VL.1 -ÜNDB
CoA Report A e r o . 184
/ 2 JAN.
ÜELF
THE COLLEGE OF A E R O N A U T I C S
C R A N F I E L D
MINIMUM HYDRODYNAMIC OILFILM THICKNESS:
AN EXPERIMENTAL AND THEORETICAL INVESTIGATION
by
CRAIIFIELD
Minimum hydrod^Tiamic oilfilm thickness: an experimental and theoretical investigation
CORRIGEHDA
Page 5^ second paragraph, line 9 should
read:-trace indicating surcession of wear. A hydrodynamic film is therefore created and
Figure 3{t>):
CoA Report Aero. 184 August, 1965.
THE COLLEGE OF AERONAUTICS
CRANFIELD
Minimum hydrodynamic oilfilm thickness: an experimental and theoretical investigation
by
B. R. Reason,* B.Sc.(Eng.) A . C . G . I .
SUMMARY
The paper deals witK an experimental and analytical investigation of the lowest limit of hydrodynanriic film thickness compatible with the condition of 'running in'.
Using the working geometry of an experimental test rig together with the two dimensional Reynolds equation., an analytical expression for the minimum hydro-dynamic film thickness has been developed.
The problem has been investigated from an experimental standpoint using a measuring system based on an air gauge and capable of detecting film thickness changes of the order of 10"* inches. Although calculations of the magnitude of the minimum hydrodynamic oilfilm thickness gave values as low as 5 x 10"* inches, the minimum value obtained experimentally was 2.5 x 1 0 " ' inches.
List of symbols 1. Introduction 2. Test apparatus
Test pads Loading system
Wear measuring system Drive unit and oil supply 3. Outline of procedure
4. Results
5. Analytical derivation of minimum hydrodynamic Derivation of an expression for 'm'
Calculation of 'm' Values of 'hm'
6. Discussion and comments
Effect of the value of 'm' on the magnitude of 'hm' The influence of 'm' on the load carried
Comparison between measured and derived values of 'hm' Effective pad movement . .
Axial p r e s s u r e profile 7. References 8. Figures 1 1 1 2 2 2 2 3 oilfilm thickness 4 6 7 8 8 8 9 9 10 10 11
LIST OF SYMBOLS b ,h ha h m ho I m •r^*t m^ X, y, z. A . B . P R U W B X, Y. a P HB X b e a r i n g width l o c a l oilfilm t h i c k n e s s m e a n oilfilm t h i c k n e s s m i n i m u m oilfilm t h i c k n e s s oilfilm t h i c k n e s s at inlet b e a r i n g length
' ° / h m
[ 1 TLO^T m 2(m - 1) Ij 1^^ _ ^j j^Logem ^ + 1 JJ r e c t a n g u l a r c o o r d i n a t e s t e s t r i g c o n s t a n t s l o c a l fluid p r e s s u r e t e s t shaft r a d i u s p e r p h e r a l v e l o c i t y of t e s t shaft b e a r i n g l o a d / u n i t width w e a r s c a r c o o r d i n a t e s *'(A + Y)'' + (B + X ) * ^''(A - Y f + (B + X)» v i s c o s i t y of l u b r i c a n t in Reyns a i r gauge r e a d i n g ( i n s . ) ( i n s . ) ( i n s . ) ( i n s . ) ( i n s . ) ( i n s . ) ( i n s . ) ( i n s . ) (lbs/in«) ( i n s . ) ( i n s / s e c . ) ( l b s / i n . ) ( i n s . ) ( i n s . ) ( i n s . ) ( l b s . s e c / i n ' ) ( i n s . )1. Introduction
The existence of extremely thin oil films between moving surfaces of machine components has long been an established fact. These films, which may be of the
order of 10"* inches or l e s s , are nevertheless exceedingly tenacious, giving, in certain circumstances, hydrodynamic lubrication with complete separation of the surfaces.
The basic requirement for an oil film to have load carrying capacity by hydro-dynamic action, is that the bearing surfaces should be capable of providing resistance to the flow of lubricant between them. It is the fluid's reaction to this resistance to flow which provides the necessary force component to oppose the applied load in both magnitude and direction.
The geometrical contact surface configurations giving r i s e to this condition may vary in form according to the nature of the bearing environment. They may occur fortuitously as, for example, in roller bearings where the contact surfaces are p r a c tically parallel (apart from a sudden constriction in the exit region) or may be a r t i -ficially 'built in' as in the case of plain journal and slider bearings where the. surfaces are designed to form a convergent wedge under dynamic conditions.
In all bearing applications, however, there is one point in the contact surface geometry where the oil film thickness is a minimum. The question of the likely niag-nitude of the oil film thickness at this point is a very pertinent one, for, apart from considerations of elastic deflections, thermal distortions or surface finish, which might engender a condition of metal-to-metal contact with attendant wear, there a r i s e s the consideration of the maximum permissible size of foreign particles. These particles occur as a contaminant in the lubricant during the normal process of wear, particularly when 'running in'. Since some of these particles will undoubtedly attempt to pass through the minimum clearance space with the lubricant, a suitably sized filter must be incor-porated into the system if continuous wear is to be avoided. The size of minimum filtration, however, has to be optimised carefully to that required to contain particles bigger than the minimum film thickness while keeping the pressure drop, and t h e r e -fore the pump work, to a minimum.
The work described in the following pages is an attempt to ascertain the order of magnitude of the minimum film thickness giving hydrodynamic conditions, which can occur during a normal process of 'running in'. It is approached, basically, from an experimental standpoint with analytical support.
2. Test apparatus
The general rig disposition is shown diagrammatically in Fig. 1. Fig. 2 and 3 show details of the film thickness measuring gauge, loading arrangement and idealised contact configuration.
Test pads. The test pads of s e m i - c i r c u l a r cross section, are mounted in troughs as shown in Fig. 4 and bear against a test shaft. The troughs themselves are fixed to rocker arms which apply the load. A small single stroke oil pump connected to the troughs allows a momentary hydrostatic oil lift to be achieved under each pad and acts against the applied load. Any asymmetric loading acting along the contact line across the pad is thereby negated, since an instability results, the pads aligning themselves to yield a symmetrical pressure profile about the contact lines centre.
2
-It is of considerable importance that the pressure disposition outlined be achieved before testing, since a constant value of axial film thickness cannot otherwise prevail asymmetric p r e s s u r e distribution, with consequent unequal side leakage, yielding a tapered film profile across the pad width.
The pads are mild steel and are loaded against a case-hardened shaft.
Loading system. Load is applied hydraulically by pressurising steel bellows with oil and is transmitted through twin cranked rocker a r m s to the pads placed on either side of the test shaft (See Fig. 2). By using two pads applying equal and opposite loads across the shaft diameter, loading of the test shaft bearings is considerably reduced and test shaft deflection eliminated. This lends itself to greater experimental accuracy. Wear measuring system. Combined pad wear is measured by an air gauge. The gauging head is shown in Fig. 2 - its operation is self explanatory. The air system supplying the gauge operates on the 'Solex' dip-tube principle (1); supply p r e s s u r e is 1. 5 p . s . i .
Maximum sensitivity for this type of system has been attained by individually testing a s e r i e s of fixed control orifices whose size approximated closely (within 5%) to that required theoretically for maximum response. Calibration was carried out in a metrology laboratory using a screw-pitch testing machine, measurement being taken over a range of 2 x 10~' inches in steps of 5 x 1 0 " ' inches incrementally and decrementally. By operating the instrument of the non-linear portion of the response curve a maximum magnification of 20,000X is possible. A travelling microscope, focussed on the m e n i s -cus of the gauge water column enabled changes of the order of 10"'inches to be detected. Drive unit and oil supply. The rig is powered by a 0. 5 H. P . D . C . shunt motor, the drive being by 'vee' belts. A hydraulic 'squish-plate' variator allows stageless control of the test shaft speed from zero to input.
Lubrication is on a closedloop system, a 12v. D,C. pump supplying high p r e s -sure oil jets to the contact area; a 5;i inch filter is incorporated.
Mean working viscosity of the oil may be estimated from temperature readings taken in the inlet and outlet zones, chromel/constantan thermocouples being employed. To reduce extraneous vibration the complete rig is mounted on a concrete bed-block. Cushioned pads of adjustable stiffness isolate the test unit from the drive equip-ment baseplate, the input drive shaft being coupled to the test shaft by a balanced barrel-coupling of rubber.
3. Outline of procedure
It is clearly important, in an investigation of this nature, to establish conclusively that conditions giving rise to minimum hydrodynamic film thickness pertain to the con-tact region.
This end might be achieved, after running a bearing under hydrodynamic con-ditions initially, by progressively increasing the duty parameter of the bearing until metallic asperity contact was imminent. In practice, however, this condition would be exceedingly difficult to realise since the point of minimum hydrodynamic approach would be almost impossible both to define and maintain; such factors as temperature variations, journal and bearing distortions and extraneous vibrations acting either singly or in combination to adversely affect the desired result.
An experiment of this nature, furthermore, would in no way simulate the 'running in' conditions occurred in practical machine parts, particularly with regard to an eventual surface finish.
With this outlook in mind the test rig was designed both to circumvent the dif-ficulties outlined and to simulate conditions analagous to those occurring in a 'running in' p r o c e s s . By loading a relatively soft pad against a hardened rotating surface a condition of pad wear can be initiated. With the progressive increase in the pad wear-s c a r a r e a , however, the mean contact p r e wear-s wear-s u r e decreawear-sewear-s if the applied load iwear-s held constant. Thus the contact, after passing through regimes of boundary and mixed friction reaches a point when the generated hydrodynamic p r e s s u r e supports the applied load entirely. In this condition asperity contact virtually ceases, the sensing gauge face indicating surcession of wear. A hydrodynamic film is therefore created and maintained since thermal equilibrium for the contact has established itself during the process of 'running-in'.
If at this point further generation of hydrodynamic p r e s s u r e is prevented by stopping the rotation of the shaft the existing oilfilms will collapse, the sensing gauge indicating the magnitude of their thicknesses before thermal contractions of the shaft or pads have any appreciable effect. The mean hydrodynamic film thickness is then displayed without extraneous influences.
4. Results
F i g . 5 shows a typical t r i a l run through three load increments, hydrodynamic conditions prevailing in all cases at the end of each run. (shown by the horizontal portion of the graph).
Test data for these t r i a l s were as
follows:-1. Estimated viscosity from thermocouple readings 5. 2 X 10"* Reyns (Lubricant straight paraffinic distillate S . A . E . 10)
2. Bellows p r e s s u r e for trials (I) 2 p. s. i. (II) 4 p . s . i . (Ill) 6 p . s . i . 3. Total load on contact area (I) 25 lbs. (II) 50 lbs. (Ill) 75 lbs. 4. Pad details. Material M.S. Contact width 0.875 inches. 5. Shaft speed 200 R. P . M . Shaft Material EN32. case hardened.
Six separate readings of the hydrodynamic oil film thickness were taken for each particular load. The results were markedly consistent for any one value of load but showed a tendency to increase with increase of load.
A working average of film thickness for all the trials was: ho = 2 . 5 X 1 0 " ' i n c h e s .
The significance of this value and its comparison with theoretically derived results will be dealt with later in the paper.
- 4
5. Analytical determination of the minimum hydrodynamic oilfilm thickness It is required to calculate the theoretical minimum oilfilm thickness 'h^n' attainable under hydrodynamic conditions from consideration of the bearing variables and the test rig geometry.
A starting point may be made from the generalised form of the Reynolds equation for an idealised bearing fluid viz.
è '•"f ^a-^C-'fj^-B^i
Now the ratio of the wear s c a r length to its width is approximately 1:20. Thus the bearing may be considered infinitely wide axially, i . e . the effect of axial side leakage of the lubricant m.ay be neglected and the p r e s s u r e becomes a function of the 'x' coordinate only. The equation therefore reduces to
d / , 3 d P ^ „ . , dh
d^ V^ d r ; = '"^B^^ <2)
Integrating ,3 d P h' ^ = 6»7„ U (h + c) . (3) dx B where ' c ' is a constant.The geometrical configuration of the test shaft and wear s c a r will be as shown in Fig. 3(a), hydrodynamic conditions just prevailing.
Since the shaft curvature is small compared to the mean oilfilm thickness the contact geometry may be represented by the simple cartesian system of Fig. 3(b) and the configuration analysed as a slider bearing.
Referring to Fig. 3(b) it is seen that the oilfilm thickness at any point along the 'x' coordinate may be expressed by the equation
h = h m [ m - (m - 1) r I (4) ^o
where m s o iim
*
Writing this equation into (2) gives an expression for the p r e s s u r e gradient U [hm [ m - (m - l ) j j + c j
f ,
^"^
^
^ . - ^ ,5,
d x , 3 hm m - (m - 1)
!T
The integration of this expression gives the p r e s s u r e at any point in the oilfilm along the 'x' coordinate, i . e .
__ r 1 ^ c 1 1 ^ ^
1 L [m - (m - 1 ) ^ ] 2hm [m - (m - 1 ) ^ J \ where c, is a constant. ' '
F r o m F i g . 3(a) it i s s e e n that the p r e s s u r e i s z e r o in the inlet and oi^tlet r e g i o n s of the b e a r i n g . Thus taking p = 0 at x = 0 and t the c o n s t a n t s c and c, m a y b e e v a l u a t e d t o yield
-2hnim
m + 1 (7)
-e
^' ~ (m^ 1) (8)
Substituting t h e s e v a l u e s into (6) and simplifying gives the final p r e s s u r e e x p r e s s i o n 6T\-QUI tn (m - 1)
V4
(m^"l
t n - -T- ( t n 1)JJ
(9)I n t e g r a t i o n of t h i s equation b e t w e e n the l i m i t s of x = * and x = 0 gives the load c a r r i e d / u n i t width, i . e . W B = 6rjgU*
( 3 /
TV-T]
dx[
mf ( m - l )
]'
I n t e g r a t i n g ^ B = 6 . B U i ^ , (m 1 I , 2(m - 1) )•TIP [ log,m - ^ ^ ^ J
(10) T h u s the m i n i m u m film t h i c k n e s s m a y be e x p r e s s e d a s lÖTj^Um* ' m \-"w;
I (11), w h e r e m* = (m^ [
l o g e m^ J ]
2(m - 1) mIt would a p p e a r f r o m c o n s i d e r a t i o n of t h i s equation that ' h m ' . the t h e o r e t i c a l m i n i -m u -m oilfil-m t h i c k n e s s , cannot be d i r e c t l y a s c e r t a i n e d s i n c e the equation c o n t a i n s a f u r t h e r unknown ' m * ' , o r m o r e e x p l i c i t l y ' m ' the r a t i o ^ o / h m
-F u l l e r (ref. 2), in a p r e v i o u s a n a l y s i s , avoids t h i s difficulty by a s s u m i n g a v a l u e of m = 2 . 0 ( b a s i n g t h i s a s s u m p t i o n on g e n e r a l working p r a c t i c e for s l i d e r b e a r i n g s ) the a r r a n g e m e n t of the t e s t r i g h e d e s c r i b e s being s i m i l a r , g e o m e t r i c a l l y , to that d e s c r i b e d in t h i s p a p e r .
It will be shown, h o w e v e r , that an a s s u m e d value of m = 2 . 0 , could, with t h i s p a r t i c u l a r g e o m e t r i c a l configuration, lead to an e r r o r of s o m e t h r e e m u l t i p l e s in p r e d i c t i n g the m a g n i t u d e of the m i n i m u m oil film t h i c k n e s s .
B y a n a l y s i n g the g e o m e t r y of the t e s t a p p a r a t u s in r e l a t i o n t o t h e e x p e r i m e n t a l w e a r s c a r c o o r d i n a t e s it h a s been found p o s s i b l e to d e r i v e an e x p r e s s i o n for ' m ' d i r e c t l y ; t h i s being finally e x p r e s s e d in t e r m s of the w e a r s c a r length ' Y ' , the t e s t shaft r a d i u s 'R' and two fixed t e s t r i g d i m e n s i o n s 'A' and ' B ' . (See F i g . 6).
U s i n g t h i s e x p r e s s i o n t e s t v a l u e s of h m nnay be d e t e r m i n e d . The method i s a s follows:
D e r i v a t i o n of an e x p r e s s i o n for ' m ' ( ° / h m )
R e f e r r i n g t o F i g . 6 it i s s e e n t h a t t h e two lengths ' o ' and ' ^ ' . m e a s u r e d f r o m the w e a r s c a r e x t r e m i t i e s at points 'I^ . P ^ ' on the t e s t shaft p e r i p h e r y to the load a i m pivotal point ' P ' , m a y be e x p r e s s e d in t e r m s of the two fixed d i m e n s i o n s ' A ' and ' B ' and the w e a r s c a r c o o r d i n a t e s ' X ' and ' Y ' , viz.
«= V(A + Y ) ^ + (Ë + X)« . (12)
and ^
B= / ( A - Y ) ' + (B + X ) ' . (13) On f o r m a t i o n of a h y d r o d y n a m i c film the t e s t pad r i s e s from i n t i m a t e contact
with the t e s t shaft, and, swinging t h r o u g h a s m a l l a r c about P , defines the film t h i c k n e s s e s at inlet and outlet of the b e a r i n g , hm and ho; t h e s e being m e a s u r e d n o r -m a l l y f r o -m the shaft s u r f a c e to the w e a r s c a r e x t r e -m i t i e s .
T h u s , s i n c e ' a ' and ' ^ ' suffer the s a m e a n g u l a r m o v e m e n t about ' P ' ( s e e F i g . 6) by s i m i l a r t r i a n g l e s
a V(A + Y)' + (B + X ) ' b " V(A - Y)»+ (B + X ) *
C o n s i d e r i n g F i g . 7 two t r i a n g u l a t i o n s m a y be m a d e e m b r a c i n g h|^;^,b, and h o , a, r e s p e c t i v e l y .
Defining 6 « 6 + ^ and 6 » 6 - ^ we have
Cos 6 = cos 6 cos é - sin 6 s i n 6 (15)
and
Cos 6 = cos 6 c o s e + sin e s i n A (16)
Now a s h m * 0, ^ •• ^ •• ^ . (The s c a r contact a n g l e - s t a t i c position). T h u s s i n c e hm will be of the o r d e r of 1 0 " ' i n c h e s o r l e s s a v e r y c l o s e a p p r o x i m a t i o n is ^ = ^ = (f, and r e w r i t i n g equations (15) and (16) in t e r m s bf l i n e a r d i m e n s i o n s on t h i s b a s i s gives
hm . (A - Y) (R - X ) (B + X) y "B~ " V(A - Y)« + (B + X)* R ' V(A - Y)S + (B + X)»' R
and h (17) Q _ (A + Y ) (R - X ) (B + X) Y ~ " / ( A + Y)a + (K H- X)» • R ^ V(A + Y ) ' + (B + X)- ' R (18)
Combining equations (17) and (18) and substituting (14) gives ho/v, - (AJ-Y)(R - X ) + Y ( B + X )
"/hm (A - Y)(R -X) - Y(B+ X) ^ ' which is the required expression.
Since 'X' is small compared with ' B ' and 'R' (0,1% of 'R') we can safely approxi-mate the equation to
ho/v, / \ R(A + Y) + BY ,,.^„. ' ^ o / h m ( = m ) = B(A - Y) - BY <'°> As a check on the validity of the equation it is seen that the ratio °/hm "• 1 • 0 as the
wear s c a r length tends towards zero (Y -• 0) i . e . the condition of simple line contact of the pad on the test shaft in the unworn state.
By substituting a value of 'Y' for any particular test condition together with constants 'A', 'B' and 'R' a value of 'm' can be obtained from equation (20). Using this value of 'm' in equation (11) a value for 'hm' may be determined for the particu-lar test conditions.
Calculation of 'm'
The relevant dimensions of the test rig are shown in Fig. 6. By simple t r i g o -, -, .-,. ^ ^-, ^. Air Gauge Reading 8 -, . -, nometry it can be shown that the ratio ^ =- = r—r- and using the approximation Y' « 2XR (since X is small compared with R) the wear s c a r dimension 'Y' may be found.
The 'Air Gauge Reading' referred to is the ordinate 2x (See Fig, 5). This will include any shaft expansion arising from heating. It was noted during preliminary trials that the bulk heating of the oil arose from churning in the oilpump. Before testing, therefore, the rig was run under no load with the hot oil jets impinging on the rotating test shaft until a stable temperature was attained. Using this technique no appreciable change in shaft temperature was observed during the trials and unwanted expansion was therefore avoided.
As an overall check on the value of the wear scar dimension (2Y), evaluated from the graph, the pads were removed after the completion of the tests and the final s c a r length measured directly from the pad using a gradicule microscope.
Test data for these trials were as follows:
1. Final dimension (2Y) measured using the graticule microscope.
(a) Upper Pad 4. 6 X 10"^ inches, (b) Lower Pad 5. 3 x 10"* inches.
(These dimensions are means, since slight variations occurred from centre to edge)
2. Air Gauge readings at hydrodynamic condition 2x (See Fig. 5)
8
-U s i n g the r e l a t i o n s h i p | r r ^ = —-r- and Y* =< 2XR gives the following
2 X o . D
v a l u e s for the s c a r length 2Y.
(2Y) (I) 3 . 6 8 X 10"* i n c h e s (II) 4 . 7 6 x lO"* i n c h e s (III) 5.06 x 10"* i n c h e s *
* ( C o m p a r e with m e a s u r e d value)
P u t t i n g in the m a c h i n e c o n s t a n t s ' A ' , ' B ' and 'R' into equation (20) gives the r e l a t i o n -ship between 'Y' and ' m ' n a m e l y
0.5000 ( 3.5000 + Y) + 1.. 5000 Y ™ " 0 . 5 0 0 0 ( 3.5000 - Y) - 1.5000 Y
Substituting the v a l u e s of 'Y' gives the following v a l u e s of ' m '
(m) (I) 1.0430. (ID 1.0559. (Ill) 1.0595.
V a l u e s of h m
Using the v a l u e s of m above and s u b s t i t u t i n g t h e s e t o g e t h e r with the t e s t d a t a into equation (11) g i v e s the following v a l u e s of ' m * ' a n d ' h m '
(m*) (1) 3.2456 X 1 0 " \ (II) 4 . 2 4 0 5 x 1 0 " ' . (Ill) 4 . 5 9 7 2 x 1 0 " ' .
(hm) (I) 5.56 X 10"'^ inches. (II) 5. 82 x 10"* inches, (111)5.26x10"* inches. (to two places).
6. D i s c u s s i o n and c o m m e n t s
Effect of the value of ' m ' on the magnitude of h m
E q u a t i o n (10) m a y be w r i t t e n in the f o r m
W ^ ( h H ) \ 6 [ j - ^ J L o g e m - 2<ELI I I ]
J J B U \ I / (m - 1)* l_ ^e m + 1 J
l e s s g r o u p —r- I "/TJ ^^ay be r e g a r d e d a s a b e a r i n g c a p a c i t y n u m b e r a n a l a g o u s to the S o m m e r f i e l d n u m b e r for plain j o u r n a l b e a r i n g s .
w h e r e the d i m e n s i o n
-W B / ' h m V
\—J~
A g r a p h of — - . \~r~J ~ ' ^ (See F i g . 8) shows t h i s n u m b e r to be e x t r e m e l y s e n s i t i v e to changes in the value of m between m = 1. 0 and m = 2 c o m p a r e d to v a l u e s of m > 2 . 0 .
F o r t h i s r e a s o n if a value of m i n i m u m film t h i c k n e s s h m i s r e q u i r e d for a p a r -t i c u l a r working condi-tion wi-thin -t h i s r e g i o n i-t is highly d e s i r a b l e -to ob-tain a value of ' m ' a p p l i c a b l e to the t e s t r i g g e o m e t r y , s i n c e the r e q u i r e d value of hm will o t h e r w i s e depend e n t i r e l y on the v a l u e of ' m ' a s s u m e d .
In the p r e s e n t i n s t a n c e if an a s s u m e d value of m = 2. 0 i s used i n s t e a d of the c a l c u l a t e d value m = 1. 0430 (condition I) the value of hm = 1. 59 x 10"* i n c h e s r e s u l t s , a value which i s n e a r l y t h r e e t i m e s l a r g e r than the figure c a l c u l a t e d f r o m the w e a r s c a r and t e s t r i g g e o m e t r y .
The influence of 'm' on the load carried
The marked influence of 'm' on the load carried by the bearing for m •• 1.0 may be judged from an examination of the p r e s s u r e curves of Fig. 9.
Considering a bearing operating under similar conditions to the test bearing with a wear s c a r length of 50 x 1 0 " ' inches and at values of m •• 1.0, the area under each particular curve is representative of the load carried. If m = 1.06 is taken as unity load (m = 1. 0595 in the test bearing) the loads carried for various values of 'm' are in the following ratios
'm' 1.02 1.04 1.06 1.08 1.10 Load WB 0,356 0.690 1.000 1.298 1.581
Thus for a change in 'm' from 1. 02 to 1.10 the corresponding change of load is nearly five times the original. This further emphasises the necessity of obtaining a value for 'm' relative to the working geometry of the test rig if the nature of the experiment is such as to allow 'm' to fall in this region.
Comparison between the measured and derived values of 'hm'
In considering the analytical approach in the determination of a magnitude for h^n it is to be borne in mind that the surfaces of the pad wear s c a r were assumed to be perfectly smooth after the process of running in. In practice, however, this would be far from the case.
During the state of wear preceding the attainment of hydrodynamic conditions the wear s c a r surfaces would become striated and grooved in a direction parallel to the plane of the shaft's rotation through the action of particle debris dragged through the contact zone, A cross section through the surface at this time might therefore have an appearance similar to Fig. 10(a).
With the approach of hydrodynamic conditions, however, the asperity peaks would tend to become flattened until, with the gradual increase of wear s c a r length, a point is reached when the total surface area is sufficient to allow a hydrodynamic film to be formed. (See Fig. 10 (b) ),
As a check on whether the configuration outlined actually appeared in practice
a Talysurf t r a c e of the wear s c a r surface was taken at the end of the trials (See Fig. 11). It is seen from this that a surface very similar to that postulated did indeed appear.
Comparison of the measured and calculated values of hm show that these dif-fered by approximately half an order of magnitude (2. 5 x 10~' inches measured; 5.5 X 10"* inches calculated.) While this discrepancy may appear large, the physical significance of this difference must be borne in mind, since, in attempting to measure o r d e r s of magnitude between 10"' inches and 10"* inches, many extraneous factors may affect the experimental result. It is of interest to note that the experimental result is the l a r g e r of the two figures, a situation which implies that either the actual oil film thickness was Initially greater than that predicted by theory or that the com-bined movement of the pads upon shut down exceeded this figure. It is shown under the heading "Axial P r e s s u r e Profile" that the former condition is hardly likely to apply and attention will therefore be directed to the latter case.
10
-Effective pad movement
As has already been outlined, in hydrodynamic conditions the applied load is entirely supported by a film of lubricant covering the complete area of contact and separating the surfaces; the reactive pressure exerted at any point being the local fluid p r e s s u r e pertaining to that point.
On collapse of the hydrodynamic film upon shutdown, metallic asperity contact will once more occur between the shaft and pads and the applied load will, in actuality, be supported by an asperity surface area signifantly less than the apparent area of the wearscar.
Bowden and Tabor (reference 3) using electrical resistivity techniques have shown this effective asperity contact area to be both a very small fraction of the gross area and directly proportional to the magnitude of the applied load. The resulting s t r e s s , based on this area, was found to be constant with varying load and of a sufficiently
high order to cause significant elastic deflections of the asperities and even plastic flow.
In the present instance because of the relatively deep grooving along the length of the wear scar (See Talysurf Trace Fig. 11) squeeze film support upon shut down would have little or no effect and the asperity peaks would deflect upon contact with the test shaft. In addition a certain amount of "bedding in" would take place before the test shaft finally stopped. The combination of these factors would, therefore, be manifested by an effective movement of the test pads in excess of the film thicknesses separating the surfaces before shutdown and this would be recorded as a direct move-ment by the air gauge sensing head.
Axial p r e s s u r e profile
F o r a perfectly smooth surface within the contact area the axial pressure profile would follow the simple curve of Fig. 12 (a), i , e , a constant magnitude of axial p r e s s u r e except for the extremities of the contact where side leakage of the lubricant causes a parabolic drop in pressure to ambient. In the practical working bearing, however, this p r e s s u r e profile would be considerably modified because of the condition of the surfaces already enumerated. In the axial direction the oil film would be thinnest at the worn tips of the asperity peaks and because of the extreme thinness of the film at these points local high peaks of p r e s s u r e would be set up varying in magnitude with the local variation of oilfilm thickness. Consistent with this condition p r e s s u r e gradients would occur from these points to the surface grooves or "valleys" (see Fig. 12 (b)). Under the influence of these steep pressure gradients the lubricant would tend to flow away from the asperity peaks into the surface "valleys" and the film thickness at the . asperities would tend to remain minimal. Since conditions of hydrodynamic lubri-cation were shown to exist at the end of each trial the conclusion must be drawn that the surfaces remained separated by a film of finite thickness with occasional local contacts between the shaft surface and asperities.
As an overall picture, axially, of contact conditions, the surface might be con-sidered to consist of a large number of bearing lands giving various local film thick-nesses with drainage grooves at certain points between asperities. Now it is clear that, under these conditions, the "local" bearing duty is considerably greater than would apply overall to a contact with perfectly smooth surfaces separated by a hydro-dynamic film. Thus the effect of the surface grooves and striations (which occur in
the practical case) would be to reduce the effective value of the working oilfilm thick-ness below that of the theoretical value calculated on the assumption of perfectly
smooth surfaces. However, since the local oil p r e s s u r e at asperity peaks may be many times the value of the maximum p r e s s u r e derived from theoretical considerations (in the present case approximately 2000 lb/in*) the effect of p r e s s u r e on the oil vis-cosity at these points cannot be ignored, the tendency being to prevent the reduction in oilfilm thickness beyond a certain minimum point.
It is concluded from these investigations that a value of minimum hydrodynamic oilfilm thickness as low as 2.5 x 10"' inches can occur under conditions of "running-in" and that this value may decrease still further with decreasing surface roughness.
7. References
1. Evans, J . C. "The Pneumatic Gauging Technique in its Application to Dimensional Measurement. "
Jnl. Inst. Prod. Engr. 1957, 36, 110.
2. Fuller, D. D. "Theory and Practice of Lubrication for Engineers. " John Wiley & Sons, Inc. New York,
3. Bowden, F . P . and Tabor, D.
"The Friction and Lubrication of Solids'.'
-AIR GAUGE OIL FILTER
LAYOUT OF TEST RIG
OIL LIFT TANK
PRESSURE GAUGES BELLOWS
FIG.
ROCKER ARMS TEST SHAFT ADJUSTER LOADING BELLOWSACTUAL CONTACT CONFIGURATION FIG.3CQ) ho-hniT
1
^-T-XT-T—<rIr
^ Lfw
h' _ ( h o - MT
a-x)
L
h = h+h«-h»[l+(m-l)jkïj|
whert m = / nn THUS h = h i i i | i n - ( m - l ) x |DEAUSED CONTACT CONRGUR«JION
FIG. 3(b)
ARR. OF HYDROSTAnC OIL LIFT
' O
i
(y
/ / . / / »c n (m) j l ^ S . l Ö ^ I N S X 2»3«lÓ^NSCURVES OF PROGRESSIVE WEAR TO HYDRODYNAMIC CONDITIONS.
1 '^"^•^
6 0 I 2 0 2«0 3 0 0 TIME ( M I N U T E S ) 360 420 480 FIG. 6 PC.- V(A^YA(B^^X)' i P.°V(A-Yfi(B^x)',A«3-5iNs, B"l-SiNs. R-OSiNS.
'T
0-20 — FIG. 7 0O5 Ï2.-«2-02, .X,.ö,+0,® °
MAX AT i n ' 2 - 2 GRAPH of y i f ) ~ m FIG. e •O 2-0 4 - 0•2500 •2000 I500 •lOOO •500 FIG 9. 2 0 X I 0 - ' , n * . 3 0
J^AA/Yr\/VA.r^'^VVA^
FIG.IOq.HYPOTHETICAL SURFACE DURING RUNNING IN FIClOb
HYPOTHETICAL SURFACE AFTER "RUNNING J N " FIG.II.
TALYSURF TRACE OF SURFACE OF PAD AFTER RUNNING IN. SCALE: VERT. SOOOM H0RIZ,20=I
FIG.I2a.
AXIAL PRESSURE PROFILE FOR SMOOTH SURFACES
t l
FIG. 12 b.
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