First Torsional Mode in a 5-Octave Marimba

Jeff La Favre
jlafavre@jcu.edu

The first torsional mode is not excited into vibration to any great degree unless the bar is struck near a corner, where the antinodes are located. Here is what Bork et al. (1999) had to say about this mode of vibration in a C3 bar they studied "This mode radiates weakly, since adjacent regions of opposite phase cancel the sound radiation at long wavelengths (here 1.06 m)." Therefore, this mode may not cause much of a problem if left untuned, because the sound waves radiating from adjacent corners of the bar tend to cancel each other, at least in the bass region of the keyboard..

Nevertheless, I decided to study the first torsional mode in my marimba bars. The first torsional mode can be determined with a strobe tuner by grasping the bar in the center and striking a corner of the bar, as illustrated in the photo below. The microphone of the strobe tuner should be positioned near the corner that is struck.

f

first tor

The table below lists the frequencies for the first torsional mode in the natural bars between C2 and D5. In all cases, the first torsional mode is the second mode of the bar, between the first transverse and second transverse modes.

First Torsional Mode for Bars of the La Favre marimba

Bar
First torsional mode as determined by strobe tuner
 
Note
Frequency (Hz)
C2 G#2 - 35 ¢ 102.2
D A#2 + 32 ¢ 119.3
E C3 + 12 ¢ 132.4
F C#3 + 47 ¢ 143.1
G F3 + 17 ¢ 177.2
A G3 + 12¢ 198.3
B A3 + 35 ¢ 225.6
C3 A#3 + 32 ¢ 238.6
D D4 - 17 ¢ 292.2
E E4 - 7 ¢ 329.8
F F4 -38 ¢ 343.3
G G#4 - 39 ¢ 408.1
A A4 + 46 ¢ 454.1
B C5 + 3 ¢ 526.6
C4 B4 + 27 ¢ 504.1
D C#5 + 23 ¢ 564.5
E D#5 + 25 ¢ 634.4
F not measured not measured
G F#5 + 33 ¢ 757.9
A G#5 - 13 ¢ 828.3
B A#5 + 36 ¢ 956.6
C5 A#5 - 2 ¢ 935.5
D C#6 - 22 ¢ 1100

Audio of the the bars struck at the corner was recorded digitally and subjected to analysis by Fast Fourier Transformation (FFT) to yield a spectrum for each bar. In most of the spectra, the first torsional mode was not evident. In four bars (E3, C4, G4, B4) a weak first torsional mode was found. The spectra for these bars are provided below.

 

fft

FFT Spectrum of E3 bar hit at bar corner, no resonator in place - sampled in the range of 0 to 186 milliseconds. Clearly the first torsional mode in this bar does not contribute an audible component to the bar timbre.

 

fft

FFT Spectrum of C4 bar hit at bar corner, no resonator in place - sampled in the range of 0 to 186 milliseconds. Clearly the first torsional mode in this bar does not contribute an audible component to the bar timbre.

 

 

fft

FFT Spectrum of G4 bar hit at bar corner, no resonator in place - sampled in the range of 0 to 46 milliseconds. Out of 23 bars tested, the first torsional mode in the G4 bar was the only one with a significant peak. During the first 46 milliseconds of vibration in this bar, the first torsional mode had an average amplitude of 130 compared to an average amplitude of 528 for the fundamental. Over the period of 0 to 186 milliseconds (see below) the first torsional mode had an amplitude of 46 compared to 344 for the fundamental. Stated another way, the first torsional mode had an amplitude approximately 25% of the amplitude of the fundamental during the period 0 to 46 milliseconds and approximately 13 % of the amplitude of the fundamental during the period of 0 to 186 milliseconds. I believe it is doubtful that the first torsional mode would be noticed by the listener, even when this bar is struck at the corner. A mallet blow at any other location on this bar would result in less excitation of the first torsional mode and clearly, under normal playing conditions, the first torsional mode will not be heard.

 

fft

FFT Spectrum of G4 bar hit at bar corner, no resonator in place - sampled in the range of 0 to 186 milliseconds.

 

FFT Spectrum of B4 bar hit at bar corner, no resonator in place - sampled in the range of 0 to 46 milliseconds. Clearly the first torsional mode in this bar does not contribute an audible component to the bar timbre.

 

You might be wondering how I could measure the first torsional mode in all the bars (C2 to D5) with the strobe tuner while the FFT analysis did not reveal the presence of the first torsional mode in most bars. The difference is accounted for by the placement of the microphone in both tests. For the strobe tuner, the microphone was placed near the bar corner struck with the mallet. For the audio recording and FFT analysis, the microphone was placed 30 inches from the bars. Therefore, in the latter case, the sound waves produced by the corners of the bar tend to cancel each other out, as stated by Bork et al. (1999). However, in the case of the strobe tuner microphone, the sound pickup is strongest from the bar corner that was struck, thus less effect of wave canceling from the other corners.

The results of these tests clearly indicate that the first torsional mode of vibration does not contribute to bar timbre in the range C2 to D5. Therefore, there is no need to be concerned with tuning this mode, at least in the range tested. There remains the possibility that the first torsional mode could contribute significantly to bars above D5. I was not able to determine the first torsional mode in bars above D5. Nevertheless, the wave traces of some bars above D5 indicate the presence of interfering vibrations that cause beating patterns in the wave form. It is possible that some of these beating patterns are due to the first torsional mode. Please check this page at a later date for testing results of bars above D5.

Even though tuning of the first torsional mode is not needed, at least in the range of C2 to D5 on my marimba, I decided to test a tuning strategy for the first torsional mode (my idea was that this strategy would also apply to the second torsional mode, which appears not to be the case). I tested a scrap C4 bar, with a fundamental of C4 -6 cents, first torsional mode of D5 exactly, and a second transverse mode of C6 - 6 cents. The bar was 1 31/32 inches wide. The width of the bar was reduced with a table saw to 1 29/32 inches wide. The tuning of the bar was then C4 -3 cents, D5 -18 cents, C6 -2 cents. The bar width was reduced again to 1 27/32 inches wide. The tuning of the bar was then C4 exactly, D5 - 45 cents, C6 +2 cents. Reducing the width of the bar should have little effect on the transverse modes and this experiment shows this to be true. But the width of the bar does have an effect on the first torsional mode of vibration.

Reducing the bar width may not be an effective strategy for tuning the second torsional mode. As part of a tuning exercise on a A2 bar with a problem second torsional mode, reducing the bar width by 1/32 inch did not improve the beating of the second torsional mode against the third transverse mode. Wedging the bar was necessary to improve the tuning of the second torsional mode.

 

Cited References

Bork, I, A. Chaigne, L.-C. Trebuchet, M. Kosfelder and D Pillot, 1999. Comparison between Modal Analysis and Finite Element Modelling of a Marimba Bar. Acust. Acta Acust. 85: 258-266.

 

 

 

 

 

Last update: 1/30/07

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