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The lubrication of parallel surface thrust bearings 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 in the Department of MECHANICAL ENGINEERING We acoept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA j MAY, 1962 . i In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying of this thesis for financial gain shall not be allowed without my written permission. Department of Mechanical Engineering The University of British Columbia, Vancouver 8, Canada. May, 1962. i i ABSTRACT The parallel surface thrust bearing has been studied both theoretically and experimentally. The general equations governing the laminar flow of a Newtonian f l u i d are presented and suitably reduced to describe the flow of lubricant through a plain collar bearing. A computer solution of the resulting equations has been obtained i n which the variations, of density and viscosity with temperature are accommodated and the circumferen- t i a l leakage of o i l from the bearing is recognised. The resulting performance curves indicate that useful load carrying capacities, produced by a 'thermal wedge' effect, are possible with parallel 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 results confirm that hydrodynamic lubrication may be achieved with a parallel surface thrust bearing. The experimental values obtained for the load carrying capaoity and the coefficient of f r i c t i o n were both less than the theoretical predictions. The discrepancies appear to be caused, for the most part, by an increase in the o i l temperature resulting from entrainment of the lubricant i n the bearing. v i i ACKNOWLEDGEMENT The experimental work described i n this report was carried out i n the Lubrication Laboratory at the University of British Columbia, and the theoretical calculations were performed at the Computing Centre i n the University of British Columbia. The use of these special f a c i l i t i e s i s gratefully acknowledged. The author would also like to thank the many people whose assistance has made this report possible. In particular, thanks are due to the followingt- Professor W.O. Richmond, for the use of the f a c i l i t i e s of the Mechanical Engineering Department of the University of Bri t i s h Columbia. Professors J. Young and C.A. Brockley, for their guidance and assistance during a l l phases of the project. The National Research Council of Canada, for sponsoring the research with N.R.C. Grant No. A1089. TABLE OF CONTENTS CHAPTER I I. l . Introduction CHAPTER II II. 1. The Governing Equations for Film Lubrication 11.2. Solution of the Governing Equations 11.3. Theoretical Performance of Bearing CHAPTER III III. 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 for Future Research APPENDIX I. The Energy Equation i n Lubrication APPENDIX II. Physical Properties of Shell Turbo 27 Oil APPENDIX III. Calibration Tests on Apparatus APPENDIX IV. Tables of Observed Results BIBLIOGRAPHY 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 li s t e d i s defined on introduction. f,0,^ Independent cylindrical co-ordinates Independent rectangular co-ordinates u,v,ur Velocity in x, y and 3 direction respectively Substantial time derivative CL Grid size A Constant B Constant C> Speoific heat at constant pressure Cv Specific heat at constant volume b Dimensionless density E Internal energy f Coefficient of f r i c t i o n Body force, direction as subscripted Ff Friction force h Film thickness H Enthalpy A Thermal conductivity M Dimensionless viscosity N Rotational speed f> Pressure P Dimensionless pressure P' Load per unit area Heat flux, direction as subscripted fr0 Inner radius of bearing r, Outer radius of bearing f„, Mean radius of bearing A Dimensionless co-ordinate t Temperature to Temperature at inlet t' Time T Dimensionless temperature Tf Friction torque U Velocity of slider Velocity, direction as subscripted W Load per pad Z Viscosity in centipoise oc Included angle of sector pad (& Constant A Dilatation • V « T P V Operator Q Dimensionless co-ordinate X Constant v i Ji V i s c o s i t y /lo V i s c o s i t y a t i n l e t p D e n s i t y po D e n s i t y a t i n l e t X S t r e s s t e n s o r 0 D i s s i p a t i o n f u n c t i o n ui) A n g u l a r v e l o c i t y CHAPTER I 1. I n t r o d u c t i o n 2 I . l . Introduction Prior to, and for some time subsequent to the introduction of Reynolds' classical paper [ l ] * i n 1886, thrust bearings were very Inefficient when compared to the journal bearing. In his quantitative analysis, Reynolds showed that a geometric restriction i n the direction of motion was necessary to produce a load oarrying f l u i d film i n a bearing. The parallel 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 lubrication region. By fortuitous coincidence, however, the journal bearing already possessed the geometry required for hydrodynamio lubrication and so operated with a lower coefficient 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 later, were designed on the t i l t i n g pad principle, as introduced by Kingsbury and Michell, and the parallel surface bearing became vi r t u a l l y obsolete. The contention was that the latter bearing could support a f l u i d film under load only to the extent of the static pressure. On several occasions, however, i t was noted that parallel surface bearings performed better than expected inasmuch as useful loads were carried with a low coefficient of f r i c t i o n . The most conclusive results were obtained by Fogg [2] who recorded load carrying capacities of the same order as a t i l t i n g pad bearing with almost the same efficiency 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 in square brackets refer to l i s t of references i n Bibliography. 3 by the introduction of radial 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 'thermal wedge' effect, analagous to the taper wedge effect which predominated i n the t i l t i n g pad bearings. 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 similar effect would be obtained by an expanding volume of lubricant flowing through a constant area. As a result of Fogg's findings, more attention was given to the parallel 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, in a revised form of the governing equation. By assuming no side leakage and linear variations of viscosity and density along the bearing, Bower concluded that a. load carrying f l u i d film could be produced by the thermal wedge effect. Cameron and Wood [k] used the revised form of Reynolds' equation i n conjunction with an energy equation tor arrive at two simultaneous partial differential equations for the pressure and temperature distributions. Using mathematically expressed variations of viscosity 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 viscosity and a linear variation of density with temperature. In this way he was able to compare the t i l t i n g pad and the parallel surface bearings on the basis of equal film thicknesses. Starting with the general equations for 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 differed from that used previously by Cameron and Wood. The resulting equations were solved by Cope for known variations of viscosity and density with temperature and for no side leakage* In this way he showed that under conditions of low variation of viscosity with temperature, large variation of density with temperature, and small film thicknesses, the thermal wedge may completely outclass the geometric wedge. Experimental results were obtained by Kettleborough [7 ] for a parallel surface bearing running at the comparatively low speed of 695 r.p.m. The coefficient of f r i c t i o n so obtained was about twice that obtained by Fogg. The work performed on parallel surface bearings has tended to be either entirely experimental or entirely theoretical, the theoretical analyses being used only to ju s t i f y the thermal wedge idea and to compare the performance of such a bearing to that of a similar t i l t i n g pad type. The f i r s t real attempt to correlate theoretical and experimental results for a given parallel surface bearing was due to Young [ 8 ] , In his quantitative analysis, Young presented three different solutions to the equations as presented by Cope, the most rigorous solution being a relaxation process, f i r s t used in lubrication by Christopherson [ 9 j , which takes account of side leakage. These different solutions were compared with each other and with the experimental results 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 coefficients 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 theoretical analysis of the f l u i d flow in a parallel 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 certain quantities can be accurately measured. It thus seems desirable to revise, and modify i f necessary, the equations governing the flow of f l u i d i n a parallel surface bearing; to obtain as sophisticated a solution to the governing equations as is reasonable; to add to the experimental data available and to compare the theoretical and experimental results so obtained. It was with these immediate objectives that the current research program was in i t i a t e d . CHAPTER II II. 1. The Governing Equations for Film Lubrication II. 2 . Solution of the Governing Equations II. 3 . Theoretical Performance of Bearing 7 I I . 1. The Grpverning Equat ions f o r F i l m L u b r i c a t i o n The equat ions g o v e r n i n g the f l o w o f the 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 the g e n e r a l equat ions g o v e r n i n g the 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 conduct ing f l u i d . The d e r i v a t i o n o f these g e n e r a l equat ions i s presented i n most t e x t s on hydrodynamics, gasdynamics and r e l a t e d f i e l d s . A c c o r d i n g l y , the g e n e r a l equat ions as presented i n [ i o ] a r e : - # = -Av-*) > ( i ) y > | | = / r - » - [ V : f j • - (2) fjfr-jfr-iV'V-^W) • - (3) i n which the symbols used a r e d e f i n e d i n the ' L i s t o f Symbols' i n the p r e l i m i n a r y pages o f the t h e s i s . These are the equat ions o f c o n t i n u i t y , mot ion and energy, r e s p e c t i v e l y , i n which the p r o p e r t i e s o f the f l u i d a r e v a r i a b l e s . The r e d u c t i o n o f these g e n e r a l equat ions to 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 the method presented by Cope [6j, a l t h o u g h the r e s u l t i n g equat ions w i l l be s l i g h t l y d i f f e r e n t . 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 , tend to become l e n g t h y and t e d i o u s . Consequent ly , i t i s proposed t o i n d i c a t e the s teps i n d i v i d u a l l y and to present the end r e s u l t s o f p e r f o r m i n g each s t e p . We b e g i n by expanding equat ions ( l ) , (2) and (3) i n c y l i n d r i c a l c o - o r d i n a t e s t i n which the components o f the heat f l u x v e c t o r q are 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 teady and l a m i n a r and t h a t the f l u i d i s Newtonian, the g e n e r a l equat ions (*f), (5) and (6) becomes 9 i n which the 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. S i n c e the s u r f a c e s i n a b e a r i n g a r e c l o s e t o g e t h e r , p a r a l l e l 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 reasonable t o assume: ( i ) That the 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 , temperature , v i s c o s i t y and thermal c o n d u c t i v i t y a c r o s s the f i l m are f a r l e s s important than t h e i r v a r i a t i o n a l o n g i t . That i s , i t i s assumed t h a t i j> » t ,JUi and A are 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 the f i l m a r e much more important than v e l o c i t y g r a d i e n t s 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 t h a t 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 the 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 to the 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 4 = 0 ( i v ) That the body f o r c e s , o t h e r than those caused by the motion o f the f l u i d , a r e n e g l i g i b l e . U s i n g the n o t a t i o n adopted, the assumption means t h a t ^ = ^ = 5 = 0 (v) That the 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 the s p e c i f i c heat remains c o n s t a n t . Consequent ly , we may w r i t e A p p l y i n g these f i v e assumptions to equat ions ( l O ) , ( l l ) and (12), they reduce to the f o r m : ±-Jir(f™)+i"k(/>v*) « 0 • • (13) 10 (1*) (15) These equat ions 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 i s d e s i r a b l e to reduce them s t i l l f u r t h e r before a t t e m p t i n g a s o l u t i o n . T h i s can be done by u s i n g a c t u a l measurements, f o r a t y p i c a l s e t o f c o n d i t i o n s , f o r v a l u e s o f ^,A , e t c . , c a l c u l a t i n g from them the orders o f magnitude o f the v a r i o u s terms i n the above equat ions and n e g l e c t i n g any terms which are found to be s m a l l . C o n s i d e r , i n p a r t i c u l a r , a 3 i n c h d iameter b e a r i n g r u n n i n g a t . 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 i n c h . These f i g u r e s a r e estimated f o r the t e s t b e a r i n g t o be u s e d . F o r these c o n d i t i o n s Table I i s compi led i n which a l l q u a n t i t i e s have been converted to the u n i t s o f l b s . , f t . , s e e s . , and ° F . TABLE I p > Ii M A r t 1.614 3000 0.00083 0.000214 12,470 0.0173 0.1 150 143 Now .-J-j-* i s o f the same o r d e r as ^jp and i s o f the same P r o c e e d i n g i n t h i s way Table I I i s c o m p i l e d . o r d e r & B ~ - t e t c . . TABLE I I 0 r r 1 £k r ~2re II dVo As? 1/5 j j k r d& -hMU) Stiff 2 x 10 J 20 x 10 J 40 x 10 3 40 x 10* 10 x 10* 5 x 10* 0.0001 x 1& 5 x 10* The above Table shows t h a t , i n the equat ions 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. The force arising from the Coriolis component of acceleration (-̂ r*-) i s about half that of the remaining terms. Although this 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 this approximation leads to considerable simplification of the equations, as w i l l be shown. In any case, the pressure of 20 lbs./ 8 0** i n . assumed here i s extremely modest, relative to the pressures of 2,000 lbs./sq. i n . which have been recorded under smaller film thicknesses. Under conditions of large pressure gradients and shear rates, the force due to Coriolis acceleration (f^j?*-) w i l l become insignificant. Similar considerations also apply to the centrifugal force (p^fr)» In the energy equation, the conductivity terms [ J l r ^ i f r " ^ ) etc J are seen to be about 10"s of the other terms and are thus negligible. The equations now become: In this form, the equations of motion (l?) may now be integrated twice with respect to ̂  to give the velocity distributions. On applying the velocity boundary conditions, at i = o , Vr — 0 , V 8 = cJr at ^ = h , Vr — 0, V,= 0 the velooity expressions are found to bei Substitution of these expressions into the equations of continuity (16) and energy (18) yieldsi 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 the 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 the g o v e r n i n g equat ions i n p o l a r c o - o r d i n a t e s . _ Ac I n the absence o f an 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 express the 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 the p r e s s u r e and temperature . I n the pressure and temperature ranges encountered, 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 both the 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 the temperature o n l y , the 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 the equat ions a v a i l a b l e t o s o l v e f o r the p r e s s u r e and temperature d i s t r i b u t i o n s a r e : oCdi-^- L A* 2\>\bi A/ 3> it} _ <JV> , MM (2.) AAlU rxyTi£m> J^CW'WJ A + "Tit (23; / « / i / 7 - X t ) (24) ^ - B f * (25) • • ( 2 2 ) 13 I I . 2 , S o l u t i o n o f the Governing E q u a t i o n s . The equat ions 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 are o b v i o u s l y too e l a b o r a t e t o permit an exact mathematical s o l u t i o n . 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 i s the r e l a x a t i o n process i n t r o d u c e d by 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 by use o f the t r a n s f o r m a t i o n s 9 « <* B These equat ions 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. (26) 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 equat ions becomes (29) lh Next, we render the dependent variables dimensionless by- introducing P, T, D and M which are defind by: 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: j o - A f / ' - A t T ) • • (32) >f = B>(uT)'ft - • (33) It now becomes convenient to call 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 l ( - ^ p ) - ^ | ^ ^ ' ' = P ^ ) - s V v 1 p • -o*) Christopherson and Southwell £l2j have shown that i f *tf is any polynomial function, the operator V* C 0 U l cL be expressed in finite-difference form by the 15 equations a 1 V^°" ^ — + terms o f o r d e r a 4 a t l e a s t t where the summation s i g n r e f e r s to 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 r a d i u s ' a 1 , whose c e n t r e i s a t the p o i n t ' c 1 . S u b s t i t u t i n g t h i s e x p r e s s i o n f o r V * i n the p r e s s u r e e q u a t i o n g i v e s s 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 F i g . 2. 3. 3 c * -a-* F I G . 2. GRID POINT NOTATION FOR RELAXATION PROCESS I n the 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)» the r e s i d u a l f o r c e s and i n f l u e n c e c o e f f i c i e n t s are d e f i n e d bys n*i h-ci 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 i s then — OiccAP a"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 ' n ' . (36) (37) 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 are calculated from equations (36) and (37). The residuals are now relaxed - that is , 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) . . (41) Jl=B[toT)'(3' • • (42) A simultaneous solution of the above equations may be obtained by first assuming an in i t ia l pressure distribution. This pressure distribution enables a first temperature distribution to be obtained using equations (40), 17 (kl) and (k2), as d e s c r i b e d tinder 'Energy E q u a t i o n ' . T h i s f i r s t temperature d i s t r i b u t i o n may then be used to 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 equat ions (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 temperature d i s t r i b u t i o n , and so o n . Due t o the f a c t t h a t the v i s c o s i t y decreases w i t h temperature , the f o r e g o i n g procedure possesses an i n h e r e n t s t a b i l i t y and the 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 to perform the f o r e g o i n g steps and t o o b t a i n the s o l u t i o n t o equat ions (39) t o (k2) i n terms o f i n p u t q u a n t i t i e s . These q u a n t i t i e s are speed, f i l m t h i c k n e s s and i n l e t o i l temperature . 18 II.3. Theoretical Performance of Bearing The test bearing, which w i l l be described i n the next chapter, has the limits O—R^O-Sand O^S^I when i t is transformed into a rectangular pad as shown i n Figure 1. A grid system was set up on the transformed bearing pad in which the grid dimension 'a' was chosen as 0.1. For this configuration, a total of 66 points existed which were to describe the pressure and temperature distributions 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 distribution, the corresponding temperature f i e l d was calculated. A revised pressure distribution was then obtained by relaxing the pressure residuals in 7 discrete stages of up to 15 passes per stage as required. At this point the residuals were reduced to about 0.1$ of their i n i t i a l value. Successive solutions were carried out for a total of 3 iterations. No modification to the pressure and temperature distributions were noticed when more than 3 iterations were performed, consequently the values obtained after 3 iterations were accepted as simultaneously satisfying equations (39) to (42), The quantities to be established from the pressure and temperature distributions are the load carrying capacity of the bearing and the corresponding coefficient of f r i c t i o n . Load Carrying Capacity For the derived pressure distribution, the load carried per pad was obtained by integration. That i s , W = / / f>rdrdB . • (43) The above integration was incorporated i n the computer program so that the load carried per pad was obtained directly for each pressure distribution. For safe operation of the bearing, the pressure distribution should be limited such thats 19 ( i ) the maximum temperature does not exceed 220°P and ( i i ) the f i l m t h i c k n e s s i s not l e s s than 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 to ensure t h a t the b a b b i t does not melt w h i l s t the second s h o u l d ensure the absence o f metal to 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 to the 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 the l o a d c a r r y i n g c a p a c i t y of the t h r u s t p a d . C o e f f i c i e n t o f F r i c t i o n 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 the 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 = %e r dr fo Le. dff = -^L-^f-rdrJe where the e x p r e s s i o n f o r %e has been s u b s t i t u t e d from equat ions (8) w i t h se t equal t o z e r o . The e x p r e s s i o n f o r i s g i v e n i n equat ions (19) and may be s u b s t i t u t e d to g i v e : f o r 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 g e t : so t h a t the t o t a l f r i c t i o n f o r c e per pad i s g i v e n b y : I n t e g r a t i n g the 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 />=£ at 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 the t r a n s v e r s e f r i c t i o n f o r c e to 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. The results are shown graphically in Figures 3 and 4 . 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. The load carrying capacity is seen to be 40 lbs. at both speeds. 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 friction. LOt\b PER PAD /^fes.) s _s t T2 COEFFICIENT OF FRICTION  CHAPTER III III. 1. Apparatus and Measurements \ I I I . l . Apparatus and Measurements ( i ) Apparatus The schematic l a y o u t of the apparatus 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 motor , a V - b e l t d r i v e and a gear 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 l o a d . 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 , which was f i t t e d t o the 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 f i x e d - p i t c h 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 i n t e r m e d i a t e s h a f t . 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 ooupled t o a R o l l s - Royce M e r l i n supercharger gear box which 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 the 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 26,000 r . p . m . A s e c t i o n a l v iew 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 | i n c h d iameter s t e e l r o d on which a 3 i n c h d iamter s t e e l d i s c had been shrunk. The oppos ing faces o f the t h r u s t d i s c were machined, and subsequent ly l a p p e d , so t h a t they were f l a t and p a r a l l e l to w i t h i n 1.0 x 10~4 i n c h and had a s u r f a c e f i n i s h o f about 12 m i c r o i n c h e s . The h i g h speed s h a f t was supported by two double row, s e l f 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 system. The c r i t i c a l speed o f the 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 s h a f t surrounded the t h r u s t d i s c . 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 which operated i n p a r a l l e l and were l o c a t e d a t 120° to 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 which were i n c o n t a c t w i t h the two oppos ing 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, which formed b e a r i n g s u r f a c e s f o r the faces o f the t h r u s t d i s c . 25 The complete c y l i n d e r assembly was torque mounted and supported by two s t e e l p i s t o n tubes which were a t t a c h e d to the l o a d i n g p i s t o n s and which were c o n c e n t r i c w i t h the h i g h speed s h a f t . The p i s t o n tubes were supported at t h e i r o u t e r ends by 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 housings were used t o admit o i l t o the t e s t s u r f a c e s . S e a l s were p r o v i d e d between the b e a r i n g h o u s i n g and the h i g h speed s h a f t , and between the h o u s i n g and the p i s t o n t u b e . The l u b r i c a n t f lowed i n the a n n u l a r space between the p i s t o n tube and the h i g h speed s h a f t and i n t o the space between the b e a r i n g s u r f a c e s . The assembled b e a r i n g i s shown i n i t s 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 the t e s t b e a r i n g , and the o u t l e t o i l was g r a v i t y f e d to another 45 g a l l o n tank which was l o c a t e d about 4 f e e t below the b e a r i n g . A motor and pump u n i t r e t u r n e d the o i l t o the upper tank a t the end o f each t e s t . 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 o f t h i s o i l are g i v e n i n Appendix I I . A s e p a r a t e o i l system s u p p l i e d l u b r i c a n t t o the gear box. 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 the gear box by means o f a gear pump. Another pump d r a i n e d the gear box sump and r e t u r n e d the o i l to the r e s e r v o i r . Both pumps were d r i v e n from the same e l e c t r i c motor . ( i i ) 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 speed, l o a d , t o r q u e , temperature and f i l m t h i c k n e s s . Speed The r o t a t i o n a l speed o f the h i g h speed s h a f t was measured d i r e c t l y by 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 instrument had a r e s o l u t i o n of 0.4$ o f the f u l l 26 s c a l e . Load The b e a r i n g l o a d was assessed by measuring the p r e s s u r e a p p l i e d to the l o a d i n g j a c k s . 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 ' pressure gauge t e s t e r which was capable o f s u p p l y i n g p r e s s u r e s o f up to 500 l b s . / s q . i n . i n increments o f 5 l b s . / s q . i n . The c a l i b r a t i o n curve f o r 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 torque t r a n s m i t t e d t o the t h r u s t pads was measured by means o f a weight pan which was k n i f e - e d g e suspended on a torque arm. The torque arm was mounted on the b e a r i n g torque r i n g and weight pan was s i t u a t e d at a d i s t a n c e o f 10 inches from the b e a r i n g c e n t e r l i n e . Temperature The temperature o f the o i l was measured a t the 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 the 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 groove . The thermocouples were mounted i n t u f n o l i n s e r t s which were screwed i n t o the 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 pad. 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 which had ranges o f 0.20mV, by increments o f 0.01 mV, and 0-100 mV by increments o f 0.05 mV. T h i s ins trument i s shown i n F i g u r e 10. Copper-constantan thermocouples were used and the c a l i b r a t i o n procedure and r e s u l t s are g i v e n i n Appendix I I I . F i l m Thickness The t h i o k n e s s o f the f l u i d f i l m i n the b e a r i n g was measured by means of a ' D a y t r o n i c ' model IO3A-8O l i n e a r d isplacement 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 .040 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 which had a h e m i s p h e r i c a l end o f r a d i u s 0.030 i n c h . The t r a n s d u c e r was screwed i n t o the 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 through the t h r u s t pad to make c o n t a c t w i t h the t h r u s t d i s c . Thus the e l e c t r i c a l output o f the t r a n s d u c e r was a measure o f the d i s t a n c e between the t h r u s t pad and the t h r u s t d i s c . 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 the t r a n s d u c e r and showed the d isplacement on a p r e - c a l i b r a t e d s c a l e . T h i s i n s t r u m e n t , which i s shown i n F i g u r e 10, had a range o f £ 0.100 i n c h and a maximum r e s o l u t i o n of 10 m i c r o i n c h e s . I n a d d i t i o n to the t r a n s d u c e r , two ' S t a r r e t t ' i0*0oo i n c h 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 lungers t o u c h i n g one o f the p i s t o n end p l a t e s . The c o n f i g u r a t i o n was such t h a t the gauges r e c o r d e d the sum o f the two f i l m t h i c k n e s s e s . The aforementioned i n s t r u m e n t s and p i e c e s o f equipment a r e 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. 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 TEST BEARING ro vo  FIG.8 TEST BEARING BEFORE ASSEMBLY F I G . 9 ASSEMBLED B E A R I N G I N P O S I T I O N 33 FIG.10 DIFFERENTIAL TRANSFORMER INDICATOR AND THERMOCOUPLE POTENTIOMETER  FIG.12 GENERAL VIEW OF APPARATUS CHAPTER IV I V . 1. E x p e r i m e n t a l Procedure I V . 2. E x p e r i m e n t a l R e s u l t s I V . 3. D i s c u s s i o n o f R e s u l t s 37 IV.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 procedure f o r the t e s t b e a r i n g was adopted to ensure proper a l ignment o f the t h r u s t s u r f a c e s p r i o r t o a s e r i e s o f t e s t s . 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 secured by b o l t i n g down the two support b e a r i n g s . With 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 unsecured , a l o a d was a p p l i e d to 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 to metal c o n t a c t between the t h r u s t pads and the t h r u s t d i s c . The r e s u l t i n g c o n t a c t p r e s s u r e caused the t e s t b e a r i n g assembly to 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 to 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 support the s t a t i o n a r y components were c a r e f u l l y s e c u r e d . Before 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 to the 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 set to read z e r o . The l o a d was then 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 were separated by about 0.010 i n c h . The s t a t i o n a r y components were torque ba lanced i n t h i s p o s i t i o n . With these p r e l i m i n a r i e s completed, the l u b r i c a t i o n system f o r the gear box was s t a r t e d and o i l was a d m i t t e d t o the t e s t s u r f a c e s . The main motor was then s t a r t e d and an i n i t i a l l o a d i n g pressure of 5 l b s . / s q . i n . was a p p l i e d to the h y d r a u l i c l o a d i n g j a c k s . A f t e r about two minutes r u n n i n g t ime the o i l 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 readings c o u l d be t a k e n . 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 r e c o r d e d . The r a t e o f o i l f l o w through the b e a r i n g and the b u l k o u t l e t temperature o f the o i l were recorded 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 repeated f o r l o a d i n g p r e s s u r e increments 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 than the minimum f i l m t h i c k n e s s 38 c r i t e r i o n , was found to 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 the b e a r i n g . At t h i s p o i n t the b e a r i n g was unloaded and the machine was s t o p p e d . Immediately the machine was at r e s t , 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 the l o a d i n g j a c k s and the t r a n s d u c e r zero r e a d i n g noted 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 then removed and the t h r u s t s u r f a c e s were separated by about 0.010 i n c h and the zero torque r e a d i n g was checked. The t e s t procedure was repeated f o r speeds r a n g i n g from 15,000 to 19,000 r . p . m . Tests a t speeds i n excess o f 19,000 r . p . m . c o u l d not be made due to the p e r s i s t e n t cage f a i l u r e of 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 gear box . Having completed a s e r i e s o f t e s t s , the t e s t b e a r i n g was d i s m a n t l e d f o r i n s p e c t i o n . The t h r u s t s u r f a c e s were found to be i n good c o n d i t i o n w i t h no evidence 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 ; the r e s u l t s o b t a i n e d were found t o be i n agreement w i t h the f i r s t se t o f r e a d i n g s . 39 IV.2. E x p e r i m e n t a l R e s u l t s The d e r i v e d r e s u l t s are presented 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 the observed r e s u l t s are g i v e n i n Appendix I V . The l o a d curves f o r the b e a r i n g are shown i n F i g u r e 13 and the c o r r e s p o n d i n g f r i c t i o n c h a r a c t e r i s t i c s are shown i n F i g u r e 14. To permit a d i r e c t comparison t o be made w i t h the t h e o r e t i c a l curves o b t a i n e d i n Chapter I I , the curves f o r 20,000 r . p . m . have been reproduced i n F i g u r e s 13 and 14. 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 . , consequent ly the former curves 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 of 16.5 l b s . per pad c o u l d not be c a r r i e d by the b e a r i n g w i t h o u t the o u t l e t o i l temperature exceeding the 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 the r e s u l t s 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 by p l o t t i n g the c o e f f i c i e n t o f f r i c t i o n / a g a i n s t the p a r a m e t e r ^ - . Such a p l o t i s shown i n F i g u r e 15 i n which the v a l u e o f Z has been taken 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 the c o e f f i c i e n t o f f r i c t i o n from the measured b e a r i n g torque and l o a d , a mean r a d i u s o f the b e a r i n g must be d e f i n e d . The t h e o r e t i c a l e x p r e s s i o n f o r the c o e f f i c i e n t o f f r i c t i o n i s g i v e n by e q u a t i o n F o r an a p p l i e d l o a d o f P ' per u n i t a r e a , the l o a d per pad W w i l l be g i v e n by the e x p r e s s i o n : (44): I f the v i s c o s i t y i s assumed constant a t some average v a l u e JUL1, then the above 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) 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 : 40 / - '7 !TrT[ , r,*-r.» / • • (^6) Prom equation (̂ 6), the mean bearing radius tm 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 relation* CS hO I.ST 2.0 ZS 3.0 3.5 -4--0 FILIA THICKNESS (in.* 16"*) FIG. IS EXPERIMENTAL LOAb CARRYING CAPACITY 6 2<7  I V . 3 . D i s c u s s i o n o f R e s u l t s An e s t i m a t e o f the a c c u r a c y o f the observed 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 the e x p e r i m e n t a l r e s u l t s w i t h the t h e o r e t i c a l p r e d i c t i o n s . F i n a l l y , the exper imenta l r e s u l t s w i l l be compared w i t h those 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 . Observed R e s u l t s F o r each nominal speed s e t t i n g , the r o t a t i o n a l speed o f the t h r u s t d i s c was found t o be independent of the a p p l i e d b e a r i n g l o a d . The accuracy o f the speed meansurements was l i m i t e d o n l y by the r e a d a b i l i t y and i n h e r e n t accuracy o f the tachometer. F o r the speeds a t which the b e a r i n g was r u n , the recorded 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$. The a c c u r a c y w i t h which the b e a r i n g l o a d was o b t a i n e d was governed by the accuracy o f the c a l i b r a t i o n curve f o r the l o a d i n g system. I n Appendix I I I the l o a d i n g curve i s g i v e n t o w i t h i n 5$» The f r i c t i o n t o r q u e , as measured, was produoed by the a c t i o n o f both t h r u s t b e a r i n g s . 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 w i t h i n an ounce o r about 5$« The probable accuracy o f the c o e f f i c i e n t o f f r i c t i o n , i f based on the r o o t mean square l a w , i s 7$. The accuracy o f the 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 by the random ' d e v i a t i o n s which were e x p e r i e n c e d . Suppose t h a t the f i l m 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 . I t was found t h a t t h i s v a l u e would p r e v a i l f o r 1 t o 2 seconds, 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 . T h i s e r r a t i c d e v i a t i o n l i m i t e d the accuracy o f the f i l m t h i c k n e s s measurements to about 10$. Accompanying the 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 the o u t l e t o i l was observed. I t was n o t i c e d t h a t i f the 45 film thickness decreased, the o i l temperature increased, the variations in temperature being of the same nature and duration as the variations in film 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 in film thickness and temperature, i t was thought that intermittent o i l starvation existed between the bearing surfaces. Evidence of localised o i l starvation was also reported by Fogg [2j. 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 resulting restriction to o i l flow at the perimeter of the bearing produced a small positive pressure there and a considerable improvement in the bearing capacity was obtained. Another factor which could contribute to the in s t a b i l i t y exhibited by the test bearing is the presence of a i r between the thrust surfaces. Air entering the bearing, either by way of the inlet 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 in the inlet 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 inlet of the next pad. This leakage could take place by either one of two mechanisms. The f i r s t mode of leakage could be attributed to carry-over from the rotating disc. That i s , since there exists a definite gap between the stationary and moving surfaces of the bearing, and since the layer of o i l next to the disc is moving with the disc, i t i s evident that some recirculation w i l l always take place. By this means i t may be visualized that the f l u i d film at the inlet to a pad w i l l be made up of entirely recirculated o i l in 46 the l a y e r adjacent to the d i s c , e n t i r e l y f r e s h o i l adjacent to the s t a t i o n a r y pad, and some form o f m i x t u r e i n the r e g i o n between these extreme l a y e r s . The second method by which o i l may l e a k from one pad t o another i s r e l a t e d t o the 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 the o u t l e t groove c o n t a i n i n g thermocouples 11 and 12 w i l l be unable t o escape u n t i l t h a t groove i s c o m p l e t e l y f i l l e d . This means t h a t the groove w i l l always be f u l l o f 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 n o n - l o a d c a r r y i n g r e g i o n i n the vertical position w i l l be f u l l o f hot o i l . By the a c t i o n o f gravity, these two r e g i o n s w i l l feed hot o i l i n t o the 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 can be o b t a i n e d by r e v i e w i n g the observed r e s u l t s i n Appendix I V . I t w i l l be noted t h a t the temperatures recorded by thermocouples 1 and 2 were h i g h e r than those recorded i n the o t h e r i n l e t grooves . S ince the average temperature r i s e across the 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 the e x p e r i m e n t a l curves tend toward the theory a t medium f i l m t h i c k n e s s e s , then f a l l away a g a i n a t low f i l m t h i c k n e s s e s . The main reason f o r the divergence a t low f i l m t h i c k n e s s e s l i e s i n the i n l e t temperatures o f the o i l . I n the t h e o r y , the i n l e t temperature was assumed constant a t 70°F, whereas average i n l e t temperatures o f up to 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 . S i n c e the thermal wedge depends on v i s c o s i t y to produce a temperature r i s e a c r o s s the b e a r i n g and s i n c e the v i s c o s i t y v a r i e s as t~*'71 , then i t f o l l o w s t h a t the i n c r e a s e d i n l e t temperature w i l l reduce 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 . A p o i n t check was made on t h i s by o b t a i n i n g a computer s o l u t i o n f o r the l o a d c a r r i e d at 15,000 r . p . m . The l o a d was found to be reduced from 25.30 l b s . 47 at 70°P to 13.85 l b s . a t 1G0°P. These f i g u r e s demonstrate the severe e f f e c t t h a t the i n l e t o i l temperature has on the 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 the l o a d observed i n the 1.0 x 10~3 t o 1.75 x 10~3 range o f f i l m t h i c k n e s s i n F i g u r e 13. I n the f i l m t h i c k n e s s range 1.75 x 10~3 to 2.5 x 10~3 the exper imenta l l o a d i s c l o s e to the t h e o r e t i c a l curve 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 the s t a t i c pressure predominates and the hydrodynamic theory does not 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 the s t a t i c pressure 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 the 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 the b e a r i n g s u r f a c e s , 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 pressure 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 than 2.5 x 10" 3 the c e n t r i f u g a l f o r c e becomes i m p o r t a n t . I n Chapter 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 the 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, the induced pressures a r e low i n the range i n q u e s t i o n and consequent ly the c e n t r i f u g a l f o r c e , which 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 to f i l m t h i c k n e s s e s o f about 1.25 x 10~3 . F o r f i l m s t h i c k e r than 1.25 x 10~3 the i n c r e a s e i n l o a d which i s p r e d i c t e d by the t h e o r y f o r an i n c r e a s e i n speed, i s more than o f f s e t by the c o r r e s p o n d i n g i n c r e a s e i n the c e n t r i f u g a l f o r c e . For f i l m s t h i n n e r t h a n 1.25 x 1 0 3 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 the 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 t h e o r y . The c o e f f i c i e n t o f f r i c t i o n curves 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 / are everywhere l e s s than the t h e o r e t i c a l p r e d i c t i o n s . 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 the i n c r e a s e d temperature i n the i n l e t o i l . S i n c e , an i n c r e a s e i n the o i l 48 temperature would r e s u l t i n a decrease i n the v i s c o s i t y and i n the c o e f f i c i e n t o f f r i c t i o n . At low v a l u e s o f f i l m t h i c k n e s s , the observed b u l k temperature o f the i n l e t o i l was about t w i c e the assumed v a l u e . The v i s c o s i t y would t h e r e f o r e be reduced by about 8 0 $ . F o r l a r g e v a l u e s o f 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 the assumed v a l u e , r e s u l t i n g i n a 30$ decrease 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 they would r e c o r d o n l y the b u l k temperature o f the 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 the f l u i d f i l m . Any such temperature g r a d i e n t would produce a f l u i d l a y e r adjaoent to the r o t a t i n g d i s c which would be a t an even h i g h e r temperature than 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 temperature of t h i s n a t u r e would cause a f u r t h e r decrease i n v i s c o s i t y and i n the c o e f f i c i e n t o f f r i c t i o n . Apart 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 t h a t any d i s c r e p - ancy i n the 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 c h a r a c t e r i s t i c s . I n the f i l m t h i c k n e s s range 1.0 x 10~3 to 1.75 x 10~3 > the e x p e r i m e n t a l l o a d i s l e s s than the t h e o r e t i c a l l o a d . F o r t h i s range , the l o a d e r r o r w i l l 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 . I n the range 1.75 x 10~3 t o 2.5 x 10~ 3 i t was observed t h a t , due to the e f f e c t o f s t a t i c pressure* the e x p e r i m e n t a l l o a d exceeded the t h e o r e t i c a l v a l u e . I n t h i s case the two e r r o r s a r e c u m u l a t i v 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 than 2.5 x 10~ 3 the exper imenta l l o a d i s a g a i n l e s s than the t h e o r e t i c a l v a l u e , due t o the 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 the 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 the 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 1 0 ~ 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 adverse 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 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 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 p a r a m e t e r i n F i g u r e 15• 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 are f o r the most p a r t i n a d i f f e r e n t range from the 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 d i f f e r e n c e i n range can be a t t r i b u t e d t o the low l o a d s c a r r i e d i n the present 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 are c l o s e to those o b t a i n e d by Young and Fogg, a l though the g r a d i e n t o f the f r i c t i o n curve 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 . d i f f e r e n t authors i s p r o b a b l y produced by the 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 , a lower c o e f f i c i e n t 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 at h i g h e r speeds. 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 , but 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 the b e a r i n g which runs a t the h i g h e r speed. F o r the same i n l e t o i l temperatures , the o u t l e t temperatures w i l l be the same so t h a t the v i s c o s i t y Z w i l l be the same f o r the two b e a r i n g s . 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 n l y have a lower c o e f f i c i e n t of f r i c t i o n , but i t w i l l have a h i g h e r v a l u e o f ^r. The f r i c t i o n c h a r a c t e r i s t i c s of the two b e a r i n g s w i l l t h e r e f o r e d i f f e r 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 combining equat ions (46) and (47)* the l o a d . The wide s c a t t e r i n the 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 by E q u a t i o n (49) shows t h a t a g e n e r a l curve c o u l d be o b t a i n e d by p l o t t i n g the c o e f f i c i e n t o f f r i c t i o n a g a i n s t the parameter J&LJJz. U n f o r t u n a t e l y , Fogg 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 d a t a t o permit such a c o r r e l a t i o n to be made. CHAPTER ? V . 1. Summary and C o n c l u s i o n s V . 2 . Suggest ions f o r F u t u r e Research 5 2 (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 of 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 thermal 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 both theory 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 t o u n d e s i r a b l e l i m i t s . T h e o r e t i c a l l y , 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 t h r u s t b e a r i n g to support hydrodynamic loads o f the o r d e r o f 30 l b s . / s q . i n . o f b e a r i n g a r e a . I n g e n e r a l , i t can be concluded t h a t f o r a given 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 o f f r i c t i o n . E x p e r i m e n t a l l y , l o a d s of 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 . T h i s l o a d c a r r y i n g c a p a c i t y i s c o n s i d e r a b l y l e s s than t h a t 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 c a p a c i t y 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 . I n t h i s r e s p e c t the e x i s t e n c e o f separate 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 they can have a d e t r i m e n t a l e f f e c t . 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 b t a i n e d are i n good agreement w i t h the r e s u l t s o b t a i n e d by o t h e r e x p e r i m e n t e r s . 53 V . 2 . Suggest ions f o r Future Reasearch I t would seem d e s i r a b l e to attempt to 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 before p e r f o r m i n g any f u r t h e r t e s t s . I n t h i s r e s p e c t i t seems necessary 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 areas of 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 o f 0.010 i n c h . 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 the b e a r i n g a t the o u t e r r a d i u s and leaves a t the i n n e r r a d i u s . 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 proper 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 pads. To permit a c c u r a t e 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 of 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 torque must be a c c u r a t e l y known. F o r s m a l l , h i g h speed b e a r i n g s , the f r i c t i o n torque i s low and the weight pan method o f o b t a i n i n g the b e a r i n g torque i s r a t h e r i n s e n s i t i v e . C o n s i d e r a b l e 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 employing a s m a l l f o r c e t r a n s d u c e r t o a c t on the torque arm. I n t h i s way r a p i d torque r e a d i n g s 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 a c c u r a c y . F i n a l l y , an automatic 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 permit 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 r e l a y s w i t c h f o r the main motor c o u l d be f i t t e d to the b e a r i n g assembly i n such a way t h a t i f meta l t o meta l c o n t a c t o c c u r s , 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 torque would cause the torque arm to depress the r e l a y b u t t o n and so s t o p the main motor . 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 will 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' will refer to Cope's analysis. To permit direct comparison with Cope's results, rectangular co-ordinates will 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> fj),t,jJ*and.Jk are functions of *x and y only 56 ( i i ) That and so on ( i i a ) That !f̂ >-§f, ̂ H ^ ^ «* 3 0 ( i i i ) That = o ( i i i a ) T h a t >ur = o ( i v ) That H = 0>t ( i v a ) That E - C v t A p p l y i n g these assumptions , t h e - e q u a t i o n s become: ^ U # + / C > ^ ) - f « f + v f ) = ^ ) + | ^ ) ^ f ^ ) V ^ f ] . (51) N e x t , we make a term by term a p p r a i s a l and hence assume t h a t : (v) c o n d u c t i v i t y terms etc. are n e g l i g i b l e (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 ( 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 With these assumptions we g e t : (fCru£+fCi,vf)~(u& + v-%)+A[(%Y+(ff] • • (52) F i n a l l y , we a p p l y the 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 the l i m i t s o f £ = o and ^ = A • These s teps 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 these equat ions i s the c a r t e s i a n form o f the energy e q u a t i o n o b t a i n e d i n Chapter I I . 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 supported by Charnes, O s t e r l e and S a i b e l f_13]. However, the l a t t e r authors 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 to mass c o n t i n u i t y . Apart from b e i n g a l i t t l e more a c c u r a t e than e q u a t i o n (53a), e q u a t i o n (53) has the advantage o f b e i n g somewhat s i m p l e r . Furthermore , e q u a t i o n (53) i n v o l v e s the s p e c i f i c heat a t constant 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 than t h a t a t constant 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 to be known f u n c t i o n s o f temperature to permit 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 . Consequent ly , 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 these q u a n t i t i e s and the temperature . V i s c o s i t i e s were measured u s i n g a Saybol t U n i v e r s a l V iscometer 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 are 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) Ibs.secYft4 (54) /jL^ISZt"*"' }t»..Sft.'/ft* (55) f i t the e x p e r i m e n t a l curves t o w i t h i n 1$ i n the temperature range encountered. The s p e c i f i c heat was assumed constant 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 Tests 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 to determine,, ( l ) the temperature versus 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 versus a p p l i e d l o a d curve f o r the l o a d i n g system. 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 at the steam p o i n t . T h i s was done by p l a c i n g the 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 crushed i c e a t the m e l t i n g p o i n t . 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 o f mercury. From t h i s i n f o r m a t i o n the e v a p o r a t i o n temperature o f the 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 at p r e s s u r e ^ , i n °C 6*>= 100.00 °C rt = b a r o m e t r i c p r e s s u r e , 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 s t a n d a r d t a b l e s , d e r i v e d from Adam's T a b l e s , the emf f o r t h i s temperature i s found to be 4.241 mV. T h i s d e v i a t i o n o f 0 . 4 $ was assumed to 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 to the i c e p o i n t , so t h a t the curve shown i n F i g u r e 17 was o b t a i n e d . Loading 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 , an i n - s i t e l o a d i n g t e s t was made. The h i g h speed s h a f t was removed from the t e s t 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 opposing 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 then 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 space. Thus a known pressure c o u l d be a p p l i e d to 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 to the t h r u s t pad faces by another 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 the gear box was then s t a r t e d up and a s e r i e s o f pressure r e a d i n g s taken f o r zero p i s t o n m o t i o n . T h i s enabled the l o a d i n g curve o f F i g u r e 18 t o be drawn, from which the maximum d e v i a t i o n was found to be 5$«  63 (tul/s<ll) SXOUr 9NIQV01 pi QSIlddV 3H()SS3Hd 6k APPENDIX IV The observed e x p e r i m e n t a l r e s u l t s are presented i n Tables I I I t o V I I . I n these T a b l e s , the 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 pressure a p p l i e d to h y d r a u l i c l o a d i n g j a c k s ( l b s . / s q . i n . ) hi m 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 = zero 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 ime St' ( c c . ) St'«= t ime i n t e r v a l to c o l l e c t o i l f l o w Q ( s e e s . ) Wr= weight a p p l i e d to torque arm a t 10 i n c h r a d i u s ( o z s . ) 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 termperature (°P) The thermocouple p o s i t i o n s are shown i n F i g u r e 7. -4" C O O - C O CM -a CM C M . 0 0 V O O N - P H H H H H O H V D LTN V D O H oo oo CM co CVJ CM C M oo -4- CVJ O N J - C O - H V O V O O N O N O N aJ H H H CM on -4- O N OO CM O J o -3- t — t — V D H H H CM on D - C O V D H O i—1' CM C O V O 0) • • • .o H H r H H CM cu OO C - on CM go o H 0 0 CM H CU o • e • o CM CM CM OO -=J" ir- OO CM V O O N c - V D CD • • o H H H CM on C O O V D O O LTN V O C — O N O O N CU • 0 • » H H r H O O on H t - IT\ V D LTN la H H OO O N C — CU • • • • H H H H CM CM CM OO H O H OO CM CL) » o » • • cvj CM CM OO V O C O C O o O J «i C O O N H o C O - • « • H H CM oo on C - C O LTN r H It O N O OO OO oo CO a a • o e H CM CM oo O C — O LTN t - C - C O H V D on CD H H CM CM on LT\ O LTN O LTN H H CM CM O N [>- H H H CM V O r H co <• • • > O N H -3- c— on H H CM -=j- LPs LTN O LT\ o a? LT\ O O O O N 0 O -d- on O N O N O CM O o H H H 6 6 6 6 6 O o OO 00 LTN H C O -=*• O OO CM CM r H H J ft LTN O LTN O LTN H H CM CM ft iH O O O N tTN H II P i i ca +5 w (D CL) CL) In CD 0) rt CD > H on CM H C M H CM CO o LTN o CM H CM V D O H on H CM C O O N O N o CM O t — LTN O N O C O c— O N O J r H d LPs O N OJ O on H H O J O J O N on L T \ O J r H -3" H O J on oo L̂ c— H CM CM t - O CM on O J CM on O N O J H LTN O N C - o on H r H O N V O O J •J ft O N on H LT\ OJ o o CM on r H on CM r H CM C M O O CM O O N O on H LTN O J O J on O J co O J H O J O O J H C O O N LTN C O 0J O J H LTN CM C M LfN r H 4 CO H H -=r •P CM PO VO H H H H OJ C7\ vo O rj H H Lf\ -4" <D • • OJ OJ OJ co ON ON o O O • • • • H H OJ CO OJ O ir\ o g -3" LTN CO • • • rH H H CM LTN OJ H VD o» O H CO • • • • H H H PH. OJ H OJ CO « H OJ LT\ c - * - • • OJ OJ OJ OJ VD OJ LTN 'ITN N CO o (U • • • a H H OJ CM LTN ON C - o 00 H H CM CM O VO r o H H on GO- 0) • • • • H H H (ri Lf\ O OJ *• H OJ VO CD e • • • OJ • OJ OJ CO O o o o O H *\ • • * • CO OJ OJ OJ r o CO OJ -4- OJ o CM Lf\ ii • o • CD OJ OJ OJ CO o CA CO O • • • • H H OJ CM LCN Lf\ o O )- • • • is o OJ co ON H H H VD OJ o • • • • ON ON VD H o ITN OJ CO a ? H ON H CO co OJ CO H H OJ <* H H H H • • • • O O O O G o OJ _ CO vo OJ > • • • OJ OJ OJ H ft • LTN o ITN O H H OJ o o en CO CQ EH co r o ITS CO OJ oo VD -P H H H H << VD CO LTN CO H OJ LTN OJ CU 9 • • « > OJ OJ OJ OJ OJ VD CO VO CO CO o CU > • • • H H OJ OJ CO CO CO CO © -=1- c o Ct) o • • • H H H OJ UN O -4" CO ». o H OJ LTN 0 • » • • H H H H ON H LTN CO H oo H • a • • OJ OJ OJ OO co o o VO N CO ON H LTN Q) a • • • rH H CM OJ CO O . ON •6 O LTN CU • • • • H H OJ OJ CM LTN H VD <n H rH CO VO CU • • • • H H H H ON co ON CO H CO LTN OJ CD • • • • OJ OJ CM co VD ON LTN ON m O H ON CU • • • • OJ OJ OJ CM LTN CO o rH »< H OJ VD OJ (U • • • • OJ OJ OJ OO H LTN O LTN CO O -4- CU • • • • H H OJ CM O O o O • • • • ON OJ OJ ON H H CO LTN O O • • • • H H VD H H H H LTN O LTN LPs & CO VD VD VO -=f co c o O O H c o H H H H si • • • * O O O O LTN OJ LTN O VD O • 0 • CM OJ OJ H ft" LTN o LTV O H rH OJ t - OO ITN t - - P . O J OO -4" V O H H H H ro H O J O N O J O O V D H • • • • O J O J O J oo O N oo O N O N • — cn O N H t - (1) • • • • H H O J O J O J OO C O C O o LTN LTN c— oo OJ • • • H H H O J O N OO C O LP\ O H O J LTN CD • » • r H H r H H LPs -4" O J O N 00 O J oo V D H • » • > O J O J O J OO LTN -4 OO C O N C O O N H LPs CD • • 0 • H H O J CvJ O O N o LP\ •0 C O L ^ H V O CD o • o •> H H O J CM -4- C O o V D in H H oo V D CD • • B • H H H H O N O N V O LP\ O J O O V D CM CD • • • • O J O J O J OO -4" O J o o m H O J LTN o CD • • • • O J O J O J oo OO LTN O J -4- M O J O O V D O J • • • • O J O J OJ oo O N H o O N _ 00 o O J CD • • • • H H O J O J H . LP\ LPs o o • • • • O N O J O O O N H H V O O -4" 00 • • • • M l O N C O 00 C O -4" LP\ LTN o a? O J O N H -4- OO O J CM O J oo -4" V D H H H H • • • • o o O O o o LTN O O V D -4- O -4" « • • • O J O J O J r H J ft LT\ o LP\ O H H O J 68 BIBLIOGRAPHY [I] R e y n o l d s , 0. "On the Theory of 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. Soc. of 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 o f 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 to Fogg [2] Cameron, A . and Wood, W . L . " P a r a l l e l Sur face Thrust B e a r i n g s " , P r o c . 6 t h . I n t e r . Cong, of App. M e c h . , 1946. [5] Shaw, M . C . "An A n a l y s i s o f the P a r a l l e l Sur face Thrust B e a r i n g " , T r a n s . Amer. Soc. Mech. E n g r s . v o l . 69, 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 , C F . "Tests on P a r a l l e l Surface Thrust B e a r i n g s " , E n g i n e e r i n g , Aug. 1955• [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 , W . E . , and L i g h t f o o t , E . N . "Transport Phenomena", John W i l e y & Sons, I n c . , I 9 6 0 . [ I I ] S o u t h w e l l , R . V . " R e l a x a t i o n Methods i n E n g i n e e r i n g S c i e n c e " , Clarendon P r e s s , O x f o r d , 1940. [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 to 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 . 214, 1952.

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