  # Modeling the tyre behaviour – part 1

### Modeling the tyre behaviour: g-g diagram.

How the tire behaves during his rolling is the main key to understand one of the most important tire diagram: the g-g diagram. This graph is determined by lateral and longitudinal forces functions of the total tire load. As we said in previous chapter, the tyres are the unique source of torques and forces due to braking, cornering, accelerations, road banking and aerodynamics.

All these forces are possible thanks to the basic grip with the ground.

Tires are too complex to consider as a whole, that’s the reason why we have to isolate and explain its characteristics separately.

First classification of forces is between lateral (generated by cornering that creates the aligning torque) and longitudinal (tractive and braking).

The lateral force is perpendicular to the direction in which the wheel is headed ignoring any inclination or camber.

For studying these forces, every tires, in every operating conditions, are associated to a vector representing the forces between tires and road. Every vector is divided into six components depending on reference axis system.

We have to start talking about the footprint, or simply “print”, the area of the tread of the tire that is in contact with the ground in any moment.

### What is GRIP.

The so called “grip” or handling between the footprint and the ground is given essentially by 2 phenomena:

• Molecular adhesion between the tread and the tarmac (or any other surface)
• Rubber elements stuck in the road between the little stones
• Elastic distortion of the rubber

The grip coefficient is defined by the relation: \begin{align} \mu=\frac{F_y}{F_z} \end{align}

where Fz is the vertical force and Fy the lateral one.

With a lateral force:

• Tire deflects and moves laterally: more is the force more the rubber deflects until the tire begin sliding where the lateral force remained more or less constant. Considering the rolling of the tire (so there is a component to the back):

There is no sliding in the footprint except near the trailing edge where, obviously, the vertical forces are low.

We see from the picture the slip angle: it’s the angle between a rolling wheel’s actual direction of travel and the direction towards which it is pointing. In other words, it’s the difference between where car is pointing (heading) and where it’s actually going.    We can understand how the vertical loads in a car during a competition or simply in the city are constantly changing due to corner, acceleration, braking etc..so how complex is study the behavior of the tire in its motion.

### Pneumatic trail: a stabilizing effect.

An effect of a lateral force is the aligning torque that’s the tendency of the tire to recover its natural rolling position and to steer around the z axis through the center of the print.

This is a stabilizing effect.

The aligning torque, like math and physics taught, is the product between a force (the lateral force) and a lever arm that in this case is called Pneumatic Trail and it’s the distance from the fore-aft center of the print to the center of action of the lateral force.

As the print is asymmetrical, the pneumatic trail is always present and so the aligning torque.

But the aligning torque is only one term of the steering torque, because there is another lever arm called mechanical trail given by the caster angle plus the position of the kingpin referred to the wheel center.

So, the steering torque can be defined as:

\begin{align} steering\,torque= aligning\,torque +mech_{trail}\cdot F_y \end{align}

Or isolating Fy

\begin{align} steering\,torque= (pneu_{trail} +mech_{trail})\cdot F_y \end{align}  