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1
A TUNABLE ELECTROMECHANICAL HELMHOLTZ RESONATOR
By
FEI LIU
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2007
2
© 2007 Fei Liu
3
To my beloved wife Hao Hu
4
ACKNOWLEDGMENTS
I would like to express my gratitude to Dr. Mark Sheplak, my advisor, for his guidance,
patience, encouragement and support during my academic pursuit in the Interdisciplinary
Microsystems Group at University of Florida. I would like to thank Dr. Lou Cattafesta, my
coadvisor, for his advice, instructive discussion and help. I also wish to express thanks to my
committee members, Dr. Toshi Nishida and Dr. Nam-Ho Kim, for their advice and valuable
feedback on this dissertation.
I would like to acknowledge the following individuals for their contributions to the
completion of my dissertation: Dr. Stephen Horowitz, for many productive discussions and his
gracious personality, and Dr. Todd Schultz, for his discussion and help with the acoustic
impedance measurements.
I am very grateful to Philip Cunio, Guiqin Wang, David Martin, Chris Bahr, Benjamin
Griffin, Brian Homeijer, Vijay Chandrasekharan, Matt Williams, Matias Oyarzun, Drew Wetzel,
Tai-An Chen and all other group mates, for their friendship, discussion, and help.
Finally, I would like to thank my parents, my parents-in-law, and my wife for their love,
understanding, and support over the years of my academic pursuit.
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TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ...............................................................................................................4
LIST OF TABLES...........................................................................................................................8
LIST OF FIGURES .........................................................................................................................9
ABSTRACT...................................................................................................................................13
CHAPTER
1 INTRODUCTION ..................................................................................................................15
Prologue..................................................................................................................................15
Background.............................................................................................................................16
Research Objectives................................................................................................................23
Research Contributions...........................................................................................................23
Outline of the Proposal ...........................................................................................................23
2 THEORETICAL BACKGROUND........................................................................................32
Helmholtz Resonator ..............................................................................................................32
Lumped Element Modeling.............................................................................................32
Model Parameter Estimation ...........................................................................................33
Electromechanical Helmholtz Resonator ...............................................................................36
Lumped Element Modeling.............................................................................................36
Tuning Behavior Analysis...............................................................................................39
Capacitive tuning of the EMHR...............................................................................39
Resistive tuning of the EMHR .................................................................................44
Inductive tuning of the EMHR.................................................................................45
3 MODELING AN EMHR USING THE TRANSFER MATRIX METHOD .........................52
Introduction.............................................................................................................................52
Characteristics of the Transfer Matrix....................................................................................55
Choice of State Variables ................................................................................................55
Properties of the Transfer Matrix Related to Special Acoustic Systems.........................58
Reciprocal system ....................................................................................................58
Conservative system.................................................................................................59
Basic Types of Elements of Acoustic Transfer Matrix Representation ..........................59
Transfer Matrix Representation for EMHR............................................................................62
Area Contraction .............................................................................................................62
Area Expansion ...............................................................................................................63
Duct Element...................................................................................................................64
Clamped Piezoelectric Backplate with Shunt Loads.......................................................66
6
Transfer Matrix of the EMHR.........................................................................................69
Transfer Matrix of the EMHR with Perforated Facesheet ..............................................70
Comparison between TM and LEM .......................................................................................71
4 EXPERIMENTAL TECHNIQUES........................................................................................83
Acoustic Impedance Measurement.........................................................................................83
Introduction .....................................................................................................................83
Theoretical Basis of the TMM ........................................................................................86
Uncertainty Analysis of the TMM ..................................................................................89
Parameter Extraction of the Piezoelectric Backplate..............................................................90
Damping Coefficient Measurement ................................................................................90
Effective Acoustic Piezoelectric Coefficient Deduction.................................................92
Experimental Setup.................................................................................................................93
Acoustic Impedance Measurement Setup .......................................................................93
Damping Measurement Setup .........................................................................................94
Parameter Extraction of the Piezoelectric Backplate ......................................................94
EMHR Construction ...............................................................................................................95
5 EXPERIMENTAL RESULTS AND DISCUSSION ...........................................................103
Evaluation of the Tuning Performance of the EMHR..........................................................103
Comparison with LEM and Transfer Matrix........................................................................106
Damping Coefficient Measurement Results.........................................................................107
Parameter Extraction of the Piezoelectric Backplate............................................................108
6 OPTIMAL DESIGN OF AN EMHR ...................................................................................122
Introduction...........................................................................................................................122
Optimizing Single Tuning Range of EMHR with Non-inductive Loads .............................124
Theoretical Background ................................................................................................124
Optimization Problem Formulation...............................................................................128
Optimization Results .....................................................................................................130
Sensitivity Analysis.......................................................................................................131
Pareto Optimization of the EMHR with Non-inductive Loads ............................................133
Optimization of the EMHR with Inductive Loads ...............................................................135
7 SUMMARY AND FUTURE WORK ..................................................................................143
APPENDIX
A NOISE LEVELS AND UNITS ............................................................................................146
B ACOUSTIC IMPEDANCE PREDICTION OF AN ORIFICE............................................149
Linear Impedance Model of Orifices....................................................................................149
Crandall’s Model...........................................................................................................149
GE Impedance Model....................................................................................................151
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End Corrections.............................................................................................................152
Effect of Nonlinearity at High Sound Pressure Levels.........................................................152
C PARAMETERS ESTIMATION FOR LEM OF THE PIEZOELECTRIC
DIAPHRAGM ......................................................................................................................160
D WAVE SCATTERING BY HELMHOLTZ RESONATOR IN A TUBE...........................163
Helmholtz Resonator as a Termination of a Circular Tube..................................................163
Helmholtz Resonator as a Termination of a Rectangular Tube............................................169
E GEOMETRIC DIMENSION OF EMHRS...........................................................................180
F COMPUTER CODES ..........................................................................................................184
Acoustic Impedance Prediction using LEM and TM ...........................................................184
Optimizing Tuning Range of an EMHR with Capacitive Loads..........................................195
Pareto Optimization Design of an EMHR with Capacitive Loads.......................................200
LIST OF REFERENCES.............................................................................................................207
BIOGRAPHICAL SKETCH .......................................................................................................215
8
LIST OF TABLES
Table page
1-1 Characteristics of the airframe noise..................................................................................25
1-2 Characteristics of propulsion noise components................................................................25
1-3 Comparison between the passive liners and adaptive liners..............................................25
1-4 Summary of typical acoustic liners....................................................................................26
2-1 End corrections for orifices or slits....................................................................................47
2-2 Parameter estimation for lumped element modeling of the EMHR ..................................48
4-1 Material properties of the piezoelectric backplate.............................................................97
4-2 Dimensions of the EMHRs ................................................................................................97
4-3 Selected loads matrix used in the experiment to tune the EMHR .....................................98
5-1 Determination of the damping coefficient of the piezoelectric plate in air .....................110
5-2 Determination of the damping coefficient of the piezoelectric plate in vacuum
chamber............................................................................................................................110
5-3 Comparison between predicted and deduced LEM parameters of the Piezoelectric
backplate ..........................................................................................................................110
6-1 Design optimization variables of the EMHR...................................................................137
B-1 Summary on some acoustic impedance models of an orifice..........................................157
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LIST OF FIGURES
Figure page
1-1 Percentage of highly annoyed persons (%) as a function of LDN for air traffic, road
traffic, and rail traffic.........................................................................................................27
1-2 Historical progress of noise reduction ...............................................................................27
1-3 Components of airframe and propulsion noise..................................................................28
1-4 Main sources of the airframe noise....................................................................................29
1-5 Shock and mixing noise components of the jet noise spectrum ........................................29
1-6 Engine noise source and comparison of noise radiation pattern from no-pass turbo
engine and by-pass turbo-fan engine .................................................................................30
1-7 Application of the acoustic liner technology in typical turbo-fan engine..........................30
1-8 An adaptive Herschel-Quincke tube ..................................................................................31
1-9 An electromechanical Helmholtz resonator.......................................................................31
2-1 A solid-walled HR under excitation of the incident wave.................................................49
2-2 Equivalent circuit representation of a solid-walled Helmholtz resonator..........................49
2-3 Schematic illustration of an EMHR...................................................................................50
2-4 Equivalent circuit representation of the EMHR.................................................................50
2-5 Simplified equivalent circuit representation for a EMHR .................................................51
2-6 EMHR with passive electrical loads is analogous to a 2DOF system ...............................51
3-1 A two-port network with reference directions for the positive direction of the current
variables indicated .............................................................................................................74
3-2 An acoustic two-port network with reference direction for the current variables
indicated.............................................................................................................................74
3-3 Three basic types of elements in an equivalent circuit representation for an acoustic
network ..............................................................................................................................75
3-4 An area contraction............................................................................................................75
3-5 A HR mounted in the side of one duct...............................................................................76
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3-6 Modeling EMHR using the transfer matrix method ..........................................................77
3-7 Elements for derivation the transfer matrix representation of the EMHR.........................78
3-8 A piezoelectric backplate and its equivalent circuit representation...................................79
3-9 An EMHR with perforated facesheet.................................................................................79
3-11 Comparison of prediction results for the normalized specific acoustic impedance of a
short- and open-circuited EMHR using TM and LEM......................................................81
3-12 Illustration of contributions to the variation in the prediction performance of the
LEM and TM .....................................................................................................................82
4-1 Illustration of the two microphone method........................................................................99
4-2 Pressure measured by a microphone..................................................................................99
4-3 Free damped vibration of a SDOF system.......................................................................100
4-4 Acoustic impedance measurement of the EMHR using TMM........................................100
4-5 The scattering by a HR mounted at the end of a PWT ....................................................101
4-6 The damping measurement for the piezoelectric backplate of an EMHR.......................102
4-7 Assembly diagram of modular EMHR (not to scale) ......................................................102
5-1 Experimental results for the normalized specific acoustic impedance of the EMHR
(Case I) as function of the capacitive loads .....................................................................111
5-2 Experimental results for the reflection coefficient of the EMHR (Case I) as function
of the capacitive load .......................................................................................................112
5-3 Experimental results for the normalized specific acoustic impedance of the EMHR
(Case I) as function of the resistive loads ........................................................................112
5-4 Experimental results for the normalized specific acoustic impedance of the EMHR
(Case I) as function of the inductive loads.......................................................................113
5-5 Experimental results for the reflection coefficient of the EMHR (Case I) as function
of the inductive load.........................................................................................................114
5-6 Experimental results of the normalized acoustic impedance of the EMHR (Case II)
for the short- and open-circuit..........................................................................................114
5-7 Comparison LEM, TR and measurement results for a short- and open-circuited
EMHR (CASE I)..............................................................................................................115
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5-8 Comparison LEM, TR and measurement results for a short- and open-circuited
EMHR (CASE I), the damping loss of the backplate is assumed to be acoustic
radiation resistance...........................................................................................................115
5-9 Comparison LEM, TR and measurement results for a short- and open-circuited
EMHR (CASE II) ............................................................................................................116
5-10 Comparison LEM, TR and measurement results for a short- and open-circuited
EMHR (CASE II), the damping loss of the backplate is assumed to be acoustic
radiation resistance...........................................................................................................116
5-11 Damping coefficient measurement for piezoelectric composite backplate (Case I) in
air .....................................................................................................................................117
5-12 Damping coefficient measurement for piezoelectric composite backplate (Case I) in
the vacuum chamber ........................................................................................................117
5-13 Determination of damping coefficient of the piezoelectric plate in air ...........................118
5-14 Curve fitting the measurement data (in air) using a 2
nd
-order system .............................118
5-15 Determination of damping coefficient of the piezoelectric plate in the vacuum
chamber............................................................................................................................119
5-16 Curve fitting the measurement data (in the vacuum chamber) using a 2
nd
-order
system ..............................................................................................................................119
5-17 Measured transverse displacement of the piezoelectric backplate due to the
application of various voltages ........................................................................................120
5-18 Predictions of the LEM and TM for short- and open-circuited EMHRs .........................121
6-1 Resonant frequency of the EMHR versus
s
κ
and
α
.......................................................138
6-2 Normalized sensitivity of the design variables at the optima for maximizing the
tuning range of
1
f
∆ ..........................................................................................................139
6-3 Illustration of the change in the optimum solution as a function of
2
R
..........................140
6-4 Normalized sensitivity of the design variables at the optima for maximizing the
tuning range of
2
f
∆ ..........................................................................................................141
6-5 Choice of the starting point for the multi-objective optimization with different
1
ε
........141
6-6 Comparison of the Pareto front obtained via the
ε
-constraint, traditional weighted
sum, and adaptive weighted sum methods.......................................................................142
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6-7 Comparison between initial (dash line) and optimal (solid line) acoustic impedance
of the EMHR with inductive loads ..................................................................................142
A-1 A-weighting to sound arriving at random incidence........................................................148
D-1 A Helmholtz resonator as termination of a circular tube.................................................178
D-2 Schematic of a rectangular tube terminated by a Helmholtz resonator ...........................178
D-3 Schematic of transform between the Cartesian coordinate system and polar system......179
E-1 Schematic of the EMHR..................................................................................................180
E-2 Engineering draft of the orifice plate...............................................................................180
E-3 Engineering draft of the orifice plate of the cavity plate .................................................181
E-4 Draft of the piezoelectric diaphragm bottom plate ..........................................................182
E-5 Draft of the piezoelectric diaphragm cap plate................................................................183
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Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
A TUNABLE ELECTROMECHANICAL HELMHOLTZ RESONATOR
By
Fei Liu
August 2007
Chair: Mark Sheplak
Cochair: Louis N Cattafesta III
Major: Aerospace Engineering
Acoustic liners are used in turbofan engine nacelles for the suppression of engine noise.
For a given engine, there are different optimum impedance distributions associated with take-off,
cut-back, and approach flight conditions. The impedance of conventional acoustic liners is fixed
for a given geometry, and conventional active liner approaches are impractical. This project
addresses the need for a tunable impedance through the development of an electromechanical
Helmholtz resonator (EMHR). The device consists of a Helmholtz resonator with the standard
rigid backplate replaced by a compliant piezoelectric composite. Analytical models (i.e., a
lumped element model (LEM) and a transfer matrix (TM) representation of the EMHR) are
developed to predict the acoustic behavior of the EMHR. The EMHR is experimentally
investigated using the standard two-microphone method (TMM). The measurement results
validate both the LEM and the TM of the EMHR. Good agreement between predicted and
measured impedance is obtained. Short- and open-circuit loads define the limits of the tuning
range using resistive and capacitive loads. There is approximately a 9% tuning limit under these
conditions for the non-optimized resonator configuration studied. Inductive shunt loads result in
a 3 degree-of-freedom (DOF) system and an enhanced tuning range of over 47% that is not
restricted by the short- and open-circuit limits. Damping coefficient measurements for a
14
piezoelectric backplate in a vacuum chamber are performed and indicate that the damping is
dominated by structural damping losses. A Pareto optimization design based on models of the
EMHR is performed with non-inductive loads. The EMHR with non-inductive loads has 2DOF
and two resonant frequencies. The tuning ranges of the two resonant frequencies of the EMHR
with non-inductive loads cannot be optimized simultaneously, so a trade-off (Pareto solution)
must be reached. The Pareto solution shows how design trade-offs can be used to satisfy specific
design requirements. The goal of the optimization of the EMHR with inductive loads is to
achieve optimal tuning of the three resonant frequencies. The results indicate that it is possible
to keep the acoustic reactance of the resonator nearly constant within a given frequency range.
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CHAPTER 1
INTRODUCTION
Aircraft noise is an inevitable environmental impact of aviation. Aircraft noise adversely
affects the quality of life of people living near an airport. As a consequence, worldwide policies
and laws have been enacted to restrict noise emission from aircraft, making noise reduction an
important goal for engineers. In this chapter, the sources of noise from aircraft and the effects of
aircraft noise on people are briefly examined. Some existing technologies to reduce aircraft
noise emission from aircraft engines are presented (e.g., acoustic liners). A new method using an
electromechanical Helmholtz resonator (EMHR) to tune the acoustic impedance of an acoustic
liner in-situ is introduced. A main goal of this thesis is to study and understand the behavior of
the EMHR via modeling and a companion experimental investigation. After the introduction,
the objectives and approaches are described. Finally, the outline of the rest of the dissertation is
given.
Prologue
Originating from the operation of military and commercial airplanes, aircraft noise
exposure can extend many miles beyond the boundaries of an airport. Aircraft noise is
potentially harmful to aircrew and passengers, ground crews and mechanics, and people working
or living in the vicinity of the airport. Problems related to aircraft noise pollution include
• Physiological effects. An increase in blood pressure or heart rate, or even muscular
contractions due to exposure to unexpected or elevated noise levels has been observed
(Broadbent 1957).
• Hearing loss. Exposure to intensive noise for long durations can result in temporary or
permanent damage to the human ear and decreases the ability to hear clearly (Chen et al.
1992; Wu and Lai 1995).
• Disturbance of sleep and communication. As shown in Figure 1-1, extensive analysis
indicates that aircraft noise is a larger annoyance than automobile and railroad noise
(Miedema and Vos 1998).
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• Distraction and reduced performance. Long term exposure to noise will impair the
ability to control attention and focus on task performance (Griefahn 2000; Kujala et al.
2004).
With more and more people suffering adverse effects from aircraft noise, public concerns
over noise emitted from aircraft have increased tremendously. As a consequence, worldwide
policies and laws have been enacted to restrict the noise emission from aircraft (FAA 2001).
Hence, modern aircraft must reduce noise emissions to satisfy new government regulations.
Figure 1-2 illustrates the historical noise level trend in commercial airplanes. This decrease is
obtained with the help of technological developments and the use of effective acoustic treatments
to aircraft.
Background
Aircraft noise is mainly generated from the airframe and propulsion systems (Smith 1989).
Propulsion noise dominates during the take-off (with or without cutback in thrust) and cruise
phases of the flight. Airframe noise, on the other hand, is dominant during approach, as shown
in Figure 1-3.
Airframe noise is generated by an aircraft in flight due to the air flow around the airframe
(Smith 1989; Crighton 1995). The fuselage, main wings, tailplane, leading edge slats, trailing
edge flaps, and landing gear with wheel bays potentially contribute to the airframe noise
signature, as shown in Figure 1-4.
Since airframe components vary in dimension from a few centimeters (e.g., the cockpit
window) to meters (e.g., wings and fuselage), the scale of turbulence induced by air flowing over
such structures differs. Thus, airframe noise is typically broadband in nature (Smith 1989;
Crighton 1995). Audible, low-frequency tones occur occasionally. These tones often arise from
the vortex-shedding mechanism at the trailing edge of the wings, cavities or other discontinuities
17
in otherwise smooth surfaces, such as landing-wheel bays (Smith 1989). Some characteristics of
the airframe noise are summarized in Table 1-1.
Another significant component of aircraft noise is generated by the propulsion system.
The components of the propulsion noise include turbomachinery noise, combustor(core) noise
and jet noise (Smith 1989; Groeneweg 1995). Fans, compressors, and turbines can each generate
significant turbomachinery-related noise. Turbomachinery noise exhibits both tonal and
broadband characteristics. The broadband noise results from sound generated in the proximity of
the surfaces of rotating or stationary blades as a result of pressure fluctuations caused by
turbulent flow (Smith 1989). The discrete tones are generated from the interaction between the
airflow perturbations in the path of a rotating blade row (Smith 1989). Fan blade passage
frequency (BPF) tones and harmonics fall into this category. BPF tones are a function of the
number of rotating blades and rotation speed (Smith 1989; Groeneweg 1995). “Buzz saw” noise
is another type of discrete tone that has been observed when the fan has a supersonic tip speed.
“Buzz saw” noise is generated at frequencies that are multiples of the rotation speed of the blade
and are distinct from the blade passing frequency (Smith 1989).
Combustor noise is generated by the combustion processes in the engine and also occurs
when the hot flow leaving the combustion chamber interacts with the turbine and exhaust nozzle
downstream. Combustor noise is usually characterized by a broadband spectrum that peaks at a
low frequency (Mahan and Karchmer 1995). The operation temperature, pressure, and physical
geometry of the hot gas path through the engine affect the combustion noise level.
Jet noise encompasses subsonic jet mixing noise and shock-associated supersonic jet noise
(Smith 1989; Lilley 1995). Subsonic jet mixing noise results from the turbulent mixing process
between the exhaust flow of the engine and the surrounding air. Supersonic jet noise is
18
associated with the expansion shocks (jet core) from the nozzle of the engine. Jet mixing noise is
broadband in nature and centered at a relatively low frequency. Supersonic jet radiates both
broadband noise and screech tones. The broadband spectrum of the shock-associated jet noise is
centered at a higher frequency than the mixing noise (Smith 1989; Lilley 1995; Shen and Tam
1998), as shown in Figure 1-5.
Figure 1-6 shows the propulsion noise source radiation patterns of a conventional turbo
engine and a modern turbo-fan engine. A summary of the characteristics of the propulsion noise
components is listed in Table 1-2. There are still other aircraft noise sources not specifically
discussed above, such as propeller noise and rotor noise from a helicopter. They warrant
consideration in the design of such aircraft.
In recognition of increasing aviation noise regulations, a vast number of techniques have
been developed or proposed to reduce or control noise emission from aircraft. Noise control in
propulsion systems attracted most of the research attention in the early era of noise reduction.
With more demanding standards, airframe noise suppression has become increasingly important.
Airframe noise reduction involves low noise gear design, development of low noise high-lift
technology, reducing of aircraft cruise drag using passive/active flow control, and other
technologies (Willshire 2001).
The development of propulsion noise reduction technologies has followed two routes:
noise source control and noise level reduction (Envia 2001). Noise source control suppresses the
noise generation mechanism. The methods include designing low noise engine components,
such as using cycle optimization design to reduce the tip speed of the fan blades, fan pressure
ratio and the velocity of the exhaust flow, and optimizing the number and spacing of the blades
to weaken the impinging wake (Smith 1989; Envia 2001). Noise level reduction, however,
19
adopts acoustic treatments to reduce noise emissions. There are many different kinds of acoustic
treatments. The use of the acoustic liners has been proven effective. Absorbent liners in the
inlet of the engine account for about 5 PNdB of noise reduction (Smith 1989). Physically,
acoustic liners provide a complex impedance boundary condition for the engine duct and thus
suppress the propagation of the noise along the duct.
Generally, the environment in which the acoustic liners have to operate is very hostile.
The typical temperatures range from -50ºC to 500ºC between the inlet airflow at high altitude
and hot gas in the exhaust system of the engine (Smith 1989). High airflow velocity in engine
inlets and fan bypass ducts results in intensive aerodynamic turbulent pressure fluctuations up to
170 dB (ref. 20 Pa
µ
) (Manglarotty 1970). Moreover, as part of the aircraft, acoustic liners must
satisfy structural integrity requirements and weight constraints. As a result, the typical acoustic
liner in service has a Helmholtz resonator structure and is comprised of a simple perforated plate
facesheet, a cellular structure such as honeycomb, and a solid backplate (usually the engine
wall), as shown in Figure 1-7. This type of acoustic liner is also called a single degree of
freedom (SDOF) resonant absorber (Motsinger and Kraft 1995). The SDOF absorber is effective
over approximately one octave of the frequency range for noise suppression. Thus, the design
parameters should be pre-tuned to the frequency band that contains the single tone of most
concern (Motsinger and Kraft 1995).
The two degree of freedom (2DOF) acoustic liner shown in Figure 1-7 is similar to a
SDOF acoustic liner. The major difference is that there are two layers of honeycomb separated
by a fibrous or perforated septum sheet. The useful bandwidth of 2DOF acoustic liner is
approximately two octaves. Thus, 2DOF liners can be used to suppress fan noise encompassing
two adjacent harmonics of the fan blade-passage frequency. Theoretically, wider attenuation
20
bandwidth can be acquired using multiple layers of honeycomb structures (Multiple DOF
system). Such a design, however, may not be realizable due to weight and space limitations of
the engine. Wider attenuation bandwidth can be also achieved by non-resonant acoustic liners,
such as bulk absorbers. Bulk absorbers have a single layer construction, similar to the SDOF
absorber, but use a fibrous mat instead of a honeycomb separator. Bulk absorbers are predicted
to work well over three octaves (Motsinger and Kraft 1995). However, bulk absorbers have not
been used in commercial engine service in the past because of structural design difficulties and
maintainability concerns due to their tendency to absorb fluids (Motsinger and Kraft 1995;
Bielak and Premo 1999).
Without grazing flow effects, the frequency range of noise suppression is fixed by the
geometry for the acoustic liners mentioned above. These types of acoustic liners are called
passive acoustic liners. These are in contrast to the active/adaptive liners, which can adjust the
performance of the acoustic liner in-situ. Active/adaptive liners have attracted more attention
from researchers recently because of their promise in suppressing the engine noise under
different operating conditions. The typical operating conditions of the engine include take-
off/cutback, cruise, and approach. When operated at different conditions, the characteristics of
the aircraft noise may be much different. Potential ways in which active/adaptive liners can
modify acoustic liner performance in-situ include using bias flow through the liner resistive
elements and changing the geometry characteristics of the liner (e.g.,the orifice dimension and
the cavity volume).
In 1957, Thurston et al. (1957) indicated that steady flow through an orifice will change its
impedance. Howe (1979) developed a model of the Rayleigh conductivity of an aperture,
through which a high Reynolds number flow passes. The interaction between the incident sound
21
wave and the mean bias flow results in the periodic shedding of vorticity. Consequently, a
portion of the acoustic energy is dissipated into heat. Howe’s work indicates that it is possible to
enhance the sound absorption using a bias flow through the orifice of the acoustic liner. Hughes
and Dowling further studied the vortex shedding mechanism for slits and circular perforations
(Hughes and Dowling 1990; Dowling and Hughes 1992). The perforated liner was
experimentally investigated by Sun and his colleagues (1999). They found that a bias flow can
enhance sound absorption and extend the effective bandwidth of the perforated liner.
Little et al. (1994) tuned their resonator by changing the neck sectional area. De Bedout et
al. (1997) developed a Helmholtz resonator with a cavity that allows a continuously variable
volume. The variable volume actuation of the cavity is realized by rotating an internal radial
wall based on a tuning (control) algorithm. Selamet et al. (2003) presented a Helmholtz
resonator with an extended neck. Their work indicates that it is possible to control the resonance
frequency of the resonator by adjusting the length of the extended neck without changing the
cavity volume. All methods above mechanically modify the geometry of the resonator-like
acoustic liner. Generally, there is an obvious need for an additional actuator, controller, and
power supply. Thus, it is difficult to apply this type of method to the acoustic liner in an engine.
In essence, the techniques described above seek to enhance the noise absorption by
modifying the impedance boundary conditions of the liner system. Moreover, it is worth noting
that all adaptive/active systems discussed above, whether by modifying the geometry of the
acoustic liner or introducing a bias flow to the liner, may require actuators, sensors, and
controllers. Consequently, such a system is often complex in comparison with a passive liner
system. A more detailed comparison between passive liners and adaptive liners is found in Table
1-3. A summary of typical acoustic liners is provided in Table 1-4. Ideally, an acoustic
22
treatment system should be robust and lightweight, have wide noise suppression bandwidth, and
have the ability to modify the performance of the system in-situ.
Another type of acoustic treatment using a Herschel-Quincke tube merits discussion. A
Herschel-Quincke (HQ) tube is essentially a hollow side-tube that travels along the axis of a
main duct (but not necessarily parallel to) and attaches to the main-duct at each of the two ends
of the tube (Stewart 1928), as shown in Figure 1-8. The HQ tube is a simple implementation of
the destructive interference principle to attenuate the sound. Historically, the HQ tube is used to
suppress the tones of special frequencies due to the limitations of constant cross-sectional areas
(Panigrahi and Munjal 2005). The limitations require that the cross-sectional areas of the
parallel ducts be equal, and the sum of the cross-sectional areas of the branch ducts be equal to
one of the entrant and exit duct (Stewart 1928; 1945). People later found that such limitations
are not necessary and removing these restrictions can enhance the attenuation bandwidth of the
HQ tube (Selamet et al. 1994; Selamet and Easwaran 1997).
The potential of the HQ tube to attenuate turbo-fan engine noise was recently investigated
by Smith et al. (2002a; 2002b). They found that the HQ tube can be used to attenuate low-
frequency broadband noise, “buzz-saw” noise, and BPF tones. The HQ tube can be designed to
work together with passive liners, which are most effective at attenuating higher-frequency
noise. They also presented an adaptive HQ tube, in which an internal throttle-plate flap tunes the
resonant frequency, as shown in Figure 1-8. Again, additional controller, sensor(s), and
actuator(s) are needed for practical implementation.
A novel method to tune the impedance of the liner system is presented in this dissertation.
The primary element of this liner is a Helmholtz resonator with the standard rigid backing
replaced by a compliant piezoelectric composite diaphragm (Sheplak et al. 2004), as shown in
23
Figure 1-9. The acoustic impedance of the resonator is adjusted and additional degrees of
freedom added via electromechanical coupling of the piezoelectric composite diaphragm to a
passive electrical shunt network.
Research Objectives
The objective of this dissertation is to investigate a compact, practical EMHR for noise
reduction applications. To complete this objective, an analytical model, i.e. transfer matrix (TM)
representation of the EMHR, is developed to predict the acoustic behavior of the EMHR. The
model is implemented in a MATLAB program and compared with the lumped elements model
(LEM) and measurement results. The EMHR is experimentally investigated using the standard
two-microphone method (TMM). The measurement results are used to validate analytical
models of the EMHR and to demonstrate the potential noise suppression applications of the
EMHR. Parameter extraction, such as for the damping coefficient and some LEM parameters of
the piezoelectric backplate of the EMHR, is implemented to improve the model performance.
Finally, an optimal design routine is developed to provide the aids in the design of the EMHR to
achieve a usable impedance range.
Research Contributions
The expected contributions of this dissertation are as follows:
• Development of a transfer matrix model (TM) and refinement of the lumped element
model (LEM) for the prediction of the acoustic impedance of the EMHR.
• Experimental validation of both TM and LEM for the EMHR.
• Development of an optimal design routine for the EMHR.
Outline of the Proposal
The organization of this dissertation is as follows. Chapter 1 introduces the background,
motivation, and technical objectives of the research. In Chapter 2, lumped element modeling a
24
Helmholtz resonator and an EMHR are presented. Chapter 2 also presents the theoretical
analysis of the electromechanical tuning behavior of an EMHR. The development of a model of
the EMHR using the transfer matrix method is detailed in Chapter 3. Chapter 4 demonstrates the
use of the standard two microphone method (TMM) to experimentally investigate the behavior
of the EMHR as well as methods (e.g., the logarithmic decrement method) implemented to
extract the model parameters. The experimental results and discussion are presented in Chapter
5, followed by the development of the design methodology using optimization techniques in
Chapter 6. The conclusion of this research project and comments on future work are given in
Chapter 7.
25
Table 1-1. Characteristics of the airframe noise (Smith 1989).
Work condition Noise source Characteristics of noise
Cruise Fuselage Main wings Broadband
Distributed low frequency peak
Fairly strong tones
Take-off
Approach
Slats
Flaps
Landing gear
Broadband
Generally omnidirectional
Table 1-2. Characteristics of propulsion noise components (Smith 1989; Hubbard 1995).
Noise component Characteristics
Fan noise Tones, blade passing frequency related
“Buzz-saw ”noise
Broadband noise
Compressor noise High frequency tones
Broadband noise
Combustor noise Broadband noise with spectrum centered at low frequency
Turbine noise High frequency tones
Broadband noise which spectrum is centered at high frequency
Jet noise Screech tones
Broadband noise
Centered at low frequency for the mixing jet noise
Centered at higher frequency for the shock-associated jet noise
Table 1-3. Comparison between the passive liners and adaptive liners (Smith 1989; Kraft et al.
1999).
Passive liners Adaptive liners
Pros Fixed geometry, simple structure Increased noise suppression frequency range or
reduced noise over variable frequency range
Cons Relative narrow suppression Frequency
range fixed by the geometry
More degree of freedoms are required
to increase the suppression frequency
range
Complex structure
Power, actuator, or controller needed
26
Table 1-4. Summary of typical acoustic liners (Nayfeh et al. 1975; Motsinger and Kraft 1995;
Parrott and Jones 1995; Bielak and Premo 1999).
Liner types Structure, Properties and Application
Perforated-plate, honeycomb
liner
Easy to manufacture
Narrow noise attenuation bandwidth (SDOF)
Increased degrees of freedom needed to widen
attenuation bandwidth
Acoustic nonlinearity
Widely used in engine duct
Resistive resonant liner
(linear liner)
Porous layer can be woven-screen metal sheets,
sintering fiber-metal sheets, etc.
Narrow noise attenuation bandwidth (SDOF)
Acoustic character is independent of sound pressure
level (SPL)
Usable in engine noise reduction
Parallel elements liner
Damping provided by the cylindrical channel
Circumvents use of the facesheet
Ceramic honeycomb is heavy and fragile
Not applicable to the engine in the past
Locally
reactive liners
a
(Ingard 1999)
Perforated-plate liner with bias
flow
Introduction of airflow, blowing or suction,
perpendicular to the acoustic liner
Ability to control acoustic impedance of liner in-situ
via bias flow
Additional actuator or controller needed
Promising way to reduce engine noise
Non-locally
reactive liners
Bulk non-resonant liner
Broad attenuation bandwidth
Poor mechanical properties (rigidity, etc.) and fluid
absorption
Not applicable to aircraft in the past
a
. Locally reactive liners are defined as liners that permit wave propagation only in the direction
normal to the acoustic liner surface, while non-locally reactive liners permit propagation in
more than one direction.
27
45 50 55 60 65 70 75 80
0
10
20
30
40
50
60
Percentage highly annoyed persons [%]
LDN
Air traffic
Road traffic
Rail traffic
Figure 1-1. Percentage of highly annoyed persons (%) as a function of LDN for air traffic, road
traffic, and rail traffic, where LDN is the average level of day/night noise measured in
decibels (dBA) over an extended period of time. Adapted from Miedema and Vos
(1998).
Figure 1-2. Historical progress of noise reduction (sideline noise level, normalized to 10
4
lb
thrust), where the noise metric is Effective Perceived Noise Level (EPNL), measured
in units of EPNdB. Adapted from Willshire (2001).
28
0
20
40
60
80
100
120
Inlet Aft fan Combustor Turbine Jet Total
airframe
Total aircraft
noise
Approach Cutback
(a)
120m
2000m
6500m
Approach
reference
Takeoff/Cutback
reference
450m
Sideline
reference
(b)
Figure 1-3. Components of airframe and propulsion noise. (a) Comparison of airframe and
propulsion noise for Boeing 747-400 with P&W 1992 technology engine during the
approach and takeoff/cutback operation with (b) the noise reference points illustrated.
Adapted from Smith (1989).
Noise level
(
EPNdB
)
29
Figure 1-4. Main sources of the airframe noise. Adapted from Smith (1989).
Shock noise
Mixing noise
Frequency
Figure 1-5. Shock and mixing noise components of the jet noise spectrum. Adapted from Smith
(1989).
30
Compressor
Jet
Turbine & core
Fan
Turbo engine Turbo-fan Engine
Figure 1-6. Engine noise source and comparison of noise radiation pattern from no-pass turbo
engine and by-pass turbo-fan engine. Adapted from Smith (1989).
Figure 1-7. Application of the acoustic liner technology in typical turbo-fan engine. Adapted
from Smith (1989).
31
Figure 1-8. An adaptive Herschel-Quincke tube. Adapted from Smith and Burdisso (2002).
PZT-backplate
Loads
Cavity
Neck
Figure 1-9. An electromechanical Helmholtz resonator.
32
CHAPTER 2
THEORETICAL BACKGROUND
In this chapter, a review of the lumped element model (LEM) of a conventional Helmholtz
resonator and an EMHR is presented. The tuning behavior of an EMHR with various electrical
loads (e.g., the EMHR with capacitive loads) is then analyzed with the aid of the LEM.
Helmholtz Resonator
As mentioned in Chapter 1, the passive acoustic liners widely used as acoustic treatments
in engine ducts have a structure similar to a Helmholtz resonator, as shown in Figure 2-1. The
name Helmholtz resonator comes from the German physician and physicist Hermmann von
Helmholtz (Rayleigh 1945). A conventional solid-walled Helmholtz resonator is comprised of a
narrow neck (i.e., a small orifice) and a cavity volume. The resonator is a SDOF system and has
a resonant frequency determined by its geometry.
Lumped Element Modeling
The modeling of a HR can be simplified if any characteristic dimension of the resonator is
small compared to the acoustic wavelength. When the wavelength is greater than all dimensions,
a distributed system can be lumped into idealized discrete elements (Rossi 1988). The linearized
continuity and Euler’s equations are thus replaced by equivalent Kirchoff’s laws for volume
velocity and pressure drop. Hence, for a Helmholtz resonator excited by an incident acoustic
pressure of low frequencies, shown in Figure 2-1, the gas trapped in the neck of the resonator is
modeled as a lumped mass element
aN
M
. The air isentropically compressed in the cavity is
modeled as an acoustic compliance
aN
C . Furthermore, the open end of the neck radiates sound,
and thus provides radiation resistance
aNrad
R , and mass
aNrad
M . Additional resistance,
aNvis
R ,
results from the thermo-viscous losses at the neck walls. The
aNrad
R and
aNvis
R comprise the
33
damping loss of the neck,
aN
R
. In the notations above, the first subscript denotes the domain
(e.g., “a” for acoustic), and the second subscript describes the element (e.g., “N” for neck). The
equivalent circuit representation for a solid-walled Helmholtz resonator is shown in Figure 2-2,
where P and Q represent the incident acoustic pressures and volumetric flow rate, respectively.
Model Parameter Estimation
In Figure 2-2, the acoustic mass,
aN
M
, is given by (Blackstock 2000)
(
)
0
2
aN
tt
M
r
ρ
π
+
∆
= , (2-1)
where
t and
r
are the thickness and the radius of the neck, respectively, t
∆
is the sum of the end
correction at the inner and outer neck ends. Note that Eq. (2-1) assumes no viscosity and the
velocity profile within the neck is constant (i.e., plug flow). The end correction in Eq. (2-1)
takes into account the entrained air near the flanged or unflanged mouth of the resonator. The
end correction,
t∆
, is given as (Ingard 1953)
inner outer
0.85 1 1.25 0.85
r
tt t r r
R
⎛⎞
∆=∆ +∆ = − +
⎜⎟
⎝⎠
, (2-2)
where
R
is the radius of the cavity. More cases of the end correction of the orifice (hole) are
listed in Table 2-1.
The cavity acoustic compliance,
aC
C , is derived assuming a uniform pressure throughout
the cavity and an isentropic compression process (Blackstock 2000),
2
00
,
aC
C
c
ρ
∀
= (2-3)
where
2
R
L
π
∀= is the cavity volume, and L is the depth of the cavity. Another method is to
treat the cavity as a short closed tube with depth L . The specific acoustic impedance of such
tube is thus given by (Blackstock 2000)
34
sp_ 0 0
cot
aC
Z
jc kL
ρ
=
− . (2-4)
If
1kL << , i.e., 0.3kL ≤ , the tangent function
cot kL kL
≈
. (2-5)
Substitution of Eq. (2-5) into (2-4) results in
2
00 00
sp_aC
cc
Z
j
kL j L
ρρ
ω
==. (2-6)
The acoustic impedance of the cavity is therefore
()
2
00
2
2
00
11
.
aC
aC
c
Z
jC
jc
jL R
ρ
ω
ωρ
ωπ
===
∀
(2-7)
Both methods yield the same results. The requirement for the first method that the process is
quasi-static is closely related to the requirement for the second method that
kL
be small
(kL<<1). However, the first method can be used for a general case, such as an irregular cavity.
The neck resistance
aN
R
is given by ( Ingard 1953)
00
00
1
222
22
2(2)
21
2
aN
c
Jkr
t
R
rr r r kr
µρ ω µρ ω
ρ
πππ
⎡
⎤
=++−
⎢
⎥
⎣
⎦
, (2-8)
where
µ
is dynamic viscosity of the air, and
(
)
1
J is the Bessel function of the first kind. The
neck resistance includes several contributions. The first term in Eq. (2-8) represents the viscous
loss in the neck wall boundary layer , which is derived assuming the fluctuating flow through the
neck due to the acoustic excitation is hydro-dynamically incompressible viscous flow (Crandall
1926). The second term represents the viscous loss at the neck ends (also refers to resistance end
correction). This approximation is adapted from Ingard (1953), which is originally derived by
empirical methods. The third term represents the radiation loss at the outer neck end. At low
frequencies,
35
2
00 00
1
2
2(2)
1
22
aN
cck
Jkr
R
rkr
ρρ
π
π
⎡⎤
=− ≈
⎢⎥
⎣⎦
. (2-9)
It is notable that Eq. (2-8) only accounts for the linear resistance when an incident wave with low
sound pressure level (SPL) excites the resonator. When the incident wave has a high SPL, an
additional nonlinear resistance term is added into Eq. (2-8), which is proportional to the particle
velocity within the neck of the resonator. Nonlinear behavior of the orifice due to a high SPL is
reviewed in Appendix B. Furthermore, notice that the assumptions for the reactance and
resistance in Eqs. (2-1) and (2-8), respectively, are different. Other advanced impedance model is
presented in Appendix B to alleviate these conflicts.
The acoustic impedance of the Helmholtz resonator is thus given by
(
)
22
0
00 00
HR HR HR
2
2
tt
ck c
ZRjX j
rj
ρ
ρρ
ω
π
πω
+∆
=+ ≈ + +
∀
(2-10)
for low frequencies, where
HR
R
and
HR
X
are the acoustic resistance and reactance of the
resonator, respectively. The resonant frequency of the resonator is the frequency at which
HR
X
vanishes,
()
2
0
0
2
c
r
f
tt
π
π
=
+
∆∀
. (2-11)
At the resonant frequency, the pressure amplification, defined as the ratio of amplitude between
the cavity pressure and the harmonic incident pressure, reaches a peak. It is clear that the
geometry of the Helmholtz resonator determines the resonant frequency. The resonant frequency
is fixed when the geometry of the resonator fixed.
*
*
No grazing flow is involved.
36
Electromechanical Helmholtz Resonator
As discussed above, the resonant frequency of a conventional Helmholtz resonator is fixed
when the device geometry is fixed. Since Helmholtz resonators work effectively near their
resonant frequency, their application to noise suppression is thus limited (Kinsler 2000). Recall
that a tunable resonant frequency is one of the desired traits of the modern Helmholtz resonator-
like acoustic liner (Chapter 1). The electromechanical Helmholtz resonator (EMHR) developed
in this dissertation is capable of tuning the acoustic impedance as well as the resonant frequency
of the resonator. As shown in Figure 2-3, an EMHR is a Helmholtz resonator, in which the rigid
backplate is replaced by, for example, a piezoelectric composite one that is attached to a passive
electrical shunt network (Sheplak et al. 2004). The acoustic impedance and the resonant
frequency of the resonator are adjusted and additional degrees of freedom are added via
electromechanical coupling of the piezoelectric composite diaphragm to a passive electrical
shunt network. Specifically, in comparison to a conventional SDOF solid-walled Helmholtz
resonator, the EMHR has two or three DOF with a variety of resistive, capacitive, and inductive
shunts.
Lumped Element Modeling
Similar to the conventional HR, at low frequencies, the dimensions of the EMHR, shown
in Figure 2-3, are much smaller than the wavelength of interest. The device components are thus
lumped into idealized discrete circuit elements (Rossi 1988). In particular, the EMHR is lumped
into two parts. The first part is similar to a conventional Helmholtz resonator described in the
previous section, where the neck of the resonator possesses both dissipative and inertial
components, and the cavity is modeled as a compliance. The second part includes the
compliance, damping loss and mass of the piezoelectric backplate and the coupling between the
acoustical and electrical energy domains.
37
Figure 2-4 shows the equivalent circuit representation for the EMHR, where the lumped
parameters are defined as follows: P and P
′
represent the incident and diaphragm acoustic
pressures, respectively. Q and Q
′
are incident and diaphragm volumetric flow rates,
respectively. In the notation below, the first subscript denotes the domain (e.g., “a” for acoustic),
and the second subscript describes the element (e.g., “D” for diaphragm).
aN
R
and
aN
M
are the
acoustic resistance and acoustic mass of the fluid in the neck, respectively.
aC
C is the acoustic
compliance of the cavity, while
aD
C and
aD
M
represent the short-circuit acoustic compliance
and mass of the piezoelectric compliant diaphragm, respectively.
aDrad
M is the acoustic
radiation mass of the diaphragm.
aD
R
is the acoustic resistance which includes the acoustic
radiation resistance and other structural damping losses. Finally,
eB
C is the blocked electrical
capacitance of the piezoelectric diaphragm (i.e., when the volumetric flow rate is zero),
φ
is the
impedance transformation factor, and
eL
Z
is the electrical load impedance across the
piezoelectric backplate. Analytical equations to estimate these parameters are listed in Table 2-
2. The parameter deduction for the backplate is presented in Appendix C.
As shown in Figure 2-4, a lossless transformer converts energy between the acoustical and
electrical domains and obeys the following transformer relations (Liu et al. 2003, 2007)
i
Q
φ
′
=
, (2-12)
and
PV
φ
′
=
. (2-13)
38
Furthermore, the transformer “transforms” the blocked electrical capacitance,
eB
C , to an
equivalent acoustic impedance,
aEB
C , and the electrical impedance,
eL
Z
, to an acoustic
impedance,
aL
Z
, by
2
eB
aEB
C
C
φ
= , (2-14)
and
2
aL eL
Z
Z
φ
= . (2-15)
Accordingly, a simplified equivalent circuit representation is obtained by converting the
electrical impedance to acoustic impedance as shown in Figure 2-5. The input acoustic
impedance (
aIN
Z
PQ= ) is then given by (Liu et al. 2003, 2007)
()
()
11
1
11
1
aL
aDrad aD aDrad
aC aD aEB aL
aIN aN aN
aL
aDrad aD aDrad
aC aD aEB aL
Z
RsMM
sC sC sC Z
ZRsM
Z
RsMM
sC sC sC Z
⎛⎞
++ + +
⎜⎟
+
⎝⎠
=+
++ + + +
+
, (2-16)
where sj
ω
= . Eq. (2-16) clearly demonstrates that modifying the shunt network
eL
Z
tunes the
acoustic impedance of the EMHR. When the shunt loads are capacitive or resistive, the EMHR
is analogous to a 2DOF system with two resonant frequencies, as shown in Figure 2-6a and 6b,
respectively. The EMHR with inductive loads in Figure 2-6c has 3DOF, and thus has three
resonant frequencies. As indicated by Eq.(2-16), the resonant frequencies of the EMHR are
tuned via adjusting the shunt loads. The tuning behavior of the EMHR with various electrical
loads is analyzed below. In particular, the tuning mechanism of the EMHR with capacitive loads
is described in detail. The tuning of the EMHR with resistive and inductive loads is analyzed in
an approximate manner because the non-capacitive tuning is not mathematically tractable. In
this case, it is more convenient to demonstrate the tuning of the EMHR numerically.
39
Tuning Behavior Analysis
Capacitive tuning of the EMHR
The EMHR with capacitive shunts is analogous to a 2DOF system with two resonant
frequencies, as shown in Figure 2-6a. Referring to Figure 2-6a, the impedance of loop 1 is
{}
1
1
1
11
11 1
1
1
LaN aN
aC
aN
aN aN aC
aC
aN aC
aN
L
wL
LL
wL L L
ZRjM
C
M
Rj MC
C
MC
R
f
f
j
Zff
j
ω
ω
ω
ω
ε
ε
⎛⎞
=+ −
⎜⎟
⎝⎠
⎛⎞
=+ −
⎜⎟
⎜⎟
⎝⎠
⎧⎫
⎛⎞
⎪⎪
=+−
⎨⎬
⎜⎟
⎪⎪
⎝⎠
⎩⎭
=∆+Ω
, (2-17)
where the quantity
1
1
2
L
aN aC
f
M
C
π
= (2-18)
is the resonant frequency of loop 1. The first term in Eq. (2-17)
1LaN aNaC
R
MC∆= (2-19)
is the dissipation factor, which is the ratio of the power dissipated to the power stored (Fischer
1955). The quantity
111LLL
f
fff
Ω
=− (2-20)
is the tuning factor which measures the deviation of the operating frequency of loop 1 from its
resonant frequency. Finally, the quantity
1wL aN aC
M
C
ε
= (2-21)
is a weighting factor by which the magnitude of the impedance of a different system which has
the same dissipation factor and the resonant frequency differs from each other.
40
Similarly, the impedance of loop 2 is
()
2
22 22 2
22
aD
L
LwL wLL L
wL L
R
ff
Zj j
Zff
εε
⎧⎫
⎡⎤
⎪⎪
=+−=∆+Ω
⎨⎬
⎢⎥
⎪⎪
⎣⎦
⎩⎭
, (2-22)
where the definitions of the quantities
2L
f
,
2wL
ε
,
2L
∆
and
2L
Ω
are analogous to their counterparts
in loop 1
()
2
2
1
2
L
aD aDrad
f
M
MC
π
=
+
, (2-23)
()
22LaD aDaDrad
R
MM C∆= + , (2-24)
222LLL
f
fff
Ω
=−, (2-25)
and
()
22wL aD aDrad
M
MC
ε
=+ , (2-26)
where
(
)
()( )
2
aEB aL aD aC
aB aL aD aB aL aD aC
CCCC
C
CCC CCCC
+
=
++++
. (2-27)
Next, for the coupled system
11 2
1
aC
PZQ Q
jC
ω
=+
, (2-28)
and
22 1
1
0
aC
Z
QQ
jC
ω
=+ , (2-29)
which leads to
2
1
12
11
aIN L
LaC
P
ZZ
QZjC
ω
⎛⎞
== −
⎜⎟
⎝⎠
. (2-30)
41
From Eqs. (2-17), (2-22) and (2-30), one has
2
11
11222
11 1
aIN
LL
wL wL wL aC L L
Z
j
ZjCj
εε ω
⎛⎞
=∆ + Ω −
⎜⎟
∆
+Ω
⎝⎠
. (2-31)
If the system is undamped or weakly damped,
12
0
LL
∆
=∆ = , (2-32)
Eq. (2-31) can be rewritten as
2
2
112
1
2
112 21
2
2
111 1
aIN
LLL
wL wL wL aC L L
L
L
Z
fff
f
jj
ZZZjCjfff
f
f
j
f
f
κ
ω
⎛⎞
⎛⎞
=Ω− = − +
⎜⎟
⎜⎟
Ω
⎛⎞
⎝⎠
⎝⎠
−
⎜⎟
⎝⎠
, (2-33)
where
2
212
aC aC
CCC
κκκ
== is the coupling factor of the system, and
1
κ
and
2
κ
are the
coupling coefficients which define the ratio of the oscillating energy stored in the coupling
elements to that stored in the total capacitance for each loop. They are given by (Fischer 1955)
(
)
()
2
1
1
1
2
11
1
aC
aC
Qdt C
C
C
Qdt C
κ
=
==
∫
∫
(2-34)
and
(
)
()
2
2
2
2
2
22
aC
aC
Qdt C
C
C
Qdt C
κ
==
∫
∫
, (2-35)
where
i
Q is volume velocity, as shown in Figure 2-6. At the resonant frequencies, Eq. (2-33)
equals zero, and the solution for the resonant frequencies is
()()()
()
2
2
22 22 2
12 12 12
2
1,2
41
2
LL LL LL
ff ff ff
f
κ
+± + −−
= . (2-36)
Furthermore, Eqs. (2-18) and (2-23) yield
42
()
()()
()
()()
()
22
12
2
2
2
2
22
11 1
4
11 1 1
4
11
1
4
1
LL
aN aC aD aDrad
aB aL aD
aN aD aDrad aC aD aDrad aB aL aD
aB aL aD
aN aC aD aDrad aB aL aD
HR D
ff
MC M M C
CCC
MMM C MM CCC
CCC
MC M M C C C
ff
π
π
α
π
α
+
⎧⎫
⎪⎪
=+
⎨⎬
+
⎪⎪
⎩⎭
⎧⎫
⎡⎤
++
⎪⎪
=+ +
⎨⎬
⎢⎥
+++
⎪⎪
⎣⎦
⎩⎭
⎧⎫
++
⎪⎪
=++
⎨⎬
++
⎪⎪
⎩⎭
=+ +
, (2-37)
where
aN
aD aDrad
M
MM
α
=
+
(2-38)
is the mass ratio between the neck and piezoelectric backplate. In addition,
11
2
HR
aN aC
f
M
C
π
=
(2-39)
is the resonant frequency of the Helmholtz resonator with solid wall instead of a piezoelectric
backplate, and
()()
1
2
aEB aL aD
D
aD aDrad aB aL aD
CCC
f
M
MCCC
π
++
=
++
(2-40)
is the resonant frequency of the piezoelectric backplate. Similarly,
(
)
22 2 2
12
1
L
LHRD
f
fff
α
−
=− −
. (2-41)
When the system is weakly coupled,
0
κ
→
, substitution of equations (2-37) and (2-41) into
(2-36) results in
() ()
2
22 22
1,2
22
11
2
HR
HR D HR D
D
HR
f
ff ff
f
f
f
αα
α
⎡⎤
++±−−
⎧
⎪
⎣⎦
==
⎨
+
⎪
⎩
. (2-42)
43
Furthermore, if the mass ratio between the neck and the piezoelectric backplate is very small as
well,
0
α
→ ,
2
22 22
1,2
2
HR D HR D
H
R
D
ff ff
f
f
f
⎡⎤
+± −
⎧
⎣⎦
==
⎨
⎩
. (2-43)
In other words, the resonant frequencies of a lightly damped EMHR with capacitive shunts
possessing weak coupling are approximately the resonant frequency of the solid-walled
Helmholtz resonator and the piezoelectric backplate. As indicated by Eq.(2-40) , the resonant
frequency of the piezoelectric backplate is adjusted with the change of capacitive loads. For
open-circuited EMHR , 0
aL
C = or 1
aL aL
ZsC
=
→∞,
[]
()
open
1
2
aEB aD
D
aD aDrad aEB aD
CC
f
M
MCC
π
+
=
+
. (2-44)
While for short-circuited EMHR,
aL
C
=
∞ or 10
aL aL
ZsC
=
→ ,
[]
()
short
11
2
D
aD aDrad aD
f
M
MC
π
=
+
. (2-45)
Moreover, for the EMHR with a capacitive load,
2
aL eL
CC
φ
= ,
[]
()()
capacitive
1
2
aEB aL aD
D
aD aDrad aEB aL aD
CCC
f
M
MCCC
π
++
=
++
. (2-46)
Clearly,
[
]
[
]
[
]
short capacitive open
DD D
ff f<<. (2-47)
or by the use of Eq. (2-43)
[
]
[
]
[
]
22 2
short capacitive open
ff f<<. (2-48)
44
Equation (2-48) demonstrates how the tunable electromechanical Helmholtz resonator works
with different capacitive shunts. The short and open loads define the limits of tuning using
capacitive loads. The second resonant frequency shifts toward the short case when the
capacitance increases.
Resistive tuning of the EMHR
As shown in Figure 2-6b, the EMHR with a resistive load is a 2DOF system. The resistive
tuning is not mathematically tractable. Thus, an approximate analysis is given as follows. When
an EMHR attached to a resistive load, the effective loads converted from the electric domain, as
shown in Figure 2-6b, is given by
2
1
1
eL
aE
eB eL
R
Z
j
CR
φω
=
+
. (2-49)
When 0
eL
R → , the EMHR is short-circuited, while the EMHR is open-circuited as
eL
R →∞.
Furthermore, in the frequency range (i.e., for small
eL
R
)
1
2
eB eL
f
CR
π
, (2-50)
Eq. (2-49) can be further approximated as
()
2
1
1
aE eL eB eL
Z
RjCR
ω
φ
=− , (2-51)
which indicates that the resistive loads effectively reduce the acoustic mass of the piezoelectric
backplate by
2
eB eL
jCR
ω
φ
− (i.e.,
2
eff
aD aD eB eL
MMjCR
ω
φ
=−
, where
eff
aD
M
is the effective
acoustic mass of the piezoelectric backplate). Thus, the resonant frequencies of the weakly
coupled EMHR are given by
2
22 22
1,2
2
HR D HR D
H
R
D
ff ff
f
f
f
⎡⎤
+± −
⎧
⎣⎦
≈=
⎨
⎩
, (2-52)
45
where
H
R
f
is given by Eq. (2-39), and the resonant frequency of the piezoelectric backplate
D
f
†
is
()
2
11
2
D
aD aDrad eB eL aD
f
M
MCRC
π
φ
=
+−
. (2-53)
In comparison with the short-circuited EMHR,
D
f
increases due to the decrease of the acoustic
mass. Moreover,
[
]
short
DD
ff≥
in Eq. (2-45). With increasing resistive loads,
D
f
moves further
away from the short-circuit resonant frequency
[
]
short
D
f toward the open-circuit resonant
frequency
[
]
open
D
f . Note that Eq. (2-53) doesn’t give that
D
f
goes to
[
]
open
D
f for the open-
circuited EMHR (
eL
R →∞). When
eL
R
is large, the assumption in Eq. (2-50) breaks down, and
Eq. (2-53) fails to predict the resistive tuning. In such a case, it is more convenient to
demonstrate the tuning of the EMHR numerically.
Inductive tuning of the EMHR
As shown in Figure 2-6c, The EMHR with an inductive load is a 3DOF system. To
simplify the problem without loss of generality, it is assumed that the EMHR with inductive
shunts has negligible damping. The effective impedance of loop 3 is given by
(
)
()
22
L3
22
eB eL
eB eL
jC j M
Z
jC j M
φω ωφ
φω ωφ
⎡⎤
⎣⎦
=
⎡⎤
+
⎣⎦
(2-54)
which can be further approximated as
2
L3 eL
Z
jM
ωφ
(2-55)
in the frequency range
†
Strictly,
D
f should be called the natural frequency.
46
1
2
E
eB eL
f
f
CM
π
=
, (2-56)
where
E
f
is the resonant frequency of loop 3. Physically, loop 3 effectively adds a mass
2
eL
M
φ
to loop 2. Similarly, the resonant frequencies of the weakly coupled EMHR are given by
2
22 22
1,2
2
HR D HR D
H
R
D
ff ff
f
f
f
⎡⎤
+± −
⎧
⎣⎦
≈=
⎨
⎩
, (2-57)
where
H
R
f
is given by Eq. (2-39), while
D
f
is
()
2
11
2
D
aD aDrad eL aD
f
M
MMC
π
φ
=
++
. (2-58)
Clearly,
[
]
short
DD
ff≤ in Eq. (2-45). With increasing inductive loads,
D
f
moves further away
from the short-circuit resonant frequency
[
]
short
D
f .
The aforementioned analysis focuses on illustration of the tuning behavior of the EMHR
with weak coupling between the solid-walled Helmholtz resonator and the piezoelectric
backplate. However, the first resonant frequency of the system can be tuned in-situ as well.
More analysis on the tuning behavior is presented in Chapter 6.
47
Table 2-1. End corrections for orifices or slits.
Items
Expression
Orifice in baffle wall (flanged)
(Blackstock 2000; Kinsler 2000)
0.85tr
∆
=
where
t
∆
is the end correction of the orifice, r is the radius of
the orifice
Orifice in free space (unflanged)
(Blackstock 2001; Kinsler 2000)
0.6tr
∆
≈
Orifice in tube
(Mechel 2002)
2
2
0.0445728 0.728326 0.177078
0.0339531 0.00810471 0.00100762
tr x x
yy xy
∆≈− − −
++ −
where
(
)
10
2log
x
rb=
,
(
)
10
logyb
λ
=
,
λ
is the wavelength and
b is the radius of the tube
Orifice array, square arrangement
(Mechel 2002)
(
)
32
0.79 1 1.47 0.47tr
σσ
∆= − +
where
2b is the hole separation distance, and
22
=4rb
σπ
is
the porosity
Orifice array, hexagonal
arrangement
(Mechel 2002)
2
2
0.0445728 0.728326 0.177078
0.0339531 0.00810471 0.00100762
tr x x
yy xy
∆≈− − −
++ −
where
(
)
10
2log
x
rb= ,
(
)
10
logyb
λ
= and 2b is the length of the
hexagon
Orifice array
(Melling 1973)
()
(
)
83tr
π
σ
∆= Ψ
where
(
)
Ψ is the Fock function
() ()
0
n
n
n
a
σ
σ
∞
=
Ψ=
∑
with the first eight coefficients are
given by
01
23
45
67
8
1 1.40925
0 0.33818
0 0.06793
0.02287 0.03015
0.01641
aa
aa
aa
aa
a
=
=−
==
==
=− =
=−
48
Table 2-2. Parameter estimation for lumped element modeling of the EMHR.
Acoustic
impedance
Description
aN
R
00
00
1
222
22
2(2)
21
2
aN
c
Jkrt
R
rr r r kr
µρ ω µρ ω
ρ
πππ
⎡⎤
=++−
⎢⎥
⎣⎦
where
1
J is the Bessel function of the first kind
Neck
(Ingard 1953)
aN
M
(
)
0
2
aN
tt
M
r
ρ
π
+
∆
=
, where
0.85 1 1.25 0.85
r
tr r
R
⎛⎞
∆= − +
⎜⎟
⎝⎠
Cavity
(Blackstock 2000)
aC
C
()
2
00
cot
aC
R
C
ckL
π
ρω
=
aD
C
()
2
0
0
2
R
aD
V
CrwrdrP
π
=
=
∫
where
()
wr is the
transverse displacement of the piezoelectric backplate, P is
the applied pressure and V is applied voltage on the
backplate
aD
M
()
()
2
2
2
0
0
2
()
R
aD A
V
M
wr rdr
π
ρ
=
=
∆∀
∫
where
A
ρ
is the
surface density of the piezoelectric backplate
aDrad
M
0
2
2
8
3
aDrad
M
R
ρ
π
=
aD
R
2
aD aDrad
aD
aD
MM
R
C
ζ
+
=
where
ζ
is the experimentally
determined damping factor
φ
()
2
0
0
2
R
aaD aD
P
dC rwrdrV C
φπ
=
=− =−
∫
Piezoelectric backplate
(Prasad et al. 2006)
eB
C
() ()
2
22
01
11
r
eB eF EM EM
p
R
CC
h
εεπ
κκ
=−= − where
r
ε
is
the relative dielectric constant of the piezoelectric
material,
0
ε
is the permittivity of free space, and
22
EM a eF aD
dCC
κ
= is the electroacoustic coupling factor
49
2r
t
L
2
R
Cavity
Orifice
Figure 2-1. A solid-walled HR under excitation of the incident wave. The scattered wave is
removed for clarity.
aN
R
aN
M
aC
C
P
I
N
Z
Q
Figure 2-2. Equivalent circuit representation of a solid-walled Helmholtz resonator.
50
2r
t
L
2
R
s
h
p
h
1
R
2
R
Piezoelectric
backplate
Cavity
Orifice
eL
Z
Figure 2-3. Schematic illustration of an EMHR.
aN
R
aN
M
aC
C
aD
C
aD
R
aDrad aD
M
M
+
:1
φ
P
P
′
eB
C
eL
Z
i
+
−
V
aI N
Z
Q
Q
′
Figure 2-4. Equivalent circuit representation of the EMHR.
51
aN
R
aN
M
aC
C
aD
C
aD
R
aDrad aD
M
M
+
P
aEB
C
aI N
Z
aL
Z
Q
Figure 2-5. Simplified equivalent circuit representation for a EMHR.
aN
R
aN
M
aC
C
aD
C
aD
R
aDrad aD
M
M
+
P
2
eB
C
φ
2
eL
C
φ
aI N
Z
1
Q
2
Q
2
eB
C
φ
2
eL
R
φ
2
eB
C
φ
2
eL
M
φ
2
aL eL
Z
jC
φω
=
2
aL eL
Z
R
φ
=
2
aL eL
Z
jM
ωφ
=
Figure 2-6. EMHR with passive electrical loads is analogous to a 2DOF system for (a)
capacitive and (b) resistive loads and a 3DOF system with an (c) inductive load.
52
CHAPTER 3
MODELING AN EMHR USING THE TRANSFER MATRIX METHOD
The purpose of this chapter is to develop an analytical model of an EMHR using the
transfer matrix method and to compare the transfer matrix of the EMHR with the existing LEM.
The chapter is organized as follows. The first section briefly reviews the application of the
transfer matrix method in acoustics. The second section introduces some essential characteristics
of the transfer matrix. In the second section, basic types of transfer matrix representation are
introduced as well. The development of the transfer matrix for different types of Helmholtz
resonators are presented in the third section. Finally, a comparison between LEM and TM is
provided.
Introduction
The transfer matrix method (also called the transmission matrix or the four-pole parameter
representation) has been widely used in electrical and structural engineering (Munjal 1987). In
both fields, the analysis of complicated systems has been greatly simplified by the use of this
method. Complex circuits or structures can be simply regarded as “black boxes” with one input
port and one output port, as shown in Figure 3-1. Matrix algebra can then be applied to the
general treatment of such networks. The interconnection of parallel, series-parallel, and parallel-
series network combinations can then be handled by simple linear addition or multiplication of
the transfer matrix (Van Valkenburg 1964).
In 1930, Stewart and Lindsay first applied the transfer matrix method to acoustic filter
design, although the name of the transfer matrix was not used in their book (Stewart and
Lindsay 1930). Later, Peterson et al. (1950) presented a dynamic analysis of the cochlea in the
human ear using the transfer matrix method. In their work, four-pole parameters were deduced
based on the equivalent T-network representation of the cochlea.
53
After this time, the transfer matrix method was more and more commonly used for
modeling a variety of acoustic problems such as Muffler/silencer design with/without mean flow
(Igarashi and Toyama 1958; Alfredson and Davies 1971),wave propagation in a duct with a
simple and/or complicated structure (Miles 1981; Munjal and Prasad 1986), and acoustic
absorbers (Parrott and Jones 1995). Alfredson et al. (1971) presented four-parameter models of a
number of silencer components, such as the sudden area change and an extended outlet and inlet
based on one-dimensional linear theory with mean flow. It was hypothesized that any possible
pressure discontinuity across the area change was caused by an entropy variation. Comparisons
between measurements and numerical predictions for these components were made, and it was
found that one-dimensional models provided acceptable accuracy for silencer design.
Miles (1981) modeled a variable area duct, such as a nozzle, as a series of exponential
tubes with each tube being short enough such that the mean flow could be regarded as uniform
along it. The transfer matrix for each sub-tube was derived by solving a constant-coefficient
acoustic state variable differential equation, and the overall transfer matrix was then obtained by
multiplying the individual duct transfer matrices.
Munjal et al. (1986) developed the transfer matrix method for a duct with an axial
temperature gradient and mean flow , and a similar analysis was performed by Peat (1988). Both
analyses were limited to ducts with small axial mean temperature gradients. Sujith (1996)
extended the transfer matrix method to a uniform duct with an arbitrarily large axial mean
temperature gradient. His analysis neglected mean flow and is thus only valid for Mach numbers
up to approximately 0.1.
Lung and Doige (1983), Munjal (1987) and Benade (1988) derived a transfer matrix
model for the case of spherical wave propagation in conical pipes without mean flow. Based on
54
a general differential equation for the propagation of sound in a variable area duct with low
Mach number flow, Easwaran et al. (1991) presented an expression for the four-pole parameter
model of the duct system. The shape of the duct was set to hyperbolic or parabolic. The conical,
exponential, catenoidal, sine and cosine ducts were shown to be special cases of a hyperbolic
duct. It was shown that the mean flow can cause a distinct change in the individual four
parameters of the matrix and a transmission loss in a variable-area duct.
Parrot et al. (Parrott and Jones 1995) used the transfer matrix method to predict surface
impedance and sound absorption for parallel-element acoustic liners. The transfer matrix,
derived from Zwikker and Kosten’s theory for propagation of sound in small tubes (Zwikker and
Kosten 1949), provided excellent agreement between predicted and measured normal-incidence
impedance for ceramic-honeycomb-structure distributed systems.
More recently, Song and Bolton (2000) used a transfer-matrix approach to estimate the
acoustic properties of rigid porous materials. In their paper, the modified two-microphone
method was used to estimate the
22× transfer matrix of porous material samples, and the result
was used to determine acoustic properties of the material, such as the characteristic impedance
and complex wave number.
Generally, the application of the transfer matrix method follows one of two paths. The
first uses a one-dimensional or quasi one-dimensional approach, while the second is based on
multi-modal theory. The former is adopted when the propagation of higher-order modes is
minimal and/or neglected. The latter is more complicated and often combines the boundary
element method (BEM) or finite element method (FEM) in acoustic applications where higher-
order modes are not neglected (Wang et al. 1993; Ji et al. 1995).
55
Characteristics of the Transfer Matrix
In the two-port network of Figure 3-1, four variables are identified: two effort variables
and two flow vector variables. A transfer matrix that relates the effort and flow state variables is
of the form (Van Valkenburg 1964)
111122
121222
VTTV
ITTI
⎡
⎤⎡ ⎤⎡ ⎤
=
⎢
⎥⎢ ⎥⎢ ⎥
−
⎣
⎦⎣ ⎦⎣ ⎦
, (3-1)
where
1
V and
1
I are the input effort and flow vector variables, respectively;
2
V and
2
I are the
output effort and flow vector variables, respectively; and
T
ij
is a matrix entry in the transfer
matrix of the two-port network. The negative sign before the output current vector variables
arises from the convention which defines the power flow as positive out of the network. In
contrast, the two-port network in acoustic applications, shown in Figure 3-2, defines the output
flow vector as positive when it is leaving the two-port network. In such cases, the transfer
matrix is given by
‡
111122
121222
TT
TT
PP
QQ
⎡
⎤⎡ ⎤⎡⎤
=
⎢
⎥⎢ ⎥⎢⎥
⎣
⎦⎣ ⎦⎣⎦
. (3-2)
Choice of State Variables
The definitions of the general state variables in different domains are listed in Table 3-1.
In acoustic applications, the effort and flow variables are acoustic pressure,
P , and volume
velocity,
Q. The volume velocity is chosen as a flow variable because this quantity will remain
continuous, even discontinuities present in a duct. However, particle velocity
U is often
‡
The convention
it
e
ω
is adopted in the context of this dissertation except for specific indication. Furthermore,
Instead of
p
′
and u
′
, the phasor of the acoustic pressure P and the phasor of the acoustic velocity U are adopted
in this chapter.
56
adopted as a flow variable. In such a case, modifications should be made for the transfer matrix
to take the area change of the acoustic network into account.
For the system shown in Figure 3-2, the
PQ
−
formulation is given by
12
11 12
12
21 22
QQ
QQ
PP
TT
QQ
TT
⎡⎤
⎡
⎤⎡⎤
=
⎢⎥
⎢
⎥⎢⎥
⎣
⎦⎣⎦
⎣⎦
, (3-3)
where
()
1, 2; 1, 2
Q
ij
Ti j==
is an entry of the PQ
−
transfer matrix, where the superscript Q
indicates that the flow variable is the volume velocity. Clearly,
2
2
11 1
0
1
Q
Q
P
TP
=
= , (3-4)
2
2
12 1
0
1
Q
P
Q
TP
=
=
, (3-5)
2
2
21 1
0
1
Q
Q
P
TQ
=
= , (3-6)
and
2
2
22 1
0
1
Q
P
Q
TQ
=
= . (3-7)
Physically,
11
1
Q
T is an open-circuit effort (pressure) gain,
12
1
Q
T is a short-circuit transfer
admittance,
21
1
Q
T is an open-circuit transfer impedance, and
22
1
Q
T is a short-circuit current
(volume velocity) gain.
The
PU−
formulation for the system shown in Figure 3-2 is
12
11 12
12
21 22
UU
UU
PP
TT
UU
TT
⎡⎤
⎡
⎤⎡⎤
=
⎢⎥
⎢
⎥⎢⎥
⎣
⎦⎣⎦
⎣⎦
. (3-8)
57
If the temperature remains constant in the acoustic network, one can also choose
00
cU
ρ
as the
flow variable, where
00
c
ρ
is the characteristic acoustic impedance at this temperature (Ingard
1999). The transfer matrix of an acoustic circuit when adopting acoustic pressure P and the
flow variable
00
cU
ρ
as the two state variables is
00 00
00 00
12
11 12
00 1 00 2
21 22
cU cU
cU cU
PP
TT
cU cU
TT
ρρ
ρρ
ρρ
⎡⎤
⎡⎤ ⎡⎤
=
⎢⎥
⎢⎥ ⎢⎥
⎣⎦ ⎣⎦
⎣⎦
. (3-9)
Note that all entries of the
00
PcU
ρ
− transfer matrix are dimensionless. For waves propagating
within a duct with a constant cross-section, the relationship between acoustic particle velocity
U
and volume velocity Q is
QUS
=
, (3-10)
where
S is the projected cross-sectional area perpendicular to the particle velocity direction.
Hence, from Eqs.(3-3)-(3-10), one has
21 22 2
11 11 12 12 2 21 22
11
QQ
UQUQ U U
TTS
TTTTST T
SS
== = =, (3-11)
and
00 00 00 00
12 2 21 22 2
11 11 12 21 22
11
QQQ
cU cU cU cU
Q
TS cT TS
TTT T T
cS S
ρρρ ρ
ρ
ρ
=== =. (3-12)
All of these three formulations can be used in an acoustic analysis. However, there are
some limitations for the application of these transfer matrices. For instance, Eq. (3-9) provides a
transfer matrix with dimensionless elements. The acoustic analysis is therefore simplified with
these dimensionless matrix elements (Ingard 1999). However, Eq. (3-9) cannot be used for a
duct with a continuously changing cross-sectional area, such as the Webster horn.
58
Properties of the Transfer Matrix Related to Special Acoustic Systems
Knowledge of properties of the transfer matrix facilitates the derivation of the transfer
matrix for some special acoustic systems (i.e., reciprocal systems). For a reciprocal system, the
transfer matrix can be determined when three independent entries are known.
Reciprocal system
A reciprocal system refers to one in which the system response is the same when the
position of the input and output terminal are interchanged (Van Valkenburg 1964; Pierce 1989).
Mathematically, a reciprocal acoustic system satisfies
21
12
21
00
PP
PP
QQ
=
=
= , (3-13)
which consequently requires that the determinant of the PQ
−
formulation of an acoustic two-
port network equals unity,
11 22 12 21
1
QQ QQ
TT TT
−
= . (3-14)
For the
00
PcU
ρ
− formulation, the determinant of the transfer matrix should be
00 00 00 00
11 22 12 21 1 2
cU cU cU cU
TT TT SS
ρρ ρρ
−=, (3-15)
where
1
S and
2
S are the projected cross-sectional areas of port 1 and port 2, respectively.
Clearly, Eqs. (3-14) and (3-15) reduce the number of independent transfer matrix parameters to
three. From a practical standpoint, a two-port network that is reciprocal will thus simplify the
analysis and permit determination of the transfer matrix when three independent elements are
known.
In some cases, an acoustic system, i.e. an acoustic duct system, may not be reciprocal when
mean flow is present or area discontinuities occur. The presence of mean flow in an acoustic
duct system causes the upstream phase speed to be different from the downstream phase speed,
59
and thus breaks down the reciprocity of the system (Munjal 1987). At an area discontinuity,
evanescent higher order modes are excited, which may result in the violation of the reciprocity of
the system (Ingard 1999).
Conservative system
If a system is conservative, the net power flux into the system is equal to zero.
Mathematically, for a conservative acoustic two-port system, it is required that
11
Q
T and
22
Q
T be
real, and
12
Q
T and
21
Q
T be purely imaginary (Van Valkenburg 1964).
Basic Types of Elements of Acoustic Transfer Matrix Representation
Dividing a complicated acoustic system into several subsystems for which the transfer
matrix model can be easily developed facilitates the derivation of the transfer matrix
representation. The overall transfer matrix of the system can then obtained by addition or
multiplication of the subsystem transfer matrices. There are three basic types of elements in a
transfer matrix representation for an acoustic network.
§
They are the distributed element, the
series lumped element, and the parallel lumped element (Munjal 1987). Their equivalent circuit
representations are shown in Figure 3-3.
For the distributed element shown in Figure 3-3a, a simple corresponding case is plane
wave propagation along a lossless duct in the x direction. The cross-sectional area of the duct is
S. The governing wave propagation equations are given by (Blackstock 2000)
11
1
jkx jkx
PAe Be
−+
=+, (3-16)
11
1
00
jkx jkx
AS BS
US e e
ZZ
−+
=−, (3-17)
22
2
jkx jkx
PAe Be
−+
=+, (3-18)
§
An exception is the transfer matrix representation for a flare tube.
60
22
2
00
jkx jkx
AS BS
US e e
ZZ
−+
=−, (3-19)
where
A
and
B
are coefficients, and
0
Z
is the characteristic specific acoustic impedance.
Letting
2
0x = and
21
lx x=−, as shown in Figure 3-3a, one obtains
2
PAB
=
+ , (3-20)
2
00
AS BS
US
Z
Z
=−, (3-21)
1
jkl jkl
PAe Be
−
=+ , (3-22)
1
00
jkl jkl
AS BS
US e e
ZZ
−
=− . (3-23)
Equations (3-20) - (3-23) result in
(
)
()
()
0
12
0
12
cos sin
sin cos
kl j Z S kl
PP
jklZS kl
QQ
⎡⎤
⎡⎤ ⎡⎤
=
⎢⎥
⎢⎥ ⎢⎥
⎣⎦ ⎣⎦
⎣⎦
, (3-24)
where
QuS= is the acoustic volume velocity. Note that
()() ( )()
2
00
cos sin sin 1kl j Z S kl j kl Z S−=
⎡⎤⎡ ⎤
⎣⎦⎣ ⎦
, (3-25)
the system is a reciprocal system. Moreover, the system is also a conservative system.
For a series lumped element shown in Figure 3-3b, one has
211
s
PPQZ
=
− , (3-26)
21
QQ
=
. (3-27)
In matrix form, one has
12
12
1
01
s
PP
Z
QQ
⎡
⎤⎡⎤
⎡⎤
=
⎢
⎥⎢⎥
⎢⎥
⎣⎦
⎣
⎦⎣⎦
, (3-28)
61
where
s
Z
is the in-line lumped complex impedance. An example of a series lumped element is
wave propagation across an area contraction, as shown in Figure 3-4. The relationships between
11
(, )PQ and
()
22
,PQ are given by Eq. (3-28), and
s
Z
is given by (Ingard 1999)
()
2
00
0
2
18
2
23
s
ckr
j
kr
Z
r
ρ
µρ ω
π
π
⎡
⎤
=++
⎢
⎥
⎢
⎥
⎣
⎦
, (3-29)
which accounts for the viscous loss and radiation impedance at the area discontinuity. More
details on the area-expansion are presented in Section 3. Clearly, such a system is not reciprocal
(
11 22 12 21
1
QQ QQ
TT TT−≠) due to the area discontinuity, and not conservative due to a portion of the
energy is dissipated at the area discontinuity.
For a parallel element as shown in Figure 3-3c, one can write the relationship between
input and output parameters directly as
21
PP
=
, (3-30)
21 1p
QPZQ
=
−+, (3-31)
or
12
12
10
11
p
PP
Z
QQ
⎡⎤
⎡
⎤⎡⎤
=
⎢⎥
⎢
⎥⎢⎥
⎣
⎦⎣⎦
⎣⎦
. (3-32)
An example of such a case is shown in Figure 3-5, where a Helmholtz resonator is used as a side-
branch to a uniform tube. The relationships between
11
(, )PQ and
(
)
22
,PQ are given by
12
12
10
11
HR
PP
QZ Q
⎡
⎤⎡ ⎤⎡⎤
=
⎢
⎥⎢ ⎥⎢⎥
⎣
⎦⎣ ⎦⎣⎦
, (3-33)
62
where
H
R
Z
is the complex acoustic impedance of the HR. Such a system is not reciprocal
(
11 22 12 21
1
QQ QQ
TT TT−≠) due to the area change at the junction and not conservative due to a portion
of the energy is dissipated by the HR.
Transfer Matrix Representation for EMHR
In this section, the transfer matrix for an EMHR mounted at the end of plane wave tube
(PWT) will be developed. As shown in Figure 3-6, the EMHR mounted at the end of the PWT is
comprised of the following four components: area contraction, area expansion, duct element and
piezoelectric backplate with shunt network. Hence, it is convenient to develop the transfer
matrix for each of these elements, and then multiply these matrices sequentially to obtain the
overall transfer matrix of the EMHR
Area Contraction
The area contraction is shown in Figure 3-7a. Due to abrupt change of the area, even with
a planar incident wave, evanescent higher order acoustic modes will be excited in the vicinity of
the discontinuity, thus the acoustic field near the discontinuity will have a transverse component.
Such effects can be taken into account in terms of plane wave variables by means of an
additional mass and resistance (Karal 1953; Eriksson 1980; T.Y.Lung and Doige 1983; Ingard
1999). Following Karal and Ingard, the relation between the acoustic variables for the area
contraction is given by
(
)
54 004444
PP c AUA
ρζ
=+ , (3-34)
00 5 5 00 4 4
cU A cU A
ρ
ρ
=
(3-35)
where
4
P and
5
P are the plane wave pressure component before and after the discontinuity. Eqs.
(3-34) and (3-35) represent continuity of the pressure and the volume velocity across the
discontinuity, respectively. The matrix representation for the area contraction is thus given by
63
544
00 5 4 5 00 4
1
0
PP
cU A A cU
ζ
ρρ
⎡⎤⎡ ⎤⎡⎤
=
⎢⎥⎢ ⎥⎢⎥
⎣⎦⎣ ⎦⎣⎦
, (3-36)
where
2
4
A
r
π
= is the cross-sectional area of the neck of the Helmholtz resonator,
2
5 tube
A
D= is
the cross-sectional area of the tube, and
4
ζ
is the normalized specific acoustic impedance given
by (Ingard, 1999a)
2
04
40
00 0
18
2
23
A
jkr
cc
ρω
ζµρω
ρ
ππ
⎛⎞
=++
⎜⎟
⎝⎠
. (3-37)
Thus, the transfer matrix for the area contraction is
4
45
1
0
AC
T
AA
ζ
⎡
⎤
=
⎢
⎥
⎣
⎦
. (3-38)
Area Expansion
Similarly, the relationship between the acoustic variables for the area expansion, shown in
Figure 3-7b, is given by (Ingard, 1999a)
(
)
32 003322
PP c AUA
ρζ
=+ , (3-39)
00 3 3 00 2 2
cU A cU A
ρ
ρ
=
, (3-40)
or in transfer matrix form,
(
)
32
233
00 3 00 2
23
1
0
PP
AA
cU cU
AA
ζ
ρρ
⎡⎤
⎡⎤ ⎡⎤
=
⎢⎥
⎢⎥ ⎢⎥
⎣⎦ ⎣⎦
⎣⎦
, (3-41)
where
2
3
A
r
π
= is the cross-sectional area of the neck of Helmholtz resonator,
2
2
A
R
π
= is the
cross-sectional area of the cavity, and
3
ζ
is the normalized specific acoustic impedance given by
2
03
30
00 0
18
2 1 1.25
23
A
jkr r
cc R
ρω
ζµρω
ρππ
⎛⎞
⎛⎞
=++−
⎜⎟
⎜⎟
⎝⎠
⎝⎠
. (3-42)
Thus, the transfer matrix for the area contraction is
64
(
)
233
23
1
0
AE
AA
T
AA
ζ
⎡
⎤
=
⎢
⎥
⎣
⎦
. (3-43)
Duct Element
Determination of the acoustic velocity and pressure fields due to sound wave propagation
in a long circular duct is a classic acoustics problem. Tijdeman (1974) reviewed this topic and
showed that the solution obtained by Zwikker and Kosten (1949) has the widest range of
validity. For plane-wave propagation in a duct with radius
R
, as shown in Figure 3-7c, the
pressure and axial velocity component are given by (Zwikker and Kosten 1949; Tijdeman 1974)
()
2
00
kx kx
c
PAeBe
ρ
γ
−Γ Γ
⎛⎞
=+
⎜⎟
⎝⎠
, (3-44)
and
(
)
()
()
32
0
0
32
0
1
kx kx
Jj s
j
Uc Ae Be
Jjs
η
γ
−Γ Γ
⎡⎤
Γ
⎢⎥
=−−
⎢⎥
⎣⎦
, (3-45)
where Γ is the complex propagation coefficient,
0
kc
ω
=
is the wave number,
rR
η
=
is the
dimensionless radius,
sR
ω
ρµ
= is the shear wave number or Stokes number,
γ
is the ratio
of specific heats, and
µ
is the dynamic viscosity coefficient of the air. The complex propagation
coefficient is given by
(
)
()
32
0
32
2
Jjs
n
Jjs
γ
Γ= , (3-46)
where
()
()
1
32
2
32
0
1
1
Jj s
n
Jj s
σ
γ
γ
σ
−
⎛⎞
−
⎜⎟
=+
⎜⎟
⎝⎠
, (3-47)
65
Pr
p
C
σµλ
==, Pr is Prandtl number and
(
)
i
J is the
th
i
order Bessel function of the first
kind.
From Eq. (3-45), the mean velocity over cross-sectional area of the duct is thus obtained by
()()
()
()
()
()
()
2
0
32
00
2
32
0
0
32
2
0
32
0
0
1
2
12
R
kx kx
R
kx kx
kx kx
UUrdr
R
cAe Be J j s
j
rdr
R
Jjs
Jjs
jc
Ae Be
Jjs
c
j
Ae Be
n
π
π
η
π
πγ
γ
γ
γ
−Γ Γ
−Γ Γ
−Γ Γ
=
⎧⎫
⎡⎤
−
Γ
⎪⎪
⎢⎥
=−
⎨⎬
⎢⎥
⎪⎪
⎣⎦
⎩⎭
Γ
=−
=−
Γ
∫
∫
, (3-48)
where the following identities of the Bessel function are used (Watson 1996)
() () ()
11
2
nn n
n
JzJz Jz
z
−+
+=
, (3-49)
() ()
01
0
a
zJ z dz aJ a=
∫
. (3-50)
Thus, from Eqs. (3-44) and (3-48), one can deduce the relationship between
()
22
,PU and
()
11
,PU
, shown in Figure 3-8, as
(
)
(
)
() ()
21
00 2 00 1
cosh sinh /
sinh cosh
PkLkLGP
cU G kL kL cU
ρρ
ΓΓ
⎡⎤
⎡⎤ ⎡⎤
=
⎢⎥
⎢⎥ ⎢⎥
ΓΓ
⎣⎦ ⎣⎦
⎣⎦
, (3-51)
where
Gj n
γ
=Γ and Γ and n are given by Eqs. (3-46) and (3-47). The same result can be
obtained for the relationship between
(
)
44
,PU and
(
)
33
,PU . Thus, the transfer matrix for the
duct element shown is given by
(
)
(
)
() ()
cosh sinh /
sinh cosh
DE
kL kL G
T
GkL kL
ΓΓ
⎡⎤
=
⎢⎥
ΓΓ
⎣⎦
. (3-52)
66
Note that such a system is reciprocal, but not conservative because a portion of the energy is
dissipated at duct. When the wave field in the duct is isentropic (
j
Γ
= and 1G = ), Eq. (3-51)
can be simplified to
(
)
(
)
() ()
21
00 2 00 1
cos sin
sin cos
PkLjkLP
cU j kL kL cU
ρρ
⎡⎤
⎡⎤ ⎡⎤
=
⎢⎥
⎢⎥ ⎢⎥
⎣⎦ ⎣⎦
⎣⎦
, (3-53)
or in the
p
Q
′
− formulation given by Eq. (3-24)
Clamped Piezoelectric Backplate with Shunt Loads
To develop the transfer matrix representation for the piezoelectric-backplate (or PZT-
backplate), it is assumed that the maximum dimension of the PZT-backplate is much less than
the bending wavelength of interest, and that only linear behavior of the PZT-backplate needs to
be considered. With these assumptions, an equivalent two-port network including a lossless
transformer can be developed for the PZT-backplate, as shown in Figure 3-8, where
1
P is the
pressure exerted on the PZT-backplate,
1
Q is the incident volume velocity, and V and
I
are the
voltage and current at the electrical port, respectively. Associated with the two-port network are
two impedances that are measurable properties of the system. One is the blocked electrical
impedance,
0
1
eB
Q
eB
V
Z
I
jC
ω
=
== , (3-54)
and the other is the short-circuit acoustic impedance
0
1
aD aD aD
aD
V
P
ZRjM
QjC
ω
ω
=
==+ + , (3-55)
67
where the definition of
eB
C ,
aD
R
,
aD
M
and
aD
C are listed in Table 2-2. Furthermore, as
indicated in Chapter 2, the lossless transformer converts energy between the acoustical and
electrical domains and obeys the transformer relations
1
I
Q
φ
′
=
(3.56)
and
PV
φ
′
=
, (3.57)
where
φ
is the impedance transformation factor. Hence, using network theory (Van Valkenburg
1964), for the two-port network shown in Figure 3-8, one has
1
eB eB
Q
I
V
j
CjC
φ
ωω
=+, (3-58)
and
11aD
PV ZQ
φ
=
+ . (3-59)
Substituting Eq. (3-58) into (3-59) results in
2
11aD
eB eB
I
PZQ
jC jC
φφ
ωω
⎛⎞
=++
⎜⎟
⎝⎠
. (3-60)
where
2
aD eB
Z
jC
φω
+ defines the open-circuit ( 0I
=
) acoustic impedance of the PZT-
backplate. On the other hand, substitution of
1
P and V for
1
Q in Eq. (3-58) from Eq. (3-59)
results in
1
22
eB aD
eB eB aD eB aD
jCZ
P
I
V
jC jCZ jCZ
ω
φ
ω
ωφωφ
⎛⎞
=+
⎜⎟
+
+
⎝⎠
, (3-61)
where
(
)
()
()
2
2
22
11 11
11
eB
eB aD
EM
eB eB aD eB aD eB eB eF
jC
jCZ
CjCZ C Z jC C C
φω
ω
κ
ωφ φω
⎡⎤
⎛⎞
=− =−=
⎢⎥
⎜⎟
++
⎝⎠
⎣⎦
. (3-62)
68
eF
C is the free electric compliance (
1
0P
=
) and
EM
κ
is the electromechanical coupling factor of
the piezoelectric backplate. Finally, from Eqs. (3-58) and (3-59), one has
2
1
eB aD aD
jCZ Z
PVI
ωφ
φ
φ
+
=−
, (3-63)
1
1
eB
jC
QVI
ω
φ
φ
=−. (3-64)
or in the matrix form of
2
1
1
.
1
eB aD aD
eB
jCZ Z
P
V
Q
I
jC
ωφ
φφ
ω
φφ
⎡⎤
+
−
⎢⎥
⎡⎤
⎡
⎤
⎢⎥
=
⎢⎥
⎢
⎥
⎢⎥
⎣
⎦
⎣⎦
−
⎢⎥
⎣⎦
(3-65)
Consequently, when the electrical boundary condition
eL
VI Z
=
− is given, the acoustic
impedance,
11
PQ, can be obtained for a given PZT-backplate as
()
(
)
()
(
)
22
1
1
11
eB aD aD eB aD eL aD
eB eB eL
j
CZ VI Z jCZ Z Z
P
QjCVI jCZ
ωφ ωφ
ωω
+− ++
==
−+
. (3-66)
The
00
PcU
ρ
− formulation of Eq. (3-65) is given by
2
1
00 1
00 00
eB aD aD
eB
DD
jCZ Z
P
V
cU
I
jcC c
AA
ωφ
φφ
ρ
ωρ ρ
φφ
⎡⎤
+
−
⎢⎥
⎡⎤
⎡
⎤
⎢⎥
=
⎢⎥
⎢
⎥
⎢⎥
⎣
⎦
⎣⎦
−
⎢⎥
⎣⎦
, (3-67)
where
2
2
13
D
AR
π
= is the effective area of the PZT-backplate due to the non-uniform
displacement of the clamped PZT-backplate.
69
Transfer Matrix of the EMHR
Next, the transfer matrix of the EMHR mounted at the end of the plane wave tube (PWT)
is obtained by multiplication of the transfer matrices of each subsystem in the order shown in
Figure 3-6
(
)
(
)
() ()
(
)
1
54
233
00 5 4 5
23
Transfer matrix for
Transfer matrix f
Transfer matrix for neck of the EMHR
area contraction
1
cosh sinh
1
0
sinh cosh
0
NNN
NN N
P
kt kt G
AA
cU A A
Gkt kt
AA
ζ
ζ
ρ
−
⎡⎤
ΓΓ
⎡
⎤
⎡⎤⎡ ⎤
=
⎢⎥
⎢
⎥
⎢⎥⎢ ⎥
ΓΓ
⎣⎦⎣ ⎦
⎣
⎦
⎣⎦
() ()
() ()
or
area expansion
2
1
00 00
Transfer matrix for cavity of the EMHR
Transfer matrix for
PZT-backpla
cosh sinh
sinh cosh
eB aD aD
CCC
CC C
eB
DD
jCZ Z
kL kL G
GkL kL
jcC c
AA
ωφ
φφ
ωρ ρ
φφ
−
⎡⎤
+
−
⎢⎥
⎡⎤
ΓΓ
⎢⎥
⎢⎥
⎢⎥
ΓΓ
⎣⎦
−
⎢⎥
⎣⎦
te
EMHR EMHR
11 12
EMHR EMHR
21 22
Transfer matrix
for the EMHR
,
V
I
V
TT
I
TT
⎡⎤
⎢⎥
⎣⎦
⎡⎤
⎡⎤
=
⎢⎥
⎢⎥
⎣⎦
⎣⎦
(3-68)
The subscript “
N ”and “C ” are for neck and cavity, respectively, and the definition of the other
parameters refers to the analysis above. Eq. (3-68) simplifies the calculation of some acoustic
characteristics of the EMHR. For example, when the electrical boundary condition is given by
eL
VI Z=− , the specific acoustic impedance of the EMHR is
EMHR EMHR
11 12
EMHR
EMHR EMHR
21 22
eL
eL
TZT
TZT
ζ
−+
=
−+
. (3-69)
For
L
Z →∞ (open circuit),
EMHR
11
EMHR
EMHR
21
T
T
ζ
= . (3-70)
For 0
L
Z → (short circuit),
70
EMHR
12
EMHR
EMHR
22
T
T
ζ
= . (3-71)
For a normal-incidence plane wave on the EMHR, the reflection coefficient
R
is
()
(
)
()()
EMHR EMHR EMHR EMHR
11 12 21 22
EMHR
EMHR EMHR EMHR EMHR
EMHR
11 12 21 22
1
1
eL eL
eL eL
TZT TZT
R
TZT TZT
ζ
ζ
−+−−+
−
==
+
−++−+
. (3-72)
For
L
Z →∞ (open circuit),
EMHR EMHR
11 21
EMHR EMHR
11 21
TT
R
TT
−+
=
−−
. (3-73)
For 0
L
Z → (short circuit),
EMHR EMHR
12 22
EMHR EMHR
12 22
TT
R
TT
−
=
+
. (3-74)
Transfer Matrix of the EMHR with Perforated Facesheet
For an EMHR with a perforated plate instead of a small neck, as shown in Figure 3-9, the
derivation of the transfer matrix is similar to the derivation above. Here, it is assumed that the
plate is thin compared to the wavelength of interest and that the phase difference of the particle
velocity between both sides of the plate can be neglected. Thus, the transfer matrix of the
perforated plate is given by
32
00 3 00 2
1
01
p
PP
cU cU
ζ
ρρ
⎡⎤ ⎡⎤
⎡⎤
=
⎢⎥ ⎢⎥
⎢⎥
⎣⎦
⎣⎦ ⎣⎦
, (3-75)
where
p
ζ
is the normalized acoustic impedance of the perforated plate given by (Ingard 1953)
(
)
()
22
0
00
2
112
222 0.85
2
p
cd
t
jk t d
cd
ρω π π
ζµρωµρω
σρ
⎡⎤
⎛⎞
⎢⎥
⎜⎟
=++++
⎜⎟
⎢⎥
⎝⎠
⎣⎦
(3-76)
71
and
()
2
dD
σ
= is the porosity of the perforated plate. The overall transfer matrix of the
EMHR with a perforated plate is thus given by
() ()
() ()
2
1
3
00 3
00 00
1
cosh sinh
01
sinh cosh
eB aD aD
p
CCC
CC C
eB
DD
jCZ Z
P
V
kL kL G
cU
I
GkL kL
jcC c
AA
ωφ
ζ
φφ
ρ
ωρ ρ
φφ
−
⎡⎤
+
−
⎢⎥
⎡⎤
ΓΓ
⎡⎤
⎡⎤ ⎡⎤
⎢⎥
=
⎢⎥
⎢⎥
⎢⎥ ⎢⎥
⎢⎥
ΓΓ
⎣⎦ ⎣⎦
⎣⎦
⎣⎦
−
⎢⎥
⎣⎦
. (3-77)
Comparison between TM and LEM
As discussed in Chapter 2, when the dimensions of the acoustic device are much smaller
than the wavelength of interest, the device components can be lumped into idealized discrete
circuit elements (Rossi 1988). In comparison with the transfer matrix representation of the
acoustic system, the LEM decouples the temporal and spatial variables associated with the
acoustic field. There is no spatial variation for a “lumped” element.
On the other hand, sometimes the LEM is no more than a reduced version of the transfer
matrix representation when the quasi-static assumption is satisfied. For example, as shown in
Figure 3-10a, a duct with a sound-soft termination reduces to the following using Eq. (3-24)
00 0
1
1
tan
as 1
jc kl l
P
jkl
QS S
ρ
ρ
ω
=≈ <<, (3-78)
where
0
lS
ρ
is the lumped acoustic mass of the short tube which is the same as
aN
M
discussed
in Chapter 2
**
. One more example, as shown in Figure 3-10b, is when the duct is ended by a
sound-hard termination, similarly, from Eq. (3-24), one has
00
1
1
2
00
cot
1
as 1
jc kl
P
kl
QS
j
c
ρ
ω
ρ
−
=≈ <<
∀
, (3-79)
**
In chapter 2,
aN
M
is the acoustic mass of the neck of the Helmholtz resonator.
72
where
Sl∀= is the volume of the tube, and
2
00
c
ρ
∀ is the lumped acoustic compliance
aC
C .
However, in some cases, more assumptions must be satisfied for the LEM to be consistent
with the transfer matrix representation. For instance, a duct terminated with a complex
boundary
T
Z
, shown in Figure 3-10c, reduces to the following using Eq. (3-24)
()
()
()
()()
()
2
00 00 00
1
100 00
cos sin cot
sin cos cot
TT
TT
Z
kl j c S kl c Z j kl S c S
P
Q j kl Z c S kl Z c j kl S
ρρ ρ
ρρ
+−+
==
++−
. (3-80)
When
1kl << , Eq. (3-80) reduces to
()()
()
()
22
00 00 00
1
100
cot ( )
cot 1 ( )
TTaC
TTaC
cZ j kl S c S Z j C c S
P
QZcjklS ZjC
ρρωρ
ρω
−+ +
=
+− +
, (3-81)
where
2
00
aC
Cc
ρ
=∀
. Furthermore, when
()()
2
00TaC
Z
jC cS
ωρ
>> , (3-82)
one has
1
1
()
1( )
TaC
TaC
ZjC
P
QZ jC
ω
ω
+
, (3-83)
which is the LEM of the system.
Figure 3-11 shows the comparison between LEM and transfer matrix prediction results for
a short-circuited EMHR. The geometry of the EMHR is listed in Table 4-2 (Case I). Both
models’ predictions match pretty well except for near the second resonant frequency of the
EMHR. The second resonant frequency of the EMHR corresponds to the resonant frequency of
the piezoelectric backplate. The acoustic impedance of the piezoelectric backplate becomes
relatively small near the resonant frequency, as shown in Figure 3-12, where Eq. (3-82) thus
breaks down. Consequently, the LEM is not consistent with the transfer matrix representation
even though the quasi-static assumption is satisfied.
73
Table 3-1. Conventional state variable definitions for the two-port network.
Electric Domain Mechanical Domain Acoustic Domain
Effort Voltage Force/ Moment Acoustic pressure
Flow Current Velocity/
Angular velocity
Volume velocity
74
1
I
2
I
1
V
2
V
Figure 3-1. A two-port network with reference directions for the positive direction of the current
variables indicated.
1
Q
2
Q
1
P
2
P
Figure 3-2. An acoustic two-port network with reference direction for the current variables
indicated.
75
1
P
0
Z
kl
1
P
2
P
S
Z
1
P
2
P
P
Z
1
Q
2
Q
2
Q
1
Q
1
Q
2
Q
2
P
Figure 3-3. Three basic types of elements in an equivalent circuit representation for an acoustic
network. (a) A distributed element. (b) A series element. (c) A parallel element.
r
Figure 3-4. Illustration of an area contraction.
76
11
,
P
Q
22
,
P
Q
Figure 3-5. A HR mounted in the side of one duct.
77
5
P
4
P
3
P
2
P
1
P
V
eL
Z
5
U
4
U
3
U
2
U
1
U
I
1 2 3 4 5
EMHR
PWT
1
1
P
U
2
2
P
U
3
3
P
U
4
4
P
U
5
5
P
U
eL
Z
,VI
1 2 3 4 5
Figure 3-6. Modeling EMHR using the transfer matrix method, where each subsystem is
denoted as: 1-area contraction; 2-duct element; 3-area expansion; 4-duct element; 5-
piezoelectric backplate.
78
55 5
,,
A
PU
44 4
,,
A
PU
(a)
33 3
,,
A
PU
22 2
,,
A
PU
(b)
22
,PU
11
,PU
(c)
Figure 3-7. Elements for derivation the transfer matrix representation of the EMHR. (a) An area
contraction. (b) An area expansion. (c) An acoustic duct system.
79
11
P
Q
,VI
eL
Z
2
R
p
h
1
R
s
h
(a)
V
1
P
1
Q
I
aD
R
aD
M
aD
C
:1
φ
eB
C
eL
Z
P
′
I
′
(b)
Figure 3-8. A piezoelectric backplate and its equivalent circuit representation. (a) A
piezoelectric backplate with shunt loads. (b) The equivalent circuit representation for
the piezoelectric backplate with the lumped-element two-port network indicated.
t
d
L
D
eL
Z
3
3
P
U
2
2
P
U
1
1
P
U
,VI
Figure 3-9. An EMHR with perforated facesheet.
80
11
,
P
U
22
,
P
U
a
M
1k
<
<
(a)
22
0PU
=
11
,
P
U
22
,
P
U
a
C
1k
<
<
(b)
22
PU
=
∞
11
,PU
22
,PU
a
C
1k
<
<
T
Z
Loading effect?
(c)
22 T
PU Z
=
Figure 3-10. Comparison between the TM and LEM. (a) Schematic of a duct terminated with
sound-soft termination. (b) Schematic of a duct with sound-hard termination. (c) Schematic of a
duct with complex termination.
81
1500 2000 2500 3000
0
5
10
15
θ
LEM
Short
TM
Short
LEM
Open
TM
Open
1500 2000 2500 3000
-5
0
5
10
15
Freq.[Hz]
χ
Figure 3-11. Comparison of prediction results for the normalized specific acoustic impedance of
a short- and open-circuited EMHR using TM and LEM.
82
1500 2000 2500 3000
10
6
10
8
10
10
10
12
10
14
10
16
Freq. [Hz]
|Z
aC
Z
aD
|
[
ρ
0
c
0
/(
π
R
2
)]
2
|Z
aC+aD
|
TM
|Z
aC+aD
|
LEM
Figure 3-12. Illustration of contributions to the variation in the prediction performance of the
LEM and TM.
83
CHAPTER 4
EXPERIMENTAL TECHNIQUES
This chapter presents the experimental setup used to evaluate the performance of the
EMHR and to validate the TM and LEM of the EMHR. The chapter is organized as follows. In
the first section, a number of methods for measuring the acoustic properties of an impedance
specimen are briefly introduced, followed by a review of the theoretical basis for the two-
microphone method (TMM). The second section presents the method for measuring the damping
coefficient of the EMHR as well as the parameter extraction of the piezoelectric backplate. The
experimental setup is detailed in the third section. The fourth section presents the sample
construction.
Acoustic Impedance Measurement
Introduction
A number of methods have been developed over the last few decades for determining in-
duct acoustic properties such as the reflection and absorption coefficient. Among these methods,
four are commonly used: the standing wave ratio method (SWM) using a probe microphone, the
two-microphone method (TMM), the single-microphone method (SMM), and the multi-point
method (MPM).
The SWM has commonly been used to measure the acoustic impedance (Lippert 1953;
Beranek 1988). The SWM is implemented by means of a plane wave tube (PWT) and a
traversable probe microphone. It is assumed that only plane waves are propagating along the
tube, thus the operating frequency of the SWM is limited to below the cut-on frequency of
higher-order propagating modes in the tube. The probe microphone is used to trace the acoustic
pressure magnitude along the center line of the duct versus the distance from the impedance
sample under test. The maximum and minimum pressures and their locations (extrema loci) in
84
the duct can thus be determined and the standing wave ratio (SWR) can be calculated. The SWR
is defined as the ratio of successive maximum and minimum pressures
max
min
P
SWR
P
=
, (4-1)
The amplitude of the reflection coefficient is readily determined from the SWR (Blackstock
2000)
1
1
SWR
R
SWR
−
=
+
. (4-2)
The phase of the reflection coefficient is calculated from the location of extrema loci. The
acoustic impedance of the sample can then be calculated from the complex reflection coefficient.
The SWM is quite accurate with a single tone source over the 0.1 to 10
00
c
ρ
impedance range
(Jones and Stiede 1997). However, the SWM is tedious and time-consuming due to the required
physical movement of the traversing microphone. Furthermore, the SWM cannot be used to
measure in-duct acoustic properties with mean flow.
Seybert and Ross (1977) introduced the TMM which is a standardized technique to
measure the impedance and absorption of acoustical materials (ASTM-E1050-98 1998). The
TMM also employs a PWT but uses two stationary, wall-mounted microphones to
simultaneously measure acoustic pressure at two known positions in the tube. Broadband
excitation is used to find the auto- and cross-spectral density functions so that the whole standing
wave pattern in the tube can be found at once. The TMM was further developed analytically and
experimentally by Chung and Blaser (1980a; 1980b). Their method uses a simple transfer
function relationship between two locations on the tube wall to decompose the acoustic wave in
a tube into its incident and reflected components. The wave decomposition leads to the
determination of the complex reflection coefficient, from which acoustic properties of the test
85
sample can be found. Chu (1986) extended the TMM by including the effect of the tube
attenuation due to thermo-viscous effects, thus allowing more freedom in choosing the positions
of the microphones. Chu (1988) also investigated the choice of microphone positions, and
indicated that a fixed choice of the microphone locations compromises the measurement
accuracy. For an accurate measurement over different frequency ranges, varied microphone
positions are preferred, as long as the separation of two microphones is not close to one-half
wavelength. Compared to the SWM, the TMM does not require traversing movement of the
microphone and thus is considerably more efficient. Furthermore, the effect of the mean flow
can be taken into account. Two major disadvantages of the TMM are the requirement of
accurate knowledge of amplitude and phase relationships between the two microphones and the
singularities associated with one-half wavelength microphone spacing. The former can be
mitigated using a sensor-switching technique (Chung and Blaser 1980) in which the
measurement of the transfer function is made with the initial microphone locations, and then the
measurement is repeated with microphone locations switched. The final measurement result is
obtained from the geometric mean of the original and switched results. Schultz et al. (2007)
indicated that there are still other factors that cause bias and precision errors for the TMM, such
as temperature uncertainty, microphone separation uncertainty, etc. Such factors can be taken
into account during uncertainty analysis for TMM.
Chu (1986) introduced the so-called single microphone method (SMM) in which a single
microphone is used to measure the sound at two locations, and then the transfer function is
derived from the measured auto- and cross-spectral densities with respect to the source signal.
The SMM is intended to eliminate the elaborate calibration procedure and any error associated
with phase mismatching found in the TMM. In essence, the SMM is an alternative
86
implementation of the TMM (Jones and Stiede 1997). A potential advantage of the SMM is its
use for high frequency measurement, where the diameter of the tube is small and space
limitations do not allow for the switching rig for two microphones. Both the SMM and the
TMM are susceptible to singularity issues.
The multiple-point method (MPM) employs multiple fixed microphones or a traversing
probe microphone to measure acoustic pressure at a minimum of three distances from the surface
of the impedance sample (Jones and Parrot 1989). A least square fitting technique is then used to
fit a wave propagation model to the point measurements. The wave propagation model includes
the effect of the mean flow and the duct wall absorption and permits reconstruction of the
standing wave pattern in the duct. The acoustic properties of the impedance sample are then
deduced (Jones and Parrot 1989; Hang and Ih 1998). The MPM can achieve very accurate
measurement results if a minimum of six pressure measurements are taken at points evenly
spaced over a distance of one-half wavelength (Jones and Parrot 1989). The physical movement
of the hardware is required for the MPM if the traversing probe microphone is adopted. The
measurement procedure is thus time-consuming. However, the multi-frequency (source) version
of the multi-point method (MPM-PR) can reduce the time necessary to complete an acoustic
impedance measurement (Jones and Stiede 1997).
Theoretical Basis of the TMM
As discussed above, the TMM is time-saving yet accurate. In this study, the TMM is used
to measure the acoustic impedance of the EMHR. An illustration of the TMM is shown in
Figure 4-1. Only plane waves are assumed to propagate along the tube, no mean flow exists, and
the effects of tube attenuation are negligible. The acoustic field inside the tube is given by
(Blackstock 2000)
87
()
(
)
(
)
,
jkx jkx
Pxf P fe P fe
+− −+
=+, (4-3)
where P
+
and P
−
denote the Fourier transforms of, respectively, the incident and the reflected
acoustic pressures at
0x
=
. The P
+
and P
−
can be resolved by measuring the complex pressure
P at two different locations, Mic.1 and Mic.2, along the PWT. The pressure spectra at two
microphone locations are expressed as
(
)
(
)
2
jkl jkl
PPfe Pfe
+−−
=+ , (4-4)
and
(
)
(
)
(
)
(
)
1
jk l s jk l s
PPfe Pfe
+
−+
+−
=+ , (4-5)
where
s is the distance between the two microphones and l is the distance from the surface of
the impedance sample to the nearest microphone. The complex reflection coefficient
R
at 0x
=
is thus given by
() ()
21 12
21
12
ˆ
ˆ
jks jks
j2k l s j2k l s
jks
jks
PP e H e
P
Ree
PePP
eH
−−
−
+
+
+
−−
== =
−
−
, (4-6)
where
12 12 11
ˆˆ
ˆ
HEGG
⎡⎤
=
⎣⎦
is the frequency response function between Mic.1 and Mic.2.
[
]
E
is
the expectation operator,
12
ˆ
G
is the estimated cross spectrum between two microphones, and
11
ˆ
G
is the estimated auto-spectrum of Mic. 1 (Bendat and Piersol 2000). The specific acoustic
impedance of the impedance sample can then be then given by (ASTM-E1050-98 1998)
00
1
1
R
Zc
R
ρ
+
=
−
. (4-7)
where
00
c
ρ
is the characteristic acoustic impedance of the medium in the tube. Eq. (4-7) is valid
only if Eqs. (4-4) and (4-5) are linearly independent (i.e.
1, or 2 0,1,2...
jks
esnn
λ
≠≠ = ).
88
As mentioned earlier, a switching-sensor technique can be used to mitigate the elaborate
calibration procedure and any error associated with phase mismatching in the TMM. What
follows is a brief presentation of the theoretical basis behind this technique. By assuming that
both microphone channel systems are linear and time-invariant and devoid of noise, one has
()
(
)
(
)
1, 2
iimi
Pf PfH f i==, (4-8)
as shown in Figure 4-2.
()
i
Pf
( 1, 2i = ) is the Fourier transform of the actual acoustic pressure at
the microphone locations.
()
mi
Hf is the frequency response associated with the first and second
microphone channels.
()
i
Pf is the Fourier transform of the measured acoustic pressure via the
microphones. When the measurement is taken with the initial microphone positions, the transfer
function from Mic. 1 to Mic. 2 is given by
()()
()()
22 11
12 2 1
12
11
11
11 11
ˆ
ˆ
ˆ
mm
o
mm
PH PH
GPP
HE
G
PP
PH PH
∗
∗
∗
∗
⎡⎤
===
⎢⎥
⎣⎦
, (4-9)
where
12
ˆ
o
H denotes the original transfer function from Mic. 1 to Mic. 2. Then the measurement
is taken with the microphone locations switched. The transfer function from Mic. 1 to Mic.2 is
now given by
()()
()()
12 21
12
21 21
ˆ
mm
s
mm
PH PH
H
PH PH
∗
∗
==
, (4-10)
where
12
ˆ
s
H denotes the switched transfer function from Mic. 1 to Mic. 2. Thus, the transfer
function from Mic. 1 to Mic. 2 can be obtained from the geometric mean (Chung and Blaser
1980)
12 2 1
12
12
11
ˆ
ˆ
ˆ
o
s
HPP
H
H
PP
∗
∗
==. (4-11)
89
Eq. (4-11) shows that the complex frequency response characteristics of the microphone system
do not affect the measurement results. In other words, the amplitude and phase characteristics of
two microphone channel systems need not match perfectly, as required when the TMM is used
(Seybert and Ross 1977).
Uncertainty Analysis of the TMM
Two types of errors are associated with the TMM: bias and random errors (ASTM-E1050-
98 1998; Seybert and Soenarko 1981; Bodén and Âbom 1986; Schultz et al. 2007). The bias
error is the fixed, systematic, or constant component of the total error. It is the same for each
measurement. The random error is the random component of the total error and has a different
value for each measurement (Coleman and Steele 1989). Random errors associated with the
TMM can be kept low by ensemble averaging and by maintaining a high coherence between the
microphones (ASTM-E1050-98 1998; Bodén and Âbom 1986). Several sources of bias error
exist; for instance, errors in microphone separation and distance from the test surface of the
specimen can cause significant bias errors associated with the TMM (Bodén and Âbom 1986).
Bias error also can arise from the tube attenuation and computational error in post-processing
(ASTM-E1050-98 1998). The tube attenuation shifts the loci of pressure minimums
asymmetrically in the standing wave pattern as the distance from the specimen increases
(ASTM-E1050-98 1998). The bias error caused by the tube attenuation can be minimized by
placing microphones as close to the specimen as possible (ASTM-E1050-98 1998).
Theoretically, the quality of the measurement results can be evaluated via the relation
between the total error and the true value of the measurand. The total error is the sum of bias
errors and random errors. However, one does not know the true value of the measurand in most
cases. Alternately, one can evaluate the “degree of goodness” of the measurement by the
90
statement that the true value of the measurand,
true
X
, lies within the interval
best X
X
U± with
%C confidence, where
best
X
is the mean value of the measurements and
X
U is the uncertainty
in
X
of the combination of bias and random errors (Coleman and Steele 1989).
Schultz et al. (2007) presented the uncertainty analysis of the TMM for acoustic impedance
testing. They employed both multivariate uncertainty analysis and the Monte Carlo method to
provide a systematic framework for computing the uncertainties of the TMM. The multivariate
method, which involves the propagation of component uncertainties, assumes small component
uncertainties that cause only linear variations in output quantities. The component uncertainties
include uncertainty in the frequency response function (FRF), uncertainty in microphone
location, temperature uncertainty, etc. The multivariate method matches the results from the
Monte Carlo method when all component uncertainties are small (
0.1%
<
) or the specimen is
sound hard. However, the component uncertainties are normally large enough to invalidate the
linear assumption. Thus, although the Monte Carlo method is more computationally intensive,
the Monte Carlo method is recommended by the authors for accurate uncertainty estimation,
Parameter Extraction of the Piezoelectric Backplate
Damping Coefficient Measurement
It is difficult to accurately model the damping loss of the piezoelectric backplate of the
EMHR. The damping loss may arise from acoustic radiation, thermo-elastic dissipation,
compliant boundaries, and other intrinsic loss mechanisms (Tilmans et al. 1992). The LEM as
well as TM employed in this study only accounts for the acoustic radiation loss. The damping
loss measurement thus provides a way to check if the acoustic radiation loss represents the
damping loss of the system. The logarithmic decrement method is used to measure the amount
of damping of the piezoelectric backplate of the EMHR. Assuming the piezoelectric backplate is
91
a SDOF system, and its free damped vibration is similar to that in Figure 4-3, then the
logarithmic decrement is defined as (Meirovitch 2001)
1
2
2
2
ln
1
x
x
πζ
δ
ζ
==
−
, (4-12)
where
ζ
is the damping coefficient of the system, and
1
x
and
2
x
are two sequential peak
displacements corresponding to the times
1
t and
1
tT
+
, where
T
is the period of the system. Eq.
(4-12) yields
()
2
2
2
δ
ζ
π
δ
=
+
. (4-13)
For small damping, such that
1
ζ
<< , Eq. (4-13) reduces to
2
δ
ζ
π
. (4-14)
The accuracy of the damping coefficient estimation in Eq. (4-13) can be improved if the peak
displacement is measured at two different times separated by a given number of periods. For
instance, let
1
x
and
1i
x
+
be the peak displacements corresponding to the times
1
t and
1i
t
+
, where
()
11
1,2,3...
i
ttiTi
+
=+ = and
T
is the period of the system. Moreover,
2
21
1
1
i
i
x
e
x
πζ ζ
−
+
=
, (4-15)
from which one has the logarithmic decrement
1
2
1
21
ln
1
i
x
ix
πζ
δ
ζ
+
==
−
. (4-16)
Rearranging and letting
1ii→−
gives
(
)
1
ln ln 1
i
xxi
δ
=
−−
. (4-17)
92
The plot ln
i
x
versus i should have the form of a straight line with the slope
δ
− if the
measurement is exact. Hence, the accuracy of the estimation for the damping coefficient can be
improved using Eq. (4-17). In particular,
N peak displacements ( , 1, 2,3...
i
xi
=
) are chosen.
ln
i
x
and 1
i
yi=− are then calculated. Then a straight line of the form
ii
zayb
=
+ (4-18)
which minimizes
()( )
22
11
ln ln
NN
ii i i
ii
x
zxayb
==
−= −−
∑∑
(4-19)
is determined. Note that
a
corresponds to
δ
−
in Eq. (4-17). The damping coefficient is thus
calculated by using Eq. (4-13)
()()
22
2
a
a
ζ
π
−
=
+−
. (4-20)
Effective Acoustic Piezoelectric Coefficient Deduction
Both LEM and TM of the EMHR assume that the piezoelectric backplate is clamped.
However, it is difficult to obtain a perfectly clamped boundary condition in practice. Thus, it is
necessary to experimentally investigate the impact of the practical boundary condition on the
prediction performance of the models. In this study, a parameter extraction method is
implemented to deduce the effective acoustic piezoelectric coefficient,
A
d , which is significant
in determining the impedance transformer factor,
φ
, and the blocked electrical capacitance,
eB
C .
As discussed in Chapter 2 as well as Appendix C,
φ
and
eB
C affect the prediction performance
of the models, especially, when the EMHR is connected with passive loads. The method to
experimentally determine
A
d is as follows (Prasad et al. 2006). First, the deformation of the
clamped piezoelectric backplate is measured at discrete points while applying a voltage to the
93
plate. Second, the deflection mode shape of the plate is reconstructed using the least square
curve-fitting method. Finally,
A
d is calculated using Eq. (C-4).
Experimental Setup
Acoustic Impedance Measurement Setup
The schematic of the experimental setup for the acoustic impedance measurement is shown
in Figure 4-4. The plane wave tube (PWT) has a cross-section of 25.4 mm by 25.4 mm and is
96.5 mm long, which permits a plane wave acoustic field at frequencies up to 6.7 kHz (Horowitz
et al. 2002).
Three Brüel and Kjær (B&K) type 4138 microphones are used simultaneously to measure
the acoustic pressure. Two microphones, labeled as Mic. 1 and Mic. 2, are flush mounted in a
rotating plug to the side of the impedance tube. The rotating plug is used to remove amplitude
and phase mismatches between the microphones as explained in Section 1. The other
microphone, labeled as Ref. Mic., is flush mounted to the end face of the impedance tube to
measure the total acoustic pressure at the entrance of the resonator. This microphone also serves
as a reference to ensure a constant SPL at the neck of the resonator. Furthermore, Mic. 2 is
mounted as close to the test specimen as possible. The distance between Mic. 2 and the test
surface of the specimen is
32.0 0.8± mm with a 95% confidence interval estimate (Schultz et al.
2007). The distance is about one and one-half times the duct dimension, and is enough to
facilitate the evanescence of higher-order modes generated from the specimen as shown in
Figure 4-5
††
. The microphone separation is
20.7 1.1
±
mm with a
95%
confidence interval
estimate (Schultz et al. 2005). Thus, there is no frequency singularities for measurement below
the frequency range of interest (
6.4 kHz< ).
††
Please see Appendix D for more details.
94
All microphones are calibrated with a Brüel and Kjær 4228 Pistonphone while connected
to a Brüel and Kjær PULSE Multi-Analyzer System Type 3560. The PULSE system serves as
the power supply and data acquisition and processing system for the microphones as well as the
signal source. A pseudo-random waveform generated from the PULSE system is fed through a
Techron Model 7540 power supply amplifier to drive a BMS H4590P compression driver. The
pseudo-random signal, which has uniform spectral density and random phase, is commonly used
to avoid leakage effects of a non-periodic signal. The driver, which can produce acoustic waves
between 200 Hz and 22 kHz, is connected to one end of the PWT via a transition piece. The
pseudo random waveform is bandpass filtered from 300 Hz to 6.7 kHz. Meanwhile, a FFT with
1000 ensemble averages is performed on each incoming microphone signal.
Damping Measurement Setup
The schematic of the experimental setup for the damping coefficient measurement is
shown in Figure 4-6. The piezoelectric backplate is placed in air or within a vacuum chamber
capable of producing a low pressure around 10 Torr (1300 Pa). A LK-G32 high accuracy laser
displacement sensor is used to measure the vibration response of the center of the plate. The
repeatability of the sensor is 0.05 µm, and the measurement range is 5 mm. A rectangular
waveform (1 Hz, 20.4Vp-p) generated from the Agilent 33120A Waveform Generator is used to
excite the piezoelectric plate. This signal is also used as the external trigger of the Tek
TDS5104B oscilloscope which acquires data from the laser displacement sensor. In addition,
1024 ensemble averages are performed.
Parameter Extraction of the Piezoelectric Backplate
The experimental setup for parameter extraction of the piezoelectric backplate is similar to
the damping measurement setup with two exceptions. The piezoelectric backplate is not placed
95
within the vacuum chamber and the transverse displacements along the radius of the backplate
are measured to reconstruct the deformation of the plate due to applied voltage.
EMHR Construction
The EMHR samples are modularly constructed, as shown in Figure 4-7. The modular
design permits the parameter studies of the orifice diameter and thickness, cavity volume, and
piezoelectric backplate geometric parameters. The sample is comprised of an orifice plate,
cavity plate, piezoelectric diaphragm cap/bottom plates and piezoelectric diaphragm. Except for
piezoelectric diaphragm, all parts are made from aluminum. The piezoelectric diaphragm is a
commercially available piezoceramic circular bender disk (APC International, Ltd, model APC
850), which consists of a circular lead zirconate titanate (PZT) 850 patch bonded to a brass shim.
A very thin silver electrode covers the PZT patch. A passive shunt network is then connected to
the diaphragm. The piezoelectric diaphragm is clamped between the cap and bottom plates,
which are bolted to the backside of the cavity plate. The orifice plate is attached to the other side
of the cavity plate when the acoustic impedance of the EMHR is measured. The orifice plate is
removed when the damping measurement and parameter extraction are implemented. Because
the piezoelectric diaphragm is not demounted between measurements, the clamped boundary
condition of the piezoelectric diaphragm is maintained for the acoustic impedance measurement,
damping measurement and other parameter extraction.
Table 4-1 summarizes the material parameters of the piezoelectric diaphragm. The
geometric parameters of the EMHR samples are listed in Table 4-2. Two samples with different
cavity dimensions are involved in this study. The dimensions of the piezoelectric diaphragm are
the same for both samples. In order to evaluate the tuning performance of the EMHR samples, a
variety of shunt networks were used to tune the EMHR. Table 4-3 shows one of the loads
97
Table 4-1. Material properties of the piezoelectric backplate.
Properties Value
Young’s Modules (
2
/Nm
)
6.3E10
Poisson ratio 0.31
Density (
3
kg m )
7700
Relative dielectric constant 1750
Piezoceramic
(APC 850)
a
Piezoelectric Strain Constant d
31
(
p
CN)
-175
Young’s Modules (
2
/Nm
)
11.0E10
Poisson ratio 0.38
Shim
(260 half hard brass)
b
Density (
3
kg m )
8530
a
. The data is adapted from APC International, Ltd
http://www.americanpiezo.com/materials/apc_properties.html
b
. The data is adapted from www.matweb.com
Table 4-2. Dimensions of the EMHRs (unit: mm, resolution 0.01 mm).
Name Case I Case II
Radius
R
2.42 2.42 Neck
Length
T
3.16 3.16
Radius
R
6.34 6.34 Cavity
Depth
L
16.42 9.38
Radius
R
1
10.05 10.05 piezoceramic
Thickness
h
p
0.13 0.13
Radius
R
2
12.41 12.41
Piezoelectric
backplate
shim
Thickness
h
s
0.19 0.19
98
Table 4-3. Selected loads matrix used in the experiment to tune the EMHR.
Resistive loads
(
Ω )
Capacitive loads
(
nF )
Inductive loads
( mH )
Nominal Measured Nominal Measured Nominal Measured
Short Short Short Short Short Short
200 199.1 10 10.3 100 101.9 (65.4)
a
2k 2.0k 47 37.8 300 304.6 (191.8)
7.5k 7.4k 100 89.7 500 507.8 (319.8)
Open Open Open Open Open Open
a
. the value in () represents the resistance of the inductive loads in
Ω
99
0
x
=
P
+
P
−
s
Figure 4-1. Illustration of the two microphone method.
()
Pf
(
)
m
Hf
(
)
Pf
Figure 4-2. Pressure measured by a microphone.
100
0
Time [s]
Amplitude
x
1
x
2
Figure 4-3. Free damped vibration of a SDOF system.
B&K Pulse System
Techron
Amplifier
Mic.2Mic.1
Ref.
Mic.
Loads
Speaker
PWT
EMHR
s
Figure 4-4. Acoustic impedance measurement of the EMHR using TMM.
101
(a)
-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
1.96
1.965
1.97
1.975
1.98
1.985
1.99
1.995
2
2.005
z/a
|p/P
+
|
ω
/
ω
0
=0.1
P
center
P
wall
P
(0,0)
-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
z/a
|p/P
+
|
ω
/
ω
0
=0.5
P
center
P
wall
P
(0,0)
-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
z/a
|p/P
+
|
ω
/
ω
0
=1
P
center
P
wall
P
(0,0)
-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
z/a
|p/P
+
|
ω
/
ω
0
=1.5
P
center
P
wall
P
(0,0)
(b)
Figure 4-5. The scattering by a HR mounted at the end of a PWT. (a) FEM simulation results of
the pressure distribution in the tube terminated with a HR. (b) Numeric results of
pressure distribution in the tube terminated with the HR under different exciting
frequencies.
Note:
center
p is the acoustic pressure along the centerline of the tube,
wall
p
is the pressure along the line
on the wall where microphones are mounted,
()
0,0
p
is the acoustic pressure distribution with assumption
that only plane waves propagate within the duct,
za
is the ratio of distance away from the test surface of
the specimen to the duct cross-section dimension, and
0
ω
is the resonant frequency of the HR. The
results show that higher order modes contribute to the pressure distribution near the specimen.
102
Figure 4-6. The damping measurement for the piezoelectric backplate of an EMHR.
Figure 4-7. Assembly diagram of modular EMHR (not to scale).
103
CHAPTER 5
EXPERIMENTAL RESULTS AND DISCUSSION
This chapter summarizes the experimental results of the normal incidence acoustic
impedance measurement for the EMHRs with different shunt loads, the damping coefficient
measurement and the parameter extraction. A comparison between the experimental results and
models predictions for the acoustic impedance of the EMHR is also provided.
Evaluation of the Tuning Performance of the EMHR
As detailed in Chapter 2, the acoustic impedance of an EMHR is tuned via adjustment of
the shunt network attached to the piezoelectric diaphragm of the EMHR. Figure 5-1 shows the
acoustic impedance measurement results for the resonator (Case I) with a variety of capacitive
loads, the measured reflection coefficient magnitude is shown in Figure 5-2. The EMHR with
capacitive loads (
1
eL eL
Z
jC
ω
= ) has 2DOF and then has two resonant frequencies (
1
f
and
2
f
),
located where the reactance of the EMHR crosses zero with positive slope (Im( ) 0
IN
Z = ). At the
resonant frequency, the particle velocity within the orifice of the EMHR is at its maximum and
there is a sharp dip in the reflection coefficient magnitude (Figure 5-2). There is also an anti-
resonant frequency (
a
f
) where the reactance of the EMHR crosses the zero with negative slope
and the particle velocity in the orifice of the EMHR goes to its minimum. As the capacitance is
increased, the second resonant frequency (
2
f
) shifts towards the short-circuit case from the
open-circuit case, as indicated by Eq.(2-48). The first resonant frequency (
1
f
) barely changes
for the EMHR (Case I). This is because of the weak coupling ( 0.17
s
κ
=
, where s indicates the
short-circuit case) between the solid-walled Helmholtz resonator and the PZT backplate for the
tested EMHR (Case I). There is an approximately 9% capacitive tuning range of the second
resonant frequency (
2
ω
∆ ) for the Case I resonator under the conditions and geometry listed in
104
Table 4-2, where the tuning range is defined as the difference between the open-circuit resonant
frequency and its respective short-circuit counterpart (e.g.,
22 2os
f
ff
∆
=−, where
2o
f
and
2
s
f
are the second resonant frequency with open- and short-circuit, respectively).
Figure 5-3 shows the measured acoustic impedance for different resistive loads across the
EMHR (Case I). Similar to the EMHR with capacitive loads, the EMHR with resistive loads has
2DOF, and thus two resonant frequencies. A trend similar to the capacitive tuning can be
observed for the resistive tuning, in which the tuning range is also defined by the short- and
open-circuit limits. When the resistance is increased, the resonant frequency moves from the
short-circuit case to the open-circuit case. Moreover, when resistive loads are attached to the
EMHR, a portion of the energy is removed from the EMHR and dissipated by the resistive loads.
In other words, resistive loads increase the system damping (Hagwood et al. 1990). As shown in
Figure 5-3, for a small resistive load, only a slight amount of energy is removed from the EMHR,
thus the additional damping due to the load is not significant. As the resistance increases, more
energy is removed and the system damping increases. The amplitude of the impedance peaks
thus continue to reduce. At the optimal resistive load, the maximum energy is extracted from the
EMHR, and the system damping becomes far larger than one of the short-circuited EMHR. The
amplitude of the impedance peaks thus reaches their minimum. As the load resistance moves
away from the optimal value, the system damping decreases with the increase of the load
resistance due to the amount of energy removed from the EMHR reduced. At high resistance
(i.e.,
eL
R →∞), the system damping goes to one of the open-circuited EMHR.
Figure 5-4 shows the results for inductive tuning of the Case I EMHR, and the measured
reflection coefficient magnitude is shown in Figure 5-5. Unlike capacitive and resistive tuning,
inductive tuning provides an additional DOF for the EMHR, resulting in a 3DOF EMHR with
105
three resonant frequencies. The third resonant frequency shifts closer to the second resonant
frequency open-circuit case as the inductance is increased. However, as a result of the 3DOF,
inductive tuning is not restricted to lie between the short- and open-circuit limits. Rather, the
second resonant frequency always lies outside the short- to open-circuit tuning range as shown in
Figures 5-4 and 5-5. Notice that the second resonant frequency of the resonator shifts to lower
frequencies as the inductance is increased. Moreover, when the inductive load is large enough,
the second resonant frequency is smaller than the first resonant frequency. There is
approximately a 47% inductive tuning range (
22
s
f
f
∆
) of the EMHR (Case I) under the
conditions and geometry listed in Table 4-2. Again, the first resonant frequency only changes
slightly due to weak coupling between the piezoelectric backplate and the Helmholtz resonator.
Furthermore, a real inductor possesses finite resistance as well (Table 4-3). Typically, the
resistance increases with increasing inductance, so the larger inductor also has a larger
resistance.
Figure 5-6 shows the experimental results of the EMHR (Case II) with short- and open-
circuits. It indicates that the first resonant frequency of the EMHR shifts due to coupling
between the piezoelectric backplate and the Helmholtz resonator is not as weak as EMHR (Case
I). There is an approximately 4.6% tuning range of the first resonant frequency, in contrast, the
tuning range of the first resonant frequency of the EMHR(Case I) is approximately 0.9%. This is
because the depth of the cavity of the EMHR (Case II) is shorter than one of the EMHR (Case I).
The acoustic compliance of the cavity consequently decreases. Hence, the coupling between
Helmholtz resonator and piezoelectric composite backplate becomes stronger ( 0.22
s
κ
= ) and
the first resonant frequency of the EMHR is not the resonant frequency of the solid-walled
106
Helmholtz resonator, as given in Eq.(2-39). The first resonant frequency is adjusted with
changing of the shunt loads of the EMHR. More details are presented in Chapter 2.
Comparison with LEM and Transfer Matrix
The comparison between measurement data and LEM and TM is shown in Figure 5-7 to
Figure 5-10. Clearly, with measured damping loss of the PZT backplate, both LEM and TM
prediction match the experimental results pretty well for the EMHR with short-circuit. The TM
gives a better prediction for the acoustic resistance than LEM does. This is because the TM
includes the viscous loss effect on the wave propagation within the cavity. There are some other
factors as presented in Chapter 3, which concern when the TM coincides with the LEM. The
predictions for the normalized specific acoustic impedance do not match the experimental data
very well for the open-circuit case. One possible explanation for the observed discrepancy is
that, for the short-circuit case, the piezoelectric backplate is electrically shorted and thus
φ
and
eB
C do not affect the acoustic impedance of the EMHR, as they do for the open-circuit case.
Hence, any inaccuracies in either
φ
or
eB
C will impact the predicted results. Some factors do
impact the piezoelectric backplate model, such as the bond layer between the piezoceramic patch
and the brass shim (the model assumes a negligible bond layer), any asymmetry in the
piezoceramic patch geometry, and imperfect clamped boundary conditions. The results also
reveal that the measured resistance is larger than the predicted results in the low and high
frequency ranges. The deviations may be caused by the fact that the reflection coefficient is
close to unity in those regions. When the reflection coefficient is near unity, the uncertainty in
the acoustic impedance measurement using the TMM will become very large (Schultz et al.
2007). The uncertainty in the acoustic impedance arises from the measurement uncertainties in
107
microphone locations (
and s in Figure 4-4), the frequency response function between the two
measurement microphones, ambient temperature and pressure.
Damping Coefficient Measurement Results
Figure 5-11 and Figure 5-12 show the damping coefficient measurement results for a
piezoelectric composite backplate in air and in a vacuum chamber. Using the logarithm
decrement method and choosing the first two peaks, the damping coefficient is calculated to be
0.026
air
ζ
=
(5-1)
for a piezoelectric backplate in air, and
0.024
vacuum
ζ
=
(5-2)
for a piezoelectric backplate in a vacuum chamber. Note that the damping coefficient in Eq.
(5-1) and (5-2) are from the first and second peak displacement of the damping oscillation of the
system. The accuracy of the estimation may be improved using more measured peak
displacements, as discussed in Chapter 4. Table 5-1 contains the measurement results for the
first twelve peak displacements of the piezoelectric plate in air. To minimize Eq.(4-19), one has
()
12 12 12
2
111
ln
ii ii
iii
y a yb xy
===
⎛⎞⎛⎞
+=
⎡
⎤
⎜⎟⎜⎟
⎣
⎦
⎝⎠⎝⎠
∑∑∑
, (5-3)
and
12 12
11
12 ln
ii
ii
ya b x
==
⎛⎞
+=
⎜⎟
⎝⎠
∑∑
. (5-4)
Inserting the values from Table 5-1 into Eqs. (5-3) and (5-4) leads to
506 66 40.7598ab
+
=−
, (5-5)
and
66 12 4.95673ab
+
=− . (5-6)
108
The solutions for Eqs. (5-5) and (5-6) are
0.09439 0.1061
ab
=
−=. (5-7)
The damping coefficient is then obtained using Eq.(4-20) and is
0.015
air
ζ
=
(5-8)
which differs significantly from the estimation in Eq. (5-1). The illustration of the determination
of damping coefficient using the method above and the comparison between numeric fittings and
measurement results are shown in Figures 5-13 and 5-14.
Following a similar procedure, the damping coefficient of the piezoelectric plate in the
vacuum chamber is
0.01
air
ζ
=
(5-9)
The data used to determine the coefficient of the piezoelectric plate in the vacuum chamber are
listed in Table 5-2. The damping coefficient found using the method above and the comparison
between numeric fittings and measurement results are shown in Figures 5-15 and 5-16.
Parameter Extraction of the Piezoelectric Backplate
Figure 5-17 shows the measured displacement of the piezoelectric backplate due to the
application of voltages. The results indicate that, when applying a voltage to the plate, the
transverse displacement of the plate is larger than one calculated by the model. This is likely due
to non-idealities in the composite plate boundary conditions. A clamped boundary is difficult to
achieve in practice and any compliance in the boundaries or in-plane compressive stress due to
mounting will result in an enhanced
A
d . Consequently, the deduced parameters such as
A
d ,
φ
and
eB
C differ from those used in the models, as listed in Table 5-3. Applying the deduced
φ
and
eB
C to the LEM and TM results in good predictions of open-circuited EMHRs, as shown in
Figure 5-18. The model prediction for the normalized specific acoustic impedance matches the
109
experimental data quite well for both short-circuit and open-circuit cases. The results indicate
that the functional form of the LEM accurately captures the physical behavior of the EMHR.
110
Table 5-1. Determination of the damping coefficient of the piezoelectric plate in air.
i
i
x
ln
i
x
i
y
2
i
y
ln
ii
yx
1 1.220 0.199 0 0 0
2 1.035 0.034 1 1 0.034
3 0.886 -0.121 2 4 -0.241
4 0.812 -0.208 3 9 -0.625
5 0.725 -0.321 4 16 -1.285
6 0.663 -0.411 5 25 -2.057
7 0.631 -0.460 6 36 -2.759
8 0.549 -0.599 7 49 -4.192
9 0.535 -0.625 8 64 -5.000
10 0.491 -0.711 9 81 -6.400
11 0.434 -0.834 10 100 -8.342
12 0.407 -0.899 11 121 -9.891
Table 5-2. Determination of the damping coefficient of the piezoelectric plate in vacuum
chamber.
i
i
x
ln
i
x
i
y
2
i
y ln
ii
yx
1 1.246 0.220 0 0 0
2 1.077 0.074 1 1 0.074
3 0.975 -0.026 2 4 -0.052
4 0.891 -0.115 3 9 -0.346
5 0.853 -0.159 4 16 -0.637
6 0.794 -0.231 5 25 -1.155
7 0.761 -0.274 6 36 -1.643
8 0.722 -0.326 7 49 -2.284
9 0.672 -0.397 8 64 -3.178
10 0.650 -0.430 9 81 -3.874
11 0.602 -0.508 10 100 -5.077
12 0.595 -0.519 11 121 -5.711
Table 5-3. Comparison between predicted and deduced LEM parameters of the Piezoelectric
backplate.
A
d
φ
eB
C
Predicted 48.3e-12 82.4 45.7e-9
Deduced 71.4e-12 122 41.2e-9
111
1500 2000 2500 3000
0
10
20
30
θ
Short
10nF
47nF
100nF
Open
1500 2000 2500 3000
-10
0
10
20
χ
Freq.[Hz]
Short
Open
Short
Open
f
1s
f
1o
f
2s
f
2o
f
ao
f
as
Figure 5-1. Experimental results for the normalized specific acoustic impedance of the EMHR
(Case I) as function of the capacitive loads.
112
1500 2000 2500 3000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Magnitude of reflection coefficient
Freq.[Hz]
Short
10nF
47nF
100nF
Open
f
2o
f
2(10)
f
2(47)
f
2(100)
f
2s
f
1s
f
1o
f
1(100)
f
1(47)
f
1(10)
Figure 5-2. Experimental results for the reflection coefficient of the EMHR (Case I) as function
of the capacitive load. Note that
2(10)
f
denotes the first resonant frequency under a
capacitive loading of 10 nF.
1500 2000 2500 3000
0
10
20
30
θ
Short
200
Ω
2k
Ω
7.5k
Ω
Open
1500 2000 2500 3000
-10
0
10
20
χ
Freq.[Hz]
Short
Open
Short
Open
f
1s
f
1o
f
2s
f
2o
Figure 5-3. Experimental results for the normalized specific acoustic impedance of the EMHR
(Case I) as function of the resistive loads.
113
1000 1500 2000 2500 3000
0
10
20
30
θ
Short
100mH
300mH
500mH
Open
1000 1500 2000 2500 3000
-10
0
10
20
χ
Freq.[Hz]
Figure 5-4. Experimental results for the normalized specific acoustic impedance of the EMHR
(Case I) as function of the inductive loads.
114
1000 1500 2000 2500 3000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Magnitude of reflection coefficient
Freq.[Hz]
Short
100mH
300mH
500mH
Open
f
3(300)
f
1(300)
f
2(300)
f
2(500)
f
3(500)
f
2(100)
f
1(500)
f
1(100)
f
2s
f
3o
f
1s
f
1o
Figure 5-5. Experimental results for the reflection coefficient of the EMHR (Case I) as function
of the inductive load. Note that denotes the third resonant frequency under an
inductive loading of 300 mH.
1500 2000 2500 3000
0
20
40
60
θ
1500 2000 2500 3000
-20
-10
0
10
20
χ
Freq.[Hz]
f
1s
f
1o
f
2s
f
2o
Short
Open
Figure 5-6. Experimental results of the normalized acoustic impedance of the EMHR (Case II)
for the short- and open-circuit.
115
10
-1
10
0
10
1
10
2
θ
1500 2000 2500 3000
-10
0
10
20
Freq.[Hz]
χ
LEM
Short
TM
Short
Data
Short
LEM
Open
TM
Open
Data
Open
Figure 5-7. Comparison LEM, TR and measurement results for a short- and open-circuited
EMHR (CASE I), the damping loss of the backplate is determined using logarithm
decrement method (
0.015
ζ
= ).
10
-2
10
0
10
2
10
4
θ
1500 2000 2500 3000
-50
0
50
100
150
Freq.[Hz]
χ
LEM
Short
TM
Short
Data
Short
LEM
Open
TM
Open
Data
Open
Figure 5-8. Comparison LEM, TR and measurement results for a short- and open-circuited
EMHR (CASE I), the damping loss of the backplate is assumed to be acoustic
radiation resistance.
116
10
-1
10
0
10
1
10
2
θ
1500 2000 2500 3000
-20
0
20
40
Freq.[Hz]
χ
LEM
Short
TM
Short
Data
Short
LEM
Open
TM
Open
Data
Open
Figure 5-9. Comparison LEM, TR and measurement results for a short- and open-circuited
EMHR (CASE II), the damping loss of the backplate is determined using logarithm
decrement method (
0.015
ζ
= ).
10
-2
10
0
10
2
10
4
θ
1500 2000 2500 3000
-400
-200
0
200
Freq.[Hz]
χ
LEM
Short
TM
Short
Data
Short
LEM
Open
TM
Open
Data
Open
Figure 5-10. Comparison LEM, TR and measurement results for a short- and open-circuited
EMHR (CASE II), the damping loss of the backplate is assumed to be acoustic
radiation resistance.
117
1 2 3 4 5 6 7 8
x 10
-3
-1.5
-1
-0.5
0
0.5
1
1.5
Time [s]
Transverse displacement [
µ
m]
Figure 5-11. Damping coefficient measurement for piezoelectric composite backplate (Case I) in
air.
1 2 3 4 5 6 7 8
x 10
-3
-1.5
-1
-0.5
0
0.5
1
1.5
Time [s]
Transverse displacement [
µ
m]
Figure 5-12. Damping coefficient measurement for piezoelectric composite backplate (Case I) in
the vacuum chamber.
118
0 2 4 6 8 10 12
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
i
lnx
i
, z
i
Data
z
i
=-0.09439y
i
+0.1061
Figure 5-13. Determination of damping coefficient of the piezoelectric plate in air.
1 2 3 4 5 6 7 8
x 10
-3
-1.5
-1
-0.5
0
0.5
1
1.5
Time [s]
Transverse displacement [
µ
m]
Data
Curve fitting 1
Curve fitting 2
Figure 5-14. Curve fitting the measurement data (in air) using a 2
nd
-order system, curve fitting
1- 0.026
ζ
= , curve fitting 2- 0.015
ζ
=
.
119
0 2 4 6 8 10 12
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
i
lnx
i
, z
i
Data
z
i
=-0.06347y
i
+0.1248
Figure 5-15. Determination of damping coefficient of the piezoelectric plate in the vacuum
chamber.
1 2 3 4 5 6 7 8
x 10
-3
-1.5
-1
-0.5
0
0.5
1
1.5
Time [s]
Transverse displacement [
µ
m]
Data
Curve fitting 1
Curve fitting 2
Figure 5-16. Curve fitting the measurement data (in the vacuum chamber) using a 2
nd
-order
system, curve fitting 1- 0.024
ζ
=
, curve fitting 2- 0.01
ζ
=
.
120
0 0.2 0.4 0.6 0.8 1
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
x 10
-6
r / R
2
Transverse Displacement [m]
Data
8
Data
10
Data
12
Data
14
(a)
0 0.2 0.4 0.6 0.8 1
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
x 10
-7
Normalized Transverse Displacement [m/V]
r / R
2
Data
8
Data
10
Data
12
Data
14
Num
1
(b)
Figure 5-17. Measured transverse displacement of the piezoelectric backplate due to the
application of various voltages.
121
10
-1
10
0
10
1
10
2
θ
1500 2000 2500 3000
-10
0
10
20
Freq.[Hz]
χ
LEM
Short
TM
Short
Data
Short
LEM
Open
TM
Open
Data
Open
(a)
10
-1
10
0
10
1
10
2
θ
1500 2000 2500 3000
-20
0
20
40
Freq.[Hz]
χ
LEM
Short
TM
Short
Data
Short
LEM
Open
TM
Open
Data
Open
(b)
Figure 5-18. Predictions of the LEM and TM for short- and open-circuited EMHRs. (a) EMHR
(Case I). (b) EMHR (Case II).
122
CHAPTER 6
OPTIMAL DESIGN OF AN EMHR
This chapter discusses the optimal design of an EMHR. The synthesis of the optimal
design problem for an EMHR aims at providing a usable impedance range. The optimal design
studies of an EMHR have been undertaken following two paths. The first path is to maximize
the tuning range of an EMHR with non-inductive loads, which is presented in Sections 2 and 3.
The second path aims to optimally tune the three resonant frequencies of the EMHR with
inductive shunts. The optimization design of the EMHR with inductive loads is presented in
Section 4.
Introduction
The acoustic impedance of an EMHR, shown in Figure 2-5, is modified in-situ by
adjusting the electrical impedance of passive shunt networks attached to the piezoelectric
backplate of the EMHR. The frequency range over which the acoustic impedance of the EMHR
can be tuned significantly is called the tuning range. Both experimental data and the theoretical
analysis presented in Chapter 5 indicate that the EMHR with non-inductive shunts has two
tuning ranges, defined by
1
f
∆ and
2
f
∆ , corresponding to the tuning range of the first and second
resonant frequencies of the EMHR, respectively. The analytical investigation indicates that the
EMHR with different geometry has different tuning ranges. The material properties of the
piezoelectric backplate also affect the tuning ranges of the EMHR. It is thus necessary to
employ an optimal design procedure to obtain a useable tuning ranges
1
f
∆
and
2
f
∆ .
Mathematically, the optimal design deals with minimizing or maximizing the objective
function under certain constraints in the form
123
{
}
()
()
0
( ) 1,2,...,
0 1,2,...,
= 0 1,2,...,
ii
i
j
JX ii n
X
gX i m
hX j n
=
≤≤
≤=
=
Minimize
such that LB UB
, (6-1)
where the design variables
jj
x
( 1,2,...,
j
jp= ) are collectively represented as the design vector
{
}
12
, ,...,
p
X
xx x= . The design variables are limited by lower and upper bounds LB , and UB ,
respectively.
()
ii
JX
is termed the objective function, and
(
)
i
gX
and
(
)
j
hX
are known as
inequality and equality constraints, respectively. The problem stated in Eq. (6-1) is called
constrained optimization problem. Some optimization problems do not have any constraints and
are called unconstrained optimization problems. Moreover, if only one objective function is
involved in the problem stated in Eq. (6-1), the process is called single-objective optimization
problem, otherwise the process is denoted by the term multi-objective optimization problem. In
general, solving a multi-objective optimization problem is much more complicated than solving
a single-objective problem. An
p
-dimensional space with each coordinate axis representing a
design variable
jj
x
( 1,2,...,jj p
=
) is denoted as the design space. Each point in the design space
is called a design point which represents a possible or impossible solution of the optimization
problem. Constraints can be functional or geometric in term of their characteristics. Constraints
limiting the performance of the system are functional constraints, while constraints defining the
physical limitation on design variables are geometric constraints. Geometric constraints are also
known as side constraints. A constraint is satisfied with a margin is known as inactive, while is
called active when it is satisfied with no margin. A design point which satisfies all constraints is
feasible or acceptable. The collection of all feasible points is named the feasible region. A
feasible point where all constraints are inactive is called a free point. Furthermore, a feasible
124
point
X
∗
is global optimum point if at this point
(
)
(
)
JX JX
∗
≤ for all
X
in the feasible region.
A feasible point
X
∗
is called local optimum point if
(
)
(
)
JX JX
ε
∗∗
≤
+ for all sufficiently
small positive and negative value of
ε
. Finally, optimization problems where both objective
function(s) and constraints are linear functions of the design variables are called linear
programming problems. In contrast, optimization problems where the objective function(s) or
one of the constraints is a nonlinear function of design variables are named nonlinear
programming. Generally, it is easier to solve a linear programming problem than to solve a
nonlinear programming (Haftka and Gurdal 1992; Rao 1996).
In this study, the optimal design of the EMHR firstly aims at maximizing the tuning
range(s) of the EMHR with non-inductive loads to satisfy a certain set of specified requirements.
Therefore, the tuning range(s) is (are) the objective function(s) of the optimal design, while the
set of specified requirements is the constraint. The optimization problem is nonlinear because
the objective function(s) and some constraints are nonlinear functions of the design variables
(Liu et al. 2006).
Optimizing Single Tuning Range of EMHR with Non-inductive Loads
Theoretical Background
As described in Chapter 2 and Chapter 5, the short- and open-circuit cases define the
tuning range of an EMHR with a non-inductive shunt network. The EMHR is a 2DOF system in
such cases and possesses two resonant frequencies, denoted by
(
)
1
,
i
f
ios= and
2i
f
, and the
subscripts
o and s denote open- and short–circuit conditions, respectively. For the short-circuit
case, the two resonant frequencies are given by
()()()
()
2
2
22 22 2
12 12 12
2
1
41
2
LLs LLs sLLs
s
ff ff ff
f
κ
+− + −−
=
(6-2)
125
and
()()()
()
2
2
22 22 2
12 12 12
2
2
41
2
LLs LLs sLLs
s
ff ff ff
f
κ
++ + −−
=
, (6-3)
where
1L
f
is the resonant frequency of loop 1 (Figure 2-6)
1
1
2
L
aN aC
f
M
C
π
= , (6-4)
and
2Ls
f
is the resonant frequency of loop 2 under short-circuit condition
()
2
1
2
Ls
aD aDrad aD aC
aD aC
f
M
MCC
CC
π
=
+
+
. (6-5)
Furthermore,
2
s
κ
is the electromechanical coupling factor for the short-circuited EMHR
()
2
12
aD aC aD aC
s
s
aD aC aC aD aC aC
CCCC
CC CCCC
κκκ
== =
++
, (6-6)
where
1
κ
and
2
s
κ
are the coupling coefficients of loop 1 and loop 2 for the short-circuit EMHR,
respectively. The coupling coefficient defines the ratio of the energy stored in the coupling
elements (
aC
C ) to that stored in the total capacitance for each loop. For loop1, the coupling
coefficient is
(
)
()
2
1
2
1
1
aC
aC
aC
Qdt C
C
C
Qdt C
κ
=
==
∫
∫
, (6-7)
where
1 aC
CC= is the total capacitance of loop 1. Similarly, for loop 2, the coupling coefficient
is
2
aD
s
aD aC
C
CC
κ
=
+
. (6-8)
126
For the open-circuit case, the two resonant frequencies are given by
()()()
()
2
2
22 22 2
12 12 12
2
1
41
2
L Lo L Lo o L Lo
o
ff ff ff
f
κ
+− + −−
=
(6-9)
and
()()()
()
2
2
22 22 2
12 12 12
2
2
41
2
LLo LLo oLLo
o
ff ff ff
f
κ
++ + −−
=
, (6-10)
where
2Lo
f
is the resonant frequency of loop 2 under open-circuit condition
()
()
2
2
1
2
Lo
aD aDrad eB aD aC
eB aD eB aD aC
f
MM CCC
CC C C C
π
φ
=
+
++
. (6-11)
Furthermore,
2
o
κ
is the electromechanical coupling factor of the open-circuit EMHR
()
2
12
2
eB aD
oo
eB aD eB aD aC
CC
CC C C C
κκκ
φ
==
++
, (6-12)
where
2o
κ
are the coupling coefficients of loop 2 for open-circuit EMHR
()
2
2
eB aD
o
eB aD eB aD aC
CC
CC C C C
κ
φ
=
++
. (6-13)
Thus, the tuning ranges of the EMHR with non-inductive loads are defined by
11 1os
f
ff
∆
=−, (6-14)
and
22 2os
f
ff
∆
=−. (6-15)
Moreover, note that
1L
f
is essentially the resonant frequency of a Helmholtz resonator with a
rigid wall instead of a piezoelectric backplate. Dividing Eqs. (6-2) and (6-3) by
2
1L
f
leads to
127
()()()()
2
2222
2
1
2
1
1141
2
ssss
s
L
f
f
α
κακ κακ
+−+ −−
= , (6-16)
and
()()()()
2
2222
2
2
2
1
1141
2
s
sss
s
L
f
f
α
κακ κακ
+++ −−
= (6-17)
where
α
is the mass ratio between the neck and piezoelectric backplate
aN
aD aDrad
M
MM
α
=
+
. (6-18)
Similarly, dividing Eqs. (6-9) and (6-10) by
2
1L
f
results in
()()()()
2
2222
2
1
2
1
1141
2
oooo
o
L
f
f
α
κακ κακ
+−+ −−
= , (6-19)
and
()()()()
2
2222
2
2
2
1
1141
2
oooo
o
L
f
f
α
κακ κακ
+++ −−
= . (6-20)
Equations (6-14) - (6-20) indicate that the tuning ranges vary with four parameters,
1L
f
(or
H
R
f
),
s
κ
,
o
κ
and
α
. An example of calculated
11
s
L
f
f and
21
s
L
f
f as function of
s
κ
and
α
is
shown in Figure 6-1. The
11oL
f
f and
21oL
f
f versus
o
κ
and
α
are expected to have similar
trend. Note that
o
κ
is less than
s
κ
because the open-circuit EMHR becomes less compliant than
the short-circuit case. For a given EMHR, the mass ratio
α
is the same for the short- and open-
circuit cases. Therefore, the dimensionless tuning range
1iL
f
f
∆
( 1, 2i
=
) can be represented by
two points on the isolines map of the
11
s
L
f
f and
21
s
L
f
f , as shown in Figure 6-1, where the
points have the same value of
α
and different value of
s
κ
. Clearly, the change of
11L
f
f∆ with
128
the change of the coupling factor is significant when
s
κ
is large, in contrast, the change of
21L
f
f∆ with the change of
s
κ
becomes significant when
s
κ
is very small. The observation
indicates that it is possible to maximize
1
f
∆
by designing the EMHR with large
s
κ
, while
2
f
∆
may be maximized by designing the EMHR with small
s
κ
.
Optimization Problem Formulation
The first optimization problem is formulated by choosing one of the tuning ranges in Eqs.
(6-14) and (6-15) as objective function. As discussed above, both tuning ranges are functions of
the geometric parameters, listed in Table 6-1, as well as the material properties of piezoelectric
composite diaphragm and air. To simplify the problem, it is assumed that the material properties
of a given piezoelectric composite disc are constants, as listed in Table 4-1. Thus, the single
objective design optimization problem seeks to maximize either tuning range,
1
f
∆ or
2
f
∆ (or to
minimize
1
f
−∆ or
2
f
−∆ ) to satisfy a certain set of constraints. The design variables are the
radius of the neck
r
, the thickness of the neck t, the radius of the cavity
R
, the depth of the
cavity
L, the radius of the shim
1
R
, the thickness of the shim
p
h , the radius of the shim
2
R
and
the thickness of the shim
s
h . The constraints are categorized as geometric constraints and
functional constraints. The side constraints include the physical bounds for the design variables.
The constraints involved in the optimization problem are: 1) Lower bounds (LB) and upper
bounds (UB):
{
}
12
,, , , , , ,
ps
LB r t R L R h R h UB≤≤
, where
LB
and UB of each design variable
are listed in Table 6-1; 2) Geometry constraints that impose physical limitations on the design
variables. The constraints are based on the size of the test apparatus, available commercial
piezoelectric benders, and size restrictions on the EMHR; 3) Frequency constraint that confines
the first short-circuit resonant frequency
1
s
f
of the EMHR to a particular range (1200 Hz to 1900
129
Hz) where noise suppression is preferred, while also prescribing an upper limit of 3000 Hz for
2
s
f
. Clearly, both the objective function and the frequency constraint are nonlinear. Thus, the
optimal design of the EMHR with non-inductive loads is a constrained nonlinear programming.
Mathematically, the single objective optimization of the EMHR with non-inductive loads is the
following:
(
)
(
)
12
212
43
11 1
0; 0
10 0; 10 0
1200 0; 1900 0; 3000 0
Minimize or
such that LB UB
ps ps
ss s
fX fX
X
RR R R
hh hh
ff f
−−
−∆ −∆
≤≤
−≤ −<
−−+ ≤ +− ≤
−≤ − ≤ − ≤
(6-21)
where
{
}
12
,, , , , , ,
ps
X
rtRLR h R h=
is the design variable vector.
The optimization problem defined in Eq. (6-21) is first implemented in MATLAB using its
fmincon function and then verified using Genetic Algorithms also provided by MATLAB
Genetic Algorithm Direct Search (GADS) toolbox. The fmincon function employs sequential
quadratic programming (SQP) for nonlinear constrained optimization. The SQP method focuses
on the solution of the Kuhn-Tucker (KT) equations. The KT equations are necessary conditions
to be satisfied at a local minimum of
(
)
JX. For convex programming problems, the KT
conditions are necessary and sufficient for a global minimum. The optimization problem is
called a convex programming problem if the objective function and the constraint functions are
convex (Rao 1996; Mathworks 2005). However, it is difficult to determine the optimization
problem set up in Eq. (6-21) is a convex programming problem or not. Thus, the optimal results
obtained using the SQP method may be local minima. Moreover, due to the high nonlinearity of
the optimization problem defined in Eq. (6-21), the solution process (i.e., convergence of the
optimal solution and optima) using SQP method is highly dependent on the initial values. The
130
Genetic Algorithms (GA) provided by MATLAB Genetic Algorithm Direct Search (GADS)
toolbox are thus adopted to explore the possibility of finding a global optimum. The GA
provides a natural search strategy and is simple to implement. At each iteration, a population of
points is generated. The objective function determines best points in the population as the
optimal solution. In consecutive iterations, the genetic algorithm randomly selects individuals
from the current population to be parents and uses them produce the children by applying
mutation and crossover rules. Over successive generations, the population evolves toward an
optimal solution (Mathworks 2005). By choosing suitable parameters for GA, it is possible to
ensure the local optimum does not dominate the population (Mathworks 2005). However, GA is
time-consuming. An alternative approach is to first employ GA with 10 to 20 generations to find
a point close to the optimal solution and then use that point as the initial condition for a gradient-
based optimization study. The GA-optimal result is then improved by setting the GA optimum
as the initial condition of the SQP optimization implemented via the fmincon function in
MATLAB.
Optimization Results
One set of the optimization results using SQP method are listed in Table 6-2. The initial
values of the design variables are chosen based on the prototype of the EMHR discussed in Liu
et al. (2003, 2007). The initial design vector satisfies all constraints and thus is a feasible initial
condition. Different initial design vectors are explored next. The result of the optimization
depends strongly on the initial values. However, the collective optimal results show that any
improvement in one tuning range (
1
f
∆ or
2
f
∆
) can only occur by compromising the other tuning
range. Both tuning ranges
1
f
∆ and
2
f
∆ can not maximized simultaneously. Table 6-2 also lists
the optimization results by combination GA and SQP methods. The results show different
131
optimum. In terms of characteristics of the optimization problem set up in Eq. (6-21), multiple
optimum are expected. This is the objective function is based on the LEM of the EMHR. Some
LEM parameters are many-to-one function. For instance, different dimensions of the cavity can
leads to the same acoustic compliance
22
00aC
CRLc
π
ρ
= only if
2
R
L is the same.
Sensitivity Analysis
The sensitivity analysis is very important for understanding which design variables are
significant drivers for a optimum solution
∗
x . It is impossible to undertake the sensitivity
analysis before the optimum solution
∗
x has been found, thus the sensitivity analysis is a post-
processing step. In fact, the sensitivity analysis with respect to design variables seeks to
determine how the objective function
J
changes as the design vector x changes. This can be
achieved by the computation of the gradient of
(
)
JX
1
2
p
Jx
Jx
J
Jx
∂∂
⎧
⎫
⎪
⎪
∂∂
⎪
⎪
∇=
⎨
⎬
⎪
⎪
⎪
⎪
∂∂
⎩⎭
(6-22)
for the single objective function, and
0
0
0
1121 1
1222 2
12
Jacobian Matrix
,,
,,
,,
J
n
n
ppnp
JxJx J x
JxJx J x
JxJx J x
∂∂∂∂ ∂ ∂
⎧⎫
⎪⎪
∂∂∂∂ ∂ ∂
⎪⎪
∇=
⎨⎬
⎪⎪
⎪⎪
∂∂∂∂ ∂ ∂
⎩⎭
(6-23)
for the multiple- objective function. The normalized sensitivity is then derived as following
using Eqs. (6-22) and (6-23)
(
)
()
% change in
% change in
jj
jj jj jj jj
XX
x
JX
JJ J
xx x x
JX
∗ ∗
∗
∗
∆∂
=
∆∂
, (6-24)
132
which is useful to compare sensitivities to the different design variables.
The sensitivity analysis results for the optima of the tuning range
1
f
∆
related to the design
variables are shown Figure 6-2. The result shows that the optimal
1
f
∆
is most sensitive to the
radius of the shim of the PZT-backplate of the EMHR (
2
R
). An increase of 1% in
2
R
will lead
to a decrease in the tuning range
1
f
∆ (the objective function) of -1.6%. Similarly, the second
most sensitive factor is the thickness of the ceramic layer of the PZT-backplate of the EMHR
(
p
h ), where an increase of one percent in
p
h will result in an increase of the tuning range
1
f
∆
by
0.52%. However, allowable changes may be restricted by the constraints, especially the
frequency constraint and some of the geometric constraints, as shown in Figure 6-3. This figure
illustrates how the optimal solution is affected by the upper constraint on
1
s
f
. When the tuning
range
1
f
∆ is maximized, there is a relative strong coupling between the piezoelectric backplate
and the cavity, the changes in the geometry of the neck and the cavity thus have some effect on
the optima.
The sensitivity analysis result for the optimization of the tuning range
2
f
∆ is shown in
Figure 6-4, which indicates that
2
R
and
p
h are also the most sensitive design parameters with
respect to maximizing
2
f
∆
. An increase of one percent in
2
R
will lead to a change of the tuning
range
2
f
∆ by -2.27%, while an increase of one percent in
p
h will result in a change of the tuning
range
2
ω
∆ by +0.73%. Moreover, the optimal solution is not sensitive to changes in the
geometry of the neck and cavity. Physically, this is because the piezoelectric backplate weakly
couples with the cavity in this case. The
2
f
is thus dominated by the piezoelectric backplate.
133
Pareto Optimization of the EMHR with Non-inductive Loads
The results of the optimizing single tuning range of EMHR with non-inductive loads
indicates that it is impossible to simultaneously maximize both
1
f
∆
and
2
f
∆
. Therefore, a trade-
off approach is pursued. Pareto optimization is thus explored to optimize both tuning rages at
the time to achieve a Pareto solution. The Pareto solution (also called a Pareto optimal) is one
where any improvement of one objective degrades at least one other objective. Three methods
are used to obtain the Pareto solution for multiple-objective optimization of the EMHR: the
ε
-
constraints method (Marglin 1967), the traditional weighted sum method (Koski 1988), and the
adaptive weighted sum (AWS) method (Kim and de Weck 2005). Mathematically, the
ε
-
constraints method is
(
)
(
)
12
11
212
43
11 1
0; 0
10 0; 10 0
1200 0; 1900 0; 3000 0
Minimize or
such that LB UB
ps ps
ss s
fX fX
X
f
RR R R
hh hh
ff f
ε
−−
−∆ −∆
≤≤
−∆ ≤
−≤ −<
−−+ ≤ +− ≤
−≤ − ≤ − ≤
(6-25)
where
{
}
12
,, , , , , ,
ps
X
rtRLR h R h=
. The tuning range
2
f
∆
is chosen to be the primary objective
function which is subject to the original constraints and an additional constraint limiting the
tuning range
1
f
∆ . The advantage of the
ε
-constraint method is that it is able to achieve the
optimal solution even in the non-convex boundary of the Pareto front (i.e., the set of the Pareto
solutions). The problem with this method is that it is generally difficult to choose a suitable
ε
.
In order to speed up the optimization, the choice of
1
ε
follows an ascending order: the result of a
previous optimization with the constraint factor
1
i
ε
is used as the starting point for the
optimization with another set of the constraint factor
1
1
i
ε
+
, as shown in Figure 6-5.
134
The traditional weighted sum method is used to convert the multi-objective optimization to
a single objective problem by using a weighted sum of the original multiple objectives
(
)
(
)
(
)
11 1 2
212
43
11 1
1
0; 0
10 0; 10 0
1200 0; 1900 0; 3000 0
Minimize
such that LB UB
ps ps
ss s
fX fX
X
RR R R
hh hh
ff f
αα
−−
−+−
⎡
⎤
⎣
⎦
≤≤
−≤ −<
−−+ ≤ +− ≤
−≤ − ≤ − ≤
(6-26)
where
1
01
α
≤≤ is the weighting coefficient. This method is easy to implement, but it has two
drawbacks. One is that an even distribution of the weighting factors does not always achieve an
even distribution of the Pareto front. The other is the weighted sum method cannot find the
Pareto solution on non-convex parts of the Pareto front, as shown in Figure 6-6 . Furthermore,
the traditional weighted sum method can produce non-Pareto points occasionally. To address
this drawback of the conventional weighted sum method: the adaptive weighted sum method
(AWS) (Kim and De Weck 2005) is adopted to search the Pareto front of the multi-objective
optimization of the EMHR. The AWS produces well-distributed solutions, finds the Pareto
optimum in non-convex regions, and excludes non-Pareto optimums. For the region unexplored
by the traditional weighted sum method, the AWS involves changing the weighting factor
adaptively rather than by using a priori weight selections and by specifying additional inequality
constraints.
The Pareto solution obtained using these three methods are shown in Figure 6-6. Clearly,
optimal solutions obtained using the weighted sum method tends to clustered together, as
mentioned above. A large portion of the Pareto front is obtained using the ε-constraints method,
which is simply ignored by the weighted sum method. This is due to the non-convex properties
of the Pareto front. The results indicate that the AWS addresses the limitations of the
135
conventional weighted sum method, and produces the Pareto optimum with a good distribution.
The result from the AWS method matches the result using the ε-constraint method very well.
As shown in Figure 6-6, the points that form the Pareto front do not distinguish themselves
from each other in the sense that none of them is “better” than any other with respect to
maximizing both
1
ω
∆ and
2
ω
∆ , because any improvement in one objective degrades the other
one. In Figure 6-6, the point A and B are corresponding to the maximized single objective
tuning ranges of
1
f
∆ and
2
f
∆ , respectively. At each point, one tuning range approaches its
maximum, while the other is essentially invariant. Between these two extrema, other points,
such as point C on the Pareto front, represents a trade-off for optimizing both tuning ranges.
Which optimal solution is attractive is determined by other design considerations. Specifically,
the Pareto solution provides the information for a designer that shows how design trade-offs can
be used to satisfy specific design requirements.
Optimization of the EMHR with Inductive Loads
The EMHR with inductive loads has 3DOF and thus has three resonant frequencies (
1
f
,
2
f
and
3
f
). One goal of the optimal design of the EMHR with inductive loads is to find an optimal
EMHR which all resonant frequencies are within the frequency range of interest and the
frequency shift between the maximum and minimum resonant frequency is minimized
(
()()
123 123
max , , min , ,
f
fff fff∆= − ). Thus, within a wide frequency range, the variation of
the reactance of the EMHR is relative small. Mathematically, the task is described as
136
(
)
212
43
11 3
0; 0
10 0; 10 0
1200 0; 1900 0; 3000 0
Minimize
such that LB UB
ps ps
ss s
fX
X
RR R R
hh hh
ff f
−−
∆
≤≤
−≤ −<
−−+ ≤ +− ≤
−≤ − ≤ − ≤
, (6-27)
where
{
}
12
,, , , , , , ,
pseL
X
rtRLR h R h Z=
is the design variables vector which also includes the
inductive shunts. The lower and upper bound of the inductive shunts are 10
mH and 150 mH ,
respectively. Due to highly nonlinearity of constraints, Genetic Algorithm provided by
MATLAB Genetic Algorithm Direct Search (GADS) toolbox is thus explored. The comparison
of the acoustic impedance between the initial and optimal EMHR is shown in Figure 6-7. The
results indicate that it is possible to keep the acoustic reactance of the resonator nearly constant
over a given frequency range.
137
Table 6-1. Design optimization variables of the EMHR (Unit: mm).
Description Symbol Lower Bound
(LB)
Upper Bound
(UB)
Neck radius
r
1.0 3.5
Neck thickness
t
1.0 4.5
Cavity depth
R
5.0 15.0
Cavity radius
L
10.0 20.0
Piezoceramic radius
1
R
1.0 25.0
Piezoceramic thickness
p
h
0.5E-1 1.0
Shim radius
2
R
1.0 25.0
Shim thickness
s
h
0.5E-1 1.0
Table 6-2. Single objective optimization results of the EMHR (Dimensions unit: mm).
SQP GA and SQP
Initial values Maximizing
1
f
∆
Maximizing
2
f
∆
Maximizing
1
f
∆
Maximizing
2
f
∆
r
2.4 3.5 3.5 3.5 3.5
t
3.2 1.0 1.0 1.0 1.0
R
6.3 6.6 13.9 7.8 13.5
L
16.4 14.2 18.8 10.1 20.0
1
R
10.1 21.1 16.2 21.3 15.0
p
h
1.2E-1 7.2E-1 6.5E-1 7.4E-1 5.6E-1
2
R
12.4 22.5 17.2 22.7 16.0
s
h
1.9E-1 2.3E-1 2.1E-1 2.3E-1 1.8E-1
1
f
∆
9.6 170.0 0.4 170.0 0.4
11
s
f
f∆
0.5% 8.8% 0.0% 8.8% 0.0%
2
f
∆
113.3 14.6 290.5 14.6 290.5
22
s
f
f∆∆
4.3% 0.5% 9.3% 0.5% 9.3%
s
κ
1.7E-1 2.4E-1 5.5E-2 2.4E-1 5.5E-2
α
5.6E-2 3.1E-2 2.0E-2 3.1E-2 2.0E-2
138
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.4
0.5
0.6
0.7
0.8
0.9
1
κ
s
f
1s
/f
L1
Increasing
α
A
B
(a)
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
1
1.5
2
2.5
3
3.5
4
4.5
κ
s
f
2s
/f
L1
Increasing
α
A
B
(b)
Figure 6-1. Resonant frequency of the EMHR versus
s
κ
and
α
. (a)
11
s
L
f
f as function of
s
κ
and
α
. (b)
21
s
L
f
f as function of
s
κ
and
α
.
139
r t R L R1 hp R2 hs
-2
-1.5
-1
-0.5
0
0.5
1
Normalized sensitivity
Figure 6-2. Normalized sensitivity of the design variables at the optima for maximizing the
tuning range of
1
f
∆ .
140
0.0215 0.022 0.0225 0.023 0.0235 0.024
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
2500
f
1s
0.0215 0.022 0.0225 0.023 0.0235 0.024
-170
-160
-150
-140
-
∆
f
1
R
2
Optimum
Figure 6-3. Illustration of the change in the optimum solution as a function of
2
R
.
141
r t R L R1 hp R2 hs
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
Normalized sensitivity
Figure 6-4. Normalized sensitivity of the design variables at the optima for maximizing the
tuning range of
2
f
∆ .
1
J
2
J
Figure 6-5. Choice of the starting point for the multi-objective optimization with different ε
1.
142
-200 -150 -100 -50 0
-300
-250
-200
-150
-100
-50
0
-
∆
f
1
-
∆
f
2
ε
-method
AWS method
Weighted sum method
A
B
C
Figure 6-6. Comparison of the Pareto front obtained via the ε-constraint, traditional weighted
sum, and adaptive weighted sum methods.
1000 1500 2000 2500 3000 3500
0
5
10
15
θ
1000 1500 2000 2500 3000 3500
-20
-10
0
10
20
χ
Freq. [Hz]
Figure 6-7. Comparison between initial (dash line) and optimal (solid line) acoustic impedance
of the EMHR with inductive loads.
Non-convex
region
143
CHAPTER 7
SUMMARY AND FUTURE WORK
A lumped element model (LEM) and a transfer matrix representation(TM) of the EMHR
have been developed to predict the acoustic behavior of the EMHR. The models have been
implemented in a MATLAB code and incorporated into the NASA ZKTL code, which converted
to MATLAB code from original FOTRAN code. In this study, LEM and TM show good
agreement within the frequency range of interest. The analysis of the tuning behavior of the
EMHR based on the LEM is validated by the experiment observations.
Experiment investigation of the EMHR has been implemented using the standard two-
microphone method (TMM). The measurement results verify both the LEM and the TM of the
EMHR. Good agreement between predicted and measured impedance was obtained. Short- and
open-circuit loads define the limits of the tuning range using resistive and capacitive loads.
There is approximately a 9% non-inductive tuning limit for the second resonant frequency under
these conditions for the non-optimized EMHR configuration studied. Inductive shunt loads
result in a 3DOF system and an enhanced tuning range of over 47% that is not restricted by the
short- and open-circuit limits. The damping coefficient measurement for the piezoelectric
backplate in the vacuum chamber is also performed. The results show that the acoustic radiation
damping loss is relative small.
Based on models of the EMHR, the Pareto optimization design of the EMHR has been
performed for the EMHR with non-inductive loads. The EMHR with non-inductive loads is
2DOF and has two resonant frequencies. Either of them can be optimally tuned by
approximately 9%. However, the results show that it is impossible to maximize both tuning
ranges simultaneously. The improvement of one tuning range degrades the other. Consequently,
a trade-off must be reached. In other words, a generally accepted Pareto solution should be
144
achieved. Three methods are explored to obtain the Pareto optimal set of the bi-objective
optimization design of the EMHR: the ε -constraint method, the traditional weighted sum
method, and the adaptive weighted sum method. Both the ε -constraint method and the adaptive
weighted sum method obtain the same Pareto front for the optimization problem, while the
weighted sum method clusters the Pareto solutions around the two single-objective design points
of the Pareto front. The Pareto solution provides the information for a designer that shows how
design trade-offs can be used to satisfy specific design requirements. The EMHR with inductive-
loads is 3DOF and has three resonant frequencies. The optimization design of the EMHR with
inductive loads aims at optimal tuning of these three resonant frequencies, i.e. to constrain three
resonant frequencies within a given range. The results indicate that it is possible to keep the
acoustic reactance of the resonator close to a constant over a given frequency range.
Future work should include the testing of sample EMHR liner designs in NASA Langley’s
dual-incidence and grazing-incidence flow facilities to evaluating both the impedance
characteristics as well as the energy reclamation abilities. Inductive tuning offers the capability
of an increased tuning range, but further work is needed regarding the design of such liners.
Specifically, system goals are required to develop the formulation of the optimal design problem.
Moreover, the optimization design of the EMHR should take into account the optimal choice of
the materials for the piezoelectric backplate to satisfy specific design requirements. Poor
electromechanical coupling limits the electromechanical tuning capabilities of the EMHR.
Novel composite material systems, like an interdigitated piezoelectric composite may
dramatically improve the electromechanical coupling (Hong et al. 2006). Such improvements
are essential to mimic both resistance and reactance of existing double layer liners. Furthermore,
145
enhanced electromechanical tuning will also enable the use of more complex passive ladder
circuits for impedance spectra shaping.
146
APPENDIX A
NOISE LEVELS AND UNITS
Level is taken on mean the quantity of noise. By definition: Level is the logarithm of the
ratio of a given quantity to a reference quantity of the same kind
(Harris 1998). The following is
the definition of the noise levels and units adopted in this dissertation
deciBel: dB, the deciBel si a measure, on a logarithm scale, of the magnitude of a
particular sound intensity by reference to a standard quantity that represents the threshold of the
hearing (Smith 1989)
Sound Pressure Level: SPL (in units of dB), a measure of the root mean square (rms)
pressure of a sound
(
)
10
20log
rms ref
SPL p p= (A-1)
where
ref
p
is a reference pressure. For air the reference pressure is 20 Pa
µ
. For water the
standard reference pressure is 1
Pa
µ
(Blackstock 2000).
Weighted Sound Pressure Level: the sound pressure level weighted to reflect human
interpretation of the loudness of sounds at different frequencies (Smith 1989). The weighting
schemes based on loudness are A-, B-, and C-weighting. The corresponding weighted sound
pressure levels are A-weighted SPL (dBA), B-weighted SPL (dBB) and C-weighted SPL (dBC).
Figure A-1 identifies the A weighting.
Perceived Noise Level: PNL (in units of PNdB), An frequency-weighted SPL obtained by
a stated procedure that combines the sound pressure levels in the 24 one-third octave bands with
midband frequencies from 50 Hz to 10 kHz
(ANSI 1994).
Effective Perceived Noise Level: EPNL, (in units of EPNdB), an internationally
recognized unit for describing the noise of a single aircraft operation. The level of
the time
147
integral of the antilogarithm of one-tenth of tone-corrected perceived noise level over the
duration of an aircraft flyover, the reference duration being 10s
(ANSI-S1.1 1994), where the
tone-corrected perceived noise level (PNLT) recognizes the impact of a discrete tone on human
annoyance by adding a penalty to the PNL (Smith 1989).
Day/night Equivalent Sound Level: DNL or LDN, dBA-based rating, twenty-four hour
average sound level but with night-time (2400-0700 and 2200-2400) penalty of 10dBA (ANSI
1994).
148
10
-1
10
0
10
1
-60
-50
-40
-30
-20
-10
0
10
Freq. [kHz]
A-weighting [dB]
Figure A–1: A-weighting to sound arriving at random incidence.
.
149
APPENDIX B
ACOUSTIC IMPEDANCE PREDICTION OF AN ORIFICE
As discussed in Chapter 1, a common component of resonant liners is the perforated metal
plate. A perforated sheet consists of arrays of orifices/holes. Understanding of the acoustic
characteristics of the orifice is critical to achieve full potential of the application of the resonant
liners.
Linear Impedance Model of Orifices
The term “linear” here means that the acoustic impedance of the orifice is independent on
the incident SPL. Conversely, the term “nonlinear” implies that the acoustic impedance of the
orifice is dependent on the incident SPL. Two orifice linear impedance models will be briefly
introduced here, one is the Crandall model (1926), the other is GE impedance model developed
by Motsinger and Kraft (1995).
Crandall’s Model
Crandall (1926) treated an orifice as a narrow tube in which a shear layer was set up near
the wall of the tube. The driving force due to pressure gradient was balanced by the inertial
force and viscous force. The equation of the motion is given as
()
0
inertial force
driving force due to
viscous force
pressure gradient
22 2
pu
dx rdr j u rdrdx rdrdx dr
xrr
πωρπ µπ
∂∂⎡∂⎤
⎛⎞ ⎛⎞
−= +−
⎜⎟ ⎜⎟
⎢
⎥
∂∂∂
⎝⎠ ⎝⎠
⎣
⎦
, (B-1)
or
0
p
jru
xrrr
µ
ωρ
∂⎡ ∂ ∂⎤
⎛⎞
−= −
⎜⎟
⎢
⎥
∂∂∂
⎝⎠
⎣
⎦
, (B-2)
where the particle velocity ,
u , is a function of the radius r . In other words, the velocity profile
in the tube only has an axial component which varies radically and keeps the same along the
length of the tube. The solution of Eq. (B-2) is
150
(
)
(
)
()
0
2
0
1
s
ss
p
xJkr
u
kJka
µ
⎡
⎤
∂∂
=− −
⎢
⎥
⎣
⎦
(B-3)
where
2
s
kj
ρ
ωµ
=−
and a is the radius of the orifice. Note that Eq. (B-3) satisfies the
boundary condition of that the velocity is finite when
0r
=
. The mean velocity over the cross-
section of the tube is thus given as
()
()
2
0
1
2
0
2
2
12
1
a
s
ss s
uurdr
a
Jka
p
kkaJkax
π
π
µ
=
⎡⎤
−
∂
⎛⎞
=−
⎢⎥
⎜⎟
∂
⎝⎠
⎣⎦
∫
. (B-4)
Hence, the specific acoustic impedance of the orifice with the thickness
t is
()
()
orifice
0
0
1
0
1
2
1
t
s
ss
p
Zdx
ux
jt
Jka
kaJ ka
ωρ
∂
=
∂
=
⎡
⎤
−
⎢
⎥
⎣
⎦
∫
. (B-5)
The specific acoustic impedance of the perforate sheet with porosity
σ
is then given as
orifice
p
Z
Z
σ
=
. (B-6)
Equation (B-5) can be further approximated based on the value of
s
ka
orifice
2
00 0 0
84
3
Z
tt
j
cca c
υ
ω
ρ
≈+ (B-7)
for
1
s
ka<
, and
2
orifice
2
2
00 0
81
11
32
9
2
s
s
ka
Z
tt
j
cca c
ka
υω
ρ
⎡
⎤
⎢
⎥
⎢
⎥
≈+++
⎢
⎥
⎢
⎥
+
⎢
⎥
⎣
⎦
(B-8)
151
for
110
s
ka≤≤, and
orifice
00 0 0
22
Z
tt
j
ccaca
ωυ ωυ
ω
ρ
⎛⎞
≈+ +
⎜⎟
⎜⎟
⎝⎠
(B-9)
for
10
s
ka< . The real portion of Eq.(B-7) is known as Poiseuille coefficient of the resistance
for laminar flow in the tube, thus Equation (B-7) is also called Poiseuille model. Eq. (B-8) is
also called Helmholtz impedance model because its real part was first determined by Helmholtz.
The approximation in Eq. (B-9) was developed by Maa (1987). It is worth noting that Crandall’s
model assumes infinite tube, thus the end correction should be added to account for finite
thickness of the perforated sheet.
GE Impedance Model
Motsinger and Kraft (1995) derived the linear resistance of the orifice assuming Poiseuille
flow within the orifice. However, a plug flow was assumed when developing the reactance of
the orifice. Furthermore, Motsinger and Kraft assumed a linear resistance equivalent to a DC
flow resistance inside the orifice, thus their linear specific resistance was independent on the
frequency. The GE linear impedance model, as Motsinger and Kraft described, is
()
(
)
orifice
2
00 0
2
8
DD
kt a
Z
t
j
ccCa C
ε
υ
ρ
+
=+
, (B-10)
where
()
0.76
D
C ≈
is the dimensionless orifice discharge coefficient, and
()
0.85 1 0.7
ε
σ
=− is
the dimensionless end correction factor and
σ
is the porosity of the perforated sheet. The
specific acoustic impedance of the perforated sheet with porosity
σ
is
()
(
)
2
00 0
2
8
p
DD
Z
kt a
t
j
ccCa C
ε
υ
ρσ σ
+
=+
. (B-11)
152
End Corrections
As mentioned above, Crandall’s model assumes infinite tube, thus the end correction
should be added to account for finite thickness of the perforated sheet. This is because the flow
of air through the orifice of finite length affects the air close to the inner and outer entries. These
nearby airs are taken into the flow and thus contribute to the total acoustic impedance of the
orifice. For a circular hole of radius
a , Lord Rayleigh (1945) presented an end correction
8
3
a
δ
π
=
, (B-12)
which corresponds to the reactive part of the radiation impedance of a circular piston of radius
a
in an infinite wall. This end correction can be used as an approximation for a flanged orifice.
However, care must be taken if it is used for an orifice has no flange or not baffled, where
0.6133a
δ
. (B-13)
Ingard (1953) did an extensive study on the design of the resonators and developed
expressions for the end corrections for some circular and rectangular geometries. Assuming
uniform velocity profile in the orifice, he developed the end correction of a concentric circular
orifice in a circular tube, a circular orifice in a square tube, and a square aperture in a square tube
as
(
)
8
11.25
3
a
δ
σ
π
=− , (B-14)
where
σ
is the porosity which defines the area ratio of the aperture and tube. Eq. (B-14) is valid
for
0.16
σ
≤ .
Effect of Nonlinearity at High Sound Pressure Levels
Sivian (1935) observed the phenomenon of increasing acoustic resistance with a
corresponding increase of the orifice velocity. Since then, nonlinear behavior of the orifice at
153
high SPL has attracted more attention from researchers such as Sivian (1935), Ingard and Labate
(1950), Ingard (1953), Bies and Wilson (1957), Ingard and Ising (1967), Zinn (1970), Melling
(1973), and Cummings (1983). The effects of high SPL incident acoustic wave on the
impedance of an orifice are that the acoustic resistance increases as a function of the SPL and the
acoustic reactance stays constant but decreases after a certain SPL.
In his paper, Sivian (1935) reported that the acoustic resistance increased with a increase in
particle velocity within the orifice while the reactance was substantially independent of the
particle velocity. Sivian tested various orifices and found that the increase in acoustic resistance
for all orifices had a similar trend, the resistance was constant when particle velocity in the
orifice was small (e.g. <0.5m/s) and the resistance appeared to be linearly proportional to the
velocity when the particle velocity in the orifice was large. Sivian presented that the increase in
acoustic resistance was an effect of the increased kinetic energy in the orifice. At high particle
velocity in the orifice, the acoustic resistance of the orifice was given by
()
()
()
()
orifice
00 0
11
00
00
16
Real
32
22
11
o
ss
ss ss
V
R
jt a j
cc
Jka Jka
cc
kaJ ka kaJ ka
ωω
ρπ
⎧⎫
⎪⎪
⎪⎪
=++
⎨⎬
⎡⎤⎡⎤
′
⎪⎪
−−
⎢⎥⎢⎥
⎪⎪
′
⎣⎦⎣⎦
⎩⎭
, (B-15)
where the first term is the acoustic resistance associated with the low particle velocity in the
orifice and the second term corresponded to the nonlinear behavior of the orifice at high particle
velocity or SPL,
o
V is the root mean square (RMS) velocity of the air particle in the orifice,
s
k
′
is
the modified Stokes wave number in the orifice
s
j
k
ω
ν
′
=−
′
, (B-16)
154
where
0
ν
µρ
′′
=
is the effective kinematic viscosity under isothermal conditions (e.g., near a
highly conducting wall) and
µ
′
is the effective absolute viscosity
2
1
1
Pr
γ
µµ
−
⎡
⎤
′
=+
⎢
⎥
⎣
⎦
, (B-17)
where
γ
is the ratio of specific heats in air and Pr is the Prandtl number in air which is a constant
equal to 0.706 over a wide range of temperatures.
Ingard and Labate (1950) thoroughly studied the effect of an incident acoustic wave on an
orifice by examining the flow patterns in and around 25 circular orifices of different thickness
and diameters. They observed the large increases in the acoustic resistance when the incident
SPL is high. They also mentioned that the acoustic mass stayed constant independent of the SPL
of an incident wave until turbulence was reached in and around the orifice.
Ingard (1953) took account of the nonlinear effect of the orifice exposed to a high SPL
incident wave using an additional resistance end correction
()
orifice
00 0
2
R
tt
cca
ωυ
ρ
=
+∆
, (B-18)
where
t∆ is the viscous and nonlinear resistance end correction
22
100
n
o
V
taa
β
⎛⎞
∆= +
⎜⎟
⎝⎠
, (B-19)
where
β
and n are coefficients which may be dependent on the frequency of the incident sound
wave,
o
V is the RMS particle velocity in the orifice. Ingard found that the nonlinear acoustic
resistance was not linearly proportional to the particle velocity (
0.7, 1.7n
β
=
= were used in the
paper). However, in his paper, the particle velocity in the orifice was less than 0.6m/s. The
nonlinear resistance was believed to be linearly proportional to particle velocity in the orifice at
155
higher SPL as indicated by Bies and Wilson (1957), they measured the orifice impedance for the
orifice in which the particle velocity was up to 50 m/s. Like Sivian (1935), they found a similar
linear dependence of resistance with particle velocity in an orifice.
Ingard and Ising (1967) investigated the acoustic nonlinearity of an circular orifice in a
plate by measuring simultaneously the particle velocity in the orifice using a hot wire and the
acoustic pressure. The particle velocity in the orifice was up to 50 m/s. When particle velocity
was above 10 m/s, it was found that the acoustic resistance was linearly proportional to the
velocity
orifice_NL
00 0
o
R
B
V
cc
ρ
= , (B-20)
where
B
is the constant which lies between 1 and 1.5 and is dependent on the fluid mechanical
behavior of the orifice, and
o
V is the amplitude of particle velocity in the orifice.
Zinn (1970) investigated the interaction between high SPL incident wave and a single
Helmholtz resonator. The flow in the entrance region of the orifice and cavity was considered in
detail using the appropriate conservation equations. His analysis indicated that the energy losses
at high SPL were due to viscous damping and the dissipation of the kinetic energy of the jets
which are periodically formed at both ends of the orifice. Zinn developed the expression for the
nonlinear resistance of an orifice
orifice_NL
2
00 0
4
3
o
D
R
V
ccC
ρπ
, (B-21)
where
o
V is the amplitude of the orifice particle velocity, and
D
C is the discharge coefficient
which was 0.61 in his paper, then Eq. (B-21) results in
orifice_NL
00 0
1.16
o
R
V
cc
ρ
(B-22)
156
Melling (1973) presented an expression for the nonlinear acoustic resistance of a
perforated plate
2
p_NL
2
00 0
0.6 1
o
D
R
V
cC c
σ
ρσ
⎛⎞
−
⎜⎟
⎝⎠
, (B-23)
where
0
V is the RMS particle velocity in the orifice. Melling found that the discharge coefficient
was a function of porosity as well as Re in the orifice. In his paper, the discharge coefficients
were approximately 0.89 and 0.96 for perforates with porosities of 7.5% and 22.5%,
respectively, to fit the measured data.
A similar expression was given by Cumming and Eversman (1983). They begun with
unsteady Bernoulli’s equation and arrived at an expression for the nonlinear acoustic resistance
22
p_NL
2
00 0
1
0.57
o
C
C
R
V
C
cC c
σ
ρσ
⎛⎞
−
⎜⎟
⎝⎠
, (B-24)
where
C
C is the contraction coefficient and is defined as the ratio between the area associated
with a
vena contracta (
vc
A ) and the physical orifice area (
o
A )
vc
o
A
C
A
C =
. (B-25)
The summary on some impedance models which include the nonlinearity of the acoustic
impedance of an orifice at high SPL is listed in Table B-1.
157
Table B-1. Summary on some acoustic impedance models of an orifice.
Model Assumption Model Description Pros
Sivian
modified
Crandall
Model
(Sivian 1935)
• Turbulence is
negligible
• Adiabatic
flow
• Uniform
velocity
profile
• No internal
dissipation
()
()
()
()
orifice
00
1
0
0
nonlinear resistance w.r.t. high SPL
1
0
2
1
2
16
3
2
1
s
ss
o
s
ss
Z
jt
c
Jka
c
kaJ ka
V
c
aj
J
ka
c
kaJ ka
ω
ρ
ω
π
=
′
⎡
⎤
−
⎢
⎥
′
⎣
⎦
+
+
⎡
⎤
−
⎢
⎥
⎣
⎦
• Particle velocity
measurement needed
Ingard Model
(Ingard 1953)
() ()
orifice
00 00 0
2
,
Z
ttjtt
ccr c
ωυ ω
δ
ρ
=+∆++
t
δ
is the mass end correction,
(
)
0
1.7 1 1.25tr
δ
σ
=− for 0.4
σ
≤
• Nonlinear behavior
prediction highly
dependent on wave
frequency and other
empirical parameters,
such as the critical
thickness
Ingard and
Ising Model
(Ingard and
Ising 1967)
orifice_NL
00 0
,
o
R
V
cc
ρ
Β
=
where
Β
is constant
which lies between 1 to 1.5.
• Prediction of
nonlinear behavior
dependent on
empiricism
Zinn Model
(Zinn 1971)
• Large
amplitude of
the particle
oscillation
compared to
the orifice
thickness
_
2
00 0
4
3
NL orifice
o
D
R
V
ccC
ρπ
=
• For
0.61
D
C = , the
model predicted
Ingard(Ingard 1953;
Ingard and Ising
1967),Bies and
Wilson (Bies and
O.B. Wilson
1957)measurement
data pretty well
• Measurement of the
orifice discharge
coefficient is
necessary and
important
158
Table B-1 (Continue)
Model Assumption Model Description Pros
Melling
modified
Crandall
Model
(Melling
1973)
• Incompressible
flow
• Poiseuille flow
within the
orifice
• Only axial
velocity
component
with radial and
axial variation
• No higher
order
harmonics
produced
nonlinearity
()
()
()
()
()
()
()
00
1
0
0
2
2
0
nonlinear resistance w.r.t. high SPL
1
0
00
2
1
1.2 1
2
16
2
3
1
p
s
ss
o
D
s
ss
Z
jt
c
Jka
c
kaJ ka
V
cC
aj
J
ka
c
kr J ka
ω
ρ
σ
σ
σ
ω
πσ
σ
=
′
⎡⎤
−
⎢⎥
′
⎣⎦
−
+
+
⎡
⎤
Ψ
−
⎢
⎥
⎣
⎦
• Interaction between
orifices included
• End correction
included
• Empirical parameter
dependent
• Particle velocity
measurement needed
Cummings
and Eversman
Model
(Cummings
and Eversman
1983)
• Zero means
flow
• Sinusoidal
incident
pressure signal
(
)
22
_
2
1
0.45 ,
c
NL orifice
C
VC
R
c
cC
σ
ρ
−
=
where
C
C is orifice contraction
coefficient which defined as the
ratio between the area associated
with a vena contracta and the
physical orifice area.
GE Impedance
Model
(Motsinger
and Kraft
1991)
• Incompressible
flow
• Poiseuille flow
within the
orifice when
developing the
linear
resistance term
• Plug flow
within the
orifice when
developing the
reactance term
()
()
()
2
00
0
linear resistance
2
nonlinear resistance w.r.t high SPL
reactance
8
1
2
2
p
D
o
D
D
Z
t
c
cCa
V
cC
kt a
C
υ
ρ
σ
σ
ε
σ
=
+
+
+
• Simple and
straightforward
• Assumptions for the
resistance and
reactance derivation
are different
• Empirical parameters
dependent
• Measurement of the
velocity at the orifice
or related parameters
required
• Frequency
independent
159
Table B-1 (Continue)
Model Assumption Model Description Pros
Kraft et al.
modified
Crandall
Model
(Kraft et al.
1999)
()
()
()
()
()
()
00
1
0
0
2
2
0
nonlinear resistance w.r.t. high SPL
1
0
0
2
1
1
2
16
2
3
1
p
s
D
ss
o
D
s
D
ss
Z
jt
c
Jka
cC
kaJ ka
V
cC
aj
J
ka
cC
kaJ ka
ω
ρ
σ
σ
σ
ω
πσ
σ
=
′
⎡⎤
−
⎢⎥
′
⎣⎦
−
+
+
⎡
⎤
Ψ
−
⎢
⎥
⎣
⎦
160
APPENDIX C
PARAMETERS ESTIMATION FOR LEM OF THE PIEZOELECTRIC DIAPHRAGM
The extraction of the model parameters for the piezoelectric backplate is more complex
due to composite plate mechanics. The piezoelectric backplate consists of an axisymmetric
piezoceramic of radius
1
R
and thickness
p
h bonded in the center of a metal shim of radius
2
R
and thickness
s
h . Up to and just beyond the first resonant mode, the one-dimensional
piezoelectric electroacoustic coupling is given by (Prasad et al. 2006)
aD a
aeF
Cd
P
dC
qV
∆∀
⎧⎫
⎧⎫ ⎧⎫
=
⎨⎬⎨ ⎬⎨⎬
⎩⎭ ⎩⎭
⎩⎭
, (C-1)
where
∆∀ is the volume displacement of the piezoelectric backplate due to the application of the
pressure
P and voltage
V
. Additionally, q is the charge stored on the piezoelectric electrodes,
a
d is the effective acoustic piezoelectric coefficient, and
eF
C is the electrical free capacitance of
the piezoelectric material. The volume displacement is calculated by integrating the transverse
displacement,
()
wr, over the whole plate
()
2
0
2
R
rw r dr
π
∆∀ =
∫
. (C-2)
Thus, the short-circuit acoustic compliance of the backplate,
aD
C , is determined by
()
2
0
0
0
1
2
aD
V
R
V
C
P
rw r dr
P
π
=
=
∆∀
=
=
∫
. (C-3)
Similarly, the effective acoustic piezoelectric coefficient,
A
d , is determined by application
of voltage to the free piezoelectric plate
161
()
2
0
0
0
1
2
A
P
R
P
d
V
rw r dr
V
π
=
=
∆∀
=
=
∫
. (C-4)
The electroacoustic impedance transformer factor,
φ
, is defined as
A
aD
d
C
φ
−
= . (C-5)
The effective acoustic mass of the piezoelectric backplate,
aD
M
, is found by equating the
total distributed kinetic energy stored in the velocity of the plate to a lumped mass as
()
2
2
0
0
2
()
R
aD A
V
M
wr rdr
V
π
ρ
=
=
∆
∫
, (C-6)
where
A
ρ
is the area density of the piezoelectric backplate. The blocked electrical capacitance,
eB
C , is related to the free electrical capacitance of the piezoelectric backplate,
eF
C , as
(
)
()
2
2
2
01
1
1
eB eF EM
r
EM
p
CC
R
h
κ
εεπ
κ
=−
=−
, (C-7)
where
r
ε
is the relative dielectric constant of the piezoelectric material,
0
ε
is the permittivity of
free space,
1
R
is the radius of the piezoceramic, and
p
h is the thickness of the piezoceramic, and
2
EM
κ
is the electroacoustic coupling factor and given by
2
2
A
EM
eF aD
d
CC
κ
= . (C-8)
The acoustic resistance,
aD
R
, of the piezoelectric backplate models acoustic resistance and
structural damping in the backplate. The damping may arise from thermo-elastic dissipation,
compliant boundaries, and other intrinsic loss mechanisms. The acoustic resistance is given by
162
2
aD aDrad
aD
aD
MM
R
C
ζ
+
=
, (C-9)
where
ζ
is an experimentally determined damping factor determined using, for example, the
logarithmic decrement method (Meirovitch 2001).
163
APPENDIX D
WAVE SCATTERING BY HELMHOLTZ RESONATOR IN A TUBE
When using the TMM to measure the acoustic impedance of the EMHR (or HR), it is
assumed that only the plane wave propagates along the tube, and two microphones should be
properly mounted certain distance away from the impedance sample. This is because when the
PWT terminated by an EMHR or HR, eventhough a plane wave mode is sent from the source,
higher order models will be excited and scattered at the interface between the tube and the HR.
If the higher order modes can reach one or two microphones, the measurement accuracy is
significantly affected. Thus, it is necessary to solve the acoustic field within the tube to
determine the proper distance away from the sample two microphones should be. After such a
distance, all higher order modes are far gone in decay.
Helmholtz Resonator as a Termination of a Circular Tube
A Helmholtz resonator terminates a circular, solid-walled tube with radius
0
R
, as shown in
Figure D-1. The lossless linear wave equation in the tube is in the form
2
2
22
0
1
0
ct
φ
φ
∂
∇
−=
∂
, (D-1)
where
φ
is the velocity potential and
0
c is the isentropic small-signal speed of sound. Moreover,
for a harmonic signal
()
,
jt
rxe
ω
φ
=Φ , the equation can be simplified as
22
0k
∇
Φ+ Φ= , (D-2)
where
0
kc
ω
= is the wave number, and
(
)
,rzΦ=Φ is the complex magnitude of the velocity
potential
φ
. To resolve Eq. (D-2), the following assumptions are taken into account
(1) The axis of HR coincides with axis of the tube.
(2) At the opening of HR, the velocity distribution is uniform and axial symmetrical.
164
(3) The acoustic filed within the tube is independent of
θ
.
Then, using the method of the separation of variables, the solution of Eq. (D-2) is given by
()
(
)
(
)
00
zz
jk z jk z
rr
A
eBeCJkrDYkr
−
Φ= + +
⎡
⎤
⎣
⎦
, (D-3)
where
A,
B
,
C
and D are coefficients.
z
k is the trace wavenumber in the z direction, and
22
rz
kkk=−
is the trace wavenumber in the r direction.
(
)
0
J
is the zeroth order Bessel
function of the first kind, and
()
0
Y is the zeroth order Bessel function of the second kind. Note
that the acoustic properties in the circular tube are finite. The coefficient
D in Eq.(D-3) thus
goes to zero due to
(
)
0
at 0
r
Ykr r→∞ → . (D-4)
Moreover, the wall of the circular tube is solid enough. The normal component of the particle
velocity thus vanishes
0
0at rR
r
∂
Φ
=
=
∂
. (D-5)
From Eqs. (D-3) and (D-5), one has
(
)
10
0
r
JkR
=
, (D-6)
which yields
00
,0,1,2,
rn
kR n
α
′
=
= … (D-7)
where
0n
α
′
is the root of
()
1
0Jx=
. The complete solution for Eq.(D-2) is the sum of all the
modes which correspond to different roots of
(
)
1
0Jx
=
0
0
0
0
zz
ik z ik z
n
nn
n
Ae Be J r
R
α
∞
−
=
⎛⎞
′
⎡⎤
Φ= +
⎜⎟
⎣⎦
⎝⎠
∑
, (D-8)
165
where
()
2
2
00zn
kk R
α
′
=−
. Furthermore, by defining
2
n
nn
AB e
β
−
=− (D-9)
where
n
β
is a complex number, Eq. (D-8) is rewritten as
[]
0
0
0
0
0
0
0
0
0
0
0
0
2sinh
zz
nznzn
n
ik z ik z
n
nn
n
ik z ik z
n
n
n
n
nzn
n
Ae Be J r
R
B
ee e J r
R
Be ikz J r
R
βββ
β
α
α
α
β
∞
−
=
∞
−−− +
=
∞
−
=
⎛⎞
′
⎡⎤
Φ= +
⎜⎟
⎣⎦
⎝⎠
⎛⎞
′
⎡⎤
=−+
⎜⎟
⎣⎦
⎝⎠
⎛⎞
′
=+
⎜⎟
⎝⎠
∑
∑
∑
. (D-10)
Hence, the particle velocity at the opening of the HR
(
)
0z
=
is given by
()
[]
[]
0
0
0
0
0
0
0
0
0
0
2cosh
2cosh
n
n
z
n
zn z n
n
z
n
zn n
n
ur
x
ik B e ik z J r
R
ik B e J r
R
β
β
φ
α
β
α
β
=
∞
−
=
=
∞
−
=
∂
=
∂
⎛⎞
′
=+
⎜⎟
⎝⎠
⎛⎞
′
=
⎜⎟
⎝⎠
∑
∑
. (D-11)
On the other hand, at the opening of the HR (
0z
=
) the particle velocity is in the form
()
00
00
0
0
Urr
uur
rrR
≤
≤
⎧
==
⎨
<
≤
⎩
(D-12)
Then, the Fourier-Bessel series expression for
(
)
ur is given by (Blackstock 2000)
()
0
0
0
0
n
n
n
ur aJ r
R
α
∞
=
⎛⎞
′
=
⎜⎟
⎝⎠
∑
, (D-13)
where
n
a is the coefficient. Comparing (D-13) with Eq.(D-11) results in
()
2cosh
n
n
n
zn
a
B
ik e
β
β
−
=
(D-14)
166
Plugging Eq (D-14) into (D-10) leads to
()
[]
00
0
0
sinh
cosh
n
zn n
n
zn
a
r
iikzJ
kR
φβα
β
∞
=
⎛⎞
′
=− +
⎜⎟
⎝⎠
∑
. (D-15)
Furthermore, to integrate both sides of Eq. (D-13) as follows
()
00
000
000
0
000
00
RR
nnn
n
n
u r J r rdr a J r J r rdr
RRR
ααα
∞
′
=
⎛⎞ ⎛⎞⎛⎞
′′
=
⎜⎟ ⎜⎟⎜⎟
⎝⎠ ⎝⎠⎝⎠
∑
∫∫
, (D-16)
One has
()
() ( )
0
00
22
00 0 0
0
2
R
nn
n
aurJrrdr
RJ R
α
α
=
∫
, (D-17)
where the following orthogonality relation is used
0
00
00
00
0
1
2
000 00
0000
0
1
0
000 10 10 010
000 0 00 0 0
0
R
nn
nn
nn nnn
nn
JrJrrdr
RR
rrrr
RJ J d
RRRR
rr r r r r
RJ J J J
RR R R R R
αα
αα
αα ααα
αα
′
′
′′
′′
⎛⎞⎛⎞
′′
⎜⎟⎜⎟
⎝⎠⎝⎠
⎛⎞⎛⎞⎛⎞⎛⎞
′′
=
⎜⎟⎜⎟⎜⎟⎜⎟
⎝⎠⎝⎠⎝⎠⎝⎠
⎡⎤
⎛⎞⎛⎞⎛ ⎞⎛⎞⎛ ⎞ ⎛
′′ ′′′
=−−
⎢⎥
⎜⎟⎜⎟⎜ ⎟⎜⎟⎜ ⎟ ⎜
′′
⎝⎠⎝⎠⎝ ⎠⎝⎠⎝ ⎠ ⎝
⎣⎦
∫
∫
()
() ()
()
() () ()
1
0
0
1
2
0
01010
00000
0
2
2
0
10
2
2
0
00 20
2
2
0
00 10 00
0
2
1
22
12
22
n
nn
n
nnn
nnnn
nnn
n
r
d
R
rrrr
RJ J d
RRRR
R
J
R
JJ
R
JJJ
R
α
αα
α
αδ
ααδ
αααδ
α
′
′
′
′
⎧ ⎫
⎡
⎤
⎞⎛ ⎞
⎪ ⎪
⎨ ⎬
⎢
⎥
⎟⎜ ⎟
⎠⎝ ⎠
⎣
⎦
⎪ ⎪
⎩ ⎭
⎛ ⎞⎛ ⎞⎛⎞⎛⎞
′
′′
=
⎜ ⎟⎜ ⎟⎜⎟⎜⎟
′
⎝ ⎠⎝ ⎠⎝⎠⎝⎠
′′
=
⎡⎤
⎣⎦
⎧⎫
′′
=−
⎨⎬
⎩⎭
⎧⎫
⎛⎞
⎪⎪
′′′
=−+
⎨⎬
⎜⎟
⎪⎪
⎝⎠
⎩⎭
∫
∫
()
2
2
0
00
2
nn
nnn
R
J
αδ
′
′
′
=
(D-18)
167
where
nn
δ
′
is the Kronecker delta function. Substitution Eq. (D-12) into (D-17) results in
()
2
0000
aUrR= , (D-19)
and
()()
00 0
10
2
000 0 0
2
1
nn
nn
Ur r
aJn
JR R
α
αα
⎛⎞⎛ ⎞
′
=≥
⎜⎟⎜ ⎟
′′
⎝⎠⎝ ⎠
. (D-20)
Using the Euler equation, the pressure in the tube is given by
()
[]
()
[]
()
[]
() ()()
0
00
0
00 0
0
0
0
00 0
0
0
2
0
00
00 0 0 0 00 0 1
2
000000
I
sinh
cosh
sinh
cosh
sinh
2
cosh
nn
zn
n
zn
nn
zn
n
nn
n
nn
p
t
ick
ar
ck ikz J
kR
ar
cikzJ
R
ikz
rr
cU r R cU J
RJ R
φ
ρ
ρ
α
ρβ
β
α
ρβ
ϑβ
β
α
ρρ
βαα
∞
=
∞
=
∂
=−
∂
=− Φ
⎛⎞
′
=− +
⎜⎟
⎝⎠
⎛⎞
′
=− +
⎜⎟
⎝⎠
+
⎛⎞ ⎛ ⎞
′
=− −
⎜⎟ ⎜ ⎟
′′
⎝⎠ ⎝ ⎠
∑
∑
[]
()
0
0
1
0
II
sinh
cosh
zn
n
n
nn
ik z
r
J
R
β
α
ϑβ
∞
=
+
⎛⎞
′
⎜⎟
⎝⎠
∑
(D-21)
where
nz
kk
ϑ
= , and the first term of Eq (D-21) is plane wave contribution while the second
term is caused by the higher modes scattered back from the HR. Furthermore, the tube is
supposed to be long enough and the exciting frequency is low, so that there is only plane wave
propagating through the whole tube, accordingly
01
n
An
=
≥ , (D-22)
which results in
n
β
=∞, then
[
]
()
sinh
lim
cosh
z
n
zn
ik z
n
ik z
e
β
β
β
→∞
⎧⎫
+
⎪⎪
=
⎨⎬
⎪⎪
⎩⎭
. (D-23)
Thus, Eq. (D-21) can be simlified as
168
()
[
]
() ()()
2
0
00000
00 0 0 00 0 1 0
0
2
1
0000000
H.O.M.
sinh
2
cosh
z
ik z
nn
n
nn n
ikz
rrr
p
cU r R cU J J e
RJ RR
β
αα
ρρ
βϑαα
∞
=
+
⎛⎞ ⎛ ⎞⎛ ⎞
′′
=− −
⎜⎟ ⎜ ⎟⎜ ⎟
′′
⎝⎠ ⎝ ⎠⎝ ⎠
∑
.
(D-24)
Furthermore, the pressure in the tube can also be expressed as
(
)
[]
()
0
00
00
H.O.M
2 sinh H.O.M
ikz ikz
pPe Pe
Pe ikz
β
β
+− −
+
=++
=− + +
(D-25)
where
0
P
+
and
0
P
−
are the incident and reflected (plane wave components) wave amplitude at
0x = , respectively, and H.O.M. represents the higher order modes, and the definition of
β
is
given by Eq. (D-9). Hence, comparing Eq (D-25) and (D-24) results in
()
()
0
2
00 0 0 0
0
0
2
cosh
cU r R
Pe
β
ρ
β
+
= (D-26)
or
()()
0
2
00 0 0 0 0 0
2coshcU P e R r
β
ρβ
+
=
. (D-27)
Furthermore, at
0z = , the relationship between the pressure and the velocity is given by
()
[
]
() ()()
()
[]
()
()
2
0
00 00
00 0 00 1 0
2
1
0000000
2
0
00 0
0
00
sinh
2
cosh
sinh
cosh
nn
n
nn n
HR HR
rr
p
crR c J J
UJRR
crR
cj
β
αα
ρρ
βϑαα
β
ρ
β
ρθ χ
∞
=
⎛⎞⎛⎞
′′
=− −
⎜⎟⎜⎟
′′
⎝⎠⎝⎠
≈−
=+
∑
, (D-28)
where
H
R
θ
and
H
R
χ
are the specific acoustic resistance and reactance of Helmholtz resonator,
respectively. Thus, from Eq (D-28), one has
() ()( )
2
000
tanh
H
RHR
Rr j
βθχ
=− + . (D-29)
169
Finally, from Eqs. (D-24)-(D-27), the pressure in the tube when it is terminated by a HR is given
by
[
]
()()
()()
0
0
00
2
00000
0000 10
2
1
0000 0 0
2sinh
21
2cosh
z
ik z
nn
n
nn n
pPe ikz
rrr
Pe R r J J e
RJ R R
β
β
β
αα
β
αα ϑ
+
∞
+
=
=− +
⎛⎞ ⎛ ⎞⎛ ⎞
′′
−
⎜⎟ ⎜ ⎟⎜ ⎟
′′
⎝⎠ ⎝ ⎠⎝ ⎠
∑
, (D-30)
Helmholtz Resonator as a Termination of a Rectangular Tube
A HR terminates a rectangular tube with cross-sectional area, ab , shown in Figure D-2.
The center of the opening of Helmholtz resonator locates at
(
)
,,0
cc
ab
. To simplify the problem,
the following assumptions are taken into account
(1) The wall of the duct is rigid.
(2) The acoustic filed in the tube is lossless and linear.
(3) At the opening of HR, the velocity distribution is uniform;
Thus, using the method of the separation of variables, the solution of Eq. (D-2) is given by
() ()
(
)
(
)
cos sin cos sin
zz
jk z jk z
xx y y
A k x B k x C k y D k y Ee Fe
−
⎡⎤
⎡
⎤
Φ= + + +
⎡⎤
⎣⎦
⎣
⎦
⎣⎦
, (D-31)
where
A
,
B
, C , D ,
E
and F are coefficients, and
x
k ,
y
k and
z
k are the trace wavenumbers
in the
x
, y and z directions. Furthermore, a rigid-wall of rectangular tube means that the
normal component of particle velocity vanishes at each solid wall
0at 0,
x
a
x
∂
Φ
==
∂
. (D-32)\
and
0at 0,yb
z
∂
Φ
==
∂
. (D-33)
Substitution of Eq. (D-31) into Eqs.(D-32) and (D-33) results in
00,1,2
x
Bkmam
π
== =… (D-34)
170
and
00,1,2
y
Dknbn
π
=
==… (D-35)
Plugging Eqs.(D-34) and (D-35) into (D-31) and collecting the coefficients results in
()
(
)
{
}
cos cos
mn mn
jk z jk z
mn mn mn
mxa nyb Ae Be
ππ
−
Φ= + , (D-36)
where
mn
k is the z -direction trace wave number
()()
22
2
mn
kkmanb
ππ
=− − . (D-37)
Physically, indices
m and n represent the number of half wave in the direction of
x
and y . The
combination
()
,mn names the acoustic mode in the duct. The complete solution for the
magnitude of the velocity potential is thus the sum of all the modes
()()
{}
00
cos cos
mn mn
jk z jk z
mn mn
mn
mxa nyb Ae Be
ππ
∞∞
−
==
Φ= +
∑∑
(D-38)
Similarly, by defining
2
mn
mn mn
AB e
β
−
=− (D-39)
one has
()() ( )
00
2cos cos sinh
mn
mn mn mn
mn
mxa nybBe jkz
β
π
πβ
∞∞
−
==
Φ= +
∑∑
, (D-40)
and
()() ( )
00
2 cos cos cosh
mn
mn mn mn mn
mn
zj mxa nybBek jkz
β
π
πβ
∞∞
−
==
∂Φ ∂ = +
∑∑
. (D-41)
Moreover, it is assumed that velocity distribution at the opening of HR (
0z =
) is
()
00
0
0otherwise
Urr
ur
≤
≤
⎧
=
⎨
⎩
. (D-42)
Thus, from Eq. (D-41) and (D-42), one has
171
()() ()
()
0
00
0
2 cos cos cosh
,
mn
mn zmn mn
z
mn
zj mxanybBek
UH xy
β
π
πβ
∞∞
−
=
==
∂Φ ∂ =
=
∑∑
, (D-43)
where
()
,Hxy is Heaviside unit step function, which equal one at the opening of HR, was zero
at other place. Furthermore, to integrate both sides of Eq. (D-43) as follows
()()
()()()() ()
0
opening
,0,0
00
cos cos
2 cos cos cos cos cosh
mn
ab
mn zmn mn
mm nn
Umxanybdxdy
j m xa n yb m xa n ybB e k dxdy
β
ππ
ππ π π β
∞∞
−
′′
==
′′
′′
=
∫∫
∑∑
∫∫
(D-44)
one has
(
)
2cosh
mn
mn mn zmn mn
RHS v jB e k
β
β
−
⎡
⎤
=
⎣
⎦
, (D-45)
where RHS is the right hand side of Eq. (D-44), and
for 0
2 for 0, 0 or 0 0
4for 0
mn
ab m n
vab mn mn
ab m n
=
=
⎧
⎪
==≠≠=
⎨
⎪
=
≠
⎩
. (D-46)
When
0mn==, the LHS of Eq.(D-44) is
2
00
LHS U r
π
= . (D-47)
Thus, the coefficient
00
B
is given by
()
00
2
00
00
00
2cosh
Ur
Bj
abe k
β
π
β
−
=− . (D-48)
For the case of which
0mn=≠ or mn
≠
but one of them equals zero, one can solve the
coefficients
mn
B
using a transformation between Cartesian coordinate system and polar system
for the opening of HR, as shown in Figure D-3
172
(
)
()
0
cos
sin
:0 2
:0
c
c
x
ra
yr b
rrr
ϕ
ϕ
ϕπ
=
+
=
+
→
≤≤
. (D-49)
Thus, the LHS of the Eq.(D-44) can be related to
()
(
)
() ()
0
0
opening
2
0
00
I
cos cos
cos cos cos sin
r
cc
LHS U m x a n y b dxdy
Umraanrbbrdrd
π
ππ
π
ϕπϕϕ
=
=++⎡⎤⎡⎤
⎣⎦⎣⎦
∫∫
∫∫
. (D-50)
Further manipulation of (I) results in
() ( )
(
)
()()
()()
[][]
[][]
Icos cos cos sin
cos cos sin
1
2
cos cos sin
cos cos cos sin
sin sin cos sin
1
2
c
cc
cc
cc
cc
cc
mr aa nr bb
mr aanr bb
mr aanr bb
maanbb mr anr b
maanbb mr anr b
πϕ πϕ
πϕ πϕ
πϕ πϕ
ππ πϕπϕ
ππ πϕπϕ
′′
=+ +⎡⎤⎡⎤
⎣⎦⎣⎦
⎧⎫
′′
++ + +
⎡⎤
⎪⎣ ⎦⎪
=
⎨⎬
′′
+− +⎡⎤
⎪⎪
⎣⎦
⎩⎭
′′ ′ ′
++−
′′ ′ ′
+++
=
[][]
[][]
[]
()
[]
()
[]
()
os cos cos sin
sin sin cos sin
cos cos cos
sin sin cos
1
2
cos cos cos
sin
cc
cc
cc
cc
cc
c
maanbb mr anr b
maanbb mr anr b
maanbb r
maanbb r
maanbb r
maa
ππ πϕπϕ
ππ πϕπϕ
ππ ϕϑ
ππ ϕϑ
ππ ϕϑ
π
⎧⎫
⎪⎪
⎪⎪
⎨⎬
′′ ′ ′
−−−
⎪⎪
⎪⎪
′′ ′ ′
−−
⎩⎭
′′ ′
+−−
⎡⎤
⎣⎦
′′ ′
+−+
⎡⎤
⎣⎦
=
′′ ′
−+−⎡⎤
⎣⎦
′′
−
[]
()
sin cos
c
nbb r
πϕϑ
⎧⎫
⎪⎪
⎪⎪
⎪⎪
⎨⎬
⎪⎪
⎪⎪
′
+⎡⎤
⎪⎪
⎣⎦
⎩⎭
, (D-51)
where
()()
22
tan
rmanbr
na mb
ππ
ϑ
′′ ′
=+
′′
=
. (D-52)
Furthermore, the integration over
ϕ
in Eq.(D-51) and
(
)
,mmnn
′
′
==
leads to
() ()()()
2
0
0
I2cos cos
cc
dmaanbbJr
π
ϕπ π π
′
=
∫
, (D-53)
173
where some identities as follows are used
() ()
()
()
()
22
00
2
0
0
coscos coscos
cos cos
cos cos
2
rdrd
rd
rd
Jr
ππ
π
π
π
ϕ
ϑϕ ϕϑϕ
ϕϕ
ϕϕ
π
−
′′
−= +⎡⎤⎡⎤
⎣⎦⎣⎦
′
=
⎡⎤
⎣⎦
′
=
⎡⎤
⎣⎦
′
=
∫∫
∫
∫
, (D-54)
() ()
()
()
22
00
2
0
sin cos sin cos
sin cos
sin cos
0
rdrd
rd
rd
ππ
π
π
π
ϕ
ϑϕ ϕϑϕ
ϕϕ
ϕϕ
−
′′
−= +
⎡⎤⎡⎤
⎣⎦⎣⎦
′
=
⎡⎤
⎣⎦
′
=⎡ ⎤
⎣⎦
=
∫∫
∫
∫
, (D-55)
() ()
0
1
cos sin
n
Jr r nd
π
θ
θθ
π
′′
=−
∫
, (D-56)
and
() ()
cos
0
cos
n
ir
n
i
Jr e nd
π
θ
θ
θ
π
−
′
′
=
∫
. (D-57)
Furthermore, the integration over
r
in Eq. (D-53) results in
()()()
()()()
()()
()()
()()
()
0
0
0
0
0
0
22
0
10
22
2cos cos
2cos cos
2cos cos
r
cc
r
cc
cc
ma a nbbJ r rdr
ma a nbb J r rdr
ma a nbbr
Jma nbr
ma nb
ππ π
ππ π
ππ π
ππ
ππ
′
′
=
=+
+
∫
∫
, (D-58)
where the identity for the Bessel function is used
174
(
)
(
)
01
rJ r dr rJ r=
∫
. (D-59)
Thus, the LHS of Eq. (D-44) is
()
(
)
()()
()()
(
)
22
0
010
22
2cos cos
cc
ma a nbbr
LHS U J m a n b r
ma nb
ππ π
ππ
ππ
=+
+
. (D-60)
Then, one can arrive at the following expression for the coefficients
mn
B
from Eqs. (D-44)
,(D-45) and (D-60)
()
(
)
(
)
()()
()()
(
)
22
0
0
10
22
cos cos
cosh
mn
cc
mn
mn zmn mn
ma a nbbr
U
B
jJmanbr
ek
ma nb
β
ππ
π
ππ
νβ
ππ
−
=− +
+
(D-61)
for
0mn=≠, or 0, 0mn=≠ , or 0, 0mn
≠
= . The complex amplitude of the velocity potential
is thus given by
()
()
()()
()
()
()()
()()
()()( )
2
00
00
00
I
22
01 0
22
0
II
sinh
cosh
cos cos
2
cosh
cos cos sinh
cc
mn zmn mn
mn
mn mn
Ur
jjkz
abk
rJ m a n b r
ma a nbb
jU
k
ma nb
mxa nyb jkz
π
φβ
β
ππ
ππ
π
νβ
ππ
ππ β
=− +
⎧⎫
+
⎪⎪
⎪⎪
−
⎨⎬
+
⎪⎪
⎪⎪
×+
⎩⎭
∑∑
,
(D-62)
where;
mn
ν
is defined by Eq. (D-46), and
zmn
k is given by Eq. (D-37). The first part of
Eq.(D-62) represents the planar incident wave with its reflected one, while the second part is for
the higher order modes excited around the discontinuity.
The pressure in the tube is thus obtained by using Euler equation
175
()
()
()()
()
()
()()
()()
()()( )
0
2
00 00
00
00
22
01 0
22
00 0
sinh
cosh
cos cos
2
cosh
cos cos sinh
cc
mn mn mn
mn
mn mn
p
t
cUr
jkz
ab
rkJ m a n b r
ma a nbb
cU
k
ma nb
mxa nyb jkz
ρ
ρπ
β
β
ππ
ππ
ρπ
νβ
ππ
ππ β
∂Φ
=−
∂
=− +
⎧⎫
+
⎪⎪
⎪⎪
−
⎨⎬
+
⎪⎪
⎪⎪
×+
⎩⎭
∑∑
.
(D-63)
The tube is long enough and the exciting frequency is low, so that there is only plane wave
propagating through the whole tube, accordingly
0
mn
A
=
(D-64)
and
mn
β
→∞ (D-65)
for
()()
,0,0mn ≠ . Hence, one has
[
]
()
sinh
lim
cosh
mn
mn
mn mn ik z
mn
ik z
e
β
β
β
→∞
⎧⎫
+
⎪⎪
=
⎨⎬
⎪⎪
⎩⎭
. (D-66)
Eq. (D-63) is then rewritten as
()
()
()()
()()
()()
()
()
()
2
00 00
00
00
0
22
22
00 0 1 0
H.O.M
sinh
cosh
cos cos
2cos
cos
mn
cc
mn mn
mn
ik z
cUr
pjkz
ab
rmaanbb
k
k
ma nb
cU J ma nbr mxa
nybe
ρπ
β
β
ππ
ν
ππ
ρπ π π π
π
=− +
⎛⎞
⎜⎟
+
⎜⎟
⎜⎟
⎜⎟
−×+
⎜⎟
⎜⎟
×
⎜⎟
⎜⎟
⎝⎠
∑∑
. (D-67)
176
Furthermore, the pressure in the tube can also be expressed as
(
)
[]
()
00
00
000
H.O.M
2 sinh H.O.M
ikz ikz
pPe Pe
Pe ikz
β
β
+− −
+
=++
=− + +
, (D-68)
where
0
P
+
and
0
P
−
are the incident and reflected plane wave amplitude at 0z = , respectively.
The definition of
00
β
is the same as Eq.(D-39). Comparing Eqs.(D-68) and (D-67) results in
()
00
2
00 0 0
0
00
2
cosh
cU r
Pe
ab
β
ρπ
β
+
= (D-69)
or
(
)
(
)
00
2
00 0 0 00 0
2coshcU P e ab r
β
ρβπ
+
= . (D-70)
At
0z =
, the relationship between the pressure and the velocity is given by
()
(
)
(
)
()()
()()
()
()
()
()
()
()
22
2
22
00 0
00 0 0 1 0
0
2
00 0 00
00
cos cos
tanh 2 cos
cos
tanh
cc
mn mn
mn
HR HR
ma a nbb
k
k
ma nb
cr
p
cJmanbrmxa
Uab
nyb
crab
cj
ππ
ν
ππ
ρπ
βρπ π π π
π
ρπ β
ρθ χ
⎧
⎫
⎪
⎪
+
⎪
⎪
⎪
⎪
⎪
⎪
=− − × +
⎨
⎬
⎪
⎪
⎪
⎪
×
⎪
⎪
⎪
⎪
⎩⎭
≈−
=+
∑∑
.
(D-71)
Thus, one has
()
(
)
(
)
2
00 0
tanh
H
RHR
ab r j
βπθχ
=− +
. (D-72)
Finally, from Eqs. (D-67) - (D-71), one has the pressure in the tube when it is terminated by a
HR
178
0
R
r
z
0
r
Figure D-1: A Helmholtz resonator as termination of a circular tube.
(
)
0,0,0
y
z
x
b
a
(
)
,,0
cc
ab
Figure D-2: Schematic of a rectangular tube terminated by a Helmholtz resonator.
179
x
z
ϕ
r
0
r
b
a
()
0,0
c
b
c
a
Figure D-3: Schematic of transform between the Cartesian coordinate system and polar system.
180
APPENDIX E
GEOMETRIC DIMENSION OF EMHRS
As discussed in Chapter 4, the EMHRs are modularly constructed, as shown in Figure E-1.
The engineering draft of the EMHR (Case I) is presented as follows.
Figure E-1. Schematic of the EMHR which consists of, from left to right, an orifice plate, cavity
plate, piezoelectric diaphragm bottom plate and piezoelectric diaphragm cap plate.
Figure E-2. Engineering draft of the orifice plate.
181
Figure E-3. Engineering draft of the orifice plate of the cavity plate.
182
Figure E-4. Draft of the piezoelectric diaphragm bottom plate.
183
Figure E-5. Draft of the piezoelectric diaphragm cap plate.
184
APPENDIX F
COMPUTER CODES
In this appendix, the MATLAB codes used to predict the acoustic impedance and to
implement the optimization design of an EMHR are presented.
Acoustic Impedance Prediction using LEM and TM
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% EMHR impedance calculation
%
% Variable Definitions
%
% -------------------------------------------------------
%
%
% Dtube: the diameter of the PWT [m]
% r: the radius of the neck (m)
% t: the thickness of the neck (m)
% R: the radius of the cavity (m)
% L: the depth of the cavity (m)
% as: the radius of the shim of the PZT backplate (m)
% ts: the thickness of the shim of the PZT backplate (m)
% ap: the radius of the piezoceramic of the PZT backplate (m)
% tp: the thickness of the piezoceramic of the PZT backplate (m)
% Es: Youngs modulus of the shim (N/m^2)
% vs:Poisson ratio of the shim
% rhos: density of the shim [kg/m^3]
% Ep: Youngs modulus of the piezoceramic (N/m^2)
% vp: Poisson ratio of the piezoceramic
% rhop: density of the piezoceramic [kg/m^3]
% d31: the piezo strain constant d31
% dp: relative dilectric constant of the piezoceramic
% damp: the damping coefficient of the PZT backplate
%
% Format for Frequency file (freq.txt)
% ---------------------------------------------------------
% f1, f2 ... fn
% where the frequencies are provided in Hz
%
%
% Format for the geometry and material properties of the EMHR input file
% (input.txt)
% ------------------------------------------------------------------
% Dtube
% r
% t
% R
% L
% as
% ts
% ap
% tp
% Es
185
% vs
% rhos
% Ep
% vp
% rhop
% d31
% dp
% damp
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function EMHR()
global rho0 t0 c0 visc Pref
global c rho freqs
disp (' ')
disp (' ')
disp ('---------------------------------------------')
disp ('EMHR impedance calculation - linear ')
disp (' ')
disp ('---------------------------------------------')
disp (' ')
file = input ('Please enter output filename: ','s');
fid=fopen(file,'w');
fprintf(fid, 'Output File is:%12s\n\n',file);
disp (' ')
% Date and Time output
b=fix(clock);
time=[num2str(b(1)),'-',num2str(b(2)),'-',num2str(b(3)),'-',...
num2str(b(4)),'-',num2str(b(5))];
fprintf(fid, 'Time (yy-mm-dd-hh-mm):%20s\n\n',time);
%Initialization of temperature, density of air
disp(' ')
disp('----Initialization of temprature, density of air----')
disp(' ')
tc = input ('Please enter temperature (C): ');
fprintf(fid,'Temperature is (C):%5.2f\n',tc);
rho = rho0*t0/(t0+tc);
c = c0*sqrt(1+tc/t0);
fprintf(fid,'Density is(kg/m^3):%5.2f\n',rho);
fprintf(fid,'Sound speed is(m/s):%5.2f\n\n',c);
disp('----------------------------------------------------')
%Frequency initial
disp (' ')
disp ('----Frequency Initialization----')
disp (' ')
ifrev = input('Frequencies evenly spaced? (Yes --1, No--2): ');
disp(' ')
if ifrev == 1
f1 = input ('Please enter beginning frequency (Hz): ');
f2 = input ('Please enter stopping frequency (Hz): ');
df = input ('Please enter delta frequency (Hz): ');
fprintf(fid,'Beginning Freq.(Hz): %12.2f\n',f1);
186
fprintf(fid,'Stopping Freq.(Hz): %12.2f\n',f2);
fprintf(fid,'Deta Freq.(Hz): %12.2f\n',df);
nf = (f2-f1)/df+1;
fprintf(fid,'# of computing points: %6.1f\n\n',nf);
freqs = f1:df:f2;
else
filefr = input ('Enter Filename containing frequencies :','s');
freqstemp = importdata(filefr);
freqs = freqstemp.data;
nf = str2num(freqstemp.textdata{1,1});
if nf ~= length(freqs)
error('Frequency input is invalid, program will be terminated' )
end
fprintf(fid,'Frequency File is:%12s\n\n',filefr);
fprintf(fid,'# of computing points: %6.1f\n\n',nf);
end
disp('----------------------------------------------------')
%Geometry initial
disp (' ')
disp ('----Geometry and Material Properties Initialization----')
disp (' ')
filein = input ('Enter Filename containing geometry and material properties
of an EMHR :','s');
Intemp = importdata(filein);
Indata = Intemp.data;
Dtube = Indata(1);
fprintf(fid,'The diameter of the PWT (m): %12.5f\n',Dtube);
r = Indata(2);
fprintf(fid,'The radius of the neck (m): %12.5f\n',r);
t = Indata(3);
fprintf(fid,'The thickness of the neck (m): %12.5f\n',t);
R = Indata(4);
fprintf(fid,'The radius of the cavity (m): %12.5f\n',R);
L = Indata(5);
fprintf(fid,'The depth of the cavity (m): %12.5f\n',L);
as = Indata(6);
fprintf(fid,'The radius of the shim of the PZT backplate (m): %12.5f\n',as);
ts = Indata(7);
fprintf(fid,'The thickness of the shim of the PZT backplate (m):
%12.5f\n',ts);
ap = Indata(8);
fprintf(fid,'The radius of the piezoceramic of the PZT backplate (m):
%12.5f\n',ap);
tp = Indata(9);
187
fprintf(fid,'The thickness of the piezoceramic of the PZT backplate (m):
%12.5f\n',tp);
%Material properties
Es = Indata(10);
fprintf(fid,'The Youngs modulus of the shim (N/m^2): %12.5f\n',Es);
vs = Indata(11);
fprintf(fid,'The Poisson ratio of the shim : %12.5f\n',vs);
rhos =Indata(12);
fprintf(fid,'The density of the shim : %12.5f\n',rhos);
Ep = Indata(13);
fprintf(fid,'The Youngs modulus of the piezoceramic (N/m^2): %12.5f\n',Ep);
vp = Indata(14);
fprintf(fid,'The Poisson ratio of the piezoceramic : %12.5f\n',vp);
rhop = Indata(15);
fprintf(fid,'The density of the piezoceramic : %12.5f\n',rhop);
d31 = Indata(16);
fprintf(fid,'The piezo strain constant : %12.5f\n',d31);
dp = Indata(17);
fprintf(fid,'The relative dielectric constant of the PZT: %12.5f\n',dp);
damp = Indata(18);
fprintf(fid,'The damping coefficient of the PZT backplate : %12.5f\n',damp);
%Shunts initial
disp (' ')
disp ('----shunts Initialization----')
flag01 = input('The shunt load is [1]--resitive, [2]--capacitive or [3]--
inductive: ');
switch flag01
case 1
disp(' ')
ZL = input('Please input the value of resitive load [Ohm]: ');
fprintf(fid,'The the value of resitive load [Ohm]: %12.5f\n',ZL);
ZL = ZL*ones(1,length(freqs));
case 2
disp(' ')
ZL = input('Please input the value of capacitive load [F]: ');
fprintf(fid,'The the value of capacitive load [F]: %12.5f\n',ZL);
ZL = 1./(j*2*freqs*pi*ZL);
case 3
disp(' ')
ZL = input('Please input the value of inductive load [H]: ');
fprintf(fid,'The the value of inductive load [F]: %12.5f\n',ZL);
ZL = j*2*freqs*pi*ZL;
end
% Compute the acoustic impedance of the EMHR
188
disp (' ')
disp ('----calculation of the acoustic impedance of the EMHR----')
disp (' ')
flag02 = input('Please choose model of the EMHR [1]--LEM [2]--TM: ');
% LEM--Lumped element model TM-- Transfer matrix model
if flag02 ==1
zetal =
LEMEMHR(r,t,R,L,as,ts,ap,tp,Dtube,Es,vs,rhos,Ep,vp,rhop,d31,dp,damp,ZL);
else
zetal = TMEMHR
(r,t,R,L,as,ts,ap,tp,Dtube,Es,vs,rhos,Ep,vp,rhop,d31,dp,damp,ZL);
end
%Result output
%Write data to output file
fprintf(fid,'Freq \t Normalized Zeta \t \r');
for inf = 1:nf
f(inf) = freqs(inf);
fprintf(fid,'%7.2f\t %12.5f\t \t %12.5f\t\t \r',...
[f(inf);real(zetal(inf)); imag(zetal(inf))]);
end
fclose(fid);
%Plot gragh for normalized resistance and reactance
figure (1)
set(gcf,'paperorientation','landscape')
set(gcf,'paperposition',[0.25 0.25 10.5 8.0])
set(gcf,'DefaultlineLinewidth',2)
h1=subplot(2,1,1);
h2=subplot(2,1,2);
subplot(2,1,1); plot(f,real(zetal))
subplot(2,1,2); plot(f,imag(zetal))
subplot(h1)
ylabel('\theta','fontsize',12)
title ('EMHR impedance calculation - linear ','fontsize',12)
grid on
subplot(h2)
ylabel('\chi','fontsize',12)
xlabel('Freq.[Hz]','fontsize',12)
grid on
disp(' ')
figure (2)
set(gcf,'paperorientation','landscape')
set(gcf,'paperposition',[0.25 0.25 10.5 8.0])
set(gcf,'DefaultlineLinewidth',2)
plot(f,real(zetal))
xlabel('Freq.[Hz]','fontsize',12)
ylabel('Normalized Resistance','fontsize',12)
title ('EMHR impedance calculation - linear ','fontsize',12)
grid on
189
figure (3)
set(gcf,'paperorientation','landscape')
set(gcf,'paperposition',[0.25 0.25 10.5 8.0])
set(gcf,'DefaultlineLinewidth',2)
plot(f,imag(zetal))
xlabel('Freq.[Hz]','fontsize',12)
ylabel('Normalized Reactance','fontsize',12)
title ('EMHR impedance calculation - linear ','fontsize',12)
grid on
disp ('End of calculation of EMHR')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Subfunction LEMEMHR
%
% This subfunction is used to compute the acoustic impedance of the EMHR
% using LEM
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function ZinEMHRLEM =
LEMEMHR(r,t,R,L,as,ts,ap,tp,Dtube,Es,vs,rhos,Ep,vp,rhop,d31,dp,damp,ZL)
global rho0 t0 c0 visc Pref
global c rho freqs
% Constants definition
mu = visc;
gama = 1.4;
% Frequency information
ww = 2*pi*freqs;
k = ww/c;
% Acoustic impedance of the neck of the EMHR
muprime = mu*(1+(gama-1)/sqrt(0.71))^2;
nu = mu/rho;
nuprime = muprime/rho;
ks = sqrt(-j*ww/nu);
ksprime = sqrt(-j*ww/nuprime);
Fs = 1-2*besselj(1,ks*r)./(ks*r.*besselj(0,ks*r));
Fsprime = 1-2*besselj(1,ksprime*r)./(ks*r.*besselj(0,ksprime*r));
ZaN = j*ww*rho/(pi*r^2).*(t./Fsprime+1.7*r./Fs);
% Acoustic impedance of the cavity of the EMHR
ZaC = -j*cot(k*L)*rho*c/(pi*R^2);
% Acoustic impedance of the PZT backplate
[MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp)
Rf0 = 1/2/pi/sqrt(MaD*CaD)
RaD = 2*damp*sqrt(MaD./CaD);
%RaD = (k*ap).^2/2*rho*csound/(pi*ap^2);
ZaD = j*ww*MaD+ 1./(j*ww*CaD)+RaD + Phi^2*ZL./(1+j*ww*CeB.*ZL);
190
ZinEMHRLEM = (ZaN + ZaC.*ZaD./(ZaC+ZaD))*(Dtube)^2/rho/c;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Subfunction TMEMHR
% This subfunction is used to compute the acoustic impedance of the EMHR
% using TM
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function ZinEMHRTR =
TMEMHR(r,t,R,L,as,ts,ap,tp,Dtube,Es,vs,rhos,Ep,vp,rhop,d31,dp,damp,ZL)
global rho0 t0 c0 visc Pref
global c rho freqs
% Constants definition
mu = visc;
gama = 1.4;
% Frequency information
ww = 2*pi*freqs;
% Geometry information
A5 = (Dtube)^2; %area of the PWT tube
A4 = pi*r^2; %area of the neck
A3 = A4;
A2 = pi*R^2; %cross-sectional area of the cavity
A1 = 1/3*pi*as^2; % effective are of the piezoelectric backplate
[MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp);
% Computing entries of the transfer matrix and acoustic impedance of the
% EMHR
for n = 1:length(freqs)
w = ww(n);
s = j*w;
k = w/c;
zeta4 = 1/(rho*c)*(sqrt(2*mu*rho*w)+rho*w^2*A4/(2*pi*c))+...
j*8*k*r/(3*pi);
zeta3 = zeta4;
GamaN=j;
GGN = 1;
[GamaC GGC] = GamaTREMHR(R,mu,rho,gama,w);
ZaD = s*MaD+1/(s*CaD)+2*damp*sqrt(MaD/CaD);
TR = [1 zeta4;0 A4/A5]*...
[cosh(GamaN*k*t) sinh(GamaN*k*t)/GGN; GGN*sinh(GamaN*k*t)
cosh(GamaN*k*t)]*...
[1 A2/A3*zeta3;0 A2/A3]*...
[cosh(GamaC*k*L) sinh(GamaC*k*L)/GGC; GGC*sinh(GamaC*k*L)
cosh(GamaC*k*L)]*...
[(s*CeB*ZaD+Phi^2)/Phi -ZaD/Phi; s*rho*c*CeB/(A1*Phi) -
rho*c/(A1*Phi)];
ZinEMHRTR(n) = (TR(1,1)*ZL(n)+TR(1,2))/(TR(2,1)*ZL(n)+TR(2,2));
191
end
ZinEMHRTR;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Subfunction Center
%
% This subfunction is used to compute the acoustic impedance of the
% piezoelectric backplate, it is modified based on the code developed by
% Guiqin Wang
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp)
% Calculate the CaD, MaD of the piezoelectric backplate
% Pressure loading
P = 1;
V = 0;
Ef = V/tp;
[wtotal wwtot wefftot] =
defEMHR(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp,P,V,Ef);
% The values of CaS, MaD, FreQ and W0
Area = pi*as^2;
CaD = abs((wtotal)/(P));
MaD = 2*pi*wwtot/(wefftot^2)/(Area^2);
% Calculate the dA
% Voltage loading
P = 0;
V = 1;
Ef = V/tp;
[wtotal wwtot wefftot] =
defEMHR(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp,P,V,Ef);
% The value of dA
dA = abs(wtotal/1);
% Calculate the Phi
Phi = dA/CaD;
% Calculate CEB
192
epsilon0 = 8.8542E-12; %permitivity of free space in F/m
dielectricconstant = dp; %relative permitivity of the piezo
epsilon=dielectricconstant*epsilon0; %absolute permitivity of the piezo
CeF = epsilon*pi*(ap^2)/tp;
kk = 1-(dA^2)/CeF/CaD;
CeB = CeF*(kk);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% ---------Subfunction defEMHR-------------
%
% The subfunction is used to compute the displacement of the piezoelectric
% backplate
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [wtotal wwtot wefftot] =
defEMHR(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp,P,V,Ef)
% Initializing shim
Esi = Es;
vsi = vs;
densitysi = rhos;
R2 = as;
tsi = ts;
% Initializing piezoelectric
tp = tp;
R1 = ap;
R21 = R1/R2;
Ep = Ep; %Youngs modulus of the piezo
vp = vp; %Poissons ratio of the piezo material
d31 = d31; %electromechanical transduction const of the piezo
densityp = rhop; %Density of the piezo
epsilon0 = 8.8542E-12; %permitivity of free space in F/m
dielectricconstant = dp; %relative permitivity of the piezo
epsilon=dielectricconstant*epsilon0; %absolute permitivity of the piezo
% Computing A,B,D for the central and annular paltes
% Constitutive Relations for isotropic circular plates
Qsi = [1 vsi; vsi 1].*(Esi/(1-(vsi^2)));
Qp = [1 vp; vp 1].*(Ep/(1-(vp^2)));
% Taking the original zxis as centre of the shim layer for
zin1 = -tsi/2 ; %distance of bottom of shim layer from reference
zin2 = tsi/2 ; %distance of interface from reference
zin3 = tp + tsi/2;
zout1 = -tsi/2 ; %distance of top of the piezo layer from reference
193
zout2 = tsi/2 ;
A_out = Qsi.*(zout2-zout1) ;
B_out = Qsi.*((zout2^2-zout1^2)/2) ;
D_out = Qsi.*((zout2^3-zout1^3)/3) ;
A_in = Qsi.*(zin2-zin1) + Qp.*(zin3-zin2) ;
B_in = Qsi.*((zin2^2-zin1^2)/2) + Qp.*((zin3^2-zin2^2)/2);
D_in = Qsi.*((zin2^3-zin1^3)/3) + Qp.*((zin3^3-zin2^3)/3);
% Computing D Mark(determinant of matrix mapping defined variables y1,y2 to
U0 theta
Dstar_in = D_in(1,1)-(B_in(1,1)^2)/A_in(1,1);
Dstar_out = D_out(1,1)-(B_out(1,1)^2)/A_out(1,1);
% Computing fictitious forces due to piezo
Mp_in = Ef* (Ep/(1-vp)) * d31 * (zin3^2-zin2^2)/2;
Np_in = Ef* (Ep/(1-vp)) * d31 * (zin3-zin2);
Mp_out = 0;
Np_out = 0;
%-----Computing coefficient matrix-------------------------------------
a_R = R1/R2;
Gamma_out = B_out(1,1)/A_out(1,1);
Gamma_in = B_in(1,1)/A_in(1,1);
AB12_in = -P*(R1^2)*(B_in(1,2)-Gamma_in*A_in(1,2))/8/Dstar_in;
AB12_out = -P*(R1^2)*(B_out(1,2)-Gamma_out*A_out(1,2))/8/Dstar_out;
BD12_in = -P*(R1^2)*(3*Dstar_in+D_in(1,2)-Gamma_in*B_in(1,2))/8/Dstar_in;
BD12_out = -P*(R1^2)*(D_out(1,2)-Gamma_out*B_out(1,2))/8/Dstar_out;
arf = 1/(a_R^4)-1;
% Compute the matrix
A11 = 1;
A12 = 0;
A13 = (1/(a_R^2))-1;
A14 = 0;
A21 = 0;
A22 = 1;
A23 = 0;
A24 = 1/(a_R^2)-1;
A31 = - B_in(1,1) - B_in(1,2);
A32 = A_in(1,1)+A_in(1,2);
A33 = B_out(1,1)*(1/(a_R^2)+1) + B_out(1,2)*(1-1/(a_R^2));
A34 = A_out(1,2)*(1/(a_R^2)-1) - A_out(1,1)*(1+1/(a_R^2));
A41 = - D_in(1,1) - D_in(1,2);
A42 = B_in(1,1) + B_in(1,2) ;
A43 = D_out(1,1)*(1/(a_R^2)+1) + D_out(1,2)*(1 - 1/(a_R^2));
A44 = B_out(1,2)*(1/(a_R^2)-1) - B_out(1,1)*(1/(a_R^2) + 1);
b1 = P*(R1^2)/8/Dstar_in + P*(R1^2)*arf/8/Dstar_out;
b2 = Gamma_in*P*(R1^2)/8/Dstar_in + Gamma_out*P*(R1^2)*arf/8/Dstar_out;
194
b3 = AB12_in + 2*Np_in + AB12_out*arf -2*Np_out;
b4 = BD12_in + 2*Mp_in + BD12_out*arf -2*Mp_out + P*(R1^2)*(3+1/(a_R^4))/8;
A = [A11 A12 A13 A14; A21 A22 A23 A24; A31 A32 A33 A34; A41 A42 A43 A44];
b = [b1 b2 b3 b4]';
% Calculate the costants
c1234 = inv(A)*b;
c1 = c1234(1);
c2 = c1234(2);
c3 = c1234(3);
c4 = c1234(4);
c6 = P*(R2^4)*(0.25-log(R2))/16/Dstar_out - 0.5*c3*(R2^2)*(0.5-log(R2));
c5 = P*(R1^4)/64/Dstar_in - 0.25*c1*(R1^2) - P*(R1^4)*(0.25-
(log(R1)/(a_R^4)))/16/Dstar_out ...
+ 0.5*c3*(R1^2)*(0.5-log(R1)/(a_R^2)) + c6;
% Composite plates;
% Computing deflections and Forces at the interface by superposition
% Finding Deflection in the central region and the annular region
% Initialize
num1 = 1500;
r = linspace(0,1,num1+1);
jj=floor(R21*num1)+1;
for i=1:num1+1
rad1=r(i)*R2;
if (i<jj+1)
w(i) = -P*(rad1^4)/(64*Dstar_in)+0.25*c1*(rad1^2)+c5;
weff(i)= w(i)*2*pi*rad1*R2/num1;
u(i) = abs(P*pi*w(i)*rad1*R2/num1);
ww(i) = (tp*densityp+densitysi*tsi)*(w(i)^2)*rad1*R2/num1;
else
w(i) = -P*(0.25*(rad1^4)-
((R2^4)*log(rad1)))/(16*Dstar_out)+0.5*c3*(0.5*(rad1^2)-
((R2^2)*log(rad1)))+c6;
weff(i)= w(i)*2*pi*rad1*R2/num1;
u(i) = abs(P*pi*w(i)*rad1*R2/num1);
ww(i) = (tsi*densitysi)*(w(i)^2)*rad1*R2/num1;
end;
end;
wwtot = sum(ww);
wtotal = sum(weff);
wefftot = wtotal/(pi*(R2^2));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The subfunction is used to compute the complex propagating
% coefficient for the TM
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [GamaTR GG] = GamaTREMHR(r,mu,rho,gama,w)
195
sig = sqrt(0.71);
s = r*sqrt(w*rho/mu);
nn = (1+(gama-1)...
/gama*besselj(2,j^(3/2)*sig*s)/besselj(0,j^(3/2)*sig*s))^(-1);
GamaTR = sqrt(besselj(0,j^(3/2)*s)*gama/besselj(0,j^(3/2)*s)/nn);
GG = j*gama/(GamaTR*nn);
Optimizing Tuning Range of an EMHR with Capacitive Loads
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% There are eight design variables to optimize the tuning range of EMHR
% x: collective variable for all eight design variables
% x(1): ->r radius of the neck of EMHR
% x(2): ->t thickness of the neck of EMHR
% x(3): ->R radius of the cavity of EMHR
% x(4): ->L depth of the cavity of EMHR
% x(5): ->ap radius of the pzt-layer of pzt-backplate of EMHR
% x(6): ->tp thickness of the pzt-layer of pzt-backplate of EMHR
% x(7): ->as radius of the shim of pzt-backplate of EMHR
% x(8): ->ts thickness of the shim of pzt-backplate of EMHR
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all
close all
%starting guess
x0 =[2.42e-3,3.16e-3,6.34e-3,9.4e-3,10.05e-3,0.13e-3,12.4e-3,0.19e-3];
%optimal initial set
options = optimset('LargeScale','off','Display','iter',...
'MaxFunEvals',2000,'TolX',1e-7,'Tolcon',1e-7);
%lowerbound for the design variables
LB =[1e-3; 1e-3; 5e-3;10e-3;1e-3;0.05e-3;1e-3;0.05e-3];
%upperbound for the design variables
UB =[3.5e-3; 4.5e-3; 15e-3;20e-3;25e-3;1e-3;25e-3;1e-3];
%linear constraints
A =[0 0 1 0 0 0 -1 0;
0 0 0 0 1 0 -1 0;
0 0 0 0 0 -1 0 -1;
0 0 0 0 0 1 0 1];
B =[-1e-3;-1e-3;-1e-4;1e-3];
disp('Please select the optimization goal:')
disp('[1]- the first resonant freq.')
disp('[2]- the second resonant freq.')
flag = input('');
196
%optimal design
switch flag
case 1
[x,fval,exitflag,output,lambda,grad] = fmincon('OPTEMHRFUN01',x0,...
A,B,[],[],LB,UB,'OPTEMHRCONNIND',options);
case 2
[x,fval,exitflag,output] = fmincon('OPTEMHRFUN02',x0,...
A,B,[],[],LB,UB,'OPTEMHRCONIND',options);
otherwise
'Catch you later'
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% The objective function to maximize f1
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function f = OPTEMHRFUN01(x)
%%
% Constants definition
rho = 1.231; % air density
csound = 343; % speed of sound at air
% Material properties of the piezo
Es =110e9; % Young's modulus of the shim
vs = 0.375; % Poisson ratio of the shim
rhos = 8530; % density of the shim
Ep = 63e9; % Young's modulus of the piezo
vp = 0.31; % Poisson ratio of the piezo
rhop = 7700; % density of the piezo
d31 = -175e-12; % d31 of the piezo
dp = 1750; % relative dilectric constant
% geometry of the EMHR
r = x(1);
t = x(2);
R = x(3);
L = x(4);
ap = x(5);
tp = x(6);
as = x(7);
ts = x(8);
% Computing the acoutic impedance of the EMHR
MaN = rho*(t+2*0.85*r)/(pi*r^2); %acoustic mass of the neck
CaC = pi*R^2*L/(rho*csound^2); %acoustic capacitance of the cavity
197
[MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp);
PhiA = Phi;
% The first resonant frequency of the short-circuited EMHR
w01s = sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2-...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
% The second resonant frequency of the short-circuited EMHR
w02s =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2+...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
% The first resonant frequency of the open-circuited EMHR
w01o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2-...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
% % The second resonant frequency of the open-circuited EMHR
w02o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2+...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
%----------- objective function ----------------
f = w02s-w02o;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% The objective function to maximize f2
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function f = OPTEMHRFUN02(x)
%%
% Constants definition
rho = 1.231; % air density
csound = 343; % speed of sound at air
% Material properties of the piezo
Es =110e9; % Young's modulus of the shim
vs = 0.375; % Poisson ratio of the shim
rhos = 8530; % density of the shim
Ep = 63e9; % Young's modulus of the piezo
vp = 0.31; % Poisson ratio of the piezo
rhop = 7700; % density of the piezo
d31 = -175e-12; % d31 of the piezo
dp = 1750; % relative dilectric constant
198
% geometry of the EMHR
r = x(1);
t = x(2);
R = x(3);
L = x(4);
ap = x(5);
tp = x(6);
as = x(7);
ts = x(8);
% Computing the acoutic impedance of the EMHR
MaN = rho*(t+2*0.85*r)/(pi*r^2); %acoustic mass of the neck
CaC = pi*R^2*L/(rho*csound^2); %acoustic capacitance of the cavity
[MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp);
PhiA = Phi;
% The first resonant frequency of the short-circuited EMHR
w01s = sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2-...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
% The second resonant frequency of the short-circuited EMHR
w02s =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2+...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
% The first resonant frequency of the open-circuited EMHR
w01o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2-...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
% % The second resonant frequency of the open-circuited EMHR
w02o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2+...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
%----------- objective function ----------------
f = w02s-w02o;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% The constraints of optimizing problem to maximize f1 or f2
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [c, ceq] = OPTEMHRCONNIND(x)
% Constants definition
rho = 1.231; % air density
199
csound = 343; % speed of sound at air
% Material properties of the piezo
Es =110e9; % Young's modulus of the shim
vs = 0.375; % Poisson ratio of the shim
rhos = 8530; % density of the shim
Ep = 63e9; % Young's modulus of the piezo
vp = 0.31; % Poisson ratio of the piezo
rhop = 7700; % density of the piezo
d31 = -175e-12; % d31 of the piezo
dp = 1750; % relative dilectric constant
% geometry of the EMHR
r = x(1);
t = x(2);
R = x(3);
L = x(4);
ap = x(5);
tp = x(6);
as = x(7);
ts = x(8);
% Computing the acoutic impedance of the EMHR
MaN = rho*(t+2*0.85*r)/(pi*r^2); %acoustic mass of the neck
CaC = pi*R^2*L/(rho*csound^2); %acoustic capacitance of the cavity
[MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp);
PhiA = Phi;
% The first resonant frequency of the short-circuited EMHR
w01s = sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2-...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
% The second resonant frequency of the short-circuited EMHR
w02s =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2+...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
% The first resonant frequency of the open-circuited EMHR
w01o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2-...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
% The second resonant frequency of the open-circuited EMHR
w02o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2+...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
%----------- nonlinear inequality constraints ----------------
c = [1200-w01s,w01s-1900,w02s-3000];
200
%----------- nonlinear equality constraints -------------------
ceq = [];
Pareto Optimization Design of an EMHR with Capacitive Loads
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% The program is used to implement Pareto optimization design of an EMHR
% using epsilon method
%
% There are eight design variables to optimize the tuning range of EMHR
% x: collective variable for all eight design variables
% x(1): ->r radius of the neck of EMHR
% x(2): ->t thickness of the neck of EMHR
% x(3): ->R radius of the cavity of EMHR
% x(4): ->L depth of the cavity of EMHR
% x(5): ->ap radius of the pzt-layer of pzt-backplate of EMHR
% x(6): ->tp thickness of the pzt-layer of pzt-backplate of EMHR
% x(7): ->as radius of the shim of pzt-backplate of EMHR
% x(8): ->ts thickness of the shim of pzt-backplate of EMHR
% x(9): ->MeL inductive loads
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
close all
clear all
global index
%starting guess
x0 =[2.42e-3,3.16e-3,6.34e-3,16.4e-3,10.1e-3,0.12e-3,12.4e-3,0.19e-3];
mm = -18:-2:-174; % set up the contraint for f1
for nn = 1:length(mm)
index = mm(nn);
%optimal initial set
options = optimset('LargeScale','off','Display','iter',...
'MaxFunEvals',2000,'TolX',1e-7,'Tolcon',1e-7);
%lowerbound for the design variables
LB =[1e-3; 1e-3; 5e-3;10e-3;1e-3;0.05e-3;1e-3;0.05e-3];
%upperbound for the design variables
UB =[3.5e-3; 4.5e-3; 15e-3;20e-3;25e-3;1e-3;25e-3;1e-3];
%linear constraints
A =[0 0 1 0 0 0 -1 0;
0 0 0 0 1 0 -1 0;
0 0 0 0 0 -1 0 -1;
0 0 0 0 0 1 0 1];
B =[-1e-3;-1e-3;-1e-4;1e-3];
[x,fval,exitflag,output,lambda,grad,hessian] =
fmincon('OPTEMHRFUN02',x0,...
201
A,B,[],[],LB,UB,'OPTEMHRCONNIND2',options);
x0 =x; % update initial value for next iteration
% Constants definition
rho = 1.231; % air density
csound = 343; % speed of sound at air
% Material properties of the piezo
Es =110e9; %89.6e9; % Young's modulus of the shim
vs = 0.375; %0.324; % Poisson ratio of the shim
rhos = 8530; %8700; % density of the shim
Ep = 63e9; % Young's modulus of the piezo
vp = 0.31; % Poisson ratio of the piezo
rhop = 7700; % density of the piezo
d31 = -175e-12; % d31 of the piezo
dp = 1750; % relative dilectric constant
% geometry of the EMHR
r = x(1);
t = x(2);
R = x(3);
L = x(4);
ap = x(5);
tp = x(6);
as = x(7);
ts = x(8);
% Computing the acoutic impedance of the EMHR
MaN = rho*(t+2*0.85*r)/(pi*r^2); %acoustic mass of the neck
CaC = pi*R^2*L/(rho*csound^2); %acoustic capacitance of the cavity
[MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp);
PhiA = Phi;
% The first resonant frequency of the short-circuited EMHR
w01s = sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2-...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
% The second resonant frequency of the short-circuited EMHR
w02s =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2+...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
% The first resonant frequency of the open-circuited EMHR
w01o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2-...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
% The second resonant frequency of the open-circuited EMHR
w02o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2+...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
202
dw011 = w01o-w01s; % the tuning range of f1
dw022 = w02o-w02s; % the tuning range of f2
% save the result of Pareto solution
Presult(1,nn) = dw011;
Presult(2,nn) = dw022;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% The constraints of optimizing problem to maximize f2 with the tuning of the
% f1 is limited
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [c, ceq] = OPTEMHRCONNIND2(x)
global index
% Constants definition
rho = 1.231; % air density
csound = 343; % speed of sound at air
% Material properties of the piezo
Es =110e9; % Young's modulus of the shim
vs = 0.375; % Poisson ratio of the shim
rhos = 8530; % density of the shim
Ep = 63e9; % Young's modulus of the piezo
vp = 0.31; % Poisson ratio of the piezo
rhop = 7700; % density of the piezo
d31 = -175e-12; % d31 of the piezo
dp = 1750; % relative dilectric constant
% geometry of the EMHR
r = x(1);
t = x(2);
R = x(3);
L = x(4);
ap = x(5);
tp = x(6);
as = x(7);
ts = x(8);
% Computing the acoutic impedance of the EMHR
MaN = rho*(t+2*0.85*r)/(pi*r^2); %acoustic mass of the neck
CaC = pi*R^2*L/(rho*csound^2); %acoustic capacitance of the cavity
[MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp);
PhiA = Phi;
% HR = 1/sqrt(MaN*CaC)/2/pi;
203
% PD = 1/sqrt(MaD*CaD)/2/pi;
w01s = sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2-...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
w01o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2-...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
w02s =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2+...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
w02o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2+...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
dw01 = w01s-w01o;
%----------- nonlinear inequality constraints ----------------
c = [dw01-(index),1200-w01s,w01s-1900,w02s-3000];
%----------- nonlinear equality constraints -------------------
ceq = [];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% The program is used to implement Pareto optimization design of an EMHR
% using weighting sum method
%
% There are eight design variables to optimize the tuning range of EMHR
% x: collective variable for all eight design variables
% x(1): ->r radius of the neck of EMHR
% x(2): ->t thickness of the neck of EMHR
% x(3): ->R radius of the cavity of EMHR
% x(4): ->L depth of the cavity of EMHR
% x(5): ->ap radius of the pzt-layer of pzt-backplate of EMHR
% x(6): ->tp thickness of the pzt-layer of pzt-backplate of EMHR
% x(7): ->as radius of the shim of pzt-backplate of EMHR
% x(8): ->ts thickness of the shim of pzt-backplate of EMHR
% x(9): ->MeL inductive loads
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
close all
clear all
global Windex
%starting guess
x0 =[2.42e-3,3.16e-3,6.34e-3,16.4e-3,10.1e-3,0.12e-3,12.4e-3,0.19e-3];
204
mm = rand(100,1); % set up the contraint for f1
for nn = 1:length(mm)
Windex = mm(nn);
%optimal initial set
options = optimset('LargeScale','off','Display','iter',...
'MaxFunEvals',2000,'TolX',1e-7,'Tolcon',1e-7);
%lowerbound for the design variables
LB =[1e-3; 1e-3; 5e-3;10e-3;1e-3;0.05e-3;1e-3;0.05e-3];
%upperbound for the design variables
UB =[3.5e-3; 4.5e-3; 15e-3;20e-3;25e-3;1e-3;25e-3;1e-3];
%linear constraints
A =[0 0 1 0 0 0 -1 0;
0 0 0 0 1 0 -1 0;
0 0 0 0 0 -1 0 -1;
0 0 0 0 0 1 0 1];
B =[-1e-3;-1e-3;-1e-4;1e-3];
[x,fval,exitflag,output,lambda,grad,hessian] =
fmincon('OPTEMHRFUN03',x0,...
A,B,[],[],LB,UB,'OPTEMHRCONNIND',options);
x0 =x; % update initial value for next iteration
% Constants definition
rho = 1.231; % air density
csound = 343; % speed of sound at air
% Material properties of the piezo
Es =110e9; % Young's modulus of the shim
vs = 0.375; % Poisson ratio of the shim
rhos = 8530; % density of the shim
Ep = 63e9; % Young's modulus of the piezo
vp = 0.31; % Poisson ratio of the piezo
rhop = 7700; % density of the piezo
d31 = -175e-12; % d31 of the piezo
dp = 1750; % relative dilectric constant
% geometry of the EMHR
r = x(1);
t = x(2);
R = x(3);
L = x(4);
ap = x(5);
tp = x(6);
as = x(7);
ts = x(8);
% Computing the acoutic impedance of the EMHR
MaN = rho*(t+2*0.85*r)/(pi*r^2); %acoustic mass of the neck
CaC = pi*R^2*L/(rho*csound^2); %acoustic capacitance of the cavity
205
[MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp);
PhiA = Phi;
% The first resonant frequency of the short-circuited EMHR
w01s = sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2-...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
% The second resonant frequency of the short-circuited EMHR
w02s =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2+...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
% The first resonant frequency of the open-circuited EMHR
w01o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2-...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
% The second resonant frequency of the open-circuited EMHR
w02o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2+...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
dw011 = w01o-w01s; % the tuning range of f1
dw022 = w02o-w02s; % the tuning range of f2
% save the result of Pareto solution
Presult(1,nn) = dw011;
Presult(2,nn) = dw022;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The objective function to maximize f1 and f2 using weighting sum
% method
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function f = OPTEMHRFUN03(x)
global Windex
%%
% Constants definition
rho = 1.231; % air density
csound = 343; % speed of sound at air
% Material properties of the piezo
Es =110e9; % Young's modulus of the shim
vs = 0.375; % Poisson ratio of the shim
rhos = 8530; % density of the shim
Ep = 63e9; % Young's modulus of the piezo
206
vp = 0.31; % Poisson ratio of the piezo
rhop = 7700; % density of the piezo
d31 = -175e-12; % d31 of the piezo
dp = 1750; % relative dilectric constant
% geometry of the EMHR
r = x(1);
t = x(2);
R = x(3);
L = x(4);
ap = x(5);
tp = x(6);
as = x(7);
ts = x(8);
% Computing the acoutic impedance of the EMHR
MaN = rho*(t+2*0.85*r)/(pi*r^2); %acoustic mass of the neck
CaC = pi*R^2*L/(rho*csound^2); %acoustic capacitance of the cavity
[MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp);
PhiA = Phi;
% The first resonant frequency of the short-circuited EMHR
w01s = sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2-...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
% The second resonant frequency of the short-circuited EMHR
w02s =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2+...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
% The first resonant frequency of the open-circuited EMHR
w01o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2-...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
% % The second resonant frequency of the open-circuited EMHR
w02o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2+...
sqrt((1/(MaN*CaC)-1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+...
4*1/(MaN*MaD*CaC^2))/2)/2/pi;
%----------- objective function ----------------
f = Windex*(w01s-w01o)+(1-Windex)* (w02s-w02o);
207
LIST OF REFERENCES
Alfredson, R. J. and Davies, P. O. A. L. (1971). "Performance of exhaust silencer components,"
J. Sound Vib.
15(2), 175–196.
ANSI (1994). S1.1. "Acoustical Terminology," (New York, American National Standards
Institute).
ASTM–E1050–98 (1998). "Impedance and Absorption of Acoustical Materials Using a Tube,
Two Microphones, and a Digital Frequency Analysis System," (ASTM International).
Benade, A. H. (1988). "Equivalent circuits for conical waveguides," J. Acoust. Soc. Am.
83,
1764–1769.
Beranek, L. L. (1988). Acoustical Measurements (Woodbury, NY, American Institute of
Physics), Chap. 7, pp. 294–353.
Bielak, G. W. and Premo, J. W. (1999). "Advanced turbofan duct liner concepts," NASA/CR-
1999-209002 (National Aeronautics and Space Administration).
Bies, D. A. and Wilson, J. O.B. (1957). "Acoustic impedance of a Helmholtz resonator at very
high amplitude," J. Acoust. Soc. Am.
29(6), 711–714.
Blackstock, D. T. (2000). Fundamental of Physical Acoustics (New York, John Wiley & Sons),
Chaps. 1, 11, and 13.
Boden, H. and Abom, M (1986). "Influence of errors on the two-microphone method for
measuring acoustic properties in ducts," J. Acoust. Soc. Am.
79(2), 541–549.
Broadbent, D. (1957). "Noise in relation to annoyance, performance and mental health," J.
Acoust. Soc. Am.
68(1), 15–17.
Chen, T. J., Chiang, H. C. and Chen, C.C. (1992). "Effects of aircraft noise on hearing and
auditory pathway function of airport employees," J. Occup. Med.
34(6), 613–619.
Chu, W. T. (1986). "Extension of the two-microphone transfer function method for impedance
tube measurements," J. Acoust. Soc. Am.
80(1), 347–348.
Chu, W. T. (1986). "Transfer function technique for impedance and absorption measurement in
an impedance tube using a single microphone," J. Acoust. Soc. Am.
80(2), 555–560.
Chu, W. T. (1988). "Further experimental studies on the transfer-function technique for
impedance tube measurements," J. Acoust. Soc. Am.
83(6), 2255–2260.
Chung, J. Y. and Blaser, D. A. (1980). "Transfer function method of measuring in-duct acoustic
properties. I. Theory," J. Acoust. Soc. Am.
68, 907–913.
Chung, J. Y. and Blaser, D. A. (1980). "Transfer function method of measuring in-duct acoustic
properties. II. Experiment," J. Acoust. Soc. Am.
68, 914–921.
208
Coleman, H. W. and Steele, W. G. (1989). Experimentation and Uncertainty Analysis for
Engineers (New York, John Wiley & Sons, Inc), pp. 1–16.
Crandall, I. B. (1926). Theory of Vibrating Systems and Sound ( New York, D. Van Nostrand
Company), pp. 229–241.
Crighton, D. G. (1995). "Airframe noise," in Aeroacoustics of Flight Vehicles: Theory and
Practice Volume 2, edited by H. H. Hubbard, (Acoustical Society of America, New York),
pp.
391–448.
Cummings, A. and Eversman, A. (1983). "High amplitude acoustic transmission through duct
terminations: Theory," J. Sound Vib.
91(4), 503–518.
De Bedout, J.M., Franchek, M.A., Bernhard, R.J., and Mongeau, L. (1997). "Adaptive-passive
noise control with self–tuning Helmholtz resonators," J. Sound Vib.
202(1), 109–123.
Dowling, A. P. and. Hughes, I. J (1992). "Sound absorption by a screen with a regular array of
slits," J. Sound Vib.
156, 387–405.
Easwaran, V. and Munjal, M. L. (1991). "Transfer matrix modeling of hyperbolic and parabolic
ducts with incompressible mean flow," J. Acoust. Soc. Am.
90(4), 2163–2172.
Envia, E. (2001). "Fan noise reduction: an overview," AIAA paper 2001-0661, in 39th
Aerospace Sciences Meeting & Exhibit (Reno, NV).
Eriksson, L. J. (1980). "Higher order mode effects in circular ducts and expansion chambers," J.
Acoust. Soc. Am.
68(2), 545–550.
FAA (2001). Advisory circular 36–1G, "Noise levels for U.S. certificated and foreign aircraft,"
(Federal Aviation Administration).
Fischer, F. A. (1955). Fundamentals of Electroacoustics (New York, Interscience Publishers,
Inc.), pp. 1–51.
Griefahn, B. (2000). "Noise induced extraaural effects," J. Acoust. Soc. Japan (E) (English
translation of Nippon Onkyo Gakkaishi)
21(6), 307–317.
Groeneweg, J. F. (1995). "Turbomachinery noise," in Aeroacoustics of Flight Vehicles: Theory
and Practice Volume 2, edited by H. H. Hubbard, (Acoustical Society of America, New
York), pp.
151–210.
Haftka, R.T. and Gurdal, Z. (1992). Elements of Structural Optimization, 3
rd
rev. and exp. ed.
(Dordrecht, The Netherlands, Kluwer Academic Publishers), Chap. 1.
Hagwood, N.W., Chung, W.H. and von Flotow, A. (1990). "Modeling of piezoelectric actuator
dynamics for active structural control," Journal of Intelligent Materials Systems and
Structures
1, 327–354.
209
Hang, S. H. and Ih, J. G. (1998). "On the multiple microphone method for measuring in-duct
acoustic properties in the presence of mean flow," J. Acoust. Soc. Am.
103(3), 1520–1526.
Harris, C. M. (1991). Handbook of Acoustical Measurements and Noise Control, 3rd ed. (New
York, McGraw–Hill), Chap. 1.
Hong, E., Trolier–McKinstry, S. Smith, R.L. Krishnaswamy, S.V. and Freidhoff, C.B. (2006).
"Design of MEMS PZT circular diaphragm actuators to generate large deflections," J.
Microelectromech. Syst.
15(4), 832–839.
Horowitz, S. B. (2001). Design and characterization of compliant backplate Helmholtz resonator,
Master thesis, University of Florida.
Horowitz, S.B., Nishida, T., Cattafesta, L., and Sheplak, M. (2002). "Characterization of a
compliant-backplate Helmholtz resonator for an electromechanical acoustic liner," Int. J.
Aeroacoust.
1(2), 183–205.
Howe, M. S. (1979). "On the theory of unsteady high Reynolds number flow through a circular
cylinder," J. Fluid Mech.
61, 109–127.
Hubbard, H. H. (1995). Aeroacoustics of Flight Vehicles: Theory and Practice (Acoustical
Society of America, New York).
Hughes, I. J. and Dowling, A. P. (1990). "The absorption of sound by perforated linings," J.
Fluid Mech.
218, 299–355.
Igarishi J, and Toyama M (1958). "Fundamentals of acoustic silencers," Technical Report 344,
Aeronautical Research Institute, University of Tokyo.
Ingard, U. (1953). "On the theory and design of acoustic resonators," J. Acoust. Soc. Am
. 25(6),
1037–1061.
Ingard, U. (1999). Notes On Duct Attenuators (N4). (Self-published, Kittery Point, NE).
Thurston, G.B., Hargrove, L.E. Jr., and Cook, B.D. (1957), "Nonlinear properties of circular
orifices," J. Acoust. Soc. Am.
29, 992–1001.
Ji, Z. L., Ma, Q., and Zhang Z. H. (1995). "A boundary element scheme for evaluation of four-
pole parameters of ducts and mufflers with low mach number non–uniform flow," J. Sound
Vib.
185(1), 107–117.
Jing, X. D. and Sun, X.F. (1999). "Experimental investigations of perforated liner with bias
flow," J. Acoust. Soc. Am.
106, 2436–2441.
Jones, M. G. (1997) "An improved model for parallel-element liner impedance prediction,"
AIAA Paper 1997–1649, in 3rd AIAA/CEAS Aeroacoustics Conference (Atlanta, GA)
.
210
Jones, M. G. and Parrot, T. L. (1989). "Evaluation of a multi-point method for determining
acoustic impedance," Mech. Syst. Signal Pr.
3(1), 15–35.
Jones, M. G. and Stiede, P. E. (1997). "Comparison of methods for determining specific acoustic
impedance," J. Acoust. Soc. Am.
101(5), 2694–2704.
Karal, F. C. (1953). "The analogous acoustical impedance for discontinuities and constrictions of
circular cross section," J. Acoust. Soc. Am.
25(2), 327–334.
Kim, I. Y. and De Weck, O. L. (2005). "Adaptive weighted-sum method for bi-objective
optimization Pareto front generation," Struct. Multidiscip. O.
29, 149–158.
Kinsler, L. E. (2000). Fundamentals of Acoustics ( New York, John Wiley & Sons), Chap. 10,
pp. 272–301.
Koski, J. (1988). "Multicriteria truss optimization," in Multicriteria Optimization in Engineering
and in the Sciences, edited by W. Stadler, (New York, Plenum Press).
Kraft, R. E., Yu, J. and Kwan H.W. (1999). "Acoustic treatment design scaling methods Volume
2 : Advanced treatment impedance models for high frequency ranges," NASA/CR-1999-
209120/Vol2, (National Aeronautics and Space Administration).
Kujala, T., Y. Shtyrov, Y., Winkler, I., Saher, M., Tervaniemi, M., Sallinen, M., Teder-Salejarvi,
W., Alho, K., Reinikainen, and K., Naatanen, R. (2004). "Long-term exposure to noise
impairs cortical sound processing and attention control," Psychophysiology
41, 875–881.
Lilley, G. M. (1995). "Jet noise classical theory and experiments," in Aeroacoustics of Flight
Vehicles: Theory and Practice Volume 1, edited by H. H. Hubbard, (Acoustical Society of
America, New York)
, pp. 211–290.
Lippert, W. K. R. (1953). "The practical representation of standing waves in an acoustic
impedance tube," Acustica
3, 153–160.
Little, E., Kashani, R., Kohler, J., and Morrison, F. (1994). "Tuning of an electro-rheological
fluid-based Helmholtz resonator as applied to hydraulic engine mounts," ASME DSC
Transp. Sys.
54, 43–51.
Liu, F., Horowitz, S.B., Cattafesta, L., and Sheplak, M. (2003). "A tunable electromechanical
Helmholtz resonator," AIAA Paper 2003-3145, in 9th AIAA/CEAS Aeroacoustics
Conference and Exhibit (Hilton Head, South Carolina).
Liu, F., Horowitz, S., Cattafesta, L., and Sheplak, M. (2006). "Optimization of an
electromechanical Helmholtz resonator," AIAA-2006-2524, 12th AIAA/CEAS
Aeroacoustics Conference (Cambridge, MA).
Liu, F., Horowitz, S., Cattafesta, L., and Sheplak, M. (2007). "A multiple degree of freedom
electromechanical Helmholtz resonator," J. Acoust. Soc. Am
. 122(1), 291-301.
211
Maa, D. Y. (1987). "Microperforated-panel wideband absorbers," Noise Control Eng. J.
29(3),
77–84.
Mahan, J. R. and Karchmer, A. (1995). "Combustion and core noise," in Aeroacoustics of Flight
Vehicles: Theory and Practice Volume 1, edited by H. H. Hubbard, (Acoustical Society of
America, New York), pp.
483–518.
Manglarotty, R. A. (1970). "Acoustic-lining concepts and materials for engine ducts," J. Acoust.
Soc. Am.
48(3), 783–794.
Marglin, S. A. (1967). Public Investment Criteria (Cambridge, MA, M.I.T. Press).
Mathworks, T. (2005). Genetic Algorithm and Direct Search Toolbox User's Guide.
Mathworks, T. (2006). Optimization Toolbox User's Guide.
Mechel, F.P. (2002). Formulas of Acoustics (Berlin, Germany, Springer–Verlag).
Meirovitch, L. (2001). Fundamentals of Vibrations. (New York, McGraw–Hill), pp. 94–98.
Melling, T. H. (1973). "The acoustic impedance of perforates at medium and high sound
pressure levels," J. Sound Vib.
29(1), 1–65.
Miedema, H. M. E. and Vos, H. (1998). "Exposure-response relationships for transportation
noise," J. Acoust. Soc.Am.
104(6), 3432–3445.
Miles, J. H. (1981). "Acoustic transmission matrix of a variable area duct or nozzle carrying a
compressible subsonic flow," J. Acoust. Soc. Am.
69(6), 1577–1586.
Morse P.M. and Ingard, K.U. (1986). Theoretical Acoustics (New Jersey, Princeton University
Press), Chap. 9.
Motsinger, R. E. and Kraft, R. E. (1995). "Design and performance of duct acoustic treatment,"
in Aeroacoustics of Flight Vehicles: Theory and Practice Volume 2: Noise Control edited
by H. H. Hubbard. Hampton, (Acoustical Society of America, New York), pp.165–206.
Munjal, M. L. (1987). Acoustics of Ducts and Mufflers. (New York, John Wiley & Sons, Inc.),
Chap. 2.
Munjal, M. L. and M. G. Prasad (1986). "On plane-wave propagation in a uniform pipe in the
presence of a mean flow and a temperature gradient," J. Acoust. Soc. Am.
80, 1501–1506.
Nayfeh, A. H., Kaiser, J. E., and Telionis, D.P. (1975). "Acoustics of aircraft engine-duct
system," AIAA J.
13(2), 130–153.
Panigrahi, S. N. and Munjal, M. L. (2005). "Plane wave propagation in generalized multiply
connected acoustic filters," J. Acoust. Soc. of Am.
118(5), 2860–2868.
212
Parrott, T. L. and M. G. Jones (1995). "Parallel-element liner impedances for improved
absorption of broadband sound in ducts," Noise Control Eng. J.
43(6), 183–195.
Peat, K. S. (1988). "The transfer matrix of a uniform duct with a linear temperature gradient," J.
Sound Vib.
123, 43–53.
Peterson, L. C. and B. P. Bogert (1950). "A dynamical theory of the cochlea," J. Acoust. Soc.
Am.
22(3), 369–381.
Pierce, A. D. (1994). Acoustics: An Introduction to Its Physical Principles and Applications
(Melville, NY, Acoustical Society of America).
Prasad, S., Gallas, Q., Horowitz, S., Homeijer, B. Sankar, B., Cattafesta, L., and Sheplak, M.
(2006). "An analytical electroacoustic model of a piezoelectric composite circular plate,"
AIAA J.
44(10), 2311–2318.
Rao, S. S. (1996). Engineering Optimization: Theory and Practice (New York, John Wiley &
Sons, Inc.), Chaps. 1 and 2.
Rayleigh, J. W. S. (1945). The Theory of Sound, Volume 2 (New York, Dover), pp. 180-187.
Rossi, M. (1988). Acoustics and Electroacoustics (Norwood, MA, Artech House), pp.
245–373.
Schultz, T., Sheplak, M., and Cattafesta, L., (2007). "Uncertainty analysis of the two-microphone
method," J. Sound Vib.
304, 91-109
Selamet, A., Dickey, N. S., and Novak, J. M. (1994). "The Herschel-Quincke tube: A theoretical,
computational, and experimental investigation," J. Acoust. Soc. Am.
96(5), 3177–3185.
Selamet, A. and Easwaran, V. (1997). "Modified Herschel-Quincke tube: attenuation and
resonance for n–duct configuration," J. Acoust. Soc. Am.
102(1), 164–169.
Selamet, A. and Lee, I. (2003). "Helmholtz resonator with extended neck," J. Acoust. Soc. Am
113(4), 1975–1985.
Seybert, A. F. and Ross, D. F. (1977). "Experimental determination of acoustic properties using a
two-microphone random-excitation technique," J. Acoust. Soc. Am
61(5), 1362–1370.
Seybert, A. F. and Soenarko, B. (1981). "Error analysis of spectral estimates with application to
the measurement of acoustic parameters using random sound fields in ducts," J. Acoust.
Soc. Am,
69, 1190–1199.
Shen, H. and Tam, C. K. W. (1998). "Numerical simulation of the generation of axisymmetric
mode jet screech tones," AIAA J.
36(10), 1801–1807.
Sheplak, M., Cattafesta, L., Nishida, T., and Horowitz, S.B. (2004) "Electromechanical acoustic
liner," U.S. Patent No. 6,782,109
Sivian, L. J. (1935). "Acoustic impedance of small orifices," J. Acoust. Soc. Am
7, 94–101.
213
Smith, J.P. and Burdisso, R.A. (2002) "Experimental investigation of the Herschel-Quincke tube
concept on the Honeywell TFE731-60," NASA/CR-2002-211431, (National Aeronautics
and Space Administration)
Smith, J.P. and Burdisso, R.A. (2002) "Experiments with fixed and adaptive Herschel-Quincke
waveguides on the Pratt and Whitney JT15D engine," NASA/CR-2002-211430, (National
Aeronautics and Space Administration)
.
Smith, M. J. T. (1989). Aircraft Noise (Cambridge, Cambridge University Press), Chaps.1, 2, and
3.
Song, B. H. and Bolton, J. S. (2000). "A transfer-matrix approach for estimating the
characteristic impedance and wave numbers of limp and rigid porous materials," J. Acoust.
Soc. Am
107(3), 1131–1152.
Steuer, R. E. (1986). Multiple criteria optimization: theory, computation, and application (New
York, John Wiley & Sons).
Stewart, G. W. (1928). "The theory of the Herschel-Quincke tube," Phys. Rev.
31, 696–698.
Stewart, G. W. (1930). Acoustics (New York, D. Van Nostrand Company).
Stewart, G. W. (1945). "The Theory of the Herschel-Quincke Tube," J. Acoust. Soc. Am.
17(2):
107–108.
Sujith, R. I. (1996). "Transfer matrix of a uniform duct with an axial mean temperature gradient,"
J. Acoust. Soc. Am
100(4), 2540–2542.
T.Y.Lung and Doige, A. G. (1983). "A time-averaging transient testing method for acoustic
properties of piping system and mufflers with flow," J. Acoust. Soc. Am
. 73(3), 867–876.
Tijdeman, H. (1974). "On the propagation of sound waves in cylindrical tubes," J. Sound Vib.
39(1), 1–33.
Tilmans, H.A.C., Elwenspoek, M., and Fluitman, J.H.J. (1992). "Micro resonant force gauges,"
Sensors and Actuators A-Physical
30, 35–53
Van Valkenburg, M. E. (1964). Network Analysis (Englewood Cliffs, Prentice–Hall), Chaps. 3
and 9.
Wang, C.N., Tse, C.C. and Chen, Y.. (1993). "Analysis of three dimensional muffler with
boundary element method," Appl. Acoust.
40(2), 91–106.
Watson, G. N. (1996). A Treatise on the Theory of Bessel Functions. (New York, Cambridge
University Press).
Willshire, B. (2001). http://www.techtransfer.berkeley.edu/aviation01downloads/willshire.pdf.
214
Wu, T. N. and Lai, J. S. (1995). "Aircraft noise, hearing ability, and annoyance," Archives of
Environmental Health
50(6), 452–456.
Zinn B. T. (1970). "A theoretical study of non-linear damping by Helmholtz resonators," J.
Sound Vib.
13(3), 347–356.
Zwikker, C. and C. W. Kosten (1949). Sound Absorbing Materials (Amsterdam, NY, Elsevier),
Chaps. I and II.
215
BIOGRAPHICAL SKETCH
Fei Liu was born in Chongqing, China, in January 1973. He received the BE degree with
an aerospace engineering major from Beijing University of Aeronautics and Astronautics
(BUAA), China in 1994. He then worked at the Department of Jet Propulsion of the BUAA as
an Assistant Lecturer from 1994 to 1998, and Lecturer and Head of the Engine Control System
Laboratory from 1998 to 2001. He came to the United States to pursue his Ph.D. study in 2001,
and entered to the Department of Mechanical and Aerospace Engineering of the University of
Florida as a graduate assistant. There he received his M.S. degree and Ph.D. degree with an
aerospace engineering major in 2003 and 2007, respectively. His research involved in designing,
modeling and testing on an electromechanical Helmholtz resonator.
He was a student member of American Institute of Aeronautics and Astronautics (AIAA)
and Acoustical Society of America.
