Identifying Lateral and Torsional Modes

Methods

Jeff La Favre

jlafavre@jcu.edu

A mallet blow on the bar side, at the end of the bar (see photo below) excites lateral and torsional modes, with little or no excitation of transverse modes. Therefore, this is a good method for examining the lateral and torsional modes without interference from transverse modes. By recording the audio of bars struck in this manner and subjecting the digital audio files to Fast Fourier Transformation, it is possible to obtain the frequencies of lateral and torsional modes. However, this data alone is not sufficient to assign specific frequencies to specific modes of vibration.

In order to identify the specific modes, I used what I will term the "salt method." The bar is supported by two blocks of foam, positioned below the nodes of the fundamental. Salt is sprinkled on the bar. Then the bar is subjected to sound at a specific frequency, using a speaker connected to a tone generator. When the bar is subjected to a sound frequency near the frequency of one of its modes, the bar will vibrate in that mode. When this happens, the salt clears from the zones of the antinodes and collects in the zones of the nodes. By observing the pattern of salt on the bar, it is possible to assign a frequency to a specific mode of vibration.

After the vibration rates of the lateral and torsional modes are known, they are compared with the vibration rates of the transverse modes. If there are lateral or torsional modes that vibrate with frequencies close to a transverse mode, the offending lateral or torsional mode must be studied further to determine if it actually is a problem or not. To make this determination, it is necessary to examine the audio spectra of bars struck at positions that can be expected during the course of playing the instrument. Obviously, the player does not strike the bar on the side at the end, as is done to determine the lateral and torsional modes. So the question remains as to whether these modes actually vibrate to any significant degree during the course of playing the instrument.

Determining the torsional and lateral modes

Hold the bar near the end of the arch, with thumb and index finger opposed, at the center of the bar width. This location coincides with nodes of the torsional modes and is near nodes of the first and second lateral modes. Strike the bar with force on the side, near the end of the bar. When using a strobe tuner to determine frequency, place the microphone near the center of the arch as in the photo above. When making an audio recording, the microphone may need to be farther from the bar, depending on the sensitivity of the recording equipment (I placed the microphone 15 inches from the bar for the recordings).

When using a strobe tuner to determine the first lateral mode, a stronger signal might be obtained by striking the side of the bar in the center of the arch. For certain bars, the mallet blow may need to be angled midway from a blow on the top surface and the side surface (i.e. at a 45 degree angle at the edge where the top surface meets the side of the bar)

A mallet blow at the center-edge is also useful in activating the second torsional mode of vibration. The bar is held the same as it is held for a strike on the bar side. A strike on the top bar surface will also activate some of the transverse modes.

A mallet blow at the edge, slightly off center, is also useful in activating the third torsional mode of vibration. The bar is held the same as it is held for a strike on the bar side. A strike on the top bar surface will also activate some of the transverse modes.

Strike Positions on the Top Surface for the Torsional Modes

A mallet blow on the bar side, at the bar end, is useful for obtaining an audio spectrum containing frequencies for lateral and torsional modes. However, it is also necessary to evaluate lateral and torsional modes with mallet blows on the top bar surface, in locations that might be struck during a performance on the instrument. The first and second lateral modes can be excited in some bars by a strike at the center-edge, the same location as illustrated below for the second torsional mode. In addition, the first and second lateral modes in some bars can be excited by a blow at the bar center, the same location as illustrated below for the fourth torsional mode.

Yellow dots mark the positions where the mallet should strike the bar. Blue dots mark the positions where you should hold the bar (lightly pinch the bar between your thumb and index finger). If the bar is in place on the instrument, you can still strike the bar at the yellow dots and expect excitation of the mode. This is due to the fact that the cord runs through the bars in a region near the nodes of the torsional modes.

Second Torsional Mode

strike

Third Torsional Mode

strike

For this mode hold the bar at the blue dot that is farther from the yellow dot you strike (i.e., if you strike one of the yellow dots on the left, hold the bar at the blue dot on the right).

Fourth Torsional Mode

strike

Fifth Torsional Mode

strike

 

Using the Speaker to Activate a Specific Mode of Vibration (the salt method)

For the transverse modes, the fourth torsional mode, and in certain cases the lateral modes, the speaker is positioned directly over the bar. The speaker is positioned at one of the central antinodes of the mode (it may be necessary to move the speaker along the bar length slightly until it is evident by salt movement that the bar is vibrating (for these photos the salt was not sprinkled on the bar).

For the torsional modes and some of the lateral modes, a shield is used so only half the bar width is exposed to the speaker (this photo is staged - in actual use, the shield is held in one hand and the speaker in the other). The shield is held as close to the bar as possible without touching it and the speaker is held just above the shield. The shield is a DVD case, but any number of objects should work. It may be necessary to move the speaker and shield along the length of the bar slightly until the correct position is found for maximum activation of the bar.

For some of the lateral modes, holding the speaker to the side at a 45 degree angle was found to be the best position to activate the bar (in this staged photo the speaker appears to be high above the bar, but in actual use, the center of the speaker was about level with the top surface of the bar).

 

 

FFT Spectra of C2 Bar Struck on the side, top center and top center-edge

The frequencies of vibration for lateral and torsional modes were determined by recording audio of the bars struck on the side at the bar end (photo above) and subjected to spectral analyses by Fast Fourier Transform (FFT). The bars were struck in the same manner utilizing a strobe tuner to confirm the frequencies obtained by FFT. The bars were then tested by the salt method to match the frequencies found with specific modes of vibration.

FFT spectrum of C2 bar struck on side at bar end - sample time 0 - 93 milliseconds. The spectrum demonstrates that a mallet strike on the side of the bar, at the bar end activates the lateral and torsional modes as follows: second torsional (598 Hz), first lateral (786 Hz), third torsional (1203 Hz), fourth torsional (1591 Hz), second lateral (1680 Hz) and fifth torsional (1949 Hz). Striking the bar on the side is useful because it activates the lateral and torsional modes we need to evaluate. However, we also need to know if these modes are active when the bar is struck at positions on the top surface, which is relevant to performance. The two spectra below represent mallet blows at locations on the bar that are struck, at least sometimes, during a performance.

 

 

FFT spectrum of C2 bar struck at center-edge - sample time 0 - 93 milliseconds. The center-edge struck bar has all the torsional and lateral modes seen in the side-struck bar. However, many of the modes are much weaker here. We must be cautious in comparing the spectra here with the first spectrum because the microphone placement was different and the force used to strike the bars may have been substantially different. Nevertheless, what we are looking for here is the relative amplitudes. The first and second lateral modes have amplitudes of 240 and 77. Compare this to amplitudes of 420 and 1061 when the bar was struck on the side. The bar should vibrate with greater amplitude in the lateral modes when struck on the side. There is a marked difference for the second lateral mode, which is the strongest mode for the side-struck bar but the weakest mode for the center-edge-struck bar. In contrast, the center-edge-struck spectrum shows that the C2 bar vibrates to a moderate degree (amplitude 240) in the first lateral mode. The strongest lateral or torsional mode in the center-edge spectrum is the third torsional at an amplitude of 352. This result suggests that the mallet blow was not exactly at the center of the bar length (the relatively strong 4th transverse mode also indicates an off-center blow). A blow slightly off center would be in the region of the antinode for the third torsional mode and slightly off the antinode for the second torsional mode (i.e. a strike at the center of the bar length should produce a stronger second torsional mode and weaker third torsional mode - see photos above for strike positions). The fourth and fifth torsional modes are weak but we can't be sure the mallet blow was in the best position to activate these modes. For reference, the peak at 665 Hz is the third transverse mode and at 1289 Hz is the fourth transverse mode.

FFT spectrum of C2 bar struck at center- sample time 0 - 93 milliseconds. With a center strike, we see that all of the lateral and torsional modes are absent except for a weak fourth torsional mode with an amplitude of 206. All of the torsional modes, except the fourth, have node lines at the center-width of the bar. So we should not expect to see them in this spectrum. The fourth torsional mode has an antinode at the center of the bar width, which is where the mallet blow was located. Therefore, we should expect to see the fourth torsional mode. Realistically, the fourth torsional mode is too weak to be audible (it is totally overpowered by the third transverse mode with an amplitude of 4988). Experienced marimba performers are aware that the bar timbre will be different if they strike the bar near the edge. This is due to the presence of lateral and torsional modes, which are absent when striking the bar near the center of the bar width, the more usual position. For reference, the peak at 665 Hz is the third transverse mode, at 1287 Hz is the fourth transverse mode, and at 2097 Hz is the fifth transverse mode.

 

Salt Patterns on a C2 Bar Exposed to Sound at Specific Frequencies

C2 bar exposed to 598 Hz from speaker

During exposure to sound from the speaker, the salt moved toward the center of the bar where it accumulated along the node line. This pattern indicates that the second torsional mode of the C2 bar vibrates at or near 598 Hz. For this mode it is necessary to shield half of the bar width from the speaker so that only one half of the bar width is exposed to the sound waves. This is done by holding the shield a small fraction of an inch above the bar.

C2 bar exposed to 786 Hz from speaker

During exposure to sound from the speaker, the salt in the center of the bar moved toward the edges and dropped off the bar. Some of the salt in the center of the width remained in place due to the fact that the crystals were trapped in grain depressions of the wood. Superficially this salt pattern resembles that of the second torsional mode. However, the movement of salt crystals is different during bar vibration. A careful examination of the two photos also reveals the difference. For the second torsional mode there is clearly an accumulation of salt at the center of the bar width. For the first lateral mode the salt does not accumulate at the center, rather it is pushed off the bar by the lateral vibrations. This pattern indicates that the first lateral mode of the C2 bar vibrates at or near 786 Hz.

In the case of the C2 bar, a shield was necessary as used for the second torsional mode. For other bars the speaker was able to activate the bar without the shield. Some bars were activated most efficiently by positioning the speaker to the side of the bar, angled at 45 degrees rather than directly over the top of the bar surface. These differences in speaker position suggest that the vibration dynamics of the first lateral mode vary between bars. Some bars may have a significant vertical element of vibration present in addition to the lateral element, while others may have a weaker vertical element.

It may seem surprising that a lateral mode could be activated by sound directed at the top bar surface. However, the first lateral mode clearly does vibrate to a certain extent in some of the marimba bars when struck with a vertical stroke. There is documentation in the scientific literature that the lateral modes can have a component of vibration in the vertical direction, which would explain why this mode is activated by a vertical strike or by exposure to sound at the proper frequency on the top bar surface (Bork et al., 1999).

C2 bar exposed to 1203 Hz from speaker

 

During exposure to sound from the speaker at 1203 Hz, the salt moved toward the center of the bar width where it accumulated. However, at the center of the bar length the salt remained in place, which indicates a node. A careful examination of the bar corners reveals the absence of salt, indicating that the corners are antinodes. This is the pattern of the third torsional mode. As for the second torsional mode, a shield is required for this mode in order for the bar to be activated by the speaker.

C2 bar exposed to 1585 Hz from speaker

During exposure to sound from the speaker at 1585 Hz, the salt accumulated at two positions parallel to the bar length as seen in the photo above. This is the only torsional mode presented on this page that has two nodes running the length of the bar. All of the other torsional modes have only one node running the length of the bar, at the center of the bar width. Unlike all of the other torsional modes, this one is activated by a mallet blow at the center of the bar. There was no need to use the shield to assist in activating the bar vibration for this mode. The speaker was positioned over the center of the bar (the spot of salt in the center marks the axis of the speaker).

C2 bar exposed to 1680 Hz from speaker

The speaker was positioned directly over the top surface of the bar in the center of the bar length. A shield was used to cover half the bar width. A salt pattern was also obtained with the speaker to the side, angled at 45 degrees. This salt pattern is not clearly diagnostic for the second lateral mode. It is likely that this bar has strong vertical and lateral elements of vibration which prevent a diagnostic salt pattern. Since all of the other modes of vibration are accounted for by clear salt patterns, the second lateral mode is assigned here by a process of elimination.

C2 bar exposed to 1957 Hz from speaker

The shield was used for this mode. The salt pattern is clearly diagnostic for the fifth torsional mode. Salt has accumulated along the center-width of the bar. The two center nodes across the bar width are also clearly marked by salt accumulation. Salt has also cleared from the corners of the bar, indicating that these are antinodes.

The various lateral and torsional modes could be confirmed by the salt method only for certain bars as follows:

Second and Third Torsional modes confirmed in bars C2 through G2

Fourth Torsional mode confirmed in bars C2 through F2

Fifth Torsional mode confirmed in bars C2 through E2

First Lateral mode confirmed in bars C2 through C4 (all bars tested)

Second Lateral mode confirmed in bars C2 through F2

These results show that the salt method has limitations and is most useful in the lowest octave (C2 - B2). The bars in the lowest octave are relatively wide, which is probably one of the factors. A wider bar surface is able to intercept more of the sound energy from the speaker. Additionally, it is well established that the higher modes become weaker, moving from the low end of the keyboard to higher notes. Thus, there is a point where the mode may become too weak to be activated by the speaker.

Even though the salt method works mostly in the lowest octave, it is still helpful in identifying modes in the next octave up when combined with spectral analysis. By graphing the trends for each mode, it is possible to extend results to the range C3 to C4 with reasonable confidence that the correct modes are assigned to the correct frequencies. The graph below is the result of an extension.

graph

 

 

 

Literature Cited

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.

 

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Last update: 3/12/07

© 2007 Jeffrey La Favre