Building a Xylophone, Part 3: Xylo-speriments
My first stop along the research highway was to find studies on how cuts along the bar affect its resonant frequencies. It is known that cutting at the center of the bar causes it to be more flexible and therefore lowers the fundamental frequency, but the relationship between cuts and higher modes is more nebulous. General tuning strategy is to make cuts progressively toward the outer ends of the bar to tune higher and higher modes.
In his 1995 experiment “Practical Tuning of Xylophone Bars and Resonators,” Ingolf Bork elucidated this relationship by discovering that cutting the bar at the antinodes of a mode had the greatest effect on lowering its frequency, while cutting at a node had minimal effect.
Next, I turned to the optimization problem of carving the bars to match desired frequencies. The batch of papers I ultimately decided to emulate used Finite Element Analysis to tackle this problem. At first those words, coupled with ‘partial differential equations,’ struck a primal fear into my heart. But upon deeper inspection, I found they were code for stuff the parameters into a computer and let magical, colorful results pop out.
Finite Element Analysis is essentially a process where a computer breaks down a 3d model into very tiny pieces (‘finite elements’), analyzes their tiny little reactions to force and disturbance, then combines all that data together to predict the reaction of the whole model. It is used often in structural engineering and industrial design to measure stress and potential breakage points, heat transfer, and resonance frequencies.
Why would structural engineers care about resonance? Often they are trying to avoid it. The classic example of a resonance problem is the ill-fated Tacoma Narrows Bridge in Tacoma, Washington — the bridge collapsed in 1940 when high winds caused it to vibrate violently at its resonant frequency. Even in small scale examples, like a machine part, the vibrations caused by resonance could cause the part to wear and weaken over time.
We can visualize the results of a Finite Element Analysis with a plot for each mode/resonant frequency showing the displacement, or movement, of the system under vibration as colors — red and blue here correspond to areas of high displacement, green zero displacement. Since the model is three-dimensional, you would need a separate plot for each dimension or a 3d animation to get a full picture of the movement for each mode.
For our purposes, we care about only the tranverse modes along the length of the bar, which have high displacement in the z-direction independent of y, and are modes 1, 4, 6, and 7 in the above diagram. As you can see, the analysis finds many other modes, some of which are transverse in other dimensions or torsional/twisting motions. Technically, those other modes could be tuned to a desirable frequency as well, but since we tend to hit a xylophone bar along the central axis of the length and not, say, on the side or the edge, those modes are unlikely to be excited.
Having reviewed the literature, there was clearly a precedent for my experiment, but there was one key distinction that threw a spooky minor chord into the works. All the FEA studies above used aluminum for their bars, creating a metallophone, not a xylophone. Aluminum is isotropic, which means its properties are the same regardless of direction, whereas wood is anisotropic, or directional. Not only that, but wood has curves, twists, and inconsistencies within the grain due to its organic growth, making it inherently unpredictable. The studies also used very precise computer-guided machines to carve the final aluminum bars.
My xylophone, on the other hand, would be hacked together by me, someone who has literally never used a saw before. I’ve sanded some crusty boat bits in my day, but that’s about the extent of my experience. So I went forward knowing that even if I could manage to get a computer result that claimed to produce heavenly frequencies, it would unlikely be replicated in reality. My mindset toward the computer phase was to see if I could implement an algorithm that predicted optimal bar shapes to a reasonable accuracy, and then use the results as a guide when actually cutting the bars.
The paper I decided to base my experiment on was a Master’s thesis by Brandon Kirkland, though I diverged in how I modeled the underside curve. In the next section, I describe my ultimate experiment setup, along with some failed strategies I littered along the way.
Next Up: Xylo-gorithms
Papers of Interest
Bork, I. (1995). Practical Tuning of Xylophone Bars and Resonators. Applied Acoustics, 103–127.
Kirkland, W. B. (2014). Topographical Optimization of Structures for Use in Musical Instruments and Other Applications. Master’s thesis, University of Alabama at Birmingham, Birmingham, Alabama.
