Sliding weights

[Ian]

Originally Posted by Jens Nordlunde

Imagine you had an arrow with a sliding weight on the haft. Before you took a shot, you pulled the weight back and when the arrow hit the target, the weight would make sure the impact was bigger than normal – would that work?

Jens:

I don't think this would have any effect, but perhaps it might if the weight shifted from the fletch to the head at the time of impact. Because we are talking about a piercing shaft, all energy would be concentrated at the point of impact, and the mass of the arrow lines up directly behind that point -- so however mass is distributed behind the point of contact would be immaterial, unless possibly if part of that mass is moving along the shaft at the moment of impact. Even then, I think the effect would be small and would need to be weighed against the effects of a rear-weighted arrow on its flight and accuracy. The small potential gain in penetrating power could well be offset by impaired accuracy.

Ian.

[Ian]

Here is what my engineering colleagues had to say about the question before us. I have translated from engineering-speak as well as I could.

The problem of a weight (mass) moving along a rod that in turn strikes another object is really quite complicated and we do not have enough information to answer your question. Here is a somewhat lay interpretation of the problem, which we have modified from The Physics of Baseball by Robert K. Adair.

Basically, the properties of a rod (sword, bat) relevant to striking another object squarely are defined by three weight distributions, or three moments.

1. The sum of the weight of each part of the sword, which is just its total mass (the zero moment)

2. The sum of the weight times distance, measured from the handle, of each piece of the sword (the first moment)

3. The sum of the weight times the square of the distance for each piece of the sword (the second moment or the moment of inertia)

There are three key positions along the sword that follow from these three moments.

a. The center of gravity
b. The center of inertia
c. The center of percussion

There are three additional factors that need to be considered.

d. The elasticity of the sword
e. The resonant frequency of the blade
f. The position of the vibrational node

Although the center of percussion and the vibrational node will be close to each other, they are not the same.

For a sword of fixed mass distribution, we can determine fairly simply the various points that correspond to the three moments.

The center of gravity is just the balance point. The center of percussion can be found by holding the sword lightly by the end of the hilt and striking the blade gently with a hammer; when the blade is struck at the center of percussion there is no detectable movement at the hilt. In most cases the center of percussion is very close to the vibrational node -- when the blade is struck at the vibrational node no vibrations are felt at the hilt.

The center of inertia can be determined by placing the sword on a frictionless surface (such as an ice rink) and pushed away. When the push is placed at the center of inertia the sword will move away without any appreciable rotation.

Each of the moments are manifest in obvious ways. The weight is felt by holding the sword vertically. The force required to hold the sword straight out in front of you at arm's length is proportional to the first moment. The force required to wave it back and forth vigorously when it is vertical is proprtional to the second moment. This second moment contributes most to the "feel" of the sword and is the factor most important to the user.

The elasticity is determined by the blade's resilience near the point of impact; a resilient blade may store energy upon impact and return that energy to the target.

The resonant frequency is a measure of the energy loss when a target is struck at a point along the blade away from the vibrational node. A higher frequency indicates a larger (i.e., longer) "sweet spot." Swords with longer blades and thicker handles will display higher vibrational frequencies and long sweet spots.

This is what we know about items that have a fixed mass distribution. When you add a varying mass distribution, the problem becomes more complex. When the weight distribution shifts, all of the moments change.

A sliding mass would create a tip-heavy sword, moving the centers of gravity, inertia and percussion away from the hilt. Depending on the fraction of the total mass that is moving and its final resting place along the blade, the respective moments may well be centered quite close to the tip, and essentially one would have a club. Such a shift in mass would likely make a clumsy and slow weapon.

We will think some more about this problem but it seems that any substantial shift in mass would produce a sword that could be difficult to control and would probably slow its action. How much of an effect would depend on the fraction of total mass that was shifting and the distance it traveled away from the hilt.

[Rivkin]

This list seems popular among the sword community - it's not the first time I've seen similar ideas expressed concerning the waves for example.

My problem is that for example it's hard for me to understand why the center of gravity is going to be a node for all waves (it should not be for at least for the waves with an odd number of halfwavelengths). Concerning the hilt, it seems more like a boundary condition to me, rather than a center of gravity. Concerning longer swords having higher frequencies and wider diaposon, it seems counter-intuitive to me - I would expect smaller swords to have larger frequencies and bigger separation in between of individual modes, but that's just my guess.

I'll be honest, I don't understand some of the ideas expressed above. Concerning the sliding mass question, again, what are the possible benefits of this construction vs. simply high momentum fixed mass weapon - nothing simple comes to mind.

[Ian]

Originally Posted by Rivkin

... My problem is that for example it's hard for me to understand why the center of gravity is going to be a node for all waves (it should not be for at least for the waves with an odd number of halfwavelengths).

The center of gravity is a balance point and the centers of inertia and percussion are located at different points along the blade. The respective centers are features of the weight distribution and mechanical properties of the sword (which mostly comprises the blade and the tang).

Waves do not originate from any of these centers. Waves are set up at the point of impact and spread out from that point. The further away the point of impact is from the vibrational node, then the more vibration will be transmitted along the blade and will be felt in the handle.

Originally Posted by Rivkin

... Concerning the hilt, it seems more like a boundary condition to me, rather than a center of gravity. Concerning longer swords having higher frequencies and wider diaposon, it seems counter-intuitive to me - I would expect smaller swords to have larger frequencies and bigger separation in between of individual modes, but that's just my guess.

The tang is continuous with the blade and would not be a boundary condition. Depending on the properties of the handle covering materials there may be some dampening of the vibration transmitted from the tang to the hand. My colleagues assure me that physics dictates that the longer the blade, the longer the sweet spot and the higher the frequency of vibrations.

Originally Posted by Rivkin

... Concerning the sliding mass question, again, what are the possible benefits of this construction vs. simply high momentum fixed mass weapon - nothing simple comes to mind.

I cannot see any real benefits to a sliding mass. That was the conclusion of my esteemed colleagues also -- they are skeptical that any greater force could be achieved, and the sliding mass would introduce unpredictable and inconsistent properties of the sword depending on how it was wielded.

Ian.

[fearn]

Hi Ian,

I think your overall conclusion is probably right--that a sliding mass is not beneficial, although it might be less troublesome in an executioner's sword.

I won't pretend to be a physics expert, but I do know a few things about swords and rods.

One thing that confused me was the difference between center of inertia and center of gravity. These are different because....?

So far as the vibrational nodes go, my limited observations are that straight swords are quite a bit like rods: the vibrational nodes are at the geometric center and the quarters. HOWEVER, the center of gravity doesn't have to be at any of these points. To give a crude example: imagine a rod two- thirds metal. It should be obvious to most people that the point of balance will be fairly close to the center of the metal part, because the wood is much lighter. In a sword with a heavy pommel and lighter blade, you can put the center of gravity and/or inertia pretty much where you want it.

So far as longer blades having bigger sweet spots due to higher vibrational frequency, I'll admit that I'm confused too. I agree that the longer blade should have a bigger sweet spot, but I'd bet a fair amount that it would have a lower frequency, just because it's longer. This is the same reason that cellos generally play lower than violins: the frequency is lower, not higher, in a longer string. I'm guessing that the word we're looking for is longer wavelength and bigger amplitude.

However, I'm still very glad that we had an engineer look at it. Now, if someone will get out there with the PVC tub and ball bearings, and find out what a sliding weight feels like when you swing it, we can all rest easily....

[Rivkin]

1. With given definitions the center of gravity and the center of inertia will be the same.

2. If its possible, I would really like to see the formula they use for sword's frequency as a function of length (do they consider it a string ? a thin and long prism ?).

3. Concerning tang not being a b.c., or even a separate body, I would prefer to hold a vastly different opinion.

4. Concerning waves propagating in swords and nodes - propagating waves usually do not have nodes. When people talk about nodes, they usually speak about standing waves, i.e. steady state solutions etc.
I suspect that the logic was that if sword can be considered a string, than a full wavelength standing wave will have a node in the middle, but it will basically be true only for even halfwavelengths mode... Plus I'm really too lazy to calculate the modes of a string with a variable mass, so I don't know how big percentage of the waves will have nodes at the center of mass.

5. Concerning the center of percussion - as far as I remember (and I remember it very poorly), the center of percussian is when you hit it, all the momentum is transfered into the rotation movement of the sword, without any daggling down or up.

[Jens Nordlunde]

Hi Ian,

Thank you for using your time on this topic, and thank you to your colleagues for their time. I had in the start expected the problem to be les complicated than it is, and I cant say that I can follow all the explanations, so I will have to read it one or two more times and see if it helps.

Fearn, I don’t have a sword with a sliding weight, only one with steel balls, and swinging that, the moving of the balls does not make much difference, it would not as the balls are not very heavy.

I think the conclusion is, like several has stated, that sliding weights on swords are non existent, and should such a sword be found, then it must have been made as an experiment – not for use.

[Ian]

Hi Kirrill:

You raise some interesting points and I will try to deal with them as best I can. My college physics is but a distant memory!

Both of my local contacts went on vacation on Friday, and will be out of the office for the next four weeks. Academics do very well with vacation time. I will do my best.

Ian.

Originally Posted by Rivkin

1. With given definitions the center of gravity and the center of inertia will be the same.

As described above, the respective moments vary with distance of the various components of mass from the handle, raised to the power of 0 (the zero moment, which is simply the total mass); with distance raised to the power 1 (the first moment); and with distance raised to the power of 2 (the second moment). The center of gravity relates to the first moment, the center of inertia relates to the second moment. If mass is distributed uniformly along the length of the rod, then I believe that the center of gravity and center of inertia will be the same. When mass is distributed unequally, then the two will be different. The difference will be demonstrated by the two tests I listed.

Originally Posted by Rivkin

2. If its possible, I would really like to see the formula they use for sword's frequency as a function of length (do they consider it a string ? a thin and long prism ?).

Need the experts for this one. I believe they modeled this as a solid rod.

The frequency we are talking about, then, is the resonant frequency of a solid rod, which (if I recall correctly) for a given diameter varies with the density of the material and its length. When we talk about a string, there is also a factor for the rigidity of the material or tension applied (a taught string resonates at a higher frequency than a slacker string). The resonant frequency is fixed for a rod of given dimensions and homogeneous construction. The amplitude of the vibration varies with the distance the rod is struck away from the resonant node.

An interesting example is the aluminum (aluminium) baseball bat, which has an outer aluminum shell and an inner core that is air-filled. Striking a ball with such a bat produces a brief, high-pitched "ching," and a lower-pitched "thunk." The higher pitched sound reflects the resonant sound of the metal shell, and the lower-pitched sound comes from resonance in the air-filled chamber.

These sounds are hard to distinguish with the human ear but apparently have been measured with sophisticated recording equipment. The low frequency sound is just a few hundred cycles per second, approaching the limits of detection for the human ear.

Originally Posted by Rivkin

3. Concerning tang not being a b.c., or even a separate body, I would prefer to hold a vastly different opinion.

For full tang construction, there should be no boundary condition because the tang is essentially an extension of the blade. This is the same situation as a baseball bat, and the handle presents no boundary condition in that example.

As I mentioned above, there may be dampening of the vibrations by materials around the tang. For partial tang construction, I am unsure how much of a boundary condition there may be. It probably varies with the width and length of the tang, and again the wrapping materials will be important in how much dampening of the vibrations might occur for the user.

Originally Posted by Rivkin

4. Concerning waves propagating in swords and nodes - propagating waves usually do not have nodes. When people talk about nodes, they usually speak about standing waves, i.e. steady state solutions etc.
I suspect that the logic was that if sword can be considered a string, than a full wavelength standing wave will have a node in the middle, but it will basically be true only for even halfwavelengths mode... Plus I'm really too lazy to calculate the modes of a string with a variable mass, so I don't know how big percentage of the waves will have nodes at the center of mass.

This one requires the experts. The test described above speaks to a property of standing waves, I think, but the center so defined also identifies the "sweet spot" which relates to properties of propagated waves also -- at least that was how it was explained to me.

With respect to analogous models, I believe that a string as we usually think of it is probably not the correct one. A string can have variable tension. If we exert enormous tension on a string, and essentially make it highly inflexible or "rigid," then we may approach a more representative model. A metal rod has a high degree of rigidity, which is essentially constant for the purposes of this discussion.

Originally Posted by Rivkin

5. Concerning the center of percussion - as far as I remember (and I remember it very poorly), the center of percussian is when you hit it, all the momentum is transfered into the rotation movement of the sword, without any daggling down or up.

Could be, but we need the experts for this one too.

[Jeff D]

Hi Rivkin

I would like to address some of your questions and hope it can add to Ian's excellent answers.

Originally Posted by Rivkin

1. With given definitions the center of gravity and the center of inertia will be the same.

Inertia is the tendency for a body at rest to stay at rest or stay in uniform motion. It requires an external force to alter this state. With our sword example I think its relevance is more in the handling of the sword than in its ability to cut as per Jen's original question. So I would agree with you that for all intense purposes it would be OK to consider it the center of mass.

2. If its possible, I would really like to see the formula they use for sword's frequency as a function of length (do they consider it a string ? a thin and long prism ?).

I am not sure what model the engineers used, however I think it would be safe to use a bar with a clamped (fixed) end (similar to a hilt) as a model. Using this model the formula is;
F(1) = 0.162 {a/(L squared)} {the square root of(Y/d)} where a=thickness, L is the length of the bar, Y is Young's module (which is a variable of the elasticity of the material), and d is the density.

3. Concerning tang not being a b.c., or even a separate body, I would prefer to hold a vastly different opinion.

Using my model I would not consider the tang in the Length formula as the hilt and tang would be the fixed portion of the bar.

4. Concerning waves propagating in swords and nodes - propagating waves usually do not have nodes. When people talk about nodes, they usually speak about standing waves, i.e. steady state solutions etc.
I suspect that the logic was that if sword can be considered a string, than a full wavelength standing wave will have a node in the middle, but it will basically be true only for even halfwavelengths mode... Plus I'm really too lazy to calculate the modes of a string with a variable mass, so I don't know how big percentage of the waves will have nodes at the center of mass.

I think again we are confusing some of the issues. The vibrations only affect Jen's question in reguards to the energy transfer. The less vibration, the more energy transfered from the kinetic energy of the swing into cutting energy. If there is a lot of vibration then a lot of energy is wasted. The vibration will obviously affect the handling as well. I think a complex bar with varing thickness, fullers etc. would set up a series of standing waves that would cancel each other out so that there will be minimal palpable vibrations.

5. Concerning the center of percussion - as far as I remember (and I remember it very poorly), the center of percussian is when you hit it, all the momentum is transfered into the rotation movement of the sword, without any daggling down or up.

I am not sure of what your question here is but I think you are correct in that the center of percussion is where the blade strikes the sword, and should there for be where the resistance of the object balances.

Hope this adds a bit.
Jeff