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Vincent Le Chevalier

Location: Paris, France
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PostPosted: Wed 27 Aug, 2008 4:17 am    Post subject: Plotting the dynamic balance of weapons         Reply with quote

Hello all!

Following up some recent discussions, I've come up with a new way to visualize the differences in dynamic balance of swords and other weapons, that is simple enough to explain in a (long-winded Wink) post. It's really something I should have done earlier; the goal here has been to remain as objective as possible.

I'll assume a certain familiarity with the concept of pivot points in the following. For a very good primer on what they are this is possibly the best post to read: (the "FOR ROTATION" part).

Warning: here I'll keep the terminology accepted in the sword context, and refer to reference point and associated pivot point instead of using the correct physics terms as Kyle did, which are pivot point and associated center of percussion. Just because I'm not sure everyone is ready yet to change their meaning for CoP...


1. Dynamic diagram of a weapon

It all started when I was looking for a way to graphically represent the relationship between pivot points on a weapon. Pivot points are the result of the interaction between geometry (location of the reference point and CoG) and the moment of inertia, and being able to graphically represent this interaction makes it all more intuitive (well it did for me Wink ).

I'll skip the demonstration but it is possible to do that by using a quantity called the radius of gyration. The radius of gyration represents how far on average the mass of the object is from the center of gravity. It tells us how the mass is spread. It is linked to the moment of inertia in this way:

J = M * k, where J is the moment of inertia, M is the total mass, and k is the radius of gyration.

It is trivial to deduce it from the location of pivot points, as it is simply the geometric average of the distances reference point - CoG, and pivot point - CoG.

Here is how the dynamic diagram looks (I took Albion's Talhoffer as an illustration, just to materialize the axis of the weapon really. This diagram is not based on actual measurements):

G is the center of gravity, H1, H2, H3 are example reference points, P1, P2, P3 are the associated pivot points.

The important thing drawn is that line OG, perpendicular to the axis of the weapon, such that the distance OG is exactly the radius of gyration. I call O the Origin of the diagram.

This particular origin gives an important property: the triangles O - reference point - pivot point all have a right angle in O. Three of these are drawn in red, green and blue. So if you have another reference point, finding the associated pivot point is just a matter of drawing a right triangle.

The circle shown is just there to illustrate how to find the point O graphically from a pair of reference and pivot points. Just draw a circle with reference-pivot as a diameter, then raise a perpendicular in G to the axis of the weapon, and voil, the intersection is your origin.

2. Comparing two weapons

Where this diagram really starts to shine is when you attempt to compare the mass distributions of two weapons. Total mass is not represented, but it is something really trivial to compare. The diagram can show, however, the subtler differences in how the total mass is distributed. My personal conviction is that how the mass is distributed is more important than how much mass there is.

Obviously, in order to compare two different mass distributions, one has to choose a common reference point between both.

Generally, for swords it is rather intuitive to choose the junction of cross and handle, in order to draw conclusions about handling. Our hand, as we grab the swords, positions itself according to this point. Thus, it seems reasonable that swords sharing similar mass distributions relative to this point will handle very closely.

So I'll be lining up the diagrams such that the axis of both weapons are aligned and the reference points end up at the same location. Here is how it can look with my ATrim Type XI and Albion Squire (the photos of the weapons are not here anymore, they would just be scrambled together anway):

You can see the horizontal axis of the weapons, the lines from CoG to Origin for both, and a line going from origin to the common reference point (that is, the cross). The only thing that seems to show is the big difference in CoG, indicating that the type XI has more mass on blade than the Squire. Big deal, I hear you say Wink

However, because we also have the radius of gyration plotted, we can do some other things. For example, everyone knows that reproducing the balance of a sword means more than adjusting CoG. That's because, as you add mass at one point, the radius of gyration also changes. If you watch just the CoG, you end up with completely different radii of gyration, which can be readily felt.

There is nevertheless a point at which you can conceivably add mass, changing both CoG and radius of gyration, ending up with the right value for both. This is found by drawing a circle as on the picture shown, passing through both origins, with its diameter on the axis of the weapons. Adding mass to the type XI at point A until the CoGs are the same would give it the same mass distribution as the Squire; conversely, adding mass at point B would turn the Squire into a type XI (mass-wise, of course).

This is theoretical, because in practice you can't really add mass on just one point and certainly not obtain a working sword like that. However it shows the potential of the diagram: it tells you exactly where and how much more of the mass should be in the blade to go from Squire to type XI. As far as I know you cannot get this information more easily. Working a bit more on the same diagram the effect of any change in the mass distribution can be figured out.

It is tempting to expand on the comparison possibilities of the diagram. Indeed once a reference point is chosen, it seems that the position of the origin gives a quite nice indicator of the dynamic balance of the weapon. If we can analyze the mass distribution thanks to it, we can probably analyze handling as well, provided that the reference point is picked on a significant spot for hand-weapon interaction.

One thing is missing handling-wise: the location of the tip. On the other hand, the vertical lines are not all that useful in the representation. I modified the diagram to look like this:

In addition to a line from origin to tip, I have drawn a line from the origin to the tip's pivot point. Of course both are perpendicular, but it precisely helps to have a right triangle somewhere for comparison. It doesn't show because the axis are confounded here, but the line for each weapon also ends at the pommel nut, showing the length of the handle.

3. Turning it into a plot of many weapons

Now in order to draw conclusions about handling, we need to draw the diagrams of many weapons with different feels. Unfortunately if we just pile them up as we did for two weapons, we end up with something completely useless:

I think we can all agree this is hardly legible Big Grin I do not even write the names of the weapons...

We have to find a trick to make the plot clearer while keeping the most relevant information. Thinking about it... The relative positions of the origins must be kept, but the superposition of diagrams is only useful when you want to compare two weapons in detail. For many weapons, we want the shape of the individual diagrams, and the positions of the origins.

So let's zoom in the area where the origins all end up (all the tips of triangle in the previous mess), and scale each diagram to make them smaller and still centered at the origins of the individual weapons. Here is what we get:

(You can get a printable pdf of it here)

I added a grid and scales along each axis for good measure (both are in cm). First thing to note is that I'm generally lazy and measuring CoG to the nearest half centimeter (either that or there is some remarkable coincidence, I'll go over my swords again to check). Similarly, the radius of gyration is only accurate to more or less half a centimeter I believe. This is due to both the imprecision in CoG and the relative difficulty of the measurement (I used exclusively the waggle test).

Some background about the weapons:
- the child foil is... the one I used as a child Happy
- the XIXth foil is from approximately 1880 (my ancestor's, actually)
- the Milanese rapier is the standard one from A&A.
- the Darkwood Armory rapier can be seen in more details here
- the Yeoman and Squire are Albion Next Gens. The measurement of the Yeoman were done by Eric Spitler
- the Napoleonic briquet is a small marine infantry weapon (it is the model "An XI" to be accurate)
- the type XI is my Angus Trim sword, actually the first real sword I've bought
- the cavalry saber is one George Turner describes in his article
- the Wushu Jian is a very flimsy training weapon probably made of aluminium, that I came across by chance while doing an internship in Beijing
- the Ninja To is that typical, a-historical straight and short blade you see in movies. Yeah, I was young and nave Wink
- I measured a number of bokens that all come up pretty close together, here you can see three of them (Boken 2 is Katori-style, the others are ordinary white oak bokens)
- there are also four iaitos, that are quite different in handling. Iaito 4 is the most lively, and of better general quality than the other three. Iaito is the most clunky in feeling

The only weapons in the bunch I have never handled are the Yeoman and the cavalry saber.

4. Interpretation

The first thing to be seen is that weapons group both horizontally and vertically, an effect that wouldn't be seen with CoG alone. For example you have a group with the three bokens and the type XI, another more loose with the Squire, Milanese rapier and iaito 4, and another one with the three other iaitos.

Handling-wise these groups are quite different. The Squire group has the lightest blade (that is, the least proportion of mass in the blade), yet cuts quickly. The type XI group has more authoritative cutters, though the bokens can be more efficiently controlled two-handed. The iaito group is actually not very pleasant, with too much mass on the blade and a very distant tip feeling, even on these shorter blades. If you were to only plot the CoG you'd never understand why the three iaitos feel so different from the rest.

The aspect of blade mass is quite accurately rendered by the angle formed by the line between the reference point and the origin. The more vertical it gets, the least mass is perceived in the blade. It changes a bit, obviously, as the grips are adjusted. For example the rapiers are meant to be fingered, which diminishes the perceived blade mass a bit.

Looking at the individual diagrams you see slightly differing shapes as well. This reflects the kind of tip control that you have on the weapon. The closer the triangle is from a right triangle, the more steady the tip is. The red line in this case is very close to the reference point line. The foils are extreme examples of this. Some weapon sacrifice this tip-steadiness, at the other hand of the scale, like the cavalry saber. The 3 iaitos do that too, but with a very diminished radius of gyration.

I can't judge the accuracy of the reproduction of Japanese swords. I strongly suspect that the 3 iaitos are over the top, and wouldn't be so close to real katanas if I could measure some. They really don't feel nice...

The short weapons are too few to see patterns, however I think there is, in reduction, the same effect as with the three first groups. The "ninja to" feels worst and ends up to the right and bottom of the others, just like the three iaitos.

The two foils might seem unnaturally distant from one another, but this is mainly a difference in scale: the child's foil is shorter, and has CoG and radius of gyration scaled accordingly to keep a diagram of the same shape.

If we were to add more weapons, and mostly quality weapons (good reproductions and antiques), I'm willing to bet we'd start to see more patterns. For example I suspect many cutting weapons end up more or less aligned along a 45 degrees line.

If this were true, it has a great potential for the classification of rapiers, in my opinion. You'd get a distinction between cutters and thrusters quite easily. I'd expect many cutters in the area of the Milanese rapier, and gradually turning into thrusters all over from Darkwood Rapier to foils, sorted by blade mass... Just an example of what we could be hoping for.


To me, this shows that many things we feel in handling are keyed to the interplay between radius of gyration and CoG. The only subjective component of the graph is the position of the common reference point. It might be inappropriate to put rapiers and other swords on the same reference because they are not held in the same way... But still, you can hold a rapier like a normal sword.

Anyway for swords that are held in much the same way, this subjective element mostly disappears. So comparing rapiers together and swords together should be possible. I can't think of another more objective reference point for the time being.

There are physical explanations of the links between diagrams and the handling properties. However they would all be based on a model of the actions of the hand, which is a good deal more subjective than I'd like. Also, it is very far from reflecting the flexibility of the grip of a good swordsman, adapting the hand position according to what he's doing.

There are things that should be added. The first that leaps to mind is total mass. You can scream in agony but for now I don't even measure it Happy I really ought to buy some kind of scale Big Grin

The nodes of vibration are also absent. I don't rightfully see what they'd bring to the analysis for now, and they'd add at least two lines for each diagram, which may not be such an improvement.


So there I am waiting for your reactions... Don't hesitate to question and split hairs Wink I know this is a lot to go through, but I'd like to see the opinions in the community before turning it into a more polished article (though some parts already are).

I originally posted this here in Michael Tinker Pearce's forums, but the traffic seems quite low at the moment and I'm impatient for comments Happy

Also, if you wish to see a weapon of yours on this diagram, it's quite easy for me to add it. You need to provide:
- distance from the reference point (junction handle-cross, remember) to the tip
- distance from the reference point to the CoG
- handle length (including pommel, though this is not so significant. Could do a chart without the pommels as well, I should think)
- at least a pair of pivot points measured as accurately as possible. Other pairs can be used to get a better estimate, or at least estimate the measurement error Wink


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Bram Verbeek

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PostPosted: Wed 27 Aug, 2008 10:31 am    Post subject:         Reply with quote

Still studying the plots, not the easiest to grasp...
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John Gnaegy

Joined: 21 Sep 2007

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PostPosted: Wed 27 Aug, 2008 11:23 am    Post subject:         Reply with quote

I don't see a geometric proof that a circle intersecting the blade at Hx and Px and a right angle determines the length of OG. It may well be true, but there's a deductive step in there missing.

Taking the same three sets of hold and pivot points, what happens if you draw three arcs with origin Px and length Px-Hx? Do the arcs intersect and if so is it at O? Those three arcs illustrate the movement accurately, if you hold the grip at H1 and move your hand to the side, the pivot point at P1 remains stationary.

Well let me state that another way. If the three arcs do intersect, then you arrive at a synthetic point in space relative to the blade, similar to point O, but arrived at differently using more intuitive arcs describing the movement. If they don't intersect then the arc method isn't useful.
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Vincent Le Chevalier

Location: Paris, France
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PostPosted: Thu 28 Aug, 2008 2:28 am    Post subject:         Reply with quote

Bram Verbeeck wrote:
Still studying the plots, not the easiest to grasp...

Hey if this matter was easy to grasp we would have had figured it out a long time ago Razz

More seriously, on the final plot at the start it is perhaps easier to just look at the groups defined by the origins of the weapons (the upper tip of the triangles). This gives you the feeling "blindfolded", because there is no notion of length to tip. Then you can compare the shapes of individual diagrams, look at the handle length, at the position of the tip's pivot point...

John Gnaegy wrote:
I don't see a geometric proof that a circle intersecting the blade at Hx and Px and a right angle determines the length of OG. It may we be true, but there's a deductive step in there missing.

Yes as I said I did not include the demonstration in order not to overload this already long post. Actually I'm not sure which part of the demonstration you miss most, anyway here is how it goes:

Let's say I build a point Ox from Hx and Px as shown on the diagram, drawing a circle with diameter HxPx and finding its intersection with the perpendicular to the axis of the weapon in G. Indeed Ox could conceivably be different for the various pairs Hx Px.

Let us try to find an expression of the length GOx in this situation.

Since the center of the circle is on the axis of the weapon (its diameter is HxPx), the triangle HxOxPx is a right triangle. Two other small triangles are also such: HxOxG and PxOxG. Let's apply the Pythagorean theorem to all these:

HxOx = HxG + GOx
PxOx = PxG + GOx
HxPx = HxOx + PxOx

Substituting the two first equalities in the last we find:

HxPx = HxG + 2 GOx + PxG

And HxPx = HxG + PxG, so

(HxG + PxG) = HxG + PxG + 2 HxG PxG = HxG + 2 GOx + PxG

And finally, after simplification,

HxG PxG = GOx(1)

Now Hx and Px are not just random points, they are associated pivot points, which means that their position relative to G is linked by the radius of gyration k in this way:

HxG PxG = k (2)

(this relation stems from Newton's equations of dynamics. The actual, accurate demonstration of how the waggle test and pendulum test indeed find these points is quite a bit longer and in my opinion well beyond the scope of this website. You'll have to trust me or alternatively George Turner but it does work out Wink )

From equations (1) and (2) it appears readily that GOx = k, for all x. Hence the construction indeed defines a unique point O for all the pairs of pivot points.

Taking the same three sets of hold and pivot points, what happens if you draw three arcs with origin Px and length Px-Hx? Do the arcs intersect and if so is it at O?

Well such arcs would intersect one another but certainly not all at the same point. It's rather easy to find an example of three pairs of pivot points that would give several intersection points. But I don't really see what we would do with the intersection point anyway...

As I said my goal was primarily to have a visual representation of the mass distribution and pivot points. I'm not really eally looking for a representation of the motion of the swords; this motion is what the user will make it, the sword is not "forced" to rotate around any particular point... It's just that the wielder has to adapt its actions to obtain the right motion.

Thanks for your thoughts!

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John Gnaegy

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PostPosted: Thu 28 Aug, 2008 1:26 pm    Post subject:         Reply with quote

Thank you Vincent, that link to George Turner's article on lays the groundwork for your comparison here. I think you lost me between figures 2 and 3, but it's interesting material to ponder.
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Vincent Le Chevalier

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PostPosted: Fri 29 Aug, 2008 11:42 am    Post subject:         Reply with quote

John Gnaegy wrote:
Thank you Vincent, that link to George Turner's article on lays the groundwork for your comparison here. I think you lost me between figures 2 and 3, but it's interesting material to ponder.

Actually George Turner's article is what started my thinking about the subject of sword balance back then... In 2003 I think? Time flies Worried Although I still consider the basic physics and simplification very valuable, there are also several conclusions I do not agree with anymore. But still probably a good read, if you add to it some of the spotlight threads here for example. If the wider context of the study is of interest to you, you can find many other pointers in this other post I made.

Coming back to the topic, when you say I lost you between figure 2 and 3, is it that you don't see the link between these two figures, or that something is unclear in the text in between, for example how is figure 2 useful? I'm willing to try to make the text clearer Happy In fact I'm considering changing some of the color scheme as well, to make related lines the same color (specifically the line from origin to tip and the line from origin to tip's pivot point).


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