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- - Tyres - - Suspension - - Driving - -

Tyres - Page 2

An important concept when talking about tyres is the ‘circle of friction’. The previous page describes the tyre dynamics of cornering and accelerating/braking (i.e.: forces acting in only one direction, either lateral or longitudinal), but in reality these forces act at the same time, and are very closely connected.


The Circle of Friction

A simple way of understanding this concept is with a simple example:
If we consider a tyre to have a finite level of total grip that it can produce (say 3000 Newtons in total), if we ‘use’ half of this up by accelerating with an acceleration that equals 1500N acting on the road in the longitudinal direction, we only have 1500N of force left to act in the lateral (cornering) direction.

If however, we accelerate to the cars full potential, and accelerate at a rate which equals 3000N acting on the road surface in the longitudinal direction, we are at the limit of the tyre’s adhesion; we cannot therefore obtain any lateral acceleration (cornering) from the tyre, and attempting to do this will result in a slide.

It is this concept that basically explains why you cannot accelerate flat out whilst cornering, and why it is easier to loose traction by accelerating mid corner than when you are travelling straight. This can be graphed, as shown below. The vertical axis is the forces in the longitudinal direction (acceleration and braking) whilst the horizontal axis is the forces in the lateral direction (cornering). The curve shows the maximum force available; driving within this curve means that you are not using the tyres to their optimum.

Traction Circle


In reality, this is not a perfect circle, as shown above. Depending on tyre type and design, the ‘circle’ shifts into an ellipse; tyres on a dragster will tend to have a profile that is tall and thin, since its designed to maximise traction in a straight line. A racing tyre will tend to be wider than they are tall, as many racing tyres can create higher cornering forces than the ‘acceleration’ forces.

If the driver is driving right on the edge of the traction circle (which indeed means he is on the limit of the tyres ability), any extra cornering input or acceleration in the longitudinal direction will result in a loss of traction.


Tyre Forces

Basic physics, and specifically Newtons laws of motion, show that the force (F) is equal to mass (m) multiplied by the acceleration (a), (equation 1), or more importantly, if we flip this round, the acceleration is the force divided by the mass. This means that for maximum acceleration, we must have maximum force, and minimum weight. This basic law explains why racing cars are as light as possible.

Equation 1: F = m x a

In the longitudinal direction, extra force is ‘easy’ to obtain, we simply add more power; bigger engine etc. Minimising weight is generally harder to do. However, extra power is useless if the tyres are not capable of transmitting this force to the road. In this case, the force (F) is equal to the mass (m) multiplied by the coefficient of friction (µ ) (equation 2),  where the coefficient of friction of the tyre is related to a number of factors, including tyre compound, road surface, tyre and road temperature etc.

Equation 2: F = m x µ

The problem comes when we equate the two. If we make both forces equal to each other, we get as shown below (equation 3):

Equation 3: m x µ = m x a

We can see that since the masses are equal, the only way to increase the amount of grip using this formula is to increase the coefficient of friction (µ). This generally means softer compound tyres, which wear out more quickly; not good for longer races. There is a way round this however. Downforce (d), which, as its name suggests, is a force acting downwards on the car as a result of the aerodynamic properties of the vehicle, and adds to the mass on the left hand side of the equation. It then becomes (equation 4):

Equation 4: (m + d) x µ = m x a

We can see that suddenly the amount of tyre force has increased, for a given mass.

(It is worth noting at this point that the addition of downforce also adds drag, thus increasing the force opposing the engines drive, and downforce also only works at higher speeds, but more on this in a later section!)

In order to get a true feeling for how fast a car can go round a corner, it is necessary to do a bit more work; clearly the tighter the corner, the less quickly you will be able to go round. We therefore need some way of measuring a cars cornering ability that is takes the corner radius into account. This is know as lateral acceleration, and is calculated as shown in equation 5, below.

Equation 5: latacc = (v x v) / R

Where latacc is the lateral acceleration, v is the vehicles velocity, and R is the radius of bend curvature. If we use the units of metres for distance, and metres per second for velocity, the lateral acceleration is given in metres per second per second (m/s/s).

Plugging numbers into this formula will give the lateral acceleration of a vehicle round a given radius bend at a given speed, and is a good way of comparing similar cars.

Many people (especially in the racing scene) talk about ‘G-force’ and ‘how many G’ they were pulling. This is simply relating the force that the vehicle achieved to the force of gravity (which is 9.8m/s/s). For example:

If a car goes round a corner and, using equation 5, it is calculated that they achieved 15m/s/s. To work out ‘how many G’ this is, divide the number by 9.8. so the calculation is :

15 / 9.8 = 1.53G.

So the cornering acceleration was 1.53G, or 1.53 times more than the force of gravity acting on the vehicle. Broadly speaking, ‘normal’ road cars on a regular road are unlikely to achieve above 1G under hard cornering, but some ‘sports’ cars may come close or even break the 1G barrier with a good road surface.

Now, to work out the cornering force…remember equation 1 all the way back up there? Well that can be used here, simply take the lateral acceleration value (latacc from equation 5) and multiply it by the mass of the vehicle, in Kilos. This gives you the cornering force achieved in Newtons. Again, this demonstrates the importance of keeping the mass to a minimum.

There is an obvious question here; if the tyre force goes up as we add more mass, surely it does not make any difference? Well, the addition of mass not only affects cornering, but it also reduces acceleration and braking performance, simply because there is more mass, so there is more energy needed to move the car. (Or more energy to get rid of if braking!) Another factor, and the key one to remember, is that:

Adding more mass does not increase the amount of grip available by the same amount.

(In bold because it is important!). For example, doubling the mass of the vehicle will not double the amount of grip produced, as it may be in the region of a 70% increase. It may seem that this is the case from equation 2, however the coefficient of friction (µ), is a strange character and does not vary linearly. We are dealing with tyre mechanics at this point which is a dark and complex subject, and particularly difficult to explain mathematically!

This is the key point to remember. It also explains why lateral weight transfer (body roll) is generally considered bad, but more on this in the next section.

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