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I try to log weight, dimensions, and deflection as a measure of longitudinal stiffness for all the soundboards that come through my shop. Very recently I put together a spreadsheet to figure out what the mass would be for each of the boards I have given a target deflection. This would be so that I could choose a top for an instrument based on it's final mass.

Given the deflection at one thickness, I can calculate the thickness at my ideal deflection at a standard width, then figure the ideal board's mass given the density of the board using:

measured (Deflection * Thickness ^ 3 / width) = target (Deflection * Thickness ^ 3 / width)
and
mass = density * length * width * thickness

I did the math for about three dozen soundboards containing sitka, englemann, german, and lutz spruce, along with a few redwood and cedar tops. There are boards with well defined tight annual rings, boards with wide coarser grain, and lots in between. A couple have moderate bearclaw figure. Cross-grain stiffness is all over the place. Density in the spruces ranged from about 6 to 8 g/in^3. What I discovered was this: the target weight for all the spruces fell within about 5 percent of each other. There are two outliers, but looking back through my notes I discovered that those samples had a good bit of runout which explains the difference. The target mass for redwood is a little more than the spruces and the cedar turns out actually quite a bit less for my target deflection.

Anyway, the striking thing to me was just how homogeneous the stiffness to weight ratio was across all the different spruce species that I have. This would imply to me that provided I am thinning my tops to a target deflection (or weight), density shouldn't really be a consideration in the selection process.

What is your experience? Am I missing something in the math?

How did you determine the ideal Young's modulus?

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Howard Klepper
http://www.klepperguitars.com

When all else fails, clean the shop.

Howard, I'm not calculating Young's Modulus per se, but the equation for Young's modulus is the basis for my math. My ideal target deflection (or Young's modulus) is arbitrary. An example might make sense... For the Young's modulus equation:

Ex = ( K*Weight*Span^3) / ( deflection*Width*Thickness^3 )

Ex, K, Weight, and Span are all constant for my deflection testing setup, so for a given board I can say that deflection*Width*Thickness^3 is going to be constant as well. A change in thickness and width should affect deflection in a predictable way:

deflection(measured)*Width(measured)*Thickness(measured)^3 = deflection(ideal)*Width(ideal)*Thickness(ideal)^3

Given an arbitrary 'ideal' deflection and width, I can solve for 'ideal' thickness. If I multiply that thickness times the measured density of the board and some arbitrary length and width, I get the theoretical mass of a board of my target stiffness:

mass = density * Length * Width * Thickness

It turns out that given an ideal deflection all of my spruce boards would end up with various thicknesses but remarkably constant mass. On the surface it would seem that for spruces, Young's modulus is more a function of density than a function of species, grain count, grain coarseness, etc.

Unless I'm missing something or made a mistake somewhere, which for me is not uncommon.

How do you determine what deflection is ideal in your samples? And what tonal characteristics do you infer from this?

I believe that stiffness does exhibit a roughly linear relationship with density for most woods...

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http://www.PeakeGuitars.com

What you have found seems to align for the most part with what others who are doing similar tests have found. You have a fairly small sample to really put a whole lot into your results. However if your findings are aligning with what others have found maybe that gives more meaningful information. Did you follow a testing method and equipment used that was based on someone elses (who has done a lot of testing) method?

I didn't notice you drawing a whole lot of conclusions (short of a corrilation between density and stiffness). You mentioned choosing your "target" stiffness was based on your finished instruments(which sounds like a reasonable thing to do).

One thing that interests me the most is cross grain stiffness, and how it effects overall stiffness.

Rich

Maybe 'ideal' is the wrong term. By 'ideal' I mean it is my own personal target deflection; today's best guess as to what stiffness will best compliment the guitar I'm building, based on what my admittedly limited experience tells me.

Indeed I know my sample base is small and so my findings could very well be coincidence or error on my part. It might also be that small differences in mass across the soundboard account for relatively large differences in tone and perhaps my calculations are not precise enough to make a useful conclusion.



I am curious... What are your thoughts to that end?

With regards to cross grain stiffness. I don't really have any conclusions. What I have found is that in curly boards cross grain stiffness rises significantly(at times almost seems as high as long grain). In straight grain with significant runnout(although no where near the level of curly) there is a loss of strength in the long grain, but cross is not effected as much. I had my wife flex a few boards, and explained what was happening with the grain. She thought maybe it was acting like corrigated cardboard, although I don't see a top or bottom sheet being spread out as in the cardboard example. That is something I don't fully understand.

Rich

I haven't done any major measuring per se, but I have handled and sifted through at least 600-800 tops (at Rivolta), and on a purely subjective level, you can get major variations in stiffness per given weight, as well as fairly major variations in weight per stiffness (ie, very, very stiff wood that's significantly lighter than other very, very stiff wood). I've got a few of both (heavier and stiff and light and stiff) Italian spruce in my big pile (70 tops) of wood, methinks I'll do some tabulation and report back when I have a little time (may be a few months)

I concur with your findings about stiffness and density, for the most part. I've been measuring the Young's modulus (E) of tops along and across the grainfor a few years, and find that the long-grain E 'tracks' the density nicely, with surprisingly little scatter. I'm a bit surprised at the small range of weights: i'd expect less dense tops to end up lighter at a given deflection stiffness, but I have not done the calculations to figure out the range to expect. It could well be pretty small. It's nice to see somebody getting similar results with different methods: makes me feel as though I'm not as crazy as all that after all.

One neat thing about all of this: it's pretty easy to find the density of a top blank, even if it's an odd shape. You just put it in a plastic bag and see how much water it displaces. From that you've got a pretty good idea of how thick to make it, provided you know what's worked for you in the past.

I'll also throw my 2 cents and corroborate whats been said about density and modulus. I find the same thing. However once in a while I'll find a top thats an outlier, it happens in both direction, it extra nice though when you find one that stiff for it weight and has good cross grain stiffness. It can and does happen. I haven't ever found that in a master grade top BTW, they seem to be purely graded on cosmetics near as I can tell.

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Jim Watts
http://jameswattsguitars.com