DOC

Large Amplitude Flexural Vibration of the Orthotropic Composite Plate Embedded with Shape Memory Alloy Fibers

By Clyde Price,2014-02-18 12:15
7 views 0
Large Amplitude Flexural Vibration of the Orthotropic Composite Plate Embedded with Shape Memory Alloy Fibers

    Large Amplitude Flexural Vibration of the

    Orthotropic Composite Plate Embedded

    with Shape Memory Alloy Fibers Availableonlineatwww.sciencedirect.com

    g

    ScienceDirect

    ChineseJournalofAeronautics20(2007)415-424

    AmplitudeFlexuralVib

    PlateEmbeddedwith

    ration

    Shape

    chinese

    JournaIof

    Aeronautics

    ;;;;;;;;;=;;;==;;;=;;;=;;

    vnvwelsevier.com/locate/cja

    oftheOrthotropicComposite

    MemoryAlloyFibers

    RenYongsheng.SunShuangshuang

    CollegeofMechanicalandElectronicEngineering.ShandongUniversityofScienceandTechnology,Qingdao266510,China

    SchoolofElectro.mechanicalEngineering,QingdaoUniversityofScienceandTechnology,Qingdao266061,China

    Received22January2007;accepted4July2007

    Abstract

    Thefreeandforcedvibrationoflargedeformationcompositeplateembeddedwithshapeme

moryalloy(SMA)fibersisinvesti-

    gated.Athermo-mechanicalconstitutiveequationofSMAproposedbyBrinsoneta1.isemployedandtheconstitutiveequationsfor

    evaluationofthepropertiesofahybridSMAcompositelaminateareobtained.Basedonthenonlineartheoryofsymmetricallylaminated

    anisotropicplates,thegoverningequationsofflexuralvibrationintermsofdisplacementandstressfunctionsarederived.TheGalerkin

    methodhasbeenusedtoconverttheoriginalpartialdifferentialequationintoanonlinearordinarydifferentialequation,whichisthen

    solvedwithharmonicbalancemethod.Thenumericalresultsshowthattherelationshipbetweennonlinearnaturalfrequencyratioand

    temperatureforthenonlinearplatehassimilarcharacteristicscomparedwiththatofthelinearone,andtheeffectsoftemperatureon

    forcedresponsebehaviorduringphasetransformationfromMartensitetoAustenitearesignificant.Theeffectsofthevolumefractionof

    theSMAfiber,aspectratioandfreevibrationamplitudeonthedynamicalbehavioroftheplatearealsodiscussed

    Keywords:smartmaterials;largedeflectionplate;composite;nonlinearvibration

    lIntroduction

    Inrecenttenyears,greatprogresshasbeen

    madeontheresearchofshapememoryalloy(SMA)

    reinforcedsmartstructuresystems.Throughthede

    signofSMAintelligentcompositestructuresystems

    withhighlyintegratedsensorsandactuators,the

    controlofcomplexflexiblestructureswithrigorous

    weightlimitationssuchasaeronauticstructureshas

    becomepossible.Forexample,thebucklingand

    vibrationcharacteristicsOfSMAreinforcedcom

    positeplateandbeamstructureshavebeengreatly

    improvedwhencomparedtotraditionalcomposite

structures,thismakesSMAbewidelyusedinbuck

    }Correspondingauthor,Te1.:+86.53288032695

    E-mailaddress:renys@sdustedu.cn

    lingandvibrationcontrol[_.

    Forthestructureswithhighstrengthandlarge flexibilityusedinaircraftdesign,thegeometric nonlinearefrectresultedfromlargedeformation cannotbeignored.Forinstance,theshapesofthe panelsofthesupersonicaircraftwillbechangeddue tothelargethermaldeflectionsinducedbyaerody

    namicheating.whichwillafrecttheaerodynamic characteristicsandreducetheflightperformanceof theaircraft.Inordertocontrolsuchkindofnonlin

    earstructures,itisnecessarytoknowtheeffectof geometricnonlinearityonthevibrationofstructures However,mostoftheresearchesonSMAreinforced compositeplatesandbeamsarebasedontheclassic smalldeflectiontheoryandonlyafewareconcern

    ingwiththelargedeflectionnonlinearityissues.

416RenYongshengeta1./ChineseJournalofAeronautics20(2007)415424

    BasedonvonKarmanplatetheory.ChuLC_j?

    establishedthenonlinearfiniteelementmodelfor theSMAfiberreinforcedcompositelargedeflection

    plateandstudiedthesuppressionoftheflutterof panelsofaircraft.ZouJingt4Jderivedtheincre

    mentalfiniteelementmotionequationforthe nonlinearcompositelaminateembeddedwithSMA fibersonthebasisofthevirtualworkprinciple,and

thebending,thermobucklingandpostbuckling

    issuesoftheSMAreinforcedcompositelaminate undertransverseloadingwerediscussed.DanoML eta1.studiedtheeffectofSMAonthesnap

    throughcharacteristicsofthenonlinearunsymmet

    ricfiber..reinforcedcompositelaminatewithlarge-- deflections.Theyestablishedtheapproximatethe

    orytoanalyzethesnapthroughcharacteristicsof

    theunsymmetricfiberreinforcedcompositelami

    nateactivatedbvSMAwires.wherethemechanica1 propertiesofthelaminatearepredictedwiththe assumedstraindisplacementfield,RayleighRitz

    methodandthevirtualworkprinciple.andthelaw ofsnapthroughvaryingwiththetemperaturewas obtainedthroughsolvingrelevantsimultaneous equationsandtheequationdescribingSMAproper

    ties.ParkJSeta1._6Jestablishedthefiniteelement modelofcompositelaminatesandsupersonicplates undernonlinearvibrationandnonlinearflutterby usingtheyonKarmanplatetheory,theoneorder

    shearingdeformationplatetheoryandthefirstorder

    pistontheory.Theboundaryproblemsofthermo

    postbucklingnonlinearvibrationandflutterwere theninvestigatedfortheSMAfiberreinforced compositeplateunderthermalandaerodynamic loading.ChoMeta1.studiedthedeformationof thenonlinearcompositeplatewithtwowayshape

    memoryeffectwiththetheoryoftheoneorder

    shearingdeformationplatewithlargedeflections

    andthethermomechanicalconstitutiveequation proposedbyLagoudaseta1.

    Basedontheaboveliteraturereview.werea

    lizethat:?mostofthecurrentresearchesonthe geometricnonlinearityoftheSMAfiber(orlayer) reinforcedcompositeplatearelimitedtousingthe finiteelementnumericalmethod[.

    andlittlework

    hasbeenreportedtousethenonlinearelastictheory togettheanalyticalvibrationsolutionfortheSMA reinforcednonlinearanisotropiclaminates;about thedescriptionofthemechanica1behaviorofSMA. somearebasedontheexpressionoftheapproxi

    mateexperimentalfitting_jJ,theothersusethe印一

    DrOximatedataderivedfromthecurvesofSMAi6-7] ,

    andthewelldevelopedandpracticalconstitutive equationofSMAbasedonthephenomenaltheory suchasBrinson'sequation[4,9ishardlyused;?the

    effectofSMAfibersonthenonlinearvibrationof compositeplatesstillneedstobeinvestigatedfur

    ther.

    Inthispaper,thefreeandforcedvibrationof theSMAreinforcedcompositelaminateswithlarge deflectionswillbestudied.Thethermomechanical

    constitutiveequationofSMAproposedbyBrinson eta1.[4andthemixturetheoryforevaluatingthe propertiesoflaminatesareemployedtoestablishthe constitutiveequationoftheSMAreinforcedcom

    positelaminateswithlargedeflections.Basedonthe

nonlineartheoryofsymmetricallylaminatedani

    sotropicplates,thetransversevibrationequation andthecompatibleequation"Nillbederivedinterms ofthetransversedeflectionandthestressfunction. TheGalerkinapproximatemethodandtheharmonic balancemethod(HBM)willbeusedtosolvethe Duffing'sdifferentialequationandtostudytheef_ fectofthecontentofSMAfibers,thenonlinearvi

    brationamplitudeandtheexcitationforceamplitude onthenaturalfrequencyandsteadystateresponse

    ofthesystem.

    2BasicTheories

    2.1Theconstitutiveequationoftheanisotr

    opiclaminatewithembeddedSMAfibers

    AstheSMAreinforcedanisotropiclaminahas thefoilOwingOff-axisstressstrainrelation~o_

    ?,

RenYongshengeta1./ChineseJournalofAeronautics20f2ooz)415-424

    theconstitutiveequationoftheSMAreinforced laminatecanbeWrittenas

    [NAT}+

    where.anddenotethestrainvectorandthe curvaturevectorofthemid-plane,respectively.The in-planeforcesandmomentsinducedbySMA(de- notedbythesubscriptr)andtemperature(denoted bythesubscriptAT)arerespectivelyexpressedas ():

    h

    /2[(k

.erk))cLz

    ()=

    h/2~

    (k)zcLz

    ?^=(?A?^?^)=

    22(er(k)erA(k))

    MM?MA)=

    fh/2(,erlger(k)er(k))TzcLz TheA,BandDinEq.(2)areexpressedas

    fll

    A=lAl2

    lAI6

    Al2

    A22

    A26

    lDl-

    D=IDl2

    LD16

    46f}l2Bl6

    26

    l,Bll22226

    6JLBI62666

    Dl2Dl6

    D22D26

    D26D66

    whereeachelementinEq.(4)isdefinedas

    (,,)=fh/2(1,Z,Z2dz

    whererepresentstheoff-axiselasticconstantsof

    theSMAfiberhybridlamina. TherecoverystressesofSMAandthestresses

inducedbytemperatureoftheklaminaaftercoor.

    dinatetransformationhavethefollowingform,re.

    spectively.

    erATx

    ,z2,z

    m'2mn

    

    ,z,

    2--

    i,l2

    Ql2

    =

    Q22

    =

    Q62

    =

    Ql6

    =

    Q26

    =

    6

    4l7

    (6)

    where(il,2,6;jl,2,6)denotestheelastic

    constantofthecompositemedium.mand,zarethe

    cosineandsinefunctionsoftheplyangleineach

    lamina,respectively.

    FromEq.(2),wehave

    M=B6.+Dt~MAr+

    .

=

    AN++A(J7,,?—N)

    (7)

    (8)

    whereA=A.B=AB.

    SubstitutingEq.(8)intoEq.(7),weget

    M=?++A(Na一?r)1+

    DMa+:BAN+(D+BB)+ BA(J7,,?rJ7,,)r+Mr(9) Withthefollowingexpression

    BA=BA:(-A)=()(10) =

    {}=f囊囊蒌]?

    {}+『誊茎茎]{}+

    雕袭J.l{?^}(12)1.IlI

    ,??????????,???????J 00

    ,...........,.........

    L

    

    ...,..................................,...L=

    ????????????

    ,?????????j

    ,????????,,??,????l ?

    22,

    

    ++

    ::

,??????,???,?????,,?J

    1??,????j

?

    4l8?

    where

    RenYongshengeta1./ChineseJournalofAeronautics20(2007)415424

    W

    ,

    ,RTyW

    ,

    J,,=2w,

    ;andWis

    thetransversedeflectionofthelaminate 2.2Transversevibrationequations Forconvenience,thestressfunction,Y)is introducedasaseparatevariableandtheinplane

    forcesareexpressedintermsof,Y)as ,

    ,Ny

    ,

    ,?=,

    (13)

    Forthesymmetricorthotropiclaminate,we haVe10,7-,:::M:

    0andQ=QwiththeexpressionofQas Q

    fEl/

    1V21EI/

    J0

Report this document

For any questions or suggestions please email
cust-service@docsford.com