Cycle Rotations in the Tour Cycle Rotations in the Tour de Francede France

Dr. Michael L. LarsenDr. Michael L. Larsen

Relevant BackgroundRelevant Background

Tour de France is a “tour of France”Tour de France is a “tour of France”Held annually since 1903 (except for Held annually since 1903 (except for

during WWI and WWII)during WWI and WWII)Length varies by year.Length varies by year.

Shortest = 2420 kmShortest = 2420 kmLongest = 5745 kmLongest = 5745 km2013 = 3360 km (about average)2013 = 3360 km (about average)

More Relevant BackgroundMore Relevant Background

Race run in “stages” (21)Race run in “stages” (21)3 Time trials (very brief – 20-50 km)3 Time trials (very brief – 20-50 km)Other stages.Other stages.

2013 details:2013 details: 7 flat stages7 flat stages 5 hilly stages5 hilly stages 6 mountain stages6 mountain stages

2013 Route2013 Route

The Question:The Question:

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

MethodologyMethodology

2 Approaches2 ApproachesApproach 1:Approach 1:

Use the distance of the race, parameters of the Use the distance of the race, parameters of the bike, gear ratios, and wheel sizes to reason out bike, gear ratios, and wheel sizes to reason out how many times the pedals turned.how many times the pedals turned.

Approach 2:Approach 2:Use the winning total time and videos of racers to Use the winning total time and videos of racers to

determine how many rotations per second. determine how many rotations per second. Multiply the two to estimate answer.Multiply the two to estimate answer.

Distance Based ApproachDistance Based Approach

Total distance = 3360 kmTotal distance = 3360 km

Wheel size: 650-700 mm diameterWheel size: 650-700 mm diameter

Gears: Gears: High gear = 53 tooth chain on 11 tooth gear. High gear = 53 tooth chain on 11 tooth gear. Low gear = 39 tooth chain on 25 tooth Low gear = 39 tooth chain on 25 tooth

sprocket.sprocket.

Distance per rotationDistance per rotation

Largest possible:Largest possible: (53/11) x (pi) x (0.7m) = 10.6 meters/rotation(53/11) x (pi) x (0.7m) = 10.6 meters/rotation

Smallest possible:Smallest possible: (39/25) x (pi) x (0.65m) = 3.2 meters/rotation(39/25) x (pi) x (0.65m) = 3.2 meters/rotation

Method One – Crude BoundsMethod One – Crude Bounds

Total distance is 3360 km = 3.36 x 10Total distance is 3360 km = 3.36 x 1066 meters.meters.

Maximum total turns = (3.36 x 10Maximum total turns = (3.36 x 1066 m) / (3.2 m) / (3.2 meters/turn) = 1.05 million turnsmeters/turn) = 1.05 million turns

Minimum total turns = (3.36 x 10Minimum total turns = (3.36 x 1066 m) / m) / (10.6 meters/turn) = 317,000 turns(10.6 meters/turn) = 317,000 turns

Best estimatesBest estimatesAssume use something close to highest Assume use something close to highest

gear on flat ground and something close to gear on flat ground and something close to lowest gear on Mountain stages. (Very lowest gear on Mountain stages. (Very approximate).approximate).Downhill slopes?Downhill slopes?Not perfectly flat/sprinting areas?Not perfectly flat/sprinting areas?

Divide into two pieces; calculate amount of Divide into two pieces; calculate amount of turns on each piece and add them up.turns on each piece and add them up.

Mountain Stages 2013Mountain Stages 2013

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

Total mountain stages: 1080 km

Total non-mountain: 2280 km

Refined EstimateRefined Estimate

2280 km (flat) x (1000 m/km) x 2280 km (flat) x (1000 m/km) x (1 rotation/(1 rotation/99 mm) = 253,000 rotations) = 253,000 rotations

1080 km (mountain) x (1000 m/km) x 1080 km (mountain) x (1000 m/km) x (1 rotation/ (1 rotation/55 mm) = 216,000 rotations) = 216,000 rotations

Total = 253,000+216,000 = 469,000 ~ Total = 253,000+216,000 = 469,000 ~

½ a million rotations½ a million rotations

Method 2: Time-BasedMethod 2: Time-Based

Winners typically finish in about 90 hours Winners typically finish in about 90 hours of real cycling time.of real cycling time.

Youtube videos suggest that most cyclists Youtube videos suggest that most cyclists rotate about 1.1 cycles/second during rotate about 1.1 cycles/second during climbs and about 1.5 cycles/second during climbs and about 1.5 cycles/second during flat parts.flat parts.

Time-Based EstimateTime-Based Estimate

90 hours x (3600 sec/hour) = 324,000 sec90 hours x (3600 sec/hour) = 324,000 sec

324,000 sec x (324,000 sec x (1.35 rot/sec1.35 rot/sec) = 437,000 ) = 437,000 rotations totalrotations total

SummarySummaryBased on distances, wheel sizes, and gear Based on distances, wheel sizes, and gear

ratios we came up with a lower-bound of ratios we came up with a lower-bound of 320,000 rotations and an upper bound of 320,000 rotations and an upper bound of 1.05 million rotations.1.05 million rotations.

A more precise estimate based on A more precise estimate based on distances, wheel sizes, and gear ratios distances, wheel sizes, and gear ratios gave us a value of about 500,000 gave us a value of about 500,000 rotations.rotations.

Based on time, we estimate 437,000 Based on time, we estimate 437,000 rotationsrotations

Major Possible Errors or Erroneous Major Possible Errors or Erroneous AssumptionsAssumptions

Crude estimation on breakdown of gear by Crude estimation on breakdown of gear by stage type.stage type.Ultimate Impact on Result – Either DirectionUltimate Impact on Result – Either Direction

Assumed constant pedaling (no coasting) Assumed constant pedaling (no coasting) in distance-based calculation.in distance-based calculation.Ultimate Impact on Result – OverestimateUltimate Impact on Result – Overestimate

Assumed pedaling at all times in time-Assumed pedaling at all times in time-based calculation. based calculation. Ultimate Impact on Result – OverestimateUltimate Impact on Result – Overestimate

Final AnswerFinal Answer We are certain the final answer lies somewhere We are certain the final answer lies somewhere

between 250k rotations and 1M rotations.between 250k rotations and 1M rotations.

Using our two methods, we found a value Using our two methods, we found a value between 400k and 500k rotations, though most between 400k and 500k rotations, though most of the errors in these methods both result in of the errors in these methods both result in overestimation.overestimation.

Consequently, we believe the answer is likely in Consequently, we believe the answer is likely in the range 300,000 < X < 500,000 and certainly the range 300,000 < X < 500,000 and certainly in the range 250,000 < X < 1,000,000in the range 250,000 < X < 1,000,000

RefinementsRefinements

Extra distance travelled due to moving Extra distance travelled due to moving around other bikers (assumed negligible).around other bikers (assumed negligible).

More sophisticated/detailed treatment of More sophisticated/detailed treatment of gear ratios based on terrain. (Current gear ratios based on terrain. (Current treatment crudely handles downhill treatment crudely handles downhill sections)sections)

Better estimate of pedaling rate based on Better estimate of pedaling rate based on more video measurements.more video measurements.

Thank You For Your AttentionThank You For Your Attention

Any questions?Any questions?