How to cycle as fast as Bradley Wiggins...

A special Tour de France guest post by Dr Phil Lightfoot - Department of Physics

Unfortunately for my employers, the best ideas always seem to come to me en route to and from work, while cycling to the University on my mountain bike.


Temporarily disconnected from the web, my thoughts are forced inward and my imagination begins to wander. Despite the fact that the design has failed to significantly evolve in over 100 years, bicycles still represent the most convenient and efficient means of human powered transportation.


Photo: Bradley Wiggins, by Surrey County Council.
Reproduced under a Creative Commons licence.
But why has bicycle design appeared to stagnate while the technology associated with motorcycles has consistently developed? What is the limiting factor and what can be done to overcome it? The answer regrettably is hitting cyclists in the face daily.

The average speed of winners of the Tour de France in the last 10 years is 24.9 mph. My highest recorded average speed along 8 miles of quiet country roads between the Physics department and a very disgruntled hungry cat stands at 19.5 mph. At first glance it may therefore seem within the realms of possibility that I could dedicate every fibre to pushing the envelope and bridging this small speed differential.

Sadly physics has other ideas.

Setting aside the world of lycra for a moment, let us consider petrol-powered transport:
  • A Bugatti Veyron sports car uses 987 horsepower (hp) to propel it to an impressive top speed of 267 mph;
  • But a Ford Fiesta needs only 99 hp to reach roughly half that speed. 
  • Perhaps the best evidence of basic physics at work was the top speed of 172 mph set by a 78 hp Moto Guzzi motorcycle in 1957 (it was almost 20 years before that speed was reached again in racing). 
  • This feat was attributed not to the power of its engine but to the aerodynamics of its design. 

Isaac Newton is largely responsible for deciphering the associated physics. Basic Newtonian mechanics states that at top speed the net force propelling a vehicle forwards must equal the net force dragging it in the opposite direction. Therefore in order to increase top speed without any increase in supplied power, the forces of drag must be reduced.

So, to relate that back to bicycles: drag associated with cycling is made up of rolling resistance and aerodynamic drag. The force Frr due to rolling resistance is given by the relatively intimidating expression (below) in which Wf and Wr refer to the weight acting through the front and rear tyres and P is the pressure within them. The constants a, b and c depend on a number of factors such as the road surface, bearing friction, tyre construction, surface temperature and the ‘knobbliness’ of the tread pattern.


The only good news is that rolling resistance can be significantly reduced by decreasing the weight of the rider and bicycle, increasing the tyre pressures, and reducing the contact patch made between the tyres and the ground. Tour de France racing tyres for example are triple the pressure and a third the width of the tractor tyres fitted to my bicycle. The riders too are approximately half my weight.

Aerodynamic drag is a result of the viscosity of air, a boundary layer forming around the bicycle which affects pressure distribution and disrupts the smooth laminar flow. This disturbance creates an area of high pressure directly in front of the rider - reducing forward velocity.

Alexander Vinokourov.jpg - Wikimedia Commons.
Used under a Creative Commons licence.
Perhaps a little surprisingly however, the shape of the rear of the bicycle/rider combination has a greater effect on drag than the shape of the front because of the turbulence created as disturbed air falls into the partial vacuum created. As is equally true for aircraft wings, the optimum shape in order to reduce aerodynamic drag is widest at 20% of its length and then tapers by 10 degrees from there rearwards. This explains some of the weird and wonderful shapes of helmets and bikes you see in professional time trials (it doesn't explain the colours however).

    
The figures above demonstrate airflow around (a) the torso of an upright commuter, (b) an unprotected head, and (c) the aero-helmets commonly used by time trial racers. Any sharp variation in profile leads to the onset of turbulence, the air unable to follow the contours of the shape generating a partial vacuum which further upsets flow. 

The total aerodynamic drag force in newtons is given by the following expression for which v is the speed in m/s and A is the projected frontal area in m2. The drag coefficient Cd  depends on the shape and surface smoothness of the object and the nature of the medium through which it travels. Cd  is 2.0 for a flat plate perpendicular to the air, 1.0 for a smooth ball, 0.3 for a sports car, 0.7 for a mountain bike and anything down to 0.2 for a race bike and professional rider.


VeloX3.jpg - Wikimedia Commons. Used under a Creative Commons licence.
Incidentally, it is always the case that a compromise must be reached between the drag coefficient and the projected frontal area, a streamlined flow necessitating a more bulbous shape. For example the current unassisted land speed record for a bicycle of 83 mph was set in 2013 by VeloX3, a recumbent design encased in a large aerodynamic faring.

This information provides us with a key insight. First however we must revisit Newton who determined the relationship between power P in watts, speed v in m/s and propulsive force Fdrive in newtons given below.


The current motor-paced record, in which a cyclist rides directly behind a powered vehicle shielding the rider from the effects of drag, stands at 167 mph. This indicates the dominance of aerodynamic drag. If we neglect rolling resistance and equate the forces of aerodynamic drag and propulsion we arrive at the following expression. Remember that if the propulsive and drag forces balance, there is no net force and so v will represent top speed.


The specific values aren't as interesting as the relationship between power and top speed which suggests that in order to double top speed, a rider must supply eight times the power. I can cycle at 20 mph and Wiggins and the other professionals can easily cycle at 25 mph; however that 5 mph difference between commuter and pro-racer corresponds to almost double the power output. Not even a grumpy cat is able to provide sufficient encouragement for me to span that divide!

Not the author's cat. Picture courtesy of www.catshaming.co.uk.

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