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Metallophone Design

December 14, 2015

When young children decide that their screaming doesn't make enough noise, they find that banging on the family pots and pans evokes louder sounds. Eventually, they find that the sound produced by banging on a metal railing is much more melodious. In my generation, long before the ubiquity of electronics in toys, there were musical toys based on mechanical impact on metal plates.

The simplest of these is the xylophone, and this same xylophone mechanism was used to simulate a piano. The sounds produced by these toy pianos were quite unlike the music produced by Schroeder in the Peanuts comic strip. Today's children are much more likely to become interested in music than children of decades ago, since their electronic toys produce melodious tones. Why, exactly, did the toys of my childhood sound the way that they did?

Toy xylophone

Toys xylophone.

(Photo by Tomasz G. Sienicki (modified), via Wikimedia Commons.)


Mechanical impact, or stroking an edge with a bow, will induce vibrations in a plate. Since these vibratory sound waves will be reflected from the edges of the plate, they will constructively and destructively interfere within the plate. Ernst Chladni (1756-1827) a German physicist and musician known as the "father of acoustics," did the first research on vibrating plates.

Whether in Chladni's 18th century or today, the advancement of science is limited by the available instrumentation. If I were to study the vibration of plates, I would use laser interferometry to map displacements, and automate the experiment using machine vision. Chladni was able to map the nodes of a plate, the places where displacement is very small, by stroking the edge with a bow, sprinkling sand on the surface, and seeing where the sand persists. Ernst Florens Friedrich Chladni (1756-1827)

Ernst Florens Friedrich Chladni (1756-1827)

There are quite a few YouTube demonstrations of Chladni patterns on vibrating plates.[1-3]

Chladni was the first to theorize that meteorites were extraterrestrial.

(An 18th century portrait, via Via Wikimedia Commons.)


The easiest type of plate to analyze is a circular plate, and since plates need to be somehow fixed, we can mount them to a post via a center hole. Chladni derived the resonant modes of such a plate in Chladni's law. The resonant frequencies f closely follow the equation,
f = C (m + 2n)p
In which m is the number of linear modes (spokes) and n is the number of radial modes (circles). The coefficient, C, is a function of the plate properties, and the exponent, p, is very nearly 2. This law applies, also, to deformed circular plates, such as cymbals, and bells.

Chladni figure - Four radial modes on a center-mounted circular plate.

Chladni figure

Four radial modes on a center-mounted circular plate.

(Created by the author using Inkscape.)


Since radial symmetry is not present in square plates, nodes in a square plate are not as simple and easy to explain as nodes in a circular plate. As can be seen in the examples below from Chladni's 1802 book, Die Akustik, the existence of reflecting boundaries that are not equidistant from the center of the plate adds considerable complexity to the nodes. While calculation of the nodes is possible with computer simulation,[4-5] an analytical solution was difficult with the tools possessed by 18th century mathematicians.

A variety of Chladni figures for a square plate

A variety of Chladni figures for a square plate from Chladni's, Die Akustik, 1802.

(Modified Wikimedia Commons image.)


The analytical solution for Chladni nodes in a square plate was discovered after years of work by French mathematician and physicist, Sophie Germain (1776-1831). This solution, contained in her paper, Recherches sur la théorie des surfaces élastiques, was such an accomplishment that she was awarded a prize from the Paris Academy of Sciences. One look at her partial differential equation will convince you that women can do math.

Sophie Germain's solution to Chladni nodes in a square plate

As we fast-forward to our computer age, we can invert this problem to ask ourselves, what shape would we need to produce a specific tone spectrum; or, how can we "tweak" an existing shape to give a better tone? Computer scientists at Harvard University, Columbia University, Disney Research, and MIT presented work at SIGGRAPH-Asia (Kobe, Japan) on using computational design to control the sound of an object by altering its shape.[6-7]

Their paper, "Computational Design of Metallophone Contact Sounds," tackles the particular problem of creating a toy glockenspiel with striking elements shaped like zoo animals (see figure).[7] Says Gaurav Bharaj, first author of the paper and a graduate student at Harvard's School of Engineering and Applied Sciences,
"Our optimization algorithm enabled us to have precise control over the sound of an object by tuning the shape of the object computationally... Through our method, we have gained control over the spectrum of frequencies and their amplitudes."[7]

An animal xylophone (zoolophone, Columbia University)

A "Zoolophone" metallophone with a variety of animal shapes. The shapes were automatically created to sound with a particular tone through use of a computer algorithm. (Changxi Zheng/Columbia Engineering image.)


The toy glockenspiel is a type of musical instrument known as an idiophone. Idiophones produce sounds in their entire structure by striking with a mallet, so they are different from drums, whose sound is produced just by their membrane. Since an idiophone's sound depends on its shape, sound design for idiophones is not straightforward, and the fallback is to use simple shapes, such as bars, and to tune these by drilling of dimples on the underside of the bars.[7]

The "zoolophone" was a severe test of the design algorithm, since it required the striking surfaces to be in the shapes of lions, turtles, elephants, giraffes, and other animals.[7] The present algorithm is an advance over previous attempts, since it optimizes both the frequency and amplitude of the tones, including addition of overtones that contribute to the timbre of the notes. Control of the timbre even allows a glockenspiel to play a chord, such as simultaneously sounding the notes C, E, and G to create the C-major triad.[7]

All this was done using a new stochastic optimization method they call Latin Complement Sampling (LCS).[7] Says Changxi Zheng, assistant professor of computer science at Columbia, and leader of the research team,
"Our discovery could lead to a wealth of possibilities that go well beyond musical instruments... Our algorithm could lead to ways to build less noisy computer fans, to erect bridges that don't amplify vibrations under stress, and to advance the construction of micro-electro-mechanical resonators whose vibration modes are of great importance."[7]

The research was supported by the National Science Foundation, Intel, the Air Force Research Laboratory, DARPA, and other sources.[7]

References:

  1. Physics Girl, "Singing plates - Standing Waves on Chladni plates," YouTube Video, April 28, 2014.
  2. Atto Gruppen, "Chladni plate experiment, round plate," YouTube Video, August 28, 2014.
  3. SBCCPhysics, Circular Centered Chladni Plate, YouTube Video, July 5, 2011.
  4. Wence Xiao, "Chladni Pattern," Basic studies in Natural Sciences, Roskilde University, May 31, 2010.
  5. Thomas Müller, "Numerical Chladni figures," European Journal of Physics, vol. 34, no. 4 (May 29, 2013), DOI: 10.1088/0143-0807/34/4/1067. Also at arXiv.
  6. Gaurav Bharaj, David I.W. Levin, James Tompkin, Yun Fei, Hanspeter Pfister, Wojciech Matusik, and Changxi Zheng, "Computational Design of Metallophone Contact Sounds," SIGGRAPH Asia 2015 (To Appear, ACM TOG, vol. 34, no. 6).
  7. Holly Evarts, "Change the shape, change the sound," Columbia University Press Release, November 2, 2015.

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