Hello all!

In keeping with my sword/math geek usual behaviour, I was crunching some ideas and concepts earlier this evening, and much to my surprise, came out with something rather simple and yet midly interesting. This is rare :)

So there has been quite a lot of discussion recently about CoP, CoB, how they interact and how they are defined, how you can find them, etc. There are currently two ways of understanding swords: the most anchored in the community is based on vibrations of the blades, and there is another, that I think is promised to a bright future, based on dynamics of the sword considered as a rigid body. In the first, vibration nodes are the key, in the second, pivot points are.

Ever since this previous post I've had in a corner of my head that both are in fact closer than it seems. Well, now I have a mathematical proof...

So tonight I took all my steel swords (Atrim Type XI, A&A Milanese rapier, Darkwood Armory rapier) and located both the hilt and blade nodes on each. On each and every sword, they were associated pivot points (within the precision of my measures, about half an inch I would say). The blade node was the pivot point of the hilt node. Then I tried that with my wooden swords (you can locate the node by the special sound they make, though it's difficult): same result. On five object so different, coincidence was beginning to be unbelievable ;)

I then decided that for good or ill I'd have to sort this out, and took my pen and paper :) The result is not as perfect as we could wish, but is already a huge step for me...

In fact, they are not exactly associated. More precisely, it depends on the shape the blade takes when it vibrates. See http://www.tinkerswords.com/harmonics.jpg for an example shape. Well, if the shape is a parabola, the equation magic happens, and the nodes are indeed associated pivot points, exactly. That's three lines of calculus if you know the definitions and the properties of pivot points and center of gravity. I can post them if anyone wants to see...

Of course, a vibrating blade, in general, does not take a parabolic shape. However, it is never so far away. The parabola is simply the most simple curve showing, as depicted, a belly and two nodes. So even though the two primary nodes are not associated pivot points, they are very, very close to being so on reasonable objects.

One could ask how this is important... Well there are several reasons:

- The definition of the CoP can be seen in a dynamic or in an harmonic sense, minimizing shock or vibrations in the hand, respectively. What the relation shows is that, on a nicely balanced sword, both are equivalent in a sense. On an harmonically balanced sword, there is one vibration node in the hilt. If the impact happens on the vibration node on the blade, two results are achieved: least vibration in the hand (since we are striking with a node), least shock in the hand (since the point of impact is the pivot point associated to part of the hand). That's why an harmonically balanced sword feels right upon impact on the node, I guess. There is more to it than just wobbling.

- There was a bit of noise after the parution of George Turner's article, that was dismissing harmonic balance in a way its supporters found a bit offensive. It turns out that as far as impacts are concerned, both sides were right :) Make an harmonically balanced sword, and the resulting pivot points will be ideally placed for rigid body impact as well.

- I never liked CoP as a statistic in reviews, because I consider it is difficult to link to handling. Turns out that if the position of the node on the hilt was published as well (a big if, it is mostly absent), I would have a near exact set of pivot points, from which I can deduce any handling property I like. Which means that if someone feels comfortable with harmonics, he will measure the nodes, if I am comfortable with pivots, I will get them, and in the end, for handling we have the same information.

That still does not give a formula for the position or modification of the nodes, only a relation. That is, I know they are associated pivots, but if I don't have one of them, I can't find them both. And if we want to play with semi-nodes and such, the link to inertia is lost. There is also the issue of the precision of these measurements... Well in theory, at least, it works.

Of course everyone will tell me that all this is known since the dawn of time, obvious to someone with experience, etc. Well, I'm still happy to have a somewhat more unified theory to explain what we observe :cool: Kind of feels like being able to tell why the sun rises every morning, not just how :D

Wishing you all good night from my part of the world...

Hi Vincent

Now all you need to do is factor in how the handle shape affects handling........*g*

Hi Vincent

If you want to be able to use a math formula to figure where the handle node is, its a bit like that old "mass distribution" thing......

How long is the blade? How thick at the base? How wide at the base? How much distal taper, and how does it taper? How much profile taper, and how does it taper? What does the contour look like { blade geometry and crossection}? Hilt length? Width and thickness of tang? Taper of tang {distal and profile}? How much does the pommel weigh, and where is the weight on it?

A lot of variables........

One thing to consider, the handle node moves from the "naked blade" to the mounted blade. Before mounting, its not unusual for it to be up on the blade proper a bit, once mounted its on the tang..........

Hi Gus!

Angus Trim wrote:

If you want to be able to use a math formula to figure where the handle node is, its a bit like that old "mass distribution" thing...... [...] A lot of variables........

I agree, but fortunately I'm not trying to do that :) I'm not aiming at finding all the properties from scratch, just from geometry and materials. I doubt it would be of much use to anyone... If you are able to have this level of detail and precision for measuring geometry, for example, it's likely that you have access to the object, and so you don't need a formula.

My original goal was to find a few parameters that give a correct idea (not exact, mind you, just correct) of the handling. At the beginning I was also looking for "the best part to cut with", but since then I figured that it depended much too heavily on considerations external to the sword. So I'm concerned with handling prior impact. Or post miss, eventually :)

For my primary goal I do not really care about vibrations, since through handling they won't happen much. Pivot points, center of gravity, on the other hand, are definitely important in defining how the weapon can be moved, how it reacts to actions.

(Btw, concerning that, I tried to make this diagram to explain how I see things. I don't know if you've seen it, it faded from view pretty quickly... But if you have comments, I'd be glad to hear them and modify the thing accordingly)

Nevertheless, due to their significance in hard impacts, vibration nodes are commonly measured, and recognized as important parameters. So the obvious counter to my approach is "you cannot just forget nodes and expect good results". Which kind of makes sense... So either I explain how the nodes have no effect on anything (well, I can't, because they do have an effect ;) ), or I prove that they are linked to the rest of the parameters, so that having them utterly wrong is not possible. I was quite happy yesterday, because it turns out that last part is actually not that hard :)

Besides, combining that with what I know of impacts and cutting, a lot of the observed effects of the nodes start to make sense, precisely because they are so closely related to pivots. Given the amount of heat that George Turner's article generated at the time, I think it's comforting to reach such a conclusion :)

That being said...

My question about the position of the handle node was not about how it moves, or how its position can be computed. I was looking for observations about its location, particularly on long-handled swords, because I do not own one. If what I'm thinking holds some water, its position can give an idea of the perceived tip control or blade mass. I'd like to know if I should keep looking this way or if there are obvious known counter-examples...

Angus Trim wrote:

One thing to consider, the handle node moves from the "naked blade" to the mounted blade. Before mounting, its not unusual for it to be up on the blade proper a bit, once mounted its on the tang..........

Indeed, and Peter Johnsson made an illustration of that on Albion's website:
http://www.albion-swords.com/articles/images/sword-dynamics.pdf

Well at the moment, I can explain the motions of all the points, save the small displacement of the blade node. Even the huge motion of the hilt node is easily explained without having to know all the details from blade geometry, weight everywhere, etc. precisely because it is tied to pivot points, whose motions are easy to predict.

Thanks for your interest!

Hi Gus!

Thanks for the story... Unfortunately I do not own any sword that shows "deminodes" as far as I understand what that means, which is not much. On the illustrations I've seen they don't look like nodes at all... Not having as much info as I would like to have on the weapons (I don't suppose you measured pivot points on your type XIXs?), it's difficult for me to draw conclusions.

Perhaps the existence of deminodes also implies that in the first mode of vibration, the blade takes a shape that is sufficiently far away from the parabola, so that the blade node and hilt node are further from being associated pivot points. This would mean that their placement is not representative of dynamic balance, explaining some of the differences. Really just throwing out some thoughts here...

Isn't there some hierarchy of the effects, though? Assuming we are cutting a hard, unyielding target:

- If the impact happens too far from the blade node, the blade will vibrate with a disturbing amplitude, thus preventing the "bite" somewhat. That's the only real reason to strike with the CoP that I can see nowadays.
- If it happens at the blade node, there is not much amplitude in vibrations anyway, so the position of the handle node is not important vibration-wise (I suspect that's why it is rarely measured). However, since it is more or less the associated pivot point of the blade node, its position is important because it determines whether the hand feels the impact force or not, or rather what amount.

So what is termed "harmonic balance" mixes two different effects: the position of the node on the blade, and the pivot point associated to this node. It just so happens, on swords that do not have funky waveforms in their first mode, that the hilt node is very close to being the said pivot point. That's how, by checking the "harmonic balance", we end up checking for an approximation of dynamic balance as well.

Problem is that deminodes could disturb this equivalence, so that part of the dynamic balance is missed when just looking at how the weapon vibrates. I doubt it would end being really far off the mark, but perceptibly, it's possible. If you have the same blade node, the same primary hilt node, but deminodes in one case and not in the other, pivot points will probably be different, so the difference would be felt not only in cutting but in handling. Whether it is possible to feel the difference without cutting is another question...

I don't know, what do you think?

Best,

Hi all!
Could anyone tell me what deminodes are?
An other question: Are aft pivot points important because there is the other hand on a longer grip?
Regards!

just a thought: if the hand grips tightly and the rest of the grip is constructed using dampening materials, will it not be possible to calculate the exact pop using the speed of sound in steel and the tone a sword generates when struck? If the first node on the blade is exactly on the hand (where you want it) and you measure the tone (in Hz) the sword generates when struck, you can simply calculate the distance where 1/2 of the sinoid would be, using 5130/(2*Hz) from the first node (5130m/s is the speed of sound in steel, if I recall correctly, on 1/2 of the sinoid would be the second node so the time would have to be half that of a complete sinoid, and thereby half the distance, Hz is simply 1/T)

the distance of the calculated pop for a tapering blade would still be the same, but you would have to calculate the average frequency a lot harder, I will look into this after I am ready working for the day.

Hi Marton,

Marton Pap wrote:

Could anyone tell me what deminodes are?

I'm not quite sure myself ;) There is an article by Michael Tinker Pearce here:
http://www.tinkerswords.com/Harmonics.html
I do not agree with everything written there, but at least there is a picture... Surely Angus could tell us more...

Marton Pap wrote:

An other question: Are aft pivot points important because there is the other hand on a longer grip?

Aft pivot point is important for handling but has nothing to do with vibrations. My interpretation is that it gives you sort of a dynamic length for the weapon. The shorter the distance between the rear of your grip and the aft pivot point, the quicker the blade will realign itself in cutting motions. Note that quicker is not necessarily better, since the position of this pivot is the result of a compromise in the design of the sword.

In general you won't find a vibration node close to the aft pivot point, I believe. Unless you consider higher harmonics that are barely visible to the eye or felt in the hand...

Bram,

Maybe it would be possible, but I think there is little point in doing so, as was said in the other thread... Because if you are able to make the experiment you describe, you can certainly grab the sword, hit the pommel or the flat, watch it vibrate and locate the CoP with all the precision you need. We don't need a millimetric position, because CoP matters for impacts, and you will rarely have the control needed to strike with such a precision, assuming your target is even that small...

The problem with the CoP is not to measure it, it's to know what it means, how to use it, what knowledge it brings us as well as what we cannot deduce from its location.

Regards

Thank you for the reply Vincent!
btw: I didn't ask the AP because of vibrations, I was just wondering about what would be the practical meaning of it (I mean correlation between the practical use and the physical meaning of pivot points), just like you said above. My first guess was the importance because of the rear hand, the second was that it tells much about the blade itself, the third was that I guess most of the parrying happens somewhere around the AP on the blade (Now I got an idea to examine the distribution of nicks on original blades to see if that might be true or not :) ) I didn't think of the realignment part what you sad, so now that is the fourth one :) . The thing that bothers me much is that I assume some correlation between the pivot points and Döbringer's advice not to grab (and pull) the pommel but I don't really know what it is :( At least I have some ideas about the forward pivot point and its part in cutting and thrusting :)
Sorry for being a bit off-topic
Regards!

I agree completely with what you say, Gus, and that's why I precised in my former post "a hard, unyielding target"... In fact I'm more and more convinced that many swords were designed not to achieve the highest performance in a very specific set of conditions, but rather to perform decently in a variety of conditions.

I have also experienced first-hand what you say about tip-cuts (or rather just behind the tip). In fact part of that can be explained with rigid body dynamics, and I think the rest is due to blade profile (given that many cutting swords are thinner at the tip). All other things being equal, a lighter target shifts the most efficient spot of the sword towards the tip. And the more mass there is in the hilt, the more dramatic the effect. So this is almost not noticeable with an axe for example.

So obviously the balance has an influence here, and we saw that a harmonic and dynamic balance are linked, so a sword that does not perform well you will perhaps see as ill-balanced harmonically, and I will perhaps see as ill-balanced dynamically, and possibly we'll both be right :)

I do wonder however if the discomfort that can be experienced with a "bad" sword is really due to vibrations or if it's the shock that is mostly felt. Or if the cut feels bad because the effect on the target is not what we expected given what our hands felt. I guess I do not have enough "bad swords" to experiment ;)

Marton,

No problem for the slight off-topic, I've done enough of it myself :D After all, the thread is about harmonics and dynamics ;)
Im afraid I can't help you much with Doebringer and the pommel. As far as I know, he is the only master to give an advice specific to that, and there are plenty of illustrations that show gripping the pommel. So I'm not sure you can reach a definitive conclusion based only on pivot points... You'll also find that the aft pivot point does not move all that much, whether you take it relative to the pommel or just in front of it. At most, it will move as much as your reference point. And the distance between them (the important thing when launching a cut) changes even less...

Regards

Hi Gus

Angus Trim wrote:

Just so we're clear, I suspect that you and I use dynamic balance differently.

Hmm yes I'm beginning to realize that as well. I can't remember when I first used that term myself :\

Angus Trim wrote:

Not to discount pivot points, but for me they're a subjective measurement, so I don't use them. I'm more interested in how a sword moves, how it "follows the point" in a thrust, linear or looping {possibly the curved thrust can be discussed using points of rotation, which are different from pivot points}, and how a sword "tracks" in the cut, or a cutting motion. How fast it recovers, etc, and so on........

Which I believe is explained entirely by actions of the user, and dynamic properties of the weapon, namely mass, center of gravity, inertia. You could add perhaps one mode of vibration if really the blade is flexible enough to start wobbling in handling (but even on my practice rapier, it's second order).

It's not that I specifically care about pivot points, but for sure they are a way to measure inertia (and that's not subjective, it's proven), and when I start to quantify all the effects you describe, I find a direct interpretation of the result in terms of where some pairs of pivot points are located. It's just so much more easy to visualize with them...

Angus Trim wrote:

I think pivot points can describe part of dynamic balance, but dynamic balance is much more than pivot points......

There are the properties I cited above, and there are actions from the user. Every motion of the weapon is driven by that. So if we don't want to pull the whole user into the study (I wish to compare swords, not users), I cannot really think of something else to add (though I'll grant you that an ergonomic study of the grip would be interesting). I'm talking of the motion here, impacts as I said somewhere already are way, way more complicated.

I think part of your opinion is caused by some kind of a restrictive view on what pivot points really are... And I have the exact same problem with the harmonic theory, in fact :) We're complementary in a way :D

Best,

Hi Vincent

I don't think we're that far apart, really, on pivot points and harmonics. What I said about pivot points and subjectivity, is the measurement, not the actual location of the pivot points. It makes sense to me that pivot points have a correlation to the nodes.

I have been able to get different readings on pivot point locations, by trying to find them with the blade held horizontally, vertically tip up, and vertically, tip down. Also by moving my hand position back on the grip. Then one night, being tired, I unintentionally "muscled" the movement, and the apparent pivot points were several inches off.........

So I don't use it....

To check dynamic balance, I go outside, do my form. Take the sword thru several different paces...... And then I cut with it, to check tracking and behavior. Then do a straight thrust into a target, and a looping thrust into a target.......

But part of the reason that I think that pivot points are only part of the answer for dynamic balance, is I don't see how they can describe things like blade and edge geometry. If the nodes and thus pivot points are in the same postion on three different swords, only real difference is the edge geometry, how can the pivot point describe the dfference in handling?

For instahce..... take three swords, everything the same except edge. One has a blunt edge, .06 inch {1.5mm}. Another has an .03 thick edge, fairly robust edge geometry. the third has a really acute edge........

Handling is similar, but is subtly different. Many folks will tell you that the sharper of the three, has a more "alive" feel in movment........but won't be able to tell you why...... I certainly can't, yet its a phenomena I've experienced a lot......

You mentioned handle, here's something that affects dynamic handling.... and thus the balance.....

How does the pivot point location describe the difference in handling between blade contours? Flattened diamond, vs hex vs fullered, if all weigh the same, and cog is the same?

Again, I can't explain it, but have experienced it........

Pretty subjective stuff, this......Some swords have been described in the past as having an "organic" feel to them. Definitely a subjective term for a subjective feel..... but I'm in line with this at times........

Pivot points are valid. Every bit as valid as nodes....... but they cannot describe a lot of what happens when a sword is in motion, no more than the nodes can.........

Because pivot points are valid, I'd like to encourage you to continue. My thought on this though, is don't expect these to tell you the whole story..... because I don't think they will......

I do not mean to hijack this thread, and consider this related when honestly contemplating Angus's position (I am interpreting it as refining design of swords by feel and handling.)

If the standard for giving the statistics of CoP/ CoB is from the guard, how appropriate is this for those long grip models such as an ATrim 1593, or Albion Munich? CoP is still a good place for a hard impact, but if the user wants docile handling, might they not appreciate the CoP/CoB being a little closer to the "strong" of the blade?

I consider the Munich to be very docile. I can execute my poor zwerchau technique as fast (actually faster) with it as I can with any other sword I have been lucky enough to use and test cut with. The leverage afforded by the long grip undoubtedly influences my impression of the sword.

My point, in such a case, is that the guard is probably not very close to the location where the wielder's perception of handling traits are likely to be derived. I suspect the relevant location is nearer to the center of the two handed grip.

This line of reasoning could be all wrong, but similarly for one handed grip I would expect the most relevant location would be closer to the center of the one handed grip (roughly 2" down along the grip from the guard, rather than right at the guard.) As far as harmonics go, I think Vincent has already stated that they do not vary as greatly within a well designed tang and grip. Balance and inertia are still fully relevant relative to where one holds the sword though, rather than at the centerline of the guard.

My "nit pick" here is that CoP is relative to where you define it. In any practical engineering calculation, CoP is defined relative to some location where a motor, gear, or possibly a wrist muscle (such as center of hand grip area on a baseball bat..actually dictated by an ASME standard for giving CoP of bats) applies a torque to make something rotate. There are an infinite number of them for any object. If you hold a blade for a "murder stroke", there is a completely new CoP for that application located down somewhere near the guard, and it depends upon where you grasped the blade. Vibrational nodes are not the whole story here either. They do matter, but you can probably relate to the concept of a baseball bat being gripped in the "wrong location" with resulting hand shock, even though it was an o.k. bat. Everything is relative to where you hold or pivot a rotating object when discussing CoP.

I consider most forum participants here to be quite sharp mentally, and capable of estimating rough rotational characteristics of swords with greatly similar geometries and grips (using only CoB/CoP and weight) regardless of where CoP has been defined. However, if you want to compare two swords where the grips are drastically different, blades similar, now the true perception in actual handling might be much harder to predict using the guard as a reference point for CoP.