1、11FTM10AGMA Technical PaperNew Methods for theCalculation of the LoadCapacity of Bevel andHypoid GearsBy B.-R. Hhn, K. Stahl, andC. Wirth, Gear Research Centre(FZG)New Methods for the Calculation of the Load Capacity ofBevel and Hypoid GearsProf.Dr.-Ing.Bernd-RobertHhn,Prof. Dr.-Ing.Karsten Stahl,an
2、d Dr.-Ing.ChristianWirth, Gear Research Centre (FZG)The 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.AbstractHypoid gears are bevel gears with non-intersecting axes where th
3、e hypoid offset is defined as the shortestdistance between these two axes. Hypoid gears are preferred over bevel gears without offset if aspects ofgearnoiseorofinstallationspaceareinfocus. Pittingandtoothrootbreakagearestillthetwomostfrequentfailure types occurring in practical applications of bevel
4、 and hypoid gears. There are several national andinternationalstandardsforthecalculationoftheloadcapacityofthesegearssuchasDIN3991,AGMA2003and ISO 10300. Butup to now these standards do not cover bevel gearswith offset(hypoid gears). For thisreasonaresearchprojectwascarriedoutatFZG(GearResearchCentr
5、e,Munich,Germany)toanalyzetheinfluence of the hypoid offset on the load capacity. A new calculation method should be developed that is inprinciple based on the current version of ISO 10300 but also valid for hypoid gears.Althoughtheloadcapacityofbevelgearshasbeeninvestigatedinseveralresearchprojects
6、attheFZG(Paul,Vollhter, Fresen), the isolated influence of the hypoid offset on pitting and bending could not be evaluatedreliably,becausethetypeoffailurewaschangingonthecorrespondingtestgearsfromtoothrootbreakagetopittingwithanincreasingoffset. Thus,neitherbendingnorpittingcouldbeinvestigatedisolat
7、edregardingtheinfluence of the hypoid offset. So, the main target of this project was the systematic investigation of theinfluence of the hypoid offset on the pitting and bending load capacity by means of two different types of testgearsthatfaileitherwithpittingortoothrootbreakageoverthewholeregarde
8、doffsetrange. Additionallytheformer test results were also taken into account to evaluate the new calculation method.Copyright 2011American Gear Manufacturers Association1001 N. Fairfax Street, 5thFloorAlexandria, Virginia 22314October 2011ISBN: 978-1-61481-009-43 11FTM10New Methods for the Calculat
9、ion of the Load Capacity of Bevel and Hypoid GearsProf. Dr.-Ing. Bernd-Robert Hhn, Prof. Dr.-Ing. Karsten Stahl,and Dr.-Ing. Christian Wirth, Gear Research Centre (FZG)IntroductionHypoid gears are bevel gears with non-intersecting axes where the hypoid offset is defined as the shortestdistance betwe
10、en these two axes. Hypoid gears are preferred over bevel gears without offset if aspects ofgear noiseor of installationspaceareinfocus. Pittingandtoothroot breakagearestillthetwomostfrequentfailure types occurring in practical applications of bevel and hypoid gears. There are several national andint
11、ernational standards for the calculation of the load capacity of these gears such as DIN 3991 1, ANSI/AGMA 2003-B972 andISO 103003. But upto nowthese standards do not cover bevelgears withoffset(hypoid gears). For this reason a research project was carried out at FZG (Gear Research Centre, Munich,Ge
12、rmany)toanalysetheinfluenceofthehypoidoffsetontheloadcapacity. Anewcalculationmethodshouldbedevelopedthat is inprinciplebasedonthecurrentversionofISO 103003 butalsovalidfor hypoidgears.AlthoughtheloadcapacityofbevelgearshasbeeninvestigatedinseveralresearchprojectsattheFZG(Paul4, Vollhter 5, Fresen 6
13、), the isolated influence of the hypoid offset on pitting and bending could not beevaluated reliably, because the type of failure was changing on the corresponding test gears from tooth rootbreakagetopittingwithan increasingoffset. Thus, neither bending nor pitting couldbe investigatedisolatedregard
14、ing the influence of the hypoidoffset. So, themain target of this project was the systematic investiga-tion of the influence of the hypoid offset on the pitting and bending load capacity by means of two differenttypes of test gears that fail either with pitting or tooth root breakage over the whole
15、regarded offset range.Additionally the former test results were also taken into account to evaluate the new calculation method.Experimental testsFor theexperimentalinvestigations twotypes of bevelgears weredesigned, onefor thepittingtests andonefor the tooth root tests. The aim of the two different
16、gear designs was to examine pitting isolated from toothrootbreakage. Thus,thegearsforthetoothroottestsweredesignedwithoffsetsa=0/15mm/31.75mmwitha relatively small module (mmn=2,22,5 mm), the gears for the pitting tests with offsetsa = 0/15 mm/31,75 mm/44 mm with abigger module(mmn=3,54,2 mm). Allwh
17、eels hadthe sameouterdiameter de2= 170 mm. The gear sets were made of 18CrNiMo7-6 case hardened and finish ground. Foreach geometry the S/N-curve was determined by approximately 20 tests.Results of the pitting testsOneachexaminedvariantpittingoccurredisolatedfromtoothrootbreakage. Besidethepittingfa
18、iluresmicropitting appeared. With increasing offset, the micro pitting area on the flank grew faster and bigger. Figure 1shows typical pitting on the pinion flanks for all four examined test gear geometries. In Figure 2 (left side) apinion flank of the geometry variant with a = 15 mm is shown after
19、11 and 27 million pinion revolutions atT1= 300 Nm.Itwasproventhattheflankformdeviationwhichoccursduetomicropittingaffectstheloaddistributionontheflankduringthelifetime. Thisleadstolocallychangingloadconditionsandthusinfluencespitting. Intestwithtorques close to the endurance limit of the gear set mi
20、cro pitting had a larger influence because of the largerun-time.Atthegeometryvariantwitha=0mm(nonhypoid)micropittingoccurredmainlyatthededendumoftheflank,whereasforthehypoidvariants(a=15mm/a=31.75mm/a=44mm)micropittingcouldbedocumentedoverthe whole active flank. The same applies to pitting.4 11FTM10
21、Figure 1. Typical pitting at the pinion flanks on the investigated test gearsFigure 2. Micropitting and pitting on pinion flank (T1= 300 Nm, a = 15 mm) (left);Initial pitting at the addendum of the pinion toe (T1= 300 Nm, a = 15 mm) (right)On the right side of Figure 2 the condition of the pinion fl
22、ank (a = 15 mm) is shown for two different runningtimes. Micropittingcouldaffectpittinginsuchwaythattheinitialpittingoccursalsoattheaddendumofthetoewhichisincontrasttohelicalgears. At commonlyusedhelicalgears micropittingoccuralmost alwaysbelowthe pitch point at the dedendum of the pinion.Althoughon
23、thewheelflanksofthehypoidvariantwithanoffseta=44mmpittingwasalsodetected,itwasnotpossible to evaluate the load capacity of the wheels because of the very few numbers of failures.Figure 3 shows the endurance limit for pitting for all tested variants. For each torquethe maximum Hertzianstresswascalcul
24、atedbymeansofaloadedtoothcontactanalysis(LTCA)withBECAL7. BECAL7(Bevel5 11FTM10Gear Calculation) was developed by the FVA (Forschungsvereinigung Antriebstechnik). It considers thedeflections between pinion and wheel, which are caused by the elasticity of the teeth, bearings, shafts andhousings. In c
25、ontrast to the increasing pinion endurance torques the occurring maximum stresses on theflanks aredecreasing. For examplethemaximum stress ontheflank of variant a = 44mm for theendurancelimit is 22% smaller than that of the non-hypoid variant.Results of the tooth root testsAllexaminedvariantsfailedb
26、ythetoothrootbreakage. Figure 4showsatoothrootbreakageonapinionthatcan be regarded as representative for all tests. The crack initiation was detected in all cases close to the30-tangent to the tooth fillet as it can be seen in Figure 5.Figure 3. Endurance limits of the test gears (pitting)Figure 4.
27、Typical tooth root breakage (a = 0 mm)Figure 5. Typical fracture surface (a = 0 mm)6 11FTM10In Figure 6 the endurance limits of the tested variants are compared. As expected the tests showed anincreasing load capacity with higher offsets. The corresponding maximum tooth root stresses are for allvari
28、ants within a range of 3.5%.New calculation method for bevel and hypoid gearsVirtual cylindrical gearInallwidely usedstandardmethods or standardcapable methods the loadcapacity is determined by meansof virtual cylindricalgears. Ideally thevirtual gears have thesame loadcapacity as the hypoidor bevel
29、gear.Whereasforbevelgearswithoutoffsetthereisinprincipalonlyonesolutionforthevirtualgears,hypoidgearscan be transferred in diverse virtual gears. One reason for that are the two different spiral angles on hypoidgears whereby the pinion spiral angel and the gear spiral angle differ due to the offset
30、angle. Equations 1through6arefor thedeterminationofthevirtualcylindricalgeargeometry. Thederivationof thefundamentalmagnitudes will be explained in the following.Reference diameter, dv(1)dv1,2=dm1,2cos1,2Tip diameter, dva(2)dva1,2= dv1,2= 2 ham1,2Root diameter, dvf(3)dvf1,2= dv1,2 2 hfm1,2Helix angl
31、e, v(4)v=m1+ m22wheredmis mean pitch diameter; is pitch angle of bevel gear.hamis mean addendum.hfmis mean dedendum.mis mean spiral angle.Figure 6. Endurance limits of the test gears (bending)7 11FTM10Base diameter, dvb(5)dvb1,2= dv1,2cosvetTransverse effective pressure angle, vet(6)vet= arctantanec
32、osvTransverse base pitch, pvet(7)pvet= mmncosvetcosvLength of path of contact, gv(8)gv=12 d2va1 d2vb1 dv1sinvet+d2va2 d2vb2 dv2sinvetwhereeis effective pressure angle;= eDfor drive side (see ISO 23509);= eCfor coast side (see ISO 23509).mmnis mean normal module.InFigure 7thehypoidschematicwiththepin
33、ionandwheelpitchconesisshown. BothpitchconesaretangenttothecommonpitchplaneT. ThesurfacelinesofeachpitchconethatarepartofthepitchplaneTenclosetheoffsetanglempinthecalculationpointP. Theshortestdistancebetweentheaxisofthepitchconesisdefinedasthehypoidoffseta. Ifa=0,thenthepitchapexesarecongruentandpi
34、nionandwheelhavethesamespiralangle.For theproposedmethodit was theaim togeneratethe virtualcylindrical gears directly from the hypoidgeargeometry without havingany stepinbetween(seeFigure 8). Furthermore themesh conditions of thevirtualgearsshouldrepresentthoseofthehypoidgearsascloseaspossible. Tofi
35、ndthesolutionofasuitablevirtualgear comprehensive investigation inter aliaby means of aloaded toothcontact analysis (LTCA) weremade.The LTCA was done by the program BECAL 7. With a special program the correlation between importantgeometry values aswellasbetweenessentialmagnitudes fortheloadcapacity
36、wereanalyzed. Somewillbeexplained in the following.Figure 7. Schematic of hypoid gears8 11FTM10Asdescribedby8and9hypoidgearsmayhaveunequalmeshconditionsondrivesideandcoastsideflankif the value of the limit pressure angle limis different from zero (for bevel gears: lim= 0). As Figure 9illustratesthen
37、ormalforcesFntDandFntCinprofilesectiononbothflanksidesareinclinedfromthepitchplaneby the (generated) pressure angles tDand tC. The projections of the zones of action are inclined by theeffective pressure angle etDand etC.Iflim 0 then the direction of the normal forces on the flank areinclinedagainst
38、thezoneofaction. Ifthegeneratedpressureanglesareequal(tD=tC, e.g.,Zyklo-Palloid,Klingelnberg) then the lengths of the zone of action for drive and coast side are unequal. Furthermore, theradius of relative curvature is different on both flank sides what effects the occurring Hertzian stresses. Tobal
39、ance the mesh conditions the generated pressure angles have to be chosen unequal regarding limaccording to ISO 23509 10. It is obvious that the effect described has to be considered in the virtualcylindrical gear, if the mesh conditions of the hypoid gear should be mapped.Thederivationofthepitchdiam
40、etersofthevirtualcylindricalcanbeseeninFigure 9whereahypoidgearwithanoffsetofa=31.75mmandanouterdiameterofde2=170mmisregarded. Thefulllinesaretheprojectionsof thetip-, root- andpitchcones of pinionandwheelinaxial planeof thevirtual gears. Althoughthe offset isconsiderable high the tip-, root- and pi
41、tch cylinders of the virtual gear according to equation 1 to 3 (dottedlines) approximates the corresponding projections of the cone in the area of contact quite well.Figure 8. Hypoid gear and its virtual cylindrical gearFigure 9. Asymmetric mesh conditions on hypoid gears9 11FTM10The length of path
42、of contact regards as mentioned before the influence of the limit pressure angle. Theexample shown in Figure 10 demonstrates how the length of path of contact is increasing on the coast sideandis decreasingonthedrivesideif theoffset is increasedandthegeneratedpressureangles arekept con-stant to n=20
43、. For an offset a = 30 mm the limit pressure angle is lim=-9.87. The values for the blacklines aredeterminedby aLTCA, thegrey lines represent thevalues for thevirtualcylindrical gears accordingto equation 8. Whereas Niemann/Winter11does not regard the effects of the limit pressure angle the newvirtu
44、al gear gives a good correlation to the LTCA.To represent themesh conditions of thehypoid gears it is not less important toapproximate theshape of thecontactzoneasgoodaspossible. However,thecomplexityofthederivationhastobelimitedinordertokeepthe method capable for a standard. ISO 10300 3 assumes to
45、have an elliptical contact zone, DIN 39911implies a rectangular contact zone. For the newmethod theshape of the contact zone is approximated by aparallelogram. Basedoncomprehensivestudiesregardingthecontact zonesof conventionalhypoidgearsitseemedtobeadequatecompromise. Theenclosedangles oftheparalle
46、logramaredependenttothehypoidoffset. For bevelgears (a = 0) theparallelogram becomesarectangle. Thederivationoftheparallelogramisbased on semi-empirical coherences that are shown in Figure 11.Whereas the face width of virtual cylindrical gears and of their pertaining bevelgears without offset haveth
47、esame size (bv= b), this is not true for hypoid gears. Before the face width bvcan be calculated, the effectiveface width bveffof the virtual cylindrical gear pair has to be determined. For that purpose, the length of thecontactpatternb2effwhichismeasuredinthedirectionofthewheelfacewidth,isused. Iti
48、sassumedthatthetheoretical zone of actionof thehypoid wheelis not arched but developed intoa parallelogram and thenpro-jectedontothecommonpitchplaneT as showninFigure 11by fat dottedlines. Thesidelines ofthis zoneofaction around mean point P are vertical to the wheel axis which in this viewcoincides
49、 withthe conedistanceRm2. Theothertwoboundarylinesareparalleltotheinstantaneousaxisofhelicalrelativemotionofthehypoidgear pair which is given by the angle mp.Figure 10. Length of path of contact over hypoid offset10 11FTM10Figure 11. Simplified zone of action for virtual cylindrical gearsThezoneof actionof thecorrespondingvirtualcylindricalgearpair isthegreatestpossibleparallelogram(fatlines in Figure 11) inscribed in the theoretical zone of action of the wheel whereby th
