1、11FTM07AGMA Technical PaperThe Effects of HelixAngle on Root Stressesof Helical GearsBy D.R. Houser, The Ohio StateUniversity and A.P. Thaler, OwensCorning FiberglasThe Effects of Helix Angle on Root Stresses of Helical GearsDonald R. Houser, The Ohio State University and Aaron P. Thaler, Owens Corn
2、ingFiberglasThe 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.AbstractThe ISOand AGMA Gear Rating Committees have for several years been comparing the results of differentrat
3、ingmethodsforseveralsetsofgearpairsthathavesimilarnormalsectionsbutdifferenthelixangles. Theanalysispresentedinthispaperusesaverysophisticatedfiniteelementcodethatwasdevelopedspecificallyfor gear and bearing contacts to analyze the example gear sets. Analyses are also performed using a moreconventio
4、nal load distribution analysis program. The results for the original gear sets show that the narrowface width gear teeth twist significantly, thus moving the load to one edge of the face width and essentiallyshowing that the example gear sets are highly unrealistic. When analyzed by the ISO and AGMA
5、 ratingmethods, the results do not reflect this twisting action. In an effort to come up with a valid comparison ofstresses for different helix angles, three adjustments using wider face widths were attempted. The firstscheme uses a wider face width with perfect involutes. Edge effects result in the
6、 peak stresses again beingneartheendsofthefacewidth. Thesecondadjustmentusesawiderfacewidthbutwithanarrowloadpatchinthe middle of the tooth pair and results in the stresses increasing with helix angle. The third method, whichusesthewidefacewidthteeththathaveleadcrownandtiprelief,givesthemostreasonab
7、leresults,withtherootstressesbeingatamaximuminthecenter regionofthetoothfacewidths. Thepaper compareseachofthe results to earlier analyses performed by others using both the AGMA and ISO calculations.Copyright 2011American Gear Manufacturers Association1001 N. Fairfax Street, 5thFloorAlexandria, Vir
8、ginia 22314October 2011ISBN: 978-1-61481-006-33 11FTM07The Effects of Helix Angle on Root Stresses of Helical GearsDonald R. Houser, The Ohio State Universityand Aaron P. Thaler, Owens Corning FiberglasIntroductionThis paper reports on the effect of helix angle on root stresses, a topic of discussio
9、n for a number of yearswithin the ISO and AGMA Gear RatingCommittees. Current ratingmethods useeither theLewis formfactor1 or the thirty degree tangent method 2 applied to the transverse tooth section to locate the position ofmaximum root stress. Corrections are then provided to account for the diag
10、onal linesof contactthat occurinhelical gear tooth contact.Even though this topic has been discussed extensively for many years, there are still disagreements on howthese factors should be calculated. Gears present significant challenges when trying to come up withchangesingeometrythatisolateoneeffe
11、ctwithoutaffecting others. Ageometry setthat supposedlyisolateshelixanglefromallothervariablesbyusingaconstantnormalcross-sectionforeachnewhelixanglegearwasproposedindocumentsoftheISORatingCommittee3,4. Inordertoachievethis,therackusedtogeneratethe normal cross-section was kept constant. However, in
12、 order to change the helix angle and still keep thegears operating at the same standard center distance, it was necessary to reduce the tooth numbers as thehelix angle was increased. Since tooth numbers must be integers, there are only a few possible helix anglesthat are possible. Table 1 shows the
13、variables that are held constant in this analysis and Table 2 shows therelationship between tooth number and helix angle for the sets.For these gear sets, Figure 1 shows the results for a variety of methods that were used to calculate the rootstresses. In this figure, the stresses are first calculat
14、ed for the spur gear (helix angle = 0) and the values fortheotherhelixanglesarenormalizedtothevaluecalculatedforthespurgearsothatallcurvesstartatavalueof 1.0 for a helix angle of zero degrees. Previous work by the ISO document authors shows calculations fortwofacewidths,oneverynarrowwiththefacewidth
15、beingequaltothemoduleandtheotherforaninfinitefacewidth. Calculations have been made using ISO 6336 method B, AGMA 2001, and a proposed Norwegianmethod3. Somemethodspredictthatincreasingthehelixanglewillcontinuallydecreaserootstresseswhileother methods show an ever increasing trend in root stresses w
16、ith helix angle.Table 1. Variables held constant in study 1Parameter Variable ValueGear ratio u 4Normal module, mm mn10.0Center distance, mm a 750.0Hob addendum, mm ha014.0Hob tip radius, mm a04.0Protuberance, mm p 0.3Normal pressure angle, degrees 20.0Face width, mm 1 mn10.0Normal tooth thickness,
17、mm t 15.74 11FTM07Table 2. Gear pairs used in study 1Helix angle, degrees Number of pinion teeth Number of gear teeth0.000 30 12014.83 29 11621.04 28 11225.84 27 10829.93 26 10433.56 25 10036.87 24 9639.95 23 9242.83 22 88Figure 1. N367 normalized tooth form factor (root stress) predictions(All data
18、 is from 3 - B: Method B from ISO 6336; N: Proposed Norwegian method)Each of the methods used to calculate the root stresses of Figure 1 use “standardstype formulas”that aretobe used in conjunction with many other factors to come up with a stress value. This papers authors thoughtthat calculations o
19、f the “real” stresses that the gears experience might shed some light on the differences ineachcalculationmethod. Therefore,anadvancedfiniteelementprogramthatisspecificallydesignedtoana-lyze gear and bearing contacts, and a more standard load distribution prediction program were employed topredict t
20、he actual root stresses for the example gear sets. This paper provides extensive analyses of theexample gear sets presented in the original N367 document and also seeks to provide a more realistic geargeometry that isolates the effects of helix angle on root stresses.Modeling methodologyEach of the
21、gear sets presented in this paper was modeled with two separate programs to evaluate the rootstresses,thefirstbeingthehighfidelity3-dimensionalfiniteelementprogramknownasTransmission3D5,6.The second program is a more conventional load distribution program (LDP) that has been developed byHouser, et a
22、l. 7,8.5 11FTM07Transmission3D is a linear finite element contact analysis program specifically designed for analyzing gearandbearingcontactsandisbasedontheCalyxcontact analysissolver. The programhas theability tomodelcomplex gear geometries including tooth micro-geometries that include lead and pro
23、file modifications. Theprogram utilizes a hybrid algorithm that combines finite element analysis with the application of asemi-analytical surface integral solution at the contact region to produce compliance terms. Thesecompliance terms are then used in a Simplex-type solver to evaluate the load dis
24、tribution across the tooth.Using Transmission3D, each of the presented gear sets is modeled as a single mesh with all non-rotationaldegreesoffreedomfixedtoground. Atoothmeshtemplate(Figure 2)withhighresolutionintherootregionisusedtoaccuratelycapture thestress gradientswithin theentire rootfillet. Ea
25、ch simulationis thenrun foronemesh cycle (one basepitch of rotation). The root stresses are found by searching the entire root fillet for themaximum principal stresses across the face width. While most tooth root stress calculation procedures onlyevaluate stresses for one tooth pair at a time, this
26、method evaluates all lines of contact simultaneously andalso includes the deflections of the entire gear blank.The second program used to model the gear sets is the Load Distribution Program (LDP). LDP is a programthat analyzes single mesh gear pairs using a finite plate compliance calculation in co
27、njunction with the inclu-sion of Hertzian deflections and deflections of the tooth base 7. The root stresses for each gear set arecomputedusingatwodimensionalboundaryelement9thathasbeen extendedto thethird dimensionusinga procedure developed by Jaramillo 10. LDP also has the ability to use finite el
28、ement created compliancefunctions as well as performing finite element calculations of the root stresses.Study 1: Narrow face width gears N367TheoriginalstudywasbasedontheN350andN367documentsfromtheISORatingCommitteeproceedings3,4 that presented 9 gear sets with varying helix angles as given in Tabl
29、e 2. The face width is equal to thenormalmodule,mn,whichprovidesaverynarrowgearwithapinionfacewidthtodiameter(F/d)ratioof0.031.It was assumed that this was done in order to essentially create a helical gear tooth form that acts like a spurgear, since one can literally define a highest point of singl
30、e tooth pair contact, evenfor thehighest helixanglegear pair.Figure 2. Transmission3D finite element mesh template6 11FTM07The gear sets of this study have perfect involute profiles, a narrow face width of 10 mm, and constantnormaltooththickness. Byholdingthenormaltooththicknessconstantandkeepingthe
31、facewidthnarrow,theeffectof the diagonal line of contact on the moment arm distance to the root centerline is reduced. This allows theresultstobenormalizedandcomparedtothespurgeargeometrywherethelinesofcontactareparalleltotheshaft axis. Analysis was performed on each of the 9 sets using both Transmi
32、ssion3D and LDP.Figure 3showsthefiniteelementmeshusedinTransmission3DwhileFigure 4showsboththepredictedloaddistributionatthehighestpointofsingletoothcontactandtherootstressesforasinglespur geartooth. Initialobservations showed that the normalized root stress never reduced and increased much more rap
33、idly thanthe 1/cos () path that is similar to the upper curve of Figure 1.Figure 3. Transmission3D model of the narrow face width meshFigure 4. Gear tooth load distribution from Transmission3D for a narrowface width spur gear tooth7 11FTM07Further investigation showed significant twisting of the loa
34、ded tooth as the helix angle increased. This twistcanbeseeninFigure 5forthe42.83helixanglegearpair. Thistwistingshiftstheloadtoonesideofthetoothandthuscausinganunusualstresspatternintherootwiththemaximumrootstresslocatedoppositetowherethe tooth is loaded. This twisting induces increased stresses in
35、the root due to the shift in load distribution toone side of the tooth causing what seem to be the highly unrealistic results shown in Figure 6.TheLDPpredictions,whichdonotincludethetwistingeffect,showincreasingrootstresswithhelixangle,butto a much lesser degree than the 1/cos() curve.Figure 5. Tran
36、smission3D load distribution and root stress patternfor the 42.833 helix angle pinionFigure 6. Predicted normalized root stress for narrow face width pinion8 11FTM07Study 2: 100 mm face width gearsWhilethepreviousgearsetspresentedintheN367documenthadanovelideaforisolatingtheeffectsofhelixangle, the
37、narrow face width prevented the analysis from producing realistic results more common to mostgear applications that have much wider face widths. To reduce the tooth twist, the face width was simplyincreased from 10 mm to 100 mm (F/d = 0.31 with the face contact ratio varied from 0 to 2.16 depending
38、onhelixangle)withallothergeometrydata beingkept thesame asfor theprevious gearsets. The piniontorquewas scaled by a factor of 10 in order to remain close to the same load per unit face width as in the previousnarrowfacewidthstudy. Perfectinvoluteprofileswereassumedsosomelocalizedtwistingisstillexpec
39、tedatthe corners while tip interference due to tooth deflection is also expected. Although not of infinite face widththatisplottedforthesecondsetofdatainFigure 1,theresultsforthisfacewidthwereexpectedtobecompar-able to the infinite face width gear pair.Again the nine gear sets were modeled and analy
40、zed using the two programs. Figure 7 shows the finite ele-mentmeshusedinTransmission3DandFigure 8showstheloadingandstressesforonemeshpositionofthe42.83 helix angle pinion. Although the tooth twist has been reduced, the predicted peak helical gear rootstressesshowninFigure 9stilloccurattheedgesofthet
41、oothfacewidthandseemabnormallyhighrelativetothe gear rating models. It is interesting to note that the highest fidelity model, Transmission3D, predicted thehigheststressessinceitstillshowsthetwistingeffectaswellashavingincreasedstressesduetothereducedtooth backing because of the angled tooth edge. T
42、he LDP FE model predicted less twist and hence lessstressandfinally,thesimpleLDPmodelthatdoesnotmodelthetwist,predictsstressesthatareslightlylower.Figure 7. Transmission3D model of the 100 mm face width mesh9 11FTM07Figure 8. Transmission3D display of load distribution and stress contour of the 42.8
43、33 helixangle, 100 mm face width pinionFigure 9. Predicted normalized root stress for 100 mm face width pinion (perfect involute)Study 3: Narrow contact patch gearsInordertoisolatetheeffectsofhelixangle,whileatthesametimeeliminatingtheedgeeffects,itwasdecidedto try analyzing a gear pair that had fea
44、tures of each of the two previous studies. In this study, the face widthwas kept at 100 mm, but a contact patch only 10 mm in width was applied down the center of the tooth. Thispatch was achieved by applying abnormally high end relief across 45% of each end of theface width,leaving10 11FTM07only10%
45、ofthefacewidthincontact. Figure 10showstheendreliefspecification. TheTransmission3Dfiniteelement mesh is the same as that used in Study 2. In order to make the gear set act like a spur gear pair, theoutside diameters were reduced so that the profile contact ratio was kept close to 1.0. Figure 11 sho
46、ws theloaddistributionandrootstresspatternforthe25helixangle pinion. Theeffects ofhelix angleon rootstressthat are shown in Figure 12 are now much better behaved, with all of the models giving somewhat similarresults and each of them roughly following the inverse of the cosine of the helix angle plo
47、t. However, if onestops to think about this a bit, the tooth normal cross-sections are very similar, but the normal load increasesby the cosine of the helix angle, so following the inverse of the helix angle trend seems reasonable.Figure 10. Flat lead crown applied to narrow contact patch gearsFigur
48、e 11. Transmission3D predicted load distribution and root stresses for the narrow contactpatch 25 helix angle pinion11 11FTM07Figure 12. Predicted normalized root stress plot for the narrow contact patch pinionStudy 4: Typical gear pairEachofthepreviouslystudiedgearpairshassomefeaturethatmakesthegea
49、rsetunrealistic. Inordertogeta more realistic gearing situation, the face width was kept at 100 mm and circular profile and lead modifica-tionswereappliedsuchthatendeffectsandtipinterferencewerereduced. Thetypicalmicro-geometryofthepinion is shown in Figure 13. The earlier finite element model was used for the Transmission3D analysis.Figure 13. Pinion micro-geometry modification12 11FTM07The contact stress distribution plot seen in Figure 14 shows complete contact across the tooth flank with arollingoffofstressattheextremesofthetoothfacewidthandpr
