning (or are cambered) to the left.

So far our bike would have steered and generally behaved as predicted if the wheels had simply been thick wooden discs. Now the tyres begin to do something: they generate what is known as camber thrust. Because the tyre can deform to a flat contact patch where it meets the ground it can, when it is leaning over, behave like a section of a large cone, lying on its side. If you roll such a cone it will turn in a circle which has the tip of the cone at its centre. If you take a tyre, lean it about 30 degrees from the vertical and roll it slowly forward, it will behave in the same way.

The bikes inertia wants it to keep travelling in a straight line. The camber thrust from the banked tyres wants it to turn in quite a tight circle. The force generated is not enough for this rate of turn, but it is a force and it does make the bike turn. The force acting on the mass of the bike gives it an acceleration to its left. The value of this equals v²/r, so the bike proceeds at whatever speed v happens to be, to turn on a radius of r.

As the bike rolls left, the reaction of the ground supporting the front tyre also tends to turn the steering to the left; the rider will feel this as a reaction in the handlebar which he can either oppose, ignore or augment.

We now have a situation which is very similar to the first one: the bike is steering left, is generating cornering thrust to the left (at both wheels this time) and has displaced its CoG (considerably further) from the line on which it is supported. As before, if the force trying to make it roll left (mgx) equals the centrifugal reaction (mv²y/r) then the bike will be balanced and will follow a circular path, radius r, at a constant speed v.

The displacement x is now a function of lean angle as well as steer angle (x=ycos(theta), where theta is the lean angle of the bike to the horizontal) and the value mgx can satisfy a much greater range of speed and radius (v²/r) values. An increasing lean angle also lowers the height of the CoG, which reduces the tendency for the bike to roll to the outside under the influence of centrifugal force. The equation now becomes: mgy cos(theta) = mv²y sin(theta)/r or v²/r = g/tan(theta) (this ignores any displacement owing to steer angle). The bikes lateral acceleration becomes 1g at an angle of lean of 45 degrees and Table 1.1 shows the speed at which curves of various radius can be taken with this acceleration and smaller acceleration values.

(table of theoretical speeds deleted)

This ignores any effect made by the steering being turned, which would displace the CoG further still. It also ignores a couple of other effects produced by the tyres.

Because the tyres are relatively wide and more or less circular in section, the contact patches move to the left as the bike rolls to the left, that is, it is no longer supported on its centreline, as the calculations assume. The implications of this are:

1) The CoG is not displaced as far as we thought it was, so the cornering force has to be reduced proportionately for a given angle of lean.

2) Bikes with a lower CoG will have to lean further to achieve the same balance as those with a higher CoG.

3) The wider the tyre, the worse it gets.

The second aspect has been hinted at: the tyre wants to run on one course but is forced away from it by the inertia of the bike. In addition the tyre contact patch can be regarded as part of a cone, that is, it has a greater radius on the outside edge than on the near side edge so the outside edge will travel correspondingly faster. It is not allowed to do this, so part of the contact patch must slip. The tyre can also run at a small angle to the one in which it is pointing and the more it is loaded, the more it is inclined to do this. The angle is called the slip angle and if the rear tyre has a different slip angle to the front, then the bike will rotate, it will turn to one side as it travels along. If the slip angle is bigger at the rear than at the front, the bike will rotate into the turn. This is called oversteer. The rider can reduce his steering effort and still hold the same rate of turn. Depending on the tyre construction and the compound from which it is made, this slip can increase the grip available. As the slip is increased, the grip also increases, reaching a peak when the slip is a few per cent higher than the tyres average speed. Beyond this peak the grip falls away again; when the tyre begins to slide or spin, the grip falls away more severely.

All this has been applied to a bike moving at steady speed. If the bike is also accelerating then it must be transmitting torque through the rear contact patch, and this will induce more slip at the rear tyre. This helps the cornering power because a certain amount of slip gives more tractive effort at the tyre and this can now be controlled using engine power as well as speed and angle of lean. It is also easier to control in this way. And it is creating more slip at the rear wheel, which leads to oversteer - which accounts for the sensation of 'drifting' when cornering under power and is the reason that bikes feel more secure and controllable in this condition.

Oversteer is a natural tendency for bikes because they have rear-wheel drive. It is also the safest condition because it 'pushes' the back tyre and, if this should spin or slide, it is easier to control than a front wheel slide. Most passenger cars, on the other hand, are set up to understeer (so that the front slip angles are greater than those at the rear). This is because the steering is more predictable - more effort equals more turn - and, if the front should lose traction, backing off the power will be enough to regain control. An oversteering car requires much more skill to control and once the back wheels lose traction the car is likely to spin whether the driver applies power or takes power off.

Factors which tend to increase the slip angle at a tyre are:

1) Flexible tread pattern

2) Flexible sidewalls

3) Flexible carcass

4) Lower coefficient of friction

5) Less pressure

6) Transmitting engine or braking torque

7) Larger section

The grip available for cornering, braking and acceleration depends mainly on the tyre compound (and the road compound too). The construction of the tyre gives it the ability to use soft compounds without overheating while the construction of the bikes frame and suspension hold the wheels in the attitude which gives maximum traction. This is the requirement for good road holding.