Holes in the F1 regulations

Once again, Formula One has tied itself into something of a knot over the question of holes. As with the double-diffuser controversy of 2009, the conflict concerns the very definition of a hole. This time, the argument pertains to the slot in the step-plane which Red Bull have incorporated in front of the rear wheel. As Craig Scarborough explains, this arguably contravenes the following regulation:

"All parts lying on the reference and step planes, in addition to the transition between the two planes, must produce uniform, solid, hard, continuous, rigid (no degree of freedom in relation to the body/chassis unit), impervious surfaces under all circumstances."

Teams such as Sauber have achieved much the same effect as Red Bull's slot, but do so by deforming the boundary of the step plane, not by creating a hole. In this respect, there is a very clear topological definition of a hole, which goes as follows:

A surface possesses a hole if there is loop through any point on the surface which cannot be continuously shrunk to a point.

If you draw a loop around a hole, then the hole prevents the loop from being shrunk to a point. This is a concept from algebraic topology, and in technical terms one can say that a surface has a hole (or holes) if the zeroth homotopy group (the 'fundamental group') is non-trivial, i.e., contains more than the identity element. A surface with a trivial fundamental group is said to be simply connected.

The slot on the Red Bull clearly prevents a loop around it from being shrunk to a point, while any loop drawn on the step-plane of the Sauber can still be shrunk to a point.

However, the actual regulation fails to provide such a clear, topological definition of a hole. In fact, it draws upon a variety of quite different concepts. The step-plane must be "uniform" (a statistical concept); "solid" (a tricky concept in physics, sometimes distinguished from a fluid by the symmetry group of the medium, or by the constitutive relationship of the medium, i.e., the expression relating stress to strain); "hard" (a concept usually defined in terms of the local elastic modulus of the material, or its resistance to indentation tests); "continuous" (a topological concept, but one which is ultimately an idealisation, given that all materials are composed of discrete particles); "rigid" (usually meaning a high elastic modulus, but here qualified by "no degree of freedom in relation to the body/chassis", so actually meaning "rigidly attached"; and "impervious", a concept pertaining to the transmissibility of other media through a material.

None of these concepts actually capture the intended requirement for the absence of holes. In particular, in topological terms, a surface with holes can be perfectly continuous. Continuous surfaces can be simply connected (without holes) or multiply connected (with holes).

As Craig Scarborough also points out, there is another regulation intended to prohibit discontinuities of all types, in all but the outer 50mm of the floor:

“In an area lying 650mm or less from the car centre line, and from 450mm forward of the rear face of the cockpit entry template to 350mm forward of the rear wheel centre line, any intersection of any bodywork visible from beneath the car with a lateral or longitudinal vertical plane should form one continuous line which is visible from beneath the car.”

Whilst all the teams, Red Bull included, seem to accept that this regulation prohibits the existence of discontinuities in all but the outer 50mm of the floor, one could argue that this is still open to interpretation. If an aperture is placed on a lateral vertical plane, created by a raised lip in the step-plane, so that the hole is not visible from below, then the intersection of a longitudinal vertical plane will still trace out a continuous line when viewed from beneath the car (i.e., when projected down onto a horizontal plane).

In conclusion, rather than adopting such a scatter-gun approach to the regulations, it might be better if the regulations defined a hole in topological terms, and then derived the empirically applicable criteria from the general definition.

A general theory of Pirelli's 2012 F1 tyres

The volatility of this season's Formula 1 season has been attributed to the temperature sensitivity of Pirelli's 2012 Formula 1 tyres. All of the teams are finding it difficult to keep the tyres in their optimum temperature band between 85 and 100 degrees C.

I'd like to propose a theory which explains what is happening. As I see it, there are two key facts, outlined by Mark Hughes in this week's Autosport, which need to be explained:

(1) Ferrari, Williams and Sauber tend to run their tyres hotter than Lotus, McLaren and Red Bull.

(2) In the past, when a tyre overheated, it would lose grip, and the temperature would then fall back into the operating band. This year, when a tyre overheats, it never recovers. Hence, a negative thermal feedback process has been replaced with a positive feedback process.

My theory depends upon the following facts about the physics of tyres:

(i) Tyre grip is generated by two mechanisms, sometimes referred to as physical grip and chemical grip. The first process involves the shear deformation of the contact patch, whilst the second involves the coefficient of friction of the tyre. The internal stress response to shear deformation depends upon the shear modulus of the tyre, which is temperature dependent, and the friction coefficient is dependent on both tyre temperature and slip velocity, (as depicted in the image below from Michelin's tyre modelling efforts). So both mechanisms by which grip is generated, are temperature dependent.

(ii) Tyre temperature is generated in a tyre by the deformation it undergoes, and by the friction associated with tyre slip.

The key difference between this year's Pirellis and last year's, is that the 2012 tyre has a flatter contact patch. I hypothesise that what this has done is to change the balance between the heat generated by deformation and the heat generated by friction, in favour of the latter. This can explain our two key facts:

Firstly, if we accept that Red Bull, Lotus and McLaren have more downforce than Ferrari, Williams and Sauber, then the former teams will tend to suffer from less tyre slip than the latter teams. This explains why the former teams now run their tyres cooler than the latter trio.

Secondly, we can now explain why thermal degradation is a positive feedback process. When a tyre overheats and loses grip, it slides more. When the balance between slip-generated heat and deformation-generated heat has been tipped in favour of the former, a car which slides more will generate ever hotter tyres, leading to runaway thermal degradation. In contrast, in 2011, when a tyre lost grip, the deformation reduced, hence the temperature returned to the optimum operating band.

It's just a theory...

Formula One's Goldilocks Enigma

Goldilocks was hungry. She tasted the porridge from the first bowl.
"This porridge is too hot!" she exclaimed.
So, she tasted the porridge from the second bowl.
"This porridge is too cold," she said.
So, she tasted the last bowl of porridge.
"Ahhh, this porridge is just right," she said happily and she ate it all up.
(Goldilocks and the Three Bears)

Something curious is happening with the tyres in Formula One this year. At first sight, it appears that if you have too much downforce, then you'll put too much energy into the tyres, and they'll suffer thermal degradation. Conversely, if you have too little downforce, then the car will slide around and/or fail to warm its tyres sufficiently. If, however, your level of downforce is 'just right', then you'll be able to optimise your speed over a race distance. It might be suggested that teams such as Genii and Williams find themselves in exactly this zone of habitable downforce.

The truth, however, will be far more complex than this. For a start, teams such as McLaren appear to move in and out of the habitable zone depending on track temperature and car set-up. And it may be that Williams and Genii were smart enough to realise at an early stage the importance of working backwards from the characteristics of the Pirelli tyres to a specification of mechanical and aerodynamic requirements.

It was Peter Wright, I think, who pointed that, unlike aircraft, the control surfaces of a racing car are provided, not by the wings, but by the tyres. The ultimate logical development of this is not merely sophisticated, team-specific tyre modelling software, but simulation tools which integrate such tyre modelling with CFD.

Now more than ever, it is the interaction between the solid visco-elastic control surfaces, the forces generated by the viscous fluid flowing over the solid geometry of the car, and the transmission of those forces by visco-elastic spring-damper systems, which will become the key to extracting maximum performance.

Jenson's rise and fall

According to the Office for National Statistics, this figure plots the number of babies named 'Jenson' each year in the UK since 1996.

Remarkably, the nominalistic statistics actually track Jenson Button's fluctuating fortunes in Formula One. There's an initial burst in 2000 as he makes his Williams debut, then a plateau through the wilderness years at Benetton, until another sharp rise as he takes on the Ferraris with one hand in 2004, (the other holding his helmet on).

There's a slight dip in 2005 as Jenson tumbles to ninth in the drivers' championship, but a commensurate gain follows in 2006 as Jenson takes his first Grand Prix victory.

In 2007 and 2008 the popularity of the name takes a sharp dive as Frome's fastest struggles with the lethargic Honda. This, however, is merely a brief furrow before a precipitous escarpment of Jensonian designation in 2009 and 2010, as he wins the World Championship, and makes a winning start at McLaren.