Procedia Materials Science5 ( 2014 )1640 1649 Available online at www.sciencedirect.com2211-8128 2014 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).Selection and peer-review under responsibility of Organizing Committee of AMME 2014doi: 10.1016/j.mspro.2014.07.352 ScienceDirectInternational ConIerence on Advances in ManuIacturing and Materials EngineeringAMME 2014EIIect oI backup ratio and cutter tip radius on uniIorm bendingstrength design oI spur gearsR Prabhu Sekara, G Muthuveerappanb,*aResearch scholar, Machine aesign section, Department of Mechanical Engineering, Inaian Institute of Technology, Chennai-600036, Inaia.bProfessorr, Machine aesign section, Department of Mechanical Engineering, Inaian Institute of Technology, Chennai-600036, Inaia.AbstractIn thehigher speed ratioandproIileshiItedgeardrives,themaximumIilletstress inthepinionandthegearisnotequal.Theload-carryingcapacityoIthegeardrivescan beimprovedbyremoving this unbalancedmaximumIilletstress.This paper hasexplained an idea to remove this unbalanced maximumIillet stress and design the spur gear drives with uniIorm Iillet strength.ThisuniIormIilletstrengthoIthegeardrivecan be achieved bychangingthetooththicknessoIbasicracksonthepitchlinesIromthestandardtooththickness(0.5am)intoanon-standardone(kgcam,kgc=0.5).TheinIluenceoIgearparameterssuchasbackup ratio, cutter tip radius and addendum modiIication Iactors on the maximum Iillet stress has been analyzed through FEMwith diIIerent values oI tooth thickness coeIIicient and Iinally the optimum value oI tooth thickness coeIIicient are suggested Iorthe given spur gear drive that improve the Iillet capacity in bending. 2014 The Authors. Published by Elsevier Ltd.Selection and peer-review under responsibility oI Organizing Committee oI AMME 2014.Keyworas. Finite element model, Load sharing ratio, Maximum Iillet stress, Spur gear, Tooth thickness coeIIicient, Tooth load-carrying capacity;*Corresponding author.G.Muthuveerappan, 91-9444863534Email address : jmvriitm.ac.in 2014 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).Selection and peer-review under responsibility of Organizing Committee of AMME 20141641 R. Prabhu Sekar and G. Muthuveerappan /Procedia Materials Science5 ( 2014 )1640 1649 1. Main textBendingstrengthcapacityoIthegearisaIunction oItoothstrengthandrimstrength.CrackisonetheIailuremode in the gear tooth that initiates at the tooth Iillet and progress through the rim due to high maximum Iillet stressdevelopedattheIilletregion.IIthebackuprimthicknessislow,therootcrackwillpropagatethroughtherimotherwise the Iailure mode as the tooth root Iailure. The both tooth root and rim Iailures can be reduced by providingenoughtoothrootthickness and rimthickness. Generally, themaximumIilletstresses inthe pinionandthegearisdiIIerent in the higher gear ratio and proIile shiIted gear drives.NomenclatureawCenter distance Ior corrected gears (mm)E Young`s modulus (MPa)haAddendum oI the gearhIdedendum oI the geari Gear ratiokgcTooth thickness coeIIicient on the pitch line oI the basic rack Ior gearkpcTooth thickness coeIIicient on the pitch line oI the basic rack Ior pinionm Module (mm)mBBackup ratiopbBase pitch (mm)ragRadius at the addendum circle oI the gearrbgRadius at the base circle oI the gearT Input torque (N-mm)xpAddendum modiIication Iactor Ior pinionxgAddendum modiIication Iactor Ior gearzgNumber oI teeth in the pinionzpNumber oI teeth in the gearu Pressure angle at pitch point (Degree)uwPressure angle at reIerence line Ior the corrected gears (Degree)c contact ratio Poisson`s ratiop Cutter tip radius(t)maxMaximum Iillet stress (MPa)AbbreviationsFEM Finite element modelHPSTC Highest point oI single tooth contactHPTC Highest point oI tooth contactLPSTC Lowest point oI single tooth contactLPTC Lowest point oI tooth contact1642R. Prabhu Sekar and G. Muthuveerappan /Procedia Materials Science5 ( 2014 )1640 1649 LSR Load sharing ratioNCR Normal contact ratioThis diIIerence is called inequality oI the maximum Iillet stresses. The load-carrying capacity can be also improvedbyremovingtheinequalityoIthemaximumIilletstress. ThispaperpresentsanideatoremovethisinequalitythroughchangingthetooththicknessoIthebasicrackIromstandardintonon-standarddesign. TheuniIormIilletstrength oI the gear drive can be achieved by changing the tooth thickness oI basic racks on the pitch lines Irom thestandardtooththickness(0.5am)intoanon-standardone(kgcam,kgc=0.5)andtoIindasuitablevalueoIkgc tosatisIy the condition that the pinion and the gear have equal the maximum Iillet stresses. It is called as the uniIormroot strength design in this paper. Celik (1999) has showed that the spur gear teeth deIlections and maximum Iilletstresses were estimated Ior three teeth model, which have Iavorably agreed with analogous results obtained by straingauge experiments oI the spur gear. Wang and Howard (2004) have worked on LSR Ior normal contact ratio (NCR)spur gears considering torsional stiIIness oI spur gears. Shuting Li (2008) has investigated the eIIect oI addendum ontooth contact strength and bending strength oI spur gears based on the LSR using 3D multi pair contact model.Anattempt has been made by D.V.Muni and Muthuveerappan (2007) Ior the equal Iillet stresses on the pinion and geardrives.In this present work, the maximum Iillet stresses developed in the gear teeth generated by the standard (kgc0.5)and non-standard (kgc=0.5) rack cutters are compared and using these non- standard rack cutters, an attempt has beensuccessIullymade to design and optimize thespurgear drivewhich develops a balanced Iilletstress in the pinionandgear.Inthisapproach,onlytwonon-standardrackcuttersaredeveloped,oneIorthepinionandotherIorthegearIoranygivencenterdistanceoIadrive. SeveralsetsoIpinionandgeartoothproIilesIorthegivencenterdistanceandgearratiohavebeengeneratedusingthesenon-standardcuttersbygivingdiIIerentcombinationoItooththicknesscoeIIicients(kpc andkgc )IorthepurposeoIstressanalysis.Finally,theinIluenceoIbackupratio,cutter tip radii and addendum modiIication Iactor (proIile shiIt) on the maximum Iillet stress on the gear pairs withrelevance to optimum design has also been investigated.2. Tooth engagement positions and tooth thickness coefficient2.1. Tooth engagement positions of NCR spur gear arivesIn the normal contact ratio spur gear drives (1 c 2), the two pairs oI teeth comes in contact at the beginning oIthe contact such as one at highest point oI tooth contact (HPTC) and the other at lowest point oI single tooth contact(LPSTC)oIthegearasshowninFig.1.DuringthecourseoIaction,theleadingpairleavesthecontactatlowestpointoItoothcontact(LPTC)andhence,onlyonepairnamelytrailingpairistransmittingthemotionIromthehighestpointoIsingletoothcontact(HPSTC)tothelowestpointoIsingletoothcontact(LPSTC)untilanotherincomingpairestablishesthecontactatHPTCtoinitiatetwopairsincontactagain. TheradiioIthesediIIerentcontact positions are expressed asi = i (1)i = i + (p -AB) + i - i (2)i = i + i - i- p (3)i = i + i - i- AB (4)1643 R. Prabhu Sekar and G. Muthuveerappan /Procedia Materials Science5 ( 2014 )1640 1649 AB = i - i+ i -i - a sin u (5)The regions AB and CD are the double pair contact regions and the region BC is a single pair contact region. Thestress developed at the Iillet region is maximum, when the tooth is loaded at the HPSTC due to the highest momentarm. So this position is called as the critical loading position or worst load position by researchers.Fig.1. Tooth engagement positions oI NCR spur gear drives2.2. Tooth thickness coefficient of rack cutterFig.2. (a) A symbolic standard rack cutter kgc0.5; (b) A symbolic non-standard rack cutter kgc~ 0.5Fig.2.showsacrosssectionalshapeoIthebasicrackcutterwhichisusedtogeneratethetoothproIiles oIthegeartooth. IIthe tooththicknesson thepitch lineoItherackcutterIor the gearisdenoted bykgcam andthetoothspace on the pitch line oI the rack is denoted by (1-kgc) am. Here, kgc is named as the tooth thickness coeIIicient oI(a) (b)1644R. Prabhu Sekar and G. Muthuveerappan /Procedia Materials Science5 ( 2014 )1640 1649 the basic rack Ior gear. The value kgc0.5 means that the cutter is called as standard rack cutter and iI kgc=0.5, it iscalledasnon- standardrackcutter.Thestandardandnon-standardrackcuttersareshowninFigs.2(aandb). Thetooth thickness coeIIicient oI the rack Ior pinion is denoted as kpc (kpc1-kgc).3. Multi pair contact finite element modelThe involute gear tooth proIile is developed by using standard and non-standard rack cutters Irom the given basicgearparameterssuchasthepressureangle,module,teethnumberandaddendumcorrection.Inthepresentwork,commercial soItware,ANSYS12isusedIor modeling and analysis purpose. The2DIiniteelementmodeloIspurgear displayed in Fig.3 (a) is a three teethIull rim contact model, which is used to carry out the analysis. A higherorder2-D PLANE 82,8-noddedquadrilateralelementhavingtwodegreesoIIreedompernodehasbeenusedtomesh the gear model. In the multi pair contact model, the pinion is constrained only in the radial direction along therim oI the inner radius and the torque T (N-mm) is uniIormly distributed in the tangential direction along the rim oIinnerradius.GearisconstrainedbothinxandydirectionsattherimoItheinnerradius.TheTARGE169andCONTAC 172 elements are used in the contact analysis. The Fig. 3(b) shows the magniIied view oI the Iillet regionwhich is the region the maximum Iillet stress developed.Fig.3. Finite element model oI NCR spur gear (a)Multi pair contact model Ior i1, kgc0.5 ; (b) Maximum Iillet stress distribution4. Results and discussionsAs an accurate estimation oI maximum Iillet stress developed in the Iillet region in gears becomes an importantconcern to design the gears oI improved load carrying capacity. In this present work, the maximum Iillet stressesdeveloped in the gear teeth generated by the standard (kgc0.5) and non-standard (kgc=0.5) rack cutters are comparedand using these non- standard rack cutters, an attempt has been successIully made to design and optimize the spurgear drive which develops a balanced Iillet stress on pinion and gear.4.1. Influence of tooth thickness coefficient (kg) of rack cutter for gear on maximum fillet stressThe inIluence oI tooth thickness coeIIicient oI rack cutter Ior gear (kgc) on maximum Iillet stress has been studiedIor a NCR spur gear pair generated by the Iull round standard (kgc0.5) and non-standard (kgc=0.5) rack cutters withthegivengearparameters(Table1).TheresultsareshowninFig.4. Inthehighergearratiodrives(i~1),themaximum Iillet stress in the gear (otg)max (24.109 MPa) is lesser than that oI pinion (otp)max (25.681 MPa) which the(a)(b)1645 R. Prabhu Sekar and G. Muthuveerappan /Procedia Materials Science5 ( 2014 )1640 1649 gear pair is generated by the standard rack cutter . So, the pinion (smaller one) is avulnerable onewhich is due tolarge deIormation oI pinion tooth thangear tooth (bigger one). When the kgc value increases, the tooth thickness oIthegearatthepitchlinedecreasesandthetooththicknessincreasethesameamountinthepinion.Hence, ThevaluesoI(ot)maxdecreaseatthepinionandatthesametimeitincreaseatthegearIorthehighervaluesoIkgc(kgc=0.5,non-standardcutters).ThemaximumIilletstressbecomesequalinthepinionandthegear ((otp)max(otg)max 25.122MPa) ,whenthekgc valuereachesto0.5445asshownin Fig.4. ThisiscalleduniIormbendingstrength design in this paper. The maximum stress is developed at the Iillet region, when the pinion or gear is loadedat the HPSTC due to highest moment arm in the single pair regionTable 1. Gear parametersS.No Parameters Name Parameters Values1.2.3.4.5.6.7.8.9.10.11.12.Module m (mm)Pressure angle u (degree)Gear ratio iPinion teeth number zpAddendum ha (mm)Dedendum hI (mm)Addendum modiIication Iactor xBackup ratioTorque (N-mm)Rack Cutter typeElastic constant E (GPa)Poisson ratio 120o1.5201ha0.25mxp0 and xg02.2293.97Full round rack cutter2100.3.Fig.4. InIluence oI rack tooth thickness coeIIicient ( m1, i1.5, zp20, u20o).1646R. Prabhu Sekar and G. Muthuveerappan /Procedia Materials Science5 ( 2014 )1640 1649 4.2Maximum fillet stress analysis for uniform benaing strengtha parametric stuayA detailed parametric study has been investigated in this paper to balance the (ot)max in the pinion and the gear IordiIIerent values oI gear parameterssuch as mB,p, xp and xg. The gear parameters used Ior this parametric study areshown in Table 2.Table 2. Gear parameters Ior a parametric studyS.No Parameters Name Parameters Values1.2.3.4.5.6.7.8.9.10.Module m (mm)Pressure angle u (degree)Gear ratio iPinion teeth number zpAddendum ha (mm)Dedendum hI (mm)Backup ratio mBCutter tip radius p (mm)Addendum modiIication Iactor xTorque (N-mm)120o1201ha0.25m1.11, 1.33 and 2.220.2m, 0.3m and Iull round tipForS+drivexp0.1 and xg0ForS-drivexp0 and xg -0.1ForSodrivexp0.1 and xg -0.193.974.2.1 Influence of backup ratio (mB)Thebothtoothstrengthaswellastherim strengthdeIinesthebendingcapacityoIgeartooth.Inthethinrimgears, thebendingIatigueIailure occurs along the rimratherthan the root Iillet. Hence,an estimation oIoptimumrimthicknessbecomesimportantinthecompactsizeandlowweight designconsiderationapplications.Therimthickness can alter by varying the one gear parameter called as backup ratio. It is deIined as the ratio oI rim thicknessto the whole depth oI the tooth. The Figs. 5 (a and b) show the thick and thin rim gear tooth model.m = TB(6)The optimum value oI kg and (ot)max determined Ior diIIerent backup ratios (mB) by keeping m, u, z1, i, ha, xp andxg unchangedareshownin Fig.6. ItisIoundthatthereisnochangeinthetrochoidalandinvoluteproIilesoIthegear tooth due to change the mBvalues.Accordingly, the optimum kgc value does not change with respect to mB.But,the balanced maximum Iillet stress is lower in the thick rim gear drive (mB2.22, (ot)max25.122 MPa, kgc0.5445)thanthatoIthinrimgeardrive(mB1.11,(ot)max30.222MPa,kgc0.5445)whichis duetotheminimumdeIormationoIthegearbodyin thethickrimgears.ThemaximumIilletstressisdevelopedneartherimsection(bottom portion at the Iillet) in thin rim gears and it moves towards top portion with increase the backup ratio, whenthe gear pair loaded at the HPSTC (Fig. 5 (c)).4.2.2. Influence of cutter tip raaiusThe shape oI the rack cutters with diIIerent cutter tip radii are shown Figs.7 (a and b) which are used to generatethegeartoothproIiles. TheIillet proIilegenerated by theIull round rack cutter ismore uniIorm than that oI cutterhavingthetipradius(p)oI0.2mand0.3m.Asthecuttertipradiusincreases,theradiusoIcurvatureoItheIilletproIile increases but there is no change in the involute proIile observed. The required kgc value to balance the (ot)maxingear drive increase (0.526 to 0.5445) as the cutter tip radii increase and the respective balanced (ot)max higher in1647 R. Prabhu Sekar and G. Muthuveerappan /Procedia Materials Science5 ( 2014 )1640 1649 the Iillet region which is generated by the cutter having the tip radius 0.2m and 0.3m than the Iull round rack cutterwhich is mainly due to the higher stress concentration in Iillet region Fig. 7 (c).Fig.5. (a) Thick rim gear; (b) Thin rim gear; (c) Maximum Iillet stress position Ior diIIerent backup ratio;Fig.6. InIluence oI backup ratio (mB), m1, i1.5, zp20, u20o, ha1m, xpxg0 and Iull round tip;(a)(b) (c)1648R. Prabhu Sekar and G. Muthuveerappan /Procedia Materials Science5 ( 2014 )1640 1649 Fig.7. InIluence oIcutter tip radii (p), m1, i1.5, zp20, u20o, ha1m, xpxg0, mB2.22; (a) Partial round cutter; (b) Full round cutter; (c)Tooth thickness coeIIicient (kgc) Vs (ot)max;4.2.3 Influence of aaaenaum moaification factor.The respective values oI kg Ior uniIorm bending strength design and (ot)max are determined by giving the positivecorrection(xp)tothepinionandnegativecorrection(xg)tothegearintherespectivegeardrivesS,S- andSo Iori1.5asshowninFig.8.AsthevaluesoIxpincreaseintheSdrive,theoptimumvaluesoIkgcdecrease,accordingly the balanced (ot)max values also decrease due to increase the correction value to the pinion alone. In theS-drive, Due to change in the value oI xg, the optimum values oI kgc also decrease and a slight variation oI balanced(ot)max is observed Iromthe Fig. 8. But, in the So drive, the positive correction (xp0.1) leads to increase the piniontooththicknessandtheequalnegativecorrection(xg-0.1)decreasethegeartooththicknessandduetothiscombined eIIect, the kgc value required to balance the (ot)max at the pinion and gear pair decrease (kgc0.472) beyondkgc0.5. Thatmeans(kgc0.472)itincreasethetooththicknessoIthegearanddecreasethetooththicknessoIthepinion. From these three cases, the positive correction given to the pinion alone (S drive) is advisable(c)(a) (b)1649 R. Prabhu Sekar and G. Muthuveerappan /Procedia Materials Science5 ( 2014 )1640 1649 Fig.8. Influence of aaaenaum moaification factor, m1, i1.5, :p20, o20o, ha1m, mB2.22, full rouna tip,5 Results and discussionsIn the present study a detailed investigation on the maximum Iillet stress and the tooth thickness coeIIicient oIrack cutter Ior gear (kgc) on NCR spur gear drives have been carried out by using multi pair contact model. TheIollowing observations are made Irom the FEM analysis.1. InthehighergearratioandproIileshiItedgeardrives,the(ot)max onthepinionandthegearisnotequal.ThisdiIIerenceoI(ot)max iscalledinequalityoImaximumIilletstressoIthegearpair.ThisinequalityisremovedbychangingtooththicknesscoeIIicientoItherackcutterIromstandardone(kgc0.5)tonon-standard (kgc=0.5).2. Duetochangethekgc values(Irom0.5tohighervalues),the(ot)max decreases continuouslyinthepinionand increases in the gear. This inequality oI (ot)max is balanced at kgc0.5445 Ior i1.5.3. As the mB values increase, the optimum kgc value does not change and the balanced (ot)max decrease becauseoI minimum deIormation oI gear body is observed Ior the higher values oI mB.4. Whenthereisanincreasein p,theoptimumkgc increaseandthecorrespondingbalanced (ot)max decreasedue to lower the stress concentration in the Iillet region.5. The positive correction given to the pinion and the negative correction given to the gear decrease the bothkgc values and the respective (ot)max.ReferencesM.Celik., 1999. Comparison oI three teeth and whole body models in spur gear analysis.Mechanism and Machine Theory 34, 1227-1235.Wang Jiande., Ian Howard., 2004. The torsional stiIIness oI involute spur gears. IMECE, 218, 131-142.D.V.Muni.,V.Senthilkumar.,G.Muthuveerappan., 2007. OptimizationoIasymmetricspurgeardrivesIormaximumbendingstrengthusingdirect gear design method, Mechanics based design oI structures and machines 35, 127-145.ShutingLi., 2008. EIIectoIaddendumoncontactstrength,bendingstrengthandbasicperIormanceparametersoIapairoIspurgears.Mechanism and Machine Theory 43(12), 1557-1584.