Indian 10urnal of Engineering & M ateri als Sc iences Vol. 10. April 2003. pp. 13 1- 137

Conical whirl instability of hybrid porous journal bearings in laminar lubrication regime

Anja ni Kumar"* & Satinder Pal Singhh

"Department or Production Eng ineering & Management. bDepartment of Mechan ica l Engineering, Regiona l Institute of Technology. l amshedpur 83 101 4. India

Received 22 l alll/arr 2002; arcepled 13 l alll/ar\, 2003

An analysi s of conica l wh irl instability of an unloaded ri gid rotor supported in hybrid porous oil journal bearings in

laminar lubrication regime is presented here. The effect of bearing feed ing parameter (/3) . aspect ratio (UD), rati o of wa ll thi ckness to journal radius (HIR) and an isotropy of porous material on stabi l i ty of rotor bearing system has been investi ­

gated. Higher va lues of /3 have been found to give better stability in con ical whirl mode and higher stability is predicted if porous bush is considered to be isotropic.

Porous journ al bearings have been widely used in in­dustry for a long ti me. Porous metal bushes are mostly used for self-act ing oil bearings and ex ternall y pressuri zed gas bearings . Hybrid porous o il bearings may be preferred to conventi onal multi-recess capil ­lary or orifice-compensated hydrostatic bearings, to avoid hi gh cost and compli cated design . But, the pu­rity of the lubricant has to be mai ntai ned for longer service of the porous bushes, because the pores of the bushes may become chocked by impuriti es in the lu­bricant. However, the porous bushes may be rep laced easily at regu lar intervals fo r better performance.

In rotor bearing system, two modes of whirl insta­bility may occur namely translatory and conical. Cy­lindrica l whirl in translatory mode occurs when a ri gid rotor is loaded symmetrically in two bearings. Coni cal whirl occurs due to non-symmetric load ing. If the transverse moment of inert ia is hi gh for two closely spaced bearings , then con ical whirl onset speed can occur before cy lindri cal whirl 1.2. For a sin­gle bearing system, a ri gid rotor in a single rigidly mounted bearing coni ca l whirl is common particularly when trans verse moment of inert ia is high . Hi rofum i3

observed th at the unstable reg ion of the conical mode began at much lower speed than did the cyl indri ca l mode for a gyroscope consisting of hydrodynamic grooved journal bearing. Marsh~ obtained the stability and gave equations or conical motion with and with­ou t gyroscopic effect using a linearized theory. Rao

* Author for correspondence : E-mai l anLkumar2001 @yahoo.co.in

and Maj umdar5 analyzed theoretically and calculated the dynamic tilt sti ffness and damp ing coefficients of externall y pressurized porous gas journal bearings. A theoretical stability analys is of conical whirl mode for sy mmetric rotating shaft model has been presented by Yoshihiro el o /.C>. The study of sti ffness and damping coefficients of a sy mmetri c rotor bearing system, in coni cal vibrational mode has also been reported in literature7

. Guha8.<J has presented a theoret ical analysis

on the con ical whirl in stability of an unloaded rigid rotor supported in porous oi l journal bearings with tangenti al velocity slip on the bearing film interface.

In the present analysis the co nical whirl instabili ty of a ri gid roto r supported in a si ngle hybrid bearing, operati ng in laminar lu brication regime is presented.

Theory and Computational Work

Governing eqllolions-·Fig. I shows schematical ly a porous hybri d bearing with the coordinate system used in the analys is. The journal rotates with a steady rotational speed w about its ax is, and undergoes whirl in a conical mode with a freq uency wp about its mean steady state posit ion. It is assu med that the mean steady state posi ti on of the unloaded journal is con­centri c. The porous material of the bush is assumed to be homogeneous but anisotropic. The tlow through the porous matrix is obtained by Darcy's law. The lubricant film in the clearance space or the bearing too is lami nar. With these assumptions, the generalized difrerential equations for porous bearings incorporat­ing the anisotropy of permeability can be written as :

132 INDIAN J _ ENG _ MATER _ SCL. APRIL 2003

- ~- - - . ~. I I ""~1~~t

:. . ."~"f-' . . ~ ~f r Y 0

(a) Y -- z PLA NF

<0) JOURNAL AT THE PLANE A - 8

( b ) X-Z P LANE Cd)

Fig. I - Beari ng geoJl1etry. coordinates and ro tation angles of journal

K O:' P' + K a2p'+K a2

p' = O ' :-12 Y a 2 La2 ux y Z

... ( I )

for porous matrix and,

O. [ h' ap 1 + a [/1 ' a/: 1 = 6U ah + 12ah a,\ /' a,\ aZ. /' a.:. ax af

12 K [OP' ] + I' Y ay F ()

... (2)

in the fi lm region. Normali zing, by putting:

- , p' p =-,

p,

- p p= - ,

p,

- y v=- , . H

2z. .: =-

L

" = " C

K - K K = x K = Z

x K ' I. K y y

The above equalions can be reduced to a non­dimensional form given as:

a2 p' 2 a2 p' 2 a2 p' K , 0 +(RIH ) 0 +(DIL) K z . , = 0 ... (3)

ae- ay- aZ.-

~[h ' Of5] + (DIL) 2 ~[h' Of5] = A. ali ao ao Oz Oz ' af)

+ lA), . aIL + plap' ] , aT a I ) F O

... (4)

where,

12 K)' R2 6 !l U R fj = A = and D = 2R I" C' H " C2

Ps

The journal axis performs peri od ic mot ions around its steady state concentri c positi on. These motions are pure rotati onal about its axis OX and OY, with am-

T 'T plitudes Re (y e' ) and Re (8 e' ), respec ti ve ly. In Fig. I these peri odic rota ti onal motions arc repre­

sented by VI, and !fly. For a first order perturbation, which is generall y valid for small ampl itude osci ll a­tions, the perturbed equat ions fo r pressures in a po­rous medium and bearing clearance and the loca l film thickness can be expressed as:

-, -, ( L ) iT - , ( L) 5: ,T - , P=Po+ - ye p,+ - (J e p , 2C 2C -

P = Po + ye' PI + (5 e' p ) ( L) 'T ( L) -T

2C 2C-

( L) 'T ( L ) -T h = l+ Z. 2C ye' cosf)+ z 2C (5e' sin O

where,

p' = p'((-), y, z, T)

P'I = P' I (e, y , z)

P' 2 = P'2 (0, y, z)

p = P (0, z, T)

PI = PI (0, ::: )

P2 = P2 (e, z )

... (5)

... (6)

Substituting Eq. (5) into Eqs (3) and (4) and co l­lec ting only the first order terms for yand 8, we get the following equat ions. For porous matrix :

a2, ( )2 a2

, (RI H )2 ,P o + D K. P o = 0 ay 2 L Z az '2

... (7)

. . . (8)

... (9) For film region:

.. . ( 10)

KUMAR & SINGH: CON ICAL WHIRL INSTAB ILITY OF JOURNAL BEA RI NGS 133

a" 17'1 + (DIL)" (J" I~I + (DIL)2 L cos fJ apo a() ~ az - 2C az

+ (DIL) - l COSe ~ =- Je. z slI1 e o L a ~ po ( LJ' 2C az" s 2C

+ { w, [ z 2~ )cos 0 1 + p [t' L, . .. ( II )

a2 a2 L a 1:2 + ( DIL)~ ':2 + (DIL) " sin e Po a(-) - a;:: - 2C az

+ (DILt - ? sIn -- = k z - cos e o L - 0 a" PII ( _ L ) 2C a7" s 2C

.[ ( LJ ' 1 [ap' o] + I 2V,s l z. 2C sIn (j + fJ ay- y=o . .. ( 12)

BOlllldary cOllditio lls - The boundary conditions fo r porous malri x are:

p 'o (- I, z) = I at - I $ z $ 1 (suppl y)

P'I (e, - I ,z) = p'2 (e, - I , z)=O

at 0 $ B $ 211: and -1 $ z $ 1

p'lJ (Y, ± I ) = P'I (O, y, ± 1) = p'" (e ,y, ± I ) = 0 at 0$(;)$211: and -1 $ y$ 0 (ambient)

ap'lJ (Y, 0) = 0 at - I $ Y $ 0 (symmetry) oz.

13'1 ((),y,0)=p'2 (e, y, 0)=0 at 0 $ 0 $ 211: and - I $ Y $ 0 (anti sy mmetry)

P'I (B , y,z) = P' I (61+2 11: , y, z)

p'~ ({], y , z) = p' ~ (61+211:, y, z)

For film reg ion:

Po (± 1)= PI (O.± I )= p" (O.± 1)= 0

at 0 -s: ()::::; 211: (ambient)

apo (0) = 0 (sy mmetry)

a::: PI ((1,0) = P2 (0, 0)=0

at 0 $ 0$211: (anti-sy mmetry)

PI (0, z) = P! (0 + 2n, :::)

132 (e, z ) = 132 (() + 2n, z )

For fi 1m-bearing interface:

p'o (0, z) = Po (z.)

P'I (0, o. z) = PI (61 ,:::)

P' 2 ((-), 0, z) = P2 (61, <:)

Simultaneous solution of Eqs (7)-( 12) sati sfyin g the above boundary conditi ons gives the steady state and dynamic pressures in the porous matrix and in the film region. The so lution has been obtained through numeri cal ca lcul ati ons using finite difference method with over relaxation scheme.

Stiffness alld damping characteristics - It can be shown that the four components each of stiffness and the damping coefficients can be computed from the fo llowing expressions by numeri ca l in tegration usi ng Simpson ' s 113 rd rule:

S xx = - Re[2 f 2r PI z cow de d Z ] o 0

Syx =-Re[2 f ()

"f PI z si ne de d l ] ()

2n ] J P~ z coif) de d z ()

SXy =- Re[2 f ()

S yy = - Re[2 f "f P2 z sine de d z o u

Dxx =-/ m [2f 2r PI z cow de d z u ()

DyX = - 1m [2 f "f PI z sine de d z () 0

[

I 2rr

. Dxy = - 1m 2 J J P 2 Z cow de d z () ()

] ]u ]u ]n

[

I 2n ] Dyy = - / m 2J J P 2 z sine de d z U

() ()

... ( 13)

For oil bearings, the va lue of A. will not affect the dynamic coeffic ients. Hence, A. = 1.0 has been used for calculati ng the coefficient.

The journal bearing system is rotationally sy mmet­ric about the z' axis since the steady state position of the unloaded journal is concentric. So the follo wing re lations should ex ist between response coefficients 7 :

Sxx = Syy; Dxx = Dyy Sxy = -SyX; Dxy = -Dyx ... ( 14)

Equations o/lIIotion - Referring to Fig. I d. the eq uations of motion of the journal for small harmonic

. I d' b . b 10 II rotatlOna Istur ances are gIven y . :

... ( 15)

134 INDIAN J . ENG. MATER. SCI. , APRIL 2003

+Syxl// x + Dyxl/l x = 0 .. . ( 16)

Being harmonic rotati ons, If/x and If/y are repre­sented as:

'1' ' iT Vlx = ye : If/y = () e

The two eq uati ons of motion can be ex pressed In non-dimensional from as:

[

( - ),\ + ,5xx + i)~Dxx ) (iV Yc + SXY + i),Dxy ) 1 ( -iVyc + SyX + i},Dyx ) (-), ~ YJ + Syy + i),Dyy )

[Yj / ,. = 0 ()

... ( 17)

Slabilitv -- At the threshold , Eq. ( 17) must allow a non-zero solution for yand 8. To test the stability of the motion , the determinant of Eq. ( 17) is set to zero, giving two equations. Substitution of Eq. ( 14) into these two equati ons gives:

, , 4 2 12 2:; - -

,{ Y~ - I , y~ (2Sxx - 2JD yX +J Y2 )-2 (Dxx + Dyx ) , ,

+ (S;x + S;x ) = 0 ( 18)

__ [ Dxx Sxx + DyX S yX ; y~ ~~,~~--~~~

- A- Dxx + JSyX ( 19)

These two equati ons are so lved simultaneously to find the threshold va lue of Y2 and A for different val­ues of 1. The rotor is just stabl e for these values of Y2 and X For a given rotor-bearing sys tem, if the nu­merical va lue of Y2 is hi gher than the above predi cted va lue, the system wi ll become unstable. The stability of the system can be studi ed by the coni cal stability parameter Y2.

Method of solution -- Putting J=O in Eqs ( 18) and ( 19) gives:

1~ Dxx Sxx + DyXSyX /~ Y~ = .. . (20)

- Dxx ~.:J :;:; :; 2 ~ :' =' /, Y~ - 2), Y2 SXX - 2 (Dxx + Dyx)+(Sxx +Syx)= O

(2 1 ) Substitution of Eq. (20) into Eq. (2 1) gives:

), 2 =( SYX l~ Dxx

Therefore, ), = - [ S yX 1 Dxx

... (22)

On the ri ght-hand side, negative sign is taken so as to make I\. positive, because numeri cal computation

gives negative value of SYX .

The va lue A and Y2 can be obtained from Eqs (20) and (2 1), as the va lues of stiffness and damping coef­ficients are known . The va lue of A for non-gyroscopic system (J=O) is the initial set value fo r solving Eqs ( 18) and ( 19) simultaneously. A trial method gives the stability parameter fo r different va lues of 1.

The solution by trial method is obtain ed as: (a) Substitute the va lue of A for J=O and the va lues of sti ffness and dampi ng coefficients into Eq. ( 19) to determine Yz fo r a particu lar va lue of J: and, (b) Sub­stitute the values of Y~ and those of the sti ffness and damping coefficients into Eq . ( 18) and eva luate the left hand side whi ch denotes the error. If the error is not zero, choose a new value of A and proceed until the error becomes zero. The erro r shoul d be negative for a stab le region '2 . ~

VI III OJ

<: o III <:

'" E '0

....-2 OJ

E o .... o 0.

1: .LJ

.2 III

o u 'c o

U

1000. 0 ~--------------------------------,

100.0

10. 0

- - - - - Guha results ( Ref. 9 )

--- Present analys',s

P =0.03 As =1. 0 H/R=0.2 Kx =1. 0 Kz = 1.0

0.03 L-____ -L ______ L-____ -'--______ '--__ ---,--'

0.0 0.1 0.2 0.3 0.4 0.49 Moment of inertio rat io of journal,]

Fig. 2 - Vari at ion o f con ical stabi li ty parameter with momenl o f inerti a ratio fo r differen t va lues of length 10 diameter rati o

KUMAR & SINGH: CONICAL WHIRL INSTABILITY OF JOURNAL BEARINGS 135

Resu lts and Discussion Fig. 2 shows a comparison of the results obtained

by present analysis with the no slip resu lts of Guha'!, whi ch gives a fairly good agreement and it checks the val idity of the results produced by the computer pro­gram.

The coni cal stability parameter (Y2) depends on pa­rameters such as bearing feeding parameter (fJ), the rati o of wa ll thickness to journal radius (HIR), aspect ratio (UD) and ani sotropy of permeability. In the pre­sent analysis, a parametric study on conical stability parameter has been made by vary ing the parameters mentioned above.

Eflect of f3 - Fig. 3 shows the variati on of conical stability parameter for different va lues of f3. It is ob­served that stability increases with an increase in bearing feeding parameter f3 (Fig. 3). Therefore for better stability , hi gher values of f3 should be preferred in conical whirl mode.

10000.0.--------------~--71

Vl Vl

~ C 0

VI C

'" E 100 .0 ;::?

N '>- 0.3

'-

'" -'" E 0 '-0 a. >.

D 0

-:n 0

. ~ c 0

U

0.03

o .2L-__ -'--__ ---' ___ ..L-__ --:-'-:--_::-'

0·0 0 .1 0·2 0·3 O·l. 0.49 Moment of inertia rati o of jou rnal,)

Fig. 3 - Variati on of conica l stabil ity parameter with moment of inert ia rati o for di fferent va lues of bearing feeding parameter

Effect of UD - It is shown that stabi lity of bear­ings in conical whirl increases with an increase in UD ratio of bearings (Fig. 4).

Effect of HIR - The computed results show ing the effect of HIR are shown in Fig. 5. The figure shows that stability improves with an increase in HIR ratio of bearings in coni cal whirl mode. An increas ing trend of stability with increas ing HIR ratio may be due to thicker wall providing more damping to the system causing an improvement in conical stability .

Effect of allisotropy of the porous /I /Oterial -Normally, porous bearings are manufactu red by the compaction of metal powder in a die by the app li ca­tion of pressure along the ax is of the die. So, the per­meability of bearings along the axial direction will be less than that in other directions. To study the effect of anisotropy, a computation was carried out by tak­ing Kx=I.0 and Kz=O.8 and the results are shown in Fig. 6.

Vl VI

~ C o VI c

'" E "0

'-.2:! '" f ~ o a.

o v 'c o

U

1000 .0,------------- - -----n

100.0

10.0

P =0.03 71 5 =1. 0 H/R =0.2 Kx =1. 0 Kz =1 .0

o .O)L..-__ ~--__,_L,---~--__:_l_;_--,--J 0.0 0.1 0.2 0.) 0.4 0.49

Moment of inert ia ra tio of journal,)

Fig. 4 - Variation of conical stability parameter w ith momen t of inertia rati o for different values of length to diameter ratio

136 INDIAN 1. ENG. MATER. SCI.. APRIL 2003

10000.0.--------------- --7TT1

u; '" ., c 0 'iii c ., E

::§ ,>-N

~

~ ., E ~ 0 a.

.?: -:a 0

Vi

-0 u 'c 0 0

1000.0

10.0

---- (3 :0.03 - --- --- (3 -0.3

L ID = 1.0

~s = 1.0 Kx "1. 0

Kz '" 1.0

II I I

11/ I II I I

II / I I

/ I / I I I /

I I

/ I

/

0.2L-. __ -L _ _ .......,.J':-__ -=7 __ --;;'-;--_~. 0.0 0.1 0.2 0.3 0.1. 0.1.9

Moment of iner t ia rat io of journal, J

Pig. 5 - Variat ion of conical stability parameler with moment of inertia ra lio for different values of HIR ratio

It is c lear from Fig. 6 that a higher stabi lity is pre­dicted if the porous bush is considered to be isotropic.

Finally , it can be concluded that for correct analy­sis of bearings in conical whirl mode, anisotropy of permeability of the porous bush should be taken into account.

Effect of J - The conical stability parameter Y2 in­creases with an increase in moment of inert ia ratio of journal J (Figs 3-6).

Conclusions For better stability of hybrid bearings in conical

whi rl mode, the value of bearing feeding parameter f3 shou ld be kept higher. The hi gher HIR ratio of bear­ings g ives better stab ility in conical whirl. The porous hybrid bearings with higher aspect ratios are more stabl e in con ica l whirl. The effect of anisotropy of the porous bush is to decrease the s tability parameter in coni ca l whirl. Therefore, anisotropy of permeability

10000 .0..------------------,

'" '" ~ c o 'Vi c ., E

::§ N .,..

.0

2 on

c u 'c o o

1000·0

- - - - - Kx =1.0, Kz = 0.8 (Anisotropy)

--- Kx =1.0 1 Kz = 1.0 (Isotropy)

L /O=1.0

As = 1. 0

H/R=0.2

/' /

/ I

I /

I /

/ /

O. 2 L,----:L,--~L,;-----"I:;----n'~-----;0~/9 o 0.1 0.2 0.3 O·/' ... Moment of inertio ratio of j ourna l, J

Fig. 6 - Variation of conical stabilit y parameter with moment of inerlia ratio for anisotropy of permeability

of the bush should be taken into consideration in the analysis of porous hybrid journal bearings particularly in conical whirl.

Nomenclature C mean radial clearance D shaft/journal diameter D iJ bearing damping coefficients (fi rst suffi x denotes

the direction of moment and the second denotes the directi on of angular velocity)

J

dimensionless bearing damping coefficients

[ ~I~?: ~~ 1 local film thi ckness dimensionless film thickness (hIC)

thi ckness of the wa ll of the porous bush polar moment of inertia of j ournal transverse moment of inerti a of j ournal (- I ) 1/2

inertia rati o (/,/1,)

KUMAR & SINGH: CONICAL WHIRL INSTABILITY OF JOURNAL BEARINGS 137

fS... KL K,

K ., K I. '

L UD p. Po

p, Po

p,

PI' P~

PI' P2

P

P

p' t ,p ' ~

P '1' p'~

I

T

U x.y . z

y,:

permeability coeftlc ients in x. y and z directions

KJ Ky, K/ Ky length of bearing aspect ratio loca l film pressure in the film region. steady state local film pressure in the fi lm region (above ambi­en t)

dimensionless Ouid pressures [p , Po J p, ps

supply pressure (above ambient ) perturbed pressures in the film region dimensionless perturbed pressures

Ouid pressure in porous matrix (above ambient ) dimension less Ouid pressure in porous matrix

perturbed Ouid pressure in porous matrix

dimension less perturbed Ou id pressures in porous

mat ri x shaft/journal radius beari ng spri ng st i ffn ess

dimension less bearing spring stiffness -- ~ [

8 C' S ] II R' L'

time dimensionless time (wpl)

shaft surface veloci ty (R w) circumferential, radial and axial coordinates dimension less rad ial and axia l coordin ates (y/H. 2 Z!L )

C reek Symbols e circu mfere nti al coordi nate 1/ absolute viscos ity of tluid

f3

y. <5

bearing feed ing parameter / [

12 K R2] C H

angu lar velocity of shaft. angu lar ve locity orjour­nal centre wh irl ratio (wr!w) bearing speed parameter (61/ URI Ps C~ ) perturbed angul ar rotati ons ofj ournul axis about x and y axes. respecti vely amplitude of perturbed angul ar rotati on of journal axis about X and Y axes, respecti vely conical stability parameter dimensionless

(8 / ,C' W2J

lpUR2 L'

References Sternlicht B & Winn L W, ASME 1 Basic Ellg. X5 (3) (1963) 503-512.

2 Sternlicht B & Winn L W. ASME 1 Basic Eng. 86 (2) (1964) 3 13-320.

3 Hirofumi M. ASME 1 Lllbr Th ee/wi. (October 1969) 609-619.

4 Marsh H, ASME 1 Lllbr Technol , (January 1969) 11 3-119. 5 Rao N S & Majumdar B C. Wear, 50 ( 1978) 20 1-2 10. 6 Yoshihiro T. Junkichi I. Hidey uki T & Atsuo S, Bli/l lSME.

25 (May 1982) 856-86 1. 7 Philips Research Reporl Supplelllenl.\·, Yo Is 1-7 ( 1972) 8-26. 8 Guha S K. Tribology Inl. 19 (2) ( 1986) 72-78. 9 GuhaS K.ASME1Tribol, 108( 1986)256-26 1.

10 Grassam N S & Powell J W, Cas Ilibricaled bearings (aut­ter-worths, London). 1964. 104- 109.

II Shames I H. Ellg illeerillg lIlechanics: Slalics and dVllalllics . 2n~ ed (Prentice-Hall of India Private Limited. New Delhi ). 197 1, 656-658.

12 Lund J W, ASME 1 Basic Eng, 89 ( 1967) 1- 12.