Analytic Guitar Models

[GregHolmberg]

Hi acoustic guitar builders--


I'm new here. I'm thinking about building my first acoustic guitar.

I've been reading Contemporary Acoustic Guitar: Design by Gore & Gilet.

I'm looking at the formulas in Chapter 2: Analytic Guitar Models and their use.

I want to use these formulas to design a new acoustic guitar. I have a body shape in mind, and I want to determine how big to make it to hit certain target resonance frequencies.

To make sure I understand how these formulas work, I'm trying to apply them to a known successful guitar to see if I get the same dimensions. I'm using Gore's Medium steel-string (490 x 390 mm) as the known successful guitar, since the book lists all the dimensions and frequencies for that.

However, I applied the formulas against this guitar, putting in the resonance frequencies and sound-hole diameter, and I did not get the same dimensions. It seems I don't understand how to apply the formulas. There is perhaps some assumption or definition I missed in the book.

Perhaps someone who understands these formulas could check my application of them, and tell me where I went wrong.

The inputs to the calculation are:

Fixed inputs:
fA: target coupled air frequency, T(1,1)1 = 90.2 Hz. page 2-18.
fC: target coupled top frequency, T(1,1)2 = 172.3 Hz. page 2-18.
⍺: sound-hole effective neck length / radius. Page 2-14. For this guitar = 1.72 (page 2-37).

Variable inputs. Builder can play with these to change the calculation output.
ft: uncoupled top frequency. 157 Hz. page 2-20.
R: sound-hole radius. This guitar's sound-hole is 95 mm, so R = 0.0475 m. Drawing 4.

From these, I calculate:

fh: uncoupled air frequency (Helmholtz). I get 114.78 Hz. Matches his number page 2-22.
fh = SQRT(fA^2 + fC^2 - ft^2)
Equ. 2.3-15

V: cavity volume. I get 0.0196 m^3.
V = R / ((2*fh/c)^2 * ⍺ * π)
Inversion of Equ. 2.3-18, where c is the speed of sound = 343.2 m/s.

According to the book, page 4-62, the guitar's top has an area of 0.144215 m^2. It has an average depth of 0.105 m (drawing 5), and so a volume of 0.0151 m^3.

This is quite a lot less than I calculated is necessary to hit these target frequencies. Where did I go wrong?

One source of error is that this is using a two-degree-of-freedom model, which ignores how the back's and the side's resonant frequencies couples with the others. Still, a volume of 196 vs. 151 seems like too big a difference to explain.

I feel like there's something I didn't understand. Like the volume of the body as used in the formulas isn't what I think it is.

Thoughts?


Greg

[Talldad]

Can I suggest you ask Trevor directly on a different forum? He is usually very accommodating.
http://www.anzlf.com/viewforum.php?f=1

[GregHolmberg]

If I try ft = 144 Hz, then I get fh = 131 Hz and V = 0.0151 m^3, which matches the actual guitar.

However, the actual guitar has a much higher ft (157Hz), and a much lower fh (115 Hz). Building a top with an uncoupled resonant frequency of 144 Hz, which then rises to 172 Hz when coupled, seems improbable.

Alternatively, I could keep ft = 157 Hz, and shrink the sound-hole to 73 mm, and get V = 0.0151, to match the actual guitar.

Another solution is to raise ⍺. To match the actual guitar's volume, it would be necessary to raise it to 2.2, but on page 2-37 it says that this is usually in the range of 1.6 to 1.8.

But the guitar as built (ft = 157 Hz, sound-hole 95 mm, ⍺ = 1.72) should't behave as it apparently does!


Greg


Note: I didn't mention it above, but I also calculated the coupling frequency, fh,t = 96 Hz. If I change ft to 144 Hz, then fh,t becomes 103 Hz. I'm not sure what this tells me, since the book doesn't describe what fh,t is useful for, but maybe someone out there knows, and this tells you something.

[GregHolmberg]

Originally Posted by Talldad

Can I suggest you ask Trevor directly on a different forum? He is usually very accommodating.
http://www.anzlf.com/viewforum.php?f=1

Yep, that does seem to be the place to talk about his book. Unfortunately, I registered nearly a week ago, and they haven't approved me yet. So I thought I'd try here. I will try there if they ever let me in!

Thank you for the suggestion.


Greg

[Alan Carruth]

Keep in mind that all models break down at some point. One of my customars designs satellites, and has software on his computer that can model the way they vibrate, to be sure that nothing is going to shake off during the launch. He told me he tried inputting a guitar design for fun, but it didn't work: it's too complex for his software. So you see, this is not rocket science; it's more complicated....

[GregHolmberg]

Alan--

Yes, the place the calculation goes off track is the last step, using the Helmholtz formula. Gore says in the book that this one is especially difficult to apply to guitars, since a guitar body is not a spherical rigid body.

He says one can fit the formula using an effective length of the neck, and the ratio of this effective length to the hole radius should be in the range of 1.6 to 1.8. He says this is supported by a paper he cites, page 2-37.

In order to make the known dimensions and uncoupled Helmholtz frequency of his Medium steel-string work, this ratio would have to be 2.2.

So there's something here I don't understand.

Greg

[Alan Carruth]

Helmholtz' original formula is, of course, itself a gross simplification. All of the mass of the system is assumed to be in the 'piston' of air in the neck, which is also rigid. The air 'spring' in the bottle is assumed to have no mass, and to be perfectly elastic. Neither of these is true, of course. Helmholtz introduced the end correction as a 'fudge factor' to get the answers to come out pretty close with his simple bottles.

More to the point here, Helmholtz' original resonators were small glass spheres; effectively rigid at the frequencies he was working at, and much denser than the air they enclosed. A guitar that had a perfectly rigid and massive top would make no sound acoustically. Wall vibrations extract energy from the pressure waves in the box, and feed energy in as well. Gore's 4-element model takes that into account up to a point, but only over a very limited frequency range. He has said that it also taxes the resources of the software to the limit.

The best models I've seen are finite element treatments that use at least several hundred elements apiece for the top, rim, back, and air. These packages, such as ANSYS, get better all the time, but they're not trivial to use. A couple of my students worked for the company that puts that out, and they said that modeling a guitar with it was a stretch at that time (maybe 8 years ago). They gave me a 'student' copy to work with and I was getting along pretty well, considering my poor math chops, until a Windows 'upgrade' crashed it.

As I understand it, a big part of Gore's objective in his model was to avoid the production of 'wolf' notes due to strong air/body resonances that coupled strongly with each other or the strings. His emphasis on light tops and low mass bridges make his guitars especially prone to that, so it's necessary to keep the low order resonances that produce large top excursions well away from played pitches. Note that he makes 'tweaks' at the end, such as fine tuning the added mass in the rim, to produce the results he wants.

Those low end resonances also do tend to define the 'character' of the sound; Dread, Jumbo, Parlor, Archtop, Classical, and so on. They seem (IMO) to be less determinative of the 'quality' of the tone, though; a lot of that resides in the higher frequency range. Unfortunately for modeling, above about 600-800 Hz there are so many resonances, and they couple so strongly, that you're in a 'resonance continuum' where it's impossible to specify things exactly, beyond saying hoe many peaks there are per octave and what the overall damping is. This is why it's probably impossible to make 'matched pairs' of guitars that sound the same; small local variations in the density and stiffness of the wood, particularly in the top, produce differences in the sound output that our ears are particularly well suited to pick up.

You might find it helpful to look at William Allen's article 'Basics of Air Resonance' (iirc) in 'American Lutherie' #1. He records a series of experiments that outline the sources of much of the variation we see. He didn't get into detailed models, but provides a decent qualitative understanding of some of the main points.

[GregHolmberg]

Alan--

Thank you for the information and the reference.

I guess the article could be found in The Big Red Book of American Lutherie Volume One, 1985-1987. Unfortunately, out of print. Available used on Amazon for $300. Googling this article specifically, I mostly find posts on forums, which are you recommending the article! Ha!

My goal is to try to estimate what body volume and sound-hole size I need to hit a certain uncoupled Helmholtz frequency (115 Hz). I seems that no one has really figured out how to calculate that, and the best option is to just mimic other guitars. It's difficult to say how that would come out if the body shape is different though. I guess I'm left with trial and error.

Greg

[Trevor Gore]

Thanks to Alan C for wading in here with his ever useful thoughts.

Greg, there are a few issues with using the formulae the way you have and I'll pick up on a couple of those.

The first one you identified yourself -

"One source of error is that this is using a two-degree-of-freedom model, which ignores how the back's and the side's resonant frequencies couples with the others. Still, a volume of 196 vs. 151 seems like too big a difference to explain."

That is going to be a problem. Alan frequently relates an experiment done by Fred Dickens where he built a very deep bodied classical guitar, measured the usual resonant frequencies then removed the back, reduced the body depth, re-attached the back and measured again. He repeated this a few times until he had a very thin bodied guitar. What he found was that the main air resonance didn't change nearly as much as might be expected, because the coupling between top and back increased as the body depth reduced, resulting in greater motion of the plates which compensated for the reduction in volume - a classical example of coupling and its effects.

Another problem is that the fA and fC figures you used are for a medium SS guitar, but the ⍺ figure you used is from a neo-classical, p 2-36.

The utility in these simple models is to demonstrate how components interact and consequently alter the characteristics of a guitar, rather than in trying to, for example, specifically determine a guitar's volume. Much more sophisticated models are required for that and even those models have fairly constrained utility.

[phavriluk]

OP mentioned that this project will be his first. I think I see way too much attention to theoretical acoustic physics and nowhere near enough attention on just plain building. Nobody's first effort is any kind of testament to theoretical magnificence. Wood has a mind of its own. I suggest OP just build the thing; he'll be plenty occupied doing that.

[tadol]

Originally Posted by phavriluk

I suggest OP just build the thing; he'll be plenty occupied doing that.

Amen! Just build - most of the best builders I know wouldn’t bother with most of that unless they had a couple dozen good instruments under their belt -

[KingCavalier]

When you build measure everything. Take notes and record everything you can, weight, deflection, tap tones etc. Then do it 10 more times, try to build the same model with the same type of woods every time. You might find even if you build to the same specs for all none will sound exactly like the others, that's just the way wood works.

Do you want to build for a player who plays lightly, say finger style, a guitar that rings on for ever or a guitar for a player who plays hard with a pick, someone who might want a fast attack and fast decay. That's where taking notes comes in. Your own observations will serve you better for the first 10 or so builds.

[redir]

If I understand what you are trying to do you are trying to reverse engineer the guitar. IOW get the dimensions of the guitar right by working backwards from it's resonance frequency? Bear in mind that Gore does things after the fact (after building the guitar) to alter the frequency. He adds weights to the sides of the guitar to accomplish this. So I don't think what you are doing is going to work.

The beauty of Gores method is that it is a model that allows you to build any design you want, within reason of course, and find the inherent properties of the materials you are working with, the dimensions of the body, and some other points, to achieve an optimal tone producing instrument.

[Alan Carruth]

Thanks for weighing in, Trevor.

GregHolmberg wrote:
"My goal is to try to estimate what body volume and sound-hole size I need to hit a certain uncoupled Helmholtz frequency (115 Hz)."

Why do you want that specific outcome? What exactly do you mean by 'uncoupled', and how does that relate to the finished instrument? I'm trying to figure out the utility and significance of that particular measurement.

Keep in mind that, depending on just what you're talking about, there are probably innumerable different designs that would get that one number, all of which would sound different.

[GregHolmberg]

So, what I'm trying to do is design an acoustic guitar with a unique shape--i.e. one that isn't a copy of some existing guitar. I want to figure out what dimensions to make it--body length, width, depth, sound-hole size.

I buy into Gore's target frequencies for a finished acoustic guitar. For example, one set of frequencies he suggests is 90 Hz (the air resonance T(1,1)1) and 170 Hz (main top resonance T(1,1)2). These are the coupled resonances, i.e. in a completed guitar with all the parts interacting (top, back, side, air).

To hit these frequencies, you have to calculate the frequencies of the air (sound-hole) and the top when they are not coupled. For example, in his Medium SS, Gore targets 115 and 157 Hz. Then when you assemble the parts, you get the 90 and 170 you want.

To get the 115 air frequency, you have to have a certain body volume and sound-hole diameter.

So the question is how to calculate that volume and diameter if you're not copying another guitar. Can they be calculated or estimated, or is it all trial and error?

I could just try to mimic Gore's Medium SS. As far as I can gather from the book, it has a top that is 0.390 m by 0.490 m, area of 0.144215 m^2, a depth of 0.095 to 0.115 m, a volume of 0.0151 m^3, and a sound-hole diameter of 0.095 m.

However, I'm not confident in some of these numbers. For example, counting the grid squares in the printed plans, I get a top area of 0.1358 m^2 (not including the head block), and therefore a volume of 0.0143 m^3. As an error-check, with a 0.095 m diameter sound-hole, this gives me value for ⍺ = 2.4, which makes me think something has gone wrong.

The thing is, since my body shape is different than the Gore Medium SS, I'm afraid that simply mimicking the body volume and sound-hole size will not produce the same frequencies. There are apparently people here who just like to build, build, build until they get it right. I'm glad you're having fun doing that, but that's not me. I have an engineering bent, and I'd like to have some level of confidence that my design is going to work before I spend hundreds of hours building it.

I would like to design size variations of this guitar (different body dimensions, sound-hole sizes, body and air resonances), and I would feel a lot better about scaling it up and down if my calculated predictions tracked real-world measurements to some degree of accuracy. At this point, my calculated predictions are so far off, I'm not confident that my scaling would produce good sounding instruments. I can't even predict the frequencies of a known successful guitar (the Medium SS) from what I think are it's dimensions, so how could I possibly predict the frequencies of a different shape?

Trevor, have you ever measured the volume of your 490x390 body? Do you know the real number for volume? What about ⍺ for the Medium SS?

Thanks,


Greg