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螺旋锥齿轮齿顶线数学建模方法（英文）.docx

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螺旋锥齿轮齿顶线数学建模方法（英文） MathematicalModelingApproachforHelicalConeGearToothTopLine Introduction Helicalgearhasbecomeoneofthemostimportantmechanismsforpowertransmission.Itiswidelyusedinvariousindustrialapplicationsduetoitsadvantagessuchashighefficiency,highloadcapacity,andlownoise.Thehelicalconegearisaspecialtypeofhelicalgearthatisusedtotransmitpowerbetweennon-intersectingshaftswithananglebetweenthem.Thehelicalconegearsarecommonlyusedinsteeringsystemsofvehiclesandmachines,andtheyarealsousedinaircraftengines.Thetoothtoplineisacriticalfeatureofagear,asitaffectsthegear’sperformance,loadcarryingcapacity,noise,andvibration.Therefore,itisimportanttounderstandthemathematicalmodelingapproachforhelicalconegeartoothtopline. MathematicalModelingApproachforHelicalConeGearToothTopLine Themathematicalmodelingapproachforhelicalconegeartoothtoplineinvolvesusingmathematicalequationstodeterminetheshapeandsizeofthegearteeth.Theshapeofthegearteethisdeterminedbytheinvolutecurve,whichisacurvemadebyapointonastringasitisunwoundfromacylinder.Theinvolutecurveisacommonlyusedcurveingeartoothdesignbecauseitensuresthatthegearsrolloneachothersmoothly.Thesizeofthegearteethisdeterminedbythemodule,whichistheratioofthepitchdiametertothenumberofteeth. Thegeartoothtoplineforhelicalconegearscanbeobtainedusingthefollowingmathematicalequations: -HelixAngle:Thehelixangleistheangleofthetoothtoplinewithrespecttotheaxialplaneofthegear.Itisgivenbytheequation: λ=tan-1(sinα/cosβ) whereαistheconeangle,andβisthepressureangle. -PitchDiameter:Thepitchdiameteristhediameterofthepitchcircle.Itisgivenbytheequation: D=mZ/cosα wheremisthemodule,andZisthenumberofteeth. -NormalDistance:Thenormaldistanceisthedistancebetweenthetoothtoplineandthepitchcircle.Itisgivenbytheequation: h=m(cosα/cosβ) -BaseCircleDiameter:Thebasecirclediameteristhediameterofthebasecircle.Itisgivenbytheequation: Db=Dcosα -ToothProfile:Thetoothprofileisdeterminedbytheinvolutecurve.Itisgivenbytheequation: r=(D/2+h)cosφ whereristhedistancefromthecenterofthegea