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英文原文AEnvelopeMethodofGearingFollowingStosic1998,screwcompressorrotorsaretreatedhereashelicalgearswithnonparallelandnonintersecting,orcrossedaxesaspresentedatFig.A.1.x01,y01andx02,y02arethepointcoordinatesattheendrotorsectioninthecoordinatesystemsfixedtothemainandgaterotors,asispresentedinFig.1.3.ΣistherotationanglearoundtheXaxes.Rotationoftherotorshaftisthenaturalrotormovementinitsbearings.Whilethemainrotorrotatesthroughangleθ,thegaterotorrotatesthroughangleτr1w/r2wθz2/z1θ,whererwandzarethepitchcircleradiiandnumberofrotorlobesrespectively.Inadditionwedefineexternalandinternalrotorradiir1er1wr1andr1ir1w−r0.ThedistancebetweentherotoraxesisCr1wr2w.pistherotorleadgivenforunitrotorrotationangle.Indices1and2relatetothemainandgaterotorrespectively.Fig.A.1.CoordinatesystemofhelicalgearswithnonparallelandnonintersectingAxesTheprocedurestartswithagiven,orgeneratingsurfacer1t,θforwhichameshing,orgeneratedsurfaceistobedetermined.Afamilyofsuchgeneratedsurfacesisgiveninparametricformbyr2t,θ,τ,wheretisaprofileparameterwhileθandτaremotionparameters.r1r1t,θx1,y1,z1x01cosθy01sinθ,x01sinθy01cosθ,p1θA,.10,,111tytxtr0,cossin,sincos0101011tytxtytxA.20,,0,,01010111xyyxrA.3cossin,sincos,,,,,1111122222zyzyCxzyxtrr202020202,sinsin,sincospyxyxA.42020202022222,sincos,sinsin,,pyxyxpxyrsincos,cossin,cossin121211CxpCxpypA.5Theenvelopeequation,whichdeterminesmeshingbetweenthesurfacesr1andr20222rrtrA.6togetherwithequationsforthesesurfaces,completesasystemofequations.Ifageneratingsurface1isdefinedbytheparametert,theenvelopemaybeusedtocalculateanotherparameterθ,nowafunctionoft,asameshingconditiontodefineageneratedsurface2,nowthefunctionofbothtandθ.Thecrossproductintheenvelopeequationrepresentsasurfacenormaland∂r2∂τistherelative,slidingvelocityoftwosinglepointsonthesurfaces1and2whichtogetherformthecommontangentialpointofcontactofthesetwosurfaces.Sincetheequalitytozeroofascalartripleproductisaninvariantpropertyundertheappliedcoordinatesystemandsincetherelativevelocitymaybeconcurrentlyrepresentedinbothcoordinatesystems,aconvenientformofthemeshingconditionisdefinedas0211111rrtrrrtr(A.7)InsertionofpreviousexpressionsintotheenvelopeconditiongivestyytxxppxC1111211cot0cot12111txCptyppA.8Thisisappliedheretoderivetheconditionofmeshingactionforcrossedhelicalgearsofuniformleadwithnonparallelandnonintersectingaxes.Themethodconstitutesageargenerationprocedurewhichisgenerallyapplicable.Itcanbeusedforsynthesispurposesofscrewcompressorrotors,whichareelectivelyhelicalgearswithparallelaxes.Formedtoolsforrotormanufacturingarecrossedhelicalgearsonnonparallelandnonintersectingaxeswithauniformlead,asinthecaseofhobbing,orwithnoleadasinformedmillingandgrinding.Templatesforrotorinspectionarethesameasplanarrotorhobs.Inallthesecasesthetoolaxesdonotintersecttherotoraxes.Accordinglythenotespresenttheapplicationoftheenvelopemethodtoproduceameshingconditionforcrossedhelicalgears.Thescrewrotorgearingisthengivenasanelementaryexampleofitsusewhileaprocedureforformingahobbingtoolisgivenasacomplexcase.TheshaftangleΣ,centredistanceC,andunitleadsoftwocrossedhelicalgears,p1andp2arenotinterdependent.Themeshingofcrossedhelicalgearsisstillpreservedbothgearrackshavethesamenormalcrosssectionprofile,andtherackhelixanglesarerelatedtotheshaftangleasΣψr1ψr2.Thisisachievedbytheimplicitshiftofthegearracksinthexdirectionforcingthemtoadjustaccordinglytotheappropriaterackhelixangles.Thiscertainlyincludesspecialcases,likethatofgearswhichmaybeorientatedsothattheshaftangleisequaltothesumofthegearhelixanglesΣψ1ψ2.FurthermoreacentredistancemaybeequaltothesumofthegearpitchradiiCr1r2.Pairsofcrossedhelicalgearsmaybewitheitherbothhelixanglesofthesamesignoreachofoppositesign,leftorrighthanded,dependingonthecombinationoftheirleadandshaftangleΣ.Themeshingconditioncanbesolvedonlybynumericalmethods.Forthegivenparametert,thecoordinatesx01andy01andtheirderivatives∂x01∂tand∂y01∂tareknown.Aguessedvalueofparameterθisthenusedtocalculatex1,y1,∂x1∂tand∂y1∂t.Arevisedvalueofθisthenderivedandtheprocedurerepeateduntilthedifferencebetweentwoconsecutivevaluesbecomessufficientlysmall.Forgiventransversecoordinatesandderivativesofgear1profile,θcanbeusedtocalculatethex1,y1,andz1coordinatesofitshelicoidsurfaces.Thegear2helicoidsurfacesmaythenbecalculated.Coordinatez2canthenbeusedtocalculateτandfinally,itstransverseprofilepointcoordinatesx2,y2canbeobtained.Anumberofcasescanbeidentifiedfromthisanalysis.iWhenΣ0,theequationmeetsthemeshingconditionofscrewmachinerotorsandalsohelicalgearswithparallelaxes.Forsuchacase,thegearhelixangleshavethesamevalue,butoppositesignandthegearratioip2/p1isnegative.Thesameequationmayalsobeappliedforthegenerationofarackformedfromgears.Additionallyitdescribestheformedplanarhob,frontmillingtoolandthetemplatecontrolinstrument.122AEnvelopeMethodofGearingiiIfadiscformedmillingorgrindingtoolisconsidered,itissuffcienttoplacep20.Thisisasingularcasewhentoolfreerotationdoesnotaffectthemeshingprocess.Therefore,areversetransformationcannotbeobtaineddirectly.iiiThefullscopeofthemeshingconditionisrequiredforthegenerationoftheprofileofaformedhobbingtool.Thisisthereforethemostcomplicatedtypeofgearwhichcanbegeneratedfromit.BReynoldsTransportTheoremFollowingHanjalic,1983,ReynoldsTransportTheoremdefinesachangeofvariableφinacontrolvolumeVlimitedbyareaAofwhichvectorthelocalnormalisdAandwhichtravelsatlocalspeedv.Thiscontrolvolumemay,butneednotnecessarilycoincidewithanengineeringorphysicalmaterialsystem.TherateofchangeofvariableφintimewithinthevolumeisvVdVttB.1Therefore,itmaybeconcludedthatthechangeofvariableφinthevolumeViscausedby–changeofthespecificvariablem/intimewithinthevolumebecauseofsourcesandsinksinthevolume,tdVwhichiscalledalocalchangeand–movementofthecontrolvolumewhichtakesanewspacewithvariableinitandleavesitsoldspace,causingachangeintimeofforρv.dAandwhichiscalledconvectivechangeThefirstcontributionmayberepresentedbyavolumeintegral.dVtVB.2whilethesecondcontributionmayberepresentedbyasurfaceintegralAdAVB.3ThereforeAVVVdAVdVtdVdtdtB.4whichisamathematicalrepresentationofReynoldsTransportTheorem.AppliedtoamaterialsystemcontainedwithinthecontrolvolumeVmwhichhassurfaceAmandvelocityvwhichisidenticaltothefluidvelocityw,ReynoldsTransportTheoremreadsdAWdVtddtdtAmVmVmVmVB.5IfthatcontrolvolumeischosenatoneinstanttocoincidewiththecontrolvolumeV,thevolumeintegralsareidenticalforVandVmandthesurfaceintegralsareidenticalforAandAm,however,thetimederivativesoftheseintegralsaredifferent,becausethecontrolvolumeswillnotcoincideinthenexttimeinterval.However,thereisatermwhichisidenticalforthebothtimes
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