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Mosley's Equations
By Bent Sorensen and Svend Damborg, Denmark
I found this great article by accident during my everyday browsing on atlasf1 website.
It throwing new light on some, until now, very logical things. Look's like Big Mosley and his team of "experts" don't think about some things in proper way. Enjoy reading!
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 looses is aerodynamical downforce, and thus grip. Here we explain the mathematics and critically discuss why we feel the FIA (ex) President's interpretations are incorrect.
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 tires or by reducing aerodynamical downforce, e.g. by banning wings. FIA (ex) 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 symbolizes 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
| mV2 | ||
| Fc= | -------- | (1) |
| R |
On a racing car this force is of course provided from the tires. The side force that a tire can deliver depends on the tire compound and construction and on how hard it is pressed against the ground. An approximate equation for the maximum frictional force is
| Ffric= | nN | (2) |
In equation (2) the symbol n stands for the friction coefficient between tires and the ground and N is the normal force on the tires; how hard they are pushed against the ground. In the mathematical language, tires 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
| Ekin= | 1 m V2 | (3) |
If the car slides or spins off, the driver usually locks up the tires, such that he utilizes the full friction of the tires. 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
| Wfric= | Ffric d | (4) |
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 tires
When the car drives through the corner the frictional force from the tires are equal to the force needed, Ffric = Fc. Setting equation (1) equal to (2) gives
| mV2 | ||
| n N= | ------- | (5) |
| R |
Multiplying by R on both sides gives
| n NR= | m V2 | (6) |
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 n N R. This gives
| Ekin= | 1/2n N R | (7) |
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 tire.
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 tires, Wfric from equation (4), equal to Ekin from equation (7) gives
| Ffric d= | 1 n N R | (8) |
By (2) we can substitute n N for Ffric. This gives
| nN d= | 1 n N R | (9) |
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 n N appears on both sides, so it cancels out. The sliding distance simply becomes
| d= | 1 R | (10) |
This is Mosley's second statement: The car stops exactly half the radius of the curve it is on when control is lost.
| Separating effects of mechanical and aerodynamical grip To derive the equations above, it was assumed that the friction force from the tires, Ffric, was the same all the time. Let us take a closer look at that assumption. How well does it fit to reality? A car that spins off onto grass or gravel may experience a smaller friction than when it ran on the tarmac, simply because the friction between rubber and tarmac is higher than between rubber and grass. If the friction during the spin is lower, the sliding distance increases, i.e. d exceeds 1/2 R. Another very important issue is aerodynamics. Before the car spins, it fully utilizes the aerodynamical downforce created by the underbody and wings. The normal force on the tires is higher than what would result for a car with no downforce. Therefore the maximum frictional force (equation (2) will be higher. But when the car spins, the direction of airflow will no longer reach the wings from the frontal direction. Also, the downforce that is produced by the underbody and diffuser at the rear end of the car is lost (it is well known that the underbody aerodynamics is very sensitive to the ride height). As a result, most of the aerodynamical downforce disappears during a spin. |
|
Figure 1. A racing car that drives through a circle having a radius R at a velocity V requires a central force Fc. As long as the tires roll, they deliver a side force Ffric (shown in blue color). |
|
Figure 2. The racing car spins off when the required centre force Fc exceeds the maximum side force that the tires can provide. The car spins off a distance d before it stops. When the tires are locked up, the frictional force from the tires Ffric (blue color) now acts in the opposite direction of sliding. |
Only the gravity force contributes to the normal force on the tires. The frictional force during breaking is thus lower than the frictional force during cornering. This can be brought into the mathematics as follows.
Before the spin, the normal force on the tires, N, comes from two sources, the gravity force (weight) of the car and the aerodynamical downforce. This can be written mathematically as
| N= | m g +Faero | (11) |
In this equation m again symbolizes 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 tires is the weight of the car
| N= | m g | (12) |
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
| n mgd= | 1/2n (m g+Faero) R | (13) |
The coefficient of friction m again appears on both sides and cancels out. By dividing both left and right hand sides by n m g we find the distance the car spins before it stops
| d= | 1 (1+Faero/m g)R | (14) |
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
| d= | 1/2 (1+1)= R | (15) |
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 tire 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 tires), while the aerodynamical grip should be decreased. In mathematical terms this is equivalent to an increase in n and a decrease in 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 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 tires, this loss of grip would not appear and closer racing would be possible.
Reducing aerodynamics and using slick tires 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?
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
