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AD-A276 235Euuml 0NASA Army Research LaboratoryContractor Report 191182 Contractor Report ARL-CR-81
Computer Program for Analysis of HighSpeed, Single Row, Angular Contact,Spherical Roller Bearing, SASHBEANVolume II: Mathematical Formulationand Analysis
ELECTE •Arun K. Aggarwal FEB 2 5 1994'•Emerson Power Transmission CorporationMcGill Manufacturing CompanyValparaiso, Indiana
September 1993
Prepared for -TIC Q4'
Lewis Research CenterUnder Contract NAS3 -25423
U.S. ARMY
ad' 06o47VDNational .Aeronautics and • • X
Space Administration RESEARCH LABORATORY
•,..• cd . .' ," .,
17 TABLE OF CONTENTS
PREFACE .................................................... 3
SLIST OF SYMBOLS ............................................ 4
[ 1.0 INTRODUCTION ............................................... 8
S2.0 FORMULATION OF MATHEMATICAL MODEL ........................... 9
t3.0 LOAD DISTRIBUTION ANALYSIS ................................. 10
3.1 COORDINATE SYSTEM AND SIGN CONVENTION ................. 10
3.2 CONTACT LOAD-DEFORMATION RELATIONSHIP ................. 12
3.3 INITIAL CLEARANCES BETWEEN ROLLER-RACEWAY LAMINAE ..... 12[3.4 THE ITERATIVE SOLUTION .................................. 14
4.0 CONTACT STRESS ANALYSIS .................................... 18
4.1 MAXIMUM AND MEAN CONTACT STRESSES ..................... 18
4.2 STRESS CONCENTRATION DUE TO EDGE LOADING .............. 20S4.3 CONTACT ELLIPSE DIMENSIONS ............................ 20
4.4 MAXIMUM SUB-SURFACE SHEAR STRESS AND DEPTH ............ 20
S4.5 ROLLER NORMAL LOADS AND OPERATING CONTACT ANGLES ...... 22
S5.0 EHD ANALYSIS ............................................... 23
6.0 FATIGUE LIFE ESTIMATION.....................................24
6. LIFE ADJUSTMENT FACTOR FOR LUBRICATION ................ 256.2 LIFE ADJUSTMENT FACTOR FOR CONSTRUCTION MATERIAL ...... 26
LIr.
7.0 INTERNAL MOTIONS AND SPEEDS ................................ 27
7.1 CAGE ROTATION SPEED .................................... 27
7.2 ROLLER ROTATION SPEEDS ................................. 29
8.0 RELATIVE SLIDING AT CONCENTRATED CONTACTS .................. 31
8.1 INNER RACEWAY CONTACTS ................................. 31
8.2 OUTER RACEWAY CONTACTS ................................. 31
9.0 HEAT GENERATION IN THE BEARING ........................... .. 32
9.1 DUE TO RELATIVE SLIDING AT CONCENTRATED CONTACTS ...... 32
9.2 TRACTION COEFFICIENT UNDER EHD CONDITIONS ............. 32
9.3 SLIDING FRICTION AT CAGE GUIDING RAILS/LANDS .......... 33
9.4 VISCOUS FRICTION TORQUE DUE TO LUBRICANT .............. 34
10.0 THERMAL ANALYSIS ........................................... 35
10.1 METHOD OF HEAT TRANSFER ANALYSIS ...................... 35
10.2 STEADY-STATE ANALYSIS .................................. 37
10.3 TRANSIENT ANALYSIS ..................................... 38
11.0 REFERENCES ................................................. 40
APPENDIX
A VOLUME OF A FULLY CROWNED SPHERICAL ROLLER ................. 43
B MOMENTS OF INERTIA OF A SPHERICAL ROLLER ................... 45
C INITIAL ESTIMATION OF INTERNAL ROTATIONAL SPEEDS ........... 46
D ROLLER GYROSCOPIC MOTION ANALYSIS .......................... 47
E LOAD-DEFORMATION RELATIONSHIP AND MATERIAL FACTORS ......... 49
2
PREFACE
This report is the second of the two volume documentation forthe bearing analysis computer program SASHBEAN. This volume,Volume II, provides the details of the underlying mathematicalformulation, analysis, and solution algorithms used for thiscomputer program. A separate volume, Volume I, provides thedetailed instructions required to successfully install andeffectively use the software for the design and analysis ofsingle row, angular contact, spherical roller bearings.
All efforts involved in the development of this software and itsdocumentation were performed by McGill Manufacturing, EmersonPower Transmission Corporation. This work was done as part ofthe Advanced Rotorcraft Transmission (ART) Program to advancethe state-of-the-art in helicopter transmissions. The ARTprogram was funded by the U.S. Army Aviation Systems Command(AVSCOM) and managed cooperatively by the AVSCOM PropulsionDirectorate and the NASA Mechanical Systems Technology Branch,both located at the NASA Lewis Research Center, Cleveland, Ohio.This work was done under a sub-contract to Sikorsky AircraftDivision of _United Technologies Corporation, the primecontractor, under NASA contract NAS3-25423.
Technical direction for this project was provided by SikorskyAircraft's representatives Mr. C.H. Keller, Jr. andMr. J.G. Kish, the Task Manager of the project. Thegovernment's technical representatives for this work wereDr. R.C. Bill, ART Program Manager and Mr. T.L. Krantz, ProjectManager for the Sikorsky ART contract.
The activities performed at McGill Manufacturing were directedby Mr. D.M. Michaels, Project Manager for the sub-contractedproject. Analytical and technical support was provided byMr. J.S. Porter, Mr. R.H. Barber, Mr. C.A. Kruse,Mr. G.A. Satkamp, Mr. A.K. Aggarwal and Mr. W.D. Nutt. Drawingand drafting aid were provided by Mr. D. Wisch andMr. T. Peterson. Typing and word processing were done byMs. C. Dodrill and Ms. B. Richards.
Acoesio;i For
NTIS CRA&IDTIC TkA6
By
Dist J Sca
3
LIST OF SYMBOLS
a - Semi-major axis of the contact ellipse (in)
A - Dimensionless material parameter for EHD analysis
b = Semi-minor axis of the contact ellipse (in)
c = Specific heat of a material (Btu/lbm.OF)
C = Initial clearance between a roller-raceway laminae (in)
CF - Centrifugal force acting on each roller (ib)
d = Distance (in)
dc - Diametral clearance (in)
D = Roller diameter, maximum (in)
E - Young's modulus of elasticity (psi)
S= Equivalent modulus of elasticity (psi)
e = Poisson's ratio of a material
F = External load on the bearing (radial/axial) (lb)
f = Friction force at a sliding contact (lb)
GM = Gyroscopic moment acting on each roller (in-lb)
h = EHD film thickness, minimum (in)
H = Rate of heat generation (Btu/hr)
Ix = Moment of Inertia of a roller about it's
longitudinal axis (in-lb sec2 )
I - Moment of Inertia of a roller about it's transverse
Iz axis (in-lb sec2)
i Ring ID; i = 1 for inner; i = 2 for outer
j = Roller ID; 1 s j s Z
1 - Lamina ID; 1 s 1 s r
K = Material constant in the load-deformation relationship
for a lamina contact
4
L = Roller length (in)
L-10 - L-10 life of a raceway (million revolutions)
LP - Lubricant parameter of the lubricant (sec)
M - Total pitching moment acting on a roller (in-lb)
N = Rotational speed of the bearing (rpm)
f = Total number of loaded laminae at a raceway
p - Mass density of a material (lbm/in3)
P = Resultant roller load (radial/axial) (lb)
Pe "Equivalent" roller load at a roller-raceway
contact (lb)
Pm "Mean" raceway load for all rollers (lb)
PC = Basic dynamic capacity of a raceway (lb)
Q = Normal load at a lamina contact (ib)
q = Normal load per unit length for a line contact (lb)
r = Radius of curvature of the contacting surface (positive
for a convex surface, negative for a concave surface)
R = Crown radius of roller or raceway
S = Contact stress (psi)
t = Time (sec)
T = Viscous torque of the lubricating fluid (in-lb)
U' Mean entrainment velocity (in/sec)
f = Dimensionless speed parameter for EHD analysis
v = Tangential velocity of a rotating point (in/sec)
W = Total width of cage rails (in)
w = Width of each lamina (in)
f = Dimensionless load parameter for EHD analysis
Z = Total number of rollers in the bearing
5
Greek Letters
S- Roller azimuth angle (radians), 0 s s 2w
T = Shear stress (psi)
r - Total number of lamina for a roller
a = Normal approach at a line contact (in)
-= Ring deflection, relative (in)
a - Roller angular displacement (pitching) (in/in)
8 Initial contact angle of the bearing (radians)
' = Operating contact angle of a lamina at
the raceway (radians)
e = D.Cosg/Pt (bearing geometry parameter)
= Roller deflection (radial/axial) (in)
"k = Kinematic viscosity of the lubricant (cStokes)
(@ operating temperature and atmospheric pressure)
a Absolute viscosity of the lubricant (lb.sec/in2 )
(@ operating temperature and atmospheric pressure)
a = Surface finish, RMS (in)
S= Lubrication film parameter from EHD analysis
Q = Angular velocity of rotation (rad/sec)
S= Coefficient of sliding friction
Special Characters
a = Weibull slope
= Material factor in the load-deformation
relationship for a line contact
S = Pressure-Viscosity coeff. of the lubricant (in 2 /lb)
6
Pt = Pitch diameter of the bearing (in)
£ = Basic dynamic capacity reduction factor
in fatigue life equation
¥ Empirical factor based on bearing type and it's
lubrication system.
f = Indicates functional relationship
Subscripts
a = Axial direction
brg = For the complete bearing
c = Composite
cg = For cage/retainer
cr = For cage rails
eff = Effective
fl = For lubricating fluid
i - Ring number; i = 1 for inner, i =2 for outer
j = Roller number; 1 s j s Z
I = L-Amina number; 1 s i s r
m = Roller mid-plane
max = Maximum value
mean = Mean value
min = Minimum value
o = Any roller
r = Radial direction
tr = Transition point
7
1.0 INTRODUCTION
Spherical roller bearings, known for their high load carryingcapacity along with their ability to perform under conditions ofmisalignments and shaft deflections, have conventionally beenrestricted to "low" to "moderate" speed applications. Speeds tothe order of 5000 rpm or 0.25 zuillion DN have typically (17] beenconsidered as the upper limits for this class of bearings.
The higher coefficient of friction exhibited by a spherical rollerbearing, as compared to that of an "equivalent" size cylindricalroller bearing, has been one of the main deterrents for theformer's use in high speed applications. In these bearings, theclose conformity of roller and raceway spherical crowns, whilelending the bearing its self-aligning and high load capacity, alsoresults in relative sliding at he concentrated contacts, resultingin the higher overall coefficients off rolling friction for thebearing.
New concepts in materials, manufacturing techniques, lubricants,component design and design tools are causing revolutions inbearing performance, and spherical roller bearings are noexception. Spherical roller bearings are now being designed anddeveloped for high load and/or high speed applications includingaerospace applications.
One such design has been developed by McGill Manufacturing, undera sub-contract from Sikorsky Aircraft Company, as part of anARMY/NASA sponsored ART project. The bearing, approaching 1.15million DN, has been tested under full load and full speedconditions.
The trend towards lightweight, high speed, and high performanceapplications, with increased requirements of reliability andsafety, has also placed a great deal of emphasis on the ability toaccurately analyze and predict the performance of any suggesteddesign. Such rigorous bearing analysis is no longer restricted bythe availability of main frame and mini computers. With thephenomenal development in the power and speed of desk top personalcomputers, such detailed analysis is now practical on personalcomputers.
The McGill computer program, SASHBEAN, based on the mathematicalformulation described in this report, provides a sophisticatedanalytical tool to design, analyze, and predict the operatingcharacteristics of single row, angular contact (including zerodegree contact angle) spherical roller bearings under high speedconditions.
8
2.0 FORMULATION OF MATHEMATICAL MODEL
To simulate the dynamic performance characteristics of a bearing,the mechanics of internal motions, load and stress distributions,lubricant flow, sliding friction, and heat generation have to bemodeled mathematically and solved for the given parameters ofbearing geometry, material, empirical factors, and external loadand speed environment.
When the bearings involved are of special design, incorporatingnon-conventional materials, and operating under extremeconditions, these mathematical models, with few simplifyingassumptions, no longer allow hand calculations. Computer programsrunning on high speed digital computers and employing efficientnumerical solution techniques are often required to solve suchcomplex mathematical models.
The mathematical model that forms the basis of the computerprogram SASHBEAN is described in detail in the following sections.
9
3.0 LOAD DISTRIBUTION ANALYSIS
As in any bearing analysis, a major effort involves modeling andsolving for the bearing's load distribution among its rollingelements for the qiven load and speed environment. The loading ofthe rings at many rolling elements poses a staticallyindeterminate problem and is often complicated to solve. Theproblem is further complicated when the high speed dynamic loadson the rolling elements, namely centrifugal forces and gyroscopicmoments, are not negligible and are fully considered.
A method, referred to here as the "LAMINA" method, has beenemployed to solve for the load distribution in the bearing underthe combined environment of externally applied loads and highspeed dynamic forces. In this technique the roller is dividedinto a number of slices (laminae), the slicing planes being normalto the roller axis. Similarly each raceway is also considered tobe made up of an equal number of laminae. As the deformation atany concentrated contact is very small, it is further assumed thatthe inter-facial shear between a loaded lamina and an adjoiningnon-loaded lamina is negligible and that the loaded lamina deformsindependently of the non-loaded one under normal loading.
3.1 COORDINATE SYSTEM AND SIGN CONVENTION
Consider a single row, (non-zero) angular contact spherical rollerbearing as shown in Figure 1. Let he outer ring be stationary andthe inner ring rotating for this formulation. In a real lifesituation, if the outer was rotating or both inner and outer wererotating, the same formulation will hold as the roller dynamicloads are estimated based on the actual rotational speed of thetwo rings.
The rollers are numbered 1 through Z sequentially in the clockwisedirection and equally spaced, as shown, The azimuth (angular)location of roller #1 is considered as 0.0 degrees and the sameincreases in the clockwise direction with the roller number.Therefore, the azimuth angle, 0, of any roller, j, is given by,
0j = 27(j-l)/Z, 1 < j 5 Z (3.1)
Let the radial load on the bearing be acting along the Y-axis,with roller #1 being directly under the load as shown in Figure 1.The relative radial deflection of the rings is thus positive alongthe positive direction of the radial load.
An axial load, applied either at he inner or outer ring, ispositive when the two rings are pressed into each other. Accord-ingly, the relative axial deflection of rings is positive when
10
this deflection causes the rings to move axially towards eachother. An axial deflection causing the rings to move away fromeach other, from the zero end-play position, is thus in thenegative direction. A positive axial load, acting along theX-axis is also shown in Figure 1.
Due to the self-aligning ability of a spherical roller bearing,any external moment loading, if present, is not considered.
MOST HEAVILY yYLOADED ROLLER OUTER
INNER
r "ýF
010
SPHERICALROLLERS
FIGURE 1
1i
3.2 CONTACT LOAD-DEFORMATION RELATIONSHIP
For a cylindrical body of finite length, L, when pressed on to aplane surfaced body of infinite extension, the normal approach,6, between the axis of the cylinder and a distant point in thesupporting body is approximately given by,
6 = 4.36E-7[A2 "7 q' 9 L*i] (3.2)
where A is a factor based on the materials of the two contactingbodies and q is the normal force per unit length of the contact-ing cylinder. Appendix E describes in detail the computationsfor the material factors for different contacting materials.
If the effective length, Lýef of the roller is subdivided(sliced) in r laminae (slices), each of width w, then equation(3.2) can be rewritten as,
S= 4.36E-7[E2- 7 q- 9 (Fw)- 1Ior or (61 . 1 1w.8 9 )
Q = (3.3)(Kr-1")
where K = (4.36E-7N2 "7 )1 - 1' is another constant and Q = q.w isthe total normal force at a lamina contact.
3.3 INITIAL CLEARANCES BETWEEN ROLLER-RACEWAY LAMINAE
Due to the difference in the radii of curvatures of roller andraceway crown profiles, there exists varying initial clearancesbetween the roller and raceway laminae. When the bearing isloaded, the contact of a roller lamina, 1, with that of a racewaylamina takes place only after this initial clearance is removed.Figure 2 shows the relative position of a roller, j, with respectto the two raceways when the bearing is held together with zeroradial and axial deflection of the rings. As the roller is sub-divided into r lamina, each of width w, the initial clearances atany lamina, 1, can be determined from geometry as follows,
Ci,l = Ro-(Ro2.dl 2)-[Ri-(Ri2_-d21 , 1 s 1 s r & i=1,2 (3.4)
Where dl = w(l-lM) is the distance of the lamina, 1, from theroller mid-plane, 1m being the roller's mid-lamina.
Having established the contact load-deformation relationship andthe initial clearances present between the roller and racewaylaminae, the load distribution analysis then reduces to determin-ing the relative axial and radial deflections of the rings whenin equilibrium under the externally applied loads and the inter-nal forces. For these ring deflections, each roller has to be in
12
operating equilibrium under the contact forces at raceways andthe dynamic loads.
As the loading of the rings at the many rolling element pointsnoses a statically indeterminate problem, further complicated bythe inclusion of the dynamic loads in the model, an iterativesolution scheme has been deployed to solve this load distributionproblem.
S~OUTER/
2.2
INNER
FIGURE 2
13
3.4 THE ITERATIVE SOLUTION
The bearing rings are assigned a small but known set of relativeradial and axial deflections. Let these deflections be %a and orrespectively. For the given ring deflections, 0a and -r, we thendetermine the radial, axial, and angular deflections of eachroller by satisfying its equilibrium under radial, axial forcesand pitching (misaligning) moments. This is also done using aniterative scheme as described below.
A roller, j, at azimuth 0 is assigned a given set of axial,radial and angular displacements as shown in Figure 3. Let a,ca, and Er be this roller's angular, axial, and radial displace-ments respectively. From geometry, for the given ring and rollerdisplacements, we can now write expressions for the normal ap-proach at each lamina as represented below,
ai,j,l = f(GeometrY, a,trC°So,•a,•r,•) - Ci,l (3.5)
where Ci,1 is the initial clearance at this lamina contact.
DISPLACED ROLLERPOSITION
ORIGINAL ROLLERPOSITION
ROLLERRADIAL DEFL.
ROLLER PITCHING(ANGULAR DEFL.)
ROLLERAXIAL DEFL.
FIGURE 3
14
A positive and non-zero 6.i• the~n indicates a loaded laminacontact. If ni - and fn2 ý'h±e the number of loaded laminae, forthe roller, j, A the innir and outer respectively, we can set upthe roller equilibrium equations in the axial, radial, and angu-lar directions as follows:
For the equilibrium of this roller in the axial direction,
w1-89 -
Pa, i j .11 6i,
For the equilibrium of this roller in the radial direction,
w . 8 9 ftirij 1 E" *l lsinsi, j, 1 (3.7)
Kfti,j'.1 1=I1
For the equilibrium of this roller in the angular direction,
w- 8 9 ftMiJ= .11 1 1ij 1 1 (3.8)' Krii, j ll '
Where . 1 is the contact angle at the raceway, i, roller, j,and lamai 1.
With the contact forces and moments now known for roller, j, its(dynamic) equilibrium condition is then established as follows:
a. Check equilibrium of this roller in axial direction:
IF I(Pa,2,j - Pa,l,j)l > Allowed Tolerance (3.9)
THEN adjust this roller's axial deflection and start over forthis roller.
b. ELSE Check equilibrium of this roller in radial direction:
IF I(Pr,2,j " Pr,l,j) - CFI > Allowed Tolerance (3.10)
THEN adjust this roller's radial deflection and start over forthis roller.
c. ELSE check equilibrium of this roller's pitching moments:
IF I(M 2 ,j + M1 ,j) - GMI > Allowed Tolerance (3.11)
THEN adjust this roller's angular displacement and start over forthis roller.
ELSE the roller, j, is fotind to be in equilibrium for the given
15
axial, radial and angular roller displacements.
This process is then repeated for all rollers in the bearing andtheir equilibrium positions determined. In actual programming,taking advantage of the bearing symmetry about a plane throughthe bearing axis, only a part of the actual number of rollers aresolved.
With the equilibrium forces for all the rollers in the bearingnow known for the assigned ring deflections, the overall bearingequilibrium equations against the externally applied radial andaxial loads are then set up as follows,
(a) Check bearing equilibrium in the radial direction:
zIF E Pr jCosoj Fr > Allowed Tolerance (3.12)1j=
THEN adjust rings' relative radial displacement and start over a-omplete new iteration.
(b) ELSE check bearing equilibrium in the axial direction:
IF , j - Fa > Allowed Tolerance (3.13)
THEN adjust rings' relative axial displacement and start over acomplete new iteration.
ELSE equilibrium of the bearing is established under the givenloading environment for these ring deflections.
The normal loads and deflections at each lamina contact are thusknown for this (equilibrium) state of the bearing.
A graphical representation (flow chart) of the iterative schemefor this load distribution analysis is shown in Figure 4.
16
iLAMINAUS ROUTINE
CONU~tTS NORMAL APPROACH jAT EACH LANtNA FOR ASSIGNMEDRING a ROLLER DELETIONS
A. 1ARKNLAINORNOTAL CI
OETEMIE RESULTANT RAG ZAL
FLGTER TRNATED RUCNACT AT ANNERJORING
RETURNFO&
FLAG FHGURE 4
17
4.0 CONTACT STRESS AMNLYSIS
4.0 MM ANDI MAN CONTACT STRESSES
Having determined the normal forces at each lamina contact, bothat inner and outer raceways, the level of contact stresses andareas are then estimated. The simplified formulation of Hertzcontact stress analysis for line contacts has been used. Theroller and the raceway laminae (slices) are considered as twocylinders of equal length (lamina width), pressed against eachother under a known normal load. As the contact deformation atany lamina contact is very small compared to the overall bodydimensions, the laminae interface shear is assumed to be negligi-bly small and thus neglected. A typical arrangement of twoparallel cylinders of equal lengths pressed against each otherunder a normal force is shown in Figure 5. The half width of thelamina line contact, b, as shown in Figure 5, is given by,
bi = Q 8 4.1)17rW (ro0-l1+ri"1)
where e and 9. are elastic constants of the two bodies based onthe respective values of modulus of elasticity and Poisson'sratio of their materials. rO and r. are the radii (with propersigns as per sign convention) of tAe roller and raceway laminaerespectively as shown in Figure 5. The maximum and mean contactstresses are then given by,
Smax = (2Q)/(irwb) (4.2)
Smean (Q)/(2wb) = S /max/4 (4.3)
In the computer program, the contact width, maximum and meancontact stresses are computed at each lamina contact. The maxi-mum of maximum and the maximum of mean contact stresses of allthe loaded lamina of the roller, j, at the inner raceway aretaken as this roller's maximum and mean contact stresses at theinner contact.
Similarly, the maximum and mean contact stresses of this rollerat the outer contact are given by the maximum of maximum and themaximum of mean lamina stresses at the outer contacts.
18
ROLLER LAMINA
INNER RING LAMINA
CONTACT STRESS DISTRIBUTION
FIGURE 5
19
4.2 S S CONCENTRATION DUE TO E LOADING
If for a roller, any of its "edge" laminae are in contact withthe raceway and thus loaded, its (roller's) contact ellipse maynot be fully contained on it's effective length and is truncatedat the loaded edge. Due to stress concentration, the level ofcontact pressure at such edge laminae is thus expected to behigher than that calculated by the above Hertz analysis.
A user specified stress concentration factor is thus applied(multiplied) to these edge laminae normal loads. Accordingly,the calculated Hertz contact stresses at these laminae contactsalso get adjusted by this stress concentration factor. Factorsof the order of 1.5 - 2.0 are typically used.
After applying the stress concentration factors, the maximum ofmaximum and the maximum of mean laminae contact stresses for eachroller both at the inner and outer raceways, are also printed outin the program output file.
4.3 CONTACT ELLIPSE DIMENSIONS
Cumulative width of the loaded laminae of a roller at the innercontacts gives us the major axis of the roller's contact ellipseat this raceway. The width of the line contact at the mostheavily loaded lamina of this roller at the same raceway gives usthe minor axis of the contact ellipse.
The axial distance of the most heavily loaded lamina mid-planefrom the roller mid-plane, being called as the eccentricity ofthe contact ellipse, is also computed and written out for eachroller.
The contact ellipse major axis, minor axis, and eccentricity foreach roller at the outer raceway contacts are also determined ina similar fashion and printed out.
4.4 MAXIMUM SUB-SURFACE SHEAR STRESS AND DEPTH
As the material below a concentrated contact is also in a stateof stress and the rolling contact fatigue failures have beenknown to originate from these subsurface points, the magnitudeand depth of this subsurface shear stress is also of importanceto a bearing analyst.
By considering the stresses caused by the normal contact load andfurther application of the principles of elasticity theory, Jones[11] has presented the expressions for the three principalstresses occurring at a point along the Z-axis, any depth belowthe contact surface as shown in Figure 6.
20
Since the surface contact pressure is maximum along the Z-axis,the three principal stresses must also attain their maximumvalues along the same axis. The maximum shear stress is thengiven by half of the maximum difference between any two principalstresses. The depth of this point (of maximum shear stress) canalso be determined.
For simplicity, the graphical method suggested by Jones [11] hasbeen used for these estimations. The graph presented in refer-ence [11], as shown in Figure 7, gives the maximum shear stressand its depth of occurrence as a function of contact ellipsedimensions ratio b/a.
With the contact ellipse dimensions and the maximum contactpressure known for each roller-raceway contact, the maximum sub-surface shear stress and its depth, are directly read from thegraph of Figure 7, in the computer program. For a completediscussion on this topic and the derivation of underlying equa-tions, the reader is referred to Jones [11] and Harris [9].
z 7
ELLIPSE a
b
z
PZ
PP
Pz
FIGURE 6
21
MAXIMUM SUB-SURFACE SHEAR STRESS/DEPTHFROM REFERENCE [11]
z-maxlb
.0. "
0*4 Sz-max/bSma
0.2.
0.0 02 0.4 0.6 0.8 1.0
b/a
FIGURE 7
4.5 ROLLER NORMAL LOADS AND OPERATING CONTACT ANGLES
The contact angle of the most heavily loaded lamina of a rollerat the inner is taken as the operating contact angle of thisroller at the inner raceway. Similarly, the contact angle of themost heavily loaded lamina of this roller at the outer is takenas the operating contact angle of this roller at the outer race-way.
The summation of the components of a roller lamina loads at theinner contacts, along the contact angle of the most heavilyloaded lamina at this raceway, gives the roller-inner contactnormal load for this roller. Similarly, the roller-outer normalload for this roller is given by the summation of lamina loadcomponents along the contact angle of the most heavily loadedlamina at the outer.
22
5.0 ZED ANALYSIS
Grubin's equation [23] has been used for EHD film thicknessestimation at each lamina contact. This is due to the fact thatthe ASME reconmmended life adjustment curve for EHD lubrication isalso based on computations using the same equation as given below:
hmin - 1. 9 5Req(f) 8 /11(A) 8 /11(9)'I/II (5.1)
hmin = Minimum thickness of the EHD film (in)
Re (RiRo)/(Ri±Ro) is the "equivalent" radius of the twocogtacting surfaces. The plus sign is for external contacts(both surfaces convex) and minus for the internal contacts(surface with larger radius of curvature is concave)
S= (naU)/(EReq) is the dimensionless speed parameter
A = (99) is the dimensionless material parameter
k = Q/(E'Reqw) is the dimensionless load parameter
U = 3(Uo+Ui) is the mean entrainment velocity (in/sec)
= = 3 1+ . is the "equivalent" modulus of
0Eo Ei elasticity (psi)
Rewriting (5.1) with definitions of various parameters, we get,
hmin = l. 9 5 Req (n aU/Req)8/11(Q/.ReqW) -1/1I (5.2)
Furthermore, in equation (5.2) the two lubricant properties, naand 9 can be combined into one parameter, known as LubricantParameter (LP), as defined below,
LP = 10 1 1 nag (5.3)
This Parameter combines both pressure and temperature-viscositycharacteristics of the lubricant and thus contains lubricant'sentire contribution to the formation of EHD film. Mobil, forexample, publishes this data for their premium lubricating oils.
Lubricant properties in the above formulation are expected at thetemperature of contacting surfaces at the EHD film inlet region.As the oil film is very thin, it is believed that it quicklyattains the inlet surface temperature. If this surface tempera-ture is unknown and cannot be easily determined, the oil outlettemperature or the average of the inlet and outlet temperaturesmay be used as a good approximation.
23
6.0 FATIGUE LIFE ESTIMATION
A method, referred to here as "Equivalent Roller Loads" method,has been used for the estimation of raceways' L-10 fatigue lives.In this method, the nonuniform and skewed load distribution alongthe roller length, as expected in high speed spherical rollerbearings, is accounted for by first calculating what is beingtermed as "Equivalent" roller loads. These are computed from theroller's individual lamina loads at a raceway using the productlaw of probability.
According to this law, the probability of survival of the entireraceway is the product of the probabilities of survival of eachindividual lamina (slice). Using the roller "Equivalent" loadsat a raceway then allows the use of formulation developed forestimating the L-10 fatigue life of a roller-raceway under radialload with line contact. Harris [9] describes in detail thederivations for the equation used and presented here for calcu-lating the roller "Equivalent" loads, from the known normallamina loads, Qi,j,l, as follows:
[l=r 9/2]2/9Pei,j =E719 i 3 (6.1)
Pe being the "equivalent" roller-raceway normal contact load atraceway, i, and roller, j,. The fatigue lives of inner and outerraceway are then given by,
L-10 = (Pc/Pm) 4 million revolutions
Where Pc is the Basic Dynamic capacity of the raceway and Pm isthe "mean" roller load at this raceway. Pc for inner and outerraceways are given by,
(1-))29/27 r 2/9
Pc,1 = 49500(f) (1+9)1/4 Esi (D) 29/27(Leff)7/9(Z) -1/4 (6.2a)
(1+9)29/27 8 2/9Pc,2 = 49500(f) I91/ (D) 2927(Leff) (Z) (6.2b),(1_e)1/ 4 osS
Where equation (6.2a) is for the inner and equation (6.2b) is forthe outer raceway. £ is a Bearing Dynamic Capacity reductionfactor based on the bearing type and is a user input to theprogram. Typically, for angular contact spherical roller bear-ings, £ lies between 0.60 and 0.85.
The "mean" roller-raceway load, Pm, is the quartic mean of theroller "equivalent" loads at this raceway and is given by,
24
1 z 1/4Pm, i "[ EPei 4j] for the rotating raceway (6.3)j-l
1i Z 1/4w
Pmii = [ EPe,i,j4m] for the stationary raceway (6.4)1ij=l
The L-10 fatigue lives of the raceways are then given by,
L_10 = (pc,i/Pm,i) 4 ; i = 1,2 (6.5)
The L-10 fatigue life of the complete bearing may then be deter-mined using the law of probability as follows,
L_10brg = (LI01- + L-102- )-al/ (6.6)
In the computer program, the overall bearing life is computed andprinted out, both before and after applying the lubrication and"material life adjustment factors to the computed raceway lives.
6.1 LIFE ADJUSTMENT FACTOR FOR LUBRICATION
In the computer program, the hmin is estimated at each concen-trated lamina contact using the user specified data on lubricantproperties at the expected surface temperatures. The minimum ofthe minimum EHD film thicknesses at each raceway is then used forcalculating the film parameter, ;, at that raceway. The compositerms surface finish of the two surfaces is calculated from theirindividual rms surface finishes as follows:
ac,i = (ao 2 + ai2 )% ; i = 1, 2 (6.7)
The film parameter, 1, for a raceway is then given by,
'i - hmin, i/c,i (6.8)
The ASME recommended curve for life adjustment factors has beenextended for lower values of i (down to I = 0.1) and used forreading the raceway life adjustment factors for lubrication. Theextension of the original ASME curve for lower values of I isbased on the collected test and field data from Sikorsky. Thisextended ASME curve has been coded into the computer programusing cubic splines interpolation and is shown in Figure 8.
25
LUBRICATION LIFE ADJUSTMENT FACTORS(ASME CURVE EXTENDED FOR LOWER LAMBDAS)
3.0I
2.5--_ _ _ -- - - 12.0 - -i ,
S 2.0 '...... '
~1.5ILI
1.0_
0.1 1.0FILM PARAMETER (LAMBDA)
FIGURE 8
For 1 - v s 10 the life adjustment factor, L/L-10, is returnedas per the curve shown in figure 8. For if < 0.i, a value of 0.27is returned and for / > 10, L/L-10 factor of 3.02 is returned.The computer program prints out both the estimated and adjustedraceways and bearing lives in the program output file.
6.2 LIFE ADJUSTMENT FACTOR FOR CONSTRUCTION MATERIAL
An appropriate life adjustment factor, based on the constructionmaterial of the bearing components, is supplied by the programuser. This factor is directly multiplied to the calculated L-10lives of the bearing raceways along with the life adjustmentfactors due to lubrication. The adjusted raceway lives are thenused in equation (6.6) to estimate the adjusted bearing life.
26
7.0 INTERNAL NOTIONS AND SPEEDS
With the operating equilibrium position of each roller in thebearing now known, the average cage and roller rotation speedsare estimated as described below.
7.1 CAGE ROTATIN SPED
Assuming no skidding at the concentrated roller-raceway contacts,the cage tangential velocity is taken as the average of thetangential velocities of the most heavily loaded points of the"controlling" rollers at the inner and outer contacts. Therollers loaded both at the inner and outer raceways are consid-ered as the rollers "controlling" the cage rotation. The rollersloaded only at one raceway are, on the other hand, considered as"controlled" (orbited) by the rotating cage.
Consider the operating position of the roller, j, at the innerraceway as shown in Figure 9. Let A be the contact center (mostheavily loaded point) and r, be the radial distance of point Afrom the bearing axis as shown. If 91 is the contact angle ofroller, j, at point A, then from geometry
ri = (AP)Cosg1 (7.1)
Where AP = OP - OA is also determined from geometry. The tangen-tial velocity of the point A, when lying on the inner raceway, isthen given by,
Vi= Qlr (7.2)
Similarly for the most heavily loaded point A at the outer race-way contact for this roller, as shown in Figure 10, we can write:
and r 2 = (AP)CosS 2 (7.3)
v2 = Q2 r 2 (7.4)
Where AP is now given by AP = OP + OA and is once again deter-mined from geometry.
Then, for no gross slip, taking the cage tangential velocity dueto roller, j, as the average of the tangential velocitiesv, and v 2 we get,
vcg = 3(vi + v 2 ) = 3(Pcg) (7.5)
The cage rotation speed due to roller, j, is then given by,
Qcg,j - (vl + v2 )/Pt (7.6)
27
ROLLER AT INNER RACEWAY,'/ RACEWAY CROWN CENTER
CONTACT CURVATURE CENTER --C
ROLLER CROWN CENTER
/ l4-, INNUAL CO ANGLE (G )
// O��OPERAING CONTACT ANGLE (f)
' 7-
00.
I~I
a F/ z,• •
NENA N N
S•wi BEARING AXIS x'
, a
FIGURE 9
28
If Zef is the number of rollers loaded at both raceways, thuscontrolling the cage rotation, the expected average cage rotationspeed is taken as the arithmetic mean of the cage speeds due toeach of the controlling rollers as given by,
Qcg (j_ cgj)/Zeff (7.7)
7.2 ROLLER ROTATION SPEEDS
Assuming pure rolling at the most heavily loaded points at theinner and outer contacts, the rotational speed of a roller, j, istaken as the average of the two speeds imparted to this roller bythe two raceways.
Once again considering the contact of a roller, j, at the inneras shown in Figure 9. The radial distance of point A, the mostheavily loaded point, from the roller axis is given by,
ro,1 = (OA)Cos(9I-8) (7.8)
For no slip at this point, the rotational speed of the roller dueto the inner raceway contact, is then given by,
0o, rol jI= (O1-Qcg.)rl (7.9)
Similarly, by considering the contact of the same roller at theouter, as shown in Figure 10, we can write,
and ro, 2 = (OA)Cos(Z-Z 2 ) (7.10)
Qo,2ro,2 = (02-gcg)r2 (7.11)
Where go 2 is the rotational speed of the jth roller due to theouter raceway contact.
The average of two roller speeds, due to inner rotation (Q 1)and outer rotation (go 2) as given by equations (7.9) and ?1.11),gives us a good estimaýe of the expected roller rotational speed.Therefore, for the roller, j,
go = %(Qo,l + go,2) (7.12)
This is true for all the rollers loaded at both raceways. Forthe rollers loaded at one raceway only, the rotational speed istaken as the speed imparted by the contacting raceway, assuming apure rolling at the most heavily loaded point of this contact.
29
ROLLER AT OUTER RACEWAY
"z xA
0
INTIAL CONTACT ANGLE r2)
OPERATIN G CONTACT4 ANGLE %~) 1
CONTACT CURVATURE CENTER ROLRCONCNE
FIGURE 10
30
8.0 RELATIVE SLIDING AT CONCENTRATED CONTACTS
Having determined the rotational speeds of the bearing cage aboutthe bearing axis, and of each roller about its own axis, we cannow estimate the magnitude and direction of expected relativesliding at each concentrated contact.
81INNER RACEWMy CONTACTS
Let Oxyz be a local coordinate system with origin at 0 as shownin Figure 9. The y-axis being perpendicular and into the planeof paper. For a point (x,y,z), lying on the contact ellipse ofthis roller at the inner raceway, the y coordinate is negligibleas compared to x and z coordinates. This is due to the fact thatthe contact ellipse has a high ellipticity, its minor axis beingvery small compared to its major axis.
Let the point (x,z) on the contact ellipse lie on the innerraceway surface. The tangential velocity of this point along they-axis is given by,
v, = -(0P-OE)QCos9 1 + xQISiniZ1 (8.1)
When the same point (x,z) is considered lying on the rollersurface, its tangential velocity along the same axis (y-axis)would be given by,
vO = (OE)QoCOs(: 1 -I) + x~oSin(i 1 -19) (8.2)
where OP and OE are determined from the known geometry. Therelative sliding velocity of this point (x,z) is then given by,
Vrel,1 = v1 - vo (8.3)
8.2 OUTER RACEWAY CONTACTS
Similarly, by considering a point (x,z) on the contact area of aroller at the outer raceway, as shown in Figure 10, we can writethe expressions for the tangential velocities of this point whenlying on the outer raceway and roller surfaces as follows,
v 2 = -(OP+OE)Q 2 Cos1R 2 + xQ2 SinJ 2 (8.4)
vo = -(OE)QoCos(Z-92) - x~oSin(8-S 2 ) (8.5)
Relative sliding velocity of this point (x,z) is then given by,
Vrel,2 = v2 - vo (8.3)
31
9.0 mEAT GEDERILTION IN THE BEARING
Total heat generation in a spherical roller bearing comes fromvarious fricticn mechanisms occurring during the bearing opera-tion. These include shear stressing of the EHD oil films due tothe relative sliding motion at the concentrated contacts, viscousfriction torque of the lubricant against the ploughing motion ofthe rolling elements through the lubricant, sliding friction atthe cage lands and rails. In a high speed bearing of this type,the major contribution to the total heat generation, under normaloperating conditions, is due to the sliding friction at theconcentrated contactsunder EHD conditions.
In the formulation for SASHBEAN computer program, heat generationdue to the three mechanisms mentioned above has been considered.These are described in detail in the following sections.
9.1 DUE TO RELATIVE SLIDING AT CONCENTRATED CONTACTS
To estimate the traction force at the sliding Elastohydrodynamiccontacts, a formulation presented by Allen et. al. [1] has beenused.
9.2 TRACTION COEFFICIENT UNDER EHD CONDITIONS
As per the above model, four parameters, namely - ambient abso-lute viscosity, pressure-viscosity coefficient, a lubricationfactor (a pseudo coefficient of friction), and a transition shearstress, can quantify the traction in a sliding EHD contact.Mathematically stated, the shear stress for a Newtonian fluid ina concentrated contact under EHD conditions is given by,
7- = [naeXp(qS)]vrel/hmin when T < Ttr and r < AflS (9.1)
= AflS when 7 > T tr and T > hflS (9.2)
Where S is the normal contact pressure at a lamina, ,r_ being thetransitional shear stress (typically 1000 psi), and Af is thelubricant factor - the pseudo coefficient of friction -typicallybetween 0.045 and .075).
Having determined the shear stress (T) in the EHD film ateach lamina contact, the total traction force for the givenlamina contact is then evaluated by integrating the shear stressover the entire contact area of this lamina contact.
With the relative sliding velocity at each lamina contact alreadyknown, the heat generation due to relative sliding is then givenby the thermal equivalent of the mechanical work done against
32
this friction force.
Rate of Work Done - Traction Force x Relative Sliding Velocity
The thermal equivalent of this mechanical work is then taken asthe rate of heat generation in the bearing and is given by,
Rate of Heat Generation - Rate of Work Done/2.5933
where the factor 2.5933 converts the mechanical work (in-lb/sec)to equivalent heat units (Btu/hr).
To estimate the sliding friction force at the concentrated con-tacts under lost lubrication conditions, it is assumed that theNewtonian viscosity relationship for the fluid shear stress nolonger holds valid. A constant (user supplied) coefficient oftraction coefficient is used for evaluating the sliding frictionforces, the mechanical work done, and the resulting heat genera-tion at each concentrated contact.
9.3 SLIDI FRICTION AT CAGE GUIDING RAILS/LANDS
The Petroff's equation [25], which provides a good approximationfor the power loss in lightly loaded journal bearings, has beenused to estimate the resisting torque of the lubricant present inthe clearances between the ring lands and the cage rails. Fol-lowing assumptions have been made for this formulation,
(a) Resultant cage-roller loads are very small.
(b) Cage is properly balanced and while rotating maintains auniform gap with the guide ring.
(c) Radial gap between the ring land(s) and the cage rail(s) isfully flooded with the lubricating fluid.
(d) Viscosity of the lubricating fluid in the gap does notchange appreciably from the bulk of lubricant in bearingcavities.
The tangential friction force at the cage-ring interface is thengiven by,
fcr = w(naWcrDcrIQcg- iI)/(dccr) (9.3)
Where i=1 for an inner guided cage and i=2 for an outer guidedcage. Wcr is the total width of the cage rails in sliding contactwith the guide ring rails, D is the mean interface diameter, (2is the angular velocity of t~e guiding ring, and dccr is thediametral clearance between the guiding ring lands and the cagerails. The resulting heat generation due to this friction power
33
loss is then given by,
Hcr - 0.1928(DcrfcrlQcg-Oil) (9.4)
9.4 VICU FRICTION TORQUE DUE TO DOUBRICANT
The viscous drag torque, experienced by the rollers movingthrough the lubricant flooded bearing cavities, has been estimat-ed using the empirical equation suggested by Harris [9] andEschmann [5] for high speed bearings as presented below,
Where, Tfl = 1.42E-_5¥(kN)2/3pt3 ; kN > 2000 (9.5)
TfL = Viscous friction torque of the lubricating fluid (in-lb)
V = A factor based on the bearing type and its lubricationsystem. As per ref. (9], V = 5 for spherical rollerbearings under oil bath lubrication.
The heat generation in the lubricant due to its viscous frictiontorque is then given by the thermal equivalent of the mechanicalwork done against this viscous drag. This is given by,
Hfl = 0.0404(N.Tfl) (9.6)
where the factor of 0.0404 converts the rate of mechanical work(in-lb./min) to equivalent rate of heat generation (Btu/hr).
34
10.0 THERMAL ANALYSIS
With the amount of heat generation at different points in thebearing now known, thermal models are then set up to perform asteady-state and transient heat transfer analysis. These analy-ses can predict the expected steady-state and time-transienttemperature maps of various points in a bearing system respec-tively. Some detailed information about the bearing application,including its supporting system, dimensional and material data,lubricant properties, and lubrication system, are required toprepare the model data for thermal analysis.
A steady-state condition would be one when a certain lubricantflow rate is maintained through the bearing system. After aninitial "warm up" time, the temperatures at different points ofthe bearing and its supporting mechanical system would havestabilized and there is no further variation in temperatures atany point throughout the system.
When a "stabilized" system is subjected to a sudden change inenvironment and its steady-state condition disturbed, the systemtemperatures would become time variant. One such change ofenvironment could be the loss of lubricant flow through thebearing system. A transient heat transfer analysis would becalled for in such a situation to predict the time-temperaturehistory of various points in the system. This information maythen be used to predict bearing's time-to-failure in the event oflubrication failure.
10.1 METHOD QF HEAT TRANSFER ANALYSIS
A method known as "Lumped-Heat-Capacity" method has been used formodeling and analyzing the system for steady-state and transienttemperatures. In this method, the whole system is consideredmade up of small elements, with the entire thermal capacity ofeach element "lumped" at its center and assuming a uniform tem-perature distribution throughout the volume of the element.
In other words, the internal resistance to heat flow within anelement is considered negligibly small compared to its externalresistance to heat flow from the surface to the surroundingelements. In general, smaller the size of elementsthe more realis-tic these assumptions are for lumped-heat-capacity analysis.
For ease of data preparation, the maximum number of elements fora single bearing system have been limited to 20 in this program.The model prepared for the steady-state and transient thermalanalyses of McGILL SB-1231 and SB-1231-1 bearings is shown inFigure 11. The element and node numbers are marked on the samefigure. The lubricating oil circuit is shown by the dotted lines.
35
THERMAL ANALYSIS MODEL FOR THETEST BEARING AND ITS SUPPORTING SYSTEM
.------ L•UBE OILPMRNIFOLD
OII
, ,a
* ID OIL
FILTER
SCAVENINGIO SCAVENGING
rr -------LUBRICATION OIL
-'oL " - - -' - - " ...... --
OIL COOLERIPUGP R -_-"
I*|I
IN WATER OUT
FIGURE 11
36
1.2 STEADY-STATE ANALYSIS
SASHBEAN Computer Program is capable of analyzing and predictingthe steady-state temperature map of an axisymmetric mechanicalsystem of any cross-section. The mechanical system is firstapproximated by an equivalent system comprised of a number ofelements of simple geometries. Each element is then representedby a node point and its node number. The temperature at thisnode point (element) may or may not be known. Heat sources (withknown heat generation rates) are assigned to any nodes represent-ing the elements of heat generation within the bearing. Theenvironment surrounding the system, also having heat transferwith the system, is also represented by one or more nodes. Asmentioned earlier, a maximum of 20 nodes are allowed by theprogram to describe the equivalent thermal model.
With the elements and their node points properly selected, theheat balance equations considering different modes of heat trans-fer are then set up for each node. Heat transfers by conduction,free and forced convection, and mass transfer are considered.Radiation heat transfer, being very small as compared to othermodes in such applications, has been neglected.
Now for a steady state condition to exist, the net flow of heatto a node i from its surrounding nodes j plus the heat generatedat node i must be equal to zero. This heat balance at node iresults in an algebraic equation with some nodal temperatures asunknowns. As this condition would'be true for all the nodes inthe system, heat balance condition at each node results in analgebraic equation for that node.
Assuming a linear relationship for free and forced convectionmodes, the resulting mathematical model is a system of linearalgebraic equations with the unknown nodal temperatures as theunknown variables. The analysis then reduces to solving thissystem of algebraic equations for the unknown variables.
Consider the heat energy flowing into node i from its surroundingnodes j. If N is the total number of nodes in the model, thenfor the steady state condition to exist,
Net heat flowing into node i = oor,
nQi + E fl(Tj-Ti) + f 2 (Tj-Ti) + f 3 (Tj-Ti) + f 4 (Tj-Ti) = 0 (10.1)
J
Where fl, f 2 , f 3 , and f 4 are the coefficients in respective heattransfer equations for various modes of heat transfer between theelements. Volume I of this documentation, the USER'S GUIDE,provides more details on various modes and their applicable
37
equations. Rewriting equation (10.1) we get,
Qi = E Fij(Tj-Ti) = 0 ; i = 1, n (10.2)where, 11
Fij - fl + f2 + t3 + f4 (10.3)
Equation (10.2) represents a system of linear algebraic equationsfor 1 s i s N. Depending upon the actual number of surroundingnodes j interacting with node i, each equation may or may nothave all the unknown variables.
For a node (say node j) with a given (known) temperature, theheat balance equation for this node is replaced by the equalityT- = T - and the unknown variable T. substituted for theTgiveniown, Igiven in the remaining equations of the system.
The Gauss-Jordan numerical scheme, with partial pivoting, hasbeen deployed to invert the coefficient matrix for the solutionof this system. Reference [3] provides a detailed discussion onthis and other numerical methods.
10.3 TRANSIENT ANALYSIS
The formulation for computing the steady-state temperatures wasbased on the condition that the net energy transfer into anynode, i, from its surrounding nodes, j, is zero. Therefore, nofurther change of element temperatures then took place at thiscondition.
For the transient formulation, on the other hand, a net energytransfer to the node i from its surrounding nodes j takes placeresulting in an increase in the internal energy of the element i.Precluding any phase changes, this increase in the internalenergy results in a temperature rise for the element i. As eachelement volume is considered to have "lumped" capacity at itsnode point, the interaction of all the elements thus determinesthe behavior of the complete system.
For the mathematical formulation of the transient problem, thenet heat energy into a node i from its surrounding node(s) j in avery small interval of time (dt) is equated to the energy re-quired to raise the temperature of the element i tn a new value.Heat transfer by conduction, free and forced conv.ctions areconsidered. To simulate the lost-lubricant condition no heattransport by mass transport of the lubricant is now available.The lubricant nodes in the steady state model are replaced by airnodes having a forced convective heat transfer with the bearingelements.
The resulting equation for the node, i, is a linear differential
38
equation of first order with a known initial value. The ele-ment's steady-state temperature, estimated prior to this analy-sis, is used as the initial value at time t = o.
If the internal energy of an element, i, is expressed in terms ofits specific heat and temperature, its rate of change with time,t, is equal to the net heat energy gained by the node, i, in thesmall interval of time, dt. Therefore we can write that,
dEi dTiq d- = (ciPivi) -- (10.4)
dt dt
Where c is the specific heat, p the mass density, and v thevolume of the element, i, material. Using an explicit finitedifference approximation we can then transform the differentialequation (10.4) into a finite difference equation as follows,
Delta (Ei) TiP+I-TiPq .. ( ciPiVi) 1(10.5)
Delta(t) Delta(t)
Where TP and TP+1 are the temperatures of element, i, at time tand t+Delta(t) respectively.
The total heat gain, q, of this element, i, from the surroundingelements, j, in the same time interval, Delta(t), by variousmodes of heat transfer, is also given by,
q = Qi + E Fij (TjP-TiP) (10.6)J
Therefore by equating (10.5) and (10.6) and solving for TiP+1 weget,
Delta(t) Delta(t)TiP+' = (Qi + Z FijTjP) + (I E. Fij]Tip (10.7)i Ci Ci i
Where, Ci = ciPiVi is the thermal capacity of the element i.
If TiP is known at time t, then T-p+1 at time t+Delta(t) can bedetermined from equation (10.7) for the element i. This processis repeated for each node in the model and this time-marchingsolution continues till the desired temperature or time limit isreached. As mentioned before, the nodal steady-state tempera-tures are used as the initial values at time t = o.
39
11.0 REFERENCES
1. Allen, C.W., Townsend, D.P., and Zaretsky, E.V., "New
Generalized Rheological Model For Lubrication of a Ball
Spinning in a Nonconforming Groove", NASA Technical Note #
D-7280, May, 1973.
.2. Beer, F.P., and Johnson Jr., E.R., "Vector Mechanics for
Enaineers - Statics and Dynamics", Fifth Edition, McGraw-Hill,
1985.
3. Charpa, S.C. and Canale, R.P., "Numerical Methods For Engineers
with Personal Computer Applications", McGraw-Hill, 1985.
4. Coe, H.H., "Thermal Analysis of a Planetary Transmission
with Spherical Roller Bearings Operating After Complete Loss of
Oil", NASA Technical Paper 2367, Sept. 1984.
5. Eschmann, Hasbargen, r-nd Weigand, "Ball and Roller Bearings -
Theory. Design, and Application", Second Edition, Revised by
Hasbargen, L. and Br.ndlein, J., John Wiley and Sons, 1985.
6. Hadden, G.B., Kleckner, R.J., Ragen, M.A., and Sheynin, L.,
"Research' Report and User's Manual for Computer Program
SHABERTH", SKF Report No. AT8YO0003, NASA CR-165365, May, 1981.
7. Hamrock, B.J., and Dowson, D., "Ball Bearing Lubrication - The
Elastohydrodynamics of Elliptical Contacts", John Wiley and
Sons, 1982.
8. Harris, T.A., and Mindel, M.H., "Rolling Elements Bearing
Dynamics", Wear, Vol. 23, pp. 311-337, 1973.
9. Harris, T.A., "Roller Bearing Analysis", Second Edition, John
Wiley and Sons, 1984.
40
10. Holman, J.P., "H Transfer", Fourth Edition, McGraw-Hill,
1976.
11. Jones, A.B., "Analysis of Stresses and Deflections", NewDeparture Engineering Data, Bristol, Conn., 1946.
12. Jones, A.B., "The Mathemapical Theory of Rolling ElementBearings", Mechanical Design and Systems Handbook, SecondEdition, Ed. Rothbart, H.A., pp. 13.1-13.75, McGraw-Hill,1990.
13. Kahaner, D., Moler C., and Nash, S., "Numerical Methods and
Software", Prentice Hall, 1989.
14. Keller Jr., C.H., and Morrison, F.R., "Development of aThrust-Carrying Cylindrical Roller Bearing for HelicopterTransmissions", Report prepared by Sikorsky Aircraft, Divi-sion of United Technologies Corp. under contract from Ap-plied Technology Laboratory, U.S. Army Research and Technol-ogy Laboratories (AVRADCOM), Sept. 1980.
15. Kellstrom, E.M., "Rolling Contact Guidance of Rollers inSpherical Roller Bearings", Paper presented at the JointASME/ASLE Lubrication Conference, Dayton, Ohio, Oct. 16-18,1979, ASME Paper #79-LUB-23.
16. Kleckner, R.J., and Dyba, G., "High Speed Spherical RollerBearing Analysis and Comparison with Experimental Perform-ance", Advanced Power Transmission Technology, NASA Confer-ence Publication 2210, pp. 239-252, 1983.
17. Kleckner, R.J., and Pirvics, J., "Spherical Roller BearingAnalysis. SKF Computer Program SPHEARBEAN. Volume I:Analytic Formulation", SKF Report No. AT81DO06, Submitted toNASA/Lewis Research Center, NASA/NTIS Report No. CR-165203,Dec. 1980.
41
18. Kleckner, R.J., and Dyba, G., "Spherical Roller BearingAnalysis. SKF Computer Program SPHEARBEAN. Volume II:
User's Manual", SKF Report No. AT81DO07, Submitted toNASA/Lewis Research Center, NASA/NTIS Report No. CR-165204,
Dec. 1980.
19. Lundberg, G. and Palmgren, A., "Dynamic Capacity of Rolling
Bearings", ACTA POLYTECHNICA, Mech. Eng. Ser. 1, No. 3, Vol.
7, 1947.
20. Lundberg, G. and Palmgren, A., "Dynamic Capacity of RollerBearings", ACTA POLYTECHNICA, Mech. Eng. Ser. 1, 1952.
21. Mobil Oil Corporation, "Mobil EHL Guidebook", Third Edition,Mobil Technical Publications, 1981.
22. Palmgren, A., "Ball and Roller Bearing EnQineering", SecondEdition, Translated by Palmgren, G., and Ruley, B., S.H.
Burbank and Co. Publisher, Philadelphia, 1946.
23. Skurka, J.C., "Elastohydrodynamic Lubrication of RollerBearings", Jr. Lubrication Technology, pp. 281-288, April1970.
24. The Society of Tribologists and Lubrication Engineers,"Starting from Scratch - Tribologv Basics", A compilation ofarticles previously published in Lubrication Engineering,
Ed. Fred Litt, STLE, Park Ridge, Illinois.
25. Wilcock, D.F., and Booser, E.R., "BearinQ Design and Appli-cation", First Edition, McGraw-Hill, 1957.
26. Winer, W.O., and Cheng, H.S., "Film Thickness, Contact
Stress and Surface Temperatures", Wear Control Handbook, pp.
81-141.
42
APPENDIX - A
VOLUME OF A FULLY CROWNED SPHERICAL ROLLER
Volume of a spherical roller, required in the estimation ofcentrifugal force and gyroscopic moment acting on a roller, isdetermined using simple integral calculus as follows.
SLet OXYZ be a coordinate frame of reference with its origin atthe roller center as shown in Fi',ire 12. The coordinates of theroller crown radius center (0') .re then given by xo = 0 and yo-(Ro-D/2). Therefore, the profile of the roller crown can berepresented by the following equation of a circle,
(X-xo)2 + (y-yo) 2 = Ro 2 (A.1)
or, y = (Ro 2 -X 2 )3 + Yo (A.2)
Now by considering the volume of a roller slice of differentialthickness, dx, at a distance x from the roller mid-plane (asshown in Figure 12) and integrating it for the full roller lengthwe get,
V=f dV = wi y 2 dx (A.3)
L L
By substituting for y = y(x) from equation (A.2) into equation(A.3) and integrating w.r.t x we get an expression for the volumeof a fully crowned spherical roller.
43
Y
L
44,
APPEMDIX - B
M3KENTS OF INERTIA OF A SPHERICAL ROLLER
Mass Moments Of Inertia (MMOI) of the roller, both about itslongitudinal and transverse axes, are required for the calcula-tions of gyroscopic moment acting on each roller. The expres-sions for these have also been derived by considering a rollerslice of differential thickness and integrating over the fulllength of a roller, as summarized below.
By referring to Figure 12 and considering the roller slice asshown, the moments of inertia of the slice about the X, Y, and Zaxes are given by,
dIx 0 My2 dm (B.1)
dIy = dIly + x 2 dm = (W2 +x2 )dm (B.2)
dIz = dIzI + x 2 dm -= (Uy 2 +x 2 )dm (B.3)
Where dIy, and dIz, are the moments of inertia of the disk abouta local coordinate axes X'Y'Z', with origin at the disk centerand parallel to the global axes OXYZ as shown in Figure 12. dmis the mass of the differential disk and is given by,
dm = pdv= pwy 2 dx (B.4)
p being is the mass density of the roller material. Roller'sspherical profile is described by y = (Ro -x ) +yo as derivedin equation (A.2) of Appendix-A with yo = - (io-D/2).
By substituting equation (B.4) into equations (B.1), (B.2), and(B.3) and performing the integration w.r.t. x for x varying from-L/2 to +L/2, we get the expressions for roller's mass moments ofinertia about its longitudinal axis (Ix) and transverse axes (Iy,Iz). Due to its symmetry about the X-axis, we get Iy = Iz.
45
APPENDIX - C
INITIAL ESTIMATION OF INTERNAL ROTATIONAL SPEEDS
To estimate the magnitudes of the roller dynamic loads, namelythe centrifugal force and gyroscopic moment, an initial estimateof the cage rotational speed about the bearing axis and eachroller's rotational speed about its own longitudinal axis wasfirst made under the following assumptions:
(a) The operating contact angle of each roller is the initialdesign contact angle of the bearing. In other words, the contactellipse at each concentrated contact is centered about the rollercenter.
(b) No gross slip occurs at any concentrated contact.
The tangential velocity of the cage is then given by the mean ofthe tangential velocities of contact center points at the innerand outer raceways. Thus,
or, Qcg = 3[fi(I-e) + ' 2 (1+e)] (C.1)
N = 3A[NI(I-e) + N2 (I+e)] (C.2)cg 2(+)
The rotational speed of each roller about its own axis, requiredfor estimating the gyroscopic moment on each roller, is thengiven by,
No = (Pt/2D) (1-e) (1+0) (N2 -NI) (C.2)
46
APPENDIX - D
ROLLER GYROSCOPIC MOTION ANALYSIS
In angular contact spherical roller bearings, the motion of eachroller is very similar to that of a gyroscope. Like a gyroscoperotor, the roller is rotating (gyroscopic spin) about its owngeometric axis, unparrallel to the bearing axis, and also orbit-ing (gyroiscopic precession) about the bearing axis along withthe cage. The gyroscopic moments seen by these rollers, oftenneglected for low speed and/or low contact angle analyses, becomequite significant in high speed applications and considerablyaffects the roller deflections and load distributions.
The formulation for the SASHBEAN Computer Program takes into fullconsideration this moment loading of the rollers. The assump-tions made for the estimation of the magnitude of this momentacting on each roller are listed below:
(a) The cage has a constant angular velocity of rotation aboutits (or bearing) axis as determined in Appendix C.
(b) Each roller has the same constant angular velocity about itslongitudinal axis as determined in Appendix C.
(c) The roller misalignment (pitching) angle at any azimuthlocation is negligibly small when compared to the designcontact angle of the bearing.
(d) The operating contact angle of each roller is the same asthe design contact angle of the bearing.
Consider a roller at any angular location (0) as shown inFigure 13. Let OXYZ be a fixed frame of reference with itsorigin, 0, at the intersection of bearing and roller axes. Leto'xyz be a rotating system of axes, attached to the roller withorigin at the roller mass center as shown in the same figure. Byconsidering the roller as a gyroscopic rotor we can see that forthis gyroscope,
Nutation Angle = Roller-Raceway Contact Angle = g
Precession Angular Velocity = Roller Orbital Velocity = 0cg
Spin Angular Velocity = Roller Rotational Velocity = go
By considering the special case of the gyroscopic motion where 9,
Qc_ and 0 remain constant, the couple (moment) required on therofler (the rotor) to sustain this motion is given by,
GM = [Iz(go+OcgCOSs) - Ix(QcgCoss)]QcgSin:9 (D.1)
47
This couple, applied to the roller by the raceways, is about anaxis perpendicular to the precession and spin axes of the gyro-scope. The moment vector is thus directed along the positive y-axis (perpendicular and pointing into the plane of the paper).The reaction couple of the roller, resisted by the raceways, isequal in magnitude and opposite in direction. For a detaileddiscussion on the topic and derivation of the above equation, thereader is referred to reference [2].
z
//!C
'Cr<
/
0 - __ _
FIGURE 13
48
APPMWIX - 9
LOAD-DEFORMATION RELATIONSHIP AND MATERIAL FACTORS
For a cylindrical body of finite length, when pressed onto aplane surfaced body of infinite extension, the normal approachbetween the axis of the cylinder and a distant point in thesupporting body, first presented by Palmgren et. al. (221, isapproximately given by,
8 4.36E-7[E2.7 Q* 9/L-81or 6 4.36E-7[J92.7q*9L*11
(E.2)
where A is a factor based on the materials of the two contactingbodies and Q = qL is the total normal force pressing the twobodies together. The material factor A is given by,
(A 1 1/3
1.643E+7- 1+E2) (E.3)E1 2
where, E'i = El/(l-e,2) and E'2 E2/(I-e2 2
(a) RELAtIONSHIP FOR STEEL QN STEEL CONTACT:
Let E. = 29.OE6 psi, E2 = 29.OE6 psi, el = 0.30, and e2 = 0.30
We get A2.7 = 1.0. Equation (E.2) then gives us,
6 = 4.36E-7q*9.L*l (E.4)
(b) RELATIONSHIP FOR QEBMIQýQN STEEL CONTACT:
Let El = 43.OE6 psi, E2 = 29.OE6 psi, el = 0.27, and e2 = 0.30
We get J92.7 = 0.872. Equation (E.2) then gives us,
6 = 3.80E-7-q*9.L*l (E.5)
49
Form ApprovedREPORT DOCUMENTATION PAGE OMB No. 0704-0188
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Cabtonc irdoniloninc.u suggaetions for reducing this burden, to Washington Headquarters Service, Deectoate to,_Infon-.lion Operateons end Repor. 1216 ,lfarsnonDavis Highlway Suite 1204, Artlingtn -A224302. and to the Offlim of Managrlment and Budge. Pageiwoik Reduction Protect (0704-0188), Washington. t)C 20603.
1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED
September 1993 Final Contractor Report4. TITLE AND SUBTITLE 5. FUNDING NUMBERS
Computer Program for Analysis of High Speed, Single Row,Angular Contact, Spherical Roller Bearing, SASHBEANVolume II: Mathematical Formulation and Analysis WU-505-63-36
6. AUTHOR(S) 1L162211A47A
Amn K. Aggarwal
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION
REPORT NUMBER
Emerson Power Transmission CorporationMcGill Manufacturing Company E-8088Valparaiso, Indiana 46383-4299
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING'MONITORINGVehicle Propulsion Directorate AGENCY REPORT NUMBER
U.S. Army Research LaboratoryCleveland, Ohio 44135-3191 NASA CR-191182and ARL-CR-81NASA Lewis Research CenterCleveland, Ohio 44135-3191
11. SUPPLEMENTARY NOTES
Project Manager, Timothy L. Krantz, (216) 433-3580.
12a. DISTRIBUTION/AVAILABILrTY STATEMENT 12b. DISTRIBUTION CODE
Unclassified -UnlimitedSubject Category 37
13. ABSTRACT (Maximum 200 words)
Spherical roller bearings have typically been used in applications with speeds limited to about 5000 r.p.m. and loadslimited for operation at less than about 0.25 million DN. However, spherical roller bearings are now being designedfor high load and high speed applications including aerospace applications. A computer program, SASHBEAN, hasbeen developed to provide an analytical tool to design, analyze, and predict the performance of high speed, single row,angular contact (including zero contact angle), spherical roller bearings. The material presented in this document is themathematical formulation and analytical methods used to develop computer program SASHBEAN. For a given set ofoperating conditions, the program calculates the bearing's ring deflections (axial and radial), roller deflections, contactareas and stresses, depth and magnitude of maximum shear stresses, axial thrust, rolling element and cage rotationalspeeds, lubrication parameters, fatigue lives, and rates of heat generation. Centifugal forces and gyroscopic momentsare fully considered. The program is also capable of performing steady-state and time-transient thermal analyses of thebearing system.
14. SUBJECT TERMS 15. NUMBER OF PAGES
51Bearing; Computer aided engineering 16. PRICE CODE
A0417. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT
OF REPORT OF THIS PAGE OF ABSTRACT
Unclassified Unclassified
NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)Prescnoed by ANSI Std. Z39-18298-102
