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Jounal of matenals processing technology 63 (1997) 881-886ExperimentExperimentExperimentExperiment andandandand studystudystudystudy intointointointo thethethethe axialaxialaxialaxial driftingdriftingdriftingdrifting ofofofof thethethethe cylindercylindercylindercylinderofofofof a a a a weldingweldingweldingwelding rollerbedrollerbedrollerbedrollerbedFenggang shen ,xide pan ,jin xueWeldingresearchinstiute,xianjiaotonguniversity.Xian.shaanxiprovince710049.P.R.chinaAbstractThe basic theory of the axial drifting of the cylinder of a welding roller bed is introduced in thepaper,and at the same time experiment and study on the mechanism of the axial drifting of thecylinder have been done on an experimental model of the welding roller bed . It is shown that themain cause of the axial drifting of the cylinder lies in the existence of a spiral angle between thecylinder and the cylinder and the roller . therelative axial motions between the roller and thecylinder are compose of spiral motion,elastic sliding and frictional sliding. The theory ofcompatible motion and non-compatible motion is put forward for the axial motions of thecylinder .the relative axial motions of the cylinder . The relative axial motion between the rollersand the cylinder is coordinated by elastic sliding and frictional sliding between themKeywords: welding roller bed; cylinder ; roller ; axial motion ; spiral angle1IntroductionIn welding production, the assembly and circular seam welding of rotary workpieces, such as aboiler, apetrochemical pressure vessel andso on,are conducted on ;a weldingrollerbed.When rotating On a welding roller bed. The cylinde will inevitably produce axial drifting due tomanufacturing, assembling tolerance of the welding roller bed and the cylinders surfaceirregularity (divergiugfrom anidealrotaryworkpiece),thusthe weldingproceduremay not be carried out successfully. It is necessary,therefore,to study the mechanismoftheaxialdriftingofthe cylinderto solve the problem of the axial driftingofthecylinder in circumferential welding. The results of the research will benefit the studyinganddesigning of antidrifting welding rollerbed. especially the analysis of the applied forces onthebed, and lead to determining the manufacturing and assemblingtoleranceofthebed,andproviding the basis of theory for the mechanical adjusting mode to avoid axialdrifting,theadjusting mode of closed circuitin the controlcircuit, and the selection oftheadjustingvalue.2.Theoretical analysis2.1. Welding roller bed and cylinderA welding roller bed is generally composed of four rollers. Driven by thedriving roller, the cylinder makes a rotary uniform motion arounditsaxis(showninFig.I), duringwhichthecircumferentialweldingprocedureis carriedout InFig.1,ais the central angle, S is the supporting distance, Lis the spanof theroller.and V, is the circular linear velocity ofthecylinder, alsonamed the welding velocity2.Theaxisofthecylinderwillbenotparallelto thatofa rolleriftherollerisdeflectedbyacertainanglefromthe dealposition,orifthecentersofthefourrollersliein theverticesofasimplequadrilateral,orifthecentersof thefourrollersarenotonthesameplane,orifthecircu- larityofthecylinderisirregularbecauseofdeviationin manufacturingandassembling.Thus.thecylinderwill nevitablymovealongitsaxiswhenrotatingonabed thecontactofthe cylinderanda rollercanbecansidered as pointcontactifcytindersaxisandrollersaxisdonot lieinthesameplane.SupposePisthepointofcontact. thecylindersnormalplaneAisdefinedbytheplaneon whicharethecylindersaxisandgeneratrix nacrossthe pointoftangencyon thecylinder(shownin fig2) makea cylinderstangent planeBacrosspointP.Thus,planeA isverticaltoplaneB. lcisa cylinders tangent across PandliesinplaneB.Iristherollerstangentacross thesamepointP,andliesinplaneBalso.Ingeneral, isdefinedastheaxialdeviationanglebetweentherol!ers axisandthecylindersaxis;isdefinedasthespiral anglebetweengeneratrixnandm. aprojectiveline obtainedbyprojectingtherollersgeneratrixmacrosspointponplaneBandisdefinedastheprojective anglebetweennandm, a projective line obtained byprojectingmonplaneA.Fig.3indicatesthattherela- tionshipamongstthethreeanglesis tan = tan2 -tan 2InFig.3,SB,S andS,arecalledthespiraldisplace- mentvector,theaxialdeviationdisplacementvectorand theprojectivedisplacementvectorrespectively.theirrelationshipbeing:Fig.2 Geometricrelationshipbetween the cylinder and anindividualrollerFig.3 Relationshipbetweentheanglevectorandthedisplacementvector2.2.2 relative axial motions relationship(1)spiral motion2.Fig.4 ComponentofaxialvelocityBecausetherollersaxisisnotparalleltothecylin- derscentralline,thereisa spiralanglebetweenVr,.and Vc,onthepointofcontact(showninFig.2).Whenthe rollerandcylinderrotatesynchronisticallyaroundtheir ownaxes,drivenbytangentialfrictionalforce.aspiral effectwilloccurbecauseofthedifferentlinearvelocity directionbetweentherollerandthecylinderatpointP ofcontactThecylinderhasacomponentofaxialvelocity,whereVcisthecircularlinearvelocityofthecylinder.is thecylindersaxialcomponentvelociryexertedbysingleroller,andjcanbe1.2,3,4,representingthefourrollers,respectively.(2)ElasticslidingBecauseoftheexistenceofaspiralangle,anaxial forceFajactsoncylinder.Whentheforceis lessthanthe maximumaxialfrictionalforcefNj(where f isthe frictionfactor,andNj isthenormalpressurebetweenasinglerollerandthecylinder),thecylinderwillslideelastically overtherolleralongtheaxialdirection23Thecomponentoftheslidingvelocityis.whereeistheelasticslidingfactorformetallicroller. e=O.OOl0.005.(3)FrictionalslidingWhenFajisgreaterthanthemaximumfrictionalforce fNj,thecylinderwillmakeafrictionalslidingoverthe roller.Theslidingresistanceis fNj3.ThecomponentofthefrictionalslidingvelocityonCylinderisVajthe magnitudeanddirectionofwhichcanbedeterminedby theuniversalrelationshipbetweenthecylinderandthefourrollersFrictionalslidingwillleadtothewearandtearofthesurfaceofthecylinderandtherollers.which is unexpectedinweldingproductionWhenthecylinderdrifts,abovethreekindsofmotiondonotoccursimultaneouslyIhereforc.theaxialdrifting velocityofthecylinderisnotthealgebraicsumofthethreecomponentsofvelocityInthecaseofelasticsliding,the axialvelocityis.2.3 axial motion of the cylinder on a welding roller bed2.3.1Axialcompatible motionUnderidealconditions,whenspiralanglesjbetween thecylinderandthefourrollersareallthesame,thatis:1=2=3=4=thecylinderwillmoveithcompatiblespiralmotion. Twocategoriescanbeclassifiedtoanalyzetheaxialmotionofthecylinder:(I)Whentheredoesnotexistanaxialcomponentdue togravity.thecylindersaxialdriftingvelocityis:Va=Vc * tan(2)Whenthereexistsanaxialcomponent of gravity Gathereexistsan axialforceonthecylinder.Now,theaxial forcesexertedonthefourrollershavethesamedirectionalAnd magnitude, the value being equal to Ga besues the component of spiral vetocity, there existcomponent of elastic on the cylinder the cylinders axial drifting velocity is2.3.2.axial non-compatible motionIngeneral.spiralanglesjbetweenthecylinderand thefourrollersarenotequaltoeachotherinsizeand direction.i .e.thegeometricrelationshipsbetweenthe cylinderandthefourrollersareallinconsistentTherefore,thecomponentsofthecylindersaxialvelocityagainstfourrollers(i.e Vc *taj) arenot identical to each another. The cylinderwillmovewith axial nompatible motionTheaxialvelocitiesofthecylinderagainsathefourrollersshouldbethesamebecausethecylinderisconsideredasarigidbodyasawholeandithasonlyone axial velocity.However.forsomeroller,Vc . tanj and thecylindersrealaxialvelocityarenotlikelytobethesame,soanaxialfrictionalforcealmostcertainlyappearsbetweenthisrollerandthecylinderThefollowingtwo categoriescanbeclassifiedtodiscussthenon-compatible axialmotionofthecylinderaccordingtoIhefrictionalforcesmagnitude:(I)Whentheaxialfrictionalforceserectedbyeach rollerandthecylinderareall less thanthemaximumaxial theactionofthecylinderagainstthefrictionalforcethe action of the cylinder against the rollersproduceselasticslidingTheaxialmotionbetween anindividualrollerandthecylinderiscoordinatedbytheirelasticslidingwhentheaxialvelocityofthecylinderisconstant,the algebraicsumofcylindersaxialforceserectedbyfourrollersshouldbezeroifrheaxialcomponentofgravityis ignored.i.e.andthereislittledifferenceamongstNj,againstthefour rollers,sothattheycanbeapproximatelyregardedas thesame.Thus:accordingtotheabovetwoequations,theaxialdriftingvelocityofthecylinderis.Where0.25Tant represents the intrinsic attributes of the welding. Other bed under thecondition that only the cylinder against all rolls produces elastic sliding this may be called thespiral rate of the cylinders spiral motion(2)Whentheaxialfrictionalforceerectedbysome rollerandthecylinderisgreaterthanthemaximumaxial frictionalforce,frictionalslidingoccursbetweenthe cylinderandthisrollerThen.themaximumaxialforceisactingonthebearingof theroller,itsvaluebeingFfmax=fFNfmaxBecauseof the esistenceofthisfrictionalsliding.the Axial motionbetween anindividual roller and the cylinderisnotcoordinatedbytheirelasticslidingNowtheaxialnon-compatiblemotionofthecylinderisdeterminedbytherelativerelationshipsbetweenthecylinderandthe fourrollers.Itisdifficulttowriteageneralcompatible equationofthecylindersaxialdriftingvelocitybecause thiskindofconditionisverycomplex.Thefollowingisfurtheranalysisanddiscussionoftheproblem Atfirst,foreaseinanalyzingproblem,thespiralangle averageisdefinedasand the relative spiral angle asArrange,1intheorderfrombigtosmllandthenfromposirivetonegative,expressedas(j).then1234Similarly,thenormalforcebetweenthecylinderanda rollercanbeexpressedasN(j).andtheaxialforceasFjfNjIngeneral,theaxialmotionofthecylinderdeterminedbythespiralangleaverage isdefinelasthe compatiblecomponentoftheaxialmotion,isvelocitybeingTheaxialmotionofthecylinderdeterminedbytherelativespiralanglejisdefinedasthenon-compatible componentofaxialmotion,itsvelocitybeingexpressed asVanAnalysisshowsthatVaisdeterminedbythe equilibriumconditionthe four roller axial forceswhenthecylindermovesalongaxialdirectionataconstantvelocity.wherenottakingintoaccountofthefunctionofgravitysaxialcomponent. Supposingthatthecylindermakesanon-compatiblecomponentofaxialmotionwiththemaximumrelativespiralangle(I).itsvelocityisThen the four axial forces can not be in equlibrium .i.eF1-(F2+F3+F4) 0Becausethereislittledifferenceamongstfournormalforces,thefouraxialfarcesarealsodeterminedbynormalforceand thefrictionfactoranyaxialforceundoubtedlybeinglessthanthesumof theotherthreeforces. Otherwise,ifthecylindermakesanon-compatiblecomponent ofaxialmotionwiththeminimumrelativespiralangle(4).its velocityis.Va”=Vc * tan(4)Similarly.fouraxialforcescannot be inequilibriumalso, i.e. :F(l)+ F(2)+ F(3) J- F(4) 0Therefore,thecylindercanonlybeapproximatelyconsideredasmakinganon-compatiblecomponentofaxial motionwiththesecondorthirdrelativespiralangle,i.e.:Inwhatevercaseasexpressedabove.whenthecylinder makeanon-compatiblecomponentofaxialmotion,the tworollers havingagreatervelocityaredrivingrollers, andtheothertworollershavingalesservelocityareresistantrollers,theequilibriumconditionofaxialforces beingoperative,i.e.:F(1)+ F(2)= F(3)+ F(4)Accordingtotheanalysisabove,andbecauseofthe unstabilityoffrictionfactorfthatisaffectedbythefactors ofload,material,conditionofthecontactsurface, andcircumstance,thenon-compatiblecomponentVaof theaxialvelocityofthecylinderis undefined. Whenthecylindermakesanon-compatibleaxialmotion,itsaxialvelocityiscomposedofacompatiblecomponentVa0andanon-compatiblecomponentVani.eVa=Va0+VanVa=Va0+VanThemostoptimaladjustmentoftheaxialmotionisto makethenon-compatiblecomponentassmallaspossible accordingtothestabilityofadjustmentanddecreaseinaxialforce. Nomatterwhetherthecylindermakescompatibleor non-compatiblemotion,supposingthatthecylinderis ideal,itsaxialvelocityisalwaysexistentanddefinable foraparticularbed,itsmagnitudeand directionreflectingthebedsinherentproperty.3.Experiment3.1.Descriphmof expermentTheexperimentalmodelis showninFig5.Experimentsweredonetostudytwofactors:thespiralangleandthe cylinderscircularlinearvelocity,whichaffecttheaxial driftingofthecylinder.Intheexperimentingprocess.the axialdisplacementSaandtheaxialdriftingvelocityVaofthecylinderweremeasuredbythevariationofthetwofactorsdescribedabove.Themeasuringmethodisshown in Fig.5,andiscarriedoutbymeansofbringingan axial displacementsensorintocontactwithoneendofthe cylinder.withthesensorbeingconnectedtoanX-Yrecordertorecordthecylindersaxialdisplacementevery5s.LinearlyregressingtheplotSa-t(texpressestime), theaveragedriftingvelocityVa,ateverydeflectinganglecanbe calculated.Beforeexperimenting.theexperimentalmodelisinitialisedasfollows:first.theheightofthefourrollersisadustedbymeansofa leveltoputthecentersofthefourrollersinthesamehorizontalplane,andatthefourvertexesoftherectangle.thentherollersaredeflectedso thattherotatingcylinderis attherelative equilibrium position.Then.thecylinderdoesnotdriftoveralong time.orperiodicallydriftovera verysmallaxialrange3.2experiment results and discussion3.2.1Effect of spiral angle(I)Fig.6showsthatchangeofVawiththevariationofThetestingconditionis:positiverotalion,Vc=35m/hL=422mm,=60”TheVa-tan4curveshowsthatVaisdirectlyproportionalto tan4when4is relativelysmall(16c ). Theslopeofthelinebeing3. 06 mm/s,Vaisnolongerdireclly proportionaltotan4when4,isgreaterthan6CThecurveisanarchedcurve.i. e . withthe incrementof4,.Va,increases.butwiththeincrementofVagraduallybecomingsmalletBecauseonlyonedrivenroller(rollerNo.4)isdeflected,i.e4 canbechangedwhilsttheothersremain zero,thecylindermakesa non-compatiblemotion. When4isrelativelysmall,Vaissmallalso.Theaxialfrictionalforcesbetweenthecylinderandrollersarelessthanthemaximumaxialfrictionalforce,andthecylinder producesanelasticslidingagainstrollers.Axialmotion betweeneachrollerandthecylinderiscoordinatedbyelasticsliding.thusVais:inthe theoretical curve, theslopeKcan becalculatedbythefollowingequation:K=3.06mm/sintheexperimentalcurve.Thus,in takingaccountoftheexperimentaltolerance,the two slopescanbeconsideredtobeapproximatelyequal.When 4isrelativelylarge,theaxialfrictionalforces betweenthecylinderandtherollersarelargerthanthe maximumaxialfrictionalForce,andcylinderproduces frictionalslidingagainsttherollersBecauseofIheexistenceofslidingfrictionalresistance.Vaisnolongerlincartyincreasedwiththeincrementoftan4Withtheincrementoftan4theincrementofV a;withgraduallybecomesmaller(2)Thefollowingthreeexperimentswerearrangedto studythecylindersnon-compatibleaxialmotionfurther, deflectingpositivelyoneroller.tworollersandthreerollersbythesamespiralangletomeasurethreecurvesbetweenSaandvTheexperimentalresultsareshownin Fig7.Withthe incrementinthenumberofdeflected rollers,Vabecomesgreater.i eVa3 Va2 Va1Whenthenumberofdrivenrollersdeflectedisvaried, thedegreeofthecylindersnon-compatibleaxialmotion willbechanged.Withtheincrementofthenumberof lollersdeflectedbythe samespiralangle.thecompatiblecomponentbecomesgreater,butthenon-compatible componentbecomessmaller.Inotherwords,thecylindersaxialmotionwillbetransformedfromnoncompatiblemotiontocompatiblemotion.Thus,Vabecomesgreateralso,ultimately,beingequaltothecompatibleaxialvelocitydeterminedbythespiralangle Now.thefourrollershavethesamespiralanyle. SothatVais:3.2.2 effect of circular linear velocityDeflectingdrivenrollerNo4toaspiralangleof+2”fromtheequilibriumposition,thecylinderwillsuffer axialdrifting,Fig.8showstheVa-Vccurve,whichlatter indicatesthatVaisdirectlyproportionaltoVc, theslope ofthecurvebeingapproximately0.00708because 4=+2istoosmall,thecylinderdoes not makefrictionalslidingagainsteachroller.Thus,therelativeaxialmotionbetweentherollerandthecylinderis completelycoordinatedbytheirelasticsliding,sothatVaisI. I. I. I.e .VaisdirectlyproportionaltoVeForthetheoretical Curve theslopeK *canbecalculatedbythefollowing equation K”=0.25tan 4=0.25tan2=0.00873 whereK=0.00708mm/sintheexperimentalcurve.Thus, intakingaccountoftheexperimentaltolerance,thetwo slopescanbeconsideredtobeapproximatelyequal.导室.4Conclusions1.Becauseofthedeviationsduetomanufacturingand assembling.thecylinderscentrallineandtherollers axisarenotparallel.i. e,theyarenotinthesame plane,andthereisaspiralangle atthcpointofcontactbetweenthecylinderandtherollerinthecircular linearvelocitydirection.Theexistenceofisthebasicreasonfortheoccurrenceofaxialdrifting.Theeffectof gravityincylindersaxialdirectionisalsoone ofrea