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| Mosley's Equations | |
| By Bent Sørensen and Svend Damborg, Denmark | |
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For safety reasons, FIA wants to prevent that cornering speed increases. There is an ongoing discussion on whether limiting the corning speed should be by cutting more and more grooves in tyres or by reducing aerodynamical downforce, e.g. by banning wings. FIA President Max Mosley has put forward several statements based on mathematics. We explain those arguments in detail and discuss the assumptions that are made in the analysis. By making a sharp distinction between aerodynamical grip and mechanical grip, we derive an equation that reveals that reducing the aerodynamical downforce in fact leads to more safety. Basic mathematical equations To understand Mosley's arguments, it is necessary to start with the governing mathematical equations. Consider a racing car that drives through a corner, for simplicity a circle. Let the letter V stands for the velocity of the car, let R be the radius of the circle (see Figure 1) and m symbolises the mass (weight) of the car. Old Isaac Newton showed that if an object turns in a circle, it must be subjected to a force pointing to the centre of the circle (if there was no centre force the direction of car would be a straight line). Denote the centre force Fc. This central force is given by
On a racing car this force is of course provided from the tyres. The side force that 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 m stands for the friction coefficient between tyres and the ground and N is the normal force on the tyres; how hard they are pushed against the ground. In the mathematical language, tyres with high grip have a high value of m. The energy that the car possess due to its velocity is called the kinetic energy, Ekin. It is
If the car slides or spins off, the driver usually locks up the tyres, such that he utilises the full friction of the tyres. The energy absorbed is the work of the frictional force. As an approximation, assume that the frictional force is constant all the time. Then the frictional work Wfric is force times distance
where the symbol d is the distance the car slides before it stops. These are the basic equations. Now we combine them. Relationship between energy of impact and grip of the tyres When the car drives through the corner the frictional force from the tyres are equal to the force needed, Ffric = Fc. Setting equation (1) equal to (2) gives
Multiplying by R on both sides gives
We note that the right hand side m V2 appears in the equation for the kinetic energy, equation (3). Thus, in equation (3) we can substitute m V2 by m N R. This gives
Recall that Ekin is the kinetic energy the car possesses when it drives 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 now that the car spins off, as shown in Figure 2. The car stops when all the kinetic energy has lost by the frictional force. Actually, energy is also lost to aerodynamical drag. This is neglected in the simple model described here. Setting the work of the frictional force of the tyres, Wfric from equation (4), equal to Ekin from equation (7) gives
By (2) we can substitute m N for Ffric. This gives
The left hand side is the energy loss during the spin, and the right hand side is the kinetic energy the car possesses before the spin. The product m N appears on both sides, so it cancels out. The sliding distance simply becomes
This is Mosley's second statement: The car stops exactly half the radius of the curve it is on when control is lost.
Before the spin, the normal force on the tyres, N, comes from two sources, the gravity force (weight) of the car and the aerodynamical downforce. This can be written mathematically as
In this equation m again symbolises the mass of the car, g is the gravity acceleration and Faero is a symbol for the aerodynamical downforce (used in connection with equation (2), the first term in (11) is sometimes called the mechanical grip, and the second term is called aerodynamical grip). During the spin, most of the aerodynamical downforce is lost. Therefore, it is reasonable to assume that all the vertical force on the tyres is the weight of the car
We proceed to insert equation (11) for the right side of equation (9), and (12) into the left hand side of equation (9). The result is
The coefficient of friction m again appears on both sides and cancels out. By dividing both left and right hand sides by m m g we find the distance the car spins before it stops
The ratio between the aerodynamical downforce and the gravity force on the car, Faero/m g, can easily be in the order of 1-2 for a Formula One racing car (it varies from one bend to the next, depending on the velocity). As an example, set Faero/m g equal to unity. Inserting this into equation (14) gives
This result shows that when aerodynamical force is included the distance to stop the car is longer than if the aerodynamical force is ignored, equation (10). In fact, for this particular example the distance to stop the car, d, (equation (15)) is now twice as long as if there was no aerodynamical downforce (equation (10)). In conclusion, increasing the aerodynamical downforce increases the spinning distance, d. Recall Mosley's argument: The faster a car is going at the point it starts to spin, the faster it will be at all points during deceleration. We can now add the following: The higher the downforce, the faster the car can drive through a curve. If it subsequently spins, the distance it slides before it stops is longer. Therefore, if the car hits a barrier on the way, it will hit the barrier harder.
Increasing tyre grip and reducing aerodynamical downforce? Some drivers have suggested that the mechanical grip should be increased (e.g. by re-introduction of slick and thus softer tyres), while the aerodynamical grip should be decreased. In mathematical terms this is equivalent to an increase in m and a decrease in Faero. If Faero is sufficiently low, such that the product m N remains lower than before, the kinetic energy (equation (7)) may still be lower than before. 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 the aerodynamical grip and increasing the mechanical grip will increase safety. In addition, this approach is likely to allow closer racing, since the current cars looses their aerodynamical downforce, because the airflow is disturbed when following another car closely. If most of the grip came from the mechanical grip of the tyres, this loss of grip would not appear and closer racing would be possible. Reducing aerodynamics and using slick tyres is the direction chosen in the Champ Cars series. On super speedways a specially designed low-downforce, high drag rear wing, called "Handford Wing", is mandatory. And... does not the Champ car series have more close racing than Formula One?
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Authors Bent Sørensen is 37 years old and has a Master of Science and a PhD in mechanical engineering. His specialization is solid mechanics. On the side, Sørensen creates Formula One technical drawings. His drawings are published in motorsports magazines worldwide. Some of his work can in the third edition of Sal Incadela's book "The Anatomy & Development of the Formula One Racing Car from 1975" (1990). Svend Damborg is an auto mechanic. In the 70's, he worked with Denmark's leading racing drivers and participated in numerous races, including nine Monte Carlo Rallies.
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| Bent Sørensen and Svend Damborg | © 1999 Atlas Formula One Journal. |
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