## Sunday, November 1, 2009

### Effect of wind and bike speed fluctuations on climbing power: Introduction

Even when climbing steep hills, wind resistance is a considerable contributor to total power. Wind resistance force, in the absence of a cross-wind, is typically modeled as proportional to the square of the relative speed of the wind relative to the bike. To get power, you multiply this force by the bike speed. The standard model, neglecting drivetrain efficiency (we assume drivetrain losses are proportional to transmitted power):

pw = ( CDA ρ / 2 ) (ssws,

where s is the bike speed, sw is the speed of the wind (positive for headwind) relative to the Earth's surface at the typical rider height (not 10 meter height, as is typically reported), CDA is the coefficient of drag times effective cross-sectional area of the rider and bike, and ρ is the air density. Too much detail: all we care about is there's a coefficient for wind resistance, which we assume is held constant, fw:

pw = fw ( ssws

Now, assuming we know the average value of s and the average value of sw, we can estimate the average value of pw and therefore total power (assuming we can model other components, specifically climbing power and rolling resistance power). But what if s and/or sw are varying?

Of course, on any real-world climb, grade varies, and at constant power, when the grade is reduced, the speed s increases. Or power varies at a constant grade, and again s varies. And sw varies, either due to changing wind itself, or changing shelter from the wind, for example the proximity of terrain, trees, houses, or other riders. Or the road isn't perfectly straight, so the component of the wind in the direction of bike motion changes. Reality is far messier than is convenient.

Before answering this question, consider the effect of speed variations on climbing and rolling resistance power. One is proportional to weight × speed × road grade, the other proportional to weight × speed × rolling resistance coefficient (assuming the standard model, for modest road grades). So they're basically indistinguishable. But in either case, if we know the average speed, since these power components are proportional to speed, we know the average power. Power fluctuations don't affect the result.

There is the whole issue of "micro-accelerations" which people argue cause increased power: accelerating a bike once takes a certain amount of energy, so accelerating it multiple times takes multiple doses of this energy, right? Well, yes if the reason we need to accelerate again is we dumped energy into our brakes. But for a typical climb we don't brake (I'm assuming no braking in this analysis) and these "microaccelerations" where power is increased are balanced by "micro-decelerations" where we gain the benefit of that power. The energy which went into acceleration during a climb is proportional to the difference in the squares of the final speed and the initial speed. What the speed did in between, whether increasing steadily or oscillating around a bit, doesn't matter.

It only matters when analyzing losses to the wind. And this is because the dependence of wind losses on bike speed and on wind speed are not simply linear, they also have seoond-order and, in the case of bike speed, third-order components. When bike speed or wind speed increases, the associated power components increase a lot. But when bike speed or wind speed decreases, these components decrease not quite as much. So fluctuations add more than they subtract from power. In other words, you need to know more than just the average bike and wind speed to model the power. The assumption of constant speeds for the bike and wind are always optimistic: it predicts the lowest possible value for average wind resistance power.

In other words, if I estimate the power it took to do a climb in a certain amount of time, and I assume constant speed for the bike and the wind, I will only underestimate that power. And underestimating my power makes me cranky.

So the point of this analysis is to find a relationship between the magnitude of fluctuations of bike speed and wind speed and the average power required to overcome wind resistance.

Okay, enough introduction. More to follow.