Applied Acoustics 89 (2015) 128–140

Contents lists available at ScienceDirect

Applied Acoustics

journal homepage: www.elsevier .com/locate /apacoust

Directionality of sound radiation from rectangular panels

http://dx.doi.org/10.1016/j.apacoust.2014.09.0060003-682X/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: pattabhib@gmail.com (P.R. Budarapu).

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, Germanyb Assystem India Private Limited, Bangalore, Indiac IFB Automotive Private Limited, Bangalore, Indiad 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 2013Received in revised form 17 August 2014Accepted 5 September 2014

Keywords:ReceptanceAcoustic radiationDirectivity

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. Thereceptance 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 computethe sound field. The point mass/spring locations are arbitrary, but chosen with the objective of attaining aunique 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 shapesand hence the sound field. The problem is more complex when the number of masses/springs areincreased and/or their values are made different. The technique of receptance method is demonstratedthrough 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 springsand 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 ofthe new modes.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The rectangular plate is one of the most widely used structuresin the industrial world. The sound radiation from vibratingunbaffled 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 commonproblem in automobiles, airplanes, industrial machinery, buildingsand electro-acoustical devices, to name a few. Understanding thesound radiation characteristics of these structures is importantfor the researchers to maintain the noise levels within the specifiedlimits. Regular or periodic excitation forces are likely to be experi-enced by the plates when they form part of a structure. The drivingforce spectrum may be composed of a single frequency alone or ofa large number of frequencies. There is usually little that can bedone to change the nature of the driving forces. Therefore,researchers are studying various techniques related to acousticradiation for making engineering systems quieter.

One of the methods is to arrange the design so that the forcesact on a nodal line for the mode shape about to be excited. Thismethod is useful when the applied forces act on a concentratedarea only. Naghshineh et al. [13] proposed material tailoring ofstructures for designing structures that radiate sound inefficientlyin light fluids. They solved the problem in two steps. In the firststep, given a frequency and overall geometry of the structure, asurface velocity distribution for minimum radiation conditionwas found. In the second step, a distribution of Young’s modulusand density distribution was found for the structure such that itexhibits the weak radiator velocity profile as one of its modeshapes. Wodtke and Lamancusa [14] discussed the use of dampinglayers in sound power minimization. They mainly concentrated onthe minimization of sound power radiated from plates under broadband excitation by redistribution of unconstrained damping layers,by assuming the total radiated sound power is represented by thepower radiated at structural resonances. Apart from the abovemethods, 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] studiedthe variation of the sound power from a baffled point-force-drivensimply supported rectangular plate subjected to a line constraint,

P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140 129

as a function of the constraint angle. Sastry et al. [17] have studiedthe sound radiation from the baffled rectangular panels attachedwith point masses and Ramaiah et al. [18,19] have estimated thesound radiation from the baffled composite panels. Xuefeng andLi [20,21] tried to reduce the sound radiation from a plate bymodifying the boundary conditions. Fahy [22] proposed thevibro-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 lowfrequencies, by appropriate placement of sensors and actuators[23,24]. In contrast, global control in the unbounded domains, suchas external radiation is still a challenge [25]. Sonti and Jones [26]have developed a curved piezo-actuator model for active vibrationcontrol of cylindrical shells. Guo and Pan [27] have demonstratedthe active noise control in free field environments. It requiresappreciable hardware and achieves reasonable broadband controlwhen the microphones and speakers are optimally located in thesound field. In the free fields, exact cancellation of sound occursonly when the secondary source is a replica of the primary andplaced at the same location, which cannot happen in practice.

In applications such as stealth in submarines and ships, onealternative might be to achieve the control over the directivity ofthe external radiated sound, rather than attenuating sound totally.Even in industrial applications it is useful to direct the sound awayfrom the work place and make the environment acceptable. Themain objective of the current work is to achieve a change in thedirectivity of a point driven rectangular plate set in a baffle byattaching point masses/springs to it. The strategy here involvesdeliberate changes in the mode shapes of the radiator in order toachieve the stated objective. The analysis of the coupled systemi.e., determination of new resonances, modes and the response, isperformed based on the receptance method [28–37]. The soundfield is estimated through the Rayleigh integral [38,39]. The devel-oped methodology has been applied to three different coupledsystems as presented in the examples. In the first example, wedemonstrate the receptance method through a steel plate attachedwith point masses. In the second and third examples, the method-ology is applied to estimate the sound directivity from a compositepanel with attached point springs and masses. The layup sequenceof 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 receptancemethod is introduced in Section 1. Details of the receptance

Fig. 1. Two systems connected at two points. (a) Two masses (system B) m1 and m2

displacements and force distribution on systems A and B, subjected to harmonic excitat

method are explained in Section 2. Section 3 explains the estima-tion of sound field using the Rayleigh integral. Numerical examplesare presented in Section 4. In the first example, acoustic directivityof the plate-mass system is studied. The size(s), location(s) and theexcitation 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. Thethird example is on estimating the acoustic directivity of acomposite panel with five attached point masses along a line, ata particular orientation. Section 5 concludes the article.

2. The receptance method

The receptance method is well developed and a detaileddescription of the method can be found in [28–37]. With the recep-tance method, vibrational characteristics of a combined system canbe estimated from the characteristics of the component systems. Afeature of the receptance method is that the receptances of thecomponent systems may be determined by any method that is suf-ficiently accurate. In this paper, the receptances are written interms of the natural frequencies and modes, which can be obtainedfrom any finite element programs or through the experiments. Theadvantage of the receptance method is that it is a pure analyticalmethod and the new mode shapes and the natural frequencies ofthe combined system are determined in terms of the old modeshapes 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 ordifferent point (location j), as given below:

aij ¼Response 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. Thenotation adopted in this paper is as follows: capital letters such asA, B, C refer to subsystems and the Greek letters a; b; c will denotethe receptances of the subsystems. The material coordinates of apoint in domain X are denoted by X, whose spatial coordinatesare denoted by x. ‘M’ indicates the external mass attached to theplate and ma is the mass per unit area of the plate. Note that fromthe 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 1and 2 respectively, shown in Fig. 1(a). Let the combined systembe subjected to harmonic excitation and a’s denote the receptance

connected at points 1 and 2 on a rectangular plate (system A). (b) Equilibriumion.

130 P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140

of system A. The force displacement relationship of system A atequilibrium (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 theamplitude of forces of system A, at locations 1 and 2, respectively.The matrix notation of Eq. (2) is given below

fXAg ¼ ½a�fFAg: 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 �fFBg: ð5Þ

Thus, a11; a22; b11 and b22 denote the drive point receptances anda12; a21; b12, and b21 denote the cross receptances. According toEq. (1) a11 is the displacement at point 1 due to a unit force appliedat point 1 and a12 is the displacement at point 1 due to a unit forceapplied 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 �fFBg: ð6Þ

and the continuity of the displacements at the point of contactyields

fXAg ¼ fXBg ð7Þ

Substituting Eqs. (3) and (5) in Eq. (7) and after simplifying usingEq. (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

½a� þ ½b�½ �fFAg ¼ 0: ð9Þ

A non-trivial solution of Eq. (9) can be found by setting thedeterminant of the receptance matrix to zero, i.e.,

½a� þ ½b�j j ¼ 0: ð10Þ

2.1. Receptance of coupled system

In this section, the receptance of the plate-mass(es)/spring(s) isestimated. Based on Eq. (1) inorder to compute the receptance of asystem, the forcing and the response functions are required to beestimated. 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 naturalfrequencies [40]. The natural frequency is a function of the systemcharacteristics and the mode shape depends on the boundary con-ditions. Therefore, Section 2.1.1 explains the calculation of naturalfrequencies and the mode shapes of the clamped rectangular metalpanel. Estimation of the natural frequencies and the mode shapesof a composite panel for the simply supported boundary conditionsare 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 ofthe plate-spring system is calculated in Section 2.1.4.

The damped harmonic response of a rectangular panelsubjected to a harmonic point force excitation at point (xp; yp) isgiven by [41],

wðx; y; tÞ ¼ 1maCmn

X1m¼1

X1n¼1

Umnðxp; ypÞFejxt

ðx2mn �x2 þ 2jfmnxmnxÞ

Umnðx; yÞ ð11Þ

where ma is the mass per unit area of the plate, F is the forceamplitude, xmn is the ðm;nÞth natural frequency, x is the drivingfrequency and Umn represent the mode shapes. The mass per unitarea (ma) is estimated as

ma ¼qh for the metal plateq1h1 þ q2h2 þ � � � 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 thenth layer of the composite plate, respectively. The mode shapes ofa rectangular panel can be expressed as the product of the modeshape 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 thebeam. Estimation of the modes for the clamped rectangular metalpanel and the rectangular simply supported composite panel areexplained in Sections 2.1.1 and 2.1.2, respectively. The constantCmn can be evaluated as,

Cmn ¼Z a

0

Z b

0U2

mndxdy: ð14Þ

If k is the equivalent viscous damping factor, the modal dampingcoefficient fmn is given by

fmn ¼k

2maxmnð15Þ

2.1.1. Modes of the rectangular metal plateThe expressions for the mode shapes (X) of a beam having the

clamped boundary conditions are given below [40],

XðxÞ ¼ cosc1xa� 1

2

� �þ

sinðc12 Þ

sinhðc12 Þ

coshc1xa� 1

2

� �for m ¼ 2;4;6; . . .

ð16Þ

where the values of c1 are obtained as roots of

tanc1

2

� þ tanh

c1

2

� ¼ 0 ð17Þ

and

XðxÞ ¼ sinc2xa� 1

2

� ��

sinðc22 Þ

sinhðc22 Þ

sinhc2xa� 1

2

� �for m ¼ 1;3;5; . . .

ð18Þ

where the values of c2 are obtained as roots of

tanc2

2

� � tanh

c2

2

� ¼ 0 ð19Þ

The functions Y in Eq. (13) are obtained by replacing x by y, a by band m by n in Eqs. (16)–(19). The natural frequencies xmn of theclamped plate are given by [40],

x2mn ¼

p4Da4q

G4x þ G4

yab

� 4þ 2

ab

� 2½lHxHy þ ð1� lÞJxJy�

� �ð20Þ

where D is expressed as

D ¼ Eh2

12ð1� l2Þ : ð21Þ

The expressions for Gx; Gy; Hx; Hy; Jx and Jy are given in Table 1.

Table 1Variables 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 ðmþ 1Þ � 0:5Gy 1.506 ðnþ 1Þ � 0:5Hx 1.248 ððmþ 1Þ � 0:5Þ2 1� 2

ððmþ1Þ�0:5Þp

� Hy 1.248 ððnþ 1Þ � 0:5Þ2 1� 2

ððnþ1Þ�0:5Þp

� Jx 1.248 ððmþ 1Þ � 0:5Þ2 1� 2

ððmþ1Þ�0:5Þp

� Jy 1.248 ððnþ 1Þ � 0:5Þ2 1� 2

ððnþ1Þ�0:5Þp

�

Fig. 2. Two springs of stiffness k1 and k2 connected at two points on a rectangularplate.

P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140 131

2.1.2. Modes of the rectangular composite plateThe 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 simplysupported boundary conditions in the analysis of composite pan-els, where the mode shapes are given by [42]

Umn ¼ sinmpx

a

� sin

npyb

� ð22Þ

The natural frequencies xcmn of the simply supported rectangular

composite plate are given by [42]

x2mn ¼

p4

qD11

ma

� 4þ 2ðD12 þ 2D66Þ

ma

� 2 nb

� 2þ D22

nb

� 4� �

ð23Þ

the constants D11; D12; D66 and D22 can be calculated as explainedin [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 systemConsider a rectangular plate attached with two point masses, at

points 1 and 2 as shown in Fig. 1(a). The cross receptances of themasses are zero, since the force on one mass does not cause theother mass to respond. Let a’s represent the receptances of theplate and b’s represent the receptances of the masses. Elementsof the receptance matrix for the plate given in Eq. (2) can beestimated in the following steps. First, the drive point receptanceaii is given by,

aii ¼xi

Fiejxt¼ 1

maCmn

X1m¼1

X1n¼1

Umnðxi; yiÞðx2

mn �x2 þ 2jfmnxmnxÞUmnðxi; yiÞ

ð24Þ

and secondly, the cross receptance aij; i – j, is given by

aij ¼xi

Fjejxt¼ 1

maCmn

X1m¼1

X1n¼1

Umnðxj; yjÞðx2

mn �x2 þ 2jfmnxmnxÞUmnðxi; yiÞ:

ð25Þ

The receptance of a mass can be estimated from the steady-state response of a mass subjected to a harmonic force input. FromNewton’s third law

M€xBi ¼ FBiekxt ð26Þ

since xBi ¼ XBiekxt , the drive point receptance of the mass bii is givenby,

bii ¼XBi

FBi¼ � 1

Mx2 : ð27Þ

The cross receptances of the mass are zero, since the force on onemass does not cause the other mass to respond. Therefore, asexplained in Section 2, the new natural frequencies (xk) of the plateattached with two point masses are obtained by solving the belowequation

a11 � 1mB1x2 a12

a21 a22 � 1mB2x2

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. Itcan also be solved graphically, by plotting the determinant as afunction of the excitation frequency x and capturing the pointswhere the determinant is zero. The above procedure can beextended to the case where N masses are attached to a plate.New natural frequencies of the plate-mass system can be obtainedby setting the determinant of the N � 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 frequencyx, is set to the new natural frequency xk and ðx; yÞ becomes thelocation of the mass. For a plate attached with N point masses therewill be N new resonances. Since the plate is constrained at Npoints, it will experience point forces at those N locations. Themagnitudes of these point forces are given by the elements ofthe eigenvector corresponding to the zero eigenvalue of the recep-tance matrix evaluated at the new natural frequency, xk. Thus, thenew kth mode shape is given by substituting xk for x in Eq. (11)with an additional summation term as shown below

Ukðx; yÞ ¼1

maCmn

X1m¼1

X1n¼1

PNi¼1Umnðxmi; ymiÞFik

ðx2mn �x2

k þ 2jfkxkxÞUmnðx; yÞ ð29Þ

where Fik is the ith element of the eigenvector of zero eigenvalue,corresponding to the kth new natural frequency and ðxmi; ymiÞ isthe location of the ith mass. The response of the plate-mass systemsubjected to a point force can now be calculated using the newmode shapes obtained from Eq. (29) as

wkðx; y; tÞ ¼1

ma

XN

k¼1

1Ck

Ukðxp; ypÞFejxt

ðx2k �x2 þ 2jfkxkxÞ

Ukðx; yÞ ð30Þ

where F is the amplitude of the force at location ðxp; ypÞ, xk the kthnatural frequency, x is the driving frequency, fk the modal dampingcoefficient, given by

fk ¼k

2maxkð31Þ

and the constant Ck is evaluated as,

Ck ¼Z a

0

Z b

0U2

kdxdy: ð32Þ

2.1.4. Receptance of the plate-spring systemConsider 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 derivedon the similar lines of the plate-mass system shown in Fig. 1. Let

132 P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140

a’s represent the receptances of the plate and b’s represent thereceptances 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 thesteady-state response of a spring subjected to a harmonic forceinput. From Newton’s third law

kxB1 ¼ FB1ejxt ð33Þsince xB1 ¼ XB1ejxt

b11 ¼XB1

FB1¼ 1

kð34Þ

Therefore, for the plate attached with two spring system, the newnatural frequencies (xc

k) are estimated by solving the equation,

a11 þ 1kB1

a12

a21 a22 þ 1kB2

¼ 0: ð35Þ

The structure of the matrix can now be extended to the case whereN 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 beestimated from Eq. (30).

3. Sound power calculation

During the normal vibrations of a rectangular plate, the normalvelocity of the acoustic medium on the surface of the plate loadedat r1 must be equal to the normal velocity of the plate v(r1), referFig. 3. Due to the acoustic perturbation on the plate surface, thegenerated acoustic pressure p(r2) at r2 can be estimated by theRayleigh’s integral [43,38] as given below

pðr2; tÞ ¼jq0x2p

ejxtZ

S1

vnðr1; tÞe�jkR

RdS1 ð36Þ

where r2 is the position vector of the observation point, r1 the posi-tion vector of the elemental surface dS1 having the normal velocityvnðr1Þ; R ¼j r2 � r1 j; q0 is the density of air, k is the acoustic wavenumber and S1 is the area of the plate. Considering a hemisphericalmeasurement surface in the far field as shown in Fig. 3, when r2 ismuch larger than the source size as defined by the larger edge of thetwo panel dimensions a and b, i.e r2 � a and a > b, R and r2 arerelated by the approximate relationship [43]

Fig. 3. A rectangular panel set in a baffle showing the details of the parameters requireintegration areas S1 and S2.

R ’ r2 � x sin h cos /� y sin h sin /: ð37Þ

The sound intensity is given by the time-averaged product of thesound 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 isalso 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 farfield is given by [43]

I ¼ j pðr; h;/;xÞj2

2q0c: ð38Þ

In otherwords, I is the time-averaged rate of energy transmissionthrough a unit area normal to the direction of propagation. Hence,the sound power Wp radiated into the semi infinite space abovethe plate is the integral of the sound intensity over the panelsurface, which is given by

Wp ¼Z

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 surfacesS1; S2 becomes equal i.e, S2 ¼ S1, then r1 and r2 would representany two arbitrary position vectors on the plate surface. Therefore,the radiated sound power from the plate can be expressed as

Wp ¼q0x4p

ZS2

ZS1

Re vðr1Þje�jkR

R

!v�ðr2Þ

" #dS1dS2 ð40Þ

where S1 and S2 are the areas on the xy plane with 0 < x < a and0 < y < b. Using the Maxwell’s reciprocity relation between sourceat r1 and receiver at r2 Eq. (40) can be simplified as

Wp ¼q0x4p

Z b

0

Z a

0

Z b

0

Z a

0vðr1Þ

sinðkRÞR

� �v�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 pointmasses is achieved for a particular size(s) and location(s) of themass(es), when the system is excited at a specific frequency. The

d to estimate the radiated sound power in the far field, along with highlighting the

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 themicrophone in the anechoic chamber to measure the sound radiation from the plate.

Fig. 5. Comparison of natural frequencies from the analytical, numerical and theexperimental models.

Table 2Comparison of natural frequencies of the plate attached with three and five-masssystem 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 0.6922 135.1 134.6 0.370 117.3 116.8 0.4263 180.0 177.5 1.389 173.5 170.4 1.7894 216.8 215.5 0.600 195.8 192.4 1.7365 233.0 230.0 1.288 211.5 209.9 0.757

Table 3Mass 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

second and third examples focussed on directing the sound from acomposite rectangular panel by attaching point springs andmasses, respectively. The considered composite panel forms partof 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

Fig. 6. Comparison of the point force response from analytical and numerical models of

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 conditionsare considered for all the edges of the plate. The plate is excitedby 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). Naturalfrequencies and the point force response of the coupled systemare estimated from the analytical model developed in Section 2.The radiated sound field is estimated based on the Rayleigh

the plate along a particular line in the x (left) and the y (right) directions, at 500 Hz.

Fig. 7. Schematic of the locations of masses (a) listed in Table 3 and (b) five infinite masses arranged along a 30� line.

Table 4Comparison of natural frequencies from analytical and numerical models, of the bareplate and the plate with five infinite masses aligned at different orientations.

Mode Withoutmasses (Hz)

Five infinite masses along

15� line (Hz) 30� line (Hz) 45� line (Hz) 60� line (Hz)

1 87.6 120.59 161.78 146.02 112.122 152.6 180.56 233.24 233.08 210.803 202.1 275.57 241.67 252.34 239.284 258.1 289.26 308.04 315.36 326.035 262.1 354.83 369.48 388.57 382.84

134 P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140

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. Asshown in Fig. 4(a) the baffled plate is excited with a piezoelectricexciter and the amplitude of the excitation is controlled througha piezoelectric controller. The response of the plate is capturedthrough the response of the piezoelectric accelerometers attached

Fig. 8. Point force response of the plate with five infinite masses oriented along 15�, 30�, 4responses close to the first natural frequency and the second row corresponds to responsangle can be observed.

at several locations of the plate. The total weight of the accelerom-eters is small compared to the weight of the plate, hence neglectedin the present work. The output signal from the accelerometer isfurther analyzed in the spectrum analyser to estimate the naturalfrequencies and the modes. The experiment is carried out in ananechoic chamber in a reverberating room, as shown in Fig. 4(b).The sound power from the plate is estimated by measuring thesound pressure levels at several identified locations in the freefield. The measured sound pressure levels are first converted tosound pressure. Secondly, the sound intensity is estimated usingEq. (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 experimentalmodel deviates for the higher order modes. The amplitudes of thedisplacement between the analytical and numerical models, dis-placements along two particular lines in the x and y directionsare 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

5� and 60� lines at different frequencies. The first row corresponds to the point forcees close to the second natural frequency. The mode bifurcation along the orientation

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 dimensionalplane 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 threedimensional plane; with permission from [17].

P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140 135

calculation are given in Table 3, where a schematic with approxi-mate locations of the masses listed in Table 3 along with the pointof excitation on the plate is shown in Fig. 7(a) and (b) shows fiveinfinite 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 naturalfrequencies 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 theanalytical and the numerical results multiplied by 100, as givenbelow

%error ¼ xanal �xnum

xanal� 100: ð42Þ

where xanal and xnum are the natural frequencies estimated fromthe analytical and numerical models, respectively. The minimumand maximum errors are observed to be 0.370% and 1.789%, occu-red while estimating the second natural frequency of the plateattached with 3 masses and the third natural frequency of the plateattached with 5 masses, respectively.

When the magnitude of the attached mass(es) reaches a largevalue, in the limit of infinity, the point of mass attachment willbe completely arrested in all degrees of freedom. Therefore, infinitemasses can be used to restrain and stiffen a particular point. In oth-erwords, infinite masses increases the stiffness of the system andhence the natural frequencies. Table 4 summarises the natural fre-quencies of the plate with five infinite masses aligned along the15�, 30�, 45� and 60� lines to the x axis. We considered 10000 kgas the infinite mass in the present study. Since the mass variableis in the denominator of Eq. (27), the drive point receptances ofthe mass (bii) in Eq. (27) will be significantly small compared tothe 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. 10. Comparison of the sound power level from the analytical, numerical andthe experimental models.

136 P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140

the natural frequencies are significantly raised by attaching theinfinite masses. Therefore, infinite masses can be used as stiffeners.

Since the infinite masses acts like restrainers, when they arealigned 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 withfive infinite masses oriented along 15�, 30�, 45� and 60� lines atvarious frequencies are plotted in Fig. 8. Each row corresponds tothe deformed configuration at a particular orientation. Theresponses are captured close to the natural frequencies, so thatthe mode bifurcation can be observed. At the natural frequenciesthe response pattern matches with the mode shape. For example,the point force responses in the first row are captured close tothe first natural frequency, so that the shape of the response willbe close to the first mode. From Fig. 8(a)–(c), it can be observed

Fig. 11. Layup sequence of different layers of

Table 5Properties of different layers of the composite plate.

Material Young’s modulus E (GPa) Density (kg/m3

Carbon Epoxy 72.40 1100Packaging 4.60 1290PLI 1.02 2540

Table 6Comparison of natural frequencies of the composite plate with zero, two, four and five-sp

Mode Without springs Two springs along 15� line

Anal. (Hz) Num. (Hz) Error % Anal. (Hz) Num. (Hz) Error %

1 62.3 62.05 0.40 95.89 93.97 2.002 140.7 139.87 0.59 160.98 160.63 0.213 173.8 174.11 �0.17 246.76 242.96 1.544 249.4 246.37 1.21 283.37 279.97 1.205 272.9 274.59 �0.61 312.97 314.58 �0.51

that the first mode is clearly restrained along the constrained line.Similarly, the second row pictures correspond to the point forceresponses 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 isemployed by varying the number, locations of the masses andthe frequency of excitation. A 50 g mass(es) are considered in thispaper. Therefore, several analytical simulations are carried out atdifferent combinations of number, locations of the masses andthe frequency of excitation, to achieve the directivity. Finally, cleardirectivity was observed in 2 cases; (1) single mass excited at1300 Hz and (2) three masses at 2500 Hz. The locations of themasses 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) andout 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 whenthe plate is attached with three point masses and excited at2500 Hz. From the intensity plots the sound field is clearly directedby attaching the masses.

Furthermore, to validate the estimated analytical sound powercalculations, sound power levels are calculated from a numericalmodel of the completely clamped plate model without attachingany 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.

the composite plate used in example 2.

) Thickness (mm) Orientation (Deg) (No. of Layers)

0.15 0/45/45/0 (4 Layers)0.20 0/0 (2 Layers)0.50 0/0 (2 Layers)

ring system with large spring stiffness from analytical and numerical models.

Four springs along 30� line Five springs along 60� line

Anal. (Hz) Num. (Hz) Error % Anal. (Hz) Num. (Hz) Error %

139.97 137.79 1.55 95.25 92.68 2.69236.43 229.91 2.75 197.27 197.66 �0.19269.37 270.14 �0.28 238.17 229.23 3.75355.63 353.07 0.72 334.78 337.23 �0.73395.89 393.34 0.64 400.67 392.52 2.03

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 alongthe 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

4.2. Example 2: Rectangular composite plate-spring system

Composite structures are extensively used in all the engineeringdisciplines 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 themetallic structure with composite structures can be aeroelasticallytailored [45,46]. In this example, we estimate the sound radiationfrom the composite rectangular panels. Fig. 11 shows the lay upsequence along with the orientation angles of different layers ofa 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 theloads on to the structure. Packaging material is used to couplethe 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 listedin 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, alignedin different orientations are listed in Table 6. Table 6 also comparesthe natural frequencies estimated from the analytical and thenumerical models and the %error estimated from Eq. (42) is alsolisted. Each spring of infinite stiffness is considered in all the calcu-lations. A spring of infinite stiffness acts as a restraint, hence thepoint of attachment acts as a node. Therefore, infinite stiffnesssprings 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 thenatural frequencies of the plate with five infinite stiffness springsaligned along lines at 15�, 30�, and 60�. Ref. [47] for the acousticradiation of a rectangular plate reinforced by springs at arbitrarylocations. From the values of natural frequencies it can be observedthat the natural frequencies are raised by attaching large stiffnesssprings. The minimum and maximum errors are observed to be�0.73% and 3.75%, occured while estimating the fourth and thirdnatural frequencies of the plate attached with five springs of infi-nite stiffness along the 60� line, respectively. Fig. 12 plots the modeshapes and the point force response of the composite rectangularpanel. The first mode at 95.89 Hz when the springs are alignedalong the 15� line is plotted in Fig. 12(a) and the second mode at

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 soundpower with the constraint angle; with permission from [18].

Table 7Comparison of the natural frequencies from the analytical and the numerical modelsof the composite plate attached with five masses along the 60� line.

Mode Analytical (Hz) Numerical (Hz)

1 54.51 54.212 118.01 116.203 149.52 150.184 218.44 218.705 245.81 241.28

138 P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140

236.43 Hz when the springs are aligned along the 30� line is shownin Fig. 12(b). Fig. 12(c) shows the first mode at 95.25 Hz when thesprings are aligned along 60� line and the point force response ofthe panel at 150 Hz with springs aligned along 60� angle is plottedin Fig. 12(d). As explained the first example, the mode bifurcationis observed in Fig. 12. Fig. 13(a) shows the acoustic intensitydistribution of the panel constrained with five springs of infinitestiffness along the 60� line at 200 Hz. The acoustic directivity is

Fig. 14. Point force response of the composite panel at 200 Hz, attached with five massmodel. The response plot in (a) is generated through the current method and the responsamplitudes of the response at several locations from both the models are observed to b

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 thecomposite rectangular panel attached with five masses along the60� line. The layers of the composite panel and their sequence isthe 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 thesecond column of Table 6. Table 7 compares the first five naturalfrequencies from the analytical and the numerical models of thecomposite plate attached with five masses along the 60� line. Thepoint force response contours of the panel at 200 Hz from theanalytical and the numerical models are plotted in Figs. 13 and14(b), respectively. The corresponding hemispherical distribution

es attached along the 60� line from (a) the analytical model and (b) the numericale plot in (b) is produced through MSC Nastran, for comparison. The pattern and thee in agreement.

Fig. 15. Hemispherical distribution of the sound intensity with five masses attached along the 60� line excited at 200 Hz (a) when projected on to a two dimensional planeand (b) in three dimensions; with permission from [19].

P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140 139

of the sound intensity in two and three dimensions is plotted inFigs. 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 completelyclamped steel plate-mass system set in a baffle, to estimate thenew natural frequencies and the structural response of the coupledsystem. The estimated structural response is validated with com-mercial software. The hemispherical acoustic field is estimatedfrom the structural response based on the Rayleigh integral. Soundpowers are compared with the results from the Sysnoise [44] forthe uncoupled system.

The methodology is further extended for a system of a plateattached with very large (infinite) masses. Infinite masses willact as restraints. When such infinite masses are attached along aline at particular orientation, they form a line constraint. Thedeveloped methodology has been validated for line constraintsalong different orientations. Two particular combinations of themass(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. Weconsidered springs of infinite stiffness, so that each spring acts asa constraint. Hemispherical acoustic field has been estimated alongwith the sound power for different orientations of the line con-straints. The acoustic directivity of the composite panel attachedwith five masses along the 60� line is studied in the third example.The directionality of the sound from the composite panels isobserved at significantly lower frequencies compared to the steelpanel.

A platform is developed to quickly perform the acoustic analysisof coupled rectangular metal and composite panels set in a bafflewith attached point masses and/or springs. In future we would liketo extend the developed technique

1. To study the acoustic field from a baffled plate attached with anactive damper like a Magneto-Rheological fluid.

2. To estimate the sound field of a rotating system such as motorwhere a drift in the predicted natural frequencies is expecteddue to the coriolis effect.

Acknowledgements

The first two authors acknowledge the support received from theFacility for Research in Technical Acoustics (FRITA), Indian Instituteof 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 supportfrom the International Research Staff Exchange Scheme (IRSES),FP7-PEOPLE-2010-IRSES, through the project ‘MultiFrac’.

References

[1] Thomas JP, Qidwai MA. The design and application of multifunctionalstructure-battery materials systems. JOM: J Miner, Met Mater Soc (TMS)2005;57(3):18–24.

[2] Williams EG. Numerical evaluation of the radiation from unbaffled finitepanels using the FFT. J Acoust Soc Am 1983;74:343–7.

[3] Oppenheimer CH, Dubowsky S. A radiation efficiency of unbaffled plates withexperimental validation. J Sound Vib 1997;199(3):473–89.

[4] Putra A, Thompson DJ. Sound radiation from rectangular baffled and unbaffledplates. Appl Acoust 2010;71:1113–25.

[5] Gomperts MC. Sound radiation from baffled, thin, rectangular plates. Acustica1977;37:93–102.

[6] Berry Alain, Guyader Jean-Louis, Nicolas Jean. A general formulation for thesound radiation from rectangular, baffled plates with arbitrary boundaryconditions. J Acoust Soc Am 1990;88(6):2792–802.

[7] Xuefeng Z, Li WL. A unified approach for predicting sound radiation frombaffled rectangular plates with arbitrary boundary conditions. J Sound Vib2010;329:5307–20.

[8] Mattei PO. Sound energy by baffled and constrained plates. J Sound Vib1995;179(1):63–77.

[9] Sonti VR. Sound radiation from a baffled rectangular plate under differentorientations of a line constraint. J Sound Vib 2003;265(1):235–43.

[10] Van de Loock J, Décultot D, Léon F, Chati F, Maze G, Rajaona DR, et al. Acousticradiation of a submerged cylindrical shell in low frequency. J Acoust Soc Am2013;133(1):26–32.

[11] Piquette JC. Direct measurements of edge diffraction from soft underwateracoustic panels. J Acoust Soc Am 1994;95(6):3090–9.

[12] Wu Zhangming, Ma Xianghong, Brett PN, Xu Jinwu. Vibration analysis ofsubmerged rectangular microplates with distributed mass loading. Proc RoySoc A 2009;465:1323–36.

[13] Naghshineh K, Koopman GH, Belegundu AD. Material tailoring of structures toachieve a minimum radiation condition. J Acoust Soc Am 1992;92(2):841–55.

[14] Wodtke HW, Lamancusa JS. Sound power minimization of circular platesthrough damping layer placement. J Sound Vib 1998;215(5):1145–63.

[15] Jog CS. Reducing radiated sound power by minimizing the dynamiccompliance. In: International union of theoretical and applied mechanics(IUTAM). Bangalore, India: Indian Institute of Science; 2000.

[16] St Pierre RL, Koopmann GH. Minimization of radiated sound power from platesusing distributed masses. In: ASME winter annual meeting, paper no. 93-WA/NCA-11. New Orleans, LA; 1993.

140 P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140

[17] Sastry YBS, Narayana TSS, Ramaiah BP. Studies on acoustic radiation fromclamped rectangular panels with attached point masses. In: Proceedings ofVETOMAC IV, international conference on engineering and technology ofmachinery. Hyderabad, India: University College of Engineering, OsmaniaUniversity; 2007.

[18] Ramaiah BP, Rammohan B, Narayana TSS. Studies on acoustic radiation fromrectangular composite panels. In: Proceedings of INCCOM IV, internationalconference on future trends in composite materials and processing. IIT Kanpur,India; 2007.

[19] Ramaiah BP, Rammohan B, Babu DS, Raghunathan R. Sound radiation from aclamped composite rectangular panel under a line constraint. In: Proceedingsof ICTACEM, 4th international conference on theoretical appliedcomputational and experimental mechanics. IIT Kharagpur, India; 2007.

[20] Xuefeng Z, Li WL. Reducing sound radiation from a plate through tuning/modifying boundary supports. In: Proceedings of the IMAC-XXVII. Orlando,Florida, USA; 2009.

[21] Li WL. Vibroacoustic analysis of rectangular plates with elastic rotational edgerestraints. J Acoust Soc Am 2006;120(2):769–79.

[22] Fahy FJ. The vibro-acoustic reciprocity principle and applications to noisecontrol. Acustica 1995;81:544–58.

[23] Elliott SJ, Curtis ARD, Bullmore AJ, Nelson PA. The active minimization ofharmonic enclosed sound fields, Part III: Experimental verification. J Sound Vib1987;117(1):35–58.

[24] Elliott SJ, Johnson ME. Radiation modes and the active control of sound power.J Acoust Soc Am 1993;94:2194–204.

[25] Cunefare KA, Koopmann GH. Global optimum active noise control: surface andfar field effects. J Acoust Soc Am 1991;90(1):365–73.

[26] Sonti VR, Jones JD. Curved piezo-actuator model for active vibration control ofcylindrical shells. Am Inst Aeronaut Astronaut J 1996;34(5):1034–40.

[27] Guo Jignan, Pan Jie. Actively created quiet zones for broad band noise usingmultiple control sources and error microphones. J Acoust Soc Am1999;105(4):2294–303.

[28] Wilken ID, Soedel W. The receptance method applied to ring stiffenedcylindrical shells: analysis of modal characteristics. J Sound Vib1976;44(4):563–76.

[29] Wilken ID, Soedel W. Simplified prediction of the modal characteristics of ring-stiffened cylindrical shells. J Sound Vib 1976;44(4):577–89.

[30] Azimi S, Hamilton JF, Soedel W. The receptance method applied to thefree vibration of continuous rectangular plates. J Sound Vib 1984;93(1):9–29.

[31] Azimi S, Hamilton JF, Soedel W. Natural frequencies and modes of cylindricalpolygonal ducts using receptance methods. J Sound Vib 1986;109(1):79–89.

[32] Soedel W. Remarks on receptance techniques. J Sound Vib 1987;115(2):365–6.[33] Allaei D, Soedel W, Yang TY. Eigenvalues of rings with radial spring

attachments. J Sound Vib 1987;121(3):547–61.[34] Huang DT, Soedel W. Study of the forced vibration of shell-plate combinations

using the receptance method. J Sound Vib 1993;166(2):341–69.[35] Soedel DT, Soedel W. Synthesizing reduced systems by complex receptances. J

Sound Vib 1994;179(5):855–67.[36] Huang DT, Ting EC. Vibration of plates with sub-structural deduction: a

reverse receptance approach. J Sound Vib 2004;271:177–207.[37] Soedel W. Vibrations of shells and plates. USA: CRC Press; 2004.[38] Lord Rayleigh. The theory of sound. London: McMillan and Co; 1894.[39] Williams EG, Maynard JD. Numerical evaluation of the Rayleigh integral for

planar radiators using the FFT. J Acoust Soc Am 1982;72(6):2020–30.[40] Leissa AW. Vibration of plates. NASA SP-160, USA; 1969.[41] Cremer L, Heckl M, Petersson BAT. Structure-borne sound: structural

vibrations and sound radiation at audio frequencies. Berlin, Heidelberg,Germany: Springer-Verlag; 2005.

[42] Jones RM. Mechanics of composite materials. London: Taylor and Francis Inc.;1998.

[43] Fahy FJ, Paolo Gardonio. Sound and structural vibration: radiation,transmission and response. Oxford, UK: Academic Press; 2007.

[44] Sysnoise. Sysnoise Rev 5.6, user’s manual. Leuven, Belgium: LMS NumericalTechnologies; 2013.

[45] Walker WP, Patil MJ, Canfield RA. Aeroelastic tailoring of flapping membranewings for maximum thrust and propulsive efficiency. In 12th AIAA aviationtechnology, integration, and operations (ATIO) conference and 14th AIAA/ISSM, paper no. AIAA 2012-5693, Indianapolis, Indiana, USA; 2012.

[46] Ramaiah BP, Rammohan B, Kumar SV, Babu DS, Raghuatnhan R. Aero-elasticanalysis of stiffened composite wing structure. Adv Vib Eng 2009;8(3):255–64.

[47] Li WL, Gibeling HJ. Acoustic radiation from a rectangular plate reinforced bysprings at arbitrary locations. J Sound Vib 1999;220(1):113–7.