Effects of Couple Stress Lubricants on Pressure and Load Capacity of Infinitely Wide Exponentially Shaped Slider Bearing

Transcription

1 Procdings of t World Congrss on Enginring and Computr Scinc 200 Vol II, Octobr 20-22, 200, San Francisco, USA Effcts of Coupl Strss Lubricants on Prssur and Load Capacity of Infinitly Wid Eponntially Sapd Slidr Baring Mobolaji H. Oladind and Jon A. Akpobi Abstract In tis papr, t prssur distribution and load capacity of an infinitly wid ponntially sapd slidr baring lubricatd wit a non Nwtonian coupl strss fluid is prsntd. Basd on Stoks micro continuum tory, t ffct of coupl strsss on t prssur distribution and load ar computd using continuous Galrkin finit lmnt mtod T domain is discrtizd into uniform ms of quadratic isoparamtric lmnts and Gauss quadratur usd to numrically intgrat t stiffnss intgrals to obtain stiffnss matrics for all lmnts wic ar subsquntly assmbld to form t global systm of quations. T rsulting global matri is solvd using Gauss Sidl itrativ scm. T numrical mtod adoptd for t solution is sown to produc convrgnt rsults wn implmntd on a squnc of succssivly finr mss. Having stablisd t rliability of t numrical mtod, paramtric studis ar carrid out to sow t ffcts of coupl strsss and aspct ratio on t prssur distribution and load. Computations put forward sow tat t ffct of coupl strsss is to nanc t load carrying capacity of t baring. It as also bn sown tat compard to t Nwtonian cas, t prssur dvlopd in t claranc zon of t baring is augmntd Kywords Eponntial slidr, finit lmnt, Rynolds, load capacity, prssur. I. INTRODUCTION In most mcanical systms wr rlativ motion occurs btwn two parts, lubricants ar introducd to rduc friction and war. T gomtry of t contacting lmnts dtrmins t sap of t lubricant film []. Various rsarcrs av considrd diffrnt configurations of t lubricating film in t claranc zon in tir analysis. T contacting surfacs can b narrowing gomtrically in linar styl as considrd by Ozalp [2]. H mployd t itrativ transfr matri approac to suggst optimum film profil paramtr 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 t claranc zon and dvlopd closd form prssions for t prformanc mtrics. Sa t al [4] as studid a slidr baring wit ponntial film ticknss profil and obtaind analytical prssions for variation of dimnsionlss prssur, M. H. Oladind is wit t Dpartmnt of Production Enginring, Faculty of Enginring Univrsity of Bnin, P.M.B 54, Nigria (mai bolajai@yaoo.com). J. A. Akpobi is wit t Dpartmnt of Production Enginring, Faculty of Enginring, Univrsity of Bnin, P.M.B 54, Nigria. (mail: alwaysjoni@yaoo.com, Tl +234 (0) ). friction, cofficint of friction and cntr of prssur. A frrofluid was usd btwn t contacting surfacs of t baring. Yurusoy [5] obtaind a prturbation solution for prssur distribution in a slidr baring wit a Powl-Eyring fluid as lubricant. Bujurk t al [6] usd a scond grad fluid in a tapr flat slidr baring similar to tat usd by Ozalp [2] and constructd a Von korman momntum intgral solution. Sa [7] computd valus for t baring caractristics of a scant sapd slidr baring using a magntic fluid as lubricant. Diffrnt typs of fluids av bn usd in t claranc zon of slidr barings and tir prformanc invstigatd as sown in t prvious works citd abov. Howvr, in ordr to nanc lubricating prformanc, t incrasing us of a Nwtonian lubricant wic as bn blndd wit long cain polymrs as bn obsrvd. Sinc t convntional micro continuum tory cannot accuratly dscrib t flow of ts kinds of fluids, various micro continuum toris av bn proposd [8]. Stoks [9] proposd t simplst micro - continuum tory wic prmits t prsnc of coupl strsss, body coupls and non symmtric tnors [0]. A numbr of rsarcrs av invstigatd t ffct of t coupl strss fluid modl on t stady stat prformanc of diffrnt slidr baring configurations using diffrnt numrical scms. In rcnt tims, most numrical work in ydrodynamic lubrication as involvd t us of t Rynolds quation and t finit diffrnc mtod []. A finit diffrnc multigrid approac was usd to invstigat t squz film bavior of porolastic baring wit coupl strss fluid as lubricant by Bujurk t al [6]. Ty rportd tat porolastic barings wit coupl strss fluid as lubricant provid augmntd prssur distribution and nsurd significant load carrying capacity. Srangi t al [2] solvd t modifid Rynolds quation tndd to includ coupl strss ffcts in lubricants blndd wit polar additivs using t finit diffrnc mtod wit a succssiv ovr rlaation scm. Ty rportd incras in load carrying capacity and rduction in friction cofficint as compard to Nwtonian lubricants. Lin [3] usd t conjugat Mtod of itration to build up t prssur gnratd in a finit journal baring lubricatd wit a coupl strss fluids. T rsults obtaind including incras in t load carrying capacity agrs wit tos obtaind by Srangi t al [2] and Bujurk t al [6]. Elsarkawy [4] providd a numrical solution for a matmatical modl for ydrodynamic lubrication of misalignd journal barings wit coupl strss fluids as lubricants using t finit diffrnc Mtod. Lin[5]

2 Procdings of t World Congrss on Enginring and Computr Scinc 200 Vol II, Octobr 20-22, 200, San Francisco, USA calculatd t stady and prturbd prssurs of a two dimnsional plan inclind slidr baring incorporating a coupl strss fluid using t conjugat gradint mtod and rportd improvd stady and dynamic prformanc compard to t Nwtonian cas spcially for igr aspct ratios. Nada and Osman [6] invstigatd t problm of finit ydrodynamic journal baring lubricatd by magntic fluids wit coupl strsss using t finit diffrnc mtod. For diffrnt coupl strss paramtrs and magntic cofficints, ty obtaind t prssur distribution. Ty concludd tat fluids wit coupl strsss ar bttr compard wit t Nwtonian cas aftr comparison of t baring static caractristics. Rcntly, Oladind and Akpobi [7] studid an infinitly wid parabolic slidr baring using finit lmnt mtod and sowd t ffct of coupl strss lubricants on t baring load. From t citd litratur it can b sn tat t ponntial slidr baring lubricatd wit coupl strss fluids as not bn givn attntion. It is tis gap tat tis papr tris to fill. In particular, tis work cntrs on t us of continuous Galrkin finit lmnt mtod to study t ffct of coupl strss on load capacity and prssur distribution on an infinitly wid ponntially sapd slidr baring.. MODIFIED REYNOLDS EQUATION T ponntial slidr baring undr considration is sown in fig. T lubricant in t claranc zon is takn to b a coupl strss fluids. T slidr baring as a lngt L and movs wit slidr vlocity U as sown in fig. Fig.: Pysical gomtry of a wid slidr baring wit an ponntial film profil. T film profil is dscribd by () [8] L U a ln( ) 2 L () Wr L is t lngt of t baring; 2 is t film ticknss at t ntry of t slidr baring and a is t film ticknss ratio (2) T film ticknss ratio is approimatd by +, wr is t profil paramtr dfind by (3). In (3), d is t souldr igt dfind as t diffrnc btwn t maimum and minimum film ticknss d (3) Calculations on slidr baring lubrication ar frquntly prformd in non dimnsional form [9, 20, 2]. Using t non dimnsional paramtrs in (4), t film ticknss profil can b cast in dimnsionlss form as sown in (5), (4) L 2 ln a (5) T dimnsionlss modifid Rynolds quation govrning t ydrodynamic film prssur for a slidr baring lubricatd by coupl strss fluid is givn by (6)[2]. d, dp f l 6 d (6) In (6), l is t dimnsionlss coupl strss paramtr. T coupl strss paramtr can b obtaind by som primnts as dscribd by stoks. It can b computd by using (7) l 2 (7) In (7), is t sar viscosity and is a nw matrial constant wit t dimnsion of momntum and is rsponsibl for t coupl strss proprty. T ffct of coupl strss is dtrmind troug t coupl strss paramtr dfind l as l. If 0, trfor l 0,and t classical form o of t Nwtonian lubricant is obtaind. f, l in (6) is dfind by (8). T function 2 3 f, l 2l 2l tan 2 l As t valu of l approacs zro, (8) is rducd to t classical form for a Nwtonian lubricant cas. T boundary conditions ar a spcification of t prssur at t nds of t barings. T boundary conditions ar givn in (9) (9) II. WEAK FORMULATION If In ordr to obtain t prssur distribution on t baring using t finit lmnt mtod, w first obtain t rsidual of t govrning quation by taking all trms on t rigt and sid to t lft and sid to obtain (0). A Galrkin (8)

3 Procdings of t World Congrss on Enginring and Computr Scinc 200 Vol II, Octobr 20-22, 200, San Francisco, USA formulation was utilizd in ordr to apply t finit lmnt mtod, d, dp R p f l 6 d (0) Multiplying (0) by a wigt function and intgrating ovr a typical lmnt wit nd nods and 3, w obtain () 3 d (, ) dp wi f l 6 d 0 Intgrating t first trm of (), w obtain (2) 3 d dp 3 dw, i dp wi f l f, l 3 dp wf i l, i,2,... n () (2) Equation () now bcoms 3 3 dw i, dp, dp f l wf l (3) 3 d - 6 =0 Now w assum a trial solution for t nodal dgr of frdom of t form of (4) p n p j j (4) j Obtaining t first drivativ of (4) and substituting into (3) wit t wigt functions st idntical to t trial functions, w obtain t Galrkin finit lmnt modl for t parabolic slidr problm sown in (5). T intgration is ovr a typical lmnt. n d j d, i dp, f l pj f l i j d 6 0 (5) i III. NUMERICAL RESULTS AND DISCUSION A. Validation of Rsults T finit lmnt rsults ar only approimat in natur and in using t rsults to prdict t load capacity, it is ssntial tat t rliability of t rsults is first amind. In numrical analysis, in particular grid point mtods, t finit lmnt solution vntually convrgs to t act solution as t ms is rfind progrssivly. Consquntly, t numrical modl is first amind for its convrgnc caractristic using mss of 4, 8 and 0 quadratic isoparamtric lmnts for and Numrical primntation sows tat t dimnsionlss prssur at t middl of t baring is for a ms of 4 quadratic lmnts, for a ms of 8 isoparamtric quadratic lmnts and for a ms of 0 isoparamtric quadratic lmnts. T finit lmnt modl applid clarly ibits convrgnc bavior. Incrasing t ms dnsity mor tan 0 quadratic lmnts only incrass t computational tim and no apprciabl ffct on t accuracy of t solution. Hnc a ms dnsity of 0 isoparamtric lmnts was usd for t paramtric studis. B. Prssur T variation of dimnsionlss prssur distribution wit dimnsionlss distanc along t baring is sown in figur 2 for a Nwtonian cas wr t coupl strss paramtr is st qual to zro. It can b dducd from t figur tat t ffct of incras in profil paramtr for an ponntially sapd slidr lubricatd wit a Nwtonian lubricant is to augmnt t prssur distribution in t baring. Tis is attributabl to t incras in wdg ffct on t baring as t profil paramtr incrass. Compard to a parabolic slidr baring of a similar profil paramtr, t ponntial slidr baring producs a smallr prssur build up in t lubricating film. Howvr, t ponntial slidr baring ibits a similar trnd in prssur build up wit incras in profil paramtr as rportd by Oladind and Akpobi (2009). Fig 3 sows t variation of dimnsionlss prssur wit distanc for an ponntial slidr baring using a non Nwtonian coupl strss fluid wit coupl strss paramtr qual 0.,0.2 and 0.3 rspctivly. T plot sows tat t prssur in t claranc zon of t baring incrass wit incras in coupl strss paramtr. Tis finding is consistnt wit tat obtaind by Oladind and Akpobi (2009) for a parabolic slidr baring. Wit incras in coupl strss paramtr from 0. to 0.3, t maimum prssur in t baring incrass by a factor approimatly qual to 2. Howvr, in contrast wit t rsults obtaind by [7] for a parabolic slidr baring, t position of t maimum prssur is not influncd by introduction of polar additivs in t Nwtonian lubricant accountd for by t coupl strss paramtr for a givn profil paramtr. Computations sow tat wit Incras in t profil paramtr, t position of t maimum prssur movs towards t it of t baring for diffrnt coupl strss paramtr. C. Load Capacity T ffct of profil paramtr on load capacity for diffrnt coupl strss paramtrs is sown in fig 4 Fig 4 sows t dimnsionlss load capacity as a function of profil paramtr for diffrnt coupl strss paramtr. Sinc t ffct of coupl strss provid an incras in t oil film prssur, t load carrying capacity is similarly influncd. In gnral, wit incras in non Nwtonian bavior, t load carrying of t baring is incrasd. T plot sows tat t incras in load capacity wit coupl strsss is gratr at igr profil paramtrs. Compard to t parabolic slidr cas wic ibits optimum profil paramtr aftr wic t bnfit drivd from t

4 Procdings of t World Congrss on Enginring and Computr Scinc 200 Vol II, Octobr 20-22, 200, San Francisco, USA introduction of polar additivs dcrass, t bnfit drivd from t addition of polar additivs incrass wit incras in profil paramtr for an ponntially sapd slidr baring. T plot also sows tat for coup strss gratr tan 0.3, t variation of dimnsionlss load capacity wit profil paramtr bcoms linar in natur. Fig 5 sows t rsults of t simulation of load capacity against coupl strss paramtr for diffrnt valus of profil paramtr. As illustratd in t grap, a igr profil paramtr for an ponntial slidr baring lubricatd wit a Nwtonian fluid brings about a igr load carrying capacity. T plot sows tat t baring load incrass wit coupl strss paramtr for a givn profil paramtr. T improvmnt in load capacity is du to t incras in t prssur gnratd in t lubricant film wit incras in profil paramtr du to wdg ffct. IV. CONCLUSION T ffct of coupl strsss and profil paramtr on t baring load and prssur profil of an infinitly wid ponntial slidr baring as bn considrd using t finit lmnt mtod. T ffct of structural and lubricant rological proprty (coupl strss) on load capacity and prssur distribution as bn prsntd. T prsnt study provids could b usd to dvlop nw dsign carts wic ar usful for baring dsign. Fig. 2: Variation of dimnsionlss Prssur wit distanc along t baring for diffrnt aspct ratios. Fig. 3: Variation of dimnsionlss Prssur wit distanc along ponntial slidr baring wit diffrnt coupl strss paramtrs.

5 Procdings of t World Congrss on Enginring and Computr Scinc 200 Vol II, Octobr 20-22, 200, San Francisco, USA Fig. 4 Variation of dimnsionlss load capacity wit profil paramtr for diffrnt coupl strss paramtrs. Fig. 5: Variation of Dimnsionlss Load Capacity wit Coupl Strss Paramtr for diffrnt Profil Paramtrs REFERENCES [] L.H. Brzins, C.E. Goodyr and P.K. Jimack,, Hig ordr discontinuous Galrkin mtod for lastoydrodynamic lubrication lin contact problms, Communications in Numrical Mtods in Enginring, Vol. 9, Issu 8, pp 7,2000. [2] A.A. Ozalp and H. Umur, Optimum surfac profil dign and prformanc valuation of inclind slidr barings, Currnt Scinc, Vol. 90, Issu, pp ,2006. [3] H. Bayrakckn and M. Yurusoy, Comparison of prssur distribution in inclind and parabolic slidr baring, Matmatical and Computational applications, Vol., Issu, pp 65-7(2006). [4] C.J. Sa and M.V. Bat, Lubrication of a porous ponntial slidr baring by frrofluid wit slip vlocity, Turkis Journal of Enginring Environmntal Scinc, Vol. 27, pp [5] M. Yurusoy, A study of prssur distribution of a slidr baring lubricatd wit Powl Eyring fluid Turkis J.Eng.Env Sci, Vol 27., pp ,2003. [6] N. M. Bujurk and B. R. Kudnati, Multigrid solution of modifid Rynolds Equation incorporating porolasticity and Coupl Strss, Journal of porous mdis, Vol. 0, issu 2, pp 25 36, [7] C.J. Sa and M.V Bat, Effct of slip vlocity in a porous scant sapd slidr baring wit a frrofluid lubricant, Fizika A 2, pp 8, [8] J.R. Lin and M.L Yu, Stady Stat prformanc of parabolic slidr barings wit a coupl strss fluid, Journal of Marin Scinc and Tcnology, Vol 2, Issu 4, pp , [9] V.K. Stoks Coupl strsss in fluids Pys Fluids, Vol. 9, pp ,966.

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