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This is to certify that the

thesis entitled

DEVELOPMENT OF AN APPROXIMATE ACOUSTIC POINT
SOURCE FOR ACOUSTICS TESTING 1

presented by

STEPHEN J. CONNOLLY

has been accepted towards fulﬁllment
of the requirements for

 

 

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—~

THE DEVELOPMENT OF AN APPROXIMATE ACOUSTIC POINT SOURCE
FOR ACOUSTICS TESTING

By

Stephen John Connolly

A THESIS

Submitted to
Michigan State University
in partial fulﬁllment of thc requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1993

ABSTRACT

THE DEVELOPMENT OF AN APPROXIMATE ACOUSTIC POINT SOURCE
FOR ACOUSTICS TESTING

By

Stephen John Connolly

The point source is a powerful modeling tool in many scientiﬁc and engineering ﬁelds. In
acoustics, the point source is an inﬁnitesimal object that displaces the surrounding ﬂuid
equally in all directions. The point source is a powerful modeling tool because in linear
systems analysis, which includes acoustics, the principle of superposition allows complex
sources of excitation to be modeled by a collection of simple sources, or point sources.
The point source can also be a powerful tool in experimentation, and models which use
simple sources can be more easily veriﬁed. A physical approximation to a point source for
experimental work is developed here. The acoustics theory dealing with point sources is
discussed, and acoustic intensity is introduced. The acoustic exciter is presented, and the
physics of the exciter are compared with the physics of the point source. An analytical
model is developed for design optimization and source calibration. The experimental
techniques used in the model veriﬁcation and parameter identiﬁcation are presented.
Sound intensity comparisons with conventional speakers clearly show that the acoustic

exciter is an excellent approximation to the point source.

ii

TABLE OF CONTENTS
List of Tables ............................................................................................................... iii
List Of Figures ............................................................................................................ iv
Introduction ................................................................................................................ 1
The Ideal Point Source, an Acoustic Monopole ........................................................... 2
The Acoustic Exciter ................................................................................................... 4
Acoustic Intensity Techniques ..................................................................................... 5
Modeling the Acoustic Exciter ..................................................................................... 6
Identiﬁcation of Model Parameters .............................................................................. 8
Experimental Results ................................................................................................... 10
Conclusions ................................................................................................................. 14
References ................................................................................................................... 16

Appendix A - Derivation of acoustic exciter system equations ...................................... l7

iii

LIST OF TABLES

W

Table 1. List of model parameters for a pair of 4" woofers and a 0.5" spacer

Page

11

iv

LIST OF FIGURES

M Page

 

Figure 1 The Radially Oscillating Sphere; an Ideal Point Volume Source Model 2

Figure 2 Intensity Field of a Loudspeaker Compared with an Acoustic Monopole 3

Figure 3a Acoustic Exciter and Intensity Measurement Guide (Photograph) 4
Figure 3b Sketch of Acoustic Exciter 5
Figure 4 Speaker Cones Moving to Imitate the Radially Oscillating Sphere 5
Figure 5 Schematic of Acoustic Exciter System Model 7

Figure 6 Measured, Analytical Volume Velocity Frequency Response Function 10
Figure 7 Radial Intensity Components (800 Hz) for Box Speaker 12
Figure 8 Mean, Standard Deviation of Intensity Magnitudes for Box Speaker 12
Figure 9 Wire Frame Animation of Intensity Vector "Displacements" at 800Hz 13
Figure 10 Vertical Projection of Data Dcpicted by Figure 9 13

Figure 11 Mean, Standard Deviation of Intensity Magnitudes for Acoustic Exciter 14

Page 1

INTRODUCTION

An omnidirectional acoustic source is needed for experimental veriﬁcation of
acoustic models and principles. The ideal point source is a basic building block in
acoustics analysis and modeling because any source of sound can be modeled as a set of
ideal point sources. Point sources are also used in acoustic ﬁnite element models, where
the driving term at an interior acoustic node is either zero or the volume acceleration of a
point volume source at that node. Similarly, the principle of acoustic reciprocity is deﬁned
in terms of ideal point volume sources, and this is the primary motivation for this work
(Fahy, 1992). An ideal point source cannot be physically realized, because it has zero
physical extent. The development of a physical approximation to the ideal point source is
the subject of the paper. The source must have ﬁnite physical size, but can approximate a

point source in the far ﬁeld by having an omnidirectional intensity pattern there.

The design problem is the development of an acouStic source that approximates an
ideal point source in the far ﬁeld, has a known volume velocity output function, and is
easily constructed from ordinary components. The volume velocity output function must
be known because the volume velocity of a point source is not an easily measured
quantity. In the mechanical domain, excitation is provided by a shaker, and that excitation
is measured with a force transducer. In acoustics, there is no such thing as an "acoustic
force transducer," so the output of the source must be inferred from some easily measured
input quantity. A candidate source must be constructed, modeled, and tested to indicate
the performance of the design. The design process involves exercises in system modeling,

acoustic intensity measurement and analysis, as well as parameter identiﬁcation.

This paper demonstrates the methods and principles used during the development
of the acoustic exciter.’ The ideal point source is examined ﬁrst, and then the acoustic

exciter is shown to be a good approximation to the point source. The principles and

 

Page 2

methods used in the development of the system model are presented. The experimental
techniques used and the methods used to process the data are also presented in detail.

THE IDEAL POINT SOURCE, AN ACOUSTIC MONOPOLE

The ideal acoustic monopole emits acoustic power equally in all directions from an
inﬁnitesimally small point in space. The ideal point volume source can in theory be
realized by a very small sphere with a periodically changing radius (Fig. 1). This model of
a point source is valid only outside the region of the sphere, and is an exact model there.

Sphere of Radius
nay-3am

R+5R.

 

Figure 1. The Radially Oscillating Sphere; an Ideal Point Volume Source Model

Two important concepts when describing an acoustic source are efficiency, and
symmetry. Efﬁciency describes how well the source converts energy into acoustic power
ﬂow, and symmetry deals with how that acoustic power flow is distributed in space. An
efﬁcient source is a volume source that does not have any surface motions that are "out of
phase." When surface motions are "out of phase," efﬁciency is reduced, and the acoustic

input to the system under test is also reduced.

The ideal point source, or acoustic monopole, is 100% efﬁcient and symmetric.
The source is 100% efﬁcient because every part of the surface of the radially oscillating

Page 3

sphere has the same velocity outward or inward, and so every part of the surface has "in
phase" motions. The source is 100% symmetric because there are surface velocities along

every imaginable axis through the center of the source.

The unenclosed speaker is not a volume source; as the cone moves back and forth,
the volume occupied by the speaker does not change. An unenclosed speaker has two
surfaces, the front and the back. As the front of the speaker cone moves "out," the back
of the speaker cone moves "in." (Fig. 2) This causes acoustic power to ﬂow entirely from
one surface to the other, and results in extremely low efﬁciency. The unenclosed speaker

is not symmetric because the resulting power flow ﬁeld is like that of a dipole.

Enclosed speakers are highly directional, and therefore unsuitable as point sources
(Fig. 2). While the efﬁciency of an enclosed speaker is high, the needed symmetry is not
present. The enclosed speaker has one emitting surface, and that surface occupies a small
percentage of the total source surface area. If the entire speaker and enclosure were an

emitting surface, with no "out of phase" cone motions, it would be suitable as a point

 

 

 

 

- H\\t// ———-' \\ f/
/: :::\\\I///"\\ Hedi/z
viii”, gr)" )4
_./ UV:
AoouetIcMonopoIe UnencIoeedLoudspedter EWLMMGI’

Figure 2. Intensity Field of a Loudspeaker Compared with an Acoustic Monopole

Page 4

THE ACOUSTIC EXCITER

The acoustic exciter must imitate the mechanics of a radially oscillating sphere. A
typical speaker cannot do this because of the high degree of directionality mentioned
above. Two speakers could do this if they were pointed in opposite directions, driven
such that the cones would move symmetrically, and the space between them sealed. If the
speakers are placed face to face, this would result in a large emission area percentage.
The acoustic exciter is just two identical speakers, bolted face to face on a spacer, and
driven in series (Fig. 3a, 3b). It was constructed using two Radio Shack 4" woofers, and a
1/2" aluminum spacer. The two speaker cones, moving symmetrically, approximate the
physics of a radially oscillating sphere (Fig. 4). As the two cones move away from one
another (Thick line), the net volume occupied by the acoustic exciter increases, and as
they move together (dashed line), the volume decreases. Because the percentage of
emitting area is near 100%, the acoustic exciter should be a good point source candidate.
The intensity ﬁeld around the source will be sampled so that the quality of the point source

approximation can be determined.

 

Figure 3a. Acoustic Exciter and Intensity Measurement Guide (Photograph)

Pages

 

Figure 3b Sketch of Acoustic Exciter

 

Figure 4. Speaker Cones Moving to Imitate the Radially Oscillating Sphere
ACOUSTIC INTENSITY TECHNIQUES

Acoustic intensity is a popular technique for determining acoustic energy
propagation patterns, because intensity is a measure of the source's input to the acoustic
ﬁeld surrounding it. Sound pressure is a measure of the acoustic ﬁeld's response to the
source's input, and depends on both the source input as well as the system under test. The

intensity ﬁeld around the source will be sampled at several locations on the surface of an

Page 6

imaginary cube, so that the radiation pattern can be determined. Volume velocity will

later be shown to be determined ﬁom these intensity measurements.

Acoustic intensity is measured using two phase calibrated microphones that are
separated by a small spacer. More information on intensity can be found in the acoustics
literature by Briiel and Kjar. Three orthogonal intensity component measurements must

be made at each point in order to obtain the full spatial intensity vector.
MODELING THE ACOUSTIC EXCITER

An analytical model for the acoustic exciter is needed for prediction of source
output, because the volume velocity of the source cannot be measured at test time. A
model would also be needed for future design optimization. Depending on the frequency
range of interest and the sound power levels needed, the correct choice of design variables
can be critical to the device's performance. The design variables are the speakers, and the
thickness of the spacer. The spacer thickness can be chosen arbitrarily, but speaker

parameters discussed in this section are a consequence of the choice of speakers.

The acoustic exciter is modeled as a two degree of freedom dynamic system,
where the degrees of ﬁ'eedom are the cone displacements. Each speaker individually is
essentially a one degree of freedom spring-mass-dashpot system driven by an electro-
magnetic coil (Radcliffe, Gogate 1992). The stiffness, damping, and mass parameters for
the cone are kcone Rcone and mcone respectively. The electrical resistance and
electromagnetic constant of the coil are Ram, and bl, respectively. When the two
speakers are placed together, and the air cavity between them is sealed, the air cavity acts
like a stiff spring, with stiffness kair (Fig. 5). The single input to the acoustic exciter is
the applied voltage, e(t). The outputs of the model are the cone displacement and

velocity, x(t), and v(t).

 

 

 

 

 

 

 

cone k cone
V R90” air no
m
H ‘ CE '\
\
kcone k cone
b' -- -- — bl
, l:‘coil F‘coil ,,

 

 

9(t)

 

Figure 5. Schematic of Acoustic Exciter System Model

The equations of motion of the speaker cones (2) are derived ﬁ'om a force balance
on each speaker cone, where the cones move horizontally with respect to gravity, as well
as a voltage sum around the current loop (Appendix A). The equation can be reduced to
one degree of freedom assuming that the cone motion is symmetric. This assumption is
valid if the speakers are exactly identical, or if the ﬁ'equency range of interest is low
enough that small amounts of asymmetry between the two speakers do not invalidate the
assumption of symmetric motion. Dissipative and stiffness terms have been collected such
that keg = kcone + 2km, and Reﬂzkconewlzlkcoﬂ. Reg is the total equivalent

mechanical viscous friction coefﬁcient

 

5‘ 0 l x O
[.]= -31. __R_g_ ]+ Em. bl (2)
V mm mm V 2 Rail-mm

Further assuming that only steady state solutions are needed, the Laplace
transformation is used to solve for cone velocity response V(s)=.£[v(t)] due to applied

voltage, B(s)=.£[e(t)] (3).

V(s) _ s-bl
E(s) 21?tm.,-(m‘m~sz-t-R.If ~s+kd)

 

(3)

Page 8

The desired solution should be for volume velocity, Q(s)=£[q(t)], so the cone
velocity solution is multiplied by the effective area of the convex surface of the backs of
the speaker cones (4).

Q(s) _ s-bl-AE
E(s) ' 2am.l (mm «and, .s+kd,)

(4)

Equation 4 is the system transfer function model for the acoustic exciter, and it will
be correlated to the measured transfer function for the purpose of identifying the model

parameters.

IDENTIFICATION OF MODEL PARAMETERS

Three parameter groups can be identiﬁed by ﬁtting the simpliﬁed model (5) to the
measured volume velocity response function (Fig. 6). The parameter groups, (6) can then

be used to solve for each parameter.

 

ks
T = 1
(s) s2+k2s+k3 (5)
C 2 O .

The seven model parameters must be determined to correlate the model to the
measured response. This can be achieved by curve ﬁtting the simpliﬁed model (5) to the
data in a least squares sense. Doing so will identify only three parameter groups, k1, k2,
k3, because (4) must be simpliﬁed so that there are the minimum number of independent

parameters.

Four other equations or parameters must be identiﬁed with other testing or
supplied information so that the identiﬁcation problem will have a unique solution.

Sometimes, the speaker manufacturer supplies many of the model parameters, but it is

Page 9

assumed here that only the free air resonance frequency, and the electrical resistance of the

coil are provided.

One equation results from the free air resonance constraint provided by the
manufacturer. The speaker manufacturer usually gives a free air resonance frequency for
the speaker. If this is not provided, it can be found by measuring the cone velocity
frequency response of a single speaker. Knowing the resonance frequency, f0, sets up a

needed constraint (7) on the solution of the identiﬁcation problem.
k
—°°"° = 21tf
J mm o (7)

Three parameters can be determined directly from lab testing. The three
parameters are Rmﬂ, kair’ and bl. The coil resistance can be either measured with an
ohmmeter, or is provided by the manufacturer. The air stiffness can be computed by
merely knowing the volume of the air cavity, the speed of sound, and the density of air (8)
(Radclifffe, Gogate 1992). SD is effective area through which the cone face acts upon the
air cavity, and is estimated to be the area of the cone face computed using a diameter that

is 87% of the advertised diameter (Beranek, 1986).
S p- c2
k - = —"——— 8
.r Va 1‘, ( )
The electromagnetic constant, bl, was found by determining the best linear
relationship between applied DC current, and zero displacement force generated at the
cone. The parameter bl is the ratio of force generated at the cone per unit of current, and

has units of MA; it is also the ratio of back emf to cone velocity, which yields units of
(8V)/m.

Page 10

EXPERIMENTAL RESULTS

Intensity was measured at twenty-four locations on the surface of an imaginary
cube 14x14x14 inches wide, inside of which was centered the acoustic exciter This cube
can be seen in Figure 3a. Those twenty-four intensity vectors were used to compute

volume velocity (9), and were compared with those of a point source.

The volume velocity output of a noise source is determined from intensity
measurements, provided that the measurements are made over the entire measm'ement

surface (9). The index, j, is the index of the intensity measurements and associated areas.

Q=§v-ndA =f-ll;(pv-n)dA =§%I-ﬁdA = Z-pl—jﬁj -ﬁj)A,. (9)

The simpliﬁed exciter model (5) was correlated to the measured volume velocity

by minimizing the model error in a least squares sense (Fig. 6).

 

 

 

 

 

 

 

 

 

 

 

 

2.5304 R . - - - Model]
2 O W ““4. ‘— Dam
. EM ‘1 l'.
Q(f) 1:. “fix
I h .
3 1513-04 J Rik
m ls ." ‘
r \ﬁ
LOB-04 7 v
" My
0513-04 Mme: , _
0.0
200 400 600 800 1000 1200
Frequency - Hz

Figure 6. Measured, Analytical Volume Velocity Frequency Response Function

Page 11

Fitting the model (5) to the data yields the values of the parameter groups, {k}.
Using these values, as well as the four other constraints discussed in the parameter

identiﬁcation section, the values for the seven parameters were determined (T able 1).

Table 1. List of model parameters for a pair of 4" woofers and a 0.5" spacer

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Value to Correlate Model Parameter Name and Units
M 8 9 Electrical Resistance of Speaker, Ohms
3mm: 4.64 Ns/m Viscous friction of Speaker, N*s/m

bl 3.75 N/A Electromagnetic Constant, N/A
inane 836 Mm Stiffness of Speaker Cone, N/m
M 0.000374 m2 Effective Areaof back of cones
.Eair 2.5589E+04 N/m Stiffness of Acoustic Cavity, N/m
, 2mm 0.000395 Kg Mass of Speaker Cone, kg

 

The needed calibration function is then T(s)=0.2222/(s7—t-1014s+8.235E+6). This
function is used to infer volume velocity output based on frequency and applied voltage.

Many times, acousticians use box speakers to approximate a point source using the
assumption that if the speaker is much smaller than the system under test, the
approximation will be a good one. This assumption will not hold if the system under test
is relatively small, and the sound power levels required are large. If the system is small,
the box speaker will need to be even smaller, and the ability to create a large amount of

Page 12

acoustical energy is reduced. For comparison purposes, four radial intensity

measurements were made surrounding a Radio Shack miniature box speaker (Fig. 7-8).

Desired Volume Veloclty Pattern

 

Figure 7. Radial Intensity Components (800 Hz) for Box Speaker

Mam cndStmdadDmtond IntersltyMwﬂtums Over Dlredlon

IsqudDevldon

 

rm 125 too am as) 315 an an (m an run 1250 um
Freqrerw-Hz

Figure 8. Mean, Standard Deviation of Intensity Vector Magnitudes for Box Speaker

The intensity data shows that the acoustic exciter approximates an ideal point

source with a high degree of accuracy (Fig. 9-11). A vertical projection of the second

Page 13

picture in Figure 9 was taken to clarify the results (Fig. 10). The comparison with a

typical box speaker shows that the acoustic exciter is a far better point source than an

ordinary speaker.

Zero Displacement Zero Displacement + Intensity Vectors

 

 

 

 

 

Figure 9. Wire Frame Animation of Intensity Vector "Displacements" at 800Hz

Desired Volume Velocly Patent

 

Figure 10. Vertical Projection of Data Dcpicted by Figure 9

Page 14

memdStmdzd Deddiond IntmsityMyihm aver Diredim

16
14
12
0.01

lsumdoauarm

888

 

.5555

10) 1% 1w 2!) am 315 41) HI) can an run 150 1100

Figure 11. Mean, Standard Deviation of Intensity Magnitudes for Acoustic Exciter

Comparing the intensity data for the box speaker and the acoustic exciter shows
that the acoustic exciter is omnidirectional, and the box speaker is not. Figure 11 shows
that the acoustic exciter is a very good point source. The standard deviation of the
acoustic exciter's intensity magnitudes is about 10% of the mean value (Fig. 9), and the
exciter can be said to be within 20% of spherical at 95% conﬁdence. The standard
deviation of the box speaker's magnitudes are around 100% of the mean (Fig. 8), and the
box speaker may not be considered a point source without moving a great distance from

the speaker.

CONCLUSIONS

A device that approximates an acoustical point source is needed for experimental
veriﬁcation of acoustic reciprocity (Fahy, 1992). This acoustical point source must emit
acoustical power equally in all directions, and also must be calibrated for volume velocity
because it cannot be measured during the test. The acoustic exciter developed in this
paper was designed and built to fulﬁll this need for a point source. An analytical model for
the acoustic exciter was developed for source calibration and future design optimization.

Acoustic intensity measurements were taken around the acoustic exciter, and used to

Page 15

Acoustic intensity measurements were taken around the acoustic exciter, and used to
determine the volume velocity calibration. These measurements were then used to identify

the model parameters and to verify the model.

It was also determined that the acoustic exciter is a much better point source than
a conventional loudspeaker. The acoustic exciter approximates a point source within 20%
at 95% conﬁdence; considering that box speakers have been used to approximate the point

source, this is a great improvement.

Page 16

REFERENCES

Gogate, S. D., Radcliffe, C.J., "Identiﬁcation and Modeling of Speaker Dynamics for
Acoustic Control Applications", ASME WAM Symposium on Active Control of Noise
and Vibration, 1992.

Beranek, L.L., Acoustics, American Institute of Physics, Inc., New York, NY, 1986

Bruel and Kjaer, Technical Review Number 3, 1982 (Sound Intensity - Theory) and
Technical Review Number 4, 1982 (Sound Intensity - Instrumentation and Applications)

Fahy, FJ., The Reciprocity Principle and Applications in Vibro-Acoustics, Second
International Congress on Recent Developments in Air and Structure-borne Sound and
Vibration, 1992.

Appendix A

 

Page 17

APPENDIX A - DERIVATION OF ACOUSTIC EXCIT ER SYSTEM EQUATIONS

First, a schematic of the acoustic exciter using ideal engineering elements is constructed

(Fig. Al)

 

 

 

 

 

 

e(t) L

 

Figure A1. Schematic of acoustic exciter using ideal engineering elements.

Ignoring the electrical domain, the equations of motion for the cones are derived from

applying Newton's laws to the cones (Al)
O " R 0 ° [C + k . -k -
moon {I + eon f1 + ecu r If I! = fl (A1)
0 menu x2 0 Reel: x2 —k-'r keen + ka'r x2 f2
In (Al), the force terms that drive the system, f, are equal to bl*icoi1 . The variable icoil

can be determined from Kirchoff's loop law for voltages (A2), where Vspl and
Vsp2 are the back emf of speaker I and speaker 2, respectively.

e(t)+2im.,Rw., +Vg, +ng = 0 (A2)
The back emf, Vspl and Vsp2 , are equal to bl‘U'c1 , and b1‘.t,, respectively (A3).

e(t) + 21",,,..,R‘,‘,,.,+bl-it1 - bl - it, = 0 (A3)

 

Page 18

The sign of the second back emf term in (A3) is negative because the current
passes through the speakers in opposite directions, because of the way in which

the terminals are wired.

The current is found by solving (A3), and the force terms in (A1) are written in
terms of bl, Rcoil, 12,, and 5:2. The coefficients of 12,, and )2, are collected into the
damping matrix of (A1), and the system of equations is determined (A4).

_b_z’ bl” bI-eU)

 

 

M“ 2T ”21?“, i. + k...+k.-. -k.-. x. _ ’ 2R“, (A4)
mm _2b1’ R + b!” i, -k.., rang” x, " bI-e(r)
“" 2R“ 2R“

Performing an eigenvalue-eigenvector analysis on the left hand side of (A4) reveals that
the eigenvectors are [1 -1]t, and [1 IF. This means that the force vector on the right hand
side of (A4) is orthogonal to the unison mode of vibration, and the opposition mode only
will be excited (x1 = 49). This simpliﬁcation allows an order reduction for the model,
and substituting x1 = -x2 into (A4) results in a one DOF system of equations (A5).

.. bl2 . _ bl
magi-[Rm +E}+(km +2k..,)x — [—ZRM, ]z(t) (A5)

The model variables need to be defined so that velocity may be found, so the
substitution of v=i into (A5) gives the equations of motion in state space form (A6),

where the dissipative and stiffness terms in (A5) have been simpliﬁed into Reff and keff.

. 0 l
[i]=[_-"_v_ __54_]:x] +[_!__ “kn (A6)
v m mm v 2R0“, m

The acoustic exciter will always operate at steady state, that is, e(t) will always be

periodic, so the solution for v is carried out using the Laplace transformation (A7).

Page 19

0 1 0
X“) k R [X09] bl
= E A7
S[V(S)] [- me‘olu — ml] V(s) + 2Reoiz 'mem (S) ( )
Solving (A7) for V(s) gives the acoustic exciter's transfer function, T(s)=V(s)/E(s) (A8).
T (s) _ V(s) _ bl . 3 (A8)

 

 

— E (s) — 2R“, "rams2 +Rgs + kl,

 

"llllllllllllllt