基于复合单元模型的轴向功能梯度梁的振动分析.pdf 14页

'中国科技论文在线http://www.paper.edu.cnVibrationanalysisofaxiallyfunctionallygradedbeamswith#thecompositeelementmodel**5LIUJike,LVZhongrong(SchoolofEngineering,SunYat-senUniversity,Guangzhou510006,P.R.China)Abstract:Thispaperpresentsasimpleyetaccurateapproachtoanalyzethefreeandforcedvibrationsofaxiallyfunctionallygraded(AFG)beams.ThegoverningequationsofthebeamisderivedbyLagrange’sequationsandthendiscretizedbythecompositeelementmethod.Numericalsimulations10arepresentedtoshowtheeffectsofYoung’smodulusvariation,massdensityvariation,andslendernessratios,onthedynamiccharacteristicsoftheAFGbeams.Thenaturalfrequenciesobtainedfromtheproposedmethodarefoundtobeincloseagreementwiththosefromfiniteelementandothermethods.TheforcedvibrationresponsesoftheAFGbeamarealsocalculatedfromtheproposedmethodandcomparedwiththoseobtainedfromthefiniteelementmethod(FEM).Timehistoriesfrom15bothmethodsarefoundtomatchverywell.ThisfurtherindicatesthecorrectnessofthepresentmethodforforcedvibrationanalysisoftheAFGbeam.Keywords:solidmechanics;compositeelement;freeandforcedvibrations;response;axiallyfunctionallygradedbeam200IntroductionFunctionallygradedmaterials(FGMs)areinhomogeneouscompositeswithmaterialpropertiesvaryingcontinuouslythroughoutthegradientdirection,whichwasfirstdevelopedinJapanin1984bySendaiGroup.FGMshavegreatadvantagesoverthetraditionalhomogeneousmaterialsduetocontinuoustransitionofmaterialproperties.Withincreasingapplicationsof25FGMs,newmethodologiestopredicttheirmechanicalbehaviorshavebeenextensivelystudiedduringthepastfewdecades.EngineersandresearchershavebeeninterestedinFGMsformany[1-3]years,becausetheyhavepromisingmechanicalandthermalproperties.Atthepresenttime,FGbeamscanbedividedintotwocategoriesbasedontheorientateddirections.Oneisthematerialconstituentsvaryinginthicknessdirection,whichhavebeen[4-6][7,8]30intensivelystudied.Theotherismaterialconstituentsvaryinginaxialdirection.Forthelatterone,theproblemofanalyzingbending,vibrationandstabilitybecomesmorecomplicatedduetothegoverningequationwithvariablecoefficients.Similarly,accordingtothematerialdistributionlaw,thefirstcommonlyusedoneforFGMsispowerlaw,theotheroneofthemostfavorablemodelsforFGMsistheexponentiallaw,inwhichmaterialpropertiesofFGMsare35assumedtovaryexponentiallyaboutspatialcoordinates.Thelatteronehasbeenwellusedtoefficientlyobtainexactsolutionsforelasticityproblems.Forexample,suchmodelwaswidely[9]appliedfordeterminingcracktipbehaviorofFGelasticbodies.ProblemsconcerningthedynamicbehaviorFGMswithmaterialconstituentsvaryinggraduallyalongcertaindirectionhave[10-20]becomeanincreasinglyactiveresearchtopic.Duetoawiderangeofapplications,the40modernaerospacestructures,fusionenergydeviseandoptoelectronicscomponentsaremadeof[21]materialsdevelopinginmorethanonedirections.Foundations:NationalNaturalScienceFoundationofChina(No.11172333,No.11272361,No.11572356);DoctoralProgramFoundationofMinistryofEducationofChina(No.20130171110039);GuangdongProvinceNaturalScienceFoundation(No.2015A030313126);GuangdongProvinceScienceandTechnologyProgram(No.2014A020218004)Briefauthorintroduction:J.KLiu,male,professor,doctoralsupervisorCorrespondanceauthor:Z.RLv,male,professor,doctoralsupervisor.E-mail:lvzhr@mail.sysu.edu.cn-1- 中国科技论文在线http://www.paper.edu.cnWithregardtothevibrationmodelling,studieshavebeenprimarilydoneinfreevibrationanalysis.AgreatnumberofresearchesaboutfirstfewnaturalfrequenciesofFGMstructures,[22-26][27]45includingbeamsandplates,havebeenreportedintheliteratures.SimsekandKocatürkinvestigatedthefreevibrationanddynamicbehaviorofFGsimply-supportedbeamunderamovingharmonicload.ThesystemoftheequationsofmotionisderivedbyLagrange’sequationsbasedupontheEuler-Bernoullitheory.Boththeexponentiallawandthepower-lawformare[28]appliedinthematerialpropertiesgradedinthethicknessdirection.HuangandLipresentedthe50freevibrationofAFGbeamswithnon-uniformcross-sectionandthusgaveguidancetobetterdesignofnon-homogeneousstructures.TheequationsofmotionsweretransformedwithvaryingcoefficientsbyFredholmintegralequations.Theeffectofflexuralrigidityandmassdensity[29]variationonnaturalfrequencieswasdemonstratedatthesametime.QianandChingcarriedoutthestaticdeformation,freeandforcedvibrationanalysisofatwo-dimensional(2D)FGcantilever[22]55beambytheuseofmeshlesslocalPetrov-Galerkin(MLPG)method.Alshorbagyetal.analyzedthefreevibrationcharacteristicsofEuler-Bernoullibeambymeansoffiniteelementmethodandtheequationsofmotionswerederivedusingvirtualworkprinciple.Inthispaper,anewmethodispresentedtoanalyzethefreeandforcedvibrationsofaxially[30-32]functionallygradedbeamsusingthecompositeelementmethod.Thecorrectnessand60accuracyoftheproposedmethodareverifiedbysomeexamplesintheexistingliteratures.Theprincipaladvantageoftheproposedmethodisthatitdoesnotneedtopartitionthebeamintouniformbeamsegments.TheproposedmethodprovidesanefficientandaccuratewaytomodeltheAFGbeamforvibrationsanalysis.1Thoery651.1compositeelementmethodAsatoolforfiniteelementmodeling,compositeelementmethodisbasicallyacombination[30]oftheconventionalfiniteelementmethodandthehighlypreciseclassicaltheory.Inthecompositeelementmethod,thedisplacementfieldiswrittenasthesumofthefiniteelementdisplacementandtheshapefunctionsfromtheclassicaltheory.Thedisplacementfieldofthe70CEMcanbeexpressedasw(x,t)w(x,t)w(x,t),(1)CEMFEMCTwherew(x,t)andw(x,t)aretheindividualdisplacementfieldsfromtheFEMandCT,FEMCTrespectively.Takingaplanarbeamelementasanexample,thefirsttermoftheCEMdisplacementfield75canbeexpressedastheproductoftheshapefunctionvectorN(x)andthenodaldisplacementvectorqw(x,t)N(x)q(t),(2)FEMTwhereq(t)[v(t),(t),v(t),(t)]and‘v’and‘’representthetransverseand1122rotationaldisplacements,respectively,andx2x3xx2x3x2x3x3x280N(x)[13()2(),2()(),3()2(),()()]LLLLLLLLL[N(x),N(x),N(x),N(x)].(3)1234ThesecondtermwCT(x,t)isobtainedbythemultiplicationoftheanalyticalmodeshapes-2- 中国科技论文在线http://www.paper.edu.cnwithavectorofNcoefficientsc(alsocalledthecdegrees-of-freedomorc-coordinates)NwCT(x,t)i(x)ci(t),(4)i185where(i=1,2,…N)istheanalyticalshapefunctionofthebeam.Differentanalyticalshapeifunctionsareusedaccordingtotheboundaryconditionsofthebeam.ThedisplacementfieldoftheCEMfortheEuler-BernoullibeamelementcanbewrittenfromEqs.(1)to(4)asw(x,t)S(x)Q(t),(5)CEM90whereSx()[NxNxNxNx(),(),(),(),(),(),...,xx()]xisthegeneralizedshape123412NTfunctionoftheCEM,Qt()[(),(),(),(),(),(),...,vttvttctctct()]isthevectorof112212Ngeneralizeddisplacements,andNisthenumberofshapefunctionsusedfromtheclassicaltheory.1.2FormulationforvibrationanalysisofAFGbeamfromthecompositeelementmethod95AsshowninFig.1,astraightuniformAFGbeamhaslengthL,widthb,thicknessh,withaCartesiancoordinatesystemO(xyz),wherexaxisistakenalongthecentralaxis,theyaxisisinthewidthdirectionandthezaxisisinthedepthdirection.xF(t)bFI100(E,)(ER,R)LLxhOIzSectionI-ILFig.1Anaxiallyfunctionallybeamunderstudy105Inthisstudy,itisassumedthatthematerialpropertiesofbeamsuchas,Young’smodulusE,massdensityρvarycontinuouslyalongthetransversaldirection.BothEandρareassumedto[10]varyaccordingtothefollowingpower-lawfunctionsEx()(EE)(1xL/)E(6)LRR()(x)(1xL/)(7)LRR110whereEandEaretheYoung’smodulioftheleftandtherightsideofthebeam,andLRLRarethemassdensitiesoftheleftandtherightsideofthebeam,isthenon-negativepower-lawcomponentwhichdictatesthematerialvariationprofilethroughtheaxisofthebeam.BasedonEuler-Bernoullibeamtheory,thepotentialenergyoftheAFGbeamcanbewrittenas21Lwxt(,)2VExI()()dx(8)Pz20x2115whereIisthemomentofinertiaofthebeamabouttheneutralaxisz.Takingintoaccountthezrotaryinertia,thekineticenergyofthebeamcanbewrittenas21LLwxt(,)221wxt(,)V()(xA)dx()(xI)dx(9)kz2200txtwhereAistheareaofthecross-section.Thepotentialenergyoftheexternalexcitationforcecanbewrittenas-3- 中国科技论文在线http://www.paper.edu.cnL120VPt()(xxwxtdxˆ)(,)(10)extF0where()istheDiracdeltafunction,xisthelocationoftheforceactingonthebeam.FTheLagrange’sequationsofthebeamforforcedvibrationcanbeexpressedasfollowsdVVVkPext()0,(i=1,2,3,...)(11)dtQQQiiiwheretheoverdotstandsforthepartialderivativewithrespectivetotime.AftersubstitutingEq.5125intoEqs.(8),(9)and(10)andthenusingtheLagrange’sequationsgivenbyEq.(11),wehavethefollowingequationsofmotion:KQMQF(t)(12)inwhichMandKarethesystematicmassandstiffnessmatricesofbeam,respectively.BytheuseofCEMformulation,MandKcanbedepictedbytotalnumberofNEdiscretizedelements,130NE(13)MMe1NE(14)KKe1wheretheelementalstiffnessmatrixoftheAFGbeamcanbeobtainedfromthefollowingequation22TLedSdSKExI()dx,e0dx22zdx[kqq][kqc]135=,(15)[k][k]cqccwherethesubmatrix[k]correspondstotheelementstiffnessmatrixfromtheFEMfortheqqbeam;thesubmatrix[k]correspondstothecouplingtermsoftheq-dofsandthec-dofs;qcsubmatrix[k]isatransposematrixof[k],andthesubmatrix[k]correspondstothecqqcccc-dofsandisadiagonalmatrix.140TheelementalmassmatrixcanbeexpressedasTTLeLdSx()dSx()MSx()()A()xSxdx()xIdx,e00dxzdx[mqq][mqc]=,(16)[m][m]cqccwherethesubmatrix[m]correspondstotheelementalmassmatrixfromtheFEMfortheqqbeam;thesubmatrix[m]correspondstothecouplingtermsoftheq-dofsandthec-dofs;qc145submatrix[m]isatransposematrixof[m],andthesubmatrix[m]correspondstothecqqcccc-dofsandisadiagonalmatrix.Thegoverningequationforfreevibrationofthebeamcanbeexpressedas2(KM)V0,(17)isthecircularfrequency,fromwhichandthenaturalfrequenciesareidentified.Theith150normalizedmodeshapesofthesteppedbeamcanbeexpressedas-4- 中国科技论文在线http://www.paper.edu.cn4NiNViiiiV4.(18)ii11Basedonthenon-dampedforcedvibrationEq.(12),theequationofmotionofthedampedforcedvibrationoftheAFGbeamcanbeexpressedasMQCQKQft(),(19)155whereCisthedampingmatrixwhichrepresentsaRayleighdampingmodel,CaaMK,(20)12whereaandaareconstantstobedeterminedfromtwomodaldampingratios.Foran12externalforceF(t)actingatthelocationxfromtheleftsupport,thegeneralizedforceFvectorf(t)canbeexpressedasT160f(t)N(x)N(x)N(x)N(x)(x)(x)F(t).1F2F3F4F1FnF(21)ThegeneralizedaccelerationQ,velocityQanddisplacementQofthesteppedbeamcanbeobtainedfromEq.(19)bydirectintegration.Thephysicalaccelerationu(x,t)isobtainedfrom165u(x,t)[S(x)]TQ.(22)Thephysicalvelocityanddisplacementcanbeobtainedinasimilarway,i.e.TTuxt(,)[()]SxQ,uxt(,)[()]SxQ.(23a,b)2Numericalsimulation2.1Freevibrationanalysisofasteel-aluminabeam[22]170ThesameAFGbeammadeofsteelandalumina(AlO)studiedbyAlshorbagyetal.is23[10,22]examinedusingtheproposedmethodandtheresultsarecomparedwiththosein.Figure1showsthegeometryofthebeamunderstudy.Table1showsthematerialpropertiesofthebeam.Therightsideofbeamispuresteelandleftsideispurealunina.Thenon-dimensionalquantitiesELL22RAusedhereare:E,,L.ratioratioEEIRRRz[22]175Table.1PropertiesofbeamsintheexperimentalstudyRef.PropertiesSteelAluminaE210GPa390GPa7800kg/m33960kg/m32.1.1OntheconvergenceoftheCEMInthiscase,theconvergenceoftheCEMforvibrationanalysisisstudied.InthecompositeelementmodeloftheAFGbeam,itisonlydiscretizedintoasinglefiniteelementanddifferentnumberofc-DOFsareexamined.Table2showstheconvergenceoffirstthreedimensionless180naturalfrequenciesofthebeam.Onecanfindthatthefirstdimensionlessnaturalfrequencyisconvergedwhenonly4c-DOFsisused.Andthenumberofc-DOFsforthesecondandthethirdis5and8,respectively.Thetotalnumberofdegreesoffreedomisonly12,whichincludes4DOFs-5- 中国科技论文在线http://www.paper.edu.cnforfiniteelementand8c-DOFs.ThisstudycaseillustratesthehighefficiencyandaccuracyofthepresentCEMmethod.185Table.2ConvergenceofthefirstthreedimensionlessnaturalfrequenciesofAFGbeamwithdifferentnumberofc-DofsNumberofc-Dofs[22]ModeRef.34578102.58212.58232.58212.58212.58212.58212.582115.16755.16895.16815.16755.16755.16755.167527.77307.82887.73727.73457.73327.73307.733032.1.2TheeffectofYoung’smodulusvariationToassesstheeffectofYoung’smodulusvariationonthevibrationcharacters,massdensityisassumedasuniforminthebeam.TheeffectofYoung’smodulusratios,slendernessratiosonthe190firstthreedimensionlessfrequenciesaretabulatedinTables3-5.Forbetterdemonstration,fivedifferentmodulusratios(Eratio=0.25,0.5,1,2,3,4)andtwoslendernessratios(L/h=20,100)areinvestigated.Itcanbeobservedthat,whenthemodulusratioEratioislessthan1,thenaturalfrequenciesincreasewithanincreaseinpowerexponent.Butonthecontrary,whenEratioisgreaterthan1,thenaturalfrequenciesdecreasewithanincreaseinpowerexponent.Variationin195slendernessratiosseemstohavelittleeffectonthedimensionlessnaturalfrequencies.TheComparisonsaremadeontheresultsobtainedfromtheproposedmethodandthoseintheexistingliteratures.Onecanfindthatthepresentresultsagreeverywellwiththepublishedwork.Table.3Thefirstdimensionlessfrequencyparameterλ1fordifferentvaluesofthemodulusratioandslendernessratioL/hEratioα=0α=0.1α=0.2α=0.5α=1α=2α=5α=1020present0.252.22032.23852.41062.58212.75332.92793.08353.1266[22]2.22032.23852.41062.58212.75332.92783.08343.1265[10]2.2203N/A2.41062.58212.75322.92783.0834N/Apresent0.52.64042.68682.72582.81482.91043.01223.10523.1316[22]2.64042.68682.72582.81482.91043.01223.10523.1316[10]2.6403N/A2.72572.81472.91043.01223.1052N/Apresent13.143.143.143.143.143.143.143.14[22]3.143.143.143.143.143.143.143.14[10]3.14N/A3.143.143.143.143.14N/Apresent23.73413.69883.66533.57583.46113.32443.19233.1531[22]3.73413.69883.66533.57583.46113.32443.19233.1531[10]3.7340N/A3.66533.57573.46113.32433.1922N/Apresent44.44064.37684.31454.13883.89373.57943.26673.1725[22]4.44064.37684.31454.13873.89373.57953.26683.1726[10]4.4406N/A4.31444.13873.89373.57943.2667N/A100present0.252.22142.32972.41182.58342.75472.92933.0853.1282[22]2.22142.32972.41182.58342.75462.92933.08493.1281present0.52.64172.68812.72712.81622.91193.01373.10683.1332[22]2.64172.68812.72712.81622.91193.01373.10673.1332present13.14153.14153.14153.14153.14153.14153.14153.1415[22]3.14153.14153.14153.14153.14153.14153.14153.1415present23.73593.70063.66713.57753.46283.3263.19383.1546-6- 中国科技论文在线http://www.paper.edu.cn[22]3.73593.70063.66713.57753.46283.3263.19393.1547present44.44284.37894.31664.14083.89573.58123.26833.1742[22]4.44284.37894.31664.14083.89573.58123.26843.1741200Table.4Theseconddimensionlessfrequencyparameterλ2fordifferentvaluesofthemodulusratioandslendernessratioL/hEratioα=0α=0.1α=0.2α=0.5α=1α=2α=5α=10200.254.43384.69934.83745.16745.4735.76756.06396.1991present[22]4.43384.69934.83745.16755.4735.76746.06366.1987[10]4.4338N/A4.83735.16745.47295.76756.0639N/Apresent0.55.27275.37525.45735.63275.80485.97406.14606.2265[22]5.27275.37525.45735.63275.80485.97396.14596.2263[10]5.2727N/A5.45725.63265.80475.97396.1459N/Apresent16.27036.27036.27036.27036.27036.27036.27036.2703[22]6.27036.27036.27036.27036.27036.27036.27036.2703[10]6.2703N/A36.27036.27036.27036.27036.2703N/Apresent27.45677.37747.30397.11766.90316.67836.44826.3364[22]7.45677.37747.30397.11766.90316.67836.44836.3365[10]7.4567N/A7.30397.11766.90306.67826.4482N/Apresent48.86768.72378.58548.21147.74007.22086.69016.4300[22]8.86768.72368.58538.21137.73997.22096.69026.4302[10]8.8675N/A8.58548.21147.73997.22086.6900N/A100present0.254.44254.67854.8475.17785.4845.77916.07606.2113[22]4.44254.67854.8475.17785.4845.77896.07566.2109present0.55.28315.38585.46805.64385.81625.98586.15816.2387[22]5.28315.38585.46815.64385.81625.98586.15806.2386present16.28276.28276.28276.28276.28276.28276.28276.2827[22]6.28276.28276.28276.28276.28276.28276.28276.2827present27.47147.39207.31837.13176.91676.69156.46106.349[22]7.47147.39197.31837.13166.91676.69156.46116.349present48.8858.74088.60238.22777.75567.23556.70356.443[22]8.8858.74078.60228.22767.75557.23566.70366.443Table.5Thefirstdimensionlessfrequencyparameterλ3fordifferentvaluesofthemodulusratioandslenderness205ratioL/hEratioα=0α=0.1α=0.2α=0.5α=1α=2α=5α=1020present0.256.66386.99487.24697.7338.17548.59939.02719.2315[22]6.66386.99497.2477.7338.17538.59899.02629.2305[10]6.6638N/A7.24687.73298.17538.59929.0270N/Apresent0.57.8898.04718.17158.43178.68148.92479.17119.2905[22]7.8898.04728.17158.43178.68148.92469.17089.2901[10]7.889N/A8.17148.43168.68138.92469.1710N/Apresent19.38179.38179.38179.38179.38179.38179.38179.3817[22]9.38179.38179.38179.38179.38179.38179.38179.3817[10]9.3816N/A9.38169.38169.38169.38169.3816N/Apresent211.15711.03410.92010.63810.32410.0019.67339.5137[22]11.15711.03310.92010.63810.32410.0019.67359.5139[10]11.156N/A10.92010.63810.32310.0009.6733N/Apresent413.26813.04412.82912.25811.56210.81510.0599.6938[22]13.26813.04312.82912.25811.56210.81510.0599.6942[10]13.267N/A12.82912.25811.56110.81410.059N/A-7- 中国科技论文在线http://www.paper.edu.cn100present0.256.66317.02587.27917.76788.21248.63819.06749.2724[22]6.66317.02597.27927.76788.21238.63779.06649.2713present0.57.92388.08278.20768.46908.71998.96439.21169.3315[22]7.92388.08268.20768.46918.71998.96429.21149.3311present19.4239.4239.4239.4239.4239.4239.4239.423[22]9.4239.4239.4239.4239.4239.4239.4239.423present211.20611.08210.96810.68610.37010.0469.71629.5558[22]11.20611.08210.96810.68510.37010.0459.71639.5559present413.32613.10112.88612.31311.61410.86410.1059.7369[22]13.32613.10112.88612.31311.61410.86410.1059.7373[1]:Alshorbagyetal.2011;2.1.3TheeffectofmassdensityvariationToanalyzetheeffectofmaterialdistribution,Young’smodulusarechosenasthesameforthebeam,i.e.,Eratio=1.Thefirstthreedimensionlessnaturalfrequenciesfordifferentratioand210L/hareshowninTables6-8.Itcanbeobservedthat,whenthemassdensityratiosratioislessthan1,thenaturalfrequenciesdecreasewithanincreaseinpowerexponent.Butonthecontrary,whenisgreaterthan1,thenaturalfrequenciesincreasewithanincreaseinpowerratioexponent.Anditisalsofindthatvariationinslendernessratioshaslittleeffectonthedimensionlessnaturalfrequencies.ThiscaseshowstheoppositetrendofthevariationofYoung’s215modulus.Table6Thefirstdimensionlessfrequencyparameterλ1fordifferentvaluesofthemodulusratioandslendernessratioL/hRratioα=0α=0.1α=0.2α=0.5α=1α=2α=5α=10200.254.44064.22364.06553.77043.52873.33123.18793.15160.53.73413.66823.61323.49233.37333.26143.17163.147713.143.143.143.143.143.143.143.1422.64042.66532.68882.75212.83662.94943.08053.124842.22032.25212.28282.36922.49522.68902.97493.09511000.254.44284.22594.06783.77253.53053.33273.18933.14910.53.73593.67013.61513.49413.37503.26293.17303.057813.14153.14153.14153.14153.14153.14153.14153.141522.64172.66662.69012.75342.83802.95103.08233.126542.22142.25312.28392.37032.49642.69052.97703.0973Table7Theseconddimensionlessfrequencyparameterλ2fordifferentvaluesofthemodulusratioandslenderness220ratioL/hRratioα=0α=0.1α=0.2α=0.5α=1α=2α=5α=10200.258.86768.40388.09377.54917.11616.75496.44856.33100.57.45677.31217.19856.96716.75686.56446.38536.310416.27036.27036.27036.27036.27036.27036.27036.270325.27275.32895.38115.51625.68175.87136.07776.193844.43384.50594.57524.76685.03195.38615.79576.05511000.258.88508.42108.11057.56477.13046.76806.46066.34260.57.47147.32677.21296.98116.77016.57726.39756.322316.28276.28276.28276.28276.28276.28276.28276.282725.28315.33935.39165.52695.69305.88316.09036.2070-8- 中国科技论文在线http://www.paper.edu.cn44.44254.51464.58414.77615.04205.39755.80866.0697Table8Thethirddimensionlessfrequencyparameterλ3fordifferentvaluesofthemodulusratioandslendernessratioL/hRratioα=0α=0.1α=0.2α=0.5α=1α=2α=5α=10200.2513.26812.55712.09311.29310.66710.1469.6949.5090.511.15710.93410.76210.41910.1149.8389.57969.46519.3829.3829.3829.3829.3829.3829.3829.38227.8897.9768.0578.2628.5058.7769.0759.23246.6346.7466.8547.1487.5438.0468.6558.9881000.2513.32612.61412.14511.34410.71510.1919.7369.5490.511.20610.98210.81010.46610.1599.8819.6219.50619.4239.4239.4239.4239.4239.4239.4239.42327.9248.0118.0928.2988.5438.8169.1169.27446.6636.7766.8847.1797.5778.0838.6969.0322.1.4ComparisononmodeshapesofAFGbeamandhomogenousbeam225Inthissection,thefirstthreemodeshapesofAFGbeamandthoseofthepuresteelbeamandpurealuminabeamarecomparedtoshowthedifferencesbetweenthem.Thepowerexponentistakenas2fortheAFGbeam.ThethreesubplotsinFig.2showthemassnormalizedfirstmodeshape,secondoneandthethirdonerespectively.Onecanfindthatineachsubplot,themodeshapeoftheAFGbeamliesbetweenthatofthepuresteelbeamandpurealuminabeam.One230canalsofindthatinthesecondandthirdsubplots,thenodalpointsofthepuresteelbeamandpurealuminabeamarethesame.ButthereisarightshiftforthenodeoftheAFGbeam.-3x1081stmode642002468101214161820Length(m)0.010.005nodalpoint02ndmodenodalpoint-0.005-0.0102468101214161820Length(m)0.013rdmodenodalpoint0.005nodalpoint0-0.005nodalpointnodalpoint-0.0102468101214161820Length(m)Figure2ComparisononthemodeshapesofAFGbeamandhomogeneousbeam(AFGbeam:blacksolidline;pureAl2O3beam:bluedot-dashline;puresteelbeam:reddashedline)2352.2Forcedvibrationanalysisofasteel-aluminabeam2.2.1Undersinusoidalexcitationforce-9- 中国科技论文在线http://www.paper.edu.cnInthiscase,forcedvibrationanalysisisconductedfortheAFGbeam.Asinusoidalforceisassumedtoactatthemid-spanofthebeam:F(t)10000sin(6t)N.Thetimestepis0.001secondandthefirsttwomodaldampingratiosaretakenas0.01and0.01incalculationthe240dampingmatrixC.Thetimedurationis12secondsincalculationtheforcedvibrationresponsesof[33]thebeamandNewmarkdirectintegrationmethodisadopted.InordertochecktheaccuracyoftheCEM,finiteelementofthebeamisalsobuiltusingMatlabfiniteelementpackage.Thebeamisdiscretizedinto50Euler-Bernoullibeamelements.Eachnodehas3DOFs,andtotallyhas153DOFs.ButthetotalnumberofDOFsforCEMisonly14.Fig.3andFig.4givethecomparisonon245theaccelerationresponsesanddisplacementresponsesatL/4andL/2ofthebeam,respectively.Itisobservedthatineachsubplotofthefigurebothtimehistorycurvesmatcheachotherverywell,thisindicatesthatproposedmethodisvalidforforcedvibrationanalysisoftheAFGbeamundersinusoidalexcitation.AccelerationresponseatL/20.3fromCEM0.2fromFEM0.1)20-0.1Acc.(m/s-0.2-0.3-0.4024681012Time(sec.)AccelerationresponseatL/40.15fromCEM0.1fromFEM0.05)20-0.05Acc.(m/s-0.1-0.15-0.2024681012Time(sec.)250Figure3ComparisononaccelerationresponsesobtainedfromCEMandFEMundersinusoidalexcitation-4DisplacementresponseatL/2x104fromCEM3fromFEM210Displ.(m)-1-2-3024681012Time(sec.)-4DisplacementresponseatL/4x102fromCEM1.5fromFEM10.50Displ.(m)-0.5-1-1.5024681012Time(sec.)Figure4ComparisonondisplacementresponsesobtainedfromCEMandFEMundersinusoidalexcitation-10- 中国科技论文在线http://www.paper.edu.cn2.2.2Underimpulsiveforce255Inthiscase,forcedvibrationanalysisunderanimpulsiveforceisconductedfortheAFGbeam.Aimpulsiveforceisassumedtoactatmid-spanofthebeam:190000(t0.05)N0.05st0.1sF(t).Fig.5andFig.6givethecomparisononthe190000(0.15t)N0.1st0.15saccelerationresponsesanddisplacementresponsesatL/4andL/2ofthebeam.Again,thetimehistorycurvesineachfigurematcheachotherverywell,thisfurtherillustratestheefficiencyand260correctnessoftheproposedmethod.AccelerationresponseatL/20.6fromCEM0.4fromFEM0.2)20-0.2Acc.(m/s-0.4-0.6-0.8024681012Time(sec.)AccelerationresponseatL/40.6fromCEM0.4fromFEM)20.20Acc.(m/s-0.2-0.4024681012Time(sec.)Figure5ComparisononaccelerationresponsesobtainedfromCEMandFEMunderimpulsiveexcitation-4x10DisplacementrepsonseatL/24fromCEM3fromFEM21Displ.(m)0-1-2024681012Time(sec.)-4DisplacementrepsonseatL/4x101.5fromCEM1fromFEM0.50Displ.(m)-0.5-1-1.5024681012Time(sec.)265Figure6ComparisononaccelerationresponsesobtainedfromCEMandFEMunderimpulsiveexcitation2.2.3ComparisononforcedvibrationresponsesofAFGbeamandhomogenousbeamInthissection,forcedvibrationresponsesoftheAFGbeamandthoseofthepuresteelbeam-11- 中国科技论文在线http://www.paper.edu.cnandpurealuminabeamarecomparedtoshowthedifferencesbetweenthem.Thepowerexponent270istakenas0.5fortheAFGbeam.ThesamesinusoidalforceasusedinSection3.2.1istakenastheexcitationforce.Fig.7(a)andFig.7(b)showthetimehistoriesoftheaccelerationresponseanddisplacementresponseoftheAFGbeam,purealuminabeamandpuresteelbeam,respectively.Onecanfindthatalthoughtheexcitationforceisthesame,themagnitudesofdynamicresponsesofdifferentbeamsaredifferent:themagnitudeofresponseofthepuresteel275beamisthelargestandthatofthepurealuminabeamisthesmallest,whilethatoftheAFGbeamisslightlylargerthanthemagnitudeofthepurealuminabeam.0.3AFGbeam(a)Al2O3beamSteelbeam0.20.10)2Acc.(m/s-0.1-0.2-0.3-0.4024681012Time(sec.)-4x105AFGbeam(b)Al2O3beam4Steelbeam3210Displ.(m)-1-2-3-4-5024681012Time(sec.)Figure7ComparisonondynamicresponsesofAFGbeamandhomogeneousbeam280(a):accelerationresponse;(b):displacementresponse3ConclusionThispaperconductsfreeandforcedvibrationanalysisfortheaxiallyfunctionallygradedEuler-Bernoullibeam.Thematerialproperties,includingtheYoung’smodulusandthemassdensity,areassumedtoaccordingtothepower-lawfunctions.Thegoverningequationsofthe285beamisderivedbyLagrange’sequationsandthendiscretizedbythecompositeelementmethod.-12- 中国科技论文在线http://www.paper.edu.cnNumericalsimulationsareconductedtofigureouttheeffectsofYoung’smodulus,massdensityratios,andslendernessratiosonthefreevibrationcharacteristicsofAFGbeam.ResultsshowthatboththeratioofYoung’smodulusandthemassdensityratiohaveanoticeableinfluenceonthedimensionlessnaturalfrequenciesoftheAFGbeam.Thenodalpointinthemodeshapesofthe290AFGbeamaredifferentfromthoseofthehomogeneousbeams.Itisalsofindthatvariationinslendernessratioshaslittleeffectonthedimensionlessnaturalfrequencies.Studyalsoshowstheforcedvibrationresponsescanbeobtainedaccuratelyandefficientlyfromtheproposedmethod.References[1]MiyamotoY.Functionallygradedmaterials:design,processing,andapplications,KluwerAcademicPublishers;2951999.[2]KapuriaS,BhattacharyyaMandKumarAN.Bendingandfreevibrationresponseoflayeredfunctionallygradedbeams:atheoreticalmodelanditsexperimentalvalidation.ComposStruct2008;82:390-402.[3]MatbulyaMS,RagbOandNassarM.Naturalfrequenciesofafunctionallygradedcrackedbeamusingthedifferentialquadraturemethod.ApplMathComput2009;215:2307-2316.300[4]BirmanVandByrdLW.Modelingandanalysisoffunctionallygradedmaterialsandstructures.ApplMechRev2007;60:195-216.[5]LiXF.AunifiedapproachforanalyzingstaticanddynamicbehaviorsoffunctionallygradedTimoshenkoandEuler-Bernoullibeams.JSoundVib2008;318:1210-1229.[6]BenattaMA,MechabI,TounsiAandBediaEAA.Staticanalysisoffunctionallygradedshortbeamsincluding305warpingandsheardeformationeffects.CompMaterSci2008;44:765-773.[7]ElishakoffIandGuedeZ.Analyticalpolynomialsolutionsforvibratingaxiallygradedbeams.MechAdvMaterStruct2004;11:517-533.[8]WuL,WangQandElishakoffI.Semi-inversemethodforaxiallyfunctionallygradedbeamswithananti-symmetricvibrationmode,JSoundVib2005;284:1190-1202.310[9]JinZHandBatraRC.Stressintensityrelaxationatthetipofanedgecrackinafunctionallygradermaterialsubjectedtoathermalshock.JThermStresses1996;19(4):317-339.[10]SimsekM,KocaturkTandAkbasSD.Dynamicbehaviorofanaxiallyfunctionallygradedbeamunderactionofamovingharmonicload.ComposStruct2012;94:2358-2364.[11]AydogduM,TaskinV.Freevibrationanalysisoffunctionallygradedbeamswithsimplysupportededges.315MaterDes2007;28(5):1651-1656.[12]YingJ,LüCF,ChenWQ.Two-dimensionalelasticitysolutionsforfunctionallygradedbeamsrestingonelasticfoundations.ComposStruct2008;84(3):209-219.[13]YangJ,ChenY.Freevibrationandbucklinganalysesoffunctionallygradedbeamswithedgecracks.ComposStruct2008;83(1):48-60.320[14]XiangHJ,YangJ.FreeandforcedvibrationofalaminatedFGMTimoshenkobeamofvariablethicknessunderheatconduction.ComposPartB:Eng2008;39(2):292-303.[15]SimsekM.StaticanalysisofafunctionallygradedbeamunderauniformlydistributedloadbyRitzmethod.IntJEngApplSci2009;1(3):1-11.[16]KitipornchaiS,KeLL,YangJ,XiangY.Nonlinearvibrationofedgecrackedfunctionallygraded325Timoshenkobeams.JSoundVib2009;324(3-5):962-982.[17]KeLL,YangJ,KitipornchaiS.Nonlinearfreevibrationoffunctionallygradedcarbonnanotube-reinforcedcompositebeams.ComposStruct2010;92(3):676-683.[18]HemmatnezhadM,AnsariRandRahimiGH.Large-amplitudefreevibrationsoffunctionallygradedbeamsbymeansofafiniteelementformulation.ApplMathModel2011;37:8495-8504.330[19]WattanasakulpongNandChaikittiratanaA.FlexuralvibrationofimperfectfunctionallygradedbeamsbasedonTimoshenkobeamtheory:Chebyshevcollocationmethod.Meccanica2015;50(5):1331-1342.[20]JinCHandWangXW.AccuratefreevibrationanalysisofEulerfunctionallygradedbeamsbytheweakformquadratureelementmethod.ComposStruct2015;125:41-50.[21]SteinbergMA.Materialsforaerospace.SciAm1986;255(4):59-64.335[22]AlshorbagyAE,EltaherMAandMahmoudFF.Freevibrationcharacteristicsofafunctionallygradedbeambyfiniteelementmethod.ApplMathModel2011;35:412-425.[23]GiuntaG,CrisafulliD,BelouettarSandCarreraE.Hierarchicaltheoriesforthefreevibrationanalysisoffunctionallygradedbeams.ComposStruct2011;94:68-74.[24]SuH,BanerjeeJRandCheungCW.Dynamicstiffnessformulationandfreevibrationanalysisoffunctionally340gradedbeams.ComposStruct2013;106:854-862.[25]RajasekaranS.Bucklingandvibrationofaxiallyfunctionallygradednonuniformbeamsusingdifferentialtransformationbaseddynamicstiffnessapproach.Meccanica2013;48:1053-1070.[26]VoTP,ThaiHT,NguyenTKandInamF.Staticandvibrationanalysisoffunctionallygradedbeamsusingrefinedsheardeformationtheory.Meccanica2014;49:155-168.345[27]SimsekMandKocatürkT.Freeandforcedvibrationofafunctionallygradedbeamsubjectedtoaconcentratedmovingharmonicload.ComposStruct2009;90(4):465-473.[28]HuangYandLiXF.Anewapproachforfreevibrationofaxiallyfunctionallygradedbeamswith-13- 中国科技论文在线http://www.paper.edu.cnnon-uniformcross-section.JSoundVib2010;329:2291-2303.[29]QianLFandChingHK.Staticanddynamicanalysisof2-Dfunctionallygradedelasticitybyusingmeshless350localPetrov-Galerkinmethod.JChinInstEng2004;27(4):491-503.[30]ZengP.Compositeelementmethodforvibrationanalysisofstructure,PartII:Element(Beam).JSoundVib1998;218(4):658-696.[31]LuZR,HuangM,LiuJK,ChenWHandLiaoWY.Vibrationanalysisofmultiple-steppedbeamswithcompositeelementmodel.JSoundVib2009;322(4-5):1070-1080.355[32]LuZRandLawSS.Dynamicconditionassessmentofacrackedbeamwiththecompositeelementmodel.MechSystSignalPr2009;23(2):415-431.[33]NewmarkNW.Amethodofcomputationforstructuraldynamics.JEngMech1959;85(3):67-94.360基于复合单元模型的轴向功能梯度梁的振动分析刘济科，吕中荣（中山大学工学院，广州510006）摘要：本文提出一种简单而准确的方法来分析轴向功能梯度（AFG）梁的自由振动和强迫振365动。梁的控制方程由拉格朗日方程导出，并通过复合单元法离散。通过数值模拟的方法来研究杨氏模量，质量密度变化和细长比对AFG梁的动态特性的影响。结果显示本文所提出的方法获得的固有频率与有限元法以及其他方法所得到的固有频率非常接近。同时比较了本文方法和有限元法得出的强迫振动响应，两者结果高度吻合。这进一步证明了本文方法对功能梯度梁振动分析的正确性。370关键词：固体力学；复合单元；自由和强迫振动；响应；轴向功能梯度梁-14-'