Univrsity of Kntucky UKnowldg Univrsity of Kntucky Mastr's Thss Graduat School 2004 VALIDATION OF FINITE ELEMENT PROGRAM FOR JOURNAL BEARINGS -- STATIC AND DYNAMIC PROPERTIES Raja Shkar Balupari Univrsity of Kntucky, balupari@ngr.uky.du Click hr to lt us know how accss to this documnt bnfits you. Rcommndd Citation Balupari, Raja Shkar, "VALIDATION OF FINITE ELEMENT PROGRAM FOR JOURNAL BEARINGS -- STATIC AND DYNAMIC PROPERTIES" 2004. Univrsity of Kntucky Mastr's Thss. 325. https://uknowldg.uky.du/gradschool_thss/325 This Thsis is brought to you for fr and opn accss by th Graduat School at UKnowldg. It has bn accptd for inclusion in Univrsity of Kntucky Mastr's Thss by an authorizd administrator of UKnowldg. For mor information, plas contact UKnowldg@lsv.uky.du.
ABSTRACT OF THESIS VALIDATION OF FINITE ELEMENT PROGRAM FOR JOURNAL BEARINGS STATIC AND DYNAMIC PROPERTIES Th analysis of baring systms involvs th prdiction of thir static and dynamic charactristics. Th capability to comput th dynamic charactristics for hydrodynamic barings has bn addd to Baring Dsign Systm BRGDS, a finit lmnt program dvlopd by Dr. R.W. Stphnson, and th rsults obtaind wr validatd. In this softwar, a standard finit lmnt implmntation of th Rynolds quation is usd to modl th land rgion of th baring with prssur dgrs of frdom. Th assumptions of incomprssibl flow, constant viscosity, and no fluid inrtia trms ar mad. Th prssur solution is intgratd to giv th baring load, and th stiffnss and damping charactristics wr calculatd by a prturbation mthod. Th static and dynamic charactristics of 60, 120 and 180 partial barings wr vrifid and compard for a lngth to diamtr L/D ratio of 0.5. A comparison has also bn obtaind for th 120 baring with L/D ratios of 0.5, 0.75 and 1.0. A 360 -journal baring was vrifid for an L/D ratio of 0.5 and also compard to an L/D ratio of 1.0. Th rsults ar in good agrmnt with othr vrifid rsults. Th ffct of providing lubricant to th rcsss has bn shown for a 120 hybrid hydrostatic baring with a singl and doubl rcss. Kywords: Rynolds quation, Journal barings, Stiffnss, Damping, Sommrfld Numbr. Raja Shkar Balupari 07/28/2004 Copyright Raja Shkar Balupari 2004
VALIDATION OF FINITE ELEMENT PROGRAM FOR JOURNAL BEARINGS STATIC AND DYNAMIC PROPERTIES By Raja Shkar Balupari Dr. Kith Rouch Dirctor of Thsis Dr. Gorg Huang Dirctor of Graduat Studis 07/28/2004
RULES FOR THE USE OF THESIS Unpublishd thsis submittd for th Mastr s dgr and dpositd in th Univrsity of Kntucky Library ar as a rul opn for inspction, but ar to b usd only with du rgard to th rights of th authors. Bibliographical rfrncs may b notd, but quotations or summaris of parts may b publishd only with th prmission of th author, and with th usual scholarly acknowldgmnts. Extnsiv copying or publication of th thsis in whol or in part also rquirs th consnt of th Dan of th Graduat School of th Univrsity of Kntucky.
THESIS Raja Shkar Balupari Th Graduat School Univrsity of Kntucky 2004
VALIDATION OF FINITE ELEMENT PROGRAM FOR JOURNAL BEARINGS STATIC AND DYNAMIC PROPERTIES THESIS A thsis submittd in partial fulfillmnt of th rquirmnts for th dgr of Mastr of Scinc in Mchanical Enginring in th Collg of Enginring at th Univrsity of Kntucky By Raja Shkar Balupari Lxington, Kntucky Dirctor: Dr. Kith Rouch, Profssor of Mchanical Enginring Lxington, Kntucky 2004 Copyright Raja Shkar Balupari 2004
Ddication To my family, for thir vrlasting lov, support and ncouragmnt
ACKNOWLEDGEMENTS I would lik to xprss dp gratitud to Dr. Kith Rouch for having givn m this opportunity to work with him. Th prsnt work would not hav bn possibl without th consistnt guidanc and patinc shown by him. I would lik to thank Dr. R. W. Stphnson from Mchcon, Inc., for his guidanc and most importantly for providing m with important data. I would also lik to thank Dr. Raymond LBau and Dr. David Hrrin for srving on my thsis committ. I would lik to thank my family for thir trmndous support and ncouragmnt to pursu highr studis. I would particularly lik to thank my brothr Ravindra Balupari for his support and guidanc through out my graduat study. I would also lik to thank all my frinds for thir assistanc during my stay hr at th Univrsity of Kntucky. iii
TABLE OF CONTENTS Acknowldgmnts...iii List of Tabls. vii List of Figurs...viii List of Fils.xi Chaptr 1. Introduction..1 1.1 Introduction...1 1.2 Typs of Journal Barings.2 1.2.1 Partial Arc Barings 2 1.2.2 Plain Journal Barings 360º.2 1.2.3 Axial Groov Barings......3 1.2.4 Prssur Dam Barings...3 1.2.5 Multi-lob Barings 3 1.2.6 Tilting Pad Barings......4 1.2.7 Hybrid Barings...4 1.3 Litratur Rviw...5 1.4 Scop of Thsis..6 2. Solution for Fluid Film Barings...9 2.1 Introduction 9 2.2 Viscosity 9 2.3 Solution of Fluid Film Barings..11 2.3.1 Assumptions..12 2.3.2 Rynolds Equation....12 2.4 Finit Elmnt Mthod 15 2.4.1 Finit Elmnt Solution of Rynolds Equation 17 3. Baring Dsign Program 25 3.1 Introduction.. 25 3.2 Ovrviw.. 25 3.3 Baring Coordinat Systm.26 iv
3.4 Solution Procdur... 27 3.5 Gnral Systm of Equations... 27 3.5.1 Eccntricity... 28 3.5.2 Film Thicknss Calculation...... 28 3.5.3 Baring Forcs and Momnts... 29 3.5.4 Hydrostatic Solution.30 3.6 Convrgnc Conditions. 30 3.6.1 Load Convrgnc.... 30 3.6.2 Prssur Convrgnc... 31 3.6.3 Rcss Flow Convrgnc.... 31 3.7 Damping and Stiffnss Calculation. 31 4. Rsults and Discussion..37 4.1 Introduction.. 37 4.2 Sommrfld Numbr 38 4.3 Partial Arc Baring.. 38 4.3.1 60 Partial Baring.... 39 4.3.2 120 Partial Baring..40 4.3.3 180 Partial Baring..40 4.4 Effct of Baring Arc...40 4.5 Effct of Lngth to Diamtr Ratio. 41 4.6 Full Journal Baring 360.... 42 4.6.1 Effct of Lngth to Diamtr Ratio 360... 43 4.7 Partial Arc Hybrid Baring..43 5. Conclusions and Futur Work...69 5.1 Conclusions..69 5.2 Futur Work. 70 Appndix 71 A1. Nomnclatur.. 71 A2.STIFFCALC Subroutin.. 72 A3. DAMPCALC Subroutin 74 A4. Sampl input fil.76 v
A5. Sampl output fil...78 Rfrncs..81 Vita...85 vi
LIST OF TABLES Tabl 4.1: Baring dsign data for 60º, 120º and 180º partial arc barings...45 Tabl 4.2: Stiffnss and damping cofficints for a 120 baring with no rcss.45 Tabl 4.3: Stiffnss and damping cofficints for a 120 baring with singl rcss...45 Tabl 4.4: Stiffnss and damping cofficints for a 120 baring with two rcsss...45 vii
LIST OF FIGURES Figur 1.1: Schmatic diagram of full plain journal barings Allair t al., 1975...7 Figur 1.2: Schmatic diagram of Partial arc baring Allair t al., 1975....8 Figur 2.1: Vlocity profil btwn two paralll plats to quantify viscosity Szri, 1999...21 Figur 2.2: Fluid film gomtry Allair t. al., 1977..21 Figur 2.3: Boundary conditions on unrolld baring surfac Allair t. al., 1975 22 Figur 2.4: Division of Fluid film into lmnts Nicholas, 1977...22 Figur 2.5: Th procss of finit lmnt analysis Bath, 1996.23 Figur 2.6: Th th two dimnsional simplx triangular lmnt Nicholas, 1977... 24 Figur 3.1: Baring Coordinat systm Stphnson, 1997.34 Figur 3.2: Sid viw and prssur profil of journal baring Allair, 1987......34 Figur 3.3: Flow chart of BRGDS Baring analysis program Stphnson, 1997...35 Figur 3.4: Th physical rprsntation of Dynamic forc cofficints of a fluid film Baring San Andrs, 2000........36 Figur 4.1: Dimnsionlss dirct stiffnss cofficints for a 60º baring with L/D=0.5...46 Figur 4.2: Dimnsionlss dirct damping cofficints for a 60º baring with L/D=0.5..46 Figur 4.3: Dimnsionlss cross-coupld stiffnss for a 60º baring with L/D=0.5...47 Figur 4.4: Dimnsionlss cross-coupld damping for a 60º baring with L/D=0.5 47 Figur 4.5: Dimnsionlss stiffnss cofficints vs. Sommrfld numbr for a 120º baring with L/D=1.0. 48 Figur 4.6: Dimnsionlss damping cofficints vs. Sommrfld numbr for a 120º baring with L/D=1.0....49 Figur 4.7: Dimnsionlss stiffnss cofficints vs. Sommrfld numbr for a 180º baring with L/D=0.5.50 Figur 4.8: Dimnsionlss damping cofficints vs. Sommrfld numbr for a 180º baring with L/D=0.5. 51 Figur 4.9: Sommrfld numbr vs. ccntricity ratio for 60º, 120º and 180º barings 52 viii
Figur 4.10: Sommrfld numbr vs. ccntricity angl for 60º, 120º and 180º barings 52 Figur 4.11: Prssur at circumfrnc vs. baring angl for 60º, 120º and 180º barings...53 Figur 4.12: Load capacity vs. ccntricity ratio for 60º, 120º and 180º barings 53 Figur 4.13: Baring flow vs. Sommrfld numbr for 60º, 120º and 180º barings 54 Figur 4.14: Dimnsionlss dirct stiffnss proprtis for 60º, 120º and 180º barings...54 Figur 4.15: Dimnsionlss dirct damping proprtis for 60º, 120º and 180º barings...55 Figur 4.16: Cross-coupld stiffnss proprtis for 60º, 120º and 180º barings..55 Figur 4.17: Cross-coupld damping proprtis for 60º, 120º and 180º barings.56 Figur 4.18: Sommrfld numbr vs. ccntricity ratio for L/D=0.5, 0.75 and 1.0...56 Figur 4.19: Sommrfld numbr vs. ccntricity angl for L/D=0.5, 0.75 and 1.0.57 Figur 4.20: Load capacity vs. ccntricity ratio for L/D=0.5, 0.75 and 1.0.57 Figur 4.21: Total flow vs. ccntricity ratio for L/D=0.5, 0.75 and 1.0..58 Figur 4.22: Dimnsionlss dirct stiffnss charactristics for L/D=0.5, 0.75 and 1.0...58 Figur 4.23: Dimnsionlss dirct damping charactristics for L/D=0.5, 0.75 and 1.0 59 Figur 4.24: Dimnsionlss cross-coupld stiffnss for L/D=0.5, 0.75 and 1.0 59 Figur 4.25: Dimnsionlss cross-coupld damping for L/D=0.5, 0.75 and 1.0...60 Figur 4.26: Dimnsionlss dirct stiffnss for a 360º baring with L/D=0.5..60 Figur 4.27: Dimnsionlss dirct damping for a 360º baring with L/D=0.5..61 Figur 4.28: Dimnsionlss cross-coupld stiffnss for a 360º baring with L/D=0.5...61 Figur 4.29: Dimnsionlss cross-coupld damping for a 360º baring with L/D=0.5.....62 Figur 4.30: Sommrfld numbr vs. ccntricity ratio for 360º baring L/D=0.5 and 1.0..62 Figur 4.31: Eccntricity angl vs. Sommrfld numbr for a 360º baring L/D=0.5 and 1.0..63 Figur 4.32: Nodal prssurs vs. angl of baring for a 360º baring L/D=0.5 and 1.0.. 63 Figur 4.33: Load capacity vs. ccntricity ratio for a 360º baring L/D=0.5 and 1.0..64 Figur 4.34: Dimnsionlss flow vs. Sommrfld numbr for a 360º baring ix
L/D=0.5 and 1.0..64 Figur 4.35: Dimnsionlss dirct stiffnss charactristics for a 360º baring L/D=0.5 and 1.0..65 Figur 4.36: Dimnsionlss dirct damping charactristics for a 360º baring L/D=0.5 and 1.0..65 Figur 4.37: Dimnsionlss cross-coupld stiffnss for a 360º baring L/D=0.5 and 1.0..66 Figur 4.38: Prssur profil at various rcss flows for a 120 baring with singl rcss... 66 Figur 4.39: Prssur profil at various journal spds for a 120 baring with singl rcss... 67 Figur 4.40: Prssur profil at various rcss flows for a 120 baring with two rcsss... 67 Figur 4.41: Prssur profil at various journal spd for a 120 baring with two rcsss...68 x
LIST OF FILES Filnam Fil siz 1. RajaThs.pdf....3.90 MB xi
CHAPTER ONE INTRODUCTION 1.1 Introduction A fluid film baring may b dfind as a baring in which th opposing or mating surfacs ar compltly sparatd by a layr of fluid lubricant Wilcock, 1979. A widly usd baring typ is th plain journal baring with application in comprssors, turbins, pumps, lctric motors and lctric gnrators. Journal Barings consists of two cylindrs rotating rlativ to ach othr. Th outr cylindr baring is stationary and th innr ring shaft rotating with an angular vlocity is calld th Journal as shown in figur 1.1. Th main purpos of th journal baring is to support th rotating machinry by providing sufficint lubrication to sparat th moving parts and to minimiz th friction du to rotation Allair, 1987. Th high-prssur fluid film in th claranc btwn th journal and th baring du to rotation of th journal provids th hydrodynamic film lubrication, and th load capacity to th baring. Th displacmnt of th shaft cntr rlativ to th baring cntr is known as ccntricity. Narly all havy industrial turbomachins us fluid film barings of som typ to support th shaft wight and control motions causd by unbalanc forcs. Th two primary advantags of fluid film barings ovr rolling lmnt barings ar thir suprior ability to absorb nrgy to dampn vibrations, and thir longvity du to th absnc of rolling contact strsss. Th damping is vry important in many typs of rotating machins whr th fluid film barings ar oftn th primary sourc of th nrgy absorption ndd to control vibrations. Fluid film journal barings also play a major rol in dtrmining rotordynamic stability, making thir carful slction and application is a crucial stp in th dvlopmnt of suprior rotor-baring systms. Journal barings ar incrdibly long-livd providd th lubricant is contaminant fr and sufficintly supplid. Thus dynamic analysis of hydrodynamic barings is important du to th forcs imposd on th shaft from machin unbalanc forcs, arodynamic forcs, and xtrnal xcitations from sals and couplings Allair, 1979. 1
1.2 Typs of Journal Barings Journal barings may b groupd into two main classs - full baring in which th baring arc compltly surrounds th journal and partial baring in which th arc is lss than 360º Raimondi and Boyd, 1958. Barings ar furthr classifid as plain journal barings, axial groov barings, prssur dam barings, multi-lob journal barings, tilting pad journal barings, and hybrid hydrostatic barings. 1.2.1 Partial Arc Barings Partial barings as shown in figur 1.2 may b furthr classifid as cntrally or ccntrically loadd dpnding upon whthr th load lin biscts th baring pad arc or divids it ccntrically. Typically th radius of th baring is gratr than th radius of th journal by an amount qual to th radial claranc. Whn th load applid is primarily unidirctional thr is no nd of full journal barings and instad a singl partial arc baring may b usd Allair, 1987. Partial arc barings ar usd in rlativly low spd applications. Th partial arc pad may also b dsignd with a hydrostatic rcss thus nabling fluid film lubrication du to both hydrostatic and hydrodynamic ffcts. Partial arc barings ar important bcaus thy form th building blocks for axial groov, multi-lob and tilting pad barings. Th prsnt work dals with th computation of dynamic charactristics of partial arc barings. 1.2.2 Plain Journal Barings 360º Plain barings ar composd of a cylindrical shaft of radius R, calld a journal, rotating with an angular vlocity, ω, about its axis in a cylindrical bushing of radius R+c and lngth L. Th cntr of th baring is labld as O b and th cntr of journal as O j as shown in figur 1.1. Undr stady stat conditions th journal cntr rmains at a constant ccntricity and attitud angl Φ for a givn load W acting on th shaft Allair t al., 1975, Allair t al., 1980. Th plain journal baring is th simplst and most common radial baring dsign, whr a plain cylindrical shll ncircls th shaft. Plain slv barings hav th highst cross coupling of all barings and ar suitabl for highly loadd or low-spd shafts. Advantags ar low cost and as of manufactur. Exampls includ automotiv crankshaft barings and low-spd or highly loadd turbo machinry applications. 2
1.2.3 Axial Groov Barings Th axial groov baring is similar to th plain journal baring, but with two or mor axial groovs addd for oil supply. As in th plain journal baring, thr is no prload and has a high tndncy for instability Allair t. al., 1975. Advantags ar low cost and as of manufactur. Axial groov barings ar vry common in many typs of commrcial machinry including turbins, gnrators, motors, pumps, and comprssors, and hav slightly bttr dynamic proprtis than do plain journal barings. 1.2.4 Prssur Dam Barings Th prssur dam baring is a fixd-gomtry baring improvs rotor dynamic stability than plain journal barings. A pockt is milld in th uppr unloadd half of th baring that nds in a "dam". A sharp prssur pak is cratd at th dam du to fluid inrtia ffcts. Th prssur pak imposs a downward load on th journal, forcing th shaft down to a gratr ccntricity that inhrntly improvs stability bcaus of th stiffnss and damping asymmtry inducd. Prssur dam barings hav rlativly high powr losss bcaus of th built-in load. Thy ar not suitabl for applications whr th load dirction changs bcaus th top half is pocktd for th dam. Th dam also rstricts thm to unidirctional opration and th oil must b kpt clan to prvnt sludg accumulation in th pockt. Manufacturing of prssur dam barings is mor difficult and xpnsiv than plain barings, as th dam gomtry is milld sparatly from th bor and must b prcis. Prssur dam barings ar usd primarily in highr spd applications to incras stability ovr othr typs of fixd-gomtry barings. 1.2.5 Multi-lob Barings Multi-lob barings ar ssntially barings with mor than on baring pad that nabl a combination of numbr of pads, rotation of baring, claranc, prload, and offst. This producs a stabilizing ffct on th shaft and can incras load capacity. Th cntr of curvatur of th pad radius R+c may b offst from th bushing/baring cntr prload. Th prload factor in this cas is givn by th ratio of distanc of cntr of curvatur from bushing cntr to th claranc in th barings Allair t al., 1975, Allair t al., 1980. Howvr, it can also consum mor powr du to th prload. Multi-lob barings can b ithr bidirctional or unidirctional, dpnding on whthr th lobs hav symmtric or asymmtric lobs. Multi-lob barings ar 3
difficult and xpnsiv to manufactur bcaus of th prcis machining oprations rquird. Thy ar commonly usd in smallr, high-spd machins rquiring high load capacity or high stability. 1.2.6 Tilting Pad barings Th tilting pad baring diffrs from th multi-lobd baring in that ach pad rotats about a pivot, nabling ach pad to hav highr dgrs of frdom corrsponding to movmnt about th pivot point. Th pad tilts such that its cntr of curvatur movs to crat a convrging pad film thicknss. Ths pads ar abl to follow th shaft motion rsulting in littl cross-coupld stiffnss and damping Allair t. al., 1975. Thy ar widly usd to stabiliz machins that hav sub synchronous vibrations. Ths barings hav highr powr losss and highr cost to manufactur. 1.2.7 Hybrid Barings In hydrostatic barings, high-prssur lubricant is fd to a rcss in th pad. Thy hav improvd high-spd load carrying capacity compard to th hydrodynamic opration. Thir dsign dpnds on th hydrodynamic ffct in addition to th hydrostatic ffct to achiv ncssary load support. Hybrid barings ar advantagous ovr pur hydrodynamic barings in that war can b avoidd at starting or stopping, thy tolrat substantial loads abov normal dsign load, withstand havy dynamic loadings which vary widly in dirction of rotation, and may allow for dsign with a smallr journal diamtr, rducing initial cost and oprating powr consumption. All th barings apart from th tilting pad barings discussd abov ar composd of fixd pad plain baring gomtry. Thus study of th plain baring is significant, as th majority of th barings possss similar charactristics as th plain baring in gomtry as wll as opration. Full barings and cntrally loadd partial arc barings of th claranc typ ar common, and it is to ths barings that th analysis prsntd in this dissrtation is applicabl. Also th dynamic analyss hav bn prformd on th fixd partial arc barings. 4
1.3 Litratur Rviw As alrady statd th dynamic analysis of hydrodynamic barings is ssntial as thy provid stability and control mchanical vibrations that occur in rotating machinry. Svral numrical tchniqus hav bn proposd to provid th solution of th fluid film lubrication problm. Raimondi and Boyd 1958 providd th numrical solution of finit journal barings for application in analysis and dsign using finit diffrnc mthod. Li 1977 and Hays 1959 providd th analytical dynamics of partial journal barings and finit journal barings rspctivly using variational mthods to solv th Rynolds quation. Th finit lmnt mthod has bn usd prominntly for som yars to continuum and fild problms Zinkiwicz, 1970. Rddi 1969 prsntd th finit lmnt solution for incomprssibl lubrication problms of complx gomtris without th loss of accuracy as th finit diffrnc mthod. Wada, t al. 1971 has succssfully usd th finit lmnt mthod for finit width and infinit width lubrication problms. Thy also showd that finit lmnts might produc mor accurat solutions for journal barings than finit diffrnc mthods. Hubnr and Bookr 1972 applid th finit lmnt mthod to th gnral lubrication problm with a systmatic dscription of procdurs. Hubnr 1975 providd th most rcnt and xtnsiv tratmnt of fluid film lubrication. Allair, t al. 1977 dvlopd a systmatic matrix approach for finit lmnts which automatically produc minimum bandwidth of algbraic quations. By organizing th labling in matrix form throughout th analysis, th solution could b asily obtaind using Gaussian limination mthods or othr mthods. Latr, Allair, t al. 1984 providd a prssur paramtr mthod for th finit lmnt solution of Rynolds quation to improv th accuracy significantly. Rao 1982 providd th mathmatical formulation of th finit lmnt mthod to th hydrodynamic lubrication problm govrnd by th Rynolds quation. Allair, t al. 1975 dvlopd a finit lmnt program FINBRG1 to comput th baring charactristics of plain, partial arc, axial groov, multi-lob and tilting pad barings. Allair, t al. 1980 also prformd th dynamic analysis of incomprssibl fluid film barings. Stphnson 1997 dvlopd Baring Dsign Systm, a program to prform baring analysis using th finit lmnt mthod. 5
1.4 Scop of Thsis 'BRGDS is an acronym for Baring Dsign Systm. Th initial program dvlopd by Dr. Stphnson has th capability to comput th baring prssur distribution, film thicknss, baring forcs and momnts for a singl partial arc baring with multipl rcsss in th pad. Th program includs turbulnc as an option. Eithr ccntricity or load can b input to th barings. If load is input, th ccntricity and attitud angl ar solvd itrativly. Th motivation for th prsnt work has bn to includ th capability to comput th damping charactristics for plain, partial arc and hybrid hydrodynamic barings and to furthr validat th rsults. Th finit lmnt solution for th partial arc and full plain journal baring including th baring damping calculations ar th highlight of this thsis. Th baring program is basd on Fortran 77. Th function STIFFCALC to comput stiffnss and DAMPCALC that has bn introducd to comput th damping charactristics of th barings ar shown in Appndix A2 and A3. Th rsults obtaind wr compard and vrifid with othr rsults. Most of th rsults usd for comparison wr obtaind from Dynamic Rotor Baring Systm program, DyRoBs. DyRoBS is a powrful and sophisticatd softwar tool authord by Wn Jng Chn for rotor dynamics including comprhnsiv baring analysis, with rsults courtsy of Stphnson R. W. from Mchcon, Inc. Th nxt chaptr of this work dals with th solution of fluid film barings, viscosity, assumptions that ar usd in th analysis and thir validity. Th Rynolds quation for fluid barings has bn shown and rvisd for finit width barings. Also th finit lmnt solution of th Rynolds quation that is th backbon of th prsnt work has bn dtaild. Chaptr thr dals with th baring dsign program BRGDS. Th gnral systms of quation, baring coordinat systm, solution procdur and convrgnc conditions usd in th program ar documntd. Howvr, th main focus is on th computation of stiffnss and damping charactristics for fluid film barings. Th rsults that ar obtaind by th analysis ar shown in chaptr four. Comparisons hav bn mad for th stiffnss and damping charactristics to succd th prsnt program for furthr analysis. Plots showing th analysis of various othr baring typs ar also shown and commnts on th rsults hav bn providd. Finally, th scop for futur work has bn outlind basd on th prsnt work. 6
Y x Baring ω o b W o j hө Ө y X Journal Ф Lin of Cntrs Load Lin Ψ Figur 1.1: Schmatic diagram of a full plain journal baring Allair t al., 1975 7
Y Lin of cntrs W ω hθ R+c O b R O j X Ф Figur 1.2: Schmatic diagram of partial arc baring Allair t. al., 1975. Copyright Raja Shkar Balupari 2004 8
CHAPTER TWO SOLUTION FOR FLUID FILM BEARINGS 2.1 Introduction Fluid film barings hav bn dfind as barings in which th opposing or mating surfacs ar compltly sparatd from ach othr by a layr of fluid lubricant. Thus it is ssntial to undrstand th fundamntal procss by which th fluid is maintaind whil supporting th load. This chaptr dals with th basic principls of fluid film barings, viscosity of fluids, th Rynolds quation and th finit lmnt intrprtation of th Rynolds quation. Th basic principl can b undrstood by considring high-spd fluid flow btwn two plan surfacs with a convrging wdg and lubricant btwn th surfacs. Th convrgnc coupld with high-spd fluid flow and fluid viscosity gnrats a high-prssur fluid film that supports th load. In th cas of a plain journal baring, th convrging wdg is formd at th bottom of th baring du to th wight of th shaft and any othr xtrnal applid load. Th fluid is pulld into th rgion undr th shaft du to th shar forcs gnratd by th shaft rotation. Th fluid is forcd into th convrging film thicknss at th bottom of th shaft producing a high-prssur film Allair, 1987. This film supports th wight of th rotating machin componnts and prvnts th shaft from touching th bushing surfac. For a givn ccntricity th fluid film has convrging and divrging gomtry, such that cavitations may occur in th divrging portion. It is thus vry important to b abl to prdict th prssur distribution and load capacity of th baring. 2.2 Viscosity Th viscosity of th fluid lubricant is its most significant physical charactristic as far as fluid film barings ar concrnd. In this analysis of hydrodynamic barings th fluid is incomprssibl, and th viscosity rmains constant throughout th flow. Th amount of prssur gnratd dpnds on th viscosity and dnsity of th fluid. Viscosity is th proprty that dfins th rsistanc of th fluid to motion. Th viscosity is du to th molcular attraction btwn th adjacnt layrs of fluid. Any applid shar forc will caus th fluid to mov, rsulting in a 9
rsistanc to flow that dpnds on th viscosity of th fluid. Whn a fluid is at rst, thr is no rsistanc to an applid shar forc. In a positiv sns, viscosity controls th fluid flow out of th baring but it is th caus of powr consumption of th baring du to lubricating shar Wilcock, 1979. A mathmatical quation to quantify viscosity can b obtaind from figur 2.1. A fluid film of thicknss H sparats a lowr plat that is stationary from an uppr plat. Th uppr plat movs with vlocity U du to application of forc F. Th fluid particl O in contact with th lowr plat has zro vlocity, and th fluid particl Q attachd to th uppr plat movs with th vlocity U. For a Nwtonian fluid th fluid vlocity incrass linarly from th stationary plat to th moving plat as shown in th figur 2.1. Th viscosity µ of any intrmdiat particl P locatd at a distanc y from th lowr plat moving with vlocity v can b obtaind from quation 2.1. Rarranging th trms w obtain quation 2.2, whr τ is th shar strss dvlopd in th fluid, A is th ara of moving plat in contact with th fluid, and U/H is th vlocity gradint btwn th plats. Th flow causd by viscosity is known as Coutt flow. UA F = µ 2.1 H τ τ µ = = 2.2 v y U H Thus dynamic or absolut viscosity has bn dfind as th forc rquird for moving a flat unit plat locatd at a unit distanc from a stationary plat by a unit vlocity whr th spac btwn th plats is filld by th fluid. Th units of absolut viscosity ar lb.s/ft 2. Fluids whos viscosity rmains constant with chang in vlocity gradint ar calld Nwtonian fluids, whil fluids that do not hav a linar rlationship ar calld non-nwtonian fluids. Th ratio of absolut viscosity to th dnsity of th fluid is calld kinmatic viscosity Szri, 1999 as in quation 2.3. Th units of kinmatic viscosity ar ft 2 /sc. µ ν = 2.3 ρ 10
11 2.3 Solution of fluid film barings Th war fr transfr of forc in hydrodynamic lubricatd barings is basd on th application of prssur in th lubricating film that balancs th xtrnal forcs on th baring. Osborn Rynolds and othrs Pkn, t al, 1983 drivd from th common Navir-Stoks quations 2.4, 2.5 and 2.6 and th continuity quation 2.9, a diffrntial quation for th calculation of th prssur distribution Szri, 1999. Th Laplacian and th divrgnc oprators ar shown in quations 2.7 and 2.8. u x u x p f z u w y u v x u u t u x + + = + + + 3 2 µ µ ρ ρ X-dirction 2.4 u y v y p f z v w y v v x v u t v y + + = + + + 3 2 µ µ ρ ρ Y-dirction 2.5 u z w z p f z w w y w v x w u t w z + + = + + + 3 2 µ µ ρ ρ Z-dirction 2.6 2 2 2 2 2 2 2 z y x + + = 2.7 z u y u x u u + + = 2.8 0 = + + + w z v y u x t ρ ρ ρ ρ 2.9 Th Navir-Stoks quations in which th fluid inrtia, body, prssur and viscous forcs ar includd ar too complicatd to yild an analytical solution for most problms. Th Rynolds quation is a simplifid vrsion of Navir-Stoks quation. Th analytical solution of th journal baring is obtaind by using th most popular form of th Rynolds quation. Th following assumptions hav to b considrd in ordr to obtain th solution.
2.3.1 Assumptions Th xact rprsntation of th fluid flow in mathmatical trms using Navir-Stoks quation is complicatd for fluid film barings. Crtain assumptions may b mad to obtain a standard quation known as Rynolds quation 2.10. Th following assumptions ar typical in th analysis of fluid film barings Camron, 1917. 1. Body forcs ar nglctd. This mans that thr ar no inrtial or othr distributd forcs acting on th fluid. Th ffct of gravity of th fluid is nglctd 2. Th prssur of th fluid is assumd constant across th thicknss of th film. 3. Th curvatur of th surfacs is larg compard to th film thicknss. Surfac vlocitis ar considrd unidirctional. 4. Thr is no slip at th boundaris. Th vlocity of th oil layr adjacnt to th boundary is th sam as that of th boundary. Th fluid attachd to th baring surfac is stationary whil th fluid nar th rotor or shaft has th sam angular vlocity as th shaft itslf. Apart from ths assumptions thr ar crtain assumptions that ar considrd to nabl simplification of th mathmatical quations. Ths assumptions ar not ssntial but mak th quations simplr Allair, 1987. 5. Th fluid lubricant usd is Nwtonian. This mans th fluid obys Nwton s law of viscosity and th fluid strss is proportional to th rat of shar. 6. Fluid film flow in th barings is laminar. 7. Th viscosity of th fluid dos not chang. 8. Th dnsity of th fluid rmains constant. 2.3.2 Rynolds Equation Rynolds quation with appropriat boundary conditions dscribs th prssur btwn two surfacs sparatd by a thin fluid film. Considr two surfacs moving rlativ to ach othr sparatd by a constant fluid film of thicknss as shown in figur 2.2. Th most gnralizd rprsntation of Rynolds quation for fluid film lubrication, dvlopd by Dowson 1962, can b rprsntd as shown in quation 2.10. 12
3 ρh p + x k x x µ 3 ρh p = y k x y µ hu 2 ρ + hv x 2 ρ ρ + h y t x U U ρ h 1 2 2 1 y V V ρ h 1 2 2 1 W + ρ 2.10 2 W 1 Whr, U 2 =U 2 i+v 2 j+w 2 k, is th vlocity of th uppr plat U 1 =U 1 i+v 1 j+w 1 k, is th vlocity of th lowr plat i, j and k ar unit vlocity vctors in th x, y, z dirctions rspctivly. U, V and W ar vlocity componnts in th x, y, z dirctions rspctivly. ρ x,y = fluid film dnsity h x,y = fluid film thicknss µ x,y = fluid film viscosity p x,y = fluid film prssur v x,y = diffusion vlocity t = tim k x = turbulnt constant k x = 12, for laminar flow k x = 12+0.0136*R 0.9, for turbulnt flow Th lft hand sid of quation 2.10 can b rwrittn as shown in quation 2.11. LHS Hr, 3 ρh = p 12 µ = i + j and for incomprssibl fluid flow x y ρ = x ρ = y ρ =0 t 2.11 Thus th right hand sid can b rwrittn as shown in quation 2.12; for slidr barings w considr quation 2.13. 13
RHS ρ h x h y t U U + ρ V V + ρh = 1 2 1 2 1 2 h = W 2 W 1 t Nglcting strtch ffcts th right hand sid can b rprsntd as quation 2.14. 2.12 2.13 RHS x y x x ρhu + ρhv ρhu ρhv + ρh = 1 2 1 1 2 2 Thus from quations 2.12 and 2.14, th full Rynolds quation can b writtn as quation 2.15. t 2.14 3 ρh p = [ ρh U U ] + [ ρh V V ] + ρh k x µ 1 2 1 2 1 2 1 2 2.15 x y t With U= 1 U U, th avrag vlocity of uppr and lowr surfacs and nglcting th body 2 [ ] 1 2 forcs F with diffusion ffcts, th Rynolds quation can b rwrittn as in quation 2.16 Rddi, 1969. 3 ρh p = k xµ t ρhu + ρh + ρv 2.16 Thus, quation 2.16 dscribs th Rynolds quation for incomprssibl and isoviscous fluid films. Body forcs ar includd in th quation. Onc th Rynolds quation is solvd and th prssur is known, th avrag fluid vlocity, u is givn by quation 2.17 and th mass flow rat q is givn by th quation 2.18. 2 h u = U P k µ x 2.17 q = ρhu 2.18 Spcifying ithr th prssur or th avrag normal fluid vlocity at ach point along th thin fluid film boundary complts th formulation of th fluid film boundary valu problm. Lt C P dnot th portion of boundary C whr th prssur is considrd known and C q rprsnt th rmaining portion of boundary whr th normal fluid vlocity or normal mass flow rat is known as shown in figur 2.3. This implis P is known on C P and q.n is known on C q, whr n is 14
th outward normal to C q. It can b shown that th prssur must b spcifid at a minimum of on point along th boundary for th solution of P to th boundary valu problm to b uniqu. It has bn shown that th prssur P that minimizs this functional whil satisfying th boundary conditions on C P is also a solution of th Rynolds quation. Th variational principl for incomprssibl, isoviscous films of any thin film gomtry is to find a prssur P x,y which minimizs th function shown in quation 2.19 and satisfis th boundary condition on C P Allair t. al., 1977. Th prssur thus obtaind is th sam as that which is th solution to th Rynolds quation 2.16. With ths quations, th thin fluid film problm is compltly formulatd xcpt for inrtia and nrgy ffcts. J 3 ρh 2kxµ t P P ρhu P + ρh + ρv P da+ q n = A C q PdC 2.19 For a fluid film journal baring th lowr surfac is fixd, fluid film viscosity and dnsity ar constant, and body forcs and diffusion forcs ar nglctd. Thus, th Rynolds quation in cylindrical coordinat systm for laminar flow of a circular baring can b rprsntd as in quation 2.20 and th corrsponding functional as quation 2.21. 1 r 2 J 3 h p [ ] + θ µ Θ 3 h p [ ] = z µ z h 6 ϖ + j θ h t 3 2 2 ρh 1 P P ρ P ρh P = + ϖ b + ϖ j h + A 2k xµ R θ y 2 θ t P Rdθdy 2.20 2.21 2.4 Finit Elmnt Mthod Finit lmnt procdurs ar vry widly usd in nginring analysis. Th procdurs ar mployd xtnsivly in th analysis of solids and structurs and of hat transfr and fluids. Finit lmnt mthods ar usful in virtually vry fild of nginring analysis. Th finit lmnt mthod is usd to solv physical problms in nginring analysis and dsign. Th physical problm typically involvs th structur or structural componnt subjctd to 15
crtain loads. Th idalization of th physical problm to th mathmatical modl rquirs crtain assumptions that togthr lad to diffrntial quations govrning th mathmatical modl. Th finit lmnt analysis provids an approximat solution to th mathmatical modl. Sinc th finit lmnt procdur is a numrical procdur it is ncssary to assss th solution accuracy. If th accuracy critria ar not mt, th numrical solution has to b rpatd with rfind solution paramtrs such as a finr msh until a sufficint accuracy is rachd. Also th choic of an approximat mathmatical modl is crucial and compltly dtrmins th insight into th actual physical problm that w can obtain by th analysis. A flow chart rprsntation of th gnral procdur for finit lmnt analysis is shown in figur 2.5 Bath, 1996. In th fluid film baring application th finit lmnt mthod rducs th fild problm in which th prssur is a continuous function of spac to a problm in which th prssur is valuatd only at crtain spcific nodal points. Btwn th nods th prssur is assumd to vary according to a simpl function calld an intrpolation function. Th problm thn bcoms on of valuating a systm of algbraic quations for unknown nodal prssurs. Th cofficints of algbraic quations dpnd upon th nodal coordinats, th intrpolation polynomials chosn and othr factors. Th finit lmnt mthod is convnintly drivd from a variational principl. In th cas of fluid film barings th variational principl has bn drivd from th Rynolds quation. Th variational principl consists of a functional in th form of an intgral ovr th baring ara. Th baring ara to b analyzd is dividd up into a finit numbr of small sub rgions, calld lmnts, of simpl gomtric shap. Th lmnts must b chosn such that thr ar no gaps or ovrlapping occurs btwn th lmnts. On th boundary of ach lmnt, a numbr of nods ar chosn whr prssur is to b valuatd. Intrior nods can b chosn but thy ar not usually mployd. An intrpolation function is chosn to approximat th prssur within ach lmnt. Th function for a givn lmnt is not valid outsid th lmnt. It is found in trms of nodal prssur and nodal coordinats. Lt us considr a polynomial of th form shown in quation 2.22 Hubnr, 1975 and Zinkiwicz, 1970. Hr it is ncssary that th lmnt has six nods placd, such that ach nod corrsponds to a constant in th polynomial. In ordr to 16
insur accuracy of th solution, th intrpolation function is usually chosn such that th prssur is continuous across th intr lmnt boundaris compatibl formulation. a + 2 2 0 a1x + a2 y + a3x + a4 xy a5 y + 2.22 Each lmnt will mak a contribution to th intgral in th functional. Ths contributions must b valuatd for ach lmnt and addd togthr. Th solution to th problm is obtaind whn th nodal prssurs ar chosn such that th functional is a minimum. Th algbraic systm of quations rsulting from a minimization procss yilds th nodal prssur. In summary th stps involvd ar 1. Exprss th problm as a functional. 2. Divid th ara into lmnts. 3. Choos intrpolation functions and nods. 4. Evaluat lmnt contributions to functional. 5. Minimiz functional. 2.4.1 Finit lmnt solution of Rynolds Equation This sction dvlops th basic thory of th finit lmnt mthod for two-dimnsional thin fluid films. Th thory is gnralizd to a finit slidr baring with rlativ surfac vlocitis as wll as a squz vlocity. Th analysis applis to any baring that can b unrolld onto a horizontal surfac such as a journal baring, partial arc baring and tilt pad barings. Considr a gnral finit lmnt slidr shown in figur 2.2. Th prssur must minimiz th function in quation 2.23 nglcting th body forcs Allair t. al., 1977. J 3 ρh 2k x µ t P = P ρhu P + ρh P da + q n A C q PdC 2.23 Hr th subscripts x and y dnot th vctor componnts in th X and Y dirctions rspctivly, th functional may b writtn as show in quation 2.24. Th boundary conditions along part of th boundary, C p, th prssur is spcifid as in quation 2.25 and along th rmaindr of th boundary, C q, th mass rat of flow is spcifid as shown in quations 2.26 and 2.27. Th valus 17
18 of p a and q a may vary around th boundary. If th prssur is spcifid around th ntir boundary, C p = C, th last intgral in quation 2.24 is simply ignord Nicholas, 1977. + + + = A y x x da P h t y P U x P U h y P x P k h P J ρ ρ µ ρ 2 2 3 2 dc P n q n q q C y y x x + + 2.24 [P] on Cp = [P a ] on Cp 2.25 [q x ] on Cq = [q ax ] on Cq 2.26 [q y ] on Cq = [q ay ] on Cq 2.27 Th fluid is dividd into a global systm of lmnts in a mannr such as that shown in figurs 2.4 and 2.6. Lt th rgion of intrst in th X-Y plan b dividd into E finit lmnts. First th baring surfac is dividd into M quadrilatrals in th y dirction and N quadrilatrals in th x dirction. This prmits th nods at th intrsctions to b labld in a rctangular matrix of ordr MxN. By xprssing th prssur and th distribution of th various forcing functions in trms of thir rspctiv nodal valus through intrpolation function, y x L i, w obtain quations 2.28 and 2.29. Hr, P r and U r ar th vctors of nodal prssurs and x-componnt vlocitis rspctivly, of lmnt. [ ],,, i i i P y x L P y x L y x p r = = 2.28 [ ],, i i i x U y x L L U y x U r = = 2.29 Each of th nodal prssurs P i, P j, P k ar labld corrspondingly and th coordinats of th nods ar dnotd by x i,y i, x j,y j, x k,y k. Within ach lmnt a linar intrpolation function is rprsntd as in quation 2.30. k k j j i i P y x L P y x L P y x L y x P,,,, + + = 2.30 Whr, [ ] 2, n n n n A y c x b a y x L + + = a n, b n, c n ar constants A = ara of th lmnt Sinc th cofficints a n, b n, c n and th ara of th lmnts ar known, only th nodal prssurs ar unknown in th lmnt. Th function is brokn up into intgrals ovr th individual lmnts as shown in quations 2.31 and 2.24.
19 = = E P J P J 1 2.31 Th scond intgral is valuatd only for th portion of th lmnt boundary that is locatd along C q. If no portion of boundary falls along C q, thn th last intgral is simply ignord. Th prssur, shar in x-dirction, shar in y-dirction, squz and boundary flow constants ar show in quations 2.32, 2.33, 2.34, 2.35 and 2.36 Rao, 1999 and Nicholas, 1977. Thus, th lmnt functional in compact form and th total functional can thus b rprsntd as in quations 2.37 and 2.38 rspctivly Bookr and Hubnr, 1972. 3 A m m n m x pmn da y L y L x L x L k h K + = µ ρ 2.32 A n X Uxn da x L hu K = ρ 2.33 A n y Uyn da y L hu K = ρ 2.34 = A n n h da L t h K ρ & 2.35 [ ] + = C q n y y x x n dc L n q n q q 2.36 { } [ ] { } { } { } { } { } 2 1 Uy T Ux T P T K P K P P K P P J = { } { } { } { } T h T q P K P + & 2.37 { } [ ] { } { } [ ] { } [ ] = = = = E Uy T E Ux T E P T K P K P P K P P J 1 1 1 2 1 { } [ ] { } [] = = + E T E h T q P K P 1 1 & 2.38 Th functional must b minimizd with rspct to all of th unknown nodal prssurs. W assum that all th nodal prssurs ar unknown vn though thr ar a fw known nodal prssurs. Thus minimization of quation 2.38, as shown in quation 2.39 rsults in quation 2.40. This quation is known as th finit lmnt formulation of Rynolds quation for fluid film journal barings Nicholas, 1977. This quation can b convnintly solvd in a rcognizd
20 matrix form as in quation 2.41. Th lmnts thus obtaind can b assmbld into an ovrall systm as in 2.42. = = = N J M I P P J J I,..., 1,2 1,2,..., 0,, 2.39 [ ] { } [ ] [ ] [ ] [ ] = = = = = + = E E h E Uy E Ux E P q K K K P K 1 1 1 1 1 & 2.40 [ ] [ ] [ ] h u p F H K U K Q P K r r & r r r & = = 2.41 [ ] F P K P r r = 2.42 Whr, th ordr of matrix P K is qual to M. Applying th prssur boundary conditions can solv th quations thus obtaind. Onc th nodal prssur P r and flow U r ar found, th baring load capacity W can b computd from 2.43. = S ds y x p W, 2.43
F Moving Plat Q U U H y P v Vlocity Profil O Stationary Plat Figur 2.1: Vlocity profil btwn two paralll plats to quantify viscosity Szri, 1999 X Y h t Squz vlocity U 2 Surfac vlocity Film thicknss, h U 1 Surfac vlocity U 1 Atmosphric prssur, P a ρhu*n=q a Mass flow rat Figur 2.2: Fluid film gomtry Allair t al., 1977. 21
Lading dg Boundary C q whr P =0 x Tailing dg X b RӨ dirction P=0 L/2 P=0 Y b Z dirction Boundary C p whr P=0 Figur 2.3: Boundary conditions on unrolld baring surfac Allair t al., 1975 X Y 11 12 13 14 N Nods 21 31 22 23 M Nods 32 M x N quadrilatral lmnts Figur 2.4: Division of fluid film into lmnts Nicholas, 1977 22
Physical Problm Chang of physical problm Mathmatical Modl Govrnd by diffrntial quations considring assumptions on Gomtry Kinmatics Matrial Law Loading Boundary conditions tc. Improv Mathmatical Modl Finit Elmnt Solution Choic of Finit Elmnts Msh Dnsity Solution Paramtrs Rprsntation of Loading Boundary conditions Rfin msh, solution paramtrs tc. Assssmnt of accuracy of finit lmnt solution of mathmatical modl Intrprtation of Rsults Rfin Analysis Dsign Improvmnts Structural optimization Figur 2.5: Th procss of finit lmnt analysis Bath, 1996. 23
Z Elmnt Ara = A nod k P k P i nod i nod j nodal prssurs P j Y X i, Y i X k, Y k X X j, Y j Linar intrpolation function Lx,y = a 0 +a 1 x+a 2 y Figur 2.6: Th th two-dimnsional simplx triangular lmnt Nicholas, 1977. Copyright Raja Shkar Balupari 2004 24
CHAPTER THREE BEARING DESIGN PROGRAM 3.1 Introduction Th program nam BRGDS is an acronym for Baring Dsign. It is a finit lmnt basd computr program for th analysis of hydrodynamic partial arc barings and barings with hydrostatic rcsss. Dr. Robrt W. Stphnson initially dvlopd th program. Additional work has bn don to includ th capability to comput th dynamic charactristics for partial arc, full and hybrid barings. In th prsnt chaptr a dtaild ovrviw of th BRGDS program, coordinat systm, flow charts for finit lmnt mthod and convrgnc conditions ar prsntd. Th main focus is on th computation of th dynamic charactristics. 3.2 Ovrviw A standard finit lmnt implmntation is usd to modl th land rgion of th baring. Th finit lmnt modl of th baring is gnratd using four-nod quadrilatral and/or thr-nod triangular lmnts. Th finit lmnt solution of th Rynolds quation is usd to obtain th prssur ovr th finit baring surfac. Th assumptions usd in th solution of Rynolds quation in this program ar incomprssibl flow, constant lubricant viscosity, and no fluid inrtia trms. Th baring pad is assumd to b unrolld or flat, so that a two-dimnsional msh rprsntation in th axial-circumfrntial plan can b usd. Th baring dg is subjctd to th boundary conditions. Th nods hav ithr prssur or flow as th only dgr of frdom. Th systm matrics ar thus dvlopd from th baring gomtry and oprating paramtrs such as journal rotational spd, lubricant viscosity and claranc. Eithr a spcifid load or a spcifid ccntricity is usd to apply loading to th baring. For a spcifid load th program itrats on th ccntricity,, and th ccntricity angl, Ф, until load quilibrium is rachd. Howvr, for spcifid ccntricity th itrations ar prformd on ccntricity angl and th baring load is calculatd. A convntional Nwton-Raphson mthod is 25
usd for adjusting th journal displacmnt to find quilibrium position. Prssurs lowr than th cavitation prssur ar st qual to th spcifid cavitation prssur. Turbulnc is includd as an option and is implmntd as a corrction factor to th laminar solution. For turbulnt solution, itrations ar prformd until prssur convrgnc is achivd sinc th corrction factors in th systm quations ar prssur dpndnt. Th hydrostatic capability of th baring is includd in th analysis by spcifying ithr flow or prssur boundary conditions on th baring land around th dg of th hydrostatic rcss. Hr, all nods around th rcss priphry ar assumd to b at th rcss prssur itslf. For spcifid rcss prssur, th rcss nods at th priphry ar st to hav th boundary condition prssur and solution is achivd. Th flow into and out of th rcss is calculatd by back substitution of th prssur solution into th systm quations. In th othr cas of rcss flow bing dfind, an itrativ solution is prformd to adjust th rcss prssur, until th flow calculatd by back substitution of prssur solution agrs with th spcifid rcss flow within a spcifid tolranc. Th prssur is thn intgratd ovr th baring to giv th load capacity and momnt on th baring about a spcifid axial location. A 2X2 Gauss quadratur routin is usd to numrically intgrat th lmnts. A thr-point intgration schm is usd to numrically intgrat th triangular lmnts Zinkiwicz and Bath. Linar stiffnss and damping cofficints ar obtaind from small prturbations of displacmnt and vlocity about th quilibrium position rspctivly. In addition, th baring film thicknss, torqu loss and fluid flow into th baring from hydrodynamic action ar calculatd. Th ffcts of th axial misalignmnt of th shaft of th baring ar not considrd in th currnt implmntation. 3.3 Baring Coordinat Systm For th purpos of th program a right handd coordinatd systm fixd in spac is st up as shown in Figurs 3.1 and 3.2. X and Y ar th horizontal and vrtical coordinats of th gomtrical cntr of th shaft. Th angular coordinat, Θ, is masurd positiv in th countrclockwis dirction, starting always from th positiv x-axis in th dirction of rotation of th shaft. Th two dimnsional nodal coordinats rquird for th baring modl ar in trms of 26
Z axial and Θ circumfrntial dgrs. Th baring radius is spcifid as input and it is usd along with Θ to corrctly dfin th circumfrntial distanc. 3.4 Solution Procdur A block diagram flow chart of th program solution procdur is show in Figur 3.3. Th squnc of stps prformd during baring analysis by BRGDS can b outlind as follows Stphnson, 1997. 1. Rad th baring information from th input fil and gnrat th baring finit lmnt modl. 2. Assign th prssur boundary conditions to th dg nod and rcss nods if any. 3. Assum initial valus of hydrostatic rcss prssurs, turbulnt corrction factors and journal position. 4. Solv th finit lmnt quation for unknown prssur. Comput th unknown flows by back substitution of prssur solution into systm quations. 5. Intgrat th prssur solution ovr th lmnts to comput th baring forcs and baring momnts. 6. Chck for forc convrgnc. If convrgnc conditions ar not satisfactory thn modify journal position and prform solution for unknown prssur stp 4 7. Rst any prssur blow th cavitation prssur to th spcifid cavitation prssur and prform stp 4. 8. Vrify prssur convrgnc for turbulnt solution if includd. If not convrgd thn updat turbulnt corrction factors and prform stp 4. 9. In th vnt of a hydrostatic rcss with spcifid rcss flow, chck rcss flow convrgnc. If convrgnc condition is not achivd updat rcss prssur and prform stp 4. 10. Display th convrgd baring solution. Prform stp 3 for mor load cass. 3.5 Gnral Systm Equations Th basic systm quations for th hydrodynamic problm in matrix form can b rprsntd as, [K p ]{p} = {q} - [K u ]{u} 3.1 27
Whr, [K p ] = systm prssur fluidity matrix {p} = nodal prssur vctor {q} = nodal flow vctor [K u ] = systm vlocity fluidity matrix {u} = nodal vlocity vctor Th systm matrics [K p ] and [K u ] ar assmbld from th individual lmnt matrics, which ar dpndnt on th film thicknss, lubricant viscosity, and lmnt gomtry Bookr t. al., 1972. Each ntry of th vctor {u} is a known valu dpndnt of th rotational spd of th journal and is qual to th rotational spd of th journal -Rω/2. Each nod contains only on unknown dgr of frdom in 3.1, ithr p or q. Not that th prssurs and flows ar scalar valus. For nods with unknown prssurs, th nt flow valus ar always zro. Dgrs of frdom ar rmovd from 3.1 at nods with spcifid prssurs. Thrfor, th right hand sid bcoms a known vctor and all unknowns in th systm ar in {p}. Th unknown flow valus at nods with spcifid prssurs can b back calculatd from th systm quations aftr th complt systm prssur solution is obtaind. 3.5.1 Eccntricity Th position of th journal in th baring is dirctly rlatd to amount of loading on th baring. Whn th journal baring is sufficintly supplid with oil and with no load acting, th journal rotats concntrically with th baring. Howvr, whn th load is applid, th journal movs to th quilibrium position or an incrasingly ccntric position, thus forming a wdg shapd oil film to support th load. As show in Figur 1.1, th distanc btwn th baring cntr and th shaft cntr is known as ccntricity. Th ratio of ccntricity to th baring claranc is dfind as ccntricity ratio, ε. If ε=0, thn thr is no load on th pad and for ε=1 th journal would touch th baring undr largr xtrnal loads. 3.5.2 Film Thicknss Calculation Th lmnts in th matrics [K p ] and [K u ] ar functions of th film thicknss, h, which in turn varis as a function of th angular location, θ, at ach nod and is calculatd from quation 3.2. 28