research article vibroacoustical analysis of multiple

Hindawi Publishing CorporationISRNMechanical EngineeringVolume 2013 Article ID 645232 13 pageshttpdxdoiorg1011552013645232

Research ArticleVibroacoustical Analysis of Multiple-Layered Structures withViscoelastic Damping Cores

Fei Lin and Mohan D Rao

Michigan Technological University Houghton MI 49931 USA

Correspondence should be addressed to Fei Lin kevinlinmtuedu

Received 2 December 2012 Accepted 16 December 2012

Academic Editors M Ahmadian R Brighenti S W Chang J Clayton P Dineva and G-J Wang

Copyright copy 2013 F Lin and M D RaoThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a modeling technique to study the vibroacoustics of multiple-layered viscoelastic laminated beams using theBiot damping model In this work a complete simulation procedure for studying the structural acoustics of the system using ahybrid numerical model is presented The boundary element method (BEM) was used to model the acoustical cavity whereas thefinite element method (FEM) was the basis for vibration analysis of the multiple-layered beam structure Through the proposedprocedure the analysis can easily be extended to another complex geometry with arbitrary boundary conditions The nonlinearbehavior of viscoelastic dampingmaterials was represented by the Biot dampingmodel taking into account the effects of frequencytemperature and different damping materials for individual layers The curve-fitting procedure used to obtain the Biot constantsfor different damping materials for each temperature is explainedThe results from structural vibration analysis for selected beamsagreewith published closed-form results and results for the radiated noise for a sample beam structure obtained using a commercialBEM software are compared with the acoustical results of the same beam by using the Biot damping model

1 Introduction

The traditional designs of free-layer constrained-layer orsandwich-layer damping treatment using viscoelastic mate-rials have been around for over forty years Recent improve-ments in the understanding and application of the damp-ing principles together with advances in materials scienceand manufacturing have led to many successful applica-tions and the development of patch damping and multiple-layered damping structures The key point in any designis to recognize that the damping material must be appliedin such a way that it is significantly strained whenever thestructure is deformed in the vibration mode under investi-gationNumerous researchers have successfully implementedthe passive constrained layer (PCL) and active constrainedlayer (ACL) systems In 1959 Kerwin [1] and Ross et al[2] presented a general analysis of viscoelastic materialstructure The damping was attributed to the extension andshear deformations of the viscoelastic layers Ditaranto [3]developed sixth-order equations of motion in terms of axialdisplacements and developed a closed-form solution MeadandMarkus [4] extended the sixth-order equations ofmotion

for transverse displacement to include various boundaryconditions A paper by Rao [5] presented the equations ofmotion of viscoelastic sandwich beams with various bound-ary conditions using the energy method The equations weresolved numerically and a practical design guideline waspresented Similar to Raorsquos theory Cottle [6] used Hamiltonrsquosprinciple to derive equations of motion The damping couldalso be increased by adding passive stand-off layer (PSOL)and slotted stand-off layer (SSOL) to the layered systemsFalugi [7] and Parin et al [8] conducted theoretical andexperimental work on a four-layered panel and a five-layeredbeam with PSOL treatment Rogers and Parin [9] and Yellinet al [10] have performed experimental investigations anddemonstrated that PSOL treatment increased the dampingsignificantly in aeronautical structures and beams Yellin andcolleagues [11 12] also developed normalized equations ofmotion for beam fully treated with PSOL using nonidealstand-off layer assumption The equations were solved usingthe method of distributed transfer functions [13]

In addition to the closed-form analytical approach manyresearchers have used the finite element method (FEM)the most popular numerical modeling method in building

2 ISRNMechanical Engineering

the numerical model of the multiple layers system In 2000Chen and Chan [14] studied four different types of integralFEMmodels with the viscoelastic coresThenumerical stabil-ity and accuracy as well as for convergence issue of these fourdifferent FEM models were demonstrated by comparing thenumerical results with those from experiments Lesieutre andLee [15] proposed a 3-node 10DOFFEMmodel for the three-layer ACL damping beam This FEM model is advantageousin active control application due to its features of nonshearlocking and adaptability to segmented constraining layers

Other than theHamilton and FEMnumerical methods inbuilding the models other researchers have proposed manyirregular modeling techniques for the numerical representa-tion of continuousdiscontinuous systems Kung and Singh[16] calculated the natural frequencies and loss factor usingthe Rayleigh-Ritz energy method and modal-strain energytechnique in modeling a 3-layer patch damping structureThese approximate modeling methods were also extended torectangular damping patch of plates and shells with visco-elastic cores Zhang and Sainsbury [17] combined the Ger-lerkin orthogonal function with the traditional finite elementmethod and successfully applied to the vibration analysis ofthe damped sandwich plates

While the FEM is used widely in the modeling of thestructure many researchers sought for proper mathematicalmodels to represent the damping behavior of the viscoelasticmaterial as well as incorporating the dampingmodel in com-mercial FEM software packages Currently many FEM com-mercial software incorporate damping models based mostlyon viscoushysteretic damping Some allow incorporation ofdamping energy dissipation in the time domain using theProny series None of these damping models however issuitable to capture the damping behavior in the frequencydomain which is the most important issue in predicting thevibro-acoustical response of complex structural systems Thedrawback of these dampingmodels raised considerable inter-est and motivation in the development of damping modelsof viscoelastic material in the frequency domain compatiblewith FEM software These damping models can be classifiedas derivative type and integral type

The ldquoFractional Derivativerdquo is essentially the representa-tive damping model in the derivative form family proposedby Bagley and Torvik [18] in 1983 This damping model notonly described the material properties of viscoelastic damp-ing but so established the closed-form equation compatiblewith the FEM technique Compared with the other integral-form models the fractional derivative is only able to capturethe relatively weak frequency-dependent information how-ever it was an important milestone in the area of dampingresearch

Lesieutre et al [20] mathematically modeled the relax-ation behavior of viscoelasticmaterial in terms of augmentingthermodynamic field (ATF) in 1989 Initially introducing asingle augment field this dampingmodel provided the abilityto represent the light-damping behavior with the applicationof a 1D viscoelastic structure In the subsequent researchusing a series of augment fields the ATF model is able tomodel the damping material of higher loss factor with theweak frequency dependence Remedying the limitation of

1D application Lesieutre and Lee [15] proposed an anelasticdisplacement field (ADF) technique in 1996 and successfullyextended its application from the 1-D problem to the 3-layersandwich beam and 3-D problems

As far as the mini-oscillator damping models are con-cerned the complex shear modulus which is a function ofboth frequency and temperature can be expressed by a seriesof mini-oscillation perturbations Biot [21] first proposed thefirst-order relaxation function with the introduction of theldquodissipative variablesrdquo into the dynamic equations using thetheory of irreversible thermodynamics In 2007 Zhang andZheng [22] utilized the Biot model to describe the dynamicbehavior of a viscoelastic structure The dimension reduc-tion technique and nonlinear curve-fitting procedure werediscussed in the paper McTavish [23] developed anothermini-oscillator damping model called ldquoGHMrdquo by the usageof Second-order relaxation function Compared with the Biotmodel the GHM model has a more complicated expressionand also requires better performance of the computationaltool

The popularity of these integral-form damping models inrecent years brought two research interests nonlinear curvefitting and dimension reduction The advanced curve-fittingtechniques in the damping models guarantee the accuracyof the numerical representation of the actual shear modulusdata from the experiment The dimension reduction tech-nique increases the computational efficiency due to the ad-ditional orders of equation in order to gain the frequencyindependence of the frequency-form damping model

Zhang et al [24] converted the nonlinear curve-fittingproblem in frequency domain with respect to the GHM pa-rameters into the constrained nonlinear optimization prob-lem The efficiency and correctness were demonstrated for acommercial viscoelastic material

Park et al [25] examined the GHM damping modelwith the application to the FEM method associated with theGuyan reduction technique The numerical example in thisresearch leads to an FEMmodel applied to theGHMdynamicequation quantitatively without increasing the number oforder

Hao and Rao [26] carried out the optimum design of athree-layer sandwich beam for the vibration analysis in 2005In this research the numerical model is a comprehensiveformulation for a three-layer unsymmetrical sandwich beamwith two different damping materials adjacent to each otherThe criterion of the optimization is to minimize the mass ofthe structure whilemaximizing the system damping In 2008Lee [27] published the semicoupled vibroacoustical analysisand optimization of a simply supported three-layer sandwichbeam The modal superposition method was used to inves-tigate the vibration problem with the fractional derivativedamping model The interior acoustical problem was studiedby BEM numerical technique and the optimization problemwas established through the appropriate sizing parameters ofthe sandwich beam

The objective of this paper is to extend the previous workby the authors [28] on the vibration analysis of a multiple-layered beam structure incorporating the Biot dampingmod-el to solve the acoustic problem to predict the radiated noise

ISRNMechanical Engineering 3

119906119894 119906119895 119883

120579119894 120579119895

119908119894 119908119895

Figure 1 Configuration of the elastic layer showing the DOF

In this paper we present a complete numerical procedurefor the vibroacoustical analysis and design for a multiple-layer laminated damping beam Results obtained from theproposed vibration analysis are compared with the previousclosed-form results to show the validity of this approachTheradiated noise spectrum at selected field point shows goodagreement between the 2-D BEM acoustical analysis andthe result without system damping calculated by commercialsoftware for a sample viscoelastic damping structure Theacoustical solution is demonstrated and the correlation be-tween sound pressure level (SPL) and the loss factor is alsohighlighted

2 FEM Modeling and the BiotDynamic Equation

The FEM modeling procedure and the establishment of theBiot dynamic equation will be discussed in this section Thestructure chosen for illustration is a seven-layer viscoelasticsandwich beamThe elastic beam and the constrained damp-ing layer are the two fundamental components in this FEM-modeling technique The concept of transfer matrix is usedto convert the local coordinates to the global coordinates inorder to assemble and construct the complete model of thesandwich damping structure with arbitrary number of layersThe Biot viscoelastic damping model will be used to describethe damping behavior Through the use of the FEM thestructure is discretized which will enable the use of the Biotdamping model for different damping layers in the structureThe reader is referred to the nomenclature for the definitionof different variables used in the derivation

21 FEMModeling of Component ITheElastic Layer Figure 1shows the elastic layer in the FEMmodel containing 2 nodesand 6 degrees of freedom (DOF)The element displacementsof each node can be expressed as followsz

120575119890

elastic = (119908119894 120579119894 119906119894 | 119908119895 120579119895 119906119895) )119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6 times 1

(1)

The stiffnessmatrix can be derived based on the followingenergy method

[Ke]119890

elastic = int1

0

119864119860

119897[120597Ne120597120585]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6 times 1

119879

[120597Ne120597120585]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1 times 6

119889120585

+ int1

0

119864119868

119897[1205972Nf1205972120585

]

119879

[1205972Nf1205971205852

]119889120585⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

6 times 6

(2)

120579119894120579119895

119895119894

11990631198951199063119894

11990611198951199061119894

119883

119908119894119908119895

Figure 2 Configuration of the constrained damping layer showingthe DOF

as the shape functions are the following

[Nf]

= [1minus31205852

+21205853

(120585 minus 21205852

+1205853

) 119897 0 31205852

minus21205853

(minus1205852

+1205853

) 119897 0]

[Ne] = [0 0 1 minus 120585 0 0 120585]

(3)

in which 120585 the local coordinate 120585 = 119909119897 120585 isin [0 1] 119897 longi-tudinal length of elastic layer 119860 cross-sectional area of theelastic layer 119864 Youngrsquos modulus of the elastic layer and 119868moment of inertia of elastic layer

Similarly the element mass matrix can be expressed as

[M]119890elastic⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6 times 6

= int1

0

119898119897([Nf]⏟⏟⏟⏟⏟⏟⏟6 times 1

119879

[Nf]⏟⏟⏟⏟⏟⏟⏟1 times 6

+ [Ne]119879

[Ne]) 119889120585 (4)

22 FEM Modeling of Fundamental Component II The Con-strained Damping Layer The FEMmodel of the constrainedlayout containing the damping layer sandwiched between twoouter layers is shown in Figure 2 This Figure illustrates eachelement consisting of 2 nodes and 8 DOF where the nodaldisplacement vector is as follows

120575119890

cons = (119908119894 120579119894 1199061119894 1199063119894 | 119908119895 120579119895 1199061119895 1199063119895)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟8 times 1

(5)

Through the introduction of transfer matrix

[TI] = [e1 e2 e3 e5 e6 e7]119879

[TIII] = [e1 e2 e4 e5 e6 e8]119879

(6)

in which each ei means the following vector ei =

( 0 0 sdotsdotsdot sdotsdotsdot 1⏟⏟⏟⏟⏟⏟⏟119894th placesdotsdotsdot sdotsdotsdot 0 )

119879

the element elastic stiffness and theelement viscoelastic stiffness matrix for this 3-layer compo-nent respectively are the following

[Ke]119890

cons = [TI]119879

sdot [Ke]elastic sdot [TI]

+ [TIII]119879

sdot [Ke]elastic sdot [TIII]

[Kv]119890

cons = int1

0

11986621198602119897

119896ℎ2

[Ne1 minus Ne3

ℎ2

+ℎ0

ℎ2

sdot1

119897sdot120597Nf1120597120585

]

119879

times [Ne1 minus Ne3

ℎ2

+ℎ0

ℎ2

sdot1

119897sdot120597Nf1120597120585

] 119889120585

[Ne1]⏟⏟⏟⏟⏟⏟⏟⏟⏟1 times 8

= [Ne]⏟⏟⏟⏟⏟⏟⏟1 times 6

[TI]⏟⏟⏟⏟⏟⏟⏟6 times 8

[Ne3] = [Ne] [TIII]

(7)


where1198602 cross-sectional area of the damping layer119866

2 long-

term shear modulus of the damping layer and 119896 correctionfactor of the shear strain energy for the rectangular cross-section 119896 = 12

Also the element mass matrix for this 3-layer componentis

[M]119890cons = [TI]119879

sdot [M]elastic sdot [TI] + [M]119890

cons2

+ [TIII]119879

sdot [M]elastic sdot [TIII]

(8)

where

[M]119890cons2=int1

0

1198982119897[Nf1]

119879

[Nf1] 119889120585 [Nf1]=[Nf] [TI]

(9)

23 FEM Modeling of a Seven-Layer Constrained DampingBeam The seven-layer sandwich beam consists of seven al-ternating layersmdashfour elastic layers and three damping layersFigure 3 shows the FEM model of a seven-layer sandwichbeam containing 2 nodes and 10 DOF and the node displace-ment vector is as follows

120575119890

7layer = ( 119908119894 120579119894 1199061119894 1199063119894 1199065119894 1199067119894 | 119908119895 120579119895 1199061119895 1199063119895 1199065119895 1199067119895 )119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟12times1

(10)

The transfermatrix to obtain the element stiffness and themass matrix when the 1st 3rd 5th and 7th layers are elasticare follows

[T1] = (e1 e2 e3 e7 e8 e9)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6times12

[T3] = (e1 e2 e4 e7 e8 e10)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6times12

[T5] = (e1 e2 e5 e7 e8 e11)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6times12

[T7] = (e1 e2 e6 e7 e8 e12)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6times12

(11)

Similarly the element stiffness and the mass matrix forthe 2nd 4th and 6th layers of the constrained damping layercan be derived through the transfer matrix

[T2] = (e1 e2 e3 e4 e7 e8 e9 e10)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟8times12

[T4] = (e1 e2 e4 e5 e7 e8 e10 e11)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟8times12

[T6] = (e1 e2 e5 e6 e7 e8 e11 e12)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟8times12

(12)

where the notation ei means

ei = (0 0 sdot sdot sdot sdot sdot sdot 1⏟⏟⏟⏟⏟⏟⏟

119894th placesdot sdot sdot sdot sdot sdot 0

)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟12 times 1

(13)

120579119894 120579119895

119895119894119883

1199067119894

1199065119894

1199063119894

1199061119894

1199063119895

1199061119895

1199065119895

1199067119895

119908119894 119908119895

Figure 3 Configuration of a seven-layer damping structure show-ing the DOF

Based on the above equations and design parameters ofeach layer the element massstiffnessdamping matrix of theseven-layer sandwich damping beam can be expressed asfollows

[Ke]119890

= T1198791 [Ke1]T1 + T1198793 [Ke3]T3

+ T1198795 [Ke5]T5 + T1198797 [Ke7]T7

[Kv]119890

= T1198792[Kv2]T2 + T1198794 [Kv4]T4 + T1198796 [Kv6]T6

[Me]119890

=

7

sum119894=1

T119879i [Mei]Ti

(14)

Thus the element matrices can be assembled to obtainthe global massstiffnessdamping matrix and can be appliedto the boundary condition through the conventional FEMtechnique Taking into the consideration of the viscoelasticdamping properties the global matrices need to be manipu-lated as a portion of the Biot dynamic equation

24 Introduction of the Biot Dynamic Equation To considerthe vibration problem numerically the dynamic equationdiscretized by FEM technique needs to be expressed by thefollowing second-order ordinary differential equation (ODE)form

Mx + Cx + Kx = f (t) (15)

The Biot viscoelastic damping model numerically rep-resents the complex shear modulus with a series of mini-oscillator perturbing terms

119904 (119904) = 119866infin

[1 +

119898

sum119896=1

119886119896

119904

119904 + 119887119896

] (16)

in which 119866infin is the long-term shear moduli 119886119896and 119887119896are

the Biot constants These parameters are positive and canbe determined by nonlinear curve fitting from the experi-mental data The curve-fitting procedure will be discussed inSection 3

Substituting the Biot damping model into (15) thedynamic equation with 119898 terms of the Biot parameters for


the first viscoelastic material and 119899 terms for the secondviscoelastic material can be developed as follows

119866infin

1Λv1 = Λ1 119866

infin

2Λv2 = Λ2

R1 = 119866infin

1Rv1Λv1 R2 = 119866

infin

2Rv2Λv2

(17)

where Rv and Λv are the eigenvector and diagonal eigen-value matrices respectively from the damping matrix CAdditionally 119886

11sdot sdot sdot 1198861119898 11988711sdot sdot sdot 1198871119898 and 119911

11sdot sdot sdot 1199111119898

denote 119898terms of the Biot parameters and the dissipative coordinatesrespectively for first viscoelastic material

Similarly 11988621sdot sdot sdot 1198862119899 11988721sdot sdot sdot 1198872119899 and 119911


denote 119899terms of the Biot parameters and the dissipation coordinatesrespectively for second viscoelastic material A detailedderivation can be found in the previous publication [28]

3 Parametric Determination of the BiotDamping Model

A curve-fitting technique is used to provide the accurateBiot constants to the dynamic equation and to establish thedynamic characteristics of the viscoelastic materials In thissection the nonlinear curve-fitting procedure for the com-plex shearmodulus in the frequency domain is converted intoa nonlinear constrained optimization problem

The complex shearmoduluswith the Biot dampingmodelform can be broken into real and imaginary parts separately

119904 (119895120596) = 119866infin

[1 +

119873

sum119894=1

1198861198941206032

1198872119894

+ 1206032] + 119895119866

infin

[1 +

119873

sum119894=1

119886119894119887119894120603

1198872119894

+ 1206032]

(18)

The Biot parametersmdash119866infin 119886119894 and 119887

119894mdashare estimated from

experimental data with the certain fitting frequency range onreal part and imaginary parts separately Generally speakingone set of the Biot parameters needs to be determined foreach ambient temperature independently In (18) 119873 is thenumber of the Biot perturbing items defining the capabilityof this numerical approximation As the Biot terms (119873) areincreased the relative error between the experimental dataand the curve-fitting result reduces

Assuming 1199091= 119866infin 1199092= 1198861 1199093= 1198871 1199094= 1198862 1199095= 1198872

with the constraint condition 119909119896 ge 0 119896 = 1 2 num thetarget equation of the optimization problem is the following

min119909

F (119909) =119875

sum119895=1

10038161003816100381610038161003816119866lowast

119895(119909) minus 119866

0119895

10038161003816100381610038161003816

2

(19)

In the target equation (19) 1198660119895

stands for the complexshear modulus from the experimental data with 119875 interestedpoints (larger than the number of unknowns) The 3M ISD-110112 viscoelastic polymer is selected in this simulationTheexperimental data is obtained by the Arrhenius empiricalequation from [19]With a specific fitting range at a particulartemperature the complex shear modulus can be synthesizedfrom one set of the Arrhenius coefficients The numberof terms (119873) in (18) needs to be determined to ensure

Table 1 The Biot constants of 3M ISD-110112 45∘C

ISD110 ISD112119866infin 55000 (Pa) 172000 (Pa)

a1 1809517 5699386303a2 1453095 0596843249a3 3221535 1000560485a4 5201026 0577694736a5 1976822a6 6561162b1 5410993 426818097b2 1093778 7026089968b3 6036544 5015607814b4 4319613 1969150769b5 2840958b6 2980672

the precision of this approximation The curve fitting of theexperimental data is accomplished using the commercialsoftware package Auto2fit on the real and imaginary partssimultaneously Using the Biot terms equal to six and fourwith respect to two commercial damping materials 3M ISD-110 and 112 respectively the results are shown in Table 1 forambient temperature (119879) equal to 45∘C and frequency rangeof 500Hz

Figures 4(a) and 4(b) show the comparison between theArrhenius data and curve-fitting data for the real and imag-inary parts respectively Figure 5 shows the relative error inthe fitting range

As shown in Figures 4(a) and 4(b) the Biot parametricdetermination technique estimates the dynamic properties of3M ISD-110112 at 45∘Cwith almost zero errorThe constantsdetermined using the above procedure along with the FEMmodel of sandwich beam will now be incorporated to solvethe complete Biot dynamic equation using the decouplingtransformation technique

4 Decoupling Transformation andDynamic Solution

In this section the algorithm used to obtain the frequencyresponse function (FRF) will be discussed with respect tothe vibroacoustical problem for a multiple-layer viscoelasticdamping structure In this research the damping matrix Din (15) does not have a proportional relationship with themass and stiffness matrixThus a decoupling transformationis needed to construct the first-order state equation byintroducing the auxiliary equationMq minusMq = 0 as follows

Ay + By = f (20)

where

A = [D MM 0 ] B = [K 0

0 minusM]

y = qq f = f0

(21)


500450400350300250200150100500

25

2

15

1

05

0

Frequency (Hz)

times106

The Arrhenius dataCurve fit data

Real

part

(a)

500450400350300250200150100500

25

2

3

15

1

05

0

Frequency (Hz)

times106


Imag

inar

y pa

rt

(b)

Figure 4 (a) Comparison between the Arrhenius and curve-fitting data for the real part of the shear modulus (3M-ISD-110 45∘C) (b)Comparison between the Arrhenius and curve-fitting data for the imaginary part of the shear modulus (3M-ISD-110 45∘C)

Here119873 is the number of DOF in theM D and Kmatricesthe DOF of A and Bmatrices is 2119873

Firstly the free vibration of (20) will be consideredAssuming f = 0 the following form of solution is obtained

(A120582 + By)Φ = 0 (22a)

or

(A120582 + By) ΨΨ120582 = 0 (22b)

where 120582matrix stands for 2119873 complex conjugate eigenvaluesincluding the natural frequencies and loss factors informa-tion

[[[

[

120582

]]]

]

=

[[[[[[

[

1205821

120582lowast

10

0 120582

119873

120582lowast

119873

]]]]]]

]

(23)

It must be noted that zero items will appear in the eigen-value matrix if the damping matrixD is not fully rankedThemode shape vector Ψ for the vector q can be extracted fromthe eigenvector matrixΦ with respect to the vector y

[Φ] = [Ψ]1

[Ψ]lowast

1sdot sdot sdot [Ψ]

119873[Ψ]lowast

119873

1205821[Ψ]1120582lowast

1[Ψ]lowast

1sdot sdot sdot 120582119873[Ψ]119873120582lowast

119873[Ψ]lowast

119873

(24)

In addition (22b) can be numerically solved by 120582Ψ =minus[A]minus1[B]Ψ using mathematical software package such asMATLAB or Mathematica

Secondly the forced vibration solution of (20) in the timedomain will be discussed Assuming f = F0 the variablesubstitution can be made by assuming 119910 = Φ119901 convertingthe state-space equation from the time space to the modal

5004003002001000

40

35

3025

20

15

105

0

minus5

Frequency (Hz)

Rela

tivel

yer

ror(

)

Error in real partError in imag part

Figure 5 Relative error between Arrhenius and curve-fitting data

space By left multiplying ofΦ119879with the substitution of 119910 weget

Φ119879AΦ +Φ119879BΦ119901 = Φ119879f (25)

The diagonal modal mass and stiffness matrix are

Φ119879AΦ =

[[[

[

Mp

]]]

]

Φ119879BΦ =

[[[

[

Kp

]]]

]

(26)

Then rewrite the equation with the diagonal mass andstiffness matrices

[[[

[

Mp

]]]

]

+[[[

[

Kp

]]]

]

119901 = Φ119879f (27)


The FRF in the frequency domain can be easily deter-mined through the complex conjugate eigenvalue matrix 120582eigenvector matrix Φ

and the modal mass matrix Mp The

modal scaling factor matrix can be calculated through thefollowing

[[[

[

Q

]]]

]

=[[[

[

Mp

]]]

]

minus1

(28)

Thus FRF can be established through the modal param-eters being expressed in partial fraction form in terms of theresidue vector and system poles as follows

[H (119895120596)] =X (119895120596)F (119895120596)

=

119873

sum119894=1

[QiΨiΨ

Ti

(119895120596 minus 120582119894)+QiΨlowast

i Ψlowast

iT

(119895120596 minus 120582lowast119894)]

(29)

The system velocity can now be obtained from the aboveequation by a simple Fourier transformation By doing so thevibration problem can be extended to an FRF-based acousti-cal problem and the combination of these two analyses is theparticle velocities information calculated by the following

X (119895120596) = 119895120596 [H (119895120596)] F (119895120596)

= 119895120596 [F (119895120596)] sdot119873

sum119894=1

[QiΨiΨ

Ti

(119895120596 minus 120582119894)+QiΨlowast

i Ψlowast

iT

(119895120596 minus 120582lowast119894)]

(30)

5 Acoustical Boundary Element Method(BEM) Analysis

51 Introduction of Acoustical BEM Theory In Section 4 thevibration problem of the multiple-layer sandwich beam issolved through the time-domain dynamic ordinary differ-ential equation of the Biot damping model with numericalanalysis by the FEM technique The vibration problem canbe extended to the acoustical problem by the semicoupledmethod the vibrationwill induce a change in sound pressureyet the sound pressure will not cause the vibration In thissection the acoustical interior problem will be numericallysolved by 2D boundary element method (BEM) technique[29] in a bounded fluid domain 119881 as shown in Figure 6

The sound pressure distribution (119901) of the time-harmonicwave in the domain 119881 satisfies the governing differentialequation well known as the Helmholtz equation associatedwith the boundary conditions on boundary Γ(= Γ

119901cup Γvn cup ΓZ)

as followsnabla2

119901 (119909) + 1198962

119901 (119909) = 0when

119901 (119909) = 1199010 119909 isin Γ

119901

vn (119909) equiv minus1

1198951199081205880

120597119901

120597119899= vn0

119909 isin Γvn

z (119909) equiv119901

vn= z0 119909 isin ΓZ

(31)

119881Γ119901

119899

Γ119885

Γ119907119899

Figure 6 Notations of 2-D BEM interior problem in a fluid domain119881

Here 119896 is equal to 120596119888 which means that the wavenumber is equal to the radiant frequency over the speed ofsound vn 1205880 z stand for the normal velocity density of thefluid 119881 (normally the air) and acoustical impedance of thefluid 119881 respectively

In this work the link between the vibration and theacoustics analysis is the normal velocity at the acousticalboundaries Recalling the dynamic solution of the decouplingtransformation the particle velocity in the time domain ateach node can be calculated through (30) if themultiple-layersandwich beam is discretized by the FEM alternatively theFRF the complex ratio between the output and input responsein the frequency domain can be determined through (29)Once the input signal is given the particle velocity of thesystem displacement versus frequency relationship can beconveniently obtained through the FRF

To solve the governing differential equation (31) in thebounded fluid domain 119881 the Helmholtz Equation can betransformed into the integral equation converting the 2-Darea integration to the 1-D curve integration around the area

119888 (120585) 119901 (120585)+intΓ

120597Ψ (120585 119909)

120597119899119901 (120585) 119889Γ=minusint

Γ

1198941205880120596Ψ (120585 119909) vn (119909) 119889Γ

(32)

in which 119888(120585) geometry-dependent coefficient normally119888(120585) = 0 when 120585 is in the domain 119881 and 119888(120585) = 05 when 120585is on the smooth boundary Γ 119901(120585) sound pressure at sourcepoint 120585 Ψ(120585 119909) 119909 is the field point and Ψ = minus(1198944)H(2)

0(119896119903)

for the 2D BEM problem 119903 the Euclidian distance between 119909and 120585 H(2)

0and the Second-type Henkel function 119899 normal

vector pointing away to the fluid domain 119881By discretizing the boundary into a series of curve-linear

elements through the introduction of the shape functions theintegral equation can be calculated numerically by solving thefollowing linear matrix

HP = 119866VN (33)

where H comes from the terms of 119888(120585) and intΓ

119901(120585)(120597Ψ(120585 119909)

120597119899)119889Γ 119866 is derived from minusintΓ

1198941205880120596vn(119909)Ψ(120585 119909)119889Γ and the

vector P andVN include sound pressure and particle velocity


Table 2 Design parameters of seven-layer structure

Length 1m Thickness 01m Number of element 12 Number of nodes 13Number of layer Height of layer Elasticviscoelastic properties Material density1st ℎ

1= 1mm 119864

1= 210GPa 120588

1= 7800 kgm3

2nd ℎ2= 08mm 119866

2 Biot 120588

2= 970 kgm3

3rd ℎ3= 1mm 119864

3= 210GPa 120588

3= 7800 kgm3

4th ℎ4= 08mm 119866

4 Biot 120588

4= 970 kgm3

5th ℎ5= 1mm 119864

5= 210GPa 120588

5= 7800 kgm3

6th ℎ6= 08mm 119866

6 Biot 120588

6= 970 kgm3

7th ℎ7= 1mm 119864

7= 210GPa 120588

7= 7800 kgm3

Anechoicboundaries

Field point(05 04)

119865 (impulse force)

1 m

06 m

Figure 7 Layout of BEM acoustical cavity problem

values both unknowns and known from the boundarycondition

Thus each set of node velocities due to the force inputresults in one set of solutions on the sound pressure byBEM discussed in this section In sum through the proposedacoustical BEM it is possible to compute the time-harmonicsound pressure distribution corresponding to each singlefrequency point in the frequency spectrum

52 Calculation Details in This BEM Analysis For this par-ticular acoustical BEM interior problem the boundary ofacoustical cavity is discretized as 18 quadratic equally spacedboundary elements The quadratic curvilinear element hasthree nodes and the interpolation between each node repre-sents the geometry of each element The shape functions areas folows

1198731=1

2120585 (120585 minus 1) 119873

2= (120585 + 1) (120585 minus 1)

1198733=1

2120585 (120585 + 1)

(34)

with respect to the following element coordinates

119909 =

3

sum119894=1

119909119894119873119894(120585) 119910 =

3

sum119894=1

119910119894119873119894(120585) (35)

where 119909119894and 119910119894are the coordinates at each nodal point and 120585

stands for the local coordinate between minus1 and 1 on a masterelement

Figure 8 Seven-layer sandwich structure with viscoelastic cores

When the seven-layered sandwich beam (119871 = 1m)is simply supported at the bottom of the acoustical cavitythe sound pressure level at the field point (119909 = 05m119910 = 04m) is calculated through this proposed method andthe calculation results are presented in Section 6 Figure 7demonstrates the detailed layout of this 2D acoustical cavityproblem The anechoic boundary condition is applied onthe inside of the acoustical cavity and the thickness of themultiple-layered beam is neglected

6 Numerical Results and Discussion

61 Design Parameter of Sandwich Beam and Vibration Anal-ysis Result In Figure 8 a seven-layer sandwich beam withviscoelastic cores is shown with the design parameters listedin Table 2

The data presented in Table 2 are used to predict thevibration performance of the system using the numericalsimulation method presented in this paper and the resultsare compared with the closed-form solution of Hao [19] Thecurve-fitting results for the damping material 3M ISD-110 at45∘C discussed earlier are selected for the shear modulus ofthe viscoelastic layers in this example The results are shownin Table 3 It shows that the simulation presented in thispaper conforms to the closed-form solution This validatesthe analysis methodology proposed in the paper

62 Frequency-Spectrum Analysis under the Arbitrary InputFigure 9 shows the transverse velocity of the middle node(node number 7) with a 10N step input in the frequencydomain vertically applied at the middle (node number 7) ofthe simply-supported seven-layer sandwich beam with thesame design parameters as the previous example The samecurve-fitting results of 3M ISD110 at the ambient temperatureof 45∘C for the shear modulus are used in this example Thispivotal result is the demonstration of extending the vibration


Table 3 Comparison of results for simply supported boundarycondition

Hao [19] FEMmodel ofthis paperNumber

of modeDampingmodel ISD110-45∘C

ArrheniusISD110-45∘C6-term Biot

1st Frequency 47443Hz 45834HzLoss factor 06248 07916

2nd Frequency 13902Hz 139489HzLoss factor 06008 06824

3rd Frequency 27661Hz 277721HzLoss factor 05317 05632

4th Frequency 461548Hz 471053HzLoss factor 04715 04681


0 50 100 150 200 250 3000

1

2

3

4

5

6

7

Frequency (Hz)

(ms

)

Transverse velocity on driving point-node number 7times10minus5

Figure 9 Transverse velocity under the impulse excitation (onmiddle node 7)

to the acoustical problem in the frequency domain when anarbitrary force is applied on the structure

63 Acoustical BEM Results Figure 10 illustrates the contourplot (119891 = 10Hz) of SPL when the seven-layer sandwich beam(using the same design parameters as before) is subjected to a10N step input in the frequency domain at the middle node

The interpolation of each elements result in Figures 10and 11 shows the continuous sound pressure distribution inthe acoustical cavity with an anechoic boundary conditionFigure 12 extracts the frequency spectrum of SPL at the filedpoint (05 04m) indicated by red dot in Figure 10 Fromthe results of Figure 12 it can be found that the dominantcontribution is due to the peak value of the first flexiblevibration mode which is in agreement with the frequency-spectrum analysis of the vibration problem

Table 4 Comparison of modal results with ANSYS simulation

ANSYS 3D FEMmodel of thisresearchNumber of

modeDampingmodel

No damping 3M-ISD110-45∘C6-term Biot

1st Frequency 64803Hz 45834HzLoss factor 07916

2nd Frequency 14338Hz 139489HzLoss factor 06824

3rd Frequency 27989Hz 277721HzLoss factor 05632

64 Validation Using a BEM Commercial Software In thissection a hybrid FEM-BEM model of a beam without theviscoelastic damping was developed using the commercialsoftware packages ANSYS ADPL and LMS Virtual LabAcoustics The harmonic vibration analysis is conducted inANSYS APDL module and the frequency spectrum of fieldpoint SPL was calculated in VirtualLab Acoustics modulefor comparison with the SPL frequency spectrum presentedin Section 5 The analysis sequence consists of the followingsteps

(a) Build the FEM model and apply appropriate bound-ary conditions in ANSYS ADPLThe 8-node elementSOLID45 (element size =10mm for each layer) wasused to build the 3D seven-layer model The designparameters are identical with the parameters in Tables1 and 2 for the comparison and the geometry bound-ary conditions are simply supported A 10 N force ateach frequency is applied at the middle nodes

(b) Conduct the harmonic vibration analysis in ANSYSADPL The harmonic analysis is used to calculate thenodal displacements for a forced vibration problemin the frequency domain The frequency range is 0ndash200Hz with a 2Hz for step size and the full methodis being utilized in this analysis The comparison ofsystem frequencies between ANSYS modal resultsand calculation results by the Biot dynamic equationis shown in Table 4 The results show that the 3Dmodel built in ANSYS APDL has good correlationwith the FEMmodel

(c) Prepare the BEM mesh in LMS Virtual Lab Pre-Acoustics module It converts from a solid FEMmodel to a skin mesh that the BEM analysis requiresThe BEM mesh can be seen as a wrap around thestructural mesh and usually the BEMmesh is coarser

(d) Calculate the sound pressure in LMS VirtuallabAcoustics module Both acoustical and structuralmeshes are imported to VL Acoustics The nodaldisplacement at each vibration mode calculated inANSYS APDL is also imported and mesh-mapped tothe acoustical skin mesh as the vibration boundaryconditionThe location of field plane and field point is


SPL contour plot with anechoic

44684467

4466

4465

44644463

4462

4461

4464459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)

BC-10 Hz

Figure 10 Contour Plot of Sound Pressure Level (in dB) when the impulse force applied is119873

4468

4467

4466

4465

44644463

44624461

446

4459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)

SPL plot with anechoic BC-10Hz

Figure 11 Element result of Sound Pressure Level in dB (10Hz)

consistent with the 2D BEM analysis in this researchThe acoustical pressure is solved over the frequencyrange from 2 to 200Hz

As shown in Figure 13 the peak frequency from the 2-DBEM calculation matches with the first dominant SPL peakobtained from the VL Acoustic result without the dampingComparing the two results it is clear that the introductionof viscoelastic damping not only causes almost a 20 dBreduction in the first peak SPL but also attenuates the soundat other peaks as well This proves that the use of viscoelasticdamping material will greatly attenuate the vibroacousticalresponse of the structure

65 Acoustical Performance for a Combination of SeveralViscoelastic Materials at Different Temperatures The temper-ature is a significant external factor affecting the performance

of viscoelastic damping material in a mechanical systemWith an increase in temperature the loss factor approachesits best performance towards the transition region and thendecreases afterwards In this example the objective is to studythe effects of both 3M ISD110 material (that has a betterdamping performance) and the 3M ISD112 over the chosentemperature between 40 and 60 degree Celsius It is of interestto study the effect of the combination of these two materialson the damping of the structure

To introduce the different viscoelastic materials theseven-layer sandwich beam (with the same parameters asin the previous example) is redesigned incorporating bothdamping materials (3M ISD110 and ISD112) This system iscompared to an identical structure with only one dampingmaterial (either 3M ISD110 or ISD112) In the system includ-ing two viscoelastic materials the outer damping layers (2ndand 6th) are 3M ISD112 and the inner damping layer (5th)


200180160140120100806040200

Frequency (Hz)

75

70

65

60

55

50

45

40

35

SPL

(dB)

SPL frequency spectrum on field point (05 04 m)

Figure 12 Nodal frequency spectrum at field point (05 04m)

200180160140120100806040200

Frequency (Hz)

SPL

(dB)

110

100

90

80

70

60

50

40

30

SPL comparison between VL acoustic result and 2D BEM result

2D BEM with viscoelastic dampingVL acoustic result without damping

Figure 13 Comparison of SPL betweenVL and 2DBEMcalculationat field point (05 04) 10N input

is the 3M ISD110 The simply supported boundary conditionis examined in this numerical example and the temperaturerange is from 40 to 60 degree CelsiusThe acoustical responseis also calculated with the step input in the frequency domain(equivalent to impulse input in the time domain) Table 5shows the first order natural frequency the system loss factorand the corresponding peak value (dB) of the sound pressurelevel over the temperature range with the simply supportedboundary condition applied to the FEMmodel

It can be seen that for the same damping material as theambient temperature is increased the value of SPL increaseswhile the loss factor decreases

Table 5 First damped frequency loss factor and SPL (simplesupported BC)

Type Temp-∘C Freq-rads Loss factor SPL-dB

3M ISD-110

40 3317 1085 503545 2882 07899 519150 2625 05847 541555 2438 0464 562460 235 0387 5745

3M ISD-112

40 3265 01946 595445 3162 01652 612550 308 01413 623055 3015 01218 629060 2961 01057 6321

3M ISD 110 amp112

40 3288 04427 543145 3076 03506 564550 293 02776 576855 2822 02252 580560 2754 0185 5967

7 Conclusions

A framework for conducting vibro-acoustical analysis formultiple-layer beam structures containing different typesof viscoelastic materials is presented in this paper Severalobservations and conclusions can be drawn from the resultsof this research

(1) The vibration section of the proposed analysis con-sists of FEM model of multiple-layered dampingbeam incorporating the Biot damping model TheFEM model of the beam structure can be extendedto more complicated damping structures using thesame procedure The nonlinear curve-fitting tech-nique accurately estimates the Biot constants TheBiot damping model can then be solved using thedecoupling transformation to yield the frequency-spectrum analysis

(2) The Biot damping model is also capable of improvinga structurersquos damping performance by adding newfeatures such as different viscoelastic materials andthe variation of operating temperature The resultobtained through the procedure of vibration analysisdiscussed in this paper compares well to the closed-form solution from a previous work The first peakfrom the frequency spectrum is the predominantcause of the vibration issue in this damping structure

(3) The direct boundary element method of analysis foracoustical cavity applied under anechoic boundarieswas chosen as the basis for predicting the particlevelocity from the frequency-spectrum analysis Theacoustical result validates the frequency-spectrumresult fromvibration analysis andhas good agreementwith the predicted SPL spectrum of the identicalsandwich beam without damping calculated by com-mercial software


Nomenclature

M Mass matrixKe Kv Elastic stiffnessviscous stiffness matrixD Damping matrixx Displacement vectorf Force vectorA B Coefficient matrix of state equationz Dissipation coordinate vector119898 119899 Number of mini-oscillators for

firstsecond type of viscoelastic material119904 Laplace variable119905 Time119864 Youngrsquos modulus119866 Shear modulusNe Nf FEM shape function of

longitudinaltransverse deflection119873 Number of DOF120588 Density of materialℎ Thickness of layer119897 Length of beamΦ Eigenvector matrix120582 Eigenvalue matrix119866infin 119886119896 119887119896 Biot constants

119901 Sound pressurev Velocity vectorvn Nodal normal component of boundary

velocity

Disclosure

The authors (D Rao and F Lin) hereby declare that they donot have any direct or indirect financial relation leading toany conflict of interests with the commercial identities (BEMsoftware FEM software Auto2fit MATLAB and Mathemat-ic) mentioned in the text of their paper

References

[1] E M Kerwin ldquoDamping of flexural waves by a constrainedvisco-elastic layerrdquo Journal of the Acoustical Society of Americavol 31 pp 952ndash962 1959

[2] D Ross E E Ungar and E M Kerwin ldquoDamping of plateflexural vibration by means of viscoelastic laminaerdquo in Struc-tural Damping-a Colloquium on Structural Damping Held at theASME Annual Meeting pp 49ndash87 1959

[3] R A Ditaranto ldquoThery of vibratory bending for elastic andviscoelastic layered finite-length beamsrdquo Journal of AppliedMechanics vol 32 pp 881ndash886 1965

[4] D J Mead and SMarkus ldquoThe forced vibration of a three-layerdamped sandwich beam with arbitrary boundary conditionsrdquoJournal of Sound and Vibration vol 10 no 2 pp 163ndash175 1969

[5] D K Rao ldquoFrequency and loss factors of sandwich beamsunder various boundary conditionsrdquo Journal of MechanicalEngineering Science vol 20 no 2 pp 271ndash282 1978

[6] E T Cottle Damping of layered beams with mixed boundaryconditions [MS thesis] Air Force Institute of Technology 1990

[7] M Falugi ldquoAnalysis of a five-layer viscoelastic constrained-layer beamrdquo in Proceedings of the Damping Workshop PaperCCB 1991

[8] M Parin L C Rogers andM Falugi ldquoPractical stand off damp-ing treatment for sheet metalrdquo in Proceedings of the DampingWorkshop Paper IBA 1989

[9] L C Rogers and M Parin ldquoExperimental results for stand-off passive vibration damping treatmentrdquo in Proceedings of theSmart Structures and Materials 1995 Passive Damping pp 374ndash383 March 1995

[10] JM Yellin I Y Shen P G Reinhall and P YHHuang ldquoExper-imental investigation of a passive stand-off layer dampingtreatment applied to an Euler-Bernoulli beamrdquo in Proceedings ofthe 1999 Smart Structures and MaterialsmdashPassive Damping andIsolation vol 3672 of Proceedings of SPIE pp 228ndash233 March1999

[11] J M Yellin I Y Shen P G Reinhall and P Y H HuangldquoAn analytical and experimental analysis for a one-dimensionalpassive stand-off layer damping treatmentrdquo Journal of Vibrationand Acoustics vol 122 no 4 pp 440ndash447 2000

[12] J M Yellin and I Y Shen ldquoAn analytical model for a passivestand-off layer damping treatment applied to anEuler-Bernoullibeamrdquo in Smart Structures and Materials 2002 Damping andIsolation Proceedings of SPIE pp 349ndash357 June 1998

[13] B Yang and C A Tan ldquoThe transfer functions of one di-mensional distributed parameter systemsrdquo Journal of AppliedMechanics vol 116 pp 341ndash349 1959

[14] Q Chen and Y W Chan ldquoIntegral finite element method fordynamical analysis of elastic-viscoelastic composite structuresrdquoComputers and Structures vol 74 no 1 pp 51ndash64 2000

[15] G A Lesieutre and U Lee ldquoA finite element for beams havingsegmented active constrained layers with frequency-dependentviscoelasticsrdquo Smart Materials and Structures vol 5 no 5 pp615ndash627 1996

[16] S W Kung and R Singh ldquoVibration analysis of beams withmultiple constrained layer damping patchesrdquo Journal of Soundand Vibration vol 212 no 5 pp 781ndash805 1998

[17] Q J Zhang andMG Sainsbury ldquoTheGalerkin elementmethodapplied to the vibration of rectangular damped sandwichplatesrdquo Computers and Structures vol 74 no 6 pp 717ndash7302000

[18] R L Bagley and P J Torvik ldquoFractional calculus-a differentapproach to the analysis of viscoelastically damped structuresrdquoAIAA Journal vol 21 no 5 pp 741ndash748 1983

[19] M Hao Vibration analysis of constrained layered beams withmultiple damping layers [PhD thesis] Michigan Tech 2005

[20] G A Lesieutre E Bianchini and A Maiani ldquoFinite elementmodeling of one-dimensional viscoelastic structures using an-elastic displacement fieldsrdquo Journal of Guidance Control andDynamics vol 19 no 3 pp 520ndash527 1996

[21] M A Biot ldquoVariational principles in irreversible thermody-namics with application to viscoelasticityrdquo Physical Review vol97 no 6 pp 1463ndash1469 1955

[22] J Zhang and G T Zheng ldquoThe Biot model and its applicationin viscoelastic composite structuresrdquo Journal of Vibration andAcoustics vol 129 no 5 pp 533ndash540 2007

[23] D J McTavish ldquoShock response of a damped linear struc-ture using GHM finite elementsrdquo in Proceedings of the 44thAIAAASMEASCEAHSASC Structures Structural DynamicsandMaterials Conference pp 1681ndash1689 April 2003 Paper 1591

[24] L Zhang H P Du Y M Shi and X Z Shi ldquoParametricdetermination for GHM of ZN-1 viscoelastic materialrdquo RareMetal Materials and Engineering vol 31 no 2 pp 91ndash95 2002


[25] C H Park D J Inman and M J Lam ldquoModel reductionof viscoelastic finite element modelsrdquo Journal of Sound andVibration vol 219 no 4 pp 619ndash637 1999

[26] M Hao and M D Rao ldquoVibration and damping analysis ofa sandwich beam containing a viscoelastic constraining layerrdquoJournal of Composite Materials vol 39 no 18 pp 1621ndash16432005

[27] D H Lee ldquoOptimal placement of constrained-layer dampingfor reduction of interior noiserdquo AIAA Journal vol 46 no 1 pp75ndash83 2008

[28] F Lin and M D Rao ldquoVibration analysis of a multiple-layeredviscoelastic structure using the biot damping modelrdquo AIAAJournal vol 48 no 3 pp 624ndash634 2010

[29] T W Wu Boundary Element Acoustics Fundamentals andComputer Codes WIT Press Ashurst UK 2000

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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Active and Passive Electronic Components

Control Scienceand Engineering

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Journal ofEngineeringVolume 2014

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VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



the numerical model of the multiple layers system In 2000Chen and Chan [14] studied four different types of integralFEMmodels with the viscoelastic coresThenumerical stabil-ity and accuracy as well as for convergence issue of these fourdifferent FEM models were demonstrated by comparing thenumerical results with those from experiments Lesieutre andLee [15] proposed a 3-node 10DOFFEMmodel for the three-layer ACL damping beam This FEM model is advantageousin active control application due to its features of nonshearlocking and adaptability to segmented constraining layers

Other than theHamilton and FEMnumerical methods inbuilding the models other researchers have proposed manyirregular modeling techniques for the numerical representa-tion of continuousdiscontinuous systems Kung and Singh[16] calculated the natural frequencies and loss factor usingthe Rayleigh-Ritz energy method and modal-strain energytechnique in modeling a 3-layer patch damping structureThese approximate modeling methods were also extended torectangular damping patch of plates and shells with visco-elastic cores Zhang and Sainsbury [17] combined the Ger-lerkin orthogonal function with the traditional finite elementmethod and successfully applied to the vibration analysis ofthe damped sandwich plates

While the FEM is used widely in the modeling of thestructure many researchers sought for proper mathematicalmodels to represent the damping behavior of the viscoelasticmaterial as well as incorporating the dampingmodel in com-mercial FEM software packages Currently many FEM com-mercial software incorporate damping models based mostlyon viscoushysteretic damping Some allow incorporation ofdamping energy dissipation in the time domain using theProny series None of these damping models however issuitable to capture the damping behavior in the frequencydomain which is the most important issue in predicting thevibro-acoustical response of complex structural systems Thedrawback of these dampingmodels raised considerable inter-est and motivation in the development of damping modelsof viscoelastic material in the frequency domain compatiblewith FEM software These damping models can be classifiedas derivative type and integral type

The ldquoFractional Derivativerdquo is essentially the representa-tive damping model in the derivative form family proposedby Bagley and Torvik [18] in 1983 This damping model notonly described the material properties of viscoelastic damp-ing but so established the closed-form equation compatiblewith the FEM technique Compared with the other integral-form models the fractional derivative is only able to capturethe relatively weak frequency-dependent information how-ever it was an important milestone in the area of dampingresearch

Lesieutre et al [20] mathematically modeled the relax-ation behavior of viscoelasticmaterial in terms of augmentingthermodynamic field (ATF) in 1989 Initially introducing asingle augment field this dampingmodel provided the abilityto represent the light-damping behavior with the applicationof a 1D viscoelastic structure In the subsequent researchusing a series of augment fields the ATF model is able tomodel the damping material of higher loss factor with theweak frequency dependence Remedying the limitation of

1D application Lesieutre and Lee [15] proposed an anelasticdisplacement field (ADF) technique in 1996 and successfullyextended its application from the 1-D problem to the 3-layersandwich beam and 3-D problems

As far as the mini-oscillator damping models are con-cerned the complex shear modulus which is a function ofboth frequency and temperature can be expressed by a seriesof mini-oscillation perturbations Biot [21] first proposed thefirst-order relaxation function with the introduction of theldquodissipative variablesrdquo into the dynamic equations using thetheory of irreversible thermodynamics In 2007 Zhang andZheng [22] utilized the Biot model to describe the dynamicbehavior of a viscoelastic structure The dimension reduc-tion technique and nonlinear curve-fitting procedure werediscussed in the paper McTavish [23] developed anothermini-oscillator damping model called ldquoGHMrdquo by the usageof Second-order relaxation function Compared with the Biotmodel the GHM model has a more complicated expressionand also requires better performance of the computationaltool

The popularity of these integral-form damping models inrecent years brought two research interests nonlinear curvefitting and dimension reduction The advanced curve-fittingtechniques in the damping models guarantee the accuracyof the numerical representation of the actual shear modulusdata from the experiment The dimension reduction tech-nique increases the computational efficiency due to the ad-ditional orders of equation in order to gain the frequencyindependence of the frequency-form damping model

Zhang et al [24] converted the nonlinear curve-fittingproblem in frequency domain with respect to the GHM pa-rameters into the constrained nonlinear optimization prob-lem The efficiency and correctness were demonstrated for acommercial viscoelastic material

Park et al [25] examined the GHM damping modelwith the application to the FEM method associated with theGuyan reduction technique The numerical example in thisresearch leads to an FEMmodel applied to theGHMdynamicequation quantitatively without increasing the number oforder

Hao and Rao [26] carried out the optimum design of athree-layer sandwich beam for the vibration analysis in 2005In this research the numerical model is a comprehensiveformulation for a three-layer unsymmetrical sandwich beamwith two different damping materials adjacent to each otherThe criterion of the optimization is to minimize the mass ofthe structure whilemaximizing the system damping In 2008Lee [27] published the semicoupled vibroacoustical analysisand optimization of a simply supported three-layer sandwichbeam The modal superposition method was used to inves-tigate the vibration problem with the fractional derivativedamping model The interior acoustical problem was studiedby BEM numerical technique and the optimization problemwas established through the appropriate sizing parameters ofthe sandwich beam

The objective of this paper is to extend the previous workby the authors [28] on the vibration analysis of a multiple-layered beam structure incorporating the Biot dampingmod-el to solve the acoustic problem to predict the radiated noise


119906119894 119906119895 119883

120579119894 120579119895

119908119894 119908119895






120575119890

elastic = (119908119894 120579119894 119906119894 | 119908119895 120579119895 119906119895) )119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6 times 1

(1)


[Ke]119890

elastic = int1

0

119864119860

119897[120597Ne120597120585]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6 times 1

119879

[120597Ne120597120585]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1 times 6

119889120585

+ int1

0

119864119868

119897[1205972Nf1205972120585

]

119879

[1205972Nf1205971205852

]119889120585⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

6 times 6

(2)

120579119894120579119895

119895119894

11990631198951199063119894

11990611198951199061119894

119883

119908119894119908119895



[Nf]

= [1minus31205852

+21205853

(120585 minus 21205852

+1205853

) 119897 0 31205852

minus21205853

(minus1205852

+1205853

) 119897 0]

[Ne] = [0 0 1 minus 120585 0 0 120585]

(3)



[M]119890elastic⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6 times 6

= int1

0

119898119897([Nf]⏟⏟⏟⏟⏟⏟⏟6 times 1

119879

[Nf]⏟⏟⏟⏟⏟⏟⏟1 times 6

+ [Ne]119879

[Ne]) 119889120585 (4)


120575119890

cons = (119908119894 120579119894 1199061119894 1199063119894 | 119908119895 120579119895 1199061119895 1199063119895)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟8 times 1

(5)


[TI] = [e1 e2 e3 e5 e6 e7]119879

[TIII] = [e1 e2 e4 e5 e6 e8]119879

(6)



119879


[Ke]119890

cons = [TI]119879


+ [TIII]119879


[Kv]119890

cons = int1

0

11986621198602119897

119896ℎ2

[Ne1 minus Ne3

ℎ2

+ℎ0

ℎ2

sdot1

119897sdot120597Nf1120597120585

]

119879


ℎ2

+ℎ0

ℎ2

sdot1

119897sdot120597Nf1120597120585

] 119889120585

[Ne1]⏟⏟⏟⏟⏟⏟⏟⏟⏟1 times 8

= [Ne]⏟⏟⏟⏟⏟⏟⏟1 times 6

[TI]⏟⏟⏟⏟⏟⏟⏟6 times 8

[Ne3] = [Ne] [TIII]

(7)



2 long-



[M]119890cons = [TI]119879


cons2

+ [TIII]119879


(8)

where

[M]119890cons2=int1

0

1198982119897[Nf1]

119879

[Nf1] 119889120585 [Nf1]=[Nf] [TI]

(9)


120575119890

7layer = ( 119908119894 120579119894 1199061119894 1199063119894 1199065119894 1199067119894 | 119908119895 120579119895 1199061119895 1199063119895 1199065119895 1199067119895 )119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟12times1

(10)


[T1] = (e1 e2 e3 e7 e8 e9)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6times12

[T3] = (e1 e2 e4 e7 e8 e10)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6times12

[T5] = (e1 e2 e5 e7 e8 e11)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6times12

[T7] = (e1 e2 e6 e7 e8 e12)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6times12

(11)


[T2] = (e1 e2 e3 e4 e7 e8 e9 e10)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟8times12

[T4] = (e1 e2 e4 e5 e7 e8 e10 e11)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟8times12

[T6] = (e1 e2 e5 e6 e7 e8 e11 e12)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟8times12

(12)




)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟12 times 1

(13)

120579119894 120579119895

119895119894119883

1199067119894

1199065119894

1199063119894

1199061119894

1199063119895

1199061119895

1199065119895

1199067119895

119908119894 119908119895



[Ke]119890

= T1198791 [Ke1]T1 + T1198793 [Ke3]T3

+ T1198795 [Ke5]T5 + T1198797 [Ke7]T7

[Kv]119890

= T1198792[Kv2]T2 + T1198794 [Kv4]T4 + T1198796 [Kv6]T6

[Me]119890

=

7

sum119894=1

T119879i [Mei]Ti

(14)



Mx + Cx + Kx = f (t) (15)


119904 (119904) = 119866infin

[1 +

119898

sum119896=1

119886119896

119904

119904 + 119887119896

] (16)






119866infin

1Λv1 = Λ1 119866

infin

2Λv2 = Λ2

R1 = 119866infin

1Rv1Λv1 R2 = 119866

infin

2Rv2Λv2

(17)











119904 (119895120596) = 119866infin

[1 +

119873

sum119894=1

1198861198941206032

1198872119894

+ 1206032] + 119895119866

infin

[1 +

119873

sum119894=1

119886119894119887119894120603

1198872119894

+ 1206032]

(18)




Assuming 1199091= 119866infin 1199092= 1198861 1199093= 1198871 1199094= 1198862 1199095= 1198872


min119909

F (119909) =119875

sum119895=1

10038161003816100381610038161003816119866lowast

119895(119909) minus 119866

0119895

10038161003816100381610038161003816

2

(19)





a1 1809517 5699386303a2 1453095 0596843249a3 3221535 1000560485a4 5201026 0577694736a5 1976822a6 6561162b1 5410993 426818097b2 1093778 7026089968b3 6036544 5015607814b4 4319613 1969150769b5 2840958b6 2980672






Ay + By = f (20)

where

A = [D MM 0 ] B = [K 0

0 minusM]

y = qq f = f0

(21)


500450400350300250200150100500

25

2

15

1

05

0

Frequency (Hz)

times106


Real

part

(a)

500450400350300250200150100500

25

2

3

15

1

05

0

Frequency (Hz)

times106


Imag

inar

y pa

rt

(b)




(A120582 + By)Φ = 0 (22a)

or

(A120582 + By) ΨΨ120582 = 0 (22b)


[[[

[

120582

]]]

]

=

[[[[[[

[

1205821

120582lowast

10

0 120582

119873

120582lowast

119873

]]]]]]

]

(23)


[Φ] = [Ψ]1

[Ψ]lowast


119873[Ψ]lowast

119873

1205821[Ψ]1120582lowast

1[Ψ]lowast


119873[Ψ]lowast

119873

(24)



5004003002001000

40

35

3025

20

15

105

0

minus5

Frequency (Hz)

Rela

tivel

yer

ror(

)




Φ119879AΦ +Φ119879BΦ119901 = Φ119879f (25)


Φ119879AΦ =

[[[

[

Mp

]]]

]

Φ119879BΦ =

[[[

[

Kp

]]]

]

(26)


[[[

[

Mp

]]]

]

+[[[

[

Kp

]]]

]

119901 = Φ119879f (27)





[[[

[

Q

]]]

]

=[[[

[

Mp

]]]

]

minus1

(28)


[H (119895120596)] =X (119895120596)F (119895120596)

=

119873

sum119894=1

[QiΨiΨ

Ti

(119895120596 minus 120582119894)+QiΨlowast

i Ψlowast

iT

(119895120596 minus 120582lowast119894)]

(29)


X (119895120596) = 119895120596 [H (119895120596)] F (119895120596)

= 119895120596 [F (119895120596)] sdot119873

sum119894=1

[QiΨiΨ

Ti

(119895120596 minus 120582119894)+QiΨlowast

i Ψlowast

iT

(119895120596 minus 120582lowast119894)]

(30)





as followsnabla2

119901 (119909) + 1198962

119901 (119909) = 0when

119901 (119909) = 1199010 119909 isin Γ

119901


1198951199081205880

120597119901

120597119899= vn0

119909 isin Γvn

z (119909) equiv119901


(31)

119881Γ119901

119899

Γ119885

Γ119907119899





119888 (120585) 119901 (120585)+intΓ

120597Ψ (120585 119909)

120597119899119901 (120585) 119889Γ=minusint

Γ

1198941205880120596Ψ (120585 119909) vn (119909) 119889Γ

(32)


0(119896119903)





HP = 119866VN (33)


119901(120585)(120597Ψ(120585 119909)


1198941205880120596vn(119909)Ψ(120585 119909)119889Γ and the





1= 1mm 119864

1= 210GPa 120588

1= 7800 kgm3

2nd ℎ2= 08mm 119866

2 Biot 120588

2= 970 kgm3

3rd ℎ3= 1mm 119864

3= 210GPa 120588

3= 7800 kgm3

4th ℎ4= 08mm 119866

4 Biot 120588

4= 970 kgm3

5th ℎ5= 1mm 119864

5= 210GPa 120588

5= 7800 kgm3

6th ℎ6= 08mm 119866

6 Biot 120588

6= 970 kgm3

7th ℎ7= 1mm 119864

7= 210GPa 120588

7= 7800 kgm3

Anechoicboundaries

Field point(05 04)


1 m

06 m





1198731=1

2120585 (120585 minus 1) 119873

2= (120585 + 1) (120585 minus 1)

1198733=1

2120585 (120585 + 1)

(34)


119909 =

3

sum119894=1

119909119894119873119894(120585) 119910 =

3

sum119894=1

119910119894119873119894(120585) (35)



















0 50 100 150 200 250 3000

1

2

3

4

5

6

7

Frequency (Hz)

(ms

)








modeDampingmodel












44684467

4466

4465

44644463

4462

4461

4464459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)

BC-10 Hz


4468

4467

4466

4465

44644463

44624461

446

4459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)









200180160140120100806040200

Frequency (Hz)

75

70

65

60

55

50

45

40

35

SPL

(dB)



200180160140120100806040200

Frequency (Hz)

SPL

(dB)

110

100

90

80

70

60

50

40

30








3M ISD-110

40 3317 1085 503545 2882 07899 519150 2625 05847 541555 2438 0464 562460 235 0387 5745

3M ISD-112

40 3265 01946 595445 3162 01652 612550 308 01413 623055 3015 01218 629060 2961 01057 6321

3M ISD 110 amp112

40 3288 04427 543145 3076 03506 564550 293 02776 576855 2822 02252 580560 2754 0185 5967

7 Conclusions






Nomenclature





velocity

Disclosure


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119906119894 119906119895 119883

120579119894 120579119895

119908119894 119908119895






120575119890

elastic = (119908119894 120579119894 119906119894 | 119908119895 120579119895 119906119895) )119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6 times 1

(1)


[Ke]119890

elastic = int1

0

119864119860

119897[120597Ne120597120585]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6 times 1

119879

[120597Ne120597120585]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1 times 6

119889120585

+ int1

0

119864119868

119897[1205972Nf1205972120585

]

119879

[1205972Nf1205971205852

]119889120585⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

6 times 6

(2)

120579119894120579119895

119895119894

11990631198951199063119894

11990611198951199061119894

119883

119908119894119908119895



[Nf]

= [1minus31205852

+21205853

(120585 minus 21205852

+1205853

) 119897 0 31205852

minus21205853

(minus1205852

+1205853

) 119897 0]

[Ne] = [0 0 1 minus 120585 0 0 120585]

(3)



[M]119890elastic⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6 times 6

= int1

0

119898119897([Nf]⏟⏟⏟⏟⏟⏟⏟6 times 1

119879

[Nf]⏟⏟⏟⏟⏟⏟⏟1 times 6

+ [Ne]119879

[Ne]) 119889120585 (4)


120575119890

cons = (119908119894 120579119894 1199061119894 1199063119894 | 119908119895 120579119895 1199061119895 1199063119895)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟8 times 1

(5)


[TI] = [e1 e2 e3 e5 e6 e7]119879

[TIII] = [e1 e2 e4 e5 e6 e8]119879

(6)



119879


[Ke]119890

cons = [TI]119879


+ [TIII]119879


[Kv]119890

cons = int1

0

11986621198602119897

119896ℎ2

[Ne1 minus Ne3

ℎ2

+ℎ0

ℎ2

sdot1

119897sdot120597Nf1120597120585

]

119879


ℎ2

+ℎ0

ℎ2

sdot1

119897sdot120597Nf1120597120585

] 119889120585

[Ne1]⏟⏟⏟⏟⏟⏟⏟⏟⏟1 times 8

= [Ne]⏟⏟⏟⏟⏟⏟⏟1 times 6

[TI]⏟⏟⏟⏟⏟⏟⏟6 times 8

[Ne3] = [Ne] [TIII]

(7)



2 long-



[M]119890cons = [TI]119879


cons2

+ [TIII]119879


(8)

where

[M]119890cons2=int1

0

1198982119897[Nf1]

119879

[Nf1] 119889120585 [Nf1]=[Nf] [TI]

(9)


120575119890

7layer = ( 119908119894 120579119894 1199061119894 1199063119894 1199065119894 1199067119894 | 119908119895 120579119895 1199061119895 1199063119895 1199065119895 1199067119895 )119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟12times1

(10)


[T1] = (e1 e2 e3 e7 e8 e9)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6times12

[T3] = (e1 e2 e4 e7 e8 e10)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6times12

[T5] = (e1 e2 e5 e7 e8 e11)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6times12

[T7] = (e1 e2 e6 e7 e8 e12)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6times12

(11)


[T2] = (e1 e2 e3 e4 e7 e8 e9 e10)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟8times12

[T4] = (e1 e2 e4 e5 e7 e8 e10 e11)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟8times12

[T6] = (e1 e2 e5 e6 e7 e8 e11 e12)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟8times12

(12)




)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟12 times 1

(13)

120579119894 120579119895

119895119894119883

1199067119894

1199065119894

1199063119894

1199061119894

1199063119895

1199061119895

1199065119895

1199067119895

119908119894 119908119895



[Ke]119890

= T1198791 [Ke1]T1 + T1198793 [Ke3]T3

+ T1198795 [Ke5]T5 + T1198797 [Ke7]T7

[Kv]119890

= T1198792[Kv2]T2 + T1198794 [Kv4]T4 + T1198796 [Kv6]T6

[Me]119890

=

7

sum119894=1

T119879i [Mei]Ti

(14)



Mx + Cx + Kx = f (t) (15)


119904 (119904) = 119866infin

[1 +

119898

sum119896=1

119886119896

119904

119904 + 119887119896

] (16)






119866infin

1Λv1 = Λ1 119866

infin

2Λv2 = Λ2

R1 = 119866infin

1Rv1Λv1 R2 = 119866

infin

2Rv2Λv2

(17)











119904 (119895120596) = 119866infin

[1 +

119873

sum119894=1

1198861198941206032

1198872119894

+ 1206032] + 119895119866

infin

[1 +

119873

sum119894=1

119886119894119887119894120603

1198872119894

+ 1206032]

(18)




Assuming 1199091= 119866infin 1199092= 1198861 1199093= 1198871 1199094= 1198862 1199095= 1198872


min119909

F (119909) =119875

sum119895=1

10038161003816100381610038161003816119866lowast

119895(119909) minus 119866

0119895

10038161003816100381610038161003816

2

(19)





a1 1809517 5699386303a2 1453095 0596843249a3 3221535 1000560485a4 5201026 0577694736a5 1976822a6 6561162b1 5410993 426818097b2 1093778 7026089968b3 6036544 5015607814b4 4319613 1969150769b5 2840958b6 2980672






Ay + By = f (20)

where

A = [D MM 0 ] B = [K 0

0 minusM]

y = qq f = f0

(21)


500450400350300250200150100500

25

2

15

1

05

0

Frequency (Hz)

times106


Real

part

(a)

500450400350300250200150100500

25

2

3

15

1

05

0

Frequency (Hz)

times106


Imag

inar

y pa

rt

(b)




(A120582 + By)Φ = 0 (22a)

or

(A120582 + By) ΨΨ120582 = 0 (22b)


[[[

[

120582

]]]

]

=

[[[[[[

[

1205821

120582lowast

10

0 120582

119873

120582lowast

119873

]]]]]]

]

(23)


[Φ] = [Ψ]1

[Ψ]lowast


119873[Ψ]lowast

119873

1205821[Ψ]1120582lowast

1[Ψ]lowast


119873[Ψ]lowast

119873

(24)



5004003002001000

40

35

3025

20

15

105

0

minus5

Frequency (Hz)

Rela

tivel

yer

ror(

)




Φ119879AΦ +Φ119879BΦ119901 = Φ119879f (25)


Φ119879AΦ =

[[[

[

Mp

]]]

]

Φ119879BΦ =

[[[

[

Kp

]]]

]

(26)


[[[

[

Mp

]]]

]

+[[[

[

Kp

]]]

]

119901 = Φ119879f (27)





[[[

[

Q

]]]

]

=[[[

[

Mp

]]]

]

minus1

(28)


[H (119895120596)] =X (119895120596)F (119895120596)

=

119873

sum119894=1

[QiΨiΨ

Ti

(119895120596 minus 120582119894)+QiΨlowast

i Ψlowast

iT

(119895120596 minus 120582lowast119894)]

(29)


X (119895120596) = 119895120596 [H (119895120596)] F (119895120596)

= 119895120596 [F (119895120596)] sdot119873

sum119894=1

[QiΨiΨ

Ti

(119895120596 minus 120582119894)+QiΨlowast

i Ψlowast

iT

(119895120596 minus 120582lowast119894)]

(30)





as followsnabla2

119901 (119909) + 1198962

119901 (119909) = 0when

119901 (119909) = 1199010 119909 isin Γ

119901


1198951199081205880

120597119901

120597119899= vn0

119909 isin Γvn

z (119909) equiv119901


(31)

119881Γ119901

119899

Γ119885

Γ119907119899





119888 (120585) 119901 (120585)+intΓ

120597Ψ (120585 119909)

120597119899119901 (120585) 119889Γ=minusint

Γ

1198941205880120596Ψ (120585 119909) vn (119909) 119889Γ

(32)


0(119896119903)





HP = 119866VN (33)


119901(120585)(120597Ψ(120585 119909)


1198941205880120596vn(119909)Ψ(120585 119909)119889Γ and the





1= 1mm 119864

1= 210GPa 120588

1= 7800 kgm3

2nd ℎ2= 08mm 119866

2 Biot 120588

2= 970 kgm3

3rd ℎ3= 1mm 119864

3= 210GPa 120588

3= 7800 kgm3

4th ℎ4= 08mm 119866

4 Biot 120588

4= 970 kgm3

5th ℎ5= 1mm 119864

5= 210GPa 120588

5= 7800 kgm3

6th ℎ6= 08mm 119866

6 Biot 120588

6= 970 kgm3

7th ℎ7= 1mm 119864

7= 210GPa 120588

7= 7800 kgm3

Anechoicboundaries

Field point(05 04)


1 m

06 m





1198731=1

2120585 (120585 minus 1) 119873

2= (120585 + 1) (120585 minus 1)

1198733=1

2120585 (120585 + 1)

(34)


119909 =

3

sum119894=1

119909119894119873119894(120585) 119910 =

3

sum119894=1

119910119894119873119894(120585) (35)



















0 50 100 150 200 250 3000

1

2

3

4

5

6

7

Frequency (Hz)

(ms

)








modeDampingmodel












44684467

4466

4465

44644463

4462

4461

4464459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)

BC-10 Hz


4468

4467

4466

4465

44644463

44624461

446

4459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)









200180160140120100806040200

Frequency (Hz)

75

70

65

60

55

50

45

40

35

SPL

(dB)



200180160140120100806040200

Frequency (Hz)

SPL

(dB)

110

100

90

80

70

60

50

40

30








3M ISD-110

40 3317 1085 503545 2882 07899 519150 2625 05847 541555 2438 0464 562460 235 0387 5745

3M ISD-112

40 3265 01946 595445 3162 01652 612550 308 01413 623055 3015 01218 629060 2961 01057 6321

3M ISD 110 amp112

40 3288 04427 543145 3076 03506 564550 293 02776 576855 2822 02252 580560 2754 0185 5967

7 Conclusions






Nomenclature





velocity

Disclosure


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











2 long-



[M]119890cons = [TI]119879


cons2

+ [TIII]119879


(8)

where

[M]119890cons2=int1

0

1198982119897[Nf1]

119879

[Nf1] 119889120585 [Nf1]=[Nf] [TI]

(9)


120575119890

7layer = ( 119908119894 120579119894 1199061119894 1199063119894 1199065119894 1199067119894 | 119908119895 120579119895 1199061119895 1199063119895 1199065119895 1199067119895 )119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟12times1

(10)


[T1] = (e1 e2 e3 e7 e8 e9)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6times12

[T3] = (e1 e2 e4 e7 e8 e10)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6times12

[T5] = (e1 e2 e5 e7 e8 e11)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6times12

[T7] = (e1 e2 e6 e7 e8 e12)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6times12

(11)


[T2] = (e1 e2 e3 e4 e7 e8 e9 e10)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟8times12

[T4] = (e1 e2 e4 e5 e7 e8 e10 e11)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟8times12

[T6] = (e1 e2 e5 e6 e7 e8 e11 e12)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟8times12

(12)




)119879

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟12 times 1

(13)

120579119894 120579119895

119895119894119883

1199067119894

1199065119894

1199063119894

1199061119894

1199063119895

1199061119895

1199065119895

1199067119895

119908119894 119908119895



[Ke]119890

= T1198791 [Ke1]T1 + T1198793 [Ke3]T3

+ T1198795 [Ke5]T5 + T1198797 [Ke7]T7

[Kv]119890

= T1198792[Kv2]T2 + T1198794 [Kv4]T4 + T1198796 [Kv6]T6

[Me]119890

=

7

sum119894=1

T119879i [Mei]Ti

(14)



Mx + Cx + Kx = f (t) (15)


119904 (119904) = 119866infin

[1 +

119898

sum119896=1

119886119896

119904

119904 + 119887119896

] (16)






119866infin

1Λv1 = Λ1 119866

infin

2Λv2 = Λ2

R1 = 119866infin

1Rv1Λv1 R2 = 119866

infin

2Rv2Λv2

(17)











119904 (119895120596) = 119866infin

[1 +

119873

sum119894=1

1198861198941206032

1198872119894

+ 1206032] + 119895119866

infin

[1 +

119873

sum119894=1

119886119894119887119894120603

1198872119894

+ 1206032]

(18)




Assuming 1199091= 119866infin 1199092= 1198861 1199093= 1198871 1199094= 1198862 1199095= 1198872


min119909

F (119909) =119875

sum119895=1

10038161003816100381610038161003816119866lowast

119895(119909) minus 119866

0119895

10038161003816100381610038161003816

2

(19)





a1 1809517 5699386303a2 1453095 0596843249a3 3221535 1000560485a4 5201026 0577694736a5 1976822a6 6561162b1 5410993 426818097b2 1093778 7026089968b3 6036544 5015607814b4 4319613 1969150769b5 2840958b6 2980672






Ay + By = f (20)

where

A = [D MM 0 ] B = [K 0

0 minusM]

y = qq f = f0

(21)


500450400350300250200150100500

25

2

15

1

05

0

Frequency (Hz)

times106


Real

part

(a)

500450400350300250200150100500

25

2

3

15

1

05

0

Frequency (Hz)

times106


Imag

inar

y pa

rt

(b)




(A120582 + By)Φ = 0 (22a)

or

(A120582 + By) ΨΨ120582 = 0 (22b)


[[[

[

120582

]]]

]

=

[[[[[[

[

1205821

120582lowast

10

0 120582

119873

120582lowast

119873

]]]]]]

]

(23)


[Φ] = [Ψ]1

[Ψ]lowast


119873[Ψ]lowast

119873

1205821[Ψ]1120582lowast

1[Ψ]lowast


119873[Ψ]lowast

119873

(24)



5004003002001000

40

35

3025

20

15

105

0

minus5

Frequency (Hz)

Rela

tivel

yer

ror(

)




Φ119879AΦ +Φ119879BΦ119901 = Φ119879f (25)


Φ119879AΦ =

[[[

[

Mp

]]]

]

Φ119879BΦ =

[[[

[

Kp

]]]

]

(26)


[[[

[

Mp

]]]

]

+[[[

[

Kp

]]]

]

119901 = Φ119879f (27)





[[[

[

Q

]]]

]

=[[[

[

Mp

]]]

]

minus1

(28)


[H (119895120596)] =X (119895120596)F (119895120596)

=

119873

sum119894=1

[QiΨiΨ

Ti

(119895120596 minus 120582119894)+QiΨlowast

i Ψlowast

iT

(119895120596 minus 120582lowast119894)]

(29)


X (119895120596) = 119895120596 [H (119895120596)] F (119895120596)

= 119895120596 [F (119895120596)] sdot119873

sum119894=1

[QiΨiΨ

Ti

(119895120596 minus 120582119894)+QiΨlowast

i Ψlowast

iT

(119895120596 minus 120582lowast119894)]

(30)





as followsnabla2

119901 (119909) + 1198962

119901 (119909) = 0when

119901 (119909) = 1199010 119909 isin Γ

119901


1198951199081205880

120597119901

120597119899= vn0

119909 isin Γvn

z (119909) equiv119901


(31)

119881Γ119901

119899

Γ119885

Γ119907119899





119888 (120585) 119901 (120585)+intΓ

120597Ψ (120585 119909)

120597119899119901 (120585) 119889Γ=minusint

Γ

1198941205880120596Ψ (120585 119909) vn (119909) 119889Γ

(32)


0(119896119903)





HP = 119866VN (33)


119901(120585)(120597Ψ(120585 119909)


1198941205880120596vn(119909)Ψ(120585 119909)119889Γ and the





1= 1mm 119864

1= 210GPa 120588

1= 7800 kgm3

2nd ℎ2= 08mm 119866

2 Biot 120588

2= 970 kgm3

3rd ℎ3= 1mm 119864

3= 210GPa 120588

3= 7800 kgm3

4th ℎ4= 08mm 119866

4 Biot 120588

4= 970 kgm3

5th ℎ5= 1mm 119864

5= 210GPa 120588

5= 7800 kgm3

6th ℎ6= 08mm 119866

6 Biot 120588

6= 970 kgm3

7th ℎ7= 1mm 119864

7= 210GPa 120588

7= 7800 kgm3

Anechoicboundaries

Field point(05 04)


1 m

06 m





1198731=1

2120585 (120585 minus 1) 119873

2= (120585 + 1) (120585 minus 1)

1198733=1

2120585 (120585 + 1)

(34)


119909 =

3

sum119894=1

119909119894119873119894(120585) 119910 =

3

sum119894=1

119910119894119873119894(120585) (35)



















0 50 100 150 200 250 3000

1

2

3

4

5

6

7

Frequency (Hz)

(ms

)








modeDampingmodel












44684467

4466

4465

44644463

4462

4461

4464459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)

BC-10 Hz


4468

4467

4466

4465

44644463

44624461

446

4459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)









200180160140120100806040200

Frequency (Hz)

75

70

65

60

55

50

45

40

35

SPL

(dB)



200180160140120100806040200

Frequency (Hz)

SPL

(dB)

110

100

90

80

70

60

50

40

30








3M ISD-110

40 3317 1085 503545 2882 07899 519150 2625 05847 541555 2438 0464 562460 235 0387 5745

3M ISD-112

40 3265 01946 595445 3162 01652 612550 308 01413 623055 3015 01218 629060 2961 01057 6321

3M ISD 110 amp112

40 3288 04427 543145 3076 03506 564550 293 02776 576855 2822 02252 580560 2754 0185 5967

7 Conclusions






Nomenclature





velocity

Disclosure


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119866infin

1Λv1 = Λ1 119866

infin

2Λv2 = Λ2

R1 = 119866infin

1Rv1Λv1 R2 = 119866

infin

2Rv2Λv2

(17)











119904 (119895120596) = 119866infin

[1 +

119873

sum119894=1

1198861198941206032

1198872119894

+ 1206032] + 119895119866

infin

[1 +

119873

sum119894=1

119886119894119887119894120603

1198872119894

+ 1206032]

(18)




Assuming 1199091= 119866infin 1199092= 1198861 1199093= 1198871 1199094= 1198862 1199095= 1198872


min119909

F (119909) =119875

sum119895=1

10038161003816100381610038161003816119866lowast

119895(119909) minus 119866

0119895

10038161003816100381610038161003816

2

(19)





a1 1809517 5699386303a2 1453095 0596843249a3 3221535 1000560485a4 5201026 0577694736a5 1976822a6 6561162b1 5410993 426818097b2 1093778 7026089968b3 6036544 5015607814b4 4319613 1969150769b5 2840958b6 2980672






Ay + By = f (20)

where

A = [D MM 0 ] B = [K 0

0 minusM]

y = qq f = f0

(21)


500450400350300250200150100500

25

2

15

1

05

0

Frequency (Hz)

times106


Real

part

(a)

500450400350300250200150100500

25

2

3

15

1

05

0

Frequency (Hz)

times106


Imag

inar

y pa

rt

(b)




(A120582 + By)Φ = 0 (22a)

or

(A120582 + By) ΨΨ120582 = 0 (22b)


[[[

[

120582

]]]

]

=

[[[[[[

[

1205821

120582lowast

10

0 120582

119873

120582lowast

119873

]]]]]]

]

(23)


[Φ] = [Ψ]1

[Ψ]lowast


119873[Ψ]lowast

119873

1205821[Ψ]1120582lowast

1[Ψ]lowast


119873[Ψ]lowast

119873

(24)



5004003002001000

40

35

3025

20

15

105

0

minus5

Frequency (Hz)

Rela

tivel

yer

ror(

)




Φ119879AΦ +Φ119879BΦ119901 = Φ119879f (25)


Φ119879AΦ =

[[[

[

Mp

]]]

]

Φ119879BΦ =

[[[

[

Kp

]]]

]

(26)


[[[

[

Mp

]]]

]

+[[[

[

Kp

]]]

]

119901 = Φ119879f (27)





[[[

[

Q

]]]

]

=[[[

[

Mp

]]]

]

minus1

(28)


[H (119895120596)] =X (119895120596)F (119895120596)

=

119873

sum119894=1

[QiΨiΨ

Ti

(119895120596 minus 120582119894)+QiΨlowast

i Ψlowast

iT

(119895120596 minus 120582lowast119894)]

(29)


X (119895120596) = 119895120596 [H (119895120596)] F (119895120596)

= 119895120596 [F (119895120596)] sdot119873

sum119894=1

[QiΨiΨ

Ti

(119895120596 minus 120582119894)+QiΨlowast

i Ψlowast

iT

(119895120596 minus 120582lowast119894)]

(30)





as followsnabla2

119901 (119909) + 1198962

119901 (119909) = 0when

119901 (119909) = 1199010 119909 isin Γ

119901


1198951199081205880

120597119901

120597119899= vn0

119909 isin Γvn

z (119909) equiv119901


(31)

119881Γ119901

119899

Γ119885

Γ119907119899





119888 (120585) 119901 (120585)+intΓ

120597Ψ (120585 119909)

120597119899119901 (120585) 119889Γ=minusint

Γ

1198941205880120596Ψ (120585 119909) vn (119909) 119889Γ

(32)


0(119896119903)





HP = 119866VN (33)


119901(120585)(120597Ψ(120585 119909)


1198941205880120596vn(119909)Ψ(120585 119909)119889Γ and the





1= 1mm 119864

1= 210GPa 120588

1= 7800 kgm3

2nd ℎ2= 08mm 119866

2 Biot 120588

2= 970 kgm3

3rd ℎ3= 1mm 119864

3= 210GPa 120588

3= 7800 kgm3

4th ℎ4= 08mm 119866

4 Biot 120588

4= 970 kgm3

5th ℎ5= 1mm 119864

5= 210GPa 120588

5= 7800 kgm3

6th ℎ6= 08mm 119866

6 Biot 120588

6= 970 kgm3

7th ℎ7= 1mm 119864

7= 210GPa 120588

7= 7800 kgm3

Anechoicboundaries

Field point(05 04)


1 m

06 m





1198731=1

2120585 (120585 minus 1) 119873

2= (120585 + 1) (120585 minus 1)

1198733=1

2120585 (120585 + 1)

(34)


119909 =

3

sum119894=1

119909119894119873119894(120585) 119910 =

3

sum119894=1

119910119894119873119894(120585) (35)



















0 50 100 150 200 250 3000

1

2

3

4

5

6

7

Frequency (Hz)

(ms

)








modeDampingmodel












44684467

4466

4465

44644463

4462

4461

4464459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)

BC-10 Hz


4468

4467

4466

4465

44644463

44624461

446

4459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)









200180160140120100806040200

Frequency (Hz)

75

70

65

60

55

50

45

40

35

SPL

(dB)



200180160140120100806040200

Frequency (Hz)

SPL

(dB)

110

100

90

80

70

60

50

40

30








3M ISD-110

40 3317 1085 503545 2882 07899 519150 2625 05847 541555 2438 0464 562460 235 0387 5745

3M ISD-112

40 3265 01946 595445 3162 01652 612550 308 01413 623055 3015 01218 629060 2961 01057 6321

3M ISD 110 amp112

40 3288 04427 543145 3076 03506 564550 293 02776 576855 2822 02252 580560 2754 0185 5967

7 Conclusions






Nomenclature





velocity

Disclosure


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










500450400350300250200150100500

25

2

15

1

05

0

Frequency (Hz)

times106


Real

part

(a)

500450400350300250200150100500

25

2

3

15

1

05

0

Frequency (Hz)

times106


Imag

inar

y pa

rt

(b)




(A120582 + By)Φ = 0 (22a)

or

(A120582 + By) ΨΨ120582 = 0 (22b)


[[[

[

120582

]]]

]

=

[[[[[[

[

1205821

120582lowast

10

0 120582

119873

120582lowast

119873

]]]]]]

]

(23)


[Φ] = [Ψ]1

[Ψ]lowast


119873[Ψ]lowast

119873

1205821[Ψ]1120582lowast

1[Ψ]lowast


119873[Ψ]lowast

119873

(24)



5004003002001000

40

35

3025

20

15

105

0

minus5

Frequency (Hz)

Rela

tivel

yer

ror(

)




Φ119879AΦ +Φ119879BΦ119901 = Φ119879f (25)


Φ119879AΦ =

[[[

[

Mp

]]]

]

Φ119879BΦ =

[[[

[

Kp

]]]

]

(26)


[[[

[

Mp

]]]

]

+[[[

[

Kp

]]]

]

119901 = Φ119879f (27)





[[[

[

Q

]]]

]

=[[[

[

Mp

]]]

]

minus1

(28)


[H (119895120596)] =X (119895120596)F (119895120596)

=

119873

sum119894=1

[QiΨiΨ

Ti

(119895120596 minus 120582119894)+QiΨlowast

i Ψlowast

iT

(119895120596 minus 120582lowast119894)]

(29)


X (119895120596) = 119895120596 [H (119895120596)] F (119895120596)

= 119895120596 [F (119895120596)] sdot119873

sum119894=1

[QiΨiΨ

Ti

(119895120596 minus 120582119894)+QiΨlowast

i Ψlowast

iT

(119895120596 minus 120582lowast119894)]

(30)





as followsnabla2

119901 (119909) + 1198962

119901 (119909) = 0when

119901 (119909) = 1199010 119909 isin Γ

119901


1198951199081205880

120597119901

120597119899= vn0

119909 isin Γvn

z (119909) equiv119901


(31)

119881Γ119901

119899

Γ119885

Γ119907119899





119888 (120585) 119901 (120585)+intΓ

120597Ψ (120585 119909)

120597119899119901 (120585) 119889Γ=minusint

Γ

1198941205880120596Ψ (120585 119909) vn (119909) 119889Γ

(32)


0(119896119903)





HP = 119866VN (33)


119901(120585)(120597Ψ(120585 119909)


1198941205880120596vn(119909)Ψ(120585 119909)119889Γ and the





1= 1mm 119864

1= 210GPa 120588

1= 7800 kgm3

2nd ℎ2= 08mm 119866

2 Biot 120588

2= 970 kgm3

3rd ℎ3= 1mm 119864

3= 210GPa 120588

3= 7800 kgm3

4th ℎ4= 08mm 119866

4 Biot 120588

4= 970 kgm3

5th ℎ5= 1mm 119864

5= 210GPa 120588

5= 7800 kgm3

6th ℎ6= 08mm 119866

6 Biot 120588

6= 970 kgm3

7th ℎ7= 1mm 119864

7= 210GPa 120588

7= 7800 kgm3

Anechoicboundaries

Field point(05 04)


1 m

06 m





1198731=1

2120585 (120585 minus 1) 119873

2= (120585 + 1) (120585 minus 1)

1198733=1

2120585 (120585 + 1)

(34)


119909 =

3

sum119894=1

119909119894119873119894(120585) 119910 =

3

sum119894=1

119910119894119873119894(120585) (35)



















0 50 100 150 200 250 3000

1

2

3

4

5

6

7

Frequency (Hz)

(ms

)








modeDampingmodel












44684467

4466

4465

44644463

4462

4461

4464459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)

BC-10 Hz


4468

4467

4466

4465

44644463

44624461

446

4459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)









200180160140120100806040200

Frequency (Hz)

75

70

65

60

55

50

45

40

35

SPL

(dB)



200180160140120100806040200

Frequency (Hz)

SPL

(dB)

110

100

90

80

70

60

50

40

30








3M ISD-110

40 3317 1085 503545 2882 07899 519150 2625 05847 541555 2438 0464 562460 235 0387 5745

3M ISD-112

40 3265 01946 595445 3162 01652 612550 308 01413 623055 3015 01218 629060 2961 01057 6321

3M ISD 110 amp112

40 3288 04427 543145 3076 03506 564550 293 02776 576855 2822 02252 580560 2754 0185 5967

7 Conclusions






Nomenclature





velocity

Disclosure


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













[[[

[

Q

]]]

]

=[[[

[

Mp

]]]

]

minus1

(28)


[H (119895120596)] =X (119895120596)F (119895120596)

=

119873

sum119894=1

[QiΨiΨ

Ti

(119895120596 minus 120582119894)+QiΨlowast

i Ψlowast

iT

(119895120596 minus 120582lowast119894)]

(29)


X (119895120596) = 119895120596 [H (119895120596)] F (119895120596)

= 119895120596 [F (119895120596)] sdot119873

sum119894=1

[QiΨiΨ

Ti

(119895120596 minus 120582119894)+QiΨlowast

i Ψlowast

iT

(119895120596 minus 120582lowast119894)]

(30)





as followsnabla2

119901 (119909) + 1198962

119901 (119909) = 0when

119901 (119909) = 1199010 119909 isin Γ

119901


1198951199081205880

120597119901

120597119899= vn0

119909 isin Γvn

z (119909) equiv119901


(31)

119881Γ119901

119899

Γ119885

Γ119907119899





119888 (120585) 119901 (120585)+intΓ

120597Ψ (120585 119909)

120597119899119901 (120585) 119889Γ=minusint

Γ

1198941205880120596Ψ (120585 119909) vn (119909) 119889Γ

(32)


0(119896119903)





HP = 119866VN (33)


119901(120585)(120597Ψ(120585 119909)


1198941205880120596vn(119909)Ψ(120585 119909)119889Γ and the





1= 1mm 119864

1= 210GPa 120588

1= 7800 kgm3

2nd ℎ2= 08mm 119866

2 Biot 120588

2= 970 kgm3

3rd ℎ3= 1mm 119864

3= 210GPa 120588

3= 7800 kgm3

4th ℎ4= 08mm 119866

4 Biot 120588

4= 970 kgm3

5th ℎ5= 1mm 119864

5= 210GPa 120588

5= 7800 kgm3

6th ℎ6= 08mm 119866

6 Biot 120588

6= 970 kgm3

7th ℎ7= 1mm 119864

7= 210GPa 120588

7= 7800 kgm3

Anechoicboundaries

Field point(05 04)


1 m

06 m





1198731=1

2120585 (120585 minus 1) 119873

2= (120585 + 1) (120585 minus 1)

1198733=1

2120585 (120585 + 1)

(34)


119909 =

3

sum119894=1

119909119894119873119894(120585) 119910 =

3

sum119894=1

119910119894119873119894(120585) (35)



















0 50 100 150 200 250 3000

1

2

3

4

5

6

7

Frequency (Hz)

(ms

)








modeDampingmodel












44684467

4466

4465

44644463

4462

4461

4464459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)

BC-10 Hz


4468

4467

4466

4465

44644463

44624461

446

4459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)









200180160140120100806040200

Frequency (Hz)

75

70

65

60

55

50

45

40

35

SPL

(dB)



200180160140120100806040200

Frequency (Hz)

SPL

(dB)

110

100

90

80

70

60

50

40

30








3M ISD-110

40 3317 1085 503545 2882 07899 519150 2625 05847 541555 2438 0464 562460 235 0387 5745

3M ISD-112

40 3265 01946 595445 3162 01652 612550 308 01413 623055 3015 01218 629060 2961 01057 6321

3M ISD 110 amp112

40 3288 04427 543145 3076 03506 564550 293 02776 576855 2822 02252 580560 2754 0185 5967

7 Conclusions






Nomenclature





velocity

Disclosure


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1= 1mm 119864

1= 210GPa 120588

1= 7800 kgm3

2nd ℎ2= 08mm 119866

2 Biot 120588

2= 970 kgm3

3rd ℎ3= 1mm 119864

3= 210GPa 120588

3= 7800 kgm3

4th ℎ4= 08mm 119866

4 Biot 120588

4= 970 kgm3

5th ℎ5= 1mm 119864

5= 210GPa 120588

5= 7800 kgm3

6th ℎ6= 08mm 119866

6 Biot 120588

6= 970 kgm3

7th ℎ7= 1mm 119864

7= 210GPa 120588

7= 7800 kgm3

Anechoicboundaries

Field point(05 04)


1 m

06 m





1198731=1

2120585 (120585 minus 1) 119873

2= (120585 + 1) (120585 minus 1)

1198733=1

2120585 (120585 + 1)

(34)


119909 =

3

sum119894=1

119909119894119873119894(120585) 119910 =

3

sum119894=1

119910119894119873119894(120585) (35)



















0 50 100 150 200 250 3000

1

2

3

4

5

6

7

Frequency (Hz)

(ms

)








modeDampingmodel












44684467

4466

4465

44644463

4462

4461

4464459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)

BC-10 Hz


4468

4467

4466

4465

44644463

44624461

446

4459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)









200180160140120100806040200

Frequency (Hz)

75

70

65

60

55

50

45

40

35

SPL

(dB)



200180160140120100806040200

Frequency (Hz)

SPL

(dB)

110

100

90

80

70

60

50

40

30








3M ISD-110

40 3317 1085 503545 2882 07899 519150 2625 05847 541555 2438 0464 562460 235 0387 5745

3M ISD-112

40 3265 01946 595445 3162 01652 612550 308 01413 623055 3015 01218 629060 2961 01057 6321

3M ISD 110 amp112

40 3288 04427 543145 3076 03506 564550 293 02776 576855 2822 02252 580560 2754 0185 5967

7 Conclusions






Nomenclature





velocity

Disclosure


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















0 50 100 150 200 250 3000

1

2

3

4

5

6

7

Frequency (Hz)

(ms

)








modeDampingmodel












44684467

4466

4465

44644463

4462

4461

4464459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)

BC-10 Hz


4468

4467

4466

4465

44644463

44624461

446

4459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)









200180160140120100806040200

Frequency (Hz)

75

70

65

60

55

50

45

40

35

SPL

(dB)



200180160140120100806040200

Frequency (Hz)

SPL

(dB)

110

100

90

80

70

60

50

40

30








3M ISD-110

40 3317 1085 503545 2882 07899 519150 2625 05847 541555 2438 0464 562460 235 0387 5745

3M ISD-112

40 3265 01946 595445 3162 01652 612550 308 01413 623055 3015 01218 629060 2961 01057 6321

3M ISD 110 amp112

40 3288 04427 543145 3076 03506 564550 293 02776 576855 2822 02252 580560 2754 0185 5967

7 Conclusions






Nomenclature





velocity

Disclosure


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











44684467

4466

4465

44644463

4462

4461

4464459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)

BC-10 Hz


4468

4467

4466

4465

44644463

44624461

446

4459

4458

00

01

01

02

02

03

03

04

04

05

05

06

06

07 08 09

Vert

ical

(m)

Horizontal (m)









200180160140120100806040200

Frequency (Hz)

75

70

65

60

55

50

45

40

35

SPL

(dB)



200180160140120100806040200

Frequency (Hz)

SPL

(dB)

110

100

90

80

70

60

50

40

30








3M ISD-110

40 3317 1085 503545 2882 07899 519150 2625 05847 541555 2438 0464 562460 235 0387 5745

3M ISD-112

40 3265 01946 595445 3162 01652 612550 308 01413 623055 3015 01218 629060 2961 01057 6321

3M ISD 110 amp112

40 3288 04427 543145 3076 03506 564550 293 02776 576855 2822 02252 580560 2754 0185 5967

7 Conclusions






Nomenclature





velocity

Disclosure


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










200180160140120100806040200

Frequency (Hz)

75

70

65

60

55

50

45

40

35

SPL

(dB)



200180160140120100806040200

Frequency (Hz)

SPL

(dB)

110

100

90

80

70

60

50

40

30








3M ISD-110

40 3317 1085 503545 2882 07899 519150 2625 05847 541555 2438 0464 562460 235 0387 5745

3M ISD-112

40 3265 01946 595445 3162 01652 612550 308 01413 623055 3015 01218 629060 2961 01057 6321

3M ISD 110 amp112

40 3288 04427 543145 3076 03506 564550 293 02776 576855 2822 02252 580560 2754 0185 5967

7 Conclusions






Nomenclature





velocity

Disclosure


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Nomenclature





velocity

Disclosure


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









research article vibroacoustical analysis of multiple

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