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I've been trying to work out the soundboard thickness I should use on my piccolo guitar, relative to a full size one.
If stiffness K=AE/L (A is sectional area, E is elastic modulus and L the length).
Rearranging gives A=KL/E.
E is a constant and I want K to stay the same.
That means that the required sectional area is directly proportional to the length of the plate.
So if the soundboard is 75% of a normal guitar, that means the cross sectional area should be 75%.
The width is 78% of a standard guitar, so the required thickness works out to be 96%.

I'm sure I've done something wrong or missed something here. Where does the rule come from that stiffness is proportional to thickness cubed?

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I think your equation only represents longitudinal stiffness - pulling in a straight line.

Bending stiffness is related to thickness cubed. Trevor's books cover it.

Kevin Looker

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I'm not a luthier.
I'm just a guy who builds guitars in his basement.
It's better than playing golf.

I thought something was wrong there! How does length relate to stiffness then? I'm sure I've seen a formula around somewhere for deflection testing which included the distance between the supports. I can't find it anywhere though.

Unfortunately Trevor's book is far too expensive for me.

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"I am always doing that which I cannot do, in order that I may learn how to do it."
Pablo Picasso

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Yeah, I don't think you really need to worry about longitudinal stiffness. There isn't really any load in that direction, so no structural concerns. Plus sound waves are produced almost entirely by bending waves in the structure, not longitudinal waves. So what you need to look at is bending stiffness, which is proportional to E*t^3/12, where E is the modulus and t is the thickness. So that's where the cubed rule comes into play.

But that doesn't really answer your question, because stiffness is only part of the equation. You aren't really designing for a given stiffness, but rather for an acceptable deflection due to the string load. So that will need to factor in the load (which I assume will be different for a piccolo than for a dread) and the span of your plate, which will also be different. I don't have the equation you need off the top of my head.

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EDIT: As usual, we are not solving a new problem. The below reference indicates that for a circular, thin, clamped plate with a point load at the center (which roughly approximates our situation) the deflection can be found from:

y = P*r^2 / 16*pi*D

where y is the deflection, P is the load, r is the plate radius, and D is the bending stiffness. So if you want y to be the same, and you know the difference in P and r, then you can calculate the needed D and thus the needed thickness from the bending stiffness formula above. This is not an exact solution for a guitar but should be close enough?

http://www.roymech.co.uk/Useful_Tables/ ... lates.html

Here is a link to several pages of beam loading equations (same as evaluating the top as a 2D cross-section.) I believe they are what you seek:

http://ruina.tam.cornell.edu/Courses/ME4735/Rand4770Vibrations/BeamFormulas.pdf

Wow - heavy maths .
I'll try and simplify the question a bit. This guitar has a small body, short scale length, but has the same tension as a standard guitar. I know a plate thickness for a normal guitar that works, so all I was trying to work out was whether the relationship between body length and top thickness (keeping stiffness equal) was linear, squared, cubed or whatever.
I hope that makes sense - I'm not very good at putting thoughts into words.

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"I am always doing that which I cannot do, in order that I may learn how to do it."
Pablo Picasso

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I don't think you want to keep the stiffness equal. If you really do, then just use the same thickness as before, since the bending stiffness does not depend on the size of the plate. However, I think what you really want is the same deflection at the bridge.

(Note, as an engineer I try to never do math in public, too much risk of embarrassment. But I suppose I'll give it a go.)

Well, from the equation I provided, and assuming you want the same deflection at the bridge:

[P1 * r1 ^2] / [16 * pi * (E1*t1^3 / 12)] = [P2 * r2 ^2] / [16 * pi * (E2*t2^3 / 12)]

and since P1 = P2 and E1 = E2 (sort of) we can reduce to:

r1^2 / t1^3 = r2 ^2 / t2^3

so t2 = (t1^3 * r2^2 / r1^2)^(1/3) where t is thickness and r is the plate radius. So use the plate thickness of you "normal" guitar as t1, and the distance from the bridge center to the edge of the lower bout of your normal guitar and your small guitar as r1 and r2, respectively. Square, multiply, divide, cubed root and you are done.

You will notice that the equation I gave and the one Hugh gave are different. His assumes that a plate can be approximated with a beam. I don't think this is correct, but I'm no expert. Perhaps an expert will be along shortly.

Yes, his books are expensive compared to "book store" books.

They are actually college texts and in no way a casual, light read.

On the other hand, I think that to take a class & learn as much as these books contain would cost well over $1000. Besides the physics behind guitar design, the books also cover building techniques from start to finish.

Kevin Looker

edit
Also keep in mind that the $245 price includes delivery from Australia which is not cheap. I think I received my books less than 4 days from ordering.

_________________
I'm not a luthier.
I'm just a guy who builds guitars in his basement.
It's better than playing golf.

Thanks very much for breaking that down for me. It makes sense to me now and I get a plate thickness of 2.2mm which sounds about right. I'll have to save that formula somewhere for future reference.



I didn't mean they were overpriced. They're definitely worth it, but I'm a college student, so have no money!

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"I am always doing that which I cannot do, in order that I may learn how to do it."
Pablo Picasso

https://www.facebook.com/FenskeGuitars

As with all of these things, it depends how many approximations you want to make.

If you want to do it with the least approximations, you need an equation for plate deflections (not beams) in an orthotropic material (one that has different Young's modulus along and across) as well as accounting for your bracing (beam equations).

But remember that you have a dynamic problem as well as a static problem, so you need to take the mass of the thing into account as well if you want to try to replicate some vibrational characteristic.

For a first shot (without too much complexity) Jake's approach is good (but I haven't checked his maths!)

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Trevor Gore, Luthier. Australian hand made acoustic guitars, classical guitars; custom guitar design and build; guitar design instruction.

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The equations I provided can be used, in fact I used them during a study on the effects of thermally treated woods to determine the modulus under a four point load. A simple beam can be approximated as the two-dimensional cross-section of a plate in this scenario, and I have been paid for my expertise in materials science as well as strength and failure analysis. If the grain happened to be in a different orientation aside from perpendicular to the load, another method would be necessary. I would say the easiest way to determine the necessary thickness in this case would be:

*Determine the string tension.
*Cut a strip from the wood you plan to use for the top and measure its deflection under a 3-point load to find its modulus.
*Using these combined data, calculate the required thickness to produce the desired deflection (which will likely turn out to be very close to standard thickness.)

I've never built or played a piccolo guitar, so I hope you will post plenty of updates along the way. Even if it is overbuilt, those high notes should ring out just fine. Full bass response seems to be one of the most common problems from overbuilding a regular guitar.

Interesting. I expect that a beam would be a good approximation for a plate in the cases you mentioned, under either three or four point loading, where the beam/plate is only supported at it's ends. The dimension "into the page" plays no part in the solution. A beam with a point load is equivalent to a plate with a line load. However, I believe a better approximation for a guitar top is a plate which is supported all the way around, at the sides as well as the ends. The stresses and thus the deflections in this case would be very different from the case with a beam. The dimension "into the page" will play an important part, as a narrow plate will give you a very different answer than a wide plate. Again I don't claim to be an expert, but the guys who wrote my engineering textbooks are, and they give different equations for plates than beams, so I think there is a difference. And our two answers above are a good example, as the plate equation has a squared dependence on the distance from the load to the support and the beam equation has a cubic dependence.

Anyhow, fun to talk some engineering on a guitar forum!

You can follow along at the new builders challenge here: viewtopic.php?f=10134&t=38819
The bass response is what I'm worried about, which is why I'm trying to work out how thin I should go on the soundboard. Thanks again for the engineering input!

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"I am always doing that which I cannot do, in order that I may learn how to do it."
Pablo Picasso

https://www.facebook.com/FenskeGuitars

From the title of this post. It sounds very X rated.

I acknowledge that there are several acceptable ways to go about calculating appropriate top thickness. The rationale behind my suggested method is simplicity. If anyone expressed a desire to optimize thickness on all of their instruments I would advise the fabrication of a fixture to do so. The top would be assembled with excess thickness, and the bracing installed as well as a bridge plate (a dummy bridge would also help, but I'm only writing this as a rough conceptual overview.) Construct a representative form of the sides of the instrument, and they will be assumed to be extremely rigid, while also having enough space to provide a clamping surface. Going a bit over-size on the top would allow the possibility to go so far as to glue it down in a few places to better supplement the clamps, which you will need a lot of. Apply load, measure deflection, remove material until ideal deflection is achieved, and that's that.

If you have access to capable software, I would personally lean towards FEM to be lazy and get things close enough for final tuning.

My personal favorite alternative is to focus more of your energy on bracing. Go with a safe top thickness, after all it's not the real load-bearing structure. Bracing will ultimately have a greater impact on tone.

I personally, respectfully, disagree with approximation based on a circular plate assuming uniform modulus. At the same time, most of this discussion has revolved around a topic I consider equivalent to swatting a fly with a sledgehammer.

I feel like I am just going to play it by ear... I am mathematically challenged, so I just tap the thing and see how it turns out.

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The other way of looking at this is the number of simplifing assumptions you have to make, and how well these assumptions are satisfied. Whether you realize it or not, when you use the beam equation, you're assuming the plate is narrow compared to it's length and that the Young's modulus is the same in all directions. Neither of these is true, so the answer is wrong, but that's not what is important.
What is important is that you've got an approximation, and by evaluating the effects of the errors in the assumptions, you have a pretty good idea of how much it could be off.

As Mr. Gore indicated, eliminating simplifing assumptions quickly escalates the complexity of the mathematics. While this approach will produce a "better" approximation, it is still an approximation.

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Amen.

I always wonder, when I see posts on forums which contain reams and reams of mathematical analysis of all descriptions ...does Wayne Henderson pay any attention to any of this stuff ? ... maybe he does ...but he doesn't strike me as an academic kind of guy who does ...it's more like it's all intuitive ...

The reason I ask, of course is because Wayne Henderson's guitars command prices way above what most luthiers can dream of ...irrespective of any academic qualifications ...

That said , I have to say that I have no idea of Wayne Henderson's academic background ...he could have a Ph.D degree for all I know ...

You are probably correct. But, I don't have Wayne Henderson's experience. And it sounds like the OP doesn't either. I don't think it would be very helpful to respond to the question with "go build guitars for 30 years, then you will know".

I agree that 30 years of experience is probably better than the approximated mathematical models given here. But the equations can be obtained in much less than 30 years. And a good understanding of the way in which guitars make sound alongside a good amount of experience might speed up that learning curve. Afterall, a guitar does not make sound by magic, it's just plain old physics.

Exactly



The soundhole is too small to get my hand inside, and I don't like thinning from the outside as there is no way of measuring the thickness.

All I wanted was a simple formula...

_________________
"I am always doing that which I cannot do, in order that I may learn how to do it."
Pablo Picasso

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I think that's an oxymoron when it comes to guitars

And re: murrmac, I think Trevor and Gerard are the first to to such an in-depth mathematical analysis of guitar building, and the number of related posts is due to the book still being new. I'm pretty good at math, but even better at intuitive calculations. I understand the concept of young's modulus, cube rule of stiffness, etc., so I use that in my decision making process, but in the end it's my fingers and ears that tell me when to stop. The range of stiffness we work in is so "human", it seems like a waste of a joyful process to make it all technical

For your instrument (11" lower bout, 15" body length, spruce, steel strings), I'd estimate 2mm + or - a bit would be good. Depends on the particular piece, as well as the exact string tension you're planning, and string height at bridge, and bracing style...

Todd, I've just closed a box, what should the top compliance look (or feel) like, can it be described?

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I fine tune the closed box on every guitar by thinning the edges of the lower bout with an air sander. After a number of guitars I am getting better at knowing when it's right by the tap tone and vibration. The binding is not on yet so you can see the thickness. I seldom go less than .090 on an OM, maybe .085 on a size 2. It's a big part of my voicing process.

I got this idea from Dana's voicing article (http://www.pantheonguitars.com/voicing.htm) and from a young man that graduated from the Red Wing MN program. He spent a summer in my shop and Dana had come to the school for several lectures and demonstrations of voicing techniques. I learned a lot from him.

I feel like I hear the top kind of "open up" at some point. That's usually where I stop. The feedback vibration under your finger as you tap is helpful as mentioned in the article and by Todd.

I've always wondered if wide top purfling like herringbone or abalone does something as you are essentially removing a significant portion of the top/lining gluing surface and replacing it with something else. When I thin the edges of a top I only use one 0.060 strip of purfling.

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[quote="Todd Stock"] I look for sensible movement out to the rim, along with consistency in the lower bout area, and thin to achieve it. Very seldom do I find myself doing anything on the bracing.quote]

Thanksfor the description, Todd. That sensible movement is the factor that I'm not sure about.
Do you, like Terence, thin the top to gain consistancy? If you need to thing the braces whats the best way of doing it through the soundhole. I tried, as you, just for grins, with a chisel but it very awkward to get a clean cut.

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