The complex case of aerodynamics and weight in racing
Moderator: robbosmans
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Sounds interesting. Does it include power through the pedal cycle as a varying input?
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Right, so;
Built an RK4 cycling simulator. Tested the following;
70kg rider, 10kg bike, 1kg of which is rotating mass@ diameter 622mm.
350W average output.
12% grade.
CDa 0.35
Air density 1.25kg/m^3
Crr 0.006
Starting at a speed of 12.1968KPH (which was chosen as it is very close to the finishing speed of the constant output model, which is 12.197kph, and therefore it removes most of any transient effects)
Running 40,000 steps to simulate 8 seconds.
Three cases were compared, perfect continuous output, 60RPM sinusoidal input, and 30RPM sinusoidal input.
Comparison is by final energy state - kinetic + gravitational potential energy. This is because the sine wave power cases are always winning on speed and losing on position, or vice versa, depending on the moment you check them at, so to pull out the overall win, you need to combine both, and that's by comparing built up energy (that hasn't been wasted to losses).
Without further ado, for the 459J of starting KE+ 2800J over 8s of input power in all 3 cases, exit condition is:
Continuous pedalling: exit speed is 12.197KPH, elevation is 3.2293m. Energy is 2993.46J.
60RPM cadence: exit speed is 11.854kph, elevation is 3.2582m. Energy is 2990.72J. Corresponds to ~0.34w of waste over 8s.
30RPM cadence: Exit speed is 11.509KPH, elevation is 3.2862m. Energy is 2987.80J. Corresponds to ~0.64w of waste over 8s.
This seems like a lot. And it doesn't make sense that this case shows a bigger disadvantage than what was calculated by hand yesterday, with all the pessimistic assumptions. So I investigated further: If I change the phase of the opening pedal stroke by 90 degrees (and adjust for the slightly raised average power), we get:
Continuous pedalling: Exit at 12.198kph, 3.2311m. 2994.9851J
60Hz: 12.210kph, 3.2299m. 2994.9878J.
Why is it suddenly ever so slightly MORE efficient?
Well, at first I thought discretization errors might be to blame. And maybe they are. But I get similar results down to a 0.000025s timestep - 40,000 steps for a single rotation of the cranks.
But then I realised, whoever exits with the lowest elevation and highest speed, has had the lowest AVERAGE speed, and thus the lowest wind drag. but is ABOUT to cop some wind. This is a function of the 8s duration, which is small compared to the speed swings. But if I turn up the duration to a few minutes I increase the discretization errors, and that does become problematic. So, I upped the duration to 40s and also dumped the cadence back to 15RPM. This magnifies the speed swings, but lowers the rate of change to reduce discretization error. We are now swinging back and forth between ~13.4 and ~10.6kph.
Now we finish 40s with 13128.857J, compared to 13132.5J for the continuous case - 0.091w over 40s. I switch back to the original phase and run 40s at 60 cadence - and we get about 2J of difference which is 0.05W.
So what have we learned? That while there are some interesting academic takeaways, the final effect is fractions of a watt regardless, somewhere between vanishingly small, irrelevantly small, and negligible
Built an RK4 cycling simulator. Tested the following;
70kg rider, 10kg bike, 1kg of which is rotating mass@ diameter 622mm.
350W average output.
12% grade.
CDa 0.35
Air density 1.25kg/m^3
Crr 0.006
Starting at a speed of 12.1968KPH (which was chosen as it is very close to the finishing speed of the constant output model, which is 12.197kph, and therefore it removes most of any transient effects)
Running 40,000 steps to simulate 8 seconds.
Three cases were compared, perfect continuous output, 60RPM sinusoidal input, and 30RPM sinusoidal input.
Comparison is by final energy state - kinetic + gravitational potential energy. This is because the sine wave power cases are always winning on speed and losing on position, or vice versa, depending on the moment you check them at, so to pull out the overall win, you need to combine both, and that's by comparing built up energy (that hasn't been wasted to losses).
Without further ado, for the 459J of starting KE+ 2800J over 8s of input power in all 3 cases, exit condition is:
Continuous pedalling: exit speed is 12.197KPH, elevation is 3.2293m. Energy is 2993.46J.
60RPM cadence: exit speed is 11.854kph, elevation is 3.2582m. Energy is 2990.72J. Corresponds to ~0.34w of waste over 8s.
30RPM cadence: Exit speed is 11.509KPH, elevation is 3.2862m. Energy is 2987.80J. Corresponds to ~0.64w of waste over 8s.
This seems like a lot. And it doesn't make sense that this case shows a bigger disadvantage than what was calculated by hand yesterday, with all the pessimistic assumptions. So I investigated further: If I change the phase of the opening pedal stroke by 90 degrees (and adjust for the slightly raised average power), we get:
Continuous pedalling: Exit at 12.198kph, 3.2311m. 2994.9851J
60Hz: 12.210kph, 3.2299m. 2994.9878J.
Why is it suddenly ever so slightly MORE efficient?
Well, at first I thought discretization errors might be to blame. And maybe they are. But I get similar results down to a 0.000025s timestep - 40,000 steps for a single rotation of the cranks.
But then I realised, whoever exits with the lowest elevation and highest speed, has had the lowest AVERAGE speed, and thus the lowest wind drag. but is ABOUT to cop some wind. This is a function of the 8s duration, which is small compared to the speed swings. But if I turn up the duration to a few minutes I increase the discretization errors, and that does become problematic. So, I upped the duration to 40s and also dumped the cadence back to 15RPM. This magnifies the speed swings, but lowers the rate of change to reduce discretization error. We are now swinging back and forth between ~13.4 and ~10.6kph.
Now we finish 40s with 13128.857J, compared to 13132.5J for the continuous case - 0.091w over 40s. I switch back to the original phase and run 40s at 60 cadence - and we get about 2J of difference which is 0.05W.
So what have we learned? That while there are some interesting academic takeaways, the final effect is fractions of a watt regardless, somewhere between vanishingly small, irrelevantly small, and negligible