Derivation of Solution for Elliptical Elastohydrodynamic. Contact Patches with Side-Slip and its Application to a. Continuously Variable Transmission

Transcription

1 T rport committ for Cristopr William Scnidr Crtifis tat tis is t approvd vrsion of t following rport: Drivation of Solution for Elliptical Elastoydrodynamic Contact Patcs wit Sid-Slip and its Application to a Continuously Variabl Transmission APPROVED BY SUPERVISING COMMITTEE: Suprvisor: Mical D. Bryant Co-suprvisor: Ofodik A. Ezkoy

2 Drivation of Solution for Elliptical Elastoydrodynamic Contact Patcs wit Sid-Slip and its Application to a Continuously Variabl Transmission by Cristopr William Scnidr, B.S. Rport Prsntd to t Faculty of t Graduat Scool of t Univrsity of Txas at Austin in Partial Fulfillmnt of t Rquirmnts for t Dgr of Mastr of Scinc in Enginring T Univrsity of Txas at Austin Dcmbr 2011

3 Drivation of Solution for Elliptical Elastoydrodynamic Contact Patcs wit Sid-Slip and its Application to a Continuously Variabl Transmission By Cristopr William Scnidr, MSE T Univrsity of Txas at Austin, 2011 SUPERVISOR: Mical D. Bryant Elastoydrodynamic lubrication (EHL) allows transfr of powr and forcs in gars and rolling barings witout surfac-to-surfac contact and is t basis for a continuously variabl transmission studid in tis rport. Prvious rsarc constructd modls and drivd solution mtods, but oftn lackd full xplanations of t approac and was usually applid to limitd and spcific cass. Tis rport prcisly dvlops t numrical solution of EHL contact and includs t mor gnral cass of lliptical contacts and sidslip. T modl and numrical mtod ar validatd on known bncmark cass and tst rsults. Sid-slip is invstigatd and t rsults sown in tis rport. Finally, t modl is usd to dtrmin t film ticknss and prssur of a contact patc undr idntical conditions to tat in a pysical driv dvlopd by Fallbrook Tcnologis in Austin, TX. A minimum film ticknss of μm is found, stting a bncmark for t maximum allowabl surfac rougnss valus to prvnt surfac-to-surfac contact. Additionally, undr normal driv conditions t film ticknss to surfac rougnss ratio is in t rang of idal valus for maximum lif. iii

4 Tabl of Contnts 1. Introduction Kinmatics Hrtz Contact Dtaild Contact Dimnsionlss Equations Discrt Equations and Scm Numrical Solution Mtod Comparison of Numrical Solution and Litratur Vnnr Cass Effct of sid-slip Fallbrook Driv Contact Conclusion Works Citd iv

5 List of Tabls Tabl 3-1. Elliptic Intgral Paramtrs... 8 Tabl 7-1: Proprtis for t two tst cass Tabl 7-2. Hamrock and Dowson stimat of minimum film ticknss Tabl 7-3. Comparison of Vnnr and Scnidr minimum and cntral film ticknsss wit constant dnsity and t prcnt diffrnc Tabl 7-4. Comparison of Vnnr and Scnidr minimum and cntral film ticknsss wit Rolands dnsity and t prcnt diffrnc Tabl 7-5. Cas 1s, variabl dnsity and Rolands viscosity. Dimnsionlss film ticknss wit varying transvrs spds Tabl 7-6. Cas 2s, variabl dnsity and Rolands viscosity. Dimnsionlss film ticknss wit varying transvrs spds Tabl 7-7. Gomtry of driv Tabl 7-8. List of paramtrs for Cas FB Tabl 7-9. Dimnsionlss film ticknss for t Fallbrook Tcnologis driv v

6 List of Figurs Figur 1-1 Fallbrook CVP. Original imag from Fallbrook Tcnologis Inc. wit modifications mad r Figur 2-1. Kinmatics of CVP. Original imag from Fallbrook Tcnologis Inc. wit modifications mad r Figur 7-1. Cas 1 (L=10, M=20), variabl dnsity, Rolands viscosity rsults. Lft is 64x64 grid siz, rigt is 128x128 grid siz. Top to bottom ar 3D plot of dimnsionlss prssur and 3D plot of dimnsionlss film ticknss Figur 7-2. Cas 1 (L=10, M=20), variabl dnsity, Rolands viscosity rsults. Lft is 64x64 grid siz, rigt is 128x128 grid siz. Top to bottom ar 2D contour plot of dimnsionlss film ticknss and 2D contour plot of dimnsionlss prssur Figur 7-3. Cas 2 (L=10, M=200), variabl dnsity, Rolands viscosity rsults. Lft is 64x64 grid siz, rigt is 128x128 grid siz. Top to bottom ar 3D plot of dimnsionlss prssur and 3D plot of dimnsionlss film ticknss Figur 7-4. Cas 2 (L=10, M=200), variabl dnsity, Rolands viscosity rsults. Lft is 64x64 grid siz, rigt is 128x128 grid siz. Top to bottom ar 2D contour plot of dimnsionlss film ticknss and 2D contour plot of dimnsionlss prssur Figur 7-5. Minimum film ticknss rror of fluid wit constant dnsity and Barus viscosity Figur 7-6. Minimum film ticknss rror of fluid wit variabl dnsity and Rolands viscosity Figur 7-7. Cas 1s wit sid slip on 128x128 grid siz. Lft is dimnsionlss prssur and rigt is dimnsionlss film ticknss. Top to bottom is v =0, 0.1 m/s Figur 7-8. Continud from prvious figur. Lft is dimnsionlss prssur and rigt is dimnsionlss film ticknss. Top to bottom is v = 0.2, 0.3, 0.4 m/s Figur 7-9. Cas 1s wit sid slip on 128x128 grid siz. Lft is dimnsionlss prssur and rigt is dimnsionlss film ticknss. Top to bottom is v =0, 0.1 m/s Figur Continud from prvious figur. Lft is dimnsionlss prssur and rigt is dimnsionlss film ticknss. Top to bottom is v = 0.2, 0.3, 0.4 m/s Figur Cas 2s wit sid slip on 128x128 grid siz. Lft is dimnsionlss prssur and rigt is dimnsionlss film ticknss. Top to bottom is v = 0, 0.1, 0.2 m/s Figur Continud from prvious figur. Lft is dimnsionlss prssur and rigt is dimnsionlss film ticknss. Top to bottom is v = 0.3, 0.4 m/s vi

7 Figur Cas 2s wit sid slip on 128x128 grid siz. Lft is dimnsionlss prssur and rigt is dimnsionlss film ticknss. Top to bottom is v =0, 0.1, 0.2 m/s Figur Continud from prvious figur. Lft is dimnsionlss prssur and rigt is dimnsionlss film ticknss. Top to bottom is v = 0.3, 0.4 m/s Figur Fallbrook Tcnologis cas (L=17.96, M=238.4), variabl dnsity, Rolands viscosity rsults. Lft sid ar dimnsionlss prssurs and rigt sid ar dimnsionlss film ticknss Figur Load lif vs. film ticknss ratio valu. T filld in circls giv pur rolling and t opn circls giv slid-roll ratio qual to vii

8 Nomnclatur a smi-maor axis of contact llips [m] b smi-minor axis of contact llips [m] c v carrir E i Young's modulus of it componnt [Pa] E p rducd modulus of lasticity [Pa] F contact normal forc [N] F x traction forc [N] G dimnsionlss matrial paramtr, G = αe p film ticknss [m] H dimnsionlss film ticknss, H = R x/a 2 0 film ticknss constant [m] H 0 dimnsionlss film ticknss constant, H 0 = 0R x/a 2 x y i k k a k b ms grid siz in t x dirction ms grid siz in t y dirction grid indx in x dirction grid indx in y dirction llips ratio, k = b/a smi-maor axis paramtr calculating siz of t Hrtz llips smi-minor axis paramtr calculating siz of t Hrtz llips K D quivalnt contact ratio factor [m] K k tst L M N p n x n y discrt dformation krnl paramtr usd to dtrmin smi-maor and smi-minor axs Mos dimnsionlss matrial paramtr Mos dimnsionlss load paramtr Numbr of plants numbr of grid points in x dirction numbr of grid points in y dirction p prssur [Pa] P dimnsionlss prssur p 0 constant in t Rolands viscosity quation, 1.96E8 viii

9 p maximum rtzian prssur, p = 3F/2πab [Pa] r rsidual R 1 ring 1 R 2 ring 2 R i' rolling contact radius of t it componnt [m] R i" transvrs contact radius of t it componnt [m] R x quivalnt transvrs contact radius of t contact pair R y quivalnt rolling contact radius of t contact pair [m] SR spd ratio s v sun u ntraining vlocity u = (u 1+u 2)/2 [m/s] u i rolling vlocity of it componnt [m/s] U s dimnsionlss spd paramtr, U s = η 0u m/e pr x v sid-slip ntraining vlocity v = (v 1+v 2)/2 [m/s] v i transvrs vlocity of t it componnt [m/s] W dimnsionlss load paramtr, W = F/E pr 2 x x lngt from contact cntr in t rolling dirction [m] X dimnsionlss coordinat, X = x/a y lngt from contact cntr in t sid-slip dirction [m] Y dimnsionlss coordinat, Y = y/b z prssur viscosity indx, typically 0.67 α prssur viscosity indx [Pa -1 ] δ cang du to rlaxation η viscosity [Pa-s] η 0 atmospric viscosity [Pa-s] dimnsionlss viscosity film ticknss ratio dimnsionlss paramtr μ traction cofficint ν i ξ poisson ratio of it componnt cofficint in Rynolds quation ix

10 ρ dnsity [kg/m 3 ] ρ 0 atmospric dnsity [kg/m 3 ] dimnsionlss dnsity ω rotational vlocity [m/s] x

11 1. Introduction On trnding mtod to incras powr train fficincy is to rplac t convntional gard transmission wit a continuously variabl transmission (CVT). T CVT allows for a continuous rang of gar ratios ltting t powr sourc run at its idal and most fficint oprating point. In automobils, t lack of discrt transitions yilds smootr and fastr acclration. T ovrall fficincy and prformanc of t vicl is improvd. Nissan is currntly t lading manufacturr tat offrs CVTs in narly all of tir modls, wit approximatly on million CVTs in srvic. Honda, Audi, and Mini fatur modls wit CVTs. Wit mor stringnt EPA standards and ig gasolin prics, CVTs will s mor us in commrcial automobils. CVTs com in svral configurations; t most popular in automobil usag is blt or pully drivn. Howvr, ty ar limitd in tir packaging and rquir a ig-prssur pump for optimal variator fficincy. Intrst in traction drivs, lik Toroidal typs using rolling traction to transmit torqu, as bn growing du to tir passiv clamping and mor flxibl packag rquirmnts. T CVT typ of intrst in tis study is a plantary similar to a Milnr CVT wit on or mor sprical plants, a sun, carrir, and two rings as sown in Figur 1-1 r trmd continuously variabl plantary (CVP). An input torqu is applid to ring 1 (R1) wic is transfrrd to t rolling sprical plant via a fluid lubricatd contact btwn ring and plant. T plant sits on a fr spinning sun (S v) and rotats about a carrir (C v). T plants transfr powr from plant to t output disk ring 2 (R2) troug anotr fluid lubricatd contact btwn t plant and ring. Du to t vry ig prssurs and typ of fluid, t lubrication bavs lastoydrodynamically. Hr t vry ig prssurs induc significant dformation in contacting solid surfacs wic nlargs t ticknss of t film witin t contact. Also t viscosity of t lubricant, wic is prssur dpndnt, can incras substantially. Tis typ of CVP is undr dvlopmnt at Fallbrook Tcnologis Inc. in Austin. Ty succssfully introducd a bicycl CVP and look to dvlop a rliabl automotiv CVP. Ts drivs nd long lif and durability comparabl to currnt automotiv transmissions, for commrcial succss. 1

12 Figur 1-1 Fallbrook CVP. Original imag from Fallbrook Tcnologis Inc. wit modifications mad r. Many factors influnc t lif of a driv, but of importanc in tis papr is t film ticknss and prssur. Mor accurat maximum prssur valus can b usd to updat strsss applid pr cycl. Mor importantly, t film ticknss in rlation to surfac rougnss is a maor factor in lif. If t surfac aspritis ar too larg, ty will allow for surfac-to-surfac contact and caus war. Rsarc also suggsts tat too larg a film ticknss advrsly affcts t lif of t rolling surfacs. Basic rlations for t film ticknss xist, but a mor sopisticatd and dtaild modl is rquird to accuratly dscrib driv conditions and dfinitivly provid answrs. Consquntly, a dtaild modl for EHL contact nds to b dvlopd. Tis modl is outlind in prvious work suc as Vnnr [1], Campos [2], and Kim [3] among otrs and is adaptd for lliptical contacts lik tos in t Fallbrook CVT. Wil tos works dscrib t numrical mtods usd, many do not xplicitly driv t quations usd. Driving and sowing tos quations r will b usful for futur work on t subct or similar problms. Tus tis papr will xplicitly sow t drivation of a numrical modl of EHL lliptical contacts, affirm its accuracy wit prviously publisd solutions, and dtrmin if t Fallbrook driv is working undr idal conditions for long lif. 2

13 2. Kinmatics T kinmatics of t CVP is discussd in tis sction. Figur 2-1 sows t dtaild CVP gomtry and trminology. Rfr to Figur 1-1 for a mor gnral prspctiv of t sam systm. Figur 2-1. Kinmatics of CVP. Original imag from Fallbrook Tcnologis Inc. wit modifications mad r. Fiv main componnts ar visibl in Figur 2-1 and Figur 1-1. T rings (R 1 and R 2), plant (spr sown), sun (S v), and carrir (C v). T angl of t carrir,, varis to cang t distanc from t plant axis of rotation to t contact points dnotd by r p2, r p1, and r ps wit α 1 and α 2 dnoting t angl btwn t vrtical axis and t point of contact for ring 1 and ring 2. r t1, r t2, r c, and r s ar t radii from t cntral axis of rotation to contact point at ring 1, ring 2, cntr of t plant, and t sun rspctivly. T radius of t plant is r p. Angular spds ar dnotd by ω 1, ω 2, ω c, and ω s for ring 1 (input spd), ring 2 (output spd), t carrir, and t sun. T kinmatics prsntd assum idal componnts wit no powr loss across intrfacs. 3

14 Spd Ratio Any powr pat may b cosn, but t most common in t automotiv CVP is cosn wit t carrir fixd, t sun fr, and t rings 1 and 2 input and output rspctivly. T kinmatic spd ratio is trfor 2 SR (2-1) 1 wit t torqu ratio simply t invrs. Gomtric rlationsips ar dfind using fixd paramtrs for plant radius (r p), sun radius (r s), and ring contact angls ( 1 and 2 ) to dfin rc rs rp t1 s p 1 r r r 1 cos (2-2) r r r 1 cos t2 s p 2 T angl cangs t driv ratio by canging t radii from plant to contact points as dfind blow, r r cos p1 p 1 rp2 rpcos 2 (2-3) r r cos ps p For an idal driv t point contact surfacs av t sam vlocity, or r r 1 t1 p p1 r r 2 t 2 p p2 (2-4) Solv on quation for ω p and substitut into t otr quation. Assuming r t1 and r t2 ar qual and using t rlations from Equation (2-3), t input and output spds ar r cos p rp 1 cos 1 (2-5) 4

15 Substituting Equation (2-5) into Equation (2-1) yilds SR k cos 2 cos 1 (2-6) Equation (2-6) sows xplicitly tat t spd ratio ffctivly t gar ratio of t CVP is a function of. Bcaus ty ar usful quantitis, t rotational spds of t plant and sun ar sown in Equation (2-7). r p 1 r t1 p1 rps sv p r s (2-7) Not tat t angular vlocity for t sun contact is ngativ. To find t forcs on ac contact patc us kinmatics, F F R1 R2 N Faxial sin P 1 2 Faxial (2-8) N sin P wr F axial is t axial forc and N p is t numbr of plants. 5

16 3. Hrtz Contact Tis sction provids a basic ovrviw of t rlvant dtails from Hrtzian contact mcanics. For a full drivation and xplanation s [1]. T CVP transmits load troug rlativly larg rolling lmnts, dscribd in Hrtz tory of bodis of gnral profil. Bodis may av a primary radius of curvatur, R x, in t rolling/traction dirction and a scondary radius of curvatur, R y, in t dirction transvrs to rolling. According to Hrtz tory, undr load t two lastic bodis dform and form an lliptical prssur distribution in t ara of contact givn by 2 2 x y p p 1 a b (3-1) Wr p is t maximum Hrtzian prssur and a and b ar t smi-axis in t coincidnt wit t x- and y-dirction rspctivly. T maximum prssur is found by intgrating t prvious quation and quating to t normal forc, F. Hnc, p 3 F (3-2) 2 ab T smi-maor and smi-minor axs of t contact llips ar givn by t following xprssions from Roark s Formula for Strss and Strain [4], a k F a b k F b 1/3 1/3 (3-3) T alpa and bta cofficints ar givn by, 1/3 KD k k E p 1/3 KD E p (3-4) K D is t quivalnt contact ratio factor givn by, 6

17 K D (3-5) R R R R ' " ' " Wr t cofficints ar dpnding on t diffrnt smi-axis wic dpnds on t rlativ radii as, wr, k k k k a b a b k k k k if k 1 tst if k 1 tst (3-6) k tst R R y x T cofficints and ar found by calculating cos wic is found by: 2 2 2K cos D 2 cos ' " ' " ' " ' " 3 R1 R1 R2 R2 R1 R1 R2 R 2 1/2 (3-7) wr is t angl of skw btwn t plans of t primary radii and t contacting bodis and is 0 if corrctly alignd. T α and β cofficints in Equation (3-4) ar found using t lookup tabl blow. 7

18 Tabl 3-1. Elliptic Intgral Paramtrs cos θ α β T llips ratio can b approximatd from [5] and [6] as, k a R y b Rx R X and R Y, t ffctiv radii of curvatur in t X and Y-dirctions, ar givn by: R R R X 2/3 ' ' R R R Y " " 1 2 Wr t subscripts 1 and 2 rfr to t rspctiv contact body and t suprscripts rfr to t rolling or transvrs dirction. T contact modulus is dfind as, (3-8) E E E P 1 2 Also of not is t traction cofficint wic is dfind as, 8

19 Fx F Mu is also known as t cofficint of friction and it dpnds on t rtz prssur, (3-9) ntrainmnt vlocity, tmpratur, fluid proprtis, and t rlativ motion of t surfacs. 9

20 4. Dtaild Contact Caractristic of Elastoydrodynamic Lubrication (EHL) problms ar ig prssur, dformation of t contacting solid surfacs, vry tin film ticknss, and ig sar rats. Viscosity and dnsity cannot b assumd constant. Classic works on t topic ar Dowson and Higginson [7] for t lin contact problm and Hamrock and Dowson [8] for t point contact. Vnnr sowd solutions for circular contacts [1]. Starting wit t gnral quations, Rynolds Equation tat govrns film prssur distribution [9] 3 3 p p ( ) ( ) 12u 12v 0 x x y y x y (4-1) wit p=0 on t boundaris and p 0 vrywr (cavitation condition). If including spin, t quation bcoms: 3 3 p p ( ) ( ) 12u y 12v x x x y y x y but sinc gnrally t contact patc sizs ar vry small, spin as a vry small ffct. T slip and sid-slip vlocitis ar givn as [3] wit spin addd, z p z z 2 2 u z u1 u2 x y cos 2 x z p z z 2 2 v z v1 v2 x y cos 2 y (4-2) T man vlocitis ar givn as, Viscosity u v u u v v (4-3) T viscosity is givn by t Barus quation [10]: 10

21 xp p 0 (4-4) Or is modld as a Nwtonian fluid using t Rolands Equation [11]: z 0xp ln p p (4-5) 0 z is t prssur-viscosity indx and p 0 is a constant (in [9]; z = 0.68 and p 0 = 1.98x10 8 ). Bair and Kottk [12] pointd out tis modl dos not captur t prssur vs log viscosity inflction wit t prssur is vry ig and can instad b found as P P P P 0 xp xp p p p p p p (4-6) A non-nwtonian fluid would av additional viscosity trms in t x and y dirction as dscribd in Kim [13], x cos and x1 0 x1 0 y sin x1 0 wr x1 is t sar strss at t wall. Dnsity For a simpl modl, t dnsity is assumd constant: 0 (4-7) For mor accuracy, t dnsity is givn as a function of prssur according to Dowson and Higginson [7]: E10 p E p (4-8) wr 0 is t dnsity at ambint conditions. 11

22 Film Ticknss T film ticknss quation is givn blow in [9] and [3] and dpnds on t location in t contact patc as wll as t lastic dformation of t bounding surfacs. T surfacs ar assumd prfctly lastic and smi-infinit. 2 2 x y 2 p( x', y ') dx ' dy ' 2 2 x y (4-9) p ( x, y) 0 2R 2 R E ( x x') ( y y ') T rigid body displacmnt, 0, is dtrmind using t forc balanc quation. Forc Balanc T load forc, F, is balancd by lubricant prssur. Tis is simply, F p x, y) dxdy ( (4-10) In t nxt sction, ow to solv tos quations is discussd but first w will nondimnsionaliz t govrning quations. Dimnsionlss Equations T dimnsionlss variabls usd ar [1]: X x / a Y y / b P p p / H R / ab x / 0 / 0 Dimnsionlss Rynolds Equation T Rynolds quation, Equation (4-1), bcoms: wr, P P ( H) v ( H) 0 X X Y Y X u Y (4-11) 3 H and 2 12u 0R x 3 ap 12

23 Wit PXY (, ) 0 on t boundaris of contact and t cavitation condition tat PXY (, ) 0 in t ntir domain. Dimnsionlss Film Ticknss T film ticknss quation, Equation (4-9), bcoms: X R Y 2 P( X ', Y ') dx ' dx ' H( x, y) H 2 R 2 ( X X ') ( Y Y ') 2 2 x 0 2 y 2 2 (4-12) wr H 0 is t initial dimnsionlss displacmnt. Dimnsionlss Forc Balanc T forc balanc quation, Equation (4-10), bcoms: 2 3 P( X, Y) dxdy (4-13) 13

24 5. Discrt Equations and Scm Using t discrtion scm similar to Vnnr [1] on a uniform grid wit ms siz dnotd by t suprscript or at points (i,). T x-componnt i rangs from 1 to n x and rangs from 1 to n y. Solving t Rynolds and Enrgy quations for prssur is difficult and complx. A computational solution mtod is rquird and MATLAB was cosn for t compilr. T finit diffrnc mtod is usd to convrt continuous into discrt functions. Som xampls ar sown blow: P x, y dxdy Doubl intgral to summations:, P P Singl drivativ using cntral diffrnc: x nx ny k1 l1 P i1, i1, x P k l x y Doubl drivativ cntral diffrnc: P P 2P P x x i1, i, i1, 2 x Tus, combining t singl and doubl drivativ wit finit diffrnc: P P P P P x x i1, i, i1, i, i1/2, 2 i1/2, 2 x x Tis mtod is usd to discrtiz t dimnsionlss gnral quations. Discrt Dimnsionlss Rynolds Equation T dimnsionlss Rynolds quation, Equation (4-11), at point (i,) is discrtizd on t grid as, wr, H H P P P i1/2, i1/2, i1/2, i1/2, i, i1/2, i1/2, P P P i, 1/2 i, 1/2 i, 1/2 i, 1/2 i, i, 1/2 i, 1/2 x v u y 0 x y (5-1) 14

25 and, i1/2, i, i1, i, 1/2 i, i, 1 /2 /2 (5-2) H 3 i, i, i, (5-3) Wit t boundary conditions P, 0 for all points on t boundary and t cavitation i condition P, 0 imposd vrywr. T Coutt or wdg trms in Equation (5-1) ar i discrtizd using an upstram scond ordr approximation on x as, H 1.5 H 2 H 0.5 H (5-4) i, i, i1, i1, i2, i2, x x for i 3. Nxt to t boundary at i 2 a first ordr approximation is usd, i, Hi, i 1, Hi1, H (5-5) x x Similarly, in t y-dirction t upstram scond ordr approximation is usd for 3 H 1.5 H 2 H 0.5 H (5-6) i, i, i, 1 i, 1 i, 2 i, 2 y and nxt to t boundary at 2 t first ordr approximation is usd, H i, i, i, 1 i, 1 y y H H (5-7) y Discrt Dimnsionlss Film Ticknss T dimnsionlss film ticknss quation, Equation (4-12), is rwrittn as dscribd in [1] as: X R Y 2 H( x, y) H K p 2 2 Y 0 2 i, i',, ' i', ' 2 RX 2 i' ' (5-8) 15

26 wr t Krnl is K i, i',, ' Y y /2 Xi x /2 Y y /2 Xi x /2 dx ' dy ' ( X X ') ( Y Y ') i 2 2 (5-9) Wic can b computd analytically as: wr Y p X p Yp Ki, i',, ' X p arcsin Yp arcsin X m arcsin X p Y p X m X X m Y m p Yp arcsin X p arcsin Ym arcsin Y p X p Ym X m Y m X m arcsin Ym arcsin Xm Ym X X X p i' i x X X X m i' i x Y Y Y p ' y Y Y Y m ' y (5-10) For a smi-infinit alf spac, t Krnl dpnds only on t diffrnc btwn t currnt position (i,) and wr t prssur acts (i', ). Tus, K is calculatd onc at i, i',, ' t start of t program and stord as Kkl, wr k i i ' and l '. T film ticknss constant H 0 is updatd according to: 3 H0, nw H0, old 1 xy P 1 (5-11) 2 i 0.1 Discrt Forc Balanc T forc balanc quation, Equation (4-13), bcoms: 2 P (5-12) 3 x y i, i 16

27 6. Numrical Solution Mtod A multigrid mtod as dscribd by Vnnr [1] is usd to solv t systm of quations and rlax t variabls. Two distinct rgims ar prsnt in t problm and ar approacd diffrntly. T variabl in t Rynolds quation varis widly ovr t domain of t problm; it is vry larg in t inlt rgion and small in t Hrtzian contact rgim. T valu of gaugs t rlativ importanc of t Poisull and Coutt trms (for larg t first two trms in Rynolds dominat; for small t last two trms ar mor important). T diffrnt rgims ar solvd diffrntly, using a Gauss-Sidl lin rlaxation for larg and Jacobi distributiv lin rlaxation for small. T notation usd in t following sction is adoptd from Vnnr [1]. First, lt Pi, and H i, dnot t currnt approximation to P i, and H i, on t uniform grid wit ms siz. Bttr approximations ar dnotd as P i, and Hi, in t sam grid aftr on rlaxation swp. Cangs ar applid along a lin simultanously and for t distributiv scm along nigboring lins. T grid is scannd along t X-dirction for 1inx in t form: on a givn lin and cangs ar solvd simultanously from a systm of quations A r (6-1) wr is a vctor of cangs to prssur i, and Rynolds quation. Lt Li, P dnot t discrt oprator on quation. T rsiduals ar tn: r is a vctor wit rsiduals i, P or t Rynolds r of t H H P P P i1/2, i1, i1/2, i1/2, i, i1/2, i1, i, 2 x r P P P i, 1/2 i, 1 i, 1/2 i, 1/2 i, i, 1/2 i, 1 x v u y 2 y (6-2) T cofficints of t A matrix ar, as dscribd in Vnnr [1], 17

28 A ik, L P 1 L P L P L P L P P 4 P P P P i, i, i, i, i, k, k 1, k1, k, 1 k, 1 (6-3) As dscribd prviously, t rlativ importanc of t trms varis wit and t problm is split into two rgims. 2 Gauss-Sidl, larg / 2 T first rgim, for larg /, Equation (6-3) is simplifid to: A ik, Li, P P k, (6-4) for 1k n bcaus t distribution for nigboring points is not rquird. For points ik 1: x A ik, 1.5 K 2 K 0.5 K v u i, ik 1,1 i1, ik 1 1,1 i2, ik 2 1,1 1.5 K 2 K 0.5 K x i, ik 1,1 i, 1 ik 1,2 i, 2 ik 1,3 y For i k: A i1/2, i1/2, i, 1/2 i, 1/2 ii, 2 2 x y 1.5 K 2 K 0.5 K v u i, 1,1 i1, 2,1 i2, 3,1 x 1.5 K 2 K 0.5 K i, 1,1 i, 1 1,2 i, 2 1,3 y For i 2: 18

29 A 1.5 K 2 K 0.5 K i1/2, i, 2,1 i1, 1,1 i2, 2,1 ii, 1 2 x x v u 1.5 K 2 K 0.5 K i, 2,1 i, 1 2,2 i, 2 2,3 y And for in x 1: A 1.5 K 2 K 0.5 K i1/2, i, 2,1 i1, 3,1 i2, 4,1 ii, 1 2 x x v u 1.5 K 2 K 0.5 K i, 2,1 i, 1 2,2 i, 2 2,3 y 2 Jacobian lin distribution, small / 2 T first rgim, for small /, uss Equation (6-3): A ik, L P 1 L P L P L P L P P 4 P P P P i, i, i, i, i, k, k 1, k1, k, 1 k, 1 bcaus du to t natur of t problm a distributiv scm is rquird. For ik 2: A ik, 1.5 K 2 K 0.5 K v u i, ik 1,1 i1, ik 1 1,1 i2, ik 2 1,1 1.5 K 2 K 0.5 K x i, 1, ik 1 i, 1 1, ik 1 1 i, 2 1, ik 2 1 y wr: K K 1 K K K K 4 k, l k, l k1, l k1, l k, l1 k, l1 Wn i k: 19

30 A 5 i1/2, i1/2, 5 i, 1/2 i, 1/2 ii, x 4 y 1.5 K 2 K 0.5 K v u i, 1,1 i1, 2,1 i2, 3,1 x 1.5 K 2 K 0.5 K i, 1,1 i, 1 1,2 i, 2 1,3 y For i 2: A 1.5 K 2 K 0.5 K 1 i1/2, i, 3,1 i1, 2,1 i2, 1,1 ii, x x v u 1.5 K 2 K 0.5 K i, 3,1 i, 1 3,2 i, 2 3,3 y For i 1: A i1/2, 1 i1/2, i1/2, 1 i, 1/2 i, 1/2 ii, x 4 x 4 y 1.5 K 2 K 0.5 K v u i, 2,1 i1, 1,1 i2, 2,1 x 1.5 K 2 K 0.5 K i, 2,1 i, 1 2,2 i, 2 2,3 y For in x 1: A i1/2, 1 i1/2, i1/2, 1 i, 1/2 i, 1/2 ii, x 4 x 4 y 1.5 K 2 K 0.5 K v u i, 2,1 i1, 3,1 i2, 4,1 x 1.5 K 2 K 0.5 K i, 2,1 i, 1 2,2 i, 2 2,3 y For in x 2: 20

31 A 1.5 K 2 K 0.5 K 1 i1/2, i, 3,1 i1, 4,1 i2, 5,1 ii, x x v u 1.5 K 2 K 0.5 K i, 3,1 i, 1 3,2 i, 2 3,3 y 7. Comparison of Numrical Solution and Litratur To tst t validity of t usr cratd computr modl, it is compard wit fully solvd modls in rlvant litratur sourcs. First, a basic validity tst for H min from Hamrock and Dowson [8], min 3.63U G W 1 R x k S (7-1) wr U F W ER P u G E 0 S (dimnsionlss spd), P ER P x 2 x (dimnsionlss load). (dimnsionlss matrial paramtrs), For dsign carts, a st of dimnsionlss variabls known as t Mos dimnsionlss paramtrs for circular contacts ar dfind blow and writtn in trms of t Hamrock and Dowson paramtrs [1], and 3/4 F us 0 3/4 2 2 S P x EPRx M W U E R u 1/4 s 0 L EP G 2U S ER P x Vnnr Cass Two tst cass prsntd in Vnnr [1] ar considrd wit t proprtis as sown blow: 1/4 21

32 Tabl 7-1: Proprtis for t two tst cass. Paramtr Cas 1 Cas 2 Unit M L U s 8.85E E-12 W 1.73E E-06 G F [N] u R x R y E p η α E E-08 [m/s] [m] [m] [1/Pa] [Pa-s] [1/Pa] Tr ar a fw problms wit Tabl 7-1 tat trow off t rsults. Spcifically, t sourc [1] givs a tabl of dimnsionlss paramtrs and uss tos to find t valu of t paramtrs listd in t abov tabl. Howvr, Vnnr s systm of quations is ovrdfind and a prfct balanc of all paramtrs is impossibl to find. For xampl, using t paramtrs in Vnnr to find L rsults in 10.2, a 2% rror. Additionally, t listd in [1] sms to b off by an ordr of magnitud. First lt s find an initial guss of t minimum film ticknss using t Hamrock and Dowson quation, Equation (7-1). For t two cass in Tabl 7-1 you gt, Tabl 7-2. Hamrock and Dowson stimat of minimum film ticknss. H min cas 1 cas E E-02 Ts cass ar run on 64x64 and 128x128 grids. For simplicity, t first solution run uss an isotrmal solvr wit incomprssibl lubricant and using t Barus viscosity quation. 22

33 Tabl 7-3. Comparison of Vnnr and Scnidr minimum and cntral film ticknsss wit constant dnsity and t prcnt diffrnc. 64x64 128x128 Vnnr SCHN Prcnt Diffrnc Cas 1 Cas 2 Cas 1 Cas 2 Cas 1 Cas 2 H c 4.819E E E E % 1.25% H min 2.988E E E E % 1.29% H c 4.936E E E E % 1.09% H min 3.052E E E E % 1.23% SCHN is t modl drivd and usd in tis rport. T prcnt diffrnc sows t diffrnc btwn t modl dscribd in tis papr and t modl in Vnnr. T rror is likly du to t aformntiond problm concrning t paramtrs in Vnnr and is witin rasonabl bounds. Using t Rolands dnsity quation rsults in t following tabl wit comparabl prcnt diffrncs. Now lt s compar a modratly mor complicatd modl, using variabl dnsity according to Equation (4-8) and Rolands viscosity as in Equation (4-5). T sam two cass from abov ar usd. Tabl 7-4. Comparison of Vnnr and Scnidr minimum and cntral film ticknsss wit Rolands dnsity and t prcnt diffrnc. Vnnr SCHN Prcnt Diffrnc Cas 1 Cas 2 Cas 1 Cas 2 Cas 1 Cas 2 64x64 128x128 H c 4.190E E E E % 3.62% H min 2.862E E E E % 0.89% H c 4.286E E E E % 2.47% H min 2.909E E E E % 1.29% Onc again t rror is rasonabl and is comparabl to t rror in Tabl 7-3. Tus t modl dscribd in tis papr is valid. T two cass for variabl dnsity ar visualizd blow wit 3D plots of prssur and film ticknss and 2D contour plots of prssur and film ticknss. 23

34 Cas 1: L = 10, M = 20: Variabl dnsity, Rolands viscosity. 64x64 and 128x128 grid sizs. Figur 7-1. Cas 1 (L=10, M=20), variabl dnsity, Rolands viscosity rsults. Lft is 64x64 grid siz, rigt is 128x128 grid siz. Top to bottom ar 3D plot of dimnsionlss prssur and 3D plot of dimnsionlss film ticknss. 24

35 Figur 7-2. Cas 1 (L=10, M=20), variabl dnsity, Rolands viscosity rsults. Lft is 64x64 grid siz, rigt is 128x128 grid siz. Top to bottom ar 2D contour plot of dimnsionlss film ticknss and 2D contour plot of dimnsionlss prssur. T maor diffrnc btwn t two grid sizs is t additional dtails in t igr rsolution rsults. All t basic structur from t 64 siz is prsnt in t 128 siz. Notic t prominnt lip on t prssur surfac plot, a caractristic of EHL. T film ticknss as a similar lip and t orsso sap common in EHL. 25

36 Cas 2: L = 10, M = 200: Variabl dnsity, Rolands viscosity. 64x64 and 128x128 grid sizs. Figur 7-3. Cas 2 (L=10, M=200), variabl dnsity, Rolands viscosity rsults. Lft is 64x64 grid siz, rigt is 128x128 grid siz. Top to bottom ar 3D plot of dimnsionlss prssur and 3D plot of dimnsionlss film ticknss. 26

37 Figur 7-4. Cas 2 (L=10, M=200), variabl dnsity, Rolands viscosity rsults. Lft is 64x64 grid siz, rigt is 128x128 grid siz. Top to bottom ar 2D contour plot of dimnsionlss film ticknss and 2D contour plot of dimnsionlss prssur. Again, t structur is narly idntical in bot grid sizs wit t igr rsolution bing mor dtaild. T prssur surfac in Figur 7-3 and Figur 7-4 mor closly rsmbls a Hrtzian prssur distribution and appars to not av t prssur spik caractristic of EHL. Howvr tr actually is a prssur spik but, according to Vnnr [1], wit incrasing load, t widt of t prssur spik xprssd in t dimnsionlss coordinat X dcrass and trfor it will only b visibl on a sufficintly fin grid. Tis assrtion is 27

38 confirmd in L [14] for a similar cas of M = 188, L = 13 on a 12% finr grid (compard to t 128x128 grid abov) is also missing t prssur spik. On a vry fin grid in Vnnr, t maximum dimnsionlss prssur spik is around 1.2 and ovr a vry small ara. Discrtization Error T cang in rror wit discrtization lvl is lookd at by comparing t minimum film ticknss valus in Tabl 7-3 and Tabl 7-4 wit tir rspctiv valus on t finst grid (512x512) prsntd in Vnnr [1]. T incomprssibl cas is prsntd in Figur 7-5 and t comprssibl cas in Figur 7-6. As xpctd, ac modl incrass in accuracy wit incrasing grid siz. Howvr, for Cas 1 t Scnidr modl drivd r prforms poorly compard to Vnnr. Tis is likly du to t prviously mntiond problm wit Vnnr s paramtrs and not an rror wit t modl itslf bcaus bot t Cas 2 rsults ar vry similar and t problm is prsnt in bot t comprssibl and incomprssibl modls. In any cas, all futur simulations ar prformd on t mor accurat 128x128 grid siz. Figur 7-5. Minimum film ticknss rror of fluid wit constant dnsity and Barus viscosity. 28

39 Figur 7-6. Minimum film ticknss rror of fluid wit variabl dnsity and Rolands viscosity. Effct of sid-slip Transvrs vlocity, or sid slip, affcts contact bavior and dtrmins t amount of pur rolling. Tis typ of contact could occur undr crtain conditions and variations of t Fallbrook driv. T traction cofficint, important to transmitting powr, is not part tis analysis. T analysis incrmntally incrass t transvrs vlocity from 0 to 50% of t rolling vlocity wit t following two cass: - Cas 1s starts wit v = 0.0 m/s and L = 10, M = 20 and incrmnts v for t cass of v = 0.1, 0.2, 0.3, and 0.4 m/s. - Cas 2s starts wit v = 0.0 m/s and L = 10, M = 200 and incrmnts v for t cass of v = 0.1, 0.2, 0.3, and 0.4 m/s. Cas 1s: Variabl dnsity and Rolands dnsity on 128x128 grid siz. A tabl of t cntral and minimum dimnsionlss film ticknsss is sown blow. T minimum film ticknss is not substantially largr for transvrs spds and t cntral film ticknss is variabl. 29

40 Tabl 7-5. Cas 1s, variabl dnsity and Rolands viscosity. Dimnsionlss film ticknss wit varying transvrs spds. Cas 1s v H c H min 4.142E E E E E E E E E E-01 And t plots ar sown blow. First t prssur and film ticknss ar sown. Not t diffrnc in scal for film ticknss from Figur 7-1 and Figur 7-2 Figur 7-7. Cas 1s wit sid slip on 128x128 grid siz. Lft is dimnsionlss prssur and rigt is dimnsionlss film ticknss. Top to bottom is v =0, 0.1 m/s. 30

41 Figur 7-8. Continud from prvious figur. Lft is dimnsionlss prssur and rigt is dimnsionlss film ticknss. Top to bottom is v = 0.2, 0.3, 0.4 m/s. 31

42 And t 2D contour plots, Figur 7-9. Cas 1s wit sid slip on 128x128 grid siz. Lft is dimnsionlss prssur and rigt is dimnsionlss film ticknss. Top to bottom is v =0, 0.1 m/s. 32

43 33

44 Figur Continud from prvious figur. Lft is dimnsionlss prssur and rigt is dimnsionlss film ticknss. Top to bottom is v = 0.2, 0.3, 0.4 m/s. As xpctd, t prssur and film ticknsss sift according to t transvrs slip. Cas 2s: Variabl dnsity and Rolands dnsity on 128x128 grid siz. T cntral and minimum film ticknss valus ar sown blow. T minimum film ticknss is gnrally largr wit transvrs slip, but t cntral film ticknss tnds to vary. 34

45 Tabl 7-6. Cas 2s, variabl dnsity and Rolands viscosity. Dimnsionlss film ticknss wit varying transvrs spds. v H c H min Cas 2s 7.629E E E E E E E E E E-02 And t plots ar sown blow. First t prssur and film ticknss 3D plots. Lft is prssur and rigt is film ticknss. 35

46 Figur Cas 2s wit sid slip on 128x128 grid siz. Lft is dimnsionlss prssur and rigt is dimnsionlss film ticknss. Top to bottom is v = 0, 0.1, 0.2 m/s. 36

47 Figur Continud from prvious figur. Lft is dimnsionlss prssur and rigt is dimnsionlss film ticknss. Top to bottom is v = 0.3, 0.4 m/s. Again, as xpctd t prssur and film ticknss surfacs sift wit incrasing transvrs spds. Nxt t 2D contour plots ar sown. Lft is prssur and rigt is film ticknss. 37

48 Figur Cas 2s wit sid slip on 128x128 grid siz. Lft is dimnsionlss prssur and rigt is dimnsionlss film ticknss. Top to bottom is v =0, 0.1, 0.2 m/s. 38

49 Figur Continud from prvious figur. Lft is dimnsionlss prssur and rigt is dimnsionlss film ticknss. Top to bottom is v = 0.3, 0.4 m/s. T incras in minimum film ticknss is mor pronouncd tan wit Cas 1s sowing tat contact undr conditions similar to Cas 2s ar mor avily influncd by sid-slip. Fallbrook Driv Contact And finally of intrst is a cas similar to t insid of t traction driv from Fallbrook Tcnologis labld as Cas FB. Paramtrs ar found undr caractristic driv conditions. T gomtry of t driv is, 39

50 Tabl 7-7. Gomtry of driv. N p 7 α 1 45 α 2 45 r p r s 21.5 mm 51 mm A typical F axial is 6140 N and ring spd as around 2000 rpm. Using t Equations in Sction 2, tis simulats on of t contacts wit t following paramtrs: Tabl 7-8. List of paramtrs for Cas FB. Paramtr Valu Unit M L F 1240 N a 6.45E-04 m b 5.42E-04 m p U s 1.84E+09 Pa 9.62E-11 For rfrnc t minimum dimnsionlss film ticknss from Hamrock and Dowson [8] is 3.93E-02 wic corrsponds to μm. T rsults of t simulation ar, Tabl 7-9. Dimnsionlss film ticknss for t Fallbrook Tcnologis driv. H c H min 9.401E E-02 And t plots ar, 40

51 Figur Fallbrook Tcnologis cas (L=17.96, M=238.4), variabl dnsity, Rolands viscosity rsults. Lft sid ar dimnsionlss prssurs and rigt sid ar dimnsionlss film ticknss. Tis dimnsionlss minimum film ticknss corrsponds to μm, a 15% diffrnc from t Hamrock and Dowson prdiction of μm. As mntiond arlir, tis is important wn considring surfac rougnss as t minimum film ticknss must b larg noug to prvnt surfac-to-surfac contact. In otr words [15], min 1.5 r q 1 r q 2 (7-2) 41

52 wr rq 1 and rq 2 ar t rms surfac rougnss valus for t two surfacs. T surfac rougnss valus in t driv ar btwn μm. T surfac rougnss valus insrtd into Equation (7-2) yild a rang of μm tat t minimum film ticknss nds to surpass. T currnt surfac rougnss valus ar sufficint. T maximum surfac rougnss allowd for tis cas is μm. T ffct of film ticknss on lif is sown in Evans [16] wit Figur T valu N was found using a Wibull plot and 50% fatigu lif. λ is t film ticknss ratio dfind as, wr σ is t composit rms rougnss dfind bfor as, c 2 2 r 0.5 q1 rq2 Figur Load lif vs. film ticknss ratio valu. T filld in circls giv pur rolling and t opn circls giv slid-roll ratio qual to In t Fallbrook driv, λ rangs from for t surfac rougnss valus of μm. According to Figur 7-16 [16], t idal λ quals around 5 wic corrsponds to a surfac rougnss of μm. T Fallbrook driv oprats nar t idal conditions for 42

53 t largr surfac rougnss undr typical conditions. Howvr in Popincanu [17], t idal λ valu varid from 5 to 10.Evn t igst film ticknss to surfac rougnss valu of 12.9 is nar t accptabl rang. Smootr surfacs cost xtra mony giving littl rason to us solids wit surfac rougnss valus of 0.1 μm or lss. T film ticknss output can also b usd to stimat t rquird lubricant flow rat to sav on t amount of fluid usd. Only a small amount of lubricant is rquird pr contact patc du to t small ara and tiny film ticknss. Taking only t widt of t contact patc, t cntral igt, and t spd of t fluid, t absolut minimum flow rat rquird is about 2.4E-08 m 3 /s pr contact patc wic totals 3.3E-07 m 3 /s for t rings. Tis is only gallons/minut. Again, tis is assuming tat all t fluid gts to t contact patc. Currntly t lub flow rat in t driv is around 3 gallons/minut so obviously most t fluid is not bing utilizd for traction (it is also usd for trmal dissipation). Tis information can b usd to dvlop a mor fficint lubrication scm. Additionally, t maximum dimnsionlss prssur valu in tis cas sligtly xcds p. T maximum P valu is wic corrsponds to a maximum prssur of 1.87 GPa instad of t Hrtzian valu of 1.84 GPa. Tis diffrnc is important wn looking at durability and t maximum strss pr cycl. Tis is also important in slcting lubricants tat can witstand tat kind of prssur, wic t currnt typ is abl to. Tr is no prssur spik in t rsults, but as mntiond for Cas 2 tr actually is on prsnt but it will only b visibl on a vry fin grid. Howvr vn wn using t 1.2 maximum dimnsionlss prssur spik valu from Cas 2, t lubricant usd is witin its oprating limits. 43

54 8. Conclusion A robust modl of lliptical lastoydrodynamic lubrication as bn dvlopd from driving t gnral quations to flsing out t numrical mtods and sowing a full numrical solution tat is quick and fficint. Prvious rsarc solutions wr duplicatd witin rasonabl rror and t uniqu condition of transvrs spd was modld to sow t ffct on prssur and film ticknss. A traction patc from a continuously variabl transmission from Fallbrook Tcnologis was modld as an lliptical EHL isotrmal contact. T film ticknss and prssur was found and t minimum film ticknss was dtrmind to b larg noug to prvnt surfac-to-surfac contact wn considring surfac rougnss but still small noug to b nar t idal film ticknss to surfac rougnss ratio for long lif. A sligtly diffrnt maximum prssur is found from t Hrtzian valu. For ig load conditions, lik in t Fallbrook driv, a vry fin grid is rquird to s t prssur spik caractristic of EHL. In t futur, t modl can b asily implmntd in a mor complt modl tat includs mor variabls and paramtrs. Of particular intrst is t inclusion of a trmal modl and non-nwtonian ffcts lik in Campos [2] and Bos [18]. Additionally, t ffct of starvd lubrication as in Vnnr [1], modling surfac rougnss, and diffrnt lubrication dlivry systm (lik oil t and oil mist) ffctivnss ar rlatd topics of intrst. 44

55 9. Works Citd [1] C. H. Vnnr and A. A. Lubrct, Multilvl Mtods in Lubrication, Amstrdam: Elsvir, [2] A. Campos, A. Sottomayor and J. Sabra, "Non-nwtonian Trmal Analysis of an EHD Contact Lubricatd wit MIL-L Oil," Tribology Intrnational, vol. 39, pp , [3] H. J. Kim, P. Ert, D. Dowson and C. M. Taylor, "Trmal lastoydrodynamic analysis of circular contacts. Part 1," Proc Instn Mc Engrs Vol 215 Part J, [4] W. C. Young and R. G. Budynas, Roark's Formula for Strss and Strain, 7t d., McGraw- Hill. [5] K. L. Jonson, Contact Mcanics, Cambridg Univrsity Prss, [6] S. H. Lowntal and E. V. Zartsky, "Dsign of Traction Drivs," NASA Rfrnc Publication 1145, [7] D. Dowson and G. R. Higginson, Elastoydrodynamic Lubrication, T Fundamntals of Rollr and Gar Lubrication, Oxford: Prgamon Prss, [8] B. J. Hamrock, Fundamntals of Fluid Film Lubrication. [9] Q. Zou, C. Huang and S. Wn, "Elastoydrodynamic Film Ticknss in Elliptical Contacts Wit Spinning and Rolling," Journal of Tribology, vol. 121, pp , Octobr [10] C. Barus, "Isotrmals, Isopistics and Isomtrics rlativ to Viscosity," Am. J. of Scinc, vol. 45, pp [11] Rolands, "Corrlational Aspcts of t Viscosity-Tmpratur-Prssur Rlationsip of Lubricating Oils," PD Tsis, Tcnical Univrsity Dlft, Dlft, T Ntrlands, [12] S. Bair, Hig Prssur Rology for Quantitativ Elasoydrodynamics, Elsvir, [13] H. J. Kim, P. Ert, D. Dowson and C. M. Taylor, "Trmal lastoydrodynamic analysis of circular contacts. Part 2: Non-Nwtonian Modl," Proc Instn Mc Engrs Vol 215 Part J, [14] R.-T. L, C.-H. Hsu and W.-F. Kuo, "Multilvl solution for trma lastoydrodynamic 45

56 lubrication of rolling/sliding circular contacts," Tribology Intrnational, vol. 28, no. 8, pp , Dcmbr [15] F. Krit and D. Y. Goswami, T CRC Handbook of Mcanical Enginring, 2nd d., CRC Prss, [16] H. P. Evans and R. W. Snidl, "Effct of film ticknss ratio on garing contact fatigu lif," IUTAM Symposium on Elastoydrodynamics and Micro-lastoydrodynamics, vol. 134, pp , [17] N. G. Popincanu, M. D. Gafitanu, S. S. Crtu, E. N. Diaconscu and L. T. Hostiuc, "Rolling Baring Fatigu Lif and EHL Tory," War, vol. 45, pp , [18] J. Bos and H. Mos, "Frictional Hating of Tribological Contact," Journal of Tribology, vol. 117, pp , January [19] H. S. Carslaw and J. C. Jagr, Conduction of Hat in Solids,