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We have finally come to the most interesting bit about our short tyres overview – the analysis of how a tyre develops longitudinal and lateral forces at the contact patch and what are the key parameters determining how it performs.

We will start with the longitudinal force **F _{x}**.

**Longitudinal Force**

First of all, we will shortly analyze what are the reasons why cars four tyres need to exchange (longitudinal) forces with the road, for example during a braking maneuver.

If we consider a car traveling in straight line and the driver wants to slow it down by applying the brakes. We recognize that the four tyres will need to exchange longitudinal forces with the road in order to produce a deceleration and to balance the inertial force connected to it.

If the car decelerates with a longitudinal acceleration **A _{x}**, to have equilibrium (in steady state) the road will have to apply an

**F**to the car given by the equation:

_{x}F_{x}= -m * A_{x}

Where **m **is the mass of the car. The sum of the forces exchanged by each tyre with the road must be equal to **F _{x}**. In our picture below, this is given by:

F_{x}= F_{xf} + F_{xr}

Where **F _{xf}** and

**F**are respectively the front and rear axle braking forces.

_{xr}In order to understand how a tyre is able to exchange longitudinal forces with the road, we will first need to define a key concept: the **slip ratio**.

**Slip Ratio and Slip Curve**

Imagine a car traveling in straight line with a certain speed and its driver starting to brake. The braking torque acting on the wheels will produce a deceleration and the wheel rotational speed will become slower than the one that the wheel would have if traveling at the same speed, but without braking.

The **slip ratio** is a measurement of the difference between vehicle forward velocity and wheel rolling speed. Its value is given by the following equation:

with **ω** being the wheel angular velocity, **R** its rolling radius and **V** vehicle’s forward velocity. Because of tyre slip (ratio), molecular adhesion and indentation take place, causing a longitudinal force at the contact patch.

SR is negative during braking, positive in case of forward acceleration and about equal to zero when the wheel rolls freely.

As we mentioned from the beginning, when we analyzed how grip forces are exchanged at the contact patch, the existence of a slip is needed for the tyre to produce planar forces. As we will see later in more details, the magnitude of the longitudinal force depends also on the vertical load acting on the tyre. Analogously to what we have seen for our simple rubber pad, we can then define a tyre longitudinal friction coefficient, as the ratio between **F _{x}** and the vertical force

**F**acting on the tyre:

_{z}For a given vertical load, the relationship between slip ratio and longitudinal force **F _{x}** can be described using a plot, often referred as

**slip curve**:

Racing tires normally reach the peak **F _{x}** value when the SR assumes values between 0.05 and 0.15. As we mentioned, the maximum available F

_{x}for a certain SR depends also on the vertical load acting on the tyre: if vertical load increases, the same does F

_{x}. As we will see later, the rate at which F

_{x}grows with F

_{z}is not linear.

If we analyze the shape of the F_{x} vs SR curve, we can identify three main regions.

The first area, **A** in the slip curve plot, is often called **linear region** because the relationship between F_{x} and SR is more or less linear. The second area, **B**, is often called **transitional** and is characterized by the slip curve bending until reaching the peak force value. Finally, the third area, **C**, is normally identified as **frictional region**, because for so high slip values, the whole tyre contact patch is slipping.

The shape of the curve is very much linked to what happens physically at contact patch when, for example, a brake torque is applied to the wheel. As the rotational speed of the wheel is smaller than the vehicle speed, the road will pull the tyre tread as soon as it comes into contact with the ground, during wheel’s rotation. Each rubber portion entering the contact area is initially in shear conditions. As it slips over the road surface to leave the contact patch, it then enters a slip situation.

Tyre’s tread can deform, while its belt is much stiffer. Consequently, when a braking torque is applied to the wheel, the road surface pulls the contact patch backward, but only the tread experiences a distortion, while the belt maintain more or less its original shape. This results in a relative motion between the bottom of the tread and the belt. This leads to a shear condition at the leading edge of the contact patch. As the tire rolls and the rubber moves toward the trailing edge of the contact patch, the tread’s stress increases and, whilst remaining sheared, the rubber enters a sliding condition.

An important point about longitudinal slip curves is the influence of tyre temperature. In very high stress / high deformation / high slip situations, rubbers temperature tends to increase very quickly; tyres coefficient of friction is highly affected by their temperatures.

The bigger the slip, the bigger the heat that is produced and that yield to a temperature increase.

In the **A** region of the slip curve above, the tread is essentially in shear and this produces moderate amount of released energy. As we move to the region **B** and **C**, the slip ratio grows and the tyre enters a true slip condition, thus releasing a much bigger amount of energy. This is in form of heat, that increases the temperature in the tread and carcass.

In an extreme situation, where the tyre is locked and the slip ratio is equal to one, this phenomenon is even more pronounced. It is not uncommon seeing locked tyres smoking, because of the rubber reaching very high temperatures.

At high temperatures, rubber hysteresis drops and the same happens to the available grip and this contributes even further to the slip curve sloping down after the peak.

One last interesting point about slip curves is that, given a certain value of F_{x}, we could find two different value of the slip ratio producing it.

We will deal about this again when describing lateral curves. Since bigger slip ratios correspond to more energy being released in form of heat, working at the SR1 will produce a smaller temperature increase in the tyre than working at SR2, given the same F_{x}. The tyre being able to exchange with the road the same F_{x} also for higher slip ratios is the reason why it is possible to see cars spinning their tires at the exit of a corner and still acceleration very quickly.

On the other hand, it is also very clear that, from a pure performance perspective, a driver’s goal should be to let the tyre work as close as possible to its peak for all the time car’s performance is limited by the amount of available grip (see situations where the available engine torque would be big enough to let the tyres spin and the driver needing to open the throttle only partially in order to avoid ineffective spinning).

In part 4 next week, we’ll look at lateral forces.