Title: Design and characterization of compliant backplate Helmholtz resonators
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Title: Design and characterization of compliant backplate Helmholtz resonators
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Language: English
Creator: Horowitz, Stephen Brian, 1977-
Publisher: University of Florida
Place of Publication: Gainesville Fla
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Publication Date: 2001
Copyright Date: 2001
 Subjects
Subject: Airplanes -- Noise   ( lcsh )
Electrical and Computer Engineering thesis, M.S   ( lcsh )
Dissertations, Academic -- Electrical and Computer Engineering -- UF   ( lcsh )
Genre: government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
theses   ( marcgt )
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Summary: ABSTRACT: To meet increasingly stringent government regulations on noise emitted from aircraft engines, improved methods are needed to reduce the level of noise emanating from aircraft engine nacelles. Noise suppression is achieved through appropriate impedance boundary conditions that reduce the propagation of engine noise. If the engine conditions change, a different impedance boundary condition is necessary to optimally suppress the noise. The ultimate goal of the research presented in this thesis is to design and build an in-situ, tunable-impedance, electromechanical acoustic liner to optimally suppress noise under changing engine conditions. To further this end, an investigation was conducted into compliant-backplate Helmholtz resonators, which will later serve as fundamental components of the electromechanical acoustic liner. This thesis presents the analytical and experimental characterization of Helmholtz resonators using isotropic metal plates as a compliant backplate. Lumped element models are developed and used to design a prototype Helmholtz resonator with a cavity backed by one of 24 different isotropic compliant backplates. Each configuration is then characterized from 1 kHz to 6.4 kHz in an impedance tube using pressure amplification and normal-incidence, acoustic impedance measurements.
Summary: ABSTRACT (cont.): The experimental results demonstrate an additional resonance and an anti-resonance in the impedance caused by adding degrees-of-freedom to a conventional Helmholtz resonator. These extrema depend in part on the resonant frequency of the compliant backplate and can later be utilized for impedance tuning by using a piezoelectric-composite backplate in place of the isotropic backplates presented in this research. Overall, the results confirm the multi-degree of freedom nature of the compliant-backplate Helmholtz resonators and their equivalence to double-layer acoustic liners.
Summary: KEYWORDS: Helmholtz resonator, acoustic liner, electromechanical, compliant, noise suppression
Thesis: Thesis (M.S.)--University of Florida, 2001.
Bibliography: Includes bibliographical references (p. 79-80).
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System Details: Mode of access: World Wide Web.
Statement of Responsibility: by Stephen Brian Horowitz.
General Note: Title from first page of PDF file.
General Note: Document formatted into pages; contains xiii, 81 p.; also contains graphics.
General Note: Vita.
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Bibliographic ID: UF00100839
Volume ID: VID00001
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DESIGN AND CHARACTERIZATION OF COMPLIANT BACKPLATE
HELMHOLTZ RESONATORS
















By

STEPHEN BRIAN HOROWITZ


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2001




























Copyright 2001

by

Stephen Brian Horowitz




























To my sweetie,
Liz















ACKNOWLEDGMENTS

Financial support for this project is provided by NASA Langley Research Center

(Grant #NAG-1-2261) and is monitored by Dr. Michael G. Jones.

I would like to thank my advisors, Dr. Mark Sheplak and Dr. Toshikazu Nishida,

for giving me some freedom in my research, while constantly pushing me to learn

subjects until I own them. Their ideas and encouragement made this research possible. I

would also like to thank Dr. Louis Cattafesta for his guidance and support throughout my

research. Additionally, I am deeply indebted to all of my fellow students in the

Interdisciplinary Microsystems Group, for their support and friendship.

Special thanks go to my family for always encouraging me to pursue my interests

and for making that pursuit possible. Finally I would like to thank my fiance, Megan

Elizabeth Elliott, for her constant love and support.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF FIGU RE S ................ ............................................. ........ .. ...... .... vii

A B ST R A C T ................x ii.............................................

CHAPTERS

1 INTRODUCTION ................................... ... .......... .. .................1.
Passive A acoustic Liners ................................... . ....... .. ................ .. 1
A daptive and A ctive A acoustic Liners ................................................... ................... 3
O their W ork in the Field .................................. .......................... .... ......... ....... 3
Electrom echanical A acoustic Liner...................................... ........................... ....... 3
2 THEORETICAL BACK GROUND ....................................................... ...................6
Conventional H elm holtz R esonator............................................................ .............. 6
Lum ped Elem ent M odel................................................... ............................... 6
Equivalent Circuit ................... .. ....................... ........... ...... ......... 10
Compliant-Backplate Helmholtz Resonator .................................................... 11
Lum ped Elem ent M odel..................... ....................................... .......................... 11
E equivalent C circuit ........... ...... ...... ........... ............................................. ................ .. 16
Mass Ratio Dependence of Coupled Resonant Frequencies............. ............. 24
3 E X PER IM E N TA L SE TU P ........................................ .............................................28
O v erview ...... ...... . ........... ......... ................................................... 2 8
M easurem ents and Equipm ent .......................... ......... ......... .................... .............. 28
Device Construction..................... .................. 33
4 EX PER IM EN TA L R E SU L TS ........................................................... .....................37
P ressu re A m plification .............................. .. .... ................................................ 37
Helmholtz Resonator with Half-Inch Diameter, Clamped Backplates ................... 37
Helmholtz Resonator with One-Inch Diameter, Clamped Backplates ................. 47
In p u t Im p ed an ce ............................... .... .. ..... ..... ....................................... 5 5
Helmholtz Resonator with Half-Inch Diameter, Clamped Backplates ................... 55
Helmholtz Resonator with One-Inch Diameter, Clamped Backplates ................. 61
D discussion of R results .............................. . .................................... .. .......... ........ .. 68
Low -Frequency M ode Shapes ............................................... ........................... 69


v









5 CON CLU SION .............. .......................... ........... ........................ 71
Sum m ary of R results ................................................................. ....... ........ .. 7 1
Future W ork.............................. ............. ...... 72
APPENDICES

A DRAWINGS ................................. .. .... ... .................. 73

B M A T H C A D C O D E .............................................................................. .................... 76

L IST O F R E FE R E N C E S .......................................................................... ....................79

BIOGRAPH ICAL SKETCH ...................... ........ ............................ ...............81
















LIST OF FIGURES


Figure Page

1: Three conventional types of passive acoustic liners (Source: Motsinger and Kraft [1]
pg 167). ............................................... ........................... 2

2: Single element of an electromechanical acoustic liner. ............................................4

3: Diagram showing (a) side view and (b) top view of a conventional Helmholtz
resonator..................................................... ................... ... ... .. .... . 7

4: Equivalent circuit representation of a conventional Helmholtz resonator ................. 11

5: (a) Magnitude and (b) phase of theoretical frequency response of a conventional
H elm holtz resonator .......................................... .. ............ ........ .... 12

6: Diagram of a compliant backplate Helmholtz resonator. .........................................12

7: Acoustical and mechanical equivalent circuit representation of a Helmholtz resonator
w ith a com pliant backplate ........................................ ......................... 16

8: Equivalent acoustic circuit representation of a Helmholtz resonator with a compliant
b ack p late. ..................................................... ................ . 19

9: Magnitude and phase of the theoretical pressure amplification of a compliant-
backplate Helmholtz resonator with a 0.0015 in. thick, 0.5 in. diameter,
aluminium backplate. ........................................... .... ... ............. 21

10: Contour plot of pressure amplification for various thickness of backplate. Bright
areas are peaks in the frequency response, while dark areas are troughs. .......22

11: Magnitude and phase of the theoretical, normalized input impedance of a compliant-
backplate H elm holtz resonator................................................ .................. 23

12: Theoretical normalized resistance and reactance of a compliant-backplate Helmholtz
resonator................ ............................. .. ............ .. ....... 23

13: Schematic of impedance tube terminated by compliant-backplate Helmholtz
resonator................ ............................. .. ............ .. ....... 29









14: Normal impedance tube showing rotating microphone plugs, and end-mounted
H elm holtz resonator ............................ ................ .................... ..... ........ 30

15: Photograph of the compliant backplate Helmholtz resonator showing compliant
backplate clamped by circular clamping ring and incident and cavity
m icrophone plugs ...................... .. .... ........................................... 34

16: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.001 in., aluminum backplate clamped at 0.5 in. diameter. ..................39

17: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.002 in., aluminum backplate clamped at 0.5 in. diameter. ..................40

18: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.003 in., aluminum backplate clamped at 0.5 in. diameter. ..................40

19: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.005 in., aluminum backplate clamped at 0.5 in. diameter. .................41

20: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.001 in., brass backplate clamped at 0.5 in. diameter ..........................41

21: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.002 in., brass backplate clamped at 0.5 in. diameter ..........................42

22: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.003 in., brass backplate clamped at 0.5 in. diameter. ...........................42

23: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.005 in., brass backplate clamped at 0.5 in. diameter ..........................43

24: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.001 in., stainless steel backplate clamped at 0.5 in. diameter ..............43

25: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.002 in., stainless steel backplate clamped at 0.5 in. diameter ..............44

26: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.003 in., stainless steel backplate clamped at 0.5 in. diameter ..............44

27: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.005 in., stainless steel backplate clamped at 0.5 in. diameter ..............45

28: Coherence between cavity and incident microphone for Helmholtz resonator with
0.001 in. thick aluminum backplate clamped at 0.5 in. diameter. ..................47

29: Coherence between cavity and incident microphone for Helmholtz resonator with
0.005 in. thick stainless steel backplate clamped at 0.5 in. diameter ..............47









30: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.002 in., aluminum backplate clamped at 0.96 in. diameter. ................48

31: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.003 in., aluminum backplate clamped at 0.96 in. diameter. ................49

32: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.005 in., aluminum backplate clamped at 0.96 in. diameter. ................49

33: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.010 in., aluminum backplate clamped at 0.96 in. diameter. ................50

34: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.002 in., brass backplate clamped at 0.96 in. diameter ........................50

35: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.003 in., brass backplate clamped at 0.96 in. diameter ........................51

36: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.005 in., brass backplate clamped at 0.96 in. diameter ........................51

37: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.010 in., brass backplate clamped at 0.96 in. diameter ........................52

38: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.002 in., stainless steel backplate clamped at 0.96 in. diameter.............52

39: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.003 in., stainless steel backplate clamped at 0.96 in. diameter.............53

40: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.005 in., stainless steel backplate clamped at 0.96 in. diameter.............53

41: Magnitude and phase of pressure amplification spectrum for Helmholtz resonator
with 0.010 in., stainless steel backplate clamped at 0.96 in. diameter.............54

42: Normalized resistance and reactance for Helmholtz resonator with 0.001 in.
aluminum backplate clamped at 0.5 in. diameter..............................55

43: Normalized resistance and reactance for Helmholtz resonator with 0.002 in.,
aluminum backplate clamped at 0.5 in. diameter.....................................56

44: Normalized resistance and reactance for Helmholtz resonator with 0.003 in.,
aluminum backplate clamped at 0.5 in. diameter..............................56

45: Normalized resistance and reactance for Helmholtz resonator with 0.005 in.,
aluminum backplate clamped at 0.5 in. diameter..............................57









46: Normalized resistance and reactance for Helmholtz resonator with 0.001 in., brass
backplate clamped at 0.5 in. diameter................................ ................57

47: Normalized resistance and reactance for Helmholtz resonator with 0.002 in., brass
backplate clamped at 0.5 in. diameter.............. ...............................................58

48: Normalized resistance and reactance for Helmholtz resonator with 0.003 in., brass
backplate clamped at 0.5 in. diameter................................ ................58

49: Normalized resistance and reactance for Helmholtz resonator with 0.005 in., brass
backplate clamped at 0.5 in. diameter.............. ...............................................59

50: Normalized resistance and reactance for Helmholtz resonator with 0.001 in., stainless
steel backplate clamped at 0.5 in. diameter. ............. ....................... ......... 59

51: Normalized resistance and reactance for Helmholtz resonator with 0.002 in., stainless
steel backplate clamped at 0.5 in. diameter. ............. .................................... 60

52: Normalized resistance and reactance for Helmholtz resonator with 0.003 in., stainless
steel backplate clamped at 0.5 in. diameter. ............. .................................... 60

53: Normalized resistance and reactance for Helmholtz resonator with 0.005 in., stainless
steel backplate clamped at 0.5 in. diameter. ............. .................................... 61

54: Normalized resistance and reactance for Helmholtz resonator with 0.002 in.
aluminum backplate clamped at 0.96 in. diameter .......................................62

55: Normalized resistance and reactance for Helmholtz resonator with 0.003 in.
aluminum backplate clamped at 0.96 in. diameter .......................................62

56: Normalized resistance and reactance for Helmholtz resonator with 0.005 in.
aluminum backplate clamped at 0.96 in. diameter .......................................63

57: Normalized resistance and reactance for Helmholtz resonator with 0.010 in.
aluminum backplate clamped at 0.96 in. diameter .......................................63

58: Normalized resistance and reactance for Helmholtz resonator with 0.002 in. brass
backplate clamped at 0.96 in. diameter.............. ............................................64

59: Normalized resistance and reactance for Helmholtz resonator with 0.003 in. brass
backplate clamped at 0.96 in. diameter.............. ............................................64

60: Normalized resistance and reactance for Helmholtz resonator with 0.005 in. brass
backplate clamped at 0.96 in. diameter.............. ............................................65

61: Normalized resistance and reactance for Helmholtz resonator 0.010 in., brass
backplate clamped at 0.96 in. diameter.............. ............................................65









62: Normalized resistance and reactance for Helmholtz resonator with 0.002 in., stainless
steel backplate clamped at 0.96 in. diameter. ............................................. 66

63: Normalized resistance and reactance for Helmholtz resonator with 0.003 in., stainless
steel backplate clamped at 0.96 in. diameter. ............................................. 66

64: Normalized resistance and reactance for Helmholtz resonator with 0.005, stainless
steel backplate clamped at 0.96 in. diameter. ............................................. 67

65: Normalized resistance and reactance for Helmholtz resonator with 0.010 in., stainless
steel backplate clamped at 0.96 in. diameter. ............................................. 67

67: Measured mode shapes for each backplate along with a theoretical mode shape for a
clamped circular plate, as given by Equation 2.10. .....................................70

68 Schematic of neck and cavity plate for 0.5 in. diameter compliant backplate
H elm holtz resonator ............................ ................ .................... ..... ........ 73

69: Schematic of incident microphone (Left) and cavity microphone (Right).................73

70: Schematic of 0.5 in. clamping ring and placement of clamping ring with incident
m icro p h o n e ................................................ ................ . 7 4

71: Schematic of cavity plate for 0.96 in. diameter compliant backplate Helmholtz
resonator............... ........ ...... . ......... ...... .......... ............. 74

72: Schematic of spacer ring, clamping ring, and placement of rings relative to incident
microphone for 0.96 in. diameter compliant backplate Helmholtz resonator..75















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

DESIGN AND CHARACTERIZATION OF COMPLIANT BACKPLATE
HELMHOLTZ RESONATORS

By

Stephen Brian Horowitz


August 2001

Chairman: Dr. Toshikazu Nishida
Cochairman: Dr. Mark Sheplak
Major Department: Electrical and Computer Engineering

To meet increasingly stringent government regulations on noise emitted from

aircraft engines, improved methods are needed to reduce the level of noise emanating

from aircraft engine nacelles. Noise suppression is achieved through appropriate

impedance boundary conditions that reduce the propagation of engine noise. If the

engine conditions change, a different impedance boundary condition is necessary to

optimally suppress the noise. The ultimate goal of the research presented in this thesis is

to design and build an in-situ, tunable-impedance, electromechanical acoustic liner to

optimally suppress noise under changing engine conditions.

To further this end, an investigation was conducted into compliant-backplate

Helmholtz resonators, which will later serve as fundamental components of the

electromechanical acoustic liner. This thesis presents the analytical and experimental

characterization of Helmholtz resonators using isotropic metal plates as a compliant









backplate. Lumped element models are developed and used to design a prototype

Helmholtz resonator with a cavity backed by one of 24 different isotropic compliant

backplates. Each configuration is then characterized from 1 kHz to 6.4 kHz in an

impedance tube using pressure amplification and normal-incidence, acoustic impedance

measurements. The experimental results demonstrate an additional resonance and an

anti-resonance in the impedance caused by adding degrees-of-freedom to a conventional

Helmholtz resonator. These extrema depend in part on the resonant frequency of the

compliant backplate and can later be utilized for impedance tuning by using a

piezoelectric-composite backplate in place of the isotropic backplates presented in this

research. Overall, the results confirm the multi-degree of freedom nature of the

compliant-backplate Helmholtz resonators and their equivalence to double-layer acoustic

liners.














CHAPTER 1
INTRODUCTION

Noise suppression within aircraft engine ducts is necessary to meet government

regulations that limit noise radiated from both commercial and private aircraft [1].

Suppression of noise is achieved by lining the engine duct with an appropriately designed

acoustic liner. The acoustic liner is designed to provide an impedance boundary

condition in the engine duct that reduces the propagation of engine noise through the

duct.


Passive Acoustic Liners

Early designs focused on single layer, passive liners, that generally consist of a

perforated faceplate and a rigid backplate, separated by a honeycomb structure [1], as

shown in Figure la. These operate together as a conventional Helmholtz resonator,

possessing a resonant frequency dependent upon the geometry of the faceplate holes and

honeycomb shaped cavity. The noise-reduction bandwidth in single-layer liners is

limited to one octave, which is tuned early in the design process to correspond to the

blade passage frequency of the intended application [1].

To meet increasingly stringent government regulations, more advanced designs

are necessary. Two approaches have generally been followed to achieve increased noise

suppression. The first approach relies on improving passive liner technology, while the

second approach uses adaptive techniques to allow for continued improvement in noise

suppression.













;',." 1 .'rLi l Kiac.. hcmfry mn lj
Kraft [ id hak1]lcp 1

l t n inle deexst fr i fredas om (SDOF).


P LI 1 1- IIf N rI P4311'm irfi" L
I ; .t L -F m I J




lay Ttlo l eree o freedom tht DOF)


j----- rol'InniF f.N --tHI*fli




(c) Bulk ah!oIref.

Figure 1: Three conventional types ofpassive acoustic liners (Source: Motsinger and
Kraft[l]pg. 167).



Several techniques exist for increasing the attenuation bandwidth and level of

maximum attenuation of passive acoustic liners, including parallel element and multi-

layer liners. Multi-layer liners provide additional degrees of freedom that can be

optimized to achieve a target impedance over a larger bandwidth than is possible with a

single-layer [2]. A two-layer liner, as shown in Figure lb for example, can achieve an

attenuation bandwidth of about two octaves [1]. A third type of passive acoustic liner,

known as a bulk absorber, contains a porous material in the cavity, as shown in Figure Ic.

A bulk absorber offers the widest bandwidth attenuation of the passive techniques,









however it cannot be used in aircraft engine ducts due to high levels of jet fuel absorption

by the porous material.

Once a passive acoustic liner is fabricated, the peak suppression frequency is

fixed, and the bandwidth of effective attenuation is limited to one octave centered around

the peak frequency. Changing engine conditions, such as at take-off and landing, pose a

problem for optimization, since the optimum impedance spectrum may vary, but the liner

impedance spectrum is fixed by the geometry of the design.


Adaptive and Active Acoustic Liners

Other Work in the Field

Various adaptive techniques have been investigated to modify the acoustic

properties of liners for improved attenuation under changing engine conditions. One

method uses a steady bias flow through the perforate faceplate to adjust the acoustic

resistance [2-6]. A second method seeks to change the density near the liner via heating

thereby controlling the resistance [2]. These techniques are adaptive and seek to improve

the attenuation characteristics of a liner by directly modifying the impedance of one or

more of the acoustic components of the liner. Another adaptive technique that has been

investigated involves using a volume-variable cavity for a Helmholtz resonator [7, 8].

Other techniques seek to reduce noise by actively canceling the incident sound field with

an out-of-phase, generated sound field [9].

Electromechanical Acoustic Liner

The liner presented in this thesis uses an alternative method of impedance tuning.

The primary element of this liner is a Helmholtz resonator containing a compliant









piezoelectric composite backplate, as shown in Figure 2. The backplate provides

acoustical-to-electrical transduction via the mechanical energy domain.





Compliant .
Compliant Piezoelectric
Backplate material
material


Figure 2. Single element of an electromechanical acoustic liner.




The impedance of this liner is not only a function of the acoustical components,

but the mechanical and electrical components as well. While this complicates the

impedance function, it provides an opportunity to tune the impedance by varying an

electrical filter network. Additionally, more degrees of freedom are added to the system

that can be optimized to improve the attenuation bandwidth. In fact, the impedance of

this electromechanical acoustic liner takes on the same form and structure as existing

multi-layer liners. The impedance of the basic electromechanical acoustic liner, with no

electrical components connected, exactly parallels a double layer liner. In this liner, the

aspects of the impedance typically caused by a second layer are instead due to

mechanical components. Because of the piezoelectric transduction, this concept can be

extended to provide as many degrees of freedom as desired, simply by adding an

appropriate electrical network of inductors and capacitors across the electrodes of the

piezoelectric material. Thus the benefits of multi -layer liners are achievable with

electromechanical acoustic liners.









The impedance of the electromechanical acoustic liner can be tuned in-situ and in

real-time. The initial goal is to have three distinct liner impedance spectra, each

optimized for a specific engine condition, such as take-off, cut-back, and landing. This

can be achieved with three separate electrical networks and a simple three-way switch to

select the appropriate network.

In order to achieve this goal, a thorough understanding of the properties of an

electromechanical acoustic liner is necessary. The fundamental element of this liner is a

compliant-backplate Helmholtz resonator. In this thesis, the theoretical analysis of

compliant-backplate Helmholtz resonators is developed and experimental verification of

the concept is presented.














CHAPTER 2
THEORETICAL BACKGROUND

Understanding the influence of individual parameters of a given system is critical

to efficient and accurate design. An intuitive and analytical understanding of the system

is necessary to achieve the desired performance specifications. Furthermore, the design

of an electromechanical acoustic liner presents a multi-domain modeling challenge.

Lumped element modeling provides an effective means of analyzing and

designing a system involving multiple energy domains. Lumped element modeling has

been used in the past for analysis of acoustic liners [10,11]. The convenience of lumped

element modeling lies in the explicit relationship between individual design parameters

and the frequency response of the system. Lumped element modeling must be used with

care, to ensure that necessary assumptions are true. In particular, the wavelength of

interest must be significantly larger than the characteristic length scale of the system, for

the lumped assumption to be valid. When this criterion is met, the lumped element

model is a reasonably accurate model of the distributed physical system. For the design

and analysis of rigid and compliant-backplate Helmholtz resonators presented in this

paper, lumped-element modeling is used extensively.



Conventional Helmholtz Resonator

Lumped Element Model

The dynamic response of a Helmholtz resonator can be conveniently modeled

using an equivalent circuit representation. This representation relates mechanical and









acoustic quantities to their electrical equivalents. In circuit theory, distributed electrical

parameters are lumped into specific components, based on how they interact with energy

[12]. Using this criterion, a resistor represents dissipation of energy, while inductors and

capacitors represent storage of kinetic and potential energy, respectively.

The techniques developed for circuit theory can be applied towards mechanical

and acoustical systems by generalizing the fundamental circuit components [13,14]. A

schematic diagram of a conventional Helmholtz resonator is shown below in Figure 3,

where Vis the cavity volume, L and S are the length and cross-sectional area of the neck,

respectively, P1 is the incident acoustic pressure, and P2 is the cavity acoustic pressure.

Both acoustic pressures are considered to be functions of the radian frequency, ca


Pil(o)


(a) V

P2(C)





Area 'S'
(b)





Figure 3: Diagram showing (a) side view and (b) top view of a conventional Helmholtz
resonator.



A conventional Helmholtz resonator can be lumped into three distinct elements.

The neck of the resonator constitutes a pipe through which frictional losses are incurred.









Additionally the air that is moving through the neck possesses a finite mass and thus

kinetic energy. Thus the neck has both dissipative and inertial components. The air in the

cavity is compressible and stores potential energy, and is therefore modeled as a

compliance.

The acoustic compliance of the cavity and effective mass of the neck can be

derived from first principles [11]. As mass flows into the bulb, the volume, V, remains

constant and so the pressure must rise, by continuity of mass.

dA = y mass flow rate (2.1)
dt dt Ls
where M is the mass in the bulb and par is the density of the medium. If the disturbance is

harmonic and isentropic then


P2' = C2Pr = (2.2)

where c is the speed of sound of the medium, co is the radian frequency, and j= N-T .

The linearized momentum equation for a lossless medium is given by

Ou
Par = -VP'. (2.3)
at
Assuming a linear pressure gradient yields


P -P2 = Pa, L, (2.4)
at
where L is the length of the neck. Substituting for P' yields the following equation.

,Qc Qol
P = Q + Q- (2.5)
joV S
Defining the volumetric flow rate as

q=-- (2.6)
Par
yields a relation between the effort PI and the flow q as shown below to be










P'= q + joa (2.7)

In the above expression, the effective compliance Ca of the cavity is


Ca = rC2 (2.8)
P ,rC Pa
The effective mass of the air in the neck is given by

M 4P kg]. (2.9)
a 3 M S 4
where L and S are the length and cross-sectional area of the neck, respectively.

The factor of 4/3 in the above expression for effective mass comes from a non-

uniform axial velocity profile in the neck, due to viscous damping. The viscous damping

represents a resistance, whose resistance can be approximated from pressure driven,

laminar pipe flow as


Ra = .L[kg] (2.10)
Ra=-~ s- ;7S

where p is the viscosity of the air.

The effective resistance and mass values of the neck are, in fact, non-linear due to

turbulence and entrance/exit effects [10]. These are a result of the high sound pressure

levels present in the engine nacelle environment. In order to keep this analysis

straightforward, these non-linear effects will be ignored in this thesis, along with any

grazing flow dependence. The non-linear effects are small at the sound pressure levels,

around 100 dBSPL, used in the experiments presented in this thesis [10].

The expression for cavity compliance given by (2.8) can be compared to an

approximation based on the exact expression for the impedance in a short closed tube as

given on page 149-50 of Blackstock [15]. The exact expression is given by










Zn PaIrC (2.11)
j tan kl

where k = -, and I is then length of the tube. Using a Maclaurin series expansion of the
c

tangent function yields


tan kkl= kl+k313 +L (2.12)
3
For kl << 1, the impedance can be approximated by keeping only the first couple of terms

in the expansion, yielding

paI c klpac
Sjkla2 3_a2
C2 ParV (2.13)
+ Jo 2
j7 3 (ra2 2
From this expression, we once again see that


Ca= c2 [ (2.14)
Pa,rC Pa
We now also have an additional mass term, given by


c_ =ParV kg (2.15)
3 (ra2 )2 M
which is equal to one-third the acoustic mass of the cavity. This correction term is small

for kl = 1 but becomes more prominent as kl increases. At kl= 1, the correction term is

33.3% of the primary term.

Equivalent Circuit

To create an equivalent circuit model for the Helmholtz resonator, we also need to

know how to connect these lumped elements. Connection rules between elements are

defined based on whether an effort-type variable or a flow-type variable is shared

between them. Whenever an effort variable, such as force, voltage or pressure, is shared

between two or more elements, those elements are connected in parallel in the equivalent









circuit. Conversely, whenever a common flow (i.e., velocity, current, or volume

velocity) is shared between elements, those elements are connected in series. These

connection rules are used to obtain the equivalent circuit representation for the Helmholtz

resonator, as shown in Figure 4.



Ra M +

P,(o) P209 Ca



Figure 4: Equivalent circuit representation of a conventional Helmholtz resonator.



The frequency response function P2/Pj, represents the pressure amplification of

the resonator. It is the ratio of cavity pressure to incident pressure. From an analysis of

the above circuit, a single resonant peak is expected in this frequency response function,

when the sum of the reactances is zero, as is given by

1
fre 2 = M (2.16)

This is shown in Figure 5, for a conventional Helmholtz resonator having a neck length

and diameter of 3.18 mm and 4.72 mm, respectively, and a cavity volume of 1950 mm3.


Compliant-Backplate Helmholtz Resonator

Lumped Element Model

In the analysis of the conventional Helmholtz resonator, it was implicitly assumed

that the walls of the cavity were rigid. In the following analysis, the effect of a compliant

wall in the cavity is examined. When one of the cavity walls is thin enough to flex under

an applied pressure, as shown in Figure 6, the compliance and mass of the thin wall must










be accounted for to accurately model the system. This introduces two additional lumped

elements.


(a) 40


20


0 ---------

-20
1000 2000 3000 4000 5000 6000
Freq [Hz]


7 -50

S-100
I.
S-150

-200
1000 2000 3000 4000 5000 6000
Freq [Hz]
Figure 5. (a) Magnitude and (b) phase of theoretical frequency response of a
conventional Helmholtz resonator.


Compliant Backplate
Cm,Mm


Figure 6. Diagram of a compliant backplate Helmholtz resonator.









By modeling the compliant backplate as a clamped circular plate, the lumped

element parameters can then be derived. The physically distributed backplate is lumped

into an equivalent mass and compliance at a single point in space. The center of the plate

(i.e., where the radius, r = 0) is chosen as the point about which the system is lumped

because of the circular geometry of the plate. The deflection of a clamped circular plate

of radius, a, and thickness, h, under a uniform pressure P is given by [13]


w(r)= wo 1- r (2.17)

where the center deflection w(0) wo is given by

Pa4
w =P (2.18)
64D
and D, the flexural rigidity, is defined as

Eh3
D= E(v2 (2.19)
12 (-v2)
Additionally, in (2.19), E is the elastic modulus, and v is the Poisson's ratio of the

material. Similarly, the differential of the plate deflection is given by


W(r ) r
dw(r) = w() = 1- dw(0) (2.20)
8w(0) a
To find the effective compliance of the backplate, the potential energy stored in

the backplate for a given displacement must first be calculated [13]. This can then be

equated to the general expression for the potential energy in a spring, where the spring

displacement is defined as the center deflection. The potential energy is then given by

ww(0)2 k
WE = (2.21)
where k is the spring stiffness.
where k is the spring stiffness.









From this relation, the effective stiffness, which is the inverse of the

effective compliance, can be extracted. The potential energy stored in a differential

element of the backplate is given by


dWE = Fdx = PdAdw(r) = P27rrdrdw(r) (2.22)
where F is the force, and the pressure P can be found from (2.18) to be

64D
P = w(0) (2.23)
a
This yields a total potential energy of

1287JD rw(O) I drdw(O)
WPE 4 f (0) 1- drfv(0)
a w(o)a
a 0 (2.24)
128rD (2 1 1 w(0)2
a 2 2 C
Thus the effective mechanical compliance of the backplate is found to be

3a2 9a (1V2) m
Cme 647rD 167rEh3 (2.25)
A similar method is used to compute the effective mass of the compliant

backplate [13]. Instead of finding the potential energy, however, the kinetic energy is

computed and equated to

1 2
WE = -mu (2.26)
2
where u is the velocity of the backplate and for harmonic motion is given by

u(r) = jow(r). (2.27)
The kinetic energy stored in a differential element of the plate is found to be

dW o = ph ( I j2r 2
dWuE = h u( 1-r 2-rdr (2.28)
2 a)
Integrating this expression over the area of the plate yields the total kinetic energy, given

by









1 2 ( Jrdr
WE = -phu(0)2 1- rdr
2o (2.29)
1 2 a2
S-u(0) ph j
2 5
This yields an effective mechanical inertance of



Mme = ph = Mac [kg]. (2.30)
The effective mass of the compliant plate is therefore equivalent to 1/5th of the actual

mass. Physically, this is due to the variation of deflection and hence kinetic energy over

the radius of the circular plate.

In addition to the compliance and mass of the membrane, another element must

now be included in the model. Since the clamped circular plate is vibrating in a medium,

the radiation impedance of the plate must be taken into account. The circular plate will

be modeled here as a piston in an infinite baffle, for the purposes of calculating a

radiation impedance. The radiation impedance of a piston in an infinite baffle is given

on page 459 of Blackstock [15] as

i 2J1 (2ka) 2K (2ka)
L 2ka + 2ka (2.31)

= PaC [R,(2ka) + jX,(2ka)]
where k = -, a is the radius of the piston, J1 is Bessel function of the first kind of order
c

one, and K1 is first-order Struve function The Maclaurin expansions of (2.31) are also

given by [15] as

(ka)2 (k< k)6
R, = -+ ()6L (2.32)
1-2 1.22.3 1-2 3 2.4
4 2ka (2ka)3 (2ka (2.33)
X= 3 > 52 32 .L (2.33)
7K 3 3 -5 3 .5 57









Under a low frequency approximation (ka <<1), only the first terms in each series are

kept, and a lumped radiation resistance and mass are given, respectively, in the acoustic

domain as


R (ka)2Parr kg_
Rrad ka 2 4
2 na im4S] m
S 2 7ra2 MS (2.34)
k2ParC kg2
2Ir m4S
and

8a pParC kg7
rad 2 4
3 r nra m\4
(2.35)
8 pr [ kg

Equivalent Circuit

The two mechanical lumped elements can easily be incorporated into the overall

equivalent circuit. The new elements are in series with each other because they both are

subject to the same motion. Additionally, the series combination of these two elements is

in parallel with the acoustic compliance. A portion of the flow entering the cavity

through the neck of the resonator will contribute to an increase in cavity pressure, while

the remainder of the flow contributes to the motion of the compliant backplate.


S Ma +1:Aeff
+ + 1 c
P co) P, ( m c aCC


Acoustical Mechanical


Figure 7: Acoustical and mechanical equivalent circuit representation of a Helmholtz
resonator i/th a compliant backplate









The equivalent circuit in Figure 7 shows the additional mechanical lumped

elements that appear in a compliant backplate Helmholtz resonator. Also, an acoustical

to mechanical transduction factor is necessary to account for the transduction of

acoustical energy to mechanical energy and vice-versa. The acoustical to mechanical

transduction factor can be defined as the effective area through which the transduction

occurs, and is given by

F
Ae, = (2.36)
P
where P is the pressure applied and F is the equivalent force. This is represented by a

transformer with the turns ratio given by Aeff For a lumped mechanical spring the force

is related to the displacement by

F= Kx (2.37)
The spring stiffness, K, is the inverse of the spring compliance, C. For a clamped circular

plate, the compliance was found to be

3a2
Cme (2.38)
647rD
The circular plate is lumped about its center, therefore the displacement indicated

in (2.37) represents the center deflection of the plate. The center deflection of the plate is

given by

Pa4
x(0) (2.39)
64D
Substituting (2.38) and (2.39) into (2.37) yields

Pa4 647rD 1
F -ra P
64D 3a2 3 (2.40)
1Area
-Area*P


The effective area is then given by









1
Aff = Area (2.41)
3
for a clamped circular plate.

Another method can be used for determining the effective area. This method

relies on conservation of volume velocity. Volume velocity, for any acousto-mechanical

interface, is defined as [13]

Q = v,dS (2.42)
where v, is the velocity normal to the surface.

For a clamped circular plate, (2.42) can be written as


Q= ofx(o ) 1- a 2 7rdr
0
= 17a2cOx(0) (2.43)
1
= Area*v(O)
3
where v(O) is the velocity at the center of the plate.

For a piston, the volume velocity relationship to velocity is given by

Q = v*Area (2.44)
Since the plate is lumped as a rigid piston moving at velocity v(O), the volume velocity of

the plate can be written as

Q = Aev (0) (2.45)
where


Af =-Area (2.46)
3
As expected, either method produces the same transduction factor. Therefore, in

addition to lumping a clamped circular plate as a rigid piston attached to a spring, the

area of transduction must be reduced by a factor of three to account for the effective

transduction of pressure to force, or velocity to volume-velocity that is actually taking









place. Physically, this factor is because the plate does not deflect like a piston, since it is

clamped at the edges. The pressure near the edges does not create as effective of a

deflection-causing force as the pressure at the center. Similarly, the total volume velocity

is reduced when compared to the piston of equal area moving with a velocity equal to the

center velocity of the plate.

The equivalent circuit shown in Figure 8 is defined strictly in terms of acoustical

parameters. The representation of the mechanical inertance and compliance of the

backplate in the acoustical energy domain requires use of the transduction factor, given

by the squared magnitude of the effective backplate area, Aef [13]. The transduction

factor is needed for conservation of energy when an impedance is reflected from one

energy domain into another.




+ +Ra
P,() P2(O Ca Cmea



Figure 8: Equivalent acoustic circuit representation of a Helmholtz resonator i/h a
compliant backplate.



The transduction of impedance from the mechanical to acoustical energy domain

is given by

ZZ
Z,= (2.47)
Aff

The acoustical equivalent circuit elements of the mechanical inertance and compliance

are given by









MAe
mea (2.48)
Aeff
2
Cmea CA2 ef (2.49)
This relationship between the acoustical and mechanical energy domains is

evident via a dimensional analysis of the two energy domains.

Relating the effective mass and compliance to their acoustical representations

yields the following expressions for the effective mass and compliance of the backplate,

in the acoustical energy domain.


Mmea = (2.50)
5A 2
9a2 _V 2) A 2
Cme9a = ff (2.51)
167 Eh3
The transfer function of the cavity pressure to the incident pressure is now given

by

s2Mmea mea + 1
P2 S(C mea +S2meaCmeaCa +Ca
(2.52)
P, s2M C +1
1 R + sA +mea
S (Cmea + S2meaCeaCa + Ca)
From this expression, the anti-resonance, which occurs at the frequency at which

the numerator equals zero, is dependent only upon the effective mass and compliance of

the backplate. This makes physical sense, as the anti-resonance of this transfer function is

due to the mechanical resonance of the backplate that prevents sound pressure from

building up in the cavity.

For a Helmholtz resonator with a compliant backplate consisting of an aluminum

shim with 1 mil thickness, but otherwise identical in geometry to the conventional

Helmholtz resonator described earlier, a frequency response function is obtained similar










to the one shown in Figure 9. The frequency response shows two resonant peaks

separated by an anti-resonance.




40
16
-8
-32
S-56
-80
1000 2000 3000 4000 5000 6000
Freq [Hz]

100

7 25
-50

S-125

-200
1000 2000 3000 4000 5000 6000
Freq [Hz]

Figure 9: Magnitude and phase of the theoretical pressure amplification of a compliant-
backplate Helmholtz resonator i ih a 0.0015 in. thick, 0.5 in. diameter, aluminium
backplate.



The frequency response depends upon the thickness and radius of the backplate.

Shown in Figure 10 is a contour plot displaying the effect of varying the thickness on the

frequency response. The anti-resonance can be seen to vary linearly with the backplate

thickness. Additionally, for thick backplates, shown near the top of the plot, the second

resonance depends more on the mechanical properties of the backplate and also varies

linearly with thickness, while the first resonance is primarily due to the acoustic elements

of the system and does not vary much with thickness. As the thickness is decreased to

where the anti-resonance approaches the first resonance, stronger coupling occurs










between the mechanical and acoustical elements, leading to a variation in both resonant

frequencies with decreasing thickness.

.0105
.0095
.0085
.0075
.0065
.0055
.0045
.0035
.0025
.0015
.0005
100 475 850 1225 1600 1975 2350 2725 3100 3475 3850 4225
Freq [Hz]

Figure 10: Contour plot ofpressure amplificationfor various thickness of backplate.
Bright areas are peaks in the frequency response, while dark areas are troughs.



The input impedance of the compliant-backplate Helmholtz resonator is given by,

1 1
sCm sC
Z = S Rea sM +R (2.53)
sime + + -
ea mea SCa
This expression, which can be derived directly from the equivalent circuit, results

from a series combination of the backplate mass and compliance in parallel with the

cavity compliance and all in series with the mass and resistance of the neck. A plot of the

magnitude and phase of the input impedance for a 0.0017 in. thick backplate is shown

below in Figure 11. The impedance is also shown in terms of resistance and reactance in

Figure 12. In the plot, the acoustic input impedance is multiplied by the area of the neck

A, to yield the specific acoustic impedance, and is then normalized by pc to yield a

non-dimensional result.














111 1--


III I IFreq [ IH-I I I -.z] -
Freq [Hz]


1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

Freq [Hz]

Figure 11: Magnitude and phase of the theoretical, normalized input impedance of a
compliant-backplate Helmholtz resonator.


Freq [Hz]


1111 I I 1 ,i II _I ,,I II ,.II'1., ,III, I. ".1 ,,I I I, ii ".f.,,,1 I I
Freq [Hz]

Figure 12: Theoretical normalized resistance and reactance of a compliant-backplate
Helmholtz resonator.


I II I I II II I









As can be seen in Figure 11 and Figure 12, there are two resonances in the

impedance. At these frequencies, the magnitude tends towards the value of the

resistance, since the phase goes to zero and the impedance is then purely resistive. Note

that resonant frequencies, in this thesis, refer to the frequencies at which the reactance

goes to zero. This corresponds to minima on an impedance plot. A pressure

amplification plot, however, indicates the acoustic response of the system, and therefore

maxima are labeled as the resonant frequencies.

In the impedance plot of Figure 11, an anti-resonance occurs between the two

resonant frequencies. It should be noted that, due to the topology of the circuit, this anti-

resonance does not coincide with the anti-resonance present in the transfer function of the

cavity to incident SPL. This can be understood by looking at the expression for the input

impedance (2.53) at the frequency at which the anti-resonance is seen in the transfer

function of the cavity to incident SPL. The transfer function heads toward zero at this

point because the reactance of the backplate, which is in parallel with the cavity

compliance heads toward zero. The total input impedance, however, does not become

purely resistive at this point, because of the mass of the neck. Instead, the anti-resonance

of the impedance occurs at a higher frequency, where the parallel combination of the

backplate impedance and the cavity compliance cancels the reactance of the mass in the

neck.

Mass Ratio Dependence of Coupled Resonant Frequencies

In the compliant backplate Helmholtz resonator, there are two resonant

frequencies, separated by an anti-resonance. The analytical expression for the two

resonances are too complex to provide much insight in general; however, when the









mechanical resonance is matched to the acoustical resonance, the analytical expression

can be greatly simplified, leading to a useful expression.

Analytical expressions for the two resonances can be found by looking at the

input impedance, and solving for the frequencies at which it equals zero. As originally

given in (2.53) and repeated here for convenience, the input impedance to the compliant

backplate Helmholtz resonator is given by

1 1
sM+SM +-
sCa sC
Z1 S ea sMa +R (2.54)
SMea + +
Semea SC
which can be simplified to

1 (sMaCaMm eCm i +s2MsCMeamea +S2MaCa +S2'm +(l)
aS (S2~MeamCeaC C +Ca +Ca)
The anti-resonant frequency, which occurs when the denominator approaches

zero, is given by


far 1= 1 V eaCmeaC Ca CCea) (2.56)
S2ll M CCmeaCea
The two resonant frequencies occur when the numerator approaches zero, and are

given by


Me,,meaCm +MaCa +MaCmea 2

1 1 2 mea Cme2a 2M l CaCmea + (2
fres 2r 2Ma[meaCaCmea 2ameaaC mea 2Mmea c mea, 2 M +Ima2a2 / C,27

2Ma C aCmea + a2Cmea
ineaC,,, inea2C,










MmeaCmea +MaC +M,Cme,,+

1 1 2 eaCaCma 2 Cmea2- 2A mea Camea C
f res2 2M Cc mea mega --a a gea (2.58)
27r 22aMmeaCaCmea a MmeaCmea 2 +M 2C2 +

2A_1CaCm +Mea Cmea2
These expressions do not provide much insight into the dependence of the

resonant frequencies on the various parameters. However, for a system in which the

uncoupled resonant frequency of the mechanical components matches the uncoupled

resonant frequency of the acoustical components, these expressions can be greatly

simplified. The matched uncoupled resonant frequencies are given by

1 1
fA fre = (2.59)
27r MC re 27 Mn^Mealmea
which yields a relationship between the acoustical and mechanical components of

MmeaCme = MalCa (2.60)
Substituting this equality into (2.57) and (2.58) greatly simplifies the expressions

for the two resonant frequencies, which can be shown to be

1
i 2 2Ma 2Ca +Cm (4CaCmea + Cme2) (2.61)

and

1

2~e 2x 2MaCa C ea + (4mea Cme2)1 (2.62)
Furthermore, we can define a mass ratio, oX that relates the acoustical to

mechanical mass, as well as the mechanical to acoustical compliance.

= a Cmea (2.63)
M Ca
Making appropriate substitutions, (2.61) and (2.62) can be expressed in terms of

the mass ratio as










fes = -2+a- [4a+ a2 (2.64)

and


fes2 =fo2 + a+ 4a+ -2 (2.65)

wherefo is the original uncoupled resonant frequency of each subsystem, given by

fo = freA = fresM (2.66)


From the above expressions, it is now evident that the resonant frequencies can be

expressed solely as a function of the original resonant frequency and the mass ratio,

alpha. Furthermore, the bandwidth between these two resonant frequencies is given by

1 1

fresh2 fresl = 2-+a+ 4ada, 2 2+ia- I 4a+2] (2.67)


Although desirable from an analytical perspective, the condition of matched

uncoupled resonance is necessary for this simplification. Most of the designs tested in

this thesis do not have uncoupled resonance frequencies that match or are close to

matching.

Additionally, by using the mass ratio, a, the expression for the anti-resonance can

also be simplified, as shown below. If the resonances are not matched, this expression

can still be used, however a would then only equal the compliance ratio, rather than both

ratios.


fr = fre + (2.68)














CHAPTER 3
EXPERIMENTAL SETUP


Overview

After developing the lumped element models used to predict frequency response

and input impedance, experiments were designed to provide verification of the models.

The verification is necessary as a first experimental step in order to demonstrate the

validity of the modeling. Initial verification was provided through the pressure

amplification frequency response function, which is a quick, easy measurement to

perform. Furthermore, in order to validate the assumption that the compliant backplate

behaves as a clamped circular plate, additional experiments were performed to measure

the mode shape.

After verification of the models, further characterization was performed through

measurement of the acoustic input impedance to the resonator. When the compliant

backplate Helmholtz resonator is later implemented as part of an electromechanical

acoustic liner, the impedance will be important in predicting the liner's effect on the

propagation of noise through an aircraft engine duct.


Measurements and Equipment

Characterization of the compliant backplate Helmholtz resonator was conducted

at the Interdisciplinary Microsystems Laboratory at the University of Florida. Twenty-

four different compliant-backplate Helmholtz resonators were tested in a normal

incidence impedance tube. The impedance tube consists of a 38 in. long, 1 in. by 1 in.









square duct, which permits characterization in a known, plane wave acoustic field at

frequencies up to 6.7 kHz.

Input impedance and pressure amplification measurements were taken for the

compliant-backplate Helmholtz resonator for a range of backplate thicknesses, radii, and

materials. Additionally, low frequency mode shapes were measured for some of the

backplates using laser vibrometry. For each set of measurements, the resonator was

mounted flush to the end of the impedance tube, as shown in Figure 13. This setup

permits simultaneous measurements of resonator impedance, pressure amplification and

mode shape, although the mode shape data was actually collected at a later time.

Mic. 2 Mic. 1 Cavity Microphone

SCompliant backplate



LIncident
Microphone

Figure 13: Schematic ofimpedance tube terminated by compliant-backplate Helmholtz
resonator.



Four Bruel and Kjaer (B&K) type 4138 microphones were used for simultaneous

acoustic pressure measurements. Two microphones were flush mounted, in a rotating

plug, to the side of the impedance tube, as shown in Figure 14. The plug allowed for

convenient microphone switching between impedance measurements to average out any

amplitude or phase calibration differences between the microphones [17]. The other two

microphones were used to measure the pressure amplification frequency response

function. One microphone was flush mounted in the side wall of the resonator cavity to

measure the cavity pressure. The second microphone was flush mounted to the end face









of the impedance tube to measure the acoustic pressure incident to the resonator. This

microphone also served as a reference to ensure a nearly constant SPL, with respect to

frequency, at the neck of the resonator for each backplate configuration. The pressure

amplification frequency response function was then obtained as the ratio of cavity to

incident acoustic pressure as a function of frequency.















Figure 14: Normal impedance tube showing rotating microphone plugs, and end-
mounted Helmholtz resonator.



The microphones were connected through Bruel and Kjaer Type 2669

preamplifiers to a Bruel and Kjaer PULSE Multi-Analyzer System Type 3560. The

PULSE system served as the power supply and data acquisition unit for the microphones,

and generated the source waveform. The source was fed through a Techron 7540 power

supply amplifier to drive a JBL Pro 2426H compression driver, which is capable of

producing acoustic waves between 1 kHz and 20 kHz. The compression driver was

connected to a tapered transition piece and mounted to the far end of the impedance tube.

The transition piece served to couple the circular throat of the compression driver to the

square duct of the impedance tube.









A pseudo random waveform was generated using the PULSE Multi-Analyzer

system. The waveform was internally bandpass filtered from 1 kHz to 6.4 kHz to avoid

exciting the compression driver beyond its intended frequency range. A 3200 bin FFT

with 8000 ensemble averages was performed on each incoming microphone signal. The

FFT spanned from 0 to 6.4 kHz, yielding a 2 Hz binwidth. Data recorded at frequencies

below 1 kHz was discarded because it was below the excitation frequency range. Using

the method above, coherence between the two microphones in the rotating plug was

typically close to unity, dropping only as low as 0.954 at frequencies where pressure

nodes occurred at one of the microphone locations. The coherence between the cavity

and incident microphones was similarly close to unity, typically residing at 0.999, except

at the resonant frequency of the backplate where it would routinely fall as low as 0.2.

This is believed to be caused by the large decrease in acoustic pressure in the cavity at the

backplate resonant frequency, along with near-field effects causing by the large

amplitude backplate deflection at these frequencies and the proximity of the cavity

microphone. Coherence plots for some of the backplates are shown in the next chapter.

To determine the normal incidence acoustic impedance, frequency response

measurements were taken using the two microphones in the rotating plug [18, 20]. The

plug was then rotated and the measurement repeated. The two measurements were then

averaged to remove any differences due to the individual microphone calibrations. This

averaged frequency response was then used with the multi-point method to determine the

acoustic impedance. The multi-point method essentially uses measurements from

multiple microphones to create an overdetermined curve fit to the standing wave pattern

inside the tube. When used with only two microphones, the multi-point method reduces









to the two-microphone method of impedance measurement [16-19]. The two-

microphone method requires less than a half-wavelength spacing between the two

microphone locations [17-19]. Although the tube is physically capable of supporting

plane waves up to 6.7 kHz, testing was limited to 6.4 kHz because of this microphone

spacing constraint. An alternative, single-microphone method [21] exists but was not

used due to greater errors associated with the technique.

Using the multi-point method, the incident and reflected acoustic pressures [18,

20] are

CE AB
DE= A (3.1)
DE A2
and


P, =- (3.2)
A
where

N N N
A=1(1), B=Y e' C= e x
jN1 j1 j (3.3)

D= e 2,x', E= e FxJ
J=l j=1
In (3.3), N is the number of microphone locations, x is the distance from the sample to

each microphone location, i = f and Pj is the measured complex acoustic pressure at

each microphone location. Furthermore, F, the plane-wave propagation constant, is

given by

F=k+i /3 (3.4)

where k = and Pv is the attenuation constant due to viscothermal dissipation, given by
c



S+(y-1) C (3.5)
4ao c pm pc,









In (3.5), ao is the ratio of the duct area to perimeter, pis the absolute viscosity, Kis the

heat conduction coefficient, c, is the constant of specific heat at constant pressure, p is

the density of the medium, and y is the ratio of specific heats.

Using the measured transfer functions of one microphone location to another, the

relative amplitudes and phases of the acoustic pressure are known, and can be inserted

into Pj above, yielding relative values for the incident and reflected acoustic pressures.

The normalized acoustic impedance can then be found as

1+R
-= =+ iz (3.6)
1-R
where


R= (3.7)

and 0 and x are the normalized resistance and reactance, respectively. This procedure

was used to obtain the normalized resistance and reactance results shown in Chapter 4.

To measure the backplate mode shapes, a Polytec PI Vibrascan Laser Vibrometer

with an OFV 055 vibrometer scanning head was used[22]. The scanning head was

controlled by a Polytec PI OFV 3001 S vibrometer controller. The backplate was

deflected under an acoustic pressure, which was supplied via an HP 33120A function

generator connected through a Crown K1 balanced current amplifier to the JBL speaker

at the end of the impedance tube. The frequency of excitation was kept below the

resonant frequency of each backplate to ensure excitation of primarily the first mode.


Device Construction

The Helmholtz resonators were constructed of modular aluminum plates, as

shown in Figure 15. The modular design allows for parts to be interchanged to test a









variety of resonator geometries. The resonators were designed to have a first resonant

frequency and most second resonant frequencies occurring within the testable frequency

range of 1 kHz to 6.4 kHz.

















Figure 15: Photograph of the compliant backplate Helmholtz resonator showing
compliant backplate clamped by circular clamping ring and incident and cavity
microphone plugs.



The front plate consists of a 4.3 in. x 2.8 in. x 0.125 in. aluminum plate. It

contains one 0.188 in. diameter, 0.125 in. deep hole that serves as the neck of the

resonator. The cavity plate, contains a 0.5 in. diameter, 0.6 in. deep hole that serves as

the resonator cavity. There were 24 different backplate configurations tested, consisting

of various combinations of material, radius, and thickness as shown in Table 1. The three

materials tested were aluminum, brass, and stainless steel, each with material properties

as shown in Table 2.

To provide proper clamping of the 0.25 in. radius compliant backplates, a 0.25 in.

thick, 1 in. diameter ring containing a 0.5 in. diameter hole was mounted to the backside

of each compliant sheet and tightened against the cavity plate. A similar ring but of 1.5










in. outer diameter with a 0.96 in. diameter hole was used to clamp the 0.5 in. radius

backplates. For the 0.5 in. radius backplates, an additional .02 in. thick ring was used

underneath each plate, to suspend the backplate above the cavity plate and prevent

contact during vibration. The rigid clamping rings allowed for an approximation of the

compliant sheet as a clamped circular plate. Schematics of all devices are shown in

Figure 68 through Figure 72 in Appendix A.




Table 1: Backplate configurations that were tested
Plate # Material Radius [in] Thick [in] Cmea [m^3/Pa] Mmea [kg/m^4]
A125-1 Aluminum 0.25 0.0013 4.618E-12 1.272E+03
AI25-2 Aluminum 0.25 0.0023 8.338E-13 2.250E+03
AI25-3 Aluminum 0.25 0.0032 3.096E-13 3.130E+03
AI25-5 Aluminum 0.25 0.005 8.116E-14 4.890E+03
AI50-2 Aluminum 0.5 0.0024 4.697E-11 5.869E+02
AI50-3 Aluminum 0.5 0.0031 2.179E-11 7.580E+02
AI50-5 Aluminum 0.5 0.0048 5.871E-12 1.174E+03
AI50-10 Aluminum 0.5 0.0105 5.609E-13 2.567E+03
Br25-1 Brass 0.25 0.0013 3.107E-12 3.974E+03
Br25-2 Brass 0.25 0.0023 5.610E-13 7.031E+03
Br25-3 Brass 0.25 0.003 2.528E-13 9.171E+03
Br25-5 Brass 0.25 0.0052 4.855E-14 1.590E+04
Br50-2 Brass 0.5 0.0021 4.717E-11 1.605E+03
Br50-3 Brass 0.5 0.0031 1.466E-11 2.369E+03
Br50-5 Brass 0.5 0.0051 3.293E-12 3.898E+03
Br50-10 Brass 0.5 0.0102 4.117E-13 7.795E+03
St25-1 Stainless Steel 0.25 0.0014 1.350E-12 3.972E+03
St25-2 Stainless Steel 0.25 0.0023 3.044E-13 6.525E+03
St25-3 Stainless Steel 0.25 0.0033 1.031E-13 9.361E+03
St25-5 Stainless Steel 0.25 0.0055 2.226E-14 1.560E+04
St50-2 Stainless Steel 0.5 0.0021 2.234E-13 1.489E+03
St50-3 Stainless Steel 0.5 0.0032 7.234E-12 2.269E+03
St50-5 Stainless Steel 0.5 0.0053 1.592E-12 3.759E+03
St50-10 Stainless Steel 0.5 0.0107 1.935E-13 7.588E+03







36

Table 2 : Material properties for backplates.
Type Density [g/cm^3] Young's Modulus [GPa] Poisson's Ratio
Aluminum 1100-H-18 2.71 69 0.33
Brass ASTM: B19 8.47 103 0.3
Stainless Steel ASTM: A666 7.86 193 0.3



Pressure amplification and impedance spectra were obtained for the Helmholtz

resonator with each of the 24 different backplate configurations, along with mode shapes

for one particular set of backplates. These data, along with a discussion of the results, are

given in the next chapter.














CHAPTER 4
EXPERIMENTAL RESULTS



Measurement results were obtained for the Helmholtz resonator with each of 24

different backplate configurations. These results consisted of pressure amplification and

impedance spectra. Additionally, mode shapes were obtained for the 0.25 in. radius,

aluminum plates, to confirm their behavior as clamped circular plates.


Pressure Amplification

The pressure amplification measurements were obtained using the cavity and

incident microphones, as described earlier in the experimental setup. The measurement

results are shown below, sorted first by diameter of clamped backplate, then material

type, followed by backplate thickness.

Helmholtz Resonator with Half-Inch Diameter, Clamped Backplates

The results obtained for the Helmholtz resonator with half-inch diameter, clamped

backplates show overall good agreement with the theory, for the two thickest backplates,

but displayed more significant discrepancies for the two thinnest backplates. This was

consistent among all three material types. In Figure 16 through Figure 27, each graph

displays the experimental results overlaid with two theoretical curves. The theoretical

curves are based on the lumped element model of the compliant-backplate Helmholtz

resonator using measured dimensions of the constructed devices. Two curves are shown

because of the significant effect caused by a small uncertainty in the measurement of









backplate thickness. One theoretical curve uses the measured dimensions with

uncertainty added to it, while the other curve used the measured dimensions with the

uncertainty subtracted from it.

The effect of the uncertainty shows up more severely in the thinnest plates, as it

comprises a larger percentage of the measured thickness. The thickness of each plate was

measured using a Sears Craftsman precision micrometer. The vernier scale on the

micrometer can theoretically provide results down to 0.0001 in.. Thickness

measurements at various places over the surface of each backplate often yielded

variations greater than this 0.0001 in. precision of the micrometer. Five thickness

measurements were taken of each backplate. From these measurements, the mean

thickness was computed along with the standard deviation for each backplate. From the

standard deviation, c, the uncertainty was calculated as

U = to (4.1)
where t = 2.776 when only five data points are taken [23]. The mean thickness and

uncertainty are shown in Table 3. The lumped element model is very sensitive to

thickness, as the compliance of the backplate depends on the thickness to the third power.

The large theoretical range shown in each graph below demonstrates this sensitivity.










Table 3 : Mean thickness and uncertainty in thickness measurement for all backplates
Plate # Mean Thickness [in] Uncertainty [in]
A125-1 1.26E-03 1.52E-04
A125-2 2.28E-03 1.24E-04
A125-3 3.32E-03 3.62E-04
A125-5 5.08E-03 3.62E-04
Br25-1 1.36E-03 6.97E-04
Br25-2 2.16E-03 1.52E-04
Br25-3 3.02E-03 1.24E-04
Br25-5 5.22E-03 3.04E-04
St25-1 1.38E-03 2.32E-04
St25-2 2.14E-03 2.48E-04
St25-3 3.26E-03 2.48E-04
St25-5 5.26E-03 2.48E-04
A150-2 2.30E-03 3.40E-04
A150-3 3.22E-03 2.32E-04
A150-5 4.86E-03 1.52E-04
A150-10 0.0105 2.32E-04
Br50-2 2.12E-03 1.24E-04
Br50-3 3.14E-03 2.32E-04
Br50-5 5.34E-03 1.52E-04
Br50-10 0.0104 2.32E-04
St50-2 2.40E-03 3.40E-04
St50-3 3.56E-03 4.21 E-04
St50-5 5.28E-03 4.12E-04
St50-10 0.0106 3.17E-04


I, ,.


1000 2000 3000 4000 5000 6000
Freq [Hz]









1000 2000 3000 4000 5000 6000
Freq [Hz]


Theory + uncertainty
Theory uncertainty
SData


Figure 16: Magnitude and phase ofpressure amplification spectrumfor Helmholtz
resonator i i/it 0.001 in., aluminum backplate clamped at 0.5 in. diameter.




















1000 2000 3000 4000 5000 6000
Freq [-Iz


Freq [Hz]


Theory + uncertainty
Theory uncertainty
Data


Figure 17. Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator i i/th 0.002 in., aluminum backplate clamped at 0.5 in. diameter.


1000 2000 3000 4000 5000 6000
Freq [Hz]


1000


3000
Freq [Hz]


Theory + uncertainty
Theory uncertainty
Data


Figure 18: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator i i/th 0.003 in., aluminum backplate clamped at 0.5 in. diameter.


~























1000 2000 3000 4000 5000 6000
Freq [Hrz







--- -------


Freq [Hz]


Theory + uncertainty
Theory uncertainty
* Data


Figure 19: Magnitude andphase ofpressure amplification spectrum for Helmholtz
resonator i itih 0.005 in., aluminum backplate clamped at 0.5 in. diameter.


-40


1000 2000 3000 4000 5000 6000
Freq [Hz]




i -- _--
A


1000


Theory + uncertainty
Theory uncertainty
Data


3000
Freq [Hz]


Figure 20: Magnitude andphase ofpressure amplification spectrum for Helmholtz
resonator in lh 0. 001 in., brass backplate clamped at 0.5 in. diameter.


!'" ".'.

























Freq [Hz]







If

iar f


3000
Freq [Hz]


4000


5000


6000


Figure 21: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator i itli 0.002 in., brass backplate clamped at 0.5 in. diameter.


Freq [Hz]








i I


2000


3000
Freq [Hz]


4000


5000


6000


Figure 22: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator i i i1i 0.003 in., brass backplate clamped at 0.5 in. diameter.


1000


2000


Theory + uncertainty
Theory uncertainty
S* Data


1000


Theory + uncertainty
Theory uncertainty
S Data























Freq [Hz]


4 .. ..


Freq [Hz]


Theory + uncertainty
Theory uncertainty
S. Data


Figure 23: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator i i hl 0.005 in., brass backplate clamped at 0.5 in. diameter.


Freq [I-z]





' l A


3000
Freq [Hz]


4000


Theory + uncertainty
Theory uncertainty
S Data


Figure 24: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator i i h 0. 001 in., stainless steel backplate clamped at 0.5 in. diameter.


1000


2000


5000


6000
























req [z]








"\ ;':


3000
Freq [Hz]


4000


Theory + uncertainty
Theory uncertainty
*+ Data


Figure 25: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator i iith 0.002 in., stainless steel backplate clamped at 0.5 in. diameter.


Freq [HIz]


1000


2000


3000
Freq [Hz]


4000


5000


6000


Theory + uncertainty
Theory uncertainty
S Data


Figure 26: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator i i tl/ 0.003 in., stainless steel backplate clamped at 0.5 in. diameter.


1000


2000


5000


6000











9CI
44




-24


Freq [Hz]
180

90





.. .. .. .. .
Freq [Hz]
Theory + uncertainty
Theory uncertainty
S Data


Figure 27: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator i it i/ 0.005 in., stainless steel backplate clamped at 0.5 in. diameter.




The results shown in Figure 16 through Figure 27 show fairly good agreement


between theory and data for the 0.003 in. and 0.005 in. thick backplates, regardless of


material. The first resonant peak is predicted well for these configurations. Additionally


the measured anti-resonance and second resonant peak match the theory fairly well for


the 0.003 in. thick backplate. The anti-resonance and second resonant peak were not


measurable for most of the 0.005 in. thick plates, as they occurred outside the testable


frequency range.


The thinner plates did follow the expected trend of lower anti-resonant and second


resonant frequencies compared to the thicker plates, however, these did not occur at


frequencies quite as low as predicted. This indicates they were stiffer than predicted and


is believed to be due to two factors, the uncertainty in their thickness measurement, and









possible in-plane tension applied as an inadvertent effect of the clamping. Many of the

stiffer-than-expected plates had a raised lip on the inside edge of the holes that allow the

clamping screws to pass through. The raised lip indicated they were catching on the

threads of the clamping screws, which may be adding tension to the plate. The in-plane

tension would raise the stiffness of the plates, thereby increasing the resonant frequency.

To further investigate this possibility, real-time pressure amplification

measurements were taken of some of the compliant backplates, while the clamping

screws were adjusted. It was found that the resonant frequency of each backplate

increased as the screws were tightened. This effect continued up to the maximum torque

that was achievable with the ratchet on hand. If no in-plane tension was occurring, then

the resonant frequency would not rise with increasing torque on the screws, therefore,

these results indicate the likely presence of in-plane tension. A redesign of the clamping

setup would be necessary to reduce or eliminate this effect.

Despite the quantitative discrepancies in the thinnest backplates, the actual

qualitative behavior of the compliant-backplate Helmholtz resonator does follow the

lumped element models of a coupled resonator system. This is evidenced by the shift in

first resonant peak due to the presence of the anti-resonance introduced by the backplate.

The coherence for these measurements typically were near one, except at

frequencies around the anti-resonance. The coherence for the pressure amplification

measurements of the Helmholtz resonator with 0.001 in. thick aluminum backplate is

shown in Figure 28. On the plates where an anti-resonance was not within the frequency

range tested, the coherence stayed above 0.98 as shown in Figure 29.













0.75

05

0.25

0

11:: .001" thick backplate
* .001" thick backplate


35Freq [Hz]4"', 41 5101 f 11
Freq [I-z]


Figure 28. Coherence between cavity and incident microphone for Helmholtz resonator
i i/t 0.001 in. thick aluminum backplate clamped at 0.5 in. diameter.


11" "" .00 151:1" thick backplate 5

* .005" thick backplate


35 fI 10 40[ 1i(i 45,6::i(i 511 55,:) ,t 50 ,:
Freq [Hz]


Figure 29: Coherence between cavity and incident microphone for Helmholtz resonator
i1 ith 0.005 in. thick stainless steel backplate clamped at 0.5 in. diameter.





Helmholtz Resonator with One-Inch Diameter, Clamped Backplates

Pressure amplification spectra were also obtained for the Helmholtz resonator

with one-inch diameter, clamped backplates. Overall, these backplates showed better

agreement with the theoretical models. The improvement is believed to be due to the

thicker plates that were used in these experiments. This improved the percentage


' ''











uncertainty in the thickness measurement, leading to more accurate predictions. The


spectra are shown below in Figure 30 through Figure 41.


40

--, -,,------------.-----------------------"-




-40

-60
1000 2000 3000 4000 5000 6000
Freq [Iz]



100



--- ,-. '


-200
1000 2000 3000 4000 5000 6000
Freq [Hz]
Theory + uncertainty
Theory uncertainty
SData


Figure 30: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator i it i 0.002 in., aluminum backplate clamped at 0.96 in. diameter.




















'A4




Freq [Hz]





" -' L -- ; - " *. .


2000


3000
Freq [Hz]


4000


5000


6000


Figure 31: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator i i hl 0.003 in., aluminum backplate clamped at 0.96 in. diameter.


Freq [Hz]


1000


2000


3000
Freq [Hz]


4000


5000


6000


Theory + uncertainty
Theory uncertainty
S Data


Figure 32: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator i i h/ 0.005 in., aluminum backplate clamped at 0.96 in. diameter.


100


100


1000


Theory + uncertainty
Theory uncertainty
S. Data


'nn
























Freq [Hz]








_',


4, ,,, ,


Freq [Hz]


Theory + uncertainty
Theory uncertainty
S. Data


Figure 33: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator ii ilt 0.010 in., aluminum backplate clamped at 0.96 in. diameter.


-41


Freq [I z]

-" -- q .-r- --..':- g 7 ~ ..





1. i
** 4.+
,, ; -


2000


3000
Freq [Hz]


4000


5000


6000


Figure 34: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator ini ilt 0.002 in., brass backplate clamped at 0.96 in. diameter.


'nn


1000


Theory + uncertainty
Theory uncertainty
" Data


i -_
























Freq [HIz]




__-7-- -"* i..;


1000


2000


3000
Freq [Hz]


Figure 35: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator i it i 0.003 in., brass backplate clamped at 0.96 in. diameter.


Freq [Hz]









I A


1000


2000


3000
Freq [Hz]


4000


5000


6000


Theory + uncertainty
Theory uncertainty
S* Data


Figure 36: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator ini tl/ 0.005 in., brass backplate clamped at 0.96 in. diameter.


fnn


Theory + uncertainty
Theory uncertainty
S. Data


4000


5000


6000
























Freq [Hz]


4, ,,, ,


Freq [Hz]


Theory + uncertainty
Theory uncertainty
S. Data


Figure 37: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator ii ilt 0.010 in., brass backplate clamped at 0.96 in. diameter.


Freq [HIz]








--a ----------
.. . ..^
____^A^. ^ ^


1000


2000


3000
Freq [Hz]


4000


5000


6000


Theory + uncertainty
Theory uncertainty
" Data


Figure 38: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator i itll 0.002 in., stainless steel backplate clamped at 0.96 in. diameter.


I-nn


i -_























Freq [Hz]








_A


3000
Freq [Hz]


4000


Theory + uncertainty
Theory uncertainty
S* Data


Figure 39: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator i i/tl 0.003 in., stainless steel backplate clamped at 0.96 in. diameter.


Freq [Hz]






_ _ T h


1000


2000


3000
Freq [Hz]


4000


5000


6000


Theory + uncertainty
Theory uncertainty
S Data


Figure 40: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator ini /ti 0.005 in., stainless steel backplate clamped at 0.96 in. diameter.


100


1000


2000


5000


6000










40
24
S


-,I




180

90
0



-1 -,


Freq [Hz]






-- *


Freq [Hz]
Theory + uncertainty
Theory uncertainty
S. Data


Figure 41: Magnitude and phase ofpressure amplification spectrum for Helmholtz
resonator ii ihl 0.010 in., stainless steel backplate clamped at 0.96 in. diameter.




Compared to the other backplates, the one-inch diameter clamped backplates

exhibited better agreement between theory and measurement. The first and second

resonant peaks and anti-resonance that fell within the testable frequency range were

predicted fairly well, for the most part. However, some of the experimentally measured

backplate resonant frequencies were lower than the predicted resonant frequencies. This

can easily result from the lack of a perfectly clamped boundary condition. Any

compliance in that boundary will lead to an increase in effective radius, resulting in a

lower resonant frequency. In addition to these larger features of the measured spectra,

smaller peaks were also visible. These smaller peaks are believed to be due to the

presence of higher-order modes and their coupled interaction with the acoustic resonance.

These peaks only occurred on the larger diameter backplates because the higher order










modes for these backplates fell within the testable frequency range and began

encroaching on the original acoustic resonant frequency near 2 kHz, where coupling with

the acoustic resonance is strongest.


Input Impedance

As previously discussed in the experimental setup, the specific acoustic

impedance at the input to the compliant-backplate Helmholtz resonator was found from

measurements of the reflection coefficient, obtained via the multi-point method.

Helmholtz Resonator with Half-Inch Diameter, Clamped Backplates

Impedance measurements were obtained for the 12 configurations that were

clamped at 0.5 in. diameter and are shown in Figure 42 through Figure 53. The

impedance data for each configuration is shown in terms of resistance (real) and

reactance (imaginary) components.








S01

000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]







2 i
1-00
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]
S* .001" thick backplate
Theory+Uncertainty
Theory-Uncertainty

Figure 42: Normalized resistance and reactance for Helmholtz resonator ii ith 0.001 in.
aluminum backplate clamped at 0.5 in. diameter.
















100






1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Frecq [Hz]


21000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]
* .002" thick backplate
Theory+Uncertainty
Theory-Uncertainty


Figure 43: Normalized resistance and reactance for Helmholtz resonator i ith 0.002 in.,
aluminum backplate clamped at 0.5 in. diameter.





100




** 0t


01
01 -------------------------------------------------
9000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]

60










21000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]
S .003" thick backplate
Theory+Uncertainty
-Theory-Uncertainty


Figure 44: Normalized resistance and reactance for Helmholtz resonator i ith 0.003 in.,
aluminum backplate clamped at 0.5 in. diameter.


















IU
i,,I -- ------------



0.1


10000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]


100


-100
1000 1500 2000 2500 3000


3500 4000 4500 5000 5500 6000
Freq [Hz]


* .005" thick backplate
Theory+Uncertainty
Theory-Uncertainty


Figure 45: Normalized resistance and reactance for Helmholtz resonator i ith 0.005 in.,

aluminum backplate clamped at 0.5 in. diameter.


0.01

1 103
1000 1500 2000 2500


3000 3500 4000 4500 5000 5500 6000
Freq [Hz]


Freq [Hz]
S* .001" thick backplate
Theory--Uncertainty
S-Theory-Uncertainty


Figure 46: Normalized resistance and reactance for Helmholtz resonator ii it/ 0.001 in.,

brass backplate clamped at 0.5 in. diameter.


1 I,,















I'- up'a


S


1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]


000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]
S .002" thick backplate
Theory+Uncertainty
Theory-Uncertainty


Figure 47: Normalized resistance and reactance for Helmholtz resonator i ithi 0.002 in.,
brass backplate clamped at 0.5 in. diameter.


0.01


1000 1500 2000 2500 3000


60
T: 0|-----------------


3500 4000 4500 5000 5500 6000
Freq [Hz]


I 1i lli, l , l il ,i l , ,i 4 1 ,1 1, 1 4 li l l i ,.i f l l... ,- i.. i, ,
Freq [HIz]
S* .003" thick backplate
Theory-Unc certainty
Theory-Uncertainty


Figure 48: Normalized resistance and reactance for Helmholtz resonator i i1th 0.003 in.,
brass backplate clamped at 0.5 in. diameter.


C~ -~--- -- --


bf -


III, % V 40r















100
e 100


0 0000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]


100









S100
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]
S .005" thick backplate
Theory+Uncertainty
Theory-Uncertainty


Figure 49: Normalized resistance and reactance for Helmholtz resonator i ilth 0.005 in.,

brass backplate clamped at 0.5 in. diameter.


1 .105
, .


1..


S1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]

200


100


20000 1500 2000 2500 3000 3500 000 500 5000 5500 6000
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000


* .001" thick backplate
Theory-+Uncertainty
S-Theory-Uncertainty


Freq [Hz]


Figure 50: Normalized resistance and reactance for Helmholtz resonator i ithi 0.001 in.,

stainless steel backplate clamped at 0.5 in. diameter.


- 3 ----


A

- -. -. . .- . .- .-.-


nf A1


i ..









60



100







S- -- - ---------- -- - -- -- -
001


0 0loo 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]






Ow


P;

50
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]
S .002" thick backplate
Theory+Uncertainty
Theory-Uncertainty


Figure 51: Normalized resistance and reactance for Helmholtz resonator ii iih 0.002 in.,

stainless steel backplate clamped at 0.5 in. diameter.





!....


10


I r:
0.1 01


1'10
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]

60

_,, I---------------------------------------------








1ili, ,, lll, ,il,,i ,,, _Al liIl ,i 41 l 4 lill D-,, Jl 5l -1f ,,. 11 4i,,
Freq [HIz]
S* .003" thick backplate
Theory+Uncertainty
Theory-Uncertainty


Figure 52: Normalized resistance and reactance for Helmholtz resonator i/ ih 0.003 in.,

stainless steel backplate clamped at 0.5 in. diameter.












100



I00


r i 1 L


1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]

100


1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]
* .005" thick backplate
Theory+Uncertainty
Theory-Uncertainty


Figure 53: Normalized resistance and reactance for Helmholtz resonator i itll 0.005 in.,
stainless steel backplate clamped at 0.5 in. diameter.







Helmholtz Resonator with One-Inch Diameter, Clamped Backplates

Impedance measurements were also obtained for the 12 configurations that were


clamped at 1 in. diameter and are shown in Figure 54 through Figure 65. The impedance


data for each configuration is shown in terms of resistance (real) and reactance


(imaginary) components.


- ~CICL~C 1
d
t













.10'


Freq [Hz]


000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]
S .002" thick backplate
Theory+Uncertainty
Theory-Uncertainty


Figure 54: Normalized resistance and reactance for Helmholtz resonator in ith 0.002 in.
aluminum backplate clamped at 0.96 in. diameter.


1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [HIz]


Freq [Hz]
S* .003" thick backplate
Theory+Uncertainty
Theory-Uncertainty


Figure 55: Normalized resistance and reactance for Helmholtz resonator i illth 0.003 in.
aluminum backplate clamped at 0.96 in. diameter.


1 ,10 3

44
1 O10
1000


1 .10,


I,,














1 .103

inn


ni


1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hiz]

151-1


S* 005" thick backplate
Theory+Uncertainty
Theory-Uncertainty


Freq [Hz]


Figure 56: Normalized resistance and reactance for Helmholtz resonator iI ith 0.005 in.

aluminum backplate clamped at 0.96 in. diameter.


1..


000 1500 2000 2500 3000 3500 4000 4500
Freq [Hz]


1500 2000 2500 3000


5000 5500 6000


3500 4000 4500 5000 5500 6000
Freq [Hz]


Figure 57: Normalized resistance and reactance for Helmholtz resonator iI ith 0.010 in.

aluminum backplate clamped at 0.96 in. diameter.


_ 1v


-100

1000


* .010" thick backplate
Theory+Uncertainty
S-Theory-Uncertainty


II
I I
I I
1


r r
Ir


-~- -


I
I I


1 .


























Freq [Hz]


1000 1500 2000 2500 3000

S .002" thick backplate
Theory+Uncertainty
Theory-Uncertainty


3500 4000 4500 5000 5500 6000
Freq [Hz]


Figure 58: Normalized resistance and reactance for Helmholtz resonator i ith 0.002 in.

brass backplate clamped at 0.96 in. diameter.


00 1500 2000 2500 3000


3500 4000 4500 5000 5500 6000
Freq [HIz]


a" 500






.---- --_-

Freq [Hz]
S* .003" thick backplate
The ory+Uncertainty
Theory-Uncertainty


Figure 59: Normalized resistance and reactance for Helmholtz resonator i ilth 0.003 in.

brass backplate clamped at 0.96 in. diameter.


! .. I


-;--r


1 .1.l -









1 10-3
4o
1 10-4
10


1000


wMww-__ -- -- -1 00 - mk--


i ,-


I .. .


O-
___________________________________________________________________________












1 .103
4


I '.


0.01


4 S
__I,_-_-_-_-_-_-_-_-_-_-_-_-_-_-


1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Frecj [liz]

__200


100


Freq [Hz]
S* .005" thick backplate
The ory+Uncertainty
Theory-Uncertainty


Figure 60: Normalized resistance and reactance for Helmholtz resonator i iOh 0.005 in.
brass backplate clamped at 0.96 in. diameter.


,-~oj

1..


0


000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [IIz]

100








-100
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]
S .010" thick backplate
Theory+Uncertainty
Theory-Uncertainty


Figure 61: Normalized resistance and reactance for Helmholtz resonator 0.010 in., brass
backplate clamped at 0.96 in. diameter.


- - - - -


I .






















0 1


0 0o000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]

_I i i. --------








S-400



Freq [Hz]
S* .002" thick backplate
Theory+Uncertainty
Theory-Uncertainty


Figure 62: Normalized resistance and reactance for Helmholtz resonator i i /hl 0.002 in.,
stainless steel backplate clamped at 0.96 in. diameter.





1 '10 I





I -
'I k------------
V, 0 1
1 10-3

1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [HIz]

200












Freq [Hz]
S* .003" thick backplate
Theory+Uncertainty
Theory-Uncertainty


Figure 63: Normalized resistance and reactance for Helmholtz resonator i ilth 0.003 in.,
stainless steel backplate clamped at 0.96 in. diameter.








67



1 ,105

I



-- -------------------

S0.1 -

0.000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [IIz]

150

100









Freq [Hz]
S .005" thick backplate
The ory+Uncertainty
Theory-Uncertainty


Figure 64. Normalized resistance and reactancefor Helmholtz resonator i iith 0.005,
stainless steel backplate clamped at 0.96 in. diameter.






10oo
f?4 1,00---------





0.1

0 0 000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

Freq [Iz]

60












Freq [Hz]
* .010" thick backplate
Theory+Uncertainty
Theory-Uncertainty


Figure 65. Normalized resistance and reactance for Helmholtz resonator i i/ih 0.010 in.,
stainless steel backplate clamped at 0.96 in. diameter.









Discussion of Results

For all 24 configurations tested, the measured specific acoustic impedance at the

input to the Helmholtz resonator was consistent with the pressure amplification

measurements. In other words, the reactance consistently passed through zero, as

expected, at the same frequency as a resonant peak in the corresponding pressure

amplification spectrum. Furthermore, the reactance reached a maximum, as expected, at

the same frequency as a measured anti-resonance in the corresponding pressure

amplification spectrum. This serves as further reassurance of the measurements, as both

types of measurements were obtained from different sets of microphones and computed

using different algorithms.

Some of the second resonant peaks were not clearly seen as a zero crossing in the

reactance. Occasionally, a significant drop in the reactance was seen, at this frequency,

but did not cross zero. This is believed to be due to the proximity of this resonance to the

anti-resonance and the limited frequency resolution of the measurements. The approach

of the reactance towards infinity followed closely in frequency by the rise from negative

infinity through zero proved difficult to measure explicitly, although the general trend is

visible in all measurements.

The microphone measurements used to obtain the impedance data exhibited good

coherence for all backplates tested. All of this coherence data was measured to be above

0.9. Shown in Figure 66 is a graph of the coherence between the two microphones in the

rotating plug when the Helmholtz resonator with a 0.001 in. thick aluminum backplate

was mounted to the termination of the impedance tube.











0.98 V[ I

S0.96

0.94

0 3 :15i:1 40 1:1 4f1 :1 50"11)f f0 f 10 3 0 I

Freq [Hz]
S .001" thick backplate

Figure 66: Coherence between rotating microphones for Helmholtz resonator with 0.001
in. thick aluminum backplate clamped at 0.5 in. diameter.

Low-Frequency Mode Shapes

The mode shapes of one set of clamped circular backplates, as measured by a

laser vibrometer, is shown in Figure 67. The figure shows normalized deflection as a

function of normalized radius. The backplates tested had a clamped diameter of .5 in.,

ranged in thickness from .001 in. to .005 in. and were all made of aluminum. The mode

shape was measured with a sinusoidal acoustic signal of 125dBSPL in the cavity at a

frequency of 1000Hz.

The measured mode shapes were seen to match fairly well to predictions for a

static clamped circular plate, thus verifying proper clamping and plate behavior. The

agreement between the theoretical mode shape of clamped circular plate and the

measured mode shapes validate the use of a clamped circular plate model in the lumped

element theory. The mode shape measurements, shown above, were not repeated for all

24 backplates, however all backplates were clamped usually a common method, and

similar results are expected.














0.8 -


0.6-


0.4-


0.2-


0
-1 -0.5 0 0.5 1
r/a
xxx Imildata
+ -+ 2 mil data
3 mil data
S5 mil data
Predicted mode


Figure 67: Measured mode shapes for each backplate iakng n ith a theoretical mode
shape for a clamped circular plate, as given by Equation 2.10.
















CHAPTER 5
CONCLUSION


Summary of Results



An analytical, lumped-element model was developed for a compliant-backplate

Helmholtz resonator. A prototype Helmholtz resonator was constructed and

characterized. Pressure amplification and normal-incidence acoustic impedance

measurements were taken with 24 different compliant backplates. The impedance

characterization serves as a basis for the design of a tunable Helmholtz resonator, using a

piezoelectric-composite plate in place of the isotropic plates presented in this thesis.

A quantitative deviation between the analytical models and the experimental work

did exist, due to unintended in-plane tension, and the presence of higher-order modes.

However, the overall qualitative behavior of the compliant-backplate Helmholtz

resonator was well predicted with theory. The interaction between the acoustic and

mechanical elements of the compliant-backplate Helmholtz resonator followed the

general trends predicted by the lumped element models despite differences in the absolute

frequency of mechanical resonance.

The quantitative differences were believed to be due to several factors. First, the

actual thickness of the compliant backplates was difficult to obtain accurately. Over the

surface of one of the "1 mil" plates, for example, the thickness was measured at various









locations from anywhere between 0.001 in. and 0.0015 in.. To improve the estimate of

the thickness, measurements were taken at several locations on the surface and averaged.

Besides inaccuracies in the thickness, other factors may have contributed. Inadvertent in-

plane tension as a result of the clamping configuration is believed to be increasing the

stiffness. The presence of damage to the plates, near the screw holes, and the continual

increase in resonant frequency with applied torque to the screws indicates this is a factor.


Future Work

The work performed to-date focused on the analytical development and

experimental verification of Helmholtz resonators possessing a compliant cavity wall. To

avoid the issue of in-plane tension, a new design for the clamping setup is necessary.

Careful consideration of the design is required to ensure that only normal forces are

applied to the boundary of the plate. Future work, beyond the redesign of the clamping

setup, will seek to extend this knowledge towards an electromechanical acoustic liner

capable of impedance tuning. To achieve this goal, a rigorous analytical model of a

piezoelectric-composite clamped circular plate is currently under development. Optimal

designs based on this analysis will then be constructed. An appropriate tunable electrical

network will then be connected to the plate for tuning the impedance. Impedance tests

will then proceed to characterize the impedance tuning range of the constructed devices.

Pending success with these devices, an array of them can be constructed to create the

electromechanical liner.


















APPENDIX A
DRAWINGS


0 0.313in.





0.600in.


Figure 68 : Schematic of neck and cavity plate for 0.5 in. diameter compliant backplate
Helmholtz resonator.


-1.500in.-





0.890in. 2
4 0.250in.


0 0.500in. 0 0.500in.
S-- 0.305in. 0 I ,0.305in.
0.500in. o 0.500i.

0 0.125in. 00.125in.
0.500in. 0.500in.


Figure 69: Schematic of incident microphone (Left) and cavity microphone (Right).


t
0.125in.


0 0.250in.




2.800in.









0.250in. 0.125in.

_-f -


1.000in.


0 0.500in.


0.542in.


Figure 70: Schematic of 0.5 in. clamping ring andplacement of clamping I ilg n ith
incident microphone.


0- 0.313in.

o0 + 0o
__ __ t n n __ _


4 4.300in.


Figure 71: Schematic of cavity plate for 0.96 in. diameter compliant backplate Helmholtz
resonator.


_ _


0.600in.
0 0.250in.-




2.800in.



t








0.250in.

1_D
--f-- 111


0 1.500in.


1.000in.


Figure 72: Schematic of spacer ring, clamping ring, andplacement of rings relative to
incident microphone for 0.96 in. diameter compliant backplate Helmholtz resonator.


0.027in.

4




















APPENDIX B
MATHCAD CODE


Constants


p :=1.165kg
3
m
:= 1.78910-5
ms

v poisson :=.33




v poisson ;= 324


v poison : =3



Impedance Tube

Atube := 1 in-1 in


Ztube ;-
Atube

Resonator Neck


air density


c :=344.m air sound speed


viscosity


Ed :=69109 Pa



9
Ed ;= 103109 Pa


Ed :=1.931011 Pa


P d :=2.71.
(cm)3

I
Pd:=8.47- g
(cm)3


d :=7.86. m
(cm)


aluminum diaphragm



brass diaphragm




stainless steel diaphragm


area of tube


Ztube =6.212-105 kg
4
m .s


acoustic impedance of tube


L:=.123-in length of neck d neck :=.186 in

d neck 2
a neck neck radius aneck =2.362*mm Area neck =a neck "
2

Leff:=L+ 1.6-a neck Leff = 6.904omm effective length of neck


Resonator Cavity

d :da cavity
dcavity :=501 in a cavity 2


a cavity = 0.25in radius of cavity











depth :=.602 in depth = 15.291mm depth of cavity

96 .
SpacerHeight :=.00 in SpacerRad :=- in
2
ExtraV:= SpacerHeight -it SpacerRad2

2 = 127.184nan2e2
Area cavity := a cavity 2 Area cavity 127.184mm2


Volume :=.- (a cavity depthh+ ExtraV Volume= 1.945


Resonator Backplate
.501
radius :=-- in radius of clamped boundary for backplate
2
Lumped Element Models Acoustic Components


spacers for 1" backplate only


-103. r 3


volume of cavity


p .Leff
M A(radius) ;=--
a neck

Volume
C a(radius) '-
2
p -c


MA(.5 in) = 458.803
4
m

4 2
C (.5 in) = 1.411.101
kg


acoustic mass of neck


acoustic compliance of cavity


f res(radius) )
2- M A(radius ) C a(radius ) res2)


S_ 40-8-1-L
Ra(radius ) 408- L
4
i a neck


A eff(radius ) ;= -radius 2
3


Ra(.5 in) = 1.82810 k
4
m .s


1.978-103 Hz resonance frequency




acoustic resistance of neck (viscous)


effective area of a/m xduction


Lumped Element Models Mechanical Components


i -(radius )2.(thick )p d
M (thick ,radius ) :=
5- A ef(radius ))2


mechanical mass of backplate


S 9 -(radius )2(1- poison 2) (A ef(radius )) mechanical compliance of
C1m(thick,radius ) '- 3backplate
16-.-Ed-((thick))3











Equivalent Circuit
Input Impedance



Zin (s, thick, radius)


ss-M n(thick, radius )+ I----
sM m(hicradiu s C m(thick, radius ) s C a(radius )
--\ f (s.M A( radius ) + R a( radius )


s M m( thick, radius


) s-C m(thick,radius ) s C a(radius


Pressure Amplification



PA(s, thick, radius ) :=
2




Backplate Deflection



W(P,s,thick,radius ) =


s-M ( thick, radius ) --
sC m(thick, radius)


s-C m(thick,radius )


PPA(s,thick,radius )

s.(s -M t radius thick, radius s1 -A eff(radius)
ms -sC m( thick, radius )


Normalized Backplate Deflection


W norm( s,thick, radius ) :=
s .(s -M m(thick, radius ) + 1 -, ) A ef( radius )
s C m(thick,radius )


Reflection coefficient

r(s, thick,radius Zin( s, thick,radius )- Ztube
Zin( s ,thick,radius ) Ztube


Absorption coefficient


alpha (s, thick, radius ) := 4


Re(Zin ( s, thick, radius ))
1 Ztube
(Re(Zin(s, thick, radius)) )2 (i( Zin (s, thick, radius ))2
\ Ztube Ztube
















LIST OF REFERENCES


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165-206.

[2] H. W. Kwan, "Bias Flow Adaptive Acoustic Liner Program Review," NASA-
LaRC, 1998.

[3] J. F. Betts, J. I. Follet, J. Kelly, and H. Wood, "An Improved Impedance Model for
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Sciences Meeting and Exhibit, Reno, NV, 2001.

[4] G. W. Bielak, J. W. Premo, and A. S. Hersh, "Advanced Turbofan Duct Liner
Concepts," NASA-LaRC NASA/CR-1999-209002, 1999.

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Bias Flow," NASA AST Lining Workshop, 1999.

[6] W. R. Watson, S. E. Tanner, and T. L. Parrott, "Optimization Method for Educing
Variable-Impedance Liner Properties," AIAA Journal, vol. 36, pp. 18-23, 1998.

[7] J. M. de Bedout, Franchek, M.A., Bernhard, R.J., Mongeau, L., "Adaptive-Passive
Noise Control with Self-Tuning Helmholtz Resonators," Journal of Sound and
Vibration, vol. 202, pp. 109-123, 1997.

[8] H. Matsuhisa, Ren, B., "Semiactive Control of Duct Noise by a Volume-Variable
Resonator," JSME International Journal, vol. 35, pp. 223-228, 1992.

[9] R. E. Kraft, B. A. Janardan, G. C. Kontos, and P.R.Gliebe, "Active Control of Fan
Noise-Feasiblity Study, Volume 1: Flyover System Noise Studies," NASA Lewis
Research Center NASA CR-195392, 1994.

[10] A. S. Hersh and B. Walker, "Fluid Mechanical Model of the Helmholtz Resonator,"
NASA Contractor Report NASA CR-2904, 1977.

[11] A. P. Dowling and J. E. FFowcs Williams, Sound and Sources of Sound.
Chichester, Eng.: Ellis Horwood Limited, 1983.










[12] J. F. Lindsay and S. Katz, Dynamics of Physical Circuits and Systems. Champaign,
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[13] M. Rossi, Acoustics andElectroacoustics. Norwood, MA: Artech House, 1988,
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[14] F. V. Hunt, Electroacoustics: The Analysis of Transduction, and Its Historical
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[15] D. T. Blackstock, Fundamentals ofPhysicalAcoustics. New York, NY: John Wiley
& Sons, Inc., 2000.

[16] W. T. Chu, "Extension of the Two-Microphone Transfer Function Method for
Impedance Tube Measurements," Journal of the Acoustical Society ofAmerica, vol.
80, pp. 347-349, 1986.

[17] J. Y. Chung and D. A. Blaser, "Transfer Function Method of Measuring In-Duct
Acoustic Properties. I. Theory," Journal of the Acoustical Society ofAmerica, vol.
68, pp. 907-921, 1980.

[18] M. G. Jones and P. E. Stiede, "Comparison of Methods for Determining Specific
Acoustic Impedance," Journal of the Acoustical Society ofAmerica, vol. 101, pp.
2694-2704, 1997.

[19] ASTM, "Standard Test Method for Impedance and Absorption of Acoustical
Materials Using a Tube, Two Microphones, and a Digital Frequency Analysis
System," ASTM E 1050-90, 1990.

[20] M. G. Jones and T. L. Parrott, "Evaluation of a Multi-Point Method for
Determining Acoustic Impedance," Mechanical Systems and Signal Processing,
vol. 3, pp. 15-35, 1989.

[21] W. T. Chu, "Transfer Function Technique for Impedance and Absorption
Measurements in an Impedance Tube Using a Single Microphone," Journal of the
Acoustical Society ofAmerica, vol. 80, pp. 555-560, 1986.

[22] Polytec-PI, Operator's Manualfor Polytec Scanning Vibrometer PSV-200.

[23] H. W. Coleman and W. G. Steele, Experimentation and Uncertainty Analysis for
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BIOGRAPHICAL SKETCH

Stephen Brian Horowitz was born on August 3, 1977, in Pompton Plains, N.J. He

attended Marjory Stoneman Douglas High School in Parkland, FL, graduating in 1995.

He received his bachelor's degree in electrical engineering from the University of Florida

in 1999. He is currently working towards his Master of Science degree in electrical

engineering at the University of Florida. He plans to continue towards a doctoral degree

at the University of Florida, concentrating his research efforts in the area of micro-

electro-mechanical systems (MEMS).




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