I wasn’t done torturing my maple test bar. As previously mentioned, xylophone bars are suspended by strings that pass through holes drilled at the nodal locations. From a practical standpoint, I needed to assess whether drilling the holes would affect the modes. If so, I would have to first “rough tune” the bar, drill the holes, and then finish tuning the bar.

Finding the Nodal Locations With the “Salt Method”

A few sites online, including the La Favre site, suggested using the “salt method” to determining the nodal locations.  This technique has you sprinkle salt on the bar in a random pattern, and then repeatedly strike the bar. Because the bar vibrates minimally at the nodal locations, the salt slowly migrates to the nodes. It’s really cool and sort of magical to watch. I didn’t take photos of the salt method applied to my maple bar, but here are some before-and-after photos of the salt migration for one of the rosewood bars.

The salt makes pretty clean little lines, clearly marking the location of the nodes. (As a brief aside, the lines are rarely perpendicular to the long axis of the bar – they are almost always angled. This is just due to the wood grain, which the MEs would refer to as “the non-isotropic nature of the material.”)

Jack and I used the salt method to determine the locations of the nodes for our maple bar and drew lines with a pencil though the middle of the salt lines. However, the results were surprising – the nodal locations determined by the salt method did not align with the “n1” locations predicted by the the tuning curves (refer back to the figure containing the bar template).

Recall that my n1 marks correspond to the location where the tuning curve crossed zero; that is, removing wood here has no effect on the modal frequency. I speculated that these zero-crossing points might be coincident with the nodal locations, reasoning that since wood removed here does not affect the fundamental, it might also identify the nodal locations. However, as noted above, that was not the case. Here is a photo with annotations showing the difference between the predicted and measured nodal locations.

The measured and computed node locations differed by about 1 cm. This disturbed me, because up until this point, all of the computer predictions had been spot on.

Making a Friend

So I went back to literature to see if I could find some clues. The Zhao paper that provided the original version of the beam software referred to a function called Drilled_Hole_Predictions.m.  A comment in the code indicated that this function would calculate the necessary location of the holes. However, the source code for this particular function was excluded from the appendix of the Zhao paper. As I have previously mentioned, comments in the source code indicate that it was derived by code written by Dr. Rodney Entwistle of Curtin University of Technology in Perth, Australia.

So I sent Dr. Entwistle a note explaining the mission that Jack and I were on and asked if I might trouble him to provide the missing function. To my great surprise, Dr Entwistle replied quickly and was very interested in our project! Being an amateur woodworker himself, and a father to four boys, I think he identified with the challenge that Jack and I had undertaken. So “Rod” and I begin a long series of correspondence that provided not only the node location function, but countless invaluable insights and pointers. Most importantly, his enthusiasm and continual encouragement kept me motivated to finish this project, even when it got tedius. Indeed, much of the material in this blog is the result of insights provided by Rod. I feel very lucky indeed to have found not just an expert who was willing to aid our pursuit, but a like-minded engineer that was a “geeked out” as I was by this engineering journey!

Back to the node location mystery. Rod pointed out that the nodal locations are not coincident with the zero-crossing computed by the tuning curves, and provided a function that computes the deflected shape of a beam. This code essentially computes the magnitude of deflection as a function of the longitudinal beam location. I used Rod’s code to compute the deflected shape curve for my maple bar, as shown in the following figure.

The two red circles mark the zero crossings – the point where the transverse beam deflection is zero (for the central axis of the bar). Relative to the longitudinal bar center, this gives the nodal locations at +/-116.3 mm. The locations that I measured, base on the salt method, were at +/- 115 mm, yielding a difference of only about 1 mm! This remarkable agreement indicated that I was indeed on the right track and ready to move forward with my rosewood bars  all thanks to Rod, his code, and his excellent guidance!

One More Gift From “The Giving Bar”

With renewed confidence afforded by the excellent correlation between the predicted and measured nodal locations, I was ready to proceed with a final experiment on my maple bar. At the top of this post, I discussed needing to assess the sensitivity of the bar frequencies to drilling the suspension holes. After the lengthy side trip into computing and measuring nodal locations, we were finally ready to perform that experiment.

I spectrally measured my bar before and after drilling the holes so that I could assess whether it had an effect on the modes. I made the holes 3/16 in diameter, which was based on the cord that I chose to suspend the bars. (You’ll hear more about the mechanics of the woodworking, the cord, and other material choices later). The top of this post contains a photo of the hole, so you can get a sense of its location and scale.

Here are the results of the spectral measurements just before, and after, the holes were drilled.

The spectral analysis shows that the fundamental frequency did not change at all, but the second partial got a bit flatter. This finding suggests that I could mostly ignore the hole effects when making my rosewood bars. However, just to be safe, I ended up cutting and tuning all of the rosewood bars a little sharp, then drilling the holes, and then performing the final tuning.

Done With Math

It seems like I have said this before, but at this point I felt like I was ready to actually begin making the rosewood bars. The maple bar experience was invaluable to bolstering my intuition and understanding of the fidelity of the math and the tuning process. I apologize if all of the previous discussion of Timoshenko beam theory, bending modes, and spectral analysis gave you a headache. If you are still with me, the rest of the posts will be your reward! We are about to build a xylophone!

March 6, 2015

If the tuning curve was right, removing wood from the ends of the bar would raise the frequency of both the first and second modes. So I decided to test this theory, thinking that it would come in handy if I overdid the sanding on one of my rosewood bars. So I began removing wood from the ends to see if the bar would get sharper.

It had been a few weeks since I tuned the bar, so before I started hacking on it, I remeasured its modes on the tuning station. Here was the result:

Weird. The fundamental had only changed a couple of cents, but the second mode got flatter! Not sure what happened here. I am speculating that it was a change in humidity, but I am not sure. Temperature can also affect the frequencies, but only weakly (more on that later). I measured the frequencies a few times and they were consistent, so it wasn’t a problem with the software. In any case, with the baseline re-established, I started removing wood from the ends. I measured the partial frequencies m times along the way. After my last sanding session, I took the following photos of the bar ends.

I removed a little over 2 mm of wood from both ends. This wood removal resulted in the following new modes:

Cool! Both the first and second partial frequencies increased quite a bit-  just as the tuning curves predicted.

More Tuning Curve Verification – Notching the Bar Center

The previous experiment verified that the tuning curves accurately predicted the sonic effects of removing wood from the ends of the bar. As previously noted, the curves also predict that wood removed from the  center of the bar will flatten the fundamental frequency, while having minimal impact on the second partial. Since this is crucial to my tuning approach, I set out to verify this prediction.

Like the previous experiment, my approach was to incrementally remove wood with the drum sander, while frequently measuring the affects on the partials. Previously, I had been using a 1-1/2 inch diameter sanding drum when removing wood. Here, however, in order to localize the wood removal, I used a 1/2 inch diameter sanding drum. All told, I removed about 2.5 mm of wood resulting in a pretty deep “notch” at the bar center. Here is a photo of the final bar and the resulting frequencies.

To summarize, the fundamental frequency dropped by 20 cents, and the second partial stayed about the same (the change of 1 cent was smaller than the measurement precision). Remarkable – the math accurately predicted the physical behavior of the bars…again!

The following figures graphically demonstrate the effect of these two experiments.

Each figure contains two graphs that plot frequency as a function of the spectral measurement number (i.e., in integer that increments each time I measure the frequencies on the tuning station). The top graph of each figure quantifies the frequency ratio for the second partial and the bottom contains the frequency ratio of the fundamental. Both ratios are relative to the desired frequency, which for this F4 bar is 350.8 Hz.

These plots qualitatively confirm the tuning curves predictions. For example, on the top figure, you can see that as I remove wood from the ends, both the fundamental and overtone frequencies increase. On the bottom figure, where wood is being removed form the center, you can see that only the fundamental gets flatter – the overtone stays more/less the same.

In the previous post, I described the math and software used to compute the undercut shape for a maple test bar. That was step 1. In this post I will describe step 2, which is actually cutting the bar out of wood.

Making the Bar

The first step in the process is cutting a piece of maple to the correct dimensions. I selected a piece of boring, straight-grain maple and cut it to the dimensions of the Yamaha F4 bar. After cutting, I measured the bar to record the exact dimensions:

Length=15.05
Width=1.502
Thickness=0.875

In order to shape the wood so that it conformed to the computed shape, I printed a full-scale paper template directly from Matlab. I’ll spare you the mundane details, but it took some trial and error to get Matlab to print precisely in full scale and to compute a calibration factor for the printer.  Here is the template that I made:

There are lots of lines on this bar, so let me elaborate a bit. The first thing you might notice is that there are multiple profile curves. The red curve is the cubic that the software computed. The other curves are included as index lines, so I could precisely determine how close I was to the red line as I was cutting the wood. They are spaced exactly 1 mm apart.

There are multiple vertical lines that warrant description. The vertical red lines labeled “n1” were intended to mark the locations of the nodal points (aka nodes) of the fundamental vibrational mode.  However, I will discuss later that this location is incorrectly marked – the actual nodes are over 1 cm away from these lines. The other vertical lines, marked an1, an2 and an3, were intended to locate the anti-nodes of the higher modes. Anti-nodes are the location where the bar deflects maximally in the transverse direction (i.e., perpendicular to the long axis of the bar). However, this terminology is erroneous – while these lines do correspond to a useful physical fiducial, they do not mark the anti-nodes. I’m afraid we have to go back to the vibration theory to flesh this out further.

“Tuning Curves” and Such

LaFavre and pretty much all of the papers do a great job discussing the vibrational modes of wooden bars. For each longitudinal mode, there is a point about which the bar bends. At this location, the bar exhibits no up and down motion as it vibrates. This fact is critical, and practical, in that it dictates the location of the mounting points for the bar. Theoretically, one could clamp the bar rigidly at exactly this point and there would be no damping of the bar due to the clamps. In practice, xylophones and marimbas suspend the bars on a string that passes through holes that are centered on the nodal locations in an attempt to minimize damping due to the suspension strings. My “n1” markings on the templates were intended to be the nodal locations of the fundamental, but were rather marking an important fiducial determined by “tuning curves” that I had computed. Let’s talk a bit more about the tuning curves, since these are critical to actually tuning the bars.

If you look back at Bork’s original paper (cited earlier), you will see plots that look kind of like this one:

This is an example of what I call a “tuning curve” plot, which was created by basically following Bork’s approach. These curves quantify the effect of removing material from the bar at various points along its length. Consider the top plot first. On the y axis, I am plotting the ratio between the unmodified fundamental frequency (i.e., the bar frequency prior to removing material) to the fundamental frequency that would result if I removed a very small amount of material at various points along its length (as plotted along the x axis). By inspection of the plot, the curve shows that the most effective way to “flatten” (reduce) the frequency of the fundamental mode is by removing material at the at the locations denoted by the two vertical lines. The middle plot shows the effect of material removal on the second mode, and the bottom plot shows the effect on the third mode.

Why is this curve important? Well, as the Bork, Zhao, and other papers note, the math that determines the bar shape only gets you part of the way to a perfectly tuned bar. This is due to limitations of the math and assumptions that we make about the wood that are not completely accurate (e.g., homogeneity, isotropy, etc.). These curves are used during the bar shaping process to tweak the frequencies of the various partials. So the steps to tuning are:

1. Compute the bar shape using the math.
2. Fabricate a bar per the computed shape, but cut proud of the computed thickness profile.
3. Measure the frequency of the partials.
4. Consult the tuning curves to determine where to remove wood.
5. Remove a tiny bit of wood at the indicated locations.
6. Repeat 3-5 ad nauseam (or at least until the bar is sufficiently tuned).

In step 2 above, the word “proud” is taken from the woodworking vernacular and means making a cut that is bit thicker or wider or longer than the final desired dimension. In this case, we cut a little proud so that we have extra material to remove as we are tweaking the bar tuning – it’s much easier to remove wood than to add it…

The tuning curve above illustrates some very important facts about bar tuning. First, notice that the most sensitive way to make the fundamental flatter is to remove wood at +/- 58.3. Removing wood at +/- 65.9 is the fastest way to flatten the second partial. Importantly, notice that there is large overlap of the curves for the first and second partials. This is unfortunate, because it means that removing wood in this area of the bar affects both the first and second partials simultaneously, making it difficult to independently tuning the partials.

However, all is not lost. Looking carefully at the curves, you can see that removing wood from the very center of the bar weakly flattens the first mode, but has negligible affect on the second partial. Bingo – we have a way to independently tune the partials! Notice that this finding is consistent with Bork, and indeed he suggests the following tuning approach  that we basically followed:

1. Carefully remove wood from the region around +/- 60 mm until the second partial is about right.
2. Remove wood from the bar center until the fundamental is perfect.

The tuning curves whisper one final secret to tuning. Notice the slight sharpening that occurs in all three modes when wood is removed from the ends of the bar? Well, this characteristic can be exploited if too much wood is inadvertently removed from the mid-portions of the bar. Here is another photo of the bars from the Kori xylophone I checked out.

Notice that some of the bars have “chamfers” at the ends? This is presumably to sharpen the bar modes.

In general, I would suggest caution when exploiting this sharpening trick; as the tuning curves illustrate, and my experiments supported, the frequencies change is a very weak function at the bar ends; a great deal of wood must be removed to appreciably sharpen the bar.

Apologies are in order for the epic discussion above, but do feel that it is necessary to really understand the underlying motivation for the hands-on tuning of the bars. Your reward for staying with me thus far is the following section where I actually cut the bar (really, no bait-n-switch this time!)

Making the Bar (No Kidding)

February 15, 2015

As I mentioned above, I printed a paper template of the computed bar shape and attached it to my hunk of maple. Here is a photo of the bar with the template attached.

At the top of the photo you can see a microphone sitting on its mount (a Kleenex box). The microphone is connected to a computer so that when the bar is struck, the frequencies of the modes can be determined. The bar is sitting on our hand-crafted “tuning jig,” which supports the bar during the spectral measurement. Two rubber-bands, which are ideally placed at the nodal locations for the fundamental, suspend the bar.

Our first sonic measurement was for the prismatic “blank,” and the second was with the template glued on (as shown in the picture) – we wanted to see if the paper template and adhesive affected the bar frequencies (they didn’t). To measure it, we put the bar on the tuning jig, gave it a whack, recorded it, and analyzed the spectral modes. For reference, here is the spectrum of my bar blank .

The analysis found two modes – the first was at 798.9 Hz and the second was at 3685.3 Hz. (It also found a little false mode at 300.1. bit that is just an artifact of the analysis and should be ignored). The target frequency for this F4 bar was 350.8 Hz, so I had a long way to go to get it tuned.

Next, I cut out the profile on the band-saw, as shown in the following photo.

(For those of you wanting to see more photos and explanation of the woodworking mechanics, please stand by. I will sort of gloss this over in the next few posts and focus on the practical realities of the math, and will later post details on the woodworking mechanics).

It may be hard to tell from the photo, but I made the first cut 3 mm proud of the computed red line. My intention of cutting way proud of the computed curve was to build some intuition regarding the sensitivity of the bar modes to wood removal. After this first cut, Jack took the bar back to the tuning jig and measured the frequency modes. After that, I used a drum sander to sand down to the 2 mm proud line and again measured the modes. After repeating the sanding and measuring steps on the 1mm line, we started the final tuning. Here is a table of the cut depth and the corresponding fundamental frequency.

This was looking encouraging; we were only about 0.5 mm proud of the computed red line, and we were only about 10 Hz sharp. At this point, we alternated between the sander and the tuning station in a effort to “walk in” to the desired. However, we could not reliably measure the frequency of the second partial. For example, here is a power spectral density (PSD) plot after we had taken a few shaves with the sander.

This data was collected by striking the bar at the center. There are strong first and third partials, but the second partial is not measurable. By accident, we hit the bar off the center – more toward one of the ends. Interestingly, we now measured energy in all three partials, as shown by the the next plot.

After discovering this, we routinely hit the bars off center to ring up energy in the second mode. (I found out later, after talking with some percussion professionals, that this knowledge is not wasted on xylophone players who are taught to hit the bar at a point that is just off center in order to yield the best timbre. However, I bet they don’t now why…)

Here were the resulting stats for this strike.

The is pretty good – the fundamental was at 352.0 Hz, which was just 6 cents sharp. The 2nd partial had a ratio of 3.016 and was 9 cents sharp. The 3rd partial, which we had decided we were not going to tune, happened in at a ration of 7.122.

Since the bar was still sharp, we continued sanding toward the profile. For this bar, the first and second partials were both sharp, so we removed wood mostly evenly across the full center portion of the bar. After a few more light sandings, we ended up with the following stats:

At this point, the fundamental was spot on, but the second harmonic was a bit sharp. Given the tuning curve above, there is really no way at this point to flatten the second mode without also flattening the fundamental. On the subsequent bars, I followed my tuning algorithm more carefully (i.e., bring the second partial into tune, or perhaps a tiny bit sharp, and then remove wood at the center to tune the fundamental) rather than just removing wood even with the fiducial lines. Here are a couple of photos of the final “tuned” bar.

It may be hard to tell from the picture, but the final cut is a small fraction of a mm above the computer prediction – I was pretty stoked! The math seemed to work!

Here’s a picture of Jack with our masterpiece:

Finding the beam theory papers was just a baby step to designing my own bars. What I needed was software to apply the math. It may have been possible to write the software from the math in the papers, but software like that can be very difficult to write and debug, especially given my very tenuous grasp on the mechanical engineering concepts that underlie it. My breakthrough moment was when I found a masters thesis on the web that was written by Mingming Zhao. The paper, called “Automatic multi-modal tuning of idiophone bars,” utilized the standard beam theory to compute the bar shapes, but most importantly, the appendix contained the actual Matlab code that applied the beam theory!

Comments in the code indicated that is was a modified version of software written by Dr Rodney Entwistle. So I looked up the Entwistle paper too, which is cited below along with a web link. This paper provided some excellent background and context for the Zhao paper.

I started by copying the Matlab functions from the PDF file of the Zhao thesis to discrete files. To confirm that I had copied the code correctly, my first task was to verify that I could repeat the results from the paper. The following is a code snippet that was modified from code in appendix G.1 of the paper. It demonstrates the convergent nature of the algorithm. It takes as arguments the target frequency (300 Hz), the desired ratios for the 2nd and 3rd partials (4.0 and 9.8), the undercut functional form (cubic) and a seed for the iterative algorithm. Here is the run and the program output (captured in Matlab comments).

bar_props.E = 17.96E9;
bar_props.G = 6.75E9; 
bar_props.rho =  837.187565; 
bar_props.width = 0.06; 
bar_props.d0 = 0.01915; 
bar_props.bar_length = 0.3; 
close all;
[test_coeffs{1},test_reqs{1}]=main('ploton',true,'bar_props',bar_props,'selected_frequency',300,'overtone_ratios',[4;9.8],'profile','cubic', 'coeffs',[6;0.3;0.003]);
%E=1.796e+10 G=6.750e+09 rho=837.2 width=0.06000 d0=0.01915 length=0.30000 profile=cubic
%seed_coeffs = [6.000000000000; 0.300000000000; 0.003000000000]
%=== Loop number: 1
%   freq_new = [129.86691; 4.53455; 12.17812]
%   freq_new = [129.86691; 4.53455; 12.17812]
%   freq_new = [129.86705; 4.53455; 12.17811]
% delta freq = [170.13310; -0.53455; -2.37811]
%=== Loop number: 2
%   freq_new = [319.67484; 4.13737; 10.36343]
%   freq_new = [319.67484; 4.13737; 10.36343]
%   freq_new = [319.67518; 4.13737; 10.36342]
% delta freq = [-19.67482; -0.13737; -0.56343]
%=== Loop number: 3
%   freq_new = [299.01530; 3.97747; 9.66143]
%   freq_new = [299.01530; 3.97747; 9.66143]
%   freq_new = [299.01562; 3.97746; 9.66142]
% delta freq = [0.98471; 0.02253; 0.13857]
%=== Loop number: 4
%   freq_new = [299.96624; 3.99930; 9.79651]
%   freq_new = [299.96623; 3.99930; 9.79651]
%   freq_new = [299.96656; 3.99930; 9.79650]
% delta freq = [0.03378; 0.00070; 0.00349]
%=== Loop number: 5
%   freq_new = [299.99992; 4.00000; 9.80000]
%   freq_new = [299.99992; 4.00000; 9.80000]
%   freq_new = [300.00024; 4.00000; 9.80000]
% delta freq = [0.00009; -0.00000; -0.00000]
%final freqs = [299.99991; 4.00000; 9.80000]
%coeffs = [10.308557568566; 0.322686204614; 0.006247284721]

In the highlighted line at the end, you can see that the algorithm converged to the desired final frequency and ratios in just a handful of iterations. It also provided the  coefficients to the cubic that provided the desired overtones.

The graphical output from this run is at the top of this post. The two figures show the computed bar shape with the bar lying upside down (i.e., with the undercut at the top of the plot). The top figure has an aspect ration of 1:1, and the bottom figure zooms the y axis to better show the curve shape. There are multiple profile because the figure redraws the computed profile after each iteration of the converging algorithm. The line corresponding to the most broad undercut came from the final iteration of the code.

This was pretty exciting! I finally had working code that could predict modal frequencies as a function of a specified bar shape.

Using the Code To Design My Bars

As previously stated, I had decided to design my bars to conform to the “double minimum” shape that I had observed on most of the commercial xylophones. To gain experience, I set out with the following plan in mind:

1. Use the software to design a “double-minimum” bar.
2. Fabricate a “test bar” out of some inexpensive wood (i.e., not Rosewood) that conformed to the computed shape.
3. Measure the frequency modes of the test bar to evaluate the accuracy of the predictions.

Step 1 – Designing My Test Bar

I chose a cubic function for the undercut profile. The choice was motivated by the fact that the cubic profile could realize the desired double-minimum shape, but the parabola would have limited the shape to a simple arch shape. The math that computes the bar shape required information about the material type. The code needs the following parameters:

• Young’s modulus (E) – This is also sometimes called the “bulk modulus” and can be thought of as the stiffness of the wood.
• The shear modulus (G) – This is harder to describe, but I found a good description here. To quote “the shear modulus is harder to talk about, but easiest to demonstrate: take a thick stack of paper (like a phone book) and with your hand on the top, push horizontally. The layers of the stack will shear and the top of the stack will move while the bottom stays put. All objects can be thought of as layers of material. How easily will these layers separate from each other? The shear modulus relates how the top layer of a material will move in relation to the bottom layer.”
• The density (rho) – This is just the mass divided by the volume of the material

I measured the density directly, by measuring a prismatic section of the wood and weighing it. I computed the density as 650 kg/m^3.

The moduli were harder to come by. I found the following reference in the Orduna-Bustamante paper previously cited: “N . E. Molin, L-E. Lindgren, and E. V. Jansson, ‘Parameters  of violin plates and their influence on the plate modes,’J . Acoust.S oc.A m. 83,
281-291 (1988). Ratios of Young’s modulus in the longitudinal direction to the shear modulus are reported as E/G=22.1 for spruce wood and E/G = 6.9 for maple wood.”

OK, so given Young’s modulus, I could compute the shear modulus. I found a value of E=12.6 GPa here. I found a second reference here that gave E=12.6 Gpa. So I split the difference and set E=12.5 Gpa.

The algorithm to compute the bar profile needs a “seed” to start its iteration. I used a photograph of the Yamaha bar to compute coefficients of a cubic that approximated the bar shape. However, there was a snafu; the algorithm failed to converge to a solution. It uses a fairly basic Newton-Raphson method of minimization to approach the solution. I am not sure why, but this had some problems. I could have probably worked out why this was the case, but I rather chose to use a Matlab minimizer called fmincon that I was familiar with. An advantage of this minimizer is that you can set constraints on the minimization, like disallowing solutions that yield a bar that is too thin. With the code modified to use fmincon, the algorithm converged! Here is the code that I ran:

%20150207 - Design for longest bar (made of Maple)
%Properies from PPT file (Maple Wood Properties)
wood_props.E = 12.5e+9; %
wood_props.G = wood_props.E/6.9;  %
wood_props.rho =  650; 
wood_props.nu=(wood_props.E/(2*wood_props.G))-1; %Poisson's ratio

load('results_dir/20150201_AllResults.mat','AllResults');
bar_num=1;
results=AllResults{bar_num};
Comment='Bar #01 (Maple)';
bp=results.inputs.bar_props;
bp.E=wood_props.E;
bp.G=wood_props.G;
bp.rho=wood_props.rho;
bp.nu=wood_props.nu;
num_elements=300;
close all    
[coeffs,freqs,inputs,opt_results]=main('ploton',true,'bar_props',bp,'selected_frequency',bp.frequency, 'coeffs',results.coeffs, ...
    'number_of_elements',num_elements, 'opt_method', results.inputs.opt_method,'lb',results.inputs.lb,'ub',results.inputs.ub,'weights',results.inputs.weights);

%delta freq = [0.00016; -0.00005; -1.51121
%final freqs = [350.81548; 3.00005; 7.51121
%coeffs = [40.559582736574; -3.643168079495; 0.015502032584]

The highlighted line shows that the algorithm gave frequency ratios for the overtones of 3.00005 and 7.51121. The first overtone was dead on, but the second overtone failed to achieve the desired ratio of 6. I tried many variations of the minimizer, but could not coerce it to yield the desired value of r3 (shorthand for the ratio of the third partial to the fundamental). I don’t know why this is the case. Perhaps it is not possible to yield the desired overtone ratio with a cubic function. In any case, as previously stated, I decided to forgo tuning the third overtone, so I didn’t work this problem further.

Here is a figure illustrating the shape of the resulting bar.

The red profile in this bar corresponds to Maple. The blue profile was computed using the properties of Rosewood. I plotted both to assess the difference between the two materials.

Yee-haw! This seems kind of reasonable. Next step – make the bar out of wood. You’ll hear more about that in the next post (if anyone is still with me…).

References

Entwistle, R.D. and McGrechan, S.R. 2007. Geometric Shape Identification for Multi-Mode Tuning of Percussion Instrument Bars. Proceedings of the 14th International Congress on Sound and Vibration. Cairns, Australia, 9-12 July 2007.

Zhao, Mingming. 2011. “Automatic multi-modal tuning of idiophone bars,” Masters Thesis,  M.Phil. Curtin University, School of Civil
and Mechanical Engineering, Department of Mechanical Engineering

Stone, B. 1992. “The receptances of beams, in closed form, including the effects of shear and rotary inertia,” Journal of Mechanical Engineering Science 206 (2): 87-94.

In the previous few posts, we spent considerable time analyzing the sound produced by commercial instruments to answer practical questions regarding tuning requirements. We came to the conclusion that  we wanted to tune the fundamental frequency to an accuracy of 2 or 3 cents, that we might not be able to accurately tune the 2nd partial on the shorter bars and that we shouldn’t worry too much about tuning the 3rd partial, since most of the commercial instruments had 3rd partial ratios that deviated greatly from the ideal value of 6.

Next, we turned our attention to the very practical problem of bar shape determination. Pictures of xylophone bars that we found online showed a wide variety of undercut shapes, as shown in the collage at the top of this post. We found examples of bars that had simple “bridge” shaped undercuts like that shown on the LaFavre site (i.e., a flat bottom with rounded ends), single scallop undercuts, double scallop undercuts, etc.. The marimbas mostly seemed to have a simple bridge shaped undercuts, but the xylophones tended to have multiple curves, at least for the bars in the lower register. We also noticed that, in general, as the bars got shorter, they all tended to use a simple single curved undercut (as opposed to multiple scallops).

The Literature

Uncertain of how to move forward, my son and I turned to our old friend – the internet. Ideally, we would find a simple formula or recipe that would allow us to deterministically shape our bars, given our 1:3:6 desired tuning ratios. Initial searches seemed encouraging. One of the first papers I found was by Orduna-Bustamante (citation below). That paper described tuning of beams to a 1:3:6 ratio by using a undercut that was defined by a simple parabola. This seemed to be the answer that I was looking for: a simple functional form of the undercut that yielded the tuning ratios I was looking for. However, I had two concerns. First, I couldn’t understand why almost all of the xylophones that I found had multiple scallops for the undercut, rather than just the simple parabola described in the paper. I mean, if a simple shape will do, why all of the fuss to cut the more complex shapes. The second concern that I had with the results is that the author stated that they were typically only good for thin beams – those that had a thickness to length ratio of < 1/30. Given that my longest bar was about 15 inches, this meant that the thickness could only be 1/2 inch. It wasn’t clear to me at the time if this was the thinnest part of the bar, or the overall thickness. In any case, when you consider that my shortest bar was about 5 inches, this meant that the math was accurate only if the bar was somewhat less than 3/16 of an inch thick. The instruments I saw and inspected had bars that were much thicker than this.

Incidentally, the Bretos papers that are cited below also indicated that a parabolic undercut could achieve a specific 1st to 2nd partial tuning ratio. It even went a little further and provided a family of graphs that demonstrated that any number of parabolic curves could obtain a specific 2nd partial ratio. That is, one could trade the width and depth of the undercut to achieve a specific ratio. If different parabolic shapes could achieve the desired ratio, then it begs the question about which shape is “best.” This got me thinking about the other properties of the bar that might be affected by the undercut shape. These are properties like:

• The frequencies of the higher overtones
• Overall amplitude of the bar for a given strike impulse
• Relative amplitude of the different tones produced by the bar (e.g., the ratio of the amplitudes between the 1st and 2nd partials)
• The rate of decay for the fundamental and overtones
• The tonal qualities of the bar as a function of where the mallet strikes the bar
• The bar thickness (e.g., some solutions may yield impractically thin bars)

Bottom line: it seemed like I had to understand the relationship of the bar shape to all of these bar properties, or just try to mimic what I had seen on other instruments – the double undercut shape. So I dug my heals in and went back to the literature.

Beam Theory

Remember in a previous post when I caveated my comments by noting that I am not a musician? Well I am not a mechanical engineer either, so the following discussion may be worth what you paid for it…

I kept looking. Along that way, I found some more papers that described computing bar modes based on the shape of the undercut (some cited below). They all discussed “beam theory” which is a mechanical engineering topic that can be used to predict the vibrational modes of long thin objects.

The mechanical engineers (MEs) define a prismatic beam as a long object (relative to its thickness) that has a uniform cross section.

There is mathematical theory that describes how beams respond to external forces and how they vibrate. The two main theories are called Bernoulli-Euler beam theory and Timoshenko beam theory. Both of these are analytic techniques that can accurately describe bending and vibrational modes in beams. Both make simplifying assumptions, so only provide an approximation to the true vibrational modes. They also require that the beams be slender and long in order to accurately predict the vibrational properties. The Timoshenko theory considers some second order effects, so is more accurate in some situations.

But What About Non-Uniform Beams?

Many of the papers that I found that describe the tuning of xylophone bars are based on Bernoulli-Euler or Timoshenko theory. However, as noted above, this theory only works for prismatic beams of uniform cross section. Of course, a shaped xylophone bar has a varying cross section, so how can this theory be used to predict the vibrational behavior of these bars? It turns out that bars of varying cross section can be approximated as a series of end-to-end prismatic beams of varying thickness. The theory can model each of the little prismatic segments accurately, and then other math is used to describe the connections between them. This technique is generally referred to as receptance sub-structuring in the ME literature.  If enough short segments are utilized, the thickness steps between the bars can be minimized, resulting in a good approximation to a smooth undercut. The bottom line is that, while this theory has some limitations, it can provide a good estimate of the vibrational properties of shaped xylophone bars. As you will see in subsequent posts, this is the basis of the approach that I used to determine the shape of my xylophone bars.

References

Chaigne, A., and Doutaut, V.
1997. “Numerical simulations of xylophones I. Time-domain modeling of the vibrating bars,” J. Acoust. Soc. Am. 101
1, 539–557.

J. Bretos, C. Santamaria, and J. Alonso Moral. “Tuning process of xylophone and marimba bars analyzed by finite element modeling and experimental measurements,” The Journal of the Acoustical Society of America, 102(6):3815-3816, 1997.

J. Bretos, C. Santamaria and J.A. Moral, “Finite Element Analysis and Experimental Measurements of Natural Eigenmodes and Random Responses of Wooded Bars used in Musical Instruments”, Applied Acoustics, 56, 141-156 (1999).

Orduna-Bustamante F., “Nonuniform beams with harmonically related overtones for use in percussion instruments,” Acoustical society of America, Vol. 90, No. 6, pp 2935-2941, 1991.