Research Article Optimization of Sound Transmission Loss through a Thin …downloads.hindawi.com/journals/sv/2014/814682.pdf · 2019-07-31 · Research Article Optimization of Sound

Research ArticleOptimization of Sound Transmission Loss through a ThinFunctionally Graded Material Cylindrical Shell

Ali Nouri and Sohrab Astaraki

Department of Aerospace Engineering Shahid Sattari Air University Tehran 73411 Iran

Correspondence should be addressed to Ali Nouri ali noriiustacir

Received 7 February 2014 Revised 10 June 2014 Accepted 22 June 2014 Published 14 July 2014

Academic Editor Gyuhae Park

Copyright copy 2014 A Nouri and S Astaraki This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Themaximizing of sound transmission loss (TL) across a functionally graded material (FGM) cylindrical shell has been conductedusing a genetic algorithm (GA) To prevent the softening effect from occurring due to optimization the objective function ismodified based on the first resonant frequency Optimization is performed over the frequency range 1000ndash4000Hz where theear is the most sensitive The weighting constants are chosen here to correspond to an A-weighting scale Since the weight of theshell structure is an important concern inmost applications the weight of the optimized structure is constrained Several traditionalmaterials are used and the result shows that optimized shells with aluminum-nickel and aluminum-steel FGMare themost effectiveat maximizing TL at both stiffness and mass control region while they have minimum weight

1 IntroductionFunctionally graded material (FGM) is a multiphase mate-rial consisting of different material components such asceramics and metals that have various mixture ratios andmicrostructures It was first produced in Japan in the mid-1980s by Yamanouchi et al [1] and Koizumi [2] FGM isa class of composite materials with continuous designedvariation of properties throughout the volume and thuscan alleviate the stress concentration found in laminatedcomposites and can significantly enhance the thermal andmechanical properties of FGM Furthermore FGM can bedesigned to fulfill particular requirements such as enhancingstiffness toughness and resistance to corrosion wear andhigh temperature by utilizing materials or material systemswith various properties Using powder metallurgy a mixtureof ceramic andmetal with continuously varying volume frac-tions can be easily manufactured Yamaoka et al [3] and Zhuet al [4] described various techniques used for the fabricationof FGMs Consequently in the last two decades FGM hasbeen developed for general use as structural componentsin extremely high temperature environments such as rocketengine componentsmilitary armour space plan body enginecomponents thermal barrier coating for turbine blades andother applications [5]

Thin circular cylindrical shells are widely used in manyengineering fields ranging from civil naval nuclear mechan-ical chemical and aerospace applications [6] Vibroacousticbehavior of FGM cylindrical shell is considered as an impor-tant subject for researchers Williamson et al [7] carriedout finite element analysis and studied the residual stressesand strains in ceramic-metal joints Moya [8] concludedthat FGMs are promising candidates for future compositesin aerospace fast computers environmental sensors and soforth Using a micromechanics model Gasik and Lilius [9]calculatedmechanical and thermal properties ofW-Cu FGMSasaki and Hirai [10] carried out experiments to estimate thefatigue resistance of a Si-graphite FGM Other researchers[11ndash15] carried out thermal stress analyses of FGMs undervarious thermal loading conditions

A cylindrical shell structure made up of a FGM is one ofthe basic structural configurations where important studieshave been done Loy et al [16 17] carried out analysis withstrain-displacement relations from Loversquos shell theory andthe eigenvalue governing equation is obtained using theRayleigh-Ritz method The results demonstrated that thefrequency characteristics are similar to those observed forhomogeneous isotropic cylindrical shells and the frequenciesare affected by the constituent volume fractions and the

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 814682 10 pageshttpdxdoiorg1011552014814682

2 Shock and Vibration

configurations of the constituent materials Noise transmis-sion through a composite cylindrical shell was studied bymany researchers Tang et al [18] considered an infinitecylindrical sandwich shell excited by an oblique plane soundwave with two independent incident angles Daneshjou et al[19] presented an analytical solution for acoustic transmissionthrough relatively thick FGM cylindrical shells using thirdorder shear deformation theory

Optimization studies of transmission loss (TL) in cylin-drical shell structures with respect to their material and geo-metric properties have been an important topic of researchfor a few decades Most of these studies have been related tocomposite materials Makris et al [20] studied transmissionloss optimization in acoustic sandwich panels The resultsindicated that the optimum TL is typically found at the valueof the core density corresponding to the lowest of the per-missible range of core densities established at the start of theoptimization searchWang et al [21] used a genetic algorithmto optimize a sandwich panel with balanced acoustic andmechanical properties at a minimal weightThe optimizationof sound transmission through FGM cylindrical shells thatwere subjected to plane sound wave has not been studiedyet In the present work an attempt is made to addressthis problem Therefore it is very important to develop anaccurate reliable analysis towards the understanding of theoptimization sound transmission through FGM cylindricalshells Therefore this paper presents a study on the opti-mization of sound transmission loss across FGM cylindricalshell using a genetic algorithm A parametric study of soundtransmission through a FGM shell excited by an incidentoblique plane sound wave was developed and consideredas objective function to determine which combinations ofdesign variables could result in a FGM shell whichmaximizesthe sound insulationThe result shows that an optimized shellwith aluminum-nickel or aluminum-steel FG has the largestTL at both stiffness and mass control region while havingminimum weight

2 Model Specification

The specific problem is an oblique plane wave impinging ona flexible FGM thin cylindrical shell of infinite length andincludes the reflection and scattering of the incident wave andthe effect of an external airflow in the x-direction (Figure 1)The angle of incidence of the wave is 120574 (measured from theaxial axis of cylinder) and the incidence wave approachesfrom the 120579 = 120587 direction The inside cavity was assumedto be anechoic and only inward-traveling waves exist Inthe analysis all waves will be assumed to have the samedependence on the axial coordinate 119911 and the cylinder will beassumed infinitely long [22] The fluid media in the externaland the internal space are defined by the density and the speedof sound 120588

1 1198881 and 120588

3 1198883 respectively

21 Functionally Graded Material In order to model FGproperties for a cylindrical shell composed of 119898 differentmaterials with a uniform thickness h and a middle reference

Incident plane wave Reflected wave External flow

Transmitted wave

Continuously graded microstructures

Ceramic phaseTransient region

Metallic phase

zR

3

1

y

x

120579

120574

Figure 1 Schematic diagram of the FGM cylindrical shell

surface the volume fraction relation119881119891changes continuously

through the thickness direction considered as [16]119898

sum

119896=1

119881119891= 1

119881119891= (

119911 + ℎ2

ℎ)

119873

(1)

where 119873 is the power-law exponent (0 le 119873 le infin) Thegeneral formulation for a FGMcomposed of two componentscan be set as [17]

119883 = (1198831minus 1198832) (

119911 + ℎ2

ℎ)

119873

+ 1198832 (2)

where 119883 can be substituted with 119864 120588 or ] in (2) to obtainthe elasticity modulus mass density and poissonrsquos ratio forthe FGMcylindrical shell respectively From these equationswhen 119911 = ℎ2 119864 = 119864

2 120588 = 120588

2 ] = ]

2 and when

119911 = minusℎ2 119864 = 1198641 120588 = 120588

1 ] = ]

1 The material properties

vary continuously from material 2 at the inner surface ofthe cylindrical shell to material 1 at the outer surface of thecylindrical shell

22 Fluid Equations Due to the existence of airflow in theexternal fluid medium the external pressure which is thesummation of the incident wave 119901119868

1and the reflected wave

119901119877

1 satisfies the following wave equation [23]

1198882

1nabla2(119901119868

1+ 119901119877

1) minus (

120597

120597119905+ 119881 sdot nabla)

2

(119901119868

1+ 119901119877

1) = 0 (3)

where nabla2 is the Laplacian operator in the cylindrical coor-dinate system The internal pressure of cavity satisfies theacoustic wave equation

1198622

3nabla2119901119879

3minus1205972119901119879

3

1205971199052= 0 (4)

where 1199011198793is the transmitted wave

Shock and Vibration 3

As a result of consideringHelmholtz equations in internaland external space and also the equality of particle velocitiesof the acoustic media and the shell the boundary conditionsof the model can be defined as

120597 (119901119868

1+ 119901119877

1)

120597119903

10038161003816100381610038161003816100381610038161003816100381610038161003816119903=119877

= minus120588(120597

120597119905+ 119881 sdot nabla)

2

119882 (5)

120597119901119879

3

120597119903

1003816100381610038161003816100381610038161003816100381610038161003816119903=119877

= minus1205881205972119882

1205971199052 (6)

23 Formulation Consider a cylindrical shell with radius119877 and thickness ℎ see Figure 1 The deformations definedwith reference to a coordinate system taken at the middlesurface are 119906 V and 119908 in the axial circumferential andradial directions respectively For a thin cylindrical shellplane stress condition is assumed and the constitutive relationis given by

120590 = [119876] 120576 (7)

The stress vector strain vector and the reduced stiffnessmatrix are defined as

120590119879= 120590119909120590120579120590119909120579

120576119879= 120576119909120576120579120576119909120579

[119876] = [

[

11987611

11987612

0

11987612

11987622

0

0 0 11987666

]

]

(8)

For isotropic materials the reduced stiffness 119876119894119895(119894 119895 = 1 2

and 6) is defined as

11987611= 11987622=

119864

1 minus V2

11987612=

V1198641 minus V2

11987666=

119864

2 (1 + V)

(9)

From Loversquos shell theory the components in the strain vectorare defined as [23 24]

120576119909= 1198901+ 1199111205811 120576

120579= 1198902+ 1199111205812 120576

119909120579= 120574 + 2119911120591 (10)

where 1198901 1198902 and 120574 are the reference surface strains and 120581

1

1205812 and 120591 are the surface curvaturesThese surface strains and

curvatures are defined as

1198901 1198902 120574 =

120597119906

1205971199091

119877(120597V120597120579

+ 119908) 120597V120597119909

+1

119877

120597119906

120597120579

1205811 1205812 120591 = minus

1205972119908

1205971199092 minus

1

1198772(1205972119908

1205971205792minus120597V120597120579)

120597V120597119909

+1

119877(1205972119908

120597119909120597120579minus120597V120597119909)

(11)

For a thin cylindrical shell the force and moment resultantsare defined as

119873119909 119873120579 119873119909120579 = int

ℎ2

minusℎ2

120590119909 120590120579 120590119909120579 119889119911

119872119909119872120579119872119909120579 = int

ℎ2

minusℎ2

120590119909 120590120579 120590119909120579 119911 119889119911

(12)

Substituting (7) with substitution from (10) into (12) theconstitutive equation is obtained as

119873119909

119873120579

119873119909120579

119872119909

119872120579

119872119909120579

=

[[[[[[[

[

11986011

11986012

0 11986111

11986112

0

11986012

11986022

0 11986112

11986122

0

0 0 11986066

0 0 11986166

11986111

11986112

0 11986311

11986312

0

11986112

11986122

0 11986312

11986322

0

0 0 11986166

0 0 11986366

]]]]]]]

]

1198901

1198902

120574

1205811

1205812

120591

(13)

where 119860119894119895 119861119894119895 and 119863

119894119895(119894 119895 = 1 2 and 6) are the extensional

coupling and bending stiffness defined as

119860119894119895 119861119894119895 119863119894119895 = int

ℎ2

minusℎ2

1198761198941198951 119911 119911

2 119889119911 (14)

According to [23 24] a thin cylindrical shellrsquos equations ofmotion in cylindrical coordinate are as follows

minus119873120579

119877+1205972119873119911

1205971199112+2

119877

1205972119872119911120579

120597119911120597120579

+1

119877[1

119877

1205972119872120579

1205971205792+120597119872119911120579

120597119911] + 119902119910= minus119872

1205972119908

1205971199052

1

119877

120597119873120579

120597120579+120597119873119911120579

120597119911

+1

119877[1

119877

120597119872120579

120597120579+120597119872119911120579

120597119911] + 119902120579= minus119872

1205972V1205971199052

120597119873119911

120597119911+1

119877

1205972119873119911120579

120597119911120597120579+ 119902119911= minus119872

1205972119906

1205971199052

119872 = int

ℎ2

minusℎ2

120588 119889119911

(15)

where 119902119911 119902120579 and 119902

119910are loads in axial circumferential and

radial direction respectively

24 Vibroacoustic Solution The harmonic plane wave 119901119868 incylindrical coordinate incident from outside of the shell tothe direction shown in Figure 1 can be expressed as [25]

119901119868(119903 119911 120579 119905)

= 1198750

infin

sum

119899=0

120576119899(minus119895)119899

119869119899(1198961119903119903) cos (119899120579) exp [119895 (120596119905 minus 119896

1119911119911)]

(16)

where 1198961is the wave number in the external medium of shell

and 119869119899is the Bessel function of the first kind of integer order


119899 120576119899is theNeumann factor119875

0is the amplitude of the incident

wave 119895 = minus1 119899 = 0 1 2 3 and120596 is the angular frequencyThe waves radiated from the shell to the outside and into thecavity 119901119877

1and 119901119879

3 can be represented as

119901119877

1(119903 119911 120579 119905)

=

infin

sum

119899=0

119875119877

11198991198672

119899(1198961119903119903) exp [119895 (120596119905 minus 119896

1119911119911 minus 119899120579)]

119901119879

3(119903 119911 120579 119905)

=

infin

sum

119899=0

119875119879

31198991198671

119899(1198963119903119903) exp [119895 (120596119905 minus 119896

3119911119911 minus 119899120579)]

(17)

where11986721198991198671119899are theHankel functions of the first and second

kind of integer order 119899 respectivelyThree components of theshell displacements can be expressed as [24]

119908 (119911 120579 119905) =

infin

sum

119899=0

119882119899exp [119895 (120596119905 minus 119896

1119911119911 minus 119899120579)]

119906 (119911 120579 119905) = 119895

infin

sum

119899=0

119880119899exp [119895 (120596119905 minus 119896

1119911119911 minus 119899120579)]

V (119911 120579 119905) = 119895infin

sum

119899=0

119881119899exp [119895 (120596119905 minus 119896

1119911119911 minus 119899120579)]

(18)

Because the traveling waves in the acoustic media and insidethe shell are driven by the incident traveling wave the wavenumbers (or trace velocities) in the 119911 direction should matchthroughout the system therefore 119896

3119911= 1198961119911

Equations (16)ndash(18) are substituted into the three equa-tions ofmotion and two boundary conditions ((5)-(6))Thesefive equations involve six variables the amplitudes of theoutgoing and incomingwaves in the exterior cavity the trans-mitted wave in the interior cavity and three displacements ofthe shell structureTherefore the solutions can be obtained asthe ratios to the one of the variables in this case the pressureamplitude of the incoming wave [18]

25 Sound Transmission Loss Sound transmission loss (STL)can be defined as the ratio of the incoming and transmittedsound powers per unit length of the cylinder Consider

STL = 10 log10

119882119868

119882119879 (19)

where 119882119879 and 119882119868 are the transmitted and incident powerflow per unit length of the shell The equations for 119882119879 and119882119868 are developed by Kim and Lee [25] However because of

the incident angle in the present studies the equation of Kimand Leersquos work should be changed to

119882119868=cos (120574) 1199012

0

12058811198881

119877 (20)

Therefore the final quantity of transmission loss can bedefined as

STL = minus10 log10120591si (21)

where

120591si =infin

sum

119899=0

119877119890 119901119879

3119899times 1198671

119899(1198963119903119877) (119895120596119882

0

119899)lowast

times 12058811198881120587

120576119899cos 1205741199012

0

(22)

3 The Optimization Problem

To optimally design a structure so that its STL must beminimal we have to deal with a multivariable constrainednonlinear optimization problem The optimization problemcan be stated asmin119891 (119883) = minusSTL (119883) 119883 = [119909

1 1199092 119909

119899] 119883 isin 119877

119899

st 119892119894(119883) = 0 119894 = 1 119898

119888119894(119883) le 0 119894 = 119898 + 1 119901

(23)

where the 119892119894are 119898 equality constraints and the 119888

119894are (p-

m) inequality constraints 119891(119909) is the objective function tobe minimized with respect to the vector 119909 of the designvariables In the present paper we study the optimizationproblem of the shell with respect to the geometric andmaterial properties The design variables for this problem arechosen as

119883 = [119877 ℎ119873]119883lb le 119883 le 119883ub (24)

where 119877 ℎ and 119873 are radius thickness (geometric designvariable) and power-law exponent respectively119883lb and119883ubare vectors which define the lower and upper bounds of thedesign variables Since the weight of the shell structure isan important concern in most applications a constraint isintroduced as follows

1198881= int

ℎ2

minusℎ2

((1205881minus 1205882) (

2119911 + ℎ

ℎ)

119873

+ 1205882)119889119911 minus 119898

0le 0

1198882=ℎ

119877minus 005 le 0

(25)

where1198980is a constant and refers to the mass per unit length

of shellA large number of optimization techniques exist for

different problems In this study the genetic algorithm isused as optimization method The genetic algorithm is amethod for solving both constrained and unconstrainedoptimization problems that is based on natural selection theprocess that drives biological evolution It repeatedlymodifiesa population of individual solutions At each step the geneticalgorithm selects individuals at random from the currentpopulation to be parents and uses them to produce the chil-dren for the next generation Over successive generations thepopulation ldquoevolvesrdquo toward an optimal solutionThe geneticalgorithm uses three main types of rules at each step to createthe next generation from the current population selectionrules select the individuals called parents that contributeto the population at the next generation Crossover rulescombine two parents to form children for the next generationMutation rules apply randomchanges to individual parents toform children


Table 1 Material and environmental properties

Material Shell materials Cavity AmbientAluminum Steel Nickel Zirconia Brass Air Air

120588(kgm3) 2760 7750 8900 5700 8500 29 29119864 (Gpa) 72 210 224 244 104 mdash mdash120592 033 03 034 03 037 mdash mdashSound speed (ms) mdash mdash mdash mdash mdash 340 340Incident angle (deg) 45Shell radius (m) 15Shell thickness (mm) 151198980(kgm) 621 (15 times 119898Al)

4 Result and DiscussionNumerical simulations have been carried out to study theoptimization problem of a FGM cylindrical shell Withproper qualitative interpretations the theoretical modeldeveloped can be used very effectively in the basic designstage of cylinder-shape vibroacoustic systems As a demon-stration of such applications design parameter studies areconducted The basic shell dimensions and simulation con-ditions used in the study are listed in Table 1

The TL of a FGM Brass-Al shell is illustrated in Figure 2As shown three important frequencies are recognized ringfrequency 119891

119903(where the wavelength of a longitudinal wave

in the shell is equal to the circumference) critical pseu-docoincidence 119891pc (spatial coincidence in radial directionbetween the wave vector projection of excitation and the shellcircumferential wave number) and coincidence frequency119891

119888

(where the trace velocity of the acoustic wave is equal to thebending wave velocity in the shell wall) The various zonescontrolled by the stiffness (0minus119891

119903) the mass (119891

119903minus119891pc) and the

coincidence (119891 gt 119891119888) and the resonant modes zone enclosed

between 119891pc and 119891119888 will be noted [26]Figure 3 shows the effect of the shell materials on the STL

Materials chosen for the comparison are aluminum steelnickel zirconia and brass as shown in Table 1 The figureshows that the brass is themost effective in the high frequencyrange and zirconia is the most effective in the low frequencyrangeThis is as expected because the density of the brass andthe stiffness of the zirconia are the largest which make themmost effective in the mass and stiffness controlled regionsrespectively The figure also shows that the aluminum whichhas the lowest stiffness is the least effective in the lowfrequency range which is again as expected because the lowfrequency range is controlled by the stiffness Steel and nickelhave the same behavior in the low and high frequency rangeand are more effective in both mass and stiffness controlledregions

Although steel and nickel are more effective the struc-tural weight of a cylindrical shell that is made of them is toohigh Since the weight of the shell structure is an importantconcern in most applications it is necessary to compromisebetween sound transmission loss of cylindrical shell andits weight The results however lead to a structure with alower stiffness than the base shell To prevent the softening

minus10

80

70

60

50

40

30

20

10

0

Frequency (Hz)

TL (d

B)

Mass controlStiffness control

fr

fpcfc

100 101 102 103 104

Figure 2 The TL of FGM shell

Frequency (Hz)100 101 102 103 104

minus20

minus10

80

70

60

50

40

30

20

10

0

TL (d

B)

AluminumSteelNickel

ZirconiaBrass

Figure 3 Effect of different materials on STL


Table 2 Effect of120573 values on the design parameters of the optimizedshell

120573 119877 (m) ℎ (mm) 119873 1198911(Hz) STL (db)

0 16 00013 0393 4 38604 158 000143 0647 55 384205 151 000151 0817 6 385706 145 000167 13 6 3844

effect from occurring due to optimization we introduce anadditional term in the objective function

min119891 (119883) = (1 minus 120573) (minusSTL (119883)) + 120573 (1198911199001minus 119891 (119883)) (26)

where 119891(119909) is the first resonant frequency of the optimizedshell 119891119900

1= 7Hz is the first resonant frequency of the

base shell and 120573 is a weighting constant Note that the firstresonant frequency of a structure is a common measure ofthe stiffness of the structureThismodified objective functionwill penalize the reduction of the stiffness of the structureThe optimization is then performed at octave band frequencyunder 1000Hz The results evaluate an equal weighting of120573 = 0 04 05 06 given to both the sound transmission lossand the stiffness of the structureThe optimization results forother values of 120573 are similar and are not presented Table 2shows the results of the optimized shell at the frequency Inthis example stainless steel and aluminum are considered asthe constituent materials of the FGM cylindrical shell

The results indicate that the optimum values are typicallyfound at the value of 120573 = 05 so it is used for otheranalyses in this study The results shown in Table 2 are theshell optimized at discrete frequency It is also common tooptimize a frequency-weighted average of the transmissionloss For practical purposes hearing encompasses a rangeof frequencies from about 16Hz to somewhat less than20000Hz However low frequencies do not affect humansvery strongly while those from 500ndash5000Hz (where the earis most sensitive) are very important

For the sound transmission class (STC)mdashthe single-number rating system which compares the sound TL of a testspecimen with a standard contourmdashthe following set of 16frequencies is used 125 160 200 250 315 400 500 630 8001000 1250 1600 2000 2500 3150 and 4000Hz [27]

This study uses the frequency range 1ndash4 KHz Howeverinstead of the 21 frequencies (150Hz steps) used then onlythe seven standardized frequencies from the STC rating inthe 1000 to 4000Hz range will be incorporated Namely thefollowing frequencies are considered 1000 1250 1600 20002500 3150 and 4000Hz The smaller number of frequenciesdoes not lessen the accuracy in the calculations of TL becausea sophisticated integration algorithm was used to find thefield-incidence average TL Also the use of these frequenciesparallels the portion of the spectrum emphasized in therecent ASTM standard on isolation between neighboringrooms [25]The lower limit of the range (1000Hz)was chosen

Table 3 Normalized weights using the A-weighting

Frequency (Hz)A-weighting dB(A) Multiplier 10dB(A)10 120573119894

1000 0 1 011561250 06 1148 013271600 1 1259 014552000 12 1318 015242500 13 1349 015593150 12 1318 015244000 1 1259 01455

Table 4 FGM combinations

FGM Inner surface Outer surfaceCase 1 Aluminum SteelCase 2 Aluminum NickelCase 3 Aluminum ZirconiaCase 4 Aluminum Brass

in part because of experimental results cited by Dym andLang [28] In this case the objective function is modified as

min119891 (119883) = (1 minus 120573) (minusSTLavg (119883)) + 120573 (1198911199001 minus 119891 (119883))

STLavg (119883) = minus10 log1010038161003816100381610038161003816120591avg (119883)

10038161003816100381610038161003816

120591avg (119883) =

119873119891

sum

119894=0

120573119894120591si (119883)

(27)

where120573119894represent weighting constants and119873

119891is the number

of frequencies [29] 120573119894are normalized so that the sum of

all the coefficients is unity The weighting constants 120573119894are

commonly chosen to correspond to anA-weighting It shouldbe mentioned here that with A-weighting the sound-levelmeter is relatively less sensitive to sounds of frequencybelow 1000 and above 4000Hz The weighting and discretefrequencies that were chosen in the bound 1ndash4 kHz areidentical to those used to defined single insulation measurein an ASTM standard [27 28]The normalized weights 120573

119894are

given in Table 3In this study based on the materials listed in Table 1

different functionally graded cylindrical shells (Table 4)should be optimized based on (27) and (24)-(25)

As the weight of structure is an important parameterin aerospace vehicles aluminum is used as inner surface ofconstituent materials because its density is lowest among themetals in Table 1 The optimized results of the all cases aretabulated in Table 5

The result shows that case 2 is the most effective for theSTL and case 3 is worse than the other cases Figure 4 showsthe variation of the STL for the all caseswith optimized designparameters of Table 5

As indicated the results are different for the optimizedshells with different kinds of FGM although STL improve-ment is achieved in all cases As obviously illustrated inFigure 4 TLs characteristics have been significantly improved


Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 1

Case 1

SteelAluminum

(a)

AluminumZirconia

Case 2

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 2

(b)

AluminumZirconia

Case 3

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 3

(c)

AluminumBrass

Case 4

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 4

(d)

Figure 4 TL as a function of frequency 119891 for the FGM shells listed in Table 4

Table 5 Optimized value of FGM combinations

Condition ℎ (mm) 119873 1198911(Hz) 119877 (m) ΣSTL at frequencies of Table 3

Case 1 13 159 55 16 30191Case 2 11 137 6 15 31289Case 3 16 084 4 18 29834Case 4 14 12 5 18 32173


Case 3Case 1Case 2 Case 4

Frequency (Hz)

Optimized shell

100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10TL

(dB)

0

Figure 5 Comparison between TL-f curves for the FGM shells listed in Table 4

minus05 minus04 minus03 minus02 minus01 050

01

02

03

04

05

06

07

08

09

1

Normalized radial distance

Volu

me f

ract

ion

Case 1Case 2

Case 3Case 4

Inner surface material

0 01 02 03 04

(a)

Outer surface material

Case 1Case 2

Case 3Case 4


01

02

03

04

05

06

07

08

09

1


Volu

me f

ract

ion

0 01 02 03 04

(b)

Figure 6 Variation of volume fraction of optimized FGM

over frequency of 1000Hz which have beenweighted accord-ing to Table 3 Figure 5 compares the TLs of FGM shells madefrom different materials with optimized design parametersfound by the GA

Since the aluminum-brass FGM shell was the heaviest itsoptimized curve is the highest on the mass control region butits behavior in the stiffness control is the worst The figureshows that case 1 and case 2 are the most effective in the low

and high frequencies range and so they should be used as thefirst selection material tabulated in Table 4 with optimizedvalue by (26) and (24) Figure 6 shows the variations of thevolume fractions in the thickness direction 119911 for all cases ofFG cylindrical shell

In Figure 6 the material properties on the inner surfaceof the all optimized cylindrical shell are those of aluminumand on the outer surface are those of as in Table 3 As 119911


increases 119911 = minus05 the amount of aluminum constituent forthe optimized case 1 and case 3 is the highest and the lowestcompared to other optimized cases respectively This meansthat the weight constraint of optimized shell at case 1 is betterthan other cases while maximum STL is achieved

5 Concluding Remarks

The optimization study of sound transmission across a FGMcylindrical shell has been presented in this paper The mainobjective of the optimization is to maximize the soundtransmission loss across the structure subjected to weight andthe thickness constraint In order to prevent the softeningeffect occurrence due to optimization an additional term isadded to the objective function based on the first resonantfrequency Several traditional materials have been chosen forinvestigation The power-law exponent shell radius thick-ness of shell and first resonant frequency have been selectedas design variables Optimization of transmission loss isperformed over the frequency range 1000ndash4000Hz wherethe ear is the most sensitive The results show that the opti-mized shell made of aluminum-nickel and aluminum-steel ismore effective on the STL in a wide frequency range whilealuminum-brass and aluminum-zircon are only effective inmass control region and stiffness control region respectivelyBased on the result of optimized shell geometries and itsvolume fraction minimum weight design with best STL isfound for aluminum-nickel and aluminum-steel functionallygraded materials

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M Yamanouchi M Koizumi T Hirai and I Shiota ldquoOn thedesign of functionally gradient materialsrdquo in Proceedings of the1st International Symposium on Functionally GradientMaterialspp 5ndash10 Sendai Japan 1990

[2] M Koizumi ldquoFunctionally gradient materials the concept ofFGMrdquo Ceramic Transactions vol 34 pp 3ndash10 1993

[3] H Yamaoka M Yuki K Tahara T Irisawa R Watanabe andA Kawasaki ldquoFabrication of functionally gradient material byslurry stacking and sintering processrdquo Ceramic TransactionsFunctionally Gradient Materials vol 34 pp 165ndash172 1993

[4] J C Zhu Z D Yin and Z H Lai ldquoFabrication andmicrostruc-ture of ZrO

2-Ni functional gradient material by powder metal-

lurgyrdquo Journal ofMaterials Science vol 31 no 21 pp 5829ndash58341996

[5] R Ansari M Darvizeh and M Hemmatnezhad ldquoVibrationanalysis of FGM cylindrical shells under various boundaryconditionsrdquo Journal of Agricultural Science and Technology vol5 no 3 pp 129ndash138 2008

[6] S R Li X H Fub and R C Batrac ldquoFree vibration of three-layer circular cylindrical shells with functionally gradedmiddlelayerrdquo Mechanics Research Communications vol 37 pp 577ndash580 2010

[7] R L Williamson B H Rabin and G E Byerly ldquoFEM study ofthe effects of inter layers and creep in reducing residual stressesand strains in ceramic-metal jointsrdquo Composites Engineeringvol 5 pp 851ndash863 1995

[8] J S Moya ldquoLayered ceramicsrdquoAdvancedMaterials vol 7 no 2pp 185ndash189 1995

[9] M M Gasik and K R Lilius ldquoEvaluation of properties of W-Cu functional gradient materials by micromechanical modelrdquoComputational Materials Science vol 3 pp 41ndash49 1994

[10] M Sasaki and T Hirai ldquoThermal fatigue resistance of CVDSiCC functionally gradient materialrdquo Journal of the EuropeanCeramic Society vol 14 pp 257ndash260 1994

[11] R C Wetherhold S Seelman and J Z Wang ldquoThe use offunctionally graded materials to eliminate or control thermaldeformationrdquo Composites Science and Technology vol 56 no 9pp 1099ndash1104 1996

[12] S Takezono K Tao E Inamura andM Inoue ldquoThermal stressand deformation in functionally graded material shells of revo-lution under thermal loading due to fluidrdquo JSME InternationalJournal Series A Mechanics and Material Engineering vol 39pp 573ndash581 1996

[13] A Makino N Araki H Kitajima and K Ohashi ldquoTransienttemperature response of functionally gradient material sub-jected to partial step wise heatingrdquo Transactions of the JapanSociety of Mechanical Engineers B vol 60 pp 4200ndash4206 1994

[14] X D Zhang D Q Liu and C C Ge ldquoThermal stress analysisof axial symmetry functionally gradient materials under steadytemperature fieldrdquo Journal of Functional Materials vol 25 pp452ndash455 1994

[15] Y Obata and N Noda ldquoSteady thermal stresses in a hollowcircular cylinder and a hollow sphere of afunctionally gradientmaterialrdquo Journal of Thermal Stresses vol 17 no 3 pp 471ndash4871994

[16] C T Loy K Y Lam and J N Reddy ldquoVibration of functionallygraded cylindrical shellsrdquo International Journal of MechanicalSciences vol 41 no 3 pp 309ndash324 1999

[17] S C Pradhan C T Loy K Y Lam and J N Reddy ldquoVibrationcharacteristics of functionally graded cylindrical shells undervarious boundary conditionsrdquoApplied Acoustics vol 61 pp 111ndash129 2000

[18] Y Y Tang J H Robinson and R J Silcox ldquoSound transmissionthrough a cylindrical sandwich shell with honeycomb corerdquo inProceedings of the 34 th AIAA Aerospace Science Meeting andExhibit 1996

[19] K Daneshjou M M Shokrieh M G Moghaddam and RTalebitooti ldquoAnalytical model of sound transmission throughrelatively thick FGM cylindrical shells considering third ordershear deformation theoryrdquo Composite Structures vol 93 no 1pp 67ndash78 2010

[20] S E Makris C L Dym and J M Smith ldquoTransmissionloss optimization in acoustic sandwich panelsrdquo Journal of theAcoustical Society of America vol 79 no 6 pp 1833ndash1843 1986

[21] T Wang S Li and S R Nutt ldquoOptimal design of acousticalsandwich panels with a genetic algorithmrdquo Journal of AppliedAcoustics vol 70 no 3 pp 416ndash425 2009

[22] L R Koval ldquoeffects of cavity resonances on sound transmissioninto a thin cylindrical shellrdquo Journal of Sound andVibration vol59 no 1 pp 23ndash33 1978

[23] M S Qatu Vibration of laminated shells and plates ElsevierAcademic Press New York NY USA 2004


[24] J N ReddyMechanics of LaminatedComposite Plates and ShellsTheory and Analysis CRC Press 2nd edition 2004

[25] J Kim and H Lee J ldquoStudy on sound transmission character-istics of a cylindrical shell using analytical and experimentalmodelsrdquo Applied Acoustics vol 64 pp 611ndash632 2003

[26] L R Koval ldquoOn sound transmission into a thin cylindrical shellunder flight conditionsrdquo Journal of Sound andVibration vol 48pp 265ndash275 1976

[27] M J CrockerHandbook of Acoustics JohnWiley and SonsNewYork NY USA 1998

[28] C L Dym and M A Lang ldquoTransmission of sound throughsandwich panelsrdquo Journal of the Acoustical Society of Americavol 56 pp 1523ndash1532 1974

[29] American Society of Testing and Materials ldquoStandard practicefor determining single-number rating of airborne sound isola-tion in multi unit building specificationsrdquo in Annual Book ofASTM Standards vol 0406 pp E597ndashE581 American Societyof Testing and Materials Philadelphia Pa USA 1984

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



configurations of the constituent materials Noise transmis-sion through a composite cylindrical shell was studied bymany researchers Tang et al [18] considered an infinitecylindrical sandwich shell excited by an oblique plane soundwave with two independent incident angles Daneshjou et al[19] presented an analytical solution for acoustic transmissionthrough relatively thick FGM cylindrical shells using thirdorder shear deformation theory

Optimization studies of transmission loss (TL) in cylin-drical shell structures with respect to their material and geo-metric properties have been an important topic of researchfor a few decades Most of these studies have been related tocomposite materials Makris et al [20] studied transmissionloss optimization in acoustic sandwich panels The resultsindicated that the optimum TL is typically found at the valueof the core density corresponding to the lowest of the per-missible range of core densities established at the start of theoptimization searchWang et al [21] used a genetic algorithmto optimize a sandwich panel with balanced acoustic andmechanical properties at a minimal weightThe optimizationof sound transmission through FGM cylindrical shells thatwere subjected to plane sound wave has not been studiedyet In the present work an attempt is made to addressthis problem Therefore it is very important to develop anaccurate reliable analysis towards the understanding of theoptimization sound transmission through FGM cylindricalshells Therefore this paper presents a study on the opti-mization of sound transmission loss across FGM cylindricalshell using a genetic algorithm A parametric study of soundtransmission through a FGM shell excited by an incidentoblique plane sound wave was developed and consideredas objective function to determine which combinations ofdesign variables could result in a FGM shell whichmaximizesthe sound insulationThe result shows that an optimized shellwith aluminum-nickel or aluminum-steel FG has the largestTL at both stiffness and mass control region while havingminimum weight

2 Model Specification

The specific problem is an oblique plane wave impinging ona flexible FGM thin cylindrical shell of infinite length andincludes the reflection and scattering of the incident wave andthe effect of an external airflow in the x-direction (Figure 1)The angle of incidence of the wave is 120574 (measured from theaxial axis of cylinder) and the incidence wave approachesfrom the 120579 = 120587 direction The inside cavity was assumedto be anechoic and only inward-traveling waves exist Inthe analysis all waves will be assumed to have the samedependence on the axial coordinate 119911 and the cylinder will beassumed infinitely long [22] The fluid media in the externaland the internal space are defined by the density and the speedof sound 120588

1 1198881 and 120588

3 1198883 respectively

21 Functionally Graded Material In order to model FGproperties for a cylindrical shell composed of 119898 differentmaterials with a uniform thickness h and a middle reference

Incident plane wave Reflected wave External flow

Transmitted wave

Continuously graded microstructures

Ceramic phaseTransient region

Metallic phase

zR

3

1

y

x

120579

120574

Figure 1 Schematic diagram of the FGM cylindrical shell

surface the volume fraction relation119881119891changes continuously

through the thickness direction considered as [16]119898

sum

119896=1

119881119891= 1

119881119891= (

119911 + ℎ2

ℎ)

119873

(1)

where 119873 is the power-law exponent (0 le 119873 le infin) Thegeneral formulation for a FGMcomposed of two componentscan be set as [17]

119883 = (1198831minus 1198832) (

119911 + ℎ2

ℎ)

119873

+ 1198832 (2)

where 119883 can be substituted with 119864 120588 or ] in (2) to obtainthe elasticity modulus mass density and poissonrsquos ratio forthe FGMcylindrical shell respectively From these equationswhen 119911 = ℎ2 119864 = 119864

2 120588 = 120588

2 ] = ]

2 and when

119911 = minusℎ2 119864 = 1198641 120588 = 120588

1 ] = ]

1 The material properties

vary continuously from material 2 at the inner surface ofthe cylindrical shell to material 1 at the outer surface of thecylindrical shell

22 Fluid Equations Due to the existence of airflow in theexternal fluid medium the external pressure which is thesummation of the incident wave 119901119868

1and the reflected wave

119901119877

1 satisfies the following wave equation [23]

1198882

1nabla2(119901119868

1+ 119901119877

1) minus (

120597

120597119905+ 119881 sdot nabla)

2

(119901119868

1+ 119901119877

1) = 0 (3)

where nabla2 is the Laplacian operator in the cylindrical coor-dinate system The internal pressure of cavity satisfies theacoustic wave equation

1198622

3nabla2119901119879

3minus1205972119901119879

3

1205971199052= 0 (4)

where 1199011198793is the transmitted wave



120597 (119901119868

1+ 119901119877

1)

120597119903

10038161003816100381610038161003816100381610038161003816100381610038161003816119903=119877

= minus120588(120597

120597119905+ 119881 sdot nabla)

2

119882 (5)

120597119901119879

3

120597119903

1003816100381610038161003816100381610038161003816100381610038161003816119903=119877

= minus1205881205972119882

1205971199052 (6)


120590 = [119876] 120576 (7)


120590119879= 120590119909120590120579120590119909120579

120576119879= 120576119909120576120579120576119909120579

[119876] = [

[

11987611

11987612

0

11987612

11987622

0

0 0 11987666

]

]

(8)



11987611= 11987622=

119864

1 minus V2

11987612=

V1198641 minus V2

11987666=

119864

2 (1 + V)

(9)


120576119909= 1198901+ 1199111205811 120576

120579= 1198902+ 1199111205812 120576

119909120579= 120574 + 2119911120591 (10)


1



1198901 1198902 120574 =

120597119906

1205971199091

119877(120597V120597120579

+ 119908) 120597V120597119909

+1

119877

120597119906

120597120579

1205811 1205812 120591 = minus

1205972119908

1205971199092 minus

1

1198772(1205972119908

1205971205792minus120597V120597120579)

120597V120597119909

+1

119877(1205972119908

120597119909120597120579minus120597V120597119909)

(11)


119873119909 119873120579 119873119909120579 = int

ℎ2

minusℎ2

120590119909 120590120579 120590119909120579 119889119911

119872119909119872120579119872119909120579 = int

ℎ2

minusℎ2

120590119909 120590120579 120590119909120579 119911 119889119911

(12)


119873119909

119873120579

119873119909120579

119872119909

119872120579

119872119909120579

=

[[[[[[[

[

11986011

11986012

0 11986111

11986112

0

11986012

11986022

0 11986112

11986122

0

0 0 11986066

0 0 11986166

11986111

11986112

0 11986311

11986312

0

11986112

11986122

0 11986312

11986322

0

0 0 11986166

0 0 11986366

]]]]]]]

]

1198901

1198902

120574

1205811

1205812

120591

(13)

where 119860119894119895 119861119894119895 and 119863



119860119894119895 119861119894119895 119863119894119895 = int

ℎ2

minusℎ2

1198761198941198951 119911 119911

2 119889119911 (14)


minus119873120579

119877+1205972119873119911

1205971199112+2

119877

1205972119872119911120579

120597119911120597120579

+1

119877[1

119877

1205972119872120579

1205971205792+120597119872119911120579

120597119911] + 119902119910= minus119872

1205972119908

1205971199052

1

119877

120597119873120579

120597120579+120597119873119911120579

120597119911

+1

119877[1

119877

120597119872120579

120597120579+120597119872119911120579

120597119911] + 119902120579= minus119872

1205972V1205971199052

120597119873119911

120597119911+1

119877

1205972119873119911120579

120597119911120597120579+ 119902119911= minus119872

1205972119906

1205971199052

119872 = int

ℎ2

minusℎ2

120588 119889119911

(15)

where 119902119911 119902120579 and 119902




119901119868(119903 119911 120579 119905)

= 1198750

infin

sum

119899=0

120576119899(minus119895)119899

119869119899(1198961119903119903) cos (119899120579) exp [119895 (120596119905 minus 119896

1119911119911)]

(16)







1and 119901119879


119901119877

1(119903 119911 120579 119905)

=

infin

sum

119899=0

119875119877

11198991198672

119899(1198961119903119903) exp [119895 (120596119905 minus 119896

1119911119911 minus 119899120579)]

119901119879

3(119903 119911 120579 119905)

=

infin

sum

119899=0

119875119879

31198991198671

119899(1198963119903119903) exp [119895 (120596119905 minus 119896

3119911119911 minus 119899120579)]

(17)



119908 (119911 120579 119905) =

infin

sum

119899=0

119882119899exp [119895 (120596119905 minus 119896

1119911119911 minus 119899120579)]

119906 (119911 120579 119905) = 119895

infin

sum

119899=0

119880119899exp [119895 (120596119905 minus 119896

1119911119911 minus 119899120579)]

V (119911 120579 119905) = 119895infin

sum

119899=0

119881119899exp [119895 (120596119905 minus 119896

1119911119911 minus 119899120579)]

(18)


3119911= 1198961119911



STL = 10 log10

119882119868

119882119879 (19)



119882119868=cos (120574) 1199012

0

12058811198881

119877 (20)



where

120591si =infin

sum

119899=0

119877119890 119901119879

3119899times 1198671

119899(1198963119903119877) (119895120596119882

0

119899)lowast

times 12058811198881120587

120576119899cos 1205741199012

0

(22)



1 1199092 119909

119899] 119883 isin 119877

119899

st 119892119894(119883) = 0 119894 = 1 119898

119888119894(119883) le 0 119894 = 119898 + 1 119901

(23)


119894are (p-


119883 = [119877 ℎ119873]119883lb le 119883 le 119883ub (24)


1198881= int

ℎ2

minusℎ2

((1205881minus 1205882) (

2119911 + ℎ

ℎ)

119873

+ 1205882)119889119911 minus 119898

0le 0

1198882=ℎ

119877minus 005 le 0

(25)












119888


119903) the mass (119891






minus10

80

70

60

50

40

30

20

10

0

Frequency (Hz)

TL (d

B)


fr

fpcfc

100 101 102 103 104


Frequency (Hz)100 101 102 103 104

minus20

minus10

80

70

60

50

40

30

20

10

0

TL (d

B)

AluminumSteelNickel

ZirconiaBrass




120573 119877 (m) ℎ (mm) 119873 1198911(Hz) STL (db)

0 16 00013 0393 4 38604 158 000143 0647 55 384205 151 000151 0817 6 385706 145 000167 13 6 3844











1000 0 1 011561250 06 1148 013271600 1 1259 014552000 12 1318 015242500 13 1349 015593150 12 1318 015244000 1 1259 01455






10038161003816100381610038161003816

120591avg (119883) =

119873119891

sum

119894=0

120573119894120591si (119883)

(27)


119891is the number




119894are







Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 1

Case 1

SteelAluminum

(a)

AluminumZirconia

Case 2

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 2

(b)

AluminumZirconia

Case 3

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 3

(c)

AluminumBrass

Case 4

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 4

(d)







Frequency (Hz)

Optimized shell

100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10TL

(dB)

0



01

02

03

04

05

06

07

08

09

1


Volu

me f

ract

ion

Case 1Case 2

Case 3Case 4


0 01 02 03 04

(a)


Case 1Case 2

Case 3Case 4


01

02

03

04

05

06

07

08

09

1


Volu

me f

ract

ion

0 01 02 03 04

(b)












References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120597 (119901119868

1+ 119901119877

1)

120597119903

10038161003816100381610038161003816100381610038161003816100381610038161003816119903=119877

= minus120588(120597

120597119905+ 119881 sdot nabla)

2

119882 (5)

120597119901119879

3

120597119903

1003816100381610038161003816100381610038161003816100381610038161003816119903=119877

= minus1205881205972119882

1205971199052 (6)


120590 = [119876] 120576 (7)


120590119879= 120590119909120590120579120590119909120579

120576119879= 120576119909120576120579120576119909120579

[119876] = [

[

11987611

11987612

0

11987612

11987622

0

0 0 11987666

]

]

(8)



11987611= 11987622=

119864

1 minus V2

11987612=

V1198641 minus V2

11987666=

119864

2 (1 + V)

(9)


120576119909= 1198901+ 1199111205811 120576

120579= 1198902+ 1199111205812 120576

119909120579= 120574 + 2119911120591 (10)


1



1198901 1198902 120574 =

120597119906

1205971199091

119877(120597V120597120579

+ 119908) 120597V120597119909

+1

119877

120597119906

120597120579

1205811 1205812 120591 = minus

1205972119908

1205971199092 minus

1

1198772(1205972119908

1205971205792minus120597V120597120579)

120597V120597119909

+1

119877(1205972119908

120597119909120597120579minus120597V120597119909)

(11)


119873119909 119873120579 119873119909120579 = int

ℎ2

minusℎ2

120590119909 120590120579 120590119909120579 119889119911

119872119909119872120579119872119909120579 = int

ℎ2

minusℎ2

120590119909 120590120579 120590119909120579 119911 119889119911

(12)


119873119909

119873120579

119873119909120579

119872119909

119872120579

119872119909120579

=

[[[[[[[

[

11986011

11986012

0 11986111

11986112

0

11986012

11986022

0 11986112

11986122

0

0 0 11986066

0 0 11986166

11986111

11986112

0 11986311

11986312

0

11986112

11986122

0 11986312

11986322

0

0 0 11986166

0 0 11986366

]]]]]]]

]

1198901

1198902

120574

1205811

1205812

120591

(13)

where 119860119894119895 119861119894119895 and 119863



119860119894119895 119861119894119895 119863119894119895 = int

ℎ2

minusℎ2

1198761198941198951 119911 119911

2 119889119911 (14)


minus119873120579

119877+1205972119873119911

1205971199112+2

119877

1205972119872119911120579

120597119911120597120579

+1

119877[1

119877

1205972119872120579

1205971205792+120597119872119911120579

120597119911] + 119902119910= minus119872

1205972119908

1205971199052

1

119877

120597119873120579

120597120579+120597119873119911120579

120597119911

+1

119877[1

119877

120597119872120579

120597120579+120597119872119911120579

120597119911] + 119902120579= minus119872

1205972V1205971199052

120597119873119911

120597119911+1

119877

1205972119873119911120579

120597119911120597120579+ 119902119911= minus119872

1205972119906

1205971199052

119872 = int

ℎ2

minusℎ2

120588 119889119911

(15)

where 119902119911 119902120579 and 119902




119901119868(119903 119911 120579 119905)

= 1198750

infin

sum

119899=0

120576119899(minus119895)119899

119869119899(1198961119903119903) cos (119899120579) exp [119895 (120596119905 minus 119896

1119911119911)]

(16)







1and 119901119879


119901119877

1(119903 119911 120579 119905)

=

infin

sum

119899=0

119875119877

11198991198672

119899(1198961119903119903) exp [119895 (120596119905 minus 119896

1119911119911 minus 119899120579)]

119901119879

3(119903 119911 120579 119905)

=

infin

sum

119899=0

119875119879

31198991198671

119899(1198963119903119903) exp [119895 (120596119905 minus 119896

3119911119911 minus 119899120579)]

(17)



119908 (119911 120579 119905) =

infin

sum

119899=0

119882119899exp [119895 (120596119905 minus 119896

1119911119911 minus 119899120579)]

119906 (119911 120579 119905) = 119895

infin

sum

119899=0

119880119899exp [119895 (120596119905 minus 119896

1119911119911 minus 119899120579)]

V (119911 120579 119905) = 119895infin

sum

119899=0

119881119899exp [119895 (120596119905 minus 119896

1119911119911 minus 119899120579)]

(18)


3119911= 1198961119911



STL = 10 log10

119882119868

119882119879 (19)



119882119868=cos (120574) 1199012

0

12058811198881

119877 (20)



where

120591si =infin

sum

119899=0

119877119890 119901119879

3119899times 1198671

119899(1198963119903119877) (119895120596119882

0

119899)lowast

times 12058811198881120587

120576119899cos 1205741199012

0

(22)



1 1199092 119909

119899] 119883 isin 119877

119899

st 119892119894(119883) = 0 119894 = 1 119898

119888119894(119883) le 0 119894 = 119898 + 1 119901

(23)


119894are (p-


119883 = [119877 ℎ119873]119883lb le 119883 le 119883ub (24)


1198881= int

ℎ2

minusℎ2

((1205881minus 1205882) (

2119911 + ℎ

ℎ)

119873

+ 1205882)119889119911 minus 119898

0le 0

1198882=ℎ

119877minus 005 le 0

(25)












119888


119903) the mass (119891






minus10

80

70

60

50

40

30

20

10

0

Frequency (Hz)

TL (d

B)


fr

fpcfc

100 101 102 103 104


Frequency (Hz)100 101 102 103 104

minus20

minus10

80

70

60

50

40

30

20

10

0

TL (d

B)

AluminumSteelNickel

ZirconiaBrass




120573 119877 (m) ℎ (mm) 119873 1198911(Hz) STL (db)

0 16 00013 0393 4 38604 158 000143 0647 55 384205 151 000151 0817 6 385706 145 000167 13 6 3844











1000 0 1 011561250 06 1148 013271600 1 1259 014552000 12 1318 015242500 13 1349 015593150 12 1318 015244000 1 1259 01455






10038161003816100381610038161003816

120591avg (119883) =

119873119891

sum

119894=0

120573119894120591si (119883)

(27)


119891is the number




119894are







Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 1

Case 1

SteelAluminum

(a)

AluminumZirconia

Case 2

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 2

(b)

AluminumZirconia

Case 3

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 3

(c)

AluminumBrass

Case 4

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 4

(d)







Frequency (Hz)

Optimized shell

100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10TL

(dB)

0



01

02

03

04

05

06

07

08

09

1


Volu

me f

ract

ion

Case 1Case 2

Case 3Case 4


0 01 02 03 04

(a)


Case 1Case 2

Case 3Case 4


01

02

03

04

05

06

07

08

09

1


Volu

me f

ract

ion

0 01 02 03 04

(b)












References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













1and 119901119879


119901119877

1(119903 119911 120579 119905)

=

infin

sum

119899=0

119875119877

11198991198672

119899(1198961119903119903) exp [119895 (120596119905 minus 119896

1119911119911 minus 119899120579)]

119901119879

3(119903 119911 120579 119905)

=

infin

sum

119899=0

119875119879

31198991198671

119899(1198963119903119903) exp [119895 (120596119905 minus 119896

3119911119911 minus 119899120579)]

(17)



119908 (119911 120579 119905) =

infin

sum

119899=0

119882119899exp [119895 (120596119905 minus 119896

1119911119911 minus 119899120579)]

119906 (119911 120579 119905) = 119895

infin

sum

119899=0

119880119899exp [119895 (120596119905 minus 119896

1119911119911 minus 119899120579)]

V (119911 120579 119905) = 119895infin

sum

119899=0

119881119899exp [119895 (120596119905 minus 119896

1119911119911 minus 119899120579)]

(18)


3119911= 1198961119911



STL = 10 log10

119882119868

119882119879 (19)



119882119868=cos (120574) 1199012

0

12058811198881

119877 (20)



where

120591si =infin

sum

119899=0

119877119890 119901119879

3119899times 1198671

119899(1198963119903119877) (119895120596119882

0

119899)lowast

times 12058811198881120587

120576119899cos 1205741199012

0

(22)



1 1199092 119909

119899] 119883 isin 119877

119899

st 119892119894(119883) = 0 119894 = 1 119898

119888119894(119883) le 0 119894 = 119898 + 1 119901

(23)


119894are (p-


119883 = [119877 ℎ119873]119883lb le 119883 le 119883ub (24)


1198881= int

ℎ2

minusℎ2

((1205881minus 1205882) (

2119911 + ℎ

ℎ)

119873

+ 1205882)119889119911 minus 119898

0le 0

1198882=ℎ

119877minus 005 le 0

(25)












119888


119903) the mass (119891






minus10

80

70

60

50

40

30

20

10

0

Frequency (Hz)

TL (d

B)


fr

fpcfc

100 101 102 103 104


Frequency (Hz)100 101 102 103 104

minus20

minus10

80

70

60

50

40

30

20

10

0

TL (d

B)

AluminumSteelNickel

ZirconiaBrass




120573 119877 (m) ℎ (mm) 119873 1198911(Hz) STL (db)

0 16 00013 0393 4 38604 158 000143 0647 55 384205 151 000151 0817 6 385706 145 000167 13 6 3844











1000 0 1 011561250 06 1148 013271600 1 1259 014552000 12 1318 015242500 13 1349 015593150 12 1318 015244000 1 1259 01455






10038161003816100381610038161003816

120591avg (119883) =

119873119891

sum

119894=0

120573119894120591si (119883)

(27)


119891is the number




119894are







Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 1

Case 1

SteelAluminum

(a)

AluminumZirconia

Case 2

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 2

(b)

AluminumZirconia

Case 3

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 3

(c)

AluminumBrass

Case 4

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 4

(d)







Frequency (Hz)

Optimized shell

100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10TL

(dB)

0



01

02

03

04

05

06

07

08

09

1


Volu

me f

ract

ion

Case 1Case 2

Case 3Case 4


0 01 02 03 04

(a)


Case 1Case 2

Case 3Case 4


01

02

03

04

05

06

07

08

09

1


Volu

me f

ract

ion

0 01 02 03 04

(b)












References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















119888


119903) the mass (119891






minus10

80

70

60

50

40

30

20

10

0

Frequency (Hz)

TL (d

B)


fr

fpcfc

100 101 102 103 104


Frequency (Hz)100 101 102 103 104

minus20

minus10

80

70

60

50

40

30

20

10

0

TL (d

B)

AluminumSteelNickel

ZirconiaBrass




120573 119877 (m) ℎ (mm) 119873 1198911(Hz) STL (db)

0 16 00013 0393 4 38604 158 000143 0647 55 384205 151 000151 0817 6 385706 145 000167 13 6 3844











1000 0 1 011561250 06 1148 013271600 1 1259 014552000 12 1318 015242500 13 1349 015593150 12 1318 015244000 1 1259 01455






10038161003816100381610038161003816

120591avg (119883) =

119873119891

sum

119894=0

120573119894120591si (119883)

(27)


119891is the number




119894are







Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 1

Case 1

SteelAluminum

(a)

AluminumZirconia

Case 2

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 2

(b)

AluminumZirconia

Case 3

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 3

(c)

AluminumBrass

Case 4

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 4

(d)







Frequency (Hz)

Optimized shell

100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10TL

(dB)

0



01

02

03

04

05

06

07

08

09

1


Volu

me f

ract

ion

Case 1Case 2

Case 3Case 4


0 01 02 03 04

(a)


Case 1Case 2

Case 3Case 4


01

02

03

04

05

06

07

08

09

1


Volu

me f

ract

ion

0 01 02 03 04

(b)












References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120573 119877 (m) ℎ (mm) 119873 1198911(Hz) STL (db)

0 16 00013 0393 4 38604 158 000143 0647 55 384205 151 000151 0817 6 385706 145 000167 13 6 3844











1000 0 1 011561250 06 1148 013271600 1 1259 014552000 12 1318 015242500 13 1349 015593150 12 1318 015244000 1 1259 01455






10038161003816100381610038161003816

120591avg (119883) =

119873119891

sum

119894=0

120573119894120591si (119883)

(27)


119891is the number




119894are







Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 1

Case 1

SteelAluminum

(a)

AluminumZirconia

Case 2

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 2

(b)

AluminumZirconia

Case 3

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 3

(c)

AluminumBrass

Case 4

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 4

(d)







Frequency (Hz)

Optimized shell

100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10TL

(dB)

0



01

02

03

04

05

06

07

08

09

1


Volu

me f

ract

ion

Case 1Case 2

Case 3Case 4


0 01 02 03 04

(a)


Case 1Case 2

Case 3Case 4


01

02

03

04

05

06

07

08

09

1


Volu

me f

ract

ion

0 01 02 03 04

(b)












References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 1

Case 1

SteelAluminum

(a)

AluminumZirconia

Case 2

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 2

(b)

AluminumZirconia

Case 3

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 3

(c)

AluminumBrass

Case 4

Frequency (Hz)100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10

TL (d

B)

0

Case 4

(d)







Frequency (Hz)

Optimized shell

100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10TL

(dB)

0



01

02

03

04

05

06

07

08

09

1


Volu

me f

ract

ion

Case 1Case 2

Case 3Case 4


0 01 02 03 04

(a)


Case 1Case 2

Case 3Case 4


01

02

03

04

05

06

07

08

09

1


Volu

me f

ract

ion

0 01 02 03 04

(b)












References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Frequency (Hz)

Optimized shell

100 101 102 103 104

minus20

minus10

70

60

50

40

30

20

10TL

(dB)

0



01

02

03

04

05

06

07

08

09

1


Volu

me f

ract

ion

Case 1Case 2

Case 3Case 4


0 01 02 03 04

(a)


Case 1Case 2

Case 3Case 4


01

02

03

04

05

06

07

08

09

1


Volu

me f

ract

ion

0 01 02 03 04

(b)












References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Optimization of Sound Transmission Loss through a Thin …downloads.hindawi.com/journals/sv/2014/814682.pdf · 2019-07-31 · Research Article Optimization of Sound