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Paper XI(iii) Theoretical and experimental orbits of a dynamically loaded hydrodynamic journal bearing
355
Paper Xl(iii)
Theoretical and experimental orbits of a dynamically loaded hydrodynamic journaI bearing R.W. Jakeman and D.W. Parkins
T h i s paper g i v e s a comparison o f t h e o r e t i c a l and e x p e r i m e n t a l o r b i t s o f a d y n a m i c a l l y loaded j o u r n a l b e a r i n g h a v i n g a p r e s s u r i s e d o i l s u p p l y t o a c e n t r a l 3600 c i r c u m f e r e n t i a l groove. The r e s u l t s o f two t h e o r e t i c a l analyses a r e presented: Methods A and B. Method B, r e f e r r e d t o as t h e Reaction Method, f e a t u r e s o i l f i l m f o r c e p r e d i c t i o n by means o f p r e computed v e l o c i t y c o e f f i c i e n t s , t h u s f a c i l i t a t i n g q u i c k e r computation. S a t i s f a c t o r y c o r r e l a t i o n o f t h e experimental r e s u l t s w i t h t h e p r e d i c t i o n s o f b o t h t h e o r e t i c a l methods i s shown. Comparisions a r e made f o r t h r e e examples i n c l u d i n q d i f f e r e n t r e l a t i v e phase and a m p l i t u d e o f t h e e x c i t a t i o n components a t b o t h once and t w i c e r o t a t i o n a l frequency. 1. INTRODUCTION 1.1 Notation
-
L i n e a r i s e d o i l f i l m displacement and v e l o c i t y c o e f f i c i e n t s have been commonly used t o model t h e i n f l u e n c e o f hydrodynamic j o u r n a l b e a r i n g s upon t h e l a t e r a l v i b r a t i o n c h a r a c t e r i s t i c s o f v a r i o u s s h a f t i n g systems. These c o e f f i c i e n t s a r e s u b j e c t t o a h i g h degree o f n o n - l i n e a r i t y which may lead to substantial errors, particularly with respect to amplitude p r e d i c t i o n , i n s i t u a t i o n s where s i g n i f i c a n t dynamic l o a d i n g i s encountered. I n more extreme cases o f dynamic loading, such as c r a n k s h a f t bearings, n o n - l i n e a r i t y renders t h e use o f a s i n g l e s e t o f displacement and velocity coefficients t o t a l l y impractical. A t i m e s t e p p i n g j o u r n a l o r b i t a n a l y s i s i s used i n these s i t u a t i o n s . Journal o r b i t analysis is inherently heavy on computing time, p a r t i c u l a r l y w i t h t h e more r i g o r o u s t y p e s o f a n a l y s i s , where o i l f i l m c h a r a c t e r i s t i c s must be computed a t each t i m e step. A c o n s i d e r a b l e r e d u c t i o n i n computing t i m e can be gained b y t h e use o f e i t h e r an approximate s o l u t i o n o f the oil film pressure distribution or pre-computed o i l f i l m data. The o b j e c t i v e o f t h e work r e p o r t e d i n t h i s paper was t o compare t h e r e s u l t s o f two journal orbit prediction methods with experimental data obtained from a t e s t r i g ( 1 ) . T h e o r e t i c a l Method A i s o f t h e r i g o r o u s type thus u s i n g numerical film pressure s o l u t i o n s a t each t i m e s t e p (2), w h i l s t Method B, r e f e r r e d t o as t h e R e a c t i o n Method, achieves a f a s t o r b i t s o l u t i o n b y t h e use o f pre-computed v e l o c i t y c o e f f i c i e n t s . Method A has been p r e v i o u s l y d e s c r i b e d i n r e f e r e n c e ( 3 ) , and t h e development o f t h e o i l f i l m f o r c e e q u a t i o n s upon which t h e R e a c t i o n Method i s based i s o u t l i n e d i n r e f e r e n c e ( 4 ) . Alignment between t h e j o u r n a l and b e a r i n g was m a i n t a i n e d f o r a l l c o n d i t i o n s covered i n t h i s paper, and t h e b e a r i n g f e a t u r e d a p r e s s u r i s e d o i l supply t o a c e n t r a l l y positioned f u l l circumferential groove. Both theoretical methods took account o f j o u r n a l i n e r t i a l f o r c e s , and Method A had an o p t i o n a l f a c i l i t y f o r modelling o i l f i l m h i s t o r y .
AXX,etc. L i n e a r i s e d o i l f i l m c o e f f i c i e n t s f o r s m a l l displacement p e r t u r b a t i o n s . BXX,etc. L i n e a r i s e d o i l f i l m c o e f f i c i e n t s f o r small v e l o c i t y perturbations. Btt, Brt,
Brr
Separate wedge and squeeze a c t i o n velocity coefficents
B r r t , B t r t I n t e r a c t i v e wedge and squeeze a c t i o n velocity coefficients. C
Radial clearance
Fr, F t
Radial,
Fex,Fey
Horizontal,
v e r t i c a l external forces.
F x , FY
Horizontal,
v e r t i c a l o i l f i l m forces.
j, i
Circumferential, p o s i t i o n reference
m
J o u r n a l mass
qn ( j , i )
N e t t o i l volume f l o w r a t e i n t o element j, i.
R
Radial journal v e l o c i t y
T
Dynamic c y c l e t i m e
t
Time f r o m s t a r t o f dynamic c y c l e and a t the s t a r t o f time step At.
Ve(j,i)
Total
volume
Vo ( j , i )
Volume
of
x, y
Horiziontal, displacement*
i,
Horizontal, vertical journal velocity*
tangential
oil
f i l m forces.
axial
e 1ement
*
oil
of in
element element
v e r t ica 1
j,
i.
j,
i.
journal
356
ic', 4;
Horizontal, vertical acceleration* Time s t e p increment
At
B
journal
Angular velocity of about b e a r i n g a x i s . *
journal
60
Equivalent journal*
velocity
E
Eccentricity
W
Journal a n g u l a r own a x i s .
*
angular
r a t i o = (x2 velocity
+
axis of
Y ~ ) ~ / ~ / C
about
its
R e f e r e r s t o t h e normal s i t u a t i o n o f a "fixed" bearing. I n t h e experimental t e s t r i g t h e s e parameters r e f e r t o t h e b e a r i n g housing s i n c e t h e j o u r n a l i s "fixed".
Suf f ixes : h,j,o
b e a r i n g housing, j o u r n a l , o i l f i l m .
P
perturbation etc.
max
maximum p e r m i t t e d value.
S
i n i t i a l l y e s t i m a t e d value. denotes c o n d i t i o n s a t t + A t , s u f f i x denotes c o n d i t i o n s a t t.
A
used
to
Axx,
compute
no
Prefix: A
denotes t h e change i n any parameter over A t e.g. Ax = xA -x
2.
BRIEF REVIEW OF PREVIOUS WORK
2.1
Factors relevant Analysis
to
Journal
Orbit
There a r e t h r e e main f a c t o r s p e r t a i n i n g to t h e j o u r n a l o r b i t a n a l y s i s methods p u b l i s h e d t o date. The v a r i o u s o p t i o n s w i t h i n t h e s e a r e o u t 1 i n e d below: (1)
(2)
(3)
O i l f i l m force derivation: a.
S o l u t i o n o f Reynolds e q u a t i o n b y s h o r t b e a r i n g approximation.
b.
Numerical f i l m p r e s s u r e s o l u t i o n f o r bearings o f f i n i t e length.
c.
Use o f pre-computed o r measured o i l f i l m properties t o f a c i l i t a t e a f a s t o r b i t solution.
d.
O i l f i l m h i s t o r y modelling.
Journal mass: a.
I n e r t i a l f o r c e s assumed t o be negligible in relation to e x t e r n a l and o i l f i l m f o r c e s , therefore journal velocity components a r e d e r i v e d t o produce o i l f i l m f o r c e s equal t o t h e e x t e r n a l f o r c e s a t each step.
b.
I n e r t i a l f o r c e s n o t neglected.
2.2
Bear inq e l a s t ic it y : a.
B e a r i n g assumed t o be r i g i d .
b.
Taken i n t o account b y i n t e r a c t i v e solution of film pressure distribution and corresponding b e a r i n g e l a s t i c deformation.
P r e v i o u s Work
One o f t h e b e s t known f a s t s o l u t i o n s i s t h e M o b i l i t y Method o f Booker(5) which f e a t u r e s option (lc). The M o b i l i t y data, upon which t h i s method depends, was o r i g i n a l l y d e r i v e d by t h e s h o r t b e a r i n g a p p r o x i m a t i o n ( l a ) , and was consequently o f l e s s e r accuracy t h a n more recent numerical solutions. Finite bearing solutions or experimental measurements may a l s o be used t o produce M o b i l i t y data, t h e r e b y s u b s t a n t i a l l y improving accuracy. T h i s method was designed f o r s i t u a t i o n s where j o u r n a l i n e r t i a l f o r c e s c o u l d be n e g l e c t e d (2a) and t h e r i g i d b e a r i n g assumption (3a), and i s t h e o r e t i c a l l y l i m i t e d t o b e a r i n g s h a v i n g c i r c u m f e r e n t i a l symmetry. The a n a l y s i s b y Holmes and Craven ( 6 ) i s one o f t h e few t o have taken account o f j o u r n a l i n e r t i a l f o r c e s (2b), t h e i r work b e i n g based on t h e s h o r t b e a r i n q approximation ( l a ) , and applied t o a r i g i d bearinq (3a).
O i l f i l m h i s t o r y m o d e l l i n g ( I d ) has a l s o r e c e i v e d v e r y l i t t l e a t t e n t i o n , t h e paper by Jones ( 7 ) g i v i n q a good account o f t h i s , b u t with the limitations o f neglecting i n e r t i a l f o r c e s (2a) and t h e r i g i d b e a r i n g assumption (3-3)
L i t t l e work has been c a r r i e d o u t on t h e modelling o f e l a s t i c i t y i n a dynamically loaded b e a r i n g (3b) due t o t h e excessive computing t i m e i n v o l v e d . The paper b y LaBouff and Booker ( 8 ) i s an example o f t h i s , and used a f i n i t e b e a r i n q s o l u t i o n ( l b ) and n e g l e c t e d i n e r t i a l f o r c e s (2a). F a n t i n o e t a1 ( 9 ) a t t a i n e d a more a c c e p t a b l e computing t i m e by u s i n g t h e s h o r t b e a r i n g a p p r o x i m a t i o n ( 1 a), b u t w i t h a consequent l o s s o f accuracy. Goenka and Oh (10) used t h e b a s i c methods o f b o t h ( 8 ) and ( 9 ) , b u t w i t h v a r i o u s r e f i n e m e n t s t o improve b o t h accuracy and computing time. 2.3
Relation o f P r e v i o u s Work
Methods
A
and
B
to
I n r e l a t i o n t o the foregoing analysis option c a t e g o r i e s , i t may be n o t e d t h a t t h e o r e t i c a l Method A i n t h i s paper used a numerical f i n i t e bearing s o l u t i o n (lb), with an o p t i o n a l f a c i l i t y f o r o i l f i l m h i s t o r y modelling (Id). Journal inertial f o r c e s were taken into consideration (2b), b u t t h e b e a r i n g was assumed t o be r i g i d ( 3 a ) . Method A i s therefore closely comparable to the theoretical work by Jones (7), and a comparison w i t h r e s u l t s t h e r e f r o m u s i n g t h e i n t e r m a i n c r a n k s h a f t b e a r i n g o f a 1.8 l i t r e 4 - s t r o k e c y c l e p e t r o l engine as a t e s t case, was g i v e n i n r e f e r e n c e ( 3 ) . The i n c l u s i o n o f inertial f o r c e s was t h e main d i f f e r e n c e between t h e above analyses, Method A h e r e i n and t h a t by Jones ( 7 ) . I n t h i s respect
357
Method A i s comparable Holmes and Craven ( 6 ) .
to
the
analysis
Method B d i f f e r e d from Method A, in that pre-computed v e l o c i t y c o e f f i c i e n t s were used i n order t o obtain a f a s t o r b i t s o l u t i o n (lc). The c o e f f i c i e n t s were d e r i v e d by a numerical f i n i t e b e a r i n g s o l u t i o n ( l b ) , b u t t h i s method negated t h e p o s s i b i l i t y o f o i l f i l m h i s t o r y m o d e l l i n g ( I d ) , f o r which no f a s t s o l u t i o n i s known t o e x i s t . A particular f e a t u r e o f Method B i s t h a t t h e v e l o c i t y coefficients used take account of the interaction o f squeeze and wedge action r e s u l t i n g f r o m t h e presence o f c a v i t a t i o n , and the a s s o c i a t e d non-1 i n e a r behaviour. In u t i l i s i n g pre-computed c o e f f i c i e n t s , Method 8 may be compared w i t h B o o k e r ' s M o b i l i t y Method (5), b u t d i f f e r s i n t h a t i t r e a d i l y a l l o w s j o u r n a l i n e r t i a l f o r c e s t o be taken i n t o account. The M o b i l i t y Method may appear t o be s i m p l e r t h a n Method B i n t h a t o n l y two parameters a r e r e q u i r e d , namely t h e M o b i l i t y Number and t h e a n g l e o f t h e squeeze p a t h r e l a t i v e t o t h e load vector. However, t h e s e two parameters are functions of both 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 angle, even f o r a c i r c u m f e r e n t i a1 1y symmetri c a l b e a r i n g Method B r e q u i r e s f i v e v e l o c i t y c o e f f i c i e n t s , b u t f o r t h e c i r c u m f e r e n t i a l l y symmetrical b e a r i n g t h e s e are f u n c t i o n s o f e c c e n t r i c i t y r a t i o only. The t o t a l amount o f pre-computed d a t a r e q u i r e d by Method B i s t h e r e f o r e s u b s t a n t i a l l y less than f o r t h e M o b i l i t y Method. I n a d d i t i o n , Method B may be extended t o cover n o n - c i r c u m f e r e n t i a l l y symmetrical b e a r i n g s b y computing t h e five velocity c o e f f i c i e n t s as f u n c t i o n s o f 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 angle.
.
3.
EXPERIMENTAL METHOD
3.1
Design o f T e s t R i q
Turnbuckle
by
F i g u r e 1 shows t h e apparatus on which t h e experimental orbits were obtained. The r o t a t i n g s h a f t i s supported a t e i t h e r end i n r o l l i n g element " s l a v e " bearings, w h i l s t t h e t e s t b e a r i n g i s mounted i n a " f l o a t i n g " housing. I n c o n t r a s t t o t h e normal p r a c t i c a l situation, i t was therefore t h e bearing housing orbits relative to the "fixed" j o u r n a l , r a t h e r t h a n j o u r n a l o r b i t s , t h a t were measured e x p e r i m e n t a l l y . Steady f o r c e s were applied separately or together in both h o r i z o n t a l and v e r t i c a l d i r e c t i o n s d i r e c t l y t o t h e t e s t b e a r i n g housing t h r o u g h t e n s i o n e d wires. R e l a t i v e displacement between t e s t b e a r i n g and journal was measured by f o u r pairs of non-contacting i n d u c t i v e transducers located on each s i d e o f t h e b e a r i n g i n t h e h o r i z o n t a l and v e r t i c a l d i r e c t i o n s . T h i s arrangement permitted calculations o f displacement a t e i t h e r b e a r i n g end o r t h e a x i a l c e n t r e plane. The t e n s i o n e d w i r e s were a t t a c h e d t o f i x e d p o i n t s l o c a t e d a t a d i s t a n c e many t i m e s ( a p p r o x i m a t e l y 20,OOO:l) greater than t h e maximum possible housing motion. This prevented housing displacement f r o m a l t e r i n g t h e d i r e c t i o n o f t h e steady f o r c e s . Stiffness of t h e s p r i n g elements i n each l o a d i n g system was made small compared t o t h a t o f t h e o i l film. This meant that test housing displacement d i d n o t a l t e r t h e magnitude o f
Steady force measurement
Electro magnetic Steady force measurement
J4,I:'
q.>
Electro magnetic vibrator-
Fig.1 Schematic Arrangement of Test Rig Loading System t h e steady f o r c e s . F i g u r e 1 shows t h a t t h e horizontal and vertical steady loading arrangements each have i n t e r m e d i a t e p u l l e y s between t h e steady f o r c e gauge and t h e t e s t b e a r i n g housing. These comprised wheels, supported by low f r i c t i o n r o l l i n q element bearings, which a l l o w e d t h e t e s t b e a r i n g housing freedom t o r o t a t e around two m u t u a l l y p e r p e n d i c u l a r t r a n s v e r s e axes w h i l s t under a l a r g e steady force. Freedom around t h e s e axes allows the bearing t o a l i g n i t s e l f w i t h the j o u r n a l l o n g i t u d i n a l axis. Moreover, this l o a d i n g arrangement e l i m i n a t e d any c o n s t r a i n t around t h e b e a r i n g c e n t r e l i n e . Hence any t o r q u e e x e r t e d b y t h e o i l f i l m was r e s i s t e d by a separate t o r q u e r e s t r a i n i n g l i n k .
A magnetic sensor i n d i c a t e d s h a f t o r i e n t a t i o n and provided a pulse for an accurate r o t a t i o n a l speed i n d i c a t o r . Dynamic b e a r i n g f o r c e s Fex, Fey, measured by p i e z o gauges, c o u l d be a p p l i e d t o t h e bearing housing either vertically or h o r i z o n t a l l y o r t o g e t h e r w i t h any r e l a t i v e phase and magnitude by t h e two e l e c t r o magnetic v i b r a t o r s . Signals f o r these e l e c t r o magnetic v i b r a t o r s and t h e i r power a m p l i f i e r s were c r e a t e d by a sinewave generator d r i v e n from t h e t e s t shaft. The v i b r a t o r connectors were designed t o impose n e g l i g i b l e c o n s t r a i n t on t h e t e s t b e a r i n g housing, t h i s c o n d i t i o n b e i n g v e r i f i e d f o r each experiment. W i t h t h e steady f o r c e o n l y a p p l i e d t o t h e t e s t bearing, plus the torque r e s t r a i n t , the housing remains f r e e t o move a small a x i a l distance along t h e shaft. This feature checked whether full convienientl y hydrodynamic c o n d i t i o n s had been e s t a b l i s h e d . However, when dynamic loads were a p p l i e d i t was found t h a t o n l y a m i c r o s c o p i c misalignment thereof was sufficient to cause an
358
unacceptably l a r g e l o n q i t u d i n a l v i b r a t i o n and torsional o s c i 1l a t i o n about an axis perpendicular t o j o u r n a l centre l i n e . To obviate t h i s , l o c a t i n q wires p a r a l l e l t o t h e b e a r i n q c e n t r e l i n e were i n t r o d u c e d f o r t h e dynamic t e s t s . They a l l o w e d t h e b e a r i n g t o move t r a n s v e r s e l y and remain p a r a l l e l t o t h e s h a f t w h i l s t p r e v e n t i n q any misalignment o r a x i a l motion. It was shown t h a t these w i r e s t r a n s m i t t e d no s t a t i c o r dynamic f o r c e s t o t h e test bearing housing in any direction perpendicular t o t h e bearing centre l i n e .
housing acceleration were recorded. Immediately a f t e r each dynamic l o a d i n g t e s t f o r c e s Fex, F were smoothly reduced t o z e r o and an e X o r i g i n " displacement time h i s t o r y recorded. T h i s accounted f o r e f f e c t s such as s m a l l o u t o f balance f o r c e s and journal runout. J o u r n a l c e n t r e l o c a t i o n was t h e n checked. Displacements due t o dynamic f o r c e s a l o n e were subsequently o b t a i n e d by s u b t r a c t i n g t h e "orgin" ordinates from those at corresponding cycle times in the i m m e d i a t e l y p r e c e d i n g d y n a m i c a l l y loaded t e s t .
3.2
4.
THEORETICAL METHOD A: JOURNAL ORB1T ANALYSI S
4.1
Introduction
T e s t R i q Equations o f M o t i o n
Equations o f m o t i o n f o r are: Fex F x = mh 'X'h + mo yo Fey
-
Fy = mh vh
+
mo
t h e housinq and o i l
yo
[11
where Fx, Fy a r e t h e o i l f i l m forces a c t i n g on t h e j o u r n a l . These f o r c e s a r e f u n c t i o n s o f t h e r e l a t i v e housing t o j o u r n a l displacements x, y, (measured d i r e c t l y by t r a n s d u c e r s ) , and r e l a t i v e v e l o c i t i e s iC, j o b t a i n e d by numerical d i f f e r e n t i a t i o n o f t h e measured x, y t i m e h i s t o r y . Accelerometers attached to the housing measured yh, j;h w i t h Xh, Yh obtained b y double n u m e r i c a l i n t e g r a t i o n . The j o u r n a l displacements w i t h r e s p e c t t o a f i x e d p o s i t i o n i n space were o b t a i n e d from:
[21 I t was
shown
moYo s 0.01
y direction.
t h a t i f -3 c X j / X h 4 +3 t h e n mhyh, and s i m i l a r l y for the
A l l experimental d a t a r e p o r t e d i n t h i s paper were found t o meet t h i s c o n d i t i o n . Morton (11) a l s o notes t h a t t h e o i l f i l m t r a n s v e r s e inertial forces may be neglected. The equations of motion may therefore be s imp1 if ied t o :
[31 3.3
T e s t B e a r i n g Data
The f o l l o w i n g d a t a d e f i n e t h e r e l e v a n t t e s t b e a r i n g dimensions and o p e r a t i n g c o n d i t i o n s used: 63.5 mm. 2x9.3 mm. lands 0.0836 mm 5.08 mm x 3600 1180. RPM. 0.0517 MPa (gauge) -0.175 MPa (gauge) 0.0186 Pa.s
Journal diameter B e a r i n g Length D iamet r a1 c 1ear ance O i l groove Journal speed O i l supply pressure C a v i t a t i o n pressure Effective viscosity 3.4
T e s t Procedure
A t each steady l o a d
-
speed combination, t e s t r i g temperatures were s t a b i l i s e d and t h e d a t a obtained f o r t h e determination o f a t t i t u d e a n g l e and e c c e n t r i c i t y . Data r e p o r t e d i n t h i s paper were a l l o b t a i n e d a t a s i n g l e v a l u e o f s t e a d y 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 angle. Time histories of the bearing housing h o r i z o n t a l and v e r t i c a l displacements r e l a t i v e t o t h e j o u r n a l , e x t e r n a l dynamic f o r c e s and
-
RIGOROUS
T h i s method i s based on t h e p r e d i c t i o n o f j o u r n a l displacement and v e l o c i t y components a t t h e end o f each t i m e s t e p by means o f displacement and velocity coefficients computed f o r t h e c u r r e n t c o n d i t i o n s . A full d e s c r i p t i o n o f t h i s method i s g i v e n i n reference (3). I n r e l a t i o n t o Method A the, t h e description "rigorous", e s s e n t i a l l y r e f e r s t o t h e use o f a n u m e r i c a l s o l u t i o n o f t h e f i l m p r e s s u r e d i s t r i b u t i o n ( 2 ) a t each o r b i t step. T h i s t y p e o f s o l u t i o n can accommodate f i n i t e l e n g t h t o diameter r a t i o s and p r e s s u r i s e d o i l feed f e a t u r e s . I n e v i t a b l y t h e term "rigorous" i s r e l a t i v e , and t h e most s i g n i f i c a n t a p p r o x i m a t i o n o f t h i s method i s c o n s i d e r e d t o be t h e r i g i d b e a r i n g assumption. As i n d i c a t e d i n t h e r e v i e w o f p r e v i o u s work, m o d e l l i n g b e a r i n g e l a s t i c i t y a t p r e s e n t r e s u l t s i n e x c e s s i v e computing t i m e u n l e s s approximate f i l m p r e s s u r e s o l u t i o n s a r e used. Bearing elasticity may be quite a p p r e c i a b l e i n some p r a c t i c a l . a p p l i c a t i o n s , n o t a b l y c o n n e c t i n g r o d bearings, b u t t h e t e s t b e a r i n g used t o o b t a i n t h e e x p e r i m e n t a l o r b i t presented i n t h i s paper, was c o n t a i n e d i n a substantial housing. Differences in the e x p e r i m e n t a l and t h e o r e t i c a l o r b i t s due t o bearing e l a s t i c i t y are t h e r e f o r e u n l i k e l y t o be s e r i o u s i n t h i s i n s t a n c e .
4.2
C a v i t a t i o n Model
A c a v i t a t i o n model which took account o f f l o w continuity, whilst assuming a constant c a v i t a t i o n pressure, was used i n Method A. D e t a i l s o f t h i s model a r e a l s o g i v e n i n reference ( 2 ) . No account i s t a k e n o f t h e n e g a t i v e p r e s s u r e s p i k e preceeding t h e r u p t u r e boundary which has been r e p o r t e d i n s e v e r a l e x p e r i m e n t a l s t u d i e s , b u t no p r a c t i c a l system f o r m o d e l l i n g t h i s f e a t u r e i s known t o e x i s t a t present. The method whereby c o n t i n u i t y i s s a t i s f i e d w i t h i n t h e c a v i t a t i o n zone i s s i m p l e and easy t o a p p l y w i t h i n a r e l a x a t i o n s o l u t i o n o f t h e f i l m pressure d i s t r i b u t i o n . Only t h e c a v i t a t i o n p r e s s u r e has t o be s p e c i f i e d , no assumptions o r i n i t i a l e s t i m a t i o n s f o r t h e l o c a t i o n o f t h e c a v i t a t i o n zone boundaries, o r t h e p r e s s u r e g r a d i e n t s a t t h e s e boundaries, a r e necessary. Furthermore, t h i s method i s eminently suitable to oil film history m o d e l l i n g , w h i c h may be d e f i n e d as t h e s t e p b y s t e p m o n i t o r i n g and u p d a t i n g o f t h e e x t e n t o f cavitation zones and the volumetric d i s t r i b u t i o n o f o i l w i t h i n them t h r o u g h o u t t h e journal orbit.
359 O i l F i l m H i s t o r y Model
4.3
A detailed description o f the o i l f i l m history model i s given i n r e f e r e n c e ( 3 ) and t h e f o l l o w i n q n o t e s o u t l i n e t h e main f e a t u r e s : The o i l f i l m i s d i v i d e d i n t o r e c t a n g u l a r elements, f o r t h e purpose o f s o l v i n g t h e f i l m pressure d i s t r i b u t i o n b y c o n s i d e r a t i o n o f f l o w continuity (2). O i l f i l m h i s t o r y modelling i s based on t h e premise t h a t i n a d y n a m i c a l l y loaded bearing, elements s u b j e c t t o c a v i t a t i o n do n o t have t o s a t i s f y f l o w c o n t i n u i t y , s i n c e t h e y may be f i l l i n g o r emptying a t any g i v e n time. D u r i n g each o r b i t t i m e s t e p t h e n e t t f l o w r a t e o f o i l t o each element i s computed, t h i s b e i n g used t o update t h e volume o f o i l w i t h i n each element a t t h e t i m e s t e p end: VoA(j,i) = Vo ( j , i )
+
qn ( j , i ) .
At
[4 1
T r a n s f e r o f a c a v i t a t i n g element t o a f u l l f i l m element occurs when e q u a t i o n [ 4 ] p r e d i c t s an o i l volume equal t o o r exceeding t h e The element volume: Vo(j,i) 3 VeJj,i). r e v e r s e t r a n s f e r may o c c u r d u r i n g t h e f i l m pressure relaxation process, when a s u b - c a v i t a t i o n p r e s s u r e i s computed f o r a f u l l film element. By means of the above processes, c a v i t a t i o n zones may expand o r c o n t r a c t i n any d i r e c t i o n a c c o r d i n g t o t h e p r e v a i l i n g c o n d i t i o n s as t h e dynamic c y c l e proceeds. 4.4
O r b i t Time Step S o l u t i o n
Since journal mass inertial f o r c e s were included, t h e o r b i t t i m e s t e p procedure was based on t h e s o l u t i o n o f t h e e q u a t i o n s o f m o t i o n f o r t h e mean c o n d i t i o n s d u r i n g each o r b i t step: (FexA-
FxA +
Fe,
- F,
)/ 2 = rn A i / A t
(FWA-FyA+ Fey- F,)/2
151
= m Ai/At
A t t h e s t a r t o f each t i m e s t e p t h e j o u r n a l displacement and v e l o c i t y components (x,y,i?,j) w i l l b e known, and t h e c o r r e s p o n d i n g o i l f i l m forces Fx, Fy can thus be computed. E x t e r n a l f o r c e components Fex, a t the end s t a r t p o i n t and FexA , FeyA hte': p o i n t can be i n t e r p o l a t e d f r o m t h e s p e c i f i e d e x t e r n a l l o a d c y c l e data. There remains 6 unknowns XA , YA , ~ C A , ?A , FXA ., FYA corresponding t o t h e end p o i n t . An a d d i t i o n a l f o u r e q u a t i o n s are t h e r e f o r e r e q u i r e d i n o r d e r t o obtain a solution. Two f u r t h e r e q u a t i o n s are p r o v i d e d by u s i n g o i l f i l m displacement and v e l o c i t y c o e f f i c i e n t s t o r e l a t e t h e o i l f i l m f o r c e changes w i t h t h e c o r r e s p o n d i n g displacement and v e l o c i t y changes d u r i n g t h e t i m e step. The displacement and v e l o c i t y c o e f f i c i e n t s a r e computed f o r t h e c o n d i t i o n s c o r r e s p o n d i n g t o t h e s t a r t p o i n t , and i t i s assumed t h a t t h e s e v a l u e s do n o t v a r y s i g n i f i c a n t l y over t h e t i m e step: +B,,Ai
[71
Ax+ A,, Ay + B,, A i + &,,A9
[81
AF, = & A x
AF, = A,,
+ A,,Ay
+B,A~
The remaining two e q u a t i o n s r e q u i r e d a r e o b t a i n e d b y r e l a t i n g t h e displacement changes during A t w i t h t h e corresponding v e l o c i t y
changes by assuming t h a t a c c e l e r a t i o n v a r i e s l i n e a r l y w i t h time during t h i s i n t e r v a l : A X = ( 2 i + i AA)t / 3
+ i(AtI2/6
[91
[lo1
by= (29 + ~ ~ ) A 1 / 3 + ~ ( b t ) ~ / 6
The d is p l acement and v e l o c i t y c o e f f i c i e n t s used i n e q u a t i o n s [ 5 ] and [ 6 ] were computed a t each o r b i t s t e p p o i n t b y t h e a p p l i c a t i o n o f displacement and v e l o c i t y p e r t u r b a t i o n s and film p r e s s u r e s o l u t i o n s t o determine t h e A critical c o r r e s p o n d i n g o i l f i l m forces. f e a t u r e o f t h e t i m e s t e p s o l u t i o n was t h e minimising of errors arising from the n o n - l i n e a r i t y of t h e s e c o e f f i c i e n t s . T h i s was achieved i n two ways: F i r s t l y , t h e d u r a t i o n of each t i m e s t e p ( A t ) was computed t o m a i n t a i n t h e s t e p displacement and v e l o c i t y changes w i t h i n c e r t a i n maximum values. These maximum changes were computed f r o m e m p i r i c a l A( ) = K1 K~.E functions o f t h e form where K1 and K2 a r e c o n s t a n t s . Secondly, t h e procedure made i n i t i a l e s t i m a t e s o f t h e s t e p and v e l o c i t y changes, these d is p l acement v a l u e s t h e n b e i n g used as t h e p e r t u r b a t i o n s t o compute t h e c o e f f i c i e n t s . Full details o f t h i s method a r e g i v e n i n r e f e r e n c e ( 3 ) .
-
5.
THEORETICAL METHOD 6: METHOD
5.1
Introduction
THE
REACTION
T h i s method achieves a s u b s t a n t i a l l y f a s t e r o r b i t a n a l y s i s by t h e use o f pre-computed velocity coefficients. The name "Reaction Method" was chosen since the velocity c o e f f i c i e n t s enable t h e t o t a l o i l f i l m f o r c e r e a c t i o n t o be e s t i m a t e d f o r any combination o f j o u r n a l v e l o c i t y and p o s i t i o n w i t h i n t h e b e a r i n g clearance. Both squeeze and wedge actions are included, together w i t h the i n t e r a c t i o n between them due t o c a v i t a t i o n . 5.2
O i l F i l m Force Equations
The o i l f i l m f o r c e equations, which f o r m t h e b a s i s o f t h e R e a c t i o n Method, a r e expressed i n p o l a r c o - o r d i n a t e terms: F t = Btt6o + B t r t
k6o
F r = Brt0o + B r r R + B r r t
.. RBo
[I11
c 121
The development of these equations is d e s c r i b e d i n d e t a i l i n r e f e r e n c e (4), and t h e p r i n c i p a l f e a t u r e s a r e as f o l l o w s : a)
The f o r c e components Fr, F t are t h e t o t a l o i l f i l m f o r c e s and n o t changes i n f o r c e f r o m an e q u i l i b r i u m osition. T h i s i s f a c i l i t a t e d by b e i n g t h e t o t a l e f f e c t i v e wedge v e l o c i t y since it incorporates both t h e angular v e l o c i t y o f t h e j o u r n a l about i t s own a x i s ( w ) and t h e angular v e l o c i t y o f t h e j o u r c a l a x i s about t h e b e a r i n g a x i s (8) i.e. 0,=0-W2 (assuming a stationary bearing).
lo
b)
The
velocity
coefficients
are
computed f o r a range o f p a r t i c u l a r v a l u e s of e c c e n t r i c i t y r a t i o and interpolation (linear or logarithmic) between the adjacent values is carried out for any given eccentricity ratio. C o n s i d e r a t i o n o f t h e dimensionless forms f o r t h e v e l o c i t y c o e f f i c i e n t s i s f u l l y d e t a i l e d i n reference (4). This indicated t h a t t h e v a l i d i t y o f dimensionless v e l o c i t y c o e f f i c i e n t s i s r e s t r i c t e d t o given values o f dimensionless s u p o l v and c a v i t a t i o n pressures, wbictj in turn are and t h e p r o d u c t funcJions o f R, 8, R . 8, according t o t h e type o f coefficient
.
Predicted o r b i t s using c o e f f i c i e n t s d e r i v e d b y R p e r t u r b a t i o n amplitudes o f 0.6, 1.2 and 1.8 mm/s d i d n o t It indicate significant differences. is therefore evident that the predicted o r b i t s are not unduly sensitive to the perturbation a m p l i t u d e f r o m which t h e v e l o c i t y coefficients a r e obtained. This means t h a t t h e p r e d i c t i o n accuracy when u s i n g dimensionless velocity c o e f f i c i e n t data i s not c r i t i c a l l y dependant on satisfying the s i m i 1a r i t y r e q u i r e m e n t s w i t h r e s p e c t to dimensionless supply and c a v i t a t i o n pressures. Reference ( 4 ) i n d i c a t e d t h a t t h e F r R curves for 8, # ? are = 0 asymptotic t o t h e c u r y e f o r €lo as t h e magnitude o f R i n c r e a s e d i n both positive and negative directions. A c c o r d i n g l y , when u s i n g t h e l i n e a r i s e d e q u a t i o n s [ll] and [12], Fr i s n o t a l l o w e d t o f a j l below t h e v a l u e c o r r e s p o n d i n g t o 8, = 0; i.e. F r E r r R.
-
The determination of velocity c o e f f i c i e n t s by t h e a p p l i c a t i o n o f v e l o c i t y perturbations i s a quasi dynamic s o l u t i o n i n t h a t i t i g n o r e s t h e dependence on p r e v i o u s c o n d i t i o n s I n o t h e r words t h e i n the o i l f i l m . R e a c t i o n Method does n o t t a k e account history, and s h o u l d of oil film t h e r e f o r e be used w i t h c a u t i o n i n s i t u a t i o n s where t h i s f a c t o r may be s i g n if ic a n t . F a s t O r b i t Time S t e p p i n g Procedure orbit time steppinq procedure was v i r t u a l l y i d e n t i c a l t o t h a t used f o r Method A. A Cartezian co-ordinate system was retained with displacement and velocity c o e f f i c i e n t s i n C a r t e z i a n terms computed by [ll] and [12] with means o f equations appropriate Polar Cartezian transformation. The most i m p o r t a n t d i f f e r e n c e i n t h e procedure concerned t h e d e t e r m i n a t i o n o f t h e t i m e s t e p duration A t . As shown i n r e f e r e n c e ( 3 ) , A t f o r each t i m e s t e p was determined t o ensure t h a t t h e changes i n the components of displacement and v e l o c i t y d u r i n g t h e s t e p were w i t h i n p r e s c r i b e d l i m i t s which were f u n c t i o n s o f eccentricity ratio. T h i s was necessary i n
-
o r d e r t o m a i n t a i n an a c c e p t a b l e accuracy o f t h e p r e d i c t e d of o i l f i l m f o r c e components a t t h e t i m e s t e p end when u s i n g l i n e a r i s e d displacement and v e l o c i t y c o e f f i c i e n t s . At was found b y p r o g r e s s i v e r e d u c t i o n o f t r i a l values u n t i l a l l t h e above l i m i t s were satisfied. I n Method A t h e i n i t i a l t r i a l v a l u e o f A t was a r b i t r a r i l y s e t equal t o about 1.5% o f t h e c y c l e t i m e and reduced by 1% f o r each successive t r i a l . T h i s approach ensured t h a t A t was w i t h i n 1% o f t h e maximum A t p e r m i t t e d b y t h e increment l i m i t c o n s t r a i n t s . The c o r r e s p o n d i n g t i m e r e q u i r e d t o compute A t was a small p a r t of t h e t o t a l computing time. W i t h t h e R e a c t i o n Method, t h e computing t i m e r e q u i r e d t o e s t a b l i s h A t became s i g n i f i c a n t . It was clearly necessary to find a satisfactory compromise between the c o n f l i c t i n g requirements t o maximise A t w i t h i n t h e g i v e n c o n s t r a i n t s , and y e t m i n i m i s e t h e computing t i m e r e q u i r e d t o determine t h i s value. After several t r i a l s , t h e optimum s o l u t i o n f o r t h e t e s t c o n d i t i o n s covered i n t h i s paper was found t o be as f o l l o w s : The i n i t i a l t r i a l A t was s e t a t 30% g r e a t e r t h a n t h e v a l u e f o r t h e p r e v i o u s t i m e step, and t h i s v a l u e was t h e n reduced by 10% f o r each successive t r i a l .
6.
D I S C U S S I O N OF RESULTS
Journal o r b i t s predicted by both t h e o r e t i c a l methods and those measured with the e x p e r i m e n t a l t e s t r i g a r e presented i n F i q u r e s The c o r r e s p o n d i n g e x t e r n a l f o r c e 2 t o 4. c y c l e d a t a i s a l s o g i v e n i n p o l a r form. E x t e r n a l f o r c e c y c l e f r e q u e n c i e s a r e a t once ( T e s t c o n d i t i o n s 1 and 3, F i g u r e s 2.and4.) and t w i c e ( T e s t c o n d i t i o n 2, F i g u r e 3) t h e j o u r n a l rotational frequency. The form of the external force cycle i s similar f o r t e s t c o n d i t i o n s 1 and 2 and r e s u l t s i n o b l i q u e o r b i t s . However, t h e e x t e r n a l f o r c e c y c l e f o r t e s t c o n d i t i o n 3 i s d i f f e r e n t and r e s u l t s i n a substantially horizontal orbit. All three t e s t c o n d i t i o n s show good agreement between t h e o r b i t s p r e d i c t e d by b o t h t h e o r e t i c a l methods. The d i f f e r e n c e s between t h e r e s u l t s o f t h e two t h e o r e t i c a l methods a r e m a i n l y due t o t h e approximations i n t r o d u c e d when f i t t i n q t h e r e l a t i v e l y s i m p l e e q u a t i o n s [ll] and [12], used i n Method B y t o p r e d i c t e d o i l f i l m f o r c e j o u r n a l v e l o c i t y data. Agreement between t h e e x p e r i m e n t a l t h o s e p r e d i c t e d by b o t h t h e o r e t i c a l However, g e n e r a l 1y s a t i s f a c t o r y . two r e g i o n s i n which s i g n i f i c a n t between the experimental and o r b i t s a r e apparent.
o r b i t s and methods i s there are differences theoretical
The l a r g e s t apparent d i s c r e c a n c y was i n t e s t c o n d i t i o n 3 where F i g u r e 4. shows t h a t b o t h t h e o r e t i c a l methods have s u b s t a n t i a l l y g r e a t e r excursions i n t h e d i r e c t i o n against r o t a t i o n . I n approachinq t h e o r b i t e x t r e m i t y a t t / T h 0.15 a s t r o n g e r o i l f i l m wedge a c t i o n w i l l be generated b y t h e a n t i c l o c k w i s e movement o f t h e j o u r n a l c e n t r e ( n e g a t i v e j). T h i s r e g i o n a l s o c o i n c i d e s w i t h n e g a t i v e R. These two e f f e c t s combine t o cause a much s t r o n g e r tendency t o c a v i t a t e i n t h e area t o t h e downstream s i d e o f t h e minimum f i l m t h i c k n e s s p o s i t i o n . It i s therefore postulated that the difference i n t h e e x p e r i m e n t a l and t h e o r e t i c a l o r b i t s i n
36 1
t h i s r e g i o n , may be due t o t h e c a v i t a t i o n pressure b e i n g s i g n i f i c a n t l y lower t h a n t h a t used f o r t h e p r e d i c t i o n o f t h i s dynamic situation. The c a v i t a t i o n p r e s s u r e used t o compute t h e t h e o r e t i c a l o r b i t s , was based on t h e v a l u e d e r i v e d by P a r k i n s ( 1 ) t o y i e l d agreement between the experimental and predicted equilibruim positions. Sensitivity o f t h e o r b i t s t o such d i f f e r e n c e s i n t h e c a v i t a t i o n p r e s s u r e would be i n c r e a s e d by t h e
Dynamic Force Cycle
80
Experimental -o-Theoretical method A .. * .. . -t) ineorericai metnoa
----
I,
Test Condition 1 External Force Data and Journal Orbit Fig 2
Experimental Theoretica I method A
low f o r c e s i n t h e c o r r e s p o n d i n g p a r t o f t h e dynamic f o r c e c y c l e . Hume and Holmes ( 1 2 ) have a l s o i n d i c a t e d t h a t t h e c o r r e l a t i o n between e x p e r i m e n t a l and t h e o r e t i c a l o r b i t s o f i s significantly a squeeze f i l m bearing, dependent upon t h e p r o v i s i o n f o r s u b s t a n t i a l l y sub-atmospheric c a v i t a t i o n pressures i n t h e t h e o r e t i c a l model. I n c o n t r a s t w i t h t h e above o b s e r v a t i o n s f o r t e s t c o n d i t i o n 3, t h e o r b i t a l movement a g a i n s t the direction o f rotation i n t e s t condition 1 ( F i g u r e 2) corresponds t o t h e b u i l d up t o maximum load. This r e s u l t s i n a mainly p o s i t i v e 6 and hence much l e s s c a v i t a t i o n t h a n t h a t experienced i n t h e corresponding p a r t o f The reduced t h e o r b i t f o r t e s t c o n d i t i o n 3. extent o f cavitation, combined w i t h t h e p r o x i m i t y t o maximum 1oad, would r e n d e r o i 1 f i l m f o r c e p r e d i c t i o n e r r o r s associated w i t h cavitation insignificant i n t h i s situation. These c o n d i t i o n s a l s o a p p l y t o t e s t c o n d i t i o n 2 ( F i g u r e 3), and e x p l a i n s why t h e s i g n i f i c a n t differences in the experimental and t h e o r e t i c a l e x t e n t o f movement a g a i n s t t h e d i r e c t i o n o f r o t a t i o n i n t e s t c o n d i t i o n 3 does n o t occur i n t h e other t e s t conditions. The second i m p o r t a n t d i f f e r e n c e between t h e t h e o r e t i c a l and e x p e r i m e n t a l o r b i t s , i s t h e s i g n i f i c a n t l y greater e c c e n t r i c i t y r a t i o o f t h e e x p e r i m e n t a l o r b i t s i n t h e v i c i n i t y o f q~ = 200 i n F i g u r e s 2 and 3, and s i m i l a r l y a t w = 400 i n F i g u r e 4. I n a l l three t e s t conditions, the location of the above d i s c r e p a n c y corresponded t o t h e r e g i o n o f maximum t o t a l load. Since t h e t h e o r e t i c a l 'models assumed a r i g i d b e a r i n g and j o u r n a l , e l a s t i c d i s t o r t i o n i s t h e most l i k e l y cause o f t h e l a r g e r experimental e c c e n t r i c i t y r a t i o s a t t h e more h i g h l y loaded p a r t s o f t h e dynamic cycle.
7 v
c -Dynamic Force Cycle
Steadv
Measured static equilibrium
-2p7 Theoretical method A Theoretical method 8
Fig 3
Test Cmdition 2 Force Data and Journal Orbit
-----
------
Test Condition 3 External Force Data and Journal Orbit Fig 1
362
Theoretical journal o r b i t s f o r t e s t condition 1 ( F i g u r e 2) were computed b o t h w i t h and w i t h o u t t h e o i l f i l m h i s t o r y model u s i n g The d i f f e r e n c e s were n e g l i g i b l e . Method A. T h i s r e s u l t was c o n s i d e r e d t o be due t o t h e combination o f a small o r b i t i n r e l a t i o n t o t h e c l e a r a n c e c i r c l e , and t o t h e e f f i c i e n t supply of oil provided by the full c i r c u m f e r e n t i a l groove.
7.
(3)
JAKEMAN R. W. "The I n f 1uence of C a v i t a t i o n on t h e n o n - l i n e a r i t y o f Velocity Coefficients in a Hydrodynamic J o u r n a l Bearing." To b e p u b l i s h e d : Leeds - Lyon T r i b o l . Symp. Leeds Sept. 1986.
CONCLU SI ON S
T h i s paper compares measured j o u r n a l o r b i t s w i t h t h e p r e d i c t i o n s o f two t h e o r e t i c a l methods. The t e s t c o n d i t i o n s used covered d i f f e r e n t forms o f e x c i t a t i o n a t b o t h once and t w i c e r o t a t i o n a l frequency.
(5)
Good agreement was shown between t h e r e s u l t s o f b o t h t h e o r e t i c a l methods, any d i f f e r e n c e b e i n g l a r g e l y due t o t h e approximations i n t r o d u c e d i n Method B i n o r d e r t o achieve f a s t e r computation. Method 6, introduces simple equations f o r t h e t o t a l o i l f i l m f o r c e components, u s i n g a new t y p e o f v e l o c i t y coefficient. An e q u i v a l e n t angular v e l o c i t y i s used which combines r o t a t i o n o f t h e j o u r n a l about i t s a x i s w i t h r o t a t i o n o f t h e j o u r n a l a x i s about t h e b e a r i n g a x i s , t h u s f a c i l i t a t i n g application to both steady and dynamic situations. The computation t i m e f o r t h i s Method was reduced b y a f a c t o r o f o v e r 300 r e l a t i v e t o t h a t f o r Method A. The e f f e c t of o i l f i l m h i s t o r y was f o u n d t o be n e g l i g i b l e i n t h e case analysed. Generally good agreement between the experimental and theoretical orbits was attained. The main differences were considered t o be r e l a t e d t h e i n f l u e n c e o f c a v i t a t i o n a t low l o a d and b e a r i n g e l a s t i c i t y a t h i g h load. 8.
REFERENCES PARKINS D.W. "Theoretical and Experimental Determination of the Dynamic Characteristics of a Hydrodynamic Journal Bearing". A.S.M.E. J n l . Lub. Tech. Vol. 101. A p r i l 1979 pp 129 139.
-
(21
BOOKER J.F. "Dynamically loaded Journal Bearings: Nume r ica 1 A p p l i c a t i o n o f t h e M o b i l i t y Method". A.S.M.E. J n l . Lub. Tech. Jan 1971. pp 168 - 176. and CRAVEN A.H. "The HOLMES R. I n f l u e n c e o f C r a n k s h a f t and Flywheel mass on t h e Performance o f Enqine Main B e a r i n g s " 1.Mech.E T r i b o l . Conv 1971 Paper C63/71 pp 80 85.
-
JONES G.J. " C r a n k s h a f t Bearings: O i l Film History" 9 t h Leeds-Lyon T r i b o l Symp.Sept.1982 pp 83 - 88. LA BOUFF G.A. and BOOKER J.F. "Dynamically Loaded Journal Bearings: A F i n i t e Element Treatment f o r R i g i d and Elastic S u r f aces". A.S.M.E./A.S.L.E. Conf. Oct. 1984 A.S.M.E. Paper 84 TRIB 11.
-
(9)
-
FANTINO B., FRENE J. and GODET M. "Dynamic Behaviour o f an E l a s t i c Theoretical Connecting Rod B e a r i n g -. S t u d i e s i n Study". SAE/SP-539 Engine Bearings and Lubrication, Paper No. 83037, Feb. 1983 pp 23 32.
-
-
GOENKA P.K. and OH K.P. "An Optimum A Connecting Rod Design Study Lubrication Viewpoint". A.S.L.E./A.S.M.E T r i bology Conference. A t l a n t a October 1985. A.S.M.E. P r e p r i n t No. 85 Trib 50.
-
ACKNOLEDGEMENTS
The a u t h o r s would l i k e t o thank t h e Head o f the College of Manufacturing, Cranfield I n s t i t u t e o f Technology and t h e Committee o f L l o y d ' s R e g i s t e r o f Shipping f o r p e r m i s s i o n t o p u b l i s h t h i s paper. They would a l s o l i k e t o thank t h e i r many c o l l e a g u e s a t b o t h C r a n f i e l d I n s t i t u t e o f Technology and L l o y d ' s R e g i s t e r o f Shipping f o r t h e h e l p and c o o p e r a t i o n throughout t h e work.
(1)
JAKEMAN R.W. "Journal O r b i t Analysis t a k i n g account o f O i l F i l m H i s t o r y and J o u r n a l Mass". Proc. o f Conf: Numerical Methods i n Laminar and T u r b u l e n t Flow. Swansea. J u l y 1985 pp 199 - 210.
JAKEMAN R.W. "A n u m e r i c a l a n a l y s i s method based on f l o w c o n t i n u i t y f o r hydrodynamic journal bearinqs". T r i b o l . I n t . Vol. 17. No. 6. Dec. 1984 pp 325 - 333.
-
-
MORTON, P.G. "Measurement o f t h e Dynamic C h a r a c t e r i s t i c s o f a Large A.S.M.E. Sleeve B e a r i n g " . Trans. Jnl. Lub. Tech. Jan. 1971. pp. 143 150.
-
HUME, B and HOLMES, R. "The Role o f Sub-Atmospheric F i l m Pressures i n t h e V i b r a t i o n Performance o f Squeeze F i l m Bearings". I. Mech. E. J n l and Mech. No. 5. 1978. Eng. Science. Vol. 20. p . 283.
Paper Xl(iii)
Theoretical and experimental orbits of a dynamically loaded hydrodynamic journaI bearing R.W. Jakeman and D.W. Parkins
T h i s paper g i v e s a comparison o f t h e o r e t i c a l and e x p e r i m e n t a l o r b i t s o f a d y n a m i c a l l y loaded j o u r n a l b e a r i n g h a v i n g a p r e s s u r i s e d o i l s u p p l y t o a c e n t r a l 3600 c i r c u m f e r e n t i a l groove. The r e s u l t s o f two t h e o r e t i c a l analyses a r e presented: Methods A and B. Method B, r e f e r r e d t o as t h e Reaction Method, f e a t u r e s o i l f i l m f o r c e p r e d i c t i o n by means o f p r e computed v e l o c i t y c o e f f i c i e n t s , t h u s f a c i l i t a t i n g q u i c k e r computation. S a t i s f a c t o r y c o r r e l a t i o n o f t h e experimental r e s u l t s w i t h t h e p r e d i c t i o n s o f b o t h t h e o r e t i c a l methods i s shown. Comparisions a r e made f o r t h r e e examples i n c l u d i n q d i f f e r e n t r e l a t i v e phase and a m p l i t u d e o f t h e e x c i t a t i o n components a t b o t h once and t w i c e r o t a t i o n a l frequency. 1. INTRODUCTION 1.1 Notation
-
L i n e a r i s e d o i l f i l m displacement and v e l o c i t y c o e f f i c i e n t s have been commonly used t o model t h e i n f l u e n c e o f hydrodynamic j o u r n a l b e a r i n g s upon t h e l a t e r a l v i b r a t i o n c h a r a c t e r i s t i c s o f v a r i o u s s h a f t i n g systems. These c o e f f i c i e n t s a r e s u b j e c t t o a h i g h degree o f n o n - l i n e a r i t y which may lead to substantial errors, particularly with respect to amplitude p r e d i c t i o n , i n s i t u a t i o n s where s i g n i f i c a n t dynamic l o a d i n g i s encountered. I n more extreme cases o f dynamic loading, such as c r a n k s h a f t bearings, n o n - l i n e a r i t y renders t h e use o f a s i n g l e s e t o f displacement and velocity coefficients t o t a l l y impractical. A t i m e s t e p p i n g j o u r n a l o r b i t a n a l y s i s i s used i n these s i t u a t i o n s . Journal o r b i t analysis is inherently heavy on computing time, p a r t i c u l a r l y w i t h t h e more r i g o r o u s t y p e s o f a n a l y s i s , where o i l f i l m c h a r a c t e r i s t i c s must be computed a t each t i m e step. A c o n s i d e r a b l e r e d u c t i o n i n computing t i m e can be gained b y t h e use o f e i t h e r an approximate s o l u t i o n o f the oil film pressure distribution or pre-computed o i l f i l m data. The o b j e c t i v e o f t h e work r e p o r t e d i n t h i s paper was t o compare t h e r e s u l t s o f two journal orbit prediction methods with experimental data obtained from a t e s t r i g ( 1 ) . T h e o r e t i c a l Method A i s o f t h e r i g o r o u s type thus u s i n g numerical film pressure s o l u t i o n s a t each t i m e s t e p (2), w h i l s t Method B, r e f e r r e d t o as t h e R e a c t i o n Method, achieves a f a s t o r b i t s o l u t i o n b y t h e use o f pre-computed v e l o c i t y c o e f f i c i e n t s . Method A has been p r e v i o u s l y d e s c r i b e d i n r e f e r e n c e ( 3 ) , and t h e development o f t h e o i l f i l m f o r c e e q u a t i o n s upon which t h e R e a c t i o n Method i s based i s o u t l i n e d i n r e f e r e n c e ( 4 ) . Alignment between t h e j o u r n a l and b e a r i n g was m a i n t a i n e d f o r a l l c o n d i t i o n s covered i n t h i s paper, and t h e b e a r i n g f e a t u r e d a p r e s s u r i s e d o i l supply t o a c e n t r a l l y positioned f u l l circumferential groove. Both theoretical methods took account o f j o u r n a l i n e r t i a l f o r c e s , and Method A had an o p t i o n a l f a c i l i t y f o r modelling o i l f i l m h i s t o r y .
AXX,etc. L i n e a r i s e d o i l f i l m c o e f f i c i e n t s f o r s m a l l displacement p e r t u r b a t i o n s . BXX,etc. L i n e a r i s e d o i l f i l m c o e f f i c i e n t s f o r small v e l o c i t y perturbations. Btt, Brt,
Brr
Separate wedge and squeeze a c t i o n velocity coefficents
B r r t , B t r t I n t e r a c t i v e wedge and squeeze a c t i o n velocity coefficients. C
Radial clearance
Fr, F t
Radial,
Fex,Fey
Horizontal,
v e r t i c a l external forces.
F x , FY
Horizontal,
v e r t i c a l o i l f i l m forces.
j, i
Circumferential, p o s i t i o n reference
m
J o u r n a l mass
qn ( j , i )
N e t t o i l volume f l o w r a t e i n t o element j, i.
R
Radial journal v e l o c i t y
T
Dynamic c y c l e t i m e
t
Time f r o m s t a r t o f dynamic c y c l e and a t the s t a r t o f time step At.
Ve(j,i)
Total
volume
Vo ( j , i )
Volume
of
x, y
Horiziontal, displacement*
i,
Horizontal, vertical journal velocity*
tangential
oil
f i l m forces.
axial
e 1ement
*
oil
of in
element element
v e r t ica 1
j,
i.
j,
i.
journal
356
ic', 4;
Horizontal, vertical acceleration* Time s t e p increment
At
B
journal
Angular velocity of about b e a r i n g a x i s . *
journal
60
Equivalent journal*
velocity
E
Eccentricity
W
Journal a n g u l a r own a x i s .
*
angular
r a t i o = (x2 velocity
+
axis of
Y ~ ) ~ / ~ / C
about
its
R e f e r e r s t o t h e normal s i t u a t i o n o f a "fixed" bearing. I n t h e experimental t e s t r i g t h e s e parameters r e f e r t o t h e b e a r i n g housing s i n c e t h e j o u r n a l i s "fixed".
Suf f ixes : h,j,o
b e a r i n g housing, j o u r n a l , o i l f i l m .
P
perturbation etc.
max
maximum p e r m i t t e d value.
S
i n i t i a l l y e s t i m a t e d value. denotes c o n d i t i o n s a t t + A t , s u f f i x denotes c o n d i t i o n s a t t.
A
used
to
Axx,
compute
no
Prefix: A
denotes t h e change i n any parameter over A t e.g. Ax = xA -x
2.
BRIEF REVIEW OF PREVIOUS WORK
2.1
Factors relevant Analysis
to
Journal
Orbit
There a r e t h r e e main f a c t o r s p e r t a i n i n g to t h e j o u r n a l o r b i t a n a l y s i s methods p u b l i s h e d t o date. The v a r i o u s o p t i o n s w i t h i n t h e s e a r e o u t 1 i n e d below: (1)
(2)
(3)
O i l f i l m force derivation: a.
S o l u t i o n o f Reynolds e q u a t i o n b y s h o r t b e a r i n g approximation.
b.
Numerical f i l m p r e s s u r e s o l u t i o n f o r bearings o f f i n i t e length.
c.
Use o f pre-computed o r measured o i l f i l m properties t o f a c i l i t a t e a f a s t o r b i t solution.
d.
O i l f i l m h i s t o r y modelling.
Journal mass: a.
I n e r t i a l f o r c e s assumed t o be negligible in relation to e x t e r n a l and o i l f i l m f o r c e s , therefore journal velocity components a r e d e r i v e d t o produce o i l f i l m f o r c e s equal t o t h e e x t e r n a l f o r c e s a t each step.
b.
I n e r t i a l f o r c e s n o t neglected.
2.2
Bear inq e l a s t ic it y : a.
B e a r i n g assumed t o be r i g i d .
b.
Taken i n t o account b y i n t e r a c t i v e solution of film pressure distribution and corresponding b e a r i n g e l a s t i c deformation.
P r e v i o u s Work
One o f t h e b e s t known f a s t s o l u t i o n s i s t h e M o b i l i t y Method o f Booker(5) which f e a t u r e s option (lc). The M o b i l i t y data, upon which t h i s method depends, was o r i g i n a l l y d e r i v e d by t h e s h o r t b e a r i n g a p p r o x i m a t i o n ( l a ) , and was consequently o f l e s s e r accuracy t h a n more recent numerical solutions. Finite bearing solutions or experimental measurements may a l s o be used t o produce M o b i l i t y data, t h e r e b y s u b s t a n t i a l l y improving accuracy. T h i s method was designed f o r s i t u a t i o n s where j o u r n a l i n e r t i a l f o r c e s c o u l d be n e g l e c t e d (2a) and t h e r i g i d b e a r i n g assumption (3a), and i s t h e o r e t i c a l l y l i m i t e d t o b e a r i n g s h a v i n g c i r c u m f e r e n t i a l symmetry. The a n a l y s i s b y Holmes and Craven ( 6 ) i s one o f t h e few t o have taken account o f j o u r n a l i n e r t i a l f o r c e s (2b), t h e i r work b e i n g based on t h e s h o r t b e a r i n q approximation ( l a ) , and applied t o a r i g i d bearinq (3a).
O i l f i l m h i s t o r y m o d e l l i n g ( I d ) has a l s o r e c e i v e d v e r y l i t t l e a t t e n t i o n , t h e paper by Jones ( 7 ) g i v i n q a good account o f t h i s , b u t with the limitations o f neglecting i n e r t i a l f o r c e s (2a) and t h e r i g i d b e a r i n g assumption (3-3)
L i t t l e work has been c a r r i e d o u t on t h e modelling o f e l a s t i c i t y i n a dynamically loaded b e a r i n g (3b) due t o t h e excessive computing t i m e i n v o l v e d . The paper b y LaBouff and Booker ( 8 ) i s an example o f t h i s , and used a f i n i t e b e a r i n q s o l u t i o n ( l b ) and n e g l e c t e d i n e r t i a l f o r c e s (2a). F a n t i n o e t a1 ( 9 ) a t t a i n e d a more a c c e p t a b l e computing t i m e by u s i n g t h e s h o r t b e a r i n g a p p r o x i m a t i o n ( 1 a), b u t w i t h a consequent l o s s o f accuracy. Goenka and Oh (10) used t h e b a s i c methods o f b o t h ( 8 ) and ( 9 ) , b u t w i t h v a r i o u s r e f i n e m e n t s t o improve b o t h accuracy and computing time. 2.3
Relation o f P r e v i o u s Work
Methods
A
and
B
to
I n r e l a t i o n t o the foregoing analysis option c a t e g o r i e s , i t may be n o t e d t h a t t h e o r e t i c a l Method A i n t h i s paper used a numerical f i n i t e bearing s o l u t i o n (lb), with an o p t i o n a l f a c i l i t y f o r o i l f i l m h i s t o r y modelling (Id). Journal inertial f o r c e s were taken into consideration (2b), b u t t h e b e a r i n g was assumed t o be r i g i d ( 3 a ) . Method A i s therefore closely comparable to the theoretical work by Jones (7), and a comparison w i t h r e s u l t s t h e r e f r o m u s i n g t h e i n t e r m a i n c r a n k s h a f t b e a r i n g o f a 1.8 l i t r e 4 - s t r o k e c y c l e p e t r o l engine as a t e s t case, was g i v e n i n r e f e r e n c e ( 3 ) . The i n c l u s i o n o f inertial f o r c e s was t h e main d i f f e r e n c e between t h e above analyses, Method A h e r e i n and t h a t by Jones ( 7 ) . I n t h i s respect
357
Method A i s comparable Holmes and Craven ( 6 ) .
to
the
analysis
Method B d i f f e r e d from Method A, in that pre-computed v e l o c i t y c o e f f i c i e n t s were used i n order t o obtain a f a s t o r b i t s o l u t i o n (lc). The c o e f f i c i e n t s were d e r i v e d by a numerical f i n i t e b e a r i n g s o l u t i o n ( l b ) , b u t t h i s method negated t h e p o s s i b i l i t y o f o i l f i l m h i s t o r y m o d e l l i n g ( I d ) , f o r which no f a s t s o l u t i o n i s known t o e x i s t . A particular f e a t u r e o f Method B i s t h a t t h e v e l o c i t y coefficients used take account of the interaction o f squeeze and wedge action r e s u l t i n g f r o m t h e presence o f c a v i t a t i o n , and the a s s o c i a t e d non-1 i n e a r behaviour. In u t i l i s i n g pre-computed c o e f f i c i e n t s , Method 8 may be compared w i t h B o o k e r ' s M o b i l i t y Method (5), b u t d i f f e r s i n t h a t i t r e a d i l y a l l o w s j o u r n a l i n e r t i a l f o r c e s t o be taken i n t o account. The M o b i l i t y Method may appear t o be s i m p l e r t h a n Method B i n t h a t o n l y two parameters a r e r e q u i r e d , namely t h e M o b i l i t y Number and t h e a n g l e o f t h e squeeze p a t h r e l a t i v e t o t h e load vector. However, t h e s e two parameters are functions of both 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 angle, even f o r a c i r c u m f e r e n t i a1 1y symmetri c a l b e a r i n g Method B r e q u i r e s f i v e v e l o c i t y c o e f f i c i e n t s , b u t f o r t h e c i r c u m f e r e n t i a l l y symmetrical b e a r i n g t h e s e are f u n c t i o n s o f e c c e n t r i c i t y r a t i o only. The t o t a l amount o f pre-computed d a t a r e q u i r e d by Method B i s t h e r e f o r e s u b s t a n t i a l l y less than f o r t h e M o b i l i t y Method. I n a d d i t i o n , Method B may be extended t o cover n o n - c i r c u m f e r e n t i a l l y symmetrical b e a r i n g s b y computing t h e five velocity c o e f f i c i e n t s as f u n c t i o n s o f 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 angle.
.
3.
EXPERIMENTAL METHOD
3.1
Design o f T e s t R i q
Turnbuckle
by
F i g u r e 1 shows t h e apparatus on which t h e experimental orbits were obtained. The r o t a t i n g s h a f t i s supported a t e i t h e r end i n r o l l i n g element " s l a v e " bearings, w h i l s t t h e t e s t b e a r i n g i s mounted i n a " f l o a t i n g " housing. I n c o n t r a s t t o t h e normal p r a c t i c a l situation, i t was therefore t h e bearing housing orbits relative to the "fixed" j o u r n a l , r a t h e r t h a n j o u r n a l o r b i t s , t h a t were measured e x p e r i m e n t a l l y . Steady f o r c e s were applied separately or together in both h o r i z o n t a l and v e r t i c a l d i r e c t i o n s d i r e c t l y t o t h e t e s t b e a r i n g housing t h r o u g h t e n s i o n e d wires. R e l a t i v e displacement between t e s t b e a r i n g and journal was measured by f o u r pairs of non-contacting i n d u c t i v e transducers located on each s i d e o f t h e b e a r i n g i n t h e h o r i z o n t a l and v e r t i c a l d i r e c t i o n s . T h i s arrangement permitted calculations o f displacement a t e i t h e r b e a r i n g end o r t h e a x i a l c e n t r e plane. The t e n s i o n e d w i r e s were a t t a c h e d t o f i x e d p o i n t s l o c a t e d a t a d i s t a n c e many t i m e s ( a p p r o x i m a t e l y 20,OOO:l) greater than t h e maximum possible housing motion. This prevented housing displacement f r o m a l t e r i n g t h e d i r e c t i o n o f t h e steady f o r c e s . Stiffness of t h e s p r i n g elements i n each l o a d i n g system was made small compared t o t h a t o f t h e o i l film. This meant that test housing displacement d i d n o t a l t e r t h e magnitude o f
Steady force measurement
Electro magnetic Steady force measurement
J4,I:'
q.>
Electro magnetic vibrator-
Fig.1 Schematic Arrangement of Test Rig Loading System t h e steady f o r c e s . F i g u r e 1 shows t h a t t h e horizontal and vertical steady loading arrangements each have i n t e r m e d i a t e p u l l e y s between t h e steady f o r c e gauge and t h e t e s t b e a r i n g housing. These comprised wheels, supported by low f r i c t i o n r o l l i n q element bearings, which a l l o w e d t h e t e s t b e a r i n g housing freedom t o r o t a t e around two m u t u a l l y p e r p e n d i c u l a r t r a n s v e r s e axes w h i l s t under a l a r g e steady force. Freedom around t h e s e axes allows the bearing t o a l i g n i t s e l f w i t h the j o u r n a l l o n g i t u d i n a l axis. Moreover, this l o a d i n g arrangement e l i m i n a t e d any c o n s t r a i n t around t h e b e a r i n g c e n t r e l i n e . Hence any t o r q u e e x e r t e d b y t h e o i l f i l m was r e s i s t e d by a separate t o r q u e r e s t r a i n i n g l i n k .
A magnetic sensor i n d i c a t e d s h a f t o r i e n t a t i o n and provided a pulse for an accurate r o t a t i o n a l speed i n d i c a t o r . Dynamic b e a r i n g f o r c e s Fex, Fey, measured by p i e z o gauges, c o u l d be a p p l i e d t o t h e bearing housing either vertically or h o r i z o n t a l l y o r t o g e t h e r w i t h any r e l a t i v e phase and magnitude by t h e two e l e c t r o magnetic v i b r a t o r s . Signals f o r these e l e c t r o magnetic v i b r a t o r s and t h e i r power a m p l i f i e r s were c r e a t e d by a sinewave generator d r i v e n from t h e t e s t shaft. The v i b r a t o r connectors were designed t o impose n e g l i g i b l e c o n s t r a i n t on t h e t e s t b e a r i n g housing, t h i s c o n d i t i o n b e i n g v e r i f i e d f o r each experiment. W i t h t h e steady f o r c e o n l y a p p l i e d t o t h e t e s t bearing, plus the torque r e s t r a i n t , the housing remains f r e e t o move a small a x i a l distance along t h e shaft. This feature checked whether full convienientl y hydrodynamic c o n d i t i o n s had been e s t a b l i s h e d . However, when dynamic loads were a p p l i e d i t was found t h a t o n l y a m i c r o s c o p i c misalignment thereof was sufficient to cause an
358
unacceptably l a r g e l o n q i t u d i n a l v i b r a t i o n and torsional o s c i 1l a t i o n about an axis perpendicular t o j o u r n a l centre l i n e . To obviate t h i s , l o c a t i n q wires p a r a l l e l t o t h e b e a r i n q c e n t r e l i n e were i n t r o d u c e d f o r t h e dynamic t e s t s . They a l l o w e d t h e b e a r i n g t o move t r a n s v e r s e l y and remain p a r a l l e l t o t h e s h a f t w h i l s t p r e v e n t i n q any misalignment o r a x i a l motion. It was shown t h a t these w i r e s t r a n s m i t t e d no s t a t i c o r dynamic f o r c e s t o t h e test bearing housing in any direction perpendicular t o t h e bearing centre l i n e .
housing acceleration were recorded. Immediately a f t e r each dynamic l o a d i n g t e s t f o r c e s Fex, F were smoothly reduced t o z e r o and an e X o r i g i n " displacement time h i s t o r y recorded. T h i s accounted f o r e f f e c t s such as s m a l l o u t o f balance f o r c e s and journal runout. J o u r n a l c e n t r e l o c a t i o n was t h e n checked. Displacements due t o dynamic f o r c e s a l o n e were subsequently o b t a i n e d by s u b t r a c t i n g t h e "orgin" ordinates from those at corresponding cycle times in the i m m e d i a t e l y p r e c e d i n g d y n a m i c a l l y loaded t e s t .
3.2
4.
THEORETICAL METHOD A: JOURNAL ORB1T ANALYSI S
4.1
Introduction
T e s t R i q Equations o f M o t i o n
Equations o f m o t i o n f o r are: Fex F x = mh 'X'h + mo yo Fey
-
Fy = mh vh
+
mo
t h e housinq and o i l
yo
[11
where Fx, Fy a r e t h e o i l f i l m forces a c t i n g on t h e j o u r n a l . These f o r c e s a r e f u n c t i o n s o f t h e r e l a t i v e housing t o j o u r n a l displacements x, y, (measured d i r e c t l y by t r a n s d u c e r s ) , and r e l a t i v e v e l o c i t i e s iC, j o b t a i n e d by numerical d i f f e r e n t i a t i o n o f t h e measured x, y t i m e h i s t o r y . Accelerometers attached to the housing measured yh, j;h w i t h Xh, Yh obtained b y double n u m e r i c a l i n t e g r a t i o n . The j o u r n a l displacements w i t h r e s p e c t t o a f i x e d p o s i t i o n i n space were o b t a i n e d from:
[21 I t was
shown
moYo s 0.01
y direction.
t h a t i f -3 c X j / X h 4 +3 t h e n mhyh, and s i m i l a r l y for the
A l l experimental d a t a r e p o r t e d i n t h i s paper were found t o meet t h i s c o n d i t i o n . Morton (11) a l s o notes t h a t t h e o i l f i l m t r a n s v e r s e inertial forces may be neglected. The equations of motion may therefore be s imp1 if ied t o :
[31 3.3
T e s t B e a r i n g Data
The f o l l o w i n g d a t a d e f i n e t h e r e l e v a n t t e s t b e a r i n g dimensions and o p e r a t i n g c o n d i t i o n s used: 63.5 mm. 2x9.3 mm. lands 0.0836 mm 5.08 mm x 3600 1180. RPM. 0.0517 MPa (gauge) -0.175 MPa (gauge) 0.0186 Pa.s
Journal diameter B e a r i n g Length D iamet r a1 c 1ear ance O i l groove Journal speed O i l supply pressure C a v i t a t i o n pressure Effective viscosity 3.4
T e s t Procedure
A t each steady l o a d
-
speed combination, t e s t r i g temperatures were s t a b i l i s e d and t h e d a t a obtained f o r t h e determination o f a t t i t u d e a n g l e and e c c e n t r i c i t y . Data r e p o r t e d i n t h i s paper were a l l o b t a i n e d a t a s i n g l e v a l u e o f s t e a d y 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 angle. Time histories of the bearing housing h o r i z o n t a l and v e r t i c a l displacements r e l a t i v e t o t h e j o u r n a l , e x t e r n a l dynamic f o r c e s and
-
RIGOROUS
T h i s method i s based on t h e p r e d i c t i o n o f j o u r n a l displacement and v e l o c i t y components a t t h e end o f each t i m e s t e p by means o f displacement and velocity coefficients computed f o r t h e c u r r e n t c o n d i t i o n s . A full d e s c r i p t i o n o f t h i s method i s g i v e n i n reference (3). I n r e l a t i o n t o Method A the, t h e description "rigorous", e s s e n t i a l l y r e f e r s t o t h e use o f a n u m e r i c a l s o l u t i o n o f t h e f i l m p r e s s u r e d i s t r i b u t i o n ( 2 ) a t each o r b i t step. T h i s t y p e o f s o l u t i o n can accommodate f i n i t e l e n g t h t o diameter r a t i o s and p r e s s u r i s e d o i l feed f e a t u r e s . I n e v i t a b l y t h e term "rigorous" i s r e l a t i v e , and t h e most s i g n i f i c a n t a p p r o x i m a t i o n o f t h i s method i s c o n s i d e r e d t o be t h e r i g i d b e a r i n g assumption. As i n d i c a t e d i n t h e r e v i e w o f p r e v i o u s work, m o d e l l i n g b e a r i n g e l a s t i c i t y a t p r e s e n t r e s u l t s i n e x c e s s i v e computing t i m e u n l e s s approximate f i l m p r e s s u r e s o l u t i o n s a r e used. Bearing elasticity may be quite a p p r e c i a b l e i n some p r a c t i c a l . a p p l i c a t i o n s , n o t a b l y c o n n e c t i n g r o d bearings, b u t t h e t e s t b e a r i n g used t o o b t a i n t h e e x p e r i m e n t a l o r b i t presented i n t h i s paper, was c o n t a i n e d i n a substantial housing. Differences in the e x p e r i m e n t a l and t h e o r e t i c a l o r b i t s due t o bearing e l a s t i c i t y are t h e r e f o r e u n l i k e l y t o be s e r i o u s i n t h i s i n s t a n c e .
4.2
C a v i t a t i o n Model
A c a v i t a t i o n model which took account o f f l o w continuity, whilst assuming a constant c a v i t a t i o n pressure, was used i n Method A. D e t a i l s o f t h i s model a r e a l s o g i v e n i n reference ( 2 ) . No account i s t a k e n o f t h e n e g a t i v e p r e s s u r e s p i k e preceeding t h e r u p t u r e boundary which has been r e p o r t e d i n s e v e r a l e x p e r i m e n t a l s t u d i e s , b u t no p r a c t i c a l system f o r m o d e l l i n g t h i s f e a t u r e i s known t o e x i s t a t present. The method whereby c o n t i n u i t y i s s a t i s f i e d w i t h i n t h e c a v i t a t i o n zone i s s i m p l e and easy t o a p p l y w i t h i n a r e l a x a t i o n s o l u t i o n o f t h e f i l m pressure d i s t r i b u t i o n . Only t h e c a v i t a t i o n p r e s s u r e has t o be s p e c i f i e d , no assumptions o r i n i t i a l e s t i m a t i o n s f o r t h e l o c a t i o n o f t h e c a v i t a t i o n zone boundaries, o r t h e p r e s s u r e g r a d i e n t s a t t h e s e boundaries, a r e necessary. Furthermore, t h i s method i s eminently suitable to oil film history m o d e l l i n g , w h i c h may be d e f i n e d as t h e s t e p b y s t e p m o n i t o r i n g and u p d a t i n g o f t h e e x t e n t o f cavitation zones and the volumetric d i s t r i b u t i o n o f o i l w i t h i n them t h r o u g h o u t t h e journal orbit.
359 O i l F i l m H i s t o r y Model
4.3
A detailed description o f the o i l f i l m history model i s given i n r e f e r e n c e ( 3 ) and t h e f o l l o w i n q n o t e s o u t l i n e t h e main f e a t u r e s : The o i l f i l m i s d i v i d e d i n t o r e c t a n g u l a r elements, f o r t h e purpose o f s o l v i n g t h e f i l m pressure d i s t r i b u t i o n b y c o n s i d e r a t i o n o f f l o w continuity (2). O i l f i l m h i s t o r y modelling i s based on t h e premise t h a t i n a d y n a m i c a l l y loaded bearing, elements s u b j e c t t o c a v i t a t i o n do n o t have t o s a t i s f y f l o w c o n t i n u i t y , s i n c e t h e y may be f i l l i n g o r emptying a t any g i v e n time. D u r i n g each o r b i t t i m e s t e p t h e n e t t f l o w r a t e o f o i l t o each element i s computed, t h i s b e i n g used t o update t h e volume o f o i l w i t h i n each element a t t h e t i m e s t e p end: VoA(j,i) = Vo ( j , i )
+
qn ( j , i ) .
At
[4 1
T r a n s f e r o f a c a v i t a t i n g element t o a f u l l f i l m element occurs when e q u a t i o n [ 4 ] p r e d i c t s an o i l volume equal t o o r exceeding t h e The element volume: Vo(j,i) 3 VeJj,i). r e v e r s e t r a n s f e r may o c c u r d u r i n g t h e f i l m pressure relaxation process, when a s u b - c a v i t a t i o n p r e s s u r e i s computed f o r a f u l l film element. By means of the above processes, c a v i t a t i o n zones may expand o r c o n t r a c t i n any d i r e c t i o n a c c o r d i n g t o t h e p r e v a i l i n g c o n d i t i o n s as t h e dynamic c y c l e proceeds. 4.4
O r b i t Time Step S o l u t i o n
Since journal mass inertial f o r c e s were included, t h e o r b i t t i m e s t e p procedure was based on t h e s o l u t i o n o f t h e e q u a t i o n s o f m o t i o n f o r t h e mean c o n d i t i o n s d u r i n g each o r b i t step: (FexA-
FxA +
Fe,
- F,
)/ 2 = rn A i / A t
(FWA-FyA+ Fey- F,)/2
151
= m Ai/At
A t t h e s t a r t o f each t i m e s t e p t h e j o u r n a l displacement and v e l o c i t y components (x,y,i?,j) w i l l b e known, and t h e c o r r e s p o n d i n g o i l f i l m forces Fx, Fy can thus be computed. E x t e r n a l f o r c e components Fex, a t the end s t a r t p o i n t and FexA , FeyA hte': p o i n t can be i n t e r p o l a t e d f r o m t h e s p e c i f i e d e x t e r n a l l o a d c y c l e data. There remains 6 unknowns XA , YA , ~ C A , ?A , FXA ., FYA corresponding t o t h e end p o i n t . An a d d i t i o n a l f o u r e q u a t i o n s are t h e r e f o r e r e q u i r e d i n o r d e r t o obtain a solution. Two f u r t h e r e q u a t i o n s are p r o v i d e d by u s i n g o i l f i l m displacement and v e l o c i t y c o e f f i c i e n t s t o r e l a t e t h e o i l f i l m f o r c e changes w i t h t h e c o r r e s p o n d i n g displacement and v e l o c i t y changes d u r i n g t h e t i m e step. The displacement and v e l o c i t y c o e f f i c i e n t s a r e computed f o r t h e c o n d i t i o n s c o r r e s p o n d i n g t o t h e s t a r t p o i n t , and i t i s assumed t h a t t h e s e v a l u e s do n o t v a r y s i g n i f i c a n t l y over t h e t i m e step: +B,,Ai
[71
Ax+ A,, Ay + B,, A i + &,,A9
[81
AF, = & A x
AF, = A,,
+ A,,Ay
+B,A~
The remaining two e q u a t i o n s r e q u i r e d a r e o b t a i n e d b y r e l a t i n g t h e displacement changes during A t w i t h t h e corresponding v e l o c i t y
changes by assuming t h a t a c c e l e r a t i o n v a r i e s l i n e a r l y w i t h time during t h i s i n t e r v a l : A X = ( 2 i + i AA)t / 3
+ i(AtI2/6
[91
[lo1
by= (29 + ~ ~ ) A 1 / 3 + ~ ( b t ) ~ / 6
The d is p l acement and v e l o c i t y c o e f f i c i e n t s used i n e q u a t i o n s [ 5 ] and [ 6 ] were computed a t each o r b i t s t e p p o i n t b y t h e a p p l i c a t i o n o f displacement and v e l o c i t y p e r t u r b a t i o n s and film p r e s s u r e s o l u t i o n s t o determine t h e A critical c o r r e s p o n d i n g o i l f i l m forces. f e a t u r e o f t h e t i m e s t e p s o l u t i o n was t h e minimising of errors arising from the n o n - l i n e a r i t y of t h e s e c o e f f i c i e n t s . T h i s was achieved i n two ways: F i r s t l y , t h e d u r a t i o n of each t i m e s t e p ( A t ) was computed t o m a i n t a i n t h e s t e p displacement and v e l o c i t y changes w i t h i n c e r t a i n maximum values. These maximum changes were computed f r o m e m p i r i c a l A( ) = K1 K~.E functions o f t h e form where K1 and K2 a r e c o n s t a n t s . Secondly, t h e procedure made i n i t i a l e s t i m a t e s o f t h e s t e p and v e l o c i t y changes, these d is p l acement v a l u e s t h e n b e i n g used as t h e p e r t u r b a t i o n s t o compute t h e c o e f f i c i e n t s . Full details o f t h i s method a r e g i v e n i n r e f e r e n c e ( 3 ) .
-
5.
THEORETICAL METHOD 6: METHOD
5.1
Introduction
THE
REACTION
T h i s method achieves a s u b s t a n t i a l l y f a s t e r o r b i t a n a l y s i s by t h e use o f pre-computed velocity coefficients. The name "Reaction Method" was chosen since the velocity c o e f f i c i e n t s enable t h e t o t a l o i l f i l m f o r c e r e a c t i o n t o be e s t i m a t e d f o r any combination o f j o u r n a l v e l o c i t y and p o s i t i o n w i t h i n t h e b e a r i n g clearance. Both squeeze and wedge actions are included, together w i t h the i n t e r a c t i o n between them due t o c a v i t a t i o n . 5.2
O i l F i l m Force Equations
The o i l f i l m f o r c e equations, which f o r m t h e b a s i s o f t h e R e a c t i o n Method, a r e expressed i n p o l a r c o - o r d i n a t e terms: F t = Btt6o + B t r t
k6o
F r = Brt0o + B r r R + B r r t
.. RBo
[I11
c 121
The development of these equations is d e s c r i b e d i n d e t a i l i n r e f e r e n c e (4), and t h e p r i n c i p a l f e a t u r e s a r e as f o l l o w s : a)
The f o r c e components Fr, F t are t h e t o t a l o i l f i l m f o r c e s and n o t changes i n f o r c e f r o m an e q u i l i b r i u m osition. T h i s i s f a c i l i t a t e d by b e i n g t h e t o t a l e f f e c t i v e wedge v e l o c i t y since it incorporates both t h e angular v e l o c i t y o f t h e j o u r n a l about i t s own a x i s ( w ) and t h e angular v e l o c i t y o f t h e j o u r c a l a x i s about t h e b e a r i n g a x i s (8) i.e. 0,=0-W2 (assuming a stationary bearing).
lo
b)
The
velocity
coefficients
are
computed f o r a range o f p a r t i c u l a r v a l u e s of e c c e n t r i c i t y r a t i o and interpolation (linear or logarithmic) between the adjacent values is carried out for any given eccentricity ratio. C o n s i d e r a t i o n o f t h e dimensionless forms f o r t h e v e l o c i t y c o e f f i c i e n t s i s f u l l y d e t a i l e d i n reference (4). This indicated t h a t t h e v a l i d i t y o f dimensionless v e l o c i t y c o e f f i c i e n t s i s r e s t r i c t e d t o given values o f dimensionless s u p o l v and c a v i t a t i o n pressures, wbictj in turn are and t h e p r o d u c t funcJions o f R, 8, R . 8, according t o t h e type o f coefficient
.
Predicted o r b i t s using c o e f f i c i e n t s d e r i v e d b y R p e r t u r b a t i o n amplitudes o f 0.6, 1.2 and 1.8 mm/s d i d n o t It indicate significant differences. is therefore evident that the predicted o r b i t s are not unduly sensitive to the perturbation a m p l i t u d e f r o m which t h e v e l o c i t y coefficients a r e obtained. This means t h a t t h e p r e d i c t i o n accuracy when u s i n g dimensionless velocity c o e f f i c i e n t data i s not c r i t i c a l l y dependant on satisfying the s i m i 1a r i t y r e q u i r e m e n t s w i t h r e s p e c t to dimensionless supply and c a v i t a t i o n pressures. Reference ( 4 ) i n d i c a t e d t h a t t h e F r R curves for 8, # ? are = 0 asymptotic t o t h e c u r y e f o r €lo as t h e magnitude o f R i n c r e a s e d i n both positive and negative directions. A c c o r d i n g l y , when u s i n g t h e l i n e a r i s e d e q u a t i o n s [ll] and [12], Fr i s n o t a l l o w e d t o f a j l below t h e v a l u e c o r r e s p o n d i n g t o 8, = 0; i.e. F r E r r R.
-
The determination of velocity c o e f f i c i e n t s by t h e a p p l i c a t i o n o f v e l o c i t y perturbations i s a quasi dynamic s o l u t i o n i n t h a t i t i g n o r e s t h e dependence on p r e v i o u s c o n d i t i o n s I n o t h e r words t h e i n the o i l f i l m . R e a c t i o n Method does n o t t a k e account history, and s h o u l d of oil film t h e r e f o r e be used w i t h c a u t i o n i n s i t u a t i o n s where t h i s f a c t o r may be s i g n if ic a n t . F a s t O r b i t Time S t e p p i n g Procedure orbit time steppinq procedure was v i r t u a l l y i d e n t i c a l t o t h a t used f o r Method A. A Cartezian co-ordinate system was retained with displacement and velocity c o e f f i c i e n t s i n C a r t e z i a n terms computed by [ll] and [12] with means o f equations appropriate Polar Cartezian transformation. The most i m p o r t a n t d i f f e r e n c e i n t h e procedure concerned t h e d e t e r m i n a t i o n o f t h e t i m e s t e p duration A t . As shown i n r e f e r e n c e ( 3 ) , A t f o r each t i m e s t e p was determined t o ensure t h a t t h e changes i n the components of displacement and v e l o c i t y d u r i n g t h e s t e p were w i t h i n p r e s c r i b e d l i m i t s which were f u n c t i o n s o f eccentricity ratio. T h i s was necessary i n
-
o r d e r t o m a i n t a i n an a c c e p t a b l e accuracy o f t h e p r e d i c t e d of o i l f i l m f o r c e components a t t h e t i m e s t e p end when u s i n g l i n e a r i s e d displacement and v e l o c i t y c o e f f i c i e n t s . At was found b y p r o g r e s s i v e r e d u c t i o n o f t r i a l values u n t i l a l l t h e above l i m i t s were satisfied. I n Method A t h e i n i t i a l t r i a l v a l u e o f A t was a r b i t r a r i l y s e t equal t o about 1.5% o f t h e c y c l e t i m e and reduced by 1% f o r each successive t r i a l . T h i s approach ensured t h a t A t was w i t h i n 1% o f t h e maximum A t p e r m i t t e d b y t h e increment l i m i t c o n s t r a i n t s . The c o r r e s p o n d i n g t i m e r e q u i r e d t o compute A t was a small p a r t of t h e t o t a l computing time. W i t h t h e R e a c t i o n Method, t h e computing t i m e r e q u i r e d t o e s t a b l i s h A t became s i g n i f i c a n t . It was clearly necessary to find a satisfactory compromise between the c o n f l i c t i n g requirements t o maximise A t w i t h i n t h e g i v e n c o n s t r a i n t s , and y e t m i n i m i s e t h e computing t i m e r e q u i r e d t o determine t h i s value. After several t r i a l s , t h e optimum s o l u t i o n f o r t h e t e s t c o n d i t i o n s covered i n t h i s paper was found t o be as f o l l o w s : The i n i t i a l t r i a l A t was s e t a t 30% g r e a t e r t h a n t h e v a l u e f o r t h e p r e v i o u s t i m e step, and t h i s v a l u e was t h e n reduced by 10% f o r each successive t r i a l .
6.
D I S C U S S I O N OF RESULTS
Journal o r b i t s predicted by both t h e o r e t i c a l methods and those measured with the e x p e r i m e n t a l t e s t r i g a r e presented i n F i q u r e s The c o r r e s p o n d i n g e x t e r n a l f o r c e 2 t o 4. c y c l e d a t a i s a l s o g i v e n i n p o l a r form. E x t e r n a l f o r c e c y c l e f r e q u e n c i e s a r e a t once ( T e s t c o n d i t i o n s 1 and 3, F i g u r e s 2.and4.) and t w i c e ( T e s t c o n d i t i o n 2, F i g u r e 3) t h e j o u r n a l rotational frequency. The form of the external force cycle i s similar f o r t e s t c o n d i t i o n s 1 and 2 and r e s u l t s i n o b l i q u e o r b i t s . However, t h e e x t e r n a l f o r c e c y c l e f o r t e s t c o n d i t i o n 3 i s d i f f e r e n t and r e s u l t s i n a substantially horizontal orbit. All three t e s t c o n d i t i o n s show good agreement between t h e o r b i t s p r e d i c t e d by b o t h t h e o r e t i c a l methods. The d i f f e r e n c e s between t h e r e s u l t s o f t h e two t h e o r e t i c a l methods a r e m a i n l y due t o t h e approximations i n t r o d u c e d when f i t t i n q t h e r e l a t i v e l y s i m p l e e q u a t i o n s [ll] and [12], used i n Method B y t o p r e d i c t e d o i l f i l m f o r c e j o u r n a l v e l o c i t y data. Agreement between t h e e x p e r i m e n t a l t h o s e p r e d i c t e d by b o t h t h e o r e t i c a l However, g e n e r a l 1y s a t i s f a c t o r y . two r e g i o n s i n which s i g n i f i c a n t between the experimental and o r b i t s a r e apparent.
o r b i t s and methods i s there are differences theoretical
The l a r g e s t apparent d i s c r e c a n c y was i n t e s t c o n d i t i o n 3 where F i g u r e 4. shows t h a t b o t h t h e o r e t i c a l methods have s u b s t a n t i a l l y g r e a t e r excursions i n t h e d i r e c t i o n against r o t a t i o n . I n approachinq t h e o r b i t e x t r e m i t y a t t / T h 0.15 a s t r o n g e r o i l f i l m wedge a c t i o n w i l l be generated b y t h e a n t i c l o c k w i s e movement o f t h e j o u r n a l c e n t r e ( n e g a t i v e j). T h i s r e g i o n a l s o c o i n c i d e s w i t h n e g a t i v e R. These two e f f e c t s combine t o cause a much s t r o n g e r tendency t o c a v i t a t e i n t h e area t o t h e downstream s i d e o f t h e minimum f i l m t h i c k n e s s p o s i t i o n . It i s therefore postulated that the difference i n t h e e x p e r i m e n t a l and t h e o r e t i c a l o r b i t s i n
36 1
t h i s r e g i o n , may be due t o t h e c a v i t a t i o n pressure b e i n g s i g n i f i c a n t l y lower t h a n t h a t used f o r t h e p r e d i c t i o n o f t h i s dynamic situation. The c a v i t a t i o n p r e s s u r e used t o compute t h e t h e o r e t i c a l o r b i t s , was based on t h e v a l u e d e r i v e d by P a r k i n s ( 1 ) t o y i e l d agreement between the experimental and predicted equilibruim positions. Sensitivity o f t h e o r b i t s t o such d i f f e r e n c e s i n t h e c a v i t a t i o n p r e s s u r e would be i n c r e a s e d by t h e
Dynamic Force Cycle
80
Experimental -o-Theoretical method A .. * .. . -t) ineorericai metnoa
----
I,
Test Condition 1 External Force Data and Journal Orbit Fig 2
Experimental Theoretica I method A
low f o r c e s i n t h e c o r r e s p o n d i n g p a r t o f t h e dynamic f o r c e c y c l e . Hume and Holmes ( 1 2 ) have a l s o i n d i c a t e d t h a t t h e c o r r e l a t i o n between e x p e r i m e n t a l and t h e o r e t i c a l o r b i t s o f i s significantly a squeeze f i l m bearing, dependent upon t h e p r o v i s i o n f o r s u b s t a n t i a l l y sub-atmospheric c a v i t a t i o n pressures i n t h e t h e o r e t i c a l model. I n c o n t r a s t w i t h t h e above o b s e r v a t i o n s f o r t e s t c o n d i t i o n 3, t h e o r b i t a l movement a g a i n s t the direction o f rotation i n t e s t condition 1 ( F i g u r e 2) corresponds t o t h e b u i l d up t o maximum load. This r e s u l t s i n a mainly p o s i t i v e 6 and hence much l e s s c a v i t a t i o n t h a n t h a t experienced i n t h e corresponding p a r t o f The reduced t h e o r b i t f o r t e s t c o n d i t i o n 3. extent o f cavitation, combined w i t h t h e p r o x i m i t y t o maximum 1oad, would r e n d e r o i 1 f i l m f o r c e p r e d i c t i o n e r r o r s associated w i t h cavitation insignificant i n t h i s situation. These c o n d i t i o n s a l s o a p p l y t o t e s t c o n d i t i o n 2 ( F i g u r e 3), and e x p l a i n s why t h e s i g n i f i c a n t differences in the experimental and t h e o r e t i c a l e x t e n t o f movement a g a i n s t t h e d i r e c t i o n o f r o t a t i o n i n t e s t c o n d i t i o n 3 does n o t occur i n t h e other t e s t conditions. The second i m p o r t a n t d i f f e r e n c e between t h e t h e o r e t i c a l and e x p e r i m e n t a l o r b i t s , i s t h e s i g n i f i c a n t l y greater e c c e n t r i c i t y r a t i o o f t h e e x p e r i m e n t a l o r b i t s i n t h e v i c i n i t y o f q~ = 200 i n F i g u r e s 2 and 3, and s i m i l a r l y a t w = 400 i n F i g u r e 4. I n a l l three t e s t conditions, the location of the above d i s c r e p a n c y corresponded t o t h e r e g i o n o f maximum t o t a l load. Since t h e t h e o r e t i c a l 'models assumed a r i g i d b e a r i n g and j o u r n a l , e l a s t i c d i s t o r t i o n i s t h e most l i k e l y cause o f t h e l a r g e r experimental e c c e n t r i c i t y r a t i o s a t t h e more h i g h l y loaded p a r t s o f t h e dynamic cycle.
7 v
c -Dynamic Force Cycle
Steadv
Measured static equilibrium
-2p7 Theoretical method A Theoretical method 8
Fig 3
Test Cmdition 2 Force Data and Journal Orbit
-----
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Test Condition 3 External Force Data and Journal Orbit Fig 1
362
Theoretical journal o r b i t s f o r t e s t condition 1 ( F i g u r e 2) were computed b o t h w i t h and w i t h o u t t h e o i l f i l m h i s t o r y model u s i n g The d i f f e r e n c e s were n e g l i g i b l e . Method A. T h i s r e s u l t was c o n s i d e r e d t o be due t o t h e combination o f a small o r b i t i n r e l a t i o n t o t h e c l e a r a n c e c i r c l e , and t o t h e e f f i c i e n t supply of oil provided by the full c i r c u m f e r e n t i a l groove.
7.
(3)
JAKEMAN R. W. "The I n f 1uence of C a v i t a t i o n on t h e n o n - l i n e a r i t y o f Velocity Coefficients in a Hydrodynamic J o u r n a l Bearing." To b e p u b l i s h e d : Leeds - Lyon T r i b o l . Symp. Leeds Sept. 1986.
CONCLU SI ON S
T h i s paper compares measured j o u r n a l o r b i t s w i t h t h e p r e d i c t i o n s o f two t h e o r e t i c a l methods. The t e s t c o n d i t i o n s used covered d i f f e r e n t forms o f e x c i t a t i o n a t b o t h once and t w i c e r o t a t i o n a l frequency.
(5)
Good agreement was shown between t h e r e s u l t s o f b o t h t h e o r e t i c a l methods, any d i f f e r e n c e b e i n g l a r g e l y due t o t h e approximations i n t r o d u c e d i n Method B i n o r d e r t o achieve f a s t e r computation. Method 6, introduces simple equations f o r t h e t o t a l o i l f i l m f o r c e components, u s i n g a new t y p e o f v e l o c i t y coefficient. An e q u i v a l e n t angular v e l o c i t y i s used which combines r o t a t i o n o f t h e j o u r n a l about i t s a x i s w i t h r o t a t i o n o f t h e j o u r n a l a x i s about t h e b e a r i n g a x i s , t h u s f a c i l i t a t i n g application to both steady and dynamic situations. The computation t i m e f o r t h i s Method was reduced b y a f a c t o r o f o v e r 300 r e l a t i v e t o t h a t f o r Method A. The e f f e c t of o i l f i l m h i s t o r y was f o u n d t o be n e g l i g i b l e i n t h e case analysed. Generally good agreement between the experimental and theoretical orbits was attained. The main differences were considered t o be r e l a t e d t h e i n f l u e n c e o f c a v i t a t i o n a t low l o a d and b e a r i n g e l a s t i c i t y a t h i g h load. 8.
REFERENCES PARKINS D.W. "Theoretical and Experimental Determination of the Dynamic Characteristics of a Hydrodynamic Journal Bearing". A.S.M.E. J n l . Lub. Tech. Vol. 101. A p r i l 1979 pp 129 139.
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BOOKER J.F. "Dynamically loaded Journal Bearings: Nume r ica 1 A p p l i c a t i o n o f t h e M o b i l i t y Method". A.S.M.E. J n l . Lub. Tech. Jan 1971. pp 168 - 176. and CRAVEN A.H. "The HOLMES R. I n f l u e n c e o f C r a n k s h a f t and Flywheel mass on t h e Performance o f Enqine Main B e a r i n g s " 1.Mech.E T r i b o l . Conv 1971 Paper C63/71 pp 80 85.
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JONES G.J. " C r a n k s h a f t Bearings: O i l Film History" 9 t h Leeds-Lyon T r i b o l Symp.Sept.1982 pp 83 - 88. LA BOUFF G.A. and BOOKER J.F. "Dynamically Loaded Journal Bearings: A F i n i t e Element Treatment f o r R i g i d and Elastic S u r f aces". A.S.M.E./A.S.L.E. Conf. Oct. 1984 A.S.M.E. Paper 84 TRIB 11.
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(9)
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FANTINO B., FRENE J. and GODET M. "Dynamic Behaviour o f an E l a s t i c Theoretical Connecting Rod B e a r i n g -. S t u d i e s i n Study". SAE/SP-539 Engine Bearings and Lubrication, Paper No. 83037, Feb. 1983 pp 23 32.
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GOENKA P.K. and OH K.P. "An Optimum A Connecting Rod Design Study Lubrication Viewpoint". A.S.L.E./A.S.M.E T r i bology Conference. A t l a n t a October 1985. A.S.M.E. P r e p r i n t No. 85 Trib 50.
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ACKNOLEDGEMENTS
The a u t h o r s would l i k e t o thank t h e Head o f the College of Manufacturing, Cranfield I n s t i t u t e o f Technology and t h e Committee o f L l o y d ' s R e g i s t e r o f Shipping f o r p e r m i s s i o n t o p u b l i s h t h i s paper. They would a l s o l i k e t o thank t h e i r many c o l l e a g u e s a t b o t h C r a n f i e l d I n s t i t u t e o f Technology and L l o y d ' s R e g i s t e r o f Shipping f o r t h e h e l p and c o o p e r a t i o n throughout t h e work.
(1)
JAKEMAN R.W. "Journal O r b i t Analysis t a k i n g account o f O i l F i l m H i s t o r y and J o u r n a l Mass". Proc. o f Conf: Numerical Methods i n Laminar and T u r b u l e n t Flow. Swansea. J u l y 1985 pp 199 - 210.
JAKEMAN R.W. "A n u m e r i c a l a n a l y s i s method based on f l o w c o n t i n u i t y f o r hydrodynamic journal bearinqs". T r i b o l . I n t . Vol. 17. No. 6. Dec. 1984 pp 325 - 333.
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MORTON, P.G. "Measurement o f t h e Dynamic C h a r a c t e r i s t i c s o f a Large A.S.M.E. Sleeve B e a r i n g " . Trans. Jnl. Lub. Tech. Jan. 1971. pp. 143 150.
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HUME, B and HOLMES, R. "The Role o f Sub-Atmospheric F i l m Pressures i n t h e V i b r a t i o n Performance o f Squeeze F i l m Bearings". I. Mech. E. J n l and Mech. No. 5. 1978. Eng. Science. Vol. 20. p . 283.