BALKANTRIB O5 5 th INTERNATIONAL CONFERENCE ON TRIBOLOGY JUNE Kragujevac, Serbia and Montenegro

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1 BLKNTRB O5 5 th NTRNTONL CONFRNC ON TRBOLOGY JUN Kragujevac, Serbia and Mntenegr NLYSS OF SURFC PRMTRS OF HLCL GRS WTH SMLLR NUMBR OF TTH Virgil tanasiu, Dumitru Lehchi Department f Mechanical ngineering, Technical University asi B-dul Cpu N.22, 66, asi, Rmania Phne/Fax : mail : vatanasi@mec.tuiasi.r bstract n the paper, the cnditin and the causes f the failure mdes f cylindrical gears are defined in selectin t the mst imprtant running surface parameters f cntacting teeth. ncluded are Hertz cntact stress, minimum film thickness, tth temperature rise. n imprved analytical mdel is used by cnsidering the lad sharing rati and specific kinematics f gear pairs. The aim f this paper is t accurately define the cnditin and causes f the failure mdes t be able t determine crrective actins fr analytical methds used in the design f cylindrical gear pairs. parametric study is cnducted fr helical gear pairs with smaller number f pinin teeth. Keywrds: helical gear, lad sharing, film thickness, tth temperature rise. 1. ntrductin The cylindrical gear pairs with parallel shafts and invlute prfile can realize high values f the gear rati by decreasing the number f pinin teeth. The experimental investigatins indicated that the majr failure mdes f these gear pairs were pitting and scring [1]. The design prcedure f these gears requires t establish the ptimum cmbinatin f gemetrical and kinematic characteristics which prvide the best perfrmance f the running surface parameters under input cnditins. This study is undertaken t investigate these failure mdes in selectin t the mst imprtant running surface parameters f cntacting teeth. ncluded are Hertz cntact stress, minimum film thickness, tth temperature rise. parametric study is cnducted t establish the ptimum amunt f the addendum mdificatin cefficients frm the viewpint f the triblgical criterin. n additin, the paper als includes an analysis fr determining tth lad sharing. 2. Kinematic cnsideratins The characteristic f helical gears is mainly invlved in the inclinatin f the cntact lines. Bth, the psitin and the length f the cntact line at time t have a significant influence n the amunt f the mesh stiffness f gears. Figure 1 shws sme f the relevant features f the meshing plane f actin fr a pair f helical gears. The meshing starts at the pint, passes thrugh pint t and finishes at pint x. n the kinematic analysis it is useful t cnsider an equivalent line f actin as shwn in Figure 1, where t z is the meshing time perid f passing a transverse base pitch f helical gears. 485

2 The psitin f the line f cntact is indicated with the crdinate X f the equivalent line f actin X = X p b (1) Base cylinder Plane f actin x Line f cntact b ε α p bt ε γ ε γ t p bt t z Figure 1: Plane f actin and the equivalent line f actin where p b is the transverse base pitch and X = ~ ε γ. t helical gears, the cntact length f a tth pair is nt a cnstant during the engagement cycle. The time - varying f the length f a cntact line is depended by the rati between transverse cntact rati ε α and verlap rati ε β. The instantaneus length f a cntact line can be expressed as X t l c ε b = α c ε csβ β b x (2) where the parameter c x = f (ε α, ε β ) [3], b represents the face-width f the gear and β b is the helix angle n the base cylinder. 3. Shared lads f meshing teeth The transmitted lad F n is shared between the meshing teeth in the regin f multiple pair cntact. The cnditin f cntact fr each pair f teeth is expressed as F F = e i ei K + = = K + (3) i and N Fi = Fn r i= N Ci = 1 (4) i= C i = F i / F n (5) where F i is the shared lad, e i is the cmpsite errr f i-th pair f teeth, and K i represents the stiffness per unit length f a pair f gear teeth [3]. By slving qs. (3) and (4) fr the case f duble pair cntact, the shared lads are expressed as KF i n KK(e i j i e) j Fi = Ki + Kj (6) F 1,2 i F n G 1,8 F 1,2 i F n G 2,8,4,4 (a) Figure 2: Variatin f lad sharing rati in meshing cycle where j = i+1. f j >, than j =. quatin (6) shws the effect f the mesh stiffness and tth errrs n the magnitude f the individual tth lad. The variatin f the shared lad factrs f the gear pairs presented in Table 1 are shwn in Figure 2 frm the beginning t the end f cntact, where, and indicate the tw tth pairs in cntact. 486

3 3. nalytical Mdels f Triblgical Parameters t is knwn that the pitting is apt t ccur first n the tth surface where the cntact pressure and relative negative sliding are the highest. This assciatin is used as a criterin fr design cmparisn Cntact pressure The cntact pressure between the meshing teeth f the invlute helical gears can be calculated by cnsidering the cntact between tw truncated cnes and is expressed as 1 1 ph = Z W n + (7) ρn1 ρn2 where Z represents the elasticity factr, W n is the nrmal lad per unit lenght, N/m; ρ - radius f curvature at the pint f cntact, m. The subscripts 1 and 2 dente the pinin and gear, respectively; the subscript n refers t nrmal plane. q. (7) is valid at any pint during the cntact cycle and the cntact pressure can be cmputed if the values f ρ n1 and ρ n2 are knwn Film Thickness The predictin f the minimum film thickness in cylindrical gears is based n the analytical mdels f the elasthydrdynamic lubricatin f line cntacts [5], [6], [1]. n the actual analysis, the fllwing mdels are cnsidered: a) The minimum film thickness in the isthermal elasthydrdynamic lubricatin f rlling line cntacts [6] can be written as Wn ηu h, = 3.7 α ρ ρ ( ).57 where: η = viscsity at ambient pressure, Ns/m 2 ; ρ = effective curvature radius, m; (8) α = pressure viscsity cefficient f lubricant, m 2 /N; = effective mdulus f elasticity, N/m 2 ; W n = nrmal lad per unit length, N/m; u = surface velcity in directin f mtin, m/s. The effect due t the thermal effects is given [6] by the frmula.4 ( ) h = h, 1+ Q /1 (9) where 2 γηu Q = kl (1) where: γ = temperature viscsity f lubricant, 1/ C; k l = thermal cnductivity f lubricant, W/(m C ). b) The lubricant film thickness between gear teeth including bth sliding and rlling mtin f cntacting surfaces [4] can be expressed as.7.6 ηu Wn α kl 2.5 n ρ γη s e h = c 2W u P d (11) where: u s = sliding velcity, m/s; γ = temperature viscsity f lubricant, 1/ C; k l = thermal cnductivity f lubricant, W/(m C ); c =.5 3; d =.3.6. n q. (11), the fllwing pressure temperature viscsity relatin is used α p γ( T T η=η e ) (12) where: T = temperature, C; T = inlet lubricant temperature, C; η = viscsity f lubricant, Ns/m 2 ; η = viscsity at ambient pressure, Ns/m 2. criterin used t determine the pssibility f surface distress is the specific film thickness λ. When λ is less than.7 bundary lubricatin prevails and lubricant, surface physical and chemical interactin, lads and temperature said t have a strng effect n distress mdes and rates Surface Temperature Rise Blk s thery states that scring will ccur 487

4 when the maximum temperature T c f cntacting teeth exceeds a critical temperature T crit, where T T T c = b + (15) n eq. (15), T b is the temperature f the tth surfaces befre meshing. The surface temperature rise T in the thermal elasthydrdynamic lubricatin f line cntacts (Taylr, 1985) can be expressed as 2µ W u T = ψ n s 1.5 p ( π v1 Ks ρs cs bh) 2µ W u T = ψ n s 2.5 G ( π v2 Ks ρs cs bh) (16) (17) where : T1 and T2 represent the temperature rise f the pinin and gear, respectively; K s = thermal cnductivity f slid, W/m C ; c s = specific heat f steel gears, J/kg C ; ρ s = density f steel gears, kg/m 3 ; b H =semiwidth f cntact; v = instantaneus tangential velcity at cntact pint; ψ = heat partitin cefficients fr pinin and gear. t can be seen frm qs. (16) and (17) that the temperature rise is directly prprtinal t the cefficient f frictin and a reasnable evaluatin f its value is essential fr the temperature predictin. The instantaneus cefficient f frictin is expressed as 2 y y µ= 3 µ m 1.5µ m l a,g l a,g where.25 WR t a µ m =.12 XRXη uη where: X R = rughness factr, (18) (19) X η = lubricatin factr ; W t = tangential frce per unit length, N/m; R a = nminal surface rughness. n q. (18), y is the distance between the cntact pint and pitch pint and l a,g represents the length f apprach path r recess path, respectively. The cmputatinal methdlgy fr the instantaneus cefficient f frictin is based n the predicted mean values at the middle length f the apprach and recess path [2]. The scring is initiated when a lcal instantaneus ttal cntact temperature in ne pint n the tth flank exceeds a critical temperature T crit. When this temperature is larger than 135 C, the scring prbability is higher. f any f the abve reach critical prprtin scring may ccur in a sense f incipient scring. 4. Results and discussin Specificatins f the analyzed gear pairs are shwn in Table 1, where: z represents the number f teeth, m n = nrmal mdule, x n = addendum mdificatin cefficient (related t m n ), x nmin = minimum value f addendum mdificatin cefficient t prevent undercutting; a w = center distance, b = face-width; β = helix angle n the pitch cylinder; l g = length f actin f apprach; ε α = transverse cntact rati, ε γ = ttal cntact rati. Test gears are made f 4Cr1 quenched and tempered steel. The tth surfaces f these gears were finishing by hbbing. The initial surface rughness R a was abut 1.2 µm. The values f addendum mdificatin cefficients are determinated frm the gemetrical cnditins and the resistance f gear teeth t surface lading [1]. 488

5 Table1. Specificatins f the gear pairs Gear z Pairs 1 z 2 x n1 L g ε α ε γ G1.737 * G * x nmin ; m n = 3 [mm]; a w = 9 [mm]; b = 3 [mm]; β =2 p H 2 G 1 16 MPa (a) p H 1 G 2 MPa Figure 3: Variatin f Hertz cntact pressure alng the equivalent line f actin The results presented in Figs. 3-5 permit t analyse the variatin f surface temperature rises, and calculated HD film thickness in relatin t maximum Hertzian stress. n these figures,,, C dente the beginning f engagement, end f engagement and pitch pint f the pinin, respectively. Figure 3 shws the distributin f the Hertz stress alng the equivalent line f actin. The cntact Hertz pressure is fund t be much imprved as the addendum mdificatin cefficient x n1 changes frm.737 t 1.1. The temperature rise n bth the pinin and gear teeth cntacting surfaces is pltted in Figure 4 as a functin f the cntacting psitin fr tw different speeds. The crrespnding film thickness is shwn in Figure 5 as a functin f tth cntact psitin. These values are cmputed by using q. (11), where c = 2.6 and d =.54. The specific film λ is less than.7 and the bundary lubricatin prevails in the case f helical gears with small number f pinin teeth. t is seem that the temperature rise fr bth the pinin and gear is higher at the beginning than the ending f the engagement. The higher temperature rise and a reduced minimum film thickness result at the ending f the engagement when the pinin speed increases frm n 1 = 18 rpm t n 1 = 34 rpm. The experimental results [1] evince that the cntact fatigue life f test gears is gverned by specific aspects f the pitting distributins n the teeth f the pinin and gear, in terms f addendum mdificatin cefficients. 489

6 Τ 1 1 n 1 = 18 [rpm] G 1 C (a) Τ 2 4 C 3 G n 1 = 18 [rpm] Τ 1 12 n 1 = 34 [rpm] G 1 C 9 6 Τ 2 8 n 1 = 34 [rpm] G 2 C (d) Figure 4: Variatin f temperature rise alng the equivalent line f actin 49

7 h µ m.6.4 n 1 = 18 [rpm] G 1 h µ m.6.4 n1 = 18 [rpm] G (a) h µ m.6.4 G 1 h.6 µ m.4 G n 1 = 34 [rpm] n 1 = 34 [ ] (c) (d) Figure 5:Variatin f film thickness alng the equivalent line f actin The experimental results [1] evince that the cntact fatigue life f test gears is gverned by specific aspects f the pitting distributins n the teeth f the pinin and gear, in terms f addendum mdificatin cefficients. Thus, at the gear pair with x n1 = x n1min and HB 1 /HB 2 = 1, the pitting cracks ccurred near the tth rt f the pinin; at the mating gear the pitting ccurred n the utside engagement zne. f the cefficient x n1 = 1.1, the pitting cracks ccurred nly n the teeth f the gear. n the mentined situatins, the pitting failure ccurred n the znes with negative specific slidings. When x n1 = 1.1 and the rati HB 2 /HB 1 >1, the pitting cracks may ccur nly n the tth surfaces f the pinin. Under a heavy lad, the surface temperature rise f the cntact zne is high and the HD film thickness decreases. When the rtatinal speed f the pinin is larger than 18 rpm, it seems that that the prbability f metallic cntact becmes high. n intermittent incipient scring ccurred n the tth surfaces f the pinin with n 1 = 34 rpm and HB 1 = HB 2 = 31. This phenmenn may ccur when the lad exceeds a critical limit value. Therefre, the initial tth surface shuld be finished as smthly as pssible in rder t btain high surface durability f these helical gears. 5. Cnclusins nalyses f the mst imprtant parameters, which are believed t act n the tth surfaces and are significant amng influences n the failure mdes are described. The prgnsis f the running surface parameters is imprved by using the shared lad in the analytical mdel f the triblgical parameters. The lad sharing rati is analytically expressed by cnsidering the mesh stiffness f each individual tth pair that is simultaneusly in cntact. The best results f the surface durability were btained when the amunt f the addendum mdificatin cefficient f the pinin was a little large than the limit f the undercut and a lw hardness pinin was cmbined with a harder gear. The cylindrical gears with high gear rati and smaller number f pinin teeth perate under cnditins f bundary lubricatin. The 491

8 prbability f scring failure is higher in the case f larger values f the pinin speed. 6. References [1] tanasiu, V.,1993, n nalysis f the Surface Failure Mdes f Helical Gears with Small Number f Pinin Teeth, Bul. nst. Plit. asi, XXXX (XL), 1 4, S.V, pp [2] tanasiu, V., Lehchi, D., 1994, Frictin Cefficients f Cylindrical Gears with Smaller Number f Teeth, Bul. nst. Plit. asi, XL(XLV), 1 4, S V, pp [3] tanasiu, V., 1998, n nalitical nvestigatin f the Time Varying Mesh Stiffness f Helical Gears, Bul. nst. Plit. asi, XLV (XL), 1 2, S V, pp [4] Drzdv, n., Tumanishvily, G.L., 1978, The lubricant layer thickness befre the scuffing f rubbing bdies,. Vestnik Mashinstrenia, N.9, pp [5]Gnsh,M.K., Pandy,R.K.,1998, Thermal lasthydrdynamic Lubricatin f Heavy Laded Line Cntacts, Jurnal f Triblgy, Vl.12, N.1,pp [6] Hamrck, B.., Jacbsn, B.O.,1983, lasthydrdynamic Lubricatin f Line Cntacts, SL Transactins, Vl. 27, 4, pp [7] Kragelski,.V., lisin, V.V., 1986, Frictin, Wear, Lubricatin, Mir Publishers, Mscw. [8] Manin, L., Play, D., 1999, Thermal Behaviur f Pwer Transmissin, Numerical Predictin, and nfluence f Design Parameters, Jurnal f Triblgy, Vl.121, N.4, pp [9] Taylr T.C., Seireg., 1985, n Optimum Design lgrithm fr Gear Systems ncrprating Surface Temperature, Jurnal f Mechanisms, Transmissins and utmatin in Design, 7, pp [1] Wang, K.L., Cheng, H.S., 1981, Numerical Slutin t the Dynamic Lad, Film Thickness and Surface Temperatures in Spur Gears, Jurnal f Mechanical Design, vl. 13, pp