I'-

Interim Report I-A,049-1S

STEADY.-STATE CHARACTERISTICS OF GAS.-LUBRICATED,SELF-ACTING, PARTIAL-ARC, JOURNAL BEARINGS

OF FINITE WIDTH

by

V. Castelli7 .• C. H. Stevenson

E. J. Gunter, Jr.

"Prepared underASTAContract Nonr-2342(00) FBA (C) MAT 41

) Task NR 062-316

April, 1963 TISIA

Jointly Supported by

DEPARTMENT OF DEFENSEATOMIC ENERGY COMMISSION

NATIONAL AERONAUTICAL AND SPACE ADMINISTRATION

Administered byOFFICE OF NAVAL RESEARCHDepartment of the Navy

THE FRANKLIN INSTITUTELABORATORIES FOR RESEARCH AND DEVELOPMENT

PHILADELPHIA PENNSYLVANIA

THE FRANKLIN INSTITUTE . Labon u for Rumah ea Dmloo"

Interim Report I-A2049-18

b

STEADY-STATE CHARACTERISTICS OF GAS-LBWRICATED,SELF-ACTING, PARTIAL-ARC, JOURNAL BEARINGS

OF FINITE WIDTH

by

V. CastelliC. H. Stevenson

E. J. Gunter, Jr.

* •Prepared under

Contract Nonr-2342(O0) FEK (c)Task ZR 062-316

April, 1963

Jointly Supported by

DEPARTMENT OF DEFENSEATOMIC ENERGY COMMISSION

NATIONAL AERONAUTICAL AND SPACE ADMINISTRATION

Administered by

OFFICE OF NAVAL RESEARCHDepartment of the Navy

Reproduction in Whole or in Part is Permittedfor any Purpose of the U. S. Government

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a

ABSTRACT

This report contains the description of a solution of theproblem of self-acting, gas-lubricated, partial-arc, finite-length journalbearings for steady-state conditions.

The results can be used for bearings of many types such ass*Tple, fixed, partial-arc journal bearings (tapered land), axial-groovedjournal bearings, pivoted-pad journal bearings, and similar geometricshapes.

Sample results are plotted, and their utilization for variousbearing types is illustrated by examples.

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TABLE OF CONTENTS

ABSTRACT*. . . . . . . .. . . . . .

NOMENCLATURE* ..... .... .... .... ... iv

1. INTRODUCTION. .*. . . . .............. 1

2o ANALYSIS* . . # . # o o & o e e o . . .. * o e 5

The Reynolds Equation . . . . . ...... . . . . . .. . 5

Convergence . * . .. . . . .0. . . ... .. . .a. .. . 9

Numerical Stability and Pressure Extrapolation . . . . . . . 13

Steady-State Bearing Characteristics, . . . . o . .. ..9. . 16

3. GENERAL CONSIDERATIONS FOR APPLICATION OF COMPUTER DATA . . . 17

4. DESIGN DATA . . . . . . . . . . .. . . . . . . . . . . . . . 20

5. USE OF FID MAPS . . . . . . . . . . . . . . . . . . . . . . 37

REFERENCES. .... .. 46

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LIST OF FIGURESFixure b

1 Schematic Diagram of a Pivoted-Pad BearingConfiguration* . . .. . e . . ... . . .. .. . . 6

2 Total Growth Indicator and Extrapolated GrowthIndicator vs. Iteration Number . .. e . ... . . 12

3 Theoretical Performance of Pivoted Partial GasJournal Bearing For A = 1.5 . . . . . . . . . .. . 22

4 Theoretical Performance of Pivoted Partial GasJournal Bearing for A - 1.5. . . . . . . ..*. . . . 23

5 Theoretical Performance of Pivoted Partial GasJournal Bearing for A =- .5 . . . . . . . . .. . 24

6 Theoretical Performance of Pivoted Partial GasJournal Bearing for A = 2.5 . . . . . . . . . . .. 25

7 Theoretical Performance of Pivoted Partial GasJournal Bearing for A = 2.5 . . ... .. . . . o. . 26

8 Theoretical Performance of Pivoted Partial GasJournal Bearing for A = 2.5 . ...... ... .. 27

9 Theoretical Performance of Pivoted Partial GasJournal Bearing for A = 3.0. .. . . . ... . .. 28

10 Theoretical Performance of Pivoted Partial GasJournal Bearing for A = 3.0. . o ..o. . . . . . . . 29

11 Theoretical Performance of Pivoted Partial GasJournal Bearing for A = 3.00 o . . a o 0 0 . . . . . 30

12 Theoretical Performance of Pivoted Partial GasJournal Bearing for A = 3.5. - o . .. .o. . ... .. 31

13 Theoretical Performance of Pivoted Partial GasJournal Bearing for A = 3.5 . . . . . # 0 . o. . .. 32

14 Theoretical Performance of Pivoted Partial GasJournal Bearing for A = 3.5. . .. . . . . .. ... 33

15 Theoretical Performance of Pivoted Partial GasJournal Bearing for A = 4.O . . .0. . . ..a. . ... 34

16 Theoretical Performance of Pivoted Partial GasJournal Bearing for A = 4.0. . . . . . . . ... . 35

17 Theoretical Performance of Pivoted Partial GasJournal Bearing for A = 4.0. . . .......... 36

18 Axial Groove Bearing Geometry .*.......... 3819 Pivoted Pad Journal Configuration with Three Shoes . 4020 Sample CL vs. e' Curves for Various Values of C'/C . 43

iii

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NOMENLATURE

A coefficient

iteration increment

B coefficient

bb total growth indicatorC* coefficient

cc coefficient

CL load coefficient = p-Wa

CLT total load coefficient

C ground clearance = R shoe- Rshaft

C' pivot circle clearance = Rpivot - Rshaft

D coefficient

Eij Reynolds difference equation coefficiente shaft eccentricity with respect to shoe circle

el shaft eccentricity with respect to pivot circle

Fij Reynolds difference equation coefficient

F frictional forcec

hH dimensionless local film thickness C

Hp dimensionless pivot film thickness

HT dimensionless trailing edge film thickness C

i grid point in 8 direction

J grid point in axial direction

k iteration number

L bearing axial width

iv

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Nomenclature (Cont'd)

i,A iteration number

M number of grid cells in e direction

N number of grid cells in axial direction

P dimensionless pressure = P/Pa

Pa ambient pressure

R shaft radius

S coefficient

T coefficient

W total load

W load component parallel to line of centers

W load component normal to line of centers

Y axial coordinate

0 shoe angular extent

0 angle between pivot and vertical axis

y relaxation factor

6 attitude angle

e eccentricity ratio = e/C

eccentricity ratio = e'/C,

dimensionless axial coordinate = y/L

0 angular coordinate6•2

A bearing number =6 w RP Ca

A bearing number based on hp, Ap A2

HVp

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NOMENCLATURE (Ooncl.)

AT bearing number based on hT, AT 2

HT

lk absolute viscosity of lubricant

t lead angle between line of centers and shoe leading edge

go same as above for first pad of grooved bearing0

g~o angle between load line and leading edge of first pad ofgrooved bearing

0 angle between shoe leading edge and pivot point

w shaft angular velocity.

vi

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1. INTRODUCTION

We have witnessed how the increasing demands of space-age

technology have fostered impressive developments in the field of com-pressible fluid-film lubrication. It is now recognized that manyproperties of gas-lubricated bearings make them attractive for certainparticular applications. Most of the very simple bearing geometries havebeen investigated to the extent that reliable design information exists,both of analytical and experimental nature. However, it is also truethat gas-lubricated bearings require accurate alignment of the shaft andbearing assembly and :iat many of them are prone to develop a self-excited

film instability.

The alignment requirement has more implications than simplythe need for initial parallelism of shaft and bearings. Subsequent

distortion of the bearing housing and supporting structures due totransient stresses and thermal gradients may produce this criticalcondition. Maintaining a high degree of alignment with some bearing-structure designs is very difficult and may in some extreme cases beimpossible.

The question of film instability can also be very serious. Ithas frequently been the major obstacle to successful bearing performance.

Th6 instability phenomenon, especially of the so-called half-frequency

variety, is most pronounced in the simplest geometries such as the 3600

Journal bearing.

Half-frequency whirl may be described as a self-excited vibrationcaused by the nonlinear hydrodynamic bearing forces. At certain threshold

speeds, this self-excited instability will cause the rotor center towhirl or precess in the same direction as shaft rotation at a rate

approximately equal to J the total rotor angular velocity. This instability

is of an altogether different nature from that of the resonance encountered

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I-A2049-18

at a rotor "critical speed", which is excited by rotor unbalance. Rotor

speeds above the initial onset of whirl will result in rapidly increasingamplitudes of the whirl orbits leading eventually to bearing failure.T•us, a properly designed gas bearing should have the half-frequency

whirl threshold above the expected operating speed range.

Information now available based on limited theoretical knowledge

and scattered experimental data, indicates that deviations from the basic

plain cylindrical geometry improve the bearing whirl stability character-

istics.

This report is concerned with the determination of basic

characteristics of partial-arc, cylindrical bearings which are somewhat

resistant to the difficulties described above. This in turn leads to the

design of bearings with three important geometries: tilting pad, axial

grooved, and fixed partial-arc (tapered land) journal bearings. All of

these geometries present favorable features with respect to the stability

problem, and the pivoted-pad bearing, in addition, can operate with

considerable misalignment. Theoretical solutions of the complete steady-

state characteristics of gas-lubricated, partial-arc bearings are not

ýresently available, and it is felt that this will make a contribution

to the advancement of the state of the art.

The film pressure generated by a self-acting, gas-lubricated

bearing is governed by the well-known isothermal Reynolds lubrication

equation. For compressible lubricants this equation is a non-linear,

non-homogeneous, partial differential equation with variable coefficients,

for which analytical solutions are available only in a few limiting

cases.

Many approximate methods are available for the solution ofReynolds' equation. One group of these methods takes advantage of the

smallness of some bearing parameter. For example, a "small 4 expansion"has been successfully attempted by the authors and furnishes data forthe region 9 < 0.4. (Note: The symbol C is the eccentricity ratio of

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I-A2049-18

tht journal in its bearing). This work has not been reported yet since

the small e region covered by the data is not of major interest at the

moment.

Solutions have also been attempted on the basis of a "small A

expansion", and their validity region would be restricted to cases whereonly a modest role is played by lubricant compressibility. (Note: The

symbol A refers to the compressibility bearing parameter). Snell used

this approach but further simplified the problem by taking the bearing

clearance to vary linearly from the leading to the trailing edge.l*

This assumption was motivated by the fact that, for cylindrical geometries,

even the "small A approximation" to Reynolds' equation presents the

difficult problem of the solution of Hill's equation.

Other methods of solution, such as those utilizing truncated

series expansions and integral fits such as Galerkin's technique, would

be very useful in obtaining close approximations to the load-carrying

capacity of a bearing with any geometry. However, the basic drawback of

all these approximate methods is that, although the overall pressure load

can be predicted quite accurately, the actual shape of the pressure pro-

file, and consequently the position of the point of application of the

load, is difficult to obtain. This argument must be considered quite

seriously in cases where the static data have to be used in dynamical

analysis. Then derivatives of the pressure profiles are necessary, and

the accuracy of the calculations depends considerably on the smoothnessof the data and on the close similarity of the approximate and exact

pressure profiles. An additional difficulty in the application of methods

such as Galerkin's is encountered in the evaluation of the expansion

coefficients. Indeed, the non-linearity of Reynolds' equation makes it

necessary to employ numerical techniques in the integration of the determining

equations.

Superscript numbers refer to the list of references at the end of thisreport.

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On the basis of these considerations the direct numerical method

of solution of the exact equation governing the problem was chosen as theone best satisfying all of the accuracy requirements. An approach for

the use of this analysis in the study of -he dynamical behavior of atilting-pad bearing arrangement is now in progress at The Franklin Institute.

In the reported analysis, the Reynolds differential equationis reduced to a finite difference equation by standard methods. The

work follows the same approach as the one used by Gross. 2 Plots of theresults are presented in a form convenient for use in design.

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2. ANALYSIS

The Reynolds Eauation

The basic theory utilizes the well discussed assumptions con-

ceming neglect of the inertia forces and temperature variations through-

out the film. Then, the Reynolds equation can be represented in the

following dimensionless form:

rPH3 R ]1 R: L PH3 2 A aPH[H L2 L [a]n -H

with the boundary conditions: (see Fig. 1)

P (e = •, rI) = 1

P (e +CV ) =TO

S(e e,n _ 1) [2]

Equation El) can be reduced, by finite difference techniques, to the

difference equation:

p2,J - 2Ei P, +F = 0 [3]

Solving equation [3] for Pi,J, the pressure at any point, yields

P E +[ Fij] [4)

where the coefficients, Ei,j and Fi,j can be expressed as:

Ei,j +()2 + i( il

+~~ 1i (9)H5

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FIG. 1. SCIHEMATIC DIAGRAM OF A PIVOTED-PAD BEARING CONFIGURATION

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I-A2049-18

F - A i-PS2D(Ae)H2 j+]Dj- a-)2

1 ~ 2 [ -P j [6)

where

In an effort to facilitate the programming, these equations were rearranged

into the following form:

F =CC[RA -A 2 -TB 2 ) [8)

E =CC [C *+ iS A + TD -- RiS.) [9]i'i i1i 2 iJi11j i~j 2 i

wvhere

A = Pli i-l.1 R A(6iJ 2 H

B P 12J -Pi.iii 2 1 idei

C* _i+l. P'i-l.i (.R)2(AO 2

i2 1TQ)(A)

iD 2C 001~+N 10)

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Equation (4) applied to all points of the bearing grid yieldsa set of simultaneous equations which can be solved by an iterative pro-

cess. In the preceding equations, the pressure at any point is definedas a function of the pressures at the four adjacent points. An arbitrarystarting value is chosen for the pressure at all grid points and theprocess is repeated until the sequential adjustment of the pressure at

each iteration is smaller than a pre-established amount.

At this point it is important to make a basic decision between

two methods of attack:

a) Fix the geometrical characteristics of the bearing

film and the relative speed and let the pressure

distribution and center of pressure be a consequence.

b) Fix the geometrical characteristics of the shaft

and bearing and the position of the center ofpressure and let the bearing be free to assume any

tilt as to comply with the specified data.

Method a) was chosen because it is much simpler to program,

converges more easily, does not get involved with double-valued problems,and usually gives an answer in half the time employed by method b). Theadopted method has the disadvantage of not producing direct data for a

specified center of pressure position as would be desirable for thesolution of a single pivoted-pad bearing problem. However, since theproduction of extensive field maps was the aim of the project, the datanecessary for the study of pivoted-pad bearings can be easily obtainedby cross-plotting.

If the production of a limited set of data for a particular

pivot pad geometry is desired, it is more convenient to program the pro-

blem according to method b).

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As is common with iterative processes, the determination and

careful control of the rate of convergence is vital to successful pro-duction runs. Moreover, if the produced data must be employed inextensive cross-plotting, smoothness and accuracy are essential. Thecriterion for truncation must be so selected as to stop all iterationprocesses at a uniform "distance" from the exact answer. Since the

exact answer is not known, fictitious criteria must be selected which,nonetheless, are required to be equally effective.

The following presentation expounds one such criterion basedon the exponential nature of the solutions generated by this iterative

technique. The convergence of the selected iterative process is determined

in the following manner:

Let the "iteration increment" at step k be defined as

aaA . A (m-l

This is, in essence, the average growth of the pressure at a non-boundary

point within the bearing. Let the "total growth indicator" at step k

be defined as

kbbk = aa [12]1=I

This indicator measures the average total growth of the pressure profileafter k iterations.

The successive iterations of the solution of Reynolds' equationas produced by either the "simultaneous displacement" or "overrelaxation"

techniques have been found to behave locally exponentially with respect

to the step number k. This fact can be used for the individual extrapolation

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of local pressure values toward the asymptotic solution in a manner akinto the "Atkin process". Moreover, it can be speculated that, if theindividual pressure points behave exponentially, also the total growthindicator might show the same effect. Then the following formula can befitted to the data:

bbA + Ae- =bb [13

where

bbA = the total growth indicator at the Ath iterationbb. = the total growth indicator after an infinite

number of stable iterations

A,B = constants

A = iteration number.

Applying Equation [13] to three sequential steps and eliminatingthe constants, one can obtain the following relation:

I bb1 - bb + 1in bb -bb A 2-A1

Sbb -bb3 ) A - A [14i

Inkbb - bb1 +1

For the particular case where the steps are successive or equally spaced

A2 - A, = A3 -A 2 [15)

then the right hand side of equation £14) becomes

A - A 1 1[6-.i• ( 16)

A3 1 1 2

Applying this to equation [14] we have

+• )2 bb - bb+bb. 1 1) bb. - bb 1

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Solving Equation [17) for bb. results in

(b2) 2 - b1 b3bb 2b2 (b1 + b [s

Note that the subscripts 1, 2 and 3 in equations [14) through [18) denote

three sequential iterations of the process, not necessarily the 1st,

2nd and 3rd iterations. (Figure 2 shows the plots of the extrapolated

growth indicator and the total growth indicator vs. the iteration number

for a typical case).

Using this approach, the convergence is defined as

bb. - bb < 19bb•

b1 1< 6 [19)

where 6 is a number produced by a predetermined convergence criterion.

The advantage afforded by this technique consist of the fact

that bb. - bbA is a measure of how "far" the iteration solution is fromthe exact one. Of course, it could be observed that the solution doesnot exactly behave according to the above-mentioned, exponential law andthat bb. is indeed bb. = bb. (k), k - iteration number. However, butfor the first few steps, bb, is a very weak function of the step number

and converges to the asymptote from above as opposed to bb which converges

to the asymptote from below.

The great advantage of this truncation method is that all pro-

cesses are stopped at approximately the same distance from the exact

answer. In contrast with this, most iteration techniques in the literature

are stopped when the iteration increment is below a given limit. Thiscasues iteration truncation errors which vary from case to case according

to the exponential behavior assumed by the solution. This technique was

experimented with at the start of the project and produced results which

could not successfully be cross-plotted except for the most regular regions

of the field maps.

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I-A2049-18ip i 11:2 lji:

12 2 22!2:: r7m:

im

24 tam

ff

i iffiNi. Il

NUT

,T I: I:! lij F."a'T i

T, 7 it it

!7

:Rfllit

'1110

it I It 4p, 11 lum Moll

.1 lift 99

X; it

it, it;:Nk;: !ý:b l: :.A IQ,t lilt

kA

fit..

41

tt r: m: vIT :i;: ::,;:;

: N-1 ý. I. 1;ýI::! .1

pi:,; li::: 11:2 i:::!q

T 1 it. i:;

:1 A;ili qijiýi

I il I FYI;.

FIG. 2 TOTAL GROWTH INDICATOR AND EXTRAPOLATED GROWTH INDICATOR va.ITERATION NUMBER

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Numerical Stability and Pressure Extrapolation

The selection of an iteration method was made along the

following lines:

(a) Simultaneous displacement techniques were chosen

over successive displacement ones due to poor

properties presented by the latter in connection

with the numerical stability program.

(b) A process which could be suitable for both over-

relaxation and underrelaxation had to be selected

due to the uncertain numerical stability behavior

of the solution.

Processes such as "simultaneous relaxation implicit by lines"

and the "alternating gradient" method (same as the relaxation implicit

by lines but in alternating directions) were tried on a full scale basis

but produced mixed results.3 On that basis they were discarded.

Then, the growth of the pressure at a particular grid point

is controlled by the following equation:

Y(m+l) . vp(m+l) + (1 - y) V" [20]iiiJ

_(m+l)where P is the assigned value of the pressure at the point i,j for

the (m+l5 th iteration. pP(M+) is the calculated value of the pressurei, j

at the point i,j for the (m+l)th iteration,, i, is the value of thei~j

pressure at the point i,j for the (m)th iteration.

From equation (201, it can be seen that for y = 1, the assigned

value of the pressure at the point i,j is identical to the calculated

value.

Numerical instability in the iterative process is detected by

an increase of the value of the iteration increment aak (as defined in

equation [11]). This term, normally behaves as a exponential and only

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when numerical instability is present does it increase rather than decrease.

Usually, instability can be eliminated by lowering sufficiently the value

of y. This attenuates the effects of erratic higher harmonics of the

solution and induces stability. As a result of lowering the value of

y, the iteration proceeds at a slower but more stable rate. Obviously

the speed of convergence is greatest when the highest value of y for

stable iteration is used. This statement holds true in relaxation processes

for most elliptic equations departing from the simple Laplace equation.

Indeed, for 2 P = 0, throughout the extensive region of numerical stability

there exists a value of y which provides the greatest rate of convergence.

This value of y is called "optimum overrelaxation factor" and the process"optimum overrelaxation". However, in equations noticeably departing

from the simple Laplace equation the stability problem dominates and

optimum overrelaxation cannot be achieved. Indeed it is only in rare

cases that any value of y greater then one can be used.

With all of the previously mentioned conditions imposed, the

number of iterations required for the solution of the system of Reynolds'

difference equations varies depending upon the bearing parameters, the

number of grid points, and the numerical stability of the iterative pro-

cess. In order to reduce the number of required iterations, the exponential

behavior of the solution was exploited. The use of the exponential

extrapolation is successful in most cases but always had to be applied

on a trial basis due to the occasional accentuation of local irregularities

into large errors. Local exponential extrapolation was accomplished by

the following equation:

p(n) + Ae(-B+,n) = P( [21)

wherepn) is the pressure of the point, i,j at the nthi, Jiteration,

P(00 is the exponential asymptote of the pressure

of the point i,j

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I-A2049-18

A,B are constants,

t is the time at the nth iteration.nUse of the same technique that was applied to equations (14) through [18)results in the following expression:

rp(2) p(1) p (3)2p(-)= . i,.I".i 1 ..1[2iJ 2P-(2) - pý1) + p(3)I 22

i,J i,j ii

The superscripts (1), (2) and (3) denote three equally spaced iterations

in this sequence at any point in the iterative process and not necessarily

the first, second and third ones. Usually P.,j is only an approximationto the exact solution of the governing difference equation and more

relaxation steps have to be applied in order to satisfy the desiredtruncation criterion.

Judicious application of the extrapolation routine has beenfound to be the most useful single expedient in the solution of this class

of problems. Extreme care must be exercised in extrapolating only gridswhich are proceeding very smoothly and which do not present any irregularities

or "seeds of instability". In view of what has been said above, it canbe concluded that the most convenient technique to follow is to adjust

grid size and relaxation factor (y) so as to start the solution on avery stable basis (values of y corresponding to very underrelaxed conditions).

A few iterations will then set the basis for extremely successful extra-

polations. The stable relaxation process can be applied again, followedafter a few steps by another extrapolation, and so forth. The establish-

ment of a proper sequence is rather empirical but important for economization

of valuable machine time in large production runs.

It is important to note that experience with extrapolation wasless successful with geometries which involved convergent-divergent filmsthan with the simply convergent or simply divergent ones. An example of

the use of pressure extrapolation can be seen in Figure 2. At the 25th

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I-A2049-18

iteration, the extrapolation routine was applied and the step in thetotal growth indicator curve is indicative of the reduction of the number

of required iterations.

Steady-State Bearin& Characteristics

Once satisfactory convergence of the pressure profile over the

bearing surface is obtained, it is possible to solve for the bearing

characteristics. The load capacity for the bearing is defined as

= L~ 2 W 2 ] (3PaR L \P• + pa•/L

where

- 1 (P-1) cos e de dn (24]PaRL _

wa (P-l) sin e de dn [25]

The frictional coefficient is defined as

F pa ..= = ,.j-I (P-l) dO d¶ [26]f A2 WLA l

These integrations are performed by Simpson's rule.

Moreover, the knowledge of the load components can be utilized

to establish the position of the center of pressure with respect to the

bearing.

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3. GEERAL CONSIDERATIONS FOR APPLICATION OF COMPUTER DATA

The characteristics of gpe-lubricated, self-acting, cylindrical,

partial-aro journal bearings have wherever possible been identified bystandard terminology. However, there are certain parameters used in thisreport that are unique to these bearings. These parameters are defined

in the text that follows:

The Pivot Circle Clearance. C'

This parameter establishes the relative position of the pivotalpoint (taken on the bearing surface) of tilting pad shoes with respect

to the shaft. We define

C' - Rpivot - Rshaft [273

where RPivot is the distance from the pivotal point to a properly selected

point fixed with respect to the bearing mounts. For the case of three

shoe arrangements Rnlvot is the radius of the circle passing through the

three pivot points, Rshaft is the shaft radius.

The Ground Clearance C

Is defined as the difference in radii of the bearing and shaft

surfaces

C- R= "ng Rehaft

The Dimensionless Pivotal Film Thickness Hp

The film thickness at the center of pressure or pivot position

is an important parameter in determining the operation of the bearing.

For a fixed shaft position it will remain a constant regardless of theattitude of the bearing pad with respect to the pivotal circle. In

agreement with the geometry of Figure 1 this film thickness can be

written as:

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I-A2049-18

Hp (C + a cos (C + O])/c [28]

The Pad Position Anrie I

C is defined as the angle drawn from the line of centers of

the bearing and shaft circles to the leading edge of the pad.

Trailing Edge Bearing Number, AT

The parameter "trailing edge bearing number" is defined alge-

braically as

A T a (R [29)

where

Cr = C + e cos (C + 0)1 [30]

AT = A [31]HT

This number more closely represents the operation of an individual shoe

from the conventional compressibility bearing number point of view.

The Pivotal Bearing Number Ar

This form of the bearing number is based on the clearance that

exists at the center of pressure, or pivot. Algebraically, it can be

expressed as

= , [321P pa hp .. . .. .. ...

where

hp = C[l + e cos (C + 0)] [33]

A = A2 134

H P

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I-A2049-18

The Clearance Ratio C'/C

This parameter is defined as the ratio of the pivotal circle

clearance to the ground clearance. It is important because it measures

the amount of preload existing due to the initial pivot circle adjust-

ment. As an example, consider a three shoe bearing system in which each

shoe has an angular extent of 1200. The film thickness at the pivotscan be expressed as

h =C' [1 + e' cosoX] ) hpl CHI * (35]

1hp2 = C' [1 + e' cos (x + 1200))= hp2 = CHp [36]

hp3 = C' E1 + 6' cos (X + 2400)] = hp3 = CHP3 [37]

Summing Equations 135J, [36) and [37] yields the following:

S+ Hp2 + =3 - (3 + eI[cos X + cos(X + 1200) + cos(X + 240•0)]] [38

Performing the trigonometric substitutions yields

C' = Hp, + HP2 + H3 39]7 3

This formula cannot be used if the spacing between pivots is not equal.

Indeed, for the more general case in which symmetry exists about a

vertical plane only, formula [29] becomes

C, HpI + H + (2 cos O)H p3

C 2(l + cos 0)

where 20 is the angle between pivots 1 and 2.

Pivotal Circle Eccentricity Ratio el

The pivotal circle eccentricity ratio is defined as the relative

location of the shaft center with reference to the pivotal circle center.

X = + -19-

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4, DESIGN DATA

The data obtained from this analysis can be presented in many

different arrangements. Three basic types of plots are presented in this

report. They consist of plots of (a) load coefficient (CL) vs. center

of pressure position (0/o); (b) pivotal film thickness (Hp) vs. center

of pressure position (0/al); (c) trailing film thickness (HT) vs. center

of pressure position (0/4).

All plots of the (a), (b) and (c) variety contain lines of

constant s and lines of constant C. Since

CL = f(c, C, A, R/L, or)

and

0/6or g(s, §P A, R/L, or)

etc., the best a two dimensional plot can do is to present CL vs. O/ct on

lines of constant c and lines of constant C while A, R/L, or must have

assigned values on each graph. It should be noticed that for compressibli

lubrication the speed parameter A and the load parameter CL cannot be

combined in a manner similar to the Sommerfeld number of incompressible

lubrication. This fact is due to the non-linearity of the governing

equation.

It is clear that the amount of information necessary to exhaust

all possible interesting geometries and running conditions can assume

colossal proportions. Our attitude was to investigate rather thoroughly

the geometries which were closely connected with other work at The

Franklin Institute and also to make the program available to anyone

interested in obtaining more data. Two values of angular extent were

investigated: or - 1200 and or - 94.51. The first angle closely approximates

symmetrical three-shoe arrangements. The second was actually used for a

practical design where the unidirectionality of the load made it preferable

to space the lower pivots 100 degrees apart rather than at an angle of 1200.

- 20 -

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I-A2049-18

For the o - 1200 runs the aspect ratio parameter R/L was givenvalues of 1/3 and 1/2. For the c - 94.5" case the values of R/L employedwere the ones dictated by the actual parts of the experimental machineunder test at The Franklin Institute (ABC Contract No. AT(30-1)-2512,Task 3); these were RL - 0.6061 and R/L - 0.4041 (corresponding to widthto length ratios of the developed surface of 1.0 and 1.5 respectively).

This report contains the part of the results which were mostcompletely organized up to the present time: the 94.50 shoes withR/L = .6061. The actual tabulation of data is very lengthy and is notcontained herein. However, it is worth mentioning that all computerdata plotted so smoothly that no point would fall off the lines presentedin the figures. Another important factor in assessing the value of thejresent solution is that actual checks of theory against experiment wereperformed with extremely favorable results. This part of the worktogether with a thorough analysis of the data and evaluations of designtechniques will be presented in a subsequent report.

Figures 3 through 17 contain the above mentioned field maps forvalues of the bearing parameter A of 1.5, 2.5, 3.0, 3.5 and 4.0. Thesevalues of A are representative of typical operating conditions. Theeffects of compressibility present in a bearing geometry are not com-pletely represented by the value of A. This is due to the fact that Ais based on the ground clearance "C" between the bearing pad and theshaft. According to the values of e and C the value of the actualclearance and particularly the value of the trailing edge clearance canbe much lower. For this reason the value of AT (the compressibilityparameter based on the trailing edge film thickness) more closely representthe extent of the compressibility effects in the bearing film.

- 21 -

THE FRANKLIN INSTITUTE *Labomuwvg for Rhawuch and Dwslopmmv

I-A2049-18

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I-A2049-18

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I-A2049-18

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THE FRANKLIN INSTITUTE • LaWboa for Rmach and

I-A2049-18

5. USE OF FIELD MAPS

The characteristics of various bearing arrangements can be

predicted by proper use of Charts 3 through 17. In all the calculations

frictional forces will be neglected in comparison to pressure forces.

All fixed geometries can be easily analyzed because, for any position

of the shaft within the bearing, the exact geometry of each bearing film

is known (I and 9). Then charts 3, 6, 9, 12 and 15 give directly the

loads and points of application of the loads in each bearing segment.Vector addition of the loads gives the total load and attitude angle.

In particular, axial groove cylindrical bearings have pads

characterized by a common value of e equal to the eccentricity ratio of

the shaft within the bearing circle. On a constant c line the pads

correspond to values of 9 starting from an arbitrary number o and

spaced at intervals equal to c (angular extent of the pads). Vector

addition of the loads will produce a total load and attitude angle

corresponding to each selection of o0

CL - f(go)

6 = attitude angle = g(Co )

In general, it will be of interest to select only the solution for which

o- = = angle between load line and leading edge of first pad

(see Fig. 18). The independence of A from the shaft position is a great

asset in these calculations. Indeed bearing parameters based on anyof the individual pad clearances would require cross-plotting of the data

to have A on one of the coordinate axes and maps of constant eccentricity.

Supplementary nomographs relating A to geometry would also be needed.

- 37 -

THE FRANKLIN INSTITUTE o Laborauoe for Researh ad DewloPrentI-A2049-18

eo

JOURNAL/

FIG. 18 AXIAL GROOVE BEARING GEOMETRY

- 38 -

THE FRANKLIN INSTITUTE • Labomti for Rmank and Dwdoomu

I-A2049-18

In the treatment of pivoted-pad bearings the geometry becomesmore complicated. Figure 19 shows this type of bearing arrangement forthe three shoe case with syuuetry about a vertical diameter.

The invarient characteristic of a working pivoted pad is theposition of its pivot and-at any journal speed - the value of A. Thismeans that all information possibly pertaining to the problem is containedon appropriate vertical lines of Figures 3 through 17. Then, for any.position of the shaft within the pivot circle (characterized by theeccentricity ratio e' and the attitude angle 6'), the pivotal clearancesSare known. One of the graphs of H. vs. 0/a (the one for the appropriatevalue of A) will translate this information into the values of c and Cfor each pad. The graphs of CL vs. 0/a will now be used to obtain the

vector loads through each of the pivots. Th3 vectorial addition of the

component loads produces the total load vector. For any eccentricity

ratio e', trial-and-error techniques have to be used in selecting a value

of 6' resulting in a total load in the wanted direction.

Considerable simplification is afforded by special geometries

such as the one with symmetry about a vertical diameter. In this case

the attitude angle is known to be zero and the trial-and-error procedure

is no longer necessary. As an example let us consider the problem of

evaluating the total CLT vs. 9' relation at any particular value of the

bearing speed parameter A. For this case let us use, (see Fig. 19)

S= 94.5* (Three equal pads)

R/L - 0.6061

A = 3.5

W - 500

Then the pivot clearances are given by

= • 2 -- £- C1 + V' cos (TT - [)J = % l + 0.643 9'3

- 39 -

THE FRANKLIN INSTITUTE 9 Laboratoria for Research and DemeloPment

I-A2049-18

-IVOT CIRCLE P O

SHAFT

h P1

PIVOTED PARTIALJOURNAL BEARING

FIG. 19 PIVOTED PAD JOURNAL CONFIGURATION WITH THREE SHOES

- 40 -

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I-A2049-18

CCH [tl+4'cos o° L [1 + e')

From the values of CL1 CL2 ' CLy the total load carrying capacity is

obtained as

CLT = 2CLl Cos 0 - CL3

LT - 1.286 CLl - CL3

The values of CLI, CL2, CL3 are obtained by (a) entering

Figure 13 with the proper values of Hp and 0/1, reading out the corres-

ponding e's and C's and (b) entering Figure 12 with O/a, € and 9 andread out CL. After step (a) the values of C 9, 0/a can be used to enterGraph 14 and read out HT. These calculations are shown in Table I for

C'values of ~- equal to 0.6, 0.8 and 1.0 and the results are plotted in

Figure 20.

An example of the use of a plot such as Figure 20 is given by

the following problem:

Given: Rotor weight = 80 lb

Rotor radius = 2 in.

Pad axial length = 3.3 in.

C - 1.5 x 10"3 in.

Angular velocity = 17,650 rpm

Find: Trailing edge film thickness, running eccentricity,

and stiffness corresponding to various values of pivot circle setting.

Procedure: CL= la 1 40 .0343L PRL 1-4.1 x2x3.3 O

41&

THE FRANKLIN INSTITUTE . Labwomutig for Roorc aid Dau4*Pwm

I-A2049-18

0000 86C800000

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1 8 00000 6 C;1300 0 0 00 0 000 86 88

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I-A2049-18

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- 43 -

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I-A2049-18

A - 2 x (2.61 x 9) x 17650/6o x 2-T(2) 2

Pc 2 14.7 x (1.5 x 10-3)2

A 3.5

Using Figure 20

iwW--1 61- HT_ h (in) LC-- asein

0.6 0.26 0.379 0.569 x 10-3 1.14 1.23 x 105

0.8 0.40 0.427 0.640 x 10-3 0.91 0.736 x 105

1.0 0.54 0.460 0.690 x 10-3 0.80 0.517 x 105

where the stiffness (per bearing) is

aw 11F PR [L

From the results it can be seen that the preload parameter

C'/C has a marked effect on the operation of pivoted-pad gas bearings.

Particularly noticeable is the effect of C'/C on the stiffness which can

be interpreted as a valuable tool for aposteriori adjustment of the

rotor critical speed. Since it is well known that the self-excited

whirl threshold speed in some multiple of the rotor critical speed

-44-

THE FRANKUN INSTITUTE • Labonvft for Reaw. h ad Dmkomaf

I-A2049-18

(approximately twice for wall attitude angles), the whirl instabilityregion can be modified within limits by proper setting of the pivotcircle clearance.

Dudley D. FullerProject Engineer

Approved by:

~12 C/9W. W. Shugarts, Jr., Manager R. DroularFriction and Lubrication Laboratory Technical Director

Franci l JacksonDirect°•'-of Labor orie"

- 45 -

THE FRANKLIN INSTITUTE • Labouiu for Ruhm and Dawkprnw

I-A2049-18

REFEmRIEES

1. L. N. Snell, "Pivoted-Pad Journal Bearings Lubricated by Gas"IGR--f/CA-285, United Kingdom Atomic Energy Authority, IndustrialGroup. 1958.

2. W. A. Gross, "Numerical Analysis of Gas-Lubricating Films"Proceedings, First International Symposium on Gas-Lubricated Bearings,Office of Naval Research, Dept. of the Navy, ACR-49, October 1959,U. S. Government Printing Office, Washington 25, D.C.

3. Forsythe and Wasow: "Numerical Methods for the Solution of PartialDifferential Equations"j, John Wiley and Son, New York.

- 46 -