Cycle Rotations in the Tour de France

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Cycle Rotations in the Tour de France

• Dr. Michael L. Larsen

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Relevant Background

• Tour de France is a tour of France
• Held annually since 1903 (except for during WWI
  and WWII)
• Length varies by year.
• Shortest 2420 km
• Longest 5745 km
• 2013 3360 km (about average)

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More Relevant Background

• Race run in stages (21)
• 3 Time trials (very brief 20-50 km)
• Other stages.
• 2013 details
• 7 flat stages
• 5 hilly stages
• 6 mountain stages

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2013 Route
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The Question

• How many times do the pedals turn for the
  Tour-de-France winner

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Methodology

• 2 Approaches
• Approach 1
• Use the distance of the race, parameters of the
  bike, gear ratios, and wheel sizes to reason out
  how many times the pedals turned.
• Approach 2
• Use the winning total time and videos of racers
  to determine how many rotations per second.
  Multiply the two to estimate answer.

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Distance Based Approach

• Total distance 3360 km
• Wheel size 650-700 mm diameter
• Gears
• High gear 53 tooth chain on 11 tooth gear.
• Low gear 39 tooth chain on 25 tooth sprocket.

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Distance per rotation

• Largest possible
• (53/11) x (pi) x (0.7m) 10.6 meters/rotation
• Smallest possible
• (39/25) x (pi) x (0.65m) 3.2 meters/rotation

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Method One Crude Bounds

• Total distance is 3360 km 3.36 x 106 meters.
• Maximum total turns (3.36 x 106 m) / (3.2
  meters/turn) 1.05 million turns
• Minimum total turns (3.36 x 106 m) / (10.6
  meters/turn) 317,000 turns

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Best estimates

• Assume use something close to highest gear on
  flat ground and something close to lowest gear on
  Mountain stages. (Very approximate).
• Downhill slopes?
• Not perfectly flat/sprinting areas?
• Divide into two pieces calculate amount of turns
  on each piece and add them up.

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Mountain Stages 2013

• Stage 8 (176 km)
• Stage 9 (165 km)
• Stage 15 (242 km)
• Stage 18 (168 km)
• Stage 19 (204 km)
• Stage 20 (125 km)

Total mountain stages 1080 km Total
non-mountain 2280 km
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Refined Estimate

• 2280 km (flat) x (1000 m/km) x (1
  rotation/9 m) 253,000 rotations
• 1080 km (mountain) x (1000 m/km) x
  (1 rotation/5 m) 216,000 rotations
• Total 253,000216,000 469,000 ½ a
  million rotations

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Method 2 Time-Based

• Winners typically finish in about 90 hours of
  real cycling time.
• Youtube videos suggest that most cyclists rotate
  about 1.1 cycles/second during climbs and about
  1.5 cycles/second during flat parts.

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Time-Based Estimate

• 90 hours x (3600 sec/hour) 324,000 sec
• 324,000 sec x (1.35 rot/sec) 437,000 rotations
  total

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Summary

• Based on distances, wheel sizes, and gear ratios
  we came up with a lower-bound of 320,000
  rotations and an upper bound of 1.05 million
  rotations.
• A more precise estimate based on distances, wheel
  sizes, and gear ratios gave us a value of about
  500,000 rotations.
• Based on time, we estimate 437,000 rotations

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Major Possible Errors or Erroneous Assumptions

• Crude estimation on breakdown of gear by stage
  type.
• Ultimate Impact on Result Either Direction
• Assumed constant pedaling (no coasting) in
  distance-based calculation.
• Ultimate Impact on Result Overestimate
• Assumed pedaling at all times in time-based
  calculation.
• Ultimate Impact on Result Overestimate

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Final Answer

• We are certain the final answer lies somewhere
  between 250k rotations and 1M rotations.
• Using our two methods, we found a value between
  400k and 500k rotations, though most of the
  errors in these methods both result in
  overestimation.
• Consequently, we believe the answer is likely in
  the range 300,000 lt X lt 500,000 and certainly in
  the range 250,000 lt X lt 1,000,000

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Refinements

• Extra distance travelled due to moving around
  other bikers (assumed negligible).
• More sophisticated/detailed treatment of gear
  ratios based on terrain. (Current treatment
  crudely handles downhill sections)
• Better estimate of pedaling rate based on more
  video measurements.

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Thank You For Your Attention

• Any questions?