Saturday, April 3, 2010

running power: part 2

Last time I came up with an analytic model for the power "wasted" in running: the power required to bob the center of mass up and down, and to life the feet and calves during each foot strike.

Unfortunately, like so many models, this one is almost useless. There's two big problems:
  1. There is no predictive model for α, the fraction of time both feet are off the ground, and
  2. there is no predictive model for hfoot, the height to raise the feet, and
  3. the optimal cadence calculation is simplistic.

Now it's true my calculation of optimal cadence matched precisely my preferred cadence, that's only because I set α to get this result. α = 70% is a fairly high fraction of the time off the ground, and the result of over 200 "wasted" watts (watts in excess of those you'd get on a bicycle, where these power terms are not present) is obviously a high number.

The model predicts as α drops, optimal cadence drops in proportion. But of course this isn't true in practice. α = 0 corresponds to walking, and you don't walk with zero cadence. The problem is I set a fixed value of hfoot, the amount to raise the feet. Simple physics suggests hfoot should be as small as possible: drag the feet skimming the ground. But the Kenyans don't run that way in the photo, so obviously there are inefficiencies to running like that which I am not capturing.

The late Ed Burke says that according to Edward Coyle, runners use around 1 J/kg/meter, assuming standard gravity of 9.8 m/sec², and assuming a metabolic efficiency of around 1 J / cal. However, this neglects wind resistance and the effect of gaining or losing altitude. So these need to be added.

Wind resistance is easy: it's the same for a runner as it is for a cyclist.

Altitude is another matter: not so obvious. For a cyclist, there is a terminal velicity on a downhill of a given slope at which the forces are in balance, and the rider will coast at a steady speed. For a runner, this obviously is not the case. Running downhill may be easier than running on level ground, but it takes work, no matter how steep the slope. Similarly, running uphill is clearly more effort than running on the flat. But it may be difficult to calculate how much harder: suppose when running, the center of mass rises, then the center of mass falls. It's possible running uphill is more efficient, for example.

I can model this by changing the relationship between α, C, and t. The body spends less time dropping than it does rising, because it's rising more than it is dropping when running uphill. But modeling the center of mass relative to the ground instead of relative to a fixed altitude, we can include the power going into the "climb" separately, and the time rising versus falling relative to the ground is again the same. The equation becomes:

h = ½ g t² ‒ v t1 grade,

where v is the runner speed, and grade is the linear approximation to the road grade. I can then proceeed as I did last time. But what is already seen in this equation is that when running uphill (positive grade), h will be less, and efficiency improves, while for running downhill, will be more, and efficiency decreases, all for the same time spent ballistic.

But I'll also need a model for hfoot. That may change with road grade. And I don't have that model, so it's hard to proceed further. More information is needed.

The Coyle model as represented by Burke is obviously oversimplistic. But running's a lot more complex than cycling, and taking it further is a nontrivial task, to say the least.

1 comment:

djconnel said...
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