Vibroacoustical Analysis of Multiple-Layered Structures with Viscoelastic Damping Cores

14 Pages·2015·2.64 MB·English

HindawiPublishingCorporation
ISRNMechanicalEngineering
Volume2013,ArticleID645232,13pages
http://dx.doi.org/10.1155/2013/645232
Research Article
Vibroacoustical Analysis of Multiple-Layered Structures with
Viscoelastic Damping Cores
FeiLinandMohanD.Rao
MichiganTechnologicalUniversity,Houghton,MI49931,USA
CorrespondenceshouldbeaddressedtoFeiLin;[e-mail]
Received2December2012;Accepted16December2012
AcademicEditors:M.Ahmadian,R.Brighenti,S.W.Chang,J.Clayton,P.Dineva,andG.-J.Wang
Copyright©2013F.LinandM.D.Rao.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,
whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
Thispaperpresentsamodelingtechniquetostudythevibroacousticsofmultiple-layeredviscoelasticlaminatedbeamsusingthe
Biotdampingmodel.Inthiswork,acompletesimulationprocedureforstudyingthestructuralacousticsofthesystemusinga
hybridnumericalmodelispresented.Theboundaryelementmethod(BEM)wasusedtomodeltheacousticalcavity,whereasthe
finiteelementmethod(FEM)wasthebasisforvibrationanalysisofthemultiple-layeredbeamstructure.Throughtheproposed
procedure,theanalysiscaneasilybeextendedtoanothercomplexgeometrywitharbitraryboundaryconditions.Thenonlinear
behaviorofviscoelasticdampingmaterialswasrepresentedbytheBiotdampingmodeltakingintoaccounttheeffectsoffrequency,
temperature,anddifferentdampingmaterialsforindividuallayers.Thecurve-fittingprocedureusedtoobtaintheBiotconstants
fordifferentdampingmaterialsforeachtemperatureisexplained.Theresultsfromstructuralvibrationanalysisforselectedbeams
agreewithpublishedclosed-formresults,andresultsfortheradiatednoiseforasamplebeamstructureobtainedusingacommercial
BEMsoftwarearecomparedwiththeacousticalresultsofthesamebeambyusingtheBiotdampingmodel.
1.Introduction for transverse displacement to include various boundary
conditions. A paper by Rao [5] presented the equations of
The traditional designs of free-layer, constrained-layer or motionofviscoelasticsandwichbeamswithvariousbound-
sandwich-layer,dampingtreatmentusingviscoelasticmate- aryconditionsusingtheenergymethod.Theequationswere
rialshavebeenaroundforoverfortyyears.Recentimprove- solved numerically, and a practical design guideline was
ments in the understanding and application of the damp- presented.SimilartoRao’stheory,Cottle[6]usedHamilton’s
ing principles, together with advances in materials science principletoderiveequationsofmotion.Thedampingcould
and manufacturing, have led to many successful applica- also be increased by adding passive stand-off layer (PSOL)
tions and the development of patch damping and multiple- and slotted stand-off layer (SSOL) to the layered systems.
layered damping structures. The key point in any design Falugi [7] and Parin et al. [8] conducted theoretical and
is to recognize that the damping material must be applied experimentalworkonafour-layeredpanelandafive-layered
in such a way that it is significantly strained whenever the beamwithPSOLtreatment.RogersandParin[9]andYellin
structure is deformed in the vibration mode under investi- et al. [10] have performed experimental investigations and
gation.Numerousresearchershavesuccessfullyimplemented demonstrated that PSOL treatment increased the damping
the passive constrained layer (PCL) and active constrained significantlyinaeronauticalstructuresandbeams.Yellinand
layer (ACL) systems. In 1959, Kerwin [1] and Ross et al. colleagues [11, 12] also developed normalized equations of
[2] presented a general analysis of viscoelastic material motion for beam, fully treated with PSOL using nonideal
structure.Thedampingwasattributedtotheextensionand stand-offlayerassumption.Theequationsweresolvedusing
shear deformations of the viscoelastic layers. Ditaranto [3] themethodofdistributedtransferfunctions[13].
developedsixth-orderequationsofmotionintermsofaxial Inadditiontotheclosed-formanalyticalapproach,many
displacements and developed a closed-form solution.Mead researchers have used the finite element method (FEM),
andMarkus[4]extendedthesixth-orderequationsofmotion the most popular numerical modeling method in building
2 ISRNMechanicalEngineering
thenumericalmodelofthemultiplelayerssystem.In2000, 1Dapplication,LesieutreandLee[15]proposedananelastic
Chen and Chan [14] studied four different types of integral displacementfield(ADF)techniquein1996andsuccessfully
FEMmodelswiththeviscoelasticcores.Thenumericalstabil- extendeditsapplicationfromthe1-Dproblemtothe3-layer
ityandaccuracyaswellasforconvergenceissueofthesefour sandwichbeamand3-Dproblems.
differentFEMmodelsweredemonstratedbycomparingthe As far as the mini-oscillator damping models are con-
numericalresultswiththosefromexperiments.Lesieutreand cerned, the complex shear modulus which is a function of
Lee[15]proposeda3-node,10DOFFEMmodelforthethree- bothfrequencyandtemperaturecanbeexpressedbyaseries
layerACLdampingbeam.ThisFEMmodelisadvantageous ofmini-oscillationperturbations.Biot[21]firstproposedthe
in active control application due to its features of nonshear first-order relaxation function with the introduction of the
lockingandadaptabilitytosegmentedconstraininglayers. “dissipativevariables”intothedynamicequationsusingthe
OtherthantheHamiltonandFEMnumericalmethodsin theory of irreversible thermodynamics. In 2007, Zhang and
buildingthemodels,otherresearchershaveproposedmany Zheng[22]utilizedtheBiotmodeltodescribethedynamic
irregularmodelingtechniquesforthenumericalrepresenta- behavior of a viscoelastic structure. The dimension reduc-
tion of continuous/discontinuous systems. Kung and Singh tion technique and nonlinear curve-fitting procedure were
[16]calculatedthenaturalfrequenciesandlossfactorusing discussed in the paper. McTavish [23] developed another
the Rayleigh-Ritz energy method and modal-strain energy mini-oscillator damping model called “GHM” by the usage
technique in modeling a 3-layer patch damping structure. ofSecond-orderrelaxationfunction.ComparedwiththeBiot
Theseapproximatemodelingmethodswerealsoextendedto model,theGHMmodelhasamorecomplicatedexpression
rectangular damping patch of plates and shells with visco- and also requires better performance of the computational
elastic cores. Zhang and Sainsbury [17] combined the Ger- tool.
lerkinorthogonalfunctionwiththetraditionalfiniteelement Thepopularityoftheseintegral-formdampingmodelsin
methodandsuccessfullyappliedtothevibrationanalysisof recentyearsbroughttworesearchinterests:nonlinearcurve
thedampedsandwichplates. fittinganddimensionreduction.Theadvancedcurve-fitting
While the FEM is used widely in the modeling of the techniques in the damping models guarantee the accuracy
structure,manyresearcherssoughtforpropermathematical ofthenumericalrepresentationoftheactualshearmodulus
modelstorepresentthedampingbehavioroftheviscoelastic data from the experiment. The dimension reduction tech-
material,aswellasincorporatingthedampingmodelincom- nique increases the computational efficiency due to the ad-
mercialFEMsoftwarepackages.Currently,manyFEMcom- ditional orders of equation in order to gain the frequency
mercialsoftwareincorporatedampingmodelsbasedmostly independenceofthefrequency-formdampingmodel.
onviscous/hystereticdamping.Someallowincorporationof Zhang et al. [24] converted the nonlinear curve-fitting
damping energy dissipation in the time domain using the probleminfrequencydomainwithrespecttotheGHMpa-
Prony series. None of these damping models, however, is rametersintotheconstrainednonlinearoptimizationprob-
suitable to capture the damping behavior in the frequency lem.Theefficiencyandcorrectnessweredemonstratedfora
domain,whichisthemostimportantissueinpredictingthe commercialviscoelasticmaterial.
vibro-acousticalresponseofcomplexstructuralsystems.The Park et al. [25] examined the GHM damping model
drawbackofthesedampingmodelsraisedconsiderableinter- withtheapplicationtotheFEMmethodassociatedwiththe
est and motivation in the development of damping models Guyan reduction technique. The numerical example in this
ofviscoelasticmaterialinthefrequencydomaincompatible researchleadstoanFEMmodelappliedtotheGHMdynamic
withFEMsoftware.Thesedampingmodelscanbeclassified equation quantitatively without increasing the number of
asderivativetypeandintegraltype. order.
The“FractionalDerivative”isessentiallytherepresenta- Hao and Rao [26] carried out the optimum design of a
tivedampingmodelinthederivativeformfamilyproposed three-layersandwichbeamforthevibrationanalysisin2005.
byBagleyandTorvik[18]in1983.Thisdampingmodelnot In this research, the numerical model is a comprehensive
onlydescribedthematerialpropertiesofviscoelasticdamp- formulationforathree-layerunsymmetricalsandwichbeam
ing but so established the closed-form equation compatible withtwodifferentdampingmaterialsadjacenttoeachother.
withtheFEMtechnique.Comparedwiththeotherintegral- Thecriterionoftheoptimizationistominimizethemassof
formmodels,thefractionalderivativeisonlyabletocapture thestructurewhilemaximizingthesystemdamping.In2008,
therelativelyweakfrequency-dependentinformation;how- Lee[27]publishedthesemicoupledvibroacousticalanalysis
ever, it was an important milestone in the area of damping andoptimizationofasimplysupportedthree-layersandwich
research. beam. The modal superposition method was used to inves-
Lesieutre et al. [20] mathematically modeled the relax- tigate the vibration problem with the fractional derivative
ationbehaviorofviscoelasticmaterialintermsofaugmenting dampingmodel.Theinterioracousticalproblemwasstudied
thermodynamic field (ATF) in 1989. Initially, introducing a byBEMnumericaltechnique,andtheoptimizationproblem
singleaugmentfield,thisdampingmodelprovidedtheability wasestablishedthroughtheappropriatesizingparametersof
torepresentthelight-dampingbehavior,withtheapplication thesandwichbeam.
of a 1D viscoelastic structure. In the subsequent research, Theobjectiveofthispaperistoextendthepreviouswork
using a series of augment fields, the ATF model is able to by the authors [28] on the vibration analysis of a multiple-
model the damping material of higher loss factor with the layeredbeamstructureincorporatingtheBiotdampingmod-
weak frequency dependence. Remedying the limitation of eltosolvetheacousticproblemtopredicttheradiatednoise.
ISRNMechanicalEngineering 3
𝑤𝑖 𝑤𝑗 𝑤𝑖 𝑤𝑗
𝑢 𝑢 𝑋
𝑖 𝑗 𝑢3𝑖 𝑢3𝑗 𝑋
𝑖 𝑗
𝜃𝑖 𝜃𝑗 𝑢1𝑖 𝑢1𝑗
𝜃𝑖 𝜃𝑗
Figure1:ConfigurationoftheelasticlayershowingtheDOF. Figure2:Configurationoftheconstraineddampinglayershowing
theDOF.
In this paper, we present a complete numerical procedure astheshapefunctionsarethefollowing:
for the vibroacoustical analysis and design for a multiple-
[N ]
layer laminated damping beam. Results obtained from the f
proposedvibrationanalysisarecomparedwiththeprevious = [1−3𝜉2+2𝜉3 (𝜉−2𝜉2+𝜉3)𝑙 0 3𝜉2−2𝜉3 (−𝜉2+𝜉3)𝑙 0],
closed-formresultstoshowthevalidityofthisapproach.The
radiated noise spectrum at selected field point shows good [N ]=[0 0 1−𝜉 0 0 𝜉],
e
agreement between the 2-D BEM acoustical analysis and
(3)
theresultwithoutsystemdampingcalculatedbycommercial
software for a sample viscoelastic damping structure. The inwhich𝜉:thelocalcoordinate,𝜉 = 𝑥/𝑙,𝜉 ∈ [0,1], 𝑙:longi-
acousticalsolutionisdemonstrated,andthecorrelationbe- tudinallengthofelasticlayer, 𝐴:cross-sectionalareaofthe
tween sound pressure level (SPL) and the loss factor is also elastic layer, 𝐸: Young’s modulusof the elastic layer, and 𝐼:
highlighted. momentofinertiaofelasticlayer.
Similarly,theelementmassmatrixcanbeexpressedas:
2.FEMModelingandtheBiot
1
DynamicEquation [⏟⏟M⏟⏟⏟⏟]⏟⏟𝑒el⏟a⏟s⏟⏟t⏟i⏟c⏟=∫ 𝑚𝑙([⏟⏟N⏟⏟f⏟⏟]⏟𝑇[⏟⏟N⏟⏟f⏟⏟]⏟+[Ne]𝑇[Ne])𝑑𝜉. (4)
0
6×6 6×1 1×6
TheFEMmodelingprocedureandtheestablishmentofthe
Biotdynamicequationwillbediscussedinthissection.The 2.2.FEMModelingofFundamentalComponentII:TheCon-
structurechosenforillustrationisaseven-layerviscoelastic strainedDampingLayer. TheFEMmodeloftheconstrained
sandwichbeam.Theelasticbeamandtheconstraineddamp- layoutcontainingthedampinglayersandwichedbetweentwo
inglayerarethetwofundamentalcomponentsinthisFEM- outerlayersisshowninFigure2.ThisFigureillustrateseach
modeling technique. The concept of transfer matrix is used element consisting of 2 nodes and 8 DOF, where the nodal
toconvertthelocalcoordinatestotheglobalcoordinatesin displacementvectorisasfollows:
order to assemble and construct the complete model of the
sandwichdampingstructurewitharbitrarynumberoflayers. {𝛿}𝑒cons =(⏟⏟𝑤⏟⏟⏟𝑖⏟⏟⏟⏟⏟𝜃⏟⏟𝑖⏟⏟⏟⏟𝑢⏟⏟⏟1⏟𝑖⏟⏟⏟⏟𝑢⏟⏟⏟3⏟⏟𝑖⏟⏟⏟⏟|⏟⏟𝑤⏟⏟⏟⏟𝑗⏟⏟⏟⏟⏟𝜃⏟⏟𝑗⏟⏟⏟⏟⏟𝑢⏟⏟1⏟⏟𝑗⏟⏟⏟⏟𝑢⏟⏟⏟3⏟⏟𝑗⏟)⏟⏟⏟𝑇⏟. (5)
TheBiotviscoelasticdampingmodelwillbeusedtodescribe 8×1
the damping behavior. Through the use of the FEM, the
Throughtheintroductionoftransfermatrix,
structureisdiscretizedwhichwillenabletheuseoftheBiot
dampingmodelfordifferentdampinglayersinthestructure. [T ]=[e e e e e e ]𝑇;
I 1 2 3 5 6 7
Thereaderisreferredtothenomenclatureforthedefinition (6)
ofdifferentvariablesusedinthederivation. [T ]=[e e e e e e ]𝑇,
III 1 2 4 5 6 8
2.1.FEMModelingofComponentI:TheElasticLayer. Figure1 in which each ei means the following vector: ei =
showstheelasticlayerintheFEMmodel,containing2nodes (00⋅⋅⋅⋅⋅⋅ ⏟⏟⏟⏟1⏟⏟⏟ ⋅⋅⋅⋅⋅⋅0)𝑇; the element elastic stiffness and the
𝑖thplace
and6degreesoffreedom(DOF).Theelementdisplacements elementviscoelasticstiffnessmatrixforthis3-layercompo-
ofeachnodecanbeexpressedasfollows:z nent,respectively,arethefollowing:
{𝛿}𝑒elastic =(⏟⏟𝑤⏟⏟⏟𝑖⏟⏟⏟⏟⏟𝜃⏟⏟𝑖⏟⏟⏟⏟𝑢⏟⏟⏟𝑖⏟⏟⏟⏟|⏟⏟⏟𝑤⏟⏟⏟𝑗⏟⏟⏟⏟⏟𝜃⏟⏟𝑗⏟⏟⏟⏟⏟𝑢⏟⏟𝑗⏟⏟)⏟⏟⏟)⏟⏟𝑇⏟. (1) [Ke]c𝑒ons = [TI]𝑇⋅[Ke]elastic⋅[TI]
6×1
+[T ]𝑇⋅[K ] ⋅[T ],
Thestiffnessmatrixcanbederivedbasedonthefollowing III e elastic III
energymethod: 1 𝐺 𝐴 𝑙 N −N ℎ 1 𝜕N 𝑇
[K ]𝑒 = ∫ 2 2 [ e1 e3 + 0 ⋅ ⋅ f1]
[K ]𝑒 = ∫1 𝐸𝐴[𝜕Ne]𝑇[𝜕Ne]𝑑𝜉 v cons 0 𝑘ℎ2 ℎ2 ℎ2 𝑙 𝜕𝜉
e elastic 0 𝑙 ⏟⏟⏟⏟⏟⏟𝜕⏟𝜉⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟𝜕⏟𝜉⏟⏟⏟⏟⏟⏟ N −N ℎ 1 𝜕N
×[ e1 e3 + 0 ⋅ ⋅ f1]𝑑𝜉,
6×1 1×6 ℎ ℎ 𝑙 𝜕𝜉
(2) 2 2
+∫1 𝐸𝐼[𝜕2Nf]𝑇[𝜕2Nf]𝑑𝜉, ⏟[⏟N⏟⏟⏟e⏟1⏟⏟]⏟=[⏟⏟N⏟⏟e⏟⏟]⏟⏟[⏟⏟T⏟I⏟⏟]⏟; [Ne3]=[Ne][TIII],
⏟⏟⏟0⏟⏟⏟⏟⏟⏟𝑙⏟⏟⏟⏟⏟⏟⏟⏟⏟𝜕⏟⏟⏟2⏟⏟𝜉⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝜕⏟⏟⏟𝜉⏟⏟2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 1×8 1×66×8
6×6 (7)
4 ISRNMechanicalEngineering
where𝐴2:cross-sectionalareaofthedampinglayer,𝐺2:long- 𝑤𝑖 𝑤𝑗
termshearmodulusofthedampinglayer,and𝑘:correction
𝑢 𝑢
factor of the shear strain energy, for the rectangular cross- 7𝑖 7𝑗
section,𝑘=1.2.
𝑢 𝑢
Also,theelementmassmatrixforthis3-layercomponent 5𝑖 5𝑗 𝑋
𝑗
is 𝑖
𝑢 𝑢
3𝑖 3𝑗
[M]𝑒 = [T ]𝑇⋅[M] ⋅[T ]+[M]𝑒
cons I elastic I cons,2 (8) 𝜃𝑖 𝑢1𝑖 𝑢1𝑗 𝜃𝑗
+[T ]𝑇⋅[M] ⋅[T ],
III elastic III
Figure3:Configurationofaseven-layerdampingstructureshow-
where
ingtheDOF.
1
[M]𝑒 =∫ 𝑚 𝑙[N ]𝑇[N ]𝑑𝜉, [N ]=[N ][T ].
cons,2 2 f1 f1 f1 f I
0
(9)
Basedontheaboveequationsanddesignparametersof
eachlayer,theelementmass/stiffness/dampingmatrixofthe
2.3. FEM Modeling of a Seven-Layer Constrained Damping
seven-layer sandwich damping beam can be expressed as
Beam. Theseven-layersandwichbeamconsistsofsevenal-
follows:
ternatinglayers—fourelasticlayersandthreedampinglayers.
Figure3 shows the FEM model of a seven-layer sandwich [K ]𝑒 = T𝑇[K ]T +T𝑇[K ]T
e 1 e1 1 3 e3 3
beamcontaining2nodesand10DOF,andthenodedisplace-
mentvectorisasfollows: +T𝑇[K ]T +T𝑇[K ]T ,
5 e5 5 7 e7 7
{𝛿}𝑒7layer =(⏟⏟⏟⏟𝑤⏟⏟⏟𝑖⏟⏟⏟⏟𝜃⏟⏟𝑖⏟⏟⏟⏟𝑢⏟⏟1⏟𝑖⏟⏟⏟⏟𝑢⏟⏟3⏟𝑖⏟⏟⏟⏟𝑢⏟⏟5⏟⏟𝑖⏟⏟⏟𝑢⏟⏟7⏟⏟𝑖⏟⏟⏟⏟|⏟𝑤⏟⏟⏟⏟𝑗⏟⏟⏟𝜃⏟⏟𝑗⏟⏟⏟⏟𝑢⏟⏟1⏟⏟𝑗⏟⏟⏟⏟𝑢⏟⏟3⏟𝑗⏟⏟⏟⏟𝑢⏟⏟5⏟⏟𝑗⏟⏟⏟⏟𝑢⏟⏟7⏟𝑗⏟⏟⏟)⏟⏟⏟𝑇⏟. [K ]𝑒 =T𝑇[K ]T +T𝑇[K ]T +T𝑇[K ]T , (14)
12×1 v 2 v2 2 4 v4 4 6 v6 6
(10)
7
Thetransfermatrixtoobtaintheelementstiffnessandthe [M ]𝑒 =∑T𝑇[M ]T.
e i ei i
massmatrixwhenthe1st,3rd,5th,and7thlayersareelastic 𝑖=1
arefollows:
Thus, the element matrices can be assembled to obtain
[T1]=(⏟⏟e⏟⏟1⏟⏟⏟⏟⏟e⏟⏟2⏟⏟⏟⏟⏟e⏟⏟3⏟⏟⏟⏟e⏟7⏟⏟⏟⏟⏟e⏟⏟⏟8⏟⏟⏟⏟e⏟⏟⏟9⏟⏟)⏟⏟𝑇⏟, theglobalmass/stiffness/dampingmatrixandcanbeapplied
6×12 to the boundary condition through the conventional FEM
technique. Taking into the consideration of the viscoelastic
[T3]=⏟(⏟e⏟⏟1⏟⏟⏟⏟⏟e⏟⏟2⏟⏟⏟⏟⏟e⏟⏟4⏟⏟⏟⏟⏟e⏟7⏟⏟⏟⏟⏟e⏟⏟8⏟⏟⏟⏟⏟e⏟⏟1⏟⏟0⏟)⏟⏟⏟𝑇⏟, dampingproperties,theglobalmatricesneedtobemanipu-
6×12 latedasaportionoftheBiotdynamicequation.
(11)
[T5]=⏟(⏟e⏟⏟1⏟⏟⏟⏟⏟e⏟⏟2⏟⏟⏟⏟⏟e⏟⏟5⏟⏟⏟⏟⏟e⏟7⏟⏟⏟⏟⏟e⏟⏟8⏟⏟⏟⏟⏟e⏟⏟1⏟⏟1⏟)⏟⏟⏟𝑇⏟, 2.4.IntroductionoftheBiotDynamicEquation. Toconsider
6×12
the vibration problem numerically, the dynamic equation
[T7]=⏟(⏟e⏟⏟1⏟⏟⏟⏟⏟e⏟⏟2⏟⏟⏟⏟⏟e⏟⏟6⏟⏟⏟⏟⏟e⏟7⏟⏟⏟⏟⏟e⏟⏟8⏟⏟⏟⏟⏟e⏟⏟1⏟⏟2⏟)⏟⏟⏟𝑇⏟. discretized by FEM technique needs to be expressed by the
followingsecond-orderordinarydifferentialequation(ODE)
6×12
form:
Similarly, the element stiffness and the mass matrix for
the2nd,4th,and6thlayersoftheconstraineddampinglayer Mẍ+Cẋ+Kx=f(t). (15)
canbederivedthroughthetransfermatrix:
The Biot viscoelastic damping model numerically rep-
[T2]=(⏟⏟e⏟⏟1⏟⏟⏟⏟⏟e⏟⏟2⏟⏟⏟⏟⏟e⏟⏟3⏟⏟⏟⏟⏟e⏟⏟4⏟⏟⏟⏟⏟e⏟7⏟⏟⏟⏟⏟e⏟⏟8⏟⏟⏟⏟⏟e⏟⏟9⏟⏟⏟⏟⏟e⏟⏟1⏟⏟0⏟⏟)⏟⏟𝑇⏟, resents the complex shear modulus with a series of mini-
oscillatorperturbingterms:
8×12
[T4]=(⏟⏟e⏟⏟1⏟⏟⏟⏟⏟e⏟⏟2⏟⏟⏟⏟⏟e⏟⏟4⏟⏟⏟⏟⏟e⏟⏟5⏟⏟⏟⏟⏟⏟e7⏟⏟⏟⏟⏟e⏟⏟8⏟⏟⏟⏟⏟e⏟⏟1⏟⏟0⏟⏟⏟⏟⏟e⏟⏟1⏟1⏟⏟)⏟⏟⏟𝑇⏟, (12) 𝑠𝐺̃(𝑠)=𝐺∞[1+∑𝑚𝑎𝑘𝑠+𝑠𝑏 ], (16)
8×12 𝑘=1 𝑘
[T6]=(⏟⏟e⏟⏟1⏟⏟⏟⏟⏟e⏟⏟2⏟⏟⏟⏟⏟e⏟⏟5⏟⏟⏟⏟⏟e⏟⏟6⏟⏟⏟⏟⏟⏟e7⏟⏟⏟⏟⏟e⏟⏟8⏟⏟⏟⏟⏟e⏟⏟1⏟⏟1⏟⏟⏟⏟⏟e⏟⏟1⏟2⏟⏟)⏟⏟⏟𝑇⏟, in which 𝐺∞ is the long-term shear moduli; 𝑎𝑘 and 𝑏𝑘 are
8×12 the Biot constants. These parameters are positive and can
be determined by nonlinear curve fitting from the experi-
wherethenotationeimeans mentaldata.Thecurve-fittingprocedurewillbediscussedin
0 0 ⋅⋅⋅ ⋅⋅⋅ ⏟⏟⏟⏟1⏟⏟⏟ ⋅⋅⋅ ⋅⋅⋅ 0 𝑇 Section3.
e =( ) .
i ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝑖⏟t⏟h⏟⏟p⏟l⏟a⏟⏟c⏟e⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (13) Substituting the Biot damping model into (15), the
dynamic equation with 𝑚 terms of the Biot parameters for
12×1
ISRNMechanicalEngineering 5
the first viscoelastic material and 𝑛 terms for the second Table1:TheBiotconstantsof3MISD-110/11245∘C.
viscoelasticmaterialcanbedevelopedasfollows:
ISD110 ISD112
𝐺∞Λ =Λ , 𝐺∞Λ =Λ , 𝐺∞ 55000(Pa) 172000(Pa)
1 v1 1 2 v2 2
(17) a1 1.809517 5.699386303
R =𝐺∞R Λ , R =𝐺∞R Λ ,
1 1 v1 v1 2 2 v2 v2 a2 14.53095 0.596843249
where Rv and Λv are the eigenvector and diagonal eigen- a3 3.221535 1.000560485
value matrices, respectively, from the damping matrix C. a4 52.01026 0.577694736
Additionally,𝑎11⋅⋅⋅𝑎1𝑚,𝑏11⋅⋅⋅𝑏1𝑚,and𝑧11⋅⋅⋅𝑧1𝑚 denote𝑚 a5 19768.22
termsoftheBiotparametersandthedissipativecoordinates, a6 6.561162
respectively,forfirstviscoelasticmaterial. b1 5.410993 4268.18097
Similarly, 𝑎21⋅⋅⋅𝑎2𝑛, 𝑏21⋅⋅⋅𝑏2𝑛, and 𝑧21⋅⋅⋅𝑧2𝑛 denote 𝑛 b2 1093.778 70.26089968
termsoftheBiotparametersandthedissipationcoordinates, b3 60.36544 501.5607814
respectively, for second viscoelastic material. A detailed b4 4319.613 1.969150769
derivationcanbefoundinthepreviouspublication[28]. b5 2840958
b6 298.0672
3.ParametricDeterminationoftheBiot
DampingModel
theprecisionofthisapproximation.Thecurvefittingofthe
experimental data is accomplished using the commercial
A curve-fitting technique is used to provide the accurate
software package Auto2fit on the real and imaginary parts
Biotconstantstothedynamicequationandtoestablishthe
simultaneously. Using the Biot terms equal to six and four
dynamiccharacteristicsoftheviscoelasticmaterials.Inthis
withrespecttotwocommercialdampingmaterials3MISD-
section, the nonlinear curve-fitting procedure for the com-
110and112,respectively,theresultsareshowninTable1for
plexshearmodulusinthefrequencydomainisconvertedinto ambienttemperature(𝑇)equalto45∘Candfrequencyrange
anonlinearconstrainedoptimizationproblem.
of500Hz.
ThecomplexshearmoduluswiththeBiotdampingmodel
Figures4(a)and4(b)showthecomparisonbetweenthe
formcanbebrokenintorealandimaginarypartsseparately:
Arrheniusdataandcurve-fittingdatafortherealandimag-
inaryparts,respectively.Figure5showstherelativeerrorin
𝑁 𝑎𝜛2 𝑁 𝑎𝑏𝜛
𝑠𝐺̃(𝑗𝜔)=𝐺∞[1+∑ 𝑖 ]+𝑗𝐺∞[1+∑ 𝑖 𝑖 ]. thefittingrange.
𝑏2+𝜛2 𝑏2+𝜛2
𝑖=1 𝑖=1 As shown in Figures 4(a) and 4(b), the Biot parametric
𝑖 𝑖
(18) determinationtechniqueestimatesthedynamicpropertiesof
∘
3MISD-110/112at45 Cwithalmostzeroerror.Theconstants
The Biot parameters—𝐺∞, 𝑎𝑖, and 𝑏𝑖 —are estimated from determinedusingtheaboveprocedurealongwiththeFEM
experimentaldatawiththecertainfittingfrequencyrange,on modelofsandwichbeamwillnowbeincorporatedtosolve
realpartandimaginarypartsseparately.Generallyspeaking, the complete Biot dynamic equation using the decoupling
one set of the Biot parameters needs to be determined for transformationtechnique.
each ambient temperature independently. In (18), 𝑁 is the
numberoftheBiotperturbingitems,definingthecapability
4.DecouplingTransformationand
ofthisnumericalapproximation.AstheBiotterms(𝑁)are
DynamicSolution
increased, the relative error between the experimental data
andthecurve-fittingresultreduces.
In this section, the algorithm used to obtain the frequency
Assuming𝑥1 =𝐺∞;𝑥2 =𝑎1;𝑥3 =𝑏1;𝑥4 =𝑎2;𝑥5 =𝑏2;... response function (FRF) will be discussed with respect to
withtheconstraintcondition:𝑥𝑘 ≥ 0;𝑘 = 1,2,...,num,the thevibroacousticalproblemforamultiple-layerviscoelastic
targetequationoftheoptimizationproblemisthefollowing: damping structure. In this research, the damping matrix D
m𝑥in F(𝑥)= ∑𝑃 󵄨󵄨󵄨󵄨󵄨𝐺𝑗∗(𝑥)−𝐺0𝑗󵄨󵄨󵄨󵄨󵄨2. (19) imna(s1s5a)nddosetsiffnnoetsshmavaetriax.pThropuos,ratiodnecaolurpellaintigotnrsahnispfowrmithatitohne
𝑗=1 is needed to construct the first-order state equation by
introducingtheauxiliaryequationMq̇−Mq̇ =0asfollows:
sheaIrnmthoedutalurgseftroemqutahteioenxp(1e9r)im, 𝐺en0𝑗tasltadnatdaswfoitrht𝑃heinctoemrepstleedx Aẏ+By=̂f, (20)
points(largerthanthenumberofunknowns).The3MISD-
where
110/112viscoelasticpolymerisselectedinthissimulation.The
experimental data is obtained by the Arrhenius empirical D M K 0
A=[ ], B=[ ],
equationfrom[19].Withaspecificfittingrangeataparticular M 0 0 −M
temperature,thecomplexshearmoduluscanbesynthesized (21)
from one set of the Arrhenius coefficients. The number y={q}, ̂f ={f}.
of terms (𝑁) in (18) needs to be determined to ensure q̇ 0
6 ISRNMechanicalEngineering
×106 ×106
2.5 3
The Arrhenius data The Arrhenius data
Curve fit data Curve fit data
2 2.5
alpart 1.5 nary part 1.25
Re 1 gi
ma 1
I
0.5
0.5
0 0
0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500
Frequency(Hz) Frequency(Hz)
(a) (b)
∘
Figure4:(a)ComparisonbetweentheArrheniusandcurve-fittingdatafortherealpartoftheshearmodulus(3M-ISD-110,45C).(b)
∘
ComparisonbetweentheArrheniusandcurve-fittingdatafortheimaginarypartoftheshearmodulus(3M-ISD-110,45C).
Here,𝑁isthenumberofDOFintheM,D,andKmatrices, 40
theDOFofAandBmatricesis2𝑁. 35
Firstly, the free vibration of (20) will be considered. 30
Assuminĝf =0,thefollowingformofsolutionisobtained: %) 25
(
or
(A𝜆+By)Φ=0, (22a) err 20
ely 15
v
or ati 10
el
Ψ R 5
(A𝜆+By){Ψ𝜆}=0, (22b) 0
−5
where𝜆matrixstandsfor2𝑁complexconjugateeigenvalues 0 100 200 300 400 500
including the natural frequencies and loss factors informa- Frequency(Hz)
tion:
Error % in real part
Error % in imag part
𝜆
1
[[[... 𝜆 ]]]=[[[[[ 𝜆∗1 ... 0 ]]]]]. (23) Figure5:RelativeerrorbetweenArrheniusandcurve-fittingdata.
[ ...] [ 0 𝜆𝑁 ]
[ 𝜆∗𝑁] space.ByleftmultiplyingofΦ𝑇withthesubstitutionof𝑦,we
get:
Itmustbenotedthatzeroitemswillappearintheeigen-
valuematrixifthedampingmatrixDisnotfullyranked.The Φ𝑇AΦ𝑝̇ +Φ𝑇BΦ𝑝=Φ𝑇̂f. (25)
modeshapevectorΨforthevectorqcanbeextractedfrom
theeigenvectormatrixΦwithrespecttothevectory: Thediagonalmodalmassandstiffnessmatrixare:
[Φ]={ [Ψ]1 [Ψ]∗1 ⋅⋅⋅ [Ψ]𝑁 [Ψ]∗𝑁 }. Φ𝑇AΦ=[[... M ]]; Φ𝑇BΦ=[[... K ]].
𝜆 [Ψ] 𝜆∗[Ψ]∗ ⋅⋅⋅ 𝜆 [Ψ] 𝜆∗[Ψ]∗ [ p ] [ p ]
1 1 1 1 𝑁 𝑁 𝑁 𝑁 (24) [ ...] [ ...]
(26)
Inaddition,(22b)canbenumericallysolvedby𝜆{Ψ} =
−[A]−1[B]{Ψ}usingmathematicalsoftwarepackagesuchas Then rewrite the equation with the diagonal mass and
stiffnessmatrices
MATLABorMathematica.
dsuobmsStaeiitcnuotnwiodinllly,cbtaehnedbfioescrmcuesadsdevedi.bbrAyastasisousnmumsionilgnugt̂fio𝑦=n=o{fF0Φ(2}𝑝,0,)thcineontvhvaeerritatiimbnlgee [[[... Mp ]]]𝑝̇ +[[[... Kp ]]]𝑝=Φ𝑇̂f. (27)
the state-space equation from the time space to the modal [ ...] [ ...]
ISRNMechanicalEngineering 7
The FRF in the frequency domain can be easily deter-
mined through the complex conjugate eigenvalue matrix𝜆,
eigenvector matrix Φ, and the modal mass matrix Mp. The Γ𝑝 𝑉 Γ𝑍
modal scaling factor matrix can be calculated through the
following:
−1
[... ] [... ]
[[ Q ]]=[[ Mp ]] . (28)
[ ...] [ ...] Γ 𝑛
𝑣𝑛
Thus,FRFcanbeestablishedthroughthemodalparam-
eters,beingexpressedinpartialfractionformintermsofthe
residuevectorandsystempolesasfollows: Figure6:Notationsof2-DBEMinteriorprobleminafluiddomain
𝑉.
X(𝑗𝜔) 𝑁 QΨΨT QΨ∗Ψ∗T
[H(𝑗𝜔)]= =∑[ i i i + i i i ].
F(𝑗𝜔) (𝑗𝜔−𝜆) (𝑗𝜔−𝜆∗)
𝑖=1 𝑖 𝑖
(29) Here, 𝑘 is equal to 𝜔/𝑐, which means that the wave
number is equal to the radiant frequency over the speed of
Thesystemvelocitycannowbeobtainedfromtheabove
equationbyasimpleFouriertransformation.Bydoingso,the sound;vn,𝜌0,zstandforthenormalvelocity,densityofthe
fluid 𝑉 (normally the air), and acoustical impedance of the
vibrationproblemcanbeextendedtoanFRF-basedacousti-
fluid𝑉,respectively.
calproblemandthecombinationofthesetwoanalysesisthe
particlevelocitiesinformationcalculatedbythefollowing: In this work, the link between the vibration and the
acoustics analysis is the normal velocity at the acoustical
Ẋ (𝑗𝜔)=𝑗𝜔[H(𝑗𝜔)]F(𝑗𝜔) boundaries.Recallingthedynamicsolutionofthedecoupling
transformation, the particle velocity in the time domain at
𝑁 QΨΨT QΨ∗Ψ∗T
=𝑗𝜔[F(𝑗𝜔)]⋅∑[ i i i + i i i ]. eachnodecanbecalculatedthrough(30)ifthemultiple-layer
(𝑗𝜔−𝜆) (𝑗𝜔−𝜆∗) sandwich beam is discretized by the FEM; alternatively the
𝑖=1 𝑖 𝑖
FRF,thecomplexratiobetweentheoutputandinputresponse
(30)
in the frequency domain, can be determined through (29).
Once the input signal is given, the particle velocity of the
5.AcousticalBoundaryElementMethod
system displacement versus frequency relationship can be
(BEM)Analysis
convenientlyobtainedthroughtheFRF.
To solve the governing differential equation (31) in the
5.1.IntroductionofAcousticalBEMTheory. InSection4,the bounded fluid domain 𝑉, the Helmholtz Equation can be
vibration problem of the multiple-layer sandwich beam is
transformed into the integral equation, converting the 2-D
solved through the time-domain dynamic ordinary differ-
areaintegrationtothe1-Dcurveintegrationaroundthearea:
ential equation of the Biot damping model with numerical
analysis by the FEM technique. The vibration problem can 𝜕Ψ(𝜉,𝑥)
𝑐(𝜉)𝑝(𝜉)+∫ 𝑝(𝜉)𝑑Γ=−∫ 𝑖𝜌 𝜔Ψ(𝜉,𝑥)v (𝑥)𝑑Γ.
be extended to the acoustical problem by the semicoupled 𝜕𝑛 0 n
Γ Γ
method:thevibrationwillinduceachangeinsoundpressure, (32)
yet the sound pressure will not cause the vibration. In this
in which 𝑐(𝜉): geometry-dependent coefficient, normally
section, the acoustical interior problem will be numerically
𝑐(𝜉) = 0when𝜉isinthedomain𝑉and𝑐(𝜉) = 0.5when𝜉
solved by 2D boundary element method (BEM) technique
[29]inaboundedfluiddomain𝑉asshowninFigure6. isonthesmoothboundaryΓ,𝑝(𝜉):soundpressureatsource
Thesoundpressuredistribution(𝑝)ofthetime-harmonic point𝜉,Ψ(𝜉,𝑥):𝑥isthefieldpointandΨ = −(𝑖/4)H(02)(𝑘𝑟)
wave in the domain 𝑉 satisfies the governing differential forthe2DBEMproblem𝑟theEuclidiandistancebetween𝑥
equation,wellknownastheHelmholtzequation,associated and𝜉,H(02) andtheSecond-typeHenkelfunction,𝑛:normal
withtheboundaryconditionsonboundaryΓ(=Γ𝑝∪Γv ∪ΓZ) vectorpointingawaytothefluiddomain𝑉.
n
asfollows: Bydiscretizingtheboundaryintoaseriesofcurve-linear
∇2𝑝(𝑥)+𝑘2𝑝(𝑥)=0,when elementsthroughtheintroductionoftheshapefunctions,the
integralequationcanbecalculatednumericallybysolvingthe
𝑝(𝑥)=𝑝 , 𝑥∈Γ followinglinearmatrix:
0 𝑝
v (𝑥)≡− 1 𝜕𝑝 =v , 𝑥∈Γ (31) HP=𝐺VN, (33)
n 𝑗𝑤𝜌0𝜕𝑛 n0 vn whereHcomesfromthetermsof𝑐(𝜉)and∫ 𝑝(𝜉)(𝜕Ψ(𝜉,𝑥)/
Γ
z(𝑥)≡ 𝑝 =z , 𝑥∈Γ . 𝜕𝑛)𝑑Γ, 𝐺 is derived from −∫Γ𝑖𝜌0𝜔vn(𝑥)Ψ(𝜉,𝑥)𝑑Γ, and the
vn 0 Z vectorPandVNincludesoundpressureandparticlevelocity
8 ISRNMechanicalEngineering
Table2:Designparametersofseven-layerstructure.
Length:1m Thickness:0.1m Numberofelement:12 Numberofnodes:13
Numberoflayer Heightoflayer Elastic/viscoelasticproperties Materialdensity
1st ℎ1=1mm 𝐸1=210GPa 𝜌1=7800kg/m3
2nd ℎ2=0.8mm 𝐺2:Biot 𝜌2=970kg/m3
3rd ℎ3=1mm 𝐸3=210GPa 𝜌3=7800kg/m3
4th ℎ4=0.8mm 𝐺4:Biot 𝜌4=970kg/m3
5th ℎ5=1mm 𝐸5=210GPa 𝜌5=7800kg/m3
6th ℎ6=0.8mm 𝐺6:Biot 𝜌6=970kg/m3
7th ℎ7=1mm 𝐸7=210GPa 𝜌7=7800kg/m3
1 m
Fieldpoint
(0.5,0.4)
m Figure8:Seven-layersandwichstructurewithviscoelasticcores.
0.6 Anechoic
boundaries
When the seven-layered sandwich beam (𝐿 = 1m)
is simply supported at the bottom of the acoustical cavity,
𝐹(impulse force) the sound pressure level at the field point (𝑥 = 0.5m,
𝑦 = 0.4m)iscalculatedthroughthisproposedmethod,and
Figure7:LayoutofBEMacousticalcavityproblem. the calculation results are presented in Section6. Figure7
demonstratesthedetailedlayoutofthis2Dacousticalcavity
problem. The anechoic boundary condition is applied on
values, both unknowns and known from the boundary the inside of the acoustical cavity, and the thickness of the
condition. multiple-layeredbeamisneglected.
Thus, each set of node velocities due to the force input
results in one set of solutions on the sound pressure by 6.NumericalResultsandDiscussion
BEMdiscussedinthissection.Insum,throughtheproposed
acousticalBEM,itispossibletocomputethetime-harmonic 6.1.DesignParameterofSandwichBeamandVibrationAnal-
sound pressure distribution corresponding to each single ysis Result. In Figure8, a seven-layer sandwich beam with
frequencypointinthefrequencyspectrum. viscoelasticcoresisshown,withthedesignparameterslisted
inTable2.
The data presented in Table2 are used to predict the
5.2. Calculation Details in This BEM Analysis. For this par-
vibration performance of the system using the numerical
ticular acoustical BEM interior problem, the boundary of
simulation method presented in this paper, and the results
acousticalcavityisdiscretizedas18quadraticequallyspaced
arecomparedwiththeclosed-formsolutionofHao[19].The
boundary elements. The quadratic curvilinear element has
curve-fittingresultsforthedampingmaterial3MISD-110at
threenodes,andtheinterpolationbetweeneachnoderepre-
∘
45 Cdiscussedearlierareselectedfortheshearmodulusof
sentsthegeometryofeachelement.Theshapefunctionsare
theviscoelasticlayersinthisexample.Theresultsareshown
asfolows:
in Table3. It shows that the simulation presented in this
1
𝑁 = 𝜉(𝜉−1), 𝑁 =(𝜉+1)(𝜉−1), paper conforms to the closed-form solution. This validates
1 2 2
theanalysismethodologyproposedinthepaper.
(34)
1
𝑁 = 𝜉(𝜉+1),
3 2 6.2. Frequency-Spectrum Analysis under the Arbitrary Input.
withrespecttothefollowingelementcoordinates: Figure9 shows the transverse velocity of the middle node
(node number 7) with a 10N step input in the frequency
3 3 domainverticallyappliedatthemiddle(nodenumber7)of
𝑥=∑𝑥𝑖𝑁𝑖(𝜉), 𝑦=∑𝑦𝑖𝑁𝑖(𝜉), (35) the simply-supported seven-layer sandwich beam with the
𝑖=1 𝑖=1
same design parametersas the previousexample. The same
where𝑥𝑖and𝑦𝑖arethecoordinatesateachnodalpoint,and𝜉 curve-fittingresultsof3MISD110attheambienttemperature
standsforthelocalcoordinatebetween−1and1onamaster of45∘Cfortheshearmodulusareusedinthisexample.This
element. pivotalresultisthedemonstrationofextendingthevibration
ISRNMechanicalEngineering 9
Table 3: Comparison of results for simply supported boundary Table4:ComparisonofmodalresultswithANSYSsimulation.
condition.
FEMmodelofthis
ANSYS3D
Hao[19] FEMmodelof Numberof Damping research
Number Damping thispaper mode model 3M-ISD110-45∘C
ofmode model ISD110-45∘C ISD110-45∘C Nodamping 6-termBiot
Arrhenius 6-termBiot
Frequency 6.4803Hz 4.5834Hz
1st
Frequency 4.7443Hz 4.5834Hz
1st Lossfactor 0.7916
Lossfactor 0.6248 0.7916
Frequency 14.338Hz 13.9489Hz
2nd
2nd Frequency 13.902Hz 13.9489Hz Lossfactor 0.6824
Lossfactor 0.6008 0.6824 Frequency 27.989Hz 27.7721Hz
3rd
Frequency 27.661Hz 27.7721Hz Lossfactor 0.5632
3rd
Lossfactor 0.5317 0.5632
Frequency 46.1548Hz 47.1053Hz
4th
Lossfactor 0.4715 0.4681
6.4. Validation Using a BEM Commercial Software. In this
Frequency 69.3118Hz 68.82259Hz section, a hybrid FEM-BEM model of a beam without the
5th
Lossfactor 0.421 0.4226 viscoelastic damping was developed using the commercial
software packages ANSYS ADPL and LMS Virtual. Lab
Acoustics. The harmonic vibration analysis is conducted in
×10−5 ANSYSAPDLmodule,andthefrequencyspectrumoffield
Transverse velocity on driving point-node number 7
7 point SPL was calculated in Virtual.Lab Acoustics module
forcomparisonwiththeSPLfrequencyspectrumpresented
6 inSection5.Theanalysissequenceconsistsofthefollowing
steps.
5
(a)BuildtheFEMmodelandapplyappropriatebound-
4 aryconditionsinANSYSADPL.The8-nodeelement
m/s) SOLID45 (element size =10mm for each layer) was
( 3
used to build the 3D seven-layer model. The design
parametersareidenticalwiththeparametersinTables
2
1and2forthecomparisonandthegeometrybound-
1 aryconditionsaresimplysupported.A10Nforceat
eachfrequencyisappliedatthemiddlenodes.
0
0 50 100 150 200 250 300 (b)Conduct the harmonic vibration analysis in ANSYS
ADPL.Theharmonicanalysisisusedtocalculatethe
Frequency (Hz)
nodal displacements for a forced vibration problem
Figure 9: Transverse velocity under the impulse excitation (on inthefrequencydomain.Thefrequencyrangeis0–
middlenode#7). 200Hzwitha2Hzforstepsize,andthefullmethod
is being utilized in this analysis. The comparison of
system frequencies between ANSYS modal results
andcalculationresultsbytheBiotdynamicequation
totheacousticalprobleminthefrequencydomainwhenan is shown in Table4. The results show that the 3D
arbitraryforceisappliedonthestructure. model built in ANSYS APDL has good correlation
withtheFEMmodel.
6.3.AcousticalBEMResults. Figure10illustratesthecontour
(c)Prepare the BEM mesh in LMS Virtual. Lab Pre-
plot(𝑓=10Hz)ofSPLwhentheseven-layersandwichbeam
Acoustics module. It converts from a solid FEM
(usingthesamedesignparametersasbefore)issubjectedtoa
modeltoaskinmeshthattheBEManalysisrequires.
10Nstepinputinthefrequencydomainatthemiddlenode.
The BEM mesh, can be seen as a wrap around the
The interpolation of each elements result in Figures 10
structuralmeshandusuallytheBEMmeshiscoarser.
and11showsthecontinuoussoundpressuredistributionin
the acoustical cavity with an anechoic boundary condition. (d)Calculate the sound pressure in LMS Virtual.lab
Figure12extractsthefrequencyspectrumofSPLatthefiled Acoustics module. Both acoustical and structural
point (0.5, 0.4m) indicated by red dot in Figure10. From meshes are imported to VL Acoustics. The nodal
the results of Figure12, it can be found that the dominant displacement at each vibration mode calculated in
contribution is due to the peak value of the first flexible ANSYSAPDLisalsoimportedandmesh-mappedto
vibration mode, which is in agreement with the frequency- the acoustical skin mesh as the vibration boundary
spectrumanalysisofthevibrationproblem. condition.Thelocationoffieldplaneandfieldpointis
10 ISRNMechanicalEngineering
SPLcontourplotwithanechoicBC-10 Hz
0.6
44.68
44.67
0.5
44.66
44.65
0.4
m) 44.64
al( 0.3 44.63
c
Verti 44.62
0.2 44.61
44.6
0.1 44.59
44.58
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Horizontal (m)
Figure10:ContourPlotofSoundPressureLevel(indB)whentheimpulseforceappliedis𝑁.
SPLplotwithanechoicBC-10Hz
0.6
44.68
44.67
0.5
44.66
44.65
0.4
m) 44.64
(
al 0.3 44.63
c
Verti 44.62
0.2 44.61
44.6
0.1 44.59
44.58
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Horizontal(m)
Figure11:ElementresultofSoundPressureLevelindB(10Hz).
consistentwiththe2DBEManalysisinthisresearch. of viscoelastic damping material in a mechanical system.
The acoustical pressure is solved over the frequency Withanincreaseintemperature,thelossfactorapproaches
rangefrom2to200Hz. itsbestperformancetowardsthetransitionregionandthen
decreasesafterwards.Inthisexample,theobjectiveistostudy
AsshowninFigure13,thepeakfrequencyfromthe2-D
the effects of both 3M ISD110 material (that has a better
BEMcalculationmatcheswiththefirstdominantSPLpeak
dampingperformance)andthe3MISD112overthechosen
obtainedfromtheVLAcousticresultwithoutthedamping.
temperaturebetween40and60degreeCelsius.Itisofinterest
Comparing the two results, it is clear that the introduction
tostudytheeffectofthecombinationofthesetwomaterials
of viscoelastic damping not only causes almost a 20dB
onthedampingofthestructure.
reductioninthefirstpeakSPLbutalsoattenuatesthesound
To introduce the different viscoelastic materials, the
atotherpeaksaswell.Thisprovesthattheuseofviscoelastic
seven-layer sandwich beam (with the same parameters as
damping material will greatly attenuate the vibroacoustical
in the previous example) is redesigned incorporating both
responseofthestructure.
damping materials (3M ISD110 and ISD112). This system is
compared to an identical structure with only one damping
6.5. Acoustical Performance for a Combination of Several material(either3MISD110orISD112).Inthesysteminclud-
ViscoelasticMaterialsatDifferentTemperatures. Thetemper- ingtwoviscoelasticmaterials,theouterdampinglayers(2nd
atureisasignificantexternalfactoraffectingtheperformance and 6th) are 3M ISD112 and the inner damping layer (5th)