英文原文AEnvelopeMethodofGearingFollowingStosic1998,screwcompressorrotorsaretreatedhereashelicalgearswithnonparallelandnonintersecting,orcrossedaxesaspresentedatFig.A.1.x01,y01andx02,y02arethepointcoordinatesattheendrotorsectioninthecoordinatesystemsfixedtothemainandgaterotors,asispresentedinFig.1.3.istherotationanglearoundtheXaxes.Rotationoftherotorshaftisthenaturalrotormovementinitsbearings.Whilethemainrotorrotatesthroughangle,thegaterotorrotatesthroughangle=r1w/r2w=z2/z1,whererwandzarethepitchcircleradiiandnumberofrotorlobesrespectively.Inadditionwedefineexternalandinternalrotorradii:r1e=r1w+r1andr1i=r1wr0.ThedistancebetweentherotoraxesisC=r1w+r2w.pistherotorleadgivenforunitrotorrotationangle.Indices1and2relatetothemainandgaterotorrespectively.Fig.A.1.CoordinatesystemofhelicalgearswithnonparallelandnonintersectingAxesTheprocedurestartswithagiven,orgeneratingsurfacer1(t,)forwhichameshing,orgeneratedsurfaceistobedetermined.Afamilyofsuchgener-atedsurfacesisgiveninparametricformby:r2(t,),wheretisaproleparameterwhileandaremotionparameters.r1=r1(t,)=x1,y1,z1=x01cos-y01sin,x01sin+y01cos,p1(A,.1)0,111tytxtr=0,cossin,sincos0101011tytxtytx(A.2)0,0,01010111xyyxr(A.3)cossin,sincos,),(1111122222zyzyCxzyxtrr202020202,sinsin,sincospyxyx(A.4)2020202022222,sincos,sinsin,pyxyxpxyrsin)(cos,cos)(sin,cossin121211CxpCxpyp(A.5)Theenvelopeequation,whichdeterminesmeshingbetweenthesurfacesr1andr2:0222rrtr(A.6)togetherwithequationsforthesesurfaces,completesasystemofequations.Ifageneratingsurface1isdenedbytheparametert,theenvelopemaybeusedtocalculateanotherparameter,nowafunctionoft,asameshingconditiontodefineageneratedsurface2,nowthefunctionofbothtand.Thecrossproductintheenvelopeequationrepresentsasurfacenormalandr2istherelative,slidingvelocityoftwosinglepointsonthesurfaces1and2whichtogetherformthecommontangentialpointofcontactofthesetwosurfaces.Sincetheequalitytozeroofascalartripleproductisaninvariantpropertyundertheappliedcoordinatesystemandsincetherelativevelocitymaybeconcurrentlyrepresentedinbothcoordinatesystems,aconvenientformofthemeshingconditionisdenedas:0211111rrtrrrtr（A.7）Insertionofpreviousexpressionsintotheenvelopeconditiongives:tyytxxppxC1111211cot)(0)cot(12111txCptypp(A.8)Thisisappliedheretoderivetheconditionofmeshingactionforcrossedhelicalgearsofuniformleadwithnonparallelandnonintersectingaxes.Themethodconstitutesageargenerationprocedurewhichisgenerallyapplicable.Itcanbeusedforsynthesispurposesofscrewcompressorrotors,whichareelectivelyhelicalgearswithparallelaxes.Formedtoolsforrotormanufacturingarecrossedhelicalgearsonnonparallelandnonintersectingaxeswithauniformlead,asinthecaseofhobbing,orwithnoleadasinformedmillingandgrinding.Templatesforrotorinspectionarethesameasplanarrotorhobs.Inallthesecasesthetoolaxesdonotintersecttherotoraxes.Accordinglythenotespresenttheapplicationoftheenvelopemethodtoproduceameshingconditionforcrossedhelicalgears.Thescrewrotorgearingisthengivenasanelementaryexampleofitsusewhileaprocedureforformingahobbingtoolisgivenasacomplexcase.Theshaftangle,centredistanceC,andunitleadsoftwocrossedhelicalgears,p1andp2arenotinterdependent.Themeshingofcrossedhelicalgearsisstillpreserved:bothgearrackshavethesamenormalcrosssectionprole,andtherackhelixanglesarerelatedtotheshaftangleas=r1+r2.Thisisachievedbytheimplicitshiftofthegearracksinthexdirectionforcingthemtoadjustaccordinglytotheappropriaterackhelixangles.Thiscertainlyincludesspecialcases,likethatofgearswhichmaybeorientatedsothattheshaftangleisequaltothesumofthegearhelixangles:=1+2.Furthermoreacentredistancemaybeequaltothesumofthegearpitchradii:C=r1+r2.Pairsofcrossedhelicalgearsmaybewitheitherbothhelixanglesofthesamesignoreachofoppositesign,leftorrighthanded,dependingonthecombinationoftheirleadandshaftangle.Themeshingconditioncanbesolvedonlybynumericalmethods.Forthegivenparametert,thecoordinatesx01andy01andtheirderivativesx01tandy01tareknown.Aguessedvalueofparameteristhenusedtocalculatex1,y1,x1tandy1t.Arevisedvalueofisthenderivedandtheprocedurerepeateduntilthedifferencebetweentwoconsecutivevaluesbecomessufficientlysmall.Forgiventransversecoordinatesandderivativesofgear1prole,canbeusedtocalculatethex1,y1,andz1coordinatesofitshelicoidsurfaces.Thegear2helicoidsurfacesmaythenbecalculated.Coordinatez2canthenbeusedtocalculateandnally,itstransverseprolepointcoordinatesx2,y2canbeobtained.Anumberofcasescanbeidentiedfromthisanalysis.(i)When=0,theequationmeetsthemeshingconditionofscrewmachinerotorsandalsohelicalgearswithparallelaxes.Forsuchacase,thegearhelixangleshavethesamevalue,butoppositesignandthegearratioi=p2/p1isnegative.Thesameequationmayalsobeappliedforthegen-erationofarackformedfromgears.Additionallyitdescribestheformedplanarhob,frontmillingtoolandthetemplatecontrolinstrument.122AEnvelopeMethodofGearing(ii)Ifadiscformedmillingorgrindingtoolisconsidered,itissuffcienttoplacep2=0.Thisisasingularcasewhentoolfreerotationdoesnotaffectthemeshingprocess.Therefore,areversetransformationcannotbeobtaineddirectly.(iii)Thefullscopeofthemeshingconditionisrequiredforthegenerationoftheproleofaformedhobbingtool.Thisisthereforethemostcompli-catedtypeofgearwhichcanbegeneratedfromit.BReynoldsTransportTheoremFollowingHanjalic,1983,ReynoldsTransportTheoremdenesachangeofvariableinacontrolvolumeVlimitedbyareaAofwhichvectorthelocalnormalisdAandwhichtravelsatlocalspeedv.Thiscontrolvolumemay,butneednotnecessarilycoincidewithanengineeringorphysicalmaterialsystem.Therateofchangeofvariableintimewithinthevolumeis:vVdVtt(B.1)Therefore,itmaybeconcludedthatthechangeofvariableinthevolumeViscausedby:changeofthespecicvariablem/intimewithinthevolumebecauseofsources(andsinks)inthevolume,tdVwhichiscalledalocalchangeandmovementofthecontrolvolumewhichtakesanewspacewithvariableinitandleavesitsoldspace,causingachangeintimeofforv.dAandwhichiscalledconvectivechangeTherstcontributionmayberepresentedbyavolumeintegral:.dVtV(B.2)whilethesec