Part 2: Rolling Resistance
Some astute readers might note there’s another factor besides drag to consider, and that’s rolling resistance from the tires.
The formula for rolling resistance is:
F = Crr*N
Coefficient of rolling resistance (Crr) and normal force (N) are both essentially constant for the point of this exercise, unless you plan on riding to the moon or losing quite a bit of weight during your ride or getting a new wheel with a new tire from the team car. If you recall from earlier:
p = F*V
Let us again, be lazy, and assign arbitrary values of 1 mystery units to Crr and N, so that simply:
p = V
Note that power here is denoted by p, because I don’t know where subscripts are. P from the previous post and p here are both power, but in different arbitrary units. In order to convert p to P, I’ll introduce a new conversion factor, constant a.
P = ap
Adding in the aerodynamic drag component, substitute V for p, we get:
P = aV + V^3
Where a is our conversion factor, V is speed in whatever units, and P is our arbitrary power units that are definitely not watts.
Now we need to determine the value of a to establish the relative relationship between the rolling resistance component (aV) and aerodynamic drag (V^3). We can do so by determining they they intersect and solving for a.
aV = V^3
a = V^2
Now we just need to know V where rolling resistance is equal to aerodynamic drag, and we can skip all the other math and estimating we would have had to do. There’s a few numbers thrown out for when tire rolling resistance and aerodynamic drag are even, oft repeated seems to be 15mph. I can only assume this is because someone misread a graph without unit labels that actually said 15km/h. Schwalbe puts it at around 18km/h (11.2mph). BikeCalculator seems to put it at around 15.1km/h (9.4mph) using default values. Both of these may be right, as the relationship between drag and rolling resistance depends on the bike, rider and tires. But they give a good ballpark figure.
Neither of these models includes drive train losses, but calculating them doesn’t really change the conclusion so I’ll skip it.
The equation for arbitrary power units for Schwalbe is:
a = (11.2)^2 = 125.44
P = 125.44V + V^3
The equation for arbitrary power units for default values for BikeCalculator is:
a = (9.4)^2 = 88.36
P = 88.36V + V^3
And since bike calculator gives us a speed of approximately 15mph for 100w, we can actually come up with a conversion factor to turn our arbitrary power units P into watts.
W ~= (V*9.4^2 + V^3)/47
Which should come within a watt of BikeCalculator with default values.
On flat land with no wind, at a speed of 7.5mph, which is equivalent to 34×32 at 90rpm, about the lowest gear you will find on a road bike, if you increase the gear ratio at a given cadence by 10% you require an additional 19.0% power. At 33.9mph, equivalent to 53×11 at 90rpm, if you increase the gear ratio by 10% you require 31.9% more power. This might be closer to the result you were expecting from the previous post on drag. That seems like quite the difference at first glance, until you realize these speeds are much slower and faster than most people will go on windless flats.
If we narrow it down to normal cruising speeds of amateur recreational riders to 15-20mph and run the same calculations with 15mph and 20mph, we get 26.6% and 28.9% respectively. Much much closer together.
Consider again, that our cog choices must be in integers and often times a limited selection of integers, and we can not arbitrarily increase gear ratio by 10% in real life unless you have a NuVinci.