arrakis math skills needed.

[bo1]

ok so i have athought about counter wieghting, i need some mathmatics to understand it better.

so it is my contention that counter wieghting my sword in the pommel is no good. important i grip the swrod right where the blade starts, index finger is 8 inches from the end of the pommel. i think that adding the wieght 8 inches below my index finger just makes for more rotational wiehgt i have to work at to get moving. but i wonder if the c.o.g. moving closer to my hand out performes the increase in rotational mass.

my theory is the the counter wieght for my personal style grip should be applied to where i grip the sword, high on the handle, near the blade start.
i dont use kite spar very much, and i prefer a 13 oz sword roughly 33" total length, balance point on my swords is usually just a couple inches towards the handle from center. i would like to get this another inch ot 2 closer, planning to add 1 oz to the counter wieght. i will put it at the logical location, once the out come of this question is finished

i know the math of arrakis will explain this and give me the answers i would like to know. feel free to take some time and give it some serious thought, i would enjoy a lengthy read of this topic.

if you do not have the equations and mathematics to back up your argument then please remain silent, i am not looking for opinion, anecdotal evidence, or some idiot that thinks they know something i need to hear.


thanks arrakis in advance.

[Thomas MacFinn]

The thing you have to remember that your sword is doing more than rotating. Before you figure in rotational forces, you need to balance the sword in the first place. Imagine your sword is a first class lever (like a see-saw).

F=m*d

Assume the total length and the balance point and the blade construction of two swords are the same. If you have a bar running up one handle and a pommel weight on the end of the 2nd handle, because the center of the weight is farther from the balance point on the pommel weight, you need less weight to balance the weight and length of the blade. On the pommel weighted sword, the total weight of the sword is less.

According to the classic hoop and disk experiment, if the total weight of the two swords was the same, the bar-weighted sword would rotate faster for the same amount of force, but in your case, the two swords don't weigh the same.

[Thomas MacFinn]

Somebody double check my assumptions, but I think if the bar ran all the way from the balance point to the center of the pommel it would have to be exactly twice as heavy as the weight in the center of the pommel to balance the same blade.

[Arrakis]

Howdy, Bo.

I'll have you an answer soon as I get done with this next leg of travelling (KY->TN->KY) and before the next round (KY->Spring War->KY).

[bo1]

that is a good point thomas, i should, for the sake of this experiment assume that the wieghts of the 2 swoords are the same. i know to always remove all variables but one to measure the results, but for some reason i failed to mention that in this instance.


thanks arrakis, i eagerly await you analysis.

[Arrakis]

Figure 1: The sword as a beam with differential weight.

Here, CoB refers to the linear center of balance along the long axis of the weapon. p, h, and b are distributed loads acting over the lengths shown and are the weights of the blade, handle, and pommel sections respectively. We can treat the (linear, constant) distributed loads as acting on a point halfway down the length of the line of distribution of the load and equivalent to the sum of the distributed load. Let equivalent point loads of each of the distributed loads be known as P, H, and B.

Let the variable x_CoB be the distance from the pommel of the weapon to the CoB of the weapon and x_B, x_H, and x_P be the distances from the left end of the weapon to P, H, and B.

Considering the sword to be a simple beam, we can find a moment balance about the Center of Balance of the weapon:

Code: Select all

B*(x_B-x_CoB) + H*(x_H-x_CoB) + P*(x_P-x_CoB) = 0                         (1)

and then solve for x_CoB:

Code: Select all

x_CoB = (B*x_B + H*x_H + P*x_P)/(B + H + P).                               (2)

If we wish to get a particular CoB for our weapon, say, through weighting the sword, we simply weight the weapon, find the original x_CoB, then:

Code: Select all

W*(x_CoB-x_CoB_NEW) + C*(x_P-x_CoB_NEW) = 0                               (3)

where C is the amount of counterweighting you're adding and W is the total weight of the sword before counterweighting. This presumes that you're adding the counterweighting in the pommel in an evenly linearly distributed fashion (because of x_P being in the occasion), but you could just as easily put the counterweighting wherever you like and simply substitute its location for x_P in Equation 3.

Equation 3, with a little manipulation, will yield:

Code: Select all

x_CoB_NEW = (W*x_CoB + C*x_P)/(W+C)                                        (4)

...

MORE TO COME TOMORROW...

[bo1]

arrakis i am wondering a couple of things having had this taste of the argument.

1 is there a way to measure the theoretical force required to move the tip given the different wieghting locations. given a vairiable hold location. i hold my sword about at the point the blade becomes the handle.

2 what about the control of stabbing, does any of this have an effect on the controlability, or perhaps its stability?

thanks again that is a good start.

[Arrakis]

*, bo, you ask the toughest questions! I'm going to have to sit down tomorrow and do some thinking and calculating. If you could see the scratch paper I've already used up...

[bo1]

i would never involve you if i didnt respect the skills you have with this type of stuff. hey if it is too much work, just tell me to go shovel, i will cease bothering you about this line of questions.

[Arrakis]

No, no, they're really interesting questions that I've never really asked myself before. I'm as interested in the answers as you are. Just bear with me as I take my sweet-ass time workin' it all out.

[bo1]

if you hurry, you get hurried answers. take your time, as i think a thorough explaination of this will change peoples opinions greatly.

[Loptr]

Howdy, Bo.

I'll have you an answer soon as I get done with this next leg of travelling (KY->TN->KY) and before the next round (KY->Spring War->KY) .

My only take away from this post.
Apparently Spring war requires a lot of personal lube........


Loptr

[bo1]

bhwaaabhwaaa, i think that is real funny. nice observation, that is good stuff right there.

that is totally sig worthy.

[Coriant]

Quote:"An analogue to Newton's Second Law is

F=M*Acm

Where F indicates the sum of all external forces on the system, and Acm indicates the acceleration of the center of mass."-Wikipedia
The point of a center of mass is that you can treat the object as though all it's mass is located there.

When you swing a sword it acts as a third-class lever with some distance separating the force from your arm and the center of mass of your sword. The shorter this distance is, the better you will be able to control the blade, both in rotation and normal swinging. If it goes to far back though then you'll hit lighter because the center of mass isn't stopped effectively from a tip far from it.

As far as where to put the weight, pretend center of mass and center of gravity are the same thing and treat it as a first-class lever balancing beam. Thomas is exactly right, the farther you get from the center of the blade in adding counterweights, the less weight you will need. Where the extra weight is however matters far less then where the actual center of mass ends up.

I realize there isn't much math here, but you don't need a lot for this. Here are some sources for more math if you want it.

http://en.wikipedia.org/wiki/Center_of_mass
http://en.wikipedia.org/wiki/First_class_lever
http://en.wikipedia.org/wiki/First_class_lever#Third-class_levers


Edit: The bar method wouldn't have to be exatly twice as heavy, because some of the weight is still in the pommel. If the center of mass of your bar were halfway between the center of the sword and the pommel then you would need twice the weight for the same balancing effect.

[Arrakis]

Ah, but the question is: Are the assumptions underlying that treatment of the system valid and sound and do they provide an adequate solution? Can Rotational Inertia and Work be ignored? Clearly, adding mass directly to the pommel increases the work you have to do when you move the sword, as you're rotating the sword about the CoM and the distance the pommel moves with the greater mass increases the work.



...
This is what's keeping me up at night?

Sigh.

[bo1]

it is a complicated dynamic, that is why a asked, i am not really as dumb as i let people think. if it was easy i would not have asked for help.

[Kenneth]

I knew there was a reason I didn't respond to this earlier....

[bo1]

kenny you are missing the point, it is not the c.o.b. i am concerned with it is the most effiecent use of weighting to get the c.o.b. i want.

the question is not what c.o.b. should i use, it is where is the best place to put the wieght?
placing it in the pommel, means that it moves the c.o.b. more effiecently towards my hand, but it is wieght further from my hand, hence it has a longer lever arm for me to over come its inertia. if i put the wieght directly where i grip my sword, the inertia is reduced to near zero to swing, but it is less effecient to move the cob where i want it. it is more an excercise in theory than anything else. i just had a question and figured i would see what poeple were able to come up with.

[Arrakis]

Alright. I think...

If you put the weight in the grip:

• You'll have to use more weight to get your balance point where you want it.
• You'll have that much more work to move with your arm
• You'll have less rotational inertia to overcome with your wrist during the secondary pivot portion of your swing.

If you put the weight in the pommel:

• You'll have less total sword weight to get your balance point
• You'll have less work to do with your arm to move the sword
• But! You'll have more rotational inertia to overcome with your wrist during the wrist-rotating portion of your swing.

So, I think it depends on whether you're looking for a quicker bulk motion (center of mass translation) or a quicker blade rotation (movement of the sword about the CoM) at the end of the snap. This is for the same balance point on a pair of swords who differ only by counterweight location and weight.

[bo1]

ok well, the weight of the swords is a constant, i am adding 2 oz counter balance, taking me to 14oz total. so really the only variable is the location of the counter wieight and its effect on the cob.


thanks so much arrakis, i realize the complexity of the question, thanks for the input and time. i owe you a perfectly functioning horn, i.e. a drink.

[Coriant]

I apologize. I misunderstood the original question. I did a quick estimate,assuming constant linear mass density for your sword (not too far off if the balance point is near the center), and in terms of oz*in your moment of inertia before weighting about the point 8 inches from the pommel is 2119. (I used the integral of r^2 times the linear mass density, where r is the distance to the balance point to each part of the sword) This doesn't mean much, but after adding 1 oz to the pommel it went to 2175 and 1/3. the difference is only about 2.5%. If you added the weight right at the 8 in. mark, the difference would be negligible.


To the second set of questions, you can get a theoretical measure of the force required to move the tip by treating it a a third-class lever, just make sure that you check once with your wrist as fulcrum and then again with either your elbow or shoulder for your swing.

One advantage to higher rotational inertia from pommel weighting is that when you sword block a heavy swing, your wrist doesn't have to put out as much force to keep your sword from getting knocked out of the way. The effect here may not be that noticeable though.

edit: sorry, I used the numbers from the first post, 13 oz first plus 1 oz to make 14.

[bo1]

thats good stuff. so really when it is all said and done, there is less than a 2% difference in force required. so do what ever the hell i want, sweet.

[Thomas MacFinn]

... Clearly, adding mass directly to the pommel increases the work you have to do when you move the sword, as you're rotating the sword about the CoM and the distance the pommel moves with the greater mass increases the work.

But you aren't rotating the sword around the center of mass. I believe in most cases you are rotating it around your index finger.

[Thomas MacFinn]

After staring at my hand moving for too long, I think the sword rotates around the center of the palm.

[Thomas MacFinn]

... This is for the same balance point on a pair of swords who differ only by counterweight location and weight.

For the slow kids in the class like myself, am I correct in assuming that you can't change just one variable? Change counterweight location and keep center of balance fixed and total weight changes. Change counterweight location and keep total weight constant and center of balance moves. Correct?

So if you have a sword that makes weight before balancing, adding a weight on the pommel to get the balance point you want will result in the lowest total weight.

On the other hand, if you have an underweight sword that is already balanced where you want it to be, where would you put the weight to have the least effect on all the forces discussed?

[Arrakis]

Thomas, yes, to that last post. Thomas, the shorter a distance your CoB has to move, the quicker it will move. Even if you're rotating the sword about some point that isn't the center of balance, the closer that point IS to the CoB, the less net work that rotation will require.

Bo, yeah, if you're keeping total weight the same, putting the weight in the pommel instead of the grip will make the center of balance lower on the sword, giving you slightly more lively tip response and a quicker swing-start, whereas putting the weight in the grip instead of the pommel will leave the center of balance a little higher and give you a stronger static block and a heftier hit once you do get the thing moving.

[bo1]

ok, thats the stuff i want, thanks. i have the full picture now, i know what i want to do then.

[Sir_Mel]

Wow, I never knew I could feel this much smarter from reading one thread.

I also don't think arrakis realized he was going to school for foam fighting theory more than anything else.

[Arrakis]

Not until first semester Senior year...