i'- · 2018-11-08 · s*tple, fixed, partial-arc journal bearings (tapered land),...
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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.
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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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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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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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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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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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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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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 o Laborazovies for Remeach and DeweloPmenz
I-A2049-18
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THE FRANKLIN INSTITUTE o Ltzborazovis for Rmawnh and Dml~opimmn
I-A2049-18
ý4- 36 -
THE FRANKLIN INSTITUTE • LaWboa for Rmach and
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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 -
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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
0.00000 .0000000 0 000000
(vwan Ic^H" \ H l 0Hcf W\ %'or\
C;C 8( C 0 00000000 0000000
;.I0 0~o o 0 0 00 0000
0'000 0
IF^t - 'Cl^Cno 00i 0
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THE FRANKLIN INSTITUTE o Laborawries for Rewrch and DmekPmm
I-A2049-18
lit11, 111 Mri:
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fillit f I.;: tlij
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FIG. 20 SAMPLE C L vs. cl CURVES FOR VARIOUS VALUES OF CI/C
- 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-
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(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
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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.
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