Friday, January 22, 2010

drivetrain losses: same speed, different gear

Last two times I looked at the results on drivetrain efficiency from having the same gear ratio, riding at the same power. This went to the question of whether it was better to ride a chainring with more or less teeth, choosing a rear cog to match cadence at the same power and speed. The answer was for low power/high cadence, you're better off in a low chainring, while at high power/low cadence you're better off with a bigger ring.

A lot of issues go into gear choice, obviously. Ted Huang told me he put a 27-tooth cog on so he could ride the Cat's Hill Criterium, which has a climb of around 15% (video from 2009 here) so he wouldn't need to front-shift at the top. The model suggests he may have been better off doing so anyway, although it fails to penalize cross-chaining in any way (consistent with the experimental evidence, which isn't very sensitive to the matter).

Ted Huang @ Cat's Hill (2007)Hmmm, Ted, that doesn't look like the big ring to me...

Anyway, another option is I'm climbing in a given chainring and I want to choose which gear to be in. Am I better grinding @ 40 rpm in a 36/12 or spinning @ 87 rpm in a 36/26? Now obviously here there are issues of physiological efficiency. I can't produce as much power at 40 rpm as I can at 75 rpm, for example. But this just considers drivetrain efficiency.

Since I know I can't maintain such good power in these low and high gears, I'm going to reduce my climbing power to 200 watts.
K = 3.37 mm
Kd = 0.0024 J/rev
T0 = 95.6 N
P = 200 watts
36/11 with 12T pulley @ 36.67 rpm: loss = 8.315W (4.157%)
36/12 with 12T pulley @ 40.00 rpm: loss = 8.039W (4.019%)
36/13 with 12T pulley @ 43.33 rpm: loss = 7.824W (3.912%)
36/14 with 12T pulley @ 46.67 rpm: loss = 7.659W (3.829%)
36/16 with 12T pulley @ 53.33 rpm: loss = 7.437W (3.718%)
36/17 with 12T pulley @ 56.67 rpm: loss = 7.367W (3.684%)
36/19 with 12T pulley @ 63.33 rpm: loss = 7.290W (3.645%)
36/21 with 12T pulley @ 70.00 rpm: loss = 7.275W (3.638%)
36/23 with 12T pulley @ 76.67 rpm: loss = 7.307W (3.654%)
36/26 with 12T pulley @ 86.67 rpm: loss = 7.418W (3.709%)

So here's an interesting result: at the lowest cadence, chain speed is low, meaning tension is high (power is the product of tension and chain speed), and drivetrain losses are high. Spicer says high tension is a good thing, right? But Spicer also says bigger rear cogs are a good thing, and Spicer was measuring each of his gears with the same tension, or different tensions at the same gear and cadence. Here's a "real-world" example: tension and cadence both depend on what gear you choose. Spicer was basically changing the "grade" of his "hill" every time he shifted in the rear. This is the point which was made by Walton and Walton using Spicer's data directly.

So despite the fact as I shift down from that ridiculous 36/11 tension is decreasing, efficiency increases. The chain is bending less at the rear cog. This only goes so far, though. Eventually, the rear pulleys start to have more of an influence, and the loss bottoms out at 36/21. Going to the 23 or 26, efficiency starts decreasing again: at the lowest gears, chain speed is increasing, increasing losses, while total tension and chain bending are changing proportionately less.

So the model produces interesting results. Neither the biggest nor the smallest gear is the best choice. Fortunately this minimum in drivetrain loss is close to my preferred climbing cadence, which is around 75 - 80 rpm. So in this example, there's no trade-off.

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