Hello all!

I have kept studying the balance of weapons and recently came up with a new result that I think some might deem interesting. This will be a long post, but I hope it will be easier to follow than some of my previous ramble ;)

Some of you might remember that I have been looking for a new graphical representation of the balance of weapons in general and swords in particular. I posted about it here. Unfortunately it's still not quite as intuitive as I had hoped ;) I'm now trying to complete it with a more intuitive description as outlined in the rest of this post.

To sum things up, the biggest problem, the thing that is very rarely measured, is the moment of inertia. It is also difficult to graph accurately yet intuitively. It represents how the mass is spread. It is adjusted in a variety of ways and is actually not all that easy to measure (it's perfectly possible but it takes a little experience in order to do it reliably).

I asked myself, how can I represent accurately the mass distribution of any sword, in the most intuitive and accurate way? Finally, since I became quite used to computations involving the dynamic properties, I decided to use the most simple equivalent object for each sword. An equivalent object shares all the dynamic properties of the original; of course for every original object, you can find several different equivalent object, just like there are several ways to achieve a similar balance on swords.

What is the most simple object that can be found? The stick :) The uniform stick, to be accurate, with the same density everywhere all along its length, no tapers, no changes in cross-section. Thus, I decided to base my equivalent object on a stick that has exactly the same length as the weapon.

Of course the stick on its own is not enough. You have to add mass somewhere to reach the same properties as the sword. There is actually a minimum mass that has to be added to the stick in order to do that. This mass is a point mass, all concentrated in one spot, whose location can be accurately computed. In theory it only works for a certain kind of objects, on others you would have to add two point masses to reach the desired properties. But I have yet to encounter a weapon that belongs to this slightly more complicated class.

To sum up: all swords (and the vast majority of weapons) have an equivalent representation showing the same properties, built as such:
- a fraction of the sword's mass is allocated to a stick of the same length
- the rest of the mass goes into a point mass located at a very specific spot, that can be computed.

The computation is actually quite involved. There are some precautions to take and the results do not really simplify. However it can still be done; I won't detail the math here as it is not really within the scope of the forum.

For example, one of my swords (a type XI by Angus Trim) is equivalent dynamics-wise to a stick weighing 541g, to which a point mass of 470g is added 14.6cm from the "pommel end". Think of it as a wooden stick with a lead washer attached somehow at the appropriate place. Of course real swords are not balanced in this way in practice, rather the blade part is tapered and a pommel added. But from the dynamic point of view it is the same. What one would feel in the hand does not change.

I applied this transformation to all the weapons I have already measured. The ratio of point mass to the total mass is variable, as can be expected. It goes from foils (70% point mass) to swords to wooden bokens (30% point mass). It represents how far away from the stick the object went in the hand of the makers; foils are more or less all adjustement and have a very low "core mass", bokens on the other end of the spectrum are just slightly tapered away from the stick. This is not a judgement of quality, it depends on what the weapon is meant to go through. It seems the most sturdy weapons tend to be close to the stick.

What I did not expect, however, was the observation that I got out of the location of the point mass. I plotted below all the weapons I have measured and handled. On the x-axis, you can read the proportion of the weapon's mass that goes into the point mass. On the y-axis, you can read the location of the point mass, in millimeters from the cross of the weapon (more accurately junction handle-cross, if applicable).

[ Linked Image ]
pdf version

As you can see the point masses are all quite close to the cross or guard. On average it's 25mm forward (roughly one inch), with a standard deviation of 50mm (two inches). This in itself this is already surprising. Remember that I never made any hypothesis involving the cross in the computation, it just happens to land around there magically...

Then I took a closer look at the results. You can see the outliers above, basically the foils have their mass a bit into the handle, and some iaitos, the longsword waster and one boken are more into the blade. It just so happens that all the weapons that have their point mass well into the blade are commonly judged ill-balanced. In general they do not move very naturally, they feel kind of clumsy and you have to fight the weapon to make it move. I decided to remove the "bad" weapons from the plot and keep essentially the quality ones (I removed the foils too, because I suspect they work differently than the rest).

And then it gets even more obvious. The average point mass position is 6.1mm into the blade. That's about the error I figure I'm making when measuring some of the properties of the swords. The standard deviation is 17mm, tighter than before...

I have been quite stunned :) To my knowledge no one ever did this kind of computation, therefore this is not something makers strive to achieve consciously. And yet, all quality swords, whatever their length (remember, there is a dagger and a short infantry saber in there), their origin, their maker, their materials, seem to obey perfectly to this principle: they are all balanced like a uniform stick with a point mass at the cross. Actually, if people were consciously trying for this, I'm not sure they would get a better result ;) I had numbers about the Albion Yeoman too, courtesy of Eric Spitler, and the mass is at the cross too. I made an educated guess about the pivot points on the Albion Munich, and it still brings a point mass at the cross. I wouldn't be all surprised if all the Albion lineup verified this. My only doubt is with the heaviest cutters. For example an axe would rather have the point mass towards the head, maybe some swords are intermediate... But the result on my type XI (a quite heavy cutter as it is) seems to go against this.

The only way to prove or disprove this is to get more measurements. I'll keep trying but I suspect this observation will remain. I cannot explain why it is so; I figure our brains are wired to manage more easily objects that are balanced like this, so that when people try to build a good sword they end up with something that fits their internal representation.



This gives a new significance to the cross as a reference point on the swords. I had started choosing it because I had a gut feeling that this was the only truly accurate geometric reference for our hand; turns out it's even more important than that, since the mass distribution seems to be tailored to the cross position. I think it can also inform us about cutting mechanics and how the swords are perfected for them. I remember Japanese swordsmen add a heavier tsuba (guard) to their swords before the most difficult cutting tests. I had never fully understood the idea. In fact it's the simplest way to keep the point mass at the guard while still adding weight... Thus they end up with a heavier sword, that strikes harder, but that is still balanced like a sword and plays nice with our internal representation.

It also means that I can start doing more accurate guesswork about quality swords. Up to now, I had to guess a pair of pivot points, and then computed the moment of inertia from that. But it was not so accurate, because the patterns followed by pivot points depend on the type of sword. Now I could just say that the sword is a uniform stick plus a point mass at the cross, and adjust the ratio of masses to get the correct CoG. In this way, I get the moment of inertia without any additional measurement compared to what is routinely given in reviews. Of course I'm making the hypothesis that the sword is actually correctly balanced, so it does not come for free, it won't work on "bad" swords :)

Well that's all for now. I'll be putting together a visual representation of all my bunch of swords very soon here... Any question, remark or suggestion is of course appreciated :)

Regards,

A relevant observation might be that the equivalent point mass (CoP for something like a simple bat) location on a uniform stick (no cross guard) is about 2/3 down from the end. (This assumes one end of the stick to be a pivot point in rotation. In real sword chopping motions, the more representative pivot might be much father up along the wielder's arm. However, it should be given this way regardless, and then corrected for various arm length and swing motion assumptions.) This is just the physics of how moment of inertia works out mathematically. It would be interesting to see how much the equivalent point mass of just the blade actually varies from this (measuring from pommel as a fair comparison with the 2/3 stick rule) to see just how much the inertia of sword rotation has actually been "engineered" historically, versus how it just happens to be that way in nature... and rather difficult to drastically alter without adding major chunks of mass (pommel/ guard) somewhere. Distal taper should move it, but how much does it really amount to on average per your significant research and calculations?

What is your moment of inertia with respect too? (What pivot point and type of motion?) A subjective aspect of this is publishing performance statistics in relation to the cross. It is an established convention. But, it does not make a lot of sense to do it this way for purposes of conveying handling. Center of grip (might require two statistics for "hand and a half" grip versus single hand grip use of the same sword) would be my preference. I would be willing to accept manufacturers' and reviewers' decisions on where center of grip should be for average people's hands. (I know I am probably beating a dead horse here, but, the subject seems to beg that I bring it up.)

Thanks for your comments Jared!

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It would be interesting to see how much the equivalent point mass of just the blade actually varies from this (measuring from pommel as a fair comparison with the 2/3 stick rule) to see just how much the inertia of sword rotation has actually been "engineered" historically, versus how it just happens to be that way in nature... and rather difficult to drastically alter without adding major chunks of mass (pommel/ guard) somewhere. Distal taper should move it, but how much does it really amount to on average per your significant research and calculations?

Actually how much mass needs to be added to a stick is already a fair measurement of how far the thing has been engineered. I mean if swords were just slightly adjusted from the simple stick we'd find about 90% of the mass or above in the stick, just like Boken 3 on my plot. As a matter of fact we don't; swords are rather 30% sticks, 70% point mass. The point mass moves the center of oscillation of the resulting object (assuming a pivot somewhere near the end of the stick) but I haven't computed how far numerically yet. My interpretation is that the concentrated mass is indeed there to adjust the centers of oscillations on the sword.

Tapering away from the stick (distal or profile) is actually a more drastic way to alter the properties. Adding chunks of mass is not practical other than for the final adjustment. That's why my equivalent "stick + mass" is not done in practice; getting this amount of concentrated mass is not easy...

However, if I'm right and most swords are indeed balanced like a stick + some mass at the cross, then the pivot point associated to the cross does not move. All the work just adjusts the center of gravity and the other pivot points (pommel in particular). Actually that's what is making me doubt my finding the most; maybe on the more thrust oriented medieval sword it does not hold true. I'd have to measure for example a type XV (Albion Talhoffer or Poitier) to make sure...

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What is your moment of inertia with respect too? (What pivot point and type of motion?)

I only store the moment of inertia around the CoG. Remember that the moment of inertia around CoG does not depend on the motion... I just use pivot points to indirectly measure it.

Also, as long as you have the moment of inertia around CoG, you have it around any point thanks to the parallel axis theorem.

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A subjective aspect of this is publishing performance statistics in relation to the cross. It is an established convention. But, it does not make a lot of sense to do it this way for purposes of conveying handling.

To be honest, for what I'm exposing in this thread, I only use the cross as the origin of length measurements. I could choose any other random point and it would just offset the results, the point mass would still land very near the cross. The computation is completely objective, and the fact that the zero point is the cross does not even comes into play. This is not meant to be immediately linked to handling either.

Where the cross really comes into play is when you start comparing swords. In order to do so, one has to choose a common origin, that is related to handling but not depending on the way you grip the sword. Otherwise you end up comparing the gripping methods, not the swords... I choose the cross as this geometric reference for several reasons. First, it is a very well defined spot, that can be located accurately and completely objectively. Second, I feel we organize our grip relative to the cross. Whether the quillon are fingered, the hand in a hammer or handshake grip, we position ourselves according to the cross generally. The lead hand at least... And the lead hand is also the one that has the finer control wheen you grip with two hands. I think the central role given to the cross in reviews is not random; it is indeed a very important point for handling.

The fact that the whole sword seems to be engineered around it, locating a point mass exactly there, gives it even more importance as I said.

And quantifying the handling of swords is certainly not a dead horse yet in my opinion :)

Regards,

Hello!

I have just finished the new plot I have been planning to do. Had to weight a bunch of my swords and restructure my program a bit but let's cut the ramble :)

Here is the thing:

[ Linked Image ]

Please click the image if you want to see a PDF at a higher resolution... (I'd like to thank Thom Ryan for the measurements done on the AT1516 and Brescia Spadona, and Eric Spitler for those on the Squire and Yeoman).

And now, what is drawn on this diagram?

Each rectangle represents a specific sword I had measurements about. The name of the weapon is written below the rectangle.

The rectangle are lined up so that reference points (I chose the cross or equivalent, such as the tsuba for Japanese types) are all on the same vertical line, drawn in red. The lengths of all the weapons are scaled by the same factor.

The area of the rectangle is directly proportional to the mass of the weapon. Inside the rectangle, you have a light grey part. This represents the fraction of mass that resides in a uniform stick on my equivalent object. The rest of the mass (in dark grey) is concentrated at a very specific point, indicated by the light grey vertical bar.

To sum up, each weapon is dynamically equivalent to a stick whose mass is represented in light grey, with the rest of the mass concentrated at the vertical light grey bar. Remember that the reference point plays no part in this equivalent object; it is just used to line up the weapons.

Now, as you can see, the light grey bar looks like a crossbar on nearly all of them :) This is not intentional at all, I have never tweaked any computation to make it look like that. I have just entered the most precise measurements I have been able to gather, and it ends up like this... As I said, if weapon makers were trying to do just that, I'm not sure they'd manage it more efficiently.

Exceptions to this tentative rule: foils, the cavalry saber, and the longsword waster (to a lesser extent the boken is also a bit far).

The point mass inside the handle for foils makes some sense. After all the guard of foils is actually a remnant of a more complete guard, complete with finger rings. The current guard is located where side rings would be, at the top of the finger rings. Thus the "virtual crossguard" of foils is also a bit inside the handle...

About the cavalry saber I don't know much. I took the measurements from the work of George Turner, and I have never held it so I don't know how it feels at all. Maybe this was not of a very high quality, maybe pure cavalry weapons can have their point mass further down the blade.

The longsword waster is in my opinion not really balanced like a sword at all. It does not move naturally in cuts, but it is light enough that one can muscle it through. I suspect the very forward point mass is an explanation of that.

If you look at the light grey area, what I'm starting to call the "core mass" of the weapon, you can see some conclusions about handling. The wasters have a low total mass but the core mass is relatively bigger. The iaito is the metal sword that has the biggest ratio of core mass, but I'd have to check on real katana to draw better conclusions. Cut&thrust weapons tend to have a lower core mass, thrusting weapons are very low, culminating in rapiers and foils.

The detailed differences between the type XI and the Squire (type XVI) are intersting. They have almost the same length, but the Squire is heavier. However, the core mass in the type XI is actually bigger. This gives the type XI a bit more punch, but gives superior maniability to the type XVI. Note that in both cases the point mass is located at the cross.


All in all, it seems to me that this diagram is a very valuable tool for assessing objectively and visually all the differences in mass distribution in a sample of weapon. We'll see how it turns out, but I guess this representation will be more intuitive to look at for most people than anything representing pivot points.

I'd be of course most grateful for any feedback, be it just about the legibility of the diagram :)

Regards,

[EDIT: if a moderator happens to read that, I never really know which subforum is more suited to that kind of thing. Many previous discussions happened in Off-Topic Talk, but then Historical Arms Talk seems to be fitting as well?)

I am trying to interpret the graphic a bit. Is the dark region an approximate envelope that the actual blade fits into (widest blade width, overall length?) I gather the light grey shade is an equivalent mass rectangular bar with no fancy tapers? The grey cross bar (equivalent area of an added point mass to the otherwise equivalent uniform bar?)

I would be interested in hearing your conclusions.
It seems to me as if many of the cross guards are located near to an ideal spot from an inertial rotation perspective. I would credit that as good design.

A question that stands out to me, looking at the darker shade and equivalent light grey region; Does the distal and profile taper appear to make that much difference to inertias? (Is there an illusion from the simplified dark grey envelope?) The impression I get from the graphic is that tapers do not look like the make much difference from an inertial perspective. This may not be surprising, as many interpret tapers and fullers as ways to improve stiffness and harmonics, not necessarily advocating huge handling benefits resulting from the tapers.

I found a discusion of this topic on another site a while back, (warning, science content as the Mythbuster boys would say) http://www.thearma.org/spotlight/GTA/motions_and_impacts.htm
Aplied the tentitive advice the auther sugests to a pair of type XVIa's I recently made, and can vouch that they do feel more "lvely" than my prievious works (where I just "guestamated" the pomel weight)
Sorry if this is off topic

good to see you are still at it Vincent. In general, I think this is a good approach to understanding the MOI - something previous discussions have brought up as a critical factor to sword handling.

I have an Arms and Armour Black Prince (type XVa) I can measure for you if you are interested... just let me know what you need me to do (PM would be fine).

I do think the idea of the position of the added mass being positioned near the cross (or more importantly perhaps, the hand/s) makes some intuitive sense - particularly of one considers motions where the sword would rotate or be re-directed.

I do think the trouble I have seen here is that I would be careful to ensure you write such that it is clear the position of this mass is not the CoG for the sword.

One question I have is the position of the point mass with respect to the CoG - how does that vary? Are there any trends? Do you see any cases where the positions wildly vary (possibly a piece of evidence debunking the idea that the position of the CoG can tell you how a sword would 'feel' when moved)?

I'll try to give this a more careful read tonight.

I'll try to answer the various questions in several posts, in order to split the thing up a bit :)

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Is the dark region an approximate envelope that the actual blade fits into (widest blade width, overall length?)

In a sense, yes... But think of it more as an average profile. Scaled appropriately it might contain the sword but it's not its purpose ;)

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I gather the light grey shade is an equivalent mass rectangular bar with no fancy tapers? The grey cross bar (equivalent area of an added point mass to the otherwise equivalent uniform bar?)

Yep, the light grey bar represents a simple uniform stick, and the grey bar a mass that would be attached to the stick. The area of the bar is not significant on this drawing, though, but the mass is exactly proportional to the dark grey area.

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It seems to me as if many of the cross guards are located near to an ideal spot from an inertial rotation perspective. I would credit that as good design.

No, because the ideal rotation point (the one that needs the least energy for the highest speed) is still the CoG. I really think the point mass location landing at the cross or near is because it is more natural for us to use that. It might have to do with our neurological structure as well...

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A question that stands out to me, looking at the darker shade and equivalent light grey region; Does the distal and profile taper appear to make that much difference to inertias? (Is there an illusion from the simplified dark grey envelope?) The impression I get from the graphic is that tapers do not look like the make much difference from an inertial perspective. This may not be surprising, as many interpret tapers and fullers as ways to improve stiffness and harmonics, not necessarily advocating huge handling benefits resulting from the tapers.

Now this idea would be completely wrong!

Tapers (both profile and distal) affect a sword's inertia enormously. Think about it: if you take a bar with no taper, and just slap a pommel on it to get a closer balance point, you will never end up with a sword... And on the diagram the point mass would be located, of course, at the pommel, because your sword-like object is not only equivalent to a mass on a stick, it is a mass on a stick, you don't need the equivalent object to tell it :)

I'll go even further: you can make a good, balanced sword with basically just tapers. This is what is done with katanas and messers, for example, see the description of the Soldat on Albion's site. But you cannot reach a desirable balance with just a pommel. Therefore, tapers are overly important in dynamic balance.

Honestly, we shouldn't even care as users. The details of tapers and pommels are only an observation of how the maker reached the balance he wished. But it's the joint effect of pommel and tapers that we are interested in; it's this joint effect that appears on my graphic. If you give me a rectangle I cannot tell you with certainty if it is a result of taper or pommel, or of how much taper. I can just see the interplay between these, and it is exactly what we feel as well.

It's just that because the research on inertial properties of swords has been so scarce, we've been trained to think as makers, having all the details of how to reach the desired balance in our head. This shouldn't be necessary... Not as far as balance is concerned, anyway.

Forgot to mention that I've been able to draw some more weapons thanks to Thom's very helpful measurements. Read about the results here: http://tinkerswords.com/forum/viewtopic.php?p=2232#2232

The most interesting new element is the A&A Cavalier Rapier. It's pretty much an outlier anyway I look at it... Might have to nag Craig Johnson a bit about how it has been designed and how it compares with originals...

If rapiers show such an amplitude of variation the whole thing might be very applicable to classification.

Yes - the A&A Cavalier rapier is a rather unique sword. I stand by my measurement that at least on mine, the pivot point for the hand is quite far out towards the tip of the blade, more so proportionally than any other sword I have owned. In my opinion the review here at MA is not quite correct in regard to some of Patrick's interpretation, again, this is just my opinion, but there is a reason why the Cavalier is listed under rapiers while the Town Guard sword is listed as a sword.... It has to do with mass balance and what the swords were built for. There is no doubt in my mind that the Cavalier rapier was purposefully built to be wielded from horseback.... given the weight and mass distribution, this is not a rapier I would want to duel someone with on foot. Its not really meant for that. It is rather heavy at 1500 g and I find that I tire rather easily when working with it, and the sword does not recover very quickly or easily from any cut. Rather the Cavalier seems to be made such that you can deliver an accurate (and powerful) thrust (given the total mass of the sword) and put the point on target despite being on horseback in all sorts of unsteady situations. Yes with its flattened diamond section it can also cut well, but I would not say that it is any better at cutting than many other rapiers that I have handled. The tapers work out such that to cut effectively with the blade you have to strike rather close to the hilt (much like a medieval Type XV blade). This reduces the blades "reach" with respect to cutting. However, with its mass and stiff cross section this is one blade that I have no doubt would thrust through a heavy buff coat like a hot knife through butter. In summary the combination of weight, section, tapers, and a pivot point out near the tip all seem to be purposefully designed for delivering powerful thrusts from horseback. my $0.02. tr

That's a very plausible explanation Thom...

Also the stout blade may be more suited for parrying especially on the battlefield. If you want a stout blade, good point control, and not far too much mass, perhaps this position of the point mass is the only possible solution.

Actually a comparison between the A&A Cavalier, Dresden and Town Guard would be a nice thing to see... All are fitted with a substantial blade and a complex guard, yet it's possible that they are not balanced the same dynamically.

Any volunteers for measuring the Dresden and Town Guard :) ?

I wonder if it may be more intuitive to change the size of the 'cross bar' that represents the point mass then, and perhaps take out the dark bar? But that may run into the issue of where the center is located - too large of a cross could obscure the placement a bit.