1、07FTM18Bevel Gear Modelby: T. KrenzerTECHNICAL PAPERAmerican Gear Manufacturers AssociationBevel Gear ModelTed KrenzerThe statements and opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.Abstra
2、ctThe paper presents a method for developing an accurate generic bevel gear model including both the facemilling and face hobbing processes. Starting with gear blank geometry, gear and pinion basic generatormachine settings are calculated. The contact pattern and rolling quality are specified and he
3、ld to the secondorder in terms of pattern length, contact bias and motion error. Based on the setup, a grid of tooth points arefoundincludingthetoothflank,filletand,ifitexists,theundercutarea.Itisproposedasthemodelforthenextgenerationofbevelgearstrengthcalculationsinthattheprocedureproducestruebevel
4、geargeometry,usesblank design parameters as input and is vendor independent except for cutter diameter.Copyright 2007American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314October, 2007ISBN: 978-1-55589-922-61Bevel Gear ModelTed KrenzerFor over a hundred ye
5、ars bevel gear tooth strengthhas been approximated using virtual spur gears.The calculation has adequately served the gear in-dustry. However with new high speed computers,strength estimatesshould bebased ona morereal-istic model. Recognizing the need for improvedbevel and hypoid strength calculatio
6、ns, finite ele-ment and boundary element strength programsbased on bevel gear geometry have been develo-ped. Most notable is Dr. Lowell Wilcoxs finite ele-ments stress TCA (tooth contact analysis). Inputtothese programs is the actual machine settings andcutter specifications as defined by vender cal
7、cula-tions. This is an excellent check on the strength ofdesigned sets. However the engineer must com-plete the entire design process and depend onvendersoftwareforinputdatabeforeatruestrengthanalysis can be made.In this paper a generic bevel gear tooth model thatcanbethenextgenerationgeartoothstren
8、gthmod-el is developed. Input is basic gear design parame-ters. Thereforecalculations canbe madeat thebe-ginning of design process. Output is basicgenerator machine settings that can be used to cal-culate a grid of pinionand gearpoints includingfilletand undercutpoints sothat finiteelement orbound-a
9、ryelementaswellasconventionalanalysis canbeapplied. The only vender input required is cutter di-ameter, blade edge radius and number of bladegroups. Eitherthefacemilling(FM)orfacehobbing(FH) generating method can be used.Thesettingsproducespecifiedtoothsurfacestothesecond order. Procedures for calcu
10、lating toothsur-facepointsaswellasfilletpointsareincluded. Sep-arate pinion setups for the drive and coast sides,rather than completing setups, are calculated. Thisis partly done so as not to provide generation meth-ods that compete with vender calculations. Due tolength constraints not all formulas
11、 can be included.A complete set of formulas can be found in theauthors book titled The Bevel Gear.Basic generatorThe basic generator is configured with a machinebase that carries a generating head and a workhead mounted on its horizontal face with the offsetperpendicular to the horizontal face.The g
12、enerating head has a cradle that carries theface mill cutter whose axis is offset from the cradleaxis by an adjustable radial distance, S, set at anangle, q, from the horizontal direction. The cutter,which has a radius, rcP,and a pressure angle, b,can have its axis tilted by an angle, i, relative to
13、 thecradle axis. The direction of the cutter tilt angle isreferenced relative to a perpendicular to the radialdistance by the swivel angle, j. The cutter phaseangle, G, about the cutter axis (not shown) is fromthe direction of tilt to the contact point.The work head holds the work in a spindle with
14、itsaxis the in horizontal plane. The work head is ro-tated by the angle, m, about the offset line that in-tersects thecradle axis. Thework isadjusted adis-tance, Xp, along its axis and a distance, Em,intheoffset direction. The work head also is adjusted adistance, XB, along the cradle axis.A timed r
15、oll relationship between the cradle andwork, Ra,is set. For face hobbing an additionaltimed roll relationship between the cutter and theworkisadded. The basicgenerator configurationisshown in Figure 1.Figure 1. Basic generator configuration2Generating processesWith face milling, the generator is set
16、up so that thecutter produces the desired spiral and pressureangles and the cutter blade tips follow the root line.Cutter blades rotate through the work piece, pro-ducing a slot as the generator and work rotatetogethertogeneratethetoothsurface. Thecutteriswithdrawn, the work piece is indexed and cyc
17、le isrepeated. See Figure 2.Figure 2. Face milling cradle/cutter setupWith face hobbing, the generator is setup to followthe root line and produce the desired spiral andpressure angles taking into account the continuousindexing motion. Cutter blades are arranged ingroups with the work indexing one t
18、ooth as eachblade group passes through the cut. The indexingmotion is superimposed onthe generatingprocess.All teeth are formed in a continuous cycle. Thelengthwise tooth form is a kinematic curve. SeeFigure 3.Figure 3. Face hobbing cradle/cutter setupGenerator vector modelThecoordinatesystemisdefin
19、edwiththeivectorinthe machine horizontal plane pointing to the right.Thejvectorisalongthemachineverticalaxispoint-ingup. Thekvectorpointsoutinthedirectionofthecradle axis. Figures 4 and 5 show a vector setup ofthe generating machine as defined below.Unit vector along the cradle axisg = (0, 0, 1)Unit
20、 vector in the offset directione = (0, 1, 0)Unit vector along the pinion axisp = cosm,0, sin mVector from machine center to cutter centerS = S (cosq, sin q,0)Unit vector along the cutter axisc = ( sinisinj, sinicosj, cosi)Cutter radial unit vectorr = ( cos i sinj, cosi cosj, sini)Unit vector along t
21、he cutting elementt =sinb isinj, sinb icosj,cosb iFigure 4. View looking along cradle axisFigure 5. View looking down on generator3Assuming some distance s along the cutting ele-ment tfrom the tip of thecutter toa pointon thecut-tingelement,thepositionvectorAfromthemachinecenter to the point isA = S
22、 + rcpr stThe position vector R from the crossing point to thepoint isR = A + Eme + Xpp XBgInput dataGear design:Pinion GearNumber of teeth n NPitch angles Mean cone distance ASpiral angle Pressure angle (drive/coast) 1/2Face width FMean dedendum bPbGDedendum angles PGClearance cMean gear slot width
23、 WGBacklash BLCutter specifications:Cutter radius rcNumber of blade groups (FH) nbCutter edge radius rePreGContact parameters (drive/coast):Pattern length factor Bd/BcContact bias angle d/cMotion error Gd/GcAnyone unfamiliar with the above terms is referredto ANSI/AGMA 2005-D03, Design Manual forBev
24、el Gears.Control factorsA portion of a gear tooth painted with marking com-poundiswipedcleanasagearisrolledwithitsmateunder light load. The pattern length factor specifiesthat portion.Althoughthecontactpatternappearstobelinecon-tact, the surfaces are mismatched which results inpoint contact. The pat
25、h of contact is the path thatpoints of contact make as the teeth role throughmesh. The input bias angle as shown in Figure 6 isdefined as the angle the path of contact makes witha perpendicular to the gear root line.Figure 6. Bias inMotion error is specified as the angular retardationof the gear in
26、micro-radians at the point wherecon-tact is transferred from one tooth to the next. Thedesired motion error is parabolic in form as seen inFigure 7 and is expressed in the formula:cp=2G10 6N22Figure 7. Motion errorEffective cutter point widthThe symbol, We, represents the effective gear cut-ter poin
27、t width and is defined as the outside gearbladepointradiusminustheinsidepointradius. Forface milling the effective point width is the gearmean slot width.We= WGIn face hobbing a half pitch of work indexing occursbetween the time when an inside blade enters thecut and an outside blade enters. Blade p
28、oint radiiare adjusted to take the half pitch indexing intoaccount.We= WG AsincosNFigure8showstheconfigurationofbothfacemillingand face hobbing cutters.4Figure 8. Face milling and face hobbingcutter configurationGear generation setupInitial calculationsdefine themean calculationpointonthe gearand th
29、egear rootcone distance. Asim-ple machine setup is used to generate the gearmember. The machine root angle is set to the gearroot angle. Blade angles are equal to the desiredpressure angles. The setup is based on a point atthe nominal cutter radius which is generally not apoint on the tooth surface
30、but gives a starting point.Process differences are taken into account. Forface milling the cutter is referenced at the point ra-dius and for face hobbing the cutteris referencedatmean height. With the face hobbing process, con-tinuousindexingissuperimposedonthegeneratingmotion. Theaddedindexingmotio
31、nmustsatisfythebasic law of gearing.Basic gear machine settingsMachine root anglem= RSliding baseXBG= bacosGVertical distanceV = rccos Horizontal distanceH = AR rcsin Cradle angletan qG=VHRadial distanceSG=Vsin qGHead setting XG=A sinG bocosGsinRRatio of roll RaG=ARRGmCutter tilt angleiG= 0Swivel an
32、glejG= 0Cutter phase angleG= 0Wherebaistheaveragededendum;isthespiralanglechangeduetothefacehobbingcontinuousin-dexing;boisthe distancefrom thegear pitchpointtothemeanpoint;ARisthegearrootconedistance;RGmis the gear radius at the mean point.Point on gear tooth surfaceUsing basic gear settings the ve
33、ctor gear model issetupsimilartothepinionmodel shownin Figures4and 5 for a point at the nominal cutter radius. Theactual mean point on the generated gear surface isfoundbyiterating onthe cradleangle position,qG,the cutter phase angle, G, and the distance alongthe cutting edge, sG.Second order gear s
34、urfaceVectors g, along the gear axis, gG,along the cradleaxis, RG, from crossing point to mean point, AG,frommachinecentertomeanpoint,nG,unitcontactnormal and tG, unit along cutter element, are usedtocalculatetherelativelinearandangularvelocitiesand accelerations between the generator and thegear.Wi
35、th the relative motion and the second order sur-face of the generating gear, the gear surface is de-fined using the formulas from the paper titled, Sec-ond Order Surface Generation, by Meriwether L.Baxter. The surface is stored as the matrixSG= aGbGbGcGwhere aG,bGand cGare the coefficients of thesec
36、-ond order surface defined by the formula:z =12ax2+ bxy +12cy2Fivequantitiesoneachthedriveandcoastsides,g,RG,nG,tG,(SG)as seen inFigure 9define thegearteeth.5Figure 9. Drive and coast side definitionDesired pinionThe pinion is placed in contact with the gear at themean point and the unit vector p, a
37、long the pinionaxis and the pinion position vector RP, are defined.The desired relative motion between the gear andpinion is calculated with the retardation term addedto the acceleration. With the relative motion be-tween the gear and the pinion andthe secondordersurfaceof thegear, thelengthwise con
38、jugatepinionsurface is defined as the surface matrix:SPo= aPobPobPocPoThe pinion surface is rotated about the contact nor-mal to give the desired path of contact on the gearsurface as defined by din the input.SP1= aPdbPdbPdcPDThe direction of the path of contact is defined withthe vectoru1= cosPtG+
39、sinPtG nGPinion generationDiamond, a third order contact characteristic, canbe a problem with the face milling process. Sinceonly second order surfaces are being held, a bladeangle change, , to control diamond is addressedwith the empirical formula. See Figure 10.FM = 0.75P+ GFH = 0Figure 10. Diamon
40、d control blade anglechangeThe normal cutter radius as calculated for the gearis changed to produce the desired pattern length.Using the iterative procedure, described below, thegenerating pitch angle and thegenerating conedis-tance are assumed and systematically changed toachieve the desired contac
41、t bias and motion error.Cradle,cutterandworkarepositionedbasedonthemachine root angle, generating cone distance,blade angles and contact normal. As setup paralleldepthteethwouldbecut. Generallyface millingde-signs are not parallel depth, so a second iterationloopis insertedto rotatethe cutterabout t
42、henormalchanging the position of the cutter axis and cutterelement until the desired root line is generated.Within this second loop the direction of the cutterpath at the deepest penetration is determined in athird loop by iteration on the cradle position.The offset direction is set perpendicular to
43、 both thegenerating gear axis and the pinion axis. The gen-erating gear position vector to the mean point isbasedonthepinionpositionvectorandthegenerat-ingconedistance. Theratioofrollbetweenthegen-erating gear and the pinion is determined by the re-quirement that the velocity in thenormal directioni
44、szero for a point of contact.6The relative motion between the pinion generatorand the pinion is now calculated. With the relativemotion and the generating gear surface, the piniontoothsurfaceisdefinedinthepinioncoordinatesys-tem.SP0= aP0bP0bP0cP0Since the desired pinion surface is in the gear coor-d
45、inatesystem,thepinionsurfaceisrotatedintothatsystem.SP1= aP1bP1bP1cP1Then the difference surfacebetween thegeneratedand desired surfaces is:1= aP1 aPD2= bP1 bPD3= cP1 cPDThevaluesofthemachinerootangleandthegener-ating cone distance are changed and the formulaswithin the loop are recalculated until 2
46、= 3=0.Pinion machine settingsTaking into account differences between the facemilling and face hobbing processes, vectors fromthe cutter center to the cutter tips and from thecradle center to the cutter center are calculated.The following are now known and define the pinionsetup:APoposition vector fr
47、om cradle center to meanpoint,RPposition vector from crossing point to meanpoint,SPposition vector from cradle center to cuttercenter,Rcposition vector from cutter axis to cutter tips,p unit vector along the gear axis,gPunit vector along the cradle axis,cP2unit vector along cutter axis,e unit vector
48、 in offset direction.The setup is converted into basic machine settingswherethemachinecenterisreferencedtothecuttercenter in the plane of the cutter tips. The angle be-tweentheperpendiculartogeneratinggearaxisandthe pinion axis in the plane of the generating gearaxis and the pinion axis is the machi
49、ne root angle,m:sinm= gP pThedifferencevectorbetweenAPoandRPiswrittenin terms ofmachine vectorswith unknownmagnitu-des. See Figure 11.APo RP= XPp + XBP1gP EmeFigure 11.The dot product with the offset gives Em.Em=APo RP eTwo linear equations result from dotting the differ-ence equation with the vectors p and gP.APo RP p = XP+ XBP1gp pAPo RP gP= XPp gP+ XBP1They are solved simultaneously for XPand XBP1:XP=APo RPp + gPsinm
