结构件间隙伸缩系统的建模与振动抑制.docx

结构件间隙伸缩系统的建模与振动抑制.docx

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结构件间隙伸缩系统的建模与振动抑制

ModelingandVibrationSuppressionforTelescopicSystemsofStructuralMemberswithClearance

PROF.DR.DIETERARNOLD

DIPL.-ING.MARTINMITTWOLLENDIPL.-ING.FRANKSCHÖNUNG,

UNIVERSITÄTKARLSRUHE,INSTITUTFÜRFÖRDERTECHNIKUNDLOGISTIKSYSTEME）

PROF.DR.-ING.JÖRGWAUERDIPL.-ING.PIERREBARTHELS

UNIVERSITÄTKARLSRUHE,INSTITUTFÜRTECHNISCHEMECHANIK

Abstract

Telescopicsystemsofstructuralmemberswithclearancearefoundinmanyapplications,e.g.,mobilecranes,rackfeeders,forklifters,stackercranes（seeFigure1）.Operatingthesemachines,undesirablevibrationsmayreducetheperformanceandincreasesafetyproblems.Therefore,thiscontributionhastheaimtoreducetheseharmfulvibrations.Forabetterunderstanding,thedynamicbehaviouroftheseconstructionsisanalysed.Themaininterestistheoverlappingareaofeachtwosectionsoftheabovedescribedsystems（seemarkingsinFigure1）whichisinvestigatedbymeasurementsandbycomputations.Atestrigisconstructedtodeterminethedynamicbehaviourbymeasuringfundamentalvibrationsandhigherfrequentoscillations,dampingcoefficients,specialappearancesandmore.Foranappropriatephysicalmodel,thegoverningboundaryvalueproblemisderivedbyapplyingHamilton’sprincipleandaclassicaldiscretisationprocedureisusedtogenerateacoupledsystemofnonlinearordinarydifferentialequationsasthecorrespondingtruncatedmathematicalmodel.Onthebasisofthismodel,acontrollerconceptforpreventingharmfulvibrationsisdeveloped.

Fig.1:

Graduatedmulti-sectionsystemswithclearance

1.Introduction

Graduatedmulti-sectionsystemsofstructuralcomponentsextendingandretractinginsideeachotherareinterestingtechnicalsystems.Theyarefound,e.g.,inmobilecranes,rackfeeders,forklifters,stackercranes（seeFigure1）.Asthemaindutyofthesemachinesisnotdrivingaroundbutloadingandunloadinggoodstoandfromracks,trucksetc.,agreatnumberofaccelerationanddecelerationoperationsoccur.Combinedwiththeextendingandretractingmotionofthesections,bendingvibrationsofthesystemperpendiculartothetelescopicaxisoccur.Thesevibrationsleadtoasignificantreductionoftheperformanceduetotherequiredwaitingtimeforrelaxationandtosafetyproblemssothatcontrolledvibrationsuppressionseemstobenecessary.

Theobjectiveofthepresentpaperistodevelopacontrollerconceptforpreventingharmfulvibrations.First,asystemwithoutclearanceandwithafixedtelescopiclengthwhichcanbecharacterizedbyatime-invariantsystemoflineardifferentialequations,isreducedtoitsdominatingmodes.Usingthisreducedmodel,aconceptofstatecontrolviapoleplacementisdesignedwhichexhibitsthedesiredeffects.Introducingaso-calledLuenbergerobserver,straightforwardmeasurementsofthemotionofthetelescopebaseandofthecontrolvariableoftheactuatoraresufficienttooperatethecontroller.Forrealtelescopicoperationsanadaptivecontrollerandobserverareintroduced.Thecontroller,developedforthereducedlinearsystemmodel,isappliedtothesignificantlymorecomplicatedsystemwithclearanceforstudyingtheinfluenceofclearanceonthevibrationsuppressionduringtelescopicmotions.

2.TestRig

AsseeninFigure1,theappearanceofgraduatedtelescopicmulti-sectionsystemswithclearancecanbeverydifferent.Neverthelessthemainproblemcanbeconcentratedtotheoverlappingareaofeachtwosections（seemarkingsinFigure1）.Onlytheorientationinspaceisdifferentasshowninthefunctionalsketchofthesystem（Figure2）.Thissketchleadsontheonehandtoanappropriatetestrigandontheotherhandtoanappropriatephysicalmodel.

Maininterest:

overlappingareaoftwosections.Difference:

orientationinspace.

Fig.2:

Sketchofthemainitem–theoverlappingarea

Thetestrig,asshowninFigure3consistsoftwosections,madeofslendersteel-beams.Thelowersectionisfixedtoarigidblockontheground;theuppersectionisconnectedtothelowersectionwithscrews.Forchoosinganassemblywithoutclearanceorwithawelldefinedclearance,aspecialsystemofslidingboltsandbushesisusedtoreducefrictiontoaminimumandtoenablefreetranslationandrotationmovement.Thistestrigismadeofelementswithaverysimplegeometry,becausetheelasticpropertiesoftheseelementsarewellknownandthemeasurementresultsarecomprehensible.

acc.tipofuppersectiontestrigwithmostsimplygeometryfor:

Stimul

stimulation

acc.tipoflowersection

€analysingdynamicbehaviour

€analysingcontrollerbehaviour

overlappingareawith/withoutclearance

Fig.3:

Testrig

Inthefollowing,twotypicalassembliesareshownintwomovies.Movie1showsthedynamicbehaviourofatwo-sectionalgraduatedsystemwithoutclearance.Ontheleftside,youcanseethewholetestrig,intheupperwindow,theoverlappingareacanbeseenandinthelowerwindow,thedevelopmentofthevibrationscanbeobserved,presentedbytheaccelerationsignalsofthesectionstips.Themovieisshowninslow-motion;thefundamentalfrequencyis0.53Hz.Asexpected,thevibrationdevelopmentisabsolutelyharmonic.

Mov.1:

Systemwithoutclearance

Movie2showsthedynamicbehaviourofatwo-sectionalgraduatedsystemwithclearance（2mm）.Ontheleftside,youcanseethewholetestrig,intheupperwindow,theoverlappingareacanbeseenandinthelowerwindow,thedevelopmentofthevibrationscanbeobserved,presentedbytheaccelerationsignalsofthesectionstips.Themovieisshowninslow-motion;thefundamentalfrequencyis0.53Hz.Asexpected,thevibration-developmentisnotlongerharmonic.Onlythecharacteristicvibrationofupperandlowersectionisinphasebalance.Eachtimethelowerendoftheuppersectionhitsthelowersection,highfrequentvibrations（e.g.6.35Hz）arestimulatedduetotheimpact.AsseeninMovie2,theaccelerationsignalofthehighfrequentvibrationsshowsanantiphasedevelopment.

Mov.2:

Systemwithclearance

3.PhysicalModell

Fromtheviewpointofmechanics,anon-linearmulti-fieldproblemofvibratingstructuralmemberswithvariablegeometryhastobeconsidered.Materialsurfaceareasofparticularcomponentsmovealongsurfaceareasofothercomponentsanddefinecomplicatedboundaryandtransitionconditions.Theclearanceproducesnon-lineareffects.InmanyapplicationsthedifferentsegmentsareslenderandcanbemodelledasBernoulli/Eulerbeamsmountedonarigidvehicleunitandcarryingatsomelocation,e.g.,attheendofthelastsection,aloadunitassumedtoberigid.Thevehicleunittogetherwiththefirstdeformablesegmentandalltheothersegments（oneofthemtogetherwiththeload）performtransversemotionsandtheextendingorretractingmotionofthesectionsissupplemented.Thecontactregionsbetweentwosectionsaremodelledasdiscretepointcontacts.Aspecialfeatureofthemodellingistointroducethereactionforcesatthecontactpointsintheformofdistributedlineloads（byusingDiracimpulsefunctions）,sothatforthecontactingsectionselementaryboundaryconditionsremain.Thecontactformulationitselftakesplaceviaone-sidedspring-damperelements.

TheprocedureisillustratedinFig.4aforatwo-sectiontelescopicbeamsystemmountedonarigidtraverse,performingatranslationalmotionaccompaniedbyanextendingmotionofthetwobeamsegmentswithdefinedclearancebetweenthem.Beam1isfixedatarigidvehicleunit;beam2carriesapointloadatitsend.ThevehicleisdrivenbyahorizontalforceFasexcitation.Thedeformationofthebeams（includingvehiclemassandload）isrepresentedbytheabsolute

displacements

w（x1,t）

andv（x2,t）.Themodelisdefinedbythefollowingparameters:

beam

lengthsl1,2,constantcross-sectionalareasA1,2,constantcross-sectionalmomentsofinertiaI1,2,

densityρandYoung’smodulusEofthetwoflexiblecomponents,massesofloadandvehicle

mLand

mT,respectively,andtelescopiclength

lA（t）.Thecontactbetweenthebeamsis

realised（seeFigure4b）viadiscretespring-dampersystemsintheformofaso-calleddisplacement

condition（notaforcecondition）[3],thegivennumbernofcontactpoints,theclearance

lS,

springstiffnessc,anddampingcoefficientd.ccanbeestimatedfromthegeometryandthematerialofthecontactpartnerswhereastheestimationofdismorecomplicated.Asthepurposeofthemodelisthecreationofacontrolconceptforvibrationsuppression,itisimportantthattheequationsofmotionstayassimpleaspossible.Inthecontrolledsystemtheclearanceplaystheroleofanexternaldisturbanceandasthecontrollerhastoworkforeverykindofcontact,averyaccurateestimationofdisnotnecessary.Intheaxialdirectionitisassumedthatthereisnofriction.Thisassumptionisjustifiableasthebearingbetweenthedifferentsegmentsisrealisedasrollerbearinginmanyapplications.Itisassumedherethattheforceflowleadsfromtheupperpartintothelowerpart.

body1

body2

spring-dampersystemc,d

Fig.4:

a）Systemmodelb）Contactformulation

4.Formulation

Boundaryvalueproblem

ApplyingHamilton’sprinciple

tt

1

δ⎰t（T-U）dt+⎰Wdt=0,

1

tvirt

（2.1）

00

thegoverningboundaryvalueproblemcanbederived.Tisthekineticenergy,UthepotentialenergyandWvirtthevirtualworkofforceswithoutpotentialoftheconsideredsystem.Thekineticenergyreads

1l1*2

1l2

*2,

2020

T=⎰ρA1wtdx1+⎰ρA2vtdx2

（2.2）

*

where

ρA1

=ρA1+mTδ（x1）

and

ρA*=ρA+mδ（x-l）

andthesymbol

δ（.）

22L22

representsDirac’sdelt