University of Puget Sound
Sound Ideas
Summer Research
Summer 2016
Vibrational Patterns in Curved Metal Plates
Samuel D. Berling
University of Puget Sound, [email protected]
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Recommended Citation
Berling, Samuel D., "Vibrational Patterns in Curved Metal Plates" (2016). Summer Research. Paper 267.
http://soundideas.pugetsound.edu/summer_research/267
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Vibrational Patterns in Curved Metal Plates
Sam Berling
Physics Department Summer 2016
Introduction
Data
The factors I investigated were (1) how plate size affected the frequency of a normal mode, (2) how
The musical saw is an instrument played by using
one hand to bend the saw into an s-shape and
using the other hand to bow the saw.
taper affects resonant frequency, (3) how curvature affected the confinement of a vibrational pattern
to a localized region on a plate, and (4) whether the stress of a blade being bent affects the resonant
frequency. I was particularly interested in the stress factor because one of the few papers on this topic
assumes stress played no role in its analysis [1].
The musical saw inspired this more general study
plates. I sought to identify factors affecting the
vibrational patterns in curved metal plates.
Plate Size
To investigate how plate size affects the
resonant frequencies of a given normal mode, I
used three plates of two widths and two lengths
[Figure 3].
I cut three plates to investigate taper, Plates 6, 7, and 8.
Plates 6 and 8 were rectangular with Plate 6 being half
the width of Plate 8. Plate 7 was tapered from the width
of Plate 8 to the width of Plate 6 [Figure 4].
3500
1400
1200
I used an Electronic Speckle-Pattern
I
the plates under certain conditions, and I used
COMSOL Mutliphysics software [Figure 2] to
Plate7
Plate8
2000
1500
Plate2
600
Plate1
horizontal nodal lines (called “(2, X)” modes) as
plate length produces a much smaller
increase in the resonant frequency of the
frequency increases.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
VerticalNodeNumber(X)
Figure 4 A plot of resonant frequencies of (2, X) Modes for a
tapered plate and two rectangular plates.
• Figure 4 shows that the resonant
frequencies of a tapered plate (Plate 7)
are in between those of the
0
0
1
my method for finding resonant frequencies and
normal modes. I focused on the modes with two
roughly the same.
are added, the difference in resonant
200
The ESPI combined with a spectrum analyzer was
lines are added, the increase remains
(2, 0) mode, but as vertical nodal lines
1000
Plate3
400
model the plates under those same conditions.
2500
Plate6
0
800
horizontal mode set but as vertical nodal
• Figure 3 illustrates that a decrease in
500
1000
Frequency(Hz)
Interferometry (ESPI) system [Figure 1] to image
Frequency(Hz)
Materials and Methods
resonant frequencies of a given
ResonantFrequenciesforPlates6,7,8for(2,X)Modes
3000
ResonantFrequenciesforPlates1,2,3for
(2,X)Modes
• Figure 3 illustrates that a decrease in
plate width significantly increases the
Taper
on the vibrational patterns of curved metal
Qualitative Results and
Analysis
2
3
VerticalNodeNumber(X)
corresponding wide (Plate 8) and narrow
Curvature
4
Figure 3 A plot of resonant frequencies for (2, X)
Modes. Plate 3 is the smallest, Plate 1 is the biggest,
and Plate 2 is in between.
these are generally the modes we hear when
To study the effect of curvature, I clamped a
rectangular plate (Plate 11) to create an s-shape,
similar to those used in the playing of a musical saw, I
then traced the shape and used a plot digitizer to
import it into Excel [Figure 5].
• Figure 6 shows that the effect stress has
on the resonant frequencies is negligible
for curved plates with none differing by
Plate11Shape1
playing an instrument such as the musical saw.
(Plate 6) rectangular plates.
0.05
more than 5%.
0.04
Position(m)
0.03
Stress
0.01
• The curvature study showed that a
0
-0.01 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-0.02
ResonantFrequenciesof(2,X)Modesfor
Plates9,10
-0.04
plate. Figure 5 shows that the pattern
Position(m)
CurvatureofPlate11Shape1
occurs roughly where the curvature is less
Plate9
Plate10
1
2
3
pattern occurs roughly at the inflection
1
0.5
point of the the double curve.
0
-0.5
0
0.1
0.2
0.3
0.4
-1
-1.5
-2
0
than 1 m-1 and that the center of the
1.5
Curvature(m-1)
1000
950
900
850
800
750
700
650
600
curved plate confines the vibrational
patterns to a particular region on the
-0.03
2
Frequency(Hz)
Figure 1 ESPI Image of a plate vibrating at its (4,3)
Mode. Mode numbers are denoted by the horizontal
(4) and vertical (3) nodal lines, excluding the ends.
White lines in ESPI images represent the nodal lines.
To study the effect of stress, I compared the
modes of a stressed curved plate (Plate 10)
and an unstressed curved plate (Plate 9)
[Figure 6].
0.02
Position(m)
4
VerticalNodeNumber(X)
References
Figure 6 A comparison of resonant frequencies of (2,
X) modes between a stressed plate (Plate 10) and
an unstressed plate (Plate 9) of the same shape.
[1] J. F. M. Scott; J. Woodhouse, “Vibration of an Elastic Strip
with Varying Curvature” Philosophical Transaction: Physical
Sciences and Engineering. Vol. 339, No. 1655 (1992): 587625.
Figure 2 A COMSOL image representing the same (4,3)
Mode as Figure 1. The blue lines in COMSOL images
represent the nodal lines.
Figure 5 The top is a digitized tracing of the shape Plate 11’s
curve. The middle is a plot of the curvature of the shape. The
bottom is an ESPI image of the plate resonating at the (2,0)
mode. The green lines denote the edges of the plate. The
orange lines denote the edges of the pattern of vibration.
The red line denotes where the curvature is zero.
Acknowledgments
I would like to thank Professor Rand Worland for mentoring
and advising me through this research process. I would also
like to express my appreciation to the Washington NASA
Space Grant Consortium for funding this research.