The dimensionless parameters normally associated with a step sector thrust bearing are the film thickness ratio, the dimensionless step location, the number of sectors, the radius ratio, and the angular extent of the lubrication feed groove. The optimum number of sectors and the parallel step configuration for a step sector thrust bearing while considering load capacity or stiffness and assuming an incompressible fluid are presented

Hamrock, B. J.

English

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NASA Technical Memorandum 83356
NASA-TM-8335619830016584
Optimum Parallel Step-Sector Bearing
Lubricated With an Incompressible Fluid
Bernard J. Hamrock
Lewis Research Center
Cleveland, Ohio
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April 1983
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Author
Th2 dimensionless parameters nOrma11y associated with a step sector thrust
bearing 2'e the film thicknpss 'atjo~ the dimensionless step )ocatj0n~ the
numt;'2' of sectcd's~ tf:P. radius r2tlo~ a,";j tr;~ 2r-i~tl_ilar ~"'t~nt c.f ihr:.::
lubrication feed grOO'dE'. T~je optimum number of sectors and trre paral1e1
step conf19uration for 2 step s~ctor thrust bearing mh 112 considering load
C2C'2Cjt~' or .stiffness an·j assuming an jncGmpres.sib1e f 1)jlj are pr""""'+-..-l
OptimumParallelStep-SectorBearingLubricated
With an IncompressibleFluid
BernardJ. Hamrock
NationalAeronauticsand Space Administration
Lewis ResearchCenter
Cleveland,Ohio 44135
Lord Rayleigh (1918)determinedthe slider bearingprofilethat yielded
a maximum load-carryingcapacityper unit width for a given minimumfilm
thicknessand bearinglength. The answer,revealedby an elegantapplica-
tion of calculusof variations,turned out to be the steppedparallel-
surfacebearing. Lord Rayleigh'sanalysisneglectedside leakageand assumed
an incompressiblefluid. Archibald(1950) includedside leakageeffects
while consideringa rectangularand sector sliderbearings lubricatedwith
an incompressiblefluid. His solutionmade use of Fourier series in pro-
viding one of the rare analyticalsolutionsto lubricationproblemswhen
side leakage is considered. Hamrock (1972) used a linearizedanalysisin
_ consideringa finite-widthrectangularstep thrust bearinglubricatedwith a
compressiblefluid. The compressibilityeffectsintroducedan additional
w parameter,namelythe compressibilitynumber A. The optimumrectangular
step bearingconfigurationfor maximum load and stiffnesswas presented.
The presenttechnicalnote will determinethe optimumnumber of sectorsand
the parallelstep configurationfor a step-sectorthrust bearingwhile con-
sideringload capacity or stiffnessand assumingan incompressiblefluid.
Figure 1 gives the configurationof the step-sectorthrust bearing.
The parametersto be used to definethe dimensionlessload and stiffnessare
l_I hi = h_h°' film thicknessrati°n = ei (ei + eo), dimen ionlessstep location
i!I :N'number of sectors
= R2/R], radius ratio
eg, aBgularextent of lubricationfeed groove,
rad
Note that the first four parametersare dimensionlessand the fifth is above
and dimensionaland is expressedin radians.
The dimensionlessload capacity in terms of the parameterslisted
accordingto the techniquedevelopedby Archibald(1950) is
2Pzho = Fm(ln _)2 [1 + e-2(-1)m+l]_z
=_ (4 + m2x2)
m=1,...
{ 1- c°sh[I1- RIB]}
1 - cosh(nB)+ (1)
x sinh(nB) sinh[(1- n)B]
where B =_ - eg
_2(_ i - 1)Km In
Fm = (3)
mx{coth[(l-n)B] + (_i)3 coth(nB)}
2_m[_-2(-1) m+l + 1] (4)
Km= m2_2 + 4(In _)2
The dimensionless load expressed in equation (1) can be written in dimen-
sional terms as
nm(RiR2)2 _ f AhsPz : _ (5)
,_h3 coth[(1- n)B] + (h + 3m=l,.., o o hs) coth(nB)
where
6°_2Km( In (_) 311 _" e-2 (-1)m_1] { 1 - cOsh{n_ I _ 1 - c°shC{ 1 I _]} ( 6)m (4 + m2_ ) sin (nB) sin [(1" n)__
A
The stiffnesscan be expressedas K = BPz/Bho. Making use of equation (5),
we can write the dimensionlessstiffnessas
_ Kh_ _ -3A(h'i- l){c°th[(l- R)B] + (_i)2 c°th(nB)} (7)
m=l,... , coth[(1- n)_] + (ITi)3coth(nB 2
Maximum Load Capacity
For radius ratios in the range 0.05 < _ < 0.95 and for angularextents
of the lubricationfeed grooveequal to _190,--_/45,_/30, and 2_/45 the
optimumvalues of the dimensionlessstep locationparameter n and film
thicknessratio hi for maximum load are
= 0.558 and \/C_i_ : 1.668 (8)(n)op t opt
The optimum number of sectors to be placed into the thrust bearing for the
range of = and eg given earlier can be expressed as
2_ (9)
(N)°pt e + 2"2411- _)
g (1 + a)
where (N)optg is roundedoff to the nearestinteger.Figure _ was obtainedby using equation (9) to show the effect of
radius ratio _ on the optimumnumber of sectorsfor angularlubrication
feed groove extentsof _/90 and 2_/45. It is observedthat, as the
radius ratio increases,the differencein the optimumnumber of sectors
increasesfor the two angularfeed groove extentsshown.
MaximumStiffness
When maximum stiffnessis considered,the only parameterthat changes
from the maximum-loadresults is the film thicknessratio. For maximum
stiffnessthe optimumfilm thicknessis
(_i)opt : 1.467 (10)
This implies that, when stiffness is to maximized, the step height hs is
reduced by 12 percent from the optimum value obtained from the maximum-load
results. The optimum step location (n)oDt remains the same at 0.558,
and the optimum number of sectors is obtained from equation (9).
References
Archibald, F. R. (1950) "A Simple Hydrodynamic Thrust Bearing," Trans. ASME,
vol. 72, no. 4, 393-400.
Hamrock, B. J. (1972) "Optimization of Self-Acting Step Thrust Bearings for
Load Capacity and Stiffness," ASLE Trans., vol. 15, no. 3, 159-170.
Lord Rayleigh (1918) "Notes on the Theory of Lubrication," Phil. Mag. J.
Sci., vol. 35, 1-12.
0i
R2
(a)
////////////////////Z
_i "_Tt'/'I_'°""4
///////"
(b)
(a)Topview.
(b)Sectionthroughasector.
Figure1. - Configurationofstep-sectorthrustbearing.
I I I I I
0 .2 .4 .6 .8 1.0
- RADIUSRATIO,a-R21R1
Figure2.- Effectofradiusratioa onoptimumnumbers
ofsectorsfor angularfeedgroovesof7rl90and27rl45.
1. ReportNo. 2. GovernmentAccessionNo. 3. Recipient'sCatalogNo.
NASA TM-83356
4. Title andSubtitle 5. ReportDate
OPTIMUM PARALLEL STEP-SECTOR BEARING LUBRICATED WITH April 1983
AN INCOMPRESSIBLE FLUID 6. PerformingOrganizationCode
506-53-1B
7. Author(s) 8. PerformingOrganizationReportNo.
Bernard J. Harnrock E-1616
10. WorkUnit No.
9. PerformingOrganizationNameandAddress
National Aeronautics and Space Administration
Lewis Research Center 11. Contractor GrantNo.
Cleveland, Ohio 44135
13. Type of ReportandPeriodCovered
12. SponsoringAgencyNameandAddress Technical Memorandum
National Aeronautics and Space Administration 14. SponsoringAgency Code
Washington, D.C. 20546
15. SupplementaryNotes
16. Abstract
The dimensionless parameters normally associated with a stop-sector thrust bearing are the film
thickness ratio, the dimensionless step location, the number of sectors, the radius ratio, and the
angular extent of the lubrication feed groove. The present paper will determine the optimum number
of sectors and the parallel step configuration for a stop-sector thrust bearing while considering load
capacity or stiffness and assuming an incompressible fluid.
17. KeyWords(Suggestedby Author(s)) 18. DistributionStatement
Hydrodynamic lubrication Unclassified - unlimited
Thrust bearings STAR Category 37
Parallel-step sector bearing
19. SecurityClessif.(ofthisreport) 20. SecurityCtassif.(of thispage) 21. No.of Pages 22. Price*
Unclassified Unclassified
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Washington, D.C.
20546
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Optimum parallel step-sector bearing lubricated with an incompressible fluid

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