1、12FTM15AGMA Technical PaperNew Methods for theCalculation of the LoadCapacity of Bevel andHypoid GearsBy C. Wirth and B.-R. Hhn,ZG- Zahnrder und GetriebeGmbH, and C. Braykoff, MANTruck S shear stresses on the surface caused by friction;5 12FTM15S thermal stresses caused by the thermal gradient;S str
2、esses caused by bending mechanism;S residual stresses.Figure 3illustratesthestresscomponentsthatinfluencethematerialexposureinaconsidered(infinitesmall)element. In Figure 3a the stress components that result from the normal load on the flank are shown. Thestresses according to the Hertzian theory ar
3、e arising out of the normal force. Due to the sliding componentsthe friction force that is tangential to the flank surface induces shear stresses. Figure 3b demonstrates theeffect of bending by a normal force that acts above the considered element (above in profile direction). Thecomponentsofthenorm
4、alforcecausenormalstresseswithanapproximatelylineardistributionoverthetooththicknessandshearstresseswithanapproximatelyparabolicdistributionandamaximuminthemiddleofthetooth. Residual stresses result from the hardening and finishing process. As an example, 20 shows com-pressive stresses are occurring
5、 in the case and are balanced by tensile stresses in the core. Unlike thestresses in Figure 3a and Figure 3b the residual stresses in Figure 3c are load independent.Oster and Hertter developed the program system “STRORHR” for the calculation of all mentioned stresscomponents on cylindrical gears. Wi
6、th this program it is possible to examine the material exposure in thesubsurface below any contact point on the flank surface. Thereby the examination direction isperpendicularto the flank surface.Stress conditions in the rolling contactIn any contact point on the flank the rolling direction x can a
7、lso be seen as the time axis. Figure 4 shows inprinciple the stress components under the surface. All volume elements in the same depth are exposed toequal stresses, but at different times. To evaluate the material exposure in a certain depth beneath the flanksurface the corresponding stresses have
8、to be regarded over the whole time axis (x-axis).However in rolling contacts a turning principal coordinate system complicates the evaluation of the materialutilization. A possibility to analyze the dynamic stresses in rolling contacts are the shear stress courses in asectional plane that is defined
9、 on the surface of the base sphere according to Figure 5.a) c)b)Figure 3. Stress conditions inside the toothFigure 4. Time-dependent stress components in a rolling contact6 12FTM15Figure 5. Base sphere with sectional plane 8Figure 6 shows for a rolling contact an example of shear stress courses in a
10、 certain sectional plane and amaterial depth of y/b0=0.3. (ti) is the time dependent graph for the projection of shear stresses in thedirectionsn2andn3(seeFigure 5). InFigure 6a,noresidualstressesareconsidered. Asaconsequencethepoint(0/0)ispartofthecourse. Atacertaintimeti,whenthecontactisstillunloa
11、ded(e.g.,thecontactpointofthe flank surfaces is still far away from the regarded volume element) 2= 3= 0 in the examined sectionalplane. During the movement of the contact over the flank surface the stress components 2and 3can bemarkedinthediagram. Ofcourseiftheinfluenceofthemovingcontactpointonthes
12、tressesat theexaminedplane is fading out the course will again reach the point (0/0). As Figure 6a shows, the instantaneous stressvector (2/3) is completely turning during one load cycle what means that it acts as an alternating load. Itsmaximum length is defined as the “maximum shear stress” max,a,
13、 the diameter of the circumscribed circle ismax,a.Figure 6bshows for thesameexaminedsectionalplane an equal load cyclebutinconsiderationof theresid-ual stresses. Unlike before the shear stresses 2and 3have discrete values even if thecontact isunloaded.However the course of the pair of values2/3issim
14、ilar, whatmeans thatmax,a=max,b. An importantfactis,thatthemaximumshearstressmax,bisdecreasingundertheinfluenceof(compressive)residualstresses(max,bb:HV(y) = HV0(y)(13)Depth range a y b.Linear interpolation of ZS(14)wherey is material depth below the contact point, mm;a is certain material depth, mm
15、;11 12FTM15HV(y) is modified local hardness in consideration of the slip influence, HV;HV0(y) is local hardness, HV;b is certain material depth, mm;ZSis factor according to 20.Residual stresses in the toothHertter demonstrated that the influence of residual stresses has to be considered in the mater
16、ial exposure(see Figure 6) for the evaluation of tooth failures. Particularly the maximum material exposure Aintis influ-encedbytheresidualstresses. Whereascompressivestresseshaveusuallyapositive effecton thematerialexposure, tensile stresses increase the material stresses 8. The total dynamic expos
17、ure Aint ais only influ-enced by means of the mean stress sensitivity. As Hertter proved the material exposure in the range of thetransitionzonefromcasetocoreaccountsforfailuremodeslikeflankbreakagethatareusuallycharacterizedby an initial crack in this region.Wirth 20 proposes to adopt the (compress
18、ive) residual stresses according to Lang 11 for the case. Duetothe balance of forces in the core tensile residual stresses have to exist. For the estimation of the residualstress distribution in the core Wirth made investigations on basis of FE-methods. Using a parabola of 4thdegree, the tensile str
19、esses can be approximated well by the balance of forces. Figure 9 shows qualitativelyin a normal section of the tooth the residual stress distribution. It is a sufficient correlation that the residualstressesintoothheightdirectionareequaltothe residualstresses inlengthwise direction. Residualstresse
20、sthat are directed orthogonal to the flank surface are neglected.Improvement of a gear set with flank breakageWheel flank breakageA decisive number of hypoid gear sets used in axle gear drives in test vehicles failed of flank breakage. Onlythe wheels were affected by this failure mode. Figure 1 (lef
21、t side) shows a flank breakage on one tooth of awheel, in Figure 1 (right side) pitting on the coastside couldbe detected. Figure 2shows onanother wheelacharacteristic flank breakage. As it can be seen the failure plane runs on both flank sides through the activetooth height.Tolearnmoreaboutthecondi
22、tionswhereandwhenflankbreakagesoccuranewtypeoftestforthestationarytestrighasbeendeveloped. Comprehensivetestrunshavebeenmade. Thegearssetshavebeentestedforadefinedloadspectrumwherethehighestloadstagewasthetorquethathasbeenconsideredinthefollowingcalculations. The gear sets failedeither bypitting orf
23、lank breakage. Pittingoccurred onthe pinionas wellason the wheel. Flank breakage was only observed on the wheel.Figure 9. Residual stress distribution in the tooth12 12FTM15Figure 10. Flank breakage on two different wheelsFigure 10 shows for a damaged wheel the investigation of the fracture surface
24、in the scanning electronmicrograph. In this case a small inclusion was detected where-from the crack propagates to the flank sur-faces. Inclusionscanberegardedasa catalystfor thecrack initiationbecause ofthe notchingeffect ofdiffer-entelasticitymodulus. Investigationsof Annast1 showedthat anAl2O3-in
25、clusion causesa stressincrease(von-Mises criterion) of approximately 30%-40%. The size and the depth beyond the surface have a relat-ively small influence. Therefore, the lower the material exposure in the core can be realized, the smaller therisk of flank breakage with initial cracks in this region
26、 will be, as Figure 10 demonstrates.Design of an improved gear setThe aim of the redesign was to develop a new gear design with a smaller material exposure to avoid flankbreakage on the one hand and pitting as far as possible on the other hand. In a first step the old design wasanalyzed with the new
27、 introduced material-physically calculation method. In the second step a new geardesign with same ratio and diameters but lower material exposure was searched by an iterative process.Table 1 contains the main geometry data of the old and the new gear design.NOTE: Because only the wheelwas affected b
28、y flank breakage allcalculations have been made forthe wheelonly!Table 1. Geometry of the examined gear setsNomenclature Symbol UnitOld design New designPinion Wheel Pinion WheelNumber of teeth z - - 8 45 8 45Pinion offset a mm 34 34Normal module mmnmm 6.134 6.122Mean pitch diameter dmmm 69.6 331.2
29、69.9 332.3Face width b mm 60.7 58.3 62.2 59.0Spiral angle 45.5 34 45.5 34Material - - - - 25MoCr4E 25MoCr4ERoughness Rz flank/tooth root Rz mm 3/16 (after run in)Total overlap ratio (underload) drive/coast- - - - 3.0/2.7 2.92/2.7Lubricant - - - - Shell Spirax ASX 75W 90Temperature of lubricant C 901
30、3 12FTM15For the calculation discrete contact points on the flank have to be chosen for the evaluation. As Figure 11shows the selected contact points are positioned in a section with considerably high load and the supposedcrack origin. To evaluate not only the risk of an initial crack at one single
31、point but also the potential of crackgrowth four different positions were examined.Table 2 contains the Hertzian stresses that were determined by means of the loaded tooth contact analysiswithBECAL10. Deformationanddeflectionsofhousing,shaftsandbearingshavebeenconsidered. Underthe same load conditio
32、ns it was possible to reduce the stresses on both flank sides in the critical area of theflank by approximately 15%. This was possible with an optimized crowning (Ease-Off) incombination withadifferent gear design (duplex instead of semi completing) and changed pressure angles.Of course the reductio
33、n of contact stresses leads in most cases to an increase in load capacity, especiallywhen the failure mode pitting is regarded. But in case of flank breakage the failure mechanism is not influ-enced only by the contact stresses but also by the material exposure deep inside the tooth. Because of ther
34、equirement to keep the amount of transferred torque by remaining the gear dimensions (and module) theflankloadintotalcannotbesignificantlyreduced. Toavoidflankbreakageithastobetheaimtoreducemater-ial exposure mainly in the core where in this case the crack initiation could be detected in several cas
35、es (seeFigure 10).As mentioned earlier in the evaluation of the material exposure in the subsurface section, especially in themiddle of the tooth thickness, the following stress components may not be disregarded:S shear stresses due to the shearing forces (flank normal forces);S tensile residual str
36、esses.TheshearstressdistributionreachesitsmaximuminthemiddleofthetooththicknessasshowninFigure 3b.Also the maximum values of the tensile residual stresses are supposed to be in this region. Whereas thedeterminationoftheshearstressesisonlyamechanicalproblem,theresidualstressesarecaused mainlybythe he
37、at treatment process. Only in the area direct beneath the surface residual stresses are influenced bythe finishing process of the gear. Due to this the residual stresses are derived by the hardness profile asdescribed earlier.Figure 11. Contact pattern and calculated contact points on the wheel flan
38、ksTable 2. Hertzian stress under the considered loadCalculationpointOld design New designDrive side Coast side Drive side Coast sideP1 1865 1951 1629 (-13%) 1692 (-13%)P2 1891 1979 1571 (-17%) 1698 (-14%)P3 1841 1961 1593 (-13%) 1651 (-16%)P4 1612 1901 1553 (-4%) 1585 (-17%)14 12FTM15Figure 12 shows
39、 for the calculations results presented in the following the assumed hardness profiles, thatare based on detailed measurements but smoothed for calculation. Because of the slightly different coolingconditions during the hardening process in profile direction of the tooth the hardness gradients and t
40、he corehardness are slightly different. The derived residual stress distributions are shown as well in Figure 12. Asitcan be seen the compressive stresses in the case are up to res400 N/mm2and are decreasing until thetransientregionofcaseandcore. Becausethecasethicknessin profiledirection ismore orl
41、ess constantthecompressiveresidualstressprofilesaresimilar. Incontrasttothatthetensileresidualstressesinthemiddleofthe tooth thickness are increasing from point P4 to P1. The reason is the mechanical balance of forces: Theseparatingforcesthatarecausedbythe compressivestresses inthe caseare approxima
42、telyconstant forP1to P4. The attracting force is represented by the tensile compressive stresses and has to have an equalamount. BecausethecoresectionbecomessmallerfromP4toP1thecorrespondingtensilestresseshavetoincrease.InFigure 13,thecalculatedmaterialexposureforpointP1toP4areshown. Theblacklinesre
43、presentthetotaldynamic exposure Aint aand the grey lines the total maximum exposure Aint.Aint acan be seen as a value to describe the material fatigue. It is based on an endurance strength (derivedfromtheVickershardness)forafailureprobabilityof50%. Pittingisatypicalfatiguefailurethatcorrelateswithth
44、etotaldynamicexposureAinta. 8,18and20showthatifmaterialexposurevaluesexceedacertainlimitin the subsurface up to a depth of y/b01( b0: half of Hertzian contact width) pitting failure occur with a highprobability. For the examined gear this decisive range is up to approximately y1 mm. As Figure 13 sho
45、wsAintaexceedsthelimitof“1”. Duetotheloadspectrumofthetestvehicleswhichhadonlyafewtimesliceswiththis considered load pitting failures on the drive side where not detected. The fatigue strength may be thereason for that.Figure 12. Profiles for hardness and residual stress15 12FTM15Figure 13. Material
46、 exposure for calculated points: old design - drive side of wheelAintrepresentsthematerialexposureconcerningyielding. AccordingtothetheoryofthecalculationmethodifAint 1 local redistribution of stress or initial cracks occur. This situation is tightened if the notching effect ofinclusions or disconti
47、nuities increases the material exposure. At this time there is no possibility provided bythe material-physically based method to regard this fact in the calculation process. Therefore the practicallimit for Aint-values that are determined for a homogenous materialshould bereduced tovalues smallertha
48、n“1” in order to be on the safe side. Because of the specifics of the hardening process discontinuities occurmore typically in the core than in the case. This is why the total maximum exposure Aintshould be limitedespecially in the core.Hertter 8 and Wirth 20found agood correlationbetween thetotal m
49、aximummaterial exposureAintandthefailure mode flank breakage: Especially high values in the material depth between the transition of case andcore as well as in the core seem to be responsible for flank breakage. It has to be mentioned that crack initi-ationsthatarecausedbyyieldinghavenoendurancelimitorfatiguestrengthforfinitelife. Accordingtotheoryonlyveryfewsingleloadcyclesareenoughforstressredistributionorcrackinitiation. Thesecracksmayhavethe ability to grow also at lower loads! Unlike fo
