Research ArticleSound Radiation and Vibration of Composite Panels Excited byTurbulent Flow Analytical Prediction and Analysis

Joana Rocha

Department of Mechanical and Aerospace Engineering Carleton University Ottawa ON Canada K1S 5B6

Correspondence should be addressed to Joana Rocha joanarochacarletonca

Received 23 November 2013 Accepted 31 March 2014 Published 28 April 2014

Academic Editor Valder Steffen Jr

Copyright copy 2014 Joana Rocha This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The present study investigates the vibration and sound radiation by panels exited by turbulent flow and by randomnoise Compositeand aluminum panels are analyzed through a developed analytical framework The main objective of this study is to identify thedifference between the vibroacoustic behaviour of these two types of panels This topic is of particular importance given thegrowing interest in applying composite materials for the construction of aircraft structures in parts where aluminum panels weretraditionally being used An original mathematical framework is presented for the prediction of noise and vibration for compositepanels Results show the effect of panel size thickness of core and thickness of face layers on the predictions Smaller compositepanels generally produced lower levels of sound and vibration than longer and wider composite panels Compared with isotropicpanels the composite panels analyzed generated lower noise levels although it was observed that noise level was amplified at certainfrequencies

1 Introduction

The shift towards composite aircraft fuselage and wingstructures and the overall use of new concepts of materialsin aircraft manufacturing carry the need of a better under-standing of structural and acoustic behaviors associated withthese types of materials The present study provides a novelanalytical framework for the prediction of the vibroacousticbehaviour of composite panels Previous models were devel-oped and validated by the author aiming at the prediction ofthe vibroacoustic behaviour of flow-excited isotropic panels[1ndash4] The present work adds a step forward to the previousmodels by advancing the predictive models towards theapplication and analysis of composite panelsrsquo behaviour Asalso shown in previous analysis the mathematical approachpresented has the ability to provide the fast and accurateprediction for various kinds of structural panels specificallyvarying panelsrsquo structural properties panelsrsquo size thicknessnumber of layers and airflow regimes With the growinginterest in composite materials in several fields the researchpresented in this study can be useful for a broad rangeof applications such as aerospace applications automotiveindustry civil engineering and high-speed boats in which

the mechanical behaviour of the material should be wellunderstood

This study considers the case of a turbofan aircraft atcruise flight conditions a situation in which interior noiselevels are governed by the transmitted turbulent boundarylayer noise into the cabin It is known that chronic noisepollution exposure can cause hearing loss and sleep andspeech interference In this context it is desired that thefuselage structure provides appropriate noise attenuationso interior noise levels are within reasonable levels Thisproper noise attenuation can only be achieved if the fuselagestructure hence fuselage panels has appropriate strengthmass and damping properties Therefore a need exists tobetter understand noise transmission properties of compositestructures such that they can be safely applied to aircraft

The panels are considered to be simply supported in theirfour edges and excited by the turbulent boundary layer (TBL)Different composite structures were studied As in [1ndash4] acombination of the Corcos model [5 6] and Efimtsov model[7] was used to provide the TBL excitation to the panelsand results obtained for the power spectral density (PSD) ofthe panelsrsquo vibrational and acoustical responses Thereforethis study provides an original approach to the dynamical

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 316481 18 pageshttpdxdoiorg1011552014316481

2 Shock and Vibration

Uinfin

h2

h2

u x

y

w z

a

b

Figure 1 Scheme of the composite panel excited by turbulent flow

response behaviour of composite structures which is typi-cally obtained only for free vibration conditions or for anacoustic sine wave harmonic excitation as discussed in moredetail in the following paragraphs Previous studies have beenconducted to understand the dynamic vibration and noiseradiation properties by composite panels however moststudies did not consider the turbulent flow excitation in theanalysis

Early studies [8 9] have been established to computethe dynamic properties of laminated plates In these studiesclosed form exact solutions have been presented to calcu-late the natural frequencies of orthotropic antisymmetricangle-ply and antisymmetric cross-ply laminated plates withsimply supported edges One of the most popular meth-ods to obtain approximate solutions for the frequencies ofan orthotropic plate is the Rayleigh-Ritz method Severalauthors [10ndash13] have investigated analyticalmethods to deter-mine the sound transmission loss of sandwich structuresconsidering the case of an infinite plate excited by a planewave Later in [14] these studies were extended for theprediction of sound transmission loss of infinite sandwichcomposite panels excited by a diffuse field Two models weredeveloped and compared a symmetrical laminate compositeand a discrete thick laminate composite

Few studies have also been published focusing on theacoustic behaviour of finite sandwich structures In [15]an analytical method is proposed to evaluate the soundinsulation behaviour of sandwich panels containing twoorthotropic cover plates and a core In [16] a variationalmodel using the Rayleigh-Ritz method is presented forthe vibroacoustic analysis of a plate covered by a free or aconstrained viscoelastic layer immersed in either a light ora heavy fluid The study in [17] presents a numerical methodfor the prediction of the acoustic and vibration responses ofsandwich panels based on the finite element formulation andcoupled to a variational boundary elementmethod to accountfor fluid loading The plates were excited by acoustic planewaves or by a diffusive sound field The main disadvantage ofthis method was the high computational time associated withthe finite element discretization procedure In a later study[18] the same authors presented a more efficient formulationbased on a modal approach to analyze the vibration andacoustic behaviours of damped sandwich panels subjected tomechanical andor acoustic harmonic excitations

The numerical techniques using either one or the com-bination of finite element method (FEM) with boundary

Uinfin

h

u x

y

w z

w z

h0

h2h3

hn

Figure 2 View of the anatomy of the composite ply layers

element method (BEM) have been widely used for thepurpose of identifying sound transmission characteristicsof composite panels for example in [19ndash21] However inthe high frequency analysis these methods often requirea large number of FEMBEM meshes ultimately resultingin high computational costs In [22] a hybrid analyticalone-dimensional finite element method is derived whichuses FEM approximation in the thickness direction andanalytical solutions in the plane directions thus reducingthe number of finite elements required In this context thepresent study addresses this problem through an analyticalapproach Empirical models are used to provide the TBL flowexcitation while the radiated sound power is calculated byintegrating the panelrsquos structural vibration response obtainedfrom the analytical model The model can be applied fora large variety of composite panels and a large number ofpanel layersrsquo configurations with the advantage of providingan accurate and fast prediction without high computationalcosts Results for the acoustical and vibration responsesare shown for composite panels excited by the turbulentboundary layer and by a randomexcitationwhich as referredearlier is a case study that was not investigated previouslyA discussion is then presented on the comparison of thevibroacoustic behaviour of composite panels compared totraditional isotropic panels

2 Theoretical Formulation

Consider the composite plate as shown in Figures 1 and 2Each layer is considered to have an arbitrary thickness ℎ

119894

and material properties which may be defined by the userThe plate is assumed to be simply supported in all fourboundaries and the (119909 119910) is its reference plane The topsurface of the sandwich panel is excited by turbulent flow orby random noise as explained in more detail in the followingsection Additionally panel displacements are assumed to besmall compared with the thickness of the panel

21 Plate Governing Equations From the theory and equa-tions of mechanics of laminated composites the displace-ment components 119906 V and 119908 at a given point in the plate

Shock and Vibration 3

2

1

u x

y

w z

h h2

hn

h0

120579x

y

Figure 3 Detailed view of the anatomy of the composite ply layers

respectively in the 119909 119910 and 119911 directions (as in Figure 2) canbe written as follows

119906 = 1199060minus 119911

120597119908 (119909 119910 119905)

120597119909

V = V0minus 119911

120597119908 (119909 119910 119905)

120597119910

119908 = 119908 (119909 119910 119905)

(1)

in which 1199060and V0denote the displacements of the midplane

of the plate in the 119909 and 119910 directions respectively Thestrains in the plate can then be written as functions of thedisplacement as follows in the vector form

(

120598119909

120598119910

120574119909119910

) = (

120597119906

120597119909120597V120597119910

120597119906

120597119910+120597V120597119909

) = (

(

1205971199060

120597119909minus 119911

1205972119908

1205971199092

120597V0

120597119909minus 119911

1205972119908

1205971199102

1205971199060

120597119910+120597V0

120597119909minus 2119911

1205972119908

120597119909120597119910

)

)

(2)

The above equations are all written in the global coordinatesystems (119909 119910 119911) Stresses in the ply or lamina coordinatesystem (1 2 3) can then be written with relation to thestresses in the plate coordinate system as follows

(

1205901

1205902

12059112

) = [119879119894](

120590119909

120590119910

120591119909119910

) (3)

in which [119879119894] is the transformation matrix for the 119894th ply

defined by

[119879119894] = [

[

cos2120579119894

sin2120579119894

2 sin 120579119894cos 120579119894

sin2120579119894

cos2120579119894

minus2 sin 120579119894cos 120579119894

minus sin 120579119894cos 120579119894sin 120579119894cos 120579119894cos2120579119894minus sin2120579

119894

]

]

(4)

and 120579119894is the angle between the plate global coordinate system

and the 119894th ply coordinate system as shown in Figure 3Similarly for strain one can write

(

1205981

1205982

12057412

) = [119879119894](

120598119909

120598119910

120574119909119910

) (5)

Additionally stress and strain components in the ply coordi-nate system are related through the reduced stiffness matrix[119876119894] as follows

(

1205901

1205902

12059112

) = [119876119894](

1205981

1205982

12057412

) (6)

in which

[119876119894] =

[[[

[

119876119894

11119876119894

120

119876119894

21119876119894

220

0 0 119876119894

66

]]]

]

(7)

with

119876119894

11=

119864119894

1

1 minus ]11989412]11989421

(8)

119876119894

22=

119864119894

2

1 minus ]11989412]11989421

(9)

119876119894

12=

]11989412119864119894

2

1 minus ]11989412]11989421

(10)

119876119894

66= 119866119894

12 (11)

where1198641198941and119864119894

2are the 119894th ply elasticitymodulus in the longi-

tudinal and transverse direction of the fibers respectively ]11989412

and ]11989421are Poissonrsquos ratios for the 119894th ply and119866119894

12is the 119894th ply

shearmodulus Combining (3) through (11) one canwrite thefollowing relationship relating stress and strain in the platecoordinate system based on each ply reduced stiffnessmatrixcomponents and transformation matrix components

(

120590119909

120590119910

120591119909119910

) = [119876119894](

120598119909

120598119910

120574119909119910

) (12)

4 Shock and Vibration

where

[119876119894] = [119879

minus1

119894] [119876119894] [119879119894] =

[[[[[[

[

119876119894

11119876119894

12119876119894

16

119876119894

12119876119894

22119876119894

26

119876119894

16119876119894

26119876119894

66

]]]]]]

]

(13)

119876119894

11= 119876119894

111198984

119894+ 2 (119876

119894

12+ 2119876119894

66)1198982

1198941198992

119894+ 119876119894

221198994

119894 (14)

119876119894

12= (119876119894

11+ 119876119894

22minus 4119876119894

66)1198982

1198941198992

119894+ 119876119894

12(1198984

119894+ 1198994

119894) (15)

119876119894

22= 119876119894

111198994

119894+ 2 (119876

119894

12+ 2119876119894

66)1198982

1198941198992

119894+ 119876119894

221198984

119894 (16)

119876119894

16= (119876119894

11minus 119876119894

12minus 2119876119894

66)1198983

119894119899119894+ (119876119894

12minus 119876119894

22+ 2119876119894

66)1198981198941198993

119894

(17)

119876119894

26= (119876119894

11minus 119876119894

12minus 2119876119894

66)1198981198941198993

119894+ (119876119894

12minus 119876119894

22+ 2119876119894

66)1198983

119894119899119894

(18)

119876119894

66= (119876119894

11+ 119876119894

22minus 2119876119894

12minus 2119876119894

66)1198982

1198941198992

119894+ 119876119894

66(1198984

119894+ 1198994

119894)

(19)

in which119898119894= cos 120579

119894and 119899119894= sin 120579

119894

By definition the equilibrium equation for the differentialplate element is defined as function of the stress and momentresultants as follows

119888 (119873119909

1205972119908

1205971199092+ 2119873119909119910

1205972119908

120597119909120597119910+ 119873119910

1205972119908

1205971199102)

+ (1205972119872119909

1205971199092+ 2

1205972119872119909119910

120597119909120597119910+119872119910

1205972119908

1205971199102) + 119891

119911=

119899

sum

119894=1

120588119894ℎ119894

1205972119908

1205971199052

(20)

in which 119891119911= minus119901ext(119909 119910 119905) is the external pressure applied to

the panel due to the fluid flow or the random excitation 120588119894is

the density of the 119894th ply and the stress resultant andmomentresultant components are defined as functions of the stresscomponents respectively as follows

(

119873119909

119873119910

119873119909119910

) =

119899

sum

119894=1

int

ℎ119894

ℎ119894minus1

(

120590119909

120590119910

120591119909119910

)119889119911

(

119872119909

119872119910

119872119909119910

) =

119899

sum

119894=1

int

ℎ119894

ℎ119894minus1

(

120590119909

120590119910

120591119909119910

)119911119889119911

(21)

After substituting the stress components defined by (12) into(21) and making use of (14) through (19) then (20) can be

transformed to the following form as function of the platedisplacement 119908

1

3

119899

sum

119894=1

(ℎ3

119894minus ℎ3

119894minus1)

times [119876119894

11(1205972119908

1205971199092)

2

+ 119876119894

22(1205972119908

1205971199102)

2

+ 2119876119894

12(1205972119908

1205971199092)(

1205972119908

1205971199102) + 4119876

119894

16(1205972119908

1205971199092)(

1205972119908

120597119909120597119910)

+ 4119876119894

26(1205972119908

1205971199102)(

1205972119908

120597119909120597119910) + 4119876

119894

66(

1205972119908

120597119909120597119910)

2

]

+1

3

119899

sum

119894=1

(ℎ3

119894minus ℎ3

119894minus1)

times [119876119894

11

1205974119908

1205971199094+ 119876119894

22

1205974119908

1205971199104+ 6119876119894

16

1205974119908

1205971199093120597119910

+ 6119876119894

26

1205974119908

1205971199091205971199103+ (2119876

119894

12+ 8119876119894

66)

1205974119908

12059711990921205971199102]

+

119899

sum

119894=1

120588119894ℎ119894

1205972119908

1205971199052= 119891119911

(22)

Each panel in the present study aims to represent thedistance between adjacent stringers and frames in an aircraftfuselage hence each individual panel is assumed to vibrateindependently In addition since panels are considered to beflat and simply supported in all four boundaries the panelrsquosdisplacement can be defined as follows [23 24]

119908 (119909 119910 119905) =

119872119909

sum

119898119909=1

119872119910

sum

119898119910=1

120572119898119909

(119909) 120573119898119910(119910) 119902119898119909119898119910

(119905) (23)

where 120572119898119909(119909) and 120573

119898119910(119910) are the spatial functions defining

the variation of119908(119909 119910 119905) with 119909 and 119910 respectively 119902119898119909119898119910

(119905)

are the temporal functions defining the variation of119908(119909 119910 119905)with time and 119872 = 119872

119909times 119872119910is the total number of plate

modes (119898119909 119898119910) considered For simply supported plates the

spatial functions may be defined as follows

120572119898119909

(119909) = radic2

119886sin(

119898119909120587119909

119886) (24)

120573119898119910

(119910) = radic2

119887sin(

119898119910120587119910

119887) (25)

Shock and Vibration 5

where 119886 and 119887 are the dimensions of the plate in the 119909 and119910 directions Also the natural frequencies for the simplysupported plate can be determined by

120596119898119909119898119910

=1205872

radicsum119899

119894=1120588119894ℎ119894

[11986311(119898119909

119886)

4

+ 11986322(119898119910

119887)

4

+2(11986312+ 211986366)(

119898119909

119886)

2

(119898119910

119887)

2

]

12

(26)

in which

119863119895119896

=

119899

sum

119894=1

int

ℎ119894

ℎ119894minus1

119876119894

1198951198961199112119889119911 (27)

Developing (22) by considering (23) to (25) making use ofthe orthogonality of the plate modes and integrating over theplate area similarly to the mathematical approach followedin [4] the equilibrium equation for an orthotropic plate iswritten in the matrix form as

[119872119901] q (119905) + [119863

119901] q (119905) + [119870

119901] q (119905) = pext (119905) (28)

where [119872119901] [119863119901] and [119870

119901] are mass damping and stiffness

119872times119872matrices respectively and q(119905) q(119905) and q(119905) are theposition velocity and acceleration vectors with componentsdefined respectively as 119902

119898(119905) = 119876

119898119890119894120596119905 119902119898(119905) = 119894119876

119898120596119890119894120596119905

and 119902119898(119905) = minus119876

1198981205962119890119894120596119905 After performing all the above

described mathematical manipulations it can be shown thateach term in (28) is defined as follows

[119872119901] = diag[

119899

sum

119894=1

120588119894ℎ119894]

[119863119901] = diag[2120585

119901120596119898119909119898119910

119899

sum

119894=1

120588119894ℎ119894]

[119870119901] =

1

3

119899

sum

119894=1

(ℎ3

119894minus ℎ3

119894minus1) 119876119894

11I (119909) + 119876

119894

22I (119910)

+ (2119876119894

12+ 8119876119894

66) I1015840 (119909) I1015840 (119910)

pext (119905) = [int

119886

0

int

119887

0

120572119898119909

(119909) 120573119898119910(119910) 119901ext (119909 119910 119905) 119889119909 119889119910]

(29)

in which 120585119901was added to account with the damping of the

plate and

I (119909) = diag [1119886(119898119909120587

119886)

4

]

I (119910) = diag [1119887(119898119910120587

119887)

4

]

I1015840 (119909) = minus1

119886(119898119909120587

119886)

2

cos [(119898119909minus 1198981015840

119909) 120587]

I1015840 (119910) = minus1

119887(119898119910120587

119887)

2

cos [(119898119910minus 1198981015840

119910) 120587]

(30)

22 Radiated Sound Power The calculation of the panelsrsquoradiated sound power in the frequency domain requires arearrangement of the time domain equations presented in(22) For this end knowing that 119902

119898= 119876119898119890119894120596119905 as previously

described one can write (28) in the following desired form inthe frequency 120596 domain

Y (120596) = H (120596)X (120596) (31)

where

Y (120596) = W (120596)

X (120596) = Pext (120596)

H (120596) = minus1205962[119872119901] + 119894120596 [119863

119901] + [119870

119901]

(32)

in which H(120596) is system frequency response matrix Y(120596) isthe response of the system to the excitationX(120596) and vectorsW(120596) andPext(120596) correspond respectively to the vectors q(119905)and pext(119905) written in the frequency domain Considering theturbulent boundary layer random excitation as a stationaryand homogeneous function the spectral density of the systemresponse is given by [25 26]

S119882119882 (120596) = Hlowast (120596) S119875119875 (120596)H

119879(120596) (33)

where superscripts lowast and 119879 denote Hermitian conjugate andmatrix transpose respectively S

119875119875(120596) is the spectral density

matrix of the random excitation Pext(120596) and S119882119882

(120596) is thespectral density matrix of the system response W(120596) Thematrix S

119875119875(120596) corresponds to the generalized power density

matrix of the turbulent boundary layer excitation defined by

Stbl (120596) = [∬

119886

0

∬

119887

0

120572119898119909

(119909) 1205721198981015840119909

(1199091015840) 120573119898119910

(119910) 1205731198981015840

119910

(1199101015840)

times 119878 (120585119909 120585119910 120596) 119889119909 119889119909

1015840119889119910119889119910

1015840]

(34)

in which 119878(120585119909 120585119910 120596) is defined by Corcos model [5 21] and

120585119909= 119909 minus 119909

1015840 and 120585119910= 119910 minus 119910

1015840 are the spatial separations in thestreamwise and spanwise directions of the plate respectivelyAs shown in [1ndash3] a combination of Corcos and Efimtsovmodels may be suitable to provide this excitation as follows

119878 (120585119909 120585119910 120596) = 119878ref (120596) 119890

minus(120572119909120596|120585119909|119880119888)119890minus(120572119910120596|120585119910|119880119888)119890

minus(119894120596120585119909119880119888)

(35)

where119880119888is the turbulent boundary layer convective velocity

120572119909= 01 and 120572

119910= 077 are empirical parameters used to

provide the best agreement with the reality [27] and 119878ref(120596)is defined using Efimtsov model [7] In [1] an analyticalform was derived for the matrix described in (34)The powerspectral density function of the plate displacement can then

6 Shock and Vibration

be defined as functions of the abovematrix respectively platearea as

119878119908119908

(1199091 1199101 1199092 1199102 120596)

=

1198722

119909

sum

11989811990911198981199092=1

1198722

119910

sum

11989811991011198981199102=1

1205721198981199091

(1199091) 1205721198981199092

(1199092)

times 1205731198981199101

(1199101) 1205731198981199102

(1199102) S119882119882(120596)1198981 1198982

(36)

and the overall power spectral density function can be foundby integrating the individual power spectral densities over theplate area displacement as following

119878119908119908 (120596) = ∬

119886

0

∬

119887

0

119878119908119908

(1199091 1199101 1199092 1199102 120596) 119889119909

1119889119909211988911991011198891199102

(37)

Analytical expressions for 119878119908119908

(1199091 1199101 1199092 1199102 120596) and 119878

119908119908(120596)

were obtained in [23] The power spectral density matrix ofthe plate velocity can be determined from the power spectraldensity matrix of the plate displacement as follows

S119881119881 (120596) = 120596

2S119882119882 (120596) (38)

Using these analytical forms the plate radiated sound powerfunction and the overall power spectral density function canbe determined respectively as follows

RSP (1199091 1199101 1199092 1199102 120596)

=

1198722

119909

sum

11989811990911198981199092=1

1198722

119910

sum

11989811991011198981199102=1

1205721198981199091

(1199091) 1205721198981199092

(1199092)

times 1205731198981199101

(1199101) 1205731198981199102

(1199102)Π(120596)1198981 1198982

RSP (120596) = ∬

119886

0

∬

119887

0

RSP (1199091 1199101 1199092 1199102 120596) 119889119909

1119889119909211988911991011198891199102

(39)

in which Π(120596)1198981 1198982

are the components of the radiationmatrix defined by

Π (120596) = S119881119881

M (120596) (40)and S119881119881

is defined in (38) andM(120596) is evaluated numericallyfrom the following equation

M (120596) = 81205880

1198880

(120596119886119887

1205873119898119909119898119910

)

2

times∬

1205872

0

cossin (

120572

2)

cossin (

120573

2)

[(120572119898119909120587)2minus 1] [(120573119898

119910120587)2

minus 1]

2

timessin 120579119889120579 119889120601(41)

where 1205880and 1198880are the density and the speed of sound of

the interior fluid respectively 120572 = (1205961198880)119886 sin 120579 cos120601 120573 =

(1205961198880)119887 sin 120579 sin120601 in which 120601 and 120579 are the angles of the

distance vector in space to the observation point [27]

3 Structural Vibration and SoundRadiation Results

The formulation presented in the previous section is usedto analyze four different rectangular composite sandwichpanels of several sizes and materials properties as shown inTable 1 Additionally four isotropic aluminum panels withsimilar sizes compared to the composite panels as shownin Table 2 are analyzed using the previous model developedby the author results are then compared with the compositepanels All the eight panels are considered simply supportedin the four boundaries and subjected to turbulent boundarylayer excitation or to random noise excitation The turbulentflow excitation is representative of Mach number 08 flightconditions as described inTable 3 and the randomexcitationrepresents a constant 40 Pascal random noise that is asound pressure level of approximately 120 dB The first tennatural frequencies obtained for the several panels analyzedare shown in Tables 4 and 5 respectively for composite andisotropic panels Cases 1 2 3 and 4 of composite panels asnoted in Table 4 are chosen for validation purposes as instudies [16 18 20] respectively It is shown that the calculatednatural frequencies are in good agreement with the previousstudies

31 Composite Panels The results presented in this sectiondiscuss the radiated sound power and the structural dis-placement levels of the composite sandwich panels studiedResults are shown for frequencies up to 1500Hz requiringa minimum number of plate modes which is dependent ofthe panel size and materials properties for proper accuracyof the predicted values Values for the radiated sound powerRSP and plate displacement power spectral density (PSD)119878119908119908

are obtained for two locations (1199091 1199101) = (05119886 05119887)

and (1199092 1199102) = (03119886 02119887) as well as overall values of these

two quantities and shown in Figures 4 and 5 As referredearlier four different panels are analyzed identified as Cases1 through 4 along the following sections and two types ofexcitation are applied (1) turbulent boundary layer excitationand (2) random noise excitation

As noted in Figure 4 and similarly to what was observedin previous results and formulation dedicated to the analysisof isotropic panels it is concluded that the overall RSPvalues differ from local RSP values especially for the TBLexcitation It is also observed that turbulent flow can beresponsible for exciting a higher number of panel modescompared to random noise with TBL and random noiseproviding dissimilar levels of panelsrsquo sound radiation Assuch to accurately predict the dynamic behaviour of a panelone should not consider the pure random noise excitation asrepresentative of the turbulent flow excitation In the overallRSP results plates of bigger dimensions are shown to havehigher RSP values both for TBL and random excitationswith Case 1 showing higher RSP values than Cases 2 3or 4 The same is not noted for the local RSP valueswith Case 3 panel showing increased RSP values for bothpositions 1 and 2with relation to the overall RSP valuesTheseresults emphasize the importance of knowing the measuring

Shock and Vibration 7

0 500 1000 1500Frequency (Hz)

minus150

minus100

minus50

0

50

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

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0

20

40

60

Radi

ated

soun

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wer

(dB

rel1

W)

(b)

0 500 1000 1500

Frequency (Hz)

minus120

minus100

minus80

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minus40

minus20

0

20

40

Radi

ated

soun

d po

wer

(dB

rel1

W)

(c)

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Frequency (Hz)

minus60

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minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

(d)

0 500 1000 1500

Frequency (Hz)

minus140

minus120

minus100

minus80

minus60

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0

20

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 1 composite TBLCase 2 composite TBL

Case 3 composite TBLCase 4 composite TBL

(e)

0 500 1000 1500

Frequency (Hz)

minus80

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 1 composite randomCase 2 composite random

Case 3 composite randomCase 4 composite random

(f)

Figure 4 Radiated sound power results for the composite panels (a) and (b) overall panel RSP (c) and (d) RSPnear surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) RSP near location (1199092 1199102) = (03119886 02119887)

position for the accurate prediction of sound and vibrationand it highlights the desired sensitivity of the developedmathematical framework to this factor Figure 5 shows thedisplacement PSD results associated with the same panels

and excitations as for the RSP results displayed in Figure 4Comparing Figures 4 and 5 it is observed that structuraldisplacement and sound radiation levels have different curvesfor the same excitation and thus panel displacement PSD

8 Shock and Vibration

minus250

minus200

minus150

minus100

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

minus200

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(b)

minus200

minus180

minus140

minus160

minus120

minus100

minus80

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(c)

minus220

minus200

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(d)

minus220

minus200

minus180

minus140

minus160

minus120

minus100

minus80

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

Case 1 composite TBLCase 2 composite TBL

Case 3 composite TBLCase 4 composite TBL

(e)

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

Case 1 composite randomCase 2 composite random

Case 3 composite randomCase 4 composite random

(f)

Figure 5 Panel displacement PSD results for the composite panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location(1199091 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

curve predictionmay not be fully representative of RSP curveeven though they would be expected to follow similar trendsIn addition as similar to RSP results it is concluded that 119878

119908119908

predictions are dissimilar whether one is assuming TBL or

random excitation Case 1 panel shows higher values of bothoverall and local plate displacement PSD predictions

The results presented here are useful in the context of theprediction of noise radiation by composite panel solutions

Shock and Vibration 9

and its possible application for example in aerospace indus-try Based on the obtained results if one would choose whichcomposite panel solution is most desirable for the purposeof manufacturing an aircraft fuselage panel a smaller panel(Case 3) or a panel with a higher aspect ratio (Case 4) wouldbe preferable for overall improved results Longer and widerpanels (ie Cases 1 and 2) are shown to greatly amplify thelow frequency range However one would need to designthe panel appropriately and with the targeted dimensionssuch that the amplification of noise and vibration at certainfrequencies would be avoided For instance Case 3 panelexhibits a general improved behaviour in terms of noiselevels but simultaneously it amplifies noise at frequenciesnear 300Hz and 1000Hz while Case 4 panel amplifies resultsnear 300Hz Comparing the size of these two panels thesurface area covered by one Case 1 panel corresponds toapproximately the surface area covered by twenty Case 3panels with the twenty Case 3 panels representing approx-imately 94 kg of total mass compared to the 676 kg of asingle Case 1 panel As such an improvement in the noiselevels comes as a tradeoff in terms of increased weightAdditionally it is shown that panels with thicker face layersand thinner core layer specifically referring to Cases 3 and 4have lower noise and vibration levels than panels with thinnerface layers and thicker core layer referring to Cases 1 and 2However as mentioned earlier Cases 3 and 4 panels withsmaller dimensions and thicker face layers have the problemof noise amplification at certain frequencies Therefore apanel which combines the properties of being shorter with athicker core layer versus thinner face layers may potentiallyshow an improved behaviour for reduced vibration and noiseradiation as well as lower noise amplification at isolatedfrequencies

32 Isotropic Panels This section addresses the RSP and 119878119908119908

predictions for the isotropic panels analyzed and results upto 1500Hz are shown in Figures 6 and 7 Four aluminumisotropic panels are investigated considering TBL or randomexcitation as follows isotropic panels Cases 3 and 4 have thesame dimensions 119886 and 119887 as the composite panels Cases3 and 4 respectively isotropic panels Cases 5 and 6 areconsidered with same dimensions 119886 and 119887 as Cases 3 and4 respectively but with increased thicknesses This approachis taken with the purpose of posterior comparison betweencomposite and isotropic panelsrsquo behaviours In this contextisotropic panels Cases 3 and 4 are both 1mm thick whileisotropic panels Cases 4 and 5 have thicknesses of 15mm and4mm respectively

As noted in Figures 6 and 7 the increase of panelthickness results in an expected shift in the panel naturalfrequencies to higher frequencies as well as a shift in themodes amplified in both RSP and 119878

119908119908results This effect is

observed in the TBL and random excitations Furthermore itis found that panels with higher aspect ratio (ie Cases 4 and6) are generally characterized by having lower levels of soundradiation and vibration As similarly noted for the compositepanels random and TBL excitations provide different RSPand 119878

119908119908predictions and overall values differ from point

values and hence it is important to specify the location in the

panel where the measurementspredictions are being takenIt is also noted that the random excitation remains as theexcitation causing higher noise and vibration levels for allcases at the higher frequency range This is also an expectedeffect as the TBL excitation decreases towards the higherfrequencies while the random excitation provides a constantexcitation over the frequency spectrum

33 Composite versus Isotropic Panels As one of the mainobjectives of this study this section presents a discussionof results for the behavior of composite sandwich panelsin comparison with the traditional isotropic panels Thefollowing four figures are shown to this end Figures 8 and9 compare results for composite panel Case 3 with isotropicpanels Cases 3 and 5 while Figures 10 and 11 comparecomposite panel Case 4 with isotropic panels Cases 4 and 6

By comparing Cases 3 and 5 panels as shown fromFigures 8 and 9 it is noted that composite panel Case 3exhibits an enhanced behaviour in comparison with isotropicpanels Cases 3 and 5 both in RSP and displacement PSDlevels However one should also note that Case 3 compositepanel shows higher levels of noise and vibration at specificisolated frequencies approximately near 300Hz and 1000Hzfor the overall RSP and for position 1 RSP results as shown inFigure 8 Considering the panel displacement PSD results inFigure 9 extra frequencies above 1000Hz are also amplifiedespecially at position 2 for both TBL and random excitationsAs described in Table 1 the composite panel Case 3 has athin core layer thus the unwanted amplification at thesefrequencies may be attenuated by increasing the core layerthickness However this modification on the panel thicknesswill increase the original panel weight of 047 kg whichalready represents heavier solution compared to Cases 3and 5 isotropic panels with weights of 03 kg and 045 kgrespectively Although the panel weight will increase if thischange is performed making this modification will in alater phase require less added acoustic insulation material tothe aircraft walls in the context of aircraft fuselage panelsrsquoapplication As such the composite panelmay represent witha slight increase in the panel weight an optimized solutionfor both sound radiation and structural vibration in almostall the frequency spectra analyzed

Similar conclusions are obtained from Figures 10 and 11dedicated to the comparison of Cases 4 and 6 panels thatis the composite panel represents an improved solution foralmost all the frequency ranges studied By analyzing thegeometric differences between composite panels Cases 3 and4 Case 4 has thicker face layers and is longer than Case 3panel having a much higher aspect ratio The amplificationof noise and vibration levels at specific frequencies is notedas in Case 3 composite panel Since face layers are thickerand this panel has a higher aspect ratio this effect is morevisible in composite Case 4 than in composite Case 3 Asobserved in results from Case 3 this could be enhancedby implementing an increase of the core layer thicknessWhile the increase in the face layer thickness is shown todecrease the levels of noise and vibration in both overalland localized predictions it also can have an impact in the

10 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

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Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

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minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

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0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

minus150

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50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 6 Radiated sound power results for the isotropic panels (a) and (b) overall panel RSP (c) and (d) RSP near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) RSP near location (1199092 1199102) = (03119886 02119887)

Shock and Vibration 11

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

(a)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(b)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(c)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus50

minus100

minus150

minus200

(f)

Figure 7 Panel displacement PSD results for the isotropic panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) 119878119908119908

near location (1199092 1199102) = (03119886 02119887)

12 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus100

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus100

minus80

minus60

minus40

minus20

0

20

40

60

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 8 Comparison of radiated sound power results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 13

0 500 1000 1500

Frequency (Hz)

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus50

minus150

minus100

minus200

minus250

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 9 Comparison of panel displacement PSD results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

14 Shock and VibrationRa

diat

ed so

und

pow

er (d

B re

l1

W)

0 500 1000 1500

Frequency (Hz)

50

0

minus50

minus100

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(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

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wer

(dB

rel1

W)

minus80

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0

20

40

60

(b)

Radi

ated

soun

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wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

100

50

0

minus50

minus100

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(c)

0 500 1000 1500

Frequency (Hz)

Radi

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soun

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(dB

rel1

W)

minus60

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0

20

40

60

80

(d)

Radi

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soun

d po

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(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

50

0

minus50

minus100

minus150

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

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(dB

rel1

W)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus80

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 10 Comparison of radiated sound power results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 15

minus50

minus100

minus150

minus250

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

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minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

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t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus40

minus60

minus80

minus100

minus120

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minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus220

minus200

Plat

e disp

lace

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t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus50

minus100

minus150

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 11 Comparison of panel displacement PSD results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b)overall panel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

16 Shock and Vibration

Table 1 Physical properties of composite panels

Variable Description units Case 1 Case 2 Case 3 Case 4119886 Length m 183 11683 0348 10119887 Width m 122 11683 03048 03120582 Aspect ratio (119886119887) 15 10 114 333ℎ1 ℎ3

Face layers thickness mm 0406 0506 0762 35ℎ2

Core layer thickness mm 64 1905 0254 051198641 1198643

Face layers Young modulus Pa 6898 times 1010

71 times 1010

6898 times 1010

66 times 1010

1198642

Core layer Young modulus Pa 457 times 105

457 times 105

267 times 106

26 times 108

]1 ]3

Face layers Poissonrsquos ratio 033 033 03 033]2

Core layer Poissonrsquos ratio 04 04 049 031205881 1205883

Face layers density kgmminus3 2768 2700 2740 26801205882

Core layer density kgmminus3 122 4806 999 1680119898 Total mass kg 676 499 047 588

Table 2 Physical properties of isotropic panels

Variable Description units Case 3 Case 4 Case 5 Case 6119886 Length m 0348 10 0348 10119887 Width m 03048 03 03048 03120582 Aspect ratio (119886119887) 114 333 114 333ℎ119901

Thickness mm 1 1 15 4119864119901

Young modulus Pa 724 times 1010

724 times 1010

724 times 1010

724 times 1010

]119901

Poissonrsquos ratio 033 033 033 033120588119901

Density kg mminus3 2800 2800 2800 2800119898 Total mass kg 030 084 045 336

Table 3 Flight test conditions and air properties

Variable Description units Value119872 Mach number 08alt Altitude m 13106119888 Speed of sound m sminus1 295119880infin

Flow speed m sminus1 23606120588 Air density kgmminus3 02622] Kinematic viscosity m2 sminus1 5422 times 10

minus5

119879 Temperature K 21665Re119909

Reynolds number 2917 times 107

1198880

Speed of sound m sminus1 3401205880

Internal air density kgmminus3 142

higher amplification of localized frequencies Therefore theface layers thickness increase should be performed carefullyby an accompanied increase in the core layer thickness whilesimultaneously considering the weight constraints for thespecific panels applications

A more detailed analysis will be required in order toformulate final conclusions about the optimum solutions forthe panel in the context of aircraft application For that theinsulation materials would also be considered to account forthe respective added weight and effect in the sound radi-ation signature However typically the acoustic insulationmaterials will only have a higher impact in decreasing the

Table 4 First 10 natural frequencies for the composite panels

Modes 119891 Hz(Case 1)

119891 Hz(Case 2)

119891 Hz(Case 3)

119891 Hz(Case 4)

1 2365 10588 4874 126582 4548 26470 11222 157933 7276 42352 13148 210194 8186 52941 19496 283355 9459 68823 21800 377416 13097 89999 26939 474967 13279 95293 30075 492388 15462 105881 33287 506329 17645 132351 36612 5585710 18191 137645 43866 62825

sound levels at higher frequency ranges and will thereforenot improve significantly the problem of higher noise levelsin the low frequency range particularly observed in theisotropic solutions As such the composite panel solutionsmay appear to be a promising replacement to the isotropicpanels especially panels like Cases 3 and 4 which are alreadycharacterized by lower levels of both RSP and 119878

119908119908in the

low frequency range In further analysis it will also beimportant to carefully address the noise amplification atisolated frequencies by identifying a good combination of

Shock and Vibration 17

Table 5 First 10 natural frequencies for the isotropic panels

Modes 119891 Hz(Case 3)

119891 Hz(Case 4)

119891 Hz(Case 5)

119891 Hz(Case 6)

1 4646 2958 6969 118322 10697 3691 16046 147643 12534 4912 18801 196494 18585 6622 27877 264885 20782 8820 31173 352826 25680 1110 38520 443997 28670 11507 43004 460298 31731 11832 47596 473309 34901 13054 52351 5221510 41816 14683 62723 58731

core and face layers in terms of both materials selection anddimensions

4 Conclusions

This study presents a mathematical method and frameworkon composite panels radiated sound and vibrationThepanelsare considered excited either by turbulent flow or by randomnoiseThe turbulent flow corresponds to the typical excitationof an aircraft panel at flight cruise conditions with a Machnumber of 08 The effect of type of panel of panel dimen-sions and of panel thickness are evaluated The turbulentflow excitation showed to excite a higher number of panelmodes compared to random noise excitation and resulted indissimilar levels of both panel vibration and radiated soundwith relation to the random excitation This effect is morevisible for composite than for isotropic panels and thereforethe pure random noise should not be used as representativeof the turbulent flow excitation in real applications

Different properties on the composite panels are shown toyield dissimilar effects in the noise and vibration predictionsThe effect of the panel size specifically its length and widthis noted to have a significant effect on the panel performancein terms of noise radiation and vibration levels Overallsmaller composite panels or panels with higher aspect ratioproduced lower levels of radiated sound and vibration thanlonger and wider composite panels Compared to isotropicpanels composite panels analyzed are shown to radiate lowernoise levels with the exception of specific natural frequencieswhich are amplified near 300Hz and 1000Hz It is shown thatthis amplification can be significant for composite panels withthicker face layers and thinner core layer and thus resultingin unwanted effect of increased sound radiated Despitethis effect this drawback may potentially be corrected byincreasing the sandwich core layer thickness Increasing thecore layer thickness will impact the panel weight and hencethe composite panels can represent a heavier solution com-pared to the isotropic panels examined On the other handthis approach may be beneficial in the design of compositepanels in the context of aircraft panel applications sincethese panels may require less acoustic insulation material

added to the aircraft walls as they already represent animproved design solution Furthermore acoustic insulationmaterials have a higher impact in decreasing sound levelsat the high frequency ranges and therefore it will notsignificantly improve the problem of higher noise levels at thelow frequency range particularly observed in the isotropicsolutions In this context advanced and optimized compositepanel configurations may be a promising replacement to thetraditional isotropic panels for aerospace applications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is grateful for the support of the National Scienceand Engineering Research Council of Canada

References

[1] J Rocha and D Palumbo ldquoOn the sensitivity of sound powerradiated by aircraft panels to turbulent boundary layer param-etersrdquo Journal of Sound and Vibration vol 331 no 21 pp 4785ndash4806 2012

[2] J Rocha A Suleman and F Lau ldquoPrediction of turbulentboundary layer induced noise in the cabin of a BWB aircraftrdquoShock and Vibration vol 19 no 4 pp 693ndash705 2012

[3] J da Rocha A Suleman and F Lau ldquoFlow-induced noise andvibration in aircraft cylindrical cabins closed-form analyticalmodel validationrdquo Journal of Vibration and Acoustics vol 133no 5 Article ID 051013 9 pages 2011

[4] J da Rocha A Suleman and F Lau ldquoPrediction of flow-inducednoise in transport vehicles development and validation of acoupled structuralmdashacoustic analytical frameworkrdquo CanadianAcousticsmdashAcoustique Canadienne vol 37 no 4 pp 13ndash292009

[5] G Corcos ldquoThe structure of the turbulent pressure field inboundary layer flowsrdquo Journal of Fluid Mechanics vol 18 pp353ndash378 1964

[6] G Corcos ldquoResolution of pressure in turbulencerdquo Journal of theAcoustical Society of America vol 35 no 2 pp 192ndash199 1963

[7] B Efimtsov ldquoCharacteristics of the field of turbulent wallpressure fluctuations at large Reynolds numbersrdquo Soviet PhysicsAcoustics vol 28 no 4 pp 289ndash292 1982

[8] J Reddy Mechanics of Laminated Composite Plates and ShellsTheory and Analysis CRC Press Boca Raton Fla USA 1945

[9] J Whitney Structural Analysis of Laminated Anisotropic PlatesTechnomic Lancaster Pa USA 1987

[10] C L Dym and M A Lang ldquoTransmission of sound throughsandwich panelsrdquo Journal of the Acoustical Society of Americavol 56 no 5 pp 1523ndash1532 1974

[11] C L Dym C S Ventres and M A Lang ldquoTransmission ofsound through sandwich panels a reconsiderationrdquo Journal ofthe Acoustical Society of America vol 59 no 2 pp 364ndash3671976

[12] S Narayanan andR L Shanbhag ldquoSound transmission througha damped sandwich panelrdquo Journal of Sound and Vibration vol80 no 3 pp 315ndash327 1982

18 Shock and Vibration

[13] C L Dym and D C Lang ldquoTransmission loss of dampedasymmetric sandwich panels with orthotropic coresrdquo Journal ofSound and Vibration vol 88 no 3 pp 299ndash319 1983

[14] S Ghinet N Atalla and H Osman ldquoDiffuse field transmissioninto infinite sandwich composite and laminate composite cylin-dersrdquo Journal of Sound and Vibration vol 289 no 4-5 pp 745ndash778 2006

[15] J Mendler ldquoComputational tool for the acoustic analysis ofsandwich panelsrdquo in Proceedings of the European Conference onNoise Control (EURO-NOISE rsquo95) Lyon France 1995

[16] O Foin J Nicolas andNAtalla ldquoAn efficient tool for predictingthe structural acoustic and vibration response of sandwichplates in light or heavy fluidrdquo Applied Acoustics vol 57 no 3pp 213ndash242 1999

[17] S Assaf and M Guerich ldquoNumerical prediction of noisetransmission loss through viscoelastically damped sandwichplatesrdquo Journal of Sandwich Structures andMaterials vol 10 no5 pp 359ndash384 2008

[18] S Assaf M Guerich and P Cuvelier ldquoVibration and acousticresponse of damped sandwich panels immersed in a light orheavy fluidrdquo Computers and Structures vol 88 pp 359ndash3842010

[19] R Zhou and M J Crocker ldquoBoundary element analyses forsound transmission loss of panelsrdquo Journal of the AcousticalSociety of America vol 127 no 2 pp 829ndash840 2010

[20] K Amichi N Atalla and R Ruokolainen ldquoA new 3D finite ele-ment sandwich plate for predicting the vibroacoustic responseof laminated steel panelsrdquo Finite Elements in Analysis andDesign vol 46 no 12 pp 1131ndash1145 2010

[21] F Franco S de Rosa and T Polito ldquoFinite element inves-tigations on the vibroacoustic performance of plane plateswith random stiffnessrdquo Mechanics of Advanced Materials andStructures vol 18 no 7 pp 484ndash497 2011

[22] Y Kim ldquoIdentification of sound transmission characteristicsof honeycomb sandwich panels using hybrid analyticalone-dimensional finite element methodrdquo in Proceedings of the35th International Congress and Exposition on Noise ControlEngineering (INTER-NOISE rsquo06) HonoluluHawaii USA 2006

[23] F Fahy Sound and Structural Vibration Radiation Transmissionand Response Academic Press London UK 1985

[24] J Blevins Formulas for Natural Frequency and Mode ShapeKrieger Publishing Malabar Fla USA 2001

[25] D Newland An Introduction to Random Vibration and SpectralAnalysis Longman New York NY USA 1984

[26] Y Lin Probabilistic Theory of Structural Dynamics McGraw-Hill Book New York NY USA 1967

[27] C E Wallace ldquoRadiation resistance of a rectangular panelrdquoTheJournal of the Acoustical Society of America vol 51 no 3 pp946ndash952 1972

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2 Shock and Vibration

Uinfin

h2

h2

u x

y

w z

a

b

Figure 1 Scheme of the composite panel excited by turbulent flow

response behaviour of composite structures which is typi-cally obtained only for free vibration conditions or for anacoustic sine wave harmonic excitation as discussed in moredetail in the following paragraphs Previous studies have beenconducted to understand the dynamic vibration and noiseradiation properties by composite panels however moststudies did not consider the turbulent flow excitation in theanalysis

Early studies [8 9] have been established to computethe dynamic properties of laminated plates In these studiesclosed form exact solutions have been presented to calcu-late the natural frequencies of orthotropic antisymmetricangle-ply and antisymmetric cross-ply laminated plates withsimply supported edges One of the most popular meth-ods to obtain approximate solutions for the frequencies ofan orthotropic plate is the Rayleigh-Ritz method Severalauthors [10ndash13] have investigated analyticalmethods to deter-mine the sound transmission loss of sandwich structuresconsidering the case of an infinite plate excited by a planewave Later in [14] these studies were extended for theprediction of sound transmission loss of infinite sandwichcomposite panels excited by a diffuse field Two models weredeveloped and compared a symmetrical laminate compositeand a discrete thick laminate composite

Few studies have also been published focusing on theacoustic behaviour of finite sandwich structures In [15]an analytical method is proposed to evaluate the soundinsulation behaviour of sandwich panels containing twoorthotropic cover plates and a core In [16] a variationalmodel using the Rayleigh-Ritz method is presented forthe vibroacoustic analysis of a plate covered by a free or aconstrained viscoelastic layer immersed in either a light ora heavy fluid The study in [17] presents a numerical methodfor the prediction of the acoustic and vibration responses ofsandwich panels based on the finite element formulation andcoupled to a variational boundary elementmethod to accountfor fluid loading The plates were excited by acoustic planewaves or by a diffusive sound field The main disadvantage ofthis method was the high computational time associated withthe finite element discretization procedure In a later study[18] the same authors presented a more efficient formulationbased on a modal approach to analyze the vibration andacoustic behaviours of damped sandwich panels subjected tomechanical andor acoustic harmonic excitations

The numerical techniques using either one or the com-bination of finite element method (FEM) with boundary

Uinfin

h

u x

y

w z

w z

h0

h2h3

hn

Figure 2 View of the anatomy of the composite ply layers

element method (BEM) have been widely used for thepurpose of identifying sound transmission characteristicsof composite panels for example in [19ndash21] However inthe high frequency analysis these methods often requirea large number of FEMBEM meshes ultimately resultingin high computational costs In [22] a hybrid analyticalone-dimensional finite element method is derived whichuses FEM approximation in the thickness direction andanalytical solutions in the plane directions thus reducingthe number of finite elements required In this context thepresent study addresses this problem through an analyticalapproach Empirical models are used to provide the TBL flowexcitation while the radiated sound power is calculated byintegrating the panelrsquos structural vibration response obtainedfrom the analytical model The model can be applied fora large variety of composite panels and a large number ofpanel layersrsquo configurations with the advantage of providingan accurate and fast prediction without high computationalcosts Results for the acoustical and vibration responsesare shown for composite panels excited by the turbulentboundary layer and by a randomexcitationwhich as referredearlier is a case study that was not investigated previouslyA discussion is then presented on the comparison of thevibroacoustic behaviour of composite panels compared totraditional isotropic panels

2 Theoretical Formulation

Consider the composite plate as shown in Figures 1 and 2Each layer is considered to have an arbitrary thickness ℎ

119894

and material properties which may be defined by the userThe plate is assumed to be simply supported in all fourboundaries and the (119909 119910) is its reference plane The topsurface of the sandwich panel is excited by turbulent flow orby random noise as explained in more detail in the followingsection Additionally panel displacements are assumed to besmall compared with the thickness of the panel

21 Plate Governing Equations From the theory and equa-tions of mechanics of laminated composites the displace-ment components 119906 V and 119908 at a given point in the plate

Shock and Vibration 3

2

1

u x

y

w z

h h2

hn

h0

120579x

y

Figure 3 Detailed view of the anatomy of the composite ply layers

respectively in the 119909 119910 and 119911 directions (as in Figure 2) canbe written as follows

119906 = 1199060minus 119911

120597119908 (119909 119910 119905)

120597119909

V = V0minus 119911

120597119908 (119909 119910 119905)

120597119910

119908 = 119908 (119909 119910 119905)

(1)

in which 1199060and V0denote the displacements of the midplane

of the plate in the 119909 and 119910 directions respectively Thestrains in the plate can then be written as functions of thedisplacement as follows in the vector form

(

120598119909

120598119910

120574119909119910

) = (

120597119906

120597119909120597V120597119910

120597119906

120597119910+120597V120597119909

) = (

(

1205971199060

120597119909minus 119911

1205972119908

1205971199092

120597V0

120597119909minus 119911

1205972119908

1205971199102

1205971199060

120597119910+120597V0

120597119909minus 2119911

1205972119908

120597119909120597119910

)

)

(2)

The above equations are all written in the global coordinatesystems (119909 119910 119911) Stresses in the ply or lamina coordinatesystem (1 2 3) can then be written with relation to thestresses in the plate coordinate system as follows

(

1205901

1205902

12059112

) = [119879119894](

120590119909

120590119910

120591119909119910

) (3)

in which [119879119894] is the transformation matrix for the 119894th ply

defined by

[119879119894] = [

[

cos2120579119894

sin2120579119894

2 sin 120579119894cos 120579119894

sin2120579119894

cos2120579119894

minus2 sin 120579119894cos 120579119894

minus sin 120579119894cos 120579119894sin 120579119894cos 120579119894cos2120579119894minus sin2120579

119894

]

]

(4)

and 120579119894is the angle between the plate global coordinate system

and the 119894th ply coordinate system as shown in Figure 3Similarly for strain one can write

(

1205981

1205982

12057412

) = [119879119894](

120598119909

120598119910

120574119909119910

) (5)

Additionally stress and strain components in the ply coordi-nate system are related through the reduced stiffness matrix[119876119894] as follows

(

1205901

1205902

12059112

) = [119876119894](

1205981

1205982

12057412

) (6)

in which

[119876119894] =

[[[

[

119876119894

11119876119894

120

119876119894

21119876119894

220

0 0 119876119894

66

]]]

]

(7)

with

119876119894

11=

119864119894

1

1 minus ]11989412]11989421

(8)

119876119894

22=

119864119894

2

1 minus ]11989412]11989421

(9)

119876119894

12=

]11989412119864119894

2

1 minus ]11989412]11989421

(10)

119876119894

66= 119866119894

12 (11)

where1198641198941and119864119894

2are the 119894th ply elasticitymodulus in the longi-

tudinal and transverse direction of the fibers respectively ]11989412

and ]11989421are Poissonrsquos ratios for the 119894th ply and119866119894

12is the 119894th ply

shearmodulus Combining (3) through (11) one canwrite thefollowing relationship relating stress and strain in the platecoordinate system based on each ply reduced stiffnessmatrixcomponents and transformation matrix components

(

120590119909

120590119910

120591119909119910

) = [119876119894](

120598119909

120598119910

120574119909119910

) (12)

4 Shock and Vibration

where

[119876119894] = [119879

minus1

119894] [119876119894] [119879119894] =

[[[[[[

[

119876119894

11119876119894

12119876119894

16

119876119894

12119876119894

22119876119894

26

119876119894

16119876119894

26119876119894

66

]]]]]]

]

(13)

119876119894

11= 119876119894

111198984

119894+ 2 (119876

119894

12+ 2119876119894

66)1198982

1198941198992

119894+ 119876119894

221198994

119894 (14)

119876119894

12= (119876119894

11+ 119876119894

22minus 4119876119894

66)1198982

1198941198992

119894+ 119876119894

12(1198984

119894+ 1198994

119894) (15)

119876119894

22= 119876119894

111198994

119894+ 2 (119876

119894

12+ 2119876119894

66)1198982

1198941198992

119894+ 119876119894

221198984

119894 (16)

119876119894

16= (119876119894

11minus 119876119894

12minus 2119876119894

66)1198983

119894119899119894+ (119876119894

12minus 119876119894

22+ 2119876119894

66)1198981198941198993

119894

(17)

119876119894

26= (119876119894

11minus 119876119894

12minus 2119876119894

66)1198981198941198993

119894+ (119876119894

12minus 119876119894

22+ 2119876119894

66)1198983

119894119899119894

(18)

119876119894

66= (119876119894

11+ 119876119894

22minus 2119876119894

12minus 2119876119894

66)1198982

1198941198992

119894+ 119876119894

66(1198984

119894+ 1198994

119894)

(19)

in which119898119894= cos 120579

119894and 119899119894= sin 120579

119894

By definition the equilibrium equation for the differentialplate element is defined as function of the stress and momentresultants as follows

119888 (119873119909

1205972119908

1205971199092+ 2119873119909119910

1205972119908

120597119909120597119910+ 119873119910

1205972119908

1205971199102)

+ (1205972119872119909

1205971199092+ 2

1205972119872119909119910

120597119909120597119910+119872119910

1205972119908

1205971199102) + 119891

119911=

119899

sum

119894=1

120588119894ℎ119894

1205972119908

1205971199052

(20)

in which 119891119911= minus119901ext(119909 119910 119905) is the external pressure applied to

the panel due to the fluid flow or the random excitation 120588119894is

the density of the 119894th ply and the stress resultant andmomentresultant components are defined as functions of the stresscomponents respectively as follows

(

119873119909

119873119910

119873119909119910

) =

119899

sum

119894=1

int

ℎ119894

ℎ119894minus1

(

120590119909

120590119910

120591119909119910

)119889119911

(

119872119909

119872119910

119872119909119910

) =

119899

sum

119894=1

int

ℎ119894

ℎ119894minus1

(

120590119909

120590119910

120591119909119910

)119911119889119911

(21)

After substituting the stress components defined by (12) into(21) and making use of (14) through (19) then (20) can be

transformed to the following form as function of the platedisplacement 119908

1

3

119899

sum

119894=1

(ℎ3

119894minus ℎ3

119894minus1)

times [119876119894

11(1205972119908

1205971199092)

2

+ 119876119894

22(1205972119908

1205971199102)

2

+ 2119876119894

12(1205972119908

1205971199092)(

1205972119908

1205971199102) + 4119876

119894

16(1205972119908

1205971199092)(

1205972119908

120597119909120597119910)

+ 4119876119894

26(1205972119908

1205971199102)(

1205972119908

120597119909120597119910) + 4119876

119894

66(

1205972119908

120597119909120597119910)

2

]

+1

3

119899

sum

119894=1

(ℎ3

119894minus ℎ3

119894minus1)

times [119876119894

11

1205974119908

1205971199094+ 119876119894

22

1205974119908

1205971199104+ 6119876119894

16

1205974119908

1205971199093120597119910

+ 6119876119894

26

1205974119908

1205971199091205971199103+ (2119876

119894

12+ 8119876119894

66)

1205974119908

12059711990921205971199102]

+

119899

sum

119894=1

120588119894ℎ119894

1205972119908

1205971199052= 119891119911

(22)

Each panel in the present study aims to represent thedistance between adjacent stringers and frames in an aircraftfuselage hence each individual panel is assumed to vibrateindependently In addition since panels are considered to beflat and simply supported in all four boundaries the panelrsquosdisplacement can be defined as follows [23 24]

119908 (119909 119910 119905) =

119872119909

sum

119898119909=1

119872119910

sum

119898119910=1

120572119898119909

(119909) 120573119898119910(119910) 119902119898119909119898119910

(119905) (23)

where 120572119898119909(119909) and 120573

119898119910(119910) are the spatial functions defining

the variation of119908(119909 119910 119905) with 119909 and 119910 respectively 119902119898119909119898119910

(119905)

are the temporal functions defining the variation of119908(119909 119910 119905)with time and 119872 = 119872

119909times 119872119910is the total number of plate

modes (119898119909 119898119910) considered For simply supported plates the

spatial functions may be defined as follows

120572119898119909

(119909) = radic2

119886sin(

119898119909120587119909

119886) (24)

120573119898119910

(119910) = radic2

119887sin(

119898119910120587119910

119887) (25)

Shock and Vibration 5

where 119886 and 119887 are the dimensions of the plate in the 119909 and119910 directions Also the natural frequencies for the simplysupported plate can be determined by

120596119898119909119898119910

=1205872

radicsum119899

119894=1120588119894ℎ119894

[11986311(119898119909

119886)

4

+ 11986322(119898119910

119887)

4

+2(11986312+ 211986366)(

119898119909

119886)

2

(119898119910

119887)

2

]

12

(26)

in which

119863119895119896

=

119899

sum

119894=1

int

ℎ119894

ℎ119894minus1

119876119894

1198951198961199112119889119911 (27)

Developing (22) by considering (23) to (25) making use ofthe orthogonality of the plate modes and integrating over theplate area similarly to the mathematical approach followedin [4] the equilibrium equation for an orthotropic plate iswritten in the matrix form as

[119872119901] q (119905) + [119863

119901] q (119905) + [119870

119901] q (119905) = pext (119905) (28)

where [119872119901] [119863119901] and [119870

119901] are mass damping and stiffness

119872times119872matrices respectively and q(119905) q(119905) and q(119905) are theposition velocity and acceleration vectors with componentsdefined respectively as 119902

119898(119905) = 119876

119898119890119894120596119905 119902119898(119905) = 119894119876

119898120596119890119894120596119905

and 119902119898(119905) = minus119876

1198981205962119890119894120596119905 After performing all the above

described mathematical manipulations it can be shown thateach term in (28) is defined as follows

[119872119901] = diag[

119899

sum

119894=1

120588119894ℎ119894]

[119863119901] = diag[2120585

119901120596119898119909119898119910

119899

sum

119894=1

120588119894ℎ119894]

[119870119901] =

1

3

119899

sum

119894=1

(ℎ3

119894minus ℎ3

119894minus1) 119876119894

11I (119909) + 119876

119894

22I (119910)

+ (2119876119894

12+ 8119876119894

66) I1015840 (119909) I1015840 (119910)

pext (119905) = [int

119886

0

int

119887

0

120572119898119909

(119909) 120573119898119910(119910) 119901ext (119909 119910 119905) 119889119909 119889119910]

(29)

in which 120585119901was added to account with the damping of the

plate and

I (119909) = diag [1119886(119898119909120587

119886)

4

]

I (119910) = diag [1119887(119898119910120587

119887)

4

]

I1015840 (119909) = minus1

119886(119898119909120587

119886)

2

cos [(119898119909minus 1198981015840

119909) 120587]

I1015840 (119910) = minus1

119887(119898119910120587

119887)

2

cos [(119898119910minus 1198981015840

119910) 120587]

(30)

22 Radiated Sound Power The calculation of the panelsrsquoradiated sound power in the frequency domain requires arearrangement of the time domain equations presented in(22) For this end knowing that 119902

119898= 119876119898119890119894120596119905 as previously

described one can write (28) in the following desired form inthe frequency 120596 domain

Y (120596) = H (120596)X (120596) (31)

where

Y (120596) = W (120596)

X (120596) = Pext (120596)

H (120596) = minus1205962[119872119901] + 119894120596 [119863

119901] + [119870

119901]

(32)

in which H(120596) is system frequency response matrix Y(120596) isthe response of the system to the excitationX(120596) and vectorsW(120596) andPext(120596) correspond respectively to the vectors q(119905)and pext(119905) written in the frequency domain Considering theturbulent boundary layer random excitation as a stationaryand homogeneous function the spectral density of the systemresponse is given by [25 26]

S119882119882 (120596) = Hlowast (120596) S119875119875 (120596)H

119879(120596) (33)

where superscripts lowast and 119879 denote Hermitian conjugate andmatrix transpose respectively S

119875119875(120596) is the spectral density

matrix of the random excitation Pext(120596) and S119882119882

(120596) is thespectral density matrix of the system response W(120596) Thematrix S

119875119875(120596) corresponds to the generalized power density

matrix of the turbulent boundary layer excitation defined by

Stbl (120596) = [∬

119886

0

∬

119887

0

120572119898119909

(119909) 1205721198981015840119909

(1199091015840) 120573119898119910

(119910) 1205731198981015840

119910

(1199101015840)

times 119878 (120585119909 120585119910 120596) 119889119909 119889119909

1015840119889119910119889119910

1015840]

(34)

in which 119878(120585119909 120585119910 120596) is defined by Corcos model [5 21] and

120585119909= 119909 minus 119909

1015840 and 120585119910= 119910 minus 119910

1015840 are the spatial separations in thestreamwise and spanwise directions of the plate respectivelyAs shown in [1ndash3] a combination of Corcos and Efimtsovmodels may be suitable to provide this excitation as follows

119878 (120585119909 120585119910 120596) = 119878ref (120596) 119890

minus(120572119909120596|120585119909|119880119888)119890minus(120572119910120596|120585119910|119880119888)119890

minus(119894120596120585119909119880119888)

(35)

where119880119888is the turbulent boundary layer convective velocity

120572119909= 01 and 120572

119910= 077 are empirical parameters used to

provide the best agreement with the reality [27] and 119878ref(120596)is defined using Efimtsov model [7] In [1] an analyticalform was derived for the matrix described in (34)The powerspectral density function of the plate displacement can then

6 Shock and Vibration

be defined as functions of the abovematrix respectively platearea as

119878119908119908

(1199091 1199101 1199092 1199102 120596)

=

1198722

119909

sum

11989811990911198981199092=1

1198722

119910

sum

11989811991011198981199102=1

1205721198981199091

(1199091) 1205721198981199092

(1199092)

times 1205731198981199101

(1199101) 1205731198981199102

(1199102) S119882119882(120596)1198981 1198982

(36)

and the overall power spectral density function can be foundby integrating the individual power spectral densities over theplate area displacement as following

119878119908119908 (120596) = ∬

119886

0

∬

119887

0

119878119908119908

(1199091 1199101 1199092 1199102 120596) 119889119909

1119889119909211988911991011198891199102

(37)

Analytical expressions for 119878119908119908

(1199091 1199101 1199092 1199102 120596) and 119878

119908119908(120596)

were obtained in [23] The power spectral density matrix ofthe plate velocity can be determined from the power spectraldensity matrix of the plate displacement as follows

S119881119881 (120596) = 120596

2S119882119882 (120596) (38)

Using these analytical forms the plate radiated sound powerfunction and the overall power spectral density function canbe determined respectively as follows

RSP (1199091 1199101 1199092 1199102 120596)

=

1198722

119909

sum

11989811990911198981199092=1

1198722

119910

sum

11989811991011198981199102=1

1205721198981199091

(1199091) 1205721198981199092

(1199092)

times 1205731198981199101

(1199101) 1205731198981199102

(1199102)Π(120596)1198981 1198982

RSP (120596) = ∬

119886

0

∬

119887

0

RSP (1199091 1199101 1199092 1199102 120596) 119889119909

1119889119909211988911991011198891199102

(39)

in which Π(120596)1198981 1198982

are the components of the radiationmatrix defined by

Π (120596) = S119881119881

M (120596) (40)and S119881119881

is defined in (38) andM(120596) is evaluated numericallyfrom the following equation

M (120596) = 81205880

1198880

(120596119886119887

1205873119898119909119898119910

)

2

times∬

1205872

0

cossin (

120572

2)

cossin (

120573

2)

[(120572119898119909120587)2minus 1] [(120573119898

119910120587)2

minus 1]

2

timessin 120579119889120579 119889120601(41)

where 1205880and 1198880are the density and the speed of sound of

the interior fluid respectively 120572 = (1205961198880)119886 sin 120579 cos120601 120573 =

(1205961198880)119887 sin 120579 sin120601 in which 120601 and 120579 are the angles of the

distance vector in space to the observation point [27]

3 Structural Vibration and SoundRadiation Results

The formulation presented in the previous section is usedto analyze four different rectangular composite sandwichpanels of several sizes and materials properties as shown inTable 1 Additionally four isotropic aluminum panels withsimilar sizes compared to the composite panels as shownin Table 2 are analyzed using the previous model developedby the author results are then compared with the compositepanels All the eight panels are considered simply supportedin the four boundaries and subjected to turbulent boundarylayer excitation or to random noise excitation The turbulentflow excitation is representative of Mach number 08 flightconditions as described inTable 3 and the randomexcitationrepresents a constant 40 Pascal random noise that is asound pressure level of approximately 120 dB The first tennatural frequencies obtained for the several panels analyzedare shown in Tables 4 and 5 respectively for composite andisotropic panels Cases 1 2 3 and 4 of composite panels asnoted in Table 4 are chosen for validation purposes as instudies [16 18 20] respectively It is shown that the calculatednatural frequencies are in good agreement with the previousstudies

31 Composite Panels The results presented in this sectiondiscuss the radiated sound power and the structural dis-placement levels of the composite sandwich panels studiedResults are shown for frequencies up to 1500Hz requiringa minimum number of plate modes which is dependent ofthe panel size and materials properties for proper accuracyof the predicted values Values for the radiated sound powerRSP and plate displacement power spectral density (PSD)119878119908119908

are obtained for two locations (1199091 1199101) = (05119886 05119887)

and (1199092 1199102) = (03119886 02119887) as well as overall values of these

two quantities and shown in Figures 4 and 5 As referredearlier four different panels are analyzed identified as Cases1 through 4 along the following sections and two types ofexcitation are applied (1) turbulent boundary layer excitationand (2) random noise excitation

As noted in Figure 4 and similarly to what was observedin previous results and formulation dedicated to the analysisof isotropic panels it is concluded that the overall RSPvalues differ from local RSP values especially for the TBLexcitation It is also observed that turbulent flow can beresponsible for exciting a higher number of panel modescompared to random noise with TBL and random noiseproviding dissimilar levels of panelsrsquo sound radiation Assuch to accurately predict the dynamic behaviour of a panelone should not consider the pure random noise excitation asrepresentative of the turbulent flow excitation In the overallRSP results plates of bigger dimensions are shown to havehigher RSP values both for TBL and random excitationswith Case 1 showing higher RSP values than Cases 2 3or 4 The same is not noted for the local RSP valueswith Case 3 panel showing increased RSP values for bothpositions 1 and 2with relation to the overall RSP valuesTheseresults emphasize the importance of knowing the measuring

Shock and Vibration 7

0 500 1000 1500Frequency (Hz)

minus150

minus100

minus50

0

50

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

(b)

0 500 1000 1500

Frequency (Hz)

minus120

minus100

minus80

minus60

minus40

minus20

0

20

40

Radi

ated

soun

d po

wer

(dB

rel1

W)

(c)

0 500 1000 1500

Frequency (Hz)

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

(d)

0 500 1000 1500

Frequency (Hz)

minus140

minus120

minus100

minus80

minus60

minus40

minus20

0

20

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 1 composite TBLCase 2 composite TBL

Case 3 composite TBLCase 4 composite TBL

(e)

0 500 1000 1500

Frequency (Hz)

minus80

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 1 composite randomCase 2 composite random

Case 3 composite randomCase 4 composite random

(f)

Figure 4 Radiated sound power results for the composite panels (a) and (b) overall panel RSP (c) and (d) RSPnear surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) RSP near location (1199092 1199102) = (03119886 02119887)

position for the accurate prediction of sound and vibrationand it highlights the desired sensitivity of the developedmathematical framework to this factor Figure 5 shows thedisplacement PSD results associated with the same panels

and excitations as for the RSP results displayed in Figure 4Comparing Figures 4 and 5 it is observed that structuraldisplacement and sound radiation levels have different curvesfor the same excitation and thus panel displacement PSD

8 Shock and Vibration

minus250

minus200

minus150

minus100

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

minus200

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(b)

minus200

minus180

minus140

minus160

minus120

minus100

minus80

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(c)

minus220

minus200

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(d)

minus220

minus200

minus180

minus140

minus160

minus120

minus100

minus80

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

Case 1 composite TBLCase 2 composite TBL

Case 3 composite TBLCase 4 composite TBL

(e)

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

Case 1 composite randomCase 2 composite random

Case 3 composite randomCase 4 composite random

(f)

Figure 5 Panel displacement PSD results for the composite panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location(1199091 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

curve predictionmay not be fully representative of RSP curveeven though they would be expected to follow similar trendsIn addition as similar to RSP results it is concluded that 119878

119908119908

predictions are dissimilar whether one is assuming TBL or

random excitation Case 1 panel shows higher values of bothoverall and local plate displacement PSD predictions

The results presented here are useful in the context of theprediction of noise radiation by composite panel solutions

Shock and Vibration 9

and its possible application for example in aerospace indus-try Based on the obtained results if one would choose whichcomposite panel solution is most desirable for the purposeof manufacturing an aircraft fuselage panel a smaller panel(Case 3) or a panel with a higher aspect ratio (Case 4) wouldbe preferable for overall improved results Longer and widerpanels (ie Cases 1 and 2) are shown to greatly amplify thelow frequency range However one would need to designthe panel appropriately and with the targeted dimensionssuch that the amplification of noise and vibration at certainfrequencies would be avoided For instance Case 3 panelexhibits a general improved behaviour in terms of noiselevels but simultaneously it amplifies noise at frequenciesnear 300Hz and 1000Hz while Case 4 panel amplifies resultsnear 300Hz Comparing the size of these two panels thesurface area covered by one Case 1 panel corresponds toapproximately the surface area covered by twenty Case 3panels with the twenty Case 3 panels representing approx-imately 94 kg of total mass compared to the 676 kg of asingle Case 1 panel As such an improvement in the noiselevels comes as a tradeoff in terms of increased weightAdditionally it is shown that panels with thicker face layersand thinner core layer specifically referring to Cases 3 and 4have lower noise and vibration levels than panels with thinnerface layers and thicker core layer referring to Cases 1 and 2However as mentioned earlier Cases 3 and 4 panels withsmaller dimensions and thicker face layers have the problemof noise amplification at certain frequencies Therefore apanel which combines the properties of being shorter with athicker core layer versus thinner face layers may potentiallyshow an improved behaviour for reduced vibration and noiseradiation as well as lower noise amplification at isolatedfrequencies

32 Isotropic Panels This section addresses the RSP and 119878119908119908

predictions for the isotropic panels analyzed and results upto 1500Hz are shown in Figures 6 and 7 Four aluminumisotropic panels are investigated considering TBL or randomexcitation as follows isotropic panels Cases 3 and 4 have thesame dimensions 119886 and 119887 as the composite panels Cases3 and 4 respectively isotropic panels Cases 5 and 6 areconsidered with same dimensions 119886 and 119887 as Cases 3 and4 respectively but with increased thicknesses This approachis taken with the purpose of posterior comparison betweencomposite and isotropic panelsrsquo behaviours In this contextisotropic panels Cases 3 and 4 are both 1mm thick whileisotropic panels Cases 4 and 5 have thicknesses of 15mm and4mm respectively

As noted in Figures 6 and 7 the increase of panelthickness results in an expected shift in the panel naturalfrequencies to higher frequencies as well as a shift in themodes amplified in both RSP and 119878

119908119908results This effect is

observed in the TBL and random excitations Furthermore itis found that panels with higher aspect ratio (ie Cases 4 and6) are generally characterized by having lower levels of soundradiation and vibration As similarly noted for the compositepanels random and TBL excitations provide different RSPand 119878

119908119908predictions and overall values differ from point

values and hence it is important to specify the location in the

panel where the measurementspredictions are being takenIt is also noted that the random excitation remains as theexcitation causing higher noise and vibration levels for allcases at the higher frequency range This is also an expectedeffect as the TBL excitation decreases towards the higherfrequencies while the random excitation provides a constantexcitation over the frequency spectrum

33 Composite versus Isotropic Panels As one of the mainobjectives of this study this section presents a discussionof results for the behavior of composite sandwich panelsin comparison with the traditional isotropic panels Thefollowing four figures are shown to this end Figures 8 and9 compare results for composite panel Case 3 with isotropicpanels Cases 3 and 5 while Figures 10 and 11 comparecomposite panel Case 4 with isotropic panels Cases 4 and 6

By comparing Cases 3 and 5 panels as shown fromFigures 8 and 9 it is noted that composite panel Case 3exhibits an enhanced behaviour in comparison with isotropicpanels Cases 3 and 5 both in RSP and displacement PSDlevels However one should also note that Case 3 compositepanel shows higher levels of noise and vibration at specificisolated frequencies approximately near 300Hz and 1000Hzfor the overall RSP and for position 1 RSP results as shown inFigure 8 Considering the panel displacement PSD results inFigure 9 extra frequencies above 1000Hz are also amplifiedespecially at position 2 for both TBL and random excitationsAs described in Table 1 the composite panel Case 3 has athin core layer thus the unwanted amplification at thesefrequencies may be attenuated by increasing the core layerthickness However this modification on the panel thicknesswill increase the original panel weight of 047 kg whichalready represents heavier solution compared to Cases 3and 5 isotropic panels with weights of 03 kg and 045 kgrespectively Although the panel weight will increase if thischange is performed making this modification will in alater phase require less added acoustic insulation material tothe aircraft walls in the context of aircraft fuselage panelsrsquoapplication As such the composite panelmay represent witha slight increase in the panel weight an optimized solutionfor both sound radiation and structural vibration in almostall the frequency spectra analyzed

Similar conclusions are obtained from Figures 10 and 11dedicated to the comparison of Cases 4 and 6 panels thatis the composite panel represents an improved solution foralmost all the frequency ranges studied By analyzing thegeometric differences between composite panels Cases 3 and4 Case 4 has thicker face layers and is longer than Case 3panel having a much higher aspect ratio The amplificationof noise and vibration levels at specific frequencies is notedas in Case 3 composite panel Since face layers are thickerand this panel has a higher aspect ratio this effect is morevisible in composite Case 4 than in composite Case 3 Asobserved in results from Case 3 this could be enhancedby implementing an increase of the core layer thicknessWhile the increase in the face layer thickness is shown todecrease the levels of noise and vibration in both overalland localized predictions it also can have an impact in the

10 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 6 Radiated sound power results for the isotropic panels (a) and (b) overall panel RSP (c) and (d) RSP near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) RSP near location (1199092 1199102) = (03119886 02119887)

Shock and Vibration 11

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

(a)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(b)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(c)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus50

minus100

minus150

minus200

(f)

Figure 7 Panel displacement PSD results for the isotropic panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) 119878119908119908

near location (1199092 1199102) = (03119886 02119887)

12 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus100

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus100

minus80

minus60

minus40

minus20

0

20

40

60

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 8 Comparison of radiated sound power results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 13

0 500 1000 1500

Frequency (Hz)

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus50

minus150

minus100

minus200

minus250

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 9 Comparison of panel displacement PSD results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

14 Shock and VibrationRa

diat

ed so

und

pow

er (d

B re

l1

W)

0 500 1000 1500

Frequency (Hz)

50

0

minus50

minus100

minus150

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

100

50

0

minus50

minus100

minus150

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

50

0

minus50

minus100

minus150

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus80

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 10 Comparison of radiated sound power results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 15

minus50

minus100

minus150

minus250

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus220

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus50

minus100

minus150

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 11 Comparison of panel displacement PSD results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b)overall panel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

16 Shock and Vibration

Table 1 Physical properties of composite panels

Variable Description units Case 1 Case 2 Case 3 Case 4119886 Length m 183 11683 0348 10119887 Width m 122 11683 03048 03120582 Aspect ratio (119886119887) 15 10 114 333ℎ1 ℎ3

Face layers thickness mm 0406 0506 0762 35ℎ2

Core layer thickness mm 64 1905 0254 051198641 1198643

Face layers Young modulus Pa 6898 times 1010

71 times 1010

6898 times 1010

66 times 1010

1198642

Core layer Young modulus Pa 457 times 105

457 times 105

267 times 106

26 times 108

]1 ]3

Face layers Poissonrsquos ratio 033 033 03 033]2

Core layer Poissonrsquos ratio 04 04 049 031205881 1205883

Face layers density kgmminus3 2768 2700 2740 26801205882

Core layer density kgmminus3 122 4806 999 1680119898 Total mass kg 676 499 047 588

Table 2 Physical properties of isotropic panels

Variable Description units Case 3 Case 4 Case 5 Case 6119886 Length m 0348 10 0348 10119887 Width m 03048 03 03048 03120582 Aspect ratio (119886119887) 114 333 114 333ℎ119901

Thickness mm 1 1 15 4119864119901

Young modulus Pa 724 times 1010

724 times 1010

724 times 1010

724 times 1010

]119901

Poissonrsquos ratio 033 033 033 033120588119901

Density kg mminus3 2800 2800 2800 2800119898 Total mass kg 030 084 045 336

Table 3 Flight test conditions and air properties

Variable Description units Value119872 Mach number 08alt Altitude m 13106119888 Speed of sound m sminus1 295119880infin

Flow speed m sminus1 23606120588 Air density kgmminus3 02622] Kinematic viscosity m2 sminus1 5422 times 10

minus5

119879 Temperature K 21665Re119909

Reynolds number 2917 times 107

1198880

Speed of sound m sminus1 3401205880

Internal air density kgmminus3 142

higher amplification of localized frequencies Therefore theface layers thickness increase should be performed carefullyby an accompanied increase in the core layer thickness whilesimultaneously considering the weight constraints for thespecific panels applications

A more detailed analysis will be required in order toformulate final conclusions about the optimum solutions forthe panel in the context of aircraft application For that theinsulation materials would also be considered to account forthe respective added weight and effect in the sound radi-ation signature However typically the acoustic insulationmaterials will only have a higher impact in decreasing the

Table 4 First 10 natural frequencies for the composite panels

Modes 119891 Hz(Case 1)

119891 Hz(Case 2)

119891 Hz(Case 3)

119891 Hz(Case 4)

1 2365 10588 4874 126582 4548 26470 11222 157933 7276 42352 13148 210194 8186 52941 19496 283355 9459 68823 21800 377416 13097 89999 26939 474967 13279 95293 30075 492388 15462 105881 33287 506329 17645 132351 36612 5585710 18191 137645 43866 62825

sound levels at higher frequency ranges and will thereforenot improve significantly the problem of higher noise levelsin the low frequency range particularly observed in theisotropic solutions As such the composite panel solutionsmay appear to be a promising replacement to the isotropicpanels especially panels like Cases 3 and 4 which are alreadycharacterized by lower levels of both RSP and 119878

119908119908in the

low frequency range In further analysis it will also beimportant to carefully address the noise amplification atisolated frequencies by identifying a good combination of

Shock and Vibration 17

Table 5 First 10 natural frequencies for the isotropic panels

Modes 119891 Hz(Case 3)

119891 Hz(Case 4)

119891 Hz(Case 5)

119891 Hz(Case 6)

1 4646 2958 6969 118322 10697 3691 16046 147643 12534 4912 18801 196494 18585 6622 27877 264885 20782 8820 31173 352826 25680 1110 38520 443997 28670 11507 43004 460298 31731 11832 47596 473309 34901 13054 52351 5221510 41816 14683 62723 58731

core and face layers in terms of both materials selection anddimensions

4 Conclusions

This study presents a mathematical method and frameworkon composite panels radiated sound and vibrationThepanelsare considered excited either by turbulent flow or by randomnoiseThe turbulent flow corresponds to the typical excitationof an aircraft panel at flight cruise conditions with a Machnumber of 08 The effect of type of panel of panel dimen-sions and of panel thickness are evaluated The turbulentflow excitation showed to excite a higher number of panelmodes compared to random noise excitation and resulted indissimilar levels of both panel vibration and radiated soundwith relation to the random excitation This effect is morevisible for composite than for isotropic panels and thereforethe pure random noise should not be used as representativeof the turbulent flow excitation in real applications

Different properties on the composite panels are shown toyield dissimilar effects in the noise and vibration predictionsThe effect of the panel size specifically its length and widthis noted to have a significant effect on the panel performancein terms of noise radiation and vibration levels Overallsmaller composite panels or panels with higher aspect ratioproduced lower levels of radiated sound and vibration thanlonger and wider composite panels Compared to isotropicpanels composite panels analyzed are shown to radiate lowernoise levels with the exception of specific natural frequencieswhich are amplified near 300Hz and 1000Hz It is shown thatthis amplification can be significant for composite panels withthicker face layers and thinner core layer and thus resultingin unwanted effect of increased sound radiated Despitethis effect this drawback may potentially be corrected byincreasing the sandwich core layer thickness Increasing thecore layer thickness will impact the panel weight and hencethe composite panels can represent a heavier solution com-pared to the isotropic panels examined On the other handthis approach may be beneficial in the design of compositepanels in the context of aircraft panel applications sincethese panels may require less acoustic insulation material

added to the aircraft walls as they already represent animproved design solution Furthermore acoustic insulationmaterials have a higher impact in decreasing sound levelsat the high frequency ranges and therefore it will notsignificantly improve the problem of higher noise levels at thelow frequency range particularly observed in the isotropicsolutions In this context advanced and optimized compositepanel configurations may be a promising replacement to thetraditional isotropic panels for aerospace applications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is grateful for the support of the National Scienceand Engineering Research Council of Canada

References

[1] J Rocha and D Palumbo ldquoOn the sensitivity of sound powerradiated by aircraft panels to turbulent boundary layer param-etersrdquo Journal of Sound and Vibration vol 331 no 21 pp 4785ndash4806 2012

[2] J Rocha A Suleman and F Lau ldquoPrediction of turbulentboundary layer induced noise in the cabin of a BWB aircraftrdquoShock and Vibration vol 19 no 4 pp 693ndash705 2012

[3] J da Rocha A Suleman and F Lau ldquoFlow-induced noise andvibration in aircraft cylindrical cabins closed-form analyticalmodel validationrdquo Journal of Vibration and Acoustics vol 133no 5 Article ID 051013 9 pages 2011

[4] J da Rocha A Suleman and F Lau ldquoPrediction of flow-inducednoise in transport vehicles development and validation of acoupled structuralmdashacoustic analytical frameworkrdquo CanadianAcousticsmdashAcoustique Canadienne vol 37 no 4 pp 13ndash292009

[5] G Corcos ldquoThe structure of the turbulent pressure field inboundary layer flowsrdquo Journal of Fluid Mechanics vol 18 pp353ndash378 1964

[6] G Corcos ldquoResolution of pressure in turbulencerdquo Journal of theAcoustical Society of America vol 35 no 2 pp 192ndash199 1963

[7] B Efimtsov ldquoCharacteristics of the field of turbulent wallpressure fluctuations at large Reynolds numbersrdquo Soviet PhysicsAcoustics vol 28 no 4 pp 289ndash292 1982

[8] J Reddy Mechanics of Laminated Composite Plates and ShellsTheory and Analysis CRC Press Boca Raton Fla USA 1945

[9] J Whitney Structural Analysis of Laminated Anisotropic PlatesTechnomic Lancaster Pa USA 1987

[10] C L Dym and M A Lang ldquoTransmission of sound throughsandwich panelsrdquo Journal of the Acoustical Society of Americavol 56 no 5 pp 1523ndash1532 1974

[11] C L Dym C S Ventres and M A Lang ldquoTransmission ofsound through sandwich panels a reconsiderationrdquo Journal ofthe Acoustical Society of America vol 59 no 2 pp 364ndash3671976

[12] S Narayanan andR L Shanbhag ldquoSound transmission througha damped sandwich panelrdquo Journal of Sound and Vibration vol80 no 3 pp 315ndash327 1982

18 Shock and Vibration

[13] C L Dym and D C Lang ldquoTransmission loss of dampedasymmetric sandwich panels with orthotropic coresrdquo Journal ofSound and Vibration vol 88 no 3 pp 299ndash319 1983

[14] S Ghinet N Atalla and H Osman ldquoDiffuse field transmissioninto infinite sandwich composite and laminate composite cylin-dersrdquo Journal of Sound and Vibration vol 289 no 4-5 pp 745ndash778 2006

[15] J Mendler ldquoComputational tool for the acoustic analysis ofsandwich panelsrdquo in Proceedings of the European Conference onNoise Control (EURO-NOISE rsquo95) Lyon France 1995

[16] O Foin J Nicolas andNAtalla ldquoAn efficient tool for predictingthe structural acoustic and vibration response of sandwichplates in light or heavy fluidrdquo Applied Acoustics vol 57 no 3pp 213ndash242 1999

[17] S Assaf and M Guerich ldquoNumerical prediction of noisetransmission loss through viscoelastically damped sandwichplatesrdquo Journal of Sandwich Structures andMaterials vol 10 no5 pp 359ndash384 2008

[18] S Assaf M Guerich and P Cuvelier ldquoVibration and acousticresponse of damped sandwich panels immersed in a light orheavy fluidrdquo Computers and Structures vol 88 pp 359ndash3842010

[19] R Zhou and M J Crocker ldquoBoundary element analyses forsound transmission loss of panelsrdquo Journal of the AcousticalSociety of America vol 127 no 2 pp 829ndash840 2010

[20] K Amichi N Atalla and R Ruokolainen ldquoA new 3D finite ele-ment sandwich plate for predicting the vibroacoustic responseof laminated steel panelsrdquo Finite Elements in Analysis andDesign vol 46 no 12 pp 1131ndash1145 2010

[21] F Franco S de Rosa and T Polito ldquoFinite element inves-tigations on the vibroacoustic performance of plane plateswith random stiffnessrdquo Mechanics of Advanced Materials andStructures vol 18 no 7 pp 484ndash497 2011

[22] Y Kim ldquoIdentification of sound transmission characteristicsof honeycomb sandwich panels using hybrid analyticalone-dimensional finite element methodrdquo in Proceedings of the35th International Congress and Exposition on Noise ControlEngineering (INTER-NOISE rsquo06) HonoluluHawaii USA 2006

[23] F Fahy Sound and Structural Vibration Radiation Transmissionand Response Academic Press London UK 1985

[24] J Blevins Formulas for Natural Frequency and Mode ShapeKrieger Publishing Malabar Fla USA 2001

[25] D Newland An Introduction to Random Vibration and SpectralAnalysis Longman New York NY USA 1984

[26] Y Lin Probabilistic Theory of Structural Dynamics McGraw-Hill Book New York NY USA 1967

[27] C E Wallace ldquoRadiation resistance of a rectangular panelrdquoTheJournal of the Acoustical Society of America vol 51 no 3 pp946ndash952 1972

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

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Electrical and Computer Engineering

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Chemical EngineeringInternational Journal of Antennas and

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International Journal of

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Navigation and Observation

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DistributedSensor Networks

International Journal of

Shock and Vibration 3

2

1

u x

y

w z

h h2

hn

h0

120579x

y

Figure 3 Detailed view of the anatomy of the composite ply layers

respectively in the 119909 119910 and 119911 directions (as in Figure 2) canbe written as follows

119906 = 1199060minus 119911

120597119908 (119909 119910 119905)

120597119909

V = V0minus 119911

120597119908 (119909 119910 119905)

120597119910

119908 = 119908 (119909 119910 119905)

(1)

in which 1199060and V0denote the displacements of the midplane

of the plate in the 119909 and 119910 directions respectively Thestrains in the plate can then be written as functions of thedisplacement as follows in the vector form

(

120598119909

120598119910

120574119909119910

) = (

120597119906

120597119909120597V120597119910

120597119906

120597119910+120597V120597119909

) = (

(

1205971199060

120597119909minus 119911

1205972119908

1205971199092

120597V0

120597119909minus 119911

1205972119908

1205971199102

1205971199060

120597119910+120597V0

120597119909minus 2119911

1205972119908

120597119909120597119910

)

)

(2)

The above equations are all written in the global coordinatesystems (119909 119910 119911) Stresses in the ply or lamina coordinatesystem (1 2 3) can then be written with relation to thestresses in the plate coordinate system as follows

(

1205901

1205902

12059112

) = [119879119894](

120590119909

120590119910

120591119909119910

) (3)

in which [119879119894] is the transformation matrix for the 119894th ply

defined by

[119879119894] = [

[

cos2120579119894

sin2120579119894

2 sin 120579119894cos 120579119894

sin2120579119894

cos2120579119894

minus2 sin 120579119894cos 120579119894

minus sin 120579119894cos 120579119894sin 120579119894cos 120579119894cos2120579119894minus sin2120579

119894

]

]

(4)

and 120579119894is the angle between the plate global coordinate system

and the 119894th ply coordinate system as shown in Figure 3Similarly for strain one can write

(

1205981

1205982

12057412

) = [119879119894](

120598119909

120598119910

120574119909119910

) (5)

Additionally stress and strain components in the ply coordi-nate system are related through the reduced stiffness matrix[119876119894] as follows

(

1205901

1205902

12059112

) = [119876119894](

1205981

1205982

12057412

) (6)

in which

[119876119894] =

[[[

[

119876119894

11119876119894

120

119876119894

21119876119894

220

0 0 119876119894

66

]]]

]

(7)

with

119876119894

11=

119864119894

1

1 minus ]11989412]11989421

(8)

119876119894

22=

119864119894

2

1 minus ]11989412]11989421

(9)

119876119894

12=

]11989412119864119894

2

1 minus ]11989412]11989421

(10)

119876119894

66= 119866119894

12 (11)

where1198641198941and119864119894

2are the 119894th ply elasticitymodulus in the longi-

tudinal and transverse direction of the fibers respectively ]11989412

and ]11989421are Poissonrsquos ratios for the 119894th ply and119866119894

12is the 119894th ply

shearmodulus Combining (3) through (11) one canwrite thefollowing relationship relating stress and strain in the platecoordinate system based on each ply reduced stiffnessmatrixcomponents and transformation matrix components

(

120590119909

120590119910

120591119909119910

) = [119876119894](

120598119909

120598119910

120574119909119910

) (12)

4 Shock and Vibration

where

[119876119894] = [119879

minus1

119894] [119876119894] [119879119894] =

[[[[[[

[

119876119894

11119876119894

12119876119894

16

119876119894

12119876119894

22119876119894

26

119876119894

16119876119894

26119876119894

66

]]]]]]

]

(13)

119876119894

11= 119876119894

111198984

119894+ 2 (119876

119894

12+ 2119876119894

66)1198982

1198941198992

119894+ 119876119894

221198994

119894 (14)

119876119894

12= (119876119894

11+ 119876119894

22minus 4119876119894

66)1198982

1198941198992

119894+ 119876119894

12(1198984

119894+ 1198994

119894) (15)

119876119894

22= 119876119894

111198994

119894+ 2 (119876

119894

12+ 2119876119894

66)1198982

1198941198992

119894+ 119876119894

221198984

119894 (16)

119876119894

16= (119876119894

11minus 119876119894

12minus 2119876119894

66)1198983

119894119899119894+ (119876119894

12minus 119876119894

22+ 2119876119894

66)1198981198941198993

119894

(17)

119876119894

26= (119876119894

11minus 119876119894

12minus 2119876119894

66)1198981198941198993

119894+ (119876119894

12minus 119876119894

22+ 2119876119894

66)1198983

119894119899119894

(18)

119876119894

66= (119876119894

11+ 119876119894

22minus 2119876119894

12minus 2119876119894

66)1198982

1198941198992

119894+ 119876119894

66(1198984

119894+ 1198994

119894)

(19)

in which119898119894= cos 120579

119894and 119899119894= sin 120579

119894

By definition the equilibrium equation for the differentialplate element is defined as function of the stress and momentresultants as follows

119888 (119873119909

1205972119908

1205971199092+ 2119873119909119910

1205972119908

120597119909120597119910+ 119873119910

1205972119908

1205971199102)

+ (1205972119872119909

1205971199092+ 2

1205972119872119909119910

120597119909120597119910+119872119910

1205972119908

1205971199102) + 119891

119911=

119899

sum

119894=1

120588119894ℎ119894

1205972119908

1205971199052

(20)

in which 119891119911= minus119901ext(119909 119910 119905) is the external pressure applied to

the panel due to the fluid flow or the random excitation 120588119894is

the density of the 119894th ply and the stress resultant andmomentresultant components are defined as functions of the stresscomponents respectively as follows

(

119873119909

119873119910

119873119909119910

) =

119899

sum

119894=1

int

ℎ119894

ℎ119894minus1

(

120590119909

120590119910

120591119909119910

)119889119911

(

119872119909

119872119910

119872119909119910

) =

119899

sum

119894=1

int

ℎ119894

ℎ119894minus1

(

120590119909

120590119910

120591119909119910

)119911119889119911

(21)

After substituting the stress components defined by (12) into(21) and making use of (14) through (19) then (20) can be

transformed to the following form as function of the platedisplacement 119908

1

3

119899

sum

119894=1

(ℎ3

119894minus ℎ3

119894minus1)

times [119876119894

11(1205972119908

1205971199092)

2

+ 119876119894

22(1205972119908

1205971199102)

2

+ 2119876119894

12(1205972119908

1205971199092)(

1205972119908

1205971199102) + 4119876

119894

16(1205972119908

1205971199092)(

1205972119908

120597119909120597119910)

+ 4119876119894

26(1205972119908

1205971199102)(

1205972119908

120597119909120597119910) + 4119876

119894

66(

1205972119908

120597119909120597119910)

2

]

+1

3

119899

sum

119894=1

(ℎ3

119894minus ℎ3

119894minus1)

times [119876119894

11

1205974119908

1205971199094+ 119876119894

22

1205974119908

1205971199104+ 6119876119894

16

1205974119908

1205971199093120597119910

+ 6119876119894

26

1205974119908

1205971199091205971199103+ (2119876

119894

12+ 8119876119894

66)

1205974119908

12059711990921205971199102]

+

119899

sum

119894=1

120588119894ℎ119894

1205972119908

1205971199052= 119891119911

(22)

Each panel in the present study aims to represent thedistance between adjacent stringers and frames in an aircraftfuselage hence each individual panel is assumed to vibrateindependently In addition since panels are considered to beflat and simply supported in all four boundaries the panelrsquosdisplacement can be defined as follows [23 24]

119908 (119909 119910 119905) =

119872119909

sum

119898119909=1

119872119910

sum

119898119910=1

120572119898119909

(119909) 120573119898119910(119910) 119902119898119909119898119910

(119905) (23)

where 120572119898119909(119909) and 120573

119898119910(119910) are the spatial functions defining

the variation of119908(119909 119910 119905) with 119909 and 119910 respectively 119902119898119909119898119910

(119905)

are the temporal functions defining the variation of119908(119909 119910 119905)with time and 119872 = 119872

119909times 119872119910is the total number of plate

modes (119898119909 119898119910) considered For simply supported plates the

spatial functions may be defined as follows

120572119898119909

(119909) = radic2

119886sin(

119898119909120587119909

119886) (24)

120573119898119910

(119910) = radic2

119887sin(

119898119910120587119910

119887) (25)

Shock and Vibration 5

where 119886 and 119887 are the dimensions of the plate in the 119909 and119910 directions Also the natural frequencies for the simplysupported plate can be determined by

120596119898119909119898119910

=1205872

radicsum119899

119894=1120588119894ℎ119894

[11986311(119898119909

119886)

4

+ 11986322(119898119910

119887)

4

+2(11986312+ 211986366)(

119898119909

119886)

2

(119898119910

119887)

2

]

12

(26)

in which

119863119895119896

=

119899

sum

119894=1

int

ℎ119894

ℎ119894minus1

119876119894

1198951198961199112119889119911 (27)

Developing (22) by considering (23) to (25) making use ofthe orthogonality of the plate modes and integrating over theplate area similarly to the mathematical approach followedin [4] the equilibrium equation for an orthotropic plate iswritten in the matrix form as

[119872119901] q (119905) + [119863

119901] q (119905) + [119870

119901] q (119905) = pext (119905) (28)

where [119872119901] [119863119901] and [119870

119901] are mass damping and stiffness

119872times119872matrices respectively and q(119905) q(119905) and q(119905) are theposition velocity and acceleration vectors with componentsdefined respectively as 119902

119898(119905) = 119876

119898119890119894120596119905 119902119898(119905) = 119894119876

119898120596119890119894120596119905

and 119902119898(119905) = minus119876

1198981205962119890119894120596119905 After performing all the above

described mathematical manipulations it can be shown thateach term in (28) is defined as follows

[119872119901] = diag[

119899

sum

119894=1

120588119894ℎ119894]

[119863119901] = diag[2120585

119901120596119898119909119898119910

119899

sum

119894=1

120588119894ℎ119894]

[119870119901] =

1

3

119899

sum

119894=1

(ℎ3

119894minus ℎ3

119894minus1) 119876119894

11I (119909) + 119876

119894

22I (119910)

+ (2119876119894

12+ 8119876119894

66) I1015840 (119909) I1015840 (119910)

pext (119905) = [int

119886

0

int

119887

0

120572119898119909

(119909) 120573119898119910(119910) 119901ext (119909 119910 119905) 119889119909 119889119910]

(29)

in which 120585119901was added to account with the damping of the

plate and

I (119909) = diag [1119886(119898119909120587

119886)

4

]

I (119910) = diag [1119887(119898119910120587

119887)

4

]

I1015840 (119909) = minus1

119886(119898119909120587

119886)

2

cos [(119898119909minus 1198981015840

119909) 120587]

I1015840 (119910) = minus1

119887(119898119910120587

119887)

2

cos [(119898119910minus 1198981015840

119910) 120587]

(30)

22 Radiated Sound Power The calculation of the panelsrsquoradiated sound power in the frequency domain requires arearrangement of the time domain equations presented in(22) For this end knowing that 119902

119898= 119876119898119890119894120596119905 as previously

described one can write (28) in the following desired form inthe frequency 120596 domain

Y (120596) = H (120596)X (120596) (31)

where

Y (120596) = W (120596)

X (120596) = Pext (120596)

H (120596) = minus1205962[119872119901] + 119894120596 [119863

119901] + [119870

119901]

(32)

in which H(120596) is system frequency response matrix Y(120596) isthe response of the system to the excitationX(120596) and vectorsW(120596) andPext(120596) correspond respectively to the vectors q(119905)and pext(119905) written in the frequency domain Considering theturbulent boundary layer random excitation as a stationaryand homogeneous function the spectral density of the systemresponse is given by [25 26]

S119882119882 (120596) = Hlowast (120596) S119875119875 (120596)H

119879(120596) (33)

where superscripts lowast and 119879 denote Hermitian conjugate andmatrix transpose respectively S

119875119875(120596) is the spectral density

matrix of the random excitation Pext(120596) and S119882119882

(120596) is thespectral density matrix of the system response W(120596) Thematrix S

119875119875(120596) corresponds to the generalized power density

matrix of the turbulent boundary layer excitation defined by

Stbl (120596) = [∬

119886

0

∬

119887

0

120572119898119909

(119909) 1205721198981015840119909

(1199091015840) 120573119898119910

(119910) 1205731198981015840

119910

(1199101015840)

times 119878 (120585119909 120585119910 120596) 119889119909 119889119909

1015840119889119910119889119910

1015840]

(34)

in which 119878(120585119909 120585119910 120596) is defined by Corcos model [5 21] and

120585119909= 119909 minus 119909

1015840 and 120585119910= 119910 minus 119910

1015840 are the spatial separations in thestreamwise and spanwise directions of the plate respectivelyAs shown in [1ndash3] a combination of Corcos and Efimtsovmodels may be suitable to provide this excitation as follows

119878 (120585119909 120585119910 120596) = 119878ref (120596) 119890

minus(120572119909120596|120585119909|119880119888)119890minus(120572119910120596|120585119910|119880119888)119890

minus(119894120596120585119909119880119888)

(35)

where119880119888is the turbulent boundary layer convective velocity

120572119909= 01 and 120572

119910= 077 are empirical parameters used to

provide the best agreement with the reality [27] and 119878ref(120596)is defined using Efimtsov model [7] In [1] an analyticalform was derived for the matrix described in (34)The powerspectral density function of the plate displacement can then

6 Shock and Vibration

be defined as functions of the abovematrix respectively platearea as

119878119908119908

(1199091 1199101 1199092 1199102 120596)

=

1198722

119909

sum

11989811990911198981199092=1

1198722

119910

sum

11989811991011198981199102=1

1205721198981199091

(1199091) 1205721198981199092

(1199092)

times 1205731198981199101

(1199101) 1205731198981199102

(1199102) S119882119882(120596)1198981 1198982

(36)

and the overall power spectral density function can be foundby integrating the individual power spectral densities over theplate area displacement as following

119878119908119908 (120596) = ∬

119886

0

∬

119887

0

119878119908119908

(1199091 1199101 1199092 1199102 120596) 119889119909

1119889119909211988911991011198891199102

(37)

Analytical expressions for 119878119908119908

(1199091 1199101 1199092 1199102 120596) and 119878

119908119908(120596)

were obtained in [23] The power spectral density matrix ofthe plate velocity can be determined from the power spectraldensity matrix of the plate displacement as follows

S119881119881 (120596) = 120596

2S119882119882 (120596) (38)

Using these analytical forms the plate radiated sound powerfunction and the overall power spectral density function canbe determined respectively as follows

RSP (1199091 1199101 1199092 1199102 120596)

=

1198722

119909

sum

11989811990911198981199092=1

1198722

119910

sum

11989811991011198981199102=1

1205721198981199091

(1199091) 1205721198981199092

(1199092)

times 1205731198981199101

(1199101) 1205731198981199102

(1199102)Π(120596)1198981 1198982

RSP (120596) = ∬

119886

0

∬

119887

0

RSP (1199091 1199101 1199092 1199102 120596) 119889119909

1119889119909211988911991011198891199102

(39)

in which Π(120596)1198981 1198982

are the components of the radiationmatrix defined by

Π (120596) = S119881119881

M (120596) (40)and S119881119881

is defined in (38) andM(120596) is evaluated numericallyfrom the following equation

M (120596) = 81205880

1198880

(120596119886119887

1205873119898119909119898119910

)

2

times∬

1205872

0

cossin (

120572

2)

cossin (

120573

2)

[(120572119898119909120587)2minus 1] [(120573119898

119910120587)2

minus 1]

2

timessin 120579119889120579 119889120601(41)

where 1205880and 1198880are the density and the speed of sound of

the interior fluid respectively 120572 = (1205961198880)119886 sin 120579 cos120601 120573 =

(1205961198880)119887 sin 120579 sin120601 in which 120601 and 120579 are the angles of the

distance vector in space to the observation point [27]

3 Structural Vibration and SoundRadiation Results

The formulation presented in the previous section is usedto analyze four different rectangular composite sandwichpanels of several sizes and materials properties as shown inTable 1 Additionally four isotropic aluminum panels withsimilar sizes compared to the composite panels as shownin Table 2 are analyzed using the previous model developedby the author results are then compared with the compositepanels All the eight panels are considered simply supportedin the four boundaries and subjected to turbulent boundarylayer excitation or to random noise excitation The turbulentflow excitation is representative of Mach number 08 flightconditions as described inTable 3 and the randomexcitationrepresents a constant 40 Pascal random noise that is asound pressure level of approximately 120 dB The first tennatural frequencies obtained for the several panels analyzedare shown in Tables 4 and 5 respectively for composite andisotropic panels Cases 1 2 3 and 4 of composite panels asnoted in Table 4 are chosen for validation purposes as instudies [16 18 20] respectively It is shown that the calculatednatural frequencies are in good agreement with the previousstudies

31 Composite Panels The results presented in this sectiondiscuss the radiated sound power and the structural dis-placement levels of the composite sandwich panels studiedResults are shown for frequencies up to 1500Hz requiringa minimum number of plate modes which is dependent ofthe panel size and materials properties for proper accuracyof the predicted values Values for the radiated sound powerRSP and plate displacement power spectral density (PSD)119878119908119908

are obtained for two locations (1199091 1199101) = (05119886 05119887)

and (1199092 1199102) = (03119886 02119887) as well as overall values of these

two quantities and shown in Figures 4 and 5 As referredearlier four different panels are analyzed identified as Cases1 through 4 along the following sections and two types ofexcitation are applied (1) turbulent boundary layer excitationand (2) random noise excitation

As noted in Figure 4 and similarly to what was observedin previous results and formulation dedicated to the analysisof isotropic panels it is concluded that the overall RSPvalues differ from local RSP values especially for the TBLexcitation It is also observed that turbulent flow can beresponsible for exciting a higher number of panel modescompared to random noise with TBL and random noiseproviding dissimilar levels of panelsrsquo sound radiation Assuch to accurately predict the dynamic behaviour of a panelone should not consider the pure random noise excitation asrepresentative of the turbulent flow excitation In the overallRSP results plates of bigger dimensions are shown to havehigher RSP values both for TBL and random excitationswith Case 1 showing higher RSP values than Cases 2 3or 4 The same is not noted for the local RSP valueswith Case 3 panel showing increased RSP values for bothpositions 1 and 2with relation to the overall RSP valuesTheseresults emphasize the importance of knowing the measuring

Shock and Vibration 7

0 500 1000 1500Frequency (Hz)

minus150

minus100

minus50

0

50

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

(b)

0 500 1000 1500

Frequency (Hz)

minus120

minus100

minus80

minus60

minus40

minus20

0

20

40

Radi

ated

soun

d po

wer

(dB

rel1

W)

(c)

0 500 1000 1500

Frequency (Hz)

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

(d)

0 500 1000 1500

Frequency (Hz)

minus140

minus120

minus100

minus80

minus60

minus40

minus20

0

20

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 1 composite TBLCase 2 composite TBL

Case 3 composite TBLCase 4 composite TBL

(e)

0 500 1000 1500

Frequency (Hz)

minus80

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 1 composite randomCase 2 composite random

Case 3 composite randomCase 4 composite random

(f)

Figure 4 Radiated sound power results for the composite panels (a) and (b) overall panel RSP (c) and (d) RSPnear surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) RSP near location (1199092 1199102) = (03119886 02119887)

position for the accurate prediction of sound and vibrationand it highlights the desired sensitivity of the developedmathematical framework to this factor Figure 5 shows thedisplacement PSD results associated with the same panels

and excitations as for the RSP results displayed in Figure 4Comparing Figures 4 and 5 it is observed that structuraldisplacement and sound radiation levels have different curvesfor the same excitation and thus panel displacement PSD

8 Shock and Vibration

minus250

minus200

minus150

minus100

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

minus200

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(b)

minus200

minus180

minus140

minus160

minus120

minus100

minus80

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(c)

minus220

minus200

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(d)

minus220

minus200

minus180

minus140

minus160

minus120

minus100

minus80

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

Case 1 composite TBLCase 2 composite TBL

Case 3 composite TBLCase 4 composite TBL

(e)

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

Case 1 composite randomCase 2 composite random

Case 3 composite randomCase 4 composite random

(f)

Figure 5 Panel displacement PSD results for the composite panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location(1199091 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

curve predictionmay not be fully representative of RSP curveeven though they would be expected to follow similar trendsIn addition as similar to RSP results it is concluded that 119878

119908119908

predictions are dissimilar whether one is assuming TBL or

random excitation Case 1 panel shows higher values of bothoverall and local plate displacement PSD predictions

The results presented here are useful in the context of theprediction of noise radiation by composite panel solutions

Shock and Vibration 9

and its possible application for example in aerospace indus-try Based on the obtained results if one would choose whichcomposite panel solution is most desirable for the purposeof manufacturing an aircraft fuselage panel a smaller panel(Case 3) or a panel with a higher aspect ratio (Case 4) wouldbe preferable for overall improved results Longer and widerpanels (ie Cases 1 and 2) are shown to greatly amplify thelow frequency range However one would need to designthe panel appropriately and with the targeted dimensionssuch that the amplification of noise and vibration at certainfrequencies would be avoided For instance Case 3 panelexhibits a general improved behaviour in terms of noiselevels but simultaneously it amplifies noise at frequenciesnear 300Hz and 1000Hz while Case 4 panel amplifies resultsnear 300Hz Comparing the size of these two panels thesurface area covered by one Case 1 panel corresponds toapproximately the surface area covered by twenty Case 3panels with the twenty Case 3 panels representing approx-imately 94 kg of total mass compared to the 676 kg of asingle Case 1 panel As such an improvement in the noiselevels comes as a tradeoff in terms of increased weightAdditionally it is shown that panels with thicker face layersand thinner core layer specifically referring to Cases 3 and 4have lower noise and vibration levels than panels with thinnerface layers and thicker core layer referring to Cases 1 and 2However as mentioned earlier Cases 3 and 4 panels withsmaller dimensions and thicker face layers have the problemof noise amplification at certain frequencies Therefore apanel which combines the properties of being shorter with athicker core layer versus thinner face layers may potentiallyshow an improved behaviour for reduced vibration and noiseradiation as well as lower noise amplification at isolatedfrequencies

32 Isotropic Panels This section addresses the RSP and 119878119908119908

predictions for the isotropic panels analyzed and results upto 1500Hz are shown in Figures 6 and 7 Four aluminumisotropic panels are investigated considering TBL or randomexcitation as follows isotropic panels Cases 3 and 4 have thesame dimensions 119886 and 119887 as the composite panels Cases3 and 4 respectively isotropic panels Cases 5 and 6 areconsidered with same dimensions 119886 and 119887 as Cases 3 and4 respectively but with increased thicknesses This approachis taken with the purpose of posterior comparison betweencomposite and isotropic panelsrsquo behaviours In this contextisotropic panels Cases 3 and 4 are both 1mm thick whileisotropic panels Cases 4 and 5 have thicknesses of 15mm and4mm respectively

As noted in Figures 6 and 7 the increase of panelthickness results in an expected shift in the panel naturalfrequencies to higher frequencies as well as a shift in themodes amplified in both RSP and 119878

119908119908results This effect is

observed in the TBL and random excitations Furthermore itis found that panels with higher aspect ratio (ie Cases 4 and6) are generally characterized by having lower levels of soundradiation and vibration As similarly noted for the compositepanels random and TBL excitations provide different RSPand 119878

119908119908predictions and overall values differ from point

values and hence it is important to specify the location in the

panel where the measurementspredictions are being takenIt is also noted that the random excitation remains as theexcitation causing higher noise and vibration levels for allcases at the higher frequency range This is also an expectedeffect as the TBL excitation decreases towards the higherfrequencies while the random excitation provides a constantexcitation over the frequency spectrum

33 Composite versus Isotropic Panels As one of the mainobjectives of this study this section presents a discussionof results for the behavior of composite sandwich panelsin comparison with the traditional isotropic panels Thefollowing four figures are shown to this end Figures 8 and9 compare results for composite panel Case 3 with isotropicpanels Cases 3 and 5 while Figures 10 and 11 comparecomposite panel Case 4 with isotropic panels Cases 4 and 6

By comparing Cases 3 and 5 panels as shown fromFigures 8 and 9 it is noted that composite panel Case 3exhibits an enhanced behaviour in comparison with isotropicpanels Cases 3 and 5 both in RSP and displacement PSDlevels However one should also note that Case 3 compositepanel shows higher levels of noise and vibration at specificisolated frequencies approximately near 300Hz and 1000Hzfor the overall RSP and for position 1 RSP results as shown inFigure 8 Considering the panel displacement PSD results inFigure 9 extra frequencies above 1000Hz are also amplifiedespecially at position 2 for both TBL and random excitationsAs described in Table 1 the composite panel Case 3 has athin core layer thus the unwanted amplification at thesefrequencies may be attenuated by increasing the core layerthickness However this modification on the panel thicknesswill increase the original panel weight of 047 kg whichalready represents heavier solution compared to Cases 3and 5 isotropic panels with weights of 03 kg and 045 kgrespectively Although the panel weight will increase if thischange is performed making this modification will in alater phase require less added acoustic insulation material tothe aircraft walls in the context of aircraft fuselage panelsrsquoapplication As such the composite panelmay represent witha slight increase in the panel weight an optimized solutionfor both sound radiation and structural vibration in almostall the frequency spectra analyzed

Similar conclusions are obtained from Figures 10 and 11dedicated to the comparison of Cases 4 and 6 panels thatis the composite panel represents an improved solution foralmost all the frequency ranges studied By analyzing thegeometric differences between composite panels Cases 3 and4 Case 4 has thicker face layers and is longer than Case 3panel having a much higher aspect ratio The amplificationof noise and vibration levels at specific frequencies is notedas in Case 3 composite panel Since face layers are thickerand this panel has a higher aspect ratio this effect is morevisible in composite Case 4 than in composite Case 3 Asobserved in results from Case 3 this could be enhancedby implementing an increase of the core layer thicknessWhile the increase in the face layer thickness is shown todecrease the levels of noise and vibration in both overalland localized predictions it also can have an impact in the

10 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 6 Radiated sound power results for the isotropic panels (a) and (b) overall panel RSP (c) and (d) RSP near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) RSP near location (1199092 1199102) = (03119886 02119887)

Shock and Vibration 11

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

(a)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(b)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(c)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus50

minus100

minus150

minus200

(f)

Figure 7 Panel displacement PSD results for the isotropic panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) 119878119908119908

near location (1199092 1199102) = (03119886 02119887)

12 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus100

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus100

minus80

minus60

minus40

minus20

0

20

40

60

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 8 Comparison of radiated sound power results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 13

0 500 1000 1500

Frequency (Hz)

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus50

minus150

minus100

minus200

minus250

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 9 Comparison of panel displacement PSD results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

14 Shock and VibrationRa

diat

ed so

und

pow

er (d

B re

l1

W)

0 500 1000 1500

Frequency (Hz)

50

0

minus50

minus100

minus150

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

100

50

0

minus50

minus100

minus150

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

50

0

minus50

minus100

minus150

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus80

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 10 Comparison of radiated sound power results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 15

minus50

minus100

minus150

minus250

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus220

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus50

minus100

minus150

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 11 Comparison of panel displacement PSD results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b)overall panel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

16 Shock and Vibration

Table 1 Physical properties of composite panels

Variable Description units Case 1 Case 2 Case 3 Case 4119886 Length m 183 11683 0348 10119887 Width m 122 11683 03048 03120582 Aspect ratio (119886119887) 15 10 114 333ℎ1 ℎ3

Face layers thickness mm 0406 0506 0762 35ℎ2

Core layer thickness mm 64 1905 0254 051198641 1198643

Face layers Young modulus Pa 6898 times 1010

71 times 1010

6898 times 1010

66 times 1010

1198642

Core layer Young modulus Pa 457 times 105

457 times 105

267 times 106

26 times 108

]1 ]3

Face layers Poissonrsquos ratio 033 033 03 033]2

Core layer Poissonrsquos ratio 04 04 049 031205881 1205883

Face layers density kgmminus3 2768 2700 2740 26801205882

Core layer density kgmminus3 122 4806 999 1680119898 Total mass kg 676 499 047 588

Table 2 Physical properties of isotropic panels

Variable Description units Case 3 Case 4 Case 5 Case 6119886 Length m 0348 10 0348 10119887 Width m 03048 03 03048 03120582 Aspect ratio (119886119887) 114 333 114 333ℎ119901

Thickness mm 1 1 15 4119864119901

Young modulus Pa 724 times 1010

724 times 1010

724 times 1010

724 times 1010

]119901

Poissonrsquos ratio 033 033 033 033120588119901

Density kg mminus3 2800 2800 2800 2800119898 Total mass kg 030 084 045 336

Table 3 Flight test conditions and air properties

Variable Description units Value119872 Mach number 08alt Altitude m 13106119888 Speed of sound m sminus1 295119880infin

Flow speed m sminus1 23606120588 Air density kgmminus3 02622] Kinematic viscosity m2 sminus1 5422 times 10

minus5

119879 Temperature K 21665Re119909

Reynolds number 2917 times 107

1198880

Speed of sound m sminus1 3401205880

Internal air density kgmminus3 142

higher amplification of localized frequencies Therefore theface layers thickness increase should be performed carefullyby an accompanied increase in the core layer thickness whilesimultaneously considering the weight constraints for thespecific panels applications

A more detailed analysis will be required in order toformulate final conclusions about the optimum solutions forthe panel in the context of aircraft application For that theinsulation materials would also be considered to account forthe respective added weight and effect in the sound radi-ation signature However typically the acoustic insulationmaterials will only have a higher impact in decreasing the

Table 4 First 10 natural frequencies for the composite panels

Modes 119891 Hz(Case 1)

119891 Hz(Case 2)

119891 Hz(Case 3)

119891 Hz(Case 4)

1 2365 10588 4874 126582 4548 26470 11222 157933 7276 42352 13148 210194 8186 52941 19496 283355 9459 68823 21800 377416 13097 89999 26939 474967 13279 95293 30075 492388 15462 105881 33287 506329 17645 132351 36612 5585710 18191 137645 43866 62825

sound levels at higher frequency ranges and will thereforenot improve significantly the problem of higher noise levelsin the low frequency range particularly observed in theisotropic solutions As such the composite panel solutionsmay appear to be a promising replacement to the isotropicpanels especially panels like Cases 3 and 4 which are alreadycharacterized by lower levels of both RSP and 119878

119908119908in the

low frequency range In further analysis it will also beimportant to carefully address the noise amplification atisolated frequencies by identifying a good combination of

Shock and Vibration 17

Table 5 First 10 natural frequencies for the isotropic panels

Modes 119891 Hz(Case 3)

119891 Hz(Case 4)

119891 Hz(Case 5)

119891 Hz(Case 6)

1 4646 2958 6969 118322 10697 3691 16046 147643 12534 4912 18801 196494 18585 6622 27877 264885 20782 8820 31173 352826 25680 1110 38520 443997 28670 11507 43004 460298 31731 11832 47596 473309 34901 13054 52351 5221510 41816 14683 62723 58731

core and face layers in terms of both materials selection anddimensions

4 Conclusions

This study presents a mathematical method and frameworkon composite panels radiated sound and vibrationThepanelsare considered excited either by turbulent flow or by randomnoiseThe turbulent flow corresponds to the typical excitationof an aircraft panel at flight cruise conditions with a Machnumber of 08 The effect of type of panel of panel dimen-sions and of panel thickness are evaluated The turbulentflow excitation showed to excite a higher number of panelmodes compared to random noise excitation and resulted indissimilar levels of both panel vibration and radiated soundwith relation to the random excitation This effect is morevisible for composite than for isotropic panels and thereforethe pure random noise should not be used as representativeof the turbulent flow excitation in real applications

Different properties on the composite panels are shown toyield dissimilar effects in the noise and vibration predictionsThe effect of the panel size specifically its length and widthis noted to have a significant effect on the panel performancein terms of noise radiation and vibration levels Overallsmaller composite panels or panels with higher aspect ratioproduced lower levels of radiated sound and vibration thanlonger and wider composite panels Compared to isotropicpanels composite panels analyzed are shown to radiate lowernoise levels with the exception of specific natural frequencieswhich are amplified near 300Hz and 1000Hz It is shown thatthis amplification can be significant for composite panels withthicker face layers and thinner core layer and thus resultingin unwanted effect of increased sound radiated Despitethis effect this drawback may potentially be corrected byincreasing the sandwich core layer thickness Increasing thecore layer thickness will impact the panel weight and hencethe composite panels can represent a heavier solution com-pared to the isotropic panels examined On the other handthis approach may be beneficial in the design of compositepanels in the context of aircraft panel applications sincethese panels may require less acoustic insulation material

added to the aircraft walls as they already represent animproved design solution Furthermore acoustic insulationmaterials have a higher impact in decreasing sound levelsat the high frequency ranges and therefore it will notsignificantly improve the problem of higher noise levels at thelow frequency range particularly observed in the isotropicsolutions In this context advanced and optimized compositepanel configurations may be a promising replacement to thetraditional isotropic panels for aerospace applications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is grateful for the support of the National Scienceand Engineering Research Council of Canada

References

[1] J Rocha and D Palumbo ldquoOn the sensitivity of sound powerradiated by aircraft panels to turbulent boundary layer param-etersrdquo Journal of Sound and Vibration vol 331 no 21 pp 4785ndash4806 2012

[2] J Rocha A Suleman and F Lau ldquoPrediction of turbulentboundary layer induced noise in the cabin of a BWB aircraftrdquoShock and Vibration vol 19 no 4 pp 693ndash705 2012

[3] J da Rocha A Suleman and F Lau ldquoFlow-induced noise andvibration in aircraft cylindrical cabins closed-form analyticalmodel validationrdquo Journal of Vibration and Acoustics vol 133no 5 Article ID 051013 9 pages 2011

[4] J da Rocha A Suleman and F Lau ldquoPrediction of flow-inducednoise in transport vehicles development and validation of acoupled structuralmdashacoustic analytical frameworkrdquo CanadianAcousticsmdashAcoustique Canadienne vol 37 no 4 pp 13ndash292009

[5] G Corcos ldquoThe structure of the turbulent pressure field inboundary layer flowsrdquo Journal of Fluid Mechanics vol 18 pp353ndash378 1964

[6] G Corcos ldquoResolution of pressure in turbulencerdquo Journal of theAcoustical Society of America vol 35 no 2 pp 192ndash199 1963

[7] B Efimtsov ldquoCharacteristics of the field of turbulent wallpressure fluctuations at large Reynolds numbersrdquo Soviet PhysicsAcoustics vol 28 no 4 pp 289ndash292 1982

[8] J Reddy Mechanics of Laminated Composite Plates and ShellsTheory and Analysis CRC Press Boca Raton Fla USA 1945

[9] J Whitney Structural Analysis of Laminated Anisotropic PlatesTechnomic Lancaster Pa USA 1987

[10] C L Dym and M A Lang ldquoTransmission of sound throughsandwich panelsrdquo Journal of the Acoustical Society of Americavol 56 no 5 pp 1523ndash1532 1974

[11] C L Dym C S Ventres and M A Lang ldquoTransmission ofsound through sandwich panels a reconsiderationrdquo Journal ofthe Acoustical Society of America vol 59 no 2 pp 364ndash3671976

[12] S Narayanan andR L Shanbhag ldquoSound transmission througha damped sandwich panelrdquo Journal of Sound and Vibration vol80 no 3 pp 315ndash327 1982

18 Shock and Vibration

[13] C L Dym and D C Lang ldquoTransmission loss of dampedasymmetric sandwich panels with orthotropic coresrdquo Journal ofSound and Vibration vol 88 no 3 pp 299ndash319 1983

[14] S Ghinet N Atalla and H Osman ldquoDiffuse field transmissioninto infinite sandwich composite and laminate composite cylin-dersrdquo Journal of Sound and Vibration vol 289 no 4-5 pp 745ndash778 2006

[15] J Mendler ldquoComputational tool for the acoustic analysis ofsandwich panelsrdquo in Proceedings of the European Conference onNoise Control (EURO-NOISE rsquo95) Lyon France 1995

[16] O Foin J Nicolas andNAtalla ldquoAn efficient tool for predictingthe structural acoustic and vibration response of sandwichplates in light or heavy fluidrdquo Applied Acoustics vol 57 no 3pp 213ndash242 1999

[17] S Assaf and M Guerich ldquoNumerical prediction of noisetransmission loss through viscoelastically damped sandwichplatesrdquo Journal of Sandwich Structures andMaterials vol 10 no5 pp 359ndash384 2008

[18] S Assaf M Guerich and P Cuvelier ldquoVibration and acousticresponse of damped sandwich panels immersed in a light orheavy fluidrdquo Computers and Structures vol 88 pp 359ndash3842010

[19] R Zhou and M J Crocker ldquoBoundary element analyses forsound transmission loss of panelsrdquo Journal of the AcousticalSociety of America vol 127 no 2 pp 829ndash840 2010

[20] K Amichi N Atalla and R Ruokolainen ldquoA new 3D finite ele-ment sandwich plate for predicting the vibroacoustic responseof laminated steel panelsrdquo Finite Elements in Analysis andDesign vol 46 no 12 pp 1131ndash1145 2010

[21] F Franco S de Rosa and T Polito ldquoFinite element inves-tigations on the vibroacoustic performance of plane plateswith random stiffnessrdquo Mechanics of Advanced Materials andStructures vol 18 no 7 pp 484ndash497 2011

[22] Y Kim ldquoIdentification of sound transmission characteristicsof honeycomb sandwich panels using hybrid analyticalone-dimensional finite element methodrdquo in Proceedings of the35th International Congress and Exposition on Noise ControlEngineering (INTER-NOISE rsquo06) HonoluluHawaii USA 2006

[23] F Fahy Sound and Structural Vibration Radiation Transmissionand Response Academic Press London UK 1985

[24] J Blevins Formulas for Natural Frequency and Mode ShapeKrieger Publishing Malabar Fla USA 2001

[25] D Newland An Introduction to Random Vibration and SpectralAnalysis Longman New York NY USA 1984

[26] Y Lin Probabilistic Theory of Structural Dynamics McGraw-Hill Book New York NY USA 1967

[27] C E Wallace ldquoRadiation resistance of a rectangular panelrdquoTheJournal of the Acoustical Society of America vol 51 no 3 pp946ndash952 1972

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Active and Passive Electronic Components

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RotatingMachinery

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

4 Shock and Vibration

where

[119876119894] = [119879

minus1

119894] [119876119894] [119879119894] =

[[[[[[

[

119876119894

11119876119894

12119876119894

16

119876119894

12119876119894

22119876119894

26

119876119894

16119876119894

26119876119894

66

]]]]]]

]

(13)

119876119894

11= 119876119894

111198984

119894+ 2 (119876

119894

12+ 2119876119894

66)1198982

1198941198992

119894+ 119876119894

221198994

119894 (14)

119876119894

12= (119876119894

11+ 119876119894

22minus 4119876119894

66)1198982

1198941198992

119894+ 119876119894

12(1198984

119894+ 1198994

119894) (15)

119876119894

22= 119876119894

111198994

119894+ 2 (119876

119894

12+ 2119876119894

66)1198982

1198941198992

119894+ 119876119894

221198984

119894 (16)

119876119894

16= (119876119894

11minus 119876119894

12minus 2119876119894

66)1198983

119894119899119894+ (119876119894

12minus 119876119894

22+ 2119876119894

66)1198981198941198993

119894

(17)

119876119894

26= (119876119894

11minus 119876119894

12minus 2119876119894

66)1198981198941198993

119894+ (119876119894

12minus 119876119894

22+ 2119876119894

66)1198983

119894119899119894

(18)

119876119894

66= (119876119894

11+ 119876119894

22minus 2119876119894

12minus 2119876119894

66)1198982

1198941198992

119894+ 119876119894

66(1198984

119894+ 1198994

119894)

(19)

in which119898119894= cos 120579

119894and 119899119894= sin 120579

119894

By definition the equilibrium equation for the differentialplate element is defined as function of the stress and momentresultants as follows

119888 (119873119909

1205972119908

1205971199092+ 2119873119909119910

1205972119908

120597119909120597119910+ 119873119910

1205972119908

1205971199102)

+ (1205972119872119909

1205971199092+ 2

1205972119872119909119910

120597119909120597119910+119872119910

1205972119908

1205971199102) + 119891

119911=

119899

sum

119894=1

120588119894ℎ119894

1205972119908

1205971199052

(20)

in which 119891119911= minus119901ext(119909 119910 119905) is the external pressure applied to

the panel due to the fluid flow or the random excitation 120588119894is

the density of the 119894th ply and the stress resultant andmomentresultant components are defined as functions of the stresscomponents respectively as follows

(

119873119909

119873119910

119873119909119910

) =

119899

sum

119894=1

int

ℎ119894

ℎ119894minus1

(

120590119909

120590119910

120591119909119910

)119889119911

(

119872119909

119872119910

119872119909119910

) =

119899

sum

119894=1

int

ℎ119894

ℎ119894minus1

(

120590119909

120590119910

120591119909119910

)119911119889119911

(21)

After substituting the stress components defined by (12) into(21) and making use of (14) through (19) then (20) can be

transformed to the following form as function of the platedisplacement 119908

1

3

119899

sum

119894=1

(ℎ3

119894minus ℎ3

119894minus1)

times [119876119894

11(1205972119908

1205971199092)

2

+ 119876119894

22(1205972119908

1205971199102)

2

+ 2119876119894

12(1205972119908

1205971199092)(

1205972119908

1205971199102) + 4119876

119894

16(1205972119908

1205971199092)(

1205972119908

120597119909120597119910)

+ 4119876119894

26(1205972119908

1205971199102)(

1205972119908

120597119909120597119910) + 4119876

119894

66(

1205972119908

120597119909120597119910)

2

]

+1

3

119899

sum

119894=1

(ℎ3

119894minus ℎ3

119894minus1)

times [119876119894

11

1205974119908

1205971199094+ 119876119894

22

1205974119908

1205971199104+ 6119876119894

16

1205974119908

1205971199093120597119910

+ 6119876119894

26

1205974119908

1205971199091205971199103+ (2119876

119894

12+ 8119876119894

66)

1205974119908

12059711990921205971199102]

+

119899

sum

119894=1

120588119894ℎ119894

1205972119908

1205971199052= 119891119911

(22)

Each panel in the present study aims to represent thedistance between adjacent stringers and frames in an aircraftfuselage hence each individual panel is assumed to vibrateindependently In addition since panels are considered to beflat and simply supported in all four boundaries the panelrsquosdisplacement can be defined as follows [23 24]

119908 (119909 119910 119905) =

119872119909

sum

119898119909=1

119872119910

sum

119898119910=1

120572119898119909

(119909) 120573119898119910(119910) 119902119898119909119898119910

(119905) (23)

where 120572119898119909(119909) and 120573

119898119910(119910) are the spatial functions defining

the variation of119908(119909 119910 119905) with 119909 and 119910 respectively 119902119898119909119898119910

(119905)

are the temporal functions defining the variation of119908(119909 119910 119905)with time and 119872 = 119872

119909times 119872119910is the total number of plate

modes (119898119909 119898119910) considered For simply supported plates the

spatial functions may be defined as follows

120572119898119909

(119909) = radic2

119886sin(

119898119909120587119909

119886) (24)

120573119898119910

(119910) = radic2

119887sin(

119898119910120587119910

119887) (25)

Shock and Vibration 5

where 119886 and 119887 are the dimensions of the plate in the 119909 and119910 directions Also the natural frequencies for the simplysupported plate can be determined by

120596119898119909119898119910

=1205872

radicsum119899

119894=1120588119894ℎ119894

[11986311(119898119909

119886)

4

+ 11986322(119898119910

119887)

4

+2(11986312+ 211986366)(

119898119909

119886)

2

(119898119910

119887)

2

]

12

(26)

in which

119863119895119896

=

119899

sum

119894=1

int

ℎ119894

ℎ119894minus1

119876119894

1198951198961199112119889119911 (27)

Developing (22) by considering (23) to (25) making use ofthe orthogonality of the plate modes and integrating over theplate area similarly to the mathematical approach followedin [4] the equilibrium equation for an orthotropic plate iswritten in the matrix form as

[119872119901] q (119905) + [119863

119901] q (119905) + [119870

119901] q (119905) = pext (119905) (28)

where [119872119901] [119863119901] and [119870

119901] are mass damping and stiffness

119872times119872matrices respectively and q(119905) q(119905) and q(119905) are theposition velocity and acceleration vectors with componentsdefined respectively as 119902

119898(119905) = 119876

119898119890119894120596119905 119902119898(119905) = 119894119876

119898120596119890119894120596119905

and 119902119898(119905) = minus119876

1198981205962119890119894120596119905 After performing all the above

described mathematical manipulations it can be shown thateach term in (28) is defined as follows

[119872119901] = diag[

119899

sum

119894=1

120588119894ℎ119894]

[119863119901] = diag[2120585

119901120596119898119909119898119910

119899

sum

119894=1

120588119894ℎ119894]

[119870119901] =

1

3

119899

sum

119894=1

(ℎ3

119894minus ℎ3

119894minus1) 119876119894

11I (119909) + 119876

119894

22I (119910)

+ (2119876119894

12+ 8119876119894

66) I1015840 (119909) I1015840 (119910)

pext (119905) = [int

119886

0

int

119887

0

120572119898119909

(119909) 120573119898119910(119910) 119901ext (119909 119910 119905) 119889119909 119889119910]

(29)

in which 120585119901was added to account with the damping of the

plate and

I (119909) = diag [1119886(119898119909120587

119886)

4

]

I (119910) = diag [1119887(119898119910120587

119887)

4

]

I1015840 (119909) = minus1

119886(119898119909120587

119886)

2

cos [(119898119909minus 1198981015840

119909) 120587]

I1015840 (119910) = minus1

119887(119898119910120587

119887)

2

cos [(119898119910minus 1198981015840

119910) 120587]

(30)

22 Radiated Sound Power The calculation of the panelsrsquoradiated sound power in the frequency domain requires arearrangement of the time domain equations presented in(22) For this end knowing that 119902

119898= 119876119898119890119894120596119905 as previously

described one can write (28) in the following desired form inthe frequency 120596 domain

Y (120596) = H (120596)X (120596) (31)

where

Y (120596) = W (120596)

X (120596) = Pext (120596)

H (120596) = minus1205962[119872119901] + 119894120596 [119863

119901] + [119870

119901]

(32)

in which H(120596) is system frequency response matrix Y(120596) isthe response of the system to the excitationX(120596) and vectorsW(120596) andPext(120596) correspond respectively to the vectors q(119905)and pext(119905) written in the frequency domain Considering theturbulent boundary layer random excitation as a stationaryand homogeneous function the spectral density of the systemresponse is given by [25 26]

S119882119882 (120596) = Hlowast (120596) S119875119875 (120596)H

119879(120596) (33)

where superscripts lowast and 119879 denote Hermitian conjugate andmatrix transpose respectively S

119875119875(120596) is the spectral density

matrix of the random excitation Pext(120596) and S119882119882

(120596) is thespectral density matrix of the system response W(120596) Thematrix S

119875119875(120596) corresponds to the generalized power density

matrix of the turbulent boundary layer excitation defined by

Stbl (120596) = [∬

119886

0

∬

119887

0

120572119898119909

(119909) 1205721198981015840119909

(1199091015840) 120573119898119910

(119910) 1205731198981015840

119910

(1199101015840)

times 119878 (120585119909 120585119910 120596) 119889119909 119889119909

1015840119889119910119889119910

1015840]

(34)

in which 119878(120585119909 120585119910 120596) is defined by Corcos model [5 21] and

120585119909= 119909 minus 119909

1015840 and 120585119910= 119910 minus 119910

1015840 are the spatial separations in thestreamwise and spanwise directions of the plate respectivelyAs shown in [1ndash3] a combination of Corcos and Efimtsovmodels may be suitable to provide this excitation as follows

119878 (120585119909 120585119910 120596) = 119878ref (120596) 119890

minus(120572119909120596|120585119909|119880119888)119890minus(120572119910120596|120585119910|119880119888)119890

minus(119894120596120585119909119880119888)

(35)

where119880119888is the turbulent boundary layer convective velocity

120572119909= 01 and 120572

119910= 077 are empirical parameters used to

provide the best agreement with the reality [27] and 119878ref(120596)is defined using Efimtsov model [7] In [1] an analyticalform was derived for the matrix described in (34)The powerspectral density function of the plate displacement can then

6 Shock and Vibration

be defined as functions of the abovematrix respectively platearea as

119878119908119908

(1199091 1199101 1199092 1199102 120596)

=

1198722

119909

sum

11989811990911198981199092=1

1198722

119910

sum

11989811991011198981199102=1

1205721198981199091

(1199091) 1205721198981199092

(1199092)

times 1205731198981199101

(1199101) 1205731198981199102

(1199102) S119882119882(120596)1198981 1198982

(36)

and the overall power spectral density function can be foundby integrating the individual power spectral densities over theplate area displacement as following

119878119908119908 (120596) = ∬

119886

0

∬

119887

0

119878119908119908

(1199091 1199101 1199092 1199102 120596) 119889119909

1119889119909211988911991011198891199102

(37)

Analytical expressions for 119878119908119908

(1199091 1199101 1199092 1199102 120596) and 119878

119908119908(120596)

were obtained in [23] The power spectral density matrix ofthe plate velocity can be determined from the power spectraldensity matrix of the plate displacement as follows

S119881119881 (120596) = 120596

2S119882119882 (120596) (38)

Using these analytical forms the plate radiated sound powerfunction and the overall power spectral density function canbe determined respectively as follows

RSP (1199091 1199101 1199092 1199102 120596)

=

1198722

119909

sum

11989811990911198981199092=1

1198722

119910

sum

11989811991011198981199102=1

1205721198981199091

(1199091) 1205721198981199092

(1199092)

times 1205731198981199101

(1199101) 1205731198981199102

(1199102)Π(120596)1198981 1198982

RSP (120596) = ∬

119886

0

∬

119887

0

RSP (1199091 1199101 1199092 1199102 120596) 119889119909

1119889119909211988911991011198891199102

(39)

in which Π(120596)1198981 1198982

are the components of the radiationmatrix defined by

Π (120596) = S119881119881

M (120596) (40)and S119881119881

is defined in (38) andM(120596) is evaluated numericallyfrom the following equation

M (120596) = 81205880

1198880

(120596119886119887

1205873119898119909119898119910

)

2

times∬

1205872

0

cossin (

120572

2)

cossin (

120573

2)

[(120572119898119909120587)2minus 1] [(120573119898

119910120587)2

minus 1]

2

timessin 120579119889120579 119889120601(41)

where 1205880and 1198880are the density and the speed of sound of

the interior fluid respectively 120572 = (1205961198880)119886 sin 120579 cos120601 120573 =

(1205961198880)119887 sin 120579 sin120601 in which 120601 and 120579 are the angles of the

distance vector in space to the observation point [27]

3 Structural Vibration and SoundRadiation Results

The formulation presented in the previous section is usedto analyze four different rectangular composite sandwichpanels of several sizes and materials properties as shown inTable 1 Additionally four isotropic aluminum panels withsimilar sizes compared to the composite panels as shownin Table 2 are analyzed using the previous model developedby the author results are then compared with the compositepanels All the eight panels are considered simply supportedin the four boundaries and subjected to turbulent boundarylayer excitation or to random noise excitation The turbulentflow excitation is representative of Mach number 08 flightconditions as described inTable 3 and the randomexcitationrepresents a constant 40 Pascal random noise that is asound pressure level of approximately 120 dB The first tennatural frequencies obtained for the several panels analyzedare shown in Tables 4 and 5 respectively for composite andisotropic panels Cases 1 2 3 and 4 of composite panels asnoted in Table 4 are chosen for validation purposes as instudies [16 18 20] respectively It is shown that the calculatednatural frequencies are in good agreement with the previousstudies

31 Composite Panels The results presented in this sectiondiscuss the radiated sound power and the structural dis-placement levels of the composite sandwich panels studiedResults are shown for frequencies up to 1500Hz requiringa minimum number of plate modes which is dependent ofthe panel size and materials properties for proper accuracyof the predicted values Values for the radiated sound powerRSP and plate displacement power spectral density (PSD)119878119908119908

are obtained for two locations (1199091 1199101) = (05119886 05119887)

and (1199092 1199102) = (03119886 02119887) as well as overall values of these

two quantities and shown in Figures 4 and 5 As referredearlier four different panels are analyzed identified as Cases1 through 4 along the following sections and two types ofexcitation are applied (1) turbulent boundary layer excitationand (2) random noise excitation

As noted in Figure 4 and similarly to what was observedin previous results and formulation dedicated to the analysisof isotropic panels it is concluded that the overall RSPvalues differ from local RSP values especially for the TBLexcitation It is also observed that turbulent flow can beresponsible for exciting a higher number of panel modescompared to random noise with TBL and random noiseproviding dissimilar levels of panelsrsquo sound radiation Assuch to accurately predict the dynamic behaviour of a panelone should not consider the pure random noise excitation asrepresentative of the turbulent flow excitation In the overallRSP results plates of bigger dimensions are shown to havehigher RSP values both for TBL and random excitationswith Case 1 showing higher RSP values than Cases 2 3or 4 The same is not noted for the local RSP valueswith Case 3 panel showing increased RSP values for bothpositions 1 and 2with relation to the overall RSP valuesTheseresults emphasize the importance of knowing the measuring

Shock and Vibration 7

0 500 1000 1500Frequency (Hz)

minus150

minus100

minus50

0

50

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

(b)

0 500 1000 1500

Frequency (Hz)

minus120

minus100

minus80

minus60

minus40

minus20

0

20

40

Radi

ated

soun

d po

wer

(dB

rel1

W)

(c)

0 500 1000 1500

Frequency (Hz)

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

(d)

0 500 1000 1500

Frequency (Hz)

minus140

minus120

minus100

minus80

minus60

minus40

minus20

0

20

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 1 composite TBLCase 2 composite TBL

Case 3 composite TBLCase 4 composite TBL

(e)

0 500 1000 1500

Frequency (Hz)

minus80

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 1 composite randomCase 2 composite random

Case 3 composite randomCase 4 composite random

(f)

Figure 4 Radiated sound power results for the composite panels (a) and (b) overall panel RSP (c) and (d) RSPnear surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) RSP near location (1199092 1199102) = (03119886 02119887)

position for the accurate prediction of sound and vibrationand it highlights the desired sensitivity of the developedmathematical framework to this factor Figure 5 shows thedisplacement PSD results associated with the same panels

and excitations as for the RSP results displayed in Figure 4Comparing Figures 4 and 5 it is observed that structuraldisplacement and sound radiation levels have different curvesfor the same excitation and thus panel displacement PSD

8 Shock and Vibration

minus250

minus200

minus150

minus100

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

minus200

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(b)

minus200

minus180

minus140

minus160

minus120

minus100

minus80

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(c)

minus220

minus200

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(d)

minus220

minus200

minus180

minus140

minus160

minus120

minus100

minus80

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

Case 1 composite TBLCase 2 composite TBL

Case 3 composite TBLCase 4 composite TBL

(e)

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

Case 1 composite randomCase 2 composite random

Case 3 composite randomCase 4 composite random

(f)

Figure 5 Panel displacement PSD results for the composite panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location(1199091 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

curve predictionmay not be fully representative of RSP curveeven though they would be expected to follow similar trendsIn addition as similar to RSP results it is concluded that 119878

119908119908

predictions are dissimilar whether one is assuming TBL or

random excitation Case 1 panel shows higher values of bothoverall and local plate displacement PSD predictions

The results presented here are useful in the context of theprediction of noise radiation by composite panel solutions

Shock and Vibration 9

and its possible application for example in aerospace indus-try Based on the obtained results if one would choose whichcomposite panel solution is most desirable for the purposeof manufacturing an aircraft fuselage panel a smaller panel(Case 3) or a panel with a higher aspect ratio (Case 4) wouldbe preferable for overall improved results Longer and widerpanels (ie Cases 1 and 2) are shown to greatly amplify thelow frequency range However one would need to designthe panel appropriately and with the targeted dimensionssuch that the amplification of noise and vibration at certainfrequencies would be avoided For instance Case 3 panelexhibits a general improved behaviour in terms of noiselevels but simultaneously it amplifies noise at frequenciesnear 300Hz and 1000Hz while Case 4 panel amplifies resultsnear 300Hz Comparing the size of these two panels thesurface area covered by one Case 1 panel corresponds toapproximately the surface area covered by twenty Case 3panels with the twenty Case 3 panels representing approx-imately 94 kg of total mass compared to the 676 kg of asingle Case 1 panel As such an improvement in the noiselevels comes as a tradeoff in terms of increased weightAdditionally it is shown that panels with thicker face layersand thinner core layer specifically referring to Cases 3 and 4have lower noise and vibration levels than panels with thinnerface layers and thicker core layer referring to Cases 1 and 2However as mentioned earlier Cases 3 and 4 panels withsmaller dimensions and thicker face layers have the problemof noise amplification at certain frequencies Therefore apanel which combines the properties of being shorter with athicker core layer versus thinner face layers may potentiallyshow an improved behaviour for reduced vibration and noiseradiation as well as lower noise amplification at isolatedfrequencies

32 Isotropic Panels This section addresses the RSP and 119878119908119908

predictions for the isotropic panels analyzed and results upto 1500Hz are shown in Figures 6 and 7 Four aluminumisotropic panels are investigated considering TBL or randomexcitation as follows isotropic panels Cases 3 and 4 have thesame dimensions 119886 and 119887 as the composite panels Cases3 and 4 respectively isotropic panels Cases 5 and 6 areconsidered with same dimensions 119886 and 119887 as Cases 3 and4 respectively but with increased thicknesses This approachis taken with the purpose of posterior comparison betweencomposite and isotropic panelsrsquo behaviours In this contextisotropic panels Cases 3 and 4 are both 1mm thick whileisotropic panels Cases 4 and 5 have thicknesses of 15mm and4mm respectively

As noted in Figures 6 and 7 the increase of panelthickness results in an expected shift in the panel naturalfrequencies to higher frequencies as well as a shift in themodes amplified in both RSP and 119878

119908119908results This effect is

observed in the TBL and random excitations Furthermore itis found that panels with higher aspect ratio (ie Cases 4 and6) are generally characterized by having lower levels of soundradiation and vibration As similarly noted for the compositepanels random and TBL excitations provide different RSPand 119878

119908119908predictions and overall values differ from point

values and hence it is important to specify the location in the

panel where the measurementspredictions are being takenIt is also noted that the random excitation remains as theexcitation causing higher noise and vibration levels for allcases at the higher frequency range This is also an expectedeffect as the TBL excitation decreases towards the higherfrequencies while the random excitation provides a constantexcitation over the frequency spectrum

33 Composite versus Isotropic Panels As one of the mainobjectives of this study this section presents a discussionof results for the behavior of composite sandwich panelsin comparison with the traditional isotropic panels Thefollowing four figures are shown to this end Figures 8 and9 compare results for composite panel Case 3 with isotropicpanels Cases 3 and 5 while Figures 10 and 11 comparecomposite panel Case 4 with isotropic panels Cases 4 and 6

By comparing Cases 3 and 5 panels as shown fromFigures 8 and 9 it is noted that composite panel Case 3exhibits an enhanced behaviour in comparison with isotropicpanels Cases 3 and 5 both in RSP and displacement PSDlevels However one should also note that Case 3 compositepanel shows higher levels of noise and vibration at specificisolated frequencies approximately near 300Hz and 1000Hzfor the overall RSP and for position 1 RSP results as shown inFigure 8 Considering the panel displacement PSD results inFigure 9 extra frequencies above 1000Hz are also amplifiedespecially at position 2 for both TBL and random excitationsAs described in Table 1 the composite panel Case 3 has athin core layer thus the unwanted amplification at thesefrequencies may be attenuated by increasing the core layerthickness However this modification on the panel thicknesswill increase the original panel weight of 047 kg whichalready represents heavier solution compared to Cases 3and 5 isotropic panels with weights of 03 kg and 045 kgrespectively Although the panel weight will increase if thischange is performed making this modification will in alater phase require less added acoustic insulation material tothe aircraft walls in the context of aircraft fuselage panelsrsquoapplication As such the composite panelmay represent witha slight increase in the panel weight an optimized solutionfor both sound radiation and structural vibration in almostall the frequency spectra analyzed

Similar conclusions are obtained from Figures 10 and 11dedicated to the comparison of Cases 4 and 6 panels thatis the composite panel represents an improved solution foralmost all the frequency ranges studied By analyzing thegeometric differences between composite panels Cases 3 and4 Case 4 has thicker face layers and is longer than Case 3panel having a much higher aspect ratio The amplificationof noise and vibration levels at specific frequencies is notedas in Case 3 composite panel Since face layers are thickerand this panel has a higher aspect ratio this effect is morevisible in composite Case 4 than in composite Case 3 Asobserved in results from Case 3 this could be enhancedby implementing an increase of the core layer thicknessWhile the increase in the face layer thickness is shown todecrease the levels of noise and vibration in both overalland localized predictions it also can have an impact in the

10 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 6 Radiated sound power results for the isotropic panels (a) and (b) overall panel RSP (c) and (d) RSP near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) RSP near location (1199092 1199102) = (03119886 02119887)

Shock and Vibration 11

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

(a)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(b)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(c)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus50

minus100

minus150

minus200

(f)

Figure 7 Panel displacement PSD results for the isotropic panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) 119878119908119908

near location (1199092 1199102) = (03119886 02119887)

12 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus100

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus100

minus80

minus60

minus40

minus20

0

20

40

60

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 8 Comparison of radiated sound power results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 13

0 500 1000 1500

Frequency (Hz)

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus50

minus150

minus100

minus200

minus250

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 9 Comparison of panel displacement PSD results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

14 Shock and VibrationRa

diat

ed so

und

pow

er (d

B re

l1

W)

0 500 1000 1500

Frequency (Hz)

50

0

minus50

minus100

minus150

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

100

50

0

minus50

minus100

minus150

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

50

0

minus50

minus100

minus150

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus80

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 10 Comparison of radiated sound power results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 15

minus50

minus100

minus150

minus250

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus220

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus50

minus100

minus150

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 11 Comparison of panel displacement PSD results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b)overall panel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

16 Shock and Vibration

Table 1 Physical properties of composite panels

Variable Description units Case 1 Case 2 Case 3 Case 4119886 Length m 183 11683 0348 10119887 Width m 122 11683 03048 03120582 Aspect ratio (119886119887) 15 10 114 333ℎ1 ℎ3

Face layers thickness mm 0406 0506 0762 35ℎ2

Core layer thickness mm 64 1905 0254 051198641 1198643

Face layers Young modulus Pa 6898 times 1010

71 times 1010

6898 times 1010

66 times 1010

1198642

Core layer Young modulus Pa 457 times 105

457 times 105

267 times 106

26 times 108

]1 ]3

Face layers Poissonrsquos ratio 033 033 03 033]2

Core layer Poissonrsquos ratio 04 04 049 031205881 1205883

Face layers density kgmminus3 2768 2700 2740 26801205882

Core layer density kgmminus3 122 4806 999 1680119898 Total mass kg 676 499 047 588

Table 2 Physical properties of isotropic panels

Variable Description units Case 3 Case 4 Case 5 Case 6119886 Length m 0348 10 0348 10119887 Width m 03048 03 03048 03120582 Aspect ratio (119886119887) 114 333 114 333ℎ119901

Thickness mm 1 1 15 4119864119901

Young modulus Pa 724 times 1010

724 times 1010

724 times 1010

724 times 1010

]119901

Poissonrsquos ratio 033 033 033 033120588119901

Density kg mminus3 2800 2800 2800 2800119898 Total mass kg 030 084 045 336

Table 3 Flight test conditions and air properties

Variable Description units Value119872 Mach number 08alt Altitude m 13106119888 Speed of sound m sminus1 295119880infin

Flow speed m sminus1 23606120588 Air density kgmminus3 02622] Kinematic viscosity m2 sminus1 5422 times 10

minus5

119879 Temperature K 21665Re119909

Reynolds number 2917 times 107

1198880

Speed of sound m sminus1 3401205880

Internal air density kgmminus3 142

higher amplification of localized frequencies Therefore theface layers thickness increase should be performed carefullyby an accompanied increase in the core layer thickness whilesimultaneously considering the weight constraints for thespecific panels applications

A more detailed analysis will be required in order toformulate final conclusions about the optimum solutions forthe panel in the context of aircraft application For that theinsulation materials would also be considered to account forthe respective added weight and effect in the sound radi-ation signature However typically the acoustic insulationmaterials will only have a higher impact in decreasing the

Table 4 First 10 natural frequencies for the composite panels

Modes 119891 Hz(Case 1)

119891 Hz(Case 2)

119891 Hz(Case 3)

119891 Hz(Case 4)

1 2365 10588 4874 126582 4548 26470 11222 157933 7276 42352 13148 210194 8186 52941 19496 283355 9459 68823 21800 377416 13097 89999 26939 474967 13279 95293 30075 492388 15462 105881 33287 506329 17645 132351 36612 5585710 18191 137645 43866 62825

sound levels at higher frequency ranges and will thereforenot improve significantly the problem of higher noise levelsin the low frequency range particularly observed in theisotropic solutions As such the composite panel solutionsmay appear to be a promising replacement to the isotropicpanels especially panels like Cases 3 and 4 which are alreadycharacterized by lower levels of both RSP and 119878

119908119908in the

low frequency range In further analysis it will also beimportant to carefully address the noise amplification atisolated frequencies by identifying a good combination of

Shock and Vibration 17

Table 5 First 10 natural frequencies for the isotropic panels

Modes 119891 Hz(Case 3)

119891 Hz(Case 4)

119891 Hz(Case 5)

119891 Hz(Case 6)

1 4646 2958 6969 118322 10697 3691 16046 147643 12534 4912 18801 196494 18585 6622 27877 264885 20782 8820 31173 352826 25680 1110 38520 443997 28670 11507 43004 460298 31731 11832 47596 473309 34901 13054 52351 5221510 41816 14683 62723 58731

core and face layers in terms of both materials selection anddimensions

4 Conclusions

This study presents a mathematical method and frameworkon composite panels radiated sound and vibrationThepanelsare considered excited either by turbulent flow or by randomnoiseThe turbulent flow corresponds to the typical excitationof an aircraft panel at flight cruise conditions with a Machnumber of 08 The effect of type of panel of panel dimen-sions and of panel thickness are evaluated The turbulentflow excitation showed to excite a higher number of panelmodes compared to random noise excitation and resulted indissimilar levels of both panel vibration and radiated soundwith relation to the random excitation This effect is morevisible for composite than for isotropic panels and thereforethe pure random noise should not be used as representativeof the turbulent flow excitation in real applications

Different properties on the composite panels are shown toyield dissimilar effects in the noise and vibration predictionsThe effect of the panel size specifically its length and widthis noted to have a significant effect on the panel performancein terms of noise radiation and vibration levels Overallsmaller composite panels or panels with higher aspect ratioproduced lower levels of radiated sound and vibration thanlonger and wider composite panels Compared to isotropicpanels composite panels analyzed are shown to radiate lowernoise levels with the exception of specific natural frequencieswhich are amplified near 300Hz and 1000Hz It is shown thatthis amplification can be significant for composite panels withthicker face layers and thinner core layer and thus resultingin unwanted effect of increased sound radiated Despitethis effect this drawback may potentially be corrected byincreasing the sandwich core layer thickness Increasing thecore layer thickness will impact the panel weight and hencethe composite panels can represent a heavier solution com-pared to the isotropic panels examined On the other handthis approach may be beneficial in the design of compositepanels in the context of aircraft panel applications sincethese panels may require less acoustic insulation material

added to the aircraft walls as they already represent animproved design solution Furthermore acoustic insulationmaterials have a higher impact in decreasing sound levelsat the high frequency ranges and therefore it will notsignificantly improve the problem of higher noise levels at thelow frequency range particularly observed in the isotropicsolutions In this context advanced and optimized compositepanel configurations may be a promising replacement to thetraditional isotropic panels for aerospace applications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is grateful for the support of the National Scienceand Engineering Research Council of Canada

References

[1] J Rocha and D Palumbo ldquoOn the sensitivity of sound powerradiated by aircraft panels to turbulent boundary layer param-etersrdquo Journal of Sound and Vibration vol 331 no 21 pp 4785ndash4806 2012

[2] J Rocha A Suleman and F Lau ldquoPrediction of turbulentboundary layer induced noise in the cabin of a BWB aircraftrdquoShock and Vibration vol 19 no 4 pp 693ndash705 2012

[3] J da Rocha A Suleman and F Lau ldquoFlow-induced noise andvibration in aircraft cylindrical cabins closed-form analyticalmodel validationrdquo Journal of Vibration and Acoustics vol 133no 5 Article ID 051013 9 pages 2011

[4] J da Rocha A Suleman and F Lau ldquoPrediction of flow-inducednoise in transport vehicles development and validation of acoupled structuralmdashacoustic analytical frameworkrdquo CanadianAcousticsmdashAcoustique Canadienne vol 37 no 4 pp 13ndash292009

[5] G Corcos ldquoThe structure of the turbulent pressure field inboundary layer flowsrdquo Journal of Fluid Mechanics vol 18 pp353ndash378 1964

[6] G Corcos ldquoResolution of pressure in turbulencerdquo Journal of theAcoustical Society of America vol 35 no 2 pp 192ndash199 1963

[7] B Efimtsov ldquoCharacteristics of the field of turbulent wallpressure fluctuations at large Reynolds numbersrdquo Soviet PhysicsAcoustics vol 28 no 4 pp 289ndash292 1982

[8] J Reddy Mechanics of Laminated Composite Plates and ShellsTheory and Analysis CRC Press Boca Raton Fla USA 1945

[9] J Whitney Structural Analysis of Laminated Anisotropic PlatesTechnomic Lancaster Pa USA 1987

[10] C L Dym and M A Lang ldquoTransmission of sound throughsandwich panelsrdquo Journal of the Acoustical Society of Americavol 56 no 5 pp 1523ndash1532 1974

[11] C L Dym C S Ventres and M A Lang ldquoTransmission ofsound through sandwich panels a reconsiderationrdquo Journal ofthe Acoustical Society of America vol 59 no 2 pp 364ndash3671976

[12] S Narayanan andR L Shanbhag ldquoSound transmission througha damped sandwich panelrdquo Journal of Sound and Vibration vol80 no 3 pp 315ndash327 1982

18 Shock and Vibration

[13] C L Dym and D C Lang ldquoTransmission loss of dampedasymmetric sandwich panels with orthotropic coresrdquo Journal ofSound and Vibration vol 88 no 3 pp 299ndash319 1983

[14] S Ghinet N Atalla and H Osman ldquoDiffuse field transmissioninto infinite sandwich composite and laminate composite cylin-dersrdquo Journal of Sound and Vibration vol 289 no 4-5 pp 745ndash778 2006

[15] J Mendler ldquoComputational tool for the acoustic analysis ofsandwich panelsrdquo in Proceedings of the European Conference onNoise Control (EURO-NOISE rsquo95) Lyon France 1995

[16] O Foin J Nicolas andNAtalla ldquoAn efficient tool for predictingthe structural acoustic and vibration response of sandwichplates in light or heavy fluidrdquo Applied Acoustics vol 57 no 3pp 213ndash242 1999

[17] S Assaf and M Guerich ldquoNumerical prediction of noisetransmission loss through viscoelastically damped sandwichplatesrdquo Journal of Sandwich Structures andMaterials vol 10 no5 pp 359ndash384 2008

[18] S Assaf M Guerich and P Cuvelier ldquoVibration and acousticresponse of damped sandwich panels immersed in a light orheavy fluidrdquo Computers and Structures vol 88 pp 359ndash3842010

[19] R Zhou and M J Crocker ldquoBoundary element analyses forsound transmission loss of panelsrdquo Journal of the AcousticalSociety of America vol 127 no 2 pp 829ndash840 2010

[20] K Amichi N Atalla and R Ruokolainen ldquoA new 3D finite ele-ment sandwich plate for predicting the vibroacoustic responseof laminated steel panelsrdquo Finite Elements in Analysis andDesign vol 46 no 12 pp 1131ndash1145 2010

[21] F Franco S de Rosa and T Polito ldquoFinite element inves-tigations on the vibroacoustic performance of plane plateswith random stiffnessrdquo Mechanics of Advanced Materials andStructures vol 18 no 7 pp 484ndash497 2011

[22] Y Kim ldquoIdentification of sound transmission characteristicsof honeycomb sandwich panels using hybrid analyticalone-dimensional finite element methodrdquo in Proceedings of the35th International Congress and Exposition on Noise ControlEngineering (INTER-NOISE rsquo06) HonoluluHawaii USA 2006

[23] F Fahy Sound and Structural Vibration Radiation Transmissionand Response Academic Press London UK 1985

[24] J Blevins Formulas for Natural Frequency and Mode ShapeKrieger Publishing Malabar Fla USA 2001

[25] D Newland An Introduction to Random Vibration and SpectralAnalysis Longman New York NY USA 1984

[26] Y Lin Probabilistic Theory of Structural Dynamics McGraw-Hill Book New York NY USA 1967

[27] C E Wallace ldquoRadiation resistance of a rectangular panelrdquoTheJournal of the Acoustical Society of America vol 51 no 3 pp946ndash952 1972

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

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Civil EngineeringAdvances in

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DistributedSensor Networks

International Journal of

Shock and Vibration 5

where 119886 and 119887 are the dimensions of the plate in the 119909 and119910 directions Also the natural frequencies for the simplysupported plate can be determined by

120596119898119909119898119910

=1205872

radicsum119899

119894=1120588119894ℎ119894

[11986311(119898119909

119886)

4

+ 11986322(119898119910

119887)

4

+2(11986312+ 211986366)(

119898119909

119886)

2

(119898119910

119887)

2

]

12

(26)

in which

119863119895119896

=

119899

sum

119894=1

int

ℎ119894

ℎ119894minus1

119876119894

1198951198961199112119889119911 (27)

Developing (22) by considering (23) to (25) making use ofthe orthogonality of the plate modes and integrating over theplate area similarly to the mathematical approach followedin [4] the equilibrium equation for an orthotropic plate iswritten in the matrix form as

[119872119901] q (119905) + [119863

119901] q (119905) + [119870

119901] q (119905) = pext (119905) (28)

where [119872119901] [119863119901] and [119870

119901] are mass damping and stiffness

119872times119872matrices respectively and q(119905) q(119905) and q(119905) are theposition velocity and acceleration vectors with componentsdefined respectively as 119902

119898(119905) = 119876

119898119890119894120596119905 119902119898(119905) = 119894119876

119898120596119890119894120596119905

and 119902119898(119905) = minus119876

1198981205962119890119894120596119905 After performing all the above

described mathematical manipulations it can be shown thateach term in (28) is defined as follows

[119872119901] = diag[

119899

sum

119894=1

120588119894ℎ119894]

[119863119901] = diag[2120585

119901120596119898119909119898119910

119899

sum

119894=1

120588119894ℎ119894]

[119870119901] =

1

3

119899

sum

119894=1

(ℎ3

119894minus ℎ3

119894minus1) 119876119894

11I (119909) + 119876

119894

22I (119910)

+ (2119876119894

12+ 8119876119894

66) I1015840 (119909) I1015840 (119910)

pext (119905) = [int

119886

0

int

119887

0

120572119898119909

(119909) 120573119898119910(119910) 119901ext (119909 119910 119905) 119889119909 119889119910]

(29)

in which 120585119901was added to account with the damping of the

plate and

I (119909) = diag [1119886(119898119909120587

119886)

4

]

I (119910) = diag [1119887(119898119910120587

119887)

4

]

I1015840 (119909) = minus1

119886(119898119909120587

119886)

2

cos [(119898119909minus 1198981015840

119909) 120587]

I1015840 (119910) = minus1

119887(119898119910120587

119887)

2

cos [(119898119910minus 1198981015840

119910) 120587]

(30)

22 Radiated Sound Power The calculation of the panelsrsquoradiated sound power in the frequency domain requires arearrangement of the time domain equations presented in(22) For this end knowing that 119902

119898= 119876119898119890119894120596119905 as previously

described one can write (28) in the following desired form inthe frequency 120596 domain

Y (120596) = H (120596)X (120596) (31)

where

Y (120596) = W (120596)

X (120596) = Pext (120596)

H (120596) = minus1205962[119872119901] + 119894120596 [119863

119901] + [119870

119901]

(32)

in which H(120596) is system frequency response matrix Y(120596) isthe response of the system to the excitationX(120596) and vectorsW(120596) andPext(120596) correspond respectively to the vectors q(119905)and pext(119905) written in the frequency domain Considering theturbulent boundary layer random excitation as a stationaryand homogeneous function the spectral density of the systemresponse is given by [25 26]

S119882119882 (120596) = Hlowast (120596) S119875119875 (120596)H

119879(120596) (33)

where superscripts lowast and 119879 denote Hermitian conjugate andmatrix transpose respectively S

119875119875(120596) is the spectral density

matrix of the random excitation Pext(120596) and S119882119882

(120596) is thespectral density matrix of the system response W(120596) Thematrix S

119875119875(120596) corresponds to the generalized power density

matrix of the turbulent boundary layer excitation defined by

Stbl (120596) = [∬

119886

0

∬

119887

0

120572119898119909

(119909) 1205721198981015840119909

(1199091015840) 120573119898119910

(119910) 1205731198981015840

119910

(1199101015840)

times 119878 (120585119909 120585119910 120596) 119889119909 119889119909

1015840119889119910119889119910

1015840]

(34)

in which 119878(120585119909 120585119910 120596) is defined by Corcos model [5 21] and

120585119909= 119909 minus 119909

1015840 and 120585119910= 119910 minus 119910

1015840 are the spatial separations in thestreamwise and spanwise directions of the plate respectivelyAs shown in [1ndash3] a combination of Corcos and Efimtsovmodels may be suitable to provide this excitation as follows

119878 (120585119909 120585119910 120596) = 119878ref (120596) 119890

minus(120572119909120596|120585119909|119880119888)119890minus(120572119910120596|120585119910|119880119888)119890

minus(119894120596120585119909119880119888)

(35)

where119880119888is the turbulent boundary layer convective velocity

120572119909= 01 and 120572

119910= 077 are empirical parameters used to

provide the best agreement with the reality [27] and 119878ref(120596)is defined using Efimtsov model [7] In [1] an analyticalform was derived for the matrix described in (34)The powerspectral density function of the plate displacement can then

6 Shock and Vibration

be defined as functions of the abovematrix respectively platearea as

119878119908119908

(1199091 1199101 1199092 1199102 120596)

=

1198722

119909

sum

11989811990911198981199092=1

1198722

119910

sum

11989811991011198981199102=1

1205721198981199091

(1199091) 1205721198981199092

(1199092)

times 1205731198981199101

(1199101) 1205731198981199102

(1199102) S119882119882(120596)1198981 1198982

(36)

and the overall power spectral density function can be foundby integrating the individual power spectral densities over theplate area displacement as following

119878119908119908 (120596) = ∬

119886

0

∬

119887

0

119878119908119908

(1199091 1199101 1199092 1199102 120596) 119889119909

1119889119909211988911991011198891199102

(37)

Analytical expressions for 119878119908119908

(1199091 1199101 1199092 1199102 120596) and 119878

119908119908(120596)

were obtained in [23] The power spectral density matrix ofthe plate velocity can be determined from the power spectraldensity matrix of the plate displacement as follows

S119881119881 (120596) = 120596

2S119882119882 (120596) (38)

Using these analytical forms the plate radiated sound powerfunction and the overall power spectral density function canbe determined respectively as follows

RSP (1199091 1199101 1199092 1199102 120596)

=

1198722

119909

sum

11989811990911198981199092=1

1198722

119910

sum

11989811991011198981199102=1

1205721198981199091

(1199091) 1205721198981199092

(1199092)

times 1205731198981199101

(1199101) 1205731198981199102

(1199102)Π(120596)1198981 1198982

RSP (120596) = ∬

119886

0

∬

119887

0

RSP (1199091 1199101 1199092 1199102 120596) 119889119909

1119889119909211988911991011198891199102

(39)

in which Π(120596)1198981 1198982

are the components of the radiationmatrix defined by

Π (120596) = S119881119881

M (120596) (40)and S119881119881

is defined in (38) andM(120596) is evaluated numericallyfrom the following equation

M (120596) = 81205880

1198880

(120596119886119887

1205873119898119909119898119910

)

2

times∬

1205872

0

cossin (

120572

2)

cossin (

120573

2)

[(120572119898119909120587)2minus 1] [(120573119898

119910120587)2

minus 1]

2

timessin 120579119889120579 119889120601(41)

where 1205880and 1198880are the density and the speed of sound of

the interior fluid respectively 120572 = (1205961198880)119886 sin 120579 cos120601 120573 =

(1205961198880)119887 sin 120579 sin120601 in which 120601 and 120579 are the angles of the

distance vector in space to the observation point [27]

3 Structural Vibration and SoundRadiation Results

The formulation presented in the previous section is usedto analyze four different rectangular composite sandwichpanels of several sizes and materials properties as shown inTable 1 Additionally four isotropic aluminum panels withsimilar sizes compared to the composite panels as shownin Table 2 are analyzed using the previous model developedby the author results are then compared with the compositepanels All the eight panels are considered simply supportedin the four boundaries and subjected to turbulent boundarylayer excitation or to random noise excitation The turbulentflow excitation is representative of Mach number 08 flightconditions as described inTable 3 and the randomexcitationrepresents a constant 40 Pascal random noise that is asound pressure level of approximately 120 dB The first tennatural frequencies obtained for the several panels analyzedare shown in Tables 4 and 5 respectively for composite andisotropic panels Cases 1 2 3 and 4 of composite panels asnoted in Table 4 are chosen for validation purposes as instudies [16 18 20] respectively It is shown that the calculatednatural frequencies are in good agreement with the previousstudies

31 Composite Panels The results presented in this sectiondiscuss the radiated sound power and the structural dis-placement levels of the composite sandwich panels studiedResults are shown for frequencies up to 1500Hz requiringa minimum number of plate modes which is dependent ofthe panel size and materials properties for proper accuracyof the predicted values Values for the radiated sound powerRSP and plate displacement power spectral density (PSD)119878119908119908

are obtained for two locations (1199091 1199101) = (05119886 05119887)

and (1199092 1199102) = (03119886 02119887) as well as overall values of these

two quantities and shown in Figures 4 and 5 As referredearlier four different panels are analyzed identified as Cases1 through 4 along the following sections and two types ofexcitation are applied (1) turbulent boundary layer excitationand (2) random noise excitation

As noted in Figure 4 and similarly to what was observedin previous results and formulation dedicated to the analysisof isotropic panels it is concluded that the overall RSPvalues differ from local RSP values especially for the TBLexcitation It is also observed that turbulent flow can beresponsible for exciting a higher number of panel modescompared to random noise with TBL and random noiseproviding dissimilar levels of panelsrsquo sound radiation Assuch to accurately predict the dynamic behaviour of a panelone should not consider the pure random noise excitation asrepresentative of the turbulent flow excitation In the overallRSP results plates of bigger dimensions are shown to havehigher RSP values both for TBL and random excitationswith Case 1 showing higher RSP values than Cases 2 3or 4 The same is not noted for the local RSP valueswith Case 3 panel showing increased RSP values for bothpositions 1 and 2with relation to the overall RSP valuesTheseresults emphasize the importance of knowing the measuring

Shock and Vibration 7

0 500 1000 1500Frequency (Hz)

minus150

minus100

minus50

0

50

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

(b)

0 500 1000 1500

Frequency (Hz)

minus120

minus100

minus80

minus60

minus40

minus20

0

20

40

Radi

ated

soun

d po

wer

(dB

rel1

W)

(c)

0 500 1000 1500

Frequency (Hz)

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

(d)

0 500 1000 1500

Frequency (Hz)

minus140

minus120

minus100

minus80

minus60

minus40

minus20

0

20

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 1 composite TBLCase 2 composite TBL

Case 3 composite TBLCase 4 composite TBL

(e)

0 500 1000 1500

Frequency (Hz)

minus80

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 1 composite randomCase 2 composite random

Case 3 composite randomCase 4 composite random

(f)

Figure 4 Radiated sound power results for the composite panels (a) and (b) overall panel RSP (c) and (d) RSPnear surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) RSP near location (1199092 1199102) = (03119886 02119887)

position for the accurate prediction of sound and vibrationand it highlights the desired sensitivity of the developedmathematical framework to this factor Figure 5 shows thedisplacement PSD results associated with the same panels

and excitations as for the RSP results displayed in Figure 4Comparing Figures 4 and 5 it is observed that structuraldisplacement and sound radiation levels have different curvesfor the same excitation and thus panel displacement PSD

8 Shock and Vibration

minus250

minus200

minus150

minus100

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

minus200

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(b)

minus200

minus180

minus140

minus160

minus120

minus100

minus80

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(c)

minus220

minus200

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(d)

minus220

minus200

minus180

minus140

minus160

minus120

minus100

minus80

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

Case 1 composite TBLCase 2 composite TBL

Case 3 composite TBLCase 4 composite TBL

(e)

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

Case 1 composite randomCase 2 composite random

Case 3 composite randomCase 4 composite random

(f)

Figure 5 Panel displacement PSD results for the composite panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location(1199091 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

curve predictionmay not be fully representative of RSP curveeven though they would be expected to follow similar trendsIn addition as similar to RSP results it is concluded that 119878

119908119908

predictions are dissimilar whether one is assuming TBL or

random excitation Case 1 panel shows higher values of bothoverall and local plate displacement PSD predictions

The results presented here are useful in the context of theprediction of noise radiation by composite panel solutions

Shock and Vibration 9

and its possible application for example in aerospace indus-try Based on the obtained results if one would choose whichcomposite panel solution is most desirable for the purposeof manufacturing an aircraft fuselage panel a smaller panel(Case 3) or a panel with a higher aspect ratio (Case 4) wouldbe preferable for overall improved results Longer and widerpanels (ie Cases 1 and 2) are shown to greatly amplify thelow frequency range However one would need to designthe panel appropriately and with the targeted dimensionssuch that the amplification of noise and vibration at certainfrequencies would be avoided For instance Case 3 panelexhibits a general improved behaviour in terms of noiselevels but simultaneously it amplifies noise at frequenciesnear 300Hz and 1000Hz while Case 4 panel amplifies resultsnear 300Hz Comparing the size of these two panels thesurface area covered by one Case 1 panel corresponds toapproximately the surface area covered by twenty Case 3panels with the twenty Case 3 panels representing approx-imately 94 kg of total mass compared to the 676 kg of asingle Case 1 panel As such an improvement in the noiselevels comes as a tradeoff in terms of increased weightAdditionally it is shown that panels with thicker face layersand thinner core layer specifically referring to Cases 3 and 4have lower noise and vibration levels than panels with thinnerface layers and thicker core layer referring to Cases 1 and 2However as mentioned earlier Cases 3 and 4 panels withsmaller dimensions and thicker face layers have the problemof noise amplification at certain frequencies Therefore apanel which combines the properties of being shorter with athicker core layer versus thinner face layers may potentiallyshow an improved behaviour for reduced vibration and noiseradiation as well as lower noise amplification at isolatedfrequencies

32 Isotropic Panels This section addresses the RSP and 119878119908119908

predictions for the isotropic panels analyzed and results upto 1500Hz are shown in Figures 6 and 7 Four aluminumisotropic panels are investigated considering TBL or randomexcitation as follows isotropic panels Cases 3 and 4 have thesame dimensions 119886 and 119887 as the composite panels Cases3 and 4 respectively isotropic panels Cases 5 and 6 areconsidered with same dimensions 119886 and 119887 as Cases 3 and4 respectively but with increased thicknesses This approachis taken with the purpose of posterior comparison betweencomposite and isotropic panelsrsquo behaviours In this contextisotropic panels Cases 3 and 4 are both 1mm thick whileisotropic panels Cases 4 and 5 have thicknesses of 15mm and4mm respectively

As noted in Figures 6 and 7 the increase of panelthickness results in an expected shift in the panel naturalfrequencies to higher frequencies as well as a shift in themodes amplified in both RSP and 119878

119908119908results This effect is

observed in the TBL and random excitations Furthermore itis found that panels with higher aspect ratio (ie Cases 4 and6) are generally characterized by having lower levels of soundradiation and vibration As similarly noted for the compositepanels random and TBL excitations provide different RSPand 119878

119908119908predictions and overall values differ from point

values and hence it is important to specify the location in the

panel where the measurementspredictions are being takenIt is also noted that the random excitation remains as theexcitation causing higher noise and vibration levels for allcases at the higher frequency range This is also an expectedeffect as the TBL excitation decreases towards the higherfrequencies while the random excitation provides a constantexcitation over the frequency spectrum

33 Composite versus Isotropic Panels As one of the mainobjectives of this study this section presents a discussionof results for the behavior of composite sandwich panelsin comparison with the traditional isotropic panels Thefollowing four figures are shown to this end Figures 8 and9 compare results for composite panel Case 3 with isotropicpanels Cases 3 and 5 while Figures 10 and 11 comparecomposite panel Case 4 with isotropic panels Cases 4 and 6

By comparing Cases 3 and 5 panels as shown fromFigures 8 and 9 it is noted that composite panel Case 3exhibits an enhanced behaviour in comparison with isotropicpanels Cases 3 and 5 both in RSP and displacement PSDlevels However one should also note that Case 3 compositepanel shows higher levels of noise and vibration at specificisolated frequencies approximately near 300Hz and 1000Hzfor the overall RSP and for position 1 RSP results as shown inFigure 8 Considering the panel displacement PSD results inFigure 9 extra frequencies above 1000Hz are also amplifiedespecially at position 2 for both TBL and random excitationsAs described in Table 1 the composite panel Case 3 has athin core layer thus the unwanted amplification at thesefrequencies may be attenuated by increasing the core layerthickness However this modification on the panel thicknesswill increase the original panel weight of 047 kg whichalready represents heavier solution compared to Cases 3and 5 isotropic panels with weights of 03 kg and 045 kgrespectively Although the panel weight will increase if thischange is performed making this modification will in alater phase require less added acoustic insulation material tothe aircraft walls in the context of aircraft fuselage panelsrsquoapplication As such the composite panelmay represent witha slight increase in the panel weight an optimized solutionfor both sound radiation and structural vibration in almostall the frequency spectra analyzed

Similar conclusions are obtained from Figures 10 and 11dedicated to the comparison of Cases 4 and 6 panels thatis the composite panel represents an improved solution foralmost all the frequency ranges studied By analyzing thegeometric differences between composite panels Cases 3 and4 Case 4 has thicker face layers and is longer than Case 3panel having a much higher aspect ratio The amplificationof noise and vibration levels at specific frequencies is notedas in Case 3 composite panel Since face layers are thickerand this panel has a higher aspect ratio this effect is morevisible in composite Case 4 than in composite Case 3 Asobserved in results from Case 3 this could be enhancedby implementing an increase of the core layer thicknessWhile the increase in the face layer thickness is shown todecrease the levels of noise and vibration in both overalland localized predictions it also can have an impact in the

10 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 6 Radiated sound power results for the isotropic panels (a) and (b) overall panel RSP (c) and (d) RSP near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) RSP near location (1199092 1199102) = (03119886 02119887)

Shock and Vibration 11

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

(a)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(b)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(c)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus50

minus100

minus150

minus200

(f)

Figure 7 Panel displacement PSD results for the isotropic panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) 119878119908119908

near location (1199092 1199102) = (03119886 02119887)

12 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus100

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus100

minus80

minus60

minus40

minus20

0

20

40

60

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 8 Comparison of radiated sound power results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 13

0 500 1000 1500

Frequency (Hz)

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus50

minus150

minus100

minus200

minus250

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 9 Comparison of panel displacement PSD results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

14 Shock and VibrationRa

diat

ed so

und

pow

er (d

B re

l1

W)

0 500 1000 1500

Frequency (Hz)

50

0

minus50

minus100

minus150

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

100

50

0

minus50

minus100

minus150

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

50

0

minus50

minus100

minus150

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus80

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 10 Comparison of radiated sound power results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 15

minus50

minus100

minus150

minus250

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus220

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus50

minus100

minus150

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 11 Comparison of panel displacement PSD results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b)overall panel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

16 Shock and Vibration

Table 1 Physical properties of composite panels

Variable Description units Case 1 Case 2 Case 3 Case 4119886 Length m 183 11683 0348 10119887 Width m 122 11683 03048 03120582 Aspect ratio (119886119887) 15 10 114 333ℎ1 ℎ3

Face layers thickness mm 0406 0506 0762 35ℎ2

Core layer thickness mm 64 1905 0254 051198641 1198643

Face layers Young modulus Pa 6898 times 1010

71 times 1010

6898 times 1010

66 times 1010

1198642

Core layer Young modulus Pa 457 times 105

457 times 105

267 times 106

26 times 108

]1 ]3

Face layers Poissonrsquos ratio 033 033 03 033]2

Core layer Poissonrsquos ratio 04 04 049 031205881 1205883

Face layers density kgmminus3 2768 2700 2740 26801205882

Core layer density kgmminus3 122 4806 999 1680119898 Total mass kg 676 499 047 588

Table 2 Physical properties of isotropic panels

Variable Description units Case 3 Case 4 Case 5 Case 6119886 Length m 0348 10 0348 10119887 Width m 03048 03 03048 03120582 Aspect ratio (119886119887) 114 333 114 333ℎ119901

Thickness mm 1 1 15 4119864119901

Young modulus Pa 724 times 1010

724 times 1010

724 times 1010

724 times 1010

]119901

Poissonrsquos ratio 033 033 033 033120588119901

Density kg mminus3 2800 2800 2800 2800119898 Total mass kg 030 084 045 336

Table 3 Flight test conditions and air properties

Variable Description units Value119872 Mach number 08alt Altitude m 13106119888 Speed of sound m sminus1 295119880infin

Flow speed m sminus1 23606120588 Air density kgmminus3 02622] Kinematic viscosity m2 sminus1 5422 times 10

minus5

119879 Temperature K 21665Re119909

Reynolds number 2917 times 107

1198880

Speed of sound m sminus1 3401205880

Internal air density kgmminus3 142

higher amplification of localized frequencies Therefore theface layers thickness increase should be performed carefullyby an accompanied increase in the core layer thickness whilesimultaneously considering the weight constraints for thespecific panels applications

A more detailed analysis will be required in order toformulate final conclusions about the optimum solutions forthe panel in the context of aircraft application For that theinsulation materials would also be considered to account forthe respective added weight and effect in the sound radi-ation signature However typically the acoustic insulationmaterials will only have a higher impact in decreasing the

Table 4 First 10 natural frequencies for the composite panels

Modes 119891 Hz(Case 1)

119891 Hz(Case 2)

119891 Hz(Case 3)

119891 Hz(Case 4)

1 2365 10588 4874 126582 4548 26470 11222 157933 7276 42352 13148 210194 8186 52941 19496 283355 9459 68823 21800 377416 13097 89999 26939 474967 13279 95293 30075 492388 15462 105881 33287 506329 17645 132351 36612 5585710 18191 137645 43866 62825

sound levels at higher frequency ranges and will thereforenot improve significantly the problem of higher noise levelsin the low frequency range particularly observed in theisotropic solutions As such the composite panel solutionsmay appear to be a promising replacement to the isotropicpanels especially panels like Cases 3 and 4 which are alreadycharacterized by lower levels of both RSP and 119878

119908119908in the

low frequency range In further analysis it will also beimportant to carefully address the noise amplification atisolated frequencies by identifying a good combination of

Shock and Vibration 17

Table 5 First 10 natural frequencies for the isotropic panels

Modes 119891 Hz(Case 3)

119891 Hz(Case 4)

119891 Hz(Case 5)

119891 Hz(Case 6)

1 4646 2958 6969 118322 10697 3691 16046 147643 12534 4912 18801 196494 18585 6622 27877 264885 20782 8820 31173 352826 25680 1110 38520 443997 28670 11507 43004 460298 31731 11832 47596 473309 34901 13054 52351 5221510 41816 14683 62723 58731

core and face layers in terms of both materials selection anddimensions

4 Conclusions

This study presents a mathematical method and frameworkon composite panels radiated sound and vibrationThepanelsare considered excited either by turbulent flow or by randomnoiseThe turbulent flow corresponds to the typical excitationof an aircraft panel at flight cruise conditions with a Machnumber of 08 The effect of type of panel of panel dimen-sions and of panel thickness are evaluated The turbulentflow excitation showed to excite a higher number of panelmodes compared to random noise excitation and resulted indissimilar levels of both panel vibration and radiated soundwith relation to the random excitation This effect is morevisible for composite than for isotropic panels and thereforethe pure random noise should not be used as representativeof the turbulent flow excitation in real applications

Different properties on the composite panels are shown toyield dissimilar effects in the noise and vibration predictionsThe effect of the panel size specifically its length and widthis noted to have a significant effect on the panel performancein terms of noise radiation and vibration levels Overallsmaller composite panels or panels with higher aspect ratioproduced lower levels of radiated sound and vibration thanlonger and wider composite panels Compared to isotropicpanels composite panels analyzed are shown to radiate lowernoise levels with the exception of specific natural frequencieswhich are amplified near 300Hz and 1000Hz It is shown thatthis amplification can be significant for composite panels withthicker face layers and thinner core layer and thus resultingin unwanted effect of increased sound radiated Despitethis effect this drawback may potentially be corrected byincreasing the sandwich core layer thickness Increasing thecore layer thickness will impact the panel weight and hencethe composite panels can represent a heavier solution com-pared to the isotropic panels examined On the other handthis approach may be beneficial in the design of compositepanels in the context of aircraft panel applications sincethese panels may require less acoustic insulation material

added to the aircraft walls as they already represent animproved design solution Furthermore acoustic insulationmaterials have a higher impact in decreasing sound levelsat the high frequency ranges and therefore it will notsignificantly improve the problem of higher noise levels at thelow frequency range particularly observed in the isotropicsolutions In this context advanced and optimized compositepanel configurations may be a promising replacement to thetraditional isotropic panels for aerospace applications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is grateful for the support of the National Scienceand Engineering Research Council of Canada

References

[1] J Rocha and D Palumbo ldquoOn the sensitivity of sound powerradiated by aircraft panels to turbulent boundary layer param-etersrdquo Journal of Sound and Vibration vol 331 no 21 pp 4785ndash4806 2012

[2] J Rocha A Suleman and F Lau ldquoPrediction of turbulentboundary layer induced noise in the cabin of a BWB aircraftrdquoShock and Vibration vol 19 no 4 pp 693ndash705 2012

[3] J da Rocha A Suleman and F Lau ldquoFlow-induced noise andvibration in aircraft cylindrical cabins closed-form analyticalmodel validationrdquo Journal of Vibration and Acoustics vol 133no 5 Article ID 051013 9 pages 2011

[4] J da Rocha A Suleman and F Lau ldquoPrediction of flow-inducednoise in transport vehicles development and validation of acoupled structuralmdashacoustic analytical frameworkrdquo CanadianAcousticsmdashAcoustique Canadienne vol 37 no 4 pp 13ndash292009

[5] G Corcos ldquoThe structure of the turbulent pressure field inboundary layer flowsrdquo Journal of Fluid Mechanics vol 18 pp353ndash378 1964

[6] G Corcos ldquoResolution of pressure in turbulencerdquo Journal of theAcoustical Society of America vol 35 no 2 pp 192ndash199 1963

[7] B Efimtsov ldquoCharacteristics of the field of turbulent wallpressure fluctuations at large Reynolds numbersrdquo Soviet PhysicsAcoustics vol 28 no 4 pp 289ndash292 1982

[8] J Reddy Mechanics of Laminated Composite Plates and ShellsTheory and Analysis CRC Press Boca Raton Fla USA 1945

[9] J Whitney Structural Analysis of Laminated Anisotropic PlatesTechnomic Lancaster Pa USA 1987

[10] C L Dym and M A Lang ldquoTransmission of sound throughsandwich panelsrdquo Journal of the Acoustical Society of Americavol 56 no 5 pp 1523ndash1532 1974

[11] C L Dym C S Ventres and M A Lang ldquoTransmission ofsound through sandwich panels a reconsiderationrdquo Journal ofthe Acoustical Society of America vol 59 no 2 pp 364ndash3671976

[12] S Narayanan andR L Shanbhag ldquoSound transmission througha damped sandwich panelrdquo Journal of Sound and Vibration vol80 no 3 pp 315ndash327 1982

18 Shock and Vibration

[13] C L Dym and D C Lang ldquoTransmission loss of dampedasymmetric sandwich panels with orthotropic coresrdquo Journal ofSound and Vibration vol 88 no 3 pp 299ndash319 1983

[14] S Ghinet N Atalla and H Osman ldquoDiffuse field transmissioninto infinite sandwich composite and laminate composite cylin-dersrdquo Journal of Sound and Vibration vol 289 no 4-5 pp 745ndash778 2006

[15] J Mendler ldquoComputational tool for the acoustic analysis ofsandwich panelsrdquo in Proceedings of the European Conference onNoise Control (EURO-NOISE rsquo95) Lyon France 1995

[16] O Foin J Nicolas andNAtalla ldquoAn efficient tool for predictingthe structural acoustic and vibration response of sandwichplates in light or heavy fluidrdquo Applied Acoustics vol 57 no 3pp 213ndash242 1999

[17] S Assaf and M Guerich ldquoNumerical prediction of noisetransmission loss through viscoelastically damped sandwichplatesrdquo Journal of Sandwich Structures andMaterials vol 10 no5 pp 359ndash384 2008

[18] S Assaf M Guerich and P Cuvelier ldquoVibration and acousticresponse of damped sandwich panels immersed in a light orheavy fluidrdquo Computers and Structures vol 88 pp 359ndash3842010

[19] R Zhou and M J Crocker ldquoBoundary element analyses forsound transmission loss of panelsrdquo Journal of the AcousticalSociety of America vol 127 no 2 pp 829ndash840 2010

[20] K Amichi N Atalla and R Ruokolainen ldquoA new 3D finite ele-ment sandwich plate for predicting the vibroacoustic responseof laminated steel panelsrdquo Finite Elements in Analysis andDesign vol 46 no 12 pp 1131ndash1145 2010

[21] F Franco S de Rosa and T Polito ldquoFinite element inves-tigations on the vibroacoustic performance of plane plateswith random stiffnessrdquo Mechanics of Advanced Materials andStructures vol 18 no 7 pp 484ndash497 2011

[22] Y Kim ldquoIdentification of sound transmission characteristicsof honeycomb sandwich panels using hybrid analyticalone-dimensional finite element methodrdquo in Proceedings of the35th International Congress and Exposition on Noise ControlEngineering (INTER-NOISE rsquo06) HonoluluHawaii USA 2006

[23] F Fahy Sound and Structural Vibration Radiation Transmissionand Response Academic Press London UK 1985

[24] J Blevins Formulas for Natural Frequency and Mode ShapeKrieger Publishing Malabar Fla USA 2001

[25] D Newland An Introduction to Random Vibration and SpectralAnalysis Longman New York NY USA 1984

[26] Y Lin Probabilistic Theory of Structural Dynamics McGraw-Hill Book New York NY USA 1967

[27] C E Wallace ldquoRadiation resistance of a rectangular panelrdquoTheJournal of the Acoustical Society of America vol 51 no 3 pp946ndash952 1972

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International Journal of

6 Shock and Vibration

be defined as functions of the abovematrix respectively platearea as

119878119908119908

(1199091 1199101 1199092 1199102 120596)

=

1198722

119909

sum

11989811990911198981199092=1

1198722

119910

sum

11989811991011198981199102=1

1205721198981199091

(1199091) 1205721198981199092

(1199092)

times 1205731198981199101

(1199101) 1205731198981199102

(1199102) S119882119882(120596)1198981 1198982

(36)

and the overall power spectral density function can be foundby integrating the individual power spectral densities over theplate area displacement as following

119878119908119908 (120596) = ∬

119886

0

∬

119887

0

119878119908119908

(1199091 1199101 1199092 1199102 120596) 119889119909

1119889119909211988911991011198891199102

(37)

Analytical expressions for 119878119908119908

(1199091 1199101 1199092 1199102 120596) and 119878

119908119908(120596)

were obtained in [23] The power spectral density matrix ofthe plate velocity can be determined from the power spectraldensity matrix of the plate displacement as follows

S119881119881 (120596) = 120596

2S119882119882 (120596) (38)

Using these analytical forms the plate radiated sound powerfunction and the overall power spectral density function canbe determined respectively as follows

RSP (1199091 1199101 1199092 1199102 120596)

=

1198722

119909

sum

11989811990911198981199092=1

1198722

119910

sum

11989811991011198981199102=1

1205721198981199091

(1199091) 1205721198981199092

(1199092)

times 1205731198981199101

(1199101) 1205731198981199102

(1199102)Π(120596)1198981 1198982

RSP (120596) = ∬

119886

0

∬

119887

0

RSP (1199091 1199101 1199092 1199102 120596) 119889119909

1119889119909211988911991011198891199102

(39)

in which Π(120596)1198981 1198982

are the components of the radiationmatrix defined by

Π (120596) = S119881119881

M (120596) (40)and S119881119881

is defined in (38) andM(120596) is evaluated numericallyfrom the following equation

M (120596) = 81205880

1198880

(120596119886119887

1205873119898119909119898119910

)

2

times∬

1205872

0

cossin (

120572

2)

cossin (

120573

2)

[(120572119898119909120587)2minus 1] [(120573119898

119910120587)2

minus 1]

2

timessin 120579119889120579 119889120601(41)

where 1205880and 1198880are the density and the speed of sound of

the interior fluid respectively 120572 = (1205961198880)119886 sin 120579 cos120601 120573 =

(1205961198880)119887 sin 120579 sin120601 in which 120601 and 120579 are the angles of the

distance vector in space to the observation point [27]

3 Structural Vibration and SoundRadiation Results

The formulation presented in the previous section is usedto analyze four different rectangular composite sandwichpanels of several sizes and materials properties as shown inTable 1 Additionally four isotropic aluminum panels withsimilar sizes compared to the composite panels as shownin Table 2 are analyzed using the previous model developedby the author results are then compared with the compositepanels All the eight panels are considered simply supportedin the four boundaries and subjected to turbulent boundarylayer excitation or to random noise excitation The turbulentflow excitation is representative of Mach number 08 flightconditions as described inTable 3 and the randomexcitationrepresents a constant 40 Pascal random noise that is asound pressure level of approximately 120 dB The first tennatural frequencies obtained for the several panels analyzedare shown in Tables 4 and 5 respectively for composite andisotropic panels Cases 1 2 3 and 4 of composite panels asnoted in Table 4 are chosen for validation purposes as instudies [16 18 20] respectively It is shown that the calculatednatural frequencies are in good agreement with the previousstudies

31 Composite Panels The results presented in this sectiondiscuss the radiated sound power and the structural dis-placement levels of the composite sandwich panels studiedResults are shown for frequencies up to 1500Hz requiringa minimum number of plate modes which is dependent ofthe panel size and materials properties for proper accuracyof the predicted values Values for the radiated sound powerRSP and plate displacement power spectral density (PSD)119878119908119908

are obtained for two locations (1199091 1199101) = (05119886 05119887)

and (1199092 1199102) = (03119886 02119887) as well as overall values of these

two quantities and shown in Figures 4 and 5 As referredearlier four different panels are analyzed identified as Cases1 through 4 along the following sections and two types ofexcitation are applied (1) turbulent boundary layer excitationand (2) random noise excitation

As noted in Figure 4 and similarly to what was observedin previous results and formulation dedicated to the analysisof isotropic panels it is concluded that the overall RSPvalues differ from local RSP values especially for the TBLexcitation It is also observed that turbulent flow can beresponsible for exciting a higher number of panel modescompared to random noise with TBL and random noiseproviding dissimilar levels of panelsrsquo sound radiation Assuch to accurately predict the dynamic behaviour of a panelone should not consider the pure random noise excitation asrepresentative of the turbulent flow excitation In the overallRSP results plates of bigger dimensions are shown to havehigher RSP values both for TBL and random excitationswith Case 1 showing higher RSP values than Cases 2 3or 4 The same is not noted for the local RSP valueswith Case 3 panel showing increased RSP values for bothpositions 1 and 2with relation to the overall RSP valuesTheseresults emphasize the importance of knowing the measuring

Shock and Vibration 7

0 500 1000 1500Frequency (Hz)

minus150

minus100

minus50

0

50

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

(b)

0 500 1000 1500

Frequency (Hz)

minus120

minus100

minus80

minus60

minus40

minus20

0

20

40

Radi

ated

soun

d po

wer

(dB

rel1

W)

(c)

0 500 1000 1500

Frequency (Hz)

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

(d)

0 500 1000 1500

Frequency (Hz)

minus140

minus120

minus100

minus80

minus60

minus40

minus20

0

20

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 1 composite TBLCase 2 composite TBL

Case 3 composite TBLCase 4 composite TBL

(e)

0 500 1000 1500

Frequency (Hz)

minus80

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 1 composite randomCase 2 composite random

Case 3 composite randomCase 4 composite random

(f)

Figure 4 Radiated sound power results for the composite panels (a) and (b) overall panel RSP (c) and (d) RSPnear surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) RSP near location (1199092 1199102) = (03119886 02119887)

position for the accurate prediction of sound and vibrationand it highlights the desired sensitivity of the developedmathematical framework to this factor Figure 5 shows thedisplacement PSD results associated with the same panels

and excitations as for the RSP results displayed in Figure 4Comparing Figures 4 and 5 it is observed that structuraldisplacement and sound radiation levels have different curvesfor the same excitation and thus panel displacement PSD

8 Shock and Vibration

minus250

minus200

minus150

minus100

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

minus200

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(b)

minus200

minus180

minus140

minus160

minus120

minus100

minus80

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(c)

minus220

minus200

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(d)

minus220

minus200

minus180

minus140

minus160

minus120

minus100

minus80

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

Case 1 composite TBLCase 2 composite TBL

Case 3 composite TBLCase 4 composite TBL

(e)

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

Case 1 composite randomCase 2 composite random

Case 3 composite randomCase 4 composite random

(f)

Figure 5 Panel displacement PSD results for the composite panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location(1199091 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

curve predictionmay not be fully representative of RSP curveeven though they would be expected to follow similar trendsIn addition as similar to RSP results it is concluded that 119878

119908119908

predictions are dissimilar whether one is assuming TBL or

random excitation Case 1 panel shows higher values of bothoverall and local plate displacement PSD predictions

The results presented here are useful in the context of theprediction of noise radiation by composite panel solutions

Shock and Vibration 9

and its possible application for example in aerospace indus-try Based on the obtained results if one would choose whichcomposite panel solution is most desirable for the purposeof manufacturing an aircraft fuselage panel a smaller panel(Case 3) or a panel with a higher aspect ratio (Case 4) wouldbe preferable for overall improved results Longer and widerpanels (ie Cases 1 and 2) are shown to greatly amplify thelow frequency range However one would need to designthe panel appropriately and with the targeted dimensionssuch that the amplification of noise and vibration at certainfrequencies would be avoided For instance Case 3 panelexhibits a general improved behaviour in terms of noiselevels but simultaneously it amplifies noise at frequenciesnear 300Hz and 1000Hz while Case 4 panel amplifies resultsnear 300Hz Comparing the size of these two panels thesurface area covered by one Case 1 panel corresponds toapproximately the surface area covered by twenty Case 3panels with the twenty Case 3 panels representing approx-imately 94 kg of total mass compared to the 676 kg of asingle Case 1 panel As such an improvement in the noiselevels comes as a tradeoff in terms of increased weightAdditionally it is shown that panels with thicker face layersand thinner core layer specifically referring to Cases 3 and 4have lower noise and vibration levels than panels with thinnerface layers and thicker core layer referring to Cases 1 and 2However as mentioned earlier Cases 3 and 4 panels withsmaller dimensions and thicker face layers have the problemof noise amplification at certain frequencies Therefore apanel which combines the properties of being shorter with athicker core layer versus thinner face layers may potentiallyshow an improved behaviour for reduced vibration and noiseradiation as well as lower noise amplification at isolatedfrequencies

32 Isotropic Panels This section addresses the RSP and 119878119908119908

predictions for the isotropic panels analyzed and results upto 1500Hz are shown in Figures 6 and 7 Four aluminumisotropic panels are investigated considering TBL or randomexcitation as follows isotropic panels Cases 3 and 4 have thesame dimensions 119886 and 119887 as the composite panels Cases3 and 4 respectively isotropic panels Cases 5 and 6 areconsidered with same dimensions 119886 and 119887 as Cases 3 and4 respectively but with increased thicknesses This approachis taken with the purpose of posterior comparison betweencomposite and isotropic panelsrsquo behaviours In this contextisotropic panels Cases 3 and 4 are both 1mm thick whileisotropic panels Cases 4 and 5 have thicknesses of 15mm and4mm respectively

As noted in Figures 6 and 7 the increase of panelthickness results in an expected shift in the panel naturalfrequencies to higher frequencies as well as a shift in themodes amplified in both RSP and 119878

119908119908results This effect is

observed in the TBL and random excitations Furthermore itis found that panels with higher aspect ratio (ie Cases 4 and6) are generally characterized by having lower levels of soundradiation and vibration As similarly noted for the compositepanels random and TBL excitations provide different RSPand 119878

119908119908predictions and overall values differ from point

values and hence it is important to specify the location in the

panel where the measurementspredictions are being takenIt is also noted that the random excitation remains as theexcitation causing higher noise and vibration levels for allcases at the higher frequency range This is also an expectedeffect as the TBL excitation decreases towards the higherfrequencies while the random excitation provides a constantexcitation over the frequency spectrum

33 Composite versus Isotropic Panels As one of the mainobjectives of this study this section presents a discussionof results for the behavior of composite sandwich panelsin comparison with the traditional isotropic panels Thefollowing four figures are shown to this end Figures 8 and9 compare results for composite panel Case 3 with isotropicpanels Cases 3 and 5 while Figures 10 and 11 comparecomposite panel Case 4 with isotropic panels Cases 4 and 6

By comparing Cases 3 and 5 panels as shown fromFigures 8 and 9 it is noted that composite panel Case 3exhibits an enhanced behaviour in comparison with isotropicpanels Cases 3 and 5 both in RSP and displacement PSDlevels However one should also note that Case 3 compositepanel shows higher levels of noise and vibration at specificisolated frequencies approximately near 300Hz and 1000Hzfor the overall RSP and for position 1 RSP results as shown inFigure 8 Considering the panel displacement PSD results inFigure 9 extra frequencies above 1000Hz are also amplifiedespecially at position 2 for both TBL and random excitationsAs described in Table 1 the composite panel Case 3 has athin core layer thus the unwanted amplification at thesefrequencies may be attenuated by increasing the core layerthickness However this modification on the panel thicknesswill increase the original panel weight of 047 kg whichalready represents heavier solution compared to Cases 3and 5 isotropic panels with weights of 03 kg and 045 kgrespectively Although the panel weight will increase if thischange is performed making this modification will in alater phase require less added acoustic insulation material tothe aircraft walls in the context of aircraft fuselage panelsrsquoapplication As such the composite panelmay represent witha slight increase in the panel weight an optimized solutionfor both sound radiation and structural vibration in almostall the frequency spectra analyzed

Similar conclusions are obtained from Figures 10 and 11dedicated to the comparison of Cases 4 and 6 panels thatis the composite panel represents an improved solution foralmost all the frequency ranges studied By analyzing thegeometric differences between composite panels Cases 3 and4 Case 4 has thicker face layers and is longer than Case 3panel having a much higher aspect ratio The amplificationof noise and vibration levels at specific frequencies is notedas in Case 3 composite panel Since face layers are thickerand this panel has a higher aspect ratio this effect is morevisible in composite Case 4 than in composite Case 3 Asobserved in results from Case 3 this could be enhancedby implementing an increase of the core layer thicknessWhile the increase in the face layer thickness is shown todecrease the levels of noise and vibration in both overalland localized predictions it also can have an impact in the

10 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 6 Radiated sound power results for the isotropic panels (a) and (b) overall panel RSP (c) and (d) RSP near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) RSP near location (1199092 1199102) = (03119886 02119887)

Shock and Vibration 11

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

(a)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(b)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(c)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus50

minus100

minus150

minus200

(f)

Figure 7 Panel displacement PSD results for the isotropic panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) 119878119908119908

near location (1199092 1199102) = (03119886 02119887)

12 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus100

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus100

minus80

minus60

minus40

minus20

0

20

40

60

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 8 Comparison of radiated sound power results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 13

0 500 1000 1500

Frequency (Hz)

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus50

minus150

minus100

minus200

minus250

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 9 Comparison of panel displacement PSD results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

14 Shock and VibrationRa

diat

ed so

und

pow

er (d

B re

l1

W)

0 500 1000 1500

Frequency (Hz)

50

0

minus50

minus100

minus150

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

100

50

0

minus50

minus100

minus150

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

50

0

minus50

minus100

minus150

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus80

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 10 Comparison of radiated sound power results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 15

minus50

minus100

minus150

minus250

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus220

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus50

minus100

minus150

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 11 Comparison of panel displacement PSD results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b)overall panel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

16 Shock and Vibration

Table 1 Physical properties of composite panels

Variable Description units Case 1 Case 2 Case 3 Case 4119886 Length m 183 11683 0348 10119887 Width m 122 11683 03048 03120582 Aspect ratio (119886119887) 15 10 114 333ℎ1 ℎ3

Face layers thickness mm 0406 0506 0762 35ℎ2

Core layer thickness mm 64 1905 0254 051198641 1198643

Face layers Young modulus Pa 6898 times 1010

71 times 1010

6898 times 1010

66 times 1010

1198642

Core layer Young modulus Pa 457 times 105

457 times 105

267 times 106

26 times 108

]1 ]3

Face layers Poissonrsquos ratio 033 033 03 033]2

Core layer Poissonrsquos ratio 04 04 049 031205881 1205883

Face layers density kgmminus3 2768 2700 2740 26801205882

Core layer density kgmminus3 122 4806 999 1680119898 Total mass kg 676 499 047 588

Table 2 Physical properties of isotropic panels

Variable Description units Case 3 Case 4 Case 5 Case 6119886 Length m 0348 10 0348 10119887 Width m 03048 03 03048 03120582 Aspect ratio (119886119887) 114 333 114 333ℎ119901

Thickness mm 1 1 15 4119864119901

Young modulus Pa 724 times 1010

724 times 1010

724 times 1010

724 times 1010

]119901

Poissonrsquos ratio 033 033 033 033120588119901

Density kg mminus3 2800 2800 2800 2800119898 Total mass kg 030 084 045 336

Table 3 Flight test conditions and air properties

Variable Description units Value119872 Mach number 08alt Altitude m 13106119888 Speed of sound m sminus1 295119880infin

Flow speed m sminus1 23606120588 Air density kgmminus3 02622] Kinematic viscosity m2 sminus1 5422 times 10

minus5

119879 Temperature K 21665Re119909

Reynolds number 2917 times 107

1198880

Speed of sound m sminus1 3401205880

Internal air density kgmminus3 142

higher amplification of localized frequencies Therefore theface layers thickness increase should be performed carefullyby an accompanied increase in the core layer thickness whilesimultaneously considering the weight constraints for thespecific panels applications

A more detailed analysis will be required in order toformulate final conclusions about the optimum solutions forthe panel in the context of aircraft application For that theinsulation materials would also be considered to account forthe respective added weight and effect in the sound radi-ation signature However typically the acoustic insulationmaterials will only have a higher impact in decreasing the

Table 4 First 10 natural frequencies for the composite panels

Modes 119891 Hz(Case 1)

119891 Hz(Case 2)

119891 Hz(Case 3)

119891 Hz(Case 4)

1 2365 10588 4874 126582 4548 26470 11222 157933 7276 42352 13148 210194 8186 52941 19496 283355 9459 68823 21800 377416 13097 89999 26939 474967 13279 95293 30075 492388 15462 105881 33287 506329 17645 132351 36612 5585710 18191 137645 43866 62825

sound levels at higher frequency ranges and will thereforenot improve significantly the problem of higher noise levelsin the low frequency range particularly observed in theisotropic solutions As such the composite panel solutionsmay appear to be a promising replacement to the isotropicpanels especially panels like Cases 3 and 4 which are alreadycharacterized by lower levels of both RSP and 119878

119908119908in the

low frequency range In further analysis it will also beimportant to carefully address the noise amplification atisolated frequencies by identifying a good combination of

Shock and Vibration 17

Table 5 First 10 natural frequencies for the isotropic panels

Modes 119891 Hz(Case 3)

119891 Hz(Case 4)

119891 Hz(Case 5)

119891 Hz(Case 6)

1 4646 2958 6969 118322 10697 3691 16046 147643 12534 4912 18801 196494 18585 6622 27877 264885 20782 8820 31173 352826 25680 1110 38520 443997 28670 11507 43004 460298 31731 11832 47596 473309 34901 13054 52351 5221510 41816 14683 62723 58731

core and face layers in terms of both materials selection anddimensions

4 Conclusions

This study presents a mathematical method and frameworkon composite panels radiated sound and vibrationThepanelsare considered excited either by turbulent flow or by randomnoiseThe turbulent flow corresponds to the typical excitationof an aircraft panel at flight cruise conditions with a Machnumber of 08 The effect of type of panel of panel dimen-sions and of panel thickness are evaluated The turbulentflow excitation showed to excite a higher number of panelmodes compared to random noise excitation and resulted indissimilar levels of both panel vibration and radiated soundwith relation to the random excitation This effect is morevisible for composite than for isotropic panels and thereforethe pure random noise should not be used as representativeof the turbulent flow excitation in real applications

Different properties on the composite panels are shown toyield dissimilar effects in the noise and vibration predictionsThe effect of the panel size specifically its length and widthis noted to have a significant effect on the panel performancein terms of noise radiation and vibration levels Overallsmaller composite panels or panels with higher aspect ratioproduced lower levels of radiated sound and vibration thanlonger and wider composite panels Compared to isotropicpanels composite panels analyzed are shown to radiate lowernoise levels with the exception of specific natural frequencieswhich are amplified near 300Hz and 1000Hz It is shown thatthis amplification can be significant for composite panels withthicker face layers and thinner core layer and thus resultingin unwanted effect of increased sound radiated Despitethis effect this drawback may potentially be corrected byincreasing the sandwich core layer thickness Increasing thecore layer thickness will impact the panel weight and hencethe composite panels can represent a heavier solution com-pared to the isotropic panels examined On the other handthis approach may be beneficial in the design of compositepanels in the context of aircraft panel applications sincethese panels may require less acoustic insulation material

added to the aircraft walls as they already represent animproved design solution Furthermore acoustic insulationmaterials have a higher impact in decreasing sound levelsat the high frequency ranges and therefore it will notsignificantly improve the problem of higher noise levels at thelow frequency range particularly observed in the isotropicsolutions In this context advanced and optimized compositepanel configurations may be a promising replacement to thetraditional isotropic panels for aerospace applications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is grateful for the support of the National Scienceand Engineering Research Council of Canada

References

[1] J Rocha and D Palumbo ldquoOn the sensitivity of sound powerradiated by aircraft panels to turbulent boundary layer param-etersrdquo Journal of Sound and Vibration vol 331 no 21 pp 4785ndash4806 2012

[2] J Rocha A Suleman and F Lau ldquoPrediction of turbulentboundary layer induced noise in the cabin of a BWB aircraftrdquoShock and Vibration vol 19 no 4 pp 693ndash705 2012

[3] J da Rocha A Suleman and F Lau ldquoFlow-induced noise andvibration in aircraft cylindrical cabins closed-form analyticalmodel validationrdquo Journal of Vibration and Acoustics vol 133no 5 Article ID 051013 9 pages 2011

[4] J da Rocha A Suleman and F Lau ldquoPrediction of flow-inducednoise in transport vehicles development and validation of acoupled structuralmdashacoustic analytical frameworkrdquo CanadianAcousticsmdashAcoustique Canadienne vol 37 no 4 pp 13ndash292009

[5] G Corcos ldquoThe structure of the turbulent pressure field inboundary layer flowsrdquo Journal of Fluid Mechanics vol 18 pp353ndash378 1964

[6] G Corcos ldquoResolution of pressure in turbulencerdquo Journal of theAcoustical Society of America vol 35 no 2 pp 192ndash199 1963

[7] B Efimtsov ldquoCharacteristics of the field of turbulent wallpressure fluctuations at large Reynolds numbersrdquo Soviet PhysicsAcoustics vol 28 no 4 pp 289ndash292 1982

[8] J Reddy Mechanics of Laminated Composite Plates and ShellsTheory and Analysis CRC Press Boca Raton Fla USA 1945

[9] J Whitney Structural Analysis of Laminated Anisotropic PlatesTechnomic Lancaster Pa USA 1987

[10] C L Dym and M A Lang ldquoTransmission of sound throughsandwich panelsrdquo Journal of the Acoustical Society of Americavol 56 no 5 pp 1523ndash1532 1974

[11] C L Dym C S Ventres and M A Lang ldquoTransmission ofsound through sandwich panels a reconsiderationrdquo Journal ofthe Acoustical Society of America vol 59 no 2 pp 364ndash3671976

[12] S Narayanan andR L Shanbhag ldquoSound transmission througha damped sandwich panelrdquo Journal of Sound and Vibration vol80 no 3 pp 315ndash327 1982

18 Shock and Vibration

[13] C L Dym and D C Lang ldquoTransmission loss of dampedasymmetric sandwich panels with orthotropic coresrdquo Journal ofSound and Vibration vol 88 no 3 pp 299ndash319 1983

[14] S Ghinet N Atalla and H Osman ldquoDiffuse field transmissioninto infinite sandwich composite and laminate composite cylin-dersrdquo Journal of Sound and Vibration vol 289 no 4-5 pp 745ndash778 2006

[15] J Mendler ldquoComputational tool for the acoustic analysis ofsandwich panelsrdquo in Proceedings of the European Conference onNoise Control (EURO-NOISE rsquo95) Lyon France 1995

[16] O Foin J Nicolas andNAtalla ldquoAn efficient tool for predictingthe structural acoustic and vibration response of sandwichplates in light or heavy fluidrdquo Applied Acoustics vol 57 no 3pp 213ndash242 1999

[17] S Assaf and M Guerich ldquoNumerical prediction of noisetransmission loss through viscoelastically damped sandwichplatesrdquo Journal of Sandwich Structures andMaterials vol 10 no5 pp 359ndash384 2008

[18] S Assaf M Guerich and P Cuvelier ldquoVibration and acousticresponse of damped sandwich panels immersed in a light orheavy fluidrdquo Computers and Structures vol 88 pp 359ndash3842010

[19] R Zhou and M J Crocker ldquoBoundary element analyses forsound transmission loss of panelsrdquo Journal of the AcousticalSociety of America vol 127 no 2 pp 829ndash840 2010

[20] K Amichi N Atalla and R Ruokolainen ldquoA new 3D finite ele-ment sandwich plate for predicting the vibroacoustic responseof laminated steel panelsrdquo Finite Elements in Analysis andDesign vol 46 no 12 pp 1131ndash1145 2010

[21] F Franco S de Rosa and T Polito ldquoFinite element inves-tigations on the vibroacoustic performance of plane plateswith random stiffnessrdquo Mechanics of Advanced Materials andStructures vol 18 no 7 pp 484ndash497 2011

[22] Y Kim ldquoIdentification of sound transmission characteristicsof honeycomb sandwich panels using hybrid analyticalone-dimensional finite element methodrdquo in Proceedings of the35th International Congress and Exposition on Noise ControlEngineering (INTER-NOISE rsquo06) HonoluluHawaii USA 2006

[23] F Fahy Sound and Structural Vibration Radiation Transmissionand Response Academic Press London UK 1985

[24] J Blevins Formulas for Natural Frequency and Mode ShapeKrieger Publishing Malabar Fla USA 2001

[25] D Newland An Introduction to Random Vibration and SpectralAnalysis Longman New York NY USA 1984

[26] Y Lin Probabilistic Theory of Structural Dynamics McGraw-Hill Book New York NY USA 1967

[27] C E Wallace ldquoRadiation resistance of a rectangular panelrdquoTheJournal of the Acoustical Society of America vol 51 no 3 pp946ndash952 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

Control Scienceand Engineering

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RotatingMachinery

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

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Advances inOptoElectronics

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

Shock and Vibration 7

0 500 1000 1500Frequency (Hz)

minus150

minus100

minus50

0

50

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

(b)

0 500 1000 1500

Frequency (Hz)

minus120

minus100

minus80

minus60

minus40

minus20

0

20

40

Radi

ated

soun

d po

wer

(dB

rel1

W)

(c)

0 500 1000 1500

Frequency (Hz)

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

(d)

0 500 1000 1500

Frequency (Hz)

minus140

minus120

minus100

minus80

minus60

minus40

minus20

0

20

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 1 composite TBLCase 2 composite TBL

Case 3 composite TBLCase 4 composite TBL

(e)

0 500 1000 1500

Frequency (Hz)

minus80

minus60

minus40

minus20

0

20

40

60

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 1 composite randomCase 2 composite random

Case 3 composite randomCase 4 composite random

(f)

Figure 4 Radiated sound power results for the composite panels (a) and (b) overall panel RSP (c) and (d) RSPnear surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) RSP near location (1199092 1199102) = (03119886 02119887)

position for the accurate prediction of sound and vibrationand it highlights the desired sensitivity of the developedmathematical framework to this factor Figure 5 shows thedisplacement PSD results associated with the same panels

and excitations as for the RSP results displayed in Figure 4Comparing Figures 4 and 5 it is observed that structuraldisplacement and sound radiation levels have different curvesfor the same excitation and thus panel displacement PSD

8 Shock and Vibration

minus250

minus200

minus150

minus100

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

minus200

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(b)

minus200

minus180

minus140

minus160

minus120

minus100

minus80

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(c)

minus220

minus200

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(d)

minus220

minus200

minus180

minus140

minus160

minus120

minus100

minus80

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

Case 1 composite TBLCase 2 composite TBL

Case 3 composite TBLCase 4 composite TBL

(e)

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

Case 1 composite randomCase 2 composite random

Case 3 composite randomCase 4 composite random

(f)

Figure 5 Panel displacement PSD results for the composite panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location(1199091 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

curve predictionmay not be fully representative of RSP curveeven though they would be expected to follow similar trendsIn addition as similar to RSP results it is concluded that 119878

119908119908

predictions are dissimilar whether one is assuming TBL or

random excitation Case 1 panel shows higher values of bothoverall and local plate displacement PSD predictions

The results presented here are useful in the context of theprediction of noise radiation by composite panel solutions

Shock and Vibration 9

and its possible application for example in aerospace indus-try Based on the obtained results if one would choose whichcomposite panel solution is most desirable for the purposeof manufacturing an aircraft fuselage panel a smaller panel(Case 3) or a panel with a higher aspect ratio (Case 4) wouldbe preferable for overall improved results Longer and widerpanels (ie Cases 1 and 2) are shown to greatly amplify thelow frequency range However one would need to designthe panel appropriately and with the targeted dimensionssuch that the amplification of noise and vibration at certainfrequencies would be avoided For instance Case 3 panelexhibits a general improved behaviour in terms of noiselevels but simultaneously it amplifies noise at frequenciesnear 300Hz and 1000Hz while Case 4 panel amplifies resultsnear 300Hz Comparing the size of these two panels thesurface area covered by one Case 1 panel corresponds toapproximately the surface area covered by twenty Case 3panels with the twenty Case 3 panels representing approx-imately 94 kg of total mass compared to the 676 kg of asingle Case 1 panel As such an improvement in the noiselevels comes as a tradeoff in terms of increased weightAdditionally it is shown that panels with thicker face layersand thinner core layer specifically referring to Cases 3 and 4have lower noise and vibration levels than panels with thinnerface layers and thicker core layer referring to Cases 1 and 2However as mentioned earlier Cases 3 and 4 panels withsmaller dimensions and thicker face layers have the problemof noise amplification at certain frequencies Therefore apanel which combines the properties of being shorter with athicker core layer versus thinner face layers may potentiallyshow an improved behaviour for reduced vibration and noiseradiation as well as lower noise amplification at isolatedfrequencies

32 Isotropic Panels This section addresses the RSP and 119878119908119908

predictions for the isotropic panels analyzed and results upto 1500Hz are shown in Figures 6 and 7 Four aluminumisotropic panels are investigated considering TBL or randomexcitation as follows isotropic panels Cases 3 and 4 have thesame dimensions 119886 and 119887 as the composite panels Cases3 and 4 respectively isotropic panels Cases 5 and 6 areconsidered with same dimensions 119886 and 119887 as Cases 3 and4 respectively but with increased thicknesses This approachis taken with the purpose of posterior comparison betweencomposite and isotropic panelsrsquo behaviours In this contextisotropic panels Cases 3 and 4 are both 1mm thick whileisotropic panels Cases 4 and 5 have thicknesses of 15mm and4mm respectively

As noted in Figures 6 and 7 the increase of panelthickness results in an expected shift in the panel naturalfrequencies to higher frequencies as well as a shift in themodes amplified in both RSP and 119878

119908119908results This effect is

observed in the TBL and random excitations Furthermore itis found that panels with higher aspect ratio (ie Cases 4 and6) are generally characterized by having lower levels of soundradiation and vibration As similarly noted for the compositepanels random and TBL excitations provide different RSPand 119878

119908119908predictions and overall values differ from point

values and hence it is important to specify the location in the

panel where the measurementspredictions are being takenIt is also noted that the random excitation remains as theexcitation causing higher noise and vibration levels for allcases at the higher frequency range This is also an expectedeffect as the TBL excitation decreases towards the higherfrequencies while the random excitation provides a constantexcitation over the frequency spectrum

33 Composite versus Isotropic Panels As one of the mainobjectives of this study this section presents a discussionof results for the behavior of composite sandwich panelsin comparison with the traditional isotropic panels Thefollowing four figures are shown to this end Figures 8 and9 compare results for composite panel Case 3 with isotropicpanels Cases 3 and 5 while Figures 10 and 11 comparecomposite panel Case 4 with isotropic panels Cases 4 and 6

By comparing Cases 3 and 5 panels as shown fromFigures 8 and 9 it is noted that composite panel Case 3exhibits an enhanced behaviour in comparison with isotropicpanels Cases 3 and 5 both in RSP and displacement PSDlevels However one should also note that Case 3 compositepanel shows higher levels of noise and vibration at specificisolated frequencies approximately near 300Hz and 1000Hzfor the overall RSP and for position 1 RSP results as shown inFigure 8 Considering the panel displacement PSD results inFigure 9 extra frequencies above 1000Hz are also amplifiedespecially at position 2 for both TBL and random excitationsAs described in Table 1 the composite panel Case 3 has athin core layer thus the unwanted amplification at thesefrequencies may be attenuated by increasing the core layerthickness However this modification on the panel thicknesswill increase the original panel weight of 047 kg whichalready represents heavier solution compared to Cases 3and 5 isotropic panels with weights of 03 kg and 045 kgrespectively Although the panel weight will increase if thischange is performed making this modification will in alater phase require less added acoustic insulation material tothe aircraft walls in the context of aircraft fuselage panelsrsquoapplication As such the composite panelmay represent witha slight increase in the panel weight an optimized solutionfor both sound radiation and structural vibration in almostall the frequency spectra analyzed

Similar conclusions are obtained from Figures 10 and 11dedicated to the comparison of Cases 4 and 6 panels thatis the composite panel represents an improved solution foralmost all the frequency ranges studied By analyzing thegeometric differences between composite panels Cases 3 and4 Case 4 has thicker face layers and is longer than Case 3panel having a much higher aspect ratio The amplificationof noise and vibration levels at specific frequencies is notedas in Case 3 composite panel Since face layers are thickerand this panel has a higher aspect ratio this effect is morevisible in composite Case 4 than in composite Case 3 Asobserved in results from Case 3 this could be enhancedby implementing an increase of the core layer thicknessWhile the increase in the face layer thickness is shown todecrease the levels of noise and vibration in both overalland localized predictions it also can have an impact in the

10 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 6 Radiated sound power results for the isotropic panels (a) and (b) overall panel RSP (c) and (d) RSP near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) RSP near location (1199092 1199102) = (03119886 02119887)

Shock and Vibration 11

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

(a)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(b)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(c)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus50

minus100

minus150

minus200

(f)

Figure 7 Panel displacement PSD results for the isotropic panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) 119878119908119908

near location (1199092 1199102) = (03119886 02119887)

12 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus100

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus100

minus80

minus60

minus40

minus20

0

20

40

60

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 8 Comparison of radiated sound power results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 13

0 500 1000 1500

Frequency (Hz)

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus50

minus150

minus100

minus200

minus250

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 9 Comparison of panel displacement PSD results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

14 Shock and VibrationRa

diat

ed so

und

pow

er (d

B re

l1

W)

0 500 1000 1500

Frequency (Hz)

50

0

minus50

minus100

minus150

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

100

50

0

minus50

minus100

minus150

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

50

0

minus50

minus100

minus150

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus80

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 10 Comparison of radiated sound power results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 15

minus50

minus100

minus150

minus250

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus220

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus50

minus100

minus150

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 11 Comparison of panel displacement PSD results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b)overall panel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

16 Shock and Vibration

Table 1 Physical properties of composite panels

Variable Description units Case 1 Case 2 Case 3 Case 4119886 Length m 183 11683 0348 10119887 Width m 122 11683 03048 03120582 Aspect ratio (119886119887) 15 10 114 333ℎ1 ℎ3

Face layers thickness mm 0406 0506 0762 35ℎ2

Core layer thickness mm 64 1905 0254 051198641 1198643

Face layers Young modulus Pa 6898 times 1010

71 times 1010

6898 times 1010

66 times 1010

1198642

Core layer Young modulus Pa 457 times 105

457 times 105

267 times 106

26 times 108

]1 ]3

Face layers Poissonrsquos ratio 033 033 03 033]2

Core layer Poissonrsquos ratio 04 04 049 031205881 1205883

Face layers density kgmminus3 2768 2700 2740 26801205882

Core layer density kgmminus3 122 4806 999 1680119898 Total mass kg 676 499 047 588

Table 2 Physical properties of isotropic panels

Variable Description units Case 3 Case 4 Case 5 Case 6119886 Length m 0348 10 0348 10119887 Width m 03048 03 03048 03120582 Aspect ratio (119886119887) 114 333 114 333ℎ119901

Thickness mm 1 1 15 4119864119901

Young modulus Pa 724 times 1010

724 times 1010

724 times 1010

724 times 1010

]119901

Poissonrsquos ratio 033 033 033 033120588119901

Density kg mminus3 2800 2800 2800 2800119898 Total mass kg 030 084 045 336

Table 3 Flight test conditions and air properties

Variable Description units Value119872 Mach number 08alt Altitude m 13106119888 Speed of sound m sminus1 295119880infin

Flow speed m sminus1 23606120588 Air density kgmminus3 02622] Kinematic viscosity m2 sminus1 5422 times 10

minus5

119879 Temperature K 21665Re119909

Reynolds number 2917 times 107

1198880

Speed of sound m sminus1 3401205880

Internal air density kgmminus3 142

higher amplification of localized frequencies Therefore theface layers thickness increase should be performed carefullyby an accompanied increase in the core layer thickness whilesimultaneously considering the weight constraints for thespecific panels applications

A more detailed analysis will be required in order toformulate final conclusions about the optimum solutions forthe panel in the context of aircraft application For that theinsulation materials would also be considered to account forthe respective added weight and effect in the sound radi-ation signature However typically the acoustic insulationmaterials will only have a higher impact in decreasing the

Table 4 First 10 natural frequencies for the composite panels

Modes 119891 Hz(Case 1)

119891 Hz(Case 2)

119891 Hz(Case 3)

119891 Hz(Case 4)

1 2365 10588 4874 126582 4548 26470 11222 157933 7276 42352 13148 210194 8186 52941 19496 283355 9459 68823 21800 377416 13097 89999 26939 474967 13279 95293 30075 492388 15462 105881 33287 506329 17645 132351 36612 5585710 18191 137645 43866 62825

sound levels at higher frequency ranges and will thereforenot improve significantly the problem of higher noise levelsin the low frequency range particularly observed in theisotropic solutions As such the composite panel solutionsmay appear to be a promising replacement to the isotropicpanels especially panels like Cases 3 and 4 which are alreadycharacterized by lower levels of both RSP and 119878

119908119908in the

low frequency range In further analysis it will also beimportant to carefully address the noise amplification atisolated frequencies by identifying a good combination of

Shock and Vibration 17

Table 5 First 10 natural frequencies for the isotropic panels

Modes 119891 Hz(Case 3)

119891 Hz(Case 4)

119891 Hz(Case 5)

119891 Hz(Case 6)

1 4646 2958 6969 118322 10697 3691 16046 147643 12534 4912 18801 196494 18585 6622 27877 264885 20782 8820 31173 352826 25680 1110 38520 443997 28670 11507 43004 460298 31731 11832 47596 473309 34901 13054 52351 5221510 41816 14683 62723 58731

core and face layers in terms of both materials selection anddimensions

4 Conclusions

This study presents a mathematical method and frameworkon composite panels radiated sound and vibrationThepanelsare considered excited either by turbulent flow or by randomnoiseThe turbulent flow corresponds to the typical excitationof an aircraft panel at flight cruise conditions with a Machnumber of 08 The effect of type of panel of panel dimen-sions and of panel thickness are evaluated The turbulentflow excitation showed to excite a higher number of panelmodes compared to random noise excitation and resulted indissimilar levels of both panel vibration and radiated soundwith relation to the random excitation This effect is morevisible for composite than for isotropic panels and thereforethe pure random noise should not be used as representativeof the turbulent flow excitation in real applications

Different properties on the composite panels are shown toyield dissimilar effects in the noise and vibration predictionsThe effect of the panel size specifically its length and widthis noted to have a significant effect on the panel performancein terms of noise radiation and vibration levels Overallsmaller composite panels or panels with higher aspect ratioproduced lower levels of radiated sound and vibration thanlonger and wider composite panels Compared to isotropicpanels composite panels analyzed are shown to radiate lowernoise levels with the exception of specific natural frequencieswhich are amplified near 300Hz and 1000Hz It is shown thatthis amplification can be significant for composite panels withthicker face layers and thinner core layer and thus resultingin unwanted effect of increased sound radiated Despitethis effect this drawback may potentially be corrected byincreasing the sandwich core layer thickness Increasing thecore layer thickness will impact the panel weight and hencethe composite panels can represent a heavier solution com-pared to the isotropic panels examined On the other handthis approach may be beneficial in the design of compositepanels in the context of aircraft panel applications sincethese panels may require less acoustic insulation material

added to the aircraft walls as they already represent animproved design solution Furthermore acoustic insulationmaterials have a higher impact in decreasing sound levelsat the high frequency ranges and therefore it will notsignificantly improve the problem of higher noise levels at thelow frequency range particularly observed in the isotropicsolutions In this context advanced and optimized compositepanel configurations may be a promising replacement to thetraditional isotropic panels for aerospace applications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is grateful for the support of the National Scienceand Engineering Research Council of Canada

References

[1] J Rocha and D Palumbo ldquoOn the sensitivity of sound powerradiated by aircraft panels to turbulent boundary layer param-etersrdquo Journal of Sound and Vibration vol 331 no 21 pp 4785ndash4806 2012

[2] J Rocha A Suleman and F Lau ldquoPrediction of turbulentboundary layer induced noise in the cabin of a BWB aircraftrdquoShock and Vibration vol 19 no 4 pp 693ndash705 2012

[3] J da Rocha A Suleman and F Lau ldquoFlow-induced noise andvibration in aircraft cylindrical cabins closed-form analyticalmodel validationrdquo Journal of Vibration and Acoustics vol 133no 5 Article ID 051013 9 pages 2011

[4] J da Rocha A Suleman and F Lau ldquoPrediction of flow-inducednoise in transport vehicles development and validation of acoupled structuralmdashacoustic analytical frameworkrdquo CanadianAcousticsmdashAcoustique Canadienne vol 37 no 4 pp 13ndash292009

[5] G Corcos ldquoThe structure of the turbulent pressure field inboundary layer flowsrdquo Journal of Fluid Mechanics vol 18 pp353ndash378 1964

[6] G Corcos ldquoResolution of pressure in turbulencerdquo Journal of theAcoustical Society of America vol 35 no 2 pp 192ndash199 1963

[7] B Efimtsov ldquoCharacteristics of the field of turbulent wallpressure fluctuations at large Reynolds numbersrdquo Soviet PhysicsAcoustics vol 28 no 4 pp 289ndash292 1982

[8] J Reddy Mechanics of Laminated Composite Plates and ShellsTheory and Analysis CRC Press Boca Raton Fla USA 1945

[9] J Whitney Structural Analysis of Laminated Anisotropic PlatesTechnomic Lancaster Pa USA 1987

[10] C L Dym and M A Lang ldquoTransmission of sound throughsandwich panelsrdquo Journal of the Acoustical Society of Americavol 56 no 5 pp 1523ndash1532 1974

[11] C L Dym C S Ventres and M A Lang ldquoTransmission ofsound through sandwich panels a reconsiderationrdquo Journal ofthe Acoustical Society of America vol 59 no 2 pp 364ndash3671976

[12] S Narayanan andR L Shanbhag ldquoSound transmission througha damped sandwich panelrdquo Journal of Sound and Vibration vol80 no 3 pp 315ndash327 1982

18 Shock and Vibration

[13] C L Dym and D C Lang ldquoTransmission loss of dampedasymmetric sandwich panels with orthotropic coresrdquo Journal ofSound and Vibration vol 88 no 3 pp 299ndash319 1983

[14] S Ghinet N Atalla and H Osman ldquoDiffuse field transmissioninto infinite sandwich composite and laminate composite cylin-dersrdquo Journal of Sound and Vibration vol 289 no 4-5 pp 745ndash778 2006

[15] J Mendler ldquoComputational tool for the acoustic analysis ofsandwich panelsrdquo in Proceedings of the European Conference onNoise Control (EURO-NOISE rsquo95) Lyon France 1995

[16] O Foin J Nicolas andNAtalla ldquoAn efficient tool for predictingthe structural acoustic and vibration response of sandwichplates in light or heavy fluidrdquo Applied Acoustics vol 57 no 3pp 213ndash242 1999

[17] S Assaf and M Guerich ldquoNumerical prediction of noisetransmission loss through viscoelastically damped sandwichplatesrdquo Journal of Sandwich Structures andMaterials vol 10 no5 pp 359ndash384 2008

[18] S Assaf M Guerich and P Cuvelier ldquoVibration and acousticresponse of damped sandwich panels immersed in a light orheavy fluidrdquo Computers and Structures vol 88 pp 359ndash3842010

[19] R Zhou and M J Crocker ldquoBoundary element analyses forsound transmission loss of panelsrdquo Journal of the AcousticalSociety of America vol 127 no 2 pp 829ndash840 2010

[20] K Amichi N Atalla and R Ruokolainen ldquoA new 3D finite ele-ment sandwich plate for predicting the vibroacoustic responseof laminated steel panelsrdquo Finite Elements in Analysis andDesign vol 46 no 12 pp 1131ndash1145 2010

[21] F Franco S de Rosa and T Polito ldquoFinite element inves-tigations on the vibroacoustic performance of plane plateswith random stiffnessrdquo Mechanics of Advanced Materials andStructures vol 18 no 7 pp 484ndash497 2011

[22] Y Kim ldquoIdentification of sound transmission characteristicsof honeycomb sandwich panels using hybrid analyticalone-dimensional finite element methodrdquo in Proceedings of the35th International Congress and Exposition on Noise ControlEngineering (INTER-NOISE rsquo06) HonoluluHawaii USA 2006

[23] F Fahy Sound and Structural Vibration Radiation Transmissionand Response Academic Press London UK 1985

[24] J Blevins Formulas for Natural Frequency and Mode ShapeKrieger Publishing Malabar Fla USA 2001

[25] D Newland An Introduction to Random Vibration and SpectralAnalysis Longman New York NY USA 1984

[26] Y Lin Probabilistic Theory of Structural Dynamics McGraw-Hill Book New York NY USA 1967

[27] C E Wallace ldquoRadiation resistance of a rectangular panelrdquoTheJournal of the Acoustical Society of America vol 51 no 3 pp946ndash952 1972

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AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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RotatingMachinery

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

8 Shock and Vibration

minus250

minus200

minus150

minus100

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

minus200

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(b)

minus200

minus180

minus140

minus160

minus120

minus100

minus80

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(c)

minus220

minus200

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(d)

minus220

minus200

minus180

minus140

minus160

minus120

minus100

minus80

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

Case 1 composite TBLCase 2 composite TBL

Case 3 composite TBLCase 4 composite TBL

(e)

minus180

minus140

minus160

minus120

minus100

minus80

minus60

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

Case 1 composite randomCase 2 composite random

Case 3 composite randomCase 4 composite random

(f)

Figure 5 Panel displacement PSD results for the composite panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location(1199091 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

curve predictionmay not be fully representative of RSP curveeven though they would be expected to follow similar trendsIn addition as similar to RSP results it is concluded that 119878

119908119908

predictions are dissimilar whether one is assuming TBL or

random excitation Case 1 panel shows higher values of bothoverall and local plate displacement PSD predictions

The results presented here are useful in the context of theprediction of noise radiation by composite panel solutions

Shock and Vibration 9

and its possible application for example in aerospace indus-try Based on the obtained results if one would choose whichcomposite panel solution is most desirable for the purposeof manufacturing an aircraft fuselage panel a smaller panel(Case 3) or a panel with a higher aspect ratio (Case 4) wouldbe preferable for overall improved results Longer and widerpanels (ie Cases 1 and 2) are shown to greatly amplify thelow frequency range However one would need to designthe panel appropriately and with the targeted dimensionssuch that the amplification of noise and vibration at certainfrequencies would be avoided For instance Case 3 panelexhibits a general improved behaviour in terms of noiselevels but simultaneously it amplifies noise at frequenciesnear 300Hz and 1000Hz while Case 4 panel amplifies resultsnear 300Hz Comparing the size of these two panels thesurface area covered by one Case 1 panel corresponds toapproximately the surface area covered by twenty Case 3panels with the twenty Case 3 panels representing approx-imately 94 kg of total mass compared to the 676 kg of asingle Case 1 panel As such an improvement in the noiselevels comes as a tradeoff in terms of increased weightAdditionally it is shown that panels with thicker face layersand thinner core layer specifically referring to Cases 3 and 4have lower noise and vibration levels than panels with thinnerface layers and thicker core layer referring to Cases 1 and 2However as mentioned earlier Cases 3 and 4 panels withsmaller dimensions and thicker face layers have the problemof noise amplification at certain frequencies Therefore apanel which combines the properties of being shorter with athicker core layer versus thinner face layers may potentiallyshow an improved behaviour for reduced vibration and noiseradiation as well as lower noise amplification at isolatedfrequencies

32 Isotropic Panels This section addresses the RSP and 119878119908119908

predictions for the isotropic panels analyzed and results upto 1500Hz are shown in Figures 6 and 7 Four aluminumisotropic panels are investigated considering TBL or randomexcitation as follows isotropic panels Cases 3 and 4 have thesame dimensions 119886 and 119887 as the composite panels Cases3 and 4 respectively isotropic panels Cases 5 and 6 areconsidered with same dimensions 119886 and 119887 as Cases 3 and4 respectively but with increased thicknesses This approachis taken with the purpose of posterior comparison betweencomposite and isotropic panelsrsquo behaviours In this contextisotropic panels Cases 3 and 4 are both 1mm thick whileisotropic panels Cases 4 and 5 have thicknesses of 15mm and4mm respectively

As noted in Figures 6 and 7 the increase of panelthickness results in an expected shift in the panel naturalfrequencies to higher frequencies as well as a shift in themodes amplified in both RSP and 119878

119908119908results This effect is

observed in the TBL and random excitations Furthermore itis found that panels with higher aspect ratio (ie Cases 4 and6) are generally characterized by having lower levels of soundradiation and vibration As similarly noted for the compositepanels random and TBL excitations provide different RSPand 119878

119908119908predictions and overall values differ from point

values and hence it is important to specify the location in the

panel where the measurementspredictions are being takenIt is also noted that the random excitation remains as theexcitation causing higher noise and vibration levels for allcases at the higher frequency range This is also an expectedeffect as the TBL excitation decreases towards the higherfrequencies while the random excitation provides a constantexcitation over the frequency spectrum

33 Composite versus Isotropic Panels As one of the mainobjectives of this study this section presents a discussionof results for the behavior of composite sandwich panelsin comparison with the traditional isotropic panels Thefollowing four figures are shown to this end Figures 8 and9 compare results for composite panel Case 3 with isotropicpanels Cases 3 and 5 while Figures 10 and 11 comparecomposite panel Case 4 with isotropic panels Cases 4 and 6

By comparing Cases 3 and 5 panels as shown fromFigures 8 and 9 it is noted that composite panel Case 3exhibits an enhanced behaviour in comparison with isotropicpanels Cases 3 and 5 both in RSP and displacement PSDlevels However one should also note that Case 3 compositepanel shows higher levels of noise and vibration at specificisolated frequencies approximately near 300Hz and 1000Hzfor the overall RSP and for position 1 RSP results as shown inFigure 8 Considering the panel displacement PSD results inFigure 9 extra frequencies above 1000Hz are also amplifiedespecially at position 2 for both TBL and random excitationsAs described in Table 1 the composite panel Case 3 has athin core layer thus the unwanted amplification at thesefrequencies may be attenuated by increasing the core layerthickness However this modification on the panel thicknesswill increase the original panel weight of 047 kg whichalready represents heavier solution compared to Cases 3and 5 isotropic panels with weights of 03 kg and 045 kgrespectively Although the panel weight will increase if thischange is performed making this modification will in alater phase require less added acoustic insulation material tothe aircraft walls in the context of aircraft fuselage panelsrsquoapplication As such the composite panelmay represent witha slight increase in the panel weight an optimized solutionfor both sound radiation and structural vibration in almostall the frequency spectra analyzed

Similar conclusions are obtained from Figures 10 and 11dedicated to the comparison of Cases 4 and 6 panels thatis the composite panel represents an improved solution foralmost all the frequency ranges studied By analyzing thegeometric differences between composite panels Cases 3 and4 Case 4 has thicker face layers and is longer than Case 3panel having a much higher aspect ratio The amplificationof noise and vibration levels at specific frequencies is notedas in Case 3 composite panel Since face layers are thickerand this panel has a higher aspect ratio this effect is morevisible in composite Case 4 than in composite Case 3 Asobserved in results from Case 3 this could be enhancedby implementing an increase of the core layer thicknessWhile the increase in the face layer thickness is shown todecrease the levels of noise and vibration in both overalland localized predictions it also can have an impact in the

10 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 6 Radiated sound power results for the isotropic panels (a) and (b) overall panel RSP (c) and (d) RSP near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) RSP near location (1199092 1199102) = (03119886 02119887)

Shock and Vibration 11

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

(a)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(b)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(c)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus50

minus100

minus150

minus200

(f)

Figure 7 Panel displacement PSD results for the isotropic panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) 119878119908119908

near location (1199092 1199102) = (03119886 02119887)

12 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus100

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus100

minus80

minus60

minus40

minus20

0

20

40

60

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 8 Comparison of radiated sound power results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 13

0 500 1000 1500

Frequency (Hz)

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus50

minus150

minus100

minus200

minus250

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 9 Comparison of panel displacement PSD results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

14 Shock and VibrationRa

diat

ed so

und

pow

er (d

B re

l1

W)

0 500 1000 1500

Frequency (Hz)

50

0

minus50

minus100

minus150

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

100

50

0

minus50

minus100

minus150

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

50

0

minus50

minus100

minus150

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus80

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 10 Comparison of radiated sound power results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 15

minus50

minus100

minus150

minus250

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus220

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus50

minus100

minus150

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 11 Comparison of panel displacement PSD results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b)overall panel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

16 Shock and Vibration

Table 1 Physical properties of composite panels

Variable Description units Case 1 Case 2 Case 3 Case 4119886 Length m 183 11683 0348 10119887 Width m 122 11683 03048 03120582 Aspect ratio (119886119887) 15 10 114 333ℎ1 ℎ3

Face layers thickness mm 0406 0506 0762 35ℎ2

Core layer thickness mm 64 1905 0254 051198641 1198643

Face layers Young modulus Pa 6898 times 1010

71 times 1010

6898 times 1010

66 times 1010

1198642

Core layer Young modulus Pa 457 times 105

457 times 105

267 times 106

26 times 108

]1 ]3

Face layers Poissonrsquos ratio 033 033 03 033]2

Core layer Poissonrsquos ratio 04 04 049 031205881 1205883

Face layers density kgmminus3 2768 2700 2740 26801205882

Core layer density kgmminus3 122 4806 999 1680119898 Total mass kg 676 499 047 588

Table 2 Physical properties of isotropic panels

Variable Description units Case 3 Case 4 Case 5 Case 6119886 Length m 0348 10 0348 10119887 Width m 03048 03 03048 03120582 Aspect ratio (119886119887) 114 333 114 333ℎ119901

Thickness mm 1 1 15 4119864119901

Young modulus Pa 724 times 1010

724 times 1010

724 times 1010

724 times 1010

]119901

Poissonrsquos ratio 033 033 033 033120588119901

Density kg mminus3 2800 2800 2800 2800119898 Total mass kg 030 084 045 336

Table 3 Flight test conditions and air properties

Variable Description units Value119872 Mach number 08alt Altitude m 13106119888 Speed of sound m sminus1 295119880infin

Flow speed m sminus1 23606120588 Air density kgmminus3 02622] Kinematic viscosity m2 sminus1 5422 times 10

minus5

119879 Temperature K 21665Re119909

Reynolds number 2917 times 107

1198880

Speed of sound m sminus1 3401205880

Internal air density kgmminus3 142

higher amplification of localized frequencies Therefore theface layers thickness increase should be performed carefullyby an accompanied increase in the core layer thickness whilesimultaneously considering the weight constraints for thespecific panels applications

A more detailed analysis will be required in order toformulate final conclusions about the optimum solutions forthe panel in the context of aircraft application For that theinsulation materials would also be considered to account forthe respective added weight and effect in the sound radi-ation signature However typically the acoustic insulationmaterials will only have a higher impact in decreasing the

Table 4 First 10 natural frequencies for the composite panels

Modes 119891 Hz(Case 1)

119891 Hz(Case 2)

119891 Hz(Case 3)

119891 Hz(Case 4)

1 2365 10588 4874 126582 4548 26470 11222 157933 7276 42352 13148 210194 8186 52941 19496 283355 9459 68823 21800 377416 13097 89999 26939 474967 13279 95293 30075 492388 15462 105881 33287 506329 17645 132351 36612 5585710 18191 137645 43866 62825

sound levels at higher frequency ranges and will thereforenot improve significantly the problem of higher noise levelsin the low frequency range particularly observed in theisotropic solutions As such the composite panel solutionsmay appear to be a promising replacement to the isotropicpanels especially panels like Cases 3 and 4 which are alreadycharacterized by lower levels of both RSP and 119878

119908119908in the

low frequency range In further analysis it will also beimportant to carefully address the noise amplification atisolated frequencies by identifying a good combination of

Shock and Vibration 17

Table 5 First 10 natural frequencies for the isotropic panels

Modes 119891 Hz(Case 3)

119891 Hz(Case 4)

119891 Hz(Case 5)

119891 Hz(Case 6)

1 4646 2958 6969 118322 10697 3691 16046 147643 12534 4912 18801 196494 18585 6622 27877 264885 20782 8820 31173 352826 25680 1110 38520 443997 28670 11507 43004 460298 31731 11832 47596 473309 34901 13054 52351 5221510 41816 14683 62723 58731

core and face layers in terms of both materials selection anddimensions

4 Conclusions

This study presents a mathematical method and frameworkon composite panels radiated sound and vibrationThepanelsare considered excited either by turbulent flow or by randomnoiseThe turbulent flow corresponds to the typical excitationof an aircraft panel at flight cruise conditions with a Machnumber of 08 The effect of type of panel of panel dimen-sions and of panel thickness are evaluated The turbulentflow excitation showed to excite a higher number of panelmodes compared to random noise excitation and resulted indissimilar levels of both panel vibration and radiated soundwith relation to the random excitation This effect is morevisible for composite than for isotropic panels and thereforethe pure random noise should not be used as representativeof the turbulent flow excitation in real applications

Different properties on the composite panels are shown toyield dissimilar effects in the noise and vibration predictionsThe effect of the panel size specifically its length and widthis noted to have a significant effect on the panel performancein terms of noise radiation and vibration levels Overallsmaller composite panels or panels with higher aspect ratioproduced lower levels of radiated sound and vibration thanlonger and wider composite panels Compared to isotropicpanels composite panels analyzed are shown to radiate lowernoise levels with the exception of specific natural frequencieswhich are amplified near 300Hz and 1000Hz It is shown thatthis amplification can be significant for composite panels withthicker face layers and thinner core layer and thus resultingin unwanted effect of increased sound radiated Despitethis effect this drawback may potentially be corrected byincreasing the sandwich core layer thickness Increasing thecore layer thickness will impact the panel weight and hencethe composite panels can represent a heavier solution com-pared to the isotropic panels examined On the other handthis approach may be beneficial in the design of compositepanels in the context of aircraft panel applications sincethese panels may require less acoustic insulation material

added to the aircraft walls as they already represent animproved design solution Furthermore acoustic insulationmaterials have a higher impact in decreasing sound levelsat the high frequency ranges and therefore it will notsignificantly improve the problem of higher noise levels at thelow frequency range particularly observed in the isotropicsolutions In this context advanced and optimized compositepanel configurations may be a promising replacement to thetraditional isotropic panels for aerospace applications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is grateful for the support of the National Scienceand Engineering Research Council of Canada

References

[1] J Rocha and D Palumbo ldquoOn the sensitivity of sound powerradiated by aircraft panels to turbulent boundary layer param-etersrdquo Journal of Sound and Vibration vol 331 no 21 pp 4785ndash4806 2012

[2] J Rocha A Suleman and F Lau ldquoPrediction of turbulentboundary layer induced noise in the cabin of a BWB aircraftrdquoShock and Vibration vol 19 no 4 pp 693ndash705 2012

[3] J da Rocha A Suleman and F Lau ldquoFlow-induced noise andvibration in aircraft cylindrical cabins closed-form analyticalmodel validationrdquo Journal of Vibration and Acoustics vol 133no 5 Article ID 051013 9 pages 2011

[4] J da Rocha A Suleman and F Lau ldquoPrediction of flow-inducednoise in transport vehicles development and validation of acoupled structuralmdashacoustic analytical frameworkrdquo CanadianAcousticsmdashAcoustique Canadienne vol 37 no 4 pp 13ndash292009

[5] G Corcos ldquoThe structure of the turbulent pressure field inboundary layer flowsrdquo Journal of Fluid Mechanics vol 18 pp353ndash378 1964

[6] G Corcos ldquoResolution of pressure in turbulencerdquo Journal of theAcoustical Society of America vol 35 no 2 pp 192ndash199 1963

[7] B Efimtsov ldquoCharacteristics of the field of turbulent wallpressure fluctuations at large Reynolds numbersrdquo Soviet PhysicsAcoustics vol 28 no 4 pp 289ndash292 1982

[8] J Reddy Mechanics of Laminated Composite Plates and ShellsTheory and Analysis CRC Press Boca Raton Fla USA 1945

[9] J Whitney Structural Analysis of Laminated Anisotropic PlatesTechnomic Lancaster Pa USA 1987

[10] C L Dym and M A Lang ldquoTransmission of sound throughsandwich panelsrdquo Journal of the Acoustical Society of Americavol 56 no 5 pp 1523ndash1532 1974

[11] C L Dym C S Ventres and M A Lang ldquoTransmission ofsound through sandwich panels a reconsiderationrdquo Journal ofthe Acoustical Society of America vol 59 no 2 pp 364ndash3671976

[12] S Narayanan andR L Shanbhag ldquoSound transmission througha damped sandwich panelrdquo Journal of Sound and Vibration vol80 no 3 pp 315ndash327 1982

18 Shock and Vibration

[13] C L Dym and D C Lang ldquoTransmission loss of dampedasymmetric sandwich panels with orthotropic coresrdquo Journal ofSound and Vibration vol 88 no 3 pp 299ndash319 1983

[14] S Ghinet N Atalla and H Osman ldquoDiffuse field transmissioninto infinite sandwich composite and laminate composite cylin-dersrdquo Journal of Sound and Vibration vol 289 no 4-5 pp 745ndash778 2006

[15] J Mendler ldquoComputational tool for the acoustic analysis ofsandwich panelsrdquo in Proceedings of the European Conference onNoise Control (EURO-NOISE rsquo95) Lyon France 1995

[16] O Foin J Nicolas andNAtalla ldquoAn efficient tool for predictingthe structural acoustic and vibration response of sandwichplates in light or heavy fluidrdquo Applied Acoustics vol 57 no 3pp 213ndash242 1999

[17] S Assaf and M Guerich ldquoNumerical prediction of noisetransmission loss through viscoelastically damped sandwichplatesrdquo Journal of Sandwich Structures andMaterials vol 10 no5 pp 359ndash384 2008

[18] S Assaf M Guerich and P Cuvelier ldquoVibration and acousticresponse of damped sandwich panels immersed in a light orheavy fluidrdquo Computers and Structures vol 88 pp 359ndash3842010

[19] R Zhou and M J Crocker ldquoBoundary element analyses forsound transmission loss of panelsrdquo Journal of the AcousticalSociety of America vol 127 no 2 pp 829ndash840 2010

[20] K Amichi N Atalla and R Ruokolainen ldquoA new 3D finite ele-ment sandwich plate for predicting the vibroacoustic responseof laminated steel panelsrdquo Finite Elements in Analysis andDesign vol 46 no 12 pp 1131ndash1145 2010

[21] F Franco S de Rosa and T Polito ldquoFinite element inves-tigations on the vibroacoustic performance of plane plateswith random stiffnessrdquo Mechanics of Advanced Materials andStructures vol 18 no 7 pp 484ndash497 2011

[22] Y Kim ldquoIdentification of sound transmission characteristicsof honeycomb sandwich panels using hybrid analyticalone-dimensional finite element methodrdquo in Proceedings of the35th International Congress and Exposition on Noise ControlEngineering (INTER-NOISE rsquo06) HonoluluHawaii USA 2006

[23] F Fahy Sound and Structural Vibration Radiation Transmissionand Response Academic Press London UK 1985

[24] J Blevins Formulas for Natural Frequency and Mode ShapeKrieger Publishing Malabar Fla USA 2001

[25] D Newland An Introduction to Random Vibration and SpectralAnalysis Longman New York NY USA 1984

[26] Y Lin Probabilistic Theory of Structural Dynamics McGraw-Hill Book New York NY USA 1967

[27] C E Wallace ldquoRadiation resistance of a rectangular panelrdquoTheJournal of the Acoustical Society of America vol 51 no 3 pp946ndash952 1972

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RotatingMachinery

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances inOptoElectronics

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Chemical EngineeringInternational Journal of Antennas and

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International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

Shock and Vibration 9

and its possible application for example in aerospace indus-try Based on the obtained results if one would choose whichcomposite panel solution is most desirable for the purposeof manufacturing an aircraft fuselage panel a smaller panel(Case 3) or a panel with a higher aspect ratio (Case 4) wouldbe preferable for overall improved results Longer and widerpanels (ie Cases 1 and 2) are shown to greatly amplify thelow frequency range However one would need to designthe panel appropriately and with the targeted dimensionssuch that the amplification of noise and vibration at certainfrequencies would be avoided For instance Case 3 panelexhibits a general improved behaviour in terms of noiselevels but simultaneously it amplifies noise at frequenciesnear 300Hz and 1000Hz while Case 4 panel amplifies resultsnear 300Hz Comparing the size of these two panels thesurface area covered by one Case 1 panel corresponds toapproximately the surface area covered by twenty Case 3panels with the twenty Case 3 panels representing approx-imately 94 kg of total mass compared to the 676 kg of asingle Case 1 panel As such an improvement in the noiselevels comes as a tradeoff in terms of increased weightAdditionally it is shown that panels with thicker face layersand thinner core layer specifically referring to Cases 3 and 4have lower noise and vibration levels than panels with thinnerface layers and thicker core layer referring to Cases 1 and 2However as mentioned earlier Cases 3 and 4 panels withsmaller dimensions and thicker face layers have the problemof noise amplification at certain frequencies Therefore apanel which combines the properties of being shorter with athicker core layer versus thinner face layers may potentiallyshow an improved behaviour for reduced vibration and noiseradiation as well as lower noise amplification at isolatedfrequencies

32 Isotropic Panels This section addresses the RSP and 119878119908119908

predictions for the isotropic panels analyzed and results upto 1500Hz are shown in Figures 6 and 7 Four aluminumisotropic panels are investigated considering TBL or randomexcitation as follows isotropic panels Cases 3 and 4 have thesame dimensions 119886 and 119887 as the composite panels Cases3 and 4 respectively isotropic panels Cases 5 and 6 areconsidered with same dimensions 119886 and 119887 as Cases 3 and4 respectively but with increased thicknesses This approachis taken with the purpose of posterior comparison betweencomposite and isotropic panelsrsquo behaviours In this contextisotropic panels Cases 3 and 4 are both 1mm thick whileisotropic panels Cases 4 and 5 have thicknesses of 15mm and4mm respectively

As noted in Figures 6 and 7 the increase of panelthickness results in an expected shift in the panel naturalfrequencies to higher frequencies as well as a shift in themodes amplified in both RSP and 119878

119908119908results This effect is

observed in the TBL and random excitations Furthermore itis found that panels with higher aspect ratio (ie Cases 4 and6) are generally characterized by having lower levels of soundradiation and vibration As similarly noted for the compositepanels random and TBL excitations provide different RSPand 119878

119908119908predictions and overall values differ from point

values and hence it is important to specify the location in the

panel where the measurementspredictions are being takenIt is also noted that the random excitation remains as theexcitation causing higher noise and vibration levels for allcases at the higher frequency range This is also an expectedeffect as the TBL excitation decreases towards the higherfrequencies while the random excitation provides a constantexcitation over the frequency spectrum

33 Composite versus Isotropic Panels As one of the mainobjectives of this study this section presents a discussionof results for the behavior of composite sandwich panelsin comparison with the traditional isotropic panels Thefollowing four figures are shown to this end Figures 8 and9 compare results for composite panel Case 3 with isotropicpanels Cases 3 and 5 while Figures 10 and 11 comparecomposite panel Case 4 with isotropic panels Cases 4 and 6

By comparing Cases 3 and 5 panels as shown fromFigures 8 and 9 it is noted that composite panel Case 3exhibits an enhanced behaviour in comparison with isotropicpanels Cases 3 and 5 both in RSP and displacement PSDlevels However one should also note that Case 3 compositepanel shows higher levels of noise and vibration at specificisolated frequencies approximately near 300Hz and 1000Hzfor the overall RSP and for position 1 RSP results as shown inFigure 8 Considering the panel displacement PSD results inFigure 9 extra frequencies above 1000Hz are also amplifiedespecially at position 2 for both TBL and random excitationsAs described in Table 1 the composite panel Case 3 has athin core layer thus the unwanted amplification at thesefrequencies may be attenuated by increasing the core layerthickness However this modification on the panel thicknesswill increase the original panel weight of 047 kg whichalready represents heavier solution compared to Cases 3and 5 isotropic panels with weights of 03 kg and 045 kgrespectively Although the panel weight will increase if thischange is performed making this modification will in alater phase require less added acoustic insulation material tothe aircraft walls in the context of aircraft fuselage panelsrsquoapplication As such the composite panelmay represent witha slight increase in the panel weight an optimized solutionfor both sound radiation and structural vibration in almostall the frequency spectra analyzed

Similar conclusions are obtained from Figures 10 and 11dedicated to the comparison of Cases 4 and 6 panels thatis the composite panel represents an improved solution foralmost all the frequency ranges studied By analyzing thegeometric differences between composite panels Cases 3 and4 Case 4 has thicker face layers and is longer than Case 3panel having a much higher aspect ratio The amplificationof noise and vibration levels at specific frequencies is notedas in Case 3 composite panel Since face layers are thickerand this panel has a higher aspect ratio this effect is morevisible in composite Case 4 than in composite Case 3 Asobserved in results from Case 3 this could be enhancedby implementing an increase of the core layer thicknessWhile the increase in the face layer thickness is shown todecrease the levels of noise and vibration in both overalland localized predictions it also can have an impact in the

10 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 6 Radiated sound power results for the isotropic panels (a) and (b) overall panel RSP (c) and (d) RSP near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) RSP near location (1199092 1199102) = (03119886 02119887)

Shock and Vibration 11

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

(a)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(b)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(c)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus50

minus100

minus150

minus200

(f)

Figure 7 Panel displacement PSD results for the isotropic panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) 119878119908119908

near location (1199092 1199102) = (03119886 02119887)

12 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus100

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus100

minus80

minus60

minus40

minus20

0

20

40

60

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 8 Comparison of radiated sound power results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 13

0 500 1000 1500

Frequency (Hz)

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus50

minus150

minus100

minus200

minus250

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 9 Comparison of panel displacement PSD results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

14 Shock and VibrationRa

diat

ed so

und

pow

er (d

B re

l1

W)

0 500 1000 1500

Frequency (Hz)

50

0

minus50

minus100

minus150

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

100

50

0

minus50

minus100

minus150

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

50

0

minus50

minus100

minus150

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus80

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 10 Comparison of radiated sound power results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 15

minus50

minus100

minus150

minus250

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus220

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus50

minus100

minus150

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 11 Comparison of panel displacement PSD results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b)overall panel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

16 Shock and Vibration

Table 1 Physical properties of composite panels

Variable Description units Case 1 Case 2 Case 3 Case 4119886 Length m 183 11683 0348 10119887 Width m 122 11683 03048 03120582 Aspect ratio (119886119887) 15 10 114 333ℎ1 ℎ3

Face layers thickness mm 0406 0506 0762 35ℎ2

Core layer thickness mm 64 1905 0254 051198641 1198643

Face layers Young modulus Pa 6898 times 1010

71 times 1010

6898 times 1010

66 times 1010

1198642

Core layer Young modulus Pa 457 times 105

457 times 105

267 times 106

26 times 108

]1 ]3

Face layers Poissonrsquos ratio 033 033 03 033]2

Core layer Poissonrsquos ratio 04 04 049 031205881 1205883

Face layers density kgmminus3 2768 2700 2740 26801205882

Core layer density kgmminus3 122 4806 999 1680119898 Total mass kg 676 499 047 588

Table 2 Physical properties of isotropic panels

Variable Description units Case 3 Case 4 Case 5 Case 6119886 Length m 0348 10 0348 10119887 Width m 03048 03 03048 03120582 Aspect ratio (119886119887) 114 333 114 333ℎ119901

Thickness mm 1 1 15 4119864119901

Young modulus Pa 724 times 1010

724 times 1010

724 times 1010

724 times 1010

]119901

Poissonrsquos ratio 033 033 033 033120588119901

Density kg mminus3 2800 2800 2800 2800119898 Total mass kg 030 084 045 336

Table 3 Flight test conditions and air properties

Variable Description units Value119872 Mach number 08alt Altitude m 13106119888 Speed of sound m sminus1 295119880infin

Flow speed m sminus1 23606120588 Air density kgmminus3 02622] Kinematic viscosity m2 sminus1 5422 times 10

minus5

119879 Temperature K 21665Re119909

Reynolds number 2917 times 107

1198880

Speed of sound m sminus1 3401205880

Internal air density kgmminus3 142

higher amplification of localized frequencies Therefore theface layers thickness increase should be performed carefullyby an accompanied increase in the core layer thickness whilesimultaneously considering the weight constraints for thespecific panels applications

A more detailed analysis will be required in order toformulate final conclusions about the optimum solutions forthe panel in the context of aircraft application For that theinsulation materials would also be considered to account forthe respective added weight and effect in the sound radi-ation signature However typically the acoustic insulationmaterials will only have a higher impact in decreasing the

Table 4 First 10 natural frequencies for the composite panels

Modes 119891 Hz(Case 1)

119891 Hz(Case 2)

119891 Hz(Case 3)

119891 Hz(Case 4)

1 2365 10588 4874 126582 4548 26470 11222 157933 7276 42352 13148 210194 8186 52941 19496 283355 9459 68823 21800 377416 13097 89999 26939 474967 13279 95293 30075 492388 15462 105881 33287 506329 17645 132351 36612 5585710 18191 137645 43866 62825

sound levels at higher frequency ranges and will thereforenot improve significantly the problem of higher noise levelsin the low frequency range particularly observed in theisotropic solutions As such the composite panel solutionsmay appear to be a promising replacement to the isotropicpanels especially panels like Cases 3 and 4 which are alreadycharacterized by lower levels of both RSP and 119878

119908119908in the

low frequency range In further analysis it will also beimportant to carefully address the noise amplification atisolated frequencies by identifying a good combination of

Shock and Vibration 17

Table 5 First 10 natural frequencies for the isotropic panels

Modes 119891 Hz(Case 3)

119891 Hz(Case 4)

119891 Hz(Case 5)

119891 Hz(Case 6)

1 4646 2958 6969 118322 10697 3691 16046 147643 12534 4912 18801 196494 18585 6622 27877 264885 20782 8820 31173 352826 25680 1110 38520 443997 28670 11507 43004 460298 31731 11832 47596 473309 34901 13054 52351 5221510 41816 14683 62723 58731

core and face layers in terms of both materials selection anddimensions

4 Conclusions

This study presents a mathematical method and frameworkon composite panels radiated sound and vibrationThepanelsare considered excited either by turbulent flow or by randomnoiseThe turbulent flow corresponds to the typical excitationof an aircraft panel at flight cruise conditions with a Machnumber of 08 The effect of type of panel of panel dimen-sions and of panel thickness are evaluated The turbulentflow excitation showed to excite a higher number of panelmodes compared to random noise excitation and resulted indissimilar levels of both panel vibration and radiated soundwith relation to the random excitation This effect is morevisible for composite than for isotropic panels and thereforethe pure random noise should not be used as representativeof the turbulent flow excitation in real applications

Different properties on the composite panels are shown toyield dissimilar effects in the noise and vibration predictionsThe effect of the panel size specifically its length and widthis noted to have a significant effect on the panel performancein terms of noise radiation and vibration levels Overallsmaller composite panels or panels with higher aspect ratioproduced lower levels of radiated sound and vibration thanlonger and wider composite panels Compared to isotropicpanels composite panels analyzed are shown to radiate lowernoise levels with the exception of specific natural frequencieswhich are amplified near 300Hz and 1000Hz It is shown thatthis amplification can be significant for composite panels withthicker face layers and thinner core layer and thus resultingin unwanted effect of increased sound radiated Despitethis effect this drawback may potentially be corrected byincreasing the sandwich core layer thickness Increasing thecore layer thickness will impact the panel weight and hencethe composite panels can represent a heavier solution com-pared to the isotropic panels examined On the other handthis approach may be beneficial in the design of compositepanels in the context of aircraft panel applications sincethese panels may require less acoustic insulation material

added to the aircraft walls as they already represent animproved design solution Furthermore acoustic insulationmaterials have a higher impact in decreasing sound levelsat the high frequency ranges and therefore it will notsignificantly improve the problem of higher noise levels at thelow frequency range particularly observed in the isotropicsolutions In this context advanced and optimized compositepanel configurations may be a promising replacement to thetraditional isotropic panels for aerospace applications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is grateful for the support of the National Scienceand Engineering Research Council of Canada

References

[1] J Rocha and D Palumbo ldquoOn the sensitivity of sound powerradiated by aircraft panels to turbulent boundary layer param-etersrdquo Journal of Sound and Vibration vol 331 no 21 pp 4785ndash4806 2012

[2] J Rocha A Suleman and F Lau ldquoPrediction of turbulentboundary layer induced noise in the cabin of a BWB aircraftrdquoShock and Vibration vol 19 no 4 pp 693ndash705 2012

[3] J da Rocha A Suleman and F Lau ldquoFlow-induced noise andvibration in aircraft cylindrical cabins closed-form analyticalmodel validationrdquo Journal of Vibration and Acoustics vol 133no 5 Article ID 051013 9 pages 2011

[4] J da Rocha A Suleman and F Lau ldquoPrediction of flow-inducednoise in transport vehicles development and validation of acoupled structuralmdashacoustic analytical frameworkrdquo CanadianAcousticsmdashAcoustique Canadienne vol 37 no 4 pp 13ndash292009

[5] G Corcos ldquoThe structure of the turbulent pressure field inboundary layer flowsrdquo Journal of Fluid Mechanics vol 18 pp353ndash378 1964

[6] G Corcos ldquoResolution of pressure in turbulencerdquo Journal of theAcoustical Society of America vol 35 no 2 pp 192ndash199 1963

[7] B Efimtsov ldquoCharacteristics of the field of turbulent wallpressure fluctuations at large Reynolds numbersrdquo Soviet PhysicsAcoustics vol 28 no 4 pp 289ndash292 1982

[8] J Reddy Mechanics of Laminated Composite Plates and ShellsTheory and Analysis CRC Press Boca Raton Fla USA 1945

[9] J Whitney Structural Analysis of Laminated Anisotropic PlatesTechnomic Lancaster Pa USA 1987

[10] C L Dym and M A Lang ldquoTransmission of sound throughsandwich panelsrdquo Journal of the Acoustical Society of Americavol 56 no 5 pp 1523ndash1532 1974

[11] C L Dym C S Ventres and M A Lang ldquoTransmission ofsound through sandwich panels a reconsiderationrdquo Journal ofthe Acoustical Society of America vol 59 no 2 pp 364ndash3671976

[12] S Narayanan andR L Shanbhag ldquoSound transmission througha damped sandwich panelrdquo Journal of Sound and Vibration vol80 no 3 pp 315ndash327 1982

18 Shock and Vibration

[13] C L Dym and D C Lang ldquoTransmission loss of dampedasymmetric sandwich panels with orthotropic coresrdquo Journal ofSound and Vibration vol 88 no 3 pp 299ndash319 1983

[14] S Ghinet N Atalla and H Osman ldquoDiffuse field transmissioninto infinite sandwich composite and laminate composite cylin-dersrdquo Journal of Sound and Vibration vol 289 no 4-5 pp 745ndash778 2006

[15] J Mendler ldquoComputational tool for the acoustic analysis ofsandwich panelsrdquo in Proceedings of the European Conference onNoise Control (EURO-NOISE rsquo95) Lyon France 1995

[16] O Foin J Nicolas andNAtalla ldquoAn efficient tool for predictingthe structural acoustic and vibration response of sandwichplates in light or heavy fluidrdquo Applied Acoustics vol 57 no 3pp 213ndash242 1999

[17] S Assaf and M Guerich ldquoNumerical prediction of noisetransmission loss through viscoelastically damped sandwichplatesrdquo Journal of Sandwich Structures andMaterials vol 10 no5 pp 359ndash384 2008

[18] S Assaf M Guerich and P Cuvelier ldquoVibration and acousticresponse of damped sandwich panels immersed in a light orheavy fluidrdquo Computers and Structures vol 88 pp 359ndash3842010

[19] R Zhou and M J Crocker ldquoBoundary element analyses forsound transmission loss of panelsrdquo Journal of the AcousticalSociety of America vol 127 no 2 pp 829ndash840 2010

[20] K Amichi N Atalla and R Ruokolainen ldquoA new 3D finite ele-ment sandwich plate for predicting the vibroacoustic responseof laminated steel panelsrdquo Finite Elements in Analysis andDesign vol 46 no 12 pp 1131ndash1145 2010

[21] F Franco S de Rosa and T Polito ldquoFinite element inves-tigations on the vibroacoustic performance of plane plateswith random stiffnessrdquo Mechanics of Advanced Materials andStructures vol 18 no 7 pp 484ndash497 2011

[22] Y Kim ldquoIdentification of sound transmission characteristicsof honeycomb sandwich panels using hybrid analyticalone-dimensional finite element methodrdquo in Proceedings of the35th International Congress and Exposition on Noise ControlEngineering (INTER-NOISE rsquo06) HonoluluHawaii USA 2006

[23] F Fahy Sound and Structural Vibration Radiation Transmissionand Response Academic Press London UK 1985

[24] J Blevins Formulas for Natural Frequency and Mode ShapeKrieger Publishing Malabar Fla USA 2001

[25] D Newland An Introduction to Random Vibration and SpectralAnalysis Longman New York NY USA 1984

[26] Y Lin Probabilistic Theory of Structural Dynamics McGraw-Hill Book New York NY USA 1967

[27] C E Wallace ldquoRadiation resistance of a rectangular panelrdquoTheJournal of the Acoustical Society of America vol 51 no 3 pp946ndash952 1972

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RotatingMachinery

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

10 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

100

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 6 Radiated sound power results for the isotropic panels (a) and (b) overall panel RSP (c) and (d) RSP near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) RSP near location (1199092 1199102) = (03119886 02119887)

Shock and Vibration 11

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

(a)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(b)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(c)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus50

minus100

minus150

minus200

(f)

Figure 7 Panel displacement PSD results for the isotropic panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) 119878119908119908

near location (1199092 1199102) = (03119886 02119887)

12 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus100

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus100

minus80

minus60

minus40

minus20

0

20

40

60

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 8 Comparison of radiated sound power results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 13

0 500 1000 1500

Frequency (Hz)

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus50

minus150

minus100

minus200

minus250

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 9 Comparison of panel displacement PSD results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

14 Shock and VibrationRa

diat

ed so

und

pow

er (d

B re

l1

W)

0 500 1000 1500

Frequency (Hz)

50

0

minus50

minus100

minus150

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

100

50

0

minus50

minus100

minus150

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

50

0

minus50

minus100

minus150

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus80

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 10 Comparison of radiated sound power results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 15

minus50

minus100

minus150

minus250

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus220

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus50

minus100

minus150

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 11 Comparison of panel displacement PSD results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b)overall panel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

16 Shock and Vibration

Table 1 Physical properties of composite panels

Variable Description units Case 1 Case 2 Case 3 Case 4119886 Length m 183 11683 0348 10119887 Width m 122 11683 03048 03120582 Aspect ratio (119886119887) 15 10 114 333ℎ1 ℎ3

Face layers thickness mm 0406 0506 0762 35ℎ2

Core layer thickness mm 64 1905 0254 051198641 1198643

Face layers Young modulus Pa 6898 times 1010

71 times 1010

6898 times 1010

66 times 1010

1198642

Core layer Young modulus Pa 457 times 105

457 times 105

267 times 106

26 times 108

]1 ]3

Face layers Poissonrsquos ratio 033 033 03 033]2

Core layer Poissonrsquos ratio 04 04 049 031205881 1205883

Face layers density kgmminus3 2768 2700 2740 26801205882

Core layer density kgmminus3 122 4806 999 1680119898 Total mass kg 676 499 047 588

Table 2 Physical properties of isotropic panels

Variable Description units Case 3 Case 4 Case 5 Case 6119886 Length m 0348 10 0348 10119887 Width m 03048 03 03048 03120582 Aspect ratio (119886119887) 114 333 114 333ℎ119901

Thickness mm 1 1 15 4119864119901

Young modulus Pa 724 times 1010

724 times 1010

724 times 1010

724 times 1010

]119901

Poissonrsquos ratio 033 033 033 033120588119901

Density kg mminus3 2800 2800 2800 2800119898 Total mass kg 030 084 045 336

Table 3 Flight test conditions and air properties

Variable Description units Value119872 Mach number 08alt Altitude m 13106119888 Speed of sound m sminus1 295119880infin

Flow speed m sminus1 23606120588 Air density kgmminus3 02622] Kinematic viscosity m2 sminus1 5422 times 10

minus5

119879 Temperature K 21665Re119909

Reynolds number 2917 times 107

1198880

Speed of sound m sminus1 3401205880

Internal air density kgmminus3 142

higher amplification of localized frequencies Therefore theface layers thickness increase should be performed carefullyby an accompanied increase in the core layer thickness whilesimultaneously considering the weight constraints for thespecific panels applications

A more detailed analysis will be required in order toformulate final conclusions about the optimum solutions forthe panel in the context of aircraft application For that theinsulation materials would also be considered to account forthe respective added weight and effect in the sound radi-ation signature However typically the acoustic insulationmaterials will only have a higher impact in decreasing the

Table 4 First 10 natural frequencies for the composite panels

Modes 119891 Hz(Case 1)

119891 Hz(Case 2)

119891 Hz(Case 3)

119891 Hz(Case 4)

1 2365 10588 4874 126582 4548 26470 11222 157933 7276 42352 13148 210194 8186 52941 19496 283355 9459 68823 21800 377416 13097 89999 26939 474967 13279 95293 30075 492388 15462 105881 33287 506329 17645 132351 36612 5585710 18191 137645 43866 62825

sound levels at higher frequency ranges and will thereforenot improve significantly the problem of higher noise levelsin the low frequency range particularly observed in theisotropic solutions As such the composite panel solutionsmay appear to be a promising replacement to the isotropicpanels especially panels like Cases 3 and 4 which are alreadycharacterized by lower levels of both RSP and 119878

119908119908in the

low frequency range In further analysis it will also beimportant to carefully address the noise amplification atisolated frequencies by identifying a good combination of

Shock and Vibration 17

Table 5 First 10 natural frequencies for the isotropic panels

Modes 119891 Hz(Case 3)

119891 Hz(Case 4)

119891 Hz(Case 5)

119891 Hz(Case 6)

1 4646 2958 6969 118322 10697 3691 16046 147643 12534 4912 18801 196494 18585 6622 27877 264885 20782 8820 31173 352826 25680 1110 38520 443997 28670 11507 43004 460298 31731 11832 47596 473309 34901 13054 52351 5221510 41816 14683 62723 58731

core and face layers in terms of both materials selection anddimensions

4 Conclusions

This study presents a mathematical method and frameworkon composite panels radiated sound and vibrationThepanelsare considered excited either by turbulent flow or by randomnoiseThe turbulent flow corresponds to the typical excitationof an aircraft panel at flight cruise conditions with a Machnumber of 08 The effect of type of panel of panel dimen-sions and of panel thickness are evaluated The turbulentflow excitation showed to excite a higher number of panelmodes compared to random noise excitation and resulted indissimilar levels of both panel vibration and radiated soundwith relation to the random excitation This effect is morevisible for composite than for isotropic panels and thereforethe pure random noise should not be used as representativeof the turbulent flow excitation in real applications

Different properties on the composite panels are shown toyield dissimilar effects in the noise and vibration predictionsThe effect of the panel size specifically its length and widthis noted to have a significant effect on the panel performancein terms of noise radiation and vibration levels Overallsmaller composite panels or panels with higher aspect ratioproduced lower levels of radiated sound and vibration thanlonger and wider composite panels Compared to isotropicpanels composite panels analyzed are shown to radiate lowernoise levels with the exception of specific natural frequencieswhich are amplified near 300Hz and 1000Hz It is shown thatthis amplification can be significant for composite panels withthicker face layers and thinner core layer and thus resultingin unwanted effect of increased sound radiated Despitethis effect this drawback may potentially be corrected byincreasing the sandwich core layer thickness Increasing thecore layer thickness will impact the panel weight and hencethe composite panels can represent a heavier solution com-pared to the isotropic panels examined On the other handthis approach may be beneficial in the design of compositepanels in the context of aircraft panel applications sincethese panels may require less acoustic insulation material

added to the aircraft walls as they already represent animproved design solution Furthermore acoustic insulationmaterials have a higher impact in decreasing sound levelsat the high frequency ranges and therefore it will notsignificantly improve the problem of higher noise levels at thelow frequency range particularly observed in the isotropicsolutions In this context advanced and optimized compositepanel configurations may be a promising replacement to thetraditional isotropic panels for aerospace applications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is grateful for the support of the National Scienceand Engineering Research Council of Canada

References

[1] J Rocha and D Palumbo ldquoOn the sensitivity of sound powerradiated by aircraft panels to turbulent boundary layer param-etersrdquo Journal of Sound and Vibration vol 331 no 21 pp 4785ndash4806 2012

[2] J Rocha A Suleman and F Lau ldquoPrediction of turbulentboundary layer induced noise in the cabin of a BWB aircraftrdquoShock and Vibration vol 19 no 4 pp 693ndash705 2012

[3] J da Rocha A Suleman and F Lau ldquoFlow-induced noise andvibration in aircraft cylindrical cabins closed-form analyticalmodel validationrdquo Journal of Vibration and Acoustics vol 133no 5 Article ID 051013 9 pages 2011

[4] J da Rocha A Suleman and F Lau ldquoPrediction of flow-inducednoise in transport vehicles development and validation of acoupled structuralmdashacoustic analytical frameworkrdquo CanadianAcousticsmdashAcoustique Canadienne vol 37 no 4 pp 13ndash292009

[5] G Corcos ldquoThe structure of the turbulent pressure field inboundary layer flowsrdquo Journal of Fluid Mechanics vol 18 pp353ndash378 1964

[6] G Corcos ldquoResolution of pressure in turbulencerdquo Journal of theAcoustical Society of America vol 35 no 2 pp 192ndash199 1963

[7] B Efimtsov ldquoCharacteristics of the field of turbulent wallpressure fluctuations at large Reynolds numbersrdquo Soviet PhysicsAcoustics vol 28 no 4 pp 289ndash292 1982

[8] J Reddy Mechanics of Laminated Composite Plates and ShellsTheory and Analysis CRC Press Boca Raton Fla USA 1945

[9] J Whitney Structural Analysis of Laminated Anisotropic PlatesTechnomic Lancaster Pa USA 1987

[10] C L Dym and M A Lang ldquoTransmission of sound throughsandwich panelsrdquo Journal of the Acoustical Society of Americavol 56 no 5 pp 1523ndash1532 1974

[11] C L Dym C S Ventres and M A Lang ldquoTransmission ofsound through sandwich panels a reconsiderationrdquo Journal ofthe Acoustical Society of America vol 59 no 2 pp 364ndash3671976

[12] S Narayanan andR L Shanbhag ldquoSound transmission througha damped sandwich panelrdquo Journal of Sound and Vibration vol80 no 3 pp 315ndash327 1982

18 Shock and Vibration

[13] C L Dym and D C Lang ldquoTransmission loss of dampedasymmetric sandwich panels with orthotropic coresrdquo Journal ofSound and Vibration vol 88 no 3 pp 299ndash319 1983

[14] S Ghinet N Atalla and H Osman ldquoDiffuse field transmissioninto infinite sandwich composite and laminate composite cylin-dersrdquo Journal of Sound and Vibration vol 289 no 4-5 pp 745ndash778 2006

[15] J Mendler ldquoComputational tool for the acoustic analysis ofsandwich panelsrdquo in Proceedings of the European Conference onNoise Control (EURO-NOISE rsquo95) Lyon France 1995

[16] O Foin J Nicolas andNAtalla ldquoAn efficient tool for predictingthe structural acoustic and vibration response of sandwichplates in light or heavy fluidrdquo Applied Acoustics vol 57 no 3pp 213ndash242 1999

[17] S Assaf and M Guerich ldquoNumerical prediction of noisetransmission loss through viscoelastically damped sandwichplatesrdquo Journal of Sandwich Structures andMaterials vol 10 no5 pp 359ndash384 2008

[18] S Assaf M Guerich and P Cuvelier ldquoVibration and acousticresponse of damped sandwich panels immersed in a light orheavy fluidrdquo Computers and Structures vol 88 pp 359ndash3842010

[19] R Zhou and M J Crocker ldquoBoundary element analyses forsound transmission loss of panelsrdquo Journal of the AcousticalSociety of America vol 127 no 2 pp 829ndash840 2010

[20] K Amichi N Atalla and R Ruokolainen ldquoA new 3D finite ele-ment sandwich plate for predicting the vibroacoustic responseof laminated steel panelsrdquo Finite Elements in Analysis andDesign vol 46 no 12 pp 1131ndash1145 2010

[21] F Franco S de Rosa and T Polito ldquoFinite element inves-tigations on the vibroacoustic performance of plane plateswith random stiffnessrdquo Mechanics of Advanced Materials andStructures vol 18 no 7 pp 484ndash497 2011

[22] Y Kim ldquoIdentification of sound transmission characteristicsof honeycomb sandwich panels using hybrid analyticalone-dimensional finite element methodrdquo in Proceedings of the35th International Congress and Exposition on Noise ControlEngineering (INTER-NOISE rsquo06) HonoluluHawaii USA 2006

[23] F Fahy Sound and Structural Vibration Radiation Transmissionand Response Academic Press London UK 1985

[24] J Blevins Formulas for Natural Frequency and Mode ShapeKrieger Publishing Malabar Fla USA 2001

[25] D Newland An Introduction to Random Vibration and SpectralAnalysis Longman New York NY USA 1984

[26] Y Lin Probabilistic Theory of Structural Dynamics McGraw-Hill Book New York NY USA 1967

[27] C E Wallace ldquoRadiation resistance of a rectangular panelrdquoTheJournal of the Acoustical Society of America vol 51 no 3 pp946ndash952 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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DistributedSensor Networks

International Journal of

Shock and Vibration 11

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

(a)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(b)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(c)

0 500 1000 1500

Frequency (Hz)

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 4 isotropic TBL

Case 5 isotropic TBLCase 6 isotropic TBL

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 4 isotropic random

Case 5 isotropic randomCase 6 isotropic random

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

minus50

minus100

minus150

minus200

(f)

Figure 7 Panel displacement PSD results for the isotropic panels (a) and (b) overall panel 119878119908119908

(c) and (d) 119878119908119908

near surface location (1199091 1199101) =

(05119886 05119887) (e) and (f) 119878119908119908

near location (1199092 1199102) = (03119886 02119887)

12 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus100

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus100

minus80

minus60

minus40

minus20

0

20

40

60

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 8 Comparison of radiated sound power results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 13

0 500 1000 1500

Frequency (Hz)

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus50

minus150

minus100

minus200

minus250

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 9 Comparison of panel displacement PSD results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

14 Shock and VibrationRa

diat

ed so

und

pow

er (d

B re

l1

W)

0 500 1000 1500

Frequency (Hz)

50

0

minus50

minus100

minus150

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

100

50

0

minus50

minus100

minus150

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

50

0

minus50

minus100

minus150

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus80

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 10 Comparison of radiated sound power results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 15

minus50

minus100

minus150

minus250

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus220

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus50

minus100

minus150

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 11 Comparison of panel displacement PSD results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b)overall panel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

16 Shock and Vibration

Table 1 Physical properties of composite panels

Variable Description units Case 1 Case 2 Case 3 Case 4119886 Length m 183 11683 0348 10119887 Width m 122 11683 03048 03120582 Aspect ratio (119886119887) 15 10 114 333ℎ1 ℎ3

Face layers thickness mm 0406 0506 0762 35ℎ2

Core layer thickness mm 64 1905 0254 051198641 1198643

Face layers Young modulus Pa 6898 times 1010

71 times 1010

6898 times 1010

66 times 1010

1198642

Core layer Young modulus Pa 457 times 105

457 times 105

267 times 106

26 times 108

]1 ]3

Face layers Poissonrsquos ratio 033 033 03 033]2

Core layer Poissonrsquos ratio 04 04 049 031205881 1205883

Face layers density kgmminus3 2768 2700 2740 26801205882

Core layer density kgmminus3 122 4806 999 1680119898 Total mass kg 676 499 047 588

Table 2 Physical properties of isotropic panels

Variable Description units Case 3 Case 4 Case 5 Case 6119886 Length m 0348 10 0348 10119887 Width m 03048 03 03048 03120582 Aspect ratio (119886119887) 114 333 114 333ℎ119901

Thickness mm 1 1 15 4119864119901

Young modulus Pa 724 times 1010

724 times 1010

724 times 1010

724 times 1010

]119901

Poissonrsquos ratio 033 033 033 033120588119901

Density kg mminus3 2800 2800 2800 2800119898 Total mass kg 030 084 045 336

Table 3 Flight test conditions and air properties

Variable Description units Value119872 Mach number 08alt Altitude m 13106119888 Speed of sound m sminus1 295119880infin

Flow speed m sminus1 23606120588 Air density kgmminus3 02622] Kinematic viscosity m2 sminus1 5422 times 10

minus5

119879 Temperature K 21665Re119909

Reynolds number 2917 times 107

1198880

Speed of sound m sminus1 3401205880

Internal air density kgmminus3 142

higher amplification of localized frequencies Therefore theface layers thickness increase should be performed carefullyby an accompanied increase in the core layer thickness whilesimultaneously considering the weight constraints for thespecific panels applications

A more detailed analysis will be required in order toformulate final conclusions about the optimum solutions forthe panel in the context of aircraft application For that theinsulation materials would also be considered to account forthe respective added weight and effect in the sound radi-ation signature However typically the acoustic insulationmaterials will only have a higher impact in decreasing the

Table 4 First 10 natural frequencies for the composite panels

Modes 119891 Hz(Case 1)

119891 Hz(Case 2)

119891 Hz(Case 3)

119891 Hz(Case 4)

1 2365 10588 4874 126582 4548 26470 11222 157933 7276 42352 13148 210194 8186 52941 19496 283355 9459 68823 21800 377416 13097 89999 26939 474967 13279 95293 30075 492388 15462 105881 33287 506329 17645 132351 36612 5585710 18191 137645 43866 62825

sound levels at higher frequency ranges and will thereforenot improve significantly the problem of higher noise levelsin the low frequency range particularly observed in theisotropic solutions As such the composite panel solutionsmay appear to be a promising replacement to the isotropicpanels especially panels like Cases 3 and 4 which are alreadycharacterized by lower levels of both RSP and 119878

119908119908in the

low frequency range In further analysis it will also beimportant to carefully address the noise amplification atisolated frequencies by identifying a good combination of

Shock and Vibration 17

Table 5 First 10 natural frequencies for the isotropic panels

Modes 119891 Hz(Case 3)

119891 Hz(Case 4)

119891 Hz(Case 5)

119891 Hz(Case 6)

1 4646 2958 6969 118322 10697 3691 16046 147643 12534 4912 18801 196494 18585 6622 27877 264885 20782 8820 31173 352826 25680 1110 38520 443997 28670 11507 43004 460298 31731 11832 47596 473309 34901 13054 52351 5221510 41816 14683 62723 58731

core and face layers in terms of both materials selection anddimensions

4 Conclusions

This study presents a mathematical method and frameworkon composite panels radiated sound and vibrationThepanelsare considered excited either by turbulent flow or by randomnoiseThe turbulent flow corresponds to the typical excitationof an aircraft panel at flight cruise conditions with a Machnumber of 08 The effect of type of panel of panel dimen-sions and of panel thickness are evaluated The turbulentflow excitation showed to excite a higher number of panelmodes compared to random noise excitation and resulted indissimilar levels of both panel vibration and radiated soundwith relation to the random excitation This effect is morevisible for composite than for isotropic panels and thereforethe pure random noise should not be used as representativeof the turbulent flow excitation in real applications

Different properties on the composite panels are shown toyield dissimilar effects in the noise and vibration predictionsThe effect of the panel size specifically its length and widthis noted to have a significant effect on the panel performancein terms of noise radiation and vibration levels Overallsmaller composite panels or panels with higher aspect ratioproduced lower levels of radiated sound and vibration thanlonger and wider composite panels Compared to isotropicpanels composite panels analyzed are shown to radiate lowernoise levels with the exception of specific natural frequencieswhich are amplified near 300Hz and 1000Hz It is shown thatthis amplification can be significant for composite panels withthicker face layers and thinner core layer and thus resultingin unwanted effect of increased sound radiated Despitethis effect this drawback may potentially be corrected byincreasing the sandwich core layer thickness Increasing thecore layer thickness will impact the panel weight and hencethe composite panels can represent a heavier solution com-pared to the isotropic panels examined On the other handthis approach may be beneficial in the design of compositepanels in the context of aircraft panel applications sincethese panels may require less acoustic insulation material

added to the aircraft walls as they already represent animproved design solution Furthermore acoustic insulationmaterials have a higher impact in decreasing sound levelsat the high frequency ranges and therefore it will notsignificantly improve the problem of higher noise levels at thelow frequency range particularly observed in the isotropicsolutions In this context advanced and optimized compositepanel configurations may be a promising replacement to thetraditional isotropic panels for aerospace applications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is grateful for the support of the National Scienceand Engineering Research Council of Canada

References

[1] J Rocha and D Palumbo ldquoOn the sensitivity of sound powerradiated by aircraft panels to turbulent boundary layer param-etersrdquo Journal of Sound and Vibration vol 331 no 21 pp 4785ndash4806 2012

[2] J Rocha A Suleman and F Lau ldquoPrediction of turbulentboundary layer induced noise in the cabin of a BWB aircraftrdquoShock and Vibration vol 19 no 4 pp 693ndash705 2012

[3] J da Rocha A Suleman and F Lau ldquoFlow-induced noise andvibration in aircraft cylindrical cabins closed-form analyticalmodel validationrdquo Journal of Vibration and Acoustics vol 133no 5 Article ID 051013 9 pages 2011

[4] J da Rocha A Suleman and F Lau ldquoPrediction of flow-inducednoise in transport vehicles development and validation of acoupled structuralmdashacoustic analytical frameworkrdquo CanadianAcousticsmdashAcoustique Canadienne vol 37 no 4 pp 13ndash292009

[5] G Corcos ldquoThe structure of the turbulent pressure field inboundary layer flowsrdquo Journal of Fluid Mechanics vol 18 pp353ndash378 1964

[6] G Corcos ldquoResolution of pressure in turbulencerdquo Journal of theAcoustical Society of America vol 35 no 2 pp 192ndash199 1963

[7] B Efimtsov ldquoCharacteristics of the field of turbulent wallpressure fluctuations at large Reynolds numbersrdquo Soviet PhysicsAcoustics vol 28 no 4 pp 289ndash292 1982

[8] J Reddy Mechanics of Laminated Composite Plates and ShellsTheory and Analysis CRC Press Boca Raton Fla USA 1945

[9] J Whitney Structural Analysis of Laminated Anisotropic PlatesTechnomic Lancaster Pa USA 1987

[10] C L Dym and M A Lang ldquoTransmission of sound throughsandwich panelsrdquo Journal of the Acoustical Society of Americavol 56 no 5 pp 1523ndash1532 1974

[11] C L Dym C S Ventres and M A Lang ldquoTransmission ofsound through sandwich panels a reconsiderationrdquo Journal ofthe Acoustical Society of America vol 59 no 2 pp 364ndash3671976

[12] S Narayanan andR L Shanbhag ldquoSound transmission througha damped sandwich panelrdquo Journal of Sound and Vibration vol80 no 3 pp 315ndash327 1982

18 Shock and Vibration

[13] C L Dym and D C Lang ldquoTransmission loss of dampedasymmetric sandwich panels with orthotropic coresrdquo Journal ofSound and Vibration vol 88 no 3 pp 299ndash319 1983

[14] S Ghinet N Atalla and H Osman ldquoDiffuse field transmissioninto infinite sandwich composite and laminate composite cylin-dersrdquo Journal of Sound and Vibration vol 289 no 4-5 pp 745ndash778 2006

[15] J Mendler ldquoComputational tool for the acoustic analysis ofsandwich panelsrdquo in Proceedings of the European Conference onNoise Control (EURO-NOISE rsquo95) Lyon France 1995

[16] O Foin J Nicolas andNAtalla ldquoAn efficient tool for predictingthe structural acoustic and vibration response of sandwichplates in light or heavy fluidrdquo Applied Acoustics vol 57 no 3pp 213ndash242 1999

[17] S Assaf and M Guerich ldquoNumerical prediction of noisetransmission loss through viscoelastically damped sandwichplatesrdquo Journal of Sandwich Structures andMaterials vol 10 no5 pp 359ndash384 2008

[18] S Assaf M Guerich and P Cuvelier ldquoVibration and acousticresponse of damped sandwich panels immersed in a light orheavy fluidrdquo Computers and Structures vol 88 pp 359ndash3842010

[19] R Zhou and M J Crocker ldquoBoundary element analyses forsound transmission loss of panelsrdquo Journal of the AcousticalSociety of America vol 127 no 2 pp 829ndash840 2010

[20] K Amichi N Atalla and R Ruokolainen ldquoA new 3D finite ele-ment sandwich plate for predicting the vibroacoustic responseof laminated steel panelsrdquo Finite Elements in Analysis andDesign vol 46 no 12 pp 1131ndash1145 2010

[21] F Franco S de Rosa and T Polito ldquoFinite element inves-tigations on the vibroacoustic performance of plane plateswith random stiffnessrdquo Mechanics of Advanced Materials andStructures vol 18 no 7 pp 484ndash497 2011

[22] Y Kim ldquoIdentification of sound transmission characteristicsof honeycomb sandwich panels using hybrid analyticalone-dimensional finite element methodrdquo in Proceedings of the35th International Congress and Exposition on Noise ControlEngineering (INTER-NOISE rsquo06) HonoluluHawaii USA 2006

[23] F Fahy Sound and Structural Vibration Radiation Transmissionand Response Academic Press London UK 1985

[24] J Blevins Formulas for Natural Frequency and Mode ShapeKrieger Publishing Malabar Fla USA 2001

[25] D Newland An Introduction to Random Vibration and SpectralAnalysis Longman New York NY USA 1984

[26] Y Lin Probabilistic Theory of Structural Dynamics McGraw-Hill Book New York NY USA 1967

[27] C E Wallace ldquoRadiation resistance of a rectangular panelrdquoTheJournal of the Acoustical Society of America vol 51 no 3 pp946ndash952 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

0 500 1000 1500

Frequency (Hz)

minus150

minus100

minus50

50

0

Radi

ated

soun

d po

wer

(dB

rel1

W)

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

0 500 1000 1500

Frequency (Hz)

minus100

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus100

minus80

minus60

minus40

minus20

0

20

40

60

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 8 Comparison of radiated sound power results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 13

0 500 1000 1500

Frequency (Hz)

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus50

minus150

minus100

minus200

minus250

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 9 Comparison of panel displacement PSD results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

14 Shock and VibrationRa

diat

ed so

und

pow

er (d

B re

l1

W)

0 500 1000 1500

Frequency (Hz)

50

0

minus50

minus100

minus150

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

100

50

0

minus50

minus100

minus150

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

50

0

minus50

minus100

minus150

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus80

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 10 Comparison of radiated sound power results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 15

minus50

minus100

minus150

minus250

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus220

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus50

minus100

minus150

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 11 Comparison of panel displacement PSD results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b)overall panel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

16 Shock and Vibration

Table 1 Physical properties of composite panels

Variable Description units Case 1 Case 2 Case 3 Case 4119886 Length m 183 11683 0348 10119887 Width m 122 11683 03048 03120582 Aspect ratio (119886119887) 15 10 114 333ℎ1 ℎ3

Face layers thickness mm 0406 0506 0762 35ℎ2

Core layer thickness mm 64 1905 0254 051198641 1198643

Face layers Young modulus Pa 6898 times 1010

71 times 1010

6898 times 1010

66 times 1010

1198642

Core layer Young modulus Pa 457 times 105

457 times 105

267 times 106

26 times 108

]1 ]3

Face layers Poissonrsquos ratio 033 033 03 033]2

Core layer Poissonrsquos ratio 04 04 049 031205881 1205883

Face layers density kgmminus3 2768 2700 2740 26801205882

Core layer density kgmminus3 122 4806 999 1680119898 Total mass kg 676 499 047 588

Table 2 Physical properties of isotropic panels

Variable Description units Case 3 Case 4 Case 5 Case 6119886 Length m 0348 10 0348 10119887 Width m 03048 03 03048 03120582 Aspect ratio (119886119887) 114 333 114 333ℎ119901

Thickness mm 1 1 15 4119864119901

Young modulus Pa 724 times 1010

724 times 1010

724 times 1010

724 times 1010

]119901

Poissonrsquos ratio 033 033 033 033120588119901

Density kg mminus3 2800 2800 2800 2800119898 Total mass kg 030 084 045 336

Table 3 Flight test conditions and air properties

Variable Description units Value119872 Mach number 08alt Altitude m 13106119888 Speed of sound m sminus1 295119880infin

Flow speed m sminus1 23606120588 Air density kgmminus3 02622] Kinematic viscosity m2 sminus1 5422 times 10

minus5

119879 Temperature K 21665Re119909

Reynolds number 2917 times 107

1198880

Speed of sound m sminus1 3401205880

Internal air density kgmminus3 142

higher amplification of localized frequencies Therefore theface layers thickness increase should be performed carefullyby an accompanied increase in the core layer thickness whilesimultaneously considering the weight constraints for thespecific panels applications

A more detailed analysis will be required in order toformulate final conclusions about the optimum solutions forthe panel in the context of aircraft application For that theinsulation materials would also be considered to account forthe respective added weight and effect in the sound radi-ation signature However typically the acoustic insulationmaterials will only have a higher impact in decreasing the

Table 4 First 10 natural frequencies for the composite panels

Modes 119891 Hz(Case 1)

119891 Hz(Case 2)

119891 Hz(Case 3)

119891 Hz(Case 4)

1 2365 10588 4874 126582 4548 26470 11222 157933 7276 42352 13148 210194 8186 52941 19496 283355 9459 68823 21800 377416 13097 89999 26939 474967 13279 95293 30075 492388 15462 105881 33287 506329 17645 132351 36612 5585710 18191 137645 43866 62825

sound levels at higher frequency ranges and will thereforenot improve significantly the problem of higher noise levelsin the low frequency range particularly observed in theisotropic solutions As such the composite panel solutionsmay appear to be a promising replacement to the isotropicpanels especially panels like Cases 3 and 4 which are alreadycharacterized by lower levels of both RSP and 119878

119908119908in the

low frequency range In further analysis it will also beimportant to carefully address the noise amplification atisolated frequencies by identifying a good combination of

Shock and Vibration 17

Table 5 First 10 natural frequencies for the isotropic panels

Modes 119891 Hz(Case 3)

119891 Hz(Case 4)

119891 Hz(Case 5)

119891 Hz(Case 6)

1 4646 2958 6969 118322 10697 3691 16046 147643 12534 4912 18801 196494 18585 6622 27877 264885 20782 8820 31173 352826 25680 1110 38520 443997 28670 11507 43004 460298 31731 11832 47596 473309 34901 13054 52351 5221510 41816 14683 62723 58731

core and face layers in terms of both materials selection anddimensions

4 Conclusions

This study presents a mathematical method and frameworkon composite panels radiated sound and vibrationThepanelsare considered excited either by turbulent flow or by randomnoiseThe turbulent flow corresponds to the typical excitationof an aircraft panel at flight cruise conditions with a Machnumber of 08 The effect of type of panel of panel dimen-sions and of panel thickness are evaluated The turbulentflow excitation showed to excite a higher number of panelmodes compared to random noise excitation and resulted indissimilar levels of both panel vibration and radiated soundwith relation to the random excitation This effect is morevisible for composite than for isotropic panels and thereforethe pure random noise should not be used as representativeof the turbulent flow excitation in real applications

Different properties on the composite panels are shown toyield dissimilar effects in the noise and vibration predictionsThe effect of the panel size specifically its length and widthis noted to have a significant effect on the panel performancein terms of noise radiation and vibration levels Overallsmaller composite panels or panels with higher aspect ratioproduced lower levels of radiated sound and vibration thanlonger and wider composite panels Compared to isotropicpanels composite panels analyzed are shown to radiate lowernoise levels with the exception of specific natural frequencieswhich are amplified near 300Hz and 1000Hz It is shown thatthis amplification can be significant for composite panels withthicker face layers and thinner core layer and thus resultingin unwanted effect of increased sound radiated Despitethis effect this drawback may potentially be corrected byincreasing the sandwich core layer thickness Increasing thecore layer thickness will impact the panel weight and hencethe composite panels can represent a heavier solution com-pared to the isotropic panels examined On the other handthis approach may be beneficial in the design of compositepanels in the context of aircraft panel applications sincethese panels may require less acoustic insulation material

added to the aircraft walls as they already represent animproved design solution Furthermore acoustic insulationmaterials have a higher impact in decreasing sound levelsat the high frequency ranges and therefore it will notsignificantly improve the problem of higher noise levels at thelow frequency range particularly observed in the isotropicsolutions In this context advanced and optimized compositepanel configurations may be a promising replacement to thetraditional isotropic panels for aerospace applications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is grateful for the support of the National Scienceand Engineering Research Council of Canada

References

[1] J Rocha and D Palumbo ldquoOn the sensitivity of sound powerradiated by aircraft panels to turbulent boundary layer param-etersrdquo Journal of Sound and Vibration vol 331 no 21 pp 4785ndash4806 2012

[2] J Rocha A Suleman and F Lau ldquoPrediction of turbulentboundary layer induced noise in the cabin of a BWB aircraftrdquoShock and Vibration vol 19 no 4 pp 693ndash705 2012

[3] J da Rocha A Suleman and F Lau ldquoFlow-induced noise andvibration in aircraft cylindrical cabins closed-form analyticalmodel validationrdquo Journal of Vibration and Acoustics vol 133no 5 Article ID 051013 9 pages 2011

[4] J da Rocha A Suleman and F Lau ldquoPrediction of flow-inducednoise in transport vehicles development and validation of acoupled structuralmdashacoustic analytical frameworkrdquo CanadianAcousticsmdashAcoustique Canadienne vol 37 no 4 pp 13ndash292009

[5] G Corcos ldquoThe structure of the turbulent pressure field inboundary layer flowsrdquo Journal of Fluid Mechanics vol 18 pp353ndash378 1964

[6] G Corcos ldquoResolution of pressure in turbulencerdquo Journal of theAcoustical Society of America vol 35 no 2 pp 192ndash199 1963

[7] B Efimtsov ldquoCharacteristics of the field of turbulent wallpressure fluctuations at large Reynolds numbersrdquo Soviet PhysicsAcoustics vol 28 no 4 pp 289ndash292 1982

[8] J Reddy Mechanics of Laminated Composite Plates and ShellsTheory and Analysis CRC Press Boca Raton Fla USA 1945

[9] J Whitney Structural Analysis of Laminated Anisotropic PlatesTechnomic Lancaster Pa USA 1987

[10] C L Dym and M A Lang ldquoTransmission of sound throughsandwich panelsrdquo Journal of the Acoustical Society of Americavol 56 no 5 pp 1523ndash1532 1974

[11] C L Dym C S Ventres and M A Lang ldquoTransmission ofsound through sandwich panels a reconsiderationrdquo Journal ofthe Acoustical Society of America vol 59 no 2 pp 364ndash3671976

[12] S Narayanan andR L Shanbhag ldquoSound transmission througha damped sandwich panelrdquo Journal of Sound and Vibration vol80 no 3 pp 315ndash327 1982

18 Shock and Vibration

[13] C L Dym and D C Lang ldquoTransmission loss of dampedasymmetric sandwich panels with orthotropic coresrdquo Journal ofSound and Vibration vol 88 no 3 pp 299ndash319 1983

[14] S Ghinet N Atalla and H Osman ldquoDiffuse field transmissioninto infinite sandwich composite and laminate composite cylin-dersrdquo Journal of Sound and Vibration vol 289 no 4-5 pp 745ndash778 2006

[15] J Mendler ldquoComputational tool for the acoustic analysis ofsandwich panelsrdquo in Proceedings of the European Conference onNoise Control (EURO-NOISE rsquo95) Lyon France 1995

[16] O Foin J Nicolas andNAtalla ldquoAn efficient tool for predictingthe structural acoustic and vibration response of sandwichplates in light or heavy fluidrdquo Applied Acoustics vol 57 no 3pp 213ndash242 1999

[17] S Assaf and M Guerich ldquoNumerical prediction of noisetransmission loss through viscoelastically damped sandwichplatesrdquo Journal of Sandwich Structures andMaterials vol 10 no5 pp 359ndash384 2008

[18] S Assaf M Guerich and P Cuvelier ldquoVibration and acousticresponse of damped sandwich panels immersed in a light orheavy fluidrdquo Computers and Structures vol 88 pp 359ndash3842010

[19] R Zhou and M J Crocker ldquoBoundary element analyses forsound transmission loss of panelsrdquo Journal of the AcousticalSociety of America vol 127 no 2 pp 829ndash840 2010

[20] K Amichi N Atalla and R Ruokolainen ldquoA new 3D finite ele-ment sandwich plate for predicting the vibroacoustic responseof laminated steel panelsrdquo Finite Elements in Analysis andDesign vol 46 no 12 pp 1131ndash1145 2010

[21] F Franco S de Rosa and T Polito ldquoFinite element inves-tigations on the vibroacoustic performance of plane plateswith random stiffnessrdquo Mechanics of Advanced Materials andStructures vol 18 no 7 pp 484ndash497 2011

[22] Y Kim ldquoIdentification of sound transmission characteristicsof honeycomb sandwich panels using hybrid analyticalone-dimensional finite element methodrdquo in Proceedings of the35th International Congress and Exposition on Noise ControlEngineering (INTER-NOISE rsquo06) HonoluluHawaii USA 2006

[23] F Fahy Sound and Structural Vibration Radiation Transmissionand Response Academic Press London UK 1985

[24] J Blevins Formulas for Natural Frequency and Mode ShapeKrieger Publishing Malabar Fla USA 2001

[25] D Newland An Introduction to Random Vibration and SpectralAnalysis Longman New York NY USA 1984

[26] Y Lin Probabilistic Theory of Structural Dynamics McGraw-Hill Book New York NY USA 1967

[27] C E Wallace ldquoRadiation resistance of a rectangular panelrdquoTheJournal of the Acoustical Society of America vol 51 no 3 pp946ndash952 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 13

0 500 1000 1500

Frequency (Hz)

minus100

minus120

minus140

minus160

minus180

minus200

minus220

minus240

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(a)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus50

minus150

minus100

minus200

minus250

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic TBLCase 3 composite TBL

Case 5 isotropic TBL

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 3 isotropic randomCase 3 composite random

Case 5 isotropic random

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 9 Comparison of panel displacement PSD results for Case 3 composite panel versus Cases 3 and 5 isotropic panels (a) and (b) overallpanel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

14 Shock and VibrationRa

diat

ed so

und

pow

er (d

B re

l1

W)

0 500 1000 1500

Frequency (Hz)

50

0

minus50

minus100

minus150

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

100

50

0

minus50

minus100

minus150

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

50

0

minus50

minus100

minus150

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus80

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 10 Comparison of radiated sound power results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 15

minus50

minus100

minus150

minus250

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus220

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus50

minus100

minus150

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 11 Comparison of panel displacement PSD results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b)overall panel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

16 Shock and Vibration

Table 1 Physical properties of composite panels

Variable Description units Case 1 Case 2 Case 3 Case 4119886 Length m 183 11683 0348 10119887 Width m 122 11683 03048 03120582 Aspect ratio (119886119887) 15 10 114 333ℎ1 ℎ3

Face layers thickness mm 0406 0506 0762 35ℎ2

Core layer thickness mm 64 1905 0254 051198641 1198643

Face layers Young modulus Pa 6898 times 1010

71 times 1010

6898 times 1010

66 times 1010

1198642

Core layer Young modulus Pa 457 times 105

457 times 105

267 times 106

26 times 108

]1 ]3

Face layers Poissonrsquos ratio 033 033 03 033]2

Core layer Poissonrsquos ratio 04 04 049 031205881 1205883

Face layers density kgmminus3 2768 2700 2740 26801205882

Core layer density kgmminus3 122 4806 999 1680119898 Total mass kg 676 499 047 588

Table 2 Physical properties of isotropic panels

Variable Description units Case 3 Case 4 Case 5 Case 6119886 Length m 0348 10 0348 10119887 Width m 03048 03 03048 03120582 Aspect ratio (119886119887) 114 333 114 333ℎ119901

Thickness mm 1 1 15 4119864119901

Young modulus Pa 724 times 1010

724 times 1010

724 times 1010

724 times 1010

]119901

Poissonrsquos ratio 033 033 033 033120588119901

Density kg mminus3 2800 2800 2800 2800119898 Total mass kg 030 084 045 336

Table 3 Flight test conditions and air properties

Variable Description units Value119872 Mach number 08alt Altitude m 13106119888 Speed of sound m sminus1 295119880infin

Flow speed m sminus1 23606120588 Air density kgmminus3 02622] Kinematic viscosity m2 sminus1 5422 times 10

minus5

119879 Temperature K 21665Re119909

Reynolds number 2917 times 107

1198880

Speed of sound m sminus1 3401205880

Internal air density kgmminus3 142

higher amplification of localized frequencies Therefore theface layers thickness increase should be performed carefullyby an accompanied increase in the core layer thickness whilesimultaneously considering the weight constraints for thespecific panels applications

A more detailed analysis will be required in order toformulate final conclusions about the optimum solutions forthe panel in the context of aircraft application For that theinsulation materials would also be considered to account forthe respective added weight and effect in the sound radi-ation signature However typically the acoustic insulationmaterials will only have a higher impact in decreasing the

Table 4 First 10 natural frequencies for the composite panels

Modes 119891 Hz(Case 1)

119891 Hz(Case 2)

119891 Hz(Case 3)

119891 Hz(Case 4)

1 2365 10588 4874 126582 4548 26470 11222 157933 7276 42352 13148 210194 8186 52941 19496 283355 9459 68823 21800 377416 13097 89999 26939 474967 13279 95293 30075 492388 15462 105881 33287 506329 17645 132351 36612 5585710 18191 137645 43866 62825

sound levels at higher frequency ranges and will thereforenot improve significantly the problem of higher noise levelsin the low frequency range particularly observed in theisotropic solutions As such the composite panel solutionsmay appear to be a promising replacement to the isotropicpanels especially panels like Cases 3 and 4 which are alreadycharacterized by lower levels of both RSP and 119878

119908119908in the

low frequency range In further analysis it will also beimportant to carefully address the noise amplification atisolated frequencies by identifying a good combination of

Shock and Vibration 17

Table 5 First 10 natural frequencies for the isotropic panels

Modes 119891 Hz(Case 3)

119891 Hz(Case 4)

119891 Hz(Case 5)

119891 Hz(Case 6)

1 4646 2958 6969 118322 10697 3691 16046 147643 12534 4912 18801 196494 18585 6622 27877 264885 20782 8820 31173 352826 25680 1110 38520 443997 28670 11507 43004 460298 31731 11832 47596 473309 34901 13054 52351 5221510 41816 14683 62723 58731

core and face layers in terms of both materials selection anddimensions

4 Conclusions

This study presents a mathematical method and frameworkon composite panels radiated sound and vibrationThepanelsare considered excited either by turbulent flow or by randomnoiseThe turbulent flow corresponds to the typical excitationof an aircraft panel at flight cruise conditions with a Machnumber of 08 The effect of type of panel of panel dimen-sions and of panel thickness are evaluated The turbulentflow excitation showed to excite a higher number of panelmodes compared to random noise excitation and resulted indissimilar levels of both panel vibration and radiated soundwith relation to the random excitation This effect is morevisible for composite than for isotropic panels and thereforethe pure random noise should not be used as representativeof the turbulent flow excitation in real applications

Different properties on the composite panels are shown toyield dissimilar effects in the noise and vibration predictionsThe effect of the panel size specifically its length and widthis noted to have a significant effect on the panel performancein terms of noise radiation and vibration levels Overallsmaller composite panels or panels with higher aspect ratioproduced lower levels of radiated sound and vibration thanlonger and wider composite panels Compared to isotropicpanels composite panels analyzed are shown to radiate lowernoise levels with the exception of specific natural frequencieswhich are amplified near 300Hz and 1000Hz It is shown thatthis amplification can be significant for composite panels withthicker face layers and thinner core layer and thus resultingin unwanted effect of increased sound radiated Despitethis effect this drawback may potentially be corrected byincreasing the sandwich core layer thickness Increasing thecore layer thickness will impact the panel weight and hencethe composite panels can represent a heavier solution com-pared to the isotropic panels examined On the other handthis approach may be beneficial in the design of compositepanels in the context of aircraft panel applications sincethese panels may require less acoustic insulation material

added to the aircraft walls as they already represent animproved design solution Furthermore acoustic insulationmaterials have a higher impact in decreasing sound levelsat the high frequency ranges and therefore it will notsignificantly improve the problem of higher noise levels at thelow frequency range particularly observed in the isotropicsolutions In this context advanced and optimized compositepanel configurations may be a promising replacement to thetraditional isotropic panels for aerospace applications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is grateful for the support of the National Scienceand Engineering Research Council of Canada

References

[1] J Rocha and D Palumbo ldquoOn the sensitivity of sound powerradiated by aircraft panels to turbulent boundary layer param-etersrdquo Journal of Sound and Vibration vol 331 no 21 pp 4785ndash4806 2012

[2] J Rocha A Suleman and F Lau ldquoPrediction of turbulentboundary layer induced noise in the cabin of a BWB aircraftrdquoShock and Vibration vol 19 no 4 pp 693ndash705 2012

[3] J da Rocha A Suleman and F Lau ldquoFlow-induced noise andvibration in aircraft cylindrical cabins closed-form analyticalmodel validationrdquo Journal of Vibration and Acoustics vol 133no 5 Article ID 051013 9 pages 2011

[4] J da Rocha A Suleman and F Lau ldquoPrediction of flow-inducednoise in transport vehicles development and validation of acoupled structuralmdashacoustic analytical frameworkrdquo CanadianAcousticsmdashAcoustique Canadienne vol 37 no 4 pp 13ndash292009

[5] G Corcos ldquoThe structure of the turbulent pressure field inboundary layer flowsrdquo Journal of Fluid Mechanics vol 18 pp353ndash378 1964

[6] G Corcos ldquoResolution of pressure in turbulencerdquo Journal of theAcoustical Society of America vol 35 no 2 pp 192ndash199 1963

[7] B Efimtsov ldquoCharacteristics of the field of turbulent wallpressure fluctuations at large Reynolds numbersrdquo Soviet PhysicsAcoustics vol 28 no 4 pp 289ndash292 1982

[8] J Reddy Mechanics of Laminated Composite Plates and ShellsTheory and Analysis CRC Press Boca Raton Fla USA 1945

[9] J Whitney Structural Analysis of Laminated Anisotropic PlatesTechnomic Lancaster Pa USA 1987

[10] C L Dym and M A Lang ldquoTransmission of sound throughsandwich panelsrdquo Journal of the Acoustical Society of Americavol 56 no 5 pp 1523ndash1532 1974

[11] C L Dym C S Ventres and M A Lang ldquoTransmission ofsound through sandwich panels a reconsiderationrdquo Journal ofthe Acoustical Society of America vol 59 no 2 pp 364ndash3671976

[12] S Narayanan andR L Shanbhag ldquoSound transmission througha damped sandwich panelrdquo Journal of Sound and Vibration vol80 no 3 pp 315ndash327 1982

18 Shock and Vibration

[13] C L Dym and D C Lang ldquoTransmission loss of dampedasymmetric sandwich panels with orthotropic coresrdquo Journal ofSound and Vibration vol 88 no 3 pp 299ndash319 1983

[14] S Ghinet N Atalla and H Osman ldquoDiffuse field transmissioninto infinite sandwich composite and laminate composite cylin-dersrdquo Journal of Sound and Vibration vol 289 no 4-5 pp 745ndash778 2006

[15] J Mendler ldquoComputational tool for the acoustic analysis ofsandwich panelsrdquo in Proceedings of the European Conference onNoise Control (EURO-NOISE rsquo95) Lyon France 1995

[16] O Foin J Nicolas andNAtalla ldquoAn efficient tool for predictingthe structural acoustic and vibration response of sandwichplates in light or heavy fluidrdquo Applied Acoustics vol 57 no 3pp 213ndash242 1999

[17] S Assaf and M Guerich ldquoNumerical prediction of noisetransmission loss through viscoelastically damped sandwichplatesrdquo Journal of Sandwich Structures andMaterials vol 10 no5 pp 359ndash384 2008

[18] S Assaf M Guerich and P Cuvelier ldquoVibration and acousticresponse of damped sandwich panels immersed in a light orheavy fluidrdquo Computers and Structures vol 88 pp 359ndash3842010

[19] R Zhou and M J Crocker ldquoBoundary element analyses forsound transmission loss of panelsrdquo Journal of the AcousticalSociety of America vol 127 no 2 pp 829ndash840 2010

[20] K Amichi N Atalla and R Ruokolainen ldquoA new 3D finite ele-ment sandwich plate for predicting the vibroacoustic responseof laminated steel panelsrdquo Finite Elements in Analysis andDesign vol 46 no 12 pp 1131ndash1145 2010

[21] F Franco S de Rosa and T Polito ldquoFinite element inves-tigations on the vibroacoustic performance of plane plateswith random stiffnessrdquo Mechanics of Advanced Materials andStructures vol 18 no 7 pp 484ndash497 2011

[22] Y Kim ldquoIdentification of sound transmission characteristicsof honeycomb sandwich panels using hybrid analyticalone-dimensional finite element methodrdquo in Proceedings of the35th International Congress and Exposition on Noise ControlEngineering (INTER-NOISE rsquo06) HonoluluHawaii USA 2006

[23] F Fahy Sound and Structural Vibration Radiation Transmissionand Response Academic Press London UK 1985

[24] J Blevins Formulas for Natural Frequency and Mode ShapeKrieger Publishing Malabar Fla USA 2001

[25] D Newland An Introduction to Random Vibration and SpectralAnalysis Longman New York NY USA 1984

[26] Y Lin Probabilistic Theory of Structural Dynamics McGraw-Hill Book New York NY USA 1967

[27] C E Wallace ldquoRadiation resistance of a rectangular panelrdquoTheJournal of the Acoustical Society of America vol 51 no 3 pp946ndash952 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

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RotatingMachinery

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Shock and VibrationRa

diat

ed so

und

pow

er (d

B re

l1

W)

0 500 1000 1500

Frequency (Hz)

50

0

minus50

minus100

minus150

(a)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus80

minus60

minus40

minus20

0

20

40

60

(b)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

100

50

0

minus50

minus100

minus150

(c)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

minus60

minus40

minus20

0

20

40

60

80

(d)

Radi

ated

soun

d po

wer

(dB

rel1

W)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

50

0

minus50

minus100

minus150

(e)

0 500 1000 1500

Frequency (Hz)

Radi

ated

soun

d po

wer

(dB

rel1

W)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus80

minus60

minus40

minus20

0

20

40

60

80

(f)

Figure 10 Comparison of radiated sound power results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b) overallpanel RSP (c) and (d) RSP near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) RSP near location (119909

2 1199102) = (03119886 02119887)

Shock and Vibration 15

minus50

minus100

minus150

minus250

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus220

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus50

minus100

minus150

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 11 Comparison of panel displacement PSD results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b)overall panel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

16 Shock and Vibration

Table 1 Physical properties of composite panels

Variable Description units Case 1 Case 2 Case 3 Case 4119886 Length m 183 11683 0348 10119887 Width m 122 11683 03048 03120582 Aspect ratio (119886119887) 15 10 114 333ℎ1 ℎ3

Face layers thickness mm 0406 0506 0762 35ℎ2

Core layer thickness mm 64 1905 0254 051198641 1198643

Face layers Young modulus Pa 6898 times 1010

71 times 1010

6898 times 1010

66 times 1010

1198642

Core layer Young modulus Pa 457 times 105

457 times 105

267 times 106

26 times 108

]1 ]3

Face layers Poissonrsquos ratio 033 033 03 033]2

Core layer Poissonrsquos ratio 04 04 049 031205881 1205883

Face layers density kgmminus3 2768 2700 2740 26801205882

Core layer density kgmminus3 122 4806 999 1680119898 Total mass kg 676 499 047 588

Table 2 Physical properties of isotropic panels

Variable Description units Case 3 Case 4 Case 5 Case 6119886 Length m 0348 10 0348 10119887 Width m 03048 03 03048 03120582 Aspect ratio (119886119887) 114 333 114 333ℎ119901

Thickness mm 1 1 15 4119864119901

Young modulus Pa 724 times 1010

724 times 1010

724 times 1010

724 times 1010

]119901

Poissonrsquos ratio 033 033 033 033120588119901

Density kg mminus3 2800 2800 2800 2800119898 Total mass kg 030 084 045 336

Table 3 Flight test conditions and air properties

Variable Description units Value119872 Mach number 08alt Altitude m 13106119888 Speed of sound m sminus1 295119880infin

Flow speed m sminus1 23606120588 Air density kgmminus3 02622] Kinematic viscosity m2 sminus1 5422 times 10

minus5

119879 Temperature K 21665Re119909

Reynolds number 2917 times 107

1198880

Speed of sound m sminus1 3401205880

Internal air density kgmminus3 142

higher amplification of localized frequencies Therefore theface layers thickness increase should be performed carefullyby an accompanied increase in the core layer thickness whilesimultaneously considering the weight constraints for thespecific panels applications

A more detailed analysis will be required in order toformulate final conclusions about the optimum solutions forthe panel in the context of aircraft application For that theinsulation materials would also be considered to account forthe respective added weight and effect in the sound radi-ation signature However typically the acoustic insulationmaterials will only have a higher impact in decreasing the

Table 4 First 10 natural frequencies for the composite panels

Modes 119891 Hz(Case 1)

119891 Hz(Case 2)

119891 Hz(Case 3)

119891 Hz(Case 4)

1 2365 10588 4874 126582 4548 26470 11222 157933 7276 42352 13148 210194 8186 52941 19496 283355 9459 68823 21800 377416 13097 89999 26939 474967 13279 95293 30075 492388 15462 105881 33287 506329 17645 132351 36612 5585710 18191 137645 43866 62825

sound levels at higher frequency ranges and will thereforenot improve significantly the problem of higher noise levelsin the low frequency range particularly observed in theisotropic solutions As such the composite panel solutionsmay appear to be a promising replacement to the isotropicpanels especially panels like Cases 3 and 4 which are alreadycharacterized by lower levels of both RSP and 119878

119908119908in the

low frequency range In further analysis it will also beimportant to carefully address the noise amplification atisolated frequencies by identifying a good combination of

Shock and Vibration 17

Table 5 First 10 natural frequencies for the isotropic panels

Modes 119891 Hz(Case 3)

119891 Hz(Case 4)

119891 Hz(Case 5)

119891 Hz(Case 6)

1 4646 2958 6969 118322 10697 3691 16046 147643 12534 4912 18801 196494 18585 6622 27877 264885 20782 8820 31173 352826 25680 1110 38520 443997 28670 11507 43004 460298 31731 11832 47596 473309 34901 13054 52351 5221510 41816 14683 62723 58731

core and face layers in terms of both materials selection anddimensions

4 Conclusions

This study presents a mathematical method and frameworkon composite panels radiated sound and vibrationThepanelsare considered excited either by turbulent flow or by randomnoiseThe turbulent flow corresponds to the typical excitationof an aircraft panel at flight cruise conditions with a Machnumber of 08 The effect of type of panel of panel dimen-sions and of panel thickness are evaluated The turbulentflow excitation showed to excite a higher number of panelmodes compared to random noise excitation and resulted indissimilar levels of both panel vibration and radiated soundwith relation to the random excitation This effect is morevisible for composite than for isotropic panels and thereforethe pure random noise should not be used as representativeof the turbulent flow excitation in real applications

Different properties on the composite panels are shown toyield dissimilar effects in the noise and vibration predictionsThe effect of the panel size specifically its length and widthis noted to have a significant effect on the panel performancein terms of noise radiation and vibration levels Overallsmaller composite panels or panels with higher aspect ratioproduced lower levels of radiated sound and vibration thanlonger and wider composite panels Compared to isotropicpanels composite panels analyzed are shown to radiate lowernoise levels with the exception of specific natural frequencieswhich are amplified near 300Hz and 1000Hz It is shown thatthis amplification can be significant for composite panels withthicker face layers and thinner core layer and thus resultingin unwanted effect of increased sound radiated Despitethis effect this drawback may potentially be corrected byincreasing the sandwich core layer thickness Increasing thecore layer thickness will impact the panel weight and hencethe composite panels can represent a heavier solution com-pared to the isotropic panels examined On the other handthis approach may be beneficial in the design of compositepanels in the context of aircraft panel applications sincethese panels may require less acoustic insulation material

added to the aircraft walls as they already represent animproved design solution Furthermore acoustic insulationmaterials have a higher impact in decreasing sound levelsat the high frequency ranges and therefore it will notsignificantly improve the problem of higher noise levels at thelow frequency range particularly observed in the isotropicsolutions In this context advanced and optimized compositepanel configurations may be a promising replacement to thetraditional isotropic panels for aerospace applications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is grateful for the support of the National Scienceand Engineering Research Council of Canada

References

[1] J Rocha and D Palumbo ldquoOn the sensitivity of sound powerradiated by aircraft panels to turbulent boundary layer param-etersrdquo Journal of Sound and Vibration vol 331 no 21 pp 4785ndash4806 2012

[2] J Rocha A Suleman and F Lau ldquoPrediction of turbulentboundary layer induced noise in the cabin of a BWB aircraftrdquoShock and Vibration vol 19 no 4 pp 693ndash705 2012

[3] J da Rocha A Suleman and F Lau ldquoFlow-induced noise andvibration in aircraft cylindrical cabins closed-form analyticalmodel validationrdquo Journal of Vibration and Acoustics vol 133no 5 Article ID 051013 9 pages 2011

[4] J da Rocha A Suleman and F Lau ldquoPrediction of flow-inducednoise in transport vehicles development and validation of acoupled structuralmdashacoustic analytical frameworkrdquo CanadianAcousticsmdashAcoustique Canadienne vol 37 no 4 pp 13ndash292009

[5] G Corcos ldquoThe structure of the turbulent pressure field inboundary layer flowsrdquo Journal of Fluid Mechanics vol 18 pp353ndash378 1964

[6] G Corcos ldquoResolution of pressure in turbulencerdquo Journal of theAcoustical Society of America vol 35 no 2 pp 192ndash199 1963

[7] B Efimtsov ldquoCharacteristics of the field of turbulent wallpressure fluctuations at large Reynolds numbersrdquo Soviet PhysicsAcoustics vol 28 no 4 pp 289ndash292 1982

[8] J Reddy Mechanics of Laminated Composite Plates and ShellsTheory and Analysis CRC Press Boca Raton Fla USA 1945

[9] J Whitney Structural Analysis of Laminated Anisotropic PlatesTechnomic Lancaster Pa USA 1987

[10] C L Dym and M A Lang ldquoTransmission of sound throughsandwich panelsrdquo Journal of the Acoustical Society of Americavol 56 no 5 pp 1523ndash1532 1974

[11] C L Dym C S Ventres and M A Lang ldquoTransmission ofsound through sandwich panels a reconsiderationrdquo Journal ofthe Acoustical Society of America vol 59 no 2 pp 364ndash3671976

[12] S Narayanan andR L Shanbhag ldquoSound transmission througha damped sandwich panelrdquo Journal of Sound and Vibration vol80 no 3 pp 315ndash327 1982

18 Shock and Vibration

[13] C L Dym and D C Lang ldquoTransmission loss of dampedasymmetric sandwich panels with orthotropic coresrdquo Journal ofSound and Vibration vol 88 no 3 pp 299ndash319 1983

[14] S Ghinet N Atalla and H Osman ldquoDiffuse field transmissioninto infinite sandwich composite and laminate composite cylin-dersrdquo Journal of Sound and Vibration vol 289 no 4-5 pp 745ndash778 2006

[15] J Mendler ldquoComputational tool for the acoustic analysis ofsandwich panelsrdquo in Proceedings of the European Conference onNoise Control (EURO-NOISE rsquo95) Lyon France 1995

[16] O Foin J Nicolas andNAtalla ldquoAn efficient tool for predictingthe structural acoustic and vibration response of sandwichplates in light or heavy fluidrdquo Applied Acoustics vol 57 no 3pp 213ndash242 1999

[17] S Assaf and M Guerich ldquoNumerical prediction of noisetransmission loss through viscoelastically damped sandwichplatesrdquo Journal of Sandwich Structures andMaterials vol 10 no5 pp 359ndash384 2008

[18] S Assaf M Guerich and P Cuvelier ldquoVibration and acousticresponse of damped sandwich panels immersed in a light orheavy fluidrdquo Computers and Structures vol 88 pp 359ndash3842010

[19] R Zhou and M J Crocker ldquoBoundary element analyses forsound transmission loss of panelsrdquo Journal of the AcousticalSociety of America vol 127 no 2 pp 829ndash840 2010

[20] K Amichi N Atalla and R Ruokolainen ldquoA new 3D finite ele-ment sandwich plate for predicting the vibroacoustic responseof laminated steel panelsrdquo Finite Elements in Analysis andDesign vol 46 no 12 pp 1131ndash1145 2010

[21] F Franco S de Rosa and T Polito ldquoFinite element inves-tigations on the vibroacoustic performance of plane plateswith random stiffnessrdquo Mechanics of Advanced Materials andStructures vol 18 no 7 pp 484ndash497 2011

[22] Y Kim ldquoIdentification of sound transmission characteristicsof honeycomb sandwich panels using hybrid analyticalone-dimensional finite element methodrdquo in Proceedings of the35th International Congress and Exposition on Noise ControlEngineering (INTER-NOISE rsquo06) HonoluluHawaii USA 2006

[23] F Fahy Sound and Structural Vibration Radiation Transmissionand Response Academic Press London UK 1985

[24] J Blevins Formulas for Natural Frequency and Mode ShapeKrieger Publishing Malabar Fla USA 2001

[25] D Newland An Introduction to Random Vibration and SpectralAnalysis Longman New York NY USA 1984

[26] Y Lin Probabilistic Theory of Structural Dynamics McGraw-Hill Book New York NY USA 1967

[27] C E Wallace ldquoRadiation resistance of a rectangular panelrdquoTheJournal of the Acoustical Society of America vol 51 no 3 pp946ndash952 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 15

minus50

minus100

minus150

minus250

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

0 500 1000 1500

Frequency (Hz)

(a)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(b)

0 500 1000 1500

Frequency (Hz)

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(c)

0 500 1000 1500

Frequency (Hz)

minus40

minus60

minus80

minus100

minus120

minus140

minus160

minus180

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(d)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic TBLCase 4 composite TBL

Case 6 isotropic TBL

minus60

minus80

minus100

minus120

minus140

minus160

minus180

minus220

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(e)

0 500 1000 1500

Frequency (Hz)

Case 4 isotropic randomCase 4 composite random

Case 6 isotropic random

minus50

minus100

minus150

minus200

Plat

e disp

lace

men

t PSD

(dB

re1

m2)

(f)

Figure 11 Comparison of panel displacement PSD results for Case 4 composite panel versus Cases 4 and 6 isotropic panels (a) and (b)overall panel 119878

119908119908 (c) and (d) 119878

119908119908near surface location (119909

1 1199101) = (05119886 05119887) (e) and (f) 119878

119908119908near location (119909

2 1199102) = (03119886 02119887)

16 Shock and Vibration

Table 1 Physical properties of composite panels

Variable Description units Case 1 Case 2 Case 3 Case 4119886 Length m 183 11683 0348 10119887 Width m 122 11683 03048 03120582 Aspect ratio (119886119887) 15 10 114 333ℎ1 ℎ3

Face layers thickness mm 0406 0506 0762 35ℎ2

Core layer thickness mm 64 1905 0254 051198641 1198643

Face layers Young modulus Pa 6898 times 1010

71 times 1010

6898 times 1010

66 times 1010

1198642

Core layer Young modulus Pa 457 times 105

457 times 105

267 times 106

26 times 108

]1 ]3

Face layers Poissonrsquos ratio 033 033 03 033]2

Core layer Poissonrsquos ratio 04 04 049 031205881 1205883

Face layers density kgmminus3 2768 2700 2740 26801205882

Core layer density kgmminus3 122 4806 999 1680119898 Total mass kg 676 499 047 588

Table 2 Physical properties of isotropic panels

Variable Description units Case 3 Case 4 Case 5 Case 6119886 Length m 0348 10 0348 10119887 Width m 03048 03 03048 03120582 Aspect ratio (119886119887) 114 333 114 333ℎ119901

Thickness mm 1 1 15 4119864119901

Young modulus Pa 724 times 1010

724 times 1010

724 times 1010

724 times 1010

]119901

Poissonrsquos ratio 033 033 033 033120588119901

Density kg mminus3 2800 2800 2800 2800119898 Total mass kg 030 084 045 336

Table 3 Flight test conditions and air properties

Variable Description units Value119872 Mach number 08alt Altitude m 13106119888 Speed of sound m sminus1 295119880infin

Flow speed m sminus1 23606120588 Air density kgmminus3 02622] Kinematic viscosity m2 sminus1 5422 times 10

minus5

119879 Temperature K 21665Re119909

Reynolds number 2917 times 107

1198880

Speed of sound m sminus1 3401205880

Internal air density kgmminus3 142

higher amplification of localized frequencies Therefore theface layers thickness increase should be performed carefullyby an accompanied increase in the core layer thickness whilesimultaneously considering the weight constraints for thespecific panels applications

A more detailed analysis will be required in order toformulate final conclusions about the optimum solutions forthe panel in the context of aircraft application For that theinsulation materials would also be considered to account forthe respective added weight and effect in the sound radi-ation signature However typically the acoustic insulationmaterials will only have a higher impact in decreasing the

Table 4 First 10 natural frequencies for the composite panels

Modes 119891 Hz(Case 1)

119891 Hz(Case 2)

119891 Hz(Case 3)

119891 Hz(Case 4)

1 2365 10588 4874 126582 4548 26470 11222 157933 7276 42352 13148 210194 8186 52941 19496 283355 9459 68823 21800 377416 13097 89999 26939 474967 13279 95293 30075 492388 15462 105881 33287 506329 17645 132351 36612 5585710 18191 137645 43866 62825

sound levels at higher frequency ranges and will thereforenot improve significantly the problem of higher noise levelsin the low frequency range particularly observed in theisotropic solutions As such the composite panel solutionsmay appear to be a promising replacement to the isotropicpanels especially panels like Cases 3 and 4 which are alreadycharacterized by lower levels of both RSP and 119878

119908119908in the

low frequency range In further analysis it will also beimportant to carefully address the noise amplification atisolated frequencies by identifying a good combination of

Shock and Vibration 17

Table 5 First 10 natural frequencies for the isotropic panels

Modes 119891 Hz(Case 3)

119891 Hz(Case 4)

119891 Hz(Case 5)

119891 Hz(Case 6)

1 4646 2958 6969 118322 10697 3691 16046 147643 12534 4912 18801 196494 18585 6622 27877 264885 20782 8820 31173 352826 25680 1110 38520 443997 28670 11507 43004 460298 31731 11832 47596 473309 34901 13054 52351 5221510 41816 14683 62723 58731

core and face layers in terms of both materials selection anddimensions

4 Conclusions

This study presents a mathematical method and frameworkon composite panels radiated sound and vibrationThepanelsare considered excited either by turbulent flow or by randomnoiseThe turbulent flow corresponds to the typical excitationof an aircraft panel at flight cruise conditions with a Machnumber of 08 The effect of type of panel of panel dimen-sions and of panel thickness are evaluated The turbulentflow excitation showed to excite a higher number of panelmodes compared to random noise excitation and resulted indissimilar levels of both panel vibration and radiated soundwith relation to the random excitation This effect is morevisible for composite than for isotropic panels and thereforethe pure random noise should not be used as representativeof the turbulent flow excitation in real applications

Different properties on the composite panels are shown toyield dissimilar effects in the noise and vibration predictionsThe effect of the panel size specifically its length and widthis noted to have a significant effect on the panel performancein terms of noise radiation and vibration levels Overallsmaller composite panels or panels with higher aspect ratioproduced lower levels of radiated sound and vibration thanlonger and wider composite panels Compared to isotropicpanels composite panels analyzed are shown to radiate lowernoise levels with the exception of specific natural frequencieswhich are amplified near 300Hz and 1000Hz It is shown thatthis amplification can be significant for composite panels withthicker face layers and thinner core layer and thus resultingin unwanted effect of increased sound radiated Despitethis effect this drawback may potentially be corrected byincreasing the sandwich core layer thickness Increasing thecore layer thickness will impact the panel weight and hencethe composite panels can represent a heavier solution com-pared to the isotropic panels examined On the other handthis approach may be beneficial in the design of compositepanels in the context of aircraft panel applications sincethese panels may require less acoustic insulation material

added to the aircraft walls as they already represent animproved design solution Furthermore acoustic insulationmaterials have a higher impact in decreasing sound levelsat the high frequency ranges and therefore it will notsignificantly improve the problem of higher noise levels at thelow frequency range particularly observed in the isotropicsolutions In this context advanced and optimized compositepanel configurations may be a promising replacement to thetraditional isotropic panels for aerospace applications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is grateful for the support of the National Scienceand Engineering Research Council of Canada

References

[1] J Rocha and D Palumbo ldquoOn the sensitivity of sound powerradiated by aircraft panels to turbulent boundary layer param-etersrdquo Journal of Sound and Vibration vol 331 no 21 pp 4785ndash4806 2012

[2] J Rocha A Suleman and F Lau ldquoPrediction of turbulentboundary layer induced noise in the cabin of a BWB aircraftrdquoShock and Vibration vol 19 no 4 pp 693ndash705 2012

[3] J da Rocha A Suleman and F Lau ldquoFlow-induced noise andvibration in aircraft cylindrical cabins closed-form analyticalmodel validationrdquo Journal of Vibration and Acoustics vol 133no 5 Article ID 051013 9 pages 2011

[4] J da Rocha A Suleman and F Lau ldquoPrediction of flow-inducednoise in transport vehicles development and validation of acoupled structuralmdashacoustic analytical frameworkrdquo CanadianAcousticsmdashAcoustique Canadienne vol 37 no 4 pp 13ndash292009

[5] G Corcos ldquoThe structure of the turbulent pressure field inboundary layer flowsrdquo Journal of Fluid Mechanics vol 18 pp353ndash378 1964

[6] G Corcos ldquoResolution of pressure in turbulencerdquo Journal of theAcoustical Society of America vol 35 no 2 pp 192ndash199 1963

[7] B Efimtsov ldquoCharacteristics of the field of turbulent wallpressure fluctuations at large Reynolds numbersrdquo Soviet PhysicsAcoustics vol 28 no 4 pp 289ndash292 1982

[8] J Reddy Mechanics of Laminated Composite Plates and ShellsTheory and Analysis CRC Press Boca Raton Fla USA 1945

[9] J Whitney Structural Analysis of Laminated Anisotropic PlatesTechnomic Lancaster Pa USA 1987

[10] C L Dym and M A Lang ldquoTransmission of sound throughsandwich panelsrdquo Journal of the Acoustical Society of Americavol 56 no 5 pp 1523ndash1532 1974

[11] C L Dym C S Ventres and M A Lang ldquoTransmission ofsound through sandwich panels a reconsiderationrdquo Journal ofthe Acoustical Society of America vol 59 no 2 pp 364ndash3671976

[12] S Narayanan andR L Shanbhag ldquoSound transmission througha damped sandwich panelrdquo Journal of Sound and Vibration vol80 no 3 pp 315ndash327 1982

18 Shock and Vibration

[13] C L Dym and D C Lang ldquoTransmission loss of dampedasymmetric sandwich panels with orthotropic coresrdquo Journal ofSound and Vibration vol 88 no 3 pp 299ndash319 1983

[14] S Ghinet N Atalla and H Osman ldquoDiffuse field transmissioninto infinite sandwich composite and laminate composite cylin-dersrdquo Journal of Sound and Vibration vol 289 no 4-5 pp 745ndash778 2006

[15] J Mendler ldquoComputational tool for the acoustic analysis ofsandwich panelsrdquo in Proceedings of the European Conference onNoise Control (EURO-NOISE rsquo95) Lyon France 1995

[16] O Foin J Nicolas andNAtalla ldquoAn efficient tool for predictingthe structural acoustic and vibration response of sandwichplates in light or heavy fluidrdquo Applied Acoustics vol 57 no 3pp 213ndash242 1999

[17] S Assaf and M Guerich ldquoNumerical prediction of noisetransmission loss through viscoelastically damped sandwichplatesrdquo Journal of Sandwich Structures andMaterials vol 10 no5 pp 359ndash384 2008

[18] S Assaf M Guerich and P Cuvelier ldquoVibration and acousticresponse of damped sandwich panels immersed in a light orheavy fluidrdquo Computers and Structures vol 88 pp 359ndash3842010

[19] R Zhou and M J Crocker ldquoBoundary element analyses forsound transmission loss of panelsrdquo Journal of the AcousticalSociety of America vol 127 no 2 pp 829ndash840 2010

[20] K Amichi N Atalla and R Ruokolainen ldquoA new 3D finite ele-ment sandwich plate for predicting the vibroacoustic responseof laminated steel panelsrdquo Finite Elements in Analysis andDesign vol 46 no 12 pp 1131ndash1145 2010

[21] F Franco S de Rosa and T Polito ldquoFinite element inves-tigations on the vibroacoustic performance of plane plateswith random stiffnessrdquo Mechanics of Advanced Materials andStructures vol 18 no 7 pp 484ndash497 2011

[22] Y Kim ldquoIdentification of sound transmission characteristicsof honeycomb sandwich panels using hybrid analyticalone-dimensional finite element methodrdquo in Proceedings of the35th International Congress and Exposition on Noise ControlEngineering (INTER-NOISE rsquo06) HonoluluHawaii USA 2006

[23] F Fahy Sound and Structural Vibration Radiation Transmissionand Response Academic Press London UK 1985

[24] J Blevins Formulas for Natural Frequency and Mode ShapeKrieger Publishing Malabar Fla USA 2001

[25] D Newland An Introduction to Random Vibration and SpectralAnalysis Longman New York NY USA 1984

[26] Y Lin Probabilistic Theory of Structural Dynamics McGraw-Hill Book New York NY USA 1967

[27] C E Wallace ldquoRadiation resistance of a rectangular panelrdquoTheJournal of the Acoustical Society of America vol 51 no 3 pp946ndash952 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

16 Shock and Vibration

Table 1 Physical properties of composite panels

Variable Description units Case 1 Case 2 Case 3 Case 4119886 Length m 183 11683 0348 10119887 Width m 122 11683 03048 03120582 Aspect ratio (119886119887) 15 10 114 333ℎ1 ℎ3

Face layers thickness mm 0406 0506 0762 35ℎ2

Core layer thickness mm 64 1905 0254 051198641 1198643

Face layers Young modulus Pa 6898 times 1010

71 times 1010

6898 times 1010

66 times 1010

1198642

Core layer Young modulus Pa 457 times 105

457 times 105

267 times 106

26 times 108

]1 ]3

Face layers Poissonrsquos ratio 033 033 03 033]2

Core layer Poissonrsquos ratio 04 04 049 031205881 1205883

Face layers density kgmminus3 2768 2700 2740 26801205882

Core layer density kgmminus3 122 4806 999 1680119898 Total mass kg 676 499 047 588

Table 2 Physical properties of isotropic panels

Variable Description units Case 3 Case 4 Case 5 Case 6119886 Length m 0348 10 0348 10119887 Width m 03048 03 03048 03120582 Aspect ratio (119886119887) 114 333 114 333ℎ119901

Thickness mm 1 1 15 4119864119901

Young modulus Pa 724 times 1010

724 times 1010

724 times 1010

724 times 1010

]119901

Poissonrsquos ratio 033 033 033 033120588119901

Density kg mminus3 2800 2800 2800 2800119898 Total mass kg 030 084 045 336

Table 3 Flight test conditions and air properties

Variable Description units Value119872 Mach number 08alt Altitude m 13106119888 Speed of sound m sminus1 295119880infin

Flow speed m sminus1 23606120588 Air density kgmminus3 02622] Kinematic viscosity m2 sminus1 5422 times 10

minus5

119879 Temperature K 21665Re119909

Reynolds number 2917 times 107

1198880

Speed of sound m sminus1 3401205880

Internal air density kgmminus3 142

higher amplification of localized frequencies Therefore theface layers thickness increase should be performed carefullyby an accompanied increase in the core layer thickness whilesimultaneously considering the weight constraints for thespecific panels applications

A more detailed analysis will be required in order toformulate final conclusions about the optimum solutions forthe panel in the context of aircraft application For that theinsulation materials would also be considered to account forthe respective added weight and effect in the sound radi-ation signature However typically the acoustic insulationmaterials will only have a higher impact in decreasing the

Table 4 First 10 natural frequencies for the composite panels

Modes 119891 Hz(Case 1)

119891 Hz(Case 2)

119891 Hz(Case 3)

119891 Hz(Case 4)

1 2365 10588 4874 126582 4548 26470 11222 157933 7276 42352 13148 210194 8186 52941 19496 283355 9459 68823 21800 377416 13097 89999 26939 474967 13279 95293 30075 492388 15462 105881 33287 506329 17645 132351 36612 5585710 18191 137645 43866 62825

sound levels at higher frequency ranges and will thereforenot improve significantly the problem of higher noise levelsin the low frequency range particularly observed in theisotropic solutions As such the composite panel solutionsmay appear to be a promising replacement to the isotropicpanels especially panels like Cases 3 and 4 which are alreadycharacterized by lower levels of both RSP and 119878

119908119908in the

low frequency range In further analysis it will also beimportant to carefully address the noise amplification atisolated frequencies by identifying a good combination of

Shock and Vibration 17

Table 5 First 10 natural frequencies for the isotropic panels

Modes 119891 Hz(Case 3)

119891 Hz(Case 4)

119891 Hz(Case 5)

119891 Hz(Case 6)

1 4646 2958 6969 118322 10697 3691 16046 147643 12534 4912 18801 196494 18585 6622 27877 264885 20782 8820 31173 352826 25680 1110 38520 443997 28670 11507 43004 460298 31731 11832 47596 473309 34901 13054 52351 5221510 41816 14683 62723 58731

core and face layers in terms of both materials selection anddimensions

4 Conclusions

This study presents a mathematical method and frameworkon composite panels radiated sound and vibrationThepanelsare considered excited either by turbulent flow or by randomnoiseThe turbulent flow corresponds to the typical excitationof an aircraft panel at flight cruise conditions with a Machnumber of 08 The effect of type of panel of panel dimen-sions and of panel thickness are evaluated The turbulentflow excitation showed to excite a higher number of panelmodes compared to random noise excitation and resulted indissimilar levels of both panel vibration and radiated soundwith relation to the random excitation This effect is morevisible for composite than for isotropic panels and thereforethe pure random noise should not be used as representativeof the turbulent flow excitation in real applications

Different properties on the composite panels are shown toyield dissimilar effects in the noise and vibration predictionsThe effect of the panel size specifically its length and widthis noted to have a significant effect on the panel performancein terms of noise radiation and vibration levels Overallsmaller composite panels or panels with higher aspect ratioproduced lower levels of radiated sound and vibration thanlonger and wider composite panels Compared to isotropicpanels composite panels analyzed are shown to radiate lowernoise levels with the exception of specific natural frequencieswhich are amplified near 300Hz and 1000Hz It is shown thatthis amplification can be significant for composite panels withthicker face layers and thinner core layer and thus resultingin unwanted effect of increased sound radiated Despitethis effect this drawback may potentially be corrected byincreasing the sandwich core layer thickness Increasing thecore layer thickness will impact the panel weight and hencethe composite panels can represent a heavier solution com-pared to the isotropic panels examined On the other handthis approach may be beneficial in the design of compositepanels in the context of aircraft panel applications sincethese panels may require less acoustic insulation material

added to the aircraft walls as they already represent animproved design solution Furthermore acoustic insulationmaterials have a higher impact in decreasing sound levelsat the high frequency ranges and therefore it will notsignificantly improve the problem of higher noise levels at thelow frequency range particularly observed in the isotropicsolutions In this context advanced and optimized compositepanel configurations may be a promising replacement to thetraditional isotropic panels for aerospace applications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is grateful for the support of the National Scienceand Engineering Research Council of Canada

References

[1] J Rocha and D Palumbo ldquoOn the sensitivity of sound powerradiated by aircraft panels to turbulent boundary layer param-etersrdquo Journal of Sound and Vibration vol 331 no 21 pp 4785ndash4806 2012

[2] J Rocha A Suleman and F Lau ldquoPrediction of turbulentboundary layer induced noise in the cabin of a BWB aircraftrdquoShock and Vibration vol 19 no 4 pp 693ndash705 2012

[3] J da Rocha A Suleman and F Lau ldquoFlow-induced noise andvibration in aircraft cylindrical cabins closed-form analyticalmodel validationrdquo Journal of Vibration and Acoustics vol 133no 5 Article ID 051013 9 pages 2011

[4] J da Rocha A Suleman and F Lau ldquoPrediction of flow-inducednoise in transport vehicles development and validation of acoupled structuralmdashacoustic analytical frameworkrdquo CanadianAcousticsmdashAcoustique Canadienne vol 37 no 4 pp 13ndash292009

[5] G Corcos ldquoThe structure of the turbulent pressure field inboundary layer flowsrdquo Journal of Fluid Mechanics vol 18 pp353ndash378 1964

[6] G Corcos ldquoResolution of pressure in turbulencerdquo Journal of theAcoustical Society of America vol 35 no 2 pp 192ndash199 1963

[7] B Efimtsov ldquoCharacteristics of the field of turbulent wallpressure fluctuations at large Reynolds numbersrdquo Soviet PhysicsAcoustics vol 28 no 4 pp 289ndash292 1982

[8] J Reddy Mechanics of Laminated Composite Plates and ShellsTheory and Analysis CRC Press Boca Raton Fla USA 1945

[9] J Whitney Structural Analysis of Laminated Anisotropic PlatesTechnomic Lancaster Pa USA 1987

[10] C L Dym and M A Lang ldquoTransmission of sound throughsandwich panelsrdquo Journal of the Acoustical Society of Americavol 56 no 5 pp 1523ndash1532 1974

[11] C L Dym C S Ventres and M A Lang ldquoTransmission ofsound through sandwich panels a reconsiderationrdquo Journal ofthe Acoustical Society of America vol 59 no 2 pp 364ndash3671976

[12] S Narayanan andR L Shanbhag ldquoSound transmission througha damped sandwich panelrdquo Journal of Sound and Vibration vol80 no 3 pp 315ndash327 1982

18 Shock and Vibration

[13] C L Dym and D C Lang ldquoTransmission loss of dampedasymmetric sandwich panels with orthotropic coresrdquo Journal ofSound and Vibration vol 88 no 3 pp 299ndash319 1983

[14] S Ghinet N Atalla and H Osman ldquoDiffuse field transmissioninto infinite sandwich composite and laminate composite cylin-dersrdquo Journal of Sound and Vibration vol 289 no 4-5 pp 745ndash778 2006

[15] J Mendler ldquoComputational tool for the acoustic analysis ofsandwich panelsrdquo in Proceedings of the European Conference onNoise Control (EURO-NOISE rsquo95) Lyon France 1995

[16] O Foin J Nicolas andNAtalla ldquoAn efficient tool for predictingthe structural acoustic and vibration response of sandwichplates in light or heavy fluidrdquo Applied Acoustics vol 57 no 3pp 213ndash242 1999

[17] S Assaf and M Guerich ldquoNumerical prediction of noisetransmission loss through viscoelastically damped sandwichplatesrdquo Journal of Sandwich Structures andMaterials vol 10 no5 pp 359ndash384 2008

[18] S Assaf M Guerich and P Cuvelier ldquoVibration and acousticresponse of damped sandwich panels immersed in a light orheavy fluidrdquo Computers and Structures vol 88 pp 359ndash3842010

[19] R Zhou and M J Crocker ldquoBoundary element analyses forsound transmission loss of panelsrdquo Journal of the AcousticalSociety of America vol 127 no 2 pp 829ndash840 2010

[20] K Amichi N Atalla and R Ruokolainen ldquoA new 3D finite ele-ment sandwich plate for predicting the vibroacoustic responseof laminated steel panelsrdquo Finite Elements in Analysis andDesign vol 46 no 12 pp 1131ndash1145 2010

[21] F Franco S de Rosa and T Polito ldquoFinite element inves-tigations on the vibroacoustic performance of plane plateswith random stiffnessrdquo Mechanics of Advanced Materials andStructures vol 18 no 7 pp 484ndash497 2011

[22] Y Kim ldquoIdentification of sound transmission characteristicsof honeycomb sandwich panels using hybrid analyticalone-dimensional finite element methodrdquo in Proceedings of the35th International Congress and Exposition on Noise ControlEngineering (INTER-NOISE rsquo06) HonoluluHawaii USA 2006

[23] F Fahy Sound and Structural Vibration Radiation Transmissionand Response Academic Press London UK 1985

[24] J Blevins Formulas for Natural Frequency and Mode ShapeKrieger Publishing Malabar Fla USA 2001

[25] D Newland An Introduction to Random Vibration and SpectralAnalysis Longman New York NY USA 1984

[26] Y Lin Probabilistic Theory of Structural Dynamics McGraw-Hill Book New York NY USA 1967

[27] C E Wallace ldquoRadiation resistance of a rectangular panelrdquoTheJournal of the Acoustical Society of America vol 51 no 3 pp946ndash952 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 17

Table 5 First 10 natural frequencies for the isotropic panels

Modes 119891 Hz(Case 3)

119891 Hz(Case 4)

119891 Hz(Case 5)

119891 Hz(Case 6)

1 4646 2958 6969 118322 10697 3691 16046 147643 12534 4912 18801 196494 18585 6622 27877 264885 20782 8820 31173 352826 25680 1110 38520 443997 28670 11507 43004 460298 31731 11832 47596 473309 34901 13054 52351 5221510 41816 14683 62723 58731

core and face layers in terms of both materials selection anddimensions

4 Conclusions

This study presents a mathematical method and frameworkon composite panels radiated sound and vibrationThepanelsare considered excited either by turbulent flow or by randomnoiseThe turbulent flow corresponds to the typical excitationof an aircraft panel at flight cruise conditions with a Machnumber of 08 The effect of type of panel of panel dimen-sions and of panel thickness are evaluated The turbulentflow excitation showed to excite a higher number of panelmodes compared to random noise excitation and resulted indissimilar levels of both panel vibration and radiated soundwith relation to the random excitation This effect is morevisible for composite than for isotropic panels and thereforethe pure random noise should not be used as representativeof the turbulent flow excitation in real applications

Different properties on the composite panels are shown toyield dissimilar effects in the noise and vibration predictionsThe effect of the panel size specifically its length and widthis noted to have a significant effect on the panel performancein terms of noise radiation and vibration levels Overallsmaller composite panels or panels with higher aspect ratioproduced lower levels of radiated sound and vibration thanlonger and wider composite panels Compared to isotropicpanels composite panels analyzed are shown to radiate lowernoise levels with the exception of specific natural frequencieswhich are amplified near 300Hz and 1000Hz It is shown thatthis amplification can be significant for composite panels withthicker face layers and thinner core layer and thus resultingin unwanted effect of increased sound radiated Despitethis effect this drawback may potentially be corrected byincreasing the sandwich core layer thickness Increasing thecore layer thickness will impact the panel weight and hencethe composite panels can represent a heavier solution com-pared to the isotropic panels examined On the other handthis approach may be beneficial in the design of compositepanels in the context of aircraft panel applications sincethese panels may require less acoustic insulation material

added to the aircraft walls as they already represent animproved design solution Furthermore acoustic insulationmaterials have a higher impact in decreasing sound levelsat the high frequency ranges and therefore it will notsignificantly improve the problem of higher noise levels at thelow frequency range particularly observed in the isotropicsolutions In this context advanced and optimized compositepanel configurations may be a promising replacement to thetraditional isotropic panels for aerospace applications

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is grateful for the support of the National Scienceand Engineering Research Council of Canada

References

[1] J Rocha and D Palumbo ldquoOn the sensitivity of sound powerradiated by aircraft panels to turbulent boundary layer param-etersrdquo Journal of Sound and Vibration vol 331 no 21 pp 4785ndash4806 2012

[2] J Rocha A Suleman and F Lau ldquoPrediction of turbulentboundary layer induced noise in the cabin of a BWB aircraftrdquoShock and Vibration vol 19 no 4 pp 693ndash705 2012

[3] J da Rocha A Suleman and F Lau ldquoFlow-induced noise andvibration in aircraft cylindrical cabins closed-form analyticalmodel validationrdquo Journal of Vibration and Acoustics vol 133no 5 Article ID 051013 9 pages 2011

[4] J da Rocha A Suleman and F Lau ldquoPrediction of flow-inducednoise in transport vehicles development and validation of acoupled structuralmdashacoustic analytical frameworkrdquo CanadianAcousticsmdashAcoustique Canadienne vol 37 no 4 pp 13ndash292009

[5] G Corcos ldquoThe structure of the turbulent pressure field inboundary layer flowsrdquo Journal of Fluid Mechanics vol 18 pp353ndash378 1964

[6] G Corcos ldquoResolution of pressure in turbulencerdquo Journal of theAcoustical Society of America vol 35 no 2 pp 192ndash199 1963

[7] B Efimtsov ldquoCharacteristics of the field of turbulent wallpressure fluctuations at large Reynolds numbersrdquo Soviet PhysicsAcoustics vol 28 no 4 pp 289ndash292 1982

[8] J Reddy Mechanics of Laminated Composite Plates and ShellsTheory and Analysis CRC Press Boca Raton Fla USA 1945

[9] J Whitney Structural Analysis of Laminated Anisotropic PlatesTechnomic Lancaster Pa USA 1987

[10] C L Dym and M A Lang ldquoTransmission of sound throughsandwich panelsrdquo Journal of the Acoustical Society of Americavol 56 no 5 pp 1523ndash1532 1974

[11] C L Dym C S Ventres and M A Lang ldquoTransmission ofsound through sandwich panels a reconsiderationrdquo Journal ofthe Acoustical Society of America vol 59 no 2 pp 364ndash3671976

[12] S Narayanan andR L Shanbhag ldquoSound transmission througha damped sandwich panelrdquo Journal of Sound and Vibration vol80 no 3 pp 315ndash327 1982

18 Shock and Vibration

[13] C L Dym and D C Lang ldquoTransmission loss of dampedasymmetric sandwich panels with orthotropic coresrdquo Journal ofSound and Vibration vol 88 no 3 pp 299ndash319 1983

[14] S Ghinet N Atalla and H Osman ldquoDiffuse field transmissioninto infinite sandwich composite and laminate composite cylin-dersrdquo Journal of Sound and Vibration vol 289 no 4-5 pp 745ndash778 2006

[15] J Mendler ldquoComputational tool for the acoustic analysis ofsandwich panelsrdquo in Proceedings of the European Conference onNoise Control (EURO-NOISE rsquo95) Lyon France 1995

[16] O Foin J Nicolas andNAtalla ldquoAn efficient tool for predictingthe structural acoustic and vibration response of sandwichplates in light or heavy fluidrdquo Applied Acoustics vol 57 no 3pp 213ndash242 1999

[17] S Assaf and M Guerich ldquoNumerical prediction of noisetransmission loss through viscoelastically damped sandwichplatesrdquo Journal of Sandwich Structures andMaterials vol 10 no5 pp 359ndash384 2008

[18] S Assaf M Guerich and P Cuvelier ldquoVibration and acousticresponse of damped sandwich panels immersed in a light orheavy fluidrdquo Computers and Structures vol 88 pp 359ndash3842010

[19] R Zhou and M J Crocker ldquoBoundary element analyses forsound transmission loss of panelsrdquo Journal of the AcousticalSociety of America vol 127 no 2 pp 829ndash840 2010

[20] K Amichi N Atalla and R Ruokolainen ldquoA new 3D finite ele-ment sandwich plate for predicting the vibroacoustic responseof laminated steel panelsrdquo Finite Elements in Analysis andDesign vol 46 no 12 pp 1131ndash1145 2010

[21] F Franco S de Rosa and T Polito ldquoFinite element inves-tigations on the vibroacoustic performance of plane plateswith random stiffnessrdquo Mechanics of Advanced Materials andStructures vol 18 no 7 pp 484ndash497 2011

[22] Y Kim ldquoIdentification of sound transmission characteristicsof honeycomb sandwich panels using hybrid analyticalone-dimensional finite element methodrdquo in Proceedings of the35th International Congress and Exposition on Noise ControlEngineering (INTER-NOISE rsquo06) HonoluluHawaii USA 2006

[23] F Fahy Sound and Structural Vibration Radiation Transmissionand Response Academic Press London UK 1985

[24] J Blevins Formulas for Natural Frequency and Mode ShapeKrieger Publishing Malabar Fla USA 2001

[25] D Newland An Introduction to Random Vibration and SpectralAnalysis Longman New York NY USA 1984

[26] Y Lin Probabilistic Theory of Structural Dynamics McGraw-Hill Book New York NY USA 1967

[27] C E Wallace ldquoRadiation resistance of a rectangular panelrdquoTheJournal of the Acoustical Society of America vol 51 no 3 pp946ndash952 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

18 Shock and Vibration

[13] C L Dym and D C Lang ldquoTransmission loss of dampedasymmetric sandwich panels with orthotropic coresrdquo Journal ofSound and Vibration vol 88 no 3 pp 299ndash319 1983

[14] S Ghinet N Atalla and H Osman ldquoDiffuse field transmissioninto infinite sandwich composite and laminate composite cylin-dersrdquo Journal of Sound and Vibration vol 289 no 4-5 pp 745ndash778 2006

[15] J Mendler ldquoComputational tool for the acoustic analysis ofsandwich panelsrdquo in Proceedings of the European Conference onNoise Control (EURO-NOISE rsquo95) Lyon France 1995

[16] O Foin J Nicolas andNAtalla ldquoAn efficient tool for predictingthe structural acoustic and vibration response of sandwichplates in light or heavy fluidrdquo Applied Acoustics vol 57 no 3pp 213ndash242 1999

[17] S Assaf and M Guerich ldquoNumerical prediction of noisetransmission loss through viscoelastically damped sandwichplatesrdquo Journal of Sandwich Structures andMaterials vol 10 no5 pp 359ndash384 2008

[18] S Assaf M Guerich and P Cuvelier ldquoVibration and acousticresponse of damped sandwich panels immersed in a light orheavy fluidrdquo Computers and Structures vol 88 pp 359ndash3842010

[19] R Zhou and M J Crocker ldquoBoundary element analyses forsound transmission loss of panelsrdquo Journal of the AcousticalSociety of America vol 127 no 2 pp 829ndash840 2010

[20] K Amichi N Atalla and R Ruokolainen ldquoA new 3D finite ele-ment sandwich plate for predicting the vibroacoustic responseof laminated steel panelsrdquo Finite Elements in Analysis andDesign vol 46 no 12 pp 1131ndash1145 2010

[21] F Franco S de Rosa and T Polito ldquoFinite element inves-tigations on the vibroacoustic performance of plane plateswith random stiffnessrdquo Mechanics of Advanced Materials andStructures vol 18 no 7 pp 484ndash497 2011

[22] Y Kim ldquoIdentification of sound transmission characteristicsof honeycomb sandwich panels using hybrid analyticalone-dimensional finite element methodrdquo in Proceedings of the35th International Congress and Exposition on Noise ControlEngineering (INTER-NOISE rsquo06) HonoluluHawaii USA 2006

[23] F Fahy Sound and Structural Vibration Radiation Transmissionand Response Academic Press London UK 1985

[24] J Blevins Formulas for Natural Frequency and Mode ShapeKrieger Publishing Malabar Fla USA 2001

[25] D Newland An Introduction to Random Vibration and SpectralAnalysis Longman New York NY USA 1984

[26] Y Lin Probabilistic Theory of Structural Dynamics McGraw-Hill Book New York NY USA 1967

[27] C E Wallace ldquoRadiation resistance of a rectangular panelrdquoTheJournal of the Acoustical Society of America vol 51 no 3 pp946ndash952 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of