Balancing swords

Let's talk about balancing swords. I can't stand swords that aren't properly balanced.

First off, a confession- this could really be titled- 'Digging around for why grip weighting is superior to pommel weighting'. Because I've never met a pommel balanced sword that felt right, regardless of where the center of mass was. And really, this is me digging to figure out why.

Second- all of the calculations basically assume the person is the dominant source of force on the weapon. This is not true with heavier styles- since gravity accelerates the same regardless of mass, heavier styles focus on taking advantage of the inertia gravity gives them. If you do similar calculations with gravity as the main focus, you end up with completely different results.

Now, with that out of the way, let's proceed. Physics is the tool we want to use for this investigation. Unfortunately, for me, physics was years ago, and I barely remember this stuff. Feel free to correct me if I'm wrong. Remember, this blog is primarily for me to figure out stuff, and other's people consumption is secondary.

The physics of a sword, are thankfully, fairly simple. Quite like the toy problems given by teachers, where a force is applied to a rod. However, most toy problems given by teachers have the pivot point fixed. Which isn't the case here- a sword has no fixed pivot point -it changes all the time. So what do we do?

The decomposition is this- into movement that moves the center of mass, and into movement that rotates the center of mass. That's Translational(moving the center of mass of the sword) and rotational(turning it).  If you are pivoting at a place other than the center of mass, according to physics, you are doing more than just rotating it- you are doing a combination of rotation, and translation.

Translational motion:

Now, translational motion, is a part of every attack, as you shift the center of mass of the sword towards an opponent.

It seems fairly simple to optimize for:
Inertia = sum of the masses
Less inertia means greater speed with the same amount of force.  People build the lightest swords they can in order to get less inertia.

However, there is another factor here- how far your applied force is from the center of mass. You see, some of your applied force will go towards rotation, and some towards translation. The closer you are to the center of mass, the more force goes to translation over rotation. This is how you can get a faster sword, by adding weight.

Rotational motion

Exchanging a vertical sword, for a horizontal one can be very powerful. Guard for reach. Normally, this is done while striking. The reverse is done while recovering. Have another picture.

Obviously to maximize the utility of rotation, you'd like the center of mass to be towards the pommel, rather than the tip, so you'll get a better increase in reach when you rotate it.

There is another reason to want it down there, but I'm a little fuzzy on the specifics of it-
While your arm speed isn't actually constant, if you pretend it is, then the closer you are to the center of mass, the more rotation you can impart. Now, why would we want to pretend our arm speed is constant? That's where I'm a little fuzzy, and just sort of handwave and say your arm has significant inertia.

Finally let's talk about the actual rotation. Rotational inertia specifically. That's how hard it is to turn the thing. The formula is: Rotational Inertia = sum of (mass * (distance to center of mass)2). So we'd like to minimize that too. We can minimize this by putting all of the weight we have to have, as close as possible to the center. (Which is why if you tuck everything in when you spin, you go faster, despite having the same energy)

Center of mass:

So, both rotational, and translation work better when we have the center of mass, down towards the hilt. This is why we weight weapons- to bring the center of mass closer to the hilt.
So, what is the center of mass? It's the place where the following is true:
Sum of (masses on the left * distance from point) = Sum of (masses to the right * distance from point)
So, that's how this thing balances- the weight on one side, cancels out the weight on the other. And the further weight is away from the center, the more it counts.

So if we want to move the center of mass, we are going to be adding mass. The question is where?

If you want to minimize your inertia(and maximize your translational movement) then you want to pile all your counterweight on the pommel, so you could use less of it to move the balance.

However, that strategy is horrible in terms of rotational inertia- rotational inertia is based on the square of the distance. In order to minimize rotational inertia, you should pile your counterweight in a crossguard. There will be considerably more of it than if you put it in your pommel, but your sword will spin faster.

Which is better?

So do we keep our inertia low, so we have better translational movement, or do we keep our rotational inertia low, so we have better rotational movement? Which is more important? I'm betting that answer is a big fat depends. So what does it depend on?

Obviously style has something to do with it. If one is more valuable for your style, you'll definitely want to favor that type of movement.

Now, lighter vs heavier weapons. Surely that affects where you want to weight the weapon? Not really. Basically you are just multiplying all of the masses by a consistent amount, and that doesn't really change anything. (That is, until you factor in gravity, which we aren't).

What does have a huge effect is the length of the weapon, vs the length of your arm.
Translational gets you about an arm's length of motion, give or take being off balance. Rotation on the other hand gets you about a sword length's of motion, give or take the location of the center of mass.

So longer weapons favor rotation, shorter favor translation. Which is true, up to a point. However, the distance^2 term in rotational inertia means much more when there is more distance. Which is why spears mostly stab, instead of hack.

So the question is, which is larger- your arm, or your blade?