Advanced Concepts of Bearing Technology,: Rolling Bearing Analysis, Fifth Edition (Rolling Bearing...
Transcript of Advanced Concepts of Bearing Technology,: Rolling Bearing Analysis, Fifth Edition (Rolling Bearing...
Rolling Bearing AnalysisF I F T H E D I T I O N
Advanced Concepts of Bearing Technology
� 2006 by Taylor & Francis Group, LLC.
� 2006 by Taylor & Francis Group, LLC.
Rolling Bearing Analysis
Tedric A. HarrisMichael N. Kotzalas
F I F T H E D I T I O N
Advanced Concepts of Bearing Technology
� 2006 by Taylor & Francis Group, LLC.
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Preface
The main purpose of the first volume of this handbook was to provide the reader with
information on the use, design, and performance of ball and roller bearings in common and
relatively noncomplex applications. Such applications generally involve slow-to-moderate
speed, shaft, or bearing outer ring rotation; simple, statically applied, radial or thrust loading;
bearing mounting that does not include misalignment of shaft and bearing outer-ring axes;
and adequate lubrication. These applications are generally covered by the engineering infor-
mation provided in the catalogs supplied by the bearing manufacturers. While catalog
information is sufficient to enable the use of the manufacturer’s product, it is always
empirical in nature and rarely provides information on the geometrical and physical justifi-
cations of the engineering formulas cited. The first volume not only includes the underlying
mathematical derivations of many of the catalog-contained formulas, but also provides
means for the engineering comparison of rolling bearings of various types and from different
manufacturers.
Many modern bearing applications, however, involve machinery operating at high
speeds; very heavy combined radial, axial, and moment loadings; high or low temperatures;
and otherwise extreme environments. While rolling bearings are capable of operating in
such environments, to assure adequate endurance, it is necessary to conduct more sophisticated
engineering analyses of their performance than can be achieved using the methods and formulas
provided in the first volume of this handbook. This is the purpose of the present volume.
When compared with its earlier editions, this edition presents updated and more accurate
information to estimate rolling contact friction shear stresses and their effects on bearing
functional performance and endurance. Also, means are included to calculate the effects on
fatigue endurance of all stresses associated with the bearing rolling and sliding contacts. These
comprise stresses due to applied loading, bearing mounting, ring speeds, material processing,
and particulate contamination.
The breadth of the material covered in this text, for credibility, can hardly be covered by
the expertise of the two authors. Therefore, in the preparation of this text, information
provided by various experts in the field of ball and roller bearing technology was utilized.
Contributions from the following persons are hereby gratefully acknowledged:
. Neal DesRuisseaux . bearing vibration and noise
. John I. McCool . bearing statistical analysis
. Frank R. Morrison . bearing testing
. Joseph M. Perez . lubricants
. John R. Rumierz . lubricants and materials
. Donald R. Wensing . bearing materials
Finally, since its initial publication in 1967, Rolling Bearing Analysis has evolved into this 5th
edition. We have endeavored to maintain the material presented in an up-to-date and useful
format. We hope that the readers will find this edition as useful as its earlier editions.
Tedric A. Harris
Michael N. Kotzalas
� 2006 by Taylor & Francis Group, LLC.
� 2006 by Taylor & Francis Group, LLC.
Authors
Tedric A. Harris is a graduate in mechanical engineering from the Pennsylvania State
University, who received a B.S. in 1953 and an M.S. in 1954. After graduation, he was
employed as a development test engineer at the Hamilton Standard Division, United Aircraft
Corporation, Windsor Locks, Connecticut, and later as an analytical design engineer at the
Bettis Atomic Power Laboratory, Westinghouse Electric Corporation, Pittsburgh, Pennsyl-
vania. In 1960, he joined SKF Industries, Inc. in Philadelphia, Pennsylvania as a staff
engineer. At SKF, Harris held several key management positions: manager, analytical ser-
vices; director, corporate data systems; general manager, specialty bearings division; vice
president, product technology & quality; president, SKF Tribonetics; vice president, engin-
eering & research, MRC Bearings (all in the United States); director for group information
systems at SKF headquarters, Gothenburg, Sweden; and managing director of the engineer-
ing & research center in the Netherlands. He retired from SKF in 1991 and was appointed as a
professor of mechanical engineering at the Pennsylvania State University at University Park.
He taught courses in machine design and tribology and conducted research in the field of
rolling contact tribology at the university until retirement in 2001. Currently, he is a prac-
ticing consulting engineer and, as adjunct professor in mechanical engineering, teaches
courses in bearing technology to graduate engineers in the university’s continuing education
program.
Harris is the author of 67 technical publications, mostly on rolling bearings. Among these
is the book Rolling Bearing Analysis, currently in its 5th edition. In 1965 and 1968, he received
outstanding technical paper awards from the Society of Tribologists and Lubrication Engin-
eers and in 2001 from the American Society of Mechanical Engineers (ASME) Tribology
Division. In 2002, he received the outstanding research award from the ASME.
Harris has served actively in numerous technical organizations, including the Anti-
Friction Bearing Manufacturers’ Association, ASME Tribology Division, and ASME Re-
search Committee on Lubrication. He was elected ASME Fellow Member in 1973. He has
served as chair of the ASME Tribology Division and as chair of the Tribology Division’s
Nominations and Oversight Committee. He holds three U.S. patents.
Michael N. Kotzalas graduated from the Pennsylvania State University with a B.S. in 1994,
M.S. in 1997, and Ph.D. in 1999, all in mechanical engineering. During this time, the focus of
his study and research was on the analysis of rolling bearing technology, including quasidy-
namic modeling of ball and cylindrical roller bearings for high-acceleration applications and
spall progression testing and modeling for use in condition-based maintenance algorithms.
Since graduation, Dr. Kotzalas has been employed by The Timken Company in research
and development and most recently in the industrial bearing business. His current responsi-
bilities include advanced product design and application support for industrial bearing
customers, while the previous job profile in research and development included new product
and analysis algorithm development. From these studies, Dr. Kotzalas has received two U.S.
patents for cylindrical roller bearing designs.
Outside of work, Dr. Kotzalas is also an active member of many industrial societies. As a
member of the ASME, he currently serves as the chair of the publications committee and as a
member of the rolling element bearing technical committee. He is a member of the awards
committee in the Society of Tribologists and Lubrication Engineers (STLE). Dr. Kotzalas has
� 2006 by Taylor & Francis Group, LLC.
also published ten articles in peer-reviewed journals and one conference proceeding. Some of
his publications were honored with the ASME Tribology Division’s Best Paper Award in
2001 and STLE’s Hodson Award in 2003 and 2006. Also, working with the American Bearing
Manufacturer’s Association (ABMA), Dr. Kotzalas is one of the many instructors for the
short course ‘‘Advanced Concepts of Bearing Technology’’.
� 2006 by Taylor & Francis Group, LLC.
Table of Contents
Chapter 1
Distribution of Internal Loading in Statically Loaded Bearings:
Combined Radial, Axial, and Moment Loadings—Flexible
Support of Bearing Rings
1.1 General
1.2 Ball Bearings under Combined Radial, Thrust, and Moment Loads
1.3 Misalignment of Radial Roller Bearings
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1.3.1 Components of Deformation
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1.3.1.1 Crowning
1.3.2 Load on a Roller–Raceway Contact Lamina
1.3.3 Equations of Static Equilibrium
1.3.4 Deflection Equations
1.4 Thrust Loading of Radial Cylindrical Roller Bearings
1.4.1 Equilibrium Equations
1.4.2 Deflection Equations
1.4.3 Roller–Raceway Deformations Due to Skewing
1.5 Radial, Thrust, and Moment Loadings of Radial Roller Bearings
1.5.1 Cylindrical Roller Bearings
1.5.2 Tapered Roller Bearings
1.5.3 Spherical Roller Bearings
1.6 Stresses in Roller–Raceway Nonideal Line Contacts
1.7 Flexibly Supported Rolling Bearings
1.7.1 Ring Deflections
1.7.2 Relative Radial Approach of Rolling Elements to the Ring
1.7.3 Determination of Rolling Element Loads
1.7.4 Finite Element Methods
1.8 Closure
References
Chapter 2
Bearing Component Motions and Speeds
2.1 General
2.2 Rolling and Sliding
2.2.1 Geometrical Considerations
2.2.2 Sliding and Deformation
2.3 Orbital, Pivotal, and Spinning Motions in Ball Bearings
2.3.1 General Motions
2.3.2 No Gyroscopic Pivotal Motion
2.3.3 Spin-to-Roll Ratio
2.3.4 Calculation of Rolling and Spinning Speeds
2.3.5 Gyroscopic Motion
2.4 Roller End–Flange Sliding in Roller Bearings
2.4.1 Roller End–Flange Contact
2.4.2 Roller End–Flange Geometry
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2.4.3 Sliding Velocity
2.5 Closure
References
Chapter 3
High-Speed Operation: Ball and Roller Dynamic Loads and Bearing
Internal Load Distribution
3.1 General
3.2 Dynamic Loading of Rolling Elements
3.2.1 Body Forces Due to Rolling Element Rotations
3.2.2 Centrifugal Force
y Taylor &
3.2.2.1 Rotation about the Bearing Axis
3.2.2.2 Rotation about an Eccentric Axis
3.2.3 Gyroscopic Moment
3.3 High-Speed Ball Bearings
3.3.1 Ball Excursions
3.3.2 Lightweight Balls
3.4 High-Speed Radial Cylindrical Roller Bearings
3.4.1 Hollow Rollers
3.5 High-Speed Tapered and Spherical Roller Bearings
3.6 Five Degrees of Freedom in Loading
3.7 Closure
References
Chapter 4
Lubricant Films in Rolling Element–Raceway Contacts
4.1 General
4.2 Hydrodynamic Lubrication
4.2.1 Reynolds Equation
4.2.2 Film Thickness
4.2.3 Load Supported by the Lubricant Film
4.3 Isothermal Elastohydrodynamic Lubrication
4.3.1 Viscosity Variation with Pressure
4.3.2 Deformation of Contact Surfaces
4.3.3 Pressure and Stress Distribution
4.3.4 Lubricant Film Thickness
4.4 Very-High-Pressure Effects
4.5 Inlet Lubricant Frictional Heating Effects
4.6 Starvation of Lubricant
4.7 Surface Topography Effects
4.8 Grease Lubrication
4.9 Lubrication Regimes
4.10 Closure
References
Chapter 5
Friction in Rolling Element–Raceway Contacts
5.1 General
5.2 Rolling Friction
5.2.1 Deformation
5.2.2 Elastic Hysteresis
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5.3 Sliding Friction
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5.3.1 Microslip
5.3.2 Sliding Due to Rolling Motion: Solid-Film or Boundary Lubrication
y Taylor &
5.3.2.1 Direction of Sliding
5.3.2.2 Sliding Friction
5.3.3 Sliding Due to Rolling Motion: Full Oil-Film Lubrication
5.3.3.1 Newtonian Lubricant
5.3.3.2 Lubricant Film Parameter
5.3.3.3 Non-Newtonian Lubricant in an Elastohydrodynamic
Lubrication Contact
5.3.3.4 Limiting Shear Stress
5.3.3.5 Fluid Shear Stress for Full-Film Lubrication
5.3.4 Sliding Due to Rolling Motion: Partial Oil-Film Lubrication
5.3.4.1 Overall Surface Friction Shear Stress
5.3.4.2 Friction Force
5.4 Real Surfaces, Microgeometry, and Microcontacts
5.4.1 Real Surfaces
5.4.2 GW Model
5.4.3 Plastic Contacts
5.4.4 Application of the GW Model
5.4.5 Asperity-Supported and Fluid-Supported Loads
5.4.6 Sliding Due to Rolling Motion: Roller Bearings
5.4.6.1 Sliding Velocities and Friction Shear Stresses
5.4.6.2 Contact Friction Force
5.4.7 Sliding Due to Spinning and Gyroscopic Motions
5.4.7.1 Sliding Velocities and Friction Shear Stresses
5.4.7.2 Contact Friction Force Components
5.4.8 Sliding in a Tilted Roller–Raceway Contact
5.5 Closure
References
Chapter 6
Friction Effects in Rolling Bearings
6.1 General
6.2 Bearing Friction Sources
6.2.1 Sliding in Rolling Element–Raceway Contacts
6.2.2 Viscous Drag on Rolling Elements
6.2.3 Sliding between the Cage and the Bearing Rings
6.2.4 Sliding between Rolling Elements and Cage Pockets
6.2.5 Sliding between Roller Ends and Ring Flanges
6.2.6 Sliding Friction in Seals
6.3 Bearing Operation with Solid-Film Lubrication: Effects
of Friction Forces and Moments
6.3.1 Ball Bearings
6.3.2 Roller Bearings
6.4 Bearing Operation with Fluid-Film Lubrication: Effects
of Friction Forces and Moments
6.4.1 Ball Bearings
6.4.1.1 Calculation of Ball Speeds
6.4.1.2 Skidding
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6.4.2 Cylindrical Roller Bearings
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6.4.2.1 Calculation of Roller Speeds
6.4.2.2 Skidding
6.5 Cage Motions and Forces
6.5.1 Influence of Speed
6.5.2 Forces Acting on the Cage
6.5.3 Steady-State Conditions
6.5.4 Dynamic Conditions
6.6 Roller Skewing
6.6.1 Roller Equilibrium Skewing Angle
6.7 Closure
References
Chapter 7
Rolling Bearing Temperatures
7.1 General
7.2 Friction Heat Generation
7.2.1 Ball Bearings
7.2.2 Roller Bearings
7.3 Heat Transfer
7.3.1 Modes of Heat Transfer
7.3.2 Heat Conduction
7.3.3 Heat Convection
7.3.4 Heat Radiation
7.4 Analysis of Heat Flow
7.4.1 Systems of Equations
7.4.2 Solution of Equations
7.4.3 Temperature Node System
7.5 High Temperature Considerations
7.5.1 Special Lubricants and Seals
7.5.2 Heat Removal
7.6 Heat Transfer in a Rolling–Sliding Contact
7.7 Closure
References
Chapter 8
Application Load and Life Factors
8.1 General
8.2 Effect of Bearing Internal Load Distribution on Fatigue Life
8.2.1 Ball Bearing Life
8.2.1.1 Raceway Life
8.2.1.2 Ball Life
8.2.2 Roller Bearing Life
8.2.2.1 Raceway Life
8.2.2.2 Roller Life
8.2.3 Clearance
8.2.4 Flexibly Supported Bearings
8.2.5 High-Speed Operation
8.2.6 Misalignment
8.3 Effect of Lubrication on Fatigue Life
8.4 Effect of Material and Material Processing on Fatigue Life
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8.5 Effect of Contamination on Fatigue Life
8.6 Combining Fatigue Life Factors
8.7 Limitations of the Lundberg–Palmgren Theory
8.8 Ioannides–Harris Theory
8.9 The Stress–Life Factor
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8.9.1 Life Equation
8.9.2 Fatigue-Initiating Stress
8.9.3 Subsurface Stresses Due to Normal Stresses Acting on the
Contact Surfaces
8.9.4 Subsurface Stresses Due to Frictional Shear Stresses
Acting on the Contact Surfaces
8.9.5 Stress Concentration Associated with Surface
Friction Shear Stress
8.9.6 Stresses Due to Particulate Contaminants
8.9.7 Combination of Stress Concentration Factors Due to
Lubrication and Contamination
8.9.8 Effect of Lubricant Additives on Bearing Fatigue Life
8.9.9 Hoop Stresses
8.9.10 Residual Stresses
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8.9.10.1 Sources of Residual Stresses
8.9.10.2 Alterations of Residual Stress Due to Rolling Contact
8.9.10.3 Work Hardening
8.9.11 Life Integral
8.9.12 Fatigue Limit Stress
8.9.13 ISO Standard
8.10 Closure
References
Chapter 9
Statically Indeterminate Shaft–Bearing Systems
9.1 General
9.2 Two-Bearing Systems
9.2.1 Rigid Shaft Systems
9.2.2 Flexible Shaft Systems
9.3 Three-Bearing Systems
9.3.1 Rigid Shaft Systems
9.3.2 Nonrigid Shaft Systems
9.3.2.1 Rigid Shafts
9.4 Multiple-Bearing Systems
9.5 Closure
Reference
Chapter 10
Failure and Damage Modes in Rolling Bearings
10.1 General
10.2 Bearing Failure Due to Faulty Lubrication
10.2.1 Interruption of Lubricant Supply to Bearings
10.2.2 Thermal Imbalance
10.3 Fracture of Bearing Rings Due to Fretting
10.4 Bearing Failure Due to Excessive Thrust Loading
10.5 Bearing Failure Due to Cage Fracture
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10.6 Incipient Failure Due to Pitting or Indentation of the
Rolling Contact Surfaces
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10.6.1 Corrosion Pitting
10.6.2 True Brinnelling
10.6.3 False Brinnelling in Bearing Raceways
10.6.4 Pitting Due to Electric Current Passing through the Bearing
10.6.5 Indentations Caused by Hard Particle Contaminants
10.6.6 Effect of Pitting and Denting on Bearing Functional
Performance and Endurance
10.7 Wear
10.7.1 Definition of Wear
10.7.2 Types of Wear
10.8 Micropitting
10.9 Surface-Initiated Fatigue
10.10 Subsurface-Initiated Fatigue
10.11 Closure
References
Chapter 11
Bearing and Rolling Element Endurance Testing and Analysis
11.1 General
11.2 Life Testing Problems and Limitations
11.2.1 Acceleration of Endurance Testing
11.2.2 Acceleration of Endurance Testing through Very
Heavy Applied Loading
11.2.3 Avoiding Test Operation in the Plastic Deformation Regime
11.2.4 Load–Life Relationship of Roller Bearings
11.2.5 Acceleration of Endurance Testing through
High-Speed Operation
11.2.6 Testing in the Marginal Lubrication Regime
11.3 Practical Testing Considerations
11.3.1 Particulate Contaminants in the Lubricant
11.3.2 Moisture in the Lubricant
11.3.3 Chemical Composition of the Lubricant
11.3.4 Consistency of Test Conditions
y Taylor &
11.3.4.1 Condition Changes over the Test Period
11.3.4.2 Lubricant Property Changes
11.3.4.3 Control of Temperature
11.3.4.4 Deterioration of Bearing Mounting Hardware
11.3.4.5 Failure Detection
11.3.4.6 Concurrent Test Analysis
11.4 Test Samples
11.4.1 Statistical Requirements
11.4.2 Number of Test Bearings
11.4.3 Test Strategy
11.4.4 Manufacturing Accuracy of Test Samples
11.5 Test Rig Design
11.6 Statistical Analysis of Endurance Test Data
11.6.1 Statistical Data Distributions
11.6.2 The Two-Parameter Weibull Distribution
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11.6.2.1 Probability Functions
11.6.2.2 Mean Time between Failures
11.6.2.3 Percentiles
11.6.2.4 Graphical Representation of the Weibull Distribution
11.6.3 Estimation in Single Samples
11.6.3.1 Application of the Weibull Distribution
11.6.3.2 Point Estimation in Single Samples: Graphical Methods
11.6.3.3 Point Estimation in Single Samples: Method of
Maximum Likelihood
11.6.3.4 Sudden Death Tests
16.3.3.5 Precision of Estimation: Sample Size Selection
11.6.4 Estimation in Sets of Weibull Data
11.6.4.1 Methods
11.7 Element Testing
11.7.1 Rolling Component Endurance Testers
11.7.2 Rolling–Sliding Friction Testers
11.7.2.1 Purpose
11.7.2.2 Rolling–Sliding Disk Test Rig
11.7.2.3 Ball–Disk Test Rig
11.8 Closure
References
Appendix
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1 Distribution of Internal Loadingin Statically Loaded Bearings:
� 2006 by Taylor & Fran
Combined Radial, Axial, andMoment Loadings—FlexibleSupport of Bearing Rings
LIST OF SYMBOLS
Symbol Description Units
A Distance between raceway groove curvature centers mm (in.)
B fi þ fo � 1
c Crown drop at end of roller or raceway effective length or
crown gap at other locations mm (in.)
C Influence coefficient mm/N (in./lb)
D Ball or roller diameter mm (in.)
Dij Influence coefficient to calculate nonideal roller–raceway
contact deformations
dm Bearing pitch diameter mm (in.)
e Eccentricity of loading mm (in.)
E Modulus of elasticity MPa (psi)
f r/D
F Applied load N (lb)
Fa Friction force due to roller end–ring flange sliding motions N (lb)
h Roller thrust couple moment arm mm (in.)
I Ring section moment of inertia mm4 (in.4)
k Number of laminae
K Load–deflection factor, axial load–deflection factor N/mmn (lb/in.n)
l Roller length mm (in.)
M Moment N�mm (lb� in.)
n Load–deflection exponent
Pd Diametral clearance mm (in.)
q Load per unit length N/mm (lb/in.)
Q Ball or roller–raceway normal load N (lb)
Qa Roller end–ring flange load in cylindrical roller bearing N (lb)
Qf Roller end–ring flange load in tapered roller bearing N (lb)
cis Group, LLC.
r Racew ay groo ve cu rvature radius mm (in.)
r Radi us to racew ay contact in tapere d roller bearing mm (in.)
rf Radi us from inner- ring axis to roll er end –flange contact in
tapered roll er be aring mm (in.)
Rf Radi us from tapere d roller axis to roller end–fl ange contact mm (in.)
< Ring radius to neutral axis mm (in.)
< Radi us of locus of raceway groove curvat ure centers mm (in.)
s Dis tance be tween loci of inner an d outer racew ay groove
curvatu re cen ters mm (in.)
u Ring radial deflection mm (in.)
U Strai n energy N � mm (lb � in.)
Z Number of balls or rollers pe r be aring row
a Mo unted contact an gle rad, 8
ao Free con tact angle rad, 8
b tan � 1 l =ðdm � DÞ rad, 8
g D cos a=dm
d Defl ection or contact de formati on mm (in.)
d1 Dis tance be tween inner and outer rings mm (in.)
D Cont act de formati on due to ideal normal load ing mm (in.)
Dc Angul ar spacing be tween rolling elem ents rad, 8
z Roll er tilt angle rad, 8
h tan � 1 l =D rad, 8
u Bea ring mis alignment angle rad, 8
l Lam ina posit ion
m Coef ficient of sliding fricti on between roll er en d and
ring flange
s Norm al contact stre ss or pressur e MPa (psi)
j Poisson ’s ratio
j Roll er skew ing an gle rad, 8
1.1 GENERAL
In most bearing applic ations, only app lied radial , axial , or co mbined radial an d axial loading s
are consider ed. How ever, unde r very heavy applie d load ing or if shafting is hollow to
mini mize wei ght, the shaft on whi ch the bearing is mounted may bend, causing a signi ficant
moment load on the bearing . Als o, the bearing housing may be nonr igid due to design
targe ted at mini mizing both size an d wei ght, causing it to ben d whi le accomm odating
moment loading . Such comb ined radial , axial , and moment loading s result in alte red dist ri-
bution of load among the bearing ’s rolling elemen t complem ent. This may cause signi ficant
chan ges in bearing deflections , co ntact stre sses, and fatigue endu rance co mpared to these
ope rating pa rameters associ ated with the sim pler load dist ributions consider ed in Chapt er 7
of the fir st volume of this ha ndbook.
In cylin drical and tapere d roller bea rings, the moment loading caused by bend ing of the shaft
resul ts in nonuni form load pe r unit lengt h along the roller–rac eway c ontacts. Misalignm ent
of the bearing inner ring on the shaft or outer ring in the housing also generates moment loading
in the bearing, causing a nonuniform load per unit length along the roller–raceway contacts.
Thus, the maximum roller–raceway contact stresses will be greater than those occurring if the
contacts are loadeduniformly along their lengths.Moreover,whenbearing rings aremisaligned,
thrust loading is induced in the rollers, causing the rollers to tilt, further exacerbating the
nonuni form ro ller–raceway co ntact loading . As seen in Chapter 11 in the first volume of this
� 2006 by Taylor & Francis Group, LLC.
r i
a o a
d o
d i
A A
Q(b)(a)
Q
r o
FIGURE 1.1 (a) Ball–raceway contact before applying load; (b) ball–raceway contact after load is applied.
handbook, fatigue life is inversely proportional to approximately the ninth power of contact
stress.Hence, a nonuniformroller–raceway contact loading can result in significant reduction in
bearing endurance.
In this chapter, methods to determine the distribution of applied loading among the
rolling elements will be established considering each of the aforementioned effects.
1.2 BALL BEARINGS UNDER COMBINED RADIAL, THRUST,AND MOMENT LOADS
When a ball is compressed by load Q, since the centers of curvature of the raceway grooves
are fixed with respect to the corresponding raceways, the distance between the centers is
increased by the amount of the normal approach between the raceways. From Figure 1.1, it
can be seen that
s ¼ Aþ di þ do ð1:1Þ
dn¼ di þ do ¼ s� A ð1:2Þ
If a ball bearing that has a number of balls situated symmetrically about a pitch circle is
subjected to a combination of radial, thrust (axial), and moment loads, the following relative
displacements of inner and outer raceways may be defined:
da Relative axial displacement
dr Relative radial displacement
u Relative angular displacement
These relative displ acements are shown in Figure 1.2.
Consider a rolling bearing be fore the applic ation of a load . Figure 1.3 sho ws the pos itions
of the loci of the centers of the inner and outer raceway groove curvature radii. It can be
determ ined from Figure 1.4 that the locu s of the center s of the inner- ring racew ay g roove
curvature radii is expressed by
<i ¼dm
2þ ri �
D
2
� �cos ao ð1:3Þ
� 2006 by Taylor & Francis Group, LLC.
drda
q
FIGURE 1.2 Displacements of an inner ring (outer ring fixed) due to application of combined radial,
axial, and moment loadings.
wher e a o is the free contact angle de termined by be aring diame tral clearan ce. From Figure 1.3
then
<o ¼ <i � A cos a o ð1: 4Þ
<i � <o ¼ A cos ao ð1: 5Þ
In Figure 1.3, c is the an gle be tween the most heavily loaded rolling elem ent an d any other
roll ing elem ent. Bec ause of symm etry 0 � c � p.
If the outer ring of the bearing is con sidered fixed in space as the load is app lied to the
bearing , then the inner ring will be displace d and the locus of inner- ring raceway groo ve radii
center s will also be displ aced as shown in Figu re 1.5. From Figure 1.5 it can be determ ined
that s , the distan ce between the center s of curvat ure of the inner- and outer-ring racew ay
groove s at any rolling element posit ion c , is given by
s ¼ ½ðA sin ao þ da þ <i u cos cÞ2 þ ðA cos ao þ dr cos cÞ2�1=2 ð1:6Þ
or
s ¼ A sin ao þ da þ<i u cos c� �2þ cos ao þ dr cos c
� �2h i1=2ð1:7Þ
where
da ¼da
Að1:8Þ
� 2006 by Taylor & Francis Group, LLC.
Q
ao
ao
y
A
0Bearing axis X
P
Y �
Z �Z
Y
Inner racewaycurvature
center locus
Outer racewaycurvature
center locus
rori
FIGURE 1.3 Loci of raceway groove curvature radii centers before applying load. (From Jones, A.,
Analysis of Stresses and Deflections, New Departure Engineering Data, Bristol, CT, 1946.)
dr ¼dr
A ð 1: 9Þ
u ¼ u
A ð 1: 10 Þ
Substi tuting Equat ion 1.7 into Equat ion 1.2 yiel ds
dn ¼ A sin ao þ da þ<i u cos c� �2þ cos a
o þ dr cos c� �2h i1 = 2
� 1
� �ð 1: 11 Þ
From Chapter 7 of the first v olume of this book, the load vs. deformati on relationshi p for a
rolling element–raceway contact is given by
� 2006 by Taylor & Francis Group, LLC.
Axis of rotationA
ri
ro
o �
o�
a �
1Pe2
1dm2
FIGURE 1.4 Radial ball bearing showing ball–raceway contact due to axial shift of inner and outer
rings.
Inner racewaycurvature
center locus
Outer racewaycurvature
center locus
Y �
X �
P�
O�
Q�
ΘZ �Z �Z
da
dr
Y Y�
s
Ro
Ri
X
yd
FIGURE 1.5 Loci of raceway groove curvature radii centers after displacement (From Jones, A.,
Analysis of Stresses and Deflections, New Departure Engineering Data, Bristol, CT, 1946.)
� 2006 by Taylor & Francis Group, LLC.
Q ¼ Kn dn ð 1: 12 Þ
In Equation 1.12, expo nent n ¼ 3 /2 for ball bearing s and 10/9 for rolle r bearing s. Subs titutio n
of Equation 1.11 into Equation 1.12 and using the form er ex ponent gives
Q ¼ KnA1:5 sin ao þ da þ <i u cos c
� �2þ cos ao þ dr cos c� �2h i1=2
�1
� �1:5
ð1:13Þ
At any ball azimuth position c, the operating contact angle is a. This angle can be determined
from
sin a ¼ sin ao þ da þ<i u cos c
sin ao þ da þ <i u cos c� �2þ cos ao þ dr cos c
� �2h i1=2 ð1:14Þ
or
cos a ¼ cos ao þ dr cos c
sin ao þ da þ<i u cos c� �2þ cos ao þ dr cos c
� �2h i1=2 ð1:15Þ
Equation 1.12 describes the normal load on the raceway acting through the contact angle.
This normal load may be resolved into axial and radial components as follows:
Qa ¼ Q sin a ð1:16Þ
Qr ¼ Q cos c cos a ð1:17Þ
If the radial and thrust loads applied to the bearing are Fr and Fa, respectively, then for static
equilibrium to exist
Fa ¼Xc ¼ �p
c ¼ 0
Qc sin a ð1:18Þ
Fr ¼Xc ¼ �p
c ¼ 0
Qc cos c cos a ð1:19Þ
Additionally, each of the thrust components produce a moment about the Y-axis such that
Mc ¼dm
2Qc cos c sin a ð1:20Þ
For static equilibrium, the applied moment M about the Y-axis must equal the sum of the
moments of each rolling element about the Y-axis (in the case of load symmetry, rolling
element thrust component moments about the Z-axis are self-equilibrating).
M ¼ dm
2
Xc ¼ �p
c ¼ 0
Qc cos c sin a ð1:21Þ
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Com bining Equation 1.13, Equat ion 1.16, and Equation 1.18 yields
Fa � Kn A1 :5 Xc¼�p
c¼ 0
sin a o þ da þ <i u cos c
� �2þ cos a o þ dr cos c
� �2h i1= 2� 1
� �1: 5
sin a o þ da þ <i u cos c
� �sin a o þ da þ <i u cos c� �2þ cos a o þ dr cos c
� �2h i1=2 ¼ 0
ð1: 22 Þ
Fr � K n A 1:5 Xc¼�p
c¼ 0
sin a o þ da þ<i u cos c
� �2þ cos a o þ dr cos c
� �2h i1=2� 1
� �1: 5
cos a o þ dr cos c
� �cos c
sin a o þ da þ<i u cos c� �2þ cos a o þ dr cos c
� �2h i1=2 ¼ 0
ð1: 23 Þ
M �dm
2Kn A 1:5
Xc¼�p
c¼0
sin a o þda þ<i u cos c
� �2þ cos a o þdr cos c
� �2h i1= 2�1
� �1: 5
sin a o þda þ<i u cos c
� �cos c
sin a o þda þ<i u cos c� �2þ cos a o þdr cos c
� �2h i1= 2 ¼ 0
ð1: 24 Þ
Thes e eq uations wer e de veloped by Jones [1].
Equation 1.22 through Equation 1.24 are simu ltaneo us nonl inear equ ations with un-
known s da, dr , and u. They may be solved by numeri cal method s; for exampl e, the New ton–
Raphson method. Havi ng obtaine d da, dr , and u, the maximu m ball load may be obtaine d from
Equat ion 1.13 for c ¼ 0.
Qmax ¼ K n A1 :5 sin ao þ da þ<i u
� �2þ cos ao þ dr
� �2h i1= 2� 1
� �1 :5
ð1: 25 Þ
Soluti on of the indica ted equati ons generally necessi tates the use of a digit al compu ter.
1.3 MISALIGNMENT OF RADIAL ROLLER BEARINGS
Altho ugh it is unde sirable, radial cyli ndrical roller bearing s and tapere d ro ller bearing s can
supp ort to a small extent the moment load ing due to mis alignment. The various types of
mis alignment are illustr ated in Figu re 1.6. Spherical roller bearing s are designe d to exclude
moment loads from actin g on the bea rings an d therefo re a re not included in this discussion.
Figure 1.7 illustr ates the misali gnment of the inner ring of a cyli ndrical roller bearing relative
to the outer ring.
To commence the analysis, it is assumed that any roller–raceway contact can be divided
into a number of ‘‘slices’’ or laminae situated in planes parallel to the radial plane of the
bearing. It is also assumed that shear effects between these laminae can be neglected owing to
the small magnitudes of the contact deformations that develop. (Only contact deformations
are considered.)
1.3.1 COMPONENTS OF DEFORMATION
In a misaligned cylindrical roller bearing subjected to radial load, at each lamina in a
crowned roller–raceway contact, the deformation may be considered to be composed of
three components: (1) Dmj due to the radial load at the roller azimuth location j, (2) cl due
to the crown drop at lamina l, and (3) the deformation due to bearing misalignment and
� 2006 by Taylor & Francis Group, LLC.
Misalignment (out-of-line)(a)
(b) Off-square or tilted outer ring
(c) Cocked or tilted inner ring
(d) Shaft deflection
FIGURE 1.6 Types of misalignments.
roller tilt at the roller azim uth location j . These componen ts are shown schema tically in
Figure 1.8.
The co mponent due to radial load is the only contact deform ation componen t consider ed
in the sim plified analytical methods presented in Chapt er 7 of the first vo lume of this book. It
needs no furt her exp lanation here.
1.3.1 .1 Cr owning
As stated previous ly, crown ing of roll ers and racew ays is acco mplished to avo id ed ge loading
that can resul t in early fatigue failu re of the roll ing compone nts. It may be accompl ished in
various forms. The simplest of these is the full circul ar pr ofile crown illustrated in Figu re 1.9.
The rollers in most spheri cal roller bearing s may be consider ed fully cro wned wheth er of
symm etrical con tour (barr el-shap ed) or of asymm etrical contour. In the latter case, the crow n
is offs et from the roller mid- length point. Full crow ning may also be applied to racew ays as
� 2006 by Taylor & Francis Group, LLC.
q
q
FIGURE 1.7 Misalignment of cylindrical roller bearing rings.
l
1
12
2 )
)
w
w
w cos yjq12
+
cλ
Δj
(l –
(l –
FIGURE 1.8 Components of roller–raceway contact deformation due to radial load, misalignment, and
crowning.
rc
lt
l
cmax
D
R
FIGURE 1.9 Schematic diagram of cylindrical roller with full circular profile crown.
� 2006 by Taylor & Francis Group, LLC.
rc
ri
lt
D
l
ro co, max
ci, max
FIGURE 1.10 Schematic diagram of uncrowned (straight profile) cylindrical roller contacting inner and
outer raceways, each with a full circular profile crown.
shown in Fi gure 1.10. This is common ly used in tapere d roller bearing s where often both the
cone and cu p racew ays are crow ned, and the rollers are not crow ned.
Most cylind rical roll er bearing s employ rollers that are crow ned only over a portion of the
roller contour; the remai ning portion is c ylindrical (the contour is somet imes called flat or
straight ). A partially crow ned cylin drical roller is illustr ated in Figure 1.11.
From Figu re 1.8, it can be seen that crown drop or crow n gap cl at a selec ted lami na is
consider ed as a negati ve de formati on; that is, no roll er–rac eway loading can oc cur at a lami na
until cl is overcome by the radial or the misali gnment deforma tion. For both the full y
crow ned or parti ally c rowned roll ers that have circular profiles, Equation 1.26 de fines cl in
terms of the roll er an d cro wn dimens ions, wher e 1 � l � k .
cl ¼cmax
2 l� 1k� 1
� �2� lsl
� �21 � ls
l
" #2l� 1
k� 1
� �2
� ls
l
� �2
> 0
02l� 1
k� 1
� �2
� ls
l
� �2
� 0
8>>>><>>>>:
9>>>>=>>>>;
ð 1: 26 Þ
For roll ers wi th circular profile parti al crow ns, blending between the stra ight an d crown ed
porti ons of the pro file is ne cessary to mini mize stress concentra tions and the resulting redu ced
fatigue life. To avoid such stress concentrations, in lieu of a circular profile, a tangential
profile might be used. In this case, the crown radius would be variable, and the crown gap at
each lamina k would be calculated using
� 2006 by Taylor & Francis Group, LLC.
l
ls
lt
rc
R
D
cmax
FIGURE 1.11 Schematic diagram of a partially crowned cylindrical roller.
cl ¼cmax
2l� 1k� 1
�� ��� lsl
1 � lsl
" #2
j 2l� 1k� 1��� ls
l> 0
0 2l� 1k� 1
�� ��� lsl� 0
8>><>>:
9>>=>>; ð1: 27 Þ
To minimiz e ed ge loading , Lundber g an d sjo vall [2] de vised a fully crowned roller ha ving a
logari thmic profi le. The crow n gap at each lamina k is calculated using
cl ¼ 0: 2 � ln 1
1:00 67 � 2l� 1
k� 1
� �2
26664
37775 ð1: 28 Þ
Subs equentl y, Reussn er [3] developed another logari thmic profile crown believ ed to be more
effe ctive. The crown gap at each lamina k for the Reuss ner crow n pr ofile is given by
cl ¼ 2 � 10 � 4 Sr w2 k2 � ln1
1 � 2l� 1
k� 1
� �6
26664
37775 ð1: 29 Þ
It is pos sible to combine roller crowning and raceway crow ning. In this case, the crow n gap at
each lami na k woul d be ca lculated as the sum of the cro wn gap s for the roller an d raceway as
follo ws:
cm l ¼ c R l þ c ml ð1: 30 Þ
In the ab ove equati on, subscri pt R refer s to the roller and m to the raceway (m ¼ i or m ¼ o).
For the be aring mis alignmen t u sh own in Figure 1.7, the effe ctive mis alignmen t at the
azimuth location of the roller cj is +1/2u cos cj. The plus sign pertains to 0 � cj �p/2; the
� 2006 by Taylor & Francis Group, LLC.
minus sign pertain s to p/2 � cj � p (assu ming symm etry of loading ab out the 0– p diame ter).
Therefor e, the total roll er–rac eway deform ation at roller location j and lamina l is given by
dlj ¼ D j �u
2l� 1
2
� �w cos cj � c l ð 1: 31 Þ
1.3.2 LOAD ON A ROLLER–RACEWAY CONTACT LAMINA
In Chapt er 6 of the first volume of this book, the follo wing equatio ns wer e given to descri be
the deformation vs. load for a roller–raceway contact:
d ¼2Q 1� j2� �pEl
lnpEl2
Q 1� j2� �
1� gð Þ
" #ð1:32Þ
d ¼ 3:84� 10�5 Q0:9
l0:8ð1:33Þ
Equation 1.32 was developed by Lundberg and Sjovall [2] for an ideal line contact. In Equation
1.32, g ¼ D cos a/dm, E is the modulus of elasticity, and j is Poisson’s ratio. Equation 1.33 was
developed empirically by Palmgren [4] from laboratory test data and pertains to the contact of a
crowned roller on a raceway. While the load–deformation characteristic of an individual
contact lamina may be described using either equation, the latter is applied here as the solution
of a transcendental equation leads to force and moment equilibrium equations of greater
complexity. Considering that the contact is divided into k laminae, each lamina of width w, the
contact length is kw. Letting q ¼ Q/l, Equation 1.33 becomes
d ¼ 3:84� 10�5q0:9 kwð Þ0:1 ð1:34Þ
Rearranging the above equation to define q yields
q ¼ d1:11
1:24� 10�5 ðkwÞ0:11ð1:35Þ
Equation 1.35 does not consider edge stresses; however, because these obtain only over very
small areas, they can be neglected with little loss of accuracy when considering equilibrium of
loading. Substitution of Equation 1.31 into Equation 1.35 gives
qlj ¼Dj � u l� 1
2
� �w cos cj � cl
1:11
1:24� 10�5ðkjwÞ0:11ð1:36Þ
Depending on the degree of loading and misalignment, all laminae in every contact may not
be loaded; in Equation 1.36, kj is the number of laminae under load at roller location j. Total
roller loading is given by
Qj ¼w0:89
1:24 � 10�5k0:11j
Xl¼kj
l¼1
Dj �1
2u l� 1
2
� �w cos cj � cl
� �1:11
ð1:37Þ
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1.3.3 EQUATIONS OF S TATIC EQUILIBRIUM
To de termine the ind ividual roller loading , it is necessa ry to satisfy the requ irements of static
equ ilibrium. Hence, for an applie d radial load,
Fr
2�
Xj ¼ Z = 2 þ 1
j ¼ 1
tj Q j cos c j ¼ 0 t j ¼ 0:5; cj ¼ 0, p
tj ¼ 1; cj 6¼ 0, pð1:38Þ
Subs tituting Equat ion 1.37 into Equation 1.38 yiel ds
0:62 � 10�5Fr
w0:89�
Xj¼Z=2þ1
j¼1
tj cos cj
k0:11j
Xl¼kj
l¼1
Dj �1
2u l� 1
2
� �w cos cj � cl
� �1:11
¼ 0 ð1:39Þ
For an applied coplanar misaligning moment load, the equilibrium condition to be
satisfied is
M
2�
Xj¼Z=2þ1
j¼1
tjQjej cos cj ¼ 0 tj ¼ 0:5; cj ¼ 0, p
tj ¼ 1; cj 6¼ 0, pð1:40Þ
where ej is the eccentricity of loading at each roller location. ej, which is illustrated in Figure 1.12,
is given by
(λ − )w
q λj
Qj
ej
l
12
12
FIGURE 1.12 Load distribution for a misaligned crowned roller showing eccentricity of loading.
� 2006 by Taylor & Francis Group, LLC.
ej ¼
Pl¼kj
l¼1
qlj l� 12
� �w
Pl¼kj
l¼1
qlj
� l
2j ¼ 3,
Z
2þ 3 ð1:41Þ
Hence,
0:62 � 10�5M
w0:89�
Xj¼Z=2þ1
j¼1
tj cos cj
k0:11j
�Xl¼kj
l¼1
�j �1
2u l� 1
2
� �w cos cj � cj
� �1:11
l� 1
2
� �w
(
� l
2
Xl¼kj
l¼1
�j �1
2u l� 1
2
� �w cos cj � cl
� �1:11)¼ 0
ð1:42Þ
1.3.4 DEFLECTION EQUATIONS
The remaining equations to be established are the radial deflection relationships. It is
necessary here to determine the relative radial movement of the rings caused by the misalign-
ment as well as that owing to radial loading. To assist in the first determination, Figure 1.13
shows schematically an inner ring–roller assembly misaligned with respect to the outer ring.
From this sketch, it is evident that one half of the roller included angle is described by
l2
D
R
dq
θj
b
q
1 2(d
m −
D ) 1 2d m
FIGURE 1.13 Schematic diagram of misaligned roller–inner ring assembly showing interference with
outer ring.
� 2006 by Taylor & Francis Group, LLC.
b ¼ tan�1 l
dm �Dð1:43Þ
and
sin b ¼ l
½ðdm �DÞ2 þ l2�1=2ð1:44Þ
The maximum radial interference between a roller and the outer ring owing to misalignment is
given by
du ¼ R cosðb� ujÞ � R cos b ð1:45Þ
where
R ¼ 0:5� ½ðdm �DÞ2 þ l2�1=2 ð1:46Þ
In developing Equation 1.45 and Equation 1.46, the effect of crown drop was investigated and
found to be negligible.
Expanding Equation 1.46 in terms of the trigonometric identity further yields
du ¼ Rðcos b cos uj þ sin b sin uj � cos bÞ ð1:47Þ
As uj is small, cos uj! 1, and sin uj! uj. Moreover, uj ¼ +u cos cj and sin b ¼ l/2R; therefore,
du ¼ �12
lu cos cj ð1:48Þ
The shift of the inner-ring center relative to the outer-ring center owing to radial loading and
clearance, and the subsequent relative radial movement at any roller location are shown in
Figure 1.14. The sum of the relative radial movement of the rings at each roller angular location
minus the clearance is equal to the sum of the inner and outer raceway maximum contact
deformations at the same angular location. Stating this relationship in equation format:
Outer-ringcenter
Inner-ringcenter
δr
δr
yj
yj
δr
δr cos
FIGURE 1.14 Displacement of ring centers caused by radial loading showing relative radial movement.
� 2006 by Taylor & Francis Group, LLC.
10,000 10,000
8,000
6,000
4,000
2,000
00 20 40 60
Distance along roller (percentage of l )(b)
80 100
1,500
1,000
500
1,500
1,000
500
100806040
(a)Distance along roller (percentage of l )
200
8,000
6,000j = 0�
j = 0�
q = 0q = 209
j = 0�
j = 30� j = 660�
630�
j = 0�
Rol
ler
load
(lb
/in.)
Rol
ler
load
(lb
/in.)
N/m
m
N/m
m
4,000
2,000
0
630�
630�
660�
6120�
660�
Radial load = 31,600 N (7,100 lb)
6180�6150�
FIGURE 1.15 Roller loading vs. axial and circumferential location—309 cylindrical roller bearing: (a)
ideally crowned rollers; (b) fully crowned rollers.
dr �1
2 l u
� �co s cj �
Pd
2� 2 Dj �
1
2 u l� 1
2
� �w cos cj � c l
� �max
¼ 0 ð 1: 49 Þ
Equation 1.39, Equat ion 1.42, and Equation 1.49 constitut e a set of Z /2 þ 3 sim ultane ous
nonlinear equati ons that can be solved for dr, u, and Dj using numeri cal analysis techni ques.
Thereaft er, the variation of roll er load per unit lengt h, and subseq uently the roller load, may
be determ ined for each roller locat ion using Equat ion 1.36 and Equat ion 1.37, respectivel y.
Using this method of digital computa tion, Harr is [5] analyze d a 309 cylindrical ro ller
bearing having the foll owing dimens ions a nd loading :
Number of rollers
� 2006 by Taylor & Francis Group, LLC.
12
Roller effective length
12.6 mm (0.496 in.)Roller straight lengths
4.78, 7.770, 12.6 mmRoller crown radius
1,245 mm (49 in.)Roller diameter
14 mm (0.551 in.)Bearing pitch diameter
72.39 mm (2.85 in.)Applied radial load
31,600 N (7,100 lb)For these co ndition s, Figure 1.15 shows the loading on various rollers for the bearing with
ideal ly crow ned rollers (ls ¼ 12.6 mm [0.496 in.]) and with fully crown ed rollers (l s ¼ 0).
Figure 1.16 sho ws the effect of roller crow ning on bea ring radial deflection as a functio n
of misalign ment.
1.4 THRUST LOADING OF RADIAL CYLINDRICAL ROLLER BEARINGS
When radial cylin drical roller be arings have fixed flanges on both inner and outer rings , they
can carry some thrust load in ad dition to radial load. The great er the amou nt of radial load
applie d, the more is the thrust load that can be carried. As sho wn by Harr is [6] and seen in
Figure 1.17, the thrust load causes e ach roller to tilt an amount zj .
0.03
ls = 4.78 mm (0.188 in.)
ls = 7.70 mm (0.303 in.)
No crown
Full crown
0.04
mm
0.05
0.06
22
20
18
16
14
Rad
ial D
efle
ctio
n (in
. � 0
.000
1)
12
10
80 5 10 15
Misalignment (min.)
20 25
24
FIGURE 1.16 Roller deflection vs. misalignment and crowning—309 cylindrical roller bearing at 31,600 N
(7,100 lb) radial load.
Again, it is assumed that a roller–raceway contact can be subdivided into laminae in
planes parallel to the radial plane of the bearing. When a radial cylindrical roller bearing is
subjected to applied thrust load, the inner ring shifts axially relative to the outer ring.
Housing
Shaft
Shaft Shaft
Qaj
Fa
Qaj
Qaj
Qaj
h
12
D2
2l
h
dT
12dT
z + h
z
h
CL CL
FIGURE 1.17 Thrust couple, roller tilting, and interference owing to applied thrust load.
� 2006 by Taylor & Francis Group, LLC.
(λ −
+Δj
Δj
−
wzj (λ −
cλ
cλ
1/2) w
1/2) w
1/2) w
1/2) w
zj (k − λ +
(k − λ +
+
−
FIGURE 1.18 Components of roller–raceway deflection at opposing raceways due to radial load, thrust
load, and crowning.
Assu ming deflections owing to roller end–fla nge con tacts are negligible , the inter ferenc e at
any axial locat ion (lamina) is
dlj ¼ Dj þ zj l � 1
2
� �w � cl , l ¼ 1, kj ð 1: 50 Þ
where cl is given by Equat ion 1.26 through Equat ion 1.30. Figu re 1.18 illu strates the
compon ent deflec tions in Equation 1.50. Subs tituting Equation 1 .50 into Equation 1.35 yiel ds
qlj ¼Dj þ zj l� 1
2
� �w� cl
1:11
1:24 � 10�5ðkjwÞ0:11ð1:51Þ
and at any azimuth cj, the total roller loading is
Qj ¼w0:89
1:24 � 10�5k0:11j
Xl¼kj
l¼1
Dj þ zj l� 1
2
� �w� cl
� �1:11
ð1:52Þ
� 2006 by Taylor & Francis Group, LLC.
1.4.1 EQUILIBRIUM E QUATIONS
To de termine roller loading , it is necessa ry to sati sfy static eq uilibrium requ irements. Hence,
for applied radial load
Fr
2�
Xj ¼ Z = 2þ 1
j ¼ 1
tj Q j cos cj ¼ 0tj ¼ 0:5; cj ¼ 0, p
tj ¼ 1; cj 6¼ 0, pð1: 53 Þ
Subs tituting Equat ion 1.52 into Equation 1.53 yiel ds
0: 62 � 10 � 5 Fr
w0 :89 �
Xj ¼ Z = 2þ 1
j ¼ 1
tj cos cj
k0 :11j
Xl¼ kj
l¼ 1
Dj þ zj l� 1
2
� �w � cl
� �1: 11
¼ 0 ð1: 54 Þ
For an ap plied centri c thrust load, the equilibrium conditio n to be satisfi ed is
Fa
2�
Xj ¼ Z =2 þ 1
j ¼ 1
tj Q aj ¼ 0 ð1: 55 Þ
At each roll er location, the thrust cou ple is ba lanced by a radial load couple caused by the
skew ed axial load distribut ion. Thus , hQaj ¼ 2Q je j and
Fa
2� 2
h
Xj ¼ Z =2 þ 1
j ¼ 1
tj Qj e j ¼ 0tj ¼ 0: 5; cj ¼ 0, p
tj ¼ 1; cj 6¼ 0, pð1: 56 Þ
wher e ej is the eccentr icity of loading indica ted in Figure 1.12 and define d by
ej ¼
Pl¼ k j
l¼ 1
qlj l� 12
� �w
Pl¼ k j
l¼ 1
qlj
� l
2 ð1: 57 Þ
Subs titution of Equat ion 1.52 and Equat ion 1.57 into Equat ion 1.56 yiel ds
0:31 � 10�5Fa
w0:89�
Xj¼Z=2þ1
j¼1
tj
k0:11j
�Xl¼kj
l¼1
Dj � zj l� 1
2
� �w � cl
� �1:11
l� 1
2
� �w� l
2
(
�Xl¼kj
l¼1
Dj � zj l� 1
2
� �w � cl
� �1:11)¼ 0
tj ¼ 12; cj ¼ 0, p
tj ¼ 1; cj 6¼ 0, p
ð1:58Þ
� 2006 by Taylor & Francis Group, LLC.
1.4.2 DEFLECTION E QUATIONS
Radi al deflection relat ionshi ps remain to be establis hed. It is necessa ry to de termine the
relative radial movem en t of the bearing rings caused by the thrust loading as well as that
due to radial loading . To assi st in this deriva tion, Figure 1.17 shows schema tically a thrust-
loaded roller–ri ng assem bly. From this sketch, a roll er angle is described by
tan h ¼ D
l ð 1:59 Þ
The maxi mum radial interfer ence between a roll er and both rings is given by
dj ¼ Dsin ðz j þ hÞ
sin h� 1
� �ð 1: 60 Þ
In developi ng the above eq uation, the effect of crown drop was foun d to be negligible .
Expa nding Equat ion 1.60 in terms of the trigonometric identity and recogni zing that z j is
small and l ¼ D ctn h yield s
dt j ¼ l z j ð 1: 61 Þ
Althou gh dtj is the radial deflection due to roller tilting, it can be similarly shown that axial
deflection owing to roller tilting is
daj ¼ Dzj ð1:62Þ
Therefore, the radial interference caused by axial deflection is
dra ¼ da
l
D ð 1:63 Þ
The sum of the relat ive rad ial movem ents of the inner and outer rings at e ach roller azim uth
minus the radial clearance is equal to the sum of the inner and outer raceway maximum
contact deformations at the same azimuth, or
da
l
D þ dr cos cj �
Pd
2� 2 Dj þ z j l� 1
2
� �w � cl
� �max
¼ 0 ð1: 64 Þ
The set of simulta neous equati ons, Equat ion 1.54, Equat ion 1.58, and Equation 1.64, can be
solved for z j , D j , dr , and da. Thereafter, the varia tion of the roll er load per unit length q and
roller load Q may be determ ined for each roller azimu th using Equation 1.51 and Equat ion
1.52, respect ively. The axial load on each roll er may be determ ined from
Qaj ¼w 0:89
3: 84 � 10 �5k0:11j h
�Xl¼ kj
l¼1
Dj þ zj l� 1
2
� �w � cl
� �1:11
l� 1
2
� �w � l
2
Xl¼kj
l¼1
Dj þ zj l� 1
2
� �w� cl
� �1:118<:
9=;
ð1:65Þ
� 2006 by Taylor & Francis Group, LLC.
1.4.3 ROLLER –RACEWAY DEFORMATIONS DUE TO SKEWING
Wh en roll ers are subjected to axial loading as sho wn in Figure 1.17, due to sliding moti ons
betw een the roll er ends and ring flanges, fricti on forces occur. For exampl e, Faj ¼ mQ aj, in
whi ch m is the coefficie nt of frictio n. In a misalign ed bearing , each roller that carries load is
squeez ed at one en d an d forced agains t the oppos ing flang e with a load Qaj , creat ing fricti on
force Faj at the roll er end. Because of F aj , a mo ment oc curs creating, in add ition to the
predo minant rolling mo tion about the roller axis, a yawi ng or skewing motion and secondary
roll er tilti ng. The tilti ng and skew ing motion s oc cur in ortho gonal planes that con tain the
roll er axis. Roller skew ing is resisted by the concave curvatu re of the out er raceway. The
resi sting forces and acco mpanyi ng deform ations alter the distribut ion of load a long both
the outer and inner raceway– roller contact s. Figure 1.19 illustrates the forces that occur on a
roll er sub jected to radial and thrust loading s. Frict ional stresses ss1j l and ss2 j l in Figu re 1.19
tend not to signifi cantly influence the roll er–rac eway normal loading s pe r unit lengt h q1jl and
q2j l on the outer and inner raceways, respectivel y.
Figure 1.20 shows the roller skewing angle jj and the roller–outer raceway loading that result.
The roller–rac eway co ntact deform ations that resul t from skew ing as de monstrated by
Harr is et a l. [7] may be de scribed by
dmj l ¼ Dmj þ w l� 1
2
� �zj þ fmj l � c l ð1: 66 Þ
In the abo ve eq uation, subscri pt m refer s to the outer and inner racew ay co ntacts; m ¼ 1 and
2, respectivel y, and de formati ons due to skew ing wmj l are g iven by
f1 j l ¼
k
2 � l�1
2 ðk þ 1Þ
��������
� �w
� �dm þ D
� 2j ð1: 67 Þ
f2jl ¼l� 1
2ðkþ 1Þ
�� ��w 2dm þD
�2j ð1:68Þ
It can be furt her seen that Equation 1.64 must become
Q2j
z
x y
Qaj
Qaj
Q1j
mQaj
mQaj
q2jl
ss 2 jl
ss1jlq1jl
FIGURE 1.19 Normal and friction forces acting on a radial and thrust-loaded roller.
� 2006 by Taylor & Francis Group, LLC.
l
D
q1j
q1j sin bj
bj
RI
xj
FIGURE 1.20 Roller–outer raceway contact showing roller skewing angle jj and restoring forces.
da
l
Dþ dr cos cj �
Pd
2� ðd1mj, max þ d2mj, maxÞ ¼ 0 ð1:69Þ
Owing to the unknown variables jj and Dmj, the latter replacing Dj, additional equilibrium
equations must be established. For equilibrium of roller loading in the radial direction,
Xm¼2
m¼1
Qmj ¼ wXm¼2
m¼1
Xl¼k
l¼1
qmjl ¼ 0 ð1:70Þ
Referring to Figure 1.20 and considering equilibrium of moments in the plane of roller
skewing,
lFaj þXm¼2
m¼1
Xl¼k
l¼1
w2 l� 1
2ðkþ 1Þ
� �smjl
�Xm¼2
m¼1
Xl¼k
l¼1
w2 l� 1
2ðkþ 1Þ
� �qmjl sin bj ¼ 0
ð1:71Þ
As the angle bj! 0, sin bj ! bj,
sin bj ¼2w
dm þDl� 1
2ðkþ 1Þ
� �jj ð1:72Þ
As indicated above, the frictional stresses, ss1jl and ss2jl, tend not to influence the roller–raceway
normal loading significantly, meaning that the frictional moment loading is rather small
� 2006 by Taylor & Francis Group, LLC.
compared with those caused by the restoring forces q1j sin bj s hown in F ig ure 1 .20 a nd r olle r
end–flange friction forces. Therefore, substituting Equation 1.72 into Equation 1.71 yields
lFa j �2w3 jj
dm þ D
Xm ¼ 2
m ¼ 1
Xl¼ k
l¼ 1
l� 1
2k þ 1ð Þ
� �2
qmj l ¼ 0 ð1: 73 Þ
Consi dering that the c ontact de formati ons due to roller radial loading are different for each
roll er–raceway contact , bearing load equilibrium equati ons, Equat ion 1.54 and Equat ion
1.58, must be changed accordi ngly; hence,
0 :62 � 10 � 5 F r
w0 :89 �
Xj ¼ Z = 2þ 1
j ¼ 1
tj cos cj
k 0 :112 j
Xl¼ k
l¼ 1
D2j þ w l� 1
2
� �zj þ f 2j l � c l
� �1: 11
¼ 0 ð1: 74 Þ
and
0:31 � 10 � 5 Fa
w0 :89 �
Xj ¼ Z = 2 þ 1
j ¼ 1
tj
k 0 :112 j
�Xl¼ k
l¼ 1
� 2j þ w l� 1
2
� �zj þ f 2j l � c l
� �1 :11
w l� 1
2
� �(
� l
2
Xl¼ k
l¼ 1
D2 j þ w l� 1
2
� �zj þ f2 j l � c l
� �1:11 )¼ 0
ð1: 75 Þ
Equat ion 1.56, Equat ion 1.69, Equat ion 1.70, and Equation 1.73 through Equation 1 . 7 5
constitute a set of simultaneous, non linear equations that may be s olved for Dmj, zj, jj, dr,
and da. Subsequently, the roller–raceway loads Qj and roller end–flange loads Qaj may be
determined.
The skewing angles determined using the earlier equations strictly pertain to full
complement bearings and bearings having no roller guide flanges. For a bearing with a
substantially robust and rigid cage, the skewing angle may be limited by the clearances
between the rollers and the cage pockets. For a bearing with guide flanges, the skewing
may be limited by the endplay between the roller ends and guide flanges. In general, the latter
situation is obtained; however, to the extent that skewing is permitted, the earlier analysis is
applicable.
1.5 RADIAL, THRUST, AND MOMENT LOADINGS OF RADIAL ROLLERBEARINGS
1.5.1 CYLINDRICAL ROLLER BEARINGS
For radial cylindrical roller bearings, it is possible to apply general combined loading. The
equations for load equilibrium defined earlier apply; however, the interference at any lamina
in the roller–raceway contact is given by
dmjl ¼ Dmj þ w l� 1
2
� �vmzj �
1
2u cos cj
� �þ fmjl � cl ð1:76Þ
where subscript m ¼ 1 refers to the outer raceway and m ¼ 2 refers to the inner raceway.
Coefficient v1 ¼ �1 and v2 ¼ þ1. The contact load per unit length is given by
� 2006 by Taylor & Francis Group, LLC.
qmj l ¼Dmj þ w l� 1
2
� �nm z j � 1
2 u cos cj
� �þ fmj l � c l
1 :11
1: 24 � 10 � 5 kj w� �0 :11
ð 1: 77 Þ
1.5.2 T APERED R OLLER BEARINGS
Similar equations may be developed for tapered roller bearings. As shown in Chapter 5 of the
first volume of this book, roller end–flange loading occurs during all conditions of applied
loading, and bearing equilibrium equations must be altered accordingly. Figure 1.21 illustrates
the geometry and loading of a tapered roller in a bearing.
Considering Figure 1.21 and establishing the following dimensions:
r2 Radius in a radial plane from the inner-ring axis of rotation to the center of the inner
raceway contact
rfz Radius in a radial plane from the inner-ring axis of rotation to the center of the roller
end–inner ring flange contact
rfx x Direction distance in an axial plane from the center of the inner raceway contact to the
center of the roller end–inner ring flange contact
The roller load equilibrium equations are
wXm¼2
m¼1
cm cos am
Xl¼k
l¼1
qmjl �Qf j cos af ¼ 0 ð1:78Þ
wXm¼2
m¼1
cm sin am
Xl¼k
l¼1
qmjl þQf j sin af ¼ 0 ð1:79Þ
Qia
QirQi
Qfa
Qf
Qo
Qor
Qfr
ai
af
ao
Qoa
FIGURE 1.21 Roller loading in a tapered roller bearing.
� 2006 by Taylor & Francis Group, LLC.
In Equation 1.78 and Equation 1.79, coeff icient c1 ¼ �1 and c 2 ¼ þ1. The eq uation for
radial plan e moment equilib rium of the roll er is
w2 Xm ¼ 2
m ¼ 1
Xl¼ k
l¼ 1
qmj l l� 1
2k þ 1ð Þ
� �� Rf Q f j ¼ 0 ð1: 80 Þ
where Rf is the radius from the roller axis of rotation to the center of the roller end–flange contact.
Equilibrium of actuating and resisting moments pertaining to roller skewing is given by
1
2 l � Qf j �
w 3 �jdm þ Dð Þ
Xm ¼ 2
m ¼ 1
Xl¼ k
l¼ 1
l� 1
2k þ 1ð Þ
� �2
qmj l ¼ 0 ð1: 81 Þ
The force an d moment eq uilibrium equati ons with respect to the bearing inner ring are as
follo ws:
Fr � wXj ¼ Z
j ¼ 1
cos cj
Xl¼ k
l¼ 1
q2 j l co s a2 � Q f j cos af
" #¼ 0 ð1: 82 Þ
Fa � wXj ¼ Z
j ¼ 1
Xl¼ k
l¼ 1
q2 j l sin a 2 þ Q f j sin a f
" #¼ 0 ð1: 83 Þ
M � wXj ¼ Z
j ¼ 1
co s cj
Xl¼ k
l¼ 1
q2j l r 2 co s a2 � Q f j r f z sin af � r f x cos afð Þ" #
¼ 0 ð1: 84 Þ
In these eq uations , the subscrip t 2 refers to the inner raceway.
1.5.3 SPHERICAL ROLLER B EARINGS
Spherical roller bearings are internally self-aligning and therefore cannot carry moment load-
ing. Moreover, for slow- or moderate-speed applications causing insignificant roller centrifugal
forces, gyroscopic moments, and friction (see Chapter 2 and Chapter 3), rollers in spherical
roller bearings will not exhibit a tendency to tilt. Therefore, the simpler analytical methods
provided in Chapter 7 of the first volume of this book will yield accurate results. For spherical
roller bearings that have asymmetrical contour rollers (for example, spherical roller thrust
bearings) roller tilting and hence skewing are not eliminated. In this case for the purpose of
analysis, the bearing may be considered a special type of tapered roller bearing with fully
crowned rollers. Then, the methods of analysis discussed in Section 1.5.2 may be applied.
1.6 STRESSES IN ROLLER–RACEWAY NONIDEAL LINE CONTACTS
In practice, the contact between rollers and raceways is rarely an ideal line contact nor is it
truly a series of independent laminae without interactions. The laminae approach used earlier
is sufficient for determining the distribution of load within the contacts as the stresses due to
truncation at the roller ends and other transitions with profile design cover very small areas.
However, as bearing fatigue life is a function of the subsurface and hence surface contact
stresses, the laminae approach is not always sufficient to estimate the contact stress distribu-
tion. Therefore, more sophisticated methods for the analysis of contact stresses are typically
performed after the load distributions of the bearing have been estimated.
� 2006 by Taylor & Francis Group, LLC.
Starting with Thomas and Hoersch [8], several researchers have advanced the contact
solution of Hertz for the nonideal situations. Using stress functions with Equation 6.7,
Equation 6.9, Equation 6.10, Equation 6.13, and Equation 6.14 in the first volume of this
book, Hartnett [9] defined the following relationship between the normal contact pressure at a
location (x0, y0) and the surface deflection at a distant point (x, y) on an elastic half space as
w x; yð Þ ¼ 1� j2
pE
� �P x0,y0ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x� x0ð Þ2þ y� y0ð Þ2q ð1:85Þ
By breaking the contact surface into several small, rectangular patches of dimensions 2g along
the y-axis and 2c along the x-axis directions with a node at the center of each patch, and
assuming constant pressure over the area, Equation 1.85 can be integrated to determine the
effect of contact pressure at a given node, i, on the deflection at another node, j. This is done
by the use of influence coefficients, Dij:
Dij ¼ xi � xj
�� ��þ c� �
lnyi � yj
�� ��þ g� �
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
yi � yj
�� ��þ g� �2þ xi � xj
�� ��þ c� �2q
yi � yj
�� ��� g� �
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
yi � yj
�� ��� g� �2þ xi � xj
�� ��þ c� �2q
264
375
þ yi � yj
�� ��þ g� �
lnxi � xj
�� ��þ c� �
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
yi � yj
�� ��þ g� �2þ xi � xj
�� ��þ c� �2q
xi � xj
�� ��� c� �
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
yi � yj
�� ��þ g� �2þ xi � xj
�� ��� c� �2q
264
375
þ xi � xj
�� ��� c� �
lnyi � yj
�� ��� g� �
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
yi � yj
�� ��� g� �2þ xi � xj
�� ��� c� �2q
yi � yj
�� ��þ g� �
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
yi � yj
�� ��þ g� �2þ xi � xj
�� ��� c� �2q
264
375
þ yi � yj
�� ��� g� �
lnxi � xj
�� ��� c� �
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
yi � yj
�� ��� g� �2þ xi � xj
�� ��� c� �2q
xi � xj
�� ��þ c� �
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
yi � yj
�� ��� g� �2þ xi � xj
�� ��þ c� �2q
264
375
ð1:86Þ
Using the influence coefficients, the interference of two bodies in contact with a given
approach d is given by
d� zj �y2
2�y
� �� 1� j2
1
pE1
þ 1� j22
pE2
� �Xi¼n
i¼1
Dijsj ¼ 0 ð1:87Þ
where zj is the drop at location j from the highest point on the body due to profiling, and the
term hd� zj� (y2 / 2)ryi ¼ 0 when the computed value is less than zero. Finally, the
equilibrium of applied contact force and the integral of the pressure over the contact yields
Q� 4gcXj¼n
j¼1
sj ¼ 0 ð1:88Þ
Equation 1.86 and Equation 1.87 allow for the nonideal contact pressure to be estimated for
any given contact geometry by varying d until Equation 1.88 is satisfied within acceptable
error limits.
� 2006 by Taylor & Francis Group, LLC.
1.7 FLEXIBLY SUPPORTED ROLLING BEARINGS
1.7.1 RING DEFLECTIONS
The preceding discus sion of dist ribution of load among the bearing rolling elem ents pertains
to bearing s that have rigidly supp orted rings. Such bearing s a re assum ed to be supporte d in
infin itely stiff (rigid) housings and on soli d shafts of rigid material . The deflections consider ed
in the determinat ion of load dist ribution were co ntact deform ations. Thi s assum ption is an
excell ent ap proximati on for mo st bearing applications .
In so me radial be aring applic ations , howeve r, the outer ring of the bearing may be
supp orted at one or tw o azim uth posit ions only, and the shaft on which the inn er ring is
posit ioned may be hollow. The con dition of two-point outer-ring suppo rt, as sho wn in Figure
1.22 and Figure 1.23, occurs in the planet gear be arings of a plan etary gear power trans mis-
sion system, and was analyzed by Jones and Harris [10]. In certain rolling mill applications,
the backup roll bearings may be supported at only one point on the outer ring or possibly at
two poin ts as shown in Figure 1.24. Thes e conditio ns wer e analyze d by Harris [11]. In certain
high-speed radial bearings, to prevent skidding it is desirable to preload the rolling elements
by using an elliptical raceway, thus achieving essentially two-point ring loading under
conditions of light applied load. The case of a flexible outer ring and an elliptical inner ring
was investigated by Harris and Broschard [12]. In each of these applications, the outer ring
must be considered flexible to achieve a correct analysis of rolling element loading.
In many aircraft applications, to conserve weight the power transmission shafting is made
hollow. In these cases, the inner-ring deflections will alter the load distribution from that
considering only contact deformation.
To determine the load distribution among the rolling elements when one or both of the
bearing rings is flexible, it is necessary to determine the deflections of a ring loaded at various
FIGURE 1.22 Planet gear bearing.
� 2006 by Taylor & Francis Group, LLC.
Gear tooth load Outer ring (integral gear)
Rolling element
Inner ring
Shaft
FIGURE 1.23 Planet gear bearing showing gear tooth loading.
points arou nd its peripher y. Thi s analys is may be achieve d by the a pplication of classical
energy methods for the bending of thin rings .
As an exampl e of the method of analys is, consider a thin ring subjected to loads of equal
magni tude equall y spaced at angles Dc (see Figure 1.25). Accor ding to Timoshenk o [13], the
Bearing
Bearing
Bearing
Bearing
Bearing
Bearing
F1
F1 F1
F2
F2 F2
F2 F1
FIGURE 1.24 Cluster mill assembly showing backup roll bearing loading.
� 2006 by Taylor & Francis Group, LLC.
Δy
Q
FIGURE 1.25 Thin ring loaded by equally spaced loads of equal magnitude.
differential equation describing radial deflection u for bending of a thin bar with a circular
center line is
d2u
df2þ u ¼ �M<2
EIð1:89Þ
where I is the section moment of inertia in bending and E is the modulus of elasticity. It can be
shown that the complete solution of Equation 1.89 consists of a complementary solution and
a particular solution. The complementary solution is
uc ¼ C1 sin fþ C2 cos f ð1:90Þ
where C1 and C2 are arbitrary constants.
Consider that the ring is cut at two positions: at the position of loading, f ¼ 12
Dc, and at
the position f ¼ 0, midway between the loads. The loads required to maintain equilibrium
over the section are shown in Figure 1.26. From Figure 1.26 it can be seen that since
horizontal forces are balanced,
Q ¼ 2F0 sin f ð1:91Þ
or
F0 ¼Q
2 sin fð1:92Þ
Q
Mo
Fo
f
O
12
Δy
FIGURE 1.26 Loading of section of thin ring between 0 � f � 12
Dc.
� 2006 by Taylor & Francis Group, LLC.
The moment at an y an gle f betw een 0 an d 12
Dc is apparen tly
M ¼ M0 � F0 <ð1 � cos fÞ ð1: 93 Þ
or
M ¼ M0 �Q <
2 sin fð 1 � cos fÞ ð1: 94 Þ
Since the section at f ¼ 0 is midway between loads, it cannot rotate. According to Castigliano’s
theorem [13] the angular rotation at any section is
u ¼ ›U
›M ð 1: 95 Þ
wher e U is the strain energy in the beam at the position of loading . Timoshenk o [13] shows
that for a c urved beam
U ¼Z f
0
M 2 <2EI
df ð 1: 96 Þ
At f ¼ 0, M ¼ M0 and since the secti on is constr ained from rotat ion,
›U
›M0
¼ 0 ¼ <EI
Z 1 =2 Dc
0
M›M
›M0
df ð 1: 97 Þ
Substi tuting Equat ion 1.94 into Equat ion 1.97 and integ rating yiel ds
M0 ¼Q<2
1
sin ð12
Dc� 2
Dc
" #ð1:98Þ
Hence,
M ¼ Q<2
cos f
sinð12
Dc� 2
Dc
" #ð1:99Þ
Equation 1.99 may be substitut ed for M in Equat ion 1.89 such that the particular solution is
up ¼Q<3
2EI
f sin f
2 sinð12
Dc� 1
Dc
" #ð1:100Þ
The complete solution is
u ¼ uc þ up ¼ C1 sin f þ C2 cos f �Q<3
2EI
f sin f
2 sinð12
Dc� 1
Dc
" #ð1:101Þ
Because the sections at f ¼ 0 and f ¼ 12
Dc do not rotate,
� 2006 by Taylor & Francis Group, LLC.
du
df
����f¼ 0
¼ 0; C1 ¼ 0
du
df
����f¼Dc=2
¼ 0; C2 ¼ �Q<3
4EI sinð12
DcÞ1
2Dc ctn
1
2Dc
� �þ 1
� �
Hence, the radial deflection at any angle f between f ¼ 0 and f ¼ 12
Dc is
u ¼ Q<3
2EI
2
Dc�
Dc cosð12
DcÞ4 sin2 ð1
2DcÞ
þ 1
2 sin ð12
DcÞ
" #cos f � f sin f
2 sinð12
DcÞ
( )ð1:102Þ
Equation 1.102 may be expressed in another format as follows:
u ¼ CfQ ð1:103Þ
where Cf are influence coefficients dependent on angular position and ring dimensions.
Cf ¼<3
2EI
2
Dc�
Dc cos ð12
DcÞ4 sin2 ð1
2DcÞ
þ 1
2 sinð12
DcÞ
" #� cos f � f sin f
2 sinð12
DcÞ
( )ð1:104Þ
Lutz [14], using procedures similar to those described earlier, developed influence coefficients
for various conditions of point loading of a thin ring. These coefficients have been expressed
in infinite series format for the sake of simplicity of use.
For a thin ring loaded by forces of equal magnitude symmetrically located about a
diameter as shown in Figure 1.27, the following equation yields radial deflections:
Qui ¼ QCijQ ð1:105Þ
where
QCij ¼ �2<3
pEI
Xm¼1m¼2
cos mcj cos mci
ðm2 � 1Þ2ð1:106Þ
Qj Qj
yi
yj
FIGURE 1.27 Thin ring loaded by forces of equal magnitude located asymmetrically about a diameter.
� 2006 by Taylor & Francis Group, LLC.
Fr FrFr
Qj
Qj
Qj
cos yj
cosy l
ylyj
FIGURE 1.28 Thin ring showing equilibrium of forces.
The negative sign in Equat ion 1.106 is used for inter nal loads and the posit ive sign is used for
exter nal loads. Equation 1.105 define s radial deflection at a ngle ci caused by Q j at posit ion
angle cj . When ro lling element loads Qj are such that a rigid body trans lation d1 of the ring
occurs in the direct ion of an ap plied load, Equat ion 1.105 is not self-suf ficient in establis hing
a solut ion; howeve r, a direct ional eq uilibrium equati on may be used in co njunction with
Equation 1.105 to determ ine the trans latory movem ent. Referri ng to Fi gure 1.28, the appro-
priate equ ilibrium equ ation is as follo ws:
Fr cos c1 � Qj cos c j ¼ 0 ð 1: 107 Þ
In the planet gear be aring a pplication demon strated in Figure 1.23, the gear tooth loads may
be resol ved into tangen tial forces , radial forces , and mo ment loads at c ¼ 90 8 (see Figure
1.29). The ring radial defle ctions at an gle ci due to tangential forces Ft are given by
Fs
Fs
Ft
FtM
M
FIGURE 1.29 Resolution of gear tooth loading on outer ring.
� 2006 by Taylor & Francis Group, LLC.
t ui ¼ t Ci Ft ð1: 108 Þ
wher e
t C i ¼2<
3
p EI
Xm ¼1
m ¼ 2
sin m p2
cos m ci
m ð m2 � 1Þ 2 ð1: 109 Þ
Equat ion 1.108 is not self -suffic ient an d an app ropriate equ ilibrium equatio n must be used to
define a rigi d ring trans lation.
The separat ing forces Fs are self -equilibra ting and thus do not cause a rigi d ring trans la-
tion. The radial defle ctions at an gles ci are given by
s ui ¼ s C i F s ð1: 110 Þ
where
sCi ¼2<3
pEI
Xm¼1m¼2
cos mp2
cos mci
ðm2 � 1Þ2ð1:111Þ
Note that Equation 1.111 is a specia l case of Equat ion 1.106 wher e pos ition angle cj is 908 and
loads Qj are external.
Similarly, the moment loads applied at c ¼ 908 are self-equilibrating. The radial deflec-
tions are given by
Mui ¼ MCiM ð1:112Þ
where
MCi ¼2<2
pEI
Xm¼1m¼2
sin mp2
cos mci
m ðm2 � 1Þ2ð1:113Þ
To find the ring radial deflections at any regular position due to the combination of applied
and resisting loads, the principle of superposition is used. Hence for the planet gear bearing,
the radial deflection at any angular position ci is the sum of the radial deflections due to each
individual load, that is,
ui ¼ sui þ Mui þ tui þ Q jui ð1:114Þ
or
ui ¼ sCiFs þ MCiM þ tCiFt þX
QCijQj ð1:115Þ
1.7.2 RELATIVE RADIAL APPROACH OF ROLLING ELEMENTS TO THE RING
A load may not be transmitted through a rolling element unless the outer ring deflects sufficiently
to consume the radial clearance at the angular position occupied by the rolling element. Fur-
thermore, because a contact deformation is caused by loading of the rolling element, the ring
� 2006 by Taylor & Francis Group, LLC.
deflections cannot be determined without considering these contact deformations. Therefore, the
loading of a rolling element at angular position cj depends on the relative radial clearance. The
relative radial approach of the rings includes the translatory movement of the center of the outer
ring relative to the initial center of that ring, whose position is fixed in space. Hence, for the planet
gear bearing, the relative radial approach at angular position cj is
di ¼ d1 cos ci þ ui ð 1: 116 Þ
From Equation 1.12, the relative radial approach is related to the rolling element load as follows:
Qj ¼K ðdj � r j Þ n
dj > r j0 dj � r j
� �ð 1: 117 Þ
wher e rj is the radial clearance at angular posit ion cj . Here, r j is the sum of Pd /2 and the
cond ition of ring ellipti city.
1.7.3 DETERMINATION OF R OLLING E LEMENT L OADS
Usin g the example of the planet gear bearing , the co mplete loading of the outer ring is shown
in Figure 1.30, which also illustrates the rigid ring translation d1. Com bining Equat ion 1.115
through Equation 1.117 yields
di � d1 cos ci � sCiFs � MCiM � tCiFt � iKXj¼Z=2þ2
j¼2
QCijðdj � rjÞn ¼ 0 ð1:118Þ
The required equilibrium equation is
Ft � iKXj¼Z=2þ2
j¼2
tjðdj � rjÞn cos cj ¼ 0 ð1:119Þ
Fs
Fs
Ft
Ft
M
M
Qj
y3
y4
i, j = 4
i, j = 3
i, j = 2d1
FIGURE 1.30 Total loading of outer ring in planet gear bearing.
� 2006 by Taylor & Francis Group, LLC.
Planet gear bearing
Rigid ring bearing
FIGURE 1.31 Comparison of load distribution for a rigid ring bearing and planet gear bearing.
con sidering the symm etry about the diame ter parallel to the load. In Equation 1.119, tj ¼ 0.5
if the rolling elem ent is locat ed at cj ¼ 08 or at cj ¼ 180 8 ; otherw ise t j ¼ 1.
Equation 1.118 and Equat ion 1.119 constitut e a set of sim ultane ous nonlinear equati ons
that may be solved by numeri cal analys is. The New ton–R aphson method is recomm end ed.
Using these methods , the unknowns dj an d hen ce Qj can be de termined at each rolling
elem ent locat ion. Figure 1.31 shows a typical dist ribution of load among the rolle rs in a
planet gear bearing compared with that of a rigi d ring be aring subject ed to a radial load of
2Ft. For the backu p roll bearing s of Figure 1.24 supporti ng individual line loads F1, Figure
1.32 compares the load distribut ion to that of a bearing that has rigid rings . Figure 1.33 sho ws
Thick ring
Thin ring
FIGURE 1.32 Comparison of load distribution of thin and thick outer rings, point-loaded backup roll
bearing.
� 2006 by Taylor & Francis Group, LLC.
0 10 20 30 40 50 60 70 80 90
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
5,000
10,000
15,000
20,000
25,000
30,000
35,000
Roller position (degrees ± )
Rol
ler
load
(Ib
)
Rol
ler
load
(N
)
14
16
20
18
12 Rollers per row
±180�
–90� +90�
+30�
222,500 N 222,500 N(50,000 Ib) (50,000 Ib)
FIGURE 1.33 Roller load vs. number of rollers and position. 222,500N (50,000 lb) at+308, inner
dimensions constant. Outer-ring section thickness increases as the number of rollers is increased and
roller diameter is subsequently decreased.
typic al load distribut ions for the backup roll bearing of Figure 1.24, whi ch sup ports paired
line loads F2. Figure 1.34 from Ref. [15] , whi ch is a photoela stic study of a sim ilarly load ed
bearing , verif ies the data in Figure 1.33.
1.7.4 F INITE E LEMENT METHODS
To spec ify ring de flections , closed form integral analytical methods as wel l as influen ce
coeff icients calculated using infinite seri es techni que s have been ind icated for ring shapes,
which are assum ed sim ple both in circumfer ence an d cross-sect ion in the previous discussions.
For more complex structures , the fini te elem ent method s of calcul ations can be used to obtain
� 2006 by Taylor & Francis Group, LLC.
FIGURE 1.34 Photoelastic study of a roller bearing supporting loads aligned at approximately +308 to
the bearing axis. (From Eimer, H., Aus dem Gebiet der Walzlagertechnik, Semesterentwurf, Technische
Hochschule, Munchen, June 1964.)
a solution whose accuracy depends only on the fineness of the grid selected to represent the
structure.
In finite element methods a function, customarily a polynomial, is chosen to define
uniquely the displacement in each element (in terms of nodal displacements). The element
FIGURE 1.35 Finite element meshes for analyzing (a) cylindrical roller bearing rings, (b) solid rollers,
(c) hollow rollers, and (d) contact zone. (From Zhao, H., ASME Trans. J. Tribol., 120, 134–139, January
1998. With permission.)
� 2006 by Taylor & Francis Group, LLC.
stiffne ss matrix is obtaine d from equ ilibrium . The sti ffness matr ix of the complete structure is
assem bled, the bounda ry c ondition s are intro duced, and solution of the resulting matrix
equati on produ ces the nodal displacement s. A digital comp uter is requir ed to solve the
displ acements and load dist ribution accurat ely in a rolling bearing mo unted on a flexible
suppo rt. Figu re 1.35 from Zhao [16] shows the grids used to analyze a flexibly mou nted
cylin drical roller bearing assum ing both solid and hollow rollers. The load distribut ion woul d
be simila r to that indica ted in Figu re 1.34.
Bourdo n et al. [17,1 8] provided a method to define stiffne ss matr ices for use in standard
finite elem ent models to analyze roll ing elem ent bearing loads and deflections , and the
loading and de flections of the mechani sms in which they are used . For flexible mechan isms
and bearing support syst ems, they de monstrated the impor tance of con sidering the ov erall
mechani cal syst em rather than only the local system in the vicin ity of the bearing s.
1.8 CLOSURE
The method s developed in this c hapter en able the calcul ation of the internal load distribut ion
of bearing s in applic ations be yond tho se co nsider ed in bearing manufa cturer s’ catal ogs as
suppo rted by the load rati ng standar ds. It must be remem bered, howeve r, that these methods
still pertain to bearing applic ations involv ing slow to mod erate ro tational speeds. At high
speeds of rotat ion, ball an d roller inert ial loading (for exampl e, centrifugal forces and
gyroscopi c moment s) influ ence the inter nal load distribut ion, also affecti ng bearing deflec-
tions, friction forces, and moments. In this chapter, the discussion of the effect of speed on
bearing performance has been limited to the determination of fatigue life in time units.
Com mencing with Chapter 3, the detailed effe cts of speed on ov erall bearing perfor mance
will be investigated.
REFERENCES
1.
� 200
Jones, A., Analysis of Stresses and Deflections, New Departure Engineering Data, Bristol, CT, 1946.
2.
Lundberg, G. and Sjovall, H., Stress and Deformation in Elastic Contacts, Pub. 4, Institute ofElasticity and Strength of Materials, Chalmers Inst. Tech., Gothenburg, Sweden, 1958.
3.
Reussner, H., Druckflachenbelastnung und Overflachenverschiebung in Walzkontact von Rota-tionkorpern, Dissertation Schweinfurt, Germany, 1977.
4.
Palmgren, A., Ball and Roller Bearing Engineering, 3rd ed., Burbank, Philadelphia, 1959.5.
Harris, T., The effect of misalignment on the fatigue life of cylindrical roller bearings havingcrowned rolling members, ASME Trans. J. Lub. Technol., 294–300, April 1969.
6.
Harris, T., The endurance of a thrust-loaded, double row, radial cylindrical bearing, Wear, 18, 429–438, 1971.
7.
Harris, T., Kotzalas, M., and Yu, W.-K., On the causes and effects of roller skewing in cylindricalroller bearings, Trib. Trans., 41(4), 572–578, 1998.
8.
Thomas, H. and Hoersch, V., Stresses due to the pressure of one elastic solid upon another, Univ.Illinois Bull., 212, July 15, 1930.
9.
Hartnett, M., The analysis of contact stress in rolling element bearings, ASME Trans. J. Lub.Technol., 101, 105–109, January 1979.
10.
Jones, A. and Harris, T., Analysis of a rolling element idler gear bearing having a deformable outerrace structure, ASME Trans. J. Basic Eng., 273–278, June 1963.
11.
Harris, T., Optimizing the design of cluster mill rolling bearings, ASLE Trans., 7, April 1964.12.
Harris, T. and Broschard, J., Analysis of an improved planetary gear transmission bearing, ASMETrans. J. Basic Eng., 457–462, September 1964.
13.
Timoshenko, S., Strength of Materials, Part I, 3rd ed., Van Nostrand, New York, 1955.6 by Taylor & Francis Group, LLC.
14.
� 200
Lutz, W., Discussion of Ref. 7, presented at ASME Spring Lubrication Symposium, Miami Beach,
FL, June 5, 1962.
15.
Eimer, H., Aus dem Gebiet der Walzlagertechnik, Semesterentwurf, Technische Hochschule,Munchen, June 1964.
16.
Zhao, H., Analysis of load distributions within solid and hollow roller bearings, ASME Trans.J. Tribol., 120, 134–139, January 1998.
17.
Bourdon, A., Rigal, J., and Play, D., Static rolling bearing models in a C.A.D. environment for thestudy of complex mechanisms: Part I—rolling bearing model, ASME Trans. J. Tribol., 121, 205–
214, April 1999.
18.
Bourdon, A., Rigal, J., and Play, D., Static rolling bearing models in a C.A.D. environment for thestudy of complex mechanisms: Part II—complete assembly model, ASME Trans. J. Tribol., 121,
215–223, April 1999.
6 by Taylor & Francis Group, LLC.
2 Bearing Component Motionsand Speeds
� 2006 by Taylor & Fran
LIST OF SYMBOLS
Symbol Description Units
a Semimajor axis of projected contact ellipse mm (in.)
b Semiminor axis of projected contact ellipse mm (in.)
dm Pitch diameter mm (in.)
D Ball or roller diameter mm (in.)
f r/D
h Center of sliding mm (in.)
n Rotational speed rpm
nm Ball or roller orbital speed, cage speed rpm
nR Ball or roller speed about its own axis rpm
Q Rolling element–raceway contact normal load N (lb)
r Raceway groove curvature radius mm (in.)
r0 Rolling radius mm (in.)
R Radius of curvature of deformed surface mm (in.)
v Surface velocity mm/sec (in./sec)
x Distance in direction of major axis of contact mm (in.)
y Distance in direction of minor axis of contact mm (in.)
a Contact angle 8, rad
b Ball pitch angle 8, rad
b0 Ball yaw angle 8, rad
g D cos a/dm
g 0 D/dm
uf Flange angle 8, rad
v Rotational speed rad/sec
vm Orbital speed of ball or roller rad/sec
vR Speed of ball or roller about its own axis rad/sec
Subscripts
f Roller guide flange
i Inner raceway
m Orbital motion
o Outer raceway
41
cis Group, LLC.
42 Advanced Concepts of Bearing Technology
r Radi al direct ion
roll Roll ing moti on
R Roll ing elem ent
RE Roll er end
s Spin ning moti on
sl Slidi ng motion betw een flange and roll er en d
x x Dir ection
z z Directi on
2.1 GENERAL
In Chapter 10 of the first volum e of this handbook, equations we re developed to calculate rolling
elem ent orbital spee d and spee d of the rolling e le ment about its own axis. These equations were
constructed using kinema tic al re lationships base d on simple rolling motion. Also, as discussed in
Cha pter 6 of the first v olume of this handbook, when a loa d occurs between a rolling eleme nt a nd
rac eway, a conta ct surfa ce i s formed. W hen the rolling element rotates relative to the defor med
surface, the simple rolling motion does n ot oc cu r; rather, a combination o f rolling a nd sliding
motions occur. Hence, a system of complex equations needs to be developed to calculate the
rolling element speeds.
Als o, f or a ngula r-c ontac t b ea ring s, if the r olling m ot io n does not oc cur o n a line e xa ctly
parallel to the raceway, a parasitic motion called spinning occurs. Such a motion is pure sliding
c ontr ibuting s ig ni fic antly to bea ring fri cti on power loss. Finall y, m otions bet wee n rolle r e nd s and
ring flanges in roller bearings are also pure sliding and can result in substantial power loss. In this
c ha pte r, the se rolling /sli di ng re la tionships wil l be disc usse d toge ther w ith the a ss ocia te d s pe eds .
2.2 ROLLING AND SLIDING
2.2.1 GEOMETRICAL C ONSIDERATIONS
The onl y co nditions that can susta in pure rolling between two co ntacting surfac es are:
FIG
� 20
1. Mathem atical line con tact under zero load
2. Line contact in whi ch the con tacting bodies are identi cal in length
3. Mathem atical point contact unde r zero load
Even when these con ditions are achieve d, it is possibl e to ha ve sliding. Sliding is then a
con dition of overall relative movem ent of the rolling body over the c ontact area.
The motion of a rolling element with respect to the raceway consists of a rotation about the
generatrix of motion. If the contact surface is a straight line in one of the principal directions,
the generatrix of motion may intersect the contact surface at one point only, as in Figure 2.1. The
of angular velocity v, which acts in the plane of the contact surface, produces rolling motion. As
indicated in Figure 2.2, the component vs of angular velocity v that acts normal to the surface
Generatrix
O
URE 2.1 Roller–raceway contact; generatrix of motion pierces contact surface.
06 by Taylor & Francis Group, LLC.
w R
w s
w
Generatrix
O
FIGURE 2.2 Resolution of angular velocities into rolling and spinning motions.
Bearing Component Motions and Speeds 43
causes a spinning motion about a point of pure rolling O. The instantaneous direction of sliding
in the contact zone is shown in Figure 2.3.
In ba ll bearing s wi th nonz ero contact an gles betwe en balls and raceway s, during operatio n
at an y shaft or outer- ring speed, a gyrosco pic moment occurs on e ach load ed ba ll, tending to
cause a slid ing motio n. In most applic ations, be cause of relative ly slow input speed s or heavy
loading , such gyroscop ic moment s and hence moti ons can be ne glected. In high-s peed
applic ations with oil-fi lm lubri cation betw een balls and racewa ys, such motion s wi ll occur.
The sliding veloci ty due to gyroscopi c motion is given by (see Figure 2.4)
vg ¼ 12
v g D ð 2: 1Þ
The sliding velocitie s caused by gyroscopi c moti on a nd spinn ing of the balls a re vector ially
additive such that at some distance h and O they cance l each other. Thus ,
vg ¼ vs h ð 2: 2Þ
and
h ¼ D
2� vg
vs
ð 2: 3Þ
The dist ance h define s the center of slid ing abo ut whi ch a rotat ion of angular velocity vs
occurs. Thi s center of sliding (spinning) may occur within or outsid e of the contact surfa ce.
Figure 2 .5 shows the patte rn of slid ing lines in the con tact a rea for simu ltaneou s rolling,
spinni ng, and gyroscopi c motio n in a ba ll bearing ope rating under a heavy load a nd at
moderat e speed . Figure 2.6, whi ch co rresponds to low-loa d an d high-s peed conditi ons
(however, not considering skidding*), indicates that the center of sliding is outside of the
O−pure rolling
FIGURE 2.3 Contact ellipse showing sliding lines and point of pure rolling.
*Skidding is a very gross sliding condition occurring generally in oil-film lubricated ball and roller bearings operating
under relatively light load at very high speed or rapid accelerations and decelerations. When skidding occurs, cage
speed will be less than predicted by Equation 8.9 for bearings with inner ring rotation.
� 2006 by Taylor & Francis Group, LLC.
Sliding velocity dueto spinning motion
Center ofrolling
A
h
O'
O
ra
wsvg
vg
v1 = wsra
vs = w
s ra
ra
Total velocity ofsliding at point A
Lateral sliding dueto gyroscopic motion
FIGURE 2.4 Velocities of sliding at arbitrary point A in contact area.
44 Advanced Concepts of Bearing Technology
con tact surface and slid ing occ urs over the entire contact surfa ce. The distance h between the
center s of co ntact and sli ding is a functi on of the magni tude of the gyroscopi c moment that
can be compen sated by con tact surface fricti on forces .
2.2.2 SLIDING AND DEFORMATION
Eve n when the generatrix of moti on apparent ly lies in the plane of the co ntact surface, as for
radial cyli ndrical roll er bearing s, sli ding on the con tact surface can occu r when a ro ller is
unde r load. In accorda nce with the Hert zian radius of the con tact surfa ce in the direct ion
trans verse to motion, the con tact surfa ce has a harmoni c mean pro file radius , whi ch means
that the co ntact surface is not plane, but general ly curved as shown in Fig ure 2.7 for a radial
bearing .* The gen eratrix of motion, parallel to the tangent plane of the cen ter of the con tact
O�
O
h
FIGURE 2.5 Sliding lines in contact area for simultaneous rolling, spinning, and gyroscopic motions—
low-speed operation of a ball bearing.
*The illustration pertains to a spherical roller under relatively light load, that is, the contact ellipse major axis does
not exceed the roller length.
� 2006 by Taylor & Francis Group, LLC.
h
O'
O
FIGURE 2.6 Sliding lines in contact area for simultaneous rolling, spinning, and gyroscopic motions—
high-speed operation of a ball bearing (not considering skidding).
Bearing Component Motions and Speeds 45
surfa ce, therefore, pierce s the con tact surfa ce at two points at which rolling occu rs. Bec ause
the rigid rolling elemen t rotates wi th a singul ar angular veloci ty abo ut its axis, surfa ce points
at different radii from the axis have different surfa ce veloci ties; only two of them that are
symm etrically dispose d about the roller geometrica l center c an exhibi t pure rolling motion . In
Figure 2.7 points within area A–A sli de back ward with regard to the direct ion of rolling an d
points out side of A–A slide forward with respect to the direct ion of roll ing. Figure 2.8 shows
the pattern of sliding lines in the ellipti cal con tact area.
If the generat rix of moti on is an gled with respect to the tangent plane at the cen ter of the
contact surfa ce, the cen ter of roll ing is posit ioned asymm etricall y in the contact elli pse and,
depen ding on the angle of the generat rix to the contact surfa ce, one poin t or two points of
intersect ion may occur at whi ch roll ing obtains . Figure 2 .9 shows the sliding lines for this
cond ition.
For a ball be aring in whi ch rolling, spinni ng, and gyroscopi c motio ns occur simu ltan-
eously, the pattern of sli ding lines in the ellipti cal contact area is as sho wn in Figu re 2.10 an d
Figure 2.11. Mo re detailed informat ion on sliding in the elliptical co ntact a rea may be fou nd
in the work by Lundberg [1].
A A
Q
R
FIGURE 2.7 Roller–raceway contact showing harmonic mean radius and points of rolling A–A.
� 2006 by Taylor & Francis Group, LLC.
A A
FIGURE 2.8 Sliding lines in contact area of Figure 2.7.
O
FIGURE 2.9 Sliding lines for rolling element–raceway contact area when load is applied; generatrix of
motion pierces contact area.
FIGURE 2.10 Sliding lines for ball–raceway contact area for simultaneous rolling, spinning, and
gyroscopic motions—high-load and low-speed operation of an angular-contact ball bearing.
46 Advanced Concepts of Bearing Technology
2.3 ORBITAL, PIVOTAL, AND SPINNING MOTIONS IN BALL BEARINGS
2.3.1 GENERAL MOTIONS
Figure 2.12 illustr ates the speed vector for a singl e ball in a bearing . The bea ring is associ-
ated wi th the coord inate syst em x, y, z with the be aring axis collinear with the x axis. In
Figure 2.12, the ba ll center O0 is displa ced angular distance c from the xz plane, and the x0
axis passi ng throu gh O 0 is distance 12 dm from, an d pa rallel to, the x axis. The bearing is seen
to rotate at speed v ab out the x axis while the ball rotat es at speed vR ab out an axis displ aced
at pitch and yaw angles b and b0 , respect ively, from the x0 axis. Hence, the ball orb its the
bearing axis at speed vm. If the balls are complet ely co nstrained by a cage, then vm is the
cage speed.
FIGURE 2.11 Sliding lines for ball–raceway contact area for simultaneous rolling, spinning, and
gyroscopic motions—low-load and high-speed operation of an angular-contact ball bearing (not con-
sidering skidding).
� 2006 by Taylor & Francis Group, LLC.
O �
O
wR
y
w
x
y
y �
z
z �
x �
½dm
Pitchcircle
b
b �
FIGURE 2.12 Ball speed vector in a nonzero ball–raceway contact.
Bearing Component Motions and Speeds 47
In the same bearing , Figure 2.13 shows a ball con tacting the outer racew ay such that the
normal force Q betw een the ball and the racew ay is dist ributed over an elliptica l surfa ce
define d by the projected major an d minor axes 2ao and 2bo , respectivel y. The radius of
curvat ure of the deform ed pressur e surfa ce as de fined by Hert z is
Ro ¼2ro D
2ro þ D ð 2: 4Þ
wher e ro is the outer racew ay groove curvat ure radius . In term s of curvat ure fo ,
Ro ¼2fo D
2fo þ 1 ð 2: 5Þ
Assu me for the pre sent purp ose that the ba ll center is fixed in space and that the outer
raceway rotates with an gular sp eed vo . (The vector of v o is pe rpendicul ar to the plane of
rotation and theref ore collin ear with the x axis.) M oreover, it can be seen from Figu re 2.12
that ba ll rotational speed vR has componen ts vx 0 and vz 0 lying in the plan e of the paper when
c¼ 0.
Because of the deforma tion at the pressur e surface define d by ao and bo , the radius from
the ball center to the raceway con tact point varies in lengt h as the co ntact ellipse is trave rsed
from þ ao to � ao. Ther efore, because of symm etry abou t the minor axis of the contact ellipse,
pure rolling motion of the ball over the raceway occurs at most at two points . The radius at
which pur e rolling occurs is define d as r 0o and must be de termined by methods of contact
deform ation an alysis.
It can be seen from Figure 2.13 that the outer racew ay has a compon ent vo cos ao of the
angular velocity vector in a direction parallel to the major axis of the contact ellipse.
Therefore, a point (xo, yo) on the outer raceway has a linear velocity v1o in the direction of
rolling as defined below:
v1o ¼ �dmvo
2� ðR2
o � x2oÞ
1=2 � ðR2o � a2
oÞ1=2 þ D
2
� �2
�a2o
" #1=28<:
9=;vo cos ao ð2:6Þ
� 2006 by Taylor & Francis Group, LLC.
Xbo
(x o, y o
)
Ro
Y
ao
wz�
w z� si
n a o
wz� cos
ao
wx� sin a
o
wo sin a
o
w x� co
s ao
w o� co
s a o
wx�
wo
ao
D2
D2
ro
√
√
− a 2o
2
R 2o − a 2
o −
Bearing axis of rotation
Outer raceway
dm
2
FIGURE 2.13 Outer raceway contact.
48 Advanced Concepts of Bearing Technology
Similarly, the ball has angular velocity components, vx0 cos ao and vz0 sin ao, of the angular
velocity vector vR lying in the plane of the paper and parallel to the major axis of the contact
ellipse. Thus, a point (xo, yo) on the ball has a linear velocity v2o in the direction of rolling
defined as follows:
v2o ¼ �ðvx0 cos ao þ vz0 sin aoÞ � ðR2o � x2
oÞ1=2 � ðR2
o � a2oÞ
1=2 þ D
2
� �2
�a2o
" #1=28<:
9=; ð2:7Þ
Slip or sliding of the outer raceway over the ball in the direction of rolling is determined by the
difference between the linear velocities of raceway and ball. Hence,
vyo ¼ v1o � v2o ð2:8Þ
or
vyo ¼ o� dmvo
2þ ðvx0 cos ao þ vz0 sin ao � vo cos aoÞ
� ðR2o � x2
oÞ1=2 � ðR2
o � a2oÞ
1=2 þ D
2
� �2
�a2o
" #1=28<:
9=; ð2:9Þ
� 2006 by Taylor & Francis Group, LLC.
Bearing Component Motions and Speeds 49
Addition ally, the ball angu lar velocity vector vR has a componen t vy0 in a direction perpen-
dicula r to the plane of the paper. This co mponent causes a sli p vx o in the direction trans verse
to the rolling, that is, in the direction of the major axis of the contact ellipse. This sli p v elocity
is given by
vxo ¼ �vy 0 ð R2o � x2
o Þ 1= 2 � ðR 2o � a2
o Þ 1 =2 þ D
2
� �2
� a2o
" #1= 28<:
9=; ð 2: 10 Þ
From Figure 2.13, it can be observed that both the ball an gular veloci ty v ectors vx 0 and v z0 ,
and the racew ay an gular veloci ty vector vo have componen ts normal to the contact area.
Hence, there is a rotation abou t a normal to the co ntact area; in other words , a spinni ng of the
outer raceway relative to the ball, the net magni tude of which is given by
vso ¼ �vo sin a o þ v x0 sin a o � vz0 cos ao ð2:11Þ
From Figure 2.12, it can be determ ined that
vx0 ¼ vR cos b co s b0 ð 2: 12Þ
vy 0 ¼ v R cos b sin b0 ð 2: 13 Þ
vz 0 ¼ vR sin b ð 2: 14 Þ
Substitution of Equation 2.12 and Equation 2.14 into Equation 2.9 through Equation 2.11 yields
vyo ¼ �dm vo
2þ ðR 2o � x2
o Þ 1 =2 � ðR 2o � a2
oÞ1=2 þ D
2
� �2
�a2o
" #1 =28<:
9=;
� vR
vo
co s b cos b0 cos ao þvR
vo
sin b sin ao � cos ao
� �v o
ð 2: 15 Þ
v xo ¼ � ðR 2o � x2o Þ
1=2 � ðR2o � a2
oÞ1=2 þ D
2
� �2
� a2o
" #1=28<:
9=;vo
vR
vo
� �co s b sin b0 ð2: 16Þ
vso ¼vR
vo
cos b cos b0 sin ao �vR
vo
sin b cos a o � sin a o
� �vo ð 2: 17 Þ
Note that at the radius of rolling r0o on the ba ll, the translation velocity of the ball is identical
to that of the outer raceway. From Figure 2.13, therefore,
dm
2co s ao
þ r0o
� �vo cos ao ¼ r 0o ðvx0 cos ao þ vz 0 sin ao Þ ð2: 18 Þ
Substi tuting Equation 2.12 and Equation 2.13 into Equation 2.18, and rearranging the terms
yields
vR
vo
¼ ðdm=2Þ þ r0o cos ao
r0oðcos b cos b0 cos ao þ sin b sin aoÞð2:19Þ
� 2006 by Taylor & Francis Group, LLC.
50 Advanced Concepts of Bearing Technology
A similar analysis may be applied to the inner raceway contact as illustrated in Figure 2.14.
The following equations can be determined:
vyi ¼ �dmvi
2� ðR2
i � x2i Þ
1=2 � ðR2i � a2
i Þ1=2 þ D
2
� �2
� a2i
" #1=28<:
9=;
� vR
vi
cos b cos b0 cos ai þvR
vi
sin b sin ai � cos ai
� �vi
ð2:20Þ
vxi ¼ � ðR2i � x2
i Þ1=2 � ðR2
i � a2i Þ
1=2 þ D
2
� �2
� a2i
" #1=28<:
9=;vi
vR
vi
� �cos b sin b0 ð2:21Þ
vsi ¼ �vR
vi
cos b cos b0 sin ai þvR
vi
sin b cos ai þ sin ai
� �vi ð2:22Þ
vR
vi
¼ �ðdm=2Þ þ r0i cos ai
r0iðcos b cos b0 cos ai þ sin b sin aiÞð2:23Þ
If instead of the ball center fixed in space, the outer raceway is fixed, then the ball center must
orbit about the center 0 of the fixed coordinate system with an angular speed vm¼�vo.
(x i, y i)
b i
w z� si
n a i
wz� cos a
iw
x� sin ai
wi sin a
i
w x� co
s ai
w i co
s ai
w iBearing axis of rotation
Innerraceway
a i
dm
2
a i
D2
− ai 2
2
√
√
Ri 2 − a
i 2 −
D2
r i�
Y
X
R i
FIGURE 2.14 Inner raceway contact.
� 2006 by Taylor & Francis Group, LLC.
Bearing Component Motions and Speeds 51
Therefor e, the inner racew ay must rotate wi th absolut e angular speed v¼vi þvm. By using
these relat ionship s, the relative angular speed s vi and v o can be describ ed in terms of the
absolut e angular speed of the inner racew ay as foll ows:
vi ¼v
1 þ r 0o ½ðdm =2Þ � r 0i cos ai �ð cos b cos b0 cos ao þ sin b sin ao Þr 0i ½ð dm =2Þ þ r 0o cos ao �ð cos b cos b0 cos ai þ sin b sin ai Þ
ð 2: 24 Þ
vo ¼�v
1 þ r 0i ½ðdm =2Þ þ r 0o cos ao �ðcos b cos b0 cos a i þ sin b sin ai Þr 0o ½ð dm =2Þ � r0i cos ai �ðcos b cos b0 cos a o þ sin b sin ao Þ
ð 2: 25 Þ
Further,
vR ¼�v
r 0o ðco s b cos b0 co s ao þ sin b sin ao Þð dm =2Þ þ r 0o cos a o
þ r 0i ð cos b cos b0 cos ai þ sin b sin ai Þð dm =2Þ � r 0i cos a i
ð 2: 26 Þ
Similarl y, if the outer racew ay rotates with ab solute angular speed v and the inn er racew ay is
stationar y, vm ¼vi and v¼vm þvo . Therefor e,
vo ¼v
1 þ r 0i ½ðdm =2Þ þ r 0o cos ao �ðcos b cos b0 cos a i þ sin b sin ai Þr 0o ½ð dm =2Þ � r0i cos ai �ðcos b cos b0 cos a o þ sin b sin ao Þ
ð 2: 27 Þ
vi ¼�v
1 þ r 0o ½ðdm =2Þ � r 0i cos ai �ð cos b cos b0 cos ao þ sin b sin ao Þr 0i ½ð dm =2Þ þ r 0o cos ao �ð cos b cos b0 cos ai þ sin b sin ai Þ
ð 2: 28 Þ
vR ¼v
r 0o ðco s b cos b0 co s ao þ sin b sin ao Þð dm =2Þ þ r 0o cos a o
þ r 0i ð cos b cos b0 cos ai þ sin b sin ai Þð dm =2Þ � r 0i cos a i
ð 2: 29 Þ
Inspect ion of the final eq uations relat ing to the relative motion s of the balls and racew ays
reveal s the followin g unknown quantities : r 0o, r 0i , b , b0 , ai , and ao . It is ap parent that an
analys is of the forces and moment s actin g on each ball will be required to ev aluate the
unknow n qua ntities . As a practi cal matter, howeve r, it is so metimes possibl e to avoid this
lengt hy pro cedure requiring digital computa tion by using the simp lifying assum ption that a
ball will roll on one raceway without spinning and spin and roll simultaneously on the other
raceway. The raceway on which only rolling occurs is called the ‘‘controlling’’ raceway.
Moreover, it is also possible to assume that gyroscopic pivotal motion is negligible; some
criteria for this will be discussed.
2.3.2 NO GYROSCOPIC PIVOTAL MOTION
In the even t that gyroscop ic rotation is mini mal, the an gle b0 approac hes 08 (see Figure 2.12).
Therefore, the angular rotation vy0 is zero and further
vx0 ¼ vR cos b ð2:30Þ
� 2006 by Taylor & Francis Group, LLC.
52 Advanced Concepts of Bearing Technology
vz 0 ¼ v R sin b ð2: 31 Þ
A second consequ ence of b0 ¼ 0 is that
vR
vo
¼ ð dm =12 Þ þ r0o cos ao
r 0o ð co s ao cos bþ sin b sin ao Þð2: 32 Þ
and
vR
vi
¼ �ðdm =2Þ þ r 0i co s ai
r 0i ð cos b cos ai þ sin b sin ai Þð2: 33 Þ
2.3.3 SPIN -TO-ROLL RATIO
Ass uming for this calculati on that ri, r o , and 12
D are essent ially equal, the ball roll ing speed
relative to the outer raceway is
vroll ¼ �vo
dm
D¼ �vo
g0ð2:34Þ
Fro m Equat ion 2.17 for negligible gyroscopi c momen t ( b0 ¼ 0),
vso ¼ vR cos b sin ao � vR sin b cos ao � vo sin ao ð2:35Þ
or
vso ¼ vR sinðao � bÞ � vo sin ao ð2:36Þ
Dividing by vroll according to Equation 2.34 yields
vs
vroll
� �o
¼ �g 0vR
vo
sinðao � bÞ þ g 0 sin ao ð2:37Þ
According to Equation 2.32, replacing 2r 0o/dm by g 0:
vR
vo
¼ 1þ g 0 cos ao
g 0ðcos b cos ao þ sin b sin aoÞð2:38Þ
or
vR
vo
¼ 1þ g 0 cos ao
g0 cosðao � bÞ ð2:39Þ
Therefore, substitution of Equation 2.39 into Equation 2.37 yields
vs
vroll
� �o
¼ �ð1þ g 0 cos aoÞ tanðao � bÞ þ g 0 sin ao ð2:40Þ
� 2006 by Taylor & Francis Group, LLC.
Bearing Component Motions and Speeds 53
Similarl y, for an inner racew ay contact,
vs
vroll
� �i
¼ ð1 � g 0 cos a i Þ tan ðai � bÞ þ g 0 sin ai ð 2: 41 Þ
2.3.4 C ALCULATION OF R OLLING AND S PINNING SPEEDS
Even assuming that gyroscopic speed vy 0 is zero, t he use of Equation 2.40 and E quation 2.41
depends on the know ledge of the ball–raceway c ontact angles ai and a o , and ball speed
vector pitch a ngle b . I n Chapter 3, means to calculate bi and b o in high- s peed, a ngular-
contact ball bearings w ill be demonstrated. Those equations assume that ball orbital speed
vm and ball speed about its own axis, v R , are known. Unfortunately, un l ess the ball speed
vector pitch a ngle b is known, the s olution of t he set of s imultaneous equations involving
contact deformations, contact angles, and ball s peeds cannot be achieved. To determine
these paramet ers i n the most elegant manner, ball–raceway friction for ces as functions of
ball and raceway speeds need to be introduced. Thi s s ituation will also be investigated later
in thi s text.
In the absen ce of using a complete set of nor mal and fri ction forces and moment balances
to solve for speeds, Jones [2] mad e the simplifying assum ption that a ball contact ing both
inner and outer raceway s rolls and spins on one of these racew ays and sim ply rolls on the
oppos ing racew ay. He ba sed this assum ption on his interpreta tion of exp erimental data
obtaine d from gas turbin e en gine main- shaft ball bea rings. The raceway on whi ch pure rolling
was assum ed to occur was called the control ling racew ay; the pheno menon was called
raceway co ntrol. Assu ming the con dition that outer racew ay control occurs, spinnin g speed
vso ¼ 0, and substitut ion of Equat ion 2.32 into Equation 2.17 yields
tan b ¼12dm sin ao
12dm cos ao þ r 0o
ð 2: 42 Þ
As r 0o � 12 D and D /dm ¼ g 0 , Equat ion 2.42 bec omes
tan b ¼ sin ao
cos ao þg0ð 2: 43 Þ
Havi ng de fined ball speed vector pitch angle b, it is possible to so lve the remai ning sp eed
equati ons.
For high-s peed operatio ns of very light ly loaded, oil-lubri cated, angular -conta ct ball
bearing s, Figure 2.15 taken from Ref . [3] indica tes that ba ll–outer raceway spinni ng sp eed
vso tends tow ard zero, appro ximating the outer racew ay control cond ition. As the applie d
thrust load increa ses to normal operating magn itudes, vso though less than vsi is substa ntial.
This allows one to infer that outer raceway control is a condition that occurs only in a very
limit ed manner for oil-lubri cated ball bearing s.
Harris [4] also investiga ted the pe rformance of thrust -loade d, soli d-film lubricated, angu-
lar-con tact ba ll be arings of the same dimens ions assuming a con stant co efficie nt of fricti on.
Figure 2.16 from that analys is demonst rates that outer racewa y control doe s not tend to
occur in that application either.
Notwithstanding the above observations, it is of interest to carry the Jones analysis [2] to
completion since it has been used for several decades with apparently little negative impact on
bearing design.
� 2006 by Taylor & Francis Group, LLC.
00
0.1
0.2
Spi
n-to
-rol
l rat
io
0.3
0.4
0.5
100 200Thrust load (lb)
Outer raceway
Inner raceway
500 1000N
1500 2000
300 400
FIGURE 2.15 Spin-to-roll ratio vs. thrust load for an oil-lubricated, angular-contact ball bearing.
54 Advanced Concepts of Bearing Technology
From Equat ion 2.24 an d Equat ion 2.25, setting b0 equ al to 0 and substitut ing for
Equat ion 2 .43, the rati o between ball an d racew ay angular velocitie s is de termined:
vR
v¼ � 1
cos ao þ tan b sin ao
1 þg0 cos ao
þ co s ai þ tan b sin ai
1 �g0 co s ai
� �g0 co s b
ð2: 44 Þ
0.4
Inner
Outer
0.3
0.2
0.1
Spi
n-to
-rol
l rat
io
0
0.1
0 2,000 4,000 6,000
Shaft speed (rpm)
8,000 10,000
FIGURE 2.16 Spin-to-roll ratio vs. shaft speed for a thrust-loaded, angular-contact ball bearing oper-
ating with a solid-film lubricant having a constant coefficient of friction.
� 2006 by Taylor & Francis Group, LLC.
Bearing Component Motions and Speeds 55
The uppe r sign pertains to outer raceway ro tation and the low er sign to inner racewa y
rotation .
Again, using the conditio n of outer raceway control as establis hed in Equat ion 2.43, it is
possibl e to determ ine the rati o of ba ll orb ital angu lar ve locity to racew ay speed . For a
rotating inner raceway vm ¼�vo ; therefore, from Equation 2.25 for b0 eq ual to 0,
vm
v¼ 1
1 þ ð1 þ g0 cos ao Þðco s ai þ tan b sin ai Þð 1 � g0 cos a i Þðco s ao þ tan b sin a o Þ
ð 2: 45 Þ
Equation 2.45 is based on the valid assum ption that r 0 � r i � D /2. Similarl y, for a rotat ing
outer raceway and by Equat ion 2.28,
vm
v¼ 1
1 þ ð 1 � g0 cos a i Þðco s ao þ tan b sin a o Þð1 þ g0 cos ao Þðco s ai þ tan b sin ai Þ
ð 2: 46 Þ
Substi tution of Equat ion 2.43 descri bing the conditio n of outer raceway control into Equa-
tion 2.45 and Equat ion 2.46 establis hes the equati ons of the requir ed ratio vm/ v. Hence, for a
bearing with rotating inner racew ay,
vm
v¼ 1 � g0 cos ai
1 þ cos ðai � a o Þð 2: 47 Þ
For a bearing with a rotat ing outer racew ay,
vm
v¼ cos ðai � ao Þ þ � 0 co s ai
1 þ cos ðai þ a o Þð 2: 48 Þ
As indicated ab ove, Equation 2.43, Equation 2.44, Equation 2 .47, and Equat ion 2.48 are
valid only when ball gyroscop ic pivota l moti on is negli gible, that is, b0 ¼ 0.
2.3.5 GYROSCOPIC MOTION
Palmgr en [5] inferred that in an oil-lubri cated, an gular-con tact ball bearing , g yroscopi c
motio ns of the balls can be prevent ed. He stated that the coefficie nt of sliding friction may
be as low as 0.02 and that gyroscopi c moti on will not oc cur if the following relationshi p is
satisfied :
Mg > 0:02QD ð 2: 49 Þ
wher e Q is the ball–racew ay nor mal load. Jone s [2] mention ed that a coeff icient of frictio n
from 0.06 to 0.07 suff ices for most ball bearing ap plications to prev ent slid ing. Both of these
statement s are inaccurate.
It has been shown that a ball in a n angu lar-contact ball bearing is capable of experiencing
both orbital speed vm about the bearing axis and speed vR about its own axis that is canted at
pitch angle b to the x0 axis. The latter axis is parallel to the be aring axis (see Figu re 2.12) . It
has been further demonstrated that sliding motion in the direction of rolling motion occurs in
the ball–raceway contacts. Additionally, owing to the nonzero contact angle, spinning motion
occurs. Given the presence of these sliding motions, it is most probable that motion initiated
� 2006 by Taylor & Francis Group, LLC.
56 Advanced Concepts of Bearing Technology
by a gyroscopi c moment will not be prevent ed. In other words , addition al sli ding in an
orthogo nal direction (gyros copic moti on) will occu r simulta neously. Subs equent an alysis
employ ing complete force a nd moment balances for each ball shows the speed of ba ll
gyroscop ic motion vy0 to be very small compared with princi pal ball speed compon ent vx
0
and relat ively small compared to vz 0 .
2.4 ROLLER END–FLANGE SLIDING IN ROLLER BEARINGS
2.4.1 ROLLER E ND–FLANGE C ONTACT
Roller be arings react with axial roller loads throug h concentra ted con tacts between roller
end s an d flange. Tape red roller bearing s and spheri cal roller bearing s (with asymm etrical
roll ers) requ ire such contact to react with the c omponent of the raceway–r oller contact load
that acts in the ro ller axial direct ion. Some cyli ndrical roll er bearing designs requir e roller
end –flange contact s to react with skewing- induced or extern ally applied roll er axial loads. As
these co ntacts experien ce sliding motion s between roll er ends a nd flang e, their contri bution to
overal l be aring fri ctional heat generat ion be comes substan tial. Fur thermo re, there are bea r-
ing failure modes associ ated with roll er en d–flang e con tact such as wear an d smear ing of the
con tacting surfa ces. Thes e failure modes are relate d to the ability of the roller end–fl ange
con tact to support the roll er axial load unde r the prevai ling speed and lubri catio n con ditions
within the contact. Both the frictio nal cha racteris tics and load-c arrying capab ility of roller
end –flange contact s a re highly depend ent on the geomet ry of the contact ing member s.
2.4.2 ROLLER E ND–FLANGE GEOMETRY
Numer ous roll er en d and flang e geomet ries have been used success fully in roller bearing
designs . Typical ly, perfor mance requir ements as well as man ufacturing con siderati ons dictate
the geo metry inco rporated into a bea ring design. Most designs use eithe r a flat (with corner
radii ) or sphere end roll er contact ing an angled flang e. The angled flang e surfa ce can be
descri bed as a por tion of a cone at an angle uf with respect to a radial plan e pe rpendicul ar to
the ring axis. This angle, known as the flange angle or flang e laybac k an gle, can be zero,
indicating that the flange surface lies in the radial plane. Examples of cylindrical roller bearing
roller end–flange geometries are shown in Figure 2.17. The flat end roller in Figure 2.17a
under zero skewing conditions contacts the flange at a single point (in the vicinity of the
intersection between the roller end flat and roller corner radius). As the roller skews, the point
of contact travels along this intersection on the roller toward the tip of the flange, as shown in
Figure 2.18b. If properl y designe d, a sphere end roller will contact the flang e on the roller end
sphere surface. For no skewing, the contact will be centrally positioned on the roller, as shown
Rolleraxis
Rolleraxis
Roller Roller
(a) (b)
q f q f
Rre
FIGURE 2.17 Cylindrical roller bearing, roller end–flange contact geometry. (a) Flat end roller.
(b) Sphere end roller.
� 2006 by Taylor & Francis Group, LLC.
(a) (b)
(c) (d)
FIGURE 2.18 Cylindrical roller bearing, roller end–flange contact location for flat and sphere end
rollers. (a) Flat end roller, zero skew angle. (b) Flat end roller, nonzero skew angle. (c) Sphere end
roller, zero skew angle. (d) Sphere end roller, nonzero skew angle.
Bearing Component Motions and Speeds 57
in Figure 2.18c. As the skewing an gle is increa sed, the con tact point moves off center an d
toward the flang e tip, as sho wn in Figure 2.18d for a flang ed inner ring. For typic al de signs,
the sphere end ro ller con tact locat ion is less sensi tive to skewing than a flat end roller co ntact.
The locat ion of the ro ller end–fl ange co ntact ha s been de termined an alytical ly [6] for
sphere end rollers co ntacting an angled flang e. Consi der the cylind rical roller be aring
arrange ment sho wn in Figure 2.19. The flanged ring co ordinat e syst em XI , YI , Z I an d ro ller
coordinat e system Xi, Yi , Zi are ind icated. The flang e contact surfa ce is mod eled as a portio n
of a cone with an apex at point C as shown in Figure 2.20. The equatio n of this cone,
express ed as a functi on of the x and y ring co ordinates is
z ¼ ½ðx � C Þ 2 ctn 2 uf � y2 �1 =2 ¼ f ð x, yÞ ð2: 50 Þ
For a point of flange surfa ce Px , Py, P z, the equati on of the surfa ce normal at P can be
express ed as
x � Px
@ f
@ x
�������x ¼ Px ,y ¼ Py
¼y � Py
@ f
@ y
��������x ¼ Px ,y ¼ Py
¼ �ðz � Pz Þ ð 2: 51 Þ
The location of the or igin of the roll er end sphere rad ius is define d as point T with coordinat es
( Tx , T y, T z) express ed in the flanged ring coo rdinate syst em. As the resul tant ro ller end–fl ange
elastic con tact force is nor mal to the end sphere surfa ce, its line of acti on must pa ss through
the sphere origin (Tx, Ty, Tz). Evaluating Equation 2.50 and Equation 2.51 at T yields the
following three equations:
Tx � Px ¼ðTz � PzÞðPx � CÞ ctn2 uf
½ðPx � CÞ2 ctn2 uf � P2y�
1=2ð2:52Þ
Ty � Py ¼ðTz � PzÞPy
½ðPx � CÞ2 ctn2 uf � P2y�
1=2ð2:53Þ
� 2006 by Taylor & Francis Group, LLC.
XI ZI
Zi
Yi
Xi rre
rre
YI q f
FIGURE 2.19 Cross-section through a cylindrical roller bearing that has a flanged inner ring.
58 Advanced Concepts of Bearing Technology
Pz ¼ ½ðPx � C Þ 2 ctn 2 uf � P2
y � 1= 2 ð2: 54 Þ
Equat ion 2.52 through Equat ion 2.54 co ntain three unknow ns ( Px, Py, Pz ) an d are suff icient
to determ ine the theoret ical poin t of contact between the roller end and flange . By intr oduc-
ing a fourt h equ ation and unknow n, howeve r, na mely the lengt h of the line from points (Tx,
Ty, T z) to (P x , P y, P z), the added ben efit of a c losed-form solution is obtaine d. The lengt h of a
line normal to the flange surface at the point ( Px, P y, Pz ), which joins this poin t with the
sphere origi n ( Tx, Ty, Tz ), is given by
D ¼ ½ðTx � Px Þ 2 þ ðTy � P y Þ 2 þ ðTz � Pz Þ 2 � 1 =2 ð2: 55 Þ
Right circular conez = f(x, y)
Contact pointP(Px, Py, Pz)
T(Tx, Ty, Tz) rs Roller end sphere
X1
q f
Z1
Zi
Yi
Xi
C
{T }1
{T }1
FIGURE 2.20 Coordinate system for calculation of roller end–flange contact location.
� 2006 by Taylor & Francis Group, LLC.
Bearing Component Motions and Speeds 59
After algebraic reduction, D is obtained from the positive root of the quadratic equation:
D ¼ �S� ðS2 � 4<TÞ1=2
2< ð2:56Þ
where values for S, <, and T are
< ¼ tan2 uf � 1
S ¼ 2 sin2 uf
cos uf
½ðTx � CÞ � tan ufðT2y þ T2
z Þ1=2�
T ¼ ½ðTx � CÞ � tan ufðT2y þ T2
z Þ1=2�
The coordinates P(Px, Py, Pz) are given by the following closed-form function of D:
Px ¼ Ty tan uf 1þ Tz
Ty
� �2" #1=2
1� D
ðT2y þ T2
z Þ1=2
" #þ c ð2:57Þ
Py ¼ Ty 1� D sin uf
ðT2y þ T2
z Þ1=2
" #ð2:58Þ
Pz ¼ Tz 1� D sin uf
ðT2y þ T2
z Þ1=2
" #ð2:59Þ
At a point of contact between the roller end and flange, D is equal to the roller end sphere
radius. Therefore, knowing the roller and flanged ring geometry as well as the coordinate
location (with respect to the flanged ring coordinate system) of the roller end sphere origin, it
is possible to calculate directly the theoretical roller end–flange contact location.
The analysis, although shown for a cylindrical roller bearing, is general enough to apply to
any roller bearing that has sphere end rollers that contact a conical flange. Tapered and spherical
roller bearings of this type may be treated if the sphere radius origin is properly defined.
These equations have several notable applications since flange contact location is of interest
in bearing design and performance evaluation. It is desirable to maintain contact on the flange
below the flange rim (including edge break) and above the undercut at the base of the flange. To
do otherwise causes loading on the flange rim (or edge of undercut) and produces higher contact
stresses and less than optimum lubrication of the contact. The preceding equations may be used
to determine the maximum theoretical skewing angle for a cylindrical roller bearing if the roller
axial play (between flanges) is known. Also, by calculating the location of the theoretical contact
point, sliding velocities between roller ends and flange can be calculated and used in an estimate
of roller end–flange contact friction and heat generation.
2.4.3 SLIDING VELOCITY
The kinematics of a roller end–flange contact causes sliding to occur between the contacting
members. The magnitude of the sliding velocity between these surfaces substantially affects
friction, heat generation, and load-carrying characteristics of a roller bearing design. The
sliding velocity is represented by the difference between the two vectors defining the linear
velocities of the flange and the roller end at the point of contact. A graphical representation of
� 2006 by Taylor & Francis Group, LLC.
C
Rc
v f
v slv RE
wR
w f
w o
rc
FIGURE 2.21 Roller end–flange contact velocities.
60 Advanced Concepts of Bearing Technology
the roll er velocity vroll and the flange veloci ty v F at their point of contact C is shown in Figure
2.21. The sli ding veloci ty vector vsl is shown as the differenc e of v RE and v f. When con sidering
roll er skewing moti ons, vsl wi ll have a co mponent in the flang ed ring axial direct ion, albeit
smal l in compari son wi th the co mponents in the bearing radial plane, if the roll er is not
subject ed to the componen ts in the be aring radial plan e. If the roll er is not subject ed to
skew ing, the contact point will lie in the plane contai ning the roll er and flang ed ring axes. The
roll er end –flange slid ing veloci ty may be calcul ated as
vs1 ¼ v f � v RE ¼ vf R c � ðvo R c þ v R rc Þ ð2: 60 Þ
wher e clockw ise rotations are co nsidere d posit ive. Var ying the posit ion of con tact poin t C
over the elast ic contact area between roller end and flang e allows the distribut ion of sliding
veloci ty to be determ ined.
2.5 CLOSURE
In this chapter, methods for c alculations of rolling and cage speeds i n ball and roller
bearings were developed f or cond itions of ro lling and spinning motions. I t will be s hown
in Chapter 3 how the dynamic loading derived from ball and r oller s peeds can significantly
affect ball bearing c ontact angles, di ametra l c learance, and s ubs e quently rolling e lement
load distribution. Moreover, spinning motions that occur i n ball bearings t end t o alter
contact area stresses, and hence they affect be aring endurance. O ther quantities affected by
bearing internal speeds are friction torque an d frictional heat generation. It is therefore
clear that accurate determinations of bearing internal speeds are necessary for analysis of
rolling be aring performance.
It will be demo nstrated subsequen tly that hyd rodynami c acti on of the lubri cant in the
contact areas can transform what is presumed to be substantially rolling motions into
combinations of rolling and translatory motions. In general, this combination of rotation
and translation may be tolerated provided the lubricant films resulting from the rolling
motions are sufficient to adequately separate the rolling elements and raceways.
� 2006 by Taylor & Francis Group, LLC.
Bearing Component Motions and Speeds 61
REFERENCES
1.
� 2
Lundberg, G., Motions in loaded rolling element bearings, SKF unpublished report, 1954.
2.
Jones, A., Ball motion and sliding friction in ball bearings, ASME Trans. J. Basic Eng., 81, 1959.3.
Harris, T., An analytical method to predict skidding in thrust loaded, angular-contact ball bearings,ASME Trans. J. Lubrication Technol., 17–24, January 1971.
4.
Harris, T., Ball motion in thrust-loaded, angular-contact bearings with coulomb friction, ASME J.Lubrication Technol., 93, 17–24, 1971.
5.
Palmgren, A., Ball and Roller Bearing Engineering, 3rd ed., Burbank, Philadelphia, 1959, pp. 70–72.6.
Kleckner, R. and Pirvics, J., High speed cylindrical roller bearing analysis—SKF Computer ProgramCYBEAN, Vol. 1: Analysis, SKF Report AL78P022, NASA Contract NAS3-20068, July 1978.
006 by Taylor & Francis Group, LLC.
3 High-Speed Operation: Balland Roller Dynamic Loads
� 2006 by Taylor & Fran
and Bearing Internal LoadDistribution
LIST OF SYMBOLS
Symbol Description Units
B fiþ fo� 1
dm Pitch diameter mm (in.)
D Ball or roller diameter mm (in.)
f r=DF Force N (lb)
Fc Centrifugal force N (lb)
Ff Friction force N (lb)
g Gravitational constant mm=sec2 (in.=sec2)
H Roller hollowness ratio
J Mass moment of inertia kg � mm2 (in. � lb � sec2)
K Load–deflection constant N=mmx (lb=in.x)
l Roller length mm (in.)
m Ball or roller mass kg (lb � sec2=in.)
M Moment N � mm (lb � in.)
Mg Gyroscopic moment N � mm (lb � in.)
M Applied moment N � mm (lb � in.)
n Rotational speed rpm
nm Ball or roller orbital speed, cage speed rpm
nR Ball or roller speed about its own axis rpm
Pd Radial or diametral clearance N (lb.)
q Roller–raceway load per unit length N=mm (lb=in.)
Q Ball or roller normal load N (lb)
Qa Axial direction load on ball or roller N (lb)
Qr Radial direction load on ball or roller N (lb)
R Radius to locus of raceway groove curvature centers mm (in.)
s Distance between inner and outer groove
curvature center loci mm (in.)
X1 Axial projection of distance between ball center and
outer raceway groove curvature center mm (in.)
cis Group, LLC.
X2 Radi al pro jection of dist ance between ba ll center and
outer racew ay groove curvat ure cen ter mm (in.)
a Cont act an gle 8, rad
b Ball attitude angle 8, rad
g ( D co s a) =dm
d Defl ection or contact de formati on mm (in.)
u Bea ring mis alignment or angular de flection 8, rad
r Mass densit y kg=mm 3 (lb z � sec 2=in.4)
f Angl e in WV plane 8, rad
c Angl e in yz plane 8, rad
v Rota tional speed rad =secvm Orbi tal speed of ba ll or roller rad =secvR Spe ed of ball or roll er ab out its own axis rad =secDc Angul ar distance betw een rolling elemen ts rad
Subscri pts
a Axi al direct ion
e Rota tion ab out an ecce ntric axis
f Roll er guide flang e
i Inne r racew ay
j Roll ing elem ent at angular location
m Cage motion and orb ital mo tion
o Oute r racew ay
r Radi al direct ion
R Roll ing elem ent
x x direct ion
z z direction
3.1 GENERAL
Dynam ic (inerti al) loading occurs betw een rolling elem ents and bearing raceways because of
roll ing elem ent orbit al speeds and speeds about their own axes. At slow -to-modera te ope rat-
ing speeds, these dynami c loads are very small compared with the ball or roll er loads caused
by the loading ap plied to the bearing . At high ope rating speeds, howeve r, these roll ing
elem ent dynami c loads, centrifugal forces , an d gyroscopi c moment s will alter the distribut ion
of the applie d loading among the ba lls or roll ers. In rolle r be arings, the increa se in loading on
the outer raceway due to roller centrifugal forces causes larger contact deformations in that
member; this effect is similar to that of increasing clearance. Increase of clearance, as
demon strated in Chapt er 7 of the fir st volume of this ha ndbook, causes increa sed maxi mum
roller load due to a decrease in the extent of the load zone. For relatively thin section bearings
supported at only a few points on the outer ring, for example, an aircraft gas turbine
mainshaft bearing, the centrifugal forces cause bending of the outer ring, also affecting the
distribution of loading among the rollers.
In high-speed ball bearings, depending on the contact angles, ball gyroscopic moments
and ball centrifugal forces can be of significant magnitude such that inner-ring contact angles
increase and outer-ring contact angles decrease. This affects the deflection vs. load charac-
teristics of the bearing and therefore also affects the dynamics of the ball bearing–supported
rotor system.
� 2006 by Taylor & Francis Group, LLC.
High speed also affects the lubrication and friction characteristics in both ball and
roller bearings. This influences bearing internal speeds, and hence rolling element dynamic
loads. It is possible, however, to determine the internal load distribution and hence contact
stresses in many high-speed rolling bearing applications with sufficient accuracy while not
considering the frictional loading on the rolling elements. This will be demonstrated in
this chapter. The effects of friction on load distribution will be considered in a later
chapter.
3.2 DYNAMIC LOADING OF ROLLING ELEMENTS
3.2.1 BODY FORCES DUE TO ROLLING ELEMENT ROTATIONS
The development of equations in this section is based on the motions occurring in an angular-
contact ball bearing because it is the most general form of rolling bearing. Subsequently, the
equations developed can be so restricted as to apply to other ball bearings and also to roller
bearings.
Figure 3.1 illustrates the instantaneous position of a particle of mass m in a ball of an
angular-contact ball bearing operating at a high rotational speed about an axis x. To simplify
the analysis, the following coordinate axes systems are introduced:
f
wR
wm
w
y
r
Ub
b'
b
z'
x'
y'
V
W dm
12dm
12dm
xO
Ball center
Axis of rotation
O'
Pitch
circle z
y
FIGURE 3.1 Instantaneous position of ball mass element dm.
� 2006 by Taylor & Francis Group, LLC.
x, y, z
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A fixed set of Cartesian coordinates with the x axis coincident with the bearing rotational axis
x’, y’, z’
A set of Cartesian coordinates with the x’ axis parallel to the x axis of the fixed set. This set ofcoordinates has its origin O’ at the ball center and rotates at orbital speed about the fixed x axis
at radius 12dm
U, V, W
A set of Cartesian coordinates with origin at the ball center O’ and rotating at orbital speed vm.The U axis is collinear with the axis of rotation of the ball about its own center. The W axis is in the
plane of the U axis and z’ axis; the angle between the W axis and z’ axis is b
U, r, f
A set of polar coordinates rotating with the ballIn addition, the following symbols are introduced:
b’
or & F
The angle between the projection of the U axis on the x’y’ plane and the x’ axis
c
The angle between the z axis and z’ axis, that is, the angular position of the ball on the pitch circleConsider that an element of mass dm in the ball has the following instantaneous location in
the system of rotating coordinates: U, r, f. As
U ¼ U
V ¼ r sin f
W ¼ r cos f
ð3:1Þ
and
x0 ¼ U cos b cos b0 � V sin b0 �Wsin b cos b0
y0 ¼ U cos b sin b0 þ V cos b0 �W sin b sin b0
z0 ¼ U sin bþW cos b
ð3:2Þ
and
x ¼ x0
y ¼ 12dm sin cþ y0 cos cþ z0 sin c
z ¼ 12dm cos c� y0 sin cþ z0 cos c
ð3:3Þ
by substitution of Equation 3.1 into Equation 3.2 and thence into Equation 3.3, the following
expressions relating the instantaneous position of the element of mass dm to the fixed system
of Cartesian coordinates can be formulated:
x ¼ U cos b cos b0 � rðsin b0 sin fþ sin b cos b0 cos fÞ ð3:4Þ
y ¼ dm
2sin cþUðcos b sin b0 cos cþ sin b sin cÞ
þ rðcos b sin f cos cþ cos b cos f sin c
� sin b sin b0 cos f cos cÞð3:5Þ
rancis Group, LLC.
z ¼ dm
2cos cþUð� cos b sin b0 sin cþ sin b cos cÞ
þ rð� cos b0 sin f sin cþ cos b cos f cos c
þ sin b sin b0 cos f cos cÞ
ð3:6Þ
In accordance with Newton’s second law of motion, the following relationships can be
determined if the rolling element position angle c is arbitrarily set equal to 08:
dFx ¼ €xxdm ð3:7Þ
dFy ¼ €yydm ð3:8Þ
dFz ¼ €zzdm ð3:9Þ
dM0z ¼f�€xx½U cos b sin b0 þ rðcos b0 sin f� sin b sin b0 cos fÞ�þ €yy½U cos b cos b0 � rðsin b0 sin fþ sin b cos b0 cos fÞ�gdm
ð3:10Þ
dM0y ¼f€xxðU sin bþ r cos b cos fÞ� €zz½U cos b cos b0 � rðsin b0 sin fþ sin b cos b0 cos fÞ�gdm
ð3:11Þ
The net moment about the x axis must be zero for constant speed motion. At each ball
location (c, b), vR (rotational speed df=dt of the ball about its own axis U) and vm (orbital
speed dc=dt of the ball about the bearing axis x) are constant; therefore, at c¼ 0,
€xx ¼ d2x
dt2¼ rv2
Rðsin b0 sin fþ sin b cos b0 cos fÞ ð3:12Þ
€yy ¼ d2y
dt2¼� 2vRvmr cos b sin f
þ v2m½�U cos b sin b0 þ rð� cos b0 sin fþ sin b cos f sin b0Þ�
þ v2Rrð� cos b0 cos fþ sin b sin b0 sin fÞ
ð3:13Þ
€zz ¼ d2z
dt2¼� 2vRvmrðcos b0 cos fþ sin b sin b0 sin fÞ
� v2m
dm
2þU sin bþ r cos b cos f
� �� v2
Rr cos b cos f
ð3:14Þ
Substitution of Equation 3.12 through Equation 3.14 into Equation 3.7 through Equation
3.11 and placing the latter into integral format yields
Fx0 ¼ �r
Z þrR
�rR
Z ðr2R�U2Þ1=2
0
Z 2p
0
€xxr dr dU df ð3:15Þ
Fy0 ¼ �r
Z þrR
�rR
Z ðr2R�U2Þ1=2
0
Z 2p
0
€yyr dr dU df ð3:16Þ
Fz0 ¼ �r
Z þrR
�rR
Z ðr2R�U2Þ1=2
0
Z 2p
0
€zzr dr dU df ð3:17Þ
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Mz 0 ¼ � r
Z þ rR
� rR
Z ð r2R � U 2 Þ 1=2
0
Z 2p
0
f�€xx½ U cos b sin b0
þ r ð cos b0 sin f� sin b sin b0 co s fÞ�þ €yy½ U cos b cos b0 � r ð sin b0 sin f
þ sin b cos b0 cos fÞ�g r d r dU df
ð3: 18 Þ
My0 ¼ � r
Z þ rR
� rR
Z ð r2R � U 2 Þ 1=2
0
Z 2p
0
f€xxð U sin bþ r co s b cos fÞ
� €zz ½ U cos b cos b0 � r ð sin b0 sin f
þ sin b cos b0 cos fÞ�g r d r dU df
ð3: 19 Þ
In Equat ion 3.15 throu gh Equat ion 3.19, r is the mass density of the ball material and rR is
the ball radius .
Perform ing the integrati ons indica ted by Equat ion 3.15 through Equat ion 3.19 establis hes
that the net forces in the x ’ and y’ direct ions are zero and that
Fz 0 ¼ 12md m v
2m ð3: 20 Þ
My 0 ¼ J vR v m sin b ð3: 21 Þ
Mz0 ¼ �J vR vm cos b sin b0 ð3: 22 Þ
wher e m is the mass of the ball and J is the mass moment of inertia, and are de fined as follows:
m ¼ 16
rpD3 ð3: 23 Þ
J ¼ 160
rp D 5 ð3: 24 Þ
3.2.2 CENTRIFUGAL FORCE
3.2. 2.1 Rotation abo ut the Bearin g Axis
Subs tituting Equat ion 3.23 into Equation 3.20 an d recogni zing that
vm ¼2p nm
60 ð3: 25 Þ
Equat ion 3 .26 yield ing the ball centri fugal force is obtaine d:
Fc ¼p3r
10800gD3n2
mdm ð3:26Þ
For steel balls,
Fc ¼ 2:26� 10�11D3n2mdm ð3:27Þ
For an applied thrust load per ball Qia and a ball centrifugal load Fc directed radially
outw ard, the ball loading is as shown in Figure 3.2. For conditio ns of equilibrium , assum ing
the bearing rings are not flexible,
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Qoa
Qor
ao
ai
Fc
Qo
Qia
QirQi
FIGURE 3.2 Ball under thrust load and centrifugal load.
Qia � Q oa ¼ 0 ð 3: 28 Þ
Qir þ Fc � Qor ¼ 0 ð 3: 29 Þ
or
Qia � Q o sin ao ¼ 0 ð 3: 30 Þ
Qia cot ai þ Fc � Q o cos ao ¼ 0 ð 3: 31 Þ
Equation 3.30 and Equation 3.31 must be solved sim ultaneo usly for unknowns Qo and ao .
Thus ,
ao ¼ co t� 1 co t ai þFc
Qia
� �ð 3: 32 Þ
Qo ¼ 1 þ cot ai þFc
Qia
� �2
� �1 =2
Qia ð 3: 33 Þ
Further,
Qi ¼Qia
sin ai
ð 3: 34 Þ
From Equation 3.32, becau se of centri fugal force Fc , it is app arent that ao <ai . ai is the
contact angle under thrust load, and ai>ao the free contact angle. This condition is discussed
in detail in Chapt er 7 in the fir st volume of this handbo ok.
� 2006 by Taylor & Francis Group, LLC.
Axi
s of
rot
atio
n
a
a
Qa
Qa
Fc
Q
Q
FIGURE 3.3 Ball loading in a 908 contact-angle, thrust ball bearing.
See Exam ple 3.1.
Thr ust ball bearing s with nom inal contact angle a¼ 90 8 ope rating at high speeds and light
loads tend to permi t the balls to override the land on both rings (washers ). The contact angle
thus de viates from 90 8 in the same direct ion on both raceways. Fr om Figure 3.3, whi ch
dep icts this conditio n,
Q ¼ Fc
2 cos að3: 35 Þ
and
a ¼ tan � 1 2Qa
Fc
� �ð3: 36 Þ
See Exam ple 3.2.
Equat ion 3.20 is not rest rictive as to geometry an d since the mass of a cyli ndrical (or nearly
cyli ndrical) roller is given by
m ¼ 14 rp D 2 l t ð3: 37 Þ
the centrifugal force for a steel roll er or biting at speed nm about a be aring axis is given by
Fc ¼ 3: 39 � 10 � 11 D 2 l t dm n2m ð3: 38 Þ
For a tapered roll er bearing , howeve r, roll er centrifugal force alters the distribut ion of load
betw een the outer raceway a nd inner- ring guide flang e. Figure 3.4 de monst rates this cond i-
tion for an applied thrust load Qia.
For equilibrium to exist,
Qia þQfa �Qoa ¼ 0 ð3:39Þ
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Fc Qo
Qor
Qi Qir
Qfa
QfrQfaf
ai
ao
Qia
Qoa
FIGURE 3.4 Tapered roller under thrust load and centrifugal force.
Qir � Q fr þ F c � Q or ¼ 0 ð 3: 40 Þ
or
Qia þ Q f sin a f � Q o sin a o ¼ 0 ð 3: 41 Þ
Qia cot ai � Q f cos af þ Fc � Q o co s ao ¼ 0 ð 3: 42 Þ
Solving Equation 3.41 and Equation 3.42 simulta neously yields
Qo ¼Qia cot a i sin af þ cos a fð Þ þ Fc sin af
sin ao þ afð Þ ð 3: 43 Þ
Qo ¼Qia cot ai sin ao � cos aoð Þ þ Fc sin ao
sin ao þ afð Þ ð 3: 44 Þ
See Exampl e 3.3.
Care must be exerci sed in operating a tapere d roller bea ring at a very high speed. At some
critical speed relat ed to the magni tude of the applied load, the force at the inner- ring
raceway con tact approach es zero, and the en tire ax ial load is carried at the roller end–
inner- ring flange contact . Becau se this contact has only slid ing motio n, very high frictio n
resul ts with attend ant high heat generat ion.
Most modern radial spherica l roll er bearing s have complem ents of symmetri cal co ntour
(barr el-shaped ) ro llers and relative ly small contact angles ; for exampl e, a� 15 8 . When the
bearing s are ope rated at a high speed , roller loading is as illustrated in Figure 3 .5.
Equilibrium of forces in the radial and axial directions gives
Qo cos ao �Qi cos ai � Fc ¼ 0 ð3:45Þ
Qo sin ao �Qi sin ai ¼ 0 ð3:46Þ
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Qo cos αo
Qo sin ao
Qo cos a i
ao
a i
a o
Qo
Qi
Fc
FIGURE 3.5 Loading of a barrel-shaped roller subjected to applied and centrifugal forces.
Solving these equations simultaneously gives
Qo ¼Fc sin ai
sinðai � aoÞð3:47Þ
Qi ¼Fc sin ao
sinðai � aoÞð3:48Þ
Therefore, it appears that roller–raceway loading is uniquely determined by roller centrifugal
loading. Clearly, in this instance the inner and outer raceway contact angles are functions of
the applied radial and thrust loadings of the bearing, and these must be determined from the
equilibrium of loading on the bearing. Doing this requires the determination of the bearing
contact deformations.
Another way to view the operation of a loaded spherical roller operating at a high speed is
to resolve the centrifugal force into components collinear with, and normal to, the roller axis
of rotation. Hence,
Fca ¼ Fc sin ao ð3:49Þ
Fcr ¼ Fc cos ao ð3:50Þ
where ao is the nominal contact angle. Equilibrium of forces acting in the radial plane of the
roller gives
Qo ¼ Qi þ Fc cos ao ð3:51Þ
and the component Fc sin ao causes the inner and outer raceway contact angles to shift slightly
from ao to accommodate the roller axial loading. In general, spherical roller bearings do not
operate at speeds that will cause significant change in the nominal contact angle. Also, consider a
double-row spherical roller bearing with barrel-shaped rollers subjected to a radial load while
rotating at high speed. The speed-induced roller axial loads are self-equilibrated within the
bearing; however, the outer raceways carry larger thrust components than do the inner raceways.
� 2006 by Taylor & Francis Group, LLC.
3.2.2 .2 Rota tion abou t an Eccentr ic Axis
Secti on 3.2.2.1 dealt with rolling elem ent centrifugal loading when the bearing rotat es about
its own axis; this is the usual case. In planetary gear transmissions, however, the planet gear
bearings rotate about the input and output shaft axes as well as about their own axes. Hence,
an additional inertial or centrifugal force is induced in the rolling element. Figure 3.6 shows a
schematic diagram of such a system. From Figure 3.6, it can be seen that the instantaneous
radius of rotation is, by the law of cosines,
r ¼ ðr2m þ r2
e � 2rmre cos cÞ1=2 ð3:52Þ
Therefore, the corresponding centrifugal force is
Fce ¼ mv2eðr2
m þ r2e � 2rmrc cos cÞ1=2 ð3:53Þ
This force Fce is maximum at c¼ 1808 and at that angle is algebraically additive to Fc. At
c¼ 0, the total centrifugal force is Fc�Fce. The angle between Fc and Fce as derived from the
law of cosines is
u ¼ cos�1 rm
r� re
rcos c
� �ð3:54Þ
Fce can be resolved into a radial force and a tangential force as follows:
Fcer ¼ Fce cos u ð3:55Þ
Rolling element
F c
F ce
r m
r e
r
wm
w e
q
y
FIGURE 3.6 Rolling element centrifugal forces due to bearing rotation about an eccentric axis.
� 2006 by Taylor & Francis Group, LLC.
Fcet ¼ F ce sin u ð3: 56 Þ
Hence, the total inst antaneous radial cen trifugal force acting on the rolling elem ent is
Fcr ¼ mv 2m r m þ m v2
e ð r m � r e cos cÞ ð3: 57 Þ
or
Fcr ¼W
g½rm ðv2
m þ v2e Þ � r e v
2e cos c� ð3: 58 Þ
wher e the positive direct ion is that taken by the constant compon ent Fc . For steel ball and
roll er elem ents, the followin g eq uations are, respect ivel y, valid :
Fcr ¼ 2:26 � 10 � 11 D 3 ½ dm ð n2m þ n2
e Þ � de n2e cos c� ð3: 59 Þ
Fcr ¼ 3: 39 � 10� 11 D2 lt ½ dm ð n2m þ n2
e Þ � de n2e cos c� ð3: 60 Þ
The inst antaneou s tangent ial compo nent of eccentr ic cen trifugal force is
Fct ¼ mv 2e r e sin c ð3: 61 Þ
For steel ba ll and roller eleme nts, respectivel y, the followi ng equatio ns pe rtain:
Fct ¼ 2: 26 � 10 � 11 D3 de n2e sin c ð3: 62 Þ
Fct ¼ 3: 39 � 10 � 11 D 2 l t de n2e sin c ð3: 63 Þ
This tangen tial force alternate s direction and tends to pro duce slid ing between the rolling
elem ent and raceway. It is theref ore resisted by a frictio nal force between the con tacting
surfa ces.
The bearing cage also undergoes this eccentr ic motio n an d if it is sup ported on the rolling
elem ents, it wi ll impos e an addition al load on the indivi dual rolling elem ents. This cage load
may be reduced by using a material of smaller mass density.
3.2.3 GYROSCOPIC MOMENT
It can usually be assumed with minimal loss of calculational accuracy that pivotal motion due
to gyroscopi c moment is negligible ; then, the angle b’ is z ero an d Equation 3.22 is of no
con sequence. The gyroscop ic moment as de fined by Equat ion 3.21 is therefore resisted
successfully by friction forces at the bearing raceways for ball bearings and by normal forces
for roller be arings. Substi tuting Equation 3.25 into Equation 3.21, the foll owing express ion is
obtained for ball bearings:
Mg ¼ 160
rpD5vRvm sin b ð3:64Þ
since
vR ¼2pnR
60ð3:65Þ
� 2006 by Taylor & Francis Group, LLC.
Z
Y
X
Z
Y
Y'
Z'
M g
wm
wR
Y'
Z'
X
FIGURE 3.7 Gyroscopic moment due to simultaneous rotation about nonparallel axes.
and
vm ¼2p nm
60 ð 3: 66 Þ
The gyrosco pic moment for steel ball bearing s is given by
Mg ¼ 4:47 � 10 � 12 D 5 nR nm sin b ð 3: 67 Þ
Figure 3.7 shows the direction of the gyroscopi c moment in a ball bearing . Accor dingly ,
Figure 3.8 shows the ball loading due to the action of gyroscopi c moment and centrifugal
force on a thrust -loade d ball bearing .
See Exampl e 3.4.
Gyrosco pic moment s also act on rolle rs in radial tapere d and spheri cal roll er bearing s an d
on the roll ers in thrust roller bearing s of all types. The rollers, howeve r, are geomet rically
constr ained from rotating due to the induced gyroscopi c mo ments. Therefor e, a gyroscopi c
moment of significan t magni tude tends to alter the distribut ion of load across the roller
contour. For steel rollers, the gyroscop ic moment s a re given by
Mg ¼ 8: 37 � 10 � 12 D 4 l t nR nm sin b ð 3: 68 Þ
3.3 HIGH-SPEED BALL BEARINGS
To determ ine the load dist ribution in a high-s peed ball bearing , consider Figure 1.2, whi ch
shows the displ acement s of a ball bearing inne r ring relative to the outer ring due to a
general ized loading syst em, includi ng radial, axial, and moment loads. Figure 3.9 sh ows the
relative angular position (azimuth) of each ball in the bearing.
� 2006 by Taylor & Francis Group, LLC.
M g
Fc
F fo
Q oQ or
Q oa
FfiQ i
Q ia
Q ir
a o
a i
FIGURE 3.8 Forces acting on a ball in a high-speed ball bearing subjected to applied thrust load.
Under zero load, the center s of the raceway groo ve c urvature radii a re separat ed by a
dist ance A as sho wn in Figure 1.1a. In Chapter 2 of the first volume of this handb ook, it was
shown that A¼BD where B¼ fiþ fo� 1. Under an applied static load, the distance between
the inner and outer raceway groove curvature centers will increase by the amount of the contact
y1 = 08
y2
y3
yj
Δy
j = 2
j = 3
j = 1
j
d m
FIGURE 3.9 Angular position of rolling elements in yz plane (radial). Dc¼ 2p=Z, cj¼ 2p=Z( j� 1).
� 2006 by Taylor & Francis Group, LLC.
deformations di and do as shown in Figure 1.1b. The line of action between the centers is
collinear with BD (A). If, however, a centrifugal force acts on the ball, then because the ball–
inner and ball–outer raceway contact angles are dissimilar, the line of action between the
raceway groove curvature centers is not collinear with BD. Rather, it is discontinuous as
indicated in Figure 3.10. It is assumed in Figure 3.10 that the outer raceway groove curvature
center is fixed in space, and the inner raceway groove curvature center moves relative to that
fixed center. Moreover, the ball center shifts by virtue of the dissimilar contact angles.
The distance between the fixed outer raceway groove curvature center and the final
position of the ball center at any ball azimuth location j is
Doj ¼ ro �D
2þ doj ð3:69Þ
Since
ro ¼ foD
Doj ¼ ð fo � 0:5ÞDþ dojð3:70Þ
Similarly,
Dij ¼ ðfi � 0:5ÞDþ dij ð3:71Þ
Final position,inner raceway groove
curvature center
A1j
Initial position,inner raceway groovecurvature center
Ball center, final position
Outer raceway groove,curvature center fixed
Ball center, initial position
X1j
X2j
d r cos y j
(f i − 0.5) D + d ij
(fo −
0.5
) D +
doj
d a + Θ i cos yj
a ij
a oj
a o
A2j
BD
FIGURE 3.10 Positions of ball center and raceway groove curvature centers at angular position cj with
and without applied load.
� 2006 by Taylor & Francis Group, LLC.
wher e do j and dij are the normal co ntact deform ations at the outer and inner racew ay contact s,
respect ively.
In accord ance with the relative axial displ acement of the inner and outer rings da an d the
relative angular displ acement s u, the axial distan ce betw een the loci of inn er and outer
racew ay gro ove curvatu re center s at any ball pos ition is
A1j ¼ BD sin a� þ da þ u< i cos cj ð3: 72 Þ
wher e <i is the radius of the locus of inner racew ay gro ove curvatu re cen ters and a8 is the
initial con tact angle be fore loading . Further, in accordan ce with a relat ive radial displacement
of the ring centers dr , the radial displacemen t between the loci of the groove curvat ure cen ters
at each ball locat ion is
A2j ¼ BD cos a� þ dr cos c j ð3: 73 Þ
Thes e da ta are intende d as an exp lanation of Figure 3 .10.
Jones [1] found it conven ient to intr oduce new variables X1 and X 2, as shown in Figure
3.10. It can be seen from Figure 3.10 that a t any ball locat ion
cos aoj ¼X2 j
ðfo � 0:5Þ D þ doj
ð3: 74 Þ
sin aoj ¼X1j
ð fo � 0: 5Þ D þ doj
ð3: 75 Þ
cos aij ¼A2 j � X2 j
ð fi � 0: 5Þ D þ dij
ð3: 76 Þ
sin ai j ¼A1j � X 1j
ð fi � 0:5Þ D þ dij
ð3: 77 Þ
Usi ng the Pytha gorean Theorem, it can be seen from Figure 3.10 that
ð A1j � X 1 j Þ 2 þ ðA2 j � X2 j Þ 2 � ½ðf i � 0:5Þ D þ dij � 2 ¼ 0 ð3: 78 Þ
X 21j þ X 22j � ½ðf o � 0: 5Þ D þ do j �2 ¼ 0 ð3: 79 Þ
Consi dering the plane pa ssing throu gh the bearing ax is and the center of a ba ll locat ed at
azim uth cj (see Figu re 3.9) , the load diagram in Figure 3.11 obtains if nonc oplanar fricti on
forces are insign ificant. Ass uming that ‘‘outer racew ay control ’’ is approxim ated at a given
ball locat ion, it can also be assum ed with littl e effe ct on calculati onal accu racy that the ba ll
gyroscop ic moment is resisted en tirely by friction force at the ball–ou ter raceway c ontact.
Othe rwise, it is safe to assum e that the ba ll gyroscopi c moment is resi sted eq ually at the ba ll–
inner and ball–outer raceway contacts. In Figure 3.11, therefore, lij¼ 0 and loj¼ 2 for outer
raceway control; otherwise, lij¼ loj¼ 1.
The normal ball loads are related to normal contact deformations as follows:
Qoj ¼ Kojd1:5oj ð3:80Þ
Qij ¼ Kijd1:5ij ð3:81Þ
� 2006 by Taylor & Francis Group, LLC.
Qoj
Fcj
M gj
a ij
a oj
Q ij
lojMgj
D
l ijM gj
D
FIGURE 3.11 Ball loading at angular position cj.
From Figure 3.11, consider ing the equilibrium of forces in the horizon tal and vertical
direction s,
Qij sin aij � Q oj sin a oj �Mg j
Dðli j cos a ij � loj cos aoj Þ ¼ 0 ð 3: 82 Þ
Qij cos ai j � Q oj cos a oj �Mg j
Dðli j sin ai j � loj sin a oj Þ þ Fc j ¼ 0 ð 3: 83 Þ
Substi tuting Equat ion 3.80, Equat ion 3.81, and Equat ion 3.74 through Equation 3.77 into
Equation 3.82 and Equation 3.83 yields
lojMgjX2j
D� Kojd
1:5oj X1j
ðfo � 0:5ÞDþ doj
þKijd
1:51j ðA1j � X1jÞ � lijMgj
DðA2j � X2jÞ
ðfi � 0:5ÞDþ dij
¼ 0 ð3:84Þ
Kojd1:5oj X2j þ lojMgjX1j
D
ðfo � 0:5ÞDþ doj
�Kijd
1:5ij ðA2j � X2jÞ þ lijMgj
DðA1j � X1jÞ
ðfi � 0:5ÞDþ dij
� Fcj ¼ 0 ð3:85Þ
Equation 3.78, Equat ion 3.79, Equation 3.84, and Equation 3.85 may be solved simu ltan-
eously for X1j, X2j, dij, and doj at each ball angular location once values for da, dr, and u are
assumed. The most probable method of solution is the Newton–Raphson method for solution
of simultaneous nonlinear equations.
The centrifugal force acting on a ball is calculated as follows:
Fc ¼ 12mdmv2
m ð3:20Þ
where vm is the orbital speed of the ball. Substituting the identity vm2 ¼ (vm=v)2v2 in
Equation 3.20, the follo wing equati on for centri fugal force is obtaine d:
Fcj ¼1
2mdmv2 vm
v
� �2
jð3:86Þ
� 2006 by Taylor & Francis Group, LLC.
wher e v is the speed of the rotating ring an d vm is the orbital speed of the ba ll at angular
posit ion cj . It shou ld be apparent that because orb ital speed is a function of contact an gle, it is
not constant for each ball locat ion.
Moreover, it must be ke pt in mind that this analysis does not co nsider friction al forces
that tend to retar d ball and he nce cage moti on. Ther efore, in a high -speed bearing , it is to be
expecte d that vm will be less than that predict ed by Equat ion 2.47 and greater than that
predict ed by Equation 2.48. Unles s the loading on the bearing is relative ly light, howeve r, the
cage speed diffe rential is usuall y insi gnificant in affectin g the accu racy of the calculati ons
ensuing in this chapter .
Gyros copic mo ment at each ball locat ion may be descri bed as foll ows:
Mg j ¼ JvR
v
� �j
vm
v
� �jv2 sin b ð3:87Þ
where b is given by Equation 2.43, vR=v by Equation 2.44, and vm=v by Equation 2.47 or
Equation 2.48.
Since Ko j , Ki j , and M gj are functions of a contact angle, Equat ion 3.74 through Equat ion
3.77 may be used to establis h these values during the iteration .
To find the values of dr, da, and u, it remains only to establish the conditions of
equilibrium applying to the entire bearing. These are
Fa �Xj¼Z
j¼1
Qij sin aij �lijMgj
Dcos aij
� �¼ 0 ð3:88Þ
or
Fa �Xj¼Z
j¼1
KijðA1j � X1jÞd1:5ij �
lijMgj
DðA2j � X2jÞ
ðfi � 0:5ÞDþ dij
" #¼ 0 ð3:89Þ
Fr �Xj¼Z
j¼1
Qij cos aij þlijMgj
Dsin aij
� �cos cj ¼ 0 ð3:90Þ
or
Fr �Xj¼Z
j¼1
KijðA2j � X2jÞd1:5ij �
lijMgj
DðA1j � X1jÞ
ðfi � 0:5ÞDþ dij
!¼ 0 ð3:91Þ
M�Xj¼Z
j¼1
Qij sin aij �lijMgj
Dcos aij
� �<i þ
lijMgj
Dri
� �cos cj ¼ 0 ð3:92Þ
or
M�Xj¼Z
j¼1
KijðA1j � X1jÞd1:5ij �
lijMgj
DðA2j � X2jÞ
� �<i
ðfi � 0:5ÞDþ dij
þ lij fiMgj
24
35 cos cj ¼ 0 ð3:93Þ
� 2006 by Taylor & Francis Group, LLC.
<i ¼ 12 dm þ ðf i � 0:5Þ D cos a� ð 1: 3Þ
Havi ng calcul ated values of X1j , X 2j , di j , and do j at each ball position, and knowi ng F a, Fr , and
M as inpu t co nditions, the values da, dr , and u may be determined by Equation 3.89, Equat ion
3.91, and Equation 3 .93. After obtainin g the prim ary unknow n qua ntities da, dr , and u, it is
then necessa ry to repeat the calcul ation of X1j , X 2j , di j , do j , and so on, until compat ible values
of the prim ary unkno wn quan tities da, dr , and u are obtaine d.
Solution of the syst em of sim ultaneo us equati ons, Equat ion 3.78, Equation 3.79, Equa-
tion 3.84, Equation 3.85, and Equation 3.89, requ ires the use of a digital comp uter. To
illustr ate the results of such a calculati on, the perfor mance of a 218 angular -conta ct ball
bearing (40 8 free contact an gle), was evaluated over a n applied thrust load range 0–44,4 50 N
(0–10,00 0 lb) and sha ft speed range 3,000–15 ,000 rpm. Figure 3.12 throu gh Figure 3.14 show
the results of the calculati ons.
070
60
50
40
30
20
10
00 2,000 4,000 6,000
Thrust load, lb8,000 10,000
10 20
Static
N � 103
30 40
Con
tact
ang
le α
i and
αo,
deg
rees
Inner raceway a i15,000 rpm
10,000 rpm
3,000 rpm
2,000 rpm
6,00
0 rp
m
10,0
00 rp
m
15,000 rpm
Outer raceway ao
6,000 rpm
FIGURE 3.12 Ball–inner raceway and ball–outer raceway contact angles ai and ao for a 218 angular-
contact ball bearing (free contact angle ao¼ 408).
� 2006 by Taylor & Francis Group, LLC.
N � 103
0 10 20 30 40
4,000
3,000
N
2,000
1,000
0
1,000
900
800
700
600
500
400
300
200
100
00 2,000 4,000 6,000
Applied thrust load, lb8,000 10,000
Bal
l nor
mal
load
Qi
and
Qo,
lbStatic loading
Qo − 6000 rpm
Qo − 10,000 rpm
Qo − 15,000 rpm
Q i − 15,000 rpm
Q i − 10,000 rpm
Q i − 6,000 rpm
FIGURE 3.13 Ball–inner raceway and ball–outer raceway contact normal loads Qi and Qo for a 218
angular-contact ball bearing (free contact angle ao¼ 408).
+0.05
−0.05
−0.10
−0.15
−0.20
0
mm
0+0.003
+0.002
+0.001
−0.001
−0.002
−0.003
−0.004
−0.005
−0.006
−0.007
−0.008
−0.0090 2,000 4,000 6,000
Applied thrust load, lb
8,000 10,000
0
10 20 30N � 103
40
da,
Axi
al d
efle
ctio
n, in
.
Static
3,000 rpm
6,00
0 rp
m10
,000
rpm
15,0
00 rp
m
FIGURE 3.14 Axial deflection da for a 218 angular-contact ball bearing (free contact angle ao¼ 408).
� 2006 by Taylor & Francis Group, LLC.
3.3.1 B ALL EXCURSIONS
For an angular -conta ct ball bea ring subject ed onl y to thrust loading , the orbital travel of the
balls occu rs in a single radial plane, whose a xial location is defined by X1j in Figure 3.10; X 1j is
the same at all ball a zimuth angles ci . For a bearing that suppo rts combined rad ial and ax ial
loads, or combined radial, axial , and moment load s, X1j is different at each ball azim uth angle
ci . Therefor e, a ball undergoe s an axial movem ent or ‘‘excu rsion’’ as it orb its the shaft or
housing center . Unless this ex cursion is accomm odated by pro viding suffici ent axial clear ance
between the ball and the cage poc ket, the cage wi ll experi ence nonuni form an d possibl y heavy
loading in the axial direct ion. This can also cause a co mplex motion of the cage, that is, no
longer simple rotation in a singl e plan e, rather includi ng an out-of -plane vibrat ion co mpon-
ent. Suc h a moti on toget her with the aforementi oned load ing ca n lead to the rapid destr uction
and seizur e of the bearing .
Under co mbined load ing, because of the varia tion in the ball– raceway con tact angles ai j
and aoj as the ball orbits the bearing axis, there is a tendency for the ball to lead (advance ah ead
of) or lag (fall behind) its centra l posit ion in the cage poc ket. The orbit al or circum ferential
trave l of the ball relat ive to the cage is, howeve r, limit ed by the cage pocket. Ther efore, a load
occurs be tween the ba ll an d the cage poc ket in the circum ferential direct ion. Under steady -state
cage rotation, the sum of these ball-cage pock et loads in the circumfer entia l direct ion is close to
zero, ba lanced only by fricti on forces. Moreover, the forces and moment s acti ng on a ba ll in the
bearing ’s plane of rotat ion mu st be in balance, includi ng accele ration or de celeration loading
and fricti on forces . To achieve this conditio n of equ ilibrium , the ball speeds, includin g orbital
speed, will be different from those calculated con sidering only kinema tic conditio ns, or even
those indica ted in Chapt er 2 , assum ing the con dition of no gyroscopi c motio n. This is a
cond ition of skiddi ng, and it will be covered in Chapt er 5.
3.3.2 L IGHTWEIGHT B ALLS
To permi t ball bearing s to operate at higher speeds, it is possibl e to reduce the adverse ball
inertial effects by redu cing the ball mass . This is esp ecially effecti ve for angular -contact ball
bearing s as the diff erential be tween the ball– inner racew ay and ball–ou ter raceway contact
angles , ai j �ao j , will be reduced. To achieve this resul t, it was first atte mpted to operate
bearing s with co mplements of hollow balls [2]; howeve r, this prove d impr actical because it
was difficul t to manu facture hollow balls that have isot ropic mechan ical propert ies. In the
1980s, hot isostaticall y presse d (HIP) sil icon nitride ceram ic was develope d as an accepta ble
mate rial for the man ufacture of rolling e lements. Bearings with balls of HIP silicon nitride,
which has a density approximately 42% that of steel and an excellent compressive strength,
are used in high-speed, machine tool spindle applic ations and are under consider ation for use
as aircraft gas turbi ne engine applic ation mains haft bearing s. Figure 3.15 through Figure 3.17
compare the bearing performan ce pa rameters for ope rations at high speed of the 218 angu lar-
contact ball bearing with steel ba lls and HIP silicon nitride balls.
Silicon nitride also has a modulus of elasticity of approxim ately 3.1 � 10 5 MPa
(45 � 10 6 psi). In a hyb rid ball bearing , that is, a bearing wi th steel rings and silicon nitr ide
balls, owin g to the higher elastic modulus of the ball material, the contact areas between balls
and raceways will be smaller than in an all-steel bearing. This causes the contact stresses to be
greater. Depending on the load magnitude, the stress level may be acceptable to the ball
material, but not to the raceway steel. This situation can be ameliorated at the expense of
increased contact friction by increasing the conformity of the raceways to the balls; for
example, decreasing the raceway groove curvature radii. This amount of decrease is specific
to each application, dependent on bearing applied loading and speed.
� 2006 by Taylor & Francis Group, LLC.
Outer raceway–steel ballsOuter raceway–silicon nitride ballsInner raceway–steel ballsInner raceway–silicon nitride balls
1,200
1,000
800
600
400
200
00 2,000 4,000 6,000
Applied thrust load, lb
8,000 10,000 12,000
Bal
l loa
d Q
0 an
d Q
i, lb
FIGURE 3.15 Outer and inner raceway–ball loads vs. bearing applied thrust load for a 218 angular-
contact ball bearing operating at 15,000 rpm with steel or silicon nitride balls.
3.4 HIGH-SPEED RADIAL CYLINDRICAL ROLLER BEARINGS
Because of the high rate of heat generation accompanying relatively high friction torque,
tapered roller and spherical roller bearings have not historically been employed for high-speed
applications. Generally, cylindrical roller bearings have been used; however, improvements in
70
60
50
40
30
20
10
00 2,000
Con
tact
ang
les a
o an
d a
i, de
gree
s
4,000
Outer raceway–steel ballsOuter raceway–silicon nitride ballsInner raceway–steel ballsInner raceway–silicon nitride balls
6,000
Applied thrust load, lb
8,000 10,000 12,000
FIGURE 3.16 Outer and inner raceway–ball contact angle vs. bearing applied thrust load for a 218
angular-contact ball bearing operating at 15,000 rpm with steel or silicon nitride balls.
� 2006 by Taylor & Francis Group, LLC.
Steel balls
0
0.003
0.002
0.001
0.000
−0.001
−0.002
−0.003
−0.004
−0.005
−0.0062,000 4,000 6,000
Applied thrust load, lb
Bea
ring
axia
l def
lect
ion,
in.
8,000 10,000 12,000
Silicon nitiride balls
FIGURE 3.17 Axial deflection vs. bearing applied thrust load for a 218 angular-contact ball bearing
operating at 15,000 rpm with steel or silicon nitride balls.
bearing internal design, accuracy of manufacture, and methods of removing generated heat
via circulating oil lubrication have gradually increased the allowable operating speeds for
both tapered roller and spherical roller bearings. The simplest case for analytical investigation
is still a radially loaded cylindrical roller bearing and this will be considered in the following
discussion.
Figure 3.18 indicates the forces acting on a roller of a high-speed cylindrical roller bearing
subjected to a radial load Fr. Thus, considering equilibrium of forces,
Q oj
Q ij
F c
FIGURE 3.18 Roller loading at angular position cj.
� 2006 by Taylor & Francis Group, LLC.
Qoj �Qij � Fc ¼ 0 ð3:94Þ
Rearranging Equation 1.33 yields
Q ¼ Kd10=9 ð3:95Þ
where
K ¼ 8:05� 104 l8=9 ð3:96Þ
Therefore,
Kd10=9oj � Kd
10=9ij � Fc ¼ 0 ð3:97Þ
Since
drj ¼ dij þ doj ð3:98Þ
Equation 3.97 may be rewritten as follows:
drj � dij
� 10=9� d10=9ij � Fc
K¼ 0 ð3:99Þ
Equilibrium of forces in the direction of applied radial load on the bearing dictates that
Fr �Xj¼Z
j¼1
Qij cos cj ¼ 0 ð3:100Þ
or
Fr
K�Xj¼Z
j¼1
d10=9ij cos cj ¼ 0 ð3:101Þ
From the geometry of the loaded bearing, it can be determined that the total radial compres-
sion at any roller azimuth location cj is
drj ¼ dr cos cj �Pd
2ð3:102Þ
Substitution of Equation 3.102 into Equation 3.99 yields
dr cos cj �Pd
2� dij
� �10=9
�d10=9ij � Fc
K¼ 0 ð3:103Þ
Equation 3.101 and Equations 3.103 represent a system of simultaneous nonlinear equations
with unknowns dr and dij. These equations may be solved for dr and dij using the Newton–
Raphson method. Having calculated dr and dij, it is possible to calculate roller loads
as follows:
� 2006 by Taylor & Francis Group, LLC.
Qi j ¼ K d10= 9ij ð 3: 104 Þ
Qoj ¼ K d10= 9ij þ Fc ð 3: 105 Þ
Roller centrifugal force can be calcul ated using Equat ion 3.38.
These equations apply to roller bearings with line or modified line contact. Fully crowned
rollers or crowned raceways may cause point contact, in which case Ki is different from Ko. These
values may be determined using Equation 3.106 also given in Chapter 7 of the first volume of this
handbook.
Kp ¼ 2:15 � 10 5 Srð Þ� 1 = 2 d�ð Þ� 3 =2 ð 3: 106 Þ
Information on high-speed roller bearings that have flexibly supported rings is given by
Harris [3].
See Exampl e 3.5.
Figure 3.19 illustrates the resul ts of the analys is for a 209 cyli ndrical roll er bearing with
zero mounted radial clearance and subject ed to applie d radial load. Figure 3.20 shows the
varia tion of bearing de flection dr with speed.
3.4.1 HOLLOW R OLLERS
Rollers can be made hollow to red uce roller centri fugal forces . Hollow roll ers are flex ible an d
great care must be exercised to assure that accuracy of shaft locat ion under the applied load is
maintained . Roller centrifugal force as a function of hollown ess ratio Di =D is g iven by
Fc ¼ 3: 39 � 10 � 11 D 2 ld m n2m ð 1 � H 2 Þ ð3: 107 Þ
Figure 3.21 taken from Ref . [4] shows the effe ct of roll er hol lowness in a high-sp eed
cylin drical roller bearing on bearing radial deflection.
For the same bearing , Figu re 3.22 illustrates the inter nal load dist ribution .
An added criteri on for evaluat ion in a bearing with hollow roll ers is the roll er ben ding
stress. Figure 3.23 sho ws the effect of roll er hollow ness on maxi mum roll er bending stress.
Practi cal limit s for roller hollown ess are indicated.
Great ca re must be given to the smoot h finishing of the insi de surface of a hollow ro ller
during man ufacturing as the stress rais ers that oc cur due to a poorly fini shed insi de surfa ce
will redu ce the allowabl e roller hollown ess ratio s still furt her than indicated in Figure 3.22.
Lightwei ght roll ers made from a ceram ic material such as silicon nitride appear feasi ble to
reduce roller centri fugal forces .
3.5 HIGH-SPEED TAPERED AND SPHERICAL ROLLER BEARINGS
Usin g digit al compu tation and methods simila r to those indica ted in Chapt er 1, the load
distribution in other types of high-speed roller bearings can be analyzed. Harris [5] indicates
all of the nece ssary equati ons. The forces actin g on a general ized roller are sho wn in Figure
3.24. In this case, roll er gy roscopi c momen t is given by
� 2006 by Taylor & Francis Group, LLC.
30015,000 rpm
1,000 rpm
10,000 rpm
15,000 rpm
10,000 rpm
1,000 rpm
1,000
500
N
0
200
100
Rol
ler
load
at i
nner
rac
eway
, lb
00 10 20 30 40 50 60
Roller location, degrees
70 80 90
FIGURE 3.19 Distribution of load among the rollers of a 209 cylindrical roller bearing with Pd¼ 0;
Fr¼ 4450 N (1000 lb); and operating at 1,000, 10,000, and 15,000 rpm shaft speed.
Mg j ¼ J vm j vR j sin 12 ðai þ ao Þ �
ð3: 108 Þ
3.6 FIVE DEGREES OF FREEDOM IN LOADING
Unti l this point, all load dist ribution calcul ation methods ha ve been limit ed to, at most, three
degrees of freedom in load ing. This has be en done in the interest of simplifying the analyt ical
methods and the unde rstand ing thereof . Every rolling bearing applie d load situatio n can be
analyze d using a system with five de grees of freedom , c onsider ing only the app lied loading .
Then every specia lize d applie d loading co ndition , for examp le, sim ple radial load, can be
analyze d using this more co mplex system. Reference [5] shows the followi ng illustr ations that
app ly to an analyt ical syst em for a ball bearing wi th five degrees of freedom in app lied loading
(see Fi gure 3.25) .
� 2006 by Taylor & Francis Group, LLC.
3.9
0.0096
0.0092
0.0088
mm
0.0084
0.0080
3.8
3.7
3.6
3.5
3.4
3.3
3.2
3.1
3.00 5,000 10,000
Shaft speed, rpm
Rad
ial d
efle
ctio
n, in
. � 0
.000
1
15,000 20,000
FIGURE 3.20 Radial deflection vs. speed for a 209 cylindrical roller bearing with Pd¼ 0 and Fr¼ 4450 N
(1000 lb).
Note the numeri cal not ation of applie d loads, that is, F1, . . . , F5, in lieu of Fa, Fr , and M.
Figure 3.26 shows the con tact angles , de formati ons, an d displac ements for the ball–racew ay
contact s at azimu th cj . Figure 3.27 shows the ball speed vector s and inertial loading for a ba ll
with its center at azim uth cj . Note the numeri cal notatio ns for raceway s; 1 ¼ o and 2 ¼ i. This
is done for ease of digit al program ming.
0 0.2 0.4 0.6 0.8
Hollowness, %
10–4
10–3
10–2
Dimensions of sample roller bearing
Z 21l c 15 mm (0.59 in.)
D 14 mm (0.55 in.)
d m 114.3 mm (4.5 in.)
p d 0.0064 mm (0.00025 in.)
d 1, M
axim
um, d
efle
ctio
n, in
.
W = 57,850 N (13,000 lb)
W = 57,850 N (13,000 lb)
W = 22,250 N (5,000 lb)
W = 22,250 N (5,000 Ib)
N = 15,000 rpm
N = 5,000 rpm
N = 15,000 rpm
N = 5,000 rpm
FIGURE 3.21 Maximum deflection vs. hollowness.
� 2006 by Taylor & Francis Group, LLC.
1,000
2,200
7 6 5 4 3 2 1 2 3 4 5 6 7 7 6 5 4 3 2 1 2 3 4 5 6 7
200
600
1,000
1,400
1,800
2,600
600
200
1,400
1,800
2,200
2,800
3,000
Roller position Roller position
Rol
ler
load
(lb
)
Rol
ler
load
, lb
Solid rollers20% Hollow80% Hollow
Solid rollers
Solid rollers
20% Hollow
20% Hollow
80% Hollow
80% Hollow
W = 57,850 N (13,000 Ib)N = 5,000 rpm
Solid rollers20% Hollow
W = 22,250 N (5,000 Ib)N = 15,000 rpm
80% Hollow
W = 57,850 NN = 15,000 rpm
W = 22,250 NN = 15,000 rpm
1210 10
8
6
42 2
46
8W
FIGURE 3.22 Roller load distribution vs. applied load, shaft speed, and hollowness.
3.7 CLOSURE
As demonstrated in the earlier discussion, analysis of the performance of high-speed roller
bearings is complex and requires a computer to obtain numerical results. The complexity can
0 0.2 0.4 0.6 0.8
Hollowness, %
W = 57,850 N (13,000 Ib)N = 15,000 rpm
W = 57,850 NN = 5,000 rpm
W = 22,250 N (5,000 Ib)N = 15,000 rpm
W = 22,250 NN = 5,000 rpm
Recommended endurance
limit SAE 8620
(689.8)
(6,898 N/mm2)106
105
104
Max
imum
ben
ding
str
ess,
psi
FIGURE 3.23 Maximum bending stress vs. hollowness.
� 2006 by Taylor & Francis Group, LLC.
(a1 + a2)
Roller
axis of
rotationM gj
y
r f
a f
a o
Q fj
Q ij
Q oj
a i
Iz
F cj
x Bearing axisof rotation
12
e2
FIGURE 3.24 Roller forces and geometry.
become even great er for ball bearing s. In this chapter as wel l as Chapt er 1 an d Chapt er 2, for
simplicity of explanation, most illustrations are confined to situations involving symmetry of
loading about an axis in the radial plane of the bearing and passing through the bearing axis
of rotation. The more general and complex applied loading system with five degrees of
freedom is, however, discussed.
The effect of lubrication has also been neglected in this discussion. For ball bearings, it has
been assumed that gyroscopic pivotal motion is minimal and can be neglected. This, of
Z
J = 1
J = z
J = 2
J = 3
Y
X
F4 (XZ-plane)
z
z
y
yx
x
12dm
F2
F3
F1
F5 (XY-plane)y2
y3
y4
FIGURE 3.25 Bearing operating in YZ plane.
� 2006 by Taylor & Francis Group, LLC.
Inner raceway groove curvaturecenter—operating location
Outer raceway groove curvature center
Ball center—operating location
BD
(f 2 − 0.5) D + δ 2j
(f 1 −
0.5
) D
+ δ 1j
A1j
A2j
α2j
Δ2 sin yj + Δ3 cos yj
Δ1 + f (Δ4 sin yj + Δ5 cos yj)
α1j
α8
X2j
X1j
Inner raceway groove curvaturecenter—initial location
Ball center—initial location
x-axis
z-axis
FIGURE 3.26 Contact angle, deformation, and displacement geometry.
M gzj
F cj
M gyj
Q1j
Q2j
a1j
a2j
w zj
w yj
w xj
wj b'jbj
wmj
Z
X
x
z
Y
Distribution of internal loading in high speed bearings
FIGURE 3.27 Ball speeds and inertial loading.
� 2006 by Taylor & Francis Group, LLC.
course, depends on the friction forces in the contact zones, which are affected to a great extent
by lubrication. Bearing skidding is also a function of lubrication at high speeds of operation.
If the bearing skids, centrifugal forces will be lower in magnitude and the performance will
accordingly be different.
Notwithstanding the preceding conditions, the analytical methods presented in this
chapter are extremely useful in establishing optimum bearing designs for given high-speed
applications.
REFERENCES
1.
� 2
Jones, A., General theory for elastically constrained ball and roller bearings under arbitrary load and
speed conditions, ASME Trans., J. Basic Eng., 82, 1960.
2.
Harris, T., On the effectiveness of hollow balls in high-speed thrust bearings, ASLE Trans., 11,209–214, 1968.
3.
Harris, T., Optimizing the fatigue life of flexibly mounted rolling bearings, Lub. Eng., 420–428,October 1965.
4.
Harris, T. and Aaronson, S., An analytical investigation of cylindrical roller bearings having annularrollers, ASLE Preprint No. 66LC-26, October 18, 1966.
5.
Harris, T. and Mindel, M., Rolling element bearing dynamics, Wear, 23(3), 311–337, February 1973.006 by Taylor & Francis Group, LLC.
4 Lubricant Films in RollingElement–Raceway Contacts
� 2006 by Taylor & Fran
LIST OF SYMBOLS
Symbol Description Units
a Semimajor axis of elliptical contact area mm (in.)
a Thermal expansivity 8C�1
A Viscosity–temperature calculation constants
b Semiwidth of rectangular contact area, semiminor axis of
elliptical contact area mm (in.)
B Doolittle parameter
C Lubrication regime and film thickness calculation constants
D Roller or ball diameter mm (in.)
dm Pitch diameter of bearing mm (in.)
E Modulus of elasticity MPa (psi)
E’ E/(1� j2) MPa (psi)
F Force N (lb)
Fa Centrifugal force N (lb)�FF F/E’<g Gravitational constant mm/sec2 (in./sec2)
G lE’G Shear modulus MPa (psi)
h Lubricant film thickness mm (in.)
h0 Minimum lubricant film thickness mm (in.)
H h/<I Viscous stress integral
J Polar moment of inertia per unit length N � sec2 (lb � sec2)
J� J/E’<i mm � sec2 (in � sec2)
kb Lubricant thermal conductivity W/m � 8C
cis Group, LLC.
(Btu/hr � in. � 8F)
K0 Bulk modulus parameter Pa � 8K
K1 Bulk modulus parameter Pa
l Roller effective length mm (in.)
L Factor for calculating film thickness reduction due to
thermal effects
M Moment N � mm (in. � lb)
n Speed rpm
p Pressure MPa (psi)
Q Force acting on roller or ball N (lb)
Q� Q/E’<
r relative occupied volume
expansion factor
R relative occupied volume m3
Ro relative occupied volume at 208C m3
R Cylinder radius mm (in.)
< Equivalent radius mm (in.)
s rms surface finish (height) mm (in.)
SSU Saybolt university viscosity sec
t Time sec
T Lubricant temperature 8C,8K (8F, 8R)
u Fluid velocity mm/sec (in./sec)
U Entrainment velocity (U1�U1) mm/sec (in./sec)
U h0U/2E’<v Fluid velocity, displacement in y direction mm/sec, mm (in./
� 2006 by Taylor & Francis Group, LLC.
sec, in.)
V Volume mm3
Vo Volume at 208C mm3
V Sliding velocity (U1�U1) mm/sec (in./sec)
V h0V/E’<w Deformation in z direction mm (in.)
y Distance in y direction mm (in.)
z Distance in z direction mm (in.)
b’ Coefficient for calculating viscosity as a function of temperature
g (D cos a)/dm
_�� Lubricant shear rate sec�1
« Strain mm/mm (in./in.)
e occupied volume expansivity 8C�1
h Lubricant viscosity cp (lb � sec/in.2)
hb Base oil viscosity (grease) cp (lb � sec/in.2)
heff Effective viscosity (grease) cp (lb � sec/in.2)
h0 Fluid viscosity at atmospheric pressure cp (lb � sec/in.2)
k Ellipticity ratio a/b
l Pressure coefficient of viscosity mm2/N (in.2/lb)
L Lubricant film parameter
vb Kinematic viscosity stokes (cm2/sec)
j Poisson’s ratio
r Weight density g/mm3/(lb/in.3)
s Normal stress MPa (psi)
t Shear stress MPa (psi)
u Angle rad�YY Factor to calculate wTS
w Film thickness reduction factor
F Factor to calculate wS
c Angular location of roller rad
v Rotational speed rad/sec
Subscripts
b Entrance to contact zone
e Exit from contact zone
G Grease
i Inner raceway film
j Roller location
m Orbital motion
NN Non-Newtonian lubricant
o Outer raceway film
R Roller
S Lubricant starvation
SF Surface roughness (finish)
T Temperature
TS Temperature and lubricant starvation
x x Direction, that is, transverse to rolling
y y Direction, that is, direction of rolling
z z Direction
m Rotating raceway
v Nonrotating raceway
0 Minimum lubricant film
1, 2 Contacting bodies
4.1 GENERAL
Ball and roller bearings require fluid lubrication if they are to perform satisfactorily for long
periods of time. Although modern rolling bearings in extreme temperature, pressure, and
vacuum environment aerospace applications have been adequately protected by dry film
lubricants, such as molybdenum disulfide among many others, these bearings have not been
subjected to severe demands regarding heavy load and longevity of operation without fatigue.
It is further recognized that in the absence of a high-temperature environment only a small
amount of lubricant is required for excellent performance. Thus, many rolling bearings can be
packed with greases containing only small amounts of oil and then be mechanically sealed to
retain the lubricant. Such rolling bearings usually perform their required functions for
indefinitely long periods of time. Bearings that are lubricated with excessive quantities of
oil or grease tend to overheat and burn up.
The mechanism of the lubrication of rolling elements operating in concentrated contact with
a raceway was not established mathematically until the late 1940s; it was not proven experimen-
tally until the early 1960s. This is to be compared with the existence of hydrodynamic lubrication
in journal bearings, which was established by Reynolds in the 1880s. It is known, for instance,
that a fluid film completely separates the bearing surface from the journal or slider surface in a
properly designed bearing.Moreover, the lubricant can be oil, water, gas, or someother fluid that
exhibits adequate viscous properties for the intended application. In rolling bearings, however, it
was only relatively recently established that fluid films could, in fact, separate rolling surfaces
subjected to extremely high pressures in the zones of contact. Today, the existence of lubricating
fluid films in rolling bearings is substantiated in many successful applications where these films
are effective in completely separating the rolling surfaces. In this chapter, methods will be
presented for the calculation of the thickness of lubricating films in rolling bearing applications.
4.2 HYDRODYNAMIC LUBRICATION
4.2.1 REYNOLDS EQUATION
Because it appeared possible that lubricant films of significant proportions do occur in the
contact zones between rolling elements and raceways under certain conditions of load and
speed, several investigators have examined the hydrodynamic action of lubricants on rolling
� 2006 by Taylor & Francis Group, LLC.
w
u = U
u = 0u = u(z)z
FIGURE 4.1 Cylinder rolling on a plane with lubricant between cylinder and plane.
bearings according to classical hydrodynamic theory. Martin [1] presented a solution for rigid
rolling cylinders as early as 1916. In 1959, Osterle [2] considered the hydrodynamic lubrica-
tion of a roller bearing assembly.
It is of interest at this stage to examine the mechanism of hydrodynamic lubrication at
least in two dimensions. Accordingly, consider an infinitely long roller rolling on an infinite
plane and lubricated by an incompressible isoviscous Newtonian fluid with viscosity h. For a
Newtonian fluid, the shear stress t at any point obeys the relationship
t ¼ h›u
›z ð4:1Þ
where @ u/ @ z is the local fluid velocity gradient in the z direction (see Figure 4.1). Because the
fluid is viscous, fluid inertia forces are small compared with the viscous fluid forces. Hence, a
particle of fluid is subjected only to fluid pressure and shear stresses as shown in Figure 4.2.
Noting the stresses of Figure 4.2 and recognizing that static equilibrium exists, the sum of
the forces in any direction must equal zero. Therefore,
XFy ¼ 0
p dz � pþ ›p
›y
� �dzþ t dy� t þ ›t
›z
� �dy ¼ 0
and
›p
›y¼ � ›t
›zð4:2Þ
p
U
dy
t
dz p + dy∂p∂y
t + dz∂t ∂z
FIGURE 4.2 Stresses on a fluid particle in a two-dimensional flow field.
� 2006 by Taylor & Francis Group, LLC.
Differentiating Equation 4.1 once with respect to z yields
›t
›z ¼ �h
›2 u
›z2 ð4:3Þ
Substituting Equation 4.3 into Equation 4.2, one obtains
›p
›y¼ h
›2u
›z2ð4:4Þ
Assuming for the moment that @p/@y is constant, Equation 4.4 may be integrated twice with
respect to z. This procedure gives the following expression for local fluid velocity u:
u ¼ 1
2h
›p
›yz2 þ c1zþ c2 ð4:5Þ
The velocity U may be ascribed to the fluid adjacent to the plane that translates relative to a
roller. At a point on the opposing surface, it is proper to assume that u¼ 0, that is, at z¼ 0,
u¼U and at z¼ h, u¼ 0. Substituting these boundary conditions into Equation 4.5, it can be
determined that
u ¼ 1
2h
›p
›yzðz� hÞ þU 1� z
h
� �ð4:6Þ
where h is the film thickness.
Considering the fluid velocities surrounding the fluid particle as shown in Figure 4.3, one
can apply the law of continuity of flow in steady state, that is, mass influx equals mass efflux.
Hence, as density is constant for an incompressible fluid
u dz� uþ ›u
›ydy
� �dzþ v dy� vþ ›v
›zdz
� �dy ¼ 0 ð4:7Þ
dy
u
dzu u + dy∂u∂y
u + dz∂u∂z
FIGURE 4.3 Velocities associated with a fluid particle in a two-dimensional flow field.
� 2006 by Taylor & Francis Group, LLC.
Therefore,
›u
›y¼ � ›v
›zð4:8Þ
Differentiating Equation 4.6 with respect to y and equating this to Equation 4.8 yields
›v
›z¼ � ›
›y
1
2h
›p
›yzðz� hÞ þU 1� z
h
� �� �ð4:9Þ
Integrating Equation 4.9 with respect to z gives
Z›v
›zdz ¼ �
Z h
0
dv ¼ 0 ¼Z h
0
›
›y
1
2h
›p
›yzðz� hÞ þU 1� z
h
� �� �dz ð4:10Þ
and
›
›yh2 ›p
›y
� �¼ 6hU
›h
›yð4:11Þ
Equation 4.11 is commonly called the Reynolds equation in two dimensions.
4.2.2 FILM THICKNESS
To solve the Reynolds equation, it is only necessary to evaluate film thickness as a function
of y, that is, h¼ h(y). For a cylindrical roller near a plane as shown in Figure 4.4, it can be
seen that
h ¼ h0 þ y2
2Rð4:12Þ
where h0 is the minimum lubricant film thickness. Substituting Equation 4.12 into Equation
4.11 gives
w
h0
h (y )
y
R
FIGURE 4.4 Film thickness h(y) in the contact between a roller and a plane.
� 2006 by Taylor & Francis Group, LLC.
›
›yh0 þ y2
2R
� �3›p
›y
" #¼ 6hUy
R ð4:13 Þ
Equation 4.13 varies only in y; hence,
d
dyh0 þ y2
2R
� �3d p
d y
" #¼ 6hUy
R ð4:14 Þ
4.2.3 LOAD SUPPORTED BY THE LUBRICANT FILM
Integration of Equation 4.14 yields pressure over the lubricant film as a function of distance y.
If both contact surfaces are considered portions of rotating cylinders, then
U ¼ U1 þ U2 ð4:15 Þ
where subscripts 1 and 2 refer to the respective cylinders. Moreover, an equivalent radius < is
defined as
< ¼ ðR�11 þ R�1
2 Þ�1 ð4:16 Þ
Note that for an outer raceway R�1 is negative. The load per unit axial length of contact
carried by the lubricant film is given by
q ¼Z
pð yÞ dy ð4:17 Þ
Considering hydrodynamic lubrication with a constant viscosity (isoviscous) fluid permits the
solution of these equations for relatively lightly loaded contacts such as those that occur in
fluid-lubricated journal bearings.
4.3 ISOTHERMAL ELASTOHYDRODYNAMIC LUBRICATION
4.3.1 VISCOSITY VARIATION WITH PRESSURE
The normal pressure between contacting rolling bodies in ball and roller bearings tends to be
of magnitude 700 MPa (100,000 psi) and higher. Figure 4.5 shows some experimental data on
viscosity variation with pressure for a few bearing lubricants. It is seen that, at a given
temperature, viscosity is an exponential function of pressure. Therefore, between the contact-
ing surfaces in a normal rolling bearing application, lubricant viscosity can be several orders
of magnitude higher than its value at atmospheric pressure.
In 1893, Barus [3] established an empirical equation for the variation of viscosity with
pressure, an isothermal relationship. The Barus equation is
h ¼ h0elp ð4:18Þ
In Equation 4.18, l, the pressure–viscosity coefficient, is a constant at a given temperature. In
1953, an ASME [4] study published viscosity vs. pressure curves for various fluid lubricants.
On the basis of the ASME data, it is apparent that the Barus equation is a crude approximation
� 2006 by Taylor & Francis Group, LLC.
1
101
102
103
Abs
olut
e vi
scos
ity, c
entip
oise
s
104
105
106
1070 200 400
N/mm2
600 800 1000
0 20 40 60 80
Pressure, psi � 1000
100 120 140 160
Siliconeat 73.9�C(165�F)
Mineral oilat 50�C(122�F)
Diesterat 54.4�C(130�F)
Mineral oilat 68.3�C(155�F)
Diesterat 73.3�C(164�F)
FIGURE 4.5 Pressure viscosity of lubricants (ASME data [5]).
because the pressure–viscosity coefficient decreases with both pressure and temperature for
most fluid lubricants. The lubricant film thickness obtained in a concentrated contact has been
established as a function of the viscosity of the lubricant entering the contact. Therefore, for the
purpose of determining the thickness of the lubricant film, the viscosity–pressure coefficient at
atmospheric pressure is utilized.
Roelands [5] later established an equation defining the viscosity–pressure relationship for
given fluids; however, including the influence of temperature on viscosity as well:
log10 hþ 1:2
log10 h0 þ 1:2 ¼ T0 þ 135
T þ 135
� �S0
1 þ p
2000
� �z
ð4:19 Þ
In Equation 4.19, pressure is expressed in kgf/cm2 and temperature in 8K; exponents S0 and z
are determined empirically for each lubricant. At high pressures, Equation 4.19 indicates
viscosities substantially lower than those produced using the Barus Equation 4.18.
Sorab and VanArsdale [6] developed an expression for viscosity vs. pressure and tem-
perature, which can be applied to several of the lubricants employed in the ASME [4] study:
lnh
h0
¼A1
p
p0
� 1
� �þ A2
T0
T� 1
� �þ A3
p
p0
� 1
� �2
þ A4
T0
T� 1
� �2
þ A5
p
p0
� 1
� �T0
T� 1
� � ð4:20Þ
� 2006 by Taylor & Francis Group, LLC.
In Equation 4.20, temperature is stated in 8K. Ref. [6] provides values of the coefficients Ai for
the various lubricants tested in Ref. [4]. As an example, the coefficients for the diester fluid
viscosity vs. pressure curve of Figure 4.5 are
A1 1.48 � 10�3
A2 11.78
A3 �7.7� 10 �8
A4 14.31
A5 2.17 � 10 �3
This fluid may be considered representative of an aircraft power transmission fluid lubricant.
Sorab and VanArsdale [6] demonstrate that Equation 4.20 is superior to the Roelands equation
in approximating the ASME viscosity–pressure–temperature data. Nevertheless each of the
approximations has only been demonstrated over the 0–1034 MPa (0–150 kpsi) pressure range
and 25–218 8C (77–425 8F) temperature range of the ASME data. Contact pressures and
temperatures in many ball and roller bearing applications are apt to exceed these ranges;
therefore, it becomes necessary to extrapolate these data substantially beyond the range of the
experimentation. This is not critical for the determination of lubricant film thicknesses.
In the estimation of bearing friction, however, lubricant viscosity at pressures higher than
1034 MPa and at temperatures greater than 2188C has a great influence on the magnitudes
of friction forces calculated and hence on the accuracy of the calculations.
Bair and Kottke [7], based on experimental studies of lubricants at high pressures (for
example, up to 2000 MPa), developed the following equation to describe absolute viscosity as
a function of pressure and temperature:
h ¼ h0exp BR0 r
V =V0 � R0r� R0
1 � R
� �� �ð4:21 Þ
where h0 is the viscosity at atmospheric pressure and 20 8C. Parameter R0, relative occupied
volume at 20 8C, and B according to Doolittle [8] are given in Table 4.1.
The occupied volume, assumed to vary linearly with temperature, is given by
r ¼ 1þ « T � T0ð Þ ð4:22Þ
where « is the occupied volume expansivity; it tends to be negative. The variation of volume
with pressure and temperature is determined from
V
V0
¼ 1 þ a T � T0ð Þ½ � 1� 1
1 þ K 00
ln 1þ p
K0
1 þ K 00� � � �
ð4:23Þ
TABLE 4.1Doolittle–Tait Parameters for T0 5 20˚C
Lubricant
h0
(Pa � sec)
a
(1/C � 1024)
«
(1/C �1023) R0 B K’0
K1(GPa)
K0
(GPa � 8K)
SAE 20 0.1089 8 �1.034 0.6980 3.520 10.40 �0.9282 580.7
PAO ISO 68 0.0819 8 �1.035 0.6622 3.966 11.38 �0.9881 580.8
Mil-L-23699 0.04667 7.42 �1.28 0.6641 3.382 10.741 �1.0149 570.8
� 2006 by Taylor & Francis Group, LLC.
where a is the thermal expansivity, K 00 is the assumed constant, and the bulk modulus varies
with temperature according to
K0 ¼ K 1 þK0
T ð4:24 Þ
where T is in 8K. Equation 4.21 through Equation 4.24 tend to give better predictions of
viscosity at elevated pressures than does Roelands [5]; however, they still tend to predict
viscosities higher than that experienced in ball and roller bearing applications.
Harris [9] introduced the use of a sigmoid curve as defined by Equation 4.25 to fit the
ASME [4] data.
h ¼ C1 þC2
1þ e� p�C3ð Þ=C4ð4:25Þ
In Equation 4.25, C1, . . . , C4 are constants determined from the curve-fitting procedure for a
given lubricant at a given temperature. Figure 4.6 illustrates the sigmoid curves for the ASME
data for a Mil-L-7808 ester-type lubricant at 37.8, 98.9, and 218.38C (100, 210, and 4258F).
The salient feature of the sigmoid viscosity vs. pressure curve is the virtually constant viscosity
value at extremely high pressures. As noted by Bair and Winer [10,11], the fluid in a high-
pressure, concentrated contact undergoes transformation to a glassy state; that is, the fluid
essentially becomes a solid during its time in the contact. It therefore appears reasonable to
assume that fluid viscosity becomes essentially constant with pressure during the fluid’s time
in the contact. To accurately predict bearing friction torque, this becomes an important
consideration for the use of a sigmoid curve to describe lubricant viscosity in the contact.
Conversely, using a sigmoid curve to approximate lubricant viscosity at atmospheric and low
pressures does not provide the accuracy of either the Roelands [5] or Sorab and VanArsdale
37.8�C (100�F)
1e+6
1e+5
1e+4
1e+3
Abs
olut
e vi
scos
ity, c
entip
oise
1e+2
1e+1
1e+00 1000 2000 3000
Pressure, MPa
4000 5000
98.9�C (210�F)218.3�C (425�F)
FIGURE 4.6 Viscosity vs. pressure and temperature for Mil-L-7808 ester-type lubricant (sigmoid curve
fit to ASME data [4]—extrapolation from 1000 to 4000 MPa). (ASME Research Committee on Lubri-
cation, Pressure–viscosity report—Vol. 11, ASME, 1953.)
� 2006 by Taylor & Francis Group, LLC.
[6] model. Either of these models may be used in the estimation of lubricant viscosity to
calculate lubricant film thickness.
4.3.2 DEFORMATION OF CONTACT SURFACES
Because of the fluid pressures present between contacting rolling bodies causing the increases
in viscosity noted in Figure 4.5, the rolling surfaces deform appreciably in proportion to the
thickness of a fluid film between the surfaces. The combination of the deformable surface
with the hydrodynamic lubricating action constitutes the ‘‘elastohydrodynamic’’ (EHD)
problem. The solution of this problem established the first feasible analytical means of
estimating the thickness of fluid films, the local pressures, and the tractive forces that occur
in rolling bearings.
Dowson and Higginson [12], for the model in Figure 4.7, used the following formulation
for film thickness at any point in the contact:
h ¼ h0 þ y2
2R1
þ y2
2R2
þ w1 þ w2 ð4:26Þ
Solid displacements w are calculated for a semi-infinite solid in a condition of plane strain. As
the width of the loaded zone is extremely small compared with the dimensions of the
contacting bodies, an approximation that w1¼w2 is valid. Hence, for the equivalent cylinder
radius,
< ¼ ðR�11 þ R�1
2 Þ�1 ð4:16Þ
and the film thickness is given by
h ¼ h0 þ y2
2Rþ w ð4:27Þ
R
U h
Q y
Q z
F o
U o
F h
h 0h
q
y
FIGURE 4.7 Forces and velocities pertaining to the equivalent roller.
� 2006 by Taylor & Francis Group, LLC.
To solve the plane strain problem, the following stress function was assumed:
F ¼ �Q
py tan�1 y
zð4:28Þ
Using this stress function, the stresses due to a narrow strip of pressure over the width ds in
the y direction are determined as follows:
sy ¼ �2y2zp ds
pðy2 þ z2Þ2ð4:29Þ
sz ¼ �2z3p ds
pðy2 þ z2Þ2ð4:30Þ
tyz ¼ �2yz2p ds
pðy2 þ z2Þ2ð4:31Þ
sy and sz are normal stresses and tyz is the shear stress. By Hooke’s law, the strains are given by
"y ¼ð1� j2Þsy
E� jð1þ jÞsz
Eð4:32Þ
"z ¼ð1� j2Þsz
E� jð1þ jÞsy
Eð4:33Þ
"yz ¼2ð1þ jÞtyz
E¼ tyz
Gð4:34Þ
where G is the shear modulus of elasticity and j is Poisson’s ratio. In plane strain,
"y ¼›v
›y, "z ¼
›w
›z, and "yz ¼
›v
›zþ ›w
›y
Using these relationships, and Equation 4.29 through Equation 4.34, it can be established
that at the surface, that is, at z¼ 0,
w ¼ � 2ð1� j2ÞpE
Z S2
S1
p ln ðy� SÞ dS þ constant ð4:35Þ
To solve for w, Dowson and Higginson [12] divided the pressure curve into segments and
represented the pressure thereunder by
p ¼ z1 þ z2S þ z3S2 ð4:36Þ
where z1, z2, and z3 are constants for that segment. Using p in this form, Equation 4.35 can be
integrated to obtain the surface deformation. This procedure, of course, is used for an
assumed pressure distribution.
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To obtain h0, the Reynolds equation is used in accordance with the pressure variation of
viscosity:
d
dyh2e�lp dp
dy
� �¼ 6h0U
dh
dyð4:37Þ
Performing the indicated differentiation and rearranging yields
h3e�lp d2p
dy2� l
dp
dy
� �2" #
þ dh
dy6uh0 þ 3h2e�lp dp
dy
� �¼ 0 ð4:38Þ
At the inlet and at the outlet of the contact,
d2p
dy2� l
dp
dy
� �2
¼ 0 ð4:39Þ
such that Equation 4.38 becomes
dh
dy6Uh0 þ 3h2e�lp dp
dy
� �¼ 0 ð4:40Þ
At the outlet end of the pressure curve, dh/dy¼ 0. This condition applies to the point of
minimum film thickness. At the inlet, Equation 4.40 is solved by
dp
dy¼ � 2h0e
lpU
h2ð4:41Þ
Thus, if viscosity and speed are known, the value of h for the point at which Equation 4.40 is
satisfied in the inlet region can be evaluated for a given pressure curve. Solving Equation 4.41
for hb (at inlet) gives
hb ¼ � 2h0elpU
ðdp=dyÞb
� �1=2
ð4:42Þ
Once hb has been determined, the entire film shape can be estimated by using the integrated
form of the Reynolds equation, that is,
dp
dy¼ �6h0e
lpU1
h2� he
h3
� �ð4:43Þ
Substitution of dp/dy from Equation 4.41 for the point at which h¼ hb determines that
he¼ 2hb/3. At other positions y, film thickness h may be determined from the following
cubic equation developed from Equation 4.43:
dp=dy
6h0elpU
h3 þ h� he ¼ 0 ð4:44Þ
At the point of maximum pressure, dp/dy¼ 0 and Equation 4.38 becomes
� 2006 by Taylor & Francis Group, LLC.
dh
dy ¼ � h3
6h0e lp U
d2 p
dy2 ð4:45 Þ
In cases of most interest, the pressure curve is predominantly Hertzian such that
p ¼ p0 1 � y
b
� �h i1=2ð4:46 Þ
where p0 is the maximum pressure and b is the semiwidth of the contact zone. Thus, at y ¼ 0,
p ¼ p0, Equation 4.45 becomes
dh
dy ¼ p0 h
3
3h0e lp0 Ub2
ð4:47 Þ
Consequently, if h is small (as it must be in a rolling bearing under load) and the viscosity is high
(as it will become because of high pressure), dh/dy is very small and the film is essentially of
uniform thickness. This result is shown by Dowson and Higginson [12], and also by Grubin [13].
4.3.3 PRESSURE AND STRESS DISTRIBUTION
In a later presentation Dowson and Higginson [12] and Grubin [13] indicated that dimen-
sionless film thickness H ¼ h/< could be expressed as follows:
H ¼ f �QQz ; �U ;U ;G�
ð4:48 Þ
where
�QQ ¼ Qz
lE 0< ð4:49 Þ
�UU ¼ h0 U
2E 0< ð4:50 Þ
G ¼ l E 0 ð4:51 Þ
E 0 ¼ E
1 � j2 ð4:52 Þ
In the expression for H and in Equation 4.47 and Equation 4.48, the equivalent radius in the
direction of rolling for a ball or roller bearing is given by
<m ¼D
21 � gð Þ ð4:53 Þ
In Equation 4.53, the upper sign refers to the inner raceway contact and the lower sign to the
outer raceway contact. The velocities with which fluid is swept into the rolling element–raceway
contacts are given by Equation 4.54 and Equation 4.55 for the inner and outer raceway
contacts, respectively.
Ui ¼dm
21� gð Þ v� vmð Þ þ gvR½ � ð4:54Þ
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−1 0
G = 5000(a) (b) G = 2500
2
4
6
8
H 3
10
5p 3
10
3
H 3
10
5
10
12
14
1
2
3
4
5
6
7
p 3
10
3
1
2
3
4
5
6
7Hertzian pressurecurve
2
4
6
8
10
12
14
1
−1 0 1 −1 0y /by /b
y /b y /b
1
−1 0 1
Q z = 3 310−4, U = 10−11 Q z = 3 310−4, U = 10−11
FIGURE 4.8 Pressure distribution and film thickness for high-load conditions. (Reprinted from Dow-
son, D. and Higginson, G., J. Mech. Eng. Sci., 2(3), 1960. With permission.)
Uo ¼dm
21 þ gð Þvm þ gvR½ � ð4:55 Þ
Dowson and Higginson [14] presented the results shown in Figure 4.8 and Figure 4.9 for
G ¼ 2500 and 5000 corresponding to bronze rollers and steel rollers, respectively, lubricated
by a mineral oil. The load �QQz ¼ 0.00003 corresponds approximately to 483 MPa (70,000 psi)
and �QQz ¼ 0.0003 corresponds approximately to 1380 MPa (200,000 psi). Dimensionless speed�UU ¼ 10 �11 corresponds to surface velocities in the order of 1524 mm/sec (5 ft/sec) for an
equivalent roller radius of 25.4 mm (1 in.) operating in mineral oil.
Note from Figure 4.8 and Figure 4.9 that the departure from the Hertzian pressure
distribution is less significant as the load increases. The second pressure peak at the outlet
end of the contact corresponds to a local decrease in the film thickness at that point.
Otherwise, the film is essentially of uniform thickness. The latter condition was confirmed
by tests conduced by Sibley and Orcutt [15].
Additionally, Dowson and Higginson [14] demonstrated the effect of distorted pressure
distribution on maximum subsurface shear stress. Figure 4.10 shows contours of tyzmax/ pmax.
Note that the shear stress increases in the vicinity of the second pressure peak and tends
toward the surface.
� 2006 by Taylor & Francis Group, LLC.
6
5
4
3p x
103
Hertzian pressurecurve
2
1
12
10
8
H x
105
6
4
2
12
10
8
H x
105
6
4
2
6
5
4
3p x
103
2
1
−2 −1 0y/b y/b
y/by/b
1
−2
(a) (b)
−1 0
= 5000Qz = 3 x 10−5, U = 10−11
1 −2 −1 0 1
−2 −1 0 1
= 2500Qz = 3 x 10−5, U = 10−11
GG
FIGURE 4.9 Pressure distribution and film thickness for light-load conditions. (Reprinted from Dow-
son, D. and Higginson, G., J. Mech. Eng. Sci., 2(3), 1960. With permission.)
0.56 0.57 0.58y/b
z/b z/b
y/b
y/b
y/z
y/b
y/z
0.20
0.40
0.60
0.80
1.00
1.20
0.20
0.40
0.60
0.80
1.00
1.20
0.59 0.60
0.01 0.01
0.4
0.3
0.03
0.04
0.05
0.02
1.0
0.8
0.6
0.5
0.4
0.02
0.03
0.04
0.05
−1.00 −0.08 −0.06 −0.04 −0.02 0 0.20
0.40.5
0.6
y = 5000
Qz = 3 x 10−5, U = 10−11
0.55
0.40 0.60 0.80 1.00 −1.00 −0.08 −0.06 −0.04 −0.02 0 0.20
0.30.40.50.55
0.6
0.40 0.60 0.80 1.00
0.61 0.62 0.60 0.61 0.62 0.63 0.64 0.65 0.66
y = 2500
Qz = 3 x 10−5, U = 10−11
FIGURE 4.10 Contours of maximum shear stress amplitude—maximum Hertz pressure. (Reprinted
from J. Mech. Eng. Sci., 2(3). 1960. With permission.)
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4.3.4 LUBRICANT FILM THICKNESS
Grubin [13] developed a formula for minimum film thickness in line contact, that is, the
thickness of the lubricant film between the protuberance at the trailing edge of the contact on
the equivalent roller surface and the opposing surface of the relative flat. The Grubin formula
is based on the assumption that the rolling surfaces deform as if dry contact occurs and is
given in a dimensionless format:
H0 ¼ 1:95 ð G �UU Þ0 :727
�QQ0 :091z
ð4:56 Þ
where H0 ¼ h0/ <y.
Based on analytical studies and experimental results, Dowson and Higginson [16] estab-
lished the following formula to calculate the minimum film thickness:
H0 ¼ 2:65 �UU0:7 G
0 :54
�QQ0 :13z
ð4:57 Þ
A significant feature of both equations is the relatively large dependency of film thickness on
speed and lubricant viscosity and the comparative insensitivity to load. Testing conducted by
Sibley and Orcutt [15] using radiation techniques seemed to confirm the Grubin equation;
however, the agreement between the Dowson and Grubin formulas is apparent. Today, the
Dowson equation is recommended as representative of line contact lubrication conditions.
Equation 4.56 and Equation 4.57 describe the minimum lubricant film thickness. The film
thickness at the center of the contact, plateau film thickness, is approximated by
Hc ¼ 43 H0 ð4:58 Þ
Archard and Kirk [17] described the minimum film thickness between two spheres as
H0 ¼ 0:84 G �UUð Þ0:741
�QQ0:074z
ð4:59Þ
Using a ball–disk test rig with a clear sapphire disk and interferometry, it is possible to obtain
photographs of the lubricant film thickness distribution in a moving ball–disk contact. Figure
4.11 shows the horseshoe pattern corresponding to the high-pressure ridge associated with the
minimum lubricant film thickness. The central or plateau film thickness is enclosed by the
horseshoe.
A more generalized formula for minimum lubricant film thickness in an elliptical area
point contact was subsequently developed by Hamrock and Dowson [18]:
H0 ¼3:63 �UU0:68
G0:49 1� e�0:68��
�QQ0:073z
ð4:60Þ
where �QQz for point contact is given by
�QQz ¼Q
E0<2ð4:61Þ
Sometimes for elliptical point contact, an equivalent line contact load is considered as follows:
� 2006 by Taylor & Francis Group, LLC.
FIGURE 4.11 Photograph of fluid-lubricated steel ball–sapphire disk contact. Interferometric fringes
indicate variation of film thickness and hence pressure. (Wedeven,L., Optical Measurements in Elasto-
hydrodynamic rolling contact bearings, Ph.D. Thesis, University of London, 1917.)
�QQe z ¼3Q
4E 0<y a ð4:62 Þ
In Equation 4.60, k is the ellipticity ratio a/ b. The central or plateau lubricant film thickness is
given by
H0 ¼2:69 �UU0 :67
G0:53 1 � 0:61e �0 :73��
�QQ0:067z
ð4:63 Þ
Kotzalas [20] conducted a study of lubricant film formation using both Roelands equation
(Equation 4.19) and a fitted sigmoid curve (Equation 4.25) to define lubricant viscosity vs.
pressure at a given temperature. He established that the calculated lubricant film thickness
distributions are substantially identical irrespective of which of the two models for viscosity
vs. pressure is used.
See Example 4.1 and Example 4.2.
4.4 VERY-HIGH-PRESSURE EFFECTS
Maximum Hertz pressures occurring in the rolling element–raceway contacts typically fall in
the range of 1000–2000 MPa (approximately 150–300 kpsi); however, in modern bearing
applications, particularly endurance tests, it is not unusual for maximum Hertz pressure to
� 2006 by Taylor & Francis Group, LLC.
reach 4000 MPa. To prevent damage to laboratory test equipment and the materials under
test, experiments used to confirm the lubricant film thickness equations provided here have
typically been confined to pressures not exceeding 1500 MPa. Venner [22] conducted EHL
analyses at high pressures and concluded that lubricant films predicted by the equations, both
minimum and central lubricant film thicknesses, are somewhat thinner than calculated by
these equations. Using a tungsten carbide ball on a sapphire disk and ultrathin film interfer-
ometry and digital techniques, Smeeth and Spikes [23] measured lubricant film thicknesses at
maximum Hertz pressures up to 3500 MPa. They confirmed Venner’s conclusions, finding
that, above contact loading of 2000 MPa, the minimum lubricant film thickness varies
inversely as dimensionless load to the 0.3 power as compared with the 0.073 power indicated
in Equation 4.60. The data shown by Smeeth and Spikes [23] might further be represented by
Equation 4.64 and Equation 4.65:
h0hp
h0
� �1 =2
¼ 1:0943 � 4:597 � 10 �12 p3max ð4:64 Þ
hchp
hcen
¼ 0:8736 � 8:543 � 10 �9 p2max ð4:65 Þ
These equations define the ratio of film thickness resulting from very high pressure to that
calculated using Equation 4.60 and Equation 4.63 for minimum and central film thicknesses,
respectively.
4.5 INLET LUBRICANT FRICTIONAL HEATING EFFECTS
At high bearing operating speeds, some of the frictional heat generated in each concentrated
contact is dissipated in the lubricant momentarily residing in the inlet zone of the contact.
This effect, examined first by Cheng [24], tends to increase the temperature of the lubricant in
the contact. Vogels [25] gives the following expression for viscosity:
hb ¼ A1eb0= TbþA2ð Þ ð4:66Þ
where Tb is in 8C and A1, A2, and b’ are parameters to be defined for each lubricant. Three
temperature–viscosity data points are required to determine A1, A2, and b’ as follows:
A1 ¼ h1e�b0= TbþA2ð Þ ð4:67Þ
A2 ¼A3T1 � T3
1� A3
ð4:68Þ
b0 ¼ T2 þ A2ð Þ T1 þ A2ð ÞT2 � T1ð Þ ln
h1
h2
� �ð4:69Þ
A3 ¼T3 � T2ð ÞT2 � T1ð Þ
ln h1=h2ð Þln h2=h3ð Þ ð4:70Þ
If only two temperature–viscosity data points are known and A2 can be fixed to 273, Equation
4.66 can be simplified to:
� 2006 by Taylor & Francis Group, LLC.
hb ¼ href e b 1 =Tb �1 =Trefð Þ ð4:71 Þ
where T is now in 8K and href is the absolute viscosity at reference temperature Tref. As Tref is
generally room temperature and as Tb is usually higher than room temperature, Equation
4.71 generally takes the form:
hb ¼ href e � A4 b ð4:72 Þ
showing that as temperature increases, lubricant viscosity decreases.
In accordance with this, it is clear that the lubricant film thickness will be reduced as a
result of temperature increase in the contact. Cheng [26] and subsequently, Murch and Wilson
[27], Wilson [28], and Wilson and Sheu [29] developed thermal reduction factors for lubricant
film thickness from numerical solutions of the thermal EHL problem for rolling–sliding
contacts. Gupta et al. [30] recommended the film thickness reduction factor in Equation 4.73.
ft ¼1 � 13 :2 p0
E
� L0 :42
1 þ 0:213 1 þ 2:23 S0:83ð ÞL0 :64 ð4:73 Þ
where p0 is the Hertzian pressure and dimensionless parameters L and S are defined as
follows:
L ¼ � ›h
›T
� �b
u1 þ u2ð Þ2
4kb
ð4:74 Þ
S ¼ 2u1 � u2ð Þu1 þ u2ð Þ ð4:75 Þ
Particularly for line contacts, Hsu and Lee [31] provided Equation 4.76.
fT ¼1
1 þ 0:0766 G0 :687 �QQ0 :447L L0 :527e0 :875 S
ð4:76 Þ
See Exam ple 4.3.
4.6 STARVATION OF LUBRICANT
The basic formulas for calculation of lubricant film thickness assume an adequate supply of
lubricant to the contact zones. The condition in which the volume of lubricant on the surfaces
entering the contact is insufficient to develop a full lubricant film is called starvation. Factors
to determine the reduction of the apparent lubricant film thickness have been developed as
functions of the distance of the lubricant meniscus in the inlet zone from the center of the
contact. As yet, no definitive equations have been developed to accurately calculate the
aforementioned distance; therefore, the meniscus distance has to be determined experimen-
tally. Figure 4.12 illustrates the concept of meniscus distance. References [33–37] give further
details about this concept.
In consideration of the meniscus distance problem, a condition of zero reverse flow is
defined. Under this condition, the minimum velocity of the point situated at the meniscus
distance from the contact center is, by definition, zero. If the meniscus distance is greater, the
latter point will have a negative velocity, that is, reverse flow. The zero reverse flow condition
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Yb
(b)
2bb
u2u2
u1 u1
Yb
(a)
FIGURE 4.12 Meniscus distance in (a) hydrodynamic and (b) elastohydrodynamic lubrication.
is therefore a quasistable situation, because no lubricant is lost to the contact owing to reverse
flow. In the case of a minimum quantity of lubricant supplied, for example, oil mist or grease
lubrication, the lubricant film thickness reduction factor owing to starvation effects, accord-
ing to Refs. [33,36], lies between 0.71 (in pure rolling) and 0.46 (in pure sliding). Castle and
Dowson [36] give the following equation for line contact:
ws ¼ 1 � e �1 :347 F0:69 �0:13
ð4:77 Þ
where
� ¼yb
b� 1
2<y
b
� �2
Hc
� �2 =3 ð4:78 Þ
It is clear that F is zero if the meniscus distance should equal b and in that case ws ¼ 0.
Accordingly, an accurate estimation of the meniscus distance is necessary to the effective
employment of a lubricant starvation factor. In the absence of this value, the condition of
zero reverse flow provides a practical limitation and a starvation factor of ws ¼ 0.70.
Thermal effects on lubricant film formation under conditions approaching lubricant
starvation are extremely significant owing to the absence of excess lubricant to help dissipate
frictional heat generation in the contacts. Accordingly, the lubricant film reduction factors for
thermal effects and starvation are not multiplicative and a combined factor is required.
Goksem et al. [33] derived the following expression for elastohydrodynamic line contact:
wTS ¼ wT 1� 1
ð4:6þ 1:15L0:6Þð0:67�QQz�YY=wTHcÞð0:52=ð1þ0:001LÞÞ
!ð4:79Þ
where L is given by Equation 4.74 and
�YY ¼ yb y2b � 1
� 1=2� ln yb þ y2b � 1
� 1=2h ið4:80Þ
For the zero reverse flow condition, the combined reduction factor for the central lubricant
film thickness is
� 2006 by Taylor & Francis Group, LLC.
wTS ¼ wT 1 � 1
4:6 þ 1:15 L0:6ð Þð0 :6345 =wT Þ 0:52 = 1þ0:001 Lð Þð Þ
!ð4:81 Þ
For point contact, Equation 4.79 through Equation 4.81 can be used in conjunction with
Equation 4.62 for equivalent line contact loading.
See Example 4.4.
4.7 SURFACE TOPOGRAPHY EFFECTS
In the methods and equations used in the calculation of lubricant film thickness thus far in
this chapter, only the macrogeometries of the rolling components have been considered; that
is, the surfaces of the components have been assumed to be smooth. In practice, each ball,
roller, or raceway surface has a roughness superimposed upon the principal geometry. This
roughness, or more correctly surface topography similar to the earth’s surface superimposed
upon the spherical surface of the planet, is introduced by the surface finishing processes
during component manufacture. In recent history, substantial manufacturing development
efforts have been expended to produce ultrasmooth rolling component surfaces. Figure 4.13
schematically illustrates a rough rolling component surface.
For a given surface, the roughness is most commonly defined by the arithmetic average
(AA) peak-to-valley distance. This is easily measurable using stylus devices such as the
Talysurf machine. Using surface-measuring devices, more extensive properties of surface
microgeometry can also be measured; see Ref. [38]. To date, AA surface roughnesses, RA,
as fine as 0.05 mm (2 min.) have been produced on ball bearing raceways approaching 600 mm
(24 in.) diameter. Balls larger than 25 mm (1 in.) diameter are routinely produced with RA
values of 0.005 mm (0.2 min.). It is, however, not certain that RA¼ 0 is an ideal microgeometry
from a lubrication effectiveness or surface fatigue endurance standpoint.
Ground M.S. RMS Surface Roughness 1.5 μm3 mm x 9 mm
1 division = 7.3 μm
1 div. = 300 μm
1 div. = 100 μm
FIGURE 4.13 Isometric view of a typical honed and lapped surface showing roughness peaks.
� 2006 by Taylor & Francis Group, LLC.
Depending on the thickness of the lubricant film relative to the roughnesses of the rolling
contact surfaces, the direction of the roughness pattern can affect the film-building capability
of the lubricant. If the surface roughness has a pattern wherein the microgrooves are
transverse to the direction of motion, this could result in a beneficial lubricant film-building
effect. Conversely, if the lay of the roughness is parallel to the direction of motion, the effect
can be to produce a thinner lubricant film. The most successful applications of rolling
bearings are those in which fluid lubricant films over the rolling element–raceway contacts
are sufficiently thick to completely separate those components. This is generally defined by
the parameter L as follows:
L ¼ h0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2r þ s2
RE
q ð4:82 Þ
In Equation 4.82, h0 is the minimum lubricant film thickness, sr the root mean square (rms)
roughness of the raceway surface, and sRE is the rms roughness of the ball or roller surface. In
general, the rms roughness value is taken as 1.25 RA.
Patir and Cheng [39] first investigated the effect of the lay of surface topography on the
lubricant film thickness generated. They developed a correction factor for lubricant film
thickness based on the distances between contact surface ‘‘hills’’ and ‘‘valleys’’ in directions
transverse and parallel to rolling motion. Tønder and Jakobsen [40] using a ball-on-disk test
rig and optical interferometry confirmed the general conclusion of Patir and Cheng that
transverse lay tends to generate thicker films than does longitudinal lay. Kaneta et al. [41] in a
similar experimental effort determined that, in the thin film region (L< 1), film thickness for
surfaces with transverse lay tends to increase with slide/roll ratio due to deformation of
asperities. When L> 3, however, deformation of asperities can be neglected.
Chang et al. [42] analytically investigated the effects of surface roughness considering the
effects of lubricant shear thinning due to frictional heating. They determined that these effects
serve to mitigate the pressure rippling influence on lubricant film thickness. Ai and Cheng
[43], considering the randomized surface roughness of Figure 4.14, conducted an extensive
analysis revisiting the influences of surface topographical lay. They generated three-dimen-
sional plots of point contact pressure and film thickness distribution for transverse, longitu-
dinal, and oblique topographical lays. Figure 4.15 through Figure 4.17 illustrate the effects
for the randomized surface roughness. They indicated that roughness orientation has a
noticeable effect on pressure fluctuation. They further noted that oblique roughness lay
induces localized three-dimensional pressure fluctuations in which the maximum pressure
may be greater than that produced by transverse roughness lay. It is to be noted that the
oblique roughness lay more likely is representative of the surfaces generated during bearing
component manufacture. Oblique surface roughness lay may also result in the minimum
lubricant film thicknesses compared with transverse or longitudinal roughness lays. Ai and
Cheng [43] further noted, however, that when L is sufficiently large such that the surfaces are
effectively separated, the effect of lay on film thickness and contact pressure is minimal.
Guangteng and Spikes [44], using ultrathin film, optical interferometry, managed to
measure the mean EHL film thickness of very thin film, isotropically rough surfaces occurring
in rolling balls on flat contacts. They found that, for L< 2, the mean EHL film thicknesses
were less than those for smooth surfaces. Subsequently, using the spacer layer imaging
method developed by Cann et al. [45] to map EHL contacts, Guangteng et al. [46] indicated
that rolling elements having real, random, rough surfaces; for example, rolling bearing
components. The mean film thicknesses tend to be less than those calculated for rolling
elements that have smooth surfaces. This implies that, in the mixed EHL regime, for example,
� 2006 by Taylor & Francis Group, LLC.
0.0002
0.0001
0.0000
Rou
ghne
ss h
eigh
t, m
m
−0.0001
−0.0002
−0.0003
−0.00040.0 2.0 4.0
Distance, mm6.0 8.0
FIGURE 4.14 Random surface roughness profile considered by Ai and Cheng. (From Ai, X. and Cheng,
H., Trans. ASME, J. Tribol., 118, 59–66, January 1996. With permission.)
L< 1.5, the mean lubricant film thicknesses will tend to be less than those predicted by the
equations given for rolling contacts with smooth surfaces. The amount of the reduction may
only be determined by testing; empirical relationships need to be developed.
4.8 GREASE LUBRICATION
When grease is used as a lubricant, the lubricant film thickness is generally estimated using the
properties of the base oil of the grease while ignoring the effect of the thickener. It has been
determined, however, by several researchers [47–50] that in a given application, owing to a
FIGURE 4.15 Pressure (a) and film thickness (b) distribution in an EHL point contact with transverse
topographical lay, random surface roughness. Motion is in the x direction. (From Ai, X. and Cheng, H.,
Trans. ASME, J. Tribol., 118, 59–66, January 1996. With permission.)
� 2006 by Taylor & Francis Group, LLC.
FIGURE 4.16 Pressure (a) and film thickness (b) distribution in an EHL point contact with longitudinal
topographical lay, random surface roughness. Motion is in the x direction. (From Ai, X. and Cheng, H.,
Trans. ASME, J. Tribol., 118, 59–66, January 1996. With permission.)
contribution by the thickener, grease may form a thicker lubricant film than that determined
using only the properties of the base oil. Kauzlarich and Greenwood [51] developed an
expression for the thickness of the film formed by greases in line contact under a Herschel–
Bulkley constitutive law in which shear stress t and shear rate _�� are related by the equation
t ¼ ty þ a _ggb ð4:83Þ
where ty is the yield stress and a and b are considered physical properties of the grease.
For a Newtonian fluid,
t ¼ h _gg ð4:84Þ
where h is the viscosity.
FIGURE 4.17 Pressure (a) and film thickness (b) distribution in an EHL point contact with oblique
topographical lay, random surface roughness. Motion is in the x direction. (From Ai, X. and Cheng, H.,
Trans. ASME, J. Tribol., 118, 59–66, January 1996. With permission.)
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The effective viscosity under a Herschel–Bulkley law is thus found by equating t from
Equation 4.83 and Equation 4.84 so that
heff ¼ty þ a _ggb
_ggð4:85 Þ
In this form, it is seen that for a> 1, heff increases indefinitely with the shear rate, and for
a< 1, heff approaches zero as the strain rate increases. Palacios et al. [49] argued that it is
more reasonable to assume that at high shear rates greases will behave like their base oils.
They accordingly proposed a modification of the Herschel–Bulkley law to the form
t ¼ ty þ a _ggb þ hb _gg ð4:86 Þ
where hb is the base oil viscosity. In this form, provided a< 1, heff approaches hb as the strain
rate approaches 1. Values of ty, a, b, and hb are given in Ref. [52] for three greases from 35
to 80 8C (95 to 176 8F).
Since viscosity appears raised to the 0.67 power in Equation 4.63, Palacios and Palacios
[52] proposed that hG, the film thickness of a grease, and hb, the film thickness of the base oil,
will be in the proportion
hG
hb
¼ heff
hb
� �0 :67
ð4:87 Þ
They proposed that this evaluation be made at a shear rate equal to 0.68 u/ hG, which requires
iteration to determine hG. Their suggested approach is to calculate hb from Equation 4.63,
determine _��¼ 0.68 u/hb, and then hG from Equation 4.87. The shear rate is then recalculated
using hG. The process is repeated until convergence occurs. The analysis was applied to line
contact, but it should also be valid for elliptical contacts with a/ b in the range of 8–10 (typical
for ball bearing point contacts).
In her investigations, Cann [53,54] notes that the portion of the film associated with
the grease thickener is a residual film composed of the degraded thickener deposited in the
bearing raceways. The hydrodynamic component is generated by the relative motion of
the surfaces due to oil, both in the raceways and supplied by the reservoirs of grease adjacent
to the raceways. She further notes that at low temperatures grease films are generally thinner
than those for the fully flooded, base fluid lubricant. This is due to the predominant bulk
grease starvation and the inability of the high viscosity, bled lubricant to resupply the contact.
At higher temperatures of operation, grease forms films considerably thicker than those
considering only the base oil. This is attributed to the increased local supply of lubricant to
the contact area due to the lower oil viscosity at the elevated temperature producing a
partially flooded EHL film augmented by a boundary film of deposited thickener.
Therefore, it can be stated that with grease lubrication the degree of starvation tends
to increase with increasing base oil viscosity, thickener content, and speed of rotation. It
tends to decrease with increasing temperature. For rolling bearing applications, the film
thickness may only be a fraction of that calculated for fully flooded, oil lubrication condi-
tions. A most likely saving factor is that as lubricant films become thinner, friction and hence
temperature increase. This tends to reduce viscosity permitting increased return flow to the
rolling element–raceway contacts. Nevertheless, depending on the aforementioned operating
conditions of grease base oil viscosity, grease thickener content, and rotational speed, lubricant
film thicknesses may be expected to be only a fraction of those calculated using Equation 4.57,
� 2006 by Taylor & Francis Group, LLC.
Equation 4.60, and Equation 4.63. According to data shown by Cann [54], fractional values
might range from 0.9 down to 0.2.
4.9 LUBRICATION REGIMES
Although this chapter has concentrated on elastohydrodynamic lubrication in rolling con-
tacts, the general solution presented for the Reynolds equation covers a gamut of lubrication
regimes; for example:
. Isoviscous hydrodynamic (IHD) or classical hydrodynamic lubrication
. Piezoviscous hydrodynamic (PHD) lubrication, in which lubricant viscosity is a func-
tion of pressure in the contact. Elastohydrodynamic (EHD) lubrication, in which both the increase in viscosity with
pressure and the deformations of the rolling component surfaces are considered in the
solution
Dowson and Higginson [55] created Figure 4.18 to define these regimes for line contact in
terms of the dimensionless quantities for film thickness, load, and rolling velocity; Equation
4.48 through Equation 4.50.
Markho and Clegg [56] established a parameter, called C1 herein, for a fixed value of G;
This factor was used to define the lubrication regime. Dalmaz [57] subsequently established
Equation 4.88 to cover all practical values of G.
C1 ¼ log10 1:5 � 106 G
5000
� �2 �QQ3z
�UU
" #ð4:88 Þ
Table 4.2 shows the relationship of parameter C1 to the operating lubrication regimes.
For calculation of the lubricant film thicknesses in rolling element–raceway contacts, only
the PHD and EHD regimes need to be considered. For calculations associated with the cage–
rolling element contacts, probably a consideration of the hydrodynamic regime is sufficient.
In this case, Martin [1] gave the following equation for film thickness in line contact:
H ¼ 4:9�UU�QQz
ð4:89 Þ
For point contact, Brewe and Hamrock [58] give
H ¼Qz
U1 þ 2 <x
3 <y
� �128
<y
<x
� �1 =2
0:131 tan �1 <y
2<x
� �þ 1:163
h iþ 2:6511
8><>:
9>=>;�2
ð4:90Þ
For the PHD regime in line contacts, data from Ref. [56] have been used to establish the
following expression for minimum film thickness:
H ¼ 10C4 � G
5000
� �0:35 ð1þC1Þð4:91Þ
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10−610−13
10−12
10−11
10−10
10−9
IHD PHD EHD
400
300
150
100
80
60
40
30
20
15
10
8
6
4
3
2
1.5
1
0.8
H = 200 � 10−6
10−5 10−4
8001,0
002,0003,0
004,000
6,0008,00010
,000
Dimensionless load, Qz
Dim
ensi
onle
ss s
peed
, U
FIGURE 4.18 Film thickness vs. speed and load for a line contact. (From Dowson, D. and Higginson, G.,
Proc. Inst. Mech. Eng., 117, 1963.)
where
C2 ¼ log10 ð618 �UU0:6617 Þ ð4:92 Þ
C3 ¼ log10 ð1:285 �UU0:0025 Þ ð4:93 Þ
and C1 is given by Equation 4.88. In Equation 4.91, C4 is given by
� 2006 by Taylor & Francis Group, LLC.
TABLE 4.2Lubrication Regimes
Parameter Limits
Lubrication
Regime Characteristics
C1 ��1 IHD Low contact pressure, no significant surface deformation
�1<C1< 1 PHD No significant surface deformation, lubricant viscosity increases with pressure
C1 � 1 EHD Surface deformation and lubricant viscosity increase with pressure
C4 ¼ C2 þ C1 C3 ðC21 � 3Þ � 0:094 C1 ð C2
1 � 0:77 C1 � 1Þ ð4:94 Þ
Dalmaz [57] also developed numerical results for point contact film thicknesses in the PHD
regime; an analytical relationship was not then established.
4.10 CLOSURE
In the earlier discussion, it has been demonstrated analytically that a lubricant film can
separate the rolling elements from the contacting raceways. Moreover, the fluid friction
forces developed in the contact zones between the rolling elements and raceways can signifi-
cantly alter the bearing’s mode of operation. It is desirable from the standpoint of preventing
increased stresses caused by metal-to-metal contact that the minimum film thickness should
be sufficient to completely separate the rolling surfaces. The effect of film thickness on
bearing endurance is discussed in Chapter 8.
A substantial amount of analytical and experimental research from the 1960s into the 21st
century has contributed greatly to the understanding of the lubrication mechanics of concen-
trated contacts in rolling bearings. Perhaps the original work of Grubin [13] will prove to be
as significant as that conducted by Reynolds during the 1880s.
Apart from acting to separate rolling surfaces, the lubricant is frequently used as a
medium to dissipate the heat generated by bearing friction as well as to remove heat that
would otherwise be transferred to the bearing from the surroundings at elevated temperat-
ures. This topic is discussed in Chapter 7.
REFERENCES
1.
� 200
Martin, H., Lubrication of gear teeth, Engineering, 102, 199, 1916.
2.
Osterle, J., On the hydrodynamic lubrication of roller bearings, Wear, 2, 195, 1959.3.
Barus, C., Isothermals, isopiestics, and isometrics relative to viscosity, Am. J. Sci., 45, 87–96, 1893.4.
ASME Research Committee on Lubrication, Pressure–viscosity report—Vol. 11, ASME, 1953.5.
Roelands, C., Correlation Aspects of Viscosity–Temperature–Pressure Relationship of LubricatingOils, Ph.D. Thesis, Delft University of Technology, 1966.
6.
Sorab, J. and VanArsdale, W., A correlation for the pressure and temperature dependence ofviscosity, Tribol. Trans., 34(4), 604–610, 1991.
7.
Bair, S. and Kottke, P., Pressure–viscosity relationships for elastohydrodynamics, Preprint AM03-1,STLE Annual Meeting, New York, 2003.
8.
Doolittle, A., Studies in Newtonian flow II, the dependence of the viscosity of liquids on free-space,J. Appl. Phys., 22, 1471–1475, 1951.
9.
Harris, T., Establishment of a new rolling bearing life calculation method, Final Report, U.S. NavyContract N68335-93-C-0111, January 15, 1994.
10.
Bair, S. and Winer, W., Shear strength measurements of lubricants at high pressure, Trans. ASME,J. Lubr. Technol., Ser. F, 101, 251–257, 1979.
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Bair, S. and Winer, W., Some observations in high pressure rheology of lubricants, Trans. ASME,
J. Lubr. Technol., Ser. F, 104, 357–364, 1982.
12.
Dowson, D. and Higginson, G., A numerical solution to the elastohydrodynamic problem, J. Mech.Eng. Sci., 1(1), 6, 1959.
13.
Grubin, A., Fundamentals of the hydrodynamic theory of lubrication of heavily loaded cylindricalsurfaces, Investigation of the Contact Machine Components, Kh. F. Ketova (ed.) [Translation of
Russian Book No. 30, Chapter 2], Central Scientific Institute of Technology and Mechanical
Engineering, Moscow, 1949.
14.
Dowson, D. and Higginson, G., The effect of material properties on the lubrication of elastic rollers,J. Mech. Eng. Sci., 2(3), 1960.
15.
Sibley, L. and Orcutt, F., Elastohydrodynamic lubrication of rolling contact surfaces, ASLE Trans.,4, 234–249, 1961.
16.
Dowson, D. and Higginson, G., Proc. Inst. Mech. Eng., 182(Part 3A), 151–167, 1968.17.
Archard, G. and Kirk, M., Lubrication at point contacts, Proc. R. Soc. Ser. A, 261, 532–550, 1961.18.
Hamrock, B. and Dowson, D., Isothermal elastohydrodynamic lubrication of point contacts—PartIII—fully flooded results, Trans. ASME, J. Lubr. Technol., 99, 264–276, 1977.
19.
Wedeven, L., Optical Measurements in Elastohydrodynamic Rolling Contact Bearings, Ph.D. Thesis,University of London, 1971.
20.
Kotzalas, M., Power Transmission Component Failure and Rolling Contact Fatigue, Ph.D. Thesis,Pennsylvania State University, 1999.
21.
Avallone, E. and Baumeister, T., Standard Handbook for Mechanical Engineers, 9th ed., McGraw-Hill, New York, 1987.
22.
Venner, C., Higher order mutlilevel solvers for the EHL line and point contact problems, ASMETrans., J. Tribol., 116, 741–750, 1994.
23.
Smeeth, S. and Spikes, H., Central and minimum elastohydrodynamic film thickness at high contactpressure, ASME Trans., J. Tribol., 119, 291–296, 1997.
24.
Cheng, H., A numerical solution to the elastohydrodynamic film thickness in an elliptical contact,Trans. ASME, J. Lubr. Technol., 92, 155–162, 1970.
25.
Vogels, H., Das Temperaturabhangigkeitsgesetz der Viscositat von Flıssigkeiten, Phys. Z., 22, 645–646, 1921.
26.
Cheng, H., A refined solution to the thermal-elastohydrodynamic lubrication of rolling and slidingcylinders, ASLE Trans., 8(4), 397–410, 1965.
27.
Murch, L. and Wilson, W., A thermal elastohydrodynamic inlet zone analysis, Trans. ASME,J. Lubr. Technol., Ser. F, 97(2), 212–216, 1975.
28.
Wilson, A., An experimental thermal correction for predicted oil film thickness in elastohydrody-namic contacts, Proc. 6th Leeds–Lyon Symp. Tribol., 1979.
29.
Wilson, W. and Sheu, S., Effect of inlet shear heating due to sliding on elastohydrodynamic filmthickness, Trans. ASME, J. Lubr. Technol., Ser. F, 105(2), 187–188, 1983.
30.
Gupta, P., et al., Viscoelastic effects in Mil-L-7808 type lubricant, Part I: Analytical formulation,Tribol. Trans., 35(2), 269–274, 1992.
31.
Hsu, C. and Lee, R., An efficient algorithm for thermal elastohydrodynamic lubrication underrolling/sliding line contacts, J. Vibr. Acoust. Reliab. Des., 116(4), 762–768, 1994.
32.
MacAdams, W., Heat Transmission, 3rd ed., McGraw-Hill, New York, 1954.33.
Goksem, P. and Hargreaves, R., The effect of viscous shear heating in both film thickness androlling traction in an EHL line contact—Part II: Starved condition, Trans. ASME, J. Lubr.
Technol., 100, 353–358, 1978.
34.
Dowson, D., Inlet boundary conditions, Leeds–Lyon Symp., 1974.35.
Wolveridge, P., Baglin, K., and Archard, J., The starved lubrication of cylinders in line contact,Proc. Inst. Mech. Eng., 185, 1159–1169, 1970–1971.
36.
Castle, P. and Dowson, D., A theoretical analysis of the starved elastohydrodynamic lubricationproblem, Proc. Inst. Mech. Eng., 131, 131–137, 1972.
37.
Hamrock, B. and Dowson, D., Isothermal elastohydrodynamic lubrication of point contact—PartIV: Starvation results, Trans. ASME, J. Lubr. Technol., 99, 15–23, 1977.
6 by Taylor & Francis Group, LLC.
38.
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McCool, J., Relating profile instrument measurements to the functional performance of rough
surfaces, Trans. ASME, J. Tribol., 109, 271–275, April 1987.
39.
Patir, N. and Cheng, H., Effect of surface roughness orientation on the central film thickness inEHD contacts, Proc. 5th Leeds–Lyon Symp. Tribol., 15–21, 1978.
40.
Tønder, P. and Jakobsen, J., Interferometric studies of effects of striated roughness on lubricant filmthickness under elastohydrodynamic conditions, Trans. ASME, J. Tribol., 114, 52–56, January
1992.
41.
Kaneta, M., Sakai, T., and Nishikawa, H., Effects of surface roughness on point contact EHL,Tribol. Trans., 36(4), 605–612, 1993.
42.
Chang, L., Webster, M., and Jackson, A., On the pressure rippling and roughness deformation inelastohydrodynamic lubrication of rough surfaces, Trans. ASME, J. Tribol., 115, 439–444, July 1993.
43.
Ai, X. and Cheng, H., The effects of surface texture on EHL point contacts, Trans. ASME,J. Tribol., 118, 59–66, January 1996.
44.
Guangteng, G. and Spikes, H., An experimental study of film thickness in the mixed lubricationregime, Proc. 24th Leeds–Lyon Symp., Elastohydrodynamics, 159–166, September 1996.
45.
Cann, P., Hutchinson, J., and Spikes, H., The development of a spacer layer imaging method(SLIM) for mapping elastohydrodynamic contacts, Tribol. Trans., 39, 915–921, 1996.
46.
Guangteng, G., et al., Lubricant film thickness in rough surface, mixed elastohydrodynamiccontact, ASME Paper 99-TRIB-40, October 1999.
47.
Wilson, A., The relative thickness of grease and oil films in rolling bearings, Proc. Inst. Mech. Eng.,193, 185–192, 1979.
48.
Mınnich, H. and Glockner, H., Elastohydrodynamic lubrication of grease-lubricated rolling bear-ings, ASLE Trans., 23, 45–52, 1980.
49.
Palacios, J., Cameron, A., and Arizmendi, L., Film thickness of grease in rolling contacts, ASLETrans., 24, 474–478, 1981.
50.
Palacios, J., Elastohydrodynamic films in mixed lubrication: an experimental investigation, Wear,89, 303–312, 1983.
51.
Kauzlarich, J. and Greenwood, J., Elastohydrodynamic lubrication with Herschel–Bulkley modelreases, ASLE Trans., 15, 269–277, 1972.
52.
Palacios, J. and Palacios, M., Rheological properties of greases in EHD contacts, Tribol. Int., 17,167–171, 1984.
53.
Cann, P., Starvation and reflow in a grease-lubricated elastohydrodynamic contact, Tribol. Trans.,39(3), 698–704, 1996.
54.
Cann, P., Starved grease lubrication of rolling contacts, Tribol. Trans., 42(4), 867–873, 1999.55.
Dowson, D. and Higginson, G., Theory of roller bearing lubrication and deformation, Proc. Inst.Mech. Eng., 117, 1963.
56.
Markho, P. and Clegg, D., Reflections on some aspects of lubrication of concentrated line contacts,Trans. ASME, J. Lubr. Technol., 101, 528–531, 1979.
57.
Dalmaz, G., Le Film Mince Visquex dans les Contacts Hertziens en Regimes Hydrodynamique etElastohydrodynamique, Docteur d’Etat Es Sciences Thesis, I.N.S.A. Lyon, 1979.
58.
Brewe, D. and Hamrock, B., Analysis of starvation on hydrodynamic lubrication in non-conform-ing contacts, ASME Paper 81-LUB-52, 1981.
6 by Taylor & Francis Group, LLC.
5 Friction in RollingElement–Raceway Contacts
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LIST OF SYMBOLS
Symbol Description Units
a Semimajor axis of contact ellipse mm (in.)
Ac True average contact area mm2 (in.2)
A0 Apparent contact area mm2 (in.2)
b Semiminor axis of contact ellipse mm (in.)
d Separation of mean plane of summits and smooth plane mm (in.)
di Raceway track diameter mm (in.)
D Rolling element diameter mm (in.)
DSUM Summit density mm�2 (in.�2)
E1, E2 Elastic moduli of bodies 1 and 2 MPa (psi)
E0 Reduced elastic modulus MPa (psi)
F Contact friction force N (lb)
F0( ), F1( ),
F3=2( ) Tabular functions for the Greenwood–Williamson model
h Lubricant film thickness mm (in.)
hc Central or plateau lubricant film thickness mm (in.)
L Roller length end-to-end mm (in.)
leff Roller effective length mm (in.)
ls Roller straight length mm (in.)
m0 Zeroth-order spectral moment, � Rq2 � s2 mm2 (min.2)
m2 Second-order spectral moment
m4 Fourth-order spectral moment mm�2 (in.�2)
n Contact density mm�2 (in.�2)
np Plastic contact density mm�2 (in.�2)
q x=aQ Contact load N (lb)
Qa Asperity-supported load N (lb)
Qf Fluid-supported load N (lb)
R Radius of deformed surface mm (in.)
R Summit sphere radius mm (in.)
Rq Root mean square (rms) value of surface profile mm (min.)
S Composite rms surface roughness for bodies 1 and 2 mm (min.)
Ss Standard deviation of summit heights for bodies 1 and 2 mm (in.)
s1, s2 Surface rms roughnesses for bodies 1 and 2 mm (min.)
t y=aT Temperature 8C (8F)
u Surface velocity mm=sec (in.=sec)
cis Group, LLC.
um Raceway surface velocity mm=sec (in.=sec)uRE Rolling element surface velocity mm=sec (in.=sec)U Rolling velocity¼ 1=2 (uREþ um) mm=sec (in.=sec)v Sliding velocity mm=sec (in.=sec)w Deflection of summit mm (min.)
wp Variable governing asperity density mm (min.)
Y Yield strength in simple tension MPa (psi)
zs Summit height relative to summit mean plane mm (in.)
�zzs Distance between surface and summit mean plane mm (in.)
z(x) Surface profile mm (in.)
a Bandwidth parameter
g Shear rate sec�1
h Absolute viscosity N-sec=m2
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(lb-sec=in.2)
L Lubricant film parameter, h=sm Friction or traction coefficient
ma Asperity–asperity friction coefficient
n1, n2 Poisson’s ratio for bodies 1 and 2
s Normal contact stress or pressure MPa (psi)
s0 Maximum normal contact stress or pressure MPa (psi)
Fo Maximum normal contact stress or pressure MPa(psi)
t Shear stress MPa (psi)
tf Shear stress due to fluid MPa (psi)
tlim Limiting shear stress in fluid MPa (psi)
tN Shear stress in Newtonian fluid lubrication MPa (psi)
f( ) Gaussian probability density function mm�1 (in.�1)
5.1 GENERAL
Ball and roller bearings were historically called antifriction bearings because of the low
friction properties associated with them. Actually, the major portion of friction associated
with rolling bearings is caused by sliding motions in the contacts between components such as
rolling elements and raceways, rolling elements and cage, roller ends and roller guide flanges,
and cage rails and inner or outer ring lands. This excludes the friction due to sliding between
bearing seals and inner or outer ring lands; this friction is generally greater than that
produced by all of the other sources of friction combined. In this chapter, the friction between
rolling elements and raceways will be investigated.
Rolling bearings are generally operated with oil lubrication; this can be accomplished
using circulating oil, bath oil, air–oil mist, or grease. Grease lubricant is an organic or
inorganic thickener containing oil that exudes from the thickener to become the predominant
lubri cant. In Chapt er 4, it was shown that the lubrican t film acts to separate the rolling
elements from the raceways. This separation can be complete or partial. With complete
separation, friction depends wholly on the properties of the lubricant at the contact temper-
atures and pressures. In the latter case, peaks or asperities from the rolling=sliding surfaces
come into contact under boundary lubrication conditions, resulting in increased friction. Thus,
it is important to establish the lubricant film thickness in each contact.
In Chapter 4, it was shown that lubricant film thickness occurring in a fluid-lubricated
(oil-lubricated), rolling element–raceway contact depends on contact geometry and load,
rolling speed, and lubricant properties. The lubricant properties, in turn, depend on the
temperature of the lubricant both within and on entering the contact. The temperatures
depen d on the friction he at generat ed and on the he at dissip ation paths availab le to the
bearing . Method s to de termine be aring tempe ratures will be discus sed in Chapt er 7; in this
chapter , it wi ll be assum ed that tempe ratures are known.
Under con ditions where fluid or grease lubri catio n is precluded , rolling be arings may also
be ope rated with soli d-film lubrican ts; for exampl e, graphit e, molybdenum disul fide, or other
compou nds. Thes e lubrican ts general ly cause rolling bearing s to ope rate with higher frictio n
and temperatur es than do fluid lubri cants. This form of lubricati on is simila r to bounda ry
lubri cation, resulting in less fricti on than direct rolling component co ntact; howeve r, heat
dissipati on capabil ity is great ly redu ced.
5.2 ROLLING FRICTION
5.2.1 DEFORMATION
The ba lls or rollers in a bearing are mainly subjected to loads perpendicular to the tangent
plane at each contact surface. Because of these normal loads, the rolling elements and
raceways are deformed at each contact, producing according to Hertz, a radius of curvature
of the common contacting surfaces equal to the harmonic mean of the radii of the contacting
bodies. For a roller of diameter D, bearing on a cylindrical raceway of diameter di, the radius
of curvature of a contact surface is
R ¼ di D
d i þ D ð 5: 1Þ
Becau se of the deform ation indica ted abo ve and because of the rolling moti on of the ro ller
over the racewa y, which requir es a tangent ial force to overco me rolling resistance , racew ay
mate rial is squeezed up to form a bulge in the forward portion of the contact as shown in
Figure 5.1. A depress ion is subsequently formed in the rear of the co ntact area. Thus , an
additio nal tangent ial force is required to overcome the resi sting force of the bulge. The bulge
is very small and the friction force is insi gnificant .
5.2.2 E LASTIC HYSTERESIS
As may be observed in the discus sion, as a rolling elemen t unde r compressive load travels over
a raceway, the material in the forward portion of the contact in the direction of rolling
undergoes compression while the material in the rear of the contact is relieved of stress. It is
recognized that as load is increasing, a given stress corresponds to a smaller deflection than
when load is de creasing (see Figure 5.2). The area between the curves in Figure 5.2 is call ed
the hysteresis loop, and it represents an energy loss (friction power loss). Generally, friction
due to elastic hysteresis is very small compared with other types of friction occurring in rolling
NT
w
FIGURE 5.1 Roller–raceway contact showing bulge due to rolling deformation.
� 2006 by Taylor & Francis Group, LLC.
Static loading
Load reversing
Strain
Stress
Energy loss
Load increasing
FIGURE 5.2 Hysteresis loop for elastic material subjected to reversing stresses.
bearings. Drutowski [1] verified this by experimenting with balls rolling between flat plates.
Friction coefficients as low as 0.0001 can be determined from the data of Ref. [1] for 12.7 mm
(0.5 in.) chrome steel balls rolling on chrome steel plates under normal loads of 356 N (80 lb).
Greenwood and Tabor [2] evaluated the rolling resistance due to elastic hysteresis. They
found that the frictional resistance is substantially less than that due to sliding if the normal
load is sufficiently large.
Drutowski [3] also demonstrated the linear dependence of rolling friction on the volume of
stressed material. In both Refs. [1,3], he further showed the dependence of elastic hysteresis on
the material under stress and the specific load on the contact area.
5.3 SLIDING FRICTION
5.3.1 MICROSLIP
If a radial cylindrical roller bearing had rollers and raceways of exactly the same lengths, if the
rollers were accurately guided by frictionless flanges, and if the bearing operated with zero
misalignment under moderate speed, then gross sliding in the roller–raceway contacts would
not occur. Gross sliding refers to the total slip of one surface over another. Depending on the
elastic properties of the contacting bodies and the coefficient of friction between the contact-
ing surfaces, microslip could occur. Using Figure 5.3, the coefficient of friction is defined as
the ratio of the tangential force F to the normal force Q. Microslip is defined as the partial
sliding of one surface relative to the other:
m ¼ F
Qð5:2Þ
Reynolds [4] first referred to microslip when, in his experiments involving rolling of an
elastically stiff cylinder on rubber, he observed that since the rubber stretched in the contact
zone, the cylinder rolled forward a distance less than its circumference in one complete
revolution about its axis. This experiment was conducted in the absence of a lubricating
medium, that is, dry contact.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 5.3 Roller between two plane surfaces—loaded by normal forces Q and tangential forces F.
Poritsky [5] de monst rated the microsli p or creep phen omenon in two dimension s con-
sider ing the actio n of a locomot ive driving wheel , also dry contact . The nor mal load betw een
contact ing cyli nders was assum ed to g enerate a parabolic stress distribut ion, sim ilar to a
Hertzian stress distribut ion, over the co ntact surfa ces as illustr ated in Figure 5.4. Supe rim-
posed on this stress dist ribution with stre sses sz was a tangen tial stre ss t x. In this case, the
local co efficient of friction in the contact is
mx ¼tx
sz
ð 5: 3Þ
Usin g this model, Poritsky demonst rated the existen ce of a ‘‘lo cked’’ region over whi ch no
slip occurs and a region of relative movem ent or slip over a co ntact area for which it was
histo rically assum ed that onl y roll ing oc curred. This is illu strated in Figu re 5.5.
U 2
U 1
Q
Q
tx
t x
s z
R 2
R 1
2b
FIGURE 5.4 Rolling under action of surface tangential stress. (From Johnson, K., Tangential tractions
and micro-slip, Rolling Contact Phenomena, Elsevier, Amsterdam, 1962, pp. 6–28. Reprinted with
permission from American Elsevier Publishing Company.)
� 2006 by Taylor & Francis Group, LLC.
Curve of completeslip
(a)
(b)
(c)
b b
b b
x
x
Lockedregion
Slipregion
FIGURE 5.5 (a) Surface tangential actions; (b) surface strains; (c) locked and microslip regions. (From
Cain, B., J. Appl. Mech., 72, 465, 1950. Reprinted with permission from American Elsevier Publishing
Company.)
Cain [6] further determ ined that in pure roll ing the locked region coinci ded with the
leadi ng edge of the con tact area. It must be emphasi zed that the locked region can only occur
when the friction co efficient is very high as be tween two unlubri cated surfaces.
Heath cote [7] determ ined that a ‘‘hard’’ ball ‘‘rolling’’ in a closely co nforming groo ve can
roll without slid ing only on two na rrow ba nds. Ultim ately, Heat hcote obtaine d a formula for
the roll ing fricti on in this sit uation. While Heath cote slip is v ery simila r to that which occurs
becau se of roll ing elemen t–racewa y de formati on, Heat hcote’ s analys is takes no acco unt of
the ab ility of the surfa ces to elastica lly de form and acco mmodat e the diff erence in surfa ce
veloci ties by different ial expansi on. Jo hnson [8] exp anded on the Heathco te analysis by slicing
an elliptical contact area, such as that in a ball–raceway contact, into differential slabs of area as
shown in Figure 5.6 and thereafter applying the Poritsky analysis for each slab. Johnson’s
analysis using elastic tangential compliance demonstrates a lower coefficient of friction;
this assumes sliding rather than microslip. Figure 5.7 shows the locked and slip regions that
obtain within the contact ellipse.
5.3.2 SLIDING DUE TO ROLLING MOTION : SOLID-FILM OR BOUNDARY LUBRICATION
5.3. 2.1 Direction of Sliding
Eve n tho ugh called rolling bearing s, the major source of fricti on during their operatio n is
slid ing. In Chapter 2, it was demonst rated that sliding occurs in most ball and roller bearing s
due to the macrog eomet ry, that is, basic internal geometry of the bearing . For a radial ba ll
bearing sub jected to a sim ple radial load, Figure 2.7 demon strates that in a singl e contact pure
roll ing can only occur at two points, designated ‘‘A.’’ At all other points along the contact,
sliding must occur in a direction parallel to rolling motion. Outside of points A, sliding occurs
� 2006 by Taylor & Francis Group, LLC.
b
u
z
Locked
Slip
Lines ofrolling
Leadingedge
Q
ba y
yb�
b�
x
a txy
FIGURE 5.6 Ball–raceway contact ellipse showing locked region and microslip region—radial ball
bearing. (From Johnson, K., Tangential tractions and micro-slip, Rolling Contact Phenomena, Elsevier,
Amsterdam, 1962, pp. 6–28. Reprinted with permission from American Elsevier Publishing Company.)
in one direct ion; between points A sli ding occurs in the oppos ite direct ion. The ellip tical
contact area showin g sli ding veloci ty direct ions may be charact erized as shown in Figure 5 .8;
it assum es that the coefficie nt of fri ction is not suffici ently great to cause the possibi lity of a
locked region. This is alwa ys the case for oil-lubri cated be arings, and it is usuall y the case for
bearing s operating effe ctively with soli d-film lubri cants such as molybden um disul fide an d
graphit e.
5.3.2 .2 Slid ing Fric tion
In Chapt er 6, the first volume of this han dbook, the normal stress at any poin t ( x, y) in the
contact was given by the equati on below :
a
b b
Lockedregion
Microslip region
Pure rolling
FIGURE 5.7 (a) Surface tangential actions; (b) surface strains; (c) locked and microslip regions. (From
Johnson, K., Tangential tractions and micro-slip, Rolling Contact Phenomena, Elsevier, Amsterdam,
1962, pp. 6–28. Reprinted with permission from American Elsevier Publishing Company.)
� 2006 by Taylor & Francis Group, LLC.
x
A A
AA
b
a
y
FIGURE 5.8 Ball–raceway elliptical contact area in a radially loaded, radial bearing. Arrows show
sliding direction.
s ¼ 3Q
2p ab1 � x
a
� �2
� y
b
� �2� �1 = 2
ð5: 4Þ
Accor ding to Equat ion 5.3 then at any point ( x, y), surfa ce fricti on shear stre ss parallel to the
roll ing direct ion is given by
ty ¼3mQ
2p ab1 � x
a
� �2
� y
b
� �2� �1= 2
ð5: 5Þ
Frict ion force parall el to the rolling direct ion is calcul ated by integ rating over the con tact area
from �a to þa and � b to þ b. Let ting q ¼ x=a and t ¼ y=b,
Fy ¼3m Q
2p ab
ðþ 1
� 1
ðþffiffiffiffiffiffiffiffi1 � q2p
�ffiffiffiffiffiffiffiffi1 � q2p
1 � q2 � t 2� �1 = 2
dt dq ¼ 3m Q
2p abI ð5: 6Þ
wher e the integ ral I is calcul ated in three pa rts as foll ows:
I1 ¼ c v1
ð� A= a
� 1
ðþ ffiffiffiffiffiffiffi1 � t2p
�ffiffiffiffiffiffiffi1 � t2p
1 � q2 � t 2� �1 =2
d t dq
I2 ¼ c v2
ðþ A =a
� A =a
ðþ ffiffiffiffiffiffiffi1 � t2p
�ffiffiffiffiffiffiffi1 � t2p
1 � q2 � t 2� �1= 2
dt dq ð5: 7Þ
I3 ¼ c v3
ðþ 1
þ A = a
ðþ ffiffiffiffiffiffiffi1� t2p
�ffiffiffiffiffiffiffi1� t 2p
1 � q2 � t2� �1= 2
dt dq
wher e cvn, the sli ding velocity direct ion coefficie nt, is þ 1 or �1 de pending on the direction of
sliding.
Equation 5.6 and Equation 5.7 are valid for operating conditions involving solid-film
lubrication and boundary lubrication where friction coefficient m can be characterized as a
constant.
� 2006 by Taylor & Francis Group, LLC.
5.3.3 S LIDING DUE TO ROLLING MOTION : FULL OIL-FILM LUBRICATION
5.3.3 .1 New tonian Lub ricant
When the lubrican t film co mpletely separat es the rolling surfa ces, New tonian fluid lubri ca-
tion is assume d, givi ng as stated in Chapter 4 the foll owing relationshi p for surfa ce frictio n
shear stress:
t ¼ h›u
›zð4:1Þ
where h is the fluid viscosity, u the fluid velocity in the direction of rolling motion, and z is the
distance into the gap between the rolling contact surfaces. Since the gap is very small compared
with the dimensions of the rolling components, Equation 4.1 can be simplified to
t ¼ hv
hð5:8Þ
where v is the sliding velocity and h is the plateau lubricant film thickness. This equation
assumes constant viscosity. Recall from Chapter 4 that h is a function of the viscosity of the
lubricant entering the contact. For a given lubricant, this viscosity is mainly dependent on
temperature. To calculate surface friction shear stress, however, the viscosity of the lubricant
in the contact must be used. Since this viscosity is not constant, the use of simple Newtonian
lubrication in rolling contact is limited to very low load applications.
5.3.3.2 Lubricant Film Parameter
The parameter L was established during the 1960s to indicate the degree to which a lubricant
film separates the surfaces in rolling ‘‘contact’’:
L ¼ h
s2m þ s2
R
� �1=2 ð5:9Þ
where sm is the root mean square (rms) roughness of the raceway and sR is the rms roughness
of the rolling element. These values are usually obtained as Ra in arithmetic average units;
rms¼ 1.25�Ra. Full-film separation can be assumed for L � 3.
5.3.3.3 Non-Newtonian Lubricant in an Elastohydrodynamic Lubrication Contact
The friction shear stress for a non-Newtonian lubricant does not occur according to Equation
4.1. Several investigators [9–12] examined the effects of non-Newtonian lubricant behavior
in the elastohydrodynamic lubrication (EHL) model. Bell [10] studied the effects of a
Ree–Eyring fluid for which the shear rate is described by Equation 5.10:
g: ¼ t0
hsin h
t
t0
ð5:10Þ
where Eyring stress t0 and viscosity h are functions of temperature and pressure. Houpert [13]
and Evans and Johnson [14] used the Ree–Eyring model for the analysis of EHL traction.
When t is small, Equation 5.10 describes a linear viscous behavior approaching that of a
Newtonian lubricant. It has been established, however, that at high lubricant shear rates, the
� 2006 by Taylor & Francis Group, LLC.
y
y
x
x
ssmax
b
a
FIGURE 5.9 Ellipsoidal surface compressive stress distribution of point contact.
non-New tonian charact eristics tend to c ause de creases in viscosit y. As ind icated, this occurs
unde r cond itions invo lving substan tial slid ing in add ition to rolling. Since the film thickne ss
that obtains is pr imarily a function of the lubrican t pro perties at the inlet to the co ntact, a
non-New tonian lubrican t will not signi ficantly infl uence lubric ant film thickne ss.
Non-N ewtonian lubricati on does, howeve r, signi ficantly influ ence fri ction in the contact.
Bec ause of fricti on, lubri cant temperatur e in the contact rise s cau sing viscos ity to dec rease.
Since pressur e increa ses greatly in, and varie s ov er, the contact , it is eviden t that Equation 4.1
beco mes
t ¼ hðT , pÞ ›u
›z ð5: 11 Þ
Ass uming that the contact areas and pressur e distribut ions are repres ented in Figure 5.9 for
point co ntact an d Figure 5.10 for line contact (as shown in Chapt er 6 in the first volume of
this handbook), Equation 5.11 defines the localized shear stress t at any point (x, y) on the
contact surface. As EHL films are very thin compared with the macrogeometrical dimensions
of the rolling components, it is appropriate to approximate Equation 5.11 as follows:
2b
smax s
yY
X
l
FIGURE 5.10 Semicylindrical surface compressive stress distribution of an ideal line contact.
� 2006 by Taylor & Francis Group, LLC.
t ¼ h T , pð Þ vh
ð 5: 12 Þ
wher e v is sli ding veloci ty an d h is the plateau lubricant film thickne ss. In Chapt er 4, severa l
equati ons wer e presen ted descri bing lubri cant viscos ity vs tempe rature and pressure . Of these,
Equation 4.21 by Bair an d Ko ttke (Ref. [7] of Chapt er 4) or Equation 4.25 reco mmended by
Harr is (Ref. [9] of Chapter 4) may be su bstitute d in Equat ion 5.12 for h (T , p) to he lp ca lculate
t with satisfactory results.
5.3.3 .4 Lim iting Sh ear Stres s
Geci m and W iner [12] and Bair and Winer [15] suggested alternati ve express ions for the
relationshi p be tween shear stre ss and stra in rate incorp orating a lim iting shear stre ss. They
propo sed that for a g iven pressur e, tempe ratur e, and degree of sli ding, there is a maxi mum
shear stre ss that can be susta ined. Based on experi mental data using a disk machi ne, Figure
5.11 from Ref . [16] shows curves of traction coeff icient vs pressur e and slide–roll ratio that
illustr ate this phe nomenon. Tr action coefficie nt is de fined as the ratio of average shear stress
to average normal stre ss. From experiments, Schipper et al. [17] indicated a range of values
for limiting shear stress; for example, 0.07< tlim=pave< 0.11.
5.3.3.5 Fluid Shear Stress for Full-Film Lubrication
Trachman and Cheng [18] and Tevaarwerk and Johnson [19] investigated traction in rolling–
sliding contacts and determined that Equation 4.1 pertains only to a situation involving
relatively low slide-to-roll ratio; for example, less than 0.003 and shown in Figure 5.11.
Following the method of Trachman and Cheng, at a given temperature and pressure it is
possible to define local contact friction as follows:
tf ¼ t�1N þ t�1
lim
� ��1 ð5:13Þ
Mean contact pressure
MPa psi
149,3001030
680
510
400
98,600
73,900
58,000
Thermal region
Slide-to-roll ratio
Tra
ctio
n co
effic
ient
Nonlinear region
Linear region
0 0.01 0.02 0.03 0.04
0.06
0.04
0.02
FIGURE 5.11 Curves of traction measured using a disk machine operating in line contact. (From
Schipper, D., et al., ASME Trans., J. Tribol., 112, 392–397, 1990. With permission.)
� 2006 by Taylor & Francis Group, LLC.
Shear rateS
hear
str
ess
Newtonian shear stress tN
Limiting shear stress t lim
Fluid shear stress t f
FIGURE 5.12 Schematic illustration of Equation 5.13. (From Houpert, L., ASME Trans., J. Lubr.
Technol., 107(2), 241, 1985. With permission.)
wher e tN is the New tonian portion of the fricti on shear stress as defined by Equation 4.1 and
tlim is the maxi mum shear stress that can be susta ined at the contact pr essure. Figure 5.12
schema tica lly demonst rates Equat ion 5.13.
5.3.4 SLIDING DUE TO ROLLING MOTION : PARTIAL OIL -FILM L UBRICATION
5.3. 4.1 Overall Surface Fricti on Shear Stress
Wh en the lubri cant film is insuf ficient to complet ely separate the surfa ces in rolling c ontact,
that is for L< 3, some of the surfa ce peaks, also called asp erities, as illustrated in Figure 5.13,
break through the lubri cant film an d con tact each other. The sli ding fricti on shear stress
during this asperi ty–asperit y interacti on occu rs in the regim e of bounda ry lubri cation and
may be calculated us ing Equation 5.5 for a ba ll–racew ay or point con tact. Only a porti on of
y
x
x
a
v
b
vx
x
v
t
FIGURE 5.13 Distributions of sliding velocity and surface friction shear stress over an elliptical area of
rolling element–raceway contact in a radially loaded, radial ball bearing.
� 2006 by Taylor & Francis Group, LLC.
the contact , howeve r, ope rates in this mann er; the remaind er of the contact su rface operate s
accordi ng to flui d-film lubri catio n; that is, Equat ion 5.13. Therefor e, as given by Harris an d
Barnsby [20], the fri ction shear stre ss acting at any point ( x, y) in the contact may be describ ed
by Equation 5.14:
t ¼ cv
Ac
A0
ma s þ 1 � Ac
A0
t� 1
N þ t� 1lim
� �� 1 ð 5: 14 Þ
wher e Ac is the area associated with asp erity–asper ity contact , A0 is the total contact area, an d
s is the normal stress or contact pressur e. Coef ficient of sliding cv ¼þ1 or � 1 dep ending on
the direction of slid ing velocity. In Equation 5.14, it is ne cessary to define values for tlim an d
m. Thes e values can only be determ ined through full-sc ale be aring testing. Based on com-
paris on of pred icted to test ed bearing he at generat ion rates, tlim can be estimat ed as 0.1 pave
and ma � 0.1 for oil-lubri cated be arings.
For an oil- lubricated, elliptical area con tact, ope rating mainly in rolling mo tion, the
sliding veloci ty and surfa ce frictio n sh ear stre ss distribut ions are ill ustrated in Fig ure 5.13.
5.3.4 .2 Fric tion Force
It can be obs erved from Figure 5.13 that fri ction shear stress t is a strong function of sliding
veloci ty v notwiths tanding the microco ntact porti on of Equation 5.14. The fri ction force
actin g ov er the contact surfa ce is obtaine d by integ ration.
Fy ¼Z
t dA ¼ ab
Zþ 1
� 1
Zþ ffiffiffiffiffiffiffiffi1 � q2p
�ffiffiffiffiffiffiffiffi1 � q2p
cv
Ac
A0
ma s þ 1 � Ac
A0
t� 1
N þ t� 1lim
� �� 1dt dq ð 5: 15 Þ
Cont act pressur e s (or p) at any point (x , y) is determined from Equation 5.4.
At a given tempe ratur e, lubri cant viscos ity in the co ntact might be calcul ated us ing
Equation 4.25:
h ¼ C1 þC2
1 þ e �ðs� C 3 Þ= C 4ð 4: 25 Þ
5.4 REAL SURFACES, MICROGEOMETRY, AND MICROCONTACTS
5.4.1 R EAL SURFACES
To calculate friction force F using Equation 5.15, it is also necessary to determine the ratio Ac=A0.
Therefore, the microgeometry of the rolling contact surfaces must be considered. In calculating
the lubricant film thickness in Chapter 4, the rolling contact surfaces are considered perfectly
smooth. The assumption is now made that the lubricant film thickness calculated using that
assumption separates the mean planes of the ‘‘rough’’ surfaces as illustrated in Figure 5.14.
The surfa ces fluctuate randomly about their mean planes in accordance with a probability
distribution. The rms value of this distribution is denoted s1 for the upper surface and s2 for
the lower surface. When the combined surface fluctuations at a given position exceed the gap
h due to the lubricant film, a microcontact occurs. At the microcontacts, the surfaces deform
elastically and possibly plastically. The aggregate of the microcontact areas is generally a
small fraction (<5%) of the nominal area of contact for 1 � L �3.
� 2006 by Taylor & Francis Group, LLC.
s1
s2
h
FIGURE 5.14 Asperity contacts through partial oil film.
A microco ntact model uses surfa ce micr ogeomet ry data to predict, at a minimum, the
den sity of microco ntacts, the real area of con tact, and the elastica lly supporte d mean load.
One of the earliest and sim plest micr ocontact models is that of Greenwo od and William son
(GW) [21]. Gen eralizat ions of this model app licable to isot ropic surfa ces have been de veloped
by Bush et al. [22] and by O’Cal laghan and Cameron [23] . Bush et a l. [24] also treated a
strong ly anisot ropic surfa ce. One of the most comprehen sive mo dels yet developed is
ASPERSIM [25], which requi r es a nine-parameter microgeometry description and
accou nts for anisotropic as well as isotropic surfaces. A c omparison of various microcon-
tact models conducted by M cCool [26] has shown t hat the GW model, despite its simplicity,
compares favorably with t he other models. Because i t i s much easier to implement than the
other models, t he GW model is the mi crocontact m odel c onsidered here.
5.4.2 GW MODEL
For the contact of real surfaces, Greenwood and Williamson [21] developed one of the first
models that specifically accounted for the random nature of interfacial phenomena. The model
applies to the contact of two flat plastic planes, one rough and the other smooth. It is readily
adapted to the case of two rough surfaces as discussed further. In the GW model, the rough
surface is presumed to be covered with local high spots or asperities whose summits are spherical.
The summits are presumed to have the same radius R, but randomly variable heights, and to be
uniformly distributed over the rough surface with a known density DSUM of summits=unit area.
The mean height of the summ its lies abov e the mean he ight of the surface as a whol e by
the a mount �zzs indica ted in Figure 5.15. The summ it height s z s are assumed to foll ow a
Gauss ian probabil ity law with a standar d deviat ion ss . Figure 5.16 shows the assumed
form for the summ it height distribut ion or probabil ity de nsity fun ction (pdf) f ( zs). It is
symmetrical about the mean summit height. The probability that a summit has a height,
measured relative to the summit mean plane in the interval (zs, zsþdzs), is expressed in terms
of the pdf as f(zs) dzs. The probability that a randomly selected summit has a height in excess
of some value d is the area under the pdf to the right of d. The equation of the pdf is
Summit meanplane
Summit heightdistribution
Surface heightdistribution
Surface meanplane
ZS
FIGURE 5.15 Surface and summit mean planes and distributions.
� 2006 by Taylor & Francis Group, LLC.
wp zsd
Contacts
f (z s)
Summit height distribution
Plastic contacts
FIGURE 5.16 Spherical capped asperity in contact.
f ðzsÞ ¼e�ðzs=2SsÞ2
Ss
ffiffiffiffiffiffi2pp ð5:16Þ
Therefore, the probability that a randomly selected summit has height in excess of d is
P ½zs > d� ¼Z 1
d
f ðzsÞ ds ð5:17Þ
This integration must be performed numerically. Fortunately, however, the calculation can be
related to tabulated areas under the standard normal curve for which the mean is 0 and the
standard deviation is 1.0.
Using the standard normal density function f(x), the probability that a summit has a
height greater than d above the summit mean plane is calculated.
P ½zs > d� ¼Z 1
d=Ss
fðxÞ dx ¼ F0
d
Ss
ð5:18Þ
where F0(t) is the area under the standard normal curve to the right of the value t. Values F0(t)
for t ranging from 1.0 to 4.0 are given in column 2 of Table CD5.1.
It is assumed that when large flat surfaces are pressed together, their mean planes remain
parallel. Thus, if a rough surface and a smooth surface are pressed against each other until the
summit mean plane of the rough surface and the mean plane of the smooth surface are separated
by an amount d, the probability that a randomly selected summit will be a microcontact is
P ½summit is a contact� ¼ P ½zs > d� ¼ F0 ðd=SsÞ ð5:19Þ
As the number of summits per unit area is DSUM, the average expected number of contacts in
any unit area is
n ¼ DSUMF0ðd=SsÞ ð5:20Þ
Given that a summit is in contact because its height zs exceeds d, the summit must deflect by
the amount w¼ zs� d, as shown in Figure 5.16.
For notational simplicity, the subscript in zs is henceforth deleted. For a sphere of radius
R elastically deflecting by the amount w, the Hertzian solution gives the contact area:
� 2006 by Taylor & Francis Group, LLC.
A ¼ pRw ¼ pRðz� dÞ ¼ pa2 z > d ð5:21Þ
where a is the contact radius.
The corresponding asperity load is
Qa ¼ 43E0R1=2w3=2 ¼ 4
3E0R1=2 ðz� dÞ3=2 z > d ð5:22Þ
where E0 ¼ [(1� v12 )=E1þ (1� v2
2)=E2]�1 and Ei, vi (i¼ 1, 2) are Young’s moduli and Poisson’s
ratios for the two bodies. The maximum Hertzian pressure in the microcontact is
s ¼ 1:5P
A¼ 2E0w1=2
pR1=2¼ 2E0
pR1=2
ðz� dÞ1=2 ð5:23Þ
Both A and Qa are functions of the random variable z. The average or expected values of
functions of random variables are obtained by integrating the function and the probability
density of the random variable over the space of possible values of the random variable.
The expected summit contact area is thus
A ¼Z 1
d
pRðz � dÞ f ðzÞ dz ð5:24Þ
which transforms to
A ¼ pRss
Z 1d=ss
x� d
Ss
fx dx ¼ pRSsF1
d
ss
ð5:25Þ
where
F1ðtÞ ¼Z 1
t
ðx� tÞ fx dx ð5:26Þ
F1(t) is also given in Table CD5.1.
The expected total contact area as a fraction of the apparent area is obtained as the
product of the average asperity contact area contributed by a single randomly selected
summit and the density of summits. Thus, the ratio of contact to apparent area, Ac=A0, is
Ac
A0
¼ pRSs DSUMF1
d
Ss
ð5:27Þ
By the same argument, the total load per unit area supported by asperities is
Qa
A0
¼ 4
3E0R1=2Ss
3=2DSUMF3=2d
Ss
ð5:28Þ
where
F3=2ðtÞ ¼Z 1
t
ðx� tÞ3=2 fðxÞ dx ð5:29Þ
F3=2(t) is also given in Table CD5.1.
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5.4.3 P LASTIC C ONTACTS
A contact ing summ it wi ll exp erience some degree of plastic flow when the maxi mum shear
stress exceeds half the yield stre ss in sim ple tension. In the contact of a sphere and a flat, the
maxi mum shear stress is related to the maximum Hertzian stre ss s0 by
tmax ¼ 0: 31 s 0 ð 5: 30 Þ
Thus , some degree of plastic deform ation is present at a co ntact if tmax > Y =2. Usi ng the
express ion for s0 Equat ion 5.23 gives
0: 31 � 2E 0 ðz � d Þ1 = 2
p R1 = 2 >
Y
2 ð 5: 31 Þ
or
z � d > 6 :4RY
E 0
2
� wp ð 5: 32 Þ
z > d þ wp ð 5: 33 Þ
Thus , any summ it whose height exceed s d þ wp will have some degree of plastic de formati on.
The prob ability of a plast ic summ it is given by the shaded area in Figure 5.16 to the right of
d þ wp . The exp ected num ber of plastic contact s pe r unit area be comes
np ¼ D SUM F 0d
Ss
þ wp
ð 5: 34 Þ
wher e
w p �wp
Ss
� 6:4R
Ss
Y
E 0
2
ð 5: 35 Þ
For fixed d=ss the degree of plastic asperi ty inter action is determ ined by the value of wp*: the
higher is wp*, the few er the plast ic con tacts. Accor dingly , GW use the invers e, 1 =wp* , as a
measur e of the plast icity of an inter face. For a given nominal pressur e Q =A0, d=Ss is found by
solvin g Equat ion 5.28 assum ing that most of the load is elastica lly suppo rted.
5.4.4 A PPLICATION OF THE GW MODEL
To use the GW model for a lubri cated co ntact, (1) the he ight d relative to the mean plane of
the summ it heights to h, the thickne ss of the lubrica nt film between the contact surfa ces, an d
(2) values of the GW parame ters R , DSUM , and ss must be establ ished. For (1), the first step is
to calculate the co mposite roughn ess rms value of the tw o su rfaces as
s ¼ s 21 þ s 22� �1 =2 ð 5: 36 Þ
When the mean plane of a rough surfa ce with this rms value is held at he ight h above a
smoot h plane, the rms value of the gap width is the same as shown in Figure 5.17, where both
surfa ces are ro ugh. It is in this sense that the surfa ce con tact of two rough surfa ces may be
trans lated into the e quivalen t contact of a rough surfa ce an d a smoot h su rface. As sh own in
Figure 5.15, the summ it and surface mean planes are sep arated by an amoun t zs.
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Circle of contact
Circle of overlap
Undeflectedshape
Deflectedshape
Contact geometry at summits
R
Summit meanplane
W = Z S– d
d z s
FIGURE 5.17 Distribution of summit heights.
For an isotropic surface with normally distributed height fluctuations, the value of zs has
been found by Bush et al. [22] to be
�zzs ¼4sffiffiffiffiffiffiffipap ð5:37Þ
The quantity a, known as the bandwidth parameter, is defined by
a ¼ m0m4
m22
ð5:38Þ
where m0, m2, and m4 are known as the zeroth, second, and fourth spectral moments of a
profile. They are equivalent to the mean square height, slope, and second derivative of
a profile in an arbitrary direction; that is
m0 ¼ Eðz2Þ ¼ s2 ð5:39Þ
m2 ¼ Edz
dx
2" #
ð5:40Þ
m2 ¼ Ed2z
dx2
!224
35 ð5:41Þ
where z(x) is a profile in an arbitrary direction x, E [ ] denotes statistical expectation, and m0 is
simply the mean square surface height. The square root of m0 or rms is sometimes referred to
as S or Rq and forms part of the usual output of a stylus-measuring device. Some profile-
measuring devices also give the rms slope, which is the same as (m2)1=2 converted from radians
to degrees. No commercial equipment is yet available to measure m4. Measurements of m4
made so far have used custom computer processing of the signal output of profile measure-
ment equipment.
Bush et al. [24] also show that the variance Ss2 of the surface summit height distribution is
related to S2, the variance of the composite surfaces, by
S2s ¼ 1� 0:8968
a
S2 ð5:42Þ
� 2006 by Taylor & Francis Group, LLC.
A summ it locat ed a distanc e d from the summ it height mean plane is at a distance h ¼ d þ�zzs ,
from the surfa ce mean plane. Thus,
d ¼ h � �zzs ð 5: 43 Þ
Usin g Equat ion 5.37 for �zzs and Equat ion 5.42 for s s gives
d
ss
¼ h =s � 4=ffiffiffiffiffiffiffipap
ðð 1 � 0 :8968 Þ=aÞ 1= 2 ð 5: 44 Þ
Equation 5.44 shows that d=ss is linea rly relate d to the lubrican t film parame ter L .
For a specif ied value of L , d=ss is calcul ated from Equation 5.44. For an isotrop ic su rface,
the two parame ters DSUM an d R may be exp ressed as (from Ref. [27])
DSUM ¼m4
6p m2
ffiffiffi3p ð 5: 45 Þ
R ¼ 3
8
ffiffiffiffiffiffip
m4
rð 5: 46 Þ
For an an isotropic surface, the value of m2 will vary with the direction in which the profi le is
taken on the surfa ce. The maxi mum and minimum values oc cur in two orthogonal ‘‘pr inci-
pal’’ directions. Sayles an d Thom as [28] recomm end the use of an equ ivalent isot ropic surfa ce
for which m2 is calcul ated as the ha rmoni c mean of the m 2 values fou nd along the princi pal
direction s. The value of m4 is simila rly taken as the ha rmonic mean of the m4 values in these
two direct ions.
5.4.5 A SPERITY-SUPPORTED AND FLUID -SUPPORTED LOADS
For a specified contact with semi axes a and b, unde r a load Q, with plate au lubri cant film
thickness h and given values of m0, m2, and m4, the load Qa carried by the asperities is
determined by first calculating Q=A0 from Equat ion 5.28 and using
Qa ¼ pabQ
A0
ð5:47Þ
The fluid-supported load is then
Qf ¼ Q�Qa ð5:48Þ
If Qa>Q, the implication is that the lubricant film thickness is larger than that calculated
using smooth surface theory. In this case, Equation 5.28 must be solved iteratively until
Qa¼Q.
See Example 5.1.
5.4.6 SLIDING DUE TO ROLLING MOTION: ROLLER BEARINGS
5.4.6.1 Sliding Velocities and Friction Shear Stresses
For roller bearings operating with predominantly rolling motion, the roller–raceway contact
friction analyses are very similar to those described for ball–raceway contacts. As indicated in
� 2006 by Taylor & Francis Group, LLC.
Z
b
l
x
x
xty
vy
l eff
l s
FIGURE 5.18 Distributions of sliding velocity and surface friction shear stress over an area of crowned
roller–raceway contact in a cylindrical roller bearing under load. Roller crowning is illustrated in the
uppermost drawing. In this contact, ideal normal stress distribution is not achieved.
Chapt er 6 in the first vo lume of this handb ook, rollers and raceways are crow ned to avoid or
mini mize edge loading , and unde r applie d load the contact surfa ce is curved in the plane
passi ng through the bearing axis of rotation and the center of roll ing contact . Pur e rolling is
define d by inst ant center s at whi ch no relat ive motion of the co ntacting elem ents occurs; that
is, the surfaces have the same velocitie s at su ch poin ts. Therefor e, in a radial , cylin drical roller
bearing having crow ned compon ents, only tw o points of pure rolling can exist on the major
axis of ea ch contact surfa ce. At all other points sliding must occur. The same is basica lly true
for the ro ller–raceway con tacts in rad ial, sp herical, and tapere d roller bearing s. Figu re 5.18
schema tica lly de picts slid ing velocitie s and surface friction shear stre sses in a crown ed
cylindrical roller–raceway contact.
5.4.6.2 Contact Friction Force
As de monstrated in Chapt er 1 and Chapt er 3, the fri ction force ov er the contact is ca lculated
by dividing the contact into n laminae; then,
. Establishing the normal stress distribution over each lamina k
. Determining the average lubricant viscosity hk using a pressure–viscosity relationship at
contact temperature. Calculating the plateau lubricant film thickness and subsequently Ac=A0 using the GW
method. Determining sliding velocities vk based on contact deformation criteria
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. Calculat ing the surfa ce friction shea r stre ss tk for each lamin a k using Equat ion 5.14
. Usin g Simps on’s rule, numeri cally integrati ng tk � A k across the co ntact, where Ak ¼ 2
bk � w, the width of a lami na
Depend ing on the geomet ries of the roll ing co mponents and the amount of normal loading
between them, sliding mo tions that acco mpany the essential rolling motion can vary in
signifi cance with regard to the frictio n g enerated due to roll ing. Genera lly, for mainl y ro lling
motio n, the amount of roll ing c ontact fricti on tends to be smal l.
5.4.7 S LIDING DUE TO SPINNING AND GYROSCOPIC MOTIONS
5.4.7 .1 Slid ing Veloci ties and Frictio n Shear S tresses
Ball bearing s that ope rate with nonzero contact angles; for exampl e, angular -conta ct an d
thrust ball bearing s, exp erience spinni ng contact moti ons, and g yroscopi c moment s that cause
gyroscopi c motion s. Nonzero contact an gle roller bearing s also experi ence spinni ng mo tions;
howeve r, gyroscopi c mo ments are resisted by nonuni form roller–rac eway loading per unit
lengt h. Spinn ing moti ons and gyroscopi c mo tions in ba ll bearing s wer e discus sed in Chapt er
2. The sliding veloci ty distribut ion and surfa ce friction shear stre ss dist ribut ion over a load ed
angular -conta ct ball be aring contact that experiences rolling, spinni ng and gyroscopi c mo-
tions is illustra ted in Figure 5.19. In Figu re 5.19, v y is the sliding velocity in the direct ion of
rolling, and v x is the slid ing veloci ty in the direct ion transverse to rolling, caused by gyro-
scopic motion; vy gives rise to friction shear stress component ty, and vx gives rise to friction
shear stress component tx. This is shown by expanding Equation 5.14 as follows:
ty ¼ cv
Ac
A0
ma s þ 1� Ac
A0
h
hvy
þ 1
tlim
� 1
ð 5:49Þ
y
b
a
x
x
ny
ty t x
nx ny
t y
FIGURE 5.19 Distributions of sliding velocity and surface friction shear stress over an elliptical area of
rolling element–raceway contact in an angular-contact ball bearing.
� 2006 by Taylor & Francis Group, LLC.
tx ¼ c vAc
A0
ma s þ 1 � Ac
A0
h
hvx
þ 1
tlim
� 1
ð5: 50 Þ
As an alternati ve to Equat ion 5.49 and Equation 5.50, Harris [29] used 3 7 sets of data of
traction c oefficient vs slid e-to-rol l rati o and L collected on a v-ri ng-single ball test rig to
generat e the foll owing empir ical relationshi p:
m ¼ �2: 066 � 10 � 3 þ 2: 612 � 10 � 6 1
Lln
h
h0
� �2
�5: 605 � 10 � 2 v
Uln
v
U
� �h ið5: 51 Þ
wher e h is the lubri cant v iscosity at contact pressur e, h0 is the lubri cant viscos ity at
atmos pheric pressur e, a nd U is the roll ing velocity. Tractio n coefficie nt m is directional ;
that is, my or mx and was developed co nsidering average normal stress over the con tact. It
might , howeve r, be co nsidered as occu rring at a point in a con tact such that ty ¼m ys and
tx ¼m xs . The lubri cant used during the v-rin g-ball testing was a Mil-L-236 99 polyole ster.
5.4. 7.2 Contact Friction Fo rce Com ponents
The fricti on force componen ts in the rolling direction Fy and in the gy roscopi c direction Fx
may be determined by integ ration ov er the contact area. Accor dingly,
Fy ¼Z
ty dA ¼ ab
Zþ 1
� 1
Zþ ffiffiffiffiffiffiffiffi1 � q2p
�ffiffiffiffiffiffiffiffi1 � q2p
cv
Ac
A0
ma s þ 1 � Ac
A0
h
h vy
þ 1
tlim
� 1
dt d q ð5: 52 Þ
Fx ¼Z
tx dA ¼ ab
Zþ 1
� 1
Zþ ffiffiffiffiffiffiffiffi1� q 2p
�ffiffiffiffiffiffiffiffi1� q 2p
cv
Ac
A0
ma s þ 1 � Ac
A0
h
hvx
þ 1
tlim
� 1
dt dq ð5: 53 Þ
Jon es [30] assum ed that gyroscopi c motion could be preven ted if the ball–racew ay fricti on
coeff icient was suffici ently great . Harris [31] de monst rated the inaccu racy of the Jones
assum ption; but, that while gyroscopi c motio n cannot be preven ted in the presence of a
gyroscop ic momen t, its speed is nevert heless ve ry smal l compared with ball speeds abou t the
two orthog onal axes.
5.4.8 SLIDING IN A T ILTED ROLLER –RACEWAY C ONTACT
In Chapt er 1, it was sh own that roll ers in cyli ndrical roll er or tapere d roller bea rings subject ed
to moment loading or misali gnment that c auses moment loading unde rgo tilt an gles zj to
accomm oda te the applie d load; the subscri pt refer s to the roller azim uth locat ion. Similarly,
cyli ndrical rollers subject ed to thrust load undergo tilt angles . Thus, the normal loading on each
con tact is nonuni form. Fi gure 5.20 depict s the slid ing velocitie s and surface frictio n shear
stresses in a crowned cylindrical roller–raceway contact over which the loaded roller is tilted.
5.5 CLOSURE
This chapter contains a generalized approach to predicting surface friction stresses and forces
for rolling element–raceway contacts; that is, both solid-film lubrication and oil-lubrication
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z
bvyvy
ty
ty
l
x
y
leff
ls
x
x
FIGURE 5.20 Distributions of sliding velocity and surface friction shear stress over an area of crowned
roller–raceway contact in a cylindrical roller bearing under load. The roller is tilted over the contact to
accommodate bearing misalignment or applied thrust load. Roller crowning is illustrated in the upper-
most drawing.
conditions are considered. In the former case, Coulomb friction is assumed and the direction
of friction shear stress at a given surface point is dictated by the direction of sliding motion at
that point. With regard to oil-lubrication, the approach is taken to predicting key perform-
ance-related parameters descriptive of real EHL contacts. These parameters include true
contact area, plastic contact area, fluid and asperity load sharing, and the relative contribu-
tions of the fluid and asperities to overall friction. It is recognized that, using more elegant
and complex analytical methods such as very fine mesh, multithousand node, finite-element
analysis together with solutions of the Reynolds and energy equations in three dimensions, it
is possible to obtain a more generalized solution with perhaps increased accuracy. Unfortu-
nately, using currently available computing equipment, such solutions would require several
hours of computational time to enable the performance analysis of a single operating
condition for a rolling bearing containing only a small complement of rolling elements.
The equations provided in this chapter for frictional shear stress are based on the
assumption of Hertz pressure (normal stress) applied over the contact. In the case of oil-
lubricated bearings, the Hertzian stress distribution is assumed to be unmodified by EHL
conditions. This assumption is sufficiently accurate for most rolling element–raceway con-
tacts in that such loading is reasonably heavy; for example, generally at least several hundred
MPa . Fur thermo re, the assum ption is made that Equation 5.14 can be applie d at every point
� 2006 by Taylor & Francis Group, LLC.
in the contact. With respect to the Coulomb fri ction compo nent of surfa ce shear stress, it is
recogn ized that surfa ce roughness peak s cau se local pressur es in excess of Hert zian values and
these wi ll cau se local ized shear stre sses in excess of those predict ed by Equat ion 5.14.
Accommodation of these variations tends to increase the computational time beyond current
engineering practicality. Therefore, for engineering purposes, frictional shear stress may be
calculated according to the average condition in each contact.
REFERENCES
1.
� 200
Drutowski, R., Energy losses of balls rolling on plates, Friction and Wear, Elsevier, Amsterdam,
1959, pp. 16–35.
2.
Greenwood, J. and Tabor, D., Proc. Phys. Soc. London, 71, 989, 1958.3.
Drutowski, R., Linear dependence of rolling friction on stressed volume, Rolling Contact Phenom-ena, Elsevier, Amsterdam, 1962.
4.
Reynolds, O., Philos. Trans. R. Soc. London, 166, 155, 1875.5.
Poritsky, H., J. Appl. Mech., 72, 191, 1950.6.
Cain, B., J. Appl. Mech., 72, 465, 1950.7.
Heathcote, H., Proc. Inst. Automob. Eng., London, 15, 569, 1921.8.
Johnson, K., Tangential tractions and micro-slip, Rolling Contact Phenomena, Elsevier, Amster-dam, 1962, pp. 6–28.
9.
Sasaki, T., Mori, H., and Okino, N., Fluid lubrication theory of rolling bearings parts I and II,ASME Trans., J. Basic Eng., 166, 175, 1963.
10.
Bell, J., Lubrication of rolling surfaces by a Ree–Eyring fluid, ASLE Trans., 5, 160–171, 1963.11.
Smith, F., Rolling contact lubrication—the application of elastohydrodynamic theory, ASMEPaper 64-Lubs-2, April 1964.
12.
Gecim, B. and Winer, W., A film thickness analysis for line contacts under pure rolling conditionswith a non-Newtonian rheological model, ASME Paper 80C2=LUB 26, August 8, 1980.
13.
Houpert, L., New results of traction force calculations in EHD contacts, ASME Trans., J. Lubr.Technol., 107(2), 241, 1985.
14.
Evans, C. and Johnson, K., The rheological properties of EHD lubricants, Proc. Inst. Mech. Eng.,200(C5), 303–312, 1986.
15.
Bair, S. and Winer, W., A rheological model for elastohydrodynamic contacts based on primarylaboratory data, ASME Trans., J. Lubr. Technol., 101(3), 258–265, 1979.
16.
Johnson, K. and Cameron, A., Proc. Inst. Mech. Eng., 182(1), 307, 1967.17.
Schipper, D., et al., Micro-EHL in lubricated concentrated contacts, ASME Trans., J. Tribol., 112,392–397, 1990.
18.
Trachman, E. and Cheng, H., Thermal and non-Newtonian effects on traction in elastohydrody-namic contacts, Proc. Inst. Mech. Eng., 2nd Symposium on Elastohydrodynamic Lubrication,
Leeds, 1972, pp. 142–148.
19.
Tevaarwerk, J. and Johnson, K., A simple non-linear constitutive equation for EHD oil films, Wear,35, 345–356, 1975.
20.
Harris, T. and Barnsby, R., Tribological performance prediction of aircraft turbine mainshaft ballbearings, Tribol. Trans., 41(1), 60–68, 1998.
21.
Greenwood, J. and Williamson, J., Contact of nominally flat surfaces, Proc. R. Soc. London, Ser. A.,295, 300–319, 1966.
22.
Bush, A., Gibson, R., and Thomas, T., The elastic contact of a rough surface, Wear, 35, 87–111,1975.
23.
O’Callaghan, M. and Cameron, M., Static contact under load between nominally flat surfaces,Wear, 36, 79–97, 1976.
24.
Bush, A., Gibson, R., and Keogh, G., Strongly anisotropic rough surfaces, ASME paper 78-LUB-16, 1978.
25.
McCool, J. and Gassel, S., The contact of two surfaces having anisotropic roughness geometry,ASLE Special Publication (SP-7), 29–38, 1981.
6 by Taylor & Francis Group, LLC.
26.
� 200
McCool, J., Comparison of models for the contact of two surfaces having anisotropic roughness
geometry, Wear, 107, 7–60, 1986.
27.
Nayak, P., Random process model of rough surfaces, ASME Trans., J. Tribol., 93F, 398–407, 1971.28.
Sayles, R. and Thomas, T., Thermal conductances of a rough elastic contact, Appl. Energy, 2,249–267, 1976.
29.
Harris, T., Establishment of a new rolling bearing fatigue life calculation model, Final Report U.S.Navy Contract N00421–97-C-1069, February 23, 2002.
30.
Jones, A., Motions in loaded rolling element bearings, ASME Trans., J. Basic Eng., 1–12, 1959.31.
Harris, T., An analytical method to predict skidding in thrust-loaded, angular-contact ball bearings,ASME Trans., J. Lubr. Techol., 93, 17–24, 1971.
6 by Taylor & Francis Group, LLC.
6 Friction Effects in RollingBearings
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LIST OF SYMBOLS
Symbol Description Units
a Semimajor axis of projected contact ellipse mm (in.)
Ac True average contact area mm2 (in.2)
A0 Apparent contact area mm2 (in.2)
A1 Ball center axial position variable mm (in.)
A2 Ball center radial position variable mm (in.)
b Semiminor axis of projected contact ellipse mm (in.)
B fiþ fo� 1
D Roller or ball diameter mm (in.)
E1, E2 Elastic moduli of bodies 1 and 2 MPa (psi)
E0 Reduced elastic modulus MPa (psi)
f r=DF Contact friction force N (lb)
Fc Centrifugal force N (lb)
FCL Friction force between cage rail and ring land N (lb)
g Gravitational constant mm=sec2 (in.=sec2)
h Lubricant film thickness mm (in.)
hc Central or plateau lubricant film thickness mm (in.)
J Mass moment of inertia kg. � mm2 (in. � lb �
cis Group, LLC.
sec2)
l Roller length end-to-end mm (in.)
leff Effective roller length mm (in.)
ls Roller straight length mm (in.)
M Moment N � mm (in. � lb)
Mg Gyroscopic moment N � mm (in. � lb)
q x=aQ Roller or ball load N (lb)
Qa Roller end–guide flange load N (lb)
QCG Cage web–rolling element load N (lb)
R Radius of deformed contact surface mm (in.)
t y=bT Temperature 8C (8F)
u Surface velocity mm=sec (in.=sec)
v Sliding velocity mm=sec (in.=sec)X1 Ball center axial position variable mm (in.)
X2 Ball center radial position variable mm (in.)
w Width of a lamina, width mm (in.)
W Lubricant flow rate through bearing cm3=mm (gal=min.)
Z Number of rolling elements
g Shear rate sec�1
da Bearing axial deflection mm (in.)
d Contact deformation mm (in.)
z 2f=(2fþ1)
z Roller tilting angle 8, rad
h Lubricant viscosity cp (lb � sec=in.2)
m Coefficient of friction for boundary or
solid-film lubrication
n1, n2 Poisson’s ratio for bodies 1 and 2
r Radius mm (in.)
j Lubricant effective density g=mm3 (lb=in.3)
j1 Lubricant density g=mm3 (lb=in.3)
j Roller skewing angle 8, rad
s Normal contact stress or pressure MPa (psi)
s0 Maximum normal contact stress or pressure MPa (psi)
t Shear stress MPa (psi)
v Rotational speed rad=secV Ring rotational speed rad=sec
Subscripts
CG Cage
CL Cage land
CP Cage pocket
CR Cage rail
g Gyroscopic motion
i Inner raceway
j Rolling element location
n Outer or inner raceway or ring, o or i
m Cage or orbital motion
o Outer raceway
R Roller
x0 x0 Direction
y0 y0 Direction
z0 z0 Direction
l Lamina
6.1 GENERAL
In Chapter 5, the sources and magni tudes of fricti on in ball–racew ay an d roll er–raceway
contacts were defined. While these are the salient considerations in the study of effects
of friction on rolling bearing performance, other sources of friction in the bearing can have
significant and even overriding effects on bearing performance. For example, the type of oil
lubrication and the amount of lubricant in the bearing, and the interaction of the cage with
� 2006 by Taylor & Francis Group, LLC.
the roll ing eleme nts and wi th piloting su rfaces on the be aring rings are impor tant sources of
fricti on. Als o, the inter actio n of integral contact seals with bearing rings will general ly
have a friction effe ct substa ntially greater than a ll of the other sou rces heretof ore indica ted.
Seal friction , howeve r, is not a topic explore d in de tail in this text .
Rolling elemen t speed s can be signifi cantly influ enced by fri ction, affecti ng rolling elem ent
centrifugal forces , gyroscop ic moment s, and be aring endu rance. Excessi ve fricti on at high
speeds can cause rolling elements to unde rgo gross sliding over the racew ays. This motio n
called skiddi ng can reduc e bearing en durance. Fr iction ca n have ancillary, but impor tant,
effects on bearing perfor mance. In rolle r bearing s, fri ction be tween roller ends and guide ring
flang es can cause rollers to skew, shorte ning bearing endu rance. All of these effec ts will be
discus sed in this ch apter.
6.2 BEARING FRICTION SOURCES
6.2.1 SLIDING IN ROLLING ELEMENT–RACEWAY CONTACTS
As indicated above, this salient feature of rolling bearing performance is discussed in detail in
Chapt er 5.
6.2.2 VISCOUS DRAG ON ROLLING ELEMENTS
In fluid-lubricated rolling bearings, during operation a certain amount of lubricant occupies
the free space within the boundaries of the bearing. Because of their orbital motion, the balls
or rollers must force their way through this fluid; the viscous fluid creates a drag force that
retards the orbital motion. The fluid within the bearing free space is a mixture of gas (usually
air) and lubricant. It is assumed that the drag caused by the gaseous atmosphere is insignifi-
cant; rather, the drag force depends on the quantity of the lubricant dispersed in the gas–
lubricant mixture. Therefore, the mixture has an effective viscosity and an effective specific
gravity. The viscous drag force acting on a ball as indicated in Ref. [1] can be approximated
by
Fv ¼cvpjD2 dmvmð Þ1:95
32gð6:1Þ
where j is the weight of the lubricant in the bearing free space divided by the volume of the
free space. Similarly, for an orbiting roller,
Fv ¼cvjlD dmvmð Þ1:95
16gð6:2Þ
The drag coefficients cv in Equation 6.1 and Equation 6.2 can be obtained from Ref. [2]
among many others.
From the testing of ball bearings operating with circulating oil lubrication, Parker [3]
established an empirical formula to estimate the percentage of the bearing free space occupied
by the fluid lubricant. Using Parker’s formula, it is possible to calculate the effective fluid
density j as indicated in the following equation:
j ¼ jlW0:37
nd1:7m
� 105 ð6:3Þ
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Inner ringland riding
Ballriding
Outer ringland riding
FIGURE 6.1 Cage types.
6.2.3 SLIDING BETWEEN THE CAGE AND THE BEARING RINGS
Three basic cage types are used in ball and roller bearings: (1) ball riding (BR) or roller riding
(RR), (2) inner ring land riding (IRLR), and (3) outer ring land riding (ORLR). These are
illustrated schematically in Figure 6.1.
BR and RR cages are usually of relatively inexpensive manufacture and are usually not
used in critical applications. The choice of an IRLR or ORLR cage depends largely on the
application and designer preference. An IRLR cage is driven by a force between the cage rail
and inner ring land as well as by the rolling elements. ORLR cage speed is retarded by
cage rail=outer ring land drag force. The magnitude of the drag or drive force between the
cage rail and ring land depends on the resultant cage=rolling element loading, the eccentricity
of the cage axis of rotation, and the speed of the cage relative to the ring on which it is piloted.
If the cage rail=ring land normal force is substantial, hydrodynamic short bearing theory [4]
might be used to establish the friction force FCL. For a properly balanced cage and a very
small resultant cage=rolling element load, Petroff’s law can be applied; for example,
FCL ¼hpwCRcndCRðvc � vnÞ
1� ðd1=d2Þco¼ 1
ci ¼ �1ð6:4Þ
where d2 is the larger of the cage rail and ring land diameters and d1 is the smaller.
6.2.4 SLIDING BETWEEN ROLLING ELEMENTS AND CAGE POCKETS
At any given azimuth location, there is generally a normal force acting between the rolling
element and its cage pocket. This force can be positive or negative depending on whether the
rolling element is driving the cage or vice versa. It is also possible for a rolling element to be
free in the pocket with no normal force exerted; however, this situation will be of less usual
occurrence. Insofar as rotation of the rolling element about its own axes is concerned, the
cage is stationary. Therefore, pure sliding occurs between rolling elements and cage pockets.
The amount of friction that occurs thereby depends on the rolling element–cage normal
loading, lubricant properties, rolling element speeds, and cage pocket geometry. The last
variable is substantial in variety. Generally, application of simplified elastohydrodynamic
theory should suffice to analyze the friction forces.
6.2.5 SLIDING BETWEEN ROLLER ENDS AND RING FLANGES
In a tapered roller bearing and in a spherical roller bearing with asymmetrical rollers,
concentrated contact always occurs between the roller ends and the inner (or outer) ring
� 2006 by Taylor & Francis Group, LLC.
(a) (b) (c)
FIGURE 6.2 Contact types and pressure profiles between sphere end rollers and flanges in a spherical
roller thrust bearing.
flange owing to a force component that drives the rollers against the flange. Also, in a radial
cylindrical roller bearing, which can support thrust load in addition to the predominant radial
load by virtue of having flanges on both inner and outer rings, sliding occurs simultaneously
between the roller ends and both inner and outer rings. In these cases, the geometries of
the flanges and roller ends are extremely influential in determining the sliding friction between
those contacting elements.
The most general case for roller end–flange contact occurs, as shown in Figure 6.2, in a
spherical roller thrust bearing. The different types of contact are illustrated in Table 6.1 for
rollers having sphere ends.
Rydell [5] indicates that optimal frictional characteristics are achieved with point
contacts between roller ends and flanges. Additionally, Brown et al. [6] studied roller end
wear criteria for high-speed cylindrical roller bearings. They found that increasing roller
TABLE 6.1Roller End–Flange Contact vs. Geometrya
Flange Geometry Type of Contact
a Portion of a cone Line
b Portion of sphere, Rf¼Rre Entire surface
c Portion of sphere, Rf>Tre Point
aRf is the flange surface radius of curvature; Rre is the roller end radius of curvature.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 6.3 Deep-groove ball bearing with integral seal.
corner radius runou t tends to increa se wear. Increas ing roll er end clear ance and l =D ratio
also tend towa rd increased ro ller wear, but are of lesser con sequence than roll er corner
radius runout.
6.2.6 SLIDING F RICTION IN S EALS
Man y roll ing be arings, particu larly grease-lubr icate d, deep -groove ba ll bearing s, are assem-
bled with integ ral seals . As illustrated in Figure 6.3, such seals general ly consis t of an
elasto meric mate rial partiall y encased in a steel or plast ic carrier. The elasto meric seali ng
bears (rides) eithe r on a ring land or on a sp ecial recess or groove cut into the inner ring as
shown in Figure 6.3. In an y case, the seal fri ction due to sliding between the elast omer and
bearing ring surface normal ly exceeds the total of all other so urces of fricti on in the bearing
unit. The technol ogy of seal fricti on de pends frequen tly on the specif ic mechani cal structure
of the seal and on the prop erties of the elastomer ic mate rial. Anal ysis of seal fri ction is not
covered in this text.
6.3 BEARING OPERATION WITH SOLID-FILM LUBRICATION: EFFECTSOF FRICTION FORCES AND MOMENTS
6.3.1 BALL BEARINGS
In Chapter 5, it was sho wn that friction in soli d-film lubri cated ball– raceway contacts could
be analyzed considering Coulomb friction; that is, surface friction shear stress t at a given
point (x, y) in the contact surface to be represented as ms, m is a coefficient of friction and s is
� 2006 by Taylor & Francis Group, LLC.
Bearing axis
Y�
Y�
z'
y�
Mgy9
x�
x�
x�
Z�
Fyi
a omQo
mQ iQ i
Qo
a iFy i
Fc
Fyo
Fy o
Z�
FIGURE 6.4 Forces and moments acting on a ball.
the normal stress at point (x, y). With this assumption, Harris [7] achieved a general solution
entailing equilibrium of forces and moments for a thrust-loaded angular-contact ball bearing. In
this case, the forces and moments acting on a bearing ball were as shown in Figure 6.4. It was
also assumed that the gyroscopic motion about the y0 axis is negligible, and the elliptical areas
of contact could be divided into two or three zones of sliding as illustrated in Figure 6.5.
Now, for the ball–raceway contacts as shown in Figure 6.5,
Bearing axis
x�
x
Outer deformedcontact surface
Inner deformedcontact surface
Line of zeroslip
Line of zeroslip
b
w R a i
a oT o1
a iT i1
a iT i2
a oT o2
2a o
2a i
ao
FIGURE 6.5 Contact areas, rolling lines, and slip directions.
� 2006 by Taylor & Francis Group, LLC.
Fy0n ¼ 2manbncn
ZTn1
�1
Zffiffiffiffiffiffiffiffi1�q2p
0
sn dt dq�ZTn2
Tn1
Zffiffiffiffiffiffiffiffi1�q2p
0
sndt dq�Zþ1
Tn2
Zffiffiffiffiffiffiffiffi1�q2p
0
sn dt dq
0BB@
1CCA ð6:5Þ
where q¼ x0=an, t¼ y0=bn, Tn1 and Tn2 define lines of rolling motion, n refers to inner or outer
ball–raceway contact, that is n¼ i or o, and sn the normal stress or pressure at any point in the
contact surface, in accordance with the following equation, is given by
sn ¼3Qn
2panbn
1� q2 � t2� �1=2 ð5:4Þ
Substituting Equation 5.4 in Equation 6.5 and integrating yields
Fy0n ¼ 3mQncn
2
3þXk¼2
k¼1
ckTnk 1� T2nk
3
� �" #
n ¼ o, i; co ¼ 1; ci ¼ �1; c1 ¼ 1, c2 ¼ �1
ð6:6Þ
From Figure 2.13 and Figure 2.14, radii rn from the ball center to points on the contact areas
are given by
rn ¼ R2n � x2
n
� �1=2� R2n � a2
n
� �1=2þ D
2
� �4
�a2n
" #1=2
n ¼ o, i ð6:7Þ
Using Equation 6.7 and Equation 5.4, the equation for friction moments is
Mx0n¼2manbncn
ZTn1
�1
Zffiffiffiffiffiffiffiffi1�q2p
0
snrn cosðanþunÞdtdq�ZTn2
Tn1
Zffiffiffiffiffiffiffiffi1�q2p
0
snrn cosðanþunÞdtdq
2664
3775
þ 2manbncn
Z1
Tn2
Zffiffiffiffiffiffiffiffi1�q2p
0
snrn cosðanþunÞdtdq
2664
3775
ð6:8Þ
In Equation 6.8, sin un¼ xn0=rn. Using the trigonometric identity,
cos ðan þ unÞ ¼ cos an cos un � sin an sin un ð6:9Þ
As un is small, cos un ) 1. Substituting into Equation 6.8 and integrating yields
Mx0n ¼ 3mQnDcn
2
3cos an þ
Xk¼2
k¼1
ckTnk 1� T2nk
3
� �cos an �
anTnk
D1� T2
nk
2
� �sin an
� �( )
n ¼ o, i; c0 ¼ 1, ci ¼ �1; c1 ¼ 1, c2 ¼ �1
ð6:10Þ
� 2006 by Taylor & Francis Group, LLC.
Similarl y,
Mz 0 n ¼ 3m Qn Dcn
� 2
3sin an þ
Xk ¼ 2
k ¼ 1
ck T nk
(
� 1 � T 2nk
3
� �sin an �
aTnk
D1 � T 2nk
2
� �cos an
� �n ¼ o, i; co ¼ 1; c i ¼ �1
k ¼ 1, 2; c1 ¼ 1; c 2 ¼ �1
ð 6: 11 Þ
Usin g Figu re 6.4, it can be establis hed that four condition s of force and moment equilibrium
abou t the x0 , y0 , and z 0 ax es must be satisfi ed togeth er wi th four ba ll posit ion equati ons
determ ined in Chapt er 3. These eight e quations must be solved for two posit ion variables, tw o
contact deformati ons, bearing axial de flection, an d sp eed vm, vx 0 , and v z 0 .
Thus, there are eight eq uations and eight unknown s; howeve r, the rolling lines Tnk, of
which there are three as shown in Figure 6.5, are functio ns of speed vm, vx0, and vz0. To
establish the required relationship, the major axes of the deformed contact surfaces as shown
in Figure 2.13 and Figure 2.14 are considered arcs of great circles defined by
ðx0n � XÞ2 þ ðz0n � ZÞ2 � ðznDÞ2 ¼ 0 ð6:12Þ
where z¼ 2f=(2fþ 1) and f¼ r=D. From Figure 2.13 and Figure 2.14, it can be determined
that the offset of the ball center from the circle center is given by the coordinates
X ¼ D
2½ð4z2
n � k2nÞ
1=2 � ð1� k2nÞ
1=2� sin an ð6:13Þ
Z ¼ D
2½ð4z2
n � k2nÞ
1=2 � 1� k2n
� �1=2� cos an ð6:14Þ
where kn¼ 2an=D. Zero sliding velocity is determined from the equations
ð�o � vmÞdm
2þ z0
� �þ vxz
0 þ vzx0 ¼ 0 ð6:15Þ
ðvm � �iÞdm
2þ z0
� �þ vxz
0 þ vzx0 ¼ 0 ð6:16Þ
Equation 6.12, Equation 6.15, and Equation 6.16 can be solved simultaneously to yield x0nk,
z0nk locations at which zero sliding velocity occurs on the deformed surface circle. It can be
shown that
Tnk ¼x0nk2þ z0nk2� �
an
sinp
2� an � tan�1 z0nk
x0nk
� �� �, k ¼ 1, 2 ð6:17Þ
Using this method Harris [7] was able to prove the impossibility of an ‘‘inner raceway
control’’ situation, even with bearings operating with ‘‘dry film’’ lubrication. Moreover, a
speed transition point seems to occur in a thrust-loaded angular-contact ball bearing at which
a radical shift of the ball speed pitch angle b must occur to achieve load equilibrium in the
� 2006 by Taylor & Francis Group, LLC.
0.424
0.422
0.420
0.418
2,0000
Orb
it–to
–sha
ft sp
eed
ratio
6,000 8,000 10,0004,000
Shaft speed (rpm)
Bearing design data
Ball diameter 8.731 mm (0.34375 in.)Pitch diameter 48.54 mm (1.9110 in.)Free contact angle 24.5�Inner raceway grove radius/ball diameter 0.52Outer raceway groove radius/ball diameter 0.52Thrust load per ball 31.6 N (7.1 1b)
FIGURE 6.6 Orbit=shaft speed ratio vs. shaft speed for a thrust-loaded angular-contact ball bearing.
(From Harris, T., ASME Trans., J. Lubr. Technol., 93, 32–38, 1971. With permission.)
bearing . Fi gure 6.6 and Figure 6.7 from Ref. [7] illustrate the resul ts of this analyt ical method
for a thrust -loade d angular -conta ct ball be aring.
Additional ly, Table 6.2 shows the corresp onding location s of rolling lines in the inner and
outer contact ellipses for this example.
38
34
32
30
28
26
24
22
20
18
16
0 2,000 4,000 6,000 8,000 10,000
36
Innerracewaycontrol
Outer raceway controlPitc
h an
gle
(deg
rees
)
Shaft speed (rpm)
FIGURE 6.7 Ball speed vector pitch angle vs. shaft speed for a thrust-loaded angular-contact ball
bearing. (From Harris, T., ASME Trans., J. Lubr. Technol., 93, 32–38, 1971. With permission.)
� 2006 by Taylor & Francis Group, LLC.
TABLE 6.2Locations of Lines of Zero Sliding in Elliptical Contact Areas of a Thrust-
Loaded Angular-Contact Ball Bearing
Shaft Outer Raceway Inner Raceway
T1 T2 T1 T2
1000 0.0001 — �0.00605 0.92123
1500 0.00183 — �0.00672 0.92376
2000 0.00129 — �0.00537 0.93140
2500 0.00047 — �0.00353 0.94272
3000 — 0.02975 0.02995 —
3500 — �0.00156 — �0.00190
4000 �0.95339 0.00156 — 0.00052
4500 �0.93237 0.00376 — 0.00064
5000 �0.91449 0.00627 — 0.00077
5500 �0.89730 0.01055 — �0.00039
Source: From Harris, T., ASME Trans., J. Lubr. Technol., 93, 32–38, 1971.
6.3.2 R OLLER B EARINGS
A sim ilar approach may be app lied to roller bearing s that have point co ntact at each raceway.
Usual ly, howeve r, roll er bearing s are de signed to operate in the line co ntact or modif ied line
contact regim e. In the form er, the area of roller–rac eway co ntact is basica lly recta ngu lar, with a
‘‘dogbone’ ’ effe ct at the lengt hwise limits. Thi s is discus sed in Chapt er 6 of the first vo lume of
this handbook. The dogbone portion of the contact occupies only a very small area and
therefore does not influence friction significantly. In modified line contact (achieved as a result
of crowned profile roller or raceway or both), the contact area is approached analytically as
elliptical in shape with the lengthwise extremities of the ellipse truncated. In both cases, the
major sliding forces acting on the contact are essentially parallel to the direction of rolling and
are principally due to the deformation of the surfaces. Thus, the sliding forces acting on the
contact surfaces of a loaded roller bearing are usually less complex than those for ball bearings.
Dynamic loading of roller bearings does not generally affect contact angles to any
significant extent, and hence the geometry of the contacting surfaces is virtually identical to
that occurring under static loading. Because of the relatively slow speeds of operation
necessitated when the contact angle differs from zero, gyroscopic moments are negligible.
In any event, gyroscopic moments of any magnitude do not substantially alter the normal
motion of the rollers. In this analysis, therefore, the sliding on the contact surface of a
properly designed roller bearing will be assumed to be a function only of the radius of the
deformed surface in a direction transverse to rolling.
To perform the analysis, it is assumed that the contact area between the roller and
either raceway is substantially rectangular, and that the normal stress at any distance from
the center of the rectangle is adequately defined by the following formula given in Chapter 6
of the first volume of this handbook:
s ¼ 2Q
plb1� t2� �1=2 ð6:18Þ
where t¼ y=b and y is the distance in the rolling direction from the centerline of the
contact. Thus, the differential friction force acting at any distance x from the center of
thecontact rectangle is given by
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dFy ¼2mQ
pl1� t2� �1=2
dt dx ð6:19Þ
Integrating Equation 6.19 between t¼+1 yields
dFy ¼mQ
ldx ð6:20Þ
Referring to Figure 6.8, it can be determined that the differential friction moment in the
direction of rolling at either raceway is given by
dMR ¼ R2 � x2� �1=2� R�D
2
� �� �dF ð6:21Þ
or
dMR ¼2mQ
pl1� t2� �1=2
R2 � x2� �1=2� R�D
2
� �� �dt dx ð6:22Þ
where R is the radius of curvature of the deformed surface. Integrating Equation 6.22 with
respect to t between the limits of +1 yields
dMR ¼mQ
plR2 � x2� �1=2� R�D
2
� �� �dx ð6:23Þ
( ) )(l + {[R2− 2
]2 2
l 22
1 12 2 − − −(R D)}
Deformed surfaceof contact
Roller
Rolleraxis
x
r1
l
cl
R
R
R2
− x
2
2
2
R2
−l 2
2 )(
FIGURE 6.8 Roller–raceway contact showing deformed surface of radius R.
� 2006 by Taylor & Francis Group, LLC.
Because of the curvature of the deformed surface, pure rolling exists at most at two points
x¼+cl=2 on the deformed surface. The radius of rolling measured from the roller axis of
rotation is r0; therefore,
Fy ¼2mQ
l
Z cl=2
0
dx�Z l
cl=2
dx
!ð6:24Þ
or
Fy ¼ mQ 2c� 1ð Þ ð6:25Þ
Also,
MR ¼2mQ
l
Z cl=2
0
ðR2 � x2Þ1=2 � R�D
2
� �� �dx�
Z l=2
cl=2
ðR2� x2Þ1=2 � R�D
2
� �� �dx
( )ð6:26Þ
or
MR ¼mQ
(R2
l2 sin�1 cl
2R� sin�1 l
2R
� �þ ð1� 2cÞ R�D
2
� �
þ cR 1� cl
2R
� �2" #1=2
�R
21� 2R
l
� �2" #1=2
)ð6:27Þ
Considering the equilibrium of forces acting on the roller at the inner and outer raceway
contacts (see Figure 6.9), Fyo¼�Fyi; therefore, from Equation 6.25 assuming mo¼mi
co þ ci ¼ 1 ð6:28Þ
Furthermore, since in uniform rolling motion the sum of the torques at the outer and inner
raceway contacts is equal to zero, therefore,
Fy o
Fy i
MR o
MR i
FIGURE 6.9 Friction forces and moments acting on a roller.
� 2006 by Taylor & Francis Group, LLC.
MR o
dm
2þ r0o
� �r0o
þ MRi
dm
2� r 0i
� �r 0i
¼ 0 ð6: 29 Þ
Fro m Figu re 6.8, it can be seen that the roll er radius of roll ing is
r 0 ¼ R 2 � cl
2
� �2" #1 =2
� R �D
2
� �ð6: 30 Þ
Hence, assum ing mo ¼mi , from Equation 6.27, Equat ion 6.29, and Equation 6.30,(R2
o
l2 sin � 1 co l
2Ro
� sin � 1 l
2Ro
� �:þ ð1 � 2co Þ Ro �
D
2
� �
þ co Ro 1 � co l
2Ro
� �2" #1 =2
�Ro
21 � l
2Ro
� �2" #1= 2
9=;
� 1 þ dm
2 R 2o � co l2
� �2h i1 =2
� Ro � D2
� � 8>><>>:
9>>=>>;
�
(R 2il
2 sin � 1 ci l
2Ri
� sin � 1 l
2Ri
� �:þ ð1 � 2ci Þ Ri �
D
2
� �
þ ci Ri 1 � ci l
2Ri
� �2" #1 =2
�Ri
21 � l
2Ri
� �2" #1 = 2
9=;
� 1 þ dm
2 R 2i �ci l2
� �2h i1= 2� Ri � D
2
� � 8>><>>:
9>>=>>; ¼ 0
ð 6: 31 Þ
Equat ion 6.28 and Equat ion 6.31 can be solved simulta neously for co and c i . No te that if Ro
and Ri , the rad ii of cu rvature of the outer an d inner co ntact surfaces, respect ively, are infinite,
the analys is does not ap ply. In this case, sliding on the co ntact surfac es is obv iated and only
roll ing occu rs.
Havin g determ ined co an d c i , one may revert to Equat ion 6.25 to de termine the ne t sliding
forces Fyo an d Fyi . Similarl y, M R o and M Ri may be calcul ated from Equat ion 6.27. Figure 6.9
shows the friction forces and moments acting on a roller.
6.4 BEARING OPERATION WITH FLUID-FILM LUBRICATION: EFFECTSOF FRICTION FORCES AND MOMENTS
6.4.1 BALL BEARINGS
6.4.1.1 Calculation of Ball Speeds
As shown in Chapt er 5, the su rface fricti on shear stresses ty0 and tx0 at a given point (x0, y0) in
the contact surface can be represented by the following equations:
� 2006 by Taylor & Francis Group, LLC.
ty 0 ¼ c vAc
A0
ma s þ 1 � Ac
A0
� �h
hvy0þ 1
tlim
� ��1
ð 5: 48 Þ
tx0 ¼ c vAc
A0
ma s þ 1 � Ac
A0
� �h
hvx0þ 1
tlim
� �� 1
ð 5:49 Þ
Means were also de monstrated to pe rmit the calcul ation of ty0 and t x 0 for a given lubricati ng
fluid and a given conditio n L of rolling co ntact surface separat ion. Figure 6 .10 shows the
force and moment loading of a ba ll in thrust-l oaded oil-lubri cated angular -conta ct ball
bearing . The coord inate syste m is the same as that used in Figu re 2.4 to describ e ball speeds.
The sli ding veloci ties in the y0 (rol ling motion ) and x 0 (gyros copic moti on) direct ions as
determ ined from Chapt er 2 are as follo ws:
vy0n ¼D
2
vn
gþ wn cnvn � vx0ð Þ cos an þ unð Þ � vz0 sin an þ unð Þ½ �
ð6:32Þ
vx0n ¼D
2wnvy0 ð6:33Þ
where
vn ¼ cn vm �Vnð Þ ð6:34Þ
wn ¼2xn
D
� �2
þ 4�2n �
2xn
D
� �2 !1=2
� 4�2n �
2an
D
� �2 !1=2
þ 1� 2an
D
� �2 !1=2
24
35
8<:
9=;
1=2
ð6:35Þ
x
x �
x �
x �
Fx 9�
Fy 9�
Fy 9�
Fu
Fu
Fy 9
Fy 9i
Fx 9i
y �
Bearing axis
Mgy 9
Mgz 9Qo
z�
y �
y �
z �
z�
ao
ai
Q i
Fc
FIGURE 6.10 Forces and moments acting on a ball in an oil-lubricated, thrust-loaded angular-contact
ball bearing.
� 2006 by Taylor & Francis Group, LLC.
un ¼ sin� 1 xn
rn
� �ð6: 36 Þ
r 0n ¼D
2 wn ð6: 37 Þ
and
�n ¼2fn
2fn þ 1 ð6: 38 Þ
In Equation 6.32 and Equation 6.33, co¼ 1 and ci¼�1.
To calculate the plateau lubricant film thickness h used in the determination of ty0 and tx0,
the entrainment velocities may be determined from the following equation:
uy0n ¼D
4
vn
gþ wn cnvn þ vx0ð Þ cos an þ unð Þ þ vz0 sin an þ unð Þ½ �
ð6:39Þ
In the calculation of ty0 and tx0, it is important to determine lubricant viscosities at the
appropriate temperatures. For calculation accuracy, it is necessary to estimate the lubricant
temperature at the entrance to each contact and in the film separating the rolling–sliding
components.
Assuming that contact loading is known, the friction forces acting over the contact areas
are given by
Fy0n ¼ anbn
Z1
�1
Zffiffiffiffiffiffiffiffi1�q2p
�ffiffiffiffiffiffiffiffi1�q2p
ty0n dt dq, n ¼ o, i ð6:40Þ
ffiffiffiffiffiffiffiffi2
p
Fx0n ¼ anbn
Z1
�1
Z1�q
�ffiffiffiffiffiffiffiffi1�q2p
tx0n dt dq, n ¼ o, i ð6:41Þ
The moments due to the surface friction shear stresses are given by
Mx0n ¼1
2Danbn
Z1
�1
Zffiffiffiffiffiffiffiffi�11�q2p
ffiffiffiffiffiffiffiffi1�q2p
ty0nwn cos ðan þ unÞ dq dt, n ¼ o, i ð6:42Þ
ffiffiffiffiffiffiffiffi1�q2p
Mz0n ¼1
2Danbn
Z1
�1
Z�ffiffiffiffiffiffiffiffi1�q2p
ty0nwn sin ðan þ unÞ dq dt, n ¼ o, i ð6:43Þ
1
ffiffiffiffiffiffiffiffi1�q2p
My0n ¼1
2Danbn
Z�1
Z�ffiffiffiffiffiffiffiffi1�q2p
tx0nwn dq dt, n ¼ o, i ð6:44Þ
� 2006 by Taylor & Francis Group, LLC.
Usin g Equation 6.40 through Equat ion 6.44, the equati ons for force an d moment equilibrium
for a bearing ball are
Qo sin ao þ Fx0 o cos ao �Fa
Z¼ 0 ð 6: 45 Þ
Xn¼ i
n ¼ o
cn Q n cos an � Fx0 n sin a nð Þ � Fc ¼ 0, n ¼ o, i; c o ¼ 1, c i ¼ �1 ð 6: 46 Þ
Xn ¼ i
n ¼ o
cn Q n sin an þ Fx0 n cos anð Þ ¼ 0, n ¼ o, i; co ¼ 1, c i ¼ �1 ð 6: 47 Þ
Xn ¼ i
n¼ o
cn Fy 0 n þ Fv ¼ 0, n ¼ o, i; c o ¼ 1, c i ¼ �1 ð 6: 48 Þ
Xn ¼ i
n ¼ o
Mz 0 n ¼ 0 ð 6: 49 Þ
Xn ¼ i
n ¼ o
My 0 n � M gy 0 ¼ 0 ð 6: 50 Þ
Xn ¼ i
n¼ o
Mz 0 n � M gz 0 ¼ 0 ð 6: 51 Þ
wher e
Mg y0 ¼ J vm v y0 ð 6: 52 Þ
Mg z0 ¼ J vm v z 0 ð 6: 53 Þ
and J is the polar moment of inertia. Viscou s drag force Fv in Equat ion 6.48 is determ ined
from Equat ion 6.1. Since only a simple thrust load is app lied, the cage speed is identical to ba ll
orbit al speed vm. Anothe r sim plification for this ex ample is an assum ption of a ball–ridi ng
cage that ha s negli gible friction betw een the c age pock ets and ba lls. The unknow n varia bles in
Equation 6.45 through Equation 6.51 are:
. Inner and outer raceway–b all contact deform ations di and do
. Ball contact angles ai and ao
. Ball speeds vx0, vy0, vz0, and vm
. Bearing axial deflection da
Hence, there are seven equations and nine unknown variables. The remaining two equations
pertain to the position of the ba ll center; as obtaine d from Chapt er 3 they are
A1 � X1ð Þ2þ A2 � X2ð Þ2� fi � 0:5ð ÞDþ di½ �2¼ 0 ð6:54Þ
X21 þ X2
2 � fo � 0:5ð ÞDþ do½ �2¼ 0 ð6:55Þ
� 2006 by Taylor & Francis Group, LLC.
In Chapt er 3 from Equation 3.72 and Equat ion 3.73, it is shown that the position varia bles A1
and A2 are given by
A1 ¼ BD sin a o þ da ð6: 56 Þ
A2 ¼ BD cos ao ð6: 57 Þ
wher e B ¼ fi þ fo � 1. Moreover, the position varia bles X1, X 2, A 1, and A2 are related to
con tact angles ai and ao , and contact de formati ons di and do as foll ows:
sin ao ¼X1
fo � 0: 5ð ÞD þ do
ð6: 58 Þ
cos ao ¼X2
fo � 0: 5ð ÞD þ do
ð6: 59 Þ
sin ai ¼A1 � X 1
fi � 0: 5ð ÞD þ di
ð6: 60 Þ
cos ai ¼A2 � X 2
fi � 0:5ð ÞD þ di
ð6: 61 Þ
This syst em of equati ons was fir st solved by Harris [8] using the sim plifying assum ption of an
isot hermal Newton ian lubri cant, adeq uately suppli ed to the ball– raceway contact s. Fig-
ure 6.11 and Figure 6.12 show the compari son of the an alytical resul ts with the experi mental
data of She vchenk o and Bolan [9] and Poplaw ski and M auriello [10] . The de viations from the
solut ion using the outer raceway control approxim ation are apparent .
6.4. 1.2 Skidding
Resul ting from the analys es by Harr is [8] as shown in Figure 6.11 and Figure 6.12, invest i-
gati on of the rolling direct ion sliding velocity, that is, vy0 as a functio n of locat ion x0 along the
major axis of the ball– inner raceway an d ball– outer raceway contacts, reveal s no ch ange in
the slid ing velocity direct ion. This means that no points of rolling moti on oc cur over the
con tacts. Thi s co ndition of gross sliding is call ed skiddi ng. An impor tant ap plication with
regard to skiddi ng is the mains haft spli t inner ring ball bearing in gas turbine en gines. This
predo minantly thrust-lo aded angular -conta ct bearing ope rates at high speeds, typic ally in the
range exceeding 2 mil lion dn (beari ng bore in mm � shaft speed in rpm) . Even though the
thrust load is high , skiddi ng tends to occu r.
Skidding resul ts in surface fri ction shear stresses of signifi cant magni tudes over the
contact areas. If the lubricant film generated by the relative motion of the ball–raceway
surfaces is insufficient to completely separate the surfaces, surface damage called smearing
will occur. An exa mple of smear ing is shown in Figure 6.13. Smeari ng is de fined as a severe
type of wear characterized by the metal tightly bonded to the surface in locations to which it
has been transferred from remote locations of the same or opposing surfaces. The transferred
metal is present in sufficient volume to connect more than one distinct asperity contact. When
the number of asperity contacts connected is small, it is called microsmearing. When the
number of such contacts is large enough to be seen with the unaided eye, it is called gross
smearing or macrosmearing.
If possible, skidding is to be avoided in any bearing application because at the very least it
results in increased friction and heat generation even if smearing does not occur. It can occur
� 2006 by Taylor & Francis Group, LLC.
1,000 2,000 3,000
9,000 rpm
4,000
2,000 4,000 6,000 8,000 10,00000.42
0.43
0.44
0.45
0.46
0.47
0.48
0.49
Shaft speed (rpm)
Cag
e–sh
aft s
peed
rat
io
Test dataRaceway control theory
Harris analysis [8]
2,114 N Thrust/ball (475 Ib)
200 400 600 800 1,000
N
0.40
0.42
0.44
0.46
0.48
0.50
0.52
Thrust load per ball (Ib)
Cag
e–sh
aft s
peed
rat
io
FIGURE 6.11 Experimental data from Ref. [9] vs. analytical data from Ref. [8] for an angular-contact
ball bearing having three 28.58-mm (1.125 in.) balls.
in high-speed bearing applications, particularly if the applied load accommodated by each
rolling element is relatively small compared with its centrifugal force. The latter causes
increased normal loads at the outer raceway contacts compared with the inner raceway
contacts. Thus, the balance of friction forces and moments requires higher friction coefficients
� 2006 by Taylor & Francis Group, LLC.
0 100 200 300 400
0 100 200 300 400
0.38
0.39
0.40
0.41
0.42
0.43
0.44
0.45
0.46
0.47 500 1,000 1,500 2,000
500 1,000 1,500 2,000
N
27,000 rpm
35,000 rpm
Raceway control theoryHarris analysis [8]Test data
Thrust load (Ib)
Cag
e–sh
aft s
peed
rat
io
0.36
0.38
0.40
0.42
0.44
0.46
Thrust load (Ib)
Cag
e–sh
aft s
peed
rat
io
N
FIGURE 6.12 Experimental data from Ref. [10] vs. analytical data from Ref. [8] for a 35-mm bore–62-
mm OD angular-contact ball bearing.
at the inner racew ay contact s to co mpensat e for the lower nor mal contact loads. In Chapt er 4,
it was shown that the lubricant film thickness generated in an oil-lubricated rolling element–
raceway contact depends on the velocities of the surfaces in contact. Considering Newtonian
lubrication as a simplified case, the surface friction shear stress is a direct function of the
sliding velocity of the surfaces and an inverse function of the lubricant film thickness. Now,
considering Equation 5.3, the coefficient is a function of sliding velocity; this is greatest at the
inner raceway contacts.
Generally, skidding can be minimized by increasing the applied load on the bearing, thus
decreasing the relative magnitude of rolling element centrifugal force to the contact load at
the most heavily loaded rolling element. Unfortunately, this remedy tends to reduce bearing
fatigue endurance. Another approach is to employ reduced mass rolling elements. These can
� 2006 by Taylor & Francis Group, LLC.
FIGURE 6.13 Raceway surface smearing damage caused by skidding: (a) 100� magnification; (b) 500�magnification.
be manufactured from silicon nitride, a rolling bearing capable ceramic that has a specific
gravity 40% that of steel. Hollow rolling elements also might be used; however, bending
stresses at the inside diameter also tend to cause earlier fatigue failure.
Skidding is also aggravated by rolling element–lubricant, rolling element–cage, and cage–
ring rail friction, each of which tends to retard motion. The most significant of these is the
viscous drag of the lubricant on the rolling elements. Therefore, a high-speed bearing
operating submerged in lubricant will skid more than the same bearing operating in oil
mist-type lubrication. In this case, a compromise is required because, in a high-speed appli-
cation, a copious supply of fluid lubricant is generally used to carry away the friction heat
generated by the bearing.
In general, a compromise between the degree of skidding and bearing endurance must be
accepted unless by making the contacting surfaces extremely smooth, the effectiveness of the
lubricant film thickness is improved to the point that skidding may occur without surface
damage.
� 2006 by Taylor & Francis Group, LLC.
6.4.2 CYLINDRICAL R OLLER BEARINGS
6.4. 2.1 Calculati on of Roller Speeds
Roller speed s in oil- lubricated, cylind rical roller bearing s can be de termined by a co nsider-
ation of the balance of fricti on forces and moment s on the individ ual roll ers and on the
bearing as a unit. Consi dering the roll er–rac eway co ntacts to be divide d into lami nae as in
Chapt er 1, the sliding velocity at a selec ted lamin a is given by
vlnj ¼ 12
dm þ c n Dl þ 23 dlnj
� �� �vnj � D l � 1
3 �lnj
� �vR j
�n ¼ o, i; co ¼ 1, c 1 ¼ �1; j ¼ 1 to Z
ð6: 62 Þ
wher e Dl is the equival ent roll er diame ter at lamin a l. It is assumed in Equat ion 6.62 that 1 =3of the elastic co ntact deform ation occurs in the roll er and 2 =3 in the racew ay. Further, to
sim plify the analysis it is assume d that the roller orbit al sp eed is con strained to equal cage
speed. This con dition occurs when roller–cage pocket clearance is very smal l in the circum-
ferent ial direction. Raceway relative speed vnj is given by
vnj ¼ c n v m �V nð Þ, n ¼ o, i; co ¼ 1, c i ¼ �1 ð6: 63 Þ
Fluid entrainment veloci ties are given by Equat ion 4.54 and Equation 4.55; mini mum
lubri cant film thickne sses are obtaine d using Equat ion 4.57. Platea u lubric ant film thick-
nesses are obtaine d using Equat ion 4.58. As for ball–racew ay contact s, the surfa ce fricti on
shear stre ss at a point on the co ntact surfa ce is obtaine d using Equat ion 5.48. In this case,
normal stress or co ntact pressur e is determ ined at each lamina l using Equation 6.50:
slnj ¼2qlnj 1 � t2
� �1= 2p bnj
ð6: 64 Þ
wher e t ¼ y=bnj and qlnj is the load per unit length on lami na l at roll er–rac eway contact nj.
The fri ction force acting over a contact is then g iven by
Fnj ¼ 2wn
Xl¼ k
l¼ 1
blnj
Z 1
0
tlnj dt ð6: 65 Þ
wher e wn is the lamina thickne ss.
Figure 6.14 shows the fricti on and normal forces actin g on a roll er in a radial ly loaded
cyli ndrical roller bearing with negligible roller end–ring guide flange fricti on.
From Figure 6.14, the following force equilibrium equations are obtained:
Xn¼i
n¼o
cnQnj � Fc ¼ 0, n ¼ o, i; co ¼ 1, ci ¼ �1; j ¼ 1 to Z ð6:66Þ
Xn¼i
n¼o
cnFnj þ Fv �QCGj ¼ 0 ð6:67Þ
where Fc is obtaine d from Equation 3.38, an d in Equat ion 6.67 F v is obtaine d from Equation
6.2. Note that if there is suffici ent clear ance between the ro ller an d the cage web, then the
roller is free to orbit at other than the cage speed. Equation 6.67 then becomes
� 2006 by Taylor & Francis Group, LLC.
Q ij
Fij
Fυ
wj
wmj
Fcj
Foj
Qoj
QCGJ
Ω i
Ωo
FIGURE 6.14 Forces acting on a roller in a radially loaded cylindrical roller bearing.
Xn ¼ i
n ¼ o
cn F nj þ F v � Q CG j ¼1
2 m
dv
dt¼ 1
2 mdm vmj
dvR j
dcð 6: 68 Þ
wher e m is the mass of the roll er.
The moment s ab out the rolle r axis due to surfa ce friction shea r stre sses are g iven by
Mnj ¼ w nXl¼ k
l¼ 1
blnj D l
Z 1
0
tlnj dt ð 6: 69 Þ
The summ ation of moment s about the ro ller a xis is
Xn ¼ i
n ¼ o
Mnj �1
2 m CG DQ CG j ¼ J vm
dvR j
dcð 6: 70 Þ
Finall y, the equilibrium of radial forces actin g on the bearing is express ed by
Xj ¼ Z
j ¼ 1
Qij cos cj � F r ¼ 0 ð 6: 71 Þ
and if the bearing ope rates at con stant speed, the sum of the moment s acti ng on the cage in
the circumfer entia l direction must equate to zero, or
dm
Xj ¼ Z
j ¼ 1
QCG j � DCR FCL ¼ 0 ð 6: 72 Þ
wher e FCL the fricti on force be tween the cage rail an d the be aring ring land is given by
Equation 6.4.
As in Chapter 3, the normal loads Qnj can be written in terms of contact deformations dnj,
and bearing radial deflection can be related to contact deformations and radial clearance.
� 2006 by Taylor & Francis Group, LLC.
Accor dingly , Equat ion 6.66, Equation 6.67, and Equat ion 6.70 through Equat ion 6.72, a set
of 3Z þ 2 simulta neous equati ons, can be solved for dr , di j , v m, vR j, and Q CGj . Ref . [1] gives
the general solution for all types of roller bea rings; that is, for five degrees of freedom in
app lied bearing loading , freedom for each ro ller to orbit at a speed other than the cage speed
( vm j inst ead of v m), an d a racew ay with an y shap e or roll er profi le.
6.4. 2.2 Skidding
Skidd ing is a pr oblem in cyli ndrical roller bearing s used to supp ort the mains haft in aircr aft
gas turbi ne engines . Thes e high-sp eed bearing s, used princi pally for locat ion, are subject ed to
very light radial load. Harr is [11], using a simp ler form of the analys is, consider ing only
isot hermal lubri cation co nditions, and neglect ing viscous drag on the rollers, ne verthe less
managed to de monst rate the adeq uacy of the analytical method. Figure 6.15, from Ref. [11],
compa res an alytical data against experi menta l data on cage speed vs. ap plied load and speed.
The analys is sho wed that skiddi ng, as indica ted by the reducti on in cage speed co mpared with
kinema tic speed, tends to decreas e as the load ap plied is increased . It also appears relat ively
insensitive to lubricant type.
Some aircraft engine manufacturers assemble their bearings in an oval-shaped or out-of-
roun d outer raceway to achieve the load dist ribution in Figure 6.16. This selec tive radial
00
1,000
2,000
3,000
500 1,000 1,500 2,000 2,500Bearing load(Ib)
Cag
e sp
eed(
rpm
)
0 2,000 4,000 6,000 8,000 10,000N
2,000 rpm
3,500 rpm
5,000 rpm
6,500 rpm
Test data
FIGURE 6.15 Cage speed vs. load and inner ring speed for cylindrical roller bearing, lubricant-diester
type according to MIL-L-7808 specification. Z¼ 36 rollers, l¼ 20 mm (0.787 in.), D¼ 19mm (0.551 in.),
dm¼ 183mm (7.204 in.), Pd¼ 0.0635 mm (0.0025 in). (From Harris, T., An analytical method to predict
skidding in high speed roller bearings, ASLE Trans., 9, 229–241, 1966.)
� 2006 by Taylor & Francis Group, LLC.
Fr
FIGURE 6.16 Distribution of load among the rollers of a bearing having an out-of-round outer ring and
subjected to radial load Fr.
preloadi ng of the bearing increa ses the maxi mum roller load an d doubles the num ber of
rollers so-loa ded. Figu re 6.17, from Ref. [11], ill ustrates the effe ct on skiddi ng of a n out -of-
round outer racewa y. Anothe r metho d to minimiz e skiddi ng is to use a few, for exampl e, three
equall y spaced hollow rollers that provide an interfer ence fit wi th the raceways unde r zero
applie d radial load and stat ic co nditions. Figure 6.18, from Ref. [12], illustr ates such an
assem bly, while Figure 6.19 and Figu re 6.20 indica te the effecti veness to minimiz e skiddi ng.
6.5 CAGE MOTIONS AND FORCES
6.5.1 I NFLUENCE OF S PEED
With respect to roll ing elem ent bearing perfor mance, cage design has be come more impor tant
as bearing rotational speeds increa se. In inst rument ba ll bearing s, unde sira ble torqu e v ari-
ations have been trace d to cage dyn amic instabil ities. In the develop ment of solid -lubricat ed
bearing s for high-s peed, high-temper ature gas turbi ne engines , the cage is a major con cern.
A key to success ful cage design is a detai led analys is of the forces a cting on the cage an d
the mo tions it undergoes. Both steady-st ate and dyna mic formu lations of varyi ng co mplexity
have been develop ed.
6.5.2 F ORCES A CTING ON THE C AGE
The prim ary forces acting on the cage are due to the interacti ons be tween the roll ing elem ent
and cage pock et ( FCP ) and the cage rail and the piloting land ( FCL ). As Figure 6.21 shows, a
roller can contact the cage on eithe r side of the poc ket, de pending on wheth er the cage is drivi ng
the roller, or vice versa . The direction of the ca ge pocket fricti on force ( FCP ) depend s on whi ch
side of the pock et the contact occurs. For an inn er land ridi ng cage, a fricti on torque ( TCL ) in the
direction of cage rotation develops at the cage–l and con tact. For an outer land ridi ng cage, a
fricti on torque tending to retard cage rotation develops at the cage–l and contact .
A lubricant viscous drag force ( fDRAG) develops on the cage surfaces resisting motion of the
cage. Centrifugal body forces (shown as FCF) due to cage rotation make the cage expand uniformly
outward radially and induce tensile hoop stresses in the cage rails.Anunbalanced force (FUB), the
magnitude of which depends on how accurately the cage is balanced, acts radially outward.
Hydrodynamic short bearing theory can be used to model the cage–land interaction as
indicated in Ref. [13]. The contact between the rolling element and cage pocket can be hydro-
dynamic, elastohydrodynamic, or elastic in nature, depending on the proximity of the two
� 2006 by Taylor & Francis Group, LLC.
0 0.005
1000
2000
Cag
e sp
eed
(rpm
)
Out-of-round (in.)
3000
0 0.1 0.2mm
6500 rpm
5000 rpm
3500 rpm
0.3
0.010 0.015
FIGURE 6.17 Cage speed vs. out-of-round and inner ring speed. Lubricant-diester type according to
MII-L-7808 specification. Z¼ 36, i¼ 1, l¼ 20 mm (0.787 in.), D¼ 14mm (0.551 in.), dm¼ 183 mm
(7.204 in.), Pd¼ 0.0635mm (0.0025 in.), Fr¼ 222.5N (50 lb). (From Harris, T., An analytical method
to predict skidding in high speed roller bearings, ASLE Trans., 9, 229–241, 1966.)
120˚
120˚
120˚
FIGURE 6.18 Cylindrical roller bearing with three preloaded annular rollers. (From Harris, T., and
Aaronson, S., An analytical investigation of skidding in a high-speed, cylindrical roller bearing having
circumferentially spaced, preloaded hollow rollers, Lub, Eng., 30–34, 1968.)
� 2006 by Taylor & Francis Group, LLC.
00 10 20 30 40
Number of rollers under load (%)
80% Hollowness
80% Hollowness
85% Hollowness
90% Hollowness
95% Hollowness
85% Hollowness
90% Hollowness
95% Hollowness
0.0254 mm (0.001 in.) Interference 0.0508 mm (0.002 in.) Interference
Bearing dimensions
Number of rollers 28Roller effective length 14.22 mm (0.56 in.)Roller diameter 17 mm (0.669 in.)Pitch diameter 182.3 mm (7.179 in.)Radial clearance 0.0064 mm (0.00025 in.)
Cag
e sp
eed
slip
(%
)
50 60 70 80 90 100 0 10 20 30 40Number of rollers under load (%)
50 60 70 80 90 100
10
20
30
40
50
60
70
80
90
100
0
Cag
e sp
eed
slip
(%
)
10
20
30
40
50
60
70
80
90
100
85% Hollowness80% Hollowness
90% Hollowness
95% Hollowness
0.0762 mm (0.003 in.) Interference
0 10 20 30 40Number of rollers under load (%)
50 60 70 80 90 100
Cag
e sp
eed
slip
(%
)
0
10
20
30
40
50
60
70
80
90
100
FIGURE 6.19 Skidding in cylindrical roller bearings having spaced preloaded hollow rollers.
bodies and the magni tude of the roll ing elem ent forces . In most cases, the roll ing elem ent–cag e
interacti on forces are small enough so that hyd rodynami c lub rication consider ations prevai l.
6.5.3 STEADY-STATE CONDITIONS
In section 6.4, it was de monst rated that analyt ical means exist to predict skidd ing in ball
and roller bearings in any fluid-lubricated application. All the calculations, even for the
least complex applications, require the use of a computer. As a spin-off from the skidding
analysis, rolling element–cage forces are determined. For an out-of-round outer raceway
� 2006 by Taylor & Francis Group, LLC.
00
2
4
6
8
10
4 8 12
Inner raceway speed (�103 rpm)
Six flex rollers
Solid rollers
Two flex rollers
Four flex rollers
Theoretical
Cag
e sp
eed
(�10
3 rp
m)
16 20 24
FIGURE 6.20 Cage speed vs. inner raceway speed: 207 roller bearing, Fr¼ 0, Pd¼�0.061mm
(�0.0024 in.), 90% hollow rollers, lubricant MIL-L-6085A at 0.85 kg=min.
cyli ndrical rolle r bearing unde r radial load , Figure 6.22, from Ref. [14] , illu strates cage web
loading for steady -sta te, centric cage rotation.
Whereas the analys is of Ref. [13] co nsidered only centric rotation in the radial plane,
Klec kner and Pirvics [15] used three degrees of freedom in the radial plane; that is, the cage
rotat ional speed an d two radial displacemen ts locat ing the ca ge center in the plan e of
rotat ion. The co rrespondi ng cage equilibrium equati ons are
XZj ¼ 1
½ð FCP j Þ sin c j � ðfCP j Þ cos cj � �W y ¼ 0 ð6: 73 Þ
FCF
fCP
FCP
FCL
FCL
TCL
TCL
fCP
FCP
FUB
fDRAG
fdrag
Cage rotation
Inner ring rotation
FIGURE 6.21 Cage forces.
� 2006 by Taylor & Francis Group, LLC.
200
−1
1
2
3 −14
−12
−10
−8
−6
−4
−2
40 60 80
Azimuth (degrees, ±)
Cag
e w
eb lo
ad (
lb)
100 140120 160 180
N
FIGURE 6.22 Cage-to-roller load vs. azimuth for a gas turbine mainshaft cylindrical roller bearing.
Thirty 12 mm� 12mm rollers on a 152.4mm (6 in.) pitch diameter. Roller i.d.=o.d.¼ 0.6, outer ring
out-of-roundness¼ 0.254mm (0.01 in.), radial load¼ 445 N (100 lb), shaft speed¼ 25,000 rpm. (From
Wellons, F., and Harris, T., Bearing design Considerations Interdisciplinary Approach to the Lubrication
of Concentrated Contacts, NASASP-237, pp. 529–549, 1970.)
XZj¼1
½ð�FCPjÞ cos cj � ðfCPjÞ sin cj� �Wz ¼ 0 ð6:74Þ
1
2dm
XZj¼1
ðFCPjÞ � TCL ¼ 0 ð6:75Þ
where Wy and Wz are the components of FCL in the y and z direction; FCPj is the cage pocket
normal force for the jth rolling element; and fCPj is the cage pocket friction force for the jth
rolling element.
The cage coordinate system is shown in Figure 6.23.
j th roller
f cpj
Fcpj
z
y
Wy
Wz TCL
yj
FIGURE 6.23 Cage coordinate system.
� 2006 by Taylor & Francis Group, LLC.
Equation 6.73 an d Equat ion 6.74 repres ent equilibrium of cage forces in the radial plane
of motio n. The summ ation of the cage poc ket normal forces and fri ction forces equilib rates
the cage–l and normal force. Equat ion 6.75 establis hes torque equ ilibrium for the cage about
its axis of rotation. The cage pocket normal forces are assumed to react at the bearing pitch
circle. The sign of the cage–land friction torque TCL depends on whether the cage is inner ring
land–riding or outer ring land–riding. In the formulations of Ref. [15], each roller is allowed
to have different rotational and orbital speeds.
6.5.4 DYNAMIC CONDITIONS
Rolling element bearing cages are subjected to transient motions and forces due to acceler-
ations caused by contact with rolling elements, rings, and eccentric rotation. In some appli-
cations, notably with very high-speed or rapid acceleration, these transient cage effects may
be of sufficient magnitude to warrant evaluation. The steady-state analytical approaches
discussed do not address the time-dependent behavior of a rolling element bearing cage.
Several researchers have developed analytical models for transient cage response [13,16–19].
Because of the complexity of the calculation involved, such performance analyses generally
require extensive time on present-day computers.
In general, the cage is treated as a rigid body subjected to a complex system of forces.
These forces may include the following:
� 20
1. Impact and frictional forces at the cage–rolling element interface
2. Normal and frictional forces at the cage–land surface (if land-guided cage)
3. Cage mass unbalance force
4. Gravitational force
5. Cage inertial forces
6. Others (that is, lubricant drag on the cage and lubricant churning forces)
Forces 1 and 2 are intermittent; for example, the cage may or may not be in contact with a
given rolling element or guide flange at a given time, depending on the relative position of the
bodies in question. Frictional forces can be modeled as hydrodynamic, elastohydrodynamic
lubrication (EHL), or dry friction, depending on the lubricant, contact load, and geometry.
Both elastic and inelastic impact models appear in the literature. General equations of motion
for the cage may be written. The Euler equations describing cage rotation about its center of
mass (in Cartesian coordinates) are as follows:
Ix _vvx � ðIy � IzÞvyvz ¼Mx ð6:76Þ
Iy _vvy � ðIz � IxÞvzvx ¼My ð6:77Þ
Iz _vvz � ðIx � IyÞvxvy ¼Mz ð6:78Þ
where Ix, Iy, Iz are the cage principal moments of inertia, and vx, vy, vz are the angular
velocities of the cage about the inertial x, y, z axes. The total moment about each axis is
denoted by Mz, My, and Mz, respectively. The equations of motion for translation of the cage
center of mass in the inertial reference frame are
m€rrx ¼ Fx ð6:79Þ
m€rry ¼ Fy ð6:80Þ
m€rrz ¼ Fz ð6:81Þ
06 by Taylor & Francis Group, LLC.
wher e m is the cage mass, rx , r y, r z de scribe the posit ion of the cage center of mass , and Fx , Fy,
Fz are the net force comp onents ac ting on the cage.
Once the cage force and momen t compon ent are determ ined, accele ration s can be com-
puted. A numeri cal integ ration of the eq uations of motion (with respect to discrete time
increm ents) will yield cage trans lational veloci ty, rotat ional velocity, and displ acement vec-
tors. In some appro aches [13,17], the cage dy namics model is solved in con junction wi th roller
and ring eq uations of motion. Othe r researc hers have devise d less cumbers ome approach es by
limit ing the cage to in-pl ane moti on [16] or by consider ing simplif ied dynami c models for the
rolling elem ents [18] .
Meeks and Ng [18] developed a c age dynami cs model for ba ll bearing s, which treat s both
ball- and ring land-gu ided cages. This model c onsider s six cag e de grees of freedom an d
inela stic co ntact between the balls and cage and be tween the cage and rings . Thi s model
was used to perfor m a ca ge design optim ization study for a solid -lubricat ed, gas turbi ne
engine bearing [19].
The results of the study indica ted that ba ll–cage pocket forces and wear are signi ficantly
affected by the combinat ion of ca ge–land and ball– pocket clear ances. Usi ng the analytical
model to identi fy more suitab le clearance values impro ved experi menta l cage perfor mance.
Figure 6.24 an d Figure 6.25 contai n typical output data from the cage dyn amics analys is.
In Figure 6.24 the cage center of mass moti on is plotted vs. time for X and Y (radial plane)
direction . The time scale relates to approx imately five shaft revolut ions at a shaft speed of
40,000 rpm. Figure 6.25 shows the plots of ball– cage poc ket normal force for tw o repres en-
tative pockets position ed ap proxim ately 90 8 apa rt.
In addition to the work of Meeks [19], Maurie llo et al. [20] succeeded in measur ing ball-to-
cage loading in a ball bearing subject ed to combined radial and thrust loading . They obs erved
impac t load ing between balls and cage to be a signi ficant fact or in high-speed bearing cage
design.
6.6 ROLLER SKEWING
Thus far in this section , roll ers have be en assum ed to run ‘‘true’’ in roller bearing s. Thi s is an
ideal situati on. Bec ause of slig htly imper fect geomet ry, there is a tendency for imbalan ce of
fricti on loading betwe en the roller–inne r racewa y and roller–o uter racew ay contact s, creat ing
a tendency for roll ers to undergo yaw motions such that each roll er’s axis of rotation assum es
an angle jj with a plan e passi ng throu gh the bearing axis of rotation. jj is called the skew ing
angle, an d the roll ers are said to skew.
In a misa ligned radial cyli ndrical roller bearing as shown schema tically in Figure 1.6,
the roll ers are ‘‘sq ueezed’’ at one end and thereby are forced against the ring guide flange. The
sliding contact be tween each roller en d an d the guide flange causes a fricti on force and hen ce a
roller skew ing moment . Depen ding on the cleara nces betw een (1) the roll er and the guide
flang e, (2) the roll er length and the cage pocket in the direct ion trans verse to rolling moti on,
and (3) the roller diame ter and the cage poc ket in the circumfer entia l direction , the ro ller
skew ing angle may be limit ed by one of these constraints . If all these clear ances are too great,
then, as indica ted in Chapter 3, the roll er skew ing angle will be limit ed by the outer racew ay
curvat ure in the direct ion of motio n. Also , as discus sed in Chapt er 3, the thrust load applie d
to rad ial cylindrical roll er bearing s with gu ide flang es on both inner and outer rings results
in roller skewing moments that are resisted by one or more of the mechanisms discussed
here. Figure 6.26 illu strates cy lindrical ro ller loading that resul ts in both roller tilti ng angle zj,
roller skewing angle jj and the roller–raceway contact normal and surface friction stresses
that ensue.
� 2006 by Taylor & Francis Group, LLC.
0.10 2.5Ball pocket clearance = 0.2 mm (0.008 in.)Race land clearance = 0.4 mm (0.016 in.)
wSHAFT = 40.000 rpm
2.0
1.5
1.0
(in.)
(�
10�
1 )(in
.) (
�10
�1 )
“X”
Def
lect
ion
(mm
x 0
.1)
0.5
0
0.80 0.88 0.96 1.04
1 Shaft rev
2 Shaft revs
3 Shaft revs
1.12 1.20
Time (s) (�10�8)
(a)
(b)Time (s) (�10�2)
1.28 1.36 1.44 1.52
54
1.60
0.80 0.88 0.96 1.04 1.12 1.20 1.28 1.36 1.44 1.52 1.60
−0.05
−1.0
−1.5
1.5
1.0
0.5
0
“Y” D
efle
ctio
n (m
m x
0.1
)
−0.5
−1.0
−1.5
0.08
0.06
0.04
0.02
0.00
−0.02
−0.04
−0.06
0.06
0.04
0.02
0.00
−0.02
−0.04
−0.06
FIGURE 6.24 Calculated cage motion vs. time. (a) Prediction of cage motion X vs. time. (b) Prediction
of cage motion, Y vs. time. (From Meeks, C., The dynamics of ball separators in ball bearings—Part II:
Results of optimization study, ASLE Paper No. 84-AM-6C-3, May 1984. With permission.)
In tapered roller bearings, even without misalignment, the rollers are forced against the
large end flange, and skewing moments occur. These are resisted by either the cage or the
outer raceway curvature in the rolling direction. In any case, the roller skewing angles tend to
be very small.
� 2006 by Taylor & Francis Group, LLC.
5012.00
10.00
8.00
LBLB
Bal
l–po
cket
forc
e (N
)
6.00
4.00
2.00
0.00
12.00
10.00
8.00
Bal
l–po
cket
forc
e (N
)
6.00
4.00
2.00
0.80 0.88 0.96 1.04 1.12 1.20 1.28 1.36 1.44 1.52 1.600.00
40
30
20
10
0
50
40
30
20
10
0
0.80 0.86 0.96 1.04 1.12 1.20
Time (sec) (310−2)
(a)
(b)
Time (sec) (310−2)
1.28 1.36 1.44 1.52 1.60
FIGURE 6.25 Calculated ball–pocket force vs. time. (a) Prediction of cage ball–pocket force vs. time
(pocket No. 1). (b) Prediction of cage ball–pocket force vs. time (pocket No. 4). (From Meeks, C., ASLE
Paper No. 84-AM-6C-3, May 1984. With permission.)
In most cases, roller skewing is detrimental to roller bearing operation because it causes
increased friction torque and friction heat generation as well as requiring a cage sufficiently
strong to resist the roller skewing moment loading.
6.6.1 ROLLER EQUILIBRIUM SKEWING ANGLE
That rollers skew until skewing moment equilibrium is achieved has implications beyond the
determination of roller end–flange load or roller end–cage load. In spherical roller bearings
containing rollers with symmetrical profiles, management of roller skewing can minimize
friction losses and corresponding friction torque. Early spherical roller bearing designs
employing asymmetrical roller profiles, because of their close osculations and primary skew-
ing guidance from cage and flange contacts, exhibit greater friction than current bearings with
� 2006 by Taylor & Francis Group, LLC.
Qoj
Qaj
Qaj
Qaj
Qaj mQaj
mQaj
mQaj
mQaj
Q ij
soj λ
s ij λ
toj λ
toj λ
z jx j
(a) (b)
(c) (d)
FIGURE 6.26 In a radial cylindrical roller bearing that has crowned rollers and subjected to combined
radial and thrust loading, (a) roller–raceway and roller end–guide flange forces, (b) roller end–guide
flange friction forces, (c) roller–raceway contact normal and surface friction stresses and roller tilting
angle, and (d) roller skewing angle as limited by the roller end–guide flange axial clearance.
symm etrical roll er designs . The tempe ratur e rise associated with fricti on limit s perfor mance in
many app lications . Desig ning the bearing s so that skew ing equ ilibrium is provided by
racew ay guidan ce alone lowers losse s and increa ses load-c arryin g capacit y. Kellst rom
[21, 22] invest igated skew ing equ ilibrium in spheri cal roll er bearing s consider ing the complex
chan ges in roll er force an d moment balance caused by roll er tilting an d skewing in the
presence of frictio n.
Any rolling element that contacts a raceway along a curved contact surface will undergo
sliding in the contact. For an unskewed roller there will be at most two points along each contact
where the sliding velocity is zero. These zero sliding points form the generatrices of a theoretical
‘‘rolling’’ cone, which represents the contact surface on which pure kinematic rolling would occur
for a given roller orientation. At all other points along the contact, sliding is present in the
direction of rolling or opposite to it, depending on whether the roller radius is greater or lesser
than the radius to the theoretical rolling cone. This situation is illustrated in Figure 6.27.
Friction forces or traction s due to sliding wi ll be orient ed to oppos e the direction of sliding
on the roller. In the absence of tangen tial roller forces from cage or flang e contact s, the roller–
raceway traction forces in each contact must sum to zero. Additionally, the sum of the inner
� 2006 by Taylor & Francis Group, LLC.
Points of rolling
Points of rolling
Surface ofrolling cone
FIGURE 6.27 Spherical roller bearing, symmetrical roller–tangential friction force directions. Motion
and force direction: � out of page; . into page.
and outer racew ay contact skew ing moment s mu st equal zero. These two conditio ns will
determ ine the posit ion of the rolling points along the contact s and thus the theoretical rolling
cone. Thes e con ditions a re met at the equilib rium skewing an gle. If the mo ments tend to
resto re the roll er to the equilibrium ske wing an gle when it is distu rbed, the eq uilibrium
skew ing ang le is said to be stabl e.
As a roll er skews relat ive to its contact ing raceway, a sliding co mponent is generat ed in the
roller axial direction an d traction forces are developed that oppos e axial slid ing. Thes e
traction forces may be benefi cial in that, if suitably orient ed, they help to carry the axial
bearing load, as indica ted in Figure 6.28.
Skewing angles that pro duce axial tract ions oppos ing the applied axial load and reducing
the roller contact load requir ed to react with the ap plied axial load are termed pos itive (Figur e
6.28a). Conver sely, skew ing angles produ cing axial tractions that add to the applie d axial
load are term ed neg ative (Figur e 6.28b) . For a positive skew ing roller, the nor mal co ntact is
reduced , and an impr ovement in con tact fatigue life achieve d.
The axial traction forces acting on the roll er also pro duce a second effe ct. Thes e forces ,
actin g in different directions on the inner an d on outer ring contact s, create a moment about
the roller an d cause it to tilt. The tiltin g motion reposi tions the inner an d outer ring contact
load distribut ions with respect to the theoret ical poin ts of rolling and distribut ion of sli ding
veloci ty. Deta iled evaluat ions [21,22] of this be havior have sh own that skewing in excess of
the equ ilibrium skewing an gle generat es a net skew ing moment oppos ing the increa sing
skew ing moti on. A roll er that skews less than the equilibrium skewing angle will generat e a
net skewing moment tending to increase the skew angle. This set of interacti ons explains the
existenc e of stable equ ilibrium skew ing an gles.
To apply this c oncept to the de sign of sphe rical roller bearing s, sp ecific design geometries
over a wide range of operating conditions must be evaluated. There are tradeoffs involved
between minimizing friction losses and maximizing contact fatigue life. Some designs may
exhibit unstable skewing control in certain operating regimes or stable skewing equilibrium
and require impractically large skewing angles.
� 2006 by Taylor & Francis Group, LLC.
Fa
fx
Fx
Fa fx fx
ΔQ
ΔQ
Q
Q
(a)
(b)
FIGURE 6.28 Forces on outer raceway of axially loaded spherical roller bearing with positive and
negative skewing. (a) Positive skewing angle. (b) Negative skewing angle.
6.7 CLOSURE
In the first volume of this handbook, ball and roller speeds were determined using kinematic
relationships; these depend on simple rollingmotion.While these speed calculations are adequate
for many applications, in this chapter, it has been demonstrated that ball and roller speeds are
functions of sliding conditions occurring in the rolling element–raceway contacts, sliding condi-
tions between the rolling elements and cage, and between the cage and bearing rings, as well as
viscous drag of the lubricant on the orbiting rolling elements. To calculate the rolling element
speeds under these conditions, it was shown necessary to create friction force and moment
balances about each rolling element and about the bearing as a unit. Solution of the system of
equations yields not only the rolling element speeds, but also the cage–rolling element forces and
cage–ring land force. This determination enables improved design of cages and bearing internal
clearances.
In cylindrical and tapered roller bearings under combined radial, axial, and moment
loadings, tendencies toward roller skewing and its effects on speeds and endurance can only
be determined by the friction force and moment balance methods introduced in this chapter.
Rolling bearing friction is manifested as temperature rises in the rolling bearing structure
and lubricant unless effective heat removal means are employed or naturally occur. In the
next chapter, means to estimate bearing internal temperatures will be discussed. It is necessary
to note that bearing performance is very sensitive to temperature as (1) alteration of internal
dimensions can affect load distribution, (2) lubricant film thickness decreases as temperature-
dependent viscosity decreases, (3) friction depends on lubricant film thickness, and (4) in
� 2006 by Taylor & Francis Group, LLC.
many cases, fatigue endurance is sensitive to the lubricant film thickness and resulting contact
surface friction shear stresses.
REFERENCES
1.
� 200
Harris, T., Rolling element bearing dynamics, Wear, 23, 311–337, 1973.
2.
Streeter, V., Fluid Mechanics, McGraw-Hill, New York, 313–314, 1951.3.
Parker, R., Comparison of predicted and experimental thermal performance of angular-contact ballbearings, NASA Tech. Paper 2275, 1984.
4.
Bisson, E. and Anderson, W., Advanced Bearing Technology, NASA SP-38, 1964.5.
Rydell, B., New spherical roller thrust bearings, the e design, Ball Bear. J., SKF, 202, 1–7, 1980.6.
Brown, P., et al., Mainshaft high speed cylindrical roller bearings for gas turbine engines, U.S. NavyContract N00140–76-C-0383, Interim Report FR-8615, April 1977.
7.
Harris, T., Ball motion in thrust-loaded, angular-contact ball bearings with coulomb friction,ASME Trans., J. Lubr. Technol., 93, 32–38, 1971.
8.
Harris, T., An analytical method to predict skidding in thrust-loaded angular-contact ball bearings,ASME Trans., J. Lubr. Technol., 93, 17–24, 1971.
9.
Shevchenko, R. and Bolan, P., Visual study of ball motion in a high speed thrust bearing, SAEPaper No. 37, January 14–18, 1957.
10.
Poplawski, J. and Mauriello, J., Skidding in lightly loaded, high speed, ball thrust bearings, ASMEPaper 69-LUBS-20, 1969.
11.
Harris, T., An analytical method to predict skidding in high speed roller bearings, ASLE Trans., 9,229–241, 1966.
12.
Harris, T. and Aaronson, S., An analytical investigation of skidding in a high-speed, cylindricalroller bearing having circumferentially spaced, preloaded hollow rollers, Lub. Eng., 30–34, January
1968.
13.
Walters, C., The dynamics of ball bearings, ASME Trans., J. Lubr. Technol., 93(1), 1–10, January1971.
14.
Wellons, F. and Harris, T., Bearing design considerations, Interdisciplinary Approach to the Lubri-cation of Concentrated Contacts, NASA SP-237, 1970, pp. 529–549.
15.
Kleckner, R. and Pirvics, J., High speed cylindrical roller bearing analysis—SKF computer programCYBEAN, Vol. I: analysis, SKF Report AL78P022, NASA Contract NAS3–20068, July 1978.
16.
Kannel, J. and Bupara, S., A simplified model of cage motion in angular-contact bearings operatingin the EHD lubrication regime, ASME Trans., J. Lubr. Technol., 100, 395–403, July 1078.
17.
Gupta, P., Dynamics of rolling element bearings—Part I–IV. cylindrical roller bearing analysis,ASME Trans., J. Lubr. Technol., 101, 293–326, 1979.
18.
Meeks, C. and Ng, K., The dynamics of ball separators in ball bearings—Part I: analysis, ASLEPaper No. 84-AM-6C-2, May 1984.
19.
Meeks, C., The dynamics of ball separators in ball bearings—Part II: results of optimization study,ASLE Paper No. 84-AM-6C-3, May 1984.
20.
Mauriello, J., et al., Rolling element bearing retainer analysis, U.S. Army AMRDL TechnicalReport 72–45, November 1973.
21.
Kellstrom, M. and Blomquist, E., Roller bearings comprising rollers with positive skew angle, U.S.Patent 3,990,753, 1979.
22.
Kellstrom, M., Rolling contact guidance of rollers in spherical roller bearings, ASME Paper 79-LUB-23, 1979.
6 by Taylor & Francis Group, LLC.
7 Rolling Bearing Temperatures
� 2006 by Taylor & Fran
LIST OF SYMBOLS
Symbol Description Units
c Specific heat W � sec/g � 8C (Btu/lb � 8F)
D Rolling element diameter mm (in.)
D Diameter m (ft)
E Thermal emissivity
f r/D
f0 Viscous friction torque coefficient
fl Load friction torque coefficient
Fa Applied axial (thrust) load N (lb)
Fr Applied radial load N (lb)
Fb Equivalent applied load to calculate friction torque N (lb)
F Temperature coefficient W � sec/8C (Btu/8F)
g Acceleration due to gravity m/sec2 (in./sec2)
Gr Grashof number
h Film coefficient of heat transfer W/m2 � 8C (Btu/hr � ft2 � 8F)
H Heat flow rate, friction heat generation rate W (Btu/hr)
J Conversion factor, 103 N � mm¼ 1 W � seck Thermal conductivity W/m � 8C (Btu/hr � ft � 8F)
L Length of heat conduction path m (ft)
M Friction torque N � mm (in. � lb)
n Rotational speed rpm
Pr Prandtl number
q Error function
Re Reynolds number
< Radius m (ft)
s Surface roughness mm (min.)
S Area normal to heat flow m2 (ft2)
T Temperature 8C (8F)
us Fluid velocity m/sec (ft/sec)
v Velocity m/sec (ft/sec)
w Weight flow rate g/sec (lb/sec)
W Width m (ft)
x Distance in x direction m (ft)
z Number of rolling elements
« Error
h Absolute viscosity cp (lb � sec/in.2)
n Fluid kinematic viscosity m2/sec (ft2/sec)
s Rolling element–raceway contact normal stress MPa (psi)
cis Group, LLC.
t Surface friction shear stress MPa (psi)
v Rotational velocity rad/sec
V Rotational velocity rad/sec
Subscripts
a Air or ambient condition
BRC Ball–raceway contact
c Heat conduction
CRL Contact between the cage rail and ring land
CPR Contact between the cage pocket and rolling element
f Friction
fdrag Viscous drag on the rolling elements
i Inner raceway
j Rolling element position
n raceway
o Oil or outer raceway
r Heat radiation
REF Roller end–flange contact
RRC Roller–raceway contact
tot Bearing total friction heat generation
v Heat convection
x x direction, transverse to rolling direction
y y direction, rolling direction
1 Temperature node 1
2 Temperature node 2, and so on
7.1 GENERAL
The overall temperature level at which a rolling bearing operates depends on many variables
among which are:
. Applied load
. Operating speeds
. Lubricant type and its rheological properties
. Bearing mounting arrangement and housing design
. Operational environment
In the steady-state operation of a rolling bearing, the friction heat generated must be
dissipated. Therefore, the steady-state temperature level of one bearing system compared
with that of another using identical sizes and number of bearings is a measure of that system’s
efficiency of heat dissipation.
If the rate of heat dissipation is less than the rate of heat generation, then an unsteady
state exists and the system temperatures will rise, most likely until lubricant deterioration
occurs, ultimately resulting in bearing failure. The temperature at which this occurs depends
greatly on the type of lubricant and the bearing materials. The discussion in this chapter is
limited to the steady-state thermal operation of rolling bearings, since this is the principal
concern of bearing users.
� 2006 by Taylor & Francis Group, LLC.
Most ball and roller bearing applications perform at relatively cool temperature levels
and, therefore, do not require any special consideration regarding thermal adequacy. This is
due to either one of the following conditions:
. The bearing friction heat generation rate is low because of light load and relatively slow
operating speed.. The bearing heat dissipation rate is sufficient because the bearing assembly is located in
a moving air stream or there is adequate heat conduction through adjacent metal.
Some applications experience adverse environmental conditions such that external heat
removal means is required. A rapid determination of the bearing cooling requirements
may then suffice to establish the cooling capability that must be applied to the lubricating
fluid. In applications where it is not obvious whether external cooling means is required, it
may be economically advantageous to analytically determine the thermal conditions of
bearing operation.
7.2 FRICTION HEAT GENERATION
7.2.1 BALL BEARINGS
Rolling bearing friction represents a power loss manifested in the form of heat generation.
The friction heat generated must be effectively removed from the bearing or an unsatisfactory
temperature condition will obtain in the bearing. In a ball–raceway contact, the friction heat
generation rate is given by
Hnyj ¼1
J
Ztnyjvnyj dAnj ¼
anjbnj
J
Zþ1
�1
Zþ ffiffiffiffiffiffiffiffi1�q2p
�ffiffiffiffiffiffiffiffi1�q2p
tnyjvnyj dt dq, n ¼ i, o; j ¼ 1�Z ð7:1Þ
where J is a constant converting N �m/sec to watts. In Equation 7.1, the surface friction shear
stress tny may be obtained directly from Equation 5.49 or from Equation 5.5 recognizing that
tnyj¼mnyj s. The values of sliding velocity vyj may be obtained from Equation 2.9 and
Equation 2.20. Similarly,
Hnxj ¼1
J
Ztnxjvnxj dAnj ¼
anjbnj
J
Zþ1
�1
Zþ ffiffiffiffiffiffiffi1�t2p
�ffiffiffiffiffiffiffi1�t2p
tnyjvnxj dq dt, n ¼ i, o; j ¼ 1�Z ð7:2Þ
where tnxj may be obtained directly from Equation 5.50 and vxj may be obtained from
Equation 2.10 and Equation 2.21. For an entire bearing, the friction heat generated in the
ball–raceway contacts is
HBRC ¼Xn¼o
n¼i
Xj¼Z
j¼1
Hynj þHxnj
� �ð7:3Þ
For an oil-lubricated bearing, in addition to the friction heat generated in the ball–raceway
contacts, friction heat is generated due to the balls passing through the lubricant in the
� 2006 by Taylor & Francis Group, LLC.
bearing free space . Usi ng Equat ion 6.1 to de fine the viscous drag force Fv, the frictio n heat
generat ion rate thereby effected is given by
Hfdrag ¼dm vm Fv Z
2J ð7: 4Þ
where dm is the bearing pitch diameter, vm is the ball orbital speed, and Z is the number of balls.
Finally, fri ction heat is generat ed due to slid ing between the cage and the inner ring land
for an inner ring pilote d c age; due to sliding betw een the cage and the outer ring land for an
outer ring pilot ed cage; and be tween the ba lls an d the cage pock ets for an y cage design
executi on. Thes e heat generat ion rates general ly tend to be small; howeve r, they may be
calcul ated using the ba ll an d cage sp eed equa tions of Chapter 2 togeth er with estimat ions of
cage rail–ri ng land load ing and cage–b all loading . These may be determined using a co mplete
fricti on force an d moment balance accord ing to Chapte r 6.
The total frictio n hea t generat ion rate is obta ined by summ ation of the compon ent heat
generat ion rates
Htot ¼ H BRC þ H fdrag þ H CRL þ H CPB ð7: 5Þ
It is noted that Htot doe s not include the friction heat generat ion rate due to the con tact
betw een integral seals and the be aring ring surfa ce. This heat compo nent will most likely be
great er than Htot as define d in Equation 7.5.
Bearing friction torque abou t the shaft can be derive d from Htot using the follo wing
equ ation:
M ¼ 10 3 �Htot
Vn
ð7: 6Þ
wher e Htot is in wat ts, fri ction torque M is in N � mm, and ring speed Vn is in rad/sec. For ring
speed in rpm,
M ¼ 9:551� 103 �Htot
nn
ð7:7Þ
7.2.2 ROLLER BEARINGS
To find the roll er–raceway co ntact fri ction heat generat ion rate, as intro duced in Chapter 1,
each contact area of effective length leff is divided into k laminae, each lamina having
thickness wn and width 2bnjl, subscript l referring to the specific lamina. Hence,
Hnj ¼2bnjwn
3kJ
Xl¼k
l¼1
Sktnjlvnjl ð7:8Þ
In Equation 7.8, Sk is the Simpson’s rule coefficient, tnjl is the average surface friction shear
stress over the lamina area 2bnjlwn, and vnjl is the sliding velocity at the lamina surface. For
cylindrical roller bearings, the sliding velocity over the lamina may be obtained from Equa-
tion 6.62. The friction heat generation rate for all roller–raceway contacts is then
HRRC ¼Xn¼o
n¼i
Xj¼Z
j¼1
Hnj ð7:9Þ
� 2006 by Taylor & Francis Group, LLC.
Usin g Equat ion 6.2 to define the viscous dra g force Fv, the fricti on he at generat ion rate
thereby effected is given by Equation 7.4.
As with ba ll bearing s, fricti on heat is generat ed due to sliding betw een cage and the inner
ring land for a n inner ring pilot ed cage, due to sliding be tween cage and the out er ring land for
an outer ring pilot ed ca ge, and between the roll ers and the cage pockets for a ny cage design
executi on. These heat generat ion rates general ly tend to be smal l; howeve r, they may be
calcul ated using the roller and cage sp eed equ ations of Chapt er 2 toget her wi th estimat ions of
cage rail –ring land loading an d cage–rol ler loading . These may be determ ined using a
complet e fri ction force and moment balance ac cording to Chapt er 6.
In addition to the abo ve-mentione d so urces of fri ction he at gen eration, in cylind rical
roller bearing s that are mis aligned or otherwis e subjected to comb ined radial and thrust
loading s, signi fican t fricti on heat generat ion can occu r be tween the roller end s an d inner
and outer ring roll er guide flanges. To estimat e the heat generat ion rates, it is first necessa ry
to calculate the roller end –flange loads Qaj us ing the analyt ical methods indica ted in
Chapt er 1 and Chapt er 3 . Then, using the methods indicated in Chapt er 6, the cage speed
vm and ro ller speeds vR j need to be estimat ed. Knowing the ring speed s, it is possibl e to
estimat e an a verage slid ing veloci ty be tween the ro ller e nds a nd ring flang e. Finall y, de pend-
ing on the lubri cation method, a c oefficient of sliding friction for the roll er end –flange
contact s needs to be assum ed or calcul ated. Genera lly, for oil- lubricated bea rings, a value
of 0.03 � m � 0.07 should be obtaine d. The fricti on heat gen eration rate for a roller en d–
flang e co ntact is then
HREF nj ¼mQa j v REF nj
J ð 7: 10 Þ
and
HREF ¼Xn ¼ o
n ¼ i
Xj ¼ Z
j¼1
HREFnj ð7:11Þ
Each roller in a tapered roller bearing experiences contact between the roller end and the large
end flang e a s indica ted in Chapt er 5 of the fir st volume of this handbo ok. In this case,
HREFj ¼mQf jvREFj
Jð7:12Þ
and
HREF ¼Xj¼Z
j¼1
HREFj ð7:13Þ
Equation 7.12 and Equation 7.13 may be used to calculate HREF for spherical roller bearings
with asymmetrical contour rollers.
For roller bearings the total friction heat generation rate, exclusive of seals, is
Htot ¼ HRRC þHfdrag þHCRL þHCPR þHREF ð7:14Þ
� 2006 by Taylor & Francis Group, LLC.
See Example 7.1 and Example 7.2.
Methods for calculating friction torque and heat generation rates may also be found in
bearing catalogs; for example, Ref [8].
7.3 HEAT TRANSFER
7.3.1 MODES OF HEAT TRANSFER
There exist three fundamental modes for the transfer of heat between masses with different
temperature levels. These are the conduction of heat within solid structures, the convection of
heat from solid structures to fluids in motion (or apparently at rest), and the radiation of heat
between masses separated by space. Although other modes exist, such as radiation to gases
and conduction within fluids, their effects are minor for most bearing applications and may
usually be neglected.
7.3.2 HEAT CONDUCTION
Heat conduction, which is the simplest form of heat transfer, may be described for the
purpose of this discussion as a linear function of the difference in temperature level within
a solid structure, that is,
Hc ¼kS
LðT1 � T2Þ ð7:15Þ
ThequantityS inEquation 7.15 is the area normal to the flowof heat between twopoints and L
is the distance between the same two points. The thermal conductivity k is a function of the
material and temperature levels; however, the latter variation is generally minor for structural
solids and will be neglected here. For heat conduction in a radial direction within a cylindrical
structure such as a bearing inner or outer ring, the following equation is useful:
Hc ¼2pkWðTi � ToÞ
lnð<o=<iÞð7:16Þ
In Equation 7.16, W is the width of the annular structure and <o and <i are the inner and outer
radii defining the limits of the structure through which heat flow occurs. If <i¼ 0, an
arithmetic mean area is used and the equation assumes the form of Equation 7.15.
7.3.3 HEAT CONVECTION
Heat convection is the most difficult form of heat transfer to estimate quantitatively. It occurs
within the bearing housing as heat is transferred to the lubricant from the bearing and from
the lubricant to other structures within the housing as well as to the inside walls of the
housing. It also occurs between the outside of the housing and the environmental fluid—
generally air, but possibly oil, water, another gas, or a working fluid medium.
Heat convection from a surface may generally be described as follows:
Hv ¼ hvSðT1 � T2Þ ð7:17Þ
where hv, the film coefficient of heat transfer, is a function of surface and fluid temperatures,
fluid thermal conductivity, fluid velocity adjacent to the surface, surface dimensions and
attitude, fluid viscosity, and density. It can be seen that many of these properties are temperature
� 2006 by Taylor & Francis Group, LLC.
depen dent. Therefor e, heat con vection is not a linea r function of tempe ratur e unless fluid
propert ies can be consider ed reasonabl y stable over a fini te tempe ratur e range.
Heat convecti on within the housing is most diff icult to descri be, and a rough approxim a-
tion will be used for the heat trans fer film coeff icient. As oil is used as a lubri cant and the
viscos ity is high, lami nar flow is assumed . Ecke rt [2] states for a plate in a laminar flow field:
hv ¼ 0: 0332 k Pr 1 = 3 us
no x
� �1= 2ð 7: 18 Þ
The use of Equation 7.18 taking us equal to bearing cage surface velocity and x equ al to
bearing pitch diame ter seems to yield worka ble values for hv, consider ing he at trans fer from
the bearing to the oil that con tacts the bearing . For heat trans fer from the hous ing insi de surfa ce
to the oil, taking us equ al to one third cage velocity and x equal to hous ing diame ter yiel ds
adequ ate results. In Equation 7.18, no repres ents kinema tic viscosit y and Pr the Pran dtl num ber
of the oil.
If coo ling co ils are submer ged in the oil sump, it is best that they be aligned parallel to the
shaft so that a lamin ar cross-flo w is obtaine d. In this case, Ecke rt [2] shows that for a cyli nder
in cross-flo w, the outsi de heat trans fer film coeff icient may be ap proxim ated by
hv ¼ 0: 06ko
D
us D
no
� �1 =2
ð 7: 19 Þ
wher e D is the outsi de diame ter of the tub e and ko is the therm al condu ctivity of the oil. It is
recomm ended that us be taken as approxim ately one fourt h of the bearing inner ring surfa ce
veloci ty.
These ap proximati ons for fil m coefficie nt are necessa rily crude. If great er accuracy is
requir ed, Ref . [2] indicates more refined methods for obta ining the film coefficie nt. In lieu of a
more e legant analys is, the values yiel ded by Equation 7.18 and Equat ion 7.19, an d Equat ion
7.20 an d Equation 7.21 that follow, shou ld suff ice for general eng ineering purpo ses.
In quiescen t air, heat trans fer by co nvection from the hous ing exter nal surface may be
approxim ated by using an outsi de film coe fficient in acc ordance with Equation 7.20 (see
Jakob an d Haw kins [3]):
hv ¼ 2: 3 � 10 � 5 ð T � Ta Þ0 :25 ð 7: 20 Þ
For forced flow of air of veloci ty us over the hous ing, Ref . [2] yiel ds:
hv ¼ 0: 03ka
D
us Dh
na
� �0: 57
ð 7: 21 Þ
wher e Dh is the approxim ate housing diame ter. Palmgr en [4] gives the followin g form ula to
approxim ate the exter nal area of a be aring hous ing or pillo w block:
S ¼ p Dh W h þ 12
Dh
� �ð 7: 22 Þ
wher e Dh is the maxi mum diame ter of the pillow blo ck and Wh is the wi dth.
The calculati ons of lubrican t film thickne ss as specified in Chapt er 4 de pend on the
viscos ity of the lubri cant entering the rolling/ sliding contact , whi le the calculati ons of tract ion
over the con tact as specif ied in Chapter 5 de pend on the viscos ity of the lubri cant in the
contact. Since lubricant viscosity is a function of temperature, a detailed performance analysis
� 2006 by Taylor & Francis Group, LLC.
of ball and roller bearings entails the estimation of temperatures of lubricants both entering,
and residing in, the individual contacts. To do this requires the estimation of heat dissipation
rates from the rotating components and rings. The coefficient of convection heat transfer for
a rotating sphere (ball) is provided by Kreith [5] as follows:
hvD
k¼ 0:33Re0:5
D Pr0:4 ð7:23Þ
where ReD, the Reynolds number for a rotating ball, is given by
ReD ¼vD2
nð7:24Þ
In Equation 7.24, D is the diameter of the ball, v is the ball speed about its own axis, and n is
the lubricant kinematic viscosity. Equation 7.23 is valid for 0.7 <Pr < 217 and
GrD < 0.1 �ReD2. The Grashof number is given by
Gr ¼ BgðTs � T1ÞD2
n2ð7:25Þ
where B is the thermal coefficient of fluid volume expansion, g the acceleration due to gravity,
Ts the temperature at the ball surface, and T1 is the fluid stream temperature. The Prandtl
number is given by
Pr ¼ hgc
kð7:26Þ
where c is the specific heat of the fluid.
For a rotating cylindrical ring or roller,
hvD
k¼ 0:19ðRe2
D þGrDÞ ð7:27Þ
In Equation 7.27, D is the outside diameter of the ring or roller. Equation 7.27 is valid for
ReD < 4 � 105.
7.3.4 HEAT RADIATION
The remaining mode of heat transfer to be considered is the radiation from the housing
external surface to the surrounding structures. For a small structure in a large enclosure, Ref.
[3] gives
Hr ¼ 5:73 «ST
100
� �4
� Ta
100
� �4" #
ð7:28Þ
where the temperature is in degrees Kelvin (absolute). Equation 7.28, nonlinear being in
temperatures, is sometimes written in the following form:
Hr ¼ hrSðT � TaÞ ð7:29Þ
� 2006 by Taylor & Francis Group, LLC.
where
hr ¼ 5:73� 10�8«ðT þ TaÞðT2 þ T2a Þ ð7:30Þ
Equation 7.29 and Equation 7.30 are useful for hand calculation in which problem T and Ta
are not significantly different. On assuming a temperature T for the surface, the pseudofilm
coefficient of radiation hr may be calculated. Of course, if the final calculated value of T is
significantly different from that assumed, then the entire calculation must be repeated.
Actually, the same consideration is true for calculation of hv for the oil film. Since ko and
no are dependent on temperature, the assumed temperature must be reasonably close to the
final calculated temperature. How close is dictated by the actual variation of those properties
with oil temperature.
7.4 ANALYSIS OF HEAT FLOW
7.4.1 SYSTEMS OF EQUATIONS
Because of the discontinuities of the structures that comprise a rolling bearing assembly,
classical methods of heat transfer analysis cannot be applied to obtain a solution describing
the system temperatures. By classical methods we mean the description of the system in terms
of differential equations and the analytical solution of these equations. Instead, methods of
finite difference as demonstrated by Dusinberre [6] must be applied to obtain a mathematical
solution.
For finite difference methods applied to steady-state heat transfer, various points or nodes
are selected throughout the system to be analyzed. At each of these points, the temperature is
determined. In steady-state heat transfer, heat influx to any point equals heat efflux; there-
fore, the sum of all heat flowing toward a temperature node is equal to zero. Figure 7.1 is a
heat flow diagram at a temperature node, demonstrating that the nodal temperature is
T2
T0
T4
T1 T3
Con
duct
ion
Con
duct
ion
Conduction Conduction
FIGURE 7.1 Two-dimensional temperature node system.
� 2006 by Taylor & Francis Group, LLC.
affe cted by the tempe ratures of ea ch of the four indica ted surroundi ng node s. (Although the
syst em depict ed in Figure 7.1 sho ws onl y four surrou nding node s, this is purely by choice of
grid and the number of node s may be great er or smal ler.) Since the sum of the hea t flows is
zero,
H1 � 0 þ H 2 � 0 þ H 3� 0 þ H4 � 0 ¼ 0 ð7: 31 Þ
For this examp le, it is assum ed that heat flow oc curs only by condu ction and that the grid is
nons ymmetri cal, making all areas S and lengt hs of flow path different . Fur thermo re, the
mate rial is assum ed nonisot ropic so that therm al co nductiv ity is different for all flow pa ths.
Subs titution of Equat ion 7.15 into Equat ion 7.31 theref ore y ields
k1 S 1
L1
ð T1 � T0 Þ þk2 S2
L2
ð T2 � T0 Þ þk3 S3
L3
ð T3 � T0 Þ þk4 S4
L4
ð T4 � T 0 Þ ¼ 0 ð7: 32 Þ
By rearr anging terms, one obtains
k1 S 1
L1
T1 þk2 S2
L2
T2 þk3 S 3
L3
T3 þk4 S4
L4
T4 �Xi ¼ 4
i ¼ 1
ki Si
L i
T0 ¼ 0 ð7: 33 Þ
or
F1 T1 þ F 2 T2 þ F3 T 3 þ F4 T4 �Xi ¼ 4
i ¼ 1
Fi T 0 ¼ 0 ð7: 34 Þ
Divid ing by S Fi yiel ds
F1
S Fi
T1 þF2
SFi
T2 þF3
SFi
T3 þF4
S Fi
T4 � T0 ¼ 0 ð7: 35 Þ
Mo re con cisely, Equat ion 7.35 may be written
fi Ti ¼ 0 ð7: 36 Þ
wher e fi are influ ence coefficie nts of tempe rature eq ual to F i / SF i. If the material were
isot ropic an d a symm etrical grid was ch osen, then f0 ¼ 1 an d the other fi ¼ 0.25.
In the e xample, only heat condu ction was illustrated. If, howeve r, he at flow be tween
points 4 an d 0 was by conve ction, then acc ording to Equation 7.17, F4 ¼ hv4 S4. For a mult i-
noda l syste m, a seri es of e quations sim ilar to Equation 7.35 may be written. If the equati ons
are linear in temperature T, they may be solved by classical methods for the solution of
simultaneous linear equations or by numerical methods (see Ref. [7]).
The system may include heat generation and be further complicated, however, by non-
linear terms caused by heat radiation and free convection. Consider the example schematic-
ally illustrated in Figure 7.2. In that ill ustration, heat is generat ed at point 0, dissipated by free
convection and radiation between points 1 and 0 and dissipated by conduction between points
2 and 0. Thus,
Hf0 þH1�0;v þH1�0;r þH2�0 ¼ 0 ð7:37Þ
� 2006 by Taylor & Francis Group, LLC.
Q Heat generated
T0 T2T1
Radiationconvection Conduction
FIGURE 7.2 Convective, radiation, and conductive heat transfer system.
The use of Equation 7.15, Equation 7.17, Equation 7.20, and Equation 7.28 gives
Hf0 þ 2:3� 10�5S1ðT1 � T0Þ1:25 þ 5:73� 10�8«S1ðT41 � T4
0 Þ þK2S2
L2
ðT2 � T0Þ ¼ 0 ð7:38Þ
or
Hf0 þ F1vðT1 � T0Þ1:25 þ F1rðT41 � T4
0 Þ þ F2ðT2 � T0Þ ¼ 0 ð7:39Þ
7.4.2 SOLUTION OF EQUATIONS
A system of nonlinear equations similar to Equation 7.39 is difficult to solve by direct
numerical methods of iteration or relaxation. Therefore, the Newton–Raphson method [7]
is recommended for solution.
The Newton–Raphson method states that for a series of nonlinear functions qi of variables Tj
qi þX ›qi
›Tj
«j ¼ 0 ð7:40Þ
Equation 7.40 represents a system of simultaneous linear equations that may be solved for «j
(error on Tj).
Then, the new estimate of Tj is
T 0j ¼ Tjð0Þ þ «j ð7:41Þ
and new values qi may be determined. The process is continued until the functions qi are virtually
zero. With a system of nonlinear equations similar to Equation 7.39, such equations must be
linearized according to Equation 7.40. Thus, let Equation 7.39 be rewritten as follows:
Hf0 ¼ F1vðT1 � T0Þ1:25 þ F1rðT41 � T4
0 Þ þ F2ðT2 � T0Þ ¼ q0 ð7:42Þ
Now,
›q0
›T0
¼ �1:25F1vðT1 � T0Þ0:25 � 4F1rT30 þ F2
›q0
›T1
¼ 1:25F1vðT1 � T0Þ0:25 þ 4F1rT31
›q0
›T2
¼ F2
ð7:43Þ
Substitution of Equation 7.42 and Equation 7.43 into Equation 7.40 yields one equation in
variables «0, «1, and «2.
� 2006 by Taylor & Francis Group, LLC.
The system of nonlinear equations is solved for T0, T1, and T2 when the root mean square
(rms) error is sufficiently small, for example, less than 0.18.
7.4.3 TEMPERATURE NODE SYSTEM
A simple system of temperature nodes that could be used to determine the temperatures in
an oil-lubricated, spherical roller bearing pillow block assembly is illustrated in Figure 7.3.
In this illustration, the dimensions of a 23072 double-row bearing are shown together with
pertinent dimensions of the pillow block. This illustration has been designed to be as simple as
possible such that all equations and methods of solution may be demonstrated. To do this, the
following conditions have been assumed:
FIGblo
� 20
1. Ten temperature nodes are sufficient to describe the system shown in Figure 7.3. Node
A is ambient temperature; nine temperatures need to be determined.
2. The inside of the housing is coated with oil and may be described by a single temperature.
3. The inner ring raceway may be described by a single temperature.
127 (5)
63.5 (2.5)
50.8
406
(16)
dia
met
er
483
(19)
dia
met
er
356
(14)
dia
met
er
533
(21)
dia
met
er
660
(26)
dia
met
er
(2)
3
2
1
406 (16)
4
5
6
7
8
89
A
A
A
A
URE 7.3 Simple temperature node system selected for analysis of a spherical roller bearing pillow
ck assembly.
06 by Taylor & Francis Group, LLC.
� 20
4. The outer ring racew ay may be descri bed by a single tempe rature.
5. The housing is symmetri cal abou t the shaft center line and vertical section A–A. Thus,
heat trans fer in the circumfer ential direction doe s not have to be consider ed.
6. The sump oil may be consider ed at a singl e tempe ratur e.
7. The shaft ends at the axial extre mities of the pillo w block are at ambie nt tempe rature.
Consideri ng the tempe ratur e node system of Figu re 7.3, the heat trans fer syst em wi th
pertin ent e quations is given in Table 7.1. The heat flow areas and lengt hs of heat flow paths
are obtaine d from the dimens ions of Figure 7.3 con sidering the location of each tempe ratur e
node. Based on Figure 7.3 and Table 7.1, a set of nine simultaneous nonlinear equations with
unknownvariablesT1 –T9 canbedeveloped.This system is nonlinear because of free convection
from the pillow block external surface to ambient air and radiation from the external surface to
structures at ambient temperature; the Newton–Raphson method may be used to obtain a
solution.
See Example 7.3.
The system chosen for evaluation was necessarily simplified for the purpose of illustration.
A more realistic system would consider variation of bearing temperature in a circumferential
direction also. This would entail many more temperature nodes and corresponding heat
transfer equations. In this case, viscous friction torque may be constant with respect to
angular location; however, friction torque due to load varies as the individual rolling element
load on the stationary ring. The latter, however, may be considered invariant with respect to
angular location on the rotating ring. A three-dimensional analysis such as that indicated by
load friction torque variation on the stationary ring should, however, show little variation in
temperature around the bearing ring circumferences so that a two-dimensional system should
suffice for most engineering applications. Of course, if temperatures of structures surrounding
or abutting the housing vary significantly, then a three-dimensional study is required.
It is not intended that the results of this method of analysis will be of extreme accuracy,
but only that accuracy will be sufficient to determine the approximate thermal level of
operation. Then, corrective measures may be taken if excessive steady-state operating tem-
peratures are indicated. In the event cooling of the assembly is required, the same methods
may be used to evaluate the adequacy of the cooling system.
Generally, the more temperatures selected, that is, the finer the heat transfer grid, the
more accurate will be the analysis.
7.5 HIGH TEMPERATURE CONSIDERATIONS
7.5.1 SPECIAL LUBRICANTS AND SEALS
Having established the operating temperatures in a rolling bearing assembly while using a
conventional mineral oil lubricant and lubrication system, and having estimated that the
bearing or lubricant temperatures are excessive, it then becomes necessary to redesign
the system to either reduce the operating temperatures or make the assembly compatible
with the temperature level. Of the two alternatives, the former is safest when considering
prolonged duration of operation of the assembly. When shorter lubricant or bearing life is
acceptable, it may be expeditious and even economical to simply accommodate the increased
temperature level by using special lubricants in the bearing operations or a bearing manufac-
tured from a high-temperature capacity steel. The latter approach is effective when space and
weight limitations preclude the use of external cooling systems. It is further necessitated in
applications in which the bearing is not the prime source of heat generation.
06 by Taylor & Francis Group, LLC.
TABLE 7.1Heat Transfer System Matrix and Heat Transfer Equations for Figure 7.3
Node A 1 2 3 4 5 6 7 8 9
1 — — — Convection
(7.17) (7.7)
Convection
(7.17) (7.7)
Convection
(7.17) (7.7)
— — Convection
(7.17) (7.7)
—
2 Conduction
(7.15)
— — Conduction
(7.16)
— — — — — —
3 Convection
(7.17) (7.7)
Conduction
(7.16)
— Conduction
(7.16)
— — — — —
4 — Convection
(7.17) (7.7)
— Conduction
(7.16)
Heat generation
(7.14)
— — — — —
5 — Convection
(7.17) (7.7)
— — — Heat generation
(7.14)
Conduction
(7.16)
6 — — — — — Conduction
(7.16)
— Conduction
(7.16)
Conduction
(7.15)
7 Convection
(7.17) (7.20)
Radiation (7.28)
— — — — — Conduction
(7.16)
— Conduction
(7.16)
Conduction
(7.15)
8 — Convection
(7.17) (7.7)
— — — — — Conduction
(7.15)
Conduction
(7.16)
Conduction
(7.15)
9 Convection
(7.17) (7.20)
Radiation (7.28)
— — — — — — Conduction
(7.15)
Conduction
(7.15)
—
�2006
by
Taylo
r&
Fra
ncis
Gro
up,L
LC
.
7.5.2 HEAT REMOVAL
For situatio ns in whi ch the bearing is the prim e sou rce of heat generat ion and in which the
ambie nt cond itions do not permi t an adeq uate rate of heat remova l, placing the bearing
housing in a moving air stre am may be suff icient to red uce ope rating tempe ratur es. This may
be accompl ished by using a fan of suff icient air moving capacit y.
Additional heat remova l capacit y may be effected by designi ng a housing with fins to
increa se the effec tive area for heat trans fer.
See Exampl e 7.4 and Exam ple 7.5.
When the bearing is not the prim e sou rce of heat generat ion, cooling of the housing in the
foregoi ng manner wi ll generally not su ffice to maintain the be aring and lubri cant co ol. In this
case, it is general ly necessa ry to co ol the lubric ant and pe rmit the lubri cant to cool the
bearing . The most effecti ve way of acc omplishing this is to pass the oil through an exter nal
heat ex changer a nd direct jets of cooled oil on the be aring. To save space when a supp ly of
moving coo lant is readily available, it may be possible to place the heat exchanger co ils
directly in the sump of the bearing housing. The co oled lubricant is then circul ated by bearing
rotation . The latter method is not quite a s efficie nt therm ally a s jet cooling althoug h bearing
fricti on torque and heat gen eration may be less by not resorting to jet lubricati on and the
atten dant ch urning of excess oil.
See Exampl e 7.6.
Several resear chers have applied these methods to effecti vely pred ict tempe ratur es in
rolling be aring applications . Initial ly, Harr is [9,1 0] applied the method to relative ly slow-
speed, spherical roller bearing s. Subseq uently , these methods have been success fully applie d
to both high-s peed ball and roll er bearing s [11–13 ]. Good agreem ent with experi mentally
measur ed tempe ratures was report ed [15] using the steady -st ate tempe ratur e calcul ation
operati on mo de of SH ABERTH [14] , a co mputer program to analyze the therm o-mec hanical
perfor mance of shaft-rol ling be aring systems. Figure 7.4 shows a noda l netwo rk model an d
the associ ated heat flow pa ths for a 35-mm bor e ball bearing . Figure 7.5 shows the agreem ent
achieve d be tween calcul ated and exp erimental ly measured tempe ratures. It must be pointed
out, howeve r, that constru ction of a therm al model that mathemati cally simulat es a bearing
accurat ely often requir es a consider able amount of effor t and he at trans fer expert ise.
7.6 HEAT TRANSFER IN A ROLLING–SLIDING CONTACT
Accur ate calculati on of lubrica nt film thickne ss and tract ion in a rolling contact depend s on
the determ ination of lubri cant viscos ity at the appropri ate tempe ratures. For lubri cant
film thickne ss, this means calculati on of the lubri cant temperatur e enteri ng the contact . For
traction, this means calcul ation of the lubric ant temperatur e for its duratio n in the con tact. In
Ref. [14], the hea t trans fer system illustr ated in Figure 7.6 was us ed.
Designating subscript k to represent the raceway and j the rolling element location, the
following heat flow equations describe the system:
Hc;2kj�1kj þHv;tout�1kj ¼ 0 ð7:44Þ
Since the lubricant is essentially a solid slug during its time in the contact, heat transfer from
the film to the rolling body surfaces is by conduction. Then, assuming that the minute slug
exists at an average temperature T3
� 2006 by Taylor & Francis Group, LLC.
Supportbearing Outer
ring
Oil sump(known temperature)
Inner ring(a)
(b)
FIGURE 7.4 Bearing system nodal network and heat flow paths for steady-state thermal analysis.
(a) Metal, air, and lubricant temperature nodes: (.) metal or air node; (8) lubricant node; (-!) lubricant
flow path. (b) Conduction and convection heat flow paths (From Parker, R., NASA Technical Paper
2275, February 1984.)
Hc ;1 kj � 2kj þ H c ;3 kj � 2kj ¼ 0 ð7: 45 Þ
The lubri cant slug is transpo rted through the contact ; it enters at tempe ratur e T30 and exits at
T30. Ther efore, for heat trans fer total ly within the slug
Hc ; 2kj � 3 kj þ H c ;4 kj � 3kj þ H gen; j þ wc T 03 kj � T 03kj
� �¼ 0 ð7: 46 Þ
Hc ;3 kj � 4kj þ H c ;5 kj � 4kj ¼ 0 ð7: 47 Þ
Hc ;4 kj � 5 kj þ H v; tout � 5kj ¼ 0 ð7: 48 Þ
Finall y, the lubri cant acts as a heat sink carryi ng heat away from the contact
Hv;6�1kj þHv;6�5kj � wcðT6 � Tl:inÞ ¼ 0 ð7:49Þ
In high-s peed bearing fri ctional perfor mance analyses such as tho se indica ted in Chapt er 6,
the rolling–sliding contact heat transfer analyses are performed thousands of times to achieve
consistent solutions. The analyses are begun by assuming a set of system temperatures.
Lubricant viscosities are then determined at these temperatures, and frictional heat gener-
ation rates are calculated. These are subsequently used to recalculate temperatures and
temperature-dependent parameters. The process is repeated until the calculated temperatures
� 2006 by Taylor & Francis Group, LLC.
440
360
320
280
Inne
r-ra
ce te
mpe
ratu
re, �
F
Inne
r-ra
ce te
mpe
ratu
re, K
400
500Predicted Experi-
mentalShaft speed,
rmp
47,50064,900
460
440
420
(a) (b)
(c) (d)
400
480
Out
er-r
ace
tem
pera
ture
, �F
Out
er-r
ace
tem
pera
ture
, K
440
360
320
280
400
500
460
400
400
420
480
500
520
460
440
420
400
480
440
480
360
320
280
Oil-
out t
empe
ratu
re, �
F
Oil-
out t
empe
ratu
re, K
400
0 0.1 0.2Total lubricant flow rate, gpm
0.3 0.4 0.5 0 0.1 0.2Total lubricant flow rate, gpm
Total lubricant flow rate, cm3/min
0.3 0.4 0.5
2000
2.0
1.5
1.0
Bea
ring
heat
gen
erat
ion,
HP
Bea
ring
heat
gen
erat
ion,
KW
0.5
1.6
1.2
0.8
0.4150010000 500
Total lubricant flow rate, cm3/min2000150010000 500
FIGURE 7.5 Comparison of predicted and experimental temperatures using SHABERTH. (a) Inner
raceway temperature. (b) Outer raceway temperature. (c) Oil-out temperature. (d) Bearing heat gener-
ation (From Parker, R., NASA Technical Paper 2275, February 1984.).
substa ntially match the assum ed tempe ratures. This method while producing more accurat e
calcul ations for bearing heat generat ions and fricti on torques req uires rather sop histicat ed
computer pr ograms for its executio n see Ref . [1,16]. For slow-s peed bearing applic ations in
which the bearing rings are rigi dly supporte d, the simp ler calcul ation methods for bearing
heat generat ions provided in Chapt er 10 of the fir st volume of this hand book will gen erally
suffice.
1-rolling element
2-rolling element surface
4-raceway surface
5-ring
3-film 6-lubricant outLubricant in
Ω
ω
FIGURE 7.6 Rolling element–lubricant–raceway–ring temperature node system.
� 2006 by Taylor & Francis Group, LLC.
7.7 CLOSURE
The tempe ratur e level at whi ch a roll ing bearing ope rates dicta tes the type an d amo unt of
lubri cant required a s well as the mate rials from the bearing compo nents that may be
fabri cated. In some app lications , the environm ent in which the bearing ope rates establis hes
the temperatur e level whereas in other applic ations the be aring is the pr ime source of heat. In
eithe r case, depend ing on the bearing material s and the enduran ce requ ired of the be aring, it
may be necessa ry to cool the bearing using the lubrican t as a co olant.
General rules cannot be formulated to determine the temperature level for a given bearing
operating under a given load at a given speed. The environment in which the bearing operates is
generally different for each specialized application. Using the friction torque formulas of Chapter
10 of the first volume of this handbook or Chapter 6 in the second volume to establish the rate of
bearing heat generation in conjunction with the heat transfer methods presented in this chapter,
however, it is possible to estimate the bearing system temperatures with an adequate degree of
accuracy.
REFERENCES
1.
� 200
Harris, T., Establishment of a new rolling bearing fatigue life calculation model, Final Report U.S.
Navy Contract N00421-97-C-1069, February 23, 2002.
2.
Eckert, E., Introduction to the Transfer of Heat and Mass, McGraw-Hill, New York, 1950.3.
Jakob, M. and Hawkins, G., Elements of Heat Transfer and Insulation, 2nd Ed., Wiley, New York,1950.
4.
Palmgren, A., Ball and Roller Bearing Engineering, 3rd Ed., Burbank, Philadelphia, 1959.5.
Kreith, F., Convection heat transfer in rotating systems, Adv. Heat Transfer, 5, 129–251, 1968.6.
Dusinberre, G., Numerical Methods in Heat Transfer, McGraw-Hill, New York, 1949.7.
Korn, G. and Korn, T., Mathematical Handbook for Scientists and Engineers, McGraw-Hill, NewYork, 1961.
8.
SKF, General Catalog 4000 US, 2nd Ed., 49, 1997.9.
Harris, T., Prediction of temperature in a rolling bearing assembly, Lubr. Eng., 145–150, April 1964.10.
Harris, T., How to predict temperature increases in rolling bearings, Prod. Eng., 89–98, December 9,1963.
11.
Pirvics, J. and Kleckner, R., Prediction of ball and roller bearing thermal and kinematic perform-ance by computer analysis, Adv. Power Transmission Technol., NASA Conference Publication 2210,
185–201, 1982.
12.
Coe, H., Predicted and experimental performance of large-bore high speed ball and roller bearings,Adv. Power Transmission Technol., NASA Conference Publication 2210, 203–220, 1982.
13.
Kleckner, R. and Dyba, G., High speed spherical roller bearing analysis and comparison withexperimental performance, Adv. Power Transmission Technol., NASA Conference Publication 2210,
239–252, 1982.
14.
Crecelius, W., User’s manual for SKF computer program SHABERTH, steady state and transientthermal analysis of a shaft bearing system including ball, cylindrical, and tapered roller bearings,
SKF Report AL77P015, submitted to U.S. Army Ballistic Research Laboratory, February 1978.
15.
Parker, R., Comparison of predicted and experimental thermal performance of angular-contact ballbearings, NASA Technical Paper 2275, February 1984.
16.
Harris, T. and Barnsby, R. Tribological performance prediction of aircraft gas turbine mainshaftball bearings, Tribol. Trans., 41(1), 60–68, 1998.
6 by Taylor & Francis Group, LLC.
8 Application Load and LifeFactors
� 2006 by Taylor & Fran
LIST OF SYMBOLS
Symbol Description Units
a Semimajor axis of projected contact ellipse mm (in.)
Ac Fatigue life reduction factor for clearance
Ac/Ao Contact area fraction in asperity–asperity contact
Asteel Fatigue life factor for steel
A1 Reliability–life factor
A2 Material–life factor
A3 Lubrication–life factor
A4 Contamination–life factor
AISO Life modification factor based on ISO systems
approach of life calculation
ASL Stress–life factor
b Semiminor axis of projected contact ellipse mm (in.)
c Simpson’s rule coefficients
C Bearing basic dynamic capacity N (lb)
CL Particulate contamination parameter
CL1 Parameter used to calculate CL
CL2 Constant used to calculate CL
CL3 Constant used to calculate CL1
dm Bearing pitch diameter mm (in.)
dr Raceway diameter mm (in.)
D Ball or roller diameter mm (in.)
e Weibull slope
Fr Applied radial load N (lb)
Fa Applied axial load N (lb)
Fe Equivalent applied load N (lb)
Flim Fatigue limit load N (lb)
FR Filter rating mm
h0 Minimum lubricant film thickness mm (min.)
I Life integral
KC Stress concentration factor due to particulate contamination
KL Stress concentration factor associated with
lubrication effectiveness
L Fatigue life
L10 Fatigue life that 90% of a group of bearings will endure revolutions� 106
L50 Fatigue life that 50% of a group of bearings will endure revolutions� 106
N Number of stress cycles
cis Group, LLC.
n Rotational speed rpm
q Load on a roller–raceway contact lamina N (lb)
qc Basic dynamic capacity for a roller–raceway contact lamina N (lb)
Q Ball or roller load N (lb)
Qc Basic dynamic capacity of a raceway contact N (lb)
r z /b
R Oil bath and grease contamination parameter
S Probability of survival
SF Composite rms surface roughness of mating surfaces mm (min.)
T Temperature 8C (8F)
u Number of stress cycles per revolution
V Volume under stress mm3 (in.3)
w Width of a roller–raceway contact lamina mm (in.)
z0 Depth of maximum orthogonal shear stress mm (in.)
Z Number of rolling elements per row
a Contact angle rad,8
bx Filter effectiveness ratio for particles of size x mm
g D cos a/dm
da Bearing axial deflection mm (in.)
dr Bearing radial deflection mm (in.)
L h0/SF
n Kinematic viscosity mm2/sec (in. 2/sec)
n1 Kinematic viscosity for adequate lubrication mm2/sec (in.2/sec)
sVM von Mises stress MPa (psi)
t0 Maximum orthogonal subsurface shear stress MPa (psi)
f Oscillation angle rad,8
c Rolling element azimuth angle rad,8
Subscripts
B Ball
i Inner ring or raceway
j Rolling element azimuth location
k Roller–raceway contact lamina location
m Raceway
n Probability of failure
o Outer ring or raceway
R Roller
RE Equivalent rotating bearing or rolling element
m Rotating raceway
n Nonrotating raceway
8.1 GENERAL
The Lundberg–Palmgren theory and the standard load and fatigue life calculations that
resulted [1–5] are only the first step toward determining the bearing fatigue lives in applica-
tions. Use of the standard methods should be limited to those applications in which the
� 2006 by Taylor & Francis Group, LLC.
intern al geo metries and roll ing c omponent mate rials of the bearing s employ ed conform to the
standar d specificat ions, and the operati ng condition s are bounde d as foll ows:
. The be aring outer ring is mounted and properl y supporte d in a rigid hous ing.
. The be aring inner ring is properl y mou nted on a nonf lexible shaft.
. The be aring is ope rated at a steady speed unde r invari ant load ing.
. Opera tional speed is suffici ently slow such that rolling elem ent centrifugal and gyro-
scopic loading s are insign ificant.. Bearing loading can be ad equately define d by a singl e radial load, a single axial load, or
a combinat ion of these.. Bearing loading does not cause signifi cant pe rmanent deform ations or mate rial trans -
form ations.. For bearing s under radial loading , mounted interna l clear ance is essent ially nil.. For angu lar-contact ba ll bearing s, nominal contact an gle is co nstant.. For roll er bearing s, uniform loading is mainta ined at each roller–rac eway c ontact.. The be aring is ad equately lubri cated.
Many app lications can be co nsidered to be included within these con ditions.
In many applic ations , these sim ple con ditions are exceed ed. For exampl e, man y app lica-
tions do not ope rate at a steady speed or load, rather , they operate unde r a load–spe ed cycle.
Furtherm ore, the bearing may supp ort, as indica ted in Chapt er 1, co mbined radial, axial , an d
moment load ings unde r which the distribut ion of intern al loading is signific antly different
from the standar d lim itations . Bea rings may ope rate at speeds that cau se sub stantial ro lling
elem ent inertial loading and variation in co ntact an gles be tween inner an d outer racew ay
contact s. These conditio ns may be address ed by ap plying the Lundber g–P almgren equati ons
in detail using co mputer program s to pe rform the c omplex calcul ations .
After the de velopm ent of the Lundb erg–Pal mgren theory, the ab ility of a lubri cant to
separat e roll ing elem ents from racew ays, as discus sed in Chapt er 4, was establ ished. This
cond ition ha s been shown to have, probab ly, the most profound effect on extend ing bearing
fatigue life compared with an y other conditio n. Improv ement s in modern bearing steel
manufa cturin g methods ha ve provided steel s of very high cleanliness and homogen eity, as
compared with the basic air- melt AISI 52100 steel used in the developm en t of the Lundber g–
Palmgr en theo ry and the standar ds. With the advent of substan tially extended life, increa sed
reliabil ity in bearing life predict ion can be consider ed.
Finally, as the improvements i n bearing manufacture and lubrication were applied, it became
apparent that, s imilar to other steel structures subjected t o cyclic l oading, bearing raceways and
rolling e lements also e xhibit an endurance limit in f atigue. T his means that in a given application, a
ball or roller bearing does not have to fail in fatigue, provided that applied loading and conditions
of oper ation are such that the bearing material fatigue limit stres s is not exceeded.
All of these c ondition s will be ad dresse d in this chap ter.
8.2 EFFECT OF BEARING INTERNAL LOAD DISTRIBUTION ON FATIGUE LIFE
8.2.1 B ALL BEARING LIFE
8.2.1.1 Raceway Life
When the distribution of load among the balls is different from that resulting from the
applied loading conditions specified in the load rating standards, it is necessary to revert to
the Lundberg–Palmgren load–life relationships as given in Chapter 11 in the f irst volume of
� 2006 by Taylor & Francis Group, LLC.
this handbook for individual ball–raceway contacts. For example, for a contact on a
rotating raceway
Lmj ¼Qcmj
Qmj
� �3
ð8:1Þ
where Qcmj is the basic dynamic capacity of the contact of ball j on the rotating raceway, and
Qmj is the load acting on the contact. It is to be noted that the capacity may be different from
point to point around the raceway because the contact angle may vary with the azimuth
angle. For a nonrotating raceway contact,
Lnj ¼Qcnj
Qnj
� �3
ð8:2Þ
It is also to be noted that the ball–raceway load may differ between raceways due to ball
inertial loading. From Equation 8.1 and Equation 8.2, it can be determined that the life of a
bearing that has a complement of Z balls is given by
L ¼Xj¼Z
j¼1
L�emj þ
Xj¼Z
j¼1
L�enj
!�1=e
ð8:3Þ
where exponent e is the slope of the Weibull distribution. It is further to be noted that the life
calculated according to Equation 8.3 does not include ball lives.
8.2.1.2 Ball Life
Notwithstanding the fact that the Lundberg–Palmgren equations are based on bearing fatigue
failure dependent only on raceway fatigue failure, there is ample evidence that balls, as well as
raceways, can succumb to fatigue failure. Assuming that in rolling bearings subjected to reason-
able levels of loading, the balls contact the raceways over defined tracks, starting with Equation
11.41 of the first volume of this handbook, an equation for basic dynamic capacity of the ball
portion of a ball–raceway contact can bedeveloped. In that equation, it is observed that for a ball,
track diameter at the rotating raceway contact dm¼D cos amj; also, dn¼D cos anj. Furthermore,
for a ball track, the number of stress cycles per ball revolution u¼ 2. Making these substitutions,
the basic dynamic capacity for the ball in a ball–raceway contact is given by
QBnj ¼ 77:92fn
2fn � 1
� �0:41
1þ cngnj
� �1:69 D1:8
cos anj
� �0:3 , n ¼ m; n ð8:4Þ
where cn¼þ1 for a ball–outer raceway contact; cn¼�1 for a ball–inner raceway contact.
Using Equation 8.4, the equation for bearing life becomes
L ¼Xj¼Z
j¼1
L�emj þ
Xj¼Z
j¼1
L�enj þ
Xj¼Z
j¼1
L�eBmj þ
Xj¼Z
j¼1
L�eBnj
!�1=e
ð8:5Þ
In using Equation 8.5, it must be recognized that bearing life is defined in revolutions of the
rotating ring. For example, for a simple rolling motion, the number of ball revolutions per inner
ring revolution, as determined from Equation 10.14 of the first volume of this handbook, is
� 2006 by Taylor & Francis Group, LLC.
nB
ni
¼ dm
2D1 � g
2nj
� �, n ¼ m , n ð 8: 6Þ
Therefor e, the ball lives indica ted in Equation 8.5 must first be divide d by the ratio of
Equation 8.6. In cases wher e ball speeds are calcul ated consider ing fricti onal effe cts, ball
speeds are calcul ated accordi ng to the method s in Chapter 2, and the ratio of Equat ion 8.6
may be replaced by the calcul ated speed rati o.
Also, in us ing Equat ion 8.5, it must be recog nized that the Weibull slope for ba ll failures
may be somew hat different from that for raceway failures . For exampl e, in a fatigue failure
investiga tion of vacuu m-indu ction-mel ted, vac uum-arc-re melted (VIMV AR) M50 steel balls,
the data of Harris [6] indica ted an average Weibull slope of 3.33. In such a case, an average
value of e may be used in Equat ion 8.5.
8.2.2 R OLLER B EARING L IFE
8.2.2 .1 Ra ceway Life
In Chapt er 1, it was shown that to de termine the distribut ion of load among the rollers
for nons tandard app lied loading , the roller–rac eway co ntacts may be divide d into a num ber
of laminae. Hence, for a roll er–raceway contact of lengt h l , if the co ntact is divided into m
laminae , each of width w, l ¼ mw and
Qmj ¼ wXk ¼ m
k¼ 1
qmkj ð 8: 7Þ
Therefor e, referring to Equat ion 8.1 and Equat ion 8.2 for ball bearing s and con sidering, as
indica ted in Chapt er 11 of the first volume of this handb ook, a fourt h power load–life
relationshi p for line contact , the foll owing equati ons may be written for the fatigu e live s of
roller–rac eway contact lamin ae:
Lm jk ¼qc mj
qmjk
� �4
ð 8: 8Þ
Ln jk ¼qc n j
qn jk
� �9= 2
ð 8: 9Þ
Accor dingly , rolle r bearing fatigue life may be calcul ated using
L ¼Xj ¼ Z
j ¼ 1
Xk¼ m
k ¼ 1
L � emjk þ
Xj ¼ Z
j¼1
Xk¼m
k¼1
L�enjk
!�1=e
ð8:10Þ
8.2.2.2 Roller Life
Similar to Equation 8 .4, the basic dynami c capacit y for a roller track at a roller–rac eway
contact lamina is given by
qcnjk ¼ 464 1þ cngnð Þ1:324w7=9 D29=27
cos anð Þ2=9ð8:11Þ
� 2006 by Taylor & Francis Group, LLC.
Roller bearing fatigue life, including the lives of the r ollers, m ay be calculated using
L ¼Xj ¼ Z
j ¼ 1
Xk ¼ m
k¼ 1
L� em jk þ
Xj ¼ Z
j ¼ 1
Xk¼ m
k ¼ 1
L� en jk þ
Xj ¼ Z
j ¼ 1
Xk¼ m
k ¼ 1
L� eR mjk þ
Xj ¼ Z
j ¼ 1
Xk ¼ m
k ¼ 1
L � eR njk
!� 1 =e
ð8: 12 Þ
As for balls, roller life must be reduced by the speed ratio for use in the above equati on.
8.2.3 CLEARANCE
The fatigu e life of a rolling be aring is strong ly depen dent on the maximum rolling e lement
load Qmax ; if Q max is signi ficantly increa sed, fatigue life is signi ficantly decreas ed. Any
parame ter that affects Qmax, theref ore, affects bearing fatigu e life. One such parame ter is
radial (diametr al) clear ance. In Chapt er 7 of the first volume of this handbo ok, the effect of
clear ance on load distribut ion in radial bearing s was exami ned. Figure 8.1 illu strates the
varia tion of load distribut ion among the roll ing eleme nts for some con ditions of radial
clear ance as de fined by the projection of the bearing load zon e on a diame ter.
The effect of c learance on bearing fatigue life may be express ed in terms of the standar d
ratin g lif e; for example , L10c ¼ A c L10 . Figu re 8.2, from Ref. [7], which gives the L10 life
reducti on fact or Ac as a function of the extent of ro lling elem ent loading , was developed by
using the load dist ribution data of Chapt er 7 of the first volume of this hand book in
accord ance with the contact life and be aring life (Equati on 8.1 through Equat ion 8.3 and
Equat ion 8.8 throu gh Equat ion 8 .10). As shown in Figure 8.1, an increa se in Qmax for rigidly
supported bearings is accompanied by a decrease in the numbers of rolling elements loaded.
Fr Fr
Fr
di
(a) e = 0.5, y l = ±90�, 0 clearance
(c) 0.5 < e <1, 90�< yl < 180�, preload
(b) 0 < e < 0.5, 0 < y l < 90�, clearance
edi
edi
ediy l
yl
y l
--
FIGURE 8.1 Rolling element load distribution for different radial clearance conditions.
� 2006 by Taylor & Francis Group, LLC.
0.1 1e
100
0.2
0.4
0.6
Ball bearings Ball bearings
Roller bearings Roller bearings
Ac
0.8
1.0
1.2
FIGURE 8.2 Fatigue life reduction factor A c based on diametral clearance.
This decreas e in load zo ne, howeve r, has less effect on the mean effective roll ing elem ent load
than does the increa se in Qmax .
See Exampl e 8.1.
8.2.4 F LEXIBLY S UPPORTED B EARINGS
If one or both rings of a rolling bearing ben d under the app lied loads such as in a planet gear
applic ation [8,9 ] or other aircraft bearing applications in which ring an d hous ing cross-
section s are optim ized for aircr aft wei ght reductio n, then load distribut ion may be con sider-
ably different from that of a rigi d ring be aring. Depending on the flex ibility of the ring an d
bearing clearance, it may be possible for a flexible ring to yield supe rior endurance charac-
teristics when compared with a rigi d ring bearing . Fig ure 8.3 from Jon es and Harr is [8] shows
the varia tion of be aring fatigue lif e with oute r ring section and clearan ce for a planet gear
bearing as shown in Figure 1.22 and Figure 1.23. The load dist ribution obt ained is illustr ated
in Figure 1.31.
When the bearing rings are flex ibly supporte d, it may be possible to alter be aring design
and obtain increa sed fatigu e life. Harr is and Br oschard [9] app lied clearance selective ly at the
planet gear bearing maxi mum load pos itions by making the bearing inner ring elliptical.
Figure 8.4 demon strates the varia tion of fatigue life with diame tral clearan ce an d out -of-
round . Out-o f-roun d is the difference betwe en the major and minor axes of the ellipti cal ring.
A furt her refer ence [10] also de monstrates that rolling bearing ring dimens ions can be
optim ized to maxi mize fatigue life.
8.2.5 HIGH -SPEED OPERATION
Opera tion at high speeds, as sh own in Chapt er 2, affects the be aring load dist ribut ion due to
the increa sed magni tude of rolling elem ent centrifugal forces and gy roscopi c moment s. The
standard methods of fatigue life calculati on [3–5] do not account for these inert ial forces an d
moment s and subsequent effects such as changes in ball bearing contact angles. Hence, the
deviation in fatigue life from that calculated according to the standard method can be
� 2006 by Taylor & Francis Group, LLC.
0500
600
700
800
900
1000
1100
0.001 0.002Diametral clearance, in.
0.003 0.004
0 0.02
Bea
ring
L 10
life,
hr
I = ∞ (rigid ring)
l = 8678 mm4
(0.02085 in.4)
l = 4339 mm4
(0.010425 in.4)
0.04mm
0.06 0.08 0.10
FIGURE 8.3 Planet gear bearing life vs. diametral clearance and outer ring cross-section moment of inertia.
con siderable. In Chapter 2, methods were developed to calcul ate load dist ribut ion in high-
speed ba ll and rolle r bearing s. Methods for using these load distribut ions in the esti mation of
fatigue life hav e been given in this chapter . Figure 8.5 demonst rates the variation of life with
load an d speed for the 21 8 angular -conta ct ba ll bearing of Fi gure 3.12 through Figu re 3.14.
0
100
200
300
400
500
L 10
life,
h 600
700 0
800
900
1000
11000 0.1
0.2032 mm (0.008 in.)0.1524 (0.006)
(0.004)0.1016
0.2
0.3048 mm 0.0.R. (0.012 in.)
0.254 mm 0.0.R. (0.010 in.)
0.0508 mm (0.002 in.)
mm0.3
0 0.002 0.004Diametral clearance, in.
0.006 0.008 0.010 0.012
FIGURE 8.4 Bearing life vs. diametral clearance and out-of-round.
� 2006 by Taylor & Francis Group, LLC.
0 1,000107
108
109
1010
1011
10120 4 8
3,000 rpm
6,000 rpm
10,000 rpm
15,000 rpm
L 10
fatig
ue li
fe, r
evol
utio
ns
According toANSI standard
12 16N
20 24 28 � 103
2,000 3,000 4,000Thrust load, lb
5,000 6,000 7,000
FIGURE 8.5 L10 life* vs. thrust load and speed; 218 angular-contact ball bearing, a¼ 408.
Note that the data shown in Figu re 8.5 do not con sider the effect of skiddi ng, whi ch resul ts in
a reducti on in ball orbit al speed , and hence reduced ba ll centrifugal and gyroscop ic loading s.
This, in turn, tends to resul t in an increa se in fatigue life; howeve r, de pending on the thickne ss
of the lubri cant films separat ing the balls from the raceways, sli ding in the ball–racew ay
contact s, with its potential de leterious effe ct on fati gue end urance, may more than eliminat e
the beneficial effect of reduced inert ial loading .
Figure 8.6 co mpares the fatigu e life of the 218 ang ular-contact ball be aring ope rating at a
high speed wi th light -weight silicon nitride balls to that of the bearing that ha s steel balls.
Whereas the silicon nitride balls ope rate wi th reduced inert ial loading , the elastic modulus of
hot isostati cally presse d (HIP) silicon nitr ide is appro ximatel y 5 0% great er than that of steel.
This results in reduced co ntact area be tween the steel racew ays an d c eramic balls; therefore,
Hertz stresses are increa sed causing a reductio n in fatigue life. Thus , the be neficial effe ct of
� 2006 by Taylor & Francis Group, LLC.
0
0e+0
1e+9
2e+9
3e+9
4e+9
5e+9
6e+9
7e+9
10,000 20,000
Applied thrust load, N
L 10
fatig
ue li
fe, r
evol
utio
ns
30,000
Steel ballsSilicon nitride balls
40,000 50,000
FIGURE 8.6 Life vs. thrust load for a 218 angular-contact ball bearing operating at approximately 1.50
million dn. (Bearing bore in millimeter times shaft speed in rpm.)
light-w eight balls is coun teracted. By decreas ing the radii of the racew ay grooves somewh at,
the Hertz stre sses may be decreas ed. This, howeve r, causes an increa se in fri ctional stresses
and higher ope rating temperatur es that may have to be accomm oda ted by co oling the
lubri cant or bearing . Opti mum bearing design may be a chieved for a given app lication by
parame tric study us ing a be aring perfor mance analysis c omputer program . It can be seen
from Figure 8.6 that there is littl e difference in the fatigu e life perfor mance of the bearing
unde r relat ively heavy loading .
Figure 8.7 shows life vs. sp eed for the 209 cyli ndrical roll er bearing of Fi gure 3.19.
Skidd ing effe cts are not includ ed in this illustr ation.
8.2.6 MISALIGNM ENT
Misalignment in nonaligning rolling bearings distorts the internal load distribution, and thus
alters fatigue life. In Chapter 1, methods were described to determine the misalignment angle in
ball and roller bearings as a function of the applied moment. In ball bearings, the load distribu-
tion from ball to ball is altered by misalignment; in roller bearings, however, the distribution of
the roller load per unit length becomes nonuniform as shown in Figure 1.8. The variable load per
unit length is given by Equation 1.36.
The analysis of roll er bearing live s indica ted in Chapte r 11 of the first volume of this
han dbook pertai ned only to bearing s that have a uni form distribut ion of load pe r unit lengt h
along the roller lengt h at each roller–rac eway con tact. As indica ted in Chapt er 1, roller–
racew ay loading varies not only from co ntact to contact , but also from lami na to lamina
along a contact. The methods define d in Chapt er 1 allow the determ ination of the load per
unit length qnjk a t each roller–rac eway lami na co ntact, where n ¼ 1 (outer racew ay) or 2 (inner
racew ay), j ¼ 1, . . . , Z, and k ¼ 1, . . . , m.
It should be apparent that misalignment can quickly lead to edge loading in the roller–
raceway contacts; edge loading of even small magnitude can rapidly diminish fatigue life. In
� 2006 by Taylor & Francis Group, LLC.
10,0005,000
High-speed bearinglife calculation
ANSI life calculation
1,000
10,000
100,000
Shaft speed, rpm
L 10
fatig
ue li
fe, h
0 15,000
FIGURE 8.7 Life vs. speed; 209 cylindrical roller bearing with zero mounted clearance supporting
44,500N (10,000 lb) radial load.
Chapter 6 of the first volume of this handbook, references were cited indicating that the magni-
tude of edge stressing can be calculated for any roller–raceway contact profile. Alternatively, the
methods defined in section 1.6 allow calculation of the contact stresses, including edge stresses, for
any roller-raceway crowning, load and misalignment combination. Figure 8.8, from Ref. [11],
shows the effect of misalignment on the life of a 309 cylindrical roller bearing as a function of roller
crowning and applied load. Table 8.1 indicates, based on experience data in manufacturers’
catalogs, maximum acceptable misalignments for the various rolling bearing types.
8.3 EFFECT OF LUBRICATION ON FATIGUE LIFE
In Chapt er 4, it was indica ted that if a roll ing be aring is adequ ately designe d and lubricated,
the rolling surfa ces can be co mpletely separat ed by a lubri cant film. Enduran ce testing of
rolling bearings as shown by Tallian et al. [12] and Skurka [13] has demonstrated the
� 2006 by Taylor & Francis Group, LLC.
00
10
20
30
40
50
60
70
80
90
100
160
140
120
100
80
60
40
20
00 5 10 15 20 255
Ideal crown
Ideal crown
Full crown
Full crown
Radial load = 31,600 N (C/2) (7,100 lb)
Radial load = 15,800 N (C/4) (3,530 lb)
ls = 7.7 mm (0.303 in.)
ls = 7.7 mm (0.303 in.)
ls = 4.8 mm (0.188 in.)
ls = 4.8 mm (0.188 in.)
Misalignment, min Misalignment, min
Per
cent
age
of s
tand
ard
life
Per
cent
age
of s
tand
ard
life
10 15 20 25
FIGURE 8.8 Life vs. misalignment for a 309 cylindrical roller bearing as a function of crowning and
applied load. (From Harris, T., The Effect of misalignment on the fatigue life of cylindrical roller
bearings having crowned rolling members, ASME Trans., J. Lubr. Technol., 294–300, April 1969.)
con siderable effect of lubrican t film thickne ss on bearing fati gue life. In Chapt er 4, methods
for estimating this lubricant film thickness were given. It was also demonstrated that lubricant
film thickness is sensitive to bearing operating speed and lubricant viscous properties.
Moreover, the film thickness is virtually insensitive to load.
The test results reported in Refs. [12,13] showed that at high operational speeds a
considerable improvement in fatigue life occurs. Moreover, a similar effect can be achieved
by using a sufficiently viscous lubricant at slower speeds. The effectiveness of the lubricant
film thickness generated depends on its magnitude relative to the surface topographies of the
contacting rolling elements and raceways. For example, a bearing with very smooth raceway
and rolling element surfaces requires less of a lubricant film than does a bearing with
relative ly rough surfa ces (see Fi gure 8.9) .
TABLE 8.1Estimated Maximum Allowable Rolling Bearing Misalignment Anglea
Bearing Type Minutes Radians
Cylindrical roller bearing 3–4 0.001
Tapered roller bearing 3–4 0.001
Spherical roller bearing 30 0.0087
Deep-groove ball bearing 12–16 0.0035–0.0047
aBased on acceptable reduction in fatigue life.
� 2006 by Taylor & Francis Group, LLC.
Roughsurfacebearing
Lowspeed
Highspeed
Lowspeed
Smoothsurfacebearing
FIGURE 8.9 Illustration of the effect of surface roughness on the lubricant film thickness required to
prevent metal-to-metal contact.
The relationshi p of lubricant fil m thickne ss to surfa ce rou ghness has been signi fied in
rolling bearing literat ure by L, whi ch utilizes the sim ple root mean square (rm s) value of
the roughn esses of the surfa ces of the contact ing bodies. Tallian [14] among many other
resear chers introd uced the use of asperi ty slopes a s wel l as pe ak height s of asperiti es. Chapt er
5, whi ch covers micro contact phe nomena, provides addition al means to evaluate the effect of
a ‘‘rough ’’ surface on con tact, an d hence bearing lub rication and perfor mance. Usi ng L,
Harr is [15] ind icated the effect of lubric ation on bearing fatigue life, as in Figure 8.10.
Accor ding to Ref . [15] , if L � 4, fati gue life ca n be ex pected to exceed standar d L10 estimat es
by at least 100%. Conver sely, if L < 1, the bearing may not attain calcul ated L10 estimat es
because of surface dist ress such as smearing that can lead to rapid fatigu e failu re of the ro lling
surfa ces. Figure 8.10 shows the v arious operatin g regions just descri bed. In Figure 8.10, the
ordinat e ‘‘per cent film’’ is a measure of the time during whi ch the ‘‘cont acting’’ surfa ces are
fully separat ed by an oil film.
Tallian [14] showe d a more definitive estimat e of rolling bearing fati gue life vs. L as did
Skurk a [13] . Bamberger et al. [16] sho w the combinat ion of the foregoi ng in Figure 8.11,
recomm ending the use of the mean curve. Expe rimen tal data ind icate that for L> 4, the L/ L10
ratios are substa ntially greater than those given in Figure 8.11 for accu rately manufa ctured
bearing s lubricated by minimal ly co ntaminated oil.
Using a microtrans ducer to measur e the pressure dist ribution in the direct ion of rolling in
an oil-lubri cated line con tact, Sc houten [17] showed that edg e stre ss in a line contact is
substa ntially reduced if an adeq uate lubri cant film separat es the co ntacting bodies. In this
situati on, the lubri cant film tends to permit an increase in fatigue life by reducing the
magnitude of normal stress at the end(s) of a heavily loaded contact.
The mean curve in Figure 8.11 is frequently used to estimate the effect of lubrication on
bearing fatigue life.
See Example 8.2 and Example 8.3.
� 2006 by Taylor & Francis Group, LLC.
Region ofincreased life
Region oflubrication-relatedsurface distress
Per
cent
film
100
80
60
40
20
00.4 0.6 1.0 2.0
Λ = function of film thickness and surface roughness
4.0 6.0 10
Region of possiblesurface distressfor bearings withsevere slidingmotions
Operating region for mostindustrial applications
FIGURE 8.10 Percent film vs. L.
Unfor tunately, if gross sli ding occu rs, the reductio n in fatigue life can be much more
severe than that pred icted in Figure 8.11. In Chapt er 11 of the first volume of this han dbook it
was shown that fatigue life is a strong function of normal stresses acting on the contacts
between mating rolling surfaces. Surface friction shear stresses augment the subsurface
stresses effected by the normal contact stresses. In fact, from Lundberg–Palmgren theory it
can be shown that for point contacts L / t0�9.3. Hence, small increases in stress cause large
decreases in life. Thus, lubricant film parameter L may only be regarded as a qualitative
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.6 0.8 1 2
Film parameter (Λ)
4
From Tallian [14]
From Skurka [13]
6 8 100
Mean curverecommended
L L 10
FIGURE 8.11 Lubrication–life factor vs. lubricant film parameter L. (From Bamberger, E., et al., Life
Adjustment Factors for Ball and Roller Bearings, AMSE Engineering Design Guide, 1971. With permission.)
� 2006 by Taylor & Francis Group, LLC.
TABLE 8.2Achem vs. Steel Type
Steel Type Achem
AISI 52100 3
M50 2
M50NiL 4
measur e of lubrica tion effecti veness. How to include the surface frictio n she ar stresses in the
predict ion of bearing fatigue life will be discus sed later in this ch apter.
8.4 EFFECT OF MATERIAL AND MATERIAL PROCESSING ON FATIGUE LIFE
In Chapter 11 of the first volume of this handbook, the effect on fatigue endurance of the basic
steel used in modern bearing manufacture was included in the bm or fcm factors in the
calculation of basic load rating C. This standard steel is assumed to be carbon vacuum degassed
(CVD) 52100, through-hardened at least to Rockwell C 58. Many roller bearings, particularly
tapered roller bearings manufactured in the United States, are however fabricated from
carburized (case-hardened) steel. Since the load and life rating methods for such bearings are
assumed to be included in the standards [35], it has been historically assumed that the
endurance performances of the CVD 52100 through-hardened steel and the basic carburizing
steels are equivalent.
To attain high-temperature, long-life performance, VIMVAR M50 tool steel was devel-
oped for aircraft gas turbine mainshaft bearing applications. This VIMVAR steel provides
excellent fatigue endurance characteristics for bearing rings and rolling elements. Because
of the necessity to operate modern gas turbine mainshaft bearings at ultrahigh speed, for
example, at 3 million dn, a carburizing version of this steel, VIMVAR M50NiL, was devel-
oped. In this case, it is intended that the ‘‘softer’’ core will arrest any fatigue cracks that
emanate in the hardened case and thus prevent through-cracking of bearing rings.
A number of specialty steels have been developed to provide superior corrosion resistance
while not sacrificing fatigue endurance properties; for example, Cronidur 30. Additionally,
ceramic materials, for example, HIP silicon nitride, are now used in the manufacture of balls
and rollers.
STLE [18] has attempted to codify the effect of some of these materials on rolling bearing
fatigue life. Moreover, STLE [18] has also separated the effects of heat treatment and
metalworking. A material–life factor Asteel has been recommended such that
L010 ¼ Asteel
C
F
� �p
ð8:13Þ
TABLE 8.3Aheattreat vs. Heat Treatment
Heat Treatment Aheattreat
Air-melt 1
Carbon vacuum degassed (CVD) 1.5
Vacuum arc remelted (VAR) 3
Double VAR 4.5
Vacuum induction melted, vacuum arc remelted (VIMVAR) 6
� 2006 by Taylor & Francis Group, LLC.
TABLE 8.4Aprocess vs. Metalworking Process
Metalworking Process Aprocess
Deep-groove ball bearing raceways 1.2
Angular-contact ball bearing raceways 1
Angular-contact ball bearing raceways—forged rings 1.2
Cylindrical roller bearings 1
wher e Asteel ¼ A chem � Aheattr eat � Aproc ess. The data in Table 8.2 through Tabl e 8.4 wer e
obtaine d from Ref . [18] .
From the tabu lar data, it can be determ ined that an angu lar-contact be aring with forged
rings manufa ctured from VIMVA R M50Ni L steel woul d be given an Asteel ¼ 28.8.
No value has been univers ally establ ished to date for HIP silicon nitride. Endura nce
testing of singl e balls in ba ll/v-ring endurance test has, howeve r, yielded high multiple s of
the endurance for steel ba lls test ed unde r the same loading con ditions. To date, owing to
relative weakne ss in tensile strength in bending test s an d extremely low coeff icient of thermal
expan sion, silicon nitride has been princi pally used for balls an d roll ers in high-pr ecision,
high-s peed applic ations ; for exa mple, machi ne tool spindl e bearing s.
8.5 EFFECT OF CONTAMINATION ON FATIGUE LIFE
Exc essive co ntaminati on in the lubricant wi ll severe ly shorte n be aring fatigue life. The
standar ds [3–5] an d manufa cturers’ catalo gs co ntain war ning statement s about this. Contam-
inants may be either particulat e or liquid , usually water. Eve n small amou nts of co ntaminan ts
have signi ficant limiting effects on bearing fatigue life.
Particulate contaminants such as gear wear metal particles, alumina, silica, and so on will
cause dents in the raceway and rolling element surfaces, which disrupt the lubricant films
that tend to separate the rolling body surfaces. This tends to locally increase the frictional
shear stresses produced in the rolling–sliding contacts. Furthermore, the raised material on the
shoulder of the dent tends to cause stress concentrations. Ville and Ne lias [19], using a two-disk
rolling–sliding test rig, demonstrated the stress concentration phenomenon. They further showed
that combined rolling–sliding motion is a more severe condition with regard to generation of
surface distress and fatigue than rolling alone. Both the film disruption and dent shoulder stress-
increasing effects accelerate the onset of rolling contact fatigue and component failure. Figure
8.12 from a study of the effects of surface topography on fatigue failure by Webster et al. [20]
indicates the relative risk of failure effected by the shoulders of dents.
Hame r et a l. [21] and Sayles et al. [22] pointed out that even relative ly soft particles can
generat e signifi cant de nting, assum ing be aring speeds and loads are sufficien tly high. They
furt her ind icate that the particle diame ter to lubrican t film thickne ss ratio ap pears to be a
critical parame ter with regard to de nting. In Figure 8.13 through Figure 8.15, Ne lias and Vi lle
[23] ch aracterize d the types of dents gen erated by hard and soft pa rticles.
Using the same roll ing–slidi ng disk endurance test rig of Ref . [19], Ne lias and Ville [23]
showe d that fatigue micr ospalling co mmence s on the surfa ce ahead of the dent in the fricti on
direct ion; see Figure 8 .16. Xu et al. [24] also noted that spalling due to de nts can init iate at
eithe r the leadi ng or trai ling edge depen ding on the direct ion of surfa ce traction.
� 2006 by Taylor & Francis Group, LLC.
Risk peaks associateddent shoulders
1.5
1.0
0.5
0.0
−0.5
−1
0
Risk peaks associatedwith subsurfaceorthogonalshear stresses
0
Ris
k
Note: z/b = 0 represents surface
0.5z /b
x /a 0.1
FIGURE 8.12 Plot showing relative risk of fatigue failure throughout raceway subsurface including
effect of dent shoulders. (From Webster, M., Ioannides, E., and Sayles, R., Proc. 12th Leeds–Lyon
Symp. Tribol., 207–226, 1986. With permission.)
Ne lias and Ville [23] also demo nstrated the dent location using trans ient elastoh ydrody-
namic lubri catio n (E HL) analys is; see Figu re 8.17. Xu et al. [24] in an a nalytical and exp eri-
menta l study present ed simila r results to those of Ne lias and Ville [23] . They also showe d that
the locat ion of spall init iation depen ds on the EHL and den t cond ition, an d that spalls can
initiat e at either the leadi ng or trai ling edge of the dent de pending on the direct ion of surfa ce
traction; see Figure 8.18.
Shoulder
FIGURE 8.13 Dent generated by a ductile metallic particle; for example M50 steel. (From Nelias, D.
and Ville, F., ASME Trans., J. Tribol., 122, 1, 55–64, 2000. With permission.)
� 2006 by Taylor & Francis Group, LLC.
FIGURE 8.14 Dent generated by hard brittle material; for example Arizona road dust. (From Nelias, D.
and Ville, F., ASME Trans., J. Tribol., 122, 1, 55–64, 2000. With permission.)
The experimental data of Sayles and MacPherson [25] demonstrated the effect of different
levels of particulate contamination on bearing fatigue life by endurance testing cylindrical
roller bearings with varying degrees of absolute lubricant filtration; for example, from 40 mm
Shoulder
(a)
FIGURE 8.15 (a) Coarse dent generated by ceramic material at slow speed; for example boron carbide or
silicon carbide at 2.51m/sec (98.8 in./sec). (b) Fine dents generated by ceramic material at high speed; for
example boron carbide or silicon carbide at 20m/sec (787.4 in./sec). (From Nelias, D. and Ville, F., ASME
Trans., J. Tribol., 122, 1, 55–64, 2000. With permission.)
Faster surface Slower surface (a) (b)
FIGURE 8.16 Surface distress (in dotted ellipses) associated with dent in rolling–sliding motion, endur-
ance tested 52100 steel components. Solid arrows signify rolling direction; dashed arrows signify
friction direction. (From Nelias, D. and Ville, F., ASME Trans., J. Tribol., 122, 1, 55–64, 2000. With
permission.)
� 2006 by Taylor & Francis Group, LLC.
50 μm 50 μm
FIGURE 8.17 For the slower surface in Figure 8.16, formation of microspalls ahead of the dent in the
sliding direction on the surface of a 52100 steel component after 60� 106 stress cycles at 3500 MPa
(5.08� 105 psi). Rolling speed is 40m/sec (1575 in./sec); slide–roll ratio¼þ0.015. Solid arrow signifies
rolling direction; dashed arrow signifies friction direction. (From Nelias, D. and Ville, F., ASME Trans.,
J. Tribol., 122, 1, 55–64, 2000. With permission.)
(0.0016 in.) down to 1 mm (0.00004 in.). Particulate matter was deemed typical of that generated
in gearboxes. Figure 8.19 is a photograph of dents incurred under the Sayles–MacPherson [25]
operati ng cond itions with 40 mm (0.0 016 in.) filtra tion. The de nts are ap proxim ately 10–30 mm
(0.0004– 0.0012 in.) long and about 2 mm (0.00008 in.) deep . Comp aring this depth wi th the
thickne ss of a goo d lubricant film (L> 1.5) , it ca n be determined that the film can easily
colla pse in the dent. Eva luation of the Sa yles–M acPherson [25] ope rating cond itions accordi ng
to the methods discus sed in Chapt er 1, Chapt er 3, and Chapt er 4, indica tes L values from
approxim atel y 0 .45 at 40 m m (0.0 016 in.) filtration to nearly 1 us ing magn etic filtra tion.
Figure 8.20 from Ref . [25] shows L50 life vs. filter ratin g. Accor ding to Figu re 8.20,
signifi cant impr ovement in life is achieve d wi th a finer lubricant filtra tion level; howeve r,
little impr ovement in life is achieve d for a filtra tion level less than 3 m m. Thus , there appears
to be a limit to fine filter effe ctiveness. Sayles –MacP herson [25] data were confirmed by
Tana ka et al. [26], who, by using seale d ball bearing s in an automot ive gearbox, manage d to
increa se fatigu e life severa l fold, compared with that of ope n (no seals or shields) bearing s in
the same ap plication. Consideri ng the lubrican t film cond itions of the test program , the data
of Figure 8.20 have been curve- fitted to the foll owing eq uation for contam ination–li fe factor:
Acontam ¼ 0: 4162 þ 3: 366ln FR =h0� �FR =h0
� 2
ð 8: 14 Þ
wher e FR is the filter ratin g.
Based on test results using 3- and 49-mm filtration, Needelman and Zaretsky [27] recom-
mend the following equation for the reduction of fatigue life due to particulate contamination:
Acontam ¼ 1:8 FRð Þ� 0 :25 ð 8: 15 Þ
It is apparent that equatio ns for fatig ue life reductio n due to pa rticulat e contam inatio n must
be app lied with care since they depend on the type of pa rticles as well as the size and on the
bearing lubricati on cond itions.
The presence of wate r in the lubri cant is thought to effect hydrogen embri ttlement of the
surfa ce steel, creat ing stre ss concentra tions and shorten ing fatigue lif e. Figure 8.21, from Ref.
[28], illustrates the life redu ction effec t.
� 2006 by Taylor & Francis Group, LLC.
2.5 0.2
0.15
0.1
0.05
0
2
1
1.5
Pre
ssur
e
Pre
ssur
e
Film
thic
knes
s
Film
thic
knes
s
0.5
0−1.5
−0.2
−0.05
−0.1−0.2
0 0.1 0.2 0.3 0.4
50 μm 50 μm
0.5
−0.1 0.1 0.20
−0.1 0.1 0.20
1.5−0.5 0.5
2.5 0.2
0.15
0.1
0.05
0
2
1
1.5
0.5
0
−1.5 1.5−0.5 0.5
2.5
2
1.5
1
0.5
0
0
−0.05
−0.1−0.2 −0.1
0 0.1 0.2 0.3 0.4 0.5
0.1 0.20
0
−0.2 −0.1 0.1 0.20
2.5
2
1.5
1
0.5
0
FIGURE 8.18 Comparison of results of numerical simulations and tests for two opposite slide–roll
ratios. The upper row shows pressure distribution and film thickness over the line contact, the middle
rows show zoom views of the film thickness around the dent and lines of constant maximum shear stress
in the metal, and the lower row shows dent area micrographs. (From Nelias, D. and Ville, F., ASME
Trans., J. Tribol., 122, 1, 55–64, 2000. With permission.)
Table 8.5 from Ref. [28] for ISO 220 circulati ng oils ind icates that the effect of water in the
lubri cant a lso varie s with the composi tion of the lubricant.
It appears that a dding 0.5% wat er to lubri cant A caused a life redu ction by a fact or of 3,
whi ch is con sistent wi th the data in Figure 8.21. The results for the remain ing lub ricant
varia nts, howeve r, demonst rate a wid e varia tion in bearing life, indica ting a signifi cant
end urance depend ency on the lub ricant composi tion as well as on the amoun t of contai ned
mois ture. Because of this , life redu ction equati ons need to be based on the combinat ion of
lubri cant type, specific composi tion, and amou nt of contai ned moisture.
8.6 COMBINING FATIGUE LIFE FACTORS
It may be observed that nons tandard loading cond itions can be accomm odated in the
estimat ion of bearing fatigue life by determ ining the bearing inter nal load distribut ion and
� 2006 by Taylor & Francis Group, LLC.
FIGURE 8.19 Denting caused by particulate contamination. (From Sayles, R. and MacPherson, P.,
Influence of wear debris on rolling contact fatigue, ASTM Special Technical Publication 771, J. Hoo,
Ed., 255–274, 1982. With permission.)
applyi ng the co ntact life equatio ns presen ted at the beginni ng of this chapter . User -friendl y
computer program s to perform the calcul ations using the eq uations and methods present ed
in Chapt er 1 through Chapt er 4 are read ily a vailable for operati on on pe rsonal compu ters.
To apply the effects of increased reliability, nonstandard materials, lubrication, and
13
12
11
10
9
8
7
6
5
4
3
2
1
01 3 8 10 20
Filter rating, μm
L 50
life 3
106
cycl
es
30 40
FIGURE 8.20 Bearing fatigue life vs. degree of lubricant filtration. (From Sayles, R. and MacPherson,
P., Influence of wear debris on rolling contact fatigue, ASTM Special Technical Publication 771, J. Hoo,
Ed., 255–274, 1982. With permission.)
� 2006 by Taylor & Francis Group, LLC.
TABLE 8.5Bearing Fatigue Life for 0.5% Water Concentration in Various Lubricants
Lubricant L10 L50
A (no water) 59.2 171.4
A 20.8 61.2
B 66.7 195.7
C 33.4 77
D 54.5 195
E 20.8 61.2
F 23.9 168
G 32.1 143
H 66.8 410
I 47.4 122
contamination, the simple approach of cascading the life factors has been most frequently
taken, and is recommended in Ref. [18] and various bearing manufacturers’ catalogs. This
approach uses the following equation:
Lna ¼ A1A2A3A4
C
F
� �p
ð8:16Þ
50
L10
L50
100106
107
5 � 107
2
5
2
200
Water conc., ppm
Life
(cy
cles
)
Film
thic
knes
s (μ
IN)
500 1000
FIGURE 8.21 Effect of water contamination on rolling bearing life. (From Barnsby, R., et al., Life
ratings for modern rolling bearings, ASME Paper 98-TRIB-57, presented at the ASME/STLE Tribology
Conference, Toronto, October 26, 1998. With permission.)
� 2006 by Taylor & Francis Group, LLC.
In the ab ove equatio n:
. A1 is the reliab ility–life fact or as determ ined from Tabl e 11.25 of the first vo lume of this
hand book.. A2 is the mate rial–li fe fact or as de termined from Table 8.2 through Table 8.4 or sim ilar
empir ical data.. A3 is the lubri cation –life fact or determ ined using Figure 8.11 or simila r empirical data.. A4 is the con taminatio n–life facto r using Equation 8.14, Equation 8.15, or simila r
empir ically derived da ta.. Lna is the adjust ed fatigue life at reliabil ity n.
This sim ple calcul ation app roach ha s been used since the 1960s when the first impr ovement s
in bearing steels and unde rstand ing of the role of lubrican t films in bearing fatigue end urance
occurred. It does not howeve r recogni ze the inter dependen cy of the various life factors.
Therefor e, it must be used judicio usly. For exampl e, the ANSI standar ds [3,4] state ‘‘It may
not be assum ed that the use of a special material , process , or de sign will overcome a defic iency
in lubri cation. Valu es of A2 great er than 1 should theref ore normal ly not be app lied if A3 is
less than 1 becau se of such defic iency.’’ The con taminati on–life facto r is strongly depe ndent
on the thickne ss of the lub ricant film co mpared with the size of forei gn particulat e matt er; in
large be arings it is far less signifi cant than in small bearing s.
8.7 LIMITATIONS OF THE LUNDBERG–PALMGREN THEORY
The Lundb erg an d Palmgr en fatigue life theory and acco mpanyi ng formu las were a signi fi-
cant developm ent in rolling be aring techn ology; howeve r, it was not possibl e to correlate the
fatigue live s of bearing surfa ces in rolling co ntact so calculated with fatigue live s of other
engineer ing struc tures. Nor was it possible to correla te roll ing contact fatigue in bearing s to
fatigue of elemen tal surfaces in rolling contact.
A major consideration in the analysis of fatigue lives of mechanical engineering structures
subjected to cyclically applied tension, bending, and torsion is the existence of an endurance limit.
This is a cyclically applied stress level that the structure can endure without succumbing to
fatigue failure. In other words, if the equivalent stresses cyclically applied to a mechanical
structure are everywhere less than the endurance limit, then the structure will survive indefinitely
without the possibility of fatigue damage. Conversely, according to the Lundberg–Palmgren
theory and the standard methods of rolling bearing fatigue life prediction derived therefrom,
irrespective of the magnitude of the applied load, rolling bearing fatigue life is finite in any
application. Innumerable modern rolling bearing applications, however, have defied this limita-
tion. Endurance data for bearings of standard design, accurately manufactured from high-
quality steel—having minimal impurities and homogeneous chemical and metallurgical struc-
tures [28]—have demonstrated that infinite fatigue life is a practical consideration in some rolling
bearing applications. Since the Lundberg and Palmgren formulas did not address the concept of
infinite fatigue life and did not relate to structural fatigue, an improvement in these formulas
beyond the application of empirical life adjustment factors was required.
As e xplained in Chapt er 11 of the first volume of this handb ook and illu strated in Figure
8.22, the Lundb erg an d Palmgr en theory consider s that a cyclically a pplied concentra ted load
results in a Hertz stress on the raceway contact surface, which in turn causes a cyclic subsurface
orthogonal shear stress. A sufficiently large magnitude of the latter stress leads to the initiation
of a fatigue crack at a point below the raceway surface where its location coincides with a weak
point in the material. The weak points are assumed to be randomly distributed throughout the
material. The subsurface crack propagates toward the surface resulting eventually in a spall
(pit). According to Lundberg and Palmgren, the large-magnitude shear stress is the range of the
� 2006 by Taylor & Francis Group, LLC.
y
yb
a
x
x
smax
s
−2.5−0.30
−0.25
−0.15
−0.05
+0.05
+0.10
+0.20
+0.25
+0.30
+0.15
0t yz
s max
−0.10
−0.20
−2.0 −1.5 −1.0 −0.5 0.5 1.0 1.5 2.0 2.50yb(a) (b)
(d) (c)
FIGURE 8.22 Basis of Lundberg–Palmgren theory: (a) cyclic Hertz stress on raceway contact surface
leads to (b) cyclic subsurface orthogonal shear stress, which leads to (c) a subsurface crack at material
weak point, which leads to (d) spall on raceway surface.
maxi mum ortho gonal shear stress, that is, 2t0; this occu rs at dep th z 0 � 0.5b below the raceway
surfa ce for bot h point and line contact.
In Chapt er 4 it was shown that oil-lub ricated co ntact pressur e distribut ions, that is, EH L
pressur e dist ribut ions, are different from the pure Hert zian pressur e distribut ion illustr ated in
Figure 8.22. M oreover, if the surfa ces are not ideal , that is, not smoot h but rather having
pertur bations or roughn ess peaks on the smoot h surfa ces, then concepts of micro-EHL as
discus sed in Chapt er 5 obtain. Additional ly, in their an alysis Lundber g and Palmgr en did not
include the effe ct of surfa ce fri ction shear stre sses; these can substan tially alter the subsurf ace
stre sses as demonst rated in Figure 8.23. In Fig ure 8.23, the subsurf ace stress de termined is
from the distorti on energy failure theory of von M ises; a simila r sit uation would occur
con sidering subsurf ace shear stre sses.
There are various opi nions concerning which sub surface stre ss effe cts rolling contact
fatigue . The dep ths below the surfac e at which maxi mum orthogo nal shear stress and maxi-
mum von Mises stre ss oc cur are somewh at differen t; the latter occurs at a dep th app roxi-
mate ly 50% deeper than the form er. W hichever stress is co nsidered most de triment al, the
effe ct of su rface shear stre ss is to bring the maximum sub surface stre ss toward the sur-
face. When the ratio t /s � 0.30 (approxi mately) , then the maximum stre ss occurs on the
� 2006 by Taylor & Francis Group, LLC.
−2.5
0.5
1.0
1.5z/b
x/b
2.0
2.5
3.0(a)
0.250.200.30
0.35
0.40
0.45
0.500.55
0.557
0.20
−2.0 −1.5 −1.0 −0.5 0
m = 0
0.5 1.0 1.5 2.0 2.5
�2.5 �2.0 �1.5 �1.0 �0.5
m = 0.250
0 0.5
0.5
x/b
0.20 0.25 0.30
0.35
0.40
0.45
0.50
0.55
0.60 0.609
0.550.598
1.0
1.0
1.5
1.5
2.0
2.0
2.5
2.5
3.0(b)
z/b
FIGURE 8.23 Lines of equal von Mises stress/smax in the material below the rolling contact surface for
(a) pure rolling with no surface friction stress (coefficient of friction m¼ 0) and (b) rolling with surface
friction stress (coefficient of friction m¼ 0.25).
surface. Figure 8.23b indicates a tendency toward this condition, showing a secondary peak
occurring in the upper-right portion of the contact. In general, shear stresses of this magni-
tude do not occur over the entire concentrated contact area in an effective EHL contact. Such
stresses could occur in micro-EHL contacts existing within the overall contact area. When the
maximum subsurface stress approaches the surface, the potential for surface-initiated fatigue
occurs. Tallian [29] considers competing modes of failure, that is, surface-initiated and
subsurface-initiated. Rigorous mathematical analysis requires the consideration of failure at
any point in the material from the surface into the subsurface, consistent with the stresses
applied to the contact surface, both normal and tangential.
The basic equation stated by Lundberg and Palmgren is
ln1
S/ Netc
0 V
zh0
ð8:17Þ
In Equation 8.17, t0 is the maximum orthogonal shear stress, z0 is the depth at which it
occurs, V is the volume of stressed material, and S is the probability of survival of the stressed
volume. Actually, Lundberg and Palmgren state that the volume under stress is proportional
to the volume of the cylindrical ring defined by
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Z0
rr
V ≈ az0 (2prr)
2a
FIGURE 8.24 Lundberg and Palmgren theory—volume of material under stress.
V ¼ 2a z0 2p r rð Þ ð8: 18 Þ
wher e rr is the racewa y radius ; thus, Lundber g and Palmgr en did not actually de fine an
effe ctive stre ss volume ; see Figure 8.24. The Lundber g an d Palmgr en propo rtionali ty is only
vali d when sim ple Hert z is applie d to a smoot h surface.
The Lundberg and Palmgren theory also does not account for the bearing operating
temperatures and their effects on material properties, also not accounting for the effect of
temperature on lubrication and hence on surface shear stresses. Furthermore, the theory does
not consider the rate at which energy is absorbed by the materials in rolling contact. Bearing
speeds are used simply to convert predicted fatigue lives in millions of revolutions to time
values. Nor are hoop stresses induced by ring fitting on shafts or in housings or by high-speed
centrifugal loading accommodated. Finally, the development of microstructural alterations
and residual stresses below the contact surfaces, induced by rolling contact, as indicated by
Voskamp [30] must be considered.
8.8 IOANNIDES–HARRIS THEORY
Consi dering the Lundber g–P almgren theory limitations , Ioan nides an d Harr is [31] de veloped
the basic eq uation
ln1
� Si
� �¼ F N, Ti � Tlimitð Þ� Vi ð8: 19 Þ
In this formu la a fati gue crack is presum ed incapabl e of getting initiat ed until the stress
criteri on Ti exceeds a thres hold value of the criteri on T limit at a given elem ental volume DV i.
It is evident that the crack thres hold criteri on Tlimit corres ponds to an endu rance limit . To
be con sistent wi th the Lundber g–Palm gren theory, the stress criteri on woul d be the orthog-
ona l shear stre ss ampli tude 2t0; howeve r, a nother crit erion, such as the v on M ises or
maxi mum shear stre ss may be used. In Equat ion 8.19, in lieu of the stre ss volume used by
Lundberg and Palmgren, that is, 2paz0d, in which d is the raceway diameter, only the
incremental volume over which Ti > Tlimit is consider ed at risk; see Figure 8.25.
Therefore, the probability of survival in Equation 8.19 is a differential value; that is, DSi.
The probability of component survival is determined according to the product law of
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z
r
dx
dz
v = S2pr dx dz
FIGURE 8.25 Risk volume in fatigue theory of Equation 8.19.
probab ility; subsequent ly, Equat ion 8.20, whi ch co rresponds to the Lundber g–P almgren
relationshi p Equat ion 8.17, is obtaine d:
ln1
S
� �� AN e
ZVR
T � Tlimitð Þc
z 0h d V ð 8: 20 Þ
wher e A is a constant pe rtaining to the overal l mate rial and z ’ is a stress- weighted average
depth to the volume at risk to fatigue. When Tlimit ¼ 0, Equation 8.20 reduces to Equat ion
8.17 if it is assum ed that T ¼ t0.
Harris an d M cCool [32] applie d the Ioann ides–Harri s theory using octahedr al shear stress
as the fatigue- initiat ing stre ss to 62 different applications involv ing deep-gr oov e and angu lar-
contact ball bearing s and cylind rical roller bearing s manu factured from CVD 52100, M50,
M50Ni L, and 8620 carburi zing steels. A value of toct,limit was determined for each mate rial.
Usin g these values, the L10 life for each app lication was ca lculated and compared agains t the
measur ed bearing fatigue life. Also, the L10 life calcul ated accordi ng to the Lundber g–
Palmgr en theory (stand ard method) was calculated and compared with the measur ed bearing
life. It was thereby determ ined by statistica l an alysis that the bearing fatigue live s calcul ated
using the Ioannides –Harr is theory were closer to the measur ed live s than wer e the live s
calcul ated using the standar d method as mo dified by the life fact ors discus sed ab ove.
Subseq uently , Harr is [33] demo nstrated the ap plication of the Ioann ides–Harri s theory in
the prediction of fatigue lives of ba lls endurance tested in ball/v- ring rigs .
To accurat ely calcul ate be aring fatigue live s using the Ioannides –Harr is theory requires:
. Select ion of a fatigue- initiat ing stre ss criteri on
. Dete rminati on an d applic ation of all resi dual, applie d, and induced stre sses acting on
the material of the rolling element–ra cewa y co ntacts. Devel opment and applic ation of a stre ss–life fact or
This was accompl ished in the Harr is and M cCool [32] invest igation using the analytical
methods define d in this text combined in ball and roll er be aring pe rformance analys is
computer pro grams TH-BBAN * and TH-R BAN.* Moreover, it should be apparent that
the co ncept of a stress–life fact or fulf ills the requir ement for the interd ependenc y of the
*FORTRAN/VISUAL BASIC computer programs developed by T.A. Harris for operation on personal computers.
� 2006 by Taylor & Francis Group, LLC.
various fati gue life-i nfluenc ing facto rs cited previous ly. As a n alte rnative to the life form ula
indica ted by Equation 8.16, resul ting from the work init iated by Ioann ides and Harr is [31],
ISO [5] established the bearing life equation format below:
LnM ¼ A1AISOL10 ð8:21Þ
where LnM is the basic rating life modified for a reliability (100� n)%, and AISO is the
integrated life factor, including all of the effects considered in the multiplicative life factors
A1 to A4 and other effects if required. In other words, AISO¼ f(A1,A2,A3,A4,Am).
ISO [5] states that the reliability–life factor A1 can be calculated using
A1 ¼ 0:95ln 100
s
ln 10090
!1=e
þ 0:05 ð8:22Þ
where S is the probability of survival in percent. This equation gives the same values of A1 as
Table 11.25 of the first volume of this handbook when Weibull slope e¼ 1.5. ISO [5] provides
the means to establish the magnitude of AISO. This will be discussed later in this chapter.
8.9 THE STRESS–LIFE FACTOR
8.9.1 LIFE EQUATION
In 1995, the Tribology Division of ASME International established a technical committee to
investigate life ratings for modern rolling bearings. The result of this effort was Ref. [34], in
which the following equation for the calculation of bearing fatigue life was established:
Ln ¼ A1ASL
C
Fe
� �p
ð8:23Þ
In the above equation, C is the bearing basic load rating as given in bearing catalogs, Fe is the
equivalent applied load, and ASL is the stress–life factor. As in the ISO Equation 8.21, A1 is
the reliability–life factor; it is not stress-dependent. ASL is calculated considering all the life-
influencing stresses acting on rolling element–raceway contacts including normal stresses,
frictional shear stresses, material residual stresses due to heat treatment and manufacturing
methods, and fatigue limit stress. In Equation 8.23, exponent p is 3 for ball bearings and 10/3
for roller bearings.
Considering nonstandard loading inwhich life is calculated for each contact, for point contacts
Lmj ¼ A1ASLmj
Qcmj
Qmj
� �p
ð8:24Þ
In the above equation, subscript m refers to the raceway contact, exponent p¼ 3 for the
rotating raceway, and p¼ 10/3 for the stationary raceway. Equation 8.24 further recognizes
that the stress–life factor ASLmj is a function of the raceway and contact azimuth location.
For line contacts,
Lmjk ¼ A1ASLmjk
qcrj
qmjk
� �p
ð8:25Þ
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In the above equati on, su bscript m refers to the raceway contact , k refers to the lamina,
expon ent p ¼ 4 for the ro tating raceway, and p ¼ 9/2 for the stat ionary racew ay.
8.9.2 F ATIGUE-I NITIATING STRESS
In Ref. [34], t he von Mises stress is considered the a ppropriate failur e-i n it iati ng st ress
criterion. The von Mises stress defined according t o E qua t ion 8.26 i s a scalar quantity
associated with the commonly used M ises–Hencky distortion e nergy t heory of fatigue
fai lure:
sVM ¼1ffiffiffi2p s x � s y
� �2þ sy � s z
� �2þ sz � s xð Þ2þ 6 t 2xy þ t
2yz þ t
2zx
� �h i1 = 2
ð 8: 26 Þ
See Ref. [35] or other mach ine design texts.
It is of interest to note that the octahedr al shear stress, a vector qua ntity, also identifi ed in
Ref. [35] as a failure- initiat ing stre ss criteri on is direct ly propo rtional in magni tude to vo n
Mises stre ss; for examp le,
toct ¼ffiffiffi2p
3sVM ð 8: 27 Þ
8.9.3 S UBSURFACE S TRESSES DUE TO NORMAL STRESSES ACTING ON THE C ONTACT S URFACES
Applied loading in all applications , that is, involv ing both standar d and nons tandard loading ,
is dist ributed ov er the rolling elem ents. The roll ing element loads that are applie d perpen-
dicula r to the con tact areas result in pressur e-type (norm al to the con tact surfa ce) stre sses. In
Chapt er 6 of the first volume of this hand book, assum ing ‘‘dr y’’ contact, eq uations to define
the magni tudes of these Hertz stresses wer e provided. In Chapte r 4, it was sho wn that, unde r
the influ ence of EH L, the nor mal stre ss distribut ion ov er the contact may be somew hat
altered from the Hert zian distribut ion. Never theless, in most roll ing bearing applic ations it
is satisfa ctory to assume the Hertzian stre ss distribut ion. On the other han d, if the ro lling
contact surfa ces are not complet ely separat ed by a lubri cant film, asp erities of these surfa ces
will co me into con tact, increa sing the con tact stresses above the Hert z stress values. A stress
concen tration facto r may be app lied to the Hertz stress to account for this pheno menon.
Equation 8.28 define s the stre ss conc entration fact or in terms of the ratio Ac /A0, the portio n
of the con tact area ov er whi ch Coulo mb fri ction occu rs:
KLn; mj ¼Qm j ; c
Qm j
Ac
A0
� �� 1
þ 1
1 � Ac
A0
� � 1 �Qm j ; c
Qmj
� �ð 8: 28 Þ
where Qm j ,c is the load c arried by the asperities a nd Q m j is the total contact normal l oad
at raceway m, azimuth location j . Subscript L refers to the s tress c on centration c aused by
an incomplete lubricant film; subscript n means that KLn,mj is applied to the normal or
Hertz stress. Ac/A0 is determined using the method of Greenwood and Williamson (see
Ref. [21] of Chapt er 5). At any point (x , y , z ) under t he contact surface, the s tresses
resulting from the Hertzian loading may be determined using the methods of Thomas
and Hoersch [36].
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In applyin g the latter, the contact surface normal stress s’ at a point ( x, y) is given by
s0 ¼ 3KL n; mj Q mj
2p amj bmj
1 � x
amj
� �2
� y
bmj
� �2" #1= 2
ð8: 29 Þ
8.9.4 SUBSUR FACE ST R ES SE S DUE TO FRICTIONAL SHEAR STRESSES ACTING ON THE CONTACT SURFACES
In most rolling be aring contacts, as discus sed in Chapt er 2, some de gree of sliding occu rs. In
angu lar-contact ball be arings, sph erical roll er bearing s, and thrust cyli ndrical roller be arings,
a substa ntial amo unt of sliding occurs. These sliding motio ns, occurri ng in relative ly he avily
loaded roll ing elem ent–ra ceway co ntacts, resul t in signi ficant frictio nal shear stresses . The
magni tude of the fricti on shear stre ss at any poin t ( x, y) on the con tact surfa ce depen ds on
the local co ntact pressure , the local sliding ve locity, the lubri cant rheologi cal prope rties, and
the topographi es of co ntact surfa ces.
Depending on the degree of contact surface separation by the lubricant film, sliding in
conjunction with the basic rolling motion may produce surface distress that can result in micro-
spalls; these can lead to macrospalls. Ne l ia s e t a l. [3 7] , c onduc ti ng e ndur anc e t est s us ing a r ol li ng –
sliding disk rig, demonstrated that smooth surfaces on 52100 and M50 steel test components,
i rr es pe cti ve o f the oc cur re nce of s lidi ng , e xpe rie nc ed no sur fa ce di st re ss. T he t es ts w er e c onduc te d
at 1500–3500 MPa (2.18–5.08 � 105 ps i) unde r l ubr ic ant f il m pa ra met er L ranging approximately
from 0.6 to 1.3. This indicates the need for finely finished rolling element and raceway surfaces,
especially in the presence of marginal lubrication. Ne lias et al. [37] noted that, in the absence of
sliding, microspall progression occurs both in the direction of sliding, and transverse to that
direction. This is shown in Figure 8.26 taken from Ref. [37]. In their test rig, the drive disk turns
faster than the follower disk, and the friction direction over the contact for the follower disk is in
the rolling direction. The friction direction over the contact of the driver disk is, however, in
t he dir ec tion oppos it e to rol ling . Fi gure 8 .2 7 s hows t ha t t he m ic roc ra cks a re de pe nde nt on t he
friction direction. It can be seen that the typical arrowhead shape is oriented in the friction
direction while crack propagation is in the direction opposite to friction. Ne lias et al. [37] further
noted that the driven surfaces were prone to greater damage than the driver surfaces.
Anothe r observat ion of Ne lias et al. [37] was that the size an d volume of the spall ed
mate rial increased with the magni tude of normal (Her tz) stress (see Figu re 8.28) . This
situation indicates that sliding damage is more severe under a heavy load than under a lighter
load, a condition that must be of concern in heavily loaded angular-contact ball bearings and
spherical roller bearings with marginal lubrication.
At any point (x, y, z) under the contact surface, the stresses resulting from the surface
shear stresses may be determined using the methods of Ahmadi et al. [38].
8.9.5 STRESS CONCENTRATION ASSOCIATED WITH SURFACE FRICTION SHEAR STRESS
To employ the methods of Ahmadi et al. [38], it is necessary to define the value of surface
friction shear stress t at each point (x, y) on the contact surface. For a contact that incurs
sliding in both the rolling and transverse to rolling directions, Equation 5.49 and Equation
5.50 can be used to define the surface friction shear stresses ty and tx when the lubricant film
is insufficient to completely separate the rolling contact surfaces:
td ¼ cv
Ac
A0
mas þ 1� Ac
A0
� �h
hvd
þ 1
tlim
� ��1
d ¼ y, x ð5:49; 5:50Þ
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(a)
(b)
FIGURE 8.26 Surfaces of M50 steel endurance test components operated at 3500 MPa (5.08� 105 psi)
under (a) simple rolling and (b) rolling and sliding. (From Nelias, D., et al., ASME Trans., J. Tribol.,
120, 184–190, April 1998. With permission.)
In applyi ng Equat ion 5.49 and Equat ion 5.50, the nor mal stress s is replac ed by s’ define d by
Equation 8.29; viscos ity is also calcul ated co nsidering s’ .Alternatively, considering only the fluid friction portion of the surface friction shear
stress, the following stress concentration factor may be applied to the latter stress:
KLf ;mj ¼ 1þ mcQmj;c
Ffmj
ð8:30Þ
where mc is the coefficient of friction associated with asperity–asperity interaction. A value
mc¼ 0.1 may be used for the lubricated contact.
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(b) Driven surface
Friction Rolling
(a) Driver surface
308
10 μm
70 μm
70 μm
Friction Rolling
FIGURE 8.27 Microcrack orientation with respect to rolling and friction directions for M50 steel
specimens tested at 3500 MPa (5.08� 105 psi). (From Nelias, D., et al., ASME Trans., J. Tribol., 120,
184–190, April 1998. With permission.)
8.9.6 STRESSES DUE TO PARTICULATE CONTAMINANTS
To determine the surface stresses associated with dents, the methods developed by Ville and
Nelias [19,23] or Ai and Cheng [39] may be applied. This requires a definition of the
contaminants involved in the application. Also, if the topography of the dented surface can
be defined, the methods of Webster et al. [20] may be applied. These methods, while effective
for laboratory investigations, typically consume many minutes and even hours of computer
time for the stress analysis of a single contact. The analysis of rolling bearing fatigue
endurance involves the iterative solution of many thousands of contacts. To include the effect
of particulate contamination in the prediction of bearing fatigue life in an engineering
application, approximations are necessary regarding the types of particles, their concentra-
tion in the lubricant, and their effects on subsurface stresses. In essence, a stress concentration
factor based on these parameters would be applied to the contact stress in the determination
of subsurface stresses.
The contamination level in an oil-lubricated application may be measured by counting
particles in the oil. This information may be used to establish a contamination parameter CL.
In bearing applications, the lubricant contains particles of widely varying sizes and properties.
� 2006 by Taylor & Francis Group, LLC.
Rolling Friction
(a) 1500 MPa (2.18 $105 psi), 5–10 μm spall size
(b) 2500 MPa (3.63 $105 psi), 20 μm spall size
(c) 3500 MPa (5.08 $105 psi), 40 μm spall size
FIGURE 8.28 Increase in microcrack size (length, depth) with normal stress for M50 steel endurance
tested under rolling and sliding conditions. (From Nelias, D., et al., ASME Trans., J. Tribol., 120, 184–
190, April 1998. With permission.)
For oil- bath-type lubri catio n, Ioan nides et al. [40] recomm end the use of the internati onal
cleanl iness cod e for hy draulic flui ds, ISO Standard 4406 [41] to codify these. The cleanliness
levels are indicated in Table 8.6.
In using Table 8.6, the following guidelines apply:
. If the filter has been validated to withstand system-operating conditions, the lowest level
(cleanest) on the left-hand side of the row associated with the specific filter rating should
be used.. If the filter has not been validated for withstanding the operating conditions of the
specific system, the highest (most contaminated) level on the right-hand side of the row
associated with the specific filter rating should be used.. If low contaminant ingress is expected, such as with a system having an air-vent filter
operating in a clean ambient environment, one level can be subtracted; that is, a move of
one level to the left is acceptable.. If high contaminant ingress is expected, such as for mobile equipment with open
reservoirs, one level should be added; that is, a move of one level to the right is
required.
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TABLE 8.6ISO 4406 Fluid Cleanliness Levels
Filter Rating b(xc)a Cleanliness Levelsb
2.5 13/10/7 14/11/9 15/12/10 16/13/11
5 15/12/10 16/13/12 16/14/12 17/15/12
7 17/14/12 18/15/12 18/16/13 19/16/14
12 18/16/13 19/16/14 20/17/14 21/18/15
22 20/17/14 21/18/15 22/19/16 23/20/17
35 22/19/16 23/20/17 24/21/18 25/22/18
aIn b(xc) � 1000, x is the particle size in mm, and 1000¼ the filtration ratio. Filtration ratio means that for a given
particle size x, the number of particles upstream of the filter is 1000� the number of downstream particles.bCode X/Y/Z; for example 13/10/7, refers to cleanliness levels whereby X is the number of particles of size� 4 mm; Y is
the number of particles of size � 6 mm; and Z is the number of particles of size � 14 mm.
. Example 1: For a full-flow 5-mm filter validated for high contaminant ingress operating
conditions, it is appropriate to start at 15/12/10 and move one level to the right; that is,
16/13/11.. Example 2: For a full-flow 12-mm filter validated for moderate contaminant ingress
operating conditions, it is appropriate to move to the highest level; that is, 21/18/15.. If operation is with two full-flow filters of the same rating in series, two levels should be
subtracted.
For use with the computer program supplied with Ref. [34] for calculation of bearing fatigue
life, Table 8.7 provides some simplified guidelines for the cleanliness level.
The earlier classifications do not account for the hardness of the particles. It has been
established, however, that in a wide scope of rolling bearing applications, there exists a similar
distribution of hard and soft particles, which produces a generally similar fatigue life-reducing
effect. Even if Table 8.6 indicates the number of particles >4 mm, >6 mm, and >14 mm, this
does not mean that just a few contaminant particles of such minute size affect the fatigue lives
of rolling bearings. The standardized figures are only a statistical measure for the existence of
critical particles.
Ioannides et al. [40] state that for circulating oil lubrication, the filtering efficiency of the
system can be used in lieu of ISO 4406 [41] to define contaminant size. This may be defined by
the filtering capacity as specified by ISO 4372 [42].
TABLE 8.7ASME Guidelines for Cleanliness Classification vs. Contamination Level
Cleanliness Classification ISO 4406 Cleanliness Level Filter Rating b(xc) (mm)
Utmost cleanliness 14/11/8 2.5–5
Improved cleanliness 16/13/10 5
Normal cleanliness 18/15/12 7
Moderate contamination 20/17/14 12–22
Heavy contamination 22/19/16 35 or coarser
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TABLE 8.8Lubricant Contamination Factor Calculation Constants
Type of Lubrication Contamination Level CL2 CL3 Restriction
Circulating oil ISO –/13/10 0.5663 0.0864
ISO �/15/12 0.9987 0.0432
ISO �/17/14 1.6329 0.0288
ISO �/19/16 2.3362 0.0216
Bath oil ISO �/13/10 0.6796 0.0864
ISO �/15/12 1.141 0.0288
ISO �/17/14 1.670 0.0133
ISO �/19/16 2.5164 0.00864
ISO �/21/18 3.8974 0.00411
Grease High cleanliness 0.6796 0.0864
Normal cleanliness 1.141 0.0432
Slight-to-typical contamination 1.887 0.0177 dm < 500mm
1.677 0.0177 dm � 500mm
Severe contamination 2.662 0.0115
Very severe contamination 4.06 0.00617
Depending on the size of the roll ing contact a reas in a be aring, sensi tivity to particulat e
contam inatio n varie s. Bal l be arings tend to be more vulnera ble than roller bea rings; con tam-
inant particles a re more harmf ul in smal l bearing s than in be arings with large rolling elemen ts.
Consid ering the foregoing and using empirical ly determ ined da ta, Ioannides et al. [40] linked
the co ntaminati on pa rameter CL to bearing size, lubri cation system, and lubri cation effect-
ivenes s. Further consider ing that solid co ntaminan ts found in bearing s are mainly hard
meta llic pa rticles resul ting from wear of the mechan ical system, they develop ed Figure
CD8.1 through Figure CD 8.14, whi ch are charts of CL vs. lub ricant effe ctiveness parame ter
k an d bearing pitch diame ter dm for v arious ISO Stan dard 4406 cleanl iness levels. For
circulati ng oil-lubri cation syst ems, filtra tion level s accordi ng to ISO 4572 are also indica ted.
The values of CL may also be obtaine d using the base eq uation for the curves pro vided
in Figure CD 8.1 through Fig ure CD8.14. This base equatio n may be obtaine d from the
appen dix of ISO Stan dard 281 [5].
CL ¼ CL1 1 � CL2
d 1= 3m
!ð 8: 31 Þ
wher e
CL1 ¼ C L3 k0 :68 d 0 :55
m CL1 � 1 ð 8: 32 Þ
Va lu es of the c ons tants CL2 and CL3 ma y b e obta ined from T a bl e 8. 8 for the v arious ISO
contamination levels. For oil-lubricated bearings, Table 8.9 gives the range of contamination
levels corresponding to the basic level given in Table 8.8. For circulating oil-lubricated bearings,
Table 8.9 also provides the b(xc) level corresponding to the basic contamination level.
In Equation 8.32 and in Figure CD8.1 through Figure CD8.14, k is defined as n/n1, where
n is the kinematic viscosity of the lubricant at the operating temperature and n1 is the
kinematic viscosity required for adequate separation of the contacts. According to ISO 281
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TABLE 8.9Contamination Ranges and b(xc) for Data of Table 8.8
Type of Lubrication Basic ISO Contamination Level ISO Contamination Range x(c) b(xc)
Circulating oil �/13/10 �/13/10, �/12/10, �/13/11, �/14/11 6 200
�/15/12 �/15/12, �/16/12, �/15/13, �/16/13 12 200
�/17/14 �/17/14, �/18/14, �/18/15, �/19/15 25 75
�/19/16 �/19/16, �/20/17, �/21/18, �/22/18 40 75
Bath oil �/13/10 �/13/10, �/12/10, �/11/9, �/12/9 — —
�/15/12 �/15/12, �/14/12, �/16/12, �/16/13 — —
�/17/14 �/17/14, �/18/14, �/18/15, �/19/15 — —
�/19/16 �/19/16, �/18/16, �/20/17, �/21/17 — —
�/21/18 �/21/18, �/21/19, �/22/19, �/23/19 — —
[5], k�L1.12 . Accor ding to ISO 28 1 [5], the refere nce viscosit y n1 may be estimat ed using
Equat ion 8.33 and Equation 8.34. Alternat ively, the ch art of Figu re CD8.15 may be used to
estimat e n1:
n1 ¼ 45,000 n� 0 :83 d � 0 :5m n < 1,000 rpm ð8: 33 Þ
n1 ¼ 4,500 n � 0: 5 d � 0: 5m n � 1,000 rp m ð8: 34 Þ
For circul ating oil, in Figure CD8.1 through Figure CD 8.4, as ind icated in footnot e a of
Table 8.6, the parame ter bx is de fined in Ref. [40] as
bx ¼Npu > x
Npd > x ð8: 35 Þ
wher e Npu is the number of parti cles ups tream of size greater than x mm, and N pd is the
numb er of pa rticles downstream of size greater than x mm. Thus, b6 ¼ 2 00 means that for
every 200 pa rticles > 6 m m upstre am of the filter, onl y 1 particle >6 mm passes through the
filter . Alt hough this is a useful method for compari ng filter perfor mance, it is not infal lible
since contam inant pa rticles may have different shapes a ccording to the applic ation.
The CL values obtaine d using Fig ures CD8.1 through Figure CD 8.9 are for oil lubrican ts
withou t additives. When the calculated bx < 1 , a high -quality lubri cant with tested and
app roved additive s may be expecte d to promot e a favora ble smooth ing of the racew ay
surfa ces dur ing run ning in. Thereby , bx may impr ove and reach a value of 1.
When co ntaminati on is not measu red or known in detail, the co ntaminati on parame ter CL
may be estimat ed using Tabl e 8.10 provided in Refs. [5,4 3].
For use in determinat ion of rolling contact fatigue life, the contam ination parame ter CL
needs to be converted to the form of a stress concentration factor to be applied to the contact
stress; for example, s’(x,y)¼Kc s(x,y). Also, the stress concentration factor may be applied to
the surface shear stress as well; for example,
t0ðx,yÞ ¼ Kctðx,yÞ
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TABLE 8.10Contamination Parameter Levels
Bearing Operation Condition CL
dm < 100mm dm � 100mm
Extreme cleanliness 1 1
Particle size of the order of lubricant film thickness
High cleanliness 0.8–0.6 0.9–0.8
Oil filtered through extremely fine filter; conditions typical of bearings
greased for life and sealed
Normal cleanliness 0.6–0.5 0.8–0.6
Oil filtered through fine filter; conditions typical of bearings greased
for life and shielded
Slight contamination 0.5–0.3 0.6–0.4
A small amount of contaminant in lubricant
Typical contamination 0.3–0.1 0.4–0.2
Conditions typical of bearings without integral seals; coarse filtering;
wear particles and ingress from surroundings
Severe contaminationa 0.1–0 0.1–0
Bearing environment heavily contaminated and bearing arrangement
with inadequate sealing
Very severe contaminationa 0 0
aIn the cases of severe and very severe contamination, failure may be caused by wear, and the useful life of the bearing
may be far less than the calculated rating life.
Barnsby et al. [34] , as derive d from Ioan nides et al. [40] , give the following eq uations for point
and line co ntacts:
KC ;point ¼ 1 þ ð1 � C 1= 3L Þ
sVM ;lim
sVM ;max
ð 8: 36 Þ
KC; line ¼ 1 þ ð1 � C 1= 4L Þ
sVM ;lim
sVM ;max
ð 8: 37 Þ
wher e sVM,max is the maxi mum value of the von Mises stress occu rring be low the contact
surfa ce, and sVM,lim is the fatig ue lim it of the vo n M ises stress for the roll ing compo nent
mate rial. Val ues for the fati gue limit stress will be discus sed later in this ch apter.
Ne lias [44] illustr ates in Figure 8.29 that for a de nted or rou gh surface the magni tude of
the maxi mum sh ear stress is strongly influenced by sliding on the surfa ce. Ne lias [44] furt her
postul ates that failure of rough or de nted surfa ces may commenc e ne ar the surfa ce; howeve r,
coales cence of microcracks may proceed inward in the direction toward the location of the
maxi mum subsurfac e stresses due to the average contact loading . Thus , the subsurf ace failure
might be initiated by the surface condition. This competition of subsurface, failure-initiating
stresses is illustrated in Figu re 8.30. Bec ause most modern ball and roller bearing s have
relatively smooth raceway and rolling element surfaces, roughness is more indicative of
dents in contaminated applications. Thus, competition for initiation of subsurface fatigue
failure would tend to occur more in applications with contamination. When calculations for
subsurface von Mises stresses (or other assumed failure-initiating stresses) indicate maximum
values approaching the surface, it may be presumed that surface pitting will most likely occur
first; however, not to the exclusion of subsurface fatigue failure depending on the amount of
operational cycles accumulated.
� 2006 by Taylor & Francis Group, LLC.
Slide-to-roll ratio, %
0.2
0.3
0.4
0.5
0.6
0.7
0.8
sHertz
tmax
0 2 4 6 8 10 12 14
FIGURE 8.29 Maximum shear stress/maximum Hertz stress vs. slide–roll ratio in the vicinity of a dent
1.5 mm deep by 40 mm wide; the dent with a shoulder 0.5 mm. (From Nelias, D., Contribution a L’etude
des Roulements, Dossier d’Habilitation a Diriger des Recherches, Laboratoire de Mecanique des
Contacts, UMR-CNRS-INSA de Lyon No. 5514, December 16, 1999. With permission.)
z /b
z /b
Mildroughness
High loadMedium loadLow load
z /bLowroughness
Highroughness
t t t
FIGURE 8.30 Competition between surface and subsurface crack growth for various loads and surface
roughnesses. Each graph represents shear stress vs. nondimensionalized depth z/b. The dashed line
represents the fatigue limit stress below which crack initiation (straight lines in inserts) does not occur
and propagation direction (arrow-tip lines in inserts). (From Nelias, D., Contribution a L’etude des
Roulements, Dossier d’Habilitation a Diriger des Recherches, Laboratoire de Mecanique des Contacts,
UMR-CNRS-INSA de Lyon No. 5514, December 16, 1999. With permission.)
� 2006 by Taylor & Francis Group, LLC.
8.9.7 COMBINATION OF STRESS CONCENTRATION FACTORS DUE TO LUBRICATION
AND CONTAMINATION
The stress concentration factors KL and KC occur due to imperfections in the contact surfaces.
These stress concentrations do not act independently; rather, their combined value is given by
KLC;mj ¼ KL;mj þ KC;mj � 1 ð8:38Þ
It can be seen that for very smooth rolling contact surfaces without dents, KLC,mj¼ 1, and for
all surfaces with no contaminants present, KLC,mj¼KL,mj.
8.9.8 EFFECT OF LUBRICANT ADDITIVES ON BEARING FATIGUE LIFE
Thus far, only the effect of the base stock lubricant has been considered with regard to fatigue life.
However, a base stock lubricant is supplied to a rolling bearing rarely only. In fact, more often
than not, with the exception of bearings that are delivered with integral seals and greased for life,
the bearing must survive with the lubricant required to maximize performance of the overall
mechanism; for example, a gear-box. Such lubricants typically contain additives to achieve one or
more of the following properties: (1) antiwear, (2) antiscuffing or extreme pressure (EP) resist-
ance, (3) antioxidation, (4) antifoaming, (5) rust/corrosion inhibition, (6) control of deposit
formations on surface through detergents, (7) demulsification to aid in separation of water,
and (8) control of sludge formation throughdispersants. Someof these additives tend to influence
fatigue endurance significantly; however, it has not been possible to specify these effects through
the use of contact stress concentration factors. Rather in Ref. [34] the effects of these additives on
life have been specified as ranges on L10 lives, as in Table 8.11.
8.9.9 HOOP STRESSES
To prevent rotation of the bearing inner ring about the shaft, and hence prevent fretting
corrosion of the bearing bore surface, the bearing inner ring is usually press-fitted to the shaft.
The amount of diametral interference, and therefore the required pressure between the ring
TABLE 8.11Estimated Bearing Life Ranges for Common Lubricant Classes
Lubricant Class Fatigue Life Range Average Fatigue Life
Industrial Lubricants
Hydraulic oils 0.6–1.0 L10 0.8 L10
Rolling bearing oils with no antiwear additive 0.8–1.4 L10 1.1 L10
Rolling bearing oils with antiwear additive 0.6–1.0 L10 0.8 L10
Turbine oils 0.6–1.0 L10 0.8 L10
Circulating oils with no antiwear additive 0.8–1.4 L10 1.1 L10
Circulating oils with antiwear additive 0.6–1.0 L10 0.8 L10
Synthetic antiwear oils 0.8–1.7 L10 1.2 L10
Gear oils 0.4–1.3 L10 0.8 L10
Automotive and Aviation Lubricants
Gear lubricants 0.3–0.7 L10 0.5 L10
Automatic transmission fluids 0.6–1.0 L10 0.8 L10
Aviation turbine oils 0.8–1.7 L10 1.2 L10
� 2006 by Taylor & Francis Group, LLC.
bore and the shaft outside diameter, depends primarily on the amount of applied loading and
secondarily on the shaft speed. The greater the applied load and shaft speed, the greater must
be the interference to prevent ring rotation. For recommendation of the magnitude of the
interference fit required for a given application as dictated only by the magnitude of applied
loading, ANSI/ABMA Standard No. 7 [45] may be consulted for radial ball, cylindrical roller,
and spherical roller bearings. For tapered roller bearings, ANSI/ABMA Standards No. 19.1
[46] and No. 19.2 [47] may be consulted. Because the ring and shaft dimensions, and materials
are defined, standard strength of materials calculations, for example, Timoshenko [48], may
be used to determine the radial stresses. The interference fit causes the ring to stretch resulting
in tensile hoop stress.
Similarly, for outer ring rotation such as in wheel bearing applications, the outer ring may
be press-fitted into the housing. In this case, compressive hoop stress and radial stress will be
induced.
Ring rotation, particularly at a high speed, gives rise to radial centrifugal stress, which in
turn causes the ring to stretch with attendant hoop stresses resisting the ring expansion. Outer
ring rotation results in tensile hoop stresses that tend to counteract the compressive hoop
stresses caused by press-fitting of the outer ring in the housing. Timoshenko [48] details the
method to calculate the tensile hoop and radial stresses associated with ring rotation.
Each of the stresses due to press-fitting or ring rotation is superimposed on the subsurface
stress field caused by contact surface stresses.
8.9.10 RESIDUAL STRESSES
8.9.10.1 Sources of Residual Stresses
Residual stress is that stress which remains in a material when all externally applied forces are
removed. Residual stresses arise in an object from any process that produces a nonuniform
change in shape or volume. These stresses may be induced mechanically, thermally, chem-
ically, or by a combination of these processes [49]. An example of such a process is as follows:
If a relatively thin sheet of malleable material such as copper is repeatedly struck with a hammer,
the thickness of the sheet is reduced, and the length and width are correspondingly increased; that
is, the volume remains constant. If the same number of equally intensive hammer blows were
uniformly delivered to the surface of a copper block several centimeters thick, the depth of
penetration of plastic deformation would be relatively shallow with respect to the block thickness.
The deformed surface layer would be restrained from lateral expansion by the bulk of subsurface
material, which experienced less deformation. Consequently, the heavily deformed surface
material would be like an elastically compressed spring, prevented from expanding to its unloaded
dimensions by its association with elastically extended subsurface material. The resulting residual
stress profile is one in which the surface region is in residual compression and the subsurface
region is in a balancing residual tension. This example is a literal description of the shot-peening
process, wherein a surface is bombarded with pellets of steel or glass. A highly desirable com-
pressive residual stress pattern is established for components that experience high, cyclic tensile
stresses at the surface during service. The magnitude of tensile stress experienced by the compon-
ent during service is functionally reduced by the amount of residual compressive stress, thereby
providing significantly increased fatigue lives for parts such as shafts and springs.
The shot-peening example illustrates the essential characteristics of a surface in which
residual stress has been induced:
� 2006 by
1. Nonuniformity of plastic deformation; the surface material is encouraged to expand
laterally.
Taylor & Francis Group, LLC.
� 2006 by
2. Subs urface mate rial, which experi ences less plast ic deform ation, is elastica lly
stra ined in tensi on as it restrains exp ansion of the surfa ce mate rial, thereby inducing
compres sive residual stress in the surfa ce region.
3. The resulting state of residu al stre ss is a reflectio n of the elast ic co mponents of strain
in the surface and sub surface regions , which are in equili brium, pro viding a bal-
anced tensi le–com pressive system.
Heat treatmen t used for harden ing rolling bearing compon ents can exert very signi ficant
influen ce over the state of resi dual stress. Depending on the steel compo sition, austeniti zing
tempe rature, que nching severity, compon ent g eometry, section thickne ss, and so forth, heat
treatment can provide either residu al compres sive stre ss or resid ual tensi le stre ss in the surfa ce
of the hardened comp onent [49–50 ]. Temperat ure gradie nts are establ ished from the sur-
face to the cen ter of a part during que nching afte r heati ng. The different ial therm al con trac-
tion associ ated with these gradie nts provides for nonuni form plastic deform ation, givin g rise
to resi dual stre sses. Additional ly, volume tric changes associated with the phase trans form-
ation occurri ng during heat treatmen t of steel occur at different times during que nching at the
part surfa ce and interior due to the thermal gradie nts establ ished. These sequenti al volume tric
changes, combined with different ial therm al contrac tions , are responsi ble for the resi dual
stress state in a hardened steel compon ent. The sequence and relative magni tudes of these
contri buting factors determ ine the stress magni tude an d wheth er the surface is in resi dual
compres sion or tensi on.
Grinding of a hardened steel componen t to finished dimens ions also affects the resi dual
surfa ce stress. Genera lly, if effects of abu sive grinding practices that generat e heat an d
produ ce micro structural alterati ons are neglect ed, it is found that the residu al stre ss effects
associ ated wi th grindi ng are confine d to mate rial within 50 m m (0.0 02 in.) of the su rface.
Good grindi ng practice, as applie d to bearing rings, produces residu al compres sive stre ss in
a shall ow surfac e layer . Grinding also involv es some plast ic deformati on of the surfa ce,
produ cing resi dual compres sion as descri bed above.
The residual stress state in a finished bearing component is therefore a function of heat
treatment and grinding. If properly ground, the residual stress in a through-hardened bearing
component will be 0 to slightly compressive. The subsurface residual stress conditions will be
determined by the prior heat treatment. In a surface-hardened component, the surface and
subsurface residual stresses will be compressive; in the core of the material, the residual stresses
will be tensile. The depth of the case must, therefore, be sufficient with regard to bearing fatigue
endurance. This depth has historically been set at approximately four times the depth of the
maximum subsurface orthogonal shear stress; see Chapter 6 of the first volume of this handbook.
8.9.1 0.2 Alteratio ns of Residua l Stress Due to Rolling Contact
As a result of cyclic stressing during rolling contact, the bearing steel experiences changes in the
microstructure. Associated with these alterations are changes in residual stress and retained
austenite; this has been reported in Refs. [30,51–55]. The forms of the changes in circumferential
direction residual stress and retained austenite profiles are illustrated in Figure 8.31. Indications
are that significant changes in residual stress and retained austenite precede any observable
alterations in microstructure. See Figure 11.4 in the first volume of this handbook.
The residual stress data of Figure 8.31 show peak values at increasing depths correspond-
ing to increasing numbers of stress cycles. A similar form is indicated for decomposition of
retained austenite, with peak effect depths slightly less than those for residual stress. The data
of Figure 8.31 for the high maximum-contact stress indicate more rapid rates of change for
both residual stress and retained austenite content.
Taylor & Francis Group, LLC.
Unrun Unrun0
−500
−1000Unrun
0
−500
−10000
10
20
30
40
50
60
70
80
90100
010
20
30
40
50
60
70
80
90
1000
0 0.1 0.2
(a) (b)
0.3Depth, mm
0.4 0.5 0.60.1 0.2 0.3
Depth, mm
0.4 0.5 0.6 0.7
Unrun
1 3 107
1 3 106
1 3 107
2 3 108
4 3 108
1 3 105 + 1 3 106
1 3 105 + 1 3 106
1 3
10
92 3
10
9
1 3 107
2 3
10
9
1 3 108
2 3 108
4 3 108
1 3 108
1 3 109
2 3 108
4 3 108
Res
idua
l str
ess,
MN
/m2
Res
idua
l str
ess,
MN
/m2
Dec
ompo
sitio
n of
reta
ined
aus
teni
te, %
Dec
ompo
sitio
n of
reta
ined
aus
teni
te, %
1 3 108
1 3 106
1 3 107
2 3 108
4 3 108
1 3
108
FIGURE 8.31 Residual stress and percent retained austenite decomposition vs. depth below raceway
surface for various numbers of inner ring revolutions of a 309 deep-groove ball bearing; the bearing ring
was manufactured from 52100 steel through-hardened to Rc 64. (a) Maximum contact stress: 3280 MPa
(475 kpsi); depth of maximum orthogonal shear stress 0.19mm (0.0075 in.); depth of maximum shear
stress 0.30mm (0.0118 in.) (b) Maximum contact stress: 3720 MPa (539 kpsi); depth of maximum
orthogonal shear stress 0.21mm (0.0083 in.); depth of maximum shear stress 0.33mm (0.0130 in.).
Harris [56] found compressive surface stresses in the range of 600MPa (87,000psi) for both
M50 and 52100 balls that had not been run. Beneath the surface, in the zone of maximum
subsurface applied stress, the compressive stress level reduced to values in the range of 70MPa
(10,000psi). When the balls were operated under normal bearing Hertz stresses, for example,
maximum 2,700MPa (400,000psi), these compressive stresses seemed to disappear, most likely,
as a result of retained austenite transformation. The slight differences in the depths at which the
peak values occur in residual stress and retained austenite decomposition imply correlation with
the maximum shear stress and the maximum orthogonal shear stress, respectively. The work of
Muro and Tsushima [53] supports the correlation of peak residual stress values with the max-
imum shear stress. There appears to be no direct relationship between retained austenite decom-
position and the generation of residual compressive stress, nor, according to Voskamp et al. [54],
any indication of which, if either, of these processes triggers microstructural alterations.
8.9.10.3 Work Hardening
It has also been observed that running-in bearing raceways under heavy loading for a short
period of time before normal operation tends to work harden the near-surface regions. This
introduces a slight compressive residual stress into the material, increasing its resistance to
fatigue. Excessive amounts of compressive stress tend to reduce resistance to fatigue.
� 2006 by Taylor & Francis Group, LLC.
8.9.11 LIFE INTEGRAL
The stresses discussed in this section each contribute to the overall subsurface stress distribution.
Using superposition and the assumption of von Mises stress as the fatigue failure-initiating
criterion, the stress tensor may be calculated for every subsurface point (x, y, z). The basic
equation of the Ioannides–Harris theory, that is Equation 8.19, may be restated as follows:
ln1
D Si
/N e sVM ;i � s VM ; lim
� �cDVi
z hið 8: 39 Þ
The ab ove equati on refers to the surviva l of volume elemen t D Vi for N stre ss cycles with
probab ility DSi . The pro bability that the entire stressed volume wi ll survi ve N stress cycles
may be de termined using the produ ct law of pro bability; that is, S ¼DS1 �D S2 � � � � � DSn.
Therefor e,
ln1
S ¼Xi ¼ n
i ¼ 1
ln1
DSi
/ N e p dr
X1¼ n
i ¼ 1
sVM; i � s VM; lim
� �cAi
z hi
" #ð 8: 40 Þ
wher e Ai is the radial plane c ross-sect ional area Dx �Dz of the vo lume elem ent on which the
effecti ve stress acts, an d dr is the racew ay diame ter. Letti ng q ¼ x/a an d r ¼ z/b, where a and b
are the semimaj or and semiminor axes, respect ively, of the co ntact ellipse (see Figure 8.22),
then Dx¼ aDq and Dz¼ bDr. Numerical integration may be performed using Simpson’s rule,
letting Dq¼Dr¼ 1/n, where n is the number of segments into which the major axis is divided.
With the indicated substitutions, Equation 8.39 becomes
ln1
S¼ Nepab1�hdr
9n2
Xj¼n
j¼1
cj
Xk¼n
k¼1
ck
sVM;jk � sVM; lim
� �crhk
" #ð8:41Þ
where cj and ck are Simpson’s rule coefficients. The number of stress cycles survived is N¼ uL,
where u is the number of stress cycles per revolution and L is the life in revolutions. Therefore,
L / upab1�hdr
9n2
Xj¼n
j¼1
cj
Xk¼n
k¼1
ck
sVM;jk � sVM;lim
� �crhk
" #( )1=e
ð8:42Þ
The above equation may be used to find the stress–life factor ASL by (1) evaluating the
equation for the stress conditions assumed by Lundberg and Palmgren, (2) evaluating
the equation for the actual bearing stress conditions occurring in the application, and (3)
comparing these. For example,
ASL ¼Lactual
LLP
¼
Pj¼n
j¼1
cj
Pk¼n
k¼1
ck
sVM;jk � sVM; lim
� �crhk
" #( )1=e
actual
Pj¼n
j¼1
cj
Pk¼n
k¼1
ck
sVM;jk
� �cLP
rhk
" #( )1=e
LP
¼ Iactual
ILP
ð8:43Þ
where I is called the life integral.
� 2006 by Taylor & Francis Group, LLC.
The accurate evaluation of I for each condition depends on the boundaries specified for
the stress volume. It was shown that earlier, because only Hertz stresses were considered in
their analysis, Lundberg and Palmgren were able to assume that the stressed volume was
proportional to padrz0, where z0 is the depth to the maximum orthogonal shear stress t0. In
the analysis of the stress–life factor, von Mises stress is used in lieu of t0, and the effective
stress is integrated over the appropriate volume. That volume is defined by the elements for
which the effective stress is greater than zero; that is, sVM,i – sVM,limit> 0. It can be
demonstrated using the Lundberg–Palmgren analysis that
L / 1
tc=e0
¼ 1
t9:30
ð8:44Þ
Considering the equivalent integrated life, Harris and Yu [57] showed that
Lij /1
t9:39ij
ð8:45Þ
Moreover, they determined that all effective stresses
sVM;i � sVM;limit < 0:6 ðsVM;i � sVM;limÞmax
influence life less than 1%. For simple Hertz loading, the life-influencing zone is illustrated in
Figure 8.32. As compared with the Lundberg–Palmgren stressed volume proportionality for
which z0/b� 0.5, for Hertz loading, the critical stressed volume stretches down to z/b� 1.6.
0
−0.2
0.2+
0.30.9
1.0
0.6
+
0.8+
0.7+ −
0.1 −
0.5+
0.4+
−0.4
−0.6
−0.8
−1
−1.2
−1.4
−1.6
−1.8
−2−1 −0.5 0
x /a
z /b
0.5 1
FIGURE 8.32 Lines of constant tyz/t0 for simple Hertz loading—shaded area indicates effective life-
influencing stresses.
� 2006 by Taylor & Francis Group, LLC.
The crit ical stressed volume is different for each roll ing elem ent–ra ceway con tact combinat ion
of applied an d resi dual stress, and it sho uld be use d in the evaluation of the life integ rals in
Equation 8.43.
8.9.12 FATIGUE LIMIT STRESS
To ev aluate the life integ rals, the value of the fatigu e limit stre ss must be known for the
bearing co mponent mate rial. Thi s can be determ ined by endurance testing of be arings or
selec ted co mponents . The test program s report ed in Refs. [32, 33] wer e extended to cover 129
bearing app lications includi ng add itional mate rials. The an alytical models to predict bearing
applic ation perfor mance an d ball/v- ring test perfor mance wer e refin ed, and perfor mance
analys es were again cond ucted, using the von Mises stre ss as the fatigue failu re-init iating
criteri on. Bas ed on this subseq uent study by Harr is [56], Table 8 .12 gives resul ting values of
fatigue limit stre ss for various mate rials.
Bo hmer et a l. [58] establis hed that the fati gue limits of steels decreas e as a functi on of
tempe rature. From their graph ical data, the followin g relat ionshi ps may be determ ined by
curve- fitting for various bea ring steels operati ng at tempe rature s exceeding 80 8 C (176 8 F):
sVM , lim ðT ÞsVM, lim ð 80Þ
� �52100
¼ 1: 165 � 2 :035 � 10 � 3 T ð 8: 46 Þ
sVM, lim ð T ÞsVM ,lim ð 80Þ
� �M50
¼ 1: 076 � 9: 494 � 10 � 4 T ð 8: 47 Þ
sVM , lim ðT ÞsVM, lim ð80 Þ
� �M50 NiL
¼ 1: 079 � 1:040 � 10 � 3 T ð 8: 48 Þ
Equation 8.46 through Equation 8.48 wer e used in the applic ation pe rformance analys es that
generated Table 8.12.
8.9.13 ISO STANDARD
In Equat ion 8.21 as present ed in Ref. [5], AISO is used to indicate the ‘‘systems approach’’ life
modification factor. Some manufacturers, for example, as in Ref. [43], have substituted their
own subscript for ‘‘ISO.’’ In this text as in Ref. [34], the integrated stress–life factor has been
designated ASL. The ISO standard [5] specifies that AISO can be expressed as a function of su/
s, the endurance stress limit divided by the real stress, which can include as many influencing
TABLE 8.12Fatigue Limit Stress (von Mises Criterion) for Bearing Materials
Material
sVM,limit
MPa (psi)
AISI 52100 CVD steel HRc 58 minimum 684 (99,200)
SAE 4320/8620 case-hardening steel HRc 58 minimum 590 (85,500)
VIMVAR M50 steel HRc 58 minimum 717 (104,000)
VIMVAR M50NiL case-hardening steel HRc 58 minimum 579 (84,000)
440C stainless steel 400 (58,000)
Induction-hardened steel (wheel bearings) 450 (65,000)
Silicon nitride ceramic 1,220 (177,000)
� 2006 by Taylor & Francis Group, LLC.
1
a XYZ
s u /s
FIGURE 8.33 AISO vs. su/s for a given lubrication condition.
stress components as necessary. AISO vs. su/s is illustrated by the schematic diagram of
Figure 8.33. While the diagram is constructed using normal stresses su and s, it can also be
based on endurance strength in shear, which is the historical criterion for calculating rolling
bearing fatigue life; for example, Lundberg and Palmgren [1,2] considered the range of the
maximum orthogonal shear stress as the failure-initiating stress. It is noted from Figure 8.33
that AISO, and hence bearing fatigue life approaches infinity as the real stress s approaches
the endurance limit stress su.
ISO [5] considers that the fatigue-initiating stress is substantially dependent on the internal
load distribution in the bearing and the subsurface stresses associated with the loading in the
most heavily loaded rolling element–raceway contact. To simplify the calculation of AISO,
ISO introduces the following approximate equivalency:
AISO ¼ fsu
s
� �� f
Flim
Fe
� �ð8:49Þ
where Flim is the statically applied load of the bearing at which the fatigue limit stress is just
reached in the most heavily loaded rolling element–raceway contact. In the determination of
Flim, the following influences are considered:
. Bearing type, size, and internal geometry
. Profile of rolling elements and raceways
. Manufacturing quality of the bearing
. Fatigue limit stress of the bearing raceway material
As for the original Lundberg–Palmgren theory and life prediction methods [1,2], rolling
element fatigue failure is not considered.
Specific means to calculate Flim for high-quality ball and roller bearings manufactured
from through-hardened 52100 steel are provided in an appendix to Ref. [5]. These are based
on a maximum contact stress; that is, Hertz stress, of 1500MPa. It is evident that the ISO
standard [5] does not apply directly to bearings manufactured from other high-quality bearing
steels.
Ioannides et al. [40] developed charts of AISO vs. CLA � Flim/Fe and k for radial ball
bearings, radial roller bearings, thrust ball bearings, and thrust roller bearings. These are
provided herein as Figure CD8.16 through Figure CD8.19. Alternatively, AISO may be
calculated using equations provided by the ISO standard [5]; for example,
� 2006 by Taylor & Francis Group, LLC.
TABLE 8.13Constants and Exponents for AISO Equation 8.50
Bearing Type Lubricant Film Adequacy x1 x2 e1 e2 e3 e4
Radial ball 0.1�L<0.4 2.5671 2.2649 0.054381 0.83 1/3 �9.3
0.4�L<1 2.5671 1.9987 0.19087 0.83 1/3 �9.3
1�L�4 2.5671 1.9987 0.071739 0.83 1/3 �9.3
Radial roller 0.1�L<0.4 1.5859 1.3993 0.054381 1 0.4 �9.185
0.4�L<1 1.5859 1.2348 0.19087 1 0.4 �9.185
1�L�4 1.5859 1.2348 0.071739 1 0.4 �9.185
Thrust ball 0.1�L<0.4 2.5671 2.2649 0.054381 0.83 1/3 �9.3
0.4�L<1 2.5671 1.9987 0.19087 0.83 1/3 �9.3
1�L�4 2.5671 1.9987 0.071739 0.83 1/3 �9.3
Thrust roller 0.1�L<0.4 1.5859 1.3993 0.054381 1 0.4 �9.185
0.4�L<1 1.5859 1.2348 0.19087 1 0.4 �9.185
1�L�4 1.5859 1.2348 0.071739 1 0.4 �9.185
AISO ¼ 0:1 1� x1 �x2
k e 1
� �e2 CL Flim
Fe
� �e3� e4
ð 8: 50 Þ
The co nstants x1 and x2 and the e xponents e 1–e 4 are given in Tabl e 8.13.
See Exampl e 8 .4 throu gh Exam ple 8.6.
8.10 CLOSURE
The Lundber g–Palm gren theory to pred ict fatigue life was a signi fican t ad vancement in the
state-of -the-art of ball an d roller be aring techn ology, affectin g the internal design an d
exter nal dimen sions for 40 years. The EHL theory, intr oduc ed by Grubi n, and furt her
advance d by scores of resear chers, initial ly affected bearing micr ogeomet ry, but late r, be cause
of the possibi lity of increa sed enduran ce toget her with improved mate rials resulted in ‘‘down-
sizing’’ of ball and roller bearing s. The Ioannides –Harr is theory, in its ability to a pply the
total stre ss patte rn to pred ict life in any bearing applica tion, and in its use of a fati gue stress
limit for roll ing be aring mate rials carri es the de velopm ent to the next plate au by substa ntially
increa sing unde rstan ding of the signifi cance of material qua lity and concen trated contact
surfa ce integ rity. It is now apparen t that a bearing , man ufactured from material that is clean
and homogen eous, whi ch ope rates with its roll ing/sliding contact s free from co ntaminan ts,
and whi ch is not overloa ded may survi ve without fatigue. In fact , Palmgr en [59] init ially
consider ed the existen ce of a fatigue limit stress; howeve r, the roll ing bearing sets that wer e
tested in the developm ent of the Lundber g–Palm gren theory failed rather comp letely unde r
the test loading , and he ab andon ed the concept. During the early 1980s, when the Ioannides –
Harr is theory was unde r developm ent, fatigu e testing consumed substa ntial calend ar time,
often requir ing 12
a year and mo re wi th no bearing failures after more than 500 mil lion
revolut ions.
This chapter convert s the Ioann ides–Harri s theory into practice. The life theory is stress-
based, as opposed to the factor-based, modified Lundberg–Palmgren theory (standard
methods [3–5] ) exempl ified by Equat ion 8.16. Rathe r, the Ioannides –Harr is theory utilizes
the base Lundberg–Palmgren life equations together with a single factor ASL that integrates
� 2006 by Taylor & Francis Group, LLC.
the effect on fatigue life of all stresses acting on the bearing contact material. An accurate life
prediction for any bearing application depends only on the successful evaluation of the
appropriate stresses. With the application of modern computers and computational methods,
these stresses are subjected to increasingly greater scrutiny. With the current availability of
powerful, inexpensive, desktop and laptop computers, engineers worldwide have the capabil-
ity to use rolling bearing performance analysis computer programs that can effectively
employ the methods described in this text for such analysis.
REFERENCES
1.
� 200
Lundberg, G. and Palmgren, A., Dynamic capacity of rolling bearings, Acta Polytech. Mech. Eng.
Ser. 1, Roy. Swed. Acad. Eng., 3(7), 1947.
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Lundberg, G. and Palmgren, A., Dynamic capacity of roller bearings, Acta Polytech. Mech. Eng.Ser. 2, Roy. Swed. Acad. Eng., 9(49), 1952.
3.
American National Standards Institute, American National Standard (ANSI/ABMA) Std. 9–1990,Load ratings and fatigue life for ball bearings, July 17, 1990.
4.
American National Standards Institute, American National Standard (ANSI/ABMA) Std.11–1990,Load ratings and fatigue life for roller bearings, July 17, 1990.
5.
International Organization for Standards, International Standard ISO 281, Rolling bearings—dynamic load ratings and rating life, 2006.
6.
Harris, T., Prediction of ball fatigue life in a ball/v-ring test rig, ASME Trans., J. Tribol., 119, 365–374, July 1997.
7.
Harris, T., How to compute the effects of preloaded bearings, Prod. Eng., 84–93, July 19, 1965.8.
Jones, A. and Harris, T., Analysis of rolling element idler gear bearing having a deformable outerrace structure, ASME Trans., J. Basic Eng., 273–277, June 1963.
9.
Harris, T. and Broschard, J., Analysis of an improved planetary gear transmission bearing, ASMETrans., J. Basic Eng., 457–462, September 1964.
10.
Harris, T., Optimizing the fatigue life of flexibly mounted, rolling bearings, Lubr. Eng., 420–428,October 1965.
11.
Harris, T., The effect of misalignment on the fatigue life of cylindrical roller bearings havingcrowned rolling members, ASME Trans., J. Lubr. Technol., 294–300, April 1969.
12.
Tallian, T., Sibley, L., and Valori, R., Elastohydrodynamic film effects on the load–life behavior ofrolling contacts, ASME Paper 65-LUBS-11, ASME Spring Lubr. Symp., NY, June 8, 1965.
13.
Skurka, J., Elastohydrodynamic lubrication of roller bearings, ASME Paper 69-LUB-18, 1969.14.
Tallian, T., Theory of partial elastohydrodynamic contacts, Wear, 21, 49–101, 1972.15.
Harris, T., The endurance of modern rolling bearings, AGMA Paper 269.01, Am. Gear Manufac.Assoc. Rol. Bear. Symp., Chicago, October 26, 1964.
16.
Bamberger, E., et al., Life Adjustment Factors for Ball and Roller Bearings, AMSE EngineeringDesign Guide, 1971.
17.
Schouten, M., Lebensduur van Overbrengingen, TH Eindhoven, November 10, 1976.18.
STLE, Life Factors for Rolling Bearings, E. Zaretsky, Ed., 1992.19.
Ville, F. and Nelias, D., Early fatigue failure due to dents in EHL contacts, Presented at the STLEAnnual Meeting, Detroit, May 17–21, 1998.
20.
Webster, M., Ioannides, E., and Sayles, R., The effect of topographical defects on the contactstress and fatigue life in rolling element bearings, Proc. 12th Leeds–Lyon Symp. Tribol., 207–226,
1986.
21.
Hamer, J., Sayles, R., and Ioannides E., Particle deformation and counterface damage whenrelatively soft particles are squashed between hard anvils, Tribol. Trans., 32(3), 281–288, 1989.
22.
Sayles, R., Hamer, J., and Ioannides, E., The effects of particulate contamination in rollingbearings—a state of the art review, Proc. Inst. Mech. Eng., 204, 29–36, 1990.
23.
Nelias, D. and Ville, F., Deterimental effects of dents on rolling contact fatigue, ASME Trans.,J. Tribol., 122, 1, 55–64, 2000.
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24.
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Xu, G., Sadeghi, F., and Hoeprich, M., Dent initiated spall formation in EHL rolling/sliding
contact, ASME Trans., J. Tribol., 120, 453–462, July 1998.
25.
Sayles, R. and MacPherson, P., Influence of wear debris on rolling contact fatigue, ASTM SpecialTechnical Publication 771, J. Hoo, Ed., 255–274, 1982.
26.
Tanaka, A., Furumura, K., and Ohkuna, T., Highly extended life of transmission bearings of‘‘sealed-clean’’ concept, SAE Technical Paper, 830570, 1983.
27.
Needelman, W. and Zaretsky, E., New equations show oil filtration effect on bearing life, Pow.Transmis. Des., 33(8), 65–68, 1991.
28.
Barnsby, R., et al., Life ratings for modern rolling bearings, ASME Paper 98-TRIB-57, presented atthe ASME/STLE Tribology Conference, Toronto, October 26, 1998.
29.
Tallian, T., On competing failure modes in rolling contact, ASLE Trans., 10, 418–439, 1967.30.
Voskamp, A., Material response to rolling contact loading, ASME Paper 84-TRIB-2, 1984.31.
Ioannides, E. and Harris, T., A new fatigue life model for rolling bearings, ASME Trans., J. Tribol.,107, 367–378, 1985.
32.
Harris, T. and McCool, J., On the accuracy of rolling bearing fatigue life prediction, ASME Trans.,J. Tribol., 118, 297–310, April 1996.
33.
Harris, T., Prediction of ball fatigue life in a ball/v-ring test rig, ASME Trans., J. Tribol., 119, 365–374, July 1997.
34.
Barnsby, R., et al., Life Ratings for Modern Rolling Bearings—A Design Guide for the Application ofInternational Standard ISO 281/2, ASME Publication TRIB-Vol 14, New York, 2003.
35.
Juvinall, R. and Marshek, K., Fundamentals of Machine Component Design, 2nd ed., Wiley, NewYork, 1991.
36.
Thomas, H. and Hoersch, V., Stresses due the pressure of one elastic solid upon another, Univ.Illinois, Bull., 212, July 15, 1930.
37.
Nelias, D., et al., Experimental and theoretical investigation of rolling contact fatigue of 52100 andM50 steels under EHL or micro-EHL conditions, ASME Trans., J. Tribol., 120, 184–190, April
1998.
38.
Ahmadi, N., et al., The interior stress field caused by tangential loading of a rectangular patch on anelastic half space, ASME Trans., J. Tribol., 109, 627–629, 1987.
39.
Ai, X. and Cheng, H., The influence of moving dent on point EHL contacts, Tribol. Trans., 37(2),323–335, 1994.
40.
Ioannides, E., Bergling, G., and Gabelli, A., An analytical formulation for the life of rollingbearings, Acta Polytech. Scand., Mech. Eng. Series No. 137, Finnish Institute of Technology, 1999.
41.
International Organization for Standards, International Standard ISO 4406, Hydraulic fluid power—fluids—method for coding level of contamination by solid particles, 1999.
42.
International Organization for Standards, International Standard ISO 4372, Hydraulic fluid power—filters—multi-pass method for evaluating filtration performance, 1981.
43.
SKF, General Catalog 4000 US, 2nd ed., 1997.44.
Nelias, D., Contribution a L’etude des Roulements, Dossier d’Habilitation a Diriger des Recherches,Laboratoire de Mecanique des Contacts, UMR-CNRS-INSA de Lyon No. 5514, December 16,
1999.
45.
American National Standards Institute, American National Standard (ABMA/ANSI) Std 7–1972,Shaft and Housing Fits for Metric Radial Ball and Roller Bearings (Except Tapered Roller
Bearings) 1972.
46.
American National Standards Institute, American National Standard (ABMA/ANSI) Std 19.1–1987, Tapered Roller Bearings-Radial, Metric Design, October 19, 1987.
47.
American National Standards Institute, American National Standard (ABMA/ANSI) Std 19.2–1994, Tapered Roller Bearings-Radial, Inch Design, May 12, 1994.
48.
Timoshenko, S., Strength of Materials, Part I, Elementary Theory and Problems, Van Nostrand,1955.
49.
Society of Automotive Engineers, Residual stress measurements by X-ray diffraction, SAE J784a,2nd ed., New York, 1971.
50.
Koistinen, D., The distribution of residual stresses in carburized cases and their origins, Trans.ASM, 50, 227–238, 1958.
6 by Taylor & Francis Group, LLC.
51.
� 200
Gentile, A., Jordan, E., and Martin, A., Phase transformations in high-carbon high-hardness steels
under contact loads, Trans. AIME, 233, 1085–1093, June 1965.
52.
Bush, J., Grube, W., and Robinson, G., Microstructural and residual stress changes in hardenedsteel due to rolling contact, Trans. ASM, 54, 390–412, 1961.
53.
Muro, H. and Tsushima, N., Microstructural, microhardness and residual stress changes due torolling contact, Wear, 15, 309–330, 1970.
54.
Voskamp, A., et al., Gradual changes in residual stress and microstructure during contact fatigue inball bearings, Metal. Tech., 14–21, January 1980.
55.
Zaretsky, E., Parker, R., and Anderson, W., A study of residual stress induced during rolling, J.Lub. Tech., 91, 314–319, 1969.
56.
Harris, T., Establishment of a new rolling bearing fatigue life calculation model, Final Report U.S.Navy Contract N00421-97-C-1069, February 23, 2002.
57.
Harris, T. and Yu, W.-K., Lundberg–Palmgren fatigue theory: considerations of failure stress andstressed volume, ASME Trans., J. Tribol., 121, 85–89, 1999.
58.
Bohmer, H.-J., et al., The influence of heat generation in the contact zone on bearing fatiguebehavior, ASME Trans., J. Tribol., 121, 462–467, July 1999.
59.
Palmgren, A., The service life of ball bearings, Zeitschrift des Vereines Deutscher Ingenieure, 68(14),339–341, 1924.
6 by Taylor & Francis Group, LLC.
9 Statically IndeterminateShaft–Bearing Systems
� 2006 by Taylor & Fran
LIST OF SYMBOLS
Symbol Description Units
a Distance to load point from right-hand bearing mm (in.)
A Distance between raceway groove curvature centers mm (in.)
D Rolling element diameter mm (in.)
dm Pitch diameter mm (in.)
Do Outside diameter of shaft mm (in.)
Di Inside diameter of shaft mm (in.)
E Modulus of elasticity MPa (psi)
F Bearing radial load N (lb)
f r/D
I Section moment of inertia mm4 (in.4)
K Load–deflection constant N/mmx (lb/in.x)
l Distance between bearing centers mm (in.)
M Bearing moment load N � mm (in. � lb)
P Applied load at a N (lb)
Q Rolling element load N (lb)
< Radius from bearing centerline to raceway groove center mm (in.)
T Applied moment load at a N � mm (in. � lb)
w Load per unit length N/mm (lb/in.)
x Distance along the shaft mm (in.)
y Deflection in the y direction mm (in.)
z Deflection in the z direction mm (in.)
a8 Free contact angle rad, 8g D cos�
dm
d Bearing radial deflection mm (in.)
u Bearing angular misalignment rad, 8Sr Curvature sum mm�1 (in.�1)
c Rolling element azimuth angle rad, 8
Subscripts
1, 2, 3 Bearing location
a Axial direction
h Bearing location
cis Group, LLC.
j Rolling elem ent locat ion
y y Directi on
z z Directi on
xy xy Plane
xz xz Plane
Sup erscript
k Appli ed load or moment
9.1 GENERAL
In some modern engineer ing ap plications of rolling bearing s, such as high-sp eed gas turbin es,
machi ne tool spindl es, and gyrosco pes, the bearing s must often be treated as integ ral to the
syst em to be able to accurat ely determ ine shaft defle ctions and dy namic shaft loading as well
as to ascertain the perfor mance of the bearing s. Chapt er 1 and Chapt er 3 detai l methods of
calcul ation of roll ing element load dist ribution for bearing s subjected to co mbinations of
radial , axial, and moment loading s. Thes e load dist ribut ions are affe cted by the shaft radial
and an gular defle ctions at the bearing . In this chap ter, equatio ns for the an alysis of bearing
loading as influenced by shaft de flections wi ll be developed.
9.2 TWO-BEARING SYSTEMS
9.2.1 RIGID SHAFT SYSTEMS
A commonl y us ed shaft–beari ng system involv es tw o angular -conta ct ball be arings or
tapere d roller be arings mounted in a back-t o-back arrange men t as illustr ated in Figure 9.1
and Figure 9.2. In these applic ations, the radial loads on the bea rings are general ly c alculated
independently using the statically determinate methods. It may be noticed from Figure 9.1
and Figure 9.2, however, that the point of application of each radial load occurs where the
line defining the contact angle intersects the bearing axis. Thus, it can be observed that a
back-to-back bearing mounting has a greater length between loading centers than does a face-
to-face mounting. This means that the bearing radial loads will tend to be less for the back-to-
back mounting.
Fal Fa2
Fr2
Pa
Frl
FIGURE 9.1 Rigid shaft mounted in back-to-back angular-contact ball bearings subjected to combined
radial and thrust loadings.
� 2006 by Taylor & Francis Group, LLC.
Fal Fa2
Fr2
Pa
Frl
FIGURE 9.2 Rigid shaft mounted in back-to-back tapered roller bearings subjected to combined radial
and thrust loadings.
The axial or thrust load carried by each be aring depends on the intern al load distribut ion
in the indivi dual bea ring. For simple thrust loading of the system, the method illustrated in
Example 9.3 may be app lied to determ ine the a xial loading in each bearing . When each
bearing must carry both radial and ax ial loads, althoug h the system is stat ically indete rmin-
ate, for systems in which the shaft may be consider ed rigid, a simplified method of an alysis
may be employ ed . In Chapter 11 of the fir st volume of this hand book, it is demonst rated that
a bearing subject ed to combined radial and axial loading may be consider ed to carry an
equival ent load de fined by
Fe ¼ XF r þ YF a ð 9: 1Þ
Loadin g fact ors X and Y are functions of the free co ntact angle, whi ch for this calcul ation is
assumed invariant with rolling element azimuth location and unaffected by applied load. This
cond ition is true for tapere d roll er bearing s; howeve r, as shown in Chapt er 1, it is onl y
approximated for ball bearings. Values for X and Y are usually provided for each ball bearing
and tapered roller bearing in manufacturers’ catalogs. Accordingly, assuming radial loads Fr1
and Fr2 are determined using statically determinate calculation methods, the bearing axial
loads Fa1 and Fa2 may be approximated considering the following conditions:
If load condition (1) is defined by
Fr2
Y2
<Fr1
Y1
and load condition (2) is defined by
Fr2
Y2
>Fr1
Y1
Pa �1
2
Fr2
Y2
� Fr1
Y1
� �
then,
Fa1 ¼Fr1
2Y1
ð9:2Þ
Fa2 ¼ Fa1 þ Pa ð9:3Þ
� 2006 by Taylor & Francis Group, LLC.
If load condition (3) is defined by
Fr2
Y2
>Fr1
Y1
Pa <1
2
Fr2
Y2
� Fr1
Y1
� �
then,
Fa2 ¼Fr2
2Y2
ð9:4Þ
Fa1 ¼ Fa2 � Pa ð9:5Þ
See Example 9.1 and Example 9.2.
9.2.2 FLEXIBLE SHAFT SYSTEMS
In the more general two-bearing shaft system, flexure of the shaft induces moment loads Mh at
non-self-aligning bearing supports in addition to the radial loads Fh. This loading system,
illustrated in Figure 9.3, is statically indeterminate in that there are four unknowns: F1, F2,
M1, and M2; but, only two static equilibrium equations. For example,
XF ¼ 0 F1 þ F2 � P ¼ 0 ð9:6ÞX
M ¼ 0 F1l �M1 þ T � P l � að Þ þM2 ¼ 0 ð9:7Þ
Considering the bending of the shaft, the bending moment at any section is given as follows:
EId2y
dx2¼ �M ð9:8Þ
where E is the modulus of elasticity, I is the shaft cross-section moment of inertia, and y is the
shaft deflection at the section. For shafts that have circular cross-sections,
I ¼ p
64ðD4
o � D4i Þ ð9:9Þ
1
F1 F2
2
T k
P ka k
l
FIGURE 9.3 Statically indeterminate two-bearing shaft system.
� 2006 by Taylor & Francis Group, LLC.
V
1
F1
x
FIGURE 9.4 Statically indeterminate two-bearing shaft system forces and moments acting on a section
to the left of the load application point.
For a cross-sect ion at 0 � x � a illustrated in Fi gure 9.4,
EId2 y
d x2 ¼ �F 1 x þ M 1 ð 9: 10 Þ
Integr ating Equat ion 9.10 yields
EIdy
dx ¼ �F1 x
2
2þ M1 x þ C 1 ð 9: 11 Þ
Integr ating Equat ion 9.11 yields
EIy ¼ �F1 x3
6þM1 x
2
2þ C1 x þ C 2 ð 9: 12 Þ
In Equat ion 9.11 an d Equation 9.12, C1 and C 2 are c onstants of integ ration . At x ¼ 0, the
shaft assum es the bearing de flection dr1 . Als o at x ¼ 0, the shaft assum es a slope u1 in
accordan ce wi th the resistance of the bearing to mo ment loading ; hence,
C1 ¼ EI u1
C2 ¼ EI dr1
Therefor e, Equation 9.11 an d Equation 9.12 be come
EIdy
dx ¼ �F1 x
2
2þ M1 x þ EI u1 ð 9: 13 Þ
and
EIy ¼ �F1 x3
6þM1 x
3
2þ EI u1 x þ EI dr1 ð 9: 14 Þ
For a cross-sect ion at a � x � l as shown in Figure 9.5,
EId2y
dx2¼ �F1xþM1 þ Pðx� aÞ � T ð9:15Þ
� 2006 by Taylor & Francis Group, LLC.
m1
F1
a k
P k
T k
V
x
FIGURE 9.5 Statically indeterminate two-bearing shaft system forces and moments acting on a section
to the right of the load application.
Integr ating Equat ion 9.15 twice yields
EIdy
dx ¼ �F1 x
2
2þ ðM1 � T Þx þ Px
x
2 � a
� �þ C3 ð9: 16 Þ
EIy ¼ �F1 x3
6þ ðM1 � T Þ x
2
2þ Px 2
x
6 � a
2
� �þ C3 x þ C 4 ð9: 17 Þ
At x ¼ l, the slope of the sh aft is u2 and the de flection is dr2 , theref ore,
EIdy
dx ¼ F1 ðl 2 � x 2 Þ
2þ ðT � M1 Þð l � xÞ þ P
2 ½ xð x � 2aÞ � l ðl � 2aÞ� þ EI u2 ð9: 18 Þ
EI y ¼� F1
6½ l 2 ð 2l � 3x Þ þ x3 � þ ð M1 � T Þ
2ð l � x Þ 2 þ P
6 ½x 2 ð x � 3aÞ
� l 2 ð 3x þ 3a � 2l Þ þ 6xla � þ EI ½dr2 � u2 ð l � xÞ� ð9: 19 Þ
At x ¼ a, singul ar co nditions of slope and defle ction occur. Ther efore at x ¼ a, Equat ion 9.13
and Equat ion 9.18 are equ ivalent as are Equat ion 9.14 and Equat ion 9.19. Solving the
resul tant sim ultaneo us eq uations yiel ds
F1 ¼P ðl � aÞ 2 ð l þ 2aÞ
l 3 � 6 Tað l � aÞ
l 3 � 6EI
l 2u1 þ u2 þ
2ðdr1 � dr2 Þl
� �ð9: 20 Þ
M1 ¼Pa ð l � aÞ 2
l 2þ T ð l � aÞð l � 3aÞ
l 2 � 2EI
l2u1 þ u2 þ
3ðdr1 � dr2 Þl
� �ð9: 21 Þ
Subs tituting Equat ion 9.20 and Equat ion 9.21 in Equation 9.6 an d Equation 9.7 yiel ds
F2 ¼Pa2ð3l � 2aÞ
l3þ 6Taðl � aÞ
l3þ 6EI
l2u1 þ u2 þ
2ðdr1 � dr2Þl
� �ð9:22Þ
M2 ¼Pa2ðl � aÞ
l2þ Tað2l � 3aÞ
l2þ 2EI
lu1 þ 2u2 þ
3ðdr1 � dr2Þl
� �ð9:23Þ
� 2006 by Taylor & Francis Group, LLC.
In Equat ion 9.20 through Equation 9.23, slope u1 and dr1 are co nsidered posit ive a nd the signs
of u2 an d dr2 may be determ ined from Equat ion 9.18 and Equat ion 9.19. The relative
magni tudes of P and T an d their directions will de termine the sense of the shaft slopes at
the bearing s. To determ ine the reaction s, it is necessa ry to develop eq uations relating bearing
misali gnment an gles uh to the mis aligning moment s M h and bearing radial deflections drh to
loads Fh. Thi s may be done by using the data of Chapte r 1 and Chapt er 3.
When the be arings are consider ed as axial ly free pin supp orts, Equat ion 9.20 and Equa-
tion 9 .22 are identical to Equat ion 4.29 an d Equation 4.30, given in the fir st volume of this
hand book for a statical ly determ inate syst em. That format is obtaine d by setting Mh ¼ dr h ¼ 0
and solving Equation 9.21 and Equat ion 9.23 sim ultane ously for u1 an d u2. Subs titution of
these values in Equat ion 9.20 and Equat ion 9.22 pro duces the resultant equati ons. If the sha ft
is very flexible and the bearing s are very rigi d wi th regard to mis alignment, then u1 ¼ u2 ¼ 0.
This substitut ion in Equat ion 9 .20 through Equation 9.23 yields the class ical solut ion for a
beam with both ends built in. The various types of tw o-bearing supp ort may be examin ed by
using Equation 9.20 through Equat ion 9.23. If more than one load or torque is applie d
between the supp orts, then by the princi ple of superposi tion
F1 ¼1
l 3
Xk¼ n
k¼ 1
P k l � ak� 2
l þ 2ak�
� 6
l 3
Xk¼ n
k¼ 1
T k ak l � ak�
� 6EI
l 2u1 þ u2 þ
2 dr1 � dr2ð Þl
� �ð 9: 24 Þ
M1 ¼1
l 3
Xk ¼ n
k ¼ 1
P k ak l � ak� 2þ 1
l 2
Xk¼ n
k¼ 1
T k l � ak�
l � 3ak�
� 2EI
l2u1 þ u2 þ
3 dr1 � dr2ð Þl
� �ð 9: 25 Þ
F2 ¼1
l 3
Xk ¼ n
k ¼ 1
Pk ak� 2
3l � 2ak�
þ 6
l 3
Xk ¼ n
k ¼ 1
T k ak l � ak�
þ 6EI
l 2u1 þ u2 þ
2 dr1 � dr2ð Þl
� �ð 9: 26 Þ
M2 ¼1
l 3
Xk ¼ n
k ¼ 1
Pk ak� 2
l � ak�
þ 1
l 2
Xk¼ n
k¼ 1
T k ak 2l � 3ak�
þ 2EI
lu1 þ 2u2 þ
3 dr1 � dr2ð Þl
� �ð 9: 27 Þ
See Exampl e 9.3 .
9.3 THREE-BEARING SYSTEMS
9.3.1 R IGID SHAFT S YSTEMS
When the shaft is rigid and the dist ance between bearing s is smal l, the infl uence of the shaft
defle ction on the distribut ion of loading among the be arings may be neglected . An ap plica-
tion of this kind is ill ustrated in Fig ure 9.6.
In this system, the angular -conta ct ball bearing s are consider ed as one double-r ow bearing .
The thrust load acti ng on the doubl e-row bearing is the thrust load Pa app lied by the be vel gear.
To ca lculate the magnitud e of the radial loads Fr and Fr3 , the effecti ve point of app lication of Fr
must be determined . Fr ac ts at the center of the doubl e-row bearing only if P a ¼ 0. If a thrust
load e xists, the line of ac tion of Fr is displaced toward the pressur e center of the rolling elem ent
row that supp orts the thrust load. This displ acement may be neglected only if the dista nce
l betw een the center of the doubl e-row ball bearing set and the roll er bearing is large co mpared
with the dist ance b. Using the X and Y factors (see Equation 9.1), pertaining to the single-
row bearing s, Figure 9.7 gives the relative dista nce b1/b as a function of the parameter FaY/Fr
(1�X). The X and Y factors for the load condition Fa/Fr> e must be selected from the bearing
catalog.
� 2006 by Taylor & Francis Group, LLC.
Fr3
l1
Pr
Fr
b1
rmp
Pa
l
o
b
FIGURE 9.6 Example of three-bearing shaft system with a rigid shaft.
See Exam ple 9.4.
9.3.2 NONRIGID SHAFT SYSTEMS
The gen eralized loading of a three-beari ng-shaf t supp ort system is illustr ated in Figure 9.8a.
This syst em may be red uced to the two syst ems of Figure 9.8b a nd analyze d accordi ng to
the methods given previou sly for a two-bear ing nonrigi d shaft system provided that
F 02 þ F 002 ¼ F2 ð9: 28 Þ
M 02 � M
00 2 ¼ M 2 ð9: 29 Þ
Hence, from Equat ion 9.24 through Equat ion 9.27,
F1 ¼1
l 31
Xk ¼ n
k ¼ 1
Pk1 ð l1 � ak
1 Þ 2 ð l 1 þ 2ak
1 Þ �6
l 31
Xk ¼ n
k ¼ 1
T k1 ak1 ð l 1 � ak
1 Þ �6El1
l 21u1 þ u2 þ
3ðdr1 � dr2 Þl1
� �ð9: 30 Þ
0.5
0.4
0.3
0.2
0.1
00 0.40.30.20.1 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4
FaY
Fr (1 − X)
b1
b
FIGURE 9.7 b1/b vs. FaY/Fr (1�X) for the double-row bearing in a three-bearing rigid shaft system.
� 2006 by Taylor & Francis Group, LLC.
a k1
a k1
a k2
a k2
P k2T k
2
P k1
P k1
P k2
T k1
21
F �2F3
F1
F1
l1
l1
l2
l2F3F2
F �2
3
T k2
(a)
(b)
1
�2
�2
3
FIGURE 9.8 (a) Three-bearing shaft system; (b) equivalent two-bearing shaft system.
M1 ¼1
l21
Xk¼n
k¼1
Pk1a
k1ðl1 � ak
1Þ2 þ 1
l21
Xk¼n
k¼1
Tk1 ðl1 � ak
1Þðl1 � 3ak1Þ �
2EI1
l12u1 þ u2 þ
3ðdr1 � dr2Þl1
� �
ð9:31Þ
F2 ¼1
l31
Xk¼n
k¼1
Pk1ðak
1Þ2ð3l1 � 2ak
1Þ þ1
l32
Xk¼n
k¼1
Pk2ðl2 � ak
2Þðl2 þ 2ak2Þ
þ 6
l31
Xk¼n
k¼1
Tk1 ak
1ðl1 � ak1Þ �
6
l32
Xk¼n
k¼1
Tk2 ak
2ðl2 � ak2Þ
þ 6EI1
l21ðu1 þ u2Þ �
I2
l22ðu2 þ u3Þ
� �
þ 12EI1
l31ðdr1 � dr2Þ �
I2
l32ðdr2 � dr3Þ
� �ð9:32Þ
� 2006 by Taylor & Francis Group, LLC.
M2 ¼1
l 21
Xk ¼ n
k ¼ 1
Pk1 ð ak
1 Þ 2 ð l 1 � ak
1 Þ �1
l 22
Xk¼ n
k¼ 1
P k2 ak2 ð l 2 � ak
2 Þ
þ 1
l 21
Xk¼ n
k¼ 1
T k1 ak1 ð 2l1 � 3ak
1 Þ �1
l 22
Xk ¼ n
k ¼ 1
T k2 ð l2 � ak2 Þð l2 � 3ak
2 Þ
þ 2EI1
l1ðu1 þ 2u2 Þ þ
I2
l2ð 2u2 þ u3 Þ
� �
þ 6EI1
l 21ðdr1 � dr2 Þ þ
I2
l 22ðdr2 � dr3 Þ
� �ð9: 33 Þ
F3 ¼1
l 32
Xk ¼ n
k ¼ 1
Pk2 ðak
2 Þ 2 ð 3l 2 � 2ak
2 Þ þ6
l 32
Xk¼ n
k¼ 1
T k2 ak2 ð l 2 � ak
2 Þ þ6EI2
l 22u2 þ u3 þ
2ðdr2 � dr3 Þl2
� �ð9: 34 Þ
M3 ¼1
l 22
Xk ¼ n
k ¼ 1
Pk2 ð ak
2 Þ 2 ð l2 � ak
2 Þ þ1
l 22
Xk¼ n
k¼ 1
T k2 ak2 ð 2l 2 � 3ak
2 Þ þ2EI2
l2u2 þ 2u3 þ
3ðdr2 � dr3 Þl2
� �
ð9: 35 Þ
An exampl e of the util ity of the general ized eq uations Equat ion 9.30 through Equation 9.35 is
the system illustra ted in Figure 9.9. For that syst em, it is assum ed that moment loads are zero
and that the differences betwe en bearing radial defle ctions are negli gibly smal l. Hence,
Equat ion 9 .30 through Equat ion 9.35 become
F1 ¼Pð l1 � aÞ2ðl1 þ 2aÞ
l31� 6EI
l21ðu1 þ u2Þ ð9:36Þ
2u1 þ u2 ¼Paðl1 � aÞ2
2EIl1ð9:37Þ
F2 ¼Pa2ð3l1 � 2aÞ
l31þ 6EI
ðu1 þ u2Þl21
� ðu2 þ u3Þl22
� �ð9:38Þ
ðu1 þ 2u2Þl1
þ ð2u2 þ u3Þl2
¼ �Pa2ðl1 � aÞ2EIl31
ð9:39Þ
F3 ¼6EIðu2 þ u3Þ
l22ð9:40Þ
aP
F1 F2 F3l1 l2
FIGURE 9.9 Simple three-bearing shaft system.
� 2006 by Taylor & Francis Group, LLC.
u2 þ 2u3 ¼ 0 ð 9: 41 Þ
Equation 9.37, Equat ion 9.39, and Equat ion 9.4 can be solved for u1, u2, and u3. Subs equent
substitut ion of these v alues in Equation 9 .36, Equat ion 9.38, and Equat ion 9.40 yiel ds the
followi ng result:
F1 ¼Pð l1 � aÞ½2l 1 ð l 1 þ l 2 Þ � að l 1 þ aÞ�
2l 21 ð l 1 þ l 2 Þð 9: 42 Þ
F2 ¼Pa ½ðl1 þ l 2 Þ 2 � a2 � l 22 �
2l 21 l 2ð 9: 43 Þ
F3 ¼� Pað l 21 � a2 Þ2 l1 l2 ð l1 þ l 2 Þ
ð 9: 44 Þ
9.3.2 .1 Rigi d Shafts
When the dist ances between bearing s are smal l or the shaft is otherwis e very stiff, the bearing
radial defle ctions determine the load distribut ion among the bearing s. From Figure 9 .10, it
can be seen that by consider ing simila r trian gles
dr1 � dr2
l1¼ dr2 � dr3
l2ð 9: 45 Þ
This iden tical relationsh ip can be obtaine d from Equation 9.30 through Equat ion 9.35 by
setting sh aft cross- section moment of inert ia I to an infinite ly large value. For a radially
loaded be aring wi th rigid rings , the maxi mum rolling elem ent load is directly propo rtional to
the applied radial load Fr, and the maximum rolling element deflection determines the bearing
radial deflection. Since rolling element load Q¼Kdn, therefore,
Fr ¼ Kdnr ð9:46Þ
Rearranging Equation 9.46,
dr ¼Fr
K
� �1=n
ð9:47Þ
l1
dr1 dr2 dr3
l2
⎣C
FIGURE 9.10 Deflection of a three-bearing shaft system with a rigid shaft.
� 2006 by Taylor & Francis Group, LLC.
Subs titution of Equat ion 9.47 in Equat ion 9.45 yiel ds
Fr1
K1
� �1= n
� Fr2
K2
� �1 =n
¼ l1
l2
Fr2
K2
� �1 = n
� Fr3
K3
� �1= n" #
ð9: 48 Þ
Equat ion 9.48 is vali d for bearing s that suppo rt a radial load only. More c omplex relation-
ships are requ ired in the presence of simu ltaneo us applie d thrust and moment loading .
Equat ion 9.48 can be solved sim ultane ously with the equ ilibrium equati ons to yiel d values
of Fr1 , F r2 , and Fr3 .
See Exam ple 9.5.
9.4 MULTIPLE-BEARING SYSTEMS
Equat ion 9.30 throu gh Equat ion 9.35 may be used to determine the bearing reactions in a
mult iple-beari ng syst em such as that sho wn in Figure 9.11 with a flex ible shaft. It is evident
that the react ion at any bearing support location h is a function of the loading exist ing at and
in between the bearing suppo rts locat ed at h � 1 and h þ 1. Therefor e, from Equation 9.30
through Equat ion 9.35, the react ive loads at each support location h are g iven as follows :
Fh ¼1
l 3h � 1
Xk ¼ p
k ¼ 1
Pkh � 1 ð ak
h � 1 Þ 2 ð 3l h � 1 � 2ah � 1 Þ
þ 1
l 3h
Xk ¼ q
k ¼ 1
Pkh ð l h � ak
h Þ 2 ð lh þ 2ak
h Þ
þ 6
l3h�1
Xk¼r
k¼1
Tkh�1a
kh�1ðlh�1 � ak
h�1Þ
� 6
l3h
Xk¼s
k¼1
Tkh ak
hðlh � akhÞ
þ 6EIh�1
l2h�1
uh�1 þ uh þ2
lh�1
ðdr;h�1 � dr;hÞ� �
� Ih
l2huh þ uhþ1 þ
2
lhðdr;h � dr;hþ1Þ
� ��
ð9:49Þ
Ph −1 Ph
Th −1 Th
h −1
Fh −1
lh −1 lh
h +1
Fh +1Fh
h
ah −1 ah k k
k
k
k
FIGURE 9.11 Multiple-bearing shaft system.
� 2006 by Taylor & Francis Group, LLC.
Mh ¼1
l 2h� 1
Xk¼ p
k¼ 1
P kh� 1 ð akh� 1 Þ
2 ð lh � 1 � akh � 1 Þ �
1
l 2h
Xk¼ q
k¼ 1
P kh akh ð l h � ak
h Þ 2
þ 1
l 2h � 1
Xk ¼ r
k ¼ 1
T kh � 1 akh � 1 ð 2l h� 1 � 3ak
h� 1 Þ
� 1
l 2h
Xk¼ s
k¼ 1
T kh ð l h � akh Þð l h � 3ak
h Þ
þ 2EIh � 1
lh� 1
uh � 1 þ 2uh þ3
lh� 1
ður; h� 1 � ur; h Þ� �
þ Ih
lh2uh þ uh þ 1 þ
3
lhðdr ;h � d r;h þ 1 Þ
� ��
ð 9: 50 Þ
For a shaft –bearin g system of n sup ports, that is, h ¼ n, Equat ion 9.49 an d Equat ion 9.50
repres ent a system of 2 n equati ons. In the most elementary case, all be arings are consider ed as
suffici ently self-al igning such that all Mh equal zero; furt hermor e, all dr, h are con sidered
negli gible compared wi th shaft defle ction. Equation 9.49 and Equat ion 9.50 thereby degen-
erate to the familiar equati on of ‘‘th ree momen ts.’’
It is evident that the so lution of Equation 9.49 and Equat ion 9.50 to obt ain bearing
react ions Mh and Fh depen ds on relationsh ips between radial load and rad ial de flection an d
moment load and mis alignment angle for each radial bearing in the syst em. Thes e relation-
ships hav e been define d in Chapt er 1 and Chapt er 3. Thus, for a very sophist icated solution to
a shaft–beari ng problem as illustrated in Figu re 9.12 one co uld co nsider a shaft that has tw o
degrees of freedom with regard to be nding, that is, defle ction in two of three princi pal
directions, sup ported by bea rings h a nd accomm oda ting load s k. At each bearing locat ion
h, one must establis h the following relation ships:
dy, h ¼ f1 ð Fx, h , Fy , h , Fz , h , M xy, h,Mxz,hÞ ð9:51Þ
dz,h ¼ f2ðFx,h,Fy,h,Fz,h,Mxy,h,Mxz,hÞ ð9:52Þ
uxy,h ¼ f3ðFx,h,Fy,h,Fz,h,Mxy,h,Mxz,hÞ ð9:53Þ
Bearing location Bearing locationh h + 1
z
y
x
Fz,h Py,h
Pz,h
Fz,h +1
Fy,h Fy,h +1
xy,hTxz,h
xy,h +1
xz,h
Txy,h
xz,h +1
k
k
k
k
FIGURE 9.12 System loading in three dimensions.
� 2006 by Taylor & Francis Group, LLC.
uxz;h ¼ f4ðFx;h;Fy;h;Fz;h;Mxy;h;Mxz;hÞ ð9:54Þ
To accommodate the movement of the shaft in two principal directions, the following
expressions will replace Equation 3.72 and Equation 3.73 for each ball bearing (see Ref. [1]):
Sxj ¼ BD sin a� þ dx þ uxz<i sin cj þ uxy<i cos cj ð9:55Þ
Szj ¼ BD cos a� þ dy sin cj þ dj cos cj ð9:56Þ
9.5 CLOSURE
For most rolling bearing applications, it is sufficient to consider the shaft and housing as rigid
structures. As demonstrated in Example 9.3, however, when the shaft is considerably hollow
and the span between bearing supports is sufficiently great, the shaft bending characteristics
cannot be considered separately from the bearing deflection characteristics with the expect-
ation of accurately ascertaining the bearing loads or the overall system deflection character-
istics. In practice, the bearings may be stiffer than might be anticipated by the simple
deflection formulas or even stiffer than a more elegant solution that employs accurate
evaluation of load distribution might predict for the assumed loading. The penalty for
increased stiffness will be paid in shortened bearing life since the improved stiffness is
obtained at the expense of induced moment loading.
It is of interest to note that the accurate determination of bearing loading in integral
shaft–bearing–housing systems involves the solution of many simultaneous equations. For
example, in a high-speed shaft supported by three ball bearings, each of which has a
complement of 10 balls, the shaft being loaded so as to cause each bearing to experience
five degrees of freedom in deflection requires the solution of 142 simultaneous equations,
most of which are nonlinear in the variables to be determined. Most likely, the system would
include some roller bearings, these having complements of 20 or more rollers per row, thus
adding to the number of equations to be solved simultaneously. Furthermore, the bearing
outer rings and inner rings may be flexibly supported as in aircraft power transmissions,
adding to the complexity of the analytical system and the difficulty of obtaining a solution
using numerical analysis techniques such as the Newton–Raphson method for simultaneous,
nonlinear equations.
REFERENCE
1.
� 2
Jones, A., A general theory of elastically constrained ball and radial roller bearings under arbitrary
load and speed conditions, ASME Trans., J. Basic Eng., 82, 309–320, 1960.
006 by Taylor & Francis Group, LLC.
10 Failure and Damage Modesin Rolling Bearings
� 2006 by Taylor & Francis Grou
10.1 GENERAL
Although ball and roller bearings appear to be relatively simple mechanisms, their internal
operations are relatively complex as witnessed by the number of pages devoted by Rolling
Bearing Analysis, 5th Ed. to the evaluation of their design and operation. It has been
established in these pages that rolling bearings can perform in many applications without
interruption of successful operation, provided:
. The bearing selected for the given application is of correct design and sufficient size.
. The bearing is properly mounted on the shaft and in the housing.
. The bearing lubrication system is of proper design; the lubricant film thicknesses
generated are sufficient to adequately separate the rolling contact surfaces; and the
amount of lubricant supplied is sufficient.. Lubrication of rolling element–cage and cage–bearing ring land interfaces is adequate.. The bearing is operated at speeds consistent with the lubrication method such that
overheating is prevented.. The bearing is protected from the ingress of contaminants.
It has also been established that, in many applications, it is possible to accommodate these
conditions.
In some applications, however, the conditions for application design functional perform-
ance and endurance are not met due to:
. Extreme operating conditions of heavy or complex loading, very high speed or
accelerations, and very high or very low operating temperatures to cite a few
and, perhaps,
. Insufficient attention to proper machinery assembly and operating practice.
Operation under such conditions will very frequently lead to early bearing failure, and
possibly early machinery failure.
As implied above, rolling bearing failure can be defined as not meeting the design
requirements of the application. Thus, failure can manifest itself as:
1. Excessive deflection
2. Excessive vibration or noise
p, LLC.
� 20
3. Unaccept ably high friction torque and tempe ratur e, or
4. Bearing seizur e
Actuall y, co nditions 1–3, singly or in combination , may lead to the last. Very likely,
con ditions 1–3 are the result of da mage to the rolling c ontact surfaces. The likelihood of such
damage in a given ap plication can, in the best circumsta nce, be pred icted and consequen tly
avoided using the an alytical techni ques contai ned in this text . In the wors t inst ance, the
reason( s) for early failure can be found through such an alyses.
The purpose of this cha pter is to e lucidate the v arious types of damage an d failure that
may occur in rolling bearing ap plications an d to connect these to the physica l phen omena
that c ause them.
10.2 BEARING FAILURE DUE TO FAULTY LUBRICATION
10.2.1 INTERRUPTION OF L UBRICANT SUPPLY TO BEARINGS
Mo st ba ll and roller bearing failu res are ca used by inter ruptio n of the lubri cant supply to the
bearing or inade quate delivery of the lubri cant to the rolling elem ent–ra ceway co ntacts in the
first place. In the aircraft gas turbine engine mainsh aft app lication, in whi ch engine failu re is
con sidered life-cr itical, ba ll and roll er bearing cages are co ated with silver. In the even t of
tempor ary loss of lubri cant supplied to the bearing s, some sil ver is trans ferred to the rolling
elem ent su rfaces, pr oviding increa sed lubri city and lower coeff icient of fri ction than steel-on-
steel. Also, in the latter instan ce, bearing s that hav e sil icon nitr ide rolling elemen ts experi ence
low er fricti on in both the roll ing elem ent–ra ceway and rolling element–cag e contact s than do
bearing s that have steel roll ing elem ents.
10.2.2 T HERMAL IMBALANCE
Duri ng ope ration of ball and roller bearing s, it is impor tant that the tempe ratur e gradie nt
betw een the bearing inn er an d outer racew ays is maint ained such that radial preloadi ng does
not oc cur. This con dition leads to increased rolling elem ent–ra ceway loading , increa sed
fricti on, and increa sed tempe ratures. If the rate of heat dissi pation from the bearing outer
ring is greater than that from the inner ring, a tempe ratur e excursi on occurs, resul ting in
bearing seizur e. Heat generat ion in other compo nents of an app lication is frequent ly greater
than that generat ed by bearing operati on; for exa mple, the heat generat ed by the windings in
an electric motor . In this case, it is impor tant that the paths for heat trans fer are designe d
such that the tempe ratur e gradie nt across the bearing does not resul t in a therma l excu rsion.
High frictio n is also caused by excess ive a mounts of sliding in a bearing . This cond ition
can occur as a result of rolling–ra cewa y co ntacts that operate in the bounda ry lubricati on
regim e. In other words , the lubri cant film thickne sses formed in the roll ing elem ent–racewa y
con tacts do not suffici ently sepa rate the rolling/ sliding compon ents, allowi ng the interacti on
of surface asperi ties on the c ontacting bodies. High fricti on also oc curs in solid-film -lubr i-
cated bearing s; for exampl e, bearing s lubricated with molybdenum disul fide.
The first stage of excess ive frictio n heat gen eration is lubri cant ox idation and deg radation.
In this case, the lubri cant changes to darker colors , e ventually beco ming black an d having
even greater fri ction; see Figu re 10.1. Lubr icant overheat ing and oxidation can also lead to
chemi cal deposit s on, and discol oration of, rolling elem ents as shown in Fi gure 10.2 as wel l as
rings as illustr ated in Figure 10.3 a nd cage in Figure 10.4.
As the bearing component temperatures increase, the hardness of bearing ring and rolling
element steels decreases, giving rise to loss of elasticity and resulting in plastic deformations
06 by Taylor & Francis Group, LLC.
FIGURE 10.1 Grease-lubricated ball bearing showing lubricant oxidation. (Courtesy NTN.)
(see Figure 10 .5 through Figu re 10.7). Ult imately , heat imbal ance failu re leads to break age of
bearing compon ents and be aring seizur e as illu strated in Figure 10.8 through Figu re 10.10.
Bearing seizur e is obv iously a co mplete loss of be aring functi on and , most likely, machi nery
functio n. This type of failu re can be catas trophi c in life-critical ap plications ; for exampl e,
automobi le wheel bearing s an d he licopter power trans mission bearing s to na me a few.
FIGURE 10.2 Hard organic coating on balls formed by grease polymerization due to high temperature
caused by sliding under high contact stresses.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.3 Discoloration and oxidation of bearing ring due to overheating of bearing during
operation.
10.3 FRACTURE OF BEARING RINGS DUE TO FRETTING
For applications involving shaft rotation, bearing inner rings are usually press-fitted or
shrink-fitted on the shaft to prevent ring rotation about the shaft due to operation under
applied loading. For outer ring rotation applications such as automobile wheel bearings, the
bearing outer ring is usually mounted in its housing with an interference fit to prevent ring
rotation relative to the housing during bearing operation. The inner ring rotation about the
shaft or the outer ring rotation relative to the housing is generally a small intermittent motion
occasioned by the circumferential spacing of the rolling elements. If the interference fitting is
insufficient to prevent this motion, a condition called fretting occurs. Fretting is a chemical
FIGURE 10.4 Machine tool ball bearing phenolic cage: (a) original color and (b) discolored and
oxidized due to overheating of bearing during operation.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.5 Tapered roller bearing—deformed cone, rollers, and cage due to heat imbalance failure.
attack on the surfa ces in relat ive moti on, and it entai ls local ized remova l of material
called fretti ng co rrosion or fretting wear. Figure 10.10 and Fig ure 10.11 sho w bearing rings
with fretti ng. This corrosi on or wear can resul t in ring cracki ng as shown in Figure 10.12.
Hence, frettin g co rrosion on bearing ring surfa ces can lead to loss of bearing function an d
potential catastrophic failure.
FIGURE 10.6 Spherical roller bearing—deformed inner raceways and rollers due to heat imbalance
failure.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.7 Cylindrical roller bearing—transformation of rollers into balls due to heat imbalance
failure.
FIGURE 10.8 Cylindrical roller bearing—breakage of cage due to heat imbalance failure.
FIGURE 10.9 Deep-groove ball bearing—breakage of cage and balls due to heat imbalance failure.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.10 Fretting corrosion in the bore of a bearing inner ring.
10.4 BEARING FAILURE DUE TO EXCESSIVE THRUST LOADING
Excessi ve thrust loading in a ball be aring can cause the balls to ride over the ring land as
shown in Fig ure 10.13. This causes the raceway area to be truncat ed resulting in much higher
contact stre ss, much higher surfa ce fricti on shear stre ss, and rapid bearing failure due to
FIGURE 10.11 Fretting corrosion on the outside diameter of a bearing outer ring.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.12 Cracking of a ball bearing outer ring due to fretting corrosion.
overhe ating. Figu re 10.14 provides a postm ortem view of the inner and outer racew ay
patte rns.
Excessi ve thrust loading in tapere d roller and sph erical roll er be arings resul ts in greatly
magni fied roll er–rac eway loading and early subsurf ace-in itiated fatigu e failu re (see late r
sectio ns).
10.5 BEARING FAILURE DUE TO CAGE FRACTURE
In Chapter 7 of the first volume of this hand book, it was indica ted that interfer ence fitti ng
of the inner ring on the shaft or the outer ring in the housing in a radial ball or roller
FIGURE 10.13 Ball bearing inner ring with rolled over left side land due to very heavy applied thrust
loading.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.14 Postmortem diagram of inner and outer raceway in a ball bearing operated with
excessive thrust loading.
bearing resul ts in the loss of radial clear ance. Also, if the inner racew ay runs hotter than the
outer raceway , then radial cleara nce is also reduced . If radial clear ance is lost co mpletely
and radial interfer ence oc curs during bearing ope ration, loading between the bea ring cage
and the roll ing elements may become excess ive and cause break age of the cag e. This is
illustr ated in Figure 10.15 and Figure 10.16. In the even t of ca ge fract ure, fragmen ts may
break off an d wedge thems elves betw een the roll ing e lements and raceways, causing
increa sed frictio n, ov erheat ing, and be aring seizur e. This can be a catas trophic-t ype failure.
Postmor tem examin ation of the bearing raceway s in such a case woul d reveal that the inner
raceway was somewhat wid er than the de sign, an d the outer raceway extend s a complet e
360 8 as sh own in Figure 10.17. Thi s ind icates excess ive radial preload as the cause of bearing
failure.
Cage fract ure can also occ ur due to excess ive mis alignment in the bea ring dur ing oper-
ation. This places high fore-an d-aft axial loading on the cage causing the breakage. The
postm ortem loading patte rns of the racew ays are sho wn in Figure 10.18.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.15 Fractured steel cages in deep-groove ball bearings: (a) ribbon-type cage and (b) ma-
chined and riveted cage.
FIGURE 10.16 Fractured machined brass in a double-row cylindrical roller bearing.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.17 Postmortem loading patterns of a deep-groove ball bearing inner and outer raceways
indicating heavy radial preloading that occurred in the bearing.
10.6 INCIPIENT FAILURE DUE TO PITTING OR INDENTATION OF THEROLLING CONTACT SURFACES
10.6.1 C ORROSION PITTING
Opera tion of a prop erly operati ng roll ing bearing entails only a small amount of frictio n
torque. As implied in Section 10.2. 2, ap plication design mu st be such as to accomm oda te the
dissipati on of bot h the heat g enerated by the applica tion and the friction heat generat ed by
the bearing s withou t signi fican t tempe ratur e rise. It was shown in Chapter 2 that, in most
rolling elem ent–ra ceway contact s, a combinat ion of ro lling and sliding motio ns oc curs. It was
furth er shown that sli ding motio n is the major cau se of rolling bearing fri ction. Inter ruptio n
of the rolling contact surfa ces by corrosi on pits or inde ntations exacerba tes this cond ition.
Figure 10.19 a nd Figure 10.20 illustrate co rrosion pitting and oxidat ion (rusting) of roll ing
contact surfaces.
Figure 10.21 demo nstrates the co rrosion of a tapere d roller bearing cone racew ay due to
mois ture in the lubrica nt. Eac h of these co nditions rep resents an interrup tion in the smoot h
surface of the rolling contact surfaces.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.18 Postmortem loading patterns of a deep-groove ball bearing inner and outer raceways
indicating significant misalignment that occurred in the bearing: (a) outer ring axis misaligned relative to
the shaft axis. (continued)
10.6.2 T RUE B RINNELLING
Brinne lling in a roll ing bearing is de fined as the plastic deform ation caused by either sud den
impac t loading during bearing operati on or he avy loading whi le the bearing is not rotating.
Figure 10.22 demon strates su ch indenta tions, typic ally locat ed at roll ing element circum fer-
entia l spacing .
10.6.3 F ALSE BRINNEL LING IN BEARING RACEWAYS
False brinn elling as illustr ated in Figure 10.23 is actual ly fretti ng wear that occurs in the
bearing raceways. It is caused by vibration that occu rs during transp ortation of the bearing
before inst allation or of the assemb led applic ation. It is also caused in the applic ation by a
vibrat ing load that resul ts in smal l amplitude oscillat ions. The lubri cant is driven from the
con tacts and fretti ng wear resul ts. As seen in Figure 10.23, the indenta tions are wi der than
those cau sed by true brinnel ling. Figure 10.24 and Figure 10.25 also depict false brinnelling
on bearing raceways.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.18 (continued) (b) Inner ring axis misaligned relative to the outer ring axis.
10.6.4 PITTING DUE TO E LECTRIC C URRENT P ASSING THROUGH THE BEARING
In ap plications involv ing elect ric motor s, if the be aring is not elect rically insul ated from the
applic ation, electric current may pass through the bearing . This current passage will form
clusters of tiny pits in the rolling surfa ce. Cont inued ope ration leads to corrugat ion of the
surfa ces c alled fluting, as shown in Figure 10.26 and Figure 10.27; the spacing of the
corrugat ions is a functi on of the be aring inter nal speeds and the frequenc y of the electrica l
current . Rolling elemen ts may also experi ence elect rical pitting as shown in Figure 10.28.
Figure 10.29 sho ws the specia l morpholog y surroundi ng a pit caused by elect rical arcing
through a bearing .
10.6.5 INDENTATIONS C AUSED BY HARD P ARTICLE C ONTAMINANTS
Disrup tions or den ts in the rolling contact surfa ces can also be cau sed by hard pa rticle
contaminants that gain ingress past seals or shields into the lubricant and into the free
space within the bearing boundaries. Such particles become trapped between the rolling
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.19 Spherical roller bearing—corrosion pitting of a roller.
elem ents and raceways and get rolled over. Thi s resul ts in relative ly deep impr essions in the
roll ing surfa ces as illustr ated in Figure 10.30 through Figure 10.32.
10.6.6 E FFECT OF P ITTING AND DENTING ON B EARING F UNCTIONAL PERFORMANCE
AND ENDURANCE
As indica ted in Chapt er 8, when a ball or ro ller bearing operate s with lubri cant films that
have thickne sses at least four tim es the composite rms roughn ess of the oppos ing rolling
con tact surfa ces, extre mely long life general ly results. Raceways in medium -size, modern,
deep -groove ball be arings are typically manufactured with raceways having surface roughness
Ra¼ 0.05 mm (2 min.) or less; the equivalent rms roughness value is 0.0625 mm (2.5 min.).
FIGURE 10.20 Spherical roller bearing—oxidation (rusting) of the inner raceways.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.21 Tapered roller bearing—corrosion on cone raceway caused by moisture in the lubricant.
FIGURE 10.22 Tapered roller bearing—brinnelling on cup raceway. (Courtesy of the Timken Company.)
FIGURE 10.23 Tapered roller bearing—false brinnelling on cup raceway. (Courtesy of the Timken
Company.)
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.24 Deep-groove ball bearing—false brinnelling on inner raceway.
Therefore, the ideal lubricant film thickness would be approximately 0.25 mm (10 min.). (Ra
for balls is only a small fraction of that for the raceways and does not significantly affect the
calculation.) Larger bearings and roller bearings generally have ‘‘rougher’’ finishes; for
example, Ra ¼ 0.25 mm (10 min.) can be representative of roller bearing raceways and rollers.
FIGURE 10.25 Tapered roller bearing—false brinnelling on cone raceway. (Courtesy of the Timken
Company.)
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.26 Tapered roller bearing—cone raceway fluting caused by electrical arcing. (Courtesy of
the Timken Company.)
FIGURE 10.27 Cylindrical roller bearing—inner raceway fluting caused by electrical arcing.
FIGURE 10.28 Tapered roller bearing—pitting of tapered rollers caused by electrical arcing. (Courtesy
of the Timken Company.)
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.29 Morphology of a pit caused by electrical arcing.
This woul d give a compo site rms roughness of 0.442 m m (17.68 m in.). In this case, an ideal
lubri cant fil m thickne ss would be app roximatel y 1.77 mm (70.7 m in.).
To determ ine the effecti veness of the lubrican t film in separat ing the rolling contact
surfa ces in the presence of a pit or dent, the de pth of the dent or de pression needs to be
con sidered. Figure 10 .33 is an elevation view of a section taken through a de nt on a raceway
surfa ce. The de pth of such a dent is typicall y in the order of severa l micro meters. This means
that the lubri cant film will tend to collap se into the de pression. Figure 10.34 is a phot ograph
taken through a trans parent disk on a ball– disk fricti on testing rig (see Chapt er 1 1). It de picts
FIGURE 10.30 Severe denting of the inner raceway of a deep-groove ball bearing.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.31 Denting of one inner raceway in a double-row spherical roller bearing.
FIGURE 10.32 Denting of rolling elements: (a) ball, (b) spherical roller, and (c) tapered roller.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.33 Elevation view of dented section of a bearing raceway.
FIGURE 10.34 Passage of a dent on the ball through the ball–disk contact of a ball–disk testing
machine: (a) the dent is entering the contact, (b) the dent is in the center of the contact, and (c) the
dent is preparing to exit the contact. (Courtesy of Wedeven Associates, Inc.)
� 2006 by Taylor & Francis Group, LLC.
the passage of a dent through the oil-lubricated ball–disk contact. It shows how the lubricant
film thickness and pressure distribution over the contact are altered by the dent. Extremely
high-pressure ridges form in the vicinity of the depression. In a bearing, these high pressures
greatly affect the surface and subsurface stresses in both the rolling element and raceway
material, providing initiation points for fatigue failures. Figure 10.35a shows a dent in a
raceway. The depression in the material is surrounded by a ridge that acts as a stress riser.
Figure 10.35b shows a fatigue spall starting at the ridge on the trailing edge of a dent. Hence,
corrosion and oxidation pits, true and false brinnelling, and hard particle contamination
dents act as locations for incipient fatigue. This can cause bearing endurance to be shorter
than that designed and may also lead to rapid failure of the bearing.
10.7 WEAR
10.7.1 DEFINITION OF WEAR
According to Tallian [1]:
Wear (of a contact component) is defined as the removal of component surface material in the
form of loose particles by the application of high tractive forces in asperity dimensions during
service.
FIGURE 10.35 Dents in a raceway: (a) depression surrounded by ridge and (b) fatigue spall formed on
the trailing edge of the dent.
� 2006 by Taylor & Francis Group, LLC.
The resul t of wear is continui ng loss of the geo metric accuracy of the rolling con tact
surfa ces, and gradual de teriorat ion of bearing functi on; for exa mple, increa sed defle ction,
increa sed friction and tempe rature, increased vibrat ion, and so fort h. In ba ll and roll er
bearing s, wear is consider ed preven table by proper atte ntion to bearing and app lication
design, manu facturing accuracy , lubri cation adeq uacy, and prevent ion of ingres s of co ntam-
inants . Ther efore, no effort is made to estimat e the life of rolling bearing s as occasio ned by
wear .
Accor ding to some bearing practiti oners, the term wear is used loosel y to include all
modes of surfa ce mate rial remova l, including pitting and spall ing. Herei n, the latter modes of
mate rial loss are not included in the definiti on of wear of roll ing bea ring mate rial.
10.7.2 T YPES OF WEAR
Mild wear is frequently calle d simply wear. Distinct ion is often made between two types of
mild wear as follo ws:
FIG
� 20
1. Adhesi ve or two -body wear occurri ng at the interface of the contact ing surfaces.
2. Abras ive or three-b ody wear occurri ng due to extra neous ha rd particles acti ng at the
interface of the con tacting surfaces.
Talli an [1] indica tes that the worn surface to the naked eye appears ‘‘featur eless, matte, and
nondi rectional’ ’ and ch aracteris tic finishing marks of the original manufa ctured surface are
worn away. He furthe r stat es that the characteris tic app earances of other define d modes of
mate rial remova l such as fret ting, micr opitting, and gall ing are distinct ly not present . In any
case, mild wear by itself , is not a mode of bearing failure, nor does it lead signi ficantly to rapid
bearing failure.
Severe wear or galling is define d as the trans fer of co mponent surfa ce material in visible
patches from a locat ion on one surface to a locat ion on the contacting surface, and pos sibly
back onto the original surface. This transfer of material takes place because of high-friction
shear forces due to sliding over the asperities of the surfaces. In rolling bearings, this severe
wear phenomenon is also called smearing. It is a welding phenomenon entailing adhesive
bondi ng between mate rial portio ns of the contact ing surfa ces. Figu re 10.36 through Figure
10.38 show bearing compo nents with smear ing on rolling co ntact surfa ces. Smearin g indica tes
URE 10.36 Cylindrical roller bearing inner raceway with smearing damage.
06 by Taylor & Francis Group, LLC.
FIGURE 10.37 Asymmetrical roller with smearing damage from spherical roller thrust bearing.
increa sed bearing fri ction and can lead to less -than-expect ed bearing endu rance. Figure 10.39
shows an en largement of a smear ed area.
10.8 MICROPITTING
Tallian [1] narrow ly de fines surfa ce distress as the plastic flow of surfa ce material due to the
applic ation of ‘‘high normal forces in asperity dimens ions.’’ Thi s su rface distress resul ts in
micro pitting, illustrated in Figu re 10.40. The implicat ion in this de finitio n is that surfa ce
distress a nd micr opitting occur during sim ple roll ing motio n; that is, in the absence of sliding.
In any case, micr opitting appears to be a severe form of surfa ce distress .
10.9 SURFACE-INITIATED FATIGUE
When the repeated ly cycled stress on a surface in roll ing co ntact with an other exceeds the
endu rance stre ngth of the mate rial, fatigue cracki ng of the surfa ce will occu r. The crack wi ll
propaga te until a large pit or spall oc curs in the surfa ce as shown in Figure 10.41. Som e
salient characteristics of such a spall are:
FIGURE 10.38 Smearing damage on the cone raceway of a tapered roller bearing.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.39 Enlarged photograph of smearing on a bearing raceway. Movement of metal is apparent.
. It is relatively shallow in depth.
. It commences at the trailing edge of the contact.
. The starting point of the arrowhead is frequently distinguishable if the fatigue spall is
detected before significant propagation.
FIGURE 10.40 Extreme surface distress (micropitting) on a ball bearing inner raceway.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.41 Surface-initiated fatigue spall in a bearing raceway.
Figure 10.42 an d Figure 10 .43 show bearing s in advanced stage s of surfa ce-initiated fatigue.
In a properl y designe d, man ufactured , applica tion-sel ected, mounted, and lubri cated
rolling bearing , the poten tial for the occurrence of surfa ce-i nitiated fati gue is virtuall y nil.
Therefor e, notw ithstandi ng Tallian’s defini tion of surface distress , a cond ition of sli ding in
margin ally lubri cated rolling elem ent–ra cewa y contact s is usua lly present when surfa ce-
initiat ed fatigue occurs. Furtherm ore, the surfa ce fricti on shear stresses during sli ding are
FIGURE 10.42 Thrust ball bearing raceway and balls with advanced stage of surface-initiated fatigue.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.43 Cylindrical roller bearing inner raceway with advanced stage of surface-initiated fatigue.
most likely increa sed by the presence of depress ions in the contact ing surfaces caused by the
aforem entione d conditi ons of:
. Corr osion or elect ric arc pitting
. Tru e brinn elling
. False brinnel ling
. Denti ng due to ha rd pa rticle co ntaminan ts
. Micropi tting
Anothe r mode of roll ing contact su rface failure is caused by hy drogen ions, which a ttack the
surfa ce material, resulting in pitt ing or spalling of the surfa ce. Figu re 10.44 depict s the
spall ed surfa ce of a be aring ball caused by hydrogen embritt lemen t. This failure mode,
whi ch is relat ively rare, is general ly associ ated wi th rolling contact surfa ce operatio n at
steady -state temperatur e above that at whi ch de gradation of the miner al oil lubri cant
commenc es. It is general ly associ ated with signifi cant diff erential or gross sli ding be tween
roll ing contact surfaces sub jected to high Hertz stre ss, the surfa ces incompl etely or margin-
ally separat ed by a miner al oil lubri cant film. The high tempe ratur e that resul ts in the
con tact cau ses the chemi cal breakdow n of the lubricant, relea sing hydrogen ions. Hydroge n
embri ttlement is also associ ated with an e nvironm ent surroundi ng the bearing , which does
not allow the hy drogen ions to easily dissi pate from the vici nity of the bearing ; for e xample,
in a well- sealed app lication.
The essent ially circular shapes of the spalled areas of Figu re 10.44 are pro bably a ssociated
with the axisymmet ric resi dual stre ss dist ribution exist ing in the bearing ba ll after heat
treatmen t and surface fini shing. As illustrated in Figure 10.45, the hydrogen ions pen etrate
the steel from the surface of the component, resulting in cracks. These in turn propagate
weakening the material until spalling occurs. Many researchers have investigated the occur-
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.44 Spalling of bearing ball surface due to hydrogen embrittlement.
rence of hydrogen embrittlement failure. In all these reported experiments, hydrogen was
introduced into the application in the presence of excessive contact stresses, and in most cases,
in the presence of elevated temperatures. In none of these cases did the production of
hydrogen ions result from lubricant chemical breakdown.
FIGURE 10.45 Cracking of steel ball surface due to penetration of hydrogen ions.
� 2006 by Taylor & Francis Group, LLC.
10.10 SUBSURFACE-INITIATED FATIGUE
As stated at the be ginning of this chap ter, each of the indica ted mo des of roll ing bearing
damage and failu re is co nsider ed avoidabl e. Under ve ry heavy loading , howeve r, even though
an a ccurately manu factured an d properl y moun ted bearing is wel l-lubrica ted, it is possible for
bearing failure to occur due to subsurfa ce-initiated fatigue. The life of a ro lling bearing from
star t of ope ration to occurrence of the first subsurf ace-in itiated spall is the basis specified in
the ISO [2] standar d a nd su pportin g nationa l standar ds for the calculati on of fatigue endu r-
ance. See Chapt er 11 of the first volume of this handbo ok. Figu re 10.46 illustr ates a ba ll
bearing racew ay with a subsurf ace-init iated spall . It is apparent that the dep th is not shall ow.
Figure 10.47 shows spalling that has propagat ed in a spheri cal roll er bearing raceway, while
Figure 10.48 indicates spalling in a tapere d roller bearing raceway due to edge loading .
Fatigue spall ing is not consider ed a catas trophi c-type fail ure. Depending on the type and
qua ntity of lubri catio n, the bearing will continue to rotate, howeve r, with ever-increa sing
fricti on (see Ref s. [3,4]). After some time, depen ding on the magni tude of loading , ope rational
speed, and lub rication effecti veness, the bearing will experien ce either excess ive v ibration or
surface friction heat generation causing the bearing to seize.
10.11 CLOSURE
This chapter detailed the various common modes of failure to which ball and roller bearings
may succumb. It is seen that most of these involve situations caused by bearing operations
outside of recommended practice. As stated previously herein and in the first volume of this
handbook, rolling bearings are rated according to their ability to resist or avoid subsurface-
initiat ed roll ing contact fatigue . The da ta of Figure 10.49, based on retur ns of failed bearing s
to manufacturers, show that the latter comprises only a small fraction of common failure
FIGURE 10.46 Subsurface-initiated fatigue spall in a ball bearing raceway.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 10.47 Subsurface-initiated fatigue spall and propagation in a spherical roller bearing raceway.
FIGURE 10.48 Subsurface-initiated fatigue spalls due to edge loading in a tapered roller bearing cone
raceway.
� 2006 by Taylor & Francis Group, LLC.
Deficient sealing18%
Subsurfacefatigue 8% Miscellaneous
2%
Inadequatelubrication 43%
Impropermounting 29%
FIGURE 10.49 Frequency of occurrence of bearing failure modes.
types. Therefore, with regard to a given bearing application, attention to proper bearing
selection, proper lubricant selection and adequate means of delivery, proper mounting
techniques, avoidance of contamination, and general adherence to good operating practice
will enable the achievement of the bearing design life.
REFERENCES
1.
� 2
Tallian, T., Failure Atlas for Hertz Contact Machine Elements, 2nd Ed., ASME Press, 1999.
2.
International Organization for Standards, International Standard ISO 281, Rolling Bearings—Dynamic Load Ratings and Rating Life, 2006.
3.
Kotzalas, M. and Harris, T., Fatigue failure progression in ball bearings, ASME Trans., J. Tribol.,123(2), 283–242, April 2001.
4.
Kotzalas, M. and Harris, T., Fatigue failure and ball bearing friction, Tribol. Trans., 43(1), 137–143,2000.
006 by Taylor & Francis Group, LLC.
11 Bearing and Rolling ElementEndurance Testing
� 2006 by Taylor & Francis Grou
and Analysis
LIST OF SYMBOLS
Symbol Description Units
A1 Reliability-life factor
ASL Stress-life factor
C Basic dynamic capacity N (lb)
f(x) Probability density function
F(x) Cumulative distribution function
Fe Equivalent applied load N (lb)
h Lubricant film thickness mm (min.)
h Hazard rate
H Cumulative hazard rate
i Failure order number
k Number of samples
kp �ln(1� p)
l Number of subgroups in a sudden death test
m Sample size of a sudden death subgroup
n Sample size
p Probability value
q(l, m, p) Pivotal function used for sudden death test analysis
r Number of failures in a censored sample
R Ratio of upper to lower confidence limits for b
R0.50 Median ratio of upper to lower confidence limits for x0.10
S Probability of Survival
t1(r, n, k) Pivotal function for testing for differences among k
estimates of x0.10
u(r, n, p) Pivotal function for setting confidence limits on xp
u1(r, n, p, k) Pivotal function for setting confidence limits on xp
using k data samples
v(r, n) Pivotal function for setting confidence limits on b
v1(r, n, k) Pivotal function for setting confidence limits on b
using k data samples
w(r, n, k) Pivotal function for testing whether k Weibull
populations have a common b
p, LLC.
x Random varia ble
xp pth percent ile of the dist ribution of the rand om varia ble x
b Weibull shape parame ter
h Weibull scale parame ter
L Lubr icant film parame ter
k Rat io of actual lubri cant v iscosity to viscos ity requir ed
s rms surfa ce roughn ess mm (m in.)
11.1 GENERAL
Simi lar to the lives of light bulbs and humans, ba ll an d ro ller be aring lives, specifically rolling
con tact (RC) fatigu e lives, are probabil istic in nature, as sho wn in Figure 11 .1. They do not
achieve a specific, uniq uely pred ictable life, notw ithstandi ng functi oning in the same envir-
onmen t. As indica ted in Chapt er 10, with prop er attention to be aring design, manufa cture,
and applic ation, all modes of rolling be aring failure can be avoided with the excep tion of RC
fatigue when contact stresses due to app lication loading exceed the be aring’s end urance
stre ngth. In that situ ation, the life of any one bearing ope rating in the app lication can differ
signi ficantly from that of an ap parently identical unit. To esti mate rolling bearing fatigue life
in a given app lication, statistica l procedures have be en establis hed for the analys is of meas-
ured end urance data.
Using the method intr oduced by Weibu ll [1] to analyze the exp erimental data accumul ated
on fatigue lives of more than 1500 ba ll and roll er bearing s, Lundber g an d Palmgr en [2,3]
develop ed form ulas and methods to enable the calcul ation of load and life ratin gs for
roll ing be arings. The an alysis by Lundber g and Palmgr en was based on the infl uence of rolling
element–raceway contact normal stresses (Hertz stresses) on RC fatigue life of the bearing
racew ays. Chapter 1 1 in the fir st vo lume of this handb ook discusses these analytical methods
Total number ofbearings failed
Total number oflamps failed
Tungstenlamps
Ballbearings
Rev
olut
ions
Hou
rs
Yea
rs
Humanlife
Total number ofdeaths
FIGURE 11.1 Comparison of rolling bearing fatigue life distribution with the distribution of the service
lives of light bulbs and life spans of human beings.
� 2006 by Taylor & Francis Group, LLC.
in detail. The spread of rolling bearing fatigue lives recorded by Lundber g and Palmgr en is
illustr ated for a typical test group in Fig ure 11.2.
From Figure 1 1.2, it is noted that tw o points on the curve are typic ally determ ined:
. L10 , the life that 9 0% of the be arings will survi ve and exceed.
. L50 , the media n life that 50% of the bearing s will survi ve and exceed.
It sho uld be observed that L10 is the bearing rating life, the lif e on which bearing selec tion is
typically based.
More recent ly, Ioannides and Harr is [4], as detai led in Chapt er 8, extended the Lundber g–
Palmgren analysis to include the influence on bearing fatigue life of all stresses in the material
in the vicinity of each contact as well as the concept of a material fatigue endurance limit
stress. The combination of the two techniques resulted in the current ISO [5] standard load
and fatigue life rating equation:
Ln ¼ A1ASL
C
Fe
� �p
ð8:23Þ
In Equation 8.23, exponent p¼ 3 for ball bearings and 10/3 for roller bearings.
The Lundberg and Palmgren experimental effort [2,3] was based on bearings manufac-
tured from 52100 steel during the 1930s and 1940s. Modern bearing manufacturing methods
have since modified and improved rolling bearing geometries. Moreover, modern 52100 steel
has been considerably improved in cleanliness and homogeneity since the time of Lundberg
and Palmgren, and modern bearings are manufactured from a variety of steels and even
ceramics. To evaluate the effects of new materials, material processing methods, and perhaps
10
5 L50
L10
10
Fat
igue
life
15
20
0.9 0.8 0.7 0.6 0.5Probability of survival, S
0.4 0.3 0.2 0.1 0
FIGURE 11.2 Distribution of fatigue lives resulting from endurance testing of a group of rolling bearings.
� 2006 by Taylor & Francis Group, LLC.
modif ied geomet ries on rolling be aring life, it is yet necessa ry to cond uct fatigue end urance
tests. These can be accompl ished through testing of co mplete bearing s or, possibl y, using
compo nents such as balls or roll ers. The spread in experimen tal fatigu e data a nd the limita-
tions of the statist ical analys is techni ques requir e that many test units be operate d for a long
time to obtain valid esti mates of bearing life. It is obv iously less exp ensive to end urance test
balls or rollers in lieu of comp lete bearing s; howeve r, the accuracy of extra polatio n of the
compo nent test results to complet e bearing s is alw ays sub ject to que stion.
This chapter discusses the concepts, methods, and philosophies of conducting endurance
tests on complete bearing assemblies as well as on elemental RC configurations.
11.2 LIFE TESTING PROBLEMS AND LIMITATIONS
11.2.1 A CCELERATION OF E NDURANCE T ESTING
The ability to use life test data colle cted on a parti cular be aring type an d size unde r a specific
set of operating conditions to predict general bearing performance requires a systematic
relation ship between app lied load an d life. This relationshi p, given by Equat ion 8.23, pro-
vides the means to use experimental life data collected under one set of test conditions to
establish projected bearing performance under a wide range of operating conditions.
The time to initiation of a RC fatigue spall in a typical application is several years;
for example, 10 years or more, assuming that applied loading is sufficiently heavy to cause
fatigue of RC surfaces. It is therefore obvious that any practical laboratory test sequence
must be conducted under accelerated conditions if the necessary data are to be accumulated
within a reasonable time. Several potential methods exist for acceleration of testing. RC
damage modes exist, however, that compete in individual bearings to produce the final
failure. Care must be exercised to ensure that the method of test acceleration does not alter
the desired failure mode of RC fatigue. Generally, two methods have been used to accelerate
endurance testing: (1) increasing the level of applied load and (2) increasing the operating
speed.
11.2.2 ACCELERATION OF ENDURANCE TESTING THROUGH VERY HEAVY APPLIED LOADING
The experimental results obtained under increased load levels can be rather easily extrapo-
lated to other conditions by using the basic load–life relationship. Thus, this is the most
widely used method of test acceleration. It is important, however, that consistency be main-
tained with the basic assumptions used in the development of the life formulas. Key among
these is that the stresses generated at and below the RC surfaces should remain within the
elastic regime. As indicated by Valori et al. [7], exceeding elastic limits of the bearing raceway
and rolling element materials will produce deviations from the basic load–life relationship.
Testing conducted in the material plastic regime produces results that are inconsistent with
bearing operating practice and cannot be reliably extrapolated. The practical maximum Hertz
stress limit for bearing endurance testing is 3300 MPa (475 psi).
11.2.3 AVOIDING TEST OPERATION IN THE PLASTIC DEFORMATION REGIME
Endurance testing of some bearings requires special consideration. For example, the outer
raceway of a self-aligning ball bearing is a spherical surface producing circular point contacts
between balls and raceways. Under very heavy applied loading, these contacts develop
stresses in the plastic regime, more rapidly than considered by the dynamic capacity calcula-
tion. Johnston et al. [8] indicated that applied load should be no greater than C/8 to prevent
substantial plastic deformation during testing of these bearings. Similarly, it is anticipated
� 2006 by Taylor & Francis Group, LLC.
that some bearing types that ha ve nonstandar d internal geomet ries co uld also experi ence
signifi cant plast ic de formati ons at low er than expecte d loads.
Cylindrica l roll er, tapere d roller, and need le roll er be arings are designe d to operate with
line con tact between rollers and raceways under most applie d load ing. Spherical roll er
bearing s, howeve r, may ope rate wi th point contact until the app lied load is suff iciently
great to cau se operation of the mo st heavil y loaded rolling elem ent in modified line contact
(see Chapt er 6 an d Chapter 11 of the first volume of this han dbook). When the load becomes
too heavy, all roller bearing s wi ll tend to exp erience edge loading and plast ic deform ations , at
least in the most heavil y loaded roller–rac eway contact s. Accord ingly, in en durance test ing of
roller bearing s, roller and racew ay geomet ries must be pro filed in the axial direction to ensure
that stress con centrations do not occu r at the contact extremit ies wi th attendan t plastic
deform ations . The pro files used in standard design roll er bearing s are often insuf ficient for
the he avy loads used in an ac celerated life test series. Edge load ing will tend to prod uce
fatigue live s that are sub stantially less than lives experi enced in field applic ations. Hence,
endu rance test resul ts gen erated unde r co ndition s involv ing edge loading co uld not be
accurat ely extra polate d to nor mal applic ations.
11.2.4 LOAD –LIFE RELATIONSHIP OF ROLLER B EARINGS
Even when ed ge stre sses do not occur, roll er bearing life unde r heavy loading doe s not tend to
follow the standar d load–l ife relationshi p. Lundb erg and Palmgr en [3] sho wed that test series
cond ucted on cyli ndrical roll er be arings indica ted a load–l ife exp onent of 4 in lieu of the
standar d 1 0/3. Act ually, the expon ent 10/3 was chosen as a comprom ise to acco mmodate the
combinat ion of line and point contact s that occurs in the ope ration of spherical roll er
bearing s. Inter pretat ion of the endu rance test data for roll er be arings need s to take this
consider ation into accou nt.
11.2.5 A CCELERATION OF E NDURANCE TESTING THROUGH HIGH -SPEED OPERATION
If all operati ng parame ters remained unchang ed, endurance test ing a bearing at higher
rotation al speed woul d sho rten the durati on of testing by generat ing a more rap id accumu-
lation of test cycles . Unfor tunate ly, the object ive of a sho rter test duratio n is general ly not
achieve d becau se fatigu e en durance is us ually cond ucted with oil film lubricati on, the lub ri-
cant generally delivered to the bea ring in c opious quantities . Ref erring to Equat ion 4.57 for
line contact s and Equation 4.60 for poin t contact s, it can be seen that the mini mum lubri cant
film thickne ss is a functi on of app roximatel y the 0.7 power of rotat ional speed. Hence, as
speed increases so does lubrican t film thickne ss. In Chapt er 8, it was de monst rated that as the
ability of the lubri cant films to separate the RC surfaces increa ses, fatigue life increa ses at a
great er rate. Ther efore, increa sing the rotational sp eed wi ll most likely increa se rather than
decreas e the dur ation of testing.
Associated with the effect of lubricant film thickness on fatigue life is the delivery of lubricant
to the rolling element–raceway contacts in sufficient quantity to enable complete lubricant films
to be generated. As the bearing speed increases, this becomes more difficult because the rapidity
of oil reflux to the contacts may not keep pace with rolling element passage. As indicated in
Chapter 4, this is called lubricant starvation. This is a particular consideration when endurance
testing is to be performed with grease-lubricated bearings.
Also, as the ope rating speed is increased, other life-i nfluenc ing effe cts occur. Und er high-
speed operati ng conditio ns, roll ing elem ent c entrifugal force increa ses. This means that the
maxi mum contact stre sses may occu r a t the outer raceway, rather than the inner raceway,
causing that component to experi ence spall ing fir st, and chan ging the e xpected failure mode.
Concur rently , as ill ustrated in Chapt er 1, for the high-s peed ope ration of rad ially load ed
� 2006 by Taylor & Francis Group, LLC.
bearing s, the number of rolling elem ents under load decreas es, increa sing the con tact stresses
on the inner raceway, but ch anging the mod e of operati on.
As discus sed in Chapt er 1, ope ration of thrust -loaded ball bearing s at high speed causes
ball– outer racew ay contact angles to decreas e an d ball– inner raceway contact angles to
increa se. This changes the fricti on charact eris tics of the bearing , whi ch also influen ce fatigue
end urance.
A pa rameter often used to express the severi ty of bea ring speed co ndition s is dN , the
prod uct of the bearing bore measur ed in mil limeter s an d the shaft speed in revo lutions per
minut e. It is usual to consider high-sp eed be aring app lications as those that have dN � 1
milli on. For be aring operati ons unde r high-s peed conditi ons, sophist icated analyt ical tech-
niques, such as tho se present ed in this text, are requ ired to reliably calculate bearing rating
live s for compari son with en durance test data.
Using high speed to accele rate bearing endu rance test program s ha s other limit ations.
Standa rd rolling bearing s have function al speed lim its be cause of the stamped meta l or
molded plast ic cage designs , whi ch are not adeq uate for high-sp eed operatio ns. Exc essive
heat generat ion rates may occur at the roll ing elemen t–racewa y contact s, which ha ve been
designe d prim arily for maxi mum load-c arrying capa bilities at lower speeds, and co mponent
precis ion may be alte red due to the dyn amic loading occurri ng during high-spee d ope ration s.
System operating effects can also produce significant life effects on high-speed bearings. For
example, insufficient cooling or the inadequate distribution of the cooling medium can create
thermal gradients in the bearings that alter internal clearances and component geometries.
Higher operating temperatures are generated at higher speeds. The test lubricants used
must then be capable of sustained extended exposure to these temperatures without suffering
degradation. The conduct of high-speed life tests requires extra care to ensure that the failures
obtained are fatigue-related and not precipitated by some speed-related performance
malfunction.
11.2.6 TESTING IN THE MARGINAL LUBRICATION REGIME
In Chapte r 8, the means to quantify the lubricati on-asso ciated effect of speed on bearing
fatigue endurance is demonstrated. Particularly in the regime of marginal lubrication, the
effect is complex owing to the interactions of rolling component surface finishes and chem-
istry, lubricant chemical and mechanical properties, lubrication adequacy, contaminant types,
and contamination levels. Testing at speeds slow enough to cause operation in the marginal
lubrication regime may indeed reduce the fatigue life in revolutions survived; however, as with
high-speed testing, the duration of testing may increase; in this case, due to the slower speed
of accumulating stress cycles. Furthermore, the above-indicated side effects must be consid-
ered in the evaluation of test results.
11.3 PRACTICAL TESTING CONSIDERATIONS
11.3.1 PARTICULATE CONTAMINANTS IN THE LUBRICANT
An individual bearing may fail for several reasons; however, the results of an endurance test
series are only meaningful when the test bearings fail by fatigue-related mechanisms. The
experimenter must control the test process to ensure that this occurs. Some of the other failure
modes that can be experienced are discussed in detail by Tallian [6]. The following paragraphs
deal with a few specific failure types that can affect the conduct of a life test sequence.
In Chapter 8, the influence of lubrication on contact fatigue life was discussed from the
standpoint of elastohydrodynamic lubrication (EHL) film generation. There are also other
� 2006 by Taylor & Francis Group, LLC.
lubri cation-rel ated effects that can affect the outco me of the test series. The first is particulat e
contam inants in the lubrican t. Depend ing on bearing size, operating speed, and lubri cant
rheology , the overal l thickne ss of the lubrican t fil m developed at the roll ing elemen t–racewa y
contact s may fall betw een 0.05 and 0.5 mm (2 and 20 min.) . Solid particles large r than the film
can be mechan ically trapped in the contact regions and damage the raceway and ro lling
elem ent surfa ces, leading to substa ntially sho rtened end urances. This ha s been amply dem-
onstra ted by Sa yles and MacP herson [9] and others .
Therefor e, filtra tion of the lubri cant to the desired level is necessa ry to en sure meani ngful
test results. The desir ed level is determ ined by the ap plication whi ch the testing pur ports to
approxim ate. If this degree of filtra tion is not provided, effects of co ntaminati on must be
consider ed when evaluat ing test resul ts. Chapt er 8 discus ses the effe ct of various de grees of
particu late contam ination , and hen ce filtra tion, on be aring fatigue life.
11.3.2 MOISTURE IN THE LUBRICANT
The moisture c ontent in the lubri cant is another impor tant consider ation. It has long be en
apparen t that qua ntities of free water in the oil cause corrosi on of the RC surfaces and thus
have a detriment al effe ct on bearing life. It has been furth er shown by Fitch [10] an d others ,
howeve r, that wat er level s as low a s 50–100 parts per million (ppm) may also ha ve a
detrimen tal effect, even wi th no ev idence of corrosi on. This is due to hydrogen embri ttlement
of the roll ing elem ent and raceway mate rial (see also Chapter 8). Mo isture con trol in test
lubrication systems is thus a major concern, and the effect of moisture needs to be considered
during the evaluation of life test results. A maximum of 40 ppm is considered necessary to
minimize life reduction effects.
11.3.3 CHEMICAL COMPOSITION OF THE LUBRICANT
Most commercial lubricants contain a number of proprietary additives developed for specific
purposes; for example, to provide antiwear properties, to achieve extreme pressure and
thermal stability, and to provide boundary lubrication in case of marginal lubricant
films. These additives can also affect bearing endurance, either immediately or after experi-
encing time-related degradation. Care must be taken to ensure that the additives included
in the test lubricant do not suffer excessive deterioration as a result of accelerated life test
conditions. Also, for consistency of results and comparing life test groups, it is a good practice
to use one standard test lubricant from a particular producer for the conduct of all general
life tests.
11.3.4 CONSISTENCY OF TEST CONDITIONS
11.3.4.1 Condition Changes over the Test Period
The statistical nature of RC fatigue requires many test samples to obtain a reasonable estimate
of life; therefore, a bearing life test sequence generally occurs over a long time. A major job of
the experimenter is to ensure the consistency of the applied test conditions throughout the test
period. The process is not simple because subtle changes can occur during this period. Such
changes might be overlooked until their effects become major, and it is too late to salvage the
collected data. The test may then have to be redone under better controls.
11.3.4.2 Lubricant Property Changes
An example of the above is that the stability of the additive packages in a test lubricant can be
a source of changing test conditions. Some lubricants are known to suffer additive depletion
� 2006 by Taylor & Francis Group, LLC.
afte r an extend ed period of operati on. The degradat ion of the add itive packa ge can alter the
RC surfa ce fricti on cond itions, altering bearing life. Gen erally, the nor mal chemi cal tests used
to evaluat e lubric ants do not determ ine the co ndition s of the ad ditive content . Therefor e, if a
lubri cant is used for en durance testing ov er a long period of time, a sample of the flui d should
be retur ned to the produ cer at regula r inter vals, for exampl e annuall y, for a detai led evalu-
ation of its con dition.
11.3. 4.3 Control of Temper ature
Adequate temperatur e controls must also be employed dur i ng the test period. The
thickness of the EHL film is sensitive to the cont act temperature. Referring to Equation
4.57 for line c ontacts and E quation 4.60 for point cont acts, i t c an be seen that the
minimum lubricant f ilm thickness is a f un ction of approximately the 0.7 pow er of
lubricant viscosity, w hich is highly sens itive t o temperature. Most test m achi nes are
located in standard industrial e nvironments where rather w ide f luctuations in ambient
temperature are experienced over a period of one year. In add ition, the heat generation
rates of individual bearings can vary as a result of the c ombined effects of normal
manufacturing tolerances. Both these conditions produce variations in operating tem-
perature levels in a lot of bearings and a ffec t the validity of the l ife data. Means m ust be
provided to monitor and control the operati ng temperat ure level of each bearing t o
achieve a degree of consistency. A tolerance level of +3 8 C (5.48 F ) is nor m ally consi dered
adequate for t he endurance test period.
11.3. 4.4 Deteri oration of Beari ng Mo unting Har dware
The c ondition of the hardware i nvolved in m ounting and dismounting of bearings
requires constant monitoring. The heavy loads used for life testing require heavy inter-
ference fits between bearing inner rings and shafts. Repeated mounting and dismounting
of bearings can damage the shaft surface, which can in turn alter the geometry of the
mounted ring. The shaft surface and the housing bore are also subject to deterioration
from fretting corrosion ( see Chapter 10). This c an produce s ignificant variations in the
geometry of the mounting surfaces, which can alter the internal bearing geometry and,
thereby, reduce bearing life.
11.3.4.5 Failure Detection
Fatigue theory considers failure as the initiation of the fatigue crack in the bulk material. To
be detectable in a practical manner, the crack must propagate to the surface and produce a
spall of sufficient magnitude to produce a marked effect on a bearing operating parameter;
for example, vibration, noise, and temperature. The ability to detect early signs of failure
varies with the complexity of the test system, the type of bearing under evaluation, and other
test conditions. There is no single method that can consistently provide the failure discrim-
ination necessary for all types of bearing tests. It is, therefore, necessary to select a method
and system that will repeatedly terminate machine operation upon the consistent occurrence
of a minimal degree of damage.
Considering the above, failure propagation rate is important. If the degree of damage at
test termination is consistent among test elements, the only variation between the experimen-
tal and theoretical lives is the lag in failure detection. In standard through-hardened bearing
steels, the failure propagation rate is quite rapid under endurance test conditions, and this is
not a major factor, considering the typical dispersion of endurance test data and the degree of
confidence obtained from statistical analysis. Care must be exercised when evaluating these
� 2006 by Taylor & Francis Group, LLC.
latter results and particular ly when co mparing the experi menta l lives with those obtaine d
from standar d steel lots .
Post-test an alysis is a de tailed examin ation of all tested bearing s using:
� 20
1. High magni fica tion optica l inspect ion
2. Highe r magni fication electron micr oscopy
3. Metallurgi cal exami nation
4. Dimensiona l examin ation
5. Chemical evaluation as requir ed
The ch aracteris tics of the failu res are examin ed to establ ish their origi ns, and the resi dual
surfa ce co ndition s are evaluat ed for indica tions of extra neous effects that may have influ-
enced bearing life. This techni que enables the experi mente r to ensure that the da ta are indeed
valid . Tallian’s ‘‘Damage Atlas’’ [11] , co ntaining numerous phot ographs of the various failure
modes, can provide va luable assi stance in this effort.
11.3. 4.6 Concurren t Test Anal ysis
When ever a bearing is remove d from the test machi ne, the experi mente r sho uld co nduct a
prelimina ry evaluat ion. Herei n, the bearing is exami ned optica lly at magnifica tions up to 30 �for indications of improper or out-of-control test parameters. Examples of indications that
may be obs erved are given in Chapt er 10. Figure 10.46 illu strates the appearance of a typic al,
subsurface fatigue-initiated spall on a ball bearing raceway. Figure 10.48 shows spalling of a
tapered roller bearing that most likely resulted from bearing misalignment. Figure 10.12
illustrates a spalling failure on a ball bearing outer ring that resulted from fretting corrosion
on the outer diameter of the ring. Figure 10.41 illustrates a more subtle form of test alteration,
where the spalling failure originated from the presence of a debris dent on the raceway
surface. The last three failures are not valid subsurface-initiated fatigue spalls and indicate
the need to correct the test methods. Furthermore, the data points need to be eliminated from
the failure data to obtain a valid estimate of the experimental bearing life.
11.4 TEST SAMPLES
11.4.1 STATISTICAL REQUIREMENTS
The statistical techniques used to evaluate the failure data require that the bearings be
statistically similar assemblies. Therefore, the individual components must be manufactured
in the same processing lot from one heat of material. Generally, it is prudent to manufacture
the total bearing assembly in this manner; however, when highly experimental materials or
processes are considered, this is often not cost effective or even possible. In those cases, the
critical element in a bearing assembly from a fatigue point of view can be used as the test
element with the other components manufactured from standard material. The effects of
failures occurring on the other parts can be eliminated during analysis of the test data. There
is some risk in this approach because it is possible that too many failures might occur on these
nontest parts, rendering it impossible to calculate an accurate life estimate for the material
under evaluation. This risk is generally small because an initial result indicating the superior
performance of an experimental process is usually sufficient to justify continued development
effort even without a firm numerical life estimate. Additional life tests would, however, be
required to establish the magnitude of the expected lot-to-lot variation before adopting a new
material or implementing a new manufacturing process.
06 by Taylor & Francis Group, LLC.
11.4.2 NUMBER OF TEST BEARINGS
Statist ical analys is pro vides a numeri cal estimat e of the value of the experi menta l life enclosed
by uppe r-boundar y and lower-bound ary estimat es at specified confide nce lim its. The precis ion
of the experi menta l life estimate can be define d by the ratio of these uppe r and low er confide nce
limit s; the experi mental aim is to mini mize this spread. The magni tude of the confide nce
inter val decreas es as the size of the test lot increa ses; howeve r, the cost of cond ucting the test
also increa ses with lot test size. Therefor e, the degree of precis ion requir ed in the test result
shou ld be establ ished during the test planning stage to define the size of the test lot to achieve
the requir ed resul ts.
11.4.3 T EST STRATEGY
The usu al method of perfor ming en durance tests is to use one large group of bearing s,
runn ing each bearing to failu re. This process is time-cons uming, but it provides the be st
experi menta l estimat es of both L10 an d L 50 lives. Prima ry inter est is, howeve r, in the magni-
tude of the e xperimental L10 , so co nsiderab le time savings can be achieve d by curtailin g the
test runs afte r a finite ope rating period equ al to at least three times the achieve d experi mental
L10 life. Also, Anders son [12] demo nstrated that saving s in test time accompan ied by increases
in precis ion of test resul ts can be ach ieved by using a sudd en death test stra tegy. In this
app roach, the en tire test lot of bearing s is divide d into subgroup s of equal sizes. Each
subgro up is then run as a unit until one bearing fails, at whi ch tim e the testing of the sub group
is terminat ed. Figure 11.3 illu strates the effect of both lot size and test strategy on the
precis ion of life test estimat es obtaina ble from an endurance testing series.
11.4.4 MANUFACTURING A CCURACY OF T EST SAMPLES
To provide an accurat e life estimat e for the varia ble unde r evaluation , the experi mente r mu st
be sure that the test bearing s are free from mate rial an d manufa cturin g defects an d that all
parts conform to establ ished dimens ional an d form toler ances. Thi s sit uation is not always
easy to a ttain since experimenta l mate rials might respond diff erently to standard manu fac-
turi ng process es, or they could requir e unique pro cessing steps that are not yet total ly de fined.
Expe rimental manufa cturing pro cesses requir e addition al verifica tion, or their use might
prod uce unexp ected variations in metallurgical or dimensional parameters. Therefore, ad-
equate test control is achieved by detailed pretest au diting of the test parts to supplem ent the
standar d in-pr ocess evaluat ions. Table 11.1 and Table 11.2 con tain lis ts of those metallurgical
and dimensional parameters considered mandatory in a typical pretest audit, as well as an
indication of the number of samples that need to be checked in each case. These lists are not
to be construed as complete; other parameters could be evaluated beneficially if time and
money permit.
11.5 TEST RIG DESIGN
Some specific characteristics are desired in an endurance test system to achieve the control
requirements of a life test series. An individual test run takes a long time; therefore, the test
machine must be capable of running unattended without experiencing variation in the applied
test parameters such as load(s), speed, lubrication conditions, and operating temperature.
The basic test system that could also be subject to fatigue, such as load–support bearings,
shafts, and load linkages, must be many times stronger than the test bearings so that test runs
can be completed with the fewest interruptions from extraneous causes. The assembly of the
� 2006 by Taylor & Francis Group, LLC.
1 2
2 6060 30
3030
15
1515
10
10
10
3
4
5
6
7
8
9
10
12
14
1
2
3
4
5
6
7
8
9
10
12
6014
1
3 4 5 6 7
7
7
6
M = G = 6
M = G = 5
M = G = 7
Sudden death test
Conventional test
NG = 2
NG = 4
NG = 10
8
8
8
8
9
N = size of test series (total number of bearings)M = number of bearing failuresG = number of groupsNG = group size (all groups are run to one failure)b = true Weibull slope
9
9
9
10 12 14 16 18 20
1 2
120
3 4 5 6 7
7
M = 6
M = 10N = 10
N = 20N = 30N = 120
M = 7M = 9
M = 12
8 9 10 12
12
12 10
10
99
9
8 8
8
12
162030
30
606030
2020 20
16
16
16
14 16 18 20
R 10b
R 10b
R 50b
R 50b
FIGURE 11.3 Effect of the lot size on confidence of life test results. (From Andersson, T., Ball Bear. J.,
217, 14–23, 1983. With permission.)
test machine should have only a minor influence on the test conditions to minimize variations
between individual test runs. For example, the alignment of the test bearings should be
automatically assured by the assembly of the test housing. If not, a simple direct means of
monitoring and adjusting this parameter must be provided. Also, since a test series requires
multiple setups, easy assembly and disassembly of the test system are desirable to minimize
� 2006 by Taylor & Francis Group, LLC.
TABLE 11.1Typical Metallurgical Audit Parameters
100% Nondestructive Tests: Ring Raceways Only
Magnaflux for near-surface materials defects
Etch inspection for surface processing defects
Sample Destructive Tests: All Components
Microhardness to 0.1 mm (0.004 in.) depth below raceway surfaces
Microstructures to 0.3 mm (0.012 in.) depth below raceway surfaces
Retained austenite levels
Fracture grain size
Inclusions ratings
turnar ound time and manpow er requiremen ts for test be aring chan ges. In addition , the test
syst em must be easy to maint ain and shou ld be capable of ope rating reliably and efficiently
for years to ensure long-term co mpatibil ity of test results. Desi gn sim plicity is a key ingredi ent
in meeting all these de mands. Sebo k and Rimrot t [13] present ed a co mprehens ive discus sion
of the design philos ophy for life test rigs ; Figure 11.4 illustrates some typical endurance test
configurations described.
The application of some of the design concepts of Figure 11.4 to actual endurance test
syst ems will be briefly address ed. Figure 11 .5 is a phot ograph of an SKF R2 rig for testing
35- to 50-mm bore ball and roller bearings under radial, axial, or combined radial and axial
loads; Figu re 11.6 is a schema tic diagra m of an SKF R3 rig, a simila r design for testing larger
bearings. The operating speed in these rigs may be varied within limits to achieve a given
test condition and bearing lubrication can be provided by grease, sump oil, circulating oil, or
air-oil mist.
Practical life test rig designs will vary, depending on the type of bearing to be tested and its
normal operati ng mode. For exampl e, Fi gure 11.7, accordi ng to Hacker [14], shows a four-
bearing test rig concept used in the testing of tapered roller bearings. In this instance, while
testing is conducted under an externally applied radial load, each bearing also sees an
TABLE 11.2Typical Dimensional Audit Parameters
100% Assembled Rings
Radial Looseness
Average and peak vibration levels
Statistical Sample of Ring Grooves
Diameter and waviness
Radius and form
Cross-groove surface texture
Statistical Sample of Balls
Diameter and out-of-round
Set size variation
Waviness
Surface texture
� 2006 by Taylor & Francis Group, LLC.
B B BB
B B B B
P
P P P P2P
2P
2P
P PP P P
D
D
D
D
D
A A AA
A A
A A
T
TT
T
T
(a)
(c) (d) (e)
(b)
d d
FIGURE 11.4 Typical bearing endurance test configurations discussed in Ref. [13]. A¼ test bearing,
B¼ load bearing, P¼ applied radial load, T¼ applied thrust load, and D¼ drive.
FIGURE 11.5 SKF R2 endurance test rig. (Courtesy of SKF.)
� 2006 by Taylor & Francis Group, LLC.
Temperature control
Load cell
Labyrinthseals of
ERC design
Separate lubrication
Speed1500/2500 r/min
Hydraulic pads for testbearing alignment
Test load
FIGURE 11.6 Schematic diagram of an SKF R3 endurance test rig. (Courtesy of SKF.)
inter nally induced thrust load . The size of the latter load is de termined by the magn itude of
the app lied radial load, the fixed axial locat ions of the bearing cu ps and co nes in the test
hous ing, an d the basic internal design of the test be arings. Figure 11.8 is a photograph of two
such test rigs . The test rig design also accomm oda tes testing of spheri cal roll er and cylin drical
roll er bearing s as indicated in Figu re 11.9. The test rig design is also applicab le to large
bearing sizes as shown in Figure 11.10.
Tests are often condu cted to define the life of bearing s in specia l applic ations. They are
frequent ly call ed life or enduran ce tests, but, more co rrectly, they are extended durati on
perfor mance test s. The same basic test practices and test rig configu rations are requir ed for
� 2006 by Taylor & Francis Group, LLC.
FIGURE 11.7 Schematic diagram of a tapered roller bearing test configuration. Four bearings are tested
simultaneously under a sudden death test strategy. (Courtesy of the Timken Company.)
FIGURE 11.8 Photograph of a four-bearing test rig. The test housing can be used to test spherical roller
and cylinder roller bearings as well as tapered roller bearings. (Courtesy of the Timken Company.)
� 2006 by Taylor & Francis Group, LLC.
FIGURE 11.9 Schematic diagram of spherical roller bearing test configuration. Four bearings are tested
simultaneously under a sudden death test strategy. (Courtesy of the Timken Company.)
FIGURE 11.10 Photograph of rigs for fatigue testing four large bearings simultaneously. These
particular rigs accommodate bearings that have 460-mm (18 in.) outside diameter. (Courtesy of the
Timken Company.)
� 2006 by Taylor & Francis Group, LLC.
FIGURE 11.11 A-frame automotive wheel hub bearing tester. (Courtesy of SKF.)
these tests, but some modifications of philosophy are required to simulate the major operat-
ing parameters of the application while achieving realistic test acceleration. An example of
this type of tester is the SKF ‘‘A-frame’’ tester developed for evaluating automotive wheel
hub bearing assemblies, see Figure 11.11. This tester simulates an automotive wheel bearing
environment by using actual automobile wheel bearing mounting hardware, combined radial
and axial loads applied at the tire periphery to produce moment loads on the bearing
assembly, grease lubrication, and forced air cooling. Dynamic wheel loading cycles equivalent
to those produced by vehicle lateral loading are applied cyclically to simulate a critical driving
sequence. Testing is conducted in the sudden death mode so that wheel hub bearing unit life
can be calculated using standard life test statistics. This test provides a way to compare the
relative performance of automotive wheel support designs using life data generated under
conditions similar to those of actual applications.
11.6 STATISTICAL ANALYSIS OF ENDURANCE TEST DATA
11.6.1 STATISTICAL DATA DISTRIBUTIONS
Many statistical distributions have been used to describe the random variability of the life of
manufactured products. Such choices can be variously justified. For example, if a product has
a reservoir of a substance that is consumed at a uniform rate throughout the product’s life,
and if the initial supply of the substance varies from item to item, according to a normal
(Gaussian) distribution, then the product life will be normally distributed. Correspondingly, if
� 2006 by Taylor & Francis Group, LLC.
the initial amoun t of the substa nce follows a gamm a distribut ion, item life will be gamm a
dist ributed.
The Weibull distribut ion is a popular pr oduct-lif e model, just ified by its pro perty of
descri bing, unde r fairly general circum stances, the way the smal lest values in large sampl es
vary among sets of large samples. Therefor e, if item life is determ ined by the smal lest life
among many potenti al failure sites, it is reasonabl e to expect that life will vary from item to
item accord ing to a Weibull distribut ion.
Anothe r pr operty that makes the Weibu ll distribut ion a reason able ch oice for some
prod ucts is that it can accou nt for a steadi ly increa sing failure rate charact eristic of wear-
out failures , a steadi ly decreas ing failu re rate charact eristic of a product that benefits from
‘‘bur n-in,’’ or a co nstant failure rate typical of prod ucts that fail due to the occurrence of a
rando m sho ck.
The two-para mete r W eibull distribut ion was adopted by Lundber g an d Pal mgren [2] to
descri be rolling be aring fatigue life on the strength of the excell ence of the empir ical fit
to bearing fatigu e life data. As sho wn in Chapter 8, when ope rating unde r moderat e load
and optim um lubri cation cond itions, a well- designe d, manufa ctured, and applie d be aring can
end ure indefi nitely without exp eriencing fatigue failure. The W eibull model cann ot descri be
this aspect of fati gue life. Never theless, under the relat ively high loads common in fatigue
testing practice, the Weibull distribution will closely approximate the observed fatigue behavior
of rolling bearing s.
11.6.2 T HE T WO-PARAMETER WEIBULL DISTRIBUTION
11.6. 2.1 Probab ility Function s
Wh en it is said that a rand om varia ble, for exampl e bearing life, follows the two -parameter
Weibu ll distribut ion, it is impl ied that the pro bability that an observed value of that variable
is less than some arbit rary value x can be express ed by
Prob ðlif e < x Þ ¼ F ð xÞ ¼ 1 � e �ðx=hÞb x, h, b> 0 ð11 : 1Þ
F (x) is known as the cumula tive dist ribution functio n (CDF ), h is the scale parame ter, and
b is the shape parame ter. F (x) may be consider ed the area unde r a curve f (x) between
0 and the a rbitrary value x. This cu rve is know n as the pr obability density functio n (pdf)
and has the form
f ðxÞ ¼ xb�1
hbe�ðx=hÞb ð11:2Þ
Figure 11.12 is a plot of the We ibull pdf for various values of b . It is noted that a wide
diversity of distribution forms are encompassed by the Weibull family, depending on the
value of b. For b¼ 1, the Weibull distribution reduces to the exponential distribution. For b
in the range of 3.0–3.5, the Weibull distribution is nearly symmetrical and approximates the
normal pdf. The ability to assume such a range of shapes accounts for the extraordinary
applicability of the Weibull distribution to many types of data.
� 2006 by Taylor & Francis Group, LLC.
b= 4.0
b= 2.0
b= 1.0
b= 12
f ( x )
x
FIGURE 11.12 The two-parameter Weibull distribution for various values of the shape parameter b.
11.6.2.2 Mean Time between Failures
The average or expected value of a random variable is a useful measure of its ‘‘central
tendency’’; it is a single numerical value that can be considered to typify the random variable.
It is defined as
EðxÞ ¼Z 1
0
x f ðxÞ dx ¼Z 1
0
xxb�1
hb
� �e�ðx=hÞbdx ð11:3Þ
The value of the integral the above equation is
EðxÞ ¼ hG1
bþ 1
� �ð11:4Þ
G( ) is the widely tabulated gamma function. Table CD11.1 gives values of G(1/bþ1) for b
ranging from 1.0 to 5.0.
In reliability theory, E(x) is known as the mean time between failures (MBTF). It
represents the average time between the failures of two consecutively run bearings; that is,
the time between the failure of a bearing and the failure of its replacement. It does not
represent the mean time between consecutive failures in a group of simultaneously running
bearings. For this latter situation, provided b � 1, MBTF will vary with the failure order
number. For example, the mean time between the first and second failures in samples of size
20 is different from the mean time between the 19th and 20th failures.
The scatter of a random variable is often characterized by a quantity called variance,
defined as the average or expected value of the squared deviation of the variable from its
expected value. Variance is given by
s2 ¼Z
x� EðxÞ½ �2f ðxÞ dx ¼Z
x� hG1
bþ 1
� �� �2xb�1
hbe�ðx=hÞbdx ð11:5Þ
The value of this integral is
s2 ¼ h2 G2
bþ 1
� �� G2 1
bþ 1
� �� �ð11:6Þ
Values of the quantity [G(2/bþ1)�G2(1/bþ1)] are given in Table CD11.1.
� 2006 by Taylor & Francis Group, LLC.
The units of varia nce s2 are the square of the units in whi ch life is measur ed; for example,
(revol utions )2 or (hr) 2. The standar d deviation or square root of s2 is often prefer red as a
measur e of scatt er be cause it is express ed in the same units as the variable itself . Neithe r the
varia nce nor the standar d de viation is cited mu ch for the Weibull dist ribution; it is more usual
to convey the magni tude of scatter by citing the values of a low percent ile an d a high
percen tile.
11.6. 2.3 Percentiles
Equat ion 11.1 gives the probabil ity that the observed value of a Weibull rando m varia ble is
less than an arbit rary value. The invers e prob lem is to find a value of x , say xp, for whi ch the
probability is a specified value p such that life will not excee d it. The term x p is defined
implicitly as
F xp
� �¼ 1� e�ðxp=hÞb ¼ p ð11:7Þ
The solution of the above equation is
xp ¼ h ln1
1 � p
� �� �1=b
ð11 : 8Þ
An impor tant specia l case in rolling bearing engineer ing is the 10th percen tile x0.10 , because it
is hist orically cu stomary that bearing s are rated by the value of their 10th percent ile life. In
bearing literat ure, x0.10 is called L10. For consistency with the statistical literature on the
Weibull distribution, x0.10 is used in this discussion. It is expressible as
x0:10 ¼ h ln1
1 � 0:10
� �� �1 =b
¼ h 0:10 54ð Þ1=b ð11 : 9Þ
The median life x0.50 is also of some interest:
x0:50 ¼ h ln1
1 � 0:50
� �� �1 =b
¼ h 0:69 31ð Þ1=b ð11 : 10 Þ
Usi ng Equation 11.8, the ratio of two percent iles, say xp and xq, is
xq
xp
¼ ln ð 1 � qÞ�1
ln ð 1� pÞ�1
" #1 =b
ð11:11Þ
Thus,
x0 :50
x0 :10
¼ ln ð1 � 0: 50 Þ�1
ln ð1 � 0: 10Þ�1
" #1=b
¼ 0: 6931
0:1054
� �1 =b
For b¼ 10/9, theref ore, x0.50 ¼ 5.45. This sup ports the rule, often quoted in the bearing
industry, that median life L50 � 5�L10, the rating life.
� 2006 by Taylor & Francis Group, LLC.
11.6. 2.4 Graphical Re presentati on of the Weibul l Distr ibution
From Equat ion 11.1, the probabil ity that a be aring survi ves a life x denoted S( x) is given by
Sð xÞ ¼ 1 � F ðx Þ ¼ e�ðx=hÞb ð 11 : 12 Þ
Taking natural logarithm s tw ice on both sides of the above equ ation leads to
ln ln1
S
� �¼ b ln ð xÞ � lnðhÞ½ � ð11 : 13 Þ
The right-han d side is a linea r functio n of ln(x). On specia l graph pa per, called Weibull
probab ility pap er, for which the ordinat e is ruled pr oportio nately to ln[ln(1/ S)] and the
abscis sa is logarithm ical ly scaled , the values of S vs. the associ ated values of x plot as straight
line. If in the design of the pa per the same cycle lengt hs are used for the logarithm ic scale on
both co ordinat e axes, the slope of the stra ight line repres entat ion will be numeri cally equal to
b. In any case, the Weibu ll shap e parame ter or Weibull slope will be relat ed to the slope of the
straight line repres entation, and in some de signs of prob ability paper an au xiliary scale is
provided to relate the shap e pa rameter and the slope.
Figure 11.13 is a plot on Weibull probabil ity coordinat es on which the dist ribution with
b¼ 1.0 an d x0.10 ¼ 15.0 is repres ented. It was co nstructed by passi ng a 45 8 line through the
point co rrespondi ng to the failu re prob ability value F ¼ 0.1 (S ¼ 0.9) a nd the life v alue
x0.10 ¼ 15.0. From this plot, the 20th may be read as the a bscissa value at whi ch a horizont al
line at the ordinate value F ¼ 0.2 inter sects the straight line. Within graphic al accuracy ,
x0.20 ¼ 32.0. Invers ely, the prob ability of failing before the life x ¼ 52.0 is read to be roughly
30%. Representing a Weibull population on probability thus offers a graphical alternative to
the use of Equat ion 11 .1 and Equat ion 11.8 for the calcul ation of prob abilities and percent -
iles. The graphical approach is sufficiently accurate for most purposes. The primary use of
probability paper is not, however, for representing known Weibull distributions, but for
estimating the Weibull parameters from life test results.
9590
80706050
40
30
458
20
10
10 32.0 52.0 100x
F (
x) (
%)
1000
86
4
2
FIGURE 11.13 Graphical representation of the Weibull population for which b¼ 1 and x0.10¼ 15.0.
� 2006 by Taylor & Francis Group, LLC.
11.6.3 E STIMATION IN S INGLE SAMPLES
11.6. 3.1 Applic ation of the Weibull Distr ibution
Thus far it ha s been assumed that the Weibull parame ters are known , and a dditionall y
requir ed quan tities such as pr obabiliti es, percent iles, expecte d v alues, varian ces, and standar d
deviat ions have been calculated in term s of these known parame ters. This is a common
situ ation in be aring applic ation engineer ing, in which, given a catalog calcula tion of x0.10
( L10 ) and the standa rd Weibull slope of b¼ 10/9, it is required to calculate the media n life, the
MBTF , and so on. In developm ental work invo lving new varia bles such as mate rials,
lubri cants, or co mponent fini shing process es, the focus is on determ ining the effe ct of these
fact ors on the W eibull parame ters. Accor dingly , a sampl e of be arings modif ied from the
standar d in some way is subject ed to testing unde r standar dized cond itions of load and speed
until some or all fail. Whe n all fail, the sample is said to be uncen sored. In a censored sample,
some bearing s are remove d from test before failure. Give n the lives to failu re or to test
suspen sion of the unfail ed bearing s, the a im is to de duce the unde rlying Weibu ll parame ters.
This process is call ed esti mation because it is recogn ized that, since life is a rando m varia ble,
identi cal sampl es wi ll result in diff erent test lives. The Weibull pa rameter values esti mated in
any single sample must themselves be regarde d a s observed v alues of random variables that
will vary from sampl e to sample accordi ng to a pro bability distribut ion know n as the
sampl ing distribut ion of the estimat e. The scatter in the sampl ing distribut ion will decreas e
as the sampl e size is increa sed. The sample size therefore affe cts the degree of precis ion
with which the parame ters are determ ined by a life test. The precision is express ed by an
unc ertainty or c onfidence interval within whi ch the parame ter value is likely to lie. An
estimat ion proced ure that resul ts in the calcul ation of a co nfidenc e interval is call ed interval
estimat ion. A pro cedure that resul ts in a single numerical value for the pa rameter is call ed
point estimat ion. Point estimat es in thems elves are virt ually useles s, because, without some
qua lification, there is no way of jud ging how precis e they are.
Accor dingly, an analyt ical techni que is given in the seq uel for compu ting inter val esti mates
of Weibull parame ters. It is recomm end ed that this techni que be sup pleme nted, howeve r, with
a point estimat e obtaine d graph ically. The graphic al ap proach to estimat ion g ives a synop tic
view of the entire dist ribution a nd offer s the opportunit y to detect anomal ies in the da ta that
cou ld easil y be overlooked if reliance is placed entirely on the analyt ical techni que .
11.6. 3.2 Point Estima tion in Single Sampl es: Graph ical Method s
Ass uming that a sampl e of n bearing s is test ed until all fail , the ordered tim es to failure are
den oted x1 < x2 < � � �< xn. If the CDF of the Weibull popul ation from which the sampl e was
draw n wer e known, it woul d foll ow that live s xi and the values F( xi ), i ¼ 1, . . . , n, woul d plot
as a stra ight line on W eibull prob ability paper. It ha s been shown that even if function F( x) is
not know n, neverth eless, F( xi ) will va ry in repeated samples ac cording to a known pdf. The
mean or exp ected value of F( xi ) ha s be en shown to equal i /(nþ 1). The media n value of F( xi ),
also know n as the media n rank, has been shown by Johnson [15] to be app roximatel y
( i � 0.3)/( n þ 0.4). The procedure then is to plot the mean or media n value of F (xi) a gainst
xi for i ¼ 1, 2, . . . , n. The traditi on in the be aring indust ry is to use the media n rather than the
mean as a plott ing posit ion ch oice, but the differen ce is smal l compared with the sampl ing
varia bility.
Table 11.3 lists the ordered lives at failure for a sampl e of size n ¼ 10, along wi th the actual
and approxim ated values of the media n ranks. Hence, the approxim ation is adequate within
the limits of graphical accuracy. The median ranks are shown plotted against the lives in
Figure 11.14.
� 2006 by Taylor & Francis Group, LLC.
TABLE 11.3Random Uncensored Sample Size of n 5 10
Failure Order Number (i) Life Median Rank (i 2 0.3)/(n 1 0.4)
1 14.01 0.06697 0.06731
2 15.38 0.16226 0.16346
3 20.94 0.25857 0.25962
4 29.44 0.35510 0.35577
5 31.15 0.45169 0.45192
6 36.72 0.54831 0.54808
7 40.32 0.64490 0.64423
8 48.61 0.74142 0.74038
9 56.42 0.83774 0.83654
10 56.97 0.93303 0.93269
The straight line fitted to the plotted points represents the graphical estimate of the entire
F(x) curve. Estimates of the percentiles of interest are then read from the fitted straight line.
For example, within graphical accuracy, the x0.10 value is estimated as 15.3. The Weibull
shape parameter, estimated simply as the slope of the straight line, is roughly 2.2.
The same graphical approach applies to right-censored data in which the censored
observations achieve a longer running time than do the failures. The full sample size n is
used to compute the plotting positions, but only the failures are plotted. When there is mixed
censoring, that is, there are suspended tests among the failures, the plotting positions are no
longer calculable by the method given because the suspensions cause ambiguity in determin-
ing the order numbers of the failures. Several alternative approaches are available for this
situation, with generally negligible difference among them. Nelson’s [16] method, known as
9590
8070605040
30
20
108
6
4
2101 100x
F(x
) (%
)
FIGURE 11.14 Probability plot for uncensored random sample of size n¼ 10.
� 2006 by Taylor & Francis Group, LLC.
TABLE 11.4Calculation of Plotting Positions for a Hazard Plot
Life Reverse Rank Hazard (h) Cumulative Hazard (H) F 5 1 2 e2H
0.569 S 10 — — —
8.910 F 9 0.1111 0.1111 0.1052
21.410 S 8 — — —
21.960 F 7 0.1429 0.2540 0.2243
32.620 S 6 — — —
39.290 F 5 0.2000 0.4540 0.3649
42.990 S 4 — — —
50.400 F 3 0.3333 0.7873 0.5449
53.270 S 2 — — —
102.600 S 1 — — —
hazard plotting, is recomm ended because it is easy to use. Col umn 1 of Table 11.4 gives the
live s of failure or test suspensio n in a sampl e of size n ¼ 10. Of the ten bearing s, r ¼ 4 have
failed, and the lives of failure are marked with an ‘‘F’’ in Table 11.4. Simi larly, the lives at test
suspen sion are marked ‘‘S’’. The lives in column 1 are in ascendi ng ord er of time on test
irre spective of whet her the bearing failed. Column 2, term ed the reverse rank by Nelson [16],
assi gns the value n to the low est time on test, the value n � 1 to the next lowest, and so on.
Col umn 3, call ed the ha zard, is the recipr ocal of the revers e rank, but is calculated only for the
failed bearing s. Col umn 4 is the cumula tive hazard and contai ns for each failu re the sum of
the hazard values in column 3 for that failure and each failure that occ urred at an earlier
runn ing time. Thus, for the second failure, the c umulative hazard is 0.2540 ¼ 0.1111 þ 0.1429.
The cu mulative ha zard can then be plott ed against life on prob ability that has been designe d
with an ex tra ‘‘hazard’’ scale . If graph paper of this type is not available, it is only necessa ry to
calculate an estimate of the plotting position applicable to ordinary paper by transforming the
cumula tive ha zard H to F ¼ 1 � e �H . Thi s compu tation is shown in co lumn 5 of Table 11.4.
Figure 11.15 shows the resultant plot. It is noted that, as in the right -censored case, only the
failures are plotted. The suspended tests have played a role, however, in determining the
plotting positions of the failures.
11.6.3.3 Point Estimation in Single Samples: Method of Maximum Likelihood
The method of maximum likelihood (ML) is a general approach to the estimation of the
parameters of probability distributions. The central idea is to estimate the parameters as the
values for which the last observed test sample would most likely have occurred.
Considering an uncensored sample of size n, the likelihood is the product of the pdf
f(x)¼ xb�1/hb exp[�(x/h)b] evaluated at each observed life value. The ML estimates of h and b
are the values thatmaximize this product. For a censored samplewith r< n failures, the likelihood
function contains, in lieu of the density function f(x), the term 1�F(x)¼ exp[�(x/h)b] evaluated
at each suspended life value. It can be shown that the ML estimate of b, denoted by a caret (^), is
the solution of the following nonlinear equation:
1
bb¼
Pi¼r
i¼1
ln xi
r�
Pi¼n
i¼1
xbbi ln xi
Pi¼n
i¼1
xbbi
ð11:14Þ
� 2006 by Taylor & Francis Group, LLC.
70
60
50
40
30
20
108
6
4
21 10 100
x
F(x
) (%
)
FIGURE 11.15 Probability plot for mixed censoring. Plotting positions are calculated based on cumu-
lative hazard.
Accor ding to McCool [17], this equatio n ha s only a single posit ive solut ion. That solut ion is
found using the New ton–R aphson method, althoug h in high ly censored cases the guess value
used to start the method might need to be modif ied to avoid convergence to a negati ve
value for b.
Having determ ined b from Equation 11.14, the ML e stimate of h is obtaine d as foll ows:
hh ¼Xi¼n
i¼1
xbbi
r
!1=bb
ð11:15Þ
The ML estimate of a general percentile is
xxp ¼ hhk1=bbp ð11:16Þ
where kp is defined as
kp ¼ �ln 1� pð Þ ð11:17Þ
Confidence limits can be set if the censoring mode corresponds to the suspension of testing
when the rth earliest failure occurs. This type of censoring is customarily called type II
censoring as contrasted to type I censoring in which testing is suspended at a predetermined
testing time. In the latter, the number of failures is predetermined by the experimenter.
The basis for confidence intervals for b is that the random function v(r, n)¼ b/b follows a
sampling distribution that depends on the sample size n and censoring number r, not on the
underlying values of h and b. Functions with this property are known as pivotal functions.
The sampling distribution of v(r, n) cannot be found analytically, but may be determined
empirically to whatever precision necessary by Monte Carlo sampling. In the Monte Carlo
� 2006 by Taylor & Francis Group, LLC.
method, rep eated sampl es are draw n by compu ter simulat ion from a Weibull distribut ion
with arbit rary parame ter values; for exampl e, b¼ 1.0 and h¼ 1.0. The M L estimat e b is
form ed for each sampl e an d divide d by the underlyin g value of b to yield a value of v ( r , n).
With typicall y 1 0,000 such values , the percent iles may be computed from the sorted set and
then equated to the exp ected value of the dist ribution.
Denotin g the 5th a nd 95th percent iles of v ( r , n) as v0.05 ( r, n) and v 0.95( r , n) leads to the
follo wing 90% c onfidence interval for b:
bb
v0 :95 ð r , nÞ< b <
bb
v0 :05 ð r , nÞð11 : 18 Þ
The raw ML esti mates of the W eibull parame ters are biased; that is, both the average and
media n of the b estimat es in an indefi nitely large numb er of samples will diff er somew hat from
the true b value for the populatio n from which the samples were draw n. It is possible to co rrect
the raw M L estimate so that eithe r its average or its media n wi ll co incide with the unde rlying
popul ation value of b. Bec ause the distribut ion of v (r , n) is not symm etrical, it is necessa ry to
cho ose whether the adjust ed estimat or sh ould be media n or mean unbiased. Median unbiased-
ness is recomm ended be cause then the ML point estimat e will ha ve the reasonabl e propert y
that it is just as likely to be larger than the unde rlying true value as to be smaller. McCool [17]
demon strated that the med ian unbi ased estimat e of b , den oted by b’ , is express ible as
bb0 ¼ bb
v0 :50 r , nð Þ ð11 : 19 Þ
Table CD11.2 gives values of v0.05( r , n), v 0.50 ( r , n), and v 0.95( r , n) for 5 � n � 30 and various
values of r .
Corr ect ing the b ias of the estim ate and setting confidence limits for a g en era l perc en tile xp
depend on the p ivotality of the random fun ction u(r, n, p) ¼ b ln ( xp / xp). Given percentiles of
u(r, n, 0.1 0) de termined by M onte C arlo sa mp ling , a 90% confi dence interv al on x0.10 can b e s et up:
xx0: 10 e � u0: 95 r ; n; 0 :10ð Þ=bb < x0: 10 < xx0 :10 e
� u 0: 05 r ; n ; 0 :10ð Þ=bb ð11 : 20 Þ
A median unbi ased estimate of x0.10 can be calcul ated as
xx 00: 10 ¼ xx0 :10 e � u 0: 50 r ; n ; 0 :10ð Þ=bb ð11:21Þ
Values of the 5th, 50th, and 95th percentiles of u(r, n, 0.10) are also given in Table CD11.2.
See Example 11.1.
11.6.3.4 Sudden Death Tests
A popular test strategy in the bearing industry is the ‘‘sudden death’’ test. In sudden death
testing, a test sample of size n is divided into l subgroups, each of size m (n¼ l�m). When the
first failure occurs in each subgroup, testing is suspended on that subgroup. When the test is
over, there are l failures, the first failures in each of the subgroups. To estimate b, these first
failures are substitut ed direct ly into Equat ion 11.14. Conf idence limit s for b are then calcu-
lated using Equation 11.18 with r¼ n¼ l. That is, the first failures are treated as members of
an uncensored sample whose size is equal to the number of subgroups. Table CD11.3 gives
� 2006 by Taylor & Francis Group, LLC.
the percent iles of v (l, l ) for 2 � l � 6. The value of x0.10 , determined by using the sampl e of first
failures an d Equat ion 11.16, is den oted as x0.10s. Accor ding to McCool [18], the ML estimat e
applic able to the complete sampl e is then calculated as follows :
xx0 :10 ¼ xx0: 10s m 1=bb ð 11 : 22 Þ
90% confidence limits for x 0.10 may be c omputed as
xx0: 10 e �u0: 95 l ; m; 0 :10ð Þ=bb < x0 :10 < xx0 :10 e
� u 0: 05 l ; m ; 0 :10ð Þ=bb ð 11 : 23 Þ
A med ian unbiased esti mate of x0.10 is calcul ated from
xx00 :10 ¼ xx0 :10 e � q 0: 50 =bb ð 11 : 24 Þ
Table CD11.3 gives values of the percent iles of the rand om functi on q( l , m, p) required for
these c alculati ons.
See Exampl e 11.2.
16.3. 3.5 Precision of Estimation: Sample Size Selection
A confidence interval reflects the uncertainty in the value of the estimated parameter due to
the finite size of the life test sample. As the sample size increases, the two ends of the confi-
dence interval approach each other; that is, the ratio of the upper to lower ends of the
confidence interval approaches 1. For finite sample sizes, McCool [19] suggested this ratio
as a useful measur e of the precis ion of estimat ion. From Equation 11.18, the co nfidenc e limit
ratio R for b estimation is
R ¼ v0 :95 r, nð Þv 0 :05 r , nð Þ ð 11 : 25 Þ
Values of R for various n and r are given in Table CD 11.2 for conventional tests and Tabl e
CD11.3 for sudden death test s. It is noted that for a given sampl e size n, the precis ion
impr oves ( R de creases) as the numbe r of failures r increa ses.
For x0.10 , the ratio of the upper to lower confidence limits contains the random variable
b. The approach taken by McCool [19] in this case was to use as a precision measure the median
value of this ratio, denoted R0.50. The expression for this median ratio contains the unknown
value of the true shape parameter b. For planning purposes, one may use an historical value
such as 10/9 or, alternatively, the value Rb
0:50 as the precision measure. Values of Rb
0:50 are given
in Table CD11.2 for conventional testing and Table CD11.3 for sudden death testing.
11.6.4 ESTIMATION IN SETS OF WEIBULL DATA
11.6.4.1 Methods
Very often an experimental study of bearing fatigue life will include the testing of several
samples, differing from each other with respect to the level of some qualitative factor under
study. A qualitative factor is distinct from a quantitative factor such as temperature or load,
which can be assigned a numerical value. Examples of qualitative factors are lubricants, cage
designs, and bearing materials.
� 2006 by Taylor & Francis Group, LLC.
McCool [20] showe d that more precis e estimat es can be made if the data in the sampl es
making up the c omplete invest igation are analyze d as a set. Thi s is possibl e if it can be
assum ed that sampl es are draw n from Weibull popul ations , whic h, althoug h they might differ
in their scale pa rameter values , none theless have a co mmon value of b.
Applicab le tabular values for carrying out the an alyses presu ppose that the sampl e size
n and the number of failu res r are the same for each sample in the set; hencefort h, this is
assum ed to be the case. It is thus assumed that k group s of size n have been tested until
the r th failu re occurred in each group . The fir st step is to de termine whet her it is plau sible
that the g roups hav e a co mmon value of b. This is done by an alyzing each grou p
indivi dually to determine the values of x0.10 and b. The large st and smallest of the k b
estimat es are then determ ined, and their ratio form ed. If the b values diff er among the
group s, this rati o woul d tend to be large. Table CD 11.4 gives the values of the 90th
percen tile of the rati o w ¼ bmax / b min for various r , n, and k. These values wer e de termined
by Monte Carlo sampling from k Weibull populati ons that had a co mmon value of b.
Ther efore, values of the ratio of the large st to the smal lest shape parame ter esti mates
exceed ing those in Table CD 11.4 wi ll occur only 10% of the time if the grou ps do have a
common value of b . Thes e values may be used as the critical values in de ciding whet her a
common b assum ption is just ified.
Havin g determined that a co mmon b assum ption is reasonabl e, this common b value can
be estimated using the data in each group, by solving the nonlinear equation
1
bb1
þ 1
rk
Xi¼k
i¼1
Xj¼r
j¼1
ln xiðjÞ �Xi¼k
i¼1
Pj¼n
j¼1
xbb1
iðjÞ ln xiðjÞ
kPj¼n
j¼1
xbb1
iðjÞ
ð11:26Þ
where b1 denotes the ML estimate of the common b value and xi(j)denotes the jth order failure
time wi thin the i th group . Conf idence limit s for b may be set a nalogousl y to Equation 11.17
as follows:
bb1
ðv1Þ0:95
< b <bb1
ðv1Þ0:05
ð11:27Þ
where v1 (r, n, k)¼ b1/b. A median unbiased estimate of b may be calculated from
bb0 ¼ bb1
ðv1Þ0:50
ð11:28Þ
Table CD11.5 gives percentiles of v1(r, n, k) needed for setting 90% confidence limits and for
bias correction of various values of n, r, and k. The scale parameter for the ith group may be
re-estimated with b1 as follows:
hhi ¼P
xbb1
iðjÞr
0@
1A
1=bb1
ð11:29Þ
The value of xpi may be estimated from
xxpi ¼ hhik1=bb1 ð11:30Þ
� 2006 by Taylor & Francis Group, LLC.
Conf idence lim its for x0.10 may be computed as follo ws:
xx0: 10 e �ðu 1 Þ0: 95 =bb1 < x0 :10 < xx0 :10 e
�ð u1 Þ 0:05 =bb 1 ð 11 : 31 Þ
where u1 ¼ b 1 ln(x 0.10/ x0.10 ) is the k sampl e general ization of u( r, n, 0.10). The media n un-
biased estimat e of x0.10 may be compu ted as
xx00 :10 ¼ xx0:10 e �ðu 1 Þ0: 50 =bb 1 ð 11 : 32 Þ
Now that x0.10 has been estimated for each group using the ML estimate b 1 of the co mmon
shape parame ter, the next questi on of interest is whet her these x0.10 values differ significantly.
That is, are the apparent differences among the x0.10 estimates real, or could they be due to
chance? To test whether the underlying true x0.10 values are all equal, the magnitude of
variation that could occur in the estimated values due to chance alone must be assessed.
This can be done by using the random function t1(r, n, k) defined by
t1 r, n, kð Þ ¼ bb1 lnxx0 :10ð Þmax
xx0:10ð Þmin
� �ð11 : 33Þ
wher e ( x0.10) max and ( x0.10 ) min are the large st and smal lest values of x 0.10 calculated amo ng the
k samples. The 90th and 95th percen tiles of t1( r , n, k) may be us ed to assess the obs erved
difference in the x0.10 values . Any two sampl es, for e xample, sample i and sampl e j, for whi ch
values the qua ntity b1 ln [( x0.10) i / (x0.10) j ] exceed s ( t 1) 0.90 , may be declar ed to diff er from each
other at the 10% level of signifi cance. Correspon dingly , using the 95th percent ile of t 1( r , n, k)
resul ts in a 5% signifi cance level test for the equali ty of the x0.10 values .
See Example 11.3.
11.7 ELEMENT TESTING
11.7.1 R OLLING C OMPONENT ENDURANCE T ESTERS
Conduct ing an endurance test seri es on full -scale bearing s is expen sive because numerou s test
sampl es are requir ed to obtain a useful experi menta l life esti mate. The identi fication of
simp ler, less costly, life test ing methods has therefore be en a long standing object ive. The
use of elem ental test con figuratio ns offer s a potenti al solut ion to this need . In this approach, a
test specim en that has a simp lified geomet ry (e.g ., flat was her, rod, or ball) is used, and RC is
developed at multiple test locations. The aim is to extrapolate the life data generated in an
element test to a real bearing application, thus saving calendar time and cost as compared
with life data generated using full-scale bearing tests. This objective has historically not been
achieved, generally because all of the operating parameters influencing fatigue life of rolling–
sliding contact s were not redu ced to stresses ; rather , as is shown in Chapter 8, they wer e
evaluated as life factors. The only stress directly evaluated in both element and full-scale
bearing endurance testing has been the Hertz or normal stress acting on the contact. Lubri-
cation, contamination, surface topography, and material effects have been evaluated as life
factors. To be able to extrapolate the life data derived from element testing to full-scale
bearing life data, it is necessary to evaluate both data sets from the standpoint of applied and
induced stresses as compared with material strength. The methods to accomplish this for full
� 2006 by Taylor & Francis Group, LLC.
bearing s are develop ed in Chapte r 8; Harr is [21] developed a simila r method for balls
end urance tested in v-ring test rigs.
Even without direct correl ation of life test data betw een elem ents and full-scale be arings,
elem ent test ing has proven useful in the ability to rank the perfor mance of various mate rials
in initial screenin g sequen ces or in adverse e nvironm ents, such as extremely low or high
tempe ratur e, oxidiz ing atmos phe res, and vacuum. Ther efore, a discussion of element life
testing techn iques is war ranted, even when the test data evaluation techni ques do not permit
direct correl ation with full-sc ale bearing life test data. Caution must alw ays be used , howeve r,
becau se the precision of the ranking process is ope n to questi on. Per formance revers als have
somet imes been experien ced when compari ng the screenin g e lement test resul ts when the
mate rials have be en retes ted in actual bea rings. Suc h reversals can be avoided if both the
elem ent test data and actual bearing test data evaluations are based on the total stress
con siderati on.
The oldest and perhaps the most widely used element test c onfiguration i s t he rolling
four-bal l machine or Barwell [22] tester developed i n t he 1950s. This system uses four
12.7-m m ( 0. 5 0 in. ) diamet er bal ls t o sim ulate an angular-contact ball bearing operating
with a vertical axis under a pure thrust l oad. On e ball is t he primary t est e lement serving
as the inner r ing of t he bearing assembl y. It is sup ported in pyramid fashion on the
remaining t hree balls, w hi ch rotate freely in a conforming cup at a prede termined contact
angle. A m odification of this test method , t he roll ing f ive-bal l test er, w as subsequently
developed at N ASA Lewis Research Center (now NASA Glenn R esearch C enter) [23].
To generate more stress cycles, t he test ball is supported by a g roup of four balls. This
system , illustrat ed in Figure 11.16 and Figure 11.17, has been used to generate an
extensive amount of life test data on standard and e xperimental beari ng steels. Acceler-
ated life testing is typically conducted at Hertz stresses of 4,138 M Pa (600,000 psi). This
loading involves some plastic deformations of the materials, m aking extrapolation of the
life test data to complete bearings unreliable. The tester m ay be used to compare R C
materials and lubricants.
Another widely used e lement test system, as illustrated in Figure 11.18, is the RC
tester developed by General Electric [24]. The test element i n this configuration i s a 4.76-
mm (0.1875 i n.) diameter r od rotating under load between two 95.25-mm ( 3.75 in.)
diameter disks. The r od can be axially repositioned t o achieve a number of R C tracks
on a single ba r . U nfortunately, this c onfiguration i s not as cost effective a s i t f irst
appears. Stress concentrations occur at the edges of t he rod c ontact unless the disks
are profiled ( crowned) in the axial direction. This significantly increases the cost of
manufacturing the disks. During operation, fatigue failures on t he rod also tend t o
damage the disk surfaces, requiring these t o be refinished at regular intervals. To achieve
accelerated t esting, Hertz stresses as gr eat as 5,517 MPa (800,000 psi) are f requently
employed. This loading is substantially in the regime of plastic deformations; hence,
extrapolation of data for prediction of bearing f atigue endurance is unreliable. T he t ester
is mainly used to compare RC m aterials.
A varia tion of an RC endurance test rig using a cy lindrical rod as the test elem ent was
descri bed by Glove r [25]. In this rig dev eloped by Federal– Mogu l–Bower and illu strated in
Figure 11.19, the load is applie d to the ro d through three balls su pported in tapere d roller
bearing outer rings (cups) . The typical ap plied Hertz stress in each contact is 4,138 M Pa
(600, 000 psi) , which , as explaine d ab ove, involv es some plastic de formati on.
Anothe r elem ent test co nfigurati on is the single ball tester developed by Pratt and W hitney,
Unit ed Tec hnologies Corporati on, for evaluat ing balls used in aircr aft gas turbi ne engine
bearing s [26] . Thi s syst em, sh own in Fi gure 11.20 and Figure 11.21, tests balls from app roxi-
� 2006 by Taylor & Francis Group, LLC.
1
2
3
4
56
qt2t1
ωl sin q
f
a
h
b
c
d
p
g
j
i
k
n
o
e
ml
FIGURE 11.16 NASA rolling five-ball test system.
mately 19–65 mm (0.75–2.50 in.) in two v-ring raceways with lubrication to simulate the
application. Ball/v-ring contact angles are typically 258 or 308 inducing spinning components
of angular velocities in the contacts, and substantial sliding. Hertz stresses in the contacts
are typically 4,000 MPa (580,000 psi), thus involving some plastic deformations. Harris [21]
developed a stress-based ball life prediction method for this system. Subsequently, endurance
FIGURE 11.17 Group of five-ball test rigs.
� 2006 by Taylor & Francis Group, LLC.
FIGURE 11.18 General Electric Polymet RC disk machine.
test data accumul ated using this rig were used in the de velopm ent of fatigue limit stre ss values
for severa l be aring RC c omponent mate rials [27].
11.7.2 ROLLING–SLIDING FRICTION TESTERS
11.7. 2.1 Purpose
The main purp ose of the element test rigs descri bed above is to accumul ate RC fati gue
end urance data in an econo mical, efficient, and rapid mann er. The relative influenc e on
bearing fatigu e endurance of mate rials, material process ing, lub ricants, and so on can be
invest igated thereby . Some of the most signifi cant stre sses that determine the extent of
bearing life are the surfa ce shear stre sses occurri ng in the roll ing elem ent–race way contacts.
Element test rigs can be designe d to investiga te the influen ce of friction on bearing en durance
and also to help quantify the magni tude of traction in the roll ing–slidi ng contact s that occur
in many rolling bearing a pplications .
11.7. 2.2 Rolling–S liding Disk Test Rig
To exp erimental ly determ ine the magnitud e of the fricti onal stre sses occu rring in EHL
con tacts, roll ing–slidi ng disk machin es have been de veloped. The device developed by Ne lias
et al. [28] is illu strated in Figu re 11.22. The disks are contoured to produce elliptical con tact
areas as ill ustrated in Figure 11.23. The motor s in Figure 11.22 may turn at diff erent speed s to
� 2006 by Taylor & Francis Group, LLC.
1. Specimen2. Ball3. Tapered bearing cup4. Ball retainer5. Compression spring6. Upper cup housing
7. Spring retainer plate8. Lower cup housing
10. Load application bolt9. Shock mount
11. Spring calibration bolt
1 23 4511 10
6
7
8
9
A A
FIGURE 11.19 Ball–rod RC fatigue test rig.
achieve the desired rolling–slidi ng moti on. Motor 2 is moun ted in hyd rostatic cylind rical
bearing s to permi t fricti on torque, and hen ce, fricti on force measur ement . The fri ction force
Ff to ap plied force W rati o is called the tract ion co efficie nt. Using the analytical method s of
Chapt er 5, the e ffective local ( x, y) frictio n coeff icients can be estimate d from the test resul ts.
In Chapt er 8, it is sho wn how the test device in Figure 11.22 has been used to determ ine the
charact eristic s of the effe ct of fricti on on fatigue of the roll ing–slidi ng con tacts in ba ll an d
� 2006 by Taylor & Francis Group, LLC.
Hydraulicloading cylinder
Load transmittingplatform
Ball
Accelerometer
Lower oil jetDrive hub
Retainer
Free wheelingloading hub
Upper oil jet
( b )
FIGURE 11.20 Photograph (a) and drawing (b) of a Pratt and Whitney single ball/v-ring test rig.
roll er bearing s. As discus sed in Chapt er 8, by equipping the test rig wi th the con taminate d
lubri cation system of Figure 11.24, Ville and Ne lias [29] invest igated the effe cts of particu late
con taminati on on rolling–sl iding con tact fatigue.
11.7. 2.3 Ball–D isk Test Rig
A ball– disk test rig, initial ly de veloped by W edeven [30] and shown in Figure 11.25, was
designe d to de termine the nature of lubri cant films in point co ntacts. Roll ing veloci ty may
be varie d by varyi ng the ball drive spindl e angle and the radius at whi ch the ball co ntacts
� 2006 by Taylor & Francis Group, LLC.
FIGURE 11.21 Schematic diagram of Pratt and Whitney single ball/v-ring test rig.
the disk. Usin g a disk elem ent of a clear mate rial such as sapph ire or glass a nd optica l
interfer ometr y, the pr essure distribut ions in Hertz point contact s could be displayed (see
Figure 4.11) . The rig has been further de veloped by Wedeven [31] with separat ely power ed
ball drive and disk drive shafts and with an air bearing sup port of the disk drive. It is thereby
possibl e to de termine con tact traction force vs. slide–roll rati o; the grap hical displ ay in Figure
11.26 was the output from the test rig. A mathe matical model of the EH L circul ar point
contact may also be developed, a nd by the matc hing analytical an d e xperimental data, the
localized friction components that comprise the traction may be determined.
Motor 2
Test disks
Stand
W
Motor 1
Hydrostatic cylindrical bearings
FIGURE 11.22 Schematic drawing of rolling–sliding disk testing device. (From Nelias, D., et al., ASME
Trans., J. Tribol., 120, 184–190, April 1998. With permission.)
� 2006 by Taylor & Francis Group, LLC.
w
z
w
Rx2 Ry2
Rx2
Ry2x
ω2
ω1
2a
2cy
FIGURE 11.23 Illustration of elliptical contact area generated by the rolling–sliding disk test device.
(From Nelias, D., et al., ASME Trans., J. Tribol., 120, 184–190, April 1998. With permission.)
The rig can be equipped with an environment chamber to allow evaluation of the traction
coefficient under conditions of high and low temperature, and high vacuum. It further permits
optical examination of the circular point contact under the effects of lubricant particulate
contamination; the photographs in Figure 10.34 were obtained using such a test rig.
11.8 CLOSURE
In Chapter 11 in the first volume of this handbook, it was demonstrated that although ball
and roller bearing fatigue life rating and endurance formulas are founded in theory, they are
Tank
3 Waygate
Test disks
Setting tank+
Magnest
Sensor
Particle counter
Pump
Head race
12 et 3 μm Cleaning filters
Oil+
Contaminants
FIGURE 11.24 Schematic diagram of a lubrication contamination system used in conjunction with a
rolling–sliding test rig. (From Ville, F. and Nelias, D., Early fatigue failure due to dents in EHL
contacts, Presented at the STLE Annual Meeting, Detroit, May 17–21, 1998. With permission.)
� 2006 by Taylor & Francis Group, LLC.
FIGURE 11.25 Ball–disk traction test rig. (From Wedeven Associates, Inc., Bridging Technology and
Application through Testing, Brochure, 1997. With permission.)
semiempirical relationships requiring the establishment of various constants to enable their
use. These constants, which depend on the bearing raceway and rolling element materials, can
be established only by appropriate testing. Because of the stochastic nature of rolling bearing
fatigue endurance, testing procedures necessarily require bearing or material populations of
sufficient size to render the test results meaningful. Sample sizing effects were discussed in
detail herein.
Historically, to establish sufficiently accurate rating formula constants, it has been
necessary to test complete bearings. With the development of stress-based life factors as
0.02
0.00
−0.02
−0.04
−0.06
−0.08
−0.10
−20 −10 0 10
% Slip
Tra
ctio
n co
effic
ient
FIGURE 11.26 Curve of traction coefficient vs. percent sliding obtained from Wedeven ball–disk test rig.
� 2006 by Taylor & Francis Group, LLC.
shown in Chapter 8, however, it i s now possible t o use element t esting methods to determine
many of these constants. For example, endurance testing of balls in v-ring test rigs may be
used to determine the basic material fatigue strengths of various materials. On the other
hand, some of the stresses that influence bearing life depend on the raceway forming and
surface finishing methods. To duplicate these effects, the exact component may need to be
endurance tested.
REFERENCES
1.
� 200
Weibull, W., A statistical theory of the strength of materials, Proc. R. Swed. Inst. Eng. Res., 151,
Stockholm, 1939.
2.
Lundberg, G. and Palmgren, A., Dynamic capacity of rolling bearings, Acta Polytech. Mech. Eng.,Ser. 1, 3(7), R. Swed. Acad. Eng., 1947.
3.
Lundberg, G. and Palmgren, A., Dynamic capacity of roller bearings, Acta Polytech. Mech. Eng.,Ser. 2, 4(96), R. Swed. Acad. Eng., 1952.
4.
Ioannides, E. and Harris, T., A new fatigue life model for rolling bearings, ASME Trans., J. Tribol.,107, 367–378, 1985.
5.
International Organization for Standards, International Standard ISO 281, Rolling Bearings—Dynamic Load Ratings and Rating Life, 2006.
6.
Tallian, T., On competing failure modes in rolling contact, ASLE Trans., 10, 418–439, 1967.7.
Valori, R., Tallian, T., and Sibley, L., Elastohydrodynamic film effects on the load life behavior ofrolling contacts, ASME Paper 65-LUBS-11, 1965.
8.
Johnston, G., et al., Experience of element and full bearing testing over several years, RollingContact Fatigue Testing of Bearing Steels, ASTM STP 771, J. Hoo, Ed., 1982.
9.
Sayles, R. and MacPherson, P., Influence of wear debris on rolling contact fatigue, ASTM STP 771,J. Hoo, Ed., 1982, pp. 255–274.
10.
Fitch, E., An Encyclopedia of Fluid Contamination Control, Fluid Power Research Center, Okla-homa State University, 1980.
11.
Tallian, T., Failure Atlas for Hertz Contact Machine Elements, 2nd Ed., ASME Press, 1999.12.
Andersson, T., Endurance testing in theory, Ball Bear. J., 217, 14–23, 1983.13.
Sebok, G. and Rimrott, U., Design of rolling element endurance testers, ASME Paper 69-DE-24,1964.
14.
Hacker, R., Trials and tribulations of fatigue testing of bearings, SAE Technical Paper 831372,1983.
15.
Johnson, L., Theory and Technique of Variation Research, Elsevier, New York, 1970.16.
Nelson, W., Theory and application of hazard plotting for censored failure data, Technometrics, 14,945–966, 1972.
17.
McCool, J., Inference on Weibull percentiles and shape parameter for maximum likelihood esti-mates, IEEE Trans. Reliab., R-19, 2–9, 1970.
18.
McCool, J., Analysis of sudden death tests of bearing endurance, ASLE Trans., 17, 8–13, 1974.19.
McCool, J., Censored sample size selection for life tests, Proc. 1973 Ann. Reliab. Maintainab. Symp.,IEEE Cat. No. 73CH0714–64, 1973.
20.
McCool, J., Analysis of sets of two-parameter Weibull data arising in rolling contact endurancetesting, Rolling Contact Fatigue Testing of Bearing Steels, ASTM STP 771, J.J. Hoo Ed., American
Society for Testing and Materials, Philadelphia, 1982, pp. 293–319.
21.
Harris, T., Prediction of ball fatigue life in a ball/v-ring test rig, ASME Trans., J. Tribol., 119, 365–374, July 1997.
22.
Barwell, F. and Scott, D., Engineering, 182, 9–12, 1956.23.
Zaretsky, E., Parker, R., and Anderson, W., NASA five-ball tester—over 20 years of research,Rolling Contact Fatigue Testing of Bearing Steels, ASTM STP 771, J. Hoo, Ed., 1982.
24.
Bamberger, E. and Clark, J., Development and application of the rolling contact fatigue test rig,Rolling Contact Fatigue Testing of Bearing Steels, ASTM STP 771, J. Hoo, Ed., 1982.
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25.
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Glover, G., A ball–rod rolling contact fatigue tester, Rolling Contact Fatigue Testing of Bearing
Steels, ASTM STP 771, J. Hoo, Ed., 1982.
26.
Brown, P., et al., Evaluation of powder-processed metals for turbine engine ball bearings, RollingContact Fatigue Testing of Bearing Steels, ASTM STP 771, J. Hoo, Ed., 1982.
27.
Harris, T., Establishment of a new rolling bearing fatigue life calculation model, Final Report U.S.Navy Contract N00421-97-C-1069, February 23, 2002.
28.
Nelias, D., et al., Experimental and theoretical investigation of rolling contact fatigue of 52100 andM50 steels under EHL or Micro-EHL conditions, ASME Trans., J. Tribol, 120, 184–190, April
1998.
29.
Ville, F. and Nelias, D., Early fatigue failure due to dents in EHL contacts, Presented at the STLEAnnual Meeting, Detroit, May 17–21, 1998.
30.
Wedeven, L., Optical Measurements in Elastohydrodynamic Rolling Contact Bearings, Ph.D. Thesis,University of London, 1971.
31.
Wedeven Associates, Inc., Bridging Technology and Application through Testing, Brochure, 1997.6 by Taylor & Francis Group, LLC.
Appendix
TABLE A.1Unit Conversion Fac
Unit
Length
Force
Torque
Temperature difference
Kinematic viscosity
Heat flow, power
Thermal conductivity
Heat convection coefficien
Pressure, stress
a English system units equ
� 2006 by Taylor & Francis Grou
All equations in the text are written in metric or standard international system units. In this
appendix, Table A.1 gives factors for conversion of Standard International system units to
English system units. Note that for the former, only millimeters are used for length and
square millimeters for area. Furthermore, the basic unit of power used herein is the watt (as
oppos ed to kilow att). To be con sistent with this , Table A.2 provides the appropri ate Englis h
system units constant for each equation in the text that has a Standard International system
units constant.
torsa
Standard International System Conversion Factor English System
mm 0.03937, 0.003281 in., ft
N 0.2247 lb
mm � N 0.00885 in. � lb8C,K 1.8 8F, 8R
mm2/sec (centistokes) 0.001076 ft2/sec
W 3.412 Btu/hr
W/mm � 8C 577.7 Btu/hr � ft � 8F
t W/mm2 � 8C 176,100 Btu/hr � ft2 � 8F
N/mm2 (MPa) 144.98 psi
al Standard International system units multiplied by conversion factor.
p, LLC.
TABLE A.2Equation Constants for SI and English System Units
Chapter Number Equation Number SI System Constant English System Constant
1 33 3.84� 10�5 4.36� 10�7
34 3.84� 10�5 4.36� 10�7
35 1.24� 10�5 8.55� 10�8
36 1.24� 10�5 8.55� 10�8
37 1.24� 10�5 8.55� 10�8
39 0.62� 10�5 4.28� 10�8
42 0.62� 10�5 4.28� 10�8
52 1.24� 10�5 8.55� 10�8
54 0.62� 10�5 4.28� 10�8
58 0.31� 10�5 2.14� 10�8
65 3.84� 10�5 4.36� 10�7
74 0.62� 10�5 4.28� 10�8
75 0.31� 10�5 2.14� 10�8
3 27 2.26� 10�11 2.11� 10�6
59 2.26� 10�11 2.11� 10�6
60 3.39� 10�11 3.17� 10�6
62 2.26� 10�11 2.11� 10�6
63 3.39� 10�11 3.17� 10�6
67 4.47� 10�12 4.18� 10�7
68 8.37� 10�12 7.83� 10�7
106 2.15� 105 3.12� 107
107 3.39� 10�11 3.17� 10�6
4 64 4.597� 10�12 1.509� 10�18
65 8.543� 10�9 4.066� 10�13
7 6 103 30.2
7 9.551� 103 288.4
18 0.0332 0.332
19 0.060 0.60
20 2.30� 10�5 0.30
21 0.030 0.30
28 5.73 0.173
30 5.73� 10�8 0.173� 10�8
8 4 77.9 5.914� 103
11 464 4.166� 104
� 2006 by Taylor & Francis Group, LLC.