Wednesday, May 27, 2015

Effect of variability in rolling resistance coefficient on cycling power

I looked at how grade variability affected average power when climbing a hill. Honestly I thought the result was going to be larger, but the reality was it was a relatively minor effect. When the hill is very gradual, for example 1%, variations in grade of a certain fraction have little effect on speed. When the hill is very steep variations in grade are more significant, but since they increase power only via wind resistance, and wind resistance is relatively unimportant (assuming still air), again variations in grade have little effect. It's only important in the middle ground where speeds are high enough that wind resistance is relatively important but where grade variations have a relatively large influence on speed.

A virtually equivalent logic applies to rolling resistance variation. A variation in rolling resistance about an average value (averaged over distance) will have the same effect as a variation in grade by the same absolute amount.

So the effect of variability in the coefficient of rolling resistance, by a given fractional amount, can be written almost by inspection by swapping rolling resistance and grade in the previous formula:

ΔP/P = 3 (σCRR / CRR)2 × αwCRR (1 - αw) / (1 + 2 αw) ]2

This will generally be less than the effect of fractional variations in grade, assuming grade is generally larger than CRR.

The definitions are the same as last time:

fmacceleration(grade + CRR)×gravitymass (and speed)-proportional power
fwmass/distance1/2 ρCDA2speed-cubed-proportional power
CRR1coefficient of rolling resistance (for example 0.4%)
CD1coefficient of wind resistance (for example 80%)
ρmass/volumemass-density of air (for example 1.1 kg/m3)
Aareaeffective cross-sectional area of bike + rider (for example 0.4 meters2)
αRR1CRR/(CRR + grade)fraction of mass-proportional power due to rolling resistance
αw11 / (1 + fmm / fws2)fraction of power due to wind resistance
αm11 / (1 + fws2 / fmm)fraction of power due to mass-proportional power

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