INDUSTRIAL 9 N D ELVGINEERIAVGCHEA1fISTRY
May, 1926
film lubrication, it did tend to lower slightly the critical point of film rupture, and decrease friction and abrasion in the region of partial lubrication. Similar conclusions are iadicated in the papers by Parsons, Koethen, and Dover in this symposium. Since the work summarized above, Barnard3 has investigated and reported on the effect of clearance ratio on the friction curves and the Pritical point, and in his present paper
+ WHITE METAAL BEARING o BRONZE B€AU/NG
‘Y F i g u r e 1-Data O b t a i n e d by Streibeck o n Bronze a n d W h i t e M e t a l Bearings n i t h a Steel S h a f t 70 M m . (2.76 I n c h e s j i n D i a m e t e r
shows that the oil circulation through a bearing is also a function of this same modulus .ZN/P. One of I)r. Dickinson’s laboratories a t the Bureau of Standards has done some very interesting work along these lines, most of which has not yet been reported. I n particular they have obtained extremely low critical points in a large bearing with very smooth surfaces, and have bem studying the effect of dirt in oil on 3
Tms
JOURYAL,
16, 3 4 i (1924).
453
the friction and the critical point, with some very interesting results. Field for Further Study Probably the work which is now most needed is a detailed study of the effect of the smoothness and structure of practical bearing surfaces on the location of the critical point, because i t can undoubtedly be greatly lowered below that generally found today. Such a lowering of the critical point would, of course, permit the use of smaller bearings and lighter oils, giving much less friction. The present uncertainty as to the precise value of the critical point in a given type of bearing, often coupled with an insufficient appreciation of the laws of lubrication, is responsible for the frequent occurrence of factors of safety in the field of lubrication many times higher than necessary, thus increasing both the cost of the bearings and friction losses. Some bearings, on the other hand, operate too near the critical point for safety. Some intensive cooperative work along these lines should in a short time put the design of bearings and the relation of lubricants on a truly scientific basis. This does not mean that the mechanism of lubrication is even yet explained in all its phases. The very interesting observations reported in the papers of Becker and Gilson raise questions which will undoubtedly puzzle tlieoriyt, fur some time to come, but should eventually lead to the formulntion of the correct modifications of lubrication theory. Acknowledgment The chairman wishes to express his appreciation to all those who have cooperated to make this symposium a success and in particular to D. P. Barnard and E. 11. Billings, who have been of great assistance in connection with the solicitation and reviewing of papers and the mechanical work involved in arranging such a symposium.
...... ........
Characteristics of Full and Partial Journal Bearings’ By H. A. S. Howarth KISGSBCRYMACHISSWORKS, PHILADELPHIA, PA.
The types of bearings to which this analysis is applied are as follows: I-Bearings bored with running clearance: (a) full or complete bearings; (b) central partial bearings in which t h e line of action of the resultant load bisects t h e bearing angle; (c) offset partial bearings in which t h e line of action of the resultant load intersects t h e bearing surface usually more t h a n half way from t h e leading edge. 11-Bearings fitted carefully to the journal. As no running clearance is provided, these bearings m u s t be less t h a n 180 degrees long in order to function. These types have been studied graphically in order t o provide a complete and thorough set of fundamental laws which m u s t govern t h e actual lubrication of all plain journal bearings. The bearing characteristics as given i n this paper represent ideal conditions in t h a t they show the maximum film thickness which may be realized for a given load. These laws may serve as a guide in the study of actual bearings in which such factors as end leakage 1 This paper, except the parts on total friction in full bearings and the viscosity-temperature conversion chart, is a resum6 of three contributions by the author to the American Society of Mechanical Engineers in 1923, 1924, and 1925, entitled “A Graphical Study of Journal Lubrication.”
and surface roughness play a modifying part and prevent the carrying power from being as great as in the ideal case. As a n example of the use of these laws is cited the question as to the advantages of employing two or more pivoted segments instead of a single partial bearing to support a journal. I t can be readily shown t h a t the film is thinner in t h e case of the bearing of two segments, t h e total arc of each bearing and other conditions being equal. I n a similar manner a three-segmented bearing is inferior t o one of two segments. I t is quite probable t h a t many bearing problems may be solved, a t least approximately, by the graphical method once its usefulness and limitations are thoroughly understood.
T
HE hydrodynainical theory of lubrication, developed
by Reynolds and amplified by Harrison2 and others, forms the basis of this analysis, and Harrison‘s equations and symbols are used so far as they apply. This graphical analysis is, therefore, a development of the twodimensional theory, originating with Reynolds, which leaves side leakage out of account, and assumes that the oil viscosity is constant throughout the film. The influence of friction 2
Trans. Phil. SOC.,22, 3, 39 (1913).
INDUSTRIAL A N D E-VGINEERING CHEMISTR Y
454
upon the direction of the resultant force of the oil film upon the bearing surfaces is neglected in the examination of partial bearings herein because of its relative insignificance for practical purposes.
VOl. 18, No. 5
c . g. s. systems, can be made from Figure 17. This figure also shows approximately how oil viscosities vary with temperature.
a
+aq
= radius of journal = radius of bearing when provided with running clear-
ance
c = eccentricity factor cq = eccentricity a t which journal runs in a bearing pro-
vided with running clearance total friction a t surface of journal per inch of axial width F’ = total friction a t surface of bearing per inch of axial width z i = film thickness a t angle 8 hl = film thickness at point of maximum pressure ho = film thickness where intersected by radius extending from 0’ through 0 to bearing surfaces 1 = axial width of bearing U = 2raN LY = r. p. m. of journal 60 p = unit pressure within the film where thickness is h, pounds per square inch R = resultant pressure of journal against bearing film R’ = resultant pressure of bearing against film U = surface velocity of journal, inches per second
F
I Figure 1-Full
Figure 2-Full Bearing with Oil Groove
Bearing
The types of bearings to which this analysis is applied, and whose characteristics are charted, are illustrated by Figures 1, 2, 5, 6, 7 , 9, 10, and 13. In all cases the journal is assumed to press downward against the bearing. These types include: I-Bearings, ance.
both full and partial, bored with running clearThe film thickness is h = q(i c COS e) (1) 11-Bearings, partial only, that are carefully fitted to the journal. The film thickness is
+
h =
E
sin 8
(2)
The center of the journal in every figure is at 0, while that of the bearing is at 0‘. The symbols used in the figures and formulas are listed below. Lengths are in inches. Viscosity conversions, from Engler, Redwood, and Saybolt into kinematic and absolute viscosities in the English and the
effi,
W CY
p E
8
& X X‘
=
2a
= load on bearing in pounds per inch of jour2al nal diameter and per inch of axial width of bearing = total load on journal (pounds) = RI = 2waZ = angle from leading edge of partial bearing to position of resultant pressure R measured in direction of journal rotation = angular (circumferential) length of partial bearing = eccentricity a t which journal runs in a fitted bearing = radial clearance = angle to point where film thickness is h and pressure is p = angle to point of maximum pressure = angle to point of minimum pressure = coefficient of friction of journal = coefficient of friction of bearing
P
SCALE
*a
AJJUMPI/OAU
Figure 4-Full
foe
c.
~ C C E N % W C / T FACTOR ~
,dl = 34&)?(/50
= coefficient of viscosity
+
Full Bearings
The characteristics of full journal bearings like Figure 1 are given in Figure 3. Curves A to F , showing the relation between load, speed, and eccentricity. were plotted from Equation 3, using the viscosities marked on the curves and the clearance ratios q/n marked below the w / N scales. (3) h
4
Curve H , plotted from (4), shows the relation between
cos el =
~i~~~~ j-cLntra1 partial Bearing. B < 180'
COS
6
=
This relation affords a means of extending the range of usefulness of Figure 3 and also of other similar charts herein. Examples 1, 2, 3, and 6 (at end of article) indicate some of the full bearing problems that may be solved by means of Figure 3. It appears from the study of full bearings that an oil supply groove may be placed, as in Figure 2, a t the point of minimum pressure. It also appears that the carrying capacity of the full bearing is independent of the pressure of the oil supply, as long as the oil film remains unbroken. Total Friction in Full Bearings It will be noted that the friction coefficients in Figure 3 are relatively very high for small eccentricities c-i. e., when the journal runs nearly concentric with the bearing. Values
-3c 2 + (4) c2
the positions of maximum and minimum pressure, and the eccentricity. Curves G and J , showing respectively the relation between the friction coefficients of the journal and the bearing to the eccentricity, were plotted from Equations 5 and 6.
(For journal) X = (For bearing) X' =
WQ
5 U Y Sp G l O 6 4 )
Bearing Characteristics Giving Total Friction instead of Friction Coefficients
(absolute), inches, pounds, seconds 4 = angle from line of action of resultant pressure R t o line passing through the centers of bearing and journal, measured in direction of rotation of journal iI. = 1 8 0 ° - ( a - P) fi
455
INDUSTRIAL A N D ENGINEERING CHEMISTRY
May, 1026
1
+ 2c
(a> {'%*I
I
Figure 6-Central
ing.
(6)
It follows from Equation 3 that, for given values of clearance ratio and eccentricity factor, the relation between load, speed, and viscosity will be as given in Equation 7. (7)
I
Partial Bear-
i3=180°
Figure 7-Central Partial Bearing. Ordinary Form
of X and A' are out of the range of Figure 3 when c is less than 0.035. The coefficients are high in this range, not because the total friction increases, but because the capacity R for a given speed falls off toward zero as c approaches zero. Some methods other than Figure 3 must therefore be employed to show the friction for small eccentricities, in order to solve the problems presented by high-speed, lightly loaded shafts and by vertical shafts carrying little or no side load.
456
ISDUSTRTAL B S D E-YGISEERTSG CHEMTSTRY
Vol. 18, N o . 5
M U
-!-C
Figure 8-Characteristics
of “Central” Partial Bearings
The total friction curws in Figure 4 show how this friction starts with a given amount for concentric running and changes as the eccentricity increases. The total friction on the journal increases, as we would expect, becoming infinite when c = 1.0 and the surfaces touch. The total friction on the bearing decreases as the eccentricity increases, becoming zero lvhen c = 1.0. Why this should be so is not so evident. It appears paradoxical and, before acceptance, would have to be verified by experiments. Hovever, as we are chiefly concerned with the friction of the revolving member, which is usually the shaft journal, the curve for journal friction should be reasonably satisfactory for practical purpose.;. The two friction curves on Figure 4 were plotted froin Equations 8 and 9 taken froin Harrison’s paper.? Figure 9-Offset ing. 9 ,
Partial Bear-
io
262
d
Jm
I Ji3
_1_”
3.9
I
O F O F F S E 7 P A R T A L B E l R h5
of a/@ Ratio w i t h Angle @ of Offset Partial
Bearing
I n addition to the friction curves, Figure 4 shows one loading (w/X) curve, which is drawn for viscosity ,u = 3.4 X Other viscosities can be taken into account according to Equation 7 . Example 7 illustrates the application of Figure 4 to the solution of full bearing problems. Partial Bearings
PartiaI bearings may be provided with running clearance or fitted. Those with running clearance are divided herein into two main classes, central and offset. Fitted partial bear-
437
ings lie in the offset class. A central partial bearing has been so called because the line of action of the resultant load on the journal bisects the bearing angle ,8. Three such bearings are illustrated in Figures 5, 6, and 7 . An offset partial bearing has been so called because the line of action of the resultant usually intersects the bearing surface at a point more than half way from the leading edge-i. e. , a > /3 2 . Three offqet bearings are illustrated in Figures 9, 10, and 13.
Central Partial Bearirigs, with Running Clearance These were chosen for study because so many inachine bearings are assumed to fall under that classification. The mathematical theory did not permit of the eaqy determination of the characteristics of these bearings when other than 180 degrees long. The curves given in Figure 8 were, therefore, the result of an extensive graphical analysis. I n Figure 8 the curves for capacity CI”, against wcentricity = , not coiiqtaiit, but varies as shown in Figure 11. The other characteristics of this iliaximum capacity series of partial bearings are given in Figure 12, the curves of which show capacity w/iV, journal friction coefficient A, and angular position 4 of the line of centers-each plotted against the eccentricity factor c . One viscosity only is used as marked. Corrections for other viscosities can be made according full and central partial beaiings. Curves for angular bearing lengths p are given from 60 to 180 degrees a t intervals of 20 degrees. The film thirkne-i a t any point is found from Equation 1, remembering that 4 on the chart shows the position of the line froin which to measure 8. Examples 5 and S illustrate the use of Figures 11 and 12.
Fitted Partial Benriiigs These bearings, as previously stated. euactly fit the journxl. No running clearance is provided. Fitted bearings must be less than 180 degrees long in order t o function. The study of
INDUSTRIAL AND ENGINEERING CHEMISTRY
458
Vol. 18, No. 5
these bearings was simplified by extending Reynolds' fundamental equation (11) to obtain a new pressure equation (12), which applies only to fitted bearings. (See Figure 13 for symbols.) ha*dx = 6pU(h--hl)
(11)
2
log tan
e
- - cos
2
01
(12)
From this point on the graphical analysis proceeded in the same manner as for full bearings and offset partial bearings. Eccentricity E is used, however, instead of the eccentricity factor c. Obviously there is an infinite series of such bearings. The study was completed only for the series with maximum capacity, which also had the lowest friction coefficient. The characteristics of these maximum capacity fitted partial bearings are given in Figure Figure 16 14, for bearings with angular lengths p varying Comparison of 120-DeaeePivoted Fitted Partial Segment Bearing Bearing and the Equivalent Double from 60 to 120 degrees by intervals of 10 degrees. One viscosity only is usedI(p = 3.4 x 10-6) and corrections for other viscosities can be made by using By comparing these bearings it is found that the journal Equation lo' each curve Of the lower set are marked friction coefficient is lowest for the fitted, next for the offset, corresponding Of P, and 6' The use Of this chart is higher for the central, but highest of all for the full bearing. explained by Examples 8 and 9. The distance of closest approach does not quite correspond, being least for the fitted, next for the central, greater for the offset, and greatest of all for the full bearing. 8 6 It must be remembered, however, that the clearance ratio was assumed to be q / a = 0.002. This ratio is not the one 4 for the minimum friction of bearings I, 11, and 111. I n bearing I, as was learned from Example 6, the best clearance ratio would be somewhere between 0.002 and 0.003 and the 5 minimum friction coefficient would be about 0.0031.
-.
'' ~
Y e
$
.. %
31? 9
P 2
$
s$
8
3.
1 b
1 .ke
8 -0
O M 3
oa40 arrr
Lo3
JCALE
fbe
rnl
d
-0
iu-.
ma
OLnr
0
(ErrruvL-/7vi
p = 3 4 x 10-0 Figure 14-Characteristics of Fitted Partial Bearings
Comparison of Partial Bearings The results obtained from Examples 3 , 4, and 8 are tabulated below for comparison: B
e
+
No. BEARINGdeg. deg. deg. Full 360 90
I
11 Offset partial I11 Central partial IV Fitted
.. . .
120 75.6 51.9 31.8 120 80 120 77.4 148 3
{ f, {& { :v e
c, 6 9 , e
1
Data from exhop h s ample 0.00153 0.00325 3
o,ooo735 o.oo19 :: ? 2 7 6 } =!:%284 = o ,30215
] 0.000716 o.oo22 O . O O O ~0.0017 ~~
4,8
3,s 8
EJect of Pivoted Segments The study of partial bearings has now proceeded far enough to answer intelligently an important question that occasionally arises in bearing designing-namely, is there a saving in friction if two or more pivoted segments are used instead of a single partial bearing to support a journal? This question can most definitely be answered by assuming both types of bearings to cover the same arc of the journal's surface. I n Figure 15 a fitted 120-degree partial bearing is shown, while in Figure 16 two pivoted fitted 60-degree segments are shown adjacent, their most favorable position. I n the single 120-degree bearing the journal must lift and shift to the left so as to form the wedge-shaped film. The eccentricity, the distance of closest approach, and the frictional coefficient can be determined by referring to Figure 14. I n the double segmental bearing the journal must lift and the segments must tip to form the wedge-shaped films. Similar data can be obtained for this bearing also from Figure 14. Example 9 shows the single bearing to be much superior to the double-pivoted segmental bearing. More than two segments would be inferior to two. Constant Viscosity Assumption
It was explained above that this graphical study was limited to the two-dimensional theory, and that the side leakage effect was left out of account. Another important assumption was that the oil viscosity p within the film was constant from one end of the film to the other. This constant viscosity assumption, while reasonably true for a slow-speed bearing or a bearing with a water-cooled shell, would not be reasonable for a high-speed bearing to which fresh oil is
May, 1926
INDUSTRIAL AND ENGINEERING CHEMISTRY
459
on the projected area of the journal is 60 pounds per square inch and the oil viscosity a t operating temperature is 150 S. U. V. Find the eccentricity, angle of maximum pressure, coefficient of friction, and nearest approach of the surfaces. w / N = 0.2. Hence from Figure 3, curve B , i t is shown that c = 0.235. Hence eccentricity = 0.00047. From curve H i t is found that €31 = 110 degrees and from curve G that X = 0.00325. The nearest approach for a full bearHence for ing is always ho = q(1-c). thiscaseho = 0.002 (1-0.235) = 0.00153 inch. Example 4 . Same data as Example 3 except that bearing is partial and central and 6 = 120 degrees. Find t, and nearest approach. w / N = 0.2. From Figure 8 the lower curve for p = 120 degrees gives c = 0.642 while the upper one gives 4 = 31.8 degrees. Since @ is less than @/2, the nearest approach is ho = q(1-c) which in this case equals 0.000716 inch. Example 5. Same data as Example 4 except that bearing is offset. Find c, +, and nearest approach. w / N = 0.2. From Figure 12 the lower curve for @ = 120 degrees gives c = 0.638, while the upper curve gives 6 = 51.9 degrees. From Figure 11 i t is found t h a t a/B = 0.63. Hence a = 75.6 deprees. The line of centers does not cut-the trailing edge of the bearing because p-a>@. Hence the nearest approach will be h, = q[l-ccos(p-a-+)]. Substitutingin this givesh, = 0.002 [ l 0.638 cos 7.5’1 = 0.000735 inch. Example 6. Full bearing same as in Example 3. Solve for q / a = 0.001, q / a = 0.003, q/a = 0.004, a n d q / a = 0.005. The results are tabulated below together with those obtained from Example 3.
+,
Figure 17
steadily supplied a t the entering end of the oil film and steadily discharged a t a higher temperature a t the other end. If the viscosity falls off as the oil passes through the film the pressure curve will change, thereby altering the location of the resultant pressure R , the probable tendency being to reduce the value of the ratio a//3 in partial bearings. The exact law of viscosity variation would have to be known in order to obtain an accurate result, true for a given case. Two assumptions, each extreme, might be made. One is that the viscosity is constant (the one used in this paper). The other is that all the heat of friction is absorbed by the oil in the film and discharged with it. It is quite probable that the graphical method will offer a ready means for a t least an approximate solution of bearing problems, assuming various rates of viscosity change through the lubricating film. Example 1. Find the journal friction coefficient for a full bearing whose diametral running clearance is 0.001 inch per inch of diameter, if the speed is 400 r. p. m., the mean pressure 100 pounds per square inch, and the oil viscosity 100 seconds S.U.V. First calculate w / N = 100/400 = 0.25. Then enter Figure 3 a t the left-hand vertical scale marked (?/a) = 0.001 and locate the point whose value is 0.25. From this point read horizontally t o curve A , then down t o the c scale where we find c = 0.12, then u p t o curve G and horizontally to the X scale corresponding with ( q / a ) = 0.001. A value of X = 0.0029 is found. I n order to determine the positions of maximum and minimum film pressure, refer to Figure 3 again. Enter its base with c = 0.12 and read up t o curve H and across t o the scales for 81 and %. The angles are found t o be as follows: = 101 degrees, and 8 2 = 259 degrees. Example 2. What clearance will give the minimum friction for a journal revolving a t 1200 r. p. m. if the unit pressure is 120 pounds per square inch and the oil viscosity is 150 S. U.V.? This must be solved by trial, but takes very little time. w / N = 120/1200 = 0.1. Assuming ( q / a ) = 0.001, X = 0.010; (?/a) = 0.002, X = 0.0057; ( q / a ) = 0.003, X = 0.0043; (?/a) = 0.004, X = 0.0041; ( q / a ) = 0.005, X = 0.0048; etc. Apparently the proper clearance is about (?/a) = 0.004. A similar problem in which w = 60 is plotted in Figure 18, showing t h a t Amin. = 0.0057. Decreasing the unit pressure w is thus found t o increase the friction coefficient. Example 3. A 2-inch journal revolves a t 300 r. p. m. in a full bearing bored 2.004 inches ( q / a = 0.002). The unit pressure
.,
n/n-
0.001 0.002 0.003 0.004 0.005
x
C
ha
0.0057 0.00325 0.0031 0.0038 0.0051
0.058 0.235 0.620 0.780 0.905
0.00094 0.00153 0.00144 0.00088
0.000475
It therefore appears that the minimum friction occurs when q/a is about 0.003, b u t t h a t greatest separation of the surfaces occurs when q / a is about 0.002. Example 7. What will be the friction torque and the horsepower lost in friction for a journal 6 inches in diameter and 9 inches wide, running a t 3600 r. p. m. in a full aom b e a r i n g bored .6.012 diameter, carrying a load of 19,500 pounds? C o m p a r e these with the same bearing carrying no load. Assume the oil viscosity t o be 3.4 X 10-8 $ a(inches, pounds, sec- d onds). w = 19,500/6 X 9 = 360 pounds ; per square inch. w / N = 360/3600 = 0.1. $ q / a = 0.002. P Entering thevertical scale for q / a = 0.002 a t left of Figure 4 with w / N = 0.1, the eccentricity factor c is found ?.to be 0.12. Passing ““.“*a 1 uDward with this t o the Figure 18-Variation of X with ?/a for a journal friction curve Case [Example Data: N = 1200; and then t o right to wSpecific = 60; p = 3.4 X 10-6 (S.U. V. = 150)l friction scale for q / a = 0.002 the value of K (scale reading) is 1.81. Multiplying as indicated in the notes at right of scales, we find that the journal friction = 1.81 X 0.63 X 3600 X 3 X 9 X 10-3 = 111 pounds. Checking this from Figure 3, the coefficient of friction is 0.0059, which gives the total o-A
t
0-
4-
0-
0,
I
-
INDUSTRIAL AIYD ENGINEERING CHEMISTRY
460
journal friction = 0.0069 X 19,500 = 115 pounds. The friction torque is 3 X 111 = 333 inch pounds per Figure 4. The power loss in journal friction is = 19 horsepower 1.81 X (3600)2 X (3)2 X 9 X If the journal carries no load = 0 and the friction scale reading is 1.75 instead of 1.81-i. e., only 3.45 per cent less than if the journal carries 19,500 pollnds. It will be found that the per cent difference is greater for a greater clearance ratio, and conversely. Exan~pZe8. A 2-inch journal runs a t 300 r. 11. m , carrying a unit load of GO pounds per square inch of projected area-i, e,, 120 pounds per inch of axial length. Find characteristics for this journal when running with radial clearance of 0.002 inch in a full bearing, in a central, in a n offset, and in a fitted 120-degree partial bearing. Assume viscosity is 3.4 x 10-6 (150 s. U. v.). For some of the characteristics see Examples 3, 4, and 5. = 0.0022, w / ~ ve 60/300 = 0.2. Hence from Figure 8, while from Figure 12, X = 0.0019. This completes the data for all cases except the fitted partial bearing, which will be studied below. The fitted 120-degree bearing can he studied from Figure 14. Entering the w / N scale with 0.2 we find from the 120-degree capacity curve t h a t CY = 77.4 degrees, 6 = 148.3 degrees, and t h a t e/a = 0.00215, X = 0.0017, and ( h o / a ) = ( s / a ) COS +. L, = 180(77.4 f 148.3 - 120) = 74.3 degrees. ( h o / ~ )= 0.00215 X 02706 = 0.000582 For comparison of results see section under “Comparison of Partial Bearings.’ ’
VoI. 18, No. 5
Example 9. Compare the friction, eccentricity, and nearest approach of surfaces for a journal 6 inches in diameter, 6 inches long, running a t 400 r. p. m., carrying a vertical load of 7200 pounds, ( a ) when supported in a 120-degree fixed fitted bearing, and ( b ) when supported in two adjacent 60-degree fitted pivoted segments. Assume the Oil viscosity p = 3.4 X 10-6 (inches, Pound% seconds). The vertical load will be 7200/6 = 1200 pounds per inch Of length Of the bearing. ( a ) The nominal unit pressure ma = l200/6 = 200. Hence W d L V = 0.5. Entering Figure 14 with this value we find ( € / a ) , = 0.0011. The nearest approach found from = O014 and Equation l3 (hole) = ( € / a ) cos l+b (13) is (ho/a)a = 0,0014 COS 74.3 degrees = 0.00038 inch. ( b ) The vertical load per linear inch (axial) is in this case divided, each segment carrying (1200/2) secant 30 degrees = 693 pounds. Each segment must be studied separately t o avoid confusion. Hence W b = 69316’ = 115.5. Wd,./N = 115.5/400 = 0.29. From Figure 14 we find ( € / a ) b = 0.0006 and that the friction coefficient X b = 0.00135 secant 30 degrees = 0,00156. The nearest approach is found to be: (hole)* = 0.0006 cos 57.1 degrees = 0.000326 (ab) Comparing the two bearings it is found that: Friction coefficient A b = 1.42 A, markedly favoring the single bearing Eccentricity ( ~ / a ) b= 0.43 ( € / a ) , markedly favoring the single bearing Nearest approach (ho/a)* = 0.86 (ho/a),also favoring the single bearing
Oil Flow in Plain Bearings By D. P. Barnard, 4th ENGINELABORATORY, ST.ANDARD OIL Co. (ISDIANA), WXITINC.,IND.
An attempt is made to present the basic laws of fluid lubrication in such a manner that they may be readily used in the correlation of test data. As is well known, these laws are founded upon the properties of the viscous flow of fluids. The work of Reynolds and Harrison has served to outline the laws governing power and film thickness. This paper describes a simple method of development of the approximate laws controlling oil flow through bearings-due both to pressure developed in the film and to oil-feed p r e s s u r e a n d presents some experimental data in substantiation of this method.
HE rate of flow of oil through plain bearings is one of
T
the most important factors entering into bearing design and lubrication. Depending upon the conditions of operation, it may be necessary to give first consideration to film thickness, friction loss, temperatures, or oil consumption. Each characteristic, however, is definitely related to the path of oil flow through the bearing, and therefore to the rate of flow. I n view of the large number of variables which determine the actual quantity of oil flowing, this feature is not ordinarily considered other than in an approximate manner. The method outlined herein will, it is believed, facilitate considerably a more complete study of oil flow and consumption in plain bearings.I I n general, the lubricant flowing through a bearing takes the following course: Oil is admitted a t a point usually mid1Tay between the ends of the bearing, where it is picked up by the revolving journal and dragged through the load supporting part of the film space. The pressure thus developed in the oil film serves to support the bearing load and also to force the lubricant toward the ends of the bearing. All of the oil fed to the bearing eventually escapes through 1 For
further details the reader is referred t o Barnard, J . S O L .Aulomo202 (1926).
l i v e Eng., 12,
end leakage due to the pressure existing in the oil film. This endwise flow may not be due entirely to pressure generated within the film itself, as any existing oil-feed pressure also increases end losses. I n studying the problem, however. it is necessary to consider separately the effects of these two pressures. The effect of the pressure generated within the film, as determined by the operating conditions, will be c o n d e r e d first. Flow Due to Pressure Developed in the Film
The primary factors controlling the pressure developed within the lubricating film are given in Table I, in which rll, L, and T denote, respectively, the physical units mass, distance, and time. Table I-Factors
Controlling Bearing Performance Xotation
Absolute oil viscosity, In centipoises Bearing load (nominal) Diameter of bearing Length of bearing Diametric clearance Journalr p m Rubbing speed (xd,? )
Z
I’ d e c .\
u
Dimensions i-1L - 1 T -1 .ML -1 T - 2 L
L
L 1-1
L1-1
As the motion of the journal tends to carry oil in the direction of rotation only, end leakage must be due solely to the pressure in the film. Furthermore, the clearance space is so narrow that all flow within it must be substantially laminar. This is especially true of the endwise component of flow. Therefore, end flow must conform to Poiseuille’s lam, which states that Flux (volume per unit time) = K [ l p e2/fiZ]
(1)
in which K is dependent on the form of the channel of flow, A p represents the pressure drop between the two points under consideration, a is the area, 1 is the length of the path of flow, and p represents the absolute viscosity of the fluid. Arranging the various bearing factors in this manner gives the relation:
