| Jörg W. wrote: |
| If pivot points (as defined in here) are just a matter of the law of the lever, don't you have to take into account that mass distribution isn't homogeneous in swords? I assume that the model gets far more complex if you include that every (infinite small) cross section segment has its own weight (due to taper e.g.). |
Ah but it is taken into account... I don't really think of pivot points in terms of lever, but in terms of inertia, and inertia takes the variation of mass distribution into account, that's the whole point.
Think of it as successive approximations. Let's say that you divide, as you said, your weapon in many small segments, each with its own position x and mass m. I don't know if it's really an accepted terminology in english, but you could speak of:
* Zero-order moment: the sum of all m for all your little segments (I'll note that as sum(m)). This is obviously the mass of the weapon, M... Gives you an idea of how much matter there is but nothing else.
* First-order moment: sum(m * x). I'm noting the multiplication by a star to clarify. Divide that by M and you get the position of the center of gravity g... This is the average location of the mass. Now you know where the mass is on average, but it could be a single mass, two masses a bit aapart a stick, whatever... But your idea is more precise already.
Right, both of those quantity are routinely measured. But you can go on:
* Second-order moment: sum(m * (x-g)²)=sum(m * (x-g) * (x-g)). This is the inertia. This is what gives you the pivot points. It represents how far on average the mass is from the center of gravity. Now you have an idea of the extension of the mass in space. You still don't know what shape it has exactly, but the idea is still more precise...
As far as dynamics are concerned, we can stop here. The laws of motion (admitting that Newton is right, but it seems to work pretty decently if you're not building a spaceship ;) ) allow to show that for any given solid object, those are the only three quantities that matter. Well in the most general case it would be more complicated, because the inertia is represented by 6 numbers arranged in a 3x3 matrix, but for object that are largely line-like, like most weapons, it's a very good approximation.
Of course you could go on with third-, fourth-order moment, and so on... But this won't add anything to your perception of the dynamic properties. Two different objects with the exact same zero-, first- and second-order moment, when acted upon by the same forces (you could have a different force on each and every tiny bit you've divided the object into if you want) will act exactly the same. So neglecting the second order moment amounts to discarding a whole part of what you could measure of weapon dynamics. And measuring it is not opening a Pandora's box of plenty of other measures because up to second order is all you need for dynamics.
The drawing I posted earlier was just an illustration of how pivot points move when you move your point of reference, not really an illustration of the physical theory behind. Note that it's not just levers, the inertia is represented, it's on the vertical axis... Pivot points are really the result of an interaction between geometry (your point of reference), on the horizontal axis and inertia on the vertical axis.
Blade harmonics don't lend themselves to such a simple analysis (well, maybe dynamics seem simple to me partly because I'm becoming a reresearcher in applied mathematics and I've been thinking about that for something like 3 years ;)). The nodes of vibration depend on mass distribution, cross section, material properties... That is, I think, what makes their relation to mass distribution (and its modifications) non obvious. But Angus can speak of that a lot better than I ever would :)
So of course there are plenty of ways to get the three moments right. The key problem for sword making is then to get them right while having a solid blade with good profile, good blade harmonics, beautiful hilt but solid, and so on. But it's art and not science at this point. However, if you have one of the moments wrong by an amount too great, it probably won't feel like an original no matter the art (though it could feel right, it will certainly not feel the same). I'm fairly certain that good sword makers have their ways, at least intuitive, to get the dynamics right even without measuring. But measuring, in my opinion, can only help...
Regards
Attachment: 46.46 KB