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.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
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- 关 键 词:
- 结构件 间隙 伸缩 系统 建模 振动 抑制
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