Conical whirl instability of turbulent flow hybrid porous journal bearings

Tribology InternationalVol. 31, No. 5, pp. 235–243, 1998 1998 Elsevier Science Ltd. All rights reserved

Printed in Great Britain0301–679X/98/$19.00+ 0.00

PII: S0301–679X(98)00026–7

Conical whirl instability ofturbulent flow hybrid porousjournal bearings

Anjani Kumar*

An analysis of conical whirl instability of an unloaded rigid rotorsupported in a turbulent flow hybrid porous journal bearing hasbeen presented, following Constantinescu’s turbulent lubricationtheory. The effect of bearing feeding parameter (b), Reynoldsnumber (Re), ratio of wall thickness to journal radius (H/R) andanisotropy of porous material on the stability of rotor-bearingsystem has been investigated. It is observed that higher values ofb gives better stability and higher stability is predicted if theporous bush is considered to be isotropic. 1998 ElsevierScience Ltd. All rights reserved.

Keywords: journal bearings, hybrid bearings, porous bearings, conicalwhirl instability, turbulent regime

Introduction

Bearings handling low viscosity lubricants such aswater, cryogenic liquids or liquid metals operate nor-mally in turbulent flow regime with large Reynoldsnumbers.

Porous journal bearings have been widely used inindustry for a long time. The main advantage of self-lubricated porous bearings is that they need no exterioroil supply once the bearing with its porous materialimpregnated with oil is installed. However, the magni-tude of the hydrodynamic pressure developed in suchbearings is not sufficient to provide substantial loadcarrying capacity. Therefore, hydrostatic or hybrid(combined hydrostatic and hydrodynamic) journal bear-ings appear to be an ideal candidate for high-speedcryogenic machinery, since its advantage is to providea significant load capacity. In conventional externallypressurized bearings the lubricant is admitted into thebearing clearance space through large number of ori-fice-compensated or capillary compensated discreteholes. In such bearings the hydrodynamic pressure isnot developed within the recess, but only in the filmland regions, resulting to an uneven distribution ofpressure in the lubricant film. Secondly, the analysis

*Production Engineering and Management Department, RegionalInstitute of Technology, Jamshedpur-831 014, IndiaReceived 19 October 1995; revised 11 September 1996

Tribology International Volume 31 Number 5 1998 235

and design of such bearings is complicated and costly.The hydrodynamic effect in the film region can be fullyutilized by using porous materials/bushes in externallypressurized lubrication. It gives rise to a more evenpressure distribution on the journal surface, as thelubricant is injected into the clearance space througha large number of pores.

Porous metal bushes are mostly used for self-actingoil bearings and externally pressurized gas bearings.Hybrid porous oil bearings may be preferred to conven-tional multi-recess capillary- or orifice-compensatedhydrostatic bearings, to avoid high cost and compli-cated design. Low viscosity fluids can flow more easilythrough the myriad of tiny and tortuous passages ofthe porous bush, but the purity of the lubricant has tobe maintained for longer service of the porous bushes,because the pores of the bushes may become chokedby impurities in the lubricant. However, the porousbushes may be replaced easily at regular intervals forbetter performance.

Porous bearings in hybrid operation have limitationslike difficulty in producing two identical bearings ofspecified permeability and porosity.

In a rotor bearing system, two modes of whirl insta-bility may occur namely translatory and conical. Cylin-drical whirl in translatory mode occurs when a rigidrotor is loaded symmetrically in two bearings. Conicalwhirl occurs due to nonsymmetric loading. For twoclosely spaced bearings, if the transverse moment ofinertia is high, then conical whirl onset speed can

Conical whirl instability of turbulent flow hybrid porous journal bearings: A Kumar

Nomenclature p91, p92 perturbed fluid pressures in porous matrixp91, p92 dimensionless perturbed fluid pressures in

C porous matrixmean radial clearanceD Rshaft/journal diameter shaft/journal radiusDij Re, Re*bearing damping coefficients (first suffix mean Reynolds number, local Reynolds

denotes the direction of moment and number (rUC/m, rUh/m)the second denotes the direction of Sij bearing spring stiffnessangular velocity) Sij dimensionless bearing spring stiffness

Dij dimensionless bearing damping S8C3Sij

mR3L3DcoefficientsS8C3Dij

mR3L3D t timeGu, Gz Tconstants in turbulent Reynolds equation dimensionless time (vpt)h Ulocal film thickness shaft surface velocity (Rv)h x, y, zdimensionless film thickness (h/C) circumferential, radial and axialH coordinatesthickness of the wall of the porous bushIp y, zpolar moment of inertia of journal dimensionless radial and axial coordinatesIt (y/H, 2Z/L)transverse moment of inertia of journali ( − 1)1/2

Greek symbolsJ inertia ratio (Ip/It)u circumferential coordinateKx,Ky,Kzm absolute viscosity of fluidpermeability coefficients inx, y and z

bdirections

bearing feeding parameterS12KyR2

C3H DKx, Kz Kx /Ky, Kz /Ky

L length of bearing v, vp angular velocity of shaft, angular velocityL /D aspect ratio of journal centrep, p0 local film pressure in the film region, l whirl ratio (vp/v)

steady state local film pressure in the ls bearing speed parameter (6mUR/psC2)film region (above ambient) cx, cy perturbed angular rotations of journal axis

p, p0about x and y axes, respectivelydimensionless fluid pressuresS p

ps

,p0

psD g, d amplitude of perturbed angular rotation of

journal axis aboutX and Y axes,ps supply pressure (above ambient)respectivelyp1, p2 perturbed pressures in the film region

g2 dimensionless conical stability parameterp1, p2 dimensionless perturbed pressuresp9 fluid pressure in porous matrix (above S8I tC3v2

mUR2L3Dambient)p9 dimensionless fluid pressure in porous

matrix

occur before cylindrical whirl1,2. For a single bearingsystem, a rigid rotor in a single rigidly mounted bearingconical whirl is common particularly when transversemoment of inertia is high. Hirofumi3 observed that theunstable region of the conical mode began at muchlower speed than did the cylindrical mode for a gyro-scope consisting of hydrodynamic grooved journalbearing. Marsh4 obtained the stability and gave equa-tions of conical motion with and without gyroscopiceffect using a linearized theory. Rao and Majumdar5

analyzed theoretically and calculated the dynamic tiltstiffness and damping coefficients of an externallypressurized porous gas journal bearings. A theoreticalstability analysis of the conical whirl mode for asymmetric rotating shaft model has been presented byYoshihiro et al.6 The study of stiffness and dampingcoefficients of a symmetric rotor bearing system, inconical vibrational mode has also been reported inliterature7. Guha8,9 has presented a theoretical analysison the conical whirl instability of an unloaded rigidrotor supported in porous oil journal bearings withtangential velocity slip on the bearing film interface.However, the effect of turbulence on conical whirlinstability is not studied by the previous investigators.

236 Tribology International Volume 31 Number 5 1998

The analysis were mainly confined to laminar flowregimes.

The effect of misalignment on the performance ofturbulent journal bearings has been presented by Safaret al.10,11 Baskharone and Hensel12, Simon and Frene13

and Childs14 studied the dynamic force and momentresponse for turbulent flow annular pressure seals.Eusepiet al.15 have shown experimentally that journalmisalignment affects considerably the bearing perform-ance in turbulent flow regime. Andres16 has presentedan analysis for calculation of the dynamic force andmoment response in turbulent flow orifice compensatedmisaligned hydrostatic journal bearings.

Anjani and Rao17 have compared the results (loadcapacity) obtained by existing turbulent lubricationtheories with experimental data and observed that Con-stantinescu’s turbulent lubrication theory is in goodagreement with the experimental results. The effect ofturbulence, eccentricity ratio and slenderness ratio onload carrying capacity and friction coefficient of plaincylindrical turbulent journal bearings have also beeninvestigated. Anjani and Rao18,19 have further investi-gated the stability of a rigid rotor supported symmetri-


cally in two equal porous journal bearings in cylindricalwhirl. Conical whirl instability of an unloaded rigidrotor supported in a rigidly mounted single hydrody-namic porous journal bearing has also been reported20

and the effect of various parameters on the stabilityof the rotor-bearing system has been investigated.

In the present analysis the conical whirl instability ofa rigid rotor supported in a single hybrid bearing,operating in turbulent flow regime is presented.

Theory and computational work

Governing equations

Fig 1 shows schematically a porous hybrid bearingwith the coordinate system used in the analysis. Thejournal rotates with a steady rotational speedv aboutits axis, and undergoes whirl in a conical mode witha frequencyvp about its mean steady state position. Itis assumed that the mean steady state position of theunloaded journal is concentric. The porous material ofthe bush is assumed to be homogeneous but aniso-tropic. The flow through the porous matrix is assumedto be viscous, laminar and governing equation of flowis obtained by Darcy’s law. The lubricant film in theclearance space of the bearing is turbulent. With theseassumptions, the generalized differential equations forporous bearings incorporating the anisotropy of per-meability can be written as follows:

Kx

∂2p9

∂x2 + Ky

∂2p9

∂y2 + Kz

∂2p9

∂z2 = 0 (1)

for porous matrix and

∂∂x Fh3Gu

m

∂p∂xG +

∂∂z Fh3Gz

m

∂p∂zG =

U2

∂h∂x

+∂h∂t

+Ky

m F∂p9

∂yGy = 0

(2)

in the film region, whereGu and Gz are turbulencecoefficients. Normalizing, by putting

p9 =p9

ps, p =

pps

, y =yH

, z =2zL

, h =hc

Kx =Kx

Ky

, Kz =Kz

Ky

, u =xR

, T = vpt, l =vp

v

the above equations can be reduced to a nondimen-sional form as given below:

Kx

∂2p9

∂u2 + (R/H)2∂2p9

∂y2 + (D/L)2Kz

∂2p9

∂z2 = 0 (3)

∂∂u Fh3Gu

∂p∂uG + (D/L)2

∂∂z Fh3Gz

∂p∂zG =

ls

12∂h∂u

+lls

6∂h∂T

+b

12 S∂p9

∂yDy = 0

(4)

where

b =12KyR2

C3H, ls =

6mURpsC2 andD = 2R


Constantinescu21 indicated the following values ofGu

and Gz

1/Gu = 12 + 0.0260(Re*)0.8265

1/Gz = 12 + 0.0198(Re*)0.741 (5)

The journal axis performs periodic motions around itssteady state concentric position. These motions arepure rotation about its axisOX and OY, with ampli-tudes Re(geiT ) and Re(deiT ), respectively. In Fig 1these periodic rotational motions are represented bycX

and cY. For a first order perturbation, which is gener-ally valid for small amplitude oscillations, the perturbedequations for pressures in a porous medium and bearingclearance and the local film thickness can beexpressed as

p9 = p90 + S L2CDgeiTp91 + S L

2CDdeiTp92

p = p0 + S L2CDgeiTp1 + S L

2CDdeiTp2 (6)

h = 1 + SzL

2CDreiT cosu + SzL

2CDdeiT sin u

where

p9 = p9(u,y,z,T) p = p(u,z,T)

p91 = p91(u,y,z) p1 = p1(u,z)

p92 = p92(u,y,z) p2 = p2(u,z)

Substituting Equation (6) into Equations (3) and (4)and collecting only the first order terms forg and d,the following equations result. For porous matrix:

(R/H)2∂2p90∂y2 + (D/L)2Kz

∂2p90∂z2 = 0 (7)

Kx

∂2p91∂u2 + (R/H)2

∂2p91∂y2 + (D/L)2Kz

∂2p91∂z2 = 0 (8)

Kx

∂2p92∂u2 + (R/H)2

∂2p92∂y2 + (D/L)2Kz

∂2p92∂z2 = 0 (9)

For film region:

Gz(D/L)2∂2p2

0

∂z2 = bS∂p90∂y Dy = 0

(10)

Gu

∂2p91∂u2 + (D/L)2Gz

∂2p1

∂z2 + (D/L)2L

2CGz cosu

∂p0

∂z

+ (D/L)2L

2CGzzcosu

∂2p0

∂z2 = −ls

12 SzL

2CD sin u

+ iFlls

6 SzL

2CD cosuG +b

12 S∂p91∂y Dy = 0

(11)

Gu

∂2p2

∂u2 + (D/L)2 Gz

∂2p2

∂z2 + (D/L)2L

2CGz sin u

∂p0

∂z

+ (D/L)2L

2CGzz sin u

∂2p0

∂z2 =ls

12SzL

2CD cosu


Fig. 1 Bearing geometry, coordinates and rotation angles of journal

+ iFlls

6 SzL

2CD sin uG +b

12S∂p29

∂y Dy = 0

(12)

Boundary conditions

The boundary conditions are as follows:

For porous matrix

p90(−1,z) = 1 at −1 # z # 1 (supply)

p91(u,−1,z) = p92(u,−1,z) = 0


at 0 # u # 2p and−1 # z # 1

p90(y,±1) = p91(u,y,±1) = p92(u,y,±1) = 0

at 0 # u # 2p and−1 # y # 0 (ambient)

∂p90∂z

(y,0) = 0 at − 1 # y # 0 (symmetry)

p91(u,y,0) = p92(u,y,0) = 0

at 0 # u # 2p and−1 # y # 0 (antisymmetry)


p91(u,y,z) = p91(u + 2p,y,z)

p92(u,y,z) = p92(u + 2p,y,z)

For film region:

p0(±1) = p1(u,±1) = p2(u,±1) = 0

at 0 # u # 2p (ambient)

∂p0

∂z(0) = (0) (symmetry)

p1(u,0) = p2(u,0) = 0

at 0 # u # 2p (antisymmetry)

p1(u,z) = p1(u + 2p,z)

p2(u,z) = p2(u + 2p,z)

For film-bearing interface:

p90(0,z) = p0(z)

p91(u,0,z) = p1(u,z)

p92(u,0,z) = p2(u,z)

Simultaneous solution of Equations (7)–(12), satisfyingthe above boundary conditions, gives the steady stateand dynamic pressures in the porous matrix and in thefilm region. The solution has been obtained throughnumerical calculations using finite difference methodwith overrelaxation scheme.

To compute the pressures numerically, the number ofdivisions along u, y, z are taken as 44, 5 and 7respectively. Initial pressure distribution, is taken to bezero at all mesh points and the iteration is started.After each iteration the computed grid pressure aremodified by over-relaxation factor. To bring the iter-ation to an end the following convergence limit isused, and the pressure distribution is obtained.

| Spn + 1m − Spn

m

Spn + 1m | # 0.001

m = 1, 2

where n is the number of iteration. In the presentwork, an over-relaxation factor of 1.3 has been takenfor quick convergence.

Stiffness and damping characteristics

It can be shown that the four components each ofstiffness and the damping coefficients can be computedfrom the following expressions by numerical inte-gration using Simpson’s 1/3rd rule.

SXX = − ReF2E1

0

E2p

0

p1z cosu du dzGSYX = − ReF2E

1

0

E2p

0

p1z sin u du dzGSXY = − ReF2E

1

0

E2p

0

p2z cosu du dzGTribology International Volume 31 Number 5 1998 239

SYY = − ReF2E1

0

E2p

0

p2z sin u du dzGDXX = − ImF2E

1

0

E2p

0

p1z cosu du dzG/l

DYX = − ImF2E1

0

E2p

0

p1z sin u du dzG/l

DXY = − ImF2E1

0

E2p

0

p2z cosu du dzG/l

DYY = − ImF2E1

0

E2p

0

p2z sin u du dzG/l (13)

For oil bearings the value ofl will not affect thedynamic coefficients. Hence,l = 1.0 has been usedfor calculating the coefficients.

The journal bearing system is rotationally symmetricabout thez9 axis since the steady state position of theunloaded journal is concentric. So the followingrelations should exist between response coefficients7:

SXX = SYY DXX = DYY

SXY = − SYX DXY = − DYX (14)

Equations of motion

Referring to Fig 1(d), the equations of motion of thejournal for small harmonic rotational disturbances aregiven by Refs 22 and 23:

I tcX + IpcYv + SXXcX

+ DXXcX + SXYcY + DXYcY = 0 (15)

I tcY − IpcXv + SYYcY

+ DYYcY + SYXcX + DYXcX = 0 (16)

Being harmonic rotations,cX and cY are representedby:

cX = geiT cY = deiT

The two equations of motion can be expressed innondimensional form as follows:

F ( − l2g2 + SXX + ilDXX)(ilJg2 + SXY + ilDXY

( − ilJg2 + SYX + ilDYX)( − l2g2 + SYY + ilDYYG

Fg

dGeiT = 0 (17)

Stability

At the threshold, Equation (17) must allow a non-zerosolution forg andd. To test the stability of the motion,the determinant of Equation (17) is set to zero, giving


two equations. Substitution of Equation (14) into thesetwo equations gives

l4g22 − l2g2(2SXX − 2JDYX + J2g2) − l2(D2

XX + D2YX)

+ (S2XX + S2

YX) = 0 (18)

g2 = FDXXSXX + DYXSYX

l2DXX + JSYXG (19)

These two equations are solved simultaneously to findthe threshold value ofg2 and l for different values ofJ. The rotor is just stable for these values ofg2 andl. For a given rotor-bearing system, if the numericalvalue of g2 is higher than the above predicted value,the system will become unstable. The stability of thesystem can be studied by the conical stability par-ameterg2.

Method of solution

Putting J = 0 in Equation (18) and (19) gives

l2g2 =DXXSXX

+ DYXSYX

DXX

(20)

l4g22 − 2l2g2SXX − l2(D2

XX + D2YX)

+ (S2XX + S2

YX) = 0 (21)

Substitution of Equation (20) into Equation (21) gives

l2 = S SYX

DXXD2

Therefore,

l = − S SYX

DXXD (22)

On the right-hand side, negative sign is taken so as tomake l positive, because numerical computation givesnegative value ofSYX.

The value ofl and g2 can be obtained from Equations(20) and (22), as the values of stiffness and dampingcoefficients are known. The value ofl for non-gyro-scopic system (J = 0) is the initial set value for solvingEquation (18) and (19) simultaneously. A trial methodgives the stability parameter for different values ofJ.The solution by trial method is obtained as follows:

1. Substitute the value ofl for J = 0 and the valuesof stiffness and damping coefficients into Equation(19) to determineg2 for a particular value ofJ.

2. Substitute the values ofg2 and those of the stiff-ness and damping coefficients into Equation (18)and evaluate the left hand side which denotes theerror. If the error is not zero, choose a new valueof l and proceed until the error becomes zero.The error should be negative for a stable region24.

Results and discussion

To check the validity of results produced by the com-puter program, data for hydrodynamic plain journalbearings operating in turbulent flow regime were gener-ated and compared with previously available results.


The program could consider plain metal journal bearingcondition by lettingb = 0 and ls = 6.0 in the gen-eralized program. The plain metal journal bearingresults thus obtained are in excellent agreement withthe published work25.

The conical stability parameter (g2) depends on par-ameters such as bearing feeding parameter (b), theReynolds number (Re), the ratio of wall thickness tojournal radius (H/R) and anisotropy of permeability. Inthe present analysis a parametric study on conicalstability parameter has been made by varying all theparameters mentioned above.

Effect of b

Fig 3 shows the variation of conical stability parameterin turbulent lubrication conditions for different valuesof b.

It is observed that stability increases with an increasein bearing feeding parameterb in turbulent lubricationregime (Fig 3). Therefore, for better stability highervalues ofb should be preferred in conical whirl mode.

Effect of Re

Figs 4–6 show the variation of conical stability par-ameter for aspect ratios 0.5, 1.0 and 1.5 respectively fordifferent values of Reynolds numbers (Re up to 30000).

It is noticed that, for all aspect ratios, stability ofbearings in conical whirl increases with an increase inRe, other factors remaining same.

Effect of H/R

The computed results showing the effect ofH/R isshown in Fig 7. The figure shows that stabilityimproves with an increase inH/R ratio of bearings inconical whirl mode. An increasing trend of stabilitywith increasingH/R ratio, may be due to thicker wallproviding more damping to the system causing animprovement in conical stability.

Effect of anisotropy of the porous material

Normally, porous bearings are manufactured by thecompaction of metal powder in a die by the applicationof pressure along the axis of the die. So, the per-meability of bearings along the axial direction will beless than that in other directions. To study the effectof anisotropy, a computation was carried out by takingKx = 1.0 and Kz = 0.8, and the results are shown inFig 8.

It is clear from Fig 8 that a higher stability is predictedif the porous bush is considered to be isotropic. For aparticular value of Re, increase in stability limit fromanisotropic to isotropic condition remains almost samefor all values of b. Finally it can be concluded thatfor correct analysis of bearings in conical whirl mode,anisotropy of permeability of the porous bush shouldbe taken into account.


Effect of J

The conical stability parameterg2 increases with anincrease in moment of inertia ratio of journalJ (Figs2–8).

Conclusions

The following conclusions may be drawn from theforegoing analysis and discussion.

1. For better stability of turbulent hybrid bearings, inconical whirl mode, the value of bearing feedingparameterb should be kept higher.

2. The conical stability parameter of turbulent hybridporous bearings is enhanced with increasing Reyn-olds number Re.

3. The higherH/R ratio of bearings gives better stab-ility in conical whirl.

4. The effect of anisotropy of the porous bush is todecrease the stability parameter in conical whirl.Therefore, anisotropy of permeability of the bushshould be taken into consideration in the analysisof porous hybrid journal bearings particularly inconical whirl.

Fig. 2 Variation of conical stability parameter withaspect ratio for different values of moment of inertiaratio, J


Fig. 3 Variation of conical stability parameter withmoment of inertia ratio for different values ofb

Fig. 4 Variation of conical stability parameter withmoment of inertia ratio for different values of Re




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Fig. 7 Variation of conical stability parameter withmoment of inertia ratio for different values ofH/R

Fig. 8 Variation of conical stability parameter withmoment of inertia ratio for anisotropy of permeability

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Conical whirl instability of turbulent flow hybrid porous journal bearings