1、09FTM08AGMA Technical PaperLoad Sharing Analysisof High Contact RatioSpur Gears in MilitaryTracked VehicleApplicationBy M. Rameshkumar,P. Sivakumar, and S. Sundaresh,Ministry of Defense andK. Gopinath,IIT MadrasLoad Sharing Analysis of High Contact Ratio Spur Gears inMilitary Tracked Vehicle Applica
2、tionM. Rameshkumar, P. Sivakumar, and S. Sundaresh, Ministry of Defense andK. Gopinath,IIT MadrasThe 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.AbstractMilitary tracked ve
3、hicles demand a very compact transmission to meet mobility requirements. A compacttransmission with low operating noise and vibration is desirable in military tracked vehicles to reduce weightand improve power to weight ratio. It is also necessary to increase the rating of existing transmissions inm
4、ilitary tracked vehicles, like prime movers, to accommodate the additional weight required for ballisticprotection. Hence, it was decided to apply high contact ratio (HCR) spur gearing concept which will reducenoise,vibration and enhance load carrying capacity,for a 35 ton trackedvehicle finaldrive.
5、 In HCRgearing,the load is shared by a minimum two pairs of teeth as in helical gears. It was decided to analyze the loadsharingofthenormalcontactratio(NCR)gearingusedinsun-planetmeshoftheexistingfinaldrive,andalso,toanalyzetheloadsharingofHCRgearingwhichwillbeusedtoreplacetheNCRgearswithoutanychang
6、eintheexistingfinaldriveassemblyexceptsun,planetandannulusgears. ThispaperdealswithanalysisofloadsharingpercentagebetweenteethinmeshfordifferentloadconditionsthroughouttheprofileforbothsunandplanetgearsofNCR/HCRgearingusingfiniteelementanalysis(FEA). Also,thepaperrevealsthevariationofbending stress,
7、 contact stress and deflection along the profile of both NCR/HCR gearing.Copyright 2009American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314September 2009ISBN: 978-1-55589-961-53Load Sharing Analysis of High Contact Ratio Spur Gears in Military TrackedVeh
8、icle ApplicationM. Rameshkumar, P. Sivakumar, and S. Sundaresh, Ministry of Defenseand K. Gopinath,IIT MadrasIntroductionContact ratio is defined as the average number ofteeth pairs in contact under static conditions andwithout errors and tooth profile modifications. Amajorityofthecurrentgearboxesfo
9、rtrackedvehicleapplications havecontact ratios rangingfrom 1.3to1.6. Then, the number of tooth engagements iseither one or two. The term High Contact Ratio(HCR)appliestogearingthathaveatleasttwotoothpairs in contact all times i.e., contact ratio of two ormore. As the percentage change in mesh stiffn
10、essfor HCR meshes is lower than the percentagechange in mesh stiffness for Normal Contact Ratio(NCR) meshes, one can expect high quality HCRgear meshes to have lower mesh inducedvibrationandnoisethanNCR gear meshes. A highpoweredcompact transmission 1 is essential to enhancethe mobility of military
11、tracked vehicles. This re-quirement is partially met by improvements appliedto NCR gearing. Literature survey indicated thatHCR gearing was designed 2 and successfullyused in helicopter transmission 3, to improvefurtherthepowertoweightratioofthetransmission.In HCR gears, since a greater number of te
12、ethsharetheload4,theconceptappearstobesimpleand has wide potential applicability, it has not beenapplied to military tracked vehicle transmissions.Figure 1. Mechanical schematic of 800 hp automatic transmission4A detailed study of the existing final drive planetarygearassemblyasshowninFigure 1ofa35t
13、onmili-tary tracked vehicle was carried out to apply theHCRconcept. Thefinaldriveisanindependentunitwhich has been isolated for separate testing toapply the concept of HCR. This paper provides anapproach to arrive at the HCR gears for the finaldriveofatrackedvehicle,adetailedanalysisofloadsharingper
14、centageof teethinmeshthroughout theprofile for both sun and planet gears of NCR/HCRgearing using finite element analysis (FEA). Bend-ing stress, contact stress and deflection of gearteeth were calculated.An attempt has also been made to prove the newconceptof HCRgearingusingANSYS software5.Evolution
15、 of design for HCR final driveThe mechanical schematic of the entire transmis-sion is shown in Figure 1. NCR gearing wasoriginally used through the transmission and finaldrive. The HCR gearing concept is applied to thefinal drive of the 800 hp automatic transmission.Thefinaldriveislocatedoutsidethem
16、aintransmis-sion and is fixed to the vehicle hull. It serves as anadditional reduction unit in multiplying the drivingandbrakingtorquefor thetracks. Thefinaldriveasshown in Figure 2 consists of three gear elementsnamely sun gear (23 teeth), planet gear (22 teeth)and annulus gear (69 teeth) of module
17、 4 mm withcontactratioof1.343. ThetransmissionLHandRHoutputs areconnectedviatoothedslidingcouplingsto the sun gears of the LH and RH final drives re-spectively. Theannulusgearsofthetwofinaldrivesare fixed andthey providethe reactiontorque. Theoutput power from theLH andRH finaldrives istak-en from t
18、he planet carriers, which are connected tothe LH and RH sprockets driving the tracks.Figure 2. Cross section of final drive5Inordertoincreasetheloadcarryingcapacityoftheexisting NCR final drive, keeping the same weightand volume envelop, it was proposed to introduceHCR gearing. Various combinations
19、of number ofteeth, module, profile correction, addendum factor,etc.,wereanalyzedtoget adesignveryclosetotheexisting NCR final drive data. The HCR gearing isdesignedinsuchawaythatonlysun,planetandan-nulus gears of the NCR final drive are changed bykeeping the same center distance (93.013 mm),face wid
20、th (64 mm) and keeping other memberssame. A minor variation in the gear reduction ratiowasnecessarysinceitisverydifficulttoachievethesameratioinview of various other constraints suchascenterdistance. Thisimpliesthat tomakeuseofthesamegearboxforHCRgearingwitharestrictionon contact ratio 2.0106, the c
21、enter distance shouldnot be altered. However, considering the variousparameters suchas toleranceontipcirclediameterand center distance, top land edge chamber, ther-mal effects, profile tip and modification, it is advis-abletoselect contactratiogreaterthan2.2wherev-er possible. Spur gears are used si
22、nce theirproductionissimplerandmoreeconomicalthanhe-lical gear. Moreover, they are freefrom axial loads.Theconcept for various factors that aid inobtainingHCRforspurgearsisevolvedfromthecontactratio(CR) given in Equation 1.Wherer1andr2aretheoperatingpitchradius ofthepinion and gear, is the operating
23、 pressure angle,m is the module and a is the addendum (based onthe operating pitch radius) which is equal to onemodule for standard gears.Since for the present investigation the center dis-tance should not be altered, a close observation ofthe equation suggests that contact ratio of spurgearingcanbe
24、increasedbyseveralways,(a)byre-ducingmodule(b)byincreasingnumberof teeth(c)byloweringthepressureangleand(d)byincreasingtheaddendum. InthispapertheHCRisobtainedbyincreasing the addendum factor and number ofteeth by reducing module. The important gear pa-rameters of both NCR and HCR gear designs aresh
25、own in Table 1.Material Properties of the gear are taken to beYoungs Modulus = 2.1 107MPa and Poissonsratio = 0.30.Finite element analysis (FEA) of NCR/HCRgear designFinite element analysis is used to study in detailvariousparameterssuchasbendingstress,contactstress, deflection etc., for both NCR an
26、d HCRgearing.CR =r1+ a2 r21cos2+r2+ a2 r22cos2r1+ r22 mcos(1)Table 1. Gear parametersSI no. Parameters NCR HCR1. Profile Involute Involute2. DIN accuracy class 7 73. Module, m 4.0 mm 2.5 mm4. Number of teeth in sun, Zs23 385. Number of teeth in planet, Zp22 366. Number of teeth in annulus, Za69 1107
27、. Profile correction in sun, Xs0.3 0.1098. Profile correction in planet, Xp0.539 0.19. Profile correction in annulus, Xa-0.3024 -0.309610. Center distance, Cd93.013 mm 93.013 mm11. Reduction, ratio, Gr4.0 3.89412. Addendum factor, Ya1.0 1.2513. Contact CR 1.343 2.010614. Facewidth, F 64 mm 64 mm6Spu
28、r gear geometryProfileofaninvolutespurgeartoothiscomposedoftwocurves. The workingportion is the involuteandthe fillet portion is the trochoid. Theoretical limitradius6isanimportantparameterwhengearkine-matics is considered. It is the radius at which theinvolute profile on a gear should start, in ord
29、er tomake use of the full length of the involute profile ofthematinggear. Thetrochoidaltoothfilletasgener-ated by a rack cutter is modeled exactly using theprocedure suggested by Earle Buckingham 7. ACcomputerlanguagecodewasdevelopedforgen-erating exact tooth profile with trochoidal fillet. Thetroch
30、oidal fillet form is generated from thededendum circle up to the limiting circle, where itmeets the involute profile at the common point oftangency and the involute profile extends up to theaddendum circle.Gear modelsThegeartoothunderconsiderationforNCRgearingis a standard one with a full depth of 2
31、.25 times themodule and the addendum of one unit module andthegeartoothforHCRgearingisafulldepthof2.75times the module and addendum of 1.25 times themodule. Each generating cutter having tip radius0.8 mm and 1.0 mm is used for the generation ofHCRandNCRgearingrespectively. Sincethegearfillet assumed
32、 as a constant radius curve in AGMAonitslayoutprocedureisnotthetruerepresentationof the spur gear geometry (withgenerated fillets), itwas decidedtoconsider thetrochoidal filletas afairmodel in this paper. AGMA 8 uses Newtonsmethod of iterations where as this paper deals withpolynomial equations for
33、direct calculations ofAGMA geometry factor withminimum process timefor computerized gear analysis.InordertomaketheperfectgeargeometryforNCR/HCRdesigndatamentionedinTable 1,aCcodeisdeveloped for generating the complete gear profileofbothNCRandHCRwithtrochoidal fillet. Thesungear (23 teeth) and planet
34、 gear (22 teeth) mesh forNCRgearingandthesungear(38teeth)andplanetgear (36 teeth) mesh for HCR gearing are gener-ated in single window FEA environment of ANSYSversion 11.0.Finite element models and meshingThe gears (bothNCR andHCR) arekept incontactby positioning at the stipulated center distance(93
35、.013 mm) with respect to the global coordinatesystemandonlytheplaneareamodelsareusedforthe FEA. Quadratic two dimensional (2D) eightnoded higher order plane strain elements (PLANE82 of ANSYS) as shown in Figure 3. To promoteconvergence of the contact solution, the finiteelement models are meshed wit
36、h a very fine meshwhere the tooth will experience the contact. Thetotal number of elements used in HCR gearing is38044andthetotal number ofnodes is115118.Thefinite element meshed model of NCR and HCRgears are shown in Figure 4 a) and b) respectively.Figure 3. PLANE 82 higher order plane strainelemen
37、ta) NCR gear b) HCR gearFigure 4. Finite element meshed model7Loading and boundary conditionsThemaximumtorque(Tc)onthecarrierofeachfinaldrivewithafactor ofsafety of1.17is44975Nmandthe sun gear is in mesh with four planet gears (N).Hence, thetorqueappliedperunit facewidthonthesun gear (38 teeth) is 4
38、5 Nm (i.e., Tc/NFGr). Theload is applied in the form of torque in clockwisedirection viewing from input (coupling) side and theplanet gear (36 teeth) is fully constrained. Boththesun and the planet gears are arrested in radialdirection with respect to local cylindrical coordinatesystem.Solution and
39、post processingLoad sharingThegear pair is rotatedas a rigidbody accordingtothe gear ratio. The solutionis repeatedfor boththegears rotated with some amount of angularincrement according to the gear ratio. Approxi-mately 45 angular increments with 0.5 degree stepareusedfor this analysisandtheanalysi
40、s iscarriedout with the help of customized APDL (ANSYSParametric Design Language) looping program.Transmission error, torsional mesh stiffness, rootstress, contact stress and load sharing ratio isobtained for all the positions. The nodal force ateach node has been obtained for each individualgear to
41、oth. By this methodology, the percentageofload sharing between teeth for both sun and planetgears at any position were determined.Accordingly, the maximum percent of tooth loadforNCR and HCR gear designs, at the tooth tip and atthehighestpointofsingletoothcontact(HPSTC)forNCR and the highest point o
42、f double tooth contact(HPDTC) for HCR gearing, were determined. Theindividual tooth loads have been determined bycomparing the total normal load to the sum of thenormal loads contributed by each pair in contactequally.It is observed from the above FEA that for NCRgearing, 100% load is applied at the
43、 HPSTC condi-tionand50%loadisappliedatthetiploadcondition.ForHCRgearing,52%loadisappliedattheHPDTCconditionandonly20%loadisappliedatthetiploadcondition.Bending stress, contact stress and deflectionIn the present study, the stresses were calculatedfor corrected gears considering the trochoidal fillet
44、form using finite element analysis, which fairly in-cludes the size and shape effects as well as that ofstressconcentrations. Theratioofthestressdeter-mined by FEA to that of modified Lewis equation isconsidered as stress correction factor.TheFEAcalculatedbendingstressinthegeartoothfillet, contact s
45、tress andvector sum deflectionweredetermined using ANSYS version 11.0 for bothNCR/HCR gear and tabulated in Table 2 andTable 3. Figure 5 shows the nodal stress plot forHPSTC in the case of NCR with the maximumstressoccurringatthetrochoidalfilletofthe22toothNCR gears. Figure 6 shows the nodal stress
46、plotHPDTC in the case of HCR with the maximumstressoccurringatthetrochoidalfilletofthe36toothHCR gears.Table 2. Results based on FEA for HPSTC/HPDTCTeethBending stress, MPa Deflection - 100% load, mm Deflection - 52% load, mmContactee100% load 52% load Deflection vector sum Deflection vector sumCont
47、actstress, MPaNCR22 608.4 - 30.94E-03 - 1504.823 673.2 - 49.15E-03 -HCR36 - 482.1 - 21.22E-03 1074.838 - 390.2 - 21.37E-03Table 3. Results based on FEA for tip loadingTeethBending stress, MPa Deflection - 50% load, mm Deflection - 20% load, mmContactee50% Load 20% load Deflection vector sum Deflecti
48、on vector sumContactstress, MPaNCR22 386.9 - 23.9E-03 - 929.623 406.7 - 32.65E-03 -HCR36 - 282.5 - 17.39E-03 914.938 - 255.5 - 17.75E-038Figure 5. Stress plot for 22 tooth NCR gearwith HPSTC loadingFigure 6. Stress plot for 36 tooth HCR gearwith HPDTC loadingAGMA ApproachAGMAspecificationdisplayscha
49、rtsforuncorrectedgears and semi-analytical method for correctedgears with constant radius fillet to calculate thegeometry factor. The mathematical procedure ofAGMA was written in the form of C computer lan-guage code in this paper so that geometry factorcan be calculated for gears with trochoid filletanalytically.Critical parametersA C code was developed to calculate geometryfactor using the mathematical procedure specifiedbyAGMA8. Theproceduretakesintoaccounttheeffects of shape of the tooth, worst load positi
