1. Directionality of sound radiation from rectangular panels
Pattabhi R. Budarapu a,⇑
, T.S.S. Narayana b
, B. Rammohan c
, Timon Rabczuk a,d
a
Institute of Structural Mechanics, Bauhaus University of Weimar, 99423 Weimar, Germany
b
Assystem India Private Limited, Bangalore, India
c
IFB Automotive Private Limited, Bangalore, India
d
School of Civil, Environmental and Architectural Engineering, Korea University, Republic of Korea
a r t i c l e i n f o
Article history:
Received 31 December 2013
Received in revised form 17 August 2014
Accepted 5 September 2014
Keywords:
Receptance
Acoustic radiation
Directivity
a b s t r a c t
In this paper, the directionality of sound radiated from a rectangular panel, attached with masses/springs,
set in a baffle, is studied. The attachment of masses/springs is done based on the receptance method. The
receptance method is used to generate new mode shapes and natural frequencies of the coupled system,
in terms of the old mode shapes and natural frequencies. The Rayleigh integral is then used to compute
the sound field. The point mass/spring locations are arbitrary, but chosen with the objective of attaining a
unique directionality. The excitation frequency to a large degree decides the sound field variations. How-
ever, the size of the masses and the locations of the masses/springs do influence the new mode shapes
and hence the sound field. The problem is more complex when the number of masses/springs are
increased and/or their values are made different. The technique of receptance method is demonstrated
through a steel plate with attached point masses in the first example. In the second and third examples,
the present method is applied to estimate the sound field from a composite panel with attached springs
and masses, respectively. The layup sequence of the composite panel considered in the examples corre-
sponds to the multifunctional structure battery material system, used in the micro air vehicle (MAV)
(Thomas and Qidwai, 2005). The demonstrated receptance method does give a reasonable estimate of
the new modes.
Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction
The rectangular plate is one of the most widely used structures
in the industrial world. The sound radiation from vibrating
unbaffled panels [2–4], baffled panels [4–9] and submerged panels
[10–12] has been the subject of active research for many years.
Particularly, the sound radiation from vibrating plates is a common
problem in automobiles, airplanes, industrial machinery, buildings
and electro-acoustical devices, to name a few. Understanding the
sound radiation characteristics of these structures is important
for the researchers to maintain the noise levels within the specified
limits. Regular or periodic excitation forces are likely to be experi-
enced by the plates when they form part of a structure. The driving
force spectrum may be composed of a single frequency alone or of
a large number of frequencies. There is usually little that can be
done to change the nature of the driving forces. Therefore,
researchers are studying various techniques related to acoustic
radiation for making engineering systems quieter.
One of the methods is to arrange the design so that the forces
act on a nodal line for the mode shape about to be excited. This
method is useful when the applied forces act on a concentrated
area only. Naghshineh et al. [13] proposed material tailoring of
structures for designing structures that radiate sound inefficiently
in light fluids. They solved the problem in two steps. In the first
step, given a frequency and overall geometry of the structure, a
surface velocity distribution for minimum radiation condition
was found. In the second step, a distribution of Young’s modulus
and density distribution was found for the structure such that it
exhibits the weak radiator velocity profile as one of its mode
shapes. Wodtke and Lamancusa [14] 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. Apart from the above
methods, Jog [15] proposed the reduction of dynamic compliance.
St Pierre and Koopmann [16] worked on the point mass attach-
ments to the structures to control the sound. Sonti [9] studied
the variation of the sound power from a baffled point-force-driven
simply supported rectangular plate subjected to a line constraint,
http://dx.doi.org/10.1016/j.apacoust.2014.09.006
0003-682X/Ó 2014 Elsevier Ltd. All rights reserved.
⇑ Corresponding author.
E-mail address: pattabhib@gmail.com (P.R. Budarapu).
Applied Acoustics 89 (2015) 128–140
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journal homepage: www.elsevier.com/locate/apacoust
2. as a function of the constraint angle. Sastry et al. [17] have studied
the sound radiation from the baffled rectangular panels attached
with point masses and Ramaiah et al. [18,19] have estimated the
sound radiation from the baffled composite panels. Xuefeng and
Li [20,21] tried to reduce the sound radiation from a plate by
modifying the boundary conditions. Fahy [22] proposed the
vibro-acoustic noise control based on the reciprocity principle.
These methods are grouped as passive methods of noise attenua-
tion and works well at high frequencies.
Complimentary to the above technique is active noise control,
which covers the low frequency range. In active noise control,
global control can be achieved for enclosed sound fields at low
frequencies, by appropriate placement of sensors and actuators
[23,24]. In contrast, global control in the unbounded domains, such
as external radiation is still a challenge [25]. Sonti and Jones [26]
have developed a curved piezo-actuator model for active vibration
control of cylindrical shells. Guo and Pan [27] have demonstrated
the 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.
Even in industrial applications it is useful to direct the sound away
from the work place and make the environment acceptable. The
main objective of the current work is to achieve a change in the
directivity of a point driven rectangular plate set in a baffle by
attaching point masses/springs to it. The strategy here involves
deliberate changes in the mode shapes of the radiator in order to
achieve the stated objective. The analysis of the coupled system
i.e., determination of new resonances, modes and the response, is
performed based on the receptance method [28–37]. The sound
field is estimated through the Rayleigh integral [38,39]. The devel-
oped methodology has been applied to three different coupled
systems as presented in the examples. In the first example, we
demonstrate the receptance method through a steel plate attached
with point masses. In the second and third examples, the method-
ology is applied to estimate the sound directivity from a composite
panel with attached point springs and masses. The layup sequence
of the composite panel considered in examples 2 and 3 corre-
sponds to the multifunctional structure battery material system,
used in the MAV [1].
The arrangement of the article is as follows: the receptance
method is introduced in Section 1. Details of the receptance
method are explained in Section 2. Section 3 explains the estima-
tion of sound field using the Rayleigh integral. Numerical examples
are presented in Section 4. In the first example, acoustic directivity
of the plate-mass system is studied. The size(s), location(s) and the
excitation frequencies are varied to arrive at a particular configura-
tion where the directionality is significant. In the second example,
directionality of the composite plate-spring system is studied. The
third example is on estimating the acoustic directivity of a
composite panel with five attached point masses along a line, at
a particular orientation. Section 5 concludes the article.
2. The receptance method
The receptance method is well developed and a detailed
description of the method can be found in [28–37]. With the recep-
tance method, vibrational characteristics of a combined system can
be estimated from the characteristics of the component systems. A
feature of the receptance method is that the receptances of the
component systems may be determined by any method that is suf-
ficiently accurate. In this paper, the receptances are written in
terms of the natural frequencies and modes, which can be obtained
from any finite element programs or through the experiments. The
advantage of the receptance method is that it is a pure analytical
method and the new mode shapes and the natural frequencies of
the combined system are determined in terms of the old mode
shapes and natural frequencies.
Receptance is defined as the ratio of response at a certain point
(location i) to the harmonic force or moment input at the same or
different point (location j), as given below:
aij ¼
Response of system A at location i
Harmonic force or moment input to system A at location j
:
ð1Þ
The response may be either a line displacement or a rotation. The
notation adopted in this paper is as follows: capital letters such as
A, B, C refer to subsystems and the Greek letters a; b; c will denote
the receptances of the subsystems. The material coordinates of a
point in domain X are denoted by X, whose spatial coordinates
are denoted by x. ‘M’ indicates the external mass attached to the
plate and ma
is the mass per unit area of the plate. Note that from
the reciprocity theorem aij ¼ aji [22].
Consider two systems, a plate (system A) and mass(es)/
spring(s) (system B), connected in the domain X at two points 1
and 2 respectively, shown in Fig. 1(a). Let the combined system
be subjected to harmonic excitation and a’s denote the receptance
Fig. 1. Two systems connected at two points. (a) Two masses (system B) m1 and m2 connected at points 1 and 2 on a rectangular plate (system A). (b) Equilibrium
displacements and force distribution on systems A and B, subjected to harmonic excitation.
P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140 129
3. of system A. The force displacement relationship of system A at
equilibrium (refer Fig. 1(b)) are given by [37]
XA1
XA2
¼
a11 a12
a21 a22
FA1
FA2
ð2Þ
where XA1 and XA2 are the displacements and FA1 and FA2 are the
amplitude of forces of system A, at locations 1 and 2, respectively.
The matrix notation of Eq. (2) is given below
fXAg ¼ ½aŠfFAg: ð3Þ
Similarly, the equilibrium equations of the system B are given by
[37]
XB1
XB2
¼
b11 b12
b21 b22
FB1
FB2
ð4Þ
and the matrix notation of Eq. (4) is
fXBg ¼ b½ ŠfFBg: ð5Þ
Thus, a11; a22; b11 and b22 denote the drive point receptances and
a12; a21; b12, and b21 denote the cross receptances. According to
Eq. (1) a11 is the displacement at point 1 due to a unit force applied
at point 1 and a12 is the displacement at point 1 due to a unit force
applied at point 2. When two such systems are joined together,
forces FA and FB become internal forces and they have to add to zero.
Therefore
fFAg ¼ ÀfFBg: ð6Þ
and the continuity of the displacements at the point of contact
yields
fXAg ¼ fXBg ð7Þ
Substituting Eqs. (3) and (5) in Eq. (7) and after simplifying using
Eq. (6), we have
ða11 þ b11ÞFA1 þ ða12 þ b12ÞFA2 ¼ 0 ð8Þ
ða21 þ b21ÞFA1 þ ða22 þ b22ÞFA2 ¼ 0
Eq. (8) can be expressed in the matrix notation as
½aŠ þ ½bŠ½ ŠfFAg ¼ 0: ð9Þ
A non-trivial solution of Eq. (9) can be found by setting the
determinant of the receptance matrix to zero, i.e.,
½aŠ þ ½bŠj j ¼ 0: ð10Þ
2.1. Receptance of coupled system
In this section, the receptance of the plate-mass(es)/spring(s) is
estimated. Based on Eq. (1) inorder to compute the receptance of a
system, the forcing and the response functions are required to be
estimated. In the present work, the panel is excited with a har-
monic point force. The harmonic point force response of a rectan-
gular panel can be expressed in terms of mode shapes and natural
frequencies [40]. The natural frequency is a function of the system
characteristics and the mode shape depends on the boundary con-
ditions. Therefore, Section 2.1.1 explains the calculation of natural
frequencies and the mode shapes of the clamped rectangular metal
panel. Estimation of the natural frequencies and the mode shapes
of a composite panel for the simply supported boundary conditions
are explained in Section 2.1.2. Section 2.1.3 is focussed on estimat-
ing the receptance of the plate-mass system and the receptance of
the plate-spring system is calculated in Section 2.1.4.
The damped harmonic response of a rectangular panel
subjected to a harmonic point force excitation at point (xp; yp) is
given by [41],
wðx; y; tÞ ¼
1
maCmn
X1
m¼1
X1
n¼1
Umnðxp; ypÞFejxt
ðx2
mn À x2 þ 2jfmnxmnxÞ
Umnðx; yÞ ð11Þ
where ma
is the mass per unit area of the plate, F is the force
amplitude, xmn is the ðm; nÞth natural frequency, x is the driving
frequency and Umn represent the mode shapes. The mass per unit
area (ma
) is estimated as
ma
¼
qh for the metal plate
q1h1 þ q2h2 þ Á Á Á þ qnhn for the composite plate
ð12Þ
where q and h are the density and thickness of the metal plate,
respectively and qn and hn are the density and thickness of the
nth layer of the composite plate, respectively. The mode shapes of
a rectangular panel can be expressed as the product of the mode
shape functions along the x and y directions [40], as given below
Umnðx; yÞ ¼ XðxÞYðyÞ ð13Þ
where X and Y are chosen as the fundamental mode shapes of the
beam. Estimation of the modes for the clamped rectangular metal
panel and the rectangular simply supported composite panel are
explained in Sections 2.1.1 and 2.1.2, respectively. The constant
Cmn can be evaluated as,
Cmn ¼
Z a
0
Z b
0
U2
mndxdy: ð14Þ
If k is the equivalent viscous damping factor, the modal damping
coefficient fmn is given by
fmn ¼
k
2maxmn
ð15Þ
2.1.1. Modes of the rectangular metal plate
The expressions for the mode shapes (X) of a beam having the
clamped boundary conditions are given below [40],
XðxÞ ¼ cosc1
x
a
À
1
2
þ
sinðc1
2
Þ
sinhðc1
2
Þ
coshc1
x
a
À
1
2
for m ¼ 2;4;6;...
ð16Þ
where the values of c1 are obtained as roots of
tan
c1
2
þ tanh
c1
2
¼ 0 ð17Þ
and
XðxÞ ¼ sinc2
x
a
À
1
2
À
sinðc2
2
Þ
sinhðc2
2
Þ
sinhc2
x
a
À
1
2
for m ¼ 1;3;5;...
ð18Þ
where the values of c2 are obtained as roots of
tan
c2
2
À tanh
c2
2
¼ 0 ð19Þ
The functions Y in Eq. (13) are obtained by replacing x by y, a by b
and m by n in Eqs. (16)–(19). The natural frequencies xmn of the
clamped plate are given by [40],
x2
mn ¼
p4
D
a4q
G4
x þ G4
y
a
b
4
þ 2
a
b
2
½lHxHy þ ð1 À lÞJxJyŠ
ð20Þ
where D is expressed as
D ¼
Eh
2
12ð1 À l2Þ
: ð21Þ
The expressions for Gx; Gy; Hx; Hy; Jx and Jy are given in Table 1.
130 P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140
4. 2.1.2. Modes of the rectangular composite plate
The damped point force response of a rectangular composite
plate due to a harmonic point force at ðxp; ypÞ is given by Eq. (11),
where ma
is estimated based on Eq. (12). We consider the simply
supported boundary conditions in the analysis of composite pan-
els, where the mode shapes are given by [42]
Umn ¼ sin
mpx
a
sin
npy
b
ð22Þ
The natural frequencies xc
mn of the simply supported rectangular
composite plate are given by [42]
x2
mn ¼
p4
q
D11
m
a
4
þ 2ðD12 þ 2D66Þ
m
a
2 n
b
2
þ D22
n
b
4
ð23Þ
the constants D11; D12; D66 and D22 can be calculated as explained
in [42]. The constant Cc
mn is estimated by using Eq. (22) in Eq.
(14). And the constant fc
mn is calculated from Eq. (15), using Eq. (12).
2.1.3. Receptance of the plate-mass system
Consider a rectangular plate attached with two point masses, at
points 1 and 2 as shown in Fig. 1(a). The cross receptances of the
masses are zero, since the force on one mass does not cause the
other mass to respond. Let a’s represent the receptances of the
plate and b’s represent the receptances of the masses. Elements
of the receptance matrix for the plate given in Eq. (2) can be
estimated in the following steps. First, the drive point receptance
aii is given by,
aii ¼
xi
Fiejxt
¼
1
maCmn
X1
m¼1
X1
n¼1
Umnðxi; yiÞ
ðx2
mn À x2 þ 2jfmnxmnxÞ
Umnðxi; yiÞ
ð24Þ
and secondly, the cross receptance aij; i – j, is given by
aij ¼
xi
Fjejxt
¼
1
maCmn
X1
m¼1
X1
n¼1
Umnðxj; yjÞ
ðx2
mn À x2 þ 2jfmnxmnxÞ
Umnðxi; yiÞ:
ð25Þ
The receptance of a mass can be estimated from the steady-
state response of a mass subjected to a harmonic force input. From
Newton’s third law
M€xBi ¼ FBiekxt
ð26Þ
since xBi ¼ XBiekxt
, the drive point receptance of the mass bii is given
by,
bii ¼
XBi
FBi
¼ À
1
Mx2
: ð27Þ
The cross receptances of the mass are zero, since the force on one
mass does not cause the other mass to respond. Therefore, as
explained in Section 2, the new natural frequencies (xk) of the plate
attached with two point masses are obtained by solving the below
equation
a11 À 1
mB1x2 a12
a21 a22 À 1
mB2x2
5.
6.
7.
8.
9.
10.
11.
12.
13.
14. ¼ 0: ð28Þ
In the current work, Eq. (28) is solved for the new natural fre-
quencies of the plate-mass system by a numerical procedure. It
can also be solved graphically, by plotting the determinant as a
function of the excitation frequency x and capturing the points
where the determinant is zero. The above procedure can be
extended to the case where N masses are attached to a plate.
New natural frequencies of the plate-mass system can be obtained
by setting the determinant of the N Â N receptance matrix to zero.
The new mode shapes of the plate-mass system can be deter-
mined from the point force response expression of the plate with-
out masses [37]. For the case of a plate attached with single mass,
Eq. (11) gives the new mode shape, when the excitation frequency
x, is set to the new natural frequency xk and ðx; yÞ becomes the
location of the mass. For a plate attached with N point masses there
will be N new resonances. Since the plate is constrained at N
points, 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 eigenvalue of the recep-
tance matrix evaluated at the new natural frequency, xk. Thus, the
new kth mode shape is given by substituting xk for x in Eq. (11)
with an additional summation term as shown below
Ukðx; yÞ ¼
1
maCmn
X1
m¼1
X1
n¼1
PN
i¼1Umnðxmi; ymiÞFik
ðx2
mn À x2
k þ 2jfkxkxÞ
Umnðx; yÞ ð29Þ
where Fik is the ith element of the eigenvector of zero eigenvalue,
corresponding to the kth new natural frequency and ðxmi; ymiÞ is
the location of the ith mass. The response of the plate-mass system
subjected to a point force can now be calculated using the new
mode shapes obtained from Eq. (29) as
wkðx; y; tÞ ¼
1
ma
XN
k¼1
1
Ck
Ukðxp; ypÞFejxt
ðx2
k À x2 þ 2jfkxkxÞ
Ukðx; yÞ ð30Þ
where F is the amplitude of the force at location ðxp; ypÞ, xk the kth
natural frequency, x is the driving frequency, fk the modal damping
coefficient, given by
fk ¼
k
2maxk
ð31Þ
and the constant Ck is evaluated as,
Ck ¼
Z a
0
Z b
0
U2
kdxdy: ð32Þ
2.1.4. Receptance of the plate-spring system
Consider a rectangular plate attached with two linear springs of
stiffness k1 and k2, respectively, as shown in Fig. 2. The displace-
ment and force relationships for the coupled system can be derived
on the similar lines of the plate-mass system shown in Fig. 1. Let
Table 1
Variables to estimate the natural frequencies of the clamped rectangular metal plate.
Variable For m = 1 or n = 1 For all the other modes
Gx 1.506 ðm þ 1Þ À 0:5
Gy 1.506 ðn þ 1Þ À 0:5
Hx 1.248 ððm þ 1Þ À 0:5Þ2
1 À 2
ððmþ1ÞÀ0:5Þp
Hy 1.248 ððn þ 1Þ À 0:5Þ2
1 À 2
ððnþ1ÞÀ0:5Þp
Jx 1.248 ððm þ 1Þ À 0:5Þ2
1 À 2
ððmþ1ÞÀ0:5Þp
Jy 1.248 ððn þ 1Þ À 0:5Þ2
1 À 2
ððnþ1ÞÀ0:5Þp
Fig. 2. Two springs of stiffness k1 and k2 connected at two points on a rectangular
plate.
P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140 131
15. a’s represent the receptances of the plate and b’s represent the
receptances of the springs. The drive point and the cross recep-
tances of the plate aii and aij are given by Eqs. (24) and (25), respec-
tively. The receptance of a spring can be estimated from the
steady-state response of a spring subjected to a harmonic force
input. From Newton’s third law
kxB1 ¼ FB1ejxt
ð33Þ
since xB1 ¼ XB1ejxt
b11 ¼
XB1
FB1
¼
1
k
ð34Þ
Therefore, for the plate attached with two spring system, the new
natural frequencies (xc
k) are estimated by solving the equation,
a11 þ 1
kB1
a12
a21 a22 þ 1
kB2
16.
17.
18.
19.
20.
21.
22.
23.
24.
25. ¼ 0: ð35Þ
The structure of the matrix can now be extended to the case where
N springs are attached, as explained in the previous section. Hence,
the new natural frequencies and the new mode shapes of the plate-
spring system can be obtained from the Eqs. (35) and (29), respec-
tively. Knowing the new natural frequencies and the mode shapes,
the point force response of the plate-spring system can be
estimated from Eq. (30).
3. Sound power calculation
During the normal vibrations of a rectangular plate, the normal
velocity of the acoustic medium on the surface of the plate loaded
at r1 must be equal to the normal velocity of the plate v(r1), refer
Fig. 3. Due to the acoustic perturbation on the plate surface, the
generated acoustic pressure p(r2) at r2 can be estimated by the
Rayleigh’s integral [43,38] as given below
pðr2; tÞ ¼
jq0x
2p
ejxt
Z
S1
vnðr1; tÞeÀjkR
R
dS1 ð36Þ
where r2 is the position vector of the observation point, r1 the posi-
tion vector of the elemental surface dS1 having the normal velocity
vnðr1Þ; R ¼j r2 À r1 j; q0 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 as shown in Fig. 3, when r2 is
much larger than the source size as defined by the larger edge of the
two panel dimensions a and b, i.e r2 ) a and a b, R and r2 are
related by the approximate relationship [43]
R ’ r2 À x sin h cos / À y sin h sin /: ð37Þ
The sound intensity is given by the time-averaged product of the
sound pressure and the particle velocity vector. In the far field,
the component of acoustic particle velocity in phase with the pres-
sure is radially directed. As a result, the sound intensity vector is
also radially directed and given by the product of the sound pres-
sure and the radial component of the particle velocity. Therefore,
for harmonic motion the time averaged sound intensity in the far
field is given by [43]
I ¼
j pðr; h; /; xÞj2
2q0c
: ð38Þ
In otherwords, I is the time-averaged rate of energy transmission
through a unit area normal to the direction of propagation. Hence,
the sound power Wp radiated into the semi infinite space above
the plate is the integral of the sound intensity over the panel
surface, which is given by
Wp ¼
Z
S1
Iðr2ÞdS2; ð39Þ
where S2 is an arbitrary surface which covers area S1, see Fig. 3. Eq.
(39) is evaluated by using Eqs. (36)–(38). When the two surfaces
S1; S2 becomes equal i.e, S2 ¼ S1, then r1 and r2 would represent
any two arbitrary position vectors on the plate surface. Therefore,
the radiated sound power from the plate can be expressed as
Wp ¼
q0x
4p
Z
S2
Z
S1
Re vðr1Þ
je
ÀjkR
R
!
vÃ
ðr2Þ
#
dS1dS2 ð40Þ
where S1 and S2 are the areas on the xy plane with 0 x a and
0 y b. Using the Maxwell’s reciprocity relation between source
at r1 and receiver at r2 Eq. (40) can be simplified as
Wp ¼
q0x
4p
Z b
0
Z a
0
Z b
0
Z a
0
vðr1Þ
sinðkRÞ
R
vÃ
ðr2Þdx1dy1
#
dx1dy2:
ð41Þ
4. Numerical examples
In this section three examples are presented. In the first exam-
ple, the sound directivity of the steel plate attached with point
masses is achieved for a particular size(s) and location(s) of the
mass(es), when the system is excited at a specific frequency. The
Fig. 3. A rectangular panel set in a baffle showing the details of the parameters required to estimate the radiated sound power in the far field, along with highlighting the
integration areas S1 and S2.
132 P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140
26. second and third examples focussed on directing the sound from a
composite rectangular panel by attaching point springs and
masses, respectively. The considered composite panel forms part
of the structure battery material used in the fabrication of MAV [1].
4.1. Example 1: Rectangular plate-mass system
In this example, we study the sound directivity of a steel plate-
mass system. The plate dimensions are 380 mm  300 mm along
the x and y directions with a thickness of 1 mm. The density,
Young’s modulus and Poisson’s ratio of the plate are 7815 kg/m3
,
205 GPa and 0.285, respectively. Clamped boundary conditions
are considered for all the edges of the plate. The plate is excited
by a point force of unit amplitude located at (100 mm, 75 mm)
from the bottom left corner of the plate, refer to Fig. 1(a). Natural
frequencies and the point force response of the coupled system
are estimated from the analytical model developed in Section 2.
The radiated sound field is estimated based on the Rayleigh
Fig. 4. Schematic of the experimental setup. (a) Connections of various equipment to the plate to measure the plate response. (b) Plate set in a baffle along with the
microphone in the anechoic chamber to measure the sound radiation from the plate.
Fig. 5. Comparison of natural frequencies from the analytical, numerical and the
experimental models.
Fig. 6. Comparison of the point force response from analytical and numerical models of the plate along a particular line in the x (left) and the y (right) directions, at 500 Hz.
Table 2
Comparison of natural frequencies of the plate attached with three and five-mass
system from analytical and numerical models.
Mode Three masses Five masses
Anal. (Hz) Num. (Hz) % Error Anal. (Hz) Num. (Hz) % Error
1 77.1 76.7 0.519 72.3 71.8 0.692
2 135.1 134.6 0.370 117.3 116.8 0.426
3 180.0 177.5 1.389 173.5 170.4 1.789
4 216.8 215.5 0.600 195.8 192.4 1.736
5 233.0 230.0 1.288 211.5 209.9 0.757
Table 3
Mass locations.
Case Locations (mm) from the bottom corner of the plate
Single mass (76, 60)
Three masses (76, 60), (76, 180) and (150, 190)
Five masses (76, 60), (76, 180), (150, 190), (250, 120) and (310, 200)
P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140 133
27. integral as discussed in Section 3. The analytical results are vali-
dated with the numerical model and the experimental model.
A schematic of the experimental model is shown in Fig. 4. As
shown in Fig. 4(a) the baffled plate is excited with a piezoelectric
exciter and the amplitude of the excitation is controlled through
a piezoelectric controller. The response of the plate is captured
through the response of the piezoelectric accelerometers attached
at several locations of the plate. The total weight of the accelerom-
eters is small compared to the weight of the plate, hence neglected
in the present work. The output signal from the accelerometer is
further analyzed in the spectrum analyser to estimate the natural
frequencies and the modes. The experiment is carried out in an
anechoic chamber in a reverberating room, as shown in Fig. 4(b).
The sound power from the plate is estimated by measuring the
sound pressure levels at several identified locations in the free
field. The measured sound pressure levels are first converted to
sound pressure. Secondly, the sound intensity is estimated using
Eq. (38). Finally, the sound power is calculated based on Eq. (39).
Fig. 5 compares the natural frequencies from the analytical,
numerical and experimental models for the bare plate. The analyt-
ical and numerical models agree with each other. The experimental
model deviates for the higher order modes. The amplitudes of the
displacement between the analytical and numerical models, dis-
placements along two particular lines in the x and y directions
are captured and compared in Fig. 6. Table 2 lists the first five nat-
ural frequencies of the plate with attached masses, from the ana-
lytical and the numerical models. Locations of masses used in the
Fig. 7. Schematic of the locations of masses (a) listed in Table 3 and (b) five infinite masses arranged along a 30° line.
Table 4
Comparison of natural frequencies from analytical and numerical models, of the bare
plate and the plate with five infinite masses aligned at different orientations.
Mode Without
masses (Hz)
Five infinite masses along
15° line (Hz) 30° line (Hz) 45° line (Hz) 60° line (Hz)
1 87.6 120.59 161.78 146.02 112.12
2 152.6 180.56 233.24 233.08 210.80
3 202.1 275.57 241.67 252.34 239.28
4 258.1 289.26 308.04 315.36 326.03
5 262.1 354.83 369.48 388.57 382.84
Fig. 8. Point force response of the plate with five infinite masses oriented along 15°, 30°, 45° and 60° lines at different frequencies. The first row corresponds to the point force
responses close to the first natural frequency and the second row corresponds to responses close to the second natural frequency. The mode bifurcation along the orientation
angle can be observed.
134 P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140
28. calculation are given in Table 3, where a schematic with approxi-
mate locations of the masses listed in Table 3 along with the point
of excitation on the plate is shown in Fig. 7(a) and (b) shows five
infinite masses arranged along the 30° line. Size(s) of the mass(es)
is taken as 50 g each. From Table 2, we observed that the natural
frequencies are decreasing with the addition of masses, as the nat-
ural frequencies are inversely proportional to the mass. The per-
centage error is estimated as the ratio of difference between the
analytical and the numerical results multiplied by 100, as given
below
%error ¼
xanal À xnum
xanal
 100: ð42Þ
where xanal and xnum are the natural frequencies estimated from
the analytical and numerical models, respectively. The minimum
and maximum errors are observed to be 0.370% and 1.789%, occu-
red while estimating the second natural frequency of the plate
attached with 3 masses and the third natural frequency of the plate
attached with 5 masses, respectively.
When the magnitude of the attached mass(es) reaches a large
value, in the limit of infinity, the point of mass attachment will
be completely arrested in all degrees of freedom. Therefore, infinite
masses can be used to restrain and stiffen a particular point. In oth-
erwords, infinite masses increases the stiffness of the system and
hence the natural frequencies. Table 4 summarises the natural fre-
quencies of the plate with five infinite masses aligned along the
15°, 30°, 45° and 60° lines to the x axis. We considered 10000 kg
as the infinite mass in the present study. Since the mass variable
is in the denominator of Eq. (27), the drive point receptances of
the mass (bii) in Eq. (27) will be significantly small compared to
the drive point receptances of the plate (aii) in Eq. (28). Hence,
the value of bii with 10,000 kg mass is very small and its contribu-
tion to the receptance matrix in Eq. (28) can be neglected. Alter-
nately, bii goes to zero as ‘M’ approaches infinity in Eq. (27).
Because of the infinite stiffness(es)/mass(es) at a particular point,
the response of the structure at that particular point will be zero.
Therefore, infinite point mass(es)/spring(s) will act as restraints.
From the values of natural frequencies it can be observed that
Fig. 9. Hemispherical distribution of sound intensity of the plate with; for the case of plate with single point mass excited at 1300 Hz (a) projected on to two dimensional
plane and (b) in three dimensional plane and for the case plate with three point masses excited at 2500 Hz (c) projected on to two dimensional plane and (d) in three
dimensional plane; with permission from [17].
P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140 135
29. the natural frequencies are significantly raised by attaching the
infinite masses. Therefore, infinite masses can be used as stiffeners.
Since the infinite masses acts like restrainers, when they are
aligned along a particular line, the line acts like a rigid boundary.
Hence the mode bifurcation across the line should be observed.
The point force response of the plate at different frequencies with
five infinite masses oriented along 15°, 30°, 45° and 60° lines at
various frequencies are plotted in Fig. 8. Each row corresponds to
the deformed configuration at a particular orientation. The
responses are captured close to the natural frequencies, so that
the mode bifurcation can be observed. At the natural frequencies
the response pattern matches with the mode shape. For example,
the point force responses in the first row are captured close to
the first natural frequency, so that the shape of the response will
be close to the first mode. From Fig. 8(a)–(c), it can be observed
that the first mode is clearly restrained along the constrained line.
Similarly, the second row pictures correspond to the point force
responses of the above plate close to the second natural frequency.
From Fig. 8(d)–(f), the second mode is restrained along the con-
strained line.
The goal of the present work is to achieve the sound direction-
ality. To achieve the objective a Guess and check method is
employed by varying the number, locations of the masses and
the frequency of excitation. A 50 g mass(es) are considered in this
paper. Therefore, several analytical simulations are carried out at
different combinations of number, locations of the masses and
the frequency of excitation, to achieve the directivity. Finally, clear
directivity was observed in 2 cases; (1) single mass excited at
1300 Hz and (2) three masses at 2500 Hz. The locations of the
masses are given in Table 3. The sound intensity contours corre-
sponding to cases 1 and 2 are plotted in Fig. 9. The acoustic inten-
sity is calculated and plotted for different in-plane angles (h) and
out of plane angles (/) values based on Eq. (38). Fig. 9(a) and (b)
shows the intensity distribution of the plate with single mass,
excited at 1300 Hz in two and three dimensions, respectively.
Fig. 9(c) and (d) plots the distribution of sound intensity when
the plate is attached with three point masses and excited at
2500 Hz. From the intensity plots the sound field is clearly directed
by attaching the masses.
Furthermore, to validate the estimated analytical sound power
calculations, sound power levels are calculated from a numerical
model of the completely clamped plate model without attaching
any masses, developed in Sysnoise [44]. On top of that, the analyt-
ical results are verified with the experiment as explained in Fig. 4.
Fig. 10 compares the sound power levels estimated from the analyt-
ical, numerical and the experimental models. The sound power lev-
els from the three models are observed to deviate by around 10 dB.
Fig. 10. Comparison of the sound power level from the analytical, numerical and
the experimental models.
Fig. 11. Layup sequence of different layers of the composite plate used in example 2.
Table 5
Properties of different layers of the composite plate.
Material Young’s 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 6
Comparison of natural frequencies of the composite plate with zero, two, four and five-spring system with large spring stiffness from analytical and numerical models.
Mode Without springs Two springs along 15° line Four springs along 30° line Five springs along 60° line
Anal. (Hz) Num. (Hz) Error % Anal. (Hz) Num. (Hz) Error % Anal. (Hz) Num. (Hz) Error % Anal. (Hz) Num. (Hz) Error %
1 62.3 62.05 0.40 95.89 93.97 2.00 139.97 137.79 1.55 95.25 92.68 2.69
2 140.7 139.87 0.59 160.98 160.63 0.21 236.43 229.91 2.75 197.27 197.66 À0.19
3 173.8 174.11 À0.17 246.76 242.96 1.54 269.37 270.14 À0.28 238.17 229.23 3.75
4 249.4 246.37 1.21 283.37 279.97 1.20 355.63 353.07 0.72 334.78 337.23 À0.73
5 272.9 274.59 À0.61 312.97 314.58 À0.51 395.89 393.34 0.64 400.67 392.52 2.03
136 P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140
30. 4.2. Example 2: Rectangular composite plate-spring system
Composite structures are extensively used in all the engineering
disciplines due to their high strength to weight ratios, ease of han-
dling and transportation. Metallic structures do not offer the flex-
ibility to achieve the required response. Replacement of the
metallic structure with composite structures can be aeroelastically
tailored [45,46]. In this example, we estimate the sound radiation
from the composite rectangular panels. Fig. 11 shows the lay up
sequence along with the orientation angles of different layers of
a panel structure used in a typical MAV [1]. Plastic Lithium Ion
(PLI) is the main battery of the structure, offering additional struc-
tural support and carbon epoxy layers resists the mechanical the
loads on to the structure. Packaging material is used to couple
the carbon epoxy layer with PLI. Combining structure and battery
(power) functions in a single material entity permits improve-
ments in system performance, which is not possible through inde-
pendent subsystem optimization.
Consider a 0.4 m  0.3 m panel, with the material properties,
thicknesses and the orientations of the individual laminas listed
in Table 5. The simply supported boundary conditions are consid-
ered on all the edges. The natural frequencies of the composite
panel attached with linear springs along a particular line, aligned
in different orientations are listed in Table 6. Table 6 also compares
the natural frequencies estimated from the analytical and the
numerical models and the %error estimated from Eq. (42) is also
listed. Each spring of infinite stiffness is considered in all the calcu-
lations. A spring of infinite stiffness acts as a restraint, hence the
point of attachment acts as a node. Therefore, infinite stiffness
springs can be used to restrain and stiffen a particular point. In oth-
erwords, infinite stiffness springs increases the stiffness of the sys-
tem and hence the natural frequencies. Table 6 summarises the
natural frequencies of the plate with five infinite stiffness springs
aligned along lines at 15°, 30°, and 60°. Ref. [47] for the acoustic
radiation of a rectangular plate reinforced by springs at arbitrary
locations. From the values of natural frequencies it can be observed
that the natural frequencies are raised by attaching large stiffness
springs. The minimum and maximum errors are observed to be
À0.73% and 3.75%, occured while estimating the fourth and third
natural frequencies of the plate attached with five springs of infi-
nite stiffness along the 60° line, respectively. Fig. 12 plots the mode
shapes and the point force response of the composite rectangular
panel. The first mode at 95.89 Hz when the springs are aligned
along the 15° line is plotted in Fig. 12(a) and the second mode at
Fig. 12. Mode shapes and response of the composite rectangular panel. (a) First mode, springs aligned along the 15° line at 95.89 Hz. (b) Second mode, springs aligned along
the 30° line at 236.43 Hz. (c) First mode, springs aligned along the 60° line at 95.25 Hz. (d) Point force response of the panel at 150 Hz with springs aligned along the 60° angle.
P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140 137
31. 236.43 Hz when the springs are aligned along the 30° line is shown
in Fig. 12(b). Fig. 12(c) shows the first mode at 95.25 Hz when the
springs are aligned along 60° line and the point force response of
the panel at 150 Hz with springs aligned along 60° angle is plotted
in Fig. 12(d). As explained the first example, the mode bifurcation
is observed in Fig. 12. Fig. 13(a) shows the acoustic intensity
distribution of the panel constrained with five springs of infinite
stiffness along the 60° line at 200 Hz. The acoustic directivity is
observed at much lower frequencies in the composite plates, com-
pared to the steel plate. Variation of the sound power with the con-
straint angle is plotted in Fig. 13(b).
4.3. Example 3: Rectangular composite plate-mass system
In this example, we want to study the acoustic directivity of the
composite rectangular panel attached with five masses along the
60° line. The layers of the composite panel and their sequence is
the same as in example 2.
Consider five point masses of 50 g each attached along the 60°
line. The natural frequencies of the bare composite plate with sim-
ply supported boundary conditions on all the edges are listed in the
second column of Table 6. Table 7 compares the first five natural
frequencies from the analytical and the numerical models of the
composite plate attached with five masses along the 60° line. The
point force response contours of the panel at 200 Hz from the
analytical and the numerical models are plotted in Figs. 13 and
14(b), respectively. The corresponding hemispherical distribution
Fig. 13. (a) Hemispherical distribution of the sound Intensity of the composite plate with five springs of infinite stiffness along the 60° line at 200 Hz. (b) Variation of sound
power with the constraint angle; with permission from [18].
Table 7
Comparison of the natural frequencies from the analytical and the numerical models
of the composite plate attached with five masses along the 60° line.
Mode Analytical (Hz) Numerical (Hz)
1 54.51 54.21
2 118.01 116.20
3 149.52 150.18
4 218.44 218.70
5 245.81 241.28
Fig. 14. Point force response of the composite panel at 200 Hz, attached with five masses attached along the 60° line from (a) the analytical model and (b) the numerical
model. The response plot in (a) is generated through the current method and the response plot in (b) is produced through MSC Nastran, for comparison. The pattern and the
amplitudes of the response at several locations from both the models are observed to be in agreement.
138 P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140
32. of the sound intensity in two and three dimensions is plotted in
Figs. 14 and 15(b), respectively.
5. Conclusions
We developed a methodology to analyze a coupled plate-mass/
spring system based on the receptance method. In the first exam-
ple, we applied the developed methodology for a completely
clamped steel plate-mass system set in a baffle, to estimate the
new natural frequencies and the structural response of the coupled
system. The estimated structural response is validated with com-
mercial software. The hemispherical acoustic field is estimated
from the structural response based on the Rayleigh integral. Sound
powers are compared with the results from the Sysnoise [44] for
the uncoupled system.
The methodology is further extended for a system of a plate
attached with very large (infinite) masses. Infinite masses will
act as restraints. When such infinite masses are attached along a
line at particular orientation, they form a line constraint. The
developed methodology has been validated for line constraints
along different orientations. Two particular combinations of the
mass(es) location(s), number and frequency of excitation are iden-
tified where the sound is significantly directed.
In the second example, the methodology is extended for a com-
posite rectangular panel set in a baffle with attached springs. We
considered springs of infinite stiffness, so that each spring acts as
a constraint. Hemispherical acoustic field has been estimated along
with the sound power for different orientations of the line con-
straints. The acoustic directivity of the composite panel attached
with five masses along the 60° line is studied in the third example.
The directionality of the sound from the composite panels is
observed at significantly lower frequencies compared to the steel
panel.
A platform is developed to quickly perform the acoustic analysis
of coupled rectangular metal and composite panels set in a baffle
with attached point masses and/or springs. In future we would like
to extend the developed technique
1. To study the acoustic field from a baffled plate attached with an
active damper like a Magneto-Rheological fluid.
2. To estimate the sound field of a rotating system such as motor
where a drift in the predicted natural frequencies is expected
due to the coriolis effect.
Acknowledgements
The first two authors acknowledge the support received from the
Facility for Research in Technical Acoustics (FRITA), Indian Institute
of Science, Bangalore, for carrying out majority of the present work.
The first two authors gratefully acknowledge the technical interac-
tions with Prof. V.R. Sonti, Department of Mechanical Engineering,
Indian Institute of Science. PRB acknowledges the financial support
from the International Research Staff Exchange Scheme (IRSES),
FP7-PEOPLE-2010-IRSES, through the project ‘MultiFrac’.
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