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Hi all,

I have been pondering a question in the back of my mind recently that I thought would make an interesting conversation here:

A lot of emphasis for steel string finger-style builders (in terms of tone) seems to be on the weight of the top plate and it's braces. In your opinion, is the weight or flexibility of the braced top the primary issue, or both?

You thoughts, please!
Peter

"Weight and Flexibility" - that sounds like a novel that Jane Austen didn't quite get round to writing.

Sorry - couldn't resist There'll be some serious discussion posters along here any minute now and the answer will be "both but it depends . . ."

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Dave White
De Faoite Stringed Instruments
". . . the one thing a machine just can't do is give you character and personalities and sometimes that comes with flaws, but it always comes with humanity" Monty Don talking about hand weaving, "Mastercrafts", Weaving, BBC March 2010

"In your opinion, is the weight or flexibility of the braced top the primary issue, or both? "

Yes.



The weight of the top has a lot to do with how much sound it can make. Given the limited horsepower available in a vibrating string, it helps to keep things light. Also, as we all know from dealing with loudspeakers, larger ones tend to sound better, all else equal. One researcher suggests that the power available from a guitar in a function of the ratio of the area of the vibrating part of the top (the lower bout, basically) and it's mass, so A/m. A higher A/m ratio gives more sound.

The main thing that limits how light you can make the top is the need to have enough stiffness to hold up under the torque of the bridge over the long term. That's mostly what the bracing is for: it certainly effects the sound, but not as much as some folks think. After all, there are all sorts of guitars with all sorts of different bracing systems that sound 'guitar-like', and some examples of almost any setup that sound darn good. How well you make it is probably more important than which pattern you use, unless you're trying to get a certain sound. It's possible to make a good guitar with almost any pattern, but if you want a Bluegrass Dread sound you should copy an old Martin.

Bigger tops may sound better, but they are usually not louder. As you increase the span of the top you have to increase the thickness and size of the bracing out of proportion to keep the stiffness up, so the ratio of A/m goes down for larger tops. It's easier to make a loud small guitar than a loud big one. Big guitars, though, tend to be more 'bass balanced', and if that's what you want it might be hard to get it any other way.

My take on things is that the quality of sound, how good the guitar is, seems to be related to how well the top and the bracing work together. All of the systems for testing tops, like deflection tests, tap tuning, Chladni patterns, and so on, tend to tell you something about that balance between the top and the braces. If that's the case, then you would tend to start by looking at the top itself, and getting the stiffness of that right.

IMO, the stiffness that counts here is the stiffness in bending along the grain: that's what's doing most of the work of keeping the bridge from pulling up. This is, of course, a product of the Young's modulus along the grain and the cube of the thickness of the top. If you get a piece that has a low Young's modulus along the grain, you can just leave it thicker and get the stiffness you need.

It turns out that the Young's modulus along the grain for softwoods pretty well tracks the density in a nice linear fashion. What this means is that you can get the same stiffness from a thin piece of dense Red spruce, or a somewhat thicker piece of less dense Engelmann, but that the Engelmann top plate will tend to weigh less, and put out more sound.

So that's what I meant by my answer. Both weight and stiffness count: you need to have a certain minimum stiffness, and that costs you a certain amount of weight, which varies depending on a lot of things. More weight or stiffness than you need will cost power, but may do other things for the sound that you like. There's no way to think of them seperately, though, IMO, and that's why the answer is 'yes'.

Just as I've been getting into deflection testing of tops, I've been wondering about this, too. Thanks, Alan, for the very interesting post on sound output and the relationship to size and A/m!

I gather that most of the premier builders thickness their tops to a specified deflection, but I wonder if it would be similarly valid to thickness tops to a specified mass. Here are some thoughts:

1) Most of the mass of the braced top is due to the top plate itself.

2) Most of the strength and stiffness of the braced top is due to the braces (for steel string designs), esp the X braces.
- Actually, most of the overall strength and stiffness are due to the "T" cross-section formed by each brace and the top wood an inch or two to the left and right of each brace. But when a "T" cross-section is put into bending, the upper leg of the T (the top) is mainly in pure tension/compression, not bending. This means that the main contribution of the top to the T's overall stiffness is a function of its area (and modulus, E*A), less so the moment of inertia of the top itself (t^3, E*I). (For the engineers, recall the parallel axis theorem... the A*d^2 term usually dominates). E*A is best specified by E*t, not E*t^3, -- so deflection testing of the top plate itself may not be the best predictor of the braced top (at least not as good as E*t). OTOH, deflection testing does get one measure of E and t in a single test, even if it is E*t^3 and not E*t.
- Exception: braces perpendicular to the top's grain won't benefit much from the "T", since cross-grain E is so low, and the T only works as a T if E is similar in both legs. In this case the high E leg (the brace) acts more as if it is alone.
- So, the longitudinal stiffness of the braced top is mostly a function of the X-brace, the thickness of the top (not t^3), and E of each.
- The lateral stiffness of the braced top is mostly a function of the lateral orientation and placement of the braces, and their E.

3) As Alan has said, mass and E are fairly proportional in wood, so to some extent it may not matter which is specified.

4) This is a sweeping generalization, but I'll put it out there for discussion: Since most vibrating systems have terms that look like sqrt(k/m), specifying k via the braces and m via the top might give better control on the end product (than specifying k for the top and braces and letting m fall where it may).

OTOH, the above arguments mostly apply to the large-scale vibration modes of the braced top (main top, long dipole...). In some smaller-scale modes where the top is unsupported by braces, the top plate vibrations are functions of its own stiffness and mass.

_________________
David Malicky

Alan,

What a great reply. I learned a lot from it. I appreciate you taking the time to share your hard-earned knowledge with us here. The A/m ratio is particularly helpful, as are your comments about the linear relationships with density in the spruces.

Anyone else want to weigh in?!?

Having fun,
Peter

David, very interesting! I will be interested to hear if anyone models their design in that way.

Peter

hi,

could somebody explain to me meaning A/m (A=.... , m=....)

best
Fric




I find, I mean I carefuly read what Allan wrote!

A=area, m=mass

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Steve Smith
"Music is what feelings sound like"

Resonance is always calculate as some form of the square root of K/m (k = stiffness, m = mass). So the ratio of the stiffness of the top to the mass of the top determines the "notes" at which it will have the strongest response (i.e. at the the resonant frequencies)

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do you get K from deflection tests ?
and mass, what kind of maas, whole SB or how you can measure lower b. mass.

I'm asking becouse on clasical thiknes of lowerb and upper b is I think different

Fric

I use Cladni patern tuning in my work, and, as I mentioned, try to get the bracing and the top working together in what I think of as an effective way, based on the shapes of those patterns. I may be full of baloney on this, of course, but it seems to work for me. In this system, you can't neglect the stiffness of the top plate itself; you're looking at wave propagation, and most of the time the waves are traveling in the top plate and not in the braces. This fits well with much of the research finding that the bracing, at most, fine tunes the sound of the instrument, while most of the tone is in the top. In this case, going for a target weight would result in low desity tops being too thick, and high density ones too thin, to work well. What it comes down to, I guess, is that while structural considerations, and thus, perhaps, t, limit how lightly you can built the guitar, there are also acoustical considerations, and these seem to depend on t^3.

It is also intersting that most studies that have looked at it find that the exact frequencies of the structural resonances of the guitar are less important than the realtionships. Having, say, the 'main top' resonance exactly on the pitch of the open G string effects the sound of that note, but not the notes around it. Placing that pitch a half semitone higher or lower will most likely make that one note stand out a bit less, but otherwise won't effect the timbre of the guitar to speak of. Obviously, larger changes in resonant pitches will have greater effects, but these are still not major unless they're really big.

OTOH, having a couple of major resonances at, or very near, the same pitch can be a real problem. I've seen an instrument that had something that sounded exactly like a string buzz due to the fact that the main top and main back resonant pitches were only 7 Hz apart. Adding a small amount of mass to the top dropped that pitch and got the spacing to 11 Hz, far enough apart to reduce the coupling and eliminate the problem, without changing the timbre of the instrument noticably.

At the moment, I'm getting decent results controlling for stiffness, ~E*t^3, but maybe I'll find out something better at some point.

Since frequency varies proportionally with the square root of the stiffness divided by the mass, for unbraced plates, I've been looking at comparing the numbers generated by frequency squared times mass. The frequency I'm looking at is the one indicative of stiffness along the grain. I can't remember the mode number. Al? So as I thickness the plate I look at the Chladni data and weight, and compare for a given model. I've just started doing this, but I think it will prove productive. I only do deflection testing on my braced tops. Al is absolutely right in that the top itself really overwhelms the bracing in regards to tone and that the balance of the top to braces is most important. Having said that I am experimenting with shifting modes around with some unusual bracing ideas. This will certainly have an affect on the tone. A simple example is the cross dipole. It is very easy to move the frequency of this mode depending on how much cross grain stiffness you add and where.

One thing that hasn't been mentioned is the affect of all this on the quickness of the response. A lot of fingerstylists like a very fast response. Less mass of course will help this, but I believe the distribution of the mass and stiffness must also play a role. I'm really becoming a fan of lighter bridges.

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Randy Muth
RS Muth Guitars Website
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The EI^3 term used to calculate beam stiffness (which is proportional to Al's Et^3) is a relation that I believe only applies to a simply restrained beam. As soon as you try to analyze a plate that is restrained along all of it's edges the problem becomes much more complex. Add to that the fact that the shape is organic...and that the material is orthotropic (i.e. the material properties may vary with direction) and the problem just gets more complex. There is one fundamental truth though and that is that the boundary stiffnesses can play a big role in the modal characteristics of a part like this...hence the little but of tuning that a lot of us do by thinning the edges of the lower bout after closing the box..

The K in the resonance equation (k/m)^.5 can be the stiffness under any type of deflection you can imagine. From simple torsional, compressive, or tensile modes all the way to the higher order more chaotic modes that you might see pop up when you do Chladni testing. K is really hard to calculate analytically using classical engineering relations. You'll usually only see relations for the most basic shapes in textbooks. FEA is the tool to use if you really want to get good numbers and to really explore this further.

Here's a link if you are interested in reading/checking out a bit of the analytical stuff:
http://www.efunda.com/formulae/solid_mechanics/plates/theory.cfm

I have all the FEA power I need at work to do this type of analysis....if I get the chance to run some stuff I'll be sure to share.

Best,
Trev

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I’m finding this thread very interesting – thanks, all! Alan, could you point me to some of the research you mentioned?

Trevor, thanks for the link. Yes, plate mechanics and the various boundary conditions make for a lot of complexity. For plates, E*I is replaced by "D" ("flexural rigidity"), but this also has a t^3 term. The boundary conditions definitely change the shape functions and overall deflection, but the mechanics of bending lead to t^3 whether biaxial or uniaxial. (If it's a 'thick' plate (Mindlin model, which accounts for shear) or deflections are large, then additional terms come in proportional to t^1.)

On the many other variables, yes, it'll be a long time before a model of the guitar can simulate the subtle compexities of a real guitar. But simpler models can be useful to control or correct for variations in wood, esp the top. I'm trying to understand what model(s) would be best for that. Maybe deflection testing on the unbraced top, or Chlandi patterns of braced but free tops are indeed the best... I just don't know at this point. Randy, your approach using f^2*m sounds interesting.

I'm wondering about a few questions:
- What aspects of tone do you all find are mostly related to the top, and what are mostly related to the bracing?
- On waves travelling in the top... For the strung guitar, after a string is plucked, to what extent are the vibration modes of the top excited by the oscillating forces at saddle (a resonance phenomenon), and to what extent do vibration waves travel/propogate from the saddle to the rest of the top (not necessarily a resonance phenom, more forcing the top like a speaker phenom). Does the latter occur in the attack phase of the note, wheras resonance takes over in the sustain (with string vibs providing energy to keep the resonance going, despite damping)?

And I'm trying to put some concepts together in my head, but don't know how to reconcile these two points (as I understand them):
1. Most of the tone is in the top. The bracing fine tunes the sound.
2. Most of the sound output of the guitar comes from the main vibration modes of the top (monopole, cross-dipole, long-dipole). Luthiers often focus on these modes to understand and control tone. These macroscopic modes involve large areas of the top and the underlying braces. I.e., for the top to assume these mode shapes, the braces must be moving with it. For voicing, Somogyi writes, "more dramatic changes can be had by attacking these braces than the top itself." (p. 114, TRG) And an analysis of the T-section (below) shows that the brace is responsible for most of the bending resistance of the top+brace, from 1/4" square braces on up.

I'm not questioning that "most of the tone is in the top" -- I trust your experience that this has real meaning. But I'm confused on how to reconcile what I understand so far.

I made a spreadsheet analyzing the T-section formed by the top and a brace, to look at how bending is manifested in each component. Snapshot is below, and link: http://www.malicky.com/davidm/Guitar.T.Inertia.xls Yes, there are a lot of simplfications, like isotropy, a brace parallel to the grain, beam theory, etc. Still I think it has something to say (in yellow):
- With a ~small brace (1/4" or 5/16" square), and assigning a generous 4" of top width to the T-section, the brace is responsible for ~3/4 of the resistance to bending of the overall T-section (77% cell). Cube rule strikes again.
- With 5/16" square braces or larger, the top and brace have more of a tension and compression profile than a bending profile (41% and 21% cells). I.e., stresses vary linearly away from the neutral axis, so while the overall "T" is in bending, the top itself is mostly in Ten-Comp. E.g., see pic below (from here: http://cnx.org/content/m27924/latest/16 ... Stress.pdf). This suggests that for bending that involves the T-section and considerable brace height, t^1 is more fitting than t^3. Thanks, Alan, for distinguishing structure and acoustic factors. Still, for the larger vibration modes (monopole...), it seems like braces would need to be active. OTOH, where braces meet each other but aren't structurally joined, the top's t^3 defines that hinge's stiffness. Maybe these hinges are related to much of the tone? And braces could be thought of as "vibration de-activators"?

Thanks for any insights.

_________________
David Malicky

david82282 asked:
"Alan, could you point me to some of the research you mentioned? "

Go to:
http://www.speech.kth.se/music/acviguit4/
and download part1.pdf through part9.pdf. That's Eric Jansson's 'Acoustics for Violin and Guitar Makers', fourth edition, and it's free. Some of it goes back a ways, and it's mostly concerned with classical guitars, but the lessons are valid.

Howard Wright's 1996 thesis: "The Acoustics and Pschycoacoustics of the Guitar", given at the University of Wales/Cardiff can be downloaded from their web site. It's a study based on cmputer modeling, so, as I have said, it has it's limits, but the main outlines are very useful. Again, free.

Both of these will have references that are helpful, of course.

"- What aspects of tone do you all find are mostly related to the top, and what are mostly related to the bracing?"

I don't think you can split it up like that. I think the character of the tone is established by big things, like model, wood selection, and so on, and the quality, how good it is within the limits of the particular character, is established by how well or poorly you make it, including getting the bracing 'right'. A Dread will always be a Dread, and a mahogany Dread will never sound exactly like a rosewood one, but there are good Dreads and bad Dreads. 'Right', as used above, is always going to be in relation to all of the other variables: the 'right' brace profile for a particular Red spruce top might be a lot different than it would be for an Engelmann top, and so on. That's the challenge.

"- On waves travelling in the top... For the strung guitar, after a string is plucked, to what extent are the vibration modes of the top excited by the oscillating forces at saddle (a resonance phenomenon), and to what extent do vibration waves travel/propogate from the saddle to the rest of the top (not necessarily a resonance phenom, more forcing the top like a speaker phenom). "

Modes is a frequency domain model, and traveling waves is time domain: both describe the same things, but one or the other may be more convenienient in a particular case. When the length of a bending wave in the top is longer than or close to the dimensions of the top then it's easier to think in terms of resonant modes (frequency response) rather than traveling waves (time domain). Which model you use in a particular case will depend on what you're looking for/at. As an example, I used the time domain model for a lot of my string paper (on my web site, under 'Acoustics') because it makes some aspects of what's happening clearer (such as how tension and transverse forces are related, and where the longitudinal wave comes from) than the usual frequency domain description, but both are equally valid. You can always get from one to the other mathematically, I'm told, if your math chops are up for it.

"I'm not questioning that "most of the tone is in the top" -- I trust your experience that this has real meaning. But I'm confused on how to reconcile what I understand so far. "

I don't see any conflict. Bracing 'shapes' the tone by effecting the mode shapes and frequencies. In general, braces tend to impose either node lines or antinodes: they add mass and stiffness in localized areas, and tend to either move a lot or stand still, depending on which predominates. Braces can thus 'quench' modes, or alter the shapes. For example, the 'cross tripole' that is an important sound producer on fan braced classical guitars up at around 500 Hz seems often to be missing, 'quenched', or shifted 'way up in pitch, on steel strings with X bracing. Sometimes steel string guitars with Martin style asymmetric tone bars will have a pair of 'diagonal dipoles' rather than the usual 'long' and 'cross' dipole modes (and sometimes the diagonals _with_ a long dipole, depending on A-1 air mode coupling). These things effect the timbre of the guitar, and are clearly related to the bracing. However, IMO, they have much less overall effect on the character of the sound than the actual choice of top wood, it's thickness and so on. What the bracing tends to do is either allow the top to work well, or keep it from doing so. In that sense, bracing is a necessary evil: we use it so that we won't have to make the top really thick and incur the weight penalty, but it needs to be done carefully so that it won't mess the top up too much. IMO

Alan and others, I want to thank you for all your valuable insight on my question!

Peter

Thanks, Alan, for your detailed reply. Those references are very helpful, and interesting.

It sounds like I misunderstood what was meant by 'most of the tone is in the top' and the traveling waves issue. I appreciate your explanation on the time v. frequency domains -- very lucid!

Thanks also for the clarifications on bracing -- nodes/antinodes, modes bracing can affect, and especially the just-right/just-enough bracing ideas... those make a lot of sense. I'm not quite seeing a complete reconciliation on the two earlier concepts, but I'm closer now. I'll look into the references next.

_________________
David Malicky

This has been a very interesting thread. Alan--thanks for the link to Eric Jansson's paper. I hadn't seen that. David--I like your ideas about the T sections. I've sometimes thought of bracing systems that way, but I'm kicking myself that I had gone on thinking of top stiffness as being related to t^3. Of course it is and it isn't. I think this idea has been discussed pretty well here. But I'm reminded that the top stiffness can't be treated independently.Thanks all.