http://atlasf1.autosport.com/

http://atlasf1.autosport.com/

Mosley’s Equations

By Bent Sorensen and Svend Damborg, Denmark

This article, originally found on the atlasf1 website, sheds new light on some previously unquestioned assumptions. It suggests that the FIA under Max Mosley and his team of experts may not have considered all aspects of the problem correctly.

ABSTRACT: Mathematics can prove that reducing aerodynamical downforce increases safety. In contrast, FIA president Max Mosley says that mathematics proves that grooves in tires increase the safety of Grand Prix racing, because (I) the energy of an impact is proportional to the grip of the tires and (II) whatever its speed, a car spins for exactly half the radius of the curve it is when control is lost. The faster a car is going at the point it starts to spin, the faster it will be at all points during deceleration, as it always stops in the same place. Therefore, if it hits a barrier on the way, the faster the car was going when it started to spin the harder it will hit the barrier. The greater grip of the tires, the faster the car would have been immediately before the spin. Hence, says Mosley, reduced grip equals increased safety, everything else being equal. However, his mathematical model ignores one point: When a car spins, it loses its aerodynamical downforce, and thus grip. Here we explain the mathematics and critically discuss why we feel the FIA (ex) President’s interpretations are incorrect.

Introduction

For safety reasons, the FIA sought to prevent increases in cornering speed. There is an ongoing discussion about whether limiting cornering speed should be achieved by cutting more grooves in tyres or by reducing aerodynamic downforce, for example by banning wings. FIA (ex) President Max Mosley put forward several statements based on mathematics. This article explains those arguments in detail and discusses the assumptions made in the analysis. By making a sharp distinction between aerodynamic grip and mechanical grip, an equation is derived that reveals that reducing aerodynamic downforce in fact leads to greater safety.

Basic Mathematical Equations

To understand Mosley’s arguments, it is necessary to start with the governing mathematical equations. Consider a racing car driving through a corner, for simplicity a circle. Let V stand for the velocity of the car, R be the radius of the circle (see Figure 1), and m symbolise the mass (weight) of the car. Newton showed that if an object turns in a circle, it must be subjected to a force pointing to the centre of the circle (without this centripetal force, the car would travel in a straight line). Denote the centre force FC. This central force is given by:

On a racing car, this force is provided by the tyres. The lateral force a tyre can deliver depends on the tyre compound and construction and on how hard it is pressed against the ground. An approximate equation for the maximum frictional force is:

In equation (2), the symbol n stands for the friction coefficient between tyres and ground, and N is the normal force on the tyres – how hard they are pushed against the ground. In mathematical terms, tyres with high grip have a high value of n.

The kinetic energy of the car, Ekin, is:

If the car slides or spins off, the driver usually locks the tyres, utilising the full friction. The energy absorbed is the work of the frictional force. Assuming the frictional force is constant, the frictional work Wfric is force times distance:

where d is the distance the car slides before it stops. These are the basic equations.

Relationship Between Impact Energy and Tyre Grip

When the car drives through the corner, the frictional force from the tyres equals the force needed: Ffric = Fc. Setting equation (1) equal to (2) gives:

Multiplying by R on both sides gives:

The right-hand side m V2 appears in the equation for kinetic energy, equation (3). Thus, in equation (3), m V2 can be substituted by n N R:

Recall that Ekin is the kinetic energy the car possesses when driving through a curve. This proves Mosley’s first statement: the energy of an impact is proportional to the grip of the tyre.

Distance a Car Slides Before It Stops

Imagine the car spins off, as shown in Figure 2. The car stops when all kinetic energy has been lost through frictional force. (Energy lost to aerodynamic drag is neglected in this simple model.) Setting the work of the frictional force, Wfric from equation (4), equal to Ekin from equation (7) gives:

By (2), n N can be substituted for Ffric:

The left-hand side is the energy loss during the spin, and the right-hand side is the kinetic energy before the spin. The product n N appears on both sides and cancels out. The sliding distance simply becomes:

This is Mosley’s second statement: the car stops exactly half the radius of the curve it was on when control was lost.

Separating Mechanical and Aerodynamic Grip

Only the gravity force contributes to the normal force on the tyres during a spin. The frictional force during braking is thus lower than the frictional force during cornering. This can be expressed mathematically as follows.

Before the spin, the normal force on the tyres, N, comes from two sources: the gravity force (weight) of the car and the aerodynamic downforce:

In this equation, m again symbolises the mass of the car, g is gravitational acceleration, and Faero is the aerodynamic downforce (the first term in (11) is sometimes called mechanical grip, and the second term is called aerodynamic grip).

During the spin, most of the aerodynamic downforce is lost. Therefore, it is reasonable to assume that all the vertical force on the tyres is simply the weight of the car:

Inserting equation (11) into the right side of equation (9) and (12) into the left-hand side gives:

The coefficient of friction n again appears on both sides and cancels out. Dividing both sides by n m g yields the distance the car spins before stopping:

The ratio between the aerodynamic downforce and the gravity force on the car, Faero/m g, can easily be in the order of 1-2 for a Formula 1 car (varying from one bend to the next depending on velocity). As an example, setting Faero/m g equal to unity and inserting into equation (14) gives:

This result shows that when aerodynamic force is included, the stopping distance is longer than when it is ignored (equation (10)). In this particular example, the stopping distance d (equation (15)) is twice as long as it would be without aerodynamic downforce (equation (10)). In conclusion, increasing aerodynamic downforce increases the spinning distance d.

Implications for Safety

Recalling Mosley’s argument: the faster a car is going when it starts to spin, the faster it will be at all points during deceleration. The following can now be added: the higher the downforce, the faster the car can drive through a curve. If it subsequently spins, the distance it slides before stopping is longer. Therefore, if the car hits a barrier on the way, it will hit the barrier harder.

Increasing Tyre Grip While Reducing Aerodynamic Downforce

Some drivers have suggested that mechanical grip should be increased (for example, by reintroducing slick and thus softer tyres) while aerodynamic grip should be decreased. In mathematical terms, this is equivalent to increasing n and decreasing Faero. If Faero is sufficiently low such that the product n N remains lower than before, the kinetic energy (equation (7)) may still be lower. Also, by equation (14), the stopping distance will be shorter when Faero/m g is smaller. The mathematics shows that the drivers are right: reducing aerodynamic grip and increasing mechanical grip will increase safety.

In addition, this approach is likely to allow closer racing, since current cars lose aerodynamic downforce when the airflow is disturbed by following another car closely. If most of the grip came from the mechanical grip of the tyres, this loss would not occur and closer racing would be possible.

Reducing aerodynamics and using slick tyres is the direction chosen by the Champ Car series. On superspeedways, a specially designed low-downforce, high-drag rear wing called the “Handford Wing” is mandatory. And does the Champ Car series not have more close racing than Formula 1?

List of symbols

d - distance the car spins before it stops

**g **- gravity acceleration

m - mass of the car

Ekin - kinetic energy of car just before spin

Faero - aerodynamical downforce

Fc - centre force required for making the car turn

Ffric - frictional force from the tires

**N **- normal force on tires

R - radius of curve the car drives through before spinning

V - velocity of car before the spin

Wfric - work of frictional force

n - friction coefficient between tires and ground