|
Trevor Gore wrote:
"A place to start is by looking at wood where:
Llong/Lcross = (Elong/Ecross)^0.25
where Llong is the length in the long grain direction, Lcross the length in the cross grain direction, likewise for the Young's modulus, E. There will be a shape effect, but I suspect the sound hole isolates the upper bout, to an extent. The relationship does suggest higher aspect ratio guitars than we're currently used to, though!
The formula above is the condition for producing rings or crosses in rectangular plates, the difference being the relative phase of the waves."
We need to keep something straight here: the 'ring+' or 'ring-and-a-half' mode is not the same thing as the 'ring' mode. In fact, you almost never get a 'closed' ring mode on a flat top guitar top. I'll do the best I can to explain this using ASCII text, but it's tricky. I wrote a a series of articles with lots of diagrams about this for 'American Lutherie' back in '91-'92 (yipes!), and they're available in re-print in the 'Big Red Book' series from the GAL.
Suppose you start with a couple of narrow bars of spruce, cut from a top half, with one bar running along the grain, an the other across. You could set these up as glockenspiel bars, supporting them about 1/5 of the way in from the ends and hitting them in the middle to excite their fundamental modes. Trevor talks a lot about this in his books, as it's a good way to find out what the Young's moduli along and across the grain of the wood are. Let's suppose your two bars end up having the same frequency: how would a plate with the length and width of those bars vibrate?
One logical prediction would be that you'd see the same sort of vibration you do in a narrow bar, with a pair of straight lines running across or along the plate, at the same pitch. Since the plate would not be able to make up it's mind which thing to do, you'd end up with four dots of glitter, if you were looking for Chladni modes, at spots 1/5 of the way in from the edges and ends. I knew one very seasoned researcher who believed that, and had a lot discussion about it when my articles came out, but that researcher was wrong. The reason lies in how plates or bars bend, and a little-cited property of materials called the 'Poisson's ratio'. Sounds fishy, I know.. (couldn't resist)
When you bend a bar or plate, most of the force that's resisting what you're doing, and trying to straighten it out, comes from tension and compression of the faces of the thing. If you support it at the ends, and place a wright in the center, then the top face is in compression and the bottom face in tension. The forces drop of pretty quickly as you go in toward the middle of the thickness of the object, and there can be shearing forces, particularly out near the ends in the center of the thickness, but we won't worry about that. Since most of the restoring force comes from tension and compression, and that's what Young's modulus is about, we can use characteristics of bent bars and plates to calculate the Young's modulus. But there's also something else going on. (My thanks to the late Oliver Rogers for the following explanation)
Suppose you take a square sheet of thin rubber and glue a wood stick across each of the opposite sides. Hold the two sticks parallel, and pull them apart. The sheet will get narrower in the middle. This makes sense: the rubber started out with a certain surface area, and it tries to retain that area; if you make it longer, it has to get narrower. If I understood Ollie correctly, the percentage of narrowing over the percentage of elongation is the 'Poisson's ratio'. If that's not exactly right, it's close.
So, look at that loaded bar as a stack of thin sheets. When you put the weight on top, and bend the thing, the bottom layer gets a little longer, so it tries to get a bit narrower. The top layer, in compression, tries to get wider. Since the layers are actually glued together the effect is that, as the center of the bar goes down, the edges in the middle go down a little bit more, and the bar becomes very slightly curved in the cross wise direction. You can see this easily in a 'bar' made out of something like foam rubber. Of course, for a narrow bar the effect is not great, but in a wider plate it can amount to something. Let's go back to our plate where the aspect ratio was set up to give the same frequency in both directions (remember?) so see what happens.
We left Pauline tied to... oh, yeah; sorry. The plate was resting on four blocks of foam about 1/5 of the way in from each edge and end. Suppose you start bending it downward at the ends. The top surface stretches, and the bottom surface compresses along the length, and the Poisson's ratio causes the center of each edge to bend upward. If you are driving the plate with a signal that's somewhere near what you'd expect a bar of that length or width to resonate at, and trow on some glitter or sawdust, the particles will gather along non-moving lines and form an X from corner to corner. This is the equivalent of having the lengthwise bar mode with the center moving 'up', and the crosswise bar mode with the center moving 'down', so we can cal l that 'out of phase'. You could, if you wanted, push both the ends and the edges down in their middles, and get a resonance going that way too. In that case you'll see a 'ring', with the middle of the plate going up as the corners go down. We could call that 'in phase'.
The thing to notice here is that the 'X' mode motion _relieves_ the stress that the Poisson's ratio puts on the plate, but to get an 'O' mode, you have to fight the Poisson's ratio, as well as the stiffness of the plate. It turns out that if you do this sort of thing with something like expanded Styrene bead board, which has the same properties in all directions, and compare a square board with a narrow bar that's as long as one side of the square, the frequencies of the 'bar' mode and the 'plate' modes will be different. If the bending mode of the bar is, say, 100 Hz, the 'X' mode of the plate might be at 97 Hz, and the 'O' mode at 103 Hz. Tee frequency difference is a measure of the Poisson's ratio of the material. (just to confuse things, wood can have six different Poisson's ratios: more information than you needed...) It's always fun to try this with different sorts of woods: mahogany has very low Poisson's ratio, and the modes end up practically on top of each other.
'Bar' and 'plate' are sort of loose definitions, but this gives you some way to define them: a 'bar' becomes a 'plate' when the lengthwise and crosswise bending mode frequencies get close enough for the Poisson's ratio to start showing up. You can tell when that happens because the node lines are no longer straight: there's some crosswise bending going on when you're trying to look at the lengthwise mode, and vice versa. When that happens, you're not looking at 'pure' Young's modulus any more, but at some mix of things, with the Poisson's ratio thrown in. In fact, one of the best ways to sort them out is precisely to 'tune' the plate's aspect ratio to give 'closed' 'O' and 'X' modes. The 'pure' bending frequencies in both directions will be simply the average of the 'X' and 'O' frequencies.
There are two problems with that:
1) you might have to trim that top half down to the point where you can't use it for a guitar any more, and
2) it can be hard to tell when you've hit the exact right aspect ratio. Are the ends of the X really in the corners, or a few millimeters out one way or the other? What if they hit on one side and not the other? You get the picture.
This is why I like to actually look at the Chladni patterns when I do this sort of testing, rather than simply holding the plate up to a mic and tapping. I don't always see straight lines, and can't always trim the plates until I do, but at least I know that the Young's modulus values I'm getting are approximations.
Now that covers the 'X' and 'O' modes. Notice that they are counterparts to each other: when you see one, you will see the other. On flat top and classical guitar plates you don't get 'closed' modes. The lengthwise bending mode usually is one of the lower pitched ones, and consists of a backwards parenthesis across the plate. [ ) ( ] The 'O' mode is not 'closed', in part because the upper transverse brace is too stiff to allow it to. It's generally either the mode below the 'ring+' or the one just below that (guitar tops are not standardized the way violins are, so they show odd modes from time to time). As near as I can tell, if you compare a guitar top with that square of foam plastic, the nearest thing on the foam to a 'ring+' weil be what you might call a 'hash' or 'tic-tac-toe' mode, with two lines across the plate, and two up and down. A trapeziod of the same sort of foam will show a ring in the middle with 'scoops' at the corners. On a guitar top the upper scoops are turned into a line across the upper bout by the stiff upper transverse bar. Usually the lower scoops are not there, but I have seen them on some Classical guitar tops, as backwards parenthesis on either side of the O in the lower bout: )O(
What a long post....
|
Login
Register
FAQ
