Equ 4.5-5

[Dominic]

Hi Trevor and Gerald, firstly, great books. They have kept me awake reading and thinking since I got them. I am sure these will become seminal works in no time and greatly advance the craft.

OK, I have been reading through the Design book and have started putting the equations into excel. But I am stuck on equation 4.5-5 for the motion of a rectangular orthotropic plate. I can’t see where you have defined (w), (ph) or ∂. Equation 4.5-5 is not used further but I still lost track. But then in equ 4.5-6 (ph) appears again. Is this density x height? I was confused because we changed the nomenclature for height to z. Anyway, appreciate the time you have volunteered to answer our questions. I am sure to have more as I progress through it.
Cheers
Dominic

[Trevor Gore]

Hi Dominic,

w is defined immediately above Equ. 4.5-5 (...For small deflections w in the z direction...)

ρ is Greek rho (density, defined bottom of page 4-59) and h = plate thickness, defined in Fig 4.5-4. The nomenclature runs through.

∂ is delta, signifying the partial derivative as in ∂/∂x, and is not normally specifically defined, as it is the standard convention.

The symbols mean the same things in Equation 4.5-6.

Rather than reproduce all of Hearmon's workings (which you can check in his papers if you wish), I jump from the 4th order partial differential equation of Equ. 4.5-5 straight to its solution (Equ. 4.5-6), which I then invert, substitute for the D's and revert to original nomenclature to give Equ. 4.5-7.

Unless you specifically want to check the derivation, in which case you'll need to consult Hearmon's papers, you need only concern yourself with Equ. 4.5-7, which is the one that counts, giving you the thickness you need to reduce a plate to in order to deliver your chosen vibrational performance, irrespective of the plate's material properties (which you have determined from Equs. 4.5-2 to 4.5-4).

Cheers, Trevor.

[jeffhigh]

A related question Trevor
I have set up a spreadsheet using equations 4.7-7 4.5-2 4.5-3 4.5-4

To confirm it was working properly, I put in some or the data from table 4.5-3 to check that I had meaningful results
I ended up with very slightly lower E results and consequently very slightly higher thickness results.

Were the results in the table calculated using 4.5-2 etc with their simplified VlcVcl of 0.02 or using figures for individual species?

thanks
Jeff

[Trevor Gore]

Hi Jeff,

I presume you meant Equ. 4.5-7.

The slight difference in values is due to me programming in a version of the Equ. between 4.5-6 and 4.5-7. In Equ. 4.5-7, all the constants in the equation above it are evaluated and rounded to 0.95977 and the difference you are getting is just the rounding error and should be of no real significance.

Excel normally works to 15 significant digits. Occasionally (and scarily) you will find that Excel will give quite different answers to the same calculation programmed accurately but in different ways. If you're doing extensive calcs it's good to be aware of this. If you're getting strange answers and you're confident about your programming, program it in a different way and see if you get the same result. As I said, scary; but ill conditioning (as it's called) is just something every programmer needs to learn to deal with!

Rgds, T.

[Dominic]

Thanks Trevor, I figured as much. It was the change in nomenclature in the middle that confused me. Again, thanks for taking the time to answer our questions. This stuff is really inspiring me.
Cheers
Dom

[Trevor Gore]

I've had a few more enquiries from people setting up spreadsheets to work out plate thicknesses (Equ. 4.5-7) which caused me to look into things a little further. The issue was in getting very slightly different answers for Young's modulus and consequently the computed plate thickness. The error is less than 0.5% (!!) in Elong and Ecross resulting in a difference in calculated plate thickness of ~0.01 to 0.02mm. In the scheme of things, this has zero impact, so don't panic(!), but it has given a few guys headaches (sorry Arnaud!) trying to figure out where the difference occurs. In a nutshell, Jeff had it right in his post above where he asked if I'd used different values for Vlc and Vcl rather than the simplified average value of 0.02 for their product. The answer turns out to be yes, I did, but there's a little more to the story, I discovered. (And there's also a bit of rounding error as previously mentioned).

First, Equ. 4.5-1 has a typo in it. The term on the bottom (3/2)^2 should read (3/2)^4, but this is only a typo in this equation and the correct formula is intrinsically used in further workings (not that a reader would pick that). Then the value for the product VlcVcl that I used to calculate the leading constant in Equ. 4.5-2 and 4.5-3 seems to have been 0.015, essentially the value for Sitka spruce, rather than the average value of 0.02 as I stated. Using the value 0.015 gives the constant quoted, i.e. 0.94625 in Equ. 4.5-2 and Equ.4.5-3. Using 0.02 gives a constant of 0.941461.

Does this difference matter? No, not at all, except for the headaches it caused. In fact, for top woods, the formula as it appears in the book is technically the more accurate, as it is for some back woods. But in practical terms it is of no consequence at all, because if you think you can thickness wood to 0.01mm tolerance you're doing a lot better than I can, and this error is well within the experimental error you're likely to accumulate in your testing procedures.

So, sorry for the headaches guys. Take an aspirin and have a lie down.

Cheers, Trevor.

[Dominic]

Thanks for the tips Trevor, Jeff and I had been working together to set up the equations and we got close enough to be happy. We assumed the small (around0.02mm) difference was from slightly different inputs in the calculation of EL as our other numbers were spot on. I also calculated the equation using the actual poison values and got very close. So it has been great fun mucking around with all the equations. I'll make a note that equ 4.5-1 needs to be changed.
Thanks Dom

[Somerby]

Hello folks - I recieved these books a few weeks ago and I am very impressed ! I am trying to develop a spread sheet to match the calculations around top thickness's and I can not get Exel to match the data in the book. I am more like 10% off! I've tried a few things around changing the way the calculations are ordered, changing precision, etc, to no avail. Tonight I am going to use my work laptop and a math analysis program as a sanity check (MATHCad).

Despite this, these books are great additions to the literature!

[Somerby]

I am answering my own posts! All is right with the world, MathCAD correctly calculates equation 4.5-7 to match the book with the identified limitations. Not sure where excel was wrong, but we can now return to the regularly scheduled programming.....

Cheers!

[Dominic]

Somerby, I found excel was very sensitive to the construction sequence of the equations. I was getting all sorts of results and Jeff and i were emailing back and forward trying to work out what was wrong. I finally stepped through it cell by cell rather than trying to write it out in one equation and I worked out how excel wants you to construct the equation. Jeff did it differently and we both got the same results so that was a good cross check.

Good to have another math geek/luthier to bounce ideas off. Are you working through any other models in the design book? Happy to discuss things and share solutions

Cheers
Dom

[Somerby]

Dominic- I am embarrassed to reveal my haphazard approach to these two volumes! Rather than proceed in a logical way, I am all over the map. This section on plate thickness's is brilliant, and I am attempting to calibrate to my previous work of simply using what is essentially the long grain modulus to set top thickness. I will go back and figure out how to make excel work as it is the easiest way to keep track of things. Not sure I am a math geek, but a geek none the less as I have an engineering background (electrical) - the math in the book is well presented and the descriptions make it easy enough to follow!

[jaggitone]

Hi Trevor,

I have a question about equation 4.5-5 and 4.5-7. I am using the numbers from sample 1 on Table 4.5-3, converting them to what I believe are the correct SI units, using the Poison's Ratio constant 0.015 (using the 0.02 constant increases the difference I'm getting), and have been playing around with the brackets. I'm still getting a different number than what is on the Table. Am I making an error with my decimal placements?
Below is the equation as I'm entering it into Excel.

Thanks--Oliver

Equation 4.5-5

Density 395.9
Plate length 0.605
f long 61.6
thickness 0.004

=0.94625*B1*(B2^4)*(B3^2)/(B4^2))
Answer 11902942470.
Rounded Answer 11.90 GPa
Should be 11.96
Difference 0.05705753
As Percentage 0.48%

Below is equation 4.5-7, is there a similar problem with my inputs?

[Trevor Gore]

I've assisted a good few other people in getting their spreadsheets to work, using the equations as per the book.

Typically, Elong comes out at 11.90GPa and the thickness at 2.71mm, which is close enough to the answers in the book and different for the reasons given higher up in the thread and insufficient difference to give a problem.

I can't see a problem (doesn't mean there isn't one!) in the way you have coded Equ.4.5-7, but we've found that Excel is notoriously bad at dealing with this equation due to ill-conditioning. I have seen the equation coded pretty much as you have it and it turns out OK, so it may even be a computer dependent thing.

All I can suggest is that you put the computation together differently, splitting it across a number of cells, and see if you can get closer. Jeff and Dom, above, had a similar problem and fixed it like that. It's a worry, though. I'm sure half the world's financial systems are run on Excel!

One thing to try is SQRT(x) rather than (x)^0.5 and x*x rather than (x)^2, as I think these are the algorithms where the major problems lie. When you've fixed it (I'm sure you will) please let us know what worked (and the machine/software combination that you're working with).

If you continue to have problems, PM me with your spreadsheet.

[J.F. Custom]

Hi Oliver.

Look, I'm not sure if this will help, but it's worth a try.

I had the same issues (very slightly incorrect answers) which did not amount to anything significant, however were doing my head in as I should have been able to get the exact answers!!

I solved my problem this way - note that I'm not sure why this worked! I did it for convenience/streamlining the equations, not to solve the issue - but it did.

It appears you have this equation (4.5-7) set up in isolation and that you input the relevant answers in position 5'B; 6'B and 7'B by hand from another excel equation file (4.5-5), correct ?

I suggest grouping the equations into one excel file. As in, make a file that is both 4.5-5 and 4.5-7 in one. With the equation that is 4.5-7 within that file, make the answers from 4.5-5 autofill in their respective positions in 4.5-7 by referring to the earlier cell numbers.

This worked for me! Note that the numbers as they appeared on screen, did not differ at all to those I input by hand as the answers, but for whatever reason when I did type them in, I'd get a slightly incorrect result. When the data is collected from a cell in the same file, the result was spot on to the tables in the book. Go figure.

Let me know if this is not clear in any way and I'll try to explain it better! Also, should you try it, I'd be interested if it worked for you too.

Jeremy.

[Dominic]

Instead of a constant of 0.95977 use this equation =(1/((PI()/2)^2*(3/2)^4))*(12*(1-0.02)) or 0.941461428. Use this number to also calculate El and Ec and use =3*(1/(PI()/2)^2) or 1.215854204 to calculate Glc. Using these numbers you get exactly what is in the book.
I figured this out looking at the equations we used during the course and found this difference. Not material in terms of the end plate thickness but nice to get the same answers.
Cheers
Dom

[Clancy]

Just to add more confusion about excel,
I did the equation both as a single block and as seperate components that were then totalled up - and got the same answer both ways.
The package I'm using is called "Microsoft Excel Starter" and was a free download that includes lots of ads & hints to 'pay to upgrade'.

[Tphil]

Ditto Craig - I messed around with brackets, replacing ^'s with *'s without changing results. My targets are coming out a little lower than the book's but only equivalent to a 0.5 change in f and that'll do me - could be what you pointed to Dominic.

BTW for iPad/Phone users, there's a handy app called "FFT Analyzer" which is easy to use, quick, accurate and can free you up from the laptop, mic, preamp, leads in the workshop. Seems to lack something in the lower frequencies but useful for quick checks in other areas.
Screenshot is of top/back traces for a spruce/blackwood soprano uke.

Trevor (Phillips)

[Trevor Gore]

Just for clarity, the modification that Dom illustrates here is the same as I was talking about here.

After the differences had been noted, I revised the modal tuning course notes. Not quite so easy to revise the pages in the book...

The spurious results that Excel throws up occasionally are still a mystery, though. Obviously if you're not getting them, nothing to worry about. All my workings are with Windows XP 2001 and Excel 2002 SP3, so maybe it's something that has been fixed with more recent software.

[Big_Al]

Hi guys I am new to this forum and this is my first post. I have been bashing away at 4.5-7 with slight differences from the results in the black book - along with a few others as I can now see. I was pleased to see the post from jaggitone with his excel figures. I was getting similar results with a very similar spreadsheet.
This is a bit nerdy but I couln't resist reproducing exactly the figures and excel expression in the post and I got a slightly different result. I post my copy here in case I missed something but it does seem to be something in the software. I applied the different constant from Domics post and got even closer to the book figures but still not exactly the same
Eqation 4.5-7


Vibrational stiffness 75
Length 0.49
width 0.39
Density 395.9
Youngs modulus along grain 11902942469.937
Youngs modulus gross grain 828243121.854775
Glc 0.91
a/b 1.25641025641026

0.941461428*B1*B2^2*B4^0.5/(B5+B8^4*B6+B8*2*(0.02857*B5+1.12*B7))^0.5*1000
Answer 2.824681683



Vibrational stiffness 75
Length 0.49
width 0.39
Density 395.9
Youngs modulus along grain 11902942469.937
Youngs modulus gross grain 828243121.854775
Glc 0.91
a/b 1.25641025641026

0.941461428*B1*B2^2*B4^0.5/(B5+B8^4*B6+B8*2*(0.02857*B5+1.12*B7))^0.5*1000
Answer 2.770798056

I am going to the bar to give my head a rest - cheers

[kiwigeo]

We should consider oursleves lucky that Microsoft don't make cars

[Trevor Gore]

Don't start me on cars and computers!! If I ever get to the end of this current saga, I'll tell you the story...

[Craig Bumgarner]

FFT Analyzer is REALLY Cool! Just scratching the surface so far but, looks great. Very easy to use. Just tapped out a 1917 00-18 Martin I'm working on. Got an average of taps over about 25 seconds (just keep tapping until the curve settles down). Peaks are about where I would expect them. I'm using it on an iPhone, a mobile FRC analyzer in one's pocket. I can see this is going to get a lot of use. Tappin' on the go! Thanks Trevor (Phillips).

[kiwigeo]

OK so I finally got started on a couple of falcate braced classicals. Unlike the recently finished instrument which was built spanish method these two these will have bolt-on necks.

The last two days have been spent dicking around with a spreadsheet to work Equation 4.4-5. Finally got thing working late last night.....talk about a major erueka moment.

Today I spent time tap testing and pinning down Young's Modulus along and across grain and GLC. The top is Engleman Spruce from Graham Hein and results were as follows:

Mass - 150gram
LL - 580mm
LC - 205mm
h - 3.9mm

F Long - 63.6Hz
F Cross - 105Hz
F diag - 47Hz

Vib stiffness - 60
a - 490mm
b - 390mm

MOS EL - 9.21GPa
MOS EC - 0.39GPa
Shear Modulus (GLC) - 0.81GPa

Target thickness works out at 2.3mm

[Trevor Gore]

That looks in the ballpark Martin, so well done!

I just looked back at this thread...

Don't start me on cars and computers!! If I ever get to the end of this current saga, I'll tell you the story...

Well, I got to the end of the saga, so here's the story...

I drive a 16 year old manual Commodore. In its day, it was a desirable car, so there were a lot of attempted break ins and a few new driver's doors. The last replacement rusted out, so I needed a new one for its rego. Genuine ones are no longer made by Holden, (over 10 years old....) so I got an alternative replacement, had it painted up and the deal was that I get the interior of the old door (with remote locking, electric windows, electric mirrors, speakers, etc. etc.) rebuilt into the new door and hung. Simple eh?

Well, the bozo who rebuilt the door didn't connect up the door wiring loom correctly - only two connectors, so not that hard...then proceeded to reeve through all the car's wiring trying to find the problem. Whilst doing so, with the ignition on, he disconnected part of the ABS. The ABS wheel sensors are inductors. Break the current to the inductor, you get a high voltage, often a spark.

So when, after two hours, they finally found the wrong connections, I started up to head home and the ABS failure light was on. 3 hours after that they still couldn't find the problem. Arranged a return appointment to get the ABS fixed. 6 hours of "diagnostic specialist" later, I was told there was an ABS problem. #$%&*@ ! Yes, I know, I said politely.

Back home and half an hour on Google, I figured they'd blown the ABS computer with the high voltage spark generated on breaking into the ABS wiring. Apparently lucky that the airbags didn't go off as well... So a new (breakers yard) ABS computer required. Whilst this was being sent to me, the instrument cluster started playing up. Random numbers being displayed, odometer failing, every conceivable warning sounding every few minutes. More time on Google suggested that the problem is due to comms between the ABS and the instrument cluster. Fix the ABS and the problems goes away.

I got the new ABS computer and fitted that. The ABS worked fine, but the instruments didn't. More time on Google; the trip computer had probably been taken out by the same high voltage spike that took out the ABS computer.

I rang a few Holden dealerships to see if there was anyone up to fixing the thing. Only one dealership had any clue about the car (being 16 years old). They put the diagnostic systems on it... nothing... stuffed trip computer.

So, the next job is to find another trip computer (which comes with the whole instrument cluster, analogue and digital with LC readout). And the next problem is finding the right instruments; my old Commodore happens to be an HSV Senator. Buckley's of finding an HSV cluster at a breakers and obviously not available new. Got a cluster for a Statesman, swapped over the instruments with HSV logo back panels and fitted it.

Finally everything works again, and the car now has a few less km on it (Holden charge $300 if you want the right km's put back on!)

Two and a half months of aggravation because one body shop tech couldn't figure that connecting red to red and black to black was a good idea.

[kiwigeo]

Tap testing results for the falcate classical:

Plate Thickness Calculator.pdf

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