混合高斯分布的极值分析

Extreme value analysis of a stochastic process features in many engineering problems.The widely used Poisson approximation can be excessively conservative if a process hasnarrowband traits, as upcrossings tend to manifest in clumps. Over the years, variousauthors have developed techniques for predicting the extremes of narrowbandGaussian processes. A bimodal process,

JournalofSoundandVibration330(2011)3458–3472

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JournalofSoundandVibration

journalhomepage:http:///locate/jsvi

ExtremevalueanalysisofbimodalGaussianprocesses

Y.M.Lown

SchoolofCivil&EnvironmentalEngineering,NanyangTechnologicalUniversity,BlockN1,NanyangAvenue,Singapore639798,Singapore

articleinfo

Articlehistory:

Received23July2010Receivedinrevisedform27January2011

Accepted29January2011HandlingEditor:L.G.Tham

Availableonline25February2011

abstract

Extremevalueanalysisofastochasticprocessfeaturesinmanyengineeringproblems.ThewidelyusedPoissonapproximationcanbeexcessivelyconservativeifaprocesshasnarrowbandtraits,asupcrossingstendtomanifestinclumps.Overtheyears,variousauthorshavedevelopedtechniquesforpredictingtheextremesofnarrowbandGaussianprocesses.Abimodalprocess,comprisingtwonarrowbandcomponents,isoftenencounteredinpractice,butrelatedstudiesarescarce.Thispaperoutlinesasemi-analyticalapproachforextendinganygivennarrowbandmethodtobimodalprocesses.Themethodissimpletouse,anditssolutionalwaysspecializestotherespectivenarrowbandresultifeithercomponentbecomesin nitesimal.Numericalsimulationsareconductedforveri cation.Theproposedbimodalapproachisfoundtobeingoodagreementwithsimulationresults,providedthattheindividualcomponentextremeshavebeenaccuratelyevaluatedusingasuitabletechnique.

&2011ElsevierLtd.Allrightsreserved.

1.Introduction

Akeyissueinthe eldofrandomvibrationistheextremevalueproblem,whichisconcernedwiththeprobabilitythatarandomstructuralresponsewillexceedaparticularthresholdwithinaprescribedtimeinterval.Anintimatelyrelatedproblem,whichisnotexplicitlyaddressedherein,isthe rstpassageproblemthatinvolvestheprobabilitydistributionofthetimetofailure.Suchisthedif cultyoftheextremevalueproblemthatthereexistsnoexactsolution,evenforthespecialcaseofastationaryGaussianprocess.Consequently,numerousapproximatetechniques(e.g.[1–5])havebeendeveloped,asurveyofwhichcanbefoundinRef.[4].

ThemostcommonapproximationistoassumethatthethresholdcrossingsareindependentandcanbemodeledasaPoissonprocess.ThePoissonapproximationhasseveraladvantageousfeatures.Itissimple,alwaysconservative,anditasymptotestowardstheexactresultasthethresholdgoestoin nity.TheconservatismofthePoissonapproximationismostseverewhentheprocessisnarrowband.Forpracticallevelsofinterest,theassumptionofindependentcrossingsisunsatisfactoryforanarrowbandprocessowingtothetendencyofcrossingstomanifestinclumps.Thatistosay,onceanupcrossingforaparticularlevelhasoccurred,thereisaninordinatelikelihoodthatmoreupcrossingswillfollowshortly.

Inaseminalpaper,Vanmarcke[1]proposedanimprovementtothePoissonapproximation,basedonacascadeofassumptions.Vanmarcke’sapproximationcanbeexpressedinanexplicitform,andhasbeenshowntobereasonablyaccurateforcertainapplications.Theformulationreliessolelyonasinglebandwidthparametertocharacterizetheautocorrelationoftheprocess.Thisisadrawback,asithasbeendemonstrated[3,4]thatprocesseswiththesamebandwidthparameter,butofdifferingspectralshapes,http://ngley[4]developedanapproach

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Tel.:+6567905265;fax:+6567910676.E-mailaddress:ymlow@ntu.edu.sg

0022-460X/$-seefrontmatter&2011ElsevierLtd.Allrightsreserved.doi:10.1016/j.jsv.2011.01.033

Extreme value analysis of a stochastic process features in many engineering problems.The widely used Poisson approximation can be excessively conservative if a process hasnarrowband traits, as upcrossings tend to manifest in clumps. Over the years, variousauthors have developed techniques for predicting the extremes of narrowbandGaussian processes. A bimodal process,

Y.M.Low/JournalofSoundandVibration330(2011)3458–34723459

thatovercomesthislimitationbyexploitingmoreoftheinformationsuppliedbytheautocorrelationfunction.Moreimportantly,themethodisstraightforwardtoapply.

Inpractice,thespectraldensityofaGaussianresponsemayexhibittwodistinctmodes,whereeachmodeisanarrowbandGaussianprocess.Onewouldexpecttheclumpingofupcrossingstobealsopresentinabimodalprocess,albeitonamoresophisticatedlevel.Abimodalresponsemayarisefromavarietyofphysicalmechanisms.Itmaybetheconsequenceofapeakintheresonantmode,andanotherpeakatthefrequencyoftheappliedforces.Anotherexampleisatwo-degree-of-freedomsystemsubjectedtowhitenoiseexcitation.Theloadingitselfcouldbebimodal;offshorestructuresoccasionallyencounterwaveswithbimodalspectra.Theimportanceofbimodalprocessesisevidencedbythenumerousstudiesonthespectralfatigueanalysisofbimodalprocesses(e.g.[6,7]).Conversely,forthebimodalextremes,onemay ndonlyonepriorinvestigationbyToroandCornell[8].ToroandCornell’s(T–C)methodisageneralizationofVanmarcke’smodel,inthesensethattheunderlyingassumptionsarebroadlysimilar.

Giventhepracticalsigni canceofbimodalprocessesandthescarcityofrelatedstudiesontheextremevalueproblem,thispaperaimstodevelopasimpleanalyticalmethodthatcanaccuratelypredictthebimodalextremes.Grantedthatthisisnotaneasytask,andnotingtheabundanceofliteratureonnarrowbandGaussianprocesses,itmakessensetoextendanexistingnarrowbandtechniquetothebimodalsituation.Inthisrespect,Langley’smethod[4]seemstobeanidealcandidateonaccountofitssimplicityandaccuracy.ItisenvisagedthatabimodalmethodbasedonanextensionofLangley’smethodwouldbemoreaccurateforawiderrangeofspectralshapesthantheT–Cformulation,whichisconstrainedbytheassumptionsinherentinVanmarcke’smodel.2.Backgroundtheoryandnarrowbandprocesses2.1.Extremevalueproblem

Considerazero-meanstationaryGaussianprocessX(t)withaone-sidedspectrumSXX(o).Theithspectralmo …… 此处隐藏：43502字，全部文档内容请下载后查看。喜欢就下载吧 ……