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The thermal wedge effect in hydrodynamic lubrication Dvorak, Frank Arthur 1964

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THE THERMAL WEDGE EFFECT IN HYDRODYNAMIC LUBRICATION •"by FRANK ARTHUR DVORAK B.Eng., Royal M i l i t a r y College, 1962 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 accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 196k In";preseriting t h i s t h e s i s 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 U n i v e r s i t y , o f 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 . a v a i l a b l e f o r reference and study. I further agree that permission f o r extensive copying of t h i s t h e s i s f o r scholarly purposes may be granted by the Head of my Department or by h i s representatives. I t i s understood that copying of t h i s t h e s i s 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 U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8 , Canada. April,'. I96U. ABSTRACT The' 'thermal wedge' e f f e c t i n hydrodynamic l u b r i c a t i o n has been studied both t h e o r e t i c a l l y - and experimentally. The f r i c t i o n theory .has been r e f i n e d to take into consideration the e f f e c t on f r i c t i o n of a v a r i a t i o n of temperature across the f i l m from the bearing to the disk surface. The t h e o r e t i c a l pressure and temperature equations formulated by Currie (12)* were used i n conjunction with the f r i c t i o n equation to obtain t h e o r e t i c a l performance curves. Tests were c a r r i e d out on.a set of bearings having three, four and f i v e pad configurations. Two d i f f e r e n t bearing ;materials were used, namely, babbitt, and D e l r i n A.F. f i b e r / r e s i n . Circumferential o i l rings were f i t t e d into the assembly to r e s t r i c t the r a d i a l flow of oil,.thereby'ensuring f u l l l u b r i c a t i o n ' o f the bearings. Experimental observations proved that a thermal gradient across the f i l m d i d e x i s t , thus j u s t i f y i n g the basic assumption In the v a r i a b l e v i s c o s i t y analysis. Experimental c o e f f i c i e n t s of f r i c t i o n compared favourably with predicted values. The agreement was b e t t e r than b y - e a r l i e r theories. The r e s u l t s Indicate that bearing performance i s improved by the use of o i l r e s t r i c t o r rings (actual load carrying ..capacities were more than doubled). •The optimum bearing .configuration was that of a four-pad babbitt bearing. Although the D e l r i n A F bearing performed much bett e r than the babbit bearings, d i f f e r e n t i a l thermal expansion of the D e l r i n produced a geometric wedge, and a bearing of t h i s material could not be considered-as a thermal wedge bearing. * Numbers i n brackets r e f e r to l i s t of references i n Bibliography. ACKNOWLEDGEMENT The experimental work presented i n t h i s t h e s i s was c a r r i e d out i n the Lubrication Laboratory, and the t h e o r e t i c a l data was compiled i n the Computing Centre of 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•like to express h i s gratitude to the fo l l o w i n g : Professor"W.6. 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. Dr. C.A. Brockley, f o r h i s guidance and continued interest,.throughout the research program. Mr. P. Hurren, and the. other technicians f o r t h e i r valuable assistance during the experimental phase of the research program. The National Research Council of Canada, f o r sponsoring the research under N.R.C. Grant No. A 1 0 6 5 -TABLE OF CONTENTS CHAPTER r l -Page 1.1 Introduction 1 CHAPTER 2 2.1 The Governing Equations f o r Film Lubrication 6 2 .2 Solution of the Governing Equations ,8 2- 3 Theoretical,Performance of .Bearing 10 CHAPTER 3 3 .1 Apparatus : 23 3- 2 Measurements 27 CHAPTER k k.l Experimental 36 k.2 Results 38 4 . 3 Discussion of Results 53 CHAPTER 5 63 5.1 Conclusions APPENDIX I. P h y s i c a l Properties of Bearing Materials 65 APPENDIX I I . Groove Configurations 67 APPENDIX I I I . P h y s i c a l Properties of S h e l l Turbo 27 O i l 71 APPENDIX IV". Instrumentation C a l i b r a t i o n s 73 APPENDIX V. Experimental Data 79 APPENDIX VI. Load Correction f o r S t a t i c O i l Supply 83 • APPENDIX VII. Dimensional Analysis Solution Qk BIBLIOGRAPHY LIST OF FIGURES Fig.Mo. Page 1 Transformation of Sector into Rectangle 8 2 Velocity-and Temperature P r o f i l e s 12 3 P a r a l l e l Plates i n Motion 12 '. k T h e o r e t i c a l Performance of Three-Pad Bearing 17 ;5 T h e o r e t i c a l Performance of Four-Pad Bearing 18 6 Theoretical:.Performance of Five-Pad Bearing 19 7 Pressure and Temperature D i s t r i b u t i o n 20 p T h e o r e t i c a l Temperature Contours 21 9 T h e o r e t i c a l Pressure Contours 22 10 Line Diagram of Apparatus 2k 11 Tnrust Pad, 3-G-roove Configuration ,.. 26a 12 Thrust Pad, ^-Groove Configuration 26b 13 Thrust Pad, 5-Groove Configuration 26c .lk Sectional View of Test Assembly 28 15 General View of Apparatus 31 16 Test Bearing Dismantled 32 17 :.Measuring Equipment 33 18 Film Thickness Probe, Disk Temperature Probe and . 3)4 Force Transducer 19 F a i l u r e of a Delrin. A F Bearing 35 20 Experimental Load Carrying Capacity - 3-ii&&- k0 •21 Experimental C o e f f i c i e n t of F r i c t i o n •- 3-Pa<3- kl 22 • Experimental Disk Temperature - 3-Pad k2 23 Experimental Load Carrying C a p a c i t y - ^-Pad U3 2k Experimental C o e f f i c i e n t .of F r i c t i o n .-• i+-Pad kk 25 Experimental Disk Temperature - U-rPad -4-5 26 Experimental Load Carrying Capacity - k6 2 7 .Experimental C o e f f i c i e n t of F r i c t i o n .- 5-Pad Ml Fig.No. Page 28 Experimental Disk Temperature - 5-?a<3- 48 29 Comparison of performance With and Without R e s t r i c t o r Rings 30 Comparison of Experimental Load Carrying Capacities 50 31 Comparison of Experimental C o e f f i c i e n t s of F r i c t i o n 51 32 Comparison of Experimental Results 52 33 Forces on an Elementary P a r t i c l e of O i l 69 34 Physical,Properties o f S h e l l Turbo 27 O i l 72 35 C a l i b r a t i o n Curve, f o r Thermistor Probe 75 36 C a l i b r a t i o n Curve f o r Force Transducer 76 37 C a l i b r a t i o n Curve f o r Copper-Constantan Thermocouples 77 38 C a l i b r a t i o n Curve f o r Loading System 78 39 T h e o r e t i c a l Load C o r r e l a t i o n 86 4.0 Correlated Load Curves 87 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. & Independent c y l i n d r i c a l co-ordinates ;•<_,/, 2 Independent rectangular co-ordinates u,v,"> Ve l o c i t y , i n x, y, and z d i r e c t i o n r e s p e c t i v e l y §5t' Substantial time de r i v a t i v e a. G r i d s i z e /!\ Constant <g Constant C./} S p e c i f i c heat at constant pressure Dimensionless density Internal energy y'- . C o e f f i c i e n t of f r i c t i o n Body force, d i r e c t i o n as subscripted s^. F r i c t i o n force /> • Film Thickness // . Enthalpy /$ Thermal Conductivity stf .Dimensionless v i s c o s i t y Rotational speed ? Pressure /p Dimensionless pressure Heat f l u x , d i r e c t i o n as subscripted yoQ Inner radius of bearing /L, Outer radius of bearing Mean radius of bearing /L± Radius of shaft /? "Dimensionless' co-ordinate £ Temperature to I n l e t temperature 2* Mean bearing temperature '' £ & Disk temperature £ ' Time y Dimensionless temperature s ^ F r i c t i o n torque (j Velocity, of s l i d e r •or V e l o c i t y , d i r e c t i o n as subscripted W Load per pad w£ Width of bearing 2* Viscosity, i n centipoise Included angle of sector pad / £ Constant ^ C o e f f i c i e n t of thermal expansion ^\ D i l a t i o n = V• 7& •^7 Operator Q Dimensidnless co-ordinate A Constant yU- V i s c o s i t y yUo V i s c o s i t y at i n l e t Density ^ .Density , at -inlet ^ Stress- tensor CO Angular v e l o c i t y <2) D i s s i p a t i o n Function • Chapter 1. 1.1 Introduction 1; 1.1 Introduction . Although" bearings had been used successfully for many years, no l o g i c a l explanation was forwarded for their operation u n t i l Tower ( l ) , i n 1883, found evidence of a considerable pressure i n a loaded ra i l r o a d bearing. Tower did not, however, give any theoretical explanation for the existence of this pressure, and no explanation was forthcoming u n t i l the introduction of Reynolds' c l a s s i c a l paper (2) i n 1886. Reynolds showed that i n order to have f l u i d f i l m lubrication, a geometric r e s t r i c t i o n or 'taper wedge' was required i n the direction of motion. In the case of r a i l r o a d or journal bearings, this wedge action i s present when the bearing runs eccentrically when loaded. However, the thrust bearings, which at that time consisted of two p a r a l l e l surfaces, did not have this geometric wedge. The lack of wedging action was borne out by the experience res u l t i n g from the use of these bearings. They operated i n the boundary lubrication region, with a high coefficient of f r i c t i o n at low speeds and high loads, and were much less e f f i c i e n t than journal bearings. On occasion, p a r a l l e l surface thrust bearings did perform better than expected, but nothing resulted from this observation, and credit for a load carrying f i l m was given to the supply pressure of the o i l . Subsequently, a l l thrust bearings were designed using the geometric wedge p r i n c i p l e , either by using tapered surfaces of fixed i n c l i n a t i o n , or t i l t i n g pads as proposed by Kingsbury and Michell. Further development of the p a r a l l e l surface thrust bearing was not carried out u n t i l 1944, when Fogg ( 3 ) , while investigating several thrust bearings, found that p a r a l l e l surface thrust bearings would carry high loads at high speeds with a low c o e f f i c i e n t of f r i c t i o n . This finding was i n opposition to past experience, since Fogg found the p a r a l l e l surface bearings to be nearly as e f f i c i e n t as Michell's t i l t i n g pad bearings. Fogg used a pair of p l a i n c o l l a r s faced with white metal. The results indicated p a r t i a l l u b r i c a t i o n , but the o i l flow was very low,.resulting i n high operating temperatures. Radial grooves were introduced i n order to ,allow greater o i l flow,.thus d i v i d i n g the bearing into sector shaped pads. Fogg was c a r e f u l to .ensure that the bearings were of a p a r a l l e l configuration. Fogg suggested that high r o t a t i v e speeds produced a high temperature r i s e over the bearing surface, and t h i s i n turn ..resulted i n thermal expansion of the l u b r i c a n t . This expansion was termed the 'thermal wedge' e f f e c t , and i s analogous to .the geometric wedge of the t i l t i n g pad or jou r n a l bearings. With the geometric wedge a constant volume of l u b r i c a n t passes through' a converging section, whereas,.with the thermal wedge, an expanding volume of l u b r i c a n t passes through-a section of constant.area. Thus an equivalent-wedging action.required to f u l f i l l hydrodynamic l u b r i c a t i o n conditions i s also present, i n . a p a r a l l e l surface bearing. Fogg's conclusions r e s u l t e d i n increased i n t e r e s t i n the p a r a l l e l . s u r f a c e thrust bearing. The suggestion of thermal•expansion of the l u b r i c a n t encouraged the replacement of the assumption of volume continuity.by that of mass con t i n u i t y . The Fogg hypothesis brought about several-approaches to the thermal wedge problem, some of which concluded that the existence of a thermal wedge e f f e c t was indeed p o s s i b l e . Other workers suggested that side leakage would o f f s e t the e f f e c t of the expansion of o i l and that thermal d i s t o r t i o n of the surfaces might r e s u l t , i n a geometric wedge, from-which .a load c a r r y i n g f i l m .could be produced. However,•no evidence was presented to substantiate e i t h e r the adverse e f f e c t s of side leakage or the r e s u l t s of thermal d i s t o r t i o n . •Bower (k) supported the idea of a thermal wedge i n a t h e o r e t i c a l develop-ment which used a. s i m p l i f i e d form of the governing equations f o r f l u i d flow. He assumed no side leakage, a l i n e a r v a r i a t i o n of density, and v i s c o s i t y along the bearing and concluded that a u s e f u l load could be c a r r i e d i f . s t r i c t tolerances were met. Cameron and Wood (5) used the general Reynolds equation with density •as a v a r i a b l e , and an energy equation based on adiabatic flow to ar r i v e at two simultaneous p a r t i a l d i f f e r e n t i a l equations i n pressure and temperature. They considered ..an exponential v a r i a t i o n of v i s c o s i t y with temperature, and a l i n e a r v a r i a t i o n of density with temperature. The r e s u l t i n g equations •were too- involved to be solved by a n a l y t i c a l methods, so. a s i m p l i f i e d version was integrated considering no side leakage. .A so l u t i o n was a r r i v e d at by Shaw ( 6).in which he considered a l i n e a r v a r i a t i o n of density with temperature, but at constant v i s c o s i t y . He compared p a r a l l e l surface and t i l t i n g pad bearings at equal f i l m thicknesses, assuming no side leakage. He suggested that, a p a r a l l e l surface thrust bearing would have a load capacity of the order of one tenth that of a t i l t i n g pad bearing. Cope (7) derived the equations f o r f i l m l u b r i c a t i o n from the general equations-of hydrodynamics. Assuming steady.flow between p a r a l l e l surfaces i n relative"motion, he -arrived-at the form of the pressure equation that had been used previously by.Cameron and Wood. Cope arrived.at a temperature equation by a c a r e f u l study of the r e l a t i v e magnitudes of various terms i n the energy equation. The r e s u l t i n g equation was d i f f e r e n t from that of Cameron and Wood by a term which considered the work done on.an element of lu b r i c a n t by .the pressure forces-exerte-dc'by the surrounding f l u i d . Cope solved a s i m p l i f i e d form of his. equations f o r no side leakage', and stated that f o r surfaces close together, with a small v a r i a t i o n of v i s c o s i t y with temperature and a large v a r i a t i o n of density.with temperature, the thermal wedge-altogether surpasses the geometric wedge. Kettleborough (8) measured the o i l f i l m pressure, the f i l m thickness and the c o e f f i c i e n t of f r i c t i o n f o r apparatus running at low speeds. The c o e f f i c i e n t of f r i c t i o n at a speed of 695'R-?M was about twice that obtained by Fogg. Nahayandi:and Osterle (9) recently suggested that the v i b r a t i o n of the s l i d e r could produce a load carrying f i l m on.a p a r a l l e l surface thrust bearing. This v i b r a t i o n , i n e f f e c t , i s supposed to produce a geometric' wedge by which a-load c a r r y i n g f i l m i s generated. More recent experiments seem to d i s p e l t h i s theory. Young (lO) made the f i r s t r e a l attempt to c o r r e l a t e theory, and experiment. He considered three methods of solving Cope 1s pressure and temperature equations, taking into account the e f f e c t of side leakage. The most rigorous method was a r e l a x a t i o n s o l u t i o n f i r s t used by Christopherson ( l l ) . Young compared the three methods with each other and with experimental r e s u l t s obtained-at speeds from 4000 to l 6 0 0 0 RPM. •Because of the large 'amount of work involved i n a t h e o r e t i c a l solution, .only point checks were made;•however, ;the r e l a x a t i o n s o l u t i o n was i n reasonable agreement with experimental r e s u l t s . The experimental c o e f f i c i e n t s of f r i c t i o n obtained-were higher than those obtained by Fogg, but lower than those obtained by Kettleborough. Currie (12) r e v i s e d the general equations governing the flow of f l u i d s i n a p a r a l l e l surface thrust bearing i n order to obtain as sophisticated a s o l u t i o n as was reasonable. The governing equations were solved by:the r e l a x a t i o n process of Christ'ppherson,r.using an IBM l 6 2 0 d i g i t a l computer. In t h i s way t h e o r e t i c a l r e s u l t s could be obtained quite r e a d i l y . Currcb'e also obtained" experimental r e s u l t s i n .the speed range 15000 to 19100 RPM. Experimental and t h e o r e t i c a l load capacity curves were i n close agreement. However, the experimental f r i c t i o n curves were much lower than theory predicted. This discrepancy was p o s s i b l y .due to a much higher operating temperature than was anticipated. I t was suspected that t h i s was* p a r t i a l l y .the r e s u l t of a carry-over of hot o i l on the bearing because of the bearing configuration. The foregoing discussion of previous wor^ suggests that.a thermal wed does e x i s t , and i t seemed desirable to reconsider and r e f i n e the theory i n v o l v i n g the c o e f f i c i e n t of f r i c t i o n , and to develop a d e f i n i t e approach to designing p a r t i c u l a r groove and pad configurations. In addition, a consideration ef d i f f e r e n t bearing materials i s desirable, i n order to optimize bearing operation. O i l r e s t r i c t o r rings w i l l also be considered i n r e l a t i o n to":the f u l l l u b r i c a t i o n of the bearings. Again theory,and experiment w i l l be compared so as to add to the av a i l a b l e information on .the thermal•wedge e f f e c t i n hydrodynamic l u b r i c a t i o n . .Thus i t was with the above ideas i n mind that the current research program was i n i t i a t e d . Chapter 2. 2 .1 The Governing Equations f o r Film Lubrication 2.2' Solution of the Governing Equations 2.3 Theoretical.Performance of Bearings 6. •2.1 The Governing Equations f o r Film Lubrication The general equations governing the flow of a viscous, incompressible, heat conducting f l u i d are w e l l known. Cope (7)> Currie ( l 2 ) and others, have derived from these equations,.the equations governing the flow of a l u b r i c a n t between p a r a l l e l : s u r f a c e s . •In-.the present work, Currie's analysis i n i t s entirety, i s used to c a l c u l a t e the t h e o r e t i c a l load capacity •and maximum o i l temperature of the p a r a l l e l surface thrust bearing. The general equations as presented by B i r d , Stewart and Lightfoot (13) are : P £>// £)p ^ ' sst' &t< r where the symbols used are defined i n the ' L i s t of Symbols' table. These equations are the equations of mass continuity, motion and energy re s p e c t i v e l y . , Currie (12) expanded these equations by,the method-presented by Cope (7)-For the flow o f l u b r i c a n t between p a r a l l e l surfaces i n close proximity and with r e l a t i v e tangential motion,.it i s reasonable to assume: ( i ) .That-.with respect to the load c a r r y i n g capacity, the v a r i a t i o n of pressure, density, temperature, v i s c o s i t y and thermal conductivity i s f a r l e s s important across the f i l m than along, i t . That i s , . i t :is assumed that p, p, t, and k.are functions of r and 0 only. ( i i ) That the velocity, gradients are only, important across the f i l m / That i s , i t i s assumed that • so that only.the f i r s t named need be retained i n groups of such terms. (1) (2) (3) 7-( i i i ) That the f l u i d v e l o c i t y perpendicular to the f i l m i s very small. That i s , i t - i s assumed that ^ a = O ( i v ) That the body forces, other than those caused by the motion of the.' f l u i d , . c a n be neglected. That i s , i t i s assumed that - ^& = =. O (v) .That the enthalpy of the f l u i d i s a function" of temperature, only and that the s p e c i f i c heat remains constant. That.is, i t i s assumed that Currie applied these -assumptions to the expanded general equations, thus g i v i n g a s i m p l i f i e d set of equations. • Further s i m p l i f i c a t i o n was effected, by,considering the r e l a t i v e magnitudes of the various terms. This was done' by taking.a t y p i c a l set of conditions, and c a l c u l a t i n g the value of the various terms, and neglecting any terms, which proved to be unimportant. In...this way. Currie a r r i v e d .at the s i m p l i f i e d equation of mass continuity, equations of motion (Reynolds' equation) and the equation of energy. Further-analysis r e s u l t e d i n two equations f o r pressure and temperature. In. the pressure range encountered, density and v i s c o s i t y •are considered-as functions of temperature only. Thus the governing -equations as presented by Currie are: 9 (/£. &P\ 4- - ^ a ( S X J 9P\ - AoJsuKd/Q (U) 2Q\C<" 39 J dA.\Ui £si) A* 3e ^ ' l[ £. ^y^t. <?6>)ae 9^% 3^ j y& = A (6) (7) 8. 2-2 Solution of"the Governing Equations As the -equations obtained were too elaborate f o r an exact a n a l y t i c a l s o l u t i o n , Currie (12) employed a numerical solution by using the rel a x a t i o n process introduced by Southwell (lU)..-and applied by.Christopherson ( l l ) . He f i r s t transformed the sector shaped pad into ,a rectangle by changing the independent'variables by the use of the r e l a t i o n s : as shown i n F i g . 1. (8) F i g . l TRANSFORMATION OF SECTOR INTO RECTANGLE He then made the dependent v a r i a b l e s dimensionless by introducing P, T, D, and M which are defined by: yU. = yU0 /!/) (9) where t 0 , y^o> and ^Mo a r e r e s p e c t i v e l y the temperature, density .and v i s c o s i t y - a t the i n l e t edge. The equations, were then .expressed i n f i n i t e ' difference form to give: / ? - / 1 = / y /0L0 '-p' ^ =- S3 ^/ - x£a7-^ (12) ^ - 3 C*-7)? us) These equations may be solved simultaneously by first.assuming a pressure distribution.' From t h i s , a temperature d i s t r i b u t i o n may be obtained using equations (11) , (12) and (13)- This f i r s t temperature d i s t r i b u t i o n establishes a second pressure d i s t r i b u t i o n using equations (lO), ( l 2 ) and ( 1 3 ) , from which a second temperature d i s t r i b u t i o n i s obtained, and so on. .Currie (12) found that'because the v i s c o s i t y decreases with temperature, .the foregoing procedure i s inherently .stable and the solutions converge r a p i d l y . An IBM 1620 d i g i t a l computer has been programmed to obtain a s o l u t i o n to equations (10) to ( 1 3 ) ,with reference to the input quantities of speed, f i l m .thickness, i n l e t temperature, sector angle and inner and outer r a d i i . 1G. 2.3 T h e o r e t i c a l Performance of Bearings The t e s t pad i n the present work- had the l i m i t s 0 ^ R ^ 0 . 7 and 0 =: 8 1.0 when transformed from a sector to a rectangle as shown i n F i g . l . 'R'. i s a v a r i a b l e depending upon the size of pad that i s under consideration, and''8' i s given a f i x e d value of 1 .0 . The g r i d system which i s set up on the rectangular pad has the g r i d dimension 'a'. Where 'a' i s a v a r i a b l e depending on the value of 'R', and has the l i m i t 0.09 ^ a - ^ 0 . 1 2 5 . a' i s also l i m i t e d i n order that i t w i l l always divide into ,'R' so ,as to give-an integer. By varying 'R' and.'a', pressure •and temperature d i s t r i b u t i o n s were obtained f o r a v a r i e t y of pad configurations. The load c a r r y i n g capacity.and the corresponding c o e f f i c i e n t of f r i c t i o n f o r each pad configuration tested was.then calculated* Load Carrying Capacity The load c a r r i e d per pad was obtained by.integrating the pressure over the surface of the pad. That i s , The computer program was adapted so that .the above i n t e g r a t i o n was c a r r i e d out d i r e c t l y , . t h u s g i v i n g the corresponding pad load for each The pressure d i s t r i b u t i o n should be l i m i t e d by two points i n order that safe operation i s ensured. These are: ( i ) The maximum temperature does not exceed the softening point of the (14) pressure d i s t r i b u t i o n . bear i n gjf m a t e r i a l used, and ( i i ) The f i l m thickness i s not -less than 0.0002 i n . The f i r s t c r i t e r i o n i s s e l f explanatory, and the second should ensure the absence of metal to metal contact. ,11. C o e f f i c i e n t of F r i c t i o n In Considering the coefficient of friction, -the following assumptions •are made: (i) .That while the variation of temperature and viscosity.across the film has been neglected in determining the load carrying capacity, it appears to be important when determining the friction forces on a fluid element. (ii) That there are ho discontinuities between the.. film temperature and the temperature of the bearing or disk surfaces. (iii) That the bearing and disk surfaces are continuous. That is, it is •assumed that there are no grooves. (iv) That the disk " is at -a,.mean temperature t^ . (v) That the pad temperature 't' is equal to the mean temperature at the bearing surface, and is expressed symbolically by: where tQ .and tm a x are the inlet and outlet temperatures respectively, at the bearing surface, as derived from the theoretical solution for the load carrying .capacity. ( v i ) That the mean radius of the bearing as defined by Young .(10 ), and Currie (12) is (vii) That the velocity, of the disk is a mean velocity. That is, it is assumed that (viii) That 7&s is a mean shear stress over the pad surface, and Is independent of 'r-' and ' 0 ' . 12. ( i x ) That the v a r i a t i o n s of v e l o c i t y and temperature across the f i l m , as presented by Hagg ( l5)> are of the form i l l u s t r a t e d by F i g . 2 . F i g . 2 VELOCITY AND TFJ4PERATUEE PROFILES Hagg (15) presented an equation f o r the c o e f f i c i e n t of f r i c t i o n i n whfiich he considered t h a t the v e l o c i t y d i s t r i b u t i o n was determined by the v i s c o s i t y v a r i a t i o n across the f i l m , which i n turn was influenced by the temperature v a r i a t i o n i n the l u b r i c a n t . The aforementioned assumptions w i l l . b e applied i n a manner s i m i l a r to that presented by Hagg. The i n i t i a l , step i s the consideration that the work done by viscous t r a c t i o n , neglecting o i l pressure gradients, i s equal to the conductive heat t r a n s f e r at a cr o s s - s e c t i o n x x (Fig.3). F i g . 3 PARALLEL PLATES IN MOTION 13-Work done by viscous = Conductive heat t r a c t i o n t r a n s f e r ((y- u. Js&z - * ( i 5 ) By " l e t t i n g c/uQ - - c/u. equation (15) becomes By assuming that the f l u i d i s Newtonian, the viscous t r a c t i o n can be represented by: The v a r i a t i o n of v i s c o s i t y with temperature used i n the f r i c t i o n c a l c u l a t i o n i s defined i n (Appendix III) as: ^ = ^ e (18) Equations (17) and (18) .may/be manipulated to giv e : o^B — — 7£<S.~*~ C/U.G ( l 9 ) ,By equating equations ( l 6 ) and ( l9)> the r e s u l t i n g equation i s of the form In. t h i s form, equation (20) may now be integrated to give the temperature d i s t r i b u t i o n . The velocity, and temperature boundary conditions are, at :1k. The constant a r i s i n g out of the i n t e g r a t i o n , may be evaluated f o r t=t-j_ .a11^ u o = The r e s u l t i n g velocity-temperature expression i s found to be: %x Now f o r t=t2-.and U Q = G the expression f o r the temperature as the surface of the disk i s = GX6, + It*. 6/* (22) S u b s t i t u t i o n of equation ( 2 l ) . i n t o equation ( 1 9 ) . y i e l d s : To make i n t e g r a t i o n easier, equation ( 2 3 ) i s transformed by the s u b s t i t u t i o n Whence i n t e g r a t i o n may be c a r r i e d out to give the viscous t r a c t i o n i n terms of v e l o c i t y , v i s c o s i t y , and temperature v a r i a b l e s . On .applying the l i m i t s , g. ^ Q ^ U o = - /? j Uo = o and employing the assumption that 6/ = yt^jui, the viscous t r a c t i o n .expression i s found to ;be: .The f r i c t i o n force on a f l u i d element due to the shearing action across the f i l m may be w r i t t e n : That i s , the t o t a l f r i c t i o n force per pad i s : 15-Now the c o e f f i c i e n t of f r i c t i o n i s defined as the r a t i o of the f r i c t i o n force across the f i l m to the t o t a l l o a d per pad. That i s , Integration and s u b s t i t u t i o n i n t o equation (26) r e s u l t s i n the formal equation f o r the determination of the c o e f f i c i e n t of f r i c t i o n . That i s , T h e o r e t i c a l Performance Curves The load capacity and c o e f f i c i e n t of f r i c t i o n were determined f o r each of the t e s t bearings by the s o l u t i o n of equations (lO) to (l4f) and equation ( 2 7 ) . In addition, pressure and temperature d i s t r i b u t i o n s were obtained f o r a given load condition. The t h e o r e t i c a l r e s u l t s are shown i n graphical form i n Fig's, k to 9- In obtaining these solutions, care was taken to ensure that the operating conditions corresponded to those that a c t u a l l y p r e v a i l e d during the experimental program. In t h i s way, the t h e o r e t i c a l r e s u l t s could be compared d i r e c t l y to experimental r e s u l t s i n a more accurate manner than had beenrojonsidered by previous workers. A second t h e o r e t i c a l s o l u t i o n was obtained f o r equation ( 2 7 ) , holding i n l e t conditions constant ( F i g . 4 ) . , The two t h e o r e t i c a l f r i c t i o n curves have been presented to demonstrate the e f f e c t i n l e t temperature has on the c o e f f i c i e n t of f r i c t i o n . A s i m i l a r s h i f t e f f e c t may be expected with respect to the load ca r r y i n g capacity f o r v a r i a b l e i n l e t conditions. In considering the t h e o r e t i c a l r e s u l t s , f o r a l i m i t i n g temperature of 235°F, the load c a r r y i n g capacity i s seen to vary from 55 l b s f o r the three-pad bearing to 105 lbs f o r the f i v e pad bearing. The corresponding f r i c t i o n c o e f f i c i e n t s vary from 0.110 to O .O53. The unusual form of the f r i c t i o n curve i n Fig. 5 may be explained by a study of equation (27) . In the thick f i l m region the decrease i n the f i l m .thickness may be more r a p i d than i s the increase i n the bearing load or the disk temperature. Hence an increase i n f r i c t i o n i s p o s s i b l e , even with :increasing :loads andi'disk temperatures. The pressure and" temperature d i s t r i b u t i o n s (Fig's. 7; 8 and 9 ) , have s l i g h t deviations from smooth contours. These deviations r e s u l t from small inaccuracies i n the r e l a x a t i o n solution to the t h e o r e t i c a l equations. The study of the t h e o r e t i c a l performance of p a r a l l e l surface bearings by the p r e v i o u s l y discussed numerical solution i s l i m i t e d by the following factors. ( 1 ) The s o l u t i o n can only be obtained e a s i l y f o r a configuration i n which the pad i s wedge shaped,"as with r a d i a l grooves, i n order that i t may r e a d i l y be transformed into .a rectangle. ( 2 ) A s o l u t i o n may be obtained f o r configurations of two, three, four and f i v e pads only, but the numerical method becomes unstable f o r a number of pads greater than f i v e . The s o l u t i o n i s unstable, because the pressure gradients become too large f o r the g r i d size' incorporated i n the solution. A-smaller g r i d size might solve t h i s problem, but i n the present work, i f a smaller grid, size i s used, the program becomes too large to be solved by the IBM 1620 computer. 3 21. 22. Chapter 3• Apparatus Measurements 23-3-. 1 Apparatus General Description A l i n e diagram of the experimental apparatus i s presented i n Fig. 1 0 . The t e s t apparatus was driven by van induction motor through a V-belt drive. The power was transmitted through a gear box to the t e s t shaft.•• The t e s t shaft was supported by two double-row, s e l f - a l i g n i n g b a l l bearings. Two p a r a l l e l surface bearings were loaded against e i t h e r side of a disk which was an i n t e g r a l part of the t e s t shaft. The load was applied through three hydraulic c y l i n d e r s . ( F i g . l 4 ) . • Component~Parts • • (1) .Induction Motor and V-Belt Drive The drive was a 15 h.p. induction motor which ran at 3*545 rpm.at f u l l load. The shaft was f i t t e d .with a'4.0/5-4 i n . v a r i - p i t c h p u l l e y . Four V-Belts were used to transmit the power to one of three interchangeable p u l l e y s of 7 - 0 , 9-0 and 12.4 i n . diameters which could be f i t t e d to the gear box input shaft. (2) Gear Box " The input shaft was connected to a R o l l s Royce Merlin supercharger gear box driven i n reverse, '.giving ,a 9• 5~~ 1 step up r a t i o , and a speed range of 11000 to 26000 rpm (Fig. 1 5 ) . (3) Thrust Disk The thrust disk.and shaft were manufactured as a s o l i d u n i t from a single piece of SAE 1020 c o l d r o l l e d s t e e l ( F i g . l 6 ) . The disk was 3 i n . • i n diameter and 3/4 i n . t h i c k . The shaft was. turned down to 5/8 in.. diameter. The unit was ground, and the disk faces were machined..and lapped f l a t and p a r a l l e l to within 0.0002 i n . Maximum run-out was 0 .001 i n . 2k. Box 25-(U) Test Bearings The t e s t hearings consisted of two s i m i l a r p a r a l l e l surface c o l l a r s . These c o l l a r s were manufactured from two hearing materials, namely D e l r i n A F f i b e r / r e s i n and Imperial Genuine Babbit.. The properties of these materials are given i n Appendix I. Two sets of these c o l l a r s were machined from a 3 i n . O.D. by 1 .'/s i n - I ' D - "by 3 / 8 i n - thick piece of D e l r i n A F f i b e r / r e s i n . Three sets of c o l l a r s were of brass with Imperial Genuine Babbit bearing faces. Grooves were machined into the bearing surface to allow easy entry of o i l , and to divide the bearing into sector pads. Bearings with a varying number of pads were used i n order to determine an optimum bearing configuration. The grooves were designed i n such a way that they ensured f u l l y flooded bearings. A complete analysis of the groove configurations can be found i n Appendix I I . ( 5) R e s t r i c t o r Rings S t e e l rings we're f i t t e d over the bearing and p i s t o n assembly, to act. as re-s t r i c t i o n s to the r a d i a l flow'•'of o i l ( F i g . l U ) . These rings were machined to give a diametral clearance o f ' 0 . 0 1 0 i n . with respect to the r o t a t i n g disk, and also to protrude l / 8 ' in.•over the disk when the disk and bearing surfaces were together. ' •"•••'-(6) Loading System The two c o l l a r s we're attached to two pistons which were h o r i z o n t a l l y opposed to each other i n s i d e a s t e e l c y l i n d e r ( F i g . l U ) . Attached to the cy l i n d e r were three hydraulic jacks operating i n p a r a l l e l , and located at 1 2 0 ° i n t e r v a l s about the c y l i n d e r . The opposing pistons had butt p l a t e s to which the jacks were pinned, thereby allowing movement of the pistons by. action of the 'jacks. 26b. 27-(7). Lubrication O i l to the t e s t bearing was g r a v i t y fed from a 4 5.gallon tank 11 feet above the t e s t assembly.'' The o i l passed through the annular space between the shaft and p i s t o n to the t e s t bearing. I t was drained to another tank situated about k f t . below'the assembly. The o i l used was Shell-Turbo 27. (The p h y s i c a l properties "are given i n Appendix I I I ) . O i l to the gear box was supplied by a gear pump,"and another pump driven by.the same motor was used to return the o i l to a r e s e r v o i r . This system ensured that the gear box was f u l l y l u b r i c a t e d at a l l times. 3". 2 Measurements The q u a n t i t i e s to be measured were speed, load, disk temperature, pad and o i l temperatures, torque, f i l m thickness and o i l supply s t a t i c pressure. (1) Speed The r o t a t i o n a l speed of the t e s t shaft was measured d i r e c t l y using.a 'Smiths' tachometer (Fig. 1 7 ) - This tachometer had speed ranges of from 0-5000 rpm, and from 0-50000 rpm with a.resolution of 0.k% of the f u l l scale. (2) .Load An .'American1 pressure gauge t e s t e r was used to apply .a given pressure to the loading jacks, from which the bearing load-,could be derived. Load increments of 5 p s i UP to a maximum load of 500 p s i were a v a i l a b l e . The c a l i b r a t i o n curve f o r the loading system i s given i n Appendix IV. (3) Disk Temperature The surface temperature of the r o t a t i n g disk was obtained by an 'Ameresco' e l e c t r i c a l surface thermometer u n i t and probe ( F i g . l 8 ) . The probe was brought into contact with the disk surface at the moment the shaft ceased r o t a t i n g . The instrument had a r e s o l u t i o n of 0.75$ of the f u l l instrument range. The c a l i b r a t i o n procedure and r e s u l t s are given i n Appendix IV. Fig. \A SECTIONAL VIEW OF APPARATUS ro CD 2 9 -(k) . O i l Temperature Temperatures were obtained by .affixing copper-constantan thermocouples to t u f n o l .inserts, which were screwed into the back of the t h r u s t . c o l l a r . Thermocouples were placed at the i n l e t , and o u t l e t end of each groove, giving the i n l e t and o u t l e t o i l temperatures. A .'Doran' potentiometer having ranges of.0.20mV by"increments of O.OlmVand 0-100mV by,increments of 0.05mV was used to determine the thermocouple p o t e n t i a l (Fig. 1 7 ) . The instrument c a l i b r a t i o n i s given i n Appendix IV. (5) Torque The t e s t bearing assembly was torque mounted on r o l l e r bearings. The f r i c t i o n torque'transmitted to the thrust c o l l a r s was measured by means of a torque arm r e s t i n g on a -.'Daytronic' model 152A-5B force transducer situated 10 i n . from the bearing centerli-ne ( F i g . l 8 ) . The transducer had an error of 1.0$>' at f u l l scale, and a l i n e a r i t y of 0.2$ of f u l l range. The instrument c a l i b r a t i o n i s given i n Appendix IV. (6) F i l m Thickness A 'Daytronic' model IO3A-8O l i n e a r displacement transducer was used to determine the operating bearing f i l m thickness (Fig. 1 8 ). The transducer had a range of + 0 . 0 4 0 i n . and was f i t t e d with.a mild s t e e l t i p of hemispherical radius 0 . 0 3 0 i n . The transducer was f i t t e d to the loading p i s t o n so that the t i p protruded through the bearing and contacted the thrust disk surface. The c a l i b r a t i o n . p r o c e d u r e i s . g i v e n i n Appendix IV. Two •'Daytronic' model 300BF d i f f e r e n t i a l transformer i n d i c a t o r s were -used to supply e x c i t a t i o n and i n d i c a t i o n f o r the force transducer and the l i n e a r displacement transducer ' (Fig.17)• Both f r i c t i o n f o r c e and f i l m thickness could be determined on p r e - c a l i b r a t e d scales. The f i l m thickness was read d i r e c t l y , and the c a l i b r a t i o n curve f o r the force transducer i s 30. given i n Appendix IV. The in d i c a t o r s had-a range of +_ 0.100 i n . and a maximum res o l u t i o n . o f 10 microinches. The' f i l m thickness was checked by two 'Starret't' l/lOOOO i n . d i a l gauges. These gauges were mounted on the s t e e l c y l i n d e r so that t h e i r plungers touched the piston'butt p l a t e s , with the r e s u l t that the d i a l readings . . were the sum of the two f i l m thicknesses. (7) Oil'Supply S t a t i c Pressure The o i l supply s t a t i c pressure was measured by means'of four pressure taps i n the:'.annulus of one of the two bearing pistons ( F i g . l U ) . These four taps were connected to a common manifold, from which an average s t a t i c pressure reading .could be taken by a Mercury.manometer. 32. L e f t , mating piston; centre, t e s t shaft and loading assembly; r i g h t , loading p i s t o n . Support hearings and loading butt p l a t e s Fig.16 TEST BEARING DISMANTLED Fig.1 8 VIEWS OF FILM THICKNESS PROBE, DISK TEMPERATURE PROBE AND FORCE TRANSDUCER 35-F i g . 1 9 FAILURE OF DELRIN A F BEARING Chapter h. h.1. Experimental Proc edure U .2 Experimental Results k.3 Discussion of Results 36. k.l' : Experimental Procedure  General Tests were c a r r i e d out on three,.four and five-grooved, white-metal bearings, and on three-grooved D e l r i n A F bearings. The objectives of the •experimental program were: (1) To c o r r e l a t e experimental c o e f f i c i e n t s of f r i c t i o n with the t h e o r e t i c a l developments presented e a r l i e r , (2) .To study/three, d i f f e r e n t grooving ..arrangements, and two d i f f e r e n t materials, . ( 3 ) To study the e f f e c t of o i l r e s t r i c t o r rings on load c a r r y i n g capacity. Procedure At the s t a r t of each test,,pressure was applied to the loading jacks i n order to, b r i n g the 'bearings and disk i n t o contact. With the surfaces i n contact, the f i l m thickness i n d i c a t o r could be zeroed. The zero p o s i t i o n of the f r i c t i o n d i f f e r e n t i a l transformer i n d i c a t o r was also recorded at t h i s time. The load was then removed and the bearings separated from the disks. O i l was admitted to ;the' test" bearings,.and the o i l temperature was recorded. . Following the foregoing.procedure, the l u b r i c a t i o n system f o r the gear box was brought into operation, and the main drive was started. An i n i t i a l pressure of 5 p s i was"applied to the loading jacks, and the assembly was allowed to reach a steady state condition. For^each load applied,.readings of thermocouple emf,. f i l m thickness, , f r i c t i o n torque, speed, o'il flow, s t a t i c pressure, and bulk .inlet and ou t l e t temperatures were recorded. To record the disk temperature, the drive had to be stopped. Considerable cooling occurred during the time required to remove the load, stop the drive and shut o f f the o i l supply. Thus, the temperature that was recorded by the thermistor was -.lower than 37-the actual temperature. Therefore, a p a r t i c u l a r procedure was developed to determine the a c t u a l disk temperature. Readings were taken of temperature versus time i n order that a cooling ,curve could be obtained. Timing commenced with, the i n i t i a l removal of the load. The elapsed time before the probe could be inserted and temperatures recorded was about 17 seconds. A-sim i l a r procedure was used f o r a l l the babbitt bearings, but procedure d i f f e r e d s l i g h t l y , f o r the D e l r i n A.F.bearings. Considerable d i f f e r e n t i a l thermal expansion was encountered i n the t e s t i n g of the D e l r i n A F bearings. Hence, 'the recorded f i l m .thickness d i f f e r e d g r e a t l y from the actu a l f i l m .thickness. The difference i n values was equal to the thermal expansion of the material. I t was therefore necessary to record a new zero f i l m thickness reading .after each load a p p l i c a t i o n i n order to determine t h i s expansion.. Each .test was continued u n t i l the o u t l e t temperature reached a maximum of 235°F. The maximum allowable temperature rather than the minimum f i l m . thickness l i m i t e d the load capacity of the bearings. This temperature l i m i t -a tion was evident i n the case of the D e l r i n A F bearings,' where the maximum continuous operating temperature, as noted i n Appendix I, was l i m i t e d to about l 8 5°F. The white-metal bearings; with a softening point of. hrJO°F, could have been run at high temperatures. However i t was f e l t that no p r a c t i c a l purpose could be achieved by doing t h i s , as.an adequate amount of information had already been obtained, and further'loading could have damaged the drive and te s t assembly. As soon, as the l a s t data were recorded, a load would again be applied to the loading jacks i n order that a 'hot' zero reading could be- taken of: the f i l m thickness..- When the assembly had cooled to . room .temperature, the zero reading would again be taken. In t h i s way,.any permanent set caused by thermal expansion would have been recorded, but none was indicated. 3 8 . When a t e s t was completed, the bearings were inspected and the equipment re-assembled. Another t e s t was then conducted, and the r e s u l t s compared with the previous test. In a l l cases there was s a t i s f a c t o r y agreement between the various sets of data. k.2 Results The experimental t e s t program was c a r r i e d out on f i v e sets of thrust bearings. The f i r s t set consisted of a three-pad D e l r i n A F bearing, and was used during i n i t i a l checking of t e s t components. This bearing f a i l e d under a heavy l o a d - a c c i d e n t a l l y applied, and the r e s u l t may be seen i n Fig. 1 9 -F a i l u r e was i n d i c a t e d by ibhe appearance of dense blue smoke. No readings of applied load or maximum o i l temperature were obtained because of the f a i l u r e . However, i t i s i n t e r e s t i n g to note that there was no p e r c e p t i b l e reduction i n speed when the f a i l u r e occurred. A f t e r dis-assembly, the t e s t shaft and other components were found to be undamaged. I t was apparent that the o i l temperature had exceeded the melting point of the material, r e s u l t i n g i n flow of the D e l r i n A F. The lack of dgmage to the equipment was probably a r e s u l t of a hydrodynamic f i l m created by the molten material. The r e s u l t s f o r a three-pad D e l r i n A F, and f o r three, four and five-pad babbitt bearings are presented i n graphical form, Fig's. 20 to 2 8 , and i n tabular form i n Appendix V. In a l l cases, the t h e o r e t i c a l curves which were presented i n Fig's, h to 6 have been reproduced so that d i r e c t comparisons could be made with the experimental r e s u l t s . In F i g ' s . 20 and 2 1 , the t h e o r e t i c a l curves.at constant i n l e t conditions and at actual i n l e t conditions have been reproduced so that the e f f e c t of introducing actual i n l e t conditions could be noted. Also, i n F i g . 2 1 , Currie's t h e o r e t i c a l curve f o r the c o e f f i c i e n t of f r i c t i o n has been i l l u s t r a t e d f o r comparison with the f r i c t i o n r e s u l t s c a l c u l a t e d by the v a r i a b l e v i s c o s i t y theory. 39-A comparison of bearing pressures with and without o i l r e s t r i c t o r rings i s shown i n Fig.2 9 - I t may be seen that the addition of r e s t r i c t o r rings increased the bearing pressure twofold. The experimental load c a r r y i n g c a p a c i t i e s f o r the various bearings tested are compared i n F i g . 3 0 , and the corresponding c o e f f i c i e n t s of f r i c t i o n , i n Fig.3 1 . Results obtained i n previous iavestigations.flof the-, thermal wedge e f f e c t may be compared d i r e c t l y to the present r e s u l t s by considering a p l o t of the c o e f f i c i e n t of f r i c t i o n , ' f , versus the parameter ZN/P where the v i s c o s i t y , Z> has-units i n centipoise, speed, N, i n rpm, and pressure, P, i n lbs.-per square inch. The corrected experimental.quantities of load and c o e f f i c i e n t of f r i c t i o n were obtained as follows. Load The bearing load was obtained by r e f e r r i n g to the load c a l i b r a t i o n curve i n Fig. § 8 . -From .the values i n the Figure, the s t a t i c load of 8 . 0 l b s . has been,"subtracted. The 8 . 0 l b . s t a t i c load capacity i s evaluated i n Appendix VI. C o e f f i c i e n t of F r i c t i o n The experimental c o e f f i c i e n t of f r i c t i o n was c a l c u l a t e d from the measured f r i c t i o n torque, the corresponding bearing load, and the mean radius r m . Both Young (10) and Currie (12) defined r m . t o be: ^~ - ( 2 8 ) In the present studies, r ^ = 1 - 5 .in., and r Q = O.6875 i n . , y i e l d i n g a value of r m = l . l ^ k i n . .Now the c o e f f i c i e n t of f r i c t i o n as previously defined i s the • r a t i o of the f r i c t i o n force to the applied bearing load. Therefore, the c o e f f i c i e n t of friction-sanay beiobtained by the expression • / = _ 2 T (29) 1*1 { k3-h5. 7 I / o / i 0 © / o / © / / 1 •o o o o »0 O i i to I \ o \ 1 A 1 / >• / / o CO o \ \ VJ 1 St M I k9. -1 n 1 ft <> m 1 1 / I' 1 - il I 1 / / 1 / / 1 / B i\ 1 a 1 1 1 < 1 I / "0 !0 0 *5 o 0 X k o I I? 1 o 1 u 5 0 . 51-5 2 . \ 53-4-3 Discussion of Results Accuracy of Measurements The r o t a t i o n a l speed of the t e s t shaft was measured d i r e c t l y by a tachometer, which had-an accuracy of +_0.4$. The accuracy of the t o t a l bearing load-was l i m i t e d by. two f a c t o r s : (1) The c a l i b r a t i o n of the load system (Appendix TV) i s l i m i t e d to.an accuracy of +_ (2) The s t a t i c pressure c o r r e c t i o n using the hydrostatic bearing theory (Appendix V i ) i s made from a s i m p l i f i e d approach which Tgriores the presence ' of grooves i n the bearing surface.. No.estimate can be given of the accuracy of t h i s method. The accuracy.of the f i l m .thickness transducer was d i r e c t l y dependent on tjr.e s t a b i l i t y of the"! bearings under load. In the present studies, the bearings exhibited stable operation, and under no circumstances did the f i l m thickness readings display the random v a r i a t i o n s as pentioned by Young (lO) and Currie ( 12) . The accuracy therefore, was l i m i t e d only.by.the s e n s i t i v i t y of the d i f f e r e n t i a l transformer i n d i c a t o r which was within _+ 3-0$-The s t a b i l i t y of operation was again apparent with the measurement of f r i c t i o n foarce, which was continuously steady-. The -accuracy of measurement was "within +_ 1 .0$. The accuracy of the measured o i l temperature was l i m i t e d by the p r e c i s i o n of the potentiometer, and by the necessity of r e c y c l i n g hot l u b r i c a n t . In the-case of the potentiometer, the accuracy was within +_ 5$ of the recorded value. While i n the case of the recycled l u b r i c a n t , a condition of steady state was considered to e x i s t i f the bulk o u t l e t temperature d i d not change more than 2°F during the time taken to record the-experimental data. 5^. •The temperature change i n the l u b r i c a n t r e s u l t e d from external mixing of previously cycled o i l , and c o l d o i l , whereby a steady i n l e t temperature was not always obtainable. The r e s u l t i n g accuracy was within. +_ 2$ of the values presented i n Appendix V. The experimental' disk temperature was determined by recording disk . temperatures versus time,.in order that a cooling curve might be p l o t t e d . As previously discussed, the disk temperature•was g r e a t l y a f f e c t e d by the load, speed,.and o i l flow, and an estimate of the "accuracy.of measurement cannot be given. The disk temperature should, therefore, be considered as an i n d i c a t i o n that-a temperature gradient does i n f a c t e x i s t between the bearing and . disk . surf aces, and that the gradient may be as much, as rJO°F. Comparison of T h e o r e t i c a l and Experimental Results' Three-pad D e l r i n A F;and Three-pad Babbitt Bearings The c o r r e l a t i o n between theory, and experiment f o r the three-pad babbitt bearing and the three-pad D e l r i n A F bearing may be seen by.a study of Fig.2 0 . The s h i f t that i s displayed by the experimental 1.load curve f o r the D e l r i n A F may be explained by -a consideration of i t s c o e f f i c i e n t of thermal expansion. The c o e f f i c i e n t f o r the D e l r i n A F.as given i n Appendix I, i s 6 . 0 k l O ' ^ in/in/°F, which'is about;. 10 times that f o r most ordinary metals. In. an actual bearing t e s t , the predicted amount of expansion of the D e l r i n . A F was '2.7 x -3 _V 10 i n . , and the actu a l amount of expansion was 2-5 x 10 J i n . The operation of the bearing d i c t a t e s that the maximum temperatures w i l l e x i s t at the periphery of the bearing, as substantiated both t h e o r e t i c a l l y / i n Fig.8 and experimentally, i n Appendix V. The maximum thermal expansion w i l l be at t h i s point. • Since the amount of expansion at any point will, depend on the "temperature gradient from the i n l e t to that point, the surface of the bearing w i l l no longer be f l a t and p a r a l l e l , but w i l l take a shape probably s i m i l a r to ;a f i x e d tapered land bearing. The tapered e f f e c t w i l l increase the load capacity f o r a given f i l m .thickness, 55-r e s u l t i n g i n the s h i f t of the D e l r i n A F l o a d - f i l m thickness curve upwards. An experimental temperature p r o f i l e was not obtained and no attempt tias been made to determine the surface p r o f i l e or to separate the load capacity a t t r i b u t e d to the ''Reynold's' e f f e c t from that of the 'thermal wedge' e f f e c t . As the present studies are di r e c t e d towards the thermal wedge e f f e c t , D e l r i n A F cannot be considered as a good experimental material-. However, as a bearing m a t e r i a l , . i t has obvious mefdt because of i t s a b i l i t y to carry substantial loads i n the th i c k f i l m region. .The load curve f o r the babbitt bearing conforms more c l o s e l y to"the pre d i c t e d curves and a d i r e c t comparison can be made between theory and experiment. Two t h e o r e t i c a l curves have been presented to demonstrate the -effect i n l e t temperature has on load capacity. The curve which i s derived from a consideration of constant i n l e t conditions, indicates higher load c a p a c i t i e s than does the curve where a c t u a l ' i n l e t temperatures are used. This e f f e c t i s consistent with the thermal wedge theorywhich p r e d i c t s that an increase i n i n l e t temperature r e s u l t s i n a reduction i n v i s c o s i t y , and a l o s s i n load capacity. The actual values of i n l e t o i l temperature and f i l m thickness have been used to determine the t h e o r e t i c a l load curve i n order that:a more f a c t u a l comparison can be made between the t h e o r e t i c a l and the actual system. I t should be noted that i f a co o l i n g system had been used to maintain the o i l supply at a constant temperature, the c o r r e l a t i o n between theory, and experiment would probably.have been better. In the f i l m thickness range 2 .8 x 10~3 i n . to.2 . 3 5 x 10~3 i n . , the experimental load i s l e s s than the t h e o r e t i c a l . The above i s caused by the adverse e f f e c t of the c e n t r i f u g a l force on the pressure f i l m . In the t h e o r e t i c a l approach, t h i s term was considered to be n e g l i g i b l e although i t s order of magnitude was about one h a l f that of the p r i n c i p l e terms. The 5 6 . assumption was necessary, i n order that a s o l u t i o n to the t h e o r e t i c a l equations was p o s s i b l e . The addition of the c e n t r i f u g a l force term would have l e f t these equations i n a form too complex f o r solution. For the f i l m thickness range 2 .3 x 10~3 j _ n . to 0-7 x 10~3 i n . , the experimental values exceed the t h e o r e t i c a l by a considerable margin, but are seen to be converging i n the t h i n f i l m .region. This phenomenon i s a r e s u l t of the i n s e r t i o n ,of o i l r e s t r i c t o r rings i n t o the bearing assembly. These rings r e s t r i c t the r a d i a l flow of o i l , and i n doing so, ensure that the bearing i s f u l l y flooded. Since the bearing i s f u l l y flooded, an optimum 'thermal wedge1 e f f e c t i s achieved. The r e s u l t i s an increase i n bearing load capacity. Fogg (3) reported that he had used c i r c u m f e r e n t i a l . o i l r ings to ensure that h i s bearings were f u l l y flooded, and he showed that they increased the load capacity considerably. The complexity of the e f f e c t of the r e s t r i c t o r rings on bearing operation has p r o h i b i t e d a t h e o r e t i c a l a nalysis of t h e i r action. • The t h e o r e t i c a l f r i c t i o n curves of Fig.2 1 include one derived from the work of Currie (12) i n order that the theory used by previous experimentors may be compared d i r e c t l y to the v a r i a b l e v i s c o s i t y theory which i s given imuthe present work. Both t h e o r e t i c a l curves are based on constant i n l e t temperatures'of rJO°F. The three-pad babbitt bearing has experimental values of c o e f f i c i e n t of f r i c t i o n i n the same range as obtained by Currie. Generally, the v a r i a b l e v i s c o s i t y approach to f r i c t i o n theory appears to be appropriate. The theory i s based on the assumption of a v a r i a t i o n of temperature and v i s c o s i t y across the f i l m from bearing to disk. As previously "mentioned, the temperature gradient has been v e r i f i e d by experiment. With respect to the curve f o r the D e l r i n A F bearing, the experimental r e s u l t s 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 . This behaviour may be explained i n part by considering that the a c t u a l mean •57-temperature of the t e s t bearing i s higher than the mean of the i n l e t and outl e t temperatures, which was used to obtain the t h e o r e t i c a l f r i c t i o n curve. •The higher temperature can be a t t r i b u t e d to,a r e - c i r c u l a t i o n of a layer of o i l adjacent to the r o t a t i n g disk. As the temperature of the layer i s higher than that of the incoming o i l , the mixing of the two w i l l r e s u l t i n a higher mean temperature. The higher temperatures give r i s e to a decrease i n v i s c o s i t y , and since <»<. , there i s a 'corresponding decrease i n f r i c t i o n . A more important consideration.with respect to the D e l r i n A F- i s that of the r e l a t i o n -f oC. //vV • The most relevant feature of the D e l r i n A F i s i t s a b i l i t y to carry s u b s t a n t i a l 1 loads at large f i l m thicknesses because of i t s d i f f e r e n t i a l thermal expansion. The l a r g e r load gives a lower c o e f f i c i e n t of f r i c t i o n .in the th i c k .film .region. Hence the experimental r e s u l t s a r e . s h i f t e d r e l a t i v e to the t h e o r e t i c a l curve. Again, with the babbitt bearing, the actu a l mean temperature i s higher than the simple mean temperature, e s p e c i a l l y i n the f i l m thickness range 2.0 x 10~3 i n . to 0-7.x 10~3 in.-, r e s u l t i n g In a lower c o e f f i c i e n t of f r i c t i o n . In t h i s region the experimental load c a p a c i t y . i s almost twice the t h e o r e t i c a l . The difference i n load capacity suggests that the experimental c o e f f i c i e n t of f r i c t i o n should be-about h a l f that of the t h e o r e t i c a l value. A comparison of r e s u l t s i n Fig.21 substantiates t h i s p r e d i c t i o n . The experimental c o e f f i c i e n t s of f r i c t i o n are greater In the f i l m region 2 .3 x 10~3in. to 2.0 x I0~ 3 i n . than are predicted. This i s converse to what i s expected, as the. experimental load c a p a c i t y . i s greater than the t h e o r e t i c a l . The co n t r a d i c t i o n can be a t t r i b u t e d to a lower disk temperature than Is pr e d i c t e d i n Fig.22. The above condition would give higher c o e f f i c i e n t s of f r i c t i o n than i s expected. In. the f i l m thickness range 3-0 x 10~3 i n . to 2 .3,xMO"3 i n . t h i s e f f e c t i s also prevalent, but even more so i s the adverse 58. e f f e c t of the c e n t r i f u g a l force which reduces the actu a l load capacity to below "that of the t h e o r e t i c a l , thus g i v i n g higher c o e f f i c i e n t s of f r i c t i o n . In Fig.2 2 , the theoretical•and experimental disk temperature, curves show the same tendency to increase with decreasing f i l m thickness. Although no attempt was made to determine the accuracy of the recorded disk temperatures, t h e i r magnitude can be accepted i f the assumption i s made that the cooling curve would-be a simple exponential r e l a t i o n . On t h i s b a s i s the cooling curve can be displayed as a st r a i g h t l i n e on a log - l o g p l o t to obtain the zero-time temperature approximation. The difference between actual and t h e o r e t i c a l values can,be explained i n two ways. (1) The complexity of the co o l i n g systems probably r e s u l t s i n errors not foreseen by the method of approximation. (2) . In t h e o r y • i t .is assumed that a l l the heat generated i s c a r r i e d away by the o i l . However i n p r a c t i c e a c e r t a i n amount of cooling w i l l take place through the disk and .shaft, r e s u l t i n g i n lower disk temperatures. Four and Five-Pad Babbitt Bearings The discussion j u s t presented f o r the three-pad babbitt bearing i s representative also of the four and five-pad bearings. The comparison between theory and experiment f o r load capacity, c o e f f i c i e n t of f r i c t i o n .and disk temperature i n Fig's. 20 through 28 i s the same f o r a l l three bearings,.and di f f e r e n c e s -are only i n magnitude. 59-Comparison of D e l r i n A F and Babbitt Bearings The operating c h a r a c t e r i s t i c s of both the D e l r i n A F and babbitt bearings have been given i n Fig's. 20 and 21 . The D e l r i n A F bearings proved to be f a r s u p e r i o r t o the babbitt bearings, both i n the load capacity, and i n the corresponding c o e f f i c i e n t of f r i c t i o n f o r a given f i l m thickness. This phenomenon i s due to the d i f f e r e n t i a l thermal expansion of the D e l r i n A F, which r e s u l t s i n an i n i t i a l l y f l a t and; p a r a l l e l surface becoming s l i g h t l y tapered, g i v i n g a 'geometric wedge' e f f e c t . T h i s , of course, increases the load capacity considerably f o r a p a r t i c u l a r f i l m thickness. The D e l r i n A F does not have as high a l i m i t i n g operating temperature as does the babbitt. However, f o r a babb i t t bearing to achieve the same load capacity, i t must operate at a much smaller f i l m thickness, with the p o s s i b i l i t y of surface to surface contact e x i s t i n g . For example, from Fig.2 0 i t may be seen that f o r a D e l r i n A F bearing operating at 150 l b . load, with a f i l m thickness of l.k' x 10~3 i n . , a corresponding babbitt bearing must operate at l e s s than than 0 . 7 x 10~3 i n . f o r the same load capacity. T he f r i c t i o n losses i n the. babbitt bearing would be approximately three times those f o r the D e l r i n . -bearing. F i n a l l y , the boundary c o e f f i c i e n t of f r i c t i o n i s much smaller •for'-' ' the D e l r i n A F than f o r the babbitt, g i v i n g an inherent f a c t o r of safety i f the bearing and disk surfaces should ever contact. R e s t r i c t o r Rings Fig.2 9 gives a comparison between two bearings, one operating with o i l r e s t r i c t o r r i n g s , and the other without. The curve f o r no rings i s taken from Currie ( 1 2 ) , and because the t e s t conditions were s i m i l a r except f o r a s l i g h t . d i f f e r e n c e i n pad area, they may be compared d i r e c t l y f o r mean bearing pressures. It has been previously mentioned that the addi t i o n of r e s t r i c t o r rings ensures that the bearing i s flooded with o i l . Therefore, 6 0 . the rings constribute to the load capacity by optimizing the thermal wedge e f f e c t which gives r i s e to greater f i l m pressures. In the f i l m thickness range 3 - 0 x 1 0 " ^ i n . to 1-75 x 1 0 " ^ in.,.more pressure i s produced without rings. This behaviour may be explained by the f a c t that Currie d i d not subtract the c o n t r i b u t i o n to bearing capacity produced by the s t a t i c pressure of the o i l supply. He i n d i c a t e d that the s t a t i c pressure was about 5 p s i . The hydrostatic bearing theory p r e d i c t s that f o r t h i s pressure the contribution to load would be about l 6 l b s . This i s equivalent to a pressure of about k p s i i n the bearing. I f t h i s c o r r e c t i o n were considered, the r e s u l t s would f a l l below that f o r r e s t r i c t o r _3 rings i n a l l cases.. Below f i l m thicknesses of 1-75 x 10 ^ i n . , the optimising e f f e c t of the r e s t r i c t o r rings increases the load capacity very r a p i d l y . i n comparison to that f o r a bearing i n which no r e s t r i c t i o n s to the r a d i a l flow of o i l are present. Another advantage of r e s t r i c t o r rings i s that they s t a b i l i z e the bearing operation. In many cases, other -experimentors such as Fogg (3)> Young ( l 0 ) , and Currie (1 2 ) reported unstable bearing operation due to intermittent o i l starvation, and the existence of a i r between the disk and bearing surfaces. The r e s t r i c t o r rings not only ensured f u l l y flooded bearings, but also reduced to a n e g l i g i b l e amount the aeration of the o i l . Comparison of Bearing Configurations The experimental r e s u l t s are given i n F i g ' s . 30 and 31- From F i g . 3 0 , i t may be seen that the four-pad bearing has the greatest load c a r r y i n g capacity of the bearings tested f o r a given f i l m thickness. This i n d i c a t e s that there i s an optimum length of pad over which the o i l can expand. I f the pad i s too long, a considerable l o s s of o i l by side leakage could e x p l a i n the lower f i l m thickness f o r a given load. I f the pad i s too short i n the c i r c u m f e r e n t i a l d i r e c t i o n , the f i l m thickness w i l l have to be smaller 61. to produce the required temperature r i s e f o r a given load. The f r i c t i o n curves of Fig . 3 1 a l s o indicate that the four-pad hearing i s the optimum design. Results f o r the maximum o i l temperatures i n Appendix V show that as the number of pads increases, the maximum o i l temperature increases. The above f i n d i n g suggests that the five-pad bearing should have the lowest c o e f f i c i e n t of f r i c t i o n . In the f i l m thickness range 2.k x 10"3 i n . to 1.75 x 10"3 i n . , the c o e f f i c i e n t of f r i c t i o n f o r the f i v e -pad bearing i s lower than that of the four-pad. But i n t h i s region,.the load capacity i s equal to, or greater than that f o r the four-pad bearing, -3 r e s u l t i n g I n a lower c o e f f i c i e n t of f r i c t i o n . From 1.75 x 10 J i n . to 0.9 x 10~3 i n . , the load capacity of the four-pad bearing i s greater, r e s u l t i n g i n i t having lower f r i c t i o n values. "After 0.9~x 10"3 i n . , the •load"capacity of the five-pad bearing converges with that of the four-pad bearing. In t h i s region,.the c o e f f i c i e n t of f r i c t i o n f o r five-pads converges and becomes 'lower than that of the four-pad bearing. At any given f i l m thickness i n the t h i n f i l m region, the load capacity of the four-pad bearing i s greater, while i t s c o e f f i c i e n t of f r i c t i o n i s lower or equal to that of the five-pad bearing. On: the foregoing basis , the four-pad bearing i s the optimum configuration. •The r e s u l t s of the'present research.are consistent with work c a r r i e d out by Kettleborough ( 8 ) . He tested bearings having two, three, four, ...five and s i x pads, and concluded that the four-pad configuration gave optimum operating conditions. General C o r r e l a t i o n of Experimental Results In order to compare the present r e s u l t s with thqse of past experiments, the c o e f f i c i e n t of f r i c t i o n has .been p l o t t e d against the parameter, ZW/P, Fig.3 2 . . The r e s u l t s f o r the babbitt bearings compare c l o s e l y with those obtained by Currie (12) and Kettleborough ( 8 ) . The 6 2 . r e s u l t s f o r the D e l r i n A F 1 bearing, while determined f o r much lower loads, give f r i c t i o n values considerably lower than Fogg's. The r e s u l t s are probably.influenced by the tapered wedge e f f e c t produced by thermal expansion. The wide d i s t r i b u t i o n of r e s u l t s as obtained by.the various authors i s probably.the r e s u l t of d i f f e r e n t operating conditions, and d i f f e r e n t methods of determining the v i s c o s i t y , Z. While Fogg's t e s t s were conducted under conditions of heavy loads and high speeds, and Kettleborough's under conditions of heavy loads and low speeds, the present tes t s were conducted under conditions of low loads and high speeds.. Each series of t e s t s , . t h e r e f o r e , . w i l l give r i s e to a d i f f e r e n t range of values of ZN/P. Also, sca t t e r w i l l r e s u l t from the methods used to determine the mean temperature,.from which Z i s evaluated. In many cases,.the mean temperature has been taken a r b i t r a r i l y , to be the sum of the i n l e t temperature plus' two t h i r d s the temperature gradient from .inlet to outlet. In the present studies,.the mean temperature i s taken as the sum of the i n l e t temperature plus one h a l f the temperature gradient. This procedure was indicated by a consideration of the t h e o r e t i c a l temperature d i s t r i b u t i o n ( F i g . 7 ) . -The average temperature over the surface was very nearly equal to that of.the. c a l c u l a t e d mean temperature. The difference i n the two methods of determining.Z may be seen i n the following example. For an i n l e t temperature of rJO°F, . and a; gradient :of TOO , the mean temperatures w i l l ' b e I 3 6 . 7°F.and 120°F r e s p e c t i v e l y . The corresponding v i s c o s i t i e s w i l l be 1 1 . 8 centipoise and T 6 . 9 centipoise r e s p e c t i v e l y . •The deviation i n values i s approximately -4-3$, and w i l l be responsible f o r a large proportion of the s c a t t e r seen i n Fi g . 3 2 . Also, an attempt has been made to correlate.; experimental r e s u l t s :by the method of dimensional.analysis. The r e s u l t s of t h i s work may be seen i n Appendix VII. Chapter 5-5-1 Conclusions 63-5-1 Conclusions The existence of a temperature gradient between the bearing and disk surfaces has been shown. This f i n d i n g v a l i d a t e s the assumption of a v a r i a t i o n i n temperature across the f i l m , . and j u s t i f i e s the variable v i s c o s i t y approach i n determining the t h e o r e t i c a l c o e f f i c i e n t of f r i c t i o n . The revised theory was found to agree more c l o s e l y with experimental r e s u l t s than previous theories. I t was concluded that i f the i n l e t temperature to the p a r a l l e l surface bearing was held constant, there would be excellent c o r r e l a t i o n between theoretical•and experimental f r i c t i o n r e s u l t s . The t h e o r e t i c a l ' r e s u l t s f o r load capacity-predict that the p a r a l l e l surface bearing should be able to carry Toads i n excesa'^of 150 l b s . , with a f i l m thickness of 0,0005 .in. The. experimental load capacity was found to be greatly influenced by the addition of c i r c u m f e r e n t i a l o i l r e s t r i c t o r rings, and 150 l b s . load was a c t u a l l y exceeded at a f i l m thickness of 0 . 0 0 0 7 .in. A comparison of r e s u l t s with and without restrife t o r rings gave r i s e to the conclusion that r e s t r i c t o r rings would increase load capacity by a considerable amount. This conclusion agrees with the work of Fogg ( 3 ) . Tests on bearings with three, four -and f i v e pads indicated that the four-pad bearing gave optimum operating conditions. This r e s u l t i s i n general agreement with Kettleborough ( 8 ) . Tests c a r r i e d out on a bearing made of D e l r i n A F f i b e r / r e s i n i n d i c a t e that D e l r i n i s an e x c e l l e n t bearing material. I t s load capacity i s much higher than that f o r a babbitt bearing at the same f i l m thickness, and the corresponding f r i c t i o n losses are; much lower. These r e s u l t s were the consequence of d i f f e r e n t i a l thermal expansion which r e s u l t e d i n the D e l r i n A F bearing taking on a tapered shape. I t was therefore concluded that while the D e l r i n A F had an excellent load capacity, i t d i d not remain 6k. f l a t and p a r a l l e l , and was unacceptable -as a pure thermal wedge bearing. The experimental f r i c t i o n r e s u l t s showed close agreement with those obtained by other-experimenters, even though much of the previous work was done i n d i f f e r e n t load and speed ranges. . Appendices 65-.APPENDIX I P h y s i c a l Properties of Bearing Materials D e l r i n A F D e l r i n A F f i b e r / r e s i n (A f o r a c e t a l , F f o r fluoro-carbon) i s a thermoplastic material c o n s i s t i n g of oriented Teflon TFE-fluorocarbon f i b e r s uniformly dispersed i n D e l r i n a c e t a l r e s i n . The dynamic c o e f f i c i e n t of f r i c t i o n f o r s t e e l s l i d i n g on unlubricated D e l r i n A F may be as low as 0.05- The actual value depends on the various conditions of load, speed and bearing temperature. ', The material has the following thermal p r o p e r t i e s : . : 1 . C o e f f i c i e n t of l i n e a r thermal expansion. • Temperature range , C o e f f i c i e n t of l i n e a r °F expansion in/in/°F < 85 h.6 x l O " 5 85 - iko 5.5 x i o - 5 lUO - -220 6 . 0 x 10-5 2. Heat Capacity, Temperature range S p e c i f i c heat °F Btu/lb°F < 32 0.2U 32 - 2U5 0 .28 2U5 - 310 0.U5 3. Thermal Conductivity 1.6 Btu/hr/ft 2/°F/in •The maximum contiguous operating temperature should be l i m i t e d to 185 °F, the softening point f o r D e l r i n A F, although on one occasion during the experimental testing,,the D e l r i n A F bearing was operated s u c c e s s f u l l y at over 250°F. '• • 66. Imperial Genuine Babbitt Imperial Genuine Babbitt i s a t i n based bearing a l l o y having an average composition of 89$ t i n , 7-5$ antimony and 3-5$ copper. The material has the following thermal p r o p e r t i e s : 1. •Melting.Point. The softening .point of the babbitt i s 470°F and the material i s t o t a l l y molten at 700°F. 2. C o e f f i c i e n t of thermal expansion. 8 - 1 0 x 1 0 - 6 in/in/°F 3- Thermal Conductivity. kOO Btu/hr/ft 2/°F/in 67-APPENDIX II Groove Configurations In previous studies, the method of choosing groove configurations and dimensions was an arbitrary.one. In.the present work, a procedure was formulated f o r the design of the grooves which took .into consideration l u b r i c a t i o n requirements. The c r i t e r i a f o r l u b r i c a t i o n requirements are: ( i ) That the quantity of l u b r i c a n t must be s u f f i c i e n t to maintain thermal equilibrium conditions below a set maximum, i n t h i s case at l e s s than 280°F, and, ( i i ) That the enti r e bearing surface must be f u l l y flooded with l u b r i c a n t . Thermal E q u i l i b r i u m The operating temperature of the bearing i s important, because d i f f e r e n t i a l expansion of the bearing may/cause c r i t i c a l changes-in the bearing clearance, and/or i t may"induce high stresses between the bearing and the metal backing pl a t e . As a f i r s t approximation, the assumption i s made t h a t . a l l of the heat generated i s transmitted to the lu b r i c a n t , and none to the bearing or.disk surfaces. That i s Heat Generated = Heat Dissipated From t h i s equation, the quantity, of l u b r i c a n t •.1Q' may be determined. (31) 68. In order to estimate -'Q.1, a t y p i c a l set of values f o r , C^o > (J o > etc. maybe used. Consider, f o r example, .the r e s u l t s of a computer • sol u t i o n f o r a 3 i n - bearing as l i s t e d i n Table I, where a l l . q u a n t i t i e s have the units of l b s . , i n . , s e c , Btu, °F. TABLE I C o (A, 7S w W6 u y A~ 0.03 .0.497 1800 • 70 280 76.71 .8125 93^0 .0283 8 . 0 0.75 From Table I, equation (31) can be evaluated, to give Q = O.I3I+ in3/sec/Pad F u l l y Flooded Bearing To determine the quantity, of l u b r i c a n t required f o r a f u l l y flooded bearing,, the following expression i s used. Q = V.A (32) where 'Y' i s a mean v e l o c i t y equal to -g-u0, and..'A' i s equal to the nominal groove length 'L^' times the f i l m thickness 'h'. From Table I, equation (32) may be evaluated to give Q = O.3375 in 3/sec/Pad On comparing the two c r i t e r i a f o r lu b r i c a n t requirements, i t has been shown that the grooves must be Targe enough to permit-a s u f f i c i e n t amount of lub r i c a n t to ensure.a f u l l y flooded bearing. Groove Dimensions The rate of o i l flow through a groove may be determined by considering the a x i a l forces on a p a r t i c l e of o i l i n the groove (Fig. 3 3 ) ' 69-Fig.3 3 FORCES ON..AN ELEMENTARY PARTICLE OF OIL For s t a t i c equilibrium, S2 ' or 9P - <£7Z- (3h) . By 'Newtons law of viscous flow •7* = Therefore equation (3*0 m a y be written 9f _ <9* ay or* (35) (36) In t h i s form, equation (36).may be integrated twice with respect•to ;'y' to give the a x i a l v e l o c i t y u). On. applying the boundary.conditions-at y = 0 , OJ = 0 y 1 - b+h, cu= 0 the v e l o c i t y r e l a t i o n i s found to be OJ = _Z <£P Y(Y-6-/>) (O7) •The quantity, of f l u i d flowing across an elemental•area i s (38) 70. By substituting-equation (37) into equation (38) and in t e g r a t i n g between the l i m i t s x. - o zfo 3c = a. the r e s u l t i n g equation.is of the form Q = -JL C t 6 * * ) 3 - (39) Now. equation (37) may be transformed to give ^ = (kO) In t h i s form equation (Uo) may be integrated to give the pressure r e l a t i o n s h i p . On applying the boundary condition the presfeure r e l a t i o n s h i p takes the form ^ - / S - 2 ju. ou -a ( L i ) Y(Y-6-/>) Since P = 0 at 2- = _ S> = zz*±L¥r> = (U2) A f t e r s u b s t i t u t i o n f o r 3P , equation (s59) becomes Q = CL#> C ^ ^ f (U3) Equation (U.3) i s now i n a suitable form f o r the s e l e c t i o n of groove dimensions. I t should be noted that i t i s much more e f f e c t i v e to a l t e r 'b* because of the cube power, rather than a l t e r i n g -.•* a' when choosing dimensions to give a p a r t i c u l a r o i l flow rate 'Q1. 71-APPENDIX I I I P h y s i c a l Properties of S h e l l Turbo 27 O i l The equations f o r the density, and v i s c o s i t y were presented by,Currie (12) as functions of temperature (Equations 6 and 7)- In order to .determine the relationship, between these•quantities and temperature,.tests were c a r r i e d out by Currie using a Saybolt Universal Viscometer f o r v i s c o s i t y measurements and a hydrometer f o r the corresponding density, measurements- Currie. found that the curves r : / 3 , = (1-727 - : 0 . 0 0 0 6 8 6 t)lbs s e c 2 / f t k (kh) = 152t" 2* 7 : !-lbs s e c / f t 2 ) f i t the experimental curves ( F i g . 3*0 to, within 1 .0$ of the operating temperature range. The value f o r the s p e c i f i c heat.as supplied by.the S h e l l O i l Company Ltd. was considered constant. ( V = 0.^97 B t u / l b G F S r The v i s c o s i t y r e l a t i o n may take an exponential form.which was used i n the ffeiption theory f o r a n a l y t i c a l convenience. The' following equation f i t s the experimental curve • (Fig.3 * 0 .to within 2 . 0 $ ,of the operating temperature range. 72. ^ 00 ^ «vi Q 73-APPENDIX IV Instrumentation C a l i b r a t i o n s 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 thermistor probe temperature c h a r a c t e r i s t i c s , (-2) the f r i c t i o n force versus meter reading curve fofc the f r i c t i o n a l torque system, (3) the f i l m thickness versus meter reading c h a r a c t e r i s t i c s f o r f i l m thickness measurement (4) the temperature versus emf c h a r a c t e r i s t i c s f o r the thermo-couples, and (5) the loading pressure versus the bearing load, f o r the loading system. (1) Thermistor C a l i b r a t i o n The Accuracy of the thermistor measurements was checked-.at the i c e point, at several points up to the steam point, and then was further-checked by heating SAE 30 l u b r i c a t i n g o i l to 260°F. Three precision,;;laboratory thermometers were used as standards. The thermistor was accurate to within. + 0.^7$ at the steam point, and was l i n e a r over the temperature tange encountered. The graphical r e s u l t i s shown i n Fig.3 5 -(2) Force Transducer C a l i b r a t i o n The transducer c h a r a c t e r i s t i c s were obtained by applying the known l i m i t i n g load to the transducer, and s e t t i n g the d i f f e r e n t i a l transformer i n d i c a t o r to read f u l l scale d e f l e c t i o n . The l i n e a r i t y of the transducer was checked oyer a range from no load to maximum-load. The transducer c a l i b r a t i o n curve was found to be l i n e a r to within 0 . 2 $ of the f u l l range value. The c a l i b r a t i o n curve i s shown i n Fig.3 6 . (3) Film" Thickness C a l i b r a t i o n The transducer was fastened to a micrometer head block, and the n u l l p o s i t i o n of the transducer was found. Movement i n e i t h e r d i r e c t i o n from the n u l l point could be measured by the micrometer head. The i n d i c a t o r scale could be set to read the r e s u l t d i r e c t l y . Ik. (k) Thermocouple C a l i b r a t i o n The "thermocouples were c a l i b r a t e d over the expected bearing temperature range. On conducting the c a l i b r a t i o n the hot junction was immersed i n . a water bath, and the c o l d junction, was placed i n an ice bath at 32°F. The water bath was slowly,heated to the b o i l i n g point. The r e s u l t i n g curve (Fig.3 7 ) i s consistent with published data on Copper-Constantan thermocouples. ( 5 ) Loading System C a l i b r a t i o n Currie ( l 2 ) obtained a relatiorf...between the load d e l i v e r e d to the thrust disks,and the pressure applied to the loading jacks by comparing one dead weight t e s t e r to, another. The comparison was made by removing the te s t shaft, and ensuring.that the space between the p i s t o n tubes was sealed. A pressure was then applied through the out l e t drain to the thrust disks by one dead weight t e s t e r , while another t e s t e r was used to pressurize the loading jacks. In t h i s way the loading curve of Fig.3 8 was established. Currie found the maximum deviation to be 5 $ - • 75-o 7 6 . 0 N fc- ^ °^ \ Q o A S I si I 5 i 77. V) >V <T) ts{ \ Q 78. TABLE I I (Speed 15000 rpm) Load ( p s i ) F i l m Thickness ( i n x 10-3) (RDG) (ZERO) O i l Flow ( c c ) (sec) F r i c t i o n Torque (RDG) . c l Temperature (°F) G 2 •••• C L Bulk Outlet c 5 5 8-5 . 2.9 U00 5-1 ,2^.0 88 78 86 100 131 •15 6.U 3-1 393 5-3 . 2 0 . 0 90 95 9h 115 15^ 25 5-9 •3-3 U07 >-5 ; l 6 . 0 108 .112 i l l :127 162 35 5.6 3-5 •U26." •5-1 13.5 118 124 122 1U1 171 .= U5 5-4 3-6 1+35' 5-3 11.0 127 131 128 150 178 55 5-6 3-8 450 h.9 .11.0 13U 139 137' 181 65 5-6 U.i U50 5-5 10 .0 Ikk 1V7 1U5 16k 192 3 Pad D e l r i n A F TABLE I I I (Speed 15000 rpm) . Load (ps i ) Film Thickness ' ( i n x 10~3) O i l (cc) Flow (sec) F r i c t i o n • Torque (RDG) c l Temperature c 2 C 3 c 4 Bulk Outlet c 5 Disk Temp. Groove Temp. o F 5 2 .82 390 • 5-3 46 82 88 91 123 131 112 15 2.14 394 4 . 9 ' hi . 87 91 " .91 118 143 -I78 127 2 5 1.82 , 372 ••3.8 37 127 .129" 128 148 .162 . 1V 6 1 148 35 .1-49 435 -4 .4 • 32 li+l+ 146 . 166 :175 188 162 ^5 1.12 .375 4 . 2 . 3 2 156 159 : . 157 184 -I92 193 172 55 0-95 4 l 0 - 5 . 1 • 32 160 165 -164 197 •200 190 170 65 O .83 330 4 . 8 32 169 -.172 173 .206 •212 203 182 75 • 0 .70 370 5-7 ,31 172 175 180 212 220 205 188 13 Pad Babbitt TABLE IV (Speed-15000 rpm) Load (ps i ) F i l m Thickness ( i n x 10-3) O i l (cc) Flow (sec) F r i c t i o n Torque (RDG) c c l Temperature °F . Cg c k c^ c 6 Bulk I n l e t c ? ."Bulk Outlet c 8 Disk Temp Op Groove Temp Op 5 •2.72 3^3 6 . 0 •' 3k 86 101 85 •89 112 98 61 150 •- Qk 15 . 2-35 320 6,8 .31. 89 106 88 92 118 105 62 l 6 l I 6 9 88 25 1 .97 •290 ""6-. 8 29 9k 112 9k 101 127 107 66 176 176 96 •35 1.62 283 7.6 25 103 125 iQk 112 1U3 115 . 7 7 8 195 182 n r . k<? 1.^3 325 8.6 .22 i l l 133 112 118 153 125 87 . 206 .190 118 55 1.20 - 300 ; 7-2 21 125 l k 5 127 13k 165 138 . 101 .216 192 128 65 0.93 . 291 7,2 20 13^ 153 137 1 U 5 I 6 9 1U8 .113 .220 200 • 138 -4 Pad Babbitt TABLE V. (Speed 15000 rpm) Load ( p s i ) • Film ( i n Thickness x 10"3) O i l (cc) Flqv (sec) F r i c t i o n Torque (RDG) c l c 2 Temperature c^ c^ c^ c 6 c 7 Bulk I n l e t c 8 c 9 • Bulk Outlet c 1 0 • Disk Temp °F Qrooy.e Temp °F 5 2.97 322 6.5 - -- 28- 83 93 93 85 99 109 138 103 61 175 -.- 89 15 2.40 335 5-9 32 .-.80 88 86 82 84 99 113 89 65 163 • 163 84 25 I . 9 6 377 7-1 • 27 •85 95 93 86 89 .106 132 94 78 .172 . 1 7 3 35 1-57 3^4 6 . 7 25 102 112 109 104 109 122 148 116 92 I87 ,172 102 45 1.28 315 . 6 .4 22-5 112 123 120 113 125 137 161 122 102 202 1.75 • 120 55 1.09 420 9 . 2 • 21..5 121 132 129 121 13k 145 171 130 107 215 189 .127 .-65 O.90 315 7-0 .. 18 .128 138 •135 127 145 154 182 137 113 226 -201 132 5 Pad Babhitt 83-APPENDIX VI Load Correction f o r S t a t i c O i l Supply The l u b r i c a t i o n system f o r the t e s t bearings operated under a s t a t i c pressure of 5 i n Hg at t h ^ entrance to the bearings. 'The supply of o i l to a c e n t r a l annulus suggested that the system could be considered as a Hydrostatic bearing, interrupted only.by the t e s t shaft running through the resevoir. The an a l y s i s ' o f F u l l e r ( l 8 ) was applied to the present problem i n order to a r r i v e at a simple, expression f o r the load support contributed by s t a t i c pressure of the o i l supply. Consider the pressure d i s t r i b u t i o n as: derived by F u l l e r , I f y - ^i/yL« \ (I^Y) The t o t a l ' load c a r r i e d by the bearing pads and the -annulus acting.as a hydrostatic bearing i s f j /^yuorflo/^ ^- 7rp* (sio~* -yiS) (^8) - > » « > - J O S u b s t i t u t i n g i n equation (47) and i n t e g r a t i n g with respect to ,'r' and 'ft' gives W - _7T sL^—yip —77~y=k,-yi%9' (49) For bearing dimensions r = 1. 500 in,., r .= O.6875 i n . and shaft dimensions r •= 0.3125 i n . , • .W.= 3-27 P s - (50) For a s t a t i c o i l pressure of 5 i n ' Hg (2.455 p s i ) , the contribution to the t o t a l bearing load a t t r i b u t e d to the hydrostatic bearing e f f e c t was ' 8 . 0 l b s . load. 8k. •APPENDIX VII Dimensional;Analysis Solution Dimensional-analysis was used to develop r e l a t i o n s h i p s between the load, the c o e f f i c i e n t of f r i c t i o n and the other operating v a r i a b l e s . The mass, length, time and temperature system of measurement was used i n conjunction with .the i n s p e c t i o n a l method of Hunsaker and Rightmire'(l6),in determining the dimensionless groups. The solutions were checked, using the Buckingham 7T Theorem as presented by Bridgman (17)• Bearing Load The following r e l a t i o n was considered i n order to obtain the bearing load-as a function of the operating v a r i a b l e s : According to the 7T theorem, the ten primary q u a n t i t i e s minus the four basic dimensions indi c a t e that there w i l l be s i x dimensionless groups. The i n s p e c t i o n a l • a n a l y s i s substantiated t h i s p r e d i c t i o n , and the s i x groups are: Geometric size f a c t o r £ Reynold's number • c ^ £ ' * ~ ^ Prandtl number l~v£-<^ * (52) Thermal expansion f a c t o r y£rZ±'7~ a Load number £ii_=^-»~ W Recovery f a c t o r _ 'f^7 . .One necessary-requirement.in obtaining'these groups'is that the dimensionless group must have some p h y s i c a l meaning, f o r example, Reynold's number, which i s a r a t i o of i n e r t i a l forces to viscous forces. A l l of these g r o u p s . f u l f i l l t h i s necessary requirement. It sliould be stressed however, that dimensional analysis can only p r e d i c t what form the groups might take, 8 5 . and cannot p r e d i c t what r e l a t i o n s h i p there w i l l he between them. Previous studies i n the f i e l d s of f l u i d dynamics and heat t r a n s f e r suggest that the most probable r e l a t i o n between the dimensionless groups w i l l be that of the product of the groups at various powers. In t h i s manner,,the load r e l a t i o n can be represented by: where 'C i s a constant. The r e l a t i o n f o r Toad was solved by considering the r e s u l t s as obtained f o r the t h e o r e t i c a l performance of the p a r a l l e l surface bearings. The logarithms of the relations, were used in-order that the equation would be l i n e a r i n form. In, t h i s manner,.simultaneous solutions, were obtained f o r the unknown q u a n t i t i e s , which i n t h i s case were the constants and the ind i c e s . , The IBM 1620 computer was programmed to carry out t h i s procedure, : and the r e s u l t of t h i s work i s : The above r e l a t i o n f o r load i s presented on l o g - l o g paper (Fig.39)• Because the r e l a t i o n considers a l l the important operating v a r i a b l e s , i t i s p o s s i b l e that i t may be used to optimize the design of the. p a r a l l e l surface bearing. For example, by e s t a b l i s h i n g speed, o i l properties, load and the temperature l i m i t a t i o n s , the mean radius and f i l m thickness can be found . by t r i a l and error. Comparison of Theory and Experiment by the Method of Dimensional Analysis The theoretical curve of Fig.39 bas been reproduced i n order that a comparison can be made between theory : and experiment. The experimental r e s u l t s are c o n s i s t e n t l y lower than the theory. This f a c t might be a r e s u l t of the method by which the theory was formulated. F i r s t , because the usual 86. 8 7. 88 . a form taken, by the dimensional. r e l a t i o n s in-.heat t r a n s f e r and f l u i d dynamics i s that of powers, i t ' h a s been assumed that the r e l a t i o n under consideration w i l l also be one of powers. Other assumptions are, of course, possi b l e , but i t was thought that the above method o f f e r e d a l o g i c a l approach. Secondly, i n determining the a c t u a l power r e l a t i o n between separate groups, more than one v a r i a b l e was tested at a time. This was done i n order to si m p l i f y the method of s o l u t i o n . Apart from the p a r t i c u l a r method of solut i o n , i t may be that the sol u t i o n was too general i n form. For example, a t h e o r e t i c a l speed range of from 5000 to -5OQOO rpm was considered, while only one speed .was employed i n the experimental t e s t s . This might give a poorer i n d i c a t i o n of the c o r r e l a t i o n between theory and experiment than would a c t u a l l y - e x i s t i f a l a r g e r speed range had been used. 8 9 . BIBLIOGRAPHY (1) .Tower, B. (2) Reynolds, 0 . (3) Fogg, A. (4) Bower, G.S. (5) Cameron,. A. (6) Shaw, M,C. " F i r s t Report on F r i c t i o n Experiments'* Proc. Inst. Mech. Engrs. (London), vol,3 ^ ^ 632, 1883. "On the Theory of Lubrication.and I t s App l i c a t i o n to Mr. Beauchamp Towfer's Experiments'", P h i l . Trans, of the Roy. Soc. of London, vol . 1 7 7 , 1886. ."Fluid F i l m Lubrication of P a r a l l e l Thrust Surfaces", Proc. Inst. Mech. vol . 1 5 5 , 1946. Contribution to Fogg ( 3 ) -and Wood, W;L. " P a r a l l e l Surface Thrust Bearings", Proc. 6 t h . Inter. Cong, of App. Mech., 1946. "An Analysis of the P a r a l l e l Surface Thrust Bearing", Trans. Amer. Soc. Mech. Engrs. v o l . 6 9 , 1947. (7) Cope, W.F. (8) Kettleborough,. C F . "The Hydrodynamical Theory of Film Lubrication Proc. Roy. Soc.,.Series A, vol . 1 9 7 , 19^9• "Tests on P a r a l l e l Surface Thrust Bearings", Engineering, ;.Aug. 1955' (9) Nahayandi, A. and Osterle, F. "The E f f e c t of V i b r a t i o n on the Load-Carrying Capacity of P a r a l l e l Surface Bearings", Paper 60 Lubs-3 Am. Soc. Mech. Engrs. (10) Young, J . "The Thermal Wedge E f f e c t i n Hydrodynamic Lubrication", Engineering Journal, v o l . 4 5 , number 3> 19^2. ( l l ) : . Christopher son, D.G. "A New Mathematical Method f o r the Solution of Film Lubrication Problems", Proc. Inst. Mech. Engrs., v ol . 1 4 6 , 1942. (12) Currie,. I.G. . "The L u b r i c a t i o n of P a r a l l e l Surface Bearings" Unpublished Thesis, U n i v e r s i t y of B r i t i s h Columbia, 1962. (13) B i r d , R.B., Stewart, W.E., and Li g h t f o o t , E.N. "Transport Phenomena",. John Wiley & Sons, Inc. i 9 6 0 . ' ' ; •' 9Q. (lU).Southwell, R.V. " R e l a x a t i o n Methods i n Engineering Science", Clarendon Press, Oxford, 19^0. (15) Hagg, A.C "Heat E f f e c t s i n L u b r i c a t i n g F i l m s " , Trans..Am. Soc. Mech. Engrs.,.vol.66, A -72 , 19hk. (l6) Hunsaker, J.C. and Rightmire, B.G. "Engineering Applications of F l u i d Mechanics Chapt . 7 , McGraw-Hill, I9U7. (17) Bridgman, P.W. . "Dimensional'Analysis", 2nd ed., Yal e U n i v e r s i t y Press,.1931-(18) F u l l e r , D.D. ."Hydrostatic L u b r i c a t i o n " , Machine Design, v o l . 1 9 , 115, 19^7-

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