Axial flow rate (per unit circumferential length) [m 2 /s] R B, R J =R Bearing Radius ~ Journal Radius [m] S

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1 NOTE 4 TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING Lctur 4 introducs th fundamnts of journal baring analysis. Th long and short lngth baring modls ar introducd. Th prssur fild in a short lngth baring is obtaind and xampls follow for th prssur profils gnratd undr various oprating conditions, namly journal rotation w/o whirling, pur circular cntrd whirl, and radial squz film motion. Nxt, th analysis focuss on dtrmining th quilibrium journal ccntricity for an applid static load. Th ommrfld numbr is a singl paramtr that prmits quick baring dsign sinc, for xampl, a larg load or a low journal angular spd or a low viscosity produc th sam oprating (larg) journal ccntricity. Th journal ccntricity and attitud angl dfining th static prformanc of th journal baring ar shown as functions of oprating spd, lubricant viscosity, and applid load. Nomnclatur C Baring radial claranc. R B -R J [m] X + Y. Journal cntr ccntricity [m] F X, F Y Fluid film raction forcs along X,Y axs, F FX + FY Fr + Fr [N] F r, F t Fluid film raction forcs along r,t axs [N] h C + cos C + X cosθ + Y sinθ. Film thicknss [m], Hh/C, J k Bookr s journal baring intgral i L Baring axial lngth [m] P Hydrodynamic prssur [N/m ] P amb Ambint prssur 0 (for simplicity of analysis) [N/m ] Q z Axial flow rat (pr unit circumfrntial lngth) [m /s] R B, R J R Baring Radius ~ Journal Radius [m] ommrfld numbr (baring dsign paramtr) [rv] t Tim [s] V X, V Y X, Y. Componnts of journal vlocity along X,Y axs[m/s] V r, V t φ,. Componnts of journal vlocity along r,t axs [m/s] V (pur) squz film vlocity [m/s] W Applid (xtrnal) static load (along X axis) [N] (X,Y) & (r,t) Coordinat systms α Angl of squz vlocity vctor with axis r ε /C. Journal ccntricity ratio Θx/R, y, z Coordinat systm on plan of baring φ F tangϕ t F. Journal attitud angl r ρ Fluid dnsity [kg/m ] μ Fluid absolut viscosity [N.s/m ] σ μ Ω LR L Modifid ommrfld numbr (short lngth baring) Ω σ 4 W C Journal angular spd (rad/s) NOTE 4. TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING. Dr. Luis an Andrés 010 1

2 NOTE 4. TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING. Dr. Luis an Andrés 010 For incomprssibl and isoviscous fluids, and in trms of th pur squz vlocity V, Rynolds quation for gnration of th hydrodynamic prssur P is ) cos( α μ μ + + V z P h z P h R Ω φ sin cos + (4.1) whr h(c+ cos) in th (r,t) systm, and V V V Ω Ω + Ω ) tan( ; sin ; cos φ α φ φ α α (4.) Rcall that () is th journal cntr ccntricity, Ω is th journal rotational spd, V r is th journal radial vlocity, and φ V t is th journal tangntial vlocity. Th boundary conditions for th prssur fild P in a plain cylindrical journal baring ar: a) Th hydrodynamic prssur and its gradints ar continuous and singl valud in th circumfrntial dirction, i.. ),, ( ),, ( t z P t z P π + (4.) b) At th baring axial nds, th prssur is ambint (P a ) Ω X V s r α OJ φ t Y Figur 4.1. Pur squz film vlocity in rotating coordinat systm

3 L L P,, t P,, t Pa (4.4) c) and as a constraint, th prssur is always qual or largr than th liquid cavitation prssur, P(, z, t) (4.5) P cav Mor physically sound and appropriat boundary conditions at th onst of th lubricant cavitation rgion ar givn latr (s Nots 6). Boundary conditions at th film rformation boundary follow latr. An analytical solution to quation (4.1) for arbitrary gomtry cylindrical barings is unknown. Most frquntly, numrical mthods ar mployd to solv Rynolds quation and thn to obtain th prformanc charactristics of baring configurations of particular intrst. Th baring prformanc charactristics as a function of th applid xtrnal load ar th journal ccntricity and attitud angl, baring flow rat, drag powr loss or friction cofficint, (tmpratur ris), and th dynamic forc cofficints (stiffnss and damping) at th oprating rotational spd Ω. Thr ar analytical solutions to Rynolds quation applicabl to two limiting gomtris of journal barings. Ths ar known as th infinitly long and infinitly short lngth journal baring modls. D journal L baring Ω Prssur fild Figur 4.. Th long baring modl In th LONG BEARING MODEL, th lngth of th baring is rgardd as vry larg, L/D, and consquntly th axial flow is ffctivly vry small, ( P/ z) 0. Th prssur profil dos not vary along th baring lngth (xcpt at its dgs), as shown in Figur 4.. For th long baring, Rynolds quation rducs to: NOTE 4. TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING. Dr. Luis an Andrés 010

4 1 h P cos( + α ) V R 1 μ (4.6) Th long baring modl givs accurat rsults for journal barings with slndrnss ratios (L/D) >. Most modrn barings in high prformanc turbomachinry applications hav a small L/D ratio, rarly xcding on. Thus, th infinitly long journal baring modl is of limitd currnt intrst. This is not th cas for squz film damprs (FDs), howvr. Th long baring modl provids a vry good approximation for tightly sald damprs vn for small L/D ratios. Anothr application is long barings supporting ship propllrs, for xampl. Th hort Lngth Journal Baring Modl In this modl, th lngth of th baring is rgardd as vry small, L/D 0, and consquntly th circumfrntial flow is ffctivly vry small, i.. ( P/ ) 0. For this limiting baring configuration, th Rynolds quation rducs to h P z 1 μ z V cos( + α) (4.7) Th short lngth baring modl provids (surprisingly) accurat rsults for plain cylindrical barings of slndrnss ratios L/D 0.50 and for small to modrat valus of th journal ccntricity, 0.75 C. Th short lngth baring modl is widly usd for quick stimations of journal baring static and dynamic forc prformanc charactristics. D journal L baring Ω L/D << 1 dp/d 0 In th short lngth baring modl, th circumfrntial prssur gradint is takn as vry small, i.. ( P/ ) 0, and hnc, th man fluid flow in th circumfrntial dirction is U Ω M x ρ h ρ R h (4.8) Axial prssur fild Fig. 4.. Th short lngth baring For film thicknss, h C + cos, dirct intgration of Equation (4.7) is straightforward. For an alignd journal, th film h is not a function of th axial coordinat. NOTE 4. TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING. Dr. Luis an Andrés 010 4

5 Th intgration rndrs th axial flow pr unit circumfrnc: h P Qz V z cos( + α) (4.9) 1μ z Not that at th baring middl plan, z0, th flow rat is nil. Furthr intgration of Equation (4.9) and applying th ambint prssur boundary condition at th baring sids lads to th following parabolic prssur fild, 6 μv cos( + α) L P(, z, t) Pa z C H or (4.10) Ω 6μ cos + ϕ sin L P(, z, t) Pa z C H with h H 1+ ε cos (4.11) C as a dimnsionlss film thicknss. ε /C is a journal ccntricity ratio; 0 ε 1, ε0.0 mans cntrd opration (typically a condition of no load support), and ε 1.0 vidncs solid contact of th journal with its baring. Not that th prssur fild in Eqn. (4.10) is parabolic in th axial dirction, with symmtry about th baring middl plan z0. Hnc th maximum hydrodynamic prssur occurs at this middl plan and quals: 6 μv cos( + α) L P(, z 0, t) Pa (4.1) 4 h In th short lngth baring modl, th circumfrntial coordinat () and th tim (t) ar not variabls but paramtrs. Consquntly, imposing prssur boundary conditions in th circumfrntial dirction is not possibl. Fortunatly, Eqn. (4.10) shows th hydrodynamic prssur is continuous and singl valud in th circumfrntial dirction No lubricant cavitation will vr occur within th baring if th magnitud of th sid (xit or discharg) prssurs P a is wll abov th liquid cavitation prssur. Howvr, if P a is low, typically ambint at 14.7 psi (1 bar), it is almost crtain that th baring will show ithr liquid or gas cavitation, and undr dynamic load conditions, air ntrainmnt (ingstion and ntrapmnt). Th cavitation modl in th short lngth baring modl sts P a 0 and disrgards any prdictd ngativ prssurs, it just quats thm to zro. This NOTE 4. TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING. Dr. Luis an Andrés 010 5

6 chop procdur although not thortically justifid sms to grasp with som dgr of accuracy th physics of th thin film flow. Hnc, if P a 0, and from Eqn. (4.1), th prssur fild is positiv, P>0, whn Thus, P>0 in th circumfrntial rgion dlimitd by cos(+α) < 0 (4.1) π π + α π π α 1 α (4.14) That is, rgardlss of th typ of journal motion, th rgion of positiv prssur has an xtnt of π (180 ) and it is cntrd (or alignd) with th pur squz vctor (V ). This important obsrvation is th basis for th infamous π film cavitation modl widly usd in th nginring litratur. Th study of a fw simpl cass for journal off cntrd motions (>0) is of intrst. Rcall from Eqn. (4.) V V V Ω cosα ; sinα φ Ω φ Ω + φ ; tan( α ) (4.) 1) Pur rotational (spinning) journal motion (Ω > 0), 0, φ 0 Thn α ½ π, V ½ Ω and P > 0 xtnds from 1 0 to π ) Pur radial squz motion without journal rotation,( Ω 0), 0, φ 0 Thn α 0, if >0, V s >0 π π and P > 0 from 1 to ; whil α π, if < 0, V s π π and P > 0 xtnds from 1 to ; ) Pur tangntial squz motion (circular whirl) without journal rotation, Ω 0 NOTE 4. TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING. Dr. Luis an Andrés 010 6

7 π Thn α if φ > 0, V s φ, And P > 0 from 1 π to π. Figur 4.4 dpicts prdictd cntrlin (at z0) prssur profils for a short lngth journal baring with th following dimnsions and oprating charactristics. Lngth L50 mm; claranc, C100 μm, rotational spd at,000 rpm (Ω14 rad/s), and lubricant viscosity μ19 cntipois ( N.s/m ). Th oil is an IO VG lubricant with a spcific gravity of Not that th midplan prssur fild P in Eqn (4.1) dos NOT show th baring radius or diamtr in it. Th rsults shown corrspond to thr journal ccntricity ratios ε/c, qual to 5%, 50% and 75% of th radial claranc (C). Th oprating conditions ar a) Journal spinning motion, (Ω14. rad/s), 0, φ 0, Maximum V ½ CΩ0.016 m/s b) Journal circular whirl, φ 14. rad/s, Ω 0, Maximum V φ 0.01 m/s c) Journal radial motion (pur squz),, 0.01 m/s, Ω φ 0, V m/s Th graphs show th rgions for positiv prssur (P>0) for ach particular cas of journal motion. Th hydrodynamic prssur incrass rapidly as th journal ccntricity incrass and as th whirl orbit radius incras. Not that circular whirl without journal spinning producs prssur magnituds twic as larg as for th cas of stady journal rotation without journal whirling. Not also that th prssur rgions in th two cass ar shiftd by 180. It is notworthy to show that pur squz film motions ( >0) gnrat prssurs much largr than thos du to th othr two typ of motions, sinc th squz film vlocity (V ) is largr. NOTE 4. TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING. Dr. Luis an Andrés 010 7

8 150 Baring cntrlin (pur spinning) Journal max vlocitis [m/s] Prssur [bars] C. Ω Cdφ. dt 0 d dt 0 a L 0.05 C /C0.5 /C0.50 /C0.75 angl (dgrs) from max. film μ rpm Baring cntrlin (tangntial vlocity) Journal max vlocitis [m/s] Prssur [bars] C. Ω 0 Cdφ. dt 0.01 d dt 0 b L 0.05 C /C0.5 /C0.50 /C0.75 angl (dgrs) from max. film μ rpm Baring cntrlin (radial vlocity) Journal max vlocitis [m/s] Prssur [bars] 500 C. Ω 0 Cdφ. dt 0 c d dt L 0.05 C /C0.5 /C0.50 /C0.75 angl (dgrs) from max. film μ rpm. 10 Figur 4.4. Cntrlin prssur profil for short lngth baring. Journal motions ar (a) pur spinning, (b) circular whirl, (c) pur radial squz motions NOTE 4. TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING. Dr. Luis an Andrés 010 8

9 Fluid film forcs for th short lngth journal baring Intgration of th prssur fild acting on th journal surfac producs a fluid film forc with radial and tangntial componnts (F r, F t ) dfind by Fr Ft L / 0 1 cos P(, z, t) R d dz (4.15) sin π π whr1 α ; α. Not that if th lubricant dos not cavitat, thn 1 0, π. ubstitution of th prssur fild, Eqn. (4.10) into th xprssion abov, givs, for P a 0, Fr 1 μ V F t C L R cos( + α) cos L d z H sin dz (4.16a) Fr μ V R L F t C 1 cos( + α ) cos d H sin (4.16b) Ω whr H(1+ε cos), V cosα ; sinα V φ. Th baring forc is typically a raction typ, i.. it balancs an applid xtrnal forc or load that acts on th journal. This xtrnal load is a fraction of th rotor wight. Rcall that rotors ar supportd on a pair of barings, in gnral. Bookr (1965) 1 dfins th following baring intgrals J jk i k j sin ( ) cos ( ) d (4.17) i H 1 and provids rcursiv analytical formulas for thir valuation. Using th dfinition abov, th fluid film raction forcs bcom Fr μv R L Ft C J J 0 11 J J 11 0 cosα sinα (4.18a) or in trms of th journal vlocity componnts 1 Bookr, J. F., 1965, A Tabl of th Journal Baring Intgral, AME Journal of Basic Enginring, pp NOTE 4. TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING. Dr. Luis an Andrés 010 9

10 Fr μ R L Ft C J J 0 11 J J 11 0 Ω φ (4.18b) Not that th fluid film forcs ar proportional to th journal cntr translation vlocitis ( Vr, Vt φ ) as wll as th journal rotational spd Ω. Th raction forcs dpnd linarly on th lubricant viscosity (μ), th baring radius R; growing vry rapidly with th ratio (L/C). Th linar transformation btwn th (r, t) and (X,Y) coordinat systms givs a rlationship for th valuation of th fluid film forcs (F X, F Y ) in th Cartsian coordinat systm FX cosφ FY sinφ sinφ Fr cosφ Ft (4.19) Equilibrium condition for a short lngth journal baring Hydrodynamic journal barings ar dsignd and built to support a static load W, hraftr alignd with th X axis (for convninc). At th quilibrium condition, dnotd by journal cntr ccntric displacmnt () with attitud angl (φ), th fluid film journal baring gnrats a raction forc balancing th applid xtrnal load W at th ratd rotational spd (Ω). Th quations of static quilibrium ar W + F F Y X 0 0 W F 0 F Y X F cosφ F r r F sinφ + F t t sinφ cosφ (4.0) For static quilibrium, 0, φ 0, so thn α ½ π, V ½ Ω; and 1 0 to π. From Eqn. (4.18b), th radial and tangntial film forcs rduc to Fr μ R L + Ft C J J 11 0 Ω (4.1) Using th formulas from Bookr s Journal Baring Intgral Tabls, and aftr som algbraic manipulations, { J + } 11 1 ε J J (4.a) ε ( 1 ε ) { ( 1 ε ) J + J } 0 1 π J J1 (4.b) ε 1 ( ε ) NOTE 4. TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING. Dr. Luis an Andrés

11 baring tatic load W Journal Rotation Ω t Y Y W F t -F r r fluid film Rotor (journal) X φ X φ Figur 4.5. Forc quilibrium for static load W Hnc, th radial and tangntial fluid film forcs for th short lngth baring ar F F r t μ R L Ω C μ R L Ω + C 4 1 ε ( 1 ε ) π ε ( ε ) (4.) Not that th baring raction forc is proportional to th journal spd Ω, th lubricant viscosity μ, and th baring radius R. Th forcs ar strong nonlinar functions of th baring lngth L and th radial claranc C, i.. ~ L /C. Th fluid film baring raction forc balancs th applid xtrnal load W. Thus, W ( F + F ) + π ( 1 ε ) ( 1 ) 1 L ε 16ε r t μ Ω R L (4.4) C 4 ε and th journal attitud angl φ (angl btwn th load and th journal ccntricity vctor ) is tangφ Ft F r π ( 1 ε ) 4 ε (4.5) Not that as th journal ccntricity ε 0, φ π /, whil as ε 1, φ 0. NOTE 4. TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING. Dr. Luis an Andrés

12 In th dsign of hydrodynamic journal barings, th baring static prformanc charactristics rlat to a singl dimnsionlss paramtr known as th ommrfld Numbr () μ N L D R W C (4.6) whr N (Ω/π) is th rotational spd in rvolutions/s. In Eqn. (4.6), th ratio (W/LD), load dividd by th baring projctd ara, is known as th baring spcific load or spcific prssur. Various journal baring configurations ar ratd by thir pak spcific prssur, for xampl, up to 00 psi for tilting pad barings and ~1,000 psi for a cylindrical journal baring. Th spcific prssur is a rlativly good indicator of th maximum (pak) film prssur within th thin film baring. In short lngth journal barings, a modifid ommrfld numbr σ) is mor physically adquat. Th paramtr is dfind as μ Ω L R L σ π ( L D) (4.7) 4W C ubstitution of Eqn. (4.4) into Eqn. (4.7) rlats σ to th quilibrium journal ccntricity ε, i.. ( 1 ε ) ( ) μ Ω LR L σ 4W C ε 16ε + π 1 ε { } (4.8) At a ratd oprating condition, σ is a known magnitud or valu sinc th baring gomtry (R, L, C), rotational spd (Ω), fluid viscosity (μ) and applid load (W) ar spcifid. Thn Eqn. (4.8) provids a rlationship to dtrmin (itrativly) th quilibrium journal ccntricity ratio ε(/c) rquird to gnrat th hydrodynamic prssur fild that producs a fluid film raction forc qual and opposit to th applid load W. Th following figurs dpict th modifid ommrfld numbr σand attitud angl φ vrsus th journal ccntricity ε(/c). Larg ommrfld numbrs σ, dnotd by ithr a small load W, a high spd Ω rotor, or a lubricant of larg lubricant viscosity μ, dtrmin small oprating journal ccntricitis or narly a cntrd opration, i.. ε 0 and φ π/ (90 ). That is, th journal ccntricity vctor is narly orthogonal or prpndicular to th applid load vctor W. NOTE 4. TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING. Dr. Luis an Andrés 010 1

13 mall ommrfld numbrs σ (larg load W, low spd Ω, or low lubricant viscosity μ) dtrmin larg journal ccntricitis, i.. ε 1.0, φ 0 (0 ). Not that th journal ccntricity vctor is narly paralll to or alignd with th applid load W. * 10 ommrfld numbr journal ccntricity (/c) High spd Low load Larg viscosity Low spd Larg load Low viscosity Figur 4.6. Modifid ommrfld numbr σ vrsus journal ccntricity ε. hort lngth journal baring μ Ω LR L σ 4 W C NOTE 4. TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING. Dr. Luis an Andrés 010 1

14 φ Attitud angl * journal ccntricity (/c) High spd Low load Larg viscosity Low spd Larg load Low viscosity Figur 4.7. Equilibrium attitud angl φ vrsus journal ccntricity ε. hort lngth journal baring NOTE 4. TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING. Dr. Luis an Andrés

15 Figur 4.8 shows th locus of journal cntr displacmnt or journal ccntricity within th baring claranc for various oprating conditions. Th journal ccntricity () approachs th claranc (C) for opration with ithr larg loads, or low rotor spds, or light lubricant viscosity, and it is alignd with th load vctor. For ithr small loads, or high shaft spds, or larg fluid viscosity (larg ommrfld numbr), th journal travls towards th baring cntr and its position is orthogonal to th applid load. This pculiar bhavior is th sourc of rotordynamic instability as will b shown shortly. y/c Low spd load, incrass, high rotor spd, load loads, larg viscosity high viscosity W load /c Journal locus Claranc circl x/c attitud angl load incrass, low spd, low viscosity spin dirction claranc circl Figur 4.8. Locus of journal cntr for short lngth baring Us th accompanying MATHCAD program to dtrmin th journal ccntricity for a journal baring configuration and spcifid load, fluid proprtis and spd oprating conditions. Th program implmnts a simpl thrmal modl and also prdicts th xit film tmpratur, baring drag powr and flow rat. NOTE 4. TATIC LOAD PERFORMANCE OF PLAIN JOURNAL BEARING. Dr. Luis an Andrés

16 Y W X Luis an Andrs Fall009 (NU) All journal barings hav a supply port (axial groov or hol) to fd cold lubricant into th film sparating th rotating journal and its th baring. Th lubricant gts hottr (incras in tmpratur) as it flows down thru th hydrodynamic wdg. om hot lubricant lavs th baring through its sids. Th spinning journal draws som hot lubricant around towards th inlt port whr it mixs with th cold stram of lubricant. Th tmpratur of th lubricant at th inlt of th film land is highr than th oil supply tmpratur. Appndix to Nots 4 : tatic load prformanc of journal baring impl lumpd paramtr thrmal analysis for prdicting th xit tmpratur and ffctiv viscosity in a short lngth journal baring

17 Y W Film thicknss hc+ cos() UΩR Nomnclatur Q : flow T : tmpratur X Qi, Ti Inlt plan ½Q, Tff ½ L FULL FILM ZONE Qπ, Tπ Cavitatd zon 0 π π -½L cavitation bubbl Not: drag (shar) only occurs in rgion [0,π[ Lt: Tff ½ (Ti + Tπ ) ½Q, Tff Tmpratur Tπ As a wighd avrag Ti UΩR

18 W Film thicknss hc+ cos() Y UΩR bubbl X Qi, Ti Inlt plan ½Q, Tff Qπ, Tπ Cavitatd zon 0 π π ½ L -½L Not: drag (shar) only occurs in rgion [0,π[ ½Q, Tff GLOBAL FLOW CONTINUITY EQN QiQ+Qπ κ 1D ENERGY TRANPORT EQUATION is a fraction of th mchanical powr is convrtd into hat and carrid away by lubricant flow Eo-Eiκ Powr Ei ρ Cp Qi Ti Eo ρ Cp Q Tff + ρ Cp Qπ Tπ

19 W Film thicknss hc+ cos() Y X ½Q, Tff UΩR Qi, Ti ½Qπ, Tπ Cavitatd zon 0 π π Not: drag (shar) only occurs in rgion [0,π[ ½Q, Tff Working with both qns. lads to κ Powr ρ Cp (Qi ½ Q) (Tπ Ti )

20 Nomnclatur Q : flow T : tmpratur λ : thrmal mixing cofficint 0.80 (TYP) Hat carry-ovr cofficint Flow balanc Enrgy balanc Q i Q supply + λ Q up Q i T i Q supply T supply + λ Q up T up UΩR Q up Qπ T up Tπ Q i T i Into th film Upstram, Q supply T supply From oil UMP tank Downstram Thrmal mixing at baring inlt groov

21 Nomnclatur Q : flow T : tmpratur E : Enrgy Flow & nrgy balanc in fd groov Q i Q supply + λ Q π Flow & nrgy balanc in film land QiQ+Qπ Eo-Eiκ Powr U RΩ Qi ~ h 0 L ( c+ ) ; U RΩ Qπ ~ h p L ( c ) Q Q Q L RΩ i π Q i T i Q supply T supply + λ Q π T π Powr ~ Ei ρ Cp Qi Ti Eo ρ Cp Q Tff + ρ Cp Qπ Tπ L π u Powr ~ Torqu ΩΩ μ R ddz; rcall u U y y h 0 0 y 0 L π Ω R ~ μ R ddz; Ω h Thrmal nrgy transport short journal baring 0 0 y 0 π Ω R L 1 Ω R L π μff d ff c μ 1+ εcos c 1 ε 0

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