1、10FTM15AGMA Technical PaperDrive Line Analysis forTooth ContactOptimization of HighPower Spiral BevelGearsBy J. Rontu and G. Szanti andE. Ms, ATA Gears Ltd.Drive Line Analysis for Tooth Contact Optimization of HighPower Spiral Bevel GearsJesse Rontu and Gabor Szanti and Eero Ms, ATA Gears Ltd.The st
2、atements 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.AbstractIt is a common practice in high power gear design to apply relieves to tooth flanks. They are meant to preventstress conce
3、ntration near the tooth edges. Gears with crownings have point contact without load. When loadis applied, instantaneous contact turns from point into a Hertzian contact ellipse. The contact area grows andchanges location as load increases. To prevent edge contact, gear designer has to choose suitabl
4、e relievesconsidering contact indentations as well as relative displacements of gear members.In the majority of spiral bevel gears spherical crownings are used. The contact pattern is set to the center ofactive tooth flank and the extent of crownings is determined by experience. Feedback from servic
5、e, as well asfrom full torque bench tests of complete gear drives have shown that this conventional design practice leads toloaded contact patterns, which are rarely optimal in location and extent. Too large relieves lead to smallcontact area and increased stresses and noise; whereas too small relie
6、ves result in a too sensitive toothcontact.Today it is possible to use calculative methods to predict the relative displacements of gears under operatingload and conditions. Displacements and deformations originating from shafts, bearings and housing areconsidered. Shafts are modeled based on beam t
7、heory. Bearings are modeled as 5-DOF supports withnon-linear stiffness in all directions. Housing deformations are determined by FEM-analysis and taken intoaccount as translations and rotations of bearing outer rings. The effect of temperature differences, bearingpreload and clearances are also inco
8、rporated.With the help of loaded tooth contact analysis (LTCA), it is possible to compensate for these displacementsand determine a special initial contact position that will lead to well centered full torque contact utilizing areasonably large portion of the available tooth flank area. At the same
9、time, crownings can be scaled to theminimum necessary amount. This systematic approach leads to minimum tooth stressing, lower noiseexcitation as well as increased reliability and/or power density as compared to conventional contact designmethod.During recent years ATA Gears Ltd. has gained comprehe
10、nsive know-how and experience in such analysesand advanced contact pattern optimization. The methodology and calculation models have been verified innumerous customer projects and case studies.Copyright 2010American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia,
11、22314October 2010ISBN: 978-1-55589-990-53Drive Line Analysis for Tooth Contact Optimizationof High Power Spiral Bevel GearsJesse Rontu and Gabor Szanti and Eero Ms, ATA Gears Ltd.IntroductionIn a majority of spiral bevel gears produced, thetooth contact is initially placed at the center of toothflan
12、k during manufacturing. Sufficient crowningsare applied to prevent the contact from reachingtooth edges under load. However, the use of largecrowning also has a down side of increasing contactstresses, since the area in contact at a particularmoment is reduced. With constantly growing de-mands for h
13、igher power density and lower noisegeneration, there are pressures for decreasingcrownings. In more general sense, there is often agreat need to optimize tooth flank topography for acertain application. This requires accurate know-ledge about behavior of tooth contact under load.Regardless of the op
14、timization goal, the change ofrelative position of bevel gears under load is an im-portant factor.When loads and temperature differences areapplied to a gear drive, the relative position of pinionand wheel changes due to deformations anddisplacements related to bearings, shafts andhousing. This caus
15、es changes in the tooth contact,significance of which is dependent on magnitudeand mutual relations of the displacements as well ascharacteristics of the tooth geometry. One of themain concerns is the spreading and movement ofcontact pattern, which ideally should be located atthe center of tooth fla
16、nk under load and cover asmuch of the flank area as possible. If the behavior ofcontact pattern is known, pre-compensation can beapplied in finish machining of tooth flanks to ensuregood running properties under load. Usually thismeans that the tooth flank topography is modifiedso that the initial c
17、ontact pattern (without load) ismoved from the center of the flank by a certainamount.Traditionally the knowledge of tooth contactbehavior has been attained through practicalexperiences. This requires time consuming andexpensive prototype testing. Alternative approachis the one based on computer sim
18、ulation, by whichsignificant cost saving are possible. During recentyears, tooth contact optimization based on LoadedTooth Contact Analysis (later LTCA) has beenapplied with good success in numerous customerprojects involving marine, industrial and automotivebevel gear applications. In LTCA the mesh
19、 of spiralbevel gears is simulated using 3D tooth geometry,taking into account the actual relative position ofgears under load. This paper describes computa-tional process used to determine how the relativeposition of bevel gears changes when load andtemperature differences are applied on a gear dri
20、ve.The process is combination of different calculationmethods and is hereafter referred to as Drive LineAnalysis. In addition to the methods usually used inDrive Line Analysis, some alternative approachesare also mentioned to provide a more generaloverview of applicable methods.Relative position of
21、bevel gearsIn nominal position the pitch cone apexes of bevelpinion and wheel (if not a hypoid gear pair) coincide.Deviation from this position (location + orientation)can be fully defined by four displacement values,which are hereafter referred as relative displace-ments. As shown in Figure 1, they
22、 consist ofdeviation of shaft angle (S), offset (E), pinion axiallocation (P) and wheel axial location (G).Figure 1. Relative displacements of bevelgears4Drive line analysisTo fully understand tooth contact behavior in acertain application, the chain of events fromassembly (tooth contact adjustment)
23、 to operatingconditions (loads and temperature differencesapplied) has to be traced. To accurately determinethe displacements of bevel gears, a detailedanalysis of the whole drive line consisting of shafts,bearings and housing, is required. The “core” ofanalysis is comprised of separate calculationm
24、odels for pinion shaft and wheel shaft, which arehereafter referred to as shaft-calculation models.These models are used to simulate deformationsand displacements of shafts and bearings. Severalcommercial software with ranging levels of capabil-ities are available for this purpose. Beam theory ispra
25、ctically always used to calculate shaft deflection,but there are significant differences in the waybearings are modeled. At the simplest level,bearings are considered as radially stiff “hinges”,which does not represent reality very well. In themore advanced software, such as used in DriveLine Analys
26、is described in this paper, bearinginternal geometry and stiffness nonlinearity areconsidered in order to accurately model the realbehavior.Bearing stiffnessGears are usually supported to gear unit housingwith rolling bearings. Bearing stiffness variessignificantly depending on type of rolling eleme
27、nt(Figure 2), affecting the displacement behavior ofthe shaft-bearing system. Another significantfactor is the internal alignment capability (Figure 3).In shaft calculation models used in Drive LineAnalysis, bearings are modeled as supports with 5degrees of freedom: two radial-, two tilt- and oneaxi
28、al direction (Figure 11). The “missing 6thDOF” isthe bearing rotation, which is naturally of no interest.Nonlinear bearing stiffness in every direction ismodeled starting from deformations of individualcontacts between rolling elements and raceways(Figure 4), also taking into account the internalcle
29、arance and operational contact angle. With thismodeling method it is possible to accurately predictthe distribution of loads and subsequently the dis-placements. Although nonlinear bearing stiffnessleads to iterative calculation, the calculation timesare minimal due to the analytical theories applie
30、d.More detailed description of the modeling theorycan be found in DIN ISO 281 Beiblatt 4 1.Figure 2. Radial stiffness of different bearingtypes 2Figure 3. Bearing internal alignmentcapabilityFigure 4. Deformation of individual elementcontact5LoadsTooth forces are considered as point loads acting ont
31、he center of the tooth flank at mean pitch diameterdm(Figure 5). Tooth force components Ft, Frand Faare calculated based on mean spiral angle m, nor-mal pressure angle nand pitch cone angle .Especially for the wheel, the axial location of theacting point of the tooth forces does not alwaysrepresent
32、the axial location where the forces areactually conveyed to the shaft. An example of suchsituation is presented in Figure 6. In shaft-calculation models this is taken into account bytransferring the tooth forces axially by a distance ofdfand correspondingly adding two additional bend-ing moments Mx
33、= Frdfand My = Ftdf. The sameprinciple is used to correctly model bearingreactions in cases with bearings with nonzeropressure angle (e.g., taper roller bearings).Figure 5. Application point of tooth forcesIn addition to tooth forces there are usually externalforces that also need to be included, su
34、ch aspropeller thrust force in marine thrusters. They areapplied to their appropriate location on the shaftusing the same principles as with the tooth forces.The weight of components is seldom important fromdeformations point of view, but might instead besignificant for other reasons discussed later
35、 in thispaper.Figure 6. Application point used in shaftcalculationBearing clearances and preloadDepending on the arrangement, bearing clearanceand pretension can have significant influence ongear displacements. In shaft calculation models,values in operating conditions are used, which of-ten differ
36、significantly from assembly values due totemperature differences. Clearances can bedivided into internal and external clearances(Figure 7).Figure 7. Examples of internal and external bearing clearances6Internal radial clearances in operating conditionsare calculated based on clearance class (e.g., C
37、N,C3, etc.), shrink fits of bearing rings and temperat-ure difference between inner and outer ring(Figure 8 and Figure 9). Both internal and externalradial clearances cause displacement of shaft, butinternal clearance also affects bearing stiffness.Therefore, if precise, external radial clearancesho
38、uld be modeled as movement of outer ring, notas increased internal clearance. However, thesignificance of this matter is small. In axial direction,internal and external clearances are basically thesame thing. Axial preload / external axial clearancein operating conditions are calculated based oninit
39、ial setting (assembly), temperature differencebetween shaft and housing, distance of bearingsand bearing pressure angle.Figure 8. Clearance reduction due totemperature difference 3Figure 9. Clearance reduction due to shrinkfit 3External radial clearances are used when bearingsneed to be free in axia
40、l direction. External axialclearances are sometimes applied to bearings in O-or X-arrangement to prevent excessive preloadingdue to temperature differences. Ideally, theseclearances should be reduced to very small valuesin operating conditions, in which case their influenceon displacements would be
41、negligible. However,because temperature differences are usually notexactly known in design phase (clearances chosenpreferably “on the safe side”, i.e., too large ratherthan too small) and gear drives are often loaded indifferent operating temperatures, consideration ofexternal clearances is also a p
42、art of Drive LineAnalysis.Deformation of gear housingDeformation of gear housing is considered throughFE-analysis, performed with commercial software.Bearing reactions obtained from preliminary shaftcalculation models are used as loadings for the FE-model. Loads are applied to the radial and axialsu
43、pport surfaces of bearings as pressuredistributions with resultant forces corresponding tothe bearing reactions (see Figure 10). Externalloads are applied if such exist.Figure 10. Example of radial bearing load onFE-model of housingAfter FE-model is solved, translation and rotationvalues of bearing
44、bores are extracted from thedisplacement results. With displacements as themain result, a relatively coarse FE-mesh(compared to e.g. stress analysis) is sufficient. Inshaft-calculation models bearing bore displace-ments are described with the same 5 degrees offreedom as bearing stiffness (see Figure
45、 11):3 translational and 2 rotational displacements areused to describe movement of one bearing bore.These values can be extracted from the FE nodedisplacements in different ways. One way is tochoose representative nodes with 90 spacing from7the support surfaces and calculate the 5 displace-ment val
46、ues from them. A more sophisticatedmethod is to place a node in the middle of thebearing bore and connect it to the cylindrical surfaceby beam elements with very small axial stiffnessand ball joint-type connections at the ends. Thisway the displacement of the center node directlyrepresent the sought
47、-after values. Similar resultscan be achieved also by fitting an un-deformedcylinder to the displacement field using a best-fitprocedure. All of the mentioned methods havebeen used successfully as a part of Drive LineAnalysis.Figure 11. Determination of bearingdisplacement from FE resultsAfter the s
48、haft calculation models are re-run usingthe bearing bore displacements, changes in bearingreactions are checked. If considerable change isobserved, the FE-model is no longer valid and theprocess is repeated. Usually no more than oneiteration is required.Effect of temperature differences to axialloca
49、tion of bevel gearsIn addition to bearing clearances and pretension,temperature differences also affect the axiallocation of bevel gears. The significance is stronglydependent on the material of the housing and thedistance between the bevel gear centerlines andaxial bearing location (Figure 12).Figure 12. Effective distance for temperaturedifference between housing and shaftsEffect of gear drive orientation duringtooth contact adjustmentUsually during the assembly of gear drives nothermal differences exist. Therefore, bearingclearances (internal a
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