Article about sword mass distribution

[Vincent Le Chevalier]

Hello all!

Finally, after more years than I care to count playing with the concepts, I've managed to finish an article about sword balance and mass distribution...

It focuses on how to measure meaningful properties directly on the sword with as little calculus as possible (perhaps an addition or a subtraction somewhere ). The properties have some intuitive interpretations in terms of handling for example, at least in my opinion... Of course I've left some things out and a more advanced interpretation is still a work in progress.

Read the article here!

Please don't hesitate to comment! It's the only way I'm going to be able to iron out the parts that are not clear...

I hope the method described can be used in reviews to introduce some more objectivity in the appreciations of handling and ease comparisons.

In appendices you will find raw data measured on swords and some mathematical demonstrations that could be of interest to the maths/physics geeks out there

Since George Turner's article, hosted by the ARMA, has more or less kicked me off in this study, it's only fair that I share it here...

Regards,

[Randall Pleasant]

Vincent

Thanks for sharing your work with us, I look forward to reading it. As with Turner's work, it will without doubt it go way over my head.

[Vincent Le Chevalier]

Hi Randall!

Thanks for sharing your work with us, I look forward to reading it. As with Turner's work, it will without doubt it go way over my head.

Actually I tried to keep things as simple and direct as possible, unless you dig into the mathematical appendix of course I do wonder if I've managed it...

Regards,

[Stacy Clifford]

I agree, thank you for sharing this with us. George's article has always been one of my favorites. I'll have to read yours when I have time to sit down and absorb it.

[Vincent Le Chevalier]

Hello!

I have just updated the page with a new format for the video, that will hopefully be supported almost everywhere...

Still at the same place!

Regards,

[Andrew F Ulrich]

First of all, thanks for not neglecting the mathematics of the issue. A pet peeve of mine is physics literature that tries to explain physics without showing any math. To me, it's sort of like teaching RMA without referencing any of the manuals.

Anyway, here's some feedback:

page 2:
"to make that kind of simplifying assumptions"
should be:
"to make those kinds of simplifying assumptions"
or
"to make that kind of simplifying assumption"

Appendix B.3:
-In equation 7, you refer to the distance from the axis of rotation to center of gravity as RG, while in equation 8 you refer to it as GR. You may want to correct this for consistency's sake.

-It would have been nice if you had said you were plugging into equation 12 to get equation 17

You might also make a small note that your measure of the precision is also limited by the first approximation you did to reduce the differential equations, and that smaller angles, while increasing accuracy in this respect, also make 'a' harder to visually discern. One thing to note in particular might be that the time derivative of theta squared becomes more significant with faster 'waggling', which you state decreases the other discrepancy.

You might also note that another discrepancy is that you have been holding it by the cross when performing the 'waggle' test, and that swords do not rotate about the cross when cutting. This becomes important when noting that the radius of gyration is dependent upon the position of the axis of rotation, namely (as per the Parallel Axis Theorem):
I + Md^2 = Mk^2,
where I is the moment of inertia about the center of mass, and d is the distance from the center of mass. So, when d changes, k changes. In fact, this becomes more of a problem when performing something like a zornhau, where, at full extension, your axis of rotation is more than an arms length away from the cross. Of course, you can possibly demonstrate this as negligible for such cuts as ones that only involve the turning of the wrist, though even then, the axis of rotation would be nearer to the center of the handle rather than to the cross. For this reason, I would personally prefer to use I as a reference rather than a two-mass system, but since you can still calculate I from what you have, I don't consider this too serious an issue.

Also, you may want to remember that when you weigh the sword that way, you must make sure the blade is completely level, otherwise your 'blade' weight will be off. With this in mind, it may not have been a bad idea to weigh the 'handle' also to make sure they added up to the total weight and account for any discrepancies.

Another thing you may want to consider is that, although it may be obvious to some, you may want to make more explicit the fact that what you refer to 'blade weight' and 'handle weight' aren't the actual weights of the blade and handle if you were to somehow disassemble and weigh them separately. For example, if you were to increase the weight of the pommel, the 'blade weight' would decrease even though the weight of the actual blade would remain the same.

I haven't done physics in a few years, so take the above with a grain of salt, but hopefully you find this helpful. Thanks for the article and especially the data on the different swords! I hope this goes on to spur another standard spec when describing a sword.

Andrew Ulrich
ARMA St. Louis.

[Andrew F Ulrich]

On second thought, Albion already posts the center of percussion of all their swords, which, along with weight, length, and center of gravity, would be sufficient for finding the moment of inertia if we only knew where the axis of rotation was located when they measured the center of percussion, that is, did they use the waggle test and if so did they hold it at the pommel, cross or elsewhere? Maybe I should call them tomorrow...

[Vincent Le Chevalier]

Thanks for the review Andrew!

I'm going to answer your previous post in more details but this is easy:

On second thought, Albion already posts the center of percussion of all their swords, which, along with weight, length, and center of gravity, would be sufficient for finding the moment of inertia if we only knew where the axis of rotation was located when they measured the center of percussion, that is, did they use the waggle test and if so did they hold it at the pommel, cross or elsewhere? Maybe I should call them tomorrow...

Actually what they call center of percussion (and what most sword enthusiasts would understand by that name) is the node of the primary vibration mode that lies on the blade. That's not the meaning in physics, I know... So, nothing to do with inertia, at least nothing as direct as center of oscillation or pivot points (which is how I used to refer to this spot. There is a rough link but it does not help if you have just one of the nodes, which is what they give.

Regards,

[Vincent Le Chevalier]

Hello again Andrew,

page 2:
"to make that kind of simplifying assumptions"
should be:
"to make those kinds of simplifying assumptions"
or
"to make that kind of simplifying assumption"

Appendix B.3:

-In equation 7, you refer to the distance from the axis of rotation to center of gravity as RG, while in equation 8 you refer to it as GR. You may want to correct this for consistency's sake.

-It would have been nice if you had said you were plugging into equation 12 to get equation 17

All corrected!

You might also make a small note that your measure of the precision is also limited by the first approximation you did to reduce the differential equations, and that smaller angles, while increasing accuracy in this respect, also make 'a' harder to visually discern. One thing to note in particular might be that the time derivative of theta squared becomes more significant with faster 'waggling', which you state decreases the other discrepancy.

Good points...

It's true that a very small angle would make the fixed point hard to discern. The idea would then be to augment the amplitude of the motion until the fixed point can be clearly seen again. Looking at the sword from the top makes it easier to spot the center of oscillation too.

About the approximations in the equations: perhaps it's not as well justified as I'd like The equations in system 11 have to be looked at as a whole; all the terms include the time derivatives, either because they are the second derivatives of something or because they are products of first derivatives. So when I neglect the derivative of theta squared, I do so without having to consider the time aspect: the 'theta squared' part is enough to make it negligible. I don't know if that clears that up or muddies the water even more

You might also note that another discrepancy is that you have been holding it by the cross when performing the 'waggle' test, and that swords do not rotate about the cross when cutting.

Ah yes but that's on purpose... This is alluded to in 2.3 actually. What I'm looking for is a straightforward way to measure inertial properties that make at least some intuitive sense and do not depend on what the user is doing. Moment of inertia about other axis (for example the shoulder or elbow, or middle of the hand, or pommel, etc.) is not something you measure that easily. As you say, you can compute it fairly easily if you know the formulas but I'm not sure that would be useful to the average sword user. Besides, many of these axis depend on the user, and that is not the goal: I don't want to have to give the length of my arm in order to describe a sword

The point H I chose (junction cross-handle) has the merit that the users nearly always grip the sword relative to that point. It allows to place the sword relative to the user, if you like. It is perfectly well defined and measurable. And by measuring directly a two-mass equivalent from that point, I have all the data needed to do further computations if I like, and yet no need to make any computation at all in order to get a first interpretation. Even though swords do not generally rotate about the cross, blade weight, cross weight and dynamic length are things you can perceive if you pay attention. I will probably develop the interpretations in a next article...

So there are no discrepancy introduced here, no additional approximation. I'm not saying swords rotate around the cross, just that it makes sense to take that point as a reference.

Also, you may want to remember that when you weigh the sword that way, you must make sure the blade is completely level, otherwise your 'blade' weight will be off. With this in mind, it may not have been a bad idea to weigh the 'handle' also to make sure they added up to the total weight and account for any discrepancies.

Good catch, I added a note to that effect. Weighing the handle can be difficult though, especially on complex hilts. If the blade is not completely oblique I don't think the error can be significant...

Another thing you may want to consider is that, although it may be obvious to some, you may want to make more explicit the fact that what you refer to 'blade weight' and 'handle weight' aren't the actual weights of the blade and handle if you were to somehow disassemble and weigh them separately. For example, if you were to increase the weight of the pommel, the 'blade weight' would decrease even though the weight of the actual blade would remain the same.

Absolutely right, I added a sentence saying just that.

I haven't done physics in a few years, so take the above with a grain of salt, but hopefully you find this helpful. Thanks for the article and especially the data on the different swords! I hope this goes on to spur another standard spec when describing a sword.

That's my hope too Thanks again for the review and feel free to play with the data! I'd like to see what interpretations and statistical facts other people can dig up...

Regards,

[Vincent Le Chevalier]

Hello all,

Just a notice that I have made some minor corrections to the article (nothing really significant) and added a word of caution in the intro, that I'll repeat here just in case:

Measuring a weapon involves some handling. Be careful when you do it with sharp blades. In fact be careful in general! I do not want people to get hurt while just trying to measure swords. If you feel your grip is not secure enough, if there are any living beings nearby (including yourself obviously) that could be hurt by a sword falling down, just lay the sword down and rest. If the sword falls and is not worth any piece of your own skin then do not attempt to catch it. Do not take chances...

The risk is nowhere near that posed by actual martial moves but better be safe than sorry...

Still at the same place!

Regards,

[Ed Rybak]

Hello. I'm continuing my questions about weapon properties, started at this thread http://www.thearma.org/forum/viewtopic.php?t=24489

Vincent, I enjoyed your article. Here are the main questions I had:

1) You measure three intrinsic properties: mass (M), position of center of gravity
(G), and radius of gyration around center of gravity (k). Is k the same as moment of inertia?

2) Fig 1 shows three different objects with different configurations, but all with the same M, G, and k. I understand your point that all three would handle and move in the same manner, but would that depend on where they were gripped?

For example, I'm guessing that it would hold true if they were all gripped at G (the center, in your figure), but what if they were gripped at an end?

3) For swords, you place one of the virtual mass positions, H, on the grip, and from there you obtain the second virtual mass position, F. I'm uncertain of the rationale behind the position H. I see you briefly discuss it as an easy point to work with, but if I may press a bit:

3a) Are you placing H right on the crossguard? Or another specific point? Is H only an arbitrary convenience, or does it matter that it be placed at some specific point relative to the hand?

3b) If H is arbitrary, am I to understand that we could freely choose some other arbitrary H (with the position of F and the virtual masses of H and F then changing in such way that M, G, and k are maintained)?

3c) If H and F can vary, is it possible to instead start with F, and from there calculate H? (Maybe not practical – but theoretically possible?)

3d) If H or F can be arbitrarily chosen, with the other position calculated to match, then what are the restrictions on the arbitrary position of H or F? Only that the arbitrarily chosen point lie somewhere between its respective end of the weapon and G?

4) Is there any reason why the same two-mass equivalence model can't be used to define other weapons – say, pole arms, axes, etc.? For example, ignoring the minor(?) complication that some of an axe's mass extends in a direction perpendicular to the main axis, could an axe also be defined by M, G, k, H, F, virtual mass at H, and virtual mass at F?

I look forward to your reply. Thank you!

[Vincent Le Chevalier]

Hello Ed!

These are all good questions!

1) You measure three intrinsic properties: mass (M), position of center of gravity
(G), and radius of gyration around center of gravity (k). Is k the same as moment of inertia?

It is related. The moment of inertia around G is equal to M k².
I like the radius of gyration because it sort of isolates the effect of the distribution of mass from the effects of the amount of mass. For example if you have a bunch of uniform sticks of the same length but of different mass, they all have the same k, but they don't all have the same moment of inertia.

2) Fig 1 shows three different objects with different configurations, but all with the same M, G, and k. I understand your point that all three would handle and move in the same manner, but would that depend on where they were gripped?
For example, I'm guessing that it would hold true if they were all gripped at G (the center, in your figure), but what if they were gripped at an end?

It would remain true. Basically the choice of the gripping point would affect what actions must be applied to move the objects in a given way, but the objects still all move in the same way given the same actions.

3) For swords, you place one of the virtual mass positions, H, on the grip, and from there you obtain the second virtual mass position, F. I'm uncertain of the rationale behind the position H. I see you briefly discuss it as an easy point to work with, but if I may press a bit:

3a) Are you placing H right on the crossguard? Or another specific point? Is H only an arbitrary convenience, or does it matter that it be placed at some specific point relative to the hand?

To be accurate, I place H at the junction between the cross and the handle. Indeed I give the reasons in the article but that part might be too terse
Consider what you need to do in order to compare weapons. You need to position the mass distribution of the weapon relative to the hand(s) holding it. Point H is the reference that must be chosen in order to position the hand(s) on the hilt.

Why am I picking that exact point? First, because it does not depend on the exact method of grip you choose. Whether you grip with a hammer grip, a handshake grip, finger the cross or not, the balance of the weapon does not change and hence the properties should not depend on it. By choosing a fixed point on the grip of the weapon I avoid that effect. On the grip of the weapon, there is only one point that is very accurately measurable on a vast array of swords, it is the junction cross-handle. The other points of the grip are not well-defined enough or common enough to serve that purpose.

Originally I was content with these reasons and started to compare weapons based on that specific point H. Then I fell onto some quite astonishing coincidence (outlined here, but I have a better understanding of that now and should really start another article when I have time ), that make me think that this point is indeed the right point or at least very close to the right point.


That being said...

3b) If H is arbitrary, am I to understand that we could freely choose some other arbitrary H (with the position of F and the virtual masses of H and F then changing in such way that M, G, and k are maintained)?

Absolutely. It is actually quite easy to compute the location of the mass at F and the values of the masses based on M,G,k, and H. This is described in Appendix B.2. So you could perfectly perform the comparison and measurements based on some other point. As I said I use the one that gives me the most meaningful result.

3c) If H and F can vary, is it possible to instead start with F, and from there calculate H? (Maybe not practical – but theoretically possible?)

Yes this would also be possible, though for swords it would be difficult to get a meaningful result I believe (and also very hard to measure).

3d) If H or F can be arbitrarily chosen, with the other position calculated to match, then what are the restrictions on the arbitrary position of H or F? Only that the arbitrarily chosen point lie somewhere between its respective end of the weapon and G?

Mathematically you can really put it wherever you want.
Physically, it's a bit hard to do the waggle test from a point that is not within the physical extent of the weapon, of course For the waggle test, you really need to have both H and F on the weapon, H to hold and F to look at.

4) Is there any reason why the same two-mass equivalence model can't be used to define other weapons – say, pole arms, axes, etc.? For example, ignoring the minor(?) complication that some of an axe's mass extends in a direction perpendicular to the main axis, could an axe also be defined by M, G, k, H, F, virtual mass at H, and virtual mass at F?

No, it can be used. Basically anything long and rigid can be analyzed with these tools. For example I have data about a decorative axe and two wooden naginata somewhere...

The big problem that I still have with these other weapons is the choice of the reference point H. For axes there is no real well-defined spot on the handle that I'd feel comfortable singling out. In that particular case I believe that the approach you suggested earlier, to start with point F on the "business end", would work better but I don't have enough data to really study the question.

In short, swords have their mass concentrated near the cross, while axes, maces and stuff like that have their mass concentrated on the business end. So I think that difference calls for a different method of analysis. Note that you don't often see people argue about the balance of these other weapons. I think it is in part because of that difference.

Thanks for the discussion!

Regards,

[Ed Rybak]

Thank you again for the detailed reply. All very helpful!

FWIW, my questions come from a geeky interest in modeling hand weapon behavior, and I'd been following a pretty similar approach to yours, right up to looking at two-mass models and even coming up with the same mass-independent property you label k (though I didn't know the term "radius of gyration" and was sifting through clunky names).

I was guessing at a lot of things (such as how to measure k if actual moment of inertia (MOI) is unknown), so I'm enjoying the hard work behind your essays.

I'm particularly interested in applying the models to rigid weapons in general, not just swords, so my questions are skewed toward that. (Forgive me if I refer to your properties "cross weight" and "blade weight" as "Hmass" and "Fmass", respectively; they're shorter to write and don't assume sword features.)

It would remain true. Basically the choice of the gripping point would affect what actions must be applied to move the objects in a given way, but the objects still all move in the same way given the same actions.

Got it. I know the point of grip will affect some other properties, such as relative pivot points (per Turner's work), but right now I'm just interested in the properties your essay discusses.

[small edit after posting: I had added questions about the above, but was just confused on a point. Section removed for simplicity.]

Absolutely. It is actually quite easy to compute the location of the mass at F and the values of the masses based on M,G,k, and H. This is described in Appendix B.2. So you could perfectly perform the comparison and measurements based on some other point. As I said I use the one that gives me the most meaningful result.

...

Mathematically you can really put [H] wherever you want.

This is of interest to me where non-swords are concerned: an axe, a staff, even a guard-less "sword" (say, a smooth bokken) all call for some arbitrarily-chosen H.

But consistency is nice for comparison purposes – it'd be nice if two people measuring the same axe didn't choose wildly different values for H! As you note, in the absence of the cross guard there's no consistent objective reference point. If I were to stubbornly insist on setting one anyway, is the butt end of the weapon a candidate (at least in theory)? How about the point G? (I believe the waggle test measurement for F then would be impossible, so it'd be impractical… but is H=G also impossible in theory? I'm blank on that.) Maybe something easy like the mid-point between the butt and G would work?

Also, this may be a dumb question, but: H and F will always be located on opposite sides of G, right?

Finally, if you've still got the kind patience, here are three big new questions:

1) The properties you discuss are nicely knowable. Measurement of M and G are obvious. H can be chosen with F then found. And you describe how to find Hmass and Fmass. Great!

But what about k? I see the mathematical definition in B.1, but how do you measure k in a real weapon? I must be missing something obvious…

2) It's interesting how you tie properties to handling characteristics: Hmass is linked to tiring weight on the arm; Fmass is felt in the wrist and approximates the mass impacting a target; dynamic length (distance from H to F) suggests the "feeling of length" of the weapon; the ratio of Fmass to M creates "blade presence"; etc. These sound very useful.

It's also interesting that H and F are mutable – that is, that we can place H here, and find the corresponding F, Hmass, and Fmass…. or we can place H there, and find a different corresponding F, Hmass, and Fmass. My understanding is that either set of properties creates an equally accurate representation of the same weapon. Sounds good.

But now a contradiction seems obvious: how can both sets of properties accurately represent the same weapon, when they result in a different mass impacting the target, different dynamic length, different blade presence, etc.?

The problem would seem to suggest that there's some ideal (optimal? correct?) pairing of H and F that most accurately represents those handling characteristics you describe; other pairings of H and F are somehow less ideal, less correct, etc.. Is that a possibility? And if so, what would define the particular ideal pairing?

3) You mention that you want to look more deeply into the consequences of properties on impact behavior. Is there an upcoming essay on the topic that we can look forward to?

Once again, thanks so much for the fascinating discussion.

[Vincent Le Chevalier]

Hi Ed,

I'm answering quickly now, I'd be glad to keep that discussion running but I'm off to the HEMAC gathering so I'll be busy playing with swords until next week

FWIW, my questions come from a geeky interest in modeling hand weapon behavior, and I'd been following a pretty similar approach to yours, right up to looking at two-mass models and even coming up with the same mass-independent property you label k (though I didn't know the term "radius of gyration" and was sifting through clunky names).

To be honest it took me a while to find that name too

But consistency is nice for comparison purposes – it'd be nice if two people measuring the same axe didn't choose wildly different values for H! As you note, in the absence of the cross guard there's no consistent objective reference point. If I were to stubbornly insist on setting one anyway, is the butt end of the weapon a candidate (at least in theory)? How about the point G? (I believe the waggle test measurement for F then would be impossible, so it'd be impractical… but is H=G also impossible in theory? I'm blank on that.) Maybe something easy like the mid-point between the butt and G would work?

Yeah it was primarily consistency I was worried about. Actually bokens are fine with that model, since generally they have some sort of mark indicating where the grip ends too. But many axes and polearms have not...
Butt of the weapon yes that would work, but the point F that it would lead to would be closer to the center of gravity, hence often further from the striking area (except on very blade-heavy tools). H=G would be problematic, it's a degenerate case: point F would have to be at infinity, and Fmass infinitely small... There is a division by zero happening in that case
Mid-point between H and G could be interesting but difficult to justify. On axes and maces it would yield a point H close to the middle of the haft, and a point F possibly beyond the tip of the weapon.

Also, this may be a dumb question, but: H and F will always be located on opposite sides of G, right?

Yes, always on opposite sides, such that HG x GF = k². They really are pivot points that George Turner wrote about, I just abandonned that terminology because none of these points are privileged locations around which the weapon pivots unless you actively make it so. If you look at his article again you should see the same relations as in mine.

But what about k? I see the mathematical definition in B.1, but how do you measure k in a real weapon? I must be missing something obvious…

You cannot directly measure it. You have to deduce it from dynamic length, Hmass and Fmass if you're interested in that specific quantity. Here is the formula:

k = l * sqrt( (Fmass * Hmass) / (Fmass + Hmass)²)

where l is the dynamic length. Hopefully I got it right I just made the calculation

If you are able to measure H, F and G, you can also use the simpler formula without in mass measurement needed:

k = sqrt(HG * GF)

It's also interesting that H and F are mutable – that is, that we can place H here, and find the corresponding F, Hmass, and Fmass…. or we can place H there, and find a different corresponding F, Hmass, and Fmass. My understanding is that either set of properties creates an equally accurate representation of the same weapon. Sounds good.

But now a contradiction seems obvious: how can both sets of properties accurately represent the same weapon, when they result in a different mass impacting the target, different dynamic length, different blade presence, etc.?
The problem would seem to suggest that there's some ideal (optimal? correct?) pairing of H and F that most accurately represents those handling characteristics you describe; other pairings of H and F are somehow less ideal, less correct, etc.. Is that a possibility? And if so, what would define the particular ideal pairing?

Yes this is a problem, and the reason why I chose to define H as I did. As long as you compare based on a common reference, the comparison remains valid. It's when you compare using two different references that things get complicated. And you cannot avoid picking a reference. H defined at the junction cross-handle really works well on swords, I found out experimentally handling my collection and comparing to the results. The ideal pairing is really the one that achieves precision (all the points are measurable and well-defined), consistency (they can be found on a vast array of weapon), and meaningfulness (the properties found can be interpreted and are in accord with the perception we have). For me having H at the junction cross-handle is a good candidate.

Is it optimal? Not yet I believe. Hmass and Fmass are not precisely what I feel when handling a weapon. I have another way to derive a better two-mass system (in the sense that I feel it fits my perceptions better), based on 'theoretical' locations of H and F according to the overall length and blade length of the weapon. But it is much more difficult to explain, and is not possible to measure directly as I describe in this article. So I fear it will remain for awhile a more contested result. It makes more assumptions if you like.

3) You mention that you want to look more deeply into the consequences of properties on impact behavior. Is there an upcoming essay on the topic that we can look forward to?

You can look forward to it but at my current rate of essay writing it might be several years before it comes up Before that one I'd really like to delve deeper on the interpretation of the properties for handling, as I said I feel I'm getting closer but I'm not exactly there yet.

In a way I find impacts a bit less interesting because they depend quite heavily on the properties of the target and of the person holding the sword. That makes them exponentially harder to analyze with any consistency. You get results but they apply to very specific circumstances...

Regards,

[Ed Rybak]

Thanks again for the comments. Please don't rush to reply, especially during what sounds like a great week planned. For my part, I've taken a few days to do some exploring, so here will try to add my own resulting observations and ideas, rather than just ask more questions.


1) Regarding the issue of finding the "right" pairing of H and F to model a weapon, given many available possible pairs:

First, I wanted to confirm for myself that it's indeed possible to have multiple pairs of H and F; that is, that there are multiple of sets of H, F, Hmass, and Fmass that model a given weapon without changing the fixed M, G, and k. The result: No problem.

For example, a weapon modeled with H 0.4m, F 0.6m, Hmass 0.5kg, and Fmass 0.5kg will have the same M (1kg), G (0.5m), and k (0.1) as a weapon modeled with H 0.45m, F 0.7m, Hmass 0.8kg, and Fmass 0.2kg. (Numbers are for model purposes; they may not represent any meaningful weapon!)

Your work at http://www.myarmoury.com/talk/viewtopic.php?t=14063 , using right triangles with the right angle "hung" on the point O in your graph, really makes this clear. Rotate the triangle a bit, and it's easy to see how H and F change without changing k; all that's needed is a little mathwork to reapportion mass between Hmass and Fmass, and M and G remain unsullied. Very nice!

So, for a given weapon, which of many sets of H, F, Hmass, and Fmass are "right"? I would think the answer is "all". That is, looking at my odd model weapon above, if Fmass can be taken to represent impact mass, then 0.5kg is the right impact mass when looking at the point F=0.6m, and 0.2kg is right when looking at a point further toward the tip, F=0.7m. At least, that sounds reasonable...

You mention more work along those lines, deriving a better "theoretical" H and F for modeling weapon feel. Would be great to see that whenever it's ready.

In the meantime, here's a thought: As I mentioned, I like the idea that properties like dynamic length can also be derived from the above values, but it's unfortunate that the properties will change with the arbitrary H and F chosen to measure; it makes dynamic length, "feeling of length", etc. seem more like guesses than objective properties.

But then again, maybe not. For swords, you've set H at the cross guard for simplicity, but it's also worth noting that this also coincides with the position of (the top of) the hand. Perhaps dynamic length, etc. should vary with the point where the weapon is gripped! If so, then the fact that dynamic length, etc. vary with the "arbitrary" selection of H (based on point of grip) is actually a feature, not a bug?

Just an idea…


2) Notes on k

You cannot directly measure it. You have to deduce it from dynamic length, Hmass and Fmass if you're interested in that specific quantity. Here is the formula:

k = l * sqrt( (Fmass * Hmass) / (Fmass + Hmass)²)

where l is the dynamic length. Hopefully I got it right I just made the calculation

At first glance I thought that's not right… but doing the calculations, it is. As I see it, there are three ways to get k:

k = l * sqrt( (Fmass * Hmass) / (Fmass + Hmass)²)
k = sqrt(HG * GF)
k = sqrt (MOI/Mass) if MOI is known (as with a rod).


3) Models

I think it's obvious that only a single rod, or only a single point, isn't enough to model swords, etc. As models that do work, you've suggested the following:

a) A single rod plus a single-point mass, per http://www.myarmoury.com/talk/viewtopic.php?t=15288
b) A two-mass model, per this thread

Just curious, when and if you have time: Have you abandoned a) in favor of b) as the better model? Or do you see them as equally valid models?

As far as I can tell, they both seem good models. In terms of actually visualizing things, though, the two-mass model b) can be a little difficult. For example, to model a staff, you have to find k based on length (easy enough) but then place two equal masses on opposite sides of G, each located the distance k away from G. Which may be perfect as a model, but it's hard to "see" the actual staff in there.

I wonder, too, if the rod + single-mass model a) has a shortcoming. It's perhaps able to model any weapon with a k equal to or lower to that of a rod – but wouldn't it be unable to model a weapon with k greater than that of a rod? (Example: Very tip-heavy weapon, counterbalanced by heavy pommel. Extreme example: barbell.) Maybe that's of zero practical concern for real weapons; just pointing it out.

Perhaps a model c) using a rod + two masses would be nice in a couple of ways: it could model any weapon as well as the two-mass model (including oddball barbell weapons : ), but would also make for easy visualization. Start with a thin rod the actual length of the weapon, and add two point masses here and here to model inertia, balance, etc… Ambitious modelers could even add a third or greater point mass to model tweaks (like adding more pommel mass) in a more direct manner than sliding the existing two virtual masses around…

Well, I'm just musing to myself here. That's a toy for elsewhere. Let me wrap up by saying that two-point model, as well as your related works, have all been great for better understanding this area of interest, and I intend to keep playing with it all. Thanks!