1. Proceedings of International and INCCOM-6 Conference
Future Trends in Composite Materials and Processing
December 12-14, 2007
Indian Institute of Technology Kanpur
STUDIES ON ACOUSTIC RADIATION FROM RECTANGULAR
COMPOSITE PANELS
B. Pattabhi Ramaiah ∗a , B. Rammohan∗, T.S.S. Narayana■
∗Scientists, Aeronautical Development Establishment (ADE), DRDO, Bangalore
■
Project Manager, NVH Group, CADES Digitech Pvt. Ltd, Bangalore
ABSTRACT
Material tailoring of structures, use of damping layers, attaching point masses are some of the methods
attempted by researchers for reducing the acoustic radiation from the vibrating panels. In applications like
submarines and ships, it may help to achieve the control over the directivity of the externally radiated sound,
in addition to attenuating it to the extent possible. In an attempt to achieve the above objective, an analytical
study is performed to estimate the radiated sound and its directivity from a simply supported rectangular
composite panel (set in a baffle) excited by a point force. Further, to obtain a change in the directivity, point
masses have been attached to the composite panel at certain locations. The aim is to identify the size, number
and locations of the point masses as well as the frequency of excitation at which there is a significant change
in directivity. The forced response analysis of the panel with attached point masses is done through the
Receptance method. The analytical results have been compared with the results obtained from the
commercially available numerical codes.
1.INTRODUCTION
The prime objective of the structural engineer is to design an airframe whose flight envelope is limited by
engine power rather than its structural limitations. One of the situations faced in a typical Aircraft industry is
quietening of a radiating panel set into vibration by various forces using a closed form solution rather than
Numerical techniques.
Material tailoring of structures [1], where the authors solved the problem in two steps, for designing structures
that radiate sound inefficiently in light fluids. In first step, given a frequency and over all geometry of the
structure, a surface velocity distribution is found that produces a minimum radiation condition. Second, a
distribution of Young’s modulus and density distribution is found for the structure such that it exhibits the
weak radiator velocity profile as one of its mode shapes.
Lamancusa and Wodtke [2] discussed the use of damping layers in sound power minimization. They mainly
concentrated on the minimization of sound power radiated from plates under broad band excitation by
redistribution of unconstrained damping layers, by assuming the total radiated sound power is represented by
the power radiated at structural resonances. Other methods include reduction of dynamic compliance, Jog [3]
and point mass attachments to structures, St.Pierre [4]. These methods are grouped as passive methods of
noise attenuation, which work well at high frequencies.
Complimentary to this is the active noise control technique that covers the lower frequency range. In active
noise control, global control can be achieved for enclosed sound fields at low frequencies, due to modal
behaviour, by appropriate placement of sensors and actuators, Elliott ad Curtis [5]. In contrast, global control
in unbounded domains, such as external radiation is a challenge. Guo and Pan [6] have demonstrated active
noise control in free field environments. It requires appreciable hardware and achieves reasonable broadband
control when the microphones and speakers are optimally located in the sound field. In the free fields, exact
cancellation of sound occurs only when the secondary source is a replica of the primary and placed at the
same location, which cannot happen in practice.
In applications such as stealth in submarines and ships, one alternative might be to achieve the control over
the directivity of the external radiated sound, rather than attenuating sound totally;
a
Corresponding Author: pattabhib@yahoo.com
1

2. Even in industrial applications it is useful to direct the sound away from the work place and make the
environment acceptable. The present work is aimed at achieving a change in directivity of a point driven
simply supported plate set in baffle by attaching point masses to it. The main strategy here involves deliberate
changes in mode shapes of the radiator in order to achieve the objective. The analysis for the point mass
attachments, determination of new resonances, modes and the response, is performed using the Receptance
method [7-9].
The natural frequencies and mode shapes of a simply supported composite plate are given in [10], in semi
analytic form. Using the natural frequencies and mode shapes, the response of the simply supported plate to
appoint force can be computed [7]. Knowing the plate response, the pressure field radiated from the plate set
in the baffle can be estimated using the Rayleigh integral and the sound power from the plate can be
calculated as explained in [12].
The mode shapes of the simply supported panel with attached point masses to it can be obtained using the
Receptance method as described in section 2. The advantage of the Receptance method is that it is totally
analytical and does not require any packages like FEM. The new mode shapes can be computed in terms of
the old modes. Once the new modes are known, the response of the plate to a point force with attached masses
can be computed. And the acoustic pressure of the new system is obtained using Rayleigh integral. The
method has the advantage that the new mode shapes and natural frequencies are expressible in terms of the
original modes and the natural frequencies. The strategy can be easily programmable in a computational
package like MatLab. Thus, the salient feature of this paper is the simplicity with which a constraint can be
implemented analytically, which affords quick physical insights and helps to make a quick decision.
2. THE RECEPTANCE THEORY
The Receptance method is well developed and a detailed description for plates can be found in [7]. With the
Receptance method, vibration characteristics of a combined system, for instance, a plate attached with a mass,
are calculated from the characteristics of the component systems, in this case the plate and the mass. A
Receptance is defined as
ijα = Deflection response of system A at location i / Harmonic force or moment input
To system A at location j (1)
The response may be either a line deflection or a slope. Usually, the subsystems are labeled A, B, C and so on,
and the Receptances are labeled γβα ,, and so on. From Maxwell’s Reciprocity theorem jiij αα = .
Figure. 2.1 shows five linear masses attached to a rectangular panel along a line. Consider the case of two
systems connected at two points as shown in Figure.2.2, the displacement and force relationships for structure
A are given by
(2)⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
=⎥
⎦
⎤
⎢
⎣
⎡
2
1
2221
1211
2
1
A
A
A
A
F
F
X
X
αα
αα
In general, for n displacements
{ } [ ]{ }AA FX α= (3)
where, XA1 and XA2 are the displacements at locations A1 and A2, respectively. FA1 and FA2 are forces at the
same locations applied to structure A. Similarly; the equations for structure B are given by
(4)⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
=⎥
⎦
⎤
⎢
⎣
⎡
2
1
2221
1211
2
1
B
B
B
B
F
F
X
X
ββ
ββ
In general for n displacements
{ } [ ]{ }BB FX α= (5)
Thus, αij and βij, i,j=1,2 are the drive point and cross receptances, having the units of displacement per unit
force. α11 is the displacement at point 1 due to a unit force at point 1, and α12 is the displacement at point 1
due to a unit force at point 2. When two such systems are joined together, the forces FA and FB become
internal forces and they have to add to zero, and the displacements have to be equal. Thus,
B
}{}{ BA FF −= (6)
2

3. and
}{}{ BA XX = (7)
Figure 2.1. Five masses attached to a plate
along a line
Figure 2.2. Two structures A and B
connected at two points 1 and 2
where the curly brackets indicate force and displacement vectors. By combining equations (1) to (7) the
following expression is obtained,
0)()(
0)()(
2222212121
2121211111
=+++
=+++
AA
AA
FF
FF
βαβα
βαβα
(8)
In general,
[ ] [ ] 0}]{[ =+ AFβα (9)
FA1= FA2= 0 being trivial solutions, the non-trivial solution is found by setting the determinant
.0
22222121
12121111
=
++
++
βαβα
βαβα
(10)
For the case of a single mass, Eqn. (10) becomes
01111 =+ βα (11)
Thus, one needs to know the α’s and the β’s of the two structures. A simply supported plate is attached with
masses is considered for the present work. Let the α’s belong to the plate and the β’s to the masses. The 2x2
receptance matrix [α] for the composite laminate is derived below.
The bending equation of the 2D orthotropic symmetric composite panel is given by [11]
( ) ( )tyxf
t
w
h
y
w
D
yx
w
DD
x
w
D pp ,,22 2
2
4
4
2222
4
66124
4
11 =
∂
∂
−
∂
∂
+
∂∂
∂
++
∂
∂
ρ
The harmonic point force damped response at point (x, y) of a rectangular plate due to a harmonic point force
at (x
(12)
p,yp) on the panel is given by [7]
∑∑= =
Φ
+−Γ
=
1 1
22
),(
)2*(
)1
),,(
m n
mn
mnmnmn
t
pp
mn
yx
j
Fex
tyxW
ωωξωω
ω∞ ∞ Φ ,( j
mn y
(13)
where is the material constant, and is given bymnΓ
(14)
∫∫Φ=Γ
a b
mnmn dxdy
0 0
2
If λ is the equivalent viscous damping factor, the modal damping coefficient mnξ is given by
3

4. mnpp
mn
h ωρ
λ
ξ
2
= (15)
mnΦ Represents the mode shapes of the plate, given by [10]
)()(),( yYxXyxmn =Φ
(16)
where X(x) and Y(y) are chosen as the fundamental mode shapes of the beam having the simply supported
boundary conditions. And the expressions for X and Y are given by [10]
⎟
⎠
⎞
⎜
⎝
⎛
=
a
xm
xX
π
sin)(
The functions Y(y) are obtained by replacing x by y and a by b and m by n in equation (16). The natural
frequencies
for m = 1,2,3, …. (17)
mnω of the Simply Supported plate are given by
( )
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
++⎟
⎠
⎞
⎜
⎝
⎛
=
4
22
22
6612
4
11
2
22
1
b
n
D
b
n
a
m
DD
a
m
D
hpp
mn
ππππ
ρ
ω (18)
Thus 11α is given by, using Equations (1) and (13)
( )
),(
),(1
1122
11
111
1
11 yx
yx
heF
X
mn
mn
mn
nmmnpp
tj
Φ
−
Φ
Γ
== ∑∑
∞
=
∞
= ωωρ
α ω
(19)
and 12α is given by,
( )
),(
),(1
1122
22
112
1
12 yx
yx
heF
X
mn
mn
mn
nmmnpp
tj
Φ
−
Φ
Γ
== ∑∑
∞
=
∞
= ωωρ
α ω
(20)
The Receptance of mass can be found by knowing the steady state response of a mass to a harmonic force
input. From Newton’s second law
(21)tj
BB
tj
B eXmeF ωω
ω 1
2
11 −=
Thus,
1
1
11
B
B
F
X
=β = 2
1
1
ωBm
− (22)
Hence, for a plate connected to two masses, from Eqn. (10), the Receptance matrix appear as,
.0
1
1
2
2
2221
122
1
11
=
−
−
ω
αα
α
ω
α
B
B
m
m
(23)
The structure of the matrix can now be extended to the case where N masses are attached. The determinant of
the NxN receptance matrix set to zero gives the new natural frequencies of the plate-mass system. The new
mode shapes of the plate-mass system can be determined from the forced response expression of the original
plate [7]. For the case of plate with single mass, Eqn. (2) gives the new mode shape, when ω, the excitation
frequency, is set to the new natural frequency and (xp, yp) is set to the coordinates of the point of mass
attachment (xm, ym). For the N mass case the characteristic equation is an NxN matrix whose determinant is set
to zero, and there will be N roots, which are the new resonances. Since the plate is constrained at N points
through N point masses, it will experience point forces at those N locations. The magnitudes of these point
forces are given by the elements of the eigenvector corresponding to the zero Eigen value of the receptance
matrix evaluated at the new natural frequency ωk. Thus, the new kth
mode shape is given by substituting ωk for
ω in Eqn. (2) with an additional summation term as follows
( )
( yx
Fyx
yx mn
kmn
kiiimn
N
i
nmmn
k ,
),(
1
),(
22
1
11
Φ
−
Φ
Γ
=Φ
∑
∑∑ =
∞
=
∞
= ωω
) (24)
4

5. where Fik is the ith
element of the eigenvector of the zero eigen value, corresponding to the kth
new natural
frequency and (xi, yi)are the location of the ith
mass.
The response of the plate-mass system to a point force can again be calculated using the new mode shapes,
Eqn. (23) as
( ) ),(
2*
),(11
),,(
22
1
yx
j
Feyx
h
tyxW k
kkk
tj
ppk
K kpp
Φ
+−
Φ
Γ
= ∑
∞
= ωωξωωρ
ω
(25)
where, F is the force amplitude at location (xp,yp) ωk the kth
natural frequency in rad/s, ξk the modal damping
coefficient, given by
kpp
k
h ωρ
λ
ξ
2
= (26)
and
(27)
dxdyk
ba
k
2
00
Φ=Γ ∫∫
3. SOUND CALCULATIONS
During the vibration of the plate, the normal velocity of the acoustic medium on the surface of the plate
loaded at r1 has to be equal to the normal velocity of the plate v(r1), in order to satisfy the requirement of the
continuity as shown in the Figure. 3.1.
Due to the acoustic perturbation on the surface of the plate, the acoustic pressure p(r2) at r2 is created and can
be obtained from Rayleigh’s integral [12]
1
0
2
1
2
),( dS
R
ev
e
j
trp
jKR
n
S
tj
−
∫= ω
π
ωρ
(28)
Where r2 is the position vector of the observation point, r1 the position vector of the elemental surface 1Sδ
having the normal velocity vn(r1), R the magnitude of the vector (r2 - r1), ρ0 is the density of air, k is the
acoustic wave number and S1 is the area of the plate. Considering a hemispherical measurement surface in the
far field, where R is much greater than the source size as defined by the larger edge of the two panel
dimensions a and b, i.e R >> a and R >> b, R and r2 are related by the approximate relationship
φθφθ sinsincossin2 yxrR −−= . (29)
Figure. 3.1. Integration areas S1 and S2 for
estimating the sound pressure field
Figure. 3.2. Integration areas S1 and S2 for
calculating the sound power S1=S2
The instantaneous acoustic intensity I (r2, t) at r2 can be expressed as,
5

6. ( ) ( ){ }trvtrptrI ,,Re
2
1
),( 2
*
212 =
(30)
where v(r2, t) is the normal velocity of the acoustic medium at r2 and * denotes it’s complex conjugate. The
intensity I, which is the time average of I(t), i.e the time averaged rate of energy transmission through a unit
area normal to the direction of propagation, is given by
( )
c
rp
I
0
2
2
,,
ρ
φθ
=
(31)
And the sound pressure level at a particular point is given by
ref
p
p
p
L log20=
(32)
Where pref is the reference pressure, which is 20 μ Pa. The sound power Wp radiated into the semi-infinite
space above the plate can be estimated from
(33)22
1
)( dSrIW
S
p ∫=
where S2 is an arbitrary surface which covers area S1 and r2 is the position vector of S2, see Figure. 3.1.
Substituting equations (32) and (34) into Eqn (36) and allowing S2 = S1 then r1 and r2 would represent any two
arbitrary position vectors on the surface of the plate as shown in Figure. 3.2. The power radiated by the plate
can then be expressed as [2]
22
0 0 0 0
112
*
1
0
)(
)sin(
)(
4
dydxdydxrv
R
kR
rvW
a b a b
p ∫∫ ∫∫ ⎥
⎦
⎤
⎢
⎣
⎡
=
π
ωρ
(34)
4. RESULTS AND DISCUSSION
The lay-up sequence of the laminates of the stiffened composite rectangular panel of size 0.4mX0.3m used for
calculations is shown in Figure. 4.1. The material properties, thicknesses and the orientations of the individual
laminas are shown in Table 1.
Carbon Epoxy, 2 Layers (00
/450
)
Packaging (00
)
PLI 1 (00
)
PLI 2 (0
Packaging (00
)
Carbon Epoxy, 2 Layers (450
/00
)
0
)
Figure. 4.1. Lay-up sequence of the stiffened Composite Rectangular Panel
Material
Youngs Modulus
E (GPa)
Density
(kg/m3
)
Thickness
(mm)
Orientation (Deg)/ No.
of Layers
Carbon Epoxy 72.40 1100 0.15 0/45/45/0 (4 Layers)
Packaging 4.60 1290 0.20 0/0 (2 Layers)
PLI 1.02 2540 0.50 0/0 (2 Layers)
Table 1. Properties of the composite panel
A comparison of the first five natural frequencies from Analytical (MatLab) model Eqn. (18) and Numerical
(Nastran) model is given in table 5.2.
4.1. ANALYSIS OF PLATE-MASS SYSTEM
The characteristic equation for multiple mass case can be obtained by extending Eqn. (23). For the purpose of
illustration two, four and five masses attached along a particular orientation mentioned in Table 2, are
considered. A comparison of the first five natural frequencies for multiple mass cases at different orientations
is given in Table 2.
6

7. After calculating the new resonances the corresponding modes and response can be obtained from
Eqns (28) and (29) respectively. A comparison of the response is provided in Figure. 3.3.
Without Masses Two Masses,
along 200
line
Four Masses,
along 400
line
Five Masses,
along 600
line
Mode
Analytical Numerical Analytical Numerical Analytical Numerical Analytical Numerical
(Hz) (Hz) (Hz) (Hz) (Hz) (Hz) (Hz) (Hz)
62.05 57.21 56.94 52.76 52.36 54.511 62.3 54.21
139.87 130.10 128.83 121.83 122.72 118.012 140.7 116.20
173.90 155.09 155.04 160.66 157.32 149.523 173.8 150.18
246.23 222.89 221.96 208.09 210.11 218.444 249.4 218.70
274.59 272.39 273.18 247.24 241.11 245.815 272.9 241.28
Table 2. Comparison of Natural Frequencies of the plate with two, four and five-mass system
From analytical and numerical models.
MATLAB NASTRAN
Figure. 4.2.1 Mode 2 with two masses
along 200
line at 130.1 Hz
Figure. 4.2.2. Mode 2 with two masses
along 200
line at 128.83 Hz
Figure. 4.2.3. Mode 4 with four masses
along 400
line at 208.09 Hz
Figure. 4.2.4. Mode 4 with four masses
along 400
line at 210.11 Hz
Figure. 4.2.5. Mode 3 with five masses
along 600
line at 149.52 Hz
Figure. 4.2.6. Mode 3 with five masses
along 600
line at 150.18 Hz
Figure 4.2: Comparison of Mode Shapes from Analytical Model and Nastran.
7

8. Figure. 4.3. Point force Response of the panel
from MatLab at 200 Hz with five masses
attached along 600
line
Figure. 4.4. Point force Response of the panel
from Nastran at 200 Hz with five masses
attached along 600
line
Figure 4.2 shows a comparison of the mode shapes from MatLab and Nastran for different conditions. A
comparison of point force response is provided in figures 4.3 and 4.4 respectively. From Table 2 and Figures
4.2, 4.3 and 4.4 it can be concluded that the analytical results are in agreement with the numerical results.
100 180 260 340 420 500 580 660 740 820 900 980 1060 1140 1220 1300 1380 1460 1540 1620 1700 1780 1860 19402000
60
65
70
75
80
85
90
95
100
105
110
Frequency (Hz)
SoundPressureLevel(dB)
Variation of Sound Pressure Level with Frequency
Figure 4.5. Variation of Sound Pressure Level (dB) with frequency (Hz)
100 180 260 340 420 500 580 660 740 820 900 980 1060 1140 1220 1300 1380 1460 1540 1620 1700 1780 1860 1940 2000
65
70
75
80
85
90
95
100
105
110
115
Frequency (Hz)
SoundPower(dB)
Variation Sound Power with Frequency
Figure 4.6. Variation of Sound Power (dB) with frequency (Hz)
8

9. Variation of Sound Pressure Level and Sound Power with frequency for the composite plate without any
masses is plotted in Figures 4.5 and 4.6 respectively. Figure 4.8 shows the Intensity distribution Eqn. (31),
of the panel constrained with five masses of infinite stiffness along 600
line and excited at 200 Hz as
compared to that of the plate without any masses shown in Figure 4.7. Figure 4.9 shows three dimensional
distribution of Intensity for the case shown in Figure 4.8. From figures 4.7 and 4.8 it can be observed that
there is a change in acoustic directivity by placing the masses. The clear change sound field can be seen
from figure 4.8. Hence it can be concluded that sound field can be directed by properly tuning the masses
and frequency of excitation. Figure 4.10 shows the variation of Sound Power (dB) Eqn. (34) with
frequency, of the composite panel with five masses placed along 600
line.
Figure. 4.7. Hemispherical Distribution of
Sound Intensity without masses excited at 200 Hz.
Figure 4.8. Hemispherical Distribution of
Sound Intensity with five masses attached along
600
line excited at 200 Hz.
100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000
70
80
90
100
110
120
130
Frequency (Hz)
SoundPower(dB)
Figure. 4.9. Hemispherical Three dimensional
Distribution of Sound Intensity with five masses
attached along 600
line excited at 200 Hz.
Figure. 4.10. Variation of Sound Power with
frequency for the case of five masses attached
along 600
line
5. CONCLUSIONS
Code in MATLAB has been developed to perform the Acoustic analysis of Laminated Composite Rectangular
Panels with point masses, for simply supported boundary conditions. Various angles are considered for the
mass location and the results are compared with the commercial packages. A particular case is observed where
there is a change in acoustic directivity after attaching the masses. This is a quick and useful method for
engineers in an industrial scenario where time is a constraint sometimes.
9

10. 6. ACKNOWLEDGEMENTS
The support provided by Mechanical Engineering Design Division, ADE, Bangalore to carry out the present
work is greatly acknowledged. We greatly acknowledge our divisional head for his valuable suggestions and
support during the progress of the present work.
7. REFERENCES
[1] Koorosh Naghshineh and Gary H. Koopman., Ashok D. Belegundu., Material tailoring of structures to
achieve a minimum radiation condition, J. Acoust. Soc. Am, 1992, 92 (2), 841-855.
[2] H.W.Wodtke., J.S.Lamancusa., Sound power minimization of circular plates through damping layer
placement, Journal of Sound and vibration, 1998, 215(5), 1145-1163.
[3] Jog, C. S., Reducing radiated sound power by minimizing the dynamic compliance, IUTAM Intenational
Symposium on Designing For Quietness, Organized by Facility for Research in Technical Acoustics (FRITA),
Indian Institute of Science, Bangalore, India, Dec. 12-14 2000.
[4] St. Pierre, R L. and Koopmann, G. H., Minimization of radiated sound power from plates using
distributed masses, presented at the ASME Winter Annual Meeting, Paper no. 93-WA/NCA-11, New Orleans,
LA, Nov. 28-Dec. 3, 1993
[5] S.J.Elliott., A.R.D. Curtis., A.J.Bullmore., and P.A.Nelson., The active minimization of harmonic enclosed
sound fields, part III: Experimental verification, Journal of Sound and vibration, 1987, 117 (1), 35-58.
[6] Jignan Guo and Jie Pan, Actively created quiet zones for broad band noise using multiple control sources
and error microphones, J. Acoust. Soc. Am, 1999, 105 (4), 2294-2303
[7] R.E.D. Bishop, D.C. Johnson, The Mechanics of Vibration, Cambridge University Press, London, 1960.
[8] W. Soedel, Vibrations of Plates and Shells, Marcel Dekker, New York, 1981.
[9] S. Azimi, J.F. Hamilton, W. Soedel, The receptance method applied to the free vibration of continuous
rectangular plates, Journal of Sound and Vibration 93 (1) (1984) 9–29.
[10] A.W.Leissa., Vibration of Plates, 1969 NASA SP-160
[11] Reddy J. N., An introduction to the Finite Element Method, second edition, McGraw-Hill, Inc, 1993.
[12] Frank. J. Fahy, Sound and Structural Vibration, Academic Press, 2006.
10