Wednesday, March 31, 2010

running power: introduction

Kenyan RunnersKenyan runners, the paradigm of form

There is a well-established relationship between cycling power and speed. Power is proportional to retarding force multipled by speed, or equivalently, work done is proportional to retarding force multiplied by unit distance traveled. The retarding force consists of the following:
  1. a rolling resistance component, proportional to weight multiplied by a "rolling resistance coefficient", which depends primarily on the tires and their air pressure
  2. a wind resistance component, which is proportional to the square of the the relative wind speed, to the air mass density, to the cross-sectional area of the cyclist and bike, and to a drag coefficient which is affected by the aerodynamic efficiency of the rider and equipment
  3. inertial force, which is proportional to the rate of change of speed.
In addition to the power dissipated to the retarding forces, power is lost in the drivetrain. This is typically modeled as a fraction of total power lost. As I've discussed before, this is a very crude approximation, as at higher powers a smaller fraction is lost to the drivetrain.

Anyway, what's lost in the model is the power associated with moving body parts around. With a bike, the body is relatively still, and while the legs are moving up and down, they're coupled, so one leg moving down while the other moves up allows the weight of the dropping leg push up on the rising leg, so in principal no work must be done by the muscles to sustain this motion. This is in part while cycling is so efficient.

Running is a different story. The legs are no longer coupled, so when I lift my foot at the rear end of my stride,my muscles need to actively work to make that happen. Even if my opposite foot is falling at the same time, the coupling between the two is likely fairly weak, so lifting and dropping the legs is going to take a toll. Similarly, each stride a runner's center of mass is moving up and down, much more so than with seated cycling. While a bouncing ball can move up and down a substantial number of times without external energy, humans don't bounce so well. So keeping that center of mass moving requires power from the muscles.

I'll post some equations next time...

2 comments:

Ron said...

If I'm thinking about this right, running legs must be strong endurance wise to give a little lift the COG everytime you land your feet on the ground, over the time period of say a marathon. I'm hoping this translates to some power on the bike. I know some friends who are both runners and cyclists and they are powerful riders. In your latest posting, you seem to joke that all your friends ride away from you on the hills lately. Could it really be all that running?

djconnel said...

Thanks for the comments, Ron! I would have posted to your Metrigear blog post by now but I've had problems with Chromium with blogger (I'm on Firefox at the moment).

My model is basically calculating how far the COM rises each step, and how far the foots/calves are raised. Not a very good one, though, as it doesn't predict how far these rise as a function of speed. But I'll post what I have.

I really do think it's the running. My legs feel stale all the time. Even when I am out of shape, my legs don't feel that way: I'm just slow and lack endurance, but typically will feel springy for at least awhile. Running just takes away my Z4-Z6. I can still fake a Z7 for a sprint, and I can grind away in Z2-Z3. But Z4-Z6: gone.