Main Bearing Stiffness Investigation

Transcription

1 Main Baring Stiffnss Invstigation B. Bllakhdhar, A. Dogui and.l. Ligir Abstract Simplifid coupld ngin block-crankshaft modls basd on bam thory provid an fficint substitut to ngin simulation in th dsign procss. Ths modls rquir accurat dfinition of th main baring stiffnss. In this papr, an invstigation of this stiffnss is prsntd. Th claranc ffct is studid using a smooth baring modl. It is manifstd for low shaft displacmnt. Th hydrodynamic assssmnt modl shows that th oil film has no stiffnss for low loads and it is infinitly rigid for important loads. Th dformation stiffnss is dtrmind using a suitabl finit lmnts modl basd on ral CADs. As a rsult, a main baring bhaviour law is proposd. This bhaviour law taks into account th claranc, th hydrodynamic sustntion and th dformation stiffnss. It nsurs proprly th transition from th configuration low rigidity to th configuration high rigidity. Kywords Claranc, dformation stiffnss, main baring bhaviour law, oil film stiffnss I. INTRODUCTION O produc a succssful crankshaft dsign rquirs an Taccurat dtrmination of th main baring loads. A coupld crankshaft ngin block quasi static modl can b usd as a dsign tool to improv th prformanc of th crankshaft and th main barings. In th bginning of ngin dvlopmnt a statically dtrminat schm was usd [1]-[]. Th raction forc of ach baring dpnds only on th throws adjacnt to that baring [3]. Aftr that, a statically indtrminat schm was considrd []-[4]. Th crankshaft bam rpos on rigid supports without baring claranc. Th load xrtd on ach throw affcts all barings. Latr, th rigid supports wr rplacd with lastic ons. In th 70 s, th main baring was modlld by non linar springs [5]. Subsquntly, a coupld quasi static crank modl and hydrodynamic baring modl was dfind [6]-[7]. Rcntly, a coupld crankshaft-block dynamics modl through hydrodynamics was stablishd [8]- [9]. But ths sophisticatd modls rquir a lot of information and important calculation rsourcs. So, th classical analysis mthods rmain attractiv and thy ar still usd in th concpt dsign phas to support th choic of th crankshaft and baring dimnsion [10]-[11]. Ths analyss us simpl approachs and giv quick and sufficintly accurat rsults. Th major nonlinar componnt in th crankshaft -ngin B. Bllakhdhar is with th Mchanical Enginring Laboratory, National Enginring School of Monastir, Univrsity of Monastir, Tunisia (phon:(+16) ; -mail: billing@yahoo.fr). A. Dogui is th dirctor of th Mchanical Enginring Laboratory, National Enginring School of Monastir, Univrsity of Monastir, Tunisia (mail: abdl.dogui@nim.rnu.tn)..l. Ligir is with Powr train Division, RENAULT SAS, 67 ru ds Bons Raisins, 9508 Ruil-Malmaison Cdx, Franc. (-mail: janlouis.ligir@rnault.com). block systm modls is th main baring. So, to nsur that th load calculation is fast and accurat a main baring bhaviour law is suggstd. Svral modls ar usd to invstigat th main baring stiffnss: a full smooth baring modl to xplor th claranc ffct, th hydrodynamic lubrication modl to study th oil film stiffnss and finit lmnts modl to dtrmin th baring dformation stiffnss. Consquntly, th stablishd bhaviour law taks into account, in th sam tim, th claranc, th hydrodynamic assssmnt and th dformation stiffnss. II.MAIN BEARING STIFFNESS Th main objctiv of th analysis of raction stiffnss of crankshaft baring carrid out in this study is to dtrmin a bhaviour law of th baring to us in a simplifid coupld crankshaft-ngin block bam modl. This bhaviour law conncts th baring raction to th shaft motion compard to th baring cntr. Th considrd baring consists of a baring cap assmbld on th crank cas. It has a hydrodynamic lubrication. Th rlativ movmnt of th axis of th shaft compard to th axis of th baring is mainly du to thr factors: - Th claranc btwn th shaft and th journal baring: th claranc affcts first of all th valu of th ccntricity du to th movmnt of hydrodynamic lubrication. It also influncs th dformations stiffnss of th baring. - Th oil film stiffnss: Th stiffnss of th oil film undr hydrodynamic prssur must b analyzd to know if it is ncssary to tak it into account or considr that this film is infinitly rigid. - Th dformation of th baring and th shaft: This dformation is not ssntially localizd in th contact zon. Thus, it dos not dpnd only on dimnsions of contact surfacs; th xtrnal gomtry of th baring can hav a considrabl influnc (function of th applid fforts). Ths analyss ar mad using analytical contact modls to invstigat th influnc of th claranc and oil film on th stiffnss of th baring and finit lmnts modls to dtrminat th dformation stiffnss. A. Th Claranc Effct Classically, a simplifid smooth baring modl is usd to dtrmin th distribution of th contact prssur in smooth barings. Hr a full smooth baring modl; without approximations; is usd to analyz th influnc of th claranc on th total rigidity of th contact shaft/baring. Th shaft is supposd to b stiff. Thrfor all th 1677

2 dformation is supposd to b undrgon by th main baring. Imposing a displacmnt to th shaft, it s supposd that th shaft imposs its form on th dformation of th baring. Th raction forc is obtaind by th intgration of th contact prssur on th ntir contact surfac. Th raction forc has th sam dirction as displacmnt (Fig.1). Baring w Notation: R : Baring radius R : Shaft radius : Radial claranc = R - R O : Baring cntr O i : Shaft cntr O O i = +U = ( x, v ); 0 = ( x, v 0 ); = ( v, u ) y O O i u o M o P v o U Fig. 1 Smooth baring modl M x Shaft Th contact prssur p in a point M of th surfac of contact is supposd to b proportional to th dformation =PM: p() = k R Using th gomtrical closings (O,O i,m o ) and (O,O i,m) it is possibl to dtrmin th angl and thus p( for a givn displacmnt U, th intgration of th contact prssur rsults in th raction forc R F. It has th sam dirction as th displacmnt: R F = ku Using th following dimnsionlss variabls: - Th rlativ displacmnt u = U/R; - Th rlativ claranc j = /R; - Th dimnsionlss raction forc f = R F /klr. W obtain: cos( ) = j ju u j (1 j)( u j) f = u sin + j(sin - ) + (j+u) 1 (sin - ) (1) 8 If th claranc is null (j=0), th prvious quation bcoms: 1 u u u f u( 4u (1 ) arccos( )) () 8 Th curvs of th figur show th volution of th dimnsionlss raction forc f vrsus th rlativ displacmnt u for diffrnt valus of rlativ claranc j. W not an important variation of th initial slop btwn th cas without claranc and that with claranc. But, for important valus of rlativ displacmnt u, th variation bcoms smallr. Th curvs of th figur 3 illustrat th volution of th slop df/du. u v Fig. Dimnsionlss raction forc f(u) df du Fig. 3 Dimnsionlss stiffnss df/du A fundamntal diffrnc is immdiatly notd btwn th xistnc or not of th claranc: at th origin (u=0), for a null claranc, initial stiffnss is qual to th unit (df/du=1). For a non null claranc (j0), vn with low valu, initial stiffnss is null (df/du=0). Whn th displacmnt incrass, th influnc of th claranc dcrass. Th lowr is th claranc, th quickr th corrsponding curv mrgs with th null claranc curv. So th claranc influncs th dformation stiffnss of th baring spcially undr th action of wak fforts. It affcts also th valu of th ccntricity du to th movmnt of hydrodynamic lubrication. B. Hydrodynamic Film Stiffnss Th hydrodynamic baring thory is usd to invstigat th oil film stiffnss. Th shaft and th baring ar supposd to b prfctly rigid. Only th oil film undrgos th dformation. Th baring is supposd to b infinitly short account of th main baring dimnsions. U is th motion of th baring cntr, is th radial claranc and = U/ is th rlativ ccntricity. For a givn rotation spd, th load on th shaft gnrats a rlativ ccntricity ; thus a shaft displacmnt U = Th curvs of th figurs 4 rprsnt th volution of th dimnsionlss baring load f and it s slop df/du according to for diffrnt valus of rlativ spd r = ( is th 0 rotation spd and th rfrnc rotation spd). 0 r f ( ; r) 16 (1 ) (3) (1 ) 1678

3 Th slop df/d rmains vry wak for lowr than 0.75 and incrass vry quickly for highr than 0.9 whatvr th rotation spd valu. Th oil film rigidity is vry low for littl loads and bcoms vry important for strong loads. So it appars that th oil film has no stiffnss for th wak loads and it is infinitly rigid for strong loads. - hxahdral lmnts with 8 nods (C3D8R) adaptd wll to solv th contact problms. Th considrd claranc is about 30m. Th crankcas and th baring caps ar mad of cast iron (Young s modulus E=10GPa, Poisson's ratio =0.7). Th shaft and th barings linrs ar mad of stl (E=10GPa, =0.7). Th avrag lin nods of th dformabl bam wr coupld to a rfrnc point. A vrtical displacmnt was imposd to this rfrnc point. So, th bam crushs th baring linr. Th figur 6 shows th baring dformation stiffnss vrsus low vrtical motion of th shaft. (a) (b) Fig. 4(a) Dimnsionlss baring load f(;r) (b) Dimnsionlss stiffnss df/d(;r) C.Th Dformation Stiffnss Th dformation stiffnss is causd both by th main baring dformation and th shaft dformation. In addition, th main baring dformation is not mainly localizd in th contact zon. So, it dos not dpnd only on dimnsions of contact surfacs. Th xtrnal gomtry of th baring and th boundary conditions can hav a considrabl influnc. To dtrmin th dformation stiffnss of th baring, a finit lmnt modl was cratd. This modl is basd on ral CADs of th crankcas and baring caps, of a four-cylindr in lin ngin (Fig.5). Fig. 5 Burst finit lmnt modl Th crankcas was mbddd at th bottom surfac (cylindr had sid). Th baring caps wr fixd to th crankcas. A dformabl bam, owing th sam diamtr as th crankshaft pivot, was considrd. Mor than 0000 lmnts dividd on two typs ar usd to dscrib th modl: - ttrahdral lmnts with 4 nods (C3D4) adaptd wll for th msh of th complx gomtris. Fig. 6 Th baring dformation stiffnss Th sam shap of th dformation rigidity as th figur 4 is found. Initially th stiffnss is null. Th ffct of th claranc disappars gradually as th movmnt of th shaft incrass. III. SUGGESTED MAIN BEARING STIFFNESS ANALYTICAL MODEL Th smooth baring modl highlights th influnc of th claranc on th baring stiffnss: th influnc of th claranc intrvns only for low shaft motion, thrfor for low fforts (Fig.3). For grat fforts, stiffnss is rlativly indpndnt of shaft motion. Th hydrodynamic lubrication modl highlights th influnc of th oil film rigidity on th ovrall rigidity of th baring: th oil film stiffnss is almost null for low ffort. Th shaft movs of 75% of th valu of th claranc for a vry wak ffort (Fig.4). This influnc bcoms vr wakr (vry important oil film rigidity) as th load incrass. Th finit lmnt modl rsults approv smooth baring modl rsults (Fig.6) and it allows us to dtrmin th dformation stiffnss K d. To illustrat th diffrnt findings, th following data corrsponding to a ral cas ar considrd: - th maximum crankshaft baring ffort is about 50kN; - th baring radius is qual to 4 mm; - th baring width is qual to mm; 1679

4 - th baring claranc is qual to 30 µm; - th oil viscosity is qual to Ns/m; - th rotation spd is qual to 3000 tr/mn; - th dformation stiffnss K d is about N/m. For a givn load F applid to th shaft, his displacmnt will b th sum of hydrodynamic displacmnt U h = obtaind by th hydrodynamic lubrication modl and th displacmnt U d =F/K d du to th lastic dformation of th baring. Th maximum rlativ ccntricity du to hydrodynamic dformation is about 0.95 which corrspond to a shaft displacmnt U of about 8.6 m. Figur 7 shows th ffort curvs: - F(U d ) corrsponding to th baring ffort vrsus th shaft cntr displacmnt du to lastic dformation. - F(U h ) corrsponding to th baring ffort vrsus th shaft cntr displacmnt du to oil film dformation. - F(U=U d +U h ) corrsponding to th baring ffort vrsus th total shaft cntr displacmnt. Fig. 7 Shaft motion/baring load curvs W suggst rgularizing th function stiffnss K=dF/dU through th considration of th oil film ffct and vntually th claranc ffct as follows: 1 K = K d ( 1 1 U ) (4) Whr: is th rgulation paramtr. Th intgration of th rgularizd function K(U) compard to th displacmnt U givs th baring load F(U): U F(U)=K d [ 1 ( 1 )ln( ) U ] (5) 1 To illustrat th quality of this rgularization, th modl (4) is idntifid starting from th curv F(U h +U d ) of figur 7. Th idntifid valus of K d, and ar prsntd in th tabl blow: Th idntifid dformation stiffnss dcrass of about 9%. W not that th suggstd analytical modl nvisags rathr prcisly th bhaviour of th main baring and nsurs proprly th transition from th configuration low rigidity to th configuration high rigidity (Fig.8). TABLE I RESULTS OF THE IDENTIFICATION Initial Fig. 8 Quality of th idntification Idntifid K d(10 8 N/m) 7,67 6,9 (m) (m) 7,5 Th mrit of th bhaviour law proposd is that it can b advantagously usful in bam crankshaft-ngin block combind modls. IV. CONCLUSION This work prsnts an invstigation of main baring stiffnss. Du to th us of smooth baring modl it was shown that th claranc ffct occurs for low loads. Du to th hydrodynamic modl it was shown that th oil film has a vry low stiffnss for low loads and bcoms vry rigid for important loads. Th main baring dformation stiffnss was dtrmind through an appropriat finit lmnt modl basd on ral CADs. Subsquntly, a main baring bhaviour law has bn proposd. This bhaviour law taks into account at th sam tim claranc ffct, hydrodynamic sustntion and dformation stiffnss. This bhaviour law is usful in simplifid bam ngin modls. REFERENCES [1] K. Radrmachr, Das instationar blastt zylindrishs glitlag, Dissrtation TH Karlsruh, 196. [] V. Gross, A. W. Hussmann, Forcs in th main barings of multicylindr ngins, SAE, 1966, [3] Y. Yilmaz, G. Anlas, An invstigation of th ffct of th countrwight configuration on main baring load and crankshaft bnding strss, Advancs in Enginring Softwar, 009, pp [4] I. Piranr, C. Pflugr, O. Bouthir, Cummings crankshaft and baring analysis procss, North Amrican MDI Usr Confrnc, 00. [5] F.V. Tinaut, A. Mlgar, B. Gimnz, L. Frnandz, H. Huidobro, A mthod to dtrmin th two componnts of th crankshaft load on a baring cap in firing ngins, S.A.E. Tchnical papr sris, 000, n

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