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LIBRARY
University

This is to certify that the

dissertation entitled

NDDAL ANALYSIS OF AN ACOLETICAL CAVITY (DUPLED
'10 A RECTANGULAR MEMBRANE

presented by

J}! W. SG'IERS

has been accepted towards fulfillment
ofthe requirements for

Master of Science degreein Mechanical Engineering

    

Ma, rprofessor
Date 5 /7 C95-

MSU is un Affirrmmve Arnom Equal Opporlumry Inslimlmn 0-1277]

 

 

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RETURNING MATERIALS:
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IDDAL ANALYSIS OF AN ACOUSTICNL CAVITY COUPLED

TO A RECTANGULAR IBIBRANE

By

Jon I. Savers

A THESIS

thnitted to
Michigan State University
in partial fnltillnent of the require-ente
for the degree of

IASTER OF SCIENCE

Depertnent of lechicnl Engineering

1984

ABSTRACT

KIDAL ANALYSIS OF AN ACDUSTICAL CAVITY (DUPL-
TO A RECTANGULAR CAVITY

By

Jon I. Sowers

In this study the Fourier and LaPlace transform solution to
the vibrational response of a cavity-backed rectangular membrane is
compared to experimental measurements found from modal analysis.
Several cavity-membrane system resonances were examined in a 575 hr to
1300 ha range. Both cavity pressure and membrane displacement mode

shapes were examined.

Results show excellent agreement with analysis. In general
membrane and cavity mode shapes matched well with the expected sine.
cosine and cosh functions of the analysis. Ihe system frequencies
when compared with predictions were well within the bounds of
experimental error. The highest system resonance examined. 1243 ha.
appeared to be a combination of two or more modes. to close to be
separated in this frequency range. Ihe membrane proved to be a

versatile tool in the study of this phenomena.

ACKNOWLEDGEMENTS

The author would like to thank Dr. Clark Radcliffe of the
department of Mechanical Engineering at lichigan State University for
his valuable suggestions during the course of this study and in the
writing there of; and also the department of Mechanical Engineering at

ISU for their support.

TABLE OF CONTENTS

Ammmem...OOOOOOOOOOCCOOOOI.0....0.0.0.0000000000000003

HomaAMEOOOOOOCOOOOOOOO...O.I0.0..OCOOOOIOOOOOOOOOOOOOOOC0.6

Chapter 1:

Chapter 2:

Chapter 3:

Chapter 4:

Chapter 5:

INTRODUCTION........................................7
BACKGROUND
Theory.............................................11
Discussion.........................................17
EXPERIIENTAL DESCRIPTION AND EROCEDURE
Description........................................24
Procedure..........................................36
RESULTS AND DISCUSSION

Membrane-cavity system.............................41
Displacement/pressure ratio comparisons............57
Cavity coupling restricted.........................58

WEBSIONSCOCO0....OCO...0.00.0.0...00.0.00000000061

Rmmm.OOOOOOOIOOOOOOOOOOOOIOOOOOOOCOOOOOO0.0.0.0.0000000062

APPENDIX A - Supplemental Theory

unbr‘na v.1601ty0000000.0...0.0.0....0.0.0.000000000064

Circnl.r --br‘n..IOOOOOOOOIOOOIOIII...I...0.00.00.00.66

Circular mubrane - tension measuruent. . . . . . . . . . . . . . .67

R.ct‘n‘u1.r .“br.n..0......OOOCOOOIOOOOOOOOOOIOO0.00.70

R.ct.n‘nl‘r °.v1ty.....OIOOOOOOOOOOCOOOOO...O...00.00.11

Bondin‘ 'tiffn°".OODOOOOOOOOOOOOIOOOO00.00.00.000000072

APPENDIX.B - Preliminary Circular Membrane Experiments........74

mnlxc-Tt‘n‘dnoat C‘libr‘tioneeeeeeeeeeeeeeeeeeeeeeeeeee79

APPENDIX D - Membrane Tension and Density Ileasurement.. .......84
API'ENDIX E - Estimates of Air Density and Sound Velocity......87
APPWDIX F - Sample Calculations..............................90
APPENDIX G - Supplimental Data.95

A,D

"U

8qr(s)
Sq:

3qr(s)

I '6
> D 'U

0-!

H

10 T8

4<

NOMENCLATURE
constants to match boundry conditions of membrane-cavity
system
constants to match boundry conditions of circular membrane
wave velocity of membrane
wave velocity of air
modulus of rigidity for a panel
constants to match boundry conditions of rectangular cavity
general force applied to membrane surface
point force applied to membrane surface

bessel function of order 1

constants used in equation for membrane-cavity system pressure
acoustic pressure inside cavity

laplace and fourier cosine transform of P

pressure on outside and inside membrane surfaces respectively
laplace transform of Pa and P,

coefficients of fourier sine expansion of Pa

laplace transform of Pig:

coefficients of fourier sine expansion of P;

laplace transform of P3qr(.)

laplacian operator 3/3; + +d/ay + 6/8:

gas constant for air

membrane tension

air temperatures 1 and 2

membrane velocity

laplace transform of V

0(a)

c(t)

pmn
qr
pm

Po

laplace and fourier cosine transform of V

constants to match boundry conditions of rectangular membrane
impedance of cavity mode mn

impedance of membrane mode qr

panel thickness

(-1)‘/’

interior point on circular membrane surface

outer radius of circular membrane or cylindrical cavity
time independent membrane or panel displacement

laplace transform of w .

laplace and fourier sine transform of w

displacement of rectangular membrane or panel mode qr
velocity potential in membrane-cavity system

laplace and fourier cosine transform of D

s)i/s

' IIC. (411+ g

I 1/C. (“:11- “8,1,8

coupling coefficient for membrane-cavity system

membrane density

air density

panel density

k zero's of bessel function 11

frequency

frequency of circular membrane mode jk
frequency of cavity mode lmn

frequency of cavity-membrane system mode n

frequency of rectangular mode qr

< >2 space time average

Chapter 1

INTRODUCTION

The vibrational response of a finite panel or membrane coupled to
an acoustical cavity is an important part of many engineering
problems. Current research examples include the transmission of
environmental noise into an occupied dwelling. environmental noise
entering the passenger compartment of a vehicle. and sound
transference between adjacent rooms. Other examples include eardrum
behavior (cavity-membrane). musical instruments such as the drum
(cavity-membrane). and stringed instruments such as violin or guitar

(cavity-panel).

Inch is known both analytically and experimentally about the
freely vibrating (uncoupled) membrane or panel and acoustical cavity
when taken seperately. but only recently has the coupling between the
two been studied closely. An early reference to membrane-cavity
coupling is a description of the kettle drum by P. I. Horse in
1948I1]. This work describes membrane motion at frequencies well
below the fundamental resonance of the cavity. lore recently several
investigators including Kihlmanl3]; Bhattacharya and Crockerl4];
Bhattacharya. Guy and CrockerIS]; have employed finite fourier cosine
and sine transforms to solve the system equations for a freely
supported (hinged) rectangular panel backed by a rectangular cavity.
Guy and BhattacharyalG] deve10ped these equations to a final correct

form. Their resulting series solution was applied to predict the

10

average pressure drOp of sound transferred through the panel into the

cavity and then compared with experimental measurements.

In the experiments of Guy and Bhattacharyal6] a speaker served as
a plane wave source whose input was monitored by a microphone placed
outside the cavity very near the panel surface. A second microphone
was placed inside the cavity at the back wall surface. The frequency
input was varied and the pressure differential between the microphones

was measured.

Their experimental results agreed well with an analytical model
they developed. Resonances and antiresonances fell close to predicted
values (0.3% to SS error). Though pressure in the cavity was measured
at only one location. the frequency range covered. 0 to 2600 hr. was
enough to include 12 system resonances. This indicates the analysis

is an accurate description of the panel-cavity system.

The intent of this investigation is to further previous
experimental work and to add to the general knowledge of
membrane-cavity coupling by providing more detailed experimental data
for comparison. The Fourier series solution of Guy and
Bhattacharyal6] readily lends itself to experimental verification by
modal analysis using any of the sophisticated computer systems now
available. In this work their analysis is compared to experimentally
determined cavity pressure/membrane displacement mode shapes and

associated frequencies of a cavity-membrane (not panel) coupled

11

system.

Chapter 2 of this report contains the background analysis on
cavity-membrane coupling. Here the basic equations are derived
followed by a discussion of their significance and predictions as
applied to modal analysis. Chapter 3 contains descriptions of the
experimental apparatus and procedural details on obtaining the
pressure/displacement mode shapes. Results for the membrane-cavity
system are presented and discussed in chapter 4 followed by

conclusions in chapter 5.

The appendices which follow discuss experimental work that was
done outside of the main body presented in the above sections. This
work is presented there to avoid distraction from the primary endeaver
of this study - the measuruent of membrane-cavity system resonances.
Topics in the appendices include supplemental theory. preliminary
experiments on the circular membrane. tension measurements .transducer

calibration and determination of membrane and air pr0perties.

12
Chapter 2 - BACKGROUND

2.1 Th
eory Membrane

 

 

 

 

 

Eigure 1‘ Mgdg; g; membgggg-ggvigy gzgggg

A representative model of the membrane-cavity system is shown in
figure 1. It consists of an acoustically reflective rectangular

cavity having a membrane at one face (x-O) under tension. T;

The acoustic velocity potential. 0. inside the cavity is given by
the wave equation.

1 a
Q’0(x.y.z.t) III-C73? D(x.y.s.t) . (1)
O

a’ a' a‘
where Q1 is the laplacian operator. —, + -—3 + _’. .
ax ay 3:

The boundary conditions are:

13

(1) Maya») = o (v) 3—0(:.0.z.t) = o
3y
(ii) a D( O) O ( ' a D( 0
at x.y.z. v1) 6y x.b.z.t) 8
(iii) a mo ( a
ax sYszst) 3 v y.2.t) (Vii) 320(3sy10at) " 0
(iv) a D( 0 a
ax l.y.2.t) (Viii) 310(Xsyscat) '3 0

where V is the panel velocity.

Boundary conditions (i) and (ii) result from zero initial
conditions at all points in the cavity. B.C. (iii) comes from
matching the membrane velocity to the air velocity at the surface of
the membrane. B.C.'s (iv) through (viii) derive from the acoustically

reflective walls. i.e. zero velocity at the wall surface.

It has been shownl6] that the solution to equation (1) may be

written as

U(x.m.n.s) 8 A coshlxu(s)] + B sinhlxc(s)] (2)

where

a(s) "%- (“;n + g“)

and

- b _ mny nu:
3 t — _
U(x.l.n.s) 0’ dt 0‘ dz 0! 0(x.y.2.t) e ‘ cos( b ) cos( c ) dy .

14

Equation (2) was found by the application of a double finite
fourier cosine transform in the y and 2 directions followed by a
laplace transform in the time domain. A and B are then determined by
applying boundry conditions (iii) and (iv) in their transformed state.

that is

— 5(0,.,n,3) . V End a— a(xs.ans$) ‘ 0 '
a: 3x

Substitution of the trinsformed pressurel6] in terms of the
velocity potential. and application of the above boundary conditions

results in

‘ : cosh[(a-x)c(s)]
P I p. s
c(s)sinh[au(s)]

 

. (3)

This is the transformed expression for the pressure inside the cavity.

The transformed membrane velocity. V. resulting from an arbitrary
pressure distribution on both membrane surfaces is developed in
appendix A. This allows for the cavity influence on the inside
surface of the membrane and a pressure input on its outside surface.

The result for the transformed velocity is

— b3 3
V - ——. ___ _
“- Z'"‘b:: ("+ mar) [PM], P'qr] “‘5’

where the coupling coefficient. flmn , 1. dgfingd .3
qr

15

rnz may nnz
5:: bc ojb 1 r‘ u ?)sin( c )cos( b )cos( c ) dy

n‘(n’-q’>(n’-:’)

 

[cos(qu)cos(mn)-1][cos(rn)cos(un)-l] . (4)

It should be noted that the solution for the membrane velocity.
V. is almost identical to the panel solution of Guy and
Bhattacharyal6]. The difference is found in the first term of the

right side of equation (A6) where in the denominator is found 49. for
thO membrane and 4Pph for the panel. Also the membrane or panel

eigenfrequencies. 'qr' are determined from their respective frequency

equations (equation (A4) for the membrane).

The substitution of the right hand side of (Al) for V in (3)

resultslG] in

 

i . 2322 2° ____:____ {P _ P ] cosh[(a-x)c(s)]
4pm q'3"B:: ("* 03,) 1qr(s) 'qr(s) c(s)sinh[au(s)]
and finally

 

z-ncosh[(a-x)u(t)]
P(x.y.z.t) ' . [1' P
25.:-; 1qr5:n (Z-‘+zqt)cosh[ac(t)]

co3("'""'Z)cos(-n£:) ejut (5)
b c

where

cothlau(t)]
a(t)

 

zmu ' 590” (6)

is the impedance of cavity mode mn.

16

zqr a j is («,1- ugr) (7)

is the impedance of membrane mode qr.

1 1/3
._ s 3
Mt) C. (“mn' u ) (8)
and
E=0.5 for m=0 K=1 for m>0
K'-O.5 for n=0 R'-1 for n)0 .

Equations (5). (6). (7) and (8) describe the wholly steady-state
harmonic solution to the pressure inside the cavity due to a known

91'0"“. input. P3. on the membrane surface. The transient part of

this solution is not considered in this study.

Two other results[6] from equation (5) give the time and spatial
averages of both. the pressure on the inside membrane surface and the

membrane velocity. normed by the input pressure:

(P, . m2...
> 16 s qr s
—:f-- xx' (—) lz———l (9)
(Pg)t q::;; 91"! ”4' lg!
<v‘>' B...

16 8 qr s
_:_:. a Z I. x p (__.) I___..z I . (10)
01% q::-: qrn ”1+ qu

These equations were found by evaluating P at x-O (inside surface of

membrane) and “NI“!!! P; has a constant amplitude. The space and

17

time averages were then squared giving the above results.

18

2.2 Discussion
2.2.1 System resonances
a. Strongest resonances

The above results show that the pressure inside the cavity and

the panel velocity are governed by three factors:

(i) the impedance of a cavity mode. Zmn

(ii) the impedance of a membrane mode. th , and

(iii) the coupling coefficient. an .

qr
The denominators of equations (9) and (10) clearly show that the
strongest resonances occur when the cavity impedence is equal to the

negative of the membrane impedence. i.e.

Z“ . u—zqr
or
P.C.m . 3 3 1/3 -9. 3 8
—— th — - > J- —-( - u ) . (11
(”:n' .02):]: °° [c.(umn ” u 0 Cr )
This condition occurs at each system natural frequency. a and

n.
substituting the material constants. dimensions and the natural
frequencies. “mu and ”qr' of an appropriate uncoupled cavity and

membrane mode will yield a resonant frequency of the system. a - an.

The expected pressure distribution at each natural frequency is
determined from (5) which shows that the cavity pressure mode shapes
in the y and 2 directions correspond to the shapes of uncoupled cavity
modes. mn. This yields only cosine variation of the pressure in the
y-z plane at the system resonances while in the x direction the
pressure distribution is described by the cosh function. The boundary
conditions imply that at the hard wall opposite the membrane the cosh
function will be finite and perpendicular to the surface (zero
velocity at the wall) and that at the membrane surface this function
will usually have a non-zero slope (non-zero velocity at panel
surface). In some cases the x direction pressure distribution is
described by the cosine function. This occurs when Emu ( u . In this
case. the cosh function of (5) with its now imaginary argument becomes

a cosine function. i.e. cosh(jx) 8 -cos(x) .

The membrane mode shapes of the system are described by the sine
functions associated with the uncoupled membrane modes. qr. This is
implied by the derivation of the transformed membrane velocity. V

(equation (A6)). The membrane frequencies become (on, the system

resonances. due to the coupling effect.

The amplitude of these system resonances are limited by the

coupling coefficient 6 Maximum coupling is shown in equation (4)

mn'
qr
to occur when m - qtl and n - rtl. For combinations where m-q and n-r

equal some multiple of 2. membrane-cavity coupling cannot exist as

demonstrated by the zero values of an or the inability to solve
qr

20

equation (11).

A simple explanation for the significance of the coupling
coefficient is found by observing the matching of membrane mode shapes
to cavity mode shapes. Figure 2 shows the crossection of membrane
mode qr - 12 in the x-z plane. Figures 3 and 4 show crossections of

cavity pressure modes in the x-z plane for lmn 8 001 and lmn 8 002.

'X

 

N00

 

mammgammfi-u

Nlo
L- o

——————d

 

.._s__Fi 111011.21”: 94.46 10 431132219991

\

1.)

Figure 1‘ Cross section 2; gavity mode 99;

suo'4r
0

 

 

Maximum coupling occurs with membrane mode 12 when lmn - 001. A

22

cross section of this membrane and cavity mode is shown in figure 5a.
Here the pressure build up in the cavity serves to add to the membrane

motion. i.e. a good matching of waveforms.

 

 

 

 

 

 

 

mammmmmmmnmmn

Zero coupling occurs with cavity mode 002. Figure 5b shows a cross
section of this coupling. Here there is poor matching of wave forms
as zero pressure occurs at maximum membrane displacement. These
functional shapes are in fact orthogonal. i.e. I;pv 4A . 0 and no
coupling occurs. Figure 6 shows cavity mode 003 combined with

membrane mode 12. There could be activity in this configuration. i.e.

I;pv dA is non zero. but the coupling coefficient will be small.

23

 

 

 

 

Figure 6‘ Cgvity mod; 90; gggplgd t9 mgmbrgne £21; 12
b. Cavity Resonance

From the expression for cavity modal impedance.

c
2“. 121.1” “00.3%“;11-
.

(”a - fig): “3)1/8] .
mn

it is evident thlt when U 3 9.3 the argument of the hyperbolic term is

zero and the cavity impedance tends to infinity. The usual standing

wave relationship for a rectangular cavity is satisfied (see Theory -

Rec. Cavity).

(31% (3)5 (EW- (2.): 1 a 0.1.2..... . (A15)
a b c C.

the cavity impedance dominates and the panel velocity approaches zero.

The uncoupled cavity resonances should be observed unchanged and

24

little panel motion will be expected at such resonances and the
pressure amplitude should not be as large as that of the resonances

occuring when qu . 'zmn-
2.2.2 Displacement/pressure ratio comparisons

For further verification of the analysis the membrane
displacement(velocity) to pressure amplitude ratio is examined for
each mode. Equation (11). the normalized and averaged membrane
velocity. is divided by (10). the corresponding pressure equation.

The resultant is a ratio of velocity to pressure:

<V’)‘ 1
(Pa): zmn

This says the ratio of membrane velocity(displacement) to cavity
pressure for each mode is inversely proportional to the cavity

impedance.

2.2.3 Other system behavior

a. At membrane resonance

Ihen the system is excited at frequency w = ”qr: a. gum-.3.

impedance. zqr. goes to zero. but in general no system resonance
occurs. The cavity impedance remains at a nearly constant nonzero

value resulting in nonzero total impedance whose slope has the same

sense as a function of 0 immediately through resonance.

25
b. Cavity Off Resonance

When the cavity modal impedance is equal to zero (i.e. the
argument of the hyperbolic term is equal to 151% 1 a 1.3.5,...) the
membrane behaves as if there is no backing cavity. Very low pressure

levels should be found in the cavity at these frequencies.
2.2.4 Panel to Membrane substitution

Because of the almost identical analytical results of the
membrane-cavity system to those of the panel-cavity system no
differences in behavior are expected when. substituting the membrane
for the panel of Guy and Dhattacharyal6]. The membrane offers a
greater degree of flexibility since its natural resonances can be
easily varied to suite the experiment. Also the membrane is much less
affected by residual stresses and the higher order effects neglected
by simplified theory. The membrane is a useful substitution for the

simply supported (hinged) panel.

26

Chapter 3 - EXPERIMENTAL DESGIPTION AND HOCEDURE

This section contains a description of the procedures and
equipment used to obtain the mode shape data on the rectangular
membrane-cavity system. First a brief introduction to the DP 5423A
Dynamic Strucural Analyzer is given followed by a description of the
input and output measurement devices. The membrane-cavity system is
then described followed by a discussion of specific experimental

techniques.

3.1 Description

3.1.1 RP 5423A Dynamic Structural Analyzer

The RP 5423A Dynamic Structural Analyser. depicted in figure 7
and shown in plates 1 and 2. is capable of performing many signal
processing operations. In these experiments transduced signals of the
input forces and output displacements/pressures from the membrane and
cavity are operated on to yield transfer functionsl8]. These transfer
functions are further manipulated to reveal the shapes. frequencies
and damping of each vibratory mode. Post processing of this modal
data allows animation of these calculated eigenfunctions. Other
useful features of the HP 5423A include tape storage and chart

recorder.

27

 

Plates 1‘ gag 24 RP 5423A s rue r analyzer 339

 

experimental a ara u

 

 

28

Impulse hammer

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Displacement
sensor
h———-
AC/DC 4** Charge
converter Amplifier
MicrOphone
rs
P - -
Lr ' ' '
Cavity-membrane system
i
Membrane motion Cavity pressure output Input excitation
output to to analyzer input 2 to anglyzor
analyzer input 2 input 1
Input 2 <Input 1 EP 5423A Dynamic
<§§§g> structural analyzer
(D C>(D
<3 O>C> C>C>C)(>
‘O C>(D
J L >—‘
Modal data: Chart
Mode shapes 0 recorder
Frequencies O
Damping
O><D CD

 

 

 

29

3.1.2. Membrane-cavity system

Figures 8. 9. 10 and plates 3. 4. 5 show the apparatus used to
create the rectangular cavity and membrane. A conventional snare
drum. 15.24 cm high and 37.8 cm diameter (freely vibrationing). with a
single "head”. Ludwig Rocker - Heavy Clear. were used for the fixture
and membrane. The box upon which the membrane was set is composed of
3/4 inch plywood and rubber strips were fastened to the bottom of the
box to form a seal between it and the base. The top of the cavity is
prepared with an outside beveled hard surface to insure solid contact
with the membrane. The inside walls of the cavity were smoothly

finished with varnish to help diminish acoustical damping.

30

0.953 cm
3/4 in plywood ( /-—‘/r

 

 

 

 

 

L 7
12.1 cm
Formica edging tapered
to form sharp edge for
solid membrane contact
Rubber

 

18.8 cm

 

 

 

 

/ stripping

 

 

 

 

K—20.2 cm-——§~

£11.91er

0

f cavity

6

 

 

31

 

__teP1n 3.. 2.113.012.1111

Angle iron
3/4 in plywood

 

 

 

M
0

 

 

 

 

 

 

Rubber stripping

Tapered rubber edge

Figure 2‘ Drawing 9; membragg giggp

32

 

fixture

and

Mgmbrgng

.4...

P

 

alarm“ Ila-rt museum-322211.!

33

 

 

 

 

 

 

 

 

 

 

 

 

 

umimwAu-lgmi .LL__..IIfln° mammals

With the membrane positioned on top of the cavity. a frame of the
same material and identical y.z dimensions as the cavity was clamped
on top to firmly fix the membrane in place. As shown in figure 9 and

inset. this frame also had rubber stripping that was cut and trimmed

34

to insure solid membrane contact with the cavity. allow free
vibration. and support the membrane along a single line around the

inner surface of the cavity.

Shown in figure 8 and plate 3 are several holes drilled into the
side of the cavity allowing insertion of the microphone for pressure
measurements. Their location was chosen to obtain a sampling of all
the pressure modes in the range of this study. The holes are plugged

with wooden dowels when not in use to ensure minimal damping.

In general it is desirable to minimize damping in both the
membrane and cavity as the analysis does not account for it and to not
reduce system resonances. Not all dissipation can be eliminated.
Some leakage in the cavity is desirable so that inside and outside
pressures are equal. thus no attempt was made to ensure a positive

seal through out the cavity.

It should be noted that the choice of wood as a construction
material was not only for its accessability. low cost and ease of
construction but perhaps more important for its inherent damping
prOperties. By having almost all structural components composed of
wood an effective isolation of boundary structure vibrations was

obtained as no such problems were observed.

35

3.1.3 Input and output measurement devices

It is necessary to excite the membrane-cavity system through the
top of the membrane as the analysis allows for no other source. An
impulse hammer was selected to create the force imput. It is ideally
suited since the force-time history of its "impulse" input can be
designed to contain a nearly continuous and constant spectrum of

frequencies over the range of interestl8].

The impulse hammer used in this study is shown in plates 1. 2 and
6. It was specially constructed to accommodate the light weight of
the membrane - generally lighter (less dense) structures require a
lighter hammerl8]. This hammer is composed of a small. 1.2 mm high by
7 mm dia. accelerometer. PCB model 303A02 SN 1672. attached to a
light wooden handle. The signal from this accelerometer passes
through a charge converter. Pm model 480C06. and then goes to the DP

5423A.

36

 

Plgtg 6‘ Impulge hgmmgr

Shown in plate 7 is the microphone and fixture used to convert
acoustic pressure into voltage (it is also shown. in place. in plates
1 and 2). The microphone. Radio Shack model 33-1056. when removed
from its supplied casing is 0.635 cm high by 0.953 cm dia.. This
"bare" micrOphone was permanently attached to a long brass tube. 0.318
cm dia.. . The tube has been mounted in a wooden plug which can be
inserted into any selected hole in the cavity wall. The brass tube
can be moved through the plug locating the microphone at selected

points within the cavity.

The microphones major requirements were to displace an
insignificant amount of air within the cavity and maintain sufficient

linearity over the experimental range to to insure accurate recording

37

of mode shapes. In addition to checking linearity (voltage output
versus pressure input)it was checked for constant output over its
frequency range as some comparisons were made of maximum modal
amplitudes. These calibration techniques and their results are

presented in appendix C.

 

Plgte 1‘ Miggophoug ggd figtg;g

To measure membrane displacement a reluctance displacement
‘sensor. laman model RD2400. with ac/dc converter. Kaman model P3000.
is employed as shown in plates 1. 2 and 8. The sensor tip. 0.635 cm
dia.. is mounted in a threaded retainer for easy adjustment. This
device was used to measure the gap between itself and a small aluminum
foil disc attached to the membrane surface. Its linearity was tested

and results can be found in appendix C.

The fixture used to adjust the median gap width. hold and

38

position the displacement sensor is also shown in plates 1. 2 and 8.
It consists of a 3/4 in. plywood beam. 4.45 cm by 48.3 cm. upon which
is mounted an aluminum yoke. The sensor height is adjusted by the two
large nuts which secure the sensor to the yoke. The typical set up
method was to adjust the gap width prior to use so that a voltage
reading of approximately half its range (see appendix C) is obtained
and then set the EP analyzer to indicate any overloads out of the

sensor's range.

 

_.a_t.eP1 LWWMLM! _¢_t_-._9_r interim—f are

3.2 Procedure

3.2.1 Membrane modal data

Twenty points on the membrane surface were identified for

measurement as shown in figure 11 and plate 4. A corner point was

selected as the output. an aluminum foil disc attached and the

39

magnetic sensor positioned above it. This point was chosen because it
does not lie on a node of any mode measured. Measurements were then
taken and transfer functions obtained upon excitation of the other 19

points with the force hammer input. Four measurements were taken at

each point and averaged as in typical impact testing procedures[8].

y 20.3 cm

I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Aluminum foil disc 7.38
/
{15 i l l l 2.7...
x——_._.
I l l I I 1):
X X X X X
I | l I l l
x x x--—-x-—-x
I l I I I 16.4 cm
I I I I I I
>'< x x x /x .
I//I ,I
V .
Striking points
MLMMMQWM

3.2.2 Cavity modal data

To measure the modal activity of the cavity the force hammer is
again used as the input. The hammer was struck on the membrane at a

corner point to input sufficient energy in all the modes of interest.

40

The micrOphoue (output) was moved from point to point in the cavity to
obtain transfer functions for each data point. Again four

measurements were averaged at each location.

The output from the microphone was taken in only two planes - one
vertical and one horizontal. Figure 12 identifies the data point
locations. These data planes were chosen to avoid pressure nodes in
the cavity and to adequately compare experimental mode shapes to
analytical functions. Advantage was taken of the symmetry of the
expected cosine functions in the horizontal plane by examining only

half of this plane.

41

 

 

 

 

 

i

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I 2.54 cm
0 o r
I
I
o J-5
/
/
/
w T——.G— -——-—C
/
// . T
/ 8.18 cm 8.18 4
”.99 cm
I

my; 1;. Cavity data aggigition 121.33.!

3.2.3 Uncoupled membrane frequencies

\

42

Besides determining the modal results for the
membrane-cavity system a second test was performed in an attempt to
measure the uncoupled membrane natural frequencies. Plate 9 shows the
cavity restricted by a sectionalized grid. constructed of cardboard.
and inserted into the cavity. This was used to eliminate modal
activity or coupling within the cavity. The cavity bottom was left
open and the membrane frequencies were then measured as described

above.

 

2.1.12 2.. Lu“ ._t_'1 1: 2:393:12:

43

Chapter 4 - Results

4.1 Rectangular Membrane-Cavity System

4.1.1 System frequencies

Table 1 compares the experimental frequency results of the
membrane-cavity system with those predicted from equation (11). the
strongest system resonances. and (A15). the natural cavity resonances.
Column 1 orders the modes according to frequency (low to high).
excluding modes 8 and 9. An asterisk indicates that mode is a natural
cavity resonance. Two alternative predictions are presented for mode
7. Column 2 gives the values for indices :3 for the couplgd
resonances and indices lmn for the natural cavity resonances. For all
system resonances in the range of this study. 1 - 0. Column 3
contains values for the coupling coefficientleq. (4)]. 8-3, for
mode 7. Columns 4 and 5 give values of “mu and “fir[°q' (A8)?ru.°d in
the prediction of system resonant frequenciesleq. (11)]. an, in
column 6. The suggested range of error in these predictions is based
upon the estimated ranges of p., p. and C.. Column 7 presents the
experimental frequencies for each mode and column 8 contains the

percent error between the predictions of column 6 and the experimental

results of column 7.

44

 

 

 

 

 

1 I 2 3 4 5 6 7 8
Mode k1)=: Ba? qnn(hz) uqr(hz) Un(hz) Un(hz) Error(5)
[exp.] [pred.] [pred.] [exp.]

00

1 11 __ 0 6371673 596_+59. 575 :31. 3.9
01

2 12 __ 906 943 " 735 'I 769 ” 2.3

. 001

3 _ __ 906 __ 869 '1 906 0 4.5
10

4 21 __ 1106 1074 " 923 " 927 " 1.1

. 010

5 __ __ 1106 __ 1031 ” 1106 ” 2.4
11

6 22 __ 1281 1274 " 111s " 1151 " 2.0
02

7 13 0.973 1316“ 1299 ~ 1220 ” M43 " 1.7

7. 93 0.530 0 1299 ” 1323 " 1243 " 6.3

3' 11° _ 964 _ 939 u I964 ,, 3.2

9' °_1_1 __ 1281 _ 1280” [1231 ” 0.01

 

 

 

 

 

 

 

 

' Natural cavity resonances

“ A predicted value of u..,

Predicted values of “n based on:
9_- 8.93:10"k¢/c-‘t47. 0.: 3.51x1o‘c-Is 1:17.
p.- 1.15x10-‘kg/cm't571 1‘ - 9.45 kg/cm £6.49.
experimental cavity resonances. Tun

1.0.9.12 .1... 99.11.221.229 .o_.f _2____ox ori-ontal 311.9 L__91__clr°di o 12.7.4.2! magnum

45

Figure 13 is a graphical presentation of. equation (11). The
right and left sides of this equation. i.e. the cavity and membrane
impedence are plotted versus the input frequency a for each cavity and
membrane mode. Along side each curve the mode index lmn or qr. is
given. Intersecting lines indicate possible resonant conditions with
a non zero value for the coupling coefficient an~ The experimental
resonant frequencies are indicated on the :zequency axis. Four

possible resonant conditions are indicated for mode 7 where membrane

mode qr - 13 could couple to more than one cavity mode.

46

 

0.0976—
0.0732—
O.O488--I
2mm
+ [—§§-1
Cl 8
qu
0.0244_.
0 ‘I
-0.0244 —.

 

 

allexp.]

7.“

wzlexp.]
I

I I
l l
l 1

wglexp.]
I

1"12010

1.1"

 

.OL.

SOJO 60(I0

w(rad)

Laurel-Lula 21W“ III-ran WWW

 

 

47

These results show a very good correspondence with theory. All
the modes except 7' were within 5% of the predicted values which is
quite good considering the possibilities of error in the material
property values and membrane tension. 0n the other hand the natural
cavity resonances exhibit more error than is expected - 2 to 5 percent
for three out of the four cavity resonances. The predicted natural
cavity resonances depend only on the velocity of sound in air and
cavity dimensions both accurate within 1%. Because of this

discrepancy the experimental. not the predicted. cavity resonances.

an. were used in predicting the coupled system resonances.

4.1.2 Membrane mode shapes

Two cross sections. one in the y direction and one in the z
direction. of the membrane mode shapes of system resonances 2.4 and 6
are dhown in figure 14. In this figure the experimental shapes are
compared to sine shape predictions. All x. values have been
normalized to the maximum displacement for that mode. Good agreement

can be seen here between experimental results and predictions.

The experimental membrane mode shapes for all seven resonances
are shown in figures 15 through 21. In each figure the mode number.
frequency. damping and coordinate axis is given. The intersection of
any two lines represents a data acquisition point. Modes 1.2.4.6 and
7 are the coupled system resonances and good correspondance can be

seen between these shapes and the expected sine shapes. The unusual

48

shapes of modes 3 and 5. which occur at the first and third cavity
reionances. can not be predicted within the analysis of this study but
will be useful for further study. The membrane mode that occurred at
the second cavity resonance of 964 hz. mode 8 of table 1. was not
examined because of its small amplitude and close proximity to mode 4
of 927 hz. Its modal amplitude data was usually obscured by its

stronger neighbor.

49

8 Theoretical shape * Normalized
0 = Experimental data displacement

y = 8.18cm ‘

  
  

20.3 cm

 

 

 

16.4 cm

 

 

Mode 2

y = 8.18cm

 

 

 

 

 

 

y = 8.18cm

 

 

 

   

Mode 6

Etggyg ;1y Comparison gt experimental mgmtrane agggy t9 tgtlyyig

 

50

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In: In:
1 2
mew mew
573.3: 707.27
0mm aura)
1.33 maze-L
I 3 I 8
Pin" 15 Figure 16
II! n
' 4
mm
022.02
mm
mus
l s

 

 

 

 

 

 

 

 

 

 

 

Figure 17 Figure 18

51

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 19

 

Figure 20

 

 

 

\l

FREDO-IZ)
l. 24

MC)
1..

I

 

I s

 

 

 

 

 

Figure 21

 

 

52

4.1.3 Cavity mode shapes

Figure 22 shows a comparison of three system cavity shapes with
predictions (equation (5)). In this figure the normalized pressure
amplitudes in the x direction for nodes 1 and 2 and the y and z
directions for mode 6 are compared to predicted shapes. Close
agreement can be seen with the expected cosine shape of mode 1. the

cosh shape of mode 2 and the cosine shapes of mode 6.

Figures 23 through 29 show the experimental pressure mode shapes
for all modes 1 through 7 in the horizontal plane only. which was
located at x - 12.1 cm. Mode. frequency. damping and coordinate axis
are given on each figure. The front and back lines correspond to data
points that are along the wall boundry. The extreme left line
contains data points approximately an inch from the wall and the
extreme right line is at the center of the cavity. The pressure
magnitude is indicated by the height or deviation of one set of lines
from the zero amplitude reference lines indicated in the figures. All
figures were nonmalized to the maximum amplitude measured for that
node. Figures 23 through 28 show very good agreement with the

expected horizontal cosine functions.

The mode shapes in the vertical plane. located at z - 6.99 cm.
are shown in figures 30 through 36. The front and back lines are at
the cavity walls and the top and bottom lines are approximately an

inch from the top (membrane surface) and bottom respectively. The

53

zero amplitude lines are indicated and the second set of lines
represents the pressure magnitude. Measurement points for the
vertical plane modes are indicated by the intersection of any two
lines. For the accurate dimensions of all data points. both vertical
and horizontal. refer to figure 12. These figures show good agreement

with the expected vertical cosine or cosh functions.

Figure 29 is difficult to interpret. Figure 36. the
corresponding vertical pressure mode. indicates this data is very near
a nodal plane. This data is further obscured by the close proximity
of cavity mode 011 (0.1;- 1281 hr). The close proximity of several
possible resonances as shown in figure 13 suggest that it is a
combination of modes. Any coupling of membrane mode qr - 13 to cavity
modes lmn 8 000.002.020.022 will yield a frequency near this resonance
with the largest coupling coefficient occuring with lmn - 002 followed
by lmn - 000. The experimental frequency lies closer to that
predicted by coupling to lmn - 002 than to 1mm - 000 but figure 37
clearly shows a cosine for the functional shape in the vertical plane

which can only occur if ”lmn is less than o This happens only with

qr'
cavity mode lmn - 000 ( ”lmn. 0). The two above combinations of
coupling (lmn - 002 and 000) are the most likely to occur. This mode

will have to be studied more closely.

54

____. Ideal shape - cosine or cosh
0 = Experimental data
’ = Normalized amplitude

  

 

 

 

 

 

 

 

l . Mode l - vertical(cosine)
y = 16.36cm
P. z 8 6.99cm
18.8 cm 0
.7 . I I 3‘ ‘
-1 qr,
2 ., 0 Mode 2 - vertical(cosh)
P‘ y - 8.18cm
z - 6.99cm
1 4h
. I I I I I . I!
18.8 cm 9.4 cm 0
Mode 6 - horizontal(cosine)
1 _ ‘ ‘ 12.1cm
P.
y = 0.0
I I I l
0.1 cm
-1

 

 

 

myrsmnmgmmmmmmm

55

 

 

 

 

p-s

FREDOIZ)
575.3

WC)
1."

 

 

[s

 

 

 

 

 

Figure 23

 

 

HEDGE)
THIRD

mm
man .J

 

Ts

 

 

 

 

 

 

 

 

MOD
mass

WC)
34...

Figure 24

 

 

Ls

 

 

 

 

 

 

HERO-I2)
828.85

UMP“)
54.431

 

is

 

 

 

 

 

Figure 25

Figure 26

56

 

 

 

 

 

FREDCHZ)
1.1: sq

DAMPCZ)
240.44

 

Fs

 

 

 

 

Figure 27

 

 

 

FREDCHZ)
1.24 KI

mm
20.7: q

 

 

 

 

 

 

 

 

1.18

mm
102.9 .I

MI

 

is

 

 

 

 

Figure 28

 

 

Figure 29

57

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IIEDGRD,
IIL44

UWPOD
44LIB

 

Ts

 

\égfi

 

 

 

Figure 31

 

 

 

I
”% "5
K/ ":1;
V,» ? I:-
/

 

 

 

 

Figure 32

 

[IMPOD
renew-u

 

I—s

 

 

 

 

 

 

Figure 33

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

iiiiiiiiiiiiii

 

 

 

 

 

 

 

 

 

 

 

 

59

4.2 Amplitude and displacement to pressure ratio comparisons

For further verification of the validity of equation (5)
comparisons are made of membrane displacement to cavity pressure ratio
of each mode. Thble 2 contains amplitude comparisons of modes 1
thourgh 7. Column 1 contains the mode number followed by its
1880°1It°d 3050 1340!: a? or lmn. in column 2. Columns 3 and 4 give
the maximum values of cavity pressure and membrane displacement for
each mode in terms of the transfer function as only relative
amplitudes are needed. The ratio of maximum displacement to maximum
pressure is given in column 5 and this can be compared to the

calculated value of the inverse cavity impedance. 1/Zin, in column 6.

60

 

 

 

 

l 2 3 4 l 5 6
Mode (l)m¥ Maximum pressure‘ Max. Mem. displ.‘ is 17‘ 1
q (transfer function) (transfer function) ress. -;:
2m
001 ‘
5 __ 0.686 0.158 0.23 0
010
3 .__ 0.683 0.409 0.60 0
ll
6 22 1.00 0.888 0.89 0.79
01
2 12 0.505 0.403 0.80 0.91
10
4 21 0.667 0.528 0.79 1.08
02
7 13 0.683 0.772 1.13 1.92
22
7 13 0.683 0.772 1.13 3.12
00
l 11 0.218 0.475 2.2 3.57

 

 

 

 

 

 

 

MLMWMWMW
‘ all amplitudes normalized to largest pressure value

‘9 no units have been calculated

The membrane displacement and pressure amplitude comparisons show
general agreement with predictions. The measured
displacement-pressure ratios for each mode 1 - 7 clearly follows the
expected trend with inverse cavity impedance 1/Z-n. lggnitudes do not
match exactly but the general trend is shown with the largest value of

the both the displacement-pressure ratio and inverse cavity impedance

61

occurring in mode 1 and the smallest ratio with the smallest inverse

cavity impedance occurring in modes 3 and 5.

4.3 Membrane with cavity coupling restricted

The results of the membrane frequency response with cavity
coupling restricted are shown in table 3. The mode number and
associated index. qr. are given in columns 1 and 2. The experimental
7313. Of ”n' the coupled system resonance. is shown in column 3.
Columns 4.5 and 6 present the experimental frequencies resulting from
varying degrees of restriction. i.e. cavity partly or wholely
compartmented. These results can be compared to the frequency
predictions for the uncoupled membrane. column 7 (based on the
experimental values of tension and pII derived in appendices D and B).
and those of the coupled membrane. column 3. Percent error is given
between the experimental frequencies of the fully restricted case and

those predicted for the uncoupled membrane.

62

 

 

 

 

1 2 3 4 5 6 7 8
Experimental 09; (hz)
Mode qr [22p ] Half Tho thirds Fully [‘9‘ O.
' Restricted Restricted Restricted pred] Error

1 11 573 604 602 601 637 5.7
2 12 767 780 848 855 943 9.3
4 21 923 925 983 986 1074 7.9
6 22 1150 1147 1177 1188 1274 6.8
7 13 1240 1220 1235 1249 1299 4.4

 

 

 

 

 

 

 

 

 

 

mnmmmmn m-rnowon 2929.93.17.99;
‘ '5 error based on 0.1, of fully restricted cavity

and predictions (columns 6 and 7)

eliminate cavity-membrane

These results show the attempt to

coupling was not completely effective. The coupling effect is

demonstrated by the approach of the membrane frequencies to those of

the coupled cavity as the cavity restriction is removed. The results

of the fully restricted case show closest agreement with predictions

of the uncoupled membrane and verify that these predictions are at

least reasonable. Total elimination of membrane-cavity coupling

proved to be impossible.

63

Chapter 5 - Conclusion

These results showed excellent agreuent with the analysis. Both
cavity pressure and membrane displacement mode shapes matched well
with the expected sine. cosine and cosh functions of theory. The
experimental system frequencies when compared to predictions were well
within the expected experimental error. Pressure/displacement ratios

also matched predicttions. lending further support to the analysis.

The pressure shape of mode 7 appeared to be a combination of more
than one system mode. Closer examination of this mode would more

precisely reveal its causes.

Modal analysis by impact testing proved to be a useful approach
to examining this syst-s behavior. The membrane is a useful
substitute for the panel in such a system. Iith improved control of
membrane tension and assymetries it will be a very versatile

experimental tool.

64

REFERENCES

.P.M. Morse. Vibration and Sound (New York-Toronto-London:

McGraw-Eill. 1948).

Leonard Meiroviteh. Analytical Methods in Vibrations (London:
Collier-MacMillan Limited. 1967)

T. Rihlman. "Sound Radiation into a Rectangular Room. Applications
to Airborne Sound Transmission in Buildings." Acoustica.
18(1967). 11-20.

M.C. Bhattacharya and M.J. Crocker. "Forced Vibration of a

Panel and Radiation of Sound into a Room." Acoustica.
22(1969/1970). 275-294.

M.C. Bhattacharya. R.'. Guy and M.J. Crocker. "Coincidence effect
with Soundwaves in a Finite Plate." Journal of Sound and Vibration.
18(1971). 157-169.

R.‘. Guy and M.C. Bhattacharya. "The Transmission of sound
Through a cavity-backed finite plate." Journal of Sound and
Vibration. 27(1973). 207-223.

Compressed Air and Gas Data. 2nd ed.. ed. C.I. Gibbs.
(Phillipsburg: Ingersoll-Rand Co.. 1969)

M. Richardson. "Modal Analysis Using Digital Test Systems."
Hewlett-Packard Co.. Santa Clara. California. Reprinted from
Seminar on Understanding Digital Control and Analysis in
Vibration Test System.

Shaums Outline of Theory and Problems of Acoustics.

Iillimm I. Seto. (New york-Toronto-London: McGraw-Rill.1971)

65
APPENDIX A - Supplimental theory

This appendix presents additional background theory for the
readers convenience. Section A.1 contains the derivation of the
transposed membrane velocity. V. Section A.2 is a description of a
freely vibrating circular membrane. The equation used in evaluating
membrane tension is derived in section A.3. Sections A.4 and A.5

describe the vibrations of a rectangular membrane and an acoustical

cavity. Bending stiffness estimations are presented in section A.6.

A.1 Membrane velocity. V

The general procedure used by Guy and Bhattacharyal6] to solve
for the 4th order displacement of a panel is applied to the 2nd order

membrane. The equation describing the balance of forces is

a.
T Q"(Ye1st) - P.‘;:3 '(Yszst) . FIy.2.t) 0 (A1)

Rere F(y.z.t) is the pressure difference between the two sides of the

mubrane . i. e

FIy.z.t) . P,(y.z.t) - P1(y.2.t) e

The LaPlace transform of (A1) becomes

T Q’;(y.z.s) - s’pl; - F,(y.z.s) - P.(y.z.s) . (A2)

Multiplying both sides of (A2) by

sin(quy/b)sin(rnz/c)

and Fourier sine transforming yields

3(m;r+ ‘3) = (bc)/(4pm) (qur(s) - P3qr(s)) (A3)

where

0‘], - .(r/p,>"’[(q/6)‘+ (.IcI’J‘“ . (A4)

Transposing (A3) and inverse Fourier sine transforming gives

 

_ P ' P
w - 2f ._1S§:‘) :9r(') sin(quy/b)siu(ruz/c) . (A5)
493.1 P-(U qr+ 8 )

The membrane velocity. V. may be written as awldt so that

V - s; .

Substituting for ; in (A5) and fourier cosine transforming yields

-<H

b
' 0 2° B-n ' a3 a [qur(s)- P,qr(')] 0 (A6)
4P. gs 3‘1 qt (3 'I' ”qr) "

This is the general solution for the membrane velocity in terms of its

LaPlace and Fourier cosine transform for a forcing function of any

time dependence.

67

A.2 Circular Membrane

 

 

MLWW

Figure A1 represents a circular membrane with a fixed boundry.

The solution[2] for the displacement function. I. is given by

"2;:

Aijj(cjkr)sin(jn) + BJij(cjkr)cos(jD) (A7)

where

61kt.- Ujkr./cn e (As)

68
This describes the usual Bessel functions. Jj, in the rgdigl dirgction
with the corresponding sine or cosine functions in the circumferential

direction. The frequency for each mode. ”qr' 1; found from eqngtion

(A8).

A.3 Circular membrane - tension measurement

3‘ I
T\:

 

 

 

 

 

MLMMQihmnmg

69

  

AY

 

  

H_——————-—

1":
1.m

B
H?
O

mmglumm-mm

Figures A2 and A3 depict the circular membrane with a known

8t1t1° point 10": Pp. at its center and under an unknown tension. T.

A balance of forces implies

Fp I Tr sinO Judo
e

0!

Fp - ZurT sine . (A9)

For small angles of 0 (applicable away from point of application of

force Fp)

sine ‘3 dY/dr (A10)

or on substitution of (A10) into (A9)

70

dY ‘ Fp/(2urT) dr .

Integrating from rm to r. :

. m
AY ,.It Fp/(2urT) dr

- Fp/(ZuT) ln(r./r-) .

Finally. solving for T:

T ' Fp/(AYZu) ln(r./r.) . (All)

This equation can now be used to evaluate tension upon substitution of

known values F. AY. r. and r..

71

A.4 Rectangular Membrane

’X

 

Eiggyg 51y Rggtangular membrang

Figure A4 depicts a freely vibrating retangular membrane with
fixed boundries of length b and c. The well known solution[2] to this

problem in terms of its mode shapes and natural frequencies is

I(y.z) a 2:,r-1'qr - Ez'r.:!qrsin(uqylb) sin(urz/c) (A12)

where

uqr- (C.u)l(bc) (b’q’+ c’r’)"’ . (A13)

These equations describe the standard sine series solution for a fixed

boundary rectangular membrane.

72

A.5 Rectangular cavity - frequency and mode shapes

 

 

5F ______ _._ -.._

 

 

 

WLWM

A rectangular cavity with all ’walls acoustically hard (no
membrane) is represented in figure A5. The standing wave solutionl9]

in terms of pressure amplitude mode shapes and frequencies is

P . EI'.’n..D1.mcos(lux/a) cos(muy/b) cos(nuz/c) (A14)

where

mm- c.[(la/a)’+(nnlb)’+(nu/c)’]‘/' . (415)

This is a series solution of cosine functions in the x.y.z directions

with zero pressure lepe (zero velocity) boundary conditions.

73
A.6 Estimation of Bending Stiffness-Rectangular Membrane

To examine the possible contributions of bending to a finite size
membrane. the differential equations of motion for two systems are
considered. They consist of a membrane under tension T and a plate of

thickness h. For the membrane we have

-TD’[w] = '9. a’lat' w . (A16)

For the plate

no‘lvl = ‘Pph a’lat’ w . (A17)

The relative contribution of bending stiffness to frequency
changes can be found by comparing the left hand sides of equations
(A16) and (A17) since they contain the plate stiffness and tension
terms. In addition only the spatial part of w(y.z.t). i.e. I(y.z).
need be considered as the left hand sides of equations (A16) and (A17)

involve only the spatial derivitives.

The modes shapes for for both the plate and membrane are

described by

'qr' qusin(qny/b) sin(rnz/c)

The maximum contribution of a’ldya or d’ldza of 'qr to equations (A16)

and (A17) occur at y = b/2 and z - c/2 and this point will be

74

evaluated to compare the right sides of (A16) and (A17).

For the first mode. q=l and rsl. the displacement function

becomes

W118 Y11siu(uy/b) sin(uz/c)

and the right sides of (A16) and (A17) become

-r(a’/ay’+ a’laz’)l;:(b/2.c/2) - 12,,a’(1/b‘+ 1/c‘) (A18)
b c l 1 l
0(a’lay'+ a’laz‘)’w,,(;;;) - n21,n‘[;;+ Egz, +';3] . (A19)

Using 1 typical value of E (55.000 lb/in' for polyethelyne).
assuming u - 0.5 and employing measured values of h. b and c these two
terms were evaluated for several modes. The ratio of bending
stiffness energy. (A19). to tension energy. (A18). never reached above
0.1% and for the first few modes typical values. are around 0.01%.
Bending stiffness need not be considered as an influence on frequency

results.

75

APPENDIX B - Preliminary circular membrane experiments

Preliminary experiments were conducted with the circular membrane
to verify proper functioning of the RP 5243A Analyzer and transducers.
check the tension capacity of the fixture. insure that membrane
resonances would overlap cavity resonances and eliminate membrane
assymetries due to dimensional instability and tension. In the
process several mode shapes were examined and compared with
predictions. This appendix begins with a brief review of modal
analysis. In section 8.2 the tension adjustment procedure is
discussed and section 8.3 presents modal results for the circular

membrane .

B. 1 Model analysi s

To measure the mode shapes and frequencies of a vibrating body;
transducers are employed to convert physical vibration (surface
motion. acoustic pressure. etc.) into electrical signals. Commonly
found transducers include accelerometers. magnetic transducers. and
microphones. Once the structure is excited (e.g. by striking with a
"hammer". using a mechanical vibrator. etc.) the signal from one of
these devices placed at some point of a vibrating body can analyzed
for frequency and amplitude. This signal added to others taken from
many locations on the structure can then be analyzed to show the

complete vibratory shape.

76

Since most analyzers have only two inputs all points on the body
can not be measured similtaniously. Tb avoid discrepencies between
repeated measurements one input is maintained as a reference by which
all other signals are compared. i.e. the signals from all structural
points are divided by a reference signal taken from one location.
These modified outputs are called transfer functions. One example of
such a method uses an "impulse" hammer to excite the structure by
striking it. The signal from this hammer is the input and the output
is taken from some other point on the structure. lith the hammer
repeatedly struck at the same location while outputs are measured at
all other locations. the hammer becomes the reference signal.
Conversely the hammer could be moved from point to point and the
output from one location used as the reference signal. Of course once
transfer functions are obtained for all structural points they can be

processed for modal information.

8.2 Tension adjustment

Preliminary experiments were necessary to check for tension
asymetries and sufficiently high tension for the rectangular
membrane-cavity system measurements. Tension asymetries were
identified by the doubling or splitting of each mode. Tension

magnitude was evaluated by derivation from the frequency results.

When tension was increased to sufficiently high values asymetry

problems became quite pronounced. Tests revealed a set of split modes

77

and the membrane had to be retensioned several times until the split
modes were eliminated. This was further compounded by some
dimensional instability in the frame and difficulty in turning the

tensioning bolts.

B.3 Modal results

A typical arrangement of the membrane. frame and transducers used
to measure circular mode shapes is shown in figure Bl. The impulse
hammer was used to excite the membrane at enough locations to verify
mode shapes and the output was measured with the magnetic transducer.
The bottom of the frame was left open to lessen the effect of cavity-

membrane coupling.

78

Aluminum foil

disc
Membrane \ Membrane
fixture \
\\
\ I Input to
E1 1, [1 analyzer

 

 

\r. r

'\\\Displacement
L. sensor

 

 

 

 

 

 

 

SW \\\\\\ £W

Output to analyzer

 

 

[ism L11 9.11.125.- 9_f Mm n 5___rotn £23. __9.sl_e£cir 2.93.12.21.02

Table Bl shows the experimental frequency results of the circular
membrane compared to predicted values. Also shown are the expected
mode shapes with corresponding values of Br., All the .xpori-ont.1
values are seen to be lower than those predicted. Measured mode
shapes are shown in figure 82 where the signs indicate relative
direction of amplitudes and dashed lines indicate the position of

nodes.

79

 

 

 

 

 

 

 

r 2 3 4 5 6 7
Mode mn Bro Shape mnn(hz) -pred. mnn(hz) - exp. i
[T = 9.45 t3] Error
cm
1 01 2 4 ® 357 231 35
2 ll 3 8 <§§E> 569 458 20
3 21 5 1 ® 758 665 12
4 02 5.5 817 748 8.4
5 31 6 3 ® 949 857 9.7
6 12 7 0 @ 1039 965 7.1

 

 

 

 

mnmgmnlrm- 011.993.9991”

m.n=0.1m..n=11m.n=2.1 m,,n-02 m.n=31
Mode 1 Mode 2 Mode 3 Mode 4 Mode 5
mmmrmmmngwmmw

Signs indicate amplitude direction and dashes are nodal lines.

 

80

Figure Bl shows significant error between predicted membrane
frequencies and experimental values. particularly in the lower modes.
It is suspected (as in the case of the rectangular cavity) that
membrane-cavity coupling is still significant. even though the bottom
of the fixture was unobstructed. The predicted fundamental frequency
of the cavity. 660 hz. is close enough to a cavity resonance to

produce this coupling.

Mode shape results confirm that these experimental frequencies
are correct. A comparison of the experimental shapes. figure 82. with
those of figure Bl show excellent agreement and the progression of

modes from low frequency to high follow the expected trend.

81

APPENDIX C - Transducer calibration

The magnetic transducer and microphone were calibrated to insure
they were of sufficient linearity. In addition precise knowledge of
the magnetic transducer calibration constant is necessary to make

absolute displacement measurements in determining membrane tension.

C.1 Magnetic inductance transducer

Figures C1 and plates Cl. C2 show the experimental set up used
for this calibration. The magnetic transducer was inserted into a
threaded fixture allowing easy rotation. The rotation of a protractor
attached to the transducer was easily converted to an accurate
measurement of the transducer's linear motion. As would be
encountered on the membrane. a circle of aluminum foil was attached to
the immobile. plastic surface below the transducer. Rotation of the
transducer changes the gap width which is measured simultaneously with

the voltage output.

82

Scale. mounted onto
retainer

Volts output

   
  

Indicator

threaded retainer- 5 -18

\ /

   
      

Treaded fixture
:/
6

Aluminum foil
disc

 

 

 

 

‘ Elam .91.. 21112.;- 91 mast—is _______transdncor _3___b_u_c 11 t ion

83

 

MQLQIMMIQWWMM

Figure CZ shows the calibration plot of the magnetic transducer.
The voltage output is plotted against the gap width which was
converted from degrees to centimeters using the conversion constant
shown. This transducer is very linear for most of its range and as
long as it is operated in the middle of its range no problems will

occur.

84

     

 

 

9 o = experimental points
—' —_.= Least sqares fit
8"
7..
6"!
Volts Slope = 53.6 v/cm
5 _ 10° = 3.92 x 10-3cn
4..
3..
2 _
1 a
I I l I

0 4.63 9.40 14.1 18.8

Gap width (cm 310*)

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C.2 Microphone

The microphone was calibrated using a GenRad calibrator. model
GR1986. This device inputs a known sound pressure level (spl).
usually measured in decibels. at selected frequencies to cover the
audio range. The voltage output was measured with the EP 5423A
analyzer set to perform an autospectrum analysis on a sinusoidal
signal. Twenty samples were averaged at each frequency and decibel

level.

85

Results are plotted in figure C3. Microphone output is shown at
three dynamic levels. 114 db. 104 db. and 94 db. The plot clearly
shows a constant output over most of the tested frequency range which
extends beyond the range of this study and a very linear drOp in

output from 114 db to 94 db. No problems are to be expected with the

use of this microphone.

 

 

o = 114 db re 20nPa
D_- 104 db
A = 94 db
30 - v V v
Volts
[db re 1mv] e .3 5
20-7
H
10 _

 

 

I I

' l 1 r
125 250 500 1000 2000 4000
Frequency (hz)

Figure 93‘ Zglttgg gytpgt gt migrophone versug trequency

86

APPENDIX D - Membrane tension and density measurement

Tension and density values for the membrane are necessary in
predicting the membrane natural frequencies (both rectangular and
circular). These frequencies are then used in equation (11) to
predict the system resonances and also in comparison with measurements
of the uncoupled membrane. This appendix discribes the tests made to

determine these two quantities. tension and density.

0.1 Tension measurement

Iith a suitably high amplitude and symmetric tension set
(inferred from the circular membrane experiments). the relationship
between applied force and membrane displacement, equation (All). was
used to determine membrane tension. As shown in figure 01 and plates
Dl. DZ the magnetic sensor. placed under the membrane. measures
displacement as known forces are applied at the center of the
membrane. A record of membrane displacement at radius r. versus force

is thus obtained.

87

App). ied weight

Weight holder

Holder snide Membrane

Foil discfi

i Voltage output

MLMOJL simmm

 

 

Plug! 21 m 224 Tension mgagurgmeng sgtn

Table 01 shows a chart of force versus voltage (displacement) at

three

radii. r

The

membrane was loaded in 100g increments up to 500g.

to determine

for each radius.

tension.

row 6. at each radius. r.. The average of the three values.

row 7. yielded a tension of 9.45 kg/cm.

voltage (displacement) was determined as the
This data was used

a linear plot with the slope given in row 5 in volts/kg

This slope is used in equation (A11) to find the

 

 

 

 

 

 

 

 

 

 

 

 

 

1 2 1 3 4
Force (kg) Voltage (V)
. r. 8 13.4 cm r, = 9.84 cm r. 8 8.73 cm
0 4.30 4.83 7.01
0.100 4.25 4.73 6.95
0.200 4.26 4.725 6.88
0.300 4.235 4.67 6.81
0.400 4.215 4.625 6.75
0.500 4.20 4.57 6.69
SIOpe (v/kg) 0.213 v/kg : 1.5% 0.534 v/kg 10.6% 0.661 v/kg 1_0.9%
T (kg/cm) 10.4 1 6.4% I 9.09 1 3.8% 8.88 i 3.5%
Tave (kg/cm) 9.45 1 7.5%
Tibia 211 8552111 £122 1521123 2322512131
This experimental value of tension. when used to predict
rectangular membrane natural frequencies. shows reasonably good
results. The comparison of predicted and experimental coupled system

 

89

frequencies. table 1. results in errors (approximately 2%) which are
less than the error introduced by the tension value and material
constants (approximately 5%). Table 3. which compares the results of
restricted cavity membrane frequencies to the predicted membrane
natural frequencies. at least confirms that the predicted tension is

within 10% of the correct value.

The percent error in this tension value results from variations
at each measurement location since the error introduced by the
measuring devices is insignificant. The assumptions used in deriving
the tension equation would also contribute insignificant error. This
variation in tension from measurement location to location could be
due to residual tension assymetries. If so. tension measurement may

be capable of eliminating these assymetries.

0.2 Density measurement

To determine the membrane density an unstretched section of
membrane material was accurately weighed and dimensioned. From these
measurements the density was found to be 8.93x10'5 kg/cm’ t 4%. The
suggested range of error is due unknown strain induced in the taut

membrane. other factors being negligible.

90
APPENDIX E - Estimates of air density and sound velocity

E.l Speed of sound

The transmission of sound in air is assumed to be an adiabatic

process. Thi' ilP1103 thlt C. is independent of pressure changesl7]
and depends on absolute temperature only. Thus the following relation

between the velocity of sound temperature:

C./c,= (TJTJ‘I' .

Using a known value of sound velocity. C“ at some tuporgtuo, I", .
new velocity. C,. was found at an estimated local temperature. T,. of
90 degrees farenhieht t 5 degrees. This equated to 3.51x10‘ cm/sec ‘t

1%.
E.2 Density

To evaluate the local atmospheric density the ideal gas law is

applied:'

The local values of atmospheric pressure and temperature along with

the gas constant for air are used to evaluate the density. p..

A typical value of 30 in Hg t 1 in Hg was chosen for the local

91

pressure. This value along with the above temperature yields a value

of 90 equal to 1.15:10“ kg/cm' 1: 55..

92

APPENDIX F - Sample calculations

F.l Tension

Equation (All).

T ‘ Fp/(AIZu) ln(r./r.)

is used to evaluate tension. T. The displacement. AI. is based on a

force. Fp. equal to 2.2 kg. At the measured radius

rm‘ 13.4 cm t.0.6%

the following values are then used:
t.- 17.4 cm t.0.5%

A! . (0.213 V/kg t 1.5%)(1/53.6 cm/V«t 0.7%)

Fp- 2.2 kg .

Thus we have

2.2 kg

 

T - 10.4 kg/cm t 6.4% .

Similarly at

f 1
(0.213 V/kg t 1.5%)(1/53.6 cm/V t 0.7%) Zn

13.4 cm t0.5%
n
[17.4 cm t0.6%]

 

93

r...“ 9.84 cm 4. 0.49.

T = 9.09 kg/cm t.3.8%.

and at

In: 8.73 cm t.0.9%

T = 8.88 kg/cm t.3.5% .

Averaging these three values of tension.

I‘m- 9.45 kg/cm 4. 7.55. .

F.2 Rectangular membrane frequencies. ”qr

The frequency equation for the rectangular membrane is

uq,- u<T/p_>"'[(q/b)'+ (:Ic)’J‘/’ .

The following membrane dimensions and properties are used:

 

 

 

b - 16.3 cm i: 0.45. 'r = 9.45 kg/cm t 1.59.
c - 20.3 cm 4. 0.45 pm- 8.93:10" kg/cm' i: 45 .
For qr - 11.
k on k
(9.45—“1:7.5su9 .805 , ")

cm s kgf 1,: 1

611- .[ _‘ ] [( >'+(
8.93x10 hz./6.3 141. 16.3 6.4.0.49.

0!

(A13)

 

i/a
)']
20.3 cmt0.4%

94

01¢ 4.002110’ rad t 6.1% = 637 hr

F.3 Cavity resonances. ”lmn

The frequency equation for the rectangular cavity is

”m' ac.[(1/.)‘+ (m/b)3+ (n/c)’1‘/’ . (A15)

The following cavity dimensions and properties are used:
a .. 18.8 cm 4». 0.41. c.- 3.51:10‘ cm/s t 11.
b s 16.3 cm t.0.4%
c I 20.3 cm.t.0.4% .

For lmn - 001.

 

1
- 3.51 10" t 11. ' "’
0.01 u( x cm/s)[(20.3 cm t.0.4%) ]

OI

u..1- 5419 rad t.1.4% or 863 hr .

17.4 llembrane-cavity system resonances. ”n

The equation predicting the system's strongest resonances is

s a
a pm (“ ' “qr,
oth[—-( 3 _ 3)1/3] -_ —( ‘ .. 3)"l3 . (11)
c C. “In 0 p.C. “I m U

The following system dimensions and properties are used:

Note:

and not on the analytical predictions.

For cavity mode 000 coupled to membrane mode

becomes

01'

s = 18.8 cm t.0.4%

‘D O
o
I I

‘D
E
II

”In:

the cavity resonances. 01.3. are based on experimental

 

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Solving this transcendental equation by iteration results in

”i

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£01=596hzt5%.

96

97

APPENDIX G - Supplimental theory

The following pages contain modal data copied directly from the
HP 5423A analyzer. "Measurement Data” shows a typical set up for the
HP 5423A analyzer For data taking. This is followed by a sample

"Frequency and Damping" table for modes 1 through 7.

Node shapes shapes were derived from the "modal' residues" given
next. e.g. cavity mode 1 in the vertical plane and membrane mode 4.
Finally two sample transfer funtions are given: first from a cavity

mode measurement and then from a membrane mode measurement.

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