A Study of the Air-Structure Interaction in a Concert Flute
by
Jennifer Froling
A Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Major Subject: SOLID MECHANICS
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
May 2012
i
CONTENTS
LIST OF TABLES ............................................................................................................ iii
LIST OF FIGURES .......................................................................................................... iv
LIST OF SYMBOLS ......................................................................................................... v
GLOSSARY ..................................................................................................................... vi
ABSTRACT .................................................................................................................... vii
1. Introduction…………………………………………………………………………….1
2. Methods………………………………………………………………………………..3
2.1 Modeling Validation Methods…………………………………………………..3
2.2 Material and Thickness Study Methods…………………………………………7
3. Results………………………………………………………………………………...11
3.1 Modeling Validation Results…………………………………………………...11
3.2 Material and Thickness Study Results………………………………………….13
4. Conclusions…………………………………………………………………………...19
5. References…………………………………………………………………………….20
ii
LIST OF TABLES
Table 2.1 Properties of Pipe Model ................................................................................... 3
Table 2.2 Properties of Air Column Model ....................................................................... 6
Table 2.3 Pipe Material Study Properties ........................................................................ 10
Table 3.1 Pipe Model Mode Results................................................................................ 11
Table 3.2 Air Column Mode Results ............................................................................... 12
Table 3.3 Pipe Material Study Results: Harmonic Pressure Amplitude .......................... 16
Table 3.4 Pipe Thickness Study Results: Harmonic Pressure Amplitude ....................... 17
Table 3.5 Pipe Material Study Results: Harmonic Frequency ........................................ 18
Table 3.6 Pipe Thickness Study Results: Harmonic Frequency ...................................... 18
iii
LIST OF FIGURES
Figure 1.1 Modern Concert Flute ...................................................................................... 2
Figure 2.1 Material and Stiffness Studies Models ............................................................. 9
Figure 3.1 Input Characteristic Impedance of Air Column ............................................. 12
Figure 3.2 Nodal Pressure Response For Baseline Model .............................................. 14
Figure 3.3 Pressure Response in Exterior Acoustic Mesh at 273 Hz .............................. 14
Figure 3.4 Pressure Response in Air Column Acoustic Mesh at 273 Hz ........................ 15
Figure 3.5 Displacement Magnitude in Pipe Model at 273 Hz ....................................... 15
iv
LIST OF SYMBOLS
a
radius [m]
A
modal coefficient
c
wave speed [m/s]
E
Young’s modulus [Pa]
f
frequency [Hz]
h
thickness [m]
I
area moment of inertia [m3]
L
length [m]
k
wave number [1/m]
m
mode number
P
pressure [Pa]
r
volumetric drag [kg/(m3s)]
Z
impedance [kg/s]
ɳ
shear viscosity [Pa∙s]
ɳB
bulk viscosity [Pa∙s]
Κ
bulk modulus [Pa]
λ
wavelength [m]
µ
mass per unit length [kg/m]
ν
Poisson ratio
ρ
density [kg/m3]
v
GLOSSARY
Acoustic
A general term referring to the behavior of vibrational energy
in a system.
Decibel (dB)
The unit of measure based on a logarithmic scale used to
measure sound pressure level.
Eigenfrequency
The frequency at which a resonance occurs in a system, or
when a small excitation in the system causes a large response.
Harmonics
A series of steady-state resonances occurring at multiples of
fundamental resonance frequency of the system.
Impedance
The resistance of a system to a harmonic excitation measured
in units of force per velocity. For example, when a system is
in a state of resonance, its impedance is at a minimum.
Mode
A term referring to the deformed state of a system at a
resonance.
Natural Frequency
See Eigenfrequency
Sound Pressure Level
The acoustic pressure in a fluid measured in decibels.
Tone
Refers to the sound produced by a system, such as a musical
instrument, that is being excited at a resonance or the
frequency at which this resonance occurs.
Wavelength
The length of an oscillatory disturbance in a solid or fluid
medium.
Wavespeed
The speed at which an oscillatory disturbance travels in a
solid or fluid medium.
vi
ABSTRACT
The objective of this project is to estimate the effect that the tubing material and
thickness have on the sound spectra produced by a modern concert flute. Since the
sound-producing mechanisms in a flute are quite complex and the number of variables
affecting the sound produced are numerous, it was necessary to focus on only these two
variables using a simplified approach. This simplified approach involved building a
Abaqus finite-element model of a cylindrical pipe with approximately the same
geometry as a concert flute and coupling it to acoustic elements, both on the inside and
outside of the pipe model. After running several analyses with five different materials
and three different pipe thicknesses, the results showed that these parameters have very
little effect on the sound produced the pipe.
vii
1. Introduction
Much attention has been paid to examining the physics of the modern concert flute,
shown in Figure 1.1, particularly the effects of the air jet produced by a flutists at the
embouchure of the instrument and the shape of tone holes, or the key holes, from which
the sound chiefly radiates from the instrument for all notes higher than the lowest note of
instrument’s range. A thorough review of the available literature on the subject,
however, shows that few attempts have been made to quantify the effect of the material
of the flute on the sound quality it produces. Even fewer published studies exist for
which finite-elements models were used to study this subject. Several experiments were
performed by John W. Coltman in which both musically skilled and unskilled listeners
were instructed to identify flutes constructed of silver, copper, and wood played by
experienced flutists. Coltman found that the listeners were able to correctly identify the
type of flute being played only about one-third of the time in 36 trials [1]. Many flutists,
however, claim that there is an appreciable different in the quality of sound produced by
flutes of different materials. The most famous flute player in modern history, JeanPierre Rampal, always played a solid gold flute and is reported to have said of the tone
of gold flutes (in contrast to silver flutes): “a little darker; the colour is a little warmer, I
like it” [2]. Whether or not this reported difference is due to the response of the edge
tone produced at the embouchure of the instrument, is due to the vibrations of the body
of the instrument, is purely psychological, or has some other explanation is still
unknown. This project examines only the possibility of an effect of the structural modes
of vibration of the instrument on the sound spectra produced, focusing on the lowest tone
that can be played by the instrument, approximately 261 Hz or, known to musicians C4
[3]. This tone corresponds to the fingering configuration in which all of the holes or
keys are covered. More than any other fingering configuration, this one causes the flute
to behavior most like an open-ended pipe, which can be modeled using the finiteelement method.
1
Modern Concert Flute
Figure 1.1
2
2. Methods
2.1 Modeling Validation Methods
The first part of this project required proving the ability of an Abaqus model to predict
the structural modes of a thin metal pipe with geometry and properties similar to that of
a modern concert flute. Most flutes of good quality are made of silver with a wall
thickness of 0.00036 meters or approximately 0.014 inches. Concert flutes are also
typically 0.612 meters or 24 inches long, measured from the center of the embouchure to
the opposite end of the instrument. The material and geometrical parameters used in the
calculation of the bending, longitudinal, and torsion modes of the pipe are given in Table
2.1 [4].
Table 2.1
Properties of Pipe Model
Parameter
Symbol
Value
L
Length of Pipe
0.612
a
Radius of Pipe
0.009
h
Thickness of Pipe
0.00036
E
Young’s Modulus of Silver
8.30E+10
ρpipe
Density of Silver
10490
ν
Poisson Ratio of Silver
0.37
Units
m
m
m
Pa
kg/m^3
NA
The first five bending modes or natural frequencies of the pipe, fn, were calculated using
equation 2.1, where I is the area moment of inertia of the pipe cross-section calculated
using equation 2.2, µ is the mass per unit length of the pipe calculated using equation
2.3, and A is the modal coefficient given for a simply-supported beam as 9.87, 39.5,
88.9, 158, and 247 for the first five bending modes [5].
𝑓𝑛,𝑏𝑒𝑛𝑑𝑖𝑛𝑔 =
3
𝐴 𝐸𝐼
√
2𝜋 𝜇𝐿4
(2.1)
𝐼 = 𝜋 𝑎3 ℎ
(2.2)
𝜇 = 2𝑎𝜋ℎ𝜌𝑝𝑖𝑝𝑒
(2.3)
The first three torsion and the first two longitudinal modes or natural frequencies of a
simply-supported pipe were calculated using Equations 2.4 and 2.6 where cp is the
compression wave speed in the pipe, given in Equation 2.5 and m is the mode number
[6].
𝑓𝑛,𝑡𝑜𝑟𝑠𝑖𝑜𝑛
𝑐𝑝 𝑚 (1 − 𝜈)
√
=
2𝐿
2
𝑬
𝒄𝒑 = √
(𝟏 − 𝝂𝟐 )𝝆𝒑𝒊𝒑𝒆
𝒇𝒏,𝒍𝒐𝒏𝒈𝒊𝒕𝒖𝒅𝒊𝒏𝒂𝒍 =
𝒄𝒑 𝒎
√(𝟏 − 𝝂𝟐 )
𝟐𝑳
(2.4)
(2.5)
(2.6)
A finite-element model was built of a simply-supported pipe with the same material and
geometric properties as given in Table 2.1 using Abaqus CAE/Standard software,
version 6.10. The pipe model was constructed using S4R elements, or four-noded
quadrilateral shell elements, with lateral boundary constraints applied at both ends. The
required mesh refinement was calculated using the flexural wave speed of a plate given
in equation 2.7 where the maximum analysis frequency is 4000 Hz. The maximum
flexural wave speed was then calculated using Equation 2.8 to determine the minimum
wavelength expected during the analysis. According to the Abaqus manuals, it is
recommended to use an element size that is approximately one sixth the size of the
smallest wavelength expected during the analysis [7].
4
𝑐𝑓 =
√2𝜋𝑓𝑚𝑎𝑥 ℎ𝑐𝑝
(2.7)
4
√12
𝜆𝑚𝑖𝑛 =
𝑐𝑓
𝑓𝑚𝑎𝑥
(2.8)
The acoustic modes of a cylindrical waveguide with one end driven by a piston and the
opposite end open to the air can be found by calculating the impedance at the input end
of the pipe. The modes of the air column inside the pipe were calculated using air
properties at 20 degrees Celsius given in Table 2.2 [8]. The wave number, k, was
calculated using equation 2.9 for a range of frequencies from 1 to 2000 Hz in 1 Hz
increments. The acoustic modes were determined by first calculating the radiation
impedance at the end of the pipe, ZmL, using equation 2.10. The impedance at the input
end of the pipe, Zm0, was calculated using equation 2.11, where S is the cross-sectional
area of the pipe [8]. The input impedance was then plotted as a function of frequency.
The minima of the impedance as function of frequency correspond to the acoustic modes
of the air column.
5
Table 2.2
Properties of Air Column Model
Parameter
Symbol
Length of Air Column
Leff
a
K
ρair
cair
Radius of Air Column
Bulk Modulus of Air
Density of Air
Velocity of Sound in Air
𝑘=
Value
Units
0.6174 m
0.009 m
1.42E+05 Pa
1.21 kg/m^3
343 m/s
2𝜋𝑓
𝑐𝑎𝑖𝑟
1
𝑍𝑚𝐿 = (𝜌𝑎𝑖𝑟 𝑐𝑎𝑖𝑟 𝑆)[ (𝑘𝑎)2 + 𝑗0.6𝑘𝑎]
4
𝑍𝑚0 = (𝜌𝑎𝑖𝑟 𝑐𝑎𝑖𝑟 𝑆)[
𝑍𝑚𝐿
)+𝑗𝑡𝑎𝑛(𝑘𝐿)
𝜌𝑎𝑖𝑟 𝑐𝑎𝑖𝑟 𝑆
]
𝑍𝑚𝐿
1+𝑗(
)tan(𝑘𝐿)
𝜌𝑎𝑖𝑟 𝑐𝑎𝑖𝑟 𝑆
(2.9)
(2.10)
(
(2.11)
A finite-element model was built in Abaqus using AC3D8, or 8-noded brick acoustic
elements, with the same approximate element size as the pipe model, which was more
than enough refinement to satisfy the six elements per wavelength guideline. The
radiation impedance boundary condition at the end of the pipe was modeled as a small
addition to the length of the air column. This effective length of the air column was
calculated using equation 2.12. Non-reflecting acoustic impedance boundary conditions
were applied at both ends of the air column model.
𝐿𝑒𝑓𝑓 = 𝐿 + 0.6𝑎
6
(2.12)
2.2 Material and Thickness Study Methods
To determine the effect of pipe material and thickness on the acoustic modes of an air
column inside the pipe, it was necessary to build a finite-element model with coupled
structural and acoustic nodal degrees of freedom and an exterior acoustic element mesh
to capture the sound radiation from the pipe. The pipe and air column were modeled
using the same geometry, element types, and element refinement as the two models
described in section 2.1. The exterior acoustic elements were modeled by creating a
larger cylinder to enclose the smaller pipe model and filling the volume between the
cylinder and the pipe model with AC3D4, or 4-noded tetrahedral acoustic elements. The
minimum required size of the elements was determined in a similar method as the pipe
model, using equation 2.13 and the six elements per wavelength guideline.
𝜆𝑚𝑖𝑛 =
𝑐𝑎𝑖𝑟
𝑓𝑚𝑎𝑥
(2.13)
The extent of the exterior cylinder was dependent on the maximum expected wavelength
calculated using equation 2.14 and the node and element number analysis maximum of
20,000 imposed by the Abaqus CAE/Standard education/teaching license. Due to these
constraints, the analysis range was limited to 250-2000 Hz and four separate models
were necessary to calculate the response of the model over this frequency range. The
four models are shown in Figure 2.1. The first model was used to analyze the response
in the frequency range 250-580 Hz, the second model was used for the frequency range
580-1100 Hz, the third model for 1100-1600 Hz, and the fourth model for 1600-2000
Hz. All analyses were run in 1 Hz increments. Each of the four models contains the
same pipe and air column models, but is different in the exterior mesh extent and
refinement. The first model, for example, has an extent in the longitudinal direction of
0.423 meters and an approximate maximum element size of 0.16 meters since the
wavelengths at the lower frequencies are quite long. The fourth model, however, has an
extent in the longitudinal direction of 0.076 and an approximate maximum element size
of 0.031 meters since the wavelengths at the higher frequencies are much shorter.
7
The geometry of the cylinder containing the exterior acoustic mesh was based on onefourth the maximum expected wavelength for the radius of the cylinder and one-third the
maximum expected wavelength plus the length of the pipe for the length of the cylinder.
This is based on guidelines also found in the Abaqus manuals indicating that the exterior
acoustic mesh be approximately one-third the maximum wavelength in the direction of
the acoustic wave where a more accurate solution is necessary, which in this case is the
longitudinal direction [7]. The four models are shown in Figure 2.1. In each of the
models, the structural nodes of the pipe were coupled to the air column nodes using tie
constraints. The nodes at the ends of the air column were also coupled to the nodes of
exterior mesh using tie constraints. Non-reflecting acoustic impedance boundary
conditions were applied the outside element faces of the exterior acoustic mesh.
𝜆𝑚𝑎𝑥 =
8
𝑐𝑎𝑖𝑟
𝑓𝑚𝑖𝑛
(2.14)
Figure 2.1
9
Although the actual damping effects of the pipe walls and air are difficult to determine,
it was necessary to introduce a small amount of damping into the model to avoid the
pressure response from going to infinity at the acoustic modes of the air column inside
the pipe. A volumetric drag coefficient was calculated for each analysis frequency using
Equation 2.15 from the Abaqus manual to account for viscous damping in the air [7].
The shear viscosity, ɳ, of air at 20 degrees Celsius is 1.85E-5 Pa*s and the bulk
viscosity, ɳB, is given by equation 2.16 [8],[9]. A damping ratio of 0.005 was added to
the pipe material [5]. This damping ratio was assumed to be similar for all the pipe
materials analyzed.
𝜌𝑎𝑖𝑟
4
𝑟(𝑓) = (2𝜋𝑓)2 (
)(𝜂𝐵 + 𝜂)
𝛫𝑎𝑖𝑟
3
(2.15)
𝜂𝐵 = 0.6𝜂
(2.16)
To perform the material and thickness studies, each of the four models was analyzed
seven times. The first analysis, or the baseline analysis, was run using the pipe material
and pipe thickness parameters given in Table 2.1. For second set of analyses, the
material studies, the pipe material parameters were then modified to reflect that of pure
gold, platinum, aluminum, and lead as given in Table 2.3 [10],[11],[12],[13]. For the
third set of analyses, the thickness parameters, the pipe material properties were returned
to that of silver and the thickness was modified twice, first to that of approximately
0.010 inches and then to 0.017 inches.
Table 2.3
Property
3
Pipe Material Study Properties
Silver Gold Platinum Aluminum
Density, kg/m
Young’s Modulus, GPa
Poisson Ratio
10490 19300
83
79
0.37
0.44
10
21450
168
0.38
Lead
2700 11340
70
16
0.35
0.44
3. Results
3.1 Modeling Validation Results
To determine the structural modes of the pipe finite-element model described in section
2.1, the Lanczos eigensolver in Abaqus CAE/Standard was used. The eigenvalue
frequencies, or natural frequencies, from the analysis were then compared to the natural
frequencies calculated using analytical methods also described in section 2.1. The
comparison between the two sets of results is shown in Table 3.1. The longitudinal and
torsion mode frequencies compare very well—the finite-element natural frequencies are
within 1% of the calculated frequencies. The bending modes did not compare as well—
within 10% of the calculated frequencies, but this was an expected result as the
analytical results were only approximate. The finite-element bending natural
frequencies are likely more accurate than those calculated using the analytical method.
Table 3.1
Pipe Model Mode Results
Mode
Calculated Natural
Frequency (Hz)
Abaqus Natural
Frequency (Hz)
72.7
Percent
Error
1st Bending
75.1
2nd Bending
300.5
288.4
4.0
3rd Bending
676.2
640.1
5.3
4th Bending
1201.9
1117.5
7.0
5th Bending
1878.9
1707.6
9.1
1st Longitudinal
2298.1
2297.9
0.0
2nd Longitudinal
4596.2
4594.4
0.0
1st Torsion
1388.3
1375.2
0.9
2nd Torsion
2776.7
2750.8
0.9
3rd Torsion
4165.0
4127.0
0.9
11
3.2
The impedance of the air column as calculated using the analytical methods described in
section 2.1 is shown as a function of frequency in Figure 3.1. The Lanczos eigensolver
in Abaqus CAE/Standard was used to determine the natural frequencies of the air
column finite-element model also described in section 2.1. The comparison of the
natural frequencies calculated using the two methods is shown in Table 3.2. The two
methods compare very well—all of the natural frequencies compare within 1%.
Input Mechanical Impedance of Air Column
40
20
Zm0, kg/s
0
-20
-40
-60
558
-80
837
1116
1395
1674
1953
279
-100
0
500
1000
Frequency, Hz
1500
2000
Figure 3.1
Table 3.2
Air Column Mode Results
Harmonic
Calculated Natural
Frequency (Hz)
Abaqus Natural
Frequency (Hz)
Percent
Error
Fundamental
Tone
st
1 Harmonic
279
277
0.56
558
555
0.57
2nd Harmonic
837
832
0.57
3rd Harmonic
1116
1109
0.58
4th Harmonic
1395
1387
0.60
5th Harmonic
1674
1664
0.61
6th Harmonic
1953
1941
0.63
12
3.2 Material and Thickness Study Results
A linear perturbation, steady-state dynamic baseline analysis was performed on the each
of the four coupled structural-acoustic models shown in Figure 2.1. The baseline
analysis used material properties of silver found in Table 2.1 and a 0.014 inch pipe
thickness. A harmonic pressure load of 100 Pa was applied at a node in the center of the
cross-section of one end of the air column. This level is an approximation of the typical
pressure levels produced by a flute player at the embouchure of the instrument [14]. The
output data measured was the pressure, P, at a single node located on the longitudinal
axis of the model and on the end of the exterior acoustic mesh boundary closest to the
output end of the pipe. This pressure was then plotted as a function of frequency for each
of the four models in the frequency range appropriate for the exterior mesh of the model.
This pressure response was converted to sound pressure level in decibel (dB) units using
equation 3.1, where Pref is equal to 20µPa [8].
SPL = 20log10 (P⁄Pref )
(3.1)
The four responses from the four models shown in Figure 2.1 were combined in the
same plot to show a continuous pressure response as a function of frequency over the
entire frequency range analyzed (see Figure 3.2). A contour plot of nodal pressure for an
elevation cut down the centerline of the exterior acoustic mesh at the fundamental
harmonic of 273 Hz is shown in Figure 3.3. For clarity, the models of the pipe and air
column inside the pipe have been removed in Figure 3.3. At the same frequency a side
view of the nodal pressure of air column inside the pipe is shown in Figure 3.4. A
contour plot of the displacement magnitude at each node at the same frequency is shown
as a side view of the pipe model in Figure 3.5. The scale on the displacement contour
plot indicates that very small displacements on the order of 10-8 meters are induced in
the pipe at this frequency.
13
Nodal Pressure Response for Baseline Model (Pa)
120
dB, re:20E-6µPa
100
80
Model 1
60
Model 2
40
Model 3
20
Model 4
0
0
500
1000
Frequency, Hz
1500
2000
Figure 3.2
Pressure in Exterior Acoustic Element Mesh at 273 Hz
Figure 3.3
14
Pressure in Air Column Acoustic Mesh at 273 Hz
Figure 3.4
Displacement Magnitude in Pipe Model at 273 Hz
Figure 3.5
15
As described in section 2.2, the models were analyzed six more times after the baseline
analysis was complete in order to evaluate the effect of different pipe materials and
thicknesses. Differences between the models were evaluated by measuring the
frequency that each harmonic occurred, or the frequency of each peak in the pressure
response, and the amplitude at each peak in sound pressure level units. These results, as
well as the percent differences in the peak amplitudes from the baseline response, are
shown in Tables 3.3-3.6.
The pressure amplitude at the fundamental tone or resonance and the first six harmonics
for the pipes of different materials are shown in Table 3.3. The smallest percent
deviation, no more than 0.035% from the baseline analysis, was found using gold
material properties. This is an expected result since the properties of gold are very
similar to that of silver. The greatest percent deviation, 2.796% from the baseline, was
found using lead material properties. This result is likely due to the very low Young’s
modulus of lead, 16 GPa, compared to that of silver, 83 GPa. The greatest difference in
sound pressure level in all of the material studies, however, is less than 3 decibels.
Table 3.3
Pipe Material Study Results:
Harmonic Pressure Amplitude (dB,ref: 20µPa) and
Percent Difference From Baseline (Silver)
Harmonic
Fundamental
Tone
Silver
Gold
%Δ
Platinum
%Δ
Aluminum
%Δ
Lead
%Δ
90.553
90.581
-0.030
90.302
0.278
90.650
0.106
92.909
-2.601
1st Harmonic
89.687
89.719
-0.035
89.403
0.317
89.792
0.117
92.195
-2.796
2nd Harmonic
96.838
96.824
0.014
96.964
-0.130
96.792
-0.048
95.085
1.810
3rd Harmonic
101.972
101.979
-0.007
101.902
0.069
101.996
0.024
102.384
-0.404
4th Harmonic
99.858
99.855
0.002
99.876
-0.018
99.850
-0.008
99.510
0.349
5th Harmonic
102.118
102.117
0.001
102.130
-0.011
102.113
-0.005
101.868
0.245
6th Harmonic
100.282
100.283
-0.001
100.276
0.006
100.284
0.002
100.250
0.032
16
The pressure amplitude at the fundamental tone and the first six harmonics for the pipes
of different thickness are shown in Table 3.4. The greatest percent deviation, 0.23%
from the baseline, was found using a thickness of 0.010 inches. A greater difference was
found by decreasing the thickness of the pipe than by increasing it. This is similar to the
result found in the material studies in that decreasing the stiffness of the pipe causes the
most change in the pressure response, although it does not change it by any significant
amount.
Table 3.4
Pipe Thickness Study Results:
Harmonic Pressure Amplitude (dB,ref: 20µPa) and
Percent Difference From Baseline (0.015'')
Harmonic
0.014''
0.010''
%Δ
0.017''
Fundamental Tone
90.553
90.738 -0.203
90.470
1st Harmonic
89.687
89.894 -0.230
89.593
2nd Harmonic
96.838
96.738
0.104
96.881
3rd Harmonic
101.972
102.020 -0.047
101.950
4th Harmonic
99.858
99.842
0.016
99.864
5th Harmonic
102.118
102.108
0.010
102.122
6th Harmonic
100.282
100.285 -0.003
100.280
%Δ
0.092
0.105
-0.045
0.022
-0.007
-0.004
0.002
The frequencies at which the fundamental tone and the first six harmonics occur for the
pipes of different materials and thickness are shown in Tables 3.5 and 3.6. The only
observed change in the fundamental and harmonic frequencies was found in the analysis
of the lead pipe at the second harmonic when the resonance frequency decreased from
819 Hz to 818 Hz. It is likely that the resonance frequencies in all of the material and
thickness studies shifted a negligible amount less than 1 Hz, which was the smallest
frequency increment used during all of the analyses.
17
Table 3.5
Pipe Material Study Results:
Harmonic Frequency (Hz) and Percent Difference From Baseline (Silver)
Harmonic
Silver Gold
Platinum
Aluminum
Lead
Fundamental Tone
273
273
273
273
273
1st Harmonic
546
546
546
546
546
2nd Harmonic
819
819
819
819
818
3rd Harmonic 1092 1092
1092
1092 1092
4th Harmonic 1365 1365
1365
1365 1365
5th Harmonic 1638 1638
1638
1638 1638
6th Harmonic 1910 1910
1910
1910 1910
Table 3.6
Pipe Thickness Study Results:
Harmonic Frequency (Hz) and
Percent Difference From Baseline (0.014’’)
0.014'' 0.010'' 0.017''
Harmonic
Fundamental Tone
1st Harmonic
2nd Harmonic
3rd Harmonic
4th Harmonic
5th Harmonic
6th Harmonic
273
546
819
1092
1365
1638
1910
18
273
546
819
1092
1365
1638
1910
273
546
819
1092
1365
1638
1910
4. Conclusions
The results of the material and thickness studies show that the properties of the pipe
finite-element model have a slight effect on the pressure response produced. The largest
difference in the response—almost 3% difference in amplitude—was found when the
pipe material was changed to lead. Much smaller differences were found with the other
pipe materials and pipe thicknesses that were evaluated. This result indicates that the
pipe must be extremely flexible or have very little stiffness for it to have any effect on
the modes of the air column inside the pipe. Even the differences found in the response
for the lead pipe are negligible and suggest that vibrations in the tubing of a flute do not
have any appreciable effect on the tone quality that the flute produces. These studies do
not prove that the material and tubing thickness of a flute have no effect on the tone
quality of a flute, but that the difference may be found in other mechanisms of the sound
production such as the air jet vibration at a flute’s embouchure or the sound radiation
from open key holes when notes higher than the fundamental tone are being played.
19
REFERENCES
[1] Coltman, John W. Effect of Material on Flute Tone Quality. The Journal of the
Acoustical Society of America, Vol 49, Part 2. July 27, 1970.
[2] “Jean-Pierre Rampal” http://en.wikipedia.org/wiki/JeanPierre_Rampal.
December 16, 2011.
[3] “ Frequencies for Equal-Tempered Scale”
http://www.phy.mtu.edu/~suits/notefreqs.html.
December 16, 2011.
[4] “Silver.” http://en.wikipedia.org/wiki/Silver.
December 16, 2011.
[5] Harris, Cyril M. and Charles E. Crede. Shock and Vibration Handbook Vols. 1-3.
McGraw-Hill Book Company, New York: 1961.
[6] Junger, Miguel and David Feit. Sound, Structures, and their Interaction 2nd Ed. MIT
Press, Cambridge, MA: November, 1986.
[7] “Abaqus 6.10 Online Documentation” © Dassault Systemes, 2010.
[8] Kinsler, Lawrence E. and Austin R. Frey. Fundamentals of Acoustics 4th Ed. John
Wiley and Sons, Hoboken, NJ: December, 1999.
[9] Pierce, Allan D. Acoustics: An Introduction to its Physical Principles and
Applications. Acoustical Society of America, Woodbury, NY: 1989.
[10] “Gold.” http://en.wikipedia.org/wiki/Gold.
March 10, 2012.
[11] “Platinum.” http://en.wikipedia.org/wiki/Platinum.
March 10, 2012.
[12] “Aluminum.” http://en.wikipedia.org/wiki/Aluminum.
March 10, 2012.
[13] “Lead.” http://en.wikipedia.org/wiki/Lead.
March 10, 2012.
[14] Fletcher, N.H. Air Flow and Sound Generation in Musical Wind Instruments.
Annual Review of Fluid Mechanics, Vol 11, 1979.
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