the dynamics analysis of a cracked timoshenko beam

JOURNALOFSOUNDANDVIBRATION(2003)1139–1153ThedynamicanalysisofacrackedTimoshenkobeambythespectralelementmethodM.Krawczuka,b,M.Palaczb,*,W.OstachowiczbaDepartmentofTechnicalSciences,UniversityofWarmiaandMazury,Oczapowskiego22,Olsztyn10-736,PolandbInstituteofFluidFlowMachinery,PolishAcademyofSciences,Fiszera14,Gda!nsk80-231,PolandReceived12March2002;accepted13June2002AbstractTheaimofthispaperistointroduceanewfinitespectralelementofacrackedTimoshenkobeamformodalandelasticwavepropagationanalysis.Theproposedapproachdealswiththespectralelementmethod.Thismethodissuitableforanalyzingwavepropagationproblemsaswellasforcalculatingmodalparametersofthestructure.Inthepaper,theresultsofthechangeinmodalparametersduetocrackappearancearepresented.Theinfluenceofthecrackparameters,especiallyofthechanginglocationofthecrack,onthewavepropagationisexamined.Responsesobtainedatdifferentpointsofthebeamarepresented.Properanalysisoftheseresponsesallowsonetoindicatethecracklocationinaverypreciseway.Thisfactisverypromisingforthefutureworkinthedamagedetectionfield.r2002ElsevierScienceLtd.Allrightsreserved.1.IntroductionInordertoimprovethesafety,reliabilityandoperationallife,itisurgenttomonitortheintegrityofstructuralsystems.Techniquesofnon-destructivedamagedetectioninmechanicalengineeringstructuresareessential[1–3].Previousapproachestonon-destructiveevaluationofstructurestoassesstheirintegritytypicallyinvolvedsomeformofhumaninteraction.Recentadvancesinsmartmaterialsandstructurestechnologyhasresultedinarenewedinterestindevelopingadvancedself-diagnosticcapabilityforassessingthestateofastructurewithoutanyhumaninteraction.ThegoalistoreducehumaninteractionwhileatthesametimemonitortheARTICLEINPRESS*Correspondingauthor.Tel.:[phone]x109;fax:[phone].E-mailaddresses:[e-mail](M.Krawczuk),[e-mail](M.Palacz),[e-mail](W.Ostachowicz).0022-460X/03/$-seefrontmatterr2002ElsevierScienceLtd.Allrightsreserved.doi:10.1016/S0022-460X(02)01387-1integrityofthestructure.Withthisinmind,manyresearchershavemadesignificantstridesindevelopingdamagedetectionmethodsforstructuresbasedontraditionalmodalanalysistechniques.Thesetechniquesareoftenwellsuitedforstructures,whichcanbemodelledbydiscretelumped-parameterelementswherethepresenceofdamageleadstosomelow-frequencychangeintheglobalbehaviourofthesystem[4–7].Ontheotherhand,smalldefectssuchascracksareobscuredbymodalapproachessincesuchphenomenaarehigh-frequencyeffectsnoteasilydiscoveredbyexaminingchangesinmodalmass,stiffnessordampingparameters.Thisisbecauseathighfrequencies,modalstructuralmodelsaresubjecttouncertainty.Thisuncertaintycanbereducedbyincreasingtheorderofthediscretemodel,however,thisincreasesthecomputationaleffortofmodal-baseddamagedetectionschemes.Thereisalsoagroupofmethods,whichutilizethermodynamicdampingforassessmentofthestructuralintegrityofvibratingstructures[8–10].Spectralanalysisisamethodforrepresentingthedynamicsolutionintheformofaseriesofsolutionsatdifferentfrequencies[11].Thespectralelementmethodreformulatesthesesolutionsasdynamicstiffnessrelations,makingitsuitableforassemblyinamanneranalogoustothefiniteelementmethod.Awidevarietyofelementshavebeendevelopedforstructuralmembersoverfiniteandsemi-infiniteregions.Thespectralelementmethodhasseveralusefulattributes.Sinceinertialpropertiesaremodelledexactly,fewelementsareneededtomodellargeregions.Thegeneralizednodesneedtobeplacedonlyatstructuraldiscontinuities.Severalspectralelementstogethercanmodelstructureshavingsectionswithdifferentthicknessesormaterialproperties[12].Thesolutionisobtainedintermsofgeneralizeddisplacements,subsequentcalculationsforvelocity,acceleration,strainandstressforanyappliedloadcanthenbefoundwithrelativelyinexpensivepost-processingcalculations.Thespectralelementmethoddirectlycomputesastructure’sfrequencyresponsefunctionandinthismannergivesadditionalinformationthatbridgesthegapbetweenmodalmethodsbasedonfreevibrationsandtimereconstructionsbasedondirectintegration[13].Spectralelementsmayalsobeusedtosolveinverseproblemssuchasforceidentification.Finally,thecoreproblemtobesolvedisincrediblysmallandisrepeatedmanytimesindata-independentmanner.Thismakesthespectralelementmethodideallysuitedforsolutiononcomputerswithmanyprocessors.Atthepresentmanyspectralmodelsofstructuresareavailableinliterature.Onecanfindspectralmodelsofrods,beamsandplateswithoutanydamage[13].ThespectralmodelofacrackedrodelementispresentedinRef.[14].Theintroducedmodelallowstheuseofpropagatingwaveforpreciselocalizationofthecrack.SpectralBernoulli–Euler’sbeamwithacrackispresentedbyRef.[15].Themodeldescribedgivesproperresultsforidentificationofcrackparametersbyanalysisofthepropagatingwave.ThereisamodelofacrackedTimoshenkobeamavailable,butitisfiniteelementmethodmodel[16].ThispaperpresentsanewfinitespectralTimoshenkobeamelementwithatransverseopenandnon-propagatingcrack.Thiselementcanbeappliedformodalanalysisandforexaminingthewavepropagationprocess.Fewnumericalexamplesarepresentedinordertoshowtheinfluenceof