Tuesday, September 1, 2009

Coast down estimation of drag and rolling resistance - Part 1

So I promised some geekery here. The goal was to do a simple test and try to estimate rolling resistance and drag coefficients. The simplest thing anyone can do is a coast down: you accelerate to speed, stop pedaling, and coast to a halt (or close to it.) So, to do the analysis, you first need to solve the equation that describes the motion:

ρ is the density of air
crr is the coefficient of rolling resistance
CdA is the drag coefficient multiplied by the frontal area of bike and rider
m is the mass of rider and bike
g is the acceleration due to gravity
v is the velocity of rider and bike

You can use separation of variables to get a rather gnarly looking expression that describes the coast down behavior of a rider in a constant position:

where I've defined:

I went out and did several coast-down tests in two positions (straight arm in drops, hands on tops,) and covering the same course forward and backward. The reason for this is that and slope will look like a change in rolling resistance, so if you take the average rolling resistance derived from the two directions, that should cancel out any error due to slope. It does NOT cancel out wind, so I did this at midnight, which is typically fairly wind-free. I downloaded the data from the powertap and used the nonlinear fitting function in Excel (blah :() to find values for crr, CdA and t0. I then used R to see if there was a statistical difference in rolling resistance and drag between the two positions I used, and to estimate the error in those values. Next time, I'll show some data.