1. Cycle Rotations in the Tour de France Dr. Michael L. Larsen

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

3. More 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 stages 7 flat stages 5 hilly stages 5 hilly stages 6 mountain stages 6 mountain stages

4. 2013 Route

5. The Question: “How many times do the pedals turn for the Tour-de-France winner”

6. Methodology 2 Approaches 2 Approaches Approach 1: 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. 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: Approach 2: Use the winning total time and videos of racers to determine how many rotations per second. Multiply the two to estimate answer. Use the winning total time and videos of racers to determine how many rotations per second. Multiply the two to estimate answer.

7. Distance Based Approach Total distance = 3360 km Total distance = 3360 km Wheel size: 650-700 mm diameter Wheel 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 sprocket. Low gear = 39 tooth chain on 25 tooth sprocket.

9. Distance 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

10. Method One – Crude Bounds Total distance is 3360 km = 3.36 x 10 6 meters. Total distance is 3360 km = 3.36 x 10 6 meters. Maximum total turns = (3.36 x 10 6 m) / (3.2 meters/turn) = 1.05 million turns Maximum total turns = (3.36 x 10 6 m) / (3.2 meters/turn) = 1.05 million turns Minimum total turns = (3.36 x 10 6 m) / (10.6 meters/turn) = 317,000 turns Minimum total turns = (3.36 x 10 6 m) / (10.6 meters/turn) = 317,000 turns

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

12. Mountain 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

13. Refined Estimate 2280 km (flat) x (1000 m/km) x (1 rotation/9 m) = 253,000 rotations 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 1080 km (mountain) x (1000 m/km) x (1 rotation/5 m) = 216,000 rotations Total = 253,000+216,000 = 469,000 ~ ½ a million rotations Total = 253,000+216,000 = 469,000 ~ ½ a million rotations

14. Method 2: Time-Based Winners typically finish in about 90 hours of real cycling time. 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. Youtube videos suggest that most cyclists rotate about 1.1 cycles/second during climbs and about 1.5 cycles/second during flat parts.

15. Time-Based Estimate 90 hours x (3600 sec/hour) = 324,000 sec 90 hours x (3600 sec/hour) = 324,000 sec 324,000 sec x (1.35 rot/sec) = 437,000 rotations total 324,000 sec x (1.35 rot/sec) = 437,000 rotations total

16. 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. 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. 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 Based on time, we estimate 437,000 rotations

17. Major Possible Errors or Erroneous Assumptions Crude estimation on breakdown of gear by stage type. Crude estimation on breakdown of gear by stage type. Ultimate Impact on Result – Either Direction Ultimate Impact on Result – Either Direction Assumed constant pedaling (no coasting) in distance-based calculation. Assumed constant pedaling (no coasting) in distance-based calculation. Ultimate Impact on Result – Overestimate Ultimate Impact on Result – Overestimate Assumed pedaling at all times in time- based calculation. Assumed pedaling at all times in time- based calculation. Ultimate Impact on Result – Overestimate Ultimate Impact on Result – Overestimate

18. Final Answer We are certain the final answer lies somewhere between 250k rotations and 1M rotations. 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. 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 < X < 500,000 and certainly in the range 250,000 < X < 1,000,000 Consequently, we believe the answer is likely in the range 300,000 < X < 500,000 and certainly in the range 250,000 < X < 1,000,000

19. Refinements Extra distance travelled due to moving around other bikers (assumed negligible). 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) 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. Better estimate of pedaling rate based on more video measurements.

20. Thank You For Your Attention Any questions? Any questions?