Dante di Pietro wrote:Anyone actually find any errors?

While his calculations don't seem to be off, but I'm not convinced the equations used are correct. I'm not a mechanical engineer, so I've haven't dealt with inertia calculations in 30 years. So his calculations of 85lbs and 24lbs don't mean much to me except that 85/24=3.5 or the head will experience 3.5 times more acceleration without a helm than with it. His padding calculations completely ignore time. Since padding distributes force over time, it's a great omission.

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"...Instead, we are almost exclusively dealing with concussions that result from diffuse axonal injury.

{pic here Image from:

http://www.neuroscientificallychallenge ... sary/axon/}

Diffuse axonal injury is mainly the result of rotational acceleration of the head because this kind of motion creates shearing forces that cut and damage axons throughout the brain. ... any attempt to protect ourselves from concussions must prevent rotational acceleration of the head."

Technically to protect the brain, we need to reduce the rotational acceleration of the head {not prevent rotation} to a below the threshold where shearing forces become dangerous*. This is important as the author tends to swap ideas about energy/momentum and force within his discussion. For example, if the head is accelerated just above the shearing threshold, doubling the mass of the head will reduce the acceleration by about half, or below the threshold of damage.

* technically the acceleration would have to be applied over a long enough period of time for parts of the axon would move enough for the shearing forces to take effect, but this is a minor point.

Using the law of conservation of momentum "the total momentum of the two objects before the collision is equal to the total momentum of the two objects after the collision". For inelastic collisions, the accelerations & forces can be high; this is pool balls colliding situation. When elastic collisions are involved the force must be summed over time, and this is calculus.

As a practical example, we know that F = ma (force equals mass times acceleration). And v=at (velocity equals acceleration times time). These assume that force is constant; therefore acceleration is constant. So to reach 60 mph, you can either floor it, and experience a lot of acceleration or just touch the gas, and slowly accelerate up to 60mph. {well my wife can slowly accelerate up to speed while I fidget}

In other words, substituting v=(F/m)t So for a car to reach 60 mph, and mass doesn't change, Ft must be a constant, or if F is halved, it must be applied for twice as long. Padding changes from the first (relatively inelastic collision/high force/high acceleration) to the second (more elastic collision; lower force, etc). Time becomes very important in these calculations.

Another way to look at it is, if I take a race car with an engine, I can get high accelerations. If I load it up with sand bags (increase mass) it will accelerate slower with the same engine. I can still get a high velocity* eventually, but not the high accelerations.

* although top speed is also reduced

So to absorb the energy in a moving weapon, the time it takes to transfer that energy is important. Without examining this in time domain, you cannot calculate the shearing forces on the axons.

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So lets look at two situations, one without a helm and with a helm, looking immediately before and after impact. Let's ignore the elastic collision caused by the bending of the rattan, compression of skin, etc. Since the times are so close together, we can assume that motion is linear {not rotational} and unaffected by the centripetal force exerted by the neck. Let's also assume that just after impact, the weapon and head move at the same speed (technically this isn't a necessary or correct assumption, but it's close enough for analysis outside the time domain).

Without a helm: mv1 + M0 = (m+M)v2 -- conservation of momentum

where m=mass weapon, M=mass head, v1=velocity weapon, v2=velocity (weapon and head), M's initial velocity is zero.

The head (M) experiences an acceleration to get up to v2

With a helm: mv1 + (H+M)0 = (m+H+M)v3 where H=mass helm; H+M initial velocity is zero

Since mv1 is the same, (m+M)v2 = (m+H+M)v3

(m+M) < (m+H+M) v2 must be greater than v3

The head (M) experiences less acceleration to get up to a lower speed v3

More specifically, if H = (m+M), then (m+M)v2 => Hv2; (m+H+M)v3 => (H+H)v3 or 2Hv3

then Hv2 = 2Hv3 or v2 = 2*v3 or v2 is twice v3; the head accelerated to half the speed because of the increased mass. Assuming the energy transfer happened over the same time period (a bad assumption, but using inelastic collisions), the head would experience half the acceleration. Lower accelerations means less damage/concussions.

Intuitively we know this because heavier helms require a different blow calibration than lighter helms.

This is only an analogy as we're ignoring the time domain analysis, but the overall effects are similar. Adding mass decreases acceleration, which decreases the shearing forces/concussions.

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Padding:

Hopefully by now you agree that force over time is more important than inelastic collision force calculations he used. So we need to look what happens over time.

The weapon hits the helm, and we'll assume an inelastic collision (or instant acceleration of the helm). So if the helm weighs 3 times the weapon, the weapon speed is reduced to one quarter of what it was. Conservation of momentum {wv1 = (w+3w)v2 or wv1=4wv2 or (wv1)/4w=v2 or v1/4=v2} So the mass of the helm has slower the overall speed.

As the weapon and helm start to move into the padding, a small force is created. {IIRC the force increases linearly with compression, or it takes twice the force to compress twice the distance ... at least initially.} Anyway only a small counter force from the head is needed for that fraction of time. This small force starts small acceleration of the head, and the padding is compressed a little bit. An equivalent counter force slows the weapon/helm a small bit.

The next fraction or time, the head is moving away from the padding (at a small velocity), but the weapon/head is moving faster. So the padding compresses more, but not twice as much because the head is moving away. So a greater force is generated, but not a great force. Since the force is lower, the acceleration is lower with less chance of a concussion. The head accelerates some more, and the weapon/helm slows some more.

This cycle repeats until the weapon speed equals the head speed.

To visualize this, imagine five square blocks stacked vertically. The height is the force, and the width is time. The force is 5 units applied over 1 unit of time. The instantaneous acceleration of the unpadded head is proportional to the force, or 5 units of acceleration.

<greatly oversimplified> Padding spreads the force over time (as described above). We can tip the five blocks over, so 1 unit of force is applied over 5 units of time. Since acceleration is proportional to force, the head accelerates at 1 unit, or one fifth the unpadded rate. -- five units is just used as a representation. Technically, the force will vary over time, not be constant. The force function would have to be integrated over time for momentum.

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Newton's first law: a body in motion will remain in motion ...

So if the head is accelerated, why doesn't it stay in motion? Where does the energy go?

The neck muscles (and some other muscles technically) exert a counter force. Muscles "absorb" the energy (technically the chemical energy in the muscles is converted into kinetic energy counter to the weapon direction). Helm mass and padding reduce the velocities give the muscles more time to react and REDUCE movement/accelerations/shearing forces/concussions. ###this is also why blindside hits are more dangerous###

So if your muscles are able to generate sufficient force at the correct time {and nothing rips}, your head wouldn't move ... no motion means no acceleration ... no acceleration means no shearing forces in the axons or no concussions. Do your neck strengthening exercises.

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So who wants to make a bunch of assumptions about rates and write up the integrals?

Notes:

elastic (inelastic) collision is used as a collision that stores (does not store) energy within the components which is released over time