Collecting 4DOF models

[GregHolmberg]

I'd like to collect here in this thread any mathematical models people have made using the four-degree-of-freedom equations. This would be composed of input values and calculated results, having been measured from and/or fit to real guitars, as shown on page 2-37 of the design book.

Unfortunately, as far as I know, there is only one such set of values in the world, on page 2-37, which is a nylon-stringed classical. I'm hoping we can collect more, maybe get some steel-stringed guitars as well. This would help folks (OK, me) design other similar guitars, starting with these as references. I'd especially like to see values for the Medium falcate-braced, live-back, steel-string guitar as built in the book.

I've started a spreadsheet of such models, one per column. If I can collect several such models, I could share it on the Internet as a Google Sheet.

So that I can iterate on the parameters faster, I've taken the Octave/MATLAB script posted by Jim Kirby here, and used Octave functions to read the parameters from the spreadsheet file, calculate all the guitar models it finds there, and update the results back into the file. I do this on Linux using the LibreOffice spreadsheet software, and I have also written a LibreOffice BASIC macro to automate this with one click in the toolbar.

I entered the data from the book into the spreadsheet, and got nearly identical results, including the plot (although I note that the coupled back frequency, fE, has a little higher dB level than in the book).

Yellow rows are physical constants, green rows are inputs, red rows are calculated results.

I've also made a few small improvements to the Octave script, as you can see in the spreadsheet above:
1. I calculate the four frequencies of the uncoupled "springs": air, top, sides, and back.
2. I calculate the monopole mobility of the top and back.
3. Using the Octave "signal" package, I find the three peaks of the coupled frequencies. This is so you con't have to zoom in on the plot and try to estimate the values.

I will share this Octave script soon. I think it needs some more testing. It should in theory work on Windows and Excel, but I don't have those to test on. There are also some bugs in the Octave "io" library (reads and writes spreadsheet files) that I'm working with the library developer on. I'm a little hesitant to post it, since setting it up involves installing a lot of stuff (Octave, three Octave packages, Java, and some Java libraries). I'm not sure I want to take on the task of supporting people getting all that working.

In the mean time, does anyone have any model values tuned to real guitars that they'd like to share?

Greg

[GregHolmberg]

I'm trying to fit the 4DOF model to what I know about the Gore falcate-braced medium steel-string guitar, built as shown in the plans included with the book (drawing No. 4).

There are bits of data on this guitar scattered throughout the book, but the main thing I don't know is the mass of the top.

Anyone built one of these and know the mass of the top?

I know the dimensions (390mm x 490mm) and I have an estimate of the area I made by counting the squares on the plan: 0.1411 m^2, not including the area of the head block.

I know from table 2.4-1 that on the Gore classical the mass of the top piston is 0.043, and the area of the top piston is 0.039, giving a density of 1.10 kg/m^2. And I know that the top on the steel-string is thicker and deeper-braced, so must be denser than that.

My current guess is that the top on the steel-string might be around 1.55 kg/m^2, giving a top mass of 0.219 kg.

Keeping the same proportions of piston area as the classical (30.35%), the steel-string would have a top piston area of 0.0428, and therefore a mass of the top piston of 0.0664.

On page 2-26 Gore says:

However, the typical equivalent mass of a top plate is around 40 grams for a conventionally built classical guitar, and less than 70 grams for a lightly built medium sized steel string guitar.

So I'm thinking that at 66.4 grams, my guess is a little low?

Maybe I should try a density of 1.63 kg/m^2, giving a mass of the top piston of 69.8 grams?

Anybody have measured weights and areas from the top of a real Gore falcate-braced medium steel-string guitar?

Thanks,


Greg

[GregHolmberg]

Surely someone out there knows the mass of the top on the falcate-braced medium steel-string?

Well, in any case, I've attempted to estimate the mass by adding up all the parts, and using their dimensions and density. Here's what I got:

Screenshot from 2021-07-24 14-53-56.png (72.28 KiB) Viewed 35138 times

This is just for the vibrating area below the transverse brace. I calculated the areas by counting squares on the drawing, not including the head and tail blocks.

So, 196.1 grams for the structure below the transverse brace, or 1.68 kg/m^2.

This seems about right. With an area for the top piston of 0.0415 m^2, that gives a mass of the top piston of 0.0698, which agrees with the quote from the book (see previous posting).

So that's what I will use in modeling the falcate-braced medium steel-string.

Greg

[GregHolmberg]

OK, my best guess for modeling the Gore Medium steel-string:

Screenshot from 2021-07-24 17-11-15.png (124.32 KiB) Viewed 35131 times

If you want to see how I came up with these numbers, here's the spreadsheet.

My technique is: I start by estimating the areas and masses by counting squares on the drawing, or making proportional calculations. I have no idea how to estimate the damping values, I just used the ones from the book. I tested these estimating techniques by using them on the nylon-string classical guitar in Fig 2.4-4, and it came out very close to the same.

Then I play with four parameters until I match the target coupled frequencies: sound hole diameter, stiffness top plate, stiffness back plate, added side mass.

I initially set the sound hole diameter to the design size (95 mm), and the stiffnesses to give monopole mobilities of 18 and 9, and added side mass to 0.

Then I select one of the sets of target coupled frequencies that the book recommends, and adjust the four parameters trying to get close to the target frequencies. Sometimes I have to change the targets if I can't get there and another target looks closer.

In this case, I ended up with targets frequencies of 95/170/214, and I was able to get pretty close to those.

My four parameters ended at: sound hole = 88 mm, Ktop = 51000, Kback = 108000, added side mass = 0.

Does anyone have real numbers for this guitar?

Thanks,


Greg

[seeaxe]

Out of interest i weighed the various bits this morning.

My soundboard weighs 196g. There's some to trim off round the edges but id expect that to not be more than 10 grams. Its Lutz with a thickness is 2.3mm, 10mm high by 5mm wide tapered braces.

Does anyone else weigh theirs and if so what sort of weights do you end up with??

The whole shebang should be about 2kg when its done, not counting any side mass i might need to add

Hi Greg, looks like you are forging a path of your own with the mathematical modelling!! This is probably no use at all, but here's the weight of my classical top with falcate bracing. I didnt get any responses re weight of the tops either.

I wasnt far off, the finished guitar ended up at 1897gms.

Cheers
Richard

[GregHolmberg]

Thanks for that, Richard.

196 g for a classical top. If your guitar is anything like the one in the plans included in the book, then I'm going to guess that your top is about 490x370 and 0.1398 m^2 in area (I counted the squares in drawing #3, including the area of the head and tail blocks).

That would put the areal density of your top at 0.196/0.1398 = 1.40 kg/m^2. Somewhat higher than modeled in the book (Fig. 2.4-4: 0.043/0.039 = 1.10), but that's only looking at the "piston" area inside the Chladni lines, so doesn't include the transverse brace, which is substantial.

I have revised my estimate above for the top of a medium falcate-braced steel-string guitar (2.72 mm of Engelmann Spruce) to use lower-density KingBilly/CF braces (400 kg/m^3 instead of 531), and now get a total of 0.1880 kg for the area below the transverse brace, and so an areal density of 1.6134 kg/m^2. Of course, I don't think I can get KingBilly Pine in the US, but that's a different issue!

It makes sense that it's denser than your 2.3mm Lutz top, since classical guitars have lower torque on the tops.


Greg

[Dave M]

It's a long story but i have been able to get an approximate weight for a falcate top. This has approx 5 mm extra round the edges and does not have to cutaway removed yet. It is a little less than 3 mm thick Came in at around 230 g.

Cheers Dave

[GregHolmberg]

It's a long story but i have been able to get an approximate weight for a falcate top. This has approx 5 mm extra round the edges and does not have to cutaway removed yet. It is a little less than 3 mm thick Came in at around 230 g.

Cheers Dave

Thanks, Dave!

Is this for a steel-string or nylon-string?

Do you have any idea of the area of the top?

Thanks,

Greg

[Dave M]

Sorry thought I had said. It is steel string, based very closely on Trevor's design in the book for a mid sized SS. Assuming you have the book you can fairly easily work out the area since it is printed on a grid of squares. (and that takes me back a bit trying to measure the area uder curves on spectra by counting squares!)

As I said it did not have the cutaway area removed yet but did have the soundhole in it. oh and it is braced.

Cheers Dave

[AKEric]

Hello,

I have data on three Falcate braced tops. All are standard OM sized, I don't have exact area measurements, but all have a 1/8" rim over hang for trimming and sound hole cut out. Tops thickness per my spreadsheet, all under 3mm. All Sitka spruce.

#1 = 205g. No cut-a-way. This was an especially light top. Best sounding one of the bunch.
#2 = 239g. With cut-a-way.
#3 = 229g. With cut-a-way.

Both 2 & 3 had slightly higher max brace height to achieve a higher set of top and air cavity frequencies.

Hope this helps.

Eric

[GregHolmberg]

Sorry thought I had said. It is steel string, based very closely on Trevor's design in the book for a mid sized SS. Assuming you have the book you can fairly easily work out the area since it is printed on a grid of squares. (and that takes me back a bit trying to measure the area uder curves on spectra by counting squares!)

As I said it did not have the cutaway area removed yet but did have the soundhole in it. oh and it is braced.

Cheers Dave

On page 4-62 in footnote 33, Gore says that the area of the top of the guitar used in table 4.5-3 (a medium steel-string with cut-away 490x390--same as in drawing #4, I assume) is 0.144215 m^2.

I got 0.1428 counting squares in drawing #4 (including the head block, tail block, and sound hole areas)--about 1% less than in the book. I counted the head block area at 0.0031 and the tail block area at 0.0010, so without those areas, the free area (not glued down) of the top I get 0.1387. If I bump that up by 1%, I get 0.1401 for the free area. The sound hole is 95mm, so an area of 0.0071 m^2, if you want to subtract that.

I'm not sure where your top will end up after trimming the edges and removing the cut-away. At its current weight, 0.230 / 0.144214 = 1.59 kg/m^3. A good reference point.

You don't say which species you used. Sitka Spruce top? I see you're in the UK, so probably not King Billy Pine for the braces. CF re-inforced?

Greg

[Dave M]

Yes Sitka. And CF under and over all braces and under the bridge plate. Braces also in Sitka. D

[GregHolmberg]

Hello,

I have data on three Falcate braced tops. All are standard OM sized, I don't have exact area measurements, but all have a 1/8" rim over hang for trimming and sound hole cut out. Tops thickness per my spreadsheet, all under 3mm. All Sitka spruce.

#1 = 205g. No cut-a-way. This was an especially light top. Best sounding one of the bunch.
#2 = 239g. With cut-a-way.
#3 = 229g. With cut-a-way.

Both 2 & 3 had slightly higher max brace height to achieve a higher set of top and air cavity frequencies.

Thanks, Eric.

I'm not sure how big an OM is. Do you have width and length for that? I think it's similar to the 490x390 in the book. OM implies steel-string, right?

If it's similiar in size, then area would be close to the number in the book, 0.144215. In which case, your tops would have areal densities of 1.42, 1.66, and 1.59.

Greg

[AKEric]

Length and width for my OM sized steel string guitars are 482mm x 382mm.

[GregHolmberg]

Length and width for my OM sized steel string guitars are 482mm x 382mm.

OK, I'm going to guess the area for the cut-away version is about 0.1390 then (vs. 0.1442, i.e. about 4% smaller), giving areal densities of 1.72 and 1.65 for the two cut-away ones.

[GregHolmberg]

I have added to my spreadsheet an estimate of the whole soundboard mass of a medium steel-string with cut-away (per drawing #4), including the transverse brace, and head and tail block areas.

I came up with 238 g. Pretty close the the numbers others have given here (229, 230, 239).

For an area of 0.144215, that's an areal density of 1.65 kg/m^3. Close to other numbers given here of 1.72, 1.65, 1.59. That gives me confidence.

I also improved my estimate for just the vibrating portion below the transverse brace. I got 198 g for an area of 0.1260, and so an areal density of 1.57.

Greg

[GregHolmberg]

I used the new 1.57 kg/m^2 number to update my 4DOF model of the medium falcate-braced steel-string.

Based on the classical in table 2.4-1, the area inside the Chladni line (i.e. the top effective piston) is about 30% of the free top area, so I'm using At = 0.0416 m^2 (compared to 0.039 for the slightly smaller classical).

Similarly, about 53% of the top becomes the outer piston (aka the "sides"--terrible name!), so As = 0.0746 m^2.

As described above, I use target frequencies of 95.2, 169.7, and 213.8 Hz, and adjust sound hole diameter, stiffness of the top and back, and added side mass.

I found that D = 88 mm, Kt = 48000 and Kb = 108000 N/m, and no added side mass gave me frequencies of 94.8, 170.0, and 213.9 Hz. Pretty close. Monopole mobility for the top and back of 17.9 and 10.3.

Not much changed due to the updated top mass. The biggest change was Kt went from 51000 to 48000.

I'm still hoping that someone who has actually built this guitar can give me real measured numbers for Kt and Kb.

Also, any insights on how to set the damping values (Ra, Rt, Rs, Rb) would be helpful.

Next, I will try to model an archtop. I want to figure out target frequencies, and top and back stiffness (Kt, Kb) and sound hole area, and see if I can maintain good monopole mobilities. Archtops are usually not nearly as deep as acoustics (75 vs 115 mm), so things could get interesting!

Greg

Screenshot from 2021-08-14 16-45-47.png (113.53 KiB) Viewed 34712 times

[GregHolmberg]

Reminder: all this data is in the spreadsheet.

[seeaxe]

Will be interesting.
How are you going to calculate your I value for a curve/recurve archtop with the normal two strut braces??
Does the fact that the "soundhole" is two F holes make a difference?
Given that the top is significantly arched, its going to act more as a shell than a beam. Beams can be analysed in 2D but shells need 3D math. Would Trevor's equations even apply?
Wouldn't you be better building an FEM?

I'm not entirely clear on what you are trying to achieve by analysing an archtop. Don't they generally rely on a pick up for sound??

Good luck anyway

[GregHolmberg]

All good points, Richard. I'll take them in order.

Yes, it would be difficult to calculate I for such a complex structure as archtops usually have. Varying thickness, complex shape, holes in inconvenient places.

I'm thinking I'll create a cylindrical top with a constant thickness. I hope this simple shape won't require FEM! The axis of the cylinder (and the grain) parallel to the strings. Wood is significantly stiffer along the grain than across it. For example, you can see in table 4.5-1 that Sitka Spruce has a ratio Elong/Ecross of 14.5. A typical archtop rise of 10 mm gives a radius of 2 m, which would give a increase of frequency of almost 3X (Equ. 1.6-2), and an increase of stiffness (K) of 8.8X. This would bring the long and cross stiffness much closer together, and also make the top resistant to cracking with humidity changes. I will press a solid piece of wood into a cylindrical mold, since it bends pretty easily in that direction of the grain. I may eventually try the "double top" method of two thin layers of wood with a layer of Nomex epoxied between--a sort of laminate that's very strong and light. It's been done on archtops before. See Steve Anderson's video.

I've sketched a number of possible symmetric bracing patterns, inspired by Gore's falcate braces. I can probably make some ballpark calculations of I using the same methods as in the book. This will get me close on dimensions, but I'm sure I'm going to have to prototype and measure many tops to hit the stiffness (Kt) target I calculate from the 4DOF model. Some luthiers have been making the braces on their archtops very minimal. See what Ken Parker is doing. Just a few mllimeters tall. I may well find out that I don't even need braces with the cylinder.

I will put the sound hole in the upper bout. Not sure about the shape yet. There's a fascinating paper on sound hole shape and power efficiency in violins here. Round is definitely not the way to go! But I think the 4DOF model can at least tell me a target area based on the target frequencies and body cavity volume. It's location and shape will alter the coupled Helmholtz frequency, so again I will have to experiment to get the frequency just right.

Many archtops are primarily electric guitars, and the hollow body adds little to the electric tone in my opinion (although some people believe it does). There is a kind of acoustic archtop that preceded pickups, the Gibson L5 being a good example. But there were also many other smaller acoustic archtops. Today these are still made by Eastman, Loar, and a few others. In addition, a Selmer is a kind of archtop, just with a very slight arch.

I'm interested in these because the force of the bridge on the top is purely downward, with no torque. The cylindrical shell is perfect for this force, and the bracing can be lighter I think, since there's no torque. The trick will be finding the right dimensions and bracing pattern.

The 4DOF model will give me targets for frequencies, top and back stiffness, sound hole area, and added side mass, derived from my designed guitar size (area, volume), so that when I get the components right and assemble them, they will couple and produce the right frequencies. I hope.

It's a challenge, but I think it will be fun.

Greg

[GregHolmberg]

OK, I've modeled a large acoustic archtop, 17" x 3.3". Pretty close to a Gibson L-5.

I came up with an 80 mm sound hole, top stiffness 55,000 N/m, back stiffness 137,000, no added side mass, producing resonant frequencies of 90, 180, and 227 Hz.

This gives me targets for the cylindrical archtop I want to build.

Let me know if anything looks unreasonable.

As usual, all the models are in this spreadsheet.

Greg

Screenshot from 2021-08-16 17-28-55.png (107.93 KiB) Viewed 34641 times

[seeaxe]

Hi Greg

How have you incorporated the curve in the top into your calculation of I?
I cant see it in the spreadsheet.

Apparently the formula for the I of a curved shell is in the later editions of Roark and Young. I gave mine away when I retired a few years ago. A 10mm rise will make a big difference. I assume you are using your I value to derive your K value??

Also, I re read your previous post and I don't understand this

A typical archtop rise of 10 mm gives a radius of 2 m, which would give a increase of frequency of almost 3X (Equ. 1.6-2), and an increase of stiffness (K) of 8.8X. This would bring the long and cross stiffness much closer together,

How does that work? Would it not be the other way around assuming you are making a cylindrical top?

Cheers
Richard

[johnparchem]

Just read through the posts. I am not sure it is addressed, does the mass of the top only include the vibrating area of the top not the weight of the top itself. That thought always kept me from weighing the top.

[GregHolmberg]

How have you incorporated the curve in the top into your calculation of I?
I cant see it in the spreadsheet.

A 10mm rise will make a big difference. I assume you are using your I value to derive your K value??

Also, I re read your previous post and I don't understand this

This would bring the long and cross stiffness much closer together

How does that work? Would it not be the other way around assuming you are making a cylindrical top?

Thanks for the interest.

I'm still working on calculating E and I for the top. The K values in the spreadsheet are target values needed to hit the coupled resonant frequency for the top and back, 89.9 and 179.8 Hz.

Next I will try to calculate E, I, brace stress, and K for the top. I will adjust the brace dimensions until I get the target value for K. I haven't yet found a formula for I of a cylindrical shell. I'm trying to work with the Equ. 1.6-2 to calculate the change in frequency from a flat panel, ω, and from there the change in stiffness, K ∝ ω^2 (ω = √(K/m)). I will continue to search for a formula for I of a cylindrical shell, though.

I got the 10mm rise from an archtop I own, and calculated the radius from the width of my design (400mm).

Using Equ. 1.6-2, thickness 2.8mm, and Poisson's ratio ν = 0.422 for Engelmann Spruce, and got ω/ω0 = 2.96 and K/K0 = 8.78.

So I can calculate K for the flat panel (no braces), then multiply by 8.78. How to then calculate K for the braced panel, I'm not sure. I really need that formula for I of a cylindrical shell.

I currently calculate K for the braced flat top using Equ. 1.3-1, which is for a beam. So that may or may not be appropriate. I'm looking at longitudinal stiffness since the falcate braces run that way, and a span of 350 mm from tail to transverse brace.

I can then adjust the brace dimensions until I get the target K needed to hit the target frequency (from the 4DOF model).

Of course, all of this is just to get me in the ballpark, and many prototype tops will have to be made with measurements of stiffness and frequency.

Regarding bringing the long-grain and cross-grain stiffness closer together, perhaps I've misunderstood how it works? The long-grain stiffness is when you bend perpendicular to the grain (i.e. pushing down on the tail and head area of the panel with a central support), right? Which is much higher than the other way (pushing down on the sides). So if I mold the cylinder shape with axis parallel to the grain (i.e. sides of the top fall away from the center under the strings), then that increases cross-grain stiffness, I think. So when I apply the downward-only force of the strings as in an archtop, it acts like a compression arch bridge. And Elong/Ecross for Engelmann Spruce goes from 14.5 to 1.7. Did I get that wrong? Am I confusing strength with stiffness?

So, obviously, I'm in over my head, and any suggestions are welcome.


Greg

[GregHolmberg]

Just read through the posts. I am not sure it is addressed, does the mass of the top only include the vibrating area of the top not the weight of the top itself. That thought always kept me from weighing the top.

Well, I've asked poeple for the weight of the whole top, since that is really the only thing that can be easily measured.

However, in the 4DOF, as seen in Table 2.4-1, the mass (0.043 kg for the "top") and area is really just the the effective "piston" area inside the Chladni lines. Likewise, the area of the "sides" is the area outside the Chladni lines (front and back). This is of course pretty much impossible to measure, and Gore's numbers just come from trying to fit the model to the real guitar. In the text he talks about a study my Jansson that attempts to measure these things from volume displacement, and notes that Jannson's number, 0.031 m^2 for the top piston area, is close to his (0.039).

In my spreadsheet, I attempt to appoximate this area in the medium steel-string by making it proportional to the ratio of the area:total-area of the classical in Table 2.4-1, and I come up with 0.0416 m^2. I then use the estimated average areal density of the vibrating area below the transverse brace (1.55 kg/m^2) to get a mass of 0.0644 kg (compared to 0.043 for the classical).

But it's all a guess. It would be great if someone with an actual falcate-braced medium steel-string could try to repeat measuring and data-fitting process that Gore did with the classical in Table 2.4-1. I don't have such a guitar, so I can only make proportional estmates.

Greg