Identifying Lateral and Torsional Modes
Methods
Jeff La Favre
jlafavre@gmail.com
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).
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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)
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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.
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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.
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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

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Third Torsional
Mode

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).
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Fourth Torsional
Mode

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Fifth Torsional
Mode

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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).
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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.
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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).
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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
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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.
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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).
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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.
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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).
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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.
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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.
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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.
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