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ROLLING BEARING ANAZYSIS
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This book is printed on acidfree paper. @ Copyright 0 2001 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada,
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate percopy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 7508400, fax (978) 7504744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 101580012,(212) 8506011, fax(212) 8506008, EMail: PERMREQ~W1LEY.COM. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold with the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional person should be sought.
Harris, Tedric A. Rolling bearing analysis / Tedric A. Harris.  4th ed. p. cm. Includes index. ISBN 0471354570 (cloth : alk. paper) 1. Roller bearings. 2. Ballbearings. TJ1071.H35 2001 621.8'22dc21 Printed in the United States of h e r i c a . 1 0 9 8 7 6 5 4 3 2 1
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Linear Motion Bearings earings for Special Applications Closure
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Tapered Roller Bearings Closure
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s t ~ ~ u t Load e d Systems
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General Load~e~ection Relationships earings under Radial Load earings under Thrust Load gs under Combined Radial and Thrust Load arings under Combined Radial, Thrust, and Moment Load isalignment of Radial Roller Bearings Thrust Loading of Radial Cylindrical Roller Bearings Radial, Thrust, and Moment Loading of Radial oller Bearings Flexibly Supported Rolling Bearings Closure
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List of Symbols General
307 308 309 313 317 330 335
olling and Sliding Orbital, Pivotal, and Spinning Motions in Ball Bearings oller EndFlange Sliding in Roller Bearings Closure
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355 Closure
Closure
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400 410 412 e
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t a ~ a t i o nof Lu~ricant urface T o p o ~ a p ~ y ~ f f e c t s rease L u ~ r i c a t i o ~
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461 463 464 472 riction in the EHL 476 Contact Closure
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~ of Friction ~ c
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~
486 olling Element
Cont~cts and Cage Forces ons and Forces
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losure
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Types of Lubricants Liquid Lubricants Grease Lubricants Polymeric Lubricants Solid Lubricants En~ironmentallyAcceptable Lubricants Seals Closure
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tatie Equivalent Load
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earing Com~onents Permissible Static Load Closure
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crostructures of Rolling Bearing Steels icrostructural Alterations Due to Rolling Contact esidual Stresses in Rolling Bearing Components of Bulk Stresses on Material Response to
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List of Symbols General Effect of Bearing Internal Load Distributi~non Fatigue Life Effect of Variable Loadingon Fatigue Life Fatigue Life of Oscillating Bearings and Fatigue Life brication on Fatigue Life aterial Processing on Fatigue Life Combining Fatigue Life Factors Li~itationsof the Lundber~PalmgrenTheory The StressLife Factor Closure
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List of Symbols
of a Lubricated ont tact
teracting Tribological Processes and Failure Modes ecommendations for ear Protection Closure
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963 963 964 968 980 997 1003 100~ 1010
1013 1014 arnped Forced ~ibrations
C~aracteristicsof Bearing ~tiffness ynamics Analysis Closure
~limi~ary ~nvesti~ation isassern~lyof Bearings xamination and Evaluationof Specific Conditions Fracto~aphy Closure
1024 1028 1033 1039 104
1043 1043 1044 1044 1049 1063 1068
and roller bearings, generically called r o Z Z ~~ ~e ~~ r i are ~ ~ coms , used machine elements. They are emplo d to permit rotary motion of, or about, shafts in simplecommercialvices such as bicycles, roller skates, and electric motors. They are also used in neering mechanisms such as aircraft gas turbines, rollin drills, ~ o s c o p e s and , power transmissions. Until appro the design and a~plicationof these bearings could be considered more art than science. Little was understood about the physical phenomena that occur during their operation. Since 1945, a date which marks the r I1 and the b e ~ n n i nof~the atomic age, scientific occurred at an e~ponentialpace. Since 1958, the date w marks the commencement of manned space travel, continually increasing demands are being madeof engineering equipment. To ascertain the effectiveness of rolling bearings in modern engineering applications, it is necessary to obtain a firm understanding of how these bearings ~erform r varied and often extremely demanding conditions of operation. ost information and data pe~tainingto the performance of rolling bearings are presented in manufacturers’ catalogs. These data are almost entirely empirical in nature, being either obtained from the testing of products by the larger bearing manufacturing companies or,more iii
likely for smaller rnanufacturing companies, based on information contained in the American National Standards Institute (ANSI) or Interrganization for Standards (ISO) publications or similar . These data pertain only to applications involvingslow speed, simple loading, and nominal operating temperatures. If an engineer wishes to evaluate the performance of bearing applications operating beyond these bounds, it is necessary to return to the basics of rolling and sliding motions over the concentrated contacts that occur in st books written on this subject was ing by Arvid Palmgren, Technical Di for many years. It explained, more completelythan had been done previously, the concept of rolling bearing fatigue life. Palmgren, together with Gustav Lundberg, Professor of Mechanical Engineering at Chalg, was the originator of rners Institute of techno lo^ in ~ o t e ~ o rSweden, the theory and formulas on which the current ANSI and IS0 standards for the calculation of rolling bearing fatigue life are based. Also, A. s’s book in two volumes,A n ~ l y s i of s Stresses and ~ e ~ ~ ~ t i o n s , xplanation of the staticloading of ball bearings. Jones, who worked in various technical capacities for New Departure sion of Motors ~orporation,MarlinRockwell Cor~oration, Fafnir ring company, and also as a consulting engineer,pioneered the use of digital computers to analyze the performance of ball and roller bearing shaftbearinghousing systems. The remainder of other early and subsequent texts on rolling bearings were, and are, largely empirical in their approaches to applications analysis. Particularly since 1 9 ~ 0much , research has been conductedinto rolling bearings and rolling contact phenomena. The use of modern laboratory equipment and transmission electron microscopes, xray diffrachigh speed digital co~putershas shed much light on the mechanical,hydrodynamic, metal cal, and chemicalphenomena involved in rolling bearing operation. significanttechnical papers been published by various engineering societiesfor ican Societyof Mechanical Engineers, the Institution e Society of Tribologists and Lubrication E ociety of Mechanical Engineersanaly~ing the performa~ce earings in exceptional a~plicationsinvolvinghighspeed, d e~traor~inary internal design and materials. Since 1960, substantial attention has been given to the mechanisms of rolling bearing lubrication and the rheology of lubricants, Notwithstanding istenco of the aforementioned literature, there remains a nee ce that presents a unified, uptodate approach to the analysis of bearing performance. That is my intention in presenting this To acco~plishthis goal, I have attempted to review the most significant technical papers and texts covering the performance of rolling bear
ings, their constituent materials, and lubrication. mathematical presentations contained in the reviewed technical literature have been condensed and simplified in this book for r ease of understanding. It should not be constru~d,howev book supplies a complete bib on rolling bearings tical analysis have bee atathat I foundmostuseful everal of the references cite own works, since in some eases these are theoriginal or are among the most s i ~ i f i c a navail t
~ ~ Z y sisi saimed at deve lling bearing operation, concepts of rolling bea basic bearing types, loading, plied loading and rollers, and con deformations. The analysis of load distribution among the rollingeleme speeds, and velocities, elastohydrodynamic lubrication, friction, temperstatistics of bearing endurance, and fatigue life are consi 1 topics depend almost entirely on the prece ossible, an attempt has been made to m iscussion, numerical stance, numerical ex ylindrical roller bear 7 spherical roller be earing are accum examples are carried out in metrl
tions presented in SI or metric system units ar system units as well. The material covered herein spans many metrx elasticity, statics, dynamics,hyd at transfer. Thus, many mathemathical sy In some cases, the same symbol has been cho Lain symbols have been lways ball or roller di ecause of the several scientific disciplines that this b treatment of each topic mayvary somewhat in scope and feasible, analytical solutions to problems have been pres other hand, empirical approaches to problems have been seemed more practical. The wedding of analytical and em~iricaltechniques is particularly evident in the chapters covering lubrication, friction, and fatigue life.
Chapter
ic
d so~etimes increase lso like to express my ylvania State University for his critical review of 1 s~ggestionsfor its modi~cation. c t to my editing, was he ater rial in thefollowing chap s, s ~ ~ j eonly Contri~utor(s)
Chapter
Topic
20
24
ar
rs remain unchange~from ofessor in theDepartment o Engineering at Penn State Great Valley; sociates, Inc., a tribological research and testingcom~any company and comm hanical Engineering at th ark, Pennsylvania, where in machine design and rolling contact tribology and conduct research,
of' my l o n ~ i m eassoci
manu~acturers aswell. I would like to express my a~~reciation to the
tion, Torrin~on, ~onnectic~t. The c o n t r i ~ ~ toofreach such.illustration is identi~ed.
TEDRICA. Professor of ~ e ~ h a n i~ngineering ~al The Pennsylvania State University Uniu~rsityPark, Pennsylvania
RIS
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After the invention of the wheel, it was learned that less effort was required to move an object on rollers than to slide the object overthe same surface. Even after lubrication was discovered to reduce the work required in sliding, rolling motionstill required less work when it could be used. For example, archeological evidence showsthat the E ~ p t i a n sca. , 2400 BC, employed lubrication, most likely water, to reduce the manpower required to drag sledges carrying huge stones and statues. The Assyrians, ca. 1100 BC, however, employed rollers under the sledges to achieve a similar result with less manpower. It was therefore inevitable that bearings using rolling motion would be developed use for in complex ~ a c h i n e r yand mechanisms. Figure 1.1depicts, in a sim~listicmanner, the evolution of rolling bearings, Dowson [l.11 provides a comprehensive presentation on the history of bearings and lubrication in general; his coverage on ball and roller bearings is extensive. though the concept of rolling motion was known and used forthousands of years, and simple forms of rolling bearings were in use ca. 50 AD during the lization, the general use of rolling bearings did not occur until the in[1.2], however, shows that Leonard0 da Vinci dustrial revolution.
d by the A ~ y r i a n sto move massive sto~esin 1100
and later, with crude cart wheals, man strived to overcome friction's drag.
r 19th century bicycles marked man's first i m ~ ~ a victov. n t
olution of rolling bearings (courtesy of SKI?).
( 1 4 ~ ~  1 ~AD) 1 9 in his Codex ~ ~ conceived d various ~ ~ forms d of pivot bearings having rolling elements and even a ball bearing with a device to space the balls. In fact, Leonardo, who among his prolific accomplishments studied friction, stated: I affirm, that if a weight of Rat surface moveson a similar plane their move~entwill be facilitated by interposing between them balls or rollers; and I do not see any difference betweenballs and rollers save the fact that balls have universal motion while rollers can move in one direction alone. But if balls or rollers touch each other in their motion, they will make the movement more difficult than if there were no contact between them, be
cause their touching is by contrary motions and this friction causes contrariwise movements. But if the balls or the rollers are kept at a distance from each other, they will touch at one point only between the load and its resistance . . . and consequently it will be easy to generate this movement. Thus did Leonard0 conceive the basic construction of the modern rolling bearing; his ball bearingdesign is shown by Fig. 1.2. The universal acceptance of rolling bearings by design engineers was initially impededby the inabilityof manufacturers to supply rolling bearings that could compete in endurance with hydrodynamic sliding bearings.Thissituation, however, has beenfavorably altered during the twentieth century, and particularly since 1960, by development of supe
1.2. ( a )Thrust ball bearing design (circa 1500) in Codex Madrid by Leonardo da Vinci [1.2];( b )Da Vincibearing with Plexiglas upper plate fabricated at Institut National des Sciences AppliquBes de Lyon (INSA) as a present for Docteur en M6canique Daniel NBlias on the occasion of his passing the requirements for “DirigerdesRecherches,” December 16, 1999.
rior rolling bearing steels and constant improvement in manufacturing, providing extremely accurate geometry, longlived rolling bearing assemblies. Initially this development was triggered by the bearing requirements for high speedaircraft gas turbines; however, competition between ball and roller bearing manufacturers for worldwide markets increased served to provide consumers substantially during the 1970s, and this has with lowcost, standard design bearings of outstanding endurance. The ~ g s all forms of bearings that utilize the rollterm roZZing ~ e ~ r i includes u ~ constrained ing action of balls or rollers to permit m i n i ~ friction, motion of onebody relative to another. ostrolling bearings are employed to permit rotation of a shaft relative to some fixedstructure. Some rolling bearings, however, permit translation, that is, relative linear motion, of a fixture in the direction provided by a stationary shaft, and a few rolling bearing designs permit a combination of relative linear and rotary motion between two bodies. This book is concerned primarily with the standardized forms of ball and roller bearings that permit rotary motion between two machine elements. These bearings will always include a complement of balls or rollers that maintain the shaft anda usually stationary supporting structure, frequently called a ~ o ~ s i n in g ?a radially or axially spacedapart relationship. Usually, a bearing may be obtained as a unit, which includes two steel rings each of which has a hardened raceway on which hardened balls or rollers roll. The balls or rollers, also called roZZing elements, are usually held in an angularly spaced relationship by a cage? whose function wasanticipated by Leonardo. The cage may also be called a se arator or retainer. , rollers, and rings of good quality, rolling bearings are normally ctured from steels that have the capability of being hardened to a high degree, at least on the surface. In universal use by the ball bearing industry is AIS1 52100, a steel moderately rich in chromium and easily hardened thro~ghout( t ~ r o ~ g ~  ~ a the r ~ mass e n e of ~ )most bearing components to 6165 Rockwell C scale hardness. This steel is also used in roller bearings by some manufacturers. ~ i n i a t u r eball bearing manufacturers, whose bearings are used in sensitive instruments such as to fabricate components from stainless steels such as r bearing manufacturers ~requently from sasehardening steels such as 310. For some specialized applicati motive wheel hub bearings, the rolling components are manufactured from inductionhardening steels. In all cases, at least the surfaces of the rolling components are extremely hard. In some high speedapplications, to m i n i ~ i z einertial loading of the balls or rollers, these components are ~abricatedfrom lightweight, high compressive s t r e n ~ h ceramic materials such as silicon nitride. Also, these ceramic rolling elements tend to
endure longer than steel at ultrahigh temperatures and in applications with dry film or minimal fluid lubrication. Cage materials, as compared to materials for balls, rollers, and rings, rally required to be relatively soft. Theymust also toweight ratio; therefore, materials as widely div ical properties as mild steel, brass, bronze, aluminum, pol~amide(nylon), polytetra~uoroethylene(teflon or PTFE), fiberglass, and plastics filled with carbon fibers finduse as cage material. n this modern age of deepspace exploration and cybersp bearings have come into use, such as gas film etic bearings, and externally pressurized (hy ese bearing types excels in some s~ecializedfield of ' ple, hydrostatic bearings are excellentfor no problem, an ample supply of pressuri me r i ~ d i t yunder heavy loading is requi be used forapplications in which loadsare light, are high, a gaseous atmosphere exists, and friction must be minowever, are not quite so li all bearings such as shown such as inertial guidance e roller bearings, such as shown in Fig, 1.4, are ill applications9and even larger slewing bear1.5, were used in tunneling machines for the el tunneling) project. s find use in diverse precision machiner~ophigh load, high temperature, dusty environ1.Q the dirty environments of earthmoving
E 1.3. Miniature ball bearing (courtesy of SKI?).
of
sm).
1.4. A large spherical roller bearing for a steel rolling mill application (courtesy
and farming (Figs. 1.7 and 1.8), the lifecritical applications in aircraft power transmissions (Fig. 1.9), and the extreme lowhigh temperature and vacuum environments of deep space (Fig. 1.10). They perform well in all of these applications. Specifically, rollingbearings have the following advantages compared to other bearing types: They operate with much less friction torque than hydrodynamic bearings and therefore considerably less power loss and friction heat generation. Starting friction torque is only slightly greater than moving friction torque. Bearing deflection is less sensitive to load fluctuation than in hydrodynamic bearings.
(b)
Large slewingbearing used in an English Channel tunneling machine. ( a ) Photograph; ( b ) schematic drawingof the assembly (courtesyof SKI?).
0
They require only small quantities of lubricant for satisfactory operation and have the potential for operation with a selfcontained, lifelong supply of' lu~ricant. y shorter axial length than conventionalhydro bearings.
1.6. Spherical roller bearings are typically used to support the ladle in a steelmaking facility (courtesy of SKI?).
.
Many ball and roller bearings must function in the high contamination environment of earthmoving vehicle operations.
.
Agricultural applications employ many bearings with special seals to assure long bearing life.
.
CH53E Sikorsky Super Stallion heavylift helicopters employ ball, cylindrical roller, and spherical roller bearings t~ansmissions in which power the main andtail rotors (cou~esyof Sikorsky Aircraft, UnitedTechnolo~esCorp.).
1.10. The ball bearings in the Lunar Excursion Module and Lunar Rover operated well in the extreme temperatures and hard vacuum on the lunar surface.
omb bin at ions of radial and thrust loads can be supported simultaneously. ~ndividualdesigns yield excellent performance overa wide loadspeed range. atisfactory performance is relatively insensitive to fluctuations in load, speed, and operating temperature.
~otwithstanding the foregoing advantages, rolling bearings have been considered to have a single disadvantage compared to hydrod~amic bearings. Tallian [1.3] defined three eras of modern rolling bearing development: an “empirical” era extending through the l~ZOs,a “classical” era lasting through the 1950s, and the“modern”era occurring thereafter. Through the empirical, classical, and even into the modern era, it was said that evenifrolling bearings are properly lubricated, properly mounted, protected from dirt and moisture, and otherwise properly operated, they will e v ~ ~ t u a lfail l y because of fatigue of the surfaces in rolling contact. Historically, as shown in Fig. 1.11,rolling bearings have been considered to have a life distribution statistically similar to that of light bulbs and human beings. esearch in the1960s [1.4]demonstrated that rolling bearings exhibit a mini mu^ fatigue life; that is, “crib deaths” due to rollingcontact ot occur when the foregoing criteria for good operation are oreover, modern manufacturing techniques enable production of bearings with extremely accurate component internal and exter
Tungsten lamps
bearings
Total number of bearings failed
t
Total number of lamps failed
Total number of deaths
1.11. Comparison of rolling bearing fatigue life distribution with those of humans and light bulbs.
nal geometries and extremely smooth rolling contact surfaces, modern steelmaking processes can provide rollingbearing steels of outstanding homogeneity with few impurities, and modern sealing and lubricant filtration methods act to minimize the incursion of harmful contaminants into the rolling contact zones. These methods, which are now being used in combination in many applications, can virtually eliminate the occurrence of rolling contact fatigue, even in some applications involving very heavy applied loading.In many lightly loaded applications, for example, most electric motors, fatigue life need not be a major design consideration. There are many different kinds of rolling bearings, and before embarking on a discussion of the theory and analysis of their operation, it is necessary to become somewhat familiar with each type. In the succeeding pages a description is given for each of the most popular ball and roller bearings in current use.
i n g l e  ~ o ~ ~ e e Conrad ~  ~ r ossembly o ~ e Ball Bearing. This ball bearing is shown in Fig. 1.12, and it is the most popular rollingbearing. The
1.12. A singlerow, deepgroove, Conradassembly,radial ball bearing.
inner and outer raceway grooves have curvature radii between 51.5 and 53% of the ball diameter for most commercialbearings. assemble these bearings, the balls are inserted between the inner and outer sings as shown by Figs. 1.13 and 1.14. The a s ~ e ~ arzgZe b ~ y4 is given as follows:
1.13. Diagram illustrating the method of assembly of a Conradtype,deepgroove ball bearing.
~ ~ o t o showing ~ a p Conradtype ~ ball bearing components just prior to snapping the inner ring to the position concentric with the outer ring.
stand m i s a l i ~ i n gloads (moment loads) of small the bearing outside surface a portion of a spher 1.15, however, the bearing can be made exter incapable of supporting a moment load. all bearing can be readily adapted with seals as shields as shown by Fig. 1.17 or both as illustrated function to keep l~bricant s and shields come in ma selective ap~lications; only as examples. form well a t high speeds provi available. Speed limits shown in mantain to bearing operation without the benefit of external cooling capability or special cooling t e c h n i ~ ~ e s . s can be obtained in different dimens Conrad assem ISO* standards. Figure 1.19 shows t ries according to ative dimensions of various ball bearing series.
.I&. A singlerow deepgroove ball bearing assembly having a sphered outer surface to make it externally aligning.
*American National Standards Institute and International Organization for Standards.
1.16. A singlerow deepgroove ball bearing having two seals to retain lubricant (grease) and prevent ingress of dirt into the bearing.
~ingle ow DeepGroove ~ i l l i n ~  ~~l so st e ~ b lBall y Bearings. This bearing as illustrated in Fig. 1.20 has a slot machined in the side wall of each of the inner and outer ring grooves to permit the assembly of more balls than the Conrad type does, and thus it has more radial loadcarrying capacity. Because the slot disrupts the groove continuity, the bearing is not recommended for thrust load applications. Otherwise, the bearing has characteristics similar to those of the Conrad type.
Dou bleRow DeepGrooue Ball earings. This ball bearing as shown in Fig. 1.21 has greater radial loadcarrying capacity than the singlerow types. Proper load sharing between the rows is a function of the geometrical accuracy of the grooves. Otherwise, these bearings behave similarly to singlerow ball bearings. ~ n s t r u ~ e nall t earings. In metric design, the standardized form of these bearings ranges in size from 1.5mm (0.0~~06in.) bore and 4mm 1574~in.) 0.d. to (0.35433in.)bore and 26mm (1.02~~2in.) o.d. e reference [1.51. ailed in reference [1.6],standa~dizedform, inch design instrument ball bearings range from 0.635mm (0.0250i~.) bore
.
A singlerow deepgrooveball bearing assembly having shields and seals, The shields are used to exclude large particles of foreign matter.
~iameter series
3 1
2
"
.
I
Size comparison of popular deepgroove ball bearing dimension series.
.
Cutaway view of a singlerow, deepgroove, filling slottype ball bearing assembly.
. A doublerow, deepgroove, radial ball bearing.
.
A delicate final assembly operation on an instrumentball bearing assembly is performed under ma~ificationin a “white room.”
earings, ~ ~ l a r  c o n t aball c t bear3 are designed to support combin ings as shown in Fig. ust loads depending on the conta thrust loads or heavy nitude. The bearing aving large contact angles can support heavier shows bearings having small and large contact thrust loads, Figure erally have groove curvature radii in therange angles, The bearing % of the ball diameter. The contact angle does not usually exceed bearings are usually mounted in pairs with the free end as shown in Fig. 11.25. These sets may be preloadedagainst other to stiffen the assembly in the axial direction. The bearin also be ~ o ~ n t in e dtandem as illustrated in Fig. 1.26 to achieve thrustcarrying capacity. ow ~ n g u l a r  ~ o n ~ a c ~earings. These bearings as depicted 27 can carry thrust either direction or a combination of radial and thrust load. earings of the rigid type are able to withstand
e
An angularcontact ball bearing.
\
(a) Small angle
. An~ular~contact ball bearings.
ALL
(a) Backtoback mounted
(b) Facetoface mount@d
Duplex pairs of an~larcontactball bearings.
I
”
(a) Nonrigid type
(b) Rlgid type
1.27. Doublerow angularcontact ball bearings.
moment loading effectively. Essentially, the bearings perform similarly to duplex pairs of singlerow angularcontact ball bearings.
~ e l ~ ~ l i g n i ~ g ~ Ball o u ~Bearings. l e  ~ o wAs illustrated in Fig. 1.28, the outer raceway of this bearing is a portion of a sphere. Thus, the bearings are internally selfaligning and cannot support a moment load. Because the balls do not conform wellto the outer raceway (it is not grooved)jthe outer raceway has reduced loadcarrying capacity. This is compensated somewhat by use of a very large ball complement that minimizes the load carried by each ball. The bearings are particularly useful in applications in which it is difficult to obtain exact parallelism between the shaft and housing bores. Figure 1.29 shows this bearing with a tapered sleeve and locknut adapter. With this arrangement the bearing does not require a locating shoulder on the shaft. ing Ball Bearings. These bearings are illustrated in Fig. 1.30. As can be seen, the inner ring consists of two axial halves such that a heavy thrust load can be supported in either direction. They may also support, simultaneously, moderate radial loading. Thebearings have found extensiveuse in supporting the thrustloads acting on high speed, gas turbine engine mainshafts. Figure 1.31 shows the compressor and turbine shaft ball bearing locations in a highperformance aircraft gas turbine engine. Obviously, both the inner andouter rings must be locked
1.28. A doublerow internally selfaligning ball bearingassembly.
up on both axial sides to support a reversing thrust load. It is possible with accurate ~~s~ grinding at the factory to utilize these bearings in tandem as shown in Fig. 1.32 to share a thrust load in a given direction.
The thrust ball bearing illustrated in Fig. 1.33 has a 90" contact angle; however, ball bearings whose contactangles exceed 45"are also classified as thrust bearings. As for radial ball bearings, thrust ball bearings are suitable for operation at high speeds. To achieve a degree of externally aligning ability, thrust ball bearings are sometimes mountedon spherical seats. This arrangement is demonstrated by Fig. 1.34. A thrust ball bearing whose contact angle is 90" cannot support any radial load.
Roller bearings are usually used for applications requiring exceptionally large loadsupportingcapability, whichcannot be feasibly obtained using
.
A doublerow internally selfaligning ball bearing assembly with a tapered sleeve and locknut adapter for simplified mounting on a shaft of uniform diameter.
1.30. A split inner ring ball bearing assembly,
1. Cutaway view of turbofan gas turbine engine showingmainshaft bearing locations (courtesyof Pratt and Whitney, United Technologies Corp.).
_..
1.32. A tandemmounted pair of split inner ring ballbearings.
a ball bearing assembly. oller bearings are usually much stiffer structures (less deflection per unit loading) and provide greater fatigue endurance than do ball bearings of a comparable size.In general, they also cost more to manufacture, and hence purchase, than comparable ball bearing assemblies. Theyusually require greater care in mounting than do ball bearing assemblies. Accuracyof alignment of shafts andhousings can be a problem in all but selfaligning roller bearings.
earings. Cylindricalroller bearings asillustrated in Fig. 1.35 have exceptionally low friction torque characteristics that
.
A 90" contactangle ball bearing assembly.
thrust
make them suitable for high speedoperation. They also have high radialloadcarrying capacity, Theusual cylindrical rollerbearing is free to float t has two rollering flanges onone ring and none on the shown. in Fig. 1.3 equipping the bearing with a guide flange on the opposing ring (illustrated by Fig. 1.37), the bearing can be made to support some thrust load. revent high stresses at the edges of the rollers the rollers are usually crowned as shown in Fig. 1.38.This crowning of rollers also g the ~ e a r i n gprotection against the effects of slight is alignment. crown is ideally designed for only one condition of loading. Crownedraceways may be used in lieu of crowned rollers. To achieve greater radialloadcarrying capacity,cylindricalroller bearings are frequently constructed of two or more rows of rollers rather than of longer rollers. This is done to reduce the tendency of the rollers to skew. Figure 1.39 shows a small doublerow cylindrical rollerbearing designed for use in precision applications. Figure 1.40 illustrates a large multirow cylindrical rollerbearing for a steel rolling mill a~plication.
earings. A needle roller bearing is a Cylindrical roller bearing having rollers of considerably greater length than diameter. This
s
.
A 90" contact angle thrust ball bearinghaving a spherical seat to make it externally aligning.
bearing is illustrated in Fig. 1.41. ecause of the geometry of the rollers, they cannot be manu~actured asaccurately as other cylindrical rollers, nor canthey be guided as well. ~onsequently,needle rollerbearin eater friction than.other cylindrical roller bearings. Needle roller bearings are designed to fit in applications in which radial space is at a premium. Sometimes to conserve space the needles bear directly on a hardened shaft. They are useful for applications in which oscillatory motion occurs or in which continuous rotation occurs but loading is light and intermittent. The bearings may be assembled without a cage, as shown in Fig. 1.42. In this fullco~plementtype bearing, the rollers are fre~uently retained by turnedunder flanges that are integral with the outer shell. The raceways are frequently hardened but not ground.
s The singlerow tapered roller bearing shown in Fig. 1.43 has the ability inations of large radial and thrust loads or to carry thrust use of the difference betweenthe innerand outer raceway contact angles, there is a force componentthat drives the tapered rollers
:ylindrical roller bearing.
, Cylindrical roller bearings withoutthrust flanges.
against the guide flange. ecause of the relatively large sliding friction generated at this flange, the bearing is not suitable for high speed operation without special attention to cooling 'and/or lubrication. Tapered roller bearing terminolo~differs somewhat from that pertaining to other roller bearings, the inner ringbeing called the cone and
Cylindrical roller bearings havingthrust flanges.
"
(b)
.
(a)Spherical roller (fully crowned); ( b )partially crowned cylindrical roller (crown radius is greatly exaggeratedfor clarity).
the outer ring the cup. epe~dingon the magnitude of the thrust load to be supported, the bearing may have a small or steep contact angle, as shown in Fig. 31.44. Since tapered roller bearing rings are s e p a ~ ~ b lthe e, bearings are mounted in pairs as indicated in Fig. 1.45, and one bearing is adjusted against the other. To achieve greater radial loadcarryingcapacity and eliminate problems of axial adjustment due to distance between bearings, tapered roller bearings may be combined as shown in Fig. 1.46 into tworow bearings. Fig. 1.47 shows a typicaldoublerow tapered roller bearing assembly for a railroad car wheel application. Doublerow bearings may also be combined into fourrow or quad bearings for exceptionally heavy radial load applications such as rolling mills. Figure 1.48 shows a quad bearing having integral seals.
. A doublerow,cylindrical roller bearing for precision machine tool spindle applications.
As with cylindrical roller bearings, tapered rollers or raceways are usually crowned to relieve heavy stresses on the axial extremities of the rolling contact members. y e~uipping thebearing with specially contoured flanges, a special cage, and lubrication holes as shown by Fig. 1.49, a tapered roller bearing can be designed to operate satisfactorily under high loadhigh speed conditions. In this case, the cage is guided by lands on both the cone rib and the cup, and oil is delivered directly by centrifugal flow to the roller endflange contacts and cage railcone land contact.
ost spherical roller bearings have an outer raceway that is a portion of a sphere; hence, the bearings, as illustratedby Fig. 1.50, are internally selfali~ing.Each roller has a curved generatrix in the direction transverse to rotation that conforms relatively closely to the inner and outer
1.40. A multirow cylindrical roller bearing for a steel rolling mill application.
1.41. Needle roller bearing, nonseparable outer ring, cage, and roller assembly (courtesy of Torrington Company, Division of Ingersoll Rand Corp.).
.
(a)
(b)
Fullcomplement needle roller bearings. (a)Drawn cup assembly with trunnionend rollers and innerring; ( 6 )drawn cup assembly with rollers retained by grease pack (courtesy of Torrington Company, Division of Ingersoll Rand Corp.).
1.43. Singlerow tapered roller bearing showing separable cup and nonseparable cone, cage, and roller assembly (courtesy of the Timken Company). Ir
raceways. This gives the bearing high loadcarrying capacity. Various executions of doublerow, spherical roller bearings are shown in Fig. 1.51. Fig. 1.51a shows a bearing with asymmetrical rollers. This bearing, similar to tapered roller bearings, has force compone~ts thatdrive the rollers against the fixed central guide flange. earings such as illustrated in Fig. 1.51b and 1.5IChave symmetrical (barrel or hourglassshape) rollers, and these force components tend to be absent except under high
Small angle
,Steep angle
1.44. Small and steep contact angle tapered roller bearings.
E 1.45. Typical mounting of tapered roller bearings.
0
speed operation. oublerow bearings having barrelshape, symmetrical rollers frequently use an axially floating central flange as illustrated by Fig. 1.51d. This eliminates undercuts in the innerraceways and permits use of longer rollers, thus increasing the loadcarrying capacity of the bearing. Roller guiding in such bearings tends to be acco~plishedby
r
e
Doublerow tapered rolling bearings. ( a )Double coneassembly; (6) double
cup assembly.
F
1.47. Sealed, greased, and preadjusted doublerow, tapered roller bearing for railroad wheel bearings (courtesy of the Timken Company).
the raceways in conjunction with the cage. In a welldesigned bearing, the rollercage loads due to roller skewing maybe m i n i m i ~ e(see ~ ChapBecause of the closeosculationbetween rollers and raceways and curved generatrices, spherical roller bearings have inherently greater
GS
LLE
.
A fourrow tapered roller bearing with integral seals for a hot strip mill application (courtesyof SKI?).
High speed tapered roller bearing with radial oil holes and manifold. The '"z"type cage is guided on the cone rib and cup lands (courtesy of the Timken Company).
friction than cylindrical rollerbearings. This is due to the degree of sliding that occurs in the rollerraceway contacts. Spherical roller bearings are therefore not readily suited for use in high speed applications. They perform wellin heavy duty applications such as rolling mills,paper mills, and power transmissions and in marine applications. ~oublerowbearings can c a m combined radial andthrust load; they cannot support moment loading. Radial, singlerow, spherical roller bearings have a basic contact angle of 0".Under thrust loading, this angle does not increase appreciably;consequently, any amount of thrust loadingmagnifies rollerraceway loading substantially. Therefore, these bearings should
.
Cutaway view of a doublerow, spherical roller bearing with symmetrical rollers and a floating guide fiange (courtesy of SKI?).
I/ I / I
(d)
I
(4
(f)
1.51. Various executions of doublerow spherical roller bearings.
not be used to carry c o ~ b i n eradial and thrust loading whenthe thrust ent of the load is relatively large compared to the radial compospecial type of' singlerow bearing has a toroidal outer raceway, this is ill~stratedby Fig. 1.5 ; it can a c c o ~ ~ o d aradial te load together with some moment load, however,little thrust load.
earings. The spherical roller th very high loadcarrying capacity
IG Singlerow, radial, toroidal roller bearing (courtesy of SKY).
153. Cutaway view of a spherical roller thrust bearing assembly (courtesy of
SKY).
osculation between the rollers and raceways. It can carry a combination thrust and radial load and is internally selfaligning. Becausethe rollers are asymmetrical, force components occur that drive the sphere ends of the rollers against a concave spherical guide flange. Thus, the bearings experience sliding friction at thisflange and do not lend themselves readily to high speed operation. oller Thrust Bearings. Because of its geometry, the cylindrical roller thrust bearing of Fig. 1.54 experiences a large amount of sliding between the rollers and raceways, also calledwashers. Thus, the bearings are limited to slow speed operation. Sliding is reduced somewhat by using multiple short rollers in each pocket rather than a single integral roller. This is illustrated by Fig. 1.55. oller Thrust Bearings. This bearing, illustrated in Fig. 1.56 has aninherent force componentthat drives each roller against theoutboard flange. The sliding frictional forces generated at the contacts between the rollers and flange limit the bearing to relatively slow speed applications. Needle Roller Thrust Bearings. These bearings, as illustrated byFig. 1.5'7, are similar to cylindrical roller thrust bearings except that needle
1.54. IG bearing.
.
Cylindricalroller
thrust
Cylindrical roller thrust bearing having two rollers in each cage pocket; the bearing has a spherical seat for external alignment capability.
1.56. Tapered roller thrust bearing. ( a ) Both washers tapered; ( b ) one washer tapered.
1.57. Needle roller thrust bearing (courtesy of Torrington Company, Divisionof Ingersoll Rand Gorp.).
rollers are used in lieu of normal size rollers. Consequently, rollerwasher sliding is prevalent to a greater degree and loading must be light. The principal advantage of the needle rollerthrust bearing is that it requires only a narrow axial space. Figure 1.58 illustrates a needle rollercage assembly that may be purchased in lieu of an entire bearing assembly.
1.58. Thrust needle rollercage assembly (courtesy of Torrington Company, Division of Ingersoll Rand Gorp.).
Bearings for linear motion, such as those used in machine tool “ways”for example, Vwaysgenerally employ only lubricated sliding action. These sliding actions are subject to relatively high stickslip friction, wear, and subsequent loss of locational accuracy. Ballbushing operating on hardened steel shafts, illustrated schematically by Fig. 1.59, provide many of the low friction, minimal characteristics o f radial rolling bearings. The ball bushing, which provides linear travelalong the shaft,limited y builtin motion. stoppers, contains three or more oblon~circuits of recirculating balls. As illustrated in Fig. 1.60, one portionof the oblong ball complement supports load on the rolling balls while the remain in^ balls operate with clearance in the return track. The ball retainer units can be fabricated relatively inexpensively of pressed steel or nylon (polyamide)material. Figure 1.61 is a p h o t o ~ a p h showing an actual unit with its components. Ballbushings of instrument quality are made to operate on shaft ~iameters assmall as 3.18 mm (0.125 in.).
FI
.
Schematic illustration of a ball bushing.
Load carryingballs I
” I
~ecircui~ring balls in ~ t e a r ~ n c e
.
Schematic diagram of a ball bushing recirculating ball set.
1. Linear ball bushing showing various components.
all bushings can belubricated with mediumheavy weight oil or with a light grease to prevent wear and corrosion. For highlinear speeds, light oils are recommended. Seals can beprovided;however,friction is increased s i ~ i ~ c a n t l y . As with radial ball bearings, life can belimited by subsurfaceinitiated fatigue of the rolling contact surfaces. A unit is usually designed to perform satisfactorily for several million units of linear travel. Since the hardened shaft is subject to surface fatigue and/or wear, provision can be made for rotating the bushing or shaft to bring new bearing surface into play.
ings were greaselubricated and, owing to the grease deterioration in this difficult application?needed to be regreased periodically. If this was not accomplished with care, inevitabl~contamination was introduced into the bearings an bearing longevity was substantia11 come thissitu ion,many bearings wereprovi greased and sealedforlife, d plex sets as shown units needed to be pressfitte into the wheel hub semblyfor the automobile m cturerand to m was made integral with the bearing outer ring as thus, the unitcould be boltedto studs on the wheel, contained unit with a flange integral with each r o use for nondriven wheels; the unit can be bolted to the nd the wheel for simple assembly. uty vehicles such as trucks, tapered roller stead of ball bearings, to the bearing unit as shown by the taselfig. 1.64. This compact, preadjuste~? unit is equipped with an inte a1 s eed sensor to the antilock braking system arings to measure applied lo
To reduce friction associated with the follower contact on cams, rolling motion maybe employed. The needle rollerbearing is particularly suited to this application because it is radially compact. Figure 1.65 shows a needle roller bearing, cam follower assembly.
.
Modernautomobilewheel preadjusted, greased and sealed€orlife,ball bearing units. ( a )Without flanges; unit is pressfitted into wheel hub and slipfitted onto the axle; ( b ) with a single flange integral with the bearing outer ring; (e) with flanges integral with bearing outer and inner rings (photographs courtesy of SKI?).
.
Modern truck wheel preadjusted, greased, and sealedforlife, Lapered roller bearing unit (courtesy of SKI?).
4. Selfcontained?tapered roller bearing with an integral speed sensor to provide signal to the antilock braking system (courtesy of the Timken Company).
rplane and helicopterpower transmission bearing a~plications are nerally characterized by the necessity to carry heavy loads at h i g ~ speed while mini~izingbearing size. The bearings are generally man~facturedfromspecialhigh s t ~ e n ~high h , quality steels e weight and outof a steel bearing itself is signi~cant,minimizing bearing side diameter aids com~actnessin engine design, allowing surro~nding engine co~ponentsto be smaller and weigh less. us, aircraft power
.
Needlerollercamfollowerassembly(courtesy Ingersoll Rand Gorp.).
of ~ o r r i n ~ o Company, n
for the turbine e n ~ n application. e
ages have i l l ~ ~ t r a t eand d described various types and e~ecutionsof ball and roller bearings. It is not to be c o n s t ~ e d that every
.
Aircraft power transmissionbearings:(left)cylindricalrollerbearing; of ~ T N ) . (right) spherical planet gear bearing (courtesy
1.67. Aircraft gas turbine engine, cylindrical roller bearing (courtesy of SKI?).
F
1.68. Aircraft gas turbine engine mainshaft bearing components: lower leftsplit inner ring ball bearing; center and upper rightcylindrical roller bearing inner and outer ring units (courtesy of FAG OEM und Handel AG).
type of rolling bearing has been described; discussion has been limited to the most popular and basic forms. For example, there are cylindrical roller bearing designs that use snap rings, instead of machined and and IS0 standards on terminology [1.7] ore common bearing designs. It is also ings are specially designed for applicadiscussed herein only to indicate that imes warranted by the application. In ail additional cost for the bearing or bearing unit; however, such cost increase is usually offset by overall ef
ficiency and cost reduction brought to mechanism and machinery design, manufacture, and operation.
1.1 D. Dowson, History of Tribology, 2nd ed., Longman, New York (1999).
1.3 T. Tallian, “Progressin Rolling Contact Technology,”SKF Report AL690007 (1969). 1.4 T. Tallian, “Weibull Distribution of Rolling Contact FatigueLife and Deviations Therefrom,”ASLE Trans. 5(1),183196 (1962). 1.5 American National Standards Institute, American National Standard ( ~ S I I ~ ~ A ) Std, 12.21992, “Instrument Ball BearingsMetric Design” (April 6, 1992). 1.6 American National Standards Institute, American National Standard ( ~ S I / ~ ~ A ) Std. 12.21992,“Instrument Ball BearingsInch Design” (April 6, 1992).
Symbol
A l3 d
D
Description
Distance between raceway groove curvature centers AID Raceway diameter Bearing pitch diameter Ball or roller nominal diameter Mean diameter of tapered roller Diameter of tapered roller at largeend Diameter of tapered roller at smallend rID effective Roller length Distance between cylindrical roller guide flanges Roller length endtoend diametral Bearing clearance Bearing free endplay
Units mm (in.) mm (in.) mm (in.) rnm (in.) mm (in.) mm (in.) mm (in.) mrn (in.) mm (in.) mm (in.) mm (in.) mm (in.)
48
ROLLING EEARING ~ C R O G ~ O ~ T ~ S
Symbol
a
C 0
i
r
Units Raceway groove curvature radius Roller corner radius Roller contour radius Spherical roller bearing diametral play Number of rolling elements Free contact angle Contact angle Tapered roller bearing flange angle Roller angle Shim angle I) cos ald,, ~ i s a l i ~ m eangle nt Curvature Curvature difference Curvature sum Osculation Angular velocity
mm (in.) mm (in.) mm (in.) mm (in.) 0 0
radlsec
S U ~ S ~ R I ~ T ~ Refers to cage Refers to outer raceway Refers to inner raceway Refers to roller
Although balland roller bearings appear to be simple mechanisms,their internal geometries are quite complex. For example,a radial ball bearing subjected to thrust loading assumes angles of contact between the balls and raceways in accordance with the relative conformities of the balls to the raceways and the diametral clearance. On the other hand, theability of the same bearing to support the thrustloading dependson the contact angles f o r ~ e dThe . same diametral clearance or play produces an axial endplay that may or may not be tolerable to the bearing user. In later chapters it will bedemonstrated how diametral clearance affects not only contact angles and endplay but also stresses, deflections, load distributions, and fatigue life, Stresses, deflections, load distribution, and life in roller ~earingsare also affected by clearance. In the determination of stresses and deflections the relative conformities of balls and rollers to their contacting raceways are of vital interest. n this chapter the principal macrogeometric relationships govern in^ the operation of ball and roller bearings shall be developed and examined.
The ball bearing can be illustrated inits most simple formas inFig. 2.1. From Fig. 2.1 one can easily see that the bearing pitch diameter is approximately equal to the mean of the bore and 0.d. or
dm = +(bore + 0.d.)
(2.1)
More precisely, however, the bearing pitch diameter is the mean of the inner and outer ring raceway contact diameters. Therefore,
Generally, ball bearings and other radial rolling bearings such as cylindrical roller bearings are designed with clearance. From Fig. 2.1, the diametral* clearance is as follows:
2.1. Radial ball bearing showing diametral clearance.
* Clearance is always measuredon a diameter; however, becausemeasurement takes place in a radial plane, it is commonly called radial clearance. Thistext will use diametral and radial clearance interchangeably.
5
R O L L ~ GB
E
~ ~ ~C RGO G E O
Table 2.1 taken from Reference 2.1 givesvalues of radial internal clearance for radial contact ball bearings under no load.
.
A 209 singlerow radial deepgroove ball bearing has the following dimensions: Inner raceway diameter, di Outer raceway diameter, do Ball diameter, D Number of balls, is Inner groove radius, ri Outer groove radius, ro
52.291 mm (2.0587 in.) 77.706 mm (3.0593 in.) 12.7 mm (0.5 in.) 9 6.6 mm (0.26 in.) 6.6 mm (0.26 in.)
Determine the bearing pitch diameter d m and diametral clearance Ipd.
P,
=
Q(52.3 i77.7)
=
65 mm (2.559 in.)
=
do  di  2D
=
77.706  52.291  2
=
0.015 mm (0.0006 in.)
X
12.7
se The ability of a ball bearing to carry load depends in large measure on the osculation of the rolling elements and raceways. Osculation is the ratio of the radius of curvature of the rolling element to that of the raceway in a direction transverse to the direction of rolling. From Fig. 2.1it can be seen that for a ball mating with a raceway, osculationis given by
+ = D
2r
Letting r
= fD,
osculation is
# =  1. 2f
(2.5)
It is to be noted that the osculation is not necessarily identical for inner and outer contacts.
s
c  m m m w m o u3 m w w mwcm
v?v?v?u?v?v?
0000000000ddddd
Determine the osculation in the 209 radial ball bearing of Example 2.1. f,,ri=
L)
6.6 = 0.52 12.7 0.962
f O
=5=" 6.6 L)
12.7

0.52 0.962
Because a radial ball bearing is generally designed to have a diametral clearance in the noload state, the bearing also can experience an axial play. Removal of this axial freedom causes the ballraceway contact to assume an oblique angle with the radial plane; hence, a contact angle different from zero degrees will occur. Angularcontactball bearings are specifically designedto operate under thrust load, and the clearance built into the unloaded bearing along with the raceway groove curvatures determines the bearing free contact angle. Figure 2.2 shows the geometry
~
.
Axis of rotation
Radial ball bearing showing ballraceway contact dueto axial shift of inner and outer rings.
4
ROLLING BEARING ~ ~ R O G E O ~ T R ~
of a radial ball bearing with axial play removed. From Fig.2.2 it can be seen that the distance between the centers of curvature 0' and 0" of the inner and outer ring grooves is
A=r,+riD ~ubstitutingr = f D yields
in which B = f, + fi  I is defined as the totalcurvature of the bearing. Also from Fig. 2.2, it can be seen that the free contact angle is the angle made by the line passing through the points of contact of the ball and both raceways and a plane perpendicular to the bearing axis of rotation. The magnitude of the free contact angle can be described as follows:
or ao = cos1 ( 1 
$1
3. A 218 an~larcontactball bearing has dimensions as follows: Inner raceway diameter, di Outer raceway diameter, do Ball diameter, D Inner groove radius, ri Outer groove radius, ro
102.79 mm (4.047 in.) 147.73 mm (5.816 in.) 22.23 mm (0.875 in.> 11.63 mm (0.4578 in.) 11.63 mm (0.4578 in.)
Determine the free contact angle of this bearing.
"= 11*63
22.23
0.5232
"11*63 
22.23
=
A
0.5232
0.5232
+ 0.5232  1 = 0.0464
= BD =
0.0464
(2.7) X
22.23
=
1.031 mm (0.0406 in.)
do  di  2D
Pa =
147.73  102.79  2
a" = cosl(l =
(2.3)
(
X
22.23
=
0.48 mm (0.019 in.)
2)
cos1 1 
=
2
X
40"
1.031
If in mounting the bearing an interference fit is used, then the diametral clearance must be diminished by the change in ring diameter to obtain the free contact angle. Hence
(
a" = cos1 1 
Pa + APd 2A
)
(2.10)
Because of diametral clearance a radial bearing is free to float axially under the condition of no load. This free endplay may be defined as the maximum relative axial movement of the inner ring with respect to the outer ring under zero load. From Fig. 2.2,
gP,
P,
=A =
sin a"
(2.11)
2A sin a'
(2.12)
Figure 2.3 shows the free contact angle and endplay versus P,/D for singlerow ball bearings. Roublerow angularcontact ball bearings are generally assembled with a certain amount of diametral play (smaller than diametral clearance). It can be determined that the free endplay for a doublerow bearing is
0 
0.12
U
0.11
0.10
0.09 0.08
0.07
3
0.06
0.05
0.04 0.03 0.02
0.01
Split inner ringball bearings as illustrated in Fig. 2.4 have inner rings that are ground with a shim between the ring halves. The width of this shim is associated with a shim angle that is obtained by removing the shim and abutting thering halves. From Fig. 2.5 it can be determined that the shim width is given by

F
~
bearing ball
ws
~Inner rings ~ of Esplit inner ring grinding. showing for shim
2.4.
2.6. Split inner ring ball bearingassembly showing shim angle.
w , = (2ri 
Since fi
=
D)sin a,
(2.14)
ri/D, equation (2.14) becomes w , = (2fi  l)D sin a,
(2.15)
The shim angle a,, and the assembled diametral play S, of the bearing accordingly dictate the free contact angle. The effective clearance P, of the bearing may be determined from Fig. 2.5 to be
P,
=
s, + (2fi  1)(1 cos a,)D
(2.16)
Thus, the bearing contact angle which is shown in Fig. 2.2 is given by
ree ~urthermore? diametral clearance can allow a ball bearing to misalign slightly under no load. The free angle of misalignment is defined as the maximum angle through which the axis of the inner ringcan be rotated with respect to the axis of the outer ring before stressing bearing components. From Fig. 2.6, using the law of cosines it can be determined that (2.18)
(2.19) Therefore 8, the free angle of misalignment?is
e = 0i + eo
(2.20)
Since the following trigonometric identity is true, cos and since
ei + cos e* = 2 cos $(ei + eo>cos +(Oi

eo)
(2.21)
ei  eo approaches zero, therefore, e = 2 COS^
(,,s
4
;
cos 00
(2.22)
or 0 = 2 cos1
Determine the free contact angle a', free endplay P,, and free angle of misalignment of the 209 radial ball bearing in Example 2.1.
BALL B E ~ I N G S
2.6. (a) Free misalignment of inner ring of singlerow ball bearing, (b) free misalignment of outer ring of singlerow ball bearing.
ROLL~G ~~~G
fi = f,
~
~
R
~
Ex. 2.2
0.52
=
O
dm = 65 mm (2.559 in.)
Ex. 2.1
Pd = 0.015 mm (0.0006 in.)
Ex. 2.1
B=fi+f*l
A
=
0.52
=
BD
=
0.04
+ 0.52  1 = 0.04 X
12.7 = 0.508
A ! ' = cos1 (1 =
(
cos1 1 
2) 2
X
0.508
Pe = 2A sin ' a =
'Os'
2
X
['
0.508
(2.12) X
0.015

sin (9'52')
=
0.174 mm (0.0069 in.)
(2 X 0.52  1) X 12.7  (0.015/4) 0.52  1)X 12.7  (0.01512)
E (65 + (2 x
 1) X 12.7  (0.015/4) + 65 (2 X(2 0.52 X 0.52  1) X 12.7 + (0.015/2)
Note "howsmall the free angle of misalignment is.
T w o bodies of revolution having different radii of curvature in a pair of
principal planes passing through the contact between the bodies may contact each other at a single point under the condition ofno applied load. Such a condition is called point contact. Figure 2.7 demonstrates this condition. In Fig. 2.7 the upper body is denoted by I and the ,lower body by 11. The principal planes are denoted by 1 and 2. Therefore, the radius of
~
O
~
2.7. Geometry of contacting bodies.
curvature of body I in plane 2 is denoted by r12.Since 7 denotes radius of curvature, curvature is defined as P";
1
(2.24)
Although radius of curvature is always of positive sign, curvature may be positive or negative, convex surfaces being positive and concave surfaces negative. To describe the contact between mating surfaces of revolution, the following definitions are used. 1. Curvature sum: 1
1
1
~ p =  +  +  + 711
r12
7111
.
1
(2.25)
7112
. Curvature difference: (2.26)
2
~ C ROLLING R O GB E~ OI N ~ ~G R ~
In equations (2.25)and (2.26)the sign convention for convex and concave surfaces is used. Furthermore, care must be exercised to see that F(p)is positive. By way of example, F(p) is determined for a ballinner raceway contact as follows (see Fig. 2.8):
("
rIIl = i d i = 1  D) 2 cos a
Let Y=
D cos a
(2.27)
dm
Then PI1 = PI2
2
=
5
For the ballouter raceway contact pI1 = p12 = 2 / D as above; however, rIIl = 21
Therefore,
(
COS dma
+ D)
3
~
PI11 =
I 2.8. Ball ~ bearing ~ geometry E
(; +J 1
PI12 =
F(P)o =
"
f0D
fo
1
l + Y
gy
4"" fo
(2.30)
l + Y
F ( p ) is always a number between 0 and 1.
le 2.5. Determine the values of curvature sum and curvature difference for the 209 radial ball bearingof Example 2.1, subjected to radial load only.
6 = fo
=
0.52
dm = 65 mm (2.559 in.)
Ex. 2.2 Ex. 2.1
4
ROLLING BEARING ~ C R O G E O ~ ~ R S
D cos a (2.27)
dm
12.7
12.7
'
0.52
cos (0") = 0.1954 65
X
+ 1 x 0*1954) = 0.202 mm' 0.1954
(5.126 in.1) (2.28)
1 2 X 0.1954 0.52 ' 1  0.1954 1 2 X 0.1954 0.52 "I 1  0.1954
."A

=
0.9399
,
" 4
(2.30)
12.7 0.52
x
1 + 0.1954 >,,,*'
=
0.1378 mm' (3.500 in.1)
(2.31)
1 2 X 0.1954 0.52 1 + 0.1954 . 1 2 X 0.1954 4"0.52 1 + 0.1954 "
=
0.9120
~ F(p),. This condition will be used to Note the when fi = f,, F ( P ) > demonstrate in a later example that an elliptical area of contact of
greater ellipticity generally exists at the inner raceway contact as opposed to the outer raceway contact. Determine the magnitude of curvature sum and curvature difference for the 218 angularcontact ball bearingof Example 2.3 subjected to light axial loading.
fi = f, = 0.5232 (102.79 = 400*
=
Ex. 2.3
+ 147.73) = 125.26 mm (4.932 in.) Ex. 2.3
I) cos a
Y=
(2.27)
dm
 22.23
X cos (40") 125.26
=
o.1359
pi=$(4"+*) 1
fi 22.23 0.5232 =
(2.28) 1Y
1  0.1359
0.108 mml (2.747 in.')
(2.29)
c _
c _
2 X 0.11359 l + 0.5232 1  0.1359 2 X 0.1359 l + 4" 0.5232 1  0.1359
=
0.9260
*It will bedemonstratedin a later chapter that contact angle increases under thrust (axial) loading. This will not be consideredin this example.
x p o D=” (4 ” ” =
fo
(2.30)
1 2+ yY )
1
” 22.23 .(4  0.5232  1 + 0.1359 0.0832 mm’ (2.114 in.’) 1
2Y
”~
(2.31) ’f
fo
l + Y
2 X 0.1359 0.5232 1 + 0.1359 1 2 X 0.1359 4”0.5232 1 + 0.1359 1
”
=
0.9038
Comparison of the F ( P ) and ~ F(p), values in Examples 2.6 and 2.7 indicates that magnitudes in the neighborhood of 0.9 are to be expected for ball bearings. Larger magnitudes of 6 and fo cause subsequently smaller valuesof F(p).
Equation (2.1) may also be used for spherical roller bearings to estimate pitch diameter. Radial internal clearance, also called diametral play, as illustrated by Fig. 2.9, is given by equation (2.32).
where ri and ro are theraceway contour radii. The diametralplay s d can be measured with a feeler gage. Table 2.2, excerpted Erom Reference 2.1, gives standard values of radial internal clearance, diametral play, under no load.
Radial spherical roller bearings are normally assembled with free diametral play and hence exhibit free endplay Pe. From Fig. 2.9, it can be seen that
7
""
"
Axis of rotation
F~~UR 2.9.E Schematic diagram of spherical roller bearing showing nominal contact angle a, diametral play S,, and endplay P,.
ro cos p
(
ro 
=
2)
cos a
(2.34)
or
p
=
[
cos1 (1 
2)
cos a
Therefore, it can be determined that
P,
=
4rJsin

sin a) + 2S, sin a
(2.35)
7. Estimate the magnitude of the free endplay P, for a 22317 spherical roller bearing having the following dimensions:
m m 0 m 0 0 m 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 PQ,Ommrx,meoOm"MPm"M"mmOooQ,o P?i l4 ?i du m M M M d d m m eo eoP rx, Q, 0_?4/iemeqm_
omooomoooooooooooooooooooo eoPrx,Omdrx,?idrx,ddrx,dueoOmO~du~mm?imdu ?id*(
d?iddmmmMMMddmmeoeo~"rx,Q,oe?i_me
d d d
omooomoooooooooooooooooooo eo"rx,Odudrx,?idrx,ddrx,dueoOmoeomrx,mm?idum ?id?idmdum~MmddmmeoeoPPrx,Q,oe?i~me
T+?id?iNmmmMMMddmmeoeo""rx,Q,
mmomoomooooooooooooooooooo dmeo"Q,?iMeoQ,mdeoQ,mmP?imomomoPeoM
m o m m m o o o m o o o o o o o o o o o o o o o o o
~ddmeorx,omd"rx,Omdeorx,?idP?idrx,Mrx,m?i ?idd?i?idummmduMMMdddmmeo"
d?i?i?idmmmmmmMMdddmmeoP
m o m m m o o o m o o o o o o o o o o o o o o o o o ~ddmeorx,OmdPrx,Omdeorx,?id"ddrx,Mrx,md
?idd?id?iT+dudummmMMmdd
3momooommooooooooooooooooo NmMMdmeoPQ,dmMdmPQ,Odudeorx,dm"Mrx,
B
3
B
3
H
8
li
7
Roller diameter, D Number of rollers per row Roller effective length, Z* Roller contour radius, R Inner raceway contour radius, ri Outer raceway contour radius, ro Bearing pitch diameter, d, Nominal contact angle, a Diametral play
[ [
p
=
cos1
p
=
cos1 (1 
(1 
25 mm 14 20.762 mm 79.959 mm 81.585 mm 81.585 mm 135.07'7 mm 12" 0.102 mm
(0.9843 in.) (0.8154 in.) (3.148 in.) (3.212 in.) (3.212 in.) (5.318 in.) (0.004 in.)
5) .] cos
)
'"O2 cos (12")] = 12.167" 2 81.585
PC= 4r,(sin p  sin a) + 28, sin a
P,
(2.34)
=
4 * 85.585(0.2108  0.2079) + 2 0.102 * 0.2079
=
0.956 mm (0.0376 in.)
(2.35)
i ~ ~ applies to spherical roller bearings in that, as The term o s c ~ Z a t also illustrated by Figs. 2.9 and 2.10, the rollers and raceways have curvature in the direction transverse to rolling. In this case, osculation is defined as follows: R pr
(2.36)
in which R is the roller contour radius.
.8. Determine the osculation at each raceway contact for the 22317 tworow spherical roller bearing of Example 2.7.
*Roller effective length is the length presumed to be in contact with the raceway under loading. Generally, I = I,  Zr,, in which I",is the roller corner radius or the grinding undercut, whichever is larger. A combination of the corner radius and grinding undercut may also be required to estimate effective length.
71
S P ~ R I ROLLER C ~ B E ~ ~ G S
FIGURE 2.10. Spherical roller bearing geometry.
+.=R  =  =79.959 ri 81.585
R ro
+i="=
79.959 81.585
o.98
=
0.98
(2.36) (2.36)
mature For spherical roller bearings with point contact between rollers and raceways, the equations for curvature sum and difference are as follows (see Fig. 2.IO):
72
(2.40)
9. Calculate the magnitudes of curvature sum and difference for the 22317 spherical roller bearing of Example 2.7.
Y=
D cos a
(2.27)
dm
=
1
11 " " ri 79.959
25 cos (12")/135.077 == 0.1810
" "
_R
=
81.585

0.00025mm'(0.0064in.')
(2.37)
0.09793 mm' (2.487 in.')
(2.38)

1
1 ro
2/(1  0.1810) 2/(1  0.1810) 1 79.959

25
X
+ 25 X
0.00025 0.00025
1 81.585
" " " "
=
0.00025 m~'(0.0064 in.')
=
0.9951.
3
GS
(2.39)
+ 25 X 0.00025)
1 =
0.068 mm' (1.726 in.')
+D l + Y
;(
:>
(2.40)

 2/(1 + 0.1810)  25 X 0.00025 = o.9929 2/(l + 0.1810) + 25 X 0.00025
Note that the magnitude of F ( p ) approaches unity for spherical roller bearings.
ite Equations (2.1)(2.3) are valid for radial, cylindrical roller bearings as well as ball bearings. Table 2.3 gives standard values of internal clearance forradial cylindrical rollerbearings. These data areexcerpted from Reference 2.1.
A 209 cylindrical rollerbearing has thefollowing dimensions: Inner raceway diameter, di Outer r~cewaydiameter, d, Roller diameter, D oller effective length, Z oller total l e n ~ hZ,, Number of rollers, i5
54.991 mm 75.032 mm 10 mm
(2.1 (2.9 (0.39~7in.)
14
ine the bearing pitch diameter dm and diametral clearance
5
E
tll
B
3 2
ri
H
ROLLING BEARD46 ~ C R O G E O ~ ~ Y
Pd
=
$(54.991 + 75.032)
=
65.011 mm (2.559 in.)
=
do  di  2D
=
75.032  54.991  2 10
=
0.041 mm (0.0016 in.)
0
Figure 2.11 illustrates a roller in a radial cylindrical roller bearing having two roller guide flanges on both inner and outer rings. In this case the roller is shown in contact with both inner and outer raceways, which would occur in the bearing load zone when simple radial loading is applied to the bearing (see Fig. 7.3). It is to be noted that clearance exists in theaxial direction betweenthe roller ends and the roller guide flanges. It can be seen from Fig. 2.11that the bearing will experience an endplay defined by
P,
=
2(2f  I,)
(2.41)
where Zf is the distance between the guide flanges of a ring and I , is the total length of the roller. As mentioned in Chapter 1 and discussed in Chapter 7, radial cylindrical roller bearings with guide flanges on both inner and outer rings can support small amounts of applied thrust load
1 2
__.
.11. Schematic drawingof a radial cylindrical roller bearing havingtwo integral roller guide flangeson the inner ring and one integral and one separable guide flange on the outer ring.
RINGS
77
ERED ROLLER
in addition to the applied radial load. Thebearing endplay influences the number of radially loaded rollers that will participate in supporting the thrust load. The endplay also influences the degree of roller skewing which can occur during bearing operation. See Chapter 14.
Most cylindricalroller bearings employcrowned rollers to avoid the stressincreasing effects of edgeloading (see Chapter 6 ) . For these rollers, even if fully crowned as illustrated by Fig. 1.31a, the contour or crown radius R is very large. Moreover, even if the raceways are crowned, R = ri = r, * a.Therefore, consideringequations (2.37) and (2.39),which describe the curvature sum for inner and outer raceway contact respectively, the difference of the reciprocals of these radii is essentially nil, and (2.42) (2.43)
be Examining equations (2.38) and (2.40), it can
seen that
=
F(p), = 1.
The nomenclature associated with tapered roller bearings is different than thatfor other types of roller bearings. For example, as indicated by Fig. 2.12, the bearing inner ringis called the cone and theouter ring the cap. It can be seen that the operation of the bearing is associated with a pitch cone; equation (2.1) can be used to describe the mean diameter of that cone. For manycalculations, this mean cone diameter will be used as the bearing pitch diameter dm.Figure 2.13 indicates dimensions and angles necessary for the p ~ r f o r ~ a n analysis ce of tapered roller bearings. From Fig. 2.13, it can be seen that ai,the inner racewayroller contact e = 3 cone angle, a,, the outer racewayroller contact angle = cup e, af,the roller large endflange contact angle = 4 cone back face d aR= roller angle. Dm= = the large end diameter of the in = the small end diameter of the roller, whichhas anendtoend length of I,.
+
78
ROLLING BEARING ~ C R O G E O ~ T R ~
1 CUP FRONT FACE CAGE
CONE BACK FACE RIB
CONE FRONT FACE RIB,
CONE BACK FACE
CONE FRONT FACE
W
a UI
I I I I
I I
CAGE CLEARANCE
2.12. Schematic drawing of tapered roller bearing indicating nomenclature.
FI
2.13. Internal dimensions for tapered roller bearing performance analysis.
CLOSURE
Tapered roller bearings are usually mounted in pairs. In general, the clearance is removed so that a Z ~ ~ e  t o  Zfiti ~ise achieved under no load. It is possible, however, fora bearing set support in^ substantially applied radial loading, that a small amount of endplay is set at room temperature mounting to achieve desired distribution of load amongthe tapered rollers under higher temperature operating conditions. Endplay in tapered roller bearings is therefore associated with the bearing pair.
From Fig. 2.13, it is seen that theouter raceway contact angle is greater than the inner raceway contact angle. Therefore, considering equations (2.37) and (2.39), the curvature sums for the inner and outer raceway contacts are given by (2.44) (2.45)
where (2.46) (2.47) .Rm
cos a,
Y o ,=
(2.48)
dm
These equations give approximate answers in the respective calculations of curvature sum since the mean radius of the roller lies in a plane slightly angled to that in which the raceway rolling radius lies. As far c~lindricalroller bearings, F ( P ) = ~ F(p), = 1.
The relationships developed in thischapter are based onlyon the macroshapes of the rolling components of the bearing. When loadis applied to the bearing, these contours may be somewhat altered; however, the undeformed geometry must be used to determine the distorted shape.
ROLLING BEARING ~ ~ R O G E O ~ T R Y
Numerical examples developedin this chapter were of necessity very simple in format. The quantity of these simple examples is justified since the results from the calculations will subsequently be used as starting points in more complex numerical examples involving stresses, deflections, friction torques, and fatigue lives.
2.1. American National Standards Institute,American National Standard ( ~ S I / ~ ~ A ) Std. 201987,“Radial Bearings of Ball, Cylindrical Roller, and Spherical Roller Types, Metric Design” (October 28, 1987). 2.2 A. Jones, Analysis of Stresses and Deflections, v01. 1, New Departure Division, General Motors Corp., Bristol, Conn., 12 (1946).
T Symbol
Description Basic inner ring width Single width of an inner ring Basic outer ring width Single width of an outer ring Basic bore diameter Bearing inner raceway diameter g outer raceway diameter diameter of a bore Single plane mean bore diameter diameter er of an outside surface ngle plane mean outside ameter ~ o m m o ndiameter asic housing bore
Units mm (in.) mm (in.) mm (in.) mm (in.) mm (in.) mm (in.) mm (in.) mm (in.) mm (in.) mm (in.) mm (in.) mm (in.) mm (in.) mm (in.)
~ E ~ E R E ~1~~~ ~ C E
Symbol
Description Outside ring 0.d. Inside ring i.d. Basic shaft diameter Modulus of elasticity Interference Radial runout of assembled bearing inner ring Radial runout of assembled bearing outer ring Length Bearing clearance Pressure Ring radius Inside radius of ring Outside radius of ring Inner ring reference face runout with bore Outside cylindrical surface runout with outer ring reference face Axial runout of assembled bearing inner ring Axial runout of assembled bearing outer ring Radial deflection Bore diameter variation in a single radial plane Mean bore diameter variation Mean outside diameter variation Outside diameter variation in single radial plane Single inner ring widthdeviation from basic ingle outer ring width deviation from basic Single bore diameter deviation from basic Single plane mean bore diameter deviation from basic for a tapered bore small end Single plane mean bore diameter deviation at large end of tapered bore Single outside diameter deviation from basic
CE
Units mm (in.) mm (in.) mm (in.) N/mm2 (psi) mm (in.) pm (in.) pm (in.) mm (in.) mm (in.) N/mm2 (psi) mm (in.) mm (in.) mm (in.) pm (in.) pm (in.) pm (in.) pm (in.) mm (in.) pm (in.) pm (in.) pm (in.) pm (in.) pm (in.) pm (in.) pm (in.) p m (in.)
pm (in.) pm (in.)
83
GENE
Symbol
Description Single plane mean outside diameter deviation from basic Clearance reduction due to press fitting of bearing in housing Clearance reduction due to press fitting of bearing on shaft Clearance increase due to thermal expansion Temperature Strain in radialdirection Strain in tangentialdirection Coefficient of linear expansion
s 0r
0t
Poisson's ratio Normal stress in radialdirection Normal stress in tangential direction
Units pm (in.) mm (in.) mm (in.) mm (in.) "C("F) mm/mm (in./in.) mm/mm (in./in.) m m / ~ m / " C(in./ in./ O F ) N/mm2 (psi) N/mm2 (psi)
all and roller bearings are usually mounted on shafks or in housings with interference fits. This is usually done to prevent fretting corrosion that could be produced byrelative movement between the bearing inner ring bore and the shaft 0.d. and/or the bearing outer ring 0.d. and the housing bore. Theinterference fit of the bearing inner ringwith the shaft is usually accomplished by pressing the former member over the latter. In some cams, however, the inner ringis heated to a controlled temperature in an oven or in an oil bath. Then the inner ring is slipped over the shaft and allowed to cool, thus accomplishing a shrink fit. Press or shrink fitting of the inner ringon the shaft causes the inner ring to expand slightly. Similarly, press fitting of the outer ring in the housing causes the former memberto shrink slightly. Thus, the bearing's diametral clearance will tend to decrease. Large amounts of interference in fitting practice can cause bearing clearance to vanish and even produce negative clearance or interference in the bearing. Thermal conditions of bearing operation can also affectthe diametral clearance. Heat generated by friction causes internal temperatures to rise. This in turn causes expansion of the shaft, housing, and bearing components. Depending on the shaft and housing materials and on the magnitude of thermal gradients across the bearing and these supporting structures, clearance can tend to increase or decrease. It is also apparent
~
E
~
~ FI!CTING ~ N
C
E
CE
that the thermal environment in which a bearing operates may have a significant effect on clearance. In Chapter 2 it was demonstrated that clearance significantly affects ball bearing contact angle. Subsequently,the effect of clearance on bearing internalload distribution and life will beinvestigated. It is therefore clear that the mechanics of bearing fitting practice is an important part of this book.
~ t a n d a ~ ddefining s recommended practices for ball and roller bearing usage were first developed in the United States by the tiFriction Bearing ~anufacturersAssociation (AE’BU), which has nowbecome the American Bearing ~anufacturersAssociation (AI3M.A). tinues the process of revising the current standards andproposing and preparin~new standards as deemed necessary by its bearing industry member companies. AI3M.Agenerated standards are subsequently proposed to the American National Standards Institute (ANSI) as United States national standards. ANSI has a committee dedicated to rolling bearing standard activities; this committee has representatives of bearing user organizations such as major industrial manufacturers and the U.S. government. Other countries have national standards organizations similar to ANSI; for example, DIN in Germany and JNS in Japan. Currently, 26 documents, so tric and English unit system parts, havepublished been as standards. Any national stand quentlyproposed be to the International Organization for Standards?andafter extendednegotiation ~ublished asinternational Standard (ISO)”with an identifying number. In this chapter, various bearing, shaft, and housing tolerance data are pted from the American National Standards. ference [3.1] defines recommended practice in fitting bearing inner d outer rings in housings. These fits are recommended ormal, and heavy loadingas defined by Fig. 3. I. Figure ations and relative m a ~ i t u d e of s the shaftbearing bore and ho~singbearing0.d. tolerance ranges. Each sha~bearingfit tolerance range is d e s i ~ a t e by d a lower cas for example, g6,h5 and so on. Similarly, ea ists of an upper cas housingbearing fit or Table 3.1 gives the forexample,6.7 r rings on shaft practicefor fitting ameter tolerance limits correspond in^ to milar data for fitting of bearing outer rings in housing in references [3. eraiice ranges on bearing bore and o.d. for various types of radial bear
ST€?,W,N A T I O ~AND ~ , ~ E ~ A T I SOT A N N~D ~ D S
PfC,
3.1. Classification of loading for ball, cylindrical roller, and spherical roller bearings.
ings. Several of these bearing types, for example,tapered roller bearings, needle roller bearings, and instrument ball bearings, exist in too many variations to include all of the appropriate tolerance tables herein. On the other hand, reference E3.101 covers a wide range of standard radial ball and rollerbearings;Tables 3.63.10 are taken from reference[3.10]. For radial ball bearings these tolerances are grouped in ABEC* classes 1, 3, 5, 7, and 9 according to accuracy of manufacturing. Accuracy improves and tolerance ranges narrow as the class number increases. Tables 3.63.10 give tolerance ranges for all ABEC classifications. Additionally, Tables 3.63.8 provide the tolerances or bore and 0.d. for radial roller bearings as well as for ballbearings. The ABEC and tolerance classes correspondin every respect to the precision cla dorsed b the ISO. Table 3.5 shows the correspondencebetween the A.NSI/ and IS0 classifications. It is further noted that inch tolerances given in Part I1 of Tables 3.63.10 are calculated from primary metric tolerances given in Part I of those tables. To define the range of interference or looseness in themounting of an inner ring on a shaft or an outer ring in a housing, it is necessary to consider combinationof the shaft, housing, and bearing tolerances. *Annular BearingEngineers’ Committee of ABMA. t Roller Bearing Engineers’ Committee of ABMA.
8
~ T E ~ R E N C F E
I AND CL ~
Interference
Shafts
Clearance
he
Interference
F
I
~ 3.2 ~. ~Graphic E representation of fits.
~CE
~
I
a
8
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*222%2
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74
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98 074
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7
LE 3.5. ANSI/AE3M.A vs IS0 Tolerance Classi~cations ABEC 1or RBEC 1 ABEC 3 or RBEC 3 ABEC 5 or RBEC 5 ABEC 7 BEC9
Normal class Class 6 Class 5 Class 4 Class 2
LE 3.6. PART 1. Tolerance Class ABEC1, RBEC1. Metric Ball and Roller Bearings [except tapered roller bearings"] of Dimensions Conforming to the Basic Plan for Boundary Dimensions of Metric Radial Bearings Given in Table 1 of [3.10].
Inner Ring (Tolerance values in micrometers) diameter series d rnm
over
incl. ~
a
0.6 2.5 10 18 30 50 80 120 180 250 315 400 500 630 800 1,000 1,250 1,600
7, 8, 9
Ah, 2.5 10 18 30 50 80 120 180 250 315 400 500 630 800 1,000 1,250 1,600 2,000
high
low
0, 1

ABS
2, 3, 4

max.
VhP
Kia
max.
max.
all high
6 6 6 8 9 11 15 19 23 26 30 34 38
10 10 10 13 15 20 25 30 40 50 60 65 70 80 90 100 120 140
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0


normal 
modifiedd low
VBs Max.
~
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
8 8 8  10  12  15 20 25 30 35 40 45 50 75  120  125  160 200
10 10 10 13 15 19 25 31 38 44 50 56 63
8 8 8 10 12 19 25 31 38 44 50 56 63
6 6 6
8 9 11 15 19 23 26 30 34 38
40
 120  120  120  120  150
200 250 300 350 400 450 500 750  1,000  1,250  1,600 2,000

250 250 250 250 380 380 500 500 500 630
12 15 20 20 20 25 25 30 30 35 40 50 60 70 80 100 120 140
ty7yyy
comomomomomooo l+cvmmd*mbocvwom I I I I I I I
0 0 0 0 0 0 0 0 0 0 0 0 0
1 *e
t;
Q)
4J
o o o m o o o o o o o o o &i mcomlloomoomooo cv m w m CD co o~cv~qo_q3 l+ l+
3.6. 2. Tolerance Class imensions ~ o n f o ~ i to n gthe Basic Plan for ~
o
etric Ball and Roller Bearings [except tapered roller bearings"] of ~ ~ id~ e an s ~ i of o Metric ~s Radial B ~ a r i n Given ~s in Table 1of 13.101.
Inner Ring (Tolerance values in .0001 in.) d'p
diameter series d mm
AdmP
over
incl.
0.6 2.5 10 18 30 50 80 120 180 250 315 400 500 630 800 1,000 1,250 1,600
2.5 10 18 30 50 80 120 180 250 315 400 500 630 800 1,000 1,250 1,600 2,000
a
high
low
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 3 3 4 4.5 6 8 10 12 14 16 18 20 30 39 49 63 79
7, 8, 
O, max.
2, 3>
4 4 4 5 6 7.5 10 12 15 17 20 22 25
3 3 3 4 4.5 7.5 10 12 15 17 20 22 25
2.5 2.5 2.5 3 3.5 4.5 6 7.5 9 10 12 13 15
43s
Vdmp max. 2.5 2.5 2.5 3 3.5 4.5 6 7.5 9 10 12 13 15
Kia mas. 4 4 4 5 6 8 10 12 16 20 24 26 28 31 35 39 47 55
all normal ~ high 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 16
47 47 47 47 59  79 98  118  138  157  177  197 295 394 492 630  787
modifiedd low 98
98 98 98  150
 150  197  197
 197 248 
VBS
max. 4.5 6 8 8 8 10 10 12 12 14 16 20 24 28 31 39 47 55
TABLE 3.6. PART 2. (Continued) Outer Ring (Tolerance values in .0001 in.) Y
0 N
VDP" Capped Bearings
Open Bearings diameter series
D mm over
a
2.5 6 18 30 50 80 120 150 180 250 315 400 500 630 800 1,000 1,250 1,600 2,000
7, 8, 9
ADmp
incl. 6 18 30 50 80 120 150 18QL 250 315 400 500 630 800 1,000 1,250 1,600 2,000 2,500
high
low
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 3 3.5 4.5 5 6 7  10  12  14  16  18 20 30 39 49 63 79 98
0, 1
2, 3, 4 max.
4 4 4.5 5.5 6.5 7.5 9 12 15 17 20 22 25 37 49 
3 3 3.5 4.5 5 7.5 9 12 15 17 20 22 25 37 49


2.5 2.5 3 3 4 4.5 5.5 7.5 9 10 12 13 15 22 30


"This diameter is included in the group. bNo values have been established for diameter series 7, 8, 9, 0 and 1. "Applies before mounting and after removal of internal or external snap ring. dThis refers to the rings of single bearings made for paired or stack mounting. "For tapered roller bearing tolerances see L3.8, 3.91.
2, 3, 4
vDm;
Kea
max.
max.
2.5 2.5 3 3 4 4.5 5.5 7.5 9 10 12 13 15 22 30
6 6 6 8 10 14 16 18 20 24 28 31 39 47 55 63 75 87 98

VCS
ACS
high
low
max.
Identical to A,, and V,, of inner ring of same bearing
mi
Q,
b"
23
!i E
E
4
3 0 0 0 0 0 0 0 0 0 0 0 0
o o o o o o o o o o o o c
3
1
lli
d
m"
on
0"
lll
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Class ABEC3, RBEC3. Metric Ball and Roller Bearings [except tapered roller bearings"] of sic Plan for Boundary Dimensions of Metric Radial Bearings Given in Table 1 of [3.10]. hner Ring (Tolerance values in .0001in.)
diameter series
ABS
d
Ah,
7, 8, 9
over
incl.
high
low
0,6 2.5 10 18 30 50 80 120 180 250
2.5 10 18 30 50 80 120 180 250 315
400 500
500 630
0 0 0 0 0 0 0 0 0 0 0 0 0
3 3 3 3 4 4.5 6 7 8.5  10  12  14  16
a
0, 1
2, 3, 4
VdmP max.
Kia max.
all high
normal
2 2 2 2.5 3 3.5 4.5 5.5 6.5 7.5 9 10 12
2 2 2 2.5 3 3.5 4.5 5.5 6.5 7.5 9 10 12
2 2.5 3 3 4 4 5 7 8 10 12 14 16
0 0 0 0 0 0 0 0 0 0 0 0 0
 16 47 47 47 47 59 79 98 118  138  157  177  197
max, 3.5 3.5 3.5 4 5 6 7.5 9 11 12 15 17 20
3 3 3 3 4 6 7.5 9 11 12 15 17 20
modified" low
98 98
98 98  150  150  197  197  197
248
VBEl max. 4.5 6 8 8 8 10 10 12 12 14 16 18 20
ce Class ABEC5, RBEC5. Metric Ball and Roller Bearings [except i n s t ~ e nbearings," t and ensions Conforming to the Basic Plan for B ~ ~ Dimensions d a ~ of Metric Radial Bearings Given. in Table 1of [3.10].
Inner Ring (Tolerance values in ~ i ~ r o ~ e ~ e r s ~ VdP
diameter series
ABS
d
mm
Ah*
7 7
over
incl.
high
low
a 0.6 2.5 10 18 30 50 80 120 180 250 315
2.5 10 18 30 50 80 120 180 250 315 400
0 0 0
5 5 5 6 8 9  10  13  15  18 23
0 0 0
0 0 0 0
0
8, 9
5 5 5 6 8 9 10 13 15 18 23
0, 1, 2737 4 Max. 4 4
4 5 6 7 8 10 12 14 18
Vhp max.
Kia max.
s d max.
Si2 max.
dl high
3 3 3 3
4 4 4
4
5 5 6 8 10 13 15
7 7 7 8 8 8 9 10 11 13 15
7 7 7 8 8 8 9 10 13 15 20
0 0 0 0 0
5 5 7 8 9 12
4
0 0 0
0 0 0
normal modd VBS low max. 40 40 80  120  120  150 200 250 300 350 400
250 250 250 250 250 250 380 380 500 500 630
5 5 5 5 5 6 7 8 10 13 15
t.r
.
~~~ntinued) Outer Ring (Tolerance values in micrometers) diameter series ~
D mm over a
2.5
6 1 3 5 80 12 150 180 250 315 400 50 63 ~
7, 8, 9
ADmp
incl.
high
low
6 18 30 50 80 120 150 180 250 315 400 500 630 8
0 0 0 0 0 0 0 0 0 0 0 0 0
5 5 6 7 9  10 11  13  15  18 20 23 28 35
0,
2, 3,
max. 5 5 6 7 9 10 11 13 15 18 20 23 28 35
4 4 5 5 7 8 8 10 11 14 15 17 21 26
vDmp
Kea
SD
8,:
max.
max.
max.
max.
3 3 3 4 5 5 6 7 8 9 10 12 14 18
5 5
8 8 8 8 8 9 10 10 11 13 13 15 18 20
8 8 8 8 10 11 13 14 15 18 20 23
~
“This diameter is included in the group. bNo values have been estab~shedfor sealed or shielded bearings. “Applies to groove type ball bearings only. dThis refers to the rings of single bearings made for paired or stack mounting. “For instrument ball bearing tolerances see K3.2, 3.31. f For tapered roller bearing tolerances see 13.8, 3.91.
6 7 8 10 11 13 15 18 20 23 25 30
25
30
A,, high
vcs
low
Identical to ABsof inner ring of same bearing
max. 5 5 5 5 6 8 8 8 10 11 13 15 18 20
" a, ' E :
a,
&I
%
cr,
i
3
>oooooooooo
o o o o o o o o o o c
0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0
oooooc3c2oo
0 0 0 0 0 0 0 0 0
d
l
I I
0
+,
4
5
~
B 0
t5
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
ooc>oooooo
~1
'I
43 U
8
(3
dE
B
0
r4
6 3
!5
r4
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
ad
ai "
E
w
6 3 B
0
w
6 3
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
1
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
11
EFFECT OF ~ E R F E R E N C EFI!t"I'mG CLEARANCE ON
NG The solution to this problem may beobtained by using elastic thick ring theory. Consider the ring of Fig. 3.3 subjected to an internal pressure p per unit length. The ring has a bore radius & and an outside radius 9j0. For the elemental area % d( % d+ the summation of forces in the radial direction is zero for static equilibrium:
(
)
crr%d# + 20, d % sin d+   cr + dcrr  % (%+ d@d+ 2 ' dmd Since d+ is small, sin higher order,
4d+
=
=
0
d+ and, neglecting small quantities of
direction, there is an elongation Corresponding to the stress in the radial u and the unit strain in the radialdirection is
F I ~ U 3.3. ~ E Thick ring loaded by internal pressure p .
1
~ ~ R F E ~ N C E
E,
F I T T ~ G
CE
du
d%
=
(3.3)
In the circumferential direction the unit strain is U
E, = 
(3.4)
9
According to plane strain theory,
Combining equations (3.33.6) yields
~ubstitutingequations (3.7) and (3.8) into (3.2) yields d2u +"" dS2
1 du %d%
u= gx2
0
(3.9)
The general solution to equation (3.9) is u = el%
+ c2Sl
(3.10)
~ubstitutingin equations (3.7) and (3.8) from (3.10) gives (3.11) (3.12) At the boundary defined by 9%= gl0, a, = p
=
0; therefore,
121
EFFECT OF ~ T E ~ E ~ FN C ~E ON CLEARANCE ~ G
(3.13)
At
=
q,ar= p
and, therefore, (3.14)
Substituting equations (3.13) and (3.14) into (3.11) and (3.12) yields (3.15) (3.16) Similarly, for a ring loaded by external pressure only, (3.17) (3.18) From equations (3.15), (3.16), and (3.5), the increase in the internal radius of a ring loaded by internal pressure p is given by (3.19) Similarly, the decrease in the external radius ternal pressure p is given by
soof a ring loaded by ex(3.20)
If a ring having elastic modulus E,, outside diameter g,,and bore I%, is mounted with a diametral interference I on a second ring having modulus E,, outside diameter 9, and bore g2, then a common pressure p develops between the rings. The radial interference is the sum of the radial deflection of each ring due to pressure p . Hence the diametral intederence is given by
~
I
= 2(u,
E
R
~ FITTING ~ ~
C
+ u,)
CE
E
(3.21)
In terms of the common diameter 9,therefore, I = P i n { 1 [
E,
(8,/9)2
+ 1+
8,/8)2 1
~,] [
+1 (9/9,)2
+1
E, (8/9,), 1
It can be seen that equation (3.22) can be used known; thus,
to determine p if I is
9
p = +E]
+[
E, 1
+ 1
(9/9,)2 (9/g2)2 
E,]
(3.23)
If the external ring is a bearing inner ring of diameter 9,and bore 8, as shown in Fig. 3.4, then the increase in 9,due to press fitting is
(3.24)
I
3.4. Schematic diagram of a bearing inner ring mounted on a shaft.
If the bearing inner ring and shaft are both fabricated from the same material, then
(3.25) For a bearing inner ring mounted on a solid shaft of the same material, diameter g2is zero and A, = I
(~)
(3.26)
By a similar process it is possible to determine the contraction of the bore of the internal ring of the assembly shown in Fig. 3.5. Thus,
For a bearing outer ring pressed into a housing of the same material,
3.5. Schematic diagram of a bearing outer ring mounted in a housing.
124
~
E
~
E
~ FITTING N C EAND C
L
~
C
(3.28)
If the housing is large compared to the ring dimensions, diameter gl approaches infinity and (3.29)
Considering a bearing having a clearance P d prior to mounting, the change in clearance after mounting is given by
The preceding equation takes no account of differential thermal expansion.
Since the pressure p between interfering surfaces is known, it is possible to estimate the amount of axial force necessary to accomplish or remove an interference fit. Because the area of shear is mgB, the axial force is given by
Fa = p7Ti3B p
(3.31)
in which ~ lis, the coefficient of friction. Accordingto Jones 13.111 the force required to press a steel ring on a solid steel shaft may be estimated by
Fa = 47100 BI
[ (~)"] 1

(3.32)
This is based on a kinetic coefficient of friction p = 0.15. Similarly; the axial force required to press a steel bearing into a steel housing is given by
[ (~)"]
Fa= 47100 CI 1 
(3.33)
SI Rolling bearings are usually fabricated from hardened steel and are generally mounted with press fits on steel shafts. In many applications, such
E
as in aircraft, however, the bearing may be mounted in a housing of a dissimilar material. Bearings are usually mounted at room temperature; but they may operate at temperatures elevated AT above room temperature. The amount of temperature elevation may bedetermined by using the heat generation and heat transfer techniques indicated in Chapter 15. Under the influence of increased temperature, materialswill expand linearly to the following equation: U =
re(T  TJ
(3.34)
in which I' is the coefficient of linear expansion in mm per mm per "C and 2, is a characteristic length. Considering a bearing outer ring of outside diameter do at temperatures To  Taabove ambient, the increase in ring outside circumference is given approximately by (3.35)
Therefore the approximate increase in diameter is (3.36)
The inner ring will undergo a similar expansion: (3.37)
Thus the net diametral expansion of the fit is given by
When the housing is fabricated from a material other than steel, the interference I between the housing and outer ring may either increase or decrease at elevated temperatures. Equation (3.39) gives the change in I with temperature:
In which r b and rh are the coefficients of expansion of the bearing and housing, respectively. For dissimilar materials the housing is likely to expand more than the bearing, which tends to reduce any interference fit. Equation (3.30) therefore becomes
If the shaft is not fabricated from the same material (usually steel) as the bearing, then a similar analysis applies.
I ~ E ~ F E ~ NF C I E ~
~
CE
G
The interference 1' between a bore and 0.d. is somewhat less than the apparent dimensional value due to the smoothing of the minute peaks and valleys of the surface. The schedule of Table 3.11 for reduction of 1' may be used. It can be seen from Table 3.11 that for an accurately ground shaft mating witha similar bore, it may be expected that thereduction on the bore diameter would be 0.0020 mm (0.00008 in.) and on the shaft possibly 0.0041 mm (0.00016 in.) or a total reduction in I of 0.0061 mm (0.00024 in.). The 209 radial ball bearing of Example 2.4 is manuEC 5 specifications. The bearing is mounted on a solid steel shaft witha k5 fit and ina rigid steel housing with a K6 fit. The nominal bearing bore is 45 mm (1.7717 in.), and the nominal 0.d. is 85 mm (3.3465in.). Determine the bearingcontact angle and free endplay under light thrust loading. Shaft tolerance range from Table 3.2 is 0.0025 mm ( 0.0127 mm (0.0005 in.) or 0.0076 mm (0.0003 in.) mean. mean tolerance from Table 3.8 is 0.004 mm (0.00016 in.), a negative value; that is, 0.004 mm (0.00016 in.) The mean interferenceon the shaft is
I
=
0.0076
+ 0.004
=
0.0116 mm (0.0005 in.)
Assuming the bearing is mounted on a ground surface, the reduction in 1' due to surface finish is approximately 0.0020 mm (0.00008 in.) (see Table 3.11) for the bearing bore and shaft. Therefore
I = di = 91 =
0.0116  2 X 0.0020 52.3 mm (2.0587 in.) di
=
0.0076mm(0.00030 in.) 2.4
Ex.
.ll. Reduction in Interference Due to Surface Condition
Finish Accurately ground surface Very smooth turned surface (2456) Machinereamed bores (4094) Ordinary accurately turned surface (94190)
Reduction 0.0001 mm
Reduction 0.0001 in.
2051 61142 102239 239483
(820)
127
EFFECT OF SURFACE FINISH
(3.26) 0.0076  = 0.0065 mm (0.00026 in.) ( ~ 3 )
=
Housing tolerance range from Table 3.3 is 0.0178 mm (0.0007 in.) to 0.0051 mm (0.0002 in.) or 0.0064 mm (0.00025 in.) mean. is 0.005 mm Bearing 0.d. mean tolerance range fromTable3.8 (0.00020 in.), a negative value, that is, 0.005 mm (0.00020 in.). The mean interference in the housing is
I
=
0.0064  0.005
=
0.0014 mm (0.00006 in.)
Assuming the bearing housing bore is accurately ground, the reduction i' due to surface finish is approximately 0.0020 mm (0.00008 in.) (see Table 3.11) for the housing bore and the bearing 0.d. Thus, the net interference in the housing is virtually zero.
& = 0.015 mm (0.0006 in) Apd
A
(3.30)
=
As
=
0.0065
=
0.508 mm (0.02 in.)

Ex. 2.4
Ah
+ 0 = 0.0065
mm (0.00026 in.) Ex. 2.4 (2.10)
=
(
cos1 1 
0.015  0.0065 2 X 0.508
=
7'25'
Pe = 2A sin a' =
2
X
0.508
(2.12) X
sin (7'25')
=
0.1312 mm (0.0052 in.)
Comparison of these values of a' and Pe with those of Example 2.4 indicates the necessity of including the effect of interference fitting in the determination of clearance. e 3.2. The 218 angularcontact ball bearing of Example 2.3 has a 90 mm (3.5433 in.) bore, a 160 mm (6.2992 in.) 0.d. and is manufactured to ABEC 7 tolerance limits. The bearing is mounted on a hollow steel shaft of 63.5 mm (2.5 in.) bore with a k6 fit and in a
12
CE
titanium housing havingan effective 0.d. of 203.2 mm (8 in.) with an M6 fit. Determine the free contact angle of the bearing. Shaft tolerance range from Table 3.2 is t0.0025 mm (0.0001 in.) to t0.0254 mm (0.0010 in.) or a +0.0140 mm (0.00055 in.) mean. Bore mean tolerance range from Table 3.9 is 0.004 mm (0.00016 in.), a negative value, that is, 0.004 mm (0.00016 in.) The mean interference on the shaft is
I
=
0.0140
+ 0.004 = 0.0180 mm
(0.00071 in.)
Assuming the bearing is mounted on a ground surface, the reduction in I due to surface finish is approximately 0.0020 mm (0.00008 in.) (see Table 3.11) for the bearing bore and shaft, therefore,
I
=
0.0180
d,
=
102.8 (4.047 mm

2
X
0.0020 = 0.0140 mm (0.00055 in.) in.)
g1 = d,
(2)[
I
A,
( 'f?'92)2
(91/92)2

:]
[
(3.25)
]
102.8 (90/63.5)2  1 90 (102.8/63,5)2  1
=
0.0140
=
0.00995 mm (0.00039 in.)
X
2.3 Ex.
Housing tolerance range from Table 3.3 is 0.033 mm (0.0013 in.) to 0.0076 mm (0.0003 in.), or a 0.0203 mm (0.0008 in.) mean. Bearing mean 0.d. tolerance range from Table 3.9 is 0.005 mm (0.0002 in.), a negative value, that is, 0.005 mm (0.OO02 in.). The mean interference is the housing is
I
=
0.0203

0.005 = 0.0153 mm (0.0006 in.)
Assuming the housing is mounted on a ground surface, the reduction in I due to surface finish is approximately 0.0020 mm (0.0008 in.) (see Table 3.11) for the bearing 0.d. and housing bore, therefore
I
=
0.0153 = 2
X
0.0020 = 0.0113 mm (0.00044 in.)
do = 147.7 mm (5.816 in.) g2 = do
For steel
2.3 Ex.
129
EFFECT OF SURFACE FINISH
E
206900 N/mm2 (30 X
=
lo6 psi)
5 = 0.3 For titanium
E
=
103500 N/mm2 (15 X
lo6 psi)
5 = 0.33
2

X
0.0113
X
(1601147.7) r
t,
(
[("")'
147.7

11
, thedepths at which significantstresses occur below the sudace arealso of interest. Again, considering only stresses caused by a concentrated force normal to the surface, Jones [6.8] after Thomas and Hoersch E6.71 gives the following equations by which to calculate the principal stresses S,, Sy’ and 8, occurring alongthe i! axis at any depth below the contact surface.
205
SUBSURPACE STRESSES
Since surface stress is maximum at the Z axis7 therefore the principal stresses must attain maximum values there (see Fig. 6.8):
s, = *A(;
(6.55)
.)

in which (6.56)
y
=
(
~
)
1
'
(6.57)
2
2 p
(6.58)
b
a;= 1 + v + &F(c$) =
lo4[
1

(1 
&(+)I

5)
(6.62) 112
sin261
d+
(6.63) (6.64)
The principal stresses indicated by the foregoing equations are graphically illustrated by Figs. 6.96.11. Since each of the maximurn principal stresses can be determined, it is further possible to evaluate the maximum shear stress on the z axis below the contact surface.By Mohr's circle (see reference[6.21),the maximum shear stress is found to be
As shownbyFig.
6.12 the maximum shear stress occurs at various
CONTACT STR,ESSAND ~ E F O ~ T I O N
St stresses occurring on element on Z axis below contact surface.
0
0.2 t
_.
b
0.4
0.6 0.8 1.o 0
0
b 2
0.2
0.4
0.6 0.8 1.o
ab 6.10. S,/am,,vs bla and zlb.
6.11. s,/am,, vs bla and zlb.
0.
0.7
0.6
0.4
0.3
0.2
0.4
6.12.
T
0.6
4 ~
~
~and ~
1.0
0.8
z,lb ~ /vs ubla. 
~
~
~
depths z, below the surface, being at 0.467b for simple point contactand b for line contact. uring the passage of a loaded rolling element over a point on the raceway surface, the maximum shear stresson the z axis varies between 0 and T ~ If the ~ ~ element . rolls in the direction of the y axis, then the shear stressesoccurring in the yz plane below the contact surface assume values from negative to positive for values of y less than and greater than zero, respectively. Thus, the maximum variation of shear stress in the y z plane at any point for a given depth is 2trYz. Lundberg and Palmgren E6.91 show that cos2 4 sin (I, sin 6 + b2 cos2 (;b
a2 tan2 6
wherein
~6.66)
y
=
(b2 + a2tan2
x
=
a tan 6 cos (I,
sin ct,
(6.67) (6.68)
Here, 6 and ct, are auxiliary angles such that
of the shear stress. Further, 6 and ct, are
which defines the amplitude related as follows:
tan2+ = t tan2 6
=
t1
(6.69)
in which t is an auxiliary parameter such that b a
 = [(t2
l)@t 
1)]1/2
(6.70)
Solving equations (6.66)(6.70) simultaneously, it is shown in reference [5.8] that (6.71) and 1 =
(t + l)(Zt  1)1/2
(6.72)
Figure 6.13 shows the resulting dist~butionof shear stress at depth x. in the direction of rolling for b l a = 0, that is, a line contact. Figure 6.14 shows the shear stress amplitude of equation (6.71) as a function of bla. Also shown is the depth below the surface at which this shear stress occurs. Since the shear stress amplitud 6.14 is greater than that of Fig. 6.12, Lundberg and sumed this shear stress,called the masimum orthogonal shear stress,to be significant in causing fatigue failure of the surfaces in rolling contact, As can be seen from Fig. 6.14, fora typical rolling bearing point contact of b l a = 0.1, the dep below the surface at which. this stress occurs is approximately 0.49b.oreover, as seen by Fig.6.13, this stress occurs
at any instant under the extremities of the contact ellipse with regard to the direction of motion, that is, at y = ir 0.9b. etallurgical research [6.10] based on plastic alterations detected in subsurf'ace material by transmission electron microscopic investigation gives indications that thesubsurface depth at which significantamounts of material alteration occur is approximately 0.75b. Assuming such plastic alteration is the forerunner of material failure, then it would appear that the m ~ i m u mshear stress of Fig. 6.12 may be worthy of consideration as thesignificant stress causing failure. Figures 6.15 and 6.16 from reference [6.10] are photomicro~aphsshowing the subsurface changes caused by constant rolling on the surface. Many researchers consider the von ~isesHencky distortion energy theory [6.11] and thescalar von Mises stress a better criterion for rolling contact failure failure. The latter stress is given by
S
CE S T ~ S S E S
6.14.
~
T
~
/ and c Fz,/b ~ ~ vs
b / a (concentrated normal load).
(6.73)
As compared to the m ~ m u m orthogonal shear stress ro which occurs at depth x. appro~imately equal to 0.5b, at y approximately equal to t 0.9b in the rolling direction, crw,max occurs at x between 0.7b and 0.8b and at y = 0. Octahedral shear stress,a vector quantity favored by some researchers, is directly proportional to aVM.
(6.74)
Figure 6.17 compares the magnitudes of ro,masimum shear stress, and roctvs depth.
.
Subsudace metallurgical structure (1300 times ma~ificationafter picral etch) showing change due to repeated rollingunder load. (a)Normal structure; ( b ) stresscycled structurewhite deformation bands and lenticular carbide formations are in evidence.
6.16. Subsurface structure (300 times magnification after picral etch) showing orientation of carbides to direction of rolling. Carbidesare thought to be weakness locations at which fatigue failure is initiated.
.
Determine the amplitude of the maximum orthogonal shear stress at the inner and outer raceways of the 218 ,angularcontact ball bearing of Example 6.1. Estimate the depths below the rolling surfaces at which these stresses occurs. ai= 2.64 mm (0.1040 in.)
x. 6.1
b. = 0.324 mm (0.01277 in.)
x. 6.1

0.1227
"
2.64
CONTACT STRESS AND D E ~ O R ~ T I O N T~ T
 orthogonal shear stress ______________ ~ maximum ~ shear stress
__________
Tact  octahedral stress shear
.rkrrnax
0.6
0.4
0.2 1_1
0
;
zlb
I
I_
0
1.o
0.5
1.5
2.0
2.5
E 6.17. Comparison of shear stresses at depths beneath the contact surface (x = y = 0).
From Fig. 6.14, 0.498, zgi bi
=
0.493
1976 N/mm2 (286,400 psi) 0.498
X
1976
2
=
491.9 N/mm2 (71,310 psi)
2.558 mm (0.1007 in.)
Ex. 6.1
0.375 mm (0.01478 in.)
Ex. 6.1
0.375 2.558
Ex. 6.1

0.1468
=
0.497,
"
From Fig. 6.14,
(fOlllSiX
zoo =
b0
0.491
aomax = 1762 N/mm2 (255,500 psi)
Ex. 6.1
EFFECT OF S ~ A C SHEAR E STRESS
700
=
0'497 2x 1762 = 438 N/mm2 (63,490 psi)
zoi = 0.493
X
0.324
=
0.160 mm (0.00630 in.)
zoo = 0.491 X 0.375 = 0.184 mm (0.00726 in.) For casehardening bearings the value of zoi and zoo can be used to estimate therequired case depth. Note that themaximum shear stress at the center of contact occursat zli = 0.763, and zlo = 0.7553, for the inner 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 more conservative to base case depth on these values. Case depth should exceed zo or z1by at least a factor of three.
In the determination of contact deformation vs load only the concentrated load applied normal to the surface need be considered for most applications. Moreover, in most rolling bearing applications, lubrication is at least adequate, and the sliding friction between rolling elements and raceways is negligible (see Chapter 14). This means that the shear stresses acting on the rolling elements and raceway surfaces in contact, that is, the elliptical areas of contact, are negligible compared to normal stresses. For the determination of bearing endurance with regard to fatigue of the contacting rolling surfaces, the surface shear stress cannot be neglected and in many cases is the most significant factor in determining endurance of a rolling bearing in a given application. Methodsof calculation of the surface shear stresses (traction stresses) will be discussed in Chapter 14. The means for determining the effect on the subsurface stresses of the combination of normal and tangential (traction) stresses applied at the surface are extremely complexrequiring the use of digital computation. Among others, Zwirlein and Schlicht [6.10] have calculated subsurface stress fieldsbasedupon assumed ratios of surface shear stress to applied normal stress. Reference E6.101 assumes that the von Mises stress is most significant with regard to fatigue failure and gives illustrations of this stress inFig. 6.18. Figure 6.19 also from reference E6.101 shows the depth at which the various stresses occur. Figure 6.19 shows that as the ratio of surface shear to normal stress increases, the maximum von Mises stress moves closer to the surface. At a ratio of ~ C J =. 0.3, the maximum von Mises stress occurs at thesurface. Various other investigators have found that if a shear stress is applied at the contact surface in addition to the normal stress, the maximum shear stresstends to increase and it is located
x/b
2.5 2.0 1.5
1
.O 0.5
0
0.5
1.0
1.5
2.0
2
0.5 1 .o z lb
JI
X
2.0
z
p=0
2.5
3.0
2.5
2.0 1.5
1.0
0.5
0
0.5
1.0
1.5
2.0
2.5
alb X
6.18. Lines of equal von Misea stress/normal applied stress for various surface shear stresses dnormal applied stress a.
E F F E C ~OF S
~
~ SC
E STRESS
6.19. Material stressing ( c q , J o  ) vs depth for different amountsof surface shear stress (do).
closer to the surface (see references [6.116.151. References [6.166.181 give indications of the effect of higher order surfaces on the contact stress solution. The references cited above are intended not to be extensive, but to give only a representation of the field of knowledge. Theforegoingdiscussion pertained to the subsurface stress field caused by a concentrated normal load appliedin combination with a uniform surface shear stress. The ratio of surface shear stress to normal stress is also called the coefficient of friction (see Chapter 14). Because of in~nitesimallysmall irregularities in the basic surface geometries of the rolling contact bodies,neither uniform normal stress fields as shown by Figs. 6.6 and 6.7 nor a uniform shear stress field are likely to occur in practice. Sayles et al. [6.19] use the model shown by Fig. 6.20 in developing an “elastic conformity factor.” Kalker [6.20] developed a mathematical model to calculate the subsurface stress distribution associated with an arbitrary distribution of shear and normal stresses over a surface in concentrated contact. Ahmadi et al. [6.21] developed a “patch” method that can be applied to determine the subsurface stresses for any concentrated contact surface subjected to arbitrarily distributed shear stresses. Using superposition, this method combined with that of Thomas and Hoersch [6.7], for example, for Hertzian surface loading, can be applied to determine the subsurface stress distributions occurring in rolling elementraceway contacts. Harris andYu E6.221, applying this method of analysis, determined that the range of maximum orthogonal shear stress, i.e., 2 ~ is~not , altered by the addition of surface shear stresses to the Hertzian stresses. Fig. 6.21 illustrates this condition.
C
I area = 71 ab I
O
~ STRESS A ~ AND ~ DEFO
I 1
s
( b)
6.20. Models for lessthanidealelastic conformity. (a)Hertzian contact model ( b )Elastic conformity envisagedwith real used in developing elastic conformity parameter. the figure roughness would bepreferentialto certain asperity wavelengths. For convenience of similar moddmws only one compliant rollingelement, whereas in practice materials if ulus were employed the deformation would be shared.
f
(a) y = 0.9b
location x
(b) y = +0.9b
6.21. Orthogonal shear stress T ~ ~ / (abscissa) c F ~ ~ vs ~ depth z / b at contact area = 0 for friction coefficients f = 0, 0.1, 0.2.
Since the Lundberg lmg+ren fatigue life theory [6.9] is based on maximum orthogonalshear stress as the fatigue failureinitiating stress, the adequacy of using that method to predict rolling b durance is called into question. Conversely, for simple directly under i.e., f = 0, maximum octahedral shear stress the center of the contact. Fig. 6.22 further shows that the magnitude of T ~ and the ~ depth ~ at , which ~ it ~occurs ~ is substantially influenced by surface shear stress. The question of which stress should be used for fatigue failure life prediction will be revisited in Chapters 18 and 23.
occurs
Basically, two hypothetical types of contact can be defined under conditions of zero load. These are a
Point contact, that is, two surfaces touch at a single point.
. 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 expands to an ellipse and the line to a rectangle in ideal line contact, that is, the bodies have equal length. Figure 6.23 illustrates thesurface compressive stress distribution which occurs in each case. When a roller of finite length contacts a raceway of greater length, the axial stress distribution along the roller is altered from that of Fig. 6.23. Since the material in the raceway is in tension at theroller ends because of depression of the raceway outside of the roller ends, the roller end compressive stress tendsto be higher than that in the center of contact. Figure 6.24 demonstrates this condition of edge loading.
(a) Hertz loading only 6.22. Orthogonal shear stress
xla.
(b) Hertz plus f = 0.2 shear stress loading
la,, (ydirection)vs depth z / b and location
C O ~ ~ STRESS C T AND D E F O R ~ T I O N
8 2
X
2 ” 
Y
x
3. Surface compressive stress distribution. (a)Point contact; ( b ) ideal line contact.
To counteract this condition, cylindricalrollers (or the raceways) may becrowned as shown by Fig. 1.38. The stress distribution is thereby made more uniform depending upon applied load. If the applied load is increased si~ificantly,edge loading will occur once again. e d contact Lundberg et al. [6.9] have defined a condition of ~ o d i ~ line for rollerracewaycontact. Thus, when the major axis (2a)of the contact ellipse is greater than the effective roller length l but less than 1.51, ~ o d i line ~ econtact ~ is said to exist. If 2a c I , then point contact exists; if 2a > 1.51, then line contact exists with attendant edge Zoa~ing.This condition may be ascertained appro~imatelyby the methods presented in the section“Surface Stresses and Deformations,” using the roller crown radius for 22 in equations (2.37)(2.40).
E OF C O ~ A C T
t ' l Tension
Tension (a)
Actual area 2b of contact\
Apparent area f of contact
Y
X
4 (c)
6.24. Line contact. (a)Roller contacting a surface of infinite length; ( b ) rollerraceway compressive stress distribution; (c) contact ellipse.
The analysis of contact stress and deformation presented in this section is based on the existence of an elliptical area of contact, except for the ideal roller under load, which has a rectangular contact. Since it is desirable to preclude edge Zoading and attendant high stress concentrations, roller bearing applications should be examined carefully according to the ~ o d i ~ Line e d contact criterion. Where that criterion is exceeded, redesign of roller and/or raceway curvatures may be necessitated. Rigorous mathematical/numerical methods have beendeveloped to calculate the distribution and magnitude of surfaces stresses in any "line" contact situation, that is, including the eEects of crowning of rollers, raceways, and combinations thereof'(see references L6.231 and [6.f214]. ~dditionall~, finite element methods (FEN) have been employed r6.251
2
C O ~ ~ STRESS C T AND DEFO
to perform the sameanalysis. In all cases, a rather great amountof time on a digital computer is required to solve a single contact situation. Figure 6.25 shows the resultof an FEM analysis of a heavily loaded typical spherical roller on a raceway. Note the slight “dogbone” shape of the contact surface. Note also the slight pressure increase where the roller crown blends into the roller end geometry. The 22317 spherical roller bearing of Example 2.9 experiences a peak roller load of 2225 N (500 lb). Estimate the type of racewayroller contact at each raceway Z = 20.71 mm (0.8154 in.)
Ex. 2.7
x p i = 0.0979 mm’ (2.487 in.’)
Ex. 2.9
0.9951
Ex. 2.9
=
x p , = 0.068 mm’
(1.726 in.”’)
F(p), = 0.9929
Ex. 2.9 Ex. 2.9
From Fig. 6.5, a:
=
10.2
a:
=
8.8
( ~ ) (113
ai = 0.0236a:
2ai
=
0.0236
=
13.64 mm (0.537 in.)
X
(6.38)
10.2 ~ ~ ~ ~ 9=) 6.828 ’ 1 3mm (0.2688 in.)
Since 2ai < Z point contact occurs at the inner raceway a,
2a,
=
0.0236a:
(
=
0.0236
8.8
=
13.3 mm (0.5236 in.)
X
113
=
6.65 mm (0.2618 in.)
Since 2a, < I , point contact occurs at the outer raceway also.
(6.38)
E OF C O ~ A C T
Distribution of maximum transverse pressure
I
I
Position along roller(mm)
Contact area plan view ~ . ~ 5Heavy . edge loaded roller bearing contact (example of nonHertzian con
tact).
5. Estimate the type of contact that occurs at each raceway of a 22317 spherical roller bearing if the peak roller load is 22,250 N (5000 lb.). At 2225 N (500 lb.),
ai = 6.828 mm (0.2688 in.)
6.4Ex.
a,
=
6.65 mm (0.2618 in.)
Ex. 6.4
I
=
20.71 mm (0.8154 in.)
Ex. 2.7
(
ai = 6.828 ~ ~ ~ ~
) 1='14.69 3
mm (0.5785 in.)
22
2ai = 29.39 mm (1.157 in.) 1.51 = 1.5 X 20.71 = 31.06 mm (1.223 in.)
Since 1 < 2ai < 1.51, modified line contact occursat the innerraceway. 113
=
14.31 mm (0.5632 in.)
2a0 = 28.6 mm (1.126 in.) Since 1 < 2a, < 1.51, modified line contact occursat the outer raceway. The circular crown shownin Fig. 1.38aresulted from the theory of Hertz E6.11 whereas the cylindrical/crowned profileof Fig. 1.3% resulted from the work of Lundberg et al. [6.5].As illustrated inFig. 6.26, eachof these surface profiles, while minimizing edge stresses, has its drawbacks. Under light loads, a circular crowned profile does not+enjoy full use of the roller length, somewhat negating the use of rollers in lieu of balls t o carry heavier loads with longer endurance (see Chapter 18). Under heavier loads,whileedge stresses are avoidedformost applications, contact stress in the center of the contact can greatly exceed that in a straight profile contact, again resulting in substantially reduced endurance characteristics. Under light loads, the partially crowned roller of Fig. 1.38b as illustrated by Fig. 6.26~experiences less contact stress than does a fully crowned roller under the same loading, Under heavy loading the partially crowned roller alsotends to outlast the fully crowned roller because of lower stress in the center of the contact; however, unless careful attention is paid to blending of the intersections of the “fiat,’ (straight portion of the profiles) and thecrown, stress concentrations can occurat the intersections with substantial reduction in endurance (see Chapter 18). When the roller axis is tilted relative to the bearing axis, both the fully crowned and partially crowned profiles tend to generate less edge stress under a given load as compared to the straight profile. After many years of investigation and with the assistance of mathematical tools such as finite difference and finite element methods as practiced using computers, a “logarithmic”profilewasdeveloped[6.261 yielding a substantially optimized stress distribution under most conditions of loading (see Fig. 6.262d). The profile is so named because it can be expressed mathematically as a special logarithmic function. Under all loading conditions,the logarithmic profile uses more of the roller length than either the fully crowned or partially crowned roller profiles.Under misalignment, edge loadingtends to be avoided under all but exceptionally heavy loads. Under specific loading (Q/ZD) from 20 to 100 N/mm2
25
(d)
6.26. Rollerraceway contact load vs length and applied load; a comparison of straight, fully crowned, partially crowned, and logarithmic profiles (from [5.241).
(290014500psi),Fig.6.2'7 taken from[6.26] illustrates the contact stress distributions attendant to the various surface profiles discussed herein. Figure 6.28, also from [6.26], compares the surface and subsurface stress characteristics for the various surface profiles.
The contact stresses between flange and roller ends may be estimated from the contact stress and deformation relationships previously presented. The roller ends are usually flat with corner radii blending into
mm'

2
1
1
6.27. Compressive stress vs length and specific roller load (&/ID)for various roller (or raceway) profiles (from i6.261).
logadthmic
B
0.1
0.2
0.3 (A) Compression stress for different pmfiles uz (B) Maximum von Mises' stress D IC) Depth at which it acts z
~
~ 6.28.~ Comparison U of R surface ~ compressive stress a;, maximum von Mises stress
am, and
depth z to the maximum von Mises stress for various roller (or raceway) profiles (from l6.261).
C O ~ ~ A STRESS C ~ AND DEFO
the crowned portionof the roller profile.The flange may also be a portion of a flat surface. This is the usual design in cylindrical roller bearings. When it is desired to have the rollers carry thrust loads between the roller ends and the flange, sometimes the flange surface is designed as a portion of a cone. In this case, the roller corners contact the flange. The angle between the flange and a radial plane is called the layback angle. Alternatively, the roller end may be designed as a portion of a sphere that contacts the flange. Thelatter arrangement, that is a sphereend roller contacting an angled flange, is conducive to improved lubrication while sacrificing some flangeroller guidance capability. In this case, some skewing control mayhave to be provided by the cage. For the case of sphereend rollers and angled flange geometry, the individual contact may be modeled as a sphere contacting cylinder. For the purpose of calculation the sphere radius is set equal to the roller sphere end radius, and the cylinder radius can be approximated by the radius of curvature of the conical flange at the theoretical point of contact. By knowing the elastic contact load, rollerflange material properties, and contact geometries, the contact stress and deflection can be calculated. This approach is only approximate, because the roller end and flange do not meet the Hertzian halfspace assumption. Also, the radius of curvature on the conical flange is not a constant but will vary across the contact width. This method applies only to contacts that are fully confinedto the spherical roller endand conical portionof the flange. It is possible that improper geometry or excessive skewing could cause the elastic contact ellipse to be truncated by the flange edge, undercut, or roller cornerradius. Such a situation is not properly modeledby Hertz stress theory and should be avoided in design because high edge stresses and poor lubrication can result. The caseof a flat end roller and angled flange contactis less amenable to simple contact stress evaluation. The nature of the contact surface on the roller, being at or near the intersection of the corner radius and end flat, is difficult to model adequately. The notion of an “effective”roller radius based on an assumed blend radius between roller cornerand end flat is suitable for approximate calculations. A moreprecisecontact stress distribution can be obtained by using finite element stress analysis technique if necessary.
he information presented in this chapter is sufficient to make a determination of the contact stress level and elastic deformations occurring in a statically loaded rolling bearing. The model of a statically loaded bearing is somewhat distorted by surface tangential stressesinduced by
rolling and lubricant action. However, under the effects of moderate to heavy loading, the contact stresses calculated herein are sufficiently accurate for the rotating bearing as well as the static bearing. The same is true with regard to the effect of “edge stresses” on roller load distribution and hence deformation. These stresses subtend a rather small area and therefore do not influence the overall elastic loaddeformation characteristic. In any event, from the Simplified analytical methods presented in thischapter, a level of loading canbe calculated against which to check other bearings at the same or different loads. The methods for calculation of elastic contact deformation are also sufficiently accurate, and these can be used to compare rolling bearing stiffness against stiffness of other bearing types.
6.1. H, Hertz, “On the Contact of Rigid Elastic Solids and on Hardness,”in ~iscellaneous Papers, MacMillan, London, 163183 (1896). 6.2. S. Timoshenko andJ. Goodier, Theory of Elasticity, 3rd ed., McGrawHill, New York (1970). 6.3. J. Boussinesq, Compt. Rend., 114, 1465 (1892). 6.4. D. Brewe and B. Hamrock, “Simplified Solution for EllipticalContact Deformation Between Two Elastic Solids,” ME Trans., J: Lub. Tech. 101(2), 231239 (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 upon Another,” Univ. Illinois Bull. 212: (July 15, 1930). 6.8. A. Jones, Analysis of Stresses and Dejlections, New Departure Engineering Data, Bristol, Conn., 1222 (1946). 6.9. A. Palmgren and G. Lundberg, “Dynamic Capacity of Rolling Bearings,”Acta Polytech. Mech. Eng. Ser. 1, R.S.A.E.E., No. 3, 7 (1947). ~ n~al~beanspruchungEin~uss g 6.10. 0.Zwirlein and H. Schlicht,‘ ‘ ~ e r k s t o ~ a n s t r e nbei von Reibung und Eigensp~nungen,” 2. W e r ~ s t o ~ e c11, h . 114 (1980). 6.11. IC. Johnson, “The Effectsof an Oscillating Tangentialforce at the Interface Between Elastic Bodies in Contact,’’ (Ph.D. Thesis, Universityof Manchester, 1954). 6.12 J. Smith and C. Liu, “Stresses Due to Tangential and Normal Loads on an Elastic Solid with Application to Some Contact Stress Problems,”ASME Paper 52A13 (December 1952). 6.13 E. Radzimovsky, “Stress Distribution and Strength Condition of Two Rolling Cylinders Pressed Together,”Univ. Illinois Eng.Experiment Station Bull., Series 408 (February 1953). 6.14. C. Liu, “Stress and Deformations Due to Tangential and Normal Loads on an Elastic Solid with Application to ContactStress,” (Ph.D. Thesis, Universityof Illinois, June 1950).
23
CON’IACT STRESS AND DEFO
6.15. M. Bryant and L. Keer, “Rough Contact Between Elastically and Geometrically IdenE J. Applied Mech. 49, 345352 (June 1982). tical Curved Bodies,”A S ~ Trans., 6.16. C. Cattaneo, “ A Theory of Second Order Elastic Contact,” Uniu. Roma Rend. Mat. Appl. 6,505512 (1947). 6.17. T. Loo, “A Second Approximation Solution on the Elastic Contact Problem,”Sei. Sin‘ ,12351246 (1958). ica 7 6.18. H. Deresiewicz, “A Note on Second Order Hertz Contact,” A S M E Trans., J. AppL Mech. 28,141142 (March 1961). 6.19. R. Sayles, G. desilva, J. Leather, J. Anderson, and P. MacPherson, “Elastic Conformity in Hertzian Contacts,”Tribology Intl. 14, 315322 (1981). 6.20 J. filker, “Numerical Calculation of the Elastic Field in a Halfspace Due to an Arbitrary Load Distributed over a Bounded Region of the Surface,” SKI? Eng. and Res. Center Report NL82D002, Appendix (June 1982). 6.21 N. Ahmadi, L. Keer, T. Mura, and V. Vithoontien, “TheInterior Stress Field Caused by Tangential Loading of a Rectangular Patch on an Elastic Half Space,”ASME Paper 86Trib15 (October 1986). 6.22 T. Harris and W. Yu, “LundbergPalmgren Fatigue Theory: Considerations of Failure Stress and Stressed Volume,”A S M E Trans., J. Tribology 121,8590 (January 1999). 6.23 I(. Kunert, “Spannungsverteilung im Halbraum bei Elliptischer Flachenpressungsverteilung uber einer Rechteckigen Drucldiache,” Forsch. Geb. Ingenieur~es27(6), 165174 (1961). 6.24. H. Reusner, “Druckflachenbelastung und Ove~achenverschiebungin Walzkontakt von Rotationkorpern (Dissertation, Schweinfurt, Germany, 1977). 6.25. B. Fredriksson, “ThreeDimensional RollerRaceway Contact Stress Analysis,” Advanced Engineering Corp. Report, Linkoping, Sweden (1980). 6.26. H. Reusner, “The Logarithmic Roller Profilethe Key to Superior Performance of Cylindrical and Taper Roller Bearings,”Ball Bearing J. 230, SKI? (June 1987).
Symbol
Description Distance between raceway groove curvature centers + f,  1,total curvature Crown drop Influence coefficient Ball or roller diameter Bearing pitch diameter Eccentricity of loading Modulus of elasticity Raceway groove radius + D Applied load Friction force due to roller endring flange sliding motions Roller thrust couple moment arm Number of rows of rolling elements Ring section moment of inertia
Units mm (in.) mm (in.) mm/N (in./lb) mm (in.) mm (in.) mm (in.) MPa (psi)
N (lb) mm (in.) mm4 (in.4) 31
Symbol
Units Axial load integral Radial load integral Moment load integral Number of laminae Loaddeflection factor; axial load deflection factor Roller length Distance between rows Moment Moment applied to bearing Loaddeflection exponent Diametral clearance Load per unit length Ball or rollerraceway normal load Roller endring flange load Raceway groove curvature radius Radius to contact in tapered roller bearing Tapered roller radius to flange contact at roller large end
N/mmn (lb/in.") mm (in.) mm (in.) N * mm (lb in.) N mm (lb in.) 9
mm (in.) N/mm (lb/in.) N(W N (1b) mm (in.) mm (in.) mm (in.) mm (in.)
of locus of raceway groove mm (in.) Distance between loci of inner and outer raceway groove curvature centers Ring radial deflection Strain energy Number of rolling elements Mounted contact angle Free contact angle tanl l / ( d m  I)) D cos a / d m eflection or contact deformation istance between inner and outer rings ontact deformation due to ideal normal gular spacing between rolling elements oad distribution factor earing misalignment angle Lamin~mposition Coefficient of sliding friction between roller end and ring flange
mm (in.) mm (in.) N mm (lb * in.) rad, rad, rad,
O O O
mm (in.) mm (in.)
mm (in.) rad, O rad, O rad, O rad, O
~E~
Symbol
Units
Description
Curvature sum skewing Roller angle Position angle Contact deformation at laminum h due to skewing roller Azimuth angle
mml (in.') rad, O rad, O mm (in.) rad, O
~ ~ ~ S ~ R I P T ~ efer to axial direction Refers to inner raceway Refers to ring angular position Refers to rolling element position Refers to rolling element position Refers to line contact Refers t o rolling element position Refers to raceway Refers to moment loading Refers to direction collinearwith normal load Refers to outer raceway Refers to point contact Refers to radial direction Refers to rolling element Refers to gear separating load Refers to gear tangential load Refers to bearing row Refers to outer raceway efers to inner raceway efers to tapered roller bearing, roller endflange contact Refers to angular location
aving deter~inedin Chapter 5 how each ball or roller in a bearing carries load, it is possible to determine how the bearing load is d i s t ~ b uted among the balls or rollers. To do this it is first necessary to devel loaddeflection relationships for rollingelements contacting raceways. using Chapters 2 and 6 these loaddeflection relationships can be de r any type of rolling element contacting any type of rac , the material presented in this chapter is completel~depe revious chapters, and a quick review might be advanta~e
ost rolling bearing applications involve steadystate rotation of either the inner or outer raceway or both; however, the speeds of rotation are usually not so great as to cause ball or roller centrifugal forces or copic moments of magnitude large enough to affect si ibution of applied load among the rolling elements. most applications the frictional forces and moments acting on the rolling elements also do not signi~cantlyinfluence this load distribution. Consequently, in analyzing the distribution of rolling element loads, it is usually satisfactory to ignore these eEects in most applications. In this chapter the load distribution of statically loaded ball and roller bearings will be investigated,
From equation (6.38) it can be seen that for a given ballraceway contact (point loading) 6
N
Q213
(7.1)
Inverting equation (7.1) and expressing it in equation format yields
Similarly, fora given rollerraceway contact (line contact) Q
K1;510/Q
(7.3)
In general then
in which n = 1.5 for ball bearings and n = 1.I1 for roller bearings. The total normal approach between two raceways under load separated by a rolling element is the sum of the approaches between the rolling element and each. raceway. Hence 6n =
Therefore,
and
si + 6,
(7.5)
=
Knsn
(7.7)
For steel ballsteel raceway contact,
K~ = 2.15
X
105
x
p2(6*)3/2
Similarly, for steel roller and raceway contact,
K~ = 7.86
X
104 ~9
For a rigidly supported bearing subjected to radial load, the radial deflection at any rolling element angular position is given by 6, = 6, cos if$  +Pd
(7.10)
in which 6, is the ring radial shift, occurring at if$ = 0 and P, is the diametral clearance. Figure 7.1 illustrates a radial bearing with clearance. Equation (7.10) may be rearranged in terms of maximum deformation as follows: 6,
in which
=
[ 2:
Smax 1   (1  cos if$)]
+z) (z) 2
(7.11)
(7.12)
From equation (7.12) the angular extent of the load zone is determined by the diametral clearance such that
h = cos1
I
(7.13)
For zero clearance, h = 90". From equation (7.41, (7.14) Therefore, from (7.11) and (7.14),
BUTION OF
~~~
LOADENG EN S T A T I C ~ L L S O ~ BE ~
Before displacement (a)
4 pd
(7.15) For static equili~rium, the applied radial load must equal the sum of the vertical components of the rolling element loads: (7.16)
or
[
*=Lt&
Fr = Qmax
It:
*=O
1
1 g (1  cos
+)In
cos
+
(7.17)
Equation (7.17) can also be written in integralform: Fr =
ZQm,
X
2T
1'" [ &
1 1   (1  cos +) 236
I"
cos t,b d+
(7.18)
(7.19) in which Jr(E) =
I+"[
22T &
1 1   (1  cos +) 2E
(7.20)
The radial integralof equation (7.20) has been evaluated numerically for various values of E. This is given in Table 7.1. From equation (7.7),
Therefore,
For a given bearing with agiven clearance under agiven load, equation (7.22) may be solved by trial and error. A value of 8, is first assumed and E is calculated from equation (7.12). This yields J,(E)from Table 7.1. If equation (7.22) does not then balance, the process is repeated. ,
Line E
Load Distribution Integral J,w  2
2
]
112
1
In (7.10~)? 15 A 5 k . For the bearing misalignment 0 shown in Fig. 7.24, the effective misalignment at roller location azimuth tc(j is rt $0 cos t,$, The plus sign per5 rt d 2 ; the minus sign pertains to + d 2 5 "C rt w tains to 0 5 (assuming symmetry of loading about the 0  w diameter). Therefore, the total rollerraceway deformation at roller location j and laminum A is given by equation (7.109).
*
*
s,
=
Aj 3 $ @A  #w cos
*

c,
h = 1, k
(7.109)
As discussed in Chapter 6, equation (6.53) is theoretical and relatescontact load to deformation for ideal line contact, while equation (6.54) i s
+
7.24. Components of rollerraceway deformation due to radial load, misalignment, and crowning.
7.25. Typical geometry of a crowned cylindrical roller showing crown radius, roller effective length, and roller straight length.
empirical and pertains to a crowned rollerraceway contact. While the loaddeformation characteristic of an individual contact laminum may be described using either equation, the former is applied here since solution of a transcendental equation leads to force and moment equilibrium equations of greater complexity. Considering that the contact is divided into k laminae, each of width w ,therefore contact length is kw . Letting q = QlZ,equation (6.54) becomes (7.110) Rearranging (7.110) to define q yields
(7.111) Equation (7.111) does not consider edge stresses; however, becausethese obtain only over very small areas, they can be neglected with little loss of accuracywhenconsidering equilibrium of loading. ~ubstitutionof equation (7.109) into (7.111) gives
Depending on the degree of loading and misalignment, all laminae in every contact may notbe loaded; in (7.IE),hj is the number of laminae under load at roller location j. Total roller loadingis given by
u ~ t i o of ~ sStatic E q u i l i ~ ~ i u m To determine the individual roller loading, it is necessary to satisfy the requirements of static equilibrium. Hence, for an applied radial load,
~ubstitutingequation (7.113) into (7.114) yields
=o
(7.115)
For an applied coplanar misaligning moment load, the equilibrium condition to be satisfied is 3Qjej cos
$
=
0 Tj
=
0.5;
$
=
I;
$ 9 0, TI (7.116)
=
0, TI
where ej is the eccentricity of loading at each roller location.ej, which is illustrated in Fig. 7.26, is given by
77 I
.L
I
F I ~ 7.26. ~ ELoad distribution for a misaligned crowned roller showing eccentricity of loading.
Hence,
The remaining equations to be established are the radial deflection relationships. It is necessary here to determine the relative radial movement of the rings caused by the misalignment as well as that owing to radial loading. To assist in thefirst determination, Fig. 7.27 shows schematically an inner ringroller assembly misaligned with respect to the
interferencg with
. From thia sketch, it ia evident that onehalf aE the roller ineluded angleis descl.ibed by
where
IAL ROLLER B
R
=
~
~
~
S
[(dm D)2 + z 2 y 2
In developing equations (7.121) and (7.122), the effect of crown drop was investigated and found to be negligible. Expanding equation (7.121) in termsof the trigonometric identity further yields 6,
=
R(cos /3 cos Oj

+ sin /3 sin Oj  cos p)
Since ej is small, cos Oj 1, and sin Oj and sin /3 = 112 R, therefore

(7.123)
Oj, Moreover Oj = rt: 0 cos lJzj
6, = k4Z0 cos lJzj
(7.124)
The shift of the inner ring center relative to the outer ring center owing to radial loading and clearance, and the subsequent relative radial movement at any roller location are shown in Fig. 7.28. The sum of the relative radial movement of the rings at each roller angular location minus the clearance is equal to the sum of the inner and outer raceway maximum contact deformations at the same angular location. Stating this relationship in equation format: d [ar ct +Z0] COS lJzj  P 2[Aj rt: 40(A  4 )COS ~ lJzj  c 2
~ = 0]
(7.125) ~ ~
Equations (7.115), (7.118), and (7.125) constitute a set of 212 + 3 simultaneous nonlinear equations that can be solved forar, 0, and Aj using numerical analysis techniques. Thereafter, the variation of roller loadper
7.88. Displacement of ring centers caused by radial loading showing relative radial movement.
2
D
I
S
~
~ OF ~II N O
~T LOADING ~ ~ IN~S T A T I C ~ Y L
unit length, and subsequently the roller load, may be determined for each roller location using equations (7.112) and (7.113), respectively. Using the foregoing method and digital computation, Harris E7.51 analyzed a 309 cylindrical roller bearing having the following dimensions and loading: Number of rollers Roller effective length Roller straight lengths Roller crown radius Roller diameter Bearing pitch diameter Applied radial load
12 12.6 mm (0.496 in.) 4.78, 7.770, 12.6 mm 1245 mm (49 in.) 14 mm (0.551 in.) 72.39 mm (2.85 in.) 31,600 N (7100 lb)
For the above conditions, Fig.7.29 shows the loading on various rollers for the bearing with ideally crowned rollers [I = 12.6 mm (0.496 in.) and with fully crowned rollers ( I = 0). Fig. 7.30 shows the effect or roller crowning on bearing radial deflection as a function of misalignment.
F
c
When radial cylindrical roller bearings have fixed flanges on bothinner and outer rings, they can carry some thrust load in addition to radial load. Thegreater the amount of radial load applied,the more thrust load that can be carried. As shown by Harris [7.6] and seen in Fig. 7.31, the thrust load causes each roller to tilt an amount 6. Again, it is assumed that a rollerraceway contact can be subdivided into laminae in planes parallel to the radial plane of the bearing. When a radial cylindrical rollerbearing is subjected to applied thrust load, the inner ring shifts axially relative to the outer ring. Assuming deflections owing to rollerendflange contacts are negligible, then the interference at any axial location (laminurn) is 6, = Aj 3 &(l h  +)Lu  CA
h = 1, kj
(7.126)
where c, is given by equations (7.108). Figure 7.32 illustrates the component deflections in equation (7.126). Substituting equation (7.126) into (7.111) yields
P
P
281
~ I S T R I ~ ~ T IOF O NIIWEXWAL L O ~ I N G IN S T A T I C ~ L S
24
O ~ I3E
~
0.08
22
20
0.05
c
E E
18 X
.r'
v
._.16
0.04
4J
8 ;F:
a"
fii .0
I, = 7.70 mm ( 0.303in. 1 14

12
\
\
0.03
,
10
8 5*
0
15
10
20
25
~ i ~ l i g n m e(min.) n~
7.30. Radial deflection vs misalignment and crowning309 bearing at 31,600 N (7100 lb)radial load.
cylindrical roller
(7.127) "
and at any azimuth 
*, the total roller loading is w0.89
1.24
X
x
A=kj
105143~1
v
[Aj
+ &(A
 $)LO
 CA]'"'
(7.128)
Housina Y
Qa)
Shaft
Shaft ” 
7.31.
“4
Shaft
Q
” ” ” ”
Thrust couple, roller tilting, and inteflerence owing to appliedthrust load.
I ~ 7.32.~ Components E of rollerraceway deflection at opposing raceways due to radial load, thrust load, and crowning.
84
~ I S T R I B ~ I OOF N ~E~~LOADING
IN STATICALLY LOADED B
~
~
~ubstitutingequation (7.128) into (7.129) yields
(7.130) For an applied centric thrust load, the equilibrium condition to be satisfied is
each roller location, the thrust couple is balanced by a radial load coupled caused by the skewed axial load distribution. Thus, hQaj = 2Qjej and F a "
2
2 j=Z/2+1
h
x
j=1
=
7Qjej = 0
7j =
0.5; 1;
3/j
=
0,
T
3/jfO?T
(7.132)
where ej is the eccentricity of loading indicated by Fig. 7.26 and defined bY
(7.133) h=l

~ubstitutionof equations (7.128) and (7.133) into (7.132) yields
(7.134)
G
S
THRUST L
O
~ OF ~ RADIAL G C
~
~
R
I
C
~~
E
~ R
O ~
~G R S
Radial deflection relationships remain to be established. It is necessary to determine the relative radial movement of the bearing rings caused by the thrust loading as well as that due to radial loading. To assist in this derivation, Fig. 7.31 shows schematically a thrustloaded rollerring assembly. From this sketch, a roller angle is described by tan q
=
D 
I
(7.135)
The maximum radial interference between a roller and both rings is given by (7.136)
In developing equation (7.136) the effect of crown drop was found to be negligible. Expanding equation (7.136) in terms of the trigonometric identity and recognizing that sj is small and I = D ctn q, yields
stj= 16
(7.137)
Whereas Stj is the radialdeflection due to roller tilting, it can be similarly shown that axial deflection owing to roller tilting is
saj= Dsj
(7.138)
Therefore, the radial interference caused by axial deflection is (7.139) The sum of the relative radial movements of the inner and outer rings at each roller azimuth minus the radial clearance is equal to the sum of the inner andouter raceway maximum contact deformationsat the same azimuth, or
~quations(7.130), (7.134), and (7,140) are a set of simultaneous equations that can be solved for 6, Aj, a,, and iSa. Thereafter, the variation of roller load per unit length and roller load may be determined for each roller a ~ i m ~using t h eq~ations(7.127) and (7.1281, respectively. The axial load on each roller may be determined from
r
Y
When rollers are subjected to axial loading as shown in Fig. 7.31, due to sliding motions between the roller ends and ring flanges friction forces occur; for exampleFaj= pQaj, in which p is the coefficient of friction. In a m i s a l i ~ e dbearing, each roller which carries load is end and forced against the opposing flangewith a load tion force Fajat the roller end. Due to F4 a moment occurs creating a yawing or skewing motion in addition to the predominant rolling motion about the roller axisand secondary rollertilting. The tilting andskewing motions occurin orthogonal planes which contain the roller axis. Friction forces acting on rolling elements are not introduced until Chapter l.4; however, roller skewingis resisted by the concave curvature of the outer raceway. The resisting forces and accompanying deformations alter the distribution of load along both the outer and inner racewayroller contacts, Figure 7.33 illustrates the forces which occur ona roller subjected to radial and thrust loading. Frictional stresses T~~~are discussed in detail in Chapter 14. Figure 7.34 shows the roller skewing angle and the rollerouter raceway loading whichresults. The rollerraceway contact deformations which result from skewing, as demonstrated by Harris et al. [7.7], may be described by equation (7.142).
4
(7 142) In (7.142), subscript m refers to the outer and innerraceway contacts; m = 1 and 2 respectively, and deformations due to skewing +mjh are given by
x
7.33. Normal and friction forces acting on a radial and t ~ r u ~ t  l o a roller. ~ed
7.34. Rollerouterracewaycontactshowing roller skewing angle 8 and restoring forces.
(7.143) (7.144)
288
D I S T ~ ~ ~ IOF O ~N E
~ LOADING A L IN STATICALLYLOADED ~
E
~
It can be further seen that equation (7.140) must become
Owing to the unknown variables 4 and A,, the latter replacing Aj, additional equilibrium equations must be established. For equilibrium of roller loading in the radial direction m=2
h=k
m=2
(7.146)
Referring to Fig. 7.34 and considering equilibrium of moments in the plane of roller skewing
lFaj+
xx
m=2 h=k
w2[A  +(k
m=1 h = l
+ l)]rmjh
m=2 A=k
(7.147)
Since the angle pj

0, sin pj

pj, therefore (7.148)
Using methods described in Chapter 14, it can be determined that the moment loading effect of rollerraceway shear stresses is rather small compared to that of the restoring and roller end forces. Therefore, substituting (7.148) into (7.147) yields
Considering that thecontact deformationsdue to roller radial loading are different for each rollerraceway contact, bearing load equilibrium equations (7.130) and (7.134) must be changed accordingly; hence
~
G
S
"0
(7.150)
and
Equations (7.132), (7.145), (7.146),(7.149), (7.150), and (7.151) constitute a set of simultaneous, nonlinear equations which may be solved for Amj, 4,4, is,,and Sa . Subsequently, the rollerraceway loads Qj and roller endflange loads Qaj may be determined. The skewingangles determined using the foregoing equations strictly pertain to full complement bearings and bearings having no roller guide flanges. For a bearing with a substantially robust and rigid cage, the skewing angle may be limited by the clearances between the rollers and the cage pockets. For a bearing with guide flanges, the skewing may be limited by the endplay between the roller ends and guide flanges. In general, the latter situationsobtain; however, to the extent that skewing is permitted, the foregoing analysis is applicable.
For radial cylindrical rollerbearings, it is possible to apply general combined loading. The equations for load equilibrium defined above apply; however, the interference at any laminum in the rollerraceway contact is given by
where subscript m = I. refers to the outer raceway and m = 2 refers to the inner raceway. Coefficient vl = 1, and v2 = + l .Contact load per unit length is given by
Similar equations may be developed for tapered roller bearings. In this case, as shown in Chapter 5 , roller endflange loading occursduring all conditions of applied loading, and the roller and bearing equilibrium equations must be altered accordingly Using Fig. 5.3 to illustrate the geometry and loading of a tapered roller in a bearing, and establishing dimensions r2, r32,and r3%as follows:
r2 r32 r3x
is the radius in a radial plane from the inner ring axis of rotation to the center of the inner raceway contact, is the radius in a radial pla e from the inner ring axis of rotation to the center of the roller endinner ring flange contact, and is the xdirection distance in an axial plane from the center of the inner raceway contact to the center of the roller endinner ring flange contact,
the roller load equilibrium equations are m=2
w
21 emcos
m=
A=k
CYm
m=2
m= 1
x
A=k
cm sin am
w
2 qmjh
(7.154)
A= 1
A= 1
qmjh+ Q~~sin a3 =
o
(7.155)
In (7.154) and (7.155), &3j is the roller endflange load,and a3is the angle that load &3j makes with a radial plane. Coefficient el = 1, and c2 = +1. The equation for radial plane moment equilibrium of the roller is
where R3 is the radius from the roller axis of rotation to the center of the roller endflangecontact. Eq~ilibriumof actuating and resisting moments pertaining to roller skewing is given by
The forceand moment equilibrium equations with respect to the bearing inner ringare as follows:
LY SUPPORTED ROLLING ~
E
~
~
G
S
(7.159)
(7.160) In equations (7.158)(7.160) for J/j 5 = 1.
=
0 or v,coefficient 7 = 0.5; otherwise
Spherical roller bearings are internally selfaligning and therefore cannot carry moment loading. Moreover, for slow or moderate speed applications causing insignificant roller inertial loading and friction, symmetrical (barrelshaped) rollers in spherical roller bearings will exhibit no tendency to tilt. Therefore, the simpler analytical methods presented earlier in this chapter will yield accurate results. For spherical roller bearings having asymmetrical rollers, however, such as spherical roller thrust bearings (Fig. 1.45), roller tilting and hence skewing is not eliminated. In this case, for the purpose of analysis, the bearing may be considered a special type of tapered roller bearing having fully crowned rollers. Then the methods of analysis discussed in the preceding section may be applied for increased accuracy.
The preceding discussionof distribution of load amongthe bearing rolling elements pertains to bearings having rigidly supported rings. Such bearings are assumed to be supported in infinitely stiff or rigid housings and on solid shafts of rigid material. The deflections considered in the determination of load distribution were contact deformations, that is, Hertzian deflections. This assumption is, in fact, an excellent approximation for most bearing applications. In some radial bearing applications, however, the outer ring of the bearing may be supported at one or two angular positions only, and the
shaft on which the inner ringis positioned may be hollow. The condition of twopoint outer ring support, as shown in Figs. 7.35 and 7.36, occurs in theplanet gear bearings of planetary gear power transmission system, and was analyzed by Jones and Harris E7.81. In certain rolling mill applications, the backup roll bearings may be supported at only one point on the outer ring or possibly at two points as shown in Fig. 7.37. These conditions were analyzed by Harris [7.9]. In certain high speed radial bearings, to prevent skidding it is desirable to preload the rolling elements by using an elliptical raceway, thus achieving essentially twopoint ring loading under conditions of light applied load. The case of a flexible outer ring andan elliptical inner ringwas investigated by Harris and Broschard E7.101. In each of the foregoing applications, the outer ring must be considered flexible to achieve a correct analysis of rolling element loading. In many aircraft applications to conserve weight the power transmission shafting is made hollow. In these cases the inner ring deflections will alter the load distribution from that considering only contact deformation. To determine the load distribution among the rolling elements when one or both of the bearing rings is flexible, it is necessary to determine
FIGURE 7.35. Planet gear bearing.
F L E ~ L ~
~ ~ ROLLING P O R T BEARINGS E ~
7.36. Planet gear bearing showing gear tooth loading.
F1GUR.E 7.37. Cluster mill assembly showing backup roll bearing loading.
€3~TION OF ~E~~
L O A D ~ GZN ~
T
~
T LOADED I ~ ~€3 L
~
the deflections of a ring loaded at various points around its periphery. This analysis may be achieved by the application of classical energy methods for the bending of thin ratings. As an example of the method of analysis, consider a thin ring subjected to loads of equal magnitude equally spaced at angles A+ (see Fig. 7.38). According to Timoshenko i7.111, the difEerentia1equation describing radial deflection u for bending of a thin bar with a circular center line is (7.161) in which I is the section moment of inertia in bending and E is the modulus of elasticity. It can be shownthat thecomplete solution of equation (7.161) consists of a complementary solution and a particular solution. The complementary solutionis u, = C, sin cf,
+ C,
cos cf,
(7.162)
in which C, and C, are arbitrary constants. Consider that the ring is cut at two positions: at theposition of loading, = $A+, and at theposition = 0, midway between the loads. The loads required to maintain equilibrium over the section are shown in Fig. 7.39.. From Fig. '7.39 it can be seen that since horizontal forces are balanced,
+
+
Qsin = 2F0
+
or
7.88. Thin ring loadedby equally spaced loads of equal magnitude.
(7.163)
1
7.39. Loading of section of thin ring between 0
2
(b
+A+.
Q Fo = 2 sin cjb
(7.164)
The moment at any angle cf, between 0 and &h,b is apparently
M
M0  Foal cos cf,)
(7.165)
(4% Mo  (1  cos cf,)
(7.166)
=
or
M
=
2 sm cf,
Since the section at cf, = 0 is midway between loads,it cannot rotate. theorem f7.111 the angular rotation at any ~ ~ c o r d to i ~Gastigliano's g section is @=:
au a"
(7.167)
in which U is the strain energy in the beam at the position of loading. ~ ~ o s h ~ E7.113 n k oshows that for a curved beam
= Mb and
since the section is constrained from rotation,
~ u b ~ t i ~ u tequation ing (7.166) into ('7.169)and inte
(7.170) Hence, (7.171) Equation (7.171) may be substituted for M in (7.161) such that the particular solution is (7.172) The complete solution is u = u,
[
Qs3 # sin (b + up = C, sin # + C, cos # + 2E1 2 sin (*A+)
1
A+
]
(7.173) Because the sections at .# = 0 and #
=
*A+ do not rotate, therefore
Hence, the radial deflection at any angle (b between # = 0 and # = i A 3 , is
(7.174) Equation (7.174) maybe expressed in another format as follows: u =
c,
(7.175)
in which C, are influence coefficientsdependent on angular position and ring dimensions.
LY $ ~ ~ O R ROLLING T E ~ ~ ~ I N G $
7
(7.176) Lutz E7.121 using procedures similar to those described above developed influence coefficients for various conditions of point loadingof a thin ring. These coefficients have been expressed in infinite series format for the sake of simplicity of use. For a thin ring loaded by forces of equal magnitude symmetrically located about a diameter as shown in Fig. 7.40, the following equation yields radial deflections:
in which (7.178) The negative sign in (7.178) is used for internal loads and the positive sign is used for external loads. Equation (7.177) definesradial deflection at angle t,bi caused by Qj at position angle t,$. When rollingelement loads Qjare such that a rigid body translation 8, of the ring occurs, in the direction of an applied load, equations (7.177) are not selfsufficient in establishing a solution; however,a directional equilibrium equation may be used in conjunction with (7.177) to determine the translatory movement. Referring to Fig. 7.41 the appropriate equilibrium equation is as follows:
I
magnitude located asymmetrically about
a diameter.
8
UTIQN OF ~E~~
~1~~~
LOADING IN STATICALLY L Q A D E ~B
~
~
N
7.41. Thin ring showing e q u i l i b ~ u of~forces,
Fr cos 3/i

Qj cos 3ij
=
0
(7.179)
In the planet gear bearing application demonstrated in Fig. 7.36 the gear tooth loads maybe resolved into tangential forces, radial forces, and moment loads at Ifi = 90" (see Fig. 7.42). The ring radial de~ections at angle Ifii due to tangential forces Ft are given by (7.180)
in which
7.42.
ring.
Resolution of gear tooth loadingon outer
~
S
mr m& 2 m(m2  I ) 2
m=m sin  cos
g i
x
2s3 =TEI m=2
(7.181)
Equations (7.180) are not selfsufficient and an appropriate equilibrium equation must be used to define a rigid ring translation. The separating forces Fs are selfequilibratin~ and thusdo not cause a rigid ring translation. The radial deflections at angles t,bi are given by sui = ,giFS
(7.182)
in which
Note that equations (7.183) are special cases of (7.178) in which position angle J/j is 90" and loads Qj are external. Similarly, the moment loads applied at sr/ = 90" are selfequilibrating. The radial deflections are given by
in which
To find the ring radial deflections at any regular position due to the combination of applied and resisting loads, the principle of superposition is used. Hence for the planet gear bearing, the radial deflection at any angular position t,bi is the sum o f the radial deflections due to each individual load, that is,
or
ui
=
,CiFS + MCiM+ tCiFt + zQCij
(7,187)
lements do the
A load maynot be transmitted through a rolling element unless the outer ring deflects suf~cientlyto consume the radial clearance at theangular position occupied bythe rolling element. Furthermore, because a contact deformatio~is caused by loading of the rolling element, the ring deflections cannot be determined without considering these contact deformations. Therefore the loading of a rolling element at angular position Jlj depends on the relative radial clearance. The relative radial approach of the rings includes the translatory movement of the center of the outer ring relative to the initial center of that ring, which position is fixed in space. Hence forthe planet gear bearing the relative radial approach at angular position +i is
sicos= 6,
Jli
+ ui
(7.188)
From equation (7.4) the relative radial approach is related t o the rolling element load as follows: Qj =r
Qj
=
IC( Sj  rj)"
sj > rj
0
Sj 5 rj
(7.189)
in which rj is the radial clearance at angular position sum of Pd/2 and the condition of ring ellipticity.
Jlj. Here rj is the
olling Elem~nt Using the example of the planet gear bearing, the complete loading of the outer ring is shown in Fig, 7.43, which alsoillustrates the rigid ring translation 6,. Combination of equations (7.187)( 7.189) yields i=Z/2+2
si  6,cos
Jli

*CiFS MciM ,CiFt 
iK j=2
*Cii(Sj
rj)" =
0
(7,190) The required equilibrium equation is (7.191) considering symmetry about the diameter parallel to the load. In equation (7.191), 3 = 0.5 if the rolling element is located at $ = 0" or at Jlj = 180"; otherwise 3 = 1.
301
FLEXIBLY S ~ ~ O R T EROLLING D BEARINGS
~
I 7.43. ~Total loading ~ ofEouter ring in planet gear bearing.
Equations (7.190) and (7.191) constitute a set of simultaneous nonlinear equations which may be solved bynumerical analysis. The NewtonRaphson method is recommended. Using this method, the unknowns Sj and hence Qj can be determined at each rolling element location. Figure 7.44 shows a typical distribution Planet gear bearing
Rigidring bearing 7.44.
bearing.
Comparison of load distribution for a rigid ring bearing and planet gear
8,000
35,000
7,000
30,000
6,000 25,000
5,000
20,000 c) v 0
.8
tc
N
4,000
0
n:
15,000
3,000
10,000 2,000
5,000 1,000
Roller position (degrees t 1 7.46. Roller load vs number of rollers and position. 222,500 N (50,000 lb) at dimensions constant. Outer ring section thickness increasesas thenumber of rollers is increased and rollerdiameter is subsequently decreased. st 30°, inner
required to solve the displacements and load distribution accurately in a rolling bearing mounted in a flexible support. Figure 7.48 from Zhao E7.141 shows the grids used to analyze a flexibly mounted cylindrical roller bearing assuming both solid and hollow rollers. The load distribution would be similar to that indicated in Fig. 7.47.
D I $ ~ I B ~ OF I OI ~N T E LOADING ~ ~ IN $ T ~ T I C ~ LLOADED Y BE~ING$
7.47. Photoelastic studyof a roller bearing supportingloads aligned at approximately + 30" to the bearing axis.
(4
7.48. Finite element meshes for analyzing (a)cylindrical roller bearing rings, (b) solid rollers, (e) hollow rollers, and ( d ) contact zone. From [7.14].
REFERENCES
Bourdon et al. E7.151 and E7.161 provide a method to define stiffness matrices for use in standard finite element models to analyze rolling bearing loads and deflections, and the loading and deflections of the mechanisms in which they are employed. For flexible mechanisms and bearing support systems, they demonstrate the importance of considering the overall mechanical system rather than only the local system in the vicinity of the bearings.
The methods developed in this chapter to calculate distribution of load among the balls and rollers of rolling bearings can be used in most bearing applications because rotational speeds are usually slow to moderate. Under these speed conditions, the effects of rolling element centrifugal forces and gyroscopic momentsare negligible. At high speeds of rotation these body forces become significant,tending to alter contact angles and clearance. Thus, they can affect the static load distribution to a great extent. In Chapter 9 the effect of these parameters on high speed bearing load distribution will be evaluated. In the foregoing discussionthe effect of load distribution on the bearing deflection has been demonstrated. Further, since the contract stresses ina bearing depend on load, maximumcontact stress ina bearing is also a function of load distribution. Consequently, bearing fatigue life that is governed by stress level is significantly affectedby the rolling element load distribution.
7.1. R. Stribeck, “Ball Bearingsfor Various Loads,”Trans. ASME 29,420463 (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. Sjovall, “The LoadDistribution within Balland Roller Bearings under Given External Radial and Axial Load,”Teknisk Tidskri?, Mek., h.9 (1933). 7.5. T. Harris, “The Effectof Misalignment on the Fatigue Life of Cylindrical Roller Bearings Having Crowned Rolling~embers,”ASME Trans,, J. Lab. Tech., 294300 (April 1969). 7.6. T. Harris, “The Endurance of a ThrustLoaded,Double Row, Radial Cylindrical Bearing,” ea^ 18,429438 (1971). 7.7. T. Harris, M. Kotzalas, and W, Yu, “On the Causes and Effects of Roller Skewing in Cylindrical Roller Bearings,”Trib. Trans., 41(4),572578 (1998). 7.8. A. Jones and T. Harris, “Analysis of a Rolling Element Idler Gear Bearing Having a Deformable Outer Race Structure,” ASME Trans., J. Basic Eng., 273278 (June 1963).
DI~TR~~TI OF’ ON ~~~~
L O A D ~ IN G ~ T A T I ~ LOADED ~ L Y I3
7.9. T. Harris, “Optimizing the Design of Cluster Mill Rolling Bearings,” ASLE Duns. 7 (Apr. 1964). 710. T. Harris and J. Broschard, “Analysis of an Improved Planetary Gear Transmission Bearing,”AS~EDuns., J Basic Eng., 457462 (Sept. 1964). 7.11. S. Timoshenko,S t r e n g t ~ofHute~ials, Part I, 3rd ed., Van Nostrand, New York (1955). 7.12. W. Lutz, Discussion of 17.81, presented at ASME Spring Lubrication Symposium, Miami Beach, Fla. (June 5, 1962). 7.13. H. Eimer, “Aus dem Gebiet der Walzlagertechnik” (Semesterentwurf, Technische Hochschule, Munich, June 1964). 7.14. ET. Zhao, “Analysis of Load Distributions within Solid and Hollow Roller Bearings,” A8ME Duns., J Tribology 120, 134139 (Jan. 1998). 7.15. A. Bourdon, J. Rigal, and D. Play, “Static Rolling Bearing Models in a C.A.D. Environment for the Study of Complex Mechanisms: Part IRolling Bearing Model,” A S ~ Duns., E J Tribology 121,205214 (April 1999). 7.16. A. Bourdon, J. Rigal, and D, Play, “Static Rolling Bearing Models in a C.A.D. Environment for the Study of Complex Mechanisms: Part 11Complete Assembly Model,” ASHE Duns., J Tribology 121,215223 (April 1999).
Symbol
Description Semimajor axis of projected contact ellipse Semiminor axis of projected contact ellipse Pitch diameter Ball or roller diameter Complete elliptic integral of the second kind r/D Center of sliding Gyratory moment Rotational speed Raceway groove radius Rolling radius Radius of curvature of deformed surface
Units mm (in.) mm (in.) mm (in.) mm (in.)
mm (in.) Nmm (in, lb) rpm mm (in.) mm (in.) 0
mm (in.)
~~~~
SPEEDS
Symbol U X
2 Y ji
x 2
a
P' P Y' Y K
0 Of
f g i m 0
R RE roll S
sl X X'
Y Y' 2
x'
Units Surface velocity Distance in xdirection Acceleration in xdirection Distance in ydirection Acceleration in ydirection Distance in xdirection Acceleration in xdirection Contact angle Angle between projection of the TJ axis on the x'y ' plane and the x' axis (Fig. 5.4) Angle between the W axis and x' axis (Fig. 5.4)
mmlsec (in./sec) mm (in.) mm/sec2 (in./sec2) mm (in.) mm/sec2 (in./sec2) mm (in.) mm/sec2 (in./sec2) rad, O rad rad
DM, D cos ald, a/b Rotational speed Flange angle
rad/sec rad, O
SU~SCRIP~S Refers to flange Refers to gyroscopic motion Refers to inner raceway Refers to orbital motion Refers to outer raceway Refers to rolling element Refers to roller end Refers to rolling motion Refers to spinning motion Refers to sliding motion on flangeroller end efers to xdirection (Fig.5.4) efers to x'direction (Fig. 5.4) efers to ydirection (Fig. 5.4) efers to y'direction (Fig. 5.4) efers to zdirection (Fig. 5.4) efers to 2'direction (Fig. 5.4)
roller bearings are used to support various kinds of loads while permitting rotational andlor translatory motion of a shaft or slider. In this book treatment has been restricted to shaft rotation or oscillation.
ROLL^^ ~ O T I O ~
Unlike hydrodynamic or hydrostatic bearings, motions occurringin rolling bearings are not restricted to simple movements. For instance, in a rolling bearing mounted on a shaft that rotates at n rpm, the rolling elements orbit the bearing axis at a speed of n, rpm, and they simultaneously revolve about their own axes at speeds of nRrpm. Additionally, the rolling motions are accompanied by a degree of sliding that occurs in the contact areas. In ball bearings, substantial amounts of spinning motion occur simultaneo~slywith rolling if the contact angles between balls and raceways are not zero, that is, for other than simple radial bearings. Also, gyroscopic pivotal motions occur,particularly in oil and greaselubricated ball bearings. In this chapter, rolling bearing internal rotational speeds and relative surface velocities, that is, sliding velocities, will be investigated and equations for their subsequent calculation will be developed.
In the case of slow speed rotation and/or an applied load of large magnitude, rolling bearings can be analyzed while neglecting dynamic effects. The resulting kinematic behavior is described in the following paragraphs. As a general case it will beinitially assumed that both inner andouter rings are rotating in a bearing having a common contact angle a (see Fig. 8.1). It is known that for a rotation about an axis, (8.1)
u = or
in which w is in radians per second. Consequently,
or
u, =
+ o,d,(I
1 y)
31
~E~~
SPEEDS AND ~ O T I O N S
F I ~ ~ R 8.1E . Rolling speeds and velocities.
271.72 60
(&=
(8.4)
in which n is in rpm, therefore
v, = 71.nodrn (1
60
+ y)
If there is no gross slip at the raceway contact, then the velocity of the cage and rolling element set is the mean of the inner and outer raceway velocities. Hence
Su~stitutingequations (8.5) and (8.6) into (8.7) yields
Since
therefore,
The angular speed of the cage relative to the inner raceway is nmi= n,
ni

(8.10)
Assuming no gross slip at the innerracewayball contact, the velocity of the ball is identical to that of the raceway at thepoint of contact. Hence,
Therefore, since n is proportional to (8.101,
ct)
and by substituting nmi as in
(8.11)
Substituting equation (8.9) for n, yields (8.12)
Considering onlyinner ring rotation, equations (8.9) and (8.12)become n, =
+ ni(I

7)
(8.13)
For a thrust bearing whose contact angle is go”, cos ac = 0, therefore, nm =
+ (ni + no)
1 d5 m nR = ;2 (no nil
.
(8.15) (8.16)
Determine the cage and ball speeds of the 209 radial ball bearing of Example 8.1 if the shaft turns at 1800 rpm.
QTIQ~S
D
12.7 mm (0.5 in.)
Ex. 2.1
dm = 65 mm (2.559 in.)
Ex. 2.1
=
a = 0" (under radial load)
Ex. 2.5
y = 0.1954
n,
=
.it n,(l  y)
=
0.5
X
(8.13)
1800(I  0.1954)
=
724.1 rpm (8.14)
 0*5x 12.7
X
1800[1  (0.1954)2] = 4430 rpm
Estimate the cage speed of the 218 angularcontact ball bearing of Example 7.5 if the shaft turns at 1800 rpm. 22.23 mm (0.875 in.)
Ex. 2.3
dm = 125.3 mm (4.932 in.)
Ex. 2.6
D
=
Ex. 6.5
a = 41.6"
Y=
D cos cx
(2.27)
dm
 22.23 cos (41.6") = 0.1327 125.3
n,
8 n;(l  y) = .it X 1800(1  0.1327) = 780.6 rpm
(8.13)
=
This estimateis satisfactory in thisapplication because of the following:
Fc = 2.26
X
D3nm2dm
=
2.26
X
X
=
18.9 N (4.247 lb)
(22.23)3(780.6)2X 125.3
Fa = 17,800 N (4000 lb) Z
=
16
Qia
=
2
(4.41)
7.5
Ex. Ex. 7.5
Ex. 7.5 =
17800 = 1113 N/ball (250 lblball) 16 
Since Fa/Z % F,,, ai (41.6") is very nearly equal to ao.
The only conditions that can sustain pure rolling betweentwo contacting surfaces are
1, ath he ma tical line contact under zero load Line contact in which the contacting bodies are identical in length ath he ma tical point contact under zero load Even when the foregoing conditions are achieved it is possible to have sliding. Sliding is then a condition of overall relative movement of the rolling body over the contact area. The motion of a rolling element with respect to the raceway consists of a rotation about the generatrix of motion. If the contact surface is a straight line in one of the principal directions, the generatrix: of motion may intersect the contact surface at one point only, as in Fig. 8.2. The component wR of angular velocity w, which acts in the plane of the contact surface, produces rolling motion. As indicated in Fig. 8.3, the component ws of angular velocity w that acts normal to the surface causes a spinning motion about a point of pure rolling 0. The instantaneous direction of sliding in the contact zone is shown in Fig. 8.4.
F
I
~ 8.2. ~ E Rollerraceway contact; generatrix of motion pierces contact surface.
8.3. Resolution of angular velocities into rollingand spinning motions.
INTE
$ ~ E E D SANI) ~ O ~ I O N $
/ 0Pure rolling
8.4.
Contact ellipse showing slidinglines and point o f pure rolling.
In ball bearings with nonzero contact angles between balls and raceways, during operation at any shaft or outer ring speed, a gyroscopic moment occurs on each loaded ball, tending to cause a sliding motion. In most applications, because of relatively slow input speeds and/or heavy loading, such gyroscopic moment^ and hence motions can be neglected. In high speed applications with an oilfilm lubrication between balls and raceways, such motion will occur. The sliding velocity due to gyroscopic motion is given by (see Fig. 8.5) ug =
8W g D
(8.17)
The sliding velocities caused by gyroscopic motion and spinning of the balls are vectorially additive such that at some distance h and 0 they cancel each other. Thus, Sliding velocity due to spinning motion
r(
f
A
Total velocity
of
sliding due copic motion
*
8.5. Velocities o f sliding at arbitrary point A in contact area.
ug = &Ish
(8.18)
and (8.19) The distance h defines the center of sliding about which a rotation of angular velocity os occurs. This center of sliding (spinning) may occur within or outside of the contact surface. Figure 8.6 shows the pattern of sliding lines in the contact area for simultaneous rolling, spinning, and gyroscopic motionin a ball bearing operating under heavy loadand moderate speed. Figure 8.7, which corresponds to low load and high speed conditions (however, not considering skidding*), indicates that thecenter of sliding is outside of the contact surface and sliding surface occurs over the entire contact surface. The distance h between the centers of contact
8.6. Sliding lines in contact area for simultaneous rolling, spinning, and gyroscopic motionslow speed operation of a ball bearing.
8.7. Sliding lines in contact area for simultaneous rolling, spinning, and gyroscopic motionshigh speed operation of a ball bearing (not considering ~ ~ ~ ~
*Skidding is a very gross sliding condition occurring generally in oilfilm lubricated ball and roller bearings operating under relatively light load at very high speed orrapid accelerations and decelerations. When skidding occurs, cage speed will be less than predicted by equation (8.9) for bearings with inner ring rotation.
~
~
31
~E~~
SPEEDS AND MOTIONS
and sliding is a function of the magnitude of the gyroscopic momentthat can be compensated by contact surface friction forces.
Even when the generatrix of motion apparently lies in the plane of the contact surface, as for radial cylindrical roller bearings, sliding on the contact surface can occur whena roller is under load. In accordance with the Hertzian radius of the contact surface in thedirection transverse to motion, the contact surface has a harmonic mean profile radius, which means that the contact surface is not plane, but generally curved as shown by Fig. 8.8 for a radial bearing.* The generatrix of motion, being parallel to the tangent plane of the center of the contact surface, therefore pierces the contact surface at two points at which rolling occurs. Since the rigid rolling element rotates with a singular angular velocity about its axis, surface points at different radii from the axis have different surface velocities only two of which being s ~ m e t r i c a l l ydisposed the roller geometrical center can exhibit pure rolling motion. In .8 points within area AA slide backward with regard to the direc
R
/
Q
8.8. Rollerraceway contact showing harmonic mean radius and points of rolling AA.
*The illustration pertains to a spherical roller under relatively light load, i.e., the contact ellipse major axis does not exceed the roller length.
ORBIT^, P ~ O T A LAND , SPI
G MOTIONS IN BALL B
tion of rolling and points outside of AA slide forward with respect to the direction of rolling. Figure 8.9 shows the pattern of sliding lines in the elliptical contact area. If the generatrix of motion is angled with respect to the tangentplane at the center of the contact surface, the center of rolling is positioned unsymmetrically in the contact ellipse and, depending on the angle of the generatrix to the contact surface, one point or two points of intersection may occurat which rollingobtains. Figure 8.10 shows the sliding lines for this condition. For a ball bearing in which rolling, spinning, and gyroscopic motions occur simultaneously, the patternof sliding lines in theelliptical contact area is as shown in Figs. 8.11 and 8.12. More detailed information on sliding in the elliptical contact area may be found in the work by Lundberg E8.41.
Figure 8.13 shows a ball contacting the outer raceway such that the normal force Q between the ball and raceway is distributed over an elliptical surface defined by projected major and minor semiaxes, a, and bo,respectively. Theradius of curvature of the deformed pressure surface as defined by Hertz is
F I G W 8.9, Sliding lines in contactarea of Fig, 7.8.
F I G ~ 8 E.10. Sliding lines for rollerraceway contact area when load is applied; generatrix of motion pierces contactarea.
E 8.11. Sliding lines for ballraceway contact area for simultaneous rolling,spinning, and gyroscopic motionshigh load and low speedoperation of an angularcontact ball bearing.
.12. Sliding lines for ballraceway contact area for simultaneous rolling, spinof an angularcontact ball ning, and gyroscopic motionslow load and high speed operation bearing (not consideringskidding).
R,
=
2r,D 2R0 + I )
(8.20)
in which r, is the outer raceway groove curvature radius. In terms of curvature f,:
R,
=
2fOB 2fo + 1
(8.21)
Assume forthe present purpose that theball center is fixed in space and that the outer raceway rotates with angular speed o,.(The vector of coo is per~endiculart o the plane of rotation and therefore collinear with oreover, it can be seen from Fig. 5.4 that ball rotational and oztlying in theplane of the paper when speed oRhas components ox, qJ = 0. Because of the deformation at the pressure surface defined by a, and bo, the radius from the ball center to the raceway contact pointvaries in length as the contact ellipse is traversed from +a, to a,. Therefore because of symmetry about the minor axis of the content ellipse, pure rolling 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 be determined by methods of contact deformation analysis. It can be seen from Figure 8.13 that the outer raceway has a component o,cos a, of the angular velocity vectorin a direction parallel to the major asis of the contact ellipse. Therefore, a point (x,, yo>on the outer
ITAL, PIVOTAL, AND S P ~ L N G MOTIONS LN BALL B E ~ L N G S
raceway has a linear velocity vlo in the direction of rolling as defined below:
Similarly, the ball has angular velocity components,wxlcos a, and wzIsin a, of the angular velocity vector wR lying in the plane of the paper and parallel to the major axis of the contact ellipse. Thus, a point (x,,y,) on the ball has a linear velocity vzo in the direction of rolling defined as follows:
32 u2, =
(cox,
X
{
cos a,
(I?: 
+ wzt sin a,)
+
 (I?: 
[
( ~ ) 2

.:I1’”)
(8.23)
Slip or sliding of the outer raceway overthe ball in thedirection of rolling is determined by the difference between the linear velocities of raceway and ball. Hence, vyo
 Ulo

(8.24)
u20
or dm@,
__
uy,  X
2
+ (w,, cos a, + wzrsin a,  wo cos a,)
{(I?: 

(I?:
+

[(
2):

1’2}
(8.25)
Additionally?the ball angular velocity vector wR has a component my, in a direction perpendicular to the plane of the paper. This component causes a slip u,, in the direction transverse to the rolling, that is, in the direction of the major axis of the contact ellipse.This slip velocity is given bY
From Fig. 8.13 it can be observed that both the ball angular velocity vectors w,? and wzt and the raceway angular velocity vector wo have components normal to the contact area. Hence, there is a rotation about a normal to the contact area, in other words a spinning of the outer raceway relative to the ball, the net magnitude of which is given by o,, = w0
sin a,
+ wXlsin a,  wzrCOS a,
(8.27)
From Fig. 5.4 it can be determined that
wyf = wR COS
p sin p’
(8.29)
oZI = wR sin
p
(8.30)
Substitution of equations (8.28) and (8.30) into (8.25), (8.26), and (8.27) yields
1
ORBITAL, PIVOTAL, AND SPINNING MOTIONS IN BALL BEXRINGS
cos /3 cos p‘ cos a,
+ WO
(8.32) cos p cos p‘ sin a,  OR sin p cos a,  sin a, Ct’O
Note that atthe radius of rolling r(:on the ball, the translational velocity of the ball is identical to that of the outer raceway. From Fig. 8.13,therefore,
(
dm
2 cos a,
+ r:) o,cos a, = r: (ox, cos a, + wzr sin a,)
~ u ~ s t i t u t i nequations g (8.28)and(8.29)into(8.34)and terms yields wR =
o,
(dm/2)+ r: cos a, r: (cos p cos p’ cos a, + sin /3 sin a,)
(8.34)
rearranging
(8.35)
A similar analysis may be applied to the inner raceway contact as illustrated in Fig. 8.14. The following equations can be determined:
(8.3’7) wsi =
(2
WR cos p cos p‘ sin ai + sin p cos ai + sin ai
mi
3
FIGURE 8.14. Inner raceway contact.
@R = mi
(dm/2) + rf cos ai rfi(cos /3 cos p’ cos ai + sin p sin ai)
(8.39)
If instead of the ball center being fixed in space, the outer raceway is fixed, then the ball center must orbit about the center 0 of the fixed coordinate system with an angular speed wm = wo. Therefore the inner raceway must rotate with absolute angular speed w = wi + wm. By using these relationships, the relative angular speeds wi and w, can be described in terms of the absolute angular speed of the inner raceway as follows: w
wi =
I +
ri[(dm/2) ri cos cri](cos p cos p’ cos CY, + sin p sin a,) ri[(dm/2)+ r: cos a,](cos p cos @’cos ai + sin @ sin ai) (8.40)
ORBITAL, P ~ O T A L , 0,=
I +
NG ~ O T I O N $IN BALL B
~
~
G
$
23
r,’[(dm/2) + r; cos a,](cos p cos p‘ cos ai + sin p sin ai) rA[(dm/2) rf cos ai](cosp cos p’ cos a, + sin p sin a,) (8.41)
Further, =
w
rA(cos p cos p’ cos a, + sin p sin a,) (dm/2)+ rA cos a,
p’ cos ai+ sin p sin ai) + r{(cos p cos (dm/2) ri cos ai (8.42)
Similarly, if the outer raceway rotates with absolute angular speedw and the inner raceway is stationary,wm = miand w = wm + w,. Therefore, w, =
Lr)
rf [(dm/2)+ r; cos a,](cos p cos p’ cos ai+ sin p sin ai) l + rA[(dm/2) ri cos aJ(cos p cos p‘ cos a, + sin p sin a,) (8.43)
o i=
I +
r;[(dm/2) rf cos a,](cos p cos p’ cos a, + sin p sin a,) rf[(dm/2)+ rA cos a,](cos p cos p’ cos ai+ sin p sin ai) (8.44) Lr)R =
w
rA(cos p cos p’ cos a, + sin p sin a,) (dm/2)+ rA cos a,
(8.45)
p’ cos ai+ sin p sin ai) + rf (cos p cos (dm/2) rf cos ai Inspection of the final equations relating the relative motions of the balls and raceways reveals the following unknown quantities: r;, ri, p’, p, aiand a,. It is apparent that analysis of the forces and moments acting oneachballwillberequired to evaluate the unknown quantities. As a practical matter, however, it is sometimes possible to avoid this lengthy procedure requiring digital computation by using the
simplifying assumptionthat a ball will roll on one raceway without spinning and spin and roll simultaneously on the other raceway. The raceway on which only rolling occurs is called the “controlling” raceway. Moreover, it is also possible to assume that gyroscopic pivotal motionis negligible; some criteria for this will be discussed.
In the event that gyroscopic rotation is minimal then the angle P’ approaches 0” (see Fig,5.4). Therefore, the angular rotation wyt is zero and further
wzl = % sin P
A second consequences of p‘
%= w,
=
(8.47)
0 is that
(d,l2) + ri cos a, ri(cos a, cos p + sin /3 sin a,)
(8.48)
(dm/2) + ri cos ai rf(cos P cos ai sin p sin ai)
(8.49)
and w 
”
wi
+
Assuming for this calculation that ri, ro, and 4 D are essentially equal, the ball rolling speed relative tothe outer raceway is
From equation (8.33) for negligible gyroscopic moment (P’ os, = % COS
p sin a, 
% sin
P cos a, 
coo
=
O),
sin a,
(8.51)
or wso=I: % sin (a, 
P) 
wo sin a,
Dividing by wrollaccording to equation (8.50) yields
(8.52)
PI^^ ~ O T I O N S EN BALL B
ORBITAL, PIVOTAL,
~
~
~
(8.53) According to equation (8.48), replacing 2rA/d, by y’: 1 + y’ cos a,
% = w,
(8.54)
COS p cos a, + sin p sin a,)
or 1 + y’ cos a, y’ cos(a,  p)
%
=
w,
(8.55)
Therefore substitution of equation (8.55) into (8.53) yields “(1 + y’ cos a,) tan (a,  p)
(~), =
+ y’
sin a,
(8.56)
sin ai
(8.57)
Similarly, for an inner raceway contact
(2)i =
(1 y’ cos ai)tan (ai p)
+ y’
Assuming now that pure.rolling occurs onlyat theouter raceway contact, therefore wBois 0, and substitution of equation (8.48) into (8.33) for this condition indicates that tan p Since rA
^L
+D and D / d ,
=
(dmsin a,)/2 (dmcos a,)/2 + r;
= y’ , equation
tan p
=
(8.58)
(8.58) becomes
sin a, cos a, + y’
(8.59)
aeevvay Control
Harris [8.5] showed that, in general, it is not possible for pure rolling that is, without simultaneous spinning motion, to occur at either the inner or outer raceway contacts as long as theballraceway contact angle is nonzero. For high speed operation of relatively lightly loaded oilfilm lubricated bearings, however, the condition of “outer raceway
S
32
I N T E SPEEDS ~ ~ AND ~ Q T I Q N S
control" tends to be approximated. Figure 8.15 taken from reference E8.51 illustrates this condition for a high speed thrustloaded aircraft gas turbine, angularcontact ball bearing. It must be noted that skidding also tends to occur at the same time. Hence, for oilfilm lubricated ball bearings (including greaselubricated ball bearings), determination of actual internal speeds and motions requires a rather sophisticated mathematical analysis. Such methods require an understanding of friction and will be discussedlater in this text. For dry filmlubricated ball bearings or for ball bearings in which a constant coefficient of friction maybe assumed in theballraceway contacts, Harris L8.61 has shown for a thrustloaded angularcontact ball bearing that, at relatively slow speed, spinning and rolling occur simultaneously at both inner and outer ballraceway contacts. For a given load, as speed is increased, a transition takesplace in which outer raceway control is approximated; however, the outer raceway contact spintoroll ratio is always nonzero (see Figs. 8.16 and 8.17). It is illustrated by Figs. 8.158.17 that the condition of "inner raceway control" is nonexistent; hence no equations for that condition are presented herein. 35 X 6 2 mm Bearing 2 = 14, D = 8.73 mm (0.34375 in.) dm = 48.54 mm (1.91 1 in.) f , = 0.51 5, f2 = 0.52 ao = 24.5O, Shaft Speed = 27500 rpm
500
4,
N 1000
1500
1
I
inner Raceway
2000
I
="""""""
0.4
.0
CI
2  0.3
0 a I
0
CI
I
.a
0.2
v)
0.1
t" Thrust Load (Ib)
8.15.
ball bearing.
Spintorollratio vs thrust load for an oilfilm lubricated angularcontact
27 Bearing Design Data Ball diameter diameter Pitch Free contact angle Inner raceway groove radiushall dia Outer raceway groove radiushall dia k s t load per ball
2*75
(8.73 mm) 0.34375 in. (48.54 mm) 10 1.91 in. 24.5 deg 0.52 0.52 (31.6 N) 7.1 lb
r
0
2,00010,000 9,000 6,000 4,000 Shaft Speed ( rpm)
E 8.16. Ballshaft speed ratio vs shaft speed for a thrustloaded, angularcontact ball bearingoperating with dry friction.
Shaft Speed (rpm)
.17. Spintorollratio vs shaft speed for a thrustloaded, angularcontact ball bearing operating with dry friction.
From equations (8.40) and (8.41), setting p’ equal to 0 and substituting for equation (8.59), the ratio between ball and raceway angular velocities is determined: cos a, + tan @ sin a, 1 + y’ cos a,
+
tl cos ai + tan @ sin ai y‘ cos @ 1  y’ cos ai
(8.60)
The upper sign pertains to outer raceway rotation and the lower sign to inner raceway rotation. Again, using the condition of outer raceway control as established in equation (8.59), it is possible to determine the ratioof ball orbital angular velocity to raceway speed. Fora rotating innerraceway wm = coo; therefore, from equation (8.41) for @’equal to 0: 1 (1 + y’ cosa,)(cos a, + tan @ sin ai> I+ (1  y’ cos ai)(cos a, + tan @ sin a,)
Urn =
w
(8.61)
Equation (8.61) is based on the valid assumption that ro ri = D/2. Similarly, fora rotating outer raceway and by equation (8.44), fi:
__
1 (1  y’ cos ai)(cos a, + tan @ sin a,) I+ (1 + y’ cos a,)(cos ai + tan @ sin ai)
”
w
(8.62)
Su~stitutionof equation (8.59) describing the condition of outer raceway control into equations (8.61) and (8.62) establishes the equations of the required ratio wJw. Hence, for a bearing with rotating inner raceway: 
&m
1  y’ cos a, 1 + cos (ai a,)
”
w
(8.63)
For a bearing with a rotating outer raceway wm 
cos (ai  a,) + y’ cos ai 1 + cos(ai + a,)
”
w
(8.64)
An indicated above, equations (8.59), (8.60), (8.63),and (8.64) are valid only when ball gyroscopic pivotal motionis negligible, that is, @’= 0. It was shown in Chapter 5 for ball thrust bearings that ball gyroscopic torque may be calculated from
G MOTIONS IN BALL B M I N G S
Palmgren C8.11 inferred that in a fluidlubricated angularcontact or thrust ball bearing, gyratory motion of the balls can be prevented if the applied loading is sufficiently great. He stated that in high speed bearings, the coefficient of sliding friction may be as low as 0.02, and that to prevent gyratory rotation, the following relationship must be satisfied: 0.02QD < M ,
(8.65)
For bearings with steel balls Q > 2.24 *
D4nRn, sin f3
(8.66)
Jones C8.21 mentioned that a coefficient of friction from 0.06 to 0.07 suffices for most ball bearing applications to prevent sliding. Neither condition is correct.Since, as shown inthis chapter, the balls have substantial rotational motions about two orthogonal axes, due to the existence of the lubricating films whichgenerally sufficiently “separate”the balls and raceways, it is not possible to prevent rotation about the third orthogonal axis. In Chapter 14, it will be shown that the friction coefficient is a function of the sliding velocity at the contact surface. Further, the ballraceway frictional forces resisting gyratory motion depend on the ratio of the sliding velocity to the lubricant film thickness, Since the latter is a function of the speed in the direction of rolling motion, the magnitude of the gyratory speed is determined by the magnitude of the gyratory moment. Jones C8.31 established a condition to determine whether outer raceway control is approximated in a given application; for example,if oaoEo
cos(a,  ao)>
(8.67)
then outer raceway control may be assumed for calculational purposes. In inequality (8.67), E is the complete elliptic integral of the second kind with modulus K = alb as defined in equation (6.32). As indicated in Chapter 14, no evidence of inner raceway control has been found in any ball bearing application; therefore, the assumption of outer raceway control may be made in the absence of more sophisticated calculation^ of ball speeds using balance of lubricated contact frictional forces an ments.
er
Roller bearings react axial roller loads through concentrated contacts between roller ends and flange. Tapered roller bearings and spherical roller bearings (with asymmetrical rollers) require such contact to react the component of the racewayroller contact load that acts in the roller axial direction. Some cylindrical roller bearing designs require roller endflange contacts to react skewinginduced and/or externally applied roller axial loads. As these contacts experience sliding motions between roller ends and flange, their contribution to overall bearing frictional heat generation becomes substantial. Furthermore, there are bearing failure modes associated with roller endflange contactsuch as wear and smearing of the contacting surfaces. These failure modes are related to the ability of the roller endflange contactto support roller axial load under the prevailing speed and lubrication conditions within the contact. Both the frictional characteristics and loadcarrying capability of roller endflange contacts are highly dependent on the geometry of the contacting members.
Numerous roller end and flange geometries have been used successfully in roller bearing designs. Typically, performancerequirements as well as manufacturing considerations dictate the geometry incorporated into a bearing design. Most designs use either a flat (with corner radii) or sphere end roller contacting an angled flange. The angled flange surface can be described as a portion of a cone at an angle Of with respect to a radial plane perpendicular to the ring axis. This angle, known as the flange angle or flange layback angle, can be zero, indicating that the flange surface lies in the radial plane. Examplesof cylindrical rollerbearing roller endflange geometries are shown in Fig. 8.18. The flat end roller in Fig. 8.18a under zero skewing conditionscontacts the flange at a single point (in the vicinity of the intersection between the roller end flat and roller corner radius). As the roller skews, the point of contact travels along this intersection on the roller toward the tip of the flange, as shown in Fig. 8.19b. If p perly designed, a sphere end roller will contact the flange on the roller end sphere surface, For no skewing the contact will be centrally ~ositionedon the roller, as shown in Fig. 8.19~. As the skewing angle is increased, the contact pointmoves off center and toward the flange tip, as shown in Fig, 8.19d for a Ranged inner ring. For typical designs sphere end roller contact locationis less sensitive to skewing than a Rat end roller contact. The location of the roller endflange contacthas been determined analytically [8.7] for sphere end rollers contacting an angled flange. Con
ING IN ROLLER ~ ~ I N G S
(4
31
t b)
8.18. Cylindrical roller bearing, roller endflange contact geometry. (a)Flat end roller. ( b ) Sphere end roller.
8.19. Cylindrical roller bearing roller endflange contact location for flat and sphere endrollers. ( a ) Flat end roller, zero skew angle. (b) Flat end roller, nonzero skew angle. ( e ) Sphere end roller, zero skew angle.( d ) Sphere end roller, nonzero skew angle.
sider the cylindrical roller bearing arrangement shown in Fig. 8.20. The flanged ring coordinate system XI, YI, Z , and roller coordinatesystem Xi, Yi, Zi are indicated. The flange contact surface is modeled as a portion of a cone with an apex at point C as shown in Fig. 8.21. Theequation of this cone, expressed as a function of the x and y ring coordinates, is
For a point of flange surface P,, Py,Pz the equation of the surface normal at P can be expressed as (8.69)
The location of the origin of the roller end sphere radius is defined as
332
J
Y FI
8.20.
Crosssection through a cylindricalroller bearing having a flanged inner
ring.
Right circular cone = ffx, Y i
FIG
.21. Coordinatesystem for calculation of rollerendflangecontactlocation.
point 1" with coordinates (Tg9 Ty9TJ expressed in the flanged ring coordinate system. Since the resultant roller endflange elastic contact force is normal to the end sphere surface, its line of action must pass t h r o u ~ h Ty,Tz). Evaluating equations ($.6$) and (8.69) at the sphere origin (Tg, I' yields the following three equations:
333
ROLLER E ~  F ~ G $E L IN ~ ROLLER ~ G BEXRINGS
T p
= x

[(E'%
C)2 ctn2 of 
(8.71)
[(Px C)2 ctn2 of  Pt]112
(8.72)

[(E'%
=

(T, E'J Py
Ty  Py =
P,
C) ctn2 of C)2 ctn2 of  P;1112
(T, P,)(P,
Equations (8.70)(8.72) contain three unknowns (Px, Py, PZ)and are sufficient to determine the theoretical point of contact between the roller end and flange. By introducing a fourth equation and unknown, however, Ty,T,)to (Px, Py, P,), the namely the length of the line from points (Tx, added benefit of closedform solution is obtained. The length of a line Py, E',), which joins this normal to the flange surface at the point (Px, Ty,TJ,is given by point with the sphere origin (T%,
After algebraic reduction, quadratic equation
6 is obtained from the positive root of the
"S
rfr (52 
9=
where values for S ,
$313
4gy)1/2
(8.74)
29%
and S are
%=
tan2 of  1
s=
2 sin2 of [(Tx  6 )  tan of(T; cos of
5=
[(Tx  C)  tan of (T;+ T,2)1/2]
+ T31'2]
The coordinatesP(P,, Py,P,) are given by the following closedform functions of 9:
(5)] 2
T Y t a n o f [ 1+
[ T, [
6 sin Of
I I
Py = Ty 1 
(T;+ T,2)1/2
P,
(27; + T
=
1
9 sin
of y 2
1/2
[l
9 sin
of
]+
(T;+ T,2)1/2
C
(8.75) (8.76) (8.77)
At a point of contact between the roller end and flange, 9is equal to the
4
~
T SPEEDS E
~
~
roller end sphere radius. Therefore, knowingthe roller and flanged ring geometry as well as the coordinate location (with respect to the flanged ring coordinate system) of the roller end sphere origin, it is possible to calculate directly the theoretical roller endflange contact location. The foregoing analysis, although shown for a cylindrical roller bearing, is general enough to apply to any roller bearing having sphere end rollers that contact a conical flange. Tapered and spherical roller bearings of this type may be treated if the sphere radius origin is properly defined. These equations have several notable applications since flange contact location is of interest in bearing design and performance evaluation. It is desirable to maintain contact on the flange below the flange rim (including edgebreak) and above the undercut at thebase of the flange. To do otherwise causes loading on the flange rim (or edge of undercut) and produces higher contact stresses and less than optimum lubrication of the contact. The precedingequations may be usedto determine the maximum theoretical skewing angle for a cylindrical roller bearing if the roller axial play (between flanges)is known. Also, by calculating the location of the theoretical contact point, sliding velocities between roller ends and flange can be calculated and used in an estimate of roller endflange contact frictionand heat generation.
oeity The kinematics of a roller endflange contactcauses sliding to occur between the contacting members. The magnitude of the sliding velocity between these surfaces substantially affects friction, heat generation, and loadcarrying characteristics of a roller bearing design. The sliding velocity is represented by the dif€erence betweenthe two vectors defining the liner velocities of the flange and the roller endat thepoint of contact. A graphical representation of the roller velocity UROLL and the flange velocity uF at their point of contact C is shown in Fig. 8.22. The sliding velocity vector us is shown as the difference of uRE and uF. When considering roller skewing motions, us will have a component in the flanged ring axial direction, albeit small in comparison to the components in the bearing radial plane. If the roller is not subjectedto skewing, the contact point will lie in the plane containing the roller and flanged ring axes. The roller endflangesliding velocity may be calculated as (8.78)
where clockwise rotations are considered positive. Varying the position of contact point C over the elastic contact area between roller end and flange allows the distribution of sliding velocity to be determined.
33 I
8.22. Roller endflange contact velocities.
In thischapter, methods for calculation of rolling and cage speedsin ball and roller bearings were developed for conditions of rolling and spinning motions. It will be shown in Chapter 9 how the dynamic loading derived from ball and roller speeds can significantly aff'ect ball bearing contact angles, diametral clearance, and subsequently rolling element load distribution. oreo over, spinning motions that occur in ball bearings tend to alter contact area stresses, and hence they affect bearing endurance. Other quantities af5ected by bearing internal speeds are friction torque and frictional heat generation. It is therefore clear that accurate determinations of bearing internal speeds are necessary for analysis of rolling bearing performance. It will be demonstrated subsequently that h ~ d r o d ~ a maction ic of the lubricant in the contact areas can transform what is presumed to be substantially rolling motionsinto combinations of rolling and translatory motions. In general, this combination of rotation and translationmay be tolerated providing the lubricant films resulting from the rolling motions are sufficient to adequately separate the rolling elements and raceways. Bearing internal design andlor bearing loading or lubrication may be modified to minimize the gross sliding motions and their potential deleterious effects. This topic will be discussed in Chapter 14.
33
INTERNU SPEEDS AND ~ O T I O N S
8.1. A Palmgren,Ball and Roller Bearing Engineering, 3rd ed.,Burbank, Philadelphia, pp. 7072 (1959). 8.2. A. €3.Jones, “Ball Motion and Sliding Frictionin Ball Bearings,”ASME J Basic Eng. 81, 112 (1959). 8.3.A.B. Jones, “A General Theory for Elastically Constrained Ball and Radial Roller Bearings under ArbitraryLoad and Speed Conditions,’’ASME J Basic Eng. 82,309320 (1960). 8.4. G. Lundberg, “Motionsin Loaded Rolling Element Bearings,”SKF unpublished report (1954). 8.5. T. A. Harris, “An Analytical Method to Predict Skidding in ThrustLoaded, Angular Contact Ball Bearings,”ASME J Lubr. Technol. 93, 1724 (1971). 8.6. TI A, Harris, “Ball Motion in ThrustLoaded, AngularContact Bearings withCoulomb Friction,” ASME J Lubr. Technol. 93, 3238 (1971). 8.7. R. Kleckner and J. Pirvics, “High Speed Cylindrical Roller Bearing AnalysisSKF Computer Program CYBEAN, Vol. 1:Analysis,” SKF Report AL78P022, NASA Contract NAS320068 (July 1978).
ST Symbol
LS Description
Units
B
fi+f,1 Ball or roller diameter d diameter m Pitch Ff Friction force F C ~ e ~ t r i f u gforce al
D
f
B
H J K Z rn ~~
9n Pd
rlD (in./sec2) mm/sec2 Gravitational constant Ball or roller hollowness ratio Mass moment of inertia ~oaddeflection N/mmx constant Roller length all or roller (lb kg mass Gyroscopic moment Applied moment Diametral clearance all or roller load
mm (in,) mm (in.) N Ob) N (1b)
kg mm2 (in. * lb * sec2) (lblin.”) mm (in.) * sec2/in.) N * mm (in. * lb) N ., mm (in. lb) mm (in.) (IN 0
~ I ~ T R I B ~ IOF ON INTE
LO~ING IN HIGH S ~ E E ~ rnGS
Symbol
Units adius to locus of raceway groove curvature centers Distance between inner and outer raceway groove curvature center loci Radial projection of distance between ball center and outer oove curvature center Axial projection of distance between ball center and outer raceway groove curvature center.
mm (in.) mm (in.) mm (in.) mm (in.) rad, rad, O
O
mm (in.) rad, rad, radlsec O O
Rotational speed Angular distance between rolling elements
rad
~ U B ~ ~ R I P T ~ Refers t o axial direction Refers to inner raceway Refers to angular position efers to cage motion and orbital on rs to outer raceway Refers to radial direction efers to rolling element efers to x direction. efers to x direction
In high speed operation of ball and roller bearings the rolling element centrifugal forces are s i ~ i ~ c a n tlarge l y compared to the forces applied to the bearing, In roller bearings this increase in loading on the outer raceway causes larger contact deformations in that member; this effect is similar to that of increasing clearance. An increase in clearance as demonstrate^ in Chapter 7 tends to increase maximum roller loading due to a decrease in the extent of the load zone. r relatively thin sec
HIGH S P E E ~ BALL B
tion bearings supported at only a few points on the outer ring; for example, an aircraft gas turbine mainshaft bearing, the centrifugal forces can cause bending of the outer ring thus affecting the load distribution among the rolling elements. In high speed ball bearings, depending on the contact angles, ball gyroscopic moments and ball centrifugal forces can be of significant magnitude such that inner raceway contactangles tend to increase and outer raceway contact angles tend to decrease. This affects the deflection vs load characteristics of the bearing and thus also affects the d ~ a m i c of s the ball bearingsuppo~edrotor system. High speed also affectsthe lubrication characteristics and thereby the friction in both ball and roller bearings. This will have an influence on bearing internal speeds, which in turn alters therolling element inertial loading, that is, centrifugal forces and gyroscopic moments.It is possible, however, to determine theinternal distribution of load, and hence stresses, in many high speed rollingbearing ap~licationswith sufficient accuracy while not considering the frictional loading of the rolling elements. This will be demonstrated in this chapter. The effects of friction, including skidding, on internal load distribution will be consideredlater,
To determine the load distribution in a high speed ball bearing, consider Fig. 7.19, which shows the displacements of a ball bearing due to a generalized loading system including radial, axial, and moment loads. Figure 9.1 shows the relative angular position of each ball in the bearing. Under zero load the centers of the raceway groove curvature radii are separated by a distance BD defined by
I
Under an applied static load, the distance between centers will increase by the amount of the contactdeformations ai plus So, as shown by Fi . The line of action betweencenters is collinear with BD.If, however, a centrifugal force acts on the ball, then because the inner and outer raceway contactangles are dissimilar, the line of action between raceway groove curvature radiicenters is not collinear with BD,but is discontinuous as indicated by Fig. 9.2. It is assumed in Fig. 9.2 that the outer acew way groove c u ~ a t u r center e is fixed in space and the inne oove curvature center moves relative to that fixed center. e ball center shifts by virtue of the dissimilar contact angles. The distance between the fixed outer raceway groove curvature center osition of the ball center at any ball locationj is
3
I
\
I
9.1. Angular position of rolling elements in yz plane (radial). A@ = 2 d 2 ,
=
2 T / Z ( j  1).
A,
=
D
ro   t Soj 2
Since ro Aoj
=f =
oD
(fo
 0.5)D+ Soj
Similarly,
aOjand Sij are the normal contact deformations at the outer and inner raceway contacts, respectively. In accordance with the relative axial displacement of the inner and outer rings Sa and the relative angular displacements, 8, the asial dis
41
HIGH SPEED BALL B E ~ ~ G §
Outer raceway groove, curvature center fixed 9.2. Positions of ball center and raceway groove curvature centers at angular
position
+, with and without applied load.
tance between the loci of inner and outer raceway groove curvature centers at any ball position is
A,
=
BD sin a" + 6,
+ 6% cos t,$
(9.4)
in which & is the radius of the locus of inner raceway groove curvature centers and cyo is the initial contact angle prior to loading. Further, in accordance with a relative radial displacement of the ringcenters tir, the radial displacement between the loci of the groove curvature centers at each ball locationis
AZj= BD cos ao + Sr cos t,$
(9.5)
The foregoing data are intended as an explanation of Fig. 9.2. Jones [9.1] found it convenient to introduce new variables .Xl and as shown by Fig. 9.2. It can be seen from Fig. 9.2 that atany ball location
34
D I S T R ~ ~ I OOF N
cos
L
~~~~
O
~ IN ~ HIGH G SPEED
X2j
aoj = (fo

0.5)D
+ aOj
Using the Pythagorean theorem, it can be seen from Fig. 9.2 that
Considering the plane passing through the bearing axis and the center of a ball located at azimuth \ctj (see Fig. 9.1),the load diagram of Fig. 9.3 obtains if noncoplanar friction forcesare insignificant. Assuming “outer raceway control” is approximated at a given ball location, then it can also be assumed with little efl’ect on calculational accuracy that the ball gyroscopic moment is resisted entirely by frictional force at the ballouter raceway contact. Otherwise, it is safe to assume that the ball gyroscopicmoment is resisted equally at the ballinner and ballouter raceway contacts. In Fig. 9.3, therefore, Aij = 0 and A, = 2 for “outer raceway control”; otherwise h, = h, = 1.
.3. Ball loading at angular position $.
3
HIGH SPEER BALL B E ~ ~ G S
The normal ball loads in accordance with equation (7.4) are related to normal contact deformations as follows: (9.12)
(9.13) From Fig. 9.3, consideringthe equilibrium of forces in the horizontal and vertical directions:
ij
sin
aij 
oj
gj (Aij cos QOjsin aOj M 
D
cos aOj+ 1M,j  (Aij sin
D
cyij
0
(9.14)
sin aoj)+ Fcj = 0
(9.15)
aij 
 A,
Aoj cos
cyoj)
=
Substituting equations (9.12) and (9.13) and (9.6) to (9.9) into (9.14) and (9.15) yields
Equations (9.10), (9,111, (9.16), and (9.17) may be solved simultaneously forXlj, . X z j , Sij, and Soj at each ball angular location oncevalues for Sa, Sr, and 8 are assumed. The most probable method of solution is the NewtonRaphson method for solution of simultaneous nonlinear equations. The centrifugal force acting on a ball is calculated as follows:
Fc = ijd,w:
(5.34)
w, is the orbital speed of the ball. Substituting the identity = (W , / W ) ~ O ~in equation (5.34), the following equation for centrifugal
in which
force is obtained:
3
D I S ~ I B ~ OF I O~ ~
E LOADING R IN 33IGH ~ SPEED ~ BE~INGS
(9.18) in which o is the speed of the rotating ring and omis the orbital speed of the ball at angular position J/j.It should be apparent that because orbital speed is a function of contact angle, it is not constant for each ball location. oreover, it must be kept in mind that this analysis does not consider frictional forces that tend to retard ball and hence cage motion. Therefore, in a high speed bearing, it is to be expected that cornwill be less than that predicted by equation (8.63) and greater than that predicted by equation (8.64). Unlessthe loading on the bearing is relatively light, n taffecting however, the cage speed differential is usually i n s i ~ i ~ c ain the accuracy of the calculations ensuing in this chapter. ~ ~ o s c o pmoment ic at each ball location maybe described as follows:
(9.19) where /3 is given by equation (8.591, qJo by equation (8.60), and om/o by equation (8.63) or (8.64). Since Koj,Kij?and Mgj are functions of contact angle, equations (9.6)(9.9) may be usedto establish these values during the iteration. To find the values of ar, aa, and 8, it remains only to establish the conditions of equilibrium applying to the entire bearing. These are
(9.20) or
j=Z
Fr j=1
or
(Qij
X;jMgj cos aij + 
D
cos J/j
=
0
(9.22)
HIGH S P E E ~ BALL BEARINGS

9[(
Qij sin aij 
j=1
hijMgi 13
~
(9.24) or
x cos $ = 0
(9.25)
& = dm + (fi 0.5)D COS ao
(6.86)
Having computed values of Xlj, X2j,Sij, and Soj at each ball position and knowing Fa, Fr, and 9~as input conditions, the values Sa, Sr, and 8 may be determined by equations (9.21), (9.23),and (9.25). Afterobtaining the primary unknown quantities Sa, S,, and 8, it is then necessary to repeat the calculation of Xlj, X2j, Sij, and Soj, and so on, until compatible values of the primary unknown quantities Sa, Sr, and 8 are obtained.
.
The 218 angularcontact ball bearing of Example 7.5 operated through the load range of 044,500 N (10,000 lb) thrust and at speeds of 3000,6000,10,000 and 15,000 rpm. such operating characteristics of the bearing as ai, ao, p, Sa, Mg,and U , I W ~ O ~ L * To obtain the answers to this study, a computer program must be developed to solve equations (9.10), (9.11), (9.16), (9.17), and (9.21) simultaneously for each loadspeedcondition. Inthese equations, which may be solvedby iterative techniques, the loaddeflection "constants" Ki and KOare functions of ai and a,, which are in turn functi of Xl and X2 according to equations (9.6)(9.9). Similarly, Fc and are functions of W ~ / Uand % / u which depend on ai and a, according to equations (8.60) and (8.61). Hence the solution is not simple and care must be exercised to include all variations in theiteration. From such a computer program, the data of Figs. 9.49.6 were developed.
3;
DI$TRIBUTIO~OF I
~LO~IN EG IN HIGH~ SPEED B~ N x 103
Thrust load, Ib
.
aiand a, vs thrust load. 218 angularcontact ball bearing,a* = 40”.
For an angularcontact ball bearing subjected only to thrust loading, the orbital travel of the balls occurs in a single radial plane, whose axial location is defined byXlj inFig. 9.2, that is, X l j is the same at all azimuth angles $. For a bearing that supports combined load, that is, radial and thrust loads and perhaps also a moment load, X l j is different at each azimuth angle $. Therefore, a ball undergoes an axial “excursion”as it orbits the shaftor housing center. Unlessthis excursion is accommodated by providing sufficient axial clearance between the ball and the cage pocket, the cage will experience nonuniform and possibly heavy loading in 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 including an outofplane vibrational component. Such motion together with the
N
9.5. Ball normal loads Q, and Qi vs thrust load for various shaft speeds. 218 angularcontact ball bearing, a” = 40”.
aforementioned loading canlead to rapid destruction and seizure of the bearing. Under combined loading, becauseof the variation in theballraceway contact angles aij and aojas a ball orbits the bearing center, there is a tendency for the ball to advance or lag its “central” position in the cage pocket. The orbital or circumferential travel of the ball relative to the cage is, however, limited by the cage pocket. Therefore, a load occurs between the ball and the pocket in the circumferential direction. Under steadystate cage rotation, the sum of these ballcage pocket loadsin the circumferential direction is close to zero, being balanced only by frictional oreover, the forces and moments acting on the ball in the ing’s plane of rotation must be in balance, including acceleration celeration loading and frictional forces. To achieve this condition of equilibrium, the ball speeds, including orbital speed, will be different fromthose calculated using the equations of Chapter 8. This condition is called ~ ~ i ~ and ~ iit will ~ g be, covered in Chapter 14.
348
D I S T R ~ ~ OF I OINTERNAL ~ LOADING IN HIGH SPEED B ~ I N G S N x 103
+0.05
0
0.05 mm
0.10
0.15
0.20
Applied thrust load, Ib
9.6. Saaxial deflection contact ball bearing, a" = 40".
vs thrust load for various shaft speeds. 218 angular
To permit ball bearings to operate at higher speeds, it is possible to reduce the adverse ball inertial effects by reducing the ball mass. This is especially effective for angularcontact ball bearings since the differential between inner and outer contact angles will be reduced. To achieve this result, it was attempted to operate bearings with complements of hollow balls [9.3];however, this proved impractical since it was dificult to manufacture balls having isotropic inertial properties. Morerecently, hot isostatically pressed (HIP) silicon nitride ceramic has been developed as an acceptable material for manufacture of rolling elements (see Chapter 16). Bearings with balls of HIP silicon nitride, which has a density approximately 425% that of steel and an excellent compressivestrength, are being usedin high speed machine tool spindle applications and areunder consideration foruse in aircraft gas turbine application main shaft bearings. Figures 9.79.9 compare bearing performance parameters for op
IAI, C ~ I ~ R ROLLER I C ~B
~
~
G
34
S
1200
 Outer raceway steel balls       Outer raceway silicon nitride  Inner raceway steel balls .. Inner raceway silicon nitride
1000
13
balls balls
6 800 I= (d
0" q
600
(d
0 
2
400
200
0 0
2000
4000
6000
8000
10000
12000
Applied thrust load, Ib 9.7. Outer and inner racewayball loads vs bearing applied thrust load for a 218 angularcontact ball bearing operating at 15,000 rprn with steel or siliconnitride balls.
erations at high speed of the 218 angularcontact ball bearing with steel balls and HIP silicon nitride balls. Silicon nitride also has a modulus of elasticity of approximately 3.1 * lo5 MPa (45 lo6 psi). In a hybrid ball bearing, Le., a bearing with steel rings and silicon nitride balls, owingto the higher elastic modulus of the ball material, the contact areas between balls and racewayswill be smaller than in an allsteel bearing. This causes the contact stresses to be greater. Depending on the load magnitude, the stress level may be acceptable to the ball material, but not to the raceway steel. This situation can be ameliorated at theexpense of increased contact friction by increasing the conformity of the raceways to the balls; for example, decreasing the raceway groove curvature radii. This amount of' decrease is specific to each application, being dependent on bearing applied loading and speed.
Because of the high rate of heat generation accompanying relatively high friction torque, tapered roller and spherical roller bearings have not historically been employed for high speed applications. Generally, cylindri
35
D I $ T R ~ ~ OF I O~E~~ ~
LO~ING IN HIGH SPEED 13:
70
60
41: Ecn %
50
8
u c
40
(6
8
...  Outer ramway silicon nitrib balls  Inner raceway  steel balls . Inner raceway  silicon nitrib balls
1
I
0 0
2000
4000
6000
800012000
10000
Applied thrust load,Ib 9.8. Outer and inner racewayball contact angle vs bearing applied thrust load for a 218 angularcontact ball bearingoperating at 15,000 rprn withsteel or silicon nitride balls.
0.003 0.002 *
e
0.001
.2
0.000
0
a,
+ e=
0.001
cn
.C5
0.003
a,
m
0.004
0.005 0.006 0
2000
4000 8000
6000
12000
10000
Applied thrust load,Ib 9.9. M a l deflection vs bearing applied thrust load for a 218 angularcontact ball bearing operating at 15,000 rprn withsteel or silicon nitride balls.
cal roller bearings have been used; however, improvements in bearing internal design, accuracy of manufacture and methods of removing generated heat via circulating oil lubrication have gradually increased the allowable operating speeds for both tapered roller and spherical roller bearings. The simplest case for analytical investigation is still a radially loaded cylindrical roller bearing and this will be considered in the following discussion. Figure 9.10 indicates the forces acting on a roller of a high speed cylindrical roller bearing subjected to a radial load Fr . Thus, considering equilibrium of forces, Qoj

Qij  Fc = 0
(9.26)
Since by equation (7.4),  KB1.11
(7.4)
therefore K y j 1 1  K B 1 : 1 1V
F
=
0
(9.27)
[K varies with roller length according to equation (7.9).1 Since Srj
= tiij
+ Boj
(9.28)
equation (9.27) may be rewritten as follows:
9.10. Roller loading at angular position
e.
D I S ~ I B U T I O NOF ~E~~
x
LOADING IN HIGH SPEED B
~
~
j =Z
Fr 
Qij COS
3/j
0
j=1
(9.30)
or (9.31) By considering the geometry of the loaded bearing, it can be determined that the total radial compression at any roller angular location 3/j is
srj= srcos 3/j  d‘2
(9.32)
~ubstitutionof equation (9.32) into (9.29) yields (9.33) Equations (9.31) and (9.33) represent a system of simultaneous nonlinear equations with unknowns 6r and ciij. As before, the NewtonRaphson method is suggested to evaluate the unknown deformations. After calculating Sf and 6ij, it is possible to calculate roller loads as follows: K6$11
(7.4)
QOj= K8$11 + Fc
(9.34)
Qij
=
Centrifugal force per roller can be calculated by using equation (5.52). The foregoingequations apply to roller bearings with line or modified line contact. Fully crowned rollers or crowned raceways maycause point contact, in which case Ki is different from KO and these values can be determined from equation (7.8). Information on high speed roller bearings having flexibly supported rings is given by Harris E9.21. ~ ~ 9.2. Z For e the 209 cylindrical roller bearing of Example 7.3 compare the load distributions at shaft speeds of 1000, 5000, 10,000, and 15,000 rpm for a rapidly applied load of 4450 N (1000 lb), if the bearing has no diametral clearance in the assembled condition.
G
S
X' = 5.869
lo5 N/mml*ll (4.799 X lo6 1b/(in.)ll1)
X
Z = 14
Ex. 7.3 Ex. 2.7
Pd = 0 A+
360"
360 14
" = = 
2
25.71" (9.29)
*=O
=o 0.007581  [0.58k1'
+
COS
? ==0.5 * = O 7j=1 *'O
(0) + 8!i1l cos (25.71")
a&" cos(l80")] = 0 10 mm (0.3937 in.)
Ex. 2.7
65 mm (2.559 in.)
Ex. 2.7
d m cos a 
(2.27)
D
65  cos (0") 10 0.1538
+
0.5
(8.13)
 y) X
ni(l

0.1538)
0.4231ni Ex. 2.7
9.6 mm (0.3780 in.) 3.39
X
3.39
X
3.788
(5.52)
1011D2Zd,ni
X
10)2(9.6)2X 65
X
(0.4231~2,)~
107nz (9.33) 
This ~ v e eight s eauations as follows:
3.788 X 107nf?__ 0 5.869 X lo5
300
3
;i,
.5
N
0
Roller location, degrees 9.11. Roller load distribution. 209 cylindrical roller bearing, Pd = 0, Fr = 4450 N (1000 lb).
Figure 9.12 shows the variation in 6r with speed.
Rollers can be made hollow to reduce roller centrifugal forces. Hollow rollers are flexible and great care must be exercised to assure that accuracy of shaft location under the applied load is satisfied. Roller centrifugal force as a function of hollowness ratio D,/D is given by
Figure 9.13 taken from reference E9.41 shows the effect of roller hollowness in a high speed cylindrical roller bearing on bearing radial deflection. Figure 9.14 for the same bearing illustrates the internal load distribution. An added criterion for evaluation in a bearing with hollow rollers is the roller bendingstress. Figure 9.15 showsthe effect of roller hollowness on maximum roller bendingstress. Practical limits for roller hollowness are indicated. Great care must be given to the smooth finishing of the inside surface of a hollow roller during ~anufacturing as the stress raisers thatoccur due to a poorly finished inside surface will reduce the allowable roller hollowness ratios still further than indicated by Fig. 9.14.
9.12. Radial deflection vs speed. 209 cylindrical roller bearing, P, = 0, Fr = 4450 N (1000 lb).
ler
Dimensions of Sample Roller Bearing
z 21
dm 114.3 m m (4.5 in.) Pd 0.0064 m m (0.00025 in.)
1, 15 m m (0.59 in.) c1 14 m m (0.55 in.)
W = 57850 N (13,000IbO N = 15,000 rpm W = 57850 N ( 13,000 Ib1 N = 5,000 rpm W = 22250 N (5,000 Ib) N = 15,000 rpm
\\ W = 22250 N (5,000 Ib) N = 5,000 rpm
104
0
0.2
0.4
0.6
0.8
Hollowness (96)
F I ~ 9.13. ~ EMaximum deflection vs hollowness.
12
W = 57,850 N (13,000Ib) N = 5,000rpm
W = 57,850 N
W = 22,250N ( 5 N = 15,000 rpm
7 6 5 4 3 2 1 2 3 4 5 6 7 Position
2 3 4 5 6 7
Roller
9.14. Roller load distribution vs applied load, shaft speed, and hollowness.
357
FrVE R E G ~ E S OF F ~ E E R O IN~LOADING
r  ( 6,898 N/mm2 1
W = 57,850 N ( 13,000Ib) N = 15,000rpm
..
= 57,850 N = 5,000rpm
W = 22,250N (5,000Ib) N = 15,000rpm
W = 22,250 N N = 5,000 rpm
 (689.8)
Recommended Endurance Limit SAE 8620
0.2
0.4
0.6
0.8
Hollowness f %)
9.15.
Maximum bending stress vs hollowness.
Lightweight rollers made from a ceramic material such as silicon nitride appear feasible to reduce roller centrifugal forces.
IC Using digital computation and methods similar to those indicated in Chapter 7, the load distribution in other types of high speed roller bearings can be analyzed. Harris E9.51 indicates all of the necessary equations. The forces acting on a generalized roller are shown by Fig. 9.16. In this case, roller gyroscopic momentis given by (9.36)
Until this point, all load distribution calculational methods have been limited to, at most, three degrees of freedom in loading. This has been
358
DISTRIBUTION OF I
~~
O E IN HIGH ~ ~ SPEED ~ B~
G
~
Bearing Axis of Rotation
F ~ ~ U 9.16. R E Roller forces and geometry.
done in the interest of simplif~ngthe analytical methods and the understanding thereof. Every rollingbearing applied load situation can be analyzed using a system with five degrees of freedom, considering only for the applied loading.Then every specialized applied loading condition, example, simple radial load, can be analyzed using this more complex system. Reference [9.5] shows the following illustrations that apply to an analytical system for a ball bearing with five degrees of freedom in applied loading (see Fig. 9.17). Note the numerical notation of applied loads, that is, F,, . . . , F5, in Figure 9.18 shows the contact angles, deformations, lieu of Fa,Fr,and a. and displacements for the ballraceway contacts at azimuth t,$. Figure 9.19 shows the ball speed vectors and inertial loading for a ball with its center at azimuth Note the numerical notations forraceways; 1 = o and 2 = i. This is done for ease of digital programming.
As demo~strated in the foregoing discussion,analysis of the pe~ormance of high speed roller bearings is complex and requires a computer to obtain numerical results. The complexity can become even greater for ball bearings. In this chapter as well as Chapters 7 and 8, for simplicity of explanation, most illustrations are confined to situations involving sym
~
35
~ I G U R E9.17. Bearing operating in YZ plane.
z  Axis
1
+ f ( A 4 sin $ j + A 5 cos $,)
ner Raceway Groove Curvature Center Operating location
360
DISTRIB~IO OF ~ INTERNAL LOADING IN HIGH SPEED B E ~ I N G S
X
FIGURE 9.19. Ball speeds and inertial loading.
metry of loading about an axis in the radial plane of the bearing and passing through the bearing axis of rotation. The moregeneral and cornplex applied loading system with five degrees of freedom is, however, discussed. The effectof lubrication has also been neglectedin thisdiscussion. For ball bearings, it has been assumed that gyroscopic pivotal motionis minimal and can be neglected. This, of course, dependsom the friction forces in the contact zones, which are affected to a great extent by lubrication. Bearing skidding is also a function of lubrication at high speeds of operation. If the bearing skids, centrifugal forces will be lower in magnitude and performance will accordinglybe dif‘ferent. Notwithstanding the preceding conditions,the analytical methods presented in this chapter are extremely useful in establishing optimum bearing designs for given high speedapplications.
9.1. A. B. Jones, “A General Theory for Elastically Constrained Ball and Radial Roller Bearings under Arbitrary Load and Speed Conditions,”A~~E Dans. ~ournaZ ofBasic Eng. 82,309320 (1960).
9.2. T. A. Harris, “Optimizing the Fatigue Life of FlexiblyMountedRolling Bearings,” Lub. Eng. 420428 (October, 1965). 9.3. T. A. Harris, “On the Effectiveness of Hollow Balls in Highspeed !t%rust Bearings,’’ M L E Dunsuet, 11, 209294 (October, 1968). 9.4. T. A. Harris and S. F. Aaronson, “An Analytical Investigation of Cylindrical Roller Bearings Having Annular Rollers,”ASLE Preprint No. 66LC26 (October 18, 1966). 9.5. T. A. Harris and M. H. Mindel, “Rolling Element Bearing Dynamics,” Wear 23, 311337 (1973).
This Page Intentionally Left Blank
Symbol
~emimajoraxis of the projected contact ellipse b Semiminor axis of the projected contact ellipse diameter dl Land D Ball or roller diameter Fforce Applied J,( 4 Radial load integral K Loadde~ectionconstant 1 effective Roller length 3Z' Moment load olling element load Number of rolling elements per row Contact angle Y D cos ald, 6 eflection or contact deformation a
i
Units
Description
mm (in.) mm (in.) mm (in.) mm (in,) N (lb) N/mm~(lb/in~) mm (in") N * mm (in. * lb) N (1W rad,
O
mm (in.)
36
BEARING ~ E ~ ~ T I PQ ~NL ,Q ~ I N AND G , S T ~ ~ S S
Symbol 8' E
a i n 0
P r R
1 2
Units Deflection rate Projection of radial load zone on bearing pitch diameter Angle of land Maximum contact stress Curvature sum Angle
mm/N (in./lb) rad, a I!?/mm2(psi) mml (in.') rad,
SUBSCRIPTS Refers t o axial direction Refers to inner raceway Refers t o direction collinear with rolling element load Refers to outer raceway Refers to preload condition Refers to radial direction Refers to ball or roller Refers to bearing 1 Refers to bearing 2
In Chapter 6 a method was developed for determining the elastic contact deformation, that is, Hertzian deformation, between a raceway and rolling element. For bearings with rigidly supported rings the elastic deflection of a bearing as a unit depends on the maximum elastic contact ~eformation in thedirection of the applied load or in the direction of interest to the designer. Because the maximum elastic contact deformation is dependent on the rolling element loads, it is necessary to analyze the load dist~butionoccurring within the bearing prior to determination of the bearing deflection. Chapters 7 and 9 demonstrated methods for evaluating the load dist~butionamong the rolling elements for rolling bearings subjected to static and dynamic loading, respectively. Again, in Chapters 7 and 9 the methods foranalyzing load distribution caused by generalized bearing loading (radial, axial, and moment loads applied simultaneously) utilized the variables Sr, Sa, and 0, which are, in fact, the principal bearing deflections. These deflectionsthat are the subjects of this chapter may be critical in determining system stability, dynamic loadingon other components, and accuracy of system o~erationin many applications.
365
DEFLECTIO~SOF BEARINGS WITH RIGID RINGS
In the beginning of Chapter 7 and somewhat in Chapter 9 some simplified methods for calculating internal load distribution were discussed. Also, in those chapters methods to determine internal load distribution for complex applied loading situations were defined. The latter, which require digital computerprograms to obtain solutions, generally use bearing deflections, radial, axial, and misali~ment,as unknown variables. These deflections are therefore determined directly from the solution of the system of nonlinear equations. For applications with relatively simple applied loading, the methods for determining bearing deflection were not defined and these will be discussed herein. It is possible to calculate the maximum rollingelement load Qmaxdue to a combination of radial and axial loads. Qmmhas attendant contact deformations a,,, and aim,, measured along the line of contact at the outer and innerraceways, respectively. Fromequation (7.4) it can be seen that (10.1) (10.2) In equations (10.1) and (10.2), n = 1.5 and n = 1.11for ball and roller bearings, respectively. The radial deflection of the bearing from Fig. 9.2 is therefore 6, = [(fi 0.5)D
+ Sirnm]COS
ai
+ [(f, 0.5)D + 60may] COS a,  BD COS
(10.3)
a"
or 6, = (fi ~ , ~ ) D ( cai o s
COS
a")
+ (f,  O.~)D(COS at,  COS a") +
+ SO,,
a,
(10.4)
sin ai + 6omor sin a,
(10.5)
COS
ai
COS
Similarly, the axial deflection is given by 6, = (fi 0.5)D(sin ai sin a")
+ (f,  0.5)D(sin a,  sin a") + tiirn, Omax
includes the effect of centrifugal loading.
In lieu of the more rigorous approach to bearing deflection outlined above, Palmgren [10.11] gives a series of formulas to calculate bearing deflection for specific conditionsof loading. For slow and moderate speed deepgroove and angularcontact ball bearings subjected to radial load which causes only radial deflection, that is, aa = 0,
sr = 4.36 X
104
Q2& D1I3cos a
(10.6)
For selfaligning ball bearings,
sr = 6.98 X
104
Q2& D1I3cos a
(10.7)
For slow and moderate speed radial roller bearings with point contactat one raceway and line contact at the other,
ar = 1.81 X
104
( 2 % ; P I 2
cos a
(10.8)
For radial roller bearings with line contact at each raceway,
ar = 7.68 X
105
Q2X lo.8cos a
(10.9)
To the values given above must be added the appropriate radial clearance and any deflection due to a nonrigid housing. The axial deflection under pure axial load, that is, ar = 0, for angularcontact ball bearings is given by
aa = 4.36 X
104
Q2& sin a
(10.10)
Q:&
(10.11)
For selfaligning ball bearings Sa = 6.98
For thrust ball bearings
X
D1I3sin a
7
DEFLECTIO~SOF BE
a,
=
5.24
X
104
213 rnax
.LSI3
sin a
(IO.12)
For radial ball bearings subjected to axial load, the contact angle a must be determined prior to using equation (10.10). For roller bearings with point contact at one raceway and line contact at the other,
aa = 1.81 X
104
QZ& sin a
(IO.13)
For roller bearings with line contact at each raceway,
a,
=
7.68
X
105
sin a
(IO.14)
1. For the 209 cylindrical roller bearing of Example 7.4 estimate the bearing radial deflection. Compare this value with amax obtained in Example 7.3assuming a diametral clearance of 0.0406 mm (0.0016 in.). rnax =
1589 N (357.1 lb)
Z = (0.3789 mm 9.6
in.)
7.4 Ex. 2.7 Ex. (10.9)
( 1589)0*9 (9.6)Oa8 cos (0')
=
7.68
=
0.00953 mm (0.000375 in.)
X
105
Total shaft movement is
=
From Example 7.3,
0.02983 mm (0.001175 in.)
amax= 0.03251 mm (0.001~8in.). For the 218 angularcontact ball bearing of Example
368
BEARING D E F ~ ~PTRIEOLNO, ~ I N G ,
AND S T ~ F ~ S S
9.1, estimate the axial deflection at 44,500 N (10,000 lb) thrust load. Compare this value against the data of Fig. 9.6. Z
16
Ex. 7.5
a" = 40"
Ex. 2.3
I) = 22.23 mm (0.875 in.)
Ex. 2.3
=
(7.26) 
44500 16 X sin (40")
=
4239 N (972.8 lb) 1721'3
(IO.IO)
4.363
=
0.064 mm (0.00252 in.)
X
104
(4239)2'3 (22.23)1/3sin (40")
=
From Fig. 9.6 it can be seen that this value is a satisfactory estimate of 6a at slow speeds. At high speed Sa will be less than this estimate.
A typical curve of ball bearing deflection vs load is shown by Fig. 10.1. It can be seen from Fig. 10.1 that asload is increased uniform15 the rate of deflection increase declines. Hence, it would be advantageous with regard to bearing deflection under load to operate above the "knee" of the loaddeflection curve. This condition can be realized by axially preloading angularcontact ball bearings. This is usually done, as shown in Fig. 10.2, by grinding stock from opposingend faces of the bearings and then locking the bearings together on the shaft. Figure 10.3 shows preloaded bearing sets before and after the bearings are axially locked togure 10.4 illustrates, graphically, the improvement in loadcharacteristics obtained by preloading ball bearings. Suppose two identical angularcontact ball bearings are placed backback or facetoface on a shaft, as shown in Fig. 10.5, and drawn tother by a locking device. Each bearing experiences an axial deflection
,
10.1. Deflection vs load characteristic for ball bearings.
Backtoback
Facetoface 10.2. Duplex sets of angularcontact ball bearings.
I
aPdue to preload Fp.The shaft is thereafter subjected to thrust load Fa, as shown in Fig. 10.5, and because of the thrust load, the bearing combination undergoes an axial deflection Sa. In this situation the totalaxial deflection at bearing 1is a, and at bearing 2,
=
ap + sa
(10.15)
Housing
Shaft
/Shim
hrust
+
10.3. (a) Duplex set with backtoback angularcontact ball bearings prior to axial preloading. The inner ring faces are ground to provide a specific axial gap. (6) Same unit as in (a)after tightening axial nut to remove gap. The contact angleshave increased. (c) Faceto face angularcontact duplex set prior to preloading. In this case it is the outer ring faces that are ground to provide the required gap. ( d ) Same set as in (c) after tightening the axial nut. The convergent contact angles increase under preloading. (e) Shim between two standardwidth bearings avoids need forgrinding the faces of the outer rings. ( f ) Precision spacers between automatically provide proper preload by making the inner spacer slightly shorter than the outer.
t
6
.4, Deflection vsload characball bearings. As the load increases, the rate of the increase of deflection decreases, therefore preloading (top line) tends to reduce the bearing deflection under additional loading.
10.5. Preloaded set of duplex bearings subjectedto Fa, an external thrust load. The computation forthe resulting deflection is complicated by the fact that the preload at bearing 1is increased by load Fawhile the preload at bearing 2 is decreased,
6, =
Sr,  sa
6 > 6a
6, = 0
SP 5s
(10.16)
sa
The total load in the bearings is equal to the applied thrust load:
Fa = F ,

F2
(10.17)
For the purpose of this analysis consider onlycentric thrust load applied to the bearing; therefore, from equation (7.33),
Fa
"
ZD2K
cos a,
Combining equations (10.15) and (10.16) yields 6,
+ s2= zap
(10.19)
ubstitution of equation (10.15)for 6, and equation (10.16)for 6, in (10.36) gives sin (a,  a") + sin (a,  a") =
2iiP cos a"
3D
(10.20)
Equations (10.18) and (10.20) may now be solved for a, and a2. Subsequent substitution of a, and a2 into equation (7.36) yields values of 6,
37
BEARING ~EFLECTION,P ~ L O ~ AND ~ GS ,T ~ ~ S S
and is,. The data pertaining to the selected preload Fpand deflection Sr, may be obtained from the following equations: (10.21)
ii,, =
BD sin (ap a") cos ap
(10.22)
Figure 10.6 shows a typical plot of bearing deflection Savs load. Note that deflection is everywhere less than that for a nonpreloaded bearing up to the load at which preload is removed. Thereafter, the unit acts as a single bearing under thrust load and assumes the same loaddeflection characteristics as given by the singlebearing curve. The point at which bearing 2 loses load may be determined graphically by inverting the singlebearing loaddeflectioncurve about the preloadpoint. This is shown 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 tapered or spherical roller bearings; hence this is not a universal practice as it is for ball bearings. Figure 10.7, however, showstapered roller bearings axially locked together in a light preload arrangement.
~ ~ . ~ .
e A duplex pair of 218 angularcontact ball bearings is mounted backtoback, as shown in Fig. 10.3. If the pair is preloaded to 4450 N (1000 lb), determine the axial deflection under 8900 N (2000 lb) applied thrust load, Compare these results with the static deflection data of Fig. 9.8.
.
Load Deflection vs load for a preloaded duplex set of ball bearings.
Housing
FIGURE 10.7. Lightly preloaded tapered roller bearings.
Z = 16
D
=
7.5
22.23 (0.875 mm
in.)
Ex. 2.3
a" = 40"
K
=
Ex.
2.3
Ex.
896.7 N/mm2 (130,000 psi)
Ex. 7.5
From the static curve of Fig. 9.4, a t 4450 N (1000 lb) ap = 40.61"
13
=
0.0464
Ex. 2.3
BD sin ( a p a") cos ap
(10.22)
~
sp= 
0.0464
=
0.0145 mm (0.0005694 in.)
X
22.23 sin (40.61"  40") cos (40.61")
This number could have been obtained from Fig. 9.6.

sin a2
8900 16 X (22.23)2X 896.7
=
(;;;:; ~
sin a1 
(10.18)
 1)1*5
(cos (40") __ 1)1'5 cos a1
sin a2 (cos (40") cos a2
 1)1'5
3
sin (al  40") cos a,
 40") 2 X 0.0145 + sin (a2 cos 0.0464 X 22.23 a2
sin ( a ,  40°) sin (a2 40") + cos a2 = 0.02805 cos a, *
Equations (a) and (b) can be solved simultaneously to yield a, and a2 and thence Sa. Alternatively, the static deflection curve of Fig. 9.6 can be used as follows to e
e

Assume values of Sa. Create tabular values of 6, = Sp + Sa (10.15) and S2 = Sp (10.16). Find F , and F2 corresponding to S, and S2, respectively. Find Fa = F ,  F2 (10.17).

Sa
mm (in.)
6, mm (in.)
mm (in.)
0.0025 (0.0001) 0.0051 (0.0002) 0.0076 (0.0003) 0.0102 (0.0004) 0.0127 (0.0005)
0.0168 (0.00066) 0.0193 (0.00076) 0.0218 (0.00086) 0.0244 (0.00096) 0.0269 (0.00106)
0.0117 (0.00046) 0.0091 (0.00036) 0.0066 (0.00026) 0.0041 (0.00016) 0.0015 (0.00006)
Fa
F, N (lb)
f14
N (1b) 2,225 (500) 4,895 (1100) 6,987 (1570) 9,345 (2100) 11,440 (2570)
5,785 (1400) 7,343 (1650) 8,678 (1950) 10,240 (2300) 11,790 (2650)
3560 (800) 2448 (550) 1691 (380) 890 (200) 356 (80)
88.
e
62
Plot Sa vs Fa and find Sa = 0.00968 mm (0.000381in.) corresponding to Fa = 8900 N (2000 lb).
From Fig. 9.6 at Fa = 8900 N (2000 lb), Sa = 0.0221 mm (0.00087 in.). Therefore, preloading of 4450 N (1000 lb) reduced Sa by 56%. If it is desirable to preload ball bearings that are not identical, equations (10.18) and (10.20) become
(10.23)
x sin a2
(
(BIDl) sin (a1 ai) cos a1
1.5

1)
sin (az ai) + (B2D2)cos __
(10.24)
a2
Equations (10.23) and (10.24) must be solved simultaneously for al and a2.As before, equation (7.36) yields the corresponding values of 6. To reduce axial deflection still further, more than two bearings can be locked together axially as shown in Fig. 10.8. The disadvantages of this system are increased space, weight, and cost. More data on axial preloading are given in reference El0.21.
in Radial preloading of rolling bearings is not usually used to eliminate initial large magnitude deflection as is axial preload. Instead, its purpose is generally to obtain a greater number of rolling elements under load and thus reduce the maximum rolling element load. It is also used to prevent skidding. Methods used to calculate maximum radial rolling element load are given in Chapter 7. Figure 10.9 shows various methods to radially preload roller bearings.
F
~ 10.8. ~Triplex~set of angularcontact E bearings?mounted two in tandem and one opposed. This arrangement provides an even higher axial stiffness and longer bearing life than with a duplex set, but requires more space.
BEAIXING DEFLECTION, ~ ~ E L O ~ I AFJD N G ,ST Housing
L""
"" Shaft
~
i
(c)
10.9. ( a ) Diametral (radial) clearance found in mostofftheshelf rolling bearings. One object of preloading is to remove this clearance during assembly. (b)Cylindrical roller bearing mounted on tapered shaft, to espand inner ring. Such bearings are usually made witha taper on the inner surface of & mm/mm. (c) Spherical rollerbearing mounted on tapered sleeve to espand the inner ring.
le 10.4. Suppose the 209 cylindrical roller bearing of Example 7.3 was manufactured with a tapered bore and was driven up a tapered shaft as in Fig. 10.9b until a negative clearance or interference 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 load zone, and the radial deflection. Compare these results with those of Example 7.3.
Fr = 0.001170
ZKn
I
0.001170 = Smax (0.00254)] 2 (Sm,,
'*11 Jr< E)
+ 0.00127)1*11Jr(e) = 0.001170
=
0.5
+ 0.000635 Smm
1, Assume E = 0.8. From Fig. 7.2, Jr = 0.266. (Smm+ 0.00127)1.11 X 0.266 = 0.001170 Smax= 0.00635 mm (0.00025 in.) 0.000635 = 0.6 , E = 0.5 + 0.00635 0.6 f 0.8 4, Assume E = 0.6. From Fig. 7.2, Jr = 0.256. 5. (S,, + 0.00127)1.11 X 0.256 = 0.001170 Smax= 0.00660 mrn (0.000260 in,) 0.000635 = 0.596 , E = 0.5 + 0,00660
.
This answer is sufficiently close to
Z
=
E
=
0.6
14
Ex.2.7
Fr = ~Qm~~Jr(E)
4450
=
14Qm,,
Q,,
=
1242 lV (279.0 lb)
amax 
X
0.256
0.00660 mm (0.000260in.)
(7.19)
3 q!Il
=
cos1 (1  2 E )
(7.12), (7.13)
Comparing results with Example 7.3 Example 7.3
e l , mm &,ax, T\J %lax,mm
+;
t0,0406 (t0.0016 in.) lb) (430.3 1915 0.032 (0.00126 in.) t50"35'
10.4Example 0.0025 (0.0001 in.) 1242 (279 lb) 0.0066 (0,00026 in.) t lOl"32'
It is sometimes desirable that the axial and radial yield rates of the bearing and its supporting structures be as nearly identical as possible. In other words, a load in eitherthe axial or radial direction shouldcause ~ ~the ~ ball c ~ ~ y identical deflections (ideally). This necessity for ~ s o e Z ~ in bearings came with the development of the highly accurate, low drift inertial gyros for navigational systems, and for missile and space guidance systems. Such inertial gyros usually have a single degree of fkeedom tilt axis and are extremely sensitive to error moments about this axis. Consider a gyro in which the spin axis (Fig. 10.10) is coincident with the x axis, the tilt axis is perpendicular to the paper at the origin 0, and the center of gravity of the spin mass is acted on by a disturbing force F in the xx plane and directed at an oblique angle 4 to the x axis,this force will tend to displace the spin mass center of gravity from 0 to 0'.
Line of applied force
10.10. Effect of disturbing force F on the center of gravity of spring mass.It is frequently desirable to obtain isoelasticity in bearings in which the displacement in any direction is in line withthe disturbing force.
If, as shown in Fig. 10.10, the displacements in the directions of the x and x axes are not equal, the force F will create an error moment about the tilt axis, In terms of the axial and radial yield rates of the bearings, the error moment % is
where the bearing yield rates 8; and 8; are in deflection perunit of force. To minimize % and subsequent drift, 8; must be as nearly equal to 8; as possiblea 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 the minimal error moments achieved through isoelasticity. In most radial ball bearings, the radial rate is usually smaller than the axial rate. This is best overcome by increasing the bearing contact angle, which reducesthe axial yield rate and increases radial yield rate. Onetoone ratios can be obtained by using bearings with contact angles that are 30" or higher. At these high angles, the sensitivity of the axialtoradial yield rate ratio to the amount of preload is quite small. It is, however, necessary to preload the bearings to maintain the desired contact angles.
Most radial ball bearings can accommodate a thrust load and function properly provided that the contact stress thereby induced is not excessively high or that the ball does not override the land. The latter condition results in severe stress concentration and attendantrapid fatigue failure of the bearing. It may therefore be necessary to ascertain for a given bearing the maximum thrust load that the bearing can sustain and still function. Thesituation in which the balls override the land will be examined first.
'
Figure 10.11 shows an angularcontact bearing under thrust in which the balls are riding at anextreme angular location without the ring lands cutting into the balls. From Fig. 10.11 it can be seen that the thrustload, which causes the major axis of the contact ellipse to just reach the land of the bearing, is the maximum permissible loadthat the bearing can accommodate with
380
BEARING ~ E F L E ~ T I O ~ ,
P ~ L AND O ~ I SN G ~,
~
Contact ellipse
Bearing axis of rotation
F
I
~ 10.11. ~ ~ Ballraceway E contact under limiting thrust load.
out overriding the corresponding land. Both the inner and outer ring lands must be considered. From Fig. 10.11 it can be determined that the angle t3, describing the juncture of the outer ring land with the outer raceway is equal to a + 4 in which a is the raceway contact angle under the load necessary to cause the major asis of the contact ellipse, that is, 2ao, to extend t o eo and CF) is the onehalf of the angle subtended by the chord 2a. The angle eo is given approximately by €lo
=
cos1 (l!5?Z$?)
(10.26)
Since the contact deformation is small, r: to the midpoint of the chord 2a0 is approximately equal to D / 2 ; therefore, sin 4 = 2a0/D or sin
2a0 (eo a ) = D
(10.27)
For steel balls contacting steel raceways, the semimajor axis of the contact ellipse is given by
S
S
L ~ T I BALL ~ G IBIEAl3ING THRUST LOAI)
( ~ ) 1 f3
(6.39)
a = 0.0236~~"
in which 2po is given by ZPO =
;
(4

l

FY
(2.30)
and a: is a function of F(p), defined by
(2.31)
Y=
D cos a
(2.27)
dm
According to equation (7.26) for a thrustloaded ball bearing, (7.26) Combining equations (6.39), (2.301, (10.27), and (7.26), one obtains
Fao = Z sin a 2po
I3
D sin (O0  a) 0.0472~~:
(10.28)
In Chapter 7 equation (7.33) was developed, defining the resultant contact angle a! in terms of thrust load and mounted contact angle. (7.33) in which K is Jones' axial deflection constant, obtainable from Fig. 7.5. Combining equation (7.33) with (10.28) yields the following relationship:
This equation may be solved iteratively for a using numerical methods.
Having calculated a, it is then possible to determine the limiting thrust load .Fao for the ball overriding the outer land from equation (7.33). Similarly, for the inner raceway
(~ ;
a;Kl/3 sin (Oi  a) (10.30) = 0.0472
I
~
1)0.5
(DXPi)1’3 (10.31)
and Xp, and .F(P)~ are determined from equation (2.28) and (2.29), respectively.
It is possible that prior to overriding of either land anexcessive contact stress may occur at the inner raceway contact (or outer raceway contact Ifaligning ball bearing). The maximum contact stress due to ball
Cmax=
3Q 
(6.47)
2mab
in which
b = 0.0236bF
(
113
(6.41)
of equations (6.41), (6.39), (6.47), (7.33) yields (10.32) permits Assuming a value of maximum permissible contact stress cmax a numerical solution for a; thereafter the limiting .Fa may be calculated = 2069 N/mm2 from equation (7.33). Presentday practice uses cmax (300,000 psi) as a practical limit for steel ball bearings. If the balls do not override the lands, however, it is not uncommon to allow stresses to exceed 3449 N/m2 (~00,000psi) for short time periods.
3
5. The outer ring land diameter of the 218 angularcontact ball bearingof Example 7.5 is 133.8 mm (5.269in.). the thrustload that will cause the balls to override the outer ring land. do = 147.7 mm (5.816 in.)
Ex. 2.3
22.23 mm (0.875 in.)
Ex. 2.3
1) =
a' = 40"
Ex. 2.3
13
Ex. 2.3
=
0.0464
Ex. 2.6
dm = 125.3 mm (4.932 in.)
T)
(10.26)
0o = cos1 (1  do  dl0 =
cosl
(
1
22.23
cos a y=Ddm
fo
(2.27)

22.23 X cos a 125.3
=
fi = 0.5232
= cos 0.1774
a
Ex. 2.3 (2.30)

=
1
2 I
X
" 22.23 ..(4  0.532  + 0.09396

I
0.1774 cos a 0.1774 cos a
0.01596 cos a + 0.1774 cos a
(a)
(2.31)
11.911 
2.089 K
=

0.3548 cos a 1 + 0.1774 cos a 0.3548 cos a 1 + 0.1774 cos a
896.7 N/mm2 (130.000 Dsi)
(b)
Ex. 7.5
384
("
cos a' 0.0472~xzK~'~ cos a
sin (0  a) =

0.5
1)
(10.29)
(D x p 0 Y 3 0.5
0.0472az sin (67'59'
(896.7)1/3
 a) =
(D 2p0)1/3 0.454az
sin (67'59'
X

a) =
("
0.5
0.7660 cos a  1)
( D xp0)1'3
Using trial and error: a
45', cos a = 0.7071. (2.132 in?) F ( p ) , = 0.9046 From Fig. 6.4, a: = 3.11 Assume a
=
. Xp, = 0.0839 mm'
0.454 a
e
sin (67'59'

a) =
X
3.11
X
("
0.7660  1)'" 0.7071
(22.23 X 0.0839)1.3
a = 48'35' Assume a = 47', cos a = 0.6820 2po = 0.0843 mm' (2.141 in?) F ( p ) , = 0.9050 From Fig. 6.4, a: = 3.12
0.454 6. sin (67'59'  a ) =
X
3.12
( 4
(b) X
(~
0.7660 0.6820

l)Oa
(22.23 X 0.0843)1'3
a = 44'8' 7, Assume a = 46', cos a = 0.6947 2po = 0.0841 mm' (2.136 in?) F ( p ) , = 0.9048 From Fig. 6.4, a: = 3.11
\
8, sin (67'59'  a) =
(22.23
X
0.0841)1/3
a = 46'21' 9. Assume a = 46'30', cos a = 0.6884 2po = 0.00842 mm (2.138 in?)
'
/
~FERENCES
F ( p ) , = 0.9049 From Fig. 6.4, a:
=
3.12 0.454
sin (67'59'  a) =
X
3.12
X
0.7660 (E 
(2.138)1'3
(4
a = 46'19'
This result is satisfactory for the purpose of this example. Use a 46'2 I '
=
1.5
(7.33) Fao
16
X
(22.23)' X 896.7
cos 40' (cos (46'21')
=
sin (46'21')
=
187,200 N (42,070 lb)
E In many engineering applications bearing deflection must be known to establish the dynamic stability of the rotor system. This consideration is important in high speed systems such as aircraft gas turbines. The bearing radialdeflection in thiscase can contribute to the system eccentricity. In other applications, such as inertial gyroscopes, radiotelescopes, and machine tools, minimizationof bearing deflection under load is required to achieve system accuracy or accuracy of manufacturing. That thebearing deflection is a function of bearing internal design, dimensions, clearance, speeds, and load distribution has been indicated in the previous chapters. However, for applications in which speeds are slow and extreme accuracy is not required, the simplified equations presented in this chapter are sufficient to estimate bearing deflection. To minimize deflection, axial or radial preloading may be employed. Care must be exercised, however, notto excessively preload rollingbearings since this can cause increased friction torque, resulting in bearing overheating and reduction in endurance.
10.1. A. Palmgren, Ball and Roller Bearing Engineering, 3rd ed., Burbank, Philadelphia, pp. 4951 (1959). 10.2. T. Harris, "HOWto Compute the Effects of Preloaded Bearings," Prod. Eng. 8493 (July 19, 1965).
This Page Intentionally Left Blank
Symbol
Description Distance to load point from righthand bearing Distance between raceway groove curvature centers Rolling element diameter Pitch diameter Outside diameter of shaft Inside diameter of' shaft Modulus of elasticity Bearing radial load Raceway groove radius + D Section moment of inertia Loaddeflection constant Distance between bearing centers Bearing moment load Applied load at a
Units mm (in.) mm (in.) mm (in.) mm (in.) mm (in.) mm (in.) N/mm2 (psi) N (lb) mm4 (in.4) N/mmx (lb/inx) mm (in.) N * mm (in. * lb) N (lb)
388
Symbol
STATICALLY ~ E T E ~ N A T S E
Description Rolling element load Radius from bearing centerline to raceway groove center Applied moment load at a Load per unit length Distance along the shaft Deflection in they direction Deflection in the x direction Free contact angle D cos ald,, Bearing radial deflection Bearing angular misalignment Curvature sum Rolling element azimuth angle
~
TSYSTEMS 
Units N (lb) mm (in.)
N mm (in. * lb) N/mm (Win.) mm (in.) mm (in.) mm (in.) rad, O mm (in.) rad, O mml (in.l) rad, O
SUBSCRIPTS Refers to bearing location Refers to axial direction Refers to bearing location Refers to rolling element location Refers to y direction Refers to x direction Refers to xy plane Refers to xz plane S~PERSC~IPT Refers to applied load or moment
In some modern engineering applications of rolling bearings, such as high speed gas turbines, machine tool spindles, and gyroscopes, the bearings often must be treated as integral to the system to be able to accurately determine shaft deflections and dynamic shaft loading as well as to ascertain the pedormance of the bearings. Chapters 7 and 9 detail methods of calculation of rolling element load distribution for bearings subjected to combinations of radial, axial, and moment loading. These load distributions are affected by the shaft radial and angular deflections at thebearing. In this chapter, equations for the analysis of bearing loading as influenced by shaft deflections will be developed.
~
A commonly used shaftbearing system involvestwo angularcontact ball bearings or tapered roller bearings mounted in a backtoback arrangement as illustrated by Figs. 11.1and 11.2. In these applications, the radial loads on the bearings are generally calculated independently using the statically determinate methods of Chapter 4. It may be noticed from Figs. 111.1and 11.2; however, that the point of application of each radial load occurs where the line defining the contact angle intersects the bearing axis. Thus, it can be observed that a backtoback bearing mounting has a greater length between loadingcenters than does a facetoface mounting. This means that the bearing radial loads will tend to be less for the backtoback mounting.
~ I 11.1. ~ Rigid shaft ~ mounted E in backtoback angularcontact ball bearings subjected to combined radial and thrust loading.
11.2. Rigid shaft mounted in backtoback tapered roller bearings subjected to combined radial and thrust loading.
The axial or thrust load carried by each bearing depends upon the internal load distribution in the individual bearing. For simple thrust loading of the system, the method illustrated in Example 11.3 may be applied to determine the axial loading in each bearing. When each bearing must carry both radial and axial load, although the system is statically indeterminate, for systems in which the shaft can be considered rigid, a simplified method of analysis may be employed. In Chapter 18, it is demonstrated that a bearing subjected to combined radial and asial loading, may be considered to carry an equivalent load defined by equation (11.1).
F,
= .XFr
+ YFa
(11.1)
Loading factors X and Y are functions of the free contact angle, assumed invariant with rolling element azimuth location and unaffected by the applied load. This condition is clearly true for tapered roller bearings; however, as shown in Chapter 7, it is only approximated for ball bearings. Values for X and Y are usually provided for each ball bearing and tapered roller bearing in manufacturers' catalogs. Accordingly, assuming the radialloads Frland Fr2are determined using the methods of Chapter 4, the bearing axial loads Faland Fa2may be approximated considering the following conditions: For load condition (1):
and for load condition (2):
then (11.2)
(I1.3) For load condition (3):
then (11.4)
(11.5) 1.1. Taperedroller bearings mounted on a shaftin a backtoback arrangement asshown in Fig. 11.2 are of different series; the smaller bearing no. 1 has an axial load factor Yl = 1.77 and for the larger bearing no. 2, Y2 = 1.91. Using static equilibrium conditions, and neglecting the applied axial loading of 1800 N, the bearing radial loads were estimated to be F,, = 3500 N and Frz = 5000 N. Estimate the thrust load carried by each bearing.
Calculate:

2
~~) $
"(Frz Y2
=
(2618  1977) = 320.5 N
Since 2618 N > 1977 N and Pa = 1800 N > 320.5 N, load condition (2) obtains. Therefore
F,,=F,,=" 2Y1
3500  988.7 N (222.2 lb) 2 1.77
(11.2)
8
Fa2= Fal + Pa = 988.7 + 1800 = 2789 N (626.7 lb)
(11.3)
Bearings 1 and 2 in the backtoback mounting of Fig. 11.1are, respectively, 318 and 218 angularcontact ball bearings, each having a 40" nominal contactangle. Assuming no influence of the applied thrust load of 225 N and using statically determinate analytical methods,the bearing radial loads are estimated to be 8900 N and 5300 N, respectively. Estimate thethrust load carried by each bearing. For 40" angularcontact ball bearings, Y
=
0.57. Therefore
392
STATICALLY ~ E T E ~ A ST E~
T
 SYS BT E ~~S
Since
Condition (1)applies.
Fal
=Z
Fr1 = "O0  7807 N (1754 lb) 2Y1 2 0.57
(11.2)
9
In the more general twobearingshaft system, flexure of the shaft induces momentloads %& at nonselfaligning bearing supports in addition to the radial loads Fk.This loading system, illustrated by Fig. 11.3, is statically indeterminate in that there arefour unknowns,Fl, F2, K , and %, but only two equilibrium equations. For example:
XF=O F1iF2P=0
(11.6)
ZM=O FIZ  &
+ T  P(Z  a ) + 91% = 0
(11.7)
Considering the bending of the shaft, thebending momentat any section is given as follows:
Fl
11.3. Statically indeterminate twobearingshaft system.
~
~
E Id "Y T dx
=
"M
(11.8)
in which E is the modulus of elasticity, I is the shaft crosssection moment of inertia, and y is the shaft deflection at the section. For shafts having circular cross sections,
For a cross section at 0
5
x
5
a illustrated in Fig. 11.4,
"Y = +,x EI d 2
+ 9lll
(11.10)
+ ntlx + c,
(11.11)
dX
Integrating equation ( 11.10) yields
E I dy
=
Flx2
"
dx
2
Integrating equation (11.11)yields
EIy
=
F1x3
+ 9111x2 + c,x + 6, 2
"
6
~
(11.12)
In equations (11.11)and (11.159,C, and C2 are constants of integration. At x = 0, the shaft assumes the bearing deflection &. Also at x = 0, the shaft assumes a slope 8, in accordance with the resistance of the bearing to moment loading; hence
11.4. Statically indeterminate twobearingshaft system forces and moments acting on a section to the left of the load application point.
e, = EI0, e, = EISrl Therefore, equations ( 11.11)and (11.12) become
Flx2
E I dy
+ %,x + Em1
(11.13)
"
2
dx
and
EIy
F x3
"
For a cross section at a
6
+$%,x23 + EI0,x + EISrl
5 x 5 Z as
(11.14)
shown in Fig. 11.5,
a)  T
(11.15)
Integrating equation (11.15) twice yields
E I dy
dx
F,x2+ (9%
"
2

T)x+ Px
(11.16)
Flx 6
At x
=
Z, the slope of the shaft is
0, and the deflection is S,,, therefore
M
11.5. Statically indeterminate twobearingshaft system forces and moments acting on a secti.on to the right of the load application.
5
+ P2 [x(x  2a)  Z(Z
 2a)l
+ E102
(11.18)
+ P6 [ x 2 ( x 3a)  Z2(3x + 3a  21) + 6x1~~1 + EI [ar2 02(Z  x)]
(11.19)
At x = a, singular conditions of slope and deflection occur. Therefore at x = a, equations (11.13) and (11.18) are equivalent as are equations (11.14) and (11.19). Solving the resultant simultaneous equationsyields
F,
=
P(Z  a)2(Z + 2a) 13

6T&

a)
13
L.
L.
(11.20) %l=
Pa(Z  a)2 Z2 
E!!! Z
 3a) + T(Z  a)(Z Z2
[ZB, +
02
+ 3(sr1Z
ar2)l
(11.21)
~ubstitutingequations (11.20) and (11.21) into (11.6) and (11.7) yields
In equations (11.20)(11.23), slope 81 and arl are considered positive and the signs of ti2 and ar2 may be determined from equations (11.18) and (I1.19). The relative magnitudesof P and T and their directionswill determine the sense of the shaftslopes at thebearings. To determine the reactions it is necessary to develop equations relating bearing misalignment angles Oh to the misaligning moments X h and bearing radial deflections arb to loads l?&. This may be done by using the data of Chapters 7 and 9.
I
In their most simple form, in which the bearings are considered as axially free pinned supports, equations (11.20) and (11.22) are identical to (4.29) and (4.30). This format is obtained by setting and Srh equal to zero and solving equations (11.21) and (11.23) simultaneously for 61 and 62. ~ubstitutionof these values into equations (11.20) and (11.22) produces the resultant equations. If the shaft is very flexible and the bearings are rigid with regard to misalignment, then 61 and 82 are approximately zero. This substitution into equations (11.20) (11.23) yields the classical solution for a beam with both ends built in. The various types of twobearing support maybeexamined by using equations (11.20)(11.23). If more than one load andlor torque is applied between the supports, then by the principle of superposition
(11.24) t
K=n
t
k=n
3ak) (11.25)
ak> +E![6,+62+ Z2 + kn
~ ~ . ~ .
2(sr1Zt
(11.26)
k=n
A pair of 209 radial ball bearings mounted on a holshaft having a 45 mm (1.772 in.) 0.d. and 40 in.) i.d. support a 13,350 IN (3000 lb) radial load actin midway between the bearings. The span between the bearing centers is 254 mm (10 in.). If each of the bearings is mounted according to the fits of Example 3.1 and their dimensions are as given by Example 2.1, e
determine the bearing load distribution, radial deflection,moment loading, and angle of misalignment. (11.9) [(45)4 (40.56)4]
=3'1416
64
=
6.855
X
lo4 mm4 (0.1647 in:)
Bearings 1 and 2 are identically loaded because load midspan, therefore, F, = F2 = 6675 N (1500 lb). %=
But,
Ejrl = Ejr2
Pa(Z
and

is applied at
a)2
Z2
e2 = 8,.
%=
6675

2
X
Hence
127 X (254  127)2 (254)2 2.069
X
X
lo5 X
6.855
X
lo4$
254 or %=
2.119
X
lo5  1.115 X
1088
f = 0.52
=
r/D,
Ex. 7.1
A+
=
40'
Ex. 3.1
A
=
0.508 mm (0.02 in.)
Ex. 3.1
(xo
=
7'30'
Ex. 2.1
L) = 12.7 mm (0.5 in.)
Ex. 2.1
dm = 65 mm (2.559 in.)
Ex. 2.1
$$ =
Since f
Ex. 2.2
dm + (ri  0.5L))COS a'
(7.86)
+ (0.52  0.5) X
65
2
=
12.7 cos (7'30')
32.75 mm (1.289 in.)
+ S, + q3 cos +)2 ao + S, COS +)211/2  1 p x (cos a' + s, cos +I cos + = 0
{[(sin a'
x
*=
Fr  KnA1.5
6675

+ (COS
+P
[(sin a' + S, + 9~3 cos +12 + (cos ao + Zr cos $121
J/=o
{[(0.1305 + 32.753 cos +)2 + (0.9914 + 3,. COS J/)2]1/2  l}1.5 *=rt?T X (0.9914 cos +) cos =0 0.3621 Kn $=o [(0.1305 + 32.753 COS J/)2 + (0.9914 + &. cos +)2]1'2
3, however, deformation of asperities can be neglected. Chang et al. [12.41] analytically investigated the effects of surface roughness considering the effects of lubricant shear thinning due frictional heating. This phenomenon will be covered in greater detail in Chapter 13. They determined that these effects serve to mitigate the “pressure rippling” influence on lubricant film thickness. Ai and Cheng E12.421 conducted an extensive analysis revisiting the influences of surface topographical lay. They generated threedimensional plots of point contact pressure and film thickness distribution for transverse, longitudinal, and oblique topographical lays. Fig. 12.1512.1’7 illustrate the effects for a randomized surface roughness. They indicated that roughness orientation has a noticeable effect on pressure fluctuation. They further noted that oblique roughness lay induceslocalized threedimensional pressure fluctuations in which the maximum pressure may be greater than that produced by transverse roughness lay. It is to be noted that theoblique roughness lay more likely is representative of the surfaces generated during bearing component manufacture. Oblique surface roughness lay may also result in theminimum lubricant film thicknesses compared to transverse or lon~tudinal roughness lays. Ai and Cheng [12.42] further note, however, that when A 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,
, .
(4
(b)
12.15. Pressure ( a )and film thickness ( b ) distribution in an EHL point contact with transverse topographical lay, random surface roughness. Motionis in the xdirection (from Ai and Cheng [12.42]).
12.16. Pressure (a)and film thickness ( b )distribution in an EHL point contact with longitudinal topographical lay, random surface roughness. Motion is in thexdirection (from Ai and Cheng l12.421).
(4
(b)
.17. Pressure ( a )and film thickness (b)distribution in an EHL point contact with oblique topograp~icallay, random surface roughness. Motion is in thexdirection (from Ai and Cheng E12.421).
isotropically roughsurface occurring in rolling ball on flat contacts. They found that, for A < 2, the mean EHL film thicknesses were less than those for smooth surfaces. Subsequently, using the spacer layer imaging method developed by Cann et al. E12.443 to map EH"Lcontacts, Guangteng et al. [12.45] indicated that rolling elements having real, random, rough surfaces; for example, rollingbearing components, the mean film
51
ASE L ~ R I C A T ~ O ~
thicknesses tend to be less than those calculated for rolling elements having smooth surfaces. This implies that, in the mixed EHL regimefor example, A < 1.5the mean lubricant film thicknesses will tend to be less than those predicted by the equations given for rolling contacts with smooth surfaces. The amount of the reduction may only be determined by testing; empirical relationships need to be developed.
When grease is used as a lubricant, the lubricant film thickness is generally estimated using the properties of the base oil of the grease while ignoring the effect of the thickener. It hasbeen determined, however, by several researchers E12.4612.491 that in a given application, owing to a contribution by the thickener, a grease may form a thicker lubricant film than that determined using only the properties of the base oil. Kauzlarich and Greenwood [12.50] developed an expression for the thickness of the film formed by greases in line contact under a Herschel~ulkley constitutive law in which shear stress r and shear rate y are related by the equation r=ry+ayP
(12.84)
where ry is the yield stress and a and p are considered physical properties of the grease. For a Newtonian fluid r = qj
(12.85)
where q is the viscosity. The effective viscosityunder a HerschelBulkley law is thus found by equating r from equations (12.84) and (12.85) so that (12.86) In thisform it is seen that for /3 > 1,qeffincreases indefinitely with shear increases. Palacios rate, andfor ,6 < 1, qeffapproaches zero as strain rate et al. E12.461 argued that it is more reasonable to assume that at high shear rates greases will behave like their base oils. They accordingly proposed a modification of the HerschelBulkley law to the form
r =
r + a
y @
+ q,y
(12.87)
where Q is the base oil viscosity. In this form, provided p < 1, qeffapproaches rlt, as strain rate approaches m. Values of ry, a, p, and q b are given in [12.51] for three greases from 3580°C (95176°F). Since viscosity appears raised to the 0.67 power in equation (12.61), Palacios et al. E12.511 proposed that h,, the film thickness of a grease, and h,, the film thickness of the base oil, will be in the proportion 0.67
(12.88) They proposed that this evaluation be made at a shear rate equal to 0.68u/hG, which requires iteration to determine h,. Their suggested approach is to calculate h, from equation (12.61), then determine y = 0.68u/h,, and then hGfrom equation (12.88). The shear rate is then recalculated using h,. The process is repeated until convergence occurs. The analysis was applied to line contact, but it should also be valid for elliptical contacts with a / b in the range of 810 (typical for ball bearing point contacts). In his investigations, Cann [12.52, 12.531notes that theportion of the film associated with the grease thickener is a “residual” film composed of the degraded thickener deposited in thebearing raceways. The hydrodynamic component is generated by the relative motion of the surfaces due to oil, both in the raceways and supplied by the reservoirs of grease adjacent the raceways. Hefurther notes that atlow temperatures grease films are generally thinner than those for the fully flooded, base fluid lubricant. This is due to the predominant bulk grease starvation and the inability of the high viscosity, bled lubricant to resupply the contact. At higher temperatures of operation, grease formsfilmsconsiderably thicker than those considering onlythe base oil. This is attributed to the increased local supply of lubricant to the contact area due to the lower oil viscosity at the elevated temperature producing a partially flooded EHL film augmented by a boundary film of deposited thickener. Therefore, it can be stated that with grease lubrication the degree of starvation tends to increase with increasing base oil viscosity, thickener content, and speed of rotation. It tends to decrease with increasing temperature. For rolling bearing applications, the film thickness may only be a fraction of that calculated for fullyflooded, oillubrication 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 elementraceway contacts. Nevertheless, depending on the aforementioned operating conditions of grease base oil viscosity, grease thickener content, and rotational speed,
453
lubricant film thicknesses may be expectedto be only a fraction of those calculated using equations (12.55),(12.581, and (12.61). According to data shown by Cann [12.53], fractional values might range from 0.9 down to 0.2.
Although this chapter has concentrated on elastohydrodynamiclubrication in rolling contacts, the general solution presented for the Reynolds equation covers a gamut of lubrication regimes; for example: Isoviscous hydrodynamic(IHD) or classical hydrodynamiclubrication Piezoviscous hydrodynamic (PHD lubrication, in which lubricant viscosity 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 component surfaces are considered in the solution Dowson and Higginson [12.54] created Fig. 12.18 to define these regimes for line contact in terms of the dimensionless antiti ties for film thickness, load, and rolling velocity; equations (12.46)( 12.48). ~arkho et al. E12.551 established a parameter, called C, herein, for a fixed value of $3 this factor was used to define the lubrication regime. Dalmaz [12.56] subsequently established equation (12.89) to cover all practical values of 9.
C,
=
log,,
[
~]
1.5 1 0 6 ( ~ ) 2
(12.89)
Table 12.1 shows the relationship of parameter C , to the operating lubrication regimes. For calculation of the lubricant film thicknesses is rolling elementraceway contacts, onlythe PHD and EHD regimes needto be considered. For calculations associated with the cagerolling element contacts, probably considerationof the hydrodynamic regimeis sufficient. In thiscase, Martin El2.11 gives the following equation for film thickness in line contact: (12.90)
For point contact, Brewe et al. [12.33] give
454
L
~
I FILMS C ~ IN ROLLING E L E ~ ~  ~ C C OE~ A~ C A T S~
R
.I. Lubrication Regimes
Parameter Limits
Lubrication Regime
Characteristics
1
IHD
1 Y/2.Using the expression for p o [(eq. 13.8)l gives 0.31 * 2E’ (X  d)li2 Y >2 TR1I2
or
(13.16)
(13.17) z>d+wp
(13.18)
Thus, any summit whose height exceeds d + w, will have some de of plastic deformation. The probability of a plastic sum the shaded area inFig. 13.3 to the right of d + w,. The e of plastic contacts per unit area becomes (13.19)
where (13.20)
d / u sthe degree of plastic asperity interaction is determi value of w;: the higher w;, the fewer plastic contacts. Acw use the inverse, l/wz, as a measure of the plasticity of an i For a given nominal pressure PIA,, d / u Bis found by solving e ~ u a t ~ o n ( 1 3 ~ 3assuming )~ that most of the load is elastically supported.
e model for a lubricated contact, (1)the height d relative ne of the summit heights to h, the thickness of the lubricant film that separates the two surfaces, must be ~ e t e r m i n ~and d, usmust be establis ) the values of the GW parameters R, e rms value of the or (1)the first step is to compute th “rough” surfaces as 0 =
(0: +
0;>1””
(13.21)
en the mean plane of a rough surface with this rms value is held at ight h above a smooth plane, the rms value of the gap width is the same as shown in Fig. 13.3, where both surfaces are rough. It is in this sense that the surface contact of two rough surfaces may be translated into the e~uivalentcontact of a rough surface and a smooth surface. As 2, the summit and surface mean planes c surface with normally ~istributedheight ~uctuations, the value of E, has been found by ush et al. [13.7] to be
CROGEO
C~OCO~ACTS

40
x, = 
G
(13.22)
The quantity a, h o w n as the bandwidth parameter, is defined by
where m,, m2,and m4 are known as the zeroth, second, and fourth spectral moments of a profile. Theyare equivalent to the mean square height, slope, and second derivative of a profile in an arbitrary direction; that is
m,
=
E (x2) = u2 (13.25) (13.26)
where x(x)is a profile in an arbitrary direction x, E [ 1 denotes statistical expectation, and m, is simply the mean square surface height. The square root of m, or root mean square (rrns) is sometimes referred to as and forms part of the usual outputof a stylus measuring device. Some of the newer profile measuring devices also give the rms slope, which is the same as (m2)1'2 converted fromradians to degrees. No commercial equipment is yet available to measure m4. ~easurementsof m4 made so far have used custom computer processing of the signal output of profile measurement equipment. Bush et al. E13.101 also show that the variance uz of the surface summit height distribution is related to c r 2 , the variance of the composite surfaces, by (13.27)
A summit located a distance d from the summit height mean plane is at a distance h = d + Z, from the surface mean plane. Thus, d=hZ,
(13.28)
Using equation (13.22) forE, and equation (13.27) for usgives (13.29)
Equation (13.29) shows that dIus is linearly related to hlu. The ratio h l u 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 lubricated. For a specified or calculated value of A, dlcr, is computed from equation (13.29) for use in the GVV model. For an isotropic surface the two SUM and 23, the average radius of the spherical caps of asperities, may be expressed as (Nayak C13.81): (13.30)
(13.31)
For an anisotropic surface, the value of m, will vary with the direction in which the profile is taken on the surface. The ma~imumand minimum values occur in two orthogonal “principal”~irections.Sayles and T h o ~ a s [13.9] recommend the use of an equivalent isotropic surface for whichm2 is computed as the harmonic mean of the m2 values found along the principal directions. The value of m4 is similarly taken as theharmonic mean of the m4 values in these two directions.
For a specified contact with semiases a and b, under a load P, with plateau lubricant film thickness h and given values of m,, m,, and m4, the a is determined by first computing PlA, from equation [13.13] and using (13.32)
The fluidsupported loadis then
If &, > P, the implication is that the lubricant film thickness is larger than computed under smooth surface theory. Inthis case, equation (13.13) could be solved iteratively until Qa= P.
4.1. An isotropic surface has roughness parameters clr2 m,
= 0.062~ pm2, m2 =
0.0018, and m4 = 1.04 X
=
pm,. Calculate
the summit density DSUM, the height of the summit mean plane above the surface mean plane, the mean summit radius R, and the standard deviation a, of the summit height distribution. From equation (13.30) the summit density is DSUM =
=
6wm2a
1.04 X 104 1.8 X X 32.65
1.77 X
pm2 (1.142 pin.2)
=
m4
(13.30)
The separation of the surface and summit mean plane is, by equations (13.22) and (13.23), (13.22) (13.23) = 2,  =
=
2.006 4
(~n=) 1/2
0.399 pm (1.571 X
in.)
The mean summit tip radius is, from equation (13.31), (13.31) =
65.2 pm (2.567 X
lom3in.)
The standard deviation of the summit height distribution is calculated from equation (13.27) to be a,
1
~)
=
[(1
=
0.186 pm (7.323 X
1/2
(13.27)
in.)
Let a steel surface having these roughness characteristics make rolling contactwith a smooth plane forming an EHL contact for which the plateau lubricant film thickness, computed from equation (12.61)
and adjusted for starvation and inlet heating, is h = 0.5 pm. Using the GW microcontact model, calculate the nominal pressure PIA,, the relative contact area &/A,, the mean real pressure PIA,, the contact density n, and, for a tensile yield strength of 2070 NImm2,the plastic contact density n,. The computed filmparameter A = 0.51(0.0625)1f2= 2.0. From equation (13.29), d  hlcr  4/(wa)lf2 CrB (I  0.8968Ia)'l2
(13.29)
"
(I  0.8968/2.006)1f2 0.544 Interpol~tingin Table 13.1 gives
F, (0.544) = 0.2935 F , (0.544) = 0.1850 F3/2
(0.544) = 0.1812
The nominal pressure is calculated from equation (13.13) with E' 1.14 X IO6N/mm2 (16.53 x lo6 psi) for steel:
=
3 X
=
x 1.14 x IO5 (0.0652)1/2(0.186 x 103)3/2
=
(13.13)
1770 X 0.1812
31.6 NImm2 (4581 psi)
From equation (14.13) the ratio of mean real contact area A, to nominal contact area A, is (13.12) = w X =
0.0652
X
0.186
X
X
1770 X 0.185
0.0125
The actual contact area thus averages only 1.25%of the nominal contact area. The mean actual pressure PIA, is
31.6 0.0125


2528 N/mm2 (3.665 X
"
lo5 psi)
From equation (13.5) the contact density n is (13.5) =
1770 X 0.2935 = 519 eontacts/mm2 (3.35 X 105/in.2)
From equation (13.20), =
6.4
( ~( )~ )
=
6.4
X
=
0.740
2
w:
(13.20)
From equation (13.19), (13.19) =
1770 F, (0.544 + 0.740)
=
1770 F, (1.284)
inter pol at in^ in Table 13.1 gives F, (1.284) = 0.100 and np = 177/ mm2 ( 114,000/in.2). If the macroeontact is elliptical with semiaxes a = 3 mm (0.01181 in.) and b = 0.33 mm (0.01299 in.) under a load of P = 3500 N (786.5 lb), the mean asperitysupported load is Q, = Tab =
(~)
= T X
3
X
0.33
X
31.6
98.3 N (22.1 lb)
The fluidsupported load is Q~ =
P
 Qa =
3500  98.3
=
3402 N (764.5 lb)
4
FRICTION IN ~ ~ ~  L ~ R ROLLING I C ~ ET EL ~ E
~
~
As indicated in Chapter 12, a Newtonian lubricant is one in which stress due to shearing of the lubricant is defined by equation (12.1). aU
7 
qax
(12.1)
This equation implies that fluid viscosity is a constant. Several investigators [13.101[ 13.131have investigated the effects of nonNewtonian lubricant behavior on the EHL model. Bell [13.ll] specifically studied the effects of a ReeEyring model, in which shear rate can be described by equation (13.34). ( 13.34)
In equation (13.34), Eyring stress T~ and viscosity q are functions of temperature and pressure. When r is small, equation (13.34) describes a linear viscousbehavior approaching that of equation (12.1). Subsequently, it has been established that the nonNewtonian characteristics of lubricants tend to cause decreases in viscosity at high lubricant shear rates. These may occurdirectly in thecontact under operating conditions involving substantial sliding in addition to rolling. It has been further established, however, that thefilm thickness which obtains over most of the contact is primarily a function of the lubricant properties at the contact inlet. At the contact inlet, pressure is substantially atmospheric; therefore, it is not anticipated that a nonNewtonian lubricant will significantly influence lubricant film thickness. NonNewtonian lubrication does, however, significantly influence friction in the contact. Due to friction, lubricant temperature in the contact rises during rolling elementraceway contact,causing lubricant viscosity to decrease. Moreover, since pressure increases greatly in, and varies over, the contact, it is evident that equation (12.11) becomes (13.34) Assuming the contact area and surface pressure distribution is as represented by Fig. 6.6 for point contact and Fig. 6.7 for line contact, then equation (13.34) defines the localized shear stress T at any point x,y on the contact surface. Since EHL films are very thin compared to the ma
crogeometrical dimensions of the rolling components, it is further appropriate to approximate equation (13.34) as follows: (13.35) where u is sliding velocity at thecontact surface point x,y, and hc is the central or plateau film thickness. Houpert E13.141 and Evans and Johnson E13.151 used the ReeEyring modelfor analysis of EHL traction. Equations (12.21)(12.23), introduced in Chapter 12, can provide the viscositypressuretemperature relationship for many common lubricants. These equations can be used in equation (13.34) in the estimation of shear stress r provided the localized temperature and pressure can themselves be estimated.
As shown in Chapter 7, owing to the macrogeometries of mating rolling componentsi.e., rolling elements and racewaysand the contact deformations of these components under load, both rollingand sliding motions occur in most rolling elementraceway contacts. Gecim and Winer air and Winer E13.161 suggested alternative espressions for the relationship between shear stress and strain rate thatincorporated a maximum or limiting shear stress. Essentially, they proposed that for a given pressure, temperature, anddegree of sliding, there is a maximum shear stress that can be sustained. Based on experimental data from a disk machine, Fig. 13.5 from Johnson and Cameron [13.17] shows curves
ean contact pressure
Siide to roil ratio
~ I ~ 13.6. ~ R Typical E curves of traction measured on a disk machine operating in line contact (from [13.171).
 L ~ R I C ~ T ~ E O ~L L ELE ~ G
of traction coefficient vs pressure and slideroll ratio, which illustrate this phenomenon. In thiscase, traction coefficient is defined as theratio of average shear stress to average normal stress. ased on experiments, Schipper et al. [13.18] indicated a range of values for limiting fluid shear stress; for ~< 0.11. ~ ~ / p ~ ~ ~ example, 0.07 < ~
rachman and Cheng [13.19] and Tevaarwerk and Johnson [13.20] investigated traction in rollingsliding contacts and found that equation (12.1) pertains only to a situation involving a relatively low slidetoroll ratio; for example, less than 0.003 as shown in Fig. 13.5. Notethat traction refers to the net frictional effect in the rolling direction. Similar to Trachman and Cheng, for a given temperature and pressure, it is possible to define local contact friction as follows: (13.36) where T~ is the ~ewtonian portion o f the €rictional shear stress as can defined by equation (12.1) and 71im is the maxim^^ shear stress that be sustained at the applied pressure, Fig. 13.6 schematically demonstrates equation (13.36). ognizing that viscosity is a function of local pressure and temperct, and since the film thickness is extremely small imensions of the rolling co~ponents,7N can be described by equation (13.35).
As indi~atedin the section on ~ i c r o ~ e o m e tand r y ~icrocontacts? when lubricant film thickness is of the same magnitude or less than the composite roughness of the rolling components, Le,,A 5 1,contact of asper
Ne~onianshear stress limitingshearstress
T,,
y,
shear rate
F I ~ U R E13.6. Schematic illustration of equation (13.36).
ities on the component surfaces becomes morefrequent. The frictionthat occurs due to sliding motions between asperities can be characterized as Coulomb friction, such that 7 ,
= Pap
(13.37)
where EA, is the Coulomb coefficientof friction and p is the local pressure. On an average basis, this frictional stress may be assumed to apply to the portion of the overall contact area associated with asperityasperity contact. If the contact area of the smooth components is defined as A,, then, according to equation (13.12), the portion of the contact associated with Coulomb friction is AJA, * A,.
Combining the stress components due to Newtonia ting shear in the fluid, and asperity interactions, [13.211 applied the following formulain thedetermination of rolling contact tractions: (13.38)
In using equation (13.38), it is necessary to define values for q i m and p. These values for can only bedetermined from testing of fullscale bearing ased on comparison of predicted to actual bearing heat generations so determined, rIim= O.Ip,,, and p = 0.1 have been found to be representative in several applications.
This chapter contains an approach to predicting key performancerelated parameters 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 friction. It is recognized that using more elegant and complex analytical methods such as very fine mesh,multithousand node, and finite element analysis together with solutions of the Reynolds and energy equations in three dimensions, it is possible to obtain a more generalized solution with perhaps increased accuracy. Unfortunately, using currently available computing equipment, such solutions would require several hours of computational time to enable the performance analysis of a single operating condition for a rolling bearing containing only a small comple
48
FRICTION IN F L ~ D  L ~ R I C ~ T ROLLING ED E L E ~ ~  ~ C ECONTACT^ ~ A Y
ment of rolling elements. The equations provided in this chapter for frictional 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 formostrollingelementraceway contacts in that such loading is reasonably heavy; for example, generally at least several hundred MPa. Furthermore, the assumption is made 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 in excess of Hertzian values and these will cause localized shear stresses in excess of those predicted by equation (13.38).Accom~odationof these variations tends to increase the computational time beyond current engineering practicality.Therefore, for engineering purposes, frictional shear stressmay becalculated according to the average condition in each contact.
13.1. J. Greenwood and J. Williamson, “Contact of Nominally Flat Surfaces,” Proc. RoyaZ SOC.London A295, 300319 (1966). 13.2. A. Bush, R. Gibson, and T. Thomas, “TheElastic Contact of a Rough Surface,” Wear 35, 87111 (1975). 13.3. M. O’Callaghan and M. Cameron, “Static Contact under Load Between Nominally Flat Surfaces,’’ Wear 36, 7997 (1976). 13.4. A. Bush, R. Gibson, and G. Keogh, “Strongly Anisotropic Rough Surfaces,” ASME Paper 78LUB16 (1978). 13.5. J. McCool and S. Gassel, “The Contactof Two Surfaces Having Anisotropic Roughness Geometry,”ASLE Special ~ubzication(SP71,2938 (1981). 13.6. J, McCool, “Comparison of Models for the Contact of Rough Surfaces,” Wear 107, 3760 (1986). 13.7. A. Bush, R. Gibson, and G. Keogh, “The Limitof Elastic Deformation in the Contact of Rough Surfaces,” Mech. Res. Cornrn. 3, 169174 (1976). 13.8. P. Nayak, “Random Process Model of Rough Surfaces,” Trans. ASME, J Lub. Technology 93F, 398407 (1971). 13.9. R. Sayles and T. Thomas, “ThermalConductances of a Rough Elastic Contact,”AppZ. Energy 2, 249267 (1976). 13.10 T. Sasaki, H. Mori, and N. Okino, “Fluid LubricationTheory of Roller Bearings Parts I and 11,” ASME Trans., J Basic Eng. 166, 175 (1963). 13.11 J. Bell, “Lubrication of Rolling Surfaces by a ReeEyring Fluid,” ASLE Trans. 5, 160171 (1963). 13.12. F. Smith, “Rolling Contact LubricationThe Application of Elastohydrod~amic Theory,” ASME Paper 64Lubs2 (April 1964). 13.13 B. Gecim and W. Winer, “A Film Thickness Analysis for Line Contacts under Pure Rolling Conditions with a NonNewtonian Rheological Model,” ASME Paper 80C2/ LUB 26 (August 8, 1980).
FERE~CES 13.14. L. Houpert, “New Results of Traction Force Calculations in EHD Contacts,” ASME Trans, J; Lub. Technology l07(2), 241 (1985). 13.15. C. Evans and IC. Johnson, “The Rheological Properties of EHI) Lubricants,” Proc. Inst. Mech. Eng. 200(C5),303312 (1986). 13.16. S. Bair andW. Winer, “A Rheological Model for Elastohydrodynamic Contacts Based on Primary Laboratory Data,” ASME Trans., J. Lub. Tech. 101(3), 258265 (1979). 13.17. IC.Johnson and R. Cameron, Proc. Inst. Mech. Eng. 182(1), 307 (1967). 13.18. D. Schipper, P. Vroegop, A. DeGee, and R. Bosma, ‘“MicroEHL in Lubricated Concentrated Contacts,’’A5”E Trans., J. Tribology 112, 392397 (1990). 13.19. E. Trachman and H. Cheng, “Thermal and NonNewtonian EEects on Traction in Elastohydrodynamic Contacts,” Proc. Inst. Mech. Eng. 2nd Symposium on Elastohydrodynamic Lubrication, Leeds, 142148 (1972). 13.20. J. Tevaamerk and IC. Johnson, “A Simple NonLinear Constitutive Equation for EHD Oil Films,” ‘Wear 35, 345356 (1975). 13.21. T. Harris and R. Barnsby, “Tribological Performance Prediction of Aircraft Turbine Mainshaft Ball Bearings,” Tribology Trans. 41(1), 6068 (1998).
This Page Intentionally Left Blank
S Symbol
a b
LS Description Semimajor axis of projected contact ellipse Semiminor axis of projected contact ellipse Basic static capacity Viscous drag coefficient Diameter Pitch diameter Roller or ball diameter Complete elliptic integral of second kind Force, friction force Centrifugal force Gravitational constant Distance between center of contact ellipse and center of spinning
Units mm (in.)
mm (in.) mm (in.) mm (in.)
mm (in.) 483
484
Symbol
FRI~TIONIN RO
Description
Units
Mass moment of inertia Effective roller length Moment Gyroscopic moment Bearing friction torque due to flange load Bearing fkiction torque due to load Bearing friction torque due to lubricant Mass Bearing rotational speed Roller or ball load Load per unit lengthor x ‘ / a Radius of curvature of contact surface Surface area y ‘/b Rolling line location on x ’ axis Cage torque Surface velocity Surface velocity Width of laminum Width of cage rail Lubricant flow rate through bearing Distance in the x direction Distance in they direction Distance in the x direction Contact angle I) cos a/dm Lubricant viscosity Angle Ellipticity parameter Coefficient of fkiction Kinematic viscosity Radius Lubricant effective density Lubricant density Normal stress Shear stress Angle Azimuth angle
kg mm2 (in. * lb * sec2) mm (in.) N mm (in. * lb) N * mm (in. lb) N mm (in. lb) N mm (in. lb) N mm (in. lb) kg(lbsec2/in.) rPm N(W N/mm (lb/in,) mm (in.) mm2 (in.2) mm (in.) N mm (in. lb) mm/sec (inhec.) mm/sec (inhec.) mm (in.) mm (in.) 9
cm3/min (gallmin.) mm (in.) mm (in.) mm (in.) rad, O cp (lb sec/in.2) rad centistokes mm (in.) g/mm3 (1b/ina3) g/mm3 (lb/in.3) N/mm2 (psi) N/mm2 (psi) rad rad, O
GENE
Symbol cn)
sz CG CL CP CR drag i
n m 0
R S u X X’
Y Y’ 2
z‘
h
Description Rotational speed Ring rotational speed
Units radlsec rad/sec
SU~SCRIPTS Refers to cage Refers to cage land Refers to cage pocket Refers to cage rail Refers to viscous friction on cage Refers to gyroscopic motion Refers to inner raceway Refers to outer or inner race.way?o or i Refers to orbital motion Refers to outer raceway efers to rolling motion Refers to spinning motion Refers to viscous friction on rolling element Refers to x direction Refers to x’ direction Refers to y direction Refers to y ’ direction Refers to z direction Refers to z’ direction Refers to laminum
It is universally recognized that friction due to rolling of nonlubricated surfaces over each other is considerably less than the dry friction encountered by sliding the identical surfaces over each other. Notwithstanding the motions of the contacting elements in rolling bearings are more complex than is indicated by pure rolling, rolling bearings exhibit considerably less friction than most fluid filmor sleeve bearings of comparable size and loadcarrying ability. A notable exception to the foregoing generalization is, of course, the hydrostatic gas bearing; however, such a bearing is not selfsustaining, as is a rolling bearing, and it requires a complex gas supply system. Friction of any magnitude represents an energy loss and causes a retardation of motion. Hence frictionin a rolling bearing is witnessed as a temperature increase and may be measured as a retarding torque, The sources of friction in rolling bearings are manifold, the principal sources being as follows:
FRI~TIONIN ROLL e
Elastic hysteresis in rolling
. Sliding in rolling elementraceway contacts due to a geometry of .
. . .
contacti~gsurfaces Sliding due to deformation of contacting elements Sliding between the cage and rolling elements and, for a landriding cage, sliding between the cage and bearing rings Viscous drag of the lubricant on the rolling elements and cage Sliding between roller ends and inner andlor outer ring flanges Seal friction
These sources of friction are discussed in the following section.
As a rolling element under compressive load travels over a raceway, the material in the forward portion of the contact surface, that is, in the direction of rolling, will undergoa compression whilethe material in the rear of the contact is being relieved of stress. It is recognized that as load is increasing, a given stress corresponds to a smaller deflection than when load is decreasing (see Fig. 14.1). The area between the curves of Fig. 14.1is called the hysteresis loop and represents an energy loss.(This Load ~ncreasing
Stress
I
14.1. Hysteresis loop for elastic material subjected to reversing stresses.
is readily determined if one substitutes force times a constant for stress and deformation times a constant for strain.) Generally, the energy loss or friction due to elastic hysteresis is small compared to other types of friction occurring in rolling bearings. Drutowski 1114.11 verified this by e~perimentingwith balls rolling between flat plates. Coefficients of rolling friction as low as 0.0001 can be determined from the reference [14.1] data for 12.7 mrn (0.5 in.) diameter chrome steel balls rolling on chrome steel plates under normal loads of about 356 N (80 lb.) Drutowski E14.21 also demonstrated the apparent linear dep of rolling frictionon the volume of significantly stressed material references [14.13 and [14.21 Drutowski further demonstrated the dependence of elastic hysteresis on the material under stress andon the specific load in the contact area.
Nominally, the balls or rollers in a rolling bearing are subjec perpendicular to the tangent plane at each contact surface. these normal loads the rolling elements and raceways are deformed at each contact, producing, accordingto Hertz, a radius off curvature of the contacting surface equal to the harmonic mean of the radii of the contacting bodies. Hence fora roller of diameter D,bearing on a cylindrical raceway of diameter d,, the radius of curvature of the contact surface is (14.1) ecause of the deformation indicated above and because of the rolling motion of the roller over the raceway, which requires a tangential force to overcome rolling resistance, raceway material is squeezed up to form efo~ard portion of the contact, as shown in Fig. 14.2. A epression is formed in the rear of the contact area. Thus, w
.2. Rollerraceway contact showing bulge due to tangential forces.
488
~RICTIONlN R O L L ~ GB
~
~
an additional tangential force is required to overcome the resisting force of the bulge.
~acroszidingdue to IzoZZing ~ o t i o ~In. Chapter 8, it was demonstrated that sliding occurs in most ball and roller bearings simply due to the macro or basic internal geometry of the bearing. Theoretically, if a radial cylindrical roller bearing had rollers and raceways of esactly the same length, if the rollers were very accurately guided by frictionless flanges, and if the bearing operated with zero misalignment, then sliding in the rollerraceway contacts wouldbeavoided. In the practical situation, however, rollers andlor raceways are crowned to avoid “edge loading,’’ and under applied loadthe contact surface is curved in theplane passing through the bearing axis of rotation and the center of “rolling” contact. Since pure rolling is defined by instant centers at which no relative motion of the contacting elements occurs, that is, the surfaces have the same velocities at such points, then even in a radial cylindrical rollerbearing, only two points of pure rolling can esist on the major axis of each contact surface. At all other points, sliding must occur. In fact, the major source of 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 esuded by a grease. Some rolling bearings are lubricated by less viscous fluids and some by dry ) . theformer cases,the lubricants such as molybdenum disulfide( ~ o S ~In coefficient of sliding friction in the contact areas, that is, the ratio of the shear force caused by sliding to the normal force pressing the surfaces together, is generally significantly lowerthan with “dry”film lu~rication. For oil and greaselubricated bearings, it was shown in Chapter 13 that the sliding friction, and hence traction, in a contact can be considi a ~ ered as composed of three components: friction due to ~ e ~ t o nfluid lubrication, friction due to a limiting shear condition, and Coulomb friction due to asperityasperity interactions. M e n the film parameter A > 3, the Coulomb friction com~onentvirtually disappears since asperities do not contact. ~ a c r o s Z i d i nDue ~ to Gyroscopic Action. In Chapter 7, for anplarcontact ball bearings, ball motions inducedby gyroscopic moments were discussed. This motion occasionspure sliding in directions collinear with the major ases of the ballraceway elliptical areas of contact. Jones L14.31 considered that gyroscopic motion can be prevented if the friction coefficient is sufficiently great; for example, as stated in Chapter 7, 0.060.07. In Chapter 12, however, it was demonstrated that for bearings operating in the full or even partial EHL regime, lubricant film thick
G
S
nesses are sufficient to cause substantial separation of the balls and raceways, and sliding motions occur overthe contacts in the rolling direction. In thepresence of the separating lubricant film, therefore, the gyroscopic moments are resisted by friction forces whosemagnitudes depend on the rates of shearing of the lubricant film in the direction of the gyroscopic moments. Therefore,ball gyroscopic motion must also occur irrespective of the magnitude of the coefficient of friction. It is further probable that gyroscopic motion also occurs in ball bearings operating with dryfilm lubrication. Palmgren [14.4] called the gyroscopic motioncreep and inexperiments he found that if the tangential force attitude was perpendicular to the direction of rolling, the relationship of the angle ,6 by which the motion of a ball deviates from the direction of rolling can be shown to be a function of the ratio of the mean tangential stress to the mean normal stress. Figure 14.3 shows for lubricated surfaces that creep becomes inl a ~ 0.08. Palmgren further deduced as a consefinite as 2 ~ ~approaches quence of creep that a ball can never remain rolling between surfaces that form an angle to each other, regardless of the minuteness of the angle. The ball, while rolling, alwaysseeks surfaces that are parallel. eynolds [14.5]first referred to microslip whenin his experiments involving the rolling of an elastically stiff cylinder on rubber he observed that since the rubber stretched in the contact zone, the cylinder rolled forward a distance less than its circumference in one complete revolution about its axis. The classicaldemonstration of the microslip or creep phenomenon was developed in two dimensions by Poritsky [14.6]. He considered the action of a locomotive driving wheel as shown in Fig.
.3. Angle of deviation from rolling motionfor a ball subjected toa tangential load perpendicular to the direction of rolling.
FRI~TIONIN ROLL
14.4. The normal load between the cylinders was assumed to generate a parabolic stress distribution over the contact surface. Superimposed on the Hertzian stress distribution was a tangential stress on the contact surface, as shown in Fig. 14.4.Using this motion Poritsky demonstrated the existence of a “locked” region over which no slip occurs and a slip region of relative movement in a contact area over which it has been historically assumed that only rolling had occurred. Cain E14.71 further determined that in rolling the “locked” region coincidedwith the leading edge of the contact area, as shown in Fig. 14.5. In general, the “locked region” phenomenon can occur only when the friction coefficient is very high as between unlubricated surfaces. Heathcote “slip” is very similar to that which occurs becauseof rolling elementraceway deformation. Heathcote i14.91 determined that a hard ball ‘6rolling”in a closely conforming groove can roll without sliding on two narrow bands only. Ultimately, Heathcote obtained a formula forthe “rolling” frictionin this situation. Heathcote’s analysis takes no account of the ability of the surfaces to elastically deform and accommodate the difference in surface velocities by differential expansion. Johnson i14.81 expanded on the Heathcote analysis by slicing the elliptical contact area into differential slabs of area, as shown in Fig. 14.6, and thereafter applying the Poi*itsky analysis in two dimensions to each slab. Generally, Johnson’s analysis using tangential elastic compliance demonstrates a lower coefficient of friction than does the Heathcote analysis, which as
Rolling under action of surface tangential stress (reprinted from t14.81 by permission of American Elsevier Publishing Company). e
Curve of complete
X
x
14.5. ( a ) Surface tangential transactions; ( b ) surface strains; (c) region of traction and microslip (reprinted from 114.81 by permission of American Elsevier Publishing Company).
14.6. Ballraceway contact ellipse showing “locked” region and microslip regionradial ball bearing(reprinted from 114.81 by permission of American Elsevier Publishing Company).
FRIC~IONIN R
O
~ E3~
G GS
sumes sliding rather than microslip. Figure 14.7 shows the "locked" and slip regions that obtain within the contact ellipse. Greenwoodand Tabor [14.10] evaluated the rolling resistance due to elastic hysteresis. It is of interest to indicate that thefrictional resistance due to elastic hysteresis as determined by Greenwood and Tabor is generally less than that due to sliding if normal load is sufficiently large.
Owing to its orbital speed, each ball or roller must overcome a viscous drag force imposed by the lubricant within the bearing cavity. It can be assumed that drag caused by a gaseous atmosphere is insi~ificant;however, the lubricant viscous drag depends upon the quantity of the lubricant dispersed in thebearing cavity. Hence, the effective fluidwithin the cavity is a gaslubricant mixture having an effective viscosity and an effective specific gravity. The viscous drag force acting on a ball as indicated in [14.111 can be approximated by
(14.2)
where is the weight of lubricant in the bearing cavity divided by the free volume within the bearing boundary dimensions, Similarly, for an orbiting roller
rollin
.
Semiellipseof contact showing sliding lines and rolling point (reprinted from L14.81 by permission of American Elsevier Publishing Company).
493
SOURCES OF FRICTIO~
(14.3)
The drag coefficients c, in equations (14.2) and (14.3) can be obtained from reference [14.12] among others.
et~een the Cage and Three basic cagetypes are used in ball and roller bearings: (1)ball riding (BR) or roller riding (RR),(2) inner ringland riding (IRLR), and (3) outer ring land riding (ORLR). Theseare illustrated schematically in Fig. 14.8. BR and RR cages are usually of relatively inexpensive manufacture and are usually not used in critical applications. The choice of an TRLR or ORLR cage depends largely upon the application and designer preference. An IRLR cageis driven by a force betweenthe cage rail and inner ring land as well as by the rolling elements. ORLR cage speedis retarded by cage raillouter ring land drag force. The magnitude of the drag or drive force between the cage rail and ring land depends upon the resultant of the cagelrolling element loading, the eccentricity of the cage axis of rotation and the speed of the cage relative to the ring on which it is piloted. If the cage raillring land normal force is substantial, hydrodynamic short bearing theory E14.131 might be used to establish the friction force FcL.For a properly balanced cage and a very small resultant cagelrolling element load, Petroff's law canbe applied; for example, (14.4)
where d2is the larger of the cage rail and ringland diameters and d, is the smaller. Inner ring land riding
Bal I riding
14.8. Cage types.
Outer ring land riding
ets At any given azimuth location, there is generally a normal force acting between the rolling element and its cage pocket. This force can be positive or negative depending upon whether the rolling element is driving the cage or vice versa. It is also possible for a rolling element to be free in the pocket with no normal force exerted; however, this situation will be of less usual occurrence. Insofar as rotation of the rolling element about its own axes is concerned, the cage is stationary. Therefore, pure sliding occurs between rolling elements and cage pockets. The amount of friction that occurs thereby depends on the rolling elementcage normal loading, lubricant properties, rolling element speeds, and cage pocket geometry. The last variable is substantial in variety. Generally, application of simplified elastoh~drod~amic theory should sufficeto analyze the f~ctionforces.
es In a tapered roller bearing and in a spherical roller bearing having asymmetrical rollers, concentrated contacts always occur between the roller ends and the inner (or outer) ring flange owingto a force componentthat drives the rollers against the flange. Also, in a radial cylindrical roller bearing, which can support thrust load in addition to the predominant radial load by virtue of having flanges on both inner and outer rings, sliding occurs simultaneously between the roller ends and both inner and outer rings. In these cases, the geometries of the flanges and roller ends are extremely influential in determining the sliding friction between those contacting elements. The most general case for roller endflange contact occurs, as shown in a spherical roller thrust bearing. The different types of llustrated in Table 14.1 for rollers having sphere ends. 141 indicates that optimal frictional characteristics are point contacts between roller ends and fla al. [14.15] studied roller end wear criteri cylindrical roller bearings. They found that increasing roller corner ra* runout tends to increase wear. Increasing roller end clearance and ratio also tend toward increased roller wear, but, are of lesser conse~uencethan roller corner radius runout.
integral seal on a ball or roller bearing generally consists of an elasartially encased in a steel or plastic carrier. This is shown in Fig. 1.16.
~
O
~ OF C FRICTION E ~
14.9. Contacttypesandpressureprofilesbetweensphereendrollers flanges in a spherical rollerthrust bearing.
and
14.1. Roller EndFlange Contact vs Geometry
Geometry Flange a b c
Portion of a cone Portion of sphere, Rf= R,, Portion of sphere, Rf FYs
Type of Contact Line Entire surface Point
“Rf is the flange surfaceradius of curvature; R,,, is the roller end radius of curvature.
The elastomeric sealing element bears either on a ring “land” or on a special recesg in a ring. In either case, the seal friction normally substantially exceeds the sum total of all other sources of friction in the bearing unit. The technology of seal friction depends frequently on the specific mechanical structure of the seal and on the elasto~eric properties. See Chapter 17 for some information on integral seals.
FRICTION IN ROLL IN^ ~
E
~
I
N
The sliding that occurs in thecontact area hasbeen discussed onlyqualitatively insofar as determination of friction forces is concerned. The analysis performed in Chapter 9 to evaluate the normal load on eachball and the contact angles took no account of friction forces in the contact other than to recognize the necessity to balance the gyroscopic moments which occur in angularcontact and thrust ball bearings. Of the many components that constitute the frictional resistance to motion in a ballraceway contact sliding is the most significant. It is further possible for the purpose of analysis to utilize a coefficient of friction eventhough the latter is a variable. Coefficient of friction in this section will be handled as a constant defined by r
(14.5)
E”=“&
where r is surface shear stress and CT is the normal stress. Jones [14.3] first utilized the methods developed in this section. In the ballraceway elliptical contact area of a ball bearing consider a differential area of d S as shown by Fig. 14.10. Thenormal stre.ss on the differential area is given by equation (6.43): (6.43)
In accordance with a sliding friction coefficient of friction p, the differential friction force at d S is given by
( ~] ) 2
[l  ( ~ ) z
v2
dS
(14.6)
The friction force of equation (14.6) has a component in the y direction = dF cos #; therefore the total friction force in the y direction due to sliding is
[
1

(z)2

cos # dy dx
( ~ ) 2 ] v 2
(14.7)
1
ICTION F O R ~ E SANIl MO
NTS IN ROLLING E L E ~ N T  ~ C E W ~ ~
C O ~4A C T S
X
F I ~ 14.10. ~ E Friction force and sliding velocities acting on contact surface.
area dS of the elliptical
Similarly, the friction force in the x direction is
[ ( 1
3PQ
$2

~)']"
sin
4 dy dx
(14.8)
Since the differential friction forcedF does not necessarily actat right angles to a radius drawn from the geometrical center of the contact ellipse, the moment of dF about the center of the contact ellipse is
or dM,
=
( x 2 + y2)v2COS (4  6 ) dF
(14.10)
in which
(14.11)
4
FRIC~IONIN ROLL^^
The total frictional moment about the center of the contact ellipse is, therefore,
x
[
1
)2(:

(~)2]1’2
cos (#  6) dy dx
(114.12)
Additionally, the moment of dB’ about the y ’ axis is (see Figs 5.4, 8.13, 8.14, and 14.10)
(14.13)
Integration of equation (14.13) over the entire contact ellipse yields
(14.14) Similarly, the frictional moment about an axis through the ball center perpendicular to the line defining the contact angle which line lies in the x‘ x’ plane of Fig. 5.4 is given by
(14.15)
Referring once again to Fig. 14.10, there are associated with area dS sliding velocities uy and u, according to equations (8.31)and (8.32) and (8.36) and (8.37) for the outer and inner raceway contacts, respectively. Also, there is associated with each contacta spinning speed cos according to equations (8.33) and (8.38). these velocities determine the angle # (see Fig. 14.10) such that
~ I C T I O FORCES ~ AND
~
0IN ROLLING ~ E L~E CF,
=
tan"
~ ~ s ~ C EC O ~ ~AT A YCT~
pus sin 8  u, pus cos 8
+ uy
(14.16)
Therefore,
(14.17) Themoments acting on a ball, bothgyroscopic and frictional, are shown in Fig. 14.11.My!and Mztmay be calculated from equations (5.35) and (5.36), respectively. Thesummation of the moments in each direction must equal zero; therefore,
MROsin a,
+ MsoCOS a, + Mzt + M E sin ai  Msi COS ai = 0
(14.18)
MsOsin a0 + MRiCOS ai + Msi sin ai= 0
(14.19)
MRo COS a,

The forces acting on a ball can be disposed as in Fig. 14.12. Fzr is the ball centrifugal force definedby equation (5.34).Fy and F, are defined by
14.11. Gyroscopic and frictional moments acting on a ball.
FRICTION IN ~ O L L I N B ~~ I N ~ S
Cent~fugal,normal and frictional forces acting on a ball. Note:
Fyiact normal to the plane of the paper.
equations (14.7) and 14.8), respectively. From Fig. 14.12 it can be seen that equation (14.20) becomes
and
Fyi+ Fyo = 0
(14.22)
Note also that equations (14.18)and (14.19)can be combined to yield
+ cos a,) + Ms,(cos a,  sin a,) + 2M~(sinq + cos ai) ~ s i ( c o ai s  sin ai)+ Mzr = 0
Mr0(sina,
(14.23)
~ i m p l i f ~ assumptions ng may be madeat this point forrelatively slow speed bearings such that ball gyroscopic moment is negligible and that outer raceway control is approximated. Although the latter is not nec
F R I C T I O ~FORCE$
N"$ IN ROLLING ELE
essarily true of slow speed bearings, the resultof calculations using these assumptions will permit the investigator to obtain a qualitative idea of the sliding zones in theballraceway contacts and an order of magnitude idea of friction in the contacts. Moreover, Q,, Qi, cy,,and ai may be determined by methods of Chapters 7 or 9. Therefore, to calculate the frictional forces and moments in the contact area, one needs only to determine the radii of rolling r: and r f . In Chapter 8 it was demonstrated that pure rolling can occur at most at two points in the contact area. If spinning is absent at a raceway contact, then all points on lines parallel to the direction of rolling and passing through the aforementioned points of pure rolling roll without sliding. The sliding velocities uyo or uyi are defined by equations (8.25) and (8.31), respectively;the distribution of sliding velocity onthe contact surf'ace is illustrated by Fig. 14.13. As in Fig. 14.13 the lines of pure rolling lieat x = fr ea. Then the frictional forces of sliding are distributed as in Fig. 14.14. Using equation (14.6) to describe the diff'erential frictional force dF, it can be seen that the net sliding frictional force in the direction of rolling at a raceway contact is
.13. Distribution of sliding velocity on the elliptical contact surface for negligible gyroscopic motion and zero spin.
5
~ICTION IN ROLLING B E ~ ~ G S
Performing the integration of equation (14.24) yields FY = k p Q ( 3 c 
c3
 1)
(14.25)
Thus, for a given value Fy obtainable from equation (14.77, the value c may be established. Referring to Fig. 8.13 or 8.14, it can be seen that the radius of rolling is given by
2
rl
=.{[l
112
(~)]

( ~ ) ] [ ( ~ )(x)] } 2
[l
2
112
+
112

(14.27)
The rolling moment about the U axis through the center of the ball as ~eterminedfrom Fig. 8.13 or 8.14 is ~~~
X
14.14.
Distribution of sliding friction forces dFy on the elliptical contact surface.
F R I C T I O ~FORCES AND ~
O
~ LN~ROLLLNG T $ E L E ~ ~ 
Rearranging equation (14.28) and converting to integral form yields
x
(id" +[ (
{[ [ (~)"]'" (~)']}" dy dx { (~)"1" (~)"]"'
/+""!""'1 b"1/[2 l  ( ~ / a ) ~ ] ~ / ~
~
)
2
u
_.
(~)2]"
[l 
__
__

(:)2
( ~ ) 2 ] v 2
+b[l(~/a)~]~~
[1
b[l(~/a)~]~~


[l 
(14.29) erforming the indicated integration and rearranging yields
sin 2r2

4r2) + sin 2r1 (sin 4r116 sin22 sin rl 
Q (nin 2r1  2 sin rl
[
(
~
)
2
1
sin2 rl]} (3c  e3  1)
(14.30) in which sin
a rl = R
ea sin T2 = 
R
(14.31) (14.32)
It is now possible to calculate r i and rf . The steps are as follows: Assume r(:= rf = r and calculate centrifugal force F, from equation (9.18);o,/w is determined from equation (8.63) or (8.64). It is recognized e
in thecalculation of and wm/wthat pitch diameter is a variable defined as follows (see Fig. 9.2): d m j = d m + 2Mf0

0.5
+ sojrcos aoj 
(fo

0.5 (14.33)
wherein
sojis obtained from equation (9.12). te Fyifrom equation (14.7), ngle # is calculated by us by using equations ( 8 . 3 ~ ) generally necessary to termined Fyi,calculate late c by using e~uation ed if wyo at x = 0 is positive, ermined c , calculate
~ t at each ndition to be satis~edis that the i n ~torque ball location must equal the output torque,
(14.34)
If equation (14.34) is not satisfied, a new value of ci, that is, r f , is assumed and the process is repeated until equation (14.34) is satisfied. f the motion of a raceway relative to the ball was merely a spinning about the normal to the center of the contact area, all other relative surface velocities being reduced to zero, the magnitude of the spinning moment as determined from equation (14.12) for # = 8 is given by
in which & is the complete elliptic integral of the second kind with modulus [1  ( l ~ l aw2. ) ~ For ] the condition of outer raceway control M8, as calculated from equation (14.23) for rolling and spinning is less than M8 as calculated from (14.35) for the outer raceway contact with only spinning motion.
IN ROLLING E ~ ~ ~  ~ CC O E~ A ~C T AS Y
FRICTION FORCES ANI) ~0~~~
31. For the 218 angularcontact ball bearing of Example 9.1, estimate thefriction torque dueto spinning aboutthe axis normal to the inner raceway contact area for the condition of 22,250 N (5000 lb) thrust load and 10,000 rpm shaft speed. Assume a coefficient of friction equal to 0.03. ai= 48.8"
D
=
Fig. 9.4
22.23 mm (0.875
d m = 125.3 mm (4.932
in.)
Ex. 2.3
in.)
Ex. 2.6
Qi = 1788 N (401.7 lb)
Fig. 9.6
Ex. fi = 0.5232
2.3
(2.27) 22.23 cos (48.8") 125.3
_.
=
0.1169 (2.28)
1 22.23 0.5232 =
1  0.1169
0.1058 mm' (2.690 in?) (2.29)

L'0.5232 + (2 X 0.1169)/(1  0.1169) 4  1/0.5232 + (2 X 0.1169)/(1  0,1169)
=
0.9244

From Fig. 6.4, a* = 3.47; b?
=
0.433
( ~ ) 113
ai = 0.0236ar =
0.0236
=
2.101 mm (0.0827 in.)
X
3.47
X
(6.39)
~RICTIONIN ROLL IN^ B K.
a? b;
=1
" 3.47
0.433
 8.01
( ~ ) v3
b;
=
0.433 = Gi =
MSi =
=
(6.45)
26,
(3.1416 X 8.01
1.022 3pQiai6i 8
(14.35)
43.19 N mm (0.382 in. * lb)
Thus far, the solution of the friction force and moment equilibrium equations has assumed that outer raceway control wasappro~imated.A more general solution was achievedby Harris [14.16] for a thrustloaded angular contact ball bearing operating with Coulomb friction.in theballraceway contacts. In this case, the forces and moments acting on a ball are shown in Fig. 14.15. Gyroscopic motion about the axis y ' is assumed negligible and the contact ellipseis divided into two or three sliding zones as shown in Fig. 14.16.
Now for the raceway contacts as shown in Fig. 14.16,
(14.36)
md~ereg = x'la,, t = y'lb,, Tnl, and Tn2define rolling lines, n refers to inner or outer ballraceway contact, that is, n = o or n = i; and unthe pressure at any point in the contact ellipse is given by
"
Bearing Axis
14.15. Forces and moments acting on a ball.
14.16. Contact areas, rolling lines, and slip directions.
8
FRICTION IN R O ~ I N G B
(14.37) ~ubstitutingequation (14.37) into (14.36) and integrating yields
Using Figures 8.13 and 8.14 to define the radii r, from the ball center to points on the inner and outer ballraceway contact areas, the equations from frictional moments are
Mxl, = 2pa,b,c,
cnrnC O S ( ( X ,

cnrncos(a,
+ 6,)dqdt
cnrncos(a,
+ 6,)
+ 0,) dq dt
I
dt dq
n =o,i c, = 1; ci = 1 where sin
(14.39)
en = x '/rn.Using the trigonometric identity cos(a,
+ 6,)
=
cos an cos On  sin a, sin
recognizing 0, is small giving cos On

en
(14.40)
1, and integrating yields
Mx'n= 3 p ~ n D c n
x [(I n
K

5)
= 0,i; = 1, 2;
cos a,

a D'nk
c, = 1; ci = 1
c1
=
1; c2
=
"1
( 'x) 1

I}
sin  an
(14.41)
imilarly,
k=2 k=l
n
= 0, i;
c,
k z 1 .9 2 .9 c 1
=
1; ci c2 9
= 1. =“j
ig. 14.15 it can be established that four condi moment equilibrium about the x’, y’, and x’ axes mu gether with four ball position equations determined in Cha eight equations must be solved for two position variables, bearing axial deflection, and speeds mm, , there are eight e~uationsand eight of which there are three as shown r 9 and wzr.To establish the requ rmed contact surfaces as shown by considered arcs of great circles defined by
where 5 = 2f/(2f + 1)and f = r / . From Figs. 8.13 and determined that the offset of the ball center from the circle center is given by the coordinates
z =23[(45E 2
k,2,)1/21
cos a,
(14.45)
an/23.Zero sliding velocity is determined from the equations (14.46) (14.47)
Equations (14.43), (14.46),and (14.47) can be solved simultaneously to yield xkk, zkk locations at which zero sliding velocity occurs on the deformed surface circle. It can be shown that
Using the foregoing method Harris [14.16] was able to prove the impossibility of an "inner raceway control" situation, even with bearings operating with "dry film" lubrication. Moreover, a speed transition point seems to occur in a thrustloade~angularcontact ball bearing at which a radical shift of the ball speed pitchangle /3 must occur to achieve load equilibrium in the bearing (see Figs. 8.16, 8.17, 14.16, 14.17, and 14.18) Additionally, Table 14.2 shows the corresponding locations of rolling lines in the inner and outer contact ellipses for this example.
A similar approach may be applied to roller bearings having point contact at each raceway. Usually, however, roller bearings, are designed to operate in the line contact or modified line contact regime (see Chapter 6) in which the area of contact is essentially rectangular, it generally being an ellipse truncated at each end of the major axis (see Fig. 6.24). In thiscase the major sliding forces onthe contact surface are essentially parallel to the direction of rolling and are principally due to the deforBearing Design Data
Ball diameter diameter Pitch contact angle Inner raceway grove rsdius/ball diameter Outer raceway groove radiuslball diameter Thrust load per ball
8.731 m m (0.34375 in.) 48.54 rnrn (1.91 10 in.) 24.5" 0.52 0.52 31.6 N (7.1 Ib)
S 0.418
.i
ct
8
0 Shaft Speed ( rpm)
14.17. Orbit/shaft speed ratio vs shaft speed.
O
2,000
4,000
6,000
8,000
10,000
Shaft Speed (rpm)
14.18. Ball speed vector pitch angle vs shaft speed.
. Shaft Speed 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
Locations of Lines of Zero Slip in Contact Ellipses Outer Raceway
T2
TI 0.0001 0.00183 0.00129 0.00047

0.95339 0.93237 0.91449 0.89730
Inner Raceway
1
0,02975 0.00156 0.00156 0.00376 0.00627 0.01055
171
0.00605 0.00672 0.00537 0.00353 0.02995 

T2 0.92123 0.92376 0.93140 0.94272
0.00190 0.00052 0.00064 0.00077 0.00039
mation of the surface. Thus, the sliding forces acting on the contact surfaces of a loaded roller bearing are usually less complex than for ball bearings. Dynamic loading of roller bearings does not generally affect contact angles, and hence geometry of the contacting surfaces in virtually iden
INGS
tical to that occurring under static loading. use of the relatively slow speeds of operation necessitated when co anglediffersfromzero degrees, gyroscopic momentsare negligible. In any event, gyroscopic moments of any magnitude do not substantially alter normal motion of the rolling elements. In this analysis therefore, the sliding on the contact surface of a properly designed roller bearing will be assumed to be a function only of the radius of the deformed contact surface in a direction transverse to rolling. To perform the analysis, it is assumed that the contact area between roller and either raceway is .substantially rectangular and that the noress at any distance from the center of the rectangle is adequately
(6.50)
Thus, the differential friction force acting at any distance x from the center of the rectangle is given by dEij
=
n1b
Integrating equation (14.49) betweeny
=
+ b yields (14.50)
Referring to Fig. 14.19, it can be determined that the differential frictional moment in the direction of rolling at either raceway is given by
or
[
1
k)] 2
v2
[(R”  x2)1’2 
dy dx
(14.52)
in which R is the radiusof curvature of the deformed surface. Integrating equation (14.52) with respect to y between limits y = rfr: b yields
~ I C T I O NFORCES AND
~
0IN ROLLING ~ E L ~E
~ ~  ~sC E W A Y
~ O ~ A C T S
t " " "
" " " "
14.19. Rollerraceway contact.
(14.53)
Because of the curvature of the deformed surface, pure rolling exists at most at two points x = ~?t:(c1)/2 on the deformed surface; the radius of rolling measured from the roller axis of rotation is r '. Thus
(14.54)
or
Fy= pQ(2c

1)
(14.55)
514
I C T I O ~IN ROLL IN^ B
Also
or
Considering the equilibrium of forces acting on the roller at the inner and outer raceway contacts (see Fig. 14.20),Fyo =  Fyi.therefore, from (14.55) assuming p,,= pi: c,
+ ci = 1
(14.58)
Furthermore, since in uniform rolling motion the sum of the torques at the outer and inner raceway contacts is equal to zero, therefore
(14.59)
14.20. Frictionforcesandmoments acting on a roller.
From Fig. 14.19, it can be seen that the roller radius of rolling is (14.60) Hence, assuming p,
{
= pi, from
2 [ R~
equations (14.57), (14.59), and (14.60):
(~)2iv2 d m

( R ,
~)}
(14.61)
Equations (14.58) and (14.61) can be solved simultaneously for c, and ci. Note that if R, and R,, the radii of curvature of the outer and inner contact surfaces respectively are infinite, the foregoing analysis does not apply. In this case sliding on the contact surfaces is obviated and only rolling occurs. Having determined e, and ci, one may revert to equation (14.55) to determine the net sliding forces Fyo and Fyi.Similarly, A I R o and A I R i may be calculated from equation (14.57).
In theanalytical development regarding rolling element and cage speeds so far, at least one location could be found in each of the rolling elementraceway contactareas that was an instantcenter; that is, at thatlocation no relative motion (sliding) occurs between the contacting surfaces. If
during bearing operation, no instant center can be found in either the inner or outer raceway contacts, particularly at the azimuth location of the most heavily loaded rolling element, then skidding is said to occur, Skidding is gross sliding of a contact surface relative to the opposing surface. Skidding results in surface shear stresses of significant magnitudes inthe contact areas. If the lubricant film generated by the relative motion of the rolling elementraceway surfaces is insufficient to completely separate the surfaces, surface damage called smearing will occur. An example of smearing is shown by Fig. 14.21. Tallian 114.171 defines smearing as a severe type of wear characterized by metal tightly bonded to the surface in locations into which it has been transferred from remote locations of the same or opposing surfaces and the transferred metal is present in sufficient volume to connect more than one distinct asperity contact. When the number of asperity contacts connected is small, it is called micro smear in^. When the number of such contacts is large enough to be seen with the unaided eye, this is called gross or macroscopic smearing. If possible, skidding is to be avoided in any application since at the very least it results in increased friction and heat generation even if smearing does not occur. Skidding can occur in high speed operation of oillubricated ball and roller bearings. Rolling element centrifugal forces in such applications tend to cause higher normal load at theouter racewayrolling element contact as compared to the inner race,wayrolling element contact at any azimuth location. Therefore, the balance of the friction forcesand moments acting on a rolling element requires a higher coefficient of friction at the inner raceway contact to compensate forthe lower normal contact load.It was shownin Chapter 12 that thelubricant film thickness generated in a fluidfilmlubricatedrollingelementraceway contact depends upon the velocities of the surfaces in contact. Moreover, considering as a simplistic case Newtonian lubrication, the surface shear stress is a direct function of the sliding velocity of the surfaces and an inverse function of the lubricant film thickness. Hence, considering equations (14.1) and (14.5), the coefficient of friction in the contact is a function of sliding speed, whichis greatest at the innerraceway contacts. Generally, skidding can be minimized by increasing the applied load on the bearing, thus decreasing the relative magnitude of the rolling element centrifugal force to the contact loadat themost heavily loaded rolling element. As will be seen in Chapter 18, this remedy will tend to reduce fatigue endurance. Therefore, a compromise between the degree of skidding allowed and bearing endurance must be accepted. Of course, by making the contacting surfaces extremely smooth, the effectiveness of the lubricant film thicknesses is improved, and skidding is more tolerable.
FRICTION I1\T ROLL
Notwithstanding, skidding is generally a high speedphenomenon caused by a difference betweeninner and outer racewayrolling element loading; it is also aggravated by any rolling element or cage loadingthat tends to retard motion. The most significant of such loadings is the viscous drag of the lubricant in the bearing cavity on the rolling elements. Therefore, a high speed bearing operating submerged in lubricant will skid more than the same bearing operating in misttype lubrication. In this case another compromise is required because, in a hi plication, a copious supply of lubricant is generally used to carry away the frictional heat generated by the bearing. Rolling elementcage friction and cagebearing ring friction as well as cagelubricant friction also affect skidding.
One of the most important applications with regard to skidding is the mainshaft angularcontact ball bearing in aircraft gas turbines. This bearing is predominantly thrust loaded, and it is therefore only necessary to divide the thrust load uniformly among the bearing balls to determine the applied load. The ball loading is shown by Fig. 14.22 for the coordinate system and eeds of Fig. 5.4.
Bearing Axis
14.22. Forces and moments acting on a ball.
The sliding velocities in they ' and x ' directions are given by equations (8.31),(8.32), (8.36), and (8.37).Thefluid entrainment velocities are given by
+ w2,qnsin(a, +
I
n = 0,i
(14.62)
where wn = cn(wm  fz,), e, = 1, ci = 1, c3 = sin'(x;/r,), (9, = tan' (X/Z), and X and is are given by equations (14.44) and (14.45). From equation (13.3~0,it can be seen that at every point along the x' axis of the contact ellipses (14.63) Using (14.63),the frictional shear stresses can be numerically evaluated at every pointin thecontact areas. It is important to determine lubricant viscosities at the appropriate temperatures. For calculational accuracy, it is necessary to estimate temperature of the lubricant at the inlet to the contact, and in the film separating the rollingsliding components. For assumed contact loading, the frictional forces acting in thecontact areas are given by
FXln= anbn J1
J"
rxln d q dt
n
= 0,i
(14.65)
The moments due to shear stresses in thecontact areas are given by
52
~ R I ~ T I OIN N ROLL~NG€3
where rn = Dpn. Hence the equations of force and moment equilibrium are sin a,
0 + Fxtocos a,  Fa z
n=i
cos an  Fxtnsin an) Fc = 0 n=o
x
n=i
en(&, sin an + Fxrncos an) = 0
(14.69)
n =o,i co = I; ci = "I.
(14.70) (14.71)
n=o
x
n=i
cnFyIn+ Fv = 0
(14.72)
x Mzln
0
(14.73)
Myln  Mgyl = 0
(14.74)
Mzln  Mgz Cage for hig~an~larcontact bearing (nylon 6,6), (d) Phenolic cage for precision ball bearing. I
his list indicates essential differences between polymericand metallic cage materials. Lubricant compatibility is rarely a factor, and loss of physical properties does not occur within bearing operation temperatures with metals. Cage design dependson the specific polymer usedin a more intimate fashion than when steel or brass is used. ~ o ~ ~ n ~p ees for r Cages. i ~ Fabricreinforced phenolic resin cages have been used for many years in highspeed bearing applications where decreased cage mass is a benefit. The low density of the material, approximately 15% that of steel, results in a low cage mass. The centrifugal force on a phenolic cage is consequently only 15%of the force acting on eds centrifugal force causes a cage to spread cage therefore offers better dimensional stabilof a phenolic resin, however, is limited to bearing xceed temperatures of 100°C (212" ther disadvantage with the phenolic resin operations to obtain the final shape. Other resins, discussed in the following p a r a ~ a p h scan , into a final shape directly, thus reducing process cost particularly nylon 6,6, have replaced phenolic in applications. The nylon 6,6 (~olyamide6,6) resin is the most wid bearing cages. It provides a low material price, desir erties, and low processing costs in one product. The material is constructed of aliphatic linkages connect polymer of molecular weight between synthesized from carbon hexamethyle adipic acid, both of which have six carbons, hencethe 6,6 designation.
is se~icrystallineand the ses dimensional
is often used with the resin at levels of 25% fill. The glass fiber gives better retention of strength and toughness at high temperatures, but with loss of flexibility. Rolling bearings selected from manufacturers’ catalogs are designed to operate in wide varieties of applications. Therefore, the strength/ toughness properties afforded to nylon 6,6 cagesby glassfiber reinforcement are required for bearing series employing such cage material. Figure 16.26from[16.32] illustrates the endurance capability of 25% glassfiberfilled nylon 6,6as a function of operating temperature. InFig. 16.26, the “black. band” indicates the spread determined with various lubricants. The lower edgeof the band is applicable for aggressive lubricants such as transmission oils (with EP additives), while the upper edge pertains to mild lubricants such as motor oils and normal greases. Table 16.5 from [I6321 indicates the strength, thermal, chemical, and structural properties of this material in the dry and conditioned states. The conditioned state is that in which some water has been adsorbed. comparison of Fig. 16.26 with Table 16.5, it can be seen that the permissi~Zeoperating temperat~reof 120°C (250°F) correspondsto a probable endurance of approximately 5000 to 10,000 hr dependingupon lubricant type. This refers to continuous operation at 120°C (250°F); operation at lesser temperatures will extend satisfactory cage performance for greater duration. ers
re
A variety of hi~htemperatureresins with and without glassfiber fill have been evaluated for use as cage materials. Included in the list are
20
.
0
100 140 T ~ ~ p e r a t u“C r~,
180
220
Life expectancy vs operating temperature for nylon 6,6 with 25% glassfiberfill (from (16.321).
SEAL
S
polybutylene terephthalate (P T),polyethylene terephthalate ( polyethersulfone (PES), polyamideimide (PAI), and etherketone (PEEK). Of these materials only PES and PEE onstrated sufficient promiseas high temperature bearing cage materials; these materials are discussed in further detail below. ~OZyet~ers~Z is~ao hightemperature ~e thermoplastic material with good strength, toughness, and impact behavior for cage applications. The resin consists of diary1 sulfone groups linked together by ether The structure is wholly omatic, providing the basis for excelle temperature properties. ing thermoplastic, it is processible using conventional molding equipment. This allows direct part production; that is, without subsequent machining or finishing. In lubricanttemperature exposure tests theresin has performed well to 170°C (338" is suitable for applications using petroleum and silicone lubricants; however, there are some problems with polymer degradation after exposure to esterbased lubricants and greases. The properties of are also Table 16.5; it can be seen that PES is not as strong as nylon it is desired to use a "snapin" type assembly of balls or rollers iececage as illustrated in Fig. 16.25, this somewhat lesser strength can result in crack formation during assembly of the bearing. ~ o z y e t ~ e r  e t ~ e r ~ise taowholly ~e aromatic thermoplastic that shows excellent physical properties to 250°C (482°F).It is particularly good for cage applications because of its abrasion resistance, fatigue strength, and toughness. It is a crystalline material and can be injection molded. Lubricant compatibility tests show excellent performance to 2QQ°C(3 and above. Tests also indicate antiwear performance equa than nylon 6,6. Table 16.5 compares the properties of PE of PES and nylon 6,6. The only known drawbackto the extensive use of PEEK as a bearing cage material is cost. This currently restricts its use to specialized applications. See [16.31],
To prevent lubricant loss and contaminant ingress, manufacturers provide bearings with sealing. The effectivenessof the sealing has a critical effect on bearing endurance. When choosing a sealing arrangement for a bearing application, rotational speed at the sealing surface, seal friction and resultant temperaturerise, type of lubricant, available volume, environmental contaminants, misalignment, and cost must all be considered. A bearing can be protected by an integral seal consisting of an elastomeric ring with a metallic support ring, the elastomer riding on an
inner ring surface (see Fig. 17.14), or by a stamped s ~ i e Z of~mild steel staked into the outer ring and approaching the inner ringclosely but not in intimate contact with it (see Fig. 17.13). hields cost less and do not increase torque for the bearing in opera. This design is usefulforexcluding s particulate contamination (150 pm). used with greased bea , it is used in bearings lubricated b ds that must pass through the bearing. The seal configuration is more expensive because of design and mate sign, it adds to bearing friction torque to a are used in greased bearings when mois tamination must be excluded, Theyare also the best choice for minimiz
ecause of the prevalence of elastomeric seals in rolling bearings, a vaety of materials has been developed to meet the requirements of difmportant properties of elastomeric seal materials include lubricant compatibility, high and lowt~mperatureperformance, wear resistance, and frictional characteristics. Table 16.6 summarizes physical properties, and Table 16.7 lists general application guidelines. In thefollowing discussionof elastomeric types, it is important to note that compounding variations starting with a particular elastomer type can lead to products of distinct properties. The general inputs to a formulated compound may be taken as follows: Elastomerbasic polymer that determines the ranges of final product properties Curing agents, activators, accezeratorsdetermine degree and rate of elastomeric vulcanization (crosslinking) PZasticizersimprove flexibility characteristics and serve as processing aids ~ntio~i~antsimproveantifatigue and antioxidation properties of product “Nitrile” rubber represents the most widely used elastomer for bearing seals. This material, consisting of copolymers of butadiene and acrylonitrile, is also knownas Buna N and NBR. Varying the ratio of butadiene to acrylonitrile has a major effect on the final product properties. The general polymer reaction can be represented as
.I
a
x
0
c,
0
Lo dc
I
0 00
0 c,
Lo cu cr3 0
dc 0 0
u
0
cr3 0 I
dc
Lo
0 0 c,
cu 0 I
dc
m
cu cu 0 u 0
dc I 0 0 0 c;,
cjr
I
0 00
m
0
u
0
dc 0
I
dc
0 0 c,
Lo dc
0
I
dc
!& w
rd
r4
SE
CH2=CHCH=CH2+CH;!=CH,
I
CN Acrylonitrile
Butadiene
Nitrile rubbers are commercially available with a range of acrylonitrile contents from 20 to 50% and containing a variety of antioxidants. Particular polymer selection will depend on lubricant lowtern requirements and thermal resistance required. rubber seal is used in many standard beari a1 cost is low compared to other elastomers oldable, which allows onestep processing of complex lip shapes. ~ubricantcompatibility with petroleumbased lubric~ntsis good for high acrylonitrile versions. This elastomer is suitable for applications to 100°C(212°F) and is therefore not indicated for hightemperature mers have been used in bearing applications. The erally based on ethyl acrylate and/or butyl acrylate, usually with an acrylonitrile comonomer present. rubbers, the higher percentage of acrylonitrile presen lubricant resistanc owever, higher acrylonitrile leve temperature properties of these rubbers. These materials are able to withstand operating temperatures up to 150°C (302°F) and, if properly formulated, show very good resistance to mineral oils and extreme pressure (EP) lubricant additives. Negative features of this material are poor water resistance, substandard strength and wear resistance for most seal applications, and high cost. Although no longer used for high temperature applications, it still is used when a low sealing force is required. Silicon rubbers are used as seal materials in some hightemperature and foodcontacting bearing applications, Silicon rubbers have a backbone structure made up of siliconoxygen linkages, which give excellent thermal resistance. A typical polymer is R
I
R
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The silicon polymer is modified by introducing different side groups, R,
into the structure in varying amounts. Typical organic substitutes are 3, the polymer is dimethylpolysiloxmethyl,phenyl, and vinyl. If an dvantages of silicon rubber seal use are hightemperature performance to 180°C (356" ) and good lowtemperature f l e ~ ~ i l i to t y 60°C he material is nontoxic and inert; hence it is chosen for food, nd medical applications. It is stable with regard to the effects of repeated high temperature. Their excellent 1ow"temperature flexibility makes these elastomers usefulforvery lowtemperature applications where sealing is required. ilicon rubbers are very expensive compared to nitrile rubbers. Lustance and mechanical strength are poor r most seal applithe whole, silicon elastomers have limit luoroelasto~ershave become increasingly popular as seal materials cellent hightemperature and lubricantcompatibilitychartypical polymer of this class is the copolymer of vinylidine fluori~eand hex uoropropylene, ~ h i c han be represented as
for bearing seal apaterials of this general type have become atures exceeding 130°C . Suitably compound fluoroelastomers w good wear resistance and water resistance for bearing seal applications. As would be expected, material cost is very high compare^ to nitrile rubbers.
Several coatings exist to improve surface characteristics o f bearing or bearing components without affecting the gross properties o f the bearing in therealm of standard bearing applications,coatings are e wear resistance, initial lubrication, sliding characteristics and cosmetic improvements. In addition, bearings operating in exnvironments o f temperature, wear, or corrosivity canbe specially
Zinc and manganese phosphate coatings are applied to finished bearings and components to provide
Increased corrosion protection by providing a porous base for preservative oils Initial lubrication during bearing runin by preventing metaltometal contacts and providing a lubricant reservoir Prepared surfaces for other surface coatingsthat is, MoS, arts are immersed in acidic solutions of metal phosphates at temperature. This produces a conversion coating integrally bonded to the bearing surface. The coated surface is now nonmetallic and nonconductive. The zinc phosphate process gives a finer structure, which may be preferred cosmetically. Themanganese phosphate process givesa heavier structure that is generally preferred for wear resistance and lubricant retention. Phosphating in itself does not provide forsubstantial improvements in rust protection. It is only when a suitable preservative is employed that full benefits are obtained.
e conversions have been used onbearings and components for Cosmetic uniformityappearance to components Lubrication during runin ust protection during extended storage lack oxide is a generic term referring to the formation of a mixture of iron oxides on a steel surface. A n advantage of the process is that no dimensional change results from the process, so tolerances can be maintained after treatment. A common approach to obtain this coating consists of treating a steel component in a highly oxidizing bath. Because the chemical process results in dissolution of surface iron, close process control is necessary to prevent objectionable surface damage. The black color is obtained from the presence of Fe&
0th electroplating and electroless disposition have long been employed in the rolling bearing industry to provide wearresistant coatings for cages. In response to aircraft bearing requirements, silver plating over a nickel or copperstruck cage is commonplace. In this case the strike metal provides an oxygen barrier t o the base metal to prevent corrosion. The silver plating offers reduced friction. Cadmium, tin, andchrome plating are also used for certain bearings and accessories.
Several coating techniques and materials somewhat reduce sliding friction and markedly improve wear and corrosion resistance in extreme environments. The techniques include physical vapor deposition ( chemicalvapordeposition (CVD)$ and specialprocess electroplating. Coating materials include titanium nitride(TiN),titanium carbide (Tic), and hard chromium. Some of these processcoating material combinations have demonstrated excellent performance on rolling contact surfaces.
In chemical vapor deposition of Tic, titanium tetrachloride is vaporized peratures in the presand allowed to react with the substrate at high ence of hydrogen and methane gas. Typically?C rocessing needs temperatures of 8501050°C (15621~22"C).Although these temperatures will promote diffusion with the substrate, the processing temperatures exceed the tempering temperatures of bearing steels requiring heat treating after coating. Postcoating heat treatment may cause dimensional distortion. This post treatment of the CVD coating diminishes the attractiveness of this process for bearing components.
The principal advantage of PVD over D is that substrate temperastrength with PVD is tures below 550°C (1022°F) are used.bond achieved with ion bom~ardmentof the substrate surface. Consequently? postcoating heat treatmentof high speed steels is not required. Therefore there hasbeen considerableinterest inapplying PVD coatings to bearing surfaces, with TiN beinga usualcoating. Excellent bonding with bearing steel and compatibilit~with a high contact stress environment have been achieved.
Super chro~eplatingtechniques have been developedthat produce coatings free of the surface cracks that characterize conventional hard chrome deposits.Increased corrosion resistance is reported for the coated bearing steel; the coating does not negatively affect the rolling contact fatigue life. Substrate temperature is below 66°C (151°F)during plating, t ,70 deforms plastically and the coating, with a reported hardness of I rather than cracking when overloaded.
CLOS~E
1
An operating rolling bearing is a system containing rings, raceways, rolling elements, cage, lubricant, seals, and ring support. In general, ball and roller bearings selectedfrom listings in manufacturers’ catalogs must be able to satisfy broad ranges of operating conditions. Accordingly, the materials used must be universal in their applicability. Throughhardened AIS1 52100 steel, nylon 6,6, lithiumbased greases, and so on, are among the materialsthat have met the testof universality for many years. Moreover, these materials as indicated in this chapter have undergone significant improvement,particularly in the past fex decades. For special applications involving extra heavy applied loading, very high speeds, high temperatures, very low temperatures, severe ambient environment, and combinations of these, the bearing system materials must be carefully matched to each other to achieve the desired operational longevity. In an aircraft gas turbine engine qainshaft bearing for example, it is insufficient that theM50 or 8450Nil bearing rings provide longterm operating capability at engine operating temperatures and speeds; rather, the bearing cage materials and lubricant must also survive for the same operating period. Therefore cages for such applications are generally fabricated from tough steel and are silver plated; nylon cages are precluded by the elevated operating temperatures andpossibly by incompatibility with the lubricant. The upper limit of bearing oper
16.27. Section through highpressure liquid oxygen fuel pump for spaceshuttle main engines (from [16.33]).
ating temperature is established by the lubricant; in most cases this is a synthetic oil according to United States military specification MILLextreme operating condition is the liquid oxygen the space shuttle main engine as shown in Fig. 1 6 . ~ 7In . this application, the bearings rotate at very high speed ~ The LOX vaporizes inthe (30,000 rpm) whilebeing Z u b r i c ~ t eby to burn up and wear notconfines of the bearing, and the bearing withstanding the initialcryogenic temperature [ 150°C ( 302”F)Iof the LOX.To achieve sufficient duration of satisfactory operation, the ball bearing cage has been fabricated from hmalon, a woven fiberglass~~ transfer es of PTFE film reinforced ~ T F E material E16.331 that Z ~ b r ~ cby from the cage pocketsto the balls. The bearing rings are fabricated from vacuummelted AISI 440C stainless steel. Target duration for bearing operation is only a few hours.
16.1. American Society for Testing and Materials, Std A.29584, High Carbon Ball and Roller Bearing Steels; Std A48579,“High Hardenability Bearing Steels.” 16.2. American Society for Testing and Materials, Std A53479, ““CarburizingSteels for AntiFriction Bearings.” 16.3. J. Braza, P. Pearson, and C. Hannigan, “The Performanceof 52100, M50, and M50NiL Steels in Radial Bearings,” SAE Technical Paper 932470 (September 1993). 16.4. E. Zaretsky, “Bearing and Gear Steels for Aerospace Applications,”NASA Technical Memorandum 102529 (March 1990). 16.5. W Trojahn, E. Streit, H. Chin, and D. Ehlert, “Progress in Bearing Performance of Advanced Nitrogen AlloyedStainless Steel,” in Bearing Steels intothe 21st Century, ed. J. Hoo, ASTM STP 1327 (1997). 16.6. H.J. Bohmer, T. Hirsch, and E. Streit, “Rolling Contact Fatigue Behavior of Heat Resistant Bearing Steels at High Operational Temperatures,” in Bearing Steels into the 21st Century, ed. J. Hoo, ASTM STP 1327, (1997). 16.7. C. Finkl, “DegassingThen and Now,” Ironand S ~ e e Z ~ a k e2632 r , (December 1981). 16.8. T. Morrison, T. Tallian, H. Walp, and G. Baile, “The Effectof Material Variables on the Fatigue Life ofAISI 52100 Steel Ball Bearings,”ASLE Trans., 5,347364 (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., 596597 (1971). 16.11. United States Steel Corp., Making, Shaping, and Treating of Steel, 9th ed., 594 (1971). 16.12. United States Steel Corp., Making, Shaping, and Treating of Steel, 9th ed., 598 (1971). 16.13. J. h e s s o n and T. Lund, “”RollingBearing Steelmaking at SKI? Steel,” Technical Report 7 (1984).
16.14. United States Steel Corp., Making, Shaping, and Treating of Steel, 9th ed., 580 (1971). 16.15. J. h e s s o n and T. Lund, “SKF Rolling Bearing SteelsProperties and Processes,” Ball Bearing J 217,3244 (1983). 16.16. American Society for Testing and Materials, Std E&81, “Standard Practice for Determining the Inclusion Content of Steel.” 16.17. J. Beswick, “Effect of Prior Cold Work on the Martensite Transformation in S M 52100,”~ e t a l l Trans. . A, 1 16.18. R. Butler, H. Bear, and T. Carter, ““Effect ofFiber Orientation on Ball Failures under RollingContact,”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 A25567,“EndQuench Test for Hardenability of Steel” (1979). 16.23.AmericanSocietyfor Metals, Atlas of Isothermal Transformation and Cooling Transformation Diagrams (1977). 16.24. T. Tallian, Failure Atlas for Hertz Contact Machine Elements, ASME Press. New York (1992). 16.25. G. Winspiar, ed., The ~anderbiltRubber Handbook, R.T. Vanderbilt, NewYork (1968). 16.26. Modern Plastics Encyclopedia, McGrawHill, New York (19851986). 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 CaseHardening Steel Provides Greater Fracture Toughness,’’Ball Bearing J, SKF, 234, 611 (July 1989). , “Turbine Engine Bearings for UltraHigh Temperatures,” Ball Bearing 34, 1215 (July 1989). 16.31. A. Olschewski,“High Temperature Cage Plastics,” Ball Bearing J, SKF, 228, 1316 (November 1986). 16.32. H. Lankamp, “Materials for Plastic Cages in Rolling Bearings,” Ball Bearing J, SKF, 227,1418 (August 1986). 16.33. R. Maurer and L. Wedeven, “Material Selection for SpaceShuttle Fuel Pumps,’’Ball Bearing J, SKF, 226,29 (April 1986). 16.34. Delta Rubber Company, Elastomer Selection Guide.
This Page Intentionally Left Blank
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Barium Carbon Calcium Fluorine Hydrogen Lithium Oxygen Phosphorous Reaction group Sulfur Viscosity index Tungsten
The primary function of a lubricant is to lubricate the rolling and sliding contacts of a bearing to enhance its performance through the prevention 45
of wear. This can be accomplished through various lubricating mechanisms such as hydrod~amic lubrication, elastohydrodynamic lubrication L), and boundary lubrication. The rolling/sliding contacts of concern are those between rollingelement and raceway, rollingelement and cage (separator), cage and supporting ring surface, and roller end and ring guide flanges. In addition to wear prevention the lubricant performs many other vital functions. Thelubricant can minimizethe frictional power lossof the bearing. It can act as a heat transfer medium to remove heat from the bearing. It can redistribute the heat energy within the bearing to minimize geometrical effects due to differential thermal expansions. It can protect the precision surfaces of the bearing components from corrosion. It can remove wear debris from the roller contact paths. It can minimize the amount of extraneous dirt entering the roller contact paths, and it can provide a damping medium for separator dynamic motions. No single lubricant or class of lubricants can satisfy all these requirements €or bearing operating conditions from cryogenicto ultrahigh temperatures, from very slowto ultrahigh speeds, and from benignto highly reactive operating environments. As for most engineering tasks, a compromise is generally exercised between performance and economic constraints. The economic constraints involvenotonly the cost of the lubricant and the method of application but also its impact on the life cycle cost of the mechanical system. Cost and performance decisions are frequently complicated because many other components of a mechanical system also need lubrication or cooling, and they might dominate the selection process.For example, an automobile gearbox typically comprisesgears, a ring synchronizer, rolling bearings of several types operating in very different load and speed regimes, plain bearings, clutches, and splines.
The selectionof lubricants is based on their flow properties and chemical properties in connection with lubrication. Additionalconsiderations, which sometimes may be of overriding importance, are associated with or retention operating temperature, environment, andthetransport properties of the lubricant in the bearing.
Liquid lubricants are usually mineral oils; that is, fluids produced from petroleumbased stocks. They have a wide range of molecular constitu
47
ents and chain lengths, giving rise to a large variation in flow properties and chemical performances. Theselubricants are generally additive enhanced for both viscousand chemical performance improvement. Overall, petroleumbased oils exhibit good performance characteristics at relatively inexpensive costs. Synthetic hydrocarbo~fluids are manufactured from petroleumbased materials. They are synthesized with both narrowly limited and specifically chosen molecular compounds to provide the most favorable properties for lubrication purposes. Most synthetics have unique properties and are made from petroleum feedstocks, but they can be made from nonpetroleum sources. Other “synthetic” fluids have unique properties and can be manufactured from nonpetroleumbased oils. These include polyglycols, phosphate esters, dibasic acid esters, silicone fluids, silicate esters, and fluorinated ethers.
Greases have two major constituents: an oil phase and a thickener system that physically retains the oil by capillary action. The thickener is normally composed of a materialwith very longtwisted and/or contorted molecules that both physically interlock and provide the necessarily large surface area to retain the oil. The resultant material behaves as a soft solid, capable of bleeding oil at controlled rates to meet the consumption demands of the bearing.
olymeric lubricants are related to greases in that these materials consist of an oil phase and a retaining matrix. They differ in one crucial point: the matrix is a true solid sponge that retains its physical shape and location in the bearing. Lubrication functions are provided by the oil alone after it has bled from the sponge. The oil content can be made higher than in greases, and a greater quantity can be installed in the void space within the bearing. This greater oil volume portends longer bearing life before all fluid is consumed by oxidation, evaporation, or leakage. The latter is particularly significant for vertical axis bearing applications. s
Solid lubricants are substituted for liquid lubricants when extreme environments such as high temperature or vacuum make liquid lubricants or greases impractical. Solid lubricants, unless melted, do not utilize the mechanism of hydrodynamic or EHL. Their performance is less predictable, and there is generally much greater heatgeneration due to friction.
Solid lubricants perform as boundary lubricants consisting of thin layers that provide lower shear strength than the bearing materials. Solid lubricants can consist of layered structures that sheareasily or nonlayered structures that deform plastically at relatively low temperatures. Graphites and molybdenum disulfide(MoS,) are common examples of materials with layered structures. Fluorides such as calcium fluoride (CaF,) are nonlayered materials that perform well at or near their melting temperatures.
Decisions in connection with the selection of lubricants must parallel decisions in connection with the supply of the lubricant to the bearing for maintaining conditions that will prevent rapid deterioration of the lubricant and bearing. h oil sump applicable to horizontal, inclined, and vertical axis arrangements provides a small pool of oil contained in contact with the bearing, as in Fig. 17.1. The liquid level in the stationary condition is arranged to just reach the lower portion of the rolling elements. Experience has shown that higher levels lead to excessive lubricant churning and resultantexcessive temperature. This churning in turn can cause premature lubricant oxidation and subsequent bearing failure. Lower liquid levels threaten oil starvation at operating speeds where windage can redistribute the oil and cut off communication with the working surfaces. Maintenance of proper level is thus very important and provision of a "sight" recommended. Oil bath systems are used at lowtomoderate speeds where grease is ruled out by short relubrication interval hot environments, or where purging of grease could cause problems. eat dissipation is somewhat better than for a greased bearing due to fluid circulation, offering im
'1. Pillow block with oil sump.
proved performanceunder conditions of heavy loadwhere contact friction losses are greater than the lubricant churning losses. This method is often used when conditionswarrant a specially formulated oil not available as a grease. A cooling coilis sometimes usedto extend the applicable temperature range of the oil bath. This usually takes the form of a watercirculating loop or, in some recent applications, the fitting of one or more heat pipes. ~ickfeedand oilring methods of raising oil from a sump to feed the bearing are not generally used with rolling bearings, but, occasionally shaft motion is used to drive a viscous pump for oil elevation, thus reinto ~ the ~ucingthe sensitivity of the system to oil level. A disc d i ~ p i n sump drags oil up a narrow groove in the housing to a scraper blade or stop that deflects the oil to a drilled passage leading to the bearing. A major limitation of all sump systems is the lack of filtration or debris entra~ment.Fitting a magnetic drain plug is advantageous for controlling ferrous particles, but otherwise sump systems are only suitable for clean conditions.
ir
As the speeds and loads on a bearing are increased, the need for deliberate means of cooling also increases. The simple use of a reservoir and ricant flow increases the heat dissipation capaa pump to supply a bilities s i ~ i ~ c a n t l ~ssure . feed permits the introduction of appro~riate heat exchange a gements. Notonlycanexcess heat beremoved, but heat can be added to assure flow under extremely cold s t a ~  u p s . s are equipped with thermostatically controlled valves to an optimum viscosity range. qually important, a circulating system can be fitted with a filtrat~on tem to remove the inevitable wear particles and extr~neousdebris. e mechanisms of debrisinduced wear and the effects of' even microscale indentations on the E L processes and the conse~uent re in fatigue life are discussed in Chapters 23 and 24. Finer filtration is being introduced in existing circulating systems with beneficial effects; however, increase pressure drops, space, weight,cost, and reli have to be considered. ~irculatingsystems are used exclusively in critical highperformance
This problem can be avoided by routing the oil to pickup scoops on the shaft with centrifugal force taking the oil via drilled passages to the inner ring, as shown in Fig. 17.2. uch of the flow passes throu slots in the bore of the inner ring, removing heat as it does so small portion of the lubricant is metered to the rolling contacts grooves between the inner ring halves. Separate drilled holes may be the cage lands. used to s u ~ p l y ~ d e ~ u aspace t e should be provided on both sides of the bearing to facilitate lubricant drainage. Often, space is at a premium, so a system of baffles can be substituted to shield the lubricant fr ~ e r m i t t i nit~to be scavenged without severe churning~ is activated at the same time as the main ~achinery,these as a dam and retain a small pool of lubricant in the bo of the bearing to provide lubrication at startupuntil the circulatin becomes established. ocarbon~asedfluids are satisfact erating at temperatures to about idation starts at room temper ure, and the lifetime of the lu~ricant s on thetemperature. dation becomes s i ~ i ~ c a n t , incipient thermal deco sition starts at about ~~~o
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discharge
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Underraceway lubricating systemsfor mainshaft bearings in an aircraft gas turbine engine. ( a ) Cylindrical roller bearing.( b )Ball bearings.
becomes a s i ~ i f i c a n problem t at about 41419°C (840"
t cover gas to exclude oxygen can extend the working range to the fluorocarbonbased fluidsare servicerication pro~ertiesof the hydrocarbon e the same thermal stability problem of hydrocarfluids, but have superior oxidation stability. Up to this time, not been able to reach the temperature limits inherent in the
ate class of lubrication arrangements can friction is essential at moderatetohigh removal is not a o the bearing as a fine spray or intain thenecessary lubricant is v i ~ u a l l yeliminated, and the volume of it can be discarded nging, cooling, and st sure to high shear stress the stability requirements of the atisfactory air quality in thework plets be reclassified and lubricant collecte has shown that the spray does not even nute ~uantitiesof lubrican
fill
with grease, the s u r r o ~
LUBRIC
If the service life of the grease used to lubricate the bearing is less than theexpected bearing life, the bearing needs to be re1 to lu~ricantdeterioration. Relubrication intervals are d bearing type, size, speed, operating temperature, greas ambient conditions associated with the application. As operating conditions become more severe, pa~icularly interms of frictional heat genoperating temperature, the bearing be relubricated ntly.Some manufacturers specify relu on intervals for their catalog beari~gs:for example, reference I 1 dations, given in the form of charts, are specific to the manu~acturer’s bearing internal designs and are generally based on goodquality, lithium soapbased greases (see “Grease Lubricants”) operating at temperatures not exceeding 70°C (158°F).It is interesting to note that for every (27°F) above 70°C (158°F)relubrication intervals must be halved. rating at temperatures lower than 70°C (158°F)tend to require ion less often; however, the lower operating temperature limit of the grease may not be exceeded [30°C (22°F) for a lithi~mbas~d grease.]. Also bearings operating on vertical shafts need to be relubricated appro~imatelytwice as often as bearings on horizontal shafts. (Rel u b ~ ~ a tinterval io~ charts aregenerally based on the latterapplication.) It is presumed that in no case is the grease upper operating temperature eded; this limit is 110°C (230°F)for a lithi e rel~bricationintervals depend on specific res such as rolling element proportions, ishes and cage confi~ration,they are different for each manufacturer even for basic bearing sizes. Therefore no such cha text; they may be found in manufacturers catalogs. given in Chapter 15 may be used to estimate the ase temperature in a given ap~lication,and the turer’s recommendations forrep~enishmentmay be relubrication interval is greater than 6 months, then all of the ase should be removed from the bearing arrangement and reease, If the interval is less than 6 months, then an I
3
In applications where conventional lubricants are not appropriate, thin solid films of soft or hard materials are applied to bearing surfaces to reduce frictionand enhance wear resistance of contacting surfaces. There are many methods of applying solid lubricants, each of which provides varying degrees of success with respect to adhesion to the substrate, thickness, and uniformity of coverage. Resinbonded solid lubricants are very commonly used. These materials usually consist of a lubricating solid and a bonding agent. The lubricating solid may bea single material or a mixture of several materials. It can be applied in a thinfilm by spraying or dipping. Dependingon the binding agent used, it may be a heatcured or aircured material. Heatcured materials are generally superior to the airdried materials. Metal surfaces are usually pretreated prior to application. Pretreatment may be 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 of thin solid films from one contacting surface to another. The interaction of rolling elements with a solidlubricated or impregnated separator transfers thinsolid filmsto the rolling elements, which in turn are transferred to the rolling contact raceways. Whenthe rubbing action against the solid lubricant occurs with sufficient load, the solid lubricant will compact itself into the existing surface imperfections. This burnishing action provides little control over film thickness and uniformity of coverage. Much greater control of solid lubricant film thickness., composition, and adhesion canbe obtained by using various electrically assisted thinfilm deposition techniques. These include ion plating, activated reactive evaporation.,dc and rf sputtering, magnetron sputtering, arccoating, and coating with highcurrent plasma discharge. Coatings of virtually all of the soft metals and hardmaterials and anumber of nonequilibrium materials can be produced with one or another of the electrically assisted, film deposition techniques. When vacuum techniques of deposition are used, the vapor of the solid lubricant species being depositedcan be reacted with process gases t o produce various synthesized compounds.
Two approaches are used to apply polymeric lubricants. The first approach is to make a suitably shaped part from a porous material, either by molding or machining, and to place it in the bearing in one or more pieces. A vacuum impregnation process then charges the material with l ~ b r i c a ~The t . need to insert the porous structure governs the amount of bearing free space that can be used. Often rivets or other fasteners
must be used to hold the pieces in place, further reducing the volume available. The second methodentails the formation of a lubricantsaturated rigid gel in thebearing itself by filling the bearing with the fluid mixture and using a curing or pol~erizationstep to effect a transition to a solid structure. Essentially all the void space in thebearing can be used. Only a very few polymers have been identified that will functionin thismanner. Further? thereappears to be a tradeoff between bleeding,shrinkage, strength, and temperature limit characteristics.
As compared to any other lubricant, in particular grease, a liquid lubriprovides the following advantages: It is easier to drainand refill, a particular advantage for applications requiring short relubricating intervals. The lubricant supply to the system can be more accurately controlled. It is suitable for lubricating multiple sites in a complex system. ecause of its ability to be used in a circulating lubricant system, it can be used in higher temperature systems where its ability to remove heat is si~ificant.
In most applications pure petroleum oils are satisfactory as lubricants. They must be free from contamination that might cause wear in the , bearing, and should show high resistance to oxidation, ~ m m i n g and deterioration by evaporation. The oil must not promote corrosion of any parts of the bearing during standing or operation. The friction torque in a liquidlubricated bearing is a fu~ctionof the bearing design, the load imposed, the viscosity and quantity of the lubricant, and the speed of operation. Only enough lubricant is needed to form a thin film over the contacting surfaces. Friction torque will increase with larger q~antitiesand with increased viscosity of the lubricant Energy loss in a bearing depends on the product of torque and spee It is dissipated as heat, causing increased temperature of the bearing and its ~ o u n t i n gstructures. The te~peraturerise will always cause a decreased viscosity of the oil and, consequently, a decrease in friction
torque from initial values. The overall heat balances of the bearing and mounting structures will determine the steadystate operating conditions, It is not possible to give definite lubricant recommendations for all bearing applications. A bearing operating throughout a wide temperature range requires a lubricant with high viscosity indexthat is, having the least variation with temperature. Verylow starting temperatures necessitate a lubricant with a sufficiently low pouring point to enable the bearing to rotate freely on startup. r specialized bearing applications involving unusual conditions, the recommendation of the bearing or lubricant manufacturer should be followed.
Mineral oil is a generic term referring to fluids produced from petroleum oils. ~hemically,these fluids consist of paraffinic, naphthenic, and arooups combined into many molecules. See Fig. 17.3.Also present stocks are traceamounts of molecules containing sulfur, oxygen, or nitro en. element all^ the composition of petroleum oils is quite con87% carbon, 1114% hydrogen, and the remainder sulfur, nind oxygen. The molecular makeup of the fluid is very complex and depends on its source. For the purpose of lubricant production, crude petroleum oils are characterized by the type of hydrocarbon distillates obtained. For this method it is c o m ~ o nt o speak of paraffinic, mixed, and naphthenic crude oils. Aromatics are generally a minor component. Depending on the source, the crude petroleum mayco ist of gasoline and light solvents, or it may consist of heavy asphalts. odern distillation, refining, and blending techniques allow the production of a wide range of oil types from a given crude stock; however, somecrude stocks are more desirable for lubricant formulation, With respect to lubricant properties, a few generalizations can be made. Paraffinic base crudes have the viscositytemperature characteristics for lubrication. Usually such crudes are low in asphalt and trace materials. The earliest commercial crude, nnsylvania, was of this type.
C
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C Naphthenic (saturated ring)
i C
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i
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I Paraffinic (saturated chain)
.3. Chemical structures of mineral oils.
Naphthenicbased crudes do not contain paraffin waxes, so they are better suited for lower temperature application. Naphthenic oils also have lower flash points and are more volatilethan comparable paraffinic oils. The performance of the base fluid for mineral oils as well as for synthetic fluid lubricants depends to a great degree on the type of additives incorporated into the system. Antioxidants, corrosion inhibitors, antifoam additives, and friction and wearminim~zingadditives employed vary depending on the specific purpose of the lubricant. The two most common lubricants used for industrial rolling bearing applications may be described as rust and oxidation (R & 0) inhibited oils and extreme pressure (EP) oils. The R & 0 oils are often used when bearings and gears share a common lubricant reservoir. These products may incorporate antifoam and antiwear agents. They are suitable for lighttomoderate loadings and for temperatures from 20 to 120°C (4 to 248°F). Extreme pressure oils usually encompass the additive package of R & 0 oil with an additional EP additive. The EP additive essentially generates a lubricating surface to prevent metaltometal contact. T w o approaches exist in formulating EP additives. The first employs an active sulfur, chlorine, or phosphorus compound to generate sacrificial surfaces on the bearing itself. These surfaces will shear rather than weld upon contact. The secondapproach uses a suspended solid lubricant to impose between 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 normal temperature range for such lubricants is 20 to 120°C (4 to 248°F). Some precautions are necessary when using EP oils of either type. EP solids will reduce internal clearances that can cause failure in certain bearings. These solids might also belost in close filtration processes. EP sulfurchlorinephosphorus compounds might becorrosive to bronze cages and accessory items.
Synthetic hydrocarbons are manufactured petroleum fluids. Being synthesized products, the particular compounds present can be both narrowly limited and specificallychosen. This allowsproduction of a petroleum fluid with the most favorable properties for lubrication purposes. 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 refined petroleum oils, allowing higher temperature performance for lubricants compounded fromthem. These materials also exhibitinherently high viscosity indexes, leading to better viscosity retention at elevated
LIQUID L ~ R I C ~ S
temperatures.Otherproperties showing improvement include flash point, pouringpoint, and low volatility. Although synthetic, the materials are compatible with petroleum products because of the compositions involved.
The most important property of a lubricating oil is viscosity. Defined as the resistance to flow, viscosity physically is the factor of proportionality between shearing stress and the rate of shearing. As described in an earlier section, increased viscosity relates to the increased ability of a fluid to separate microsurfaces under pressure, the fundamentalprocess of lubrication. For bearing applicationsviscosity is usually measured kinematically perASTM specification I)445. This method measures thepassage time required under the force of gravity for a specified volume of liquid to pass through a calibrated capillary tube. A related concept of importance is viscosity index: (VI), which is an arbitrary number indicating the effect of temperature on the kinematic viscosity viscosity for a fluid. The higher theVI for an oil, the smaller the change will be with temperature. For typical paraffinic base stocks VI is 8595. Polymer additions may be made to petroleum base stocks to obtain VI of 190 or more. The shearing stability of these additives is questionable, and VI generally deteriorates with time. Many synthetic base stocks have V I S far in excess of mineral oils, as Table 17.1 shows. The method of calculating VI from measured viscosities is described in ASTM specification D567.
Figures 17.4 and 17.5 can be used to derive a minimum acceptable viscosity for an application. Figure 17.4 indicates the minimum required viscosity as a function of bearing size and rotational speed for a petroleumbased lubricant. The viscosity of a lubricating oil decreases with increasing temperature. Therefore, the viscosity at the operating temperature rather than theviscosity at the standardizedreference temperature of 40°C (104°F) must be used. Figure 17.5 can be used to determine the actual viscosity at the operating temperature if the viscosity grade (VG) of the lubricant is known.
.
A bearing having a pitch diameter of 65 mm (2,559 in.) operates at a speed of 2000 rpm. As shown in Fig. 17.4, the intersection of dm = 65 mm with the oblique line representing 2000 rpm yields a minimum required kinematic viscosity of 13 mm2/sec (0.02 in2/sec), assuming that the operating temperature is 80°C (176°F); in Fig. 17.5 the intersection between 80°C (176°F) and the required vis
0
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*
u3oomcu acmmm
ci m m m
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v v v v
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ocu3* ri
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1000
500
200
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20
a
10
5
3 10
20
50
500 100
200
1000
dm =: Pitch diameter (rnm)
17.4. Minimum required lubricant viscosity versus bearing pitch diameter and speed, dm = (bearing bore + bearing o.d.1 + 2, v, = required lubricant viscosity for adequate lubrication at theopening temperature.
cosity of 13 mm2/sec (0.02 in.2/sec) is between the oblique lines for VG46 and VG68. Therefore, a lubricant with minimumviscosity grade VG46 should be used; that is, a minimum lubricant viscosity of 46 mm2/sec (0.0'7 in.2/sec) at standard reference temperature of40°C (104°F). m e n determining operating temperature,it must be kept in mind that oil temperature is usually 311°C (520°F) higher than bearing housing temperature. If a lubricant with higher than required viscosity is selected, an improvement in bearing fatigue life can be expected; however, since increased viscosity raises the bearing operating temperature, there is frequently a practical limit to the lubrication improvement that can
L ~ R I C AND ~ SL ~ R I ~ A T TEC I ~ N F

NOTE Viscosity classification numbers are al I S 0 3448 accordingto i n t ~ n a t i ~Standard 1975 for oils having a viscosity index of 95. Approximate equivalent SAE viscosity grades are shown in parentheses.
F
I 1’7.6. ~Viscosit~tem~erature ~ ~ chart.
be obtained by this means. For exceptionally low or high speeds, for critical loading conditions, or for unusual lubrication conditions, the bearing manufacturer should be consulted.
Many types of “synthetic” fluids have been developed in response to lubrication requirements not adequately addressed by petroleum oils. These areas include extreme temperature, fire resistance, low volatility, and high viscosity index. Table 1’7.1lists some typical properties of various lubricantbase stocks and indicates application areas for finished productsof each type. As for petroleum oils, many additive chemistries have been developed to provide property enhancement. Using synthetic lubricants requires a thorough understandin application requirements involved. The favorable properties shown by some synthetics are obtained only with unsuitable ~ e r f o ~ a n charaece
1
teristics in such areas asloadcarrying ability and highspeed operation. Also, many very hightemperature fluids, principally developed for military applications, have short service lives compared to commercial re~uirements. “Po1yglyco1s7’are oftenused as synthetic lubricant bases in wateremulsion fluids. These products are linear polymers of the general formula shown by Fig.17.6: R, R’, R” are alkyl groups, and R’ maybe hydrogen. This class of fluids includes glycols, polyethers, and polyalkylene glycols. Properties of the class include excellent hydrolytic stability, high viscosity index, and low volatility. “Phosphate esters” 9 (tertiary) have properties that make them useful as lubricants. They are generally represented as shown in Fig. 17.7: R, R’, and R” are organic groups. Phosphate esters have poor hydrolytic stability and lowviscosity index. Their outstanding characteristic is fire resistance, and assuch these fluids are often used as hydraulic fluids in hightemperature applications. Dibasic acid esters represent a family of synthetic base stocks widely used in aircraft gas turbine engine applications and as a basis for lowvolatility lubricants. They are synthesized by reacting aliphatic dicarboxylic acids (adipic to sebacic) with primary branch alcohols (butyl to octyl). Someare available from such natural sources as castor beans and animal tallow. Characteristic properties of these fluids are low volatility and high viscosity index. Poly01 esters formed by linking dibasic acids through a polyglycol center have beenfound suitable as highfilm strength lubricants. Blends of dibasic esters, complex esters with suitable antiwear additives, VI improvers, and antioxidates are used to form the current generation of aircraft gas turbine engine lubricants. Generally, these products show excellent viscositytemperature relationships, good lowtemperature properties, and acceptable hydrolysis resistance. Elastomeric seals used with these materials must be chosen carefully because they chemically attack standardrubbers. Silicone fluids (organosiloxanes) exhibit outstanding viscosity retention with temperature and are functional in conditions of extreme heat ROCH~CHOR”
I
R’
FI
.
Chemical structure of a polyglycol.
0 II
ROPOR‘ OR”
17.7. Chemical structure of phosphate esters.
L ~ R I C ~ T TEC I O ~
.
Chemical structure of a silicone
ese fluids are the basis for many high temperature, 204°C ts. The general structure of silicone fluids may be shown represents methyl, phenyl, or some other organic group. favorable viscositytemperature characteristics, both thermal and oxidation resistance are excellent. As a family, these fluids also exhibit low volatility and good hydrolytic stability. These materials are inert towards most elastomers and polymers as long as very high temperatures are avoided. Oxygen exposurewith hightemperature use, however, can result in gelation and loss of fluidity. The lubrication properties of the base oils are not impressive compared to other lubricating fluids. Typical applications of these materials as lubricants are electric motors, brake fluids, oven preheater fans, plastic bearings, and electrical insulating fluids. icate esters represent a mating of the previous two lubricant fluid . As a class, these fluids possess good thermal stability and low volatility. These materials areused in hightemperature hydraulic fluids and lo~volatilitygreases. ~luorinatedpolyethers as a class represent the highesttemperature lubricating fluidscommerciallyavailable.Although distinct chemical versions are marketed, all of these fluids are fully fluorinated and completely free of hydrogen. This structural characteristic makes them inert to most chemical reactions, nonflammable, and extremely oxidation reroducts from these oils show very low volatility and excellent resistance to radiationinduced polymerization. The products are essentially insoluble in common solvents, acids, and bases. Density is approximately double that of petroleum oils. Products of this chemical family are used to lubricate rolling bearings at extremely high temperatures"C Other applications areas are in high that is, ~ 0 4  ~ ~ 0(~00~00°F). vacuums, corrosive environments, and oxygenhandling systems. The cost of these lubricants is very high. Table 17.2 gives characteristics of synthetic oils compared to those of mineral oils.
rease is a thickened oil that allows localizationof the lubricant to areas of contact in thebearing. rolling bearing grease is a suspension of fluid
L ~ R I C ~ T TEC I O ~
dispersed into a soap or nonsoap thickener,with the addition of a variety of performan~eenhancingadditives. Grease provides lubricant by bleeding; that is, when the moving parts of a bearing come in contact with grease, a small quantity of thickened oil will adhere to the bearing surfaces. The oil is gradually d oxidation or lost by evaporation, centrifugal force, and so on; the grease near the bearing will be de~leted. everal differing viewpointscurrently exist concern in^ the mechanism of grease operation. Until recently, grease was looked upon as merely a sponge holding oil near the working contacts. As these contacts consumed oilbywayof evaporation and oxidation, a replenish~entflow an equilibrium as long as the supply lasted. and microflow lubrication assessment techni ckener phase plays rather complex roles in both the development of a separating film between the surfaces and in themodulation of the replenishing flows. The manner in which the thickener controls oil outflow, reabsorbs fluid thrown from the contacts, and acts as a trap for debris is little understood at this time. The mechanism is not steady but is characterized by a series of identi~ableev reases offer the following advantages compared to
. .
.
intenance is reduced because there is no oil level to maintain. ew lubricant needs to be added less frequently. bricant in proper quantity is confined to the housing. enclosures can therefore be simplified. Freedom from leakage can be accomplished9 avoidin tion of products in food, textile and chemical industries. The efficiency of labyrinth “seals” is improved, and better sealing is offered for the bearing in general. The friction torque and temperature rise are generally more favorable.
omb io^. The procedures described in the following are available from the American Society forTesting and Materials (ASTM). The determination of the resistance of lubricating greases to oxidation when stored under static conditions fora long time is described by ASTM specification D942. A sample is oxidized in a “bomb”heated to 99°C (210°F) and filled with oxygen at 0.76 N/mm2 (I10 psi). Pressure is observed and recorded at stated intervals. The degree of oxidation after a given period of time is determined by the corresponding decreasein oxygen pressure.
~ r o ~Point. ~ i nDropping ~ point is the temperature at which a grease becomes a liquid and is sometimes referred to as the melting point. The test is performed per ASTM specification D566. ~ u a ~ o r a t i oLoss. n The method of determining evaporation loss is described by ASTM specification I)972. Thesample in an evaporation cell is placed in a bath maintained at the desired test temperature [usually 99149’6 (210300”F)l. Heated air is passed over the cell surface for 22 hr. The evaporation loss is calculated from the sample weight loss. ~ l a Point. s ~ Flash point is the lowest temperature at which an oil gives off inflammable vapor by evaporation, per ASTM specification D566. L o ~  T e ~ ~ e r a t uTorque. re Lowtemperature torque is the extent to which a lowtemperature grease retards therotation of a slowspeed ball bearing when subjected to subzero temperature. The method of testing is described by ASTM specification D1478. Oil ~ e ~ a r a t i o nThis . is the tendency of lubricating grease to separate oil during storage in both conventional and cratered containers, as described by ASTM specification D1742;the sample is determined by supporting on a 74pm sievesubjected to 0.0017 N/mm2 (0.25 psi) air pressure for 24 hr at 25°C (77°F). Any oil seepage drains into a beaker and is weighed. Penetr~tion. The penetration is determined at 25°C (77°F)by releasing a cone assembly froma penetrometer and allowing the cone to drop into the grease for 5 sec. The greater is the penetration, the softer is the grease. Worked penetrations are determined immediately after working the sample for 60 strokes in a standard grease worker. Prolonged penetrations areperformed after 100,000strokes in a standard grease worker. A common grease characteristic is described by NLGI (National Lubricating Grease Institute) grade assigned, as shown in Table 17.3. rolling bearing applications employ an NLGI 1, 2, or 3 grade grease. Pour Point. Pour point is the lowest temperature at which an oil will pour or flow. The pour point is measured under the conditions in AST specification D97. The pour point together with measured lowtemperature viscosities givesan indication of the lowtemperature serviceability of an oil. Viscosity, Viscosity Index. The values of viscosity and VI generally refer to the base oil values of these properties as discussed in “Liquid Lubricants.”
. NLGI Grades 000 00 0 1 2
3 4 5 6
NLGI PenetrationGrades Penetration (60 Strokes) 445475 400300 355385 3 10340 265295 220250 175205 130160 85115
esistance. Water washout resistance is the resistance of a lubricatinggreasetowashout by water from * tested at 38°C *C as described by AST ation I)1264. (100°F) and ~ ~ . 5 (145"F),
Thickener composition is critical to grease performance, particularly oilbleed in^ with respectto temperature capabilit~, waterresistance, and characteristics. Thickeners aredivided into two broad classes: soaps and a compound of a fatty acid and a metal. Common nonsoaps. Soap refers to sometal include aluminum, barium, calcium, lithium, and dium. majority of commercial greasesaresoaptype,withlithium being the most widely used. ~ i t ~ i usoap m greasesLithium soaps are divided into two types: 12hydroxystearate and complex. The latter material is derived from organic acid component^ and permits higher temperatureperformance. The upper operating temperature limitof the usual lithiumbasedgreaseisapproximately 110°C (230°F). For a lithium complexbased grease the upper temperature limit is extended to 140°C (284°F). Conversely, the lower operating temperature limits are 30°C (22°F) and 20°C (4"F), respectively. Highquality lithium soap greases of both types have excellent service histories in rolling bearings and have been usedextensively in prelubricated; that is, sealed and grease~~orZi~e applications. Lithiumbased products have found acceptance as multipurpose greases and have no serious deficiencies except in severe temperature or loading extremes. C a Z c i soap ~ ~ greasesThe oldest of the metallic soap types,calciumbased greases, has undergone several important technical changes.
In the first formulations, substantial water (0. to stabilize the finished product. Loss of water sistency; as such, grease upper temperature o only 60°C (140°F) [ dingly; the lower gardless of temperature9evaporation Lure is only 10°C occurs, requiring frequency relubrication of the bearing. ely, the ability of the grease to entrain water is of some e; such greases have been widely used in food rocessing ts, water pumps, and wet applications in general. this of formulation has been made obsolete by newer pr with er temperature performance. he latest develop ncalciumthick~ned greases is the calcium n the soap is modified by adding an complexbased greas t product results having upper an and a substantially ature limits of 130°C (266°F) and rmance of these greases in rolling bea m. Although high temperature and sure) characteristics have been exhibited, there are some problems with excessive grease thickening in use, causing an eventual loss of lubrication to the bearing. o soap greasesSodium ~ ~ u soap~ greases were developedto provide an increase in the limited temperature capability of early calcium inherent problem with this thickener is poor sistance; however, small amounts of water are emulsified into the grease pack, which helps to protect metal surfaces from rusting. The upper operating temperature limit for such greases is only 80°C (176°F).The loweroperating tem is 30°C (22°F). Sodiumbase greases have been superceded by more waterresistant products in applications such as electric motors and front wheel bearings. Sodium complexbase greases have subsequently beendeveloped having upper and lower operating temperature limits of 1140°C (284°F) and 20°F (4"F), respectively. ~luminum comple~ greasesAluminum stearate used in rolling bearings, but aluminum compl being used more often. Greases formed from the complex soap perform favorably on waterresistance tests; however, the upper operating temperaturelimit is somewhat low at Ii0"C (230°F) compared to other types of highquality greases. The lower operating temperature limit is satisfactory at 30°C (22°F). These greases find use in rolling mills and foodprocessing plants. onsoap~ase greasesOrganic thickeners, including ureas, amides, and dyes, are used to provide higher temperature capability than
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8
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I AND C L~~ RSI C ~ T I OTEC N
is available with metallic soap thickeners. Improved oxidationstability over metallic soaps occurs becausethese materials do not catalyze base oil oxidation. Dropping points forgreases of these types are generally above 260°C (500°F)with generally good low temperature properties. The most popular of these thickeners is polyurea, which is extensively usedin hightemperature ball bearing greases for the electric motor industry. The recommended upper operating temperature limit for polyureabase grease is 140°C(284°F); the lower temperature limit is 30°C (22°F).
Inorganic thickeners include various clays such as bentonite. Greases made from a clay base do not have a melting point, so service temperature depends on the oxidation and thermal resistance of the base oil. These greases find use in special military and aerospace applications requiring very high temperature performance for short intervals, for example, greater than 170°C (338°F).On the other hand, the recommended upper temperature limit for continuous operation is only 130°C (266°F); the lower temperature limit is 30°C (22°F).
Grease ~ o ~ ~ ~ t i b i ~Mixing i t y , greases of differing thickeners and/or base oils can produce an incompatibility and loss of lubrication with eventual bearing failure. When differing thickeners are mixedthat is, soap and nonsoap or differing soap typesdramatic changes in consistency can result, leading to a grease either too stiff to lubricate properly or too fluid to remain in the bearing cavity. Mixing greases of digering base oilsthat is, petroleum and siliconeoilscanproduce a twocomponent fluidphase that will not provide a continuous lubrication medium. Early failures can be expected under these conditions. The best practice t o follow is to not mix lubricants but rather purge bearing cavities and supply lines with new lubricant until previous product cannot be detected before starting operation.
A polymeric lubricant uses a matrix or spongelike material that retains its physical shape and location in the bearing. Lubrication functions are provided by the oil alone after it has bled from the sponge. Ultrahigh molecular weight polyethylene forms a pack with generally good performance properties, but it is temperature limited to about 100°C (212"F), precluding its use in many applications within the temperature capability of standard rolling bearings. Some higher temperature materials, such as polymethylpentene, form excellent porous structures but arerelatively expensive and sufferfromexcessively shrinkage. Fillers and blowing agents, tools of the plastic industry, interfere with the oilflow
behavior, and contribute little in this situation.Other solutions must be developed. Figure 17.9 shows bearings filled with polymeric lubricant. Successful application has been achieved where a bearing must operate under severe acceleration conditionssuch as those occurring planetary transmissions. Bearing rotational speed about its own axis may be moderate, but the centrifu~ng action due to the planetary motion is strong enough to throw conventional greases out of the bearing despite the presence of seals. When polymerically lubricated bearings are substituted~life improvements of two orders of magnitude are not uncommon. Such situations occur in cablemaking,tirecordwinding, and textile mill applications. h o t h e r major market for polymer lubricants is food processing. Food machinery must be cleaned f r e q ~ e n toften l ~ daily, using steam, caustic, or sulfamic acid solutions. Thesedegreasing fluids tend to remove lubricant from the bearings, and it is standard practice to follow every cleaning procedure with a relubrication sequence. Polymer lubricants have proven to be highly resistant to washout by such cleansing methods, hence the need for regreasing is reduced. Thereservoir effect of polymer lubricants has been exploitedto a degree in bearings normally ~ubricated
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~ 17.9.~ Polymerlubricated ~ E rolling bearing.
by a circulating oil stress where there can be a delay in the oil reaching a critical location. The same effect has been used t o provide a backup in case the oil supply system should fail. igh occupancy ratio of the void space by the polymer minimizes rtunity for the bearing to “breathe” as temperature change. Corrosion due to internal moisture condensation is therefore reduced. all ferrous surfaces are very close to the pack, conditions are optimum sing vapor phase corrosioncontrol additives in the formulations. espite these advantages polymeric lubricants have somespecific drawbacks. There tends to be considerable physical contact betweenthe pack and the moving surfaces of the bearing. This leads to increased frictional torque, which produces more heat in the bearing. In conjunction with thermal insulatingproperties of the polymer and its inherently limited temperature tolerance, the speed capability is reduced. Moreover, compared to grease, the solid polymer is relatively incapable of entrapping wear debris and dirt particles.
Solid lubricants are used where conventionallubricants are not suitable. Extreme environment conditions frequently make solid lubricants a prehoice of lubrication. Solid lubricants can survive temperatures e the decomposition temperatures of oils. They can also be used in chemically reactive environments. The disadvantages of solid lubricants are (1)high coefficient of friction, (2) inability to act as a coolant, (3) finite wear life, (4) difficult replenishment, and (5) little d a ~ p i n g effect for controllingvibrational instabilities of rolling elements and se any c o ~ m o nsolid lubricants, such as graphite and molybdenum diounds that shear eas’ has weak van der ferred planes of th ial a characteristic between sulfur bon oxidizes at approximately 39 coefficient of friction. the oxides can be he low friction associated with graphite depends on intercalation with gases, liquids, or other substances. r example, the presence of absorbed water in graphite imparts goo ubricating qualities. Thus, aphite has deficiencies as a lubricant except when used in an n ironment containing contaminants such as gases and water proper additives graphite can beeffe e up to 649°C (1200 S, in that it is a ungsten disulfide ( ~ S , )is similar to layered lattice solid lubricant. It does not need absorbable vapors to develop 1ow“shearstren~h characteristics.
Other “solid” lubricating materials are solid at bulk temperatures of the bearing but melt from frictional heating at points of local contact, giving rise to a lowshearstrength film. This melting may be very localized and of very short duration. Soft oxides,such as lead monoxide ( are relatively nonabrasive and have relatively low friction coefficients, especially at high temperatures where their shear strengthsare reduced. At these temperatures deformation occursby plastic flow rather than by brittle fracture. elted oxides can forma glaze on the surface. This glaze can increase or decrease friction,depending on the of the glaze within the contact region.Stable fluorides such as lithium fluoride (LiF,), calcium fluoride (CaF,), and barium fluoride (BaF,) also lubricate well at high temperatures but over a broader range than lead oxides.
~oncernsfor the environment have led to the development of more environmentally friendly or environ~entally acce~table lubricants. gradability and low ecoto~icityare required for these lubricant many countries now have specific requirements for branding lubricants as environmentally friendly. Initially9twocycle engine marine and forest applications were targeted for use of biode~adablelubricants. This use has now been extendedto include hydraulic fluids, engine oils in general and greases. For example, environmentally friendly grease products are available as rope lubricants and rail lubricants. iode adability and low ecotoxicity of a lubricant depend on the base egradable fluids include vegetable oils, synthetic esters, poglycols, and some polyalphaolefins. The susceptability of a substance to be biodegraded by microorganisms is a measure of its biodegradability. Biodegradability can be partial, resulting in the loss of some specific process such as splitting an ester linkage (prima~ybiodegradation) or complete, resulting in the total breakdown of the substance into simple compounds such as carbon dioxide and water (ultimate biode~adation).There is currently no standard method accepted forassessing an environmentally acceptable lubricant, and several methods are in The ASTM is currently addressing this problem. d is one of the more widely usedtests; but,countri established their own certification tests. For example, in German lu‘79test to obtain certification a d, Sweden, Denmark, Norway, have their own certification requirements and environmental labels. Ecotoxicity tests involve different types of aquatic species that form the aquatic food chain. Testing the toxicity of a lubricant on bacteria,
L
water fleas (invertebrates), and trout (vertebrates) or other species may be required depending on the application and state and country regulatory requirements. Development of lubricating oils and greases in the future and worldwide will require significant consideration of not only health and safety issues, but environmental requirements as well.
Once the general method of lubricating a bearing has been determined, the question of a suitable sealing method frequently needs to be addressed. A seal has two basic tasks. It must keep lubricant where it belongs and keep contaminating materials from the bearing and its lubricant. This separation must be accomplished between surfaces in relative motion, usually a shaft or bearing inner ring and a housing. The seal must not only accommodate rotary motion, but it must also accommodate eccentricities due to runouts, bearing clearance, misali~ments, and deflections. The selection of a seal design depends on the category of lubricant employed (grease, oil, or solid). Also, the amount and nature of the contaminant that must be kept out needs to be assessed. Speed, friction, wear,ease of replacement, and economics governthe final choice. earings run under a great variety of conditions, so it is necessary to judge which seal type will be sufficiently effectivein each particular circumstance.
Grease is the simplest lubricant to seal. The fluidity of the oil has been deliberately reduced by blending with the thickener. The stiff nature of ease means that it requires little in the way of constraint yet at the same time readily plugs small spaces. Givena suitably small gap, grease can form layers on the opposing moving surfaces, effectively closingthe gap completely. This principle is used in labyrinthtype designs. Because it has anoily consistency,dirt or dust particles that penetrate a seal are caught by the grease and prevented from entering the bearing. The wicking type of oil delivery to the bearing means that the particles are permanently kept out of circulation unless the grease becomes stirred in some way. Some seals make physical contact between the surfaces. A film of grease provides the necessary continuous supply of lubricant to establish hydrodynamic separation with its attendant low friction and wear.
SEALS
73
eals wit Oils are more difficult to seal than greases. They will flow through the smallest gaps if there is any hydraulic head. Either the possibility of a head developing must be prevented by the cavity design,or running gaps must be eliminated by use of a movable seal lip. Oils are excellent as dirt traps, but they lack the ability to keep the dirt out of circulation unless backed up by a filter system. Dirt can only bekept out by positive gap elimination.
Only the powder and reactive gas forms of solid lubricants pose sealing challenges. Even then, since they are essentially oncethrough systems, some leakage is tolerable unless the material is toxic. Furthermore sealing o f such lubricants can pose grave difficulties, particularly since they are almost exclusively usedat extremely high temperatures at which gas dynamic behaviorincreases. Zero gap conditions are thennecessary with extra provision made to prevent the powder or soottype exhaust products from compacting and causing separation.
La~yri~ Seals. t ~ Labyrinth seals consist of an intricate series of narrow passages that protect well against dirt intrusion. An example is shown in Fig. 17.10. This type is suitable for use in pillow blocks or other assemblies where the outer stationary structure is separable. The inner part is free to float on the shaft so that it can position itself relative to the fixed sections. The mechanism of sealing is complex, being associated with turbulent flow fluid mechanics. It is reasonably effective with liquids, greases, and gases, provided that thereis no continuous static head across the assembly.
F ~ G U ~17.10. E Bearing housing with labyrinth seal.
s
It is normal practice to add grease to the labyrinth, making the gaps even smaller than can be achieved mechanically due to tolerance stackirt has virtually no chance of penetrating such a system without becoming ensnared in the grease. A further advantage accrues at regreasing. Spent lubricant can purge readily through the labyrinth and flush the trapped debris with it. The relatively moving parts are separatedby a finite gap, so wear, in the absence of' large bridging particles of dirt, is essentially nonexistent. Likewise, frictional losses are extremely low, The number of convolutions of the labyrinth passage can be increased with the severity of the dirt exclusion requirements. Separate flingers and trash guards or cutters may be added on the outboard side to deal with wet or fibrous contaminants thatcould damage or penetrate thelabyrinth. Figure 17.11 shows a ball bearing with an integral labyrinth seal and outboard flingerring.
17.11. Deepgroove bearing assembly with integral labyrinth seal and flinger ring.
emicircular pieces of felt, pressed into trapezoidal section grooves in thehousing, lightly contact the shaftsurface, as shown in Fig. 17.12. Inexpensive and simple to install and replace, the greaseladen felts keep dirt out of the enclosure; however, the dirt entrapped in the felt fibers cancause serious shaft surface wear. Also, the felt can become compacted, eventually leaving an air gap. Friction is often high and difficult to control. For these reasons, felt seals, though once popular, are not currently in significant use.
S ~ i e Z ~As~ Fig. , 17.13 shows, a shield takes up very little axial space and can usually be accommodatedwithin the standardboundary dimensions of the bearing. The near knife edge standing just clear of the ring land is, in effect, a singlestage labyrinth seal. Effective enough to keep all but the most fluidgreases in thebearing, the shield can be considered as a modest dirt excluder, suitable for use in most workplace environments. Under harsher conditions it must be backedup with extra guards. Special greases or acceptance of leakage and reduced lubricant life are necessary when shielded bearings are used in vertical axis applications. The absence of contact friction permits these bearings to be used at the highest speed allowed by the mode of lub~cationand type of lubricant. Z a ~ ~ o ~ eLip r i cSeaZs. The narrow gap between a shield and an inner ring groove or chamfer can be closed by a carefully designed section of elastomer (nitrile rubber for general purposes). Figure 17.14 illustrates a typical configuration. The flexible material makes rubbing contact with the ring and establishes a barrier to the outward flow of lubricant or the ingress of contaminants. When the bearing is in motion, the elastomer must slide over the metal surface, and a frictional drag is produced, which even for a welldesigned seal, is generally greater than the fric
17.12. Bearing pillow block with felt seals.
67
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17.13. Radial ball bearing with shields.
17.14. Radial ball bearing with integral single lip seal.
tional torque of the bearing. Often moreimportant is the seal ~ r e a ~ a ~ a y torque, which can be several times the running torque. Considerable research has been devotedto finding bothelastomers and seal designs that achieve a suitable balance between sealing efficacy, lip or ring wear, and frictional torque.
77
The lip of the seal must bear on the ring with sufficient pressure to follow the relative motions of the runningsurface caused by eccentricities and roundness errors. This pressure is achieved by a slight interference fit, producing a dilation of the seal. The spring rate of the lip governs the speed at which the lip can respond to the running errors without a gap being formed through which fluid can pass. Higher bearing speeds demand better runningaccuracies. Spring rate is regulated by the elastic properties of the seal material and the design of the bending section. At first glance, even though a lubricant is present, no hydrod~amic lift would be expected onthe lip, due to the axial symmetry. has established, both theoretically and experimentally, that a very thin, stable dynamic film persists over much of the operating regime. The mechanism of sealing is a complex one involvingthe elastomeric lip, the counterface, and the grease, or at least the oil phase of the grease. Figure he seal of Fig. 17.14 composed of a molded annulus of polo a thin steel disc. The disc provides mechanical support against minor pressure differences that can occur acrossthe seal and also assures a slight compression of the polymer against the outer ring recess, thus creating a fluidtight static seal at that point. The inside diameter of the disc, in conjunction with the waisted section of the molding, defines the flexure point of the lip itself and is located so that the deflected lip bears against the counterface groove with suitable pressure and at a predetermined angle. This angle of contact producesappropriate convergence and divergence on either side of the contact, which helps the sealing function appreciably: The lip pressure induced by the interference between the seal and its counterface is sufficient to prevent fluid leakage under static conditions. To function adequately,the elastomer must exhibit specific properties. eyond compatibilitywith the common types of lubricating oils and swell that can be accommodatedby the seal configuration, it must survive the frictional heating at the lip and heat from the bearing or its environment without hardening, cracking, or otherwise aging. To survive startup and the presence of dirt, it must have wear and abrasion resistance. Care must be taken when forming the elastomer and its fillers that the final cured product does not promote corrosion of the counterface under humid conditions. 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 bearing must be positioned to wet the seals upon assembly. In most cases the grease volume is sufficient to require a period of working whenthe bearing first operates. This is followed by channeling, and the formation of grease packs against the inside surfaces of the seals. Operational sufficiency is then assured.
17.15. Lip seal construction showing interference with bearing inner ringseal groove and retention in outer ring groove.
L develops under the lip, which progressively lowers the seal torque by changing the friction from Coulomb to viscous shear, Wear is thereby greatly reduced. Currently, there are twoschools of thought concerning lubricant film formation. One approach ascribes the film to asperities on the seal lip, producing localized hydrodynamicpressure ~uctuations, as illustrated in Fig. 17.16. Cavitation downstream from each asperity limits the negative pressure remaining to separate the surfaces. The c o u ~ t e ~ a i l i view n g invokes the viscoelastic properties of the seal material and theinability of the elastomer to follow precisely the radial motions of the counterface produced by eccentricity and outofroundness. Both of these mechanisms may be valid and function simultan~ously and essentially independently of one another. The first is governed by
( a 1 LOW SPEED ~ L l D l ~ (b) ~ HIGH SPEED SLIDING NO C A V I ~ ~ T I O ~
IT^ C A V ~ T A T ~ O ~
17.16. Inducing hydrodynamicseparation of sealing surfaces by asperities.
the microgeometry of the lip as modified by wear, abrasion scratches, thermal and installation distortions, and possibly inhomogeneities in elastomer properties. The second is a byproduct of manufacturing process characteristics and nonrotary displacements of the inner ring. Seal torque arises from four sources: adhesion betweenasperities, abrasion, viscous shearing of the film, and hysteresis in theelastomer. The last two are strongly influenced by temperature and so tend to be selflimiting; otherwise they all depend not only on the application but on the detail installation itself. Methods for exact prediction of torque and operating temperature have not yet been devised.In seemingly identical conditions onesealed bearing frequently runs cooler than another, or one will leak slightly and another will not. Much work needs to be done to predict seal performance in a given application. The primary task of the single lip seal is to contain grease. It can exclude moderate dust as found in typical home or commercial atmospheres, and it finds a great many suitable applications. Some dusts, such as from wood sanders or lint accumulating on the bearings in textile machinery, have the ability t o wick considerable amounts of oil throu~h the lip film, which shortens bearing life. In these situations and where there is heavy exposure to dirt, particularly waterborne dirt, such as in automotive uses, additional protection in the form of dust lips and flingers should be provided. Figure 17.17 shows an example of a doubleli~ seal.
Garter Seals. Similar to many respects to the lip seal, the garter seal uses a hoop spring or garter to apply an essentially constant inward pressure on the lip. As shown in Fig. 17.18, the arrangement requires more axial space than is available in a bearing of standard envelope
el7.17. Doublelip seal. Inside lip is for fluid retention; outside lip is for dust exclusion.
17.18. Garter seal Xsection. showing retaining spring.
1
dimensions. Either extra wide rings must be used, or the seal must be fitted as a separate entity in the assembly. The sprin~induce~ pressure gives a very positive sealing effect and is used to contain oil rather than grease, for two reasons: Oil can be thrown or pumped by an operating bearing with considerable velocity, a lip h seal, and the lip itself requires sufficient t o cause leakage t ~ r o u ~
a generous supply of oil for lubrication and removal of frictional heat. Relieved of the need to provide the closing force, the elastomer section can be designed to hinge Ereely so that relatively large amplitudes of shaft eccentricity can be accommodated. Thestrictly radial natureof the spring force precludes the use of anything other than a cylindrical counterface surface. Axial floating of the shaft is therefore accommodate^ well. The design lends itself to molding, and artificial asperities and other film generating devices can readily be formed in the elastomer. Fi 17.19 shows an example of a helical rib pattern intended not only to enhance the oil film thickness but to act as screw pump to minimize leakage.
~ e r r o ~ ~Seals. i ~ i c Magnetic fluids are a recent introduction to the ar~ tbasisenal of tools available to the sealing engineer. A f e r r o l u b ~ c ais cally a dispersion of very fine particles of ferrite in oil. The particles are in diameter and are coated with a molecular dispersing typically 100 !A agent to prevent coalescence. Brownian movement inhibits sedimentation. The result is a lubricant that responds to magnetic fields. Figure 17.20 shows a twostage seal, each stage composed of several gaps across which is suspended a ferrofluid film. This type of seal has proved very effective where bearing and shaft systems penetrate a vac
F I ~ 17.19. ~ E Radial seal with "quarter moon" projections moulded on the lip to develop a hydrodynamic lubricant film during operation.
Pole blocks
"A
17.20. Ferromagnetic fluid shaR seal.
uum enclosure and in computer disc drive spindle assemblies where absolutely clean internal conditions must be maintained. Its ability to withstand pressure gradients to 0.345 N/mm2 (50 psi) (by multista~ng)and accept high eccentricities with 100%fluid tightness makes the ferrofluidic seal essentially unique. Two things prevent greater application. The ferrite increases the apparent viscosity of the fluid, and viscous heating limits the speed capability. Thegreatest drawback is the need to introduce magnets into the system. Tramp iron is attracted to the seals unless considerable conventionalsealing is applied outboard, negating much of the seal's advantages.
Following the design and manufacturer of a rolling element bearing, the technology associated with creating and maintaining the internal environment of the bearing during its operation is the single mosti m p o ~ a n t factor connected with its performance and life. This environment is intimately associated with the lubricant selected, its means of application, and the method of sealing. In this chapter, a brief overvi given to each of these important considerations. No attempt has been made to provide an exhaustive study of lubricant types, means of lubrication, or means of sealing. It remains for the reader to explore each of these topics to the depth required by the individual application.
17.1. SKI?, General Catalogue 4000US Second Edition (199701).
Symbol
~escriptio~
Units
aterial factor for ball bearings, constant Semimajor axis of projected contact mm (in.) mensionless semimajor axis aterial factor for roller bearings with line contact Semiminor axis of projected contact ellipse imensionless semiminor axis Rating factor for contemporary material asic dynamic capacity of a bearing raceway or entire bearing Exponent on T~ iameter
mm (in.)
N (1b) mm (in.)
684
F ~ T I LIFE: G ~ L ~ B E R G  P m ~ O~ RGY AND ~ ~ RATING S
Symbol
Description
diameter
Pitch Ball or roller diameter odulus of elasticity Weibull slope Probability of failure Applied radial load Applied axial load Equivalent applied load rlD Material factor Factor combining the basic dynamic capacities of the separate bearing raceways Exponent on zo Number of rows Factor relating mean load on a rotating raceway to Qmax Factor relating mean load on a nonrotating raceway to Qmax Radial load integral Axial load integral A constant Fatigue life The fatigue life that 90% of a group of bearings will endure The fatigue life that 50%of a group of bearings will endure Effective roller length Length of rolling path Number of revolutions Rotational speed Number of bearings in a group Orbital speedof rolling elements to relative inner raceway Ball or roller load Basic dynamic capacity of a raceway contact Equivalent rolling element load Roller contour radius groove Raceway radius Probability of survival TOhllaX
Number of stress cycles per revolution
T
~
Units mrn (in.) mm (in.) N/mm2 (psi) N (lb) N (lb) N (1b)
revolutions
X
IO6
revolutions mm (in.) mm (in.) revolutions rpm
X
IO6
rPm N (lb)
N (1b) N (1b) mm (in.) mm (in.)
~
S
85
LIST OF SYMBOLS
Units
Symbol Volume under stress Rotation factor ~2(0.5)/~1(0.5) Radial load factor Axial load factor Number of rolling elements per row Depth of maximum orthogonal shear stress Contact angle L1 cos ald, Factor describing loaddistribution Zolb Capacity reduction factor Reduction factor to account for edge loading and nonuniform stress distribution on the rolling elements Reduction factor used in conjunction with a loadlife exponent n = 10.3 Normal stress Maximum orthogonalsubsurface shear stress Position angle of rolling element ~imitingposition angle Spinning speed Rolling speed Curvature sum Curvature difference SUBSCRIPTS Refers to axial direction Refers to a single contact Refers to an equivalent load Refers to inner raceway Refers to a rolling element location Refers to line contact Refers to a rotating raceway Refers to nonrotating raceway Refers to the outer raceway Refers to the radial direction Refers to probability of survival s Refers to rolling element Refers to body I Refers to body 11
mm3 (in.3)
mm (in.) rad,'
N/mm2 (psi) Nlmm2 (psi) rad,' rad,' radlsec rad/sec mml (in.l)
It has been considered that if a rolling bearing in service is properly lubricated, properly aligned, kept free of abrasives, moisture, and corrosive reagents, and properly loaded,then all causes of damage are eliminated saveone, material fatigue. Historically,rolling bearing theory postulated that no rotating bearing can give unlimited service, because of the probability of fatigue of the surfaces in rolling contact.As indicated in Chapter 6, the stresses repeatedly acting on these surfaces can be extremely high as compared to other stresses acting on engineering structures. In the lattersituation, some steels appear to have an endurance limit, as shown in Fig. 18.1. This endurance limit is a level of cyclically applied, reversing stress, which, if not exceeded, the structure will accommodatewithout fatigue failure. The endurance limit for structural fatigue has been established by rotating beam and/or torsional testing of simple bars for various materials. In Chapter 23, the concept of a fatigue endurance limit for rolling bearings will be discussedin detail as well as thecorrelation of structural fatigue with rolling contactfatigue. In thischapter, the concept of rolling contact fatigue and its association with bearing load and life ratings is covered. Rolling contact fatigue is manifested as a flaking off of metallic particles from the surface of the raceways and/or rolling elements. For well lubricated, properly manufactured bearings, this flaking usually commences as a crack below the surface and is propagated to the surface, eventually forming a pit or spa11 in the loadcarrying surface. Lundberg et al. El8.11 postulated that it is the maximum orthogonal shear stress T~ of Chapter 6 that initiates the crack and that this shear stress occurs
Logarithm of number of stress cycles to fatigue failure
N
at depth x. below the surface. Figure 18.2 is a photograph of a typical fatigue failure in a ball bearing raceway. Figure 18.3, taken from reference E18.21, indicates the typical depth in a spalled area. Not all ,researchers accept the maximum orthogonal shear stress as the s i ~ i ~ c a stress n t initiating failure. Another criterion is the Von Mises distortion energy theory, which yields a scalar “stress” level of similar magnitude to the double amplitude; that is, of the maximum
18.2. Rolling bearing fatigue failure.
18.3. Characteristics of a fatiguespalledarea. Photographs of a typical fatigue spall showing sections cut through spall.
688
FATIGUE LIF'E
L ~ B E R G  P ~ ~ G THEORS R E ~ AND RATING
STAND~~S
orthogonal shear stress. Moreover, the subsurface depth at which this value is a maximum is approximately 50% greater than for ro.According to references [18.2]? this greater depth for failure initiation appears to be verified. Lundberg et al. [18.1] postulated that fatigue cracking commences at weak points below the surface of the material. Hence, changing the chemical composition, metallurgical structure, and homogeneity of the steel can significantly affect the fatigue characteristics of a bearing, all other factors remaining the same. In referring to weak points, one does not include macroscopic slag inclusions, which cause imperfect steel for bearing fabrication and hence premature failure. Rather microscopic inclusions and metallurgical dislocations that are undetectable except by laboratory methods are possibly the weak points in question. Figure 18.4, taken from reference E18.21, shows a firacture failure at weak points developed during rolling. This type of experimental study tends to confirm the LundbergPalmgren theory insofar as failure that initiates at weak points. That the weak points are those at a specified depth below the rolling contact surface, rather than at other depths or even at the surface, will be discussed later.
Even if a population of apparently identical rolling bearings is subjected to identical load, speed, lubrication, and environmental conditions, all the bearings do not exhibit the same life in fatigue. Instead the bearings fail according to a dispersion such as that presented in Fig. 18.5. Figure 18.5 indicates that thenumber of revolutions a bearing may accomplish with 100% probability of survival, that is, S = 1, in fatigue is zero. Alternatively?the probability of any bearing in the population having infinite endurance is zero. Forthis model, fatigue is assumed to occur when the first crack or spa11 is observed on a loadcarrying surface. It is apparent, owing to the time required for a crack to propagate from the subsurface depth of initiation to the surface, that a practical fatigue life of zero is not possible. This will be discussedin greaterdepth later; however, for the purpose of discussing the general concept of bearing fatigue life, Fig. 18.5 is appropriate. ince such a life dispersionexists, bearing manufacturers have chosen to use one or two points (or both) on the curve to describe bearing endurance. These are 11
Llo the fatigue life that 90% of the bearing population will endure. LSothe median life, that is, the life that 50% of the bearing population will endure.
Probability of survival, S
18.5. Rolling bearing fatigue life distribution.
In Fig. 18.5, L,, = 5L10 approximately. This relationship is based on fatigue endurance data for all types of bearings tested and is a good rule of thumb when more exact information is unavailable. The probability of survival S is described as follows: (18.1)
in which % is the number of bearings that have successively endured L, revolutions of operation and % is the totalnumber of bearings under test. Thus if 100 bearings are being tested and 12 bearings have failed in fatigue at LIZrevolutions, the probability of survival of the remaining bearings is S = 0.88. Conversely, a probability of failure may be defined as follows: S = l  S
(18.2)
Bearing manufacturers almost universally refer to a “rating life” as a measure of the fatigue endurance of a given bearing operation under given load conditions. This rating life is the estimated L,, fatigue life of a large population of such bearings operating under the specified loading.
In fact, it is not possible to ascribe a given fatigue life to a solitary bearing application. One may howeverrefer to the reliability of the bearing. Thus, if for a given application using a given bearing, a bearing manufacturer will estimate a rating life, the manufacturer is, in effect, stating that the bearing will survive the rating life (Llorevolutions) with 90% reliability. Reliability is therefore synonymous with probability of survival. Fatigue life is generally stated in millions of revolutions. As an alternative it may be and frequently is given in hours of successful operation at a given speed. An interesting aspect of bearing fatigue is the life of multirow bearings. As an example of this effect, Fig. 18.6 shows actual endurance data of a group of singlerow bearings superimposed on the dispersion curve
Number of bearings failed, I%;
18.6. Fatigue life comparison of a singlerow bearing to a tworow bearing. A group of singlerow bearings programmed for fatigue testing, was numbered by random selection,no. 130, inclusive. Theresultant lives, plotted individually, give the upper curve. m e lower curve results if bearings nos. 1and 2, nos. 3 and 4,nos. 5 and 6, and so on were considered tworow bearingsand the shorter life of the two plotted as the life of a tworow bearing.
F A ~ I LIFE G ~ L ~ B E R G  P ~ ~ GTHEORY R E ~ AND
QS
DS
of Fig. 18.5. Next consider that the testbearings are randomly grouped in pairs. The fatigue life of each pair is evidently the least life of the pair if one considersa pair is essentially a doublerow bearing. Note from Fig. 18.6 that the life dispersion curve of the paired bearings falls below the curve for the single bearings. Thus, the life of a doublerow bearing subjected to the same specified loadingas a singlerow of identical design is less than the life of a singlerow bearing. Hence in thefatigue of rolling bearings, the product law of probability E18.31 is in effect. en one considers the postulated cause of surface fatigue, the physical truth of this rule becomes apparent. If fatigue failure is, indeed, a fbnction of the number of weak points in a highly stressed region, then as the region increases in volume, the number of weak points increases and the probability of failure increases although the specific loading is unaltered. This phenomenon is further explained by Weibull [18.4, 18.53.
In a statistical approach to the static failure of brittle engineering materials, Weibull [18.5] determined that the ultimate strength of a material cannot be expressed by a single numerical value and that a statistical distribution was required for this purpose. The application of the calculus of probability led to the fundamental law of the Weibull theory: (18.3)
E~uation(18.3) describes the probability of rupture 3 due to a given distribution of stress CT overvolume 13 in which n(u) is a material characteristic. Weibull's principal contribution is the determination that structural failure is a function of the volume under stress. The theory is based on the assumption that the initial crack leads to a break. In the fatigue of rolling bearings, experience has demonstrated that many cracks are formed belowthe surface that do not propagate to the surface. hus Weibul19stheory is not directly applicable to rolling bearings. Lundberg et al. [18.1] theorized that consideration ought to be given to the fact that the probability of the occurrence of a fatigue break should be a function of the depth zo below the loadcarrying surface at which the most severe shear stress occurs. The Weibulltheory and rolling bearing statistical methods are discussed in greater detail in Chapter 20. According to Lundberg et al. [18.1]let r(n)be a function that describes the condition of material at depth x after n loadings. Therefore dr(n)is the change in thatcondition after a small number of d n subsequent load
ings. The probability that a crack will occur in the volume element A3 at depth a: for that change in condition is given by
s(n) = g[r(n)]dr(n)AQ
(18.4)
Thus, the probability of failure is assumed to be proportional to the condition of the stressed material, the change in thecondition of the stressed material, and,the stressed volume, Themagnitude of the stressed volume is evidently a measure of the number of weak points under stress. In accordance with equation (18.4), S(n) = 1  $n) is the probability that thematerial will endure at least n cycles of loading. The probability that the material will survive at least n + dn loadings is the product of the probabilities that it will survive n load cycles and that the material will endure the change of condition dr(n).In equation format, that is
Rearranging equation (18.5) and taking the limit as dn approaches zero yields (18.6)
Integrating equation (18.6) between 0 and N and recognizing that AS(0) = 1gives (18.7)
or
I
(18.8)
By the product law of probability, it is known that the ~roba~ility S ( ~ the entire volume 3 will endure is
S ( N ) = A,S(N)
X
A,S(N) *
* *
(18.9)
Combining equations (18.8) and (18.9) and taking the limit as A 3 approaches zero yields
(18.10)
Equation (18.10) is similar in form to Weibull's function equation (18.3) except that G[I'(n)]includes the effect of depth z on failure. Alternatively, (18.10) could be written as follows: 1 In  = f(ro,N , Z,]Q S
(18.11)
in which rois the maximum orthogonalshear stress,zo is the depth below the loadcarrying surface at which this shear stressoccurs, and N is the number of stress cycles survived with probability 5. It can be seen here that ro and zo could be replaced by another stressdepth relationship. Lundberg et al. [18.1] empirically determined the following relationship, which they felt adequately matched their test results: (18.12) Furthermore, the assumption was made that the stressed volume was effectively bounded by the width 2 a of the contact ellipse, the depth zo, and the length g of the path, that is, T?
 azo%
(18.13)
~ubstitutingequations (18.12) and (18.13) into (18.11) gives 1 ln S

(18.14)
Today it is known that a lubricant film fully separates the rolling elements from the raceways in an accurately manufactured bearing that is properly lubricated. In thissituation, the surface shear stress in rolling a contact is generally negligible. Considering the operating conditions and the bearings used by Lundberg and Palmgren in the 1940s to develop their theory, it is probable that surface shear stresses of magnitudes greater than zero occurred in the rolling elementraceway contacts. It has been shown by many researchers that, if a surface shear stress occurs in addition to the normal stress, the depth at which the ma~imum subsurface shear stress occurswillbecloser to the surface than zo. Hence, the use of zo in equations (18.12)( 18.14), must be questioned considering the LundbergPalmgren test bearings and probable test conoreover, if zo is in question, then the use of a in the stressed
volume relationship must be reconsidered. This problem is covered in Chapter 23. If the number of stress cycles N equals uL, in which u is the number of stress cycles per revolution and L is the life in revolutions, then
More simply, for a given bearing under a given load, 1 ln=AL~
(18.16)
s
or
In In
1  = e In L, S
+ ln A
(18.17)
Equation (18.17) defines what is called a Weibull distribution of rolling bearing fatigue life. The exponent e is called the Weibull slope. It has been found experimentally that between the L, and LG0lives of the bearing life distribution, the Weibull distribution fits the test data extremely well (see Tallian [18.61). From equation (18.17) it can be seen that In In l/s vs InI; plots as a straight line. Figure 18.7 shows a Weibull plot of bearing test data. It should be evident from the foregoing discussion and Fig. 18.7 that theWeibull slopee is a measure of bearing fatigue life dispersion. Fromequation (18.17) it can be determined that theWeibull slope for a given test group is given by
Life, revolutions x lo6
18.7. Typical Weibull plot for ball bearings (reprinted from [18.1]).
In e =
In (l/s,) ln (1/S2) L In 2 L2
(18.18)
in which (L,, S,) and (I;,, s,) are any two points on the best straight line passing through the test data.This best straight line may be accurately determined from a given set of endurance test data by using methods of estreme value statistics as described by Lieblein [18.7]. ~ccordingto Lundberg et al. [18.1, 18.81, e = 10/9 for ball bearings and e = 918 for roller bearings. These values are based on actual bearing endurance data from bearings fabricated from throughhardened AIS1 52100 steel. Palmgren [l8.9] states that for commonly used bearing steels, e is the range 1.11.5. For modern, ultraclean, vacuumremelted steels, values of e in the range 0.7 to 3.5 have been found. The lower value of e indicates greater dispersion of fatigue life. At L = Ll0, s = 0.9. Setting these values into equation (18.17) gives
In In
1 = e 1n Llo + 1nA 0.9
(18.19)
Eliminating A between equations (18.17) and (18.19) yields 1 In =
s
( ~ )
“ 1 In0.9
(18.20)
or 1 = 0.1053 ln 
s
(~~
(18.21)
Equation (18.21) enables the estimation of L,, the bearing fatigue life at reliability S (probability of survival) once the Weibull slope e and “rating life” have been det~rminedfor a given application. The equation is valid between S = 0.93 and S = 0.40a range that is useful for most bearing . a~plications Ze ~ 8 . ~A . 209 radial ball bearing in a certain application yields a fatigue life of 100 million revolutions with 90% reliability. “hat fatigue life would be consistent with a reliability of 95%?
7
1 In  = 0.1053
(18.21)
s
10f9
1 ln = 0.1053 0.95 106) (10,"; L,
=
52.2
X
lo6 revolutions
E ~ ~ 18.~2. ~Of aZgroup @ of 100 ball bearings on a given application 30 have failed in fatigue. Estimate the L life which may be expected of the remaining bearings. At the moment 70 bearings remain, the relative number of surviving bearings is 70 sa="
100

0.70
The corresponding consumed lifeis obtained from 1 In  = 0.1053 Sa
( ~ ) 1019
(18.21)
1 In = 0.1053 0.70
M e r additional L,, life of the surviving 70 bearings has been attained, the number of surviving bearings is 0.9 x 70 or 63. The relative number of surviving bearings is 0.63. The correspondingtotal life is given by. 1 In  = 0.1053 %b
In
1 0.63
~
=
(2)
1019
0.1053
Li0 = L,

La
le 1$.~. A group of ball bearings has an Llo life of 5000 hr in
~
~
T LIFE: I GL ~ ~~ E R G  P ~ ~ G THEORY R E N ANI)
a given application. The bearings have been operated for 10,000 hr and some have failed. Estimate theamount of additional Llo life that can be expected from the remaining bearings. The relative number of bearings attaining or exceeding lifeLa is Sa and
After the additional Llo life is attained, the relative number of bearings remaining is s, = 0.9Sa corresponding to life Lb.
(2
In 1 = 0.1053 $b
since 8, = 0.9Sa, 1 1 In = 1nsb 0.9
+ ln1 sa
Therefore, 1 In 
sa
+ 0.1053 ==
(~~
By subtraction
or (L& + The additional Llo life is given by
LiO
=
 La
X
0.1053
(18.21)
In equation (6.71) it was established that at point contact (6.71)
More simply, ro =
Tvmax
(18.22)
in which T is a function of the contact surface dimensions, that is, bla (see Fig. 6.14). From equation (6.47) the maximum compressi~e stress within the contact ellipse is (6.47)
Furthermore, from equations (6.38) and (6.40) a and b are (6.38) (6.40)
in which (18.23)
By equation (6.58)
in which [ is a function of bla per equation (6.72) and (6.14). ~ubstitutingequations (6.47) and (6.40) into (18.15)yields (18.24)
Letting d equal the raceway diameter, then 2 =
and
TG?
(18.25) Rearranging equation (18.25)
From equations (6.38) and (6.40),
Q
EO~P 3a*(b*)2

"
ab2
(18.27)
Create the identity
and substituting equations (18.27) and (18.28) into (18.26) yields In
S
Chl
[
Ed>=P
]
3a*(b*)2
(c+h1)/2
):(
(ch1)/2
( ~ )
(ch+1)/2
dD2hueLe (18.29)
~ubstitutingequation (6.38) for the semimajor axis a in point contact into equation (18.29):
(18.30) Rearranging equation (18.30) gives
Equation (18.31) can be further rearranged. Recognizing that the probability of survival s is a constant for any given bearing application,
~
~ CAPACITY ~ (eh+2)/3
( ~ ) Letting T
IzI
=
\
ILIFE OF CA ROLL^^ CONTACT
[
T'dD 2hue
(18.32)
ghl('*)cl(b*)c+hl
Tl and J
= J1
when bla
=
1, then
I
 [(~~ ($)
]
h (1L ) x p ) ( 2 ~ + h  2 ) / 3 (,*)cl('*)c+hlE
(18.33) f)" ( 3  h )
Further rearrangement yields (18.34)
in which A, is a material constant and
[(g)'($)
(L)xp)(2e+h2)/3 3/(ch+2)
h1
@
=
x
(,*)cl(b*)c+hl
(18.35)
For a given probability of survival the basic dynamic capacity of a rolling elementraceway contactis defined as thatload whichthe contact will endure for one million revolutions of a bearing ring. Hence, basic dynamic capacityof a contact is Qc = A1@j(2c+h5)/(ch+2)
(18.36)
P'or a bearing of given dimension,by equating equation (18.34)to (18.36), one obtains QL(3e)/(ch+2)
=
Qc
(18.37)
or
( ~ )
(eh+2)/(3e)
1; =
(18.38)
Thus for an applied loadQ and a basic dynamic capacity Qc (of a contact), the fatigue life in millions of revolutions may be calculated. Endurance tests of ball bearings [18.1] have shown the loadlife exponent to be very close to 3. Figure 18.8 is a typical plot of fatigue life vs load for a ball bearing. The adequacy of the value of 3 was substantiated through statistical analysis by the U.S. National Bureau of Standards [18.11].Equation (18.38) thereby becomes
10
5 4
5u
3
$ 2 0
A
1
0.5
0
Life, revolutions x lo6
18.8. Load vs life for ball bearings (reprinted from [lS.l]>.
(18.39) This equation is also accurate for roller bearings having point contact. Since e = 10/9 for point contact, therefore
ch=8
(18.40)
Evaluating the endurance test data of approximately 1500 bearings, Lundberg et al. [18.1] determined that c = 31/3 and h = 713. Substituting the values for c and h into equations (18.35) and (18.36), respectively, gives
= A1(9D1.'
(18.42)
Recall that for a rollerraceway point contactin a roller bearing,
(2.38, 2.40) Therefore, I)
I)
Y
D
 ZpF(p) = 1   + 7 +2 2 R  1 . 4  y 2r
Also, from equation (2.37),
(18.43)
C C A F ' A C I ~AND LIFE OF A ~
O
L CONTACT L ~ ~
D D Y 2ip=1.+~"2 2R I Ty
D 2r
(18.44)
Adding equations (18.43) and (18.44) gives (18.45)
Subtracting equation (18.43) from (18.44)yields
:(
:)
[I  F ( p ) l  2ip = D   2
(18.46)
From equation (18.45), (18.47)
At this point in the analysis define SZ as follows: (18.48)
Let
Also recognize that d in (18.41) is given by
d = d&
T y)
(18.50)
Therefore, substituting equations (18.49) and (18.50) into (18.41) yields
Lundberg et al. [18.1]determined that within the range corresponding to ball and roller bearings, very nearly is given by
$2,
Figure 18.9 from reference [18.1]establishes the validity of this assumption. ~ubstituting(18.52) and (18.47) into (18.51) gives
di>
= 0.0706
[~1 ~] 222
X
r
( ~ ) 0.3
0.41
(1
7)1.39
(18.53)
~4l’~
The number of stress cycles u per revolution is the number of rolling elements which pass a given point (under load) on the raceway of one ring while the other ring has turned through one complete revolution. Hence fromChapter 8 the number of rolling elements passing a point on the inner ringper unit time is
=
0.5Z(l
+ y)
(18.54)
For the outer ring, U, =
0,5Z(l  7)
(18.55)
or
in which the upper sign refers to the innerring and the lower signrefers to the outer ring.
F I G ~ E
L18.11).
Substitution of equation (18.56) into (18.53) gives the following expression for @: 0.41
@ =
0*089
(1 T (1
($x 5)
741.39 0.3 ,,,)1/3
( ~ )
2lf3 (18.57)
Combining equation (18.57) with (18.42) yields an equation for Q,, the basic dynamic capacityof a point contact, in terms of the bearing design parameters:
2R
0.41
(1 ,,,)1.39 (L)o'3 D1.821/3 (18.58) (1 "r cos cy
Test data of Lundberg et al. El8.11 resulted in an average value A = 98.1 in mm N units (7450 in in. * lb units) for bearings fabricated from 52100 steel throughhardened to Rockwell C = 61.764.5. This value strictly pertains to the steel quality and manufacturing accuracies achievableat that time, that is, up to approximately 1960. Subsequent improvements in steelmaking and manufacturing processes have resulted in significant increases in this ball bearing material factor. This situation will be discussed in detail later in thechapter. 0
Equation (18.29) is equally valid forline contact. It can be shown forline approaches the limit 2/71= contact that as bla approaches zero, (a*)(~*)2 Therefore, the following expression can bewritten for line contact:
(18.59) In a manner similar to that used for point contact, it can be developed that
in which
It can be further established that (18.63) and
QC
L)29/2717/9~V4
=l3
(18.64)
in which l3 = 552 in mm N units (49,500 in in. lb units) for bearings fabricated from throughhardened 52100 steel. As for ball bearings, the material factor for roller bearings has undergone substantial increase since the investigations of Lundberg and Palmgren. This situation will be covered in detail later in thechapter. For line contact it was determined that 4
( ~ ) 4
L
=
(18.65)
and further, from Lundberg et al. [18.8],that c  h + l 4 2e Since e
=
(18.66)
9/8 for line contact, from (18.66)
which is identical for point contact,establishing that c and h are material constants. Some roller bearings have fully crowned rollers such that "edge loading" does not occurunder the probable maximum loads,that is, modified line contact occurs under such loads. Under lighter loading, however, point contact occurs. For such a condition, equation (18.64) should yield the same capacity value as equation (18.58). Unfortunately,this is a deficiency in the original LundbergPalmgren theory owing to the calculational tools then available. This situation can be rectified for the sake of continuity by utilizing the exponent % in lieu of (and % in lieu of $) in equation (18.64). Also, the value of constant l3 becomes 488 in mm * N units (43800 in in. lb units) for roller bearings fabricated from
+
throughhardened 52100 steel. Again, this material factor strictly perera. tains to the roller bearings of the Lundberg~alm~en
According to the foregoing analysis, the fatigue life of a rolling elementraceway point contact subjected to normal load Q may be estimated by
(~) 3
L
=
(18.39)
in which I; is in millions of revolutions and
D1*8%1’3(18.58) For ball bearings this equation becomes
in which tKe upper signs referto the inner raceway contact andthe lower signs refer to the outer raceway contact. Since stress is usually higher at the inner raceway contact than at the outer raceway contact, failure generally occurs on the inner raceway first. This is not necessarily true for selfaligning ball bearings for whichstress is high on the outer raceway, it being a portion of a sphere. A rolling bearing consists of a plurality of contacts. For instance, a point on the inner raceway of a bearing with inner ring rotation may experience a load cycle as shown in Fig. 18.10. Although the maximum load and hence masimum stress is s i ~ i f i c a n in t causing failure, the statistical nature of fatigue failure requiresthat the load history be considered. Lundberg et al. E18.11 determined empirically that cubic mean
* Palmgren recommended reducing this constant to 93.2 (7080) for
singlerow ball bearings and to 88.2 (6700)for doublerow, deepgrooveball bearings to account for inaccuracies in raceway groove form owing to the manufacturing processes at that time. Subsequent improvements in the steel quality and in the manufacturingaccuracies have seen the material factor increase significantlyfor groovetype bearings. This increaseis accommodated by a factor b,, that augments the aboveindicated material factors; this is discussed in detail later in this chapter.
QmtW
x3 1 Revolution 
18.10, Tmical load cycle for a point on inner raceway of a radial bearing.
load fits the test datavery well for point contact. Hence for a ring which rotates relative to a load, (18.68) In the terms of the angular disposition of the rolling element, (18.69) The fatigue life of a rotating raceway is therefore calculated as follows:
( ~ ) 3
L P
=
(18.70)
Each point on a raceway that is stationary relative to the applied load is subjected to virtually a constant stress amplitude. Only the space between rollingelements causes the amplitude to fluctuate with time. From equation (18.31) it can be determined that the probability of survival of any given contact point on the nonrotating raceway is given by
According to the product law of probability, the probability of failure of the ringis the product of the probability of failure of the individual parts; hence since 3e = (e  h + 2113,
709
F A T I G ~LIF'E OF A ROLLlNG BEABING
(18.72) in which Qeuis defined as follows:
In discrete numerical format, equation (18.73) becomes (18.74) From equations (18.74) and (18.39),the fatigue life of a nonrotating ring may be calculated by
LV
=
( ~ ) 3
(18.75)
To determine the life of an entire bearing the lives of the rotating and nonrotating (inner and outer or vice versa) raceways must be statistically combined accordingto the product law. The probability of survival of the rotating raceway is given by (18.76) Similarly for the nonrotating raceway 1 In  = KvL;
(18.77)
S V
and for the entire bearing 1 In  = (Kp+ KJL"
s
Since $, = S,,
=
S,the combination of equations (18.76)(18.78) yields 1; = (1;;" +
Since e
=
(18.78)
10/9 for point contact equation (18.79) becomes
( 18.79)
1; =
(1;;l.ll
+ L;l.l1)0.9
(18.80)
Based on the preceding development,it is possible to calculate a rolling bearing fatigue life in point contact if the normal load is known at each rolling element position. These data may be calculated by methods established in Chapters 7 and 9. It is seen that the bearing lives determined according to the methods given above are based on subsurfaceinitiated fatigue failure of the raceways. Ball failure was not consideredapparently because it was not frequently observed in the LundbergPalmgren fatigue life test data.It was rationalized that, because a ball could changerotational axes readily,the entire ball surface was subjected to stress, spreading the stress cycles over greater volume consequently reducing the probability of ball fatigue failure prior to raceway fatigue failure. Some researchers have since observed that in most applications, each ball tends to seek a single axis of rotation irrespective of original orientation prior to bearing operation. This observation tends to negate the LundbergPalmgreln assumption. It is perhaps more correct to assume that Lundberg and Palmgren did not observe significant numbers of ballfatigue failures because at that time the ability to manufacture accurate geometry balls of good metallurgical parameters exceeded that for the corresponding raceways. The ability to accurately manufacture raceways of good quality steel has consistently improved since that era, and for many ball bearings of current manufacture the incidence of ballfatigue failure in lieu of racewayfatigue failure, particularly in heavily loadedbearing applications, is frequently observed. Obviously, the accuracy of ball manufacture, and ball materials and processing has also improved; however, the gap has narrowed significantly. It is now clear that bearing fatigue life based upon ballfatigue failure also needs to be considered.
~~.~
Ze The 209 radial ball bearing of Example 7.1 is operated at a shaft speed of 1800 rpm. Estimate theLl0 life of the bearing.
.Z=9
2.5
2.1
Ex.
Ex. y = 0.1954 L) = mm 12.7
f
=
0.52
(0.5 in.)
Ex. 2.1 Ex. 2.2
F ~ T I LIFE G ~ OF A ~ O L L I ~BG E
~
~
11
G
0.41
=
93.2 X
(0.1954)0.3( 12.7)1*8(9)*3
=
7058 N (1586 lb)
=
93.2 X
=
OS4'
(1+ 0.1954)1.39 (1  0.1954)1'3
(0.1954)0.3(12.7)1.8(9)1'3
13,970 N (3140 lb)
From Example 7.1,
0" 4536 N 40" 2842 N 80" 58 N 120" 0 160" 0
(1019 lb) (638.6 lb) (13.0 lb)
Since the inner ring rotates,
Qei
=
(;E
(18.68)
Q:)1'3
=
{$[(4536)3 + 2 X (2842)3 + 2 x (58)3 + 0
=
2475 N (556.3 lb)
+ 0]}1/3
3
(18.70)
=
(
~
)
=
23.2 3 million revolutions
7121
FATIGUE LIFE L ~ B E R G  P ~ M GTHEORY ~ N AND RATING S T A N D ~ ~ S
(18.74) =
{&'((4536)10/3 +2 X
=
(58)10/3 +
X
(2842)10/3+ 2
o + 01p3
2605 N (585.3 lb) 13970 2605
Lo = 
(18.75)
154.4 million revolutions
+ L,l.ll)0.9 L " (Ll~l*ll
(18.79)
=
[(t23.2)l*11+ (154.4)1*11]0*Q X lo6
=
20.9
X
lo6 revolutions
or L = 20*9 x lo6= 60 X 1800
hr
In lieu of the foregoing rigorousapproach to the calculation of bearing fatigue life, an approximate method was developed by Lundberg et al. [lS.l] for bearings having rigidly supported rings and operating at moderate speeds. It was developed in Chapter 7 that (7.15) and n = 1.5 for point contact. This equation may be substitute^ into equation (18.69) forQePto yield
*This Llo fatigue life was calculated according to the basic LundbergPalmgren theoryand is based upon the standard bearing materials and manufactu~ngprocesses employeduntil appro~imately1960. To be able to compare the numerical exampleresults with the graphical data of LundbergPalmgren as well as with graphic data generated to demonstrate the eEects of nonstandard load distributions, all numerical examples in this and the next two sections are calculated usingthe basic Lundberg~almgren theory. In Chapter 23 the eflects on fatigue life of subsequent improvements in materials and manufactu~ngprocesses are discussed.
F A T ~ G LIFE ~ OF A ~ O L L BEABZRIG ~ G
or Qey
(18.82)
= QmaxJl
Similarly for the nonrotating ring,
or
Table 18.1 gives values of J , and J 2 for point contact and various values for E . Again referring to Chapter 7, equation (7.66) states for a radial bearing
18.1. J , and J2 for Point Contact SingleRow Bearings €
J1
€11
0 0 0.72330.1 0.6925 0.5 0.4275 0.5 0.2 0.4806 0.6 0.6231 0.5983 0.4 0.5150 0.6215 0.3 0.5986 0.3 0.7 0.5411 0.4 0.5625 0.4 0.9 0.5808 0.6 1.0 0.5970 0.7 0.8 0.6104 0.6248 0.9 1.0 0.6372 0.6652 1.24 0.7064 1.67 2.5 0.7707 75 5 00 1
DoubleRow Bearings J2
€1
0 0.4608 0.5100 0.5427 0.5673 0.6331 0.8 0.6105 0.5875 0.6045 0.6196 0.6330 0.6453 0.6566 0.682 1 0.7190 0.7777 0.8693 1
J1
0.2 0.1 0.64530.6248 0 0.6566 0.6372
JZ
714
F ~ T I LIFE: G ~ L
~
~
E
Fr
=
R
~
P
~
~
G
R
2QmmJr cos a
~ ING N s
s (7.66)
Setting F, = C,, the basic dynamic capacity of the rotating ring (relative to the applied load),and substituting for Qmmaccording to equation (18.82) gives
c, =
Jr
Qcp
cos a Jl
(18.85)
Basic dynamic capacity is defined here as that radial load that 90% of a group of apparently identical bearing rings will survive for one million revolutions. Table7.1 and 7.4 give values of J,. Similarly? forthe nonrotating ring
C,
=
QCV 2 cos a Jr
(18.86)
Jz
At
E =
0.5, which is a nominal value for radial rolling bearings, C,
=
0.407Qc, 2 COS a
(18.87)
C,
=
O.389Qc, 2 COS a
(18.88)
Again, the product lawof probability is introduced to relate bearing fatigue life of the components. From equation (18.31) it can be established that (18.89)
Similarly? 1
(18.90)
1 In  = (K,
s
+ K,)C10’3
(18.91)
Combining equations (18.89)( 18.91)determines
C
= (C;10/3 + c,10/3)0.3
(18.92)
FATIGUE LIFE OF A ROLLING BEARING
in which C is the basic dynamic capacity of the bearing. Rearrangement of equation (18.91) gives
A similar approach may be taken toward calculationof the effect of a plurality of rows of rolling elements. Consider that a bearing with point contact has two identical rows of rolling elements, each row being loaded identically. Then for each row the basic dynamic capacity is 6 , and the basic dynamic capacityof the bearing is 6. From equation (18.93),
Hence, a tworow bearing does not have twice the basic dynamic capacity of a singlerow bearing because of the statistical nature of fatigue failure. In general, for a bearing with point contact having a plurality of rows i of rolling elements,
of one row.Equations (18.85) in which Ck is the basic dynamic capacity and (18.86) can now be rewritten as follows: (18.95)
or C,
=
0.407Qc,i0*72COS a
C,
=
Jr Qc,i 0*7.Z COS a 
(E =
0.5)
(18.96) (18.97)
J2
C,
=
0.389Qc,i0.72COS a
(E
=
0.5) (18.98)
~ubstitutionof QCfrom equation (18.58)into (18.95) givesthe following expression for basic dynamic capacityof a rotating ring: 0.3('
cos
a)0.7 z U 3 D 1.8 Jl
(18.99)
(18.100) For the nonrotating ring, 0.3(i cos
a)0.7
g2l3D 1.8 5 J2
(18.1031)
(18.102) According to equation (18.93) the basic dynamic capacityof the bearing assembly is as follows for E = 0.5:
c = fc(i
cos a)0.7z213D1.8* (18.103)
in which
(18.104) Generally, it is the innerraceway that rotates relative t o the load and therefore
(18.105) For ball bearings, equation (18.105) becomes
*ANSI [18.101 recommends using D raised to the 1.4 power in lieu of 1.8 for bearings having ballsof diameter greater than 25.4 mm (1in.).
717
F A T I G LIFE ~ OF A ROLLING BEARING
f , = 39.9* {I
+
[ ("y),.i. (fi ") 1.04 1 + Y
fo
2f0  I  1
x 2L
}
0.41 10/3 0.3 ~
(18.106) Equation (18.103)in conjunction with (18.106)is generally considered valid for ball bearings whose rings and balls are fabricated from AIS1 52100 steel heat treated at least to Rockwell C 58 hardness throughout. If the hardness of the bearing steel is less than Rockwell C 58, a reduction in the bearing basic dynamic capacity accordingto the following formula may be used:
C'
=
c
RC (=)
36
(18.107)
in which RC is the Rockwell C scale hardness. By using equations (18.103) and (18.106) the basic dynamic capacity? of a radially loaded bearing may be calculated. The pertinent Ll0 fatigue life formula is given below: (18.108) id which Fe is an equivalent radial load which willcause the same Ll0 fatigue life as the applied load. From equation (7.66) it can be seen that (7 66)
in which Fr is an applied radial load and Qmmis the maximum rolling element load. For a rotating ring, from equation (18.82) therefore,
*According to Palmgren 118.91 this factor can be as low as 37.9 (2880) for singlerowball bearings and 35.9 (2730) for doublerow deepgrooveball bearings to accountfor manufacturing inaccuracies. $The term basic d y n a ~ i ccapacity was created by Lundberg and Palmgren Il8.11. ANSI C18.10,18.121 uses the term basic load rating and IS0 [18.131 using basic dynamic load rating. These terms are interchangeable.
(18.109) in which QeEL is the mean equivalent rolling element load in a combined loading defined by Jr.At E = 0.5 [see Chapter 7 and equations (18.82) and (18.84)] loading is ideal and purely radial; therefore, (18.110) in which FeEL is the equivalent radial load. Similarly, for a nonrotating ring (18,111) The fatigue life of the rotating ringmay be described by (18.112) [see equations (18.20) and (18.89)( 18.91)]. Similarly,the nonrotating ring, (18.113) For the bearing, (18.114) Com~iningequations (18.112)(18.114) yields (18.115)
to (18.110) gives ~ q u a ~ i (18.109) on (18.11~) Similarly,
19
F A T I G LIFE ~ OF BEARXNG A RQLLJHG
(318.117) Substituting for Fep and Fewin equation (18.115) yieldsthe following expression for equivalent radial load:
(18.118)
In terms of an asial load Fa applied to a radial bearing:
(see Chapter 7 for evaluation of J,). In a manner similar to that developed for a radial load Fr,
Fa Qep
Fa
=
Jl
X 
Ja
=
J2
X
Ja
(18.120) (18.121)
Combining equations (18.110) and (18.120) yields (18.122) Similarly; equations from (18.111)and (18.12l), (18.123) Substitutin~equations (18.122) and (18.123) for Fepand Fev, respectively, in equation (18.115) gives
(18,124) In equations (318.118) and (18.124), for inner ring rotation, that is,
720
FATIGUE LIFE:
L ~ 3 E R G  P ~ MTHEORY G ~ ~ AND RATING S
T
~
with load stationary relative to the outer ring, C, = Ci and C, = C,. For pure radial displacement of the bearing rings ( E = 0.5); therefore,
[(E)3.33(E) ]
3.33 0.3
Fe
=
+
~r
(18.125)
For outer ring rotation, that is, with the inner ring stationary relative to load, C, = uC, and C, = CJu, in which u = ~ ~ ( 0 . 5 ) / ~ ~ (For 0 . 5this ). case in pure radial load,
Fe = VFr
(18.126)
in which
=
[
(
3.33 0.3
~
~
.
3
+
( ~ ]) 3
(18.127)
The factor V , which is a rotation factor, can be rearranged as follows
v=u
(18.128)
When Ci/Co approaches 0, then V = u = 1.04: for point contact. In the other extreme, when Ci/C, becomes infinitely large, V = l/u = 0.962 for point contact. Figure 18.11 shows the variation of V with CJC, for outer ring rotation. For most applications V = 1is sufficiently accurate. BothANSI[18.10] and IS0 [18.13]neglect the rotation factor and simply recommend the following equation for equivalent radial load:
Fe = X F r
+ YFa
Values of X and Y are given in Table 18.2 for radial ball bearings.
For a bearing loaded in pure thrust every rolling element is loaded ,equally as follows:
~
S
FATIGUE LIFE OF A ROLLING BEARING
CilCo
FIGURE 18.11. Rotation factor V vs CJC,.
For both the rotating and stationaryraceways, the mean equivalent rolling element load is simply Q as defined by equation (7.26). Setting Fa= C,, therefore,
C,,
=
Qe,Z sin a
(18.130)
C,,
=
Q,,Z sin a
(18.131)
Hence, by equation (18.58),
(18.132)
(18.133) In equations (18.132) and (18.133) the upper signs refer to an inner raceway and the lower signs to an outer raceway. The basic dynamic capacity of an entire thrust bearing assembly is given by
7
e,
3.1
* 3.1
* 3.1
*
a,
co
2 0
co
2
e
a,
00
2
rl
u3
03
2
d
2
d
@
2
u3
0
d d d d d
c(9
0
2
$3
0
m
0 u3
$3
0
d
724
C,
F A T I G LIFE: ~ L
=
98.1 {l +
~
B
E
R
[(,>,“’ (2
] }
0.41 3.33 0.3
X
l k y
P THEORY ~ ~ GANI) ~ RATING N S T A N I ) ~ ~
~
2rv  D, 2r,  D
x (2r,2r’ D)o*41yo.3(1 T y)1.39 (cos (1 + y)O33
tan &067D1.8
a ) O s 7
(18.134)
For ball bearings with inner ring rotation equation (18.134) becomes 3.33 0.3
1
(18.135) Lundberg et al. [18.1]recommended a reductionin the materialconstant to accommodate inaccuracies in manufacturing that cause unequal internal load distribution. Hence, equation (18.135) becomes
C,
=
88.2*(1  0.33 sin a)
0.41
x 
! ;2(

1)
(1
y)1*39 (cos
+ yy.33
a)0.7
tan d0.67D1.8 (18.136)
1x1 (18.136) as recommended by Palmgren [l8.9], the term (1  0.33 sin a) accounts for reduction in C, caused by added friction due to spinning (presumably). Thefollowing is the formula forbasicdynamiccapacity: C,
= f,(cos
a ) O s 7
tan d W 3 D 1.8T
(18.137)
for which it is apparent that (appro~imately)
*This value can be as high as 93.2 (7080)for thrustloaded angularcontact ball bearings. TAN81 [18.10] recommends using I) raised to the 1.4 power in lieu of 1.8 for bearings having ballsof diameter greater than 25.4 rnm (1in.).
FATIGUE L I m OF A ROLLING BEARING
f, = 88.2(1  0.33 sin a)
0.41
(18.138)
X
For thrust bearings with a 90" contact angle
6 , = fcZ2/3111*8
(18.139)
in which (approximately) f, = 59.1
[
1+
(b X fo
)
1.36 0.3 ~
26  1
2fi

1
(18.140)
For thrust bearings having i rows of balls in which %k is the number of rolling elements per row and C, is the basic dynamic capacity perrow, the basic dynamic capacityC, of the bearing may be determined as follows: (18.141)
As for radial bearings, the Llo life of a thrust bearing is given by 3
=
( ~ )
(18.142)
in which Feais the equivalent asial load. As before,
Fea = X F r
+ YF,
X and Y as recommended by ANSI are given by Table 18.3. Ze 18.5 The 218 angularcontact ball bearing of Example 9.1 is operated at 10,000 rpm under a 22,250 N (5000 lb) thrust load. Estimate theLlo fatigue life of the bearing for inner raceway rotation.
Z
=
16
Ex. 7.5
D
=
22.23 mm (0.875 in.)
Ex. 2.3
E 18.3. X and Y Factors for Ball Thrust Rearings Single Direction Bearings F ~ / F ~e
Double Direction Bearings"
Fa/Fr5 e
Fa/Fr > e
Rearing Type
x
Y
x
Y
x
Y
e
Thrust ball bearings with contact angleb a = 45" a = 60" a = 75"
0.66 0.92 1.66
1 1 1
1.18 1.90 3.89
0.59 0.54 0.52
0.66 0.92 1.66
1 1 1
1.25 2.17 4.67
~~
"Double direction bearings are presumed to be symmetrical. *For a = 90": Fr = 0 and Y = 1.
Ex. 2.6
d m= 125.3 mm (4.932 in.)
fi
= f, =
Ex. 2.3
0.5232
ai= 48.8"
Ex. 9.4
a, = 33.3"
Ex. 9.4
Qi
=
1788 N (401.7 lb)
Ex. 9.6
&,
=
2241 N (503.7 lb)
Ex. 9.6 (2.27)
 22.23 cos (48.8") = 0.1169
125.3
D 1*8ZV3(18.6'7) =
93.2
=
17,040 N (3830 lb)
Oe4'
(1 0. 1169)1.39 (1+ 0.1169)V3
*Strictly speaking, an angularcontact ball bearing with a' < 40" is classified as a radial bearing and would be rated by using f, = 93.2.
27
FATIGUE LIFE OF A ROLLING B M I N G
According to Fig. 9.5 at 110,000 rpm and 22,250N (5000lb)thrust load, this bearing operation approximates "outer raceway control." To account for spinning, the inner raceway capacity is reduced by a factor of (1  0.33 X sin ai). Qci(l  0.33 sin ai) 17,040[1  0.33 sin (48.8")] = 12,810 N (2879 lb) (18.70) Under pure thrust Qei = Q i.
(
Li = 1 ~ ~
=)
368 3
million revolutions
D cos a, Yo =

dm 22.23 cos (33.3") = 0.1483 125.3 (1 + (1
(
Qc = 93.2 2fo2f0  1)O"' =
93.2
(
1.8zl/3
L)
(18.67)
cosyoa. )0*3
(1  0.1483)1/3
cos (33.3") =
26,880 N (6040 lb) 3

(18.75)
(
~
~
=
"
L, =
or
1724 ~ million ~ revolutions ) 3
(L,171.11 + j;l.l1)0.9
=
[(368)111+ (1724)1.11]0*9 X
=
315
X
lo6 revolutions
(18.79)
lo6
728
FATIGmLIFE L ~ B E R G  P ~ G mO ~ R N Y AND ~
L=
315 X lo6 10,000 x 60
=
T $ T ~A N DG~ $
525 hr
The Llo fatigue life of a rollerraceway line contact subjected to normal load Q may be estimated by
( ~ ) 4
L
=
(18.65)
in which L is in millions of revolutions and =
552
1;,29/27l7/92  1f4
(18.64)
The upper signs refer to an inner raceway contact and the lower signs refer to an outer raceway contact. To account for stress concentrations due to edge loadingof rollers and noncentered roller loads, Lundberg et al. [18.8] introduced a reduction factor h such that
Based on their test results, the schedule of Table 18.4 for hi and h, was developed. Variation in h for line contact is probably due to method of roller guiding, for example, in some bearings rollers are guided by flanges that are integral with a bearing ring; other bearings employ roller widing cages. In lieu of a cubic mean roller load for a raceway contact, a quartic mean will be used such that
The difference between a cubic mean load and a quartic mean load is substantially negligible. The fatigue life of the rotating raceway is
LE 18.4. Values of hi and A, Contact Inner Line contact Modified line contact
Raceway 0.410.56 0.60.8
Outer Raceway 0.380.6 0.60.8
7~~
F A T I G ~LIFE OF A ROLLEYG BEARING 4
L P
=
( ~ )
(18.145)
As with pointcontact bearings, the equivalent loading of a nonrotating raceway is given by Q,, =
(z
1j=z
Qf)
l/4t?
=
j=1
(L [* 2T
u4.5
Q$5
d+)
(18.146)
0
The life of the stationary raceway is
( ~ )
I;,=
4
(18.147)
As with pointcontact bearings, the life of a roller bearing having line contact is calculated from
Thus, if each roller loadhas been determined by methods of Chapters 7 and 9, the fatigue life of the bearing may beestimated by using equations (18.144) and (18.148).
~~.~
Ze Assuming modified line contact, estimate the Llo fatigue life of the 209 cylindrical rollerbearing of Example 7.3. Assume also inner raceway rotation.
Z
=
14
Ex. 2.7
D
=
10 mm (0.3937 in.)
Ex. 2.7
d m = 65 mm (2.559 in.)
I
=
+
0
25.71" 51.42"
180"
9.6 mm (0.378 in.)
Q* 1915 N (430.3 lb) 1348 N (302.9 lb)
0 0
Ex. 2.7 Ex. 2.7
Ex. 7.3
D cos a Y=
(2.27)
d m
10 65
=
From Table 18.4, use hi (,i
=
5524 (

(1 +
0.1538
A, = 0.61 (see also Table 18.10),
=
y)2g1272/9D291271 7/9z 114
(18.143)
y)W4
=
552
=
6381 N (1434 lb)
X
=
0.61
(18.144) =
{A[( 1 9 ~+)2(~13.4~3)~ +0+
=
1095 N (246 lb)
Li =
=
*
0])114
( ~ ) 4
(
=
QCo= 552A, =
552
X
(18.145) 1155 million revolutions
+ y)29/272 1 9 2~91271719z 114 (1  y)ll4
(18.143)
0.61
(9.6)719( 14rV4 =
9621 N (2162 lb) (18.146)
=
{&[(1915)4*5 + 2(1348)4*5+ 0
=
1148 N (258 lb)
+
* * *
OI)v4.5
31
FATIGUE LIFE OF A ROLLING BFXRING
( ~ ) 4
Lo
=
(18.147)
(L19/8+ L;9/8)8/9
(18.148)
=
[(1155)9/8+ (4937)g’8]8’gX
=
9.85 X
lo6
lo8 revolutions
To simplify the rigorous methodof calculating bearing fatigue life just outlined, an approximate method was developed by Lundberg et al. [18.1, 18.81 for roller bearings having rigid rings and moderate speeds. In a manner similar to point contact bearings, (18.149)
(18.150) (18.151)
J2=
(1
i’”[l  1 (1  cos +) 271“ *1 2E ~
g
d 5i l , ~
(18.152)
Table 18.5 gives values of J , and J2as functions of E. As for pointcontactbearings, equations (7.66), (18.85),and (18.86) are equally valid for radial roller bearings in line contact. Therefore at E = 0.5,
6,
=
O.377i7/’Q,2 cos a
(18.153)
C,
=
0.363i7/9&,2cos a
(18.154)
According to the product law of probability, (18.155) The reduction factor h accounting for edge loading may be applied to the entire bearing assembly. For line contact at one raceway and point contact at the other h = 0.54 if a symmetrical pressure distribution similar to that shown by Fig. 6.233is attained along the roller length. Figure 18.12, taken from reference E18.81 shows the fit obtained to the test data
LE 18.6. J1 and J2 for Line Contact Single Row €
J1
0 0.5287 0.5772 0.30.6079 0.7 0.6309 0.6495 0.9 0.6653 0.6792 0.6906 0.7028 0.7132 0.7366 0.7705 0.8216 0.8989 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.25 1.67 2.5 5 00
Double Row JZ
0 0.5633 0.6073 0.6359 0.6571 0.6744 0.6888 0.7015 0.7127 0.7229 0.7323 0.7532 0.7832 0.8301 0.9014
€1
€11
J,
J2
0.5 0.6
0.5 0.4
0.8
0.2 0.1
0.7577 0.6807 0.6806 0.6907 0.7028
0.7867 0.7044 0.7032 0.7127 0.7229
1
while using h = 0.54. Table 18.6 is a schedule for h for bearing assemblies. Using the reduction factor, A, the resulting expression for basic dynamic capacity of a radial roller bearing is
C
=
{ [ (1' 1);
207h 1 + X
1.04
~
'43/10"]
"}
2/9 ?2/9(
1
?)29/27
(1 1 y)**
(iZ cos a ) 7 / 9 Z 3 / 4 D 29/27
(18.156)
In most bearing applications the inner raceway rotates and
X
(iZ cos a)7/gZ3/4D29/27
(18.157)
As for pointcontact bearings an equivalent radial load can be developed and
FATIGUE LIFE OF A ROLLINGB ~ I N G 1
0.6 0.5 0.4
0.3 t,
i$ 0.2 0.1
0.05 Llo Fatigue life, revolution x lo6
18.121. L,,vs FIC for roller bearing. Test pointsare for an SKI? 21309 spherical roller bearing.
18.6. ReductionFactor h
Contact Condition
h Range
raceways Line contact at both Line contact at one raceway Point contact at otherraceway Modified line contact
0.40.5 0.50.6 0.60.8
The rotation factor V is given. by 219
V=v 1
+
(~) 912
(18.159)
in which u = J2(0.5)/J1(0.5).Figure 18.11 shows the variation of Vvvith CJC, for both point and line contact. ANSI [18,12] gives the same formula for equivalent radial load for radial roller bearings as for radial ball bearings. (Rotation factor V is once again neglected.)
Fe = XFr
+ YFa
(18.129)
X and Y for spherical selfaligning and tapered roller bearings are given in Table 18.7. The life of a roller bearing in line contact is given by (18.160)
For thrust bearings, Lundberg et al. [18.8] introduced the reduction factor q, in addition to A, to account for variations in raceway groove dimensions which may cause a roller from experiencing the theoretical uniform loading: 
Fa
(7.26)
"
Z sin
a!
According to reference [18.8], forthrust roller bearings q = 1  0.15 sin
(18.161)
a!
ons side ring the capacity reductions due to h and q, for thrust roller bearings in line contact, the following equations may be used for thrust bearings in which a! If: 90": C,
=
552Aqy2"
(1 7 y)29/27 ( I cos a!)7/9 tan a.Z3/4D29/27(18.162) (1 rt y)*4
In equation (18.162) the upper signs refer to the inner raceway?that is, k = i; the lower signs refer to the outer raceway, that is, k = 0. For thrust roller bearings in which a! = 90";
18.7. X and Y for Radial Roller Bearings Fa/Fr5 1.5 tan a X Singlerow bearing Doublerow bearing
1 1
Y 0 0.45 ctn a
Fa/Fr> 1.5 tan a
X 0.4 0.4 0.67
Y ctn a 0.67 ctn a
FATIGUE ROLLING LIFE OF A
BEARING
~~~
c,i = c,,  46ghy2/Y17/9D29/2723/4
(18.163)
I_
Equations (18.162) and (18.163) may be substituted into (18.155) to obtain the basicdynamiccapacity of a bearing row in thrust loading. Equation (18.164) may be used to determine the basic dynamic capacity in thrustloading for a thrust roller bearing having i rows and 2, rollers in each row:
1
(18.164)
The fatigue life of a thrust roller bearing can be calculated by the following equation: I; =
(~)*
(18.165)
According to ANSI E18.131 the equivalent thrust load may be estimated bY
Fea= mr4 YF,
(18.166)
Table 18.8 gives values of X and Y .
If a roller bearing contains rollers and raceways having strai then line contact obtains at each contact and the formulations of the id. If, however, the rollers have a curved preceding two sections are profile (crowned; see Fig. 1. of smaller radius than one or both of the conforming raceway profiles or if one or both raceways have a convex: profile and the rollers have straight profiles, then depending on the an18.8. X and Y for Thrust Roller Bearings
Contact Type
Bearing Single direction Double direction
a a a a a
< 90" 90" < 90" < 90" < 90" =
FJFr 5 1.5 tan Fr = 0 Fa/Fr > 1.5 tan FJFr 5 1.5 tan Fa/Fr > 1.5 tan
X
Y
a
0 0
1
cx
tan a 1.5 tan a tan a
a cx
1 1 0.67 1
gular position of a roller and its roller load, oneof the contact conditions in Table 18.9 will occur. Of the contact conditionsin Table 18.9, optimum rollerbearing design for any given application is generally achieved when the most heavily loaded roller is in modified line contact. As stated inChapter 6 thiscondition produces the most nearly uniform stress distribution along the roller profile, and edge loading is precluded. It is also stated in Chapter 6 that a logarithmic profile roller can produce an even better load distribution over a wider range of loading; however,this roller profile tends to be special. The more usual profile is that of the partially crowned roller. It should be apparent that theoptimum crownradii or osculations necessary to obtain modified line contact can only be evaluated for a given bearing after the loading has been established. Series of bearings, however, are often optimized by basing the crown radii or osculations on an estimated load, for example, 0.5C or 0.25C, in which C is the basic dynamic capacity. Depending on the applied loads, bearings in such a series may operate anywhere from pointto line contact at the most heavily loaded roller. Because it is desirable to use one rating method for a given roller bearing, and because in any given rollerbearing application it is possible to have combinations of line and point contact, Lundberg and Palmgren [18.8] estimated the fatigue life should be calculated from 1;
=L
(18.16"l)
(~)10'3
Note that 3 < 10/3 < 4. In equation
C
(18.168)
= VC,
C, is the basic dynamic capacity in line contact as calculated by equations (18.157) or (18.163). TABLE 18.9. RollerRacewayContact Inner at
Raceway Condition 1 2 3 4
5
Line Line Point Modified line Point
2ai > 1.51 2ai > 1.51 2a, < 1 1 5 2ai 5 1.51 2ai < 1
Outer Raceway Line Point Line Modified line Point
aaiis the semimajor axis of inner raceway contact ellipse. is the semimajor axis of outer raceway contact ellipse.
aob
Load
2a0 > 1.51 2a0 < 1 2a0 > 1.51 1 5 2a0 5 1.51
Heavy Moderate Moderate Moderate
2a0 Light Tlimitis considered at risk. See Fig. 23.36.
Therefore, the probability of survival in equation (23.48) is a diff'erentia1 value; that is, ASi. The probability of component survival is determined accordingto the product lawof probability; subsequently,equation (23.49), whichcorresponds to theLundbergPalm~en relationship (23.47), is obtained. (23.49)
in which A is a constant pertaining to the overall material and z ' is a stressweighted average depth to the volume at risk to fatigue. When Tlimit= 0, equation (23.49) reduces to (23.47) if it is assumed T = T,. In E23.341, equation (23.49) was applied to fatigue data for rotating beams in bending, beams in cyclically reversed torsion, and flat beams in reversed bending. Figures 23.37 and 23.38 illustrate that equation (23.49) applies to structural fatigue data also. To fit the equation to the test data of Figs. 23.37 and 23.38, it was only necessary to establish a single point on one curve for one specimen; that point is identified by an asterisk. Thereafter, all other computed points followed. Harris and McCool [23.35] applied the IoannidesHarris theory using octahedral shear stress as the fatigueinitiating stress to 62 different applications involvingdeepgroove and angularcontact ball bearings and cylindrical roller bearings manufactured from CVD 52100, M50, M5O~iL,and 8620 carburizing steels. A value of q,ct,limit was determined for each material. Using these values, the L,, life for each application was calculated and compared against the measured bearing fatigue life. Also, the Ll0 life calculated according to the LundbergPalmgren theory (standard method) wascalculated and compared to the measured bearing life. It was thereby determined by statistical analysis that the bearing fatigue lives calculated using the IoannidesHarris theory were closerto the measured lives than were the lives calculated using the standard
I
908
PLICATION LOAD
I
340 320 300 280 260 320
n . .
60
~3.3~.Application of equation (23.34) to rotating beam fatigue test data (from E23.341).
method as modifiedby the life factors discussed above. ~u~sequently, Harris i23.361 demonstrated the application of the Ioannidesory in the prediction of fatigue lives of balls endurance tested in balllvring rigs (see Fig. 19.17). To accurately calculate bearing fati e lives using the Ioannidesarris theory requires Selection of a fati~einitiatingstress c~terion ter~inationand ap~licationof all residual, applied, and induced on the material of the rolling elementrace~aycontacts elopment and ap~licationof a stresslife factor
I O ~ ~ E S  THEORY ~ I S
909
150 180 17 160 150
1
I I 1 Illlit
I I I I I1111
I I 1 IlllU
180 170
160 150
.
Application of equation (23.34) to torsionbeam
fatiguedata
(from
i23.341).
This was accomplished in the Harris and McCool E23.351 investigation using the analytical methods defined in this text combined in ball and roller bearing performance analysis computer programs THBBAN* and RBAN.* Moreover, it should be apparent thatthe concept of a stresslife factor fulfills the requirement for the interdependency of the various fatigue lifeinfluencing factors cited previously.
* F O R T ~ / BASIC ~ S computer ~ ~ programs developed by the authorfor operation on personal computers.
~ P L I C ~ T LOAD I O ~ AND LIF'E FACTOR^
In 1995, the TribologyDivision of ASME International established a technical committee to ipvestigate life ratings for modern rolling bearings. The intended result of the committee's efforts wasto establish and disseminate a common method for the prediction of fatigue lives in rolling bearing applications. In E23.321, the committee established the following equation for the calculation of bearing fatigue life: (23.50)
In (23.50), C is the bearing basic loadrating asgiven in bearing catalogs, Fe is the equivalent applied load, and a s L is the stresslife factor, which comprises all lifeinfluencing stresses acting on rolling elementraceway contacts, including normal stresses, frictional shear stresses, material methods, and residual stresses due to heat treatment and manufact~ring fatigue limit stress. It is clear that equation (23.50) may be modified by the reliabilitylife factor q,which is not stressrelated. Accordingly, (23.51)
In (23.51), exponent p = 3 for ball bearings and 1013 for roller bearings. Considering nonstandard loading in which lifeis calculated for each contact, for point contacts P
(23.52)
In ( 2 3 . ~ ~subscript ), rn refers to the raceway contact, exponentp = 3 for the rotating raceway, and p = 10/3 for the stationary raceway. Equation (23.52) further recognizes that the stresslife factor Q;sLmj is a function of the raceway and contact azimuth location. For line contacts,
(23.53)
In ( 2 ~ . ~ 3subscript ), rn refers to the raceway contact, k refers to the laminum, exponent p = 4 for the rotating raceway, and p = 9/2 for the stationary raceway.
The ASME committee selectedthe von Mises as theappropriate failureinitiating stress criterion. The vonMises stress definedaccording to equation (23.54) is a scalar quantity
(23.55) associated with the commonly used MisesHencky distortion energy theory of fatigue failure. See Juvinall andMarshek [23.37] or other machine design texts. It is of interest to note that theoctahedral shear stress, avector quantity, selected in [23.35] as the failureinitiating stress is directly proportional in m a ~ i t u d eto von Mises stress; for example,
Applied loading in all applications; that is, involving both standard and nonstandard loading, is distributed over the rolling elements. The rolling element loads which are applied perpendicular to the contact areas result inpressuretype (normal to the contact surface) stresses. In Chapter 6, assuming “dry” contact, equations to define the m a ~ i t u d e sof rtz stresses were provided. In Chapter 12, it was shown that, under the influence of elast~hydrodynami~ lubrication, ~istri~utio over n the contact may be somewhat altered distribution. Nevertheless, in most rolling bearing ap ian stress dis * isfactory to assume the z ) under the contact su may be determi As indicated ab methods accounts only
In most rolling bearing contacts, as discussed in Chapters 13 and 114, ee of sliding occurs. n an~larcontactball bearings, spherical
PLICATION LOAD ANI) LlN3 FACTORS
roller bearings, and lightly loaded high speed cylindrical rollerbearings, a substantial amount of sliding occurs. These sliding motions, occurring in relatively heavily loaded rolling elementraceway contacts, result in significant frictional shear stresses. The magnitude of the frictional shear stress at any point (x,y ) on the contact surface depends on the local contactpressure, the local sliding velocity, the lubricant rheological properties, and the topographies of contact surfaces. Depending on the degree of contact surface separation by the lubricant film, sliding in conjunction with the basic rolling motion may produce surface distress which can result in microspalls; these can lead to macrospalls. N6lias et al. [23.37], conductingendurance tests using a rollingsliding disk rig, demonstrated that smooth surfaces on 52100 and M50 steel test components, irrespective of the occurrence of sliding, esperienced no surface distress. The tests were conducted at 15003500 MPa (2.185.08 * lo5 psi) under lubricant film parameter A. ranging approsimately from 0.61.3. This indicates the need for finely finished rolling element and raceway surfaces, especially in the presence of marginal lubrication. N6lias et al. [23.39] noted that, in the absence of sliding, all progression occurs bothin the direction of sliding and transthat direction. This is shown in Fig. 23.39 taken from [23.39]. their test rig, the driver disk turns faster than the follower disk, and the friction direction over the contact for the follower disk is in the rolling direction. The friction direction over the contact of the driver disk is, however, in the direction opposite to rolling. Figure 23.40 shows that the microcracks are dependent on the friction direction. It can be seen that the typical arrowhead shape is oriented in the friction direction while crack propagation is the direction oppositeto friction. N6liaset al. C23.391 further noted that thedriven surfaces were proneto greater damage than the driver surfaces. Another observation of N6lias et al. [23.39]Gas that the size and volume of the spalled material increased with the magnitude of normal ertz) stress. See Fig. 23.41. This situation indicates that sliding damage is more severe under heavy loadthan under lighter load, a condition
.
(5.08 *
Surfaces of M50 steel endurance test components operated at 3500 MPa
lo5 psi) under ( a )simple rolling and ( b )rolling and sliding (from E23.391).
(a) Driver surface Rollin
Friction
(b) Driven surface FIG 8.40, Microcrack orientationwithrespect to rollingandfrictiondirectionsfor M50 steel specimens tested at 3500 MPa (5.08 * lo6 psi) (from i23.391).
that must be of concern in heavily loaded an~larcontactball bearings and spherical roller bearings with marginal lubrication. At any point (x, y, x ) under the contact surface, the stresses resulting from the surface shear stresses may be determined using the method^ of Ahmadi et al. E23.401.
To determine the surface stresses associated with dents, the methods developed by Ville and Ndias E23.231,[23.271 or Ai and Cheng may be applied. This requires a definition of the contaminants i in the application. Also, if the topo~aphyof the dented surface can be the methods of ~ e b s t e et r al. [23.24] may be applied. , while effective forlaboratory investigations, typically co many minutes and even hours of computer time for the stress analysis of a single contact. The analysis of rolling involves the iterative solution of many thous the effect of pa~iculatecontamination in ering application, approxi~ations articles, their concentration in the
P L I C ~ ~ I OLO N
Rollin
Friction
(a) 1500 MPa (2. 18 $IO5 psi), 510 pmspall size
(b) 2500 MPa (3.63
o5psi), 20 pm spa11size
(c) 3500 MPa (5.08 $IO5 psi), 40 pm spa11 size 1. Increase of microcrack size (length, depth) with normal stress for M50 steel endurance tested under rolling and sliding conditions (from [23.39]).
their effects on subsurface stresses. In essence, a stress concentration parameters would be applied to the contact stress ed application may be meanformation may be used to earing a~plications, nal cleanliness code for hydraulic codify these. The cleanliness clas ssifications do not account for the hardness of the established, however, that ina wide scopeof rolling tions, there exists a similar distribution of har
. IS0 4406 Fluid Cleanliness Classes
~
Number of Particles per 100 ml of Oil Over 5inpm >500,000 >250,000 >130,000 >64,000 >32,000 >16,000 >8,000 >4,000 >2,000
>1,000 >1,000 >500 >250
Size 21,000,000 2500,000 2250,000 2130,000 264,000 232,000 2 16,000 28,000
24,000 22,000 22,000 2 1,000 3500
Over 15inpm >64,000 >32,000 >16,000 >8,000 >4,000 >2,000
>1,000 >500 >250 >130 >64 >32 >32
size
Code
2130,000
264,000 232,000 216,000 28,000
24,000 22,000 21,000 2500 2250 2130
264 264
~
20117 19116 18115 17/14 16/13 15112 14111 13110 1219 1118 1117 1016 916
les, which produces a generally similar fatigue lifer if Table 23.8 indicates the number of particles > 5 is does not mean that just a few contaminant pa minute size affect the fatigue lives of rolling bearings. The figures are only a statistical measure for the existence of cles. The codenumber in column 5 of Table 23.8raised to yields the limiting particle numbers in columns 2 and 4; for example, 220= 1 lo6 and 217 = 1.3 lo5. Ioannides et al. E23.421 also state that for circulating oil lubrication, the filtering efficiency of the system can be used in lieu of I E23.431 to define contaminant size. This may be defined by the capacity as specified by IS0 4572 E23.441. epending on the size of the rolling contact areas in a bearing, sensitivity to particulate contamination varies. Ball bearings tend to be more vulnerable than roller bearings; contaminant particles are more harmful in small bearings than in bearings with large rolling e onsidering the foregoing and using empirically determin nides et al. E23.421 linked the contamination parameter size, lubrication system, and lubrication effectiveness. Further considering that solid contami~antsfound in bearings are mainly hard meta * particles resulting from wear of the mechanical system, they develo .50, which are charts of 6, vs lubrication parameter K for andard 4406 cleanliness levels. For circulating o filtration levels accordingto IS0 4572 are also in 23.50, K is defined as vlv, where v is the nematic viscosity of the lubricant at the operating temperature and vl is the kine*
9
1 .o
0.9 0.8 0.7 0.6
0.5 0.4
0.3 0.2 0.1
2. C, vs
K
and dmfor filtered circulating oilIS0
13/10p,
= 200 (from
f23.423).
matic viscosity required for adequate separation of the contacts. If v, occurs at A = I, then A = K O  ~ . or cir~ulatingoil in Figs. 23.4223.45, the ~arameterBe is defined in
is the number of pa~icles s the n u m ~ e rof p
17 CI 1 .o
0.9
0.8 0.7 0.6
0.5 0.4
0.3
0.2 0.1
CL vs
K
and dm for filtered circulating oilIS0
17/14&5
2
75 (from
45. CLvs
K
and dm for filtered circulating oilIS0
19/16p,,
2
75 (from
23.44. 123.421).
1 .o
0.9 0.8
0.7 0.6 0.5 0.4
0.3
0.2 0.1
E23.421).
pm. Thus, p6 = 200 means that for every 200 particles > 6 pm upstream of the filter, only 1 particle > 6 pm passes through the filter. though this is a useful method for comparing filter performance, it is not infallible since contaminant particles may have ~ifferentshapes according to the a ~ ~ l i c a t i o n .
contamination in the system after mounting contamination which ~enetratesto the bearing during operation = contamination generated in the system.
= =
PLICATION LOAD AND LIFE FACTORS
0.9 0.8
0.7 0.6
0.5 0.4 0.3
0.2 0.1
0.3 0.3 0.5 0.7 0.9 1.1 1.3 1.5
23*46. CLvs
K
1.7 1.9 2.1
2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9
and dmfor oil bathIS0
‘IC
13/10”22 = 3 (from [23.42]).
1.o
2000
0.9 0.8 0.7 0.B 0.5
.”50
0.4
0.3 0.2
nm
0,;
0.1 0.3 0.5 0.7 0.9
1.1 1.3 1.5
23.47. C, vs
K
t.7 1.9 2.i 2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9
and dmfor oil bathIS0
15/1222
= 45 (from [23.421).
These subparameters are given integer values ranging from I to 5 ; 1 represents the least contamination. 1 2 ,is influenced by the bearing mounting environment. It is improved by careful flushing of the system after mounting. RBdepends on the efficiency of the sealing arrangement; separate regreasing improves this. The dirtiness of the environment needs to be considered. R, normally ranges from I to 2; the lower value is used for very clean applications when RA + R B < 5. R, = 3 is used forwellmachinedground gears; R, = 3 to 5 when bearin~sand gears are lubricate^ with the same oil. The C,values obtained using Figs. 23.4223.50 are for lubricant withen the calculated K < 1, a high quality lubricant with approved additives may be expected to promote a favorable smoothing of the raceway surfaces during running in. Thereby, K may improve and reach a value of‘ I.
~
THE STRESSLIFE FACTOR CL 1 .o
0.9 0.8
0.7
0.6 0.5 0.4
0.3 0.2 0.1
0.1 0.3 Ct, 5 0.7 0.9 1.1 7.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.93.1
23,48. C, vs
K
3.3 3.5 3.7 3.9
and d,for oil bathIS0 17/1422
=
CL
67 (from 123.421).
/
0.5 0.4
0.3 0.2 0.1
23.50. C, vs
K
and d,for oil bathIS0 21/1822
=
1115 (from f23.421).
PLICATION L
~~~
O AMD LIFE FACTORB
When contamination is not measured or known in detail, the contamination parameter C,may be estimated using Table 23.9 given in f23.321. In general, the stress concentration factor to be applied to a point contact normal stress has been analytically and empirically determined , Also, the stress conas Kc = l/CL1’3;for example, ~ ’ ( xy ), = Kc ~ ( xy). centration factor may be applied to the surface shear stress aswell; for example, +(x, y ) = Kc ?(x,y). Roller (line contact) bearings have been determined to be more tolerant of particulate contamination than ball (point contact) bearings; in thiscase a normal stress concentration factor Kc = I/CL1’*appears to be satisfactory for many applications. N6lias [23.45] illustrates in Fig. 23.51 that for a dented or “rough” s d a c e the magnitude of the maximum shear strees is strongly influenced by sliding on the surface. N6lias [23.45] further postulates that failure of rough or dented surfaces may commencenear thesurface; however, coalescence of microcracks may proceed inward in the direction toward the location of the maximum subsurface stresses due to the average contact loading. Thus, the subsurface failure might be initiated by the surface condition. This competition of subsurface failureinitiating stresses is illustrated by Fig. 23.52. Because most modern balland roller bearings have relatively smooth raceway and rolling element surfaces, roughness is more indicative of dentsin contaminated applications. Thus, competition for initiation of subsurface fatigue failure would tend to occur more in applications with contamination. When calculations for subsurface Von Mises stresses (or other assumed failureinitiating
. Contamination Parameter Levels Bearing Operation Condition
ost cleanliness fine filtering of oil, very cleanoil baths, or bearings greased and sealed in a clean environment
G
,
0.95 0.8
ion, clean oil bath, or bearings greased and shielded 0.6
Good sealing adapted t o environment, cleanliness during mounting, recommended oil changeintervals observed
0.10.5 Inadequately cleaned cast housing, unsatisfactory sealing, wear particles entering oil system eavy contamination Cast housing not cleaned, particles from machiningleft in housing, foreignmatter penetrates into bearing, water penetration into bearing, corrosion
0 (We ealcul~tionsin
1
""""_____"""""__ " " " " " " _ " " " " " . _
" " " " " . _ _ " " " " . _ _ _
0
2
4
6
8
1 0 1 2 1 4
slidetoroll ratio (%) 29.51, M ~ i m shear u ~ stress/m~imumHertz stress vs slidelroll ratio in the vicinity of a dent 1.5 pm deep by 40 pm wide; the dent having a shoulder 0.5 pm (fhm N6lias 123.451).
stresses) indicate maximum values approaching the surface, it may be presumed that surface pitting will most likely occur first; however, not to the exclusion of subsurface fatigue failure depending on the amount of operational cycles accumulated.
To prevent rotation of the bearing inner ring about the shaft, andhence prevent fretting corrosion of the bearing bore surface, the bearing inner ring is usually pressfitted to the shaft. The amount of diametral interference, and therefore the required pressure between the ring bore and the shaft outside diameter, depends primarily on the amount of applied loading and secondarily on the shaft speed. Thegreater the applied load and shaft speed, the greater must be the interference to prevent ring rotation. For recommendation of the magnitude of the interference fit required for a given application as dictated only by the magnitude of applied loading, SI/^^ Standard No. 7 L23.461 may be consulted for radial ball, cylindrical roller, and spherical roller bearings. For tapered roller bearings, SI/^^ Standard No. 19.1 [23.47] and 19.2 [23.48] may be consulted. Since the ring and shaft dimensions and materials are defined, standard strength of materials calculations; e.g., Timoshenko i23.491 maybeused to determine the radial stresses. The interference fit causes the ringto stretch, resulting in tensile hoop stress. Similarly, forouter ring rotation such as in wheel bearing applications, the outer ring may be pressfitted into the housing. In this case, compressive hoop stress and radial stresswill be induced. Ring rotation, particularly at high speed, gives rise to radial centrifugal stress, which in turn causes the ringto stretch with attendant hoop
~PLICATIO LO ~
stresses resisting the ring expansion. uter ring rotation results in tensile hoop stresses which tend to counteract the compressive hoopstresses caused by press fitting of the outer ring in the housing. Timoshenko [23.49] details the method to calculate the tensile hoop andradial stresses associated with ring rotation. Each of the stresses due to press fitting and/or ring rotation is superimposed onthe subsurface stress field causedby contact surface stresses.
Heat treatment of bea n introduce a differential stress distribution in the nearsurface region which influences fatigue life. For example, the case regionof carburized bearing components contains compressive stresses, mainly in the circumferential direction of a ring. As distance beneath the contact surface increases, the compressive stress undergoes a transition to the tensile stress field necessary to keep the in equilibrium. Fortunately, the tensile stress region is sufficiently below the subsurface zone influenced by the surface Hertzian and frictional stresses that it doesn’t influencefatigue. The grinding and surface finishing processes which produce the surface topographies or microgeometries of bearing rolling contact components introduce residual stresses which may be detrimental to bearing endurance. If the processes are abusively applied, accidentallyor intentionally to achieve rapid component production, the induced residual stresses can be rather high and tensile. Tloskamp [23.50] conductedstudies of the magnitude of residual stresses in run and unrun AIS1 52100 steel ball bearing raceways. In an unreported endurance test program for bearing balls, the author found compressive surface stresses in the range of 600 MPa (~7,000psi) for both M50 and 52100 balls, whichhad not been run. Beneath the surface, in the zone of maximum subsurface applied stress, the compressive stress level reduced to values in the range of 70 NIPa (10,000 psi). When the balls were operated under normal bearing Hertz stressesfor example, maximum 2700 MPa (400,000 psi)these compressive stresses seemed to disappear, most likely a result of retained austenite transformation. On the other hand, it has been observed that r ~ n n i n ~  bearing in raceways under heavy loading for a short period of time prior to normal operation tends to work harden the nearsurface regions. This introduces slight compressive residual stress into the material, increasing its resistance to fatigue. Excessive amounts of compressive stress tend to reduce resistance to fatigue.
The stresses discussed in this section each contribute to the overall subsurface stress distribution. Using superposition and the assumption of
LOWLoad
Medium Load
igh Load
LOW Roughness
Mild Roughness
High Roughness
.
Competition between surface and subsurface crack growth for various loads and surface roughnesses. Each graph representsshear stress vs nondimensionalized the fatigue limitstress below which crackinitiation depth zlb. The dashed line represents (straight lines in inserts) does not occur and propagation direction (arrowtip lines ininserts) (from N6lias 123.451).
ises stress as thefatigue failureinitiating criterion, the stress tenalculated for everysubsurface point (x,y, x). The basicequaarris theorythat is, (23.48)may be restated as follows:
Equation (23.5$) refers to the survival of volume element AVi for N stress cycles with probability ASi. The probability that the entire stressed volume will survive N stress cycles may be determined using the product law of probability; that is, S = AS, A&2 . . . yn. Therefore,
where Aiis the radial plane crosssectional area Ax A x of the volume
9
~ ~ L I C A T I OLOAD N AND LIFE FACTORS
element on which the effective stress acts, and dr is the raceway diameter. Letting q = xla and r = zlb, where a and b are the semimajor and semiminor axes respectively of the contact ellipse (see Chapter 61, then Ax = aAq and Az = bAr. Numerical integration may be performedusing Simpson's rule, letting Aq = Ar = l l n , where n is the number of segments into which the major axis is divided. With the indicated substitutions, equation (23.59) becomes
where cj and ck are theSimpson's rule coefficients. Thenumber of stress cycles survived is iV = uL,in which u is the number of stress cycles per revolution and L is the life in revolutions. Therefore
Equation (23.61) may be used to find the stresslife factor %L by (1) evaluating the equation for the stress conditions assumed by Lundberg and Palmgren, (2) evaluating the equation for the actual bearing stress conditions occurringin the application, and (3) comparing these. For example,
(23.62)
where I is called the life integral. The accurate evaluation of I for each condition depends upon the boundaries specified forthe stressesvolume. In Chapter 18, it was shown that, becauseonly Hertz stresses wereconsidered in their analysis, Lundberg and Palmgren were able to assume the stressed volume was orthogonal proportional to aa dgo,where zo is the depth to the m a ~ m u m shear stress roeIn the analysis of the stresslife factor, von Mises stress is used in lieu of ro,and the effective stress is integrated over the appropriate volume. That volume is defined by the elements for which the effective stress is greater than zero; that is > 0. It can be demonstrated using the LundbergPalmgren analysis of Chapter 18 that
THE $TRE$$L~FACTOR
(23.63) Considering the equivalent integrated life, Harris andYU E23.511 showed that
(23.64) Moreover, they determined that all effective stresses
influence life less than 1 percent. For simple Hertz loading, the lifeinfluencing zone is illustrated by Fig. 23.53.
F I G ~2 E3.53. Lines of constant r y z / rfor o simple Hertz loadingshaded area indicates effective lifeinfluencing stresses.
As compared to the LundbergPal~gren stressed, for which zo/b = 0.5 (see Chapter 6), for Hertz loading, the critical stressed volume stretches down to z / b = 1.6. The critical stressed volume is different for each rolling elementraceway contact combinationof applied and residual stress, and it should be used in the evaluation of the life integrals in equation (23.62).
To evaluate the life integrals, the value of the fatigue limit stress must be known for the bearing component material. This can be determined by endurance testing of bearings or selected components. Based on endurance test datafrom 62bearing applications and theuse of octahedral shear stress as the fatigue failureinitiating criterion, Harris andMcCool E23.351 indicated some values for various bearing steels. In vring rig, ball endurance testing E23.3631, the fatigue limit stress values obtained for CVD52100 and V I W B M50 steel bearings were shown to be adequate. The test programs reported in E23.351 and E23.361 were extended to cover 129 bearing applications and additional materials. The analyticalmodels to predict bearing application performance and ball/ vring test performance were refined and performance analyses were again conducted, using the vonMises stressasthe fatigue failureinitiating criterion. Based on this subsequent study, Table 23.10 gives resulting values of fatigue limit stress for various materials. Bohmer et al. E23.521 established that thefatigue limits of steels decrease as a function of temperature. From their graphical data, the following relationships maybe determined by curvefitting for various bearing steels operating at temperatures exceeding 80°C (176°F):
Equations (23.63)(23.65)wereused inthe analyses which generated Table 23.10.
applicationperformance
In [23.5], in a systems approach to bearing fatigue life calculation, the The subscript “XYZ~’is meant to stresslife factor is presented as “axye)).
7
THE STRESSLIFE FACTOR
.
Fatigue Limit Stress (Von Mises Criterion) for Bearing Steel Steel
C W 52100 Carburized steel (common types) V I W m M50 M5ONiL Inductionhardened steel (wheel bearings)
680 (99,000) 590 (86?000) 720 (104,000) 510 (74,000) 450 (65,000)
“indicate that a manufacturer or organization can selectany combi~ation and number of letters”. In this test, theintegrated stresslife factor has been designated %L. The IS0 Standard E23.51 also states “diagrams or equations can be made up expressing QYzas a function of cr,/a; the endurance stress limit divided by the real stress”and shows the schematic diagram of Fig. 23.54. Figure 23.54 illustrates how &yz asymptotically approachesinfinity as the real stress cr approaches the fatigue limit stress uU.Ioaniides et al. [23.42] presented equation (23.66) as a means to determine t&L.
Values of parameter M and exponent rn in equation (23.66) are given as a function of lubrication effectiveness in Table 23.11.In equation (23.66), F is the applied load and Ffim is the fatigue limit load, which is given for each bearing in a catalog; for example,see [23.53]. It should be apparent, that equation (23.66) accommodates onlythe applied Hertzian stresses. Values of parameter q b r g and exponents ut and c / e are given in Table
1
F
I
a“/@
~ 23.54. ~ E axyzvs cU/a for a given lubrication condition (from 123.51).
928
LOAD APPLICATION
AND LIFE FACTORS
TABLE 23.11. M and m vs Lubrication Effectiveness Parameter
K
(from
E23.401) 0.1
.
tensitic matrix;, const ents which interfe~ewith the normal p nee, fatigue is usually i ge en~ompasses the and featu~eless.This
stage may only be evide
steels is shown in
.
Fatigue. (a)~ubsurfaceinitiatedfatigue showing relatively smooth first stage region (magnification 1OOX); ( b )characteristic fatiguefeatures at 1500X magnification; ( e ) fatigue features at 6000X magnification.
a
ES
analysis of fatigue failures in bearing steels does not appear to be feasible at this time.
The various mechanisms of crack propagation discussed occur in combinations exhibiting two or more of the identiffing features simultaneously. Often these occurrences involve modes associated with material properties and microstructure. An example of this condition observedin bearing steels involves mixtures of quasicleavage andintergranular modes, both of which are lowenergy mechanisms. Stress conditions favoring both mechanisms are apparently equal. Strain rate affects fracture mechanisms. Mixtures of fracture modes are observed in transition regions between stable crack propagationand unstable crack propagation. Bearing components that fail by fatigue eventually attain a critical crack size and fracture by quasicleavage. Transition zones betweenthe fatigue and quasicleavage zones sometimes display dimples intermingled in the cleavage facets. These transition zones are relatively narrow. Evaluation of failurecontaining mixed modesof cracking in theorigin area depends on which mode is dominant.
An overview of the more commonconditions and damage leading to bearing failures has been presented. Details regarding mechanisms should be referred to in the appropriate chapters. It should not be construed that theexamples citedhere are allinclusive; for example, cage problems and wear patterns have not been addressed. Considerable variation may be observed within the examples used.Illustrations andphotographs are presented to depict representative features. Laboratory work, such as metallography, stress determinations, phase identi~cation,microprobe analysis, and so on, should be conducted to verify and support visual observations.
27.1. T.Tallian, “Rolling Contact Failure Control Through Lubrication,”€‘roc. last. Meek Eng. 282,205236 (196768). 27.2. J. Mohn, H. Hodgen, H. Munson, and W. Poole, “Improvement of the Corrosion Resistance of Turbine Engine Bearings.” ~ W ~  T R  8 4  2 0 (1984). 14 27.3. C. Rowe and L. Armstrong, “Lubricant ERects on RollingContact Fatigue,” ASLE Trans. 23,2339 (January 1982).
27.4. G. Lundberg andA. Palmgren, “Dynamic Capacityof Roller Bearings,”Acta ~ o Z ~ t e c ~ . ~ e c Eng. ~ . Ser. 2, RSAEE, No. 4,96 (1952). 27.5. J. Martin, S. Borgese, and A. Eberhardt, “Microstructural Alterations of Rolling Bearing Steel UndergoingCyclic Stressing,”ASME Paper 65WA/CF4 (1965). 27.6. R. Osterlund, 0. Vingsbo, L. Vincent,and P. Guiraldeng, “Butterflies in Fati~ed Ball Bea~ngsFor~ationMechanisms and Structures,”Seand. J: ~etaZZ.11 (1982). 27.7. S. Way, “Pitting Due to Rolling Contact,”J: AppZ. Mech. 2, A49A58 (1935).
This Page Intentionally Left Blank
APPENDIX
All equations in the text are written in metric system units. In this appendix, Table A.l gives factors for conversion of metric system units to English system units. Note that for the former, only millimeters are used for length and square millimeters for area with the exception of viscosity, which being in centistokes is square centimeters per second. Furthermore, the basic unit of power used herein is the watt (as opposed to kilowatt). Consistent with the foregoing, Table A.2 provides the appropriate English system units constant for each equation in the text having a metric system units constant.
TABLE A.l. Unit Conversion Factorsa Conversion Factor
Unit
Metric System
Length Force Torque Temperature difference Kinematic viscosity
mm N mmN "C, OK cm2/sec (centistokes) W W/mm "C W/mm "C
0.03937, 0.003281 0.2247 0.00885 1.8 0.001076
in., ft lb in. * lb
3.412 577.7 176,100
Btu/hr Btu/hr R O Btu/hr ft2
N/mm2
144.98
psi
Heat flow, power Thermal conductivity Heat convection coefficient Pressure, stress a

English System
OF,
O R
ft2/sec
F
O F
English system units equal metric system multiplied by conversion factor.
1071
A.2. Equation Constants for Metric and English System Units Chapter Number
Equation Number
3
5
6
7
8 10
Metric System Constant
33 41 52 75
47100 47100 2.26 * lo'' 3.39 1011 2.26
76
3.39 10l1
80 81 39 41 43 52 54 8 9 110 111 113 115 118 127 128 134 141 150 151 153 66 6 7 ' 8
9 10 11 12 13 14 28 29 30 32
4.47 8.37 0.0236 0,0236 2.79 104 3.35 103 3.84 105 2.15 105 7.86 104 3.84 105 1.24 105 1.24 105 3.84 105 3.84 105 1.24 105 1.24 105 1.92 105 3.84 105 3.84 105 1.92 105 1.24 105 2.24 4.36 104 6.98 104 1.81 104 7.68 105 4.36 104 6.98 104 5.24 104 1.81 104 7.68 0.0472 0.0472 0.0472 1.166 I
I
En~lishSystem Constant
lo6 lo6
6.83 6.83
.I1 IO+ .17
.I1 106 jJ.17 4.18 107 7.83 107 0.0045 0.0045 1.01 105 2.78 104 4.36 107 3.12 107 1.14 107 4.36 107 8.71 lo' 671 lo' 4.36 107 4.36 * 107 8.71 lo' 8.71 10"' 2.18 lo' 4.36 107 4.36 107 2.18 lo' 8.71 lo' 2.09 105 8.71 lo' 2.53 105 4.33 * ~m 107 8.71 lo' 253 105 1.90 105 4.33 8.71 0.0090 0.0090 0.0090 &.24 105 I
Chapter Number
Equation Number
Metric System Constant
English System Constant
14
78 103 104 1 7 8 9 10 17 19 58 67 99 100 101 102 104 105 106 132 133 134 135 136 138 140 143 156 157 162 163 169 170 171 1 2 3 4 6 7 9 12 50 51
105 107 1.60 109 1.047 lo* 0.0332 0.060 2.30 lo' 0.030 5.73 5.73 lo+ 98.1 98.1 98.1 39.9 98.1 38.2 39.9 39.9 39.9 98.1 98.1 98.1 98.1 88.2 88.2 59.1 552 207 207 552 469 207 552 469 1.30 107 5.25 107 2.52 107 6.03 23.8 23.8 44.0 220 1.274 * 32900
8624 1.45 2.32 103 0.0404 0.332 0.60 0.30 0.30 0.173 0.173 7450 7450 7450 3030 7450 2900 3030 3030 3030 7450 7450 7450 7450 6700 6700 4490 49500 18600 18600 49500 42100 18600 49500 42100 6.20 2.50 1.20 10l1 1.98 1017 3440 3440 6379 32150 4.62 lo' 4.77
15
18
21
26
AE3EC, 85 tolerance classes, 99 i4BMA, 14,45,84 Adhesive wear, 943,947, 1046 A.FBMA, 84 Agricultural applications, 9 Aircraft gas turbine application, 4, 25, 43, 348, 523, 621 AISI, 580 AISI 8620: microstructure, 838 AISI 52100 steel, 4, 581, 598, 603 fatigue: dynamic capacity constant: line contact, 706 point contact, 705 endurance calculation exponents, 696 life, 739, 894 Weibull slope, 696 hardness, 836 retained austenite, 836 toughness, 614 ultimate strength, 614 MSI 440C steel, 4, 583, 613, 620 Aluminum, 5 Angularcontactball bearings, 19 angle, 20 automotive wheel application, 41 backtoback arrangement, 21, 369 ball friction forces due to gyroscopic moment, 176 doublerow, 19, 22 diametral play, 55 duplex set, 369 facetofacearrangement, 21, 369 groove curvature radii, 19 limiting thrust load, 379 selfaligning,22 singlerow, 19 split inner ring, 22 tandem arrangement, 21 triplex set, 375 Annealing of steel, 598, 613 ANSI, 14, 45, 84 load rating standards, 741, 825 Antifriction bearings, 533 Asperityasperity Coulomb friction, 478 ASTM, 664 Asymmetrical roller loading, 159 Automotive wheelhub bearings, 4, 41 Axial deflection, 245 ball bearings under thrust load, 247 duplex set of ball bearings, 371 high speed angularcontact ball bearing, 350 Jones' constant, 246, 381
Axial loading of rollers: applied roller thrust loading, 177 cylindrical roller bearing flanges, 178 skewing, 179 Axial preloading, 368 Backtobackbearing arrangement, 21, 368 Backup roll bearing, 293 load distribution in, 302 Bainite, 607 Ball bearings, 11 axial preloading, 368 basic dynamic capacity: radial bearings, 741 thrust bearings, 741 clearance effect on fatigue life, 867 Conrad assembly, 12, 13 assembly angle, 12 contact angle under combined radial and thrust loading, 259 coulomb friction (operating with), 506 curvature, 62 da Vinci, 3 dimension series, 17 doublerow, 15, 18 fillingslot, 15, 18 free angle of misalignment, 58 free contact angle, 55 free endplay, 55 friction in ballraceway contacts, 496 friction forces and moments (in), 519 friction torque, 540 applied load (due to), 540 viscous drag (due to), 542 groove curvature radii, 12 high speed, 339 instrument bearings, 15 internal load distribution effect on fatigue life, 864 internal load patterns, 1051 loci of groove curvature centers, 266 radial, 11 seals and shields, 17 shielded bearing, 16 singlerow, deepgroove,11 static load ratings, 825 stiffness, 1028 surface treatment for components, 638 skidding, 518 Ball bushing, 40 Ball excursions, 346 Ball loading, 342 friction forces, 518 gyroscopic moment, 174 induced, 158 normal to raceway, 343
static, 157 stress cycles per ball revolution, 865 Balls: dimensional audit parameters, 778 endurance testing: NASA fiveball endurance tester, 782 Pratt & Whitney vring/ball endurance tester, 783 fatigue failure, 710 hollow, 348 sapphire, 621 silicon nitride, 348, 623 speeds, 520 traction test rig, 790 viscous drag on, 493 Ball speed components, 318, 520 Ball surface velocities, 319 Barus equation, 424 Basic electric furnace processing of steel, 584 Basic dynamic capacity: line contact radial bearings, 732 line contact thrust bearings, 735 point contact raceways, 714 point contact radial bearings, 715 point contact thrust bearings, 720 radial ball bearings, 741 radial roller bearings, 741 radial roller bearings with point and line contact, 738 thrust ball bearings, 742 thrust roller bearings, 750 thrust roller bearings with point and line contact, 739 Basic static load ratings, 825 permissible static load, 831 shakedown, 852 Bearing deflections, 6 combined radial, thrust, and moment loading in ball bearing, 267 radial, 235 rigid ring bearings, 365 stiffness, 1029 contact angle effects, 1037 preload effects, 1037 shaft bending effects, 1038 speed effects, 1034 Bearing disassembly, 1044 Bearing failure investigation, 1043 Bearing frequencies, 993 Bearing heat generation: roller skewing effect, 177 Bearing loading: cantilever beam support, 142 classification, 85 concentrated radial loading, 135 concentrated radial and moment loading, 143 friction torque (due to), 540 internal due to rotation about eccentric axis, 172 internal used to determine failure cause, 1050 load classification, 85
multiple bearingshaft systems, 410 permissible static load, 831 shaft supported by three bearings: nonrigid shaft system, 404 rigid shaft system, 400 shaft supported by two bearings: statically determinate system, 135 statically indeterminate system: flexible shaft, 392 rigid shaft, 389 X and Y factors, 390 Bearing noise, 964 Bearings with integral sensors, 43 Bearing vibrations, 964 Belt loads, 138 Bevel gear: loading, 144 speeds, 150 spiral bevel gears, 145 BG42 steel, 581, 613, 620 Biodegradable lubricants, 671 Bore, 49 Boussinesq, 189 Brass for cages, 5 tensile strength, 625 Brinelling, 1052 Bronze for cages, 626 Cage, 4 ball riding, 493 bronze, 626 deepgroove ball bearing, 12 forces, 515, 529 land riding, 493 friction torque, 532 low carbon steel, 625 materials, 5,625 motions and forces, 529, 533 polymeric materials, 626 skewing control, 179 sliding friction, 493 speed, 309, 526 Camfollower application, 44 Cantilever beam support, 142 Carbides in steel, 212 orientation after overrolling, 213 Carburizing steel, 580, 598, 608 fatigue life of bearings, 894 fracture (effect on), 855 residual stress, 616 toughness, 615 Casehardening: depth, 215 fatigue life (effect on), 740, 894 residual stress, 616 steel, 580 Castigliano's theorem, 295 Centric thrust load, 245 Centrifugal force, 165 ball, 166, 343 roller, 169, 355 rotation about eccentric axis, 172
EX
Centrifugal force (Continued ) spherical roller loading, 171 tapered roller loading, 169 Ceramic rolling elements, 4, 348, 621 Chain drive loads, 138 Chemical vapor deposition (CVD), 640 Circular crown profile of roller, 224 Circulating oil lubrication, 649 Clean steel, 589, 593 macroinclusions, 593 nonmetallic inclusions, 593 Clearance, 49, 73 effect of interference, 86, 124 effect on contact angle, 54, 245 surface finish effect, 126 tables, 51 Coatings, 638 black oxide, 639 phosphate, 638 Coefficient of friction, 217, 329, 496 solid lubricants, 670 Combined radial and thrust load, 10 deepgroove ball bearing, 12 doublerow bearings, 262 singlerow bearings, 256 Combining fatigue life factors, 903 Composite shear stress on contact surface, 475 Conditionbased maintenance, 1005 Cone, 28,77 Conrad assembly, 12, 13 Consumable electrode vacuum melting of steel, 583 Contact: asperity and fluidsupported load, 472 deformation (elastic), 195, 234, 340 rollerraceway skewing effect, 286 dynamic capacity, 699 elastohydrodynamiclubrication friction, 476 ellipse, 193 fatigue life, 699 flangeroller end, 330 heat transfer (in), 574 lubricated, structural elements of, 936 nearsurface region, 938 permanent deformation: line contact 824 point contact, 821 stresses, 185 concentration factors due to contaminant denting, 920 due to crowning, 225 maximum compressive, 195 surface shear stresses, 476, 523 concentration factors due to contaminant denting, 920 Contact angle, 53, 66 centrifugal force effect onball, 166 combine radial and thrust loading of ball bearings, 260 high speed angularcontact ball bearing, 350 thrust load effect, 245
Contaminants, 11 Contamination: cleanliness (is0 4406)classes, 915 dents, 1055 discoloration, 1058 fatigue life (effecton), 896 hydrolysis, 1057 INSA life test system, 789 life factor, 899 life testing considerations, 769 stress concentration factor for contact, 920 Conversion factors ( ~ e t r i c / ~ n g l iunits sh system), 1071 Corrosive wear, 946,1048 false brinelling, 1054 Coulomb friction: asperityasperity sliding, 478 ball bearings, 506 Crack propagation, 855 Crank mechanism loading, 141 Cronidur 30 steel, 581 Crown drop, 273 Crowned rollers, 26, 29 crown drop, 273 geometry, 275 insufficient crowning effect, 1052 logarithmic profile, 224 Cup, 29, 77 Curvature, 60 difference, 61, 71, 79 sum, 61, 71,79 total, 54 Cylindrical roller bearings, 25 axial loading through flanges, 178 axially floating, 26 basic dynamic capacity: radial bearings, 742 clearance, 73 combined radial, thrust and moment loading, 289 curvature difference, 77 curvature sum, 77 deflection: combined radial and thrust loading, 285 radial loading and misalignment, 277 doublerow, 26 endplay, 73, 76 friction torque due to roller endflange contacts, 543 high speed, 349 load classification,85 load distribution: high speed, 354 radial loading and misalignment, 272 radial and thrust loading, 280 misalignment, 272 failure, 773 fatigue life (effecton), 874 multirow, 31 pitch diameter, 73 roller skewing control, 330 sudace treatment for components, 638 thrust flanges, 29 Cylindrical roller thrust bearings, 38
1077 Damage Atlas, 771 Damped forced vibrations, 1015 Dark etching region of overrolled AIS1 52100 steel, 839, 841 Decarburization of steel, 596 Deepgroove ball bearing, 11 spherical outside surface, 14 Deflections: radial ball bearings, 366 roller bearings, 277, 366 selfaligningball bearings, 366 thrust bearings, 367 Deformation: bands in subsurface, 212 brinelling, 1052 contact, 234 rolling, 487 skewing effect in rollerraceway contact, 286 surface, 189 Delamination, 943 Dents, 620 brinelling, 1052 c o n t ~ i n a t i o n(due to), 897 lubrication in thevicinity of, 956 Diametral play, 55, 66 Differential expansion, 124 Dimensional audit parameters, 778 Dimensional instability of components, 856 Dimension series, 17 DIN, 84 Disassembly of bearings, 1044 Distortion energy theory of f&ilure, 210 Distributed load systems, 153 dN, 621 life testing considerations, 767 VIWA€iM50 NiL effect, 895 Doublerow ball bearings, 15 Dowson, 1, 428, 434, 453 Dryfilm lubrication, 418 silicon nitride components, 624 Duplex ball bearings, 370 backtoback arrangement, 368 facetofacearrangement, 368 Dynamic loading: crankreciprocatingload, 141 eccentric rotor, 140 rolling elements, 161 Earthmoving applications, 8 Eccentric rotor, dynamic loading,140 Eddy current testing of steel components, 595 Edge loading, 26, 219 fatigue life effect, 872, 1052 life testing considerations, 766 Elastic hysteresis in rolling, 486 Elastohydrod~amic lubrication: cage friction, 494 contact deformation, 427 firiction, 476 fluid entrainment velocity, 432, 523 isothermal, 424 lubricant film thickness, 433
MicroEHL, 937 pressure and stress distribution, 431 viscosity variation with pressure, 424 Elastomeric seal materials, 632 lip seals, 675 Electric motor applications, 11 Element test rigs, 779 Elliptical area of contact, 62 Elliptical eccentricity ratio (ellipticity), 194 Elliptic integrals, 193 Endplay, 53, 66, 73, 78 Endurance testing: bearing test rigs, 778 element test rigs, 779 INSA contaminationlifetest system, 789 sudden death testing, 779 Weibull distribution analysis, 807 test samples, 772 theoretical basis, 764 English units system equation constants, 1072 Environmentally acceptable lubricants, 671 Epicyclic power transmission, 144, 151 Equivalent axial load: line contact bearings, 735 point contact bearings, 725 static, 830 Equivalent cylinder radius, 423 Equivalent radial load: line contact bearings, 733 point contact bearings, 718 static, 830 Equivalent radii, 194 Ester lubricants, 661 Evolution of rolling bearings, 1 Externally aligning bearings: radial ball, 14 cylindrical roller thrust, 38 Extreme environment coatings for components, 640 Extreme pressure (EP) additives in lubricants: constituents, 656 effect on seal materials, 637 Facetofacebearing arrangement, 21, 368 Failure: electric arc damage, 1060 fatigue, 10, 686 interacting modes on surface, 953 investigation, 1043 internal load distribution, 1050 rolling element tracking, 1049 mechanisms, 1044 corrosion, 1048 cracking, 1047 lubricant deficiency, 1046 mechanical damage, 1044 wear, 1045 Failure probability, 689 fracture, 831 False brinelling, 1054
1 Fatigue failure, 10, 686 balls, 710 cracking, 688 fracture toughness (effect on propagation), 855 delamination, 949 life distribution, 10 modes, 618 pitting, 949 propagation: detection, 1008 life testing considerations, 771 subsurface cracks, 855 traction coefficient of failed surface, 1008 roller bearing misalignment (due to), 773 spall, 688, 772, 773 stress cycles per revolution, 695, 704 stressed volume of material, 694 subsurface, 619 wear (classified as), 948 Fatigueinitiating stress, 911 Fatigue life: AIS1 52100 steel, 595 basic dynamic capacity: point contact raceways, 714 point contact radial bearings, 715 point contact thrust bearings, 720 radial ball bearings, 742 radial roller bearings with point and line contact, 738 thrust ball bearings, 742 thrust roller bearings, 750 thrust roller bearings with point and line contact, 739 c o m b ~ i n glife factors, 903 contamination effect on, 896 denting, 897 life factor, 899 dispersion, 688 doublerow bearings, 691 element testing, 779 equivalent axial load: line contact bearings, 735 point contact bearings, 725 equivalent radial load: line contact bearings, 733 point contact bearings, 717 failure probability, 689 fatiguei~tiating stress, 911 hardness (effect on), 717 high speed bearings, 868 hoop (ring) stresses (effect on), 921 integral, 922 internal load distribution (effect on), 864 IS0 Standard, 926 limit stress, 904 temperature effect on, 926 values for steels, 927 line contact, 704 radial bearings, 728 load (effect of), 702 I;, life, 688 lubrication effect on, 890 material effect on, 894
materiallife factor, 895 material processing effect on, 894 maximum orthogonal shear stress, 694 median 688 life, minimum life, 10 oscillating bearings, 879 point contact, 701 radial bearings, 707 thrust bearings, 720 point contact bearings, 717 prestressing (effect on),850 radial roller bearings with point and line contact, 735 rating life, 696, 764 reliability (effect on life calculation), 886 life factor, 890, 910 residual stress (effect on),850 rotation factor, 720 sinusoidal loading (effect on),878 steel composition and processing (effecton), 739 stresslife factor, 910 testing, 764 ballrod rolling contact fatigue tester, 787 design considerations, 777 elements, 779 General Electric Polymet rolling contact endurance tester, 786 INSA rollingslidingdisc endurance tester, 788 NASA fiveball endurance tester, 782 Pratt & Whitney vring/ball endurance tester, 783 sample size selection, 810 SKF Aframe automotive wheel hub bearing endurance tester, 779 SKF R2 endurance tester, 781 sudden death, 779 tapered roller bearing endurance tester, 782 variable loading (effect on), 874 water effect on fatigue life, 902 Weibull distribution, 692 application, 800 estimation in data sets, 811 graphical representation of twoparameter distribution, 798 slope, 695 sudden death test analysis, 807 Fatigue limit stress, 904 temperature effect on, 926 Fatigue strength of steel, 614 Ferrofhidic seals, 681 Fiberglass, 5 Fillingslot ball bearings, 15 Filtration of lubricant, 649 fatigue life effect, 898 Finite element method of analysis, 221, 302 Fit: classification, 86 linetoline, 78 standards for practice, 84 tolerance classes, 85 Fiveball endurance test rig, 782
EX
Five degrees of freedom in loading, 357 Flaking, 1061 Flamehardening steel, 611 Flange: bearing friction torque due to roller end contacts (with), 543 layback angle, 228 roller end contact stress, 225 roller end geometry, 330 sliding at roller ends, 494 Flexibly supported bearings, 291 fatigue endurance, 868 Flinger, 674 Fluid entrainment velocities, 432, 523 Fluorinated ether lubricants, 662 Fluting, 1060 FractographF 1063 cleavage, 1065 fatigue, 1066 intergranular fiacture, 1065 microvoid coalescence, 1063 mixed modes, 1068 scanning electron microscopy, 1063 Fracture of bearing components, 831,853 Fracture toughness (effect on crack propagation), 855 Free angle of misalignment, 58 Free contact angle, 55 Free endplay, 55 Frequencies in bearing operation, 993 Fretting, 1046, 1050 Friction: balldisc traction test rig, 788 coulomb, 478 elastic hysteresis, 486 e l a s t o h y d r o d ~ ~lubrication, ic in, 476 forces in ballraceway contacts, 496 forces and moments in rollerraceway contacts, 510 gyroscopic motion(resistance to), 342 heat generation, 553 Heathcote slip, 490 limiting shear stress in lubricant, 477 microslip (due to), 489 moments in ballraceway contacts, 497 seal, 494 shear stresses in ballraceway contacts, 519 silicon nitride components, 624 torque, 6 applied load (due to), 540 total on bearing, 544 viscous drag (due to), 542 traction coefficient, 477 spalled surface, 1008 viscous drag (due to), 492 Friction wheel drive loads, 139 Full complement bearings, 27 Fully crowned roller, 29, 224 Garter seals, 679 Gear forces, 136 herringbone gears, 137
planetary gears, 137, 292 spur gears, 137 Gear train speeds, 151 General Electric Polymet rolling contact test rig, 782, 786 Generatrix of motion, 313, 316 Graphite, 648 Grease lubricants, 662 properties, 664 thickeners, 666 compatibility, 668 Grease lubrication, 652 advantages, 664 lubricant film thickness, 451 relubrication, 652 Groove curvature radii, 12 effect on contact angle, 53 instrument ball bearings, 16 Grubin, 433 Gyroscope bearings, 378 Gyroscopic moment, 174, 344 Gyroscopic motion, 324, 329 Hardenability of steel, 603 Hardness: fatigue life (effect on), 717, 739 Rockwell C,4 testing methods, 615 Harmonic mean radius, 487 Hazard, 802 Health Usage and Monitoring System (HUMS), 1007 Heat: conduction, 556 convection, 558 ball (from rotating), 560 roller (from rotating), 560 dissipation, 569 flow analysis, 561 generation, 553 radiation, 560 removal methods: air cooling of housing, 570 cooling of lubricant, 571 under raceway cooling, 574 transfer: modes, 556 rolling elementraceway contact (in), 574 temperature nodes, 561 Heathcote slip, 490 Heat treatment of steel, 597 Helical gear loading, 143 Helicopter applications, 9, 43 Hertz, 185 High speed ball bearings, 339 ball excursions, 346 fatigue life, 868 High speed cylindrical roller bearing, 349 fatigue life, 869 High temperature: heat removal, 569 polymers for cages, 629 Hollow rollers to control skidding, 525
Hooke’s law, 187 Hoop (ring) stress effect on fatigue life, 921 Hot isostatically pressed (HIP) silicon nitride, 623 Housing, 4 tolerance range classification,93 Hydrodynamic bearings, 6 Hydrodynamic lubrication, 419 pressure distribution, 423 Reynold’s equation, 419 Hydrostatic bearings, 5 Hypoid gear loading, 147 Ideal line contact, 219 Indentations, cause of fatigue, 620 Induced loading: ball, 158 Induction hardening steel, 610 Influence coefficients forring bending, 296 INSA contaminationlifetest system, 789 Instrument ball bearings, 15 Interference fit: effect on clearance, 55, 119 surface finish effect, 126 IoannidesHarris fatigue life theory, 906,931 ISO, 14, 45, 84 load rating standards, 741 Isoelasticity, 378 Jet oil lubrication, 650
JNS,84 Labyrinth seals, 673 Lambda parameter, 448 fatigue life (effect on),891 life testing considerations, 767 Leonard0 Da Vinci, 1 Life factors combined, 903 Life testing: accelerated, 765 confidence in results, 776 contamination effects, 769 INSA life test system, 789 mounting and dismounting effects, 770 plastic deformations (effect on), 766 practical considerations, 768 speed considerations, 767 theoretical basis, 764 Lightly loaded applications, 11 Lightweight balls, 348 Lightweight rollers, 357 Linear motion bearings, 4, 40 Line contact, 190 basic dynamic capacity, 706 radial roller bearings, 732 definition, 219 deformation (elastic), 202, 234 “dogbone” shape, 222 fatigue life, 704 radial bearings, 728 ideal, 219 lubricant film thickness, 434 modified, 220
permanent deformation, 824 semi width, 202 Line of contact, 157 Linetoline fit, 78 Liquid lubricants, 654 mineral oils, 655 Loading: bearing, 135 classification, 85 combined radial and thrust, 256 doublerow bearings, 262 combined radial, thrust, and moment: ball bearings, 266 cylindrical roller bearings, 289 spherical roller bearings, 291 tapered roller bearings, 290 five degrees of freedom, 357 limiting thrust load in radial ball bearings, 379 radial, 235 Load ratings: standards, 741 Load zone: combined radial and thrust loading, 258 fatigue life effect,864 radial load, 235 Low carbon steel for cages, 625 Llo fatigue life, 688, 717, 761 Lubricant: environmentally acceptable, 671 esters, 661 film thickness, 422, 428, 523, 694 contact inlet frictional heating effect, 441 contact shear stresses, 476 fatigue life (effecton), 891 line contact, 434 point contact, 437 starvation effect, 444 surface topography effect, 446 very high pressure effect, 440 filtration effect on fatigue life, 898 fluorinated ethers, 662 functions, 646 glassy state in contact, 427 greases, 647, 662 properties, 664 thickeners, 666 compatibility, 668 high temperature considerations, 569 liquid, 646 advantages, 654 guidelines for use, 654 mineral oils, 655, 657 properties, 658 synthetic oil properties, 660 polyglycols, 661 polymeric, 647,668 quantity, 7 selection, 657 solid, 647, 670 starvation, 444 synthetic hydrocarbons, 656 types, 646
EX viscosity index, 657 Lubrication, 360 bath, 648 boundary (wear), 941 circulating oil, 649 contact structural elements, 937 dents (inthe vicinity of), 956 fatigue life (effect on), 890 grease, 651 jet, 650 limiting shear stress in elastohydrodynamic lubrication, 477 methods, 648 nonNewtonian, 476 oil sump, 648 oncethrough,651 polymeric, 653 regimes, 453 solid, 653 wear, 939 LundbergPalm~enfatigue life theory, 208, 219,688,692,694, 794,931 casehardening steel bearings (application to), 740 limitations of the theory, 863, 904 stresslife relationship, 894 Macrogeometry, 48 Marquenc~ng,607 Martensite, 606, 612 Material effect on fatigue life, 894 Materiallife factor, 895 Maximum compressivestress: line contact, 202 point contact, 195 M a ~ m u mlikelihood method in statistics, 804 Maximum orthogonal shear stress, 694 Maximum rolling element load, 238, 242, 717 Mean time between failures, 795 Mechanical properties of steel, 614 Median fatigue life, 688, 761 MeltingRefining (MR) method for vacuum degassing of steel, 588, 595 Metallur~: audit parameters, 777 structure of steel, 212 M5ONiL steel, 581, 620 M50 steel, 581, 620 Microcontacts, 464 Greenwood ~illiamsonmodel, 465 plastic contacts, 469 Microslip, 489 Mineral oil lubricants, 655 iniature ball bearings, 5, 16 isalignment fatigue failure, 773, 1052 fatigue life (effecton), 870 limitations per bearing type, radial roller bearings, 272 types, 273 Modifie~line contact, 220 basic dynamic capacity factors, 739 Modulus of elasticity, 187
HIP silicon nitride, 349 shear, 188,429 Moisture: corrosion, 1058 life testing (eEect on), 769 Molybdenum disulfide, 418, 488, 648, 670 Mounting: locknut adapter, 24 tapered sleeve, 24 Multiple bearingshaft systems, 410 NASA fiveball endurance test rig, 782 Nearsurface region of contact, 938 Needle roller bearing, 26 cam follower application, 44 thrust bearing, 39 Newtonian fluid, 419, 451 NewtonRaphson method: contact angle change determination, 247 heat transfer temperature calculations, 564 high speed ball loading, 343 load distribution calculations for ball bearings, 272 Nitrile rubber for seals, 632 Noise, 964 sensitive bearing applications, 965 Normal approach between raceways, 234 Nylon (polyamide) 6,6 for cages, 628 Octahedral shear stress, 211 Oil bath lubrication, 648 Oncethrough lubrication, 651 Orbital motion, 317 speed, 309,328 Orthogonal shear stress, 209, 218 maximum orthogonal shear stress, 694 Oscillatory motion, 27 fatigue life of bearings (due to), 879 Osculation, 50, 70 Outofround raceway, 525 Outside diameter (o.d.1, 49 Overriding ring land, 379 Oxygen in steel, 583 PalmgrenMiner rule, 874 Partially crowned roller, 29, 224 Permanent deformations, 820 brinelling, 1052 line contacts, 824 point contacts, 820 shakedown, 852 Permissible static load, 831 Photoelastic study of roller bearing, 304 Physical vapor deposition (PVD), 640 Pitch diameter, 49, 66, 73 Pitting, 949, 1059 Planet gear bearing, 44, 152 load distribution, 292, 301 loads, 172 Planet gear speeds, 152 Plastic deformations: calculation (of), 820 life testing (effect on), 766
Plastic deformations (Continued) residual stresses (associated with), 843 measurement method, 844 shakedown, 852 wear (associated with), 947 Plating processes for components, 639 chemical vapor deposition, 640 physical vapor deposition, 640 thin dense chrome (TDC), 640 Point contact, 190, 219 deformation (elastic), 234 dynamic capacity, 699, 705 fatigue life, 699, 701 radial bearings, 707 lubricant film thickness, 436 permanent deformation, 821 Poissons’ ratio, 188, 429 Polyetheretherketone (PEEK) material for cages, 631 Polyethersulfone(PES) material for cages, 631 Polyglycol lubricants, 661 Polymeric lubricants, 647, 668 Pol~etrafluoroethylene(PTFE), 5 Powder metal components, 621 Pratt & Whitney vring/ball endurance test rig, 783 Preloading, 368 deflection, 372 isoelasticity, 378 radial, 375 Pressfitting, 83 force, 124 hoop stresses, 124 fatigue life (effecton), 921 Prestressing (effect on fatigue life), 850 Probability of survival, 688 Progression of failure, 1008 Pulley loads, 138 Pyrowear 675 steel, 582
’
Raceway control, 325 Raceways: crowning, 26, 30 dimensional audit parameters, 778 loci ofgroove curvature centers, 268,339 roller deformation components, 272 speed components, 318 surface velocities, 319 Radial clearance, 49 Radial deflection of roller bearing, 279 Radial load distribution, 235 effect of clearance, 239 Radial load integral, 237 Radial preloading, 375 Radii of curvature, 193 deformed surface, 318 Railroad car wheel application, 29 Rating life, 696, 764 RBEC, 85 tolerance classes, 99 Reliability, 691, 761 fatigue (as function of), 886
life factor, 890 Residual stress, 616, 843 alteration with overrolling, 848 fatigue life (efTect on), 850 measurement, 844 Retainer, 4 Reynolds equation, 419 Ring: deflections due to pressure, 121 fracture, 853 carburized steel (effect on), 855 integral flange, 42, 494 land, 379 radial shift, 235 stresses due to fit, 120 Roelands equation, 425 Roller bearings, 23 clearance effect on fatigue life, 867 fatigue endurance, 25 flexibly supported bearings, 868 internal load distribution effect on fatigue life, 866 misalignment, 272 radial deflection, 279 radial, 25 maximum roller load, 242 static load ratings, 826 Rollerraceway: contact laminum: load, 274, 280 deformations due to skewing, 286 heat transfer, 574 Rollers: axial loading, 177 centrifugal force, 355 cornerflange contact, 228 crowning, 26 deformation components, 272 eccentricity of loading, 276, 284 endring flange contact: sliding, 494 stress, 225 geometry, 275 hollow, 355, 525 logarithmic profile, 224 skewing, 26, 177, 179,534 tilting, 177, 280 viscous drag on, 492 Rolling bearings, 4 Rolling elements: centrifugal force, 165 dynamic loading, 161 maximum load, 238 rotational speed, 311 sliding in cage pocket, 494 types, 4 Rollingmill application, 31 Rolling motion, 309 deformation due to, 487 elastic hysteresis, 486 pure, 316, 501 sliding and, 313 Rotation about eccentric axis (forces), 172
Rotation factor V, 720, 733 Rotor dynamics, 1014 critical speed, 1039 synchronous response, 1039 shaft whirl, 1042 vibration mode shapes, 1040 Scoring, 1054 Scuffing, 944 Seals, 14 deepgroove ball bearing, 15 elastomeric lip, 675 ferrofluidic, 681 flinger (with), 674 friction, 494 functions, 672 garter, 679 greased bearing, 672 high temperature, 637 labyrinth, 673 materials, 631 oillubricated bearings (for), 673 shields (with), 17, 675 solidlubricatedbearings (for), 673 torque, 679 Selfaligning: ball bearings, 22 deflections, 366 spherical roller bearings, 30 spherical roller thrust bearings, 37 Semi axes of contact ellipse, 192 Semi width of line contact, 202 Separator, 4 Shaft: concentrated radial loading, 135 speeds, 150 tolerances, 87 whirl, 1042 Shakedown, 833,852 Shields, 14, 16, 675 Shrink fitting, 83 Silicon carbide, 621 Silicon nitride, 621 balls, 348, 896 fatigue life effect at high speed, 872 fracture toughness, 624 rollers, 357 tensile strength, 624 Silicon rubber for seals, 637 Skewing, 26, 177,228 angle, 286, 537 damage due to, 1049 flangerollerend contact and, 330 roller axial loading, 179 rollerraceway d e f o ~ a t i o n s286 , SKI? Aframe automotive wheel hub bearing endurance test rig, 779 Skidding motion, 315, 347, 360, 515 ball bearings (in), 518 cylindrical roller bearings (in), 523 outofround raceway(to control), 525 Slewing bearings, 5 , 7
Sliding friction: cage, 493 distribution of forces in ballraceway contacts, 502 gyroscopic motion(due to), 488 rolling motion (in), 488 skewing, effect of, 179 sphericalroller bearings, 35 tapered roller bearings, 28 Sliding motion, 313 deformation (cause), 316 Sliding velocity: ball bearing inner raceway, 321 ball bearing outer raceway, 320 ballraceway contacts (in), 498 distribution in ballraceway contacts, 501 gyroscopic motion, 314 roller endflange, 334 spinning motion, 314 Smearing, 516,944, 1046 Smoothness of bearing operation, 832 Solid lubrication, 653, 670 Space vehicle applications, 10 Spall, 688, 772, 773, 1061 Specific loading, 224 Speeds: shaft, 150 Spherical roller bearings, 6, 30 asymmetrical rollers, 32 barrelshaped rollers, 32 clearance, 68 contact angle, 66 curvature diflerence, 71 curvature sum, 71 free endplay, 66 high speed, 357 hourglassshaped rollers, 32 load classification, 85 osculation, 70 pitch diameter, 66 planet gear bearing, 44 roller skewing, 34, 537 singlerow, 35, 37 sliding friction, 35 steelmaking (in), 8 surface treatment for components, 638 Spherical roller loading: centrifugal force, 171 static, 159 Spherical roller thrust bearings, 37 Spinning motion, 313 ball bearings, 317 frictional moment (in), 504 Spinroll ratio, 324 Spiral bevel gear loading, 145 Split inner ring ball bearings, 22 diametral play, 56 shim, 56 shim angle, 57 tandem arrangement, 25 Spur gears: loads, 137 speeds, 150
EX
Standards, 84 interference fits, 921 Starvation of lubricant, 444 Static equivalent load, 828 Static load ratings, 825 permissible static load, 831 Statistical analysis: endurance test samples, 772 hazard, 802 life testing considerations, 769 masimum likelihood method, 804 mean time between failures, 795 product law of probability, 693, 714 sample size selection, 810 Weibull distribution, 692 twoparameter, 795 graphical representation, 798 percentiles, 797 probability functions, 795 shape parameter, 799 sudden death test analysis, 807 Steel: AISI 52100,4, 581, 598, 603 fatigue life properties, 696 AISI 440C, 4,583 annealing, 598, 613 austenite, 612, 836 bainite, 607 banding, 619 basic electric furnace processing, 584 carbonitriding, 609 casehardening, 4, 580, 608 fatigue life effects, 741, 894 cleanliness, 593 macroinclusions, 594 oxygen content, 594 cobalt alloys: L605, 621 Stellite 3, 621 Stellite 6, 621 composition (effect onfatigue life), 739 dimensional instability of components, 856 fatigue failure modes, 618 subsurface"initiated, 619 surfaceinitiated, 620 fatigue limit stress values, 927 fatigue strength, 614 grain size, 604 hardenabilit~,603 hardening methods, 605 heat treatment, 597 mechanical properties affected by, 614 continuous cooling transformation (cct) curves, 601 timetemperaturetransformation (TTT) curve, 601 high temperature, 569 induction heating, 610 inhomogeneities, 619 low carbon for cages, 625 machinability, 596 marq~enching,607
Martensite, 606, 612 materiallife factor, 895 melting methods, 582 electroslag refining, 583, 591 vacuum arc remelting, 590 vacuum degassing, 583, 585 fatigue life (effect on), 739 vacuum induction melting, 589 metallurgical characteristics, 593 audit parameters, 777 cleanliness, 593 quality, 593 banding, 619 decarburization, 596 inhomogeneities, 619 macroinclusions, 619, 688 nonmetallic, 842 porosity, 596 segregation, 595, 604 sulfide inclusions, 619 microstructure, 596, 836 alterations due to rolling contact, 837 butterfiies, 841, 1061 dark etching region of overrolled AISI 52100 steel, 839, 841 white etching bands, 842, 1061 carbides, 596 porosity, 619 Poissons' ratio, 188* processing methods (effects on), 597 fatigue life (effect on), 739 products: forms, 592 inspection, 592 eddy current, 595 macroetching, 596 ultrasonic testing, 595 quality, 593 raw materials, 583 residual stresses, 843 alteration with overrolling, 848 fatigue life (effecton), 850 retained austenite, 612, 614, 836 alteration with overrolling, 848 stainless, 4, 583, 613 structure, 596 subsurface structure after overrolling, 213 surface hardening, 4, 580, 608 tempering, 613, 618 thermal treatment for structural stability, 612 through~hardening,4, 580 ma~ensite,606 tool steels, 620 types, 579 Stiffness of bearings, 1028 contact angle effects, 1037 preload efiects, 1037 shaft bending effects, 1038 speed effects, 1034 Strain, 187 Stress concentr~tions: contaminant denting (due to), 920
crowning, 224 Stress cycles per revolution, 695, 704 Stressed volume: IoannidesHarris theory, 907 LundbergPalmgren theory, 694 Stresses: fatigueinitiating, 907 hertz, 186,623 hoop stress effect on fatigue life, 921 material processing (due to), 922 octahedral shear, 211,907 residual, 843 subsurface, 204 comparison of shear stresses, 214 maximum shear, 205 orthogonal shear, 209 principal, 205 surface: normal, 189 shear, 215 Von Mises, 210, 907 Stresslife factor, 910,931 Stribeck, 238 Subsurface metallurgical structure of 52100 steel, 212 Subsurface stresses: frictional shear stresses on contact surface (due to), 911 hertz stress (due to), 911 Sudden death endurance testing, 775 Weibull distribution analysis, 807 Surface damage: adhesion, 1055 brinelling, 1052 decarburization, 620 false brinelling, 1054 grinding burns, 620 inception, 950 indentations, 620 interacting modes of failure, 953 marks, 620 scoring, 1054 smearing, 516 Surface finish: effect on clearance, 126 Surface hardening steel, 608 flamehardening, 611 induction heating, 610 residual stress, 616 Surfaceinitiated fatigue, 620 Surface shear stresses, 215 ballraceway contacts (in), 496 composite shear stress, 479 contaminant (particulate) effect on, 913 coulomb friction in asperity contact (due to), 478 Surface topography: egect on lubricant film thickness, 446 honed and lapped surface, 822 rough surfaces, 464 fatigue life (effecton), 891 Survival probability, 688, 691, 709, 907
Talyrond, 981 Tandem bearing arrangement, 21 Tapered roller bearings, 27 cone angle, 77 cup angle, 77 doubledirection, 35 doublerow, 29 endplay, 78 endurance test rig, 782 fourrow, 35 high speed, 357 misalignment, 272, 1052 pitch diameter, 77 roller: angle, 77 endflange contact geometry, 334 static loading, 160 small and steep angles, 33 surface treatment for components, 638 thrust bearing, 39 Tapered roller loading: static,, 160 tapered shaft mounting, 376 Tapered sleeve mounting, 24, 376 Temperature: effect on clearance, 125 expansion of rings, 125 nodes in heat transfer system, 561 surface (in wear), 940 viscosity variation (with), 660 Tempering of steel, 613, 618 Test rigs: ballrod rolling contact fatigue tester, 787 design considerations, 777 General Electric P o l p e t rolling contact endurance tester, 786 INSA rollingslidingdisc endurance tester, 788 NASA fiveball endurance tester, 782 Pratt & Whitney vring/ball endurance tester, 783 SKI? Aframe automotive wheel hub bearing endurance tester, 779 SKI? I22 endurance tester, 781 tapered roller bearing endurance tester, 782 Theory of elasticity, 185 Thermal gradient, 83 Thermal imbalance failure, 775, 1048 Thermal treatment of steel for structural stability, 612 Thin dense chrome (TDC)plating for components, 640 Thin ring deflections, 294 Three bearingshaft systems: nonrigid shaft, 404 rigid shaft, 400 Throughhardening steel, 580 Thrust ball bearings, 23 deflections, 367 effect of centrifugal force on contact angle, 168 limiting load, 379 Thrust carried on roller ends, 228
1 Thrust loading: centric, 245 eccentric, 249 excessive contact stress in radial ball bearings, 382 radial cylindrical roller bearings, 280 Thrust load integral, 251 double direction bearings, 255 single direction bearings, 251 Thrust roller bearings, 37 deflections, 367 Titanium carbide, 621 coating, 640 Titanium nitride coating for components, 640 Tolerances: classes, 85 B E C , 99 ~ S I l vs BISO,~98 RBEC, 99 housing bore limits, 94 shaft tolerance range classification,87 Tool steels, 620 Traction stresses, 215 Tribological processesassociated with wear, 939 Triplex set of angularcontact ball bearings, 375 Truck wheel application, 43 Tungsten carbide, 621 Two bearingshaft systems, statically indeterminate: fiesible shaft, 392 rigid shaft, 389 Ultrasonic testing of steel components, 595 Under raceway coolingof bearing, 574, 650 Unit (Metric/English units system) conversion factors, 1071 Vacuum arc remelting of steel, 590 fatigue endurance Weibull slope, 696 Vacuum degassing, 583 fatigue life (effecton), 739 Vacuum induction melting of steel, 589 Variable loading eRect on fatigue life, 875 Vibration, 964 causes in bearings: geometrical imperfections, 970 nonroundness, 980 waviness, 971, 980 testing, 986 variable elastic compliance, 969 coupled motion, 1020 damped forced, 1015 detection of failed bearings, 997 condition monitoring, 1003 health usage and monitoring system, 1008 microsensors, 1007 shock pulse method, 1005 frequencies in bearing operations, 993 multidegreeoffreedomsystem, 1024 natural frequencies in bearings, 996 resonant, 996
EX
role of bearings in machine, 968 sensitive bearing applications, 965 smoothness of bearing operation, 832 testing of bearings, 991 waviness (relationship to), 994 V I W m steels, 581, 591 Viscosity: Barus equation, 424 index, 657 kinematic, 543 Roelands equation, 425 selection for application, 657 specification, 657 variation with pressure, 424 ASME study, 425 lubricant glassy state, 427 sigmoid curvefit to ASME data, 426 variation with temperature, 660 Viscous drag: cage (on), 530 balls (on), 492 bearing friction torque, 542 rollers (on), 524 Volume under stress, 694, 925 Von Mises stress, 210, 215, 923 Water contamination effect on fatigue life, 902 Waviness, 980 testing, 986 Wear, 936, 1045 adhesive, 944 corrosive, 946 delamination, 949 failure classification, 936 phenomenologicalview, 949 pitting, 949 processes, 936, 942 protection, 955 roller endflange, 330 smearing, 944, 1046 tribological processes, 939 Wedeven balldisc test rig, 790 Weibull distribution, 692, 794 application, 800 maximum likelihoodmethod, 804 slope, 695, 706, 799 sudden death test analysis, 807 twoparameter, 795 graphical representation, 798 mean time between failures, 795 probability functions, 795 shape parameter, 799 White room, 16, 19 Worm gears: loading, 149 speeds, 150 XRay diffraction, 844 X and Til factors: point contact radial bearings, 720 point contact thrust bearings, 725 radial roller bearings, 734 static loading, 830