PERTANIKA14(3),359-371 (1991)

COMMUNICATION VII

A Subroutine Package for Solving Hydrodynamic Lubrication Problems

ABSTRAK

Kertas-kerja ini membentangkan satu perisian subrutin untuk menyelesaikan masalah pelincir hidrodinamik yangberoperasi dibawah saput sesuhu. Analisis dan algoritma pengiraanjuga diterangkan. Kaedah unsur terhingga telahdigunakan untuk menyelesaikan kedua-dua kes aliran lamina dan gelora. Satu contoh galas jara susuk tiga cupingdigunakan untuk menunjukkan penggunaan perisian subrutin ini.

ABSTRACT

The paperpresents a subroutinepackage that solves theproblem ofhydrodynamic lubrication operatingunder isothermalfilm. The underlying analysis and computational algorithm are described. The finite element method was used forsolving thepressure equationfor cases ofboth laminar and turbulentflow. A representative example ofa three-lobeprofilebore bearing is used to demonstrate the application of the subroutine package..

INTRODUCTION

The need to reduce machinery failure andmaintenance costs while simultaneously increasingtheir power output and efficiency has led tonumerous studies on ways of improvingperformance, reliability and life expectancyofcriticalmachine elements. This has resulted in muchdevelopmentand advancement in design technologyin the area of highly loaded rotating machineryparticularly for elements such as bearings, camfollowers and gear systems.

Advances in computer techniques in analysishave enabled the efficient numerical solution oflubrication problems. These include principallypowerful numerical tools such as the finite differ­ence method, the finite cell method and recentlythe finite element method. The most commonlyused approach is the finite difference scheme. Thismethod gives a point-wise approximation to thegoverning equations. The method, obtained bywriting difference equations for an array of gridpoints, becomes more and more accurate as morepoints are considered. However, the finite differ­ence scheme becomes inconvenient to use whenirregular geometry or unusual boundary conditionsare encountered. When complex geometricalconfigurations and abrupt changes in field proper­ties are involved, the finite difference method be­comes inherently difficult to apply because of theneed to employ irregular meshes and special auxil­iary conditions to implement the boundary condi­tions.

One of the newest and increasingly popularnumerical techniques available today is the finiteelement method. This approach originated overtwenty years ago in the aircraft industry as aneffective means for analysing complex air framestructures (Zeinkewicz 1977). The method hasbeen developed to include structural mechanics,metal forming, fluid mechanics and fluid filmlubrication.

The use of the finite element technique forsolving hydrodynamic lubrication problems wasoriginally applied to field problems byZeinkewiczand Cheng (1968). Booker and Huebner (1972)adopted the concept of a variational approachand a direct solution method for an infinitely longhydrodynamic bearing. Other researchers haveexpanded and developed the method. Theseinclude the works conducted by Gethin (1988)and more recently by Basri (1990) which are moreuser friendly.

This paper describes an analysis, the associ­ated computational algorithm, and a resultingsubroutine package developed to solve hydrody­namic lubrication problems.

Theoretical BasisThe Reynolds equation is fundamental to theanalysis ofhydrodynamic lubrication. By assumingthat the bearing runs aligned and acountingisoviscous lubricant in the film, the turbulentReynolds equation may be written (Ng and Pan1975)

~ (~ op] + l..(~ op] = llUdhoX kx ox oz kz OZ dx (1)

SHAHNORBASRI

The package consists of seven subroutines thatcompute the plain, cylindrical and multi-lodebearings, and a calling program. The programs arewritten in Fortran 90, are double precision, andself-contained. Parameters are passed between thecalling program and subroutines throughCOMMON blocks.

In equation (1), the quantIties kx and kz areincluded to model non-laminar film and formoderate Reynolds number « 5000) are given byTaylor (1923)

kx 1 = 12.0 + 0.0039 ReT 1.06 (2)Gx

kz = _1_ = 12.0 + 0.0021 ReTI.06 (3)Gz

For turbulent flow the turbulent correction factorsof Gz and Gx are included to give the volumetricflow terms as

The Subroutine Package andSolution Procedures

(6)dP = 0dZ

Mter the above boundary conditions were setup in the numerical model, the Reynolds orpressure equation was solved numerically. Eachlobe ofthe bearingwas divided into a finite number

The subroutines print the detailed solution at eachtime step, the intermediate results of iteration,and diagnostics in cases of failure. To facilitateplotting the results, the summary output is writtenin single precision into a separate file. These sets ofsource programs are about 4000 lines long.

The following set of boundary conditions wasprescribed during the computational process:

(i) the condition that lubricant feed pressure isnegligible in comparison with excursion inthe film is reflected by the condition p(O, x) =

O. The position of the boundary where thiscondition holds is at the oil supply groove andit is also applicable at the downstream end ofthe pad which is also at a supply groove.

(ii) the pressure is ambient (i. e zero) along thebearing edge, then p(z, L/2) =0

(iii) when appropriate and ca"itation occurs, at thetrailing edge the Swift-Steiber condition wasapplied i. e. at the cavitation boundary, zerogradient is satisfied by:

(5)

(4)h 3 dp--Gx-12jl dX

h 3 dp--Gz-12jl dX

Uh

2Qx

Qz

L2

CI • Al - A

81

61

L

Fig.]: Bearing geometry and nomenclature for the numerical model

360 PERTANlKA VOL. 14 NO.3, 1991

ASUBROUTINE PACKAGE FOR SOLVING HYDRODrNAMIC LUBRICATION PROBLEMS

of eight noded isoparametric elements of theserendipity family as shown in Figure 2 andthroughout this study a mesh comprising 80elements was used. Four elements were used in thehalf bearing length with twenty elements beingemployed in the direction of shaft rotation. Thefine circumferential division was necessary to en­able accurate calculation of the Swift-Steiberboundary condition.

The solutions for pressure distribution in thelubricant flow were obtained using an iterativescheme and the strategic steps in the solutionprocedure were the following:

(i) Define bearing geometry R, 1\, m, D, 8tand 8

and assume an initial bearing attitude angle <po(ii) Set-up initial pressure boundary conditions

and the finite element mesh.(iii) Calculate film thickness in each lobe of the

bearing.

(iv) Solve the Reynolds or pressure equation.(v) Continue the iteration for the pressure

generation within the clearance gap of thebearing (i. e. steps iii and iv) until theagreement between two successive iterationsat all points within the finite element mesh isbetter than 0.5%.

(vi) When step v has converged, by appropriateintegration, the parameters which are of

Qp =0___________ 9.?:.... _

p=o

p=OFig.2: The finite element, mesh and associated boundary conditions

PERTANlKA VOL. 14NO.3, 1991

ap = p =0ax

361

SHAHNORBASRI

andRef -Reel

Reel ~ Ref ~ Rec2 and ReT = RefRec2-Recl

in terest to the bearing designer (load carryingcapacity, power loss and side leakage) arecalculated by a simple summation over thebearing lobes.

The computer program developed for theturbulent calculation allows the assumption ofeither laminar, transition or turbulent flow withinthe gap depending on the magnitude of the localfilm Reynolds number. The change offlow type iscarried out automatically in the course of theiteration process, whenever the local film Reynoldsnumber value falls in the limit defined by thefollowing bands (Frene and Constan tinescu 1975)

Ref < Reel and ReT =0 (Laminar) (7)

Ref ~ Rec2 and Ref =ReT (Turbulent) (8)

Verification ofthe Numerical ModelsBefore proceeding with the parametric study, toconfirm the basis of the mathematical model usedin this paper, the computed global performancedata for the basic bearing were compared withpublished data from Lund and Thomsen 1978 andFlack and Allaire 1982 and are presented in Table1 and Figure 3.

It can be seen clearly that the baseline modeldeveloped (Table 1) shows complete agreementwith published data which confirms its foundationwith regards to this work.

Figure 3 illustrates a comparison of. thecomputed dependence of eccentricity ratio onSommerfeld number and is compared withpublished results by Flack and Allaire (1982) forvarious loading directions. Again, the graphicalpresentation shows the agreement of the presentmodel with established work.

Sample Computation and DiscussionFrom a design viewpoint, performance trends areessential to enable bearing selection for a particularengineering application. Therefore in this section,steady state design characteristics of a three-lobeprofile bore bearing are presented.

(10)

............................. (9)

( )

1/2

41.2 ~bReel

(Transition)

where

while the establishment offully turbulent flow thefilm Reynolds number is at twice Reel, i. e.

Rec2 = 2 Reel (11)

TABLElGlobal performance data

E S \jI

1 2 1 2

0.100 1.234 1.234 60.09 60.240.310 0.353 0.355 61.00 61.200.429 0.181 0.182 59.46 60.020.702 0.085 0.086 55.23 55.700.806 0.054 0.055 51.68 51.920.882 0.034 0.036 47.19 47.19

H Q2 1 2

1.424 1.432 0.134 0.1351.478 1.479 0.147 0.1471.574 1.577 0.165 0.1681.794 1.796 0.189 0.1912.016 2.208 0.207 0.2092.290 2.293 0.232 0.231

I-data from Lund and Thomsen 1978 2-present data

Parametric StudyA systematic series of calculations was completedfor a range of present and loading vectors forsymmetric and tilted configurations. The bearinghad the following geometric details:

TABLE 2Bearing geometries

R =37.41 mm l\=37.52 mm

Cb/R =0.0003

e = 20°g

e = 0° and 40°t

Figures 4 and 5 show sets of dimensionlessmid-plane pressure distribution in the symmetricand tilted three-lobe configuration for various load­ing vectors and for laminar turbulent calcula­tions. Both figures show that for the loading direc­tion of -60° and 60°, the predicted mid-planepressure distribution is identical but indexed in arotational sense. This is as expected and alsoconfirms the basis of the numerical model. Whenturbulent flow is modelled, clearly the filmpressures are higher and this is reflected in the

362 PERTANlKA YOL.14NO.3, 1991

A SUBROUTINE PACKAGE FORSOLVING HYDRODYNAMIC LUBRICATION PROBLEMS

aLOADING DIRECTION -30

1·6 i.6

1·. 1.4

J.2 1.Z

E E10 j

i.e

·8, C.B

·6 ~ C.6,

.. C.4

·2 C.Z

C.C

0.4 ~O.Z

O.C ~-'------.L- ' '

Z

aLOADING DIRECTION 30

C.B

1.2

0.6

i.E

6Z 3

a. LOADING DIRECTION 0

75Z

1/50 1/s"

-------- FLACK S ALLAIRE PRESENT

Ftg. 3: Dependence ofeccentricity ratio on Sommerfeld numberfor m=O. 879L/D=0.5 and C/R=0.0015: a comparison

with published work

-20(AI LAMINAR

0 20-60 60I I I I I

0Pmax=0.57 Pmax=-l.06 Pmax=1.68 PmaX'50.67

3.04.05.0

131 TIJRBULEIfi'

0~ 07 A A Y0\I "

ic.~)I I \: ' \. , \ II • 1 1 11

\~~ \~\ \ II

.' \.----: ,/

~/,&/>~' ~/

~oPmax= 1.072.0

Pmax=0.73 Pmax= 3.0 Pmax=4.73 Pmax=0.731.68 4.0

5.0

Fig. 4: Dimensionlessfilm pressure distribution: N = 20000 rev/min; m = 0.8; symmetric bearingelCb = 0.8

global bearing behaviour as shown in Figure 6. For

a loading directly on-pad (81

= 0°), the pressure ismore significant in lobe 2 and almost negligible inlobes 1 and 3. For the symmetric bearing (Figure 4)

it is apparent that the pressure is maximised when

the loading vector is directed such that 81= 20°

clockwise, while for 40° tilted configuration (Figure5) the pressure is maximised at a loading vector of

PERTANIKA VOL. 14NO.3, 1991 363

SHAHNORBASRI

Prnax= 1.34Prnax= 1.17

2.03.0

Prnax= 4.01.58 5.0

fAJ l..J.HDWloI

-20,

Prnax=4.66Prnax= 1.34

Prnax= 1.35 Prnax=4.89 Pmax= 1.23 Prnax=1.35

Fig. 5: Dimensionless film pressure distribution: N = 20000 rev/min; m = 0.8; tilted geometry e/Cb

= 0.8

-20° counterclockwise. Under these conditions,the load carrying ability assumes its largest value.This occurs because the change in the filmthickness in the loaded lobe is such that the Rey­nolds boundary condition is positioned at thedownstream end oflobe 2. This condition is similarto that for the plain cylindrical bore geometrywhere the load-earrying ability is optimised whenthe loadingvector is in the direction which is nearlyorthogonal to the grooves. From the designviewpoint, this condition can ,be exploited tooptimise the bearing performance on assembly byorienting the bearing appropriately.

Figure 6 illustrates the effects of including theturbulent flow correction factors in the pressureequation over a range ofloading directions. It canbe seen that the bearing global behaviour isidentical in form to that for a laminar calculationbutwith enhanced load carrying capacitydue to theincrease in pressure generation. 'Power loss andleakage increased, the extent of which alsodepends on the nominal film Reynolds number.The shear stresses are higher when the flow is

turbulent and this affects the higher power loss inthe film under this flow condition. The flow ishigher due to steeper pressure gradients generatedat the edge of the film but, similar to the laminarflow condition, is independentofloading directionalthough the balance of flow from the differentlobes changes with loading direction.

Figures 7(a) and (b) illustrate the load carryingability for different tilt angles (8

t) and presets

(m) respectively for a range ofloading directions.Figure 7(a) clearly shows a maximum value for acombination oftiltangle and loading direction and

Figure 7(b) shows that for tilt angle 8 t of 40° presetsaffects load carrying ability but the maximum loadcarrying ability still occurs at the same loading di­rection. The position for maximum (Figure 7(a))lies where the loading direction is at -20° for tiltangle of 40°, 20° for symmetric bearing and 0° fortilt angle of 20°. Figure 7(b) illustrates the combi­nation effect ofpreset (m) and loading directionfor a fixed eccentricity ratio of0.8 and tilt angle of40°. A maximum is demonstrated for a loading di­rection of -20° and as with normal operation, load

364 PERTANIKA VOL. 14NO.3, 1991

A SUBROUTINE PACKAGE FORSOLVING HYDRODYNAMIC LUBRICATION PROBLEMS

z "6

·····;.·······i········:-·······; ·······t········;········;·······

. . . . . .........................................................· . . .· . . .· . . .· . . .· . . .· . . .· . . .· . .· . .· . .· .

-=::-=-:...,..=-=-=-=----_.:..-_---_...:.._-_:::::-:_::::-::::-- -- --. . .. , ,

·60 ·50 -40 ·30 ·20 ·10 o 10 20 30 40 50 60

LOADING DIRECTION

. -: ; .

· .· ...............................................................· . . . . ... ...., '.".. . ... . ... .· . . .

. : ! : ";' i' .· .· .· .: ....... --:--.,.. . ' ......

.. , ~ ·i/o/,f, j.. ······t·~·,< \ ) \ ~ ) \ .: .,.,/ :~ : :

-~~r ~~--~---~---~--~--

·50 ·40 ·30 ·20

.. .. ~ [ i : ~ ; ~ ·1'······ \, ~ ~ ,. r;;·.:L~ijiH~~ ~I ..: 1- TURBULENTI ••

....... ~ : j -; ·..t.. ·· .. ·~ j ~, :: ~ .. , ~ ... .

a., ... . . . .. .

110 ••••••• -: •••••••• : ••••••• : ••••••• -: •••••••• : ••••••• : ••••••••: •••••••• : ••••••• : •••••••• ; ••••••• : •••••••

: :: . .:::. . . .,. : : : .. ::

---~--~--~--~--~--~--~---:---~--~--~-_.

·50 -40 ·30 ·20 ·10 0 10 20 30 40 50

LOADING DIRECTION

Fig. 6: Comparison between laminar and turbulent bearing behaviour at different loading direction: m = 0.85, N =

20000 rev/min; e/Cb

= 0.8; re =393

carrying ability increases with increased preset.As explained previously, the maximum load occursat this position due to thinner film in loaded lobe

2 and when Reynolds pressure boundary conditionis positioned at the downstream end of this lobe, itconsequently maximises the pressure generation.

PERTANIKA VOL. 14NO.3, 1991 365

.. : ~ ..

SHAHNORBASRl

","'":,",, ': /:,/:\:/:\. :

- • TILTED 40' " • / • ,. •... --IILTED20,I .••\f. ..... :.....\; ...... ;......

- SYMMETRIC,' ,(,: 'i .i I :\ : :'\, I . \ . .

I I : \ :. ' : \: \: : ,': : I : \: : :13: 3 ...... : ...... J ..... ~.......:,. .... : ...\ .. : .... . : ...\\.~...... ~.

: '. / : i : \: ,:/ : : /: '( -(

: /: . /: : .. \ : :\ : : :., . .". . ,,' ',,' . ....;":'1...... .:.... :,,:.r. ... ..:.......:..... i "'" . .: .-::_~-- ~-' . . .~ .... .r':~'':''~~~~~7~i-'' ·~S~

. .

·20 ·\0 o 10 20 30

13:

LOADING DIRECTION

(Al

.. .. .. .. .. .. .... ; : : :" -:" - ..

.:.. .. .. .. ....: .. : : ~ : :- : .

. .. ... .. .. ... .. .. .. .. .. .. .."T" ·~·1······~··· ; ro ~ •••••• -:- ••••• ";" ~' •••••

:" : ,: .: : ..../? .....: l\~ ...... 1 • .. ··i .. ·.. ··;. .. ····:· .. ·.. ·:·· .. ·..;· .. ···

: / ~ ~ \: :: / : ,--~, : ~: . . . . ... 'I"'" ••. :,.. : ..••.,.~ •. , .•• , , ..•......•••••....•.•.. ,. .••.... " ,.. .... " ...

: " ~/': : ~,'; : : : : :"." ,,"'" . .. .. ,., . .. .. .. ..;.;-~ ~ ~ ~ ~ : ..~ ~ ~ ~ : .

..". ... -r- :' : : : to, _+-_=- .. ...;...--~, ~: : ~ : -.,-~---+---~_:_-.:=----=--~---

·20 ·\0 o 10 20

-20-60

LOADING DIRECTION

(8)o 20 ~o 80

I iLl I ! I

0000000Fig. 7: Load carrying capacity at different loading direction: N = 20000 rev/min; e/Cb =0.8

Figure 8 shows the power loss in the bearingfilm. The losses in the bearing vary significantlywith tilt angle and presets and are dependent onloading direction in a manner similar to the load

carrying capacity. The condition of maximum lossoccurs when loading vector is positioned at -20°for 40° tilt, 0° for 20° tilt and 20° for symmetricconfiguration. This occurs due to the high shear

366 PERTANlKA VOL. 14NO.3.1991

ASUBROUTINE PACKAGE FOR SOLVlNG HYDRODYNAMIC LUBRICATION PROBLEMS

G2 r------------------~-----------:~-,

60 •••••• ! ; : ; ; /.~".! ; ; ; ; ..: / '\ :

58 ::~:t~~g~ .~ ~.I.. ~ ~:\ ~ ~ ~ ; .. ~ ..: - SYMMETRIC: . 'I : :, : : : :

: •• ::::1:::.:.]:.:.::1;/:I~)<1::::::::.:.::!··:::~.<. t::::.t•••::.t: ••:••: : i :/ :\ : ',:" : :: : I : /. : \ : • ': ::52 .••••• ~ ••.••• ~..•. ,.-:•••. ~.: ••••••:.~\.•• : ••••. : ••••.• ~ .••.•• ~ .••..~ •.•.••• : •.....

: :,': "': : \: : : ::-..: :• • I ;;' • • \. • • • ,. •

50 ······'·.:..:.~:,.r ·:·······~······~····.-.~·· ~ ~ _: :""-, .. ~ .-~~ ", .. .." ~---

~e ..j / ~ ~ = ;·"' ··i .. ····~······~ .. ······ = ... , .. ". .. .~ .:.;.:.:.- ~ ,'" .; .....·1·····~f:-~~ ...~~:.··::..:~~:.:::.::~.:..:.:.~·

.... ...... .4..'--...:.--..;...-...;..--.;...-...;..-------..;...-...,;.------~

·20 ·10 o 10 20 30 ~o 50

LOADING DIRECTION(AI

70._----------------------------,.. .. .. .. ,.. .

.~ : : -: -: ~ ~ .

." ..,. .. .: .:.. .. . .. .. .

. '" i i· : : .: ; .... ..,. .

65

60

......;-': . : : : . :II 55 - •• /, ~ ..:/ , :: :~ :, .. ..

. /: . :, .. -~.-:.-~-~-50 ···;·}I.··.:··r·-:-::~ ···~····\.·:······;······;······;··· ~ ~ .:, j.' : ',: , : : : : :

/, , . : ~, :': : : : :: ....": ,/: : :,:'....._: : : :

~5 =~~~~~>'l~~"" .~ ~ ~ ...'~'f<~~~~~~:~~=t~~J::l~~·• • I • • • • •

~ i-_--:..__..;.:.-_~.__.;..._--:.__..:..__....__....__....__.... .__...·20 ·10 o 10 20 30

LOADING DIRECTION

(6)

-60 -20 o 20 BD

l L I I I J I

0000000Fig. 8: Power loss at different loading direction: N = 20000 rev/min; e/C

b= 0.8

stress in lobe 2 which is fully wetted with a com­paratively thin film is this region.

Dimensionless leakage is shown in Figure 9for

a range of tilt angles (9a) and presets (9b). Thecumulative flow is independent of loadingdirection for the full range of tilt angles. However,

PERTANIKA VOL. 14NO.3, 1991 367

SHAHNORBASRI

. ..., ..,. . . . . . . . . .---~---~---~---~----~---~---~----~---~---~---~----· . . . .... . .· . .. . .. . . .. . .. .

3.0 ······1···eIT~g~ rt······~······j······j······1······i·······t······~·.....: I:: ~~METRJC : : : : : : : :

2.5 •••••• ~ •••••••:•••••••:••••••• :••••••• ~ •••••• ! .. ··.··i·······:·······:.. ·····-:·······~······..

3.5 r-------------------------------.

· .. .102.0 •••••• : ...... : ...... ~....... :- ...... : : : : ~ :- : ••....· . . . . . . . . . .· .. . .. . .. . . .. . .

--~---:---:---:---;--:--~--T--~-~--""t--· . . . . . . . . . .1.5 •••••• ~ •••••• -: •••••••:••••••• :- •••••• :- •••••• : ~ •.•••• ~ -: :- •••... ~ •••••.· . . .. . . .. . .. .. .· . .. .· .. .. .· .. .. .· . ... ...,· . " .....1.0 ······,······..:;·······:-······:·······i······i ······i······,·······:······.:······:;······.. .. ... ... ... .. .0.5 L-_...;.__..;..._...;.__..;..._.....;__~__.:..._...;.__.:..._...;.__.:..._...J

·20 ·10 o 10 20 30 40 50

LOADING DIRECTION

(A)

5.0 r------------------------------~

4.5 ~':':-:-:-~·...,r:.:-~;:~iJ;":',..·~-:.,..,.,.,..,.r·...,.·-:~t~,......,..i~':':-:-:-~....,~':'~,.......,.....,~':':-:-:-.~...,~':',..,.: --111-08 : : : : : : : :: -111-0.7 : : : : : : : :· .

4.0 •••••• , ••• '" '.0 : ••••••• : r•••.•. i \ : : : ···i .. . .. .. . . ..· . . . . . . . . . .

••••••• , , '" 0 ••••• _ •••••• '" ••••• " •••••••· . . . .. ..· .. . . . . . . .--~--~--:--~--~--~-~--~--~-~--~--· .. .. .

: :..::::::3.0 •••••• ~ _: •••••••:••••••• ~ •••••• ~ •••••• ! ~ ~ _: : ~ .· . . . . . .. ..· . .

: : :

10 3.5

. .. .· . . . . . . .2.5 ••••.• \ •••••••:•••••••:••••••• :•••••• ,; ••• , •• i , .. , , : .; ; ... ... ... .. .

" .," ." ." .2.0 l=====:;===.....__.... ...._....;__.i-_.i-_~=~=d

·20 ·10 o 10 20 30 .0 50

LOADING DIRECTION(8)

-fjIJ o 20 60

I i ~~ L I Iooooot){QFig. 9: Hydrodynamic leakage at different loading direction: N = 20000 rev/min; e/Cb = 0.8

the flow is increased when the tilt angle and presetare increased. This is due to the fact that when the

preset is increased, the lobe radius (R j ) is also

increased. For example, a bearing with an aspectratio of 1.0, clearance ratio of 0.003, a shaft radiusof 37.5 mm and preset of 0.8 will produce a lobe

368 PERTANlKA VOL. 14 NO.3, 1991

A SUBROUTINE PACKAGE FOR SOLVING HYDRODYNAMIC LUBRICATION PROBLEMS

radius equal to 38 mm; and a bearing with a presetof 0.7 will have a lobe radius of 37.78 mm. This

shows that lobe clearance (C1) is proportional to

the bearing preset and therefore as the presetincreases, the flow increases too. For this reason,the load-earrying capacity at high presets decreases,leading to improvement in heat removal char­acteristics.

To complete this setofdata, Figure 1Oshows thechange in attitude angles for laminar flow

conditions and various loading directions. FigurelOra) shows the bearing attitude angle for threetypes ofprofile configurations. For the tilted three­lobe bearing (Figure 10(b)) the journal centre locitrends tend to shift towards positive attitude anglewhen position of loading is between -60° and -20°.However, between-20° to 10°, the attitude angle isnegative. In view of rotordynamic bearingdesign, closeness between the line of centres andthe load line reduces the susceptibility to film

.", ·\0 '0 '0 JO '0 50

. . ... , ' .. . .

LOADING DIRECTION

(Al

.0 r---=~-----------:"-~---:---:---:---.

//~.::..~~, : '\,JO ...... :.... \'

: '\

~ .Hj+\~~.!.. :+++ 0'1~ .. .,\. . . . . '/. .

: : : ,\ : .. // : :~ 10 ; ; ~_.\.\.:. : : : ~ • .: :. : ..

~ ::: \\: . . . '7' .: : : ,;..: :~.

~ ~ ! \::~l~:.~:;~~-'0

-50 ..., <Xl .2lJ -'0 'D 'D 30 'D 50

LOADING DIRECTION

(8)

-£0 -~o -20 20 ~o 80

I I ---L-_-' I I

0000000Fig. 10: Attitude angle at different loading direction: N = 20000 rev/min; e/C

b= 0.8

PERTANlKA VOL. 14 NO.3, 1991 369

SHAHNORBASRI

* computed at m =0.85, E =0.8

TABLE 4Dimensionless power loss H

TABLE 5Dimensionless Side Leakage Q

Faculty ofEngineeringUniversiti Pertanian Malaysia,43400 UPM, Serdang, SelangorDarulEhsan,Malaysia.

5000246.0246.4253.1250.6247.6246.0246.0

284.9288.5311.6291.1288.1285.2284.9

100073.1174.4685.4877.6774.6773.4473.11

82.8484.6997.3287.7684.3783.0382.84

Re50050.8452.4461.4154.0051.8250.9850.84

56.4458.5868.0560.6858.0256.4556.44

LID = 0.5 Req 0 500 1000 5000-60 8.190 9.134 9.345 9.869-40 8.187 9.133 9.435 9.874-20 8.167 9.121 9.399 9.854

0 8.176 9.120 9.414 9.85220 8.180 9.125 9.426 9.85740 8.184 9.129 9.433 9.86260 8.190 9.134 9.436 9.869

LID = 1.0

-60 4.548 5.494 5.841 6.541-40 4.545 5.487 5.832 6.450-20 4.537 5.484 5.794 6.428

0 4.543 5.489 5.815 6.43220 4.546 5.491 5.832 6.44040 4.547 5.493 5.840 6.44860 4.548 5.494 5.841 6.451

LID = 0.5e 0-60 38.82-40 40.80-20 46.64

o 40.6820 39.1540 38.6560 38.82

LID = 1.0

-60 42.01-40 44.78-20 51.03

o 44.1820 42.4540 41.9660 41.01

REFERENCES

BASRI,S.B. 1990. ThermalAnalysisofA Three-LobeProfileBoreBearing, PhD Thesis, University ofWales, U.K.

BOOKER,j.F and K.H. HUEBNER. 1972. Application ofFinite Elements to Lubrication: An Engineering

*computed at m = 0.85, E =0.8

CONCLUSION

A finite element model and a subroutine packagehave been developed and presented to predict theperformance characteristics of hydrodynamicbearings. The numerical procedure is robust, easyto program and, since it uses fewer computationsper iteration, is quite efficient. The underlyinganalysis is simplified by the assumption of constantdensity and viscosity lubricant. The algorithm thesubroutine package has been successfully appliedto a three-lobe profile bore bearing operating forvarious loading directions. Theoretical results arepresented in nondimensional form and dimensionalexamples are given, including a comparison withpublished data. The agreement with publishedwork is satisfactory.

SHAHNOR BASRlDepartment ofMechanical and Systems Engineering

TABLE 3Dimensionless Load W = 1I S

L/D=0.5 Re

e 0 500 1000 5000-60 1.03 1.07 1.30 3.48-40 1.60 1.64 1.93 4.74-20 2.45 2.66 3.17 7.83

0 1.22 1.31 1.65 4.5520 0.94 1.00 1.25 3.4440 0.90 0.95 1.17 3.1860 1.03 1.07 1.30 3.48

L/D=1.0

-60 1.45 1.59 2.10 6.32-40 2.27 2.37 2.95 8.35-20 3.29 3.70 4.65 12.58

0 1.72 1.98 2.67 8.2920 1.33 1.52 2.05 6.3740 1.62 1.42 1.91 5.8660 1.45 1.59 2.10 6.32

* computed at m =0.85, E =0.80

induced whirl.To complete the set of data, Tables 3 to 5

present a small selection ofdimensionless perform­ance characteristics for tilted profile bore configu­ration so that precise values ofthe data are availablefor utilisation in computed-based design proce­dures similar to ESDU 8403land and ESDU 85028.It can be seen from the tables presented that evenat moderate values of Reynolds numbers, turbu­lence can have a dramatic effect on the values oftheglobal bearing performance compared with lami­nar operation particularly on the power loss.

370 PERTANlKA VOL. 14 NO.3, 1991

A SUBROUTINE PACKAGE FOR SOLVING HYDRODYNAMIC LUBRICATION PROBLEMS

Approach. Trans ASME (JOLT) 24(4): 313-323.

FLACK, R. D. and P.E. ALLAIRE. 1982. An Experimentaland Theoretical Examination of the StaticCharacteristics of Three Lobe Bearings. Trans.ASLE25(1): 88-91.

FRENE,J. andV.N. CONSTANTINESCU. 1975. OperatingCharacteristics ofJournal Bearings in TransitionRegime. In Proceedings ofLeeds-Lyon Symposium onTribology, eds. Dowson, Godet and Taylor.

GETHIN, D.T. 1988. Finite Element Approach toAnalysing Thermohydrodynamic Lubrication inJournal Bearings.Tribology International 21 (2): 67­75.

LUND, J.W. and KK THOMSEN. 1978. A CalculationMethod and Data for Dynamic Coefficients of Oil­LubricatedJournal Bearings, Special Publication ofthe American Society of Mechanical Engineers,No. 100118, New York.

NG, C.W. and C.H.T. PAN. 1975. ALinearized TurbulentLubrication Theory Trans. ASMEJournal ofBasicEngineering 87: 264-269.

TAYLOR, G.!. 1923. Stability of a Viscous LiquidContained between Two Rotating Cylinders, Phil.Trans. Roy. Soc. London, 223(A): 289-343.

ZEINKEWICZ, D.C. 1977. The Finite Element Method, 3rdEdition, McGraw-Hill Publication.

ZEINKEWICZ, D.C. and Y.K CHENG. 1968. FiniteElements in the Solution ofField Problems. TheEngineers 24: 507-510.

(Received 2 May, 1991)

NOTATION

Cb

Base clearance between the journal and

bearing

Lobe clearance

co Rotational speed (rad/s)

Reynolds number for onset of turbulentflow

Sommerfeld number

Loading direction

Tilt angle

Attitude angle

Angular position

Eccentricity ratio (=e/Cb

)

Lobe eccentricity ratio (=e/Cb

)

Lubricant molecular viscosity

Lubricant kinematic viscosity

Lubricant density

Shear stress component

Couette shear stress

Rec2

a

£

P't

'tc

Bush torque reaction

Bush torque reaction in the cavitatedregion

V Sliding velocity

e Eccentricity

hI Local film thickness in the bearing lobe

k Turbulent correction factor

m Preset

p Film pressure

aI' a 2, a 3 Position of film commencement in eachbearing lobe

Groove angle

Journal Diameter

Base Circle Diameter

Turbulent viscosity coefficient

Power loss

Dimensionless power loss

Bearing length

Rotational Speed (rev/min)

Lubricant feed pressure

Leakage

Dimensionless leakage

Carried over lubricant

Lubricant feed flow

Lubricant at lobe inlet

Flow per unit width in x-direction

Flow per unit witdth in z-direction

Journal radius

Base Circle radius

Lobe radius

Nominal Reynolds Number (pVC b))l

Turbulent Reynolds Number

Local film Reynolds number

Reynolds number for onset of transition

flow

Db

Gx' Gz

H

H

L

N

Pf

QQ

~o

C4Q;nQ"<4R

~R

I

Re

ReT

Ref

Reel

PERTANIKA VOL. 14NO.3, 1991 371