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Film lubrication of journal bearings Trant, Geoffrey Allan 1933

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B . C y U j B R A R Y  I CAT.  m^JtLJl^^Fs\  I 'FIDS. LUBRIOATIOH of JOURNAL B E M M ^  Geoffrey A l l a n Trant.  -0O0-  A Thesis submitted f o r the Degree of MASTER OP APPLIED SCIENCE i n the Department of MECHANICAL and ELECTRICAL ENGINEERING.  -0OO0-  THE UNIVERSITY o f BRITISH COLUMBIA. A p r i l j 1933  I  .••'} N  0  1  CONTENTS. Preface. ., Page. Part 1. Definitions. .. 1. Cohesion..........................;.. 3. Adhesion. 3. Oiliness,....................<• .3, Viscosity....... * i . . . . v 4. Part 2. Formation of the .Film.........................................9. F u l l Bearings..... ............................. ..9. P a r t i a l Bearings ( Boswall ) 19 Gl©3.x*s.n.c©• • • ••»* * * * • * *«••*•««• • •» • *. • •««* * *. • • * 10* Part 3.  P a r t i a l "bedded bearing. Observation of f i l m formation in....13. Some theories of f i l m l u b r i c a t i o n 16. Introduction to Reynolds' Theory. 16. Sommerfeld's Theory......................................... 23. Michell's . 25. Harrison's " ......................................... 27. 0s.3TcLu.XXo s ••••••••••••••*•••••••••••••••••••••••••• 31 • IB o SWSLX X s ••*••••••••••••••••••••«•••••••*••»• • • • • • 35 • Tests f o r Pressure and Friction...............................41. Description of machine used i n Tests f o r O i l Film Pressure *............42. Tests f o r F r i c t i o n i n O i l Film. 46. Readings. ]^C*©S SHI*© e * * * m «• • • •««* • •» • • • •»• • • • • • • * • • *• • • • • • • * • • • » • • • 4r7 » - FriCIjXOII* 4:8 e Calibration of Thermo-Couple. .51. Viscosity 51. Curves. • l?r©s su.i*© ««•» • • •«•« • *««***«*«• • •« •« • • • • • •» • * * • * • »• • • • • • • * • * n  Part 4.  •  Sx*xc"tsiori* 1  Part 5.  • < » « • • *  * . * • • • • • • • • < • « .  • » * « * . » « • • • • . « * . • • ' • • • * • * • * • • . * • * • •  •• • • • * • • * • • • * • • • * •• • • • • * *• ••«> • • • • * »•«•• * •  Thermo-Couple. 85. ViscosiL*ty• • • • • • ••• •••••• •••••• •• • • • • •«• • 8fo* Conclusion.. ... .87. : ..Film Pressure • • ^.............. ••».•••. • •........ •...».... ••.«.• • 87. Friction..........; ........90.  Preface.  In t h i s essay I have not attempted anything so ambitious as the development of any theory of my own on the subject of l u b r i c a t i o n , but  my  purpose has been to enquire into the manner i n which pressure i s d i s t r i b u t e d i n o i l films i n bearings, and how  the c o e f f i c i e n t of f r i c t i o n v a r i e s under  d i f f e r e n t operating conditions; and to give some account of c e r t a i n experiments and tests that I have made with the object of gaining knowledge  on  these points. For convenience,  I have d i v i d e d the essay into f i v e parts, the  f i r s t part dealing with d e f i n i t i o n s ; the second with o i l f i l m formation; the t h i r d with theories that have been developed by c e r t a i n well known authorit i e s ; the fourth with my own experiments;  and the f i f t h with my conclusions  and c r i t i c i s m s . I have b r i e f l y outlined the theories of such writers on the subj e c t as Sommerfeld, M i c h e l l , Harrison, Oardullo and Boswall.  This i n order  to f i n d how f a r t h e i r assumptions are borne out by the r e s u l t s of my own i n vestigations. I take t h i s opportunity to make to Dr. H. Vickers, Head of the Dept. of Mech. and Elec. Engineering, to Prof. P. W. Vernon and to Dr. H. F. G. Letson of the University of B r i t i s h Columbia, my grateful aknowledgments f o r the help and advice that they have so k i n d l y given me. Geoffrey A. Trant. Vancouver, B.C. •• April,;. 1935.  FILM LUBRICATION of JOURNAL BEARINGS.  --00OO00— Part 1.—Lubrication  may  Definitions.  "be defined as the introduction,  between  two  r e l a t i v e l y moving "surfaces, of a substance f o r the purpose of reducing the f r i c t i o n a l resistance that would develop i f the moving surfaces actual contact.  The  substance may  were i n  wholly^ or only p a r t i a l l y , prevent con-  tact of the surfaces; the reduction i n f r i c t i o n depending upon the completeness of t h e i r separation, change i n t h e i r character due physical,  to chemical, or  action of the l u b r i c a t i n g substance, the f r i c t i o n a l  resistance  developed by the surfaces moving over the l u b r i c a t i n g substance, and upon the resistance developed within that substance i t s e l f . Where the separation of the rubbing surfaces the l u b r i c a t i o n  that  occurs i s known as  where there i s complete separation •film' l u b r i c a t i o n .  is  incomplete  'boundary l u b r i c a t i o n ' ,  the l u b r i c a t i o n i s  I t has been pointed out  ^  and  'floatation'  or  that a remarkable f e a t -  ure of l u b r i c a t i o n i s that boundary l u b r i c a t i o n and  film lubrication  opposed to each other i n a l l important c h a r a c t e r i s t i c s .  are  In boundary l u b r i -  cation there i s s t a t i c f r i c t i o n , the f r i c t i o n a l resistance being equal to that of k i n e t i c f r i c t i o n unless the state of the s o l i d s (rubbing surfaces) themselves i s altered by the stresses,  and  the resistance varies as some  inverse function of the v i s c o s i t y of the lubricant, and the area and the r e l a t i v e v e l o c i t y . i s absent, and  In f i l m l u b r i c a t i o n , s t a t i c f r i c t i o n  the resistance to r e l a t i v e  the v i s c o s i t y of the lubricant,  that  ( l ) Diet, of Applied Physics, p.572.  i s independent of  is  motion to say,  varies d i r e c t l y with i t depends upon the  internal f r i c t i o n of the lubricant, and increases as the velocity of the relative motion increases. So completely are the relations of these two divisions of lubrication opposed to one another, that i t i s probable that  i f there  could be introduced a single pure chemical substance between two clean solid rubbing surfaces so as  gradually to increase the thickness  of the  layer, we should meet with a discontinuity of state such that, at a certain c r i t i c a l film thickness, 'boundary* conditions would disappear and give way to 'floation'. The only link that connects the two types of lubrication i s that both depend for effective operation upon the property possessed by lubricants of adhering  to the surfaces with which they are i n contact. (l)tbat  R.O.Boswell states:  /in boundary lubrication adhesion practi-  cally controls the action since any tendency lubricant  for the bond holding the  to the surfaces to break down would inevitably lead to metallic  contact and consequent seizure, while with film lubrication  the viscosity  of the lubricant and the relative movability of the surfaces are the controlling factors. He points out, however, that adhesion s t i l l has an important influence since (a) the maintenance of the film depends upon the ability of the journal to drag in the requisite quantity of lubricant; (b) the pressure conditions i n the film cannot be effectively established or maintained i f there i s any possibility of slip occurring between the lubrioant and the metallic surfaces i t serves to separate; and (c) the surfaces w i l l be subject to boundary lubrication conditions when the journal i s f i r s t set i n motion and when the film breaks down just before the journal comes to rest. In the last preceding paragraph reference was made to two (l).Proc. Inst. Mech. Eng. 1932. Vol.122.  3»> properties of lubricants,namely, adhesion and viscosity,and i t seems to me desirable that* before proceeding further with my subject, I should here give definitions of these and of other properties and discuss b r i e f l y the theories relating to them. The properties that have effects on lubrication are cohesion, adhesion, oilimess and viscosity. These I shall deal with i n the order named. Cohesion. Cohesion i s the property that holds together the molecules of which an o i l i s composed. This property i s of high importance in lubrication because i t resists the destruction or break-down of the film. Adhesion,, Adhesion i s the property that causes a lubricant to wet, or adhere to, the solid surfaces with which i t comes i n contact. This i s due to the potential energy at the surface of the lubricant having an attract i o n for the solid surfaces, such attraction resulting/adhesion,that i s , surface tension with respect to the solid. . -  • •  Oiliness.  '  • (1)' .••  Oiliness has been defined  ' a s the property which causes a  difference i n f r i c t i o n when two lubricants having the same viscosity at atmospheric pressure and at the true temperature of the film are tested under external conditions. The lubricant giving the lesser friction i s then considered as possessing the greater oiliness.  This means, not that  two oils,to be comparable i n respect to oiliness, must have the same v i s cosity at the same temperature, but that to compare the oiliness of the two oils the tests must be made at such temperatures as w i l l make the viscosities the same. Of the r e a l nature of oiliness and of the manner i n which i t  (l)  A. S. M. E. 1930, M.S.P. 52 - 12, p. 152.  functions in reducing f r i c t i o n very l i t t l e has yet been learned. Herschel experimented i n an endeavor to determine just what oiliness implies. He ran six types of o i l through his machine and also six samples of the same o i l treated with Oleic Acid. The absolute viscosities were then determined at approximately the temperature of the tests as a "basis for plotting the curves, showing the observed coefficients of f r i c t i o n as functions of viscosity.  The curves obtained showed a slight decrease i n f r i c -  tion with increasing viscosity,and a distinct reduction i n friction,averaging about 20$, due to the addition of Oleic Acid.  In these experiments  i t was intended to eliminate,as far as possible, the influence of viscosity by choosing for the tests so low a speed that the coefficient of friction 'f w i l l be at i t s maximum value. The existence of such a maximum has been found and i t l i e s far below the speed corresponding to the more familiar minimum point.  So far as  the intrusion of purely hydrodynamic action i s  concerned i t may be inferred that when * f passes through a maximum on the 1  speed diagram i t must likewise do so on the ZN/P diagram and that therefore Bf/M  = 0 when "cVf/clN = 0  where Z = absolute viscosity, and N = R.P.M.  The coefficient of f r i c t i o n when plotted against ZN/P per unit area) showed a very wide divergence between o i l s .  (P = load  Now, i f i t were  not for the existence of oiliness, that i s , some property of the lubricant, other than viscosity,that can influence f r i c t i o n , a l l the curves would be expected to coincide.  Accordingly,the differences between the ordinates  of the curves can be taken as a measure of the degree of oiliness. Viscosity. Many writers on the subject of lubricants have given good de(2)  finitions of this property.  Mr.FiE.Cardullo  (l) A. Si M. E. 1931, P.M.E. 53 - 4, p.23 (2) A, S. M. E. 1930, M.S.P. 52 - 12, p.143  defines i t  as resistance  to flow', and points out that experiments show that this resistance i s of the nature of a resistance to a shearing movement,and i s proportional to the rate of shearing. It differs,in i t s nature, from the resistance to flow due to eddying and to other inertia effects, since the resistance due to inertia effects i s proportional to the square of the velocity of flow,while the resistance due to viscosity i s proportional to the f i r s t power of the velocity. (1)  Maxwell states  that the coefficient of viscosity of a  fluid i s measured "by the tangential force on a unit area of either of two horizontal planes at unit distance apart,one of which i s fixed while the other moves with unit velocity, the space between being f i l l e d with the viscous  fluid.  . (2). ' Newton assumed  that for a f l u i d moving i n parallel layers,  the shearing stress at a point at which the velocity gradient,in a direction perpendicular to the layers, i s dv/dy would be directly proportional to the velocity gradient; that i s to say, that the relation between the viscosity of the fluid and the relative motion would be  f = yodv/dy where  i s the v i s c o s i t y of the f l u i d . (3)  r  There i s an interpretation, illustrated i n F i g .-—1 , ,of >ythe coefficient of v i s c o s i t y of a f l u i d according  to  the Kinetic Theory of Matter, which i s b r i e f l y as follows:- A B represents a plane i n a f l u i d , parallel to the direction of motion, and at right angles to A B  there i s a definite vel-  Fig.l..  Aocity B. These lines represent the magnitude gradient shown by the lines parallel oftothe velocity of layers of the fluid relative to the velocity of A B, so that the molecules immediately (i)-Dlct.of^Appli^ (3) *' • "  "  Physics, p.342.  6.  above A B  are moving faster than those immediately below.  Some of these  molecules w i l l cross A B from the upper to the lower layer and an equal number w i l l pass upwards, to replace them. Thus the layer just above A B i s continually losing momentum and that below i s continually gaining i t . The effect i s to bring into existance a definite shearing stress on the plane A B which constitutes the viscous drag. The internal f r i c t i o n of fluids consists therefore of a transfer of motion from one layer to another, but this transfer does not proceed without a loss of energy, since the translatory motion of the layers i s transformed into heat. Most of the methods used to measure viscosity depend upon a relation between  the motion of the f l u i d and variations of i t s internal  pressure, and as this has a close connection with the theory of viscous flow of lubricants i n bearings, I shall here give a brief summary of the equations of motion of a Viscous fluid, as set forth i n Glazebrook's Dictionary of Applied Physics,Vol.1 :"Assuming the truth of the Newtonian Hypothesis, the equations of motion of the f l u i d can be obtained as follows:- If we imagine three planes to be drawn through any point 'P* i n the f l u i d perpendicular to the axes 'XV , 'Y' and 'Z' respectively,the three components of the stress per unit area exerted across the f i r s t of these planes may be denoted by Pxx, Pxy and x z p  respectively; those of the stress across the plane perpendicular to 'Y' by Pyx, Pyy and Pyz; and those of the stress across the plane perpendicular to 'Z' by zx, Pzy and -Paz. p  It follows at once that,considering an element dx, dy, dz, having i t s centre at *P',and taking moments, we getiPyz = Pzy, Pzx = Pxz, Pxy = Pyx. Also, i f P i , 2 , 3 be the principal stresses at 'P*, i t can be shown:p  p  (a) That x x * Pyy + Pzz = l + 2 + P3 - - - - - p  p  p  - - - - (l.)  • 7.  i . e. the arithmetic mean of the normal pressures on any three mutually perpendicular planes through the point 'P i s the same and equal to 'p'(say) 8  (b) That the values of the stresses i n terms of * p t h e coefficient of viscosity ^ i ' , and the rates of distortion are given by the expressions:Pxx = P  P  w  P  - 2 / 5 ^  + |y  +  l l )  +  ^ a l "  = - p - 2/3u( • •  "•  • ) + 2u  z z = - p » 2/5ja( •• +•  "+  " ) +2fx g  ) - - - - .- - - - - (2.) {  j  where 'u , 'v' and 'w* are the component© of velocity of 'P' i n the 'X', 1  'I' and Also:-  'Z' directions. y z = z y = )i( ^  P  p  +Is ^  (  y £ . * £ > p  x y = y x = u( § | t | y ) p  ;-(«•) (  She condition for laminar motion, i.e» amotion i n which the f l u i d moves i n a system of parallel planes, the velocity being, i n direction, everywhere the same, and i n magnitude proportional to the distance from some fixed plane of the system - i s s-u = ay, v s 0, w = 0 from which i t follows that,the axis of 'X* being taken i n the direction of motion,and the velocity being proportional to the distance from the plane 'ZX' we get:p  x x s Pyy = Pzz = -p,  Pyz = 0, z x = 0, Pxy= y& p  'a being the rate of distortion. 8  The stresses i n different fluids under similar conditions of motion,will be proportional to the corresponding values of Jn', but,if we wish to compare their effects i n modifying the existing motion, we have to take account of the ratio of these stresses to the inertia of the fluid. Prom this point of view, the determining quantity i s the ratio  pjp»  which i s  . ' 8.  denoted by the special symbol U, viscosity.  .  called the Kinematic coefficient of  The equations of motion are obtained by considering the forces  acting on a rectangular element having i t s centre at 'P',  Thus, resolving  forces parallel to 'X', the difference of the normal pressures i s :~ ) dxdydz.  The tangential tractions on the ZX' faces amount to f  ( Pyz/dy) dxdydz, and those on the XI a  9  8  faces are ( Pzx#la)dxdydz. If, therea  fore, 'X', 'I , and 'Z' are the components of the external forces per unit 1  mass, we have s-  « P  Where D a Dt  Substituting  *P_-  Dt  5p  ^  $y  4 + ujl + v i  3k . ay  at  ^Pzz AZ  + wil  Jz  the values of Pxx, Pyy and Pzz  given aboue, we have:-  )  and so for £l Dt where  0=  +  fe  and  + |2  ^ Dt and  -  ( 5 tf . 1  j \ + *  + *  -  (4.)  I  Part 2 —  For  '  9.  Formation of the Film.  effective film lubrication an abundant supply of o i l to  the bearing i s necessary,for i t i s important that there be no rupture of the  film, Otherwise metallic contact w i l l ensue and result i n wear which,  with perfect f i l m lubrication, should be entirely absent. The o i l may be supplied either by pressure or by gravity, but in either case the manner i n which the film i s formed i s the same. At Fig. 2 i s shown a cross section of  Load  a journal bearing with exaggerated clearance. With the journal at rest the clearance space i s completely f i l l e d with lubricant and the posi t i o n of the journal i s at the bottom of the bearing. V/hen the j aural i s set i n motion the o i l that adheres to i t i s carried around Fig.2.  with it,ana,as the other side of the o i l film i s adhering to the bearing, the journal begins to to climb up.  A 'wedging  8  effect i s produced as  the o i l i s drawn under the journal with the r e s u l t t h a t  ?  so long as the  clearance space i s kept f i l l e d , the journal w i l l float upon a film of o i l . The foregoing i s a brief description of the process of the formation of o i l film i n complete journal bearings,and,as the experiments that I have carried out - and shall later give account of - were i n connection with a partial bearing, I feel that I cannot do better than quote here Mr.R.O.Boswall's remarks on the subject of film formation i n partial bearings.  (l)Proc. Inst.Mech.Eng. 1932, Vol.122, p.430  ,  —  •  '  •  '  '  '  '  '10,'.  ••  "The process of f i l m formation takes place under very r a p i d l y changing speed conditions, and i t i s only p o s s i b l e to make a conjecture as to what a c t u a l l y happens i n the i n i t i a l stages.  I t would appear that when  the speed i s low, conditions correspond to those associated with greasy l u b r i c a t i o n and that t h i s type of l u b r i c a t i o n r a p i d l y develops i n t o the • f l u i d f i l m type as the speed increases, provided conditions are s a t i s f a c t ory f o r the proper formation of the f i l m .  That i s to say, the journal  must be able to take up a p o s i t i o n . r e l a t i v e to the brass such that a continuous f i l m of lubricant i s maintained between the surfaces under conditions i n which (a) The resultant f i l m pressure i s equal to the external load transmitted by the journal. •(b) The l i n e of a c t i o n of t h i s resultant f i l m pressure coincides with the l i n e of a c t i o n of the load. F a i l u r e to comply with these two requirements w i l l lead to greasy l u b r i cation or possibly to a combination of f i l m and greasy l u b r i c a t i o n conditions. The i n i t i a l movements of the journal appear to have some i n fluence upon the f i n a l running conditions, and the following explanations may help to v i s u a l i z e the sequence of events which immediately follows the commencement of journal r o t a t i o n and ultimately leads to the establishment of an e f f e c t i v e pressure f i l m . Two types of "brass w i l l be considered: larger . . . .  (a) clearance brasses  which are f i n i s h e d to a s l i g h t l y / r a d i u s than the journal;  (b) bedded  brasses which have the same radius as the journal. (a) Clearance Brasses.- The p o s i t i o n when the journal i s stationary Is shown i n occur.  F i g . 3, and t h e o r e t i c a l l y only l i n e contact should  Actually owing to a c e r t a i n amount of l o c a l d i s t o r t i o n produced  11. lay the s t a t i c pressure, there w i l l be surface contact over a small ares. On each side of t h i s , however, there w i l l be a clearance space between the surfaces which converge'on the i n l e t side and diverge on the o u t l e t side. Fd,g. 3.  Fig. 4.  As soon as the journal commences to rotate i t w i l l  experience  a momentary tendency to r o l l towards the i n l e t edge of the brass.  This  motion i s almost immediately counteracted by the introduction of lubricant which i s dragged i n by adhesion to the journal and establishes a state of s o l i d f i l m or greasy l u b r i c a t i o n over the small contact area.  This reduces  the f r i c t i o n a l resistance at t h i s point and causes the journal to s l i p back into the p o s i t i o n indicated i n F i g . 4.  At t h i s stage the l a y e r of l u b r i -  cant between the surfaces w i l l be extremely t h i n , and during the s l i p p i n g process a c e r t a i n amount of abrasion w i l l be inevitable. This phase of the motion i s r a p i d l y passed through, and  the  lubricant begins to accumulate i n the convergent entry portion of the clearance space.  Very l i t t l e l u b r i c a n t however, can escape past the greasy' l u -  b r i c a t i o n area, and as the speed increases the greater part of the l u b r i cant drawn i n by the journal w i l l be forced to escape along the sides of the brass.  Tin doing t h i s i t w i l l be compelled to traverse a convergent  passage and f i l m pressure conditions w i l l consequently be produced i n the entry p o r t i o n of the clearance space. This has the e f f e c t of displacing,the greasy l u b r i c a t i o n area towards the outlet edge of the brass and the journal w i l l s h i f t into a  X2o.  p o s i t i o n similar to that shown i n Pig. 5,  The e c c e n t r i c i t y or distance be-  tween the centres of journal and brass w i l l be very l i t t l e changed and the journal centre w i l l simply swing around the f i x e d c e n t r e C ' of the brass i n the d i r e c t i o n of journal rotation.  This motion i s represented by the  small angular displacement of the l i n e of centres.  l e t p o r t i o n of the clearance space and the consequent d i s t r i b u t i o n of the load over a greater area w i l l necessarily reduce the i n t e n s i t y of pressure over the area where greasy l u b r i c a t i o n s t i l l occurs.  This encourages the  production of a thicker f i l m i n t h i s region, and as the speed continues to increase and more lubricant i s introduced, t h i s small section of the bearing area w i l l become completely flooded.  This enables f l u i d f i l m conditions  to be established between the convergent and divergent portions of the clearance space and a continuous f i l m w i l l ultimately be obtained provided conditions are favourable f o r i t s formation. During the f i n a l stages i n the process of f i l m formation the f i l m thickness increases and the e c c e n t r i c i t y decreases.  The l i n e of  centres also continues i t s angular motion and the journal centre commences to t r a v e l along a s p i r a l path. (b)Bedded Brasses*- I f the brass i s bedded to the surface o f the journal the conditions leading to the formation of the f i l m are somewhat d i f f e r e n t .  In the stationary p o s i t i o n the load w i l l i n a l l probab-  • . .  ,- 13.  i l i t y be concentrated a t a number of high points which have been l e f t on the surfaces owing to imperfections i n t h e i r mechanical f i n i s h .  When the  journal commences to rotate a c e r t a i n amount of abrasion i s bound to occur. This w i l l be almost immediately dragged i n by the journal.  checked by the greasy layer of lubricant  As the speed increases excess lubricant i s i n -  troduced which tends to c o l l e c t by the a c t i o n of surface tension i n the region of the high points and gradually spreads u n t i l a complete greasy layer i s formed across the surface o f the brass. at  The f r i c t i o n conditions  this stage w i l l correspond to those associated with greasy l u b r i c a t i o n  and w i l l therefore depend upon the chemical rather than the p h y s i c a l c h a r a c t e r i s t i c s of the lubricant. A c e r t a i n amount of l u b r i c a n t w i l l be squeezed away l a t e r a l l y and the greasy layer w i l l tend to t h i n out towards the outlet edge of the brass.  This w i l l cause the journal to take up a p o s i t i o n i n which the l i n e  of centres i s i n c l i n e d at an angle to i t s o r i g i n a l V e r t i c a l p o s i t i o n as shown i n Fig. 6.  The convergency of the clearance space provides one of the  essential conditions f o r f i l m formation, and as the. speed oontinues to i n crease and further lubricant i s introduced a pressure f i l m w i l l be formed provided conditions are suitable.  That i s to say, provided the journal can  take up a p o s i t i o n which enable the necessary resultant f i l m pressure and centre of pressure conditions to be established." In the f i r s t sentence of h i s remakks quoted above, Mr. Boswall states that ' i t i s only possible to make a conjecture as to what a c t u a l l y happens i n the i n i t i a l stages' of f i l m formation, I would be i n c l i n e d to go s t i l l farther and say that what a c t u a l l y happens i n a l l the stages of the formation of the f i l m i s conjectural. My opinion,based upon the r e s u l t s of experiments made with a bedded brass bearing ( p a r t i a l ) above the journal, i s that what a c t u a l l y  14. occurs i s as follows: When the journal i s at rest, there i s , except for high points l e f t i n finishing, metallic contact,or at most a very thin greasy layer,between the "bearing and the journal throughout the whole area of the former. As soon as the journal commences to rotate some of the o i l adhering to i t is carried under the bearing, momentarily producing greasy lubrication. "At this stage the position of the bearing i s as shown i n Pig. 7.  The slight t i l t of the bearing at this point i s the natural consequence of i t s tendency,through motion of the journal.  adhesion, to follow the direction of  As o i l continues to be drawn under the bearing at"  the inlet edge a wedge-like pressure film i s formed there.  This pressure  overcomes the vertical load at that point causing the bearing to t i l t eccentrically as shown i n Pig. 8 , and producing film lubrioation over part of the surface of the bearing (shown i n the figure as AB') while the remainder -C B'/ s t i l l has greasy lubrication. As this 4harB action continues,the point of application of the resultant pressure 'R' of the wedge-like pressure film moves i n the direction of rotation owing to the increasing pressure i n the o i l film that i s forming, the result being that the area A B• continually increases while that of C B'continually decreases.  This continues until the  f i n a l stage of the formation of the film i s reached, that is,when the point B' reaches the point C, and we have a complete pressure film between the journal and the bearing.  15. The p o s i t i o n o f the hearing at t h i s f i n a l stage i s shown i n Pig.  9.  I t w i l l be observed that the l i n e of a c t i o n of the resultant  pressure 'R' Has moved toward the outlet side of the bearing, and that there i s a balance between the resultant f i l m pressure load  'W.  'R'  and the v e r t i c a l  This balance o f forces  w i l l maintain the bearing i n the posi t i o n shown and continuance of the pressure f i l m w i l l be assured so long as  (a) there i s an abundant  supply of o i l ; (b) the side leak-  Pig. 9.  age i s not excessive; (c)the temperature of the bearing i s kept within suitable l i m i t s ;  and (d) both load and speed are properly proportioned.  —oOo—  16.  Part 3.—  Some Theories of Film Lubrication.  The pioneers i n the development of theories connected with the problem of film lubrication were, 0. Reynolds and N. P. Petroff. Mr. Reynolds published his theory of the viscous flow  In 1886  of lubricants i n  bearings, and i t later became known that Mr. Petroff, i n the sar-e year, and independently of Mr. Reynolds, developed a practically identical theory. The theory has become known as the  'Reynolds Theory of Film Lubrication, ••  Since that time many eminent men have continued the work, and have amplified the Reynolds Theory.  To the work of some of these I shall  have occasion to refer i n the following pages. For a clear understanding of mathematical analyses of the viscous flow of lubricants i n journal bearings i t w i l l be well to proceed practically from f i r s t principles, and to consider f i r s t the flow of viscous liquids between t¥Q parallel and adjacent planes.  And here I may say that for the  information contained i n the next twelve pages I have made frequent application to  Chapter 5, Part B, of 'Mechanical Properties of Fluids.' by Lamb. In Fig. 10  Z= 0 and  Z = h are two parallel planes the distance  •'h • between which i s very small Now the components of velocity nor- , mal to the planes must be negligible, and therefore both the rate of shear and the rate of momentum i n the  'Z'  direction are small. Hence, the f l u i d pressure 'p' does not vary in that direction  Fig. 10.  and,therefore,is a function of 'x' and 'y' only. Also, the rates of change  ' •• 17. •-• from the values of the finite velocity components 'u* and 'v* i n the fluid to their values, known to be zero, on the walls are rapid compared with their rates of change i n the 'X and 'Y' directions. 8  Considering a rect-  angular element between the planes Z = 0 and Z = h, the viscous tractions on i t s lower face i n the directions i n which 'X'and 'Y' increase w i l l be:-  ~- dxdy and - n u i*I d x d y ; and the corresponding tractions on the a az upper face w i l l bw :p.{ ^ + ||*dzj dxdy and p.[ | l + | ~ d z j dxdy r  z  A  Now, the sums of. these tractions added to the difference of the f l u i d pressure on the faces parallel to the ZyY and Z X planes are equal to the rates of increase of the momentum of the element i n the X and Y d i rections. Therefore: - £  J55 <j dxdydz = jpdxdydz ~ ........ fdV -dp , • du . dz* ax •* dt  /. \  7  dv dp dv . . . .^ ^ = ~ + ^ ~ ......  and similarly  The rates of increase of velocity a.u • order of the" product of small terms u  ~ L and ™ dt dt dv  and 7 ^  ./ \ 1ift  being of the  we may safely neglect  the momentum terms, reducing the equations ( l ) and (la) to:. au _ a? _ aV a " ^ az* ax • az -ay-'--" d  u  =  /Y  P  ?  The pressure p' being independent of *z' we can integrate 9  equation (2) directly as follows:-  au = 1  | E  (  Z  .  +  Q  /  /  )  „ _i ( z + cz +c) /» dx 2 * z  1  Llarly sxmxxarxy  v | v =jx =  t *2 f X  +  d  '  Z  ^  Substituting i n the limitsju = v = 0 when- ».z-.-::= :Q:- thereforeOc^s d^= 0 2.  3.  and  u = v = 0 when  z=h  § * <? = § h  <P* =  0  substituting these values of 'o* and 'd' f e get :u = I^£z(£^)  and  v = i $ L z ( ^ )  (« h  and the resultant Velocity of the f l u i d at any point i s : -  (>*•*)*«£'( ( g p + (§*) j*«i5^i  .........(4.)  and i s i n the direction of the most rapid f a l l of pressure. The total flow across a width  dy  from plane Z = 0 to plane  Z = h i n the increasing direction of X i s :U dy = dy/udz = ^ .•.U =  12^  Now,  f c.\  3p  h  — "° i i  S null s.x*Xy V  /(z*-zh) dz  &*p ••••••••••••«•••••••••«•«««•••••••«•••••««•••( 6« jj dy  since the amount of the f l u i d that flows out must the same  as that that flows i n i t follows that :h( —  + 0)  dxdy = 0  . . Ox*  ay  or from (5) and (6)  a  We may now consider what would he the effect i f the parallel planes had relative motion. Suppose plane Z=h to "be moving parallel to plane Z=0 with component velocities u,and v, i n the imposed on the f l u i d velocities of magnitude ^' and  u  X and Y directions. and  v  There w i l l be super-  of (3) uniform rates ot shear  ^', and the components of velocity at Z w i l l become  u' = ± f e z £ | = h l  +  u,§  andv' = l | ^ | = h l  but neither the pressure nor the relation  +  v,|  + S_E = 0 w i l l be affected. dx  dy  a  We have now reached the point where we can develop a general differential equation determining the value of the pressure 'p' at any point. When dealing with bearings to which the Reynolds Theory i s applic-  19. able i t i s customary to imagine one of the surfaces as continuous, or un-liraited i n dimension in the direction of the relative motion, while the other surface i s regarded as essentially discontinuous i n the same direction. In the Pig.11* X and Y are the axes of co-ordinates i n directions  at right  angles to each other along the surface of the continuous element, and Z the co-ordinate axis normal to this surface*,  The surfaces  of the contin-  uous and discontinuous elements are assumed to be nearly parallel, and the distance between them *h' to be small i n comparison with their r a d i i of curvature.  We suppose the  discontinuous surface to be moving with component of velocities 'uj  Pig. 11.  and v/ i n the X and Y directions,parallel to the continuous surface at XY. 1  The problem of finding the pressure and motions i s the same as before, except that here the surfaces are not parallel, and for this reason, as shown by equations (5) and (6), the rate of increase of volume of fluid i n the element due to the rates of change of pressure and of film thickness i n the X and Y directions i s :~ •A/ v  \  d  ( h  clp_  (8 )  and the rate at which the fluid passes out of the element as the result of the shearing deformation due to the movement of the upper surface over the lov/sx* xs •  dxdy  clydx • •••••«»•••••••••••••••••••-•••••'« (9«)  Because of the movement of the upper plane the volume of the element i s diminishing at the rate :-  .1  ,j  a+  ~| ( y  dxdy + v,  +  ( u ^ *  dydx . Consequently :-  dh  s  o  (&.)  20. Y/hich i s the general differential equation from which, "by integration,we can determine the value of the pressure 'p at any point i n the fluid. 9  We are now i n position to consider the case of inclined planes unlimited i n one direction, and a solution of this problem w i l l afford us a general view of the nature of Reynold s Theory of Lubrication. 1  In Fig. 12 the lower plane Z= 0 i s unlimited i n the direction of X and Y, . while the other plane, which i s also unlimited i n the direction of Y, extends only from line X=0.  x = a, to x = a and i t intersects the plane I = 0 on the fc>  It w i l l be seen that the  distance between the planes i s proportional to x and that therefore h = cx where c i s the tangent between the planes.  Assuming that the  upper plane moves over the lower with velocity 'uj i n the direction ' of X while the other plane i s sta-  Fig.12.  tionary, that i s 'v = 0', and that the whole i s immersed i n fluid, then the pressure both i n front of and behind ,the moving plane w i l l be constant and equal toTT . As none•of .the conditions vary i n the dh Hy  and C\D are both zero. dy  and, integrating, we get:-  Y  direction  Therefore equation (10$ becomes  h  +  dx  6 uu, (h-h)=0 '  (ll. )  where 'hj i s the value of h' where |E = 0 r  Therefore  ||= 6  (  §;)  = - 6 i£i( i - ~' ) since h = cx a  C  A ..X •  Now, as |£ i s positive when h < h,, and negative when h > h,, i t i s seen p has a maximum value at(x = x ) between x = a, and x = a. (  21. Integrating, we get :- p Since  =6^r'(  "f^<T )  |  (12.)  A  p = IT both when x = a, and when x = a , then:A  " = ^ < i . - f f - ) = ^ ( i - f c - ) 2a_<_a. Therefore x. = ; ................. (13.) al+a * Hence the point of maximum pressure i s nearer to a, than to a , and 6  A  6  A  2  t  &  ( a, a  C A  2  a> ap j +  Then by substituting values for x, and A  i n (12. ) we find that:-  By this equation we can determine the pressure at a l l points between the two planes. The total upward pressure exerted on the upper plane, per unit width i n the  Y  direction i s represented by the equation  V(a7^~aJ = 6 ^ ' ( log,f V 2 | ^ and i s dependent only on the ratio of a ^ to  :  )  ...(150  a, for a given value of c,  the mean pressure being = 6 ^ (  a^- a,  1 a  n  loz*«-  c* a^.- a, x  )...  (16. )  a^ + a, '  The total friction, or resistance to the motion, of the upper plane per unit width i s 1F =^/u g'dx = ^ / ^ L dx - y i f log, | . which, £ike  P, i s dependent only on the ratio of a^ to a, and c.  The coefficient of friction i s represented by :-  (17. )  22. By applying such reasoning to the case of actual bearings,we may gain an insight into Reynolds' Theory.  Imagining the lower plane  Z = 0 to be replaced by the surface  of a cylinder,with axis parallel to the Y which extends from  axis, and the upper plane,  x = a, to x = a^,to be replaced by a curved surface,  which, at every point of co-ordinates  x, y, measured respectively c i r -  cumferentially from a generating line of the cylinder corresponding  to  X = 0, and a x i a l l y from a circumferential circle to the cylinder Corresponding to Y = 0, i s at the same normal distance 'h' from the cylinder as are the two planes from one another, the above results w i l l apply. Such an ideal case i s illustrated i n Pig. 13. in which the cylinder may be regarded as .the journal of an axle  _____  or shaft, and the upper surface as the bearing surface.  \ /^5><Cr  The results, which  '\  have been calculated above, as to the fluid pressure, obviously w i l l be the  /  same i f , instead of the bearing mov-  V  ing i n the direction of X with a  \! '  /  's'  \ \ ^  Pig. 13.  linear velocity 'u', the "journal revolves i n the opposite direction with the same surface velocity.  In actual practice bearings are not of unlimited width, but the r e s u l t s , as calculated, w i l l apply with a f a i r degree of accuracy to the middle portions of a bearing the dimension of which, i n the direction transverse to the relative motion, i s not less than two or three times that i n the direction of motion.  In such middle portions the lubricant  w i l l tend to flow in the direction of motion of the journal, while i n the lateral portions i t w i l l tend to flow towards the nearest side because of decreasing pressure i n that direction, and therefore the theoretical con-  23. ditions w i l l not hold true for the lateral portions of bearings. If,by some means or other,the ends of the bearing could be closed, so that there would be no leakage of o i l , then the flow of o i l would be i n the direction of motion throughout the bearing and the conditions assumed for unlimited surfaces would be realizes, provided that the bearing i s of such form that »h' = 'ox'. Sommerfeld s simplification of the Reynolds Theory may be 1  b r i e f l y set forth as follows:In fig.14.  0 and 0* are the centres of a journal and a semi-  cylindrical bearing having r a d i i  r and r + dr respectively,both of i n -  f i n i t e length i n the direction of their axes, d Let 0 0' = __ and —  = a , 'a' therefore has  different values for different cases, varying from one, when the bearing and the journal are i n contact, to i n f i n i t y when they are concentric.  Let |f be the angle between 0 0 '  and the vertical, and 0 be an angular co-ordinate measured from the direction  Fig. 14.  0 0 , the 8  +  J L  and JT - J2T,  co-ordinates for the ends of the bearing being then the linear co-ordinate  X i n the direction of motion of j>he bearing  relatively to the journal i s a constant and equals - r 0 .  Then from  equation ( l l . ) we find that J J J | = 6 ;m.  ...(19)  h  ~^ ' h  and since p = constant 7 T when 0 = $ - IT and when 0 = $ + I T then /  h — h' 2 j | 6,0 = S^uu.ry  = o  and also, i f 'p' i s the f l u i d pressure, and q ° the circumferential 8  traction per unit width at 0, and P the total load on the bearing per unit width, then P cos ^ - / (p -TT) cos 0 rd0 - / q sin 0 r&0 = 0 , _  JJ.  T  a.  24.  sr  and  P sin f/ ( p - TT) sin jZf rdjtf + / q cos 0 rd$ = 0/ "but z^r * - r • , ( p -Ti) COS 0 <5J0 = \( V - ") sin 0 )  since /  p -TT) sin 0 C * -Y^§£.  and / < ( p -TT) sin 0.d0  r  and since p = rr,both when $ = # + § terms not under the integral vanish  / j % and  JL q  _ - /|j> sin 0 djZT  ) cos 0d$ = - ~  A O S  and when 0 = tfl - % , then the  and therefore:sin * j  ,  9 n  v  - q ) sin #d0 = - P cos ^ ) r  _a  Now, from equations (l?) and (19)  of which the last term can be neglected since *h' i s very small compared with 'r'. So equations (20.) may be rewritten as 6 jxa, r / and  h  ° 'sin 06.0 ~ - ~ cos h  g  C?l 1  6 uu, r / h - h,cos 0&0 = P sin I. «• h r 3  J  and since  h = d + er cos 0 = «r( a + cos'0 ) ( a + cos ,0 )  h• = then these equations become  (  a  +  co 0)  cos V  S  0  ( 2 2 > )  ... • • ^  the limits being from  /  h  ^  / ( a + cos 0) u  - 2ZI  t  o  * r  ,v  6;ua*r u, x  U,+  These can be readily integrated, and from the results we are able to find the pressure and the coefficient of f r i c t i o n 'f•= M where 'M* i s the . Pr total moment acting on the bearing.  • .  • 25. Another modification of the Reynolds Theory, and a more import-  ant one since i t applies to actual hearings and takes into account the transverse flow of the f l u i d under pressure to the sides as well as i n the direction of the motion of the journal, i s that worked out "by Michell.  His solu-  tion i s very long so I shall here give only some indication of his method of procedure and a few working formulae and constants. In Pig.15. A B 0 D i s a rectangular plate i n the plane (length of plate = a^- a and width = b )  Z = Cx  Pig.15.  /  sliding i n the direction of X with velocity u,.  The pressure i s assumed  to be uniform everywhere eacept i n the space between the plate A B C D and the infinite fixed plate i n the plane Z = 0, that i s , the boundary conditions of the plate A B C D when x = a, or x = a  are p = Inconstant • &  for a l l values of y, and also when y = 0 or y= b  for a l l values of x . The pressure p between the two plates satisfies the differential equation  _ Ah h = cx and  or, since  = 0,  then _k_R + I ___ + A_B 6 ^ . i ••« i »• •• t ax" x dx dy* c*x* This equation may be written i n the form of +  &  ax*  +  x \ dy, /  =  0  (23.)  *• i  ( s i n ^ * i s i n ^ H . . . . . i U i n 2L*_J .. ) = 0  crx-* v  b  3  b  m  b  since the sum of the series i n the bracket, for a l l of y with which we are concerned, namely y = 0  to y = b i s  J^.  Por a solution of this differential equation so as to give a function of x and y  p as  i t i s assumed- that p =71 + p^ + p^ + ..+ ^+ ... (24. )  HUH  , xn whxch  A = m  5  mix  ' b The integer m can have only odd values because symmetrical on both sides of y = — . 2 L sin Thus p -7T=_> B „ ___—  where m  mtrx  •  •  •  • • •  b  _L£  24-*—- as K and  dN  b  an*  b  \N ^ j T  •  b  = - ^,m*>3m b^-N  dy*  i s odd.  as N, then Hi*  = ___Z* _LP- = £_!__.* < C 1 cL^m  ax*-  p -TT must be  h  If for brevity we write dx  "26.  sin rn^Ty g JZElL... » m being a function of x only.  The coefficient of sin S^JL  TO~/T ( d. ^m b N &N* fl  +  N s  ±  n  2  "a_P"  N*  )  s  i  n  ~V"  dJm IN"  2 ^m v .. m'ffy f  "IF"  } s  i  n  b"  m^_ b  i n equation (23.) i s  1 &^m _ /  N  dN ~  1  +  1  ,  I  $r*)  m  ~  /'  =  \•  0 ........... (25. )  And, as a l l such coefficients must vanish, the factor within the brackets may be equated to zero, of which equation particular integrals are the Bessel's Functions  I,(N) and K, (N), and the complete integral may be writ-  ten i n either form ? m  or  =  I , (N) + B  m  K (N) 7  K ( 1 + |  ^m = A ' I (N) * B ' K , (N) - K ( N + 3N + 5»3~ i f  the second form being useful when N The coefficients Aj_,  on these two lines.  )  6  (26. )  + . . . . . ) . . . (27.)  i s very large and i s asymtotic.  A^, B and B^ are to be determined to make m  vanish for x = a and x = a and hence p y  +J--,,*  f  m  w i l l vanish for a l l values of  These coefficients can be determined only arith-  metically, numerical values being assigned to the quantities The coefficients  a, , a,,, and b.  etc. having been calculated, the values of  the pressure (p) for as many points of x and y as may be desired, are  27. also calculated arithmetically, and when 'p i s known, the total fluid 1  pressure supporting the bearing may be determined by arithmetical or graphi c a l summation from the relation P / / pdxdy »»••....,..,.»«,...».,,.,»...,, ..»..».»».•....».. (28.) =  v, /a  The f r i c t i o n traction, from equation (17.), i s P sy^-logg, ^* per unit width, and log ^ 4 S i ' 1/  •py. _  = /  for the whole bearing,  A~  S f, '»  lo  (29. )  <  ?  A being i t s area = b ( a -a,).  A few numerical formulae are given below :The total pressure on a square bearing i n which  a -a,= a  = b is  ±  p , • 0669 JUU, A  2  being, by comparison with equation (15.) only 0*421 of the total pressure on a portion, equal i n area, length and inclination, of a plane of infinite width, thus showing the effect of the escape of o i l from the sides of the bearing. The position of the centre of pressure for the f i n i t e square bearing i s at a distance  0*42 a, from the rear edge, as compared with  0*431 a, i n the case of the infinite bearing. The coefficient of f r i c t i o n  £  i s 10*3 c.  Harrison has added much to the knowledge of the subject, simplifying the Reynolds Theory and developing one of his own.  His equations are  for a f u l l bearing and he makes certain assumptions which may be briefly set forth as follows :-  v  (l) ' •  In the equations of motions of the film the effects of gravity of inertia of the f l u i d may be neglected as compared with the internal stresses arising from the shearing of the lubricant. Dictionary f Applied Physics, p377 Q  Also, on account of the  28, the thinness of the film, i t s curvature may "be neglected, so that the same equations hold whether the surfaces are plane or cylindrical. As "before, letting any point i n the f l u i d and p  u  and v "be the component velocities at  the pressure at that point, then the equations  of motion are :— 7  ay  :  a  V  (i)  u  uW  (2)  where JJL = the coefficient of viscosity and V*= ^~*  +  ^(J*.;  and the equation of continuity of the f l u i d i s s-  3a + & = o dx  (3)  dy  The "boundary conditions are :~ u = U, v = 0, where y = 0 ) u=o,,.v=0 where  h  "  (4)  y =h  i s the variable distance between the surfaces and i s a function  of x. Since the surfaces are nearly parallel, pared with  u  and the rate of variation of u  v w i l l be small com-  i n the direction of x  w i l l be small compared with i t s rate of variation i n the direction of y. Therefore equations  in  (l) and (2) become :-  (e)  » o  ay From (6) we see that  p  i s independent of y  and so we are able to i n -  tegrate (5) giving us, on substitution of the limits, , - l / | y (y- h ) U +  Now from (3)  / s S dy = - ( v ) = 0 / dx 0 h  ...(7)  29. Therefore, substituting, i n the value of ^  , from. (7) and integrating,  ( i | g ) = 6 ^ 1 1 ^ ; i.e. h |g J 6^U ( h . h,)  we get  ...<8)  where h i s the value of h for which dx Pig. 16. represents a section through a journal and i t s bearing. 0  i s the centre of the journal, which i s of  radius ' a ' , and 0* i s the centre of the bearing,of radius 'a + n', and 00' = cn where o<1. Therefore the value of h = n( 1 + c cos 0 ) where © = POO'. Then writing x = a 0 » h = n( 1 + o cos 0, ) we get 7  dp, _  1  ,u U a^c ( cos e - cos 9.)  6  de  °'n*{  i + o cos 9 y-  6  (  ••••  )  W  On integrating we find :-  (l-c*+ S(l+c cose))) + ( l - c ' ) ~ ^ 2 ( l - c )-(l+c oose)(2+o*)] t a n ~ ( / l ^ ~ tan § ) } l+o \  (10)  1  Now, since ient  p  o f tan  can have only one value for any given value of ©, the coeffic_ 1  (/T-Q  tan ^) i s  and therefore ,  3c + (2+c) COB 0, = 0  (11)  This equation determines the points at which substituting this value o f 8, i n equation (10) we get:p = c • 6^ P  H a o Bin-9  (2 + c cos @)  = 0, and on de ( )  n'(2 + c*)( 1 + c cos e r  12  '  Prom this equation i t i s seen that the positions of maximum and minimum pressure are equidistant from the point of nearest approach, and that the one rises as much above the value of the pressure a±t that point as the f a l l s below i t .  The total vertical force on the journal, due to the pressure,  so. acts downwards through  0  and i s given by  B = / p sin e ad6 = I f ^ J i U a ^ c .  i  ( ) 15  Now the total force,due to the viscous drag on the surfaoe of the .journal, must, "by symmetry, act through 0' and i s given "by S = / f sin 0 a d 6. A  =  * •  8  • *  feu  h  (7  - JifcfLe)*>  >  t h e r e f o r e  (2  * - ^ <  2  - G ^ f e * >  •  <»>  There i s also exerted on the journal a couple which i s represented by  4»>itJa*(l+-2o*K . '6  Letting f  '  =  . '  ( } 15  f be the viscous drag on the surface of the bearing, then 1  "/^ t y ) h ~  \  y =  "^  h  )  f r o m  (7) therefore there must be  corresponding couples and forces R* aoting upwards through 0' on the bearing and these are equal to R, so ,. . A g B J U  < ) 16  M' . . ^ g . t V ^ * Now  S and  smaller order than  R  •  (  1  7  )  S* are not equal and opposite, but being of a  and B', may be neglected.  The inequality of M and  M* however, i s an essential characteristic of the equality of the force systems  (R M), (R M), R acting at 0 and R  1  at 0'.  The coefficient of f r i c t i o n for the joarnal i s given by :-  A =| = £-LL+ Ra 3 a  c  and that for the bearing by  =R'a X = I^t£l Sac  It should noted that results calculated from Harrisons theory are based on the assumptions that there i s a continuous film of lubricant around the journal, and that the pressure distribution i s symmetrical about a horizontal axis.  51. In the United States l i t t l e was done,in the way of a systematic study of the problems connected with lubrication, before the end of the last century.  Since that time however, many able men i n that country have,  by study, experiment and research, made valuable additions to the l i t e r a ture on the subject.  Among these are such men as  S. Timoshenko, R.C.Heck,  G.B.Karelitz, H.A.S.Howarth, P.E.Cardullo and E.O.Waters. In this short essay I can refer to the work of only one of these men, P. E. Cardullo, and to his only briefly. In his treatise on lubrication published i n the A. S. M. E., Cardullo went into the matter at much length and developed equations,using rectangular co-ordinates, but he seems to have ignored, more or less,  the  theory of viscous flow, although he has made use of ideas expressed by such men as Reynolds, Sommerfeld and Harrison.  He starts from f i r s t prin-  ciples and states that the foroe required to maintain a uniform velocity of  V inches per second of one layer of liquid of area S sq. in. at a  distance  T ins. away from a fixed layer, also  5*  — .  • ^  IYI  •  *  •  •  •  -  •  »  •  S •  »  »  sq.in, i n area i s •  •  •  •  •  •  »  •  •  •  •  )  T  where  K = the absolute viscosity of the liquid. FT TT —  f o\  '" '  ••  111,1  •«•» • • • • • • • • • • • » • • • \ & J  s v Pig.17. shows the oross section of a bearing of radius  r + o  with, revolving i n i t , a journal of radius  r , where  c  i s half the  clearance for a running f i t .  Point  i s the centre of the bearing, and i s the centre of the journal.  a  b  The  Pig.17.  journal and the bearing are not concentric, their centres being separated by the distance  (a ~b) which i s  . 32.  designated by  e c  where  c i s a peoper fraction and i s termed the eccen-  t r i c i t y ratio of the bearing.  The thickness of the film varies from point  to point around the bearing and this variable quantity i s indicated by the symbol  x. Assuming that the load aots vertically through the line  and that the line of centres  A B  also that the o i l i s supplied at  makes an angle  w  with the line  Z N Z Nj  G, the point of maximum thickness of the  film, then, as the journal revolves there w i l l be, i n the o i l , a tendency to revolve with the journal, and also a tendency to remain stationary with the bearing.  There i s thus set up a relative motion which continually  shears the o i l , and the velocity of any point i n the o i l film w i l l depend upon i t s relative distance from the surfaces of the moving journal and the stationary bearing.  At the point  G  and at the ends of the bearing the  average velocity of the o i l w i l l be one half that of the journal, and at a l l other points i n the o i l the velocity w i l l be less than one half that of the journal due to the fact that the pressures, which vary from point  to  point i n the o i l , tend to oppose or accelerate the rotation of the film of oil. To simplify -his theory Cardullo assumed that the velocity in the plane of rotation of any point i n the o i l film, i s proportional td> i t s distance from the surface of the bearing, anc'i that, at the surface of the journal, i t i s equal to the velocity of the journal; the average velocity of the film therefore being one half that of the journal. If we consider an element of the lubricant, distant 0 from the line  A B, thenits thickness i n a radial is  '  x = c ( 1 + e cos 6)  (3)  After the element has moved through an angular distance in the film thickness w i l l be :-  d 9 the change  d x = - e c sin 9 d 9 .......... .(4)  33. Then i f the length of the journal i s L and the width of the element i s d s i n each case, the change in volume w i l l he :-  (5)  dU = L dx ds = - L ds ec sin 0 dS  the negative sign indicating that the volume decreases as 6 increases. This change in volume represents the quantity of liquid that i s squeezed the ends of the hearing.  out at  This action results in an axial component of  flow which we shall now investigate. If 2 w i s assumed to he the angular velocity of the journal, in radians per second, then w w i l l he the angular of the lubricant. The time required for the element of lubricant, which has already been considered to move  through the angular distance  d6, w i l l be ~  .  Dividing the  w  change in the volume by twice the area of the element multiplied by the time, the mean value of the axial component of velocity with which the o i l flows from the end of the journal'iwill be found as follows :L ds ec sin9 dQ w (p.) dU d6 2c(l + e cos6)dsd9 2x ds w obtain for the mean axial velocity of the lubricant Simplifying this we at any point at the ends of the bearing :- =— L _ _e—sin0 Y * 1 + e cosG  /n\ \f)  w  Let us now consider the effect of this axial flow of the lubricant in producing a hydrostatic pressure sufficient to l i f t the journal. Fig.18  represents an axial section  of the lubricant, through the element that we have already considered.  A  very small portion of this section i s taken, namely, that portion bounded by  S and S', of axial length dz,  width, i n a circumferential direction ds  and thickness i n a radial direction  Fig. 18. dy. The difference in pressure  34, between the two ends of the elements i s dp. As the thickness of the element i s x, and the distance from the surface of the journal to the surface  S  of the element i s y, the distance from the  centre of the f i l m to the surface  S  of the element i s (  - y ). At the  centre of the f i l m the a x i a l v e l o c i t y i s a maximum,diminishing to zero at the surfaces of the journal and the hearing, and the a x i a l shearing force at the centre of the f i l m p a r a l l e l to  S  i s zero.  At  S  the shearing  force i s represented by  I = d p. d s ( | - y) _ ( 2L - y\ dp A  d s d z  A  Yo)  n • •" > dz  But,from equation ( l ) the shearing stress i s :A;  T  And therefore, substituting f o r  T  the value of dy, and f o r V  dv, which i s the difference i n v e l o c i t y between the surfaces F _ ir <3v Therefore  (  K d v = az  v  x  v = V when y = y K V = f§  By solving f o r V tant  y  and S', (a)  , >, dp r  ( ~ ~ y > dy 2 '  Then , i n t e g r a t i n g equation (lO) between the l i m i t s and  S  the value  (lO) of v = 0  we get -  when y = 0,  (11)  we are able to f i n d the a x i a l v e l o c i t y at any pointqdis-  from the bearing i . e .  —- P ( -Q- — *L ) ............................... (12) K dz * 2 In order to f i n d the mean v e l o c i t y from the centre of the f i l m to the surV  —  face o f e i t h e r the journal or the bearing, that Is from we write  y  =/ ^L_£ =  | / V dy  Substituting (12) i n (13) we get :-  y = 0  to y = x (15)  35.  A V  I /  s  ( 3SL  - X * \ av  (14.)  dp & "* Kxdz ./^ ( xydy — y dy)  ........................ (15)  So  Integrating and putting i n the l i m i t s we f i n d V  = Ixfe < Axaz  * V 4  =  3 ^  d  ?  1  2  X  K d  *  = - i ^  z  1  2  K  d  dp.  (16) V  z  Now equation (?) gives us the mean a x i a l v e l o c i t y of the element of the lubricant d i s t a n t ^ point  z  inches from the centre of the bearing, and so, at a  inches, from the edge of the bearing the quantity  ^ - z  may be  substituted f o r the quantity ^ and the following equation r e s u l t s :w e sin 9 (17) he cos b which gives the mean a x i a l v e l o c i t y of the lubricant a t any point on the surface o f the bearing between the elements of maximum and minimum clearance. Nov/ equating the equations c ( 1 + e cos 6 ) f o r x  (16) and (l7.) and substituting the quantity  we get  s  • s  c d p( 1,+ e cos e)  —— \f, v  .  a  '- = w c e s i n 6  v  Integrating between the l i m i t s of p = 0  / L \ , r^ \ ( £ - z ) dz .... (18) a  and p = p, z = 0 and  z =z  3 w  e  h  and  a  5  v  P c{  e  1 + e cos 6j _  ,  •  , >  solving t h i s f o r p we get  •  P = 6K 2 ^ e ^ E ^ L _ ^ ( Lz - z*) • c*( cosG/  ....(20)  v  e  This equation gives the value f o r the hydrostatic pressure cant a t any point distant angle 9 and d i s t a n t  p  of the l u b r i -  from the. element of maximum clearance,  z" from the nearest end of the bearing, the values of L,  e, K, w, 6, and c being as previously given. Mr. E. 0. Boswall, who, i n c o l l a b o r a t i o n with Mr. J . C. B r i e r ley, has written a paper on'Film Lubrication of the Journal Bearing' (*) deals . ( l ) Proc. Inst. Mech. Eng. 1932. Vol.122, pp 423  et seq.  36. with the matter i n a very different manner, making use of the "Method of Dimensions" and the "Principle of Similarity".  The advantages he claims for  this method are, f i r s t , that i t enables a functional relationship to be established between any given set of quantities by means of a dimensionally homogeneous equation which expresses any one of .the quantities as a function of the remainder; and; secondly, that i t can be applied for the purpose of indicating the connection between results obtained from models or small scale apparatus and the operating conditions for the full-scale machine. According to Mr. Boswall,the operating conditions, after the film has formed, w i l l depend upon :W  the total load transmitted by the journal,  U  the surface speed of the journal,  Z  the viscosity of the lubricant,  R  the radius of the journal,  r  the clearance between the journal and the bearing  B  the transverse width of the bearing,  Jtf ' the angle subtended by the journal d  the position of the line of action of the load from the inlet edge of the bearing,  S  the specific heat of the lubricant, and the viscosity-temperature characteristics of the lubricant The attitude of the journal with relation to the bearing w i l l  depend upon (a) the eccentricity, 'e' or distance between the centres of and .bearing journal^ and (b) the angular position *©• of the line of centres, measured from the inlet edge of theb'earing; and must be such that the resultant film pressure i s equal to the load and i t s line of action passes through a point at an angular distance 'd' from the inlet edge of the bearing and thus coincides  with the line of action of the load.  37. It i s assumed that the journal can take up a certaM definite position with relation to the bearing, and that the principal factors that determine that position are :-The load transmitted by the journal, the surface speed of the journal, the inlet viscosity of the lubricant, the dimensions of the bearing surfaces, the position of the line of action of the load, and, to a less extent,.the viscosity changes that occur in the film owing to the increase i n the temperature of the lubricant as i t flows between the bearing surfaces. In applying the "Method of Dimensions' a non-dimensional term indicated by '•(' i s introduced to represent the viscosity changes and the viscosity-temperatare conditions for the lubricant. Using the notation described, i t i s now possible to express the eccentricity, or journal displacement along the line of centres,tentatively, as:where  e = f ( W; Z, U, B, E, r, Jf, d, (-)  (l)  f denotes a function of the terms within the brackets. Applying dimensional methods to this, we find  that the function-  a l relationship between the eccentricity and the quantities upon which i t depends must be : —  e = R f, (c)  where (C) i s used to represent  (2)  (ZUB/W,B/R,r/R,^,d,(• ) i n which a l l the terms  are non-dimensional. This serves to show that the eccentricity i s directly proportional to R  and,so far as load, speed, viscosity and transverse width are  concerned, depends upon the value of the product ZUB/W, and upon the values of the terms B/R, r/R,  and d.  Dealing similarly with the angle  © which fixes the position of  the line of centres with respect to the inlet edge of the bearing, we find that, since the same set of known quantities i s involved :-  38. To f i n d the center of pressure: Since the l i n e s of action of load and resultant f i l m pressure must coincide, the centre of pressure w i l l he a t an angular distance 'd' from the i n l e t edge of the hearing, and, r e arranging equations.(2) and (3) i t w i l l he seen that l>y :-  .  or by :-  r/R, e/R, f,  .)•  d = f (ZUB/W, B/R, r/R, 0, Jf,  )  * = f (ZUB/W, B/R, a  6  *d' can he expressed  Combining these to eliminate the term ZUB/W we get :~  d = f (B/R, e  r/R, e/R, 0,  (> ).........  (4r)  Hence the p o s i t i o n of the centre o f pressure i s a function of the geometrical  shape of the f i l m and the manner i n which the v i s c o s i t y changes. Temperature conditions : Letting  M  denote the moment required  to drive the journal, then :-  IvI —' ^tlR. JT^ (0^•••••••••••e* and the work expended  • • • • • ••••••••••*•«• ( 5 )  per second, expressed i n energy units, i s :-  E = WU f ( c ) . . . . . .  (6)  5  This energy i s transmitted d i r e c t l y to the lubricant, and the r i s e i n temperature w i l l depend upon the quantity of lubricant passing through the clearance space and upon i t s s p e c i f i c heat. It w i l l now be assumed, therefore, that the s p e c i f i c heat term S the  represents volumetric heat expressed i n energy units i n order to avoid introduction of the mechanical equivalent of heat, i . e , S _  "k i n energy u n i t s per unit volume r i s e i n temperature This introduces an additional temperature unit, denoted by nea  and the dimensions f o r volumetric heat Letting  Q  w i l l take the form  M.L  T  t  represent the quantity of lubricant that passes  through the clearance space per second, we f i n d that:Q = URB f ^ C )  - . (7)  t,  39. and that i f the average rise of temperature of the o i l i s denoted "by t i t s heat content expressed i n energy units per second w i l l he :-  H — U R B St^  f4(c)•»••»•••••••.......»,....,,...,,(8)  Assuming that no appreciable, loosof heat takes place through conduction from the film to the bearing surfaces, equation (8) w i l l also represent the energy expended i n driving #he journal> and i t follows that :W/RBSt = _V(c) a  or  t  a  *~ R B S ^^  )  (9)  Frictional Resistance can be expressed by means of the usual coefficient of friction, or i n terms of the shear stress to which the lubricant is subjected as i t passes between the surfaces.  The coefficient of f r i c t i o n  i s given by:~~  V H  —  ^O^»««»««»««9o««909««c  • • • • • • • • • • • •  (_L0)  force The shear stress i s determined by the tangential/applied at the surface of the journal, and i s :F = jx W  or W f, (C)  (ll)  and, the area of the film being I B jl , the average intensity of shear stress on the lubricant at the journal surface w i l l be :F - jx W/R B $ or W/R B # f (c) a  y  (12)  and i s therefore directly proportional to the load. From the foregoing i t w i l l be seen difficulties  that there are mathematical  connected with a numerical solution of the problem* for we  have no information regarding the relation between the term  (r , which  represents the viscosity change conditions, and the known operating conditions, and such information can be gained  only by making some suitable assumption.  By rearrangement of equation (9) however, i t is possible to express the unknown term  as :- { = f (W/RBSt,ZUB/W,B/R,r/R,#,d).. (l3) a  and this makes i t possible to express the various operating conditions, i n cluding the average temperature rise, i n the terms of thefunction :-  ;•-  e e M Q u  = R f, (D) = f*(D.) — WR f $ = URB f (D) = f,(D) v  ( ) \ ( )  f 14:)  Where (D) i s used to represent ( Z U B/W, W/R B St ,B/R, 0, r/R, d ). a  The terra W/R B Sta appears here because of i t s association with temperature conditions.. It may therefore be inferred that, although direct load has an influence/upon the shear stress to which the lubricant i s subjected, i t has influence of only secondary character upon journal attitude and coefficient of friction, these conditions being determined principally by the magnitude of the term Z U B/W. Using angular, instead of surfacei speed somewhat simplifies matters, and i t i s also convenient to represent W/RB by the symbol P. Therefore, i f the rate of rotation i s denoted by N, the term become  Z N/P  Z U B/W  will  and the function (D) w i l l take the form ( Z N/P, P/Sta, B/R, r/R,  d)  This form for the function (D) applies only to clearance bearings.  In the case of bedded bearings r = 0 and the geometrical term r/R  w i l l disappear.  41.  Part 4.— Tests for Pressure and Friction.  I shall now describe some tests that I have made with a view to learn something of the distribution of pressure i n the o i l film, and of the manner in which the coefficient of f r i c t i o n changes under varying conditions of load,temperature and viscosity. The machine used i n these tests i s described by i t s makers the Tinius Olsen Testing Machine Company, of Philadelphia, Pa., as the Latest Improved O i l Testing Machine.  This machine i s designed as a form of dyna-  mometer which provides a means of measuring frictional resistance directly. The journal i s 3f ins. in. diameter by  ins. long and the bearing block,  which i s on the upper portion of the journal, i s 2 ins. wide.  The load i s  applied, and adjusted, by means of a worm screw and a system of levers connected with the loading yoke at the top of the machine, and i s indicated on a dial.  The bearing block supports the loading yoke on a knife-edge, and _  therefore the load i s not a distributed load. At both ends of the lower portion of the loading yoke are knife-edges from which depends the system of levers already mentioned. A balance lever, calibrated i n pounds, and carrying a sliding weight, i s attached at right angles to the loading yoke, and as the yoke i s free to move, on i t s knife-edges, either in, or against, the direction of motion of the journal, the friction that causes such movement i s weighed on the balance arm.  The reading, in pounds, on the balance arm,  divided by the total load on the bearing w i l l give the coefficient of f r i c tion for an o i l under known conditions of load, speed and temperature. The machine was designed originally for testing lubricants for f r i c t i o n , and bearing metals for durability; and, as I shall shortly describe, I found i t necessary to make certain alterations in i t before  •  • • •  '  '  • • 42.  s a t i s f a c t o r y tests of o i l f i l m pressure and f r i c t i o n were possible. The f a c t of the existence of pressure i n lubricants between bearings and journals was discovered accidentally by Mr. Beauchamp, i n the year 1883, while he was making f r i c t i o n tests with a p a r t i a l bearing. With a view of attaching a lubricator, he had d r i l l e d a hole v e r t i c a l l y  through  the bearing block, and, on replacing the block on the journal, without the l u b r i c a t o r attached, he observed,that,with h i s machine i n operation, o i l flowed through the hole. tained a reading of bearing was only  Connecting a pressure guage to the hole he  ob-  200 l b s . per sq. i n . while the average load on the  100 l b s , per sq. i n .  My experiments were made with a view to a s c e r t a i n the pressure d i s t r i b u t i o n throughout  the bearing surface.  Experiments were at f i r s t  made with the bearing block that was supplied with the machine, but without s a t i s f a c t o r y r e s u l t s as, although the machine would run f r e e l y under a quarter load (1250 l b s . ) , the bearing would seize i f that load were exceeded. O i l grooves were then cut i n the bearing and, although these improved matters to the extent that f u l l load operation of the machine became possible, such operation was not suitable f o r my purpose, i n that the l u b r i c a t i o n obtained d i d not emerge beyond boundary conditions.  I t was then concluded  that the point of admission of the o i l , which point was only h a l f an inch from the centre l i n e of the bearing, was too l a t e to permit of the  forma-  tion of a wedge of o i l s u f f i c i e n t to l i f t the bearing o f f the journal. I therefore had an e n t i r e l y new bearing bearing block made, with the same bearing surface area as before, but with a projecting l i p on  the  i n l e t side and,above the l i p , an o i l well with an outlet to the under side of the l i p , f i n . from the i n l e t edge of the bearing. A-portion of the under side of the p r o j e c t i n g l i p was  scraped away to provide a space between the  •.'43. '  l i p and the journal, which space, being kept f i l l e d with o i l from the well, would provide for the journal what vrauld amount practically to a bath of o i l , A hearing surface, as nearly as possible perfect, was then obtained by extreme care i n the application of the usual methods. Fig.19  shows a cross section view  of this bearing block set i n position on the journal. From this i t w i l l be seen that,in addition to the features I have described, provision was made for water-cooling of the block by a flow of water through two Fig. 19.  holes d r i l l e d longitudinally through the block. The holes drilled  Oil  through the block for the purpose of making  Ocst  tests  for pressure were made with a Ho.60 d r i l l , and a'plan of these holes i s shown i n Fig. 20 which  e.  shows the bearing surface of the block divid-  Fig.20.  ed equally both lengthwise and' crosswise by the lines lines on each side of the line  X - X  X - X  and Y -Y.  Two  and parallel thereto, and three lines  to the right of line Y - Y and parallel to that line, give at their intersections twenty points for test holes arranged i n four rows,1, 2, 3, 4, across the bearing, and five rows, A, B, G, D,& E, lengthwise. It w i l l be seen that only one half of the bearing was used for test holes, i t being assumed that the distribution of pressure in the film would be the same on  44.  "both sides of the centre of the bearing.  I may also explain that the twenty  test holes were not a l l made at once, but i n sets of four as required. At the upper surface of the bearing block the test holes were enlarged and tapped for copper tubing connections with a header provided with eight needle valves and connected with three guages for high, medium and low pressure readings. My f i r s t tests with the aparatus as I have described were s t i l l far from satisfactory. 2500 lbs.,  It was observed that, with a loading i n excess of  the bearing ran hot i n spite of the water cooling. This was found  to be due to the fact that with a partial bearing the rotating journal would carry i n under the bearing with the o i l some a i r which, finding an outlet through the o i l inlet, would prevent an adequate supply of o i l from ing the bearing.  reach-  This defect was rectified by tapping the o i l inlet hole  and connecting i t by means of flexible tubing with an o i l resevoir (in which o i l was maintained at a constant level) at an elevation just sufficient  to  overcome the a i r pressure and to insure an abundant supply of o i l to the bearing.  On the next page w i l l be found two photographs of the machine,  equipped as I have described above, one taken from the drive side and the other from the opposite side. With the machine thus further improved I now made a test, for pressure in the o i l film, at each hole i n the *C line i n turn.  I f i r s t ap-  plied a quarter load (1250 lbs.) and took readings of the load, R. P. M., inlet o i l temperature, inlet water temperature, outlet water temperature and o i l film pressure i n Ins. per sq. in. Similar tests were made under  half,  three-quarter and f u l l loads, but other conditions were kept constant. These tests were a l l made with Penzoil o i l of rating S.A..E. 60, and were repeated with the same brand of o i l of rating 40 and again with the same brand  of  rating 20. When tests with this set of holes were completed, the holes  46.' were plugged at the surface of the hearing, a new set of holes was drilled and the bearing was again carefully surfaced.  Tests similar to those just  described were then made with this set of holes.  This process was repeated  again and again u n t i l a l l the sets of holes had been tested, the oils used throughout being the same as those used i n the f i r s t tests. After these tests for film pressure had been made, friction tests were carried out, with the same oils, and,as nearly as possible, under identi c a l conditions of load, speed,temperatures of cooling water,and inlet o i l temperature, the readings for f r i c t i o n being taken from the balance arm scale of the machine as described on page 41. For f r i c t i o n under other conditions of operation a further set of tests was carried out.  For this purpose a Thermo-Gouple of chromel and alu-  mel wires was made, passed through holes i n the bearing block and brazed to the bearing surface which was then recovered again in the manner above described. The  Thermo-Couple was then connected, through a cold junction placed  i n a jar of ice, with a Potentiometer and calibrated through a range of temperatures from  75 F, to 116 P.  This set of tests was made (a) With speed and o i l film temperature constant and the load varied, and (b) with speed and load constant and the temperature of the o i l film varied. The object of the (a) tests was to ' ,o'f •• •• .• ascertain the effect/change of load on the coefficient of friction, and that of the (b) tests to find the effect .on the coefficient of friction of change of temperature in  the o i l film  In addition to the tests above described, the same three oils were tested for changes of viscosity .through a range of temperatures from 100  P. to 210 F., the tests being made with a Saybolt Viscosimeter. The readings taken during the making of a l l the tests are listed  below, and the curves plotted from these readings are shown i n the drawings  47. and blueprints that follow. Film Pressure Tests. Oil Load. lbs.  Speed. r.p.m.  1250 2500 3750 5000  445  1250 2500 3750 5000  Temperatures Row. Oil in. Water in. Water out. 72 F,  tt  tt  w  : It  l» . "'  it ti  43  tl  1250 2500 3750 5000  tt  ti  it  n tt  ' tt  it  •70 • it  It  it  it  60  • ; .11 '  46 P.  tt  it  1250 2500 3750 5000  1250 2500 3750 5000  ..Penzoil S. A. E." 60,  43.5 it  tt  it  it  It  tt  it  it  76 it  w  II  it  it  It  it  it  It  ts  It ,-H: • ' '  tt  1250 2500 3750 5000  it ii  '  tt  it it  74 it  it  tt  tt  1250 2500 3750 5000  tt  63 M  1250 2500 3750 5000  it  1250 2500 3750 5000  tt  it tt  tt  tt  tt  tt  46  72  II  tt  43 tt tt ti  45.5 ti  it  tt  tt  ti  it  it  it  ti  76  it  is  it  it  it  it  it  tt  it  «••  A. tt tt tt  B,  , .43.5 /.44 45 46 47,5 48.5 50 47 48 48.5 48 48 49 50 51  (Penzoil S. A. E. 40) • 48 50 .'.'.52. i 54 ,  tl  66  48 P. 50 52 54  it  tt  •  ts  C . II  it it  D. it II  it  E. ti  it it  A. '» "  44.5 45.5 47 48 •  B.  44 45.5 47 48.5  0.  44 45 46 47  D.  it  it ii  it it  it  ti  it II  Pressure Holes. 2. 1. 4. . 3. 236 558 875 1200  195 355 530 700  170 220 265 320  145 175 205 250  700 525 500 1500 1300 1100 2325 2125 1500 3150 2950 1900  300 350 405 450  370 415 505 560  200 280 370 450  500 1150 1700 2510  370 475 750 950 1475 1130 1950 1500  170 235 270 305  500 825 1150 1500  300 200 650 600 1000 950 1350 1200  100 150 195 270  770 640 1180 990 1550 1320 1915 1640  230 190 545 340 875 520 1150 690  175 225 270 325  150 195 200 240  700 1550 2400 3225  550 650 1050 1450 2500 1550 3100 2050  290 345 400 455  740 1020 1340 1600  600 820 1060 1250  290 405 500 610  160 240 320 410  550 1190 1725 2300  380 500 770 1000 1500 1160 1950 1500  170 255 270 305  48.  Loa d. lbs.  Speed. r.p.m.  (Penzoil S.A.E.40 continued).. Temperatures. Row. Pressure Holes. Oilin. Water in. Water out 2. 1. 3. 4..  1250 2500 3750 5000  445  72 P.  1250 2500 3750 5000 1250 2500 3750 5000 1250 2500 3750 5000  II  u  •• • It If  ti it  II It li II . IS Ml II II II II II  1250 2500 3750 5000  (Penzoil S.A. E. 20) 46 • 48 50 iII> 52 ti 54  63  43  ii  . ••  it  72  43.5  it  ti n  it  it it  76  tl  it  II  ti  II u  ti ti  II  II ' II II II  tt tt  it II  w  1250 2500 3750 5000  II II  ti  • • . • II.'  48 p.: 49 50 51  II  73  II  46 P.  72  46  ti it  tt  it ii  it  E.  it it  II  A.  it  ti  tt  44.5 46 47 48  B.  45 47 49 51  c.  1! 11  H tt  ""  II It  44: 46 47 48  D.  48 49 50  E.  it it II  it  ••SI/';"'  it ti  500 300 175 850 640 575 1150 1010 900 1525 1300 1175  90 150 215 275  233 205 180 550 350 225 875 525 275 1205 725 325  148 180 205 250  650 1450 2290 3050  550 500 1450- 1175 2275 1700 2950 2200  295 350 410 450  760 990 1245 1490  660 870 1085 1250  320 480 630 820  160 240 330 410  525 510 390 1100 975 760 1650 1475 1140 2200 1900 1490  150 210 250 290  525 800 1100 1550  105 155 210 260  275 625 1000 1325  225 610 925 1180  Friction Tests. Penzoil S. A. E. 60. Load. lbs. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000  Speed. r.p.nu v .445 it ti. • V  it II  u  it it it  '' . Ml . It  Friction Temperature Water in. Water out, Force. 7.75 lbs. 47 F. 46 P. <3£3©  Oil in. 62 P. ti 18  II tt II  It II It It II  n  it '•-  tt  it it ti II  it II H  47.5 48 48.5  it it II  48. 75 49 ti 49. 5  8. 975 13.42 16.47 18.96  18.96 19.15 21.85 23.13 24.85 25.41  Coeff. o Pricti  -  .01795 .01342 .0109 .00948 .0076 .00658 .00621 .00578 .00552 .00508  49.  Friction Tests, Continued. Penzoil S.A.E. 40. Load. lbs. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 •500 1000 1500 2000 2500 3000 5500 4000 4500 5000  Load. lbs. 0 1000 2000 3000 4000 5000  Speed. r. p.m. 445  Oil in. 72 F.  " •i.M""-.:;-' 'i  it  tt  1!  ti  11  tt  It  n  II  it  It  it  II  tt • 'tt  It ,• . .19.'  it  II  •it  It  tt  II  it  II  Temperatures. Friction Water in. Water out. Force. 46 F. 47 F. 7,42 lbs. ti 47. 5 8.11 1! 48 10.8 •if. . 48.5 12,05 ii 13.05 • *".' 49 it 49.5 13.55 it 50 14.04 tt 50. 5 14.55 it 51 14.96 it 51* 5 13. 75 it 52 16.35  It' tt  11  11  It  It  !»  tt  It H  IS  til  tl  it  Speed. r.p.m. 445 II  it tt ti it.  0 1000 2000 3000 4000 5000  It  0 1000 2000 3000 4000 5000  tt  ^  Penzoil S.A.E. 20. 46 F. 47 F. 4.5 tt 47.5 5.075 it 6.89 48 it 7.575 48.5 tt 49 7.9 it 7.9 49.5 II. '•. 49.5 8.3 50 8.72 50 9.5 ti 10.02 50 it 10.675 51 '«.'••  Penzoil S.A.E,60. Oil in. Oil Film Temperature. Millivolts. Fahr. 104 1.5 71 F, • 104 n 104 1.5 tt 104 1.5 ts 104 1.5 It 104 1.5 72.5  (Penzoil S.A.E. 40) 79 F. .86 ii  it  II  ti  II  it  it  II  tt  it  t«  ti '  it  tl  it  8 II  tl  71  it tt  (Penzoil S.A.E. 20) 79 F. .86 ti n.. •  tl  tt  II  ti  It  ti  It  tt  It  It  it  It .• . ••  11  it  tl  Friction Force. 8.05 11.12 14.9675 21.63 28. 75 34, 975  Coeff, of Frlctic .0162 ,0108 .00803 .•00653 .00542 •. 00468 .00416 .00374 .60306 .00327  .01015 .00689 .00505 .00395 .00316 .00277 .00249 .00237 .00229 .00214  Coeff of Fricti ••  .  .01112 ,00748 .00721 .0071875 .006995  5.75 10 12.55 15.0575 19.562 20.75  .01000 .008275 .0050191 .0048905 .00415  3.175 4 8 12 16.135 20  . 004 .004 .004 .0040339 .004  • -  -  50.  Load. lbs. 1250 M  Speed. r.p.m. 445 it  tt  1! •ti  ti  t": tt It  li  2500  II  II  II  ti  • II  it  tt  tt  it «  Friction Tests, Continued. Penzoil S.&.E. 60 Oil i n . Oil film Temperature. Friction Millivolts. Pahr. force. 71 P. .84 77.5 13 it •1.0 85 11.5 II 1.25 96 9.675 ti 1.5 104 8. 375 it 1,75 111 7.3 it 2.0 115 5.5  it  •«'••'  II  tt  3750  II  tt  . "  ti  «  tt  it  II  it  tt  it  tt  ti  It  it  II  II  1250 ti  •  II'.  It  - '.•«'•'  tt  78 P. li  »  II  it  II  It  It  it  II  tt  II  tt tt  It  it  it  It  tl  II  II  tt  it  II  it  2500  " it • .•  It  If  /.•ti _/  tl  It  3750  It  tt  It  M  it  tt  II  it.  It  II  tt  It  It  tt  II  II  1250  u  84 P.  it  it  • tt  it  tt  it  n  it  tt  it  it  it  n  t»  .95  Ooeff.of Friction. .0104 .00921 .00774 .0067 .00584 .0044  1.5 1.75 2.0 2.15  83 96 104 111 115 117  17.2 _L3« 325 10.04 8. 72 •6. 25 4.6  • 00688 • 00534 .00401 .00348 .0025 .00184  1.14 1.5 1.3 1.75 2 2« 25  92 104 98 111 115 119  17.4 13.55 15.45 10.435 7.0 5.0  .00464 .00361 .00412 •. 00286 .001865 .001332  10 8.45 5.25 4r« 25 3« 3 2.25  ,008 .00676 .0042i ; .0037 .00264 .0018  74.5 85 . 96 104 111 _L__.2e 5  13.0 11.0 8.45 5.625 3. 875 2.5  .0052 .0044 .00338 .00225 .00155 .001  80: 94 104 111 115 119  16. 25 13.75 9.8 7.3 5.875 5.0  .00433 .00367 .00261 .001945 .001567 .001332  7.6 6.0 4.55 3.1 2,4 1.55  .00608 ,0048 .00364 .00248 .00192 .00124  1. 25  Penzoil S.A . £ . 40 . 65 67, 5 .9 80 92.5 1.15 98 1. 3 104 1.5 107 1.6 . 78 1.0 1,25 1.5 1,75 ••1.85 .9 1.2 1.5 1. 75 2.0 2* H  Penzoil S.A.E, 20 65 .6 78 .85 87 1.05 91 IL» 2L_L5 96 1.25 101.5 1.4  51. Load. lbs. 2500 n  Speed*  Oil in.  445 :  «  84 tt  '  tt  tt  it  tt;  It  it  t! '  It  tt  tt  It;  .' '•"  .tt  It  tt  »  tt  tt  It  It  3750 'tt''"--''-;;''.'  •• tt t!  It  .6 .85 1.05  it  65 : 78 87 93 98 103  •.:';/«•  • • .  Penzoil S.A.E» 20. Oil Film Temperature. Millivolts. Fa.hr. .6 65 .85 78 1.05 • 87 1.15 93 98 IL© 2 * 1.45 103  1® 3 1.45  tt  Friction Force. 9.25 7.3 K  Coeff. of Friction. .0037 .00292 .002.,. .00152 .0008 ..00064  C\  •3.8 2. 0 : 1. 6 10. 65 8.9 7.7 6.85  .00284 .00237 .00205 .001825 .001532 .00107  4.15  Calibration of Thermo—Couple. Millivolts .75 .91 1.05 1.12 1.305 1.43 1.67 1.83 2.0  =  Fahrenheit 73 81 87 91 97.5 102 109 112 116  Viscosity Bests.' Temperature. Fahrenheit.  Time i n seconds. S.A.E. 60. S..A.E. 40. S.A.E. 20.  100  1100  605  387  130  425  280  194  160  235  159  102  . 200  105  90  65  100  81  58  —o|o—  2» cl»  54.  Oil  Penzoil, S. A. E. 60  Load....3750 lbs. Scale...1 i n . v e r t i c a l = 510 lbs.  2.b.  55.  Oil....Penzoil, S. A. E. 60. Load...3750 l b s .  56.  3. a.  Oil.... Penzoil, S. A. E. 60. Load....2500 lbs. Scale...1 in. vertical = 510 lbs. Bearing twice f u l l size.  57.  •3* 13«  Oil  Penzoil, S. A . E. 60.  Load.... 25-00 lbs. Scale...1 in.vertical = 510 lbs. Bearing twice f u l l size.  o  4.  a.  58.  Oil....Penzoil, S. A . E. 60. Load...1250 l b s . Scale.. 1 i n . v e r t i c a l = 510 lbs. Bearing twice f u l l  size.  4.b.  59.  Oil  Penzoil, S. A. E. 60.  Load....1250 l b s . Scale...1 i n . V e r t i c a l = 510 l b s . Bearing twice f u l l size.  3. a.  64.  Oil  Penzoil, S. A. E. 40.  Load....2500 l b s . Scale...1 i n . v e r t i c a l = 510 l b s . Bearing twice f u l l  size.  3*  D«  65.  Oil  Penzoil, S. A. E. 40.  Load....2500 lbs. Scale...1 in. vertical = 510 lbs. Bearing twice f u l l size.  4 • 3.»  66.  Oil  Penzoil, S. A. 3. 40.  Load....1250 l b s . Scale...1 i n . V e r t i c a l = 510. l b s . Bearing twice f u l l  size.  4.b.  67.  Oil  Penzoil, S . A. E. 40.  Load....1250 l b s . Scale...1 i n . v e r t i c a l = 510.lbs. Bearing twice f u l l  size.  3. a.  72.  Oil  Penzoil, S. A. E. 20.  Load....2500 lbs. Scale...1 in. vertical = 510 lbs. Bearing twice f u l l size.  3.13.  73.  Oil  Penzoil, S. A. E. 20.  Load....2500 lbs. Scale...1 in. vertical = 510 lbs. Bearing twice f u l l  4. a.  74.  Oil  Penzoil, S. A. E. 20.  Load....1250 l b s . Scale...1 i n . v e r t i c a l = 510 l b s . Bearing twice f u l l  size.  4.b.  75.  Oil  Penzoil, S. A. E. 20.  Load....1250 lbs. Scale...1 in. vertical = 510 lbs. Bearing twice f u l l size.  I  87.  Part 5.-- Conclusion.  In this concluding part of my essay X.;.shall offer a few explanatory remarks i n respect of the results of my experiments and tests,as given i n the l i s t s of readings taken and the curves plotted therefrom; state the conclusions that,II -consider, are justified by these results; and show wheri n some of my conclusions are at variance with assumptions made by certain authorities. Film Pressure.  Prom my readings and curves i t w i l l be seen that  the point of maximum pressure i n the o i l film l i e s on a line that bisects the bearing transversely, but somewhat to the outlet side of a line that bisects i t longitudinally, i.e. at, or near, Hole l.B., this being the point of the greatest accumulated wedge action of the o i l .  Reference to Curves l a . ,  which are plotted isometrically, and are characteristic of a l l the longitudinal .curves, will show that there i s a decrease of pressure from 3150 lbs. per sq.in. at Hole l.B. to 450 lbs. per sq.in. at Hole 4.B. and a further decrease to zero at the side of the bearing.  Prom such a decrease of pressure  (27001bs.per sq.in.) i t must follow that there i s a considerable velocity i n the o i l film i n that direction. This proves that there i s flow i n two direcrotation of tionsj(a)in the direction of/ the journal; and (b) i n a direction at right angles to the direction of rotation of the journal. If pressure 6urves 1 a. and l b . were combined to give a surface of pressure i t would be seen that, from the point of greatest pressure, lines of decreasing pressure would iate i n a l l directions. It does not necessarily follow that there  rad-  i s flow  in a l l directions, but only i n those directions that are not contrary to the direction of motion of the journal.  This evidence that  there are aide  flows  i n more than/two directions i n the o i l film, and that these flows have considerable velocity, seems to me to be sufficient to show that Reynolds, Sommerfeld, and Harrison were hardly justified, when-theorising regarding . infinite bearings, i n assuming flow i n the direction of rotation only, and, that Michell s theory - a continuation of that of Reynolds - while more nearly 1  correct because he applies i t to a f i n i t e bearing and because he takes .side flow into consideration, would not work out exactly with actual bearings for the reason that he neglects the effects of flow in a l l directions but two,viz. the direction of rotation of the journal, and the direction at right angles to the-direction of rotation. Mr. Cardullo, i n his work on Film Lubrication explains at some length that the average velocity of the o i l film i n the bearing i s less than one-half that of the journal. He assumes (see p.52) that at the point where there i s maximum film thickness and at the ends of the bearing the average circumferential velocity of the o i l i s one-half that of the journal, and at a l l other points less than one-half.  If this be so, the average velocity  through-out the film must be much less than one-half that of the journal, and this suggests a question whether he i s justified i n assuming, as he does,for a basis of his calculations, an average o i l film velocity equal to one-half the journal velocity. Also, i n his mathematical theory, Mr. Cardullo makes is an ssumption that/opposite to that made by Reynolds, Sommerfield, and Harrison, for, while those writers developed their theories on the assumption of flow i n the direction of rotation only, he bases his calculations entirely on the assumption of flow i n the direction at right angles to the direction of motion. Mr. R. 0. Boswall states ( p. 427, Proc. Inst. Mech. Eng. 1932) tfhaVthe- magnitude of a pressure film w i l l depend on the viscosity of the of the lubricant,the speed of the journal surface, the thickness of the film.  . .  • •  and the relative inclination of the bearing surfaces.  ••89.  In my tests, however,  I found that, i n only one set of holes out of five was there any appreciable difference i n o i l film pressure when oils of different viscosities were used. There was a marked decrease of pressure when,"by shutting off the cooling water, the bearing was allowed to become heated, and this leads me to suppose \ that Mr. Boswall' s  mention of viscosity as one of the factors governing o i l  film pressure magnitude, may have been intended to apply to changes i n viscosity of the same o i l resulting from temperature changes, and this ularly as he does not include load  partic-  i n his l i s t of governing factors.  If the diagrams of pressure curves at right angles to the direction of motion be examined, i t w i l l be seen that a l l the curves are roughly parabolic i n shape but rise rather sharply i n the section representing the centre portion of the bearing, showing that the load acts mainly at that portion. This i s because the load i s a concentrated,and not a distributed load.  If the load were evenly distributed over thevhole bearing the curves  obtained would be parabolic. In the diagrams of pressure curves i n the direction of motion i t w i l l be noticed that the curves rise to high points on the inlet side of the • centre,then drop slightly to the centre; this drop being followed immediately by a sharp rise to the peak on the outlet side of the centre.  The f a l l to  the centre may be attributable either to a deflection of the journal due to the central loading, or to an imperfection in the bearing surface along the line bisecting the bearing longitudinally.  I am inclined to attribute the  irregularity to the f i r s t suggested cause for the reasons that, with one quarter loading the curves do not show this drop, and that the curves No. 4 holes do not show the drop even under f u l l loading. A l l the curves show that the film pressure varies directly as the load.  for  ^• /'-! V^'^:^v^':^:'' - '':y'V V'v .v-v-v: r  v  :  i;;  Friction Tests.  ;  ];••'•  :>•  ..V"  V - • /  ^90. ' • :  From the curves showing the manner i n which  the coefficient of friction varies with the load, under,conditions  of oper-  ation approximating those of the pressure tests, i t w i l l be seen that the heaviest o i l -S. A. E. 60 -ogives the highest coefficient of friction, while the o i l No.40 comes next, and No.20 gives the lowest coefficient of friction. It w i l l also be seen that with a l l three the coefficient decreases rapidly with increased loads up to  3000 lbs., but that from that point of loading  upwards the curves show a tendency to flatten out.  The decrease in the co-  efficient of friction with increase of load i s a natural result, for i n measuring the friction i n o i l films i t i s the internal friction of the o i l that i s measured,that i s to say, i t s viscosity, and increased loads cause increased temperatures with corresponding decreased viscosities, and hencS decreased coefficient of friction. Another effect of increased load is to lessen the thickness of the o i l film and consequently i t s internal friction w i l l also be less. In the diagrams showing the results of friction tests made under constant o i l film temperature and varying load conditions, i t will be observed that with the two heavier oils the coefficient of friction decreases  very  rapidly under loads up to 2000 lbs.,but,that with further increase of load the friction curve tends to become very nearly horizontal. This also i s natur a l since, with temperature constant,viscosity i s also constant, and we are here measuring changes i n the internal resistance in the film resulting from changes in i t s thickness due to loading.  With the light o i l , however, there  i s no perceptible decrease i n the coefficient of friction with an increase of load. This for the reason that,with this o i l , the film produced i s very thin even under a load as small as 500 lbs., and additional loading w i l l not cause appreciable reduction in the thickness of the film until an excessive load i s applied when the film w i l l break down entirely.  91.  In the curves plotted from readings taken under the third set of conditions, namely, constant load and varied temperatures, straight lines are, more or less,nearly approached, 65 P. to  This i s "because the range of temperatures,.  120 P. within which the readings were taken, coincides with the  range of temperatures within which the readings for viscosity, when plotted, gave lines approximately straight. In conclusion I wish to say that,.' in commenting, as I have above, on some of the assumptions made and conclusions drawn "by certain writers on the subject, I had no intention of implying that their knowledge i s deficient or that my own i s complete, but merely a desire to show that much investigation remains to be made into the matter of film lubrication before its complete emergence from the realm of conjecture can be regarded as accomplished.  —ooOoo—  

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