# Trouble modeling runner’s swing-leg as double compound pendulum.

1. Nov 21, 2007

### tedjan

There is a problem in biomechanics that relates to a human runner’s gait (1) and how much power they consume. At each speed a runner chooses a Preferred Step Frequency or PSF (2). If an athlete tries to run with a step frequency that differs from their PSF at that speed, their oxygen consumption increases (3). There is no physics based model to explain this phenomenon.

I thought I could make headway on the problem by treating the swing leg, not the stance leg, as a double compound pendulum hung from the hip (1). Unfortunately, the top of the hip or pivot for the upper pendulum is not fixed. Instead, the hip pivot is being accelerated in the horizontal and vertical directions by the changing forces and angles of the stance leg (4). The accelerations are time dependent. They vary greatly while the other leg, the stance leg, is in contact with the ground.

I think that there are two ways of modeling the swing leg from its trailing position at toe off , thorough its position next to the stance leg at mid-stance to its forward swing position just before touch down. The swing leg can be treated as a double compound pendulum hung from the hip in two possible reference frames. In the laboratory reference frame it can be modeled as if it is hung from a jerking moving pivot. In the Center of Mass reference frame – the CM is close to the hip pivot – it can be treated as if it hung from a fixed pivot in a non-inertial accelerating CM frame.

I thought it would be easier to treat the swing leg in the hip or CM frame and apply the accelerations of the hip support directly to the masses as in m[1,2]*a[x,y].

Maybe this is the wrong approach, but I am having a great deal of difficulty with this problem. I can’t write the Legrangian because the potential energy is not constant since the vertical acceleration changes from zero at touch down to about 2-g at mid-stance. At the same time there is a horizontal acceleration on the masses. It increases from zero at touch down to a positive peak and then decreases to zero at mid-stance. It then does the reverse from mid-stance to toe-off.

I’ve been working on this problem for several months with little to show. I was able to push a fixed but flexed leg (5) through the swing cycle and confirm that the swing time for the fixed but flexed leg came very close to the swing time for the leg while running.

I would greatly appreciate any suggestions on solving this problem.

Ted

(1) http://members.aol.com/EasyExperiments/GaitCycle/GaitCycle.gif
(2) http://members.aol.com/EasyExperiments/PSFVsSpeed.gif
(3) http://members.aol.com/EasyExperiments/SmithAndHunter.gif
(4) http://members.aol.com/EasyExperiments/Accelerations.gif
(5) http://members.aol.com/EasyExperiments/BentKneeLeg.jpg