no

. Q

Arg

a Bl

. 4,

Received in revised form

16 March 2011

Accepted 22 March 2011Available online 5 May 2011

Keywords:

Regular perturbation

Hydrodynamic lubrication

Reynolds equation

Journal bearings

neous partial differential equation of elliptical type that, in

mostthis

shortbear-hat is

form analytical solutions exist. The signicance of the ISJB and

Contents lists available at ScienceDirect

.els

Tribology Int

Tribology International 44 (2011) 10891099length JBs contemplate these ideal limit cases. For example, oneE-mail address: gvignolo@plapiqui.edu.ar (G.G. Vignolo).ILJB solutions lie not only in the fact that they are analytical butalso in that they indicate trends, and often establish upper andlower limits to a JB performance [3]. Most of the attempts thathave been done to solve the complete Reynolds equation for nite

0301-679X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.triboint.2011.03.020

n Corresponding author at: Planta Piloto de Ingeniera Qumica (PLAPIQUI),

UNS-CONICET, CC 717 Baha Blanca, Argentina.tinuity equation, previously combined with the NavierStokesequations, into the lm thickness. The result is a nonhomoge-

not considered. In both these cases, the Reynolds equationreduces to a linear ordinary differential equation for which closedto have analytical solutions, even approximate ones, to under-stand the overall behavior of the system and the dominant effects.

When the thin-lm lubrication approximation is considered,the pressure in the lubricant lm is described by the Reynoldsequation [2]. This equation is obtained by integrating the con-

concentric cylinders separated by a uid lm [1]. The twowidely used approximations of the Reynolds equation fortype of bearings are the ones known as the innitelyjournal bearing (ISJB) [5] and the innitely long journaling (ILJB) [6], depending on the derivative of the pressure tThe understanding of hydrodynamic lubrication began in the19th century and many advances have been done since then [1].Nevertheless, the exact analytical solution of the governingequations is still not possible in most applications, and numericalmethods are required to solve them. However, it is always useful

However, approximate solutions have been obtained using elec-trical analogies, mathematical summations, relaxation methods,and numerical and graphical methods [1,3,4].

The present work is focused on journal bearings (JBs). Thehydrodynamically lubricated JBs are bearings that develop load-carrying capacity due to the relative motion of two quasi-1. IntroductionThe understanding of the behavior of hydrodynamic bearings requires the analysis of the uid lm

between two solid surfaces in relative motion. The differential equation that governs the movement of

this uid, called the Reynolds equation, arises from the integration over the lm thickness of the

continuity equation, previously combined with the NavierStokes equation. An order of magnitude

analysis, which is based on the relative value of the dimensions of the bearing, produces two

dimensionless numbers that govern the behavior of the system: the square of the aspect ratio, length

over diameter (L/D)2, and the eccentricity ratio (Z). An analytical solution of the Reynolds equation canonly be obtained for particular situations as, for example, the isothermal ow of Newtonian uids and

values of L/D-0 or L/D-N. For other conditions, the equation must be solved numerically.The present work proposes an analytical approximate solution of the Reynolds equation for

isothermal nite length journal bearings by means of the regular perturbation method. (L/D)2 is used

as the perturbation parameter. The novelty of the method lays in the treatment of the Ocvirk number as

an expansible parameter.

The zero-order solution of the Reynolds equation (obtained for L/D-0), which matches the Ocvirk

solution, may be used to describe the behavior of nite length journal bearings, up to L/D1/81/4, andrelatively small eccentricities. The rst-order solution obtained with the proposed method gives an

analytical tool that extends the description of pressure and shear-stress elds up to L/D1/2 and Z1/2(or combinations of larger eccentricities with smaller aspect ratios, or vice versa). Moreover, the friction

force and load-carrying capacity are accurately described by the proposed method up to L/D1 and Zvery near to 1.

& 2011 Elsevier Ltd. All rights reserved.

general, requires considerable numerical effort to be solved.Article history:

Received 11 November 2010Approximate analytical solution to Reynite length journal bearings

Gustavo G. Vignolo a,b,c,n, Daniel O. Barila c, Lidia Ma Planta Piloto de Ingeniera Qumica (PLAPIQUI), UNS-CONICET, CC 717 Baha Blanca,b Engineering Department, Universidad Nacional del Sur (UNS), Alem 1253, 8000, Bahc Universidad Nacional de la Patagonia San Juan Bosco (UNPSJB), Ruta Prov. No. 1, km

a r t i c l e i n f o a b s t r a c t

journal homepage: wwwlds equation for

uinzani a

entina

anca, Argentina

9005 Comodoro Rivadavia, Argentina

evier.com/locate/triboint

ernational

Y radial coordinateZ axial coordinate

G.G. Vignolo et al. / Tribology International 44 (2011) 108910991090approach is the one that assumes the existence of a homogeneoussolution plus a particular one given by the ILJB solution [3]. In thatcase, the homogeneous solution, if obtainable, may be regarded asa correction factor of the ILJB for nite length JBs [3,7].

Another approach frequently used to solve the Reynoldsequation contemplates the use of asymptotic methods [810].These methods have been used to approximate the solution touid lm lubrication problems like those of slider, step, partial,and JBs. For example, asymptotic results have been obtained for

Nomenclature

D journal bearing diameter, D2RF magnitude of the load-carrying capacityFf friction force, Ff fFFx X-component of the load-carrying capacityFy Y-component of the load-carrying capacityH lm thickness, H cecosyL journal bearing lengthO Ocvirk number, O mURPp Rc

2 LR

2O0 zero-order Ocvirk numberO1 rst-order Ocvirk numberP pressurePEXT external pressurePp projected loadPREF characteristic pressureR journal bearing radiusS Sommerfeld number, S mURPp Rc

2T dimensionless shear stress, T @u@y h t hcmU

U surface journal velocitycharacteristic tangential

velocityV characteristic radial velocityVY tangential velocityVy radial velocityinnitely long slider and step squeeze-lm bearings using differ-ent asymptotic expansions to take into account the effects of thetrailing edge and to reach smaller values of the squeeze andbearing numbers [1114]. Innitely long slider bearings andslider bearings with a discontinuity in the lm slope at highbearing numbers have also been analyzed with similar methods[15,16]. The general nite width gas slider bearing has also beenstudied in order to nd a formal explicit uniformly valid asymp-totic representation of the pressure with a low order error overthe entire bearing [1719]. In the case of short slider bearings, thedescriptions have been done by rectifying the innitely shortbearing theory and performing an asymptotic analysis to correctthe pressure eld near both, the leading and the trailing edges ofthe slider [20,21], and by solving an EulerLagrange equationusing a small aspect ratio singular perturbation approach [22].The innitely short bearing theory has been also applied tocylindrical partial-arc short bearings using a matched asymptoticperturbation method to correct the inaccuracies caused by theazimuthal edge pressure boundary conditions [20,21,23]. Thenite length JB with high-eccentricity has been analyzed in termsof inner and outer asymptotic expansions to get the pressuredistribution inside the uid lm, load-carrying capacity, andfrictional loss [24]. Furthermore, several authors have contem-plated the effect of nonzero inertia and curvature, not present inthe Reynolds equation, on the ILJB using different series expan-sions methods [2529]. Regular perturbation series expansionshave been applied to extend the range of applicability of the ISJBand to quantify the accuracy of this ideal limit solution [21].When all the previously mentioned methods and approaches areconsidered, one thing that has to be kept in mind is that theReynolds equation is a multi-parameter equation, and conse-quently, the different expanded solutions present limitations andrestrictions associated to the chosen parameters.

The constant development towards higher speed, higher per-formance, but smaller size machinery has established the trend tothe use of shorter bearings [30]. For that reason, the ISJB approx-imation has received much attention over the years. Furthermore,if an isothermal and incompressible Newtonian uid is considered,

c journal bearing clearancee eccentricityf friction coefcienth dimensionless lm thickness h Hc 1ZcospYp dimensionless pressure, p PPEXTPpp0 zero-order dimensionless pressurep1 rst-order dimensionless pressurep0 dimensionless pressure, p0 PPEXTPREFu dimensionless tangential velocity, u VYUuc dimensionless linear tangential velocity, uc Uhv dimensionless radial velocity, v VyVw dimensionless axial velocity, w VzWy dimensionless radial coordinate, y Ycz dimensionless axial coordinate, z ZLY dimensionless tangential coordinate, Y yp XRp xpe perturbation parameter, e LD

2Z eccentricity ratio, Z ecm uid viscosityt shear stressVz axial velocityW characteristic axial velocityX tangential coordinatethis approximation yields a simple pressure eld, which is afunction of both, the tangential and axial coordinates, and satisesall pressure boundary conditions [1,3]. This solution is frequentlyapplied in the areas of rotor-bearing dynamics, dampers, and shaftseals, even though the accuracy of the ISJB solution depends onhow close the real conditions are to the assumed ones. In thatsense, the nonzero aspect ratio (length over diameter) andeccentricity are the most important factors that deviate the realsolution from the ISJB approximation [21,31]. In order to getmore accurate results, specic analytic developments have beencarried out to perform dynamic analyses. In general, these devel-opments use different combinations of short and long bearingsolutions [32,33].

The purpose of the present work is to provide a new analyticalsolution valid to nite length JBs obtained as an extension of theISJB approximation by means of the regular perturbation method.In the following section, the governing equations as well as theresulting equations from the order of magnitude analysis arepresented. Then, the proposed approach to the regular perturba-tion method is discussed, and nally the predicted variables arecompared to the results from numerical solutions as well as otheranalytical approaches.

2. Governing equations

Fig. 1 displays a scheme of the JB together with the associatedcoordinate system and some dimensions. The journal, of radius R,turns inside a bearing of length L at an angular speed O. The loci

G.G. Vignolo et al. / Tribology International 44 (2011) 10891099 1091of the journal and the bearing are separated by a distance, e,called eccentricity. The maximum value that e can reach, c, is thedifference between bearing and journal radii. The gap betweenthem, H, is fully lled with an incompressible uid, the lubricant.Although both, journal and bearing, are cylinders, the ow of thelubricant in the gap can be studied in rectangular coordinates{X,Y,Z}, ignoring the curvature, because R is several orders ofmagnitude larger than c. Dimensionless coordinates Y, y, andz are dened by means of the characteristic values pR, c,and L as

Y XpR

, y Yc

, z ZL

1

For isothermal ow, the dimensionless variables of interest arethe tangential velocity, u, axial velocity, w, radial velocity, v, and

0

Fig. 1. Geometry and system of coordinates.pressure, p , dened as

u VYU

, v VyV

, w VzW

, p0 PPEXTPREF

2

where U, V, W, and PREF are the characteristic values of thecorresponding variables.

The dimensionless mass balance equation:

0 UpR@u

@Y V

c

@v

@yW

L

@w

@z3

determines the values of VUc/R and WUL/R if the three termsare of similar order of magnitude.

The dimensionless momentum conservation equations can belargely simplied neglecting the terms that are of an order ofmagnitude c/R, since this ratio normally has values of the orderof 1/1000. Accordingly, the nal expressions of the Y-, y- andz-direction momentum balances are, respectively

0 PREFpR@p0

@Ym U

c2@2u

@y20 PREF

c

@p0

@ym V

c2@2v

@y2

0 PREFL

@p0

@zmW

c2@2w

@y24

The main assumption of the ISJB approximation, known asthe Ocvirk solution [5], is that the pressure gradient in theY-direction can be neglected when compared to the axial one.This means that the tangential speed prole should be linear, likein Couette ow, and that PREF can be estimated from thez-direction momentum balance and dened as

PREF mUR

R

c

2 LR

25

In this case, the dimensionless mass and momentum equationstake the following form:

0 1p@u

@Y @v

@y @w

@z6

0 1p

L

R

2 @p0@Y

@2u

@y20 @p

0

@y c

L

2 @2v@y2

0 @p0

@z @

2w

@y2

7On the other hand, the main assumption of the ILJB, known as

the Sommerfeld solution [6], is that the axial pressure gradientis much smaller than the tangential one. No axial speed prole isthen expected. The value of PREF can be then calculated from theY-direction momentum equation, and dened as PREFmU/R(R/c)2.In both, the ISJB and the ILSB cases, the pressure gradient in theradial direction can be neglected because the momentumconservation equations in the y-direction are (c/R)2 or (c/L)2 timessmaller, respectively, than the equations in the other twodirections.

2.1. Reynolds equation

The so called Reynolds equation [2] is obtained combining themass and momentum balances and integrating the mass balancewithin the thickness of the lm considering non-slip boundaryconditions over the walls for all velocity components. If h is thedimensionless gap, dened as hH/c, the dimensionless expres-sion of the complete Reynolds equation is

dh

dY 1

6pL

R

2 @@Y

h3@p0

@Y

p

6h3

@2p0

@z28

when the dimensionless pressure PREF is dened according toEq. (5) for short JBs.

Clearly, the use of the ISJB hypothesis in Eq. (8) makes the termthat includes the (L/R)2 factor negligible. The resulting simpliedequation has only two terms and it is analytically solvable usingappropriate boundary conditions for the pressure. It is clear fromthe comments above that an extension of the two ideal cases (ISJBand ILJB) to nite length JBs should consider the fact that theorder of magnitude of pressure changes with aspect ratio.

2.2. Sommerfeld and Ocvirk numbers

Traditionally, the behavior of JBs has been related to the Bearingor Sommerfeld number [1,6], S, or the capacity or Ocvirk number[5,34], S(L/D)2. In the following sections, a slightly modied denitionof the Ocvirk number, O, is used, in which the square of the length-to-diameter ratio is replaced by the square of the length-to-radiusratio. The resulting used expressions are

S mURPp

R

c

2, O S L

R

29

where Pp is a mean pressure dened as the ratio between the load-carrying capacity and the projected area, 2RL. Thus, the Sommerfeldnumber represents the ratio between PREF for ILJBs and the meanpressure Pp, while the Ocvirk number corresponds to the ratiobetween PREF for ISJBs (Eq. (5)) and Pp. Traditionally, these numbersare dened using the shaft speed in RPS [1,3,35], which introduces afactor of 2p with respect to the denitions of both S and O in Eq. (9).

If the Sommerfeld and the Ocvirk numbers are introduced in

the Reynolds equation, the dimensionless pressure changes from

terms in the pressure expansion are determined by a recursionformula. However, while the correction terms in this series should

G.G. Vignolo et al. / Tribology International 44 (2011) 108910991092decrease in importance as the order increases, this does nothappen neither near the minimum gap nor at values of eccen-tricity close to unity. This occurs because this is a multipleparameter problem. To solve it, the authors proposed a morestringent scaling of the Reynolds lubrication equation consideringthe aspect ratio and the eccentricity parameter to be in the sameorder of magnitude. Consequently, they pointed out that thePoiseuille ow component in the sliding direction becomesimportant near the minimum gap region when the eccentricityis sufciently large for any given value of aspect ratio.

However, if only one length scale is considered, like in the rstpart of the work of Buckholz and Hwang [21], more than onevariable must be taken in account in the series expansion to keepin balance the order of magnitude of the terms of the differentialequations. For example, in addition to the pressure, other authors[25,29] have expanded the stream function or the Reynoldsnumber, but none of these approaches were applied to ISJBs.

In the present paper, an extension of the short bearing approx-p0 to p, which is given by

p PPEXTPp

10

and Eq. (8) becomes

Odh

dY S L

R

2 dhdY

16p

L

R

2 @@Y

h3@p

@Y

p

6h3

@2p

@z211

This equation shows that Smust tend to innity as (L/R)2 tendsto zero in order to keep the equality between the two dominantterms. Consequently, when applied to ISJBs, the solution of thetwo remaining terms of Eq. (11) can only be used for bearingsworking at very high Sommerfeld numbers. Furthermore, sincethe larger the Sommerfeld number, the smaller the eccentricity[1], when a bearing works at high Sommerfeld number, theeccentricity tends to zero, and vice versa. Then, the ISJBs solutionis also accurate for very small eccentricities.

3. Finite length JBs and proposed analytical solution

Although in their work, Dubois and Ocvirk [5] suggested thatthe ISJB approximation could be used for aspect ratios up to one,the following research demonstrated that this assumption is onlyjustied for bearings with L/D smaller than 1/8, for all eccen-tricities. In practice the assumption is used for L/D up to 1/2 andeccentricities up to 0.75 [35].

The analysis of nite length JBs may be done by shorteningthe ILJB or lengthening the ISJB. The former approach can beapplied considering an edge effect on the extremes of the bearingwith a thickness of order of magnitude R/L. Each edge is treatedas a boundary layer by means of functional analysis, and thesolution of this zone is then matched to that of the center zoneof the bearing [7].

The extension of the solution from ISJBs to nite lengthbearings can be done by applying the Regular PerturbationMethod [36]. Perturbation approaches have historically beenviewed as a viable method for analyzing higher order effects.Buckholz and Hwang [21] proposed an analysis of the Reynoldsequation for short bearings by introducing a regular perturbationexpansion for the pressure. The mathematical problem wasdetermined by an ordering of the pressure terms of the seriesaccording to powers of the bearing aspect ratio. The expression ofthe zero-order solution is the Ocvirk solution while higher orderimation to nite length JBs is proposed using a regular perturbationmethod with the perturbation parameter, e, dened as

e LD

212

where D is the diameter of the journal (D2R). The difference withrespect to previous methods is that the balance between the termsof the Reynolds equation is kept by expanding not only thepressure but also the Ocvirk number. In that way, the resultingpressure eld should be able to describe JBs with higher eccentri-cities and higher aspect ratios than previous methods in whichonly the pressure was expanded. Thus,

p p0L

D

2p1O LD

4-p p0ep1Oe2 13

and

OO0L

D

2O1O LD

4-O O0eO1Oe2 14

Introducing these expansions into the Reynolds equation(Eq. (11)) and keeping the terms up to order e, gives

O0eO1dh

dY 2

3pe@

@Yh3

@p0@Y

p

6h3

@2p0ep1@z2

15

To solve this equation, the dimensionless gap, h, can becalculated with

h Hc 1ZcospY 16

where Z is the eccentricity ratio dened as

Z ec

17

The zero-order solution (p0 and O0) is the solution of the ISJB(L/D0), known as the Ocvirk solution. This solution gives thepressure eld

p0 3O01

4z2

ZsinpY

1ZcospY 3 18The boundary conditions used to obtain this equation are

p00 at z1/2, and qp0/qz0 at half the length (z0). Thispressure eld automatically satises the conditions along theazimuthal boundaries for a JB, that is, p00 at Y0 and Y2.

The rst-order solution of Eq. (15) gives

p1 2a1ZsinpYa2Z2 sin2pY1=4a2Z3 sin3pY

a3 cospYa4 cos2pYa5 cos3pYa6 cos4pYa7 cos5pYa819

where

a1 O04524z216z427Z2 3O1

214z24Z2

a2 O0524z216z46O114z2a3 80Z120Z310Z5 a4 80Z240Z4 a5 40Z35Z5a6 10Z4 a7 Z5 a8 1680Z230Z4 20

In order to calculate O0 and O1, it is necessary to recall that theOcvirk number denition (Eq. (9)) includes the mean pressure, Pp,dened as

Pp F2RL

21

where the load-carrying capacity, F, is the magnitude of the loadapplied to the journal. If the system is equilibrated, then load hasthe same value as the force done by the lubricant to the journal.Consequently, F can be obtained integrating the uid pressureeld over the journal area (or bearing area, since c5R).

Another fact that has to be considered is that the lm thickness

is convergent in 0oYo1 and divergent in 1oYo2. The pressure

cities, and in the data for larger aspect ratios in the region ofZ-0. As the aspect ratio increases, the curves separate as the

G.G. Vignolo et al. / Tribology International 44 (2011) 10891099 1093eld given by Eq. (18) reaches positive values in the rst region andnegative ones in the second one. Given the limited capability ofliquids to support negative pressures, the method frequently usedto avoid the unrealistic sub-ambient pressures of the divergentzone when calculating F is to ignore them. This approach, known asthe p lm approach, is the most frequently used among analy-tical studies of JBs and it is also adopted in this work.

According to Fig. 1

F2 F2x F2y 22

where

Fx PpLRpZ 1=21=2

Z 10pcospYdzdY 23

and

Fy PpLRpZ 1=21=2

Z 10psinpYdzdY 24

Replacing Eqs. (23) and (24) into Eq. (22) and combining withEq. (21) gives

4

p2Z 1=21=2

Z 10p0ep1cospYdzdY

" #2

Z 1=21=2

Z 10p0ep1sinpYdzdY

" #225

which can be expressed as

4

p2Z 1=21=2

Z 10

p0 cospYdzdY" #2

Z 1=21=2

Z 10

p0 sinpYdzdY" #2

2eZ 1=21=2

Z 10

p0 cospYdzdYZ 1=21=2

Z 10

p1 cospYdzdY" #

2eZ 1=21=2

Z 10

p0 sinpYdzdYZ 1=21=2

Z 10

p1 sinpYdzdY" #

Oe2

26O0 and O1 can then be obtained by solving the expressions of

order-zero and order-e in Eq. (26), respectively. This proceduregives

O0 4k2

Z4Z2p2k4I20

q 27and

O1 2

5O0

3Z21Z22p2k5I0b27Z2Is11kZ2p2k5I0b4Z2Is11

28

with

k 1Z2 b Z4Is12ZIs13 I0 Z 10

sin2pY1ZcospY3

dY

Is11 Z 10

sin2pYdenom

dY Is12 Z 10

sinpYsin2pYdenom

dY

Is13 Z 10

sinpYsin3pYdenom

dY

denom a3 cospYa4 cos2pYa5 cos3pYa6 cos4pYa7 cos5pYa8 29

The coefcients ai are dened in Eq. (20).The proposed method will be identied as the P&O-perturba-

tion in the rest of the paper making reference to the fact the bothpressure and Ocvirk number are expanded in order to extend theISJB solution to describe nite length JBs.

For comparison reasons, two other perturbation expansions

will be considered in the paper in which only the pressure iseccentricity increases, the Ocvirk number calculated with theP&O-perturbation method being the one that follows more closelythe numerical solution of Eq. (11). Additionally, it may be noticedthat, agreeing with the hypotheses of the Ocvirk solution, itsaccuracy deteriorates when the Ocvirk number decreases. Thedistance between the ISJB and the P&O-perturbation solutions isdue to the value of O1. The Ocvirk number from the P-perturba-tion, which is always larger than O0eO1, also shows an improve-ment at low eccentricities, in the range of applicability of theOcvirk solution, but gives non-real results outside this range.

Once the Ocvirk number has been calculated, the load-carryingcapacity, F, can be obtained from Eq. (9) as follows:

F 2mLUO

R

c

2 LR

230

where O is the Ocvirk number given by different solutions inexpanded. In one of them, p0 and O0 are the zero-order solution,like in the P&O-perturbation, and p1 is calculated using O0. Thismethod, which will be identied as P0-perturbation, produces apressure solution that does not satisfy Eq. (26) but is of order e.Furthermore, since the P0-perturbation solution is equivalent tothat of the P&O-perturbation method with O10, the comparisonbetween these two solutions will clearly show the effect of O1 inthe proposed method. In the other case, the Ocvirk number iscalculated introducing the expanded expression of the pressure(p0ep1) in Eq. (26). This approach gives an O that is a function ofe and that will affect the zero-order term of the pressure, whichwill not match that of the ISJB solution when more than one termis considered. This method, which produces a pressure that is notof order e, is equivalent to the one of Buckholz and Hwang [21]when only two terms are considered. The predictions of this lastmethodology will be identied as P-perturbation.

4. Results

In the following sections, the results from the P&O-perturba-tion method are discussed and compared to the Ocvirk solution(ISJB solution), the numerical exact solution of Eq. (11), and theother two perturbation expansions previously commented. Thecomparison is carried out for L/D up to 1 and eccentricitiesbetween 0 and 1. The Ocvirk number is analyzed in the rstsection, followed by the discussion of the pressure proles.Finally, the shear stress proles and friction force results areanalyzed.

4.1. Ocvirk number and load carrying capacity

Fig. 2 displays the value of the Ocvirk number as a function ofthe eccentricity ratio for four different values of L/D. The analy-tical results from the ISJB approximation (O0) and the P-perturba-tion (O) and P&O-perturbation (O0eO1) methods are presentedtogether with the numerical results of Eq. (11) and the resultsfrom Raimondi and Boyd [37]. The difference between Raimondiand Boyds numerical solution and that of the full Reynoldsequation arises from the boundary condition applied to pressure.Raimondi and Boyd used the Reynolds boundary condition [1,38]while the solution of Eq. (11) is obtained in this paper usingGumbels (or p) boundary condition [38], which is also used in theperturbation methods.

As expected, all results agree in the limit of small aspect andeccentricity ratios. This can be seen in the data for L/D1/8,where the curves are practically indistinguishable at all eccentri-Fig. 2. In the case of the P&O-perturbation, an analytical solution

0.0

1E-2

1E-1

1E+0

1E+1

1E+2

O

L/D = 1/4

1E-1

1E+0

1E+1

1E+2

O

L/D = 1

0.2 0.4 0.6 0.8 1.0

G.G. Vignolo et al. / Tribology International 44 (2011) 1089109910940.0

1E-2

1E-1

1E+0

1E+1

1E+2

OL/D = 1/8

0.462.4

2.8

3.2

1E-1

1E+0

1E+1

1E+2

O

L/D = 1/2

0.2 0.4 0.6 0.8 1.0

0.48 0.50 0.52

ISJB (O0)P-pert. (O) P&O-pert. (O0+O1)Numerical sol. of Eq. (11) Raimondi & Boyd [37]of F may be obtained using OO0eO1 and the expressions of O0and O1 are given by Eqs. (27) and (28).

4.2. Pressure proles

Fig. 3 displays the values of p0(Y) and p1(Y) as a functionof eccentricity ratio calculated with Eqs. (27) and (28) at z0.p0 is also the Ocvirk solution and the zero-order term of theP0-perturbation method. This function has a sinusoidal depen-dency with the azimuthal position when Z-0 (as a result of theway the dimensionless pressure has been dened) and displays amaximum value that gradually shifts towards the region ofminimum gap as the eccentricity increases. The rst-ordersolution of the Reynolds equation obtained with the P&O-perturbation method gives curves of p1(Y) that tend to increasethe zero-order solution at small angles and to reduce it atlarge angles. As expected, this effect is small at small eccentri-cities and gets dramatic as the eccentricity increases. The pre-dicted pressure (p0ep1) will be slightly larger than in the ISJBfor Yo0.4 regardless of the aspect and eccentricity ratios(for L/Do1). At larger angles, the larger the eccentricity, thesmaller the aspect ratio (and vice versa) that will keep thecorrective term to be of smaller order of magnitude thanthe zero-order pressure term. For certain combinations of aspectand eccentricity ratios, the pressure may become negative nearthe minimum gap.

As an example, Fig. 4 shows the pressure proles predictedby the ISJB approximation and the different perturbation methodsas a function of the azimuthal direction, calculated at z0

1E-20.0

1E-20.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0

Fig. 2. Ocvirk number as a function of eccentricity ratio for different aspect ratios.

Fig. 3. Dimensionless zero- and rst-order functions of the expanded pressureaccording to the proposed P&O-perturbation method at z0. The curves arepresented with steps of 0.1 in the value of eccentricity.

G.G. Vignolo et al. / Tribology International 44 (2011) 10891099 1095and the selected conditions L/D0.5 and Z0.5. The exactsolution of Eq. (11), obtained numerically, is also displayed inthat plot.

As it may be appreciated, all approximate solutions arequalitatively correct although the prediction of the P&O-pertur-bation is the one closest to the numerical solution, at least up tothe vicinity of the maximum pressure. This method, althoughlocates pmax a little before the position given by the numericalmethod (difference in YmaxE0.03), is the one that gives theclosest value (underpredicts the peak pressure by just 1.3%).The ideal Ocvirk solution, on the other hand, overestimates the

0.0

0

1

2

3

p

ISJB

P0-pert.

P-pert.

P&O-pert.

Numerical

0.2 0.4 0.6 0.8 1.0

Fig. 4. Dimensionless pressure proles predicted by the different methods at z0for L/D0.5 and Z0.5.maximum pressure by 8.6% and locates it after its position(difference with respect to the numerical solution ofYmaxE0.03).The P0-perturbation method, which uses O0 in the calculation ofp1, largely underestimates the peak pressure by 33% because p1reaches relatively large and negative values. The P-perturbationsolution, on the other hand, overestimates the peak pressure by12%. Both these methods place the maximum pressure before itslocation (difference inYmaxE0.05). The results displayed in Fig. 4show that the P&O-perturbation method is the one that, ingeneral, better captures the physics of the ow, at least at theaspect and eccentricity ratios considered in this gure. Thismethod predicts a pressure eld that practically matches theexact solution up toY0.4, stays the closest up toY0.6, and itis not far away from it at larger angles. When the sum of thesquare differences between the pressure predicted by the differ-ent methods and the numerically calculated are considered for0oYo1, the smallest value is that of the P&O-perturbationmethod followed by the ISJB approximation.

A more complete analysis of the capabilities and limitations ofthe proposed method can be done from Fig. 5. This gure displaysthe pressure proles calculated at z0 as a function of journalaspect ratio (for Z0.5) and eccentricity ratio (for L/D0.5) atthree different azimuthal positions. Within the analyzed ranges,the results show that, the zero-order solution matches the exactnumerical results at all azimuthal positions only at very smallaspect and eccentricities ratios (L/Do0.1 and Zo0.05). ForZ0.5, the P0-perturbation method makes an improvement overthe Ocvirk solution only at small values of Y, regardless the

numerical results in larger ranges of aspect and eccentricity ratios

than the ISJB method. The rst-order solution obtained with theproposed P&O-perturbation method matches the exact numericalresults up to L/D of at least 0.4 and improves the matching outsidethis range at practically all tested conditions.

The friction force, Ff, which corresponds to the integration ofthe uid shear stress over the journal area, can be also dened asthe product of a friction coefcient, f, and the load-carryingcapacity,

Ff Z 20

Z 1=21=2

t hpRLdYdz fF 33

Accordingly, a friction coefcient, f R=c L=R 2, can be obtainedcombining Eq. (33) with Eq. (30) and considering tmU=c@u=@y and that, according to the p lm hypothesis, in thedivergent zone of the uid lm, 1oYo2, a Couette type of owexists (linear velocity prole). That is,

fR L 2

1pOZ 1 Z 1=2 @u dYdz 1pO

Z 2 Z 1=2 1dYdz 34aspect ratio (Fig. 5, left), while for L/D0.5, it worsens the ISJBsolution at small eccentricities for all Ys and at practically alleccentricities ratios for Y40.5, always underestimating thepressure. The P-perturbation method, on the other hand, givesgood results only at low aspect (L/Do0.3) and eccentricities(Zo0.2-0.3) ratios, regardless of the angular position. Outsidethese ranges, it overestimates the pressure to nally diverge tonon-real values. Then, according to Fig. 5, the ISJB approximationgives, in general, better predictions of p(Y) than the P- andP0-perturbation methods, at least at the selected conditions. TheP&O-perturbation, on the other hand, largely improves the pre-dictions of the ISJB method. The pressure calculated with theproposed method agrees with the numerical results in largerranges of aspect and eccentricity ratios and produces realisticvalues outside those ranges. Furthermore, p(Y) practicallymatches exactly the numerical results for L/Do0.35 at all Ys(for Z0.5) and for small Ys at all Zs and for Zo0.35 at largerYs (L/D0.5).

4.3. Friction force

Another useful variable to analyze is the shear stress. Thedimensionless shear stress at the moving wall is dened as

T @u@y

h

thc

mU 31

which corresponds to the integration of the dimensionlessY-Direction Momentum Balance at the moving wall, after apply-ing no-slip boundary conditions, that is

T @u@y

h

1pOL

R

2 @p@Y

h

2 1

h

!32

Fig. 6 shows the shear stress prole at the moving wallcalculated at the same conditions than the pressure proles ofFigs. 4 and 5. The displayed results demonstrate that the cap-abilities and limitations of the proposed method to predict both,pressure and shear stress proles, are the same. As expected, thezero-order solution of the Reynolds equation describes thebehavior of very short JBs and small eccentricities. The P0- andP-perturbation methods give the same shear stress proles, atleast in the range of real values of O of the P-perturbation. This isbecause p(Y) has the same analytical expression in both cases(pp0ep1O), differing only in the value of O. According toFig. 6, these methods give values of T that approach those of thec R 2 0 1=2 @y h 2 1 1=2 h

G.G. Vignolo et al. / Tribology International 44 (2011) 1089109910960.0L/D

0.0

0.5

1.0

p = 0.2

1.5

2.0

2.5

p

= 0.5

0.2 0.4 0.6 0.8 1.0which can be combined with Eq. (32) to give the followingexpression:

fR

c

L

R

2 e

Z 10

Z 1=21=2

h@p

@YdYdz 1

2pOZ 20

Z 1=21=2

1

hdYdz 35

The dominant term in Eq. (35) is the second integral-term inthe right-hand side (Couette ow), which establishes that thefactor f R/c is of order (R/L)2. Furthermore, this integral-term(evaluated with O0) corresponds to the prediction of f R=c L=R

2of the ISJB approximation (L/D-0) and to the zero-order term ofthe P0- and P&O-perturbations (using O0) and of the P-perturba-tion (using O). The corrective rst-order terms of the P0- andP-perturbations are given by the rst integral-term evaluatedusing the zero-order expression of the pressure. The P&O-pertur-bation method contemplates an additional rst-order term, theone obtained from replacing O by (O0eO1) in the second integral.

Fig. 7 displays the friction coefcient as a function of eccen-tricity for four different aspect ratios. The gures also include theexact numerical solution of the Reynolds equation and the resultsfrom Raimondi and Boyd. At L/D1/8, the predicted frictioncoefcients are practically indistinguishable at all eccentricities.

0.0L/D

0.5

1.0

0.0L/D

1.0

1.5

2.0

2.5

3.0

3.5

p

= 0.7

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

Fig. 5. Dimensionless pressure proles at three different tangential positions as a f0.0

0.0

0.5

1.0

1.5

p

= 0.2

ISJBP0-pert. P-pert. P&O-pert. Numerical

1.0

1.5

2.0

p

= 0.5

0.2 0.4 0.6 0.8 1.0The same occurs at low eccentricities and larger aspect ratiosalthough, as expected, the range of agreement of the predictionsdecreases as L/D increases. The rst-order term in theP0-perturbation expansion (generated by the azimuthal variationof p) slightly increases the value of the friction coefcient, whichis still too low compared with the exact value (mainly at large L/D).The P-perturbation, extends the range of accuracy of the ISJB,mainly up to aspect ratios of about 0.5, but predicts non-realresults at values of eccentricity that abruptly decrease asthe aspect ratio increases. The best predictions are those of theP&O-perturbation method, that fall extremely close to the exactsolutions. The accuracy of this method in the calculation ofthe friction coefcient extends up to L/D1 and e practically 1.Undoubtedly, the P&O-perturbation, with the expansion of both,the pressure and the Ocvirk number (which is a dimensionlessmean pressure), is the one that better captures the physics of theJB ow.

Finally, following a criteria similar to the one of Pandazarasand Petropoulos [39], Fig. 8 presents iso-operational curves usingthe Ocvirk number as the operational parameter. The guredisplays eccentricity and friction coefcient as a function ofaspect ratio at O1, 10, and 100. Fig. 8(left) shows that different

0.0

0.0

0.5

0.0

0.0

1.0

2.0

3.0

p

= 0.7

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

unction of dimensionless length (left, Z0.5) and eccentricity (right, L/D0.5).

pert

G.G. Vignolo et al. / Tribology International 44 (2011) 10891099 10973.0ISJB P0-pert. y P-values of O determine different ranges of eccentricity that shouldbe used for the range of aspect ratio considered in this work.Similarly, Fig. 8(right) shows that those values of O determinedifferent ranges of friction coefcient. As expected, as the Ocvirk

0.00.0

1.0

2.0

T

L/D = 0.5 - = 0

P&O-pert.Numerical

0.00.5

1.0

1.5

2.0

2.5

T

= 0.2 = 0.5

0.0L/D

1

2

3

4

T

= 0.5 = 0.5

0.0L/D

0

1

2

3

4

T

= 0.7 = 0.5

0.2 0.4

0.2 0.4 0.6 0.8 1.0L/D

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

Fig. 6. Dimensionless shear stress proles as a function of tangential position (top, Z. number decreases, the eccentricity increases and the frictioncoefcient decreases.

The gure shows that, at low L/D, all curves match at allOcvirk numbers. However, as the aspect ratio increases, the

.5

0.0

1.0

1.1

1.2

1.3

1.4

T

= 0.2 - L/D = 0.5

0.0

1

2

3

4

T

= 0.5 - L/D = 0.5

0.0

1

2

3

4

5

T

= 0.7 - L/D = 0.5

0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

0.5, L/D0.5), dimensionless length (left, Z0.5) and eccentricity (right, L/D0.5).

G.G. Vignolo et al. / Tribology International 44 (2011) 1089109910981E+1

1E+2

1E+3

c (L

/R)2

L/D = 1/8

ISJBoperational curves separate. The ISJB approximation (L/D-0)predicts constant values for each O while the other techniquesdisplay curves with values that increase as L/D increases. Itcan also be appreciated that, the lower the value of O, the largerthe separation. In agreement with the results already displayedin the previous gures, the P&O-perturbation method is the onethat follows closely the numerical solution of the Reynoldsequation.

0.0

1E-1

1E+0

f R/ P0-pert.

P-pert. P&O-pert. Numerical sol. of Eq. (11) Raimondi & Boyd [37]

0.0

1E-1

1E+0

1E+1

1E+2

1E+3

f R/c

(L/R

)2

L/D = 1/2

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

Fig. 7. Dimensionless friction coefcient as a functio

0.01E-2

1E-1

1E+0

O = 100

O = 10

O = 1

0

ISJBP0-pert.P&O-pert. Numerical

0.5 1.0L / D

Fig. 8. Dimensionless eccentricity (left) and friction coefcient (right)1E+1

1E+2

1E+3

c (L

/R)2

L/D = 1/45. Conclusions

An analytical tool to perform nite journal bearing calculationsis introduced. The novelty of the proposed regular perturbationmethod lies in that both, the pressure and the Ocvirk number, areexpanded. This idea is supported by the fact that the Ocvirknumber is a dimensionless mean pressure and that the order ofmagnitude of the pressure varies with aspect ratio. The square of

0.0

1E-1

1E+0

f R/

0.0

1E-1

1E+0

1E+1

1E+2

1E+3

f R/c

(L/R

)2

L/D = 1

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

n of eccentricity ratio for different aspect ratios.

.0L / D

1E+0

1E+1

1E+2

1E+3

O = 1

O = 10

O = 100

0.5 1.0

f R/c

(L/R

)2

as a function of aspect ratio for three constant Ocvirk numbers.

the aspect ratio is then used as the perturbation parameter.Pressure and shear-stress, as well as the Ocvirk number and thefriction coefcient (total friction force over load-carrying capa-city) were calculated and analyzed as a function of azimuthalposition, if applies, and aspect and eccentricity ratios.

The zero-order solution of the proposed perturbation methodcorresponds to the ISJB approximation (L/D-0). This approxima-tion correctly describes the behavior of short journal bearings upto approximately the following limits: Zo1 for L/D1/8; Zo0.6for L/D1/4; Zo0.2 for L/D1/2; and Z-0 for L/D1. The rst-order solution of the proposed method extends the limits up toZ0.9 for L/D1/4; Z0.6 for L/D1/2; and Z0.4 for L/D1. Inthe case of variables that correspond to the integration ofpressure or stress elds, like friction coefcient and load-carrying

[14] Di Prima RC. Asymptotic methods for an innitely long step slider squeezebearing. ASME J Lubr Technol 1973;95:20816.

[15] Di Prima RC. Higher order approximations in the asymptotic solution of theReynolds equation for slider bearings at high bearing numbers. ASME J LubrTechnol 1969;91:4551.

[16] Schmitt JA, Di Prima RC. Asymptotic methods for an innite slider bearingwith a discontinuity in lm slope. ASME J Lubr Technol 1976;98:44652.

[17] Eckhaus W, De Jager EM. Asymptotic solutions of singular perturbationproblems for linear differential equations of elliptic type. Arch Rational MechAnal 1966;23:2686.

[18] Grasman J. On singular perturbation and parabolic boundary layers. J EngMath 1968;2:16372.

[19] Di Prima RC. Asymptotic methods for a general nite width gas sliderbearing. ASME J Lubr Technol 1978;100:25460.

[20] Schuss Z, Etsion I. On the solution of lubrication problems involving narrowcongurations. ASLE Trans 1981;24(2):18690.

[21] Buckholz RH, Hwang B. The accuracy of short bearing theory for Newtonianlubricants. ASME J Tribol 1986;108:739.

[22] Rohde SM, Li DFA. Generalized short bearing theory. ASME J Lubr Technol1980;102:27882.

[23] Buckholz RH, Lin J, Pan CHP. On the role of axial edge effects and cavitation inlubrication for short journal bearings. ASME J App Mech 1984;52(2):26773.

G.G. Vignolo et al. / Tribology International 44 (2011) 10891099 1099Consequently, the treatment of the Ocvirk number as anexpansible variable produces noticeable improvements in theanalytical description of the ow elds of journal bearings, andeven more remarkable ones in the calculations of load-carryingcapacity and friction coefcient.

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[32] Hirani H, Athre K, Biswas S. Dynamically loaded nite length journalbearings: analytical method of solution. ASME J Tribol 1999;121(4):84452.

[33] Bastani Y, de Queiroz MA. New analytic approximation for the hydrodynamicforces in nite-length journal bearings. ASME J Tribol 2010;132(1). (014502-1-9).

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[36] Bender CM, Orszag SA. Advanced mathematical methods for scientists andengineers. New York: McGraw Hill; 1978.

[37] Raimondi A, Boyd JA. Solution for the nite journal bearing and its applica-tion to analysis and design, parts I, II, and III. ASLE Trans 1958;1(1):159209.

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[39] Pandazaras C, Petropoulos G. Tribological design of hydrodynamic slidingjournal bearings. Formulating new functional charts. Ind Lubr Tribol 2005;57:411.[24] Tayler AB. A uniformly valid asymptotic solution of Reynoldss equation: thecapacity (and even the Ocvirk number itself) the improvementin the range of accuracy is larger. The friction coefcient is, forexample, very well described even for bearings with aspect ratioup to 1 and medium to high eccentricities.

Approximate analytical solution to Reynolds equation for finite length journal bearingsIntroductionGoverning equationsReynolds equationSommerfeld and Ocvirk numbers

Finite length JBs and proposed analytical solutionResultsOcvirk number and load carrying capacityPressure profilesFriction force

ConclusionsReferences