rolling bearing analysis

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Library ofCongresB Cataloging-in-Publication Data:Harris, Tedric A.

Rolling bearing analysis / Tedric A. Harris. - 4th ed.p. em.

Includes index.ISBN 0-471-35457-0 (cloth: alk. paper)1. Roller bearings. 2. Ball-bearings.

TJ1071.H35 2001621.8'22-dc21 00-038171

Printed in the United States of America.

10 9 8 7 6 5 4 3 2

PREFACE xiii

1. ROLLING BEARING TYPES AND APPLICATIONS 1Introduction to Rolling Bearings 1Ball Bearings 11Roller Bearings 23Linear Motion Bearings 40Bearings for Special Applications 41Closure 44

2. ROLLING BEARING MACROGEOMETRY 47List of Symbols 47General 48Ball Bearings 49Spherical Roller Bearings 66Radial Cylindrical Roller Bearings 73Tapered Roller Bearings 77Closure 79

vi CONTENTS3. INTERFERENCE FITTING AND CLEARANCE 81

List of Symbols 81General 83Industrial, National, and International Standards 84Effect of Interference Fitting on Clearance 86Press Force 123Differential Expansion 123Effect of Surface Finish 125Closure 130

4. BEARING LOADS AND SPEEDS 133List of Symbols 133General 134Concentrated Radial Loading 135Concentrated Radial and Moment Loading 143Shaft Speeds 150Distributed Load Systems 153Closure 153

5. BALL AND ROLLER LOADS 155List of Symbols 155General 157Static Loading 157Dynamic Loading 161Roller Axial Loading in Radial Bearings 177Closure 181

6. CONTACT STRESS AND DEFORMATION 183List of Symbols 183General 185Theory of Elasticity 185Surface Stresses and Deformations 189Subsurface Stresses 204Effect of Surface Shear Stress 215Type of Contact 218Roller End-Flange Contact Stress 225Closure 228

7. DISTRIBUTION OF INTERNAL LOADINGIN STATICALLY LOADED BEARINGS 231List of Symbols ?~1

CONTENTS vii

General 233Load-Deflection Relationships 234Bearings under Radial Load 235Bearings under Thrust Load 245Bearings under Combined Radial and Thrust Load 256Ball Bearings under Combined Radial, Thrust, and

Moment Load 266Misalignment of Radial Roller Bearings 272Thrust Loading of Radial Cylindrical Roller Bearings 280Radial, Thrust, and Moment Loading of Radial

Roller Bearings 289Flexibly Supported Rolling Bearings 291Closure 302

8. INTERNAL SPEEDS AND MOTIONS 307List of Symbols 307General 308Simple Rolling Motion 309Rolling and Sliding 313Orbital, Pivotal, and Spinning Motions in Ball Bearings 317Roller End-Flange Sliding in Roller Bearings 330Closure 335

9. DISTRIBUTION OF INTERNAL LOADING INHIGH SPEED BEARINGS 337List of Symbols 337General 338High Speed Ball Bearings 339High Speed Radial Cylindrical Roller Bearings 349High Speed Tapered and Spherical Roller Bearings 355Five Degrees of Freedom in Loading 358Closure 360

10. BEARING DEFLECTION AND PRE LOADING 363List of Symbols 363General 364Deflections of Bearings with Rigid Rings 365Preloading 368Limiting Ball Bearing Thrust Load 379C,lmmrp •..•n ••

viiiCONTENTS

11. STATICALLY INDETERMINATE SHAFT-BEARINGSYSTEMS 387List of Symbols 387General 388Two-Bearing Systems 389Three-Bearing Systems 400Multiple-Bearing Systems 410Closure 412

12. LUBRICANT FILMS IN ROLLINGELEMENT-RACEWAY CONTACTS 415List of Symbols 415General 418Hydrodynamic Lubrication 419Isothermal Elastohydrodynamic Lubrication 424Very High Pressure Effects 440Inlet Lubricant Frictional Heating Effects 441Starvation of Lubricant 444Surface Topography Effects 446Grease Lubrication 4151Lubrication Regimes 4154Closure 4156

13. FRICTION IN FLUID-LUBRICATED ROLLINGELEMENT-RACEWAY CONTACTS 481List of Symbols 461General 468Microgeometry and Microcontacts 464Asperity- and Fluid-Supported Load 472Friction in the EHL Contact 478Closure 479

14. FRICTION IN ROLLING BEARINGS 488List of Symbols 488General 485Sources of Friction 486Friction Forces and Moments in Rolling Element-Raceway

Contacts 496Skidding and Cage Forces 5115Cage Motions and Forces 529

CONTENTS ixRoller Skewing 534Bearing Friction Torque 540Closure 547

15. ROLLING BEARING TEMPERATURES 551List of Symbols 551General 552Heat Generation 553Heat Transfer 556Analysis of Heat Flow 561High Temperature Considerations 569Heat Transfer in a Rolling-Sliding Contact 574Closure 577

16. BEARING STRUCTURAL MATERIALS 579General 579Rolling Bearing Steels 579Steel Manufacture 582Effects of Processing Methods on Steel Components 597Heat Treatment of Steel 597Rolling Contact Fatigue: Modes and Causes 618Materials for Special Bearings 620Cage Materials 625Seal Materials 632Surface Treatments for Bearing Components 638Closure 641

17. LUBRICANTS AND LUBRICATION TECHNIQUES 645List of Symbols 645General 645Types of Lubricants 646Lubrication Methods 648Liquid Lubricants 654Grease Lubricants 662Polymeric Lubricants 668Solid Lubricants 670Environmentally Acceptable Lubricants 671Seals 672Closure 682

x CONTENTS18. FATIGUE LIFE: LUNDBERG-PALMGREN

THEORY AND RATING STANDARDS 683List of Symbols 683General 686Fatigue Life Dispersion 688Weibull Distribution 692Dynamic Capacity and Life of a Rolling Contact 699Fatigue Life of a Rolling Bearing 707Effect of Steel Composition and Processing on Fatigue Life 739Load Rating Standards 742Closure 761

19. BEARING ENDURANCE TESTING AND ELEMENTTESTING METHODS 763List of Symbols 763General 763Theoretical Basis of Life Testing 764Practical Testing Considerations 768Test Samples 772Test Rig Design Considerations 777Element Testing 779Rolling-Sliding Contact Friction Testing 784Closure 791

20. STATISTICAL METHODS TO ANALYZE ENDURANCE 793List of Symbols 793General 794The Two-Parameter Weibull Distribution 795Estimation in Single Samples 800Estimation in Sets of Weibull Data 811Closure 816

21. PERMANENT DEFORMATION AND BEARINGSTATIC CAPACITY 819List of Symbols 819General 820Calculation of Permanent Deformation 820Static Load Rating of Bearings 825Static Equivalent Load 828

CONTENTS x

Fracture of Bearing Components 83JPermissible Static Load 83JClosure 83~

22. MATERIAL RESPONSE TO ROLLING CONTACT 83fList of Symbols 831:General 831:Microstructures of Rolling Bearing Steels 83(Microstructural Alterations Due to Rolling Contact 834Residual Stresses in Rolling Bearing Components 84~Effects of Bulk Stresses on Material Response to

Rolling Contact 85~Closure 854

23. APPLICATION LOAD AND LIFE FACTORS 86]List of Symbols 86]General 86~Effect of Bearing Internal Load Distribution on Fatigue Life 864Effect of Variable Loading on Fatigue Life 874Fatigue Life of Oscillating Bearings 81£Reliability and Fatigue Life 88EEffect of Lubrication on Fatigue Life 89CEffect of Material and Material Processing on Fatigue Life 894Effect of Contamination on Fatigue Life 89ECombining Fatigue Life Factors 90~Limitations of the Lundberg-Palmgren Theory 904Ioannides- Harris Theory 90EThe Stress-Life Factor 90~Closure 93]

24. WEAR 936List of Symbols 931':General 93EStructural Elements of a Lubricated Contact 937Tribological Processes Associated with Wear 939Phenomenological View of Wear 949Interacting Tribological Processes and Failure Modes 953Recommendations for Wear Protection 955Closure 958

xii CONTENTS

25. VIBRATION, NOISE, AND CONDITION MONITORING 963List of Symbols 963General 963Vibration and Noise-Sensitive Applications 964The Role of Bearings in Machine Vibration 968Nonroundness Effect and Its Measurement 980Detection of Failing Bearings in Machines 997Failure Detection-Condition Monitoring 1003Condition-Based Maintenance 1005Closure 1010

26. ROTOR DYNAMICS AND CRITICAL SPEEDS 1013List of Symbols 1013General 1014Damped Forced Vibrations 1015Coupled Vibratory Motion (Rigid Shaft) 1020Multi-Degree-of-Freedom System (Flexible Shaft) 1024Bearing Stiffness 1028Characteristics of Bearing Stiffness 1033Rotor Dynamics Analysis 1039Closure 1042

27. INVESTIGATION AND ANALYSIS OF BEARINGFAILURES 1043General 1043Preliminary Investigation 1043Disassembly of Bearings 1044Failure Mechanisms 1044Examination and Evaluation of Specific Conditions 1049Fractography 1063Closure 1068

APPENDIX 1071INDEX 1074

Ball and roller bearings, generically called rolling bearings, are com-monly used machine elements. They are employed to permit rotary mo-tion of, or about, shafts in simple commercial devices such as bicycles,roller skates, and electric motors. They are also used in complex engi-neering mechanisms such as aircraft gas turbines, rolling mills, dentaldrills, gyroscopes, and power transmissions. Until approximately 1940,the design and application of these bearings could be considered moreart than science. Little was understood about the physical phenomenathat occur during their operation. Since 1945, a date which marks theend of World War II and the beginning of the atomic age, scientific pro-gress has occurred at an exponential pace. Since 1958, the date whichmarks the commencement of manned space travel, continually increas-ing demands are being made of engineering equipment. To ascertain theeffectiveness of rolling bearings in modern engineering applications, it isnecessary to obtain a firm understanding of how these bearings performunder varied and often extremely demanding conditions of operation.

Most information and data pertaining to the performance of rollingbearings are presented in manufacturers' catalogs. These data are al-most entirely empirical in nature, being either obtained from the testingof products by the larger bearing manufacturing companies or, more

xiii

xiv PREFACE

likely for smaller manufacturing companies, based on information con-tained in the American National Standards Institute (ANSI) or Inter-national Organization for Standards (ISO) publications or similarpublications. These data pertain only to applications involving slowspeed, simple loading, and nominal operating temperatures. If an engi-neer wishes to evaluate the performance of bearing applications oper-ating beyond these bounds, it is necessary to return to the basics ofrolling and sliding motions over the concentrated contacts that occur inrolling bearings.

One of the first books written on this subject was Ball and RollerBearing Engineering by Arvid Palmgren, Technical Director of ABSKFfor many years. It explained, more completely than had been done pre-viously, the concept of rolling bearing fatigue life. Palmgren, togetherwith Gustav Lundberg, Professor of Mechanical Engineering at Chal-mers Institute of Technology in Goteborg, Sweden, was the originator ofthe theory and formulas on which the current ANSI and ISO standardsfor the calculation of rolling bearing fatigue life are based. Also, A.Burton Jones's book in two volumes, Analysis of Stresses and Deflections,gave a good explanation of the static loading of ball bearings. Jones, whoworked in various technical capacities for New Departure Ball BearingsDivision of General Motors Corporation, Marlin-Rockwell Corporation,and Fafnir Ball Bearing company, and also as a consulting engineer, pi-oneered the use of digital computers to analyze the performance of balland roller bearing shaft-bearing-housing systems. The remainder ofother early and subsequent texts on rolling bearings were, and are,largely empirical in their approaches to applications analysis. Particu-larly since 1960, much research has been conducted into rolling bearingsand rolling contact phenomena. The use of modern laboratory equipmentsuch as scanning and transmission electron microscopes, x-ray diffrac-tion devices, and high speed digital computers has shed much light onthe mechanical, hydrodynamic, metallurgical, and chemical phenomenainvolved in rolling bearing operation. Many significant technical papershave been published by various engineering societies-for example, theAmerican Society of Mechanical Engineers, the Institution ofMechanicalEngineers, the Society of Tribologists and Lubrication Engineers, andthe Japan Society of Mechanical Engineers-analyzing the performanceof rolling bearings in exceptional applications involving high speed,heavy load, and extraordinary internal design and materials. Since 1960,substantial attention has been given to the mechanisms of rolling bear-ing lubrication and the rheology of lubricants. Notwithstanding the ex-istence of the aforementioned literature, there remains a need for areference that presents a unified, up-to-date approach to the analysis ofrolling bearing performance. That is my intention in presenting thisbook.

To accomplish this goal, I have attempted to review the most signifi-cant technical papers and texts covering the performance of rolling bear-

PREFACE xv

ings, their constituent materials, and lubrication. The concepts andmathematical presentations contained in the reviewed technical litera-ture have been condensed and simplified in this book for rapidity andease of understanding. It should not be construed, however, that thisbook supplies a complete bibliography on rolling bearings. Only thosedata that I found most useful in practical analysis have been referenced.Several of the references cited are my own works, since in some casesthese are the original or are among the most significant available on theparticular subject.

The format ofRolling Bearing Analysis is aimed at developing for thereader a basic understanding of rolling bearing operation. Thus, the in-itial chapters discuss the simplest concepts of rolling bearings, such asbasic bearing types, geometries, applied loading, loading of single ballsand rollers, and contact stresses and deformations. Then, the complexanalysis of load distribution among the rolling elements, componentspeeds, and velocities, elastohydrodynamic lubrication, friction, temper-atures, statistics of bearing endurance, and fatigue life are considered.Several topics depend almost entirely on the preceding discussions. Asnearly as possible, an attempt has been made to maintain continuity ofpresentation. To amplify the discussion, numerical examples are pre-sented in most chapters. For instance, numerical examples deal with a209 radial ball bearing, a 209 cylindrical roller bearing, a 218 angular-contact ball bearing, and a 22317 spherical roller bearing in many chap-ters. Analytical data for each bearing are accumulated as the readerprogresses through the book. The examples are carried out in metric orstandard international (SI) system units (millimeters, Newtons, seconds,°C, and so on); however, the results are also given parenthetically inEnglish system units. In the Appendix, the numerical constants for equa-tions presented in SI or metric system units are provided in Englishsystem units as well.

The material covered herein spans many scientific disciplines, such asgeometry, elasticity, statics, dynamics, hydrodynamics, statistics, andheat transfer. Thus, many mathemathical symbols have been employed.In some cases, the same symbol has been chosen to represent differentparameters in different chapters. To help avoid confusion, a list of sym-bols is presented at the beginning of most chapters. In the interest ofclarity, however, certain symbols have been retained for singular usage.For example, D is always ball or roller diameter, dm is always bearingpitch diameter, and a is always contact angle.

Because of the several scientific disciplines that this book spans, thetreatment of each topic may vary somewhat in scope and manner. Wherefeasible, analytical solutions to problems have been presented. On theother hand, empirical approaches to problems have been used where itseemed more practical. The wedding of analytical and empirical tech-niques is particularly evident in the chapters covering lubrication, fric-tion, and fatigue life.

PREFACE xvii

particularly in the area of rolling contact fatigue. This has afforded methe opportunity to continue development of the Ioannides-Harris fatiguelife theory; the results of this development are presented in Chapter 23.This material represents not only the results of my research, but alsothe substantial collaborative efforts of the ASME Tribology DivisionTechnical Committee on Life Ratings for Modern Rolling Bearings. Inaddition to myself, contributing significantly to the results of this com-mittee are the following members: Roger Barnsby, Pratt and Whitney,United Technologies Corporation; Dr. Stathis Ioannides, SKF Engineer-ing and Research Centre, the Netherlands; Dr. Thomas Losche, FAGBearings, Germany; Dr. Kikua Maeda, NTN, Japan; Dr. Yasuo Mura-kami, NSK, Japan; Harvey Nixon and Michael Hoeprich, the TimkenCompany; and Dr. Martin Webster, Mobil Oil Company.

As stated previously, the material presented herein exists substan-tially in other publications, The purpose of this text is to concentratethat knowledge in one place for the benefit of both the student and therolling bearing user who need or want a broader understanding of thetechnical field and/or product. The references provided at the end ofeachchapter enable the curious reader to go into further detail.

Because of my longtime association with the SKF company, as withthe previous editions of this text, several of the illustrations in this 4thedition have previously appeared in SKF publications; for such illustra-tions, appropriate references are identified. In this edition, however, Ihave included photographs and illustrations from other rolling bearingmanufacturers as well. I would like to express my appreciation to thefollowing companies for contributing photographic material: FAG OEMund Handel AG, Schweinfurt, Germany; NSK Corporation; NTN BearingCorporation of America; the Timken Company, Canton, Ohio; TorringtonBearings Division, Ingersoll Rand Corporation, Torrington, Connecticut.The contributor of each such illustration is identified.

TEDRIC A. HARRIS

Professor of Mechanical EngineeringThe Pennsylvania State UniversityUniversity Park, Pennsylvania

INTRODUCTION TO ROLLING BEARINGS

After the invention of the wheel, it was learned that less effort was re-quired to move an object on rollers than to slide the object over the samesurface. Even after lubrication was discovered to reduce the work re-quired in sliding, rolling motion still required less work when it could beused. For example, archeological evidence shows that the Egyptians, ca.2400 BC, employed lubrication, most likely water, to reduce the man-power required to drag sledges carrying huge stones and statues. TheAssyrians, ca. 1100 BC, however, employed rollers under the sledges toachieve a similar result with less manpower. It was therefore inevitablethat bearings using rolling motion would be developed for use in complexmachinery and mechanisms. Figure 1.1 depicts, in a simplistic manner,the evolution of rolling bearings. Dowson [1.1] provides a comprehensivepresentation on the history of bearings and lubrication in general; hiscoverage on ball and roller bearings is extensive. Although the conceptof rolling motion was known and used for thousands of years, and simpleforms of rolling bearings were in use ca. 50 AD during the Roman civi-lization, the general use of rolling bearings did not occur until the in-dustrial revolution. Reti [1.2], however, shows that Leonardo da Vinci

1

4 ROLLING BEARING TYPES AND APPLICATIONS

rior rolling bearing steels and constant improvement in manufacturing,providing extremely accurate geometry, long-lived rolling bearing assem-blies. Initially this development was triggered by the bearing require-ments for high speed aircraft gas turbines; however, competition betweenball and roller bearing manufacturers for worldwide markets increasedsubstantially during the 1970s, and this has served to provide consumerswith low-cost, standard design bearings of outstanding endurance. Theterm rolling bearings includes all forms of bearings that utilize the roll-ing action of balls or rollers to permit minimum friction, constrainedmotion of one body relative to another. Most rolling bearings are em-ployed to permit rotation of a shaft relative to some fixed structure. Somerolling bearings, however, permit translation, that is, relative linear mo-tion, of a fixture in the direction provided by a stationary shaft, and afew rolling bearing designs permit a combination of relative linear androtary motion between two bodies.

This book is concerned primarily with the standardized forms of balland roller bearings that permit rotary motion between two machine el-ements. These bearings will always include a complement of balls orrollers that maintain the shaft and a usually stationary supporting struc-ture, frequently called a housing, in a radially or axially spaced-apartrelationship. Usually, a bearing may be obtained as a unit, which in-cludes two steel rings each of which has a hardened raceway on whichhardened balls or rollers roll. The balls or rollers, also called rolling el-ements, are usually held in an angularly spaced relationship by a cage,whose function was anticipated by Leonardo. The cage may also be calleda separator or retainer.

Balls, rollers, and rings of good quality, rolling bearings are normallymanufactured from steels that have the capability of being hardened toa high degree, at least on the surface. In universal use by the ball bearingindustry is AISI 52100, a steel moderately rich in chromium and easilyhardened throughout (through-hardened) the mass of most bearing com-ponents to 61-65 Rockwell C scale hardness. This steel is also used inroller bearings by some manufacturers. Miniature ball bearing manu-facturers, whose bearings are used in sensitive instruments such asgyroscopes, prefer to fabricate components from stainless steels such asAISI 440C. Roller bearing manufacturers frequently prefer to fabricaterings and rollers from case-hardening steels such as AISI 3310, 4118,4620, 8620, and 9310. For some specialized applications, such as auto-motive wheel hub bearings, the rolling components are manufacturedfrom induction-hardening steels. In all cases, at least the surfaces of therolling components are extremely hard. In some high speed applications,to minimize inertial loading of the balls or rollers, these components arefabricated from lightweight, high compressive strength ceramic materi-als such as silicon nitride. Also, these ceramic rolling elements tend to

INTRODUCTION TO ROLLING BEARINGS 5

endure longer than steel at ultrahigh temperatures and in applicationswith dry film or minimal fluid lubrication.

Cage materials, as compared to materials for balls, rollers, and rings,are generally required to be relatively soft. They must also possess goodstrength-to-weight ratio; therefore, materials as widely diverse in phys-ical properties as mild steel, brass, bronze, aluminum, polyamide (nylon),polytetrafluoroethylene (teflon or PTFE), fiberglass, and plastics filledwith carbon fibers find use as cage material.

In this modern age of deep-space exploration and cyberspace, manydifferent kinds of bearings have come into use, such as gas film bearings,foil bearings, magnetic bearings, and externally pressurized (hydrostatic)bearings. Each of these bearing types excels in some specialized field ofapplication. For example, hydrostatic bearings are excellent for applica-tions in which size is no problem, an ample supply of pressurized fluidis available, and extreme rigidity under heavy loading is required. Self-acting gas bearings may be used for applications in which loads are light,speeds are high, a gaseous atmosphere exists, and friction must be min-imal. Rolling bearings, however, are not quite so limited in scope. Con-sequently, miniature ball bearings such as shown in Fig. 1.3 are foundin precision applications such as inertial guidance gyroscopes and highspeed dental drills, large roller bearings, such as shown in Fig. 1.4, areutilized in metal rolling mill applications, and even larger slewing bear-ings, as illustrated in Fig. 1.5, were used in tunneling machines for the"Chunnel" (English Channel tunneling) project.

Moreover, rolling bearings find use in diverse precision machinery op-erations; for example, the high load, high temperature, dusty environ-ment of steel-making (Fig. 1.6), the dirty environments of earthmoving

and farming (Figs. 1.7 and 1.8), the life-critical applications in aircraftpower transmissions (Fig. 1.9), and the extreme low-high temperatureand vacuum environments of deep space (Fig. 1.10). They perform wellin all of these applications. Specifically, rolling bearings have the follow-ing advantages compared to other bearing types:

• They operate with much less friction torque than hydrodynamicbearings and therefore considerably less power loss and friction heatgeneration .

· Starting friction torque is only slightly greater than moving frictiontorque.

• Bearing deflection is less sensitive to load fluctuation than in hydro-dynamic bearings.

(b)

FIGURE 1.5. Large slewing bearing used in an English Channel tunneling machine. (a)

Photograph; (b) schematic drawing of the assembly (courtesy of SKF).

. They require only small quantities of lubricant for satisfactory opera-tion and have the potential for operation with a self-contained, life-longsupply of lubricant.

. They occupy shorter axial length than conventional hydrodynamicbearings.

• Combinations of radial and thrust loads can be supported simultane-ously.

• Individual designs yield excellent performance over a wide load-speedrange.

• Satisfactory performance is relatively insensitive to fluctuations inload, speed, and operating temperature.

Notwithstanding the foregoing advantages, rolling bearings have beenconsidered to have a single disadvantage compared to hydrodynamicbearings. Tallian [1.3] defined three eras of modern rolling bearing de-velopment: an "empirical" era extending through the 1920s, a "classical"era lasting through the 1950s, and the "modern" era occurring thereafter.Through the empirical, classical, and even into the modern era, it wassaid that even if rolling bearings are properly lubricated, properlymounted, protected from dirt and moisture, and otherwise properly op-erated, they will eventually fail because of fatigue of the surfaces in roll-ing contact. Historically, as shown in Fig. 1.11, rolling bearings havebeen considered to have a life distribution statistically similar to that oflight bulbs and human beings.

Research in the 1960s [1.4]demonstrated that rolling bearings exhibita minimum fatigue life; that is, "crib deaths" due to rolling contactfatigue do not occur when the foregoing criteria for good operation areachieved. Moreover, modern manufacturing techniques enable produc-Gionof bearings with extremely accurate component internal and exter-

nal geometries and extremely smooth rolling contact surfaces, modernsteel-making processes can provide rolling bearing steels of outstandinghomogeneity with few impurities, and modern sealing and lubricant fil-tration methods act to minimize the incursion of harmful contaminantsinto the rolling contact zones. These methods, which are now being usedin combination in many applications, can virtually eliminate the occur-rence of rolling contact fatigue, even in some applications involving veryheavy applied loading. In many lightly loaded applications, for example,most electric motors, fatigue life need not be a major design considera-tion.

There are many different kinds of rolling bearings, and before em-barking on a discussion of the theory and analysis of their operation, itis necessary to become somewhat familiar with each type. In the suc-ceeding pages a description is given for each of the most popular balland roller bearings in current use.

BALL BEARINGS

Radial Ball Bearings

Single-Row Deep-Groove Conrad Assembly Ball Bearing. This ball bear-ing is shown in Fig. 1.12, and it is the most popular rolling bearing. The

inner and outer raceway grooves have curvature radii between 51.5 and53% of the ball diameter for most commercial bearings.

To assemble these bearings, the balls are inserted between the innerand outer rings as shown by Figs. 1.13 and 1.14. The assembly angle 1>is given as follows:

1> = 2(Z - 1) D/dm (1.1)

in which Z is the number of balls, D is ball diameter, and dm is pitchdiameter. The inner ring is then snapped to a position concentric withthe outer ring, the balls are separated uniformly, and a riveted cage asshown in Fig. 1.14 or a plastic cage as illustrated by Fig. 16.25a is in-serted to maintain the separation. Because of the high osculation and anappropriate ball diameter and ball complement to substantially fill thebearing pitch circle, the deep-groove ball bearing has comparatively highload-carrying capacity when accurately manufactured from good-qualitysteel and operated in accordance with good lubrication and contaminant-exclusion practices. Although it is designed to carry radial load, it per-forms well under combined radial and thrust load and under thrustalone. With proper caged design, deep-groove ball bearings can with-


stand misaligning loads (moment loads) of small magnitude. By makingthe bearing outside surface a portion of a sphere as illustrated in Fig.1.15, however, the bearing can be made externally self-aligning and,thus, incapable of supporting a moment load.

The deep-groove ball bearing can be readily adapted with seals asshown in Fig. 1.16 or shields as shown by Fig. 1.17 or both as illustratedby Fig. 1.18. These components function to keep lubricant in the bearingand exclude contaminants. Seals and shields come in many different con-figurations to serve general or selective applications; those shown inFigs. 1.16-1.18 should be taken only as examples. In Chapter 17, sealsare discussed in greater detail.

Deep-groove ball bearings perform well at high speeds provided ade-quate lubrication and cooling are available. Speed limits shown in man-ufacturers' catalogs generally pertain to bearing operation without thebenefit of external cooling capability or special cooling techniques.

Conrad assembly bearings can be obtained in different dimension se-ries according to ANSI and ISO* standards. Figure 1.19 shows the rel-ative dimensions of various ball bearing series.

Single-Row Deep-Groove Filling-Slot Assembly Ball Bearings. Thisbearing as illustrated in Fig. 1.20 has a slot machined in the side wallof each of the inner and outer ring grooves to permit the assembly ofmore balls than the Conrad type does, and thus it has more radial load-carrying capacity. Because the slot disrupts the groove continuity, thebearing is not recommended for thrust load applications. Otherwise, thebearing has characteristics similar to those of the Conrad type.

Double-Row Deep-Groove Ball Bearings. This ball bearing as shown inFig. 1.21 has greater radial load-carrying capacity than the single-rowtypes. Proper load sharing between the rows is a function of the geo-metrical accuracy of the grooves. Otherwise, these bearings behave sim-ilarly to single-row ball bearings.

Instrument Ball Bearings. In metric design, the standardized form ofthese bearings ranges in size from 1.5-mm CO.05906-in.)bore and 4-mmCO.15748-in.) o.d. to 9-mm (0.35433-in.) bore and 26-mm (1.02362-in.) o.d.See reference [1.5].As detailed in reference [1.6], standardized form, inchdesign instrument ball bearings range from 0.635-mm CO.0250-in.)bore

and 2.54-mm (O.IOO-in.)o.d. to 19.050-mm (O.7500-in.)bore and 41.275-mm C1.6250-in.)o.d. Additionally, instrument ball bearings have extrathin series that range up to 47.625-mm C1.8750-in.)o.d. and thin seriesthat range up to lOO-mmC3.93701-in.)o.d. Those bearings having lessthan 9-mm CO.3543-in.)o.d. are classified as miniature ball bearings ac-cording to [1.6]; such bearings can use balls as small as O.6350-mmCO.0250-in.)diameter. Figure 1.3 illustrates this type of bearing. Theyare fabricated according to more stringent manufacturing standards,such as for cleanliness, than are any of the bearings previously described.This is because minute particles of foreign matter can significantly in-crease the friction torque and negatively affect the smooth operation ofthe bearings. For this reason, they are assembled in a white room asillustrated in Fig. 1.22.

Groove radii of instrument ball bearings are usually not smaller than57% of the ball diameter. The bearings are usually fabricated from stain-less steels since corrosion particles will seriously deteriorate bearing per-formance.

Angular-Contact Ball Bearings

Single-Row Angular-Contact Ball Bearings. Angular-contact ball bear-ings as shown in Fig. 1.23 are designed to support combined radial andthrust loads or heavy thrust loads depending on the contact angle mag-nitude. The bearings having large contact angles can support heavierthrust loads. Figure 1.24 shows bearings having small and large contactangles. The bearings generally have groove curvature radii in the rangeof 52-53% of the ball diameter. The contact angle does not usually exceed40°. The bearings are usually mounted in pairs with the free endplayremoved as shown in Fig. 1.25. These sets may be preloaded against eachother to stiffen the assembly in the axial direction. The bearings mayalso be mounted in tandem as illustrated in Fig. 1.26 to achieve greaterthrust-carrying capacity.

Double-Row Angular-Contact Ball Bearings. These bearings as depictedin Fig. 1.27 can carry thrust load in either direction or a combination ofradial and thrust load. Bearings of the rigid type are able to withstand

moment loading effectively. Essentially, the bearings perform similarlyto duplex pairs of single-row angular-contact ball bearings.

Self Aligning Double-Row Ball Bearings. As illustrated in Fig. 1.28, theouter raceway of this bearing is a portion of a sphere. Thus, the bearingsare internally self-aligning and cannot support a moment load. Becausethe balls do not conform well to the outer raceway (it is not grooved), theouter raceway has reduced load-carrying capacity. This is compensatedsomewhat by use of a very large ball complement that minimizes theload carried by each ball. The bearings are particularly useful in appli-cations in which it is difficult to obtain exact parallelism between theshaft and housing bores. Figure 1.29 shows this bearing with a taperedsleeve and locknut adapter. With this arrangement the bearing does notrequire a locating shoulder on the shaft.

Split Inner Ring Ball Bearings. These bearings are illustrated in Fig.1.30. As can be seen, the inner ring consists of two axial halves suchthat a heavy thrust load can be supported in either direction. They mayalso support, simultaneously, moderate radial loading. The bearings havefound extensive use in supporting the thrust loads acting on high speed,gas turbine engine mainshafts. Figure 1.31 shows the compressor andturbine shaft ball bearing locations in a high-performance aircraft gasturbine engine. Obviously, both the inner and outer rings must be locked

up on both axial sides to support a reversing thrust load. It is possiblewith accurate flush grinding at the factory to utilize these bearings intandem as shown in Fig. 1.32 to share a thrust load in a given direction.

Thrust Ball Bearings

The thrust ball bearing illustrated in Fig. 1.33 has a 90° contact angle;however, ball bearings whose contact angles exceed 45° are also classifiedas thrust bearings. As for radial ball bearings, thrust ball bearings aresuitable for operation at high speeds. To achieve a degree of externallyaligning ability, thrust ball bearings are sometimes mounted on sphericalseats. This arrangement is demonstrated by Fig. 1.34.A thrust ball bear-ing whose contact angle is 90° cannot support any radial load.

ROLLER BEARINGS

General

Roller bearings are usually used for applications requiring exceptionallylarge load-supporting capability, which cannot be feasibly obtained using

a ball bearing assembly. Roller bearings are usually much stiffer struc-tures (less deflection per unit loading) and provide greater fatigue en-durance than do ball bearings of a comparable size. In general, they alsocost more to manufacture, and hence purchase, than comparable ballbearing assemblies. They usually require greater care in mounting thando ball bearing assemblies. Accuracy of alignment of shafts and housingscan be a problem in all but self-aligning roller bearings.

Radial Roller Bearings

Cylindrical Roller Bearings. Cylindrical roller bearings as illustratedin Fig. 1.35 have exceptionally low friction torque characteristics that

make them suitable for high speed operation. They also have high radial-load-carrying capacity. The usual cylindrical roller bearing is free to floataxially. It has two roller-guiding flanges on one ring and none on theother, as shown in Fig. 1.36. By equipping the bearing with a guide flangeon the opposing ring (illustrated by Fig. 1.37), the bearing can be madeto support some thrust load.

To prevent high stresses at the edges of the rollers the rollers areusually crowned as shown in Fig. 1.38. This crowning of rollers also givesthe bearing protection against the effects of slight misalignment. Thecrown is ideally designed for only one condition ofloading. Crowned race-ways may be used in lieu of crowned rollers.

To achieve greater radial-load-carrying capacity, cylindrical rollerbearings are frequently constructed of two or more rows of rollers ratherthan of longer rollers. This is done to reduce the tendency of the rollersto skew. Figure 1.39 shows a small double-row cylindrical roller bearingdesigned for use in precision applications. Figure 1.40 illustrates a largemultirow cylindrical roller bearing for a steel rolling mill application.

Needle Roller Bearings. A needle roller bearing is a cylindrical rollerbearing having rollers of considerably greater length than diameter. This

bearing is illustrated in Fig. 1.41. Because of the geometry of the rollers,they cannot be manufactured as accurately as other cylindrical rollers,nor can they be guided as well. Consequently, needle roller bearings haverelatively greater friction than other cylindrical roller bearings.

Needle roller bearings are designed to fit in applications in which ra-dial space is at a premium. Sometimes to conserve space the needlesbear directly on a hardened shaft. They are useful for applications inwhich oscillatory motion occurs or in which continuous rotation occursbut loading is light and intermittent. The bearings may be assembledwithout a cage, as shown in Fig. 1.42. In this full-complement-type bear-ing, the rollers are frequently retained by turned-under flanges that areintegral with the outer shell. The raceways are frequently hardened butnot ground.

Tapered Roller Bearings

The single-row tapered roller bearing shown in Fig. 1.43 has the abilityto carry combinations of large radial and thrust loads or to carry thrustload only. Because of the difference between the inner and outer racewaycontact angles, there is a force component that drives the tapered rollers

the outer ring the cup. Depending on the magnitude of the thrust loadto be supported, the bearing may have a small or steep contact angle, asshown in Fig. 1.44. Since tapered roller bearing rings are separable, thebearings are mounted in pairs as indicated in Fig. 1.45, and one bearingis adjusted against the other. To achieve greater radial load-carrying ca-pacity and eliminate problems of axial adjustment due to distance be-tween bearings, tapered roller bearings may be combined as shown inFig. 1.46 into two-row bearings. Fig. 1.47 shows a typical double-rowtapered roller bearing assembly for a railroad car wheel application.Double-row bearings may also be combined into four-row or quad bear-ings for exceptionally heavy radial load applications such as rolling mills.Figure 1.48 shows a quad bearing having integral seals.

As with cylindrical roller bearings, tapered rollers or raceways areusually crowned to relieve heavy stresses on the axial extremities of therolling contact members.

By equipping the bearing with specially contoured flanges, a specialcage, and lubrication holes as shown by Fig. 1.49, a tapered roller bear-ing can be designed to operate satisfactorily under high load-high speedconditions. In this case, the cage is guided by lands on both the cone riband the cup, and oil is delivered directly by centrifugal flow to the rollerend-flange contacts and cage rail-cone land contact.

Spherical Roller Bearings

Most spherical roller bearings have an outer raceway that is a portionof a sphere; hence, the bearings, as illustrated by Fig. 1.50, are internallyself-aligning. Each roller has a curved generatrix in the direction trans-verse to rotation that conforms relatively closely to the inner and outer

raceways. This gives the bearing high load-carrying capacity. Various ex-ecutions of double-row, spherical roller bearings are shown in Fig. 1.51.

Fig. 1.5Ia shows a bearing with asymmetrical rollers. This bearing,similar to tapered roller bearings, has force components that drive therollers against the fixed central guide flange. Bearings such as illustratedin Fig. 1.5Ib and 1.5Ic have symmetrical (barrel- or hourglass-shape)rollers, and these force components tend to be absent except under high

speed operation. Double-row bearings having barrel-shape, symmetricalrollers frequently use an axially floating central flange as illustrated byFig. 1.5Id. This eliminates undercuts in the inner raceways and permitsuse of longer rollers, thus increasing the load-carrying capacity of thebearing. Roller guiding in such bearings tends to be accomplished by

friction than cylindrical roller bearings. This is due to the degree of slid-ing that occurs in the roller-raceway contacts. Spherical roller bearingsare therefore not readily suited for use in high speed applications. Theyperform well in heavy duty applications such as rolling mills, paper mills,and power transmissions and in marine applications. Double-row bear-ings can carry combined radial and thrust load; they cannot support mo-ment loading. Radial, single-row, spherical roller bearings have a basiccontact angle of 0°. Under thrust loading, this angle does not increaseappreciably; consequently, any amount of thrust loading magnifiesroller-raceway loading substantially. Therefore, these bearings should

ROLLER BEARINGS 37

not be used to carry combined radial and thrust loading when the thrustcomponent of the load is relatively large compared to the radial compo-nent. A special type of single-row bearing has a toroidal outer raceway,this is illustrated by Fig. 1.52; it can accommodate radial load togetherwith some moment load, however, little thrust load.

Thrust Roller Bearings

Spherical Roller Thrust Bearings. The spherical roller thrust bearingshown in Fig. 1.53 has a very high load-carrying capacity due to high


osculation between the rollers and raceways. It can carry a combinationthrust and radial load and is internally self-aligning. Because the rollersare asymmetrical, force components occur that drive the sphere ends ofthe rollers against a concave spherical guide flange. Thus, the bearingsexperience sliding friction at this flange and do not lend themselves read-ily to high speed operation.

Cylindrical Roller Thrust Bearings. Because of its geometry, the cylin-drical roller thrust bearing of Fig. 1.54 experiences a large amount ofsliding between the rollers and raceways, also called washers. Thus, thebearings are limited to slow speed operation. Sliding is reduced some-what by using multiple short rollers in each pocket rather than a singleintegral roller. This is illustrated by Fig. 1.55.

Tapered Roller Thrust Bearings. This bearing, illustrated in Fig. 1.56has an inherent force component that drives each roller against the out-board flange. The sliding frictional forces generated at the contacts be-tween the rollers and flange limit the bearing to relatively slow speedapplications.

Needle Roller Thrust Bearings. These bearings, as illustrated by Fig.1.57, are similar to cylindrical roller thrust bearings except that needle


LINEAR MOTION BEARINGS

Bearings for linear motion, such as those used in machine tool "ways"-for example, V-ways-generally employ only lubricated sliding action.These sliding actions are subject to relatively high stick-slip friction,wear, and subsequent loss of locational accuracy. Ball bushing operatingon hardened steel shafts, illustrated schematically by Fig. 1.59, providemany of the low friction, minimal characteristics of radial rolling bear-ings.

The ball bushing, which provides linear travel along the shaft, limitedonly by built-in motion stoppers, contains three or more oblong circuitsof recirculating balls. As illustrated in Fig. 1.60, one portion of the oblongball complement supports load on the rolling balls while the remainingballs operate with clearance in the return track.

The ball retainer units can be fabricated relatively inexpensively ofpressed steel or nylon (polyamide) material. Figure 1.61 is a photographshowing an actual unit with its components. Ball bushings of instrumentquality are made to operate on shaft diameters as small as 3.18 mm(0.125 in.).

Ball bushings can be lubricated with medium-heavy weight oil or witha light grease to prevent wear and corrosion. For high linear speeds, lightoils are recommended. Seals can be provided; however, friction is in-creased significantly.

As with radial ball bearings, life can be limited by subsurface-initiatedfatigue of the rolling contact surfaces. A unit is usually designed to per-form satisfactorily for several million units of linear travel. Since thehardened shaft is subject to surface fatigue and/or wear, provision canbe made for rotating the bushing or shaft to bring new bearing surfaceinto play.

BEARINGS FOR SPECIAL APPLICATIONS

Automotive Wheel Bearings

Angular-contact ball bearings for automobile wheels used to be individ-ual bearings mounted in duplex sets as shown by Fig. 1.25. On assemblyin the vehicle, these had to be adjusted to eliminate bearing endplay. Thesame was true for tapered roller, truck wheel bearing sets. To excludecontaminants from the bearings, external seals were required. The bear-


ings were grease-lubricated and, owing to the grease deterioration in thisdifficult application, needed to be regreased periodically. If this was notaccomplished with care, inevitably contamination was introduced intothe bearings and bearing longevity was substantially curtailed. To over-come this situation, many bearings were provided as preadjusted,greased and sealed-for-life, duplex sets as shown by Fig. 1.62a. Theseunits needed to be press-fitted into the wheel hubs. To simplify the as-sembly for the automobile manufacturer and to minimize size, a flangewas made integral with the bearing outer ring as shown by Fig. 1.62b;thus, the unit could be bolted to studs on the wheel. Subsequently, a self-contained unit with a flange integral with each ring, as shown by Fig.1.62c, came into use for nondriven wheels; the unit can be bolted to thevehicle frame and the wheel for simple assembly.

For heavier duty vehicles such as trucks, tapered roller bearings (Fig.1.63) are used instead of ball bearings.

Function can also be added to the bearing unit as shown by the ta-pered roller bearing unit in Fig. 1.64. This compact, preadjusted, self-contained bearing unit is equipped with an integral speed sensor toprovide a signal to the anti-lock braking system (ABS). Sensors are alsoplaced in rolling bearings to measure applied loading.

Cam Follower Bearings

To reduce friction associated with the follower contact on cams, rollingmotion may be employed. The needle roller bearing is particularly suitedto this application because it is radially compact. Figure 1.65 shows aneedle roller bearing, earn follQwerassembly.

Aircraft Gas Turbine Engine and Power Transmission Bearings

Airplane and helicopter power transmission bearing applications aregenerally characterized by the necessity to carry heavy loads at highspeed while minimizing bearing size. The bearings are generally man-ufactured from special high strength, high quality steels. While weightof a steel bearing itself is significant, minimizing bearing width and out-side diameter aids compactness in engine design, allowing surroundingengine components to be smaller and weigh less. Thus, aircraft power

train bearings have slimmer rings, as illustrated by the cylindrical rollerbearing in Fig. 1.66. Moreover, bearings are made integral with othercomponents to save weight. This is shown by the planetary gear trans-mission, spherical roller planet bearing of Fig. 1.66. The gas turbine en-gine cylindrical roller bearing of Fig. 1.67 has a slender, hollowed out,flange integral with the bearing outer ring. The flange is bolted to theengine frame for ease of assembly.

Figure 1.68 shows a gas turbine mainshaft, split inner ring ball bear-ing having an outer which bolts to the housing assembly; it also depictscylindrical roller bearing inner and outer ring units specially fabricatedfor the turbine engine application.

type of rolling bearing has been described; discussion has been limitedto the most popular and basic forms. For example, there are cylindricalroller bearing designs that use snap rings, instead of machined andground flanges. ANSI!ABMA and ISO standards on terminology [1.7]and [1.8] illustrate many of the more common bearing designs. It is alsoapparent that many rolling bearings are specially designed for applica-tions. Some of these have been discussed herein only to indicate thatspecial design bearings are sometimes warranted by the application. Ingeneral, special bearing designs entail additional cost for the bearing orbearing unit; however, such cost increase is usually offset by overall ef-

46ROLLING BEARING TYPES AND APPLICATIONS

ficiency and cost reduction brought to mechanism and machinery design,manufacture, and operation.

REFERENCES1.1 D. Dowson, History of Tribology, 2nd ed., Longman, New York (1999).1.2 L. Reti, "Leonardo on Bearings and Gears," Scientific American, 224(2), 101-110

(1971).

1.3 T. Tallian, "Progress in Rolling Contact Technology,"SKF Report AL690007 (1969).1.4 T. Tallian, "Weibull Distribution of Rolling Contact Fatigue Life and Deviations There-

from," ASLE Trans. 5(1), 183-196 (1962).

1.5 American National Standards Institute, American National Standard (ANSI/ABMA)Std. 12.1-1992, "Instrument Ball Bearings-Metric Design" (April 6, 1992).

1.6 American National Standards Institute, American National Standard (ANSI/ABMA)Std. 12.2-1992, "Instrument Ball Bearings-Inch Design" (April 6, 1992).

1.7 American National Standards Institute, American National Standard (ANSI/ABMA)Std. 1-1990, "Terminology for Anti-Friction Ball and Roller Bearings and Parts" (July24, 1990).

1.8 International Organization for Standards, International Standard ISO 5593, "RollingBearings-Vocabulary" (1984-07-01).

LIST OF SYMBOLS

Symbol Description UnitsA Distance between raceway groove curvature

centers mm (in.)B AIDd Raceway diameter mm (in.)dm Bearing pitch diameter mm (in.)D Ball or roller nominal diameter mm (in.)Dm Mean diameter of tapered roller mm (in.)Dmax Diameter of tapered roller at large end mm (in.)Dmin Diameter of tapered roller at small end mm (in.)f riDl Roller effective length mm (in.)If Distance between cylindrical roller guide mm (in.)

flangesIt Roller length end-to-end mm (in.)Pd Bearing diametral clearance mm (in.)Pe Bearing free endplay mm (in.)

48ROLLING BEARING MACROGEOMETRY

Symbol Description Unitsr Raceway groove curvature radius mm (in.)rc Roller corner radius mm (in.)R Roller contour radius mm (in.)Sd Spherical roller bearing diametral play mm (in.)Z Number of rolling elementsa 0

Free contact angle 0

a Contact angle 0

af Tapered roller bearing flange angle 0

aR Roller angle 0

as Shim angle 0

y D cos a/dm() Misalignment angle 0

P Curvature mm-I (in.-I)F(p) Curvature difference"Lp Curvature sum mm-I Cin.-I)~ Osculationw Angular velocity rad/sec

SUBSCRIPTSc Refers to cage0 Refers to outer racewayi Refers to inner racewayr Refers to roller

GENERAL

Although ball and roller bearings appear to be simple mechanisms, theirinternal geometries are quite complex. For example, a radial ball bearingsubjected to thrust loading assumes angles of contact between the ballsand raceways in accordance with the relative conformities of the balls tothe raceways and the diametral clearance. On the other hand, the abilityof the same bearing to support the thrust loading depends on the contactangles formed. The same diametral clearance or play produces an axialendplay that mayor may not be tolerable to the bearing user. In laterchapters it will be demonstrated how diametral clearance affects not onlycontact angles and endplay but also stresses, deflections, load distribu-tions, and fatigue life. Stresses, deflections, load distribution, and life inroller bearings are also affected by clearance.

In the determination of stresses and deflections the relative conform-ities of balls and rollers to their contacting raceways are of vital interest.In this chapter the principal macrogeometric relationships governing theoperation of ball and roller bearings shall be developed and examined.

TAPERED ROLLER BEARINGS 77

in addition to the applied radial load. The bearing endplay influences thenumber of radially loaded rollers that will participate in supporting thethrust load. The endplay also influences the degree of roller skewingwhich can occur during bearing operation. See Chapter 14.

Curvature

Most cylindrical roller bearings employ crowned rollers to avoid thestress-increasing effects of edge-loading (see Chapter 6). For these roll-ers, even if fully crowned as illustrated by Fig. 1.31a, the contour orcrown radius R is very large. Moreover, even ifthe raceways are crowned,R = ri = ro ~ 00. Therefore, considering equations (2.37) and (2.39),whichdescribe the curvature sum for inner and outer raceway contact respec-tively, the difference of the reciprocals of these radii is essentially nil,and

TAPERED ROLLER BEARINGS

Pitch Diameter

The nomenclature associated with tapered roller bearings is differentthan that for other types of roller bearings. For example, as indicated byFig. 2.12, the bearing inner ring is called the cone and the outer ring thecup. It can be seen that the operation of the bearing is associated witha pitch cone; equation (2.1) can be used to describe the mean diameterof that cone. For many calculations, this mean cone diameter will be usedas the bearing pitch diameter dm' Figure 2.13 indicates dimensions andangles necessary for the performance analysis of tapered roller bearings.From Fig. 2.13, it can be seen that O'i, the inner raceway-roller contactangle = t cone angle, 0'0' the outer raceway-roller contact angle = t cupangle, O'f, the roller large end-flange contact angle = t cone back facerib angle, and O'R = roller angle. Dmax = the large end diameter of theroller, and Dmin = the small end diameter of the roller, which has an end-to-end length of it.

80 ROLLING BEARING MACROGEOMETRY

Numerical examples developed in this chapter were of necessity verysimple in format. The quantity of these simple examples is justified sincethe results from the calculations will subsequently be used as startingpoints in more complex numerical examples involving stresses, deflec-tions, friction torques, and fatigue lives.

REFERENCES2.1. American National Standards Institute, American National Standard (ANSI /ABMA)

Std. 20-1987, "Radial Bearings of Ball, Cylindrical Roller, and Spherical Roller Types,Metric Design" (October 28, 1987).

2.2 A. Jones, Analysis of Stresses and Deflections, vol. 1, New Departure Division, GeneralMotors Corp., Bristol, Conn., 12 (1946).

LIST OF SYMBOLS

Symbol Description Units

B Basic inner ring width mm (in.)B. Single width of an inner ring mm (in.)C Basic outer ring width mm (in.)C. Single width of an outer ring mm (in.)d Basic bore diameter mm (in.)d· Bearing inner raceway diameter mm (in.)

1

do Bearing outer raceway diameter mm (in.)d. Single diameter of a bore mm (in.)drop Single plane mean bore diameter mm (in.)D Basic outside diameter mm (in.)D. Single diameter of an outside mm (in.)

surfaceDrop Single plane mean outside mm (in.)

diameter~l Common diameter mm (in.)~lh Basic housing bore mm (in.)

82 INTER ••.•~RPJN('II rrI'I'DfG AND CLEARANCE


'J\ Outside ring o.d. mm (in,)'J!2 Inside ring i.d. mm (in,)'J\ Basic shaft diameter mm (in,)E Modulus of elasticity N/mm8 (psi)I Interference maa (in.)Kia Radial runout of assembled bearinll ~m (in.)

inner ringKea Radial runout of assembled bearinll ~ (in,)

outer ringS) Length mm (in,)Pd Bearing clearance mm (in.)p Pressure N/mmll (psi)~R Ring radius _Cin.)H~ Inside radius of ring mm (in.)~Ro Outside radius of ring mm (tn,)Sd Inner ring reference face runout ~m (in.)

with boreSD Outside cylindrical surface runout ~m (in.)

with outer ring reference faceSia Axial runout of assembled bearing ~m (tn.)

inner ringSea Axial runout of assembled bearing ~m (in,)

outer ringu Radial deflection mm (in,)Vdp Bore diameter variation in a single ~m (tn.)

radial planeVdmp Mean bore diameter variation ~m (in.)VDmp Mean outside diameter variation ~m Un,)VDp Outside diameter variation in ~m Un,)

single radial planeLlBs Single inner ring width deviation ~m Un.)

from basicLles Single outer ring width deviation ~m (in.)

from basic ,

Llds Single bore diameter deviation ~m (tn.)from basic

Lldmp Single plane mean bore diameter ~m Un.)deviation from basic for a taperedbore small end

Lld1mp Single plane mean bore diameter ~m (tn.)deviation at large end of taperedbore

LlDs Single outside diameter deviation ~m (in.)from basic

GENERAL 83


LlDmp Single plane mean outside /Lm (in.)diameter deviation from basic

Llh Clearance reduction due to press mm (in.)fitting of bearing in housing

Lls Clearance reduction due to press mm (in.)fitting of bearing on shaft

Llt Clearance increase due to thermal mm (in.)expanSIOn

T Temperature °C(OF)Er Strain in radial direction mm/mm (in.!in.)Et Strain in tangential direction mm/mm (in.!in.)r Coefficient of linear expansion mm/mm;oC (in.!

in. ;OF)g Poisson's ratio(Ir Normal stress in radial direction N/mm2 (psi)(It Normal stress in tangential N/mm2 (psi)

direction

GENERAL

Ball and roller bearings are usually mounted on shafts or in housingswith interference fits. This is usually done to prevent fretting corrosionthat could be produced by relative movement between the bearing innerring bore and the shaft o.d. and/or the bearing outer ring o.d. and thehousing bore. The interference fit of the bearing inner ring with the shaftis usually accomplished by pressing the former member over the latter.In some cases, however, the inner ring is heated to a controlled temper-ature in an oven or in an oil bath. Then the inner ring is slipped overthe shaft and allowed to cool, thus accomplishing a shrink fit.

Press or shrink fitting of the inner ring on the shaft causes the innerring to expand slightly. Similarly, press fitting of the outer ring in thehousing causes the former member to shrink slightly. Thus, the bearing'sdiametral clearance will tend to decrease. Large amounts of interferencein fitting practice can cause bearing clearance to vanish and even pro-duce negative clearance or interference in the bearing.

Thermal conditions of bearing operation can also affect the diametralclearance. Heat generated by friction causes internal temperatures torise. This in turn causes expansion of the shaft, housing, and bearingcomponents. Depending on the shaft and housing materials and on themagnitude of thermal gradients across the bearing and these supportingstructures, clearance can tend to increase or decrease. It is also apparent

84 INTERFERENCE FITTING AND CLEARANCE

that the thermal environment in which a bearing operates may have asignificant effect on clearance.

In Chapter 2 it was demonstrated that clearance significantly affectsball bearing contact angle. Subsequently, the effect of clearance on bear-ing internal load distribution and life will be investigated. It is thereforeclear that the mechanics of bearing fitting practice is an important partof this book.

INDUSTRIAL, NATIONAL, AND INTERNATIONAL STANDARDS

Standards defining recommended practices for ball and roller bearingusage were first developed in the United States by the Anti-FrictionBearing Manufacturers Association (AFBMA), which has now becomethe American Bearing Manufacturers Association (ABMA).ABMA con-tinues the process of revising the current standards and proposing andpreparing new standards as deemed necessary by its bearing industrymember companies. ABMA-generated standards are subsequently pro-posed to the American National Standards Institute (ANSI) as UnitedStates national standards. ANSI has a committee dedicated to rollingbearing standard activities; this committee has representatives of bear-ing user organizations such as major industrial manufacturers and theU.S. government. Other countries have national standards organizationssimilar to ANSI; for example, DIN in Germany and JNS in Japan. Cur-rently, 26 documents, some having metric and English unit system parts,have been published as ANSI!ABMAstandards.

Any national standard may subsequently be proposed to the Inter-national Organization for Standards, and after extended negotiationpublished as "International Standard (ISO)"with an identifying number.In this chapter, various bearing, shaft, and housing tolerance data areexcerpted from the American National Standards.

Reference [3.1] defines recommended practice in fitting bearing innerrings to shafts and outer rings in housings. These fits are recommendedin terms oflight, normal, and heavy loading as defined by Fig. 3.1. Figure3.2 shows the designations and relative magnitudes of the shaft-bearingbore and housing-bearing o.d. tolerance ranges. Each shaft-bearing fittolerance range is designated by a lower case letter followedby a number,for example, g6, h5 and so on. Similarly, each tolerance range symbol forhousing-bearing fit consists of an upper case letter followedby a number,for example, G7 or H6. Table 3.1 gives the ANSI/ABMA-recommendedpractice for fitting inner rings on shafts; Table 3.2 shows the shaft di-ameter tolerance limits corresponding to the recommended fit. Tables 3.3and 3.4 yield similar data for fitting of bearing outer rings in housingbores.

ANSI!ABMA in references [3.2-3.10] also provide standards for tol-erance ranges on bearing bore and o.d. for various types of radial bear-

ings. Several of these bearing types, for example, tapered roller bearings,needle roller bearings, and instrument ball bearings, exist in too manyvariations to include all of the appropriate tolerance tables herein. Onthe other hand, reference [3.10] covers a wide range of standard radialball and roller bearings; Tables 3.6-3.10 are taken from reference [3.10].For radial ball bearings these tolerances are grouped in ABEC* classes1, 3, 5, 7, and 9 according to accuracy of manufacturing. Accuracy im-proves and tolerance ranges narrow as the class number increases. Ta-bles 3.6-3.10 give tolerance ranges for all ABEC classifications.Additionally, Tables 3.6-3.8 provide the tolerances or bore and o.d. forradial roller bearings as well as for ball bearings. The ABEC and RBECttolerance classes correspond in every respect to the precision classes en-dorsed by the ISO. Table 3.5 shows the correspondence between theANSI!ABMA and ISO classifications. It is further noted that inch tol-erances given in Part II of Tables 3.6-3.10 are calculated from primarymetric tolerances given in Part I of those tables.

To define the range of interference or looseness in the mounting of aninner ring on a shaft or an outer ring in a housing, it is necessary toconsider combination of the shaft, housing, and bearing tolerances.

•••Annular Bearing Engineers' Committee of ABMA.tRoller Bearing Engineers' Committee of ABMA.

CLOSUREThe important effect of bearing fitting practice on diametral clearancehas been demonstrated for ball bearings in the numerical examples. Be-cause the ball bearing contact angle determines its ability to carry thrustload and the contact angle is dependent on clearance, the analysis of thefit-up is important in many applications. The numerical examples hereinwere based on mean tolerance conditions. In many cases, however, it isnecessary to examine the extremes of fit.

Although only the effect of fit-up on contact angle has been examined,it is not to be construed that this is the only effect of significance. Later,the sensitivity of other phases of rolling bearing operation to clearancewill be investigated.

The thermal conditions of operation have been shown to be of no lesssignificance than the fit-up. In precision applications, the clearance mustbe evaluated under operating conditions.

Tables 3.6-3.10 contain tolerance limits on radial and axial runout aswell as the tolerance limits on mean diameters. Runout affects bearingperformance in subtle ways such as through vibration as discussed inChapter 25.

REFERENCES3.1. American National Standards Institute, American National Standard (ANSI IABMA)

Std 7-1995, "Shaft and Housing Fits for Metric Ball and Roller Bearings (ExceptTapered Roller Bearings) Conforming to Basic Boundary Plans" (October 27, 1995).

132INTERFERENCE FI'ITING AND CLEARANCE

3.2.j\merican National Standards Institute, American National Standard (ANSI /ABMA)Std 12.1-1992, "Instrument Ball Bearings Metric Design" (April 6, 1992).

3.3.j\merican National Standards Institute, American National Standard (ANSI /ABMA)Std 12.2-1992, "Instrument Ball Bearings Inch Design" (April 6, 1992).

3.4.j\merican National Standards Institute, American National Standard (ANSI /ABMA)Std 16.1-1988, "Airframe Ball, Roller, and Needle Bearings-Metric Design" (Novem-per 17, 1988).

3.5.j\merican National Standards Institute, American National Standard (ANSI/ ABMA)Std 16.2-1990, "Airframe Ball, Roller, and Needle Bearings - Inch Design" (December20, 1990).

3.6.American National Standards Institute, American National Standard (ANSI /ABMA)Std 18.1-1982, "Needle Roller Bearings Radial Metric Design" (December 2, 1982).

3.7.American National Standards Institute, American National Standard (ANSI /ABMA)Std 18.2-1982, "Needle Roller Bearings Radial Inch Design" (May 14, 1982).

3.8.American National Standards Institute, American National Standard (ANSI/ABMA)Std 19.1-1987, "Tapered Roller Bearings Radial Metric Design" (October 19, 1987).

3.9.American National Standards Institute, American National Standard (ANSI /ABMA)Std 19.2-1994, "Tapered Roller Bearings-Radial Inch Design" (May 12, 1994).

310. American National Standards Institute, American National Standard (ANSI/ABMA). Std 20-1987, "Radial Bearings of Ball, Cylindrical Roller, and Spherical Roller Types,

Metric Design" (October 28, 1987).3 11. A. Jones, Analysis of Stresses and Deflections, New Departure Division, General Mo-

. tors Corp., Bristol, Conn., 161-170 (1946).

LIST OF SYMBOLS


a Distance to load point from right-hand mm (in.)bearing center

e Gear train valueF Bearing radial load N (lb)g Gravitational constant mm/sec2 (in.lsec2)

h Thread pitch of worm at the pitch radius mm (in.)H Power watts (HP)I Distance between bearing centers mm (in.)I Length of connecting rod mm (in.)n Speed rpmN Number of teeth on gearP Applied radial direction load N (lb)Pp Force applied on piston pin N (lb)Pn Inertial force due to reciprocating masses N (lb)Pel Centrifugal force acting on connecting rod N (lb)

bearing due to rotating masses

134 BEARING LOADS AND SPEEDS

Symbol Description UnitsPcc Centrifugal force acting on crankshaft N (lb)

bearing due to rotating massesr Crank radius mm (in.)rp Gear pitch radius mm (in.)T Applied moment load N . mm (lb . in.)w Applied load per unit length NImm (lb/in.)WI Weight of reciprocating parts N (lb)W2 Weight of connecting rod including bearing N (lb)

assembliesW2, Weight of reciprocating portion of N (lb)

connecting rodW2" Weight of rotating portion of connecting rod N (lb)W3 Weight of crank pin and crank webs with N (lb)

balance weightsx Distance along shaft mm (in.)Z Number of threads on worm, teeth on worm

wheely Bevel gear cone angle 0, radA Lead angle of worm at the pitch radius 0, rad4> Gear pressure angle 0, radi/J Gear helix angle 0, rad

SUBSCRIPTS1,2,3 Refers to bearing location1,2 Refers to driving or driven geara Refers to axial directionr Refers to radial directiont Refers to tangential direction

SUPERSCRIPTk Refers to location of applied load or moment

GENERALThe loading a rolling bearing supports is usually transmitted to the bear-ing through the shaft on which the bearing is mounted. Sometimes, how-ever, the loading is transmitted through the housing that encompassesthe bearing outer ring; for example, a wheel bearing. In either case, andin most applications, it is sufficient to consider the bearing as simplyresisting the applied load and not as an integral part of the loaded sys-tem. This condition will be covered in this chapter together with defini-tion of the loads transferred to the shaft-bearing system by some commonpower transmission components.

More generally, for a train of several gears, the train value e is the ratioof the product of the pitch radii of the driving gears to the product of thepitch radii of the driven gears. (It is noted that the numbers of teeth canbe substituted for the pitch radii.) Hence, the output shaft speed can bedirectly determined.

Planetary gear or epicyclic power transmissions are designed toachieve substantial speed reduction in a compact space. In its simplestform, the epicyclic transmission is shown schematically by Fig. 4.19, inwhich R refers to the ring gear, P to the planet gear, and S to the sungear. The sun gear typically is connected to the input shaft, and theoutput shaft is connected to the arm. In general, there are three or moreplanets; therefore, each planet gear shaft transmits one-third or less ofthe input power. A single planet is shown in Fig. 4.19 for the purpose ofanalysis of speeds. Using Fig. 4.19, it can be seen that the speed of the

156 BALL AND ROLLER LOADS


Qa Axial direction load on ball or roller N (lb)r Radius mm (in.)U Coordinate direction distance mm (in.)V Coordinate direction distance mm (in.)W Coordinate direction distance mm (in.)x Coordinate direction distance mm (in.)x Acceleration in x-direction mml sec2 (in.! sec2)

x' Coordinate direction distance mm (in.)y Coordinate direction distance mm (in.)y Acceleration in y-direction mml sec2 (in.! sec2)

y' Coordinate direction distance mm (in.)z Coordinate direction distance mm (in.)Z Acceleration in z-direction mml sec2 (in.! sec2)

z' Coordinate direction distance mm (in.)a Contact angle 0, rad{3 Angle between Waxis and z' axis rad{3' Angle between projection of the U

axis on the x'y' plane and the x' axis rad"Y D cos aldm

"Ys Roller skewing angle 0, rad~ Roller tilting angle 0, rad() Angle radp Mass density kg/mm3 (lb . sec2 •

in. -4)

cP Angle in WV plane radIf1 Angle in yz plane radWm Orbital angular velocity of ball or

roller rad/ secWR Angular velocity of ball or roller

about its own axis radl sec

SUBSCRIPTSa Refers to axial directione Refers to rotation about an eccentric

axISf Refers to guide flangei Refers to inner racewayJ Refers to rolling element at location jm Refers to orbital rotation0 Refers to outer racewayR Refers to rolling elementr Refers to radial direction

STATIC LOADING 157

GENERAL

The loads carried by ball and roller bearings are transmitted throughthe rolling elements from one ring to the other. The magnitude of theloading carried by the individual ball or roller depends on the internalgeometry of the bearing and on the type of load impressed on it. In ad-dition to applied loading, rolling elements are subjected to dynamic load-ing due to speed effects. Bearing geometry also affects the dynamicloading. The object of this chapter is to define the rolling element loadingin ball and roller bearings under varied conditions of bearing operation.

and the component Fe sin (1'0 causes the inner and outer raceway contactangles to shift slightly from (1'0 to accommodate the roller axial loading.In general, spherical roller bearings do not operate at speeds that willcause significant change in the nominal contact angle. Also, consider adouble-row spherical roller bearing having barrel-shaped rollers sub-jected to a radial load while rotating at high speed. The speed-inducedroller axial loads are self-equilibrated within the bearing; however, theouter raceways carry larger thrust components than do the inner race-ways.

Rotation about an Eccentric Axis. The foregoing section dealt with roll-ing element centrifugal loading when the bearing rotates about its ownaxis. This is the usual case. In planetary gear transmissions, however,the planet gear bearings rotate about the input and/or output shaft axesas well as about their own axes. Hence an additional inertial or centrif-ugal force is induced in the rolling element. Figure 5.9 shows a schematicdiagram of such a system. From Fig. 5.9 it can be seen that the instan-taneous radius of rotation is by the law of cosines

Axial Loading Effects in Radial Roller Bearings

Axial loading of the rollers in a radial roller bearing can significantlyaffect bearing performance. In cylindrical and tapered roller bearings aroller axial load is reacted between the roller end and a guide flange.The combination of load and sliding in this contact can cause the bearingto generate excessive heat and, under certain conditions, lead to wearand smearing of the contacting surfaces. Conversely, with proper man-agement of roller end-flange design and lubrication, bearing heat gen-eration and axial load-carrying capacity can be optimized. Axial rollerloads interact with roller radial forces and ring deflections to determineroller tilting and skewing motions that influence roller-raceway contactstresses and sliding velocity distributions. Roller tilting and skewingsubstantially affect roller bearing heat generation, friction torque, androlling contact fatigue life. Many modern spherical roller bearing designsdo not use flanges as the primary source of roller guidance. Such designsrely on proper management of roller tilting and skewing motions viaroller-raceway traction forces to provide roller guidance while minimiz-ing bearing heat generation and friction torque.

External Applied Roller Thrust Loading

Many radial roller bearing designs have the ability to carry applied axialor thrust load in addition to their predominant radial loading. Cylindri-

178 BALL AND ROLLER LOADS

cal roller bearings can carry thrust loading by virtue of flanges fixed toinner and outer rings. The angular-contact roller arrangement oftaperedroller bearings reacts applied thrust load over the combination of roller-raceway contact surfaces and roller end-flange contacts. Spherical rollerbearings react applied thrust loading through their roller-raceway con-tacts (and flanges if asymmetric rollers are employed). Axial loading ofradial roller bearings causes alteration of internal load distribution androller response that can significantly affect bearing performance.

Cylindrical roller bearings with fixed inner and outer ring flanges, asshown in Fig. 5.12a, can carry axial load through contacts between theroller end faces and the flanges. The couple produced on the roller by theaxial roller end-flange forces Qaj results in a tilting ~ about the center ofthe roller. As the roller tilts, the roller-raceway load distribution shiftsasymmetrically, as represented in Fig. 5.13. The roller-raceway load dis-

BOLLER AXIAL LOADING IN RADIAL BEARINGS 179

tribution of Fig. 5.13 can be compared with the ideal load distributionin Fig. 6.23b.

Roller Axial Loading Due to Roller Skewing

In roller bearings skewing is defined as an angular rotation of the rolleraxis (in a plane tangent to its orbital direction) with respect to the axisof the contacting ring. The magnitude of the skewing motion may beexpressed as a skewing angle "YS' The skewing angle of a roller at a givenazimuth position might be different for inner and outer ring contacts ifmisalignment exists between inner and outer rings. In general, the skew-ing angle will vary with roller azimuth position 1jJ.

Skewing is caused by forces acting on the roller, which result in amoment loading about an axis oriented in the bearing radial directionand passing through the roller at its midpoint. These skewing forces areoften due to asymmetrical distribution of tangential friction forces at theroller-raceway contacts arising from asymmetrical normal loading and/or sliding velocity distributions along the contacts. Normal and frictionalforces acting on the roller due to contact with guide flanges (includingroller applied axial loads) and cage may contribute to, or serve to limit,roller skewing motions. Operating conditions associated with asymmet-ricalload distributions include misalignment of inner and outer rings incylindrical roller bearings and tapered roller bearings and applied rolleraxial loading in cylindrical roller bearings. In high-speed roller bearingoperation the dynamic effects of roller mass unbalance and impact load-ing between roller and flange or cage can become significant causativefactors in roller skewing. Forces that give rise to roller skewing motionsmay also, by virtue of their points of application or radial force compo-nents, be coupled to the roller tilting action. Quantification of rollerskewing and coupled tilting behavior require computer programs de-signed for this purpose.

As an illustration of roller skewing, consider the cylindrical rollerbearing roller subjected to radial and axial applied loads shown in Fig.5.14. The applied axial load Qaj causes the roller to tilt through the angle'j' The tilting motion gives rise to the asymmetrical contact load distri-bution in Fig. 4.14a. Assuming no gross sliding or skidding in the appli-cation, the tilting also causes sliding motion to occur along the inner andouter raceway contacts. Roller deformations cause the sliding velocity onthe uncrowned roller to be zero at only one point along each racewaycontact. The tangential sliding velocity distributions for the unskewedroller are represented by Fig. 5.14b. Positive and negative values obtaincorresponding to sliding in the z direction. Tangential friction forces inboth contacts are related to the magnitude of the normal contact loadand magnitude and direction of tangential sliding velocity. The asym-metrical tangential friction force distribution shown in Fig. 5.14c results.This distribution causes a skewing moment about the y axis. The skew-

FIGURE 5.14. Roller loading and tangential sliding velocity in a cylindrical roller bearingwith applied axial and radial load. (a) Applied roller loads and roller load distribution. (b)

Tangential sliding velocity distribution. (c) Tangential traction force distribution and skew-ing moments. (d) Roller-flange contact forces and resultant axial friction force (view fromouter ring).

ing moment tends to cause the roller to skew and generate an axialfriction force at both contacts, the resultant of which, Qa, is shown inFig. 5.14d. Also contributing to the roller skewing moment are the fric-tional forces generated at the flange contacts.

In the illustration the skewing moment and axial friction force mustbe reacted by the flange contacts. In principle, a skewing angle "Ys may

REFERENCE 181

be achieved whereby roller skewing moment and axial force are in equi-librium with the flange contact forces. Note that, in general, the locationsof the roller-flange contacts, the roller-raceway normal load distribution,sliding velocities, and tangential and axial friction forces are all func-tions of the roller skewing angle. The skewing angle at which this forcebalance is obtained is known as an equilibrium skewing angle.

As bearing operational speed increases, dynamic effects become sig-nificant. Brown et al. [4.1] investigated the problem of roller axial loaddue to skewing in cylindrical roller bearings under high-speed conditions.Their analytical and experimental work highlighted the detrimental ef-fects of unbalance forces due to roller corner radius runout. Analyticalmodels were developed for roller impact loading on flange and cage. Theeffect of bearing design parameters was empirically correlated with ob-served roller end wear.

CLOSURETo analyze rolling bearing performance in a given application, it is usu-ally necessary to determine the load on individual balls or rollers. Howwell the balls or rollers accept the applied and induced loads will deter-mine bearing endurance. For example, light radial load applied to a 90°contact angle thrust bearing can cause the bearing to fail rapidly. Simi-larly, thrust load applied to a 0° contact angle, radial ball bearing islargely magnified according to the final contact angle that obtains. InChapters 7 and 9 this book will concern itself with the distribution ofload among balls and rollers. It will be shown that the manner in whicheach rolling element accepts its load will determine in large measure theloading of all others. Moreover, in angular-contact ball bearings the ballloading can affect ball and cage speeds significantly. In high-speed rollerbearings if roller loading is too light, rolling motion may be preemptedby skidding. The material in this chapter is therefore fundamental toeven the most rudimentary analysis of a rolling bearing application.

REFERENCE5.1. P. Brown, D. Robinson, L. Dobek, and J. Miner, "Mainshaft High-Speed, Cylindrical

Roller Bearings for Gas Turbine Engines," U.S. Navy Contract NOOOI40-76-C-0383Interim Report (1978).

LIST OF SYMBOLS


a Semimajor axis of the projected contact mm (in.)

a* Dimensionless semimajor axis of contactellipse

b Semiminor axis of the projected contactellipse mm (in.)

b* Dimensionless semiminor axis of contactellipse

E Modulus of elasticity NImm2 (psi)&, Complete elliptic integral of the second kind&,(</1) Elliptic integral of the second kindJ Complete elliptic integral of the first kindJ(</1) Elliptic integral of the first kindF Force N (lb)

G Shear modulus of elasticity N/mm2 (psi)

1 Roller effective length mm (in.)

Q Normal force between rolling element andraceway N (lb)

183

184 CONTACT STRESS AND DEFORMATION

Symbol Description Unitsr Radius of curvature mm (in.)S Principal stress N/mm2 (psi)u Deflection in x direction mm (in.)U Arbitrary functionv Deflection in y direction mm (in.)V Arbitrary functionw Deflection in z direction mm (in.)x Principal direction distance mm (in.)X Dimensionless parametery Principal direction distance mm (in.)y Dimensionless parameterz Principal direction distance mm (in.)Zl Depth to maximum shear stress at x = 0,

y=O mm (in.)Zo Depth to maximum reversing shear stress

y =F 0, x = 0 mm (in.)Z Dimensionless parameter'Y Shear strain8 Deformation mm (in.)8* Dimensionless contact deformationE Linear strain~ zlb, roller tilting angle 0, rad() Angle rad1} Auxiliary angle radK albA Parameter~ Poisson's ratio(J Normal stress N/mm2 (psi)T Shear stress N/mm2 (psi)v Auxiliary angle radcf> Auxiliary angle rad or °F(p) Curvature difference"Lp Curvature sum mm-l (in.-l)

SUBSCRIPTSi Refers to inner raceway0 Refers to outer racewayr Refers to radial directionx Refers to x directiony Refers to y directionz Refers to z directionyz Refers to yz plane

TIlEORY OF ELASTICITY 185


xz Refers to xz planeI Refers to contact body III Refers to contact body II

GENERAL

Loads acting between the rolling elements and raceways in rolling bear-ings develop only small areas of contact between the mating members.Consequently, although the elemental loading may only be moderate,stresses induced on the surfaces of the rolling elements and racewaysare usually large. It is not uncommon for rolling bearings to operatecontinuously with normal stresses exceeding 1380 NImm2 (200,000 psi)compression on the rolling surfaces. In some applications and duringendurance testing normal stresses on rolling surfaces may exceed 3449N/mm2 (500,000 psi) compression. Since the effective area over whichload is supported rapidly increases with depth below a rolling surface,the high compressive stress occurring at the surface does not permeatethe entire rolling member. Therefore, bulk failure of rolling members isgenerally not a significant factor in rolling bearing design; however, de-struction of the rolling surfaces is. This chapter is therefore concernedonly with the determination of surface stresses and stresses occurringnear the surface. Contact deformations are caused by contact stresses.Because of the rigid nature of the rolling members, these deformationsare generally of a low order of magnitude, for example 0.025 mm (0.001in.) or less in steel bearings. It is the purpose of this chapter to developrelationships permitting the determination of contact stresses and de-formations in rolling bearings.

THEORY OF ELASTICITY

The classical solution for the local stress and deformation of two elasticbodies apparently contacting at a single point was established by Hertz[6.1] in 1881. Today, contact stresses are frequently called Hertzian orsimply Hertz stresses in recognition of his accomplishment.

To develop the mathematics of contact stresses, one must have a firmfoundation in principles of mechanical elasticity. It is, however, not apurpose of this text to teach theory of elasticity and therefore only arudimentary discussion of that discipline is presented herein to demon-strate the complexity of contact stress problems. In that light consideran infinitesimal cube of an isotropic homogeneous elastic material sub-

It is apparent from equation (6.16) that as r approaches 0, ar becomesinfinitely large. It is further apparent that this condition cannot existwithout causing gross yielding or failure of the material at the surface.

Hertz reasoned that instead of a point or line contact, a small contactarea must form, causing the load to be distributed over a surface, andthus alleviating the condition of infinite stress. In performing his anal-ysis, he made the following assumptions:

1. The proportional limit of the material is not exceeded, that is, alldeformation occurs in the elastic range.

2. Loading is perpendicular to the surface, that is, the effect of surfaceshear stresses is neglected.

3. The contact area dimensions are small compared to the radii ofcurvature of the bodies under load.

4. The radii of curvature of the contact areas are very large comparedto the dimensions of these areas.

The solution of theoretical problems in elasticity is based on the as-sumption of a stress function or functions that singly or in combinationfit the compatibility equations and the boundary conditions. For stress

SUBSURFACESTRESSESHertz's analysis applied only to surface stresses caused by a concentratedforce applied perpendicular to the surface. Experimental evidence indi-cates that failure of rolling bearings in surface fatigue caused by thisload emanates from points below the stressed surface. Therefore, it is ofinterest to determine the magnitude of the subsurface stresses. Since thefatigue failure of the surfaces in contact is a statistical phenomenon de-pendent on the volume of material stressed (see Chapter 18), the depthsat which significant stresses occur below the surface are also of interest.

Again, considering only stresses caused by a concentrated force nor-mal to the surface, Jones [6.8] after Thomas and Hoersch [6.7] gives thefollowing equations by which to calculate the principal stresses Sx, Sy,and Sz occurring along the Z axis at any depth below the contact surface.

depths z, below the surface, being at 0.467b for simple point contact and0.786b for line contact.

During the passage of a loaded rolling element over a point on theraceway surface, the maximum shear stress on the z axis varies betweeno and Trnax' If the element rolls in the direction of the y axis, then theshear stresses occurring in the yz plane below the contact surface assumevalues from negative to positive for values of y less than and greaterthan zero, respectively. Thus, the maximum variation of shear stress inthe yz plane at any point for a given depth is 2Tyz.

Lundberg and Palmgren [6.9] show that

Figure 6.13 shows the resulting distribution of shear stress at depth Zo

in the direction of rolling for bfa = 0, that is, a line contact.Figure 6.14 shows the shear stress amplitude of equation (6.71) as a

function of bfa. Also shown is the depth below the surface at which thisshear stress occurs. Since the shear stress amplitude indicated by Fig.6.14 is greater than that of Fig. 6.12, Lundberg and Palmgren [6.9] as-sumed this shear stress, called the maximum orthogonal shear stress, tobe significant in causing fatigue failure of the surfaces in rolling contact.As can be seen from Fig. 6.14, for a typical rolling bearing point contactof bfa = 0.1, the depth below the surface at which this stress occurs isapproximately 0.49b. Moreover, as seen by Fig. 6.13, this stress occurs

at any instant under the extremities of the contact ellipse with regardto the direction of motion, that is, at y = ± 0.9b.

Metallurgical research [6.10] based on plastic alterations detected insub-surface material by transmission electron microscopic investigationgives indications that the subsurface depth at which significant amountsof material alteration occur is approximately 0.75b. Assuming such plas-tic alteration is the forerunner of material failure, then it would appearthat the maximum shear stress of Fig. 6.12 may be worthy of consider-ation as the significant stress causing failure. Figures 6.15 and 6.16 fromreference [6.10] are photomicrographs showing the subsurface changescaused by constant rolling on the surface.

Many researchers consider the von Mises- Hencky distortion energytheory [6.11] and the scalar von Mises stress a better criterion for rollingcontact failure failure. The latter stress is given by

For case-hardening bearings the value of ZOi and ZOo can be used toestimate the required case depth. Note that the maximum shear stressat the center of contact occurs at Zli = 0.76bi and ZlO = 0.755bo for theinner and outer raceways, respectively (see Fig. 6.12). Hence Zli =

0.246 mm (0.00867 in.) and Zlo = 0.281 mm (0.01108 in.). It is moreconservative to base case depth on these values. Case depth shouldexceed Zo or Zl by at least a factor of three.

EFFECT OF SURFACE SHEAR STRESS

In the determination of contact deformation vs load only the concen-trated load applied normal to the surface need be considered for mostapplications. Moreover, in most rolling bearing applications, lubricationis at least adequate, and the sliding friction between rolling elementsand raceways is negligible (see Chapter 14). This means that the shearstresses acting on the rolling elements and raceway surfaces in contact,that is, the elliptical areas of contact, are negligible compared to normalstresses.

For the determination of bearing endurance with regard to fatigue ofthe contacting rolling surfaces, the surface shear stress cannot be ne-glected and in many cases is the most significant factor in determiningendurance of a rolling bearing in a given application. Methods of calcu-lation of the surface shear stresses (traction stresses) will be discussedin Chapter 14. The means for determining the effect on the subsurfacestresses of the combination of normal and tangential (traction) stressesapplied at the surface are extremely complex requiring the use of digitalcomputation. Among others, Zwirlein and Schlicht [6.10]have calculatedsubsurface stress fields based upon assumed ratios of surface shearstress to applied normal stress. Reference [6.10] assumes that the vonMises stress is most significant with regard to fatigue failure and givesillustrations of this stress in Fig. 6.18.

Figure 6.19 also from reference [6.10] shows the depth at which thevarious stresses occur. Figure 6.19 shows that as the ratio of surfaceshear to normal stress increases, the maximum von Mises stress movescloser to the surface. At a ratio of or/ a = 0.3, the maximum von Misesstress occurs at the surface. Various other investigators have found thatif a shear stress is applied at the contact surface in addition to the nor-mal stress, the maximum shear stress tends to increase and it is located

closer to the surface (see references [6.11-6.15]. References [6.16-6.18]give indications of the effect ofhigher order surfaces on the contact stresssolution. The references cited above are intended not to be extensive, butto give only a representation of the field of knowledge.

The foregoing discussion pertained to the subsurface stress fieldcaused by a concentrated normal load applied in combination with a uni-form surface shear stress. The ratio of surface shear stress to normalstress is also called the coefficient of friction (see Chapter 14). Becauseof infinitesimally small irregularities in the basic surface geometries ofthe rolling contact bodies, neither uniform normal stress fields as shownby Figs. 6.6 and 6.7 nor a uniform shear stress field are likely to occurin practice. Sayles et al. [6.19] use the model shown by Fig. 6.20 in de-veloping an "elastic conformity factor."

Kalker [6.20] developed a mathematical model to calculate the sub-surface stress distribution associated with an arbitrary distribution ofshear and normal stresses over a surface in concentrated contact. Ah-madi et al. [6.21] developed a "patch" method that can be applied todetermine the subsurface stresses for any concentrated contact surfacesubjected to arbitrarily distributed shear stresses. Using superposition,this method combined with that of Thomas and Hoersch [6.7], for ex-ample, for Hertzian surface loading, can be applied to determine thesubsurface stress distributions occurring in rolling element-raceway con-tacts. Harris and Yu [6.22], applying this method of analysis, determinedthat the range of maximum orthogonal shear stress, i.e., 21'0' is not al-tered by the addition of surface shear stresses to the Hertzian stresses.Fig. 6.21 illustrates this condition.

TYPE OF CONTACT 219

Since the Lundberg-Palmgren fatigue life theory [6.9] is based onmaximum orthogonal shear stress as the fatigue failure-initiating stress,the adequacy of using that method to predict rolling bearing fatigue en-durance is called into question. Conversely, for simple Hertzian loading,i.e., f = 0, maximum octahedral shear stress 'Toct,max occurs directly underthe center of the contact. Fig. 6.22 further shows that the magnitude of'Toct,max and the depth at which it occurs is substantially influenced bysurface shear stress.

The question of which stress should be used for fatigue failure lifeprediction will be revisited in Chapters 18 and 23.

TYPE OF CONTACT

Basically, two hypothetical types of contact can be defined under condi-tions of zero load. These are

1. Point contact, that is, two surfaces touch at a single point.2. Line contact, that is, two surfaces touch along a straight or curved

line of zero width.

Obviously, after load is applied to the contacting bodies the point expandsto an ellipse and the line to a rectangle in ideal line contact, that is, thebodies have equal length. Figure 6.23 illustrates the surface compressivestress distribution which occurs in each case.

When a roller of finite length contacts a raceway of greater length, theaxial stress distribution along the roller is altered from that of Fig. 6.23.Since the material in the raceway is in tension at the roller ends becauseof depression of the raceway outside of the roller ends, the roller endcompressive stress tends to be higher than that in the center of contact.Figure 6.24 demonstrates this condition of edge loading.

To counteract this condition, cylindrical rollers (or the raceways) maybe crowned as shown by Fig. 1.38. The stress distribution is therebymade more uniform depending upon applied load. If the applied load isincreased significantly, edge loading will occur once again.

Lundberg et al. [6.9] have defined a condition of modified line contactfor roller-raceway contact. Thus, when the major axis (2a) of the contactellipse is greater than the effective roller length l but less than 1.5l,modified line contact is said to exist. If 2a < l, then point contact exists;if 2a > 1.5l, then line contact exists with attendant edge loading. Thiscondition may be ascertained approximately by the methods presentedin the section "Surface Stresses and Deformations," using the rollercrown radius for R in equations (2.37)-(2.40).

The analysis of contact stress and deformation presented in this sec-tion is based on the existence of an elliptical area of contact, except forthe ideal roller under load, which has a rectangular contact. Since it isdesirable to preclude edge loading and attendant high stress concentra-tions, roller bearing applications should be examined carefully accordingto the modified line contact criterion. Where that criterion is exceeded,redesign of roller and/or raceway curvatures may be necessitated.

Rigorous mathematical/numerical methods have been developed tocalculate the distribution and magnitude of surfaces stresses in any"line" contact situation, that is, including the effects of crowning of roll-ers, raceways, and combinations thereof (see references [6.23] and [6.24].Additionally, finite element methods (FEM) have been employed [6.25]

The circular crown shown in Fig. 1.38a resulted from the theory of Hertz[6.1]whereas the cylindrical/crowned profile of Fig. 1.38b resulted fromthe work of Lundberg et al. [6.5]. As illustrated in Fig. 6.26, each of thesesurface profiles, while minimizing edge stresses, has its drawbacks. Un-der light loads, a circular crowned profile does not enjoy full use of theroller length, somewhat negating the use of rollers in lieu ofballs to carryheavier loads with longer endurance (see Chapter 18). Under heavierloads, while edge stresses are avoided for most applications, contactstress in the center of the contact can greatly exceed that in a straightprofile contact, again resulting in substantially reduced endurance char-acteristics.

Under light loads, the partially crowned roller of Fig. 1.38b as illus-trated by Fig. 6.26c experiences less contact stress than does a fullycrowned roller under the same loading. Under heavy loading the par-tially crowned roller also tends to outlast the fully crowned roller becauseof lower stress in the center of the contact; however, unless careful at-tention is paid to blending of the intersections of the "flat" (straight por-tion of the profiles) and the crown, stress concentrations can occur at theintersections with substantial reduction in endurance (see Chapter 18).When the roller axis is tilted relative to the bearing axis, both the fullycrowned and partially crowned profiles tend to generate less edge stressunder a given load as compared to the straight profile.

After many years of investigation and with the assistance of mathe-matical tools such as finite difference and finite element methods as prac-ticed using computers, a "logarithmic" profile was developed [6.26]yielding a substantially optimized stress distribution under most condi-tions of loading (see Fig. 6.262d). The profile is so named because it canbe expressed mathematically as a special logarithmic function. Under allloading conditions, the logarithmic profile uses more of the roller lengththan either the fully crowned or partially crowned roller profiles. Undermisalignment, edge loading tends to be avoided under all but exception-ally heavy loads. Under specific loading (QIlD) from 20 to 100 N/mm2

(2900-14500 psi), Fig. 6.27 taken from [6.26] illustrates the contactstress distributions attendant to the various surface profiles discussedherein. Figure 6.28, also from [6.26], compares the surface and subsur-face stress characteristics for the various surface profiles.

ROLLER END-FLANGE CONTACT STRESS

The contact stresses between flange and roller ends may be estimatedfrom the contact stress and deformation relationships previously pre-sented. The roller ends are usually flat with corner radii blending into


thecrownedportionofthe roller profile. The flange may also be a portionofaflatsurface. This is the usual design in cylindrical roller bearings.Whenit is desired to have the rollers carry thrust loads between therollerends and the flange, sometimes the flange surface is designed asaportionof a cone. In this case, the roller corners contact the flange.Theangle between the flange and a radial plane is called the laybackangle.Alternatively, the roller end may be designed as a portion of aspherethat contac.tsthe flange. The latter arrangement, that is a sphere-endroller contactmg an angled flange, is conducive to improved lubri-cationwhile sacrificingsome flange-roller guidance capability. In thiscasesome skewingcontrol may have to be provided by the cage.

F~r the case of sphere-end rollers and angled flange geometry, theindividualcontact may be modeled as a sphere contacting cylinder. Forthepurpose of calculation the. sphere radius is set equal to the rollersphereend radius, and the cY~lIlder radius can be approximated by theradiusof curvature ofthe comcal flange at the theoretical point of con-tact.By knowing the elas~ic contact load, roller-flange material proper-tiesand contact geometrIes, the contact stress and deflection can becal~ulated.This approach is only approximate, because the roller endandflange do not meet the Hertzian half-space assumption. Also, theradiusof curvature on the conical flange is not a constant but will varyacrossthe contactwidth. This method applies only to contacts that arefullyconfinedto th~ spherical roller end and conical portion of the flange.Itispossible that Im~roper geometry or excessive skewing could causetheelasticcontactellIpse to be truncated by the flange edge, undercut,orrollercorner radius.Such a situation is not properly modeled by Hertzstresstheory andshouldbe avoided in design because high edge stressesandpoorlubricationcan result.

Thecase ofa fiat end roller and angled flange contact is less amenabletosimplecontactstress evalua~ion. The nature of the contact surface ontheroller,beingat or near the Intersection of the corner radius and endflatis difficult to model adequately. The notion of an "effective" rollerradiusbased onan assumed blend radius between roller corner and endflatis suitable for approximate calculations. A more precise contactstressdistribution canbe obtained by using finite element stress analysistechniqueif necessary.

CLOSURE

Theinformationpresented in this chapter is sufficient to make a deter-minationof the contact stress level and elastic deformations occurringina statically loadedrolling bearing. The model of a statically loadedbearingis somewhatdistorted by surface tangential stresses induced by

REFERENCES 229rolling and lubricant action. However, under the effects of moderate toheavy loading, the contact stresses calculated herein are sufficiently ac-curate for the rotating bearing as well as the static bearing. The sameis true with regard to the effect of "edge stresses" on roller load distri-bution and hence deformation. These stresses subtend a rather smallarea and therefore do not influence the overall elastic load-deformationcharacteristic. In any event, from the simplified analytical methods pre-sented in this chapter, a level of loading can be calculated against whichto check other bearings at the same or different loads. The methods forcalculation of elastic contact deformation are also sufficiently accurate,and these can be used to compare rolling bearing stiffness against stiff-ness of other bearing types.

REFERENCES6.1. H. Hertz, "On the Contact of Rigid Elastic Solids and on Hardness," in Miscellaneous

Papers, MacMillan, London, 163-183 (1896).6.2. S. Timoshenko and J. Goodier, Theory of Elasticity, 3rd ed., McGraw-Hill, New York

(1970).6.3. J. Boussinesq, Compt. Rend., 114,1465 (1892).6.4. D. Brewe and B. Hamrock, "Simplified Solution for Elliptical-Contact Deformation

Between Two Elastic Solids," ASME Trans., J. Lub. Tech. 101(2),231-239 (1977).

6.5. G. Lundberg and H. Sjovall, Stress and Deformation in Elastic Contacts, Pub. 4,Institute of Theory of Elasticity and Strength of Materials, Chalmers Inst. Tech.,Gothenburg (1958).

6.6. A. Palmgren, Ball and Roller Bearing Engineering, 3rd ed., Burbank, Philadelphia(1959).

6.7. H. Thomas and V. Hoersch, "Stresses Due to the Pressure of One Elastic Solid uponAnother," Uniu. Illinois Bull. 212 (July 15, 1930).

6.8. A. Jones, Analysis of Stresses and Deflections, New Departure Engineering Data,Bristol, Conn., 12-22 (1946).

6.9. A. Palmgren and G. Lundberg, "Dynamic Capacity of Rolling Bearings," Acta Poly-tech. Mech. Eng. Ser. 1, R.SAE.E., No.3, 7 (1947).

6.10. O. Zwirlein and H. Schlicht, "Werkstoffanstrengung bei Walzbeanspruchung-Einflussvon Reibung und Eigenspannungen," Z. Werkstofftech. 11, 1-14 (1980).

6.11. K. Johnson, "The Effects of an Oscillating Tangential force at the Interface BetweenElastic Bodies in Contact," (Ph.D. Thesis, University of Manchester, 1954).

6.12. J. Smith and C. Liu, "Stresses Due to Tangential and Normal Loads on an ElasticSolid with Application to Some Contact Stress Problems," ASME Paper 52-A-13 (De-cember 1952).

6.13. E. Radzimovsky, "Stress Distribution and Strength Condition of Two Rolling Cylin-ders Pressed Together," Uniu. Illinois Eng. Experiment Station Bull., Series 408 (Feb-ruary 1953).

6.14. C. Liu, "Stress and Deformations Due to Tangential and Normal Loads on an ElasticSolid with Application to Contact Stress," (Ph.D. Thesis, University of Illinois, June1950).


6.15. M. Bryant and L. Keer, "Rough Contact Between Elastically and Geometrically Iden-tical Curved Bodies," ASME Trans., J. Applied Mech. 49, 345-352 (June 1982).

6.16. C. Cattaneo, " A Theory of Second Order Elastic Contact," Univ. Roma Rend. Mat.Appl. 6,505-512 (1947).

6.17. T. Loo, "A Second Approximation Solution on the Elastic Contact Problem," Sci. Sin-ica 7, 1235-1246 (1958).

6.18. H. Deresiewicz, "A Note on Second Order Hertz Contact," ASME Trans., J. Appl.Mech. 28, 141-142 (March 1961).

6.19. R. Sayles, G. deSilva, J. Leather, J. Anderson, and P. MacPherson, "Elastic Conform-ity in Hertzian Contacts," Tribology Intl. 14, 315-322 (1981).

6.20. J. Kalker, "Numerical Calculation of the Elastic Field in a Half-Space Due to anArbitrary Load Distributed over a Bounded Region of the Surface," SKF Eng. andRes. Center Report NL82D002, Appendix (June 1982).

6.21. N. Ahmadi, L. Keer, T. Mura, and V.Vithoontien, "The Interior Stress Field Causedby Tangential Loading of a Rectangular Patch on an Elastic Half Space," ASME Paper86-Trib-15 (October 1986).

6.22. T. Harris and W.Yu, "Lundberg-Palmgren Fatigue Theory: Considerations of FailureStress and Stressed Volume," ASME Trans., J. Tribology 121, 85-90 (January 1999).

6.23. K. Kunert, "Spannungsverteilung im Halbraum bei Elliptischer Fliichenpressungs-verteilung tiber einer Rechteckigen Druckfliiche," Forsch. Geb. Ingenieurwes 27(6),165-174 (1961).

6.24. H. Reusner, "Druckfliichenbelastung und Overfliichenverschiebung in Wiilzkontaktvon Rotiitionkorpern (Dissertation, Schweinfurt, Germany, 1977).

6.25. B. Fredriksson, "Three-Dimensional Roller-Raceway Contact Stress Analysis," Ad-vanced Engineering Corp. Report, Linkoping, Sweden (1980).

6.26. H. Reusner, "The Logarithmic Roller Profile-the Key to Superior Performance ofCylindrical and Taper Roller Bearings," Ball Bearing J. 230, SKF (June 1987).

LIST OF SYMBOLS


A Distance between raceway groovecurvature centers mm (in.)

B {; + fo - 1, total curvaturec Crown drop mm (in.)C Influence coefficient mm/N (in. lIb)D Ball or roller diameter mm (in.)dm Bearing pitch diameter mm (in.)e Eccentricity of loading mm (in.)E Modulus of elasticity MPa (psi)f Raceway groove radius -7- DF 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 Number of rows of rolling elementsI Ring section moment of inertia mm4 (in.4)

231

232 DISTRIBUTION OF INTERNAL LOADING IN STATICALLY LOADED BEARINGS

Symbol Description UnitsJa Axial load integralJr Radial load integralJm Moment load integralk Number oflaminaeK Load-deflection factor; axial load deflection

factor N/mmn (lb/in.n)

l Roller length mm Cin.)L Distance between rows mm Cin.)M Moment N . mm (lb . in.)~')]l Moment applied to bearing N . mm (lb . in.)n Load-deflection exponentPd Diametral clearance mm Cin.)q Load per unit length N/mm (lb/in.)Q Ball or roller-raceway normal load N (lb)Qa Roller end-ring flange load N (lb)r Raceway groove curvature radius mm Cin.)r Radius to contact in tapered roller bearing mm Cin.)R3 Tapered roller radius to flange contact at

roller large end mm Cin.)~R Ring radius to neutral axis mm Cin.)~R Radius of locus of raceway groove

curvature centers mm Cin.)s Distance between loci of inner and outer

raceway groove curvature centers mm Cin.)u Ring radial deflection mm Cin.)U Strain energy N . mm (lb . in.)Z Number of rolling elementsa Mounted contact angle rad, °aO Free contact angle rad, °f3 tan-1lfCdm - D) rad, °y D cos a/dm8 Deflection or contact deformation mm Cin.)81 Distance between inner and outer rings mm Cin.)d Contact deformation due to ideal normal

loading mm Cin.)dif; Angular spacing between rolling elements rad, °E Load distribution factor( Roller tilt angle rad, °YJ tan-1 lfD rad, °() Bearing misalignment angle rad, °A Laminum positionJL Coefficient of sliding friction between roller

end and ring flange

GENERAL 233


"Lp Curvature sum mm-1 Cin.-1)

~ Roller skewing angle rad, °<I> Position angle rad, °<I>A Contact deformation at laminum A due to

roller skewing mm Cin.)if; Azimuth angle rad, °

SUBSCRIPTSa Refer to axial directioni Refers to inner racewayi Refers to ring angular positionJ Refers to rolling element positionk Refers to rolling element positionI Refers to line contactm Refers to rolling element positionm Refers to racewayM Refers to moment loadingn Refers to direction collinear with normal

load0 Refers to outer racewayp Refers to point contactr Refers to radial directionR Refers to rolling elements Refers to gear separating loadt Refers to gear tangential load1,2 Refers to bearing row1 Refers to outer raceway2 Refers to inner raceway3 Refers to tapered roller bearing, roller end-

flange contactif; Refers to angular location

GENERAL

Having determined in Chapter 5 how each ball or roller in a bearingcarries load, it is possible to determine how the bearing load is distrib-uted among the balls or rollers. To do this it is first necessary to developload-deflection relationships for rolling elements contacting raceways. Byusing Chapters 2 and 6 these load-deflection relationships can be devel-oped for any type of rolling element contacting any type of raceway.Hence, the material presented in this chapter is completely dependenton the previous chapters, and a quick review might be advantageous atthis point.


Most rolling bearing applications involve steady-state rotation of ei-ther the inner or outer raceway or both; however, the speeds of rotationare usually not so great as to cause ball or roller centrifugal forces orgyroscopic moments of magnitude large enough to affect significantly thedistribution of applied load among the rolling elements. Moreover, inmost applications the frictional forces and moments acting on the rollingelements also do not significantly influence this load distribution. Con-sequently, in analyzing the distribution of rolling element loads, it isusually satisfactory to ignore these effects in most applications. In thischapter the load distribution of statically loaded ball and roller bearingswill be investigated.

BEARINGS UNDER COMBINED RADIAL AND THRUST LOAD

Single-Row BearingsIf a rolling bearing without diametral clearance is subjected simultane-ously to a radial load in the central plane of the rollers and a centricthrust load, then the inner and outer rings of the bearing will remainparallel and will be relatively displaced a distance 8a in the axial direc-tion and 8r in the radial direction. At any regular position IjJ measuredfrom the most heavily loaded rolling element, the approach of the ringsIS

81/1 = 8a sin ex + 8r cos ex cos IjJ (7.58)

Figure 7.13 illustrates this condition. At IjJ = 0 maximum deflection oc-curs and is given by


The foregoing equations were developed by Jones [7.2].Equations (7.104)-(7.106) are simultaneous nonlinear equations with

unknowns Da, Dr' and e. They may be solved by numerical methods; forexample, the Newton-Raphson method. Having obtained Da, D" and e,the maximum rolling element load may be obtained from equation (7.95)for 1/1 = o.

Qmax = KnAn{[(sin (\'0 + 8a + UI)J)2 + (cos (\'0 + 8r)2]1/2 - 1}n (7.107)

Solution of the indicated equations generally necessitates the use of adigital computer. In certain cases, however-for example, applicationswith simple radial, simple thrust or radial and thrust loading with nom-inal clearance-the simplified methods presented in the beginning of thischapter will probably provide sufficiently accurate calculational results.

MISALIGNMENT OF RADIAL ROLLER BEARINGS

Although it is usually undesirable, radial cylindrical roller bearings andtapered roller bearings can support to a small extent the moment loadingdue to misalignment. The various types of misalignment are illustratedin Fig. 7.22. Clearly, spherical roller bearings are designed to exclude allmoment loads on the bearings and therefore are not included in thisdiscussion.

Figure 7.23 illustrates the misalignment of a cylindrical roller bearinginner ring relative to the outer ring.

To commence the analysis, it is assumed that any roller-raceway con-tact can be subdivided into a number of laminae situated in planes par-allel to the radial plane of the bearing. It is also assumed that sheareffects between these laminae can be neglected owing to the small mag-nitudes of the contact deformations that develop. (Only contact defor-mations are considered.)

Components of Deformation

In a misaligned cylindrical roller bearing subjected to radial load, at eachlaminum in a crowned roller-raceway contact, the deformation may beconsidered to be composed of three components: (1) Llmj due to the radialload at roller azimuth location j, (2) C A due to the crown drop at laminumA, and (3) deformation due to the bearing misalignment and roller tilt atroller azimuth locationj. These components are illustrated schematicallyin Fig. 7.24.

The component due to radial load was the only component consideredin the simplified analysis previously discussed; it needs no further ex-


unit length, and subsequently the roller load, may be determined foreach roller location using equations (7.112) and (7.113), respectively.

Using the foregoing method and digital computation, Harris [7.5]an-alyzed a 309 cylindrical roller bearing having the following dimensionsand loading:

Number of rollers 12Roller effective length 12.6 mm (0.496 in.)Roller straight lengths 4.78, 7.770, 12.6 mmRoller crown radius 1245 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 (7100 lb)

For the above conditions, Fig. 7.29 shows the loading on various rollersfor the bearing with ideally crowned rollers [l = 12.6 mm (0.496 in.) andwith fully crowned rollers (l = 0).

Fig. 7.30 shows the effect or roller crowning on bearing radial deflec-tion as a function of misalignment.

THRUST LOADING OF RADIAL CYLINDRICAL ROLLERBEARINGS

When radial cylindrical roller bearings have fixed flanges on both innerand outer rings, they can carry some thrust load in addition to radialload. The greater the amount of radial load applied, the more thrust loadthat can be carried. As shown by Harris [7.6] and seen in Fig. 7.31, thethrust load causes each roller to tilt an amount ~.

Again, it is assumed that a roller-raceway contact can be subdividedinto laminae in planes parallel to the radial plane of the bearing. Whena radial cylindrical roller bearing is subjected to applied thrust load, theinner ring shifts axially relative to the outer ring. Assuming deflectionsowing to roller-end-flange contacts are negligible, then the interferenceat any axial location (laminum) is

DAj = Llj + ~p A - t)w - cA A = 1, kj (7.126)

where CA is given by equations (7.108). Figure 7.32 illustrates the com-ponent deflections in equation (7.126). Substituting equation (7.126) into(7.111) yields

Roller-Raceway Deformations Due to Roller Skewing

When rollers are subjected to axial loading as shown in Fig. 7.31, due tosliding motions between the roller ends and ring flanges friction forcesoccur; for example Faj = J.LQaj, in which J.L is the coefficient of friction. Ina misaligned bearing, each roller which carries load is squeezed at oneend and forced against the opposing flange with a load Qaj, creating fric-tion force Faj at the roller end. Due to Faj a moment occurs creating ayawing or skewing motion in addition to the predominant rolling motionabout the roller axis and secondary roller tilting. The tilting and skewingmotions occur in orthogonal planes which contain the roller axis. Frictionforces acting on rolling elements are not introduced until Chapter 14;however, roller skewing is resisted by the concave curvature of the outerraceway. The resisting forces and accompanying deformations alter thedistribution of load along both the outer and inner raceway-roller con-tacts. Figure 7.33 illustrates the forces which occur on a roller subjectedto radial and thrust loading. Frictional stresses TmjA are discussed in de-tail in Chapter 14.

Figure 7.34 shows the roller skewing angle ~ and the roller-outerraceway loading which results.

The roller-raceway contact deformations which result from skewing,as demonstrated by Harris et al. [7.7], may be described by equation(7.142).

Spherical Roller Bearings

Spherical roller bearings are internally self-aligning and therefore can-not carry moment loading. Moreover, for slow or moderate speed appli-cations causing insignificant roller inertial loading and friction,symmetrical (barrel-shaped) rollers in spherical roller bearings will ex-hibit no tendency to tilt. Therefore, the simpler analytical methods pre-sented earlier in this chapter will yield accurate results. For sphericalroller bearings having asymmetrical rollers, however, such as sphericalroller thrust bearings (Fig. 1.45), roller tilting and hence skewing is noteliminated. In this case, for the purpose of analysis, the bearing may beconsidered a special type of tapered roller bearing having fully crownedrollers. Then the methods of analysis discussed in the preceding sectionmay be applied for increased accuracy.

FLEXIBLY SUPPORTED ROLLING BEARINGS

Ring Deflections

The preceding discussion of distribution of load among the bearing roll-ing elements pertains to bearings having rigidly supported rings. Suchbearings are assumed to be supported in infinitely stiff or rigid housingsand on solid shafts of rigid material. The deflections considered in thedetermination of load distribution were contact deformations, that is,Hertzian deflections. This assumption is, in fact, an excellent approxi-mation for most bearing applications.

In some radial bearing applications, however, the outer ring of thebearing may be supported at one or two angular positions only, and the


shaft on which the inner ring is positioned may be hollow.The conditionof two-point outer ring support, as shown in Figs. 7.35 and 7.36, occursin the planet gear bearings ofplanetary gear power transmission system,and was analyzed by Jones and Harris [7.8]. In certain rolling mill ap-plications, the back-up roll bearings may be supported at only one pointon the outer ring or possibly at two points as shown in Fig. 7.37. Theseconditions were analyzed by Harris [7.9]. In certain high speed radialbearings, to prevent skidding it is desirable to preload the rolling ele-ments by using an elliptical raceway, thus achieving essentially two-point ring loading under conditions of light applied load. The case of aflexible outer ring and an elliptical inner ring was investigated by Harrisand Broschard [7.10].In each of the foregoing applications, the outer ringmust be considered flexible to achieve a correct analysis of rolling ele-ment loading.

In many aircraft applications to conserve weight the power transmis-sion shafting is made hollow. In these cases the inner ring deflectionswill alter the load distribution from that considering only contact defor-mation.

To determine the load distribution among the rolling elements whenone or both of the bearing rings is flexible, it is necessary to determine


of load among the rollers in a planet gear bearing compared to that of arigid ring bearing subjected to a radial load of 2Ft. For the backup rollbearings of Fig. 7.37 supporting individual line loads Fl, Fig. 7.45 com-pares the load distribution to"that of a bearing having rigid rings. Figure7.46 shows typical load distributions for the backup roll bearing of Fig.7.37 which supports paired line loads F2• Fig. 7.47 from [7.13], which isa photoelastic study of a similarly loaded bearing verifies the data of Fig.7.46.

Finite Element Methods

In the foregoing to specify ring deflections, closed form integral analyticalmethods as well as influence coefficients calculated using infinite seriestechniques have been indicated for ring shapes, which are assumed sim-ple both in circumference and cross section. For more complex structures,the finite element method of calculation can be used to obtain a solutionwhose accuracy depends only upon the fineness of the grid selected torepresent the structure.

In finite element methods a function, customarily a polynomial, is cho-sen to define uniquely the displacement in each element (in terms ofnodal displacements). The element stiffness matrix is obtained fromequilibrium. The stiffness matrix of the complete structure is assembled,the boundary conditions are introduced, and solution of the resultingmatrix equation produces the nodal displacements. A digital computer is

required to solve the displacements and load distribution accurately ina rolling bearing mounted in a flexible support. Figure 7.48 from Zhao[7.14] shows the grids used to analyze a flexibly mounted cylindricalroller bearing assuming both solid and hollow rollers. The load distri-bution would be similar to that indicated in Fig. 7.47.

REFERENCES 305

Bourdon et al. [7.15] and [7.16] provide a method to define stiffnessmatrices for use in standard finite element models to analyze rollingbearing loads and deflections, and the loading and deflections of themechanisms in which they are employed. For flexible mechanisms andbearing support systems, they demonstrate the importance of consider-ing the overall mechanical system rather than only the local system inthe vicinity of the bearings.

CLOSUREThe methods developed in this chapter to calculate distribution of loadamong the balls and rollers of rolling bearings can be used in most bear-ing applications because rotational speeds are usually slow to moderate.Under these speed conditions, the effects of rolling element centrifugalforces and gyroscopic moments are negligible. At high speeds of rotationthese body forces become significant, tending to alter contact angles andclearance. Thus, they can affect the static load distribution to a greatextent. In Chapter 9 the effect ofthese parameters on high speed bearingload distribution will be evaluated.

In the foregoing discussion the effect of load distribution on the bear-ing deflection has been demonstrated. Further, since the contractstresses in a bearing depend on load, maximum contact stress in a bear-ing is also a function of load distribution. Consequently, bearing fatiguelife that is governed by stress level is significantly affected by the rollingelement load distribution.

REFERENCES7.1. R. Stribeck, "Ball Bearings for Various Loads," Trans. ASME 29, 420-463 (1907).7.2. A. Jones, Analysis of Stresses and Deflections, New Departure Engineering Data,

Bristol, Conn. (1946).7.3. J. Rumbarger, "Thrust Bearings with Eccentric Loads," Mach. Des. (Feb. 15, 1962).7.4. H. Sjoviill, "The Load Distribution within Ball and Roller Bearings under Given Ex-

ternal Radial and Axial Load," Teknisk Tidskrift, Mek., h.9 (1933).7.5. T. Harris, "The Effect of Misalignment on the Fatigue Life of Cylindrical Roller Bear-

ings Having Crowned Rolling Members," ASME Trans., J. Lub. Tech., 294-300 (April1969).

7.6. T. Harris, "The Endurance of a Thrust-Loaded, Double Row,Radial Cylindrical Bear-ing," Wear, 18, 429-438 (1971).

7.7. T. Harris, M. Kotzalas, and W. Yu, "On the Causes and Effects of Roller Skewing inCylindrical Roller Bearings," Trib. Trans., 41(4), 572-578 (1998).

7.8. A. Jones and T. Harris, "Analysis of a Rolling Element Idler Gear Bearing Having aDeformable Outer Race Structure," ASME Trans., J. Basic Eng., 273-278 (June1963).


7.9. T. Harris, "Optimizing the Design of Cluster Mill Rolling Bearings," ASLE Trans. 7(Apr. 1964).

7.10. T. Harris and J. Broschard, "Analysis of an Improved Planetary Gear TransmissionBearing," ASME Trans., J. Basic Eng., 457-462 (Sept. 1964).

7.11. S. Timoshenko, Strength of Materials, Part I, 3rd ed., Van Nostrand, New York (1955).7.12. W. Lutz, Discussion of [7.8], presented at ASME Spring Lubrication Symposium, Mi-

ami Beach, Fla. (June 5, 1962).7.13. H. Eimer, "Aus dem Gebiet der Walzlagertechnik" (Semesterentwurf, Technische

Hochschule, Munich, June 1964).7.14. H. Zhao, "Analysis of Load Distributions within Solid and Hollow Roller Bearings,"

ASME Trans., J. Tribology 120, 134-139 (Jan. 1998).7.15. A. Bourdon, J. Rigal, and D. Play, "Static Rolling Bearing Models in a CAD. Envi-

ronment for the Study of Complex Mechanisms: Part I-Rolling Bearing Model,"ASME Trans., J. Tribology 121, 205-214 (April 1999).

7.16. A. Bourdon, J. Rigal, and D. Play, "Static Rolling Bearing Models in a C.A.D. Envi-ronment for the Study of Complex Mechanisms: Part II-Complete Assembly Model,"ASME Trans., J. Tribology 121, 215-223 (April 1999).

LIST OF SYMBOLS


a Semimajor axis of projected contactellipse mm (in.)

b Semiminor axis of projected contactellipse mm (in.)

dm Pitch diameter mm (in.)D Ball or roller diameter mm (in.)c; Complete elliptic integral of the second

kindf riDh Center of sliding mm (in.)Mg (}yratory moment N-mm (in.. lb)n Rotational speed rpmr Raceway groove radius mm (in.)r' Rolling radius mm (in.)R Radius of curvature of deformed

surface mm (in.)

307

308 INTERNAL SPEEDS AND MOTIONS


v Surface velocity mm/sec (in.!sec)x Distance in x-direction mm (in.)x Acceleration in x-direction mml sec2 (in.! sec2)

y Distance in y-direction mm (in.)y Acceleration in y-direction mml sec2 (in.! sec2)

z Distance in z-direction mm (in.)Z Acceleration in z-direction mml sec2 (in. Isec2)

0' Contact angle rad,o{3' Angle between projection of the U axis

on the x'y' plane and the x' axis (Fig.5.4) rad

{3 Angle between the W axis and z' axis(Fig. 5.4) rad

y' Dldm

y D cos 0'1dm

K albw Rotational speed radl secOf Flange angle rad, 0

SUBSCRIPTSf Refers to flangeg Refers to gyroscopic motioni Refers to inner racewaym Refers to orbital motion0 Refers to outer racewayR Refers to rolling elementRE Refers to roller endroll Refers to rolling motions Refers to spinning motionsl Refers to sliding motion on flange-roller

endx Refers to x-direction (Fig. 5.4)x' Refers to x' -direction (Fig. 5.4)y Refers to y-direction (Fig. 5.4)y' Refers to y' -direction (Fig. 5.4)z Refers to z-direction (Fig. 5.4)z' Refers to z' -direction (Fig. 5.4)

GENERAL

Ball and roller bearings are used to support various kinds of loads whilepermitting rotational and/or translatory motion of a shaft or slider. Inthis book treatment has been restricted to shaft rotation or oscillation.

IMPLE ROLLING MOTION 309

Unlike hydrodynamic or hydrostatic bearings, motions occurring in roll-ing bearings are not restricted to simple movements. For instance, in arolling bearing mounted on a shaft that rotates at n rpm, the rollingelements orbit the bearing axis at a speed of nm rpm, and they simul-taneously revolve about their own axes at speeds of nR rpm. Additionally,the rolling motions are accompanied by a degree of sliding that occursin the contact areas. In ball bearings, substantial amounts of spinningmotion occur simultaneously with rolling if the contact angles betweenballs and raceways are not zero, that is, for other than simple radialbearings. Also, gyroscopic pivotal motions occur, particularly in oil- andgrease-lubricated ball bearings. In this chapter, rolling bearing internalrotational speeds and relative surface velocities, that is, sliding veloci-ties, will be investigated and equations for their subsequent calculationwill be developed.

ROLLING AND SLIDING 313

ROLLING AND SLIDING

Geometrical Considerations

The only conditions that can sustain pure rolling between two contactingsurfaces are

1. Mathematical line contact under zero load2. Line contact in which the contacting bodies are identical in length3. Mathematical point contact under zero load

Even when the foregoing conditions are achieved it is possible to havesliding. Sliding is then a condition of overall relative movement of therolling body over the contact area.

The motion of a rolling element with respect to the raceway consistsof a rotation about the generatrix of motion. If the contact surface is astraight line in one of the principal directions, the generatrix of motionmay intersect the contact surface at one point only, as in Fig. 8.2. Thecomponent WR of angular velocity w, which acts in the plane of the con-tact surface, produces rolling motion. As indicated in Fig. 8.3, the com-ponent W

sof angular velocity w that acts normal to the surface causes a

spinning motion about a point of pure rolling O. The instantaneous di-rection of sliding in the contact zone is shown in Fig. 8.4.

In ball bearings with nonzero contact angles between balls and race-ways, during operation at any shaft or outer ring speed, a gyroscopicmoment occurs on each loaded ball, tending to cause a sliding motion.In most applications, because of relatively slow input speeds and/orheavy loading, such gyroscopic moments and hence motions can be ne-glected. In high speed applications with an oil-film lubrication betweenballs and raceways, such motion will occur.

The sliding velocity due to gyroscopic motion is given by (see Fig. 8.5)

The distance h defines the center of sliding about which a rotation ofangular velocity Ws occurs. This center of sliding (spinning) may occurwithin or outside of the contact surface. Figure 8.6 shows the pattern ofsliding lines in the contact area for simultaneous rolling, spinning, andgyroscopicmotion in a ball bearing operating under heavy load and mod-erate speed. Figure 8.7, which corresponds to low load and high speedconditions (however, not considering skidding*), indicates that the centerof sliding is outside of the contact surface and sliding surface occurs overthe entire contact surface. The distance h between the centers of contact

*Skidding is a very gross sliding condition occurring generally in oil-film lubricated balland roller bearings operating under relatively light load at very high speed or rapid accel-erations and decelerations. When skidding occurs, cage speed will be less than predictedby equation (8.9) for bearings with inner ring rotation.


and sliding is a function of the magnitude of the gyroscopicmoment thatcan be compensated by contact surface friction forces.

Sliding and DeformationEven when the generatrix of motion apparently lies in the plane of thecontact surface, as for radial cylindrical roller bearings, sliding on thecontact surface can occur when a roller is under load. In accordance withthe Hertzian radius of the contact surface in the direction transverse tomotion, the contact surface has a harmonic mean profile radius, whichmeans that the contact surface is not plane, but generally curved asshown by Fig. 8.8 for a radial bearing.* The generatrix of motion, beingparallel to the tangent plane of the center of the contact surface, there-fore pierces the contact surface at two points at which rolling occurs.Since the rigid rolling element rotates with a singular angular velocityabout its axis, surface points at different radii from the axis have differ-ent surface velocities only two of which being symmetrically disposedabout the roller geometrical center can exhibit pure rolling motion. InFig. 8.8 points within area A-A slide backward with regard to the direc-

ORBITAL, PIVOTAL, AND SPINNING MOTIONS IN BALL BEARINGS 317

tion of rolling and points outside of A-A slide forward with respect tothe direction of rolling. Figure 8.9 shows the pattern of sliding lines inthe elliptical contact area.

If the generatrix of motion is angled with respect to the tangent planeat the center of the contact surface, the center of rolling is positionedunsymmetrically in the contact ellipse and, depending on the angle ofthe generatrix to the contact surface, one point or two points of inter-section may occur at which rolling obtains. Figure 8.10 shows the slidinglines for this condition.

For a ball bearing in which rolling, spinning, and gyroscopic motionsoccur simultaneously, the pattern of sliding lines in the elliptical contactarea is as shown in Figs. 8.11 and 8.12. More detailed information onsliding in the elliptical contact area may be found in the work by Lund-berg [8.4].

ORBITAL, PIVOTAL, AND SPINNING MOTIONS IN BALLBEARINGS

General MotionsFigure 8.13 shows a ball contacting the outer raceway such that thenormal force Q between the ball and raceway is distributed over an el-liptical surface defined by projected major and minor semiaxes, ao andb

o, respectively. The radius of curvature ofthe deformed pressure surface

as defined by Hertz is

Assume for the present purpose that the ball center is fixed in space andthat the outer raceway rotates with angular speed Wo0 (The vector ofW

ois perpendicular to the plane of rotation and therefore collinear with

the x axis.) Moreover, it can be seen from Fig. 5.4 that ball rotationalspeed WR has components Wx' and Wz' lying in the plane ofthe paper whenI/I = O.

Because of the deformation at the pressure surface defined by ao andb

o, the radius from the ball center to the raceway contact point varies in

length as the contact ellipse is traversed from +ao to -ao· Thereforebecause of symmetry about the minor axis of the content ellipse, purerolling motion of the ball over the raceway occurs at most at two points.The radius at which pure rolling occurs is defined as r~ and must bedetermined by methods of contact deformation analysis.

It can be seen from Figure 8.13 that the outer raceway has a compo-nent Wo cos 0'0 of the angular velocity vector in a direction parallel to themajor axis of the contact ellipse. Therefore, a point (xo, Yo) on the outer

If instead of the ball center being fixed in space, the outer raceway isfixed, then the ball center must orbit about the center 0 of the fixedcoordinate system with an angular speed Wm = - WOo Therefore the innerraceway must rotate with absolute angular speed W = Wi + Wm' By usingthese relationships, the relative angular speeds Wi and Wo can be de-scribed in terms of the absolute angular speed of the inner raceway asfollows: Inspection of the final equations relating the relative motions of the

balls and raceways reveals the following unknown quantities:r~, r{, {3',{3,ai and ao• It is apparent that analysis of the forces and mo-ments acting on each ball will be required to evaluate the unknownquantities. As a practical matter, however, it is sometimes possible toavoid this lengthy procedure requiring digital computation by using the

Raceway ControlHarris [8.5] showed that, in general, it is not possible for pure rollingthat is, without simultaneous spinning motion, to occur at either theinner or outer raceway contacts as long as the ball-raceway contact an-gle is nonzero. For high speed operation of relatively lightly loaded oil-film lubricated bearings, however, the condition of "outer raceway


:ontrol" tends to be approximated. Figure 8.15 taken from reference [8.5]llustrates this condition for a high speed thrust-loaded aircraft gas tur-)ine, angular-contact ball bearing. It must be noted that skidding also;ends to occur at the same time.

Hence, for oil-film lubricated ball bearings (including grease-ubricated ball bearings), determination of actual internal speeds and:notions requires a rather sophisticated mathematical analysis. Such:nethods require an understanding of friction and will be discussed laterIn this text.

For dry film-lubricated ball bearings or for ball bearings in which aconstant coefficient of friction may be assumed in the ball-raceway con-tacts, Harris [8.6] has shown for a thrust-loaded angular-contact ballbearing that, at relatively slow speed, spinning and rolling occur simul-taneously at both inner and outer ball-raceway contacts. For a givenload, as speed is increased, a transition takes place in which outer race-way control is approximated; however, the outer raceway contact spin-to-roll ratio is always nonzero (see Figs. 8.16 and 8.17). It is illustratedby Figs. 8.15-8.17 that the condition of "inner raceway control" is non-existent; hence no equations for that condition are presented herein.

Jones [8.2] mentioned that a coefficient of friction from 0.06 to 0.07 suf-fices for most ball bearing applications to prevent sliding. Neither con-dition is correct. Since, as shown in this chapter, the balls havesubstantial rotational motions about two orthogonal axes, due to the ex-istence ofthe lubricating films which generally sufficiently "separate" theballs and raceways, it is not possible to prevent rotation about the thirdorthogonal axis. In Chapter 14, it will be shown that the friction coeffi-cient is a function of the sliding velocity at the contact surface. Further,the ball-raceway frictional forces resisting gyratory motion depend onthe ratio of the sliding velocity to the lubricant film thickness. Since thelatter is a function of the speed in the direction of rolling motion, themagnitude of the gyratory speed is determined by the magnitude of thegyratory moment.

Jones [8.3] established a condition to determine whether outer race-way control is approximated in a given application; for example, if

then outer raceway control may be assumed for calculational purposes.In inequality (8.67), B is the complete elliptic integral of the second kindwith modulus K = a/b as defined in equation (6.32). As indicated inChapter 14, no evidence of inner raceway control has been found in anyball bearing application; therefore, the assumption of outer raceway con-trol may be made in the absence of more sophisticated calculations ofball speeds using balance of lubricated contact frictional forces and mo-ments.

J30 INTERNAL SPEEDS AND MOTIONS

ROLLER END-FLANGE SLIDING IN ROLLER BEARINGS

Roller End-Flange Contact

Roller bearings react axial roller loads through concentrated contactsbetween roller ends and flange. Tapered roller bearings and sphericalroller bearings (with asymmetrical rollers) require such contact to reactthe component of the raceway-roller contact load that acts in the rolleraxial direction. Some cylindrical roller bearing designs require roller end-flange contacts to react skewing-induced and/or externally applied rolleraxial loads. As these contacts experience sliding motions between rollerBndsand flange, their contribution to overall bearing frictional heat gen-Bration becomes substantial. Furthermore, there are bearing failuremodes associated with roller end-flange contact such as wear and smear-ing of the contacting surfaces. These failure modes are related to theability of the roller end-flange contact to support roller axial load underthe prevailing speed and lubrication conditions within the contact. Boththe frictional characteristics and load-carrying capability of roller end-flange contacts are highly dependent on the geometry of the contactingmembers.

Roller End-Flange Geometry

Numerous roller end and flange geometries have been used successfullyin roller bearing designs. Typically, performance requirements as well asmanufacturing considerations dictate the geometry incorporated into abearing design. Most designs use either a flat (with corner radii) orsphere end roller contacting an angled flange. The angled flange surfacecan be described as a portion of a cone at an angle Of with respect to aradial plane perpendicular to the ring axis. This angle, known as theflange angle or flange layback angle, can be zero, indicating that theflange surface lies in the radial plane. Examples of cylindrical roller bear-ing roller end-flange geometries are shown in Fig. 8.18. The flat endroller in Fig. 8.18a under zero skewing conditions contacts the flange ata single point (in the vicinity of the intersection between the roller endflat and roller corner radius). As the roller skews, the point of contacttravels along this intersection on the roller toward the tip of the flange,as shown in Fig. 8.19b. If properly designed, a sphere end roller willcontact the flange on the roller end sphere surface. For no skewing thecontact will be centrally positioned on the roller, as shown in Fig. 8.19c.As the skewing angle is increased, the contact point moves offcenter andtoward the flange tip, as shown in Fig. 8.19d for a flanged inner ring.For typical designs sphere end roller contact location is less sensitive toskewing than a flat end roller contact.

The location of the roller end-flange contact has been determined an-alytically [8.7] for sphere end rollers contacting an angled flange. Con-


roller end sphere radius. Therefore, knowing the roller and flanged ringgeometry as well as the coordinate location (with respect to the flangedring coordinate system) of the roller end sphere origin, it is possible tocalculate directly the theoretical roller end-flange contact location.

The foregoing analysis, although shown for a cylindrical roller bear-ing, is general enough to apply to any roller bearing having sphere endrollers that contact a conical flange. Tapered and spherical roller bear-ings of this type may be treated if the sphere radius origin is properlydefined.

These equations have several notable applications since flange contactlocation is of interest in bearing design and performance evaluation. Itis desirable to maintain contact on the flange below the flange rim (in-cluding edge break) and above the undercut at the base of the flange. Todo otherwise causes loading on the flange rim (or edge of undercut) andproduces higher contact stresses and less than optimum lubrication ofthe contact. The preceding equations may be used to determine the max-imum theoretical skewing angle for a cylindrical roller bearing if theroller axial play (between flanges) is known. Also, by calculating the lo-cation of the theoretical contact point, sliding velocities between rollerends and flange can be calculated and used in an estimate of roller end-flange contact friction and heat generation.

Sliding Velocity

The kinematics of a roller end-flange contact causes sliding to occur be-tween the contacting members. The magnitude of the sliding velocitybetween these surfaces substantially affects friction, heat generation,and load-carrying characteristics of a roller bearing design. The slidingvelocity is represented by the difference between the two vectors definingthe liner velocities of the flange and the roller end at the point of contact.A graphical representation of the roller velocity VROLL and the flange ve-locity VF at their point of contact C is shown in Fig. 8.22. The slidingvelocity vector Vs is shown as the difference of VRE and VF. When consid-ering roller skewing motions, Vs will have a component in the flangedring axial direction, albeit small in comparison to the components in thebearing radial plane. If the roller is not subjected to skewing, the contactpoint will lie in the plane containing the roller and flanged ring axes.The roller end-flange sliding velocity may be calculated as

VsI = VF - VRE = wfRe - (woRe + wRrJ (8.78)

where clockwise rotations are considered positive. Varying the positionof contact point C over the elastic contact area between roller end andflange allows the distribution of sliding velocity to be determined.

CLOSUREIn this chapter, methods for calculation of rolling and cage speeds in balland roller bearings were developed for conditions of rolling and spinningmotions. It will be shown in Chapter 9 how the dynamic loading derivedfrom ball and roller speeds can significantly affect ball bearing contactangles, diametral clearance, and subsequently rolling element load dis-tribution. Moreover, spinning motions that occur in ball bearings tend toalter contact area stresses, and hence they affect bearing endurance.Other quantities affected by bearing internal speeds are friction torqueand frictional heat generation. It is therefore clear that accurate deter-minations of bearing internal speeds are necessary for analysis of rollingbearing performance.

It will be demonstrated subsequently that hydrodynamic action of thelubricant in the contact areas can transform what is presumed to besubstantially rolling motions into combinations of rolling and translatorymotions. In general, this combination of rotation and translation may betolerated providing the lubricant films resulting from the rolling motionsare sufficient to adequately separate the rolling elements and raceways.Bearing internal design and/or bearing loading or lubrication may bemodified to minimize the gross sliding motions and their potential del-eterious effects. This topic will be discussed in Chapter 14.


REFERENCES8.1. A Palmgren, Ball and Roller Bearing Engineering, 3rd ed., Burbank, Philadelphia, pp.

70-72 (1959).8.2. A. B. Jones, "Ball Motion and Sliding Friction in Ball Bearings," ASME J. Basic Eng.

81, 1-12 (1959).8.3. A. B. Jones, "A General Theory for Elastically Constrained Ball and Radial Roller

Bearings under Arbitrary Load and Speed Conditions," ASME J. Basic Eng. 82, 309-320 (1960).

8.4. G. Lundberg, "Motions in Loaded Rolling Element Bearings," SKF unpublished report(1954).

8.5. T. A. Harris, "An Analytical Method to Predict Skidding in Thrust-Loaded, AngularContact Ball Bearings," ASME J. Lubr. Technol. 93, 17-24 (1971).

8.6. T. A. Harris, "Ball Motion in Thrust-Loaded, Angular-Contact Bearings with CoulombFriction," ASME J. Lubr. Technol. 93, 32-38 (1971).

8.7. R. Kleckner and J. Pirvics, "High Speed Cylindrical Roller Bearing Analysis-SKFComputer Program CYBEAN, Vol. 1: Analysis," SKF Report AL78P022, NASA Con-tract NAS3-20068 (July 1978).

LIST OF SYMBOLS


B t;+fo-1D Ball or roller diameter mm (in.)dm Pitch diameter mm (in.)Ff Friction force N (lb)

Fe Centrifugal force N (lb)

f rIDg Gravitational constant mm/sec2 (in.lsec2

)

H Ball or roller hollowness ratioJ Mass moment of inertia kg' mm2 (in .. lb . sec2

)

K Load-deflection constant N Immx (lb/in.X)

1 Roller length mm (in.)m Ball or roller mass kg (lb . sec2/in.)

Mg Gyroscopic moment N . mm (in. ·lb)0n. Applied moment N . mm (in.. lb)

Pd Diametral clearance mm (in.)

Q Ball or roller load N (lb)

337

338 DISTRIBUTION OF INTERNAL LOADING IN HIGH SPEED BEARINGS


~R Radius to locus of raceway groovecurvature centers mm (in.)

s Distance between inner and outerraceway groove curvature centerloci mm (in.)

X2 Radial projection of distancebetween ball center and outerraceway groove curvature center mm (in.)

Xl Axial projection of distancebetween ball center and outerraceway groove curvature center. mm (in.)

a Contact angle rad, 0

(3 Ball attitude angle rad, 0

'Y D cos a/drn8 Deflection or contact deformation mm (in.)0 Angular misalignment of bearing rad, 0

fjJ Angular location rad, 0

w Rotational speed rad/ sec6.fjJ Angular distance between rolling

elements rad

SUBSCRIPTSa Refers to axial directioni Refers to inner raceway) Refers to angular positionm Refers to cage motion and orbital

motion0 Refers to outer racewayr Refers to radial directionR Refers to rolling elementx Refers to x directionz Refers to z direction

GENERAL

In high speed operation of ball and roller bearings the rolling elementcentrifugal forces are significantly large compared to the forces appliedto the bearing. In roller bearings this increase in loading on the outerraceway causes larger contact deformations in that member; this effectis similar to that of increasing clearance. An increase in clearance asdemonstrated in Chapter 7 tends to increase maximum roller loadingdue to a decrease in the extent of the load zone. For relatively thin sec-

HIGH SPEED BALL BEARINGS 339

tion bearings supported at only a few points on the outer ring; for ex-ample, an aircraft gas turbine mainshaft bearing, the centrifugal forcescan cause bending of the outer ring thus affecting the load distributionamong the rolling elements.

In high speed ball bearings, depending on the contact angles, ball gy-roscopic moments and ball centrifugal forces can be of significant mag-nitude such that inner raceway contact angles tend to increase and outerraceway contact angles tend to decrease. This affects the deflection vsload characteristics of the bearing and thus also affects the dynamics ofthe ball bearing-supported rotor system.

High speed also affects the lubrication characteristics and thereby thefriction in both ball and roller bearings. This will have an influence onbearing internal speeds, which in turn alters the rolling element inertialloading, that is, centrifugal forces and gyroscopic moments. It is possible,however, to determine the internal distribution of load, and hencestresses, in many high speed rolling bearing applications with sufficientaccuracy while not considering the frictional loading of the rolling ele-ments. This will be demonstrated in this chapter. The effects of friction,including skidding, on internal load distribution will be considered later.

HIGH SPEED BALL BEARINGS

To determine the load distribution in a high speed ball bearing, considerFig. 7.19, which shows the displacements of a ball bearing due to a gen-eralized loading system including radial, axial, and moment loads. Fig-ure 9.1 shows the relative angular position of each ball in the bearing.

Under zero load the centers of the raceway groove curvature radii areseparated by a distance BD defined by

BD = (fo + t; - l)D (2.7)

Under an applied static load, the distance between centers will increaseby the amount of the contact deformations 8i plus 80, as shown by Fig.7.18. The line of action between centers is collinear with BD. If, however,a centrifugal force acts on the ball, then because the inner and outerraceway contact angles are dissimilar, the line of action between racewaygroove curvature radii centers is not collinear with BD, but is discontin-uous as indicated by Fig. 9.2. It is assumed in Fig. 9.2 that the outerraceway groove curvature center is fixed in space and the inner racewaygroove 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 centerand the final position of the ball center at any ball location j is

Having computed values of Xlj' X2j, Dij, and Doj at each ball position andknowing Fa, F" and ~~ as input conditions, the values Da, D" and 0 maybe determined by equations (9.21), (9.23), and (9.25). After obtaining theprimary unknown quantities Da, D" and 0, it is then necessary to repeatthe calculation of Xlj' X2j, Dij, and Doj, and so on, until compatible valuesof the primary unknown quantities Da' Dr' and 0 are obtained.

Example 9.1. The 218 angular-contact ball bearing of Example 7.5is to be operated through the load range of 0-44,500 N (10,000 lb)thrust and at speeds of 3000, 6000, 10,000 and 15,000 rpm. Determinesuch operating characteristics of the bearing as !Yi, !Yo, {3,Qi' Qo, nm,Da, Mg, and w/ Wroll'

To obtain the answers to this study, a computer program must bedeveloped to solve equations (9.10), (9.11), (9.16), (9.17), and (9.21)simultaneously for each load-speed condition. In these equations,which may be solved by iterative techniques, the load-deflection "con-stants" K

iand Ko are functions of !Yi and !Yo, which are in turn functions

of Xl and X2

according to equations (9.6)-(9.9). Similarly, Fe and Mgare functions of wm/ W and ~/ W which depend on !Yi and !Yo accordingto equations (8.60) and (8.61). Hence the solution is not simple andcare must be exercised to include all variations in the iteration. Fromsuch a computer program, the data of Figs. 9.4-9.6 were developed.

Ball ExcursionsFor an angular-contact ball bearing subjected only to thrust loading, theorbital travel of the balls occurs in a single radial plane, whose axiallocation is defined by Xl) in Fig. 9.2, that is, Xl) is the same at all azimuthangles if1J. For a bearing that supports combined load, that is, radial andthrust loads and perhaps also a moment load, Xl) is different at eachazimuth angle 1/1). Therefore, a ball undergoes an axial "excursion" as itorbits the shaft or housing center. Unless this excursion is accommodatedby providing sufficient axial clearance between the ball and the cagepocket, the cage will experience nonuniform and possibly heavy loadingin the axial direction. This can also cause a complex motion of the cage,that is no longer simple rotation in a single plane, but rather includingan out-of-plane vibrational component. Such motion together with the

aforementioned loading can lead to rapid destruction and seizure of thebearing.

Under combined loading, because of the variation in the ball-racewaycontact angles O'i) and 0'0) as a ball orbits the bearing center, there is atendency for the ball to advance or lag its "central" position in the cagepocket. The orbital or circumferential travel of the ball relative to thecage is, however, limited by the cage pocket. Therefore, a load occursbetween the ball and the pocket in the circumferential direction. Understeady-state cage rotation, the sum of these ball-cage pocket loads in thecircumferential direction is close to zero, being balanced only by frictionalforces. Moreover, the forces and moments acting on the ball in the bear-ing's plane of rotation must be in balance, including acceleration or de-celeration loading and frictional forces.

To achieve this condition of equilibrium, the ball speeds, includingorbital speed, will be different from those calculated using the equationsof Chapter 8. This condition is called skidding, and it will be covered inChapter 14.

Lightweight BallsTo permit ball bearings to operate at higher speeds, it is possible to re-duce the adverse ball inertial effects by reducing the ball mass. This isespecially effective for angular-contact ball bearings since the differentialbetween inner and outer contact angles will be reduced. To achieve thisresult, it was attempted to operate bearings with complements of hollowballs [9.3];however, this proved impractical since it was difficult to man-ufacture balls having isotropic inertial properties. More recently, hotisostatically pressed (HIP) silicon nitride ceramic has been developed asan acceptable material for manufacture of rolling elements (see Chapter16). Bearings with balls of HIP silicon nitride, which has a density ap-proximately 42% that of steel and an excellent compressive strength, arebeing used in high speed machine tool spindle applications and are underconsideration for use in aircraft gas turbine application main shaft bear-ings. Figures 9.7-9.9 compare bearing performance parameters for op-

erations at high speed of the 218 angular-contact ball bearing with steelballs and HIP silicon nitride balls.

Silicon nitride also has a modulus of elasticity of approximately 3.1 .105 MPa (45' 106 psi). In a hybrid ball bearing, i.e., a bearing with steelrings and silicon nitride balls, owing to the higher elastic modulus of theball material, the contact areas between balls and raceways will besmaller than in an all-steel bearing. This causes the contact stresses tobe greater. Depending on the load magnitude, the stress level may beacceptable to the ball material, but not to the raceway steel. This situ-ation can be ameliorated at the expense of increased contact friction byincreasing the conformity of the raceways to the balls; for example, de-creasing the raceway groove curvature radii. This amount of decrease isspecific to each application, being dependent on bearing applied loadingand speed.

HIGH SPEED RADIAL CYLINDRICAL ROLLER BEARINGS

Because of the high rate of heat generation accompanying relatively highfriction torque, tapered roller and spherical roller bearings have not his-torically been employed for high speed applications. Generally, cylindri-..

done in the interest of simplifying the analytical methods and the un-derstanding thereof. Every rolling bearing applied load situation can beanalyzed using a system with five degrees of freedom, considering onlythe applied loading. Then every specialized applied loading condition, forexample, simple radial load, can be analyzed using this more complexsystem. Reference [9.5] shows the following illustrations that apply toan analytical system for a ball bearing with five degrees of freedom inapplied loading (see Fig. 9.17).

Note the numerical notation of applied loads, that is, Fl, ... , F5, inlieu of Fa, Fr, and ~ll. Figure 9.18 shows the contact angles, deformations,and displacements for the ball-raceway contacts at azimuth ifIJ. Figure9.19 shows the ball speed vectors and inertial loading for a ball with itscenter at azimuth IJij• Note the numerical notations for raceways; 1 = 0

and 2 = i. This is done for ease of digital programming.

CLOSUREAs demonstrated in the foregoing discussion, analysis ofthe performanceof high speed roller bearings is complex and requires a computer to ob-tain numerical results. The complexity can become even greater for ballbearings. In this chapter as well as Chapters 7 and 8, for simplicity ofexplanation, most illustrations are confined to situations involving sym-

metry of loading about an axis in the radial plane of the bearing andpassing through the bearing axis of rotation. The more general and com-plex applied loading system with five degrees of freedom is, however,iiscussed.

The effect of lubrication has also been neglected in this discussion. Forball bearings, it has been assumed that gyroscopic pivotal motion is min-imal and can be neglected. This, of course, depends on the friction forcesin the contact zones, which are affected to a great extent by lubrication.Bearing skidding is also a function of lubrication at high speeds of op-Bration. If the bearing skids, centrifugal forces will be lower in magni-tude and performance will accordingly be different.

Notwithstanding the preceding conditions, the analytical methods pre-sented in this chapter are extremely useful in establishing optimumbearing designs for given high speed applications.

REFERENCES9.1. A. B. Jones, "A General Theory for Elastically Constrained Ball and Radial Roller

Bearings under Arbitrary Load and Speed Conditions," ASME Trans. Journal of BasicEng. 82, 309-320 (1960).

REFERENCES 361

9.2. T. A. Harris, "Optimizing the Fatigue Life of Flexibly-Mounted Rolling Bearings," Lub.Eng. 420-428 (October, 1965).

9.3. T. A. Harris, "On the Effectiveness of Hollow Balls in High-Speed Thrust Bearings,"ASLE Transact. 11,209-294 (October, 1968).

9.4. T. A. Harris and S. F. Aaronson, "An Analytical Investigation of Cylindrical RollerBearings Having Annular Rollers," ASLE Preprint No. 66LC-26 (October 18, 1966).

9.5. T. A. Harris and M. H. Mindel, "Rolling Element Bearing Dynamics," Wear 23, 311-337 (1973).

LIST OF SYMBOLS


a Semimajor axis of the projected contact mm (in.)ellipse

b Semiminor axis of the projected mm (in.)contact ellipse

d1 Land diameter mm (in.)D Ball or roller diameter mm (in.)F Applied force N (lb)Jr( E) Radial load integralK Load-deflection constant N/mmx(lb/in.X

)

I Roller effective length mm (in.)~11! Moment load N . mm (in. ·lb)Q Rolling element load N (lb)Z Number of rolling elements per rowa Contact angle rad, 0

'Y D cos a/dmD Deflection or contact deformation mm (in.)

.•...•

GENERAL

[n Chapter 6 a method was developed for determining the elastic contact:leformation, that is, Hertzian deformation, between a raceway and roll-Lngelement. For bearings with rigidly supported rings the elastic deflec-tion of a bearing as a unit depends on the maximum elastic contact:leformation in the direction of the applied load or in the direction ofinterest to the designer. Because the maximum elastic contact deforma-tion is dependent on the rolling element loads, it is necessary to analyzethe load distribution occurring within the bearing prior to determinationDfthe bearing deflection. Chapters 7 and 9 demonstrated methods forevaluating the load distribution among the rolling elements for rollingbearings subjected to static and dynamic loading, respectively.

Again, in Chapters 7 and 9 the methods for analyzing load distributioncaused by generalized bearing loading (radial, axial, and moment loadsapplied simultaneously) utilized the variables 8r, 8a, and e, which are, infact, the principal bearing deflections. These deflections that are the sub-jects of this chapter may be critical in determining system stability, dy-namic loading on other components, and accuracy of system operation inmany applications.

DEFLECTIONS OF BEARINGS WITH RIGID RINGS 365

DEFLECTIONS OF BEARINGS WITH RIGID RINGS

In the beginning of Chapter 7 and somewhat in Chapter 9 some simpli-fied methods for calculating internal load distribution were discussed.Also, in those chapters methods to determine internal load distributionfor complex applied loading situations were defined. The latter, whichrequire digital computer programs to obtain solutions, generally usebearing deflections, radial, axial, and misalignment, as unknown varia-bles. These deflections are therefore determined directly from the solu-tion ofthe system of nonlinear equations. For applications with relativelysimple applied loading, the methods for determining bearing deflectionwere not defined and these will be discussed herein.

It is possible to calculate the maximum rolling element load Qrnax dueto a combination of radial and axial loads. Qrnax has attendant contactdeformations 80max and Dimax measured along the line of contact at theouter and inner raceways, respectively. From equation (7.4) it can be seenthat

PRELOADING

Axial Preloading

A typical curve of ball bearing deflection vs load is shown by Fig. 10.l.It can be seen from Fig. 10.1that as load is increased uniformly, the rateof deflection increase declines. Hence, it would be advantageous withregard to bearing deflection under load to operate above the "knee" ofthe load-deflection curve. This condition can be realized by axially pre-loading angular-contact ball bearings. This is usually done, as shown inFig. 10.2, by grinding stock from opposing end faces of the bearings andthen locking the bearings together on the shaft. Figure 10.3 shows pre-loaded bearing sets before and after the bearings are axially locked to-gether. Figure 10.4 illustrates, graphically, the improvement in load-deflection characteristics obtained by preloading ball bearings.

Suppose two identical angular-contact ball bearings are placed back-to-back or face-to-face on a shaft, as shown in Fig. 10.5, and drawn to-gether by a locking device. Each bearing experiences an axial deflection

FIGURE 10.3. (a) Duplex set with back-to-back angular-contact ball bearings prior toaxial preloading. The inner ring faces are ground to provide a specific axial gap. (b) Sameunit as in (a) after tightening axial nut to remove gap. The contact angles have increased.(c) Face-to face angular-contact duplex set prior to preloading. In this case it is the outerring faces that are ground to provide the required gap. (d) Same set as in (c) after tight-ening the axial nut. The convergent contact angles increase under preloading. (e) Shimbetween two standard-width bearings avoids need for grinding the faces of the outer rings.(f) Precision spacers between automatically provide proper preload by making the innerspacer slightly shorter than the outer.

FIGURE 10.4. Deflection vs load charac-teristics for ball bearings. As the load in-creases, the rate of the increase of deflectiondecreases, therefore preloading (top line)tends to reduce the bearing deflection underadditional loading.

Figure 10.6 shows a typical plot of bearing deflection 8a vs load. Notethat deflection is everywhere less than that for a nonpreloaded bearingup to the load at which preload is removed. Thereafter, the unit acts asa single bearing under thrust load and assumes the same load-deflectioncharacteristics as given by the single-bearing curve. The point at whichbearing 2 loses load may be determined graphically by inverting thesingle-bearing load-deflection curve about the preload point. This isshown by Fig. 10.6.

Since roller bearing deflection is almost linear with respect to load,there is not as much advantage to be gained by axially preloading ta-pered or spherical roller bearings; hence this is not a universal practiceas it is for ball bearings. Figure 10.7, however, shows tapered roller bear-ings axially locked together in a light preload arrangement.

Example 10.3. A duplex pair of 218 angular-contact ball bearings ismounted back-to-back, as shown in Fig. 10.3. If the pair is preloadedto 4450 N (1000 lb), determine the axial deflection under 8900 N (2000lb) applied thrust load. Compare these results with the static deflec-t,ion rl::'!t,::,!of FilL 9.R.

Equations (10.23) and (10.24) must be solved simultaneously for al anda2• As before, equation (7.36) yields the corresponding values of 8.

To reduce axial deflection still further, more than two bearings can belocked together axially as shown in Fig. 10.8. The disadvantages of thissystem are increased space, weight, and cost. More data on axial pre-loading are given in reference [10.2].

Radial Preloading

Radial preloading of rolling bearings is not usually used to eliminateinitial large magnitude deflection as is axial preload. Instead, its purposeis generally to obtain a greater number of rolling elements under loadand thus reduce the maximum rolling element load. It is also used toprevent skidding. Methods used to calculate maximum radial rolling el-ement load are given in Chapter 7. Figure 10.9 shows various methodsto radially preload roller bearings.

FIGURE 10.9. (a) Diametral (radial) clearance found in most-off-the-shelf rolling bear-ings. One object of preloading is to remove this clearance during assembly. (b) Cylindricalroller bearing mounted on tapered shaft, to expand inner ring. Such bearings are usuallymade with a taper on the inner surface of -b mm/mm. (c) Spherical roller bearing mountedon tapered sleeve to expand the inner ring.

Example 10.4. Suppose the 209 cylindrical roller bearing of Exam-ple 7.3 was manufactured with a tapered bore and was driven up atapered shaft as in Fig. 10.9b until a negative clearance or interfer-ence of 0.00254 mm (0.0001 in.) resulted. For a radial load of 4450 N(1000 lb), determine the maximum roller load, the extent of the loadzone, and the radial deflection. Compare these results with those ofExample 7.3.

Preloading to Achieve Isoelasticity

It is sometimes desirable that the axial and radial yield rates of thebearing and its supporting structures be as nearly identical as possible.In other words, a load in either the axial or radial direction should causeidentical deflections (ideally). This necessity for isoelasticity in the ballbearings came with the development of the highly accurate, low driftinertial gyros fOT navigational systems, and for missile and space guid-ance systems. Such inertial gyros usually have a single degree offreedomtilt axis and are extremely sensitive to error moments about this axis.

Consider a gyro in which the spin axis (Fig. 10.10) is coincident withthe x axis, the tilt axis is perpendicular to the paper at the origin 0, andthe center of gravity of the spin mass is acted on by a disturbing forceF in the xz plane and directed at an oblique angle cf> to the x axis, thisforce will tend to displace the spin mass center of gravity from 0 to 0'.

where the bearing yield rates 8; and 8~ are in deflection per unit of force.To minimize ~)n and subsequent drift, 8; must be as nearly equal to

8~ as possible-a requirement for pinpoint navigation or guidance. Also,from Fig. 10.10 it can be noted that improving the rigidity of the bearing,that is, decreasing 8; and 8~ collectively, reduces the magnitude of theminimal error moments achieved through isoelasticity.

In most radial ball bearings, the radial rate is usually smaller thanthe axial rate. This is best overcome by increasing the bearing contactangle, which reduces the axial yield rate and increases radial yield rate.One-to-one ratios can be obtained by using bearings with contact anglesthat are 30° or higher.

At these high angles, the sensitivity of the axial-to-radial yield rateratio to the amount of preload is quite small. It is, however, necessaryto preload the bearings to maintain the desired contact angles.

LIMITING BALL BEARING THRUST LOAD

General ConsiderationsMost radial ball bearings can accommodate a thrust load and functionproperly provided that the contact stress thereby induced is not exces-sively high or that the ball does not override the land. The latter con-dition results in severe stress concentration and attendant rapid fatiguefailure of the bearing. It may therefore be necessary to ascertain for agiven bearing the maximum thrust load that the bearing can sustainand still function. The situation in which the balls override the land willbe examined first.

Thrust Load Causing Ball to Override Land

Figure 10.11 shows an angular-contact bearing under thrust in whichthe balls are riding at an extreme angular location without the ring landscutting into the balls.

From Fig. 10.11 it can be seen that the thrust load, which causes themajor axis of the contact ellipse to just reach the land of the bearing, isthe maximum permissible load that the bearing can accommodate with-

CLOSUREIn many engineering applications bearing deflection must be known toestablish the dynamic stability of the rotor system. This consideration isimportant in high speed systems such as aircraft gas turbines. The bear-ing radial deflection in this case can contribute to the system eccentricity.In other applications, such as inertial gyroscopes, radiotelescopes, andmachine tools, minimization of bearing deflection under load is requiredto achieve system accuracy or accuracy of manufacturing. That the bear-ing deflection is a function of bearing internal design, dimensions, clear-ance, speeds, and load distribution has been indicated in the previouschapters. However, for applications in which speeds are slow and ex-treme accuracy is not required, the simplified equations presented in thischapter are sufficient to estimate bearing deflection.

To minimize deflection, axial or radial preloading may be employed.Care must be exercised, however, not to excessively preload rolling bear-ings since this can cause increased friction torque, resulting in bearingoverheating and reduction in endurance.

STATICALLY INDETERMINATESHAFT -BEARING SYSTEMS

LIST OF SYMBOLS


a Distance to load point from right-hand mm (in.)bearing

A Distance between raceway groove mm (in.)curvature centers

D Rolling element diameter mm (in.)dm Pitch diameter mm (in.)':IJo Outside diameter of shaft mm (in.)':1\ Inside diameter of shaft mm (in.)E Modulus of elasticity N/mm2 (psi)F Bearing radial load N (lb)f Raceway groove radius -:-DI Section moment of inertia mm4 (in.4)

K Load-deflection constant N /mmx (lb/inX)

l Distance between bearing centers mm (in.)®1l Bearing moment load N· mm (in. ·lb)P Applied load at a N (lb)

387

'188 STATICALLY INDETERMINATE SHAFI'-BEARING SYSTEMS


Q Rolling element load N Ob)R Radius from bearing centerline to mm (in.)

raceway groove centerT Applied moment load at a N . mm (in. ·lb)

w Load per unit length N Imm Ob/in.)

x Distance along the shaft mm (in.)

y Deflection in the y direction mm (in.)

z Deflection in the z direction mm (in.)oP Free contact angle rad,o

'Y D cos aldm

{) Bearing radial deflection mm (in.)() Bearing angular misalignment rad, 0

2,p Curvature sum mm-I (in.-I)

IJ1 Rolling element azimuth angle rad,o

SUBSCRIPTS1,2,3 Refers to bearing locationa Refers to axial directionh Refers to bearing locationJ Refers to rolling element locationy Refers to y directionz Refers to z directionxy Refers to xy planexz Refers to xz plane

SUPERSCRIPTk Refers to applied load or moment

GENERAL

In some modern engineering applications of rolling bearings, such ashigh speed gas turbines, machine tool spindles, and gyroscopes, the bear-ings often must be treated as integral to the system to be able to accu-rately determine shaft deflections and dynamic shaft loading as well asto ascertain the performance of the bearings. Chapters 7 and 9 detailmethods of calculation of rolling element load distribution for bearingssubjected to combinations of radial, axial, and moment loading. Theseload distributions are affected by the shaft radial and angular deflectionsat the bearing. In this chapter, equations for the analysis ofbearing load-ing as influenced by shaft deflections will be developed.

TWO-BEARING SYSTEMS 389

TWO-BEARING SYSTEMS

Rigid Shaft Systems

A commonly used shaft-bearing system involves two angular-contactball bearings or tapered roller bearings mounted in a back-to-back ar-rangement as illustrated by Figs. 11.1 and 11.2. In these applications,the radial loads on the bearings are generally calculated independentlyusing the statically determinate methods of Chapter 4. It may be noticedfrom Figs. 11.1 and 11.2; however, that the point of application of eachradial load occurs where the line defining the contact angle intersectsthe bearing axis. Thus, it can be observed that a back-to-back bearingmounting has a greater length between loading centers than does a face-to-face mounting. This means that the bearing radial loads will tend tobe less for the back-to-back mounting.

390 STATICALLY INDETERMINATE SHAFT-BEARING SYSTEMS

The axial or thrust load carried by each bearing depends upon theinternal load distribution in the individual bearing. For simple thrustloading of the system, the method illustrated in Example 11.3 may beapplied to determine the axial loading in each bearing. When each bear-ing must carry both radial and axial load, although the system is stati-cally indeterminate, for systems in which the shaft can be consideredrigid, a simplified method of analysis may be employed. In Chapter 18,it is demonstrated that a bearing subjected to combined radial and axialloading, may be considered to carry an equivalent load defined by equa-tion (11.1).

Fe = XFr + YFa (11.1)

Loading factors X and Yare functions of the free contact angle, assumedinvariant with rolling element azimuth location and unaffected by theapplied load. This condition is clearly true for tapered roller bearings;however, as shown in Chapter 7, it is only approximated for ball bear-ings. Values for X and Yare usually provided for each ball bearing andtapered roller bearing in manufacturers' catalogs. Accordingly,assumingthe radial loads Frl and Fr2 are determined using the methods of Chapter4, the bearing axial loads Fal and Fa2 may be approximated consideringthe following conditions:

In equations (11.20)-(11.23), slope 01 and 8r1 are considered positiveand the signs of O2 and 8r2 may be determined from equations (11.18)and (11.19). The relative magnitudes of P and T and their directions willdetermine the sense of the shaft slopes at the bearings. To determine thereactions it is necessary to develop equations relating bearing misalign-ment angles Oh to the misaligning moments 9TLh and bearing radial de-flections 8rh to loads Fh' This may be done by using the data of Chapters7 and 9.

396 STATICALLY INDETERMINATE SHAFI'-BEARING SYSTEMS

In their most simple form, in which the bearings are considered asaxially free pinned supports, equations (11.20) and (11.22) are identicalto (4.29) and (4.30). This format is obtained by setting ~ and ark equalto zero and solving equations (11.21) and (11.23) simultaneously for (Jl

and (J2' Substitution of these values into equations (11.20) and (11.22)produces the resultant equations. If the shaft is very flexible and thebearings are rigid with regard to misalignment, then (Jl and (J2 are ap-proximately zero. This substitution into equations (11.20) (11.23) yieldsthe classical solution for a beam with both ends built in. The varioustypes of two-bearing support may be examined by using equations(11.20)-(11.23). If more than one load and/or torque is applied betweenthe supports, then by the principle of superposition

Example 11.3. A pair of 209 radial ball bearings mounted on a hol-low steel shaft having a 45 mm (1.772 in.) o.d. and 40.56 mm (1.597in.) i.d. support a 13,350 N (3000 lb) radial load acting on the shaftmidway between the bearings. The span between the bearing centersis 254 mm (10 in.). If each of the bearings is mounted according to thefits of Example 3.1 and their dimensions are as given by Example 2.1,


Additionally, Figs. 11.6-11.8 show Dr' 0, and 0ll for various combina-tions of shaft hollowness and span between bearings.

THREE-BEARING SYSTEMS

Rigid-Shaft Systems

When the shaft is rigid and the distance between bearings is small, theinfluence of the shaft deflection on the distribution of loading betweenthe bearings may be neglected. An application of this kind is illustratedby Fig. 11.9.

In this system, the angular-contact ball bearings are considered as onedouble-row bearing. The thrust load acting on the double-row bearing isthe thrust load Pa applied by the bevel gear. To calculate the magnitudeof the radial loads Fr and Fr3, the effective location of Fr must be deter-mined. Fr acts at the center of the double-row bearing only if Pa is nil.

FIGURE 11.7. Bearing misalignment angle vs shaft hollowness and span, 209 radial ballbearing, 13,350 N (3000 lb) at midspan.

If a thrust load exists, the line of action of the radial load Fr is displacedtoward the pressure center of the rolling element row which supports thethrust load. This displacement may be neglected only if the distance Ibetween the center of the double-row ball bearing set and the roller bear-ing is large compared with the distance b. Using the X and Y factors (seeChapter 18) pertaining to the single-row bearings, Fig. 11.10 gives therelative distance b1/b as a function ofFaY/ Fr(1 - X). The X and Y factorsfor Fa/ Fr > e should be used.

Example 11.4. In the bearing mounting of Fig. 11.9, the tangentialgear load is Pt = 7000 N, the radial plane gear separating force isPr = 2300 N, and the axial load Pa = 2000 N applied at the gear mean


Fl = 24,030 N (5400 lb)

F2 = 14,240 N (3200 lb)

F3 = 6230 N (1400 lb)

MULTIPLE-BEARING SYSTEMS

Equations (11.30)-(11.35) inclusive may be used to determine the bear-ing reactions in a multiple bearing system such as that shown in Fig.11.14 having a flexible shaft. It is evident that the reaction at any bear-ing support location h is a function of the loading existing at and inbetween the bearing supports located at h - 1 and h + 1. Therefore,from equations (11.30)-(11.35), the reactive loads at each support loca-tion h are given as follows: For a shaft-bearing system of n supports, that is, h = n, equations

(11.49) and (11.50) represent a system of 2n equations. In the most ele-mentary case, all bearings are considered as being sufficiently self-aligning that all ~')]lh equal zero; furthermore, all 8rh are considerednegligible compared to shaft deflection. Equations (11.49) and (11.50)thereby degenerate to the familiar equation of "three moments."

It is evident that the solution of equations (11.49) and (11.50) to obtainbearing reactions ~1f(h and Fh depends on relationships between radialload and radial deflection and moment load and misalignment angle foreach radial bearing in the system. These relationships have been definedin Chapters 7 and 9. Thus, for a very sophisticated solution to a shaft-bearing problem as illustrated in Fig. 11.15 one could consider a shafthaving two degrees of freedom with regard to bending, that is, deflection


in two of three principal directions, supported by bearings h and accom-modating loads k. At each bearing location h, one must establish thefollowing relationships:

8y,h = fl(Fx,h Fy,h' Fz,h' ~Xy,h' 0llxz,h) (11.51)

8z,h = f2(Fz,h' Fy,h' Fz,h' ~llxy,h' ~xz,h) (11.52)

0zy,h = f3(Fx,h' Fy,h' Fz,h' ~nxy,ho 0llxz,h) (11.53)

0zz,h = fiFx,h' Fy,h' Fz,h' ~,h' ~xz,h) (11.54)

To accommodate the movement of the shaft in two principal directions,the following expressions will replace equations (9.4) and (9.5) for eachball bearing (see Jones [11.11]):

Sxj = BD sin aP + 8x + Oxz~J"4sin 'h + Ox/!"4 cos IJij (11.55)

Szj = BD cos (x0 + 8y sin 'h + 8j cos IJij (11.56)

CLOSUREFor most rolling bearing applications, it is sufficient to consider the shafta,ndhousing as rigid structures. As demonstrated by Example 11.3, how-ever, when the shaft is considerably hollow and/or the span betweenbearing supports is sufficiently great, the shaft bending characteristicscannot be considered separately from the bearing deflection character-istics with the expectation of accurately ascertaining the bearing loadsor the overall system deflection characteristics. In practice the bearingsmay be stiffer than might be anticipated by the simple deflection for-mulas or even stiffer than a more elegant solution that employs accurateevaluation of load distribution might predict for the assumed loading.The penalty for increased stiffness will be paid in shortened bearing lifesince the improved stiffness is obtained at the expense of induced mo-ment loading.

It is of interest to note that the accurate determination of bearingloading in integral shaft-bearing-housing systems involves the solutionof many simultaneous equations. For example, a high speed shaft sup-ported by three ball bearings, each ofwhich has a complement of 10balls,the shaft being loaded such as to cause each bearing to experience fivedegrees of freedom in deflection, requires the solution of 142 simulta-neous equations, most of which are nonlinear in the variables to be de-termined. Most likely, the system would include some roller bearings,these having complements of 20 or more rollers per row, thus adding tothe number of equations to be solved simultaneously. Furthermore, the

REFERENCE 413

bearing outer rings and/or inner rings may be flexibly supported as inaircraft power transmissions, adding to the complexity of the analyticalsystem and difficulty of obtaining a solution using numerical analysistechniques such as the Newton-Raphson method for simultaneous, non-linear equations.

REFERENCE11.1. A. Jones, "A General Theory for Elastically Constrained Ball and Radial Roller Bear-

ings under Arbitrary Load and Speed Conditions," ASME Trans., J. Basic Eng. 82,309-320 (1960).

LUBRICANT FILMS INROLLING ELEMENT-RACEWAY CONTACTS

LIST OF SYMBOLS

Symbol Description Unitsa Semimajor axis of elliptical

contact area mm (in.)A Viscosity-temperature

calculation constantsb Semiwidth of rectangular

contact area, semiminor axisof elliptical contact area mm (in.)

C Lubrication regime and filmthickness calculationconstants

D Roller or ball diameter mm (in.)dm Pitch diameter of bearing mm (in.)E Modulus of elasticity N/mm2 (psi)E' EI(l - e) N/mm2 (psi)f riD for ball bearingF Force N (lb)

415

416 LUBRICANT FILMS IN ROLLING ELEMENT-RACEWAY CONTACTS

Symbol Description Units~c Centrifugal force N (lb)F F/E' ~Rg Gravitational constant mm/ sec2 (in. / sec2)§ AE'G Shear modulus N/mm2 (psi)h Film thickness mm (in.)hO Minimum film thickness mm (in.)H hhRI Viscous stress integralJ Polar moment of inertia per

unit length N . sec2 (lb . sec2)J J/E'~R; mm . sec2 (in . sec2)kb Lubricant thermal W/m' °C (Btu/hr' in ..

conductivity OF)1 Roller effective length mm (in.)L Factor for calculating film

thickness reduction due tothermal effects

M Moment N . mm (in .. lb)n Speed rpmp Pressure N/mm2 (psi)Q Force acting on roller or ball N (lb)Q Q/E'YI.R Cylinder radius mm (in.)~R Equivalent radius mm (in.)s rms surface finish (height) mm (in.)SSU Saybolt university viscosity seet Time seeT Lubricant temperature °C, oK(OF,OR)u Fluid velocity mm/sec (in.lsec)U Entrainment velocity (U1 +

U2) mm/sec (in.lsec)U TJoU/2E'HI.v Fluid velocity, displacement in

y direction mm/sec, mm (in.lsec, in.)V Sliding velocity (U1 - U2) mm/sec (in.lsec)V TJoV/E'YI.w Deformation in z direction mm (in.)y Distance in y direction mm (in.)z Distance in z direction mm (in.){3' Coefficient for calculating

viscosity as a function oftemperature

LIST OF SYMBOLS 417Symbol Description Units

'Y D cos a/dml' Lubricant shear rate sec-1r Surface roughness orientation

parameteri1 Surface roughness parameter mm (in.)€ Strain mm/mm (in.lin.)TJ Lubricant viscosity cp (lb . sec/in.2)

'1h, Base oil viscosity (grease) cp (lb . sec/in.2)

TJeff Effective viscosity (grease) cp (lb . sec/in.2)

TJo Fluid viscosity at atmosphericpressure cp (lb . sec/in.2)

K Ellipticity ratio a/bA Pressure coefficient of

viscosity mm2/N (in.2/lb)A Lubricant film parameterl'J, Kinematic viscosity stokes (cm2/sec)~ Poisson's ratiop Weight density g/mm3 (lb/in.3)

(J Normal stress N/mm2 (psi)T Shear stress N/mm2 (psi)() Angle rady Factor to calculate 'PTs'P Film thickness reduction

factor<I> Factor to calculate 'Ps

'"Angular location of roller rad

w Rotational speed rad/ see

SUBSCRIPTSb Refers to entrance to contact

zonee Refers to exit from contact

zoneG Refers to greasei Refers to inner raceway filmJ Refers to roller locationm Refers to orbital motionNN Refers to non-Newtonian

lubricant0 Refers to outer raceway filmR Refers to rollerS Refers to lubricant starvationSF Refers to surface roughness

(finish)



T Refers to temperatureTS Refers to temperature and

lubricant starvationx Refers to x direction, that is,

transverse to rollingy Refers to y direction, that is,

direction of rollingz Refers to z directionIL Refers to rotating racewayv Refers to nonrotating raceway0 Refers to minimum lubricant

film1,2 Refers to contacting bodies

GENERAL

Ball and roller bearings require fluid lubrication if they are to performsatisfactory for long periods of time. Although modern rolling bearingsin extreme temperature, pressure, and vacuum environment aerospaceapplications have been adequately protected by dry film lubricants, suchas molybdenum disulfide among many others, these bearings have notbeen subjected to severe demands regarding heavy load and longevity ofoperation without fatigue. It is further recognized that in the absence ofa high temperature environment only a small amount of lubricant isrequired for excellent performance. Thus many rolling bearings can bepacked with greases containing only small amounts of oil and then bemechanically sealed to retain the lubricant. Such rolling bearings usuallyperform their required functions for indefinitely long periods of time.Bearings that are lubricated with excessive quantities of oil or greasetend to overheat and "burn" up.

The mechanism ofthe lubrication of rolling elements operating in con-centrated contact with a raceway was not established mathematicallyuntil the late 1940s; it was not proven experimentally until the early1960s. This is to be compared with the existence of hydrodynamic lubri-cation in journal bearings, which knowledge was established by Reynoldsin the 1880s. It is known, for instance, that a fluid film completely sep-arates the bearing surface from the journal or slider surface in a properlydesigned bearing. Moreover, the lubricant can be oil, water, gas, or someother fluid that exhibits adequate viscous properties for the intendedapplication. In rolling bearings, however, it was only relatively recentlyestablished that fluid films could, in fact, separate rolling surfaces sub-jected to extremely high pressures in the zones of contact. Today, the

HYDRODYNAMIC LUBRICATION 419

existence of lubricating fluid films in rolling bearings is substantiated;in many successful applications, these films are effective in completelyseparating the rolling surfaces. In this chapter, methods will be pre-sented for the calculation of the thickness of lubricating films in rollingbearing applications.

HYDRODYNAMIC LUBRICATION

Reynold's Equation

Because it appeared possible that lubricant films of significant propor-tions do occur in the contact zones between rolling elements and race-ways under certain conditions of load and speed, several investigatorshave examined the hydrodynamic action oflubricants on rolling bearingsaccording to classical hydrodynamic theory. Martin [12.1] presented asolution for rigid rolling cylinders as early as 1916. In 1959, Osterle[12.2]considered the hydrodynamic lubrication of a roller bearing assem-bly.

It is of interest at this stage to examine the mechanism of hydrody-namic lubrication at least in two dimensions. Accordingly, consider aninfinitely long roller rolling on an infinite plane and lubricated by anincompressible isoviscous Newtonian fluid having viscosity 'Y/. For a New-tonian fluid, the shear stress T at any point obeys the relationship

in which auf az is the local fluid velocity gradient in the z direction (seeFig. 12.1). Because the fluid is viscous, fluid inertia forces are small com-pared to the viscous fluid forces. Hence a particle of fluid is subjectedonly to fluid pressure and shear stresses as shown in Fig. 12.2.

in which Q' is the supported load per unit length of the cylinder.Unfortunately, the film thicknesses indicated by subsequent calcula-

tions are far smaller than the composite surface roughness achievableon rolling bearings. Thus, it is apparent that hydrodynamic lubricationby an isoviscous fluid alone cannot explain the existence of a satisfactoryfluid film in the contact zone between rolling element and raceway. Theforegoing analysis represents a limiting solution for light loads.

ISOTHERMAL ELASTOHYDRODYNAMIC LUBRICATION

Viscosity Variation with PressureAccording to the sample calculations of compressive contact stress inChapter 6, the normal pressure between contacting rolling bodies islikely to be of the magnitude of 700 N Imm2 (100,000 psi) and higher. Innormal fluid film bearing applications pressures ofthis magnitude do notexist and, consequently, it is usually assumed that viscosity is unaffectedby pressure. Figure 12.5 shows some experimental data on viscosity var-iation with pressure for different bearing lubricants. It is to be notedthat fluid viscosity is an exponential function of pressure such that be-tween the contacting surfaces in a loaded rolling bearing assembly, vis-cosity can be several orders of magnitude greater than its value at zeropressure.

In 1893, Barus [12.4] established an empirical equation to describethe variation of viscosity with pressure for a given liquid at a given tem-perature; an isothermal relationship. Barus's equation is usually statedas follows:

In equation (12.20), the viscosity-pressure coefficient A is a constant atthe given temperature. In 1953, an ASME [12.5] study published viscos-ity vs pressure curves for various fluid lubricants. Based on the ASMEdata, it is apparent that the Barus relationship is a crude approximationsince the viscosity-pressure coefficient tends to decrease with both pres-sure and temperature for most fluid lubricants. It has been established,however, that the lubricant film thickness that obtains in a concentratedcontact is a function of the mechanical properties of the lubricant enter-ing the contact. Therefore, for the purpose of determining the thickness

This fluid may be considered representative of an aircraft power trans-mission fluid lubricant. Sorab and VanArsdale [12.7] demonstrate thatequation (12.22) is superior to the Roelands equation in approximatingthe ASME viscosity-pres sure-temperature data. Nevertheless each ofthe approximations has only been demonstrated over the 0-1034 MPa(0-150 kpsi) pressure range and 25-218°C (77-425°F) temperaturerange of the ASME data. Contact pressures and temperatures in manyball and roller bearing applications exceed these ranges; therefore, itbecomes necessary to extrapolate these data substantially beyond therange of the experimentation. This is not critical for the determinationof lubricant film thicknesses. In the estimation of bearing friction, how-ever, lubricant viscosity at pressures higher than 1034 MPa and at tem-peratures greater than 218°C has a great influence on the magnitudesof friction forces calculated and hence on the accuracy of the calculations.

Harris [12.8]introduced the use of a sigmoid curve as defined by equa-tion (12.23) to fit the ASME [12.5] data.

ISOTHERMAL ELASTOHYDRODYNAMIC LUBRICATION 427

procedure for a given lubricant at a given temperature. Figure 12.6 il-lustrates the sigmoid curves for the ASME data for a Mil-L-7808 ester-type lubricant at 37.8, 98.9, and 218.3°C (100, 210, and 425°F). Thesalient feature of the sigmoid viscosity vs pressure curve is the virtuallyconstant viscosity value at extremely high pressures. As noted by Bairand Winer [12.9, 12.10], the fluid in a high pressure, concentrated con-tact undergoes transformation to a glassy state; i.e., the fluid essentiallybecomes a solid during its time in the contact. It therefore appears rea-sonable to assume that fluid viscosity becomes essentially constant withpressure during its time in the contact. To accurately predict bearingfriction torque, this becomes an important consideration for the use of asigmoid curve to describe lubricant viscosity in the contact. Conversely,using a sigmoid curve to approximate lubricant viscosity at atmosphericand low pressures does not provide the accuracy of either the Roelands[12.6] or Sorab et al. [12.7] model. Either of these models may be usedin the estimation of lubricant viscosity to calculate lubricant film thick-ness.

Deformation of Contact Surfaces

Because of the fluid pressures present between contacting rolling bodiescausing the increases in viscosity noted on Fig. 12.5, the rolling surfaces


deform appreciably in proportion to the thickness of a fluid film betweenthe surfaces. The combination of the deformable surface with the hydro-dynamic lubricating action constitutes the "elastohydrodynamic" prob-lem. The solution of this problem established the first feasible analyticalmeans of estimating the thickness of fluid films, the local pressures, andthe tractive forces that occur in rolling bearings.

Dowson and Higginson [12.11] for the model of Fig. 12.7 used thefollowing formulation for film thickness at any point in the contact:

Consequently, if h is small (as it must be in a rolling bearing under load)and viscosity is high (as it will become because of high pressure), dh/dyis very small and the film is essentially of uniform thickness. This resultis shown by Dowson et al. [12.11], and also by Grubin [12.12].

Pressure and Stress Distribution

In a later presentation Dowson et al. [12.13] indicated that dimensionlessfilm thickness H = hlYl. could be expressed as follows:

Dowson et al. [12.13] presented the results shown by Figs. 12.8 and 12.9for ~ = 2500 and 5000 corresponding to bronze rollers and steel rollersrespectively lubricated by a mineral oil. The load Qz = 0.00003 corre-sponds to approximately 483 N/mm2 (70,000 psi) and Qz = 0.0003 cor-responds to 1380 n/mm2 (200,000 psi) approximately. Dimensionlessspeed U = 10-11 corresponds to surface velocities on the order of 1524mm/sec (5 ft/sec) for an equivalent roller radius of 25.4 mm (1 in.) op-erating in mineral oil.

Note from Figs. 12.8 and 12.9 that the departure from the Hertzianpressure distribution is less significant as load increases. The secondpressure peak at the outlet end of the contact corresponds to a localdecrease in the film thickness at that point. Otherwise, the film is es-

FIGURE 12.8. Pressure distribution and film thickness for high load conditions (re-printed from [12.13]by permission of the Institution of Mechanical Engineers).

sentially of uniform thickness. The latter condition was confirmed bytests conducted by Sibley and Orcutt [12.14].

Additionally, Dowson et al. [12.13] demonstrated the effect of the dis-torted pressure distribution on maximum subsurface shear stress. Figure12.10 shows contours of TYZmjPmax' Note that the shear stress increasesin the vicinity of the second pressure peak and tends towards the surface.This condition was indicated in Chapter 6.

Lubricant Film Thickness

Grubin [12.12] developed a formula for minimum film thickness in linecontact, that is, the thickness of the lubricant film between the protu-berance at the trailing edge ofthe contact on the equivalent roller surfaceand the opposing surface of the relative flat. The Grubin formula is based

A significant feature of both equations is the relatively large dependencyof film thickness on speed and lubricant viscosity and the comparativeinsensitivity to load. Testing conducted by Sibley et al. [12.14] using ra-diation techniques seemed to conform the Grubin equation; however, theagreement between the Dowson and Grubin formulas is apparent. Today,the Dowson equation is recommended as representative of line contactlubrication conditions.

VERY HIGH PRESSURE EFFECTS

As indicated previously, maximum Hertz pressures occurring in the roll-ing element-raceway contacts typically fall in the range of 1000-2000

INLET LUBRICANT FRICTIONAL HEATING EFFECTS 441

MPa (approximately 150-300 kpsi), however, in modern bearing appli-cations, particularly endurance tests, it is not unusual for maximumHertz pressure to reach 4000 MPa. To prevent damage to laboratory testequipment and the materials under test, experiments used to confirmthe lubricant film thickness equations provided above have typicallybeen confined to pressures not exceeding 1500 MPa. Venner [12.21]con-ducted EHL analyses at high pressures and concluded that lubricantfilms predicted by the equations, both minimum and central lubricantfilm thicknesses, are somewhat thinner than calculated by these equa-tions. Using a tungsten carbide ball on a sapphire disk and ultrathinfilm interferometry and digital techniques, Smeeth and Spikes [12.22]measured lubricant fill thicknesses at maximum Hertz pressures up to3500 MPa. They confirmed Venner's conclusions, finding that, above con-tact loading of 2000 MPa, minimum lubricant film thickness varies in-versely as dimensionless load to the 0.3 power as compared to the 0.073power indicated by equation (12.58). The data shown by Smeeth et al.[12.22] can be represented by equations (12.64) and (12.65). These equa-tions define the ratio of film thickness resulting from very high pressureto that calculated using equations (12.58) and (12.61) for minimum andcentral film thicknesses respectively.

INLET LUBRICANT FRICTIONAL HEATING EFFECTS

At high bearing operating speeds, some of the frictional heat generatedin each concentrated contact is dissipated in the lubricant momentarilyresiding in the inlet zone of the contact. This effect, examined first byCheng [12.23], tends to increase the temperature of the lubricant in thecontact. Vogels [12.24] gives the following expression for viscosity:

'YJb = Ale/3' /(1\,+A2) (12.66)

In which Tb is in °C and AI' A2' and f3 are parameters to be defined for.ach lubricant. Three temperature-viscosity data points are required todetermine AI, A2' and f3 as follows:

STARVATION OF LUBRICANT

The basic formulas for calculation of lubricant film thickness assume anadequate supply of lubricant to the contact zones. The condition in whichthe volume of lubricant on the surfaces entering the contact is insuffi-cient to develop a full lubricant film is called starvation. Factors to de-termine the reduction of the apparent lubricant film thickness have beendeveloped as functions of the distance of the lubricant meniscus in theinlet zone from the center of the contact. As yet, no definitive equationshave been developed to accurately calculate the aforementioned distance;therefore, the meniscus distance has to be determined experimentally.Figure 12.12 illustrates the concept of meniscus distance. References[12.32-12.36] give further detail about this concept.

In consideration of the meniscus distance problem, a condition of zeroreverse flow is defined. Under this condition, the minimum velocity of thepoint situated at the meniscus distance from the contact center is, bydefinition, zero. If the meniscus distance is greater, the latter point willhave a negative velocity, that is, reverse flow. The zero reverse flow con-dition is therefore a quasistable situation because no lubricant is lost tothe contact owing to reverse flow.In the case of a minimum quantity oflubricant supplied, for example, oil mist or grease lubrication, the lubri-cant film thickness reduction factor owing to starvation effects, according

It is clear that c:P is zero if the meniscus distance should equal b and inthat case 'Ps = o. Accordingly, the accurate estimation of the meniscusdistance is necessary to the effective employment of a lubricant starva-tion factor. In the absence of this value, the condition of zero reverse flowprovides a practical limitation and a starvation factor of 'Ps = 0.70.

Thermal effects on lubricant film formation under conditions ap-proaching lubricant starvation are extremely significant owing to the ab-sence of excess lubricant to help dissipate frictional heat generation inthe contacts. Accordingly,the lubricant film reduction factors for thermaleffects and starvation are not multiplicative and a combined factor isrequired. Goksem et al. [12.32] derived the following expression for elas-tohydrodynamic line contact:

SURFACE TOPOGRAPHY EFFECTS

In the methods and equations used in the calculation of lubricant filmthickness thus far only the macrogeometries of the rolling componentshave been considered; i.e., the surfaces of the components have beenassumed to be smooth. In practice, each ball, roller or raceway surfacehas a roughness superimposed upon the principal geometry. This rough-ness, or more correctly surface topography similar to the earth's surfacesuperimposed upon the spherical surface of the planet, is introduced bythe surface finishing processes during component manufacture. In recent

SURF ACE TOPOGRAPHY EFFECTS447

history, substantial manufacturing development efforts have been ex-pended to produce ultra-smooth rolling component surfaces. Figure 12.13schematically illustrates a rough rolling component surface.

For a given surface, the roughness is most commonly defined by thearithmetic average (AA) peak-to-valley distance. This is easily measur-able using stylus devices such as the Talysurf machine. Using surface-measuring devices, more extensive properties of surface microgeometrycan also be measured; see McCool [12.37].Some ofthese will be discussedin Chapter 13. To date, AA surface roughnesses, RA, as fine as 0.05 J.Lm(2 J.Lin.)have been produced on ball bearing raceways approaching 600mm (24 in,) diameter. Balls larger than 25 mm (1 in,) diameter are rou-tinely produced with RA values of 0.005 J.Lm(0.2 J.Lin.).It is, however, notcertain that R

A= 0 is an ideal microgeometry from a lubrication effect-

iveness or surface fatigue endurance standpoint.Depending on the thickness of the lubricant film relative to the rough-

nesses of the rolling contact surfaces, the direction of the roughness pat-tern can affect the film-building capability of the lubricant. If the surfaceroughness has a pattern wherein the microgrooves are transverse to thedirection of motion, this could result in a beneficial lubricant film-building effect. Conversely, if the lay of the roughness is parallel to thedirection of motion, the effect can be to produce a thinner lubricant film.The most successful applications of rolling bearings are those in whichfluid lubricant films over the rolling element-raceway contacts are suf-ficiently thick to completely separate those components. This is generallydefined by the parameter A as follows:

where Lly and Llx are "correlation" lengths pertaining to distances between"hills" and "valleys" on a surface in the directions transverse and parallel(longitudinal) to motion, respectively.

T0nder and Jakobsen [12.39], using a ball-on-disk test rig and opticalinterferometry, confirmed Patir and Cheng's general conclusion thattransverse lay tends to generate thicker films than does longitudinal lay.Kaneta et al. [12.40] in a similar experimental effort determined that,in the thin film region (A < 1), film thickness for surfaces with transverse

SURFACE TOPOGRAPHY EFFECTS 449

lay tends to increase with slide/roll ratio due to deformation of asperi-ties. When A > 3, however, deformation of asperities can be neglected.

Chang et al. [12.41] analytically investigated the effects of surfaceroughness considering the effects of lubricant shear thinning due fric-tional heating. This phenomenon will be covered in greater detail inChapter 13. They determined that these effects serve to mitigate the"pressure rippling" influence on lubricant film thickness. Ai and Cheng[12.42] conducted an extensive analysis revisiting the influences of sur-face topographical lay. They generated three-dimensional plots of pointcontact pressure and film thickness distribution for transverse, longitu-dinal, and oblique topographical lays. Fig. 12.15-12.17 illustrate the ef-fects for a randomized surface roughness.

They indicated that roughness orientation has a noticeable effect onpressure fluctuation. They further noted that oblique roughness lay in-duces localized three-dimensional pressure fluctuations in which themaximum pressure may be greater than that produced by transverseroughness lay. It is to be noted that the oblique roughness lay more likelyis representative of the surfaces generated during bearing componentmanufacture. Oblique surface roughness lay may also result in the min-imum lubricant film thicknesses compared to transverse or longitudinalroughness lays. Ai and Cheng [12.42] further note, however, that whenA is sufficiently large such that the surfaces are effectively separated,the effect of lay on film thickness and contact pressure is minimal.

Guangteng et al. [12.43], using ultrathin film, optical interferometry,managed to measure the mean EHL film thickness of very thin film,

FIGURE 12.17. Pressure (a) and film thickness (b) distribution in an EHL point contactwith oblique topographical lay, random surface roughness. Motion is in the x-direction (fromAi and Cheng [12.42]).

isotropically rough surface occurring in rolling ball on flat contacts. Theyfound that, for A < 2, the mean EHL film thicknesses were less thanthose for smooth surfaces. Subsequently, using the spacer layer imagingmethod developed by Cann et al. [12.44] to map EHL contacts, Guang-teng et al. [12.45] indicated that rolling elements having real, random,rough surfaces; for example, rolling bearing components, the mean film

GREASE LUBRICATION 451

thicknesses tend to be less than those calculated for rolling elementshaving smooth surfaces. This implies that, in the mixed EHL regime-for example, A < 1.5-the mean lubricant film thicknesses will tend tobe less than those predicted by the equations given for rolling contactswith smooth surfaces. The amount of the reduction may only be deter-mined by testing; empirical relationships need to be developed.

GREASE LUBRICATION

When grease is used as a lubricant, the lubricant film thickness is gen-erally estimated using the properties of the base oil of the grease whileignoring the effect of the thickener. It has been determined, however, byseveral researchers [12.46-12.49] that in a given application, owing to acontribution by the thickener, a grease may form a thicker lubricant filmthan that determined using only the properties of the base oil. Kauzlar-ich and Greenwood [12.50] developed an expression for the thickness ofthe film formed by greases in line contact under a Herschel-Bulkleyconstitutive law in which shear stress T and shear rate 'Yare related bythe equation

T = Ty + a yfJ (12.84)

where Ty is the yield stress and a and {3are considered physical prop-erties of the grease.

For a Newtonian fluid

T = TlY (12.85)

where TI is the viscosity.The effective viscosity under a Herschel-Bulkley law is thus found by

equating T from equations (12.84) and (12.85) so that

In this form it is seen that for {3> 1, Tleff increases indefinitely with shearrate, and for {3< 1, Tleff approaches zero as strain rate increases. Palacioset al. [12.46] argued that it is more reasonable to assume that at highshear rates greases will behave like their base oils. They accordinglyproposed a modification of the Herschel-Bulkley law to the form

They proposed that this evaluation be made at a shear rate equal to0.68u/hG, which requires iteration to determine hG' Their suggestedapproach is to calculate hb from equation (12.61), then determine l' =

0.68u/hb, and then hG from equation (12.88). The shear rate is then re-calculated using hG' The process is repeated until convergence occurs.The analysis was applied to line contact, but it should also be valid forelliptical contacts with a/b in the range of 8-10 (typical for ball bearingpoint contacts).

In his investigations, Cann [12.52, 12.53] notes that the portion of thefilm associated with the grease thickener is a "residual" film composedof the degraded thickener deposited in the bearing raceways. The hydro-dynamic component is generated by the relative motion of the surfacesdue to oil, both in the raceways and supplied by the reservoirs of greaseadjacent the raceways. He further notes that at low temperatures greasefilms are generally thinner than those for the fully flooded, base fluidlubricant. This is due to the predominant bulk grease starvation and theinability of the high viscosity, bled lubricant to resupply the contact. Athigher temperatures of operation, grease forms films considerablythicker than those considering only the base oil. This is attributed to theincreased local supply of lubricant to the contact area due to the loweroil viscosity at the elevated temperature producing a partially floodedEHL film augmented by a boundary film of deposited thickener.

Therefore, it can be stated that with grease lubrication the degree ofstarvation tends to increase with increasing base oil viscosity, thickenercontent, and speed of rotation. It tends to decrease with increasing tem-perature. For rolling bearing applications, the film thickness may onlybe a fraction of that calculated for fully flooded, oil lubrication conditions.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 ofgrease base oil viscosity, grease thickener content, and rotational speed,

LUBRICATION REGIMES 453

lubricant film thicknesses may be expected to be only a fraction of thosecalculated using equations (12.55), (12.58), and (12.61). According to datashown by Cann [12.53], fractional values might range from 0.9 down to0.2.

LUBRICATION REGIMES

Although this chapter has concentrated on elastohydrodynamic lubrica-tion in rolling contacts, the general solution presented for the Reynoldsequation covers a gamut of lubrication regimes; for example:

• Isoviscous hydrodynamic (IHD) or classical hydrodynamic lubrication• Piezoviscous hydrodynamic (PHD lubrication, in which lubricant vis-

cosity is a function of pressure in the contact• Elastohydrodynamic (EHD) lubrication, in which both the increase of

viscosity with pressure and the deformations of the rolling componentsurfaces are considered in the solution

Dowson and Higginson [12.54] created Fig. 12.18 to define these regimesfor line contact in terms of the dimensionless quantities for film thick-ness, load, and rolling velocity; equations (12.46)-(12.48).

Markho et al. [12.55] established a parameter, called C1 herein, for afixed value of Hl; this factor was used to define the lubrication regime.Dalmaz [12.56] subsequently established equation (12.89) to cover allpractical values of I'J.

Table 12.1 shows the relationship of parameter C1 to the operating lu-brication regimes.

For calculation of the lubricant film thicknesses is rolling element-raceway contacts, only the PHD and EHD regimes need to be considered.For calculations associated with the cage-rolling element contacts, prob-ably consideration of the hydrodynamic regime is sufficient. In this case,Martin [12.1] gives the following equation for film thickness in line con-tact:

where

C2 = log10(618Uo.6617) (12.93)

C3 = loglO(1.285Uo.0025) (12.94)

and C1 is given by equation (12.89). In equation (12.92), C4 is given by

C4 = C2 + C1Ciq - 3) - O.094C1 (q - O.77C1 - 1) (12.95)

Dalmaz [12.56] also developed numerical results for point contact filmthicknesses in the PHD regime; an analytical relationship was not thenestablished.


CLOSUREIn the foregoing discussion it has been demonstrated analytically that alubricant film can serve to separate the rolling elements from the con-tacting raceways. Moreover, the fluid friction forces developed in the con-tact zones between the rolling elements and raceways can significantlyalter the bearing's mode of operation. It is desirable from the standpointof preventing increased stresses caused by metal-to-metal contact thatthe minimum film thickness should be sufficient to completely separatethe rolling surfaces. The effect of film thickness on bearing endurance isdiscussed in Chapter 23.

A substantial amount of analytical and experimental research fromthe 1960s through the 1990s has contributed greatly to the understand-ing of the lubrication mechanics of concentrated contacts in rolling bear-ings. Perhaps the original work of Grubin [12.12] will prove to be assignificant as that conducted by Reynolds during the 1880s.

Besides acting to separate rolling surfaces, the lubricant is frequentlyused as a medium to dissipate the heat generated by bearing friction aswell as remove heat that would otherwise be transferred to the bearingfrom surroundings at elevated temperatures. This topic is discussed inChapter 15.

REFERENCES12.1. H. Martin, "Lubrication of Gear Teeth," Engineering 102, 199 (1916).

12.2. J. Osterle, "On the Hydrodynamic Lubrication of Roller Bearings," Wear 2, 195(1959).

12.3. B. Sternlicht, P. Lewis, and P. Flynn, "Theory of Lubrication and Failure of RollingContacts," ASME Trans., J. Basic Eng. 213-226 (1961).

12.4. C. Barus, "Isothermals, Isopiestics, and Isometrics Relative to Viscosity," Amer. J.Science 45, 87-96 (1893).

12.5. ASME Research Committee on Lubrication "Pressure-Viscosity Report-Vol. II,"ASME (1953).

12.6. C. Roelands, "Correlation Aspects of Viscosity-Tern perature-Pressure Relationshipof Lubricating Oils" (Ph.D. thesis, Delft University of Technology, 1966).

12.7. J. Sorab and W. VanArsdale, "A Correlation for the Pressure and Temperature De-pendence of Viscosity," Tribology Trans. 34, 4, 604-610 (1991).

12.8. T. Harris, "Establishment of a New Rolling Bearing Life Calculation Method," FinalReport, U. S. Navy Contract N68335-93-C-Ol11 (January 15, 1994).

12.9. S. Bair and W. Winer, "Shear Strength Measurements of Lubricants at High Pres-sure," Trans. ASME, J. Lubrication Technology, Ser. F 101, 251-257 (1979).

12.10. S. Bair and W. Winer, "Some Observations in High Pressure Rheology of Lubri-cants," Trans. ASME, J. Lubrication Technology, Ser. F 104, 357-364 (1982).

12.11. D. Dowson and G. Higginson, "A Numerical Solution to the ElastoHydrodynamicProblem," J. Mech. Eng. Sci. 1(1), 6 (1959).

REFERENCES457

12.12. A. Grubin, "Fundamentals of the Hydrodynamic Theory of Lubrication of HeavilyLoaded Cylindrical Surfaces," in Investigation of the Contact Machine Components,ed. Kh. F. Ketova (Translation of Russian Book No. 30, Chapter 2), Central ScientificInstitute of Technology and Mechanical Engineering, Moscow (1949).

12.13. D. Dowson and G. Higginson, "The Effect of Material Properties on the Lubricationof Elastic Rollers," J. Mech. Eng. Sci. 2(3) (1960).

12.14. L. Sibley and F. Orcutt, "Elasto-hydrodynamic Lubrication of Rolling Contact Sur-faces," ASLE Trans. 4, 234-249 (1961).

12.15. D. Dowson and G. Higginson, Proc. Inst. Mech. Eng., Vol. 182, Part 3A, 151-167

(1968).12.16. G. Archard and M. Kirk, "Lubrication at Point Contacts," Proc. Royal Soc. Ser. A

261,532-550(1961)12.17. B. Hamrock and D. Dowson, "Isothermal Elasto-hydrodynamic Lubrication of Point

Contacts-Part III-Fully Flooded Results," Trans. ASME, J. Lubrication Technol-ogy 99, 264-276 (1977).

12.18. L. Wedeven, "Optical Measurements in Elasto-hydrodynamic Rolling Contact Bear-ings," Ph.D. Thesis, University of London (1971).

12.19. M. Kotzalas, "Power Transmission Component Failure and Rolling Contact FatigueProgression," Ph.D. Thesis, Pennsylvania State University (1999).

12.20. E. Avallone and T. Baumeister, Marks Standard Handbook for Mechanical Engi-neers, 9th ed., McGraw-Hill, New York (1987).

12.21. C. Venner, "Higher Order Multilevel Solvers for the EHL Line and Point ContactProblems," ASME Trans., J. Tribology 116, 741-750 (1994).

12.22. M. Smeeth and H. Spikes, "Central and Minimum Elastohydrodynamic Film Thick-ness at High Contact Pressure," ASME Trans., J. Tribology 119,291-296 (1997).

12.23. H. Cheng, "A Numerical Solution to the Elastohydrodynamic Film Thickness in anElliptical Contact," Trans. ASME, J. Lubrication Technology 92, 155-162 (1970).

12.24. H. Vogels, "Das Temperaturabhiingigkkeitsgesetz der Viscositiit von Fhissigkeiten,"Phys. Z. 22, 645-646 (1921).

12.25. H. Cheng, "A Refined Solution to the Thermal-Elastohydrodynamic Lubrication ofRolling and Sliding Cylinders," ASLE Trans. 8(4),397-410 (1965).

12.26. L. Murch and W. Wilson, "A Thermal Elastohydrodynamic Inlet Zone Analysis,"Trans. ASME, J. Lubrication Technology 97(2), 212-216 (1975).

12.27. A. Wilson, "An Experimental Thermal Correction for Predicted Oil Film Thicknessin Elastohydrodynamic Contacts," in Thermal Effects in Tribology: Proceedings ofthe 6th Leeds-Lyon Symposium on Tribology, 1979 (1980).

12.28. W. Wilson and S. Sheu, "Effect of Inlet Shear Heating Due to Sliding on Elastohy-drodynamic Film Thickness," Trans. ASME, J. Lubrication Technology 105(2), 187-

188 (1983).12.29. P. Gupta, H. Cheng, D. Zhu, N. Forster, and J. Schrand, "Viscoelastic Effects in Mil-

L-7808 Type Lubricant, Part I: Analytical Formulation," Tribology Trans. 35(2),269-274 (1992).

12.30. C. Hsu and R. Lee, "An Efficient Algorithm for Thermal Elastohydrodynamic Lu-brication under Rolling/Sliding Line Contacts," J. Vibration, Acoustics and Relia-bility in Design 116(4), 762-768 (1994).

12.31. W. MacAdams, Heat Transmission, 3d ed., McGraw-Hill, New York (1954).12.32. P. Goksem and R. Hargreaves, "The Effect of Viscous Shear Heating in Both Film

Thickness and Rolling Traction in an EHL Line Contact-Part II: Starved Condi·tion," Trans. ASME, J. Lubrication Technology 100, 353-358 (1978).


12.33. D. Dowson, "Inlet Boundary Conditions," Leeds-Lyon Symposium (1974).12.34. P. Wolveridge, K. Baglin, and J. Archard, "The Starved Lubrication of Cylinders in

Line Contact," Proc. Inst. Mech. Eng. 185, 1159-1169 (1970-71).12.35. P. Castle and D. Dowson, "A Theoretical Analysis of the Starved Elastohydrody-

namic Lubrication Problem," Proc. Inst. Mech. Eng. 131, 131-137 (1972).12.36. B. Hamrock and D. Dowson, "Isothermal Elastohydrodynamic Lubrication of Point

Contact-Part IV: Starvation Results," Trans. ASME, J. Lubrication Technology 99,15-23 (1977).

12.37. J. McCool, "Relating Profile Instrument Measurements to the Functional Perform-ance of Rough Surfaces," Trans. ASME, J. Tribology 109,271-275 (April 1987).

12.38. N. Patir and H. Cheng, "Effect of Surface Roughness Orientation on the CentralFilm Thickness in EHD Contacts," in Elastohydrodynamics and Related Topics:Pro-ceedings of the 5th Leeds-Lyon Symposium on Tribology, 1978, 15-21 (1979).

12.39. K. T!iJnderand J. Jakobsen, "Interferometric Studies of Effects of Striated Rough-ness on Lubricant Film Thickness Under Elastohydrodynamic Conditions," Trans.ASME, J. Tribology 114, 52-56 (January 1992).

12.40. M. Kaneta, T. Sakai, and H. Nishikawa, "Effects of Surface Roughness on PointContact EHL," Tribology Trans. 36,(4),605-612 (1993).

12.41. L. Chang, M. Webster, and A. Jackson, "On the Pressure Rippling and RoughnessDeformation in Elastohydrodynamic Lubrication of Rough Surfaces," Trans. ASME,J. Tribology 115, 439-444 (July 1993).

12.42. X. Ai and H. Cheng, "The Effects of Surface Texture on EHL Point Contacts," Trans.ASME, J. Tribology 118,59-66 (January 1996).

12.43. G. Guangteng and H. Spikes, "An Experimental Study of Film Thickness in theMixed Lubrication Regime," in Elastohydrodynamics '96: Fundamentals and Appli-cations in Lubrications and Traction: Proceedings of the 23rd Leeds-Lyon Sympo-sium on Elastohydrodynamics, 1996, 159-166 (1997).

12.44. P. Cann, J. Hutchinson, and H. Spikes, "The Development of a Spacer Layer Im-aging Method (SLIM) for Mapping Elastohydrodynamic Contacts," Tribology Trans.39, 915-921 (1996).

12.45. G. Guangteng, P. Cann, A. Olver, and H. Spikes, "Lubricant Film Thickness inRough Surface, Mixed Elastohydrodynamic Contact," ASME Paper 99-TRIB-40 (Oc-tober 1999).

12.46. A. Wilson, "The Relative Thickness of Grease and Oil Films in Rolling Bearings,"Proc. Inst. Mech. Eng. 193,185-192 (1979).

12.47. H. Miinnich and H. Glockner, "Elastohydrodynamic Lubrication of Grease-Lubricated Rolling Bearings," ASLE Trans. 23, 45-52 (1980).

12.48. J. Palacios, A. Cameron, and L. Arizmendi, "Film Thickness of Grease in RollingContacts," ASLE Trans. 24,474-478 (1981).

12.49. J. Palacios, "Elastohydrodynamic Films in Mixed Lubrication: An Experimental In-vestigation," Wear 89, 303-312 (1983).

12.50. J. Kauzlarich and J. Greenwood, "Elastohydrodynamic Lubrication with Herschel-Bulkley Model Greases," ASLE Trans. 15, 269-277 (1972).

12.51. J. Palacios and M. Palacios, "Rheological Properties of Greases in EHD Contacts,"Tribology Int. 17, 167-171 (1984).

12.52. P. Cann, "Starvation and Reflow in a Grease-Lubricated Elastohydrodynamic Con-tact," Tribology Trans. 39(3),698-704 (1996).

12.53. P. Cann, "Starved Grease Lubrication of Rolling Contacts," Tribology Trans. 42(4),867-873 (1999).

12.54. D. Dowson and G. Higginson, "Theory of Roller Bearing Lubrication and Deforma-tion," Proc. Inst. Mech. Eng. 117 (1963).

REFERENCES 459

12.55. P. Markho and D. Clegg, "Reflections on Some Aspects of Lubrication of Concen-trated Line Contacts," Trans. ASME, J. Lubrication Technology 101,528-531 (1979).

12.56. G. Dalmaz, "Le film mince visquex dans les contacts hertziens en regimes hydro-dynamique et elastohydrodynamique" (Docteur d'Etat es Sciences thesis, I.N.S.A.Lyon, 1979).

FRICTION IN FLUID-LUBRICATED ROLLINGELEMENT-RACEWAYCONTACTS

LIST OF SYMBOLS


a Semimajor axis of contactellipse mm (in.)

Ae True average contact area mm2 (in.)2

Ao Apparent contact area mm2 (in.)2b Semiminor axis of contact

ellipse mm (in.)d Separation of mean plane of

summits and smooth plane mm (in.)

DSUM Summit density mm-2 (in.)-2E1, E2 Elastic moduli of bodies 1 and

2 MPa (psi)E' Reduced elastic modulus MPa (psi)Fo( ), F1( ), Tabular functions for theF3/2( ) Greenwood-Williamson modelh Lubricant film thickness mm (in.)he Central or plateau lubricant

film thickness mm (in.)

461

462 FRICTION IN FLUID-LUBRICATED ROLLING ELEMENT-RACEWAY CONTACTS

Symbol Description Unitsmo Zeroth-order spectral moment,

== R2 == S2 /Lm2 (/Lin.2)q

m2 Second-order spectral momentm4 Fourth-order spectral moment mm -2 (in. -2)n Contact density mm -2 (in. -2)np Plastic contact density mm -2 (in. -2)

P Local contact pressure MPa (psi)Po Maximum contact pressure MPa (psi)p Applied load N (lb)Qa Asperity-supported load N (lb)Qf Fluid-supported load N (lb)R Summit sphere radius mm (in.)Rq Root mean square (rms) value

of surface profile /Lm (/Lin.)T Temperature °C (OF)v Sliding velocity mm/sec (in./sec)W Deflection of summit /Lm (/Lin.)wp Variable governing asperity

density /Lm (/Lin.)y Yield strength in simpletension MPa (psi)

Zs Summit height relative tosummit mean plane mm (in.)-Distance between surface andZs

summit mean plane mm (in.)z(x) Surface profile mm (in.)a Bandwidth parameter'Y Shear rate sec-I

TJ Absolute viscosity N-sec/m2 (lb-sec/in.2)A Lubricant film parameter, his/La Coulomb friction coefficientVI' v2 Poisson's ratio for bodies 1

and 2(TI' (T2 Standard deviation of summit

heights for bodies 1 and 2 mm (in.)(T Standard deviation for

summit heights for compositesurface mm (in.)

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)

GENERAL 463

Symbol Description Unitscp( ) Gaussian probability density

function mm-I (in.-I)

GENERAL

In its full complexity, a rolling element-raceway contact cannot be rep-resented by a simple analytical expression. The combined action of anapplied load and kinematic constraints produces some combination ofrolling, sliding, and spinning motions. These motions act to draw lubri-cant into the contact where, its properties altered by the pressure andtemperatures that vary throughout the contact region, it forms a filmthat serves to separate the contacting bodies to an extent depending onboth the microgeometry ofthe bodies, and the properties of the lubricant.When the separating film is small relative to the composite surfaceroughness, a myriad of microcontacts of highly irregular shapes formswithin the macrocontact, causing pressure, temperature, and film thick-ness perturbations on a microscale. Moreover, these microcontacts maydeform plastically as well as elastically with the result that the micro-geometry varies with time.

Sliding and spinning motions on the macrocontact act to shear theseparating lubricant film and, if separation is only partial, to drag themicrocontacts across each other. A tangential force is produced fromthese combined effects. This tangential or traction force alters the stressdistribution in the solids and is a critical factor in determining fatiguelife. The magnitude of the fluid contribution to the traction depends onthe lubricant properties under the locally variable pressure and temper-ature and shear rates that prevail in the macrocontact. The contributionto the traction caused by the sliding microcontacts will depend on thelocal film conditions or the nature of the surface boundary films thatresult from oxidation and additives present in the lubricant.

As discussed in Chapter 12, several researchers have attempted tomodel the effect of surface roughness, i.e., microgeometry, on the thick-ness of the lubricant films in rolling/sliding concentrated contacts. Theseefforts have frequently included the estimation of fluid friction and itseffect on the lubricant film thickness. In general, it has been determinedthat the friction and the resultant localized temperature rise in the con-tact has little effect on lubricant film thickness; as indicated in Chapter12, lubricant film thickness depends on events occurring at the inlet tothe contact and not in the contact proper. In these analyses, the indicatedsolutions have been obtained by numerical analysis, most recently usingfinite element methods requiring meshes of several thousand nodes andseveral minutes to hours of calculation even with high speed digital cal-

464 FRICTION IN FLUID·LUBRICATED ROLLING ELEMENT-RACEWAY CONTACTS

culation. While this approach is useful for research purposes, it does notsuffice for use in the determination of ball and roller bearing perform-ance in practical engineering applications.

This chapter describes an approach that synthesizes state-of-the artmodels for lubricant film thickness and asperity load sharing into a prac-ticable, analytical description of a real, rolling element-raceway contact.

MICROGEOMETRY AND MICROCONTACTS

"Rough" Surfaces

In calculating the lubricant film thickness in Chapter 12, it is assumedthat the surfaces are perfectly smooth. The assumption is now made thatwhen the surfaces are rough the lubricant film thickness, calculated asif the surfaces were smooth, separates the mean planes of the roughsurfaces, as shown in Fig. 13.1.

The surfaces fluctuate randomly about their mean planes in accord-ance with a probability distribution. The root-mean-square (rms) valueof this distribution is denoted lT1 for the upper surface and lT2 for thelower surface. When the combined surface fluctuations at a given posi-tion exceed the gap h due to the lubricant film, a microcontact occurs.At the microcontacts the surfaces deform elastically and possibly plas-tically. The aggregate of the microcontact areas is generally a small frac-tion «5%) of the nominal area of contact.

A microcontact model uses surface microgeometry data to predict, ata minimum, the density of microcontacts, the real area of contact, andthe elastically supported mean load. One of the earliest and simplestmicrocontact models is that of Greenwood and Williamson (GW) [13.1].Generalizations of this model applicable to isotropic surfaces have beendeveloped by Bush et al. [13.2] and by O'Callaghan and Cameron [13.3].Bush et al. [13.4] also treated a strongly anisotropic surface. One of themost comprehensive models yet developed is ASPERSIM [13.5], whichrequires a nine-parameter microgeometry description and accounts foranisotropic as well as isotropic surfaces. A comparison of various micro-

MICROGEOMETRY AND MICROCONTACTS465

contact models conducted by McCool [13.6] has shown that the GWmodel, despite its simplicity, compares favorably with the other models.Because it is much easier to implement than the other models, the GWmodel is the microcontact model recommended here.

GW ModelFor the contact of real surfaces Greenwood and Williamson [13.1] devel-oped one of the first models that specifically accounted for the randomnature of interfacial phenomena. The model applies to the contact of twoflat plastic planes, one rough and the other smooth. It is readily adaptedto the case of two rough surfaces as discussed further below. In the GWmodel the rough surface is presumed to be covered with local high spotsor asperities whose summits are spherical. The summits are presumedto have the same radius R, but randomly variable heights, and to beuniformly distributed over the rough surface with a known density DSUM

of summits/unit area.The mean height of summits lies above the mean height of the surface

as a whole by the amount Zs indicated in Fig. 13.2. The summit heightsZs are assumed to follow a Gaussian probability law with a standarddeviation lTs' Figure 13.3 shows the assumed form for the summit heightdistribution or probability density function (pdf) f(zs)' It is symmetricalabout the mean summit height. The probability that a summit has aheight, measured relative to the summit mean plane in the interval (zs,Zs + dzs) is expressed in terms of the pdf as f(zs) dzs· The probability thata randomly selected summit has a height in excess of some value d isthe area under the pdf to the right of d. The equation of the pdf is

This integration must be performed numerically. Fortunately, however,the calculation can be related to tabulated areas under the standardnormal curve for which the mean is 0 and the standard deviation is 1.0.

Using the standard normal density function <f>(x), the probability thata summit has a height greater than d above the summit mean plane iscalculated.

where Fo(t) is the area under the standard normal curve to the right ofthe value t. Values Fo(t) for t ranging from 1.0 to 4.0, are given in column2 of Table 13.1.

It is assumed that when large flat surfaces are pressed together, theirmean planes remain parallel. Thus, if a rough surface and a smoothsurface are pressed against each other until the summit mean plane ofthe rough surface and the mean plane of the smooth surface are sepa-rated by an amount d, the probability that a randomly selected summitwill be a microcontact is

P[summit is a contact] = P[zs > d] = Fo (d/ (Ts) (13.4)

Since the number of summits per unit area is DSUM' the average expectednumber of contacts in any unit area is

n = DSUMFO (d/ (Ts) (13.5)

Given that a summit is in contact because its height Zs exceeds d, thesummit must deflect by the amount w = Zs - d, as shown in Fig. 13.4.

For notational simplicity the subscript on Zs is henceforth deleted. Fora sphere of radius R elastically deflecting by the amount w, the Hertziansolution gives the contact area

For fixed dl as the degree of plastic asperity interaction is determined bythe value of w~: the higher w~, the fewer plastic contacts. Accordingly,GW use the inverse, 1/ w~, as a measure of the plasticity of an interface.For a given nominal pressure P1Ao, dl as is found by solving equation(13.13), assuming that most of the load is elastically supported.

Application of the GW Model to a Lubricated Contact of TwoRough Surfaces

To use the GW model for a lubricated contact, (1) the height d relativeto the mean plane of the summit heights to h, the thickness of the lu-bricant film that separates the two surfaces, must be determined, and(2) the values of the GWparameters R, DSUM, and as must be established.For (1) the first step is to compute the composite rms value of the two"rough" surfaces as

When the mean plane of a rough surface with this rms value is held ata height h above a smooth plane, the rms value of the gap width is thesame as shown in Fig. 13.3, where both surfaces are rough. It is in thissense that the surface contact of two rough surfaces may be translatedinto the equivalent contact of a rough surface and a smooth surface. Asshown in Fig. 13.2, the summit and surface mean planes are separatedby an amount zS'

For an isotropic surface with normally distributed height fluctuations,the value of Zs has been found by Bush et a1. [13.7] to be


Equation (13.29) shows that dl as is linearly related to hi a. The ratiohi a is also referred to as the lubricant film parameter A. When A > 3,contacts are few and the surfaces may be considered to be well lubri-cated.

For a specified or calculated value of A, dl as is computed from equa-tion (13.29) for use in the GW model. For an isotropic surface the twoparameters DSUM and R, the average radius of the spherical caps of as-perities, may be expressed as (Nayak [13.8]):

For an anisotropic surface, the value of m2 will vary with the directionin which the profile is taken on the surface. The maximum and minimumvalues occur in two orthogonal "principal" directions. Sayles and Thomas[13.9] recommend the use of an equivalent isotropic surface for which m2

is computed as the harmonic mean of the m2 values found along theprincipal directions. The value of m4 is similarly taken as the harmonicmean of the m4 values in these two directions.

ASPERITY- AND FLUID-SUPPORTED LOAD

For a specified contact with semiaxes a and b, under a load P, with pla-teau lubricant film thickness h and given values of mo, m2, and m4, theasperity load Qa is determined by first computing PIAo from equation[13.13] and using

The fluid-supported load is then

Qf = P - Qa (13.33)

If Qa > P, the implication is that the lubricant film thickness is largerthan computed under smooth surface theory. In this case, equation(13.13) could be solved iteratively until Qa = P.

Example 14.1. An isotropic surface has roughness parameters a2 =

mo = 0.0625 JLm2, m2 = 0.0018, and m4 = 1.04 X 10-4 JLm-2• Calculate


and adjusted for starvation and inlet heating, is h = 0.5 /Lm. Usingthe GW microcontact model, calculate the nominal pressure PIAo, therelative contact area AclAo, the mean real pressure PIAc, the contactdensity n, and, for a tensile yield strength of 2070 N/mm2, the plasticcontact density np'

The computed film parameter A = 0.5/(0.0625)1/2 = 2.0. From equa-tion (13.29),


FRICTION IN THE EHL CONTACT

Non-Newtonian Lubrication

As indicated in Chapter 12, a Newtonian lubricant is one in which stressdue to shearing of the lubricant is defined by equation (12.1).

This equation implies that fluid viscosity is a constant. Several investi-gators [13.10]-[13.13] have investigated the effects of non-Newtonian lu-bricant behavior on the EHL model. Bell [13.11] specifically studied theeffects of a Ree-Eyring model, in which shear rate can be described byequation (13.34).

In equation (13.34), Eyring stress 70 and viscosity TJ are functions oftem-perature and pressure. When 7 is small, equation (13.34) describes alinear viscous behavior approaching that of equation (12.1). Subse-quently, it has ·been established that the non-Newtonian characteristicsof lubricants tend to cause decreases in viscosity at high lubricant shearrates. These may occur directly in the contact under operating conditionsinvolving substantial sliding in addition to rolling. It has been furtherestablished, however, that the film thickness which obtains over most ofthe contact is primarily a function of the lubricant properties at the con-tact inlet. At the contact inlet, pressure is substantially atmospheric;therefore, it is not anticipated that a non-Newtonian lubricant will sig-nificantly influence lubricant film thickness.

Non-Newtonian lubrication does, however, significantly influence fric-tion in the contact. Due to friction, lubricant temperature in the contactrises during rolling element-raceway contact, causing lubricant viscosityto decrease. Moreover, since pressure increases greatly in, and variesover, the contact, it is evident that equation (12.1) becomes

Assuming the contact area and surface pressure distribution is as rep-resented by Fig. 6.6 for point contact and Fig. 6.7 for line contact, thenequation (13.34) defines the localized shear stress 7 at any point x, yonthe contact surface. Since EHL films are very thin compared to the ma-

FRICTION IN THE EHL CONTACT 477

crogeometrical dimensions of the rolling components, it is further appro-priate to approximate equation (13.34) as follows:

where v is sliding velocity at the contact surface point x, y, and he is thecentral or plateau film thickness.

Houpert [13.14] and Evans and Johnson [13.15] used the Ree-Eyringmodel for analysis of EHL traction. Equations (12.21)-(12.23), intro-duced in Chapter 12, can provide the viscosity-pressure-temperaturerelationship for many common lubricants. These equations can be usedin equation (13.34) in the estimation of shear stress 7 provided the lo-calized temperature and pressure can themselves be estimated.

Limiting Shear Stress

As shown in Chapter 7, owing to the macrogeometries of mating rollingcomponents-i.e., rolling elements and raceways-and the contact de-formations of these components under load, both rolling and sliding mo-tions occur in most rolling element-raceway contacts. Gecim and Winer[13.13] and Bair and Winer [13.16] suggested alternative expressions forthe relationship between shear stress and strain rate that incorporateda maximum or limiting shear stress. Essentially, they proposed that fora given pressure, temperature, and degree of sliding, there is a maximumshear stress that can be sustained. Based on experimental data from adisk machine, Fig. 13.5 from Johnson and Cameron [13.17] shows curves


of traction coefficient vs pressure and slide-roll ratio, which illustratethis phenomenon.

In this case, traction coefficient is defined as the ratio of average shearstress to average normal stress. Based on experiments, Schipper et al.[13.18] indicated a range of values for limiting fluid shear stress; forexample, 0.07 < Tlim/Pave < 0.11.

Shear Stress in the Lubricant

Trachman and Cheng [13.19] and Tevaarwerk and Johnson [13.20] in-vestigated traction in rolling-sliding contacts and found that equation(12.1) pertains only to a situation involving a relatively low slide-to-rollratio; for example, less than 0.003 as shown in Fig. 13.5. Note that trac-tion refers to the net frictional effect in the rolling direction. Similar toTrachman and Cheng, for a given temperature and pressure, it is pos-sible to define local contact friction as follows:

Tr = (Tji;, + TN1)-1 (13.36)

where TN is the Newtonian portion of the frictional shear stress asdefined by equation (12.1) and Tlim is the maximum shear stress that canbe sustained at the applied pressure. Fig. 13.6 schematically demon-strates equation (13.36).

Recognizing that viscosity is a function of local pressure and temper-ature in the contact, and since the film thickness is extremely smallcompared to the dimensions of the rolling components, TN can be de-scribed by equation (13.35).

Shear Stress due to Coulomb Friction

As indicated in the section on Microgeometry and Microcontacts, whenlubricant film thickness is of the same magnitude or less than the com-posite roughness of the rolling components, i.e., A ::::;1, contact of asper-

CLOSURE 479

ities on the component surfaces becomes more frequent. The friction thatoccurs due to sliding motions between asperities can be characterized asCoulomb friction, such that

Ta = JLaP (13.37)

where JL is the Coulomb coefficient of friction and P is the local pressure.On an average basis, this frictional stress may be assumed to apply tothe portion of the overall contact area associated with asperity-asperitycontact. If the contact area of the smooth components is defined as Ao,then, according to equation (13.12), the portion of the contact associatedwith Coulomb friction is AJAo . Ao.

Composite Shear Stress

Combining the stress components due to Newtonian fluid friction, limi-ting shear in the fluid, and asperity interactions, Harris and Barnsby[13.21] applied the following formula in the determination ofrolling con-tact tractions:

In using equation (13.38), it is necessary to define values for Tlim and JL.These values for can only be determined from testing offull-scale bearingapplications. Based on comparison of predicted to actual bearing heatgenerations so determined, Tlim = O.lPave and JL = 0.1 have been foundto be representative in several applications.

CLOSURE

This chapter contains an approach to predicting key performance-relatedparameters descriptive of real EHL contacts, including contact density,true contact area, plastic contact area, fluid and asperity load sharing,and the relative contributions of the fluid and asperities to overall fric-tion. It is recognized that using more elegant and complex analyticalmethods such as very fine mesh, multi-thousand node, and finite elementanalysis together with solutions of the Reynolds and energy equationsin three dimensions, it is possible to obtain a more generalized solutionwith perhaps increased accuracy. Unfortunately, using currently availa-ble computing equipment, such solutions would require several hours ofcomputational time to enable the performance analysis of a single op-erating condition for a rolling bearing containing only a small comple-


ment of rolling elements. The equations provided in this chapter forfrictional shear stress are based on the assumption of Hertz pressure(normal stress) applied, unmodified by EHL conditions, to the contact.This assumption is sufficiently accurate for most rolling element-raceway contacts in that such loading is reasonably heavy; for example,generally at least several hundred MPa. Furthermore, the assumption ismade that equation (13.38) can be applied at every point in the contact.With respect to the Coulomb friction component of surface shear stress,it is recognized that surface roughness peaks cause local pressures inexcess of Hertzian values and these will cause localized shear stressesin excess of those predicted by equation (13.38).Accommodation of thesevariations tends to increase the computational time beyond current en-gineering practicality. Therefore, for engineering purposes, frictionalshear stress may be calculated according to the average condition in eachcontact.

REFERENCES13.1. J. Greenwood and J. Williamson, "Contact of Nominally Flat Surfaces," Proc. Royal

Soc. London A295, 300-319 (1966).13.2. A. Bush, R. Gibson, and T.Thomas, "The Elastic Contact of a Rough Surface," Wear

35, 87-111 (1975).13.3. M. O'Callaghan and M. Cameron, "Static Contact under Load Between Nominally

Flat Surfaces," Wear 36, 79-97 (1976).13.4. A. Bush, R. Gibson, and G. Keogh, "Strongly Anisotropic Rough Surfaces," ASME

Paper 78-LUB-16 (1978).13.5. J. McCool and S. Gassel, "The Contact of Two Surfaces Having Anisotropic Rough-

ness Geometry," ASLE Special Publication (SP-7), 29-38 (1981).13.6. J. McCool, "Comparison of Models for the Contact of Rough Surfaces," Wear 107,

37-60 (1986).13.7. A. Bush, R. Gibson, and G. Keogh, "The Limit of Elastic Deformation in the Contact

of Rough Surfaces," Mech. Res. Comm. 3, 169-174 (1976).13.8. P. Nayak, "Random Process Model of Rough Surfaces," Trans. ASME, J. Lub. Tech-

nology 93F, 398-407 (1971).13.9. R. Sayles and T.Thomas, "Thermal Conductances of a Rough Elastic Contact," Appl.

Energy 2, 249-267 (1976).13.10. T. Sasaki, H. Mori, and N. Okino, "Fluid Lubrication Theory of Roller Bearings Parts

I and II," ASME Trans., J. Basic Eng. 166,175 (1963).13.11. J. Bell, "Lubrication of Rolling Surfaces by a Ree-Eyring Fluid," ASLE Trans. 5,

160-171 (1963).13.12. F. Smith, "Rolling Contact Lubrication-The Application of Elastohydrodynamic

Theory," ASME Paper 64-Lubs-2 (April 1964).13.13. B. Gecim and W. Winer, "A Film Thickness Analysis for Line Contacts under Pure

Rolling Conditions with a Non-Newtonian Rheological Model," ASME Paper 80C2/LUB 26 (August 8, 1980).

REFERENCES 48113.14. L. Houpert, "New Results of Traction Force Calculations in EHD Contacts," ASME

Trans, J. Lub. Technology 107(2), 241 (1985).13.15. C. Evans and K. Johnson, "The Rheological Properties of EHD Lubricants," Proc.

Inst. Mech. Eng. 200(C5), 303-312 (1986).13.16. S. Bair and W.Winer, "ARheological Model for Elastohydrodynamic Contacts Based

on Primary Laboratory Data," ASME Trans., J. Lub. Tech. 101(3), 258-265 (1979).13.17. K. Johnson and R. Cameron, Proc. Inst. Mech. Eng. 182(1),307 (1967).13.18. D. Schipper, P. Vroegop, A. DeGee, and R. Bosma, "Micro-EHL in Lubricated Con-

centrated Contacts," ASME Trans., J. Tribology 112,392-397 (1990).13.19. E. Trachman and H. Cheng, "Thermal and Non-Newtonian Effects on Traction in

Elastohydrodynamic Contacts," Proc. Inst. Mech. Eng. 2nd Symposium on Elasto-hydrodynamic Lubrication, Leeds, 142-148 (1972).

13.20. J. Tevaarwerk and K. Johnson, "A Simple Non-Linear Constitutive Equation forEHD Oil Films," Wear 35,345-356 (1975).

13.21. T. Harris and R. Barnsby, "Tribological Performance Prediction of Aircraft TurbineMainshaft Ball Bearings," Tribology Trans. 41(1), 60-68 (1998).

FRICTION INROLLING BEARINGS

LIST OF SYMBOLS


a Semimajor axis of projectedcontact ellipse mm (in.)

b Semiminor axis of projectedcontact ellipse mm (in.)

CsBasic static capacity N (lb)

CvViscous drag coefficient

d Diameter mm (in.)

dm Pitch diameter mm (in.)

D Roller or ball diameter mm (in.)5, Complete elliptic integral of

second kindF,f Force, friction force N (lb)

Fe Centrifugal force N (lb)

g Gravitational constant mml sec2 (in.! sec2)

h Distance between center ofcontact ellipse and center ofspinning mm (in.)

483

484 FRICTION IN ROLLING BEARINGS


I Mass moment of inertia kg . mm2 (in.. Ib . sec2)

l Effective roller length mm (in.)M Moment N . mm (in. ·lb)Mg Gyroscopic moment N· mm (in. ·lb)Mf Bearing friction torque due to

flange load N . mm (in.. Ib)M1 Bearing friction torque due to

load N . mm (in. ·lb)My Bearing friction torque due to

lubricant N . mm (in. ·lb)m Mass kg (lb· sec2/in.)n Bearing rotational speed rpmQ Roller or ball load N (lb)q Load per unit length or x' /a N/mm (lb/in.)R Radius of curvature of contact

surface mm (in.)S Surface area mm2 (in.2)

t y'/bT Rolling line location on x' axis mm (in.)T Cage torque N· mm (in. ·lb)u Surface velocity mm/sec (in.lsec.)v Surface velocity mm/sec (in.lsec.)W Width of laminum mm (in.)WCR Width of cage rail mm (in.)C1~ Lubricant flow rate through

bearing cm3/min (gal/min.)x Distance in the x direction mm (in.)y Distance in the y direction mm (in.)z Distance in the z direction mm (in.)a Contact angle rad, 0

'Y D cos a/dmT/ Lubricant viscosity cp (lb . sec/in.2)

() Angle radK Ellipticity parameterJL Coefficient of frictionvo Kinematic viscosity centistokesp Radius mm (in.)g Lubricant effective density g/mm3 (lb/in.3)

gb Lubricant density g/mm3 (lb/in.3)

(J Normal stress N/mm2 (psi)T Shear stress N/mm2 (psi)4J Angle radIjJ Azimuth angle rad, 0

GENERAL485


w Rotational speed rad/ see0 Ring rotational speed rad/ see

SUBSCRIPTSCG Refers to cageCL Refers to cage landCP Refers to cage pocketCR Refers to cage raildrag Refers to viscous friction on cageg Refers to gyroscopic motioni Refers to inner racewayn Refers to outer or inner raceway, 0 or im Refers to orbital motion0 Refers to outer racewayR Refers to rolling motionS Refers to spinning motionv Refers to viscous friction on rolling elementx Refers to x directionx' Refers to x I directiony Refers to y directiony' Refers to y' directionz Refers to z directionz' Refers to z' directionA Refers to laminum

GENERAL

It is universally recognized that friction due to rolling of nonlubricatedsurfaces over each other is considerably less than the dry friction en-countered by sliding the identical surfaces over each other. Notwith-standing the motions of the contacting elements in rolling bearings aremore complex than is indicated by pure rolling, rolling bearings exhibitconsiderably less friction than most fluid film or sleeve bearings of com-parable size and load-carrying ability. A notable exception to the fore-going generalization is, of course, the hydrostatic gas bearing; however,such a bearing is not self-sustaining, as is a rolling bearing, and it re-quires a complex gas supply system.

Friction of any magnitude represents an energy loss and causes a re-tardation of motion. Hence friction in a rolling bearing is witnessed as atemperature increase and may be measured as a retarding torque.

The sources of friction in rolling bearings are manifold, the principalsources being as follows:


1. Elastic hysteresis in rolling2. Sliding in rolling element-raceway contacts due to a geometry of

contacting surfaces3. Sliding due to deformation of contacting elements4. Sliding between the cage and rolling elements and, for a land-

riding cage, sliding between the cage and bearing rings5. Viscous drag of the lubricant on the rolling elements and cage6. Sliding between roller ends and inner and/ or outer ring flanges7. Seal friction

These sources of friction are discussed in the following section.

SOURCES OF FRICTION

Elastic Hysteresis in Rolling

As a rolling element under compressive load travels over a raceway, thematerial in the forward portion of the contact surface, that is, in thedirection of rolling, will undergo a compression while the material in therear of the contact is being relieved of stress. It is recognized that asload is increasing, a given stress corresponds to a smaller deflection thanwhen load is decreasing (see Fig. 14.1). The area between the curves ofFig. 14.1 is called the hysteresis loop and represents an energy loss. (This

SOURCES OF FRICTION487

is readily determined if one substitutes force times a constant for stressand deformation times a constant for strain.) Generally, the energy lossor friction due to elastic hysteresis is small compared to other types offriction occurring in rolling bearings. Drutowski [14.1] verified this byexperimenting with balls rolling between flat plates. Coefficients of roll-ing friction as low as 0.0001 can be determined from the reference [14.1]data for 12.7 mm (0.5 in.) diameter chrome steel balls rolling on chromesteel plates under normal loads of about 356 N (80 lb.)

Drutowski [14.2] also demonstrated the apparent linear dependenceof rolling friction on the volume of significantly stressed material. In bothreferences [14.1] and [14.2]Drutowski further demonstrated the depend-ence of elastic hysteresis on the material under stress and on the specificload in the contact area.

Rolling and DeformationNominally, the balls or rollers in a rolling bearing are subjected to loadsperpendicular to the tangent plane at each contact surface. Because ofthese normal loads the rolling elements and raceways are deformed ateach contact, producing, according to Hertz, a radius off curvature of thecontacting surface equal to the harmonic mean of the radii of the con-tacting bodies. Hence for a roller of diameter D, bearing on a cylindricalraceway of diameter di, the radius of curvature of the contact surface is

Because of the deformation indicated above and because of the rollingmotion of the roller over the raceway, which requires a tangential forceto overcome rolling resistance, raceway material is squeezed up to forma bulge in the forward portion of the contact, as shown in Fig. 14.2. Asubsequent depression is formed in the rear of the contact area. Thus,


an additional tangential force is required to overcome the resisting forceof the bulge.

Sliding Friction in Rolling Element-Raceway Contacts

Macrosliding due to Rolling Motion. In Chapter 8, it was demonstratedthat sliding occurs in most ball and roller bearings simply due to themacro or basic internal geometry of the bearing. Theoretically, if a radialcylindrical roller bearing had rollers and raceways of exactly the samelength, if the rollers were very accurately guided by frictionless flanges,and if the bearing operated with zero misalignment, then sliding in theroller-raceway contacts would be avoided. In the practical situation,however, rollers and/or raceways are crowned to avoid "edge loading,"and under applied load the contact surface is curved in the plane passingthrough the bearing axis of rotation and the center of "rolling" contact.Since pure rolling is defined by instant centers at which no relative mo-tion of the contacting elements occurs, that is, the surfaces have the samevelocities at such points, then even in a radial cylindrical roller bearing,only two points of pure rolling can exist on the major axis of each contactsurface. At all other points, sliding must occur. In fact, the major sourceof friction in rolling bearings is sliding.

Most rolling bearings are lubricated by a viscous medium such as oil,provided either directly as a liquid or indirectly exuded by a grease. Somerolling bearings are lubricated by less viscous fluids and some by drylubricants such as molybdenum disulfide (MoS2). In the former cases, thecoefficient of sliding friction in the contact areas, that is, the ratio of theshear force caused by sliding to the normal force pressing the surfacestogether, is generally significantly lower than with "dry" film lubrication.

For oil and grease-lubricated bearings, it was shown in Chapter 13that the sliding friction, and hence traction, in a contact can be consid-ered as composed of three components: friction due to Newtonian fluidlubrication, friction due to a limiting shear condition, and Coulombfriction due to asperity-asperity interactions. When the film parameterA > 3, the Coulomb friction component virtually disappears since asper-ities do not contact.

Macrosliding Due to Gyroscopic Action. In Chapter 7, for angular-contact ball bearings, ball motions induced by gyroscopic moments werediscussed. This motion occasions pure sliding in directions collinear withthe major axes of the ball-raceway elliptical areas of contact. Jones [14.3]considered that gyroscopic motion can be prevented if the friction coef-ficient is sufficiently great; for example, as stated in Chapter 7, 0.06-0.07. In Chapter 12, however, it was demonstrated that for bearingsoperating in the full or even partial EHL regime, lubricant film thick-

SOURCES OF FRICTION 489

nesses are sufficient to cause substantial separation ofthe balls and race-ways, and sliding motions occur over the contacts in the rolling direction.In the presence of the separating lubricant film, therefore, the gyroscopicmoments are resisted by friction forces whose magnitudes depend on therates of shearing of the lubricant film in the direction of the gyroscopicmoments. Therefore, ball gyroscopic motion must also occur irrespectiveof the magnitude of the coefficient of friction. It is further probable thatgyroscopic motion also occurs in ball bearings operating with dry-filmlubrication.

Palmgren [14.4] called the gyroscopic motion creep and in experimentshe found that if the tangential force attitude was perpendicular to thedirection of rolling, the relationship of the angle f3 by which the motionof a ball deviates from the direction of rolling can be shown to be afunction of the ratio of the mean tangential stress to the mean normalstress. Figure 14.3 shows for lubricated surfaces that creep becomes in-finite as 27m/O"m approaches 0.08. Palmgren further deduced as a conse-quence of creep that a ball can never remain rolling between surfacesthat form an angle to each other, regardless of the minuteness of theangle. The ball, while rolling, always seeks surfaces that are parallel.

Microslip. Reynolds [14.5] first referred to microslip when in his exper-iments involving the rolling of an elastically stiff cylinder on rubber heobserved that since the rubber stretched in the contact zone, the cylinderrolled forward a distance less than its circumference in one completerevolution about its axis. The classical demonstration of the microslip orcreep phenomenon was developed in two dimensions by Poritsky [14.6].He considered the action of a locomotive driving wheel as shown in Fig.


14.4. The normal load between the cylinders was assumed to generate aparabolic stress distribution over the contact surface. Superimposed onthe Hertzian stress distribution was a tangential stress on the contactsurface, as shown in Fig. 14.4. Using this motion Poritsky demonstratedthe existence of a "locked" region over which no slip occurs and a slipregion of relative movement in a contact area over which it has beenhistorically assumed that only rolling had occurred. Cain [14.7] furtherdetermined that in rolling the "locked" region coincided with the leadingedge of the contact area, as shown in Fig. 14.5. In general, the "lockedregion" phenomenon can occur only when the friction coefficient is veryhigh as between unlubricated surfaces.

Heathcote "slip" is very similar to that which occurs because of rollingelement-raceway deformation. Heathcote [14.9] determined that a hardball "rolling" in a closely conforming groove can roll without sliding ontwo narrow bands only.Ultimately, Heathcote obtained a formula for the"rolling" friction in this situation. Heathcote's analysis takes no accountof the ability of the surfaces to elastically deform and accommodate thedifference in surface velocities by differential expansion. Johnson [14.8]expanded on the Heathcote analysis by slicing the elliptical contact areainto differential slabs of area, as shown in Fig. 14.6, and thereafter ap-plying the Poritsky analysis in two dimensions to each slab. Generally,Johnson's analysis using tangential elastic compliance demonstrates alower coefficient of friction than does the Heathcote analysis, which as-


sumes sliding rather than microslip. Figure 14.7 shows the "locked" andslip regions that obtain within the contact ellipse. Greenwood and Tabor[14.10] evaluated the rolling resistance due to elastic hysteresis. It is ofinterest to indicate that the frictional resistance due to elastic hysteresisas determined by Greenwood and Tabor is generally less than that dueto sliding if normal load is sufficiently large.

Viscous Drag

Owing to its orbital speed, each ball or roller must overcome a viscousdrag force imposed by the lubricant within the bearing cavity. It can beassumed that drag caused by a gaseous atmosphere is insignificant; how-ever, the lubricant viscous drag depends upon the quantity of the lubri-cant dispersed in the bearing cavity. Hence, the effective fluid within thecavity is a gas-lubricant mixture having an effective viscosity and aneffective specific gravity. The viscous drag force acting on a ball as in-dicated in [14.11] can be approximated by


Sliding between Rolling Elements and Cage Pockets

At any given azimuth location, there is generally a normal force actingbetween the rolling element and its cage pocket. This force can be posi-tive or negative depending upon whether the rolling element is drivingthe cage or vice versa. It is also possible for a rolling element to be freein the pocket with no normal force exerted; however, this situation willbe of less usual occurrence. Insofar as rotation of the rolling elementabout its own axes is concerned, the cage is stationary. Therefore, puresliding occurs between rolling elements and cage pockets. The amountof friction that occurs thereby depends on the rolling element-cage nor-mal loading, lubricant properties, rolling element speeds, and cagepocket geometry. The last variable is substantial in variety. Generally,application of simplified elastohydrodynamic theory should suffice to an-alyze the friction forces.

Sliding between Roller Ends and Ring Flanges

In a tapered roller bearing and in a spherical roller bearing having asym-metrical rollers, concentrated contacts always occur between the rollerends and the inner (or outer) ring flange owing to a force component thatdrives the rollers against the flange. Also, in a radial cylindrical rollerbearing, which can support thrust load in addition to the predominantradial load by virtue of having flanges on both inner and outer rings,sliding occurs simultaneously between the roller ends and both inner andouter rings. In these cases, the geometries of the flanges and roller endsare extremely influential in determining the sliding friction betweenthose contacting elements.

The most general case for roller end-flange contact occurs, as shownby Fig. 14.9, in a spherical roller thrust bearing. The different types ofcontact are illustrated in Table 14.1 for rollers having sphere ends.

Rydell [14.14] indicates that optimal frictional characteristics areachieved with point contacts between roller ends and flanges. Addition-ally Brown et al. [14.15] studied roller end wear criteria for high speedcylindrical roller bearings. They found that increasing roller corner ra-dius runout tends to increase wear. Increasing roller end clearance andliD ratio also tend toward increased roller wear, but, are of lesser con-sequence than roller corner radius runout.

Seals

An integral seal on a ball or roller bearing generally consists of an elas-tomer partially encased in a steel or plastic carrier. This is shown in Fig.1.16.

The elastomeric sealing element bears either on a ring "land" or on aspecial recess in a ring. In either case, the seal friction normally sub-stantially exceeds the sum total of all other sources of friction in thebearing unit. The technology of seal friction depends frequently on thespecific mechanical structure of the seal and on the elastomeric proper-ties. See Chapter 17 for some information on integral seals.


FRICTION FORCES AND MOMENTS IN ROLLINGELEMENT-RACEWAY CONTACTS

Ball Bearings

The sliding that occurs in the contact area has been discussed only qual-itatively insofar as determination of friction forces is concerned. Theanalysis performed in Chapter 9 to evaluate the normal load on each balland the contact angles took no account of friction forces in the contactother than to recognize the necessity to balance the gyroscopic momentswhich occur in angular-contact and thrust ball bearings. Of the manycomponents that constitute the frictional resistance to motion in a ball-raceway contact sliding is the most significant. It is further possible forthe purpose of analysis to utilize a coefficient of friction even though thelatter is a variable. Coefficient of friction in this section will be handledas a constant defined by

FIGURE 14.12. Centrifugal, normal and frictional forces acting on a ball. Note: Fyo andFyi act normal to the plane of the paper.

equations (14.7) and 14.8), respectively. From Fig. 14.12 it can be seenthat equation (14.20) becomes

t D(Fxi + Fxo) - M = 0 (14.21)

and

Fyi + Fyo = 0 (14.22)

Note also that equations (14.18) and (14.19) can be combined to yield

-Mro(sin ao + cos ao) + Mso(cos ao - sin ao)

+ MR;Csinai + cos a) - Msi(cosai - sin a) + Mz' = 0 (14.23)

Simplifying assumptions may be made at this point for relatively slowspeed bearings such that ball gyroscopic moment is negligible and thatouter raceway control is approximated. Although the latter is not nec-

FRICTION FORCES AND MOMENTS IN ROLLING ELEMENT-RACEWAY CONTACTS 501

essarily true of slow speed bearings, the result of calculations using theseassumptions will permit the investigator to obtain a qualitative idea ofthe sliding zones in the ball-raceway contacts and an order of magnitudeidea of friction in the contacts. Moreover, Qo> Qi, ao, and ai may be de-termined by methods of Chapters 7 or 9. Therefore, to calculate thefrictional forces and moments in the contact area, one needs only to de-termine the radii of rolling r ~ and r:.

In Chapter 8 it was demonstrated that pure rolling can occur at mostat two points in the contact area. If spinning is absent at a racewaycontact, then all points on lines parallel to the direction of rolling andpassing through the aforementioned points of pure rolling roll withoutsliding. The sliding velocities uyo or uyi are defined by equations (8.25)and (8.31), respectively; the distribution of sliding velocity on the contactsurface is illustrated by Fig. 14.13. As in Fig. 14.13 the lines of purerolling lie at x = ± ca. Then the frictional forces of sliding are distributedas in Fig. 14.14. Using equation (14.6) to describe the differential fric-tional force dF, it can be seen that the net sliding frictional force in thedirection of rolling at a raceway contact is


tical to that occurring under static loading. Because of the relatively slowspeeds of operation necessitated when contact angle differs from zerodegrees, gyroscopicmoments are negligible. In any event, gyroscopic mo-ments of any magnitude do not substantially alter normal motion of therolling elements. In this analysis therefore, the sliding on the contactsurface of a properly designed roller bearing will be assumed to be afunction only of the radius of the deformed contact surface in a directiontransverse to rolling.

To perform the analysis, it is assumed that the contact area betweenroller and either raceway is substantially rectangular and that the nor-mal stress at any distance from the center of the rectangle is adequatelydefined by

Equations (14.58) and (14.61) can be solved simultaneously for Co and Ci'

Note that if Ro and Ri' the radii of curvature of the outer and innercontact surfaces respectively are infinite, the foregoing analysis does notapply. In this case sliding on the contact surfaces is obviated and onlyrolling occurs.

Having determined Co and Ci, one may revert to equation (14.55) todetermine the net sliding forces Fyo and Fyi' Similarly, MRo and MRi maybe calculated from equation (14.57).

SKIDDING AND CAGE FORCES

In the analytical development regarding rolling element and cage speedsso far, at least one location could be found in each ofthe rolling element-raceway contact areas that was an instant center; that is, at that locationno relative motion (sliding) occurs between the contacting surfaces. If


during bearing operation, no instant center can be found in either theinner or outer raceway contacts, particularly at the azimuth location ofthe most heavily loaded rolling element, then skidding is said to occur.Skidding is gross sliding of a contact surface relative to the opposingsurface. Skidding results in surface shear stresses of significant magni-tudes in the contact areas. If the lubricant film generated by the relativemotion of the rolling element-raceway surfaces is insufficient to com-pletely separate the surfaces, surface damage called smearing will occur.An example of smearing is shown by Fig. 14.21. Tallian [14.17] definessmearing as a severe type of wear characterized by metal tightly bondedto the surface in locations into which it has been transferred from remotelocations of the same or opposing surfaces and the transferred metal ispresent in sufficient volume to connect more than one distinct asperitycontact. When the number of asperity contacts connected is small, it iscalled microsmearing. When the number of such contacts is large enoughto be seen with the unaided eye, this is called gross or macroscopicsmearing.

If possible, skidding is to be avoided in any application since at thevery least it results in increased friction and heat generation even ifsmearing does not occur. Skidding can occur in high speed operation ofoil-lubricated ball and roller bearings. Rolling element centrifugal forcesin such applications tend to cause higher normal load at the outer race-way-rolling element contact as compared to the inner raceway-rollingelement contact at any azimuth location. Therefore, the balance of thefriction forces and moments acting on a rolling element requires a highercoefficient of friction at the inner raceway contact to compensate for thelower normal contact load. It was shown in Chapter 12 that the lubricantfilm thickness generated in a fluid film-lubricated rolling element-raceway contact depends upon the velocities of the surfaces in contact.Moreover, considering as a simplistic case Newtonian lubrication, thesurface shear stress is a direct function of the sliding velocity of thesurfaces and an inverse function of the lubricant film thickness. Hence,considering equations (14.1) and (14.5), the coefficient of friction in thecontact is a function of sliding speed, which is greatest at the inner race-way contacts. Generally, skidding can be minimized by increasing theapplied load on the bearing, thus decreasing the relative magnitude ofthe rolling element centrifugal force to the contact load at the most heav-ily loaded rolling element. As will be seen in Chapter 18, this remedywill tend to reduce fatigue endurance. Therefore, a compromise betweenthe degree of skidding allowed and bearing endurance must be accepted.Of course, by making the contacting surfaces extremely smooth, the ef-fectiveness of the lubricant film thicknesses is improved, and skidding ismore tolerable.


Notwithstanding, skidding is generally a high speed phenomenoncaused by a difference between inner and outer raceway-rolling elementloading; it is also aggravated by any rolling element or cage loading thattends to retard motion. The most significant of such loadings is the vis-cous drag of the lubricant in the bearing cavity on the rolling elements.Therefore, a high speed bearing operating submerged in lubricant willskid more than the same bearing operating in mist-type lubrication. Inthis case another compromise is required because, in a high speed ap-plication, a copious supply of lubricant is generally used to carry awaythe frictional heat generated by the bearing. Rolling element-cage fric-tion and cage-bearing ring friction as well as cage-lubricant friction alsoaffect skidding.

Skidding in Ball Bearings

One of the most important applications with regard to skidding is themainshaft angular-contact ball bearing in aircraft gas turbines. Thisbearing is predominantly thrust loaded, and it is therefore only neces-sary to divide the thrust load uniformly among the bearing balls to de-termine the applied load.

The ball loading is shown by Fig. 14.22 for the coordinate system andball speeds of Fig. 5.4.

Mgy' = Jwmwy, (14.76)

Mgz' = Jwmwz' (14.77)

and J is the polar moment of inertia. Fv in equation (14.72) is determinedfrom equation (14.2). A ball-riding case with negligible friction in theball pockets is assumed. Since only a simple thrust load is assumed, cagespeed is identical to ball orbital speed Wm' The unknowns in equations(14.69)-(14.75) are inner and outer raceway-ball contact deformations,ball contact angles or position variables, bearing axial deflection, and ballspeeds, Wx" wy" Wz" and Wm' Hence, there are nine unknowns and sevenequations. The remaining two equations pertaining to ball position areobtained from Chapter 9. The solution of the equations requires the useof a computer. These equations were first solved by Harris [14.18] usingthe simplifying assumption of an isothermal Newtonian lubricant, ade-quately supplied to the ball-raceway contacts.

Figures 14.23 and 14.24 show the comparison of the analytical resultswith the experimental data of Shevchenko and Bolan [14.19] and Po-plawski and Mauriello [14.20]. Note the deviations from the outer race-way control approximation.

Parker [14.21] established an empirical formula to estimate the per-centage of the bearing "free space" occupied by fluid lubricant. UsingParker's formula it is possible to calculate the effective fluid density ~ inequation (14.3) and hence Fu in equation (14.72). The effective densityso determined is given by equation (14.78).

where FCL is given by equation (14.4).As in Chapter 7, normal loads Q n) can be written in terms of contact

deformations, and bearing radial deflection can be related to contact de-formations and radial clearance. Accordingly, equations (14.82), (14.83),(14.85), (14.86), and (14.87), a set of 3Z + 2 equations, can be solved forSr, Si)' Wm' wi' and QCG/ Reference [14.11] gives the general solution forall types of roller bearings (and ball bearings); that is, for five degrees offreedom in applied bearing loading, freedom for each roller (and ball) toorbit at a speed other than cage speed (wm) instead of wm), and any shapeof raceway and/ or roller.

Harris [14.22] using a simpler form of the analysis, considering onlyisothermal lubrication conditions and neglecting viscous drag on the roll-ers, nevertheless managed to demonstrate the adequacy of the analyticalmethod. Figure 14.26, taken from reference [14.22], compares analyticaldata against experimental data on cage speed vs applied load and speed.The analysis further indicated that skidding tends to decrease as appliedload is increased and is relatively insensitive to the type of lubricant.

Several aircraft engine manufacturers assemble their bearings in an"out-of-round" outer raceway to achieve the load distribution ofFig. 14.27as a means of minimizing skidding. This artificial loading of the bearingincreases the maximum roller load and doubles the number of the rollers80 loaded. Figure 14.28, taken from reference [14.22] illustrates the effecton skidding of an out-of-round outer raceway. Another method to mini-mize skidding is to use a few, for example, three, equally spaced hollowrollers that provide an interference fit with the raceways under zero ra-dial load and static conditions. Figure 14.29, from reference [14.23], il-lustrates such an assembly, while Fig. 14.30 indicates the effectivenessto minimize skidding.

Figure 14.31, taken from the commentary to reference [14.23], con-firms the adequacy of the analytical method by showing a high degreeof skidding for 2-4 90% hollow rollers tested in a 207 cylindrical rollerbearing.

CAGE MOTIONS AND FORCES

Influence of SpeedWith respect to rolling element bearing performance, cage design hasbecome more important as bearing rotational speeds increase. In instru-ment ball bearings undesirable torque variations have been traced tocage dynamic instabilities. In the development of solid-lubricated bear-ings for high-speed, high-temperature gas turbine engines, the cage is amajor concern.

A key to successful cage design is a detailed analysis of the forcesacting on the cage and the motions it undergoes. Both steady-state anddynamic formulations of varying complexity have been developed.

Forces Acting on the CageThe primary forces acting on the cage are due to the interactions betweenthe rolling element and cage pocket (Fcp) and the cage rail and the pi-loting land (FcL)' As Fig. 14.32 shows, a roller can contact the cage oneither side of the pocket, depending on whether the cage is driving theroller, or vice versa. The direction of the cage pocket friction force (Fcp)

depends on which side of the pocket contact occurs. For an inner landriding cage, a friction torque (T CL) in the direction of cage rotation de-velops at the cage-land contact. For an outer land riding cage, a frictiontorque tending to retard cage rotation develops at the cage-land contact.

A lubricant viscous drag force ({DRAG) develops on the cage surfacesresisting motion of the cage. Centrifugal body forces (shown as FCF) dueto cage rotation make the cage expand uniformly outward radially andinduce tensile hoop stresses in the cage rails. An unbalanced 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 [14.24]. The contact between the rollingelement and cage pocket can be hydrodynamic, elastohydrodynamic, orelastic in nature, depending on the proximity of the two bodies and themagnitude of the rolling element forces. In most cases the rollingelement-cage interaction forces are small enough that hydrodynamic lu-brication considerations prevail.

Steady-State Conditions

In a previous section it was demonstrated that analytical means exist topredict skidding in ball and roller bearings in any fluid-lubricated ap-plication. All of the calculations, even for the least complex application,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 outerraceway cylindrical roller bearing under radial load, Fig. 14.33, from ref-

Equations (14.88) and (14.89) represent equilibrium of cage forces inthe radial plane of motion. The summation of the cage pocket normalforces and friction forces equilibrate the cage-land normal force. Equa-tion (14.90) establishes torque equilibrium for the cage about its axis ofrotation. The cage pocket normal forces are assumed to react at the bear-ing pitch circle. The sign of the cage-land friction torque TCL depends onwhether the cage is inner ring land-riding or outer ring land-riding. Inthe formulation of [14.26] each roller is allowed to have different rota-tional and orbital speeds.

Dynamic Conditions

Rolling element bearing cages are subjected to transient motions andforces due to accelerations caused by contact with rolling elements, rings,and eccentric rotation. In some applications, notably with very highspeed or rapid acceleration, these transient cage effects may be of suffi-cient magnitude to warrant evaluation. The steady-state analytical ap-proaches discussed do not address the time-dependent behavior of rollingelement bearing cage. Several researchers have developed analyticalmodels for transient cage response [14.24, 14.27-1430]. Due to the com-plexity of the calculations involved, such performance analyses generallyrequire extensive time on present-day computers.

In general, the cage is treated as a rigid body subjected to a complexsystem of forces. These forces may include the following:

1. Impact and frictional forces at the cage-rolling element interface2. Normal and frictional forces at the cage-land surface (if land-

ITl1i(h~d cage)

CAGE MOTIONS AND FORCES 533

3. Cage mass unbalance force4. Gravitational force5. Cage inertial forces6. Others (i.e., lubricant drag on the cage and lubricant churning

forces)

Forces 1 and 2 are intermittent. For example the cage might or mightnot be in contact with a given rolling element or guide flange at a giventime, depending on the relative position of the bodies in question. Fric-tional forces can be modeled as hydrodynamic, EHL, or dry friction, de-pending on the nature of the lubricant, contact load, and geometry. Bothelastic and inelastic impact models appear in the literature. Generalequations of motion for the cage may be written. The Euler equationsdescribing cage rotation about its center of mass (in Cartesian coordi-nates) are as follows:

Ixwx - (ly - Iz) wywz = Mx (14.91)

Iywy - (lz - Ix) WzWx = My (14.92)

Izwz - (Ix - Iy) wxwy = Mz (14.93)

where Ix, Iy, Iz are the cage principal moments of inertia, and wx' wY' Wzare the angular velocities of the cage about the inertial x, y, z axes. Thetotal moment about each axis is denoted by Mx, My, and Mz, respectively.The equations of motion for translation of the cage center of mass in theinertial reference frame are

mrx = Fx (14.94)

mry = Fy (14.95)

mrz = Fz (14.96)

where m is cage mass, rx, ry, rz describe the position of the cage centerof mass, and Fx, Fy, Fz are the net force components acting on the cage.

Once cage force and moment components are determined, accelera-tions can be computed. Numerical integration ofthe equations of motion(with respect to discrete time increments) will yield cage translationalvelocity, rotational velocity, and displacement vectors. In some ap-proaches [14.24], [14.28] the cage dynamics model is solved in conjunc-tion with roller and ring equations of motion. Other researchers havedevised less cumbersome approaches by limiting the cage to in-planemotion [14.27] or by considering simplified dynamic models for the roll-ing elements [14.29].


Meeks and Ng [14.29] developed a cage dynamics model for ball bear-ings, which treats both ball- and ring land-guided cages. This model con-siders six cage degrees of freedom and inelastic contact between ballsand cage and between cage and rings. This model was used to performa cage design optimization study for a solid-lubricated, gas turbine en-gine bearing [14.30].

The results of the study indicated that ball-cage pocket forces andwear are significantly affected by the combination of cage-land and ball-pocket clearances. Using the analytical model to identify more suitableclearance values improved experimental cage performance. Figures 14.35and 14.36 contain typical output data from the cage dynamics analysis.

In Fig. 14.35 the cage center of mass motion is plotted versus time forX and Y (radial plane) directions. The time scale relates to approximatelyfive shaft revolutions at a shaft speed of 40,000 rpm. Figure 14.36 showsplots of ball-cage pocket normal force for two representative pockets po-sitioned approximately 90° apart.

In addition to the work of Meeks [14.30], Mauriello et al. [14.31] suc-ceeded in measuring ball-to-cage loading in a ball bearing subjected tocombined radial and thrust loading. They observed impact loading be-tween balls and cage to be a significant factor on high speed bearing cagedesign.

ROLLER SKEWING

Thus far in this section, rollers have been assumed to run "true" in cy-lindrical, spherical, and tapered roller bearings. In fact, due to slightlyimperfect geometry there is an inevitable tendency for unbalance of fric-tionalloading between the roller-inner raceway and roller-outer racewaycontacts, and thus a tendency for rollers to skew. Additionally, in a mis-aligned radial cylindrical roller bearing, as indicated schematically inFig. 7.23, rollers are "squeezed" at one end and thereby forced againstthe "guide" flange. The latter causes a roller end-flange frictional forceand hence a roller skewing moment that must be substantially resistedby the cage. In tapered roller bearings, even without misalignment, therollers are forced against the large end flange and skewing momentsoccur. The thrust load applied to radial cylindrical roller bearings, asdiscussed in Chapter 7, results in a roller skewing moment that is aug-mented by unbalance of raceway-roller friction forces, as indicated inFig. 14.37.

In most cases roller skewing is detrimental to roller bearing operationbecause it causes increased friction torque and frictional heat generationas well as necessitating a cage strong enough to resist the roller momentloading.

FIGURE 14.36. Calculated ball-pocket force vs time. (a) Prediction of cage ball-pocketforce vs time (pocket No.1). (b) Prediction of cage ball-pocket force vs time (pocket No.4)(from [14.30]).

Equilibrium Roller Skewing Angle

The notion that rollers skew until skewing moment equilibrium isachieved has implications beyond those of roller end-flange load deter-mination. In spherical roller bearings with symmetric roller profiles,proper management of roller skewing can reduce frictional losses and

FIGURE 14.37. Normal, axial, and frictional loading of a roller at azimuth IjIj; in a radialcylindrical roller bearing subjected to radial and thrust applied loading.

corresponding friction torque. Early spherical roller bearing designs em-ploying asymmetrical roller profiles, because of their close osculationsand primary skewing guidance from cage and flange contacts, exhibitgreater friction than current bearings with symmetrical roller designs.The temperature rise associated with friction limits performance inmany applications. Designing the bearings so that skewing equilibriumis provided by raceway guidance alone lowers losses and increases load-carrying capacity. Kellstrom [14.32, 14.33] investigated skewing equilib-rium in spherical roller bearings considering the complex changes inroller force and moment balance caused by roller tilting and skewing inthe presence of friction.

Any rolling element that contacts a raceway along a curved contactsurface will undergo sliding in the contact. For an unskewed roller therewill be at most two points along each contact where the sliding velocityis zero. These zero sliding points form the genera trices of a theoretical"rolling" cone, which represents the contact surface on which pure kin-ematic rolling would occur for a given roller orientation. At all otherpoints along the contact, sliding is present in the direction of rolling oropposite to it, depending on whether the roller radius is greater or lessthan the radius to the theoretical rolling cone. This situation is illus-trated in Fig. 14.38.

Friction forces or tractions due to sliding will be oriented to opposethe direction of sliding on the roller. In the absence of tangential rollerforces from cage or flange contacts, the roller-raceway traction forces in

FIGURE 14.38. Spherical roller bearing, symmetrical roller-tangential friction force di-rections. Motion and force direction: 0 out of page; • into page.

each contact must sum to zero. Additionally, the sum of the inner andouter raceway contact skewing moments must equal zero. These two con-ditions will determine the position ofthe rolling points along the contactsand thus the theoretical rolling cone. These conditions are met at theequilibrium skewing angle. If the moments tend to restore the roller tothe equilibrium skewing angle when it is disturbed, the equilibriumskewing angle is said to be stable.

As a roller skews relative to its contacting raceway a sliding compo-nent is generated in the roller axial direction and traction forces aredeveloped that oppose axial sliding. These traction forces may be bene-ficial in that, if suitably oriented, they help to carry the axial bearingload, as indicated in Fig. 14.39.

Those skewing angles that produce axial tractions opposing the ap-plied axial load and reducing the roller contact load required to react theapplied axial load are termed positive (Fig. 14.39a). Conversely, thoseskewing angles producing axial tractions that add to the applied axialload are termed negative (Fig. 14.39b). For a positive skewing roller thenormal contact loading is reduced, and an improvement in contact fa-tigue life achieved.

The axial traction forces acting on the roller also produce a secondeffect. These forces, acting in different directions on the inner and onouter ring contacts, create a moment about the roller and cause it to tilt.

FIGURE 14.39. Forces on outer raceway of axially loaded spherical roller bearing withpositive and negative skewing. (a) Positive skewing angle. (b) Negative skewing angle.

The tilting motion respositions the inner and outer ring contact loaddistributions with respect to the theoretical points of rolling and distri-bution of sliding velocity. Detailed evaluations [14.32, 14.33] of this be-havior have shown that skewing in excess of the equilibrium skewingangle generates a net skewing moment opposing the increasing skewingmotion. A roller that skews less than the equilibrium skewing angle willgenerate a net skewing moment tending to increase the skew angle. Thisset of interactions explains the existence of stable equilibrium skewingangles.

To apply this concept to the design of spherical roller bearings, specificdesign geometries over a wide range of operating conditions must beevaluated. There are tradeoffs involved between minimizing frictionlosses and maximizing contact fatigue life. Some designs may exhibitunstable skewing control in certain operating regimes or stable skewingequilibrium and require impractically large skewing angles. Computerprograms that predict spherical roller bearing performance contribute tomore accurate evaluations. See Fig. 14.40, which shows the possibletradeoffs between frictional power loss and calculated fatigue life for a

FIGURE 14.40. Study offrictional power loss vs calculated fatigue life of spherical rollerbearing with equilibrium skewing control. Q / P = ratio of bearing power loss to appliedload. 0

0= outer raceway osculation. 0, = inner raceway osculation.

bearing design using skewing control. Results are shown for several val-ues of outer and inner raceway osculation.

BEARING FRICTION TORQUE

Torque Due to Applied Load

Exclusive of an analytical approach to determine bearing friction torque,Palmgren [14.4] empirically evaluated bearing friction torque due to allmechanical friction phenomena with the exception of friction owing tothe quantity of lubricant contained within the bearing boundary dimen-sions; that is, within the bearing cavity. Data were compiled on eachbasic bearing type. Palmgren [14.4] gave the following equation to de-scribe this torque:

Ml = flF{3dm (14.97)

in which fl is a factor depending upon bearing design and relativebearing load. For ball bearings,

fl = z(F/Cs)Y (14.98)

BEARING FRICTION TORQUE 541

in which Fs is static equivalent load and Cs basic static load rating (theseterms are explained in Chapter 21 covering plastic deformation andstatic capacity). Table 14.3 gives appropriate values ofz andy. Values ofC

sare generally given in manufacturers' catalogs along with data to en-

able calculation of Fs. The internal designs of roller bearings havechanged both from macrogeometrical and microgeometrical bases sincethe publication by Palmgren [14.4]. Therefore, Table 14.4 as updatedaccording to data from [14.34]gives empirical values of fl for roller bear-ings. For modern design, double-row, radial spherical roller bearings,SKF [14.34] uses the formula:

Ml = flFadb (14.99)

in which constant fl and exponents a and b depend upon the specificbearing series. As the internal design ofthese bearings is specific to SKF,the catalog [14.34] should be consulted to obtain the required values offl, a, and b.


F f3 in equation (14.97) depends on the magnitude and direction of theapplied load. It may be expressed in equation form as follows for radialball bearings:

Ff3 = 0.9 Fa ctn a - O.lFr (14.100)

or

Ff3 = Fr (14.101)

Of equations [14.100], the one yielding the larger value of F f3 is used. Fordeep groove ball bearings, with nominal contact angle 00

, the first equa-tion can be approximated by

Ff3 = 3Fa - O.lFr (14.101)

For radial roller bearings,

Ff3 = 0.8Fa ctn a (14.102)

Ff3 = Fr

Again, the larger value of F f3 is used. For thrust bearings, either ball orroller, F f3 = Fa'

These values of torque as calculated from equation (14.97) appear tobe reasonably accurate for bearings operating under reasonable load andrelatively slow speed conditions. (Harris [14.35] used these data suc-cessfully in the thermal evaluation of a submarine propeller shaft thrustbearing assembly.)

Viscous Friction TorqueComplex methods for calculating viscous friction forces in lubricated roll-ing bearings were indicated in Chapter 12, which dealt with elasto-hydrodynamic lubrication. In lieu of those methods to estimate frictiontorque, a simpler, empirical method was developed to cover standardbearing types.

For bearings that operate at moderate speeds and under not-excessiveload, Palmgren [14.4] determined empirically that viscous friction torquecan be expressed as follows:

BEARING FRICTION TORQUE543

M = 1O-7f.(v n)2/3d3 van 2: 2000 (14.103)v 0 0 m

M = 160 X 1O-7f.d3 van ::; 2000 (14.104)v a ill

in which Va is given in centistokes and n in revolutions per minute. Inequations (14.103) and (14.104), fa is a factor depending upon type ofbearing and method oflubrication. Table 14.5 as updated in [14.34] givesvalues of fa for various types of bearings subjected to different conditionsof lubrication. Equations (14.103) and (14.104) are valid for oils havinga specific gravity of approximately 0.9. Palmgren [14.4] gave a more com-plete formula for oils of different densities. For grease-lubricated bear-ings, kinematic viscosity Va refers to the oil within the grease, and theequation is approximately valid shortly after the addition of lubricant.

Radial cylindrical roller bearings with flanges on both inner and outerrings can carry thrust load in addition to the normal radial load. In thiscase, the rollers are loaded against one flange on each ring. The bearingfriction torque due to the roller end motions against properly designedand manufactured flanges is given by

TABLE 14.5. Valuesof fo vs Bearing Type and LubricationType ofLubrication

Oil Bath(vertical shaft)

BearingType Grease Oil Mist Oil Bath or Oil Jet

Deepgrooveballa 0.7-2b 1 2 4Self-aligningballe 1.5-2b .0.7-1b 1.5-2b 3-4b

Thrust ball 5.5 0.8 1.5 3Angular-contactballa 2 1.7 3.3 6.6

Cylindricalrollerwith cagea 0.6-1b 1.5-2.8b 2.2-4b 2.2-4b,d

full complement 5-10b - 5-10b -Spherical rollere 3.5-7b 1.7-3.5b 3.5-7b 7-14b

Tapered rollera 6 3 6 8_lOb,d

Needle roller 12 6 12 24Thrust cylindricalroller 9 - 3.5 8

Thrust spherical roller - - 2.5-5b 5_lOb

Thrust needle roller 14 - 5 11cUse 2 x fo value for paired bearings or double row bearings.bLower values are for light series bearings; higher values are for heavy series bearings.'Double row bearings only.dFor oil bath lubrication and vertical shaft, use 2 X fo·


Mr = frFadm (14.105)

Values of fr are given in Table 14.6 when Fa/Fr :::;0.4 and the lubricantis sufficiently viscous.

Total Friction Torque

A reasonable estimate of the friction torque of a given rolling bearingunder moderate load and speed conditions is the sum of the load frictiontorque, viscous friction torque, and roller end-flange friction torque, ifany, that is,

M = Ml + Mv + Mr (14.106)

Since Ml and Mv are based on empirical formulas, the effect of rollingelement-cage pocket sliding friction is included.

For high speed ball bearings for which friction due to spinning motionsbecomes important, the equations previously given should enable a cal-culation of friction torque. This torque should be added to that of equa-tion (14.106). It must be remembered also that equation (14.106) doesnot account for friction torque due to seals, which in most instances farexceeds the friction torque of the bearing alone.

Example 14.2. Estimate the total friction torque for a 209 cylindri-cal roller bearing rotating at 10,000 rpm and supporting a radial loadof 4450 N (1000 lb). The bearing is lubricated by a mineral oil bath,the oil having a kinematic viscosity of 20 centistokes.

dm = 65 mm (2.559 in.) Ex. 2.7

D = 10 mm (0.3937 in.) Ex. 2.7

'Y= 0.1538 Ex. 2.7

Z = 14 Ex. 2.7

l = 9.6 mm (0.378 in.) Ex. 2.7

TABLE 14.6. Valuesof ff for Radial CylindricalRollerBearingsType ofLubrication

Bearing Type Grease Oil

With cage,optimumdesign 0.003 0.002With cage,other designs 0.009 0.006Full complement,single row 0.006 0.003Full complement,doublerow 0.015 0.009

BEARING FRICTION TORQUE 545

From Table 14.4, assume fl = 0.0003 for a medium series bearinghaving a cage.

Ml = flF{3dm (14.97)

= 0.0003 X 4450 X 65

= 86.78 N . mm (0.7677 in .. lb)

von = 20 X 10,000 = 200,000 (14.103)

Mv = 10-7 fo(von)2/3d::n

For oil bath lubrication from Table 14.5, assume fo = 3 for a mediumseries bearing,

Mv = 10-7 X 3 X (200,000)2/3 X (65)3

= 281.8 N . mm (2.493 in .. lb)

M = Ml + Mv + Mr (14.106)

= 86.8 + 281.8 + 0 = 368.6 N . mm (3.261 in .. lb)

Example 14.3. Estimate the rolling friction torque and viscous fric-tion torque ofthe 219 angular-contact ball bearing operating at a shaftspeed of 10,000 rpm and a thrust load of 22,250 N (5000 lb). Thebearing is jet lubricated by a highly refined mineral oil having a kin-ematic viscosity of 5 centistokes at operating temperature.

dm = 125.3 mm (4.932 in.) Ex. 2.6

D = 22.23 mm (0.875 in.) Ex. 2.3

a = 400 (nominal) Ex. 2.3

'Y= 0.1359 Ex. 2.6

f = 0.5232 Ex. 2.3

Z = 16 Ex. 2.5

Cs = 'PsiZD2 cos a (21.8)


From Table 21.2 at "Y= 0.1359, 'Ps = 15.48

Cs = 15.48 X 1 X 16 X (22.23)2cos 40°

= 93,760 N (21,070 Ib)

Fs = XsFr + YsFa (21.15)

From Table 21.2, Xs = 0.5; Ys = 0.26 for 0' = 40°,

Fs = 0.5 X 0 + 0.26 X 22,250

= 5785 N (1300 Ib)

fl = z(FJCs)Y (14.98)

From Table 14.3, z = 0.001; y = 0.33 for 0' = 40°

fl = 0.001 (5,785/93,760)°·33

= 0.0003988

Ff3 = 0.9Fa ctn 0'0 - O.lFr (14.100)

= 0.9 X 22,250 ctn 40° - 0.1 X 0

= 23,860 N (5363 Ib)

M1 = flFf3dm (14.97)

= 0.0003988 X 23,860 X 125.3

= 1192 N . mm (10.55 in .. Ib)

From Table 14.5, fo = 6.6 for oil jet lubrication

von = 5 X 10,000 = 50,000

Mv = 1O-Yo(von)2/3d'!n (14.103)

= 10-7 X 6.6 X (50,000)2/3X (125.3)3

= 1762 N . mm (15.59 in .. Ib)

M = M1 + Mv + Mf (14.106)

= 1192 + 1762 + 0

= 2954 N . mm (26.13 in. ·lb)

REFERENCES 547

CLOSURERolling bearings are sometimes called antifriction bearings to emphasizethe small amount of frictional power consumed during their operation.Notwithstanding, it has been shown in this chapter that the rolling pro-cess does involve frictional power losses from various sources. Recentbasic research had done much to define the mechanics of rolling friction,and for certain ideal conditions of rolling, estimates of rolling frictiontorque can be made. The operation of industrial rolling bearings thatemploy curved raceways, cages, and seals is, however, far from ideal inthat sources of frictional power loss other than rolling are present in thebearings. Therefore, although it is important to understand the mechan-ics of rolling friction, empirical data are usually required to define fric-tion torque of rolling bearing assemblies. These empirical data arepresented in the previous section.

Rolling bearing friction is manifested as temperature rises in the roll-ing bearing structure and lubricant unless effective heat removal meth-ods are employed or naturally occur. When excessive temperature leveloccurs, the rolling bearing steel suffers loss in its ability to resist rollingsurface fatigue and the lubricant undergoes deterioration such that it isineffective. Subsequently, rapid bearing failure may be anticipated. Bear-ing thermal analysis and methods of heat removal are discussed furtherin Chapter 15.

Rolling bearing friction also tends to retard motion. In sensitive con-trol systems such as those employing instrument bearings, torque dueto bearing friction can significantly affect rotor speed.

REFERENCES14.1. R. Drutowski, "Energy Losses of Balls Rolling on Plates," Friction and Wear, Elsev-

ier, Amsterdam, 16-35 (1959).14.2. R. Drutowski, "Linear Dependence of Rolling Friction on Stressed Volume," Rolling

Contact Phenomena, Elsevier, Amsterdam, (1962).14.3. A. Jones, "Motions in Loaded Rolling Element Bearings," ASME Trans., J. Basic

Eng., 1-12 (1959).14.4. A. Palmgren, Ball and Roller Bearing Engineering, 3rd ed., Burbank, Philadelphia,

34-41 (1959).14.5. O. Reynolds, Phi/os. Trans. Royal Soc. London, 166, 155 (1875).14.6. H. Poritsky, J. Appl. Mech., 72, 191 (1950).14.7. B. Cain, J. Appl. Mech., 72,465 (1950).14.8. K. Johnson, "Tangential Tractions and Micro-slip," Rolling Contact Phenomena, El-

sevier, Amsterdam, 6-28 (1962).


14.9. H. Heathcote, Proc. Inst. Automobile Eng. London, 15,569, (1921).14.10. J. Greenwood and D. Tabor, Proc. Phys. Soc. London, 71,989 (1958).14.11. T. Harris, Rolling Element Bearing Dynamics," Wear, 23, 311-337 (1973).14.12. V. Streeter, Fluid Mechanics, McGraw-Hill, New York, 313-314 (1951).14.13. E. Bisson and W. Anderson, Advanced Bearing Technology,NASA SP-38 (1964).14.14. B. Rydell, "New Spherical Roller Thrust Bearings, the E Design," Ball Bearing J.,

SKF, 202,1-7 (1980).14.15. P. Brown, L. Dobek, F. Hsing, and J. Miner, "Mainshaft High Speed Cylindrical

Roller Bearings for Gas Turbine Engines," U. S. Navy Contract NOOI40-76-C-0383,Interin Report FR-8615 (April 1977).

14.16. T. Harris, "Ball Motion in Thrust-Loaded, Angular-Contact Ball Bearings with Cou-lomb Friction," ASME Trans., J. Lub.Tech., 93, 32-38 (1971).

14.17. T.Tallian, G. Baile, H. Dalal, and O. Gustafson, Rolling Bearing Damage Atlas, SKFIndustries, Inc., King of Prussia, PA, 119-143 (1971).

14.18. T. Harris, "An Analytical Method to Predict Skidding in Thrust-Loaded, Angular-Contact Ball Bearings," ASME Trans., J. Lub.Tech., 93,17-24 (1971).

14.19. R. Shevchenko and P. Bolan, "Visual Study of Ball Motion in a High Speed ThrustBearing," SAE Paper No. 37 (January 14-18, 1957).

14.20. J. Poplawski and J. Mauriello, "Skidding in Lightly Loaded, High Speed, Ball ThrustBearings," ASME Paper 69-LUBS-20 (1969).

14.21. R. Parker, "Comparison of Predicted and Experimental Thermal Performance ofAngular-Contact Ball Bearings," NASA Tech. Paper 2275 (1984).

14.22. T. Harris, "An Analytical Method to Predict Skidding in High Speed Roller Bear-ings," ASLE Trans., 9, 229-241 (1966).

14.23. T. Harris and S. Aaronson, ''An Analytical Investigation of Skidding in a High Speed,Cylindrical Roller Bearing Having Circumferentially Spaced, Preloaded AnnularRollers," Lub. Eng., 30-34 (January 1968).

14.24. C. Walters, "The Dynamics of Ball Bearings," ASME Trans., J. Lub. Tech., Vol. 93,(1), 1-10 (January 1971).

14.25. F. Wellons and T. Harris, "Bearing Design Considerations," Interdisciplinary Ap-proach to the Lubrication of Concentrated Contacts, NASA SP-237, 529-549 (1970).

14.26. R. Kleckner and J. Pirvics, "High Speed Cylindrical Roller Bearing Analysis-SKFComputer Program CYBEAN, Vol. I: Analysis," SKF Report AL78P022, NASA Con-tract NAS3-20068 (July 1978).

14.27. J. Kannel and S. Bupara, "A Simplified Model of Cage Motion in Angular-ContactBearings Operating in the EHD Lubrication Regime," ASME Trans., J. Lub. Tech.,100, 395-403 (July 1978).

14.28. P. Gupta, "Dynamics of Rolling Element Bearings-Part I-IV Cylindrical RollerBearing Analysis," ASME Trans., J. Lub. Tech., 101, 293-326 (1979).

14.29. C. Meeks and K. Ng, "The Dynamics of Ball Separators in Ball Bearings-Part I:Analysis," ASLE Paper No. 84-AM-6C-2 (May 1984).

14.30. C. Meeks, "The Dynamics of Ball Separators in Ball Bearings-Part II: Results ofOptimization Study," ASLE Paper No. 84-AM-6C-3 (May 1984).

14.31. J. Mauriello, N. Lagasse, A. Jones, and W. Murray, "Rolling Element Bearing Re-tainer Analysis," U. S. Army AMRDL Technical Report 72-45 (November 1973).

14.32. M. Kellstrom, and E. Blomquist, "Roller Bearings Comprising Rollers with PositiveSkew Angle," U. S. Patent 3,990,753 (1979).

REFERENCES 54914.33. M. Kellstrom, "Rolling Contact Guidance of Rollers in Spherical Roller Bearings,"

ASME Paper 79-LUB-23 (1979).14.34. SKF, General Catalog 4000 US, 2nd ed. (1997).14.35. T. Harris, "Prediction of Temperature in a Rolling Contact Bearing Assembly," Lub.

Eng., 145-150 (April 1964).

ROLLING BEARINGTEMPERATURES

LIST OF SYMBOLS


c Specific heat W . secl g . °C (Btu/lb . OF)D Rolling element diameter mm (in.)~l Diameter m (ft)/::; Thermal emissivityF Temperature coefficient W . sec;oC (Btu;oF)g Acceleration due to gravity m/sec2 (in.lsec2)

Gr Grashof numberh Film coefficient of heat transfer W1m2• °C (Btu/hr . ft2 •

OF)H Heat flow W (Btu/hr)J Conversion factor, 103 N . mm =

1 W . seck Thermal conductivity W1m . °C (Btu/hr . ft . OF)f! Length of heat conduction path m (ft)M Friction torque N . mm (in. ·lb)n Rotational speed rpm

551

552 ROLLING BEARING TEMPERATURES


Pr Prandtl numberq Error functionRe Reynolds number~R Radius m (ft)S Area normal to heat flow m2 (ft2)T Temperature °C (OF)US Fluid velocity m/sec (ft/sec)v Velocity m/sec (ft/sec)w Weight flow rate g/sec (lb/sec)~li') Width m (ft)x Distance in x-direction m (ft)E ErrorTJ Absolute viscosity cp (lb . sec/ in.2)v Fluid kinematic viscosity m2/sec (ft2/sec)w Rotational velocity rad/ sec0, Rotational velocity rad/ sec

SUBSCRIPTSa Refers to air or ambient conditionc Refers to heat conductionf Refers to frictionJ Refers to rolling element position0 Refers to oilr Refers to heat radiations Refers to spinning motionv Refers to heat convection1 Refers to temperature node2 Refers to temperature node, and so on

GENERAL

The temperature level at which a rolling bearing operates is a functionof many variables among which the following are predominant:

1. bearing load2. bearing speed3. bearing friction torque4. lubricant type and viscosity5. bearing mounting and/or housing design6. environment of operation

HEAT GENERATION 553In the steady-state operation of a rolling bearing, as for any other

machine element, whatever heat is generated internally is dissipated.Therefore, the steady-state temperature level of one bearing system com-pared to another system using identical sizes and number of bearings isa measure of the relative ability of that system's efficiency of heat dis-sipation.

Of course, if the rate of heat dissipation is less than the heat gener-ation rate, then an unsteady state exists and the system temperaturewill rise until lubricant distress occurs, with ultimate bearing failure.The temperature level at which this occurs is determined largely by thetype of lubricant and the bearing material. This dissertation is limitedto the steady-state thermal operation of rolling bearings since this is acommon concern of bearing users regarding satisfactory operation.

Most rolling bearing applications perform at temperature levels thatare relatively cool and therefore do not require any special considerationregarding thermal adequacy. This is due to either one of the followingconditions:

1. The bearing heat generation rate is low because of light load and/or .relatively slow speed.

2. The ability to remove heat from the bearing is sufficient because oflocation of the bearing assembly in a moving air stream or becauseor adequate heat conduction through adjacent metal.

Some applications occur in certain adverse environmental conditionssuch that it is certain that external cooling is required. A rapid deter-mination of the bearing cooling requirements may then suffice to estab-lish the cooling capability that must be applied to the lubricating fluid.In other applications it is not obvious whether external cooling is re-quired, and it may be economically advantageous to establish analyti-cally the thermal conditions of bearing operation.

HEAT GENERATION

Although rolling bearings have been called antifriction bearings, never-theless, they exhibit a small amount of friction during rotation. Thisshould be evident since if friction were not present, the rolling elementswould slip on the rotating ring rather than roll.

Friction in a rolling bearing as in most other mechanisms representsa wasteful power dissipation manifested in the form of heat generation.This frictional power must be effectively removed or an unsatisfactorytemperature condition will obtain in the bearing.

ANALYSIS OF HEAT FLOW 561

Hr = hr8(T - Ta) (15.18)

in which

hr = 5.73 X 10-8 I:~(T + Ta)(T2 + T;) (15.19)

Equations (15.18) and (15.19) are useful for hand calculation in whichproblem T and Ta are not significantly different. Upon assuming a tem-perature T for the surface, the pseudofilm coefficient of radiation hr maybe calculated. Of course, if the final calculated value of T is significantlydifferent from that assumed, then the entire calculation must be re-peated. Actually, the same consideration is true for calculation of hy forthe oil film. Since ko and Vo are dependent upon temperature, the as-sumed temperature must be reasonably close to the final calculated tem-perature. How close is dictated by the actual variation of those propertieswith oil temperature.

ANALYSIS OF HEAT FLOW

Systems of Equations

Because of the discontinuities of the structures that comprise a rollingbearing assembly, classical methods of heat transfer analysis cannot beapplied to obtain a solution describing the system temperatures. By clas-sical methods is meant the description of the system in terms of differ-ential equations and the analytical solution of these equations. Instead,methods of finite difference as demonstrated by Dusinberre [15.5] mustbe applied to obtain a mathematical solution.

For finite difference methods applied to steady-state heat transfer, var-ious points or nodes are selected throughout the system to be analyzed.At each of these points, temperature is determined. In steady-state heattransfer, heat influx to any point equals heat effiux; therefore, the sumof all heat-flowing toward a temperature node is equal to zero. Figure15.2 is a heat flow diagram at a temperature node; demonstrating thatthe nodal temperature is affected by the temperatures of each of the fourindicated surrounding nodes. (Although the system depicted by Fig. 15.2shows only four surrounding nodes, this is purely by choice of grid andthe number of nodes may be greater or smaller.) Since the sum of theheat flows is zero, therefore

H1-O + H2-0 + H3-0 + H4-0 = 0 (15.20)

For this example, it is assumed that heat flow occurs only by conductionand that the grid is nonsymmetrical, making all areas 8 and lengths offlow path different. Furthermore, the material is assumed nonisotropic

van = 20 . 350 = 7000

My = 10-7 fa(van)2/3d'1r, = 10-7 • 7 . (7000)2/3(444.5)3 (14.103)

= 2.250 . 104 N . mm (199.1 in . lb)

M = M1

+ My = 5.499 . 104 + 2.250 . 104

= 7.749· 104 N . mm (685.6 in . lb)

Hf

= 1.047. 10-4 nM = 1.047· 10-4.350.7.749. 104 (15.1)

= 2840 W (9687 Btu/hr)

Since this problem is for illustrative purposes, it has been designed tobe as simple as possible such that all equations and methods of solutionmay be demonstrated. Therefore, the following conditions will be as-sumed:

1. Nine temperature nodes are sufficient to describe the system shownby Fig. 15.4.

2. The inside of the housing is coated with oil and may be describedby a single temperature.

3. The inner ring raceway may be described by a single temperature.4. The outer ring raceway may be described by a single temperature.

5. The housing is symmetrical about the shaft centerline and verticalsection A-A. Thus, heat transfer in the circumferential directionneed not be considered.

6. The sump oil may be considered at a single temperature.7. The shaft ends at the extremities of the housing are at ambient

temperature.

Considering the temperature nodes indicated in Fig. 15.4, the heattransfer system is that indicated by Table 15.1. Table 15.1 also indicateswhich equations are used to determine heat flow,film coefficient of heattransfer, and rate of heat generation. The heat flow areas and lengths of


flow path are obtained from the dimensions of Fig. 15.4, considering thelocation of each temperature node.

Based on Table 15.1 and Fig. 15.4, a set of nine simultaneous, nonlin-ear equations with unknown variables TCT9 can be developed. Since thissystem is nonlinear in temperature because of free convection and ra-diation from the housing to ambient, the Newton-Raphson method ofequations (15.29) will be used to obtain a solution. The final values oftemperature are shown in the proper location in Fig. 15.5.

The system chosen for evaluation was necessarily simple. A morerealistic system would consider variation of bearing temperature in acircumferential direction also. For this case, viscous torque may beconsidered constant with respect to angular position; however, loadtorque varies as the individual rolling element load on the stationaryring but may be considered invariant with respect to angular position onthe rotating ring. A three-dimensional analysis such as that indicated byload torque variation on the stationary ring should, however, show little

HIGH TEMPERATURE CONSIDERATIONS 569

variation in temperature around the bearing rings so that a two-dimensional system should suffice for most engineering applications. Ofcourse, if temperatures of structures surrounding or abutting the hous-ing are significantly different, then a three-dimensional study is re-quired. A three-dimensional study will require a computer.

It is not intended that the results of the foregoing method of analysiswill be of extreme accuracy, but only that accuracy will be sufficient todetermine the approximate thermal level of operation such that correc-tive measures may be taken in the event excessive steady-state operatingtemperatures are indicated. Moreover, in the event that cooling of theassembly is required, the same methods may be used to evaluate theadequacy of the cooling system.

Generally, the more temperature nodes selected or the finer the grid,the more accurate will be the analysis.

HIGH TEMPERATURE CONSIDERATIONS

Special Lubricants and Steels

Having established the operating temperatures in a rolling bearing as-sembly while using a conventional mineral oil lubricant and lubricationsystem, and having estimated that the bearing and/or lubricant temper-atures are excessive, it then becomes necessary to redesign the systemto either reduce the operating temperatures or make the assembly com-patible with the temperature level. Of the two alternatives, the formeris safest when considering prolonged duration of operation of the assem-bly; however, when shorter finite lubricant life and/or bearing life areacceptable, it may be expeditious and even economical to simply accom-modate the increased temperature level by using special lubricants and/or bearing steels. The later approach is effective when space and weightlimitations preclude the use of external cooling systems and is furthernecessitated in many applications in which the bearing is not the primesource of heat, such as in aircraft gas turbine engines.

Heat Removal

For situations in which the bearing is the prime source of heat and inwhich the ambient conditions surrounding the housing do not permit anadequate rate of heat removal, simply placing the housing in a movingair stream may be sufficient to reduce operating temperatures. For in-stance, placing the housing of Fig. 15.4 in a fanned air stream of 15.2m/sec (50 ft/sec) velocity will create a heat transfer coefficient of about2.386 X 10-5 W/mm2 . °C (4.2 Btu/hr . ft2 . OF)on the outside surface ofthe housing, which is approximately 4 times that for the natural con-


vection system and gives a maximum bearing temperature of 168°C(334°F) as opposed to 221°C (429°F) obtained for free convection. Figure15.6 shows the remaining system temperatures. Thus, in this case, thesystem temperatures can be significantly reduced if a fan is used to cir-culate air over the housing.

If a fan is used, increased heat transfer from the housing to the airstream may be effected by placing fins on the housing. This increasesthe effective area for heat transfer from the housing. Consider that theexternal area of the housing of Fig. 15.4 is doubled by use of fins. Usingthe film coefficient of 2.386 X 10-5 WImm2 • °C (4.2 Btu/hr . ft2

• OF)forthe moving air stream and 2 times the external housing area yields amaximum bearing temperature of 149°C(300°F). Figure 15.7 shows othersystem temperatures.

When the bearing is not the prime source of heat, cooling of the hous-ing will generally not suffice to maintain the bearing and lubricant cool.

FIGURE 15.7. Temperature distribution, air velocity 15.2 m/sec (50 ftlsec) finned hous-ing.

For example, consider a shaft temperature of 260°C (500°F) instead of48.9°C (120°F). With the aforementioned moving air system in operation,the maximum bearing temperature for the reference system is 196°C(385°F). (Figure 15.8 shows other system temperatures.) Thus, it is nec-essary to cool the lubricant and permit the lubricant to cool the bearing.The most effective way of accomplishing this is to pass the oil throughan external heat exchanger and direct jets of the cooled oil on the bear-ing. To save space when a supply of moving coolant is readily available,it may be possible to place heat exchanger coils directly in the sump. Thecooled lubricant is then circulated by bearing rotation. The latter methodis not quite as efficient thermally as jet cooling although bearing frictiontorque and heat generation may be less by not resorting to jet lubricationand the attendant churning of excess oil. The adequacy of either systemof cooling may be demonstrated approximately by assuming that oil tem-

FIGURE 15.8. Temperature distribution, outside air velocity 15.2 m/see (50 fUsee).

perature is maintained at an average of 60°C (140°F)with shaft temper-ature of 260°C(500°F)as above and ambient temperature of49°C(120°F)in quiescent air. Maximum bearing temperature is thereby suppressedto 93°C (200°F), which would appear to be a satisfactory operating level.(see Fig. 15.9 for other temperatures.)

It was intended to demonstrate in the foregoing discussion that it ispossible to estimate with a reasonable degree of accuracy the tempera-tures occurring in an oil-lubricated rolling bearing assembly. Further-more, if the bearing and oil temperatures so calculated are excessive, itis possible to determine the type and degree of coolingcapability requiredto maintain a satisfactory temperature level.

Several researchers have applied the foregoing methods to effectivelypredict temperatures in rolling bearing applications. Initially, Harris[15.8, 15.9] applied the method to relatively slow speed spherical roller

FIGURE 15.9. Temperature distribution, oil is cooled to 66°C (150°F) average tempera-ture.

bearings. Subsequently, these methods have been successfully applied toboth high speed ball and roller bearings [15.10-15.12].

Good agreement with experimentally measured temperatures hasbeen reported [15.14]using the steady-state temperature calculation op-tion of SHABERTH, a computer program to analyze the thermomechan-ical performance of shaft-rolling bearing systems. Figure 15.10 shows anodal network model and the associated heat flow paths for a 35-mm-bore ball bearing. Figure 15.11 shows the agreement achieved betweencalculated and experimentally measured temperatures. It must bepointed out, however, that construction of a thermal model that accu-rately models a bearing often requires a considerable amount of effortand heat transfer expertise.

(D)

FIGURE 15.10. Bearing system nodal network and heat flow paths for steady-state ther-mal analysis. (a) Metal, air, and lubricant temperature nodes: • metal or air node; 0 lu-bricant node; . ~ lubricant flow path. (b) Conduction and convection heat flow paths (from[15.14]).

HEAT TRANSFER IN A ROLLING-SLIDING CONTACT

As indicated previously, accurate calculation of lubricant film thicknessand traction in a rolling contact depends on the determination of lubri-cant viscosity at the appropriate temperatures. For lubricant film thick-ness, this means calculation of the lubricant temperature entering thecontact. For traction, this means calculation of the lubricant temperaturefor its duration in the contact. In Reference [15.14], the heat transfersystem illustrated by Fig. 15.12 was used. Designating subscript k torepresent the raceway and j the rolling element location, the followingheat flow equations describe the system.

In high speed bearing frictional performance analyses such as thoseindicated in Chapter 14, the rolling-sliding contact heat transfer anal-yses are performed thousands of times to achieve consistent solutions.The analyses are begun by assuming a set of system temperatures. Lu-bricant viscosities are then determined at these temperatures, and fric-tional heat generation rates are calculated. These are subsequently usedto recalculate temperatures and temperature-dependent parameters.

REFERENCES 577

The process is repeated until the calculated temperatures substantiallymatch the assumed temperatures. This method, while producing moreaccurate calculations for bearing heat generations and friction torques,requires rather sophisticated computer programs for its execution. Forslow speed bearing applications in which the bearing rings are rigidlysupported, the simpler calculations for bearing heat generations illus-trated by Examples 14.2 and 14.3 will usually suffice.

CLOSUREThe temperature level at which a rolling bearing operates dictates thetype and amount of lubricant required as well as the materials fromwhich the bearing components may be fabricated. In some applicationsthe environment in which the bearing operates establishes the temper-ature level whereas in other applications the bearing is the prime sourceof heat. In either case, depending on the bearing materials and the en-durance required of the bearing, it may be necessary to cool the bearingusing the lubricant as a coolant.

General rules cannot be formulated to determine the temperaturelevel for a given bearing operating under a given load at a given speed.The environment in which the bearing operates is generally different foreach specialized application. Using the friction torque formulas of Chap-ter 14 to establish the rate of bearing heat generation in conjunctionwith the heat transfer methods presented in this chapter, however, it ispossible to estimate the bearing system temperatures with an adequatedegree of accuracy.

REFERENCES15.1. E. Eckert, Introduction to the Transfer of Heat and Mass, McGraw-Hill, New York

(1950).15.2. M. Jakob and G. Hawkins, Elements of Heat Transfer and Insulation, 2nd ed., Wiley,

New York (1950).15.3. A. Palmgren, Ball and Roller Bearing Engineering, 3rd ed., Burbank, Philadelphia

(1959).15.4. F. Kreith, "Convection Heat Transfer in Rotating Systems," Adu. in Heat Transfer,

5, 129-251 (1968).15.5. G. Dusinberre, Numerical Methods in Heat Transfer, McGraw-Hill, New York (1949).

15.6. G. Korn and T. Korn, Mathematical Handbook for Scientists and Engineers,McGraw-Hill, New York (1961).

15.7. SKF, General Catalog 4000 US, 2nd ed., 49 (1997).

15.8. T. Harris, "Prediction of Temperature in a Rolling Bearing Assembly," Lubr. Eng.,145-150 (April 1964).


15.9. T. Harris, "How to Predict Temperature Increases in Rolling Bearings," Prod. Eng.,89-98 (Dec. 9, 1963).

15.10. J. Pirvics and R. Kleckner, "Prediction of Ball and Roller Bearing Thermal andKinematic Performance by Computer Analysis," in Adu. Power Transmission Tech.,NASA Conference Publication 2210, 185-201 (1982).

15.11. H. Coe, "Predicted and Experimental Performance of Large-Bore High Speed Balland Roller Bearings," in Adu. Power Transmission Tech., NASA Conference Publi-cation 2210, 203-220 (1982).

15.12. R. Kleckner and G. Dyba, "High Speed Spherical Roller Bearing Analysis and Com-parison with Experimental Performance," in Adu. Power Transmission Tech., NASAConference Publication 2210, 239-252 (1982).

15.13. W. Crecelius, "User's Manual for SKF Computer Program SHABERTH, SteadyState and Transient Thermal 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).

15.14. R. Parker, "Comparison of Predicted and Experimental Thermal Performance ofAngular-Contact Ball Bearings," NASA Tech. Paper 2275 (February 1984).

15.15. T. Harris and R. Barnsby, "Tribological Performance Prediction of Aircraft Gas Tur-bine Mainshaft Ball Bearings," Tribology Trans 41(1), 60-68 (1998).

BEARING STRUCTURALMATERIALS

GENERAL

The functional performance and endurance of a "dimensionally perfect"bearing with ideal internal geometries and surfaces, correct mounting,and preferential operating conditions is significantly influenced by thecharacteristics of its materials. Major criteria to be considered for sat-isfactory bearing performance include material selection and processingwith resultant physical properties. This chapter contains brief descrip-tions ofvarious bearing steel analyses, melting practices, manufacturingprocess variables, and the influence of these factors on the physical andmetallurgical and properties with respect to bearing performance. It alsocontains discussion concerning metallic and nonmetallic materials usedfor cages, seals, and shields.

ROLLING BEARING STEELS

Types of Steels for Rolling Components

Rolling bearing steels, from their inception, were selected on the basi~of hardenability, fatigue strength, wear resistance, and toughness. Amer·

57{

580 BEARING STRUCTURAL MATERIALS

ican Iron & Steel Institute (AIS!) 521000 steel, an alloy machinable inits annealed condition and exhibiting high hardness in the heat-treatedstate, was introduced around 1900 and is still the most-used steel forball bearing plus most roller bearing applications. For large bearingsizes, particularly with respect to cross-sectional thickness, modificationsto this basic analysis incorporating silicon, manganese, and molybdenumwere introduced. Carburizing steel came into being when the taperedroller bearing was introduced. Over the years more demanding productrequirements promoted the introduction of high speed steels and stain-less steels for high temperature operating conditions and corrosion re-sistance.

Through-Hardening Steels

The largest tonnage of bearing steels currently produced is the categoryof through-hardening steels. Table 16.1 lists common grade designationsand respective chemical compositions for this family of alloys.

Through-hardening steels are classified as hypereutectoid-type steelswhen containing greater than 0.8% carbon by weight and essentially con-taining less than 5% by weight of the total alloying elements. Assumingsatisfactory material availability, the bearing producer selects the ap-propriate grade of steel based on bearing size, geometry, dimensionalcharacteristics, specific product performance requirements, and manu-facturing methods and associated costs.

Case-Hardening Steels

Table 16.2 outlines the grade designations and corresponding chemicalcompositions of the common carburizing steels.

These steels are classified as hypoeutectoid steels; their carbon con-tents are generally below 0.80%. Carburizing steels are alloyed withnickel, chromium, molybdenum, and manganese to increase hardenabil-ity. The higher harden ability grades are used in applications requiringring components of heavier cross section. Carbon is diffused into the sur-face layer ofthe machined components to approximately 0.65-1.10% car-bon during the heat treatment operation to achieve surface hardnessescomparable to those attained with through-hardening grades of steel.

Steels for Special Bearings

Demands for good bearing performance under hostile operating condi-tions have consistently increased. The aerospace industry, in particular,requires products capable of operating at increasingly higher speeds,heavier. loads, and with high reliability. Other challenging applicationsinclude performing in corrosive atmospheres, at cryogenic temperatures,

ROLLING BEARING STEELS 581

TABLE 16.1. Chemical Composition of Through-Hardening Bearing Steels

Composition (%)

Gradea C Mn Si Cr Mo

ASTMb-A295 (52100) 0.98 0.25 0.15 1.30 -ISOc grade 1,683/XVII 1.10 0.45 0.35 1.60 0.10ASTM-A295 (51100) 0.98 0.25 0.15 0.90 -

DINd 105 Cr4 1.10 0.45 0.35 1.15 0.10ASTM-A295 (50100) 0.98 0.25 0.15 0.40 -

DIN 105 Cr2 1.10 0.45 0.35 0.60 0.10ASTM-A295 (5195) 0.90 0.75 0.15 0.70 -

1.03 1.00 0.35 0.90 0.10ASTM-A295 (K19526) 0.89 0.50 0.15 0.40 -

1.01 0.80 0.35 0.60 0.10ASTM-A295 (1570) 0.65 0.80 0.15 - -

0.75 1.10 0.35 - 0.10ASTM-A295 (5160) 0.56 0.75 0.15 0.70 -

0.64 1.00 0.35 0.90 0.10ASTM-A485 grade 1 0.95 0.95 0.45 0.90 -

ISO Grade 2,683/XVII 1.05 1.25 0.75 1.20 0.10ASTM-A485 grade 2 0.85 1.40 0.50 1.40 -

1.00 1.70 0.80 1.80 0.10ASTM-A485 grade 3 0.95 0.65 0.15 1.10 0.20

1.10 0.90 0.35 1.50 0.30ASTM-A485 grade 4 0.95 1.05 0.15 1.10 0.45

1.10 1.35 0.35 1.50 0.60DIN 100 Cr Mo 6 0.92 0.25 0.25 1.65 0.30ISO-grade 4,683/XVII 1.02 0.40 0.40 1.95 0.40

a Phosphorus and sulfur limitation for each alloy is 0.025% maximum (each element).b American Society for Testing Materials [16.1].cInternational Organization for Standards.

and in hard vacuum. To achieve satisfactory bearing operation in thecritical applications associated with these conditions, it is necessary tominimize adverse effect on fatigue life due to undesirable nonmetallicinclusions. This required substantial changes in steel-melting practice.Vacuum induction-melted, vacuum arc-remelted (VIMVAR)alloy steelssuch as M50 and BG42 were developed to provide the required highreliability in resistance to fatigue. M50NiL [16.3, 16.4]was developed asa high temperature operation, carburizing steel to provide through-cracking resistance for very high speed bearing inner rings. More re-cently, Cronidur 30 [16.5, 16.6], a steel rich in nitrogen and chromium,has been introduced to satisfy requirements for high temperature oper-ation with both corrosion resistance and high reliability fatigue resis-tance. Furthermore, in many aircraft power transmissions, to minimizeweight and space, bearing inner raceways are made integral with a one-


TABLE 16.2. ChemicalCompositionof Carburizing Bearing SteelsComposition(%)

Gradea C Mn Si Ni Cr Mo

SAEb 4118 0.18 0.70 0.15 - 0.40 0.080.23 0.90 0.35 - 0.60 0.15

SAE 8620, ISO 12 0.18 0.70 0.15 0.40 0.40 0.15DIN 20 NiCrMo2 0.23 0.90 0.35 0.70 0.60 0.25SAE 5120 0.17 0.70 0.15 - 0.70AFNORc18C3 0.22 0.90 0.35 - 0.90 -

SAE4720, ISO 13 0.17 0.50 0.15 0.90 0.35 0.150.22 0.70 0.35 1.20 0.55 0.25

SAE4620 0.17 0.45 0.15 1.65 - 0.200.22 0.65 0.35 2.00 - 0.30

SAE 4320, ISO 14 0.17 0.45 0.15 1.65 0.40 0.200.22 0.65 0.35 2.00 0.60 0.30

SAE E9310 0.08 0.45 0.15 3.00 1.00 0.080.13 0.65 0.35 3.50 1.40 0.15

SAEE33lO 0.08 0.45 0.15 3.25 1.40 -0.13 0.60 0.35 3.75 1.75

KRUPP 0.10 0.45 0.15 3.75 1.35 -0.15 0.65 0.35 4.25 1.75 -

"Grades are listed in ASTM A534 [16.2]; phosphorus and sulfur limitation for each alloyis 0.025% maximum (each element).b Society of Automotive Engineers.<French Standard.

piece gear-shaft component. Thus, the steel used must satisfy thefatigue-resistance requirements for the gear as well as the shaft andsupporting bearings. Pyrowear 675 carburizing steel is frequently em-ployed in such applications. Table 16.3 lists the chemical compositionsof several special bearing steels.

STEEL MANUFACTURE

Melting Methods

During the past 30 years high quality bearing steels have been meltedby the electric arc furnace melting process. During the oxidizing periodin the furnace cycle, impurities such as phosphorus and some sulfur areremoved from the steel. Further refining removes dissolved oxides andother impurities that might negatively affect the steel's performance. Un-fortunately, this furnace practice by itself does not remove undesirable

STEEL MANUFACTURE 583

TABLE 16.3. ChemicalCompositionof Special Bearing SteelsTypicalComposition(%)

Grade C Mn Si Cr Ni V Mo W N

M50 0.80 0.25 0.25 4.00 0.10 1.00 4.25 -

BG-42 1.15 0.50 0.30 14.50 - 1.20 4.00 -

440-C 1.10 1.00 1.00 17.00 -- - 0.75 -

CBS-60 0.20 0.60 1.00 1.45 - - 1.00 -

CBS-lOOO 0.15 3.00 0.50 1.05 3.00 0.35 4.50 -

VASCOX-2 0.22 0.30 0.90 5.00 - 0.45 1.40 -

M50-NiL 0.15 0.15 0.18 4.00 3.50 1.00 4.00 1.35Pyrowear 675 0.07 0.65 0.40 13.00 2.60 0.60 1.80 -

EX-53 0.10 0.37 0.98 1.05 - 0.12 0.94 2.13Cronidur 30 0.31 - 0.55 15.2 - - 1.02 - 0.38

Note: M50NiL and Pyrowear 675 are surface-hardening steels.

gases absorbed by the molten steel during melting and entrapped duringsolidification. Although vacuum ladle degassing was patented in 1943[16.7], it was not until the 1960s through the 1970s that cost-effectivevacuum degassing production facilities were used to principally removeoxygen and hydrogen and further improve material quality. These pro-cesses provide bearing steels with the good machinability, hardenability,and homogeneity required for manufacturing economy and good productperformance.

The advent of high-performance, aircraft gas turbine engines and thecorresponding demand for premium-quality bearing steels led to the de-velopment of sophisticated induction and consumable electrode vacuummelting techniques. Electroslag remelting capability was developed inconcert with these two vacuum processes. These special melting tech-niques have significantly influenced the development of higher-alloy toolsteels for elevated temperature and corrosion-resistant bearing applica-tions.

Raw MaterialsThe demand upon the steel industry to provide higher-quality alloy steelspromoted the development of the cold charge in the basic lined, electricarc furnace. These furnaces let high-alloy scrap and lower-grade alloyscrap be mixed with plain carbon scrap; economical operation and prod-uct quality are achieved through proper selection and weight control ofthese materials. Knowing the exact chemistry of the scrap charge re-duces the consumption of more costly alloy additions to the melt andminimizes introduction of undesirable "tramp" alloying elements. An


abundance of certain trace metallic elements, occurring unintentionallyin an AISI 52100 steel, has shown negative correlation on fatigue life inball bearings [16.8].

The furnace melter selects scrap that, when melted, contains a lowerpercentage of the specified alloying elements for the specific grade anal-ysis and permits final adjustment with selected alloying and carbon ad-ditions. The furnace charge must not contain elements that are not tobe part of the final heat chemistry. Raw material selection for a furnacecharge is also judged by physical size and weight. Light scrap takes upfurnace volume. Heavy scrap reduces protection to furnace walls and roofduring meltdown. A proper balance and distribution of scrap must bemaintained to enable proper thermal and electrical distribution duringthe initial stage of meltdown. Variation from these parameters can pro-duce "off-analysis" heat. Careful attention must also be given to slag-making materials, such as lime, silica, and fluorspar, to assure consistentquality and slag performance.

Basic Electric Furnace Process

The basic electric arc furnace (Fig. 16.1) [16.9] is circular, lined withheat-resistant brick, and contains three electrodes in a removable roof.The charge is blended to provide efficient melting, the electrodes arelowered, and arcing begins. A layer of complex slag is produced thatcovers the molten surface layer and absorbs impurities from the steel.During this oxidizing period a carbon boil occurs, which produces gases


from the molten bath. This "complex" slag is then replaced with a "re-ducing" slag to decrease oxygen levels.

Because the molten metal in the refining cycle is less active than inthe oxidizing cycle, furnaces may be equipped with inductive stirrers.These stirrers generate a magnetic field that imparts a circulatory mo-tion to the molten bath, enhancing both temperature control and ho-mogeneity of chemical composition throughout the melt. When thechemical composition of the molten steel has been adjusted to the desiredrange and the proper pouring temperature has been reached, the "heat"is "tapped" or removed from the furnace. Material not to be vacuumtreated is then ready for "teeming" into ingot molds.

Vacuum Degassing of Steel

Material quality produced in the basic electric arc furnace process can beimproved through vacuum degassing by various methods, including theladle, stream, D-H (Dortmund-Harder), and R-H (Ruhrstahl-Heraeus)processes. In conjunction with these refining practices, inert gas shroud-ing may be incorporated with both bottom pouring and uphill teeming.These methods economically reduce undesirable gases and remove non-metallic inclusions from the molten steel.

Ladle Degassing. The ladle degassing process requires a ladle ofmoltenmetal to be placed in a sealed, evacuated chamber. Gases resulting frompressure differentials cause turbulence and a moderate stirring action inthe molten bath. Ladle degassing units may be equipped with inductionstirrers and/or injection devices for bubbling inert gases such as argonor helium through the molten metal to further agitate the bath. See Fig.16.2 [16.10]. With this technique the quantity of metal exposed to thevacuum is greatly increased, which allows small alloy additions to bemade to the melt.

Stream Degassing. The molten metal in stream degassing is pouredfrom an intermediate or "pony" ladle, as illustrated in Fig. 16.3 [16.11],into the vacuum chamber containing an empty ladle. As the moltenmetal enters the vacuum chamber, the stream explodes into fine sprayor droplets that fall into a second, or teeming, ladle. When this ladle isfull, the vacuum seal is broken and the molten metal is poured into theingot molds.

D-H (Dortmund-Horder) Process. The D-H or Dortmund-Harder pro-cess is a cyclic type of vacuum operation. This practice necessitates po-sitioning the furnace ladle under a refractory-lined vessel, which movesalternately up and down. As the vacuum vessel descends into the moltenmetal, the steel enters the nozzle, or snorkel, is lifted up into the vacuum


chamber, and is agitated vigorously as degassing begins. As the degassedsteel flows back into the ladle, it is mixed with the remaining steel andthe cycle is repeated. A total cycle requires approximately 20 sec and isrepeated 40 to 60 times during the entire degassing operation. Heatlosses occurring during the cycle are compensated by a graphite, electri-cally resistant, heating element positioned in the upper part of the vac-uum chamber. Vacuum-sealed hoppers allow alloys to be added duringthe operation. When the vacuum degassing cycle is completed, the vac-uum vessel is purged with an inert gas so that the accumulated flam-mable gas in the vessel will not be ignited before the nozzle is raisedabove the surface level of the liquid steel and the ladle is removed.

R-H (Ruhrstahl-Heraeus) Process. The R-H (Ruhrstahl-Heraeus) vac-uum chamber (Fig. 16.4) [16.12] straddles the ladle containing the mol-


ten metal. Two vertical legs extending downward from the base of thevacuum chamber are submerged just below the surface level of the metalin the holding ladle. The vacuum chamber is evacuated, and a pressureimbalance is created by flowing inert gas into one leg extension. Theresulting pumping action circulates the molten metal between the hold-ing ladle and the vacuum chamber. Outgassing of the molten metal oc-curs as it enters the vacuum chamber. The circulation process continuesuntil the specified gas content is obtained.

Ladle Furnace

Ladle metallurgy of molten steel is conducted in a ladle furnace outsidethe normal constraints of the initial electric arc melting furnace. Theladle furnace is equipped with independent electrodes for both temper-ature control and electromagnetic stirrers for bath circulation. Thereforethere is no need to superheat steel in the electric arc furnace to compen-sate for subsequent temperature drops experienced in standard ladle de-gassing practices as previously described.

Lance injection permits powdered alloys to be inserted deep into theladle. Argon is used as a carrier gas for these powders; the resultingbubbling action helps disperse particles uniformly throughout the moltenbath. Lance injection combined with wire feeding provides inclusionshape control, reduction of sulfur content, and improvement of fluidity,chemical homogeneity, and overall microcleanliness.

Ladle furnace technology permits very rapid meltdown of scrap in theelectric arc furnace and improved refining capability in a subsequentladle furnace operation. This system generates improved product qualitywith a correspondingly improved economy in steel melting.

M-R Process

Consistent demands for cleaner bearing steels have resulted in the melt-ing and refining process, termed the M-R process. This method employsa twin-shell, electric arc melting furnace in parallel with a SKF-ASEAladle-refining unit.* The twin-shell furnace has two vessels with oxy-fuelburners and two roofs-that is, one contains graphite electrodes, and theother has no electrodes. Melting occurs in one furnace while chargingand preheating of scrap can be carried out in the other shell. In themelting furnace carbon and phosphorus contents are adjusted to valuesbelow the final maximum limits. The furnace is then tapped, and theladle of molten metal is transferred to the ASEA ladle-refining furnacecontaining an independent electrode roof. The equipment provides manymetallurgical options, including vacuum degassing, desulfurizing, deox-

*This method was developed by SKF Steel AB in conjunction with ASEA in Sweden.


idizing, and adjusting the chemical composition of the molten steel. Theadditional ability to induction stir under conditions of close temperaturecontrol permits precipitation deoxidation with aluminum, resulting insteels with very low contents of oxygen and nonmetallic inclusions. Thesequence of steps in the melting and refining process is illustrated inFig. 16.5 [16.13].

Methods for Producing Ultrahigh-Purity Steel

Vacuum Induction Melting. Yet more sophisticated steel melting pro-cesses were introduced during the 1960s for producing ultrahigh-puritysteels, also called "clean" bearing steels, which are essentially free fromdeleterious nonmetallic-type inclusions.

In vacuum induction melting, selected scrap material containing fewimpurities and comparable in chemical composition to the alloy gradebeing melted is charged into a small electrical induction furnace. The


furnace (Fig. 16.6) [16.14] is encapsulated in a large vacuum. chambercontaining sealed hoppers strategically located for adding required al-loys.

Outgassing of the melt occurs early in a very rapid meltdown andrefining period. After the melting cycle, furnace tilting and pouring themolten metal into ingot molds take place. The molds are automaticallymanipulated into and out of the pouring position while still within thevacuum-sealed chamber. This vacuum induction melting furnace processwas among the first vacuum processing methods employed in manufac-turing premium aircraft quality bearing steels. One of its primary func-tions today is to provide electrodes used to produce ultrahigh-purity,vacuum arc remelted steels.

Vacuum Arc Remelting. Vacuum technology for bearing steel alloy pro-duction, as described in the foregoing sections on vacuum degassing andinduction vacuum melting, provided a way to reduce the gas content andnonmetallic inclusions in steel. Steel electrodes, melted in furnaces usingvacuum technology can be remelted by still more sophisticated tech-niques, such as the consumable electrode vacuum melt practice, to pro-vide material for bearings requiring the utmost reliability. This process,illustrated in Fig. 16.7, involves inserting an electrode of the desiredchemical composition into a water-cooled, copper mold in which a vacuumis created.

An electrical arc is struck between the bottom face of the electrodeand a base plate of the same alloy composition. As the electrode is con-sumed under extremely high vacuum conditions, it is automatically low-ered and the voltage is controlled to maintain constant meltingparameters. Because the solidification pattern is controlled, the remeltedproduct is essentially free from center porosity and ingot segregation.The product has improved mechanical properties, particularly in thetransverse direction. Aircraft bearing material specifications for criticalapplications specify the VIM-VARsteel melting practice.

Electroslag Refining. The electroslag refining process (ESR) is very sim-ilar to that of the consumable electrode vacuum melting process exceptthat a liquid slag bath positioned at the base of the electrode providesthe electrical resistance required for melting. The slag bath is eitherintroduced into the furnace chamber in the molten state or provided asa powder slag that will quickly melt upon striking an arc between theelectrode and base plate. Refinement of the molten steel occurs as steeldroplets pass through the slag bath. Control of slag composition permitsremoval of sulfur and oxygen and undesirable impurities. The resultingingot solidification pattern reduces porosity, minimizes segregation, andprovides for improved physical properties in the transverse and longi-tudinal directions.


Steel Products

Product Forms. Rolling bearings cover a broad spectrum of sizes withring components varying in both cross-sectional configurations and ma-terial grades. Balls or rollers vary in size and shape to accommodatetheir mating ring components. Generally, these components are manu-factured from forgings, tubing, bar stock, or wire. The bearing manufac-turer orders from the steel producer the form and condition of the rawmaterial best suited for the selected method of processing and that willmeet bearing performance plus customer requirements.

Regardless of the melting practice employed, the resulting ingots arestripped from the molds, homogenized in soaking pits, rolled into bloomsor billets, and subsequently conditioned to remove surface defects. Billetsare reheated and hot-worked into bars, tubes, forgings, or rolled rings.Additional cold-working operations convert the hot-rolled tubes and barsinto cold-reduced tubes, bars, and cold-drawn wire. As an ingot passesback and forth between the rolls on a blooming mill, its cast structure isbroken up and refined. Continued hot-working operations elongate andbreak up nonmetallic inclusions and alloy segregation. Hot mechanicalworking operations also permit plastic deformation of the material intodesired shape or form. The subsequent cold-working of material resultsin induced stresses and improved machinability. Cold-working also pro-duces changes in the mechanical properties, improved surface finish, andcloser tolerances. Dimensional tolerances and eccentricities of cold-worked tubes are far superior to those of hot-rolled tubes.

Most mill product forms require thermal treatment at the mill beforeor after final finishing so that the forms are ready for machining or form-ing. The thermal treatment may include annealing, normalizing, orstress relieving. The product is then straightened, if necessary, inspected,and readied for transport.

Product Inspection. The steel producer performs two major functions tosatisfy customer product quality requirements. First, an inspection planis implemented to ensure that during the various manufacturing stagesthe product meets the specified quality limits. Second, the product istested in its final form to ensure that it conforms to predetermined in-ternal and external standards. Nondestructive testing methods, includ-ing hot acid etching, magnetic particle, eddy current, ultrasonic, andhydrostatic pressure tests, in conjunction with standard dimensionaltesting of the product are successfully used in statistical process control(SPC) and final inspection programs.

Numerous industrial, military, aerospace, or other independent cus-tomer specifications usually form part of the purchaser's requirement forbearing quality steels. These standards necessitate testing to satisfy heatcharacteristics, including chemical analyses, hardenability, macrotech


and microinclusion ratings. Special product requirements include surfacedefects, microstructure, hardness, and dimensional tests.

Steel Metallurgical Characteristics

Quality Requirements. Cleanliness, segregation, and microstructure aresteel product characteristics that influence bearing manufacture andsubsequent product performance. The cleanliness of steel pertains tononmetallic inclusions entrapped during ingot solidification that cannotbe subsequently removed. Segregation pertains to an undesirable non-uniform distribution of alloying elements. Microstructure of the millproduct relates directly to its suitability for machining and/or formingby the bearing manufacturer. The producer therefore incorporates testsearly in the manufacture of the product and before shipment to be surethat pertinent mill and customer quality requirements are met.

Cleanliness. Steel quality with respect to nonmetallic inclusions de-pends on the initial raw material charge, selection ofthe melting furnacetype/practice, and control of the entire process, including teeming andconditioning of the ingot molds. Exogenous inclusions result from erosionor breakdown of furnace refractory material or from other dirt particlesoutside the melt that are entrapped during tapping and teeming. Indig-enous inclusions are products of deoxidation occurring within the melt.Inclusions less than 0.5 mm (0.020 in.) long are considered microinclu-sions; larger ones are considered macroinclusions. Nonmetallic inclu-sions are classified by their composition and morphology as sulfides,aluminates, silicates, and oxides. Occasionally, nitrides are included inrating steel cleanliness.

It is well establishes through testing that nonmetallic inclusions aredetrimental to rolling contact fatigue life. The data further indicate thathard, brittle-type inclusions, based on their size, shape, and distribution,are more detrimental than soft deformable-type particles such as sul-fides. By encapsulating harder, nonmetallic particles and forming a co-coonor cushion around them so that they do not become points of stressconcentration under cyclicloading, sulfides act beneficially. Sulfide inclu-sions can also enhance machinability by performing as a lubricant. Be-cause aluminates and silicates have sharp corners and are very brittle,they can act as stress raisers and initiate early fatigue failures. Globularoxides are inclusions formed with such elements as calcium. These par-ticles are very hard and brittle and are considered to be most detrimentalto machinability and rolling contact fatigue life.

Since approximately 1960, consistent efforts have been conducted toimprove the quality of the bearing steels, particularly improvement ofcleanliness. As indicated by Figs. 16.8 and 16.9, it was not uncommon


for AISI 52100 steel to have high oxygen content; for example, 35 partsper million (ppm) and substantial amounts of macroinclusions. Duringthe 1970s, the acid open hearth furnace was introduced, which improvedcleanliness; that is, oxygen content in the form of oxides was reduced to20 ppm, and macroinclusions over a period of time were reduced to vir-tually nil. By 1982, the M-R process as shown by Fig. 16.8, and subse-quently similar vacuum degassing processes, lowered the oxygen contentto 10 ppm while maintaining a low level of macroinclusions. As shownby Fig. 16.10, the decrease of oxygen content from 35 to 10 ppm affordedat least a tenfold improvement in bearing fatigue life.

Many methods have been devised for detecting or quantitatively de-fining nonmetallic inclusions. One of the most popular involves micro-scopic examination at 100x magnification of polished specimens of apredetermined size from specific ingot locations and comparing the worstfield observed against standard photographs weighted on a numberedrating system [16.12]. Material specifications containing this method ofevaluation stipulate the testing frequency and the acceptance or rejectionlimits. Industry standards also cite product acceptance tests such as frac-ture, magnaflux stepdown (AMS 2300/AMS 2301), and visual rating-counting indications on machined bar surfaces to determine materialquality with respect to nonmetallic inclusion content.

Premium-quality aircraft bearing steel with respect to detection ofnonmetallic inclusions is automatically checked on a 100% bar productbasis by employing both eddy current and ultrasonic test procedures.

Segregation. Nonuniform solidification rate of molten metal within aningot mold might lead to segregation of alloying constituents. Because ofthe rapid freezing rate occurring in the mold-ingot interface, this out-


ermost portion or shell will solidify first and form columnar crystals. Atthe centermost portion of the ingot, which cools at a much slower rate,the grains are equiaxed. Because solidification is not spontaneousthroughout the ingot, the molten metal that freezes last will also becomericher in alloying elements such as manganese, phosphorus, and sulfur,because the elements have inherently lower melting points.

Segregation is only slightly improved by thermal treatment and hot-working. Precautions must be taken in the melting and pouring prac-tices, such as incorporating special molds designed to control rates offreezing and thereby prevent or minimize segregation during solidifica-tion.

Macroetching of properly prepared billet or bar slices, employing di-lute, hot hydrochloric acid solution, which preferentially attacks thesenumerous alloying constituents, is used to reveal material segregation.

Structure. Both macroscopic and microscopic methods of inspection areused to evaluate steel structure. Discs cut from the ends of bars or billetsfor macroscopic examination are prepared according to industry stan-dards, acid etched, and examined with the unaided eye or under mag-nification generally not exceeding lOx. Although numerous etchingreagents are available, the generally recommended and accepted solutionis hot dilute, hydrochloric (muriatic) acid.

In addition to the detection of alloy segregates, material may be mi-crostructurally evaluated for other objectionable characteristics such as"pipe," porosity, blow holes, decarburization, excessive inclusions, cracks,and banding. The aerospace industry uses hot acid etching and forginginspection to ensure conformance of grain flow patterns to a previouslytype-tested product.

Microscopic examination of steels involves a more detailed study ofthe structure at magnifications generally between lOOx and 1000x. Nu-merous reagents are available to help identify and rate specific micro-constituents. The steel producers' manufacturing processes incorporatingthermal cycles influence the resulting microstructure of the finished millproduct. Because microstructures are a reflection of material physicalproperties, microstructural ratings are contained within specification orpurchase order requirements. Bars and tubes fed into automatic singleor multiple spindle machines for turning into ring components must bein a soft annealed condition. Carbides should be uniform in size and welldistributed throughout a ferrite matrix. Tool life may be expected to in-crease as the sizes of the "spheroidal" carbides increase. Conversely, pres-ence of "lamellar" carbide will adversely affect machinability and toollife.

When choosing the optimum microstructure for maximum machina-bility oflow-carbon, carburizing steel, a "blocky" microstructure offerriteand pearlite, should be selected. A very soft annealed structure in these

HEAT TREATMENT OF STEEL 597

low-carbon grades, such as AISI 8620, is appropriate for cold-formingoperations but is considered gummy and unsatisfactory in machiningoperations. Each material grade in its finished mill product form mustexhibit the proper microstructure and hardness so that it can be econom-ically converted to its designated configuration.

EFFECTS OF PROCESSING METHODS ON STEELCOMPONENTS

Many of the mechanical properties of finished bearing components aredeveloped by manufacturing methods that dictate the form and conditionfor raw material. Generally, raw material is produced by either hot- orcold-reduction processes and furnished as tubing, bars, wire, and forg-ings. Cold-reduction for producing bars, tubing, balls, and rollers ofAISI52100 will lower both the austenite transformation temperature duringheating for hardening and the martensite start temperature upon cooling[16.17]. The resulting fracture grain size of a ring produced from a cold-reduced tube will be finer than that of an identical ring from a hot-rolledannealed tube. Although the volume change for the hot-rolled and cold-reduced components is the same, the ring from the cold-reduced tubewill have a smaller diameter after heat treatment.

Bars and tubes are elongated during manufacture and display direc-tional properties; that is, the mechanical properties are different in thelongitudinal direction compared to the transverse direction. Forging thebar into ring components provides a more homogeneous product. Ringrolling might provide beneficial grain flow conforming to the rolling con-tact surface. Bearing endurance tests demonstrate that end grain is det-rimental to rolling contact fatigue life [16.18].

Raw material intended for machining operations before heat treat-ment should be received in a readily machinable condition. The materialshould have sufficient stock to render an "as-machined" component freefrom carburization, decarburization, and/or other surface defects.

HEAT TREATMENT OF STEEL

Basic Principles

Heat treatment of bearing steel components necessitates heating andcooling under controlled atmospheric conditions to impart desired ma-terial characteristics and properties such as hardness, a diffused high-carbon surface layer, high fracture toughness or ductility, high tensilestrength, improved machinability, proper grain size, or reduced stressstate. Specific thermal cycles that produce these material characteristics


are called annealing, normalizing, hardening, carburizing, tempering,and stress relieving. Selective thermal cycles provide distinctive micro-structures such as bainite, martensite, austenite, ferrite, and pearlite.

Iron and carbon are the basic constituents in bearing steels along withspecific amounts of manganese, silicon, or other alloying elements suchas chromium, nickel, molybdenum, vanadium, or tungsten. Bearingsteels have a distribution in carbon content from 0.08% minimum (AISI3310) to 1.10% maximum (AISI 52100). Beginning with ingot solidifica-tion, bearing steels take on a crystalline structure. These crystals arecomposed of atoms placed at fixed locations within a unit cell. Spacingsremain constant at fixed temperature. Although there are 14 differentspace lattice types, the bearing metallurgist deals primarily with onlythree: body-centered cubic (bcc), face-centered cubic (fcc), and body-centered tetragonal (bct). See Fig. 16.11.

These types of three-dimensional cells have different physical and me-chanical properties because of differences in atomic spacings; they alsohave a different solubility for atoms of other alloying elements. An atomof one or more such alloying elements residing in the high-carbon bearingsteel may be substituted for an iron atom. Elements with very smallatomic radius, such as carbon, which is about one eighth the size of iron,can be placed in the interstitial spaces in the lattice.

Pure iron has a bcc structure at room temperature and an fccstructurewithin a specific elevated temperature range. The temperature upon


heating or cooling, at which the atoms shift from one unit-cell type toanother, is called a transformation temperature. These alterations canbe observed in the time-temperature cooling curve for iron, as shown byFig. 16.12.

Pure iron is bcc below 912°C (1673°F) and fcc above. When carbon isadded to iron, the transformation temperature is lowered and extendedover a broader temperature range. This information is displayed in theiron-carbon phase diagram in Fig. 16.13.

Because bearing steels rarely exceed 1.1%carbon and their heat treat-ment does not exceed a metal temperature of 1302°C (2375°F) [for T-1(C-0.70, Cr-4.00, W-1.00, V-18.00)], only a section of the iron-carbonphase diagram, in Fig. 16.14, will be required for further discussion.

Carbon is dissolvable in molten iron just as salt is dissolvable in water.It is this action occurring in solid solution that enables alteration of me-chanical properties of steel. High-carbon-chromium bearing steels, as re-ceived from the steel producer, are generally in a soft, spheroidizedannealed condition suitable for machining. The microstructure consistsof spheroidal carbide particles in a ferritic matrix. This mixture offerriteand carbide that exists at room temperature transforms to austenite atapproximately 727°C (1340°F). The austenite is capable of dissolving farlarger quantities of carbon than that contained within the ferrite. Byaltering the cooling rate from the austenitizing temperature, the distri-


bution of the resulting ferrite and carbide can be modified, thus givinga wide variation in resultant material properties.

Based on carbon content, steel can be put into three categories: eu-tectoid, hypoeutectoid, and hypereutectoid. Eutectoid steels are thosecontaining 0.80% carbon, which upon heating above 727°C (1340°F) be-come 100% austenite. This composition upon cooling from the austeniticrange to approximately 727°C (1340°F) simultaneously forms ferrite andcementite. This product is termed pearlite, and it will revert to austeniteif it is reheated to slightly above 727°C (1340°F).

Hypoeutectoid steels are those containing less than 0.80%carbon. Theiron-carbon diagram indicates that for a 0.40% carbon steel approxi-mately 843°C (1550°F) is required to dissolve all the carbon into the aus-tenite. Under conditions of slow cooling, ferrite separates from theaustenite until the mixture reaches 727°C (1340°F). At this point theremaining austenite, containing 0.80% carbon, transforms into pearlite.The resulting microstructure is a mixture of ferrite and pearlite. Thepearlite will dissolve into solid solution when it is reheated to approxi-mately 727°C (1340°F).At temperatures above 727°C (1340°F) the ferritewill dissolve into austenite.

The iron-carbon diagram indicates the existing phases when very slowheating and cooling rates are enacted.

Time-Temperature Transformation Curve

The time-temperature transformation diagram is an isothermal trans-formation diagram. Steel will transform when cooled rapidly from theaustenitizing temperature to a lower temperature than the minimum atwhich the austenite is stable. Diagrams for various grades of steel havebeen developed at specific austenitizing temperatures to depict the timerequired for the austenite to begin to transform and to be completelytransformed at any constant temperature studied. Figure 16.15 [16.19]shows an isothermal time-temperature transformation (TTT) diagramfor a typical high-carbon steel (AISI 52100). The shape and the positionof the curves change with increased alloy content, grain size of the aus-tenite, and austenitizing temperature. Figure 16.16 [16.20] depicts theTTT diagram for a typical alloy steel (AISI 4337).

Continuous Cooling Transformation Curves

A eutectoid steel, upon slow cooling, will transform to pearlite at ap-proximately 727°C (1340°F). If this same steel specimen is quenched intoa liquid medium controlled at a temperature just below 727°C (1340°F),a coarse pearlite structure will result. As the temperature of the holdingmedium is lowered, however, the diffusion of carbon atoms is decreasedand the lattice spacing of ferrite and cementite is reduced, thus produc-

FIGURE 16.15. Time-temperature transformation diagram for AISI 52100 steel (from[16.19]).

ing a pearlitic microstructure. These microstructures indicate that theformation of pearlite is a nucleation and growth process. At still lowertemperatures carbon atoms move more slowly, and the resulting trans-formation product is bainite, which consists of ferrite needles containinga fine dispersion of cementite. Under still further cooling, the trans-formation product martensite is formed, which consists of a very fine,needlelike structure. Martensite forms athermally involving a shearmechanism in the microstructure; it is not a product of isothermal trans-formation. The quenching must be done very rapidly into a medium suchas molten salt or oil at a controlled temperature to prevent the austenitefrom converting to a soft transformation product such as pearlite.

The TTT diagrams reveal the microstructures that form at a singleconstant temperature; however, steel heat treatment uses rapid cooling,

and transformation occurs over a range of temperatures. Continuouscooling transformation (CCT) curves have been developed to explain theresulting transformations. Figure 16.17 is a cooling transformation dia-gram for AISI 52100 steel.

An AISI 52100 steel Jominy test bar 25.4 mm (1 in.) in diameter by76.2 or 101.6 mm (3 or 4 in.) long may be used to explain the value ofthe CCT diagram. The piece is austenitized at 843°C(1550°F)and, whileheld vertically, is sprayed with a stream of water on the lower end face.The cooling rate then varies from the quenched surface to the extremeopposite end, which cools much slower. Microstructures can then be cor-related with various cooling rates occurring along the length of the bar.

Hardenability

Hardness should not be confused with harden ability. Hardness is resis-tance to penetration and is normally measured by an indenter of fixedgeometry applied under static load in a direction perpendicular to thematerial surface being tested. Hardenability pertains to the depth ofhardness achievable in an alloy. The alloying elements in the steel, aswitnessed by the movement of the isothermal transformation curves tothe right on the TTT diagram, permit additional cooling time from theaustenitizing temperature to the point of martensite transformation.

FIGURE 16.17. Continuous cooling transformatio:1 diagram for AISI 52100 steel.

This positive effect of alloy additions to steel readily explains the needfor the numerous modifications of the basic AISI 52100 for varying sec-tion thicknesses of the bearing components.

Hardenability is also influenced by the effect of grain size and thedegree of hot-working. The hardenability of a coarse-grained steel ismuch greater than that of a fine-grained steel. Hot-working of materialinto progressively smaller bar sizes correspondingly reduces the harden-ability spread found in ingots and blooms by reducing the segregation ofcarbon and other alloying elements normally experienced during ingotsolidification. Higher austenitizing temperatures and longer soakingtimes at temperatures that promote grain coarsening also enhance har-denability by permitting more carbon to go into solid solution.

Because hardenability is a measure of the depth of hardness achievedunder perfect heat treatment parameters, it is possible to quench barproducts varying in diameter and to measure the resulting cross-sectional hardness patterns to determine hardenability. Grossman[16.21] defined the ideal critical size of a bar processed in this mannerto be one in which the core hardens in an "ideal quench" to 50% mar-tensite and fully to 100%martensite at the surface. Ideal quench is onein which the surface of the heated test specimen instantaneously reaches


the quench medium temperature. Quenching identical specimens intomedia of less severity reduces the extent of hardening. Under these con-ditions Grossman defined the smaller bar diameter that hardened to 50%martensite in the core, as the "actual critical diameter." This varianceled to the development of severity quench curves (H-values) relating toboth the ideal critical diameter and the actual critical diameter.

The Jominy end-quench harden ability test is standardized [16.22]with respect to specimen geometry, apparatus, water temperature, andflow rate such that all results can be rated on a comparison basis. Thehardness ratings at 1.6-mm (0.0625-in.) intervals when plotted againstspecimen length provide a curve indicative of the harden ability of analloy.End-quench harden ability data normally incorporate both the max-imum and minimum hardenability limits anticipated under specificheattreatment parameters [16.23].

Hardening Methods

Heat treatment practices used for bearing components are eitherthrough-hardening or surface-hardening. Heat treatments for thethrough-hardening martensitic grades are substantially comparable toone or another in that they necessitate heating (to an austenitizingtemperature), quenching, washing, and tempering. Time-temperatureparameters, primarily based on weight and cross-sectional thickness ofthe part being processed, have been established for the various through-hardening bearing alloys.

Ring components, particularly of large diameter and thin sectionthickness, require elaborate means of handling to minimize physicaldamage. In furnace construction, precautions are taken to avoid massloading and excessive weight, which could adversely influence the ge-ometry of the parts during heating or quenching.

Furnace manufacturers generally use natural gas or electricity astheir heat source for equipment. Arrangements for protective atmo-spheres are normally provided to minimize carburization or decarburi-zation of the high-carbon-chromium steel parts during processing. Fur-naces of comparable construction and processing capability are alsoselected for the heat treatment of carburizing grades of steel except thatthe atmosphere is controlled to provide the carbon potential necessaryfor carbon to diffuse into the steel.

Precise, uniform furnace temperatures are maintained and controlled,providing exact reproducibility of processing cycles.Adequate quenchingfacilities are provided for salt, oils, water, or synthetic-type quenchants.Temperature control, agitation, and fixtures are used independently orin combination to reduce distortion in the heat-treated components.

Induction heating, often using synthetic-type quenchants, can be usedfor automated heat treatment of special bearing components. This can


be a selective-type heat treatment in which only the rolling contact sur-face is hardened.

After hardening, the parts are washed to remove all quenchant resi-due before tempering. Tempering furnaces are generally electric or gasfired. Parts may be either batch loaded or automatically transportedthrough these units.

Many types of furnaces are being used to process bearing components:for example, roller hearths, rotary drum, rotary hearth, shaker hearth,batch/pit type, conveyor belt/cast link, and pusher tray. In addition, au-tomated salt lines using programmable hoists are in operation for steelsrequiring austenitizing temperatures of 802-1302°C (1475-2375°F).

Through-Hardening, High-Carbon-Chromium Bearing Steels

General Treatment. The high hardness and high strength required forthrough-hardening bearing steels are achieved by first austenitizing ata temperature sufficiently high to provide carbon solution and then cool-ing sufficiently fast into the bainite or martensite temperature ranges toavoid the formation of undesirable soft constituents. Heat treatment ofthese steels generally involves temperatures of approximately 802-871°C (1475-1600°F), uniform soaking, and quenching into a medium ofsalt, water, or synthetic oil controlled between approximately 27 and230°C (80-445°F). The resulting "as-quenched" hardness range for mar-tensite-hardened components is normally Rockwell C (Re) 63-67; forbainite-hardened components, 57-62. Although bainite-hardened com-ponents do not require subsequent thermal treatment, martensite-hardened components are tempered.

Martensite. The martensite start (MS) temperature is lowered as theaustenitizing temperature and the time at temperature are increased,permitting more carbon to go into solid solution. Correspondingly, thetendency exists for more austenite to be retained during the martensitetransformation. The morphology of the resultant martensite also de-pends on the dissolved carbon content: high amounts of dissolved carbonare associated with plate martensite formation, and low amounts pro-mote a tendency to form lath martensite.

High austenitizing temperatures also have the tendency to coarsenthe material grain size. This condition is evident by both visual and low-power magnification of fracture surfaces. Properly heat-treated, high-carbon-chromium grades of steel show a fine, "silkylike" appearance onfracture faces.

After quenching, components are washed and tempered to relievestresses and improve toughness. Tempering at temperatures at or


slightly above the MS point will also transform retained austenite tobainite. The penalty for tempering at higher temperatures is loss ofhard-ness, which can adversely affect load-carrying capacity and endurance ofthe bearing component. Components of lower hardness are also moreprone to handling and functional surface damage than their harder coun-terparts are.

Marquenching. Quenching into a low-temperature medium [49-82°C(120-180°F)] can produce thermal shock and nonuniform phase trans-formation stresses. Components with nonuniform cross sections and/orsharp corners can warp or fracture. Transformation stresses may be re-duced by quenching the part into a hot oil or hot salt medium controlledat a temperature between 177 and 218°C(350-425°F), in the uppermostportion of the martensite transformation range. Temperature equaliza-tion throughout the cross section of the component permits uniformphase transformation to progress during subsequent air cooling to roomtemperature. Although the as-quenched hardness is normally Re 63-65,tempering cycles for martempered parts are similar to those used instraight martensite hardening operations.

Bainite. Bainite hardening is an "austempering"-type heat treatmentin which the component is quenched from the austenitizing temperatureto a temperature slightly above the MS temperature, which is the lowerbainite transformation zone. Molten salt baths between 220 and 230°C(425-450°F) are normally used for this type of heat treatment. Watercan be added to the quench bath to achieve the critical quench rate, thusavoiding the formation of undesirable soft constituents. Bainite-hardening grades of steel are again selected on the basis of componentcross-sectional area. The higher the harden ability is, the greater the per-missible cross-sectional area or thickness of a given component. As alloycontent increases, the "nose" and "knee" of the transformation curve arepushed further to the right, which lengthens the time for bainite trans-formation to begin.

These alloys normally require 4 hours or more for complete transfor-mation to bainite. Hardness values of 57-63 Re are achieved on com-ponents processed in this manner. Subsequent tempering is not required.Quenching into molten salts and holding at these temperatures signifi-cantly reduce stresses induced due to thermal shock and phase transfor-mation.

Bainite hardening produces components with small compressive sur-face stresses, in contrast to martensite hardening, which produces smalltensile stresses in the as-quenched surface layers. A bainite microstruc-ture is coarser, with a more "feathery" needle than that produced instraight martensite hardening.


Surface Hardening

Methods. Surface hardening is done by altering the chemical composi-tion of the base material-for example, by carburizing or carbonitrid-ing-or by selectively heat treating the surface layer of a givenhigh-carbon bearing steel component. Induction and flame-hardeningpractices are used to fabricate production bearings. Laser beam and elec-tron beam processes are also possible, depending on the hardness depthrequired.

Surface hardening of bearing steels produces well-defined depths ofhigh surface hardness and wear characteristics. High residual compres-sive stresses, present in the surface layer, enhance rolling and bendingfatigue resistance. The surface layer is supported by a softer and toughercore which tends to retard crack propagation.

Carburizing. The carburizing source or medium (gas, liquid, or solid)supplies carbon for absorption and diffusion into the steel. The sameprecautions followed for through-hardening furnace operations are fol-lowed in carburizing to minimize handling damage, reduce part distor-tion, and provide process economy.The normal carburizing temperaturerange is 899-982°C (1650-1800°F) with the carbon diffusion rate in-creasing with temperature. Therefore, it is easier to control narrow casedepth ranges at the lower carburizing temperatures.

Based on the alloy steel being processed, time, temperature, and at-mospheric composition determine the resulting carbon gradient. The re-sulting carbon content affects the hardness, amount of retainedaustenite, and microstructure of the carburized case. The hardness pro-file and compressive stress field depend on the carbon profile.

Although the practice of quenching directly from the carburizing fur-nace is used to heat treat bearing components, it is general practice toreharden carburized components to develop both case and core propertiesand, at the same time, to employ fixture-quenching devices to reduce partdistortion.

Based on the grade of steel being carburized, the carbon potential ofthe furnace atmosphere must be adjusted so that large carbides and/ora carbide network are not formed. Those alloying elements such as chro-mium which lower the eutectoid carbon content are most likely to formglobular carbides. Carbon can be further precipitated to the grain bound-aries if the steel is then slow cooled before quenching. These grainboundary carbides and/or the carbide network can reduce mechanicalproperties.

Choosing the bearing material not only involves considering propersurface hardness and microstructure, but it also must incorporate coreproperties to prevent case crushing. Resistance to case crushing is gen-erally provided by increasing subsurface strength. Therefore, a materialwith a section thickness and hardenability that will provide a core hard-


ness Rc 30 to 45 is selected. Carburizing grades should be fine grainedto minimize sensitivity to grain growth at high carburizing tempera-tures.

Direct quenching from the carburizing furnace has the advantage thatone can obtain a case microstructure free of soft constituents, such asbainite, while using a leaner alloy steel. This heat treatment practiceoffers less part distortion than is experienced in reheating and quench-ing, particularly if the temperature is lowered to 816-843°C (1500-1550°F) before quenching. Adversely, this practice can produce partswith too much retained austenite and possible microcracking. The excessaustenite in the case could permit plastic deformation of components un-der heavy load; microcracking could provide initiation points for fatigue.Microcracking can be minimized by keeping the carbon content in theas-carburized component lower than eutectoid level. Reheating at thelower austenitizing temperature and quenching tend to reduce micro-cracking.

Gas carburizing is common to the roller bearing industry because gasflow rates and carbon potential of the atmosphere may be accuratelycontrolled. Gases present in furnace atmospheres include carbon dioxide,carbon monoxide, water vapor, methane, nitrogen, and hydrogen.

Over a period of time at a predetermined temperature, the specifiedcase depth is established. This effective case depth (ECD) is generallydefined as the perpendicular distance from the surface to the farthestpoint where the hardness drops to Rc 50. Normal ECDs for bearing com-ponent range between 0.5 and 5 mm (0.020-0.200 in.) with a surfacecarbon content between 0.75 and 1.00%.

Carburized components are tempered after quenching to increasetheir toughness. Cold-treating might be introduced to transform retainedaustenite to martensite. Additional tempering is then required.

Carbonitriding. Carbonitriding is a modified gas-carburizing process.Because of the health hazards and ecological problems in disposing ofcyanide salts, the preferred method is to use a gaseous atmosphere. Atan elevated temperature an atmosphere is generated having a given car-bon potential to which ammonia is added. Nitrogen and carbon are dif-fused into the steel forming the hard, wear-resistant case. Because thesehard carbonitrided cases are generally shallow in nature, ranging fromapproximately 0.07 to 0.75 mm (0.003-0.03 in.), produced at furnacetemperatures ranging from 788 to 843°C (1450-1550°F), the case-coreinterface is easily differentiated. These same beneficial shallow casecharacteristics can also be achieved in components requiring excessivelyheavy case depths. In this instance the parts are generally carburized tothe heavy case depths and then reheated in a carbonitriding atmosphere.

Ammonia added to the carburizing atmosphere dissociates to formnascent nitrogen at the work surface. The combination of the carbon andnitrogen being adsorbed into the surface layer of the steel lowers the


critical cooling rate of the steel; that is, the hardenability of the steel issignificantly increased by the nitrogen. This characteristics permitslower-cost materials, such as AISI 1010 and 1020, to be processed to thedesired high hardness by oil quenching and thus minimizes distortionduring heat treatment.

All parameters being constant, the carbonitrided component will evi-dence a more uniform case depth than that produced by carburizing.Because nitrogen lowers the transformation temperature, carbonitridedcomponents will have more retained austenite than carburized compo-nents of the same carbon content. These high levels of austenite may bereduced by increasing the carbonitriding temperature, controlling thesurface concentrations to approximately 0.70-0.85% carbon, keeping theammonia content at a minimum during processing, and introducing adiffusion cycle before quenching.

The presence of nitrogen in the carbonitrided case also enhances re-sistance to tempering. Carbonitrided components are tempered in the190-205°C (375-400°F) range to increase toughness and maintain a min-imum hardness of Rc 58.

Induction Heating. Induction heating is a means for rapidly bringingthe surface layer of a high-carbon-Iow-alloy bearing steel component intothe austenitic temperature range, from which it can be quenched directlyto martensite. Induction heating is accomplished by passing an alter-nating current through a work coil or inductor. A concentrated magneticfield is then induced within the coil. This magnetic field will in turninduce an electrical potential in a part placed within the coil. Since apart represents a closed circuit, the induced potential establishes an elec-trical current within the part. Heating of the part is then the result ofthe material's resistance to the flow of induced current.

Power generating equipment is selected according to frequency re-quirements. Motor generators have historically been used to providemedium-frequency ranges from 1-10 kHz and to provide deep, hardenedsurface layers. These units are currently being replaced by solid-stateinverters using silicon-controlled-rectifier (SCR) switching devices. Radiofrequency generators provide frequencies ranging from 100 to 500 kHzfor very shallow case depth requirements.

The chief factors influencing the success of the induction-heating op-eration are frequency selection, power density, heating time, and cou-pling distance:

Frequency selection-the size of the part and the depth of heatingdesired dictate the frequency requirement.

Power density-the watts available per square mm of inductor surfaceinfluence the depth to which a part can be surface hardened.


Heating time-the heating time required to bring the part to temper-ature is a critical factor with respect to overheating and resultingcase depth.

Coupling distance-the coupling distance is defined as the distancebetween the coil and the part surface.

Quenching of induction-hardened components is generally accom-plished by either a spray or immersion method. Spray quenching in-volves a pressure deposition of the quenchant onto the component by aseries of holes machined into the inductor or by a separate quench ring.The immersion method necessitates dropping the part out ofthe inductorinto an agitated quench bath. The required physical and metallurgicalproperties in high-carbon-chromium bearing steel can be achieved by us-ing a synthetic quench in lieu of water and/or oil. Concentrations maybe adjusted to provide maximum quenchability while minimizing thetendency for cracking.

All surface-hardened components require tempering after a quench-ing. Although the case depth may be similar to those achieved by car-burizing, a steeper hardness gradient exists in the case-core transitionzone.

Properly induction-hardened AISI 52100 steel bearing componentswill generally achieve hardness values of Rc 65-67 as quenched. If thepart before heat treatment is in the annealed condition, the microstruc-ture of the as-hardened surface zone will consist of fine spheroidal car-bide and a matrix of untempered martensite. When it is examined forfractures, a fine grain size can be seen.

Flame-Hardening. Flame-hardening is used primarily in the heat treat-ment of very large rings of high-car bon-low-alloy steel components morethan 1 m (approximately 3 ft) in diameter. A combustible gas is mixedwith oxygen to fire a cluster of burners directed to selectively heat thering component as it rotates at a fixed rate through the impinging flame.The depth of the heat-affected zone is a function of the dwell time of thepart at the heat source. The rotating part, upon reaching the properaustenitic temperature, is water quenched. The core material, being inthe unaffected heat zone, remains in the annealed condition. Subsequenttempering is mandatory to relieve stresses and increase the ductility ofthe as-quenched component.

Flame-hardening is not a capital intensive process from an equipmentstandpoint. It is very versatile for selective hardening and rapidly adapt-able for changing ring sizes with varying cross-sectional configurationsand thicknesses.

The progressive zone heating method means an overlap will occur af-ter 360° are completed. The resulting overtempering effect in the heat


sink zone will result in a spot of lower hardness. Precautions must betaken to minimize the thermal and transformation stresses at this over-lap point to prevent cracking.

Thermal Treatment for Structural Stability

Knowledge of size and shape changes of rolling bearing components oc-curring in heat treatment is critical to subsequent manufacturing oper-ations and to the component's functional suitability. Basic high-carbon-low-alloy steel with a bcc structure expands rapidly (Fig. 16.18) as it isheated to approximately 727°C (1340°F)due to the coefficient of thermalexpansion. At this critical temperature the material, as previouslystated, undergoes a phase transformation to an fcc structure (i.e., aus-tenite), resulting in component shrinkage. The specificvolume of austen-ite is less than that of ferrite. If the material is heated to still highertemperatures within the austenite range, it continues to increase in vol-ume due to the coefficient of thermal expansion. Conversely, when rap-idly cooled, the material shrinks to the martensite transformationtemperature. The martensite, formed as the part continues to contractwhile cooling to room temperature, is a bct structure. The resulting in-crease in volume, due to this transformation occurring at such low tem-peratures, stresses the material. Because it is virtually impossible inproduction heat treatment to achieve complete transformation from theaustenitic (fcc) structure to the untempered martensite (bct), varying


amounts of austenite, depending on the severity of the quench, are re-tained in the as-quench microstructure. Components must be thermallytreated to reduce the residual stresses and to provide required structuralstability.

Dimensional changes occurring in bearing steels essentially dependon precipitation of fine carbide from martensite and decomposition ortransformation of retained austenite. Because changes can also be in-duced during bearing operation, due to the temperature or stress envi-ronment, the manufacturer must select the appropriate heat treatmentto provide required stability. Tempering of high-carbon-chromium steelsgenerally occurs in the range 66-260°C (150-500°F). At these tempera-tures fine carbide is precipitated, and the tempered martensite remainsessentially bct with some shrinkage. Tempering in the range 205-288°C(400-550°F) results in a time-temperature-dependent decomposition ofretained austenite to bainite and volume increase. Loss of hardness athigh temperatures is prevented by tempering below 260°C (500°F).

The annealed microstructures of high-speed steels, providing maxi-mum machinability, contain numerous hard metallic carbides, such astungsten, molybdenum, vanadium, or chromium, imbedded in a soft fer-ritic matrix. Unlike the high-carbon-chromium steels, temperatures farabove the critical temperature must be attained to dissolve the desiredamount of these hard carbide particles. Carbide precipitation is avoidedby rapidly cooling the steel from the austenitizing temperature into themartensite transformation temperature range. After further cooling toroom temperature, the structure normally contains 20-30% retainedaustenite. Heating to temperatures required for tempering high-carbon-chromium steel produces only slight tempering of the martensite. Be-tween 427 and 593°C (800-1l0°F), "secondary hardening" occurs; thatis, the austenite is conditioned and subsequently transforms to marten-site upon cooling back through the MS temperature transformationrange. Multiple tempering at these high temperatures is required to com-plete the transformation of austenite to martensite and to precipitatevery fine alloy carbides, which are responsible for the secondary hard-ening phenomenon and provide for the high-temperature hardness re-tention characteristic of high-speed steels.

Subzero treatments are often used after the initial quench or inter-mittently between tempering cycles to complete austenite-to-martensitetransformation upon cooling. However, because cold-treatment sets uphigh internal stresses in the as-quenched components, it is generallyrecommended that cold-treating be practiced only after the first temper-ing cycle.

Corrosion-resistant steels, for example, AISI 440C and BG42 (AMS5749), are generally heat-treated incorporating deep freezing immedi-ately after rapid cooling from the austenitizing temperature. AISI 440Cmay be subsequently multiple tempered at approximately 149°C(300°F)


or 316°C (600°F), depending on the product hardness requirements. Be-cause of its alloy composition BG42 is heat-treated according to standardpractices for high-speed steels-that is, multiple tempering at 524°C(975°F) incorporating refrigeration cycles.

Retained austenite, present in the case microstructure of case-carburized steels, is a relatively soft constituent providing some toler-ance for stress concentrations arising from inclusions, handling damage,and surface roughness. Case properties are preserved by generally tem-pering bearing components from 135-196°C (285-385°F). The core is sta-ble at normal bearing operating temperatures.

Mechanical Properties Affected by Heat Treatment

Elasticity. The elastic properties of rolling bearing steel are not signif-icantly affected by heat treatment. Hence the modulus of elasticity atnormal temperatures is 202 kN Imm2 (29.3 X 106 psi) for both through-hardened and case-hardened steels.

The limit of elastic behavior-that is, the stress under maximum uni-axial loading giving insignificant plastic deformation or permanent set-is described for rolling bearing steels by a 0.2% offset yield strength-that is, 0.2% remaining plastic strain. Figure 16.19 illustrates thatstrength properties tend to decline as transformation temperature in-creases for a given rolling bearing steel composition.

Ultimate Strength. Ultimate strength is the stress at which the samplebreaks in the uniaxial test described before; it is significantly affected byheat treatment. For through-hardened AISI 52100 ultimate strength formartensitic steel generally lies between 2900 and 3500 N/mm2 (420-510ksi). For the best case-hardening bearing steels, for example, AISI 8620,ultimate strength approximates 2600 N Imm2 (380 ksi).

Fatigue Strength. Fatigue strength is determined in a cyclic push-pullor reversed bending test as the maximum stress that can be enduredwith no failure before accumulation of 10 million cycles. These data de-pend strongly on heat treatment, surface finish and treatment, test con-ditions, and so on. Accordingly, it is difficult to generalize and nonumerical values are given herein. It is best to test the individual steel.

Toughness. Two test methods are used to determine toughness of bear-ing steels: the fracture toughness test and the impact test. In the first,a plain stress value Krc is measured in N/mm2_mI/2; this is the stressrelated to the defect size that can be tolerated without incipient struc-tural failure. For martensitic AISI 52100, Krc falls between 15 and 22,

depending on heat treatment. A slight increase in Krc occurs as temper-ature increases. Case-hardening steel tends to have greater fracturetoughness than through-hardening steel. A Krc value of 60 is not uncom-mon for surface-hardened steel.

The second test-the impact of a hammer blow of defined energy ona sample-measures the energy absorbed in breaking the sample. Formartensite-hardened AISI 52100 this is only 4.5 J (3.3 ft-Ib) comparedto 172 J (127 ft-Ib) for the soft annealed material.

Hardness. The manner in which carbon is distributed in steel dictatesthe resulting hardness and mechanical properties. Although carbonmakes by far the greatest contribution to hardness, increasing the alloycontent also increases hardness.

Hardness, a material's resistance to penetration and, hence, wear, canbe measured by static or dynamic methods. Static testing involves ap-plying a load through a penetrator of defined geometry. Depending onthe type of hardness tester employed, either the depth of penetration orthe size of the indentation becomes the measurement of the materialhardness. See Fig. 16.20.

Dynamic testing involves bouncing a diamond-tipped hammer from aspecific height onto the surface of the test specimen. The resultant re-

FIGURE 16.20. Hardness tests. (a) Rockwell Rc; indentation body: diamond cone; load150 kgf (including 10-kgf preload), indentation depth for Rc 63: 74 ILm, hardness testingrange: Rc 20-67. (b) Vickers; indentation body: diamond pyramid 136°, indentation depth:td-for V 782 = Rc: 22 ILm, hardness testing range: up to V 2000. (c) Brinell; Indentationbody: hardened steel ball (D), hard metal ball (D), Indentation depth: --t;d, Hardness test-ing range: up to 400 B (-42.5 Rc), up to 600 B (-57 Rc).

bound height is a measure of material hardness. The scleroscope is theonly piece of equipment based on the dynamic test principle.

Hardness attainable is a function of carbon content, as shown by Fig.16.21. In general, as hardness is increased, toughness decreases for agiven alloy steel.

Residual Stress. Stresses induced in a component through fabricationor thermal treatments are totally eliminated upon uniform heating andsoaking in the austenite temperature range. Quenching ofthe componentcan once again generate tremendous internal stresses in the part.Through-hardening of the martensitic high-carbon steels may producesurface tensile stresses that can produce part distortion or even cracking.Surface-hardening heat treatments, including carburizing, carbonitrid-ing, induction or flame-hardening, generally produce parts showing sur-face compressive stresses. Regardless of the heating method selected foraustenitizing, subsequent thermal cycles with or without subzero treat-ment can appreciably alter the established stress state in the as-quenched parts.

The stresses induced in a through-hardened component duringquenching are principally the result of temperature variances and non-uniform phase transformations. Bearing rings, essentially being thinhoops of varying cross-sectional thickness, are prone to both size and

shape changes. Fixture quenching, employed to retain the components'as-machined dimensional characteristics, may hamper quench mediumflow and induce additional nonuniform stress distribution in the partbecause of mechanical restraints that do not adapt to size or shapechanges. The machined undercuts, grooves, filling slots, oil holes, andflanges have sharp corners and recesses provide additional focal pointsfor stress concentration.

The high-carbon-chromium bearing steels under recommended aus-tenitizing temperatures have an MS temperature range of approximately204-232°C (400-450°F). Increasing the carbon content and various al-loying elements in this family of steels will tend to depress the MS tem-perature. Coarse-grained materials will also have a lower MS point thanfine-grained materials of the same chemistry. Therefore, austenitizing atvery high temperatures will reduce the MS point into the range wherethe material is less able to adjust plastically. The higher austenitiz-ing temperatures will also result in higher thermal gradients occurringin the parts during quenching. Subzero treatments that permit theaustenite-to-martensite transformation to be further completed couldcause high stress levels, which can crack the parts, Bainitic heat treat-ment or martempering will appreciably reduce transformation stressesduring quenching.

It is standard practice in the rolling bearing industry to coolmartensite-hardened parts to room temperature from the quench-


ing bath, wash the parts, and subject them to a tempering cycle. Alow-tempering temperature for a long time is equivalent to a high tem-perature for a short time from the standpoint of reducing the residualcomponent stresses. This sequence of operation may be interrupted bya subzero treatment following the washing operation to permit the com-pletion of the austenite-to-martensite transformation. Parts processedin this manner are very prone to cracking because of the resulting highresidual stress state. These parts must be tempered as soon as theywarm to room temperature.

Surface-hardening heat treatments, by a diffusion process altering thecomposition of the material or by the rapid heating of a selected surfacearea of a homogeneous steel, are developed and controlled to providesurface compressive stresses with normal counterbalancing tensilestresses in the core. Induction surface hardening of an appropriate ma-terial to the proper case depth results in the maximum compressivestress being located at the case-core transition zone. The magnitude ofthis compressive stress in a surface-hardened high-carbon steel alloywillnormally be less than that produced in a carburized part at its point ofmaximum compressive stress, which is at the approximate midpointof the total case depth. This point corresponds to the carbon content ofapproximately 0.50%.

Tempering of as-quenched, surface-hardened components with orwithout support of subzero treatment will generally reduce the retainedaustenite level and modestly alter the level of compressive stresses.

ROLLING CONTACT FATIGUE: MODES AND CAUSES

Failure Modes

Contact fatigue failure in rolling bearings occurs when local materialstresses exceed the local fatigue limit; cracks are initiated and then prop-agated. Even if the stresses induced by cyclic loading between rollingelements and raceways are generally below the fatigue limit, additionalstresses can be caused locally by material inhomogeneities and defectsacting as stress concentrators. These stresses are superimposed on thosearising due to normal bearing operation. Inhomogeneities occurring fromthe steelmaking process are distributed throughout the entire material:for example, slag inclusions and pores. They can also be generated bythe manufacturing process, where they are mainly limited to surfacezones: for example, ring marks, scratches, and grinding burns. The dif-ferent types ofmaterial inhomogeneities and defects are the reason thatbearings fatigue in two observable modes. In one mode, failure is causedby a stress-raising inhomogeneity in the subsurface region where thenormal stresses induced by cyclic loading are maximum. In the othermode, failure is initiated by an inhomogeneity or defect in the surface,

ROLLING CONTACT FATIGUE: MODES AND CAUSES 619

thus increasing the surface stresses. Tallian [16.24] provides a compen-dium of results concerning rolling bearing failure modes and causes.

Subsurface Failures

Inhomogeneities. Material inhomogeneities that can lead to subsurface-initiated failures are macro- and microinclusions, pores, and bandings.The stresses surrounding these inhomogeneities depend on the nature,size, distribution, shape, and interface between the inhomogeneity andthe base material.

Macroinclusions. Macroinclusions are impurities originating from therefractory material because of their large size [above 0.5 mm (0.02 in.)],high brittleness, very low cold deformability, and irregular shape. Theyalways cause early failure when situated in the surface or subsurfacezones.

Microinclusions. All oxide-type slag microinclusions are brittle andpractically impossible to deform. They occur either in stringers or in glob-ular shape. Above 40 /-tm (0.0016 in.) diameter for the globular shapetheir negative influence on life is great; see Fig. 16.10. High tensilestresses can occur in the matrix surrounding the inclusions, owing to thedifference in thermal contraction between matrix and particle duringquenching. They are called tessellated stresses. A relationship exists be-tween the size and size distribution of oxides, on the one hand, and thetotal oxygen content of the steel, on the other hand; the higher the ox-ygen content, the bigger the maximum size and the total number of ox-ides.

Sulfide Inclusions. Sulfide inclusions are relatively soft and easily cold-deformed. Therefore, high stress peaks surrounding sulfid~ inclusionscan be partially reduced by plastic deformation. In some cases sulfidessurround the oxides and, by plastic deformation, close the voids so thatthe oxides become less dangerous. The allowable size and distribution ofsulfides are greater than those of oxides.

Pores. Pores are generated by gas bubbles enclosed during steel solid-ification and often occur as clusters. If their surfaces are not oxidized,they are closed during warm- and cold-processing of the steel. In theoxidized condition of the surface they have a negative influence on fa-tigue life and, regardless size and shape, can be treated similarly toglobular-type slag inclusions.

Banding. High local differences in steel chemical composition (segre-gations) caused by improper solidification conditions can lead to different


component microstructures after hardening. The high stresses betweenmicrostructural zones, originating from the different MS points, and thereduced fatigue limits, can lead to early fatigue failures.

Surface-Initiated Failures

In addition to the foregoing defects, other material defects in the sur-faces, such as high-tempered or rehardened grinding burns, decarburizedlayers, and marks can lead to surface-initiated failure.

Grinding Burns. Grinding burns occur under improper grinding con-ditions of hardened components. The material is then locally tempered,yielding insufficient hardness, or it is rehardened. The heat-affected zonevaries between a few microns and a few hundred microns. In the lattercase tensile stresses up to 1200 N/mm2 (175 ksi) have been measured.

Decarburization. Decarburization is the result of heat treatment in anoxygen rich atmosphere. The carbon content in the surface is therebydepleted. The different MS points in the base material and the surfacelayer give rise to residual tensile stresses.

Marks and Indentations. Marks and indentations occur because of in-correct handling of components during manufacturer or mounting. Dur-ing bearing operation the normally induced stresses can add to theresidual tensile stress caused around oxides, decarburized zones, band-ings, or areas of grinding burns. The resulting stresses can get so highthat fatigue can be initiated. The voids around marks, indentations, andsimilar defects are discontinuities in the surface material from whichfatigue can also begin.

MATERIALS FOR SPECIAL BEARINGS

For most rolling bearing applications, the through-hardening steel AISI52100 and case-hardening steels described in Table 16.2 are sufficient toprovide good performance characteristics such as fatigue and wear re-sistance, appropriate fracture toughness, and consistently reliable me-chanical properties. The advent of the aircraft piston engine, however,created the demand for long-lived endurance at higher operating tem-peratures. This demand was met by using the tool steels Ml, M2, MI0,and Tl. These steels lost their prominence in the 1950s with the intro-duction of vacuum-melted M50 to meet the needs of aircraft gas turbineengine bearings. In many applications, particularly instrument ball bear-ings, corrosion resistance became important and this requirement wasmet by using AlSI 440C and BG42 stainless steels generally at the sac-

MATERIALS FOR SPECIAL BEARINGS 621

rifice of fatigue endurance when compared to bearings fabricated fromAlSI 52100. Because oflight applied loading, however, fatigue enduranceis not a major consideration in such applications. The chemical compo-sitions of some of the foregoing steels are given in Table 16.3.

The exploration of space and the continuing development of theaircraft gas turbine engine provided the demand for yet increaseddevelopment of exotic materials. Examples are sapphire for balls, precip-itation-hardening stainless steels, and nickel-based superalloys. Addi-tionally, the nuclear power industry created the need for cobalt alloyssuch as L-605, Stellite-3, and Stellite-6. Powder metal-forming tech-niques have now provided the means to create steels of differential prop-erties; for example, extremely hard, corrosion-resistant surfacescombined with tough, high strength substrates.

The requirement for aircraft gas turbine engine mainshaft ball androller bearings to operate at ever-increasing speeds initiated the searchfor a relatively high temperature capability, fracture-tough steel. Be-cause of the bearing ring hoop stresses caused by ring centrifugalstresses and rolling element centrifugal forces at high speeds, fatiguespalls under such conditions can lead to fracture of rings fabricated fromthrough-hardening steels such as M50. Thus, the operating speeds ofaircraft gas turbine engines were limited to approximately 2.4 milliondN (bearing bore in mm X shaft speed in rpm). With the developmentof M50-Nil, a case-hardening derivative of M50 whose chemical compo-sition is shown in Table 16.3, this limitation has been overcome. Figure16.22 from Spitzer [16.29] shows the effect of the higher fracture tough-ness of M50-Nil on speed capability.

The need for bearings to operate at ultrahigh temperatures has trig-gered the development of cemented carbides and ceramics as rollingbearing materials. Materials such as titanium carbide, tungsten carbide,silicon carbide, sialon, and particularly silicon nitride are being investi-gated. At elevated temperatures, these materials retain hardness, havecorrosion resistance, and provide some unique properties, some ofwhichare advantageous, such as low specific gravity for silicon nitride. Con-versely, other properties of these materials, such as extremely high elas-tic modulus and low thermal coefficient of expansion for silicon nitrideas compared to steel, create significant bearing design problems thatmust be overcome if these materials are to succeed for use in rollingbearing structure components, particularly rings. Ceramic materialssuch as silicon nitride, as illustrated by Fig. 16.23, commence life aspowders that after a series of processes, principal among which is hotisostatic pressing, are transformed into highly engineered bearing com-ponents.

Table 16.4, excerpted from Pallini [16.30],gives significant mechanicalproperties and allowable operating temperatures for several of the ma-terials described above. Considering the low density and high elastic

MATERIALS FOR SPECIAL BEARINGS 623

TABLE 16.4. Properties of Special Bearing Structural MaterialsElasticModulus Coefficient

Rockwell N/mm2 of ThermalHardness C Max Useful X 103 Expansion

(room Temperature Specific (psi x Poisson's 10~6/oCMaterial temp) °C (OF) Gravity 106) Ratio (10~6 I"F)

440C 62 260(500) 7.8 200(29) 0.28 1O.1(100°C)stainless (5.61)steel

M50 tool 64 320(600) 7.6 190(28) 0.28 12.3(300°C)steel (6.83)

M2 tool 66 480(900) 7.6 190(28) 0.28 12.3(300°C)steel (6.83)

T5 tool 65 560(1050) 8.8 190(28) 0.28 11.3steel (6.28)

T15 tool 67 590(1100) 8.2 190(28) 0.28 11.9steel (6.61)

Titanium 67 800(1470) 6.3 390(57) 0.23 10.7carbide (5.94)cermet

Tungsten 78 815(1500) 14.0 533(77.3) 0.24 5.9carbide (3.28)

Silicon 78 1200(2200) 3.2 310(48) 0.26 2.9nitride (1.61)

Silicon 90 1200(2200) 3.2 410(59) 0.25 5.0carbide (2.78)

Sialon 78 1300(2372) 3.3 288(42) 0.23 3.0201 (1.67)

modulus of hot isostatically pressed (HIP) silicon nitride (Si3N4) as com-pared to steel, Figs. 9.7-9.9 compared performance parameters of a 218angular-contact ball bearing having HIP silicon nitride balls with thoseof the same bearing having steel balls. It can be seen that, at high speed,there is a tradeoff between reduced ball load and increased contact Hertzstress. Figure 16.24 from [16.30] indicates the frictional properties ofHIPsilicon nitride when used with various types of lubricants. It is apparentthat for sustained low friction operation, bearings with HIP silicon ni-tride balls require oil lubrication. It is usual for standard ball bearingsto have raceway groove curvature radii rill = 0.52D-0.53D, where D isthe ball diameter and subscript m refers to the raceway. For applicationsuling bearings with HIP silicon nitride balls, it is possible to reduce oneor both ofthese radii to, for example, 0.51OD-0.515D, thereby decreasingHertz stresses. Unfortunately, this action also causes increases in contactfriction; therefore, ball bearings using HIP silicon nitride balls may have

their designs optimized for Hertz stress (and hence fatigue life) or fric-tion. Considering the basically straight contour raceways, this option isnot available for cylindrical or tapered roller bearings.

Figure 16.24 further shows that irrespective of fluid or dry-film lubri-cation, the friction coefficient of HIP silicon nitride is dependent on theoperating temperature and environment. It has been demonstrated, how-ever, that when lubricant flow is interrupted, steel bearings with HIPsilicon nitride balls will sustain operation longer without seizure as com-pared to the same bearings with steel balls.

While the compressive strength of HIP silicon nitride is excellent, thetensile strength is only about 30% that ofM50 steel. The fracture tough-ness is also only a small percentage of that of M50 steel, let aloneM50NiL steel. Furthermore, although in rolling contact under heavy loadthe material tends to fail by surface fatigue and even tends to havelonger fatigue life than steel, any disruption of the surface can lead torapid crumbling of the surface under continued operation. Means for fail-ure detection is therefore an important consideration in life-critical ap-plications.

CAGE MATERIALS 625

CAGE MATERIALS

Material Types

The generally stated function of the rolling bearing cage is to maintainthe rolling elements at properly spaced intervals for assembly purposes.It is sometimes inferred that, in normal bearing operation, the cage isnot necessary; rather it "goes along for the ride"; that is, it is not a highlystressed component requiring the strength of the accompanying ringcomponents. There are more exceptions to this statement than examples.For example, mainshaft and accessory aircraft gas turbine engine bear-ings require AISI 4340 steel (AMS 6414 or AMS 6415) cages supplied inthe hardness range of Rc 28-35. These cages are also silver plated (AMS2410 or AMS 2412) to provide corrosion resistance and added lubricity.In many bearing applications not only do the rolling elements contactthe cage pockets, but the cages themselves are either inner ring or outerring land "riding."

Although cages are manufactured from many types of material, in-cluding aluminum, S-Monel, graphite, nylon, and cast iron, the majorbearing product lines use brass or steel. In ball bearings principally, butalso in some roller bearings, plastics are replacing these metals.

Low-Carbon Steel

Plain low-carbon strip steel, suitable for cold-forming (0.1-0.23% C) isused in the bulk fabrication of pressed, two-piece, or finger-retention-typesteel cages. Two-piece cages are joined by mechanical lock joints, rivets,or welds. The material has a tensile strength of 300-400 N/mm2 (44-58ksi). The AISI 4340 machined steel cages previously mentioned for air-craft applications have approximately 0.4% C for increased strength. Ad-ditionally, low-carbon steel tubes and forgings are used to make cagesfor bearing applications that need unique features for lubrication orgreater material strength. Many steel cages are surface hardened orphosphate coated to provide improved wear characteristics.

Brass

Brass cages are generally manufactured from continuously cast rounds,centrifugally cast cylinders, sand castings, or sheet metal/plate. Becausehigh tensile strength, 300-380 N/mm2 (44-55 ksi), alpha brass has poormachinability but is readily capable of deep drawing (ductility increasingwith increasing zinc content up to 38%), it is used to cold-form one-piececages. When zinc is increased from 38 to 46%, a mixture of alpha andbeta phases is formed. Ultimate strength is higher with the higheramounts of zinc. Adding phosphorus and/or aluminum provides alloysthat can be centrifugally cast, readily machined, drilled, or broached for


making ball and roller bearings. Other nonferrous brass alloys may becentrifugally cast, but they are hot-worked by upsetting or ring rollingto meet specific product requirements. Cage blanks may also be producedby extruding the centrifugally cast billet.

Bronze

Silicon-iron-bronze (Cu: 91.5%, Zn: 3.5%, Si: 3.25%, Mn: 1% and Fe:1.20%)is an alloy recommended for ball and roller bearing cages oper-ating at temperatures up to 316°C (600°F). The as-cast billet materialmust be extensively hot-worked and extruded to promote optimum ma-terial properties.

Polymeric Cage Materials

Advantages and Disadvantages. The use of polymer, particularly nylon(polyamide) 6,6, as a cage material is widespread in many rolling bearingapplications. Polymeric cages have the following advantages over metal-lic cages in both production and operation:

1. Processing of polymeric materials often allows one-step fabricationof complex designs, thus eliminating the machining operations nec-essary to produce a comparable metallic retainer and saving money.

2. Plastic cages tend to be free from the debris that accompanies theproduction of metallic cages. The increased cleanliness contributesto reduced bearing noise.

3. Polymers are more flexible than metals. This is advantageous incage assembly and in bearing operation under some difficult load-ing conditions.

4. Favorable physical properties of polymeric materials lead to cageperformance advantages in many applications; for example, lowdensity (reduced cage weight), good chemical resistance, low fric-tion and damping properties for low torque and quiet running.

The primary disadvantage of polymer usage is the deterioration ofinitial properties of the material due to temperature, lubricant, and en-vironmental exposure. Polymer deterioration causes loss of strength andflexibility, which is important to the cage function during bearing oper-ation. Bearing rotation causes centrifugal forces to act on the cage, whichdeform it radially. Misalignment of the inner and outer rings can causelarge stresses on the cage during bearing operation. Hence, loss of cagestrength can lead to failure. Therefore polymeric material candidatesmust be evaluated for the rate and degree of deterioration under condi-tions of extreme temperature, lubricant exposure, and other environ-mental factors.

CAGE MATERIALS 627

Rolling Bearing Plastic Cages. Some examples of polymeric cage de-signs are shown in Fig.16.25. Properties of cage polymeric materials are

Low coefficient of thermal expansionGood physical property retention, especially strength and flexibility,

throughout the temperature range of operationCompatibility with lubricants and environmental factorsDevelopment of a suitable cage design to minimize friction and provide

proper lubrication


This list indicates essential differences between polymeric and metal-lic cage materials. Lubricant compatibility is rarely a factor, and loss ofphysical properties does not occur within bearing operation temperatureswith metals. Cage design depends on the specific polymer used in a moreintimate fashion than when steel or brass is used.

Polymeric Types for Cages. Fabric-reinforced phenolic resin cages havebeen used for many years in high-speed bearing applications where de-creased cage mass is a benefit. The low density of the material, approx-imately 15% that of steel, results in a low cage mass. The centrifugalforce on a phenolic cage is consequently only 15% of the force acting ona steel cage. At high speeds centrifugal force causes a cage to spreadradially. The low-density cage therefore offers better dimensional stabil-ity at high speeds. Use of a phenolic resin, however, is limited to bearingapplications that do not exceed temperatures of 100°C(212°F)continuousand 120°C (248°F) peak. Another disadvantage with the phenolic resinis the necessary ofmachining operations to obtain the final shape. Otherresins, discussed in the following paragraphs, can be injection moldedinto a final shape directly, thus reducing process cost. Resins of this type,particularly nylon 6,6, have replaced phenolic in many rolling bearingapplications.

The nylon 6,6 (polyamide 6,6) resin is the most widely used plastic forbearing cages. It provides a low material price, desirable physical prop-erties, and low processing costs in one product. The material is con-structed of aliphatic linkages connected by amide linkages to form apolymer of molecular weight between 25,000 and 40,000. Nylon 6,6 issynthesized from carbon hexamethylenediamine and adipic acid, both ofwhich have six carbons, hence the 6,6 designation.

heatH2N(CH2)6NH2+ HOOC(CH2)4COOH---+

-[NH(CH2)6NHOC(CH2)4CO]x- + H2O

The material is semicrystalline and thermoplastic. It possesses manydesirable properties for cage applications: strength, toughness, abrasionresistance, chemical resistance, and impact resistance. The resin is some-what hygroscopic (to 3%), and absorbed water causes dimensionalchanges that must be considered during cage design.

Product modifications containing additives are abundant for nylon 6,6.The variations provide improved physical properties, environmental in-ertness, and improved processing characteristics. Being thermoplastic, itis an injection-moldable resin allowing direct production of complex cageshapes with obvious cost advantages. In general, resin compatibility withlubricants is very good. Cages formed from this resin exhibit a high de-gree of flexibility, which allows easy assembly and operation under mis-alignment ofthe inner and outer bearing rings. Glass fiber reinforcement

CAGE MATERIALS 629

is often used with the resin at levels of 25% fill. The glass fiber givesbetter retention of strength and toughness at high temperatures, butwith loss of flexibility.

Rolling bearings selected from manufacturers' catalogs are designedto operate in wide varieties of applications. Therefore, the strength/toughness properties afforded to nylon 6,6 cages by glass-fiber reinforce-ment are required for bearing series employing such cage material. Fig-ure 16.26 from [16.32] illustrates the endurance capability of 25%glass-fiber-filled nylon 6,6 as a function of operating temperature. In Fig.16.26, the "black band" indicates the spread determined with variouslubricants. The lower edge of the band is applicable for aggressive lubri-cants such as transmission oils (with EP additives), while the upper edgepertains to mild lubricants such as motor oils and normal greases. Table16.5 from [16.32] indicates the strength, thermal, chemical, and struc-tural properties of this material in the dry and conditioned states. Theconditioned state is that in which some water has been adsorbed. Bycomparison of Fig. 16.26 with Table 16.5, it can be seen that the per-missible operating temperature of 120°C (250°F) corresponds to a proba-ble endurance of approximately 5000 to 10,000 hr depending uponlubricant type. This refers to continuous operation at 120°C (250°F); op-eration at lesser temperatures will extend satisfactory cage performancefor greater duration.

SEAL MATERIALS 631

polybutylene terephthalate (PBT), polyethylene terephthalate (PET),polyethersulfone (PES), polyamideimide (PAD, and polyether-etherketone (PEEK). Of these materials only PES and PEEK have dem-onstrated sufficient promise as high temperature bearing cage materials;these materials are discussed in further detail below.

Polyethersulfone is a high-temperature thermoplastic material withgood strength, toughness, and impact behavior for cage applications. Theresin consists of diaryl sulfone groups linked together by ether groups.The structure is wholly aromatic, providing the basis for excellent high-temperature properties. Being thermoplastic, it is processible using con-ventional molding equipment. This allows direct part production; that is,without subsequent machining or finishing. In lubricant-temperature ex-posure tests the resin has performed well to 170°C(338°F).The materialis suitable for applications using petroleum and silicone lubricants; how-ever, there are some problems with polymer degradation after exposureto ester-based lubricants and greases. The properties of PES are alsoshown in Table 16.5; it can be seen that PES is not as strong as nylon6,6. When it is desired to use a "snap-in" type assembly of balls or rollersin a one-piece cage as illustrated in Fig. 16.25, this somewhat lesserstrength can result in crack formation during assembly of the bearing.

Polyether-etherketone is a wholly aromatic thermoplastic that showsexcellent physical properties to 250°C (482°F). It is particularly good forcage applications because of its abrasion resistance, fatigue strength, andtoughness. It is a crystalline material and can be injection molded. Lu-bricant compatibility tests show excellent performance to 200°C (392°F)and above. Tests also indicate antiwear performance equal to or betterthan nylon 6,6. Table 16.5 compares the properties of PEEK with thoseof PES and nylon 6,6. The only known drawback to the extensive use ofPEEK as a bearing cage material is cost. This currently restricts its useto specialized applications. See [16.31].

SEAL MATERIALS

Function, Description, and Illustration

To prevent lubricant loss and contaminant ingress, manufacturers pro-vide bearings with sealing. The effectiveness of the sealing has a criticaleffect on bearing endurance. When choosing a sealing arrangement fora bearing application, rotational speed at the sealing surface, seal fric-tion and resultant temperature rise, type of lubricant, available volume,environmental contaminants, misalignment, and cost must all be consid-ered.

A bearing can be protected by an integral seal consisting of an elas-tomeric ring with a metallic support ring, the elastomer riding on an


inner ring surface (see Fig. 17.14), or by a stamped shield of mild steelstaked into the outer ring and approaching the inner ring closely but notin intimate contact with it (see Fig. 17.13).

Shields cost less and do not increase torque for the bearing in opera-tion. This design is useful for excluding gross particulate contamination(150 /Lm). Often used with greased bearings, it is used in bearings lu-bricated by liquids that must pass through the bearing. The seal config-uration is more expensive because of design and materials. Dependingon seal lip design, it adds to bearing friction torque to a greater or lesserextent. Seals are used in greased bearings when moisture and all con-tamination must be excluded. They are also the best choice for minimiz-ing grease purging.

Elastomeric Seal Materials

Because of the prevalence of elastomeric seals in rolling bearings, a va-riety of materials has been developed to meet the requirements of dif-fering applications. Important properties of elastomeric seal materialsinclude lubricant compatibility, high- and low-temperature performance,wear resistance, and frictional characteristics.

Table 16.6 summarizes physical properties, and Table 16.7 lists gen-eral application guidelines.

In the followingdiscussion of elastomeric types, it is important to notethat compounding variations starting with a particular elastomer typecan lead to products of distinct properties. The general inputs to a for-mulated compound may be taken as follows:

Elastomer-basic polymer that determines the ranges of final productproperties

Curing agents, activators, accelerators-determine degree and rate ofelastomeric vulcanization (cross-linking)

Plasticizers-improve flexibility characteristics and serve as process-ing aids

Antioxidants-improve antifatigue and antioxidation properties ofproduct

"Nitrile" rubber represents the most widely used elastomer for bearingseals. This material, consisting of copolymers of butadiene and acrylo-nitrile, is also known as Buna Nand NBR. Varying the ratio ofbutadieneto acrylonitrile has a major effect on the final product properties.

The general polymer reaction can be represented as


into the structure in varying amounts. Typical organic substitutes aremethyl, phenyl, and vinyl. If R=CH3, the polymer is dimethylpolysilox-ane.

Advantages of silicon rubber seal use are high-temperature perform-ance to 180°C (356°F) and good low-temperature flexibility to -60°C(-76°F). The material is nontoxic and inert; hence it is chosen for food,beverage, and medical applications. It is stable with regard to the effectsof repeated high temperature. Their excellent low-temperature flexibilitymakes these elastomers useful for very low-temperature applicationswhere sealing is required.

Silicon rubbers are very expensive compared to nitrile rubbers. Lu-bricant resistance and mechanical strength are poor for most seal appli-cations. On the whole, silicon elastomers have limited usefulness.

Fluoroelastomers have become increasingly popular as seal materialsbecause of excellent high-temperature and lubricant-compatibility char-acteristics. A typical polymer of this class is the copolymer of vinylidinefluoride and hexafluoropropylene, which an be represented as

Materials of this general type have become common for bearing seal ap-plications at temperatures exceeding 130°C (266°F). Suitably compoundfluoroelastomers show good wear resistance and water resistance forbearing seal applications. As would be expected, material cost is veryhigh compared to nitrile rubbers.

SURFACE TREATMENTS FOR BEARING COMPONENTS

Coatings in General

Several coatings exist to improve surface characteristics of bearing orbearing components without affecting the gross properties of the bearingmaterial. Within the realm of standard bearing applications, coatings areused to provide wear resistance, initial lubrication, sliding characteris-tics and cosmetic improvements. In addition, bearings operating in ex-treme environments of temperature, wear, or corrosivity can be speciallytreated.

Phosphate Coating

Zinc and manganese phosphate coatings are applied to finished bearingsand components to provide

SURFACE TREATMENTS FOR BEARING COMPONENTS 639

Increased corrosion protection by providing a porous base for preser-vative oils

Initial lubrication during bearing run-in by preventing metal-to-metalcontacts and providing a lubricant reservoir

Prepared surfaces for other surface coatings-that is, MoS2

Parts are immersed in acidic solutions of metal phosphates at tem-perature. This produces a conversion coating integrally bonded to thebearing surface. The coated surface is now nonmetallic and nonconduc-tive. The zinc phosphate process gives a finer structure, which may bepreferred cosmetically. The manganese phosphate process gives a heavierstructure that is generally preferred for wear resistance and lubricantretention. Phosphating in itself does not provide for substantial improve-ments in rust protection. It is only when a suitable preservative is em-ployed that full benefits are obtained.

Black Oxide

Black oxide conversions have been used on bearings and components for

Cosmetic uniformity appearance to componentsLubrication during run-inRust protection during extended storage

Black oxide is a generic term referring to the formation of a mixture ofiron oxides on a steel surface. An advantage of the process is that nodimensional change results from the process, so tolerances can be main-tained after treatment.

A common approach to obtain this coating consists of treating a steelcomponent in a highly oxidizing bath. Because the chemical process re-sults in dissolution of surface iron, close process control is necessary toprevent objectionable surface damage. The black color is obtained fromthe presence of Fe304'

Plating Processes

Both electroplating and electroless disposition have long been employedin the rolling bearing industry to provide wear-resistant coatings forcages. In response to aircraft bearing requirements, silver plating overa nickel- or copper-struck cage is commonplace. In this case the strikemetal provides an oxygen barrier to the base metal to prevent corrosion.The silver plating offers reduced friction. Cadmium, tin, and chrome plat-ing are also used for certain bearings and accessories.


Extreme Environment Coatings

Several coating techniques and materials somewhat reduce sliding fric-tion and markedly improve wear and corrosion resistance in extremeenvironments. The techniques include physical vapor deposition (PVD),chemical vapor deposition (CVD), and special process electroplating.Coating materials include titanium nitride (TiN), titanium carbide (TiC),and hard chromium. Some of these process-coating material combina-tions have demonstrated excellent performance on rolling contact sur-faces.

Chemical Vapor Deposition

In chemical vapor deposition of TiC, titanium tetrachloride is vaporizedand allowed to react with the substrate at high temperatures in the pres-ence ofhydrogen and methane gas. Typically,CVDprocessing needs tem-peratures of 850-1050°C (1562-1922°C). Although these temperatureswill promote diffusion with the substrate, the processing temperaturesexceed the tempering temperatures of bearing steels requiring heattreating after coating. Postcoating heat treatment may cause dimen-sional distortion. This post treatment of the CVDcoating diminishes theattractiveness of this process for bearing components.

Physical Vapor Deposition

The principal advantage of PVD over CVD is that substrate tempera-tures below 550°C (1022°F) are used. High bond strength with PVD isachieved with ion bombardment of the substrate surface. Consequently,postcoating heat treatment ofhigh speed steels is not required. Thereforethere has been considerable interest in applying PVD coatings to bearingsurfaces, with TiN being a usual coating. Excellent bonding with bearingsteel and compatibility with a high contact stress environment have beenachieved.

Special Process Chromium Electrodeposition

Super chrome-plating techniques have been developed that produce coat-ings free of the surface cracks that characterize conventional hardchrome deposits. Increased corrosion resistance is reported for the coatedbearing steel; the coating does not negatively affect the rolling contactfatigue life. Substrate temperature is below 66°C (151°F)during plating,and the coating, with a reported hardness of Rc 70 deforms plasticallyrather than cracking when overloaded.

CLOSURE 641

CLOSURE

An operating rolling bearing is a system containing rings, raceways, roll-ing elements, cage, lubricant, seals, and ring support. In general, balland roller bearings selected from listings in manufacturers' catalogsmust be able to satisfy broad ranges of operating conditions. Accordingly,the materials used must be universal in their applicability. Through-hardened AISI 52100 steel, nylon 6,6, lithium-based greases, and so on,are among the materials that have met the test of universality for manyyears. Moreover, these materials as indicated in this chapter have un-dergone significant improvement, particularly in the past few decades.

For special applications involving extra heavy applied loading, veryhigh speeds, high temperatures, very low temperatures, severe ambientenvironment, and combinations of these, the bearing system materialsmust be carefully matched to each other to achieve the desired opera-tionallongevity. In an aircraft gas turbine engine mainshaft bearing forexample, it is insufficient that the M50 or M50-Nil bearing rings providelong-term operating capability at engine operating temperatures andspeeds; rather, the bearing cage materials and lubricant must also sur-vive for the same operating period. Therefore cages for such applicationsare generally fabricated from tough steel and are silver plated; nyloncages are precluded by the elevated operating temperatures and possiblyby incompatibility with the lubricant. The upper limit of bearing oper-


ating temperature is established by the lubricant; in most cases this isa synthetic oil according to United States military specification MIL-L-23699 or MIL-L-7808.

An example of an extreme operating condition is the liquid oxygen(LOX) turbopump for the space shuttle main engine as shown in Fig.16.27. In this application, the bearings must rotate at very high speed(30,000 rpm) while being lubricated by LOX. The LOX vaporizes in theconfines of the bearing, and the bearing tends to burn up and wear not-withstanding the initial cryogenic temperature [-150°C (-302°F)] of theLOX. To achieve sufficient duration of satisfactory operation, the ballbearing cage has been fabricated from Armalon, a woven fiberglass-reinforced PTFE material [16.33] that lubricates by transfer ofPTFE filmfrom the cage pockets to the balls. The bearing rings are fabricated fromvacuum-melted AISI 440C stainless steel. Target duration for bearingoperation is only a few hours.

REFERENCES16.1. American Society for Testing and Materials, Std A295-84, High Carbon Ball and

Roller Bearing Steels; Std A485-79, "High Hardenability Bearing Steels."16.2. American Society for Testing and Materials, Std A534-79, "Carburizing Steels for

Anti-Friction Bearings."

16.3. J. Braza, P. Pearson, and C. Hannigan, "The Performance of 52100, M-50, and M50-NiL Steels in Radial Bearings," SAE Technical Paper 932470 (September 1993).

16.4. E. Zaretsky, "Bearing and Gear Steels for Aerospace Applications," NASATechnicalMemorandum 102529 (March 1990).

16.5. W.Trojahn, E. Streit, H. Chin, and D. Ehlert, "Progress in Bearing Performance ofAdvanced Nitrogen Alloyed Stainless Steel," in Bearing Steels into the 21st Century,ed. J. Hoo, ASTM STP 1327 (1997).

16.6. H.-J. Bohmer, T. Hirsch, and E. Streit, "Rolling Contact Fatigue Behavior of HeatResistant Bearing Steels at High Operational Temperatures," in Bearing Steels intothe 21st Century, ed. J. Hoo, ASTM STP 1327, (1997).

16.7. C. Finkl, "Degassing-Then and Now," Iron and Steelmaker, 26-32 (December1981).

16.8. T.Morrison, T. Tallian, H. Walp, and G. Baile, "The Effect of Material Variables onthe Fatigue Life ofAISI 52100 Steel Ball Bearings," ASLE Trans., 5,347-364 (1962).

16.9. United States Steel Corp., Making, Shaping, and Treating of Steel, 9th ed., 551(1971).

16.10. United States Steel Corp., Making, Shaping, and Treating of Steel, 9th ed., 596-597 (1971).



16.13. J. Akesson and T. Lund, "Rolling Bearing Steelmaking at SKF Steel," TechnicalReport 7 (1984).

REFERENCES 643


16.15. J. Akesson and T. Lund, "SKF Rolling Bearing Steels-Properties and Processes,"Ball Bearing J. 217, 32-44 (1983).

16.16. American Society for Testing and Materials, Std E45-81, "Standard Practice forDetermining the Inclusion Content of Steel."

16.17. J. Beswick, "Effect of Prior Cold Work on the Martensite Transformation in SAE52100," Metall. Trans. A, 15A, 299-305 (1984).

16.18. R. Butler, H. Bear, and T.Carter, "Effect of Fiber Orientation on Ball Failures underRolling-Contact," NASA TN 3933 (1975).

16.19. SKF Steel, The Black Book, 194 (1984).16.20. SKF Steel, The Black Book, 151 (1984).

16.21. M. Grossman, Principles of Heat Treatment, American Society for Metals (1962).16.22. American Society for Testing and Materials, Std A255·67, "End-Quench Test for

Hardenability of Steel" (1979).

16.23. American Society for Metals, Atlas of Isothermal Transformation and CoolingTransformation Diagrams (1977).

16.24. T. Tallian, Failure Atlas for Hertz Contact Machine Elements, ASME Press. NewYork (1992).

16.25. G. Winspiar, ed., The Vanderbilt Rubber Handbook, R.T. Vanderbilt, New York(1968).

16.26. Modern Plastics Encyclopedia, McGraw-HilI, New York (1985-1986).16.27. Metal Finishing Guidebook and Directory 85, Metals and Plastics Publications,

Hackensack, N.J. (1985).

16.28. A. Graham, Electroplating Engineering, 3rd ed., Van Nostrand Reinhold, New York(1971).

16.29. R. Spitzer, "New Case-Hardening Steel Provides Greater Fracture Toughness," BallBearing J., SKF, 234, 6-11 (July 1989).

16.30. R. Pallini, "Turbine Engine Bearings for Ultra-High Temperatures," Ball BearingJ., SKF, 234, 12-15 (July 1989).

16.31. A. Olschewski, "High Temperature Cage Plastics," Ball Bearing J., SKF, 228,13-16 (November 1986).

16.32. H. Lankamp, "Materials for Plastic Cages in Rolling Bearings," Ball Bearing J.,SKF, 227, 14-18 (August 1986).

16.33. R. Maurer and L. Wedeven, "Material Selection for Space Shuttle Fuel Pumps," BallBearing J., SKF, 226, 2-9 (April 1986).

16.34. Delta Rubber Company, Elastomer Selection Guide.

646 LUBRICANTS AND LUBRICATION TECHNIQUES

of wear. This can be accomplished through various lubricating mecha-nisms such as hydrodynamic lubrication, elastohydrodynamic lubrication(EHL), and boundary lubrication. The rolling/sliding contacts of concernare those between rolling element and raceway, rolling element and cage(separator), cage and supporting ring surface, and roller end and ringguide flanges.

In addition to wear prevention the lubricant performs many other vi-tal functions. The lubricant can minimize the frictional power loss of thebearing. It can act as a heat transfer medium to remove heat from thebearing. It can redistribute the heat energy within the bearing to mini-mize geometrical effects due to differential thermal expansions. It canprotect the precision surfaces of the bearing components from corrosion.It can remove wear debris from the roller contact paths. It can minimizethe amount of extraneous dirt entering the roller contact paths, and itcan provide a damping medium for separator dynamic motions.

No single lubricant or class oflubricants can satisfy all these require-ments for bearing operating conditions from cryogenic to ultrahigh tem-peratures, from very slow to ultrahigh speeds, and from benign to highlyreactive operating environments. As for most engineering tasks, a com-promise is generally exercised between performance and economic con-straints. The economic constraints involve not only the cost of thelubricant and the method of application but also its impact on the lifecycle cost of the mechanical system.

Cost and performance decisions are frequently complicated becausemany other components of a mechanical system also need lubrication orcooling, and they might dominate the selection process. For example, anautomobile gearbox typically comprises gears, a ring synchronizer, roll-ing bearings of several types operating in very different load and speedregimes, plain bearings, clutches, and splines.

TYPES OF LUBRICANTS

Selection Criteria

The selection of lubricants is based on their flowproperties and chemicalproperties in connection with lubrication. Additional considerations,which sometimes may be of overriding importance, are associated withoperating temperature, environment, and the transport or retentionproperties of the lubricant in the bearing.

Liquid Lubricants

Liquid lubricants are usually mineral oils; that is, fluids produced frompetroleum-based stocks. They have a wide range of molecular constitu-

TYPES OF LUBRICANTS 647

ents and chain lengths, giving rise to a large variation in flowpropertiesand chemical performances. These lubricants are generally additive en-hanced for both viscous and chemical performance improvement. Overall,petroleum-based oils exhibit good performance characteristics at rela-tively inexpensive costs.

Synthetic hydrocarbon fluids are manufactured from petroleum-basedmaterials. They are synthesized with both narrowly limited and specif-ically chosen molecular compounds to provide the most favorable prop-erties for lubrication purposes. Most synthetics have unique propertiesand are made from petroleum feedstocks, but they can be made fromnon-petroleum sources. Other "synthetic" fluids have unique propertiesand can be manufactured from non-petroleum-based oils. These includepolyglycols, phosphate esters, dibasic acid esters, silicone fluids, silicateesters, and fluorinated ethers.

GreasesGreases have two major constituents: an oil phase and a thickener sys-tem that physically retains the oil by capillary action. The thickener isnormally composed of a material with very long twisted and/ or contortedmolecules that both physically interlock and provide the necessarilylarge surface area to retain the oil. The resultant material behaves as asoft solid, capable of bleeding oil at controlled rates to meet the con-sumption demands of the bearing.

Polymeric Lubricants

Polymeric lubricants are related to greases in that these materials con-sist of an oil phase and a retaining matrix. They differ in one crucialpoint: the matrix is a true solid sponge that retains its physical shapeand location in the bearing. Lubrication functions are provided by theoil alone after it has bled from the sponge. The oil content can be madehigher than in greases, and a greater quantity can be installed in thevoid space within the bearing. This greater oil volume portends longerbearing life before all fluid is consumed by oxidation, evaporation, orleakage. The latter is particularly significant for vertical axis bearingapplications.

Solid Lubricants

Solid lubricants are substituted for liquid lubricants when extreme en-vironments such as high temperature or vacuum make liquid lubricantsor greases impractical. Solid lubricants, unless melted, do not utilize themechanism of hydrodynamic or EHL. Their performance is less predict-able, and there is generally much greater heat generation due to friction.


Solid lubricants perform as boundary lubricants consisting of thin layersthat provide lower shear strength than the bearing materials. Solid lu-bricants can consist oflayered structures that shear easily or nonlayeredstructures that deform plastically at relatively low temperatures. Graph-ites and molybdenum disulfide (MoS2) are common examples ofmaterialswith layered structures. Fluorides such as calcium fluoride (CaF

2) are

nonlayered materials that perform well at or near their melting temper-atures.

LUBRICATION METHODS

Oil Sump or Bath

Decisions in connection with the selection of lubricants must paralleldecisions in connection with the supply of the lubricant to the bearingfor maintaining conditions that will prevent rapid deterioration of thelubricant and bearing. An oil sump applicable to horizontal, inclined, andvertical axis arrangements provides a small pool of oil contained in con-tact with the bearing, as in Fig. 17.1.

The liquid level in the stationary condition is arranged to just reachthe lower portion of the rolling elements. Experience has shown thathigher levels lead to excessive lubricant churning and resultant excessivetemperature. This churning in turn can cause premature lubricant oxi-dation and subsequent bearing failure. Lower liquid levels threaten oilstarvation at operating speeds where windage can redistribute the oiland cut off communication with the working surfaces. Maintenance ofproper level is thus very important and provision of a "sight" glass isrecommended.

Oil bath systems are used at low-to-moderate speeds where grease isruled out by short relubrication intervals, hot environments, or wherepurging of grease could cause problems. Heat dissipation is somewhatbetter than for a greased bearing due to fluid circulation, offering im-

LUBRICATION METHODS 649

proved performance under conditions ofheavy load where contact frictionlosses are greater than the lubricant churning losses. This method isoften used when conditions warrant a specially formulated oil not avail-able as a grease. A coolingcoil is sometimes used to extend the applicabletemperature range of the oil bath. This usually takes the form of a water-circulating loop or, in some recent applications, the fitting of one or moreheat pipes.

Wick-feed and oil-ring methods of raising oil from a sump to feed thebearing are not generally used with rolling bearings, but, occasionallyshaft motion is used to drive a viscous pump for oil elevation, thus re-ducing the sensitivity of the system to oil level. A disc dipping into thesump drags oil up a narrow groove in the housing to a scraper blade orstop that deflects the oil to a drilled passage leading to the bearing. Amajor limitation of all sump systems is the lack of filtration or debrisentrapment. Fitting a magnetic drain plug is advantageous for control-ling ferrous particles, but otherwise sump systems are only suitable forclean conditions.

Circulating Oil Systems

As the speeds and loads on a bearing are increased, the need for delib-erate means of cooling also increases. The simple use of a reservoir anda pump to supply a lubricant flow increases the heat dissipation capa-bilities significantly. Pressure feed permits the introduction of appropri-ate heat exchange arrangements. Not only can excess heat be removed,but heat can be added to assure flow under extremely cold start-ups.Some systems are equipped with thermostatically controlled valves tokeep the oil in an optimum viscosity range.

Equally important, a circulating system can be fitted with a filtrationsystem to remove the inevitable wear particles and extraneous debris.The mechanisms of debris-induced wear and the effects of even micro-scale indentations on the EHL processes and the consequent reductionsin fatigue life are discussed in Chapters 23 and 24. Finer filtration isbeing introduced in existing circulating systems with beneficial effects;however, increased pressure drops, space, weight, cost, and reliabilityhave to be considered.

Circulating systems are used exclusively in critical high-performanceapplications, of which the main shaft support bearings of an aircraft gasturbine engine constitute prime examples. Subjected to heavy thrusts atnear limiting speeds, the angular-contact ball bearings generate consid-erable frictional heat, particularly at the inner raceway contacts. Thisheat must be removed effectively together with leakage heat conductedto the bearing cavity from the surrounding engine components.

Heavily loaded bearings running at moderate speeds can be suppliedwith oil jets aimed at the rolling elements. At higher speeds, bearingwindage deflects the jets, and lubrication and coolingbecome ineffective.


This problem can be avoided by routing the oil to pickup scoops on theshaft with centrifugal force taking the oil via drilled passages to theinner ring, as shown in Fig. 17.2. Much of the flowpasses through axialslots in the bore of the inner ring, removing heat as it does so. Only asmall portion of the lubricant is metered to the rolling contacts throughgrooves between the inner ring halves. Separate drilled holes may beused to supply the cage lands.

Adequate space should be provided on both sides of the bearing tofacilitate lubricant drainage. Often, space is at a premium, so a systemof baffles can be substituted to shield the lubricant from the windage,permitting it to be scavenged without severe churning. When the lubri-cant pump is activated at the same time as the main machinery, thesebaffles act as a dam and retain a small pool of lubricant in the bottomof the bearing to provide lubrication at start-up until the circulating flowbecomes established.

Hydrocarbon-based fluids are satisfactory for circulating lubricant sys-tems operating at temperatures to about 274°C (525°F).Hydrocarbon ox-idation starts at room temperature, and the lifetime of the lubricantdepends on the temperature. Oxidation becomes significant at 175°C(347°F), incipient thermal decomposition starts at about 300°C (572°F),


and becomes a significant problem at about 449°C (840°F). Use of aninert cover gas to exclude oxygen can extend the working range to thelatter limit. Beyond 449°C (840°F), fluorocarbon-based fluids are service-able, but conspicuously lack the lubrication properties ofthe hydrocarbonlubricants. They have the same thermal stability problem of hydrocar-bon-based fluids, but have superior oxidation stability. Up to this time,they have not been able to reach the temperature limits inherent in thetool steels.

Once-Through Systems

A separate class oflubrication arrangements can be used when minimumbearing friction is essential at moderate-to-high speeds and where loadsare sufficiently low that heat removal is not a major concern. Lubricantis delivered to the bearing as a fine spray or an air-entrained mist injust sufficient quantities to maintain the necessary lubricant films in thecontacts. Lubricant churning is virtually eliminated, and the volume oflubricant is so small that it can be discarded after a single passagethrough the bearing. Scavenging, cooling, and storage facilities are un-necessary. The one-time exposure to high shear stresses and/or temper-atures relaxes the oxidation and the stability requirements of the fluidto some extent. The necessity for satisfactory air quality in the workplacerequires that the exhaust droplets be reclassified and lubricant collectedbefore discharge.

Recent work has shown that the spray does not even need to be con-tinuous. Trace injection of minute quantities of lubricant at intervals ofup to one hour is sufficient to keep precision spindle assembles runningat friction torque levels unobtainable by any other method.

Grease Application

In the majority of rolling bearing applications grease lubrication can beemployed. In ease of application, grease has some advantages comparedto oil in that it is easily retained, and it also helps seal the bearingoperating surfaces from particulate contaminants and moisture. Greaselubrication is, however, in general restricted to relatively slower speedapplications owing to reduced capability for frictional heat dissipation ascompared to oil; hence, limiting speeds as shown in bearing catalogs areless for grease lubrication than for oil lubrication. Moreover, care mustbe exercised when charging a bearing with grease. Toomuch grease willcause a rapid temperature rise and potential bearing seizure. Therefore,while the bearing free space may be filled with grease, the surroundingspace in the housing in general should only be partially (30-50%) filled.For very slow speed operation, to provide maximum corrosion protection,the housing may be completely filled.


If the service life of the grease used to lubricate the bearing is lessthan the expected bearing life, the bearing needs to be relubricated priorto lubricant deterioration. Relubrication intervals are dependent uponbearing type, size, speed, operating temperature, grease type, and theambient conditions associated with the application. As operating condi-tions become more severe, particularly in terms of frictional heat gen-eration and operating temperature, the bearing must be relubricatedmore frequently. Some manufacturers specify relubrication intervals fortheir catalog bearings: for example, reference [17.1]. Such recommen-dations, given in the form of charts, are specific to the manufacturer'sbearing internal designs and are generally based on goodquality, lithiumsoap-based greases (see "Grease Lubricants") operating at temperaturesnot exceeding 70°C (158°F). It is interesting to note that for every 15°C(27°F) above 70°C (158°F) relubrication intervals must be halved. Bear-ings operating at temperatures lower than 70°C (158°F) tend to requirerelubrication less often; however, the lower operating temperature limitof the grease may not be exceeded [-30°C (-22°F) for a lithium-basedgrease.]. Also bearings operating on vertical shafts need to be relubri-cated approximately twice as often as bearings on horizontal shafts. (Re-lubrication interval charts are generally based on the latter application.)It is presumed that in no case is the grease upper operating temperaturelimit exceeded; this limit is HO°C (230°F) for a lithium-based grease.

Because relubrication intervals depend on specificbearing internal de-sign features such as rolling element proportions, working surface fin-ishes and cage configuration, they are different for each manufacturereven for basic bearing sizes. Therefore no such charts are given in. thistext; they may be found in manufacturers catalogs. Of course, the meth-ods given in Chapter 15 may be used to estimate the operating bearinggrease temperature in a given application, and the grease manufac-turer's recommendations for replenishment may be employed.

If the relubrication interval is greater than 6 months, then all of theused grease should be removed from the bearing arrangement and re-placed with new grease. If the interval is less than 6 months, then anincremental grease charge may be added according to the bearing man-ufacturer's recommendation. The 6 month limit is understood to be arough guideline.

Relubrication interval requirements vary significantly according to thetypes ofgrease used and even where apparently similar greases are used.For small ball bearings, the relubrication interval is often longer thanthe life of the bearing application, and relubrication is not normally re-quired. Ball bearings fitted with seals that are lubricated for life areoften used in such cases. Where there is definite risk of contamination,the recommended relubrication intervals should be reduced. This alsoapplies to applications in which the grease is required to seal againstmoisture.


Solid Lubrication Application

In applications where conventional lubricants are not appropriate, thinsolid films of soft or hard materials are applied to bearing surfaces toreduce friction and enhance wear resistance of contacting surfaces. Thereare many methods of applying solid lubricants, each of which providesvarying degrees of success with respect to adhesion to the substrate,thickness, and uniformity of coverage.

Resin-bonded solid lubricants are very commonly used. These mate-rials usually consist of a lubricating solid and a bonding agent. The lu-bricating solid may be a single material or a mixture of several materials.It can be applied in a thin film by spraying or dipping. Depending on thebinding agent used, it may be a heat-cured or air-cured material. Heat-cured materials are generally superior to the air-dried materials. Metalsurfaces are usually pretreated prior to application. Pretreatment maybe chemical or mechanical; the latter tends to increase the surface area,which gives the binder greater holding power.

The application of solid lubricants frequently relies on the transfer ofthin solid films from one contacting surface to another. The interactionof rolling elements with a solid-lubricated or impregnated separatortransfers thin solid films to the rolling elements, which in turn are trans-ferred to the rolling contact raceways. When the rubbing action againstthe solid lubricant occurs with sufficient load, the solid lubricant willcompact itself into the existing surface imperfections. This burnishingaction provides little control over film thickness and uniformity of cov-erage.

Much greater control of solid lubricant film thickness, composition,and adhesion can be obtained by using various electrically assisted thin-film deposition techniques. These include ion plating, activated reactiveevaporation, dc and rf sputtering, magnetron sputtering, arc coating, andcoating with high-current plasma discharge. Coatings of virtually all ofthe soft metals and hard materials and a number of nonequilibrium ma-terials can be produced with one or another of the electrically assisted,film deposition techniques. When vacuum techniques of deposition areused, the vapor of the solid lubricant species being deposited can be re-acted with process gases to produce various synthesized compounds.

Application of Polymeric Lubricants

Two approaches are used to apply polymeric lubricants. The first ap-proach is to make a suitably shaped part from a porous material, eitherby molding or machining, and to place it in the bearing in one or morepieces. A vacuum impregnation process then charges the material withlubricant. The need to insert the porous structure governs the amountof bearing free space that can be used. Often rivets or other fasteners


must be used to hold the pieces in place, further reducing the volumeavailable.

The second method entails the formation of a lubricant-saturated rigidgel in the bearing itself by filling the bearing with the fluid mixture andusing a curing or polymerization step to effect a transition to a solidstructure. Essentially all the void space in the bearing can be used. Onlya very few polymers have been identified that will function in this man-ner. Further, there appears to be a tradeoff between bleeding, shrinkage,strength, and temperature limit characteristics.

LIQUID LUBRICANTS

Advantages Over Other Lubricants

As compared to any other lubricant, in particular grease, a liquid lubri-cant provides the following advantages:

1. It is easier to drain and refill, a particular advantage forapplications requiring short relubricating intervals.

2. The lubricant supply to the system can be more accurately con-trolled.

3. It is suitable for lubricating multiple sites in a complex system.4. Because of its ability to be used in a circulating lubricant system,

it can be used in higher temperature systems where its ability toremove heat is significant.

Guidelines for Use

In most applications pure petroleum oils are satisfactory as lubricants.They must be free from contamination that might cause wear in thebearing, and should show high resistance to oxidation, gumming, anddeterioration by evaporation. The oil must not promote corrosion of anyparts of the bearing during standing or operation.

The friction torque in a liquid-lubricated bearing is a function of thebearing design, the load imposed, the viscosity and quantity of the lu-bricant, and the speed of operation. Only enough lubricant is needed toform a thin film over the contacting surfaces. Friction torque will in-crease with larger quantities and with increased viscosity of the lubri-cant.

Energy loss in a bearing depends on the product of torque and speed.It is dissipated as heat, causing increased temperature of the bearingand its mounting structures. The temperature rise will always cause adecreased viscosity of the oil and, consequently, a decrease in friction

LIQUID LUBRICANTS 655

torque from initial values. The overall heat balances of the bearing andmounting structures will determine the steady-state operating condi-tions.

It is not possible to give definite lubricant recommendations for allbearing applications. A bearing operating throughout a wide tempera-ture range requires a lubricant with high viscosity index-that is, havingthe least variation with temperature. Very low starting temperaturesnecessitate a lubricant with a sufficiently low pouring point to enablethe bearing to rotate freely on start-up. For specialized bearing appli-cations involving unusual conditions, the recommendation of the bearingor lubricant manufacturer should be followed.

Mineral Oils

Mineral oil is a generic term referring to fluids produced from petroleumoils. Chemically, these fluids consist of paraffinic, naphthenic, and aro-matic groups combined into many molecules. See Fig. 17.3. Also presentin crude stocks are trace amounts ofmolecules containing sulfur, oxygen,or nitrogen. Elementally, the composition of petroleum oils is quite con-stant: 83-87% carbon, 11-14% hydrogen, and the remainder sulfur, ni-trogen, and oxygen. The molecular makeup of the fluid is very complexand depends on its source.

For the purpose oflubricant production, crude petroleum oils are char-acterized by the type ofhydrocarbon distillates obtained. For this methodit is common to speak of paraffinic, mixed, and naphthenic crude oils.Aromatics are generally a minor component. Depending on the source,the crude petroleum may consist of gasoline and light solvents, or it mayconsist of heavy asphalts. Modern distillation, refining, and blendingtechniques allow the production of a wide range of oil types from a givencrude stock; however, some crude stocks are more desirable for lubricantformulation.

With respect to lubricant properties, a few generalizations can bemade. Paraffinic base crudes have the viscosity-temperature character-istics for lubrication. Usually such crudes are low in asphalt and tracematerials. The earliest commercial crude, Pennsylvania, was of this type.


Naphthenic-based crudes do not contain paraffin waxes, so they are bet-ter suited for lower temperature application. Naphthenic oils also havelower flash points and are more volatile than comparable paraffinic oils.

The performance of the base fluid for mineral oils as well as for syn-thetic fluid lubricants depends to a great degree on the type of additivesincorporated into the system. Antioxidants, corrosion inhibitors, anti-foam additives, and friction and wear-minimizing additives employedvary depending on the specific purpose of the lubricant. The two mostcommon lubricants used for industrial rolling bearing applications maybe described as rust and oxidation (R & 0) inhibited oils and extremepressure (EP) oils. The R & 0 oils are often used when bearings andgears share a common lubricant reservoir. These products may incorpo-rate antifoam and antiwear agents. They are suitable for light-to-moderate loadings and for temperatures from -20 to 1200e (-4 to24BOF).

Extreme pressure oils usually encompass the additive package of R &o oil with an additional EP additive. The EP additive essentially gen-erates a lubricating surface to prevent metal-to-metal contact. Two ap-proaches exist in formulating EP additives. The first employs an activesulfur, chlorine, or phosphorus compound to generate sacrificial surfaceson the bearing itself. These surfaces will shear rather than weld uponcontact. The second approach uses a suspended solid lubricant to imposebetween two otherwise contacting surfaces.

Extreme pressure oils are used where bearing (or associated gear)loadings are high or where shock loadings may be present. The normaltemperature range for such lubricants is -20 to 1200e (-4 to 24BOF).Some precautions are necessary when using EP oils of either type. EPsolids will reduce internal clearances that can cause failure in certainbearings. These solids might also be lost in close filtration processes. EPsulfur-chlorine-phosphorus compounds might be corrosive to bronzecages and accessory items.

Synthetic Hydrocarbons

Synthetic hydrocarbons are manufactured petroleum fluids. Being syn-thesized products, the particular compounds present can be both nar-rowly limited and specifically chosen. This allows production of apetroleum fluid with the most favorable properties for lubrication pur-poses. One commercially important type is the polyalphaolefin fluids,which have been widely used as turbine lubricants, as hydraulic fluids,and in grease formulations.

These fluids show improved thermal and oxidation stability over re-fined petroleum oils, allowing higher temperature performance for lu-bricants compounded from them. These materials also exhibit inherentlyhigh viscosity indexes, leading to better viscosity retention at elevated

LIQUID LUBRICANTS 657

temperatures. Other properties showing improvement include flashpoint, pouring point, and lowvolatility. Although synthetic, the materialsare compatible with petroleum products because of the compositions in-volved.

Viscosity of Lubricants

The most important property of a lubricating oil is viscosity. Defined asthe resistance to flow,viscosity physically is the factor of proportionalitybetween shearing stress and the rate of shearing. As described in anearlier section, increased viscosity relates to the increased ability of afluid to separate microsurfaces under pressure, the fundamental processoflubrication. For bearing applications viscosity is usually measured kin-.ematically per ASTMspecification D-445. This method measures the pas-sage time required under the force of gravity for a specified volume ofliquid to pass through a calibrated capillary tube.

A related concept of importance is viscosity index (VI), which is anarbitrary number indicating the effect of temperature on the kinematicviscosity for a fluid. The higher the VI for an oil, the smaller the viscositychange will be with temperature. For typical paraffinic base stocks VI is85-95. Polymer additions may be made to petroleum base stocks to ob-.tain VI of 190 or more. The shearing stability of these additives is ques-tionable, and VI generally deteriorates with time. Many synthetic basestocks have VIs far in excess of mineral oils, as Table 17.1 shows. Themethod of calculating VI from measured viscosities is described in ASTM:specification D-567.

Selection of Proper Viscosity for Petroleum Oil Lubricants

Figures 17.4 and 17.5 can be used to derive a minimum acceptable vis-cosity for an application. Figure 17.4 indicates the minimum requiredviscosity as a function of bearing size and rotational speed for a petro-Ileum-based lubricant. The viscosity of a lubricating oil decreases withincreasing temperature. Therefore, the viscosity at the operating tem-perature rather than the viscosity at the standardized reference temper-:ature of 400e (l04°F) must be used. Figure 17.5 can be used to determinethe actual viscosity at the operating temperature if the viscosity grade(VG)of the lubricant is known.

Example 17.1. A bearing having a pitch diameter of 65 mm (2.559in.) operates at a speed of 2000 rpm. As shown in Fig. 17.4, the inter-section of dm = 65 mm with the oblique line representing 2000 rpmyields a minimum required kinematic viscosity of 13 mm2/sec (0.02in2/sec), assuming that the operating temperature is BOoe (176°F);inFig. 17.5 the intersection between BOoe (176°F) and the required vis-

cosity of 13 mm2/sec (0.02 in.2/sec) is between the oblique lines forVG46 and VG68.Therefore, a lubricant with minimum viscosity gradeVG46 should be used; that is, a minimum lubricant viscosity of 46mm2/sec (0.07 in.2/sec) at standard reference temperature of 40°C(104°F).When determining operating temperature, it must be kept inmind that oil temperature is usually 3-11oC (5-20°F) higher thanbearing housing temperature.

If a lubricant with higher than required viscosity is selected, animprovement in bearing fatigue life can be expected; however, sinceincreased viscosity raises the bearing operating temperature, there isfrequently a practical limit to the lubrication improvement that can

and cold. These fluids are the basis for many high temperature, 204°C(400°F), lubricants. The general structure of silicone fluids may be shownas in Fig. 17.8: R represents methyl, phenyl, or some other organic group.

In addition to favorable viscosity-temperature characteristics, boththermal and oxidation resistance are excellent. As a family, these fluidsalso exhibit low volatility and good hydrolytic stability. These materialsare inert towards most elastomers and polymers as long as very hightemperatures are avoided. Oxygen exposure with high-temperature use,however, can result in gelation and loss of fluidity. The lubrication prop-erties of the base oils are not impressive compared to other lubricatingfluids. Typical applications of these materials as lubricants are electricmotors, brake fluids, oven preheater fans, plastic bearings, and electricalinsulating fluids.

Silicate esters represent a mating of the previous two lubricant fluidtypes. As a class, these fluids possess good thermal stability and lowvolatility. These materials are used in high-temperature hydraulic fluidsand low-volatility greases.

Fluorinated polyethers as a class represent the highest-temperaturelubricating fluids commercially available. Although distinct chemicalversions are marketed, all of these fluids are fully fluorinated and com-pletely free of hydrogen. This structural characteristic makes them inertto most chemical reactions, nonflammable, and extremely oxidation re-sistant. Products from these oils show very low volatility and excellentresistance to radiation-induced polymerization. The products are essen-tially insoluble in common solvents, acids, and bases. Density is approx-imately double that of petroleum oils. Products of this chemical familyare used to lubricate rolling bearings at extremely high temperatures-that is, 204-260°C (400-500°F). Other applications areas are in highvacuums, corrosive environments, and oxygen-handling systems. Thecost of these lubricants is very high.

Table 17.2 gives characteristics of synthetic oils compared to those ofmineral oils.

GREASE LUBRICANTS

General Conditions of Use

Grease is a thickened oil that allows localization ofthe lubricant to areasof contact in the bearing. A rolling bearing grease is a suspension of fluid


dispersed into a soap or nonsoap thickener, with the addition of a varietyof performance-enhancing additives.

Grease provides lubricant by bleeding; that is, when the moving partsof a bearing come in contact with grease, a small quantity of thickenedoil will adhere to the bearing surfaces. The oil is gradually degraded byoxidation or lost by evaporation, centrifugal force, and so on; in time theoil in the grease near the bearing will be depleted.

Several differing viewpoints currently exist concerning the mechanismof grease operation. Until recently, grease was looked upon as merely asponge holding oil near the working contacts. As these contacts con-sumed oil by way of evaporation and oxidation, a replenishment flowmaintained an equilibrium as long as the supply lasted. Research usingoptical EHL and microflowlubrication assessment techniques has shownthat the thickener phase plays rather complex roles in both the devel-opment of a separating film between the surfaces and in the modulationof the replenishing flows. The manner in which the thickener controlsoil outflow, reabsorbs fluid thrown from the contacts, and acts as a trapfor debris is little understood at this time. The mechanism is not steadystate, but is characterized by a series of identifiable events.

Greases offer the following advantages compared to fluid lubricants:

1. Maintenance is reduced because there is no oil level to maintain.New lubricant needs to be added less frequently.

2. Lubricant in proper quantity is confined to the housing. Design ofenclosures can therefore be simplified.

3. Freedom from leakage can be accomplished, avoiding contamina-tion of products in food, textile and chemical industries.

4. The efficiency of labyrinth "seals" is improved, and better sealingis offered for the bearing in general.

5. The friction torque and temperature rise are generally more favor-able.

Grease Property Definitions

Bomb Oxidation. The procedures described in the following are avail-able from the American Society for Testing and Materials (ASTM).Thedetermination of the resistance of lubricating greases to oxidation whenstored under static conditions for a long time is described by ASTM spec-ification D-942. A sample is oxidized in a "bomb" heated to 99°C (210°F)and filled with oxygen at 0.76 N/mm2 (110 psi). Pressure is observed andrecorded at stated intervals. The degree of oxidation after a given periodof time is determined by the corresponding decrease in oxygen pressure.

GREASE LUBRICANTS 665

Dropping Point. Dropping point is the temperature at which a greasebecomes a liquid and is sometimes referred to as the melting point. Thetest is performed per ASTM specification D-566.

Evaporation Loss. The method of determining evaporation loss is de-scribed by ASTM specification D-972. The sample in an evaporation cellis placed in a bath maintained at the desired test temperature [usually99-149°C (210-300°F)]. Heated air is passed over the cell surface for 22hr. The evaporation loss is calculated from the sample weight loss.

Flash Point. Flash point is the lowest temperature at which an oil givesoff inflammable vapor by evaporation, per ASTM specification D-566.

Low-Temperature Torque. Low-temperature torque is the extent towhich a low-temperature grease retards the rotation of a slow-speed ballbearing when subjected to subzero temperature. The method of testingis described by ASTM specification D-1478.

Oil Separation. This is the tendency of lubricating grease to separateoil during storage in both conventional and crate red containers, as de-scribed by ASTM specification D-1742; the sample is determined by sup-porting on a 74-JLm sieve subjected to 0.0017 N Imm2 (0.25 psi) airpressure for 24 hr at 25°C (77°F). Any oil seepage drains into a beakerand is weighed.

Penetration. The penetration is determined at 25°C (77°F) by releasinga cone assembly from a penetrometer and allowing the cone to drop intothe grease for 5 sec. The greater is the penetration, the softer is thegrease. Worked penetrations are determined immediately after workingthe sample for 60 strokes in a standard grease worker. Prolonged pene-trations are performed after 100,000 strokes in a standard grease worker.A common grease characteristic is described by NLGI (National Lubri-cating Grease Institute) grade assigned, as shown in Table 17.3. Mostrolling bearing applications employ an NLGI 1, 2, or 3 grade grease.

Pour Point. Pour point is the lowest temperature at which an oil willpour or flow.The pour point is measured under the conditions in ASTMspecification D-97. The pour point together with measured low-temperature viscosities gives an indication of the low-temperature serv-iceability of an oil.

Viscosity, Viscosity Index. The values of viscosity and VI generally referto the base oil values of these properties as discussed in "Liquid Lubri-cants."


TABLE 17.3. NLGIPenetration GradesNLGIGrades Penetration (60Strokes)000 445-47500 400-3000 355-3851 310-3402 265-2953 220-2504 175-2055 130-1606 85-115

Water Washout Resistance. Water washout resistance is the resistanceof a lubricating grease to washout by water from a bearing tested at 38°C(lOO°F)and 79.5°C (145°F), as described by ASTM specification D-1264.

Types of Grease Thickeners

Thickener composition is critical to grease performance, particularlywith respect to temperature capability, water-resistance, and oil-bleedingcharacteristics. Thickeners are divided into two broad classes: soaps andnonsoaps. Soap refers to a compound of a fatty acid and a metal. Commonmetals for soaps include aluminum, barium, calcium, lithium, and so-dium. The great majority of commercial greases are soap type, with lith-ium being the most widely used.

Lithium soap greases-Lithium soaps are divided into two types: 12-hydroxystearate and complex. The latter material is derived fromorganic acid components and permits higher temperature perform-ance. The upper operating temperature limit of the usual lithium-based grease is approximately llO°C (230°F). For a lithiumcomplex-based grease the upper temperature limit is extended to140°C (284°F). Conversely, the lower operating temperature limitsare -30°C (-22°F) and -20°C (-4°F), respectively. High-qualitylithium soap greases of both types have excellent service historiesin rolling bearings and have been used extensively in prelubricated;that is, sealed andgreased-for-life applications. Lithium-based prod-ucts have found acceptance as multipurpose greases and have noserious deficiencies except in severe temperature or loading ex-tremes.

Calcium soap greases-The oldest of the metallic soap types, calcium-based greases, has undergone several important technical changes.

GREASE LUBRICANTS 667

In the first formulations, substantial water (0.5-1.5%) was neededto stabilize the finished product. Loss of water destroys grease con-sistency; as such, grease upper temperature operating limiting isonly 60°C (140°F) [Correspondingly, the lower operating tempera-ture is only -lOoC (14°F).] Regardless of temperature, evaporationof water occurs, requiring frequency relubrication of the bearing.Alternately, the ability of the grease to entrain water is of someadvantage; such greases have been widely used in food processingplants, water pumps, and wet applications in general. Today, thistype of formulation has been made obsolete by newer products withbetter temperature performance.

The latest development in calcium-thickened greases is the calciumcomplex-based grease. Herein the soap is modified by adding an acetate,and a substantially different product results having upper and loweroperating temperature limits of 130°C (266°F) and -20°C (-4°F), re-spectively. Performance of these greases in rolling bearings is sometimesless than optimum. Although high temperature and EP (extreme pres-sure) characteristics have been exhibited, there are some problems withexcessive grease thickening in use, causing an eventual loss of lubrica-tion to the bearing.

Sodium soap greases-Sodium soap greases were developed to providean increase in the limited temperature capability of early calciumsoap greases. An inherent problem with this thickener is poorwater-washout resistance; however, small amounts of water areemulsified into the grease pack, which helps to protect metal sur-faces from rusting. The upper operating temperature limit for suchgreases is only 80°C (176°F). The lower operating temperature limitis -30°C (-22°F). Sodium-base greases have been superceded bymore water-resistant products in applications such as electric mo-tors and front wheel bearings. Sodium complex-base greases havesubsequently been developed having upper and lower operatingtemperature limits of 140°C(284°F) and -20°F (-4°F), respectively.

Aluminum complex greases-Aluminum stearate greases are seldomused in rolling bearings, but aluminum complex-base greases arebeing used more often. Greases formed from the complex soap per-form favorably on water-resistance tests; however, the upper oper-ating temperature limit is somewhat low at llO°C (230°F)comparedto other types of high-quality greases. The lower operating temper-ature limit is satisfactory at -30°C (-22°F). These greases find usein rolling mills and food-processing plants.

Non-soap-base greases-Organic thickeners, including ureas, amides,and dyes, are used to provide higher temperature capability than


is available with metallic soap thickeners. Improved oxidation sta-bility over metallic soaps occurs because these materials do not cat-alyze base oil oxidation. Dropping points for greases of these typesare generally above 260°C (500°F)with generally good low temper-ature properties. The most popular of these thickeners is polyurea,which is extensively used in high-temperature ball bearing greasesfor the electric motor industry. The recommended upper operatingtemperature limit for polyurea-base grease is 140°C (284°F); thelower temperature limit is -30°C (-22°F).

Inorganic thickeners include various clays such as bentonite. Greasesmade from a clay base do not have a melting point, so service tempera-ture depends on the oxidation and thermal resistance of the base oil.These greases find use in special military and aerospace applicationsrequiring very high temperature performance for short intervals, for ex-ample, greater than 170°C(338°F).On the other hand, the recommendedupper temperature limit for continuous operation is only 130°C (2660F);the lower temperature limit is -30°C (-22°F).

Grease Compatibility. Mixing greases of differing thickeners and/orbase oils can produce an incompatibility and loss of lubrication witheventual bearing failure. When differing thickeners are mixed-that is,soap and nonsoap or differing soap types-dramatic changes in consis-tency can result, leading to a grease either too stiff to lubricate properlyor too fluid to remain in the bearing cavity. Mixing greases of differingbase oils-that is, petroleum and silicone oils-can produce a two-component fluid phase that will not provide a continuous lubrication me-dium. Early failures can be expected under these conditions. The bestpractice to follow is to not mix lubricants but rather purge bearing cav-ities and supply lines with new lubricant until previous product cannotbe detected before starting operation.

POLYMERIC LUBRICANTS

A polymeric lubricant uses a matrix or spongelike material that retainsits physical shape and location in the bearing. Lubrication functions areprovided by the oil alone after it has bled from the sponge. Ultrahighmolecular weight polyethylene forms a pack with generally good per-formance properties, but it is temperature limited to about 100°C(2120F),precluding its use in many applications within the temperature capabil-ity of standard rolling bearings. Some higher temperature materials,such as polymethylpentene, form excellent porous structures but are rel-atively expensive and suffer from excessively shrinkage. Fillers andblowing agents, tools of the plastic industry, interfere with the oil-flow

POLYMERIC LUBRICANTS 669

behavior, and contribute little in this situation. Other solutions must bedeveloped.

Figure 17.9 shows bearings filled with polymeric lubricant. Successfulapplication has been achieved where a bearing must operate under se-vere acceleration conditions such as those occurring planetary transmis-sions. Bearing rotational speed about its own axis may be moderate, butthe centrifuging action due to the planetary motion is strong enough tothrow conventional greases out of the bearing despite the presence ofseals. When polymerically lubricated bearings are substituted, life im-provements of two orders of magnitude are not uncommon. Such sit-uations occur in cablemaking, tire-cord winding, and textile millapplications.

Another major market for polymer lubricants is food processing. Foodmachinery must be cleaned frequently, often daily, using steam, caustic,or sulfamic acid solutions. These degreasing fluids tend to remove lubri-cant from the bearings, and it is standard practice to follow every clean-ing procedure with a relubrication sequence. Polymer lubricants haveproven to be highly resistant to washout by such cleansing methods,hence the need for regreasing is reduced. The reservoir effect of polymerlubricants has been exploited to a degree in bearings normally lubricated


by a circulating oil stress where there can be a delay in the oil reachinga critical location. The same effect has been used to provide a backup incase the oil supply system should fail.

The high occupancy ratio of the void space by the polymer minimizesthe opportunity for the bearing to "breathe" as temperature change. Cor-rosion due to internal moisture condensation is therefore reduced. Sinceall ferrous surfaces are very close to the pack, conditions are optimumfor using vapor phase corrosion-control additives in the formulations.

Despite these advantages polymeric lubricants have some specificdrawbacks. There tends to be considerable physical contact between thepack and the moving surfaces of the bearing. This leads to increasedfrictional torque, which produces more heat in the bearing. In conjunc-tion with thermal insulating properties of the polymer and its inherentlylimited temperature tolerance, the speed capability is reduced. Moreover,compared to grease, the solid polymer is relatively incapable of entrap-ping wear debris and dirt particles.

SOLID LUBRICANTS

Solid lubricants are used where conventional lubricants are not suitable.Extreme environment conditions frequently make solid lubricants a pre-ferred choice of lubrication. Solid lubricants can survive temperatureswell above the decomposition temperatures of oils. They can also be usedin chemically reactive environments. The disadvantages of solid lubri-cants are (1) high coefficient of friction, (2) inability to act as a coolant,(3) finite wear life, (4) difficult replenishment, and (5) little dampingeffect for controlling vibrational instabilities of rolling elements and sep-arator components.

Many common solid lubricants, such as graphite and molybdenum di-sulfide (MoS2), are layered lattice compounds that shear easily along pre-ferred planes of their structure. MoS2 has weak van der Waals forcesbetween sulfur bonds, giving the material a characteristic relatively lowcoefficient of friction. MoS2 oxidizes at approximately 399°C (750°F) inair; the oxides can be abrasive.

The low friction associated with graphite depends on intercalationwith gases, liquids, or other substances. For example, the presence ofabsorbed water in graphite imparts good lubricating qualities. Thus,pure graphite has deficiencies as a lubricant except when used in anenvironment containing contaminants such as gases and water vapor.With proper additives graphite can be effective up to 649°C (1200°F).

Tungsten disulfide (WS2) is similar to MoS2 in that it is a type oflayered lattice solid lubricant. It does not need absorbable vapors to de-velop low-shear-strength characteristics.

ENVIRONMENTALLY ACCEPTABLE LUBRICANTS 671

Other "solid" lubricating materials are solid at bulk temperatures ofthe bearing but melt from frictional heating at points of local contact,giving rise to a low-shear-strength film. This melting may be very local-ized and of very short duration. Soft oxides, such as lead monoxide (PbO),are relatively nonabrasive and have relatively low friction coefficients,especially at high temperatures where their shear strengths are reduced.At these temperatures deformation occurs by plastic flowrather than bybrittle fracture. Melted oxides can form a glaze on the surface. This glazecan increase or decrease friction, depending on the "viscosity" ofthe glazewithin the contact region. Stable fluorides such as lithium fluoride (LiF2),calcium fluoride (CaF2), and barium fluoride (BaF2) also lubricate wellat high temperatures but over a broader range than lead oxides.

ENVIRONMENTALLY ACCEPTABLE LUBRICANTS

Concerns for the environment have led to the development of more en-vironmentally friendly or environmentally acceptable lubricants. Biode-gradability and low ecotoxicity are required for these lubricants, andmany countries now have specific requirements for branding lubricantsas environmentally friendly. Initially, two-cycle engine marine and forestapplications were targeted for use of biodegradable lubricants. This usehas now been extended to include hydraulic fluids, engine oils in generaland greases. For example, environmentally friendly grease products areavailable as rope lubricants and rail lubricants.

Biodegradability and low ecotoxicity of a lubricant depend on the basestock. Biodegradable fluids include vegetable oils, synthetic esters, po-lyalkylene glycols, and some polyalphaolefins. The susceptability of asubstance to be biodegraded by micro-organisms is a measure of its bio-degradability. Biodegradability can be partial, resulting in the loss ofsome specific process such as splitting an ester linkage (primary biode-gradation) or complete, resulting in the total breakdown of the substanceinto simple compounds such as carbon dioxide and water (ultimate bio-degradation). There is currently no standard method accepted for assess-ing an environmentally acceptable lubricant, and several methods are inuse worldwide. ASTM is currently addressing this problem. The CEC L-33-A-93method is one of the more widely used tests; but, countries haveestablished their own certification tests. For example, in Germany lu-bricants must pass their RAL-UZ79 test to obtain certification as a "BlueAngel Eco-label" product. England, Sweden, Denmark, Norway, Canada,and Poland each have their own certification requirements and environ-mental labels.

Ecotoxicity tests involve different types of aquatic species that formthe aquatic food chain. Testing the toxicity of a lubricant on bacteria,


water fleas (invertebrates), and trout (vertebrates) or other species maybe required depending on the application and state and country regula-tory requirements.

Development of lubricating oils and greases in the future and world-wide will require significant consideration of not only health and safetyissues, but environmental requirements as well.

SEALS

Functions of Seals

Once the general method of lubricating a bearing has been determined,the question of a suitable sealing method frequently needs to be ad-dressed. A seal has two basic tasks. It must keep lubricant where itbelongs and keep contaminating materials from the bearing and its lu-bricant. This separation must be accomplished between surfaces in rel-ative motion, usually a shaft or bearing inner ring and a housing. Theseal must not only accommodate rotary motion, but it must also accom-modate eccentricities due to run-outs, bearing clearance, misalignments,and deflections. The selection of a seal design depends on the categoryof lubricant employed (grease, oil, or solid). Also, the amount and natureof the contaminant that must be kept out needs to be assessed. Speed,friction, wear, ease of replacement, and economics govern the final choice.Bearings run under a great variety of conditions, so it is necessary tojudge which seal type will be sufficiently effective in each particular cir-cumstance.

Seals with Greased Bearings

Grease is the simplest lubricant to seal. The fluidity of the oil has beendeliberately reduced by blending with the thickener. The stiff nature ofgrease means that it requires little in the way of constraint yet at thesame time readily plugs small spaces. Given a suitably small gap, greasecan form layers on the opposing moving surfaces, effectively closing thegap completely. This principle is used in labyrinth-type designs.

Because it has an oily consistency, dirt or dust particles that penetratea seal are caught by the grease and prevented from entering the bearing.The wicking type of oil delivery to the bearing means that the particlesare permanently kept out of circulation unless the grease becomes stirredin some way.

Some seals make physical contact between the surfaces. A film ofgrease provides the necessary continuous supply oflubricant to establishhydrodynamic separation with its attendant low friction and wear.

SEALS 673

Seals with Oil-Lubricated Bearings

Oils are more difficult to seal than greases. They will flow through thesmallest gaps if there is any hydraulic head. Either the possibility of ahead developing must be prevented by the cavity design, or running gapsmust be eliminated by use of a movable seal lip. Oils are excellent asdirt traps, but they lack the ability to keep the dirt out of circulationunless backed up by a filter system. Dirt can only be kept out by positivegap elimination.

Seals for Solid-Lubricated Bearings

Only the powder and reactive gas forms of solid lubricants pose sealingchallenges. Even then, since they are essentially once-through systems,some leakage is tolerable unless the material is toxic. Furthermore seal-ing of such lubricants can pose grave difficulties, particularly since theyare almost exclusively used at extremely high temperatures at which gasdynamic behavior increases. Zero gap conditions are then necessary withextra provision made to prevent the powder or soot-type exhaust prod-ucts from compacting and causing separation.

Types of Seals

Labyrinth Seals. Labyrinth seals consist of an intricate series ofnarrowpassages that protect well against dirt intrusion. An example is shownin Fig. 17.10. This type is suitable for use in pillow blocks or other as-semblies where the outer stationary structure is separable. The innerpart is free to float on the shaft so that it can position itself relative tothe fixed sections. The mechanism of sealing is complex, being associatedwith turbulent flow fluid mechanics. It is reasonably effective with liq-uids, greases, and gases, provided that there is no continuous static headacross the assembly.


It is normal practice to add grease to the labyrinth, making the gapseven smaller than can be achieved mechanically due to tolerance stack-ups. Dirt has virtually no chance of penetrating such a system withoutbecoming ensnared in the grease. A further advantage accrues at re-greasing. Spent lubricant can purge readily through the labyrinth andflush the trapped debris with it.

The relatively moving parts are separated by a finite gap, so wear, inthe absence of large bridging particles of dirt, is essentially nonexistent.Likewise, frictional losses are extremely low.The number of convolutionsof the labyrinth passage can be increased with the severity of the dirtexclusion requirements. Separate flingers and trash guards or cuttersmay be added on the outboard side to deal with wet or fibrous contam-inants that could damage or penetrate the labyrinth. Figure 17.11 showsa ball bearing with an integral labyrinth seal and outboard flinger ring.

SEALS 675

Felt Seals. Semicircular pieces of felt, pressed into trapezoidal sectiongrooves in the housing, lightly contact the shaft surface, as shown in Fig.17.12. Inexpensive and simple to install and replace, the grease-ladenfelts keep dirt out of the enclosure; however, the dirt entrapped in thefelt fibers can cause serious shaft surface wear. Also, the felt can becomecompacted, eventually leaving an air gap. Friction is often high and dif-ficult to control. For these reasons, felt seals, though once popular, arenot currently in significant use.

Shields. As Fig. 17.13 shows, a shield takes up very little axial spaceand can usually be accommodated within the standard boundary dimen-sions of the bearing. The near knife edge standing just clear of the ringland is, in effect, a single-stage labyrinth seal. Effective enough to keepall but the most fluid greases in the bearing, the shield can be consideredas a modest dirt excluder, suitable for use in most workplace environ-ments. Under harsher conditions it must be backed up with extra guards.Special greases or acceptance of leakage and reduced lubricant life arenecessary when shielded bearings are used in vertical axis applications.The absence of contact friction permits these bearings to be used at thehighest speed allowed by the mode of lubrication and type of lubricant.

Elastomeric Lip Seals. The narrow gap between a shield and an innerring groove or chamfer can be closed by a carefully designed section ofelastomer (nitrile rubber for general purposes). Figure 17.14 illustratesa typical configuration. The flexible material makes rubbing contact withthe ring and establishes a barrier to the outward flowoflubricant or theingress of contaminants. When the bearing is in motion, the elastomermust slide over the metal surface, and a frictional drag is produced,which even for a well-designed seal, is generally greater than the fric-

tional torque of the bearing. Often more important is the seal breakawaytorque, which can be several times the running torque. Considerable re-search has been devoted to finding both elastomers and seal designs thatachieve a suitable balance between sealing efficacy,lip or ring wear, andfrictional torque.

SEALS 677

The lip of the seal must bear on the ring with sufficient pressure tofollowthe relative motions ofthe running surface caused by eccentricitiesand roundness errors. This pressure is achieved by a slight interferencefit, producing a dilation of the seal. The spring rate of the lip governsthe speed at which the lip can respond to the running errors without agap being formed through which fluid can pass. Higher bearing speedsdemand better running accuracies. Spring rate is regulated by the elasticproperties of the seal material and the design of the bending section.

At first glance, even though a lubricant is present, no hydrodynamiclift would be expected on the lip, due to the axial symmetry. Recent workhas established, both theoretically and experimentally, that a very thin,stable dynamic film persists over much of the operating regime. Themechanism of sealing is a complex one involving the elastomeric lip, thecounterface, and the grease, or at least the oil phase ofthe grease. Figure17.15 shows the seal of Fig. 17.14 composed of a molded annulus of pol-ymer bonded to a thin steel disc.

The disc provides mechanical support against minor pressure differ-ences that can occur across the seal and also assures a slight compressionof the polymer against the outer ring recess, thus creating a fluid-tightstatic seal at that point. The inside diameter of the disc, in conjunctionwith the waisted section of the molding, defines the flexure point of thelip itself and is located so that the deflected lip bears against the coun-terface groove with suitable pressure and at a predetermined angle. Thisangle of contact produces appropriate convergence and divergence on ei-ther side of the contact, which helps the sealing function appreciably.The lip pressure induced by the interference between the seal and itscounterface is sufficient to prevent fluid leakage under static conditions.

To function adequately, the elastomer must exhibit specific properties.Beyond compatibility with the common types oflubricating oils and swellthat can be accommodated by the seal configuration, it must survive thefrictional heating at the lip and heat from the bearing or its environmentwithout hardening, cracking, or otherwise aging. To survive start-up andthe presence of dirt, it must have wear and abrasion resistance. Caremust be taken when forming the elastomer and its fillers that the finalcured product does not promote corrosion ofthe counterface under humidconditions. The range of candidate materials, their chemical structures,and physical properties are discussed in Chapter 16.

Lip seals require the presence of lubricant, for if allowed to run dry,wear and failure are usually rapid. The grease charge for the bearingmust be positioned to wet the seals upon assembly. In most cases thegrease volume is sufficient to require a period ofworking when the bear-ing first operates. This is followed by channeling, and the formation ofgrease packs against the inside surfaces of the seals. Operational suffi-ciency is then assured.

FIGURE 17.15. Lip seal construction showing interfer-ence with bearing inner ring seal groove and retention inouter ring groove.

In operation, EHL develops under the lip, which progressively lowersthe seal torque by changing the friction from Coulomb to viscous shear.Wear is thereby greatly reduced. Currently, there are two schools ofthought concerning lubricant film formation. One approach ascribes thefilm to asperities on the seal lip, producing localized hydrodynamic pres-sure fluctuations, as illustrated in Fig. 17.16. Cavitation downstreamfrom each asperity limits the negative pressure remaining to separatethe surfaces. The countervailing view invokes the viscoelastic propertiesof the seal material and the inability of the elastomer to followpreciselythe radial motions of the counterface produced by eccentricity and out-of-roundness.

Both of these mechanisms may be valid and function simultaneouslyand essentially independently of one another. The first is governed by

the microgeometry of the lip as modified by wear, abrasion scratches,thermal and installation distortions, and possibly inhomogeneities inelastomer properties. The second is a by-product of manufacturing pro-cess characteristics and nonrotary displacements of the inner ring.

Seal torque arises from four sources: adhesion between asperities, ab-rasion, viscous shearing of the film, and hysteresis in the elastomer. Thelast two are strongly influenced by temperature and so tend to be self-limiting; otherwise they all depend not only on the application but onthe detail installation itself. Methods for exact prediction of torque andoperating temperature have not yet been devised. In seemingly identicalconditions one sealed bearing frequently runs cooler than another, or onewill leak slightly and another will not. Much work needs to be done topredict seal performance in a given application.

The primary task of the single lip seal is to contain grease. It canexclude moderate dust as found in typical home or commercial atmo-spheres, and it finds a great many suitable applications. Some dusts,such as from wood sanders or lint accumulating on the bearings in textilemachinery, have the ability to wick considerable amounts of oil throughthe lip film, which shortens bearing life. In these situations and wherethere is heavy exposure to dirt, particularly waterborne dirt, such as inautomotive uses, additional protection in the form of dust lips and flin-gers should be provided. Figure 17.17 shows an example of a double-lipseal.

Garter Seals. Similar to many respects to the lip seal, the garter sealuses a hoop spring or garter to apply an essentially constant inwardpressure on the lip. As shown in Fig. 17.18, the arrangement requiresmore axial space than is available in a bearing of standard envelope

dimensions. Either extra wide rings must be used, or the seal must befitted as a separate entity in the assembly.

The spring-induced pressure gives a very positive sealing effect andis used to contain oil rather than grease, for two reasons: Oil can bethrown or pumped by an operating bearing with considerable velocity,sufficient to cause leakage through a lip seal, and the lip itself requires

SEALS 681

a generous supply of oil for lubrication and removal of frictional heat.Relieved of the need to provide the closing force, the elastomer sectioncan be designed to hinge freely so that relatively large amplitudes ofshaft eccentricity can be accommodated. The strictly radial nature of thespring force precludes the use of anything other than a cylindrical coun-terface surface. Axial floating of the shaft is therefore accommodatedwell.

The design lends itself to molding, and artificial asperities and otherfilm generating devices can readily be formed in the elastomer. Figure17.19 shows an example of a helical rib pattern intended not only toenhance the oil film thickness but to act as screw pump to minimizeleakage.

Ferrofluidic Seals. Magnetic fluids are a recent introduction to the ar-senal of tools available to the sealing engineer. A ferrolubricant is basi-cally a dispersion of very fine particles of ferrite in oil. The particles aretypically 100 A in diameter and are coated with a molecular dispersingagent to prevent coalescence. Brownian movement inhibits sedimenta-tion. The result is a lubricant that responds to magnetic fields.

Figure 17.20 shows a two-stage seal, each stage composed of severalgaps across which is suspended a ferrofluid film. This type of seal hasproved very effective where bearing and shaft systems penetrate a vac-

682LUBRICANTS AND LUBRICATION TECHNIQUES

uum enclosure and in computer disc drive spindle assemblies where ab-solutely clean internal conditions must be maintained.

Its ability to withstand pressure gradients to 0.345 N/mm2 (50 psi)(bymultistaging) and accept high eccentricities with 100%fluid tightnessmakes the ferrofluidic seal essentially unique. Two things preventgreater application. The ferrite increases the apparent viscosity of thefluid, and viscous heating limits the speed capability. The greatest draw-back is the need to introduce magnets into the system. Tramp iron isattracted to the seals unless considerable conventional sealing is appliedoutboard, negating much of the seal's advantages.

CLOSUREFollowing the design and manufacturer of a rolling element bearing, thetechnology associated with creating and maintaining the internal envi-ronment of the bearing during its operation is the single most importantfactor connected with its performance and life. This environment is in-timately associated with the lubricant selected, its means of application,and the method of sealing. In this chapter, a brief overview has beengiven to each of these important considerations. No attempt has beenmade to provide an exhaustive study of lubricant types, means of lubri-cation, or means of sealing. It remains for the reader to explore each ofthese topics to the depth required by the individual application.

REFERENCES17.1. SKF, General Catalogue 4000US Second Edition (1997-01).

686 FATIGUE LIFE: LUNDBERG-PALMGREN THEORY AND RATING STANDARDS

GENERAL

It has been considered that if a rolling bearing in service is properlylubricated, properly aligned, kept free of abrasives, moisture, and cor-rosive reagents, and properly loaded, then all causes of damage are elim-inated save one, material fatigue. Historically, rolling bearing theorypostulated that no rotating bearing can give unlimited service, becauseof the probability of fatigue of the surfaces in rolling contact. As indicatedin Chapter 6, the stresses repeatedly acting on these surfaces can beextremely high as compared to other stresses acting on engineeringstructures. In the latter situation, some steels appear to have an endur-ance limit, as shown in Fig. 18.1. This endurance limit is a level of cy-clically applied, reversing stress, which, if not exceeded, the structurewill accommodate without fatigue failure. The endurance limit for struc-tural fatigue has been established by rotating beam and/or torsionaltesting of simple bars for various materials.

In Chapter 23, the concept of a fatigue endurance limit for rollingbearings will be discussed in detail as well as the correlation ofstructuralfatigue with rolling contact fatigue. In this chapter, the concept of rollingcontact fatigue and its association with bearing load and life ratings iscovered.

Rolling contact fatigue is manifested as a flaking off of metallic par-ticles from the surface of the raceways and/or rolling elements. For welllubricated, properly manufactured bearings, this flaking usually com-mences as a crack below the surface and is propagated to the surface,eventually forming a pit or spall in the load-carrying surface. Lundberget al. [18.1J postulated that it is the maximum orthogonal shear stressTo of Chapter 6 that initiates the crack and that this shear stress Occurs

GENERAL 687

at depth Zo below the surface. Figure 18.2 is a photograph of a typicalfatigue failure in a ball bearing raceway. Figure 18.3, taken from refer-ence [18.2], indicates the typical depth in a spalled area.

Not all researchers accept the maximum orthogonal shear stress asthe significant stress initiating failure. Another criterion is the VonMises distortion energy theory, which yields a scalar "stress" level ofsimilar magnitude to the double amplitude; that is, 2To, of the maximum


orthogonal shear stress. Moreover, the subsurface depth at which thisvalue is a maximum is approximately 50% greater than for TO' Accordingto references [18.2], this greater depth for failure initiation appears tobe verified.

Lundberg et al. [18.1] postulated that fatigue cracking commences atweak points below the surface of the material. Hence, changing thechemical composition, metallurgical structure, and homogeneity of thesteel can significantly affect the fatigue characteristics of a bearing, allother factors remaining the same. In referring to weak points, one doesnot include macroscopic slag inclusions, which cause imperfect steel forbearing fabrication and hence premature failure. Rather microscopic in-clusions and metallurgical dislocations that are undetectable except bylaboratory methods are possibly the weak points in question. Figure 18.4,taken from reference [18.2], shows a fracture failure at weak points de-veloped during rolling. This type of experimental study tends to confirmthe Lundberg-Palmgren theory insofar as failure that initiates at weakpoints. That the weak points are those at a specified depth below therolling contact surface, rather than at other depths or even at the sur-face, will be discussed later.

FATIGUE LIFE DISPERSION

Even if a population of apparently identical rolling bearings is subjectedto identical load, speed, lubrication, and environmental conditions, allthe bearings do not exhibit the same life in fatigue. Instead the bearingsfail according to a dispersion such as that presented in Fig. 18.5. Figure18.5 indicates that the number of revolutions a bearing may accomplishwith 100% probability of survival, that is, S = 1, in fatigue is zero. Al-ternatively, the probability of any bearing in the population having in-finite endurance is zero. For this model, fatigue is assumed to occur whenthe first crack or spall is observed on a load-carrying surface. It is ap-parent, owing to the time required for a crack to propagate from thesubsurface depth of initiation to the surface, that a practical fatigue lifeof zero is not possible. This will be discussed in greater depth later; how-ever, for the purpose of discussing the general concept of bearing fatiguelife, Fig. 18.5 is appropriate.

Since such a life dispersion exists, bearing manufacturers have chosento use one or two points (or both) on the curve to describe bearing en-durance. These are

1. LlO the fatigue life that 90% of the bearing population will endure.2. L50 the median life, that is, the life that 50% of the bearing popu-

lation will endure.

FATIGUE LIFE DISPERSION 691

In fact, it is not possible to ascribe a given fatigue life to a solitary bear-ing application. One may however refer to the reliability of the bearing.Thus, if for a given application using a given bearing, a bearing manu-facturer will estimate a rating life, the manufacturer is, in effect, statingthat the bearing will survive the rating life (LIO revolutions) with 90%reliability. Reliability is therefore synonymous with probability of sur-vival.

Fatigue life is generally stated in millions of revolutions. As an alter-native it may be and frequently is given in hours of successful operationat a given speed.

An interesting aspect of bearing fatigue is the life of multirow bear-ings. As an example of this effect, Fig. 18.6 shows actual endurance dataof a group of single-row bearings superimposed on the dispersion curve


of Fig. 18.5. Next consider that the test bearings are randomly groupedin pairs. The fatigue life of each pair is evidently the least life of the pairif one considers a pair is essentially a double-row bearing. Note from Fig.18.6 that the life dispersion curve of the paired bearings falls below thecurve for the single bearings. Thus, the life of a double-row bearing sub-jected to the same specified loading as a single-row of identical design isless than the life of a single-row bearing. Hence in the fatigue of rollingbearings, the product law of probability [18.3] is in effect.

When one considers the postulated cause of surface fatigue, the phys-ical truth of this rule becomes apparent. If fatigue failure is, indeed, afunction of the number of weak points in a highly stressed region, thenas the region increases in volume, the number of weak points increasesand the probability of failure increases although the specific loading isunaltered. This phenomenon is further explained by Weibull [18.4, 18.5].

WEIBULL DISTRIBUTION

In a statistical approach to the static failure of brittle engineering ma-terials, Weibull [18.5] determined that the ultimate strength of a mate-rial cannot be expressed by a single numerical value and that astatistical distribution was required for this purpose. The application ofthe calculus of probability led to the fundamental law of the Weibulltheory:

Equation 08.3) describes the probability of rupture ~f due to a given dis-tribution of stress a over volume tJ in which n(a) is a materialcharacteristic. Weibull's principal contribution is the determination thatstructural failure is a function of the volume under stress. The theory isbased on the assumption that the initial crack leads to a break. In thefatigue of rolling bearings, experience has demonstrated that manycracks are formed below the surface that do not propagate to the surface.Thus Weibull's theory is not directly applicable to rolling bearings. Lund-berg et al. [18.1] theorized that consideration ought to be given to thefact that the probability of the occurrence of a fatigue break should be afunction of the depth Zo below the load-carrying surface at which themost severe shear stress occurs. The Weibull theory and rolling bearingstatistical methods are discussed in greater detail in Chapter 20.

According to Lundberg et al. [18.1]let f(n) be a function that describesthe condition of material at depth z after n loadings. Therefore df(n) isthe change in that condition after a small number ofdn subsequent load-


gular position of a roller and its roller load, one of the contact conditionsin Table 18.9 will occur.

Of the contact conditions in Table 18.9, optimum roller bearing designfor any given application is generally achieved when the most heavilyloaded roller is in modified line contact. As stated in Chapter 6 this con-dition produces the most nearly uniform stress distribution along theroller profile, and edge loading is precluded. It is also stated in Chapter6 that a logarithmic profile roller can produce an even better load dis-tribution over a wider range of loading; however, this roller profile tendsto be special. The more usual profile is that of the partially crownedroller. It should be apparent that the optimum crown radii or osculationsnecessary to obtain modified line contact can only be evaluated for agiven bearing after the loading has been established. Series of bearings,however, are often optimized by basing the crown radii or osculations onan estimated load, for example, 0.5C or 0.25C, in which C is the basicdynamic capacity. Depending on the applied loads, bearings in such aseries may operate anywhere from point to line contact at the most heav-ily loaded roller.

Because it is desirable to use one rating method for a given rollerbearing, and because in any given roller bearing application it is possibleto have combinations of line and point contact, Lundberg and Palmgren[18.8] estimated the fatigue life should be calculated from

FATIGUE LIFE OF A ROLLING BEARING 737

If both inner and outer raceway contacts are line contacts and A =0.45 to account for edge loading and nonuniform stress distribution,curve 1 of Fig. 18.13 shows the variation ofload with life by using equa-tion (18.160) and the fourth power slope. Assuming u = 1.36 and usingequation (18.167), curve 2 illustrates the approximation to curve 1. Theshaded area shows the error which occurs when using the approxima-tion. Points A on Fig. 18.13 represents 5% error.

If outer and inner raceway contacts are point contacts for loads arbi-trarily less than 0.21C (L = 100 million revolutions), then for A = 0.65curve 1 of Fig. 18.14 shows the load-life variation of the bearing. Notethe inverse slope of the curve decreases from 4 to 3 at L = 100 millionrevolutions. Curve 2 of Fig. 18.14 shows the fit obtained while usingequation (18.167) and jJ = 1.20.

Lastly, if one raceway contact is line contact and the other is pointcontact, curve 1 of Fig. 18.15 shows load-life variation for A = 0.54.Transformation from point to line contact is arbitrarily assumed to occurat L = 100 million revolutions. Curve 2 of Fig. 18.15 shows the fit ob-tained while using equation (18.167) and jJ = 1.26.

In Fig. 18.13, using equation (18.167), fatigue lives between 150 and1500 million revolutions have a calculational error less than 5%. Simi-larly, in Fig. 18.14 lives between 15 and 2000 million revolutions haveless than 5% calculational error, and in Fig. 18.15 lives between 40 and10,000 million revolutions have less than 5% calculation error. Since theforegoing ranges represent probable regions of roller bearing operation,Lundberg et al [18.8] considered that equation (18.167) was a satisfac-tory approximation by which to estimate fatigue life of roller bearings.

EFFECT OF STEEL COMPOSITON AND PROCESSING ONFATIGUE LIFE

All of the equations pertaining to basic dynamic capacity of a racewaycontact Qc, or bearing raceway Cj or Co, or of an entire bearing C devel-oped thus far are based on bearings fabricated from AISI 52100 steelhardened at least to 58 Rockwell C. This is air-melted, air-cast steelwhose chemical composition is shown in Table 16.1. Moreover, the equa-tions are based upon steel processing methods and manufacturing meth-ods that existed at the time the original load rating standards werecreated. Equation (18.107) gives a recommended reduction in basic dy-namic capacity if the steel is not as hard as 58 Rockwell C. Additionally,if the bearing material hardness is greater than 58 Rockwell C as occa-sioned by a special heat treatment, the bearing fatigue life can tend tobe greater than that predicted by the standard life rating formulas. Asshown in Fig. 18.16 from reference [18.14], this effect is diminished ifthe bearing operates at elevated temperature.

Developments in the processing of rolling bearing steel have been con-tinuous since 1960, the date corresponding to the introduction of carbonvacuum-degassed (CVD) steel in the United States. Improvements inmelting practices have yielded bearing steels of significantly increasedfatigue endurance capability. For instance, AISI 52100 steel melted in a

vacuum has fewer impurities and therefore tends to yield increased roll-ing bearing fatigue life. Walp et al. [18.15]determined that the presenceof traces ofmetallic impurities such as aluminum, copper, and vanadiumis detrimental to fatigue life. Remelting of vacuum-melted steel whileusing a consumable electrode furnace tends to produce a more homoge-neous steel, which inherently has a better rsistance to fatigue. Somerolling bearing steels afford increased corrosion resistance in certain ap-plications however at the expense of potentially decreased fatigue en-durance. In Chapter 16, the various bearing steels and steelmanufacturing processes are discussed in detail. When rolling bearingsare fabricated of special steels, the expected increased or decreased fa-tigue lives may be estimated using life adjustment factors discussed laterin Chapter 23.

Koistinen [18.16]demonstrated a two-stage heat-treating process thatforms a surface layer in compression conducive to increased fatigue en-durance. The rationale for the increased endurance is discussed in Chap-ter 23. Additionally, for AISI 52100 and M50 bearing steels, ring androlling element forming processes that tend to eliminate end grain in thebearing rolling contact surfaces have been used to manufacture rollingbearings of increased endurance capability.

The endurance characteristics of case-hardened bearing steels mustalso be considered. These steels, covered in detail in Chapter 16, have

LOAD RATING STANDARDS 741

the advantage of a tough, fracture-resistant core as well as a fatigue-resistant, hard surface layer (case). Since the information in Chapter 6on subsurface shear stress indicates the critical stress occurs close to theraceway or rolling element surface, case-hardened steels are appropriatefor rolling bearings. In fact the fracture-resistant core is essential inmany applications. The Lundberg-Palmgren method has historicallybeen applied to case-hardened steel bearings also; however, the effect ofthe compressive stresses resulting from heat treatment is not properlyaccommodated. In Chapter 23 life calculation for case-hardened steelbearings is discussed.

One may question the value of the data developed in this chapterconcerning LlO fatigue life and basic dynamic capacity since these dataare based only upon air-melted AISI 52100 steel of a type no longeracceptable in modern industrial practice. The answer is threefold: (1) itis important to understand the origin of the load rating standards incurrent worldwide use in order to use them effectively and accurately,(2) a comparison may be conducted between the endurance characteris-tics of similar and dissimilar bearings of different manufacturers on thebasis of geometry alone, and (3) the equations may be used to optimizerolling bearing design for any given application.

LOAD RATING STANDARDS

To accommodate the improvements in bearing geometrical accuracies af-forded by modern manufacturing methods and the improvements in themodern basic rolling bearing steel chemistries and metallurgies, ANSIand ISO standards have included in the formulas for basic load rating*"a rating factor for contemporary, normally used material and manufac-turing quality." ISO [18.13] uses the bm factor directly in the equationsfor basic dynamic radial load rating* and basic dynamic axial load rat-ing;* for example, equation (18.103) for radial ball bearings becomes:

C = bmfc(i cos a)O.7Z2/3D1.8t (18.172)

ISO [18.13] gives values of the factor bm which may be applied to theformulas for basic dynamic capacity for each of the various executions of

*The terms basic loading rating, basic dynamic radial load rating (or basic dynamic axialload rating), and basic dynamic capacity may be used interchangeably. The last term wasthat created by Lundberg and Palmgren.tANSI [18.10] recommends using D raised to the 1.4 power in lieu of 1.8 for bearingshaving balls of diameter greater than 25.4 mm (1 in,). In this case, for metric units cal-cualtion of basic load rating, fern values must be multiplied by 3.647; that is, fern = 3.647X fern (tabular value).

REFERENCES 761

CLOSUREThe rolling bearing industry was among the first to use fatigue life as adesign criterion. As a result, the space-age term reliability, which is syn-onymous with probability of survival, is familiar to rolling bearing man-ufacturers and users. The concepts ofLlO or rating life and L50 or medianlife are used as yardsticks of bearing performance. By means of the ANSIand ISO load rating standards, the rolling bearing industry establishedrelatively uncomplicated methods to evaluate the rating life. These stan-dards enjoy worldwide acceptance, and they can be applied to comparethe adequacy of diverse bearing types from different manufacturers foruse in most engineering applications.

In certain applications, however, the simple use of ANSI or ISO for-mulas to rate rolling bearing performance can lead to inaccurate esti-mates of fatigue life. These applications include those involving highspeed, flexible bearing ring support structures, extremely slow speed,and unusual loading conditions. For these situations, methods to esti-mate fatigue life will be presented in Chapter 23. These methods rely onthe basic Lundberg-Palmgren theory; however, they typically requiremore complex analyses of bearing contact angles, internal load distri-bution, internal speeds, lubrication, and so on. Such analyses require theuse of a computer.

The ANSI and ISO standard methods terminate with the calculationof bearing rating life based on a bearing fabricated from a good quality,hardened bearing steel; however, they indicate how life may be influ-enced by such parameters as alternate rolling component materials, lu-brication, reliability, and combinations of these. Modern improvementsin rolling bearing steels and methods of bearing manufacture have suc-ceeded in yielding bearings capable of substantially increased endurance.In many applications, modern rolling bearings will not experience rollingcontact fatigue failure. In other words, they may be said to have infinitelife. These concepts are also presented in detail in Chapter 23.

REFERENCES18.1. G. Lundberg and A. Palmgren, "Dynamic Capacity of Rolling Bearings," Acta Poly-

tech. Mech. Eng. Ser. 1, R.S.A.E.E., No.3, 7 (1947).18.2. T. Martin, S. Borgese, and A. Eberhardt, "Microstructural Alterations of Rolling

Bearing Steel Undergoing Cyclic Stressing," ASME Preprint 65-WA / CF-4, WinterAnnual Meeting, Chicago (November 1965).

18.3. P. Hoel, Introduction to Mathematical Statistics, 2nd ed., Wiley, New York (1954).18.4. W.Weibull, "AStatistical Representation of Fatigue Failure in Solids,"Acta Polytech.

Mech. Eng. Ser. 1, R.S.A.E.E., No.9, 49 (1949).18.5. W. Weibull, "A Statistical Theory of the Strength of Materials," Proc. R. Swedish

Inst. Eng. Res., No. 151, Stockholm (1939).


18.6. T. Tallian, "Weibull Distribution of Rolling Contact Fatigue Life and DeviationsTherefrom," ASLE Trans. 5(1) (April 1962).

18.7. J. Lieblen, "ANew Method ofAnalyzing Extreme-Value Data," Technical Note 3053,NACA (January 1954).

18.8. G. Lundberg and A. Palmgren, "Dynamic Capacity of Roller Bearings," Acta Poly-tech. Mech. Eng. Ser. 2, R.S.A.E.E., No.4, 96 (1952).

18.9. A. Palmgren, Ball and Roller Bearing Engineering, 3rd ed., Burbank, Philadelphia(1959).

18.10. American National Standards Institute, American National Standard (ANSI 1AFBMA) Std 9-1990, "Load Ratings and Fatigue Life for Ball Bearings."

18.11. J. Lieblein and M. Zelen, "Statistical Investigation of Fatigue Life ofBall Bearings,"National Bureau of Standards, Report No. 3996 (March 28, 1955).

18.12. American National Standards Institute, American National Standard (ANSI 1AFBMA) Std 11-1990, "Load Ratings and Fatigue Life for Roller Bearings."

18.13. International Organization for Standards, International Standard ISO 28111,"Rolling Bearings-Dynamic Load Ratings and Rating Life-Part I; CalculationMethods" (1999).

18.14. E. Zaretsky, E. Bamberger, T. Harris, W. Kacmarsky, C. Moyer, R. Parker, and J.Sherlock, Life Adjustment Factors for Ball and Roller Bearings, ASME EngineeringDesign Guide (1971).

18.15. H. Walp, T. Morrison, T. Tallian, and G. Baile, "The Effect of Material Variables onthe Fatigue Life of AISI 52100 Steel Ball Bearings," ASLE Trans. 5(2) (1962).

18.16. D. Koistinen, "Heat Treated Steel Article," U.S. Patent No. 3117041 (January 7,1964).

764 BEARING ENDURANcE TESTING AND ELEMENT TESTING METHODS

service that can be achieved from a bearing in a specific application. Asindicated in Fig. 1.11, the ability to make these types of predictions ishampered since rolling contact fatigue is probabilistic, similar to humanlife and the service life of light bulbs. Therefore, the life of anyone bear-ing operating in a specific environment can differ significantly from thatof an apparently identical unit. This distribution of rolling bearing fa-tigue data is illustrated in Fig. 18.5.

Over the years, statistical procedures have been established for theanalysis of bearing fatigue data. A "rating" life system has been definedthat uses two specific points on the failure distribution curve to comparethe fatigue endurance of bearing designs. These points are the LlO life,or the life that 90% of the bearings can be expected to survive, and theLso life, which 50% of the bearings can be expected to exceed. Concur-rently, testing procedures have been developed to measure bearing fa-tigue life (endurance). As shown in Chapter 18, these experimentaltechniques have been utilized to collect a quantity of data which allowedthe derivation of theoretical life prediction formulas based on a calcu-lated bearing basic dynamic capacity, the equivalent radial load appliedto the bearing, and a number of environmental and manufacturing_related factors. This basic concept is routinely used to select bearingsthat will yield the desired performances in specific applications.

Bearing endurance tests are used to evaluate bearing materials, newheat treatment processes, and improved forming or surface finishingtechniques. The spread in experimental fatigue data and the limitationsof the statistical analysis techniques require that many bearings betested for a long time to obtain valid estimates of bearing life. Testingcosts are directly related to the manufacturing costs of the test specimensand the length of the test process; therefore, conducting a life test serieson full bearing assemblies is expensive. Effort has thus been expendedto develop life testing techniques using simple partial bearings or singlerolling contacts.

This chapter discusses the concepts and philosophies of conductingendurance tests on complete rolling bearing assemblies and on elementalrolling contact configurations.

THEORETICAL BASIS OF LIFE TESTING

The ability to use life test data collected on a particular bearing typeand size under a specific set of operating conditions to predict generalbearing performance requires a systematic relationship between appliedload and life. The basic form of this relationship was defined by Lundbergmd Palmgren [19.1Jas equation 09.1).

As indicated in Chapter 23, this equation has undergone several modi-fications since it was first proposed; however, for the purposes of thisdiscussion, the basic format is used. In equation 09.1), exponent p = 3for ball bearings and 10/3 for roller bearings. The form of the load-liferelationship for ball bearings was graphically illustrated in Fig. 18.8 asa straight line on a log-logchart. This formula provides the means to useexperimental life data collected under one set of test conditions to estab-lish projected bearing performance under a wide range of conditions.

The time to initiation of a fatigue-originated spalling failure on a roll-ing contact in a typical application is about 10 to 15 years. It is thereforeobvious that any practical laboratory test sequence must be conductedunder accelerated conditions if the necessary data are to be accumulatedwithin a reasonable time. Several potential methods exist for test accel-eration. Rolling contact fatigue modes exist, however, that compete inindividual bearings to produce the final failure [19.2]. Care must betaken to ensure that the method of test acceleration does not change thefailure mode. In an endurance test-that is, a test series conducted toestablish the experimental life of a lot of bearings for comparison withtheoretical lives and/or other test data-this means that the primaryfailure mode must remain fatigue related. Generally, there have beentwo methods of test acceleration used in most endurance testing: increas-ing the applied load levels and/or increasing the operating speed.

The experimental results obtained under increased load levels can berather easily extrapolated to other conditions by using the basic load-lifeformula. Thus, this is probably the most widely used method of life testacceleration. It is important to note that consistency must be maintainedwith the basic assumptions used in the derivation of the life formulas.The stresses generated at and below the bearing raceways should remainwithin the elastic regime. Exceeding elastic stress levels will producemodifications to the load-life relationship as reported in [19.3Jand illus-trated in Fig. 19.1.

Intuitively, it is surprising that such extreme loads can produce ap-parent life increases of significant magnitude over theoretical predic-tions. This phenomenon is, however, readily explained by the plasticdeformations produced in the raceways, creating a conforming contactpattern that then significantly reduces the existing stress levels. Testingin this regime produces results that are inconsistent with operating prac-tice and cannot be reliably extrapolated. The practical load limit for test-ing acceleration is usually considered to be that load that produces amaximum Hertz stress of approximately 3300 N/mm2 (475 ksi).

Some cases require special consideration in general life testing appli-cation. One involves testing self-aligning ball bearings. The spherical

geometry of the outer ring raceway needed to provide the self-aligningcapability produces circular point contacts between balls and racewaywith very high stress levels. As loads are increased, these contacts de-velop plastic stresses more rapidly than is considered by the dynamiccapacity calculation. It has been shown in past experiments [19.4] thatthe load should be no greater than C/Fe = 8.0 to prevent substantialplastic deformations in these bearings. Similarly, it would be anticipatedthat other types of bearings having nonstandard internal geometriescould also experience significant plastic deformations at lower than an-ticipated loads. This situation must be evaluated before initiating anendurance test on specially designed bearings.

Roller bearings generally have contact ellipses that are much longerthan they are wide; that is, they have line rather than point contacts.Roller and raceway geometries must be profiled in the axial direction toensure that stress concentrations do not occur at the edges of these con-tacts and thus produce substantial plastic deformations there. The pro-files used in standard bearings are often insufficient for the heavy loadsused in an accelerated life test series. Therefore care must be taken toensure that life reduction effects due to edge loading are not experiencedduring the testing process.

Even when edge stresses do not occur, roller bearings exhibit behaviorunder high loads that does not coincide with the stated load-life rela-tionship. Life test series conducted on cylindrical roller bearings [19.5]have shown that under loads with C/Fe less than about 4.5, the load-liferelationship of these bearings conforms to a power law function with anexponent closer to 4 rather than the stated 10/3 value. Modification fac-tors presented in [19.5] allow the normal catalog capacities to be con-verted for use in the latter relationship to yield a more reliable estimateof the rating life. Also, extra care must be used when calculating the life-modifying effects of the lubricant film. Experimental indications are thatroller bearings tend to generate EHL films thicker than those generally

THEORETICAL BASIS OF LIFE TESTING 767

calculated. These thicker films will then produce greater than expectedlife enhancements. To insure the proper interpretation of life test data,these effects must be established before extrapolating the data to sets ofoperating conditions containing different speeds or lubricants.

The second usual method of accelerating life testing is by increasingoperating speed. Operating a rolling bearing assembly faster produces amore rapid accumulation of stress cycles, but not necessarily a shortertest time. The increased operating speed also produces thicker EHLfilms, which then enhance bearing endurance. The life enhancement ef-fect may overshadow the increased stressing rate, and test times will beincreased.

As the operating speeds are increased to even higher speed levels,other life-confounding effects can come into play. Under these operatingconditions ball or roller centrifugal loads are increased, which causesincreased outer raceway loading, increased clearance, and more concen-trated inner raceway loading that reduce assembly life, as shown by Fig.23.4.

The magnitude of this effect increases rapidly, since centrifugal forcevaries with the square of the speed. Bearing size also has a major effect,producing variations in both rolling element mass and radius of rotation.These effects are particularly important in angular-contact ball bearings,where the centrifugal forces alter the outer ring contact angle, producingeven more significant life alteration effects. A parameter often used toexpress the severity of bearing speed conditions is dN, the product of thebearing bore diameter expressed in millimeters and the rotational speedof the bearing in revolutions per minute. It is normal to consider high-speed bearings as those with dN values of 1 million or more. For bearingoperation under high-speed conditions, sophisticated analytical tech-niques are required to reliably calculate bearing rating lives for compar-ison with collected life data.

Using speed to accelerate bearing test programs has its limitations.Functional speed limitations exist in standard bearing designs becauseof the stamped metal or molded plastic cage designs, which are inade-quate for high-speed operation. Excessive heat generation rates may oc-cur at the raceway contacts, which have been designed primarily formaximum load-carrying capabilities at lower speeds, and component pre-cision may be altered due to the dynamic loading ofhigh-speed operation.System operating effects can also produce significant life effects on high-speed bearings. Some effects are insufficient cooling or the inadequatedistribution of the cooling medium, creating thermal gradients in thebearings that affect internal clearances and geometries. Higher operat-ing speeds will also produce higher bearing operating temperature levels.The lubricants used then must be capable of sustaining extended expo-sure to these high temperatures without suffering degradation. The con-

BEARING ENDURANCE TESTING AND ELEMENT TESTING METHODS

duct of high-speed life tests requires extra care to ensure that thefailures obtained are fatigue related and not precipitated by some speed-originated performance malfunction.

In Chapter 12 it was shown that rolling contacts in ball and rollerbearings operate in the EHL regime and that the thickness of the lubri-cant film generated by EHL is a strong function of bearing internalspeeds, increasing as these speeds are increased. Furthermore, in Chap-ter 23 it is shown that the fatigue life of a rolling bearing is a functionof the thicknesses of the lubricant films it generates compared to theroughnesses of the rolling contact surfaces. Bearing fatigue life tends toincrease as the surfaces in rolling contact are effectively separated bylubricant films. The adequacy of lubrication is currently expressed as A,the ratio of the lubricant film thickness to the composite rms roughnessof the contacting surfaces. Values of A in excess of 3 tend to yield ex-tended bearing fatigue life. To ensure an adequate lubricant film, a suf-ficient lubricant supply must be available to preclude lubricantstarvation (see Chapter 12). Lubricant starvation is a particular consid-eration when performing endurance testing with grease lubrication athigh speeds.

In Chapter 23, the means to quantify the lubrication-associated effectof speed on bearing endurance is demonstrated. Particularly in the re-gime of marginal lubrication, the effect is complex owing to the inter-actions of rolling component surface finishes and chemistry, lubricantchemical and mechanical properties, lubrication adequacy, contaminanttypes, and contamination levels. Considering adequate lubricant supplyand minimal contamination, increasing testing speed to the point thatcomplete separation of rolling contact surfaces is achieved has tended toextend test duration appreciably, thus thwarting the desired accelerationof endurance testing. Testing at speeds slow enough to cause operationin the marginal lubrication regime can achieve shortened testing time;however, the above-indicated side effects must be considered in the eval-uation of test results.

PRACTICAL TESTING CONSIDERATIONS

An individual bearing may fail for several reasons; however, the resultsof an endurance test series are only meaningful when the test bearingsfail by fatigue-related mechanisms. The experimenter must control thetest process to ensure that this Occurs.Some of the other failure modesthat can be experienced are discussed in detail by Tallian [19.2]. Thefollowingparagraphs deal with a few specificfailure types that can affectthe conduct of a life test sequence.

In Chapter 23, the influence of lubrication on contact fatigue life isdiscussed from the standpoint of EHL film generation. There are also

PRACTICAL TESTING CONSIDERATIONS 769

other lubrication-related effects that can affect the outcome of the testseries. The first is particulate contaminants in the lubricant. Dependingon bearing size, operating speed, and lubricant rheology, the overallthickness of the lubricant film developed at the rolling element-racewaycontacts may fall between 0.05 and 0.5 JLm (2-20 JLin.).Solid particleslarger than the film can be mechanically trapped in the contact regionsand damage the raceway and rolling element surfaces, leading to sub-stantially shortened endurances. This has been amply demonstrated bySayles and MacPherson [19.6] and others.

Therefore, filtration of the lubricant to the desired level is necessaryto ensure meaningful test results. The desired level is determined by theapplication which the testing purports to approximate. If this degree offiltration is not provided, effects of contamination must be consideredwhen evaluating test results. Chapter 23 discusses the effect of variousdegrees of particulate contamination, and hence filtration, on bearingfatigue life.

The moisture content in the lubricant is another important consider-ation. It has long been apparent that quantities of free water in the oilcause corrosion of the rolling contact surfaces and thus have a detrimen-tal effect on bearing life. It has been further shown by Fitch [19.7] andothers, however, that water levels as low as 50-100 parts per million(ppm) may also have a detrimental effect, even with no evidence of cor-rosion. This is due to hydrogen embrittlement of the rolling element andraceway material. See also Chapter 23. Moisture control in test lubri-cation systems is thus a major concern, and the effect of moisture needsto be considered during the evaluation of life test results. A maximumof 40 ppm is considered necessary to minimize life reduction effects.

The chemical composition of the test lubricant also requires consid-eration. Most commercial lubricants contain a number of proprietary ad-ditives developed for specific purposes; for example, to provide antiwearproperties, to achieve extreme pressure and/or thermal stability, and toprovide boundary lubrication in case of marginal lubricant films. Theseadditives can also affect the endurance of rolling bearings, either im-mediately or after experiencing time-related degradation. Care must betaken to ensure that the additives included in the test lubricant will notsuffer excessive deterioration as a result of accelerated life test condi-tions. Also for consistency of results and comparing life test groups, it isgoodpractice to utilize one standard test lubricant from a particular pro-ducer for the conduct of all general life tests.

The statistical nature of rolling contact fatigue requires many testsamples to obtain a reasonable estimate of life. A bearing life test se-quence thus needs a long time. A major job of the experimentalist is toensure the consistency of the applied test conditions throughout the en-tire test period. This process is not simple because subtle changes canoccur during the test period. Such changes might be overlooked until

770 BEARING ENDURANCE TESTING AND ELEMENT TESTING METHODS

their effects become major. At that time it is often too late to salvage thecollected data, and the test must be redone under better controls.

For example, the stability of the additive packages in a test lubricantcan be a source of changing test conditions. Some lubricants have beenknown to suffer additive depletion after an extended period of operation.The degradation of the additive package can alter the EHL conditions inthe rolling contacts, altering bearing life. Generally, the normal chemicaltests used to evaluate lubricants do not determine the conditions of theadditive content. Therefore if a lubricant is used for endurance testingover a long time, a sample of the fluid should be returned to the producerat regular intervals, say annually, for a detailed evaluation of its condi-tion.

Adequate temperature controls must also be employed during the test.The thickness of the EHL film is sensitive to the contact temperature.Most test machines are located in standard industrial environmentswhere rather wide fluctuations in ambient temperature are experiencedover a period of a year. In addition, the heat generation rates of individ-ual bearings can vary as a result of the combined effects of normal man-ufacturing tolerances. Both of these conditions produce variations inoperating temperature levels in a lot of bearings and affect the validityof the life data. A means must be provided to monitor and control theoperating temperature level of each bearing to achieve a degree of con-sistency. A tolerance level of ± 3C is normally considered adequate forthe endurance test process.

The deterioration of the condition of the mounting hardware used withthe bearings is another area requiring constant monitoring. The heavyloads used for life testing require heavy interference fits between thebearing inner rings and shafts. Repeated mounting and dismounting ofbearings can produce damage to the shaft surface, which in turn canalter the geometry of a mounted ring. The shaft surface and the bore ofthe housing are also subject to deterioration from fretting corrosion. Fret-ting corrosion results from the oxidation of the fine wear particles gen-erated by the vibratory abrasion of the surfaces, which is accelerated bythe heavy endurance test loading. This mechanism can also produce sig-nificant variations in the geometry of the mounting surfaces, which canalter the internal bearing geometry. Such changes can have a major ef-fect in reducing bearing test life.

The detection of bearing failure is also a major consideration in a lifetest series. The fatigue theory considers failure as the initiation of thefirst crack in the bulk material. Obviously there is no way to detect thisoccurrence in practice. To be detectable the crack must propagate to thesurface and produce a spall of sufficient magnitude to produce a markedeffect on an operating parameter of the bearing: for example, noise, vi-bration, and/or temperature. Techniques exist for detecting failures inapplication systems. The ability of these systems to detect early signs of

PRACTICAL TESTING CONSIDERATIONS 771

failure varies with the complexity of the test system, the type of bearingunder evaluation, and other test conditions. Currently no single systemexists that can consistently provide the failure discrimination necessaryfor all types of bearing life tests. It is then necessary to select a systemthat will repeatedly terminate machine operation with a consistent min-imal degree of damage.

The rate of failure propagation is therefore important. If the degreeof damage at test termination is consistent among test elements, the onlyvariation between the experimental and theoretical lives is the lag infailure detection. In standard through-hardened bearing steels the fail-ure propagation rate is quite rapid under endurance test conditions, andthis is not a major factor, considering the typical dispersion of endurancetest data and the degree of confidence obtained from statistical analysis.This may not, however, be the case with other experimental materialsor with surface-hardened steels or steels produced by experimental tech-niques. Care must be used when evaluating these latter results and par-ticularly when comparing the experimental lives with those obtainedfrom standard steel lots.

The ultimate means of ensuring that an endurance test series wasadequately controlled is the conduct of a post-test analysis. This detailedexamination of all the tested bearings uses high-magnification opticalinspection, higher-magnification scanning electron microscopy,metallur-gical and dimensional examinations, and chemical evaluations as re-quired. The characteristics of the failures are examined to establish theirorigins and the residual surface conditions are evaluated for indicationsof extraneous effects that may have influenced the bearing life. This tech-nique allows the experimenter to ensure that the data are indeed valid.The "Damage Atlas" compiled by Tallian et al. [19.8] containing numer-ous black and white photographs of the various bearing failure modescan provide guidance for these types of determinations. This work wassubsequently updated by Tallian [19.9], now including color photographsas well.

The post-test analysis is, by definition, after the fact. To provide con-trol throughout the test series and to eliminate all questionable areas,the experimenter should conduct a preliminary study whenever a bear-ing is removed from the test machine. In this portion of the investigationeach bearing is examined optically at magnifications up to 30x for in-dications of improper or out-of-control test parameters. Examples of thetypes of indications that can be observed are given in Figs. 19.2-19.6.

Figure 19.2 illustrates the appearance of a typical fatigue-originatedspall on a ball bearing raceway. Figure 19.3 contains a spalling failureon the raceway of a roller bearing that resulted from bearing misalign-ment, and Fig. 19.4 contains a spalling failure on the outer ring of a ballbearing produced by fretting corrosion on the outer diameter. Figure 19.5illustrates a more subtle form of test alteration, where the spalling fail-

ure originated from the presence of a debris dent on the surface. Figure19.6 gives an example of a totally different failure mode produced by theloss of internal bearing clearance due to thermal unbalance of the sys-tem.

The last four failures are not valid fatigue spalls and indicate the needto correct the test methods. Furthermore, these data points would needto be eliminated from the failure data to obtain a valid estimate of theexperimental bearing life.

TEST SAMPLES

Specific requirements have been established for a test sample to be usedin an endurance test sequence. The statistical techniques used to eval-uate the failure data require that the bearings be statistically similarassemblies. Therefore the individual components must be manufacturedin the same processing lot from one heat of material. Generally, it isconsidered prudent to manufacture the total bearing assembly in thismanner; however, when highly experimental materials or processes areconsidered, this is often not cost effective or even possible. In those casesthe inner ring, the most critical element in a bearing assembly from afatigue point of view, can be used as the test element, with the othercomponents being manufactured from standard material. The effects offailures occurring on the other parts can be eliminated during analysisof the test data. There is some risk in this approach because it is possiblethat too many failures could occur on these nontest parts, rendering it

impossible to calculate an accurate life estimate for the material undeJevaluation. In the cases cited, however, this risk is small because aninitial result indicating the superior performance of an experimental pro.cess is usually sufficient to justify continued developmental effort evenwithout a firm numerical estimate. In any case, additional life testEwould be required to establish the magnitude of the expected lot-to-lotvariation before adopting a new material or implementing a new man-ufacturing process.

The number of bearings to be tested and the test strategy to be em-ployed must also be carefully considered. Statistical analysis provides anumerical estimate of the value of the experimental life enclosed by up-per- and lower-bound estimates at specified confidence levels. The pre-cision of the experimental life estimate can be defined by the ratio 01these upper and lower confidence limits, and the experimental aim is tominimize this spread. The magnitude of the confidence interval de-

creases as the size of the test lot is increased; the cost of conducting thetest series also increases with test lot size. Therefore, the degree of pre-cision required in the test result should be established during the plan-ning stages to define the size of the test lot to achieve the required result.

The test strategy employed also affects testing precision. The classicalmethod of performing endurance tests is to use one large group runningeach individual bearing to failure. This process is time consuming, butit provides the best experimental estimates of both the L10 and L50 lives.Primary interest is, however, in the magnitude of the experimental L10,

so considerable time savings can be achieved by curtailing the test runsafter a finite operating period equal to at least three times the achievedexperimental L10 life. Recently it has been shown that additional savingsin test time accompanied by increases in test precision can be obtainedby using a sudden death test strategy [19.10]. In this test approach theoriginal test lot is subdivided into smaller groups of equal sizes. Eachsubgroup is then run as a unit until one bearing fails, at which time thetesting of that subgroup is terminated. Figure 19.7 illustrates the effectof both lot size and test strategy on the precision of life test estimatesobtainable from an endurance testing series.

Toprovide an accurate life estimate for the variable under evaluation,the experimenter must be sure that the test bearings are free from ma-terial and manufacturing defects and that all parts conform to estab-lished dimensional and form tolerances. Although this is an obviousrequirement, it is not always easy to attain. Experimental materialsmight respond quite differently to standard manufacturing processes, orthey could require unique processing steps that are not yet totally de-

TEST RIG DESIGN CONSIDERATIONS 777TABLE19.1. TypicalMetallurgicalAudit Parameters100% Nondestructive Tests: Ring Raceways Only

Magnafluxfor near-surfacematerials defectsEtch inspectionfor surfaceprocessingdefects

Sample Destructive Tests: All Components

Microhardnessto 0.1 mm (0.004 in.) depth belowracewaysurfacesMicrostructuresto 0.3 mm (0.012 in.) depth belowracewaysurfacesRetained austenite levelsFracture grain sizeInclusionsratings

fined. Experimental manufacturing processes require additional verifi-cation, or their use might produce unexpected variations in relatedmetallurgical or dimensional parameters. Therefore, adequate test con-trol is achieved by detailed pretest auditing of the test parts to supple-ment the standard in-process evaluations. Tables 19.1 and 19.2 containlists of those metallurgical and dimensional parameters considered man-datory in a typical pretest audit, as well as an indication of the numberof samples that need to be checked in each case.

These lists are not complete. Other parameters could be evaluatedbeneficially if time and money permit.

TEST RIG DESIGN CONSIDERATIONS

Some specific characteristics are desired in an endurance test system toachieve the control requirements of a life test series. An individual testrun uses a long time, so the test machine must be capable of runningunattended without experiencing variation in the applied test parame-ters, such as load(s), speed, lubrication conditions, and operating tem-perature. The basic test system components that could also be subject tofatigue, such as load support bearings, shafts, and load linkages, shouldbe many times stronger than the test bearings so that test runs can becompleted with the fewest interruptions from extraneous causes. Theassembly of the test machine should have a minor influence on the ex-perimental test conditions to minimize variations between individualtest runs. For example, the alignment of the test bearing relative to theshaft should be automatically ensured by the assembly of the test hous-ing. If not, a simple direct means of monitoring and adjusting this pa-rameter must be provided. Again, since a test series requires multiplesetups, easy assembly and disassembly of the test system is desirable tominimize turnaround time and manpower requirements for test bearingchanges. In addition, the test system must be easy to maintain and

778 BEARING ENDURANCE TESTING AND ELEMENT TESTING METHODS

TABLE 19.2. TypicalDimensionalAuditParameters

100%Assembled Bearings

Radial loosenessAverageand peak vibration levels

Statistical Sample of Ring Grooves

Diameter and wavinessRadius and formCross-groovesurface texture

Statistical Sample of Balls

Diameter and out-of-roundSet sizevariationWavinessSurfacetexture

should be capable of operating reliably and efficiently for years to ensurethe long-term compatibility of test results. Basically, design simplicity isa key ingredient in meeting all of these demands. A more comprehensivediscussion of the design philosophy of life test rigs has been publishedin [19.11]. Figure 19.8 illustrates some of the typical endurance test rigconfigurations discussed there.

The application of these design concepts to actual endurance test sys-tems will be briefly addressed. Figure 19.9 is the schematic of an SKFrig for testing 35-50-mm-boreball and roller bearings under radial, ax-ial, or combined loads. Figure 19.10 is an actual photograph.

Operating speed may be varied within limits to achieve a given testcondition 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 bear-ing to be tested and its normal operating mode. For example, Fig. 19.11illustrates a four-bearing test rig concept used in the life testing of ta-pered roller bearings [19.12]. In this instance, while testing is conductedunder an externally applied radial load, the bearing also sees an inter-nally induced axial load. The size of this latter load is a function of themagnitude ofthe applied radial load, the fixed axial locations of the bear-ing cups and cones in the test housing, and the basic internal design ofche test bearings. Figure 19.12 shows a rig using this design concept.

Tests are often conducted to define the life of bearings as used in spe-~ific applications. They are frequently called life or endurance tests, but,more correctly, they are extended duration performance tests. The same

basic test practices and rig considerations are required for these tests,but some modifications of philosophy are required to simulate the majoroperating parameters of the application while achieving realistic testacceleration. An example of this type of tester is the SKF-developed"A-frame" tester shown in Fig. 19.13, which is used for evaluating au-tomotive wheel hub assemblies.

This tester simulates an automotive wheel bearing environment byusing actual car mounting hardware, combined radial and axial loadsapplied at the tire periphery to produce moment loads on the bearingassembly, grease lubrication, and forced-air cooling. Dynamic wheel load-ing cycles equivalent to those produced by vehicle lateral loading areapplied cyclically to simulate a critical driving sequence. Testing is con-ducted in the sudden death mode so that hub unit life in this simulatedenvironment can be calculated from standard life test statistics. This testprovides a way to compare the relative performance of automotive wheelsupport designs using life data generated under conditions similar tothose of actual applications.

ELEMENT TESTING

Because numerous test samples are required to obtain a useful experi-mental life estimate, conducting an endurance test series on full-scalebearings is expensive. The identification of simpler, less costly life testing

methods has therefore been a longstanding objective. The use of elemen-tal test configurations offers a potential solution to this need. In thisapproach, a test specimen having a simplified geometry (e.g., flat washer,rod, or ball) is used and rolling contact is developed at multiple testlocations. The ambition is to extrapolate the life data generated in anelement test to a real bearing application, thus saving calendar time andcost as compared to life data generated using full-scale bearing tests.This objective has historically not been achieved, generally because allof the operating parameters influencing fatigue life of rolling-slidingcontacts were not reduced to stresses; rather, as is shown in Chapter 23,they were evaluated as life factors. The only stress directly evaluated inboth element and full-scale bearing endurance testing has been theHertz or normal stress acting onthe contact. Lubrication, contamination,mrface topography, and material effects have been evaluated as life fac-;ors. To be able to extrapolate the life data derived from element testing,0 full-scale bearing life data, it is necessary to evaluate both data setsrom the standpoint of applied and induced stresses as compared to ma-erial strength. The methods to accomplish this for full bearings are de-

veloped in Chapter 23; Harris [19.13] developed a similar method forballs endurance tested in V-ring test rigs.

Even without direct correlation of life test data between elements andfull-scale bearings, element testing has proven useful in the ability torank performance of various materials in initial screening sequences orin adverse environments, such as extremely low or high temperature,oxidizing atmospheres, and vacuum. Therefore, discussion of element lifetesting techniques is warranted, even when the test data evaluationtechniques are not such as to permit direct correlation with full-scalebearing life test data. Caution must always be used, however, becausethe precision of the ranking process is open to question. Performancereversals have sometimes been experienced when comparing the screen-ing element test results when the materials have been retested in actual

bearings. Such reversals can be avoided if both the element test data andactual bearing test data evaluations are based on the total stress consid-eration.

The oldest and perhaps most widely used element test configurationis the rolling four-ball machine developed in the early 1950s [19.14].Thissystem uses four 12.7-mm (0.5-in.) diameter balls to simulate an angular-contact ball bearing operating with a vertical axis under a pure thrustload. One ball is the primary test element serving as the inner ring ofthe bearing assembly. It is supported in pyramid fashion on the remain-ing three balls, which rotate freely in a conforming cup at a predeter-mined contact angle. A modification of this test method, the rollingfive-ball tester, was subsequently developed at NASA Lewis ResearchLaboratories [19.15], and it uses four balls in the intermediate position.This latter system, illustrated in Fig. 19.14 has been used to generatean extensive amount of life test data on standard and experimental bear-ing materials.

Another widely used element test system is the RC (rolling contact)tester developed at General Electric [19.16] (Fig. 19.15). The test elementin this configuration is a 4.76-mm (0.1875-in.) rod rotating under loadbetween two 95.25-mm (3.75-in.) diameter discs. The rod can be axiallyrepositioned to achieve a number of rolling contact tracks on a single test

specimen. Unfortunately, this configuration is not as cost effective as itfirst appears. Stress concentrations will occur at the edges of the rodcontact unless the discs are profiled in the axial direction. This signifi-cantly increases the cost of manufacturing the discs. During operation,fatigue failures on the rod also tend to damage the disc surfaces, requir-ing that these be refinished at regular intervals.

An interesting modification of this element test concept was developedto eliminate the need for the expensive test discs [19.17]. In this version,illustrated in Fig. 19.16, three standard balls supported in standard ta-pered roller bearing outer rings serve the function of the discs and in-crease the number of test contacts. The test specimen is a cylindrical rodwhich can be used for several tests.

Another element test configuration is the single-ball tester developedby Pratt and Whitney, United Technologies Corporation for evaluatingballs used in aircraft gas turbine engine bearings [19.18]. This systemshown in Fig. 19.17 tests balls from approximately 19-65 mm (0.75-2.50in.) in two V-ring raceways with lubrication to simulate the application.It is this system for which Harris [19.13] developed a stress-based balllife prediction method. The fatigue limit stress values (see Chapter 23)

described and defined in Chapter 8, do nevertheless occur in the contacts.The surface frictional shear stresses accompanying the sliding motionsneed to be included in the evaluation of fatigue endurance. To experi-mentally determine the magnitude of the frictional stresses occurring inEHL contacts, rolling-sliding disc machines have been developed. Thedevice developed by Nelias et al. [19.20] is illustrated by Fig. 19.18. Thediscs are contoured to produce elliptical contact areas as illustrated byFig. 19.19. The motors in Fig. 19.18 may turn at different speeds toachieve the desired rolling-sliding motion. Motor 2 is mounted in hy-drostatic cylindrical bearings to permit friction torque, and hence, fric-tion force measurement. The friction force Ff to applied force W ratio iscalled the traction coefficient. Using the analytical methods of Chapter13, the effective local (x, y) friction coefficients can be estimated from thetest results. In Chapter 23, it is shown how the test device of Fig. 19.18has been used to determine the characteristics of the effect of friction onfatigue of the rolling-sliding contacts in ball and roller bearings. Byequipping the test rig with the contaminated lubrication system of Fig.19.20, Ville and Nelias [19.21] investigated the effects of particulate con-tamination on rolling-sliding contact fatigue. This will be discussed fur-ther in Chapter 23.

A ball-disc test rig, initially developed by Wedeven [19.21] and shownby Fig. 19.21, was designed to determine the nature oflubricant films inpoint contacts. Rolling velocity may be varied by varying the ball drivespindle angle and the radius at which the ball contacts the disc. Usinga disk element of a clear material such as sapphire or glass and optical

ture, and high vacuum. As shown by Fig. 19.23, it further permits opticalexamination of the circular point contact under the effects of lubricantparticulate contamination.

CLOSUREIn Chapter 18, it was demonstrated that although ball and roller bearingfatigue life rating and endurance formulas are founded in theory, theyare semiempirical relationships requiring the establishment of variousconstants to enable their use. These constants, which depend upon thebearing raceway and rolling element materials, can be established onlyby appropriate testing. Because ofthe stochastic nature of rolling bearingfatigue endurance, testing procedures necessarily require bearing and/or material populations of sufficient size to render the test results mean-ingful. Sample sizing effects are discussed in detail in Chapter 20.

Historically, to establish sufficiently accurate rating formula con-stants, it has been necessary to test complete bearings. With the devel-opment of stress-based life factors as shown in Chapter 23, however, itis now possible to use element-testing methods to determine many ofthese constants. For example, endurance testing of balls in V-ring testrigs may be used to determine the basic material fatigue strengths ofvarious materials. On the other hand, some ofthe stresses that influencebearing life depend on the raceway forming and surface finishing meth-ods. To duplicate these effects, the exact component may need to be en-durance tested.


tech. Mech. Eng. Ser. 1, RSAEE, No.3, 7 (1947).19.2. T. Tallian, "On Competing Failure Modes in Rolling Contact," ASLE Trans. 10,418-

439 (1967).

792BEARING ENDURANCE TESTING AND ELEMENT TESTING METHODS

19.3.R. Valori, T. Tallian, and L. Sibley, "Elastohydrodynamic Film Effects on the LoadLife Behavior of Rolling Contacts," ASME Paper 65-LUBS-ll (1965).

19.4.G. Johnston, T. Andersson, E. Van Amerongen, and A. Voskamp, "Experience ofElement and Full Bearing Testing over Several Years," in Rolling Contact FatigueTesting of Bearing Steels, ASTM STP 771, ed. J. J. C. Hoo, American Society forTesting and Materials, Philadelphia (1982).

19.5.G. Lundberg and A. Palmgren, "Dynamic Capacity of Roller Bearings," Acta Poly-tech. Mech. Eng. Ser. 2, RSAEE, No.4, 96 (1952).

19.6.R. Sayles and P. MacPherson, "Influence of Wear Debris on Rolling Contact Fatigue,"in Rolling Contact Fatigue Testing of Bearing Steels, ASTM STP 771, ed. J. Hoo,255-274 (1982).

19.7.E. Fitch, An Encyclopedia of Fluid Contamination Control, Fluid Power ResearchCenter, Oklahoma State University (1980).

19.8.T. Tallian, G. Baile, H. Dalal, and O. Gustafsson, Rolling Bearing Damage: A Mar.

phological Atlas, SKF Industries, Philadelphia (1974).19.9.

T. Tallian, Failure Atlas for Hertz Contact Machine Elements, ASME Press, NewYork (1992).

19.10. T. Andersson, "Endurance Testing in Theory," Ball Bearing J. 217, 14-23 (1983).19.11. G. Sebok and U. Rimrott, "Design of Rolling Element Endurance Testers," ASME

Paper 69-DE-24 (1964).

19.12. R. Hacker, "Trials and Tribulations of Fatigue Testing of Bearings," SAE TechnicalPaper 831372 (1983).

19.13. T. Harris, "Prediction of Ball Fatigue Life in a Ball/V-Ring Test Rig," ASME Trans.,J. Tribology 119, 365-374 (July 1997).

19.14. F. Barwell and D. Scott, Engineering 182, 9-12 (1956).

19.15. E. Zaretsky, R. Parker, and W.Anderson, "NASA Five-Ball Tester-over 20 Years ofResearch," in Rolling Contact Fatigue Testing of Bearing Steels, ASTM STP 771, ed.J. J. C. Hoo, American Society for Testing and Materials, Philadelphia (1982).

19.16. E. Bamberger and J. Clark, "Development and Application of the Rolling ContactFatigue Test Rig," in Rolling Contact Fatigue Testing of Bearing Steels, ASTM STP771, ed. J. Hoo (1982).

19.17. D. Glover, "A Ball-Rod Rolling Contact Fatigue Tester," in Rolling Contact FatigueTesting of Bearing Steels, ASTM STP 771, ed. J. J. C. Hoo, American Society forTesting and Materials, Philadelphia (1982).

19.18. P. Brown, G. Bogardus, R. Dayton, and D. Schulze, "Evaluation of Powder-ProcessedMetals for Turbine Engine Ball Bearings," in Rolling Contact Fatigue Testing ofBearing Steels, ASTM STP 771, ed. J. J. C. Hoo, American Society for Testing andMaterials, Philadelphia (1982).

19.19. T. Harris and J. McCool, "On the Accuracy of Rolling Bearing Fatigue Life Predic-tion," ASME Trans., J. Tribology 118,297-310 (April 1996).

19.20. D. NeJias, M.-L. Dumont, F. Couhier, G. Dudragne, and L. Flamand, "Experimentaland Theoretical Investigation of Rolling Contact Fatigue of 52100 and M50 Steelsunder EHL or Micro-EHL Conditions," ASME Trans. J. Tribology 120, 184-190(April 1998).

19.21. F. Ville and D. Nelias, "Early Fatigue Failure Due to Dents in EHL Contacts," Pre-sented at the STLE Annual Meeting, Detroit (May 17-21, 1998).

19.22. L. Wedeven, "Optical Measurements in Elastohydrodynamic Rolling Contact Bear-ings" (Ph.D. dissertation, University of London, 1971).

19.23. Wedeven Associates, Inc., "Bridging Technology and Application through Testing,"Brochure (1997).

794STATISTICAL METHODS TO ANALYZE ENDURANCE

SymbolDescription

tIer, n, k)Pivotal function for testing for differences among kestimates of XO.IOu(r, n, p)Pivotal function for setting confidence limits on x

puI(r, n, p, k)

Pivotal function for setting confidence limits on xp

usingk data samples

vCr, n)Pivotal function for setting confidence limits on /3vI(r, n, k)Pivotal function for setting confidence limits on /3usingk data samples

w(r, n, k)Pivotal function for testing whether k Weibullpopulations have a common /3x A random variablexpThe pth percentile of the distribution of the randomvariable x

/3The Weibull shape parameterTJThe Weibull scale parameter

GENERAL

Many statistical distributions have been used to describe the randomvariability of the life of manufactured products. Such choices can be var-iously justified. For example, if a product has a reservoir of a substancethat is used up at a uniform rate through the product's life, and if theinitial supply of the substance varies from item to item according to anormal (Gaussian) distribution, then the product life will be normallydistributed. Correspondingly, if the initial amount of the substance fol-lows a gamma distribution, item life will be gamma distributed.

The Weibull distribution is a popular product life model generally jus-tified by its property of describing, under fairly general circumstances,the way that the smallest values in large samples vary among sets oflarge samples. Thus if item life is determined by the smallest life amongmany potential failure sites, it is reasonable to expect that life will varyfrom item to item according to a Weibull distribution.

Another property that makes the Weibull distribution a reasonablechoice for some products is that it can account for a steadily increasingfailure rate characteristic of wear-out failures, a steadily decreasing fail-ure rate characteristic of a product that benefits from "burn-in," or aconstant failure rate typical of products that fail due to the OCcurrenceof a random shock.

The two-parameter Weibull distribution was adopted by Lundberg andPalmgren [20.1J to describe bearing fatigue life on the strength of theexcellence of the empirical fit to bearing fatigue life data. As will bedescribed in Chapter 23, because of improvements in both the materialand the methods of finishing bearings, it has been found recently that

800STATISTICAL METHODS TO ANALYZE ENDURANcE

native to the use of equations (20.1) and (20.8) for the computation ofprobabilities and percentiles. The graphical approach is sufficiently ac-curate for most purposes. The primary use of probability paper is not,however, for representing known WeibuII distributions but for estimatingthe WeibuII parameters from life test results.

ESTIMATION IN SINGLE SAMPLES

Application of the WeibuU Distribution

Up to this point it has been assumed that the WeibuII parameters areknown, and additionaIIy required quantities such as probabilities, per-centiles, expected values, variances, and standard deviations have beencalculated in terms of these known parameters. This is a common situ-ation in bearing application engineering, in which, given a catalog cal-culation of XO.IO (LlQ) and the standard WeibuII slope of f3 = 1.1, it isrequired to compute the median life or the MTBF, and so on. In devel-opmental work involving new variables such as materials, lubricants, orfinishing processes, the focus is on determining the effect of these factorson the Weibull parameters. Accordingly, a sample of bearings modifiedfrom standard in some way is subjected to testing under standardizedconditions of load and speed until some or all fail. When all fail, thesample is said to be uncensored. In a censored sample some bearings areremoved from test prior to failure. Given the lives to failure or to testsuspension for the unfailed bearings, the aim is to deduce the underlyingWeibull parameters. This process is called estimation, because it is rec-ognized that, since life is a random variable, identical samples wiII resultin different test lives. The Weibull parameter values estimated in anysingle sample must themselves therefore be regarded as observed valuesof random variables that wiII vary from sample to sample according to aprobability distribution known as the sampling distribution of the esti-mate. The scatter in the sampling distribution wiII decrease with samplesize. Sample size thus affects the degree of precision with which theparameters are determined by a life test. The precision is expressed byan uncertainty or confidence interval within which the parameter valueis likely to lie. An estimation procedure that results in the computationof a confidence interval is called interval estimation. A procedure thatresults in a single numerical value for the parameter is called point es-timation. Point estimates in themselves are virtually useless, since with-out some qualification there is no way of judging how precise they are.

Accordingly, an analytical technique is given in the sequel for com-puting interval estimates of WeibuII parameters. It is recommended thatthis technique be supplemented, however, with a point estimate obtainedgraphically. The graphical approach to estimation gives a synoptic view

ESTIMATION IN SINGLE SAMPLES 801

of the entire distribution and offers the opportunity to detect anomaliesin the data that could easily be overlooked if reliance is placed entirelyon an analytical technique.

Point Estimation in Single Samples: Graphical Methods

Assume that a sample of n bearings is tested until all fail. The orderedtimes to failure are denoted Xl < X2 < ... Xn' Consider the CDF of theWeibull population from which the sample was drawn. If this functionwere known, it would follow that the lives Xi and the values F(xi), i =

1, ... , n, would plot as a straight line on Weibull probability paper. Ithas been shown that even though the function F(x) is not known, none-theless F(xi) wiII vary in repeated samples according to a known pdf. Themean or expected value of F(x) has been shown to equal i/(n + 1). Themedian value of F(xi), also known as the median rank, has been shownby Johnson [20.2] to be approximately (i - 0.3)/(n + 0.4). The procedurethen is to plot the mean or median value of F(xi) against Xi for i = 1, 2,... , n. The tradition in the bearing industry is to use the median ratherthan the mean as a plotting position choice, but the difference is smallcompared to the sampling variability.

Table 20.2 lists the ordered lives at failure for a sample of size n =

10, along with the actual and approximated values of the median ranks.Hence the approximation is adequate within the limits of graphical ap-proximation. The median ranks are shown plotted against the lives inFig. 20.3.

The straight line fitted to the plotted points represents the graphicalestimate of the entire F(x) curve. Estimates of the percentiles of interestare then read from the fitted straight line. For example, to within graph-ical accuracy the XO.lO value is estimated as 15.3. The Weibull shape pa-

TABLE 20.2. Random Uncensored Sample Size of n = 10

Failure Order Median i - 0.3No. (i) Life Rank n + 0.4

1 14.01 0.06697 0.067312 15.38 0.16226 0.163463 20.94 0.25857 0.259624 29.44 0.35510 0.355775 31.15 0.45169 0.451926 36.72 0.54831 0.548087 40.32 0.64490 0.644238 48.61 0.74142 0.740389 56.42 0.83774 0.83654

10 56.97 0.93303 0.93269

rameter, estimated simply as the measured slope of the straight line, isroughly 2.2.

The same graphical approach applies for right-censored data in whichthe censored observations achieve a longer running time than the fail-ures do. The full sample size n is used to compute the plotting positions,but only the failures are plotted. When there is mixed censoring-thatis, there are Suspended tests among the failures-the plotting positionsare no longer computable by the method given since the Suspensionscause ambiguity in determining the order numbers of the failures. Sev-eral alternative approaches are available for this situation, with gener-ally negligible differences among them. The method known as hazardplotting due to Nelson [20.3] is recommended because it is easy to use.Column 1 of Table 20.3 gives the lives of failure or test suspension in asample of size n = 10. Of the 10 bearings, r = 4 have failed, and thelives at failure are marked with an "F" in Table 20.3. Similarly the livesat test Suspension are indicated by "S." The lives in column 1 are inascending order of time on test irrespective ofwhether the bearing failed.Column 2, termed the reverse rank by Nelson [20.3], assigns the valuen to the lowest time on test, the value n - 1 to the next lowest, and soon. Column 3, called the hazard, is the reciprocal of the reverse rank,but is computed only for the failed bearings. Column 4 is the cumulativehazard and contains for each failure the sum of the hazard values incolumn 3 for that failure and each failure that occurred at an earlierrunning time. Thus for the second failure the cumulative hazard is

ESTIMATION IN SETS OF WEIBULL DATA 811

For XO.10 the ratio of the upper to lower confidence limits contains therandom variable f3. The approach taken in [20.6] in this case is to useas a precision measure the median value of this ratio, denoted RO.50' Theexpression for this median ratio contains the unknown value of the trueshape parameter f3. For planning purposes one may use an historicalvalue such as 1.1 or, alternatively, the value R g.50 as the precision mea-sure. Values of Rg.50 are given in Table 20.4 for conventional testing andTable 20.5 for sudden test testing.

ESTIMATION IN SETS OF WEIBULL DATA

Methods

Very often an experimental study of bearing fatigue life will include thetesting of several samples, differing from each other with respect to thelevel of some qualitative factor under study. A qualitative factor is dis-tinct from a quantitative factor, such as temperature or load, which canbe assigned a numerical value. Examples of qualitative factors includelubricants, cage designs, or bearing materials.

It was shown in [20.7] that more precise estimates can be made if thedata in the samples making up the complete investigation are analyzedas a set. This is possible if it can be assumed that the samples are drawnfrom Weibull populations, which, although they might differ in theirscale parameter values, nonetheless have a common value of f3.

Applicable tabular values for carrying out the analyses presupposethat the sample size n and the number of failures r are the same foreach sample in the set, so henceforth this is assumed to be the case. Itis thus assumed that k groups of size n have been tested until the rthfailure occurred in each group. The first step is to determine whether itis plausible that the groups have a common value of f3. This is done byanalyzing each group individually to determine the values of XO.10 and f3.The largest and smallest of the k f3 estimates are then determined, andtheir ratio formed. If the f3 values did differ among the groups, this ratiowould tend to be large. Table 20.6 gives the value of the 90th percentileof the ratio w = {3maJ {3min for various r, n, and k. These values weredetermined by Monte Carlo sampling from k Weibull populations thatdid have a common value of f3. Thus, values of the ratio of largest tosmallest shape parameter estimates exceeding those in Table 20.6 willoccur only 10% of the time if the groups do have a common value of f3.These values may be used as the critical values in deciding whether acommon f3 assumption is justified.

Having determined that the common f3 assumption is reasonable, thiscommon f3 value can be estimated using the data in each group, by solv-ing the nonlinear equation

Ml6S1'.o\.'tISTICAL METHODS TO ANALYZE ENDURANcEGroupNo.

1 2 3 4 5Raw ML estimate .xOlD

4.84 2.81 3.80 4.87 6.34Median unbiased estimate .xb.l0

4.55 2.65 3.57 4.58 5.97Lowerconfidencelimit

3.25 1.89 2.55 3.27 4.25Upper confidencelimit

6.13 3.56 4.82 6.18 8.04From Table 20.7 the 90th perc€!o.tileoft

1(10, 10,5) is 1.26. The small-

est ratio of raw .x0.1O values that will be significant, denoted SSR (short-

est significant ratio) is defined implicitly by

1.26 = {31 In (SSm = 2.480 In (SSR).Solving for SSR gives

SSR = e}(p(_1.2_6)= 1.66.2.48

Thus groups for which the ratio of raw ML estimates of X

O

.lO exceeds1.66 may be declared different. This wonld include Groups 5 and 3,because their X

O.10 estimates are in the ratio 6.34/3.80 = 1.67 > 1.66.

Note that the confidence limits on x overlap for these two groups.In general, if the confiden~e lill},itsd~o not overlap, the groups differ.However the groups may d'ffer •• in the present case even though theconfidence limits do overlap.

CLOSURE

Chapter 19 explored the reasons fOe and concepts and methods o( en-durance testing of ball and roller hea;';ngS and components. In this chap.tel' the means to relate such test rei'!\.1.ltsto applications of standard andspecial bearing products were cover~d. The applicability of the Weibulldistribution to such test data was de"'onstrated. It is further shown that,having specified a Weibull populatio" for example, by a catalog calcu-lation of theoretical life, it is p088ibl; to calculate other characteristicsthat may occasionally be of interest, such as the mean time betweenfailures.

The methodology for forming a graphical estimate of a Weibnll popu-lation using a censored or an uncenso>ed data sample was described andillustrated.

Examples were given for using the ~ethod of maximum likelihood forthe POint and interval estimation of t"e Weibull parameters from eithertyPe II censored or sudden de~th test Mmples ..

Finally, the Pnx:edure was g>v~n fOr <>nalyzingWeibull samples m.sets.rhis pnx:edure g>ves more pre~"e est'mated parameters because ,t ex-racts information from the entIre set ~f data.

REFERENCES 817REFERENCES20.1. G. Lundberg and A. Palmgren, "Dynamic Capacity of Rolling Bearings," Acta Poly-

tech. Mech. Eng. Ser. 1, R.S.A.E.E., No.3, 7 (1947).20.2. L. Johnson, Theory and Technique of Variation Research, Elsevier, New York (1970).20.3. W. Nelson, "Theory and Application Hazard Plotting for Censored Failure Data,"

Technometrics 14, 945-966 (1972).20.4. J. McCool, "Inference on Weibull Percentiles and Shape Parameter for Maximum

Likelihood Estimates," IEEE Trans. Reliab. R-19, 2-9 (1970).20.5. J. McCool, "Analysis of Sudden Death Tests of Bearing Endurance," ASLE Trans. 17,

8-13 (1974).20.6. J. McCool, "Censored Sample Size Selection for Life Tests," Proceedings 1973 Annual

Reliability and Maintainability Symposium, IEEE Cat. No. 73CH0714-64 (1973).20.7. J. McCool, "Analysis of Sets of Two-Parameter Weibull Data Arising in Rolling Con-

tact Endurance Testing," in Rolling Contact Fatigue Testing of Bearing Steels, ASTMSTP 771, ed. J. J. C. Hoo, American Society for Testing and Materials, Philadelphia,293-319 (1982).

820PERMANENT DEFORMATION AND BEARING STATIC CAPACITY

Symbol Description Unitsa Contact angle

0

'}' D cos a/dm8s Permanent deformation mm (in.)TJ Hardness reduction factorp Curvature

mm -1 (in. -1)a Yield or limit stressN/mm2 (psi)'Ps Load rating factor

SUBSCRIPTSa Refers to axial directioni Refers to inner racewayIp Refers to incipient plastic flow of material0 Refers to outer racewayr Refers to radial directions Refers to static loading

GENERAL

Many structural materials exhibit a strain limit under load beyondwhich full recovery of the original elemental dimensions is not possiblewhen the load is removed. Bearing steel loaded in compression behavesin a similar manner. Thus when a loaded ball is pressed on a bearingraceway, an indentation may remain in the raceway and the ball mayexhibit a "flat" spot after load is removed. These permanent deforma-tions, if they are sufficiently large, can cause excessive vibration andpossibly stress concentrations of considerable magnitude.

CALCULATION OF PERMANENT DEFORMATION

In practice, permanent deformations of small magnitude OCcureven un-der light loads. Figure 21.1 taken from reference [21.4J shows a verylarge magnification of the contacting rolling element surfaces in a typicalball bearing both in the direction of rolling motion and transverse to thatdirection.

Figure 21.2, also from reference [21.4J, shows an isometric view of aground surface having spatial properties similar to honed and lappedraceway surfaces. Noting the occurrence of "peaks and valleys" even witha finely finished surface, it is apparent that prior to distributing a loadbetween rolling element and raceway over the entire contact area thusgiving an average compressive stress a = Q/A, the load is distributedonly over the smaller area of contacting peaks, giving a much larger

REFERENCES 83~

permanent deformations result in increased friction, noise, and vibra-tion. Chapter 25 discusses the noise and vibration phenomenon in sub.stantial detail. In this chapter, the discussion centered on bearing staticload ratings, which, if not exceeded while the bearing was not rotatingwould preclude permanent deformations of significant magnitude. ThEratings were based on a maximum allowable permanent deformation oj0.000ill. Subsequently, it was determined that for various types of balland roller bearings, this deformation could be related to a value of rollingelement-raceway contact stress. In accordance with this stress, basicstatic load ratings are developed for each rolling bearing type and size.

Generally, a load of magnitude equal to the basic static load ratingcannot be continuously applied to the bearing with the expectation ofobtaining satisfactory endurance characteristics. Rather, the basic staticload rating is based on a sudden overload or, at most, one of short du-ration compared to the normal loading during continuous operation. Ex-ceptions to this rule are bearings that undergo infrequent operation ofshort duration, for example, bearings on doors of missile silos or damgate bearings. For these and simpler applications, bearing design maybe based on basic static load rating rather than on endurance of fatigue.In Chapter 22, the concept of a shakedown limit stress is discussed,which pertains to raceway subsurface microstructural alterations thatoccur during rotation while the current static load ratings are based upondamage during nonrotation. Because of relatively slow speeds of rotationand infrequent operation, neither vibration nor surface fatigue may beas significant in such applications as excessive plastic flow of subsurfacematerial. The bearings could thus be sized to eliminate or minimize suchplastic flow and ultimately bearing failure.

REFERENCES21.1. A. Palmgren, Ball and Roller Bearing Engineering, 3rd ed., Burbank, Philadelphia

(1959).

21.2. American National Standard, ANSI / AFBMA Std 9-1990, "Load Ratings and FatigueLife for Ball Bearings."

21.3. American National Standard, ANSI / AFBMA Std 11-1990, "Load Ratings and Fa-tigue Life for Roller Bearings."

21.4. R. S. Sayles and S. Y. Poon, "Surface Topography and Rolling Element Vibration,"Precis. Eng. 137-144 (1981).

21.5. International Standard ISO 76, "Rolling Bearings-Static Load Ratings" (1989).21.6. G. Lundberg, A. Palmgren, and E. Bratt, "Statiska Biirformagan hos Kullager och

Rullager," Kullagertidningen 3 (1943).

836 MATERIAL RESPONSE TO ROLLING CONTACT

Materials-oriented investigations have been primarily phenomenolog-ical in nature; that is, attempts have been made to carefully control testconditions and to correlate material response characteristics with testconditions and bearing performance. Subsequent analyses of the metal-lurgical mechanisms involved are prerequisites for establishing reliableanalytical models for predicting performance.

This chapter reviews the microstructural alterations resulting fromrolling contact and the attendant changes in residual stress state. Ob-served effects of residual and applied bulk stresses, combined with roll-ing contact stress, on bearing performance are discussed in terms offatigue life, failure mode, and dimensional stability.

MICROSTRUCTURES OF ROLLING BEARING STEELS

The microstructure of martensitically hardened and tempered AISI52100 steel is shown in Fig. 22.1. The microstructure consists primarilyof plate martensite [22.1], with 5-8 vol.% of (Fe,Cr)3C type carbides[22.2], and up to 20 vol.%retained austenite, depending on austenitizingand tempering conditions. Tempered hardness is generally Rc 58-64. Thelower values of retained austenite content and hardness are character-istically associated with higher tempering temperatures.

MICROSTRUCTURAL ALTERATIONS DUE TO ROLLING CONTACT 837

The hardened and tempered microstructures ofthe near-surface (case)and subsurface (core) regions of carburized bearing steel (e.g., AISI8620), are shown in Fig. 22.2. The case microstructure is predominantlymartensitic (plate type). The volume fractions of carbide and retainedaustenite vary widely, depending on the carburizing conditions, steelanalysis, and tempering procedures used.

The microstructure of the core region-that is, the subsurface mate-rial unaffected by carburizing-is also primarily martensitic, but, due tothe low carbon content (gradually 0.2 wt.% or less), is of the lath typerather than plate martensite. Characterization of these two martensitemorphologies is detailed in [22.1] and [22.3].

With the relatively rare exceptions of single-stress-cycle bearing fail-ure (e.g., fracture from heavy impact loading and wear), all material-related bearing failures involve accumulation of irreversible plasticdeformation during cyclic stressing. The latter is the classical definitionof metallurgical fatigue. The variety of bearing failure manifestationsarises from the conditions that promote fatigue initiations and the man-ner in which the operating conditions sustain propagation of the fatiguemechanisms. Fatigue failure may be initiated by an exogenous inclusionin the bearing steel, by mechanical disruption of the integrity of therolling contact surface (e.g., debris dents or scratches), or by a corrosionpit, to name just a few. The subsequent bearing failure, which by defi-nition precludes continued functionality, results from progression of fa-tigue damage from the initiating cause.

Specifications for microstructural characteristics, hardness, depth ofhardening, amount of retained austenite, and so forth, to a large extenthave been based upon bearing performance experiences in the field andthe results of bearing endurance tests conducted under controlled oper-ating conditions. Several decades of collective experience provide today'sphenomenologically based compendium of materials knowledge concern-ing requirements for and responses to rolling contact stressing. Usingthis experience to provide bearing steel quality heat treatment, manu-facturing procedures, and operating conditions that minimize the likeli-hood of experienced early-failure modes has permitted the study ofmaterial response to long-term rolling contact stressing. The objectivesof such studies have been oriented toward gaining insight into the fun-damental mechanisms involved in rolling contact fatigue and to assessfatigue life potential under "ideal" operating conditions. Alterations ofthe microstructure that have been observed under these conditions aredescribed in the next section.

MICROSTRUCTURAL ALTERATIONS DUE TOROLLING CONTACT

Marked alteration of the near-surface microstructure of endurance-tested bearing inner rings has been reported in the literature since 1946

MICROSTRUCTURAL ALTERATIONS DUE TO ROLLING CONTACT 839

[22.4-22.13].The alterations are principally characterized by differencesin the etching response of the microstructure in the region just beneaththe raceway surface (see Fig. 22.3), and are most heavily concentratedat a depth corresponding to the maximum shear stress associated withthe Hertzian stress field in the contact [22.5, 22.6].

Three aspects of microstructural alterations have been described[22.7, 22.8] and chronologically characterized [22.8]: the dark etchingregion (DER), DER + 30°bands, and DER + 30°bands + 80°bands. Thestructural changes obtained as a function of stress level and number ofinner ring revolutions are diagrammed in Fig. 22.4. Optical micrographsof the structural alterations, in parallel sections, are shown in Fig. 22.5.

The first alteration is the formation of the DER. Transmission electronmicroscopy identified the DER as consisting of a ferritic phase, contain-ing an inhomogeneously distributed excess carbon content (equivalent tothat of the initial martensite) mixed with residual parent martensite. Astress-induced process of martensitic decay is indicated [22.8]. The sec-ond manifestation of altered microstructure [22.8, 22.9] is the formationof white etching, disc-shape regions offerrite, about 0.1-0.5}.Lm (40-200}.Lin.)thick, and inclined at an angle of approximately 30° to the racewaycircumferential tangent. These regions are sandwiched between carbide-

rich layers. The third feature, initially reported in [22.7], is a second setof white etching bands, considerably larger than the 30° bands and in-clined at 80° to the raceway tangent in parallel sections. These disc-shaped regions are abut 10 J-tm(0.0004 in.) thick and consist of severelyplastically deformed ferrite [22.8].

As shown in Fig. 22.5, both the 30° and 80° bands incline toward thesurface in the direction of rolling element motion. Reversing the directionof rotation reverses the orientation of the bands. The characteristic an-gular orientations of these white etching bands have not been satisfac-torily explained. Crystallographic texturing in the near-surface region ofball bearing inner rings after high-stress operation has, however, beenobserved by Voskamp [22.6]. Angular characteristics of the texturing arereported to be consistent with white etching band orientations, suggest-ing that the bands have a crystallographically determined nature.

Moreover, the crystal alignment appears to accentuate the plane ofleast resistance to fracture; that is, the cube planes of body-centered-cubic (bcc) Fe become aligned parallel to the raceway surface with theso-called [110] direction parallel to the direction of rolling.

Hardness has been reported to increase slightly in the early stages oftesting and then to decrease markedly in the region associated with mi-crostructural alteration [22.8, 22.13). The number of stress cycles re-quired to produce the sequence of events leading to microstructuralalteration may be significantly reduced by increasing the temperature ofthe test specimen [22.13]. Localized changes in retained austenite con-

FIGURE 22.5. Optical micrographs of structural changes in 309 deep-groove ball bearinginner rings (parallel section). (a) DER in early stage. (b) Fully developed DER and 30°bands. (c) DER, 30° bands, and 80° bands. AISI 52100 steel (from [22.8]).

tent and residual stress level are also associated with these microstruc-tural alterations. See next section.

Another manifestation of microstructural alteration found in bearingrolling contact components that have experienced substantially heavyloading is commonly called a "butterfly," because ofwing-like emanationsfrom a "body" composed of a nonmetallic inclusion. An example of a but-terfly is shown by Fig. 22.6. Characteristically, as shown in [22.14-22.17]and illustrated by Fig. 22.7, after natal acid etching the butterfly wingsappear white in contrast to the surrounding matrix of martensite andare oriented at an angle of 40-45° to the raceway track in a directiondetermined by the direction of the friction force acting on the surface.They can occur at depths well below the depth of the maximum shearstress and are characteristically associated with microcracks runningalong the smooth edges of the wings.

Butterflies form around oxide, silicate, and titanium nitride particles,but not in conjunction with manganese sulfide or carbide particles [22.15,

22.16,22.18]. Wing development depends on stress level and the numberof stress cycles [22.16]. Butterflies generally form only under very heavyloading. A comprehensive characterization of the microstructural fea-tures of butterfly wings [22.19] concluded that they consist of a disper-sion of ultra-fine-grained ferrite and carbide, very similar in nature andformation mode to the 30° and 80° white etching bands described earlier.Further, the wings are probably initiated by preexisting cracks associ-ated with nonmetallic inclusion bodies. [22.14-22.16,22.19]. Subsequentcrack and wing growth proceed together.

White etching bands and butterflies are striking manifestations ofhigh stress, high cycle, rolling contact. While it has been difficult to pos-itively identify them as failure-initiating characteristics, Nelias [22.17]has indicated that fatigue failure does not seem to OCCurin their absence.

RESIDUAL STRESSES IN ROLLING BEARING COMPONENTS 843

Therefore, the stress below which they do not occur after a substantialnumber of cycles might be identified as the lower value of a fatigue limitstress. See Chapter 23.

RESIDUAL STRESSES IN ROLLING BEARING COMPONENTS

Sources of Residual Stresses

Residual stress is that stress which remains in a material when all ex-ternally applied forces are removed. Residual stresses arise in an objectfrom any process that produces a nonuniform change in shape or volume.These stresses may be induced mechanically, thermally, chemically, or bycombinations of these processes [22.20]. If a relatively thin sheet of mal-leable material, such as copper, is repeatedly struck with a hammer, thethickness of the sheet is reduced, and the length and width are corre-sponding increased, preserving constancy of volume. Ifthe same numberof equally intensive hammer blows were uniformly delivered to the sur-face of a copper block several inches thick, the depth of penetration ofplastic deformation would be relatively shallow with respect to the blockthickness. The deformed surface layer would be restrained from lateralexpansion by the bulk of subsurface material, which experienced lessdeformation. Consequently, the heavily deformed surface material wouldbe like an elastically compressed spring, prevented from expanding toits unloaded dimensions by its association with elastically extended sub-surface material. The resulting residual stress profile is one in which thesurface region is in residual compression and the subsurface region is ina balancing residual tension. This example is a literal description of theshot-peening process, wherein a surface is bombarded with pellets ofsteel or glass. A highly desirable residual-stress pattern is establishedfor components that experience high, cyclictensile stresses at the surfaceduring service. The magnitude of tensile stress experienced by the com-ponent during service is functionally reduced by the amount of residualcompressive stress, thereby providing significantly increased fatiguelives for parts such as springs and shafts.

The shot-peening example illustrates the essential characteristics ofa surface in which residual stress has been induced:

1. Nonuniformity of plastic deformation-that is, being near-surfaceonly-encourages the surface material to expand laterally.

2. Subsurface material, which experienced less plastic deformation, iselastically strained (in tension, in this example) as it restrains ex-pansion of the surface material, thereby inducing residual com-pressive stress in the surface region.


3. The resulting state of residual stress is a reflection of the elasticcomponents of strain in the surface and subsurface regions, whichare in equilibrium, providing a balanced tensile-compressive sys-tem.

Heat treatment, such as is used for hardening rolling bearing compo-nents, can exert very significant influence over the state of residualstress. Depending on the steel analysis, austenitizing temperature,quenching severity, component geometry, section thickness, and so forth,heat treatment can provide either residual compressive stress or residualtensile stress in the surface of the hardened component [22.20, 22.21].Temperature gradients are established from the surface to the center ofa part during quenching in a hardening treatment. Differential thermalcontraction associated with these gradients provides for nonuniformplastic deformation, giving rise to residual stresses. Additionally, volu-metric changes associated with the phase transformations taking placeduring heat treatment of steel occur at different times during quenchingat the part surface and interior due to the thermal gradients established.These sequential volumetric changes, combined with differential thermalcontractions, are responsible for the residual-stress state in a hardenedsteel component. The sequence and relative magnitudes of these contrib-uting factors determine the stress magnitude and whether the surface isin residual tension or compression.

Grinding of a hardened steel component to finished dimensions alsoaffects the residual surface stress. Generally, neglecting the effects ofabusive grinding practices that generate excessive heat and produce mi-crostructure alterations, it is found that the residual-stress effects as-sociated with grinding are confined to material within the first 50microns (0.002 in.) of the surface. Good grinding practice, as applied tobearing rings, produces circumferential residual compressive stress in ashallow surface layer. Grinding also involves some plastic deformation ofthe surface, producing residual compression as described earlier.

The residual-stress state in a finished bearing ring is therefore a func-tion of heat treatment and grinding. If properly ground, the residualstress in a through-hardened bearing ring will be 0 to slightly compres-sive. The subsurface residual stress conditions will be determined by theprior heat treatment.

Measuring Residual Stress

The most widely used method to precisely determine residual stress incrystalline materials is x-ray diffraction. X-ray diffraction equipment andtechniques are well developed and described in the literature [22.19,22.22, 22.23].

All metals, being crystalline solids, consist of atoms arranged inplanes precisely positioned in terms of interplanar distances. In a single


crystal of a metal the orientation of these planes of atoms is consistenteverywhere within the crystal. Most metallic objects of interest here arenot single crystals but polycrystalline; that is, they consist of many crys-tals. Each crystal or grain in the microstructure of a polycrystallinemetal is delineated from its neighbors by mismatch in the orientation ofthe crystallographic planes. The region of mismatch or disorder betweenneighboring grains is called the grain boundary.

In the unstressed condition the distances between crystallographicplanes assume equilibrium values. If elastically stressed in tension (i.e.,a tensile stress component perpendicular to the planes), the interplanardistance is increased. In compression the distance is decreased. Conse-quently, if the equilibrium interplanar distance, the stressed interplanardistance, and the orientation of the planes with respect to the stress axisare known, the elastic strain conditions are defined. Multiplying thestrain by the elastic modulus for the material being studied provides thevalue of residual stress. X-ray diffraction is used to measure interplanardistances. The technique is therefore used to measure elastic strain, fromwhich the associated residual stress is calculated.

The relationship stating the conditions that must be met for x-raydiffraction to occur was first formulated by Bragg [22.24] in 1912 and isknown as Bragg's law:

A = 2d sin () (22.1)

where A is the wavelength of the x-rays used, d is the interplanar spac-ing, and ()is the angle of incidence of the x-ray beam to the diffractingplanes. What Bragg's law states is that for a given x-ray wavelength Aand interplanar spacing d, there is an angle of incidence ()such that thex-rays penetrating the specimen surface will experience constructive in-terference and emerge from the surface at an angle ()to the planes ofspacing d. With appropriate detection equipment the emergent (dif-fracted) x-rays can be detected, and the precise diffraction angle can bedetermined.

The orientations of a polycrystalline specimen and a particular familyof crystallographics planes (in randomly oriented grains) to the x-raybeam in residual-stress determination are shown in Fig. 22.8.

Initially, the specimen is oriented such that the normal to the speci-men surface (A) and the normal to the diffracting planes (B) are coinci-dent (Fig. 22.8a); that is, the diffracting planes are parallel to thespecimen surface. The diffraction angle, ()is determined for these planes.

The specimen is then rotated to an orientation shown in Fig. 22.8b.In this orientation the normal to the diffracting planes makes an angle1/1 with the normal to the specimen surface. These planes, being at anonzero angle to residual stress acting in the direction parallel to thespecimen surface, are elastically separated from their equilibrium spac-ing d. Residual compressive stress gives a smaller value of d. According


to equation (22.1), for fixed Aa decrease in d requires an increase in thevalue of sin O-that is, a larger diffraction angle o. Conversely, residualtensile stress gives a larger value of d, corresponding to a smaller o.Therefore, comparing the 0 values obtained with the two specimen ori-entations-that is, Fig. 22.8a,b-will show an increase or decrease, in-dicating residual compression or residual tension, respectively. Themagnitude of the change in 0 is related to the magnitude of residualstress by a calculated stress factor [22.20, 22.22, 22.23].

Values of residual stress as functions of depth below the surface (i.e.,residual stress profiles) are obtained by successive material removal andx-ray residual-stress determinations. Material removal is most appro-priately performed by electrochemical means. Mechanical removal ofma-terial would introduce alterations to the residual stress profile beingmeasured, and therefore may not be employed.

Fixed installation x-ray equipment is most often used to obtain theresidual stress measurements. The measurements may be obtained us-ing the Ruud-Barrett position-sensitive scintillation detector [22.25].This detector operates by converting incident x-rays into light by meansof a cadmium-zinc sulfide scintillation coating on one end of a pair ofcoherent, flexible fiber optic bundles. These transport the light to an elec-tronics package, which amplifies the intensity of the light and convertsit into an electrical signal. This is shown schematically by Fig. 22.9. Thedevice is interfaced with a computer, which stores and processes the largeamount of data generated. The software package incorporated in thecomputer uses a number of algorithms to correct electronic and mechan-


ical hardware inconsistencies as well as x-ray focusing errors. Specificdata-fitting algorithms are applied for x-ray background and peak loca-tion. Accurate residual stress measurements using the device can be ob-tained to a depth of approximately 0.0016 mm (0.0006 in.) below thesurface of most metals. Therefore, successive electrochemical removal ofsurface layers is required to obtain residual stress profiles versus depth.

Alteration of Residual Stress Due to Rolling Contact

Associated with the microstructural alterations resulting from rollingcontact stressing, significant changes in residual stress and retained aus-tenite contact have been reported [22.6, 22.10, 22.11, 22.13, 22.26, 22.27].

FIGURE 22.10. (b) Maximum contact stress: 3720 N Imm2; depth of maximum orthogonal

shear stress: 0.21 mm; depth of maximum unidirectional shear stress: 0.33 mm (from[22.26]).

The forms of the changes in tangential residual stress and retained aus-tenite content profiles are illustrated in Fig. 22.10.

The low and high load values correspond to the two levels of maximumcontact stress indicated in Fig. 22.4, depicting the number of revolutionsof which the various microstructural alterations occur. Comparison ofFigs. 22.4 and 22.10 indicates that significant changes in residual-stressprofile and retained austenite content precede any observable alterationsin microstructure. The residual-stress data of Fig. 22.10 show peak val-ues at increasing depths corresponding to increasing numbers of stresscycles. A similar form is indicated for decomposition of retained austen-ite, with peak effect depths being slightly less than for residual stress.The data in Fig. 22.10 for the high maximum-contact stress indicatemore rapid rates of change for both residual stress and retained austen-ite content. Subsequent testing conducted by the author on VIMVARM50 steel balls loaded and run at 2760-4000 N/mm2 maximum Hertzstress demonstrated the same tendency. This testing further indicated,


however, that at the lower loading, the residual stress tends to dissipateafter a significant amount of running time. This implies that for bearingsoperated under common industrial applied loading situations; for ex-ample, less than 2000 N/mm2 maximum Hertz stress, residual stressdue to mechanical working during manufacture does not tend to affectendurance.

The slight differences in the depths at which the peak values occur inresidual stress and retained austenite decomposition imply correlationwith the maximum shear stress and maximum orthogonal shear stress,respectively. Earlier work [22.13] supports the correlation of peak resid-ual stress values with the maximum shear stress. There appears to beno direct relationship between retained austenite decomposition and thegeneration of residual compressive stress, nor any indication of which, ifeither, of these processes triggers microstructural alterations [22.26].

Effects of Residual Stress on Rolling Contact Fatigue Life

Experimental studies on the fatigue life of rolling bearing componentshave indicated a positive effect of residual compressive stress over a"zero" stress state [22.27-22.30, 22.34, 22.35]. Another investigation[22.31] indicated that in addition to the positive effect of compressivestress superimposed on Hertzian contact stresses, there was a definitenegative effect of superimposed tensile stress. In [22.22] the negativeeffect of tensile stress was demonstrated, but it was also concluded thatthere was no advantage of high residual compressive stress over a zerostress state superimposed on the Hertzian stress field. Another studyrevealed that bearings tested with inner ring raceways at two levels ofresidual compressive stress showed no significant difference in life[22.33].

The variety of methods used to induce the residual stresses in com-ponents for rolling contact fatigue testing included prestressing of theinner rings by running them in bearings at a load higher than the sub-sequent test load (thereby inducing subsurface residual compression indescribed previously) [22.27,22.29,22.32], bulk loading of test elementsby shrink-fitting rings on a shaft or press-fitting into a housing [22.31],and altering the chemistry of the surface during heat treatment to pro-vide residual compression in the quenched and tempered surface [22.33,22.34].

As discussed earlier, the subsurface residual compressive stress gen-erated in a bearing ring during high-contact-stress operation is accom-panied by changes in hardness, microstructure, and crystallographictexture. Therefore, prestressing by high-stress operation before testingcould introduce significant factors to rolling contact fatigue life and toresidual compressive stress.


Bulk loading of rolling contact test specimens by heavy interference-fitting on shafts or in housings provides stress profiles across the speci-men section that are quite different from self-contained, balanced,residual-stress profiles within a freestanding component. Although sucha test scheme might accurately indicate performance trends for bearingapplications in which bulk ring loading is similarly experienced, it is notclear that such interact with a Hertzian stress field in the same manneras a true residual state of stress.

Alteration of surface chemistry by infusion of nitrogen or carbon toprovide residual surface compressive stress also changes the microstruc-tural characteristics, the mechanical properties of the surface region,and, perhaps, physical properties such as friction coefficient. Conse-quently, resolution of the separate influence of residual stress is ob-scured.

As indicated previously, Voskamp [22.6] determined that realigningcrystals, caused by "over-rolling," causes a plane of "weakness" alignedparallel to the raceway to form below the raceway. Moreover, he postu-lated that residual tensile stresses created by such over-rolling tend tocause fatigue failure to occur with propagation beneath and parallel tothe raceway. Voskamp [22.6] further shows the photographic evidence ofsuch failures in highly loaded bearings; for example, 5200 MPa (750 ksi)maximum contact stress.

Voskamp and Mittemeijer [22.36], in a further effort to establish theeffect of residual stresses on bearing fatigue life, identified three stagesof material response:

1. A material strengthening stage (shakedown) during which a de-crease occurs for the plastically deformed material volume

2. An effectively stationary (stable) stage during which no change oc-curs in the plastically deformed volume

3. A final stage during which material softening accompanied by pro-nounced plastically deformed material volume increase occurs,leading to fatigue failure.,

They suggest that carbon diffusion induced by local temperature in-creases occurring during rolling contact is the key mechanism leading tofatigue damage. They further suggest that the probability for crack ini-tiation and growth at nonmetallic inclusions is enhanced. Finally, theyimply that a higher initial load applied to bearings during the shake-down stage tends to prolong endurance by modifying the 2nd and 3rdstage responses to rolling contact; this being the result of increased workhardening.

Regardless of direct or indirect association with rolling contact fatiguelife, there is general agreement that a residual or applied bulk compres-


sive residual stress is a more desirable situation in a rolling bearingcomponent than a residual or applied bulk tensile. It further appearsthat excessive residual compressive stress can also lead to early bearingcomponent fatigue failure.

Shakedown

Shakedown is described as a self-stabilization of material under a cycli-cally applied load of such magnitude that the material yield stress hasbeen exceeded. Thus, a permanent change has occurred in the materialbelow the rolling contact surfaces. Plastic flow of material has occurredin the structure, generally within a limited region. As a result of theplastic deformations produced during one load cycle, residual stressesOCCurafter the load is removed, keeping the material in equilibrium.During the next load cycle, the residual stresses act together with thestresses caused by the externally applied load. If the load is not tooheavy, the amount of plastic flow is less than during the previous cycle.If the load causes stresses in excess of the shakedown limit, however, theplastic flow continues and, in fact, spreads until failure OCcurs.

In Chapter 21, static capacities and loading were based on permanentdeformations Occurring in nonrotating bearings. Subsequently duringbearing rotation, the indentations or rolling contact surface deformationswere considered to impair bearing endurance and/or cause undue vibra-tion. The permanent deformations Occurring during the shakedown pro-cess are the result of rolling contact during normal bearing rotation, andthey do not eventually impair the rolling contact surface unless theshakedown limit has been exceeded. It is conceivable therefore that abearing "static capacity" criterion could be based on the shakedown limit.

It is possible to apply the distortion energy yield criterion; that is, thevon Mises yield criterion (see reference [22.37]) to bearing steel. Exper-imental investigations have indicated a variation of the von Mises yieldlimit with heat treatment parameters. Yhland [22.38J states that for nor-mal heat-treated through-hardened carbon chromium steel; that is, AISI52100, as measured in tension tests, the von Mises yield limit stress isin the region of 1800-2000 N/mm2 (260,000-290,000 psi). Rydholm[22.39J developed a method to calculate the shakedown limit consideringthe foregoing yield criterion. For a well-lubricated bearing operating inline contact, the shakedown limit is approximately 2.31 times the yieldstress in simple tension. For a point contact bearing, the shakedown limitis approximately 2.77 O"yield.It is further possible to evaluate a situationin which no plastic flow Occurs; that is, loading is relatively light andsubsurface stresses are therefore low. In this case, for a well-lubricatedbearing operating in line contact the "incipient plastic flow" limit is1.790"yield; for point contact bearings the limit is 1.560"yield. Figure 22.11taken from reference [22.38J shows a interesting comparison of the


fects, on bearing operation, and failure manifestations. However, signif-icant consequences are associated with them, involving both dimensionalinstability and catastrophic fracture of bearing rings.

For many decades a peculiar failure manifestation has been observedin bearings installed with heavy interference fit between the inner ringbore and shaft. Many commonly experienced with heavily loadedthrough-hardened roller bearings, the failure is characterized bythrough-section fracture of the inner ring on an axial plane. A typicalfracture surface is shown in Fig. 22.12.

The fracture surface is predominantly planar and characteristicallyexhibits a semicircular region of stable fatigue crack propagation origi-nating at a raceway surface. The remaining portion of the fracture Sur-face is characteristic of unstable crack propagation-that is, rapidfracture. Spalling might or might not be associated with the axially ori-ented crack on the raceway surface. Such failures are almost exclusivelyexperienced with through-hardened bearing rings. The traditional rem-edies for this failure experience have been to reduce the interference fit(if this can be done without permitting relative motion between the innerring bore and shaft surface during operation) or to use a case-hardenedinner ring.

This type offailure assumed greater significance when it was observedin aircraft gas turbine engine bearings. This observation prompted thefirst published analysis of the mechanisms involved [22.35, 22.40, and22.41]. The described characteristics of the failures are identical to thoseindicated here. The circumferential tensile stresses associated with innerring failures were induced centrifugally (due to high-speed rotation)

EFFECTS OF BULK STRESSES ON MATERIAL RESPONSE TO ROLLING CONTACT 855

rather than from heavy shaft fit, but the magnitude of 172-207 NImm2

(25-30 ksi) agrees closely with hoop stress values calculated for fit-induced ring fractures. The failure scenario outlined in [22.41] describesthe role of circumferential tensile stress of relatively modest magnitudein producing bearing ring fracture.

In classical subsurface-initiated fatigue failure, a crack initiates belowthe surface at a stress raiser such as a nonmetallic inclusion or carbidecluster. The crack propagates radially outward toward the surface. It alsopropagates radially inward but, in the absence of circumferential tensilestress, does not reach significant depth. During continued bearing op-eration, this crack participates in the formation of a spall. In the pres-ence of circumferential tensile stress of sufficient magnitude [172NImm2

(25 ksi) or greater], radially inward stable crack propagation continuesto the point at which the critical crack size is reached. The critical cracksize is defined by the magnitude of the circumferential tensile stress andthe plane strain fracture toughness of the bearing steel. When the criticalcrack size is reached, a rapid through-section fracture occurs. Rapid frac-ture occurs on a plane perpendicular to the circumferential tensile stress.

Carburized materials provide both residual compressive stress in thehigh-hardness surface region and fracture toughness that increases withdecreasing hardness from the surface to the core region. Centrifugally orfit-induced circumferential tensile stress should be offset to some degreeby residual circumferential compression. The increased fracture tough-ness accommodates larger crack size. The individual contributions fromthese sources to the successful use of carburized inner rings in heavyinterference fit applications have not been described in the open litera-ture.

In [22.31], rolling contact fatigue experiments were performed withthrough-hardened inner ring specimens containing 1000 N/mm2 (145ksi) circumferential tensile stress. From the foregoing discussion it is notsurprising that the inner ring failed by axial fracture with only minuteindications of fatigue crack propagation (the critical crack size at thislevel of circumferential tension is estimated to be about 0.127 mm (0.005in.). Perhaps of greater significance is the comparison of the runningtimes to fracture for the high-tensile-stressed rings to running times ac-cumulated by the "zero-stress" baseline. Ring fractures were experiencedin 20 to 30 hr in the stressed ring tests, whereas baseline rings ran for440 to 960 hr with no failures. This indicates that significant life reduc-tion could be associated with stress conditions that promote ring fracture(e.g., bulk tensile loading).

The results of bearing tests performed with circumferentially tensile-stressed inner rings [345 NImm2 (50 ksi)] indicated both very early fail-ure and radial crack propagation in through-hardened AISI M50 bearingsteel [22.42]. Similarly tested bearings, with inner rings made of a car-burizing version of AISI M50 (i.e., M50 Nil), completed substantially


longer test times with no indications of failure. The residual compressivestress in the carburized case is cited as the reason for improved perfor-mance.

Dimensional Instability

Dimensional instability of bearing components in service, and particu-larly growth of bearing inner rings, is a known problem. For many yearsit was common knowledge that dimensional instability of hardened andtempered steel was due to retained austenite being transformed to mar-tensite or bainite, depending on the thermal exposure conditions. Volumechanges associated with the transformation of retained austenite, how-ever, have not always been of sufficient magnitude to account for ob-served dimensional changes. Additionally, substantial dimensionalchanges of hardened and tempered bearing steel components have beenreported with no reduction in retained austenite content [22.43, 22.44].

Overhaul statistics from U.S. railroads indicate that inner ring growthof carburized railway axle bearings is the major cause of bearing rejec-tion at overhaul [22.45].Metallurgical phase transformations [22.45]andaccumulation of microplastic deformation [22.46] have been cited ascauses. A detailed investigation of retained austenite and residual-stressprofiles in railway journal bearings that exhibited inner ring boregrowth, however, showed no alteration of either retained austenite con-tent or residual-stress profile [22.43]; instead, ring growth correlatedwith retained austenite content and bearing operating temperature.Rings with higher retained austenite content exhibited more boregrowth. Increased operating temperature produced increased growth.The absence of any indications of microstructural alteration or residual-stress buildup lead to the conclusion that a creep mechanism is involvedin producing the observed change in inner ring dimensions, as opposedto metallurgical phase change or microplastic deformation.

Data in [22.44] are consistent with the findings in [22.43]. A cleverlydesigned fixture was used to determine the dimensional stability of hard-ened AISI 52100 steel containing 0-15% retained austenite. Testing wasperformed from -34 to 74°C (-30-165°F) with constant applied tensilestress levels of 0, 69, and 138 N/mm2 (10 and 20 ksi). Test times up to1700 hr were used. The results indicated that dimensional change(length increase) increased with increasing amounts of retained austen-ite. More dramatically, for a given austenite content and test tempera-ture (which was well below the specimen tempering temperature), largeincreases in specimen length were associated with the application of rel-atively modest tensile stress. See Fig. 22.13. Also, no reduction in re-tained austenite content was detected, even after tests resulting insignificant length extensions. A creep mechanism similar to that pro-posed in [22.42] could be indicated. Additionally, these data imply thatassessment of growth potential via unstressed thermal exposure tests

may provide an underestimate for an application such as a heavy inter-ference-fitted bearing ring. Further, increasing interference fit to com-pensate for such an underestimate may result in more rapid loss of fit.See Fig. 22.13. Since this can be done without decomposition of trans-formation of the initial retained austenite content, the potential for con-tinued growth is preserved. Clearly this would not be the case if growthwas experienced at the expense of retained austenite content. This couldbe a significant consideration when making judgments pertaining to suchmatters as bearing refurbishment.

Retained austenite content remains a primary consideration in termsof bearing ring dimensional stability, with definite correlations betweengrowth potential and initial austenite content. The mechanisms by whichdimensional change is effected, however, are not understood. Publishedwork [22.47] indicates desirable contributions of retained austenite torolling bearing performance. Consequently, it may be technologically im-prudent to totally ignore possible performance advantages in pursuit ofaustenite-free dimensional stability.

CLOSUREBecause of the extremely high rolling element-raceway contact stressesthat occur during operation of many ball and roller bearings, the micro-structure of the bearing material, that is, steel, undergoes significant

H58MATERIAL RESPONSE TO ROLLING CONTACT

change. This phenomenon has been investigated for many years, and yetit has not thus far been possible to quantitatively relate such micro-structural changes to the imminence of rolling contact fatigue. It isknown that not only do the applied stresses cause such microstructuralalterations, but also bearing operating temperatures well below steeltempering temperatures significantly affect the rate and amount of mi-crostructural change. One type ofmicrostructural change, that is, shake-down,

appears to be related to the yield strength of the steel. Ashakedown limit might be established as one bearing operating criterion.Investigation in this important area of material science is continuing.

REFERENCES22.1.

J. van de Sanden, "Martensite Morphology of Low Alloy Commercial Steels," Prac-tical Metallogr. 17, 238-248 (1980).22.2.

J. Rescalvo, "Fracture and Fatigue Crack Growth in 52100, M50 and 18-4-1 BearingSteels," Ph.D. thesis, Department of Materials Science and Engineering, Massachu-setts Institute of Technology (June 1979).

22.3.G. Krauss, Principles of Heat Treatment of Steel, American Society for Metals, 61-75 (1980).

22.4.A. Jones, "Metallurgical Observations of Ball Bearing Fatigue Phenomena," Proc.ASTM 46, 1 (1946).

22.5.T. Tallian, "On Competing Failure Modes in Rolling Contact," ASLE Trans. 10,418-439 (1967).

22.6.A. Voskamp, "Material Response to Rolling Contact Loading," ASME Trans., J. Tri-bology 107, 359-366 (1985).

22.7.T. Lund, Jernkontorets Ann. 153, 337 (1969).

22.8.H. Swahn, P. Becker, and O. Vingsbo, "Martensite Decay During Rolling ContactFatigue in Ball Bearings," Metallurgical Trans. A 7A, 1099-1110 (Aug. 1976).22.9.J. Martin, S. Borgese, and D. Eberhardt, "Microstructural Alterations of RollingBearing Steel Undergoing Cyclic Stressing," Trans. ASME 59, 555-557 (Sept. 1966).

22.10. A Gentile, E. Jordan, and A. Martin, "Phase Transformations in High-Carbon HighHardness Steels Under Contact Loads," Trans. AIME 233,1085-1093 (June 1965).

22.11. J. Bush, W. Grube, and G. Robinson, "Microstructural and Residual Stress Changesin Hardened Steel Due to Rolling Contact," Trans. ASM 54,390-412 (1961).

22.12. M. Kuroda, Trans. Jpn. Soc. Mech. Eng. 26, 1256-1270 (1960).

22.13. H. Muro, and N. Tsushima, "Microstructural, Microhardness and Residual StressChanges Due to Rolling Contact," Wear 15, 309-330 (1970).

22.14. H. Styri, Proc. ASTM 51,682-700 (1951).

22.15. R. Tricot, J. Monnot, and L. Luansi, Metals Eng. Quart. 12,39-42 (1972).22.16. W.Littmann and R. Widner, "Propagation of Contact Fatigue from Surface and Sub-

surface Origins," J. Basic Eng. 88, 624-636 (1966).

22.17. D. Nelias, "Contribution a l'etude des roulements," (Dossier d'Habilitation a Dirigerdes Recherches, Laboratoire de Mecanique des Contacts, UMR-CNRS-INSA de LyonNo. 5514, December 16, 1999).

22.18. L. Uhrus, "Clean Steel," Iron and Steel Institute, London, 104-109 (1963).

REFERENCES 85922.19. P. Becker, "Microstructural Changes Around Non-Metallic Inclusions Caused by

Rolling-Contact Fatigue of Ball-Bearing Steels," Metals Technology, 234-243 (June1981).

22.20. "Residual Stress Measurements by X-Ray Diffraction," SAE J784a, 2nd ed. Societyfor Automotive Engineers, New York (1971).

22.21. D. Koistinen, "The Distribution of Residual Stresses in Carburized Cases and TheirOrigins," Trans. ASM 50,227-238 (1958).

22.22. B. Cullity, Elements of X-ray Diffraction, Addison-Wesley, Reading, Mass. (1959).22.23. C. Gazzara, "The Measurement of Residual Stress with X-ray Diffraction," Rept.

AD-A130 614, Army Material & Mechanics Res. Center (May 1983).22.24. W. Bragg, "The Diffraction of Short Electromagnetic Waves by a Crystal," Proc.

Camb. Phil. Soc. 17, 43 (1912).

22.25. C. Ruud and D. Carpenter, Operators Manual for the D-lOOOAStress Analyzer, Den-ver X-Ray Instrument, Inc., Colorado (1985).

22.26. A. Voskamp, R. Osterlund, P. Becker, and O. Vingsbo, "Gradual Changes in ResidualStress and Microstructure During Contact Fatigue in Ball Bearings," Metals Tech-nology, 14-21 (Jan. 1980).

22.27. E. Zaretsky, R. Parker, and W. Anderson, "A Study of Residual Stress Induced Dur-ing Rolling," J. Lub. Tech. 91, 314-319 (1969).

22.28. R. Scott, R. Kepple, and M. Miller, "The Effect of Processing-Induced Near-SurfaceResidual Stress on Ball Bearing Fatigue," in Rolling Contact Phenomena, ed. J. B.Bidwell, Elsevier, 301-316 (1962).

22.29. E. Zaretsky, R. Parker, W.Anderson, and S. Miller, "Effect of Component DifferentialHardness on Residual Stress and Rolling-Contact Fatigue," NASA TND-2664 (1965).

22.30. C. Foord, C. Hingley, and A. Cameron, "Pitting of Steel Under Varying Speeds andCombined Stresses," J. Lub. Tech. 91,282-290 (1969).

22.31. R. Kepple and R. Mattson, "Rolling Element Fatigue and Macroresidual Stress,"J. Lub. Tech. 92, 76-82 (1970).

22.32. W. Littmann, Discussion to Reference 31, J. Lub. Tech. 92, 81 (1970).22.33. C. Stickels and A. Janotik, "Controlling Residual Stresses in 52100 Bearing Steel

by Heat Treatment," Metal Progress, 34-40 (Sept. 1981).22.34. D. Koistinen, "The Generation of Residual Compressive Stresses in the Surface Lay-

ers of Through-Hardening Steel Components by Heat Treatment," Trans. ASM 57,581-588 (1964).

22.35. J. Clark, "Fracture Failure Modes in Lightweight Bearings," AIAA, Journal Aircraft12, No.4 (1975).

22.36. A. Voskamp and E. Mittemeijer, "The Effect of the Changing Microstructure on theFatigue Behaviour During Cyclic Rolling Contact Loading," in MicrostructuralChanges During Rolling Contact Fatigue, Metal Fatigue in the Subsurface Region ofDeep Groove Ball Bearing Inner Rings, A. Voskamp, Proefschrift ter verkrijging vander graad doctor aan de Technische Universiteit Delft (January 8, 1997).

22.37. M. Spotts and T. Shoup, Design of Machine Elements, 7th ed., 123-127, PrenticeHall, Englewood Cliffs, N.J. (1998).

22.38. E. Yhland, "Static Load-Carrying Capacity," Ball Bearing Journal (SKF), 211, 1-8(1982).

22.39. G. Rydholm, "On Inequalities and Shakedown in Contact Problems, LinkopingStud-ies in Science and Technology," Dissertation No. 61, Linkoping, Sweden (1981).

22.40. E. Bamberger, E. Zaretsky, and H. Signer, "Endurance and Failure Characteristicsof Main-Shaft Jet Engine Bearings at 3 X 106 DN," ASME Trans. J. Lub. Tech. 95(4)(1976).

860MATERIAL RESPONSE TO ROLLING CONTACT

22.41. E. Bamberger, "Materials for Rolling Element Bearings," presented at ASME-ASLEInternational Lubrication Conference, San Francisco, Calif. (August 1980).

22.42. J. Clark, "Fracture Tough Bearings for High Stress Applications," presented atAlAA/SAE/ ASME/ ASEE 21st Joint Propulsion Conference, Monterey, Calif. (July8-10, 1985).

22.43. A. Voskamp and B. Schalk, "Ring Growth in Case Hardened Railway Journal RollerBearings," presented at the 2nd International Heavy Haul Railway Conference,Pueblo, Color. (September 1982).

22.44. E. Mikus, T. Hughel, J. Gerty, and A. Knudsen, "The Dimensional Stability of aPrecision Ball Bearing Material," Trans. ASM 52,307-315 (1960).

22.45. J. McGrew, A. Krawler, and G. Moyar, "Reliability of Railroad Roller Bearings,"ASME Trans. J. Lub. Tech. 99, 30-40 (1977).

22.46. R. Steel, Discussion to Reference 45, ASME Trans. J. Lub. Tech. 99, 39 (1977).22.47. J. Seehan and M. Howes, "The Effect of Case Carbon Content and Heat Treatment

on the Pitting Fatigue of 8620 Steel," SAE Conf. Congress, No. 720268 (January1972).

---- •• ~~<U..I fil'lU LU'E FACTORSSymbolDescription

UnitsFa Applied axial loadN (lb)Fr Applied radial loadN (lb)Fe Equivalent applied loadN (lb)F1im Fatigue limit loadN (lb)FR Filter rating/LmhO

Minimum lubricant film thickness/Lm(/Lin.)Kc

Stress concentration factor due to particulatecontamination

L Fatigue life106 revLID Rating life106 revLso Median life106 revn SpeedrpmN Number of stress cyclesq

Load on a roller-raceway contact laminumN (lb)qc

Basic dynamic capacity for a roller-racewaycontact laminum

N (lb)Q Rolling element loadN (lb)Qc

Basic dynamic capacity for a ball-racewaycontact

N (lb)r zlbR

Oil bath and grease contaminationparametery,Probability of survivalSFComposite rms surface roughness of matingsurfaces

/Lm(/Lin.)T Temperature°C (OF)u Stress cycles per revolutionV Stressed volumemm3 (in.3)

wWidth of a roller-raceway contact laminum

mm (in.)Zo Depth at which TO occursmm (in.)Z

Number of rolling elements per bearing rowa Contact angle0, rad13x

Filter effectiveness ratio for particles of sizex /Lm

°a Bearing axial deflectionmm (in.)Or Bearing radial deflectionmm (in.)y Dldm cos a

A h0I SFv kinematic viscosity

mm2/sec(in.2/sec)VI

kinematic viscosity for adequate lubricationmm2/sec(in.2/sec)(TVM von Mises stressMPa (psi)

GENERAL863

Symbol Description UnitsTO Maximum orthogonal shear stress MPa (psi)1> Oscillation angle

0, radif; Rolling element azimuth angle0, rad

SUBSCRIPTSb Refers to balli Refers to inner ring or racewayJ Refers to rolling element azimuth locationk Refers to roller-raceway contact laminum

locationn Refers to nonrotating racewayr Refers to rotating racewayRE Refers to equivalent rotating bearing or

rolling element

GENERAL

The Lundberg-Palmgren theory and the standard load and fatigue lifecalculations which resulted [23.1-23.5] are only the first step towarddetermining the bearing fatigue lives in applications. Use of the standardmethods should be limited to those applications in which the internalgeometries and rolling component materials of the bearings employedconform to the standard specifications, and the operating conditions arebounded as follows:

• The bearing outer ring is mounted and properly supported in a rigidhousing.

• The bearing inner ring is properly mounted on a nonflexible shaft.• The bearing is operated at a steady speed under invariant loading.• Operational speed is sufficiently slow such that rolling element cen-

trifugal and gyroscopic loading are insignificant.• Bearing loading can be adequately defined by a single radial load, a

single axial load, or a combination of these.• Bearing loading does not cause significant permanent deformations or

material transformations.• For bearings under radial loading, mounted internal clearance is es-

sentially nil.

• For angular-contact ball bearings, nominal contact angle is constant.• For roller bearings, uniform loading is maintained at each roller-

raceway contact.• The bearing is adequately lubricated.

864 APPLICATION LOAD AND LIFE FACTORS

Many applications can be considered to be included within these condi-tions.

In many applications, these simple conditions are exceeded. For ex-ample, many applications do not operate at steady speed or load, rather,under a load-speed cycle. Furthermore, the bearing may support, as in-dicated in Chapter 7, combined radial, axial, and moment loading underwhich distribution of internal loading is significantly different from thestandard limitations. Bearings may operate at speeds which cause sub-stantial rolling element inertial loading and variation in contact anglesbetween inner and outer raceway contacts. These conditions may be ad-dressed by applying the Lundberg-Palmgren theory in detail using com-puter programs to perform the complex calculations.

Since the development of the Lundberg-Palmgren theory, the abilityof lubricant to separate rolling elements from raceways, as discussed inChapter 12, was established. This condition has been shown to haveprobably the most profound effect on extending bearing fatigue life com-pared to any other. Improvements in modern bearing steel manufactur-ing methods have provided steels of very high cleanliness andhomogeneity, as compared to the basic air-melt AISI 52100 steel used inthe development of the Lundberg-Palmgren theory and standards. Withthe advent of substantially extended life, increased reliability bearinglife prediction can be considered.

Finally, as the improvements in bearing manufacture and lubricationwere applied, it became apparent that, similar to other steel structuralmembers subjected to cyclic loading, bearing raceways and rolling ele-ments also exhibit an endurance limit in fatigue. This means that in agiven application, a ball or roller bearing does not have to fail in fatigueproviding applied loading and conditions of operation are such that thebearing material fatigue limit stress is not exceeded.

All of the foregoing conditions will be addressed in this chapter.

EFFECT OF BEARING INTERNAL LOAD DISTRIBUTION ONFATIGUE LIFE

Ball Bearing Life

When the distribution of load among the balls is different from that re-sulting from the applied loading conditions specified in the load ratingstandards, it is necessary to revert to the Lundberg-Palmgren load-liferelationships for individual ball-raceway contacts. For example, for acontact on a rotating raceway:

steel. This results in reduced contact area between the steel racewaysand ceramic balls; therefore, Hertz stresses are increased, causing re-duction in fatigue life. Thus, the beneficial effect of lightweight balls iscounteracted. By decreasing the radii of the raceway grooves somewhat,the Hertz stresses may be decreased. This, however, causes an increasein frictional stresses and higher operating temperatures which may haveto be accommodated by cooling the lubricant or bearing. Optimum bear-ing design may be achieved for a given application by parametric studyusing a bearing performance analysis computer program. It can be seenfrom Fig. 23.5 that there is little difference in the fatigue life perform-ance of the bearing under relatively heavy loading.

Figure 23.6 shows life vs speed for the 209 cylindrical roller bearingof Example 9.2. Skidding effects are not included in these illustrations.

Misalignment

Misalignment in nonaligning rolling bearings distorts the internal loaddistribution, and thus alters fatigue life. In Chapter 7, methods weredescribed to determine the misalignment angle in ball and roller bear-ings as a function of the applied moment. In ball bearings, the load dis-tribution from ball to ball is altered by misalignment; in roller bearings,however, the distribution of roller load per unit length becomes nonuni-form as shown by Fig. 7.26. The variable load per unit length is givenby equation (7.112).

[23.20], if A is numerically equal to or greater than 4, fatigue life can beexpected to exceed standard LlO estimates by at least 100%.On the otherhand, if A is less than unity, the bearing will probably not attain thecalculated LlO estimates because of surface distress such as smearing orpulling which can lead to rapid fatigue failure of the rolling surfaces.Fig. 23.23 shows the various operating regions just described. In Fig.23.23, the ordinate, that is, percent film, is a measure of the time per-centage during which the "contacting" surfaces are fully separated by anoil or lubricant film.

Tallian [23.19] showed a more definitive estimate of rolling bearingfatigue life vs A as did Skurka [23.18]. Bamberger et al. [23.12] showthe combination of the foregoing in Fig. 23.24, recommending the use ofthe mean curve. Experimental data indicate that for A > 4, the LILlOratios given by Fig. 23.24 are substantially greater for accurately man-ufactured, bearings lubricated by minimally contaminated oil.

Using a microtransducer to measure the pressure distribution in anoil-lubricated line contact in the direction of rolling, it was shown in[23.21] that the edge stress in a line contact is substantially reduced ifill adequate lubricant film separates the contacting rolling bodies. Thus,n this situation, the lubricant film tends to permit an increase in fatigueife by reducing the magnitude of normal stress at the end(s) of heavilyoaded contact.

Example 23.5.In the 209 cylindrical roller bearing of Example 12.1,

the rollers have surface finishes of 0.102 JLm (4-JLin.)rms and the

896APPLICATION LOAD AND LIFE FACTORS

TABLE 23.6. C4,rocess vs MetalworkingProcessMetalworkingProcess

('~rocessDeep-grooveball bearing raceways

1.2Angular-contactball bearing raceways1Angular-contactball bearing raceways-forgedrings

1.2Cylindricalroller bearings1

mined that an angular-contact bearing with forged rings manufacturedfrom VIMVARM50NiL steel would be given an C:tgteel = 28.8.

No value has been universally established to date for hot isostaticl'illypressed silicon nitride. Endurance testing of single balls in ball/v-ringendurance test rigs (see Chapter 20) has, however, yielded high multiplesof the endurance for steel balls tested under the same loading conditions.To date, owing to relative weakness in tensile strength in bending testsand extremely low coefficient of thermal expansion (see Chapter 16), sil-icon nitride has been principally used for ball and rollers in high preci-sion, high speed applications; for example, machine tool spindle bearings.

EFFECT OF CONTAMINATION ON FATIGUE LIFE

Excessive contamination in the lubricant will severely shorten bearingfatigue life. The standards [23.3-23.5] and manufacturers' catalogs con-tain warning statements about this. Contaminants may be either partic-ulate or liquid, usually water. Even small amounts of contaminants have3ignificant limiting effects on bearing fatigue life.

Particulate contaminants such as gear wear metal particles, alumina,;ilica, and so on will cause dents in the raceway and rolling elementmrfaces, which disrupt the lubricant films which tend to separate the'olling body surfaces. This tends to locally increase the frictional shear:tresses produced in the rolling-sliding contacts. Furthermore, the'aised material on the shoulder of the dent tends to cause stress concen-rations. Ville and Nelias [23.23] using a two-disk, rolling-sliding testig, demonstrated the stress concentration phenomenon. They furtherhowed that combined rolling-sliding motion is a more severe conditionrith regard to generation of surface distress and fatigue than rollinglone. Both the film disruption and dent shoulder stress-increasing ef-)cts accelerate the onset of rolling contact fatigue and component fail-reo Figure 23.25 from a study of the effects of surface topography onLtigue failure by Webster et al. [23.24] indicates the relative risk oflilure effected by the shoulders of dents. Hamer et al. [23.25-23.26]Jinted out that even relatively soft particles can generate significantmting assuming bearing speeds and loads are sufficiently high. They


dependent on the thickness of the lubricant film compared to the size offoreign particulate matter; in large bearings it is far less significant thanin small bearings.

LIMITATIONS OF THE LUNDBERG-PALMGREN THEORY

Lundberg and Palmgren formulas represented a significant developmentin rolling bearing technology; however, it was not possible to correlatethe fatigue of bearing surfaces in rolling contact so calculated to struc-tural fatigue. Nor was it possible to correlate rolling contact fatigue inbearings to fatigue of elemental surfaces in rolling contact. Moreover,materials tested for structural fatigue have typically exhibited a fatiguelimit as shown by curves similar to Fig. 18.I.

According to Fig. 18.1, for cyclic loading less than the fatigue limit,fatigue, for all practical purposes, does not occur. On the contrary, rollingbearing applications according to the standard methods of calculationswere characterized by a finite fatigue life in any application. Innumer-able modern rolling bearing applications, however, have defied this lim-itation. Data for bearings of standard design, accurately manufacturedfrom high-quality steel-that is, having minimal impurities and homo-geneous chemical and metallurgical structures [23.32], have demon-strated that infinite fatigue life is a practical consideration in somerolling bearing applications. Since the Lundberg and Palmgren formulasdid not address the concept of a possible infinite fatigue life and did notrelate to structural fatigue, an improvement in these formulas beyondapplication of some empirical life adjustment factors was required.

The Lundberg and Palmgren theory considers that a fatigue crackbegins at a point below the surface in rolling contact, at which point alarge-magnitude orthogonal shear stress coincides with a weak point inthe material. Such weak points are assumed to be randomly distributedthroughout the material. As demonstrated in Chapter 6, the orthogonalshear stress results from a concentrated load applied normal to the sur-faces in contact, giving a Hertzian surface stress distribution similar tothat of Fig. 6.6 for point contact. Figure 6.13 shows the orthogonal shearstress distribution in the subsurface material.

Knowledge of pressure distributions in EHL contacts has demon-strated that such pressure distributions can be substantially differentfrom the pure Hertzian distribution indicated in Fig. 6.6. Figure 12.10shows just how different an EHL pressure distribution can be comparedto the Hertzian distribution.

Moreover, if the surfaces are nonideal-that is, not smooth but havingperturbations on the smooth surface-then concepts of micro-EHL, asdiscussed in Chapter 13, obtain. Additionally, in their analysis Lundbergand Palmgren did not accommodate surface shear stresses, which can

922APPLICATION LOAD AND LIFE FACTORS

stresses resisting the ring expansion. Outer ring rotation results in ten-sile hoop stresses which tend to counteract the compressive hoop stressescaused by press fitting of the outer ring in the housing. Timoshenko[23.49] details the method to calculate the tensile hoop and radialstresses associated with ring rotation.

Each of the stresses due to press fitting and/or ring rotation is super-imposed on the subsurface stress field caused by contact surface stresses.

Stresses Due to Material Processing

Heat treatment of bearing components can introduce a differential stressdistribution in the near surface region which influences fatigue life. Forexample, the case region of carburized bearing components contains com-pressive stresses, mainly in the circumferential direction of a ring. Asdistance beneath the contact surface increases, the compressive stressundergoes a transition to the tensile stress field necessary to keep thering in equilibrium. Fortunately, the tensile stress region is sufficientlybelow the subsurface zone influenced by the surface Hertzian and fric-tional stresses that it doesn't influence fatigue.

The grinding and surface finishing processes which produce the sur-face topographies or microgeometries of bearing rolling contact compo-nents introduce residual stresses which may be detrimental to bearingendurance. If the processes are abusively applied, accidentally or inten-tionally to achieve rapid component production, the induced residualstresses can be rather high and tensile. Voskamp [23.50]conducted stud-ies of the magnitude of residual stresses in run and unrun AISI 52100steel ball bearing raceways. In an unreported endurance test programfor bearing balls, the author found compressive surface stresses in therange of 600 MPa (87,000 psi) for both M50 and 52100 balls, which hadnot been run. Beneath the surface, in the zone of maximum subsurfaceapplied stress, the compressive stress level reduced to values in therange of 70 MPa 00,000 psi). When the balls were operated under nor-mal bearing Hertz stresses-for example, maximum 2700 MPa (400,000psi)-these compressive stresses seemed to disappear, most likely a re-sult of retained austenite transformation. On the other hand, it has beenobserved that running-in bearing raceways under heavy loading for ashort period of time prior to normal operation tends to work harden thenear-surface regions. This introduces slight compressive residual stressinto the material, increasing its resistance to fatigue. Excessive amountsof compressive stress tend to reduce resistance to fatigue.

Life Integral

The stresses discussed in this section each contribute to the overall sub-surface stress distribution. Using superposition and the assumption of

CLOSURE 931

CTVmlimit = 750 MPa (109,000 psi). Based on the above values, the cal-culated LlO = 304.1 . 106 revolutions. Thus, L1 = 0.21 . 304.1 . 106 =63.87 . 106 revolutions.

The calculations using TH-BBAN tend to be more precise thanthose of Examples 2.6 and 2.7. They use the total stress pattern actingon the ball-raceway contacts, not just the Hertz stresses. The calcu-lational results do not include fatigue of the ball surfaces.

CLOSUREThe Lundberg-Palmgren theory to predict fatigue life was a significantadvancement in the state-of-the-art of ball and roller bearing technology,affecting the internal design and external dimensions for 40 years. TheEHL theory, introduced by Grubin and further advanced by scores ofresearchers, initially affected bearing microgeometry, but later, becauseof the possibility of increased endurance together with improved mate-rials resulted in "downsizing" of ball and roller bearings. The Ioannides-Harris theory, in its ability to apply the total stress pattern to predictlife in any bearing application, and in its use of a fatigue stress limit forrolling bearing materials, carries the development to the next plateau bysubstantially increasing understanding of the significance of materialquality and concentrated contact surface integrity. It is now apparentthat a bearing, manufactured from material that is clean and homoge-neous, which operates with its rolling/sliding contacts free from contam-inants, and which is not overloaded, may survive without fatigue. In fact,Palmgren [23.54] initially considered the existence of a fatigue limitstress; however, the rolling bearings sets which were tested in the de-velopment of the Lundberg-Palmgren theory failed rather completelyunder the test loading, and he abandoned the concept. During the early1980s, when the Ioannides-Harris theory was under development, as in-dicated by [23.55] and other modern bearing fatigue investigations, fa-tigue testing consumed substantial calendar time, often requiring! yearand more with no bearing failures after more than 500 million revolu-tions.

This chapter converts the Ioannides-Harris theory into practice. Thelife theory is stress-based, as opposed to the factor-based, modified Lund-berg-Palmgren theory (Standard methods [23.3-23.5]) exemplified byequation (23.46). Rather, the Ioannides-Harris theory utilizes the baseLundberg-Palmgren life equations (23.15), (23.1)-(23.2), or (23.8)-(23.10) together with a single factor CfsL which integrates the effect onfatigue life of all stresses acting on the bearing contact material. Anaccurate life prediction for any bearing application only depends on thesuccessful evaluation of the appropriate stresses. With the application of


modern computers and computational methods, these stresses are beingsubjected to increasingly greater scrutiny. With the current availabilityof powerful, inexpensive desktop and laptop computers, engineers world-wide have the capability to use rolling bearing performance analysiscomputer programs which can effectively employ the methods describedin this text for such analysis.


tech. Mech. Eng. Ser. 1, Royal Swedish Acad. Eng., No.3, 7, (1947).23.2. G. Lundberg and A. Palmgren, "Dynamic Capacity of Roller Bearings," Acta Poly-

tech. Mech. Eng. Ser. 2, Royal Swedish Acad. Eng., No.9, 49, (1952).23.3. American National Standards Institute, American National Standard (ANSI I

ABMA) Std. 9-1990 "Load Ratings and Fatigue Life for Ball Bearings" (July 17,1990).

23.4. American National Standards Institute, American National Standard (ANSI IABMA) Std. 11-1990 "Load Ratings and Fatigue Life for Roller Bearings" (July 17,1990).

23.5. International Organization for Standards, International Standard ISO 281 11,"RoIling Bearings-Dynamic Load Ratings and Rating Life (2000).

23.6. T. Harris, "Prediction of Ball Fatigue Life in a Ball IV-Ring Test Rig," ASME Trans.,J. Tribology, Vol. 119, 365-374 (July 1997).

23.7. T. Harris, "How to Compute the Effects of Preloaded Bearings," Prod. Eng., 84-93(July 19, 1965).

23.8. A. Jones and T. Harris, "Analysis of RoIling Element Idler Gear Bearing Having aDeformable Outer Race Structure," ASME Trans., J. Basic Eng., 273-277 (June1963).

23.9. T. Harris and J. Broschard, "Analysis of an Improved Planetary Gear TransmissionBearing," ASME Trans., J. Basic Eng., 457-462 (September 1964).

23.10. T. Harris, "Optimizing the Fatigue Life of Flexibly Mounted, Rolling Bearings," Lu-brication Eng., 420-428 (October 1965).

23.11. T. Harris, "The Effect of Misalignment on the Fatigue Life of Cylindrical RollerBearings Having Crowned RoIling Members," ASME Trans., J. Lubr. Tech., 294-300 (April 1969).

23.12. E. Bamberger, T. Harris, W. Kacmarsky, C. Moyer, R. Parker, J. Sherlock, and E.Zaretsky, Life Adjustment Factors for Ball and Roller Bearings, ASME EngineeringDesign Guide (1971).

23.13. A. Palmgren, Ball and Roller Bearing Engineering, 3rd Ed., Burbank, Philadelphia(1959).

23.14. L. Houpert, "Bearing Life Calculation in Oscillatory Applications," Tribology Trans.,Vol. 42, 1, 136-143 (1999).

23.15. T. TaIlian, "Weibull Distribution of Rolling Contact Fatigue Life and DeviationsTherefrom," ASLE Trans., 5 (1) (April 1962).

23.16. T. Harris, "Predicting Bearing Reliability," Mach. Des., 129-132 (January 3, 1963).23.17. T. TaIlian, L. Sibley, and R. Valori, "Elastohydrodynamic Film Effects on the Load-

Life Behavior of RoIling Contacts," ASME Paper 65-LUBS-11, ASME Spring Lubri-cation Symposium, NY (June 8, 1965).

REFERENCES 933

23.18. J. Skurka, "Elastohydrodynamic Lubrication of Roller Bearings," ASME Paper 69-LUB-18 (1969).

23.19. T. TaIlian, "Theory of Partial Elastohydrodynamic Contacts," Wear, 21, 49-101(1972).

23.20. T.Harris, "The Endurance of Modern RoIling Bearings," AGMAPaper 269.01, Amer-ican Gear Manufacturers' Ass'n Roller Bearing Symposium, Chicago (October 26,1964).

23.21. M. Schouten, Lebensduur van Overbrengingen, TH Eindhoven (November 10, 1976).23.22. STLE, Life Factors for Rolling Bearings, E. Zaretsky, Ed. (1992).23.23. F. Ville and D. Nelias, "Early Fatigue Failure Due to Dents in EHL Contacts," Pre-

sented at the STLE Annual Meeting, Detroit (May 17-21, 1998).23.24. M. Webster, E. loannides, and R. Sayles, "The Effect of Topographical Defects on

the Contact Stress and Fatigue Life in Rolling Element Bearings," Proc. 12th Leeds-Lyon Symposium on Tribology, 207-226 (1986).

23.25. J. Hamer, R. Sayles, and E. loannides, "Particle Deformation and Counterface Dam-age When Relatively Soft Particles Are Squashed between Hard Anvils," TribologyTrans., Vol. 32, 3, 281-288 (1989).

23.26. R. Sayles, J. Hamer, and E. loannides, "The Effects of Particulate Contaminationin Rolling Bearings-A State of the Art Review," Proc. Inst. Mech. Eng., Vol. 204,29-36 (1990).

23.27. F. Ville, Contamination of Oil by Solid Particles, Indentation Process and SurfaceFatigue, Ph.D. thesis, Laboratoire de Mecanique des Contacts, UMR CNRS-INSANo. 5514, Lyon, France (November 16, 1998).

23.28. G. Xu, F. Sadeghi, and M. Hoeprich, "Dent Initiated Spall Formation in EHL RollinglSliding Contact," ASME Trans, J. Tribology, Vol. 120, 453-462 (July 1998).

23.29. R. Sayles and P. MacPherson, "Influence of Wear Debris on RoIling Contact Fatigue,"ASTM Special Technical Publication 771, J. Hoo, Ed., 255-274 (1982).

23.30. A. Tanaka, K. Furumura, and T. Ohkuna, "Highly Extended Life of TransmissionBearings of 'Sealed-Clean' Concept," SAE Technical Paper, 830570 (1983).

23.31. W.Needelman and E. Zaretsky, "New Equations Show Oil Filtration Effect on Bear-ing Life," Power Transmission Design, Vol. 33,8,65-68 (1991).

23.32. R. Barnsby, T. Harris, S. Ioannides, W. Littmann, T. Losche, Y. Murakami, W. Nee-delman, H. Nixon, and M. Webster, "Life Ratings for Modern Rolling Bearings,"ASME Paper 98-TRIB-57, Presented at the ASME/STLE Tribology Conference, To-ronto (October 26, 1998).

23.33. T. TaIlian, "On Competing Failure Modes in RoIling Contact," ASLE Trans, 10,418-439 (1967).

23.34. E. loannides and T. Harris, "ANew Fatigue Life Model for Rolling Bearings," ASMETrans., J. Tribology, Vol. 107, 367-378 (1985).

23.35. T. Harris and J. McCool, "On the Accuracy of Rolling Bearing Fatigue Life Predic-tion," ASME Trans., J. Tribology, Vol 118, 297-310 (April 1996).

23.36. T. Harris, "Prediction of Ball Fatigue Life in a Ball/V-Ring Test Rig," ASME Trans.,J. Tribology, Vol. 119, 365-374 (July 1997).

23.37. R. Juvinall and K. Marshek, Fundamentals of Machine Component Design, 2nd Ed.,Wiley, New York (1991).

23.38. H. Thomas and V. Hoersch, "Stresses Due the Pressure of One Elastic Solid uponAnother," Univ. Illinois, Bull., 212 (July 15, 1930).

23.39. D. Nelias, M.-L. Dumont, F. Couhier, G. Dudragne, and L. Flamand, "Experimentaland Theoretical Investigation of Rolling Contact Fatigue of 52100 and M50 SteelsUnder EHL or Micro-EHL Conditions," ASME Trans. J. Tribology, Vol. 120, 184-190 (April 1998).


23.40. N. Ahmadi, L. Keer, T. Mura, and V.Vithoontien, "The Interior Stress Field Causedby Tangential Loading of a Rectangular Patch on an Elastic Half Space," ASMETrans., J. Tribology, Vol 109, 627-629 (1987).

23.41. X. Ai and H. Cheng, "The Influence of Moving Dent on Point EHL Contacts," Tri-bology Trans., Vol. 37,2,323-335 (1994).

23.42. E. loannides, G. Bergling, and A. Gabelli, "An Analytical Formulation for the Lifeof Rolling Bearings," Acta Polytechnica Scandinavica, Mech. Eng. Series No. 137,Finnish Institute of Technology (1999).

23.43. International Organization for Standards, International Standard ISO 4406, "Hy-draulic Fluid Power-Fluids-Method for Coding Level of Contamination by SolidParticles" (1987).

23.44. International Organization for Standards, International Standard ISO 4372, "Hy-draulic Fluid Power- Filters-Multi-Pass Method for Evaluating Filtration Per-formance" (1981).

23.45. D. Nelias, Contribution a L'etude des Roulements, Dossier d'Habilitation a Dirigerdes Recherches, Laboratoire de Mecanique des Contacts, UMR-CNRS-INSA de LyonNo. 5514 (December 16, 1999).

23.46. American National Standards Institute (1972), American National StandardCARMA/ ANSI) Std 7-1972, "Shaft and Housing Fits for Metric Radial Ball andRoller Bearings" (Except Tapered Roller Bearings)? (1972).

23.47. American National Standards Institute (1987), American National Standard(ABMA / ANSI) Std 19.1-1987, "Tapered Roller Bearings-Radial, Metric Design" (Oc-tober 19, 1987).

23.48. American National Standards Institute (1975), American National Standard(ABMA / ANSI) Std 19.2-1994, "Tapered Roller Bearings-Radial, Inch Design" (May12, 1994).

23.49. S. Timoshenko, Strength of Materials, Part I, Elementary Theory and Problems, VanNostrand (1955).

23.50. A. Voskamp, "Material Response to Rolling Contact Loading," ASME Paper 84-TRIB-2 (1984).

23.51. T.Harris and w.-K. Yu, "Lundberg-Palmgren Fatigue Theory: Considerations ofFail-ure Stress and Stressed Volume,"ASME Trans., J. Tribology, Vol. 121, 85-89 (1999).

23.52. H.-J. Bohmer, T. Losche, F.-J. Ebert, and E. Streit, "The Influence of Heat Genera-tion in the Contact Zone on Bearing Fatigue Behavior," ASME Trans., J. Tribology,Vol. 121, 462-467 (July 1999).

23.53. SKF, General Catalog 4000 US, 2nd Ed. (1997).

23.54. A. Palmgren, "The Service Life of Ball Bearings," Zeitschrift des Vereines DeutscherIngenieure, 68(14) 339-341 (1924).

23.55. T. Andersson, "Endurance Testing in Theory," Ball Bearing J., 217, 14-23 (1983).

936WEAR

GENERAL

The forces transmitted in a bearing give rise to stresses of varying mag-nitudes between surfaces in both rolling and sliding motion. As a resultof repeated loads in concentrated contacts, changes Occur in the contactsurfaces and in the regions below the surfaces. These changes cause sur-face deterioration or wear. Wear is the loss or displacement of materialfrom a surface. Material loss may be loose debris. Material displacementmay occur by local plastic deformation or the transfer of material fromone location to another by adhesion. When wear has progressed to thedegree that it threatens the essential function of the bearing, the bearingis considered to have failed. Through experience and detailed failureanalysis, the bearing engineer recognizes distinct classes of failure. Theyare listed in Table 24.1.

These failures are defined without presupposing the exact mechanismby which they Occur.They are defined in engineering terms based on adescription of observations. The observations and their classifications re-flect the remaining evidence of a complicated sequence of events involv-ing many physical and chemical processes that preceded it, includingthose in the manufacturing of the original surfaces. Associated with thephysical and chemical interactions on the surfaces are several mecha-nistic wear processes. They are listed in Table 24.2.

Wear prevention is accomplished by forming lubricating films byhydrodynamic lubrication, elastohydrodynamic lubrication (EHL), andboundary lubrication. During the surface life of a bearing, the lubrication

TABLE24.1. Bearing FailureClassificationDue to Wear

MildmechanicalwearAdhesivewearSmearingCorrosive(tribochemical)wearPlastic flowSurface indentationAbrasivewearSurfacedistressPittingFatigue spalling

TABLE24.2. MechanisticWearProcessesAdhesion Plastic flowChemicalreaction Fatigue

STRUCTURAL ELEMENTS OF A LUBRICATED CONTACT 937

and wear processes are interactive and competitive. The topic of wearcannot be divorced from the topic of lubrication, and it is essential thateach contact area within a bearing be considered as a tribological system.The tribological interactions of a system are described schematically inFig. 24.1.

Numerous technical options exist for improving wear performancethrough materials, lubricant base stocks, additives, finishing processes,and surface modification technologies. An appropriate bearing design fora particular application is derived from the synergistic assembly of manytribological contributions.

STRUCTURAL ELEMENTS OF A LUBRICATED CONTACT

Load-carrying capacity is derived from the integral strength of four gen-eral regions of a lubricated contact; these are indicated in Fig. 24.2.

The EHL/micro-EHL region is created by the generation of an EHLfilm, which on a global scale is derived from the hydrodynamic pressuregenerated in the inlet region of the contact; on a local scale it is derivedfrom the micro-EHL action associated with the local topography of thesurfaces. The EHL/micro-EHL region is typically less than 1 JLm (40JLin.)thick.

The surface film region contains the outer layers of the surface, whichconsist of surface oxides, adsorbed films, and chemical reaction films de-rived from the lubricant and its additives. It is usually less than 1 /Lm(40 }.tin.)thick.

The near-surface region contains the inner layers of the surface, in-cluding a finely structured and highly worked Bielby layer as well asother deformed layers. These deformed layers, which have a differentmicrostructure than the material below them, may arise from surfacepreparation techniques, such as grinding and honing, or they may beinduced during operation (e.g., run-in). Hardness and residual stress canvary significantly, and they could be substantially different from the ma-terial below. The near-surface region may extend to 50 /Lm (0.002 in.)below the surface.

For concentrated contacts a subsurface region can be defined, whichmay be 50-1000 }.tm(0.002-0.04 in.) below the surface. This region isnot significantly affected by the mechanical processes that produce thesurface or the asperity-induced changes that occur during operation. Itsmicrostructure and hardness may still be different from the bulk mate-rial below it, and significant residual stresses might still be present.These stresses and microstructures, however, are the result of macropro-cesses such as heat treating, surface hardening, and forging. For typicalHertzian contact pressures the maximum shear stress is located withinthe subsurface region. In other words, the detrimental global contactstresses are communicated to the subsurface region where fatigue be-gins. This fatigue is called subsurface-initiated.

Between the near-surface region and the subsurface region is a "qui-:!scent zone" that resides below the surface where the local asperity and

TRIBOLOGICAL PROCESSES ASSOCIATED WITH WEAR 939surface defect stresses are not significant and the stress field from themacroscopic Hertzian contact stress is not yet appreciable. This zone isquiescent from the point of view of stress and the accumulation of plasticflow and fatigue damage. The existence of the quiescent zone is impor-tant with regard to rolling contact fatigue. It inhibits the propagation ofcracks between the stress field in the near-surface region and the stressfield in the subsurface region.

With regard to rolling contact fatigue, major material improvementshave been made that reduce the risk of subsurface-initiated fatigue. Therisk of surface-initiated fatigue is now a more dominant factor.

TRIBOLOGICAL PROCESSES ASSOCIATED WITH WEAR

Global and Local Processes

The lubricated contact system can be characterized by global processesassociated with the lubricated contact as a whole, and by local processesassociated with local features of the system-that is, those derived fromthe topography of the surface, the microstructure of the underlying ma-terials, or the presence of wear debris. An inherent practical problemwith the control or prediction of wear is that the failure process of wearis initiated generally on a local level, but it is influenced greatly by theinteraction of both global and local processes. Some of the global andlocal processes are definable on reasonably good scientific grounds; how-ever, their interactions in a real system seem to be the important quan-tity that is missing. This section discusses some of the importanttribological processes in connection with wear.

Lubrication Processes

EHL. The formation of an EHL film contributes to wear reduction byreducing the local stresses between the surfaces and by creating a lu-bricant film easy to shear. The global pressure and elastic shape are verysimilar to the Hertzian condition for dry contact, giving rise to threereasonably well-defined regions shown schematically in Fig. 24.3.

The formation of an EHL film is derived from the hydrodynamic pres-sure generated in the inlet region. EHL is on excellent quantitativegrounds, which allows the oil film thickness to be predicted from theviscous properties of the lubricant, the geometry of the contact system,and the operating conditions. This has proven to be a very useful designtool for predicting the lubrication regime for various operating condi-tions; however, it is not sufficient to predict wear. This is partly becauseEHL is primarily an inlet phenomenon; that is, its major action occursin a region displaced from the Hertzian region where the more local

events involved in wear initiation take place. The severity of these localevents can be significantly influenced by the EHL process by the thick-ness of the EHL film. It determines what may be called the "degree ofasperity interaction."

Thus, the EHL process is viewed as a quantitative foundation uponwhich a predictive capability of wear can be established by incorporatingseveral less quantifiable processes.

Surface Temperature. Surface temperature is not a lubrication processbut a key link between lubrication and wear because it significantly in-fluences the viscous properties of the lubricant that control the thicknessof the EHL film, and it is a major driving force in the formation of chem-ical reaction films. Additionally, it determines the rate of lubricant deg-radation and influences the strength of surface films as well as the flowproperties of the material in the near-surface region. Consequently, it isnot surprising that the total temperature level is a frequently used cri-terion for failure.

From a simplistic point of view the total temperature (T) is the sumof a bulk temperature (Tb) of the bearing component and the flash tem-perature (Tf) associated with the instantaneous temperature rise derivedfrom the friction within the lubricated contact. Flash temperature maylrise from the traction of the lubricant film as well as from the energyiissipated from the adhesion, plastic flow of surface films, and defor-nation of the material within the near-surface region. The global mag-litude of Tf can be predicted if simplifying assumptions about the:oefficient of friction and convection heat transfer are made.

Vlicro-EHL. The lambda ratio, A = hi a, a most useful engineering[uantity, is the ratio of the EHL film thickness h to the average combinedoughness height a of the interacting surfaces. It is a simple way oflescribing the degree of asperity interaction. Thus, when A > 3, fatiguelfe is much greater than for lower A, because local asperity stresses have

TRIBOLOGICAL PROCESSES ASSOCIATED WITH WEAR 941

been significantly reduced. Its connection with surface-initiated fatigueseems to be more obvious than failure modes associated with wear. Thelatter failure modes generally appear at low A (e.g., A < 1) where, un-fortunately, it loses much of its meaning.

When a is on the same order of magnitude as h, the surface topog-raphy becomes intimately involved in the lubrication process itself in theform ofmicro-EHL. This comes about from a global standpoint where theorientation of the topographical features can influence the average filmthickness. It also comes about from a local standpoint through the gen-eration of micro-EHL pressures associated with topographical features,as shown in Fig. 24.4.

Measurements [24.2] imply that the energy dissipation due to frictionbecomes concentrated at specific local topographical sites, giving rise tolocal stresses and temperatures even without physical contact.

Boundary Lubrication. It is well known that surface films are impor-tant to boundary lubrication because they prevent adhesion and providea film that is easy to shear. These films may be in the form of oxides,adsorbed films from surfactants, and chemical reaction films from otheradditives. These surface films are schematically shown in Fig. 24.5.

Their reactions and interactions are complex. Most studies on thesubject have focused on the chemical identification or phenomenologicaleffect of surface films, but little is known about the mechanism of pro-tection, the means of removal, or the rate of reformation. At high tem-

peratures the oxidation of the base fluid can contribute to surface filmformation. There have been many studies on the catalytic effect of metalson the bulk oxidation of fluids. Similar oxidative processes can occurunder the thermal stress environment in the contact region where in-termediate oxidation species can react with the surface or organo-metallic material. These reactions can influence boundary lubrication inseveral ways, such as by corrosive wear, by competition with other ad-ditives, or by forming polymeric material-that is, a friction polymer.

The contribution of surface films in preventing wear is complex. Thetime and spatial distribution of the various surface films within the con-tact seems to be important, particularly with regard to the accumulationof material (including debris of all sorts) in depressions and the forma-tion of films at asperity sites. In view of the complexity of surface films,one wonders what the real lubricating "juice" is in a real system.

Wear Processes

Interactions. Perhaps the most important quantity in connection withwear is the deformation attributes of the near-surface region. It is un-fortunate that there is little understanding of near-surface mechanicalproperties or the attributes needed to complement the various lubricat-ing mechanisms to improve wear resistance. To maintain surface integ-rity, the near-surface region must prevent microfracture and maintain aviable surface finish even in the presence of plastic flow.

The interaction of tribological processes in an asperity encounter isshown schematically in Fig. 24.6.

The severity of interaction is reflected in the normal load N, which isinfluenced by the thickness of the EHL film h. The shear force F is in-fluenced by the various surface films and micro-EHL films, along withthe flow properties in the near-surface regions. The exact mechanismwhereby shear stress is applied to the near-surface region is not known.This could come about through metal-to-metal adhesion but it is alsopossible to have sufficient compressive and shear stresses applied locallythrough a thin lubricant film.

In any case, the severity of interaction is important to the initiationand propagation of wear. It will determine whether the result is (1) abenign elastic encounter, (2) a further accumulation of plastic fracturesites that can lead to the generation of wear particles (e.g., microspalling,mild mechanical wear, and delamination), (3) oxidative or corrosive wear,or (4) the advancing of adhesive transfer, which can lead to smearing.The wear processes associated with these events are discussed in con-nection with a description of commonly recognized wear modes.

Wear can be defined in terms of four "mechanistic wear processes":adhesion, plastic deformation, fatigue, and chemical reactions [24.3].Therecognized wear modes in bearing technology, such as "smearing" or "pit-ting," are not singly connected to these wear processes but are associatedwith the interaction of the processes both simultaneously and sequen-tially. The importance of connecting the wear process with the commonlyaccepted failure mode is associated with engineering decisions requiredto overcome wear problems through lubrication, material selection, de-sign, or allowable operating conditions.

944WEAR

Adhesion. Under high normal and tangential stresses, boundary filmstrapped between contacting asperities can be stretched until they rup-ture, which allows the formation of metal-to-metal contact on an atomicscale and gives rise to strong adhesive or welded junctions.

With relative motion between the contacting asperities, junctiongrowth occurs by plastic deformation. Fracture ultimately takes place,and it can occur at a location other than the original interface, resultingin material transfer from surface to surface. The formation and ruptureof adhesive junctions is accompanied by very high local temperaturesthat can form reaction films on the newly formed surface and change themechanical properties of the underlying material. ''Adhesive wear" occurswhen the adhesive transfer of material is the important or controllingmechanism.

Smearing. Smearing is adhesive wear on a large scale, which occursbetween rolling element and raceway when sliding is substantial. Thesevere plastic deformation that accompanies smearing is shown in Fig.24.7.

Smearing is sometimes called scuffing or galling. The precise mecha-nism of smearing is not well understood but it does involve the grossfailure of the surface and is accompanied by an increase in friction andcontact temperature. A current view of smearing [24.4] is that underconditions yet to be defined it is a gradual breakdown in the lubricationof interacting asperities, the nature of which may be boundary, micro-EHL, or a mixture of the two. Although the final smearing mode mayrepresent the gross breakdown of various lubricating films and the near-surface region, it may be triggered by the deterioration in surface topog-raphy as a result of adhesive wear or local plastic flow.

Chemical Reaction. The mechanistic wear processes associated with fa-tigue and plastic flow are the result of material deformation caused bystress. A significant part of wear and its control involves chemical re-action processes with the environment. The environment is defined asthat portion of the contact system that is not an intrinsic part of thesurfaces. The environment includes the surrounding atmosphere as wellas the lubricating films.

Pure chemical reactions should be distinguished from "tribochemical"reactions, which are a consequence of the tribological interactions be-tween the contacting surfaces. "Corrosion" results from reaction of thesurface with the ambient environment under the prevailing ambient con-ditions; tribochemical behavior is activated by mechanical interaction ofthe contacting surfaces. Corrosion often occurs on bearing componentsbecause of improper handling or storage resulting from the absence orremoval of a protective film. An example of corrosion is shown in Fig.24.8.

FIGURE 24.7. Smearing. (a) Smeared material crossing preexisting finishing marks ofa honed roller. (b) Replica electron micrograph showing adhesive wear [A], original ma-chining marks [B], and microcracks [CJ. (c) Etched metallographic cross section throughsmeared area showing white etching bands at surface attributed to rehardening as a resultof overheating (from [24.13]).

Preventing adhesive wear is done by forming tribochemical films.These films may be formed from oxygen in the atmosphere or from an-tiwear or extreme pressure additives in the lubricant. "Tribochemicalwear" generally involves a continuous process of surface film formationand removal. The formation process involves chemical reaction or ad-sorption of chemical species on the surface. The removal process resultsfrom mechanically induced crack formation and abrasion of the reactionproducts in the contact. The process introduces "clean," that is, activated,local areas where new tribochemical films can be formed and subse-quently removed. The tribochemical process introduces thermal and me-chanical activation of the near-surface region, which can cause (1)greater chemical reactivity as a result of increased asperity temperatureand (2) changes in the microstructure and mechanical properties of the

near-surface layer due to high local temperatures and mechanical work-mg.

Under favorable operating conditions tribochemical reactions may beassociated with "mild wear." Mild wear is associated with low wear ratesand smooth surfaces frequently characterized by oxidation of the sur-faces and subsequently removed-that is, oxidative wear [24.5, 24.6].Unfavorable operating conditions can produce "severe wear," where thesurfaces are extensively disturbed and may be characterized by extensiveadhesion and plastic flow rather than oxidative wear. Severe wear canbe prevented by increasing the rate of chemical reactions to form pro-tective surface films at the same rate as clean activated local areas aregenerated. In this way a balance can be obtained between adhesive wearand "chemical wear." "Corrosive wear" is a term used when chemicalwear dominates the adhesive wear mode by a wide margin. The rate of~hemical wear is controlled by additive composition and concentration.'\n optimum additive formulation is achieved when there is a balance)etween adhesive and chemical wear for a given degree of contact se-

TRIBOLOGICAL PROCESSES ASSOCIATED WITH WEAR 947

verity [24.7].This balance between adhesive and chemical wear is shownschematically in Fig. 24.9.

Plastic Deformation. Depending on geometry, relative hardness, andload, the shape of a contacting surface can be permanently deformed, onboth a macroscopic and a microscopic scale, as a result of plastic defor-mation. On a macroscopic scale the overload of rolling elements understatic conditions can cause "Brinell marks" or distort the entire rollingtrack under operating conditions. A much more destructive macroscopicplastic deformation process occurs when the thermal balance betweenthe rolling element bearing components becomes unstable because moreheat is generated than is removed. A thermal runaway can cause thebearing materials to soften and flow plastically until the entire bearinggeometry has been destroyed.

Almost all wear processes involve plastic flow on a microscopic scale.The plastic deformation that occurs from overrunning of hard particlessuch as contaminants and wear debris, is "denting." Figure 24.10 is anexample.

"Plowing" occurs when there is displacement of material by a hardparticle under the presence of sliding or combined rolling/sliding condi-tions. See Fig. 24.11.

"Abrasive wear" occurs when the plastic deformation leads to materialremoval and wear debris. The interaction of hard rolling elements withsofter separator materials often lead to abrasive wear, as shown in Fig.24.12.

General plastic deformation of asperities and ridges on rolling contactsurfaces is generally referred to as "surface distress," or at least the in-itial stages of surface distress. The final stages of surface distress involve

the loss of material through microfracture and pitting. Figures 24.13 and24.14 are examples of surface distress.

Fatigue.The final mechanistic wear process is fatigue. Fatigue is caused

by cyclically repeated stresses on the contact surface, which eventuallyintroduce permanent damage within the material. Damage begins as acrack. After repeated stress cycles, cracks can propagate and eventuallylead to loss of surface material. Fatigue may initiate and propagate fromthe macrostresses induced in the subsurface region, resulting in "spall-ing" characterized by relatively large craters. Fatigue can also be initi-ated in the near surface region as a result of microstresses fromasperities or surface defects, such as dents, grooves, nicks, and scratches.If the combined micro- and macrostress fields propagate cracks throughthe quiescent zone and into the subsurface region, surface-initiated fa-

tigue spalling can occur. "Pitting" and delamination" occur when crackpropagation is confined to the near-surface region. These processes areassociated with the final stages of surface distress discussed above. Themicrostructural material changes and theory for spalling fatigue are dis-cussed further in Chapters 18, 22, and 23.

PHENOMENOLOGICAL VIEW OF WEAR

The previous section reviewed the tribological processes associated withlubrication and wear. In a real system these processes interact and com-pete with one another in a complicated way so that the contribution ofthe individual processes to the overall picture is not very clear. Under-standing the parts of the tribological system is important for selectingbearing materials, lubrication, bearing selection, life prediction, design

decisions, and failure analysis. From an engineering standpoint it is es-sential to have a phenomenological view of wear in addition to an un-derstanding of the constitutive parts of the tribological processes. In thisway the complicated tribological processes can be reduced to a descrip-tion of simpler observed behavior as a result of external operating con-ditions. For example, the wear rate for a given system can be observedas a function of time, loads, velocities, temperatures, and lubricant filmthicknesses. The phenomenological approach is useful if the behavior isorderly with respect to the controlling variables. Figure 24.15 is a sche-matic representation for unlubricated, boundary lubricated, and fluidfilm lubricated systems [24.8].

REFERENCES 961

and chemical properties significantly different from the bulk material.Third, chemical reaction processes occur in parallel with stress-inducedmechanical deformation and ultimately are interdependent. Fourth, thedescription of wear modes is generally confused by the proliferation ofterminology and a lack of definitive connection between wear mode andwear process.

Consequently, the subject of wear has generally been presented not inquantitative terms with specific materials of construction but in termsof tribological processes associated with specific structural elements of alubricated contact. Nevertheless, in [24.14] a method to predict wear lifein ball and roller bearings was presented. Since it is possible to eliminatesignificant wear in rolling bearings by effective lubrication, sealing, and!or shielding in most applications, the need to be able to predict wearrate in a bearing application is far less important than the need to pre-vent wear. Consequently, efforts by most major rolling bearing manufac-turers have been aimed at prevention, and the approach in [24.14] hasnot been generally used in industry.

REFERENCES24.1. H. Czichos, "Importance of Properties of Solids to Friction and Wear Behavior,"

Tribology in the 80's, NASA Conference Publication 2300, Vol. I, Sessions 1 to 4,Proceedings of an International Conference held at NASA-Lewis Research Center,Cleveland, Ohio (April 18-21, 1983).

24.2. L. Wedeven and C. Cusano, "Elastohydrodynamic Contacts-Effects of Dents andGrooves on Traction and Local Film Thickness," NASA TP 2175 (June, 1983).

24.3. A. Dorinson and K. Ludema, Mechanics and Chemistry in Lubrication, TribologySeries, 9, Elsevier (1985).

24.4. A. Dyson and L. Wedeven, "Assessment of Lubricated Contacts-Mechanisms ofScuffing and Scoring," NASA TM-83074 (February, 1983).

24.5. J. Lancaster, "The Formation of Surface Films at the Transitions between Mild andSevere Metallic Wear," Proc. Roy. Soc. A. 273, 466-483 (1963).

24.6. T. Quinn, "NASA Interdisciplinary Collaboration in Tribology-A Review of Oxi-dation Wear," NASA Contractor Report 3686.

24.7. C. Rowe, "Lubricated Wear," in Wear Control Handbook, ASME, 143-160 (1980).24.8. M. Peterson, "Design Considerations for Effective Wear Control," in Wear Control

Handbook, ASME, 413-473, (1980).24.9. A. Begelinger, A. deGee, and G. Solomon, "Failure of Thin Film Lubrication-

Function Oriented Characterization of Additives and Steels," ASLE 23, 23-34(1980).

24.10. T. Tallian, "Rolling Contact Failure Control Through Lubrication," Proc. Inst. Mech.Engrs. 182,205-236 (1967-68).

24.11. R. Parker, "Correlation of Magnetic Perturbation Inspection Data with Rolling El-ement Bearing Fatigue Results," ASME Trans. J. Lub. Tech. 97, Ser. F, No.2, 151-158 (Apr. 1975).

962WEAR

24.12. L. Wedeven, "Influence of Debris Dent on EHD Lubrication," ASLE 21, 41-52(1978).

24.13. T. TalIian, G. Baile, H. Dalal, and O. Gustafsson, Rolling Bearing Damage Atlas,SKF Industries, Inc., Revere Press, Philadelphia (1974).

24.14. J. Brandlein, P. Eschmann, L. Hasbargen, and R. Weigand, Ball and Roller Bear-ings-Theory, Design, and Application, 3rd ed., 205-212, Wiley, New York (1995).

964VIBRATION, NOISE, AND CONDITION MONITORING

Symbol DescriptionUnitsZ Number of balls or rollers

Or Radial deflectionmm (in.)A Wavelengthmm (in.)() Angular measurerad

SUBSCRIPTSc Refers to cagei

Refers to inner ring, shaft of raceway0 Refers to outer ring or raceway

GENERAL

This chapter provides a practical overview of bearing vibration. Whererelevant, reference is also made to noise, sometimes resulting from ex-cessive bearing vibration. Common bearing applications in which noiseand/or vibration are important are described.

Machine vibration or noise levels, whether excessive or not, are af-fected by bearings in three ways: as a structural element defining in parta machine's stiffness; as a generator of vibration by virtue of the wayload distribution within the bearing varies cyclically; as a vibration gen-erator because of geometrical imperfections from manufacturing, instal-lation or wear and damage after continued use.

Illustrations of manufacturing and installation problems are shown,in some cases with the use of vibration measurement taken from ma-chines after bearing installation. Descriptions are also given of methodsused in rolling bearing factories to evaluate bearing component quality,control manufacturing processes, and minimize bearing vibration.

Detection of progressive bearing deterioration in operating machineryby vibration measurements has become more economical and reliable inrecent years. Some aspects of such machinery monitoring are considered.

VIBRATION AND NOISE-SENSITIVE APPLICATIONS

i;ignificance of Vibration and Noise

n many cases objectionable airborne noise from a machine results fromneasurable vibration of machine components. Correlation between bear-ng noise and machine vibration measurements has been reported [25.1,5.2J. Therefore, with respect to rolling bearings, the terms "noise" andvibration" usually denote similar and related phenomena. Their relativenportance to the bearing user may differ, depending on where and howle machine is used.

VIBRATION AND NOISE-SENSITIVE APPLICATIONS 965

Regardless of which seems to be more important in a particular ap-plication, noise and vibration may both be indicators in new machinesof quality problems with bearings, machine components, or assemblymethods. Such problems can limit the functional capabilities of the ma-chine, and they can reduce the potential useful life of the bearings or themachine itself In cases where a machine has successfully performed itsfunction and is approaching the normal time for repair or replacement,the first indication may come from increased levels of noise or vibration.

Noise-Sensitive Applications

The application that has been the major driving force for reduced noiseis that of small and medium electric motors, primarily utilizing deep-groove ball bearings. Figure 25.1 shows such an application. The outerring of the bearing at the left end of the motor is free to move axiallyunder controlled thrust load of a spring to remove axial clearance fromwithin the bearing. This allows for thermal expansion of the shaft andmotor assembly without loss of preload while simultaneously preventingexcessive bearing loads or distortion of motor components.

Quiet running characteristics of the electric motor are required in of-fice equipment and household appliances where noise may be an irritant.Noise is also a problem in building heating and air conditioning systems,where motor or fan support bearing noise can be transmitted and am-plified through duct work or air columns. Also included in this categoryare drive systems of elevators, using larger electric motors with deep-

966 VIBRATION, NOISE, AND CONDITION MONITORING

groove ball bearings and cylindrical roller bearings and spherical rollerbearings in pillow blocks to support cable sheaves. Aside from irritation,excessive noise in the latter application might make passengers con-cerned. Automotive applications also requiring quiet running perform-ance include alternators (deep-groove ball bearings and needle rollerbearings), transmissions, differentials (tapered roller bearings), and fans.

Objectional noise might be characterized by volume or sound level andpitch or frequency. Sound from a machine may be more irritating if aparticular frequency is dominant. Possibly even more objectionable areintermittent or transient sounds that vary with time in either pitch orvolume at regular or irregular intervals. Such effects might be more eas-ily heard than measured, since common measuring methods may be ac-quiring data over time periods that are long compared to short-durationtransient sounds or vibrations. In addition, transient sound or vibrationis sometimes most significant when a machine is coming up to operatingspeed or coasting down. Even in vacuum cleaners or dishwashers thiseffect is sometimes heard.

Qualitative audible evaluation of airborne sound from machines suchas electric motors is performed as a routine inspection in many cases.Audible evaluation is also performed by processing the output of a vi-bration measurement transducer through a loudspeaker. This is usefulfor detecting transients and also in some cases for identifying the causeof excessive measured vibration. The vibration parameter monitoredcould be velocity or acceleration rather than displacement, which over-emphasizes low-frequency vibration.

The United States Navy has made extensive demands on bearingmanufacturers with respect to bearing vibration and noise reduction aswell as boundary dimension and running accuracy tolerances [25.3].Thisstems in part from the requirement to make submarines more difficultto detect by monitoring sound transmitted through water. At the sametime, improvements in reliability and reduced maintenance costs areachieved. Extensive research efforts on bearing vibration have beensponsored by the U.S. Navy [25.4].

Vibration-Sensitive Applications

Applications where bearing and machine vibration are more importantthan noise fall into two categories. In some cases the machine must becapable of high running and positioning accuracies to function properly.In other cases the major concerns are safety, if vibration causes cata-strophic failure, and the economic impact of reduced machine utilizationand increased repair cost if vibration foreshortens the life of components.

Not only is noise intrinsically less important than vibration in thesecategories, but it may also be incapable of indicating a significant prob-lem. This would occur if the predominant frequency of high-amplitudevibration falls outside the audible range; for example, rotating imbalance

VIBRATION AND NOISE-SENSITIVE APPLICATIONS 967

in a machine running at 1800 rpm (30 Hz). In addition, abnormal noisemight be undetectable because of ambient noise where the machine op-erates or because of normal noise from the process the machine performs.

Bearing applications where machine accuracy might be affected byvibration include machine tools. Grinding spindles often must be capableof producing components with size and two- or three-point roundnesswithin a micron (4 X 10-5 in.). Figure 25.2 shows a grinding wheel spin-dle using precision double-row, cylindrical roller bearings and a double-direction, angular-contact ball thrust bearing to achieve high radial andaxial stiffnesses. The cylindrical roller bearings have tapered bores foraccurately controlling preload. Precision angular-contact ball bearings inmatched sets are also widely applied in spindles.

In addition to size control and roundness, precision spindles must becapable of producing even finer levels of geometrical accuracy such asrelatively low levels of surface roughness and circumferential wavinessamplitudes of much less than a micron (4 X 10-5 in.). Vibration cancontribute to excessive roughness or waviness and can also produce chat-ter, a more severe form of waviness that can cause permanent metallur-gical damage to hardened steel parts.

Other machines in which vibration might prevent the required accu-racy from being achieved include rolling mills for sheet steel, paper, andchemical films. Computer disc drives are a further example, requiringnonrepeatable bearing runout accuracy of no more than one quarter toone half of a micron (1-2 X 10-5 in.) for the spindle and head combined.Similarly, gyroscope bearings require good dynamic running accuracy aswell as very low torque levels.

Cases where running accuracy is not as important as safety and ma-chine reliability often involve machines that are producing or transmit-ting high horsepower, have massive rotating components, and are


running at high speeds relative to the size of the equipment. Eccentricmass produces large and potentially destructive forces in these applica-tions. As discussed in Chapter 26, such equipment may operate at speedsabove resonant frequencies, so large amplification of vibration could oc-cur as equipment is run up to speed. Examples include compressors,pumps, and turbines.

Applications in this section are examples where machine noise or vi-bration is important. More demanding applications continually arise, re-quiring greater accuracy, higher speeds and loads, and improvedreliability. Therefore, bearing manufacturers have continuously empha-sized improvement of bearing quality with respect to noise and vibrationthrough ongoing development of machines and methods for manufactur-ing and inspection.

THE ROLE OF BEARINGS IN MACHINE VIBRATION

Bearing Effects on Machine Vibration

Rolling bearings have three effects with respect to machine vibration.The first effect is as a structural element that acts as a spring and alsoadds mass to a system. As such, bearings define, in part, the vibrationresponse of the system to external time-varying forces. The second andthird effects Occurbecause bearings act as excitation sources, producingtime-varying forces that cause system vibration. In one case this exci-tation is inherent in the design of rolling bearings and cannot be avoided.In the other case these forces result from imperfections, which usuallyare avoidable.

Structural Element

With sufficient load, the bearing is a stiff structural member of a ma-chine. It is a spring whose deflection varies nonlinearly with force, incontrast to the usual linear spring characteristics assumed in dynamicmodels, such as the single-degree-of-freedom spring-mass-damper modeldiscussed in Chapter 26. As a first approximation, it may be adequateto estimate machine vibration response by considering the bearing as alinear spring. In this case a bearing spring constant is determined bytaking the slope of the force-deflection curve of the bearing at the normaloperating load. The approximation may be insufficient in cases requiringprecise knowledge of transient vibration response, particularly near ma-chine resonant frequencies. In these cases extensive mathematical mod-eling and experimental modal analysis are performed, both of which arebeyond the objectives of this chapter. If it is sufficient to consider bearingstiffness as a constant, under a specific set of operating conditions, thenthis approximation can be derived from equations in Chapter 10.

THE ROLE OF BEARINGS IN MACHINE VIBRATION 969

Bearing stiffness increases with increasing load, a characteristic re-ferred to as a "hardening" spring. Larger nominal operating loads orbuilt-in preload would result in smaller variations in dynamic bearingdeflection when subjected to a particular dynamic load variation. Simi-larly, increased bearing stiffness raises the value of a resonant frequencyassociated with this spring, since a resonant frequency is inversely pro-portional to the square root of stiffness. Moreover, radial stiffness de-creases with increasing contact angle, whereas the reverse is true foraxial stiffness. Therefore, response to dynamic load variation will dependstrongly on the direction of such loads relative to that of the nominalload that governs contact angle.

Since the bearing "spring" is nonlinear, it is evident that sinusoidaldeviations from the nominal load will not cause sinusoidal bearing de-flection. When the load is greatest, the increase to nominal bearing de-flection will be less than the decrease from nominal bearing deflectionwhen the load is at its lowest value. Iflarge dynamic fluctuations in loadare experienced, say in a radially loaded bearing, then it is possible forthe load zone to alternate from the bottom to the top of the outer raceway.If the bearing has radial internal clearance, there is the possibility ofessentially no loading at all on the outer raceway for brief instances.Such conditions could arise because of external loading or conditionswithin the bearing.

Variable Elastic Compliance

The second effect of bearings on machine vibration occurs because bear-ings carry load with discrete elements whose angular position, with re-spect to the line of action of the load, continually changes with time. Thismere change of position causes the inner and outer raceways to undergoperiodic relative motion even if the bearing is geometrically perfect.Analysis of this motion is described in [25.4]. The following example il-lustrates how variable elastic compliance vibration occurs.

Example 25.1 Consider a 204 radial ball bearing with eight 7.938-mm (0.3125-in.) balls. The bearing supports a 4450-N (lOOO-lb)radialload. Figure 25.3 shows the bearing at two different times. In Fig.25.3a ball 1 is located directly under the load, and balls 1, 2, and 8carry the load. In Fig. 25.3b balls 1 and 8 straddle the load symmet-rically, and balls 1, 2, 7, and 8 carry the load. Obviously, the radialdeflection is different in each situation.

With methods from Chapter 6 the radial deflection in the first caseis estimated to be 0.04323 mm (0.001702 in.). In the second case thedeflection is approximately 0.04353 mm (0.001714 in.). The bearingdeflection is less in the former arrangement. The position of the ballset in Fig. 25.3a gives a stiffer bearing at that instant. The shaft andinner raceway have approached closer to the outer raceway in the time

FIGURE 25.3. (a) Angular position of ball set, time = O. (6) Angular position, time =t[lIZ (cage rotation frequency)].

it takes for one half of the bal1 spacing to pass a point on the outerraceway to reach the position shown in Fig. 25.3b. The shaft wiU re-turn to its original position as bal1 1comes under the load line. Thisfrequency of vibration is therefore equal to the cage rotational fre-quency multiplied by the number ofbal1s; that is, the frequency of thisvibration OCcursat the frequency of bal1s passing the outer raceway.

This example iUustrates vertical elastic compliance vibration. Hori-zontal motion also Occurs,at the same vibrational frequency, as the bal1set assumes angular positions that are asymmetrical with respect to theload line. Both vertical and horizontal vibration amplitudes are nonsi-nusoidal as a result of the nonlinear deflection characteristics. The ex-istence of this type of vibration, which OCcurseven with a geometrical1yperfect bearing, is one reason why bearing damage detection is best per-formed by monitoring frequencies other than the fundamental bearingfrequencies.

Geometrical Imperfections

The first effect that bearings have on machine vibration arises from ge-ometrical imperfections. These imperfections are always present to var-ying degrees in; manufactured components. Sayles and Poon [25.6Jdiscuss three mechanisms by which imperfections in bearings cause vi-bration: waviness (Fig. 25.4) and other form errors causing radial or axialmotion of raceways; microslip together with asperity coUisions and en-trained debris that break through the lubricant film and shocks due tolocal elastic deformations caused by summits that do not break the lu-bricant film.

FIGURE 25.4. Vibration from raceway waviness.

The local elastic contacts are of approximately the same size orsmal1er than Hertzian contact areas. At any instant there may be onlya few such summits in the Hertzian deformation zone, depending on thetype of bearing component finishing processes employed; for example,honing and lapping. Elastic deformations of the type discussed occur rap-idly, and the time separating one such contact from the next is brief. Amajor contribution to bearing vibration in the higher frequencies, forexample, above 10,000 Hz, is thought to be the result of such deforma-tions. Due to their impulsive nature, however, they are capable of excit-ing lower-frequency resonances.

Control1ing component waviness and other types of errors from man-ufacturing, distortion, or damage occurring while the bearing is assem-bled to the machine, is a high priority. The effects of such form errors onmachine vibration or noise can be significant.

Waviness Model

Figure 25.4 represents a bearing with waviness on the outer raceway. Itis assumed that the bearing supports a mass and that the outer ring isrigidly supported by a housing. If no waviness is present on the surfaceof the bearing raceways, a force balance in the vertical direction is

FB - Mg = 0 (25.1)

If waviness is present, then for an approximation it wiUbe assumed thatthe mass wiU move up and down as a rigid body, with reaction forceproduced in the bearing as a result of the acceleration of the mass. Inthis case the force balance is


FB + I1FB - Mg = My (25.2)

For waviness that can be approximated as sinusoidal the equations are

y = A sin (27Tft) (25.3)and

y = -A(27Tf)2sin (27Tft) (25.4)

The frequency f is the rate at which balls pass over a complete wavecycle. The assumption of only vertical motion of the mass implies twoconditions: (1) that the wave peaks are always in phase with balls, and(2) that variation of the ball set angular position within the load zonehas no influence on the direction of motion. For illustrative purposes,these simplifying assumptions will suffice to demonstrate the importanceof relatively small form errors.

Combining equations (25.2) and (25.4) and rearranging,

FB + I1FB = Mg - MA(27Tf)2 sin (27Tft) (25.5)

For sufficient waviness amplitude and passage frequency, the right sideof the equation can vanish, in which case the bearing force (left side ofthe equation) vanishes, or it can become negative, in which case thebearing produces a negative force to restrain the motion of the mass. Inthis case the load zone would alternate from the bottom to the top of theouter raceway. If the bearing has clearance, it could become unloaded ineither direction at some instant. The followingexample gives an estimateof waviness amplitude that would cause this condition.

Example 25.2 For a bearing with 50 waves on the circumference ofthe outer raceway, estimate the amplitude of the waviness that couldcause sufficient acceleration to momentarily unload the bearing. Theshaft speed is 1800 rpm (30 Hz), and the cage speed is 11 rps.

The rate at which any ball passes over a wave cycle is the productof the cage speed and the number of waves per circumference of theouter raceway-in this case, 550 wave cycles per second. The conditionfor bearing unloading would Occurwhen

Mg = MA(27Tf)2

or

Bearing raceway waviness of this amplitude and frequency are in ex-cess of acceptable levels. Although wavy components of this type rarelyoccur, they can occur due to improper manufacturing procedures or man-ufacturing machine malfunctions. Examples of excessively wavy compo-nents are presented later on.

Example of Electric Motor Vibration

Three examples are discussed to reveal sources of noise and vibrationproblems in small electric motors with newly installed bearings. The firstexample shows measurements of bearing distortion that, occurring as aresult of assembly to faulty machine components, produced waviness onthe inner raceway. The second example shows the effect of improper as-sembly, and the third illustrates the effect of a defective bearing on motorvibration.

Example 25.3 Assembly of bearings in housings or on shafts withpoorly controlled geometry can distort the bearing components andproduce wavy running surfaces that affect the machine vibration ornoise. Johansson [25.6] discusses effects of inner raceway distortionon electric motor vibration.

Figure 25.5a shows a circumferential trace of a motor shaft-bearingjournal where a 6 mm bore deep-groove ball bearing was mounted.The motor was rejected after assembly for audible noise and for vi-bration as determined by hand-turning the armature while it was sup-ported by the bearing. The shaft speed in the power tool applicationwas 23,000 rpm. The shaft exhibits a three-point out-of-roundness con-dition of approximately 24 /Lm (0.001 in.). Shaft diameter tolerancefor the particular application would normally be held within a totalspread, from one shaft to another, of 8 /Lm(0.0003).

Figure 25.5b shows traces of the bore and ball groove after disas-sembly from the shaft, with both surfaces being less than 1 /Lm(0.00004 in.) out-of-round. Figure 25.5c shows a trace of the ballgrooveas mounted on the shaft. This indicates that the raceway in themounted condition exhibited 16 /Lm(0.0006 in.) of three-point out-of-

FIGURE 25.5a. Motor shaft circumference (each radial division is 2 JLm) (from [25.7]).

round. Note the two local imperfections on the raceway. The largest isapproximately 2 /Lm CO.00008 in.) deep and is located on one of thelobes. The origin of this defect was not determined, although it prob-ably occurred during press-fit assembly or from damage during run-ning.

The bearing has six balls, so the three high points on the distortedinner raceway could be in contact with three balls, while the otherthree balls carry no load. This effect would lower the bearing stiffness,either axial or radial, which depends on the number of balls. On theother hand, larger individual ball loads are expected to be generatedduring parts of the rotational cycle of the shaft, tending to raise stiff-ness, but simultaneously causing large axial vibration.

The cage operating speed can be calculated by using equations ofChapter 8 to be approximately 138 Hz. The cage speed relative to theinner raceway is 245 Hz, so the rate at which a wave cycle passes aball is 735 Hz. The rate at which any of the high points passes fromone ball to the next is equal to the product of the cage speed relativeto the inner raceway and the number of balls (1470), harmonically

\VI

FIGURE 25.5b. Inner raceway after disassembly (each radial division i3 1 JLm) (from[25.7]).

related to the wave passage frequency, because the number of balls isa multiple of the number of waves. Accordingly,there is the potentialfor large-amplitude vibrations with two fundamental frequencies (735and 1470 Hz) well into the audible range. In addition, numerous highharmonics of each would be expected, with the potential for excitationof various structural resonances in the motor.

Example 25.4 This example illustrates a loose assembly that con-tributes to noise and vibration. In this case airborne sound measure-ment in the form of frequency spectrum analysis is used to identifythe source of the problem.

The digital frequency spectrum analyzer is a computer-based in-strument that transforms time-sampled data into the frequency do-main through Fourier series analysis. Knowledge of dominantfrequencies in vibration signals can often reveal sources of a specificproblem. An additional benefit of the technique is that storage and

FIGURE 25.5c. Inner raceway mounted on shaft (from [25.7]).

documentation of data are facilitated. Figure 25.6a shows a simplifiedtime signal, which might be obtained as a voltage signal from a trans-ducer such as an accelerometer or eddy current displacement sensor.This signal might be displayed on an oscilloscope. The same signalcan be sampled and Fourier transformed into the frequency domain,as shown in Fig. 25.6b, to show the amplitude as a function of fre-quency. In this case the time signal consists of only a few frequencies,which in the frequency spectrum show up distinctly.

Figure 25.7a shows the frequency spectrum of a vacuum cleanermotor rejected for noise after assembly. The spectrum was obtainedfrom airborne noise measurement utilizing a microphone. The ampli-tude scale is uncalibrated and logarithmic; each vertical division rep-resents an amplitude increase of a factor of 10, with lower frequenciesbeing attenuated, as is common in sound measurement. The normaloperating speed of the motor is 20,000 rpm. Neither the sound spec-trum nor qualitative audible evaluation revealed anything abnormalat this speed. When the motor was turned off and was coasting down,however, a distinct rumble was heard at a speed subsequently esti-

mated to be around 10,000 rpm. The spectrum of Fig 25.7a was thenobtained with the motor running at that speed. It seemed likely thatsome system resonance was occurring as the motor coasted down andpassed a critical speed.

The spectrum shows distinct frequency peaks, determined to be165.5, 325, 487.5, 650, 975, and 1137.5 Hz. A similar good motor (e.g.,Fig. 25.7b) shows only a peak at 975. Rotor endplays on the motorswere 0.094 mm (0.0037 in.) for the motor of Fig. 25.7a and 0.0508 mm(0.002 in.) for the other motor. The peak at 975 Hz for the good motorwas expected to be the blade pass frequency on the motor fan, whichhas six blades. This means that the actual running speed was 9750rpm. In this case the measured spectral peaks on the noisy motor wereall harmonically related to the running speed. Harmonics of the run-ning speed usually occur when mechanical looseness, which could becaused by improper bearing mounting [25.8], exists.

Looseness would result in low stiffness, lower resonant frequency,and "play" in the system. The imbalance force, rotating at the shaft

THE ROLE OF BEARINGS IN MACHINE VIBRATION 979

speed, could then produce significant vibration amplitude at that fre-quency. The harmonics probably result from either of two effects: (1)directional stiffness variation or (2) shocks occurring if the load zoneshifts from one side of the bearing to the other in an unstable manner.

The noisy motor was disassembled. A spring clip, which retains thebearing outer raceway in a plastic housing, had been improperlyseated during assembly, resulting in a loose bearing mount. Repairand reassembly reduced the noise to an acceptable level. Diagnosis ofthis problem could have been made with vibration transducers ratherthan sound measurement. The audible evaluation of the transient vi-bration during coast-down also provided a clue to the source of theproblem.

Example 25.5 Noise emitted from a small electric motor with a de-fective bearing is discussed. Figure 25.8 shows vibration spectra of twosmall electric motors (3600 rpm) on the same plot. These data wereobtained by screwing a small accelerometer into a nut glued to theend cover. The frequency range investigated was to 10,000 Hz, cov-ering the most important part of the audible frequency rate at usualmachine operating speeds.

The rms (root mean square) vibration amplitude at each frequencyis plotted in volts dB, where the dB value is equal to 20 loglo(voltage/reference voltage). In this case the reference voltage is 1.0. Each major


division on the plot is equal to 10 dB, with the amplitude scale rangingfrom -40 dB (0.01 V) full scale down to -120 dB (10-6 V). The accel-erometer output is 0.010 V/g of acceleration.

One motor is seen to have vibration amplitudes from 5 to 25 dBhigher than the other motor over much of the frequency range. Themotor with the higher vibration also gave torque readings more thana factor of 2 higher than the other motor. Frequency spectra at variouspoints on the motors were taken with the same results. Spectra inother frequency ranges were taken with no conclusive results regard-ing the origin of the vibration. Qualitative evaluation of audible noiseindicated a low-amplitude clicking sound that repeated at a fairly highrate.

The motors were disassembled, the shafts and housings werechecked for geometry and found to be normal, and the bearing torqueswere measured. The bearings from the motor with high vibration hadrubbing seals, whereas the other set had noncontacting seals. The fourbearings were vibration tested, with seals removed, on a standardbearing test apparatus discussed in the next section. It was found thatone of the bearings from the motor with high vibration also gave highreadings on the inspection tester. Spectrum analysis indicated har-monics of the frequency at which balls pass a point on the outer race-way. Examination of the outer raceway revealed a defect thatappeared to be related to manufacturing and had escaped detection infinal inspection vibration testing.

The discussion and examples of this section have viewed several waysin which bearings can affect or cause machine vibration. The irregular-ities on bearing surfaces that exist from manufacturing, assembly intothe machine, or from deterioration after long-term use provide a sourceof forced excitation to rotating machine members or structural compo-nents that can increase stress, accelerate wear, increase frictional losses,and possibly cause catastrophic machine failure. Other forms of bearingdistortion or imperfections occurring as a result of assembly include mis-alignment (housing centers out of line from each other or not parallel),Brinell damage, contamination by debris, and bell-mounted housingspreventing axial movement of the outer ring (e.g., in small electric mo-tors with spring preload).

~ONROUNDNESSEFFECTANDITSMEASUREMENT

l!oundness and Waviness

}enerally, a part is said to be round in a specific cross section if there~xists a point within that x-section from which all other points on the

NONROUNDNESS EFFECT AND ITS MEASUREMENT 981

periphery are equidistant. The first-mentioned point is of course the cen-ter of the circle, and the x-section is a perfect circle as in Fig. 25.9a. Ifthe x-section is not a perfect circle as in Fig. 25.9b, it is said to be out-of-round with the "out-of-roundness" specified as the difference in dis-tance of points on the periphery from the center. Thus, out-of-roundnessin Fig. 25.9b is r1 - r2• In addition to the basic profile of Fig. 25.9b, anirregular profile similar to Fig. 25.9c is usually present in manufacturedmachine elements, and this includes rolling bearing raceways and rollingelements. The irregular surface of Fig. 25.9c is of substantial importanceto bearing frictional performance and endurance; this was discussed inChapters 13 and 18. The lobed surface of Fig. 25.9b is also significant,as it is a causative factor of bearing vibration. The important feature iscalled waviness, that is, the number of lobes per circumference.

Waviness can occur in the machining process. A round bar or ring-type element is compressed at the points of contact in a chuck, three jawor five jaw, causing stresses in the part. The part is then "turned" orground perfectly circular; however, when it is released from the chuck,the stresses are released, and the part becomes lobed. Centerless grind-ing also causes waviness where the original bar stock is irregular; per-haps due to the previous machining operation.

Waviness is measured by equipment such as the Taylor-Hobson Tal-yrond; Talyrond traces are shown in Fig. 25.10. They were also shownin Fig. 25.5(a)-(c).

For a tapered roller bearing mounted in an SKF VKL tester, shownschematically by Fig. 25.11, Yhland [25.9] examined the correspondencebetween waviness and the resulting vibration spectrum. For a bearingwith Z rolling elements and if p and q are integers equal to or greaterthan 1 and 0, respectively, then for vibrations in the radial directionmeasured at a point on the o.d. of the outer ring, the vibration circularfrequencies as functions of inner ring, outer ring, and roller waviness aregiven in Table 25.1.

and r = sin 90, approximating a sinusoidal wave.This type of discrete frequency waviness is an excitation source for

vibration as well as generation of dynamic force variations on bearingcomponents. Defects of the magnitude shown in Fig. 25.13 are rareand are easily detected when they occur. Waviness of relatively lownumber (usually odd) of waves per circumference occurs because ofinaccuracies in grinding machine tooling or setup involving slidingcontact shoe supports for the workpiece as it is ground. Contact of twohigh points simultaneously on two shoes causes more material to beremoved by the wheel at a position opposite the shoes; conversely, lowpoints on the shoes result in less material being removed, producinghigh points on either side of the wheelwork contact zone. This condi-tion is detectable with conventional in-process gaging used to controldiameter, provided that gages are set up for three-point diameter mea-surement rather than for two point.

Other types of imperfections can result from machine malfunction. SeeFig. 25.14, which shows a circumferential trace of a spherical roller. Thiscomponent was ground "on-centers" and does not show a waviness pat-tern of the type seen before. A machine control system malfunction, how-ever, allowed the roller to be released from the grinding station beforethe wheel was retracted, with the result that the roller contacted thewheel and produced a localized flat spot on the roller surface over ap-proximately 5% of the circumference to a maximum depth of 18 J.1-m(0.00072 in.).

In contrast to waviness from incorrect setup, three-point diametermeasurement is not likely to reveal this imperfection unless measure-ments are made around the entire circumference.

More subtle defects can also occur.They may be characterized by muchsmaller deviations from true geometrical form and require more detailedcomponent inspection, such as waviness testing or vibration testing ofassembled bearings.

Examples of such defects are shown in Figs. 25.15 and 25.16. Thesetraces show, respectively, a spherical roller and a cylindrical roller withlower-amplitude and higher-frequency waviness than the examples of

Fig. 25.13. The roller of Fig. 25.15 has 36 waves with an average peak-to-valley amplitude of approximately 1 /Lm (0.00004 in.). The roller ofFig. 25.16 has over 100 waves with a peak-to-valley amplitude of lessthan 0.5 /Lm(0.00002 in.). Both rollers were identified as causes of noiseand vibration in assembled bearings. The cylindrical roller bearing cor-responding to Fig. 25.16 had been installed in a large electric motor. Themotor emitted a periodic audible noise at slow running speed. With astopwatch, the repetition rate of the noise could be associated with eachrevolution of the cage. Subsequent investigation on a test rig traced thenoise to this particular roller.

Waviness Testing

Component inspection for waviness ahs been performed for many yearsand is described in the literature [25.9,25.10]. This inspection is used toassess the degree of radial deviations from a true circle on the circum-ference of a component. This is accomplished by rotating the componenton a hydrodynamic spindle and applying a contacting transducer per-pendicular to the surface of the component. The transducer is a stylusthat follows the radial deviations and produces a voltage output propor-

tional to the instantaneous radial rate of change of the displacement ofthe stylus; that is, the signal from the transducer is proportional to ve-locity. This proportionality exists over a wide frequency range, such as10,000 Hz, which allows reasonably high test speeds to be used.

The voltage signal from the transducer is amplified and bandpass fil-tered into three or more bands typically 2.5 octaves wide. The filter bandscombined with the selected testing speed encompass a broad range ofwaviness frequencies. Frequencies shown in the previous componenttraces extend to approximately 100 waves per circumference. Wavinesstesting equipment and procedures in use cover a range well beyond this.

The rms value of the signal in each filter band is obtained and com-pared to specifications to determine acceptance or rejection of the lot ofcomponents being inspected and to provide information for corrective ac-tion on the manufacturing processes. Frequency spectrum analysis isalso becoming widely applied as instruments become less expensive andmore suitable for use in the factory.

The following discussion illustrates some reasons why velocity mea-surement has several advantages over displacement for evaluating bear-ing component quality. The cylindrical roller of Fig. 25.16 was rotated on

a waviness testing machine, and the amplified output of the velocitytransducer was analyzed with a frequency spectrum analyzer. Resultsare shown in the plot of Fig. 25.17.

The abscissa covers a frequency range of 0-2000 Hz, and the rollerwas tested at 720 rpm (12 Hz), so this frequency range would detect adominant wave pattern up to 166 waves per circumference (2000/12).The ordinate gives rms voltage in dB referenced to 1 "Y,ranging from-10 dB full scale (0.3162 V) to -90 dB (3.162 X 10-5 V). Nominal cali-Jration for the velocity transducer and amplifier is 3.0 p,V1p,in.-sec. A:ursor mark is indicated at the peak, with corresponding coordinates>rinted below the frequency axis. The peak occurs at 1250 Hz, with rmslmplitude at -23.43 dB. Since roller test speed was 12 Hz, the frequencyIt which peak Occurs corresponds to approximately 104 waves per cir-umference.

Example 25.7. For the test arrangement previously indicated, de-termine the rms radial velocity of the predominant waviness and thenumber of waves per circumference. Also estimate its average peak-to-valley amplitude in microns.

NONROUNDNESS EFFECT AND ITSMEASUREMENT 991

components of diverse size and configuration. Additionally, transducersand instrumentation for velocity measurement systems do not require asgreat a dynamic range as displacement measurement. Numerical speci-fications of similar magnitude are also more easily developed and ap-plied.

Some types of defects that can arise on components are very local innature. Detection of these defects may not be feasible with wavinesstesting, which may only acquire data from one or two circumferences oncomponents being tested. Balls have numerous potential axes of rotation.Therefore, visual component inspection and vibration testing of assem-bled bearings provides more definitive assurance of final bearing quality.

Vibration Testing

Aside from defects that are not discovered in waviness testing, vibrationtesting of the assembly allows the detection of damage occurring in as-sembly, such as binding or excessively loose case, Brinell damage to race-ways or scuffing of balls, and distortion of raceways from incorrectinsertion of seals or shields with bearings tested after grease insertion.

Certain types of geometrical problems may also be detectable in vi-bration testing. These can include, for example, oversized rolling ele-ments, improper cross-groove form on raceways, or groove run out to sidefaces of the raceways. In addition, testing can reveal contamination bydirt or inferior grease quality.

Figure 25.18 shows a manually operated vibration testing apparatusfor relatively small bearings, for example, up to 100 mm o.d. Similarequipment is in use for larger diameter bearings, and automatic versionsare implemented on production lines. The main elements of the systemare the test station and the vibration signal analysis instrument. Thetest station consists of a hydrodynamic spindle, an air cylinder for ap-plying load to the bearing being tested, and an adjustable slide for po-sitioning the velocity transducer. The spindle is belt driven by the motormounted beneath the stand. A schematic representation of the system isshown in Fig. 25.19.

The inner raceway of the bearing mounts on a precision arbor fastenedto the spindle, which rotates at 1800 rpm. A specified thrust load is ap-plied to the side face of the nonrotating outer ring. The tip of the velocitytransducer is lightly spring-loaded on the outer diameter of the outerring. The loading tool (not shown) consists of a thin-walled steel ringmolded into a neoprene annulus. The ring contacts the side face of theouter ring. The tool and load combinations are sufficiently compliant toallow radial motion of the outer ring to occur as balls roll over wavysurfaces or defects in the ball grooves. The voltage signal from the trans-ducer is input to the analysis instrument, which amplifies the signal,

bandpass filters it, and displays the rms velocity values in each band.The three frequency bands are 50-300 Hz, 300-1800 Hz, and 1800-10,000 Hz. Larger bearings are tested at slower rotational speed (700rpm) with correspondingly lower filter bands: 20-120 Hz, 120-700 Hz,md 700-4000 Hz.

The basic testing technique has been successfully used by bearingnanufacturers and customers for many years. Numerous refinementslave been made during this time, and development work continues inhe area of vibration measurement. Such efforts include investigationsf alternative transducer design and system calibration procedures,ifferent methods of applying load to the test bearings, increased appli-9.tionof statistical methods in setting product specifications and ana-'zing test results, and implementation of supplementary methods ofgnal analysis.

Bearing Frequencies

The calculation of the fundamental pass frequencies of rolling contactbearings is used to establish component waviness testing speeds and fil-ter bands that coincide with vibration measurement bands. In addition,knowledge of these frequencies is useful, though not always essential, inmachinery condition monitoring. Derivation of these equations is pre-sented in Chapter 8. Results are given here for the case of a stationaryouter ring and rotating inner ring, as is used in the vibration test systemdescribed before and is most often the case in typical bearing applica-tions. Assuming no skidding of rolling elements, the frequencies of in-


50-300 Hz 300-1800 Hz 1800-10,000HzOuter raceway 4-26 27-156 157-868Inner raceway 2*-16 17-97 97-541Balls 2-5 6-30 31-167* Includingtwo-andthree-pointout-of-roundness.

Similar values are obtained for the range of ball bearings sizes tested inthis manner: for example, deep-groove ball bearings in the 2 and 3 series.Waviness testing procedures are established to correlate with averagewaviness ranges over a wide range of bearing sizes. In addition, therange of waviness orders tested in either vibration or waviness mea-surement corresponds approximately to wavelengths of the size of thesmall axis of the Hertzian contact ellipse in typical applications such aselectric motors [25.9].

Other Factors in Vibration Testing

Defects other than waviness can contribute to bearing vibration or noise.Some of them can be difficult to detect with the conventional three-bandinspection method. Such defects include local defects on raceways orballs, dirt, grease with improper constituents or properties, and cageswith incorrect clearance or geometry. Some of these defect types mayproduce brief disturbances spaced widely apart in time, which, as a con-sequence, have only a small effect on the average measured vibration inthe inspection bands. For example, a single localized defect on the innerraceway or on an individual ball will be remote from the transducer lo-cation during most of the time that it takes for the inner raceway (orcage in the case of a defective ball) to make one revolution. Such defectsimpacting the outer ring will momentarily excite its various natural fre-quencies. The lowest of these natural frequencies is a rigid body mode(individual balls act as springs) and higher natural frequencies beingmodes of outer ring flextural vibration. These modes result from bendingof the outer ring into shapes that have an integral number of lobes, asanalyzed in [25.4]. Resonant vibration of the outer ring amplifies theeffects of these local defects, and the resonant vibration can be used todetect their presence. Therefore vibration measurement is supplementedwith the use of a peak detector instrument whose functions are shownschematically in Fig. 25.20. Although the figure shows peaks of relativelyconstant amplitude, which might be the case for an outer raceway defect,maximum peak values are obtained and evaluated because they can varywith time in the case of ball and inner raceway defects.

DETECTION OF FAILING BEARINGS IN MACHINES

Vibration analysis is one of the most common methods used to evaluatethe condition oj bearings in an operating machine. As previously shown,such measurements may be used for machines with bearings in new con-dition as well as for machines whose bearings are deteriorating and ap-proaching the end of their useful lives. If a machine's vibration responseto known excitation forces has been determined through techniques suchas finite element analysis and modal analysis, then vibration measure-ments during its in-service operation can define the dynamic character-istics of the forces acting on the machine.

Vibration data can also be used to infer forcing characteristics andcondition of machine components, including bearings. General methodsfor evaluating data include one or more of the following:

1. Comparison of data with guidelines developed empirically on sim-ilar types of equipment [25.8, 25.11, 25.12]

2. Comparison of data from similar or identical machines in servicewithin the same factory

3. Trending of data from one machine over time


4. Evaluation of data in an absolute sense with no prior history. Forexample, by evaluating time signals or frequency spectra to asso-ciate vibration with specific machine components

Many machine problems can be traced to faults other than damagedbearings. If a moderately detailed vibration analysis capability is notavailable, however, bearings are often replaced unnecessarily.

The beginning of progressive bearing damage, which can be calledincipient failure, is often characterized by a sizeable local defect on oneof the components. When this occurs, subsequent rolling over of the dam-age will produce repetitive shocks or short-duration impulses. It can besurmised that such impulses might appear, if they could be measured,as those in Fig. 25.21a and b.

Figure 25.21a could represent, for example, the effect of successiverolling elements passing over a damaged area on the outer raceway. Sim-ilarly, Fig. 25.21b might represent the effect of inner raceway damageinteracting with several rolling elements in the load zone of a radiallyloaded bearing without preload. In this case the damage enters the loadzone once per revolution of the shaft. The location of the rolling elementswith respect to the load zone will vary somewhat from one shaft revo-

DETECTION OF FAILING BEARINGS IN MACHINES 999

lution to the next. If a sensor were placed on the bearing housing tomeasure the resulting vibration from the series of impacts, it may showa response as in Fig. 25.21c. This vibration corresponds to lightly dampedoscillation of some system natural frequency greater than the repetitionfrequency of the train of impacts. It could, for example, be a resonanceof the bearing outer ring or of the housing or sensor itself. Such resonantresponse is excited by harmonics that exist in the periodic nonsinusoidalforcing function. Figure 25.22 shows the time history of an electricalsignal representing a pulse train with a fundamental frequency of 160Hz; Fig. 25.23 is the frequency spectrum of that signal. It contains allharmonics of the fundamental.

Impulsive occurrences in bearings, therefore, can cause system vibra-tion at many frequencies that can be harmonically related. Forcing har-monics that are near system resonant frequencies can cause significantlyamplified vibration response compared to the vibration at nonresonantfrequencies. In the early states of failure the impulse might have littleeffect on the amplitude of vibration at the fundamental bearing passfrequencies. In addition, significant normal machine vibration could oc-cur at these lower frequencies, so small changes in vibration amplitudeinitially may be difficult to detect. Higher-order harmonics, with spacingrelated to specific component frequencies, however, might be detectableat higher frequencies if the sensor and mounting method provide suffi-cient response at the higher frequencies. Small accelerometers stud-mounted to electrically isolated nuts and glued to a surface on themachine work satisfactorily. Magnetic mounting is faster but it requiresa better surface and frequency is lower.The followingexample illustratesthe effect of local bearing damage on vibration response.

Example 25.9. A test rig was used to run two 205 ball bearings,each mounted on a pillow block at opposite ends of a shaft. The shaft

DETECTION OF FAILING BEARINGS IN MACHINES 1001

{bpir = Zfci = 9 x 16.97 = 152.7 Hz (25.16)

Figure 25.24 shows spectra of the two bearings on the same plot. Thefrequency span is 0-10,000 Hz and full-scale amplitude is -20 dB.Data were obtained with a stud-mounted accelerometer, 0.010 V/g.

The vibration amplitudes of the damaged bearing are 20 dB greaterthan those of the undamaged bearing at most frequencies above 3000Hz. The spectrum of the damaged bearing also shows peaks, about 10of them in each 1000 segment, whose spacing corresponds to Zfc. Bet-ter resolution of peak spacing would require using a narrower fre-quency span in regions of the spectrum or an instrument with finerresolution. Nevertheless, the figure illustrates the major effect oflocalbearing damage on vibration in the higher-frequency regions.

Figure 25.25, taken on the damaged bearing over 2500 Hz, clearlyshows harmonics of the ball passage frequency over the outer racewayfrom 500 to 1250 Hz. Amplitudes of harmonics from 700 to 1200 Hzwere approximately 10 dB greater than vibration amplitudes of theundamaged bearings in this range. Below 700 Hz, amplitudes of thetwo bearings were the same. Depending on the presence of othersources of machine vibration and the magnitude of bearing damagethat exists, it might also be possible to successfully identify a problemat low frequencies.

Figure 25.26 shows part of a digitized time sample taken from thedamaged bearing, indicating some perturbation at a spacing corre-sponding to lIZfc sec. The previous illustration considers only a singlelocal damage that can be associated with a specific bearing componentwith frequency spectrum analysis. Numerous cases of such failure de-tection are reported in the literature [25.13, 25.14].

Most forms of damage preceding bearing failure will result in pro-gressive wear and roughening of component surfaces and irregularrunning geometry. Such irregularities may produce vibration that canclearly be identified with specific components, or they could producevibration whose amplitude varies randomly in time and frequency con-tent. In either case machine vibration measurement in one form oranother can be used to periodically assess bearing condition.

A particularly damaging form of failure is caused by electrical cur-rent passing through bearings in large motors. Arcing erodes the bear-ing and creates numerous large damaged areas. Sample data showdetection of such damage in a 1200-rpm, 1000-hp vertical pump motorusing axial vibration on the pump motor base. The top bearing is a170-mm (6.69-in.) bore, 40° angular-contact ball bearing. In contrastto an 800-hp pump with rotating imbalance, discussed earlier, vibra-tion spectra indicated harmonics, as see in Fig. 25.27a.

Accurate evaluation of harmonics was performed with cepstrumanalysis [25.15], which determines periodicity within a spectrum, al-

lowing identification to "zoom" analysis of frequency bands within thespectrum. Harmonics with a spacing of 101.7 Hz were identified, andthey are corresponded to the outer raceway ball pass frequency. Thebearing was removed and replaced after less than nine months of op-eration. Data from the pump after rebuild are shown in Fig. 25.27b,and a photograph (Fig. 25.28) illustrates the nature of damage on theouter raceway. Analysis and means of prevention of this type of dam-age are presented in [25.16-25.18].

FAILURE DETECTION-CONDITION MONITORING

Aside from evaluation of vibration spectra to identify specific machinefrequencies, data can be obtained or analyzed by other means to trendthe onset of failure. Mathew and Alfredson [25.19]present a comprehen-sive evaluation of vibration parameters over the life of bearings run toadvanced stages of damage progression or failure. Conditions underwhich bearings were tested include bearing components with initial dam-age, contained lubrication, overload condition leading to cage collapse,and sudden loss of lubrication. Parameters that might be obtained withrelatively low cost instrumentation include peak acceleration, rms accel-eration over a broad frequency band, and shock pulse data. The lastevaluates vibration at a frequency corresponding to the resonant fre-quency of the accelerometer (32 Hz). Other parameters were calculatedby performing arithmetic operations on two frequency spectra, one of

which was usually the initial spectrum obtained when tests were begun.The calculated parameters were then trended. Statistical functions, in-cluding probability density, skewness, and kurtosis were also evaluated.The results indicated that several parameters evaluated from frequencyspectra were successful trend indicators, generally providing a 30-dB in-crease or more by the time a test run was completed. One such parameteris simply obtained by subtracting the initial spectrum from each newspectrum and computing the rms of the resulting spectrum. This valueis then trended over the duration of tests.

Aside from computations of trend parameters from complete vibrationspectra, the shock pulse method was reported to provide successful de-tection for all tests except the case of total lubrication loss. For the suc-cessful tests the shock pulse values are estimated to have increased 40dB. In the test with lubrication loss, seizure occurred in 2 hr. This sug-gests that lower-frequency vibration may be a better initial indicatorthan higher-frequency vibration unless the components have time to un-dergo sufficient gradual distress to be detectable in the high-frequencyregime.

CONDITION·BASED MAINTENANCE

In the foregoing sections, it has been demonstrated that monitoring bear-ing vibrations and comparing the vibration signals against a baseline for


satisfactory bearing operation may be used as a means to detect im-pending bearing failure. According to definitions of bearing failure dis-cussed thus far-for example, initial spalling or pitting of rolling contactsurfaces-the occurrence of abnormal signals may indicate that bearingfailure has already Occurred. On the other hand, the bearing, althoughrunning rough and with increased friction, generally will continue torotate after initial surface damage, permitting effective machinery use.Eventually, the rolling contact surfaces will be completely destroyed andthe machinery will cease to function due to bearing seizure or excessivevibratory loading and component fracture. These last conditions repre-sent potential catastrophe from the minimum standpoint of unscheduledmachinery downtime and excessive costs, or worse, from the standpointof loss of human life in life-critical applications. The latter would include,for example, air transport applications and applications which handlehazardous fluids. From the time at which excessive vibration signal isexperienced to the time at which the machinery no longer functions rep-resents a duration in which action may be taken to prevent catastrophicevents.

Historically, many applications have relied on preventive maintenanceto minimize unscheduled downtime due to bearing failure. Based on cal-culations of bearing endurance, either from fatigue of rolling contact sur-faces or other wear phenomena, or based on past experience of bearingfailures, periodical stoppages of machinery are scheduled, during whichbearings are inspected and replaced. Frequently, inspection does not oc-cur and rolling bearings are simply replaced. The problem with this pro-cedure, in addition to the cost of taking equipment out of service andlosing production and revenue, is that the bearings which had been inoperation were mostly likely not prone to failure; however, they mightbe replaced with bearings which could fail. Once a rolling bearing hasexperienced sustained operation, it has passed the period in which birthdefects cause early failures, and under proper mounting, applied load,speed, and lubrication conditions, it will continue to operate without fail-ure. Thus, presuming proper operation, it is usually best to allow thebearing to run without interruption once an initial operating period hasbeen successfully achieved.

Maintenance is considered the largest controllable cost in modern in-dustry. Based on bearing condition monitoring, which provides opera-tional information on impending failure, and prognostic knowledge of theduration of effective bearing performance, taking failure-prevention ac-tion after the first signals of impending bearing failure have been re-ceived, but prior to occurrence of catastrophic events is a morecost-effective procedure than preventive maintenance; unnecessary ma-chinery downtime is avoided. Of course, condition-monitoring sensorsand techniques must be proven reliable, and life prognostication methodsmust be proven sufficiently accurate. This procedure is called condition-based maintenance (CBM).

CONDITION-BASED MAINTENANCE 1007

CBM is a relatively new concept, and additional information is re-quired for its effective implementation. The initial CBM consideration isthat, more often than not, the actualload-speed-temperature operatingconditions experienced by the bearing are significantly different from thedesign. Bearing types and sizes are selected based on a design duty cycleof the machinery. Therefore, at any instant, prediction of remaining bear-ing life should be based on actual accumulated conditions of operation.In describing the Health Usage and Monitoring System (HUMS) tech-nology under development for helicopter maintenance, Cronkhite [25.20]illustrated the potential for increased endurance of helicopter mechani-cal components; see Fig. 25.29.

Design conditions tend to be conservative to assure reliable operation,and the lower shaded area on Fig. 25.29 indicates that actual life willtend to exceed design life. Load and speed-sensing equipment are readilyavailable to enable the acquisition of such data. Miniature microcom-puters are also available at reasonable cost to enable detailed evaluationof the acquired data. With regard to prediction of bearing life to theinitial spall, the actualload-speed-temperature operation ofthe bearingmust be accommodated in the analysis.

Instead of considering failure at the occurrence ofthe initial spall, theability to detect incipient spalling becomes important. This knowledgewould most likely indicate additional time in which to take failure-prevention action. With the continued development of micro-sized pres-sure, temperature, and ultrasound sensors which can be embedded inclose proximity to, or directly in, the bearing, it appears probable thateffective means to sense incipient fatigue failure will eventually be avail-able. Figure 23.30 shows a stress pin, a miniature pressure sensor, em-bedded in the outer ring of a tapered roller bearing. This sensor does not

(b)

FIGURE 25.30.Stress pins inserted in the cup of a tapered roller bearing (a) locations

to determine axial stress distribution; (b) circumferential locations to determine distribu-tion of load among the rollers-showing wireless connection of analog/digital converter fortransmission of signal (courtesy of Oceana Sensor Technologies, Virginia Beach, Virginia).

impair bearing function, and may be used to determine whether bearingloading conforms to design. If bearing loading is substantially in excessof design, this is an indication of failure occurrence or incipient failure.

Additionally, since bearing operation will continue after the OCcur-rence of the initial spall, algorithms to establish the time available foreffective operation after this event can be developed. Kotzalas [25.21],using a ball/v-ring rig, progressed ball spalls past the initial surfaceflaking until the entire track was eventually destroyed. Figure 25.31shows spall progression. Figure 25.32 illustrates the voltage signal froman accelerometer mounted on the test head. It can be seen that vibrationloading remains rather low for more than 1.5 hours, even under theheavy loading, high speed operation of the test condition. Using a ball-

disk test rig to measure traction coefficient of the failed balls, Kotzalas[25.21] determined that an effective lubricant film was generated evenin the presence of gross spalling, and it is presumed this film was in-strumental in assuring the continued operation of the test ball. The in-crease in vibration after 100 minutes points to the breakdown of thelubricant film, metal-to-metal contact, component temperature rise, andeventual seizure or fracture, depending on heat dissipation paths. Figure25.33 shows the effect of increased load on the time from initial spalling

to component failure. It can be seen that Hertz stress magnitude, andhence load magnitude, has a profound effect on the duration of spallprogression. Moreover, comparing Fig. 25.33a (37.8°C OOO°F»with Fig.25.32 (71.1°C(160°F», it may be determined that lubricant film thicknessand heat dissipation rate have significant effect on the rate of spall pro-gression; the lower the temperature, the longer is the duration of spallprogression to component failure.

Kotzalas [25.21] also correlated the ball traction coefficient with thedegree of spall progression, and degree of spall progression with the ac-celerometer signal. Thus, it was possible to correlate friction with theaccelerometer signal. Therefore, accelerometer signal, so correlated,might be used to indicate friction, which may be used in an on-boardcomputer program to more accurately estimate remaining bearing life.

CLOSUREThis chapter provides an indication of how bearings can affect the vibra-tion of machines, as a result of either inherent design characteristics orimperfections and deviations from ideal running geometry within thebearing. Some examples illustrated that such imperfections and geomet-ric deviations can occur during bearing component manufacture, duringassembly of a bearing into a machine, or from bearing deterioration dur-ing operation. Each can have a pronounced effect on machine vibration,either by altering stiffness properties or by acting as a source of forcesto directly generate vibration.

REFERENCES 1011

It was also shown that detection of vibration frequencies and ampli-tudes is used as a means to determine the health of bearings in machin-ery. Using condition-monitoring, it is possible to detect bearing fatiguefailure in machinery and to determine the location of the failed bearing.Recognizing that bearing function does not cease with the initial rollingcomponent surface spall and using the prognostic methods associatedwith condition-based maintenance techniques, it is possible to estimatehow long the machine may be expected to continue to function reason-ably.

REFERENCES25.1. T. Tallian and O. Gustafsson, "Progress in Rolling Bearing Vibration Research and

Control," ASLE Paper 64C-27 (October 1964).25.2. R. Scanlan, "Noise in Rolling-Element Bearings," ASME Paper 65-WA/MD-6 (No-

vember 1965).25.3. Military Specification Mil-B-17931D (Ships), "Bearings, Ball, Annular, for Quiet Op-

eration" (April 15, 1975).25.4. O. Gustafsson, T. Tallian et a!., "Final Report on the Study of Vibration Character-

istics of Bearings," US. Navy Contract Nobs-78552, US. Navy Index No. NE 071200 (December 6, 1963).

25.5. R. Sayles and S. Poon, "Surface Topography and Rolling Element Vibration," inPrecision Engineering, IPC Business Press (1981).

25.6. L. Johansson, "Bearing Noise in Electric Motors," Ball Bearing J. 200 (1979).25.7. J. Hyer and D. Sileo, "Some Practical Considerations in the Selection and Use of

Ball Bearings in Small Electric Motors," Small Motor Manufacturers Association(March 1985).

25.8. J. Mitchell, Machinery Analysis and Monitoring, PennWell, Tulsa, Chap. 9 and 1091981).

25.9. E. M. Yhland, "Waviness Measurement-An Instrument for Quality Control in Roll-ing Bearing Industry," Proc. Inst. Mech. Eng., 182, Pt. 3K, 438-445 (1967-68).

25.10. O. Gustafsson and U Rimrott, "Measurement of Surface Waviness of Rolling-Element Bearing Parts," SAE Paper 195C (June 1960).

25.11. International Organization for Standardization, "Acoustics, Vibration and Shock,"ISO Standards Handbook 4 (1980).

25.12. S. Norris, "Suggested Guidelines for Forced Vibration in Machine Tools for Use inProtective Maintenance and Analysis Applications," in Vibration Analysis to Im-prove Reliability and Reduce Failure, ASME H00331 (September, 1985).

25.13. J. Taylor, "Identification of Bearing Defects by Spectral Analysis," ASME J. Mech.Design 102 (April 1980).

25.14. J. Taylor, "AnUpdate of Determination ofAntifriction Bearing Condition by SpectralAnalysis," Vibration Institute (April 1981).

25.15. R. Randall, "Cepstrum Analysis and Gearbox Fault Diagnosis," Bruel & Kjaer Ap-plication Note 233-80.

25.16. E. Wallin, "Prevention of Bearing Damage Caused by the Passage of Electric Cur-rent," Ball Bearing J. 153 (1968).


25.17. S. Andreason, "Passage of Electric Current Through Rolling Bearings," Ball BearingJ. 153 (1968).

25.18. A. Boto, "Passage of Electric Current Through a Rolling Contact," Ball Bearing J.153 (1968).

25.19. J. Mathew, and R. Alfredson, "The Condition Monitoring of Rolling Element Bear-ings Using Vibration Analysis," ASME Paper 83-WA/NCA-l (November 1983).

25.20. J. Cronkhite, "Practical Application of Health and Usage Monitoring (HUMS) toHelicopter Rotor, Engine, and Drive Systems", Paper presented at the AmericanHelicopter Society 49th Annual Forum, St. Louis, Mo. (May 19-21,1993).

25.21. M. Kotzalas, "Power Transmission Component Failure and Rolling Contact FatigueProgression" (Ph.D. thesis in Mech. Eng., Pennsylvania State Univ., August 1999).

1014ROTOR DYNAMICS AND CRITICAL SPEEDS

Symbol Description Unitsm Mass kg (lb-sec2Iin.)Pd Bearing internal diametral

clearance mm (in.)Q Ball normal load N (lb)r Radius of gyration mm (in.):;, Stiffness N/mm (lblin.)t Time secx Displacement (lateral) mm (in.)Xo Displacement amplitude mm (in.)Z Number of rolling elements(I' Contact angle (loaded bearing) 0, rad(1'0 Contact angle (unloaded bearing) 0, rad8 Elastic deflection mm (in.){} Angular displacement or

inclination0, rad

Bo Angular displacement amplitude 0, rad¢ Phase angle

0, radA Eigenvalue( Damping factorW Frequency rad/ secWn Natural or critical frequency rad/ secv Frequency of vibration

SUBSCRIPTS1,2 Refers to first or second natural

frequencya Refers to axial directionc Refers to criticali Refers to ith frequency or ith valuelr Refers to inner raceway0 Refers to amplitude valueor Refers to outer racewayr Refers to radial directionx Refers to coordinate directiony Refers to coordinate directionz Refers to coordinate direction

GENERAL

In 1948, DenHartog [26.1] published the first text to address the math-ematical principles of vibrating motion in mechanical systems. Theseprinciples have been built upon and extended over the years to providevital tools to machine designers. With the advent of digital computer

DAMPED FORCED VIBRATIONS 1015

techniques, sophisticated design tools have been created that provide theability to predict critical speeds and rotor behavior in high-speed, shaft-bearing systems. The specific topic of rotor-bearing dynamics rose in so-phistication and importance to such a degree that in 1965 the U.S. AirForce sponsored the construction of a 10-part design guide on the subject[26.2]. The guide was later updated in 1978 [26.3].

The majority of this book deals with the technologies of rolling bearingdesign ranging through materials, performance prediction, lubrication,and fatigue life. This chapter deals with the one characteristic of rollingbearings that directly influences shaft-bearing dynamic behavior-stiffness. The subject of bearing-rotor systems interaction will be ad-dressed in a threefold manner. First, the basics of mechanical vibrationto form a foundation for understanding the analytics involved in rotordynamics are presented. Second, the concept of bearing stiffness and thenature of its behavior in the rotor system environment are considered.Finally, a brief overview of rotor dynamic analysis is given.

DAMPED FORCED VIBRATIONS

The basic principles of flexible rotor dynamics stem from the mathemat-ical representation of damped forced vibrations [26.4]. Figure 26.1ashows the system as a mass with viscous damping being forced by a

COUPLED VIBRATORY MOTION (RIGID SHAFT)

Consider a free-body diagram that can be constructed to represent a rigidbeam on elastic supports [26.5]. It would be represented by a large un-symmetrical body supported by unequal springs. Such a system is shownin Fig. 26.6. The mass of the body is m, and its mass center is locatedat G. To define the motion of the body, a coordinate x will be used tospecify linear position and a coordinate (J to specify angular position.Both will be measured from some reference position or neutral plane.


The combined load-preload effect on bearing radial stiffness warrantsfurther explanation [26.6]. In general, the radial stiffness versus radialload curve for an angular-contact bearing is composed of three distinctbehavior regions. Figure 26.19 illustrates this for a typical angular-contact ball bearing with heavy preload. In the region of light radial loadthe stiffness is nearly constant. The reason is that in this region, thepreload is dominant and the radial load is not sufficient to effect achange. In the middle region or region of medium radial load, there is aslight drop in stiffness to some minimum value. Here the radial load isnearly equal to, and then surpasses, the preload. This causes greaterload sharing among the elements, therefore reducing total stiffness. Thethird region shows the effect of the radial load dominating the system,resulting in a linearly increasing stiffness.

Shaft Bending Effects

The interactive nature of the shaft-bearing system is terms of its me-chanical behavior dictates that shaft deflections influence bearing de-flection and, hence, stiffness. One such shaft deflection is angulardeflection or misalignment, which results from the bearing resistance toapplied moments. Figure 26.20 illustrates a cylindrical roller bearingexperiencing shaft misalignment. It is obvious that shaft bending (mis-alignment) will influence the load distribution. The general trend is thatas shaft stiffness decreases, bending increases, resulting in increasedbearing stiffness.

FIGURE 26.20. Misalignment of bearing rings.

ROTOR DYNAMICS ANALYSIS

Critical Speed

The major objective of rotor dynamics analysis is to allow developmentof rotating machinery that will be free from vibrational problems detri-mental to its performance. This is generally a two-step process consistingof a critical speed analysis and a synchronous response analysis. Criticalspeed analysis is the process of determining the natural frequencies of arotor-bearing system and identifying the mode shapes associated withthem. This must be done to insure that the machine operating speed islocated at a frequency safely spaced from any undamped natural fre-quencies. If allowed to operate at or near a natural frequency, a machinemay enter an unstable and destructive vibration situation.

Synchronous Response

Synchronous response analysis allows the further examination of therotor-bearing system behavior as a function of operating speed. It relatesrotor displacement and bearing loads to operating frequency. Both of theforegoing analytical techniques are accomplished by computer programsconstructed using the mathematical principles reviewed in this chapter.In general, the programs mathematically model the shaft-bearing systemas a series of rigidly connected, multi span beams supported on springs.Division of the rotor into segments allows for the proper modeling ofvariations, such as cross-sectional thickness, material, location of gearsor discs, and location of bearing supports.


Figure 26.21 is an example of a shaft-bearing system model. Sectionmass can be either distributed across the element or lumped at the endsof the element. The problem is then solved by a transfer matrix tech-nique. The transfer matrix is derived directly from the differential equa-tion describing the dynamic behavior of each beam segment.

Mode Shapes

Associated with each identified natural frequency (critical speed) is anormalized mode shape. An estimation of the relative displacement ofeach shaft station is computed. Figure 26.22 illustrates two commonbending modes associated with systems involving stiff bearings. These

ROTOR DYNAMICS ANALYSIS 1041

are the shapes with which the system would vibrate if excited at thecorresponding critical speed. Rotor-bearing systems can be excited intovibration modes by any number of periodic forces. The most commoncause of periodic forces in rotating machinery is mass unbalance. Massunbalance may be a result of inadequate balancing techniques, shaftbending due to gravity, debris deposits, or unstable rotor structures.

Analyzing the effects of unbalance forces on shaft vibration and bear-ing loads is done by synchronous response analysis. By solving the dif-ferential equation of motion for each shaft segment, with the harmonicdriving forces being represented by the unbalance mass, vibration am-plitude can be computed. Then by sweeping through a range of frequen-cies (operating speeds) synchronous response plots for each shaft stationcan be constructed. Figure 26.23 illustrates the typical results of a syn-chronous response. They identify the location of the critical operatingspeeds and allow the determination of safe operating frequencies basedon severity of vibration amplitude (Fig. 26.23a) and bearing load (Fig.26.22b ).

FAILURE MECHANISMS 1045

faces are marred, stress conditions can be imposed that introduce a po-tential to reduce bearing life significantly. Abusive handling can inducenicks and dents that are harmful, particularly when located in regionstracked by rolling elements. Displaced metal particles generated by nicksand scuffing-type damage introduce secondary effects when they dislodgeand indent the raceway.

Permanent indentation created by rolling element overload is calledbrinelling, a type of damage destined to result in failure. Brinelling mayoccur by an overload mechanism, such as dropping the bearing or im-proper mounting techniques. Initial signs of brinelling are signaled bynoisy bearing operation.

Raceways may be damaged when they are subjected to vibratory mo-tion while rolling elements are not rotating. This type of damage, calledfalse brinelling, can occur before and after mounting on equipment. Falsebrinelling damage has been observed in bearings that were subjected tovibration during transit as well as on equipment that lay idle for a periodof time.

Wear Damage

Wear generally results in gradual deterioration of bearing components,which in turn leads to loss of dimensions and other associated problems.Failure by wear does not mean that bearings will be removed solely be-cause of change in fit or clearances. Secondary conditions arising fromwear can become the predominant failure mechanism. Lubricants maybe affected or become contaminated to the degree where lubrication isseverely diminished. Stress raisers could be generated that may serve assites for crack initiation.

Adhesive wear is described in Chapter 24 and is involved in removalof material and possible transfer to mating components. Under properlylubricated conditions, mating components' microscopic asperities couldyield and be flattened by cold work. Under these conditions the bearingmight function adequately for its projected life. When lubricating con-ditions become inadequate, however, increased friction results in metal-to-metal contact, giving rise to localized deformation and frictionwelding. Operating forces cause increased plastic deformation by tearingthe locally friction-welded regions from the matrix. One component isnow pitted, and the other contains the transferred metal. This conditioncould be progressive, depending on operating conditions. Generally,lighter adhesive damage is described as scuffing and scoring, whereasmore gross damage is described as seizing and galling.

Figure 27.2 shows a cylindrical roller bearing raceway which has worndue to sliding motions between rollers and raceways. Figure 27.3 showssmearing damage leading to increased friction, high metal temperatures,material softening, and plastic movement of the raceway metal.

FAILURE MECHANISMS 1047

ponents [27.1]. Improperly lubricated bearings produce varying condi-tions, which lead to progressive contact surface deterioration andreduced life.

Initial stages of wear involve the plastic deformation of grinding fur-row asperities, which, in subsequent cold-working, fracture to produceextremely fine platelets of steel. This stage may also be accompanied byadhesion of microscopic asperities that delaminate and pass on into thelubricant. These cold-worked, hard particles, which also contain carbides,serve as abrasive media. After a time, original grinding furrows in therolling element tracks are worn smooth to produce a glazed condition.Continued operation will generally lead to a deterioration stage thatmanifests itself as a frosted condition and, sequentially, to scuffing. Mi-croscopic pits and crevices created by this adhesion mechanism serve asstress raisers for the initiation of microspalls. Figure 27.4 illustrates thefrosted surface condition and micropitting.

In a lubrication system where the quantity of lubricant to vital areasis too low, bearing component temperatures increase. This in turn in-creases lubricant bulk temperature, decreasing viscosity and effectingincreased friction that makes the situation progressively worse. Bearingsurface degradation will be accelerated, and the surfaces will discolor asthe process progresses. Figure 27.5 shows a cylindrical roller bearing,which has undergone gross sliding in the absence of EHL films suffi-ciently thick to adequately separate the rollers and inner raceway in thecontacts. Figure 27.6 shows a needle roller bearing which has experi-enced roller skewing, thermal excursion with attendant overheating anddiscoloration, cage fracture, and seizure.

Crack Damage

Cracking of bearing components may originate as a function ofoperatingstress conditions via overload or cyclic loading (fatigue). Additionally,manufacturing-related cracks may derive from the steelmaking processand/or working processes. With the exception of cracks arising from in-clusions and hydrogen in steels, cracks associated with steelmaking andprimary working processes seldom survive through secondary workingprocesses.

Cracks that arise from bearing manufacture secondary working pro-cesses frequently are associated with heat treatment and grinding. Thenature of the processes are such that sequential operations tend to pro-mote rapid crack propagation in bearing steels, if any are present duringmanufacture. An obvious exception is cracking that began at the finalgrinding stages. In view of the foregoing, cracking problems encounteredare usually operationally related as compared to manufacturing-related,occurring via cyclic loading. Stress states may be complex, arising fromcombined effects of component residual stress state, static stresses im-posed by mounting, and stresses superimposed by applied loads. Some

1054 INVESTIGATION AND ANALYSIS OF BEARING FAILURES

indicative of the roller. An example of this condition is shown in Fig.27.12.

False Brinell Marks

False brinelling is a condition that resembles true brinelling, but it isgenerated by a different mechanism. True brinelling is generated byplastic deformation of the steel, whereas false brinelling is generated bya corrosion-wear mechanism. Nonrotating bearings subjected to vibra-tion wear by fretting corrosion between rolling elements and raceways.Corrosion products accumulate, which then proceed to accelerate wearby abrasion. Surface irregularities created by wear serve as initiationsites for spalls during subsequent operation. The EHL film may also beaffected locally, effecting marginal lubricating conditions. Figure 27.13depicts false brinelling.

Score Marks

Careless handling is the primary cause for scores and digs that marbearing surfaces. Depressions of this type plastically deform the steel,usually displacing metal that is subsequently cold-worked by the rolling

FIGURE 27.13. False brinelling of a tapered roller bearing cup raceway; the insert showsthe corrosion products in the "brinell" mark (courtesy of the Timken Company).

elements. Eventually, the cold-worked slivers fracture, producing frag-ments in the system that may be coined in the roller path; crowns aregenerated around the indentations, which are subject to the effects dis-cussed under brinelling.

Notched regions created by the scores function as stress raisers. Thesharpness of the notches and their locations with respect to operatingforces are important factors affecting the propensity for crack initiation.Stress intensity is greater for sharply notched surface discontinuities.

Adhesion Damage

Adhesion damage manifests itself as a buildup of metal on a component,resulting from metal transfer from the interacting component. In turn,this interacting component, if examined during early stages of progress,would contain corresponding pitted regions where metal had been pulled.Figure 27.14 displays the characteristic features associated with metaltransfer.

Lubricant Contamination

Particle Dents. Particulate matter suspended in the lubricant results indeformation of rolling elements and raceways by indentation. The natureof the indentations corresponds to the hardness of the particles. Particledents initiating from relatively soft materials have shallow, smooth fea-tures, whereas dents from hard materials display depths and configu-rations conforming to the particles. Figure 27.15 shows a ball bearinginner raceway which has been severely dented by hard particles. Dentsare accompanied with the crown features discussed under brinelling, thecrown heights corresponding to the volume of metal displaced. The crown

EXAMINATION AND EVALUATION OF SPECIFIC CONDITIONS 1061

Electric arcing also causes metallurgical properties to be altered tosignificant depths, since it effects intense, highly localized temperatureswhich melt the surface. Heat is rapidly dissipated as it is conducted intothe mass. Temperature transitions, as evidenced metallurgically, can ex-hibit remelting, austenitization, and retempering. Surface and near-surface regions are markedly affected. Initial surface zones may consistof brittle untempered martensite followed by high temperature, retem-pered zones. As temperatures dissipate with depth, the effects are lesspronounced. When arc events are random, the effects are more easilyidentified in the microstructure. A single arc generates a hemisphericallyshaped, affected region consisting of the zones discussed above. Succeed-ing arc events superimpose heat-treatment effects on the previously af-fected region. An example of a surface and microstructure in a flutedregion is shown by Fig. 27.22.

Arcing also alters the previously existing stress field. Rehardenedzones exhibit high residual compressive stresses, whereas adjacent re-tempered zones counteract these stresses with residual tensile stresses.Under cyclic loading this tensile stress zone is vulnerable to fatigue in-itiation and propagation.

Spalling

Spalling, sometimes called flaking, is a stable, stress-related crack mech-anism caused by subcritical cyclic loading. Spalls may surface-initiatedor subsurface-initiated. The processes are called fatigue. Surface-initiating spalling usually occurs by progressive deterioration of rollingcontact surfaces when EHL films are insufficiently thick to adequatelyseparate the surfaces. Several of the previously described mechanismsalso exhibit conditions that could potentially culminate in surface-initiated spalling. When inadequately lubricated, rolling contact surfacesbecome glazed (predominantly in the presence of gross sliding), and thenbecome microscopically pitted. SEM examination reveals these pits to bemicroscopic spalls. Continued operation results in propagation ofthe mi-crospalls as well as continued initiation of new spall sites until the sur-face appears frosted. Figure 27.23 shows a typical surface-initiated spall.

Subsurface-initiated spalling is associated with stress concentrationsusually within the depth of the maximum shear stress for properly lu-bricated rolling element-raceway contacts, over which Hertz stresses arepredominant [27.4].Constituents in the material matrix, particularlyox-ide-type inclusions, intensify the stress locally.Regions surrounding theinclusion are strained, eventually initiating microscopic cracks. Thesemicroscopic regions often manifest themselves as white etched areas,commonly called butterflies [27.6].This was illustrated by Fig. 22.6. The

FRACTOGRAPHY

Scanning Electron Microscopy

The foregoing discussions involved macroscopic features identified withthe various failure mechanisms. Macroscopic features, however, do notalways represent the failure-initiating mechanism. It is important thatthe failure mode be identified at the origin area, preferably the initiationsite itself. This is not always possible with conventional optical micro-scope techniques due to the inherent loss of focusing ability at highermagnifications. A SEM provides greater depth of focus and the ability tostudy irregular surfaces. Identifying modes of initiation at crack originswill often provide data leading to the cause of failure, despite damageincurred in other regions. These data will also provide an insight to thetypes and relative magnitudes of the applied loads. Four basic modes offracture will be presented with regard to how they appear in bearingsteels. Rolling bearings may be manufactured from various grades ofsteel, but only those that are termed "bearing steels" are considered here.These steels consist of high-carbon and low-alloy compositions.

Microvoid Coalescence

Under abnormal loading conditions in bearings, microvoid coalescence,commonly called dimples, represent the ductile mode of cracking due to

FRACTOGRAPHY 1065

of failure are sometimes observed in bearing flange fracture when thrustloads are excessive.

Cleavage

Cleavage is a rapid overloading mode of failure resulting in bearing frac-tures. A fracture may initiate by cleavage, or it may initiate by a differentmode and propagate by cleavage. In other words, a crack may initiateand propagate by one or more modes until it reaches a critical size. Atthis point the crack will expand rapidly; that is, fracture occurs. Thisunstable crack propagation occurrence is related to the steel's fracturetoughness property, which is influenced by composition, microstructure,temperature, and loading rate. If a bearing steel is impacted with suffi-cient force, cleavage will be observed across the section thickness, in-cluding the initiation site.

Cleavage is a low energy fracture that propagates transgranularlyalong specific crystallographic planes. It appears as flat planes thatchange orientation from grain to grain. Fan-shaped features are evidenton the facets. These features arise from second-order planes, giving theappearance of steps, and are called river patterns. These patterns aretypically forked, indicating the direction of crack propagation toward theconverging feature within a grain.

Matrix carbides interfere with the normal cleavage progression, andbearing steels contain numerous carbides that are precipitated withinthe prior austenite grains during the temper treatment. Therefore, bear-ing steels do not exhibit the normal cleavage features. The crack appearsto propagate around carbides and develop smaller cleavage facets withingrains. This condition is called quasicleavage, which is displayed in Fig.27.26. Hence, cleavage is more difficult to identify in bearing steels.Quasicleavage is indicative of unstable crack propagation in bearingsteels occurring due to sudden overload.

Intergranular Fracture

Intergranular fracture is a low energy mode of cracking that starts atgrain boundaries. This condition is an embrittlement mechanism, whichreduces more tightly bonded grain boundary energy areas. Embrittle-ment of bearing steels has been caused by improper heat treatment,whereby a brittle phase is precipitated at the grain boundaries. This hasbeen shown to occur by the precipitation of phosphorous at the prioraustenite grain boundaries. Quench cracks exhibit this behavior.

Hydrogen gas can also embrittle the steel and cause cracks to progressintergranularly in the affected region. Hydrogen may be dissolved duringthe melting process and diffuse to form gas pockets during solidification.

1068INVESTIGATION AND ANALYSIS OF BEARING FAILURES

analysis of fatigue failures in bearing steels does not appear to be fea-sible at this time.

Mixed Modes

The various mechanisms of crack propagation discussed occur in com-binations exhibiting two or more of the identifying features simultane-ously. Often these occurrences involve modes associated with materialproperties and microstructure. An example of this condition observed inbearing steels involves mixtures of quasicleavage and intergranularmodes, both of which are low-energy mechanisms. Stress conditions fa-voring both mechanisms are apparently equal.

Strain rate affects fracture mechanisms. Mixtures of fracture modesare observed in transition regions between stable crack propagation andunstable crack propagation. Bearing components that fail by fatigueeventually attain a critical crack size and fracture by quasicleavage.Transition zones between the fatigue and quasicleavage zones sometimesdisplay dimples intermingled in the cleavage facets. These transitionzones are relatively narrow.

Evaluation of failure-containing mixed modes of cracking in the originarea depends on which mode is dominant.

CLOSUREAn overview of the more common conditions and damage leading to bear-ing failures has been presented. Details regarding mechanisms shouldbe referred to in the appropriate chapters. It should not be construedthat the examples cited here are all-inclusive; for example, cage problemsand wear patterns have not been addressed. Considerable variation maybe observed within the examples used. Illustrations and photographs arepresented to depict representative features.

Laboratory work, such as metallography, stress determinations, phaseidentification, microprobe analysis, and so on, should be conducted toverify and support visual observations.

REFERENCES27.1. T. Tallian, "Rolling Contact Failure Control Through Lubrication," Proe. Inst. Meeh.

Eng. 182, 205-236 (1967-68).

27.2. J. Mohn, H. Hodgen, H. Munson, and W. Poole, "Improvement of the Corrosion Re-sistance of Turbine Engine Bearings." AFWAL-TR-84-2014 (1984).

27.3. C. Rowe and L. Armstrong, "Lubricant Effects on Rolling-Contact Fatigue," ASLETrans. 23, 23-39 (January 1982).

REFERENCES 1069

27.4. G. Lundberg and A. Palmgren, "Dynamic Capacity of Roller Bearings," Aeta Polyteeh.Meeh. Eng. Ser. 2, RSAEE, No.4, 96 (1952).

27.5. J. Martin, S. Borgese, and A. Eberhardt, "Microstructural Alterations of Rolling Bear-ing Steel Undergoing Cyclic Stressing," ASME Paper 65-WA/CF-4 (1965).

27.6. R. Osterlund, O. Vingsbo, L. Vincent, and P. Guiraldeng, "Butterflies in Fatigued BallBearings-Formation Mechanisms and Structures," Scand. J. Metall. 11 (1982).

27.7. S. Way, "Pitting Due to Rolling Contact," J. Appl. Meeh. 2, A49-A58 (1935).

ABEC, 85Axial loading of rollers:tolerance classes, 99

applied roller thrust loading, 177ABMA, 14, 45, 84cylindrical roller bearing flanges, 178Adhesive wear, 943, 947, 1046 skewing, 179AFBMA, 84

Axial preloading, 368Agricultural applications, 9Aircraft gas turbine application, 4, 25, 43,

348,523,621Back-to-back bearing arrangement, 21, 368AISI, 580Back-up roll bearing, 293AISI 8620:

load distribution in, 302microstructure, 838 Bainite, 607AISI 52100 steel, 4, 581, 598, 603 Ball bearings, 11fatigue:

axial preloading, 368dynamic capacity constant: basic dynamic capacity:line contact, 706 radial bearings, 741

point contact, 705 thrust bearings, 741endurance calculation exponents, 696 clearance effect on fatigue life, 867life, 739, 894 Conrad assembly, 12, 13Weibull slope, 696 assembly angle, 12

hardness, 836contact angle under combined radial andretained austenite, 836 thrust loading, 259toughness, 614 coulomb friction (operating with), 506ultimate strength, 614 curvature, 62

AISI 440C steel, 4, 583, 613, 620 da Vinci, 3Aluminum, 5 dimension series, 17Angular-contact ball bearings, 19 double-row, 15, 18

angle, 20 filling-slot, 15, 18automotive wheel application, 41 free angle of misalignment, 58back-to-back arrangement, 21, 369 free contact angle, 55ball friction forces due to gyroscopic free end play, 55

moment, 176 friction in ball-raceway contacts, 496double-row, 19, 22 friction forces and moments (in), 519

diametral play, 55 friction torque, 540duplex set, 369 applied load (due to), 540

face-to-face arrangement, 21, 369 viscous drag (due to), 542groove curvature radii, 12groove curvature radii, 19high speed, 339limiting thrust load, 379instrument bearings, 15self-aligning, 22internal load distribution effect on fatiguesingle-row, 19

life, 864split inner ring, 22internal load patterns, 1051tandem arrangement, 21loci of groove curvature centers, 266triplex set, 375radial, 11Annealing of steel, 598, 613seals and shields, 17ANSI, 14, 45, 84shielded bearing, 16load rating standards, 741, 825single-row, deep-groove, 11Antifriction bearings, 533static load ratings, 825Asperity-asperity Coulomb friction, 478stiffness, 1028ASTM, 664surface treatment for components, 638Asymmetrical roller loading, 159skidding, 518Automotive wheel hub bearings, 4, 41

Ball bushing, 40Axial deflection, 245Ball excursions, 346ball bearings under thrust load, 247Ball loading, 342duplex set of ball bearings, 371

friction forces, 518high speed angular-contact ball bearing,gyroscopic moment, 174350induced, 158Jones' constant, 246, 381normal to raceway, 343

'074

INDEX 1075

static, 157 multiple bearing-shaft systems, 410stress cycles per ball revolution, 865 permissible static load, 831

Balls: shaft supported by three bearings:dimensional audit parameters, 778 non-rigid shaft system, 404endurance testing: rigid shaft system, 400

NASA five-ball endurance tester, 782 shaft supported by two bearings:Pratt & Whitney v-ring/ball endurance statically determinate system, 135

tester, 783 statically indeterminate system:fatigue failure, 710 flexible shaft, 392hollow, 348 rigid shaft, 389sapphire, 621 X and Y factors, 390silicon nitride, 348, 623 Bearing noise, 964speeds, 520 Bearings with integral sensors, 43traction test rig, 790 Bearing vibrations, 964viscous drag on, 493 Belt loads, 138

Ball speed components, 318, 520 Bevel gear:Ball surface velocities, 319 loading, 144Barus equation, 424 speeds, 150Basic electric furnace processing of steel, 584 spiral bevel gears, 145Basic dynamic capacity: BG42 steel, 581, 613, 620

line contact radial bearings, 732 Biodegradable lubricants, 671line contact thrust bearings, 735 Bore, 49point contact raceways, 714 Boussinesq, 189point contact radial bearings, 715 Brass for cages, 5point contact thrust bearings, 720 tensile strength, 625radial ball bearings, 741 Brinelling, 1052radial roller bearings, 741 Bronze for cages, 626radial roller bearings with point and line

contact, 738thrust ball bearings, 742 Cage, 4thrust roller bearings, 750 ball riding, 493thrust roller bearings with point and line bronze, 626

contact, 739 deep-groove ball bearing, 12Basic static load ratings, 825 forces, 515, 529

permissible static load, 831 land riding, 493shakedown, 852 friction torque, 532

Bearing deflections, 6 low carbon steel, 625combined radial, thrust, and moment materials, 5, 625

loading in ball bearing, 267 motions and forces, 529, 533radial, 235 polymeric materials, 626rigid ring bearings, 365 skewing control, 179stiffness, 1029 sliding friction, 493

contact angle effects, 1037 speed, 309, 526preload effects, 1037 Cam-follower application, 44shaft bending effects, 1038 Cantilever beam support, 142speed effects, 1034 Carbides in steel, 212

Bearing disassembly, 1044 orientation after overrolling, 213Bearing failure investigation, 1043 Carburizing steel, 580, 598, 608Bearing frequencies, 993 fatigue life of bearings, 894Bearing heat generation: fracture (effect on), 855

roller skewing effect, 177 residual stress, 616Bearing loading: toughness, 615

cantilever beam support, 142 Case-hardening:classification, 85 depth, 215concentrated radial loading, 135 fatigue life (effect on), 740, 894concentrated radial and moment loading, residual stress, 616

143 steel, 580friction torque (due to), 540 Castigliano's theorem, 295internal due to rotation about eccentric Centric thrust load, 245

axis, 172 Centrifugal force, 165internal used to determine failure cause, ball, 166, 343

1050 roller, 169, 355load classification, 85 rotation about eccentric axis, 172

1076INDEX

Centrifugal force (Continued)Contaminants, 11spherical roller loading, 171 Contamination:tapered roller loading, 169

cleanliness (iso 4406) classes, 915Ceramic rolling elements, 4, 348, 621 dents, 1055Chain drive loads, 138discoloration, 1058Chemical vapor deposition (CVD), 640fatigue life (effect on), 896Circular crown profile of roller, 224 hydrolysis, 1057Circulating oil lubrication, 649INSA life test system, 789Clean steel, 589, 593life factor, 899macroinclusions, 593life testing considerations, 769nonmetallic inclusions, 593stress concentration factor for contact, 920Clearance, 49, 73

Conversion factors (Metric/English unitseffect of interference, 86, 124system), 1071effect on contact angle, 54, 245

Corrosive wear, 946, 1048surface finish effect, 126false brinelling, 1054tables, 51

Coulomb friction:Coatings, 638asperity-asperity sliding, 478black oxide, 639ball bearings, 506phosphate, 638

Crack propagation, 855Coefficient of friction, 217, 329, 496Crank mechanism loading, 141solid lubricants, 670Cronidur 30 steel, 581Combined radial and thrust load, 10 Crown drop, 273deep-groove ball bearing, 12 Crowned rollers, 26, 29double-row bearings, 262 crown drop, 273single-row bearings, 256 geometry, 275

Combining fatigue life factors, 903 insufficient crowning effect, 1052Composite shear stress on contact surface, logarithmic profile, 224475 Cup, 29, 77Condition-based maintenance, 1005 Curvature, 60Cone, 28, 77 difference, 61, 71, 79Conrad assembly, 12, 13 sum, 61, 71, 79Consumable electrode vacuum melting of total, 54

steel, 583 Cylindrical roller bearings, 25Contact: axial loading through flanges, 178asperity and fluid-supported load, 472 axially floating, 26deformation (elastic), 195, 234, 340 basic dynamic capacity:

roller-raceway skewing effect, 286 radial bearings, 742dynamic capacity, 699 clearance, 73elastohydrodynamic lubrication friction, 476 combined radial, thrust and momentellipse, 193 loading, 289fatigue life, 699 curvature difference, 77flange-roller end, 330 curvature sum, 77

deflection:heat transfer (in), 574combined radial and thrust loading, 285lubricated, structural elements of, 936radial loading and misalignment, 277near-surface region, 938

double-row, 26permanent deformation:endplay, 73, 76line contact 824friction torque due to roller end-flangepoint contact, 821

contacts, 543stresses, 185high speed, 349concentration factors due to contaminantload classification, 85denting, 920load distribution:due to crowning, 225

high speed, 354maximum compressive, 195radial loading and misalignment, 272surface shear stresses, 476, 523radial and thrust loading, 280concentration factors due to contaminant

misalignment, 272denting, 920failure, 773Contact angle, 53, 66fatigue life (effect on), 874centrifugal force effect on ball, 166

multi-row, 31combine radial and thrust loading of ballpitch diameter, 73bearings, 260roller skewing control, 330high speed angular-contact ball bearing,surface treatment for components, 638350thrust flanges, 29thrust load effect, 245

Cylindrical roller thrust bearings, 38

INDEX 107

Damage Atlas, 771 Micro-EHL, 937Damped forced vibrations, 1015 pressure and stress distribution, 431Dark etching region of overrolled AISI 52100 viscosity variation with pressure, 424

steel, 839, 841 Elastomeric seal materials, 632Decarburization of steel, 596 lip seals, 675Deep-groove ball bearing, 11 Electric motor applications, 11

spherical outside surface, 14 Element test rigs, 779Deflections: Elliptical area of contact, 62

radial ball bearings, 366 Elliptical eccentricity ratio (ellipticity), 194roller bearings, 277, 366 Elliptic integrals, 193self-aligning ball bearings, 366 Endplay, 53, 66, 73, 78thrust bearings, 367 Endurance testing:

Deformation: bearing test rigs, 778bands in subsurface, 212 element test rigs, 779brinelling, 1052 INSA contamination-life test system, 789contact, 234 sudden death testing, 779rolling, 487 Weibull distribution analysis, 807skewing effect in roller-raceway contact, test samples, 772

286 theoretical basis, 764surface, 189 English units system equation constants,

Delamination, 943 1072Dents, 620 Environmentally acceptable lubricants, 671

brinelling, 1052 Epicyclic power transmission, 144, 151contamination (due to), 897 Equivalent axial load:lubrication in the vicinity of, 956 line contact bearings, 735

Diametral play, 55, 66 point contact bearings, 725Differential expansion, 124 static, 830Dimensional audit parameters, 778 Equivalent cylinder radius, 423Dimensional instability of components, 856 Equivalent radial load:Dimension series, 17 line contact bearings, 733DIN, 84 point contact bearings, 718Disassembly of bearings, 1044 static, 830Distortion energy theory of failure, 210 Equivalent radii, 194Distributed load systems, 153 Ester lubricants, 661dN, 621 Evolution of rolling bearings, 1

life testing considerations, 767 Externally aligning bearings:VIMVAR M50 NiL effect, 895 radial ball, 14

Double-row ball bearings, 15 cylindrical roller thrust, 38Dowson,1,428,434,453 Extreme environment coatings forDry-film lubrication, 418 components, 640

silicon nitride components, 624 Extreme pressure (EP) additives inDuplex ball bearings, 370 lubricants:

back-to-back arrangement, 368 constituents, 656face-to-face arrangement, 368 effect on seal materials, 637

Dynamic loading:crank-reciprocating load, 141eccentric rotor, 140 Face-to-face bearing arrangement, 21, 368rolling elements, 161 Failure:

electric arc damage, 1060Earthmoving applications, 8 fatigue, 10, 686Eccentric rotor, dynamic loading, 140 interacting modes on surface, 953Eddy current testing of steel components, 595 investigation, 1043Edge loading, 26, 219 internal load distribution, 1050

fatigue life effect, 872, 1052 rolling element tracking, 1049life testing considerations, 766 mechanisms, 1044

Elastic hysteresis in rolling, 486 corrosion, 1048Elastohydrodynamic lubrication: cracking, 1047

cage friction, 494 lubricant deficiency, 1046contact deformation, 427 mechanical damage, 1044friction, 476 wear, 1045fluid entrainment velocity, 432, 523 Failure probability, 689isothermal, 424 fracture, 831lubricant film thickness, 433 False brinelling, 1054

1078INDEX

Fatigue failure, 10, 686 material-life factor, 895balls, 710material processing effect on, 894cracking, 688maximum orthogonal shear stress, 694fracture toughness (effect on median (L50) life, 688

propagation), 855 minimum life, 10delamination, 949 oscillating bearings, 879life distribution, 10 point contact, 701modes, 618 radial bearings, 707pitting, 949thrust bearings, 720propagation:

point contact bearings, 717detection, 1008 prestressing (effect on), 850life testing considerations, 771 radial roller bearings with point and linesubsurface cracks, 855 contact, 735traction coefficient of failed surface, 1008 rating life, 696, 764roller bearing misalignment (due to), 773 reliability (effect on life calculation), 886spall, 688, 772, 773 life factor, 890, 910

stress cycles per revolution, 695, 704 residual stress (effect on), 850stressed volume of material, 694 rotation factor, 720subsurface, 619sinusoidal loading (effect on), 878wear (classified as), 948steel composition and processing (effect on),Fatigue-initiating stress, 911 739Fatigue life: stress-life factor, 910AISI 52100 steel, 595 testing, 764

basic dynamic capacity:ball-rod rolling contact fatigue tester, 787point contact raceways, 714 design considerations, 777point contact radial bearings, 715 elements, 779

point contact thrust bearings, 720General Electric Polymet rolling contactradial ball bearings, 742

endurance tester, 786radial roller bearings with point and line INSA rolling-sliding disc endurancecontact, 738 tester, 788thrust ball bearings, 742NASA five-ball endurance tester, 782thrust roller bearings, 750Pratt & Whitney v-ring/ball endurancethrust roller bearings with point and line tester, 783contact, 739 sample size selection, 810combining life factors, 903SKF A-frame automotive wheel hubcontamination effect on, 896

bearing endurance tester, 779denting, 897SKF R2 endurance tester, 781life factor, 899 sudden death, 779dispersion, 688tapered roller bearing endurance tester,double-row bearings, 691 782element testing, 779 variable loading (effect on), 874equivalent axial load: water effect on fatigue life, 902line contact bearings, 735 Weibull distribution, 692point contact bearings, 725 application, 800equivalent radial load: estimation in data sets, 811line contact bearings, 733 graphical representation of two-point contact bearings, 717

parameter distribution, 798failure probability, 689 slope, 695fatigue-initiating stress, 911 sudden death test analysis, 807hardness (effect on), 717 Fatigue limit stress, 904high speed bearings, 868 temperature effect on, 926hoop (ring) stresses (effect on), 921 Fatigue strength of steel, 614integral, 922 Ferrofluidic seals, 681internal load distribution (effect on), 864 Fiberglass, 5ISO Standard, 926 Filling-slot ball bearings, 15limit stress, 904 Filtration of lubricant, 649temperature effect on, 926 fatigue life effect, 898values for steels, 927

Finite element method of analysis, 221, 302line contact, 704 Fit:radial bearings, 728 classification, 86load (effect of), 702 line-to-line, 78

LlO life, 688 standards for practice, 84lubrication effect on, 890 tolerance classes, 85material effect on, 894 Five-ball endurance test rig, 782

INDEX 1079Five degrees of freedom in loading, 357 planetary gears, 137, 292Flaking, 1061 spur gears, 137Flame-hardening steel, 611 Gear train speeds, 151Flange: General Electric Polymet rolling contact test

bearing friction torque due to roller end rig, 782, 786contacts (with), 543 Generatrix of motion, 313, 316

layback angle, 228 Graphite, 648roller end contact stress, 225 Grease lubricants, 662roller end geometry, 330 properties, 664sliding at roller ends, 494 thickeners, 666

Flexibly supported bearings, 291 compatibility, 668fatigue endurance, 868 Grease lubrication, 652

Flinger, 674 advantages, 664Fluid entrainment velocities, 432, 523 lubricant film thickness, 451Fluorinated ether lubricants, 662 relubrication, 652Fluting, 1060 Groove curvature radii, 12Fractography, 1063 effect on contact angle, 53

cleavage, 1065 instrument ball bearings, 16fatigue, 1066 Grubin, 433intergranular fracture, 1065 Gyroscope bearings, 378microvoid coalescence, 1063 Gyroscopic moment, 174, 344mixed modes, 1068 Gyroscopic motion, 324, 329scanning electron microscopy, 1063

Fracture of bearing components, 831, 853Hardenability of steel, 603Fracture toughness (effect on crack

propagation), 855 Hardness:Free angle of misalignment, 58 fatigue life (effect on), 717, 739Free contact angle, 55 Rockwell C, 4Free endplay, 55 testing methods, 615Frequencies in bearing operation, 993 Harmonic mean radius, 487Fretting, 1046, 1050 Hazard, 802Friction: Health Usage and Monitoring System

ball-disc traction test rig, 788 (HUMS), 1007coulomb, 478 Heat:

elastic hysteresis, 486 conduction, 556elastohydrodynamic lubrication, in, 476 convection, 558forces in ball-raceway contacts, 496 ball (from rotating), 560forces and moments in roller-raceway roller (from rotating), 560

contacts, 510 dissipation, 569gyroscopic motion (resistance to), 342 flow analysis, 561heat generation, 553 generation, 553Heathcote slip, 490 radiation, 560

limiting shear stress in lubricant, 477 removal methods:microslip (due to), 489 air cooling of housing, 570moments in ball-raceway contacts, 497 cooling of lubricant, 571seal, 494 under raceway cooling, 574shear stresses in ball-raceway contacts, transfer:

519 modes, 556silicon nitride components, 624 rolling element-raceway contact (in), 574torque, 6 temperature nodes, 561

applied load (due to), 540 Heathcote slip, 490total on bearing, 544 Heat treatment of steel, 597viscous drag (due to), 542 Helical gear loading, 143

traction coefficient, 477 Helicopter applications, 9, 43spalled surface, 1008 Hertz, 185

viscous drag (due to), 492 High speed ball bearings, 339Friction wheel drive loads, 139 ball excursions, 346Full complement bearings, 27 fatigue life, 868Fully crowned roller, 29, 224 High speed cylindrical roller bearing, 349

fatigue life, 869High temperature:

Garter seals, 679 heat removal, 569Gear forces, 136 polymers for cages, 629

herringbone gears, 137 Hollow rollers to control skidding, 525

1080 INDEX

Hooke's law, 187 permanent deformation, 824Hoop (ring) stress effect on fatigue life, 921 semi width, 202Hot isostatically pressed (HIP) silicon nitride, Line of contact, 157

623 Line-to-line fit, 78Housing, 4 Liquid lubricants, 654

tolerance range classification, 93 mineral oils, 655Hydrodynamic bearings, 6 Loading:Hydrodynamic lubrication, 419 bearing, 135

pressure distribution, 423 classification, 85Reynold's equation, 419 combined radial and thrust, 256

Hydrostatic bearings, 5 double-row bearings, 262Hypoid gear loading, 147 combined radial, thrust, and moment:

ball bearings, 266Ideal line contact, 219 cylindrical roller bearings, 289Indentations, cause of fatigue, 620 spherical roller bearings, 291Induced loading: tapered roller bearings, 290

ball, 158 five degrees of freedom, 357Induction hardening steel, 610 limiting thrust load in radial ball bearings,Influence coefficients for ring bending, 296 379INSA contamination-life test system, 789 radial, 235Instrument ball bearings, 15 Load ratings:Interference fit: standards, 741

effect on clearance, 55, 119 Load zone:surface finish effect, 126 combined radial and thrust loading, 258

Ioannides-Harris fatigue life theory, 906, 931 fatigue life effect, 864ISO, 14, 45, 84 radial load, 235

load rating standards, 741 Low carbon steel for cages, 625Isoelasticity, 378 LlO fatigue life, 688,717,761

Lubricant:Jet oil lubrication, 650 environmentally acceptable, 671JNS, 84 esters, 661

film thickness, 422, 428, 523, 694Labyrinth seals, 673 contact inlet frictional heating effect, 441Lambda parameter, 448 contact shear stresses, 476

fatigue life (effect on), 891 fatigue life (effect on), 891life testing considerations, 767 line contact, 434

Leonardo Da Vinci, 1 point contact, 437Life factors combined, 903 starvation effect, 444Life testing: surface topography effect, 446

accelerated, 765 very high pressure effect, 440confidence in results, 776 filtration effect on fatigue life, 898contamination effects, 769 fluorinated ethers, 662

INSA life test system, 789 functions, 646mounting and dismounting effects, 770 glassy state in contact, 427plastic deformations (effect on), 766 greases, 647, 662practical considerations, 768 properties, 664speed considerations, 767 thickeners, 666theoretical basis, 764 compatibility, 668

Lightly loaded applications, 11 high temperature considerations, 569Lightweight balls, 348 liquid, 646Lightweight rollers, 357 advantages, 654Linear motion bearings, 4, 40 guidelines for use, 654Line contact, 190 mineral oils, 655, 657

basic dynamic capacity, 706 properties, 658radial roller bearings, 732 synthetic oil properties, 660

definition, 219 polyglycols,661deformation (elastic), 202, 234 polymeric, 647, 668"dogbone" shape, 222 quantity, 7fatigue life, 704 selection, 657

radial bearings, 728 solid, 647, 670ideal, 219 starvation, 444lubricant film thickness, 434 synthetic hydrocarbons, 656modified, 220 types, 646

INDEX 1081

viscosity index, 657 HIP silicon nitride, 349Lubrication, 360 shear, 188, 429

bath, 648 Moisture:boundary (wear), 941 corrosion, 1058circulating oil, 649 life testing (effect on), 769contact structural elements, 937 Molybdenum disulfide, 418, 488, 648, 670dents (in the vicinity of), 956 Mounting:fatigue life (effect on), 890 locknut adapter, 24grease, 651 tapered sleeve, 24jet, 650 Multiple bearing-shaft systems, 410limiting shear stress in elastohydrodynamic

lubrication, 477 NASA five-ball endurance test rig, 782methods, 648 Near-surface region of contact, 938non-Newtonian, 476 Needle roller bearing, 26oil sump, 648 cam follower application, 44once-through, 651 thrust bearing, 39polymeric, 653 Newtonian fluid, 419, 451regimes, 453 Newton-Raphson method:solid, 653 contact angle change determination, 247wear, 939 heat transfer temperature calculations, 564

Lundberg-Palmgren fatigue life theory, 208, high speed ball loading, 343219, 688, 692, 694, 794, 931 load distribution calculations for ball

case-hardening steel bearings (application bearings, 272to), 740 Nitrile rubber for seals, 632

limitations of the theory, 863, 904 Noise, 964stress-life relationship, 894 sensitive bearing applications, 965

Normal approach between raceways, 234

Macrogeometry, 48 Nylon (polyamide) 6,6 for cages, 628

Marquenching, 607Martensite, 606, 612 Octahedral shear stress, 211Material effect on fatigue life, 894 Oil bath lubrication, 648Material-life factor, 895 Once-through lubrication, 651Maximum compressive stress: Orbital motion, 317

line contact, 202 speed, 309, 328point contact, 195 Orthogonal shear stress, 209, 218

Maximum likelihood method in statistics, 804 maximum orthogonal shear stress, 694Maximum orthogonal shear stress, 694 Oscillatory motion, 27Maximum rolling element load, 238, 242, 717 fatigue life of bearings (due to), 879Mean time between failures, 795 Osculation, 50, 70Mechanical properties of steel, 614 Out-of-round raceway, 525Median (L50) fatigue life, 688, 761 Outside diameter (o.d.), 49Melting-Refining (M-R) method for vacuum Overriding ring land, 379

degassing of steel, 588, 595 Oxygen in steel, 583Metallurgy:

audit parameters, 777 Palmgren-Miner rule, 874structure of steel, 212 Partially crowned roller, 29, 224

M50NiL steel, 581, 620 Permanent deformations, 820M50 steel, 581, 620 brinelling, 1052Microcontacts, 464 line contacts, 824

Greenwood Williamson model, 465 point contacts, 820plastic contacts, 469 shakedown, 852

Microslip, 489 Permissible static load, 831Mineral oil lubricants, 655 Photoelastic study of roller bearing, 304Miniature ball bearings, 5, 16 Physical vapor deposition (PVD), 640Misalignment: Pitch diameter, 49, 66, 73

fatigue failure, 773, 1052 Pitting, 949, 1059fatigue life (effect on), 870 Planet gear bearing, 44, 152limitations per bearing type, 874 load distribution, 292, 301radial roller bearings, 272 loads, 172types, 273 Planet gear speeds, 152

Modified line contact, 220 Plastic deformations:basic dynamic capacity factors, 739 calculation (of), 820

Modulus of elasticity, 187 life testing (effect on), 766

1082INDEX

Plastic deformations (Continued) life factor, 890residual stresses (associated with), 843 Residual stress, 616, 843

measurement method, 844 alteration with overrolling, 848shakedown, 852 fatigue life (effect on), 850wear (associated with), 947 measurement, 844

Plating processes for components, 639 Retainer, 4chemical vapor deposition, 640 Reynolds equation, 419physical vapor deposition, 640 Ring:thin dense chrome (TDC), 640 deflections due to pressure, 121

Point contact, 190, 219 fracture, 853deformation (elastic), 234 carburized steel (effect on), 855dynamic capacity, 699, 705 integral flange, 42, 494fatigue life, 699, 701 land, 379

radial bearings, 707 radial shift, 235lubricant film thickness, 436 stresses due to fit, 120permanent deformation, 821 Roelands equation, 425

Poissons' ratio, 188, 429 Roller bearings, 23Polyetheretherketone (PEEK) material for clearance effect on fatigue life, 867

cages, 631 fatigue endurance, 25Polyethersulfone (PES) material for cages, flexibly supported bearings, 868

631 internal load distribution effect on fatiguePolyglycollubricants, 661 life, 866Polymeric lubricants, 647, 668 misalignment, 272Polytetrafluoroethylene (PTFE), 5 radial deflection, 279Powder metal components, 621 radial, 25Pratt & Whitney v-ring/ball endurance test maximum roller load, 242

rig, 783 static load ratings, 826Preloading, 368 Roller-raceway:deflection, 372 contact laminum:isoelasticity, 378 load, 274, 280radial, 375

deformations due to skewing, 286Press-fitting, 83 heat transfer, 574force, 124 Rollers:hoop stresses, 124 axial loading, 177fatigue life (effect on), 921 centrifugal force, 355Prestressing (effect on fatigue life), 850 corner-flange contact, 228Probability of survival, 688 crowning, 26Progression of failure, 1008deformation components, 272Pulley loads, 138eccentricity of loading, 276, 284Pyrowear 675 steel, 582end-ring flange contact:

sliding, 494Raceway control, 325 stress, 225Raceways: geometry, 275

crowning, 26, 30 hollow, 355, 525dimensional audit parameters, 778 logarithmic profile, 224loci of groove curvature centers, 268, 339 skewing, 26,177, 179,534roller deformation components, 272 tilting, 177, 280speed components, 318 viscous drag on, 492surface velocities, 319 Rolling bearings, 4

Radial clearance, 49 Rolling elements:Radial deflection of roller bearing, 279 centrifugal force, 165Radial load distribution, 235 dynamic loading, 161

effect of clearance, 239 maximum load, 238Radial load integral, 237 rotational speed, 311Radial preloading, 375 sliding in cage pocket, 494Radii of curvature, 193 types, 4

deformed surface, 318 Rolling-mill application, 31Railroad car wheel application, 29 Rolling motion, 309Rating life, 696, 764 deformation due to, 487RBEC, 85 elastic hysteresis, 486

tolerance classes, 99 pure, 316, 501Reliability, 691, 761 sliding and, 313

fatigue (as function of), 886 Rotation about eccentric axis (forces), 172

INDEX 1083

Rotation factor V, 720, 733 Sliding friction:Rotor dynamics, 1014 cage, 493

critical speed, 1039 distribution of forces in ball-racewaysynchronous response, 1039 contacts, 502shaft whirl, 1042 gyroscopic motion (due to), 488vibration mode shapes, 1040 rolling motion (in), 488

skewing, effect of, 179spherical roller bearings, 35

Scoring, 1054 tapered roller bearings, 28Scuffing, 944 Sliding motion, 313Seals, 14 deformation (cause), 316

deep-groove ball bearing, 15 Sliding velocity:elastomeric lip, 675 ball bearing inner raceway, 321ferrofluidic, 681 ball bearing outer raceway, 320flinger (with), 674 ball-raceway contacts (in), 498friction, 494 distribution in ball-raceway contacts, 501functions, 672 gyroscopic motion, 314garter, 679 roller end-flange, 334greased bearing, 672 spinning motion, 314high temperature, 637 Smearing, 516, 944, 1046labyrinth, 673 Smoothness of bearing operation, 832materials, 631 Solid lubrication, 653, 670oil-lubricated bearings (for), 673 Space vehicle applications, 10shields (with), 17, 675 Spall, 688, 772, 773, 1061solid-lubricated bearings (for), 673 Specific loading, 224torque, 679 Speeds:

Self-aligning: shaft, 150ball bearings, 22 Spherical roller bearings, 6, 30deflections, 366 asymmetrical rollers, 32spherical roller bearings, 30 barrel-shaped rollers, 32spherical roller thrust bearings, 37 clearance, 68

Semi axes of contact ellipse, 192 contact angle, 66Semi width of line contact, 202 curvature difference, 71Separator, 4 curvature sum, 71Shaft: free endplay, 66

concentrated radial loading, 135 high speed, 357speeds, 150 hourglass-shaped rollers, 32tolerances, 87 load classification, 85whirl, 1042 osculation, 70

Shakedown, 833, 852 pitch diameter, 66Shields, 14, 16, 675 planet gear bearing, 44Shrink fitting, 83 roller skewing, 34, 537Silicon carbide, 621 single-row, 35, 37Silicon nitride, 621 sliding friction, 35

balls, 348, 896 steel-making (in), 8fatigue life effect at high speed, 872 surface treatment for components, 638

fracture toughness, 624 Spherical roller loading:rollers, 357 centrifugal force, 171tensile strength, 624 static, 159

Silicon rubber for seals, 637 Spherical roller thrust bearings, 37Skewing, 26, 177,228 Spinning motion, 313

angle, 286, 537 ball bearings, 317damage due to, 1049 frictional moment (in), 504flange-roller end contact and, 330 Spin-roll ratio, 324roller axial loading, 179 Spiral bevel gear loading, 145roller-raceway deformations, 286 Split inner ring ball bearings, 22

SKF A-frame automotive wheel hub bearing diametral play, 56endurance test rig, 779 shim, 56

Skidding motion, 315, 347, 360, 515 shim angle, 57ball bearings (in), 518 tandem arrangement, 25cylindrical roller bearings (in), 523 Spur gears:out-of-round raceway (to control), 525 loads, 137

Slewing bearings, 5, 7 speeds, 150

1084INDEXStandards, 84

Martensite, 606, 612interference fits, 921material-life factor, 895Starvation of lubricant, 444melting methods, 582Static equivalent load, 828

electroslag refining, 583, 591Static load ratings, 825vacuum arc remelting, 590permissible static load, 831vacuum degassing, 583, 585Statistical analysis:

fatigue life (effect on), 739endurance test samples, 772vacuum induction melting, 589hazard, 802

metallurgical characteristics, 593life testing considerations, 769audit parameters, 777maximum likelihood method, 804cleanliness, 593mean time between failures, 795quality, 593product law of probability, 693, 714

banding, 619sample size selection, 810decarburization, 596Weibull distribution, 692inhomogeneities, 619two-parameter, 795macroinclusions, 619, 688graphical representation, 798

nonmetallic, 842percentiles, 797porosity, 596probability functions, 795segregation, 595, 604shape parameter, 799sulfide inclusions, 619sudden death test analysis, 807

microstructure, 596, 836Steel:alterations due to rolling contact, 837AISI 52100, 4, 581, 598, 603

butterflies, 841, 1061fatigue life properties, 696dark etching region of overrolled AISIAISI 440C, 4, 583

52100 steel, 839, 841annealing, 598, 613white etching bands, 842, 1061austenite, 612, 836

carbides, 596bainite, 607porosity, 619banding, 619Poissons' ratio, 188basic electric furnace processing, 584processing methods (effects on), 597carbonitriding, 609

fatigue life (effect on), 739case-hardening, 4, 580, 608products:fatigue life effects, 741, 894

forms, 592cleanliness, 593inspection, 592macroinclusions, 594

eddy current, 595oxygen content, 594macroetching, 596cobalt alloys:ultrasonic testing, 595L-605, 621

quality, 593Stellite 3, 621raw materials, 583Stellite 6, 621residual stresses, 843composition (effect on fatigue life), 739

alteration with overrolling, 848dimensional instability of components, 856fatigue life (effect on), 850fatigue failure modes, 618

retained austenite, 612, 614, 836subsurface-initiated, 619alteration with overrolling, 848surface-initiated, 620

stainless, 4, 583, 613fatigue limit stress values, 927structure, 596fatigue strength, 614subsurface structure after overrolling, 213grain size, 604surface hardening, 4, 580, 608hardenability, 603tempering, 613, 618hardening methods, 605thermal treatment for structural stability,heat treatment, 597

612mechanical properties affected by, 614through-hardening, 4, 580continuous cooling transformation (cct)

martensite, 606curves, 601tool steels, 620time-temperature-transformation (TTT)types, 579curve, 601

Stiffness of bearings, 1028high temperature, 569contact angle effects, 1037homogeneity, 11preload effects, 1037induction heating, 610shaft bending effects, 1038inhomogeneities, 619speed effects, 1034low carbon for cages, 625

Strain, 187machinability, 596Stress concentrations:marquenching, 607

contaminant denting (due to), 920

INDEX 1085crowning, 224 Talyrond,981

Stress cycles per revolution, 695, 704 Tandem bearing arrangement, 21Stressed volume: Tapered roller bearings, 27

Ioannides-Harris theory, 907 cone angle, 77Lundberg-Palmgren theory, 694 cup angle, 77

Stresses: double-direction, 35fatigue-initiating, 907 double-row, 29hertz, 186, 623 end play, 78hoop stress effect on fatigue life, 921 endurance test rig, 782material processing (due to), 922 four-row, 35octahedral shear, 211, 907 high speed, 357residual, 843 misalignment, 272, 1052subsurface, 204 pitch diameter, 77

comparison of shear stresses, 214 roller:angle, 77maximum shear, 205end-flange contact geometry, 334orthogonal shear, 209static loading, 160principal, 205

small and steep angles, 33surface:surface treatment for components, 638normal, 189thrust bearing, 39shear, 215

Tapered roller loading:Von Mises, 210, 907static, 160Stress-life factor, 910, 931tapered shaft mounting, 376Stribeck, 238

Tapered sleeve mounting, 24, 376Subsurface metallurgical structure of 52100Temperature:steel, 212

effect on clearance, 125Subsurface stresses:expansion of rings, 125frictional shear stresses on contact surfacenodes in heat transfer system, 561(due to), 911surface (in wear), 940hertz stress (due to), 911viscosity variation (with), 660Sudden death endurance testing, 775

Tempering of steel, 613, 618Weibull distribution analysis, 807 Test rigs:Surface damage:ball-rod rolling contact fatigue tester, 787adhesion, 1055design considerations, 777brinelling, 1052General Electric Polymet rolling contactdecarburization, 620 endurance tester, 786false brinelling, 1054 INSA rolling-sliding disc endurance tester,grinding burns, 620 788inception, 950 NASA five-ball endurance tester, 782indentations, 620 Pratt & Whitney v-ring/ball enduranceinteracting modes of failure, 953 tester, 783marks, 620 SKF A-frame automotive wheel hubscoring, 1054 bearing endurance tester, 779smearing, 516 SKF R2 endurance tester, 781

Surface finish: tapered roller bearing endurance tester, 782effect on clearance, 126 Theory of elasticity, 185

Surface hardening steel, 608 Thermal gradient, 83flame-hardening, 611 Thermal imbalance failure, 775, 1048induction heating, 610 Thermal treatment of steel for structuralresidual stress, 616 stability, 612

Surface-initiated fatigue, 620 Thin dense chrome (TDC) plating forSurface shear stresses, 215 components, 640

ball-raceway contacts (in), 496 Thin ring deflections, 294composite shear stress, 479 Three bearing-shaft systems:contaminant (particulate) effect on, 913 non-rigid shaft, 404coulomb friction in asperity contact (due rigid shaft, 400

to), 478 Through-hardening steel, 580Surface topography: Thrust ball bearings, 23

effect on lubricant film thickness, 446 deflections, 367honed and lapped surface, 822 effect of centrifugal force on contact angle,rough surfaces, 464 168

fatigue life (effect on), 891 limiting load, 379Survival probability, 688, 691, 709, 907 Thrust carried on roller ends, 228

1086INDEX

Thrust loading: role of bearings in machine, 968centric, 245 sensitive bearing applications, 965eccentric, 249 smoothness of bearing operation, 832excessive contact stress in radial ball testing of bearings, 991

bearings, 382 waviness (relationship to), 994radial cylindrical roller bearings, 280 VIMVAR steels, 581, 591

Thrust load integral, 251 Viscosity:double direction bearings, 255 Barus equation, 424single direction bearings, 251 index, 657

Thrust roller bearings, 37 kinematic, 543deflections, 367 Roelands equation, 425Titanium carbide, 621 selection for application, 657coating, 640 specification, 657

Titanium nitride coating for components, 640 variation with pressure, 424Tolerances: ASME study, 425classes, 85 lubricant glassy state, 427ABEC, 99 sigmoid curve fit to ASME data, 426ANSI/ ABMA vs ISO, 98 variation with temperature, 660RBEC, 99 Viscous drag:housing bore limits, 94 cage (on), 530shaft tolerance range classification, 87 balls (on), 492

Tool steels, 620 bearing friction torque, 542Traction stresses, 215 rollers (on), 524Tribological processes associated with wear, Volume under stress, 694, 925

939 Von Mises stress, 210, 215, 923Triplex set of angular-contact ball bearings,

375Water contamination effect on fatigue life,Truck wheel application, 43

902Tungsten carbide, 621Waviness, 980Two bearing-shaft systems, statically testing, 986indeterminate: Wear, 936, 1045flexible shaft, 392

adhesive, 944rigid shaft, 389corrosive, 946delamination, 949

Ultrasonic testing of steel components, 595 failure classification, 936Under raceway cooling of bearing, 574, 650 phenomenological view, 949Unit (Metric/English units system) pitting, 949

conversion factors, 1071 processes, 936, 942protection, 955

Vacuum arc remelting of steel, 590 roller end-flange, 330fatigue endurance Weibull slope, 696 smearing, 944, 1046

Vacuum degassing, 583 tribological processes, 939fatigue life (effect on), 739 Wedeven ball-disc test rig, 790

Vacuum induction melting of steel, 589 Weibull distribution, 692, 794Variable loading effect on fatigue life, 875 application, 800Vibration, 964 maximum likelihood method, 804

causes in bearings: slope, 695, 706, 799geometrical imperfections, 970 sudden death test analysis, 807

nonroundness, 980 two-parameter, 795waviness, 971, 980 graphical representation, 798

testing, 986 mean time between failures, 795variable elastic compliance, 969 probability functions, 795

coupled motion, 1020 shape parameter, 799damped forced, 1015 White room, 1,;, 19detection of failed bearings, 997 Worm gears:

condition monitoring, 1003 loading, 149health usage and monitoring system, speeds, 150

1008micro-sensors, 1007 X-Ray diffraction, 844shock pulse method, 1005 X and Y factors:

frequencies in bearing operations, 993 point contact radial bearings, 720multi-degree-of-freedom system, 1024 point contact thrust bearings, 725natural frequencies in bearings, 996 radial roller bearings, 734resonant, 996 static loading, 830

rolling bearing analysis

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