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The lubrication of parallel surface thrust bearings Currie, Iain George 1962

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THE LUBRICATION OP PARALLEL SURFACE THRUST BEARINGS  by IAIN GEORGE CURRIE  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  i n the Department of MECHANICAL ENGINEERING  We acoept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  j  MAY, 1962  .  i  In presenting this thesis i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the University of B r i t i s h Columbia, I agree that the l i b r a r y s h a l l make i t f r e e l y available f o r reference and study.  I further agree that permission f o r extensive copying of this thesis  f o r scholarly purposes may be granted by the Head of my Department or by his representatives.  It i s understood that copying of this thesis f o r  f i n a n c i a l gain s h a l l not be allowed without my written permission.  Department of Mechanical Engineering The University of B r i t i s h Columbia, Vancouver 8, Canada. May,  1962.  ii ABSTRACT  The p a r a l l e l surface thrust bearing has been studied both t h e o r e t i c a l l y and experimentally.  The general equations governing the  laminar flow of a Newtonian f l u i d are presented and s u i t a b l y reduced to describe the flow of lubricant through a p l a i n c o l l a r bearing.  A computer  solution of the r e s u l t i n g equations has been obtained i n which the variations, of density and v i s c o s i t y with temperature are accommodated and the circumferent i a l leakage of o i l from the bearing i s recognised.  The r e s u l t i n g performance  curves indicate that useful load carrying capacities, produced by a  'thermal  wedge' e f f e c t , are possible with p a r a l l e l surface thrust bearings. A series of tests was carried out on a three inch diameter bearing operating at speeds ranging from  15,000 to 19,000 r.p.m.  The  r e s u l t s confirm that hydrodynamic l u b r i c a t i o n may be achieved with a p a r a l l e l surface thrust bearing.  The experimental values obtained f o r the load  carrying capaoity and the c o e f f i c i e n t of f r i c t i o n were both less than the t h e o r e t i c a l predictions.  The discrepancies appear to be caused, f o r the  most part, by an increase i n the o i l temperature r e s u l t i n g from of the lubricant i n the bearing.  entrainment  vii  ACKNOWLEDGEMENT  The experimental work described i n t h i s report was c a r r i e d out i n the Lubrication Laboratory at the University of B r i t i s h Columbia, and the t h e o r e t i c a l calculations were performed at the Computing Centre i n the University of B r i t i s h Columbia.  The use of  these s p e c i a l f a c i l i t i e s i s g r a t e f u l l y acknowledged. The author would also l i k e to thank the many people whose assistance has made this report possible.  In p a r t i c u l a r , thanks are  due to the followingtProfessor W.O. Richmond, f o r the use of the f a c i l i t i e s of the Mechanical Engineering  Department of the University of B r i t i s h Columbia.  Professors J . Young and C.A. Brockley,  f o r t h e i r guidance and  assistance during a l l phases of the project. The National Research Council of Canada, with N.R.C. Grant No. A1089.  f o r sponsoring the  research  TABLE OF CONTENTS  CHAPTER I I. l . Introduction CHAPTER I I I I . 1. The Governing Equations f o r Film Lubrication 11.2. Solution of the Governing Equations 11.3. Theoretical Performance of Bearing CHAPTER I I I I I I . l . Apparatus and Measurements CHAPTER IV IV. 1. Experimental Procedure IV.2. Experimental Results IV. 3 . Discussion of Results  •  , '  CHAPTER V V. l . Summary and Conclusions V . 2 . Suggestions f o r Future Research APPENDIX APPENDIX APPENDIX APPENDIX  I. II. III. IV.  BIBLIOGRAPHY  The Energy Equation i n Lubrication Physical Properties of S h e l l Turbo 27 O i l C a l i b r a t i o n Tests on Apparatus Tables of Observed Results  iv LIST OF FIGURES Fig. No.  Title  Page  1  Transformation of Sector into Rectangle  13  2  Grid Point Notation for Relaxation Process  15  3  Theoretical Performance at 15,000 r.p.m.  21  4  Theoretical Performance at 20,000 r.p.m.  22  5  Schematic Layout of Apparatus  28  6  Sectional View of Test Bearing  29  7  Details of Thrust Pad  30  8  Test Bearing Before Assembly  31  9  Assembled Bearing in Position  32  10  Differential Transformer Indicator and Thermocouple Potentiometer  33  11  Displacement Transducer with Special Tip  34  12  General View of Apparatus  35  13  Experimental Load Carrying Capacity  41  14  Experimental Coefficient of Friction  42  15  Comparison of Experimental Data  43  16  Physical Properties of Shell Turbo 2? Oil  59  17  Calibration Curve for Copper-Constantan Thermocouples  62  18  Calibration Curve for Loading System  63  V LIST OF SYMBOLS  The following symbols have been used throughout the text. Any symbol not l i s t e d i s defined on introduction. f,0,^ u,v,ur  Independent c y l i n d r i c a l co-ordinates Independent rectangular co-ordinates V e l o c i t y i n x, y and 3 d i r e c t i o n respectively Substantial time derivative  CL A B C> Cv b E f Ff h H A M N f> P P' fr r, f„, A t to t' T Tf U 0  W Z oc (& A  V Q X  Grid size Constant Constant Speoific heat at constant pressure S p e c i f i c heat at constant volume Dimensionless density Internal energy Coefficient of f r i c t i o n Body force, d i r e c t i o n as subscripted F r i c t i o n force Film thickness Enthalpy Thermal conductivity Dimensionless v i s c o s i t y Rotational speed Pressure Dimensionless pressure Load per unit area Heat f l u x , d i r e c t i o n as subscripted Inner radius of bearing Outer radius of bearing Mean radius of bearing Dimensionless co-ordinate Temperature Temperature at i n l e t Time Dimensionless temperature F r i c t i o n torque V e l o c i t y of s l i d e r Velocity, d i r e c t i o n as subscripted Load per pad V i s c o s i t y i n centipoise Included angle of sector pad Constant Dilatation • V « T P Operator Dimensionless co-ordinate Constant  vi  Ji /lo p po X 0 ui)  Viscosity V i s c o s i t y at i n l e t Density Density at i n l e t Stress tensor Dissipation function Angular v e l o c i t y  CHAPTER I 1.  Introduction  2 I . l . Introduction P r i o r to, and f o r some time subsequent to the introduction of Reynolds'  c l a s s i c a l paper [ l ] * i n 1886,  thrust bearings were very  I n e f f i c i e n t when compared to the journal bearing.  In his quantitative  analysis, Reynolds showed that a geometric r e s t r i c t i o n i n the d i r e c t i o n of motion was necessary to produce a load oarrying f l u i d f i l m i n a bearing. The p a r a l l e l surface thrust bearing, which at that time was operating at low speed and with high loads imposed on i t , had no such geometric  restriction.  Due to these severe running conditions and the lack of o i l grooves, or other means of aiding o i l entry to the bearing surfaces, the thrust bearing was then operating i n the boundary l u b r i c a t i o n region.  By fortuitous  coincidence, however, the journal bearing already possessed the geometry required f o r hydrodynamio l u b r i c a t i o n and so operated with a lower c o e f f i c i e n t of f r i c t i o n than the thrust bearing. As a oonsequence of Reynolds' theory, thrust bearings of the early twentieth century, and l a t e r , were designed on the t i l t i n g pad p r i n c i p l e , as introduced by Kingsbury and M i c h e l l , and the p a r a l l e l surface bearing became v i r t u a l l y obsolete.  The contention was that the l a t t e r bearing could  support a f l u i d f i l m under load only to the extent of the s t a t i c pressure. On several occasions, however, i t was noted that p a r a l l e l surface bearings performed better than expected inasmuch as useful loads were carried with a low c o e f f i c i e n t of f r i c t i o n . were obtained by Fogg [2] who  The most conclusive results  recorded load carrying capacities of the same  order as a t i l t i n g pad bearing with almost the same e f f i c i e n c y as the t i l t i n g pad bearing.  His tests were carried out at comparatively high speeds and  the annular thrust surfaces were divided into a number of sector shaped pads  * Numbers i n square brackets r e f e r to l i s t of references i n Bibliography.  3 by the introduction of r a d i a l o i l grooves i n the stationary surfaces. These o i l grooves, together with circumferential o i l seals, ensured that the bearing surfaces were always supplied with o i l . Fogg explained this apparent hydrodynamic action as a wedge' e f f e c t , analagous t i l t i n g pad bearings.  'thermal  to the taper wedge effect which predominated i n the That i s , instead of having a constant volume of  lubricant flowing through a diminishing area, Fogg postulated that due to the thermal expansion of the lubricant a s i m i l a r e f f e c t would be obtained by an expanding volume of lubricant flowing through a constant area. As a r e s u l t of Fogg's findings, more attention was given to the p a r a l l e l surface thrust bearing,  most of this attention being  quantitative i n nature. Bower [3] used the equation of mass continuity, as opposed to volume continuity, i n a revised form of the governing equation. no side leakage and  By assuming  l i n e a r variations of v i s c o s i t y and density along the  bearing, Bower concluded that a. load carrying f l u i d f i l m could be produced  by  the thermal wedge e f f e c t . Cameron and Wood [k]  used the revised form of Reynolds'  equation i n conjunction with an energy equation tor a r r i v e at two p a r t i a l d i f f e r e n t i a l equations f o r the pressure and temperature  simultaneous distributions.  Using mathematically expressed variations of v i s c o s i t y and density with temperature,  these equations were then solved on the assumption  of no side  leakage from the bearing. Shaw £5] obtained a solution to the revised governing equation by assuming no side leakage, constant v i s c o s i t y and a l i n e a r v a r i a t i o n of density with temperature.  In t h i s way he was able to compare  the t i l t i n g pad and the p a r a l l e l surface bearings on the basis of equal f i l m thicknesses.  Starting with the general equations f o r the steady flow of a viscous f l u i d , Cope [ 6 j arrived at the revised form of the pressure equation, but obtained an energy equation which d i f f e r e d from that used previously by Cameron and Wood.  The r e s u l t i n g equations were solved by Cope  variations of v i s c o s i t y and density with temperature  f o r known  and f o r no side leakage*  In t h i s way he showed that under conditions of low v a r i a t i o n of v i s c o s i t y with temperature,  large v a r i a t i o n of density with temperature,  and small f i l m  thicknesses, the thermal wedge may completely outclass the geometric wedge. Experimental r e s u l t s were obtained by Kettleborough [ 7 ] f o r a parallel  surface bearing running at the comparatively low speed of 695 r.p.m.  The c o e f f i c i e n t of f r i c t i o n so obtained was about twice that obtained by Fogg. The work performed on p a r a l l e l surface bearings has tended to be either e n t i r e l y experimental or e n t i r e l y t h e o r e t i c a l , the t h e o r e t i c a l analyses being used only to j u s t i f y the thermal wedge idea and to compare the performance of such a bearing to that of a s i m i l a r t i l t i n g pad type.  The  f i r s t r e a l attempt to correlate theoretical and experimental r e s u l t s f o r a given p a r a l l e l surface bearing was due to Young [ 8 ] ,  In h i s quantitative  analysis, Young presented three d i f f e r e n t solutions to the equations as presented by Cope, the most rigorous solution being a relaxation process, f i r s t used i n l u b r i c a t i o n by Christopherson [ 9 j , which takes account of side leakage.  These d i f f e r e n t solutions were compared with each other and with  the experimental r e s u l t s obtained from a thrust bearing operating i n the speed range 4,000 to 16,000 r.p.m.  The relaxation solution gave f a i r l y  good agreement at the points checked but theoretical performance curves could not be obtained this way due to the large amount of work involved i n even one solution.  The experimental c o e f f i c i e n t s of f r i c t i o n were higher  than those obtained by Fogg, but lower than those obtained by Kettleborough. The v a l i d i t y of the assumptions which accompany the  t h e o r e t i c a l analysis of the f l u i d flow i n a p a r a l l e l surface thrust bearing, evidently depends on the nature of the operating conditions i n the test bearing i t s e l f .  Also, the theoretioal results obtained can be successfully  compared with the experimental results only i f c e r t a i n quantities can be accurately measured.  I t thus seems desirable to revise, and modify i f  necessary, the equations governing the flow of f l u i d i n a p a r a l l e l surface bearing;  to obtain as sophisticated a solution to the governing equations as  i s reasonable;  to add to the experimental data available and to compare the  theoretical and experimental r e s u l t s so obtained.  I t was with these  immediate objectives that the current research program was i n i t i a t e d .  CHAPTER II. 1. II. 2 . II. 3 .  II  The Governing Equations for Film Lubrication Solution of the Governing Equations Theoretical Performance of Bearing  7 I I . 1.  The Grpverning E q u a t i o n s f o r F i l m L u b r i c a t i o n The e q u a t i o n s g o v e r n i n g t h e f l o w o f t h e l u b r i c a n t i n a; p a r a l l e l  s u r f a c e b e a r i n g w i l l be o b t a i n e d from t h e g e n e r a l e q u a t i o n s g o v e r n i n g t h e f l o w o f a v i s c o u s , c o m p r e s s i b l e , heat c o n d u c t i n g f l u i d .  The d e r i v a t i o n o f t h e s e  g e n e r a l e q u a t i o n s i s p r e s e n t e d i n most t e x t s on h y d r o d y n a m i c s , gasdynamics and related fields.  A c c o r d i n g l y , t h e g e n e r a l e q u a t i o n s as p r e s e n t e d i n [ i o ]  are:-  >  (i)  y>||= r-»-[V fj  •  - (2)  fjfr-jfr-iV'V-^W)  •  -  #  = -Av-*) :  /  (3)  i n which the symbols u s e d a r e d e f i n e d i n the ' L i s t o f Symbols' i n t h e p r e l i m i n a r y pages o f t h e t h e s i s . m o t i o n and e n e r g y , r e s p e c t i v e l y ,  These a r e t h e e q u a t i o n s o f c o n t i n u i t y , i n w h i c h t h e p r o p e r t i e s o f the f l u i d  are  variables. The r e d u c t i o n o f t h e s e g e n e r a l e q u a t i o n s t o t h e i r f i n a l form w i l l be c a r r i e d out by t h e method p r e s e n t e d b y Cope [6j, r e s u l t i n g e q u a t i o n s w i l l be s l i g h t l y d i f f e r e n t .  although the  The a l g e b r a i c m a n i p u l a t i o n s ,  a l t h o u g h s t r a i g h t f o r w a r d , t e n d t o become l e n g t h y and t e d i o u s .  Consequently,  i t i s proposed t o i n d i c a t e the s t e p s i n d i v i d u a l l y and t o p r e s e n t the end r e s u l t s o f p e r f o r m i n g each We b e g i n c y l i n d r i c a l co-ordinates t  step. by expanding e q u a t i o n s ( l ) , (2)  and (3)  in  i n w h i c h t h e components o f the heat f l u x v e c t o r q a r e g i v e n bys —  ]  >  •  •  (7)  F o r a Newtonian f l u i d we have?  Thus assuming the f l o w i s s t e a d y and t h e g e n e r a l e q u a t i o n s (*f),  l a m i n a r and t h a t t h e f l u i d i s N e w t o n i a n ,  (5) and (6) becomes  9 i n w h i c h t h e c o n t i n u i t y e q u a t i o n ( l O ) has been r e a r r a n g e d t o i t s more u s u a l form. Since the surfaces  i n a bearing are close together,  parallel  and i n r e l a t i v e t a n g e n t i a l m o t i o n , i t i s r e a s o n a b l e t o assume: ( i ) That t h e v a r i a t i o n o f p r e s s u r e ,  d e n s i t y , t e m p e r a t u r e , v i s c o s i t y and  t h e r m a l c o n d u c t i v i t y a c r o s s the f i l m a r e f a r l e s s i m p o r t a n t t h a n t h e i r v a r i a t i o n along i t .  That i s , i t i s assumed t h a t  i j> » t ,JUi and A a r e f u n c t i o n s o f r and & o n l y . ( i i ) That the v e l o c i t y g r a d i e n t s a c r o s s t h e f i l m a r e much more i m p o r t a n t t h a n v e l o c i t y gradients p a r e l l e l to i t .  S y m b o l i c a l l y , t h i s assumption s t a t e s  that  and so o n , so t h a t o n l y the f i r s t named need be r e t a i n e d i n groups o f such t e r m s . ( i i i ) That t h e f l u i d v e l o c i t y p e r p e n d i c u l a r t o t h e f i l m i s v e r y s m a l l so t h a t we may w r i t e V ( i v ) That t h e body f o r c e s , are n e g l i g i b l e .  4  =  o t h e r t h a n t h o s e caused by the m o t i o n o f the  U s i n g t h e n o t a t i o n a d o p t e d , the a s s u m p t i o n means  ^ = ^=5 (v)  0  =  fluid, that  0  That t h e e n t h a l p y o f the f l u i d i s a f u n c t i o n o f temperature o n l y and t h a t t h e s p e c i f i c heat remains c o n s t a n t .  C o n s e q u e n t l y , we may w r i t e  A p p l y i n g t h e s e f i v e assumptions t o e q u a t i o n s  ( l O ) , ( l l ) and  (12), t h e y reduce t o the f o r m :  ±-Jir(f™)+i"k(/> *) v  « 0  •  • (13)  10  (1*)  (15) These e q u a t i o n s a r e s t i l l r a t h e r e l a b o r a t e and i t  is  d e s i r a b l e to reduce them s t i l l f u r t h e r b e f o r e a t t e m p t i n g a s o l u t i o n . can be done by u s i n g a c t u a l measurements, values of  , etc.,  ^,A  This  f o r a t y p i c a l set of c o n d i t i o n s , f o r  c a l c u l a t i n g from them the o r d e r s o f magnitude o f  the  v a r i o u s terms i n the above e q u a t i o n s and n e g l e c t i n g any terms w h i c h a r e found t o be s m a l l .  Consider, i n p a r t i c u l a r , a 3 i n c h diameter b e a r i n g running at  . 15,000 r . p . m . w i t h a l i g h t l u b r i c a t i n g o i l a t a temperature o f 150°P and f i l m t h i c k n e s s o f 0.001 be u s e d .  inch.  These f i g u r e s a r e estimated f o r t h e t e s t b e a r i n g t o  For these c o n d i t i o n s Table I i s compiled i n which a l l q u a n t i t i e s  have been c o n v e r t e d t o the u n i t s o f l b s . ,  ft.,  s e e s . , and ° F .  TABLE I  Ii  >  p  3000  1.614  M 0.000214  0.00083  12,470  A  r  t  0.0173  0.1  150  Now .-J-j-* i s o f the same o r d e r as ^jp and order & B ~ -  t  143  i s o f the same  etc..  P r o c e e d i n g i n t h i s way T a b l e I I i s c o m p i l e d . TABLE I I 1 £k r ~2re  0  r 2 x 10  J  r  20 x 10  J  40 x 10  II  dVo  1/5  As? 3  40 x 10*  jjk  r d&  10 x 10*  5 x 10*  -hMU)  Stiff  0.0001 x 1&  5 x 10*  The above T a b l e shows t h a t , i n the e q u a t i o n s o f m o t i o n , the i n e r t i a terms (f^Te"  e t c  «)  a  r  e  about one t w e n t i e t h o f the p r i n c i p a l terms and may be  11 neglected without serious error. component of acceleration  The force a r i s i n g from the C o r i o l i s  (-^r*-) i s about  h a l f that of the remaining terms.  Although t h i s term i s v i r t u a l l y of the f i r s t order, i t w i l l be neglected here also since t h i s approximation leads to considerable s i m p l i f i c a t i o n of the equations, as w i l l be shown.  In any case, the pressure of 20 l b s . / * * i n . 80  assumed here i s extremely modest, r e l a t i v e to the pressures of 2,000 lbs./sq. i n . which have been recorded under smaller f i l m thicknesses.  Under conditions  of large pressure gradients and shear rates, the force due to C o r i o l i s acceleration  (f^j?*-) w i l l become i n s i g n i f i c a n t .  Similar considerations also  (p^fr)»  apply to the c e n t r i f u g a l force  In the energy equation, the conductivity terms  [Jlr^ifr"^)  are seen to be about 10" of the other terms and are thus n e g l i g i b l e . s  etc J  The  equations now become:  In t h i s form, the equations of motion ( l ? ) may now be integrated twice with respect to ^ to give the v e l o c i t y d i s t r i b u t i o n s .  On applying the v e l o c i t y  boundary conditions, at  i= o ,  V — 0 ,  V = cJr  at  ^= h ,  V — 0,  V,=  r  r  8  0  the v e l o o i t y expressions are found to bei  Substitution of these expressions into the equations of c o n t i n u i t y (16) and energy (18) y i e l d s i  12  I n t e g r a t i n g w i t h r e s p e c t t o ^ between t h e l i m i t s ^ = 0 and ^ = A and s i m p l i f y i n g , we o b t a i n t h e g o v e r n i n g e q u a t i o n s i n p o l a r c o - o r d i n a t e s .  _  Ac  •  •  ( ) 22  I n t h e absence o f a n e q u a t i o n o f s t a t e f o r l i q u i d s , i t i s u s u a l t o e x p r e s s t h e d e n s i t y and v i s c o s i t y as f u n c t i o n s o f t h e p r e s s u r e and t e m p e r a t u r e .  I n the  p r e s s u r e and temperature ranges e n c o u n t e r e d , i t i s s u f f i c i e n t l y a c c u r a t e t o assume t h a t b o t h t h e d e n s i t y and v i s c o s i t y a r e f u n c t i o n s o f t h e t e m p e r a t u r e o n l y , t h e v a r i a t i o n b e i n g g i v e n by»  /> = / \ f / - X t )  .  . (24)  A  .  •  = B f  e  (25)  Thus t h e e q u a t i o n s a v a i l a b l e t o s o l v e f o r t h e p r e s s u r e and t e m p e r a t u r e distributions are:  oCdi-^-  AAlU  L  A* 2\>\bi A/ 3> it} _ <JV> , MM  r yTi£m> x  J^CW'WJ  A  +  /«/i/7-Xt) ^ - B f *  "Tit  .)  (2  (23;  (24) (25)  13 II.2,  S o l u t i o n of the Governing Equations. The e q u a t i o n s o b t a i n e d i n the p r e v i o u s s e c t i o n a r e  too e l a b o r a t e  to permit an exact mathematical s o l u t i o n .  obviously  Thus any s o l u t i o n  w i l l have t o be o b t a i n e d by n u m e r i c a l means and the method t o be used here  is  t h e r e l a x a t i o n p r o c e s s i n t r o d u c e d b y S o u t h w e l l [ l l ] and a p p l i e d t o l u b r i c a t i o n t h e o r y by C h r i s t o p h e r s o n [ 9 ]  .  The f i r s t s t e p w i l l be t o change the independent v a r i a b l e s use o f the  by  transformations 9 «  <* B  These e q u a t i o n s map the s e c t o r shaped pad i n t o a r e c t a n g u l a r shaped pad as shown i n F i g u r e  1.  F I G . 1.  TRANSFORMATION OF SECTOR INTO RECTANGLE  I n terms o f the new c o - o r d i n a t e s R and  , the g o v e r n i n g e q u a t i o n s becomes  (26)  lh  Next, we render the dependent variables dimensionless byintroducing P, T, D and M which are defind by:  (29)  Ji = >UM where to ,yo a n d a r e respectively the temperature, density and viscosity at the inlet edge.  Substitution of equations (29) into equations (2k), (25),  (27) and (28) yields:  jo-Af/'-AtT) >f = B>(uT)'  ft  •  • (32)  -  • (33)  It now becomes convenient to c a l l equation (30) the pressure equation, equation (31) the energy equation, and treat them individually. Pressure Equation Making use of the identity:  in which the operator V*now  s t a n d B  for  +  »  t  n  e  pressure equation  may be written in the form:  v (-^p)-^|^^'' = P ^ ) - s V v p l  1  •  -o*)  Christopherson and Southwell £l2j have shown that i f *tf is any polynomial function, the operator V *  C0U  l c L be expressed in finite-difference form by the  15 equations a V^°" ^  —  1  + terms o f o r d e r a  where the summation s i g n r e f e r s radius  4  at least  t  t o f o u r p o i n t s e q u a l l y spaced on a c i r c l e o f  ' a , whose c e n t r e i s a t t h e p o i n t 1  'c . 1  Substituting this  expression  f o r V * i n the pressure equation givess  where the s u b s c r i p t s now r e f e r t o the p o i n t s on a g r i d network as shown i n Fig.  2.  3. 3  c  *  -a-*  F I G . 2.  GRID POINT NOTATION FOR RELAXATION PROCESS  I n t h e r e l a x a t i o n s o l u t i o n o f e q u a t i o n (35)» t h e r e s i d u a l f o r c e s influence coefficients  and  a r e d e f i n e d bys  (36)  h-ci  n*i  (37) The e f f e c t o f any change A P on the r e s i d u a l f o r c e s  is  then  — OiccAP "t each p o i n t which i s a l t e r e d CUNAP a t each o f the f o u r s u r r o u n d i n g p o i n t s a  'n'.  16 For a given temperature distribution, D and M may be calculated using equations (29).  Then, for any assumed distribution of P ,  the residuals and influence coefficients and (37).  are calculated from equations (36)  The residuals are now relaxed - that i s , the values of P are  adjusted in such a way that the residuals are reduced to a negligible value. This procedure gives the pressure distribution corresponding to the given temperature distribution. Energy Equation In terms of the grid notation established, the energy equation (31) may be written in finite-difference form to givei  For an assumed pressure distribution and a known variation of temperature along the inlet edge of the pad, the corresponding temperature may be obtained throughout the bearing by successive applications of equation (38) across the grid. Simultaneous Solution for Pressure and Temperature To summarize, the equations to be solved for the pressure and temperature distributions ares  jz=A(l-*t°T)  Jl=B[toT)' ' (3  .  . (41)  •  • (42)  A simultaneous solution of the above equations may be obtained by f i r s t assuming an i n i t i a l pressure distribution.  This pressure distribution  enables a f i r s t temperature distribution to be obtained using equations  (40),  17 (kl)  and (k2), as d e s c r i b e d tinder ' E n e r g y E q u a t i o n ' .  This f i r s t  temperature  d i s t r i b u t i o n may t h e n be u s e d t o o b t a i n a second p r e s s u r e d i s t r i b u t i o n from equations  (39)» ( ^ l ) and ( ^ 2 ) , as d e s c r i b e d under ' P r e s s u r e E q u a t i o n ' .  The  second p r e s s u r e d i s t r i b u t i o n w i l l l e a d t o a second t e m p e r a t u r e d i s t r i b u t i o n , and so o n .  Due t o t h e f a c t t h a t the v i s c o s i t y d e c r e a s e s w i t h t e m p e r a t u r e ,  t h e f o r e g o i n g p r o c e d u r e p o s s e s s e s an i n h e r e n t s t a b i l i t y and t h e s o l u t i o n s converge a f t e r a few i t e r a t i o n s . An IBM 1620 computer has been programmed t o p e r f o r m the f o r e g o i n g s t e p s and t o o b t a i n t h e s o l u t i o n t o e q u a t i o n s of input q u a n t i t i e s . oil  temperature.  (39) t o (k2) i n terms  These q u a n t i t i e s a r e s p e e d , f i l m t h i c k n e s s and i n l e t  18 II.3. Theoretical Performance of Bearing The test bearing, which w i l l be described i n the next chapter, has the l i m i t s O—R^O-Sand as shown i n Figure 1.  O^S^I when i t i s transformed into a rectangular pad  A g r i d system was set up on the transformed bearing pad  i n which the g r i d dimension  'a' was chosen as 0.1.  For t h i s configuration,  a t o t a l of 66 points existed which were to describe the pressure and temperature d i s t r i b u t i o n s corresponding to equations (39) to (42). For an a r b i t r a r i l y chosen i n i t i a l pressure d i s t r i b u t i o n , the corresponding temperature f i e l d was calculated.  A revised pressure  d i s t r i b u t i o n was then obtained by relaxing the pressure residuals i n 7 discrete stages of up to 15 passes per stage as required.  At this point the  residuals were reduced to about 0.1$ of t h e i r i n i t i a l value. solutions were carried out f o r a t o t a l of 3 i t e r a t i o n s .  Successive  No modification to  the pressure and temperature d i s t r i b u t i o n s were noticed when more than 3 i t e r a t i o n s were performed, consequently the values obtained a f t e r 3 i t e r a t i o n s were accepted as simultaneously s a t i s f y i n g equations (39) to (42), The quantities to be established from the pressure and temperature d i s t r i b u t i o n s are the load carrying capacity of the bearing and the corresponding c o e f f i c i e n t of f r i c t i o n .  Load Carrying Capacity For the derived pressure d i s t r i b u t i o n , the load carried per pad was obtained by integration. W=/  /  That i s , f>rdrdB  .  •  (43)  The above integration was incorporated i n the computer program so that the load carried per pad was obtained d i r e c t l y f o r each pressure d i s t r i b u t i o n . For safe operation of the bearing, the pressure d i s t r i b u t i o n should be limited such thats  19 (i)  the maximum temperature does n o t exceed 220°P and  (ii)  t h e f i l m t h i c k n e s s i s not l e s s t h a n 0.0002 i n c h .  The f i r s t c r i t e r i o n i s i n t e n d e d t o ensure t h a t t h e b a b b i t does n o t m e l t w h i l s t t h e second s h o u l d ensure the absence o f m e t a l t o m e t a l c o n t a c t .  The b e a r i n g  l o a d c o r r e s p o n d i n g t o t h e s m a l l e s t l i m i t i n g p r e s s u r e d i s t r i b u t i o n was c o n s i d e r e d t o be t h e l o a d c a r r y i n g c a p a c i t y o f t h e t h r u s t p a d . C o e f f i c i e n t of  Friction C o n s i d e r i n g the f r i c t i o n f o r c e on a f l u i d element t o be due  t o t h e s h e a r i n g a c t i o n a c r o s s the f i l m , we may w r i t e :  dFf = % r dr fo e  Le.  where the e x p r e s s i o n f o r %e set equal to z e r o . be s u b s t i t u t e d t o  for  -^L-^f-rdrJe  dff =  has been s u b s t i t u t e d from e q u a t i o n s  The e x p r e s s i o n f o r  i s given i n equations  (8) w i t h (19) and may  give:  any p o i n t i n the f l u i d f i l m .  I n p a r t i c u l a r , a t ^-0 we  get:  so t h a t the t o t a l f r i c t i o n f o r c e p e r pad i s g i v e n b y :  I n t e g r a t i n g t h e f i r s t term i n t h i s e x p r e s s i o n w i t h r e s p e c t  t o 6 and n o t i c i n g  t h a t />=£ a t 0="* and 8=0:  The c o e f f i c i e n t  o f f r i c t i o n i s d e f i n e d as the r a t i o o f t h e t r a n s v e r s e f r i c t i o n  f o r c e t o the a p p l i e d b e a r i n g l o a d . That i s , J  W  20  Equation (44) was also incorporated in the computer program. Bearing: Performance Curves Using equations (39) to ( 4 4 ) , the load carrying capacity and the coefficient of friction for the test bearing were obtained at speeds of 15,000 and 20,000 r.p.m. 4.  The results are shown graphically in Figures 3 and  In calculating these results, the inlet o i l was assumed to be at a  temperature of 70°F and the physical properties of the lubricant as given in Appendix II were used. both speeds.  The load carrying capacity is seen to be 40 lbs. at  In fact, for any selected limiting temperature, the load  carrying capacity is seen to be the same.  The only difference between the  limiting conditions is the film thickness at which the limiting load is carried.  This fact can be understood by considering the action of the  thermal wedge.  The load carried depends on the temperature rise across the  bearing and hence the density change.  The necessary temperature rise may be  achieved at low speed with thin fluid films or at high speed with greater film thicknesses.  However, the high speed condition has the advantage of having  a lower coefficient of f r i c t i o n .  LOt\b PER  PAD  s  T2  COEFFICIENT OF  /^fes.) _s  FRICTION  t  CHAPTER III III. 1.  \  Apparatus and Measurements  I I I . l . Apparatus and Measurements ( i ) Apparatus The s c h e m a t i c l a y o u t o f the a p p a r a t u s i s shown i n F i g u r e 5 . The d r i v e system c o n s i s t e d o f an e l e c t r i c m o t o r , a V - b e l t d r i v e and a g e a r box.  The 15 h . p . i n d u c t i o n motor r a n a t 3,545 r . p . m .  under f u l l  load.  The V - b e l t d r i v e c o n s i s t e d o f one 4 . 0 / 5 . 4 i n c h v a r i - p i t c h p u l l e y , w h i c h was f i t t e d t o t h e motor s h a f t ,  and one o f t h r e e i n t e r c h a n g e a b l e  fixed-pitch  p u l l e y s o f 7 . 0 , 9*0 and 12.4 i n c h d i a m e t e r s , which c o u l d be f i t t e d t o the intermediate shaft.  The i n t e r m e d i a t e s h a f t was d i r e c t o o u p l e d t o a R o l l s -  Royce M e r l i n s u p e r c h a r g e r g e a r box w h i c h had a 9»5*1 s t e p up r a t i o .  With  t h i s d r i v e system t h e t e s t b e a r i n g c o u l d be r u n a t almost any speed from 11,000 t o 2 6 , 0 0 0 r . p . m . A s e c t i o n a l v i e w o f the t e s t b e a r i n g assembly i s shown i n F i g u r e 6.  The h i g h speed s h a f t c o n s i s t e d o f a |  w h i c h a 3 i n c h d i a m t e r s t e e l d i s c had been s h r u n k .  i n c h d i a m e t e r s t e e l r o d on The o p p o s i n g f a c e s o f  t h e t h r u s t d i s c were m a c h i n e d , and s u b s e q u e n t l y l a p p e d , so t h a t t h e y were f l a t and p a r a l l e l t o w i t h i n 1.0 x 10~ 12 m i c r o i n c h e s .  4  i n c h and had a s u r f a c e f i n i s h o f about  The h i g h speed s h a f t was s u p p o r t e d by two d o u b l e row,  self  a l i g n i n g , b a l l - t y p e b e a r i n g s which were l u b r i o a t e d by an o i l d r i p s y s t e m . The c r i t i c a l speed o f t h e h i g h speed s h a f t assembly was about 10,000 r . p . m . A s t e e l c y l i n d e r c o n c e n t r i c w i t h the h i g h speed surrounded the t h r u s t d i s c .  shaft  T h i s c y l i n d e r c a r r i e d t h r e e matched h y d r a u l i c  j a c k s w h i c h o p e r a t e d i n p a r a l l e l and were l o c a t e d a t 120° t o each o t h e r around the o u t s i d e o f the o y l i n d e r .  The j a c k s were connected t o two  t r i a n g u l a r end p l a t e s w h i c h were i n c o n t a c t w i t h t h e two o p p o s i n g l o a d i n g p i s t o n s i n the s t e e l c y l i n d e r .  The l o a d i n g p i s t o n s c a r r i e d two s i m i l a r  b a b b i t t e d t h r u s t c o l l a r s , as shown i n F i g u r e 7, w h i c h formed b e a r i n g f o r t h e f a c e s o f the t h r u s t d i s c .  surfaces  25 The complete c y l i n d e r assembly was t o r q u e mounted and s u p p o r t e d b y two s t e e l p i s t o n tubes w h i c h were a t t a c h e d t o t h e l o a d i n g p i s t o n s and w h i c h were c o n c e n t r i c w i t h t h e h i g h speed s h a f t .  The p i s t o n  tubes were s u p p o r t e d a t t h e i r o u t e r ends b y two r o l l e r b e a r i n g s as shown i n F i g u r e 8. The r o l l e r b e a r i n g h o u s i n g s were used t o admit o i l t o t h e t e s t surfaces. shaft,  S e a l s were p r o v i d e d between t h e b e a r i n g h o u s i n g and t h e h i g h speed  and between t h e h o u s i n g and t h e p i s t o n t u b e .  The l u b r i c a n t flowed i n  the a n n u l a r space between t h e p i s t o n tube and t h e h i g h speed s h a f t and i n t o t h e space between t h e b e a r i n g s u r f a c e s . its  The assembled b e a r i n g i s shown i n  t e s t p o s i t i o n i n F i g u r e 9. O i l was s u p p l i e d from a 45 g a l l o n t a n k , which was l o c a t e d  about 11 f e e t above t h e t e s t b e a r i n g , and t h e o u t l e t o i l was g r a v i t y f e d to a n o t h e r 45 g a l l o n t a n k which was l o c a t e d about 4 f e e t below t h e b e a r i n g . A motor and pump u n i t r e t u r n e d t h e o i l t o t h e upper tank a t t h e end o f each test.  The o i l used was S h e l l Turbo 27;  the p h y s i c a l p r o p e r t i e s of t h i s  o i l a r e g i v e n i n Appendix I I . A separate  o i l system s u p p l i e d l u b r i c a n t t o t h e g e a r b o x .  O i l from a r e s e r v o i r was p r e s s u r e f e d t o t h e g e a r box b y means o f a g e a r pump.  A n o t h e r pump d r a i n e d t h e g e a r box sump and r e t u r n e d t h e o i l t o t h e  reservoir. (ii)  B o t h pumps were d r i v e n from t h e same e l e c t r i c m o t o r .  Measurements The e s s e n t i a l q u a n t i t i e s t o be measured under t e s t were  l o a d , t o r q u e , temperature and f i l m  speed,  thickness.  Speed The r o t a t i o n a l speed o f t h e h i g h speed s h a f t was measured d i r e c t l y b y use o f a ' S m i t h ' s ' tachometer which had speed ranges from 0-5,000 and 0-50,000 r . p . m .  The i n s t r u m e n t had a r e s o l u t i o n o f 0.4$ o f t h e f u l l  26 scale. Load The b e a r i n g l o a d was a s s e s s e d by m e a s u r i n g t h e a p p l i e d t o the l o a d i n g j a c k s .  pressure  The p r e s s u r e was o b t a i n e d from an ' A m e r i c a n '  p r e s s u r e gauge t e s t e r w h i c h was c a p a b l e o f s u p p l y i n g p r e s s u r e s o f up t o l b s . / s q . i n . i n increments of 5 l b s . / s q . i n .  The c a l i b r a t i o n c u r v e f o r  500 the  l o a d i n g system i s g i v e n i n Appendix I I I . Torque The f r i c t i o n t o r q u e t r a n s m i t t e d t o t h e t h r u s t pads was measured by means o f a w e i g h t pan w h i c h was k n i f e - e d g e arm.  suspended on a t o r q u e  The t o r q u e arm was mounted on t h e b e a r i n g t o r q u e r i n g and weight pan  was s i t u a t e d a t a d i s t a n c e o f 10 i n c h e s from t h e b e a r i n g c e n t e r l i n e . Temperature The t e m p e r a t u r e o f the o i l was measured a t t h e i n l e t and o u t l e t o f each s e c t o r o f the t h r u s t pad a t t h e p o i n t s shown i n F i g u r e 7. Two thermocouples were p l a c e d i n each o i l g r o o v e .  The thermocouples were  mounted i n t u f n o l i n s e r t s w h i c h were screwed i n t o t h e b r a s s b a c k i n g p l a t e o f the t h r u s t p a d .  Thermocouple p o t e n t i a l was measured by a  'Doran'thermocouple  p o t e n t i o m e t e r w h i c h had ranges o f 0.20mV, by i n c r e m e n t s o f 0.01 mV, and 0-100 mV by i n c r e m e n t s o f 0.05 mV.  T h i s i n s t r u m e n t i s shown i n F i g u r e 1 0 .  C o p p e r - c o n s t a n t a n thermocouples were used and the c a l i b r a t i o n p r o c e d u r e and r e s u l t s a r e g i v e n i n Appendix I I I . Film Thickness The t h i o k n e s s o f t h e f l u i d f i l m i n t h e b e a r i n g was measured by means o f a ' D a y t r o n i c ' model IO3A-8O l i n e a r d i s p l a c e m e n t t r a n s d u c e r , which i s shown i n F i g u r e 11.  T h i s t r a n s d u c e r had a range o f  0 . 0 4 0 i n c h and was  f i t t e d w i t h a s p e c i a l s t e e l t i p w h i c h had a h e m i s p h e r i c a l end o f r a d i u s 0 . 0 3 0 inch.  The t r a n s d u c e r was screwed i n t o t h e l o a d i n g p i s t o n crown, as shown i n  27 F i g u r e 5 , and the p l u n g e r t i p passed t h r o u g h the t h r u s t pad t o make w i t h the t h r u s t d i s c .  contact  Thus the e l e c t r i c a l o u t p u t o f t h e t r a n s d u c e r was a  measure o f t h e d i s t a n c e between t h e t h r u s t pad and the t h r u s t  disc.  A ' D a y t r o n i o ' model 300BF d i f f e r e n t i a l t r a n s f o r m e r i n d i c a t o r s u p p l i e d the e x c i t a t i o n f o r t h e t r a n s d u c e r and showed t h e d i s p l a c e m e n t on a pre-calibrated scale.  T h i s i n s t r u m e n t , w h i c h i s shown i n F i g u r e 1 0 , had a  range o f £ 0.100 i n c h and a maximum r e s o l u t i o n o f 10 m i c r o i n c h e s . I n a d d i t i o n to the transducer,  two ' S t a r r e t t '  i * oo 0  0  d i a l gauges were mounted on the s t e e l c y l i n d e r w i t h t h e i r p l u n g e r s one o f the p i s t o n end p l a t e s . r e c o r d e d the sum o f t h e two f i l m  inch touching  The c o n f i g u r a t i o n was such t h a t t h e gauges thicknesses.  The a f o r e m e n t i o n e d i n s t r u m e n t s and p i e c e s o f equipment shown i n t h e i r t e s t p o s i t i o n s i n F i g u r e  12.  are  28  ELECTRIC MOTOR  ^ ^ V - B E L T  DRIVE  FIG. 5  SCHEMATIC LMOVT OF APPARATUS  ROLLER BEARING  ISTON END PLATE  COUPLING  DOUBLE ROW BALL BEARING  LOADING PISTON  TORQUE RING  FIG.6 SECTIONAL VIEW  OF T E S T  BEARING  vroo  FIG.8  TEST BEARING BEFORE ASSEMBLY  FIG.9  ASSEMBLED BEARING  IN  POSITION  33  FIG.10  DIFFERENTIAL TRANSFORMER INDICATOR AND THERMOCOUPLE POTENTIOMETER  FIG.12  GENERAL VIEW OF APPARATUS  CHAPTER IV I V . 1. I V . 2. I V . 3.  E x p e r i m e n t a l Procedure Experimental Results Discussion of Results  37 I V . 1 . E x p e r i m e n t a l Procedure A p a r t i c u l a r assembly p r o c e d u r e f o r the t e s t b e a r i n g was adopted t o ensure p r o p e r a l i g n m e n t o f the t h r u s t s u r f a c e s of t e s t s .  p r i o r to a  series  W i t h the e n t i r e b e a r i n g assembled but u n b o l t e d , the h i g h speed  s h a f t was s e c u r e d by b o l t i n g down the two s u p p o r t b e a r i n g s .  W i t h the  n o r m a l l y s t a t i o n a r y p a r t s o f the t e s t b e a r i n g assembly s t i l l u n s e c u r e d , l o a d was a p p l i e d t o the t h r u s t s u r f a c e s ,  so c a u s i n g m e t a l t o m e t a l  between the t h r u s t pads and the t h r u s t d i s c .  a  contact  The r e s u l t i n g c o n t a c t  pressure  caused t h e t e s t b e a r i n g assembly t o a l i g n i t s e l f w i t h the t h r u s t d i s c .  With  the l o a d s t i l l a p p l i e d t o the b e a r i n g , the two r o l l e r b e a r i n g s which s u p p o r t the s t a t i o n a r y components were c a r e f u l l y  secured.  B e f o r e b e g i n n i n g each t e s t ,  a p r e s s u r e o f 25 l b s . / s o . . i n . was  a p p l i e d t o t h e l o a d i n g j a c k s to b r i n g the t h r u s t s u r f a c e s  i n t o c o n t a c t and  the d i f f e r e n t i a l t r a n s f o r m e r i n d i c a t o r and d i a l gauges were s e t t o r e a d z e r o . The l o a d was t h e n removed from the b e a r i n g and the t h r u s t s u r f a c e s s e p a r a t e d by about 0.010  inch.  were  The s t a t i o n a r y components were t o r q u e  balanced i n t h i s p o s i t i o n . W i t h t h e s e p r e l i m i n a r i e s c o m p l e t e d , the l u b r i c a t i o n system f o r t h e g e a r box was s t a r t e d and o i l was a d m i t t e d t o t h e t e s t s u r f a c e s .  The  main motor was t h e n s t a r t e d and an i n i t i a l l o a d i n g p r e s s u r e o f 5 l b s . / s q . i n . was a p p l i e d t o the h y d r a u l i c l o a d i n g j a c k s . t i m e the o i l taken. recorded.  A f t e r about two minutes r u n n i n g  temperature i n the b e a r i n g s t a b i l i z e d and t e s t r e a d i n g s c o u l d be  Readings o f speed,  t o r q u e , f i l m t h i c k n e s s and o i l temperature were  The r a t e o f o i l f l o w through the b e a r i n g and t h e b u l k o u t l e t  temperature o f the o i l were r e c o r d e d as supplementary i n f o r m a t i o n . The f o r e g o i n g procedure was r e p e a t e d f o r l o a d i n g p r e s s u r e i n c r e m e n t s o f 5 l b s . / s q . i n . u n t i l the o u t l e t o i l temperature was about 220°P. The maximum temperature c r i t e r i o n ,  r a t h e r t h a n the minimum f i l m  thickness  38 c r i t e r i o n , was found t o l i m i t the l o a d c a r r y i n g c a p a c i t y o f t h e b e a r i n g . t h i s p o i n t t h e b e a r i n g was u n l o a d e d and t h e machine was s t o p p e d . t h e machine was a t r e s t ,  At  Immediately  a l o a d i n g p r e s s u r e o f 25 l b s . / s q . i n . was q u i c l y  a p p l i e d t o t h e l o a d i n g j a c k s and the t r a n s d u c e r z e r o r e a d i n g n o t e d w h i l e the b e a r i n g components were s t i l l h o t .  The l o a d was t h e n removed and t h e t h r u s t  s u r f a c e s were s e p a r a t e d by about 0.010  i n c h and t h e z e r o t o r q u e r e a d i n g was  checked. The t e s t p r o c e d u r e was r e p e a t e d f o r speeds r a n g i n g from 15,000 t o 19,000 r . p . m .  T e s t s a t speeds i n excess o f 19,000 r . p . m . c o u l d  not be made due t o t h e p e r s i s t e n t cage f a i l u r e o f a r o l l e r b e a r i n g i n the h i g h speed end o f the g e a r b o x . H a v i n g completed a s e r i e s o f t e s t s , dismantled f o r i n s p e c t i o n .  t h e t e s t b e a r i n g was  The t h r u s t s u r f a c e s were found t o be i n good  c o n d i t i o n w i t h no e v i d e n c e o f m e t a l t o m e t a l c o n t a c t under r u n n i n g c o n d i t i o n s . The b e a r i n g was reassembled and the p r e v i o u s t e s t s r e p e a t e d ; o b t a i n e d were found t o be i n agreement w i t h t h e f i r s t s e t o f  the  results  readings.  39 IV.2. Experimental R e s u l t s The d e r i v e d r e s u l t s a r e p r e s e n t e d i n g r a p h i c a l form i n t h i s s e c t i o n and t h e observed r e s u l t s a r e g i v e n i n Appendix I V .  The l o a d  f o r t h e b e a r i n g a r e shown i n F i g u r e 13 and t h e c o r r e s p o n d i n g  friction  characteristics  comparison  a r e shown i n F i g u r e 14.  To p e r m i t a d i r e c t  curves  t o be made w i t h t h e t h e o r e t i c a l c u r v e s o b t a i n e d i n Chapter I I , t h e c u r v e s f o r 20,000 r . p . m . have been reproduced i n F i g u r e s 13 and 1 4 .  The t h e o r e t i c a l  curves f o r a speed o f 15,000 r . p . m . l i e v e r y c l o s e t o those f o r 20,000 r . p . m . , c o n s e q u e n t l y t h e former c u r v e s have been o m i t t e d .  E x p e r i m e n t a l l y , i t was  found t h a t l o a d s i n excess o f 16.5 l b s . p e r pad c o u l d n o t be c a r r i e d b y t h e b e a r i n g w i t h o u t t h e o u t l e t o i l temperature e x c e e d i n g t h e s p e c i f i e d l i m i t o f 220°P. The e x p e r i m e n t a l r e s u l t s o b t a i n e d may be compared w i t h t h e r e s u l t s obtained by other i n v e s t i g a t o r s by p l o t t i n g the c o e f f i c i e n t /  against  the parameter^-.  Such a p l o t i s shown  of f r i c t i o n  i n F i g u r e 15 i n which t h e  v a l u e o f Z has been t a k e n i n c e n t i p o i s e , N i s i n r . p . m . and P ' i s i n l b s . / s q . i n . To o b t a i n t h e c o e f f i c i e n t  o f f r i c t i o n from t h e measured  b e a r i n g t o r q u e and l o a d , a mean r a d i u s o f t h e b e a r i n g must be d e f i n e d . t h e o r e t i c a l expression f o r the coefficient  The  o f f r i c t i o n i s given by equation  (44):  I f t h e v i s c o s i t y i s assumed c o n s t a n t a t some average v a l u e JUL , t h e n t h e above 1  e q u a t i o n may be i n t e g r a t e d t o g i v e :  (45) F o r an a p p l i e d l o a d o f P ' p e r u n i t a r e a , t h e l o a d p e r pad W w i l l be the  expression:  S u b s t i t u t i n g f o r W i n e q u a t i o n (45) y i e l d s :  given by  40  / -'7 TrT[ r,*-r.» / !  ,  Prom equation (^6), the mean bearing radius t  m  •  •  (^ )  i s defined by«  For the bearing tested, Pi » 1.5 and fo = 0.6875 so that K » = 1.144 inches. Having defined the mean radius of the bearing, the experimental  coefficient  of f r i c t i o n may be obtained from the measured f r i c t i o n torque per pad by using the r e l a t i o n *  6  CS  hO  I.ST  FILIA  2.0  THICKNESS  FIG. IS EXPERIMENTAL  ZS  (in.*  3.0  3.5  16"*)  LOAb CARRYING CAPACITY  -4--0  6  2<7  IV.3.  Discussion of Results An e s t i m a t e o f t h e a c c u r a c y o f t h e o b s e r v e d r e s u l t s w i l l be  made, t o g e t h e r w i t h some g e n e r a l comments.  T h i s d i s c u s s i o n w i l l be f o l l o w e d  by a comparison o f t h e e x p e r i m e n t a l r e s u l t s w i t h t h e t h e o r e t i c a l p r e d i c t i o n s . Finally, other  t h e e x p e r i m e n t a l r e s u l t s w i l l be compared w i t h t h o s e o b t a i n e d by  investigators.  Observed R e s u l t s F o r each n o m i n a l speed s e t t i n g , t h e r o t a t i o n a l speed o f t h e t h r u s t d i s c was found t o be independent o f t h e a p p l i e d b e a r i n g l o a d .  The  a c c u r a c y o f the speed meansurements was l i m i t e d o n l y b y t h e r e a d a b i l i t y and inherent accuracy o f the tachometer.  F o r t h e speeds a t which t h e b e a r i n g  was r u n , t h e r e c o r d e d f i g u r e s may be c o n s i d e r e d as b e i n g a c c u r a t e t o w i t h i n 0.2$. The a c c u r a c y w i t h which t h e b e a r i n g l o a d was o b t a i n e d was governed b y t h e a c c u r a c y o f t h e c a l i b r a t i o n c u r v e f o r t h e l o a d i n g s y s t e m . I n Appendix I I I t h e l o a d i n g c u r v e i s g i v e n t o w i t h i n 5$» The f r i c t i o n t o r q u e , as measured, was produoed by t h e a c t i o n of both t h r u s t bearings. within friction,  The b a l a n o i n g system was found t o be s e n s i t i v e t o  an ounce o r about 5$«  The p r o b a b l e a c c u r a c y o f t h e c o e f f i c i e n t  of  i f based on t h e r o o t mean square l a w , i s 7$. The a c c u r a c y o f t h e f i l m t h i c k n e s s r e c o r d i n g s was governed b y  the random ' d e v i a t i o n s w h i c h were e x p e r i e n c e d . t h i c k n e s s was n o m i n a l l y 0.0010 i n c h . p r e v a i l f o r 1 t o 2 seconds,  Suppose t h a t t h e f i l m  I t was found t h a t t h i s v a l u e would  then i t would f a l t e r b r i e f l y t o about 0.0009 i n c h  and r e t u r n t o 0.0010 i n c h a f t e r a s m a l l o v e r s h o o t .  This e r r a t i c deviation  l i m i t e d t h e a c c u r a c y o f t h e f i l m t h i c k n e s s measurements t o about 10$. Accompanying t h e v a r i a t i o n i n f i l m t h i c k n e s s , a v a r i a t i o n i n the temperature o f t h e o u t l e t o i l was o b s e r v e d .  I t was n o t i c e d t h a t i f the  45 f i l m thickness decreased, the o i l temperature increased, the v a r i a t i o n s i n temperature being of the same nature and duration as the variations i n f i l m thickness.  The accuracy of the temperature measurements were  consequently  limited to be within an estimated figure of jfo of the recorded values. From the foregoing description of the variations i n f i l m thickness and temperature, i t was thought that intermittent o i l starvation existed between the bearing surfaces. was also reported by Fogg [ 2 j .  Evidence of l o c a l i s e d o i l starvation  To ensure that the thrust pads were  operating f u l l y flooded, Fogg f i t t e d circumferential o i l seals to his test bearing.  The r e s u l t i n g r e s t r i c t i o n to o i l flow at the perimeter of the  bearing produced a small p o s i t i v e pressure there and a considerable improvement i n the bearing capacity was obtained. Another f a c t o r which could contribute to the i n s t a b i l i t y exhibited by the test bearing i s the presence of a i r between the thrust surfaces.  A i r entering the bearing, either by way of the i n l e t o i l or  through the outlet grooves, would create a region which would move across the bearing and which would support l i t t l e or no load. The observed results show that the temperature i n the i n l e t o i l grooves rose to very high values even although the supply o i l was at 70°F.  The high temperatures recorded would appear to be caused by leakage  of the outlet o i l from one pad into the i n l e t of the next pad.  This leakage  could take place by e i t h e r one of two mechanisms. The f i r s t mode of leakage could be a t t r i b u t e d to carry-over from the r o t a t i n g disc.  That i s , since there exists a d e f i n i t e gap between  the stationary and moving surfaces of the bearing, and since the layer of o i l next to the disc i s moving with the disc, i t i s evident that some r e c i r c u l a t i o n w i l l always take place.  By t h i s means i t may be v i s u a l i z e d that the f l u i d  f i l m at the i n l e t to a pad w i l l be made up of e n t i r e l y r e c i r c u l a t e d o i l i n  46 the l a y e r adjacent pad,  to t h e d i s c , e n t i r e l y f r e s h o i l adjacent  t o the s t a t i o n a r y  and some form o f m i x t u r e i n t h e r e g i o n between t h e s e extreme  layers.  The second method b y which o i l may l e a k from one pad t o a n o t h e r i s r e l a t e d t o t h e p a r t i c u l a r t h r u s t pad c o n f i g u r a t i o n employed. R e f e r r i n g t o F i g u r e 7, i t w i l l be seen t h a t o i l e n t e r i n g t h e o u t l e t c o n t a i n i n g thermocouples 11 and 12 w i l l be u n a b l e t o escape u n t i l i s completely f i l l e d .  groove  that  groove  T h i s means t h a t t h e groove w i l l always be f u l l  of hot  o i l under t e s t c o n d i t i o n s .  S i m i l a r l y , the non-load c a r r y i n g region i n the  vertical position w i l l be f u l l  o f hot o i l .  By t h e a c t i o n o f gravity, these  two r e g i o n s w i l l f e e d h o t o i l i n t o t h e i n l e t groove c o n t a i n i n g thermocouples 1 and 2.  V e r i f i c a t i o n o f t h i s s t a t e o f a f f a i r s c a n be o b t a i n e d b y r e v i e w i n g  t h e observed r e s u l t s i n Appendix I V .  I t w i l l be n o t e d t h a t t h e temperatures  r e c o r d e d b y thermocouples 1 and 2 were h i g h e r t h a n t h o s e r e c o r d e d i n t h e o t h e r i n l e t grooves.  S i n c e t h e average temperature r i s e a c r o s s t h e o f f e n d i n g pad  i s v e r y s m a l l , i t s l o a d c a r r y i n g c a p a c i t y w i l l be g r e a t l y i m p a i r e d . 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 R e s u l t s R e f e r r i n g t o F i g u r e 13, i t i s seen t h a t t h e e x p e r i m e n t a l curves t e n d toward t h e t h e o r y a t medium f i l m t h i c k n e s s e s , t h e n f a l l again a t low f i l m t h i c k n e s s e s .  away  The main r e a s o n f o r t h e d i v e r g e n c e a t l o w  f i l m t h i c k n e s s e s l i e s i n t h e i n l e t temperatures o f t h e o i l .  I n the theory,  t h e i n l e t temperature was assumed c o n s t a n t a t 70°F, whereas average i n l e t temperatures o f up t o 150°F were r e c o r d e d e x p e r i m e n t a l l y .  Since the thermal  wedge depends on v i s c o s i t y t o produce a temperature r i s e a c r o s s t h e b e a r i n g and s i n c e t h e v i s c o s i t y v a r i e s as  t~*' 71 , t h e n i t f o l l o w s t h a t t h e i n c r e a s e d  i n l e t temperature w i l l reduce t h e l o a d c a r r y i n g c a p a c i t y o f t h e b e a r i n g .  A  p o i n t check was made on t h i s b y o b t a i n i n g a computer s o l u t i o n f o r t h e l o a d c a r r i e d a t 15,000 r . p . m .  The l o a d was found t o be reduced from 25.30 l b s .  47 a t 70°P t o 13.85  l b s . a t 1G0°P.  These f i g u r e s demonstrate t h e s e v e r e  effect  t h a t t h e i n l e t o i l temperature has on t h e b e a r i n g performance and e x p l a i n the r e d u c t i o n i n t h e l o a d observed i n the 1.0 x 10~  3  t o 1.75  x 10~  3  range o f  film  t h i c k n e s s i n F i g u r e 13. I n t h e f i l m t h i c k n e s s range 1.75  x 10~  3  t o 2.5 x  10~  the  3  e x p e r i m e n t a l l o a d i s c l o s e t o the t h e o r e t i c a l c u r v e and a c t u a l l y exceeds the t h e o r y i n some p a r t s .  I n t h i s range t h e s t a t i c p r e s s u r e predominates and  the hydrodynamic t h e o r y does n o t account f o r the s t a t i c p r e s s u r e .  The v a l u e  o f t h e s t a t i c p r e s s u r e was about 5 l b s . / s q . i n . but s i n c e the a c t u a l v a l u e depends on t h e l o s s e s i n the s u p p l y l i n e and a t e n t r y t o t h e b e a r i n g  surfaces,  no attempt has been made t o s u b t r a c t l o a d c a r r i e d by the s t a t i c p r e s s u r e from the t o t a l b e a r i n g l o a d . F o r f i l m t h i c k n e s s e s g r e a t e r t h a n 2.5 x 10"  3  f o r c e becomes i m p o r t a n t .  the c e n t r i f u g a l  I n C h a p t e r I I i t was shown t h a t the c e n t r i f u g a l  term i n t h e d i f f e r e n t i a l e q u a t i o n was r e l a t i v e l y s m a l l f o r moderate o r h i g h hydrodynamic p r e s s u r e s .  However, t h e i n d u c e d p r e s s u r e s a r e low i n t h e range  i n q u e s t i o n and c o n s e q u e n t l y t h e c e n t r i f u g a l f o r c e , w h i c h i m p a i r s the l o a d c a r r y i n g c a p a c i t y o f the b e a r i n g , becomes r e l a t i v e l y i m p o r t a n t .  The e f f e c t  o f the c e n t r i f u g a l f o r c e can be seen t o extend down t o f i l m t h i c k n e s s e s about 1.25  x 10~ . 3  F o r f i l m s t h i c k e r t h a n 1.25  x 10~  3  of  the i n c r e a s e i n l o a d  w h i c h i s p r e d i c t e d by t h e t h e o r y f o r an i n c r e a s e i n s p e e d , i s more t h a n o f f s e t by t h e c o r r e s p o n d i n g i n c r e a s e i n t h e c e n t r i f u g a l f o r c e . t h i n n e r t h a n 1.25  x 10  3  For f i l m s  an i n c r e a s e i n speed produces a h i g h e r l o a d c a r r y i n g  c a p a c i t y f o r t h e same f i l m t h i c k n e s s , as p r e d i c t e d by the The c o e f f i c i e n t  theory.  o f f r i c t i o n c u r v e s shown i n F i g u r e 14 i n d i c a t e  t h a t the e x p e r i m e n t a l v a l u e s o f / a r e everywhere l e s s t h a n t h e t h e o r e t i c a l predictions.  The l a r g e s t s i n g l e f a c t o r c a u s i n g t h i s d i s c r e p a n c y i s  i n c r e a s e d temperature i n the i n l e t o i l .  Since  the  , an i n c r e a s e i n t h e o i l  48 temperature would r e s u l t i n a d e c r e a s e i n t h e v i s c o s i t y and i n t h e of f r i c t i o n .  coefficient  A t low v a l u e s o f f i l m t h i c k n e s s , t h e observed b u l k t e m p e r a t u r e  o f t h e i n l e t o i l was about t w i c e the assumed v a l u e . t h e r e f o r e be reduced by about 8 0 $ .  The v i s c o s i t y would  For l a r g e values of f i l m t h i c k n e s s the  observed i n l e t temperature was 10 F above t h e assumed v a l u e , r e s u l t i n g i n a 30$ d e c r e a s e i n v i s c o s i t y .  The thermocouples were l o c a t e d i n such a way  t h a t t h e y would r e c o r d o n l y the b u l k temperature o f t h e i n l e t o i l .  From  the  p r e v i o u s d i s c u s s i o n on r e c i r c u l a t i o n o f the l u b r i c a n t i t seems p o s s i b l e t h a t a temperature g r a d i e n t e x i s t e d a c r o s s t h e f l u i d f i l m .  Any such t e m p e r a t u r e  g r a d i e n t would produce a f l u i d l a y e r adjaoent t o t h e r o t a t i n g d i s c w h i c h would be a t an even h i g h e r temperature t h a n t h a t observed e x p e r i m e n t a l l y . A l o c a l i n c r e a s e i n t e m p e r a t u r e o f t h i s n a t u r e would cause a f u r t h e r d e c r e a s e i n v i s c o s i t y and i n t h e c o e f f i c i e n t o f  friction.  A p a r t from v i s c o s i t y c o n s i d e r a t i o n s , f°<v7"  8 0  ancy i n t h e l o a d c h a r a c t e r i s t i c s w i l l i n f l u e n c e the f r i c t i o n I n t h e f i l m t h i c k n e s s range 1.0 x 10~ i s l e s s t h a n the t h e o r e t i c a l l o a d .  3  3  t o 2.5 x 10~  3  3  For t h i s range, the l o a d e r r o r w i l l  i t was observed t h a t ,  I n the range 1.75  due t o the e f f e c t o f s t a t i c  the e x p e r i m e n t a l l o a d exceeded t h e t h e o r e t i c a l v a l u e . errors are cumulative!  characteristics.  t o 1.75 x 10~ > t h e e x p e r i m e n t a l l o a d  t h e r e f o r e p a r t i a l l y compensate f o r the v i s c o s i t y e r r o r . 10~  t h a t any d i s c r e p -  x  pressure*  I n t h i s case the two  F o r f i l m t h i c k n e s s e s g r e a t e r t h a n 2.5 x 10~  3  the  e x p e r i m e n t a l l o a d i s a g a i n l e s s t h a n t h e t h e o r e t i c a l v a l u e , due t o t h e a c t i o n o f c e n t r i f u g a l f o r c e , so t h a t a p a r t i a l c o r r e c t i o n i s once more o b t a i n e d . The t h e o r y p r e d i c t s t h a t an i n c r e a s e i n speed w i l l be accompanied by a r e d u c t i o n i n t h e c o e f f i c i e n t o f f r i c t i o n .  I n F i g u r e 14 i t  w i l l be n o t i c e d t h a t t h e converse i s t r u e f o r f i l m t h i c k n e s s e s g r e a t e r t h a n 1.25  x 10~ .  adverse  3  T h i s c o n t r a d i c t i o n of the t h e o r y may be a t t r i b u t e d to the  i n f l u e n c e o f c e n t r i f u g a l f o r c e on l o a d c a p a c i t y , which has been  49 p r e v i o u s l y d i s c u s s e d , and the dependence o f the c o e f f i c i e n t  o f f r i c t i o n on  the l o a d .  C o r r e l a t i o n o f E x p e r i m e n t a l Data The c o e f f i c i e n t p a r a m e t e r i n F i g u r e 15•  o f f r i c t i o n has been p l o t t e d a g a i n s t  the  The e x p e r i m e n t a l r e s u l t s o b t a i n e d a r e f o r  the  most p a r t i n a d i f f e r e n t range from t h e e x i s t i n g p u b l i s h e d r e s u l t s .  The  difference  present  i n range can be a t t r i b u t e d t o t h e low l o a d s c a r r i e d i n the  s e r i e s o f t e s t s and the c o r r e s p o n d i n g h i g h v a l u e o f v i s c o s i t y .  I n the r e g i o n  o f o v e r l a p , the e x p e r i m e n t a l r e s u l t s a r e c l o s e t o t h o s e o b t a i n e d by Young and Fogg, a l t h o u g h t h e g r a d i e n t o f the f r i c t i o n c u r v e i s c l o s e r t o t h a t o b t a i n e d by K e t t l e b o r o u g h . The wide s c a t t e r i n t h e e x p e r i m e n t a l r e s u l t s as o b t a i n e d b y d i f f e r e n t a u t h o r s i s p r o b a b l y produced by t h e d i f f e r e n c e  i n o p e r a t i n g speeds.  I n s e c t i o n I I . 3 i t was p o i n t e d out t h a t f o r a g i v e n l o a d c a r r y i n g c a p a c i t y , lower c o e f f i c i e n t  a  o f f r i c t i o n c o u l d be o b t a i n e d by o p e r a t i n g a t h i g h e r s p e e d s .  That i s , f o r two i d e n t i c a l b e a r i n g s c a r r y i n g i d e n t i c a l l o a d s , b u t o p e r a t i n g a t d i f f e r e n t speeds, a lower c o e f f i c i e n t  o f f r i c t i o n w i l l be o b t a i n e d from t h e  b e a r i n g which runs a t t h e h i g h e r s p e e d .  F o r the same i n l e t o i l  temperatures,  t h e o u t l e t temperatures w i l l be t h e same so t h a t t h e v i s c o s i t y Z w i l l be t h e same f o r the two b e a r i n g s . o n l y have a l o w e r c o e f f i c i e n t ^r.  T h i s means t h a t the h i g h speed b e a r i n g w i l l  not  o f f r i c t i o n , b u t i t w i l l have a h i g h e r v a l u e o f  The f r i c t i o n c h a r a c t e r i s t i c s o f t h e two b e a r i n g s w i l l t h e r e f o r e  i f p l o t t e d on F i g u r e 15. A method o f o b t a i n i n g a more g e n e r a l c o r r e l a t i o n o f e x p e r i m e n t a l d a t a i s i n d i c a t e d by c o m b i n i n g e q u a t i o n s  (46) and (47)*  differ  E q u a t i o n (49) shows t h a t a g e n e r a l c u r v e c o u l d be o b t a i n e d by p l o t t i n g the coefficient  of f r i c t i o n against  the parameter J&LJJz.  and K e t t l e b o r o u g h do not g i v e s u f f i c i e n t be made.  Unfortunately,  Fogg  d a t a t o p e r m i t such a c o r r e l a t i o n t o  CHAPTER ? V . 1. V. 2.  Summary and C o n c l u s i o n s Suggestions f o r Future Research  52 (Y.l.  Summary and C o n c l u s i o n s The e x i s t e n c e  o f a l o a d c a r r y i n g f l u i d f i l m due to the a c t i o n  o f a t h e r m a l wedge has been demonstrated b o t h t h e o r e t i c a l l y and e x p e r i m e n t a l l y . I n b o t h t h e o r y and experiment i t was found t h a t the maximum temperature o f  the  l u b r i c a n t reached r e s t r i c t i v e v a l u e s b e f o r e the f i l m t h i c k n e s s was reduced  to  undesirable  limits. Theoretically,  i t seems p o s s i b l e f o r a p a r a l l e l s u r f a c e  thrust  b e a r i n g t o support hydrodynamic l o a d s o f the o r d e r o f 30 l b s . / s q . i n . o f bearing area.  In general,  i t can be c o n c l u d e d t h a t f o r a g i v e n l o a d c a p a c i t y  a h i g h o p e r a t i n g speed i s a s s o c i a t e d w i t h a low c o e f f i c i e n t  of  friction.  E x p e r i m e n t a l l y , l o a d s o f about 1 2 l b s . / s q . i n . o f b e a r i n g a r e a were s u p p o r t e d .  This load c a r r y i n g c a p a c i t y i s considerably l e s s than that  o b t a i n e d by o t h e r i n v e s t i g a t o r s .  The f a c t o r which l i m i t e d the l o a d  capacity  o f the b e a r i n g was r e c i r c u l a t i o n o f the l u b r i c a n t which r e s u l t e d i n h i g h i n l e t temperatures  to the b e a r i n g .  In t h i s respect  the e x i s t e n c e  of  s e p a r a t e o i l o u t l e t grooves f o r each t h r u s t pad appear t o be o f l i t t l e o r no v a l u e and i f used i n c e r t a i n c o n f i g u r a t i o n s effect.  The f r i c t i o n c h a r a c t e r i s t i c s  the r e s u l t s o b t a i n e d by o t h e r  t h e y can have a d e t r i m e n t a l  o b t a i n e d a r e i n good agreement w i t h  experimenters.  53 V . 2 . Suggestions  f o r F u t u r e Reasearch I t would seem d e s i r a b l e t o attempt t o develop the l o a d  c a r r y i n g c a p a c i t y o f the p a r a l l e l s u r f a c e t h r u s t b e a r i n g b e f o r e any f u r t h e r t e s t s .  I n t h i s respect  i t seems n e c e s s a r y  performing  to r e s t r i c t  the  r a d i a l f l o w o f o i l a t the p e r i p h e r y o f the b e a r i n g t o ensure an adequate s u p p l y o f l u b r i c a n t reaches a l l a r e a s o f the b e a r i n g s u r f a c e s .  One method  o f a c h i e v i n g t h i s i s t o f i t c i r c u m f e r e n t i a l s e a l s as d e s c r i b e d by Fogg. The s e a l s used by Fogg were J i n c h t h i c k and had a d i a m e t r a l c l e a r a n c e 0.010  of  inch. A l t e r n a t i v e l y , the same e f f e c t c o u l d be a c h i e v e d by r e v e r s i n g  the o i l f l o w so t h a t the o i l e n t e r s leaves at the i n n e r r a d i u s .  the b e a r i n g a t the o u t e r r a d i u s and  W i t h t h i s type o f f l o w ,  the r a d i a l o i l grooves  c o u l d be wedge shaped which would a l l o w f o r the p r o p e r f l o w o f o i l i n t o the b e a r i n g and a t the same time c o n v e n i e n t l y f u r n i s h s e c t o r shaped t h r u s t To p e r m i t a c c u r a t e  pads.  e x p e r i m e n t a l v a l u e s o f the c o e f f i c i e n t  f r i c t i o n t o be o b t a i n e d , the b e a r i n g t o r q u e must be a c c u r a t e l y known. s m a l l , h i g h speed b e a r i n g s , method o f o b t a i n i n g  t h e f r i c t i o n t o r q u e i s low and the weight  the b e a r i n g t o r q u e i s r a t h e r i n s e n s i t i v e .  I n t h i s way r a p i d t o r q u e  c o u l d be o b t a i n e d w i t h a h i g h degree of  For pan  Considerable  improvement i n s e n s i t i v i t y c o u l d be o b t a i n e d by e m p l o y i n g a s m a l l t r a n s d u c e r t o a c t on the t o r q u e arm.  of  force  readings  accuracy.  F i n a l l y , an a u t o m a t i c shut-down d e v i c e c o u l d be i n s t a l l e d which would p e r m i t t e s t s t o be made i n the t h i n f i l m r e g i o n .  A relay  switch  f o r the main motor c o u l d be f i t t e d t o the b e a r i n g assembly i n such a way t h a t i f metal to metal contact  occurs,  the r e s u l t i n g i n c r e a s e d f r i c t i o n t o r q u e  would cause the t o r q u e arm to depress motor.  the r e l a y b u t t o n and so s t o p  the main  APPENDICES  i  55  APPENDIX I The Energy Equation in Lubrication. Two different forms of energy equation are to be found in the literature - for example, the equation used by Cope [ 6 j differs from that used by Christopherson [ 9 ] .  In Chapter II, the energy equation obtained agrees  with that obtained by Christopherson.  The reason for this w i l l be shown by  repeating the procedure of Chapter II and at the same time repeating Cope's derivation.  In the following analysis, equations and assumption numbers  with the subscript 'a' w i l l refer to Cope's analysis.  To permit direct  comparison with Cope's results, rectangular co-ordinates w i l l be used. For the steady, laminar flow of a Newtonian fluid we had:  but H=E+-fr Substituting this expression into the previous equation, we get:  but from the continuity equation,  Substituting this expression in the above equation we get:  Thus we have two identical forms of the energy equation,  The second of these equations was obtained by Cope. As before, we assume (i)  That f> ,p, t,JU. and  are functions of x and y only  (ia) That j> j),t,jJ*and.Jk are functions of *x and y only f  56 (ii)  That  (iia)  That  (iii)  That  and so on  !f^>-§f, ^ H ^ ^  «*  30  = o  ( i i i a ) T h a t >ur = o (iv)  That H = 0 > t  ( i v a ) That E  -Cvt A p p l y i n g these assumptions, the-equations  ^U#  +  /  C>^)-f«f vf) = +  ^ )  +  become:  | ^ ) ^ f ^ ) V ^ f ]  .  N e x t , we make a term by term a p p r a i s a l and hence assume ( v ) c o n d u c t i v i t y terms (va) c o n d u c t i v i t y terms  ^^M.'j  etc.  are n e g l i g i b l e  etc  are n e g l i g i b l e  (51)  that:  ( v i a ) d i l a t i o n terms/J^-^ etc.) a r e n e g l i g i b l e W i t h t h e s e assumptions we g e t :  (fCru£+fCi,vf)~(u&  +  v-%) [(%Y (ff] +A  +  •  • ( ) 52  F i n a l l y , we a p p l y t h e v e l o c i t y d i s t r i b u t i o n e x p r e s s i o n s ,  and  i n t e g r a t e w i t h r e s p e c t t o ^ between t h e l i m i t s o f £ = o and ^ = A •  These  s t e p s r e s u l t i n the e q u a t i o n s :  The f i r s t o f t h e s e e q u a t i o n s i s t h e c a r t e s i a n form o f t h e energy e q u a t i o n obtained i n Chapter I I . supported by Charnes,  E q u a t i o n (53a) was o b t a i n e d by Cope and was l a t e r O s t e r l e and S a i b e l  f_13].  However, t h e l a t t e r a u t h o r s  used the R e y n o l d s ' e q u a t i o n i n t h e i r d e r i v a t i o n and R e y n o l d s , i n t u r n , used  the concept o f volume c o n t i n u i t y as opposed t o mass c o n t i n u i t y . A p a r t from b e i n g a l i t t l e more a c c u r a t e t h a n e q u a t i o n e q u a t i o n (53) has t h e advantage o f b e i n g somewhat s i m p l e r .  (53a),  Furthermore,  e q u a t i o n (53) i n v o l v e s t h e s p e c i f i c heat a t c o n s t a n t p r e s s u r e which i s more r e a d i l y o b t a i n a b l e t h a n t h a t a t c o n s t a n t volume.  58 APPENDIX I I P h y s i c a l P r o p e r t i e s o f S h e l l Turbo 27 O i l The d e n s i t y and v i s c o s i t y o f the l u b r i c a n t have t o be known f u n c t i o n s o f temperature t o p e r m i t a s o l u t i o n o f the g o v e r n i n g e q u a t i o n s . C o n s e q u e n t l y , t e s t s were c a r r i e d out t o o b t a i n an a c c u r a t e g r a p h i c a l r e l a t i o n s h i p between t h e s e q u a n t i t i e s and the t e m p e r a t u r e .  V i s c o s i t i e s were  measured u s i n g a S a y b o l t U n i v e r s a l V i s c o m e t e r and d e n s i t i e s were measured u s i n g a hydrometer.  The r e s u l t s a r e shown g r a p h i c a l l y i n F i g u r e 16.  The  curves s p~(\.lZ7  ~0-ooo(>%G t)  /jL^ISZt"*"' fit  }t»..Sft.'/ft*  Ibs.secYft  4  (54)  ( ) 55  the e x p e r i m e n t a l c u r v e s t o w i t h i n 1$ i n the temperature range e n c o u n t e r e d .  The s p e c i f i c heat was assumed c o n s t a n t at the v a l u e s u p p l i e d by the S h e l l O i l Company L t d .  Cf, — 0-497  Bto/lb.'F  59  APPENDIX I I I C a l i b r a t i o n T e s t s on Apparatus C a l i b r a t i o n t e s t s were c a r r i e d out t o determine,, ( l )  the  temperature v e r s u s emf c h a r a c t e r i s t i c f o r the thermocouples and (2) the l o a d i n g p r e s s u r e v e r s u s a p p l i e d l o a d c u r v e f o r the l o a d i n g s y s t e m . Thermocouple C a l i b r a t i o n The emf c h a r a c t e r i s t i c o f the thermocouples was o b t a i n e d by a check a t the steam p o i n t .  T h i s was done by p l a c i n g t h e hot j u n c t i o n i n a  hypsometer and the c o l d j u n c t i o n i n a f l a s k c o n t a i n i n g water and c r u s h e d a t the m e l t i n g p o i n t .  ice  I n t h i s manner the emf was measured as 4.258 mV w i t h  the barometer r e a d i n g 739»8mm  of mercury.  From t h i s i n f o r m a t i o n t h e  e v a p o r a t i o n temperature o f t h e steam was c a l c u l a t e d from the r e l a t i o n :  where th = temperature o f wet steam a t p r e s s u r e ^ ,  i n °C  6*>= 100.00 °C t = barometric pressure,  r  i n mm o f mercury  Thus the s a t u r a t i o n temperature was c a l c u l a t e d t o be 210.64°F and from standard t a b l e s ,  d e r i v e d from Adam's T a b l e s ,  found t o be 4.241 mV.  t h e emf f o r t h i s temperature  is  T h i s d e v i a t i o n o f 0 . 4 $ was assumed t o be l i n e a r l y  d i s t r i b u t e d from the steam p o i n t t o the i c e p o i n t , so t h a t t h e c u r v e shown i n F i g u r e 17 was  obtained.  L o a d i n g System C a l i b r a t i o n To o b t a i n a r e l a t i o n between the p r e s s u r e a p p l i e d t o the l o a d i n g j a c k s and the l o a d d e l i v e r e d t o the t h r u s t s u r f a c e s , l o a d i n g t e s t was made.  an i n - s i t e  The h i g h speed s h a f t was removed from the  test  b e a r i n g and the p i s t o n tubes s e a l e d so t h a t the space between the two o p p o s i n g p i s t o n s was p r e s s u r e t i g h t .  The o i l o u t l e t was t h e n u t i l i z e d t o  61 admit p r e s s u r i z e d o i l from a gauge t e s t e r i n t o t h i s s p a c e . pressure  Thus a known  c o u l d be a p p l i e d t o the l o a d i n g j a c k s by one gauge t e s t e r and a  known p r e s s u r e a p p l i e d t o the t h r u s t pad f a c e s by a n o t h e r t e s t e r .  The  complete d r i v e system up t o and i n c l u d i n g t h e gear box was t h e n s t a r t e d up and a s e r i e s o f p r e s s u r e r e a d i n g s  taken f o r zero p i s t o n motion.  This  e n a b l e d the l o a d i n g c u r v e o f F i g u r e 18 t o be drawn, from w h i c h the maximum d e v i a t i o n was found t o be 5$«  63  (t l/s<ll) u  SXOUr 9NIQV01 pi QSIlddV 3H()SS3Hd  6k APPENDIX I V  The observed e x p e r i m e n t a l r e s u l t s a r e p r e s e n t e d i n T a b l e s to V I I .  I n these Tables,  t h e f o l l o w i n g symbols have been u s e d :  f>u = l o a d i n g p r e s s u r e a p p l i e d t o h y d r a u l i c l o a d i n g j a c k s hi  m  (lbs./sq.in.)  f i l m t h i c k n e s s r e a d i n g ( i n s x 10" ) 3  Iii = z e r o r e a d i n g w i t h b e a r i n g components hot ( i n s x I O ) 3  Q. = o i l f l o w c o l l e c t e d i n t i m e St'  (cc.)  St'«= t i m e i n t e r v a l t o c o l l e c t o i l f l o w Q ( s e e s . ) W r = weight a p p l i e d t o t o r q u e arm a t 10 i n c h r a d i u s (  ozs.)  6h = thermocouple e . m . f . f o r p o s i t i o n i n d i c a t e d by s u b s c r i p t (mV) ft'=» b u l k o i l o u t l e t t e r m p e r a t u r e  (°P)  The thermocouple p o s i t i o n s a r e shown i n F i g u r e 7.  III  -a  -P  co  aJ  -4"  CM H  O O  H H  CVJ  CM  CVJ VO  ON VO  H  go  CD  CU  la  CM  oo  -4-  J ON  CO ON  ON  OO t—  CM t—  H  H  H  CM  DO •  i—1'  H  H  CO CM • rH  VD CO • H  cu  OO  H•  CM  CM  ir-  OO  VO • H  • H  CO VO • H  O C—  H H  t H • H  CM O  It CO  VD CM  -3-  H  «i  oo  oo  oo  0  H  H  CM  on  VD ON • rH  OO O  LTN ON  OO  on  IT\ OO  VD ON • H  LTN C— • CM  OO OO • OO  H CM  »  cvj  CM  CM  VO CO H  CO ON • H  CO H  CON  CO O  «  O H  OJ  tTN  CO  ON  o o  H  II  P  CM  ii ca  o LTN  on  LT\  o  on rH  o  OJ rH  on  CM  w  (D  CM  -3"  H  rH CM  CM  OJ  CM  H  CL)  on  oo  CM  In  CD  O  H  on 0)  L^  O ON  H  O  c—  CL)  oo  CD t -  CM  CM  on  on  on  H  LTN  H  CM  CM  CM  OJ  CO ON  O  t -  OJ  CM  OJ  on  o  OJ  on  co  CM  CM  OJ  O t—  ON  ON  rt  e  LTN VD  O O  on  H OJ  LT\  O  LTN  O  LTN  LTN  CM  CM  ON  [>-  ON  OJ  H  OJ H  co  O  LTN  CO  on -=j-  CO  ON  ON  C-  LTN CO  <•  ON  LPs LT\ 0O  ON  J  OJ  H  CM  ft  LT\  H  OJ  OJ CO  oo  C— CO  H  •  rH OO  O C-  CM  on  CM  •  on  CM  on  VD  oo  H  H  a?  o  ON  H  O ON  Ho  CM  LTN OO • CM  O  »  o  O  H  CM  VD  H  on CM  H  .o  c-  •  iH  H VO  CM ON  o  ft  +5  on  -=J"  CD  H  OJ VD  CM  CM  H  on  CM  e  o  H  H ON  on  00  H  a  -  CM • OO  C-  »  a  CM ON H  LTN  CM  CU  CL)  VD  CO VO H  H  o  CU  O00 H  H  -4-  0)  CO CM. H  H  VO •  H  rH •  >  c—  H H  -3H  CM  LTN O  O O  LT\ O  ON  O  O  o  6  6  O H  -d-  CM  o  c—  ON  on  ON OJ  rH  H 6  H 6  H 6  CO  OO -=*•  00  LTN O  OO  CM  CM  rH  H  LTN  O H  LTN H  O CM  LTN CM  o  O  o  on  H  0J  rH  OJ H  LPs ON  ON VO  LTN CM  OJ  OJ  CM  d > H  •J  ft  LfN rH  4  •P  rj  <D  CO  CM H  H  C7\  vo  Lf\  -4"  OJ  OJ  OJ  co  o  O  ON  o O  CO  •  •  •  o»  «  N  (U  CO  OJ  O LT•N  ir\ CO  o  rH  H  • H  LTN O• H  OJ H• H  H CO • H  VD  OJ H* OJ  H OJ OJ  OJ LT•\ OJ  CO  VD  OJ CO  LTN o  'ITN  H  OJ  CM  o  00  CM  CM  VO  ro  -3"  •  H  O H• H  •  LTN  H H • H  CO  He OJ• OJ • OJ O o O H • • OJ OJ  Cii D  CO o • OJ  *\  is  <*  _  o en  •  CQ  CU  VD H9  CO OJ  LTN  • •  LT«N  OJ>  OJ  OJ  OJ  OJ  CU  OJ CO  VD CO  CO  VO  •  •  H  H  OJ  OJ  ©  Ct)  CO  CO  CO  CO  o  •  •  CM ». 0  •  PH.  OJ  N  Q)  a  on •  GO-  (ri  O VO • OJ  OJ  o  o  •  •6  CU  H  UN  H  O H» H  -4"  •  OJ• H  CO LTN • H  ON  H  LTN  CO  o  •  oo  •  a  OJ  OJ  OJ  co  o  o  COa rH  •  H  OJ  H  •  OO  CM  VO LTN • OJ  •  H  O O • OJ  . ON LTN • OJ  LTN  H  •  ON • H  H  •  rH  H  H  H  VD VO • H  ON H• OJ  co  CO  ON LTN  CO OJ  OJ  CM  co  VD O • OJ  ON H • OJ  LTN  ON ON •  LTN H  CO OJ  VD  OJ  OJ  OJ  OO  LTN  H CO  LTN O  -4-  H  H  OJ  CM  CM  CD  •  co •  H  <n H•  CU  •  o  CO  H  CO  ON  H  -=1-  H  c•  •  >  CO VD H  CO m  CU  •  •  CO  •  •  •  OJ CM o OJ  -4-  OJ  •  H  CO• H  O • OJ  CM  LCN  Lf\  o  O  O  O  o  O  o  H  OJ H  H  co  ON  ON  OJ H  OJ H  ON  VD  OJ  o  CO  LTN  O  O  ON  ON  VD H  H  H H  H H  VD H  o H CO  ITN  CO H CO  LTN CO  O VD  LTN  co  OJ ON OJ  -=f  co  H H  H H  OJ H  H  O  O  H  •  •  •  *  Lf\  OJ  (U  •  CU  CO  CA  •  •  O  CO  G vo  o OJ  OJ  OJ  OJ  OJ  H  • LTN  o H  ITN H  O OJ  •  O  •  •  •  •  •  • O  •  »<  •  • O  •  >  ft  H  • ro  •  a?  ro oo  ITS  H  OJ  o  )-  <<  •  OJ  Lf\  *•  O  H  H  CD  •  VO H  H  C-  0)  -P  OJ H  ON  g  -=r  H  H•  OJ H  co  H  PO H  &  si  •  •  EH  ft"  VO  H  •  •  •  •  •  H  •  OJ  CM  o  rH  •  •  •  •  H  •  OJ  •  O  •  •  •  LPs  VD co co  H  *  O  O  O  LTN VD  OJ  LTN O  O  CM  OJ  OJ  H  LTN  o H  LTV  O OJ  •  O  0  rH  •  -P  t . OJ H  ro  OJ • OJ  •— (1)  o  ON  cn  •  ON O  •  rH  O  CO o  -4H  •  •  OJ  -4" H  • OJ OO OJ • OJ  ON  CD H  M l  a?  •  H  •  c— •  H  OO H  CO OJ  •  »  rH  H  oo »  ON  •  OJ VD • OJ OO H  t VO H  •  oo •  OJ  LP\ LTN H  ON H >  OO  CO LPs  •  0  H  OJ  CvJ  ON L^ • H  o  LP\ VO  H  o  •>  OJ  CM  CO H  o  H  H  VD VD • H  •  oo B  ON OO • OJ  VO VD • OJ  OJ OJ  o  LTN  •  •  LP\ CM  •  OO  o o  •  oo  OJ  OJ  LTN OO  OJ VD • OJ  -4-  o o  • OJ  ON OJ • OJ  o  •  OJ  H  00 • H  OJ •  oo  LPs  o  OJ H  OO H  ON  VO • ON  O • CO  -4"  00  00  CO  •  •  •  •  •  •  -4"  LP\  -4-  LTN ON OJ  o  OO  oo  -4"  VD H •  OJ  OJ H  H  o  o  O  o -4-  OJ  • OJ  LTN O • OJ  LT\  o  LP\ H  o  VD  «  J  CO  H  ON  . LP\  •  ft  CO  -4  ON OJ  H  ON t • OJ  LTN CO • H  CD  _  ON H • OJ  oo  OJ  H  M  oo  OJ  CD  m  OJ  •  -4"  •  H  CD  OJ  LPs OJ  CD  in  ON H  OO LTN  CD  •0  OJ VD  OJ LTN H  N CD  H OO  ON • H  OJ  00  -4"  •  H  ITN  OO OO H  H  •  H  •  H CM  O OO  -4" • rH  O OJ  68 BIBLIOGRAPHY  [I] R e y n o l d s , 0.  "On the Theory o f L u b r i c a t i o n and I t s A p p l i c a t i o n t o M r . Beauchamp Tower's E x p e r i m e n t s " , P h i l . T r a n s , o f the Roy. S o c . o f London, v o l . 177, 1886.  [2] Fogg, A .  " F l u i d F i l m L u b r i c a t i o n of P a r a l l e l Thrust S u r f a c e s " , P r o c . I n s t . Mech. E n g r s . , v o l . 155, 1946.  [3] Bower, G . S .  C o n t r i b u t i o n t o Fogg [2]  Cameron, A . and Wood, W . L . " P a r a l l e l Surface Thrust Bearings", I n t e r . Cong, o f App. M e c h . , 1946.  Proc. 6th.  [5] Shaw, M . C .  "An A n a l y s i s o f the P a r a l l e l S u r f a c e T h r u s t B e a r i n g " , T r a n s . Amer. Soc. Mech. E n g r s . v o l . 6 9 , 1947.  [6] Cope, W . F .  "The Hydrodynamioal Theory of F i l m L u b r i c a t i o n " , P r o c . Roy. S o c , s e r i e s A , v o l . 197, 1949.  [7] K e t t l e b o r o u g h ,  CF.  " T e s t s on P a r a l l e l S u r f a c e T h r u s t E n g i n e e r i n g , Aug. 1955•  Bearings",  [8] Young, J .  "The Thermal Wedge E f f e c t i n Hydrodynamic L u b r i c a t i o n " , E n g i n e e r i n g J o u r n a l , v o l . 45, number 3, 1962.  [9] C h r i s t o p h e r s o n , B . G .  "A New M a t h e m a t i c a l Method f o r the S o l u t i o n o f F i l m L u b r i c a t i o n P r o b l e m s " , P r o c . I n s t . Mech. E n g r s . , v o l . 146, 1942. ...  [lO] B i r d , R . B . , S t e w a r t ,  [II]  Southwell, R.V.  W . E . , and L i g h t f o o t , E . N . " T r a n s p o r t Phenomena", J o h n W i l e y & Sons, I960. " R e l a x a t i o n Methods i n E n g i n e e r i n g C l a r e n d o n P r e s s , O x f o r d , 1940.  Inc.,  Science",  [l2j C h r i s t o p h e r s o n , D . G . , and S o u t h w e l l , R . V . " R e l a x a t i o n Methods A p p l i e d t o E n g i n e e r i n g P r o b l e m s " , P r o c . Roy. S o c , s e r i e s A , v o l . 168, 1938. [13] Charnes, A . , O s t e r l e ,  F . , and S a i b e l , E . "On the Energy E q u a t i o n f o r F l u i d - F i l m L u b r i c a t i o n " , P r o c . Roy. S o c , s e r i e s A , v o l . 1952.  214,  

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