Equation 1.7-1

[Craig Bumgarner]

I'm a bit stumped on the units for Equation 1.7-1, both for input and results. I inferred from the text that K is the load in Newtons over the resulting deflection in mm. I don't understand the units of the results shown in Fig. 1.7-8, s/KG^-3, in particular what is "s". Elsewhere "s" is pluck length, but I don't see that in Eq. 1.7-1 so the units of the result confuse me.

I'm trying to relate measured Specific Mobility of one of my guitars with the chart in fig 1.7-8, and in calculation, I'm off by a factor of e+5, so I'm thinking maybe I have a units problem. The other possibility is I'm using a lot of weight (2.2KG) compared to what is suggested in the book. My deflection is measured to .001" and I can easily see variations of .001" to 002" with the addition of 500 grams, but maybe my load is out of range somehow.

Thanks,

[Craig Bumgarner]

I don't understand the units of the results shown in Fig. 1.7-8, s/KG^-3, .....

Well, first, that should be s/KG x 10 ^-3, but I still don't understand the units.

Second, maybe it would be a help if I posted the data I working with:

• Deflection load: 5 pounds, I input 22.241 Newtons
• Deflection: .007", I input .1778mm
• My result for K (load/deflection) is 125.0899
• Uncoupled Main top frequency: 230hz (just a guess at this point, have not actually measured it. The coupled top peak of this Selmer style guitar is 258hz and the main air peak is ~ 98hz, established by Freq Response Curve in VA-11).
• Using my K figure and 230hz to calculate m at .00005989

When I run K and m through equation 1.7-1 for Specific Mobility, I get 11.55. Somehow, based on the Specific Mobility numbers on the Y axis of Fig 1.7-8 and -9, and the s/Kg x 10^-3 units for these numbers on the Y axis, I would have expected a number like .01155. The Gore falcate in 1.7-8 has a SM if .018 (or 18 x 10^-3) right?

Not very good at higher math so my error could be most anywhere. I am using a spreadsheet and it seems to work as I would expect with simple numbers.

[Trevor Gore]

Having spent all of 5 seconds on this (!), the problem is likely the "zero" count.

I suspect that you have entered values in millimetres. To be true to SI units, they should be entered in metres.

So, 0.007" = 0.1778mm, but should be entered as metres, i.e. 0.0001778 metres. Millimetres is just a shorthand way of saying 10^-3 metres. SI units are based on the metric kilogram (sort of inconsistent because of the "kilo"), metre, second set of units, which despite the minor inconsistency of the kilo-gram, is way more sensible than imperial units and is why everyone but the USA uses them.

Try checking the zero count and see how you go.

N. B. To avoid running out of accuracy (called ill-conditioning, due to mixing very big numbers with very small numbers in Excel) it pays to take the factors of ten off to "one side" and put them back in at the end of the calc. Just be sure to put the correct number of zeros back in!

[Barry Daniels]

I am working on this spreadsheet too. Inputing Craig's numbers using what I think are the correct units, I get an s of 113. Seem to be one digit off.

[Trevor Gore]

Again, without a lot of checking, I would suspect a factor of ~10 out might be due to confusing load and mass. A 1 kilogram mass applies a load of 9.8 newtons.

[Steve]

Craig - I've just plugged your numbers in & I'm getting a result of 0.01156 s/kg (or, 11.56 x 10-3 s/kg). I get the same answer as you did if I leave the deflection in mm (instead of converting to metres).
I reckon that the s refers to seconds (from pulling apart K's Newtons into kg*metres/second^2).
Steve

[Trevor Gore]

I reckon that the s refers to seconds

Yes. Of course. I thought that was obvious (using the SI metre, kilogram, second or m, k, s system), but clearly not, if you're not used to SI units. s is seconds unless otherwise stated.

Apologies to Craig for not addressing that specifically in his first post.

[Barry Daniels]

Got it. I was not converting to Newtons. Darn SI units.

[Craig Bumgarner]

Okay, I'm going to assume then that my calculation is basically right. Being metric, the zero count will depend on whether one uses Newtons, Kg, mm or Meters. From here out, I just need to be consistent. As the graph in Figures 1-7.8 and 1-7.9 uses a scale of 0-30 on the Y axis, I'll use what gives me a similar order of magnitude. As Kg is in the units, I'll use Kg for the load and centimeters for the deflection.

I tightened up my measurements a bit. In my deflection method of old, I used weights from 5 pounds to 20 pounds (2000 to 9000 grams) and measured deflection in thousandths of inches. I have lots of records like this from my own guitars and known good examples, so I am hopeful my measurements will correlate with the much lighter loads (400 - 1500 grams) used in the book.

I put load and deflection numbers for one guitar, just a body at this point, into Excel and here are the results. I'm happy to see the Specific Mobility numbers are reasonably close for the various loads. Hopefully, this suggest some degree of accuracy and consistency across the various loads.

As a separate issue, I'm surprised by the measured uncoupled top frequency which I take to be 278hz. This is the highest peak on the FRC (below), but note there is a second, lower peak just before it at 232hz. From the book, I had expected the uncoupled top main to be lower than the coupled top main (258hz), but this isn't what I got (maybe I read the book wrong, I can't find the reference now). In this case, it actually went up. Maybe something wrong in my testing? (plugged hole with a 16mm thick styrofoam plug, nice tight fit to sound hole). Anyhow, FRC curves of coupled and uncoupled top below, comments appreciated.

BTW, the high main monopole frequency seems to characteristic of this Selmer style guitar. Part of why I'm being finicky about all this is because these guitars do seem different so while the principals in the book apply, the numbers I'm getting are different.

Uncoupled top FRC

Coupled top FRC

[Craig Bumgarner]

On second thought, maybe better to use Newtons and mm:

Specific Mobility, screen shoot 2.jpg (38.43 KiB) Viewed 32625 times

Thanks for everyone's help on this!

[Barry Daniels]

I wonder if that double peak on the uncoupled top frequency is abnormal? It looks like a remnant of the double peak is also present on the coupled top curve.

[Trevor Gore]

On second thought, maybe better to use Newtons and mm....

Definitely!

There is a difference between mass and load. In engineering, mass is a measure of inertia, (units of kilograms) and load is a force measured in units of newtons. Force = mass * acceleration (Newton's 2nd Law) so the load on planet earth of a mass of 1 kg is (mass * acceleration due to gravity) = 1kg * 9.8 metres per second per second = 9.8 newtons. A mass of 1 kg is still a mass of 1 kg on the moon, but it has a different weight because the acceleration due to gravity is different.

One of the problems with imperial units is that loads, weights (a load) and mass often get confused, frequently with the different units being referred to by the same name. e.g. pounds can be found referring to pounds mass or pounds force (as well as being a unit of currency!). Using SI units properly takes care of these ambiguities.

Looking at your mobility numbers, my initial impression is that they are wrong!

I know what 17.0 ms/kg (that's milliseconds per kilogram, same as s/kg *10^-3) looks like. My limited experience with Selmer type guitars is that they are a lot stiffer, as you have found. Loading with 20 pounds on one of my flat-tops would be pretty radical. Also, I can't see why their effective mass would be proportionately less, which it would need to be to get the mobility number back up. I would expect the effective mass (including the bridge) to be similar to that of a flat top, or greater. So it may be worth checking your figures yet one more time.

Also, have a read of Section 2.3.7. Whilst those findings seem to hold on all the flat top guitars I've measured (classical and steel string) they may not be valid on a Selmer type build because of the quite different relative masses and stiffnesses of the components. That doesn't mean they're not valid for Selmer style guitars, I don't know. You'll have to satisfy yourself, one way or the other. The other thing about your plots is that if the top is stiffer than the back, once uncoupled, the frequencies might move in the opposite direction to that expected (repulsion effect).

When you have it all sorted, maybe you should write a Selmer chapter.

[Craig Bumgarner]

Okay, thanks for these ideas, sorry if I'm being a pest. I too have been skeptical of the mobility number as the Selmer tops for the same reasons you state, they do seem stiffer than most and the mass is probably not a lot less. I say probably as I really don't know what the tops on other styles weight. My tops, ready to glue to the sides, weigh in at about 250-260 grams.

I would guess my tops to be a bit lighter than a flat-top SS guitar, based not on actual measurement of FT-SS guitars, but on supposition (danger, danger). Mine have considerably less bracing, both in number and dimension, and often a cedar top that is not much thicker than what I commonly see on FT-SS guitars. No bridge patch and a bridge that typically weighs in at < 12 grams. That said, I have no idea what the range for top weight is for guitars other than my own. Much of the Selmer stiffness comes from the considerable arch both in the long and cross directions. The arch is ~10mm measured at the outer edges of the top with a straight edge resting on the bridge area, ~ 7 foot radius if you think that way. I suppose this contributes substantially to stiffness without adding mass. I will say Selmer guitars are loud compared to most guitars and this is one of their primary desirable characteristics as they were developed in the 1930s and 40s to play acoustically, both rhythm and solo in jazz ensembles of 4-5 players, often in crowded, noisy rooms.

To work through my calculation of mobility again, I'm thinking these are the elements:

Data: Measured DEFLECTION in mm, under a measured LOAD in Newtons. Uncoupled top MAIN RESONANCE FREQUENCY (f), in hertz. I'm pretty confident about the deflection and load, at least at the higher loads I use, as I have done them many times and results have been very consistent. My loads are applied w/ barbell weights which might be off a little bit, but not much (I just weighed one, it is 23 grams short of the 5lbs (2268 grams) it is labeled as). The uncoupled top main is just what I read off the FRC in VA, tapping an unstrung body with a plugged sound hole and I have little experience with this, so I could easily being doing something wrong here, especially as my main top resonances seem so much higher than other guitars. I did an FRC on a decent dreadnaught guitar, however, and it was right in the ball park with the book for dreads, suggesting my FRC setup is okay.

Calculations, based on the book, in Excel formula format:

• Specific Mobility = 1/(SQRT(K*m))
• K = LOAD/DEFLECTION
• m = K/((2*3.1416*f)^2), where f is the uncoupled main resonance of the top by way of recorded tappings in VA.

Interpretation: Lot of variables here and it is the area I am least confident of. The main concern seems to be whether the Selmer guitar is too different from flat top guitars to be compared directly. If my data and calculations are sound, however, I guess I am willing to go with it as is and just say viva la difference. I'm not trying to "outdo" other guitar types, just trying to measure good ones in style and try to get mine to compare.

Thanks for your patience and help, I'm not at all fluent with the physics and mathematics of your book, but I do find it very interesting and much of it make a great deal of sense to me in a general way.

[Craig Bumgarner]

I see in my last Excel screen shot, the formatting of the column for m was only to 4 places. It still calculates correctly, but looks better when formatted to 8 places.

Specific Mobility, screen shot 3.jpg (39.16 KiB) Viewed 32577 times

[jeffhigh]

I have no idea what the SM Numbers for a Selmer would be. I have not tested my one build, but the top is definitely lightweight and stiff compared to a flattop at only 2mm thick and with a high bent arch (pliage).

[Trevor Gore]

I've got my time cut out for me over the next few days, Craig, but I'll run your numbers through my program later and see what I get. Your mass numbers look very small compared to mine. You might want to check your zero count there. If I'm reading this right, your numbers are suggesting 3.2milligrams (!) as the effective mass of the top (column H), if your units are kilograms.

[Trevor Gore]

I've just run your data through my program and get much the same answer as you, Craig. So it seems that the Selmer guitars do have a very low effective mass! Your intermediate numbers look odd, though!

[Steve]

I've just run your data through my program and get much the same answer as you, Craig. So it seems that the Selmer guitars do have a very low effective mass! Your intermediate numbers look odd, though!

It looks like K has been calculated in N/mm (instead of N/m) which in turn has modified m, which looks like it may have given a result in s/kg x 10-3 to line up with the units in the graph?
Steve

[Craig Bumgarner]

It looks like K has been calculated in N/mm (instead of N/m) which in turn has modified m, which looks like it may have given a result in s/kg x 10-3 to line up with the units in the graph?
Steve

Yes, that's right. In order to get the zero count right compared to the graph, I used Newtons and millimeters for load and deflection.

Trevor, thanks for running my numbers through your program and checking my calculations. I'm still skeptical about the data I'm imputing, but can fine tune that as time goes along. For establishing K for instance, I notice that no matter how solidly I chock the guitar, there is 2-3 thousandths of play before applying load. Can't feel it, but if I push down on the edge of the guitar, the dial indicators register, which of course they should not. Some small amount of this may be flex in the sides but at least some of it is back lash in the chocking. This wasn't to important to me in past deflection measurements as I wasn't using them for anything other than comparison against similar measurements on other guitars and if all the measurements are overstated by .002", it really didn't matter, especially when I was looking at deflections in the .038" range. But .002" makes a considerable difference in SM, especially if the load and deflection to calculate K are small. I need to cook up a better (and faster) means of chocking and measuring for K.

Also, still not sure how valid that 278hz uncoupled top main really is. This is without bridge, neck and strings. That double peak bothers me too. I'll report back once I get it string up in a week or two. I ran the FRCs on an older guitar of mine last night, with bridge, strung up, well played in. It is a Selmer style as well, but more heavily built than what I build now. The uncoupled top main came in at 230hz, the coupled top at 240hz and the uncoupled back at 230hz as well which seems rather suspicious. Deflection with 5 lbs. was .007", so running this through the spreadsheet gives a SM of 11.5. If SM relates to volume as I understand from the book, the lower SM is not too surprising as this guitar has good tone but is not as loud as what I build now.

[Craig Bumgarner]

I'm Ba-ack! Sorry to be a pest, but I've got a couple questions about MM and target monopole peak resonance frequency. If I may.....

What is the basis for target monopole peak resonance frequencies? Good examples? From what I understand, which, granted, isn't much, the higher the frequency, the higher the stiffness or the lower the (effective) mass. If deflection under load is already pretty high, then a higher frequency must mean lower effective mass and a high MM. Correct? And if so, why isn't a higher frequency (assuming no scale note conflicts) always a good thing. In reading the book, the effort seems to usually be to get the frequency down, not up.

The Selmer style guitar I just boxed looks like this (no bridge, neck, etc.):

Monopole Mobility.jpg (23.1 KiB) Viewed 32344 times

IF....., that is, I assume the uncoupled top monopole freq. is 258Hz. But look, if you will, at this graph.

uncoupled top.jpg (92.17 KiB) Viewed 32344 times

How would you interpret the double peak? Which would you use in the MM calculation? I get these double peaks a lot. I'm guessing peak means the highest in amplitude, 258Hz.

Aside.... Any idea what is going on at 440Hz - 520Hz?

BTW, my 400mm x 470mm thicknessed cedar top (3.3mm) and profiled top weighed 158 grams before bracing and the braces below the sound hole in total weighed in at 35 grams.

Thanks,

[DarwinStrings]

In reading the book, the effort seems to usually be to get the frequency down, not up.

I think Craig that getting the frequency down is because it is a bit easier than up, so you have a target to aim at, you build a bit above then once you have strung up you drop things a little to exactly where you want them. You have to decide on which frequency you want for the air or top etc. You may choose say 151Hz for air and 226Hz for the top, you then aim a bit higher in the build and drop it post build unless you can build it bang on your choice without adjustment.

Aside.... Any idea what is going on at 440Hz - 520Hz?

From what I understand you will have to Chladni test to be sure of those peeks unless you know them well enough to guess and I reckon all the tea leaves would just keep falling of that pliadge end of your style of guitar, might make things difficult. Good luck.

Jim

[Trevor Gore]

What is the basis for target monopole peak resonance frequencies? Good examples?

Yes. Good examples for flat tops in specific sizes. Check out Table 22-1.

If deflection under load is already pretty high, then a higher frequency must mean lower effective mass and a high MM. Correct?

Yes, if you have a higher frequency for the same stiffness.

why isn't a higher frequency (assuming no scale note conflicts) always a good thing.

Because, generally, you still want the guitar to sound "in genre" for the type, which means hitting specific modal resonances. If you don't mind a Dreadnought sounding like a Parlour you can do it differently, of course. If you pitch the main resonances higher it's like turning the tone control to treble. You gain treble and loose bass.

In reading the book, the effort seems to usually be to get the frequency down, not up.

It's much easier to drop frequencies than raise them. Hence building a little stiff then dropping stiffness or adding mass to achieve target.

But look, if you will, at this graph...

I'm guessing, but two possibilities come to mind:

1) The split peak is something to do with the back. On flat-tops, when the sound hole is plugged, the strength of the coupling of the back to the top via the air is similar in strength but opposite in phase to the coupling via the sides and so they cancel. Therefore when you look at the spectral response of a top tap, you see only one peak - that of the uncoupled top. This relationship, which is just fortuitous for flat tops, may not happen with Selmer style guitars, so you see a residual back peak, which means the top is not uncoupled from the back so you get a "peculiar" result for the top frequency which will knock-on into the mobility result.

2) Because of the pliage in the top, you may have a long dipole very close in frequency to the main monopole. I can't see these two modes being strongly coupled, which is good, because if this is the case, when you figure which is the monopole you should still get a good number for mobility.

So you'll have to figure out which modes the peaks represent, which should be a bit of fun with Chladni patterns unless you have a handy holographic interferometry facility you can bludge!

[Craig Bumgarner]

I've been thinking about this overnight and figured if the stiffness is about where I want it, then mass is the next major thing I have to play with. This morning, I added 71 grams to the bridge area with magnets. This dropped the un-coupled top from 258 to 180Hz. The weight was chosen at random, and obviously quiet a lot, ~ 30% of the total braced top, but wanted to exaggerate to see what would happen. Pretty dramatic change, at least on the graphs. Plugging 180Hz in the MM calculation reduces MM to 16.7. (all this in on a closed body, no bindings, no bridge, no neck, no finish).

I also noticed that the coupled top FRC has more peaks with similar amplitude, albeit less amplitude, as opposed to what I see without weights. I'll be interested to repeat this once the guitar is finished to see what difference I can hear.

coupled top with no added weight.jpg (78.21 KiB) Viewed 32308 times

coupled top with 71g at bridge.jpg (86.92 KiB) Viewed 32308 times

[Craig Bumgarner]

Trevor,

I was writing my previous post when you posted, so mine went up after yours and without reading yours.

Thanks to both you and Jim for your explanations, they are very helpful. I'll have to mull over that bit about back & top coupling some, but I get the general idea, thanks.

Yes, Chladni patterns seem to be the next step in understanding the FRCs. I'm stuck on getting a suitable tone generator, any suggestions these days?

[Craig Bumgarner]

Okay, I got it figured out. Software: NCH Tone Generator, free download, ordered a cheap power amp and speaker. I'll be reading the tea leaves within the week and hope to figure out what those pesky peaks really are.