基于环境激励的时域分解法

Anewmethodtoextractmodalparametersusingoutput-onlyresponses--ByeongHwaKima,NorrisStubbsb,TaehyoParka,ContentsBackgroundTheoryNumericalExampleConclusion1、BackgroundThetypicalmethod:Figure.1Peak-peakingmethod1、BackgroundFigure.2FDDmethodleft:targeteigenfrequencies.Right:Measuredeigen-frequencies0.16710.16831.04721.04892.93232.93575.74605.8363left:targetdamping.Right:Measureddamping0.00500.00910.00500.00460.00500.00490.00500.00471、BackgroundDrawbacksofthetypicalmethod:1、theresultisdeterminedbythetimeorfrequencysamples,butalsowiththenumberofsensorlocations.2、it’shardtocomputethenaturalfrequencies,modalshapeandthedampingratio.3、Accordingtofirstpoint,thelargeamountofnumericalcomputationretardsthereal-timeimplementation.2、TheoryConsiderthesimplebeamasshowninFig.1.Thepx1outputdisplacementprofilevector,y(t),causedbyanarbitraryloadattimetcanbespannedbymodeshapesasfollows:(1)2、TheoryUndertheassumptionthatndominantandwell-separatedpolesareresolvedinthediscreteaccelerationresponsewithinthefoldingfrequency:(2)(3)2、TheoryAccordingtotheFiltertheory,wecangettheithfilteredsdofaccelerationresponse:Thep×1noisevector,𝜀𝑓(𝑘);canbespannedbyitsbasesatthesampletimek:(4)(5)(6)2、TheoryIfNsamplesaremeasured,thematrixformbecomes：Consideracross-correlation𝐸𝑖oftheithmode-isolatedaccelerationtimehistorysignals(8)(9)2、TheoryFortheorthogonalbases,itcanbeshownthat:wherethep×pmatrix,U≡𝜑𝑖,𝜓1,⋯,𝜓𝑝−1,isasingularvectormatrixof𝑌𝑖,andthep×pdiagonalmatrix,Ω≡𝑞𝑖,𝜎1,⋯,𝜎𝑝−1,isasingularvaluematrixof𝑌𝑖.(10)(11)(12)(13)2、TheoryTheunderlyingassumptionintheorderofthesingularvaluesisthat𝑞𝑖𝜎1⋯𝜎𝑝−1.Thedesiredithun-dampedmodeshape,𝜑𝑖,whichisa𝑃×1realvector,canbeobtainedbytakingthefirstsingularvectorafteraSVD,of𝐸𝑖.Pre-multiplicationofthetransposeoftheidentifiedithmodeshapeonEq.(8)yieldsBecausethenoisebasesareassumedtobeorthogonaltothemodalbasesThissignalcontainsasingle-outputsdofsystemthatrepresentstheithmodalbehavioroftheaccelerationfortheentiresetofpsignals.Therefore,theauto-spectrumof𝐶𝑖containsonepeak.Thefrequencyatthepeakisthedesireddampednaturalfrequencyoftheithmode.3、NumericalExampleConsiderthesimplesupportedbeamshowninFig.1.Thelengthofthebeamis30m,and11accelerometers(p=11)areplacedonthebeamwithanequalspacingof3m.Thesecondmomentofcross-sectionalarea,Young’smodulus,anddensityofthebeamare1.0m4,20GPa,and2.7×103kg/m3,respectively.Thefiniteelement(FE)modelconsistsof100linearelements(101nodes),eachwithauniformlengthof0.3m.FromtheFEanalysis,theun-dampednaturalfrequenciesforthefirst,second,andthirdbendingmodesare1.98,7.90,and17.78Hz.Thedampingratiois1.5%.Givenaunitpulseload3、NumericalExamplethefirstthreepeaksrangefrom1to3Hz,from7to9Hz,andfrom17to19Hz3、NumericalExampleFirst,Athird-orderButterworthdigitalband-passfilterisdesignedforeachfrequencyrange.Second,theoutputenergycorrelationmatrix,𝐸𝑖,isconstructed.ThefinalstepintheextractionofthenaturalfrequencyforeachmodeistocomputeasinglerepresentativesdoftimehistoryinEq.(15).TheERADCmethodidentifiesthefirstthreenaturalfrequenciesare1.98,7.90,and17.78Hz,respectively.ThecorrespondingdampingratiosfromtheERADCmethodareexactly1.5%forallthethreemodes.ThenextstepinvolvesperformingaSVDanalysisoneachoftheenergycorrelationmatrices.Thefirstcolumnvectorofthesingularvectormatrix,U,isdesignatedastheun-dampedmodeshape4、ConclusionSecond,theformulationoftheproposedmethodincludesnoassumptionfortheinputs.Therefore,theproposedmethodcanextractundampedmodeshapesforthearbitraryinputsrelatedtotheextractionofmodeshapes.Third,themodeisolationtaskcompletedbyusingadigitalband-passfilterreducesthenormallyhighlevelofoperatorinteraction.Therefore,theproposedmethodcanbeappliedtoanautomatedon-linehealth-monitoringsystemFirst,theSVDmatrixsizeoftheproposedmethodreliesonthenumberofsensorsinsteadofthenumberoftimesamplesorfrequencysamples.Therefore,theproposedapproachisadequatetoobtainspatiallyrefinedmodeshapes.ThankYou!!