MODELLING OF ELASTO-HYDRODYNAMIC LUBRICATION PROBLEMS IN GEARS

Transcription

1 MODELLING OF ELASTO-HYDRODYNAMIC LUBRICATION PROBLEMS IN GEARS AMIN AMANI Faculty of Industral Desgn Engneerng, Delft Unversty of Technology Landbergstraat 15, 68 CE Delft, The Netherlands MOHAMMAD RAJABALINEJAD Faculty of Industral Desgn Engneerng, Delft Unversty of Technology Landbergstraat 15, 68 CE Delft, The Netherlands CHRISTOS SPITAS Faculty of Industral Desgn Engneerng, Delft Unversty of Technology Landbergstraat 15, 68 CE Delft, The Netherlands Abstract For hghly loaded contacts, lubrcant pressure can cause elastc deformaton of the surface that s n the same order as the flm thckness of lubrcant. When ths occurs, the nfluence of deformaton on lubrcaton performance becomes an mportant parameter. Contacts operatng under ths condton are n the regme of elasto-hydrodynamc lubrcaton (EHL).In addton, numercal calculatons of EHL problems are hgh-prced belongng to the large quantty of numercal computatons needed as a result of ther contact domans beng much bgger than those of nvolute gears and bearngs [1,]. To amplfy the effortlessness of smulatons, Homotopy Perturbaton Method (HPM)[3-6,13-17] s used. HPM s an effectve and relatvely uncomplcated methodology for solvng nonlnear thermal problem for lubrcaton of gears. In ths paper, we contrast HPM n related to hgher-order approxmate solutons for the EHL. It s concluded that the results of HPM provde accurate soluton, as an easy methodology for computng the nonlnear dfferental equatons for EHL. KEYWORDS : Elasto-hydrodynamc, Gear lubrcaton, Homotopy Perturbaton Method INTRODUTION One of the most perplexed applcatons of trbology s the tooth contact of gears. For desgn of gears, there s no sutable replacement for vrtual occurrence. In assocaton wth the many types of gears, the most commonly used are spur gears. Because loads dstrbuton on the gears are channelzed through lubrcated contacts of Hertzan as elasto-hydrodynamc (EHL) contacts, the acton of lubrcaton n spur gears has a large extent conjontly [7,8].

2 Because of the elastc deformaton, the results n hgh contact pressures due to the contactng lubrcated domans n gears wll be generated by the hgh pressures n contact zones [9]. Uncomplcated and smple and drect numercal solutons for solvng of lne contact problems n gears cannot establsh hgh convergence at the poston of loads dstrbuton [1]. There are some numercal methodologes appled comprehensvely for provdng the results of EHL problems such as : nverse methods, forward-teratve methods, mult-grd methods and Newton-Raphson methods. In the group of these methods, the Newton-Raphson methodology conventonally obtans a methodcal approach and a better concdence n nonlnear system nvestgaton [11]. Because applcaton of the Reynolds equaton as the man equaton n EHL problem, and also lack of analytcal soluton, the Reynolds equaton frequently solved numercally. Some of teratve methodologes are comparatvely smple to perform than straghtforward methods. Due to complexty of Reynolds equaton as a second-order nonlnear partal dfferental equaton, t s dffcult to be solved by teratve methods [1]. Many analytcal methods have been used to solve nonlnear problem n elastohydrodynamc lubrcaton. For example, perturbaton methods [,3] are mentoned to be the most common tools n nonlnear analyss of engneerng problems. These methods are contnuously beng amplfed and appled to ever more complcated problems. However, tradtonal perturbaton methods have many shortcomngs and are not vald for strongly nonlnear equatons. To overcome the shortcomngs, many new technques have been proposed n the lterature, for example: Homotopy Perturbaton Method [4-6,13-17], Lndstedt-Poncaré [18], Harmonc Balance Method [19], Varatonal Iteraton Method [- 1], and Varatonal Approach Method [3,]. The man focus of ths paper s on a comparatve study of applcaton of the Homotopy Perturbaton Method (HPM) wth other avalable technques n soluton of nonlnear thermal problem of a elasto-hydrodynamc lubrcaton n gears. We use Homotopy perturbaton method (HPM) to observe the approxmate analytcal soluton of nonlnear dfferental equaton of elasto-hydrodynamc lubrcaton n gears. In the numercal methodologes, stableness and concdence should be consdered so as to keep off abnormalty or ncompatble results. Wth the formulaton of the elasto-hydrodynamc lubrcaton problem t can be supported that the system of equatons can be cut down to some parameter problem by presentng approprate transformaton varables. FUNDAMENTAL OF THE HOMOTOPY BERTURBATION METHOD (HPM) HPM s combned by the classcal perturbaton technque and homotopy technque [3-4,13-17] whch removes the restrcton of the classcal perturbaton methods. For explanaton of the basc dea for the presented method as a method to solve the nonlnear dfferental equatons, we consder the followng nonlnear dfferental equaton: A ( u) f ( r) =, r Ω (1) where A s operator, f s a known functon and u s a sought functon. Assume that operator A can be wrtten as: A u = L u + N u, r Ω, () ( ) ( ) ( ) where L s the lnear operator, N s the nonlnear operator. Hence, Eq. (1) can be rewrtten as follows:

3 L( u) N ( u) f ( r) + =, r Ω, (3) In case the nonlnear Eq. (1) has no small parameter, We defne an operator H as: H ( ν, ε ) = ( 1 ε ) L( ν ) L( u ) + ε A( ν ) f ( r) = (4) Where ν ( r, ε ) : Ω [,1] R and n Eq. (4), ε [,1] s called homotopy parameter, and u s the frst approxmaton that satsfed the boundary condton. Accordng to the HPM, the assumpton of approxmaton soluton of Eq. (4) can be expressed as a power seres of p: 3 v = v + εv1 + ε v + ε v (5) Settngε = 1result n the approxmate soluton for Eq. (4) and eventually the best approxmaton for soluton s: u = lm v = v + v + v +... (6) p 1 1 Convergence of the Eq. (6) s happened for most of the cases, and also the rate of convergence s related to the selecton of operator A. The LUNRICATION MODEL There s a great deal alteraton n the pressure n EHL lubrcant problem over very small length, therefore usng a specal accurate model for lubrcaton can be most mportant. Due to ths sgnfcance many researcher are currently workng on the model to analyze the behavor of ol n durng the lubrcant process. The Reynolds equaton For the dstrbuton of pressure because of an appled load for a specal geometry the Reynolds equaton defned that was derved from the equatons of Naver-Stokes. The Reynolds equaton n Cartesan coordnates and steady-state stuaton s: 3 3 ρh p ρh p (7) ( ) + ( ) = ( ρuh) + ( ρwh) x 1η x 1η y x y Where p s the pressure, h s flm thckness,η s the vscosty of the lubrcant, ρ s the densty of the lubrcant, x s the coordnate at rollng drecton [1], u = ( u + u ) / s veloctes of lubrcant because of speeds of surface a and b n x drecton and also w = ( wa + wb ) / s veloctes of lubrcant because of speeds of surface a and b n y drecton. Reynolds nvestgaton n second order The classcal nvestgaton to the second-order solvaton of the steady-state sothermal lne contact problem requres some equatons. The lne contact hydrodynamc n only steady-state results s: 3 ρh P (8) ( ) = u ( ρh) X 1η X X The relatonshp between densty-pressure Pressure and temperature nfluence on vscosty are sgnfcant. Wth any changes n pressure, the alteraton of densty s small wth regard to the change n vscosty quantty. It can be wrtten for dmensonless densty as:.6 p ρ ρ 1 (9) = p where ρ s the densty at zero pressure. a b

4 The relatonshp between vscosty-pressure The vscosty of lubrcant s very mportant n EHL contacts. Barus equaton ntroduces the relaton of vscosty on pressure that s: η = η exp( α p) (1) where η s the absolute vscosty at a constant temperature and at p, and the parameter ofα s the coeffcent of pressure-vscosty n lubrcant and t s temperature dependent. The formulaton of Elastcty equaton Because of the pressure whch s grew n the flud of lubrcatng, the elastc equaton defned as normal dsplacements to the domans. The total flm thckness s represented by: n x (11) h = h + + fj p j R j= 1 where h s the ntal flm thckness, x / R s the separaton due to the geometry, where and R s the radus of relatve curvature equvalent radus defned by 1/ R = 1/ R1 + 1/ R here R 1, R represent the radus of curvature of the ntersectng areas and the last part of the equaton, n xout fj p as the total elastc deformaton of the surface s p( x )ln( x x ) dx j π E xn j= 1. Non-dmensonalsaton Non-dmensonalsaton s recommended for solvng the numercally calculaton. It permts the concdental modellng of all knd of numercal solvng and lends to wde generalty to the results [1,3-4]. The Reynolds equaton for thermal and steady state condtons n nondmenson model can be wrtten as [11]: 3 ρh p ua + u (1) b ( ) 1um ( ρh) = ;, um = x η x x wth these dmensonless parameters: p x hr h R P = ; X ; H ; ; ; H ; p = ρ η ρ η b = b = ρ = η = (13) b H That b s half Hertzan contact length, ph s maxmum Hertzan pressure, and fnally dmensonless Reynolds equaton can be rewrtten as: 3 ρh P 1η umr (14) x ( ) λ ( ρh ) = ;, λ = 3 X η X X b PH Wth the followng boundary condtons: x = x, p = ; and x = x, p = p / x = (15) n out And also h d as elastc deformaton at any pont x on the area s determned by: xout hd ( x) = p( x )ln( x x ) dx π E xn E s the equvalent Young's modulus gven by: 1 ν1 1 ν = + E E E 1 Where E 1, E the elastc modul and ν 1, ν are Posson ratos of the two surface materals. that dmensonless form s: (16) (17)

5 1 X out (18) H d ( x) = P( X )ln( X X ) dx π X n The flm thckness of EHL for lne contact s: h( x) = h + h ( x) + h ( x) (19) g d Thus, for any pont X the shape of flm s gven by: X 1 X () out H ( X ) = H + P( X )ln( X X ) dx π Xn H contans the constant term of Hd ( x ) as ntal flm thckness. Flm thckness equaton s represented as: M X 1 (1) H ( X ) = H + D, k Pk π k = 1 Soluton to EHL by Homotopy perturbaton method (HPM) For the values of parameters n the presented equaton, we use references [,7,11]. Let us construct the homotopy map for Eq. (14), as follows: d d d H ( P, ε ) = (1 ε ) P(x) -.86x P(x) P(x) x + dx dx dx d d + ε P(x).7344P(x) P(x) +... dx dx Substtutng n P( x) = P ( x) + ε P1 ( x) + ε P ( x) as pressure to Eq. () and rearrangng Eq.() based on powers of å-terms, we have: d d d ε : P (x) x P (x).86x P (x) (3) dx dx dx : ( P (x) ) d P (x) ( P (x) ) d x P (x) x d ε + + P 1(x) x+... dx dx dx d d d d ε : P 1(x) P (x) P (x) P 1(x) P 1(x) P (x)+... dx dx dx dx Wth the boundary condtons and n the same manner, the rest of components were obtaned usng the maple package. Accordng to the HPM, we can conclude that: PPr essure = lm P ( x ) = P ( x ) + P 1( x ) + P ( x ) +... (6) ε l Therefore, substtutng the values of P ( x ), P( x) and 1 P ( ) x from Eqs. (3,4,5) n to Eq. (6) yelds: () (4) (5) P( x) =.115x ( x + 1.5) ( x + 1.5) (7) ( x + 1.5) ( x + 1.5) ( ) ( x + 1.5) 1 + O ( x + 1.5) The results are shown n fgures 1 and, where P = p / ph s as dmensonless pressure and X = x / b s as dmensonless locaton. For the accuracy estmaton of the approxmate analytcal approach, a comparson between HPM and a few other methods n the ndcated references [7,9].Comparng the results, HPM presents a good result and ts power and capablty s one of the advantages of ths method. In ths work, MAPLE package was used n the mathematcal calculatons.

6 Fgure 1 : Pressure dstrbutons obtaned usng HPM methods across the entre contact. Fgure :Comparson wth Data from Ref.[7,9]. CONCLUSION In ths study, one of the fast computatonal methodology as Homotopy Perturbaton Method (HPM) was successfully utlzed to solve the EHL problems. In ths work, HPM s appled for the analyss to solve the Reynolds equaton. The acheved results ndcate that the present method can be easly extended to the analyss of two-dmensonal thermal and sothermal EHL problems such as fnte lne contact and pont contact problems. The results are compared wth a few other numercal methods [7,9] and show good correspondence. Applyng HPM to Reynolds equaton has some advantages comparng to the other methods: t overcomes the dffcultes arsng n the calculaton Newton-Raphson

7 method. Also HPM does not requre small parameters such as classcal perturbaton technques n the equaton, so that the lmtatons of the tradtonal perturbaton methods can be elmnated, and thereby the calculatons are smple and straghtforward. REFERENCES [1] Sptas,V., Sptas,C. Numercal and expermental comparatve study of strengthoptmsed AGMA and FZG spur gears, Acta Mechanca,193(1-)(7) [] Yang, J.B., Chen, C.W. Analytcal and numercal calculatons for thermal elastohydrodynamc lubrcaton problems of helcal W-N gears, Wear, 145() (1991) 1-5. [3] Aman, A., Ganj, D. D., Ahmad Jebell, A., Shahab, M., Safara Nosar, N. Applcaton of He s varatonal approach method for perodc soluton of strongly nonlnear oscllaton problems, Internatonal Journal of Appled Mathematcs and Computaton, (3)(1) [4] Lu,H.K. Applcaton of homotopy perturbaton methods for solvng systems of lnear equatons, Appled Mathematcs and Computaton, 17(1) (11) [5] Rajabalnejad, M., Meester, L., VanGelder, P.H.A.J.M., Vrjlng, J.K. Dynamc bounds coupled wth Monte Carlo smulatons, Relablty Engneerng & System Safety, 96() (1) [6] Slota, D. The applcaton of the homotopy perturbaton method to one-phase nverse Stefan problem, Internatonal Communcatons n Heat and Mass Transfer, 37(1) [7] Coc, C.A.B. An elastohydrodynamc lubrcaton model for helcopter hgh-speed transmsson components,phd thess, The Unversty of Toledo,December 4. [8] Mhalds, A., Panagotds, K. Transent thermo-elastohydrodynamc lubrcaton of gear teeth, Lubrcaton Scence 15(4) (3) [9] Goodyer, C.E. Adaptve numercal methods for elastohydrodynamc lubrcaton, PhD thess,the Unversty of Leeds School of Computng, May 1. [1] Hughes,T.G., Elcoate, C.D., Evans, H.P. A novel method for ntegratng frst- and second-order dfferental equatons n elastohydrodynamc lubrcaton for the soluton of smooth sothermal, lne contact problems, Internatonal Journal for Numercal Methods n Engneerng, 44(1999) [11] Khan, H., Snha, P., Saxena, A. A smple algorthm for thermo-elasto-hydrodynamc lubrcaton problems, Internatonal Journal of Research and Revews n Appled Scences, 1(3)(9) [1] Wang, N., Chang, S.H., Huang,H.C. Comparson of teratve methods for the soluton of compressble-flud Reynolds equaton, Journal of Trbology,133(11) [13] Bazar, J., Eslam, M. A new homotopy perturbaton method for solvng systems of partal dfferental equatons, Computers & Mathematcs wth Applcatons, 6(1) (11) [14] Moghm, S.M., Ganj, D.D., Hossen, H.M., Jalaal,M. Homotopy perturbaton method for nonlnear MHD Jeffery Hamel problem, Computers & Mathematcs wth Applcatons, 61(8) (11) [15] He, J-Huan. Homotopy perturbaton method: a new nonlnear analytcal technque,

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