Bending Geometry Factor For Profile Corrected Involute Gear Tooth With Trochoidal Fillet

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1 Bending Geomey Faco Fo Poile Coeced Involue Gea Tooh Wih Tochoidal Fille S.P.Ganesan and G. Muhuveeappan 1 Comba Vehicles Reseach and 1 Machine Design Secion Developmen Esablishmen DRDO Mechanical Engineeing Depamen Chennai IIT Chennai Keywods: Spu Gea, Poile Coecion, Tochoid, Geomey Faco, Bending Sess. ABSTRACT The poile o he ille places a vey impoan ole on he acual ille bending sess. The AGMA appoximaes he ille cuvaue wih a consan adius on is layou pocedue wheeas his pape evaluaes bending sess in gea aking acual Tochoidal ille geneaed by he ack cue. The geomey aco o he uncoeced gea is calculaed and compaed wih ha o AGMA. The wok is exended o he coeced gea consideing a case whee he cue ooh deph is.5 imes module and he cue adius is 0. imes he module. Acual gea sess is deemined using Finie Elemen Mehod wih loading boh a he Tip Load (HPTC) and Highes Poin o Single Tooh Conac (HPSTC). 1. INTRODUCTION Fo high powe ansmission like ha used in Bale ank, which equies compacness and abiliy o ansmi high oque wih inie lie o geas, which may in one way be achieved by using Epicyclic gea ain and poile coeced geas. Hence in such ciical aeas he gea ooh sesses need o be exacly deemined consideing he acual gea ooh geomey wih Tochoidal ille. Aled Lewis was he is peson o calculae he bending sess in 189 assuming gea ooh o be a uniom sengh paabolic canileve beam neglecing he adial componen o oce and he sess concenaion a he ille o gea ooh.

2 * Coespondence: Phone: , ax: In his pape he advances made in Compuaional mehod using Finie Elemen Analysis and analyical mehod is used in evaluaing he bending geomey aco o poile coeced spu gea aking exac Tochoidal ille geneaed wih ack cues. The load applied a he ip o gea ooh as well as he load a he highes poin o single ooh conac ae consideed in he pesen wok o evaluae he Geomey aco. The complee gea poile is geneaed by a VC++ pogam and he sess coecion aco is evaluaed using ANSYS 6.0 FEA package. The J aco obained o he sandad gea is compaed wih AGMA [4] J aco and he wok is exended o poile coeced gea.. GENERATION OF GEAR TROCHOID PROFILE: The Fig 1 shows geneaion o Tochoidal ille by he ip o he basic ack wih a 0 (shap cue). Coodinaes o he ochoid ae calculaed using Analyical mehod as shown in Eq (1) o (7). The ype o Tochoidal ille changes wih paamees o he cue like ype o cue (Pinion o Rack), edge adius (a), addendum (b), pessue angle and poile coecion equied in he gea. Fig 1: Geneaion o Tochoidal ille. an ψ an 1 [ b ] [ b ] [ b ] [ b ] [ b ] Eq (1) Eq ()

3 R + a a sin ψ Eq (3) ' ' x y δ + cos 1 a R sin ψ Eq (4) " R * Sin ( ) Eq (6) R * Cos ( ) Eq (7) + " Eq (5) 3. CALCULATION OF BENDING STRESS: The heoeical bending sess accoding o Wiled Lewis [1] omula is given by WT σ Eq (8) h * m* Y Modiied Fom aco (Y) is obained by consideing he gea ooh as a beam, ixed a one end and loaded a he ohe. Modiied Fom Faco calculaed om he geomeic layou o he gea ooh poile. The above Eq (8) doesn include he sess coecion eec. Consideing he sess coecion eec he Eq (8) is modiied as WT * K > W σ σ T Eq (9) h h * m* Y * m* J Y Whee J Eq (10) K in which 1 Y c Eq (11) / cosαa 6h anαa m * * ± cosα The gea paamee a he ciical secion like hickness(), eecive momen am (h), adius o cuvaue ( ) ae calculaed analyically om he geomey o gea. And accoding o AGMA sess coecion aco (K ) is obained om empiical elaion aived om expeimens conduced by Dolan & Bohame [3] o Ф 0 deg is given in Eq (1) as K e * R h Eq (1)

4 In his pape sess coecion aco K is aken as he aio o acual sess (obained om he Finie Elemen Mehod using Ansys 6.0 package) o he heoeical sess om he Lewis Eqn. K σ ansys σ Eq (13) lewis The Eq (13) akes cae o ceain pacical eecs such as a) Eecive sess concenaion. b) Locaion o load. c) Size eec. d) End o ooh eec. 4. FINITE ELEMENT ANALYSIS To deemine he sess coecion aco he acual sess in he gea ooh ille and he coodinae o he ciical poins a which i occus ae deemined hough Ansys 6.0 sowae. Fo solving he gea secion is consideed as a plain sain case. The suace and ille poile geneaed using a paicula cue ae made use o in Finie Elemen Analysis. The Rim hickness aco K B is aken as 1.0 and a backup aio m B o 1.0 based on he Dago s [6] analysis o gea ooh bending aigue sengh so ha he im hickness doesn have any eec on he bending sengh o gea ooh. Fig shows he meshing o he gea ooh wih he Tip loading and Fig 3 shows he nodal sess plo in he deomed shape wih maximum sess occuing a he Tochoidal ille o he gea ooh.

5 Fig : Meshing o Gea Tooh 5. RESULTS AND DISCUSSION Fig 3: Nodal Sess Plo in deomed shape The Bending geomey aco ae evaluaed o poile coeced gea geneaed wih ack ype cues, boh condiions whee he load being applied a he Tip o ooh and load being applied a he Highes poin o single ooh conac. The Bending geomey aco accoding o AGMA (empiical elaion o K e ) is calculaed using he Flowcha in AGMA [5] and is compaed wih esuls obained om analyical cum inie elemen appoach (Ansys). Pinion wih 0, 30, 40 eeh sepaaely meshed wih geas o vaious numbe o eeh anging om 1 o 75 eeh whee consideed o analysis o evaluae he Bending Geomey aco wih and wihou coecion in he gea. Cue speciicaion: Tip adius a 0.0 imes module Coecion X 0.1, 0., 0.3 Tooh deph.5 imes module This analysis is exended o Highes Poin o Single ooh Conac (HPSTC) and Tip Load (HPTC). Fig 4 shows J aco obained om he pesen mehod closely ageeing

6 Fig 4 J-Faco o uncoeced gea (Tip) wih ha o AGMA [5] value, o Tip Loading (HPTC) wih a maximum deviaion o 4%. Fig 5 shows J aco o coeced gea wih HPTC loading. Fig 6 shows he J aco o poile coeced gea wih HPSTC Loading. Fig 7 shows he compaison beween HPTC and HPSTC Loading.

7 Fig 5 J-Faco o coeced gea (Tip) Fig 6 J-Faco o coeced gea (hpsc) Fig 7 HPSC & Tip Loading Compaison

8 6. PRACTICAL SIGNIFICANCE OF PAPER: i. The ille adius is assumed as smooh o calculaing he geomey aco in AGMA [4], whee he acual shape o he ochoid is consideed in his pape. ii. The cue geomey geneaing he gea like cue edge adius, poile coecion, pessue angle and whole deph aco could be vaied as pe he as pe he equiemen o deemining he Geomey aco o he Gea. iii. The laes advances made in he compuaional mehod is used in he evaluaing he sess coecion aco. NOMENCLATURE a Cue edge adius o ack cue. b Dis be pich line o cue & cene o ue edge adius Face widh h Eecive momen am J Bending Geomey aco K Sess Coecion aco K e Sess Concenaion aco om Phooelasic mehod K B Rim hickness aco m Module m B Backup aio o he gea im R, R Radius veco o he ochoid and oo ille. Pich cicle adius o gea. Radius o ochoid Radius o cuvaue a he ciical poin Thickness a he ciical secion w Widh a he Ciical secion W Tangenial applied load x,y Coodinaes o he geneaed Tochoid Y Modiied Lewis om aco Z Numbe o he Teeh in Gea wheel σ b Bending sess φ Pessue angle δ Angle be cene o gea ooh & cene o ochoid Angle be adius veco and cene line o ochoid ψ Angle be he adius veco & angen o ochoid

9 Subscip c p g Ansys Lewis Compessive ensile Pinion Gea om Ansys om Lewis Equaion REFERENCES 1. Lewis, W., Invesigaion o he sengh o gea eeh, Poc. o he Enginees Club, Philadelphia, PA, 1893, PP Eale Buckingham, Analyical Mechanics o Gea Geomey, New Yok, Dove Publicaions, Dolan, T.J. and Boghame, E.L., A Phooelasic Sudy o he Sesses in Gea Tooh Filles, Univesiy o Illinois, Engineeing Expeimen Saion, Bullein No. 335, AGMA inomaion shee 908-B89, Geomey acos o deemining he piing esisance and bending sengh o spu, helical and heingbone gea eeh. 5. AGMA inomaion shee 918-A93, A summay o Numeical Examples demonsaing he Pocedues o Calculaing Geomey Facos o spu and Helical geas. 6. Dago, R.J., An impovemen in he Convenional Analysis o Gea Tooh Bending aigue Sengh, AGMA P4.4, and Ocobe Mei. H.E, Gea Engineeing, Piman pess, Newyok, Xiaogen Su, Donald R. House, Chaaceisics o ochoids and hei applicaion o deemining gea eeh ille shapes, Mechanism and Machine Theoy, Volume 35, 000,