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What Trevor Gore said...
I learned to do vibration testing from Mort Hutchins, Carleen's husband, and he learned from Dan Haines, who did a pretty large study of tonewood properties early on. Haines settled on vibration testing to minimize the effects of creep: if you leave a sample loaded for a while it tends to keep moving, where in a vibrating object creep should just become part of the damping. Many deflection testers will load the member, zero the gauge, and then unload it, looking at the change immediately, to minimize this. As Gore also points out, things like runout (and built-in stress?) can give you a different deflection in one direction than the other.
Another advantage of vibration testing is that you can get damping numbers directly. It's likely that small differences in damping that you can't hear are not too important in use, but, again, it's nice to have numbers.
I also wonder if the equations used to calculate Young's modulus from deflection testing are any more accurate than the ones used in vibration tests. It could be said that with proper supports a static test eliminates the crosswise bending that is one thing that messes up vibration tests, but the Poisson's ratio that couples between cross and long bending is still there, and should have an effect.
The McIntyre and Woodhouse articles in the Catgut 'Newsletter' talked about ways to test rectangular plates and correct for the inaccuracies of single-mode tests using data from other modes. The lowest frequency mode of a rectangular plate will normally be a 'torsion' mode, with node lines parallel to the plate edges. The restoring force for this comes from two shear moduli, which normally work together in any complex bending mode of a flat plate, so it's more or less kosher to just lump them. Thus this 'T' mode gives you a means of determining those moduli.
If the aspect ratio of the plate is such that the lengthwise and crosswise bending modes would come in at the same frequency (the travel time for a bending wave is the same in both directions), you will see a closed 'X' and closed 'O' mode. Normally the 'O' mode pitch is higher than the 'X' mode. What's happening here is that the normal 'bar' modes, which would be at the same frequency in both directions, are coupled by the Poisson's ratio of the material. If you visualize the plate bending along the grain such that the ends are moving 'down' as the center moves 'up', the upper surface is stretching along the length of the plate, while the lower surface is being compressed. When you stretch an elastic material it gets narrower in the middle, and the ratio of the narrowing over the change in length is the Poisson's ratio. Since the upper surface of the plate is getting longer, it's trying to get narrower in the middle, and lower surface is trying to get wider, so the plate 'wants' to curl upward across the grain as it's being bent downward along the grain (this is easily seen in bending a piece of sponge or hexcell, which have a high Poisson's ratio). This is the shape of the 'X' mode, so the Poisson's ratio of the material helps to relieve the bending stress, giving a lower pitch than you'd get from bars the same length and thickness of the same material. In the 'O' mode the Poisson's stress adds to the restoring force, and raises the pitch. Thus the difference between the two pitches is a direct measurement of the Poisson's ratio.
(Whether Poisson's ratio matters directly in tone production is an interesting question. Oliver Roger's computer modeling study of violin 'free' plate modes determined that it was not a major factor, but in arched plates the Poisson's ratio is overwhelmed by the effect of arching in determining the X:O mode ratio. I have not done a large study of Poisson's ratios of guitar wood, but I will note that Honduras mahogany has about the lowest Poisson's ratio of any I've looked at. Walnut is much more similar to soft maple in that respect than cherry, and I've had much better luck with walnut in arched plates than cherry. Whether that means anything....)
By getting readings of various modes in this way it should be possible to factor things in and correct for the simplifications in the usual models to get more exact numbers if you want them. You still can't do anything about measurement errors, variations in material properties from place to place, humidity, and so on, but what the heck...
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