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# Drop test physics IV

Tuomas Pöysti 2021

In the previous part, I managed to extract displacement data from a load cell dataset sampled from a small drop test. The original data was something like this:

And the displacement data looks as follows (both have time in seconds as the horizontal axis):

## The lowest point

What have we learned this far? I think one of the interesting things is that my center of gravity dips some 23 mm below the zero level. This is (part of) the stretch in the system that allows for the energy to be absorbed.

By the way, notice that the lowest point is reached a bit later than the maximum peak of force (or acceleration). The greatest acceleration is 31.5 m/s^2 @ 0.00 s and the lowest point of my center of gravity is -23 mm @0.03 s. Not sure if this is just inaccuracy, but as far as I know, this would be perfectly possible.

In fact, it even makes sense. There certainly is a spring component and a damper component in the system, and the total “braking” force is their sum. The spring part by definition reaches it’s maximum at the lowest point of displacement, but the damper part is proportional to velocity, which is naturally drops as the lowest point gets closer.

If this is the case, the greatest forces are experienced on the way down, not while reaching the lowest point. The difference is just a curiosity, though.

## Where is the spring?

How did the cowstail manage to stretch over 20 mm or almost 5%? I guess it did not. The system has to be seen as a whole. In the picture below, the system on the left represents a rigid climber attached to a rigid cowstail. The next one considers the cowstail a spring, but assumes a rigid climber – this would be a sufficient model for the DMM’s widely cited article. In the third one, the climber’s mass is reduced closer to a center of gravity, which is attached to the cowstail spring by another spring. This secondary spring is effectively my body, in a complex way.

I guess the diagram might be a bit leading, since I draw the cowstail spring as way stiffer looking than the one inside my body. I’m just assuming this is the case – not sure if I felt the same if I had done more core exercises!

While picturing a harness-supported human body flexing so that it’s center of gravity shifts downwards, one again has to remember to see the system as a whole. Of course the leg loops may bury themselves deeper into the softness of the climber’s thighs, but there’s probably a lot more to it.

For example: the human head is around 8% of the whole body weight. That is, if one managed to bow their head 10 cm down, it would result in their center of gravity shifting about 8 mm down. Consider the shoulders, arms, legs and all the loose soft tissue doing the same!

I am going to study this some more, and I’ll be surprised if I’m to find anything else than that the spring was me, practically speaking.

## Power

Back to basic physcs! Power is the rate of energy (or energy’s derivative over time, if we want to drop some impressive terms). The SI unit of power is W (watt). As we already knew, energy is the product of force and displacement. Since velocity is the derivative of displacement over time, power is

P = Fv

Or as a plot:

This shows that the cowstail-body combination has brief maximum power of 3kW while arresting the fall. What is this piece of information good for? Nothing, probably.

## Force vs displacement

This is the one I was waiting for the most, the plot of force against displacement. I think it’s lovely:

It points out the hysteresis in the system, showing how the energy is dissipated. The two straight portions resemble some kind of spring-like behavior. The trianglular area in the force-displacement plane equals an 87.7 J work:

But how much was my kinetic energy? That’s easy now that we have velocity data and know my mass (almost too well at this point):

The maximum total kinetic energy is about 138 J. It seems the 87.7 J chunk is over 64% of the whole kinetic energy to be absorbed! That is, if this data is correct, most of the kinetic energy was absorbed before my center of gravity reached the zero level of displacement. As we defined in the earlier parts, this is the level that the center of gravity settles at the end of the data, after the oscillating has ended.

## The displacement oddity

Doesn’t that sound a bit weird? In fact, how can my center of gravity have the same displacement several times during the test, with the force varying from 0 to almost 2.5 kN?

A simple answer is that there’s an error in the calculations. But I think we don’t need to go there, yet.

This might be explained by the same “think it as a whole” mantra. My center of gravity probably moves a lot with respect to the harness, whether I’m falling or just slowly getting suspended by the harness. While doing the test, I dropped myself by hanging on a crimp block that was next to my load cell anchor. That is, my arms and shoulders were high, and my back was probably quite vertical. Most likely the first thing that happens when the cowstail starts to catch is them shifting downwards and back.

It must be possible to shift one’s center of gravity quite much by taking different poses while supported by a harness. Think about flexing your knees and so on.

Whatever this kind of tests ever suggest about falling on a cowstail, there will always be a great deal of variance in postures and the body’s readiness to take the hit.

## Further studies

Depending on how interesting I find this in the future, I’ll probably be testing and posting. If you’d like to learn more, please ask me or otherwise show that this matters, it may have a huge effect on how much I care! I generally do this thing solely for myself, so it’s completely possible that tomorrow I’m more interested about ethics or cooking or running and leave this for years.

I already know that I should conduct a control test using a rigid mass. Also, it might be useful to have a quick release link to drop me, that might change how the center of gravity shifts. Also, that way it would be possible to record the exact moment of release and to have control over drop height.

And when it comes to drop height, I need to be able to take a bit harder falls. Maybe find someone brave to show me how it’s done!