Hydrodynamic Fluid Film Bearings and Their Effect on the Stability of Rotating Machinery

Transcription

1 Their Effec on he Sabiliy of Roaing Machinery Lui San André Ma-Child Tribology Profeor Turbomachinery Laboraory, Texa A&M Univeriy College Saion, T USA Lanandre@mengr.amu.edu ABSTRACT The lecure inroduce he baic principle of hydrodynamic lubricaion and he fundamenal equaion of Claical Lubricaion Theory. The analyi proceed o derive he aic and dynamic performance characeriic of hor lengh cylindrical journal bearing, wih applicaion o he dynamic forced performance of a rigid roor uppored on plain bearing. In a radial bearing, he Sommerfeld number define a relaionhip beween he aic load and he journal eccenriciy wihin he bearing. Thi deign parameer how he aic performance of he bearing a roor peed increae. Roordynamic force coefficien are inroduced and heir effec on he abiliy of a roor-bearing yem horoughly dicued. Cro-coupled force coefficien are olely due o journal roaion, and he magniude (and ign) of he cro-iffne deermine roordynamic abiliy. The whirl frequency raio (WR) relae he whirl frequency of ubynchronou moion o a hrehold peed of inabiliy. The deired WR i null; however, plain cylindrical bearing how a whirl raio of ju 0.50, limiing he operaion of roaing machinery o haf peed below wice he yem fir criical peed. The analyi conclude wih a review of pracical (in ue) journal bearing configuraion wih highligh on heir major advanage and diadvanage, including remedie o reduce or enirely avoid ubynchronou whirl inabiliy problem. 1.0 UNDAMENTS O LUID ILM BEARING ANALSIS igure 1 depic an idealized geomery of a fluid film bearing. The major characeriic of a lubrican film, and which allow a major implificaion of i analyi, i ha he hickne of he film (h) i very mall when compared o i lengh (L) or o i radiu of curvaure (R), i.e. (h/l) or (h/r) <<< 1. y (U,V) urface velociie V Vy Vz U h(x,z, Vx z x Lz y Lx h << Lx,Lz y V h(x,z, U Vz Vy Vx Lx x igure 1: Geomery of low Region in a Thin luid ilm Bearing (h << Lx, Lz). San André, L. (006) Hydrodynamic luid ilm Bearing and. In Deign and Analyi of High Speed Pump (pp ). Educaional Noe RTO-EN-AVT-14, Paper 10. Neuilly-ur-Seine, rance: RTO. Available from: hp:// RTO-EN-AVT

2 A plain cylindrical journal bearing, ee igure, comprie of an inner roaing cylinder (journal) of radiu R J and an ouer cylinder (bearing) of radiu R B (>R J ). The wo cylinder are cloely paced and he annular gap beween he wo cylinder i filled wih ome lubrican. The radial clearance c (R B -R J ) i very mall. In mo fluid film bearing wih incompreible liquid, c/r B 0.001; while for ga film bearing, C/R B , ypically. D J R J D B R B igure : Schemaic View of a Cylindrical Bearing. A a conequence of he mallne in film hickne, he effec of he film curvaure are negligible in he operaion of a journal bearing. The analyi of he flow equaion define he circumferenial flow Reynold number (Re) a ρ U c Re * & Re* Re (c/l * ) (1a) µ baed on he characeriic peed (U * ΩR). Re denoe he raio beween fluid ineria (advecion) force and vicou-hear force. And, ρ ω c Re (1b) µ i he queeze film Reynold number repreening he raio beween emporal fluid ineria force due o ranien moion a a characeriic frequency (ω) and vicou-hear force. luid ineria effec in hin film flow are of imporance only in hoe applicaion where boh Reynold number are larger han ONE 1, i.e. Re *, Re >> 1. The film hickne o characeriic lengh raio (c/l * ) in hin film flow i ypically very mall. Thu, fluid ineria erm are o be reained for flow wih Reynold number of he order []: Re > (L * /c) x 1 > 1, 000 for (c/l * ) () Claical lubricaion i baed on he aumpion ha fluid ineria effec are negligible, i.e. (Re, Re * ) 0, in mo pracical applicaion; and hence, rendering effecively an ineriale fluid. 1 In acualiy, Re > 1 for eady uper laminar flow in hin film bearing, a demonraed in [1]. In hin film flow, raniion o urbulence i due o inabiliy of hear driven parallel flow. A raniion o urbulence iniiaed by he appearance of Taylor vorice generaed by cenrifugal force i more peculiar o configuraion wih large clearance, i.e. no common in hin film bearing. The acceped Reynold number for flow urbulence in journal bearing i Re c ρωrc/µ >,000 [] RTO-EN-AVT-14

3 The hin film laminar flow of an incompreible, ineriale and iovicou fluid i governed by he following equaion coninuiy and momenum ranpor equaion: ( v ) ( v y ) ( v ) x x + y + z z 0 P x v y P z v y x x ; 0 + ; 0 + µ µ () where v x,y,z are he velociy componen and he film preure (P) i uniform acro he film hickne (h). The momenum equaion eablih a quai-aic balance of preure and vicou force. The lubricaion equaion denoe low flow condiion wih ime appearing a a parameer, no an independen variable. Table 1 preen he circumferenial flow Reynold number (Re) for a ypical journal bearing applicaion operaing wih differen fluid. The example bearing i a inch (76 mm) diameer (R J ) journal and he clearance o radiu raio (c/r J ) i 0.001, a ypical value for journal bearing. Two roaional peed of 1,000 and 10,000 rpm (Ω104.7 and 1047 rad/) are noed in he able. The calculaed Reynold number (Re) how ha bearing applicaion wih mineral oil and (even) air do no need o include fluid ineria effec, i.e. Re < 1,000. However, proce fluid applicaion uing waer, R14a refrigeran and cryogenic fluid how large Reynold number a a peed of 10,000 rpm. luid Table 1: Imporance of luid Ineria Effec on Several luid ilm Bearing Applicaion (c/r J )0.001, R J 8.1 mm (1.5 inch) Abolue vicoiy (µ) lbm.f. x 10-5 Kinemaic vicoiy (ν) cenioke Re a 1,000 rpm Re a 10,000 rpm Air Thick oil 1, Ligh oil Waer ,588 Liquid hydrogen ,05 Liquid oxygen ,94 Liquid nirogen ,477 R14 refrigeran ,96 Noe ha curren bearing applicaion uing proce liquid o replace mineral oil may operae a peed well above 10,000 rpm. Incidenally, he operaing peed of cryogenic urbopump i on he order of 0 70 krpm, and fuure applicaion (currenly in he work) will operae a peed cloe o 00 krpm! Incidenally, proce ga and liquid eal, iolaing region of high and low preure in a ypical compreor or pump, have larger radial clearance han load uppor fluid film bearing. or example, in waer neck-ring and inerage eal in pump, R/C ~ 50, and hu fluid ineria effec are of imporance even a relaively low roaional peed (~1,000 rpm and larger). The opic of eal i analyzed in he nex lecure. 1.1 Oher luid Ineria Effec luid ineria wihin hin film flow domain can be afely ignored in mo convenional (oil lubricaed) bearing applicaion. However, fluid ineria effec may alo be of grea imporance a he inle o he film and dicharge from he film ecion in a ypical pad bearing or eal applicaion, ee igure. Depending on he flow condiion upream of a udden conracion or a udden enlargemen, a fracion of he dynamic preure head, ypically given a (½ ρ U ), i lo or recovered. RTO-EN-AVT

4 U U P P ~ ½ ρu P igure : Preure Drop & Rie a Sudden Change in ilm Thickne. Sudden preure loe are ypical a he edge of a pocke in a hydroaic bearing and a he inle plane of annular preure eal. The ame phenomenon alo occur a he leading edge of a bearing pad in high peed iling pad bearing. A udden preure recovery i alo quie ypical a he dicharge ecion of a preurized annular or labyrinh eal. Noe ha he imporance of fluid ineria effec may be rericed only o he inle and dicharge ecion, and may no be relevan wihin he hin film flow domain..0 RENOLDS EQUATION AND KINEMATICS O JOURNAL MOTION Lubricaed cylindrical bearing are low fricion, load bearing uppor in roaing machinery. Thee fluid film bearing alo inroduce vicou damping ha aid in reducing he ampliude of vibraion in operaing machinery. igure 4 how a chemaic view of a cylindrical bearing. The journal pin wih angular peed (Ω) and i cener (O J ), due o dynamic load, alo decribe ranlaional moion wihin he bearing clearance. The bearing or houing i aionary in mo applicaion. Noable excepion are hoe of floaing ring journal bearing and auomoive reciprocaing engine uppor rod bearing RTO-EN-AVT-14

5 xrθ Noe: film gap enlarged for decripion purpoe y A Θ θ h Ω e OB OJ Journal φ Bearing r Θθ+φ Clearance CR B -R J OB e ilm hickne: h C + e co(θ) or h C + e co(θ) + e in(θ) e e OJ φ r igure 4: Schemaic View of a Cylindrical Journal Bearing. ixed Coordinae Syem (θ,z) and Moving Coordinae Syem (Θ,z)..1 Reynold Equaion for Journal Bearing The mallne of hi raio allow for a Careian coordinae (xrθ, y, z) o be locaed on he bearing urface (ee igure 4). Then, in Claical Lubricaion, Reynold equaion decribe he generaion of hydrodynamic preure (P) wihin he bearing. Thi equaion arie from inegraion of he momenum equaion () acro he film hickne and ubiuion ino he coninuiy equaion []: Ω 1 ρ h P ρ h P { ρ h} + { ρ h} + (4) Θ R Θ 1 µ Θ z 1 µ z in he flow domain {0 Θ π, -½ L z ½ L}, where h(θ, z, ) i he film hickne, L i he bearing axial lengh, U ΩR J i he journal urface peed, and (ρ, µ) denoe he lubrican deniy and vicoiy, repecively. RTO-EN-AVT

6 The boundary condiion for he hydrodynamic preure in a plain cylindrical bearing are : a) he preure i coninuou and periodic in he circumferenial direcion, i.e. P(Θ, z,) P(Θ + π, z, ) (5) b) he preure equal he dicharge or amopheric value (P a ) on he bearing ide, i.e. P(Θ, ½ L, ) P(Θ, - ½ L, ) P a (6) A a conrain, he hydrodynamic preure need o be greaer han he liquid caviaion preure everywhere in he flow domain, i.e. P P cav in 0 Θ π, - ½ L z ½ L (7) Here P cav repreen he lubrican auraion preure or he ambien preure needed for releae of diolved gae. In pracice, no diincion i made beween hee wo value ince hydrodynamic film preure could be one o wo order of magniude larger han he ambien value. Conider he journal and bearing o be aligned and he journal cener o have an eccenriciy diplacemen e ( c). The film hickne i h c + eco(θ ) (8) Thi formula i accurae for (c/r) raio a large a The film hickne derived aume rigid bearing and journal urface, uniform axial and azimuhal clearance and no journal mialignmen.. Kinemaic of Journal Moion The journal cener O J i diplaced a diance (e) from he bearing cener O B. Thi diance i known a he journal eccenriciy and may vary wih ime depending upon he impoed exernal load on he bearing. The journal eccenriciy canno exceed he bearing clearance, oherwie olid conac and poenial caarophic failure may occur. The eccenriciy componen in he (, ) fixed coordinae yem are: e e co(φ); e e in(φ) (9) where φ i known a he bearing aiude angle, and Θθ+φ. Then, he film hickne alo equal h c + e coθ + e in Θ e inθ (10) and h Θ e in Θ + e coθ; h e coθ + e in Θ (11) where (. ) denoe differeniaion wih repec o ime, i.e. ( / ). Subiuion of he film hickne gradien ino Reynold equaion (4) give he following lubricaion equaion for an incompreible and iovicou fluid: The imple journal bearing model doe no accoun for feeding hole or axial groove for upply of he lubrican ino he bearing. A more deailed dicuion on lubrican caviaion and i phyical model can be found in [] RTO-EN-AVT-14

7 1 R h P h P + e Θ 1µ Θ z 1µ z + e Ω co Θ + e Ω e in Θ (1) An alernaive form of Reynold equaion arie when uing he angular coordinae (θ). Thi angle ar from he locaion of maximum film hickne. A coordinae yem wih radial and angenial (r, ) axe i convenienly defined wih he uni radial vecor along he line joining he bearing and journal cener. Recall ha e e co(φ); e e in(φ), and e e + e. The journal cener velociie in he (, ) and (r, ) coordinae yem are relaed by he ranformaion: OB e V e e coφ inφ inφ e coφ eφ (1) OJ φ r V r Noe ha V e, V e φ are he radial and angenial r componen of he journal ranlaional velociy. rom he film hickne h c + e coθ, i follow Tranlaional velociie of journal cener h h einθ θ θ e coθ e inθ e coθ + e φ inθ V r coθ + V inθ (14) Thu, Reynold equaion (4) for an incompreible and iovicou fluid i alo expreed a 1 R h P h P Ω + e coθ + e φ inθ θ 1µ θ z 1µ z (15) Equaion (15) i of paricular imporance ince i allow u o realize an imporan phyical phenomenon. Conider he journal cener o decribe circular cenered orbi wih a fixed ampliude or radiu, e. Hence de/d0. urhermore, if he frequency of whirl equal o 50% of he roaional peed; φ Ω / ; hen he righ hand ide of Eqn. (15) i null; and hence he preure i zero, P0 wihin he bearing film land. There i no generaion of hydrodynamic preure, hu reuling in a udden lo of load uppor capabiliy. The 50% ped whirl phenomenon i he bai of roordynamic inabiliy, a explained laer.. Bearing Reacion orce Once he preure field i obained, fluid film force acing on he journal urface, ee igure 5, are calculaed by inegraion of he preure field acing on he journal urface. An equal oppoing force ac on he bearing a well. The bearing reacion force are expreed in he fixed (, ) coordinae yem and moving (r, ) coordinae yem a RTO-EN-AVT

8 10-8 RTO-EN-AVT-14 ( ) dz d R z P L Θ Θ Θ Θ in co,, 0 0 π ; ( ) dz d R z P L r θ θ θ θ π in co,, 0 0 (16) θ P.coθ P.inθ P r θ Θ P journal r igure 5: luid ilm orce Acing on Journal Surface. The relaionhip beween he fluid film force in boh coordinae yem i given by: r φ φ φ φ co in in co (16.b) The fluid film force are generic funcion of he journal roaional peed (Ω) and he journal cener ranlaional velociie, i.e. ( ) Ω Ω,,, φ α α α e e e e ; α, or r, (17) An analyical oluion of Reynold equaion for arbirary geomery cylindrical bearing i no feaible. Mo frequenly, numerical mehod are employed o olve Reynold equaion and o obain he performance characeriic of bearing configuraion of paricular inere. There are analyical oluion o Reynold equaion applicable o wo limiing geomerie of journal bearing. Thee are known a he infiniely long and infiniely hor lengh journal bearing model []. In he LONG BEARING MODEL, ee igure 6, he lengh of he bearing i very large, L/D, and conequenly he axial flow i effecively very mall, i.e. ( P/ z) 0.

9 L bearing D journal Ω Axial preure field igure 6: The Long Bearing Model. or large L/D raio, Reynold equaion reduce o: 1 R h P θ 1 µ θ Ω Θ { h} + { h} (18) Thi bearing model give accurae reul for journal bearing wih lenderne raio (L/D) >. Mo modern bearing in high performance urbomachinery applicaion have a mall L/D raio, rarely exceeding one. Thu, he infiniely long journal bearing model i of limied curren inere. Refer o [] for deail on he analyical oluion of Eqn. (18). Thi i no he cae for queeze film damper (SD), however, ince he long bearing model provide a very good approximaion for ighly ealed damper even for mall L/D raio [4]..0 STATIC LOAD PERORMANCE O SHORT LENGTH BEARINGS In hi mo ueful bearing model, ee igure 7, he bearing lengh i hor, L/D 0, and conequenly he circumferenial flow i effecively mall, i.e. ( P/ θ) 0. or hi limiing bearing configuraion, Reynold equaion reduce o Ω h P { h} + { h} Θ z 1 µ z (19) L bearing D journal Ω L/D << 1 dp/dθ 0 Axial preure field igure 7: Shor Lengh Bearing Model. RTO-EN-AVT

10 The hor lengh bearing model provide (urpriingly) accurae reul for plain cylindrical bearing of lenderne raio L/D 0.50 and for mall o moderae value of he journal eccenriciy, e 0.75 c [4]. The hor lengh bearing model i widely ued for quick eimaion of journal bearing aic and dynamic force performance characeriic. Inegraion of equaion (19) lead o he preure diribuion P( θ, z, ) P a Ω 6 µ e coθ + e φ inθ z C H L (0) wih Hh/c 1 + ε co(θ) a he dimenionle film hickne, and ε e/c i he journal eccenriciy raio; [0 ε 1], ε 0 mean cenered operaion (ypically a condiion of no load uppor), and ε 1.0 evidence olid conac of he journal wih i bearing. No lubrican caviaion will occur if he exi or dicharge preure P a i well above he liquid caviaion preure. However, if P a i low, ypically ambien condiion a 1 bar, i i almo cerain ha he bearing will caviae or how air enrainmen when he oule plenum i no flooded wih lubrican. The caviaion model in he hor lengh bearing imply neglec any prediced negaive preure and equae hem o zero. Thi chop procedure alhough heoreically no well juified grap wih ome accuracy he acual phyic [5]. Hence, if P a 0, and from equaion (1), he preure field P>0, when co(θ+α) < 0. Thu, P>0 in he circumferenial region limied by π π θ + α π π α θ1 θ θ α (1) Tha i, regardle of he ype of journal moion, he region of poiive preure ha an exen of π (180 ); hu hen he infamou π film caviaion model widely ued in he lieraure. luid film reacion force on he journal are evaluaed by inegraion of he preure field acing on he journal urface. Wih P a 0, he radial and angenial force ( r, ) are given by r µ R L c J J 0 11 J J 11 0 e Ω e φ () where he J are inegral defined in analyical form by Booker [6]. Noe ha he fluid film force are proporional o he journal cener ranlaional velociie ( e, e φ ) a well a he journal roaional peed (Ω). The reacion force depend linearly on he fluid vicoiy and he bearing radiu and grow rapidly wih he raio (L/C). Hydrodynamic journal bearing are deigned (and implemened) o uppor a aic load W, hereafer aligned wih he axi for convenience, ee igure 8. A he equilibrium condiion, denoed by a journal cener eccenric diplacemen (e) wih an aiude angle (φ), he hydrodynamic bearing generae a reacion force balancing he applied exernal load a he raed roaional peed (Ω). The equaion of aic equilibrium are W W 0 coφ r r inφ + inφ coφ () RTO-EN-AVT-14

11 bearing Saic load W Journal Roaion Ω e W - r r fluid film φ Roor (journal) φ igure 8: orce Equilibrium for Saically Applied Load. or aic equilibrium, e 0, φ 0,and θ 1 0 o θ π. rom equaion (), he aic radial and angenial film reacion force are µ R L Ω ε µ R L Ω π ε r ; + (4) c ( 1 ε ) c 4( 1 ε ) / igure 9 depic he radial and angenial force for a ypical hor lengh bearing. The force are proporional o he lubrican vicoiy and roor urface peed (ΩR), he lengh (L ), and inverely proporional o he radial clearance (c ). Mo imporanly, he bearing force grow rapidly (non-linearly) wih he journal eccenriciy (εe/c) Saic orce for hor lengh bearing Radial and Tangenial force [N] r journal eccenriciy (e/c) * igure 9: Radial and Tangenial orce for Shor Lengh Bearing. µ0.019 Pa., L0.05 m, c0.1 mm,, 000 rpm, L/D0.5. The exernal load (W) i balanced by he fluid film reacion force. Thu, W ( + ) + π ( 1 ε ) ( 1 ) 1 L ε 16ε r µ Ω R L (5) c 4 ε RTO-EN-AVT

12 and he journal aiude angle φ i obained from angφ r π ( 1 ε ) 4 ε (6) Noe ha a he journal eccenriciy ε 0, φ π /, while a ε 1, φ 0..1 Deign of Hydrodynamic Bearing Selecion of Operaing Eccenriciy In he deign of hydrodynamic journal bearing, he bearing aic performance characeriic are relaed o a unique dimenionle parameer known a he Sommerfeld Number (S) defined a S µ N L D W R c (7) where N (Ω/π) i he roaional peed in revoluion/ec. In pracice, he pecific load or preure i known a he raio of applied load o bearing projeced area, i.e. (W/LD). In hor lengh journal bearing, a modified Sommerfeld number (σ) i defined and relaed o (S) by [5, 7]: µ Ω L R L σ π S ( L D) (8) 4W C Subiuion of Eqn. (8) ino Eqn. (5) relae he modified Sommerfeld number o he equilibrium operaing journal eccenriciy (e), i.e. ( 1 ε ) { 16ε + π ( 1 )} µ Ω L R L σ 4W c ε ε A a raed operaing condiion, σ i known ince he bearing geomery (R, L, c), roaional peed (Ω), fluid vicoiy (µ) and applied load (W) are known. Then, equaion (9) provide a relaionhip o deermine (ieraively) he equilibrium journal eccenriciy raio ε (e/c) required o generae he fluid film reacion force balancing he exernally applied load W. igure 10 and 11 depic he modified Sommerfeld number and aiude angle v. journal eccenriciy, repecively. Large Sommerfeld (σ) number; i.e. denoing mall load, high peed Ω or large lubrican vicoiy, deermine mall operaing journal eccenriciie or nearly cenered operaion, ε 0, φ π/ (90 ). Tha i, he journal eccenriciy vecor i nearly orhogonal o he applied load. (9) 10-1 RTO-EN-AVT-14

13 high peed mall load, large vicoiy Sommerfeld number for hor journal bearing Sommerfeld number Modified Sommerfeld number journal eccenriciy (e/c) low peed large load, low vicoiy igure 10: Modified Sommerfeld (σ) Number veru Journal Eccenriciy high peed mall load, large vicoiy aiude angle journal eccenriciy (e/ c) low peed large load, low vicoiy igure 11: Equilibrium Aiude Angle veru Journal Eccenriciy. Small Sommerfeld (σ) number, i.e. denoing large load, low peed Ω or low lubrican vicoiy, deermine large operaing journal eccenriciie, ε 1.0, φ 0 (0 ). Noe ha he journal eccenriciy vecor i nearly parallel o he applied load. igure 1 how he journal diplacemen wihin he bearing clearance for differen operaing condiion. The journal eccenriciy approache he clearance for large load, low haf peed or ligh lubrican vicoiy, and i i aligned wih he load vecor. or mall load, high peed or large lubrican vicoiie RTO-EN-AVT

14 (large Sommerfeld number), he journal ravel oward he bearing cener and i poiion i orhogonal o he applied load. Thi peculiar behavior i he ource of roordynamic inabiliy a will be hown horly. ey/c Low load, high peed, large vicoiy W load e/c Journal locu Clearance circle ex/c aiude angle load increae, low peed, low vicoiy pin direcion clearance circle igure 1: Locu of Journal Cener for Shor Lengh Bearing. 4.0 DNAMICS O A RIGID ROTOR SUPPORTED ON SHORT LENGTH BEARINGS igure 1 depic a ymmeric rigid roor of ma M, and upporing a aic load ( o W) along he axi. The roor i mouned on wo idenical plain hydrodynamic journal bearing. The equaion of moion of he roaing yem a conan roaional peed (Ω) are given by [5]: M + M u Ω in( Ω) + o M + M u Ω co( Ω) (0) where u i he magniude of he imbalance vecor, () and () are he coordinae of he roor ma cener, and (, )are he fluid film bearing reacion force. Since he roor i rigid, he cener of ma diplacemen are idenical o hoe of he journal bearing, i.e. ( ) e ( ), ( ) e ( ) RTO-EN-AVT-14

15 Saic load Rigid haf Dik M Journal bearing dik o u Ω Clearance circle e igure 1: Rigid Roor Suppored on Journal Bearing. (u) Imbalance, (e) Journal Eccenriciy. We are inereed on he roor dynamic behaviour for mall ampliude moion abou he equilibrium poiion defined by: O, 0, e, e or e, φ (1) o O O O O O where (e o,φ o ) denoe he aic equilibrium journal eccenriciy and aiude angle, repecively. The bearing aic reacion force aify o - ro o φ o r o + O O 0 0 o 0 O O ro ro coφ O O inφ + O O coφ inφ Small ampliude journal moion abou he equilibrium poiion, a repreened in igure 14, are defined a: O O () e e + e ( ), e e + e ( ), or + ( ), + ( ) (a) O O O O or converely, e( ) e + e( ), φ( ) φ + φ( ) (b) O O wih d e d d e d e e.. d e e d d e e d (c) RTO-EN-AVT

16 φ o o e e e o e clearance circle e e e φ r igure 14: Small Ampliude Journal Moion abou an Equilibrium Poiion. The journal dynamic diplacemen in he (r, ) coordinae yem are relaed o hoe in he (, ) yem by he linear ranformaion e e coφo inφo inφo e( ) coφ O eo φ( ) (4) Similar relaionhip hold for he journal cener velociie and acceleraion. Noe ha he aumpion of mall ampliude moion require e, e << c, i.e., he journal dynamic diplacemen are maller han he bearing clearance. Recall ha he fluid film force are general funcion of he journal cener diplacemen and velociie, i.e. [ e ( ), e ( ), e ( ), e ( )], α. Now, expre he bearing reacion force a a α α, Taylor Serie expanion around he aic journal poiion (e o, e o ), i.e. O O (5) 4.1 Definiion of Dynamic orce Coefficien in luid ilm Bearing luid film bearing iffne (K ij ) ij, and damping (C ij ) ij, force coefficien are defined a K ij i j ; C ij i i, j, (6) j or example, K - / correpond o a iffne produced by a fluid force in he direcion due o a journal aic diplacemen in he direcion. By definiion, hi coefficien i evaluaed a he equilibrium RTO-EN-AVT-14

17 poiion wih oher journal cener diplacemen and velociie e o zero. The negaive ign in he definiion enure ha a poiive magniude iffne coefficien correpond o a reoraive force. The force coefficien (K, K ) are known a he direc iffne erm, while (K, K ) are referred a cro-coupled. igure 15 provide a picorial repreenaion of he bearing force coefficien a mechanical parameer. Kxx, Cxx Kyy Cyy Kxy, Cxy K ij - i / j journal C ij - i /( j /δ) Kyx, Cyx bearing igure 15: The Phyical Repreenaion of Siffne and Damping Coefficien in Lubricaed Bearing. Ineria or added ma coefficien {M ij } ij, can alo be defined a i M ij ;, are j journal cener acceleraion. Ineria coefficien are of paricular imporance in uper laminar and urbulen flow fluid film bearing and annular eal. The ineria force coefficien or apparen mae have a ound phyical inerpreaion. Thee coefficien are alway preen in a fluid film bearing. Ineria coefficien can be of large magniude, in paricular for dene liquid. However, he effec of ineria force on he dynamic repone of roor-bearing yem i only of imporance a large exciaion frequencie, i.e. high queeze film Reynold number. (Thi fac alo hold for mo mechanical yem ubjeced o fa ranien moion). Wih he given definiion, he bearing reacion force are repreened a i,j, where { } ( ) ( ) O O K K K K C C C C (7) where o o ½W and o 0. Noe ha he defined force coefficien allow he repreenaion of he dynamic fluid film bearing (or eal) force in erm of he fundamenal mechanical parameer {K, C, and M}. However, hi doe no mean ha hee force coefficien mu be accordance wih acceped phyic grounded knowledge. or example, he vicou damping coefficien may be negaive, i.e. nondiipaive, or he iffne coefficien non-reoraive. rom Eqn. (0), he linear equaion for mall ampliude moion of he roor-bearing yem are M O O C + M C C C K + K K K M u Ω coω in Ω (8) RTO-EN-AVT

18 The lieraure preen he force coefficien in dimenionle form according o he definiion: k ij c cω Kij ; cij Cij i,j, (9) 0 0 where o i he aic load applied on each bearing (along he direcion). [Noe ha he oal aic load W o i hared by he wo bearing in a ymmeric roor moun]. Lund [8] derived fir he analyical formula for he hor bearing force coefficien. igure 16 and 17 depic he dimenionle force coefficien, iffne and damping, a funcion of he journal eccenriciy and of he modified Sommerfeld number (σ), repecively. In he figure, boh repreenaion are neceary ince a ime he journal eccenriciy i known a priori; while mo ofen, he deign parameer, i.e. he Sommerfeld number, i known in advance. In general, he phyical magniude of he iffne and damping coefficien increae rapidly (nonlinearly) a he journal eccenriciy increae RTO-EN-AVT-14

19 100 Siffne coefficien x (c/o) Siffne K (c/o) kxx kyy kyx kxy keq journal eccenriciy (e/c) high peed low peed mall load < > large load large vicoiy mall vicoiy 100 Damping coefficien x (c Omega/o) Damping: C (c Omega/o) cxx cyy cyx cxy Modified Sommerfeld number igure 16: Dimenionle Siffne and Damping Coefficien v. Journal Eccenriciy (ε) for Shor Journal Bearing. RTO-EN-AVT

20 100 Siffne coefficien x (C/o) Siffne K (C/o) kxx kyy kyx kxy keq Modified Sommerfeld number low peed high peed large load < > mall load low vicoiy large vicoiy 100 Damping coefficien x (c Omega/o) Damping: C (c Omega/o) cxx cyy cyx cxy Modified Sommerfeld number igure 17: Dimenionle Siffne and Damping Coefficien v. Sommerfeld Number (σ) for Shor Journal Bearing RTO-EN-AVT-14

21 Noe ha he dimenionle force coefficien do no repreen he acual phyical rend. or example, a e o 0, K K 0, bu he dimenionle k k have non zero value. Thi peculiariy follow from he definiion of dimenionle force coefficien uing he applied load ( o ). Recall ha, a e o 0, he aic load o i alo naugh. 4. Dynamic orce Coefficien for Journal Cenered Operaion, i.e. No Applied Load A he journal cener approache he bearing cener, e o 0, φ o 90 o, and from he formula preened, K µ Ω R L π Ω µ R L π K k c; C C c (40) c 4 c Thu, a he cenered journal poiion a hydrodynamic bearing offer no direc (uppor) iffne bu only cro-coupled force. A mall load applied on he bearing will caue a journal diplacemen in a direcion orhogonal (ranvere) o he load, a hown in he chemaic view below. Thi behaviour i common o all fluid film journal bearing of rigid geomery. Non-roaing rucure Roaing rucure Ω The ignificance of he cro-coupled effec in fluid film bearing 5.0 ROTORDNAMIC STABILIT O RIGID ROTOR SUPPORTED ON SHORT LENGTH BEARINGS The linearized equaion of moion are wrien in dimenionle form a [5, 9] p x c c x k k x + + co( τ ) p δ (41) y c c y k k y in( τ ) where u x, y, τ Ω, δ, d C M ( ' ) ; p c c c dτ c ij are he dimenionle iffne and damping force coefficien. Ω o i a dimenionle ma, and k ij and I i of inere o deermine if he roor-bearing yem i able for mall ampliude journal cener moion (perurbaion) abou he equilibrium poiion. To hi end, e he imbalance parameer δ 0 in he equaion above o obain, RTO-EN-AVT

22 p x c c x k k x (4) y c c y k k y 0 If he roor-bearing yem i o become unable, hi will occur a a hrehold peed of roaion (Ω ) and he roor will perform (undamped) orbial moion a a whirl frequency (ω ). Thee moion, aifying equaion (4), are of he form: jω jωτ jω jωτ x Ae Ae ; y B e B e ; j 1 (4) where ω ω Ω i known a he whirl frequency raio, i.e. he raio of he roor whirl or preceional frequency o he roor one peed of inabiliy. Subiuion of (4) ino equaion (4) lead o [5]: p ω k + k + + jω c jω c k p ω + + k jω c + jω c A 0 B 0 (44) In Eqn. (44), he deerminan, mu be zero for a non-rivial oluion of he homogenou yem. Afer algebraic manipulaion, he real and imaginary par of render [5,9] p C M (45) k c + k c c k c k ω S ω keq c + c o and ( k k )( k k ) k k eq eq ω ω (46) c c c c Ω 5.1 Threhold Speed, Criical Ma, Equivalen Siffne and Whirl requency Raio or a given value of journal eccenriciy (ε o ), i.e. a given Sommerfeld number (σ ), one evaluae Eqn. (45) o obain he equivalen iffne k eq, and hen Eqn. (46) o ge he whirl frequency raioω S. Thi ubiuion hen yield p ω (~criical ma), which in urn render he one peed of inabiliy Ω. k eq S igure 18 and 19 depic he whirl frequency raio (ω/ω) and he dimenionle hrehold peed of inabiliy (p ) veru equilibrium journal eccenriciy and modified Sommerfeld number, repecively. Noe ha for near cenered journal operaion, i.e. large Sommerfeld number, he whirl frequency i 0.50, i.e. half-ynchronou whirl. Recall ha in a mechanical yem, an equivalen damping raio > 0 caue he aenuaion of moion induced by mall perurbaion from an equilibrium poiion. A null damping raio bring he yem ino uained periodic moion wihou decay or growh, hu denoing he hrehold beween abiliy and inabiliy (ampliude growing moion) RTO-EN-AVT-14

23 1 Whirl frequency raio Whirl frequency raio Modified Sommerfeld number 1 Whirl frequency raio Whirl frequency raio journal eccenriciy (raio) high peed low peed mall load < > large load large vicoiy mall vicoiy igure 18: Whirl requency Raio v. Sommerfeld Number (σ) and Journal Eccenriciy (ε). RTO-EN-AVT

24 Dim [-] hrehold peed (p) 6 (p) hrehold peed inabiliy 4 unable Modified Sommerfeld number Dim [-] hrehold peed (p) (p) hrehold peed inabiliy 6 4 unable journal eccenriciy (e/c) high peed low peed mall load < > large load large vicoiy mall vicoiy igure 19: Threhold Speed of Inabiliy (p ) v. Sommerfeld Number (σ) and Journal Eccenriciy (ε). On he oher hand, if one aume ha he curren roaional peed (Ω) i he one peed of inabiliy, hen from he relaion above i follow he large magniude of ½ yem ma (M) o make he roorbearing yem unable. Thi ma i known a he criical ma, M c, and correpond o he limi ma which he yem can carry dynamically. If he roor ma i equal o or larger han wice M c, hen he yem will become unable a he raed peed Ω 4. The whirl frequency raio (WR), ω Ω, i he raio of he roor whirl frequency o he one peed of inabiliy. Noe ha hi raio, a given in equaion (46), depend only on he fluid film bearing 4 Recall ha each bearing carrie half he aic load, and alo half he dynamic or ineria load (.M c C Ω ) RTO-EN-AVT-14

25 characeriic and he equilibrium eccenriciy. The WR i independen of he roor characeriic (roor ma and flexibiliy) [5]. Reference [10] preen an analyi including fluid ineria effec, more applicable o annular preure eal and bearing handling proce fluid of large deniy. The parameer k eq i a journal bearing (dimenionle) equivalen iffne, alo depiced in igure 16 and 17. rom he definiion of hrehold peed and whirl raio, p M Ω ( C ) andω ω Ω, hen o Thu, he whirl or preceional frequency i ω K eq (47) C o M keq K eq ω ωn (48) M i.e., he whirl frequency equal he naural frequency of he rigid roor uppored on journal bearing. or operaion cloe o he concenric poiion, ε o 0, i.e. large Sommerfeld number (no load condiion), he force coefficien are, ee equaion (40), k k 0 ; c c ; k k ; c c 0 (49) eq ( k c c k ) c k + 0 ω k and 0.50 a ε 0 Ω c (50) Thi value of whirl frequency raio (WR) i a characeriic of hydrodynamic plain journal bearing. The WR how ha a he one ped of inabiliy he roor whirl a i naural frequency equal o 50% of he hrehold roaional peed. urhermore, under no exernally applied load, o 0, a in verically urbomachinery, he bearing poee no uppor iffne, i.e. K eq 0 and he yem naural frequency (ω n ) i zero, i.e. he roor-bearing yem mu whirl a all operaing peed. Noe ha if K 0, i.e. he bearing doe no have cro-coupled effec, hen he WR 0, i.e. no whirl occur and he yem i alway dynamically able. Cro-coupled effec are hen reponible for he inabiliie o commonly oberved in roor mouned on journal bearing. If he whirl frequency raio i 0.50, hen he maximum roaional peed ha he roor-bearing yem can aain i ju, i.e., wice he naural frequency (or oberved rigid roor criical peed). ω Ω max ω ω n (51) 0.50 igure 18, 19 and 0 depic he whirl frequency raio, he dimenionle hrehold peed (p ) and he criical ma (p ) veru he Sommerfeld number and equilibrium journal eccenriciy. The reul demonrae ha a rigid-roor uppored on plain journal bearing i STABLE for journal eccenriciy raio ε > 0.75 (mall Sommerfeld number) for all L/D raio. Noe ha he criical ma and whirl frequency raio are nearly invarian for operaion wih journal eccenriciie (ε o ) below RTO-EN-AVT

26 0 Criical ma Dimenionle Criical Ma unable Modified Sommerfeld number 0 Dimenionle Criical ma Dimenionle Criical Ma unable journal eccenriciy (e/c) high peed low peed mall load < > large load large vicoiy mall vicoiy igure 0: Criical Ma (m c p ) v. Sommerfeld Number (σ) and Journal Eccenriciy (ε). Keep in mind ha increaing he roaional peed of he roor-bearing yem deermine larger Sommerfeld number, and conequenly, operaion a maller journal eccenriciie for he ame applied aic load. Thu, operaion a ever increaing peed will evenually lead o a roor dynamically unable yem a he analyi reul how. 5. Effec of Roor lexibiliy on Sabiliy of Syem A imilar analyi can be performed conidering roor flexibiliy [5, 11]. Thi analyi i more laboriou hough raighforward. The analyi how ha roor flexibiliy doe no affec he whirl frequency raio. However, he one peed of inabiliy i dramaically reduced ince he naural frequency of he roorbearing yem i much lower. The relaionhip for he hrehold peed of inabiliy of a flexible roor i: 10-6 RTO-EN-AVT-14

27 K ro M bearing p f p T 1+ keq C (5) where he ub index f denoe a flexible roor. K ro i he roor iffne on each ide of he mid dik hown in he graph, and T o K i he roor aic ag or elaic deformaion a midpan. ro The elaic haf and bearing are mouned in erie, i.e. he bearing and haf flexibiliie add (reciprocal of iffnee), and hu he equivalen yem iffne i lower han ha of he bearing, and herefore he yem naural frequency decreae ignificanly. igure 1 depic he hrehold peed of inabiliy (p f ) for a flexible roor mouned on plain hor lengh journal bearing. Noe ha he more flexible he roor i, he lower he hrehold peed of inabiliy. If he fluid film bearing are deigned oo iff (low Sommerfeld number), hen he naural frequency of he roor-bearing yem i ju (K ro /M) 1/, irrepecive of he bearing configuraion. Threhold peed (p) for flexible roor 6 Threhold peed (p) 4 unable rigid T/c0.1 T/c1 T/c10 Modified Sommerfeld number low peed high peed large load < > mall load low vicoiy large vicoiy igure 1: Threhold Speed of Inabiliy (p ) for lexible Roor veru Sommerfeld Number (σ). Saic Sag (Γ/c) Varie. 5. Phyical Inerpreaion of Dynamic orce for Circular Cenered Whirl The bearing dynamic force in he radial and angenial are RTO-EN-AVT

28 r d K K rr r K K r e C e φ C rr C C 0 r e 0 r e φ (5) Conider circular journal moion of ampliude e a a forward frequency (ω), a hown in igure. A he cenered poiion, he bearing ha no direc iffnee, only cro-coupled iffne and direc damping, i.e. K rr K C r C r 0 (54) K K r K r Ω C ; C C C rr µ R L C π whirl orbi (ω) -(C ω + K r ) e Roor pin e r -(C r ω + K rr ) e igure : orce Diagram for Circular Cenered Whirl Moion. And hu, he radial and angenial force become r d 0; ( C ω K ) e (55) d r A deabilizing force will drive he journal in he direcion of he forward whirl moion, i.e. >0 if he equivalen damping (C eq ) i negaive (ee igure ), i.e. 1 ( C K r ) Ceq < 0 ω (56) 10-8 RTO-EN-AVT-14

29 RTO-EN-AVT whirl orbi Cro-coupled force K r e Damping force - C ω e Roor pin igure : orce Driving and Rearding Roor Whirl Moion. A he hrehold peed of inabiliy, K r C Ω. Thu, unable forward whirl moion occur for roor peed Ω ω. In he (,) coordinae yem, e co(ω) and e in(ω). Thu, he bearing dynamic force become Ω + Ω Ω + C e C e C d d ω ω ω ω ω ω ω ω 1 ) co( ) in( 1 ) co( ) in( (57) Noe ha (, ) oppoe he forward whirl moion for journal peed Ω < ω. or larger roor peed, he bearing force become poiive and aid o he growh of he forward whirl ampliude of moion, a hown graphically in igure 4.

30 whirl orbi -K -K Energy of orbi: Area_orbi (K -K ) K >0, K <0 <0, >0 igure 4: Repreenaion of ollower orce from Cro-Coupled Siffnee. The work performed by he bearing force during a full orbial period (Tπ/ω) equal [9] E ( ) d ( ω e) d ( ) + d ( ) E ( π e )( C ω K ) Area C ω r orbi d ( C ω K eq r ) ω e T (58) Noe ha E<0 i equivalen o negaive work, i.e. energy removed or diipaed from he roor-bearing yem. However when E>0, i.e. for Ω ω, he fluid film bearing add "energy" ino he roor-bearing yem hu driving he whirl moion forward. rom hi dicuion one can eaily deduce ha roor-bearing evidencing whirl orbi wih kewed area (harp ellipoid) will be le prone o roordynamic inabiliy, ee igure 5. Thi ype of dynamic repone i obained by deign and conrucion of a bearing generaing (direc) iffne aymmery, a given in muliple pad bearing configuraion (ellipic, muliple-lobe wih preload, preure-dam bearing). However, hee bearing are limied o fixed orienaion aic load and roor pin in only one direcion RTO-EN-AVT-14

31 Energy of orbi: Area x (K -K ) igure 5: Influence of Bearing Aymmery on Whirl Orbi. 5.4 Experimenal Meauremen of Roor-Bearing Syem Inabiliy The archival lieraure i abundan in experimenal and field decripion of evere inabiliie induced by fluid film bearing on roaing machinery. A an example of e conduced a he auhor laboraory on a high peed e rig, igure 6 depic recorded ampliude of moion veru haf peed in a rigid roor uppored on plain journal bearing. The diplacemen meauremen correpond o roor moion along he verical and horizonal plane (LV, LH). The curve wih larger ampliude denoe he oal ampliude of moion while he oher in ligh color how he filered ynchronou (1) moion wih low roll compenaion. The paage hrough a well-damped criical peed i eviden a ~ 8.5 krpm. A he haf peed increae, he ampliude of moion decreae. However, a a haf peed ~ wice he criical peed, he roor become violenly unable wih large ampliude moion nearly equalling he bearing clearance. igure 6: Ampliude of Roor Moion veru Shaf Speed. Experimenal Evidence of Roordynamic Inabiliy. RTO-EN-AVT

32 igure 7 depic he waerfall of he verical haf moion. The graph how he frequency conen of he vibraion ignal a he roor accelerae. The ynchronou moion are denoed by he 1 line. The whirl frequency raio i 0.50 a he one of he evere ubynchronou moion. A he peed increae, he whirl frequency lock a he yem naural frequency. Thi phenomenon i known a oil-whip. The roor wa everely damaged upon compleion of he experimen. igure 7: Waerfall of Recorded Roor Moion Demonraing Subynchronou Whirl. 6.0 CLOSURE Compreor, urbine, pump, elecric moor, elecric generaor and oher roaing machine are commonly uppored on fluid film bearing. In he pa, mo applicaion implemened common cylindrical plain journal bearing. A machine have achieved higher peed and larger power, roor dynamic inabiliy problem uch a oil whirl have brough he need o implemen oher bearing configuraion. Cuing axial groove in he bearing o upply oil flow ino he lubricaed urface generae ome of hee geomerie. Oher bearing ype have variou paern of variable clearance (preload and offe) o creae a pad film hickne ha ha rongly converging and diverging region, hu generaing a direc iffne for operaion even a he journal cenered poiion. Variou oher geomerie have evolved a well, uch a he iling pad bearing, which allow each pad o pivo, and hu o ake i own equilibrium poiion. Thi feaure uually reul in a rongly converging film region for each loaded pad and he near abence of cro-coupled iffne coefficien. Table and ummarize ome of he advanage and diadvanage of variou bearing in condened form. igure 8 how graphical keche for ome of he bearing configuraion below. Reference [1, 1, 14] offer imporan echnical informaion on he deign, operaion and abiliy conideraion for he mo common fluid film bearing ued in indurial applicaion, wih emphai in pump and compreor RTO-EN-AVT-14

33 Table : ixed Pad Non-Pre Loaded Journal Bearing Bearing Type Advanage Diadvanage Commen Plain Journal 1. Eay o make. Low Co 1. Mo prone o oil whirl Round bearing are nearly alway cruhed o make ellipical bearing Parial Arc 1. Eay o make. Low Co. Low horepower lo 1. Poor vibraion reiance. Oil upply no eaily conained Bearing ued only on raher old machine Axial Groove 1. Eay o make. Low Co 1. Subjec o oil whirl Round bearing are nearly alway cruhed o make ellipical or muli-lobe loaing Ring 1. Relaively eay o make. Low Co 1. Subjec o oil whirl (wo whirl frequencie from inner and ouer film (50% haf peed, 50% [haf + ring] peed) Ued primarily on high peed urbocharger for PV and CV engine Ellipical 1. Eay o make. Low Co. Good damping a criical peed 1. Subjec o oil whirl a high peed. Load direcion mu be known Probably mo widely ued bearing a low or moderae roor peed Offe Half (Wih Horizonal Spli) 1. Excellen uppreion of whirl a high peed. Low Co. Eay o make 1. air uppreion of whirl a moderae peed. Load direcion mu be known High horizonal iffne and low verical iffne - may become popular - ued ouide U.S. Three and our Lobe 1. Good uppreion of whirl. Overall good performance. Moderae co 1. Expenive o make properly. Subjec o whirl a high peed Currenly ued by ome manufacurer a a andard bearing deign RTO-EN-AVT

34 Table : Pad Journal Bearing wih Sep, Dam or Pocke, Tiling Pad Bearing Bearing Type Advanage Diadvanage Commen Preure Dam (Single Dam) 1. Good uppreion of whirl. Low co. Good damping a criical peed 4. Eay o make 1. Goe unable wih lile warning. Dam may be ubjec o wear or build up over ime. Load direcion mu be known Very popular in he perochemical indury. Eay o conver ellipical over o preure dam Muli-Dam Axial Groove or Muliple-Lobe 1. Dam are relaively eay o place in exiing bearing. Good uppreion of whirl. Relaively low co 4. Good overall performance 1. Complex bearing requiring deailed analyi. May no uppre whirl due o non bearing caue Ued a andard deign by ome manufacurer Hydroaic 1. Good uppreion of oil whirl. Wide range of deign parameer. Moderae co 1. Poor damping a criical peed. Require careful deign. Require high preure lubrican upply Generally high iffne properie ued for high preciion roor NON-IED PAD JOURNAL BEARINGS Bearing Type Advanage Diadvanage Commen Tiling Pad Journal Bearing lexure Pivo, Tiling Pad Bearing 1. Will no caue whirl (no cro coupling) 1. High Co. Require careful deign. Poor damping a criical peed 4. Hard o deermine acual clearance 5. Load direcion mu be known Widely ued bearing o abilize machine wih ubynchronou nonbearing relaed exciaion oil Bearing 1. Tolerance o mialignmen.. Oil-free 1. High co. Dynamic performance no well known for heavily loaded machinery. Prone o ubynchronou whirl Ued mainly for low load uppor on high peed machinery (APU uni) 10-4 RTO-EN-AVT-14

35 θ θ 0 W W (a) Ellipical bearing (b) -pad bearing θ θ W (c) 4-pad bearing W (d) 5-pad bearing Tiling pad bearing lexure pivo hydroaic pad bearing igure 8: Schemaic View of Variou Radial luid ilm Bearing Configuraion. REERENCES [1] Turbulence in luid ilm Bearing, L. San André, Lecure Noe (#8) in Modern Lubricaion, hp://phn.amu.edu/tribgroup, 00. [] Tribology ricion, Lubricaion & Wear, A. Szeri, Hemiphere Pub, RTO-EN-AVT

36 [] Caviaion in Liquid ilm Bearing, L. San André, Lecure Noe (#6) in Modern Lubricaion, hp://phn.amu.edu/tribgroup, 00. [4] Effec of luid Ineria on inie Lengh Sealed Squeeze ilm Damper, L. San André & J.M. Vance, ASLE Tranacion, 0,, pp. 84-9, [5] Turbomachinery Roordynamic, (chaper ), D. Child, John Wiley & Son, Inc., 199. [6] A Table of he Journal Bearing Inegral, J.. Booker, ASME Journal of Baic Engineering, pp. 5-55, [7] Roordynamic of Turbomachinery, J.M. Vance, J., Wiley Iner-Science Pub., [8] Self-Excied, Saionary Whirl Orbi of a Journal in a Sleeve Bearing, J. Lund, Ph.D. Thei, Renelaer Polyechnic Iniue, Troy, N.., [9] Dynamic of Simple Roor-luid ilm Bearing Syem, L. San André, Lecure Noe (#5) in Modern Lubricaion, hp://phn.amu.edu/tribgroup, 00. [10] Effec of Eccenriciy on he orce Repone of a Hybrid Bearing, L. San André, STLE Tribology Tranacion, 4, 4, pp , [11] The Sabiliy of an Elaic Roor in Journal Bearing wih lexible Suppor, J, Lund, ASME Journal of Applied Mechanic, pp , [1] Deign of Journal Bearing for Roaing Machinery, P. Allaire & R.D. lack, Proc. of he 10 h Turbomachinery Sympoium, TAMU, pp. 5-45, [1] luid ilm Bearing undamenal and ailure,. Zeidan & B. Herbage, Proc. of he 0 h Turbomachinery Sympoium, TAMU, pp [14] undamenal of luid ilm Journal Bearing Operaion and Modeling, M. He & J. Byrne, Proc. of he 4 h Turbomachinery Sympoium, TAMU, pp , RTO-EN-AVT-14