The Analysis of the Lubrication of a Flexible Journal

8/2/2019 The Analysis of the Lubrication of a Flexible Journal

1/6

H. D. Conway 1H. C. Lee

IBM Sy. lam Do lopmenlD iv ision. End lco ll . N . Y.

Introd uction

Reprinted from the October 1975Journal of Lubrication Technology

The A na lys is o f th e Lub ric a tio n o fa F lex ib le Jou rna l B ea ringAn analysis is made of the effects of flexibility on the lubrication of a loaded journalbearing. Numerical and graphical results are presented for (a) the pressure distributionsill the oil film. (b) the thickness oariation of the film, (c) the magnitude and direction ofthe load applied to the bearing. and (d) the frictional forces in the bearing. The oil isfirst assumed to be isoviscous and the aboue data are found to depend on the values ofthe two independent parameters Co = 6U"oR2t(J - I'02)/Ec3 and ( = elc (see Nomencla-ture). The analysis is then extended to the case of a pressure-dependent viscosity" =" o e " P oil. and the data found to depend on the values of the three independent parame-ters C o. e, and (j = Eeoltt l - "02).

One of the most important lubrication problems to be encoun-tered in engineering practice must surely be that which concernsthe journal bearing. It is for this reason that this bearing has beenthe subject of numerous experimental and theoretical investiga-tions dating back to a publication, in 1849, by von Pauli 1 1 1 . 2 Avery interesting historical survey of the work from that date up torecent times (the latest article quoted is dated 1965) is given in thehook by Cameron (2].In the journal bearing, a circular shaft rotates inside a cylindri-

cal sleeve-the hearing itself. The shaft and sleeve bearing are sep-arated by oil, and, under load, the centers of these two componentsare not coincident, being separated by a distance known as the ec-centricity. The main problems encountered are the calculation ofthe pressure distribution in the oil film, the thickness variation ofthe film, the magnitude and direction of the load, and the friction-al forces in the bearing.Quite naturally, the oil film calculations require a knowledge of

the boundary conditions and these are still in some doubt, Calcula-tions to date are usually made using one of three sets of boundaryconditions known, respectively, as the Sommerfeld. Half-Summer-feld, and Reynolds conditions. These conditions will not be givenhere since they are discussed in detail by Cameron [2].Suffice it to say that the boundary conditions used in the

present investigation are those of Reynolds. These are probablythe most correct but they are also the most complicated. The pres-sure is assumed to be zero at a certain point denoted (I = 0, to rise

J Dept. of Theoretical and Applied Mechanics, Cornell University, Ithaca,N.Y., and Consultant, IB~f Corp., Endicott. N.Y.1"umbers in brackets designate Heferences at end o r paper.Contributed by the Lubricntion Division for puhlication in the .JOUR-NAL OF LUBHICATION TECHNOLOGY. Manuscrip! received at AS:\IEHeadquarters, .July 1~. 1975. Paper No. 75-LubM.

Journal of Lubrication Technology

to a maximum value and then to fall to zero at another point J =")'.The value of d p / d J is also taken as zero at J = ,'. The value of theangle")' is initially unknown.Journal bearing calculations are usually made assuming that the

shaft is long compared with its diameter, thus enabling the prob-lem to be taken as one-dimensional. This assumption will also bemade here. Alternatively, hearing calculations have been made (byOcvirk and DuBois [3]) assuming that the bearing length is smallcompared with the diameter.Most of the calculations which have been carried out in the past

have been made on the assumption that both shaft and sleevebearings are rigid and consequently do not deform. Despite thissimplification, the resulting analyses are far more complicatedthan for disc or Michell pads.In the present study we shall take into account deformations of

the bearing in the following manner. Assume, as is the case inmany examples, that the bearing is essentially a thin tube sur-rounded by a relatively rigid housing. Since the periphery of rhebearing is much larger than its thickness, the radial deformationsof the latter at a point may be assumed to be proportional to thepressure at the point. The corresponding effect on the oil filmthickness is then easily allowed for. Alternatively, the bearing maybe thought of as a so-called Winkler foundation, and modeled as aseries of closely-spaced springs. each of which can deflect indepen-dently of its neighbors. This model has been found to be very valu-able in other lubrication problems studied by Conway and Engel14.5].The shaft in the present analysis is assumed to be rigid and its

deformations ignored. This is a reasonable assumption in manycases because not only does the compact shape of the shaft causeits deflections to be small compared with those of the bearing, butthe modulus of elasticity of t.he former is usually much bigger thanthat of the latter.

OCTOBER 1975 / 599


2/6

w.

w,

HOI,I,ing 840';1'10

Fig. 1 Geometry 01Journal bearing

In the first part of the study we shall assume that the viscosityof the oil is independent of pressure. That is, the oil is isoviscous. Anumerical iteration scheme will then be used to solve the equationsinvolved. This analysis will then be extended to the case of an oilwhose viscosity increases with pressure. Isothermal conditions willbe assumed to prevail throughout the investigations.Subsequent to the completion of the above study, the important

investigation by Oh and Huebner [6 J of the lubrication of a fini te-length flexible bearing came to our attention. A special feature ofthis latter study is the use of the finite-element technique, andconsequently it can be applied to particular bearings of quite com-plicated geometry. On the other hand, the idealization in the studypresented here enables us to focus our attention on several specificparameters-three in the case of a variable viscosity lubricant-and the effects on journal bearing performance produced by theirvariation.Oh and Huebner also reference the work of several other investi-

gators, including that of Higginson [ 7 J . Higginson has used a simi-lar method to ours for predicting the response of the journal. Hiswork is l imited to an isoviscous lubricant and the method of analy-sis differs from that adopted here. Hooke, Brighton and O'Donog-hue [8 J also assume an isoviscous oil in their analysis of the journalbearing but estimate the bearing deflections more accurately.However, i t must be remembered that the actual boundary condi-tions between the bearing shell and housing are always open toquestion. Hooke, et al. [8 J came to the conclusion that, for Pois-son's ratio II = 0.28 and 0.4, the bearing response model used byHigginson [7 J and in the present article compares well with themore accurate deformation calculat ions. However, for materialssuch as rubber, for which II '" 0.5, the results are significantly dif-ferent, even for thin shells.Analysis(a) Constant Viscosity Oil. Fig. 1 shows a sketch of a lubricat-

ed journal bearing and shaft, together with pert inent dimensions.Reynolds' equation for the constant viscosity (II = 1)0) oil film of aninfinitely long bearing (end leakage ignored) is [ 2 J

(1)

where p is the pressure at angle 8, U is the peripheral velocity ofthe shaft, h = h(8) is the oil film thickness and he is the thicknessof the latter at the point where dp/d8 = 0 and 8 = 1. Taking intoaccount the radial compression of the bearing of radial thickness t ,modulus of elasticity E and Poisson's ratio" the variable oil filmthickness is

\(2)where c = (R , - R ,}) and, = eccentricity elc. Under plain strainconditions (axial strain = 0, [4, 5]), "02 is a quantity between ,,2(tangential stress = 0) and 2,,2/(1 - ,.) (tangential strain = 0; bear-ing honded to rigid housing [7). Note that p = 0 at 8 = 0 and h =cO + d = (Rl - R2 + e).Differentiating (2) we obtain

dp E dhde = 1(1 - Ill) [de + e sin Ii1 (3)Substituting (3) in (1) it follows that

dh _ 6Ur!nRzl(1 - v02) (h - he) ~ . I : i _ 0 (4)d8 E 113 ' e SID -Equation (4) may now be integrated to give

6 U1) R / (I - V 2) f 8h - he = e (cos fJ - cos y) + 0 2E 0 y h(~) - he18~)X d~ (5)

We now apply the Reynolds boundary condit ions p = 0 at 8 = 0and p = dp/dO = 0 at 0 = ,),(>11'). Thus since f 0"'(dp/dO)d8 = 0 wehave. from (1), r (h -; he ) de = 0o h (6)Introducing the nondimensional fi lm thickness ratio H (h-h ./h . equations (5) and (6) are writ ten, respectively, in the fol-lowing forms

E ~ (8 U(e d~ (7)U(fJ) = (helel (cos e - cos y) + ~- Jo [1 + H(~)P

(y H(~) d~Jo [1 + H(~)P = 0 (8)Note that in equation (7) the integral limit is changed to (0,8) be-cause of equation (8). It is now necessary to solve equation (7)subject to the constraint condition (8). The pressure distribution isobtained from equation (2) and is writ ten in nondimensional formas

- c2 II/)=6 R p=Co{=[1+1I(8)]-(1+cos9l} (9)U7)o 2 c_________ NoIDenclature- __c = R, - R2e = eccentricity (see Fig. 1)f = coefficient of friction =F /W1= 6RzI/c1 1 = coefficient of friction for complete filmregion/2 = coefficient of friction for cavitated re-gion

h = film thicknessh; = film thickness at 8 = ') ' where dp/dO=0

P = pressure

p = pc 2/6U1JoR2q = reduced pressure = =e=v]t = thickness of bearing (see Fig. 1)Co = 6U I1oR2 t(I - v02)/Ec3E = modulus of elasticity of hearingF = frictional force per unit hearing loadH = (h - hJ/h.R I= radius of bearing (see Fig. 1)R2 = radius of shaft (see Fig. 1)U =peripheral velocity of shaftW = bearing load per unit length =v'W,l+ w/W,,y = load components in x,y directions

(per unit bearing length)IVx,y IV , c2Wx/6U"oR22 , IV y

= c2Wy/6Ul1oR22a = constant in II = 71oe"PfJ = Eea]t (1- 1'02)')'=value of 8where dp/dO =0E=e/c,.=Poisson's ratio of bearing"0 = a quantity between" (tangential stress= 0) and v'2.,2/1 "I' (tangential strain =

0)II =viscosity

600 / OCTOBER 1975 Transactions of the ASME


3/6

Since P(..,,) = Hb) = 0, it followsthathclc = 1 + E COS;

The load components W % and W " per unit bearing length in the xand)' directions, respectively, are then given in nondimensionalform as

ii"~= 6~2l~ 2 = C " P cos 0 a o . 1I~ = C " P sin 0 d O(ll).'1)0 2 . 0 . 0The numerical iteration scheme adopted for the solution of

equations (7) and (8) is as follows. We first select values of Co =6U"oR21 (l - I102)/Ec3 and t = elc, both of which are composed ofknown constants, These are also independent of one another sinceCo does not contain e as ( does. We also assume a value of ")"> rwhich determines the value of h.Jc from equation (10). Now, start-ing with H(E ) = 0, new values of H(O ) are successively iteratedusing equation (7) until convergence is satisfactory for H(O) . Theintegral in (8) is then evaluated to see if it is zero. If it is not, ")"(and accordingly he/el is varied until it is. The pressure distribu-tion and load components are then found from equations (9) and(11), respectively.For numerical evaluation, it is necessary to replace the integral

by a numerical scheme such as the trapezoidal method. This meth-od is convenient since it allows any number of intervals and non-uniform subinterval sizes. Thus wewrite equation (7) aslll(O) = 110(0) ;l _ _ f_ o __ _ '\ ' T ,ll( 0;)'IJ!(O;) = HoU)j ) + < V c P 7 : 0 [1 + H(Oj )P ' i= 1,2, .... V' , H (O . \. ) = 0 (12)where

EH o ( O ) = (helC) (cos 0 - cos ,)\ ~ eJ2 , j = 0

TJ = c (~OJ + ~OJ.l)j2, j = 1,2 .... i-I M;/2. j = i

For the iteration mentioned above, we may express equation (12)asIfnl(Oj) = HMj )

C ' T H 1n - I ) ( O )+ ( l z )c )" > E l1 ~ llln-Il(Jj) P ' i = 1,2 .... ,N (14)where Hln-ll(IJN) =O . Alsolet

ER =. ~a". 11I1nl(lJj) - F"-1)(Oj) I1':!1 ... , \ _1 I

EN = H 1 n ) ( o , v ) = Hlr.l(r) (15)Thus, starting with n = I, the iteration is repeated until 11 conver-gence ('R ., 0) is reached for an assumed value of")"= ON. Then, 'IVis used to search for the correct value of l'corresponding to select-ed values of Co and e, The pressure lind load components are, re-spectively,

andN1 1 \ = 6TiJ(O,) cos 8i

Jsll.v

W y =6 Tjp(OJ) si n e,iaO

The frictional forces in the hearing may now be computed.These forces consist of two parts, namely the friction in the com-


(10)plete oil film region 0


4/6

I 0.75

0.'

0.3

o.~

0.1

20 ~ 60 SO 100 120 1:1 160 180 200 no 2:1 260Go.. .~1 ' : c I c o g ) 'v / ( W, /-. W F;/rn7.S 0.860 l.m O.5J1 0.7112 13.081

0.1 242.7 e.ess 1.931 0.519 0.706 13.3220.~5 24 '. , ~ 0.91" :.643 0 .(96 0.710 13.7890.5 Uol.3 O.9S1! 1.330 O."~ 0.717 14.80:

Fig. 2 Graphs of p and h/h" versus 8. load. frlcllonal coefficient and otherdata. for f = 0.25 and for various values of Co

whence on integrationII - he = e(cos 0 - cos )-)

6UR 1 t (1 )'0 - - ! ! L . . . " [h-c(l+feo.8) J II - II+ '210 E - "1\" I l!t(l-vO) (~)de (28)r ~

Equation (28) is written in nondimensional form by writing H = (h- h,)/h, . and hence by noting that hJe = c/e + cos 'YEH{O) = (h./c)(cos 0 - cos '1)

Co 8 at ~ ~ . eosr..eosOJ I1(~)+~ I ('de (29)(h.Jc) 'r [1 + HWr .,where Co = 6 11 01 hU t(l - ,,2 )/E c3 ~ = Eea/tO - ,.2) and f = elc.The constraint equation is ohtained as follows. From (24)dp h II --2L[h-CI!Hc0s8IJ'1 II6l.'R ""( - ..) - 6('R HI-l'02) . (" - e)dO = 2110L' h3_. 2110e ---,;r-and since f 01 (dp/d/l)d/l = 0 and writing H = (h - hJ/h. as before.we get

where again It = Eealt (1 - .,,2). Equation (29) must now be solvedsubject to the constraint condition (:10). Note that both Co and {3contain only known constants. In addition they may be chosen in-dependently of one another since, for example. Co contains '10. R2 .and U whereas { 3 does not. Equations (29) and (30) are the variableviscosity counterparts of the isoviscous equations (i) and (8); thelatter may be obtained from the former merely by writing {J =O.The solution of equation (29) subject. to the constraint condition

(30) is carried out in II similar manner to the isoviscous case.Values of Co, (3 and. = elc are selected. Also selected is a value of'Y > 11".rom which h.Jc is found from equation (0). These valuesare substituted until convergence is satisfactory for H(O). The inte-

602 / OCTOBER 1975

~.G . - O.S !VIo.2.5

0.8 z. o0.7

0.6 I.S

0.5

0.' 1.0

0.3

0.2 O.S

0.1

I.S

1.0

0.;

20


5/6

P2.0

LS

1.6

1."

1.2

1.0

0.8

0.6

0.4

0.2

20 40 60 e o 100 120 140 160 180 200 2206 D~rec\

C" r (dog.) he/c W i / o W . W f f203.4 0.312 1.033 2.076 O.78i 5.891

0.01 2OS.8 0.325 1.002 2.042 0.781 5.9~70.05 213.3 0.373 0.922 1.895 0.754 6.2530.10 219.7 0.423 0.870 1.no o.zn 6.755

Fig. 4 Graphs 01 p and IVh. versus 8, load, lrictional coefficient and otherdata, lor e = 0_75 and lor various values 01 C o

20 40 60 so 100 )20 140 160 180 10 0 n o 140Q OegrH'

1 (d09') ne/':.. W}/.w. ' I ' , ' '.if0 2'2.7 0.885 1.937 0."9 0.706 13.322

1.42.9 c.sse 1.896 0.5;0 0.708 12.6402"3.7 O.aB' 1.837 0.630 0.712 11, lSI2.8 0.69' l.n7 0.7>7 0.716 9._

Fig. 5 Graphs 01 p and n/n; versus 0, load, Iricllonal coefficient and otherdata, lor < = 0.25 and Co = 0.1 and for various values 01 {JFigs. 5 and 6 for various values of # = Eea/l (1 - "O~) . It is alwaysexpedient in iterations to use best initial values of H(IJ), which forvariable viscosity would be tbe values with { 3 = 0 or solutions withnearest values of /3 .As seen from Figs. 2, 3, and 4, the graphs of (J and h/h . are quite

sensitive to variations in the value of Co. especially for the largervalues of the eccentricity ratio < = elc. The maximum value of the


e "q.75, C.=O .O Ihih, P

2.0

1.8

1.6

I.'1.2

1.0

0.8

0.6

0.4

0.2

IVh, (~ ~ 0 - 10)

o 20 40 60 80 100 120 140 160 180 200 220g Degren

r (dog.) he.!':: W,l -Wa W I,If- f205.8 0.325 1.002 2.042 0.781 5.947205.8 0.325 1.000 2.059 0.781 5.898205.9 0.325 0.981 2. ,., 0.781 5.680

10 206.0 0.326 0.961 2.252 0.781 5.405Fig. 6 Graphs of Ii and h/h. versus 8. load, Irlcllonat coefficient and otherdata, for f = 0.75 and C o = 0.01 and tor various values 01 {J

20 normalized pressure p decreases with increases of Co as does alsothe maximum value of h/h . ..As expected, the values of the normal-ized load also decrease with increase of Co.Hearing flexibility is shown to reduce the frictional force F = /W

1 , because the pressure gradients dp/dfJ decrease in the complete filmzone and the cavitated zone (2r - 1') is also decreased. However,the load W is also reduced and the net result is an increase in both[1 and l: and hence the overall friction coefficient / = /J + f2 as

, 0 shown in the data given in Figs. 2, 3, and 4.This is contrary to the findings of Higginson 1 7 J . However,

Higginson's statement that flexibility reduces the frictional coeffi-cient appears to be based on much larger values of Co than ours. It

c. i seems possible that there is a reversal in the trend of increasingfrictional coefficients as Co is increased ami that beyond certainvalues of Co the frictional coefficients decrease.Referring now to the variable viscosity" = -qoe"P results given in

Figs. 5 and 6, it is seen that the film thickness curves h/h . . arequite insensitive to changes in the values of {3 = Ef'a/t(l - V()2).However, the maximum pressure ratio increases rapidly with in-crease of ;3 .The frictional force is found to change but little with increase of

n, and hence fl. However the load W is found to increase substan-tially with increase of 1 3. Hence the net result is a decrease of /J andf~ (and hence /l with increase of { 3 as shown in the data given inFigs. 5 and 6.References

\'011 Pauli, F. A . "Uher dell Widerstand rlpr Zapfenreibung," Kunstund Cewerbeblut t des Polytechnischen Verein des Konigreich Bayern, vol,8/9, 18-19,pp. 4&2-469. See also Cameron, A.. Principle; of Lubrication,Wil"y, New York, 1966,pp. 263-264.2 Cameron, A., Principlv. of Lubrication, Wiley, New York, 1966, PI'.

~82-289.:1 Ocvirk, F. W., and Dulsois, G. B. "Analytical Derivation and Experi.

mental Evaluation of Short Bearing Approximat ions of Full Journal Bear-ings," N.A.C.A. ']'{.ch. Report No. 1157, 1953.

OCTOBER 1975 / 603


6/6

4 Conway, H. D~and Engel, P. A., "The Elastohydrodynamic Lubrica-t ion of a Thin Layer," ,JOURNAL OF LUBRICATION TECHNOLOCY.TRANS. AS~IE, Series F, Vol.95,r-;o.3, July 1973, pp. 381-385.5 Conway, H. D., and Engle, P. A., "A Grubin-type Formula for theElastohydrodynamic Lubrication of a Thin Elastic Layer," JOURNAL OFLCBRICATION TECHNOLOGY, TRANS. ASME, Series F, Vol. 96, Apr.1974, No.2, pp. 3O

Documents

The Analysis of the Lubrication of a Flexible Journal