# Thread: Trying to physically verify equation for Spring Mass model

1. ## Trying to physically verify equation for Spring Mass model

I wanted to experimentally verify the step frequency’s (SF) dependence on Body Mass (BM). It appears in an equation I use to estimate leg-stiffness (kleg) for scenarios that I run through the model.

The expression is for the frequency of a mass bouncing vertically on a spring. This is similar to a subject running in-place with close to zero aerial phase.

The equation,

SF=(1/(2*PI))*SQRT(kleg/BM)

implies that if the BM increases, the SF should decrease according to 1/SQRT(BM).

If a subject while running very slowly, adds 20% to their BM using a weight jacket, then their step frequency should decrease by SQRT(1/1.2) or 8.7%. .

I have tried this myself, but I don’t have access to a weight jacket. I could only hand-hold an additional 10% of my BM while I ran slowly. Then, my SF dropped by 3%; I expected it to decrease by 5%.

I have looked for research on this issue, but I could not find any. Has anyone ever worked in this area? Or, are there any published articles on this topic?

Ted

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## Re: Trying to physically verify equation for Spring Mass model

I would guess it's because the weights are not near the centre of mass, Ted. You are, in effect, using halteres:

3. ## Re: Trying to physically verify equation for Spring Mass model

Chris, that's fascinating - I had never known about the jumping weights before but it reminds me of a training run I did years ago for the Pike Peak marathon - I found that I could run downhill super fast on the hairpin course by carrying a gallon jug of water and swinging it every time I went around a curve so I never had to slow down to stay on the track around the bends. Wonderful, Thank You!

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## Re: Trying to physically verify equation for Spring Mass model

Ha! I would like to see a video of that, Edi. And surely beer would work even better than water?

5. ## Re: Trying to physically verify equation for Spring Mass model

I ran for years with the Tampa Bay Hash House Harriers but that was just a vacation ... very fun going from running at sea-level to running around about 8,000 feet for a week. Almost all the serious Pikes Peak running made it up there before me but I beat just about everyone on the way back down. I ran with a water jug because that way I still had a full jug when I reached the end ...

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## Re: Trying to physically verify equation for Spring Mass model

Ted,

Could your leg stiffness have increased by 3.5% to go with the increase in mass of 10% to give a resultant 3% increase in SF?

Cheers
Allan

7. ## Re: Trying to physically verify equation for Spring Mass model

Chris: Thank you for the video link. Marcus Essa is truly an impressive athlete. When I did my initial experiments I held the hand weights tightly against my ilium, so there would only be a small but --- I am sure present -- halters effect.

Allan: I am not sure that bio-mechanical leg-stiffness could be related to the BM. All the physical systems I have dealt with followed Hooke’s Law. That and Newton’s equations are at the foundation of the spring mass model.

SFvsBM.png

I redid this experiment several times. With a weight belt tightly secured around my ilium, I able to get SF measurements (blue diamonds) that agreed with the theoretical values (red squares). I was surprised that the results were so close – experiments have NEVER worked-out that way in the past!

If anyone attempts do this experiment, they may want to use an 8-pocket weight-belt with 2 lb. (4.4 Kg) weights. They should also add extra supports or blocks under their tread-board to keep it from flexing during the stance phase. I performed my experiment at 3 mph (1.33 m/s, 20 min/mile, 12.4 min/km) just above the walk-run transition speed.

Ted

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## Re: Trying to physically verify equation for Spring Mass model

Hi Ted
I wondered if you had looked at this thread on the site and if so what you thought of the ideas -

Ground reaction forces and gait

I wonder if shortening the effective arm length during gait would increase step frequency -just a hunch . With regard to your first post ,would holding weights reduce arm swing amplitude compared to a non weight holding set up .

I don't have the expertise or equipment to look the idea of pelvic acceleration feeding energy into the bodies pendulums (arms legs ) to produce body segment co-ordination but if you did it could be interesting .

Regards

Gerry

9. ## Re: Trying to physically verify equation for Spring Mass model

Hi Gerry,

Yes I had followed at that very interesting discussion. Physics and modeling mechanical systems is my area, so part of discussion was beyond me.

The SM model functions in the sagittal plane, so it does not include rotational considerations about the vertical axis. I could not guess on the coupling between arm length and step frequency. I believe that the hip-hugging weight belt avoided that issue.

Ted

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## Re: Trying to physically verify equation for Spring Mass model

Hi Ted, We did not address this issue directly, but we do present some data in the paper below that may be of help to you. We had a variety of conditions where we varied body weight and body mass independently during treadmill running (the bodyweight increase condition would match closest to what you are doing, I believe). Best of luck on an interesting problem!

Chang YH, Huang HW, Hamerski CM, Kram R. (2000) The independent effects of gravity and inertia on running mechanics. J Exp Biol 203: 229-238.
http://jeb.biologists.org/content/203/2/229

cheers,

Young-Hui

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