interpretation of data

[benvaneven]

I made calculations for top thickness and the predicted panel mass.
For the calculation of the predicted panel mass I used the formula for the volume of a cylinder (I measured the topcontour with a tape measure). I then used the formula mass = density*volume.
If my frequency measuring and my calculations for the predicted panel mass are correct, I think the top panel will not make a great classical guitar with a top thickness of 2,8mm and a mass of 211g.
Too much mass for a responsive guitar.
I can play with the vibrational stiffness and lower it from 60 to 50 but I think according to the theory in the book, the frequency of the top monopole will be lower than the aimed 190Hz-200Hz and the guitarsound will loose the emphasize on the treble.
I consider this built as a experiment so I will not be disappointed with a less good guitar. I just want to be able to predict more or less how the guitar will turn out based on the data I collected.
So my question is: Am I thinking in the right direction?

Measured Top data
mass (g) 182
Ll (mm) 525
Lc (mm) 197,5
h (mm) 4,1

F long (Hz) 68
F cross (Hz) 138
F diag (Hz) 58

Density (Kg/m3) 428,1156736 kg/m3

Calculations
El 8422680490 8,42268049
Ec 694739289,1 0,694739289
Glc 1119907640 1,11990764
El/Ec 12,12351255

vibrational stiffness 60
a (top length) 499 0,499
b (lower bout width) 360 0,36
a^2 0,249001
a/b 1,386111111
(a/b)^4 3,691409108
(a/b)^2 13,6265012
Density^0,5 20,69095632

h 2,830450887

predicted top mass
top contour (mm) 1480
top height (mm) 2,830450887
2pi 6,283185307
r= contour/2pi 235,5493158
pi 3,141592654
v = pi*r^2*h 493365,9697
m = density*v (g) 211,2177044

[Trevor Gore]

For the calculation of the predicted panel mass I used the formula for the volume of a cylinder (I measured the topcontour with a tape measure). I then used the formula mass = density*volume.

...

...

predicted top mass
top contour (mm) 1480
top height (mm) 2,830450887
2pi 6,283185307
r= contour/2pi 235,5493158
pi 3,141592654
v = pi*r^2*h 493365,9697
m = density*v (g) 211,2177044

There's a problem with the way you calculated the panel area. Using an equivalent radius derived from the perimeter length (contour) will not give you the answer you want. For example:

Take a square, side length 10; contour (perimeter length) 40; area 100

Take a circle perimeter length 40 = 2*pi*r ; r=20/pi ; area = pi*r^2 = pi*(20/pi)^2 = 400/pi = 127.3

A circle always encloses the largest area for the smallest perimeter length.

Best to use a method like this to calculate the panel area and hence the mass.

I haven't checked the rest of your calculations.

[benvaneven]

I used the method for calculating the panel mass you proposed.
The panel mass is 169,9g and the target thickness 2,8mm.
I continue this build with this plate and for the next guitar I search for a lower density plate.

Ll (mm) 525
Lc (mm) 197,5
h (mm) 4,1
mass (g) 182
Density (Kg/m3) 428,1156736
F long (Hz) 68
F cross (Hz) 138
F diag (Hz) 58
El 8,42268049
Ec 0,694739289
El/Ec 12,12351255
Glc 1,11990764
vibrational stiffness 60
a (top length) (mm) 499
b (lower bout width) (mm) 360
Target thickness (mm) 2,830450887
Panel mass (g) 169,8841031

[Trevor Gore]

I continue this build with this plate and for the next guitar I search for a lower density plate.

You report a density of 428.1 kg/m^3 with a El of 8.42 GPa. Again, I haven't checked your calculations but I'll assume you're correct. Certainly, this is not a great piece of wood and you have recognised that. Typical spruce is ~400 kg/m^3 and ~12 GPa. Very good spruce is ~350 kg/m^3 and ~10 GPa. You will be starting a long way behind if you use your current piece of wood. I would suggest finding a piece that is closer to average properties, then at least you'll know it is not the wood that is killing your guitar!

Let's hope you can find some better wood without too much trouble.

[benvaneven]

Would be great if you can check the calculations.
I did copy and paste, don't know if this is workable.
I have other tops avaible. I will rectangle and thickness them and see what comes out.
Many thanks for your time and advice.

Target thickness
Measured Top data
mass (g) 182
Ll (mm) 525
Lc (mm) 197,5
h (mm) 4,1

F long (Hz) 68
F cross (Hz) 138
F diag (Hz) 58

Density (Kg/m3) 428,1156736

Calculations
El 8,42268049
Ec 0,694739289
Glc 1,11990764
El/Ec 12,12351255

vibrational stiffness 60
a (top length) 499
b (lower bout width) 360

Target thickness 2,827830661

Panel mass
measured data
L (mm) 497
W (mm) 187
H (mm) 2,4
mass (g) 236
mass template (g) 178

calculations
density 1058,041655 kg/m3

area template mm2 70098,05932
Volume 1/2top 198225,4414
volume top 396450,8828
Panel mass 169,7268367

[Trevor Gore]

Would be great if you can check the calculations.

I checked using the data in your first post.

I get the same El, Ec, G and density as you, but for f=60 I get a target thickness of 2.52mm rather than the 2.83mm you got.

You need to check your coding for equation 4.5-7. Check your results using the numbers in Table 4.5-3.

(Your (a/b)^2 and (a/b)^4 numbers look distinctly odd )

[benvaneven]

I corrected my coding and checked the results using the numbers of table 4.5.3
I have the same numbers. I am struggling with the units, i have to divide the target thickness by 1000 to get the result in mm. the same for panel mass. Should invest some time why that is. Nevertheless being able to calculate the correct panel thickness and mass is a big step forward. Thanks a lot for your assistance.
Is there also a way i can check my calculations for the flexural rigidity?

Target thickness
Measured Top data
mass (g) 182
Ll (mm) 525
Lc (mm) 197,5
h (mm) 4,1

F long (Hz) 68
F cross (Hz) 138
F diag (Hz) 58

Density (Kg/m3) 428,1156736

Calculations
El 8,42268049
Ec 0,694739289
Glc 1,11990764
El/Ec 12,12351255

vibrational stiffness 60
a (top length) 499
b (lower bout width) 360

a^2 0,249001
a/b 1,386111111
(a/b)^4 3,691409108
(a/b)^2 1,921304012
Density^0,5 20,69095632

Target thickness 2,520152416

Panel mass
measured data
L (mm) 497
W (mm) 187
H (mm) 2,4
mass (g) 236
mass template (g) 178
calculations
density 1058,041655 kg/m3

area template 0,070098059 m2
Volume 1/2top 0,176657794
volume top 0,353315587
Panel mass 151,2599405

[Trevor Gore]

Keeping count of the zeros is often a source of error. Everything should be in SI units (meters, kilograms, seconds). Stress/pressure is in Pascals, but we work in GPa (10^9 * Pascals). Your answer should come out in meters, which then need multiplying by 1000 to get millimetres. Applied loads (when used) should be in Newtons. The weight exerted by a mass of 1 kg is 9.81 Newtons.

To check your flexural rigidity calcs, see Equ. 4.4-2 and the line of text immediately following it on p 4-38.

[benvaneven]

I found a piece of spruce with a density of 436 kg/m^3 and Elong 14,1 Gpa. The target thickness comes at 2 mm and the estimated panel mass at 126 g. I think the stiffness allows to reduce the mass to acceptable numbers, so it seems to me this is a better piece of spruce then the one i mentioned before. Or should i search for a better piece?

[Trevor Gore]

The new piece has typical density for its Young's modulus, whereas the original piece had rather high density for its Young's modulus, so at least you're now starting on a more level playing field.

The best wood (in terms of making the lowest mass tops) is usually the lowest density wood, because adding thickness (to increase flexural rigidity, E*I) adds stiffness faster than it adds mass, due to the cubed thickness term in the second moment of area, I.

The search for "best wood" is perennial. The trick is to make the best of the wood you have, which is what these techniques are all about. You should be OK with this piece if you've had enough of searching.