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- Part 1 – Tyre as a vertical spring
- Part 2 – Road Interaction & Contact Patch Grip
- Part 3 – Longitudinal Forces
In many aspects the way tyres develop lateral forces is similar to the way it develops longitudinal ones. Before we dig into the analysis of how a tyre is able to exchange a lateral force Fy with the ground, we need to understand why this should happen in order for the car to negotiate a corner at a certain speed.
If we consider a car, traveling around a bend at a forward speed V, in order for the car to remain on the desired trajectory (for sake of simplicity, we will assume for now that the car moves on a circular path, with radius R), the road will have to apply on the car a lateral force Fy.
In particular, said Ay the lateral acceleration experienced by the car, we can write (in steady state):
Fy = -m * Ay
This means that, to have equilibrium, the four tyres will have to each produce a lateral force so that the sum of all four tyres lateral forces will be equal to Fy.
Fy = Fyfr + Fyfl + Fyrl + Fyrr
The main difference between lateral and longitudinal forces generation regards the key parameter that we need to consider, normally called slip angle for cornering forces.
If we consider a car negotiating a corner, the main way that the driver has to point the car in the right direction is the angle at which the front wheels are steered.
If the car is traveling at high enough speed to generate a relevant lateral acceleration, the path followed by the front wheels will not be exactly the one given by the rim middle plane, but a different one. The two directions will differ from each other by a certain angle, called slip angle. This angular difference induces friction between tyre and road and this generates a lateral centripetal force.
Similarly to the slip ratio, the slip angle is also linked to tyre’s structure flexibility and to a deformation of the tread in contact with the road when external forces are acting.
As we saw in our example with a rubber pad pressed against a surface (Part 2), an external force acting on the tread will deform it. Initially with no slide but only with local micro-movements (slip) that are the key elements for forces to be exchanged with the road.
Also for lateral forces, we can define a coefficient of friction, given by the ratio between the lateral force itself and the vertical load acting on the tyre:
As we will see later on, the bigger the Fz, the bigger the Fy. Differently from a simple rubber pad, the coefficient of friction µ doesn’t remain constant and decreases as the vertical load increases.
The relationship between slip angle and cornering force (at a given vertical load) can be described using a slip curve that looks similar to the one we already analyzed for longitudinal forces. The main difference is that the horizontal axis now shows slip angle values, while the vertical one refers to the lateral force.
Typical race tyres can produce peak cornering force anywhere between 3 and 10 degrees, mostly depending on tyre materials and construction.
For the lateral slip curve, as we seen for the longitudinal one, we can define the three regions A (linear), B (transitional) and C (frictional) with very similar characteristics.
Again, the key point is that after the tyre has reached the maximum achievable lateral force, any increase of the slip angle will produce a reduction of the magnitude of the force itself, reducing car’s cornering abilities.
As for the longitudinal force, the shape of the lateral slip curve is strictly connected to what happens at the contact patch when the road exerts a cornering force on the tread.
Given a tyre, subject to a certain vertical load and traveling at a certain slip angle α (which is the angle between the tyre and the rim own directions), each tread portion reaching the leading edge of the contact patch will initially locate itself vertically with respect to the road surface. As it gets closer to the trailing edge it will be subject to a distortion as it tries to remain parallel to wheel’s direction. When this distortion reaches a critical point, our tread portion will begin to slip. So, as we also saw for longitudinal slip, the tread will initially be subject to shear and only afterwards it will start to slip, before leaving the contact patch.
This means that at the leading edge of the contact patch or for small slip angles, the tyre will mainly be sheared in a perpendicular direction with respect to the tangent of its path. For bigger slip angles, or when it is closer to the trailing edge, our rubber portion will slip. The slip region gets bigger and bigger as the slip angle increase, until when the whole contact patch could be actually slipping (typically this happens for slip angles close to the peak one).
As we already saw analyzing longitudinal slip curves, also for lateral ones, we could find two distinct value of the slip angle that would allow to the same cornering force.
As for the slip ratio, bigger slip angles also correspond more energy being released in heat form. This means that, given a certain lateral force Fy, if we achieve it with the bigger slip angle SA2, we will automatically produce a temperature increase compared to if we would achieve it with the smaller slip angle SA1.
This feature of a tyre is what is also connected to the car sliding and still being somehow manageable. This same feature is something often used in drifting or rallying to control the car also past the peak slip angle point.
Again, a driver’s goal is anyway to always target for the peak lateral force, in order to maximize the cornering performance.
In general, a tyre having the peak at smaller slip angles will be more efficient, dissipating less energy in form of heat and reacting more responsively to drivers input. It will also be a less forgiving tyre, making it harder to drive for a non professional driver. It will also generate less resistance to forward motion, as the component of the cornering force pointing longitudinally with respect to the car will be smaller.
On the other hand, as smaller slip angles tend to be associated with cooler temperatures, it could be also harder to bring this tyre up to temperature (or, on the other hand, it could be a tyre that does not overheat too easily).
One additional important point about lateral forces is that the Fy normally has its application point seating a bit behind the contact patch center.
The distance between the lateral force and the contact patch center is often referred as Pneumatic Trail. This metrics is very important: because of it, a moment is produced by the Fy and it acts again the steering action, trying to realign the wheel in its forward direction. This torque is often called Self aligning Torque.
Self Aligning Torque
Beside giving a feedback to the driver in the form of a reaction torque at the steering wheel that he has to overcome, this torque is also responsible for a global effect on the car that reduces its capability to quickly change direction. We will come back to this in another article.
What is important to mention here is that the pneumatic trail changes with the slip angle and it is normally bigger for very small slip angles, reducing its length as the slip angle gets bigger.
We can plot the self aligning torque with respect to the slip angle using a similar plot to what we have seen for lateral forces:
An interesting feature of self aligning torque is that, given a certain vertical load it reaches its peak value at a smaller slip angle than lateral force. This means that, considering a driver approaching a corner and gradually increasing the steering angle, in absence of other effects, the driver could get steering wheel feedback that gets dropping long before the maximum cornering force has been reached.
Despite this being something that all drivers have to keep in mind and learn to handle, there are often other effects that produce larger steering torques than the pneumatic trail. These can overcome the effects of pneumatic trail. We will deal with these aspects in a separate article, regarding suspension geometry.
Part 5 next week – load sensitivity