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Jounalofm atenalsprocesingtechnolgy63(197)81-86ExperimentandstudyintotheaxildriftingofthecylinderofaweldingrolerbedFengangshen,xidepan,jinxueWeldingresarchinstiute,xianjiaotnguniversity.Xian.shanxiprovince71049.P.R.chinaAbstractThebasictheoryoftheaxildriftingofthecylinderofaweldingrolerbedisintroducedinthepaer,andathesam etim eexperim ntandstudyonthem chanism oftheaxildriftingofthecylinderhavebendoneonaexperim ntalm odeloftheweldingrolerbed.Itishownthathem aincauseoftheaxildriftingofthecylinderliesinthexistenceofaspiralnglebetenthecylinderandthecylinderandtheroler.therelativeaxilm otinsbetwentherolerandthecylinderarecom pseofspiralm otin,elasticslidingandfrictionalsliding.Thetheoryofcom patiblem otinandno-com patiblem otinisputforwardfortheaxilm otinsofthecylinder.therlativeaxilm otinsofthecylinder.Therlativeaxilm otinbetwentherolersandthecylinderiscordinatedbyelasticslidingandfrictionalslidingbetwenthemKeywords:weldingrolerbed;cylinder;roler;axilm otin;spiralngle1IntroductionInweldingproduction,theasem blyandcirularseam weldingofrotaryworkpiecs,uchasboiler,apetrochem icalpresurevselandson,areconductedon;aeldingrolerbed.WhenrotaingOnaweldingrolerbed.Thecylindewilinevitablyproduceaxildriftingduetom anufacturing,asem blingtoleranceoftheweldingrolerbedandthecylindersurfaceiregularity(divergiufrom anidealrotaryorkpiec),thustheweldingprocedurem aynotbecariedoutsucesfuly.Itisnecsary,therfore,tostudythem chanism oftheaxildriftingofthecylindertoslvetheproblem oftheaxildriftingofthecylinderincirum ferntialwelding.Thersultsofthersearchwilbenfithestudyinganddesignigofantidriftingeldingrolerbed.especialytheanlysioftheapliedforcesonthebed,andleadtodetrm inigthem anufacturingandasem blingtoleranceofthebed,andprovidingthebasioftheoryforthem chanicaladjustingm odetoavoidaxildrifting,theadjustingm odeofclosedciruitinthecontrolciruit,andtheslectionftheadjustingvalue.2.Theoreticalanlysi2.1WeldingrolerbedandcylinderAweldingrolerbedisgenralycomposedofourolers.Drivenbythedrivngroler,thecylindermakesarotaryuniformotionarounditsaxis(howninFig.I),duringwhichthecircumferntialweldingprocedureiscariedoutInFig.1,aisthecntralngle,Sisthesuportingdistance,Listhespanoftheroler.andV,isthecircularlinearvelocityofthecylinder,alsonamedtheweldingvelocity2.Theaxisofthecylinderwilbenotparleltothatofaroleriftherolerisdeflctedbyacertainanglefrom thedealpositon,orifthecentrsofthefourrolerslieintheverticesofasim plequadrilateral,orifthecentrsofthefourrolersarenotonthesam eplane,oriftheciru-larityofthecylinderisiregularbecauseofdeviationinm aufacturingandasem bling. Thus.thecylinderwilnevitablym ovealongitsaxiswhenrotaingonabedthecontactofthecylinderandarolercanbecansiderdaspointcontactifcytindersaxisandrolersaxisdonotlieinthesam eplane.SuposePisthepointofcontact.thecylindersnorm al planeAisdefinedbytheplaneonwhicarethecylindersaxisandgenratrixnacrosthepointoftangencyonthecylinder(showninfig2)m akeacylinderstangentplaneBacrospointP.Thus,planeAisverticaltoplaneB.lcisacylinderstangentacrosPandliesinplaneB.Iristherolerstangentacrosthesam epointP,andliesinplaneBalso.Ingenral, isdefinedastheaxildeviationanglebetwentherol!ersaxisandthecylindersaxis; isdefinedasthespiralnglebetwengenratrixnandm .aprojectivelineobtainedbyprojectingtherolersgenratrixm acrospointponplaneBand isdefinedastheprojectiveanglebetwennandm ,aprojectivelineobtainedbyprojectingm onplaneA.Fig.3indicatesthattherela-tionshipam ongst thethreanglesistan =tan2 -tan2InFig.3,SB,S andS ,arecaledthespiraldisplace-m nt vector,theaxildeviationdisplacem nt vectorandtheprojectivedisplacem nt vectorrespectively.theirrelationshipbeing:Fig.2Geom tricrelationshipbetwenthecylinderandanindivdualrolerFig.3Relationshipbetwentheanglevectorandthedisplacem nt vector2.2relativeaxilm otinsrelationship(1)spiralm otin2.Fig.4Com pnet ofaxilvelocityBecausetherolersaxisisnotparleltothecylin-derscentralline,therisaspiralangle betwenVr,.andVc,onthepointofcontact(showninFig.2).Whentherolerandcylinderrotaesynchronisticalyaroundtheironaxes,drivenbytangentialfrictionalforce.aspiralefctwilocurbecauseofthediferntlinearvelocitydirectionbetwentherolerandthecylinderatpointPofcontactThecylinderhasacom pnet ofaxilvelocity,wherVcisthecirularlinearvelocityofthecylinder. isthecylindersaxilcom pnet velociryexrtedbysingleroler,andjcanbe1.2,3,4,represntigthefourrolers,respectively.(2)ElasticslidingBecauseoftheexistenceofaspiralangle,anaxilforceFajactsoncylinder.Whentheforceislesthanthem axim um axilfrictionalforcefNj(wherfisthefrictionfactor,andNjisthenorm al presurebetwenasinglerolerandthecylinder),thecylinderwilslideelasticalyovertheroleralongtheaxildirection23Thecom pnet oftheslidingvelocityis.whereistheelasticslidingfactorform etalicroler.=O.l0.5.(3)FrictionalslidingWhenajisgreatrthanthem axim um frictionalforcefNj,thecylinderwilm akeafrictionalslidingovertheroler.TheslidingresitanceisfNj3.Thecom pnet ofthefrictionalslidingvelocityonCylinderisVajthem agnitudeanddirectionofwhiccanbedetrm inedbytheuniversalrelationshipbetwenthecylinderandthefourrolersFrictionalslidingwilleadtotheearandtearofthesurfaceofthecylinderandtherolers.whicisunexpectdinweldingproductionWhenthecylinderdrifts,abovethrekindsofm otindonotocursim ultaneouslyIherforc.theaxildriftingvelocityofthecylinderisnotthealgebraicsum ofthethrecom pnetsofvelocityInthecaseofelasticsliding,theaxilvelocityis.2.3axilm otinofthecylinderonaweldingrolerbed2.3.1Axialcom patiblem otinUnderidealconditons,whenspiralangles jbetwenthecylinderandthefourrolersarealthesam e, thatis: 1= 2= 3= 4=thecylinderwilm oveithcom patiblespiralm otin.Twocategoriescanbeclasifedtoanlyzetheaxilm otinofthecylinder:(I)Whentherdoesnotexistanaxilcom pnet duetogravity.thecylindersaxildriftingvelocityis:Va=c*tan(2)Whentherexistanaxilcom pnetofgravityGatherexistanaxilforceonthecylinder.Now,theaxilforcesexrtedonthefourrolershavethesam edirectionalAndm agnitude,thevaluebeingequaltoGabesuesthecom pnetofspiralvetocity,therexistcom pnetofelasticonthecylinderthecylindersaxildriftingvelocityis2.3.2.axilno-com patiblem otinIngenral.spiralangles jbetwenthecylinderandthefourrolersarenotequaltoeachotherinsizeandirection.i.e.thegeom tricrelationshipsbetwenthecylinderandthefourrolersarealinconsitentTherfore,thecom pnetsofthecylindersaxilvelocityaginstfourrolers(i.eVc*ta j)arenotidenticaltoeachanother.Thecylinderwilm ovewithaxilnom patiblem otinTheaxilvelocitesofthecylinderaginsathefourrolersshouldbethesam ebecausethecylinderisconsiderdasarigdbodyasawholeandithasonlyoneaxilvelocity.Howevr.forsom eroler,Vc.tan jandthecylindersrealaxilvelocityarenotlikelytobethesam e,soanaxilfrictionalforcealm ost certainlyapearsbetwenthisrolerandthecylinderThefolwingtwocategoriescanbeclasifedtodiscustheno-com patibleaxilm otinofthecylinderacordingtoIhefrictionalforcesm agnitude:(I) Whentheaxilfrictionalforceserctedbyeachrolerandthecylinderareallesthanthem axim um axiltheactionofthecylinderaginstthefrictionalforcetheactionfthecylinderaginstherolersproduceselasticslidingTheaxilm otinbetwenaindivdualrolerandthecylinderiscordinatedbytheirelasticslidingwhentheaxilvelocityofthecylinderisconstant,thealgebraicsum ofcylindersaxilforceserctedbyfourrolersshouldbezeroifrheaxilcom pnet ofgravityisignored.i.e.andtherislitledifernceam ongst Nj,aginstthefourolers,sothattheycanbeaproxim atelyregardeasthesam e. Thus:acordingtotheabovetwoequations,theaxildriftingvelocityofthecylinderis.Wher0.25Tantrepresntstheintrinsicatributesofthewelding.Otherbedundertheconditonthatonlythecylinderaginstalrolsproduceslasticslidingthism aybecaledthespiralrateofthecylinderspiralm otin(2)Whentheaxilfrictionalforceerctedbysom erolerandthecylinderisgreatrthanthem axim um axilfrictionalforce,frictionalslidingocursbetwenthecylinderandthisrolerThen.them axim um axilforceisactingonthebearingoftheroler,itsvaluebeingFfm ax=fFNfm axBecauseofthesitenceofthisfrictionalsliding.theAxialm otinbetwenaindivdualrolerandthecylinderisnotcordinatedbytheirelasticslidingNowtheaxilno-com patiblem otinofthecylinderisdetrm inedbytherelativerelationshipsbetwenthecylinderandthefourrolers.Itisdificulttowriteagenralcom patibleequationofthecylindersaxildriftingvelocitybecausethiskindofconditonisverycom plex. Thefolwingisfurtheranlysianddiscusionoftheproblem Atfirst,foreaseinanlyzingproblem , thespiralangleaverageisdefinedasandtherlativespiralngleasArange,1intheorderfrom bigtosm l andthenfrom posirvetonegative,expresdas(j).then1234Sim larly, thenorm al forcebetwenthecylinderandarolercanbeexpresdasN(j).andtheaxilforceasFjfjIngenral,theaxilm otinofthecylinderdetrm inedbythespiralangleaverageisdefinelasthecom patiblecom pnet oftheaxilm otin, isvelocitybeingTheaxilm otinofthecylinderdetrm inedbytherelativespiralanglejisdefinedastheno-com patiblecom pnet ofaxilm otin, itsvelocitybeingexpresdasVanAnalysi showsthatVaisdetrm inedbythequilbriumconditonthefouroleraxilforceshenthecylinderm ovesalongaxildirectionataconstantvelocity.whernottakingintoacountofthefunctionofgravitysaxilcom pnet.Suposingthatthecylinderm akesano-com patiblecom pnet ofaxilm otinwiththem axim um relativespiralangle(I).itsvelocityisThenthefouraxilforcescanotbeinequlibrium .ieF1-(F2+F3+F4)0Becausetherislitledifernceam ongst fournorm al forces,thefouraxilfarcesarealsodetrm inedbynorm al forceandthefrictionfactoranyaxilforceundoubtedlybeinglesthanthesum oftheotherthreforces.Otherwise,ifthecylinderm akesano-com patiblecom pnetofaxilm otinwiththem inim um relativespiralangle(4).itsvelocityis.Va”=Vc*tan(4)Sim larly. fouraxilforcescannotbeinequilbrium also,i.e.:F(l)+F(2)+F(3)J-F(4)0Therfore,thecylindercanonlybeaproxim atelyconsiderdasm akingano-com patiblecom pnet ofaxilm otinwiththesecondorthirdrelativespiralangle,i.e.:Inwhatevrcaseasexpresdabove.whenthecylinderm akeano-com patiblecom pnet ofaxilm otin, thetworolershavingagreatrvelocityaredrivngrolers,andtheothertworolershavingalesrvelocityareresitantrolers,theequilbrium conditonofaxilforcesbeingoperative,i.e.:F(1)+F(2)=F(3)+F(4)Acordingtotheanlysiabove,andbecauseoftheunstabiltyoffrictionfactorfthatisafectdbythefactorsofload,m aterial, conditonofthecontactsurface,andcirum stance, theno-com patiblecom pnet Vaoftheaxilvelocityofthecylinderisundefined.Whenthecylinderm akesano-com patibleaxilm otin, itsaxilvelocityiscom psedofacom patiblecom pnetVa0andano-com patiblecom pnetVani.eVa=a0+anaa0VanThem ost optim al adjustm ent oftheaxilm otinistom aketheno-com patiblecom pnet assm al asposibleacordingtothestabiltyofadjustm ent anddecraseinaxilforce.Nom aterwhetrthecylinderm akescom patibleorno-com patiblem otin, suposingthatthecylinderisideal,itsaxilvelocityisalwaysexistentanddefinableforaparticularbed,itsm agnitudeandirectionreflctingthebedsinherntproerty.3.Experim nt3.1.Descriphm ofexperm ntTheexperim ntal m odel ishowninFig5.Experim ntswerdonetostudytwofactors:thespiralangleandthecylinderscirularlinearvelocity,whicafecttheaxildriftingofthecylinder.Intheexperim ntigproces.theaxildisplacem nt SaandtheaxildriftingvelocityVaofthecylinderwerm easuredbythevaritionofthetwofactorsdescribedabove.Them easuringm ethodisshowniFig.5,andiscariedoutbym eansofbringi anxialdisplacem nt sensorintocontactwithoneendofthecylinder.withthesensorbeingconectdtoanX-Yrecordertorecordthecylindersaxildisplacem nt evry5s.LinearlyregresingtheplotSa-t(texpresstim e),theaveragedriftingvelocityVa,tevrydeflctinganglecanbecalulated.Beforeexperim ntig. theexperim ntal m odel isinitalisedasfolws:first.theheightofthefourrolersisadustedbym eansofalevltoputthecentrsofthefourrolersinthesam ehorizontalplane,andatthefourvertxesoftherectangle.thentherolersaredeflctedsothattherotaingcylinderisattherelativequilbrium positon.Then.thecylinderdoesnotdriftoveralongtim e.orperiodicalydriftoveraverysm al axilrange3.2experim ntresultsandiscusion3.2.1Efectofspiralngle(I)Fig.6showsthatchangeofVawiththevaritionofThetestingconditonis:positverotalion,Vc=35m /hL=42m m , =60”TheVa-tn 4curveshowsthatVaisdirectlyprortionaltotan 4when 4isrelativelysm al (16c).Theslopeofthelinebeing3.06m m /s, Vaisnolongerdireclyprortionaltotan 4when 4,isgreatrthan6CThecurveisanarchedcurve.i.e.withtheincrem nt of 4,.Va,increase.butwiththeincrem nt ofVagradualybecom ingsm alet Becauseonlyonedrivenroler(rolerNo.4)isdeflcted,i.e 4canbechangedwhilsttheothersrem ainzero,thecylinderm akesano-com patiblem otin.When 4isrelativelysm al, Vaissm al also.Theaxilfrictionalforcesbetwenthecylinderandrolersarelesthanthem axim um axilfrictionalforce,andthecylinderproducesanelasticslidingaginstrolers.Axialm otinbetweneachrolerandthecylinderiscordinatedbyelasticsliding.thusVais:inthetheoreticalcurve,theslopeKcanbecalulatedbythefolwingequation:K=3.06m m /s intheexperim ntal curve.Thus,intakingacountoftheexperim ntal tolerance,thetwoslopescanbeconsiderdtobeaproxim atelyequal.When 4isrelativelylarge,theaxilfrictionalforcesbetwenthecylinderandtherolersarelargerthanthem axim um axilfrictionalForce,andcylinderproducesfrictionalslidingaginsttherolersBecauseofIheexistenceofslidingfrictionalresitance.Vaisnolongerlincartyincreasedwiththeincrem nt oftan 4Withtheincrem nt oftan 4theincrem nt ofVa;withgradualybecom esm aler(2)Thefolwingthreexperim ntswerarngedtostudythecylindersno-com patibleaxilm otinfurther,deflctingpositvelyoneroler.tworolersandthrerolersbythesam espiralangletom easurethrecurvesbetwenSaandvTheexperim ntal resultsareshowninFig7.Withtheincrem nt inthenum berofdeflctedrolers,Vabecom esgreatr.ieVa3Va2Va1Whenthenum berofdrivenrolersdeflctedisvaried,thedegreofthecylindersno-com patibleaxilm otinwilbechanged.Withtheincrem nt ofthenum ber oflolersdeflctedbythesam espiralangle.thecom patiblecom pnet becom esgreatr,buttheno-com patiblecom pnet becom essm aler.Inotherwords,thecylindersaxilm otinwil betransform edfromnocom patiblem otintocom patiblem otin. Thus,Vabecom esgreatralso,ultim ately, beingequaltothecom patibleaxilvelocitydetrm inedbythespiralangle Now.thefourrolershavethesam espiralanyle .SothatVais:3.2.efctofcirularlinearvelocityDeflctingdrivenrolerNo4toaspiralangleof+2”from theequilbriumpositon,thecylinderwilsuferaxildrifting,Fig.8showstheVa-ccurve,whiclaterindicatesthatVaisdirectlyprortionaltoVc,theslopeofthecurvebeingaproxim ately0.708because 4=+2istosm al, thecylinderdoesnotm akefrictionalslidingaginsteachroler.Thus,therelativeaxilm otinbetwentherolerandthecylinderiscom pletlycordinatedbytheirelasticsliding,sothatVaisI. e.VaisdirectlyprortionaltoVeForthetheoreticalCurvetheslopeK*canbecalulatedbythefolwingequationK”=0.25tan 4=0.25tan2=0.873wherK=0.708m m /s intheexperim ntal curve.Thus,intakingacountoftheexperim ntal tolerance,thetwoslopescanbeconsiderdtobeaproxim atelyequal.导室.4Conclusions1.Becauseofthedeviationsduetom anufacturingandasem bling. thecylinderscentrallineandtherolersaxisarenotparlel.i.e,theyarenotinthesam eplane,andtherisaspiralangle atthcpointofcontactbetwenthecylinderandtherolerinthecirularlinearvelocitydirection.Theexistenceof isthebasicreasonfortheocurenceofaxildrifting. heefctofgravityincylindersaxildirectionisalsooneofreasonsfordrifting.2.Therelativeaxilm otinsbetwenanindivdualrole