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 #NAG12261) 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
CompliantBackplate 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 HalfInch Diameter, Clamped Backplates ................... 37
Helmholtz Resonator with OneInch Diameter, Clamped Backplates ................. 47
In p u t Im p ed an ce ............................... .... .. ..... ..... ....................................... 5 5
Helmholtz Resonator with HalfInch Diameter, Clamped Backplates ................... 55
Helmholtz Resonator with OneInch 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 compliantbackplate Helmholtz
resonator................ ............................. .. ............ .. ....... 23
13: Schematic of impedance tube terminated by compliantbackplate Helmholtz
resonator................ ............................. .. ............ .. ....... 29
14: Normal impedance tube showing rotating microphone plugs, and endmounted
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 insitu, tunableimpedance, electromechanical acoustic liner to
optimally suppress noise under changing engine conditions.
To further this end, an investigation was conducted into compliantbackplate
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 normalincidence, acoustic impedance
measurements. The experimental results demonstrate an additional resonance and an
antiresonance in the impedance caused by adding degreesoffreedom 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
piezoelectriccomposite backplate in place of the isotropic backplates presented in this
research. Overall, the results confirm the multidegree of freedom nature of the
compliantbackplate Helmholtz resonators and their equivalence to doublelayer 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 noisereduction bandwidth in singlelayer 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. Multilayer liners provide additional degrees of freedom that can be
optimized to achieve a target impedance over a larger bandwidth than is possible with a
singlelayer [2]. A twolayer 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 takeoff 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 [26]. 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 volumevariable cavity for a Helmholtz resonator [7, 8].
Other techniques seek to reduce noise by actively canceling the incident sound field with
an outofphase, 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
acousticaltoelectrical 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
multilayer 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 insitu and in
realtime. The initial goal is to have three distinct liner impedance spectra, each
optimized for a specific engine condition, such as takeoff, cutback, and landing. This
can be achieved with three separate electrical networks and a simple threeway 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
compliantbackplate Helmholtz resonator. In this thesis, the theoretical analysis of
compliantbackplate 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 multidomain 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 compliantbackplate Helmholtz resonators presented in this
paper, lumpedelement 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 crosssectional 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= NT .
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 crosssectional 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, nonlinear 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 nonlinear effects will be ignored in this thesis, along with any
grazing flow dependence. The nonlinear 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 14950 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 onethird 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 efforttype variable or a flowtype 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.
CompliantBackplate 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
S100
I.
S150
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( 1r 2rdr (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
Su(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 firstorder Struve function The Maclaurin expansions of (2.31) are also
given by [15] as
(ka)2 (k< k)6
R, = + ()6L (2.32)
12 1.22.3 12 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 viceversa. 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 acoustomechanical
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 volumevelocity 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
deflectioncausing 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 antiresonance, 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 antiresonance 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 antiresonance.
40
16
8
32
S56
80
1000 2000 3000 4000 5000 6000
Freq [Hz]
100
7 25
50
S125
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 antiresonance 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 antiresonance 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 compliantbackplate 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
nondimensional result.
111 1
III I IFreq [ IHI 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
compliantbackplate 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 compliantbackplate
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 antiresonance 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 antiresonance 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 antiresonance 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 antiresonance
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 antiresonance. 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 antiresonant 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 antiresonance 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 compliantbackplate 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
compliantbackplate 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 compliantbackplate 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 MultiAnalyzer 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 MultiAnalyzer
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 nearfield 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 multipoint method to determine the
acoustic impedance. The multipoint 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 multipoint method reduces
to the twomicrophone method of impedance measurement [1619]. The two
microphone method requires less than a halfwavelength spacing between the two
microphone locations [1719]. 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, singlemicrophone method [21] exists but was not
used due to greater errors associated with the technique.
Using the multipoint 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 planewave 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+(y1) 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)
1R
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]
A1251 Aluminum 0.25 0.0013 4.618E12 1.272E+03
AI252 Aluminum 0.25 0.0023 8.338E13 2.250E+03
AI253 Aluminum 0.25 0.0032 3.096E13 3.130E+03
AI255 Aluminum 0.25 0.005 8.116E14 4.890E+03
AI502 Aluminum 0.5 0.0024 4.697E11 5.869E+02
AI503 Aluminum 0.5 0.0031 2.179E11 7.580E+02
AI505 Aluminum 0.5 0.0048 5.871E12 1.174E+03
AI5010 Aluminum 0.5 0.0105 5.609E13 2.567E+03
Br251 Brass 0.25 0.0013 3.107E12 3.974E+03
Br252 Brass 0.25 0.0023 5.610E13 7.031E+03
Br253 Brass 0.25 0.003 2.528E13 9.171E+03
Br255 Brass 0.25 0.0052 4.855E14 1.590E+04
Br502 Brass 0.5 0.0021 4.717E11 1.605E+03
Br503 Brass 0.5 0.0031 1.466E11 2.369E+03
Br505 Brass 0.5 0.0051 3.293E12 3.898E+03
Br5010 Brass 0.5 0.0102 4.117E13 7.795E+03
St251 Stainless Steel 0.25 0.0014 1.350E12 3.972E+03
St252 Stainless Steel 0.25 0.0023 3.044E13 6.525E+03
St253 Stainless Steel 0.25 0.0033 1.031E13 9.361E+03
St255 Stainless Steel 0.25 0.0055 2.226E14 1.560E+04
St502 Stainless Steel 0.5 0.0021 2.234E13 1.489E+03
St503 Stainless Steel 0.5 0.0032 7.234E12 2.269E+03
St505 Stainless Steel 0.5 0.0053 1.592E12 3.759E+03
St5010 Stainless Steel 0.5 0.0107 1.935E13 7.588E+03
36
Table 2 : Material properties for backplates.
Type Density [g/cm^3] Young's Modulus [GPa] Poisson's Ratio
Aluminum 1100H18 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 HalfInch Diameter, Clamped Backplates
The results obtained for the Helmholtz resonator with halfinch 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 compliantbackplate 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]
A1251 1.26E03 1.52E04
A1252 2.28E03 1.24E04
A1253 3.32E03 3.62E04
A1255 5.08E03 3.62E04
Br251 1.36E03 6.97E04
Br252 2.16E03 1.52E04
Br253 3.02E03 1.24E04
Br255 5.22E03 3.04E04
St251 1.38E03 2.32E04
St252 2.14E03 2.48E04
St253 3.26E03 2.48E04
St255 5.26E03 2.48E04
A1502 2.30E03 3.40E04
A1503 3.22E03 2.32E04
A1505 4.86E03 1.52E04
A15010 0.0105 2.32E04
Br502 2.12E03 1.24E04
Br503 3.14E03 2.32E04
Br505 5.34E03 1.52E04
Br5010 0.0104 2.32E04
St502 2.40E03 3.40E04
St503 3.56E03 4.21 E04
St505 5.28E03 4.12E04
St5010 0.0106 3.17E04
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 [Iz]
' 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 antiresonance and second resonant peak match the theory fairly well for
the 0.003 in. thick backplate. The antiresonance 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 antiresonant 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 inplane tension applied as an inadvertent effect of the clamping. Many of the
stifferthanexpected 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 inplane
tension would raise the stiffness of the plates, thereby increasing the resonant frequency.
To further investigate this possibility, realtime 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 inplane tension was occurring, then
the resonant frequency would not rise with increasing torque on the screws, therefore,
these results indicate the likely presence of inplane 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 compliantbackplate 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 antiresonance introduced by the backplate.
The coherence for these measurements typically were near one, except at
frequencies around the antiresonance. 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 antiresonance 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 [Iz]
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 OneInch Diameter, Clamped Backplates
Pressure amplification spectra were also obtained for the Helmholtz resonator
with oneinch 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.
Inn
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 oneinch diameter clamped backplates
exhibited better agreement between theory and measurement. The first and second
resonant peaks and antiresonance 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 higherorder 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 compliantbackplate Helmholtz resonator was found from
measurements of the reflection coefficient, obtained via the multipoint method.
Helmholtz Resonator with HalfInch 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
100
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [Hz]
S* .001" thick backplate
Theory+Uncertainty
TheoryUncertainty
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
TheoryUncertainty
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
TheoryUncertainty
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
TheoryUncertainty
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
TheoryUncertainty
STheoryUncertainty
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
TheoryUncertainty
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
TheoryUnc certainty
TheoryUncertainty
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
TheoryUncertainty
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
STheoryUncertainty
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
TheoryUncertainty
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
TheoryUncertainty
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
TheoryUncertainty
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 OneInch 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
TheoryUncertainty
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
TheoryUncertainty
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]
1511
S* 005" thick backplate
Theory+Uncertainty
TheoryUncertainty
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
STheoryUncertainty
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
TheoryUncertainty
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
TheoryUncertainty
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 
1 103
4o
1 104
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
TheoryUncertainty
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
TheoryUncertainty
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. 
S400
Freq [Hz]
S* .002" thick backplate
Theory+Uncertainty
TheoryUncertainty
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
1 103
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Freq [HIz]
200
Freq [Hz]
S* .003" thick backplate
Theory+Uncertainty
TheoryUncertainty
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
TheoryUncertainty
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
TheoryUncertainty
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 antiresonance 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
antiresonance 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.
LowFrequency 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, lumpedelement model was developed for a compliantbackplate
Helmholtz resonator. A prototype Helmholtz resonator was constructed and
characterized. Pressure amplification and normalincidence 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
piezoelectriccomposite 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 inplane tension, and the presence of higherorder modes.
However, the overall qualitative behavior of the compliantbackplate Helmholtz
resonator was well predicted with theory. The interaction between the acoustic and
mechanical elements of the compliantbackplate 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 todate focused on the analytical development and
experimental verification of Helmholtz resonators possessing a compliant cavity wall. To
avoid the issue of inplane 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
piezoelectriccomposite 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
:= 1.789105
ms
v poisson :=.33
v poisson ;= 324
v poison : =3
Impedance Tube
Atube := 1 in1 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 =6.212105 kg
4
m .s
acoustic impedance of tube
L:=.123in 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+ 1.6a 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_ 4081L
Ra(radius ) 408 L
4
i a neck
A eff(radius ) ;= radius 2
3
Ra(.5 in) = 1.82810 k
4
m .s
1.978103 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)
ssM 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
) sC m(thick,radius ) s C a(radius
Pressure Amplification
PA(s, thick, radius ) :=
2
Backplate Deflection
W(P,s,thick,radius ) =
sM ( thick, radius ) 
sC m(thick, radius)
sC 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
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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
electromechanical systems (MEMS).
