Simple Experiments, Part 1: The Relative Importance of E_{x} Compared With E_{y} For Top Compliance and Frequency

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Isotropic materials have physical properties which are the same in all directions. Orthotropic materials have properties which are different when measured at right angles to one another. Wood is highly orthotropic. For more discussion and data see J. Bodig and B. A. Jayne, Mechanics of Wood and Wood Composites, Van Nostrand Reinhold Co., New York, 1982, 712 pp.
It is well known that the parallel to grain value of Young's Modulus E_{x} is 520 times greater than the perpendicular to grain value, E_{y}. This observation can be easily validated by taking a square soundboard of uniform thickness and flexing it parallel and perpendicular to grain. As described in other pages of this website ( www.ukuleles.com/Technology/woodprop.html , www.ukuleles.com/Technology/statmeas1.html, and www.ukuleles.com/Technology/dynmeas1.html the actual values of Young's Modulus can also be calculated for a given piece of wood from a series of simple physical measurements.
Graham Caldersmith has outlined an approach for free and semisupported rectangular wooden plates which utilizes Ex, Ey and Ex,y along with the associated Poisson's ratios to predict the first 56 sets of modal frequencies (Caldersmith, G. W., Vibrations of Rectangular Orthotropic Plates, Acustica, Vol. 56, 1984, pp. 144152). I have put his equations on a spreadsheet and also explained in more detail about measuring E_{x}, E_{y} and E_{x,y} dynamically. At the time of this writing, I have been unable to find such a model for fixed circular plates, hence my empirical study below and in Simple Experiment 2.
I measure the Mechanical Compliance of each top after it has been glued to the sides. I was curious about the relative importance of parallel to grain vs. crossgrain stiffness of the top in controlling this compliance. I also wanted to know if either of the two stiffness' had primary control over the first fundamental top resonance.
I began with three different types of softwood for tops as shown in Table 1. The samples are listed in order of increasing overall stiffness.
Table 1. Calculated Values, Measured Data for Three Different Tone wood Samples
E_{x}, psi 
E_{y}, psi 
E_{x}/E_{y} 
DiskCompl, in 
E_{disk}, calc 

plate#1, Red cedar 
6.32E+05 
2.93E+04 
22 
0.024 
2.39E+05 
plate#2, Redwood 
1.26E+06 
1.17E+05 
11 
0.012 
4.77E+05 
plate#3, Doug fir 
2.43E+06 
1.93E+05 
13 
0.0082 
7.20E+05 
The calculated values of E_{x}, E_{y} and the E_{x}/E_{y} ratio for each tone wood sample are shown in columns 2, 3 and 4.
In order to grossly approximate a soundboard attached to a set of sides, each board was glued to the end of a piece of 1/4" thick PVC pipe. The pipe had an i.d. of 8.19" and depth of 1.5". The center glue line of each tone wood plate was centered on the pipe. Excess material protruding from the outer edges was trimmed off. The Mechanical Compliance was then measured for each plate and is shown above in column 5. The average Young's Modulus for the top based on mechanical compliance was then calculated using equation 1 from Marks, L. S. Mechanical Engineers' Handbook, McGrawHill, 1916, pp. 421:
E_{disk}, calc = (0.6825 * r^{ 2}
* P )/(pi * t^{3} * f)
(1)
where r is the radius of the disk ( 4.09"); P, the applied load (1.37 lbs); t, the thickness (0.096"); and f the deflection, inches (column 5). Column 6 shows the calculated E_{disk} values which are ~ a third of the E_{x} values.
A simple linear regression was run in order to see if there was any simple correlation or predictability using E_{x} and E_{y} to predict E_{disk}. The results astonished me! Using equation 2, it was possible to predict E_{disk} with remarkable accuracy.
E_{disk} (regression), psi = 0.058*E_{x} + 2.3*E_{y} + 134000 (2)
r^{2} = 1.00
I realize that the sample set is very small (you can fit any three points to a circle, right?), but the results are intriguing enough to warrant further investigation.
At this point, I began to wonder about the relationship between measured frequency on the plates and the values of E_{x}, E_{y} and E_{disk}. Tom Irvine has programmed a wonderfully useful set of calculations on his website www.vibrationdata.com . Using his "circular.exe" program for the conditions of a fixed disk, I then input the values for E_{x}, E_{y} and E_{disk}^{ } and obtained the frequencies in Table 2, denoted f_{calc} . The variable f_{meas} , the first fundamental top frequency, was obtained by recording the tap tone from the center of the plate and deconvolving it into its frequency spectrum using the software program Spectra Plus . The measured tap tone first fundamental stays the same for each case and is repeated down column 3 for comparison convenience.
Table 2. Calculated vs. Measured Frequency Data for Western Red Cedar

f_{calc} Hz 
f_{meas} Hz 
% Difference 
E_{x} 
420 
235 
81 
E_{y} 
90 
235 
61 
E_{disk} 
251 
235 
12 
_{ }Note that the value of Young's Modulus estimated by mechanical means actually is the best predictor of the first fundamental. Tables 3 and 4 show similar data but for redwood and Douglas fir.
Table 3. Calculated vs. Measured Frequency Data for redwood

f_{calc} Hz 
f_{meas} Hz 
% Difference 
E_{x} 
526 
295 
78 
E_{y} 
160 
295 
46 
E_{disk} 
325 
295 
10 
Table 4. Calculated vs. Measured Frequency Data for Douglas fir

f_{calc} Hz 
f_{meas} Hz 
% Difference 
E_{x} 
634 
327 
101 
E_{y} 
160 
327 
49 
E_{disk} 
345 
327 
10 
But where are we going with all this? I suppose what I'm trying to eventually understand is the effects of the bridge and bracing of various designs on the distribution of resonant frequencies. So I needed to look at the "simplest" case first, that of a plate with no bracing or bridge.
Please proceed to Simple Experiment, Part 2.
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