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- Part 1 – Tyre as a vertical spring
- Part 2 – Road Interaction & Contact Patch Grip
- Part 3 – Longitudinal Forces
- Part 4 – Lateral Forces
- Part 5 – Load Sensitivity

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We have now analyzed a bit of the main aspects of tyres behavior, including how a tyre produces longitudinal and lateral forces and load sensitivity.

However, a question still remains: how does a tyre behave, when it had to exchange with the road lateral and longitudinal forces at the same time?

Any time a tyre interacts with the road with a planar force having both a lateral and a longitudinal component, we talk about **combined grip**.

A typical situation where combined grip is used by the driver is when he/she starts to steer, while still pushing on the brake pedal (trail braking), or when he/she accelerates out of a corner while the car has not yet completely been realigned. Both of these situations take place regularly on a track if a driver explores the full potential of car and tyres.

In general, it is common for the car to negotiate a section of a track and experience both a lateral and longitudinal acceleration, exchanging with the road forces with both a lateral and a longitudinal component. From a pure mathematical perspective, a force can be represented by a vector, which is an entity defined by a magnitude (force’s intensity) and a direction (if we talk about planar forces, a combination of a lateral and longitudinal portion). A vector (and, hence, a force) is normally represented graphically with an arrow, this making much easier also for who is not too versed with mathematic to grasp its meaning.

The picture above describes a pure braking situation on the left side and a pure cornering situation, on the right side.

From a more rigorous perspective, we can imagine that the two vectors/forces refers to a known and fixed vertical load.

In both of these cases, the tyre can exploit its full potential in one direction only, hence maximizing either car’s lateral or longitudinal acceleration.

Since we now know that tyre forces can be represented as vectors, we can also conclude that, in cases where a combination of a lateral and a longitudinal force are exchanged with the road, the magnitude of the combined force can be obtained combining (with a vector sum) lateral and longitudinal components.

In a first approximation, useful here to explain the concept of combined grip, we can imagine the maximum possible magnitude of lateral and longitudinal force that a tyre can exchange with the road (for a fixed vertical load) to be the same. This will also mean that if we ask the tyre to provide at the same time a lateral and a longitudinal force, the resultant force vector (which is, as we saw, the vector sum of lateral and longitudinal component) cannot have a bigger magnitude than the one of the pure lateral / longitudinal force maximum magnitude.

If all of this is true, we can then define the maximum combined force for a given vertical load by using a circle, with its radius given by the maximum pure longitudinal /pure lateral force:

This means, on one side, that the magnitude of our maximum combined force is the same as the maximum magnitude of the pure lateral or longitudinal force. But this also means that, in combined grip situations, the maximum allowed lateral and longitudinal forces will be smaller than in pure cornering or in pure braking:

The circle defining the maximum combined force the tyre can exchange with the road is often called the **friction circle**.

The reality of things is actually a bit more complex that what we explained, as very often, for a given vertical load, the maximum magnitude of the longitudinal force a tyre can achieve is not necessarily the same as the maximum magnitude of the lateral one. This is why engineers very often talks about friction ellipse, instead of friction circle.

Another important point to mention is that, depending on tyre construction and the goals that want to be achieved with a certain design, it is possible that the shape of the envelope defining the maximums, is not a regular shape and cannot be defined either with a circle or an ellipse. However, this goes beyond the scope of this short overview.

It is also interesting to analyze how a slip curve changes its shape, in a combined grip situation.

As we have seen, anytime we have a non-zero slip ratio and slip angle at the same time, the amount of longitudinal and lateral force that the tyre can produce drops.

The shape of a plot of longitudinal force with respect to slip ratio, when the tyre also experiences a slip angle, depends very much on tyre construction. In general, anyway, the bigger the slip angle, the smaller the available longitudinal force for a given slip ratio.

This effect of having a bigger slip angle is stronger at smaller slip ratios and tends to lose intensity as we move toward bigger and bigger slip ratios, up to 1 (or -1).

We can identify a similar trend if we look at the plot of lateral force with respect to slip angle, when the tyre also experiences a slip ratio.

For a given slip angle, the tyre will produce smaller and smaller cornering forces, as the slip ratio gets bigger.

It is also interesting to cross these plots, looking for example at how the curve of lateral force with respect to slip ratio looks like, for growing slip angles:

We see the same tendencies we already identified, with the lateral force (at a given slip angle) dropping as the slip ratio increase. This plot gives maybe a better idea of how much cornering potential the tyre loses if the slip ratio increases, with the lateral forces coming very close to zero if the slip ratio gets to the value of one.

Similar tendencies can be seen if we look at the plot of longitudinal force with respect to slip angle, with varying slip ratio.

It is interesting to notice that the longitudinal force drops much slower with respect to the slip angle, compared to what the lateral force does with respect to the slip ratio. This tells us that, while it is difficult to steer when a front tyre experiences high slip ratios (or, in an extreme case, is locked), it is still possible brake the car (up to a point) even when the front tyres are steered. The reason for this is that, normally, pretty small slip ratios are required to produce high braking forces.

The final part next week will look at cambers, temperatures and pressures.