A Comparative study of Load Capacity and Pressure Distribution of Infinitely wide Parabolic and Inclined Slider Bearings

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1 Procdings of th World Congrss on Enginring 2 Vol II WCE 2, Jun 3 - July 2, 2, London, U.K. A Comparativ study of Load Capacity and Prssur Distribution of Infinitly wid Parabolic and Inclind Slidr Barings Mobolaji H. Oladind and John A. Akpobi Abstract A mathmatical modl for th hydrodynamic lubrication of infinitly wid inclind and parabolic slidr barings with coupl strss lubricants is prsntd. A numrical solution for th mathmatical modl using finit lmnt schm is obtaind using thr nods isoparamtric quadratic lmnts for both configurations of barings. Stiffnss intgrals obtaind from th wak form of th govrning quations wr solvd using gauss quadratur to obtain a finit numbr of stiffnss matrics. Th global systm of quations was obtaind for th barings and solvd using gauss sidl itrativ schm with a convrgnc critrion on -. Numrical primnts indicat that whn th slidr baring is modld to includ coupl strss paramtr, th finit lmnt mthod producs stabl and convrgnt rsult whil th finit diffrnc tchniqu dos not. Computations rval that whn compard using similar profil and coupl strss paramtr, gratr prssur builds up in a parabolic slidr compard to an inclind slidr indicating gratr wdg ffct in th parabolic slidr. It has also bn shown that whn load carrying capacity is th yardstick for comparison, th parabolic slidr barings is suprior to th inclind cas irrspctiv of th profil and coupl strss paramtr. Ind Trms Hydrodynamic lubrication, parabolic slidr, inclind slidr, finit lmnt. INTRODUCTION In most mchanical systms whr rlativ motion occurs btwn two parts, lubricants ar introducd to rduc friction and war. Th gomtry of th contacting lmnts dtrmins th shap of th lubricant film []. Various rsarchrs hav considrd diffrnt configurations of th lubricating film in th claranc zon in thir analysis. Th contacting surfacs can b narrowing gomtrically in linar styl as considrd by Ozalp [2]. H mployd th itrativ transfr matri approach to suggst optimum film profil Dr M. H. Oladind is with th Dpartmnt of Production Enginring, Faculty of Enginring, Univrsity of Bnin. P.M.B 54, Bnin City Nigria (-mail: moladind@unibn.du). Dr J. A. Akpobi is with Dpartmnt of Production Enginring, Faculty of Enginring, Univrsity of Bnin. P.M.B 54, Bnin City Nigria. ( alwaysjohni@yahoo.com, Phon: ). paramtrs for rducd friction cofficint. Bayrakpkn t al [3] carrid out a comparativ study of inclind and parabolic slidr barings using a non-nwtonian fluid in th claranc zon of th slidr barings. H dvlopd clos form prssions for th prformanc charactristics of th barings. Shah t al [4] studid a slidr baring with ponntial film thicknss profil and obtaind analytical prssions for variation of dimnsionlss prssur, friction, cofficint of friction and cntr of prssur. A frrofluid was usd btwn th contacting surfacs of th baring. Yurusoy [5] obtaind a prturbation solution for prssur distribution in a slidr baring with a Powl-Eyring fluid. as lubricant. Bujurk t al [6] usd a scond grad fluid in a tapr flat slidr baring similar to that usd by Ozalp [2] and constructd a Von korman momntum intgral solution. Shah [7] computd valus for th baring charactristics of a scant shapd slidr baring using a magntic fluid lubricant. Diffrnt typs of fluids hav bn usd in th claranc zon of slidr barings and thir prformanc invstigatd as shown in th prvious works citd. Howvr, in ordr to nhanc lubricating prformanc, th incrasing us of Nwtonian lubricant which has bn blndd with long chain polymrs has bn obsrvd. Sinc th convntional micro continuum thory cannot accuratly dscrib th flow of ths kinds of fluids, various micro continuum thoris hav bn proposd. [8]. Stoks [9] proposd th simplst micro - continuum thory which prmits th prsnc of coupl strsss, body coupls and non symmtric tnors []. A numbr of rsarchrs hav invstigatd th ffct of th coupl strss fluid modl on th stady stat prformanc of diffrnt slidr baring configurations using diffrnt numrical schms. In rcnt tims, most numrical work in hydrodynamic lubrication has involvd th us of th Rynolds quation and th finit diffrnc mthod []. A finit diffrnc multigrid approach was usd to invstigat th squz film bhavior of porolastic baring with coupl strss fluid as lubricant by Bujurk t al [6]. Thy rportd that porolastic barings with coupl strss fluid as lubricant provid augmntd prssur distribution and nsurd significant load carrying capacity..sarangi t al [2] solvd th modifid Rynolds quation tndd to includ coupl strss ffcts in lubricants blndd with polar additivs using th finit diffrnc mthod with a succssiv ovr rlaation schm. Thy rportd incras in load carrying capacity and rduction in friction cofficint as compard to Nwtonian lubricants. Lin [3] usd th conjugat mthod of itration to ISBN: ISSN: (Print); ISSN: (Onlin) WCE 2

2 Procdings of th World Congrss on Enginring 2 Vol II WCE 2, Jun 3 - July 2, 2, London, U.K. build up th prssur gnratd in a finit journal baring lubricatd with a coupl strss fluids. Th rsults obtaind including incras in th load carrying capacity agr with thos obtaind by Sarangi t al [2] and Bujurk t al [6]. Elsharkawy [4] providd a numrical solution for a mathmatical modl dscribing th hydrodynamic lubrication of misalignd journal barings with coupl strss fluids as lubricants using th finit diffrnc mthod. Lin [5] calculatd th stady and prturbd prssur of a two dimnsional plan inclind slidr baring incorporating a coupl strss fluid using th conjugat gradint mthod and rportd improvd stady and dynamic prformanc compard to th Nwtonian cas spcially for highr aspct ratios. Nada and Osman [6] invstigatd th problm of finit hydrodynamic journal baring lubricatd by magntic fluids with coupl strsss using th finit diffrnc mthod. For diffrnt coupl strss paramtrs and magntic cofficints, thy obtaind th prssur distribution. Thy concludd that fluids with coupl strsss ar bttr compard with th Nwtonian cas aftr comparison of th baring static charactristics. Th opn litratur is rplt with slidr baring dsign with coupl strss fluids as lubricants using finit diffrnc mthod as th numrical tool for analysis as can b dducd from th litratur citd. Prvious rsarchrs sm not to hav ploitd th applicability of finit lmnt mthods in slidr baring dsign. Th finit lmnt mthod is probably th most accurat and vrsatil, but tnds to b vry tim consuming and rquirs high knowldg, not assssabl to th common dsignr [7], hnc it s obvious absnc in th prusd litratur. It is this gap that th prsnt papr sks to fill. In particular, this work cntrs on th us of continuous Galrkin finit lmnt mthod for carrying out a comparativ study of prssur distribution and baring load of infinitly wid parabolic and inclind slidr barings lubricatd with coupl strss fluids. II. MODIFIED REYNOLDS EQUATION Th gomtry of parabolic and inclind slidr barings undr considration ar shown in figs. and 2 rspctivly. Th lubricant in th claranc zon is takn to b a coupl strss fluids. Th slidr baring has a lngth L and movs with a vlocity U as shown in th figs. and 2 h i d L U Fig. : Baring gomtry of a parabolic shapd slidr h hm hi Fig. 2: Baring Gomtry of inclind shapd slidr Th oil film profil for th parabolic slidr in non dimnsionalisd form is as shown in (). Th corrsponding film profil quation for th inclind slidr baring is shown in (2) h h h h 2 = m + p = m + ( 2 + ) () m ( ) h= h + δ Whr h m is th minimum film thicknss at th it of th slidr and δ rprsnts th profil paramtr of th baring. Th continuity and momntum quation for a slidr baring can b writtn in non dimnsional form as in (3) and (4) rspctivly. u w + = z (3) 2 4 p u u = l 2 4 z z (4) p = z (5) Calculations on slidr baring lubrication ar frquntly prformd in non dimnsional form [8, 9, 2]. W dfin th following non dimnsional paramtrs. z u l =, z =, u =, w = w (6) L h U Uh U L 2 pho l ( η / μ) P =, l = = μul ho ho (7) In ths quations u and w rprsnts th non dimnsional vlocity componnts in and z dirctions rspctivly. p is th non dimnsional prssur. μ is th shar viscosity and η is a nw matrial constant with th dimnsion of momntum and is rsponsibl for th coupl strss proprty. Its valu can b dtrmind by som primnts as discussd by Stoks. Th 2 hm (2) ISBN: ISSN: (Print); ISSN: (Onlin) WCE 2

3 Procdings of th World Congrss on Enginring 2 Vol II WCE 2, Jun 3 - July 2, 2, London, U.K. 2 η dimnsion of l = is of lngth. Th lngth could b μ idntifid as th charactristic matrial lngth or th molcular lngth of th polar suspnsions in a non polar fluid. Th ffcts of coupl strss ar thrfor dominatd through th dimnsionlss coupl strss paramtr l l = h. If η =, o thrfor l =,and th classical form of th Nwtonian lubricant is obtaind. Th boundary conditions ar th no slip conditions and th non coupl strss conditions. Th non dimnsional modifid Rynolds quation govrning th hydrodynamic film prssur is givn by d (, ) dp f h l = 6 dh (8) d d d f h, l dfind by Whr th function is ( ) 2 3 h f ( h, l ) = h 2l h 2l tanh 2 l As th valu of l approachs zro, (9) is rducd to th classical form for a Nwtonian lubricant cas.. WEAK FORMULATION Obtain th rsidual of th govrning quation by taking all trms on th right hand sid to th lft hand sid to obtain q.. A Galrkin formulation was utilizd in ordr to apply th finit lmnt mthod [2]. d (, ) dp dh Rp = f( h, l ) 6 d d d () Multiplying () by a wight function and intgrating ovr a typical lmnt with nd nods and 3, w obtain 3 d dp dh wi f ( h, l ) 6 d= d d d () Intgrating th first trm of (), w obtain th quation blow. 3 d dp wi f ( h, l ) d d d (2) 3 3 dw = i dp dp f ( h, l ) + wi f ( h, l ) d d d i =, 2,... n Eq. now bcoms (9) 3 (, ) dwi f h l d ( ) dp d 3 3 dp dh + wf h, l - 6 d d d (3) Now w assum a trial solution for th nodal dgr of frdom of th form of (8). n p = p ϕ (4) j j j= Obtaining th first drivativ of (4) and substituting into (3) with th wight functions st idntical to th trial functions, w obtain th Galrkin finit lmnt modl for th parabolic slidr problm shown in (5). Th intgration is ovr a typical lmnt as shown in th quation. n j= dϕ j dϕ i dp dh f ( h, l ) pj + f ( h, l ) ϕi 6 ϕid d d = d d IV. BOUNDARY CONDITIONS (5) Th boundary conditions ndd to solv (5) ar th spcification of th prssur at th nd of th baring. Th prssurs at th nds of th baring ar st to. V. SOLUTION METHODOLOGY Th dimnsionlss physical domain, is dividd into uniform quadratic lmnts of lngthδ. This rsults to a constant transformation to a local lmnt co ordinat systm in which th diffrntial and / ar givn by Δ ξ (6) Δ ξ (7) Whr ξ is th natural co-ordinat systm in an lmnt. Th prssur in an lmnt is approimatd by basis or trial functions φ I ovr ach lmnt and is givn by (8) p 3 ( ξ ) p ϕ ( ξ) = (8) j= j j Th transformd lmnt intgrals obtaind from th wak formulation of th govrning quation shown in (5) ar valuatd numrically by Gauss Quadratur by making us of th transformd intgral shown in (9) ( ) 3 dϕξ dϕξ ( ) (9) K = ( f (, l( ξ ))) J ( ξ) d( ξ) J ( ξ) dξ J ( ξ) dξ J ( ξ ) is th Jacobian and is dfind by th prssion blow ISBN: ISSN: (Print); ISSN: (Onlin) WCE 2

4 Procdings of th World Congrss on Enginring 2 Vol II WCE 2, Jun 3 - July 2, 2, London, U.K. ( ) 3 d dϕi ξ J ( ξ ) = = i dξ i= dξ Th lmnt stiffnss intgrals ar valuatd itrativly using thr gauss points and wights according to th prssion in (2) + n I( ξ ) dξ % wnli( ξnl ) (2) l= Aftr writing th global intgral as a sum of th individual lmnt stiffnss intgrals, a systm of algbraic quations is obtaind. Boundary conditions ar imposd on th global systm of quations rsulting in a condnsd matri which is solvd by gauss sidl itrativ schm to obtain th prssur solution. Paramtric studis ar carrid out to dtrmin th ffct of baring paramtrs on th prssur distribution and baring load VI. NUMERICAL RESULTS AND DISCUSSION In th following sction, th rsults of th finit lmnt basd simulation of th slidr baring configurations undr considration ar prsntd. Th validity of th rsults of th finit lmnt simulation is amind. In particular, w invstigat th convrgnc charactristics of th rsults obtaind using th finit lmnt mthod and compar sam with thos obtaind using th finit diffrnc mthod. VII. VALIDATION OF RESULTS TABLE I NODAL PRESSURES OBTAINED USING FINITE ELEMENT METHOD FOR INFINITELY WIDE INCLINED SLIDER BEARING WITH COUPLE STRESS PARAMETER Nod position lmnts 2 lmnts 4 lmnts 8 lmnts ISBN: ISSN: (Print); ISSN: (Onlin) WCE 2

5 Procdings of th World Congrss on Enginring 2 Vol II WCE 2, Jun 3 - July 2, 2, London, U.K. TABLE II NODAL PRESSURES OBTAINED USING FINITE DIFFERENCE METHOD FOR INFINITELY WIDE INCLINED SLIDER BEARING WITH COUPLE STRESS PARAMETER Nod position lmnts 2 lmnts 4 lmnts 8 lmnts Th rsults prsntd in tabl I show that th convrgnc rat of th solution varis spatially along th baring lngth. Thr ar points which hibit fast convrgnc rat whras othr convrgs lss quickly. For a msh of tn () quadratic lmnts, it can b sn that th finit lmnt solution convrgs at points.2..3,.4 and.7. Sinc ths points convrg quickly, w can apply adaptiv msh rfinmnt to th othr aras to acclrat th convrgnc rat. For a msh of 8 lmnts, th solution convrgs for all th points considrd. From th standpoint of cost of computation, it is unconomical to simulat th baring with gratr numbr of lmnts. Th global convrgnc of th finit lmnt mthod dmonstrats that th mthod can b usd to corrctly simulat th hydrodynamic schm of inclind slidr baring whn th lubricant is a coupl strss fluid. Tabl II shows th rsults obtaind using th finit diffrnc mthod for an infinitly wid inclind slidr baring with coupl strss fluid. It can b sn that thr is no vidnc of convrgnc of th solution at all points vn with a msh dnsity of 8 lmnts. Th solution obtaind point wis appars to b incrasing without bound. Th rason for this is th non linarity introducd by th coupl strss paramtr into th govrning quation. Th finit diffrnc mthod is thrfor unsuitabl for simulating this class of barings with coupl strss fluids. Th bhaviour of th finit lmnt and finit diffrnc rsults for parabolic slidr baring configurations hav also bn amind and a conclusion similar to th inclind cas has bn obtaind. In ordr to rval th caus of th instability of th finit diffrnc schm, th simulation was carrid out without th us of a coupl strss paramtr. This is quivalnt to modling th baring with Nwtonian lubricants. Th rsults shown in tabl III wr obtaind for an inclind slidr baring. TABLE III NODAL PRESSURES OBTAINED USING FINITE DIFFERENCE METHOD FOR INFINITELY WIDE INCLINED SLIDER BEARING WITHOUT COUPLE STRESS PARAMETER Nod position lmnts 2 lmnts 4 lmnts 8 lmnts Tabl III shows that th finit diffrnc mthod convrgs whn th inclind slidr baring is modld without th us of coupl strss paramtr. It is concludd that in th prsnt cas whr th barings ar modld to includ coupl strss paramtr, th finit diffrnc mthod is unsuitabl. VIII. PARAMETRIC STUDIES To provid information for nginrs involvd with slidr baring dsign, comparison of th prformanc of th two baring profils is mad. First, comparison of th prssur distribution undr th sam baring paramtrs is invstigatd through numrical primnts. Tabl IV shows th dimnsionlss prssur obtaind for th parabolic and inclind slidr barings rspctivly forδ =, ISBN: ISSN: (Print); ISSN: (Onlin) WCE 2

6 Procdings of th World Congrss on Enginring 2 Vol II WCE 2, Jun 3 - July 2, 2, London, U.K. l =.2 minimum film thicknss =, and a uniform msh of lmnts Fig 3 shows th dimnsionlss prssur gnratd undr th sam conditions for inclind and parabolic slidr barings with coupl strss. Th graph shows that a lowr maimum prssur is gnratd using th sam baring paramtrs for an inclind slidr baring with coupl strss compard to a parabolic baring. Th maimum dimnsionlss prssur gnratd for parabolic slidr baring is.34 compard to.3 for th inclind slidr baring. Undr th magnitud of th paramtrs considrd, th parabolic slidr baring gnrats a load capacity of.2 compard to.9 for th inclind slidr baring. For dimnsionlss distanc up to.8 th prssur gnratd in th parabolic slidr baring is gratr than that in th inclind slidr baring, but downstram from.8 to th nd of th baring, th prssur in th lattr is gratr. To obsrv th prformanc of th two barings without coupl strss, fig. 4 is prsntd. Fig. 4 shows th dimnsionlss prssur distribution of th two barings without coupl strss for δ =. Th graph shows that th parabolic slidr baring rtains its highr maimum prssur as in th non Nwtonian lubricant cas. Howvr, th ffct of simulating th two barings without coupl strss is to dcras th prssur gnratd from.34 to.28 and from.3 to.25 for th parabolic and inclind slidr barings rspctivly. Incrasing th profil paramtr from (fig. 4) to.4(fig. 5) rsults in a dcras in th maimum prssur gnratd in th parabolic slidr in contrast to th inclind cas whr it has a positiv ffct. Incrasing th profil paramtr is th sam as incrasing th wdg ffct which ultimatly rsults in gratr prssur in th lubricating film of th inclind slidr baring. Th nt rsult of simulating th barings without coupl strss is to dcras th load carrying capacity as shown in fig. 4 Fig. 4 shows that th load carrying capacity for a parabolic slidr baring is highr than that for th inclind cas for th sam coupl strss paramtr. Parabolic slidr barings ar thrfor suprior at highr coupl strss paramtrs than at lowr coupl strss rgims Dimnsionlss Prssur (p) inclind parabolic Dimnsionlss distanc () Fig. 3: Dimnsionlss prssurs for infinitly wid parabolic and inclind slidr baring with coupl strss Dimnsionlss Prssur (p) Dimnsionlss distanc () Parabolic Inclind Fig. 4: Dimnsionlss prssur against dimnsionlss distanc for inclind and parabolic slidr without coupl strss. δ = ISBN: ISSN: (Print); ISSN: (Onlin) WCE 2

7 Procdings of th World Congrss on Enginring 2 Vol II WCE 2, Jun 3 - July 2, 2, London, U.K. Dimnsionlss Prssur (p) Dimnsionlss distanc () Parabolic Inclind Fig. 5: Dimnsionlss prssur against dimnsionlss distanc for infinitly wid inclind and parabolic slidr baring. Without coupl strss ( δ =.4 ) Dimnsionlss load capacity Parabolic Slidr Inclind Slidr Coupl strss (l) Fig. 6: Dimnsionlss Load capacity against dimnsionlss coupl strss for parabolic and inclind slidr barings TABLE IV DIMENSIONLESS PRESSURE AT SELECTED POINTS ALONG THE BEARING FOR PARABOLIC AND INCLINED SLIDER BEARINGS Distanc( ) Prssur(Inclind) Prssur(Parabolic) ISBN: ISSN: (Print); ISSN: (Onlin) WCE 2

8 Procdings of th World Congrss on Enginring 2 Vol II WCE 2, Jun 3 - July 2, 2, London, U.K. IX. CONCLUSION A comparativ study of th prssur distribution and load capacity of infinitly wid parabolic and inclind slidr barings has bn prsntd. Finit lmnt mthod was usd to discrtiz th govrning quation with th associatd boundary conditions. It has bn shown that whn Rynolds quation is modld to includ coupl strsss, finit diffrnc mthod fails to produc convrgnt solution to th govrning quation. Th infinitly wid parabolic slidr baring has bn shown to b suprior in trms of baring load as a rsult of th gratr prssur gnration rsulting from incrasd wdg ffct. REFERENCES [] L. H. Brzins, C. E. Goodyr. and P.K. Jimack, High ordr discontinuous Galrkin mthod for lastohydrodynamic lubrication lin contact problms, Communications in Numrical Mthods in Enginring, vol. 9(8), 2, pp 7 [2] A. A. Ozalp and H. Umur Optimum surfac profil dign and prformanc valuation of inclind slidr barings, Currnt Scinc, vol. 9 ( ), 26, pp [3] H. Bayrakckn and M. Yurusoy, Comparison of prssur distribution in inclind and parabolic Slidr Baring, Mathmatical and Computational applications, vol. (), 26, pp [4] C. J. Shah, and M.V. Bhat, Lubrication of a porous ponntial slidr baring by frrofluid with slip vlocity, Turkish Journal of Enginring Environmntal Scinc, vol. 27, 23, pp [5] M. Yurusoy, A study of prssur distribution of a slidr baring lubricatd with Powl Eyring fluid, Turkish J.Eng.Env Sci, vol. 27, 23, pp [6] N. M. Bujurk and B.R.Kudnati, Multigrid solution of modifid Rynolds quation incorporating porolasticity and Coupl Strss, Journal of Porous Mdia, vol. (2), 27, pp [7] C. J. Shah, and M.V.Bhat,, Effct of slip vlocity in a porous scant shapd slidr baring with a frrofluid lubricant, Fizika A 2, 23, pp. 8. [8] J.R. Lin, and M.L.Yu, Stady stat prformanc of parabolic slidr barings with a coupl strss fluid, Journal of Marin Scinc and Tchnology, vol. 2(4), 24, pp [9] V. K. Stoks, Coupl strsss in fluids, Phys Fluids, vol. 9, 966, pp [] J. R. Lin, Effcts of coupl Strss on th Lubrication of Finit Journal Baring, War, vol. 26, 997, pp [] B. P. Mitidirri, Advancd modling of lastohydrodynamic Lubrication, Doctoral Thsis, Tribology Sction and thrmo fluids sction, Dpartmnt of Mchanical Enginring, Imprial Collg, London, 25, pp 9. [2] M. Snangi, B. C. Majumda,, and A.S. Skhar, Elastohydrodynamically lubricatd ball barings with coupl strss fluids, Part : Stady stat analysis, Tribology transactions, vol. 48(3), 25, pp [3] R. Linj, Squz film charactristics of finit Journal barings; coupl strss fluid mod, Tribology Intrnational, vol. 3(4), 998, pp [4] A. A. Elsharkawy, Effcts of misalignmnt on th prformanc of finit journal barings lubricatd with coupl strss fluids, Intrnational Journal of Computr Applications in Tchnology (IJCAT), vol. 2(3), 24. [5] J. R. Lin, Drivation of dynamic coupl-strss Rynold s quation of sliding-squzing surfacs and numrical solution of plan inclind slidr barings, Tribology Intrnational, vol. 36(9), 23, pp [6] G. S. Nada, and T. A. Osman, Static prformanc of finit hydrodynamic journal barings lubricatd by magntic fluids with coupl Strsss, Tribology lttrs, vol. 27(3), 27, pp [7] F. Paulo, J. C. Pimnta and A. Jorg, Journal barings subjctd to dynamic loads: Th Analytical Mobility Mthod, Mcanica Eprimntal, vol. 3, 26, pp ISBN: ISSN: (Print); ISSN: (Onlin) WCE 2