bcm brewe non linear stability

8/3/2019 BCM Brewe Non Linear Stability

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.-

NASATechnical Memorandum 102309

AVSCOMTechnical Report 87-C-26

Stability of a Rigid Rotor Supportedon Oil-Film Journal BearingsUnder Dynamic Load

B. C . MajumdarLewis Research Center

Cleveland, Ohio

and

D.E. BrewePropulsion DirectorateU .S. Army Aviation Research and Technology Activity-A VSCOM

Lewis Research CenterCleveland, Ohio

Prepared for theNational Seminar on BearingsMadras, India, September 17- 18, 1987

USARMAVlATlOSYSTEMS COMMANDAVIATION RLT ACTIVITY

~

{NASA-TN-102309) STAB IL ITY OF A R I G I D ROTOR N89-27114

SUPPORTED O N OIL-FILM JOURNAL BEARINGS UNOEB

DYIAl¶IIC LOAD (H A S& , Lewis Research C e n t e r )C S C L 20D Unclas

G3/34 0 2 2 4 1 1 613 P



STABILITY OF A R I G I D ROTOR SUPPORTED ON OIL-FILM

JOURNAL BEAR INGS UNDER DYNAM IC LOAD

B.C. Majumdar*N a t i o n a l A e ro n a u t i c s a nd Space A d m i n i s t r a t i o n

Lewis Research Center

C leve land , Oh io 44135

and

D.E. BreweP r o p u l s lo n D i r e c t o r a t e

Lewis Research CenterC leve land, Ohi o 44135

U .S . Army A v i a t i o n R es ea rc h a nd Te ch no lo gy A c t i v i t y - AVSCOM

SUMMARY

Most p u b l i s h e d work r e l a t i n g to d y n a m i c a l l y l oa d ed j o u r n a l b e a r i n g s a r ed i r e c t e d t o d e te rm in in g t h e m in lmu m f i l m t h i c k n e s s f rom t h e p r e d l c t e d j o u r n a lt r a j e c t o r i e s . These d o n o t g i v e any i n f o r m a t i o n a b o ut t h e su bs yn ch ro no us w h i r ls t a b i l i t y o f j o u r n a l b e a ri n g systems s i n c e t h e y d o n o t c o n s i d e r t h e e q u a t i o n so f mo t io n . I t i s , however , necessary to know whe the r th e b ea r i ng sys tem opera-t i o n i s s t a b l e o r n o t u n de r s uc h an O p e ra t i n g c o n d i t i o n .

\r

I

The purpose o f t h e p r e s e n t p ap er I s to a na ly ze t h e s t a b i l i t y c h a r a ct e r i s -t i c s o f the sys tem. A l i n e a r i z e d p e r t u r b a t i o n t h eo r y ab ou t t h e e q u i l i b r i u mp o i n t can p r e d i c t t h e t h r e s h o l d of s t a b i l i t y ; h o w e v e r i t d o e s n o t i n d i c a t ep o s t w h i r l o r b i t d e t a i l . The l i n e a r i z e d method may i n d i c a t e t h a t a b e a r i n g i su n s t a b l e f o r a g i v e n o p e r a t i n g c o n d i t i o n whereas t h e n o n l i n e a r a n a l y s i s mayi n d i c a t e t h a t I t forms a s t a b l e l i m i t c y c l e . For t h i s re a so n , a n o n l i n e a rt r a n s i e n t a n a l y s i s o f a r i g i d r o t o r s up po rt ed o n o i l j o u r n a l b e a r i n g s u nd er( 1 ) a u n i d i r e c t i o n a l c on s t an t lo ad , ( 2 ) a u n i d i r e c t i o n a l p e r i o d i c l oa d , and( 3 ) v a r i a b l e r o t a t i n g l o a d ar e p er fo rm ed .

I n t h i s p a pe r , t h e h yd ro dy na mi c f o r c e s a r e c a l c u l a t e d a f t e r s o l v i n g t h et ime -d ep e nd en t R e yn olds e q u a t i o n b y a f i n 1 e d i f f e re n c e me th od w i h a succes-s i v e o v e r r e l a x a t i o n scheme. U s i n g t h e se f o r c e s , e q u a t i o n s o f m o t i o n a r e s o l v e dby th e fo u r th -o rd e r Runge-Kut ta me thod t o p r e d i c t t he t r a n s i e n t b e ha v io r o f t h er o t o r . W i t h t h e a i d o f a h ig h- sp ee d d i g i t a l c om pu te r a nd g r a p h i c s , t h e j o u r n a lt r a j e c t o r i e s a r e o b t ai n e d f o r s e ve ra l d i f f e r e n t o p e r a t i n g c o n d i t i o n s .

I NTRODUCT ION

There are two p r i n c i p a l a pp ro ac he s b y means o f w h i c h on e c an a n a l y z e t h ew h i r l i n s t a b i l i t y o f a r o t o r s upp or te d on f l u i d f i l m b e a r i n g s .( 1 ) l i n e a r i z e d ( p e r t u r b a t i o n ) method and ( 2 ) n o n l i n e a r t r a n s i e n t a n a l y s i s . I n

These are :

* N a t i o n a l R e se a rc h C o u n c i l - NASA R e se a rc h A s s o c la te ; p re s e n t a d d re s s :I n d ia n I n s t i t u t e o f Technology, Dept . o f Mechan ica l Eng ine e r ing , Kharagpur -721302 , Ind ia .



t h e l i n e a r i z e d m eth od a s m a l l p e r t u r b a t i o n o f t h e j o u r n a l c e n t er ab ou t i t se q u i l i b r i u m p o i n t i s g iv e n.mined a f t e r s o l v i n g t h e b a s i c d i f f e r e n t i a l e q u a t io n . These c o e f f i c i e n t s a r et h e n u se d i n t h e e q u a t i o n s o f m ot i on to f i n d t h e c r i t i c a l mass pa ra m et er andw h i r l r a t i o . The mass pa r am et e r, a f un c t i on o f r o t o r speed, i s a measure ofs t a b i l i t y . The n o n l i n e a r t r a n s i e n t a n a l y s i s , on t h e o t h e r h and, g i v e s t h ej o u r n a l l o c u s a nd from t h i s on e c an know a b o u t t h e s ys te m s t a b i l i t y .

The s t i f f n e s s and dam pin g c o e f f i c i e n t s a r e d e t e r -

S t a b i l i t y a n a l y s i s o f f i n i t e j o u r n a l b e a r in g s by t h e l i n e a r i z e d method hasbeen g i ve n by A l l a i r e ( r e f . 11, whe reas Ake r s , M l chae l son , and Cameron ( r e f . 2)s t u d i e d t h e same b e a r i n g c o n f i g u r a t i o n s u s i n g a n o n l i n e a r t r a n s i e n t a p p ro ac h.I n r ef e re n ce 2 i t was shown t h a t u nd er c e r t a i n o p e r a t i n g c o n d i t i o n s t h e j o u r n a lmo t io n was bounded and co ul d form a l i m i t c y c l e .

The aim of t he p r esen t pape r i s to s tu d y t h e o r e t i c a l l y t he s t a b i l i t y c ha r-a c t e r i s t i c s o f f i n i t e o i l j o u r n a l b e a r i n g s u n d er dy na mic l o a d u s i n g a n o n l i n e a rt r a n s i e n t m e t h o d . A f e w p a p er s ( r e f s . 3 to 5) d e a l w i t h t h e d y n a m i c a l l y l o a d e dbea r i ngs t o p r e d i c t th e jo u r n a l l ocus . As t h es e d o n o t c o n s i d e r t h e e q u a t i o n so f m ot i on for t h e p r e d i c t i o n o f t he p o s i t i o n o f j o u r n a l c e n t e r , t h e y c a nn o t

i n d i c a t e w h et he r t h e b e a r i n g sy st em i s s t a b l e or n ot . However , thes e ar e use-f u l f o r es t i m a t i ng m i n i m um f i l m t h i c k n e s s of d y n a m i c a l l y l o a d e d b e a r i n g s . Thedynamica l equat i ons o f m ot i on a r e so l ved by f ou r t h - o r d e r Runge -Ku tt a m et hod t of i n d e c c e n t r i c i t y r a t i o and a t t i t u d e a ng le and t h e i r d e r i v a t i v e s for t h e n e x tt i m e s t e p . These values a r e t h e n i n t r o d u c e d i n t h e two-d imens iona l t ime-dependent Reynolds equat ion t o f i n d t h e h yd ro dy na mi c f o r c e s .a bo ve ap pr oa ch a n o n l i n e a r t r a n s i e n t a n a l y s i s o f a r i g i d rotor on o i l j o u r n a lb e a r i n g s u n d e r ( 1 ) a u n i d i r e c t i o n a l c o n st a n t l o a d , ( 2 ) a u n i d i r e c t i o n a l p e r i -o d i c l o a d, and ( 3 ) v a r i a b l e r o t a t i n g l o a d i s per fo rmed. A number o f t r a j e c t o -r i e s h av e be en o b t a i n e d w i t h t h e a i d o f a h igh -speed d i g i t a l com pu te r andg r a p h i c s .

F o l l o w in g t h e

C

D

NOMENC LATURE

r a d i a l c l e a r a n c e

j o u r n a l d i a m e t e r

e c c e n t r i c i t y

hydrodynamic fo rces

F r = FrC/,,RL, Fe = FeC2/q,R3L (dimensionless)

f i l m t h i c k n e s s , 6 = h / C ( d i m e n s i o n l e s s )

mass parameter , M = MCo2/Wo d i m ens i on l ess )

f i l m p r e s s u r e , = pC2/,,R3L

j o u r n a l r a d i u s

d i m ens i on l ess t i m e , T = w p t

-

-

2



t time

x , z, e, 2

w , ii'09 '0

E eccentricity ratio, E = e/C (dimensionless)

0 absolute viscosity of oil

coordinates, 8 = x/R, 2 = z/(L/2) (dimensionless)

load, W = WC2/qwR3L (dimensionless)

steady-state load, flo = WC2/qwR3L (dimensionless)-

'0 atti tude angle

angular coordinates at which film commences and cavitates, respec-tively, from minimum film thickness

91 92

819 82 81 = w + 91, 82 = w + 92

aw angular velocity of journal

whirl ratio, Q = wP/w (dimensionless)

angular velocity of whirl

THEORY

The basic differential equation for pressure dist ribution in the bearingclearance under dynamic condi tions can be written as (See fig. 1 . )

Equation I1 1 when nondlmensionalized with the following substitutions:

8 = x / R , z = z/(L/2), h = h/C, p = pC2/qwR2, T = wpt, and R = w ~ / w , will readas

The film thickness (dimensionless) i s

-h = 1 + E COS 8 ( 3 )

with the use of equation ( 3 1 , equation (2) can be written as

where (b = aT/aT and E = aE/aT. In this study we have used the Reynolds

boundary conditions, which are given by:

3



and

where 81 and 02 a r e t h e a n g u l a r c o o r d i n a t e s a t w hi ch t h e f i l m commences andre forms, r e s p e c t i v e l y .

E q u a t i o n ( 4 ) i s s ol ve d n u m e r i c a l ly f o r p re ss ur e b y a f i n i t e d i f f e r e n c em eth od w i t h a s u c c e s si v e o v e r r e l a x a t i o n s cheme s a t i s f y i n g t h e a b ov e b ou n d ar yc o n d i t i on s . I n i t i a l l y , E and (b aye se t equa l t o z e r o t o o b t a i n t h e s te ad y -s t a t e h y dr od yn am ic f o r c e s Fr and Fe.

The fo rces a re computed from

and (6)

where

- Fr c2 - F8C2

3nd Fe =-UR L

Fr =-OUR L

R e l e a s in g t h e j o u r n a l from t he s t ea dy - s t a t e po s i t i o n , one can com pu te'p, E , and (r, f o r t h e n e x t t i me s t e p by s o l v i n g t h e f o l l o w i n g e q u a t i o n s o fmot 1on :

E ,

MC[$-- ~($9~1Fr t W C O S 'p

The d i m ens i on l ess f o r m o f e q u a t i o n s ( 7 ) and (8) a r e :

cos 'p = 02 . 2 'r

M Q 2 E - RQ ET - -

wO

(7)

(9)

4



- 2.. 2 - F0

wO

MR E + 2flR EQ - F+ s i n Q = 0 (10)

where

- woc2and Wo =-M C u 23

= -wO qw R L

T h e s te a d y -s ta te l o a d ios o b ta in e d by l e t t i n g E and (b equa l t o z e r o .

'p. These a r e yo ly ed by us in g a fou r th -o r de r Runge-Ku t ta me thod f o r c o n s t a n tv a lu e s o f R , M , F r , a n d Fe.

Equa t ions ( 9 ) a nd ( 1 0 ) a r e s ec on d- or de r d i f f e r e n t i a l e q u at i o n s i n E an d

I n t he f o l l o w i n g s e c t i o n t h r e e t yp es o f l o a d a r e c o n s i d e r e d .

A U n i d i r e c t i o n a l C o n s t a n t L o a d

Assuming t h e i n i t i a l c o n d i t i o n s g iv e n I n t a b l e I , the hyd rodynamic fo rcesu n de r s t e a d y - s t a t e c o n d i t i o n a r e f o u nd .o b t a i n E , & , Q , and (b f o r t h e s u b s e q u e n t t ime s te p . Now the new va lue o fE , E , an d (9 a r e i n t r o d u c e d i n eq u at i on s ( 3 ) an d ( 4 ) to d e te rm in e t h e h y d ro -d yn am ic f o r c e s . T he se f o r c e s a l o n g w i t h t h e s te a d y -s ta te l o a d , mass p a ra mete rand w h i r l r a t i o a r e u t i l i z e d f o r t h e s o l u t i o n of e q u a t i o n s (9) an d ( 1 0 ) . Theprocess i s r ep ea te d u n t i l we g e t a t r a j e c t o r y t h a t d e sc r ib e s t h e s t a t u s o f t h esystem.

Equa t ions o f m o t i o n a r e s o l v ed to

A U n i d i r e c t i o n a l P e r i o d i c Load

T h e a p p l i e d l o a d was assumed t o be

- -W = W [ l + sin(: T)] ( 1 1 )

where Wo i s t h e a p p l i e d s te a dy -s ta te l o a d c g n s i s t e n t w i t h t h e e c c e n t r i c i t y ,= 0.8. A t each t i me s t ee a new va lue o f W w a s c a l c u la te d from equa-

-i o n ( 1 1 ) . Fo r v a lu e s o f M and R g iv en i n t a b l e I a n d u s in g W , F andFe, e q u a t i o n s (9) and (10) were so lved f o rs te p . The r e s t o f t h e p ro c e d u re i s r e p e a te d as d e s c r i b e d .

E , E , Q , an d (9 fo r t h e n e x t t i m e

V a r i a b l e R o t a t i n g L oad

T h i s t y p e o f l o ad l ng i s i n t e r e s t i n g s in c e I t p e r t a i n s t o an eng ine bear-The data used f o r t h e a na l y s is r e f e r to the Ruston and Hornsby 6 VEB-Xn g .

M K I I I c o n n e c t i n g r o d b e a r i n g ( r e f . 5) an d i s l i s t e d i n t a b l e I . The p o l a r l o a dd iag ram i s r e p re s e nt e d by f i g u r e 2 .p o s i t i o n s o f c ra n k a n g le t h e r e a d e r may r e f e r t o append ix 4 . 1 o f t h e p a p e r( r e f . 5). The t ime s tep A T = n / 1 8 was t a ke n i n t h i s ca se to m a t c h w i t h t h e

Fo r t h e ma g n i t u d e o f l oa d a t d i f f e r e n t

5



g i v e n a p p l ie d l od d a t l o o c r a n k a n g l e i n t e r v a l .n o n d i m e n s i o n a l i z e d u s i n g t h e e x p r e s s i o n W = WC2/,aR3L.d uc ed i n t h e e q u a t i o n s o f m o t i o n t o d e te r mi ne t h e p o s i t i o n o f t h e j o u r n a l ce n-t e r ( E , E , 9, nd 4). The hydrodynamic forces F r and Fe were, however,c om pute d a f t e r s o l v i n g e q u a t i o n ( 4 ) .

The r e s u l t a n t a p p l i e d l o a d wasT h i s l o a d was i n t r o -

RESULTS AND DISCUSSION

To a s su re o u r s e l v e s t h a t o u r f o r m u l a t i o n i s c o n s i s t e n t w i t h r ef er en ce 2 ,o u r r e s u l t s a r e s hown i n f i g u r e 3 ( a ) a nd co mp are d t o t hose o f r e f e r ence 2 showni n f i g u r e 3 ( b ) . The c om pa ri so n i n d i c a t e s v e r y go od a gr ee me nt for t h e c o n d i -t i o n s s t a t e d i n t h e f i g u r e . I n f i g u r e 4 (a ) a t y p i c a l th re e -d im e n si on a l p r es -s u r e d i s t r i b u t i o n f o r a p a r t i c u l a r t i me s t e p i s shown. From t h i s f i g u r e onecan s e e t h e c a v i t a t e d r e g i o n and t h e v a r i a t i o n o f p r e ssu r e th r oughou t t he com-p l e t e c l e a r a n c e space o f t h e b e a r i n g . F i g u r e 5 g i v e s t h e j o u r n a l l o c u s whent h e b e a r i n g i s s t a b l e u n de r th e a c t i o n o f c o n st a nt u n i d i r e c t i o n a l l o ad . I nf i g u r e 3, t h e j o u r n a l t r a j e c t o r y w i t h i n t h e c le a ra n c e c i r c l e i s shown for anu n s t a b l e b e a r i n g . The j o u r n a l l o c u s r e ac h e s a l i m i t c y c l e . T h i s t y p e o f

ob se rv a t i o n has been made by many workers ( r e f s . 6 t o 8 ) w h i l e d e a l i n g w i t hs h o r t and i n f i n i t e l y l o n g b e a ri n gs .

Hav i ng com par ed t he r esu l t o f t h e p re s e nt s o l u t i o n w i t h t h a t o f Akerse t a l . ( r e f . 2 1 , an attempt was made to s tu dy t h e e f f e c t o f p e r i o d i c a nd v a r i a -b l e l oa d . F i g u r e 6 shows t he j ou r n a l l ocus fo r a p e r i o d i c l o a d s u p e r i m p o s e d o nt he con s t an t l oa d g i v en t h e same as t hose o f a u n i d i r e c t i o n a l c o n s t a n t l o a d .From t hese two f i g u r e s i t may be seen t h a t a be ar in g wh ich i s s t a b l e u n d e r t h ea c t i o n of a u n i d i r e c t i o n a l c o n s t a n t l o a d ca n be made u n s t a b l e when a p e r i o d i cl o a d i s a p p l i e d . I n t h e l a t t e r case th e j o u r n a l l o c u s r ea ch es a l i m i t c y c l e .

T h e j o u r n a l c e n t e r t r a j e c t o r y o f t h e c o n n e c t in g r o d b e a r i n g ( v a r i a b l er o t a t i n g l oa d) i s shown i n f i g u r e 7 . T h e t r a j e c t o r y i s ve r y com p l ex : i t n e i -

t h e r t e n d s t o go t o an e q u i l i b r i u m p o i n t n o r t o r e a c h a l i m i t c y c l e .m e n t i o n e d t h a t t h e v a l u e o f the minimum f i l m t h i c k n e s s f o r t h i s case exceedst h a t o b t a i n e d b y o t h e r s u s i n g t h e m o b i l i t y m eth od o f s o l u t i o n . The m o b i l i t ym eth od ( r e f . 3 ) , h o we ve r, d oe s n o t c o n s i d e r t h e s t a b i l i t y o f t he sys t em .

The p r e s e n t s o l u t i o n d oe s n o t t a k e a c c o un t of t h e t r a n s p o r t o f f l u i dt h ro u g h t h e c a v i t a t e d r e g i o n and c on s e q u en t ly d oes n o t t a k e f u l l a c co u n t o f t h eo i l f i l m h i s t o r y . A subsequent an a l ys i s has been unde r t aken t h a t w i l l i n c l u d eth e o i l f i l m h i s t o r y e f f e c t s f o r t hese dynam i c cond i t i ons and w i l l be compareda t a l a t e r d a t e. A p r o p e r a c c o u n t i n g o f the mass f l o w i n t i m e i s b e l i ev e d t obe i m p o r t a n t i n d y n a m i c a l l y l o ad e d b e a r i n g s an d s h o u ld i n f l u e n c e t h e t r a j e c t o r yo f t h e r o t o r i n a more r e a l i s t i c way.

I t may be

CONCLUSIONS

From t h e n o n l i n e a r t r a n s i e n t a n a l y s i s o f an o i l - f i l m j o u r n a l b e a r i n g un de rd i f f e r e n t dyna mic l o a ds w i t h Re yn old s t y p e bo un d ar y c o n d i t i o n s , t h e f o l l o w i n gc o n c lu s io n s a r e e v i d e n t .

6



1 . A lt h o ug h t h e a n a l y s i s I s c o s t l y I n t e r m s o f computer t ime, i t g i v e s t h eo r b i t a l t r a j e c t o r y w i t h i n t h e cl ea ra nc e c i r c l e wh ic h i s n o t o b t a in a b l e u s in g as i m p l i f i e d ( l i n e a r i z e d ) t h eo ry .

b e a r i n g t h eo r y, t h e j o u r n a l l o c u s for a f i n i t e b ea r in g ends i n a l i m i t c y c l e2 . A s i t i s shown by o t h e r s w h i l e d e a l i n g w i t h s h o r t a nd i n f i n i t e l y l o n g

for an unstab le sys tem.3 . A s t a b l e j o u r n a l w i t h a c o n s t a n t u n i d l r e c t i o n l o a d ca n be made u n s t a b l e

b y s u p e rimp o sin g a p e r i o d i c l o a d o n t h e s ys te m.

4 . A m o r e c o r r e c t ( c a v i t a t e d ) b ou nd ar y c o n d i t i o n u s i n g t h e p r e h i s t o r y o ff i l m may show some i n t e re s t i n g phenomena w h ic h a re n o t r e v e a le d i n t h e p r es e ntmethod o f s o l u t l o n .

REFERENCES

1 . A l l a i r e , P .E . , D e si g n o f J o u r n a l B e a r i n gs for H ig h -s p e e d R o ta t i n g

Mach ine ry , fundamen ta ls o f t h e D e s ig n of F l u i d F i l m B e a r i n g s , ASMEP u b l i c a t i o n , New York, 1979, pp. 45-83.

2 . Ake rs , A . , Mic h a e l s o n , S . , and Cameron, A . , S t a b i l i t y Co nto urs f o r aW h i r l i n g F i n i t e J o u r n al B ea ri ng , J . Lub. Tech., Tran s. ASME, S e r i e s F ,v o l . 9 3 , n o . 1 , 1971, pp. 177-190.

3. Booker, J.F., Dynamica l l y Loaded Jou rna l Bear ings - M o b i l i t y M ethod o fS o l u t i o n , J . B a s i c E n gr . T ran s . ASME, S e r i e s D, v o l . 1 8 7 , no. 3, 1965,p. 537.

4 . H o r s n e l l , R. and McCa l l ion , H . , P r e d i c t i o n o f Some Journa l Bear ingC h a r a c t e r i s t i c s U nd er S t a t i c a nd D ynam ic L o a d in g , P r o c . L u b r i c a t i o n an d

Wear Con ven tio n, I n s t . Mech. Eng rs., London, 1963, pp. 126-138.

5. Campbel l , J. Love, P.P., M a r t i n , F . A . , and Re f ique , S . C . , B e a r i n g s forA r e v i e w o f t h e P r e s e n t S t a t e o f T h e o r e t i c a le c i p r o c a t i n g M a c h i ne r y:

Exper imen t and Ser v ic e Knowledge , Con f . Lu b r ic a t io n and Wear, P roc . I n s t .Mech. Engrs . , v o l . 182, (P ar t 3A), 1967-68, pp. 51-74.

6 . K i r k , R . a nd G u e n te r , E . J. , T ra n s l e n t J o u r n a l B e a r i n g A n a l y s i s , NASA

CR-1549, June 197 0.

7 . Holmes., R . , T h e V i b r a t i o n o f a R i g i d S h a f t o n S h o r t S l e e v e B e ar i n g s , J.Mech . Engr. Sc i . , v o l . 2 , no . 4 , 1960, p. 337.

8. Jakobsen, K . a n d C h r l s t e n s e n , H . , N o n li n e ar T ra n s i e n t V i b r a t i o n s i nJ o u r n a l B e a r i n g s , P ro c . I n s t . Mech. E n g r . v o l . 183, P a r t 3P, 1968-69,pp. 50-56.

7



C o n d i t i o n L/D I Eo 11 . 1 I Q o(RAD/S) D ( M ) C /R q ( P a - s )

V a r i ab 1e 1 0 . 5 6 2 I 0 . 7 2 4 1 5 . I 1 .0 I 6 2 . 8 4 1 0 . 2 I 0 . 0 0 0 8 I 0 . 1 5 Ir o t a t i n g

U n i d i r e c t i o n a lc o n s t a n t

p e r i o d i cU n i d i r e c t i o n a l

F I G U R E 1. - A SCHEMATIC DIAGRAM OF JOURNAL BEARING.

--- ------ ----.O 0 . 8 5 . 0.5 -----

--- ------ ----. O 0 . 8 5. 0 . 5

-----

8



R E L A T I V E T O C Y L I N D E R A X I S

F I G U R E 2. - POLAR LOAD DIAGRAn OF ENGINE BEARING.

( A ) O B T A I N ED F R OM T H E P RE S EN T K T H O D S O L U T I ON .

(B) OBTAINED BY AKERS ET AL. [21 .

F I G U R E 3 . - JOURNAL CENTER TRAJECTORY. (L/D = 1.0. E = 0 . 5 , 3 30°,

M = 10.24. n = 0.5).

9



F I G U R E 4. - A T Y P I C A L P R E S S U R E D I S T R I B U T I O N ( L / D = 1.0, Eo = 0.8,-M = 5. R = 0.5. T = 0).

F I G U R E 5 , - JOURNAL CENTER TRAJECTORY FOR A UNIDIRECTIONAL CONSTANT

LOAD (L/D = 1 . 0 . ~ ~0.8. R = 5 . n = 0.5).

F I G U R E 6 . - JOURNAL CENTER TRAJECTORY FOR A UNIDIRECTIONAL PERIODIC F I G U R E 7 . - JOURNAL CENTER TRAJECTORY FOR VARIABLE ROTATING LOAD.

LOAD (L/D = 1.0, Eo = 0.8, K = , R = 0.5).

10



National Aeronautics andSpace AdInlnlStrdllOn

7. Key Words (Suggested by Author@))

Journal bearings; Stability; Dynamic loads;Bearings; Dynamics

Report Docum entation Page

18. Distribution Statement

Unclassified-UnlimitedSubject Category 34

-2. Government Accession No.

NASA TM-102309. Report No.

9. Security Classif. (of this report)

Unclassified

AVSCOM TR 87-C-26.

4. Title and Subtitle

Stability of a Rigid Rotor Supported on Oil-Film Journal

Bearings Under Dynamic Load

20. Security Classif. (of this page) 1 21. NO of ; y e s 22. Price'

Unclassified 1 A03

- - ~ _ -7. Author@)

B.C Majumdar and D.E. Brewe

9. Performing Organization Name and Address

NASA Lewis Research CenterCleveland, Ohio 44135-3191andPropulsion DirectorateU.S.Army Aviation Research and Technology Activity-AVSCOM

Cleveland, Ohio 44135-31272. Sponsoring Agency Name and Address

National Aeronautics and Space AdministrationWashington, D.C . 20546-0001

andU.S. Army Aviation Systems CommandSt. Louis, Mo. 63120-1798

3.Recipient's Catalog No.

5. Report Date

6. Performing Organization Code

8. Performing Organization Report No.

E-3727

10. Work Unit No.

505-63-81

11 . Contract or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

5. Supplementary Notes

Prepared for the National Seminar on Bearings Madras, India, September 17- 18, 1987. B.C. Majumdar, National Research Council-

NASA Research Associate; present address: Indian Institute of Technology, Department of Mechanical Engineering, Kharagpur-721302,

India. D.E. Brewe, Propulsion Directorate, U.S. A m y Aviation Research and Technology Activi ty-AVSCOM.

6. Abstract

Most published work relating to dynamically loaded journal bearings are directed to determining the minimum filmthickness from the predicted journal trajectories. These do not give any information about the subsynchronouswhirl stability of journal bearing systems since they do not consider the equations of motion. It is, however,necessary t o know whether the bearing system operation is stable or not under such an operating condition. Thepurpose of the present paper is to analyze the stability characteristics of the system. A linearized perturbation

theory about the equilibrium point can predict the threshold of stability; however it does not indicate postwhirlorbit detail. The linearized method may indicate that a bearing is unstable for a given operating condition whereasthe nonlinear analysis may indicate that it forms a stable limit cycle. For this reason, a nonlinear transientanalysis of a rigid rotor supported on oil journal bearings under ( I ) a unidirectional constant load, (2) a unidirec-

tional periodic load, and (3) variable rotating load are performed. Ln,this paper, th e hydrody namic forces arecalculated after solving the time-dependent Reynolds equation by a finite difference method with a successiveoverrelaxation scheme. Using these forces, equations of motion are solved by the fourth-order Runge-Kutta

method to predict the transient behavior of th e rotor. With the aid of a high-speed digital computer and graphics,the journal trajectories are obtained for several different operating conditions.

'4

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bcm brewe non linear stability

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