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Thread: Polar Moment in swords

  1. #1
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    Polar Moment in swords

    I'm making this post 'sticky' like the other posts on cutting and harmonics, as I believe it too is fundamental to sword performance.

    OK- I'm stronger on the concept than the specifics, and in fact have been using this term innapropriately! Thinking about it, when I have referred to a sword as having a "High Polar Moment' I believe that I should actually have been saying 'Low Polar Moment.' This isn't going to be a perfect description- we're still sussing some of this out...

    So what the heck is 'Polar Moment?' With regards to swords the 'Poles' are the tip and tip of the pommel. If the sword's tip is 36" from the strong hand (the point of rotation) then the opposite pole is the tip of the pommel- usually only a few inches from the POR. The type and amount of distal and profile taper and the weight of the hilt (particularly the pommel) are what determines a swords 'Polar Moment.'

    A case ito illustrate the point are two swords that I made recently- the Falchion and the Type XVII. Both swords have recently been seen on this forum so pics should be easy to find. The Falchion has a great deal of mass near the point because the blade actually gets wider towards the point- reverse profile taper. It does distal taper significantly, but not enough to off-set this expansion in width. The Type XVII has extreme profile taper and has a hex cross section for much of it's length. The Falchion weighs 2lb14oz and the Type XVII weighs 3lb4oz. Yet the Type XVII is universally percieved as being 'lighter' than the falchion. Both swords were handled by dozens of people and they all had the same reaction. This is because of Polar Moment. The Falchion with it's heavy tip and pommel has a high Polar Moment- it's mass is distributed more equally along it's length. The short distance from the POR (Point of Rotation) to the pommel decreases this effect but not significantly as there is relatively so much mass at the opposite pole (the tip of the sword.) This makes the sword slow in rotation. On the contrary the type XVII with it's extreme profile taper has only a tiny percentage of it's overall weight at the tip, and again the weight of the pommel is relatively close to the POR. This means that the greatest percentage of the sword's mass is located near the POR- the sword rotates easier as most of the mass that you are moving is close to the POR so you have good 'leverage.' Another way to say this is that the sword has a small 'Moment of Inertia' (MOI) towards the tip. The Type XVII has a Low Polar Moment, making it fell handier- thus it is percieved as lighter because it is easier to move and change direction of the point. Savvy?

    So- take two swords with the same length of blade and the same weight and Center of Gravity (COG) For argument lets say that both swords have 32" blades and balance four inches from the cross. One sword is made from 3/16 inch stock and has little profile and no distal taper with a heavy pommel to draw the COG back to four inches. The classic 'sharpened bar' type of sword. The other sword is 3/8" thick at the base but has a good amount of profile and distal taper. This sword doesn't need a heavy pommel to draw the COG back because of the distal taper, so the swords are of equal weight. Despite both swords weighing the same and having the same center of gravity the second sword will feel lively in the hand, and the first sword will feel like a club. Since it has little profile or distal taper it has a much greater percentage of it's mass farther from the POR- it has a high 'Moment of Inertia' (MOI) at the tip compared to the other sword, so the tip moves slower making the sword feel heavy and clumsy. This sword has a High Polar Moment. The second sword's mass is concentrated near the POR- it has a low MOI at the point or a Low Polar Moment. It feels lively and quick in the hand, and most people will tell you that it is lighter than the first sword even if intellectually they know the swords are the same weight! The second sword could in fact be heavier than the first sword and still feel lighter to most people because of the ease of moving the point.

    This is also why a shorter sword often feels 'handier' than a longer sword of the same weight- It has a shorter 'Moment Arm' which moves the tip's mass closer to the POR- reducing it's Polar Moment.

    A note- when we say the tip moves easier and faster, what we are talking about is accelleration. An axe can move bloody fast at the end of the swing, but the mass of the head being at the end of the Momeent Arm means that it accellerates more slowly.

    The practical effects of this from what Gus and I have observed- the sword feels lively in the hand, tends to be more forgiving in it's Harmonic Balance and tends to have a long COP. All good things...

    OK- now for the questions and corrections...
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    Differ on the terms

    but for the most part, I agree. What people feel is called torque. Hold a five pound weight at your side and you feel five pounds. Hold that same weight straight out in front of you and you feel about 120 inch/pounds in you shoulder. Torque is in this case, mass times the distance from the joint or POR (middle finger?) Any mass aft or rearward of this point will act as a counter balance for the blade, reducing torque. A well balanced heavier XVII can be livier than a lighter falchion!

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    Re: Differ on the terms

    Originally posted by Michael Bouse
    but for the most part, I agree. What people feel is called torque. Hold a five pound weight at your side and you feel five pounds. Hold that same weight straight out in front of you and you feel about 120 inch/pounds in you shoulder. Torque is in this case, mass times the distance from the joint or POR (middle finger?) Any mass aft or rearward of this point will act as a counter balance for the blade, reducing torque. A well balanced heavier XVII can be livier than a lighter falchion!
    Very well and thoughtfully said- I hadn't thought of it in terms of torque but I think that you are spot-on.
    Tinkerswords.com Fine knives, swords and daggers in the style of the European Middle Ages and Viking Era

    "Then, one night as my car was going backwards through a cornfield an ninety miles per hour, I had an epiphany..."

    Luke 22:36 Then said he unto them, But now, he that hath a purse, let him take it, and likewise his scrip: and he that hath no sword, let him sell his garment, and buy one.

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    Re: Polar Moment in swords

    Originally posted by Michael Tinker Pearce


    OK- now for the questions and corrections...
    So a "broad" generalization is that a sword with HPM can be thought more "hack and slash", and one with LPM as more "cut and thrust".
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    Re: Re: Polar Moment in swords

    Originally posted by D. Opheim
    So a "broad" generalization is that a sword with HPM can be thought more "hack and slash", and one with LPM as more "cut and thrust".
    Broadly speaking I suppose you could say that- at least about straight double- edged swords.
    Tinkerswords.com Fine knives, swords and daggers in the style of the European Middle Ages and Viking Era

    "Then, one night as my car was going backwards through a cornfield an ninety miles per hour, I had an epiphany..."

    Luke 22:36 Then said he unto them, But now, he that hath a purse, let him take it, and likewise his scrip: and he that hath no sword, let him sell his garment, and buy one.

  6. #6
    I have a quibble with this, b/c of the implication in handling, which is that a blade with a low polar moment out by the point is necessarily handier.

    My experience comes from swords and axes, but particularly axes. If you want to make a long-handled axe sing sweetly in your hand, you taper the shaft so more of the weight is out by the head... aka make the balance "worse." A long-axe with a tapered shaft handles MUCH more quickly in the hand than one with a uniform shaft.

    I have no trouble whatsoever with tip-heavy swords that are well-made. They're just as quick... they simply require a different method of handling in the hand. Its still a lever: so long as one end moves, the other must also.

    Where the difference in moment occurs seems much less important to me for handling as that the weapon clearly possesses a moment differentiation, rather than having it be fairly uniform like a stick or crowbar.
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    Good thread Tinker.

    I would like to add another type of sword in your 2 sword comparison. Same size and weight as your 2 examples, but with you and Gus call "tip mass", say a leaf blade. And, If you can remember my reverse tapered fuller sword, I would like to compare that also in your polar movement equation. The last 2 swords I just mentioned, have distal taper, like your good example. How will these types of swords compare to the sharpened bar in acceleration, polar movement?

    BTW, missed you at Gus' last Sunday. Had a ton of questions, just ask Gus Thanks, Jerry
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  8. #8
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    Originally posted by Russ Mitchell
    I have a quibble with this, b/c of the implication in handling, which is that a blade with a low polar moment out by the point is necessarily handier.

    My experience comes from swords and axes, but particularly axes. If you want to make a long-handled axe sing sweetly in your hand, you taper the shaft so more of the weight is out by the head... aka make the balance "worse." A long-axe with a tapered shaft handles MUCH more quickly in the hand than one with a uniform shaft.

    I have no trouble whatsoever with tip-heavy swords that are well-made. They're just as quick... they simply require a different method of handling in the hand. Its still a lever: so long as one end moves, the other must also.

    Where the difference in moment occurs seems much less important to me for handling as that the weapon clearly possesses a moment differentiation, rather than having it be fairly uniform like a stick or crowbar.
    A former axe-man myself, I know that they can be quick- but the high polar-moment does affect the ability to change direction of the cut. Certainly you can work with this and achieve good results. Low Polar Moment increases tip accelleration, not necessarily peak velocity of the cut. Also- a 'tip heavy' sword (with the COG relatively far forward) can still have a Low Polar Moment- I have handled numerous Tulwars that balance 6-8 inches from the hilt, but the distal taper and all-steel handle reduce PM to an extent that they still feel quite lively in the hand.
    Tinkerswords.com Fine knives, swords and daggers in the style of the European Middle Ages and Viking Era

    "Then, one night as my car was going backwards through a cornfield an ninety miles per hour, I had an epiphany..."

    Luke 22:36 Then said he unto them, But now, he that hath a purse, let him take it, and likewise his scrip: and he that hath no sword, let him sell his garment, and buy one.

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    Originally posted by Jerry Bennett
    Good thread Tinker.

    I would like to add another type of sword in your 2 sword comparison. Same size and weight as your 2 examples, but with you and Gus call "tip mass", say a leaf blade. And, If you can remember my reverse tapered fuller sword, I would like to compare that also in your polar movement equation. The last 2 swords I just mentioned, have distal taper, like your good example. How will these types of swords compare to the sharpened bar in acceleration, polar movement?

    BTW, missed you at Gus' last Sunday. Had a ton of questions, just ask Gus Thanks, Jerry
    Sorry to have misssed you, too! I had my neice's 18th birthday celebration to attend that day however. My phone number is 206-772-6720- give a call we can talk or meet for your 'ton' of questions.
    Tinkerswords.com Fine knives, swords and daggers in the style of the European Middle Ages and Viking Era

    "Then, one night as my car was going backwards through a cornfield an ninety miles per hour, I had an epiphany..."

    Luke 22:36 Then said he unto them, But now, he that hath a purse, let him take it, and likewise his scrip: and he that hath no sword, let him sell his garment, and buy one.

  10. #10
    do you mean is that your type XII has smaller moment of inertia (I guess it's what you mean by polar moment ?) with the respect to the axis going perpendicular to the sword's plane through somewhere in the hilt ?

  11. #11
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    Re: Polar Moment in swords

    Originally posted by Michael Tinker Pearce
    So- take two swords with the same length of blade and the same weight and Center of Gravity (COG) For argument lets say that both swords have 32" blades and balance four inches from the cross. One sword is made from 3/16 inch stock and has little profile and no distal taper with a heavy pommel to draw the COG back to four inches. The classic 'sharpened bar' type of sword. The other sword is 3/8" thick at the base but has a good amount of profile and distal taper. This sword doesn't need a heavy pommel to draw the COG back because of the distal taper, so the swords are of equal weight. Despite both swords weighing the same and having the same center of gravity the second sword will feel lively in the hand, and the first sword will feel like a club. Since it has little profile or distal taper it has a much greater percentage of it's mass farther from the POR- it has a high 'Moment of Inertia' (MOI) at the tip compared to the other sword, so the tip moves slower making the sword feel heavy and clumsy. This sword has a High Polar Moment. The second sword's mass is concentrated near the POR- it has a low MOI at the point or a Low Polar Moment. It feels lively and quick in the hand, and most people will tell you that it is lighter than the first sword even if intellectually they know the swords are the same weight! The second sword could in fact be heavier than the first sword and still feel lighter to most people because of the ease of moving the point.
    i'm confused a bit here i think... i'm not sure how two swords with the same weight and same COG could 'feel' much differant. wouldn't the tip-heaviness or lack thereof be a factor of the COG, and therefor play the biggest role in high or low polar moment? unless we are talking the bare blades themselves being even, and the fittings changing the COG.
    it just seems to me that a sword with low polar moment would have a COG closer to the hand, and vice-versa with a high polar moment.

    also... polar i can understand, but where does the moment come from? i kept thinking in my head "movement"...

    good informative post. hopefully i can understand better what it all means soon. ; )

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    Moment

    Moment is short for moment of inertia. Yes it deals with movement over time. At rest, the sword will have a moment. As you start moving it, the moment of inertia changes. Mass x V xV. As the speed increases, the inertia goes way up. A thing at rest wants to stay at rest, a thing in motion wants to stay in motion.
    That said, changing the mass' shape, lenght, CG ect effects its handling charactorists.

  13. #13

    Re: Moment

    Originally posted by Michael Bouse
    Moment is short for moment of inertia. Yes it deals with movement over time. At rest, the sword will have a moment. As you start moving it, the moment of inertia changes. Mass x V xV. As the speed increases, the inertia goes way up. A thing at rest wants to stay at rest, a thing in motion wants to stay in motion.
    That said, changing the mass' shape, lenght, CG ect effects its handling charactorists.
    Moment of inertia (I) does not change, unless the axis of rotation are changed. It's defined as M*R*R, for one particular axis and a point mass like body, and in general it's a 3x3 matrix.

    What changes is called an angular momentum, which is I*omega, where omega is the angular velocity, omega=V/R, so it's mvr. Kinetic energy is I*omega*omega/2, so this one is quadratic in velocity.

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    Sure it changes

    MxVxV If you go from 5ft/sec to 10 ft/sec the net effect if 25 to 100. As for KE, KE= 1/2mvv or 1/2mwwrr., where w=radians and r = radias. Okay, angular momentum for a complex thing, say one inch segents of your sword, is given as Inertia=the sum of m1r1r1+ m2r2r2= m3r3r3 ect. KE=1/2(sum mrr)ww. Don't know if this site is ready for radians or omega/ Hope you like crunching numbers! I've done it but don't love it.

  15. #15

    Re: Sure it changes

    Originally posted by Michael Bouse
    MxVxV If you go from 5ft/sec to 10 ft/sec the net effect if 25 to 100. As for KE, KE= 1/2mvv or 1/2mwwrr., where w=radians and r = radias. Okay, angular momentum for a complex thing, say one inch segents of your sword, is given as Inertia=the sum of m1r1r1+ m2r2r2= m3r3r3 ect. KE=1/2(sum mrr)ww. Don't know if this site is ready for radians or omega/ Hope you like crunching numbers! I've done it but don't love it.
    1. I think that the cross product of a vector with itself will be 0, so mxvxv will always be 0 (or it's understood as mvv ?)
    2. mxvxv has nothing to do with "moment of inertia".
    Here
    http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html
    you can find formulas for moments of inertia for different bodies, moment of inertia is a characteristic of object's material and shape, not its motion.

    Concerning moments of inertia for "complex things", I believe it's actually an integral that can be resolved to summation in case if the body is represented as point masses.

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    Symbols cloud things up

    Sometimes we forget what the symbols represent.
    KE=1/2mvv in the linear.
    In the angular or circular, vv is replaced with (radius x radians) squared. Lets take a radius of one meter and a rotation of 1/4 turn in one second(1.57 radians/sec). The product is 1.57 meters/sec and squared its 2.965 meters/sec. There is the velosity.

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    wow, this thread suddenly got all math-sciency.
    i uh... i think i'll just take it at face value that you guys are right.
    physics is awesome and all, i just don't have the head for remembering all the formulae and such.
    thanks for trying though. i think i need to go cut something now.

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    I stand corrected

    Went back and checked the books. For rigid object the Moment of Inertia doesn't change. The rate of rotation does change the KE

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    I should note- I am making empirical observations here and I have to leave the math to you physics people. I am using the terms of modern physics because they fit the empirical data but when you get into the math and formulas I can get lost pretty quick...

    Anyway- the observed effects are as stated. It's observable, it's consistant and repeatable. I just can't use math to tell you why...
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    "Then, one night as my car was going backwards through a cornfield an ninety miles per hour, I had an epiphany..."

    Luke 22:36 Then said he unto them, But now, he that hath a purse, let him take it, and likewise his scrip: and he that hath no sword, let him sell his garment, and buy one.

  20. #20
    This is a very interesting subject. I have done a little research after reading this and I think the term, rather than "polar momentum" would be "moment of inertia" (http://en.wikipedia.org/wiki/Moment_of_inertia).

    I've tried to take the idea a little further with a thought experiment in the following thread under Chinese Swords and Swordsmanship:

    http://forums.swordforum.com/showthr...threadid=56669

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    as a 'sword-swinger' with a little bit of 'sword-maker' knowledge: My gladius with a relatively light pommel and waisted(leaf-shaped, sorta) blade and a balance point about dead-on at the top of the hand-guard, has good 'acceleration' but poor 'bite' if you both fence and sword-and-board, you know what I mean. I can get there faster, but don't do as much damage when I get there.

    disagree! go on, disagree! I DARE you!
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    How could I possibly disagree with your observations about your sword? The COG does seem a bit close for good 'bite.' It is possible to make a sword with the COG farther forward and still preserve a low polar moment however.

    A question was asked some time ago about whether LPM swords would be easier to deflect than swords not posssessing this quality- the answer is that yes, they would. The potential for greater accelleration and energy in the strike off-sets this to some degree- also if the blade is deflected recovery or the 'flow' into a counter will be quicker as well. Preferences seem to have varied in period as well, as both types of sword are present. Personally I find that longsword techniques work better with a sword that rotates easily, and I like an extremely agile sword for Saviolo's rapier methods- but then I like to cut, so that's no surprise!
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  23. #23
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    Quote Originally Posted by D. Opheim View Post
    So a "broad" generalization is that a sword with HPM can be thought more "hack and slash", and one with LPM as more "cut and thrust".
    I think this is only true up to a point. In thrusting you want the point to lag the hand slightly and not feel like it has a mind of its own. Add to this the need for "presence" to work around the opponent's blade rather than being worked by the opponent's balde, and I've found a slightly less lively sword with a COG between 5 1/2" to 6 1/2" to be a much better thruster, esp. when wrapping a finger.

    Mike
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    Quote Originally Posted by Sam Anderson View Post
    as a 'sword-swinger' with a little bit of 'sword-maker' knowledge: My gladius with a relatively light pommel and waisted(leaf-shaped, sorta) blade and a balance point about dead-on at the top of the hand-guard, has good 'acceleration' but poor 'bite' if you both fence and sword-and-board, you know what I mean. I can get there faster, but don't do as much damage when I get there.

    disagree! go on, disagree! I DARE you!
    I think there is a momentum distribution thing going on in this concept of "bite" that exists in addition to harmonics. If the total momentum of the sword from the point of contact to the tip is close to the total momentum from the point of contact to the grip (or possibly including the arm to the elbow), the sword will bite true. If the tip has more momentum, it will try to rotate the sword around the cut rather then cut straight in. If the base has more momentum, it will want to recoil more. This cannot be divorced from harmonics becuse hitting away from the center of momentum is that provides the force to create displacement to begin with.

    Mike
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    Quote Originally Posted by Michael Stora View Post
    In thrusting you want the point to lag the hand slightly and not feel like it has a mind of its own. Add to this the need for "presence" to work around the opponent's blade rather than being worked by the opponent's balde, and I've found a slightly less lively sword with a COG between 5 1/2" to 6 1/2" to be a much better thruster, esp. when wrapping a finger.

    Mike
    Granted I have had over a year to test out the theory and have learned quite a bit.
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