A Methodology for Dynamic Coefficients and ...

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A Methodology for Dynamic Coefficients and Nonlinear Response of Multi-Lobe Journal Bearings a

a

T. V. V. L. N. Rao , S. Biswas & K. Athre

b

a

Indian Institute of Technology Delhi, Industrial Tribology Machine Dynamics and Maintenance Engineering Center, New Delhi, 110 016, India b

Indian Institute of Technology Delhi, Department of Mechanical Engineering, New Delhi, India, 110 016 Published online: 25 Mar 2008.

To cite this article: T. V. V. L. N. Rao , S. Biswas & K. Athre (2001): A Methodology for Dynamic Coefficients and Nonlinear Response of Multi-Lobe Journal Bearings, Tribology Transactions, 44:1, 111-117 To link to this article: http://dx.doi.org/10.1080/10402000108982433

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A Methodology for Dynamic Coefficients and Nonlinear Response of Multi-Lobe Journal ~ e a r i n ~ s ' T. V. V. L. N.RAO and S.BISWAS Indian Institute of Technology Delhi Industrial Tribology Machine Dynamics and Maintenance Engineering Center New Delhi, India 1 I0 016

and

Downloaded by [University Technology Petronas] at 21:38 06 May 2013

K.ATHRE Indian Institute of Technology DelhelhE Department of Mechanical Enginwring New Delhi, India 110 016

A nleri~doIogyfor prediction of dynarnic coeflcien?s and nonlinear simrrlntion of multi lobe bearings is developed. This simple fornrula~ion t l ~ ais f an exlension of Reason and Norung (E982) uppmnch. An eusy to implementjinira pertrrrbation method is used ro cvalunre st~JFnesxand damping coemients. In order to obtain tl~aJ~ridJilm forces, the oil film pressure generated in the bearing is .wived rising n semi analyrical pressure model. Four mulri lobe jolinral bearirlgs, naomeEy: W-aria1 groove, el I iprical, tlime-lobe and u@er bearings am studied in terns of non dimensional st~frlcs.7 ottd dantping coefficients. The paramerers are: U D ralios of 0.5 nnd 1.0,pre-londfdfncros of 0.5 and o&r fnctar of 0.5; except for the offset Bearitlg,for tvhicf~ihc offserfactor is 1.0. The resulrs are pmvided in gmplzical form nnd compared with existing Final manuscript approved June 6,2000 Revlew led by ltzhak Green

nirmericol techniques. AII the coeftcients sE1olv /ha injT14ence of Sommerj-eld (long) bearing mlution ar high eccentricity ratios. Results also indicate the applicable mnge of recipmcul mluiior~ship combining the Ocvirk (shortJ and Somrnctfeid bearing sol~rtions for all multi Iobt bearing configurations. The journaI center trajectories obtained am compared wirh numarieal ~ s E/su and the effect of s t a b f i i ~on rotor beating sysrem is srudied for both baIanced and unbalanced m r o ~Ir is concluded thar rhc a p p m i mate bearing sohtions which are predomirzuntly applied lo cylirldrical haring confiigumfion, can o l ~ be o successfilly implemenfed for noncirctrlor proJles.

KEY WORDS Multi lobe Journal Bearing; Stiffness and Damping Coefficients;Transient Response X-direction

NOMENCLATURE b$ ' Bq

C D e, , EU

fr f,* Far-FY

"

k. H Kc

=darnping coefticient. ij = x.y, Ndrn; B,.= bil = adinl clear~nce,m =journal diameter, rn = unbalance cccenrricity, rn; EU= e,/C

= peru1rbe.djournal cenm diaplacernenr and velocity in Y-

K Z

= coordinate dong axial dimtion: Z = d = offset factor

C W

direction

a 6 @

= bearing Sorccs along venical and horizontal directions. N: F, = f,hw IWC)~RL, E, = f j r p (RIC)~ RL =oil film thicknass, m: H=WC = stiffness coeficient, ij = x.y, Ndm; K.. = k.U C/W '1

= width of the baring, m m, M = rotor moss, kg; ~ = r n ~ o ' / w = pressure in the oil filrn,~lm'; P = p/qw (wc)~ P. P I? P, P, = non-dimensional dynamic pressures for finite, Ocvirk and

L

AY,A 9

Sammerfeld bearings respectively

r]

8

w

S

R

= preload factor =journal eccentricity ratio and attitude angle

= oil vixosity, ~slrn' = angular coordinates fmrn venicaI load direction (Fig. I) = journal angular velocity. radls = angular velocity of whirl = whirl ratio, o$w

Subscript

R

=journal ndius, rn

s

S

= Sommerfeld number = time, sec; T= t a,P =static load, N; w = w h w ( W C ~ R L

I X. Y. x.

= steady-state position = Ocvirk (short) bearing parameter = Sommerfeld (long) bearing parameter = displacements mnid velocities pressure gradients

P 1, 2

= haring lobe number = leading and waiting sdge of each lobe

r, 7n; W WS

Ax. A X

= non dimensional speed parameter: ""Y? = perturbed journal ccnter displacement and velmity in

o

T.b0,S.BISWAS AND K.A M E

Allaire, et. a]., (1980). Also the whirl speed ratiw of the transient orbits are determined using Fast Fourier Transform analysis. I n this paper. a semi-analytical technique is developed to evaluate the dynamic coefficients and transient response of muIti lobe bearings. The aim i s to provide a simple methodology for obtaining results comparable to the nurnerica! solution.


METHODOLOGY

Flg. 1-deometry ol muH4 l o b bearings [Lund and Thornsen, 1978).

Reason and Narang (1982) proposed harmonic combination of short (Ocvirk) and long (Sornmerfeld) bearing approximations under steady-state conditions. Rao, et al., (2000) implemented the Reason and Narang approach for unsteady conditions of plain cylindrical bearings to evaluate linearized stability and journal transient response. The same methodology is now extended to multi Iobe bearings. The geometrical configumtions of multi lobe bearings (Lund and Thornsen, 1978) are shown in Fig. 1. The preload factor and offset factor o f the rnulti lobe bearing geometry i s also described by Jones. et al.. (1981).

INTRODUCTION The determination of stiffness and damping caeficients is important, since they can be used to represent the forces developed in the bearing due to small amplitude motion of journal a b u t its equilibrium position. These cmfficients are used to cnlculate the stability khavior of rotors. Linearized analysis i s suitable for predicting b r i n g behavior for small amplitude motions itbout equilibrium position, but fails to accurately provide information For large ninplitudes of journal orbits above the threshold speed. The nonlinear transient motion is performed to obtain the limit cycle journal orbits. Multi lobe bearings are widely used in high speed rotating ~n;lchineryto suppress instability under lightly loaded conditions. In the ~nultilobe bearing analysis, the approximate solution o f the Reynolds equalion is the long bearing approximation using an end leakage correction factor (Warner, 1963) which results in a marginal e m r for bearings with both low length to diameter ratios and eccentricity mtios. ti, et al. (1979) and Jones, et al., (1981) develop n series solution based an the variational principles, The rariational approximations are widely used to obtain analytical solutions to the Reynolds equation. Lund and Thornsen (1 978) provide dava of stiffness and damping coefficients for two axial, elliptical, three Iobe und offsel bearing canfigurntions using the infinitesi~ n nperlurblrtion l method. Design data for three lobe bearings in the hminar and turbulent regimes of operation are presented for various U D ratios (Malik, et al,, 1983) and the solution o f the Rcynolds equation i s obtained by the Gaierkin's method. Kostnewsky, et al., (1996) compare the numerically evaluated and experimentally predicted steady-state parameters as welI as stiffness and damping coefficients for ~ o - a x i a lgroove bearings. The uncenuintias for the dynamic coefficients are also provided. The theomtical steady-state performance analysis o f the three lobe k r i n g s was first studied by Pinkus (1959). Kirk and Gunter (1976) conducted extensive studies on the nonlinear journal response of plain cylindrical bearing. The effect of unbalance above and below the threshold s p e d and transient response of the journnl motion using series solution o f the Reynolds equation based on variational principle is studied by Li, et al., (1980) and

Governing Equations Under dynamic conditions, for each lobe, the two dimensional Reynolds equation in non dimensional form in fixed co-ordinate system (Jones, et al., 1981), foran aligned journal bearing isgiven as:

+

where H = 1 + X case Y sine The finite bearing pressure expression for each lobe is written as:

where~=~(o,bX,d~,d~.~P For a small amplitude motion of the jaurnal center, the nondimensional dynamic pressure in Eq.121 i s a function of the slatic equilibrium pasirion as well as the perturbation displacements and velocities parameters. In the case o f rnu1ti lobe bearings or panial arc bearings, Ocvirk (short) bearing model is not applicable as the boundary conditions are applied only at the axial ends of the bearings (Giberson, 1971). However meaningful results and trends can be studied using short bearing solution. Falkenhagen, et a]., (1972) numerically integrate long bearing pressure expressions to evaluate linear stability of vertical rotor system. In Eq. (21 of the present methodology, long (Sornmerfeld) bearing solution is evaluated numerically using successive ever-relaxation method, while analytical solution of short (Ocvirk) bearing given in Eq. [3] is employed.

A Methodology for Dynamic Coefficients and Nonlinear Response OF Multi-Lobe Journal Bearings

Boundary Conditlons

are:

The boundmy conditions for numerical evaluation of long bearing approximation and analytical evaluation of short bearing solution are: {(B,Z)=O; at 8 = 8 , and 8,; 5(6,2)?0for 8=#,to 8,

=Ofor@=#, to^+#

[41

where 8, and O2 are the leading and trailing edges for each lobe, and these are with reference to the Imd line as shown in Fig. I .


Hydrodynarnlc Forces The fluid film forces acting on the journal are obtained by integration of finire baring pmssure [Eq.21 and can be expressed as

Somrnerkld Number Under steady-state conditions, the load capcity, attitude angle and Sommerfeld number are:

Semi-Analytical Solutian The methodology for evaluating stiffness and damping coeficients i s as fottows:

I. The bearing eccentricity ratio (€1and the attitude angie (4) are given. The eccentricity ratio (E,) and the attitude angle ($,) for each lobe are calcu!ated (Lund and Thomsen,

1938). 2. Using Eq. [2] the finite bearing non-dimensional pressure P is evaluated in each lobe. The fluid film forces generated by each lobe are then evaluated (Eq.151). The fluid film forces in vertical and horizontal directions in each bearing lobe are added to obtain the bearing fluid film forces. 3. The linearized sliffness and damping coeficients are obtained from Eq. [7]using finite perturbation technique. Finite values of displacements and velwiries are cansidered about static equilibrium position and the bearing forces for each perturbed position are evaluated as described in step 2,

The journal transient response is evaluated using following steps: Stiffness and Damping Coefflclents The finite perturbation method is used to calculate stiffness and damping coefficients by approximation of force coefficients in which small displacements (AX,AY) and velocities (A x , A y ) . are taken about the equilibrium position. The dynamic coefficients K, and KyX can be expressed as:

I. Initial values of the journal center position and velocity are given in a fixed co-ordinate system. 2. The fluid film pressure (Eq. 221) is evaluated for each lobe under dynamic conditions. The bearing forces are evaluated by summation of forces from individual lobes IEq. [5]). 3. Using fourth order Runge Kutta method, the equations of journal motion (Eqs. [8]-[9J) are solved to evatuate the journal center position and velocity at the next time step. 4. The steps 2 and 3 are repeated until a definite pattern of journal trajectory is obtained.

RESULTS AND DISCUSSION Similarly, the coefficients ( ~ , . K , , ) are evaluated using while for (B,, BYx) and (BXY,Bw) finite perturbation of the derivates (-$-%) and (-9.3)respectively, are required, Qiu and Teu (1996) indicate that for two percent of the perturbation amplitudes and velocities, the coefficients obtained by Finite perturbation method are very close to the infinitesimal perturbation method. However. in the present method, by trial and error, using a value of 0,03 as the perturbation to the displacements and velocities, AX, dY, A x , d 3, the dynamic coefficients obtained are found to k close to numerical results (Lvnd and Thomsen, 1978; Kostrtewsky, et al., 19%).

I-%.-%),

Equations of Journal Motion The non-dimensional equations of rigid journal motion in fixed coordinate system with hydrodynamic and unbalance forms

Dynamlc Coefflclents To determine the validity of the proposed method, stiffness and damping coefficients as a function of journal eccentricity ratio (Figs. 2-5) stre compared with numerical data (Lundand Thomsen, 1978) for UD ratios of 0.5 and l .O. The pre load factor is 0.5 for two axial. elliptical, three lobe and offset bearings while offset factor is 0.5 for two axial, elliptical, three lobe bearings and I .O for offset bearing, The stiffness and damping coefficients for a two axial groove bearing are depicted in Figs. 2(a) and Z(b). All four stiffness coefficients obtained from the present method agree well with the numerical solution at high eccentricity ratios. The coefficient KYX deviates at low eccentricity ratios and the deviation from the numerical method increases with LA3 ratio. The long bearing

T.RAO,S. BISWAS AND K.AWRE

Numcricot methoa lLund.r9781 L h 21.0 Lfo:0.5 ..A+ P rmsqnt mmtkod LID 1.0 Up 10.5 0 er 6 0

.

1


12

Mumetlcal method lCund 19781 C/O ml.0 L I 0 ~ 0 . 5m rn b 4 Present mthd t l ~ = l O u0.0-5 o rn e o

- 0 2

FIB. 2--Sttffneas and damplng coefflclentsof two exlaf groove bearing.

Numerical method 1Luna 1970) LID.l.0 U~r0.Sm.r. Prcsant method

l/~=l.P 30)

L ~ e O m 5 0 0 a o

I

.

Numfrlcol merhoO f u n d l 9 B l LID=1.0 L/o*0& 1

rn

Present method Llo :l.Q C/O -0.5 a

0

m

a

Flg. 4-Stlffness end damplng cmtflcients ol thm l o b bearlng.

Nurnerlml method lLund 19'181 110.1.0 Q F O . ~ - - ~ * Present mcfnod L/~=1.0 L/~sP.sO*ao

.- ..

Humeritol method lCvrrd T978t Cf0: 0 5 L1~~1.0 P ~ C S M method I L/D=O.S 0 e 0 1/0=14 301 f

'

5q

Fig. !5-StFfmess

end demplng eoafflclants of Met bearing.

Fig, +Stiffness and damplng eoeffleknt8 of elllptlcal hearlng.

prcssurc solution closely approximates the numerical solution at both high eccentricity ratios and V D ratios while the short bearing prcssrlre solution i s close to 'the numerical solution at low occenaicity ratios. The prediction of stiffness and damping coefficients using the present m e t h d is according to the harmonic pressure relationship given in Eq. 121, in which accurate solution of long benring approximation i s employed. The cross damping coemcients evaluated using the present method are unequal, but arc close to numerical results, while rhe direct horizontnl damping term Byy deviates from numerical solution at low eccentricity mtios and with the increase in UD ratios. Figurcs 3ta) and 3(b) indicate the dynamic coeflcients Sot an cllipticol bearing configuration. There is no significant variation of dikmping coefficients with U D ratio. The cross coupling d a m p itip terms are also unequal but are close to numerical solution. The niuximum change in !he damping cwficients (Bx,, B ,, and RYX) WCUTS at low eccentricity ratios. Unlike the two-axial groove and elliptical karing configuntions, the stiffness and damping coefficients (Figs. 4(a) and 4@)) of rllrec lobe bearing indicates a different trends of results. The direct vertical stiffness coefficient Kxx using the present methodology is leas thnn the numerical solution (Lund and Thornsen,

1978) and the divergence increases with

LID ratio. The stiffness

coemcients KYX,KYYare lower and K,

is higher than the numerical solution at low eccentricity ratios. The cross damping terms ,,B , By, are unequal at lower eccentricity ratios and for UD ratios o f 0.5 and 1.0, while the coefficients Bxx, Byy are close to the numerical salution. The stiffness and damping coefficients for the offset bearing configuration are depicted in Figs. 5(a) and 5(b). In contrast to the results obtained for the two axial, elliptical and three lobe bearing configurations, the offset bearing results show the divergence o l K , using the present method v i s - h i s numerical soPution nt lower eccentricity and higher W D ratios. The direct damping coefficients Bxx, Byy also show marginal change From numerical solution unlike the results of three lobe bearing. Comparison o f theoretically obtained eccentricity mtio, attitude angle, stiffness and damping coeficients for two-axial groove bearing using the present m e t h d is made with the experimental and numerical work o f Kosmwsky, et el., (1996). Using the present method, the Sornrnerfeld number is given as input parameter for the same test bearing geometry, operating conditions and coordinate system details used by Kusrrzewsky, el a!., (1996). Table 1 presents the comparison o f eccentricity ratio and

k Methodology for Dynamic Coeficienta and Nonlinear Response of Multi-Lobe Journal Bearings


TABLE I ~ O W P A R ~ S O N OF ECCE~ICITYRATIOAND A ~ D ANGLE E FOR Two-AXIALGROOVE BEARING ( K O ~ E W S KetY&I.., 1996) For Ua=0.5: 6=Q -4.5

Fig. G(a)-Stlffn~s mffielents i w two RXCIgroove Ibearlng.

Crr CPY

F ~ Q B(b)-Damplng . coeffielmlsIw lwu axlal growe bearing.

attitude angle for two-axial groove bearing. The eccentricity ratio obtained using the present method is close ta the results from numerical method (Kostnewsky, et a]., 1996). Though the attitude angle is close lo the experimental value at low Sommerfeld numbers, the deviation increases at high Sommerfeld numbers.Figure 6 gives the camparitan of stiffness and damping coefficients with numerical and experimental values for two-axial groove bearing. From Figs. 6(a) and 6(b), the stiffness coefficients Kxx, K , K, are lower than numerical method while Kyy is higher than the numerical method. The comparison of damping coefficients obtained from presenl rnethcd with numerical method also show a sirniIar variation as predicted for stiffness coefficients.

Journal Transient Response The results of the two-axial groove bearing are compared with

Flg. 7--0rbItal response for two axlal groove Wrlng.

the numerical simulations provided by Muller-Karger and Granados (1997). The static operating conditions described by the Sommerfeld number are used for comparison of the journal response obtained from numerical method (Muller-Karger and Granados, 1997) and the present approach. For the Sommerfeld number, S=0.2813, a transient analysis is sirnulrtted for the nondimensional synchronous dynamic force amplitude, Fd.28, with two orientation angles of 0" and 90'. The same geometry (Pad arc=15O0, Gmve=30', UD=0.5) of two-axial bearing configuration is used. As the orientation of the dynamic force is changed, the orbits change shape and size as shown in Fig. 7. The magnitude of the orbit i s similar for both the methods while there is a shift in the initial position and orientation of the orbits and this is due to the difference in Sommerfeld number and attitude angle evaluated using the presea method. The results of the elliptical, three-lobe and offset bearings are compared for the journal transient response (Li, et a]., 1980; Allaire. et al.. 1980). Figure 8 indicates the nonlinear trajectory of el tiptical, three-lobe and offset bearing configurations as the journal i s released from the bearing center with zero initial velocities. The journal speed is fixed and is below the linearized stability threshold. For elliptica!, three-lobe and offvet bearings, both the present method and numerical solution predict journal center tnjectories approaching static equilibrium position. and [he deviation between of the orbits increases at steady state position. The shaft speed below the stability threshold for an unbalance eccentricity ratio (EUa.25) is analyzcd (Fig. 9) for cIlipticrrl,

T. RAO,S. BISWAS AND K.ATHRE

"""I

w54m k b d 5

BJ Li o .I. Wort. IW

I

---

hm-idF-r


Flg. &Belaneed rotor 8taWe ]journal orblts.

three lobe and offset bearing configurations. The limiting motion of the orbits are established for all the cases due to the presence of synchronous unbalanceexcitation, which results in higher dynamic fluid film forces transmitted to the benring surface. The effect of unbalance is ro increase the size of the orbits which become nonlinear. With the addition of unbalanced mass both the methods result in steddy srnte synchranous orbits around the equilibrium position.

CONCLUSIONS For mulri lobe journal bearings, a semi-analytical method tising a combinalion of unsteady Ocvirk and Sommerfeld bearing solurions i s employed to evaluate stiffness and damping coeficients, and transient journal response. A finite perturbation ~ppmachis used ro evaluate dynamic coefficients, The resuits are ~wnipared for diffesenl values of eccentricity ratios and also Sonimerfeld numbers, The dynamic ceefflcients obtained by the present method agree well with those obtained by ahe numerical infinitesimal perturbation method at higher eccentricity ratios. Better predictions for highly preloded three lobe haring are obrained conipnred to two-axial groove bearing configurations. Thc nonlinear dynnrnic response is analyzed using a synchronous excitntion Force for two-axial groove bmring. The journal trajectories For elliptical, three lobe and offset bearings are obtained using the present method for position perturbation and unbalance excitation, and these are compared with numerical results. With

he presenl semi-analytical approach, reasonable accuracy is obtained, and it is suitable For analysis of journal center trajecto-

ry. REFERENCES 11) Allairc, 1'. E., l i . D, E and Choy, K.C,'Tranxicnl Unklmce Rekponsc or Four Mulli Lohc Journal Beudng,"' ASME Jour: OJ h b c Tcch.. IOZ. pp m M 7 . f 1980). (21 Falkcnhngcn. G. L.Gunter. E. I.and Schuttm. E b,"Stnbilily and TmnsImt Molion or a Vcniwl 'Fhm M Bcarinb System:' ASME lour. of Eng. for Itt[lrt.*ly, pp 665-673, (1972). 13) GihcMan. M. E. Dir;cushion to Akcrs, A., Michaclson. S. and ndmeron, A,, "Stnbili~yConrmim of Whirling Fioilc Journal Bcoring:' ASME Jorrr oJLrbr Ec11.:9.3#pp l f 7- 1m. ( 197 1 ). (4) Joncr. W. 13.. RIIITCII. L. E.. Allmite. P. E.,nnd Gunter. E. I.. "Rapid Soluliun to Rcyrmfcls F?ualio~lTor Applic;rlinn to Nonlinear Transient Analyris of Rcxible Raor-ll~iringSync~ns."NASA LcRC, NASA Gnnl No. NSG-3177.(1981).

Flg. M o t o r synchronous journal orblts.

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A Methodology for Dynamic Coeficients and Nonlinear Response of Multi-Lobe Journal Bearings

(7) Li, D. F.. AIIaIte. P. E and Barrett, L. L,"Analytiwl Dynamics or Pnninl Journal Bcarings wilh Applicalions." ASLE Tmnr., 22. pp W-1 12, ( 1 979). (8)Li, D. F,. Choy, K. C. nnd Allaim, P. E. "Stnblliry and Tmnsicnt Chmcrcristics or Four Muhi Cobe Beaing Configumlhns," ASME JOUK oJLubc Tmb., 102, pp 291-299. (1980). 19) Lund, S. W. md Rtomren. K. K.. "A filmlation M e w m d Dnta for the Dynamic CoeTlicients of Oil Lubrimted Jwml Bcnrings," Topics in Fluid Film Bearing a d Rntor Bearing Systen~.ASME, New York, pp 1-28, ( 1978). (EO) Molik. M.. Sinhuwn. R. nnd Chndn. M. "Design Datn Tor Three Lobc Be;lnn@."ASLE Tmr~x..24.3, pp 345-353. (1981). 111) Mullet-Kqcr, C. M. and Gmados. A. L.. "Owivation of Hydrodynamic Bearing Cocmcicnis Using Minimum Square M e W , " ASME JOIIC J Trib.. 119. pp 802-807, (1977).


(12) Pfnkus, O., "Analysis nnd Chmneriaics ofThrce Lobe Bcmingr."'ASMEJuuc 81, pp 49-55. (1959). Of5& (131 QIU. 2.L nrrd Tieu. A. K.. 'The Effm OF Pmurbatian Ampliludes on Eight

Force Coeficiems o i Jwmal Bearin&" Ttib. Trans., 39, Z pp 469475. (1996). T. V. V. L. N., Hirani, H.. Biswu, S.. and Aihre, K.. ''An Analytical Appmch to Evaluale Dynnmic C o c m c i m and Nonlinear Trnn~ienrAnalysis oTa Hydmdynnrnic Journal Bearing:' Trih T m . ,43. 1. pp 109-115. (2003). (15) Reason, B. R. nnd N m g . I.P.. "Rapid k i g n md Pcdormnncc Evalvdtion OF S k d y Stae Journal Bearings ATcchniquc Amcnable to Prqmrnmble Hand Calculstws," ASLE T m . ,25,pp4294M. (1982). ( 1 6 ) Wmcr, P C., "Swic and Dynamic Propcnics or Panjal Journal Bearings," ASME JOII~: OJBasic En#., pp 247.257, (1963). (14) Rao,

-