1. Proceedings of ICTACEM 2007
International conference on Theoretical, Applied, Computational and Experimental Mechanics,
December 27-29, 2007, IIT Kharagpur, India.
ICTACEM-2007/107
SOUND RADIATION FROM A BAFFLED COMPOSITE
RECTANGULAR PANEL UNDER A LINE CONSTRAINT
B. Pattabhi Ramaiah●
, B. Rammohan●
, D. Satish Babu●
, R. Raghunathan■
●
Scientists, Aeronautical Development Establishment, DRDO, Bangalore, India.
■
Group Director, Aeronautical Development Establishment, DRDO, Bangalore, India.
ABSTRACT
In the present day scenario Composite structures are extensively used especially in
Unmanned Aerial Vehicles (UAV). Material tailoring of structures, use of damping layers;
point masses are some of the methods attempted by the researchers for reducing the acoustic
radiation from radiating panels. In the present work the influence of a line support under
various orientations of a Point-force driven simply supported composite rectangular laminate
set in an infinite baffle is studied. An attempt has been made here to estimate the Acoustic
Radiation using a closed form solution rather than FEM/BEM methods. The challenge is to
arrive at a quick solution not involving extensive computations, expensive packages and
time.
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.
In the present work the influence of a line support under various orientations of a Point-force
driven simply supported composite rectangular laminate set in an infinite baffle is studied.
The line constraint is approximated by attaching infinitely stiff springs along the intended
line. Receptance method [4–6] is used to arrive at the new natural frequencies and mode
shapes. 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.
The natural frequencies and mode shapes of a simply supported composite plate are given in
[8], 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 a baffle can be estimated using the Rayleigh
integral and the sound power from the plate can be calculated [2].
The mode shapes of the simply supported panel with springs attached 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 springs can be computed. And the
acoustic pressure of the new system is obtained using Rayleigh integral.
1

2. 2. THE RECEPTANCE THEORY
The Receptance method is well developed and a detailed description for plates can be found
in [5]. With the Receptance method, vibration characteristics of a combined system, for
instance, a plate attached with a spring, are calculated from the characteristics of the
component systems, in this case the plate and the spring. 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 = .
Fig. 2.1 shows five linear springs attached to a rectangular panel along a line. Consider the
case of two systems connected at two points as shown in Fig.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)
Fig 2.1. Five springs attached to a plate
along a line
Fig 2.2. Two structures A and B connected
at two points 1 and 2
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,
2

3. 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,
}{}{ BA FF −= (6)
and
}{}{ BA XX = (7)
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 spring, 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 springs is considered for the present work. Let the α’s belong to the plate and
the β’s to the springs. 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 [7]
( ) ( )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 [5]
∑∑
∞
=
∞
=
Φ
+−
Φ
Γ
=
1 1
22
),(
)2*(
),(1
),,(
m n
mn
mnmnmn
tj
ppmn
mn
yx
j
Feyx
tyxW
ωωξωω
ω
(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
mnpp
mn
h ωρ
λ
ξ
2
= (15)
mnΦ Represents the mode shapes of the plate, given by [8]
)()(),( 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
⎟
⎠
⎞
⎜
⎝
⎛
=
a
xm
xX
π
sin)( for m = 1,2,3, …. (17)
3

4. 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 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 spring can be found by knowing the steady state response of a spring to a
harmonic force input. From Newton’s second law
(21)tj
BB
tj
B eXKeF ωω
111 =
Thus,
1
1
11
B
B
F
X
=β =
1
1
BK
(22)
Hence, for a plate connected to two springs, from Eqn. (10), the Receptance matrix appear as,
.0
1
1
2
2221
12
1
11
=
+
+
B
B
K
K
αα
αα
(23)
The structure of the matrix can now be extended to the case where N springs are attached.
The determinant of the NxN receptance matrix set to zero gives the new natural frequencies
of the plate-spring system. The new mode shapes of the plate-spring system can be
determined from the forced response expression of the original plate Ref. [2]. For the case of
plate with single spring, 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 spring attachment (xm, ym). For the N spring 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 springs, 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)
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
spring.
The response of the plate-spring system to a point force can again be calculated using the
new mode shapes, Eqn. (23) as
4

5. ( ) ),(
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)
3. SOUND CALCULATIONS
dxdyk
ba
k
2
00
Φ=Γ ∫∫
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 Fig. 3.1.
Fig. 3.1. Integration areas S1 and S2 for
estimating the sound pressure field
Fig. 3.2. Integration areas S1 and S2 for
calculating the sound power S1=S2
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
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)
The instantaneous acoustic intensity I (r2, t) at r2 can be expressed as,
( ) ( ){ }trvtrptrI ,,Re
2
1
),( 2
*
212 =
(30)
5

6. 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
Fig. 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
Fig. 3.2. The power radiated by the plate can then be expressed as
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 Fig. 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
)
Fig. 4.1. Lay-up sequence of the stiffened Composite Rectangular 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.
Youngs Modulus
E (GPa)
Density
(kg/m
Thickness
(mm)
Orientation (Deg)/
No. of Layers
Material 3
)
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
4.1. Analysis of Plate-Spring System
The characteristic equation for multiple spring case can be obtained by extending Eqn. (23).
For the purpose of illustration two, four and five springs attached along a particular
orientation mentioned in Table 2, are considered. A comparison of the first five natural
frequencies for multiple spring 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 Fig. 3.3.
Without Springs Two Springs,
along 150
line
Four Springs,
along 300
line
Five Springs,
along 600
lineMode
Analytical
(Hz)
Numerical
(Hz)
Analytical
(Hz)
Numerical
(Hz)
Analytical
(Hz)
Numerical
(Hz)
Analytical
(Hz)
Numerical
(Hz)
1 62.3 62.05 95.89 93.97 139.97 137.79 95.25 92.68
2 140.7 139.87 160.98 160.63 236.43 229.91 197.27 197.66
3 173.8 174.11 246.76 242.96 269.37 270.14 238.17 229.23
4 249.4 246.37 283.37 279.97 355.63 353.07 334.78 337.23
5 272.9 274.59 312.97 314.58 395.89 393.34 400.67 392.52
Table 2. Comparison of Natural Frequencies of the plate with two, four and five-spring system
From analytical and numerical models.
MATLAB NASTRAN
Fig. 4.2.1 Mode 1 along 150
line
At 97.82 Hz
Fig. 4.2.2. Mode 1 along 150
line
At 96.36 Hz
Fig. 4.2.3. Mode 2 along 300
line
At 237.5 Hz
Fig. 4.2.4. Mode 2 along 300
line
At 231.32 Hz
Fig. 4.2.5. Mode 1 along 600
line
At 238.17 Hz
Fig. 4.2.6. Mode 3 along 600
line
At 229.23 Hz
Fig 4.2: Comparison of Mode Shapes from Analytical Model and Nastran.
7

8. Fig. 4.3. Point force Response of the panel
from MatLab at 150 Hz with 600
constraint
angle
Fig. 4.4. Point force Response of the panel
from Nastran at 150 Hz with 600
constraint
angle
Figure 4.5 shows the Intensity distribution Eqn. (31), of the panel constrained with five
springs of infinite stiffness along 600
line 200 Hz. Figure 4.6 shows the variation of Sound
Power Eqn. (34), of the panel for constrained with five springs of infinite stiffness for
various angles at 200 Hz.
Fig. 4.5. Hemispherical Distribution of
Sound Intensity at 200 Hz.
Fig. 4.6. Variation of Sound Power with
constraint angle at 200 Hz.
5. CONCLUSIONS
Code in MATLAB has been developed to perform the Acoustic analysis of line constrained
Composite Rectangular Panels with springs of large stiffness, for simply supported boundary
conditions. Various constraint angles are considered for the study and the results are
compared with the commercial packages. This is a quick and useful method for engineers in
an industrial scenario where time is a constraint sometimes.
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.
8

9. 7. REFERENCES
[1] K. Naghshineh, G. Koopmann, A. Belegundu, Material tailoring of structures to achieve a
minimum radiation condition, Journal of the Acoustical Society of America 92 (2, pt. 1)
(1992) 841–855.
[2] R.L.St. Pierre, G.H. Koopmann, 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, November 28–December 3, 1993.
[3] H.W. Wodtke, J.S. Lamancusa, Sound power minimization of circular plates through
damping layer placement, Journal of Sound and Vibration 215 (5) (1998) 1145–1163.
[4] R.E.D. Bishop, D.C. Johnson, The Mechanics of Vibration, Cambridge University Press,
London, 1960.
[5] W. Soedel, Vibrations of Plates and Shells, Marcel Dekker, New York, 1981.
[6] 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.
[7] Reddy J. N., An introduction to the finite element method, second edition, McGraw-Hill,
Inc, 1993.
[8] A.W.Leissa., Vibration of plates, 1969 NASA SP-160
9