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Vincent Le Chevalier

Posted: Sat 12 May, 2007 11:18 am Post subject: A theory for balancing 


Hello all!
Well, this will be something of a long post, but I hope some will find the time and patience to read through...
Lately I've been thinking about ways to modify an object's balance to achieve a given set of dynamic properties. For example, it could mean taking a sword and a stick of the same length, and modify the stick, adding and removing mass along its length, until the resulting object has the same dynamic balance as the sword.
Until now I have just explored the theoretical possibilities of such operations. For example, is it possible to go from a stick to any other balance? What is the cost of that (i.e. how heavy the modifications of the stick are)? Of course the thing could have practical applications as well, such as building wasters or even prototypes to clearly show the differences in handling.
The process is akin (though less complicated) to what swordsmiths do when they are building weapons and adjusting their balance. Peter Johnsson was kind enough yesterday to give a few hints about how he does that in his thread "Homage to the medieval sword". For all I know, many respected sword makers have a similar approach. But their task is more difficult since they have to address not only the dynamic balance, but also the harmonic balance, while keeping a functional blade geometry.
My goal in this is to stay away as much as possible from empirical knowledge, and instead start from a sound basis of equations from physics, find kind of an "optimal" way, then adapt and interpret to explain how and why the empirical ways actually work.
The first stage is obviously to determine what makes up the dynamic balance. The general consensus is that center of gravity alone is not an accurate representation of it. As was discussed already in previous threads ("Use of the term CoP and controversy" and "Balance points, pivot points, and nodes on the sword" (http://www.myArmoury.com/talk/viewtopic.php?t=8088) for example), dynamic balance is a result of the interplay between geometry (position of tip and pommel, positions of the hands, essentially) and mass distribution (that can be summed up accurately by center of gravity and inertia, and is seen through the positions of pivot points). So in the end, the goal is to reproduce what does not depend on the user: length, center of gravity, inertia. Fortunately, there are several ways to reach the same COG and inertia (infinitely many, in fact), so we are not bound to reproduce the mass distribution in every detail.
The starting point I chose is a simple stick (uniform mass across its length). It really could be anything, I just choose this as a relatively "neutral" mass distribution. It should have the same length as the "target" weapon I wish to reproduce.
Now the question is, where should I add or remove mass to bring COG and inertia closer to those of the target. This means being able to figure out how those two quantities move when I modify the mass distribution. This is not obvious at first, since whenever you add a lump of mass somewhere on the object, both the inertia and the COG change. This is where theory comes into play.
I have found a rather simple geometrical construction that allows me to find what the dynamic properties become upon adding a specific mass distribution on the object. This is of course based on the equations that define center of gravity and inertia, but seeing that geometrically is a great aid for intuitive visualization (geometrically it's circles intersecting, analytically it's solving quadratic equations... not the most attractive of all things). And it works both ways, that is:
* Given the position and amount of mass added, I can find where the COG and inertia move to
* Given the target COG and inertia, I can find where to add mass and how much. Let's call S the spot where I should add a point mass
From here there are two cases.
* Either S is on the stick. The mass added in S is in this case the minimum mass that should be added to achieve the target balance. That is, modifying the mass elsewhere only means that a greater modification will be made.
* Either S is outside the stick. In this case, obviously, you won't be able to add mass there... And the minimum modification becomes the addition of two point masses, one at each end of the stick, in a specific proportion of course.
I suspect case 1 is encountered most of the time. Case 2 is still a possibility though, for example with leaf blades maybe it could happen. In each case, the fact that the added mass is minimal rises from the properties of my circles.
Of course this approach, while theoretically interesting, is not really manageable in practice, for several reasons. First, doing a point mass with sufficient mass will be difficult (at some stage it will not be a point mass anymore, because it's too big and has a nonnull inertia). Second, the very place where the mass should be added is not always convenient, for example it could be somewhere on the handle. It gives the minimum modification in terms of mass, but not of difficulty...
Fortunately, the geometrical approach also allows to see the effect of adding or removing mass according to a given distribution, for example shaving mass uniformly off a portion of the stick. And indeed, in many cases, it is possible to determine what length of stick must be modified to have the exact same effect as the point mass in S. This addresses the first problem encountered earlier, as no point mass is needed anymore. However, it is still a heavy modification of the original stick, and is not necessarily structurally sound (the risk is to end up with an overly thin blade part, and a steep step in the middle).
So the last possibility is proceeding in two steps. First, shave mass off the blade part (the portion found earlier) and possibly more at the tip than in the middle, to preserve strength. Do that until the pivot point of the pommel is in the right place. Then, add mass to the pommel until the rest of the properties line up. Adding mass in the pommel makes sense in this case because it really is the most extreme point you can add mass to, so it has the greatest effect possible. And it will not modify its associated pivot point that was adjusted in the previous step (that's a basic property of pivot points). The first step makes the heaviest mass modification, but does it over a great length of the object. The second step makes a slight modification, but concentrated in one spot.
I was very satisfied to note that the twostep process seems very similar to what Peter described. Indeed, blade modification and shaping makes most of the work, and the pommel only helps for the final adjustment. I believe Angus Trim also expressed a similar thing, that most of the balance was achieved by the blade, and not by the pommel, and that dryhandling the unmounted blade is already informative. It also explains why acting on the pommel cannot make everything. In fact, changing the pommel only allows to reach some specific set of properties, but far from all...
I think I'll try all that in the real world now, take a broomstick and try to make it feel like my type XI. I think that wrapping steel wire on one end could provide a way to get something close to an adjustable point mass for the pommel. Sure enough it will not cut as well, but I'd like to see what I can get
I'll probably add a section about that in my everexpanding article about dynamic balance as well, with full figures and diagrams, but I'd really like to hear reactions from the members here. What else do you feel should be explained? What applications would you like to see?
This forum has been a great help in the development of my little theories about dynamic balance, and I felt I should share... I hope someone will manage to read through the whole of my rambling, and then tell me that I'm not clear
Kindest regards

Vincent
Ensis Sub Caelo
Last edited by Vincent Le Chevalier on Mon 14 May, 2007 6:35 am; edited 2 times in total




Angus Trim

Posted: Sat 12 May, 2007 2:09 pm Post subject: 


Hi Vincent
Yeah, the blade handled unmounted gives me the best idea of where I'm going to go with it........very true......
Good luck on your experiment.........
It should be instructive......... but in the long run, I'm bettin' you're going to want to deal with a blade that can be modified. Simply put, the distal taper is in my view, the single most important part of balancing a blade, and the finished sword. But it doesn't work alone.........
Distal taper, profile taper, tang taper and length ratio with blade, blade length, and width, fullering, crossection, and changes of crossection down the blade, and edge geometry..........
The lighter swords/ length are the most vulnerable to minor changes, changing the dynamic properties........
Handle length and pommel shape and weight..........
Mass distribution? Yeah, that's the easy way of expressing things, but subtle things on the blade/ tang can make a big difference, that might not be all that obvious when finding where the mass is distributed.........
But a stick can definitely be a good point to start with.........
swords are fun




Vincent Le Chevalier

Posted: Mon 14 May, 2007 1:48 am Post subject: 


Hi Gus,
Thanks for your encouragements...
Angus Trim wrote:  Mass distribution? Yeah, that's the easy way of expressing things, but subtle things on the blade/ tang can make a big difference, that might not be all that obvious when finding where the mass is distributed......... 
Well technically mass distribution is not necessarily the easy way, since you could represent a different mass at each and every point of the weapon with it. The key point is that all this detail is not dynamically perceptible (because the laws of motion do not depend on all the details), so that several objects can be dynamically equivalent even if they do not have the exact same mass distribution.
But then this is just the dynamics, and as I said your problem as a smith is more difficult, because dynamics are just part of what makes a good weapon... Not being a smith, I have decided to narrow the scope of my study
Aside from the practical application of building prototypes or such, one thing that I will investigate further is this possibility to represent any dynamic balance with just a stick and one or two point masses. Even if not doable in real life, there could be a potential for visualizing the differences between weapons. I'll try that on some of my own weapons to see what it brings up (possibly nothing, but it's worth a try at least...).
Thanks again for stopping by!

Vincent
Ensis Sub Caelo




Alberto Dainese

Posted: Mon 14 May, 2007 6:04 am Post subject: 


Hi Vincent,
thank you for your post, I found it very interesting and I am eager to read the article you're working on.
I just want to add my two cent and, perhaps, insert a little difficulty. In your analysis you assume the stick to be "geometrically" stiff, I mean it doesn't flex, but in *real world* if you make a blade thinner, it would flex. I think that this is something to take into account when looking to dynamic balance. Or am I totally wrong? (really much more than possible ) How do you cope with this?




Vincent Le Chevalier

Posted: Mon 14 May, 2007 2:42 pm Post subject: 


Hi Alberto,
To be honest, I'm eager to see my article completed too... Unfortunately I only have a limited time, and each time I write something in it I've got a new idea, which makes it pretty difficult to finish of course
Hopefully, this summer...
Concerning the point you raised:
Alberto Dainese wrote:  In your analysis you assume the stick to be "geometrically" stiff, I mean it doesn't flex, but in *real world* if you make a blade thinner, it would flex. I think that this is something to take into account when looking to dynamic balance. Or am I totally wrong? (really much more than possible) How do you cope with this? 
In what I've done so far, I don't take flexion into account, indeed. More exactly, in theory, I consider that the weapon is a straight 1D rigid body, so just mass (or more exactly density) varying according to the position on a straight line (the axis of the weapon).
This of course is not reality. It is just a model. The question is, is the model close enough to reality to allow for some interesting deductions... Personally I think it is:
* even considering curvature, most weapons are really longer in one of their dimensions.
* most weapons don't flex very much during handling. Impacts would be another story... But for now I'm interested in handling.
Dynamically, flexion essentially acts as a spring: if you stop the handle dead, the blade keeps moving for some time, and energy is temporally stored into the blade, instead of being absorbed through the hand all in one shot (note that the energy will come back anyway, as the blade will not vibrate indefinitely). But I think that the variability in handling created by the mass distribution is way greater than that introduced by flexion.
Actually, if I was to add something in my model, it would be a second dimension (to take curvature into account), but not flexion. Flexion is considerably more difficult to model and measure, and in my experience the effect is quite low compared to mass distribution.
Anyhow the 1D rigid model already brings enough interesting results by itself, so I'll try to finish working on that properly before adding complexity. Some form of Occam's razor, I think...
Regards

Vincent
Ensis Sub Caelo




Alberto Dainese

Posted: Tue 15 May, 2007 1:24 am Post subject: 


All Right Vincent, thank you for clarification
Good luck with your work !





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