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ROLLING BEARING ANAZYSIS
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This book is printed on acid-free 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 per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012,(212) 850-6011, fax(212) 850-6008, E-Mail: 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 0-471-35457-0 (cloth : alk. paper) 1. Roller bearings. 2. Ball-bearings. TJ1071.H35 2001 621.8'22-dc21 Printed in the United States of h e r i c a . 1 0 9 8 7 6 5 4 3 2 1
00-038171
Linear Motion Bearings earings for Special Applications Closure
1 11 23 40 41 44
Tapered Roller Bearings Closure
47 48 49 66 73 77 79 V
81
123
1 Closure
130
134 135
s t ~ ~ u t Load e d Systems
3 0 153 153
157 157
Closure
181
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
233 34 35 45 256
List of Symbols General
307 308 309 313 317 330 335
olling and Sliding Orbital, Pivotal, and Spinning Motions in Ball Bearings oller End-Flange Sliding in Roller Bearings Closure
266 272 80 2 ~ 9 291 302
355 Closure
Closure
e
400 410 412 e
418 419 424 440 441 444 446
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 ~
456
Closure e
461 463 464 472 riction in the EHL 476 Contact Closure
479 483 485
o
~ of Friction ~ c
~
~
486 olling Element-
Cont~cts and Cage Forces ons and Forces
496 515 529 E”
c
losure
Closure e
Closure
Types of Lubricants Liquid Lubricants Grease Lubricants Polymeric Lubricants Solid Lubricants En~ironmentallyAcceptable Lubricants Seals Closure
643.
645 645 646 648 654 662 668 670 671 672 682
e
4
4
e
820 820
tatie Equivalent Load
i
earing Com~onents Permissible Static Load Closure
831 831 832
e
crostructures of Rolling Bearing Steels icrostructural Alterations Due to Rolling Contact esidual Stresses in Rolling Bearing Components of Bulk Stresses on Material Response to
835 835 836 37 843 853 857
Closure e
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 Stress-Life Factor Closure
861 863 864 874 879 886 890 894 89 903 904 906 909 931
e
List of Symbols
of a Lubricated ont tact
teracting Tribological Processes and Failure Modes ecommendations for ear Protection Closure
936 937 939
955 958
i e
Closure
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,Marlin-Rockwell 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 shaft-bearing-housing 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, x-ray 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 societies-for ican Societyof Mechanical Engineers, the Institution e Society of Tribologists and Lubrication E ociety of Mechanical Engineers-analy~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, up-to-date 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, long-lived 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 low-cost, 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 spaced-apart 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 61-65 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 sase-hardening steels such as 310. For some specialized applicati motive wheel hub bearings, the rolling components are manufactured from induction-hardening 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 -to-weight 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 deep-space 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 life-critical applications in aircraft power transmissions (Fig. 1.9), and the extreme low-high 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 self-contained, life-long 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.
.
CH-53E Sikorsky Super Stallion heavy-lift 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 load-speed 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 steel-making 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 single-row, deep-groove, Conrad-assembly,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 Conrad-type,deepgroove ball bearing.
~ ~ o t o showing ~ a p Conrad-type ~ 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 single-row deep-groove 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 single-row deep-groove ball bearing having two seals to retain lubricant (grease) and prevent ingress of dirt into the bearing.
~ingle- ow Deep-Groove ~ 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 ble-Row Deep-Grooue Ball earings. This ball bearing as shown in Fig. 1.21 has greater radial load-carrying capacity than the single-row types. Proper load sharing between the rows is a function of the geometrical accuracy of the grooves. Otherwise, these bearings behave similarly to single-row ball bearings. ~ n s t r u ~ e nall t earings. In metric design, the standardized form of these bearings ranges in size from 1.5-mm (0.0~~06-in.) bore and 4-mm 1574~-in.) 0.d. to (0.35433-in.)bore and 26-mm (1.02~~2-in.) o.d. e reference [1.51. ailed in reference [1.6],standa~dizedform, inch design instrument ball bearings range from 0.635-mm (0.0250-i~.) bore
.
A single-row deep-grooveball 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 deep-groove ball bearing dimension series.
.
Cutaway view of a singlerow, deep-groove, filling slot-type ball bearing assembly.
. A double-row, deep-groove, 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 thrust-carrying 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 angular-contact ball bearing.
\
(a) Small angle
. An~ular~contact ball bearings.
ALL
(a) Back-to-back mounted
(b) Face-to-face mount@d
Duplex pairs of an~lar-contactball bearings.
I-
”
(a) Nonrigid type
(b) Rlgid type
1.27. Double-row angular-contact ball bearings.
moment loading effectively. Essentially, the bearings perform similarly to duplex pairs of single-row angular-contact 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 self-aligning 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 load-carrying 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 high-performance aircraft gas turbine engine. Obviously, both the inner andouter rings must be locked
1.28. A double-row 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 load-supportingcapability, whichcannot be feasibly obtained using
.
A double-row internally self-aligning 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 tandem-mounted 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 self-aligning 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 radialload-carrying 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 radial-load-carrying 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 double-row 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 full-co~plement-type bearing, the rollers are fre~uently retained by turned-under flanges that are integral with the outer shell. The raceways are frequently hardened but not ground.
s The single-row 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 load-carryingcapacity and eliminate problems of axial adjustment due to distance between bearings, tapered roller bearings may be combined as shown in Fig. 1.46 into two-row bearings. Fig. 1.47 shows a typicaldouble-row tapered roller bearing assembly for a railroad car wheel application. Double-row bearings may also be combined into four-row or quad bearings for exceptionally heavy radial load applications such as rolling mills. Figure 1.48 shows a quad bearing having integral seals.
. A double-row,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 load-high 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 end-flange contacts and cage rail-cone 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 self-ali~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)
Full-complement needle roller bearings. (a)Drawn cup assembly with trunnion-end rollers and innerring; ( 6 )drawn cup assembly with rollers retained by grease pack (courtesy of Torrington Company, Division of Ingersoll Rand Corp.).
1.43. Single-row 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 load-carrying capacity. Various executions of double-row, 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 hourglass-shape) 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. ouble-row bearings having barrel-shape, 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 load-carrying capacity of the bearing. Roller guiding in such bearings tends to be acco~plishedby
r
e
Double-row tapered rolling bearings. ( a )Double coneassembly; (6) double
cup assembly.
F
1.47. Sealed, greased, and preadjusted double-row, tapered roller bearing for railroad wheel bearings (courtesy of the Timken Company).
the raceways in conjunction with the cage. In a well-designed bearing, the roller-cage 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 four-row 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 roller-raceway 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. ~ouble-rowbearings can c a m combined radial andthrust load; they cannot support moment loading. Radial, single-row, 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 roller-raceway loading substantially. Therefore, these bearings should
.
Cutaway view of a double-row, 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 double-row 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' single-row 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 load-carrying capacity
IG Single-row, radial, toroidal roller bearing (courtesy of SKY).
1-53. 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 self-aligning. 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, roller-washer 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 roller-cage assembly that may be purchased in lieu of an entire bearing assembly.
1.58. Thrust needle roller-cage assembly (courtesy of Torrington Company, Division of Ingersoll Rand Gorp.).
Bearings for linear motion, such as those used in machine tool “ways”for example, V-ways-generally employ only lubricated sliding action. These sliding actions are subject to relatively high stick-slip friction, wear, and subsequent loss of locational accuracy. 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 built-in 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 medium-heavy 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 subsurface-initiated 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 grease-lubricated 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 sealed-for-life, d plex sets as shown units needed to be press-fitte 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 anti-lock 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-€or-life,ball bearing units. ( a )Without flanges; unit is press-fitted into wheel hub and slip-fitted 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 sealed-for-life, Lapered roller bearing unit (courtesy of SKI?).
4. Self-contained?tapered roller bearing with an integral speed sensor to provide signal to the anti-lock 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 right-cylindrical 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),183-196 (1962). 1.5 American National Standards Institute, American National Standard ( ~ S I I ~ ~ A ) Std, 12.2-1992, “Instrument Ball Bearings-Metric Design” (April 6, 1992). 1.6 American National Standards Institute, American National Standard ( ~ S I / ~ ~ A ) Std. 12.2-1992,“Instrument Ball Bearings-Inch 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 end-to-end 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 single-row radial deep-groove 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 mwc-m
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 no-load state, the bearing also can experience an axial play. Removal of this axial freedom causes the ball-raceway contact to assume an oblique angle with the radial plane; hence, a contact angle different from zero degrees will occur. Angular-contactball 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 ball-raceway 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,+ri-D ~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 = cos-1 ( 1 -
$1
3. A 218 an~lar-contactball 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" = cos-l(l =
(2.3)
(
X
22.23
=
0.48 mm (0.019 in.)
2)
cos-1 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" = cos-1 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 single-row ball bearings. Rouble-row angular-contact 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 double-row 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 cos-1
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 single-row ball bearing, (b) free misalignment of outer ring of single-row 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 ! ' = cos-1 (1 =
(
cos-1 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 = - + - + - + 7-11
r12
7-111
.
1
(2.25)
7-112
. 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 ball-inner 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 ball-outer 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 angular-contact 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) 1-Y
1 - 0.1359
0.108 mm-l (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
=
[
cos-1 (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 P-Q,Ommrx,meoOm"MP-m"M"mmOooQ,o P?-i l-4 ?-i du m M M M d d m m eo eoP- rx, Q, 0_?4/-iemeqm_
omooomoooooooooooooooooooo eoP-rx,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~MmddmmeoeoP-P-rx,Q,oe?-i~me
T+?-id?-iNmmmMMMddmmeoeo""rx,Q,
mmomoomooooooooooooooooooo dmeo"Q,?-iMeoQ,mdeoQ,mmP-?-imomomoP-eoM
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,OmdP-rx,Omdeorx,?-id"ddrx,Mrx,md
?-idd?-id?-iT+dudummmMMmdd
3momooommooooooooooooooooo NmMMdmeoP-Q,dmMdmP-Q,Odudeorx,dm"Mrx,
B
3
B
3
H
8
l-i
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
=
cos-1
p
=
cos-1 (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 two-row 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
r-i
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 stress-increasing effects of edge-loading (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 raceway-roller contact e = 3 cone angle, a,, the outer raceway-roller contact angle = cup e, af,the roller large end-flange 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 anendto-end 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. 20-1987,“Radial Bearings of Ball, Cylindrical Roller, and Spherical Roller Types, Metric Design” (October 28, 1987). 2.2 A. Jones, Analysis of Stresses and Deflections, 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 0-r
0-t
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 ti-Friction 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.A-generated 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 shaft-bearing bore and ho~sing-bearing0.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 housing-bearing 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 3.6-3.10 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 3.6-3.10 give tolerance ranges for all ABEC classifications. Additionally, Tables 3.6-3.8 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 3.6-3.10 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
I4
oooooF4
*222%2
*322%
000000
I
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000 d0
a,
u3 M
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9
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rn
8 iii
9
z
9
z
9
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i
7
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00
d
c:
l l
c:
00
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2
2
a
I
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or(
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orc
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wdr
+ + w+ m+ w+ m+ w+ m+ w+ m+
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0
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Nc-
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m
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om s
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m 0 0
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+
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s m
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dim + I
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++
mdi
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oc-
I "TI,':
+ ---"-
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---"--"-1 $24: $24: m r ' r+' +
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I
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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 ABEC-1, RBEC-1. 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 mcoml-loomoomooo 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
l-li
d
m"
on
0"
l-ll
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Class ABEC-3, RBEC-3. 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 ABEC-5, RBEC-5. 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
r-4
6 3
!5
r-4
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 (3.3-3.6) 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%
+ c2S-l
(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 (24-56) Machine-reamed bores (40-94) Ordinary accurately turned surface (94-190)
Reduction 0.0001 mm
Reduction 0.0001 in.
20-51 61-142 102-239 239-483
(8-20)
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)
=
(
cos-1 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 angular-contact 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 t-0.0025 mm (0.0001 in.) to t-0.0254 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 <
-
0.3
+
206900 103500
~
+1
)
2
+ 0.33
( ~ ) 2
-1
i
(3.27) =
0.0064 mm (0.00025 in.) (3.30)
A P d =,A8 - Ah =
-0.00995
-
0.0064 = -0.01635 mm (-00064 in.)
Ex.2.3
Pd = 0.483 mm (0.019 in.) (2.10) =
c0s-l
(
1-
0.483 - 0.01635) 2 X 1.031
=
39019,
~ ~ 3.3.~ TheZinnere ring of the 218 angular-contact ball bearing operates at a mean temperature of 148.9"C (300°F)and the outer ring is at 121.1"C (250°F). Consideringthat the bearing was assembled at 21.1'6 (70°F) and considering the press fits of Example 3.2, what free contact angle will occur? For steel
3130
I' = 11.7 X For titanium I' = 8.5 X
mm/mm/'C (6.5 mm/mm/'C (4.7
in./in./'F)
X
X
in./in./"F) Ex. 2.3
di = 102.8 mm (4.047) in.)
Ex. 2.3
do = 147.7 mm (5.816 in.) Because of differential expansion,
AT
- di(Ti -
= rb[do(Fo - Fa) =
11.7
=
0.0191 mm (0.00075 in.)
X
Ta)]
10"6(147.7 X 100 - 102.8
X
127.8)
(3.38)
The outer ring and housing havedifferent rates of expansion Ex. 3.2 $h = 160 mm (6.2992 in.)
AI
= (I'b
-
I'h)gh(To -
Ta)
(3.39)
=
(11.7 - 8.5) X
=
0.0508 mm (0.0020 in.)
Ah
=
0.0064 mm (0.000~5in.)
Ex. 3.2
I
=
0.0113 mm (0.00044 in.)
Ex. 3.2
X
160 X (121.1 - 21.1)
I=I+AI =
0.0113 + 0.0508
=
Ah = o*062 X 0.00635 0.0113
0.062 mm (0.00244 in.)
=
0.0348 mm (0.00137 in,)
A, = 0.00995 mm (0.00039 in.) APd = AT - A, - Ah =
0.0193. - 0.00995 =
Ex. 3.2
-
(3.40)
0.0348
-0.0257 mm (-0.00101 in.)
Pd = 0.483 mm (0.019 in.)
Ex. 2.3
A
Ex. 2.3
=
1.031 mm (0.0406 in.)
(2.10) =
(
cos-1 1 -
0.483 - 0.0257 2 X 1.031
=
38'54'
131
REFEREN~ES
4. The 209 ball bearing of Example 3.1 has a nominal width of 19 mm (0.7480 in.). What force is required to accomplish the shaft press fit in Example 3.1?
2-
=
0.0076 mm (0.00030 in.)
(Ex. 3.1
5B8
=
45 mm (1.7717 in.)
Ex. 3.1
=
52.3 mm (2.0587 in.)
Ex. 3.1 (3.32)
=
4.71
X
lo4 X
r
19 X 0.0076 1 L
=
1766 N (397 lb)
The important effect of bearing fitting practice on diametral clearance has been demonstrated for ball bearings in the numerical examples. Because the ball bearing contact angle determines its ability to carry thrust load and the contact angle is dependent on clearance, the analysis of the fit-up is important in many applications. The numerical examples herein were based on mean tolerance conditions. In many cases, however, it is necessary to examine the extremes of fit. Although onlythe effect of fit-up on contact angle has been examined, it is not to be construed that thisis the only effect of significance. Later, the sensitivity of other phases of rolling bearing operation to clearance will be investigated. The thermal conditions of operation have been shown to be of no less significance than thefit-up. In precision applications, the clearance must be evaluated under operating conditions. Tables 3.6-3.10 contain tolerance limits on radial and axial runout as well as the tolerance limits on mean diameters. Runout affects bearing performance in subtle ways such as through vibration as discussed in Chapter 25.
3.1. American National Standards Institute, American National Standard( ~ ~ ~ Std 7-1995, “Shaft and Housing Fits for Metric Ball and Roller Bearings (Except Tapered Roller Bearings) Conformingto Basic Boundary Plans” (October 27, 1995).
I
~
3.2. American National Standards Institute, American National Standard ~ S I / ~ ~ A ) Std 12.1-1992, “Instrument Ball Bearings Metric Design” (April6, 1992). 3.3. American NationalStandards Institute, American National Standard ( ~ S I I ~ ~ A ) Std 12.2-1992, “Instrument Ball Bearings Inch Design” (April6, 1992). 3.4. American NationalStandards Institute, American National Standard ~ S I f ~ ~ A ) Std 16.1-1988, “Airframe Ball, Roller,and Needle Bearings-Metric Design” (November 17, 1988). 3.5. h e r i c a n National Standards Institute, American National Standard ( ~ S ~ / ~ ~ A - Inch Design” (December Std 16.2-1990, “Airframe Ball, Roller, and Needle Bearings 20, 1990). 3.6. American NationalStandards Institute, American National Standard ( ~ S I l ~ ~ ) Std 18.1-1982,“Needle Roller Bearings Radial Metric Design” (December 2, 1982). 3.7. American NationalStandards Institute, American National Standard( ~ S I I ~ ~ A ) 14, 1982). Std 18.2-1982,“Needle Roller Bearings Radial Inch Design” (May 3.8. American NationalStandards Institute, American National Standard ~ S I / ~ ~ A ) 19, 1987). Std 19.1-1987, “Tapered Roller Bearings Radial Metric Design” (October 3.9. American NationalStandards Institute, American National Standard ( ~ S I f ~ ~ A ) Std 19.2-1994,“Tapered Roller Bearings-Radial Inch Design” (May 12, 1994). 3.10. American NationalStandards Institute, American National Standard ( ~ S I f ~ ~ A ) Std 20-1987, “Radial Bearingsof Ball, Cylindrical Roller, and Spherical Roller Types, Metric Design” (October28, 1987). s , Departure Division, GeneralMo3.11. A. Jones, Analysis of Stresses and ~ e ~ e c t i o nNew tors Corp., Bristol, Conn,, 161-170 (1946).
ST Synbol
a
S Description Distance to load point fromright-hand bearing center Gear train value Bearing radial load Gravitational constant Thread pitch of worm at thepitch radius Power Distance between bearing centers Length of connecting rod Speed Number of teeth on gear Applied radial direction load Force applied on piston pin Inertial force due to reciprocating masses Centrifugal force acting on connecting rod bearing due to rotating masses
Units mm (in.)
N (W mm/sec2 (in./sec2) mm (in.) watts (HI)) mm (in.) mm (in.) rPm
133
Symbol pcc
radiusr rP
T
X
is
Y h
- #+
Units
Centrifugal force acting on crankshaft bearing due to rotating masses Crank Gear pitch radius Applied moment load Applied per load unit length Weight of reciprocating parts Weight of connecting rod including bearing assemblies Weight of reciprocating portion of connecting rod Weight of rotating portion of connecting rod Weight of crank pin and crank webs with balance weights along Distance shaft Number of threads on worm, teeth on worm wheel Bevel gear cone angle Lead angle of worm at the pitch radius Gear pressure angle Gear helix angle
N (lb) mm (in.) mm (in.) N mm (lb in.) N/mm (lblin.) N (lb) N (lb) N (1b)
N (lb) N (lb) mm (in.)
O,
O, O, O,
rad rad rad rad
SUBSCRIPTS Refers to bearing location Refers to driving or driven gear Refers to axial direction Refers to radial direction Refers to tangential direction k
SUPERSCRIPT Refers to location of applied load or moment
The loading a rolling bearing supports is usually transmitted to the bearing through the shafton which the bearing is mounted. Sometimes,however, the loading is transmitted through the housing that encompasses the bearing outer ring; for example, a wheel bearing. In either case, and in most applications, it is sufficient to consider the bearing as simply resisting the applied load and not as an integral partof the loaded system. This condition will be covered in this chapter together with definition of the loads transferred to the shaft-bearing system by some common power transmission components.
135
The most elementary rolling bearing-shaft assembly is that shown by Fig. 4.1 in which a concentrated load is supported between two bearings. This load may be caused by a pulley, gear, piston and crank, electric motor, and so on. Generally; the shaft is relatively rigid, and bearing misalignment due to shaft bending is negligible. Thus, the system is statically determinate; that is, the bearing reaction loads F may be determined from the equations of static equilibrium. Hence
Z F = O Fl+F2-P=0
EM=O F,Z
-
F2(Z - a ) = 0
Solving equations (4.2) and (4.4) simultaneously yields
Fl=P(l-;)
(4.5)
a F2 = P -
(4.6)
Z
For an overhanging load as shown in Fig. 4.2, equations (4.5) and (4.6) remain valid if the distances measured to the left of the left-hand bearing support are considered negative.
4.1. ~ o - ~ ~ a ~ n system. ~ - s h ~ f t
BEARING LOADS ANL) SPEEDS
Ei: FI~URE 4.2. ~ o - ~ e a ~ n g -system, s h a ~ overhung load.
Equations (4.5) and (4.6)therefore become
F,
=
P (1
T
);
If the number of loads Pk act on the shaft as shown in Fig. 4.3, the magnitudes of the bearing reactions may be obtained by the principle of superposition. For these cases k=n
(4.9) k=l k=n
(4.10) k=l
Equations (4.9) and (4.10) are valid for loads applied in the same plane. If loading is applied in difl‘erent planes, then the loads must be resolved into orthogonal components; for example,Pt and P,”(assuming the shaft axis is aligned with the x-direction). Accordingly, the bearing radial reactions .Fly, Fk,F2y,and F2z will be determined. Then
Among the most common machine elements used in combi~ationwith rolling bearings in power transmissions are involute form spur gears.
F I ~ 4.3. ~ E Two-bearing-shaft system, multiple loading.
These gears are used to transmit powerbetween parallel shafts. As shown by machine design tests (for example, Spotts and Shoup 14.11, Juvinall and Marshek [4.2], Hamrock et al. [4.3], and several others), load is transmitted normal to the flanks of the gear teeth at anangle 4 to a tangent to the gear pitch circle at thepoint of contact. This normal load P can be resolvedinto a tangential load Ptand a separating or radial load Pr.Figure 4.4 illustrates the loads transmitted by spur gears at the gear pitch radius rp.The tangential load Pt can be determined from the power relationship (4.13) H = 2vnPtrP
The separating force is calculated using equation (4.12).
.4.
Loads transmitted by a spur gear.
138
Equations (4.13) and (4.14) are also valid for herringbone gears, which also transmit power between parallel shafts. These gears are illustrated by Fig. 4.5. Loading of planet gears in planetary gear transmissions is illustrated by Fig. 4.6. It can be seen that the total radial load acting on the shaft is 2 Pt. Also, the separating loads are self-equilibrating; that is, they cancel each other. In Chapter 7 the separating loads are shown to cause bearing outer ring bending, affectingdistribution of load among rollers.
Belts and pulley arrangements also produceradial loading, as illustrated by Fig.4.7. It can beseen from Fig.4.7 that theload appliedto the pulley
-.-
F3i 4.5. Loads t r a n s ~ i t ~bydherringbone a gear. J/ is the helix angle.
4.6.
Planet gear loading.
CONCE
EL) RARIAL LOADING
~
13
I 4.7. ~ Belt-and-pulley ~ arrangement. ~ E
shaft is a function of the sum of the tensions in the belt and the drive and driven pulley diameters. See references C4.11-[4.3] for means to determine the belt tension loads. Due to belt expansion and variation in the transmitted power, the belt is generally preloaded more than is theoretically necessary. The radial load on the shaft may be approxi~ated by (4.15)
I ) = flf2,'
where I)' is the theoretical pulley load. If the belt cross section is very large, then E" = f3A
(4.16)
where A is the cross-sectional area. Values of f l , f2, and f3 are given by Table 4.1. In Table 4.1 the higher f l values are appropriate when belt speed is low; the higher f i values should be used when the center distance is short and the operating conditions are not favorable.
Figure 4.8 schematically illustrates the loads in a friction wheel drive. In thiscase a coefficient of friction must be determined that is a function
LE 4.1. Factors for Belt and Chain Drive Calculations Type of Drive
fi
fi
f3
leather Flat belt with tension pulley Flat leather belt without tension pulley Fabric belt, rubberized canvas belt, nylon belt Balata belt TI-belt Steel belt Chain
1.75-2.5
1.0-1.1
550
2.25-3.5
1.0-1.2
800
1.5-2.0
1.0-1.2 1.0-1.2 1.1-1.5
275
4.0-6.0
I
-
BEAREVG LOADS ANI) SPEEDS
FIGURE 4.8. Schematic illustration of friction wheel loading.
of the application. The combinationsof materials and operating environments are too numerous for friction coefficient values to be included herein. It is suggested that a mechanical engineering handbook; for example, Avallone and Baumeister i4.41 be consulted.
ue to an Eccentric In some applications dynamic loadingis generated due to rotation of an eccentric mass. This is illustrated by Fig. 4.9.The force P d generated by the weight W located at distance e from the axis of rotation is given by (4.18)
where g is the gravitational acceleration and w is the speed of rotation in radlsec. The force P d is constant with regard to a rotating shaft angular position, and translates into a similar condition with regard to bearing loading. This condition must be considered in the evaluation of fatigue life (see Chapter 18).
FIG
.9. Rotor with eccentric mass.
141
C Q N C E N ~ T RADIAL E~ LQADING
ue to a Crankeehanism Reciprocating mechanisms, such as pistons in internal combustion engines, air compressors, and axial pumps, may employ rollipgbearings in several different locations. Each of these bearings experiences dynamic loading associated with the reciprocating motion. Figure 4.10 is a schematic representation of a crank mechanism. In Fig. 4.10 three bearing locations are indicated: (1) crankshaft support bearings, (2) connecting rod-crank bearing, and (3) the piston pin bearing. In this mechanism, usually 30-40% of the connecting rod is considered as reciprocating; therefore
w2,= (0.3 - (4.19) 0.4)w2 W21= (0.7 - 0.6)W2
(4.20)
All forces acting from left to right in Fig. 4.10 are considered positive; they are negative when acting in the opposite direction. The reciprocating masses comprise the piston and piston pin and the reciprocating portion of the connecting rod assembly. Therefore, the inertial force due to reciprocating masses is
Pil =
-(W1
r + W2') rco2(cos a + 7 cos (2a)
g
(4.21)
At top dead-center ( a = 0), Pilreaches a maximum of (4.22)
and at bottom dead-center ( a = 180')
4.10. Schematic diagram of crank mechanism.
(4.23)
The centrifugal force acting on the large-end bearing of the connecting rod is given by (4.24)
The centrifugal force transmitted to the crankshaft bearings is
In equation (4.25) rl is distance between the crankshaft axis and the avity (CG) of weight W.; the minus sign is used when the crank-pin and the CG are on opposite sides of the cranksha~t axis. The external force Ppand the inertial force Pixhave a c o m ~ o nline of action and may be combined such that the resultant is I") = Pp 4-
.Pi,
(4.26)
In some applications only a single bearing i s used to support the shafi as a cantilever subjected to load in a radial plane. As i n ~ i c ~ t by e d Fig. 4.11, in thiscase the b~aring must also support a moment or m i s a l i ~ i n ~ mome~tload. The equations for bearing radial and m o ~ e n load t are
143 k=n
(4.27)
F = Z P k k=l k=n
M
=
2 Pkak
(4.28)
k= 1
In a power transmission system involving helical, bevel, spiral bevel, hypoid, or worm gearing, in addition to the radial loads, thrust loads are transmitted between gears. These thrust loads occur at distances from the shaft axis and thereby create concentrated moments that act on the shaft. This is illustrated schematically by Fig. 4.12. For these cases, the equations defining radial loading of the bearings are
F,=&
ZPk k=n 1=1
(
41') 9"
1T"
"
(4.29) (4.30)
-
Helical gears also are mostly used to transmit power between parallel shafts; the radial andaxial loads transmitted are illustratedby Fig. 4.13. As before for spur gears, the tangential load component Pt is determined from equation (4.13). The separating load is given by
4.13. Two-bearing-shaft system with concentrated radial and moment loads.
144
BEARING LOADS AND SPEEDS
Loads transmitted by a helical gear.
tan Cp P,. = P, cos J/
(4.31)
where Cp is the pressure angle and J/ is the helix angle. The axial load component P, is given by
Pa = Pt tan IC/
(4.32)
Helical gears can also transmit power between crossed shafts. In this case, once P, is determined using equation (4.13), equations (4.31) and (4.32) can be used to determine Pal,Prl,Pa2,and P,.. for gears 1 and 2, which have helix angles +l and +2.
Bevel gears are used to transmit power between shafts whose axes of rotation intersect. Thus, the shaft axes lie in a common plane. As with helical gears, bevel gears, radial and axial loads are transmittedbetween the mating gear teeth. Because the axial load occurs at a distance rp from the shaft axis, a concentrated moment T = rpP , is created. Figure 4.14 shows the loads transmitted by a straight bevel gear. As for spur gears, the tangential load component P, is determined from equation (4.13); however,a mean pitch radius of the teeth-for example, r,-must be used. The radius to the pitch circle of the large end of the bevel gear is called the back cone radius rb. Using the half cone angle y and the gear tooth flank length Z, , it can be determined that
r,
=
I
rb - - sin y 2
(4.33)
The s e ~ a r a t i (radial) n~ andaxial loads are given by equations (4.34) and (4.35).
45
F
I
~ 4.14. ~ ELoads transmitted by a straight bevel gear.
Pr = Pt tan # cos y
(4.34)
Pa = Pt tan # sin y
(4.35)
+
Spiral bevel gears, similar to helical gears, have a helix angle as indicated by Fig. 4.15.In thiscase the axial and radialloads trans~itted between teeth are affected by the helix angle, the direction of the helix, and thedirection of rotation of the gear. Furthermore, a distinction must be made betweendriving and driven gears. Considering the ~ r ~ v i ~ g g e ~ r and for the followinghelix and rotational directions, the gear forces transmitted are given by equations (4.36) and (4.37).
Right-hand helix (RH) Left-hand helix (LH) Clockwiserotation(CW)Counterclockwiserotation(CCW)
Pal = t' (-sin cos
+
Prl = t' (sin cos
+
+ cos y1 + tan 4 sin yl)
(4.36)
+ sin y1 + tan ct) cos yl>
where subscri~t1refers to the driving gear, and yl is the hal~-con~ angle.
14
4.1Ei.
Loads transmitted by a spiral bevel
gear.
RH-CCW
* *
LH-CW
Pal = t‘ (sin cos
+ cos y1 + tan
sin
3/11
prl = pt (- sin t,b sin y1 + tan # cos yl) cos
(4.38) (4.39)
Considering the driven gear (subscript 2):
RH-CW
Pa2 =
~
t‘
cos
LH-CCW
*
(sin tc/ cos y2 + tan
sin y2)
Pt (-sin $ sin y2 + tan # COS Pr2= cos J,
y2)
(4.40) (4.41)
CORTCE
RH-CCW
LH-CW
pt (-sin sl/ cos Pa, = cos sl/
Pr2= t‘ (sin cos sl/
y2
+ tan 4 sin y2)
+ sin y2 + tan
cos y2)
(4.42) (4.43)
Hypoid gears are used to transmit power between shafts whose ases of rotation do not intersect; that is, the shaft axes lie in diEerent planes. Because of this arrangement, illustrated by Fig.4.16, a substantial amount of sliding occurs betweencontacting gear teeth, and a coefficient of sliding friction must be defined. For hypoidgears properly lubricated with a mineral or synthetic oil, the coefficient of friction p = 0.1 is representative. Similar to spiral bevel gears, the loading of the individual gears depends on the helix direction together with the direction of rotation. The equations for d r i u i gear ~ ~ loads are given as functions of P, the resultant gear tooth load, which is defined by equation (4.44).
4.16. Hypoid gear mesh.
14
P=
cos # cos
pt t,b1
RH-CW .pal
= P(--COS
4 sin t/t1 cos y1 + sin
LH-CCW
sin yl + p
Prl = P(cos # sin 11/1 sin y1 + sin # cos yl
"CCW
(4.44)
+ p sin t,b1
- p
COS $1 COS 71)
cos
+lsin
yl)
(4.46)
LH-CW
cos CJ, sin t/tl cos yl + sin sin yl - p COS +l COS 71) Prl = P ( - cos # sin +lsin y1 + sin # cos y1 + p COS q1 sin yl>
pal=
The driven gear tooth loads are given by equations (4.49)-(4.52).
RH-CW
(4.45)
LH-CCW
(4.47) (4.4
49
RH-CCW
Pa2= P(-cos (13 sin Pr2 = P(cos (13 sin
t,h2
cos y2 + sin (13 sin y2 +
t,h2 sin
y2
EA,
cos
t,h2
cos y2) (4.51)
+ sin (13 cos y2 - p cos t,h2 sin y 2 )
(4.52)
Worm gearing, which may be regarded as a case of 90" crossed helical gears, is used to effect substantial reduction in speed within a single reduction set. Even morethan in hypoid gears, substantial sliding occurs between the worm screw and the worm wheel teeth. Figure 4.17 illustrates the loads transmitted in a worm gear drive. The lead angle h of the worm thread at the pitch radius rplof the worm is defined in terms of the threadpitch h at thatpoint and thefriction coefficientby equation (4.53). tan A
=
h 2Wpl
(4.53)
Then for the worm
I
Schematic of tooth loading in a worm gear !
drive.
BEARING LOADS AND SPEEDS
"1
=
sin (b pt1 cos (b sin A + p cos h
(4.54)
Pal
=
cos (b - p tan h Ptl p + cos (b tan h
(4.55)
For the wormwheel, the following relationships are true: Pt2 = Pal, Pr2= Prl, and Pa, = Ptl.
In some applications only the input shaft speed is given; however, performance of both input and output shaftbearings must be evaluated. In general, simple kinematic relationships relying on instant center concepts are used to determine the input-output speed relationship. Using the friction wheel shownin Fig. 4.8, at thepoint of contact, the condition of no slippage is assumed, and therefore the surface velocity u of wheel 1equals that of wheel 2. Since u = or, (4.56)
The same relationship holds for pulleys and spur and helical gears. For straight and helical bevel gears, the following equation pertains: (4.57)
where rmland rm2 are the mean pitch radii of gears 1 and 2. For hypoid gears (4.58)
For worm gearing (4.59)
where Z, is the number of threads on the worm and is, is the number of teeth on the worm wheel. To achieve substantial speed reductions, gears are frequently combined in units called gear trains; Fig. 4.18 illustrates a simple train com-
S
SPEEDS
input
output
4.18. Simplefour-gear train.
posed of four gears. The train value is defined as the ratio of the output to input speeds; that is,
It can be further demonstrated (see Ref [l]etc.) that (4.61) More generally, for a train of several gears, the train value e is the ratio of the product of the pitch radii of the driving gears to the product of the pitch radii of the driven gears. (It is noted that thenumbers of teeth can be substitute^ for the pitch radii.) Hence, the output shaftspeed can be directly determined. Planetary gear or epicyclicpower transmissions are designed to achieve substantial speed reduction in a compact space. In its simplest form, the epicyclic transmission is shown schematically by Fig. 4.19, in which R refers to the ring gear, I> to the planet gear, and S to the sun gear. The sun gear typically is connected to the input shaft, and the output shaftis connected to the arm. In general, there are three or more planets; therefore, each planet gear shaR transmits one-third or less of the input power. A single planet is shown in Fig. 4.19 for the purpose of analysis of speeds. Using Fig. 4.19, it can be seen that the speed of the
1
4.19. Schematic diagram of a simple epicyclic power transmission.
sun gear relative to the arm is nsA = ns - nA. Furthermore, the speed of the ring gear relative to the arm is nRA= nR - nA. Therefore e = nSA
= nR nS
- nA - nA
(4.62)
Now, by holding the arm stationary, allowing the ring gear to rotate as in a simple gear train, and applying equation (4.61),
(4.63) The minus sign in equation (4.63) denotes that the ring gear (output) rotates in the opposite direction with regard to the sun gear (input). Equating (4.62) and (4,63), setting nR = 0, and rearranging yields the expression for the output shaft speed. Subsequently, the planet gear shaft speed can be determined.
(4.64)
Planetary gear transmission confi~rations are too numerous and varied to provide equations to calculate speeds for each case; however, the calculational method indicated herein is universally valid and may be applied.
Sometimes the load is distributed over a portion of the shaft as in a rolling millapplication. In general this type of loading maybe illustrated by Fig. 4.20. If the loading is irregular, then it may be considered as a series of loads Pk,each acting at its individual distance ab from the lefthand bearing support. Equations (4.9) and (4.10) may then be used to evaluate reactions F , and F2. For a distributed load for which the load per unit length w 'may be represented by a continuo~sfunction; for example, w = w, sin x or w = w,(l + bx), equations (4.9) and (4.10) become (see Fig. 4.21)
F,
=
la; ); (1 -
wdx
(4.65) (4.66)
In this chapter, methods and equations have been provided to calculate bearing loads in statically determinate shaft-bearing systems. These methods are adequate for performance analysis of ball and roller bearings in most applications. Also, methods and equations have been provided to estimate the loading transmitted through the shaft to its bearing supports resulting from various common power transmission, machine elements. Many applications are, however, moresophisticated than those covered in this chapter. These may involve statically indeterminate sys-
BEARING LOADS AND SPEEDS
7 a 2 -
1
tems, covered in Chapter11,andlor increased complexity of system loading, which can only be determined by detailed evaluation of the specific application.
4.1. M. Spotts and T. Shoup, Design of Machine Elements, 7th Ed., Prentice Hall, Englewood Cliffs, N.J. (1998). ls ~ o m ~ o n e Design, nt 2nd ed., 4.2. R. Juvinall and I(. Marshek, ~ u n ~ a m e ~ tofa Machine Wiley, New York (1991). 4.3. B. Harnrock, €3. Jacobson, and S. Schrnid, ~ u n ~ a m e n t aof z sM a c ~ i n eElements, McGraw-Hill, New York (1999). 4.4. E. Avalloneand T. Baumeister, Marks’ Standard ~ a n d ~ o for o kM e c ~ a ~ i c aEngineers, Z 9th ed., ~ c G r a ~ - H i New l l , York (1987).
Symbol
Units
Description
Ball or roller diameter diameter Pitch Force Centrifugal force Gravitational (in./sec2) mm/sec2 constant Mass moment of inertia Roller length Moment Gyratory moment Mass of ball or roller Rotational speed Orbital ball or roller speed, cage speed Ball or roller speed about its own axis Ball or roller normal load Radial direction load on ball or roller
mrn (in.) mm (in.) N (lb) N Ob) kg mm2 (in. lb * sec2) mm (in.) N mm (in. lb) N mm (in. * lb) kg (lb * sec2/in.) rPm rPm rpm N (W N (lb) 155
n
15
BALL AND R O L ~ R LOADS
Symbol &a
r
u
v
w
X
x X‘
Y Y
Y‘ z 2 x‘ a
P P’ Y YB
Y 6 P
# @ urn OR
a e
Units Axial direction load on ball or roller Radius Coordinate direction distance Coordinate direction distance Coordinate direction distance Coordinate direction distance Acceleration in x-direction Coordinate direction distance Coordinate direction distance Acceleration in y-direction Coordinate direction distance Coordinate direction distance Acceleration in x-direction Coordinate direction distance Contact angle Angle between W axis and z’ axis Angle between projection of the 7 7 axis on the x’y’ plane and the x’ axis D cos aid, Roller skewing angle Roller tilting angle Angle Mass density Angle in W plane Angle in yx plane Orbital angular velocity of ball or roller Angular velocity of ball or roller about its own axis SU~SCRIP~S Refers to axial direction Refers to rotation about an eccentric EEiS
f i j m 0
r
Refers to guide flange Refers to inner raceway Refers to rolling element at location j Refers to orbital rotation Refers to outer raceway Refers t o rolling element Refers t o radial direction
N (lb) rnm (in.) mm (in.) rnrn (in.) mm (in.) mm (in.) mm/sec2 (in./sec2) mm (in.) mm (in.) mm/sec2 (in./sec2) mm (in.) rnm (in.) mm/sec2 (in./sec2) mm (in.) rad rad O,
rad rad rad rad kg/mm3 (lb sec2 in.-*) rad rad
O, O,
rad/sec rad/sec
STATIC L
O
~
~
7
G
The loads carried by ball and roller bearings are transmitted through the rolling elements from one ring to the other. The magnitude of the loading carried by the individual ball or roller depends on the internal geometry of the bearing and on the type of load impressed on it. In addition to applied loading, rollingelements are subjected to dynamic loading due to speedeffects. Bearing geometryalsoaffects the dynamic loading. The objectof this chapter is to define the rolling element loading in ball and roller bearings under varied conditions of bearing operation.
A rolling element can support a normal load along the line of contact between the rolling element and the raceway (see Fig. 5.1). If' a radial lr is applied to the ball of Fig. 5.1, then the normal load supported by the ball is
Hence a thrust load of m a ~ i t u d e QP
I
,I
Radially loaded ball.
or Qa
=
Qr tan a
(5.3)
is induced in the assembly.
.
The 209 radial ball of Example 3.1 is subjected to a thrust (axial) load of 445 N (100 lb) per ball. What is the ~ a ~ i t u d e of the induced ball radial load assuming the contact angle* is not changed by the thrust load? a
=
Ex. 3.1
7'25'
Qa = Qr tan a
Qr
=
(5.3)
3419 N (768 lb)
This result indicates the degree of thrust load amplificationat small contact angles. The 218 an~lar-contactball bearing of Example 3.3 is subjected to a thrust load of 2225 N (500 lb) per ball. What is the magnitude of the normal ball load that is induced assuming the contact angle* is not changed by the thrust load? a = 38'54'
2225 = Q sin (38'54')
Q
=
3543
N (796.2 lb)
It can be seen that a bearing with a 40' contact angle can support a thrust load better than a bearing with an 7'25' contact angle. Equations (5.2) and (5.3) are also valid for spherical roller bearings using symmetrical rollers. For a double-row spherical roller bearing under an applied radial load the inducedroller thrust loads are selfequilibrating (see Fig. 5.2).
*This assumption is not accurate in this case and is made only to illustrate a point.
STATIC LO^^^
\
/
5.2. Radially loaded symmetrical rollers.
Qia =
Qir
tan ai
For static e~uilibrium the sum of forces in anydirection is equal tozero; therefore,
or
160
BALL AND ROLLER LOADS
5.3. Radially loaded asymmet-
rical roller.
Qir tan ai + Qf sin af-
Solving equations (5.8) and (5.9) for Q, and Qfyields
Qo
= Qir
Qf
=
Qir
(sin af+ tan ai COS af) sin (a, + cxf)
(5.10)
(sin a, - tan ai cos CY,) sin (a, + af)
(5.11)
The thrust load induced by the applied radial load is (5.12)
Under an applied thrust load Qia the tain, considering static equilibrium:
Qo
Qf
= Qia
=LI
Qia
follow in^ equations of load ob-
(cos af + ctn ai sin ai) sin (a, + af)
(5.13)
(ctn aisin a, - cos a,) sin (a, + af)
(5.14)
A 90000 series steep-angle tapered roller bearing has the following dimensions: ai = 22"
a, = 29"
D
=
22.86 mm (0.9000 in.) (mean)
I
=
30.48 mm (1.2 in.)
dm = 142.24 mm (5.600 in.) (mean) af =
64" (with bearing axis)
If the most heavily loaded roller supports a thrust of 22,250 N (5000 lb), what is the magnitude of the maximum load onthe guide flange?
Qf = Qia =
(ctn aisin a, - cos a,) sin (a, + af)
22250
(ctn 22" sin 29" - cos 29") sin (29" + 64")
(5.14) =
7245 N (1628 lb)
Compare this load with the maximum normal load on the cone. Qia Qi = -
sin ai
-
22250 sin 22"
Fig. 5.3 -
59,410 N (13,350 lb)
"
olling Element The development of equations in this section is based on the motions occurring in an angular-contact ball bearing because it is the most general form of rolling bearing. Subsequently, the equations developed can be so restricted as to apply to other ball bearings and also to roller bearings. Figure 5.4 illustrates the instantaneous position of a particle of mass in a ball of an angular-contact ball bearing operating at high rotational
w X
5.4.
Instantaneous position of ball mass element dm.
speed about an axis x. To simplify the analysis the following coordinate axes systems are introduced:
A fixed set of Cartesian coordinates with the x axis coincident with the bearing rotational axis. A set of Cartesian coordinates with the x' axis parallel to the x axis of the fixed set. This set of coordinates has its origin 0' at the ball center and rotates at orbital speed about the fixed x axis at radius dm. A set of Cartesian coordinates with origin at the ball center 0' and rotating at orbital speed am.The 77 axis is collinear with the axis of rotation of the ball about its own center. TheW axis is in theplane of the U axis and x' axis; the angle between the W axis and z' axis is ,6. A set of polar coordinates rotating with the ball.
+
In addition to the foregoing coordinate systems, the following symbols are introduced:
3
D
p’ The angle betweenthe projection of the U axis on the x’y’plane and the x’ axis. The angle betweenthe z axis andx’ axis, that is, the angular position of the ball on the pitch circle.
tl,
Consider that an element of mass d m in the ball has the following instantaneous location in the system of rotating coordinates: U, r, #. Since
u=u V = r sin #
W
=
(5.15)
r cos 4
and
x’ = U cos p cos p’
-
V sin p’
-
W sin /3 cos p‘
y‘ = U cos p sin p‘
+ V cos p‘
-
W sin /3 sin p‘
z‘ =
U sin p
(5.16)
+ W cos p
and
x
=
x‘
y =
$ dm sin tl, + y‘ cos tl, + z‘ sin tl,
z =
$ dm cos tl, - y‘ sin .IC/ + z’ cos sl/
(5.17)
therefore, by substitution of equations (5.15) into (5.16) and thence into (5.17), the following expressions relating the instantaneous position of the element of mass d m to the fixed system of Cartesian coordinates can be formulated.
x
=
y
=
U cos p cos p’ - r(sin p’ sin #
2
sin tl, +
+ sin p cos p‘
cos #)
(5.18)
cos p sin p’ cos tl, + sin p sin $1
+ r (cos p sin # cos tl, + cos /3 cos # sin tl, -
sin p sin p‘ cos # cos tl,)
(5.19)
x = - d m cos tl, + U(-cos p sin p’ sin IC/ + sin p cos tl,) 2
+ r(--cos p’ sin # sin tl, + cos p cos # cos tl, + sin p sin p’ cos # cos tl,)
(5.20)
In accordance with Newton’s second law of motion, the following re-
lationships can be determined if the rolling element position angle @ is arbitrarily set equal to 0”:
The net moment about the x axis must be zero forconstant speed motion. At each ball location (@,p), % (rotational speed d # / d t of the ball about its own axis U - 0‘) and wm (orbital speed ~~/~~of the ball about the bearing axis x) are constant; therefore, at @ = 0:
(5.26)
d “Y Y = - -d t - -2%wmr cos /3 sin #
+ uL[-U cos p sin p’ + r(-cos p’ sin (b + sin p cos # sin p’)] + wkr( -cos p’ cos # + sin p sin p’ sin 4) (5.2’7) d 2z z=”- -2%wmr(cos p’ cos 41) + sin p sin p’ sin #I dt
-
w;r cos p cos Ip
(5.28)
~ubstitutionof equations (5.26) to (5.28) into equations (5.21) to (5.25) and placing the latter into integral format yields
65
(5.29) (5.30) (5.31)
+ r (cos p’ sin 6, - sin p sin p‘ cos st))] + y [ U cos p cos p’ - r(sin p’ sin st) + sin p cos p’ cos #)I )rdrdU d6,
(5.32)
Io lo2“ {a[u + (r;-U2)1/2
My, =
sin p
-p -rR
-
r cos p cos st)]
%[Ucos p cos p’ - r(sin p’ sin 6,
+ sin
cos p’ cos 6,)l)rd6, dU dr
(5.33)
In equations (5.29)-(5.33) p is the mass density of the ball material and rR is the ball radius. Performing the integrations indicated by equations (5.29)-(5.33) establishes that the net forces in thex’ and y‘ directions are zero and that
(5.34) (5.35)
(5.36) in which m is the mass of the ball and J is the mass moment of inertia. rn and J are defined as follows:
Rotation abouttheBearing (5.34) and recognizing that
m
=
J
=
Q pvD3 pvD5
(5.37) (5.38)
Axis. ~ubstitutingequation (5.37) into
BALL AND ROLLER LOADS
Wm
=
2rnm 60
(5.39)
equation (5.40) yielding ball centrifugal force is obtained:
F,
=
r 3 p D3nidm 10800g
(5.40)
For steel balls,
F,
=
2.26
X
10-11D2nidm
(5.41)
For an applied thrust load per ball Qia and a ball centrifugal load F, directed radially outward, the ball loading is as shown in Fig. 5.5. For conditions of equilibrium, assuming the bearing rings are not flexible, (5.42) (5.43)
or
r
~
~ 5.5. GBall under ~ thrust R load ~ and centrifugal load.
D
Qia - Q, sin a, = 0 Qia ctn ai + Fc - Q, cos a,
=
0
(5.44) (5.45)
In equations (5.44) and (5.45), Q, and a, are unknown; therefore, these equations must be solved simultaneously for Q, and a,. Thus, a,
=
ctn-1 (ctn
ai
+
+ (ctn
ai
+
Q, = [1
2) 2)] 2
(5.46) 112 Qia
(5.47)
Further, Qi
=
Qia ~
sin ai
(5.48)
It is apparent from equation (5.46) that because of centrifugal force Fc, a, < ai. The quantity ai is the contact angle under thrust load and it is greater than the free contact angle a'. This condition will be discussed in Chapter 7.
4. If the 218 angular-contact ball bearing of Example 5.2 has a cage speed of 5000 rpm, what outer ring contact angle will ob2225 N (500 lb)
Ex. 5.2
38'54'
Ex. 3.3
22.23 mm (0.875 in.)
Ex. 2.3
125.3 mm (4.932 in.)
Ex. 2.6 (5.41)
2.26
X
10-193n24,
2.26
X
10-11(22.23)3(5000)2 X 125.3
775.2 N (174.2 lb) ctn-1 (ctn a,
+
2)
ctn 38"54'
+2225 775.2)
(5.46) 32012'
Compare the normal ball loads at the inner and outer raceways.
BALL ANI) ROLLER LOADS
Qi = 3543 N (796.2 lb)
Ex. 5.2
2)] 2
Q0= [ I + (ctn = =
ai
+
[
1 + (ctn 38'54'
1/2
(5.47)
&ia 2
+775*2) 2225
1
112
X
2225
4176 N (938 lb)
Ball thrust bearings with nominal contact angle a = 90" operating at high speed and light load tend to permit the balls to override the land on both rings. The contact angle thus deviates from 90" in the same direction on both raceways (see Fig. 5.6). From Fig. 5.6,
& = - PC 2 cos a a = tan-1
( ~ )
(5.49)
(5.50)
Ze 5.5. A 90" nominal contact angle ball thrust bearing has a cage speed of 5000 rpm. The bearing that has a 127 mm (5 in.) pitch diameter and 12.7 mm (0.5 in.) balls, supports a thrust load of 445 N (100 lb) per ball. What contact angle obtains?
/ 5.6. Ball loading in a ball thrust bearing.
F~ = 2.26
X
10-193n3,
=
2.26
X
10-11(12.7)3(5000)2X 127
=
146.7 N (32.97 lb)
(5.41)
(5.50)
=
2 x 445 tan-1 146.7
=
80’38’
(
)
Equation (5.34) is not restrictive as to geometry and since the mass of a cylindrical (or nearly cylindrical) roller is given by
the centrifugal force for a steel roller rotating about a bearing axis is given by
Fc = 3.39
X
10-11R2Z,d,n~
(5.52)
For a tapered roller bearing, however, rollercentrifugal force alters the distribution of load between the outer raceway and central guide flange. Figure 5.7 demonstrates this condition for an applied thrust ia. For equilibrium to exist,
(5.53) (5.54) or ia
+ Qf sin af - Q, sin a, = 0 COS
af+ Fc -
Q,
COS
a, = 0
Solving equations (5.55) and (5.56) simultaneously yields
f =
ia(ctnaisin a, + cos a,) sin (a, + af)
+ Fc sin a,
(5.55) (5.56)
17
5.7. Asymmetrical roller under thrust load and centrifugal load.
If the tapered roller bearing of Example 5.3 has a cage speed of 1700 rpm, what is the guide flange load due to the most heavily loaded roller? Qia = 22,250 N (5000 lb) I) = 22.86 mm (0.9000
dm= 142.2 (5.600 mm
f =
in.)
x. 5.3
lo-11~2zt~m~~
(5.52)
3.39
10-11(22.86)2 X 30.48
=
221.9 IN (49.86 lb)
&,(ctn a, sin a, - cos a,) sin (a,+ af)
X
142.2(170~)2
+ .Fc sin a,
- 2225O(ctn 22" sin 29" - cos 29") + 221.9 sin 29" sin (29" + 64") =
Ex. 5.3
Ex. 5.3
=
=
X
in.)
30.48 mm (1.2 in.)
I,
PC= 3.39 x
Err. 5.3
7351 N (1652 lb)
What is the maximum normal load at theouter raceway?
(5.58)
171
0
-
,(ctn aisin af+ cos (xcuf) + Fc sin af sin (a, + af)
- 22250(ctn 22" sin 64" + cos 64") sin (29" + 64") =
(5.57)
+ 221.9 sin 64"
59,500 N (13,370 lb)
Care must be exercised in operating a tapered roller bearing at very high speed. At some critical speed related to the magnitude of the applied load, the force at theinner ringraceway contact approaches zero and the entire axial load is carried at the flange-roller end contact. Since this contact has only sliding motion, veryhigh friction results with a~tendant high heat generation. Most modern radial spherical roller bearings have complements of barrel-shaped rollers and relatively small contact angles; for example, (x 5 15". When the bearings are operated at high speed, roller loading is as illustrated in Fig. 5.8. Equilibri~mof forces in the radial and axial directions gives (5.58) (5.59) Solving these equations simultaneously yields
5.8. Loading of barrle-shaped (spherical)rollersubjected to applied and centrifbgal forces.
172
QO
=
Fc sin ai sin (ai- a,)
(5.60) (5.61)
Therefore, it appears that roller-raceway loading is uniquely determined by roller centrifugal loading. Clearly,in thisinstance the innerand outer raceway contact angles are functions of the bearing applied radial and thrust loading, and these must be determined from the equilibrium of loading on the bearing. To do this requires the determination of the bearing contact deformations.This will be discussedin subsequent chapters. Another way to view the operation of a loaded spherical roller operting at high speed is to resolve the centrifugal force into components collinear with, and normal to, the roller axis of rotation. Hence
Fca = Fc sin ao
(5.62)
Fcr= Fc cos ao
(5.63)
where ao is the nominal contact angle. ~quilibriumof forces acting in the radial plane of the roller gives
&,
=
Qi
+ Fc cos ao
(5.64)
and thecomponent Fc sin aocauses the inner andouter raceway contact angles to shift; slightly from ao to accommodate the roller axial loading. In general, spherical roller bearings do not operate at speeds that will cause s i ~ i f i c a nchange t in the nominal contact angle. Also, consider a double-row spherical roller bearing having barrel-shaped rollers subjected to a radial load while rotating at high speed. The speed-induced roller axial loads are self-equilibrated within the bearing; however, the outer raceways carry larger thrust co~ponents than do the inner raceways.
~ o t a t ~ about o n a n ~ c c e n t Axis. ~ i c The foregoing section dealt with rolling element centrifugal loading when the bearing rotates about its own axis. This is the usual case. In planetary gear transmissions, however, the planet gear bearings rotate about the input and/or output shaft axes as well as about their own axes. nce an additional inertial or centrifrollin lement. F i ~ r 5.9 e shows a schematic ugal force is in~uced in the am of such a system. From Fig. 5.9 it can be seen that the instantaneous radius of rotation is by the law of cosines
F I ~ 5.9. ~ E Rolling element centrifugal forces due to bearing rotation about an eccentric axis.
r
=
(ri
+ rz
- 2r,r, cos
+)1/2
(5.65)
Therefore, the corresponding centrifugal force is
+
This force F,, is maximum at = 180" and at thatangle is algebraic~lly additive to F,. At = 0 the total centrifugal force is Fc - Fee. The angle between F, and F,, as derived from the law of cosines is
+
F,, can be resolved into a radial force and a tangential force as follows:
174
BALL AND ROLLER LOADS
Fcer= Fce cos 0
(5.68)
Feet = FCesin 8
(5.69)
Hence the total instantaneous radialcentrifugal force acting on the rolling element is
Fc. = rnw;r, + rnw:(r,
- re cos $)
(5.70)
or
(5.71) in which the positive direction is that taken by the constant component Fc. For steel ball and roller elements the following equations are respectively valid:
FCr= 2.26
X
FCr= 3.39
X
+ n:) - den: cos $1 10-11D2Zt[d,(n& + n:) - den:cos $1
10-11.Z)3[d,(n;
(5.72) (5.73)
The instantaneous tangentialcomponent of eccentric centrifugal force is
Fct= rnw :re sin $
(5.74)
For steel ball and roller elements, respectively, the following equations pertain:
Fct= 2.26
X
10-11D3den: sin $
(5.75)
Fct= 3.39
X
10-11D2Ztden~ sin $
(5.76)
This tangential force alternates direction and tends to produce sliding between the rolling element and raceway. It is therefore resisted by a frictional force between the contacting surfaces. The bearing cage'also undergoes this eccentric motionand if it is supported on the rolling elements, it will impose additional load on the individual rolling elements. This cage load may be reduced by using a material of smaller mass density.
It can usually be assumed with minimal loss of calculational accuracy that pivotal motiondue to gyroscopic moment is negligible; then the an-
gle p' is zero and equation (5.36) is ofno consequence. The gyroscopic moment as defined by equation (5.35) is therefore resisted successfully by friction forcesat the bearing raceways for ball bearings and by normal forces for rollerbearings. ~ubstitutingequation (5.39) into (5.35) the following expressionis obtained for ball bearings: (5.77) since (5.78)
and 0,
=
2vnm 60
(5.79)
The gyroscopic moment forsteel ball bearings is given by
Mg = 4.47
X
10-12D5nRnm sin p
(5.80)
Figure 5.10 showsthe direction of the gyroscopic moment in a ball bearing. Accordingly, Fig. 5.11 shows the ball loading due to the action of gyroscopic moment and centrifugal force ona thrust-loaded ball bearing.
2 IGURE 6.10. Gyroscopic moment due to s i ~ u l t a ~ e o rotation us about nonparallel axes.
176
BALL AJ9D ROLLER LOADS QW
F
I
~ 5.~11. EForces acting on a high speed ball under applied thrust load.
~ 5.7. In ~ the ball ~ thrust~ bearingZ of Example e 5.5, the balls rotate at a speed of 50,000 rpm about axes perpendicular to the bearing axis, that is, /3 = 90'. What is the gyroscopic moment that must be resisted by raceway friction forces and what coefficient of friction is required t o resist gyroscopic pivotal rotation?
~
D mm = 12.7 n,,
=
Ex. 5.5
(0.5 in.)
Ex. 5.5
5000 rpm
M g = 4.47
X
10-12D5nR72, sin /3
=
4.47
X
10-12(12.7)5X 50,000
=
0.369 N m (3.265 in lb)
(5.80) X
5000
Fc = 146.7 N (32.97 lb) a = $0'38'
&=-
li'C
2 cos a -
146*7 = 450.8 N (101.3 lb) 2 cos (80'38')
X
1
Ex. 5.5 Ex. 5.5 (5.49)
177
ROLLER AXIAL LONXNG IN l&WIAL BEARINGS
(Fig. 5.11
p(450.8
+ 450.8) X - 0.369 x 2
10-3 = p =
o 0.0645
Gyroscopic moments also act on roller bearings. The rollers, however, are geometrically constrained from rotating; therefore, a gyroscopic momentof significant magnitude tends to alter thedistribution of load across the roller contour. For steel rollers the gyroscopic moment is given by
nil, = 8.37 X
10-12D4Z,nRn,sin p
(5.81)
Axial LoadingEffects in Radial Roller Bearings Axial loading of the rollers in a radial roller bearing can significantly affect bearing performance. In cylindrical and tapered roller bearings a roller asial load is reacted between the roller end and a guide flange. The combination of load and sliding in thiscontact can cause the bearing to generate excessive heat and, under certain conditions, lead to wear and smearing of the contacting surfaces. Conversely, with proper management of roller end-flange design and lubrication, bearing heat generation and axial load-carrying capacity can be optimized. Axial roller loads interact with roller radial forces and ring deflections t o determine roller tilting and skewing motionsthat influence roller-raceway contact stresses and sliding velocity distributions. Roller tilting and skewing substantially affect roller bearing heat generation, friction torque, and rolling contactfatigue life. Many modernspherical roller bearing designs do not use flanges as theprimary source of roller guidance. Such designs rely on proper management of roller tilting and skewing motions via roller-raceway traction forces to provide roller guidance while minimizing bearing heat generation and friction torque.
Many radial roller bearing designs have the ability to carry applied axial or thrust load in addition to their predominant radial loading. Cylindri-
17
BALL AND ROLmR L
cal roller bearings can carry thrust loading by virtue of flanges fixed to inner andouter rings. The angular-contact roller arrangement of tapered roller bearings reacts applied thrust load overthe combination of rollerraceway contact surfaces and roller end-flange contacts. Spherical roller bearings react applied thrust loading through their roller-raceway contacts (and flanges if asymmetric rollers are employed). Axial loading of radial roller bearings causes alteration of internal load distribution and roller response that can significantly affectbearing performance. Cylindrical roller bearings with fixed inner and outer ring flanges, as shown in Fig. 5.12a, can carry axial load through contacts between the roller end facesand theflanges. The couple produced on the roller by the axial roller end-flange forcesQajresults in a tilting 4 about the center of the roller. As the roller tilts, the roller-raceway load distribution shifts asymmetrically, as represented in Fig. 5.13. The roller-raceway load dis-
(a)
(b)
(e)
5.12. Applied thrust loading on radial roller bearings. (a)Cylindrical roller. ( b ) Tapered roller. (e) Spherical roller.
5.13. Roller tilting displacement and contact load distribution due to combined axial and radial load on a cylindrical roller bearing.
ROLLER, AXIAL L
O
~ IN ~ RADIAL G BEARINGS
79
tribution of Fig. 5.13 can be compared with the ideal load distribution in Fig. 6.23b.
In roller bearings skewing is defined as anangular rotation of the roller axis (in a plane tangent to its orbital direction) with respect to the axis of the contacting ring. The magnitude of the skewing motion may be expressed as a skewing angle 3/s. The skewingangle of a roller at a given azimuth position might be different for inner and outer ring contacts if misalignment exists between inner andouter rings. In general, the skewing angle will vary with roller azimuth position +. Skewing is caused by forces acting on the roller, which result in a moment loading about an axis oriented in the bearing radial direction are and passing through the roller at its midpoint. These skewing forces often due to asymmetrical distribution of tangential friction forcesat the roller-raceway contacts arising from asyxtmetrical normal loading and/ or sliding velocity distributions along the contacts. Normal and frictional forces acting on the roller due to contact with guide flanges (including ial loads) and cage may contribute to, or serve to limit, otions. Operating conditions associated with asymmetrical load distributions include misalignment of inner andouter rings in cylindrical rollerbearings and tapered roller bearings and applied roller axial loading in cylindrical roller bearings. In high-speed roller bearing operation the dynamic effects of roller mass unbalance and impact loading between roller and flange or cage can become significant causative factors in roller skewing. Forces that give rise to roller skewing motions may also, by virtue of their points of application or radial force components, becoupled to the roller tilting action. ~uantificationof roller skewing and coupled tilting behavior require computer p r o ~ a m sdesigned for this purpose. As an illustration of roller skewing, consider the cylindrical roller bearing roller subjected to radial and axial applied loads shown in Fig, 5.14. The appliedaxial load QG causes the roller to tilt through the angle 4.The tilting motion gives rise to the asymmetrical contact load distribution in Fig. 4.14a.Assuming no gross sliding or skidding in the application, the tilting also causes sliding motion to occur alongthe inner and outer raceway contacts. Roller deformations cause the sliding velocity on the uncrowned roller to be zero at only one point along each raceway contact. The tangential sliding velocity distributions for the unskewed roller are represented by Fig. 5,14b. Positive and negative values obtain correspond in^ to sliding in the z direction. Tangential friction forces in both contacts are related to the magnitude of the normal contact load and magnitude and direction of tangential sliding velocity. The asymresults, metrical tangential friction forcedistribution shown in Fig. 5 . 1 4 ~ This distribution causes a skewing moment about the y axis. The skew-
180
BALL AND ROLLER LOADS
Outer ring X
Inner ring .-
out of page
(a)
X
i
Msi
(c)
Roller loadingand tangential sliding velocity in a cylindrical roller bearing with applied axial and radial load. (a)Applied roller loads and roller load distribution. ( b ) Tangential slidingvelocity distribution. (e) Tangential traction force distribution and skewing moments. (d) Roller-flange contact forcesand resultant axial friction force (view from outer ring). 5.14.
ing moment tends to cause the roller to skew and generate an axial friction force at both contacts, the resultant of which, Q,, is shown in Fig. 5.14d. Also contributing to the roller skewing moment are the fiictional forces generated at the flange contacts. In the illustration the skewing moment and axial friction force must be reacted by the flange contacts. In principle, a skewing angle y8 may
be achieved whereby roller skewing momentand axial force are in equilibrium with the flange contact forces. Note that, in general, the locations of the roller-flange contacts, the roller-raceway normal loaddistribution, sliding velocities, and tangential and axial friction forces are all functions of the roller skewing angle. The skewing angle at which this force balance is obtained is known as an equilibrium skewing angle. As bearing operational speed increases, dynamic effects become significant. Brown et al. [4.1] investigated the problem of roller axial load due to skewing in cylindrical rollerbearings under high-speed conditions. Their analytical and experimental work highlighted the detrimental effects of unbalance forces due to roller corner radius runout. Analytical models were developed for rollerimpact loading on flange and cage. The effect of bearing design parameters was empirically correlated with observed roller end wear.
To analyze rolling bearing performance in a given application, it is usually necessary to determine the load on individual balls or rollers. How well the balls or rollers accept the applied and induced loads willdetermine bearing endurance. For example, light radial load applied to a 90” contact angle thrust bearing can cause the bearing to fail rapidly. Similarly, thrust load applied to a 0” contact angle, radial ball bearing is largely magnified according to the final contact angle that obtains. In Chapters 7 and 9 this book will concern itself with the distribution of load among balls and rollers. It will be shown that themanner in which each rollingelement accepts its load willdetermine in large measure the loading of all others. Moreover, in an~lar-contactball bearings the ball loading can affectball and cage speeds significantly. In high-speed roller bearings if roller loading is too light, rolling motion may be preempted by skidding. The material in this chapter is therefore fundamental to even the most rudimentary analysis of a rolling bearing application.
5.1. I?. Brown, D. Robinson, L. Dobek, and J. Miner, “Mainshaft High-speed, Cylindrical Roller Bearings for Gas Turbine Engines,” U.S. Navy Contract NOOOl40-76-C-0383 Interim Report (1978).
This Page Intentionally Left Blank
LS Symbol
Description Semimajor axis of the projected contact Dimensionless semimajor axis of contact ellipse Semiminor axis of the projected contact ellipse Dimensionless semiminor axis of contact ellipse Modulus of elasticity Complete elliptic integral of the second kind Elliptic integral of the second kind Complete elliptic integral of the first kind Elliptic integral of the first kind Force Shear modulus of elasticity Roller effective length Normal force between rolling element and raceway
Units mm (in.)
mm (in.) N/mm2 (psi)
N (1b) N/mm2 (psi) mm (in.)
183
184
Symbol
CONTACT STBESS AND D E F O ~ T I O N
Description Radius of curvature Principal stress Deflection in x direction Arbitrary function Deflection in y direction Arbitrary function Deflection in x direction Principal direction distance Dimensionless parameter Principal direction distance Dimensionless parameter Principal direction distance Depth to maiimum shear stressat x = 0, y = o Depth to maximum reversing shear stress y z0 , x = 0 Dimensionless parameter Shear strain Deformation Dimensionless contact deformation Linear strain zlb, roller tilting angle Angle Auxiliary angle a/b Parameter Poisson’s ratio Normal stress Shear stress Auxiliary angle Auxiliary angle Curvature difference Curvature sum S U ~ S C ~ ~ ~ S Refers to inner raceway Refers to outer raceway Refers to radial direction Refers to x direction Refers to y direction Refers to X direction Refers to yz plane
Units mm (in.) N/mm2 (psi) mm (in.) mm (in.) mm (in.) mm (in.) mm (in.) mm (in.) mm (in.) mm (in.) mm (in.) *, rad rad rad
N/mm2 (psi) N/mm2 (psi) rad rad or *
is6
THEORS OF ELASTICITS
Spbol xz
I I1
1
Description
Units
Refers to xz plane Refers to contact body I Refers to contact body I1
Loads acting between the rolling elements and raceways in rolling bearings develop only small areas of contact between the mating members. Consequently, although the elemental loading may only be moderate, stresses induced on the surfaces of the rolling elements and raceways are usually large. It is not uncommon for rolling bearings to operate continuously with normal stresses exceeding 1380 N/mm2 (200,000 psi) compression on the rolling surfaces. In some applications and during endurance testing normal stresses on rolling surfaces may exceed 3449 N/mm2 (500,000 psi) compression. Since the effective area over which load is supported rapidly increases with depth below a rolling surface, the high compressive stress occurring at the surface does not permeate the entire rolling member. Therefore, bulk failure of rolling members is generally not a significant factor in rolling bearing design; however, destruction of the rolling surfaces is. This chapter is therefore concerned only with the determination of surface stresses and stresses occurring near the surface. Contact deformations are caused by contact stresses. Because of the rigid nature of the rolling members, these deformations are generally of a low order of magnitude, for example 0.025 mm (0,001 in.) or less in steel bearings. It is the purpose of this chapter to develop relationships permitting the determination of contact stresses and deformations in rolling bearings.
The classical solution forthe local stress and deformation of two elastic bodies apparently contacting at a single point was established by Hertz i6.11 in 1881. Today, contact stresses are frequently called Hertzian or simply Hertz stresses in recognition of his accomplishment. To develop the mathematics of contact stresses, one must have a firm foundation in principles of mechanical elasticity. It is, however, not a purpose of this text to teach theory of elasticity and therefore only a rudimentary discussion of that discipline is presented herein to demonstrate the complexity of contact stress problems. In that light consider an infinitesimal cube of an isotropic homogeneous elastic material sub-
186
CONTACT STRESS AND ~ E F O R ~ T I O N
jected to the stresses shown in Fig. 6.1. Considering the stresses acting in the x direction and in the absence of body forces, static equilibrium requires that axdx dy
+ rw dx dz + T= dx dy - (rw
+~
d
y d x)d z
( +2 ) ax
-dx dz dy
- (rm
+2
d x ) dxdy
=
0
(6.1)
Therefore,
Similarly, for the y and z directions, respectively,
2 I
.""_.
/ I
Y
X
F
~ 6.i. ~ Stresses~acting E on an infinitesimal cube of material under load.
QRY OF E ~ T I C I ~
Equations (6.2)--(6.4)are the equations of equilibrium in Cartesian coordinates. Hooke’s law for an elastic material states that within the proportional limit a
E‘==.-
E
in which E is strain and E is the modulus of elasticity of the strained material. If u, v, and w are the deflections in the x,y, and x directions, then
Ey -
av aY
If instead of an elongation or compression the sides of the cube undergo relative rotation such that the sides in the deformed conditions are no longer mutually perpendicular, then the rotational strains are given as
” au
Yxy =
yxz
-
du
av
+ ajt
aw
dz + -ax
When a tensile stress a, is applied to two faces of a cube, then in addition to an extension in the x direction, a contraction is produced in the y and x directions as follows:
CONTACT STRESS AND ~ E ~ O ~ T I O N E, =
-
Ey -
a; E
Sa, -E
In equation (6.8)’ 6 is Poisson’s ratio; for steel 6 0.3. Now the total strain in each principal direction due to the action of normal stresses a,, ay,and a, is the total of the individual strains. Hence
Equations (6.9) wereobtained by the method of superposition. In accordance with Hooke’s law, it can further be demonstrated that shear stress is related to shear strain as follows:
(6.10)
in which G is the modulus of elasticity in shear andit is defined (6.11) One further defines the volume expansion of the cube as follows: E = E,
+ E y + E,
(6.12)
189
S n F A C E STRESSES AND ~ E F O R ~ T I O N S
Combining equations (6.9),(6.11), and (6.12)one obtains for normal stresses
:(-+-+(~~
ffx=2G
ffy=2G
1-25 1-25
4 4
(6.13)
Finally, a set of “compatibility” conditions canbe developed by differentiation of the strainrelationships, both linear androtational, and substituting in the equilibrium equations (6.2)-(6.4): v2u
+--1 - 12 5 a a€ = 0 x
v2v
1 a€ +--0 1-25ay
v2w
1 at: +--0 1-2582
(6.14)
in which
a2
a2
a2
ax2
ay2
az2
v2=-+-+-
(6.15)
Equations (6.14) represent a set of conditions that by using the known stresses acting on a body must be solved to determine the subsequent strains and internal stresseso f that body. See Timoshenko and Goodier C6.21 for a detailed presentation of the foregoing.
STRESSES
DEFQ
Using polar coordinatesrather than Cartesian, Boussinesq C6.31 in 1892 solved the simple radial distribution of stress within a semiinfinite solid as shown in Fig. 6.2. With the boundary condition of a surface free of shear stress, the following solution was obtained for radial stress:
CONTACT STRESS AND ~ E F O ~ T I O ~
x FIGURE 6.2. Model for Boussinesq analysis.
2 F cos
Q"--"-
r
m
e
(6.16)
It is apparent from equation (6.16) that as r approaches 0, 0;. becomes infinitely large. It is further apparent that this condition cannot exist without causing gross yielding or failure of the material at thesurface. Hertz reasoned that instead of a point or line contact, a small contact area must form, causing the load to be distributed over a surface, and thus alleviating the condition of infinite stress. In performing his analysis, he made the following assumptions: The proportional limit of the material is not exceeded, that is, all defor~ationoccurs in the elastic range. Loading is perpendicular to the surface, that is, the effect of surface shear stresses is neglected. 3. The contact area dimensions are small compared to the radii of curvature of the bodies under load. The radii of curvature of the contact areas arevery large compared to the dimensions of these areas. e
The solution of theoretical problems in elasticity is based on the assumption of a stress function or functions that singly or in combination fit the com~atibilityequations and the boundary conditions. For stress
SURFACE STR.ESSES AND ~ E F O ~ T I O N S
distribution in a semiinfinite elastic solid, Hertz introduced the assumptions:
x = bX
y = Y-
(6.17)
b
in which b is an arbitrary fixed length and hence, X , Y , and Z are dimensionless parameters. Also,
(6.18)
in which c is an arbitrary length such that deformations u / c , u / c , and w l c are dimensionless. U and V are arbitrary functions of X and Y only such that
v2u= 0 v2v= 0
(6.19)
Furthermore, b and c are related to U as follows:
bE
-= C
a2U
-2 aZ2
(6.20)
Theforegoing assumptions, which are partly intuitive and partly based on experience, when combined with elasticity relationships (6.7), (6.10), (6.12), (6.13),and (6.14) yield the following expressions:
CONTACT STRESS AND D E F O ~ ~ O ~
z""a2v
-V = X
".=z"" a2V 0-0
2 -av
a2U
axz ax2
a 0
a2U
az
aV
2-
az
aY2aY2
(6.21)
in which
E'rom the preceding formulas, the stresses and deformations maybe determined for a semiinfinite body limited by the xy plane on which rm = ryZ= 0 and a; is finite on the surface, that is, at X = 0. Hertz's last assumption was that the shape of the deformed surface was that of an ellipsoid of revolution. The function V was expressed as follows:
in which so is the largest positive root of the equation
X2 IC2
+ s;
+ - +Y- 2= 1+
s;
z2 5;
(6.23)
and K
=
alb
(6.24)
Here, a and b are the semimajor and semiminor axes of the projected
193
S W A C E S’IYWSSES AND ~ E ~ O ~ T I O N S
elliptical area of contact. For an elliptical contact area, the stressat the geometrical center is 3Q 2vab
(6.25)
= --
00
The arbitrary length c is defined by c = - 3& 4vGa For the special case K
= m,
(6.26)
then 0-0
2Q vb
(6.27)
= --
c = -Q
(6.28)
TG
Since the contact surface is assumed to be relatively small compared to the dimensions of the bodies, the distance between the bodies may be expressed as z = - x2 +-
Y2
(6.29)
2ry 2r,
in which rx and ry are the principal radii of curvature. Introducing the auxiliary quantity F ( p ) as determined by equation (2.26),this is found to be a function of the elliptical parameters a and b as follows: (K2
F(p) =
+ l)G
(IC2
-
23
(6.30)
- l)G
in which 3 and G are thecomplete elliptic integrals of the first and second kind, respectively. (6.31) (6.32) By assuming values of the elliptical eccentricity parameter
K,
it is pos-
194
sible to calculate corresponding values of F(p) and thus create a table of K VS
F(p).
Brewe and Hamrock [6.4], using a least squares method of linear regression, obtained simplified approximations for K, 3,and 6. These equations are:
i~) 0.636
K
= 1.0339
(6.33) (6.34)
-
1.5277 + 0.6023 In
(2)
(6.35)
For 1 5 K 5 10, errors in calculation of K are less than 3%, errors on 6 are essentially nil except at K = I and vicinity where they are less than 2%,and errors on tF are essentially nil except at K = I and vicinity?where are defined they are less than 2.6%. The directional equivalent radii 9, by
K1 = PXI + PXII
(6.36)
where subscript x refers to the direction of the major axis of the contact and y refers to the minor axis direction. Recall that F(p) is a function of curvature of contacting bodies.
It was further determined that (6.38) (6.39) (6.40) (6.41)
STJRFACE STRESSES AND DEFO
(6.42) =
2.79 X 10-46*&2/3Ep1/3(for steel bodies)
(6.43)
in which S is the relative approach of remote points in the contacting bodies and
(6.44) (6.45) (6.46)
Values of the dimensionless quantities a*, b*, and 15" as functions of F ( p ) are given in Table 6.1. The values of Table 6.1 are also plotted in Figs. 6.3-6.5. For an elliptical contact area, themaximum compressivestress occurs at the geometrical center. Themagnitude of this stress is
pmax
-
3Q 2~ab
(6.47)
The normal stress at other points within the contact area is given by equation (6.48) in accordance with Fig. 6.6:
(6.48)
The foregoing equations (6.30)-(6.43) of surface stress and deformation apply to point contact. ~ x u ~ 6~.1.Z For e the 218 angular-contact ball bearing of Example
5.2 determine the maximum normal contact stresses and contact deformations at the inner and outer raceways.
196
CONTACT STRESS AND ~ E F O ~ T I O ~
LE 6.1. Dimensionless Contact Parameters 0 0.1075 0.3204 0.4795 0.5916
1 1.0760 1.2623 1.4556 1.6440
1 0.9318 0.8114 0.7278 0.6687
1 0.9974 0.9761 0.9429 0.9077
0.6716 0.7332 0.7948 0.83495 0,87366
1,8258 2.011 2.265 2.494 2.800
0.6245 0.5881 0.5480 0.5186 0.4863
0.8733 0.8394 0.7961 0.7602 0.7169
0.90999 0.93657 0.95738
3.233 3.738 4.395
0.4499 0.4166 0.3830
0.6636 0.6112 0.5551
0.97290
5.267
0.3490
0.4960
0.983797
6.448
0.3150
0.4352
0.990902 0.995112 0.997300 0.9981847
8.062 10.222 12.789 14.839
0.2814 0.2497 0.2232 0.2072
0.3745 0.3176 0.2705 0.2427
0.9989156 0.9994785 0.9998527 1
17.974 23.55 37.38
0.18822 0.16442 0.13050 0
0.2106 0.17167 0.11995 0
co
Q = 3543 N (796.2 lb)
Ex. 5.2
a = 38" 54'
Ex. 3.3
I) = 22.23 mm (0.875 in.)
Ex. 2.3
dm = 125.3 mm (4.932 in.)
Ex. 2.6 (2.27)
-
22.23 cos (38" 54') = 0.1381 125.3
Since this value is only slightly larger than the values of y obtained
197
660
860
L 60
zoo WO 900 800
01'0
ZI'O $1'0
91'0
81'0
ozo ZZO
PZO
920 820
200
CONTACT STRESS AND ~ E F O ~ T I O N
Y
x
F
I
~ 6.~ 6. E Ellipsoidal surface compressive stress distribution of point contact.
in Example 2.6, it is sufficient for the purpose of this example to use , and F(p), obtained in that example, Hence, values of 2pi, F ( P ) ~2po, Xpi = 0.108 mm-l (2.747 in?)
F ( P )= ~ 0.9260 Xpo = 0.0832 mm-l (2.114 in.”’) F(p), = 0.9038
From Fig. 6.4,
a? = 3.50, b j
=
0.430, 6;
=
0.630
( ~ ) 113
ai = 0.0236aj =
0.0236
=
0.0236bj
=
0.0236 “ X 0.430
=
0.324 mm (0.01277 in.)
X
3.50
(6.39) =
( ~ ) (
2.64 mm (0.1040
in.)
113
bi
X
-
(6.41)
201
SURFACE STRESSES AND D E F O ~ ~ O N S
c i r n a x
=
3Q 27ra,bi
-
3 x 3543 2 7 ~X 2.64 X 0.324
=
1976 N/mm2 (286,400 psi) (compression)
(6.47)
Si = 2.79
X
10-46fQ2f3Zp;’3
=
2.79
X
lom4X
=
0.0195 mm (0.000766 in.)
0.630 X (3543)2’3(0.108)1f3
(6.43)
At the outer raceway, from Fig. 6.4, a$
=
3.10, bz
=
0.455, 6: = 0.672
( ~ ) 113
a, = 0.0236aZ =
0.0236
X
3.10
(6.39) =
X
2.56 mm (0.1007 in.)
( ~ ) (~~~~2)1f3 1/3
bo = 0.0236bz =
0.0236
=
0.3754 mm (0.01478 in.)
X
cornax
0.455
(6.41)
X
3&
=
27raob,
-
3 x 3543 2 7 ~X 2.56 X 0.3754
=
1762 N/mm2 (255,500 psi) (~ompr~ssion)
(6.47)
6, = 2.79 X 10-46$Q2f3Zp~/3 =
2.79
=
0.01902 mm (0.000749 in.)
X
X
0.672
X
(3543)2’3(0.0832)1’3 (6.43)
Note that cbaxis greater than cornax. This is true for most ball and roller bearings in static loading.
202
CO~ACT STRESS AND DEFO
For ideal line contact to exist, the length of body I must equal that of body 11. Then K approaches infinity and the stress distrib~tion in the contact area degenerates to a sernicylindrical form as shoyn by Fig. 6.7. For this condition,
(6.49) (6.50) (6.51)
For steel roller bearings the semiwidth of the contact surface may be appro~imatedby
b = 3.35
X
(
~
~
2
(6.52)
The contact deformation for a line contact condition was determined by Lundberg and Sjovall E6.51 to be
i
6.7. Semic~lind~ical surface compressive stress distribution o f ideal line con-
tact.
SmFACE STRESSES ANI) D E F O ~ T I O ~ S
03
(6.53)
Equation (6.53) pertains to an ideal line contact. In practice, rollers are crowned as illustratedby Fig. 6.26b,c, and d. Based on laboratory testing of crowned rollers loaded against raceways, Palmgren I6.61 developed equation (6.54) for contact deformation.
In addition to Hertz [6.l] and Lundberg and Sjovall I6.51, Thomas and Hoersch E6.71 analyzed stresses and deformations associated with concentrated contacts. These references provide more complete information on the solution of the elasticity problems associated with concent~ated contacts. 2. Estimate the maximum normal contact stress anddeformation at theinner raceway of the 90000 series tapered roller bearing of Example 5.3:
Q,= 59,410 N (13,350 lb)
Ex. 5.3
CY; =
22"
Ex. 5.3
L) =
22.86 mm (0.9000 in.)
Ex. 5.3
1 = 30.48 mm (1.2 in.)
Ex. 5.3
dm = 142.2 mm (5.600 in.)
Ex. 5.3
L)
cos CY;
- 22.86 -
(2.27)
X cos(22") = 0.1490 142.2
(2.37)
=
Assuming that neither rollers nor raceways are crowned, that is, R ri = and ideal line contactoccurs,
204
CONTACT SmESS AND D E F O ~ T I O N
xpi =
bi
1 D(1 -
ri)
-
2 = 0.1028 mm-’ (2.611 in.-’) 22.86(1 - 0.1490)
=
3.35
X
10-3
=
3.35
X
10-3
=
0.461 mm (0.01815 in.)
T
(6.52)
( ~ ) 1 ’ 2
59410 30.48 X 0.1028 (6.49)
2 X 59410 = 2692 N/mm2 (390,200 psi) X 30.48 X 0.461
This roller is very heavily loaded and probably requires crowning to avoid edge loading (see Figs. 6.24 and 6.25):
(6.54) ==
3.84
X
X
(59410)0*9= 0.0491mrn(0.001932 (30.48)0*s
in.)
Hertz’s analysis applied onlyto surface stresses caused by a concentrated force applied perpendicular to the surface. Experimental evidence indicates that failure of rolling bearings in surface fatigue caused by this load emanates from points below the stressed surface. Therefore, it is of interest to determine the magnitude of the subsurface stresses. Since the fatigue failure of the surfaces in contact is a statistical phenomenon dependent on the volume of material stressed (see Chapter l8>, 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. 6.9-6.11. 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
=
t-1
(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 sub-surf'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 ~ises-Hencky 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 structure-white 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 case-hardening 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 [6.11-6.151. References [6.16-6.181 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 element-raceway 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 less-than-idealelastic 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 failure-initiating 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,, (y-direction)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 roller-racewaycontact. 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 raceway-roller 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 non-Hertzian 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. Roller-raceway contact load vs length and applied load; a comparison of straight, fully crowned, partially crowned, and logarithmic profiles (from [5.241).
(2900-14500psi),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 flange-roller guidance capability. In this case, some skewing control mayhave to be provided by the cage. For the case of sphere-end 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, roller-flange 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 half-space 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 load-deformation 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, 163-183 (1896). 6.2. S. Timoshenko andJ. Goodier, Theory of Elasticity, 3rd ed., McGraw-Hill, New York (1970). 6.3. J. Boussinesq, Compt. Rend., 114, 1465 (1892). 6.4. D. Brewe and B. Hamrock, “Simplified Solution for Elliptical-Contact Deformation Between Two Elastic Solids,” ME Trans., J: Lub. Tech. 101(2), 231-239 (1977). 6.5. G. Lundberg and H.Sjovall, Stress and Deformation in Elastic Contacts, Pub. 4, Institute of Theory of Elasticity and Strength of Materials, Chalmers Inst. Tech., Gothenburg (1958). 6.6. A. Palmgren, Ball and Roller Bearing Engineering, 3rd ed., Burbank, Philadelphia (1959). 6.7. H. Thomas and V. Hoersch, “Stresses Due to the Pressure of One Elastic Solid upon Another,” Univ. Illinois Bull. 212: (July 15, 1930). 6.8. A. Jones, Analysis of Stresses and Dejlections, New Departure Engineering Data, Bristol, Conn., 12-22 (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~beanspruchung-Ein~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 . 1-14 (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 52-A-13 (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, 345-352 (June 1982). tical Curved Bodies,”A S ~ Trans., 6.16. C. Cattaneo, “ A Theory of Second Order Elastic Contact,” Uniu. Roma Rend. Mat. Appl. 6,505-512 (1947). 6.17. T. Loo, “A Second Approximation Solution on the Elastic Contact Problem,”Sei. Sin‘ ,1235-1246 (1958). ica 7 6.18. H. Deresiewicz, “A Note on Second Order Hertz Contact,” A S M E Trans., J. AppL Mech. 28,141-142 (March 1961). 6.19. R. Sayles, G. desilva, J. Leather, J. Anderson, and P. MacPherson, “Elastic Conformity in Hertzian Contacts,”Tribology Intl. 14, 315-322 (1981). 6.20 J. filker, “Numerical Calculation of the Elastic Field in a Half-space 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 86-Trib-15 (October 1986). 6.22 T. Harris and W. Yu, “Lundberg-Palmgren Fatigue Theory: Considerations of Failure Stress and Stressed Volume,”A S M E Trans., J. Tribology 121,85-90 (January 1999). 6.23 I(. Kunert, “Spannungsverteilung im Halbraum bei Elliptischer Flachenpressungsverteilung uber einer Rechteckigen Drucldiache,” Forsch. Geb. Ingenieur~es27(6), 165-174 (1961). 6.24. H. Reusner, “Druckflachenbelastung und Ove~achenverschiebungin Walzkontakt von Rotationkorpern (Dissertation, Schweinfurt, Germany, 1977). 6.25. B. Fredriksson, “Three-Dimensional Roller-Raceway Contact Stress Analysis,” Advanced Engineering Corp. Report, Linkoping, Sweden (1980). 6.26. H. Reusner, “The Logarithmic Roller Profile-the 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 end-ring 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 Load-deflection factor; axial load deflection factor Roller length Distance between rows Moment Moment applied to bearing Load-deflection exponent Diametral clearance Load per unit length Ball or roller-raceway normal load Roller end-ring 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 tan-l 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
mm-l (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 load-deflection relationships for rollingelements contacting raceways. using Chapters 2 and 6 these load-deflection 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 steady-state 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 ball-raceway contact (point loading) 6
N
Q213
(7.1)
Inverting equation (7.1) and expressing it in equation format yields
Similarly, fora given roller-raceway 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 ball-steel 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 = cos-1
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,<E)
Point Contact Contact
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
l/Z 0.1156 0,1590 0.1892 0.2117 0.2288 0.2416 0.2505
E
1/z 0.1268 0.1737 0.2055 0.2286 0,2453 0.2568 0.2636
Contact Contact
0.8 0.9 1.0 1.25 1.67 2.5 5.0 G3
0.2559 0.2576 0.2546 0.2289 0.1871 0.1339 0.0711 0
0.2658 0.2628 0.2523 0.2078 0.1589 0.1075 0.0544 0
238
D I $ T R ~ ~ I QOF N ~E~~
LQADING
IN STATICALLY LOADED ~ ~ I N G $
t
F
I
~ 7.2. ~ E J,(E)vs E for radial bearings.
Figure 7.2 gives values of Jrvs E . Figure '7.3 shows radial load distribution for various values of E . Here, E is the ratio of the projected load zone on the bearing diameter. For ball bearings under pure radial load and zero clearance Stribeck C7.11 concluded that
Qrnax
=
4.37 Fr is cos a
(7.23)
Accounting for nominaldiametral clearance in thebearing, one may use the following approximation.
(7.24)
.
The209 radial ball bearing of Example 2.1 experiences a radial load of 8900 N (2000 lb). Determine the loading at each ball location.
(a) c = 0.5, $ 1 = t90", 0 clearance
(b) 0
(e) 0.5 < c < 1, 90" <$ r
< t < 0.5, 0 < $ 1
< 180", preload
~ I ~ 7U .3. RRolling ~ element load distribution for different amounts of clearance.
Z=9
Ex. 2.1
Pd = 0.0150 mm (0.0006 in.)
Ex. 2.1
dm = 65 mm (2.559 in.)
Ex. 2.1
I>
=
fi =
x x
12.7 mm (0.5 in.) fo
=
0.52
Ex. 2.2
y = 0.1954
Ex. 2.5
pi = 0.202 mm-' (5.126 in.-')
Ex. 2.5
I"'(P)~= 0.9399 p, = 0.138 mm-' (3.500 in.?)
F(p), = 0.9120
From Fig. 6.4,
Ex. 2.1
Ex. 2.5 Ex. 2.5 Ex. 2.5
240
DISTRIBUTION OF INTERNALLOADING IN STATICALLYLOADED ~~~~S
8; = 0.602,
6:
=
0.658
=
2.15
=
1.026 X IO6N/mm1.5(2.951 X IO7 l b / i ~ ~ . l . ~ )
K~~= 2.15
X
X
IO5 X (O.202)-li2 (0.602)-3/2
105
x p,-1i2a3-3i2
(7.8)
(7.22)
€+2) 2
="I--) 2
(a)
(7.12)
0.015
Equations (a) and(b) may be solved by trial and error or iteration. 6, = 0.06041 mm (0.002379 in.) E =
0.438
(7.19)
Q,,
=
4536 N (1019 1b)
(7.15) =
a*
4536 (1.142
360" 360" = --
-0.9397
__
4oo
"------"
z
0
0.7660
1c/ - 0.142)1.5
COS
9
1
t- 40 t- 80 t- 120 t- 160
4536 (1019 lb) 2846 (638.6 lb) 6 1 (13.7 lb) 0 0
0.1737 -0.5000
le 7.2. By using Stribeck's equation (7.23), determine the load distribution for the 209 radial ball bearing of the preceding example. 4.37 F,
&Inax = ZGE -
4.37 X 8900 9 x cos (0")
=
4321 N (971.2 lb)
Corresponding to equation (7.23),
E
(7.23)
= 0.5
(7.15) =
4321 COS^.^
* %
0" (650.7 2897 t- 40" k 80" t- 120" t- 160"
4321 (971.2 lb) lb) (70.1 313 lb) 0 0
+
For radial roller bearings with zero clearance under pure radial load, it can also be determined that 4.08 F,
_.
(7.25)
- Z-ZE
Equation (7.24) is also a valid approximation for radial roller bearings with nominal clearance.
3. The 209 cylindrical rollerbearing of Example 2.7 supports a radial load of 4450 N (1000 lb). Determine the loading at each roller location and the extent of the load zone.
(0.3780mm
14
Ex. 2.7
65 mm (2.559 in.)
Ex. 2.7
0.041 mm (0.0016 in.)
Ex. 2.7
9.6
Ex. 2.7
in.)
7.86
X
104 z8/9
7.86
X
104(9.6)8'9
5.869
Kn =
X
lo5 N/mml*ll(4.799 X lo6 lb/in.'.'')
r
):(
(7.9)
1 1/1.11
7 1.11 1/1.11
+
(~)
(0.5)1*11 X 5.869 2.720
ZKn(6, 4450
=
X
X
lo5
lo5 N/mrn1.l1 (2.222 x lo6 lb/in.'.")
- +pd)""C&(
14 X 2.720
X
(7.22)
E)
IO5
(6, - 0.0205)1.11~r(~) = 0.001169
(a)
€=!(1-2)
(7.12)
2
= i ( 1 - w0.041 )=0.5-2 Eq~ations (a) and (b) may be solved by trial and error.
0.01025
4
(b)
B
E
~ ~ ~ E~
S R
43
Sr = 0.0320 mm (0.00126 in.) E
=
0.1824
Jr(0.1824) = 0.165
Fr 4450
(7.19)
= 2Qrn~~r(€)
=
14 X QrnmX 0.165
Qrna = 1926 N (432.8 lb) (7.15)
['
-2
1 (1 - cos *) 0.1824
=
1926
=
1926 (2.741 COS @ - 1.741)1.11
0" 25.71' 51.42' 77.13" 102.84' 128.55" 154.26' 180"
X
1 0.9010 0.6237 0.2227 -0.2227 -0.6237 -0.9010 -1 *I
=
cos-'
= COS-'
( ~ ) (x
0.041 o , o ~ 2 0 )=
1926 (432.8 lb) 1355 (304.7 lb) 0 0 0 0 0 0
(7.13) rfi 50'10'
By using equation (7.24) to determine the load distribution which occurs for the 209 cylindrical roller bearing of the preceding example.
rnax
5Fr -2 cos a -
x 4450 = 1589 N (357.1 lb) 14 X cos (0')
(7.24)
24
DIST~IBUTIONOF INTERlWL LOADING IN S T A T I C ~ L LOADED ~ BE~INGS
From Fig. 7.2, E
=
0.28
(7.15)
=
1589 (1.786 COS
+ - 0.786)'*''
+ 0 25.71' 5 I .42" 77.13" 102.84" 128.55" 154.26" 180"
1589 (357.1 lb) 1280 (287.6 lb) 461 (103.6 lb) 0 0 0 0 0
=
cos-1
( ~ )
(7.13)
since
E=ql-g) 2
+I =
=
(7.12)
cos-l (1 - 2 E ) cosv1(1 - 2
X
0.28)
= rir 63'54'
For lightly loaded rolling bearings, the appro~imation of equation (7.24) is not adequate to determine maximum rolling element load and it should not be used in that instance.
45
GS
ST L
Thrust ball and roller bearings subjected to a centric thrust load have the load distributed equally among the rolling elements. Hence
&=-
F a
(7.26)
is sin a
In equation (7.26), a is the contact angle that occurs in the loaded bearing. For thrust ball bearings whose contact angles are nominally less than 90", the contact angle in the loaded bearing is greater than the initial contact angle a" that occurs in the nonloaded bearings. This phenomenon is discussed in detail in the following paragraphs.
In the absence of centrifugal loading, the contact angles at inner and outer raceways are identical; however, they are greater than those in the unloaded condition. In the unloaded condition, contact angle is defined bY -?d cos ao = 1 - 2191)
(7.27)
in which -?d is the mounted diametral clearance. A thrust load Fa applied to the inner ring as shown in Fig. 7.4 causes an axial deflection 6,. This axial deflection is a component of a normal deflection along the line of contact such that from Fig. 7.4, (7.28) Since G;!
=
Kn6k5, =
(-
Kn(B1))1-5
1.5
cos ao - 1) cos a
(7.29)
Substitution of equation (7.26) into (7.29) yields 1.5 Fa
isKn(191))1.5
(7.30)
Since Kn is a function of the final contact angle a, equation (7.30) must
-Shifted of inner ring
position
Q
"
7.4.
Angular-contact ball bearing under thrust load.
be solved by trial and errorto yield an exact solution for a. Jones f7.21, however, defines an axial deflection constant K as follows:
(7.31)
+
in which y = (D cos cx)/d, and g( y ) refers to the inner raceway and g( - y) refers to the outer raceway. Jones f7.21further indicates that the sum of g( y ) and g( - y) remains virtually constant for all contact angles
+
ing dependent only on total curvature is related to Kn as follows: I
ence,
. The mial deflectionconstant
6 5 UNDER THRUST LOAD 1.5 Fa
(7.33)
Taking K from Fig. 7.5, equation (7.33) may be solvednumerically by the ~ e w t o n - R a ~ h s omethod. n The equation to be satisfied iteratively is 1.5
Fa ~ZD2K a'=a+
cos ao cos a (cos a - 1) ~
1.5
+
("----
cos CYo 1.5 tan2 a - 1) cos a
0.5
cos a* (7.34)
Equation (7.34) is satisfied when a' - a is essentially zero. The asial deflection Sa corresponding to Sn may also be determined from Fig. 7.6 as follows:
700,000
.-
500,000
300,000 ~00,000 100,000 0
F B
I = f,
~ 7.5. ~ EAsia1deflection = rlD [7.21).
+ fi - 1,f
constant K vs total curvature B for ballbearings.
aa = (SD + an) sin a
-
SD sin a"
(7.35)
~ubstituting6n from equation (7.28) yields
a,
=
SD sin ( a - a")
(7.36)
cos a
Figure 7.6 presents a series of curves for the rapid calculation of the change in contact angle ( a - a"), and axial deflection as functions of initial contact angle and t = Fa/ZD2K. Ze 7.5. The 218 angular-contact ball bearing of Example 2.3 experiences a statically applied thrust load of 17,800 N (4000 lb). Determine the contact angle, normal ball load,and axial deflection of the bearing considering a ball complement of 16.
2.3
Ex. 3
=
0.0464
a" = 40"
D From Fig. 7.5, K
=
=
22.23 mm (0.875
2.3 in.)
896.7 N/mm2 (130,000 psi)
Ex. 2.3 Ex.
1.5
(7.33) 17800 16(22.23)2X 896.7
Fa ZD2K
sin a
"
a'=a+
1.5
cos a" cos a (cos a - 1)
(2
-- 1)1.5
+ 1.5 tan2 a
(-
cos a" cos a
0.5
-
1)
cos a" (7.34)
=a+
Equation (b) is solved byassuming values of a. The iteration is continued until the absolute value of a' - a approaches zero. a is determined to be 0.7260 radians (41.6"). This result is sufficiently accurate for the illustrative purpose intended here. A similar result could have been obtained by using Fig. 7.6 at t = 0.0025 and a" = 40". (7.26) -
17800 = 1676 N (376.6 lb) 16 X sin (41.6")
( a - a") sa= BD sin cos a -
0.0464
X 22.23 sin (1.6") (0.00152 mm = 0.0386 cos (41.6")
(7.36) in,)
SingleDirectionBearings. Figure 7.7 illustrates asingle-row thrust bearing subjected to an eccentric thrust load. Taking 1(/ = 0 as the position of the maximum loaded rollingelement, then
7.7. 90"ball thrust bearing under an eccentric load.
8, =
sa + $0 dmcos tr/
(7.37)
Also, Smax=
sa + $0 dm
(7.38)
From equations (7.37) and (?.38), one may develop the familiar relationship
s,
=
[ 2:
smax1 - - (1 - cos *)
1
(7.39)
in which
.=l(l+Z) 2
(7.40)
The extent of the zone of loading is defined by (7.41)
As before,
Static equilibrium requires that
Fa =
&+sin a
(7.43)
x4
(7.44)
*=0
%, =
eFa =
&*dmsin a cos II/
*=0
Equations (7.43) and (7.44) may also be written in terms of thrust and moment integrals as follows:
in which (1 - cos +) %=
eFa = 4& m ~ d m J m (sin E)a
I'
d+
(7.46) (7.47)
in which (7.48)
2E
Table 7.2 as shown by Rumbarger L7.31 gives values of Ja(e)and J,(E) as functions of 2eIdm,.Figures 7.8 and 7.9 yield identical data in graphical format. Assuming that the contact angle remains constant a t 41.6", determine what the magnitude would be of the maximum ball load in the 218 angular-contact ball bearing of Example 7.5 if the 17,800 N (4000 lb) thrust load was applied a t a point 50.8 mm (2 in.) distant from the bearing's axis of rotation. dm= 125.3 mm (4.932 in.)
2e - 2 X 50.8 dm 125.3
"
=
From Fig. 7.8, Ja= 0.285, Jm= 0.233,
0.8110
E =
0.525
2.6 Ex.
5%
~ ~ S T R ~ B U OF T ~INTERNAL ON LOADING IN STATICALLY LOADED B E ~ ~ G $ , J,(E) and Jm(e)for
Single-Row Thrust Bearings
Contact Line Contact Point 2e
2e
0 0.1 0.2 0.3
1.0000 0.9663 0.9318 0.8964
1/ z 0.1156 0.159 0.1892
1/x 0,1196 0,1707 0.2110
1.0000 0.9613 0.9215 0.8805
1l z 0.1268 0.1737 0.2055
1/x 0.1319 0.1885 0.2334
0.4 0.5 0.6 0.7
0.8601 0.8225 0.7835 0.7427
0.2117 0.2288 0.2416 0.2505
0.2462 0,2782 0.3084 0.3374
0.8380 0.7939 0.7488 0.6999
0.2286 0.2453 0.2568 0.2636
0.2728 0.3090 0.3433 0.3766
0.8 0.9 1.o 1.25
0.6995 0.6529 0.6000 0.4338
0.2559 0.2576 0.2546 0.2289
0.3658 0.3945 0,4244 0.5044
0.6486 0.5920 0.5238 0.3598
0.2658 0.2628 0.2523 0.2078
0.4098 0.4439 0.4817 0.5775
1.67 2.5 5.0
0.3088 0.1850 0.0831 0
0.1871 0.1339 0.0711 0
0,6060 0.7240 0,8558 1.0000
0.2340 0.1372 0.0611 0
0.1589 0.1075 0.0544 0
0.6790 0.7837 0.8909 1.0000
co
Ex. 7.5
is = 16 Fa = ZQmaJa(~) sin a 17800 = 16 X Qma
X
0.285
(7.45) X
sin (41.6")
QmaX= 5878 N (1321 lb)
An identical result is obtained by using J,(E) in equation (7.47),
Figure 7.10 demonstrates a typical distribution of load in a 90" thrust bearing subjected to eccentric load.
Double Direction Bearings. The following relationships are valid for a two-row double-direction thrust bearing:
3
(7.49) (7.50)
It can also be shown that €1
+ €2 = I
(7.51)
and (7.52) Considering equation (7.4), equation (7.52) becomes
DI$TRIBUTION OF I N T E ~ ~
7.9.
L O IN~$ T I ~NTG I C ~ LLYO ~ E D B E ~ I ~ G $
J,(E),J,(E),E vs 2eld, for line-contact"thrust bearings.
55
B E ~ I N UNDER ~ S THRUSTLOAD
(7.53)
In equation (7.53), n = 1.5 for ball bearings, and n = 1.11 forroller bearings. From conditions of equilibrium one may conclude that
Fa = Fal- Fa2 = ZQmm,Jasin a
(7.54)
in which (7.55) %=
+ % = -ijZQmm,d,Jm
(7.56)
sin a
in which (7.57)
Table 7.3 below gives values of Jaand Jmas functions of 2eldm for tworow bearings. Figures 7.11 and 7.12 give the same data in graphical format.
Line Contact
ContactPoint 2e €1
€2
dJam
&ma,, Jm
dm
Qmxl
2e Ja
Jm
&ma,
0.50 0.51 0.60 0.70
0.50 0.49 25.72 0.40 2.046 0.30 1.092
0.4577 0.4476 0.3568 0.3036
0 0.0174 0.1744 0.2782
1,000 0.941 28.50 0.544 2.389 0.281 1.210
0.4906 0.4818 0.4031 0.3445
0 0.0169 0.1687 0.2847
1.000 0.955 0.640 0.394
0.80 0.90 1.0 1.25
0.20 0.10 0 0
0.800 0.671 0.600 0.434
0.2758 0.2618 0.2546 0.2289
0.3445 0.3900 0.4244 0.5044
0.125 0.037 0 0
0.823 0.634 0.524 0.360
0.3036 0.2741 0.2523 0.2078
0.3688 0.4321 0.4817 0.5775
0.218 0.089 0 0
0.309 0.185 0.083 0
0.1871 0.1339 0.0711 0
0.6060 0.7240 0.8558 1.0000
0.234 0.137 0.061 0
0.1589 0.1075 0.0544 0
0.6790 0.7837 0.8909 1.0000
~0
1.67 0 2.5 0 5.0 0 0
~0
~~~
DISTR~UTIONOF INTERNALLOADING IN STATICALLYLOADED B
~
~
0.9
0.8
0.7 r)
E 0.6
2cu
z
2 0.5 w
M
w
4
0.4
t 3
0.3
0.2 0.1
0
Jm, J,, E
0.2 0.4 0.6 0.8 I
1.2 1.4 1.6 2e/drn
1.8 2 2.2 2.4
QmmJQmmlvs & / d m for double-row point-contact thrust
~ E , ~ ,
bearings.
ER CO
If a rolling bearing without diametral clearance is subjected simultaneously to a radial load in the central plane of the rollers and a centric thrust load, then the inner and outer rings of the bearing will remain parallel and will be relatively displaced a distance aa in the axial direction and 6, in the radial direction. At any regular position II/ measured from the most heavily loaded rollingelement, the approach of the rings is 8,
=
aa sin a +
6, cos a cos ~!,t
(7.58)
Figure 7.13 illustrates this condition. At II/ = 0 maximum deflection occurs and is given by
G
S
F I G ~7 E.12. J,, J,, el, e2, Qmax2/Qmaxl vs 2e/d, for double-row line-contactthrust bearings.
cos
F I G U ~ E7.13. Rolling bearing displacements due to combined radial and axial. loading.
25
D I S T ~ ~ OF ~ I O N ~T E LOADING ~ ~ IN S T A T I C ~ YLOADED ~
Smax=
sasin a + 8, cos a
E
(7.59)
~ombiningequations (7.58) and (7.59) yields
[
s,, = srnm1 - l (1 - cos $1 2E
1
(7.60)
This expression is identical in form to equation (7.11), however,
satan a
(7.61)
8r
It s h o ~also l ~ be apparent that
[
Q,, = Qmax
1, -
- (1 - cos $)
E;
(7.62)
~
As in equation (7.4), n = 1.5 for ball bearings and n = 1.11for roller bearings. For static equilibrium to exist, the summation of rolling element forces in each direction must equal the applied load in that direction.
x
*=+h
Fr =
Q,, cos a cos II/
(7.63)
+= -44
(7.64)
in which the limiting angle is defined by
$1
=
cos-l
(
-
satan
sr
a
(7.65)
Equations (7.63) and (7.64) may be rewritten in terms of a radial integral and thrust integral, respectively.
where
~
(7.67) and (7.68) where (7.69) The integrals of equations (7.67) and (7.69) wereintroduced by Sjovall 117.41. Table 7.4 gives values of these integrals for point and line contact as functions of Fr tan a/Fa. Note that the contact angle a is assumed identical for all loaded balls or rollers. Thus the values of the integrals are approximate; however, they are sufficiently accurate for most calculations. Using these integrals,
LE 7.4. J,(E)and Ja(E ) for Single-Row Bearings
0 0.2 0.3 0.4
1 0.9318 0.8964 0.8601
1/ z 0.1590 0.1892 0.2117
1/ z 0.1707 0.2110 0.2462
1 0.9215 0.8805 0.8380
1/ z 0.1737 0.2055 0.2286
1/x 0.1885 0.2334 0.2728
0.5
0.8225
0.2288
0.2782
0.7939
0.2453
0.3090
0.6 0.7 0.8 0.9
0.7835 0.7427 0.6995 0.6529
0.2416 0.2505 0.2559 0.2576
0.3084 0.3374 0.3658 0.3945
0.7480 0.6999 0.6486 0.5920
0.2568 0.2636 0.2658 0.2628
0.3433 0.3766 0.4098 0.4439
0.2523
0.4817
.0.2078 0.1589 0.1075 0.0544 0
0.5775 0.6790 0.7837 0.8909 1
0.5238 1 0.4244 0.2546 0.6000 1.25 1.67 2.5 5 co
0.4338 0.3088 0.1850 0.0831 0
0.2289 0.1871 0.1339 0.0711 0
0.5044 0.6060 0.7240 0.8558 1
0.3598 0.2340 0.1372 0.0611 0
Fr anax
=
(7.70)
J,(E)Z cos a
or (7.71) Figures 7.14 and 7.15 also givevalues of J,, J,, and E vs F, tan aIF, for point and line contact, respectively. ~ 7.7. The ~ 218 angular-contact ~ ~ ball Z bearing e of Example 7.5 is subjected to a radial load of 17,800 N (4000lb) and a thrust load of 17,800 Pa (4000 lb). Estimate the normal load on each ball if the contact angle can be presumed to remain constant at 40".
~
1.o
0.9 0.8
0.7
0.6
0.4
0.3
0.2
0.1
0
~1~~~
0.2
0.4
0.6 0.8 F,. tan alF,,
1.0
7.14. J,(E), JJE), E vs Fr tan a / F a for point-contact bearings.
0.9
0.8 0.7
0.6 W
i \-.
.s" 0.5 i
%
0.4
0.3
0.2
0.1
0
FIG
0.2
0.4
0.6 0.8 1.2 1.0 E; tan aJF,
7.16. J,(E),J,(E),E vs Fr a / F , for line-contact bearings.
Fr tan Fa
From Fig. 7.14, J ,
=
cy
- 17800 tan (40") = o,8391 17800
0.221, J a = 0.263,
E =
0.455,
53 = 16
Ex. 7.5 (7.70)
-
17800 0.221 X 16 X cos (40")
=
6571 N (1477 lb)
The same result could have been obtained by using equation (7.71).
262
D I $ T R I ~ ~ I OOF N INFERNAL LOADING IN $TATICALLS LOADED ~ ~ I N G $
cos-1 (1 - 2€) COS-'
(1 - 2
X
(7.61), (7.65) 0.455) = t- 84'47' (7.62)
[
2
X
6571(1.099 COS 360 -360 2 16
"
1.5
1
6571 1 -
=
0.455 Q - 0.0989)1.5
22.5" (22'30')
cos II,
0 22.5 45 67.5 90 112.5 135 157.5 180
1 0.9239 0.7071 0.3827 0 -0.3827 -0.7071 -0.9239 -1
6571 (1477 lb) 5765 (1296 lb) 3670 (824.2 lb) 1200 (269.6 lb) 0 0 0 0 0
Let the indices 1 and 2 designate the rows of a two-row bearing having zero diametral clearance. Then (7.72) (7.73)
Sal = - 4 3 2
~ubstitutingthese conditions into equations (7.59) and (7.60) yields (7.74) €1
+ €1
=
1
(7.75)
Equation (7.75) pertains only if both rows are loaded. If only one row is loaded, then €1
2 1,
€2
=
0
(7.76)
B E ~ ~ UNDER G S C
O RADIAL~ANI3 THRUST ~ LOAD~
It is further clear from equation (7.4) that =
(>:”
(7.77)
Qmm,
The laws of static equilibrium dictate that (7.78) (7.79)
As before,
F,
=
ZQm,,,J,
(7.80)
COS cy
Fa = ZQmm,Jasin
(7.81)
cy
in which (7.82) (7.83) Table 7.5 gives values of J , and J a as functions of F, tan a/Fa. Figures 7.16 and 7.17 give the same data ingraphical format for point and line contact, respectively.
LE ‘7.5. J a and J , for Double-Row Bearings
0.5 0.5
00
0.6 0.4 2.046 0.7 0.3 1.092 0.8 0.2 0.8005 0.9 0.1 0.6713 1.0 0
0.6000 0.2546 0.4244
0.4577 0
1
1
0,35680.17440.5440.477 0.3036 0.2782 0.281 0.212 0.2758 0.3445 0.125 0.078 0.2618 0,3900 0,037 0.017 0
00
0.4906 0
1
1
0
0
2.389 0.4031 0.1687 0.640 0.570 1.210 0.3445 0.2847 0.394 0.306 0.8232 0.3036 0.3688 0.218 0.142 0.6343 0.2741 0.4321 0.089 0.043 0
0.5238 0.2523 0.4817
264
D
I
S
~
~ OF ~ INTERNAL I O ~ LOADING IN S T ~ T I C A L LLOADED ~ ~ E ~ I N G S
.4
7.16. J,, Jayel, eZyQmax2/Qmaxl, Frz/Frlvs Fr tan a/Fa for double-rowpointcontact bearings.
Ze 7.8. The 22317two-row spherical roller bearings of Example 2.7 supports a 89,000 N (20,000 lb) radial load and a 22,250 N (5000 lb) thrust load simultaneously. Estimate the roller load distribution.
Ex. 2.8
a = 12"
Z = 14
2.7
Ex.
Fr tan a - 89000 X tan (12") = 0.8502 22250 Fa From Fig. 7.17, J , = 0.303, Ja = 0.370, el = 0.220, Fr2/Fr1 = 0.143
=
0.8, ez = 0.2,
F I ~ 7.~ 17. EJ,, Ja7 eZ7Qmax2/Qmaxlt Fr2/Fr1 vs F,tan a / F afor double-row line-contact bearings.
89000 = 14 X Qmml
Qm,
X
0.303 x cos (12")
=
21450 N (4819 lb)
=
0.220
=
~ 1 4 5 0 ( 0 . 6 ~COS 5 t,b
X
21450 = 4719 N (1060 lb)
+ 0.375)1.11
DISTR~UTIO OF ~
~~~
=
LOADING IN S T ~ T I LOADED ~ ~ S El
~~~~
cos-1 (1 - 2€,)
(1- 2
X
0.8) = +- 126'52'
(7.62)
[
4719 I
+12
-
1 (1 - cos 2 x 0.2
~
=
4719(2.5 COS
=
cos-l (1-
==:
$1
+ - 1.5)1*11
2E2)
COS-^ (1 - 2
X
0.2)
I=:
+- 53'8'
A + = - 360' =--
360 - 25.71" 14
0 25.71 51.42 77.13 102.84 128.55 154.26 180
1 (4819 2140 0.9010 19980 (4488 0.6237 15930 (3578 10250 0.2227 -0.2227 (964 4321 -0.6237 -0.9010 -1
Z
(25'43')
lb) lb) lb) (2299 lb) lb) 0 0 0
4719 (1060 lb) 3442 (773 lb) 204 (46 lb) 0 0 0 0 0
If a ball is compressed by a load Q, then since the centers of curvature of the raceway grooves are fixed with respect to the corresponding raceway, the distance between the centers is increased by the amount of the normal approach between the raceways. From Fig. 7.18 it can be seen that
s =A
+ si + ti,
s* = Si" so= s " A
(7.84) (7.85)
If a ball bearing that has a number of rolling elements situated symmetrically about a pitch circle is subjected to a combination of radial, thrust, and moment loads, the following relative displace~entsof inner and outer ring raceways may be defined:
(a)
(b)
7.18. (a)Ball-raceway contact before loading; ( b ) ball-raceway contact under load.
Sa relative axial displacement Sr relative radial displacement e relative angular misalignment
These relative displacements are shown in Fig. 7.19. Consider a rolling bearing prior to the application of load. Figure 7.20 shows the positions of the loci of the centers of the inner andouter race-
7.19.Displacements of an inner ring (outer ring fixed) due to combined radial, asial, and moment loading.
268
D I S T R I B ~ I O NOF INTERNALLOADING IN STATICUY LOADED B E ~ I N ~ S
Bearing axis x
7.20.
Lociof raceway groove curvature radii centers before loading (reprinted
from [7.21).
way groove curvature radii. It can be determined from Fig. 2.2 that the locus of the centers of the inner ring raceway groove curvature radii is expressed by
in which ao is the free contact angle determined by bearing diametral clearance. From Fig. 7.20, then
S&= tni
-
tni
- A COS
% =A
COS
a"
(7.87)
a"
(7.88)
+
In Fig. 7.20, is the angle between the most heavily loaded rolling element and any other rolling element. Because of symmetry 0 5 5 71: If the outer ring of the bearing is considered fixed in space as load is applied to the bearing, then the inner ring will be displacedand thelocus of inner ringraceway grooveradii centers will also be displaced as shown in Fig. 7.21. From Fig. 7.21 it can be determined that s, the distance between the centers of curvature of the inner and outer ring raceway grooves at any rolling element position is given by
+
+,
s
=
[(A sin
ao
+ Sa + $Ii8 cos
or
Y printed from [7.2]).
+)2
+ (A cos
cyo
+ 6, cos +)2]1'2
(7.89)
270
D I S T R I B ~ I O NOF ~E~~
s = A[(sin ao
+ 8, +
LOADING IN S T A T I C ~ L SLOADED B
cos +))" +
ao +
(COS
8, COS
(7.90)
in which 6a aa = A
-
(7.91) (7.92) (7.93)
~ubstitutingequation (7.90) into (7.85) yields
+ (cos ao + 8, cos @)2]1/2
- I} (7.94)
~ubstitutionof equation (7.94) into (7.4) gives
Q = KnAn{[(sin ao + Sa +
giecos
+)2
+ (cos ao + 3,
cos
@)2,1/2
-
1p (7.95)
+,
At any rolling element position the operating contact angle is a. This contact angle can be determined from sin a
=
sin ao + Sa + 9xi3 cos sl/ [(sin ao + Sa + 9xi3 cos $1' + (cos ao + S, cos
(7.96)
+)'I'/'
or __
cos a
=
[(sin ao + Sa +
+
cos ao + 6, cos si3cos +12 + (COS ao + 3,
COS
+)211/2
(7.97)
Equation (7.95) describesthe normal load acting through contact angle a. This normal load maybe resolved into axial and radial components as follows:
Q, = Q sin a
&,
=
& cos + cos a
(7.98) (7.99)
If the applied radial and thrust loads on the bearing are F, and Fa,respectively, then for static equilibrium to exist:
*
$=0 *=+n
~dditional~y, each o f the thrust components produces a momeat about the Y axis (moments about the 2 axis are self-equ~librati~~), w]nich is given by
272
I ) I S T R ~ ~ OF I OI~~ E R N A L LOADING IN STATICALLY LOADED B M I N G S
The foregoing equations were developed by Jones E7.21. Equations (7.104)-( 7.106)are simultaneous nonlinear equations with unknowns Sa, Sr, and 6. They may be solved by numerical methods; for example, the Newton-Raphson method. Having obtained Sa, &, and 6, the maximum rollingelement load may be obtained from equation (7.95) for !,t = 0. Lllax
=
KnA"{[(sina'
+ sa+
+ (cos a' + 8r)2]1'2- 1)"
(7.107)
Solution of the indicated equations generally necessitates the use of a digital computer. In certain cases, however-for example, applications with simple radial, simple thrust or radial andthrust loading with nominal clearance-the simplified methodspresented in the beginning of this chapter will probably provide sufficientlyaccurate calculational results.
Although it is usually undesirable, radial cylindrical rollerbearings and tapered roller bearings can support to a small extent the moment loading due to misalignment, The various types of misalignment are illustrated in Fig. 7.22. Clearly, spherical roller bearings are designed to exclude all moment loads on the bearings and therefore are not included in this discussion. Figure 7.23 illustrates themisalignment of a cylindrical rollerbearing inner ring relative to the outer ring. To commence the analysis, it is assumed that any roller-raceway contact can be subdivided into a number of laminae situated in planes parallel to the radial plane of the bearing. It is also assumed that shear effects between these laminae can be neglected owing to the small magnitudes of the contact deformations that develop. (Only contact deformations are considered.)
In a misaligned cylindrical roller bearing subjected to radial load, at each laminum in a crowned roller-raceway contact, the deformation may be considered to be composed of three components: (1)Amj due to the radial load at roller azimuth location j, (2) e, due to the crown dropat laminum A, and (3) deformation due to the bearing misalignment and roller tilt at roller azimuth location j. These componentsare illustratedschematically in Fig. 7.24. The component due to radial load was the only component considered in the simplified analysis previously discussed; it needs no further ex-
"".
??z%%%# € is alignment (out-of-line) (a)
i%t&%
0
0
-
-
0
fl
0
Off-square or tilted outer ring (b)
?3%zm
?mm
%?z7%%m Cocked or tilted inner ring (c)
Shaft deflection (d)
7.22. Types of misalignment.
planation. The componentdue to crowning can be defined by consider in^ the roller crowning shown by Fig. 7.25. From Fig. 7.25, it can be seen that crown drop cAat a selected laminum is considered as a negative deformation; Le., no roller-raceway loadingcan occur at a laminum until cAis overcome bythe radial and/or the misalignment deformation. Equation (7.108) defines cAin terms of the roller and crowning ~ a r a m e t ~ r s .
FIGURE 7-23. Misalignment of cylindrical roller bearing rings.
0
"C
( h - $)w 5
I
-
I,
~
2
(7.108)
c, = 0
1 - 1, 2
"C
(A - $)w
-2
[
~
I
+ I, 2
1 + I, 2
( ~] ) [Rz 2
c, =
5
112
-
25
(A - $)w
5
-
4)
((A - $>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 roller-raceway 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 roller-raceway 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 roller-raceway contact. While the load-deformation 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 ring-roller assembly misaligned with respect to the
interferencg with
. From thia sketch, it ia evident that one-half 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 roller-raceway 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 roller-end-flange 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
4-J
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 crowning-309 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
10-5143~1
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 roller-raceway 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 thrust-loaded roller-ring 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 raceway-roller 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 roller-outer raceway loading whichresults. The roller-raceway 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. Roller-outerracewaycontactshowing 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 roller-raceway 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 roller-raceway 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 roller-raceway loads Qj and roller end-flange 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 roller-raceway 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 end-flange 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 end-inner ring flange contact, and is the x-direction distance in an axial plane from the center of the inner raceway contact to the center of the roller end-inner ring flange contact,
the roller load equilibrium equations are 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 end-flange 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 end-flangecontact. 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 self-aligning and therefore cannot carry moment loading. Moreover, for slow or moderate speed applications causing insignificant roller inertial loading and friction, symmetrical (barrel-shaped) 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 two-point 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 back-up 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 back-up 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 self-sufficient 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 self-sufficient and an appropriate equilibrium equation must be used to define a rigid ring translation. The separating forces Fs are self-equilibratin~ 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 self-equilibrating. 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 29,420-463 (1907). 7.2, A. Jones, Analysis of Stresses and Deflections, New Departure Engineering Data, Bristol, Conn. (1946). 7.3. J. Rumbarger, “Thrust Bearings with Eccentric Loads,”Mach. Des. (Feb. 15, 1962). 7.4. H. 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., 294-300 (April 1969). 7.6. T. Harris, “The Endurance of a Thrust-Loaded,Double Row, Radial Cylindrical Bearing,” ea^ 18,429-438 (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),572-578 (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., 273-278 (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). 7-10. T. Harris and J. Broschard, “Analysis of an Improved Planetary Gear Transmission Bearing,”AS~EDuns., J Basic Eng., 457-462 (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, 134-139 (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 I-Rolling Bearing Model,” A S ~ Duns., E J Tribology 121,205-214 (April 1999). 7.16. A. Bourdon, J. Rigal, and D, Play, “Static Rolling Bearing Models in a C.A.D. Environment for the Study of Complex Mechanisms: Part 11-Complete Assembly Model,” ASHE Duns., J Tribology 121,215-223 (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.) N-mm (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 x-direction Acceleration in x-direction Distance in y-direction Acceleration in y-direction Distance in x-direction Acceleration in x-direction 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 flange-roller end efers to x-direction (Fig.5.4) efers to x'-direction (Fig. 5.4) efers to y-direction (Fig. 5.4) efers to y'-direction (Fig. 5.4) efers to z-direction (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 grease-lubricated 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 innerraceway-ball 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 angular-contact 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 Roller-raceway 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 $
/ 0-Pure 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 oil-film 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 motions-low speed operation of a ball bearing.
8.7. Sliding lines in contact area for simultaneous rolling, spinning, and gyroscopic motions-high speed operation of a ball bearing (not considering ~ ~ ~ ~
*Skidding is a very gross sliding condition occurring generally in oil-film lubricated ball and roller bearings operating under relatively light load at very high speed 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 A-A slide backward with regard to the direc-
R
/
Q
8.8. Roller-raceway contact showing harmonic mean radius and points of rolling A-A.
*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 A-A 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 roller-raceway contact area when load is applied; generatrix of motion pierces contactarea.
E 8.11. Sliding lines for ball-raceway contact area for simultaneous rolling,spinning, and gyroscopic motions-high load and low speedoperation of an angular-contact ball bearing.
.12. Sliding lines for ball-raceway contact area for simultaneous rolling, spinof an angular-contact ball ning, and gyroscopic motions-low 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 theball-raceway 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 thrust-loaded aircraft gas turbine, angular-contact ball bearing. It must be noted that skidding also tends to occur at the same time. Hence, for oil-film 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 film-lubricated ball bearings or for ball bearings in which a constant coefficient of friction maybe assumed in theball-raceway contacts, Harris L8.61 has shown for a thrust-loaded angular-contact ball bearing that, at relatively slow speed, spinning and rolling occur simultaneously at both inner and outer ball-raceway contacts. For a given load, as speed is increased, a transition takesplace in which outer raceway control is approximated; however, the outer raceway contact spinto-roll ratio is always nonzero (see Figs. 8.16 and 8.17). It is illustrated by Figs. 8.15-8.17 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.
Spin-to-rollratio vs thrust load for an oil-film lubricated angular-contact
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. Ball-shaft speed ratio vs shaft speed for a thrust-loaded, angular-contact ball bearingoperating with dry friction.
Shaft Speed (rpm)
.17. Spin-to-rollratio vs shaft speed for a thrust-loaded, angular-contact 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 fluid-lubricated angular-contact 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 ball-raceway 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 raceway-roller contact load that acts in the roller axial direction. Some cylindrical roller bearing designs require roller endflange contacts to react skewing-induced 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 end-flange contactsuch as wear and smearing of the contacting surfaces. These failure modes are related to the ability of the roller end-flange contactto support roller axial load under the prevailing speed and lubrication conditions within the contact. Both the frictional characteristics and load-carrying 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 end-flange 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 end-flange 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 end-flange contact geometry. (a)Flat end roller. ( b ) Sphere end roller.
8.19. Cylindrical roller bearing roller end-flange contact location for flat and sphere end-rollers. ( a ) Flat end roller, zero skew angle. (b) Flat end roller, nonzero skew angle. ( 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 rollerend-flangecontactlocation.
point 1" with coordinates (Tg9 Ty9TJ expressed in the flanged ring coordinate system. Since the resultant roller end-flange 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 closed-form 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 closed-form 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 end-flange 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 end-flange contactcauses sliding to occur between the contacting members. The magnitude of the sliding velocity between these surfaces substantially affects friction, heat generation, and load-carrying characteristics of a roller bearing design. The sliding velocity is represented by the 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 end-flangesliding 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 end-flange 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. 70-72 (1959). 8.2. A. €3.Jones, “Ball Motion and Sliding Frictionin Ball Bearings,”ASME J Basic Eng. 81, 1-12 (1959). 8.3.A.B. Jones, “A General Theory for Elastically Constrained Ball and Radial Roller Bearings under 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 Thrust-Loaded, Angular Contact Ball Bearings,”ASME J Lubr. Technol. 93, 17-24 (1971). 8.6. TI A, Harris, “Ball Motion in Thrust-Loaded, Angular-Contact Bearings withCoulomb Friction,” ASME J Lubr. Technol. 93, 32-38 (1971). 8.7. R. Kleckner and J. Pirvics, “High Speed Cylindrical Roller Bearing Analysis-SKF Computer Program CYBEAN, Vol. 1:Analysis,” SKF Report AL78P022, NASA Contract NAS3-20068 (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 ~oad-deflection 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 bearing-suppo~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 ball-inner and ball-outer 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 Newton-Raphson 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 angular-contact ball bearing of Example 7.5 operated through the load range of 0-44,500 N (10,000 lb) thrust and at speeds of 3000,6000,10,000 and 15,000 rpm. 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 load-speedcondition. Inthese equations, which may be solvedby iterative techniques, the load-deflection "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. 9.4-9.6 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 angular-contact ball bearing,a* = 40”.
For an angular-contact 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 out-of-plane vibrational component. Such motion together with the
N
9.5. Ball normal loads Q, and Qi vs thrust load for various shaft speeds. 218 angular-contact ball bearing, a” = 40”.
aforementioned loading canlead to rapid destruction and seizure of the bearing. Under combined loading, becauseof the variation in theball-raceway 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 steady-state cage rotation, the sum of these ball-cage 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. Sa-axial 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 angular-contact 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 9.7-9.9 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 raceway-ball loads vs bearing applied thrust load for a 218 angular-contact ball bearing operating at 15,000 rprn with steel or siliconnitride balls.
erations at high speed of the 218 angular-contact 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 all-steel bearing. This causes the contact stresses to be greater. Depending on the load magnitude, the stress level may be acceptable to the ball material, but not to the raceway steel. This situation can be ameliorated at 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 raceway-ball contact angle vs bearing applied thrust load for a 218 angular-contact 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 angular-contact 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 Newton-Raphson 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.)l-l1)
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)
10-11D2Zd,ni
X
10)2(9.6)2X 65
X
(0.4231~2,)~
10-7nz (9.33) -
This ~ v e eight s eauations as follows:
3.788 X 10-7nf?__ -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
10-4
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 ball-raceway 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. 82,309-320 (1960).
9.2. T. A. Harris, “Optimizing the Fatigue Life of Flexibly-MountedRolling Bearings,” Lub. Eng. 420-428 (October, 1965). 9.3. T. A. Harris, “On the Effectiveness of Hollow Balls in High-speed !t%rust Bearings,’’ M L E Dunsuet, 11, 209-294 (October, 1968). 9.4. T. A. Harris and S. F. Aaronson, “An Analytical Investigation of Cylindrical Roller Bearings Having Annular Rollers,”ASLE Preprint No. 66LC-26 (October 18, 1966). 9.5. T. A. Harris and M. H. Mindel, “Rolling Element Bearing Dynamics,” Wear 23, 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 Load-de~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) mm-l (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 deep-groove and angular-contact ball bearings subjected to radial load which causes only radial deflection, that is, aa = 0,
sr = 4.36 X
10-4
Q2& D1I3cos a
(10.6)
For self-aligning ball bearings,
sr = 6.98 X
10-4
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
10-4
( 2 % ; P I 2
cos a
(10.8)
For radial roller bearings with line contact at each raceway,
ar = 7.68 X
10-5
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
10-4
Q2& sin a
(10.10)
Q:&
(10.11)
For self-aligning ball bearings Sa = 6.98
For thrust ball bearings
X
D1I3sin a
7
DEFLECTIO~SOF BE
a,
=
5.24
X
10-4
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
10-4
QZ& sin a
(IO.13)
For roller bearings with line contact at each raceway,
a,
=
7.68
X
10-5
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
10-5
Total shaft movement is
=
From Example 7.3,
0.02983 mm (0.001175 in.)
amax= 0.03251 mm (0.001~8in.). For the 218 angular-contact 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
10-4
(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 load-deflection curve. This condition can be realized by axially preloading angular-contact 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 angular-contact ball bearings are placed back-back or face-to-face 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.
Back-to-back
Face-to-face 10.2. Duplex sets of angular-contact 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 back-to-back angular-contact 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) Face-to face angular-contact 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 standard-width 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 load-deflection characteristics as given by the single-bearing curve. The point at which bearing 2 loses load may be determined graphically by inverting the single-bearing load-deflectioncurve 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 angular-contact ball bearings is mounted back-to-back, 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 angular-contact 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 most-off-the-shelf 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
=
cos-1 (1 - 2 E )
(7.12), (7.13)
Comparing results with Example 7.3 Example 7.3
e l , mm &,ax, T\J %lax,mm
+-;
t-0,0406 (t-0.0016 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 possible-a requirement for pinpoint navigation or guidance.Also, from Fig.10.10 it can be noted that improving the rigidity of the bearing, that is, decreasing 8; and 8; collectively, reduces the magnitude of 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. One-to-one ratios can be obtained by using bearings with contact angles that are 30" or higher. At these high angles, the sensitivity of the axial-to-radial 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 angular-contact 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. ~ ~ Ball-raceway 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 one-half of the angle subtended by the chord 2a. The angle eo is given approximately by €lo
=
cos-1 (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 thrust-loaded 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 If-aligning 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). Present-day 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 = cos-1 (1 - do - dl0 =
cos-l
(
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. 49-51 (1959). 10.2. T. Harris, "HOWto Compute the Effects of Preloaded Bearings," Prod. Eng. 84-93 (July 19, 1965).
This Page Intentionally Left Blank
Symbol
Description Distance to load point from right-hand 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 Load-deflection 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 mm-l (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 shaft-bearing system involvestwo angular-contact ball bearings or tapered roller bearings mounted in a back-to-back 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 back-to-back bearing mounting has a greater length between loadingcenters than does a faceto-face mounting. This means that the bearing radial loads will tend to be less for the back-to-back mounting.
~ I 11.1. ~ Rigid shaft ~ mounted E in back-to-back angular-contact ball bearings subjected to combined radial and thrust loading.
11.2. Rigid shaft mounted in back-to-back 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 back-to-back 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 back-to-back mounting of Fig. 11.1are, respectively, 318 and 218 angular-contact 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" angular-contact 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 two-bearing-shaft system, flexure of the shaft induces momentloads %& at non-self-aligning 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 F1i-F2--P=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 two-bearing-shaft system.
~
~
E Id "Y T dx
=
"M
(11.8)
in which E is the modulus of elasticity, I is the shaft cross-section moment of inertia, and y is the shaft deflection at the section. For shafts 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 two-bearingshaft 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 two-bearing-shaft system forces and moments acting on a secti.on to the right of the load application.
5
+ P-2 [x(x - 2a) - Z(Z
- 2a)l
+ E102
(11.18)
+ P-6 [ 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 two-bearing 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 + k-n
~ ~ . ~ .
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
<ar
x
Note that Sa
=
0, therefore,
+-
s,
4dmKnA1*'
+
[(sin a' + ;T, + %3 cos +)2 (cos a' + cos +)21112
*=O
(b)
sa= 0.
{[(sina' + ;T, + %3cos +I2 + X(cos(sina' a'+ + S,cos++ %3 ) y 2 - 1)1.5 COS +) COS
*=rtw
-
(7.105)
+
s,
(7.106)
=o { T(0.1305
+ 32.753 cos +)2
+ (0.9914 + Sr cos
*'tP
%L -
11.77 Kn
*=2
0
+)2]1/2 - l}1.5 (0,1395 + 32.753 cos $1 cos =0 [(0.1305 + 32.75 3 cos $1' + (0.9914 + Zr cos +)2]1/2
X
+
(c)
In the preceding equations (b) and (e), Kn is determined as follows:
in which
and
2yl
L p , =1) 1(4--+ fi - 1 - Y To determine
oneneeds F ( p ) ,
(2.29)
Y=-
1) cos A!
dm
(2.27)
A similar procedure is followed to determine KO.It can be seen that Ki and KOdepend on the contact angle.If this is significantly different at each ball location, that is, at $ = 0", 40°, 80", 120°, and 160", Kn is different at each location and must be included in the summation terms of equations (b) and (c). (Example 7.1 indicates the method of calculation of ICn when a is constant.) In equations (b) and (c),
"
isr=-="=:
ar
A
1.969
-t ) =0- = - =
A
1.969
0.508 8,
(7.92)
0.508 0
(7.93)
Equations (a)-(c) constitute a set of simultaneous nonlinear equations in unknowns Sr, 0, and 9~By using the Newton-Raphson method the equations may be solved. The answers are
ar = 0.02073 mm (0.000816 in.) 8 = O0l1' %=
6.804 x
lo4 N
0
mm (602 in. * lb)
The corresponding loaddistribution and contact angles are
0 40" 80" 120" 160"
18'36' 16"18' 10'3' 0 0
3540 (795.5 lb) 2213 (497.2 lb) 194 (43.6 lb) 0 0
00
STATICALLY ~ E T E ~ I N A TS E
~
- S Y S TB E~S
Additionally, Figs. 11.6- 11.8 show Sr, 8, and %Z for various combinations of shaft hollowness and spanbetween bearings.
T
M e n the shaft is rigid and the distance between bearings is small, the influence of the shaft deflection on the distribution of loading between the bearings may be neglected. An application of this kind is illustrated by Fig. 11.9. In this system, the an~lar-contact ball bearings are considered as one double-row bearing. The thrust load acting on the double-row bearing is the thrustload Pa applied by the bevel gear. To calculate the magnitude of the radial loads Fr and Fr3, the effective location of Fr must be determined. Fr acts at the center of the double-row bearing only if Pa is nil. Q.~ZQ
I
1
1
0.0018 -
‘ 40.05
1 50.8mm (2in.)
0.04
0.03
0.02
0.01
0
-0.01
0 F
.
10
20
30
40 50 60 ~oliowness,percent
70
0
90
100
Bearing radial deflection vs shaft hollowness and span, 209 radial ball bearing, 13,350 N (3000 lb) at rnidspan.
~
~
0.001
0
20
40 60 Hollowness, percent
80
100
F
~ 11.7. ~Bearing ~ misalignment E anglevs shaft hollowness and span, 209 radial ball bearing, 13,350 N (3000 lb) at midspan,
If a thrust load exists, the line of action of the radialload Fr is displaced toward the pressure center of the rolling element row which supports the thrust load. This displacement may be neglected only if the distance I between the center of the double-row ball bearing set andthe roller bearing is large compared with the distance b. Using the X and Y factors (see Chapter 18) pertaining to the single-row bearings, Fig. 11.10 gives the relative distance b,/b as a function of Fay/Fr(1- Jl). The X and Y factors for Fa/Fr > e should be used. In the bearing mounting of Fig. 11.9, the tangential gear load is Pt = 7000 N, the radial plane gear separating force is Pr = 2300 N, and theaxial load Pa = 2000 N applied at the gear mean
Hollowness, percent
E 11.8. Bearing moment load vs shaft hollowness and span, 209 radial ball bearing 13,350 N (3000 lb) at midspan.
b"--b
I-
11.9. Three-bearing-sha~system with rigid shaft.
Fa y F A 1 -x>
11.10. b,lb vs F,YIFr(l - X ) for a double-row bearing.
pitch radius rrnp= 75 mm. The 200 series, angular-contact ball bearings have 40" contact angles. The span I = 125 mm, and the gear center is located a distance I , = 50 mm from the center of the cylindrical roller bearing. Estimate the loads acting on each bearing in the mounting. The span between the angular-contact ball bearing pressure centers is b = 75 mm. For a 40" contact angle bearing, X = 0.35 and Y = 0.57 (see Chapter 18). Assuming Pr and Pa act in thexy plane, and Pt acts in thexx plane, where the shaft axis is coincident with the x axis, and considering the an~lar-contactbearing set asa double-row bearing-that is, force Fr acts at the symmetry point 0-then the loading on that bearing is as follows: (4.30) =
1 50 Pr + rrnp Pa = -* 2300 125 I I
Frz= P =
Fr
75 +125
*
2000
=
a
-
I
50 Pt 4 - = 7000 * 1 125
=I:
2800 N
+ F:z)1'2 = (21202 + 28002)1'2= 3512 N
=
2120 N
(F;
(4.12)
4
STATICALLY ~ E T E ~ A SHAFTT E
Fay __ 2300 0.57 = 0.574 Fr(l- X ) 3512(1 - 0.35) From Fig. 11.10,b, = 0.23b. Therefore, F, acts closer to the external load Pr than was assumed, it must be greater than3512 N, and b, will be somewhat greater than 0.23b. Assume b, = 0.27b = 0.27 50 = 13.5 mm. The distance between forces Fr3and Fris 125 - 0.5 75 + 13.5 = 101 mm. Recalculating the ball bearing loads: 1 "p, + %pa
Fv
Frz= Pt A! I F,
1
1
=
(F;
==
=
50 2300 101
- e
+75 101
*
2000
=
2624 N
50 7000 * - = 3465 N 101
+ F & ) l I 2 = (26242 + 346Ij2)lI2= 4346 N (976.6 lb) Fay - 2300 0.57 = o.463 Fr(1 - X ) 4356(1 - 0.35)
From Fig. 11.10, 6, =s 0.27b, therefore the solution is adequate. The load on the cylindrical roller bearing is obtained as follows: (4.29)
F 3 v = ( l - + ) P , . - - rmP P1a = =
1"-
2300
-
101 75 * 2000
324 N
F2== (1 =
( 15001)
+)
Pt = (1 -
""> 101
*
7000
=
3535 N
(4.7)
(F&,+ F&.J1I2 = (3242 + 35352)1/2= 3550 N (797.7 lb) (4.12)
The generalized loading of a three-bearin~-sha~support system is illustrated by Fig. 11.11a.
This system may be reduced to the two systems of Fig. 11.l l b and analyzed according to the methods given previously for a two-bearing, nonri~d-shaftsystem provided that
9% -
q=
Hence from equations (11.24)-( ll.27),
6EI "
IS 1
k=n
- 2EI [26,
4
k=n
+ O2 +
(11.32)
(11.33)
(11.34)
07
(11.35)
An example of the utility of the generalized equations (11.30)-( 11.35) is the system illustrated by Fig. 11.12. For that system it is assumed that moment loads are zero and that the diflerences between bearing radial deflections are negligibly small. Hence, equations (11.30)--(11.35) become F,
=
P(1,
20,
-
a)2(11+ 2a) 1::
6EI 14 (01 + 02)
(11.36)
"
+ 02 = Pa(ll 2EI1,
(11.37)
(11.39)
F3 =
6EI(O2 + 03) 1;
(11.40)
e2 + 203 = 0
(11.41)
Equations (11.37), (11.39), and (11.4) can besolvedfor 64, 02, and 03. Subsequent substitution of these values into (11.36), (11.3$),and (11.40) yields the following result:
.",%
F2
F1
F
12
-
I 11.12.~ Simple~three-bearing-shaft ~ system.
F3
40
STATICALLY~ E T E R ~ A ST E
Fl
(11.42)
=
F2 =
Pa[(Zl- z212
- a2 -
(11.43)
2z :z2
F3 =
(11.44)
Rigid Shafts. When the distances between bearings are small or the shaft is otherwise very stiff;the bearing radial deflections determine the load distribution among the bearings. From Fig. 11.13 it can be seen that by considering similar triangles
This identical relationship can be obtained from equations (11.30)(11.35) by setting I to an infinitely large value. From equation (7.4),
For a bearing with rigid rings, the maximum rolling element load is directly proportional to the applied load and the maximum rolling element deflection determined the bearing deflection; therefore,
F
=
Ira:
(11.46)
Rearranging equation (11.46), (11.47)
Substitution of equation (11.47) into (11.45) yields
12
h
FIGURE 11.13. Deflection of three-bearing-shaft system due to rigid shaft.
Equation (11.48) is valid for bearings that support a radial load only. More complex:relationships are required in thepresence of simultaneous thrust andlor moment loading. Equation (11.48) can be solved simultaneously with the equilibrium equations to yield values of F,, F2, and F3.
.3
127 m m
127 m m (5in.)
(5in.)
1.5. Three identical ball bearings are mounted on a shaft and located 127 mm (5 in.) apart. A load of 44,500 N (10,000 lb) is supported 50.8 mm (2 in.) from the centrally located bearing. Determine the radial load on each bearing if the shaft may be considered rigid. ZFr=O F,
+ F2 + F3 - 44,500 = 0 XM3=0
F,
X
254
+ F2 X
127
-
44500
X
177.8 = 0
254F1 3- 127F2 - 7.912 X
But . K l = IC2
=
K3 and I,
=
I,. Therefore,
F 0 . 6 7 __ 1
2F0.67 + F0.67 2
3
=
0
Solving equations (a)-(c) simultaneously yields
lo6 = 0
$ T A T I C ~ L Y~ E T E
G $YSTE~S
F, = 24,030 N (5400 lb)
F2 = 14,240 N (3200 lb) F3 = 6230 N (1400 lb)
Equations (11.30)-(11.35) inclusive may be used to determine the bearing reactions in a multiple bearing system such as that shown in Fig. 11.14 having a flexible shaft. It is evident that thereaction at any bearing support location h is a function of the loading existing at and in between the bearing supports located at h - 1 and h + 1. Therefore, from equations (11*30)-(11.35),the reactive loads at each support loeation h are given as follows: F h
(11.49)
1
k=IJ
1
k=a
(11.50) For a shaft-bearing system of n supports, that is, h = n, equations 1.50) represent a system of 2n equations. In the most elementary case, all bearings are considered as beingsufficientlyselfaligning that all 9xh equal zero; furthermore, all Srh are considered negligible compared to shaft deflection. Equations (11.49) and (11.50) thereby degenerate to the familiar equation of “three moments.” It is evident that thesolution of equations (11.49) and (11.50)to obtain bearing reactions and Fh depends on relationships between radial load and radial deflection and moment load and misalignment angle for each radial bearing in thesystem. These relationships have been defined in Chapters ‘I and 9. Thus, for a very sophisticated solution to a shaftbearing problem as illustrated in Fig. 11.15 one could consider a shaft having two degrees of freedom with regard to bending, that is, deflection earing location
earing I~atioR
h+l
h
2
h
h+l
x
pih
Fy,h+l
11.15. System loading in three dimensions.
412
STATICALLY ~ E T E ~ ~ A S T E~
- SYSTEMS B
in two of three principal directions, supported by bearings h and accommodating loads k. At each bearing location h7 one must establish the following relationships:
To accommodate the movement of the shaft in two principal directions, the following expressions will replace equations (9.4) and (9.5) for each ball bearing (see Jones El1.111):
Sxj = L3.D sin a' Szj = BD cos a'
+ 6%+ Om& sin Itj + + tiy sin Itj + tij cos Itj
cos JIj
(11.55) (11.56)
For most rollingbearing applications, it is sufficient to consider the shaft and housing as rigid structures. As demonstrated by Example 11.3,however, when the shaft is considerably hollow and/or the span between bearing supports is sufficiently great, the shaft bending characteristics cannot be considered separately from the bearing deflection characteristics with the expectation of accurately ascertaining the bearing loads or the overall system deflection characteristics. In practice the bearings may be stiffer than might be anticipated by the simple deflection formulas or even stiffer than a more elegant solution that employs accurate evaluation of load dist~butionmight predict for the assumed loading. The penalty for increased stiffness will be paidin shortened bearing life since the improved stiffness is obtained at the expense of induced moment loading. It is of interest to note that the accurate determination of bearing loading in integral shaft-bea~n~-housingsystems involves the solution of many simultaneous equations. For example, a high speed shaft supported by three ball bearings, each of which has a complement of 10 balls, the shaft; being loaded such as to cause each bearing to experience five degrees of freedom in deflection, requires the solution of 142 simultaneous equations, most of which are nonlinear in the variables to be determined. Most likely, the system would include some roller bearings, these having complements of 20 or more rollers per row, thus adding to the number of equations to be solved simultaneously. Fu~hermore, the
~
REFERENCE
413
bearing outer rings and/or inner rings may be flexibly supported as in aircraft power transmissions, adding to the complexity of the analytical system and difficulty of obtaining a solution using numerical analysis techniques such as theNewton-Raphson method for simultaneous, nonlinear equations.
11.1. A. Jones, “A General Theory for Elastically Constrained Ball and Radial Roller Bearings under Arbitrary Load and Speed Conditions,” ASME Truns., J. Basic Eng. 82, 309-320 (1960).
This Page Intentionally Left Blank
Symbol
a A b
c L)
dm E E'
f
F
Units Semimajor axis of elliptical contact area ~iscosity-temperature calculation constants Semiwidth of rectangular contact area, semiminor axis of elliptical contact area Lubrication regime and film thickness calculation constants Roller or ball diameter Pitch diameter of bearing Modulus of elasticity E / ( l - 52) r/L) for ball bearing Force
mm (in.)
mm (in.)
mm (in.) mm (in.) N/mm2 (psi) N/mm2 (psi)
416
LUBRICANT FILMS IN ROLLING E L E ~ N T - ~ C E ~CONTACTS AY
Symbol
Description Centrifugal force FlE' %, Gravitational constant @' Shear modulus Film thickness ~ i n i m u mfilm thickness hl% Viscous stress integral Polar moment of inertia per unit length
JAY& Lubricant thermal conductivity Roller effective length Factor for calculating film thickness reduction due to thermal effects ~oment Speed Pressure Force acting on roller or ball &/E'% Cylinder radius Equivalent radius rms surface finish (height) Saybolt university viscosity Time Lubricant temperature Fluid velocity Entrainment velocity (U, + u2)
7)oU/2Et% Fluid velocity, displacement in y direction Sliding velocity ( V , - U2) ~0V/Et% Deformation in X direction Distance in y direction Distance in x direction Coefficient for calculating viscosity as a function of temperature
Units
mm/sec2 (in./sec2) N/mm2 (psi) mm (in.) mm (in.)
N * sec2 (lb sec2) mm sec2 (in sec2) Wlm * "C (Btulhr * in.
0
OF)
mm (in.)
mm (in.) mm (in.) mm (in.) sec see "C, OK ( O F , OR) mmlsec ( i n h e c ) mmlsec (in./sec) mmlsec, mm (inhec, in.) mmlsec (inhec) mm (in.) mm (in.) mm (in.)
417
Symbol Y Y
I: A E
‘I rh, ’Ieff
‘IO K
h
A vb
s P
0-
7
e
-
Y cp
Q,
Ict w
b e
G i j m
NN 0
R S SF
Units
Description
aid, Lubricant shear rate Surface roughness orientation parameter Surface roughness parameter Strain Lubricant viscosity Base oil viscosity (grease) Effective viscosity (grease) Fluid viscosity at atmospheric pressure Ellipticity ratio a / b Pressure coefficient of viscosity Lubricant film parameter Kinematic viscosity Poisson’s ratio Weight density Normal stress Shear stress Angle Factor to calculate ‘prs Film thickness reductiotn factor Factor to calculate cps Angular location of roller Rotational speed
L) cos
SUBSCRIPTS Refers to entrance to contact zone Refers to exit from contact zone Refers to grease Refers to inner raceway film Refers to roller location Refers to orbital motion Refers to non-Newtonian lubricant Refers to outer raceway film Refers to roller Refers to lubricant starvation Refers to surface roughness (finish)
sec-l mm (in.) mm/mm ( i d i n . ) cp (lb sec/in.2) cp (lb s e c / h 2 ) cp (lb s e c / h 2 ) cp (lb sec/ine2) mm2/N ( h 2 / l b ) stokes (cm2/sec) g/mm3 (lb/in.3) N/mm2 (psi) N/mm2 (psi) rad
rad rad/sec
41
L
~
R
I FILMS C ~ IN ROLLING E L E ~ N T - ~ C EC ~O A ~ A~C T S
Symbol T TS X
Y
Units Refers to temperature Refers to temperature and lubricant starvation Refers to x direction, that is, transverse to rolling Refers t o y direction, that is, direction of rolling Refers to x direction Refers to rotating raceway Refers to nonrotating raceway Refers to minimum lubricant film Refers to contacting bodies
Ball and roller bearings require fluid lubrication if they are to perform satisfactory for long periods of time. Although modern rolling bearings in extreme temperature, pressure, and vacuum environment aerospace applications have been adequately protected by dry film lubricants, such as molybdenum disulfide among many others, these bearings have not been subjectedto severe demands regarding heavy loadand longevity of operation without fatigue. It is further recognized that in the absence of a high temperature environment only a small amount of lubricant is required for escellent performance. Thus many rolling bearings can be packed with greases containing only small amounts of oil and then be mechanically sealedto retain the lubricant. Such rollingbearings usually perform their required functions for indefinitely long periods of time. Bearings that are lubricated with escessive quantities of oil or grease tend to overheat and “burn” up. The mechanismof the lubrication of rolling elements operating in concentrated contact with a raceway was not established mathematically until the late 1940s; it was not proven experimentally until the early 1960s. This is to be compared with the existence of hydrodynamic lubrication in journalbearings, which knowledge was established by Reynolds in the 1880s. It is known, for instance, that a fluid film completely separates thebearing surface from the journalor slider surface in a properly designed bearing. Moreover, the lubricant can be oil, water,gas, or some other fluid that eshibits adequate viscous properties for the intended application. In rolling bearings, however, it was only relatively recently established that fluid films could, in fact, separate rolling surfaces subjected to estremely high pressures in the zones of contact. Today, the
existence of lubricating fluid films in rolling bearings is substantiated; in many successful applications, these films are effective in completely separating the rolling surfaces. In this chapter, methods will be presented for the calculation of the thickness of lubricating films in rolling bearing applications.
Because it appeared possible that lubricant films of significant proportions do occur in the contact zones between rolling elements and raceways under certain conditions of load and speed, several investigators have examined the hydrodynamic actionof lubricants on rolling bearings according to classical hydrodynamic theory. Martin E12.11 presented a solutionforrigidrollingcylinders as early as 1916. In 1959, Osterle E12.21 considered the hydrodynamic lubrication of a roller bearing assembly' It is of interest at this stage to examine the mechanism of hydrodynamic lubrication at least in two dimensions. Accordingly, consider an infinitely long roller rolling on an infinite plane and lubricated by an incompressible isoviscous Newtonian fluid having viscosity q. For a Newtonian fluid, the shear stressr at any point obeys the relationship au
(12.1)
r=-77az
in which aulaz is the local fluid velocity gradient in the z direction (see Fig. 12.1). Because the fluid is viscous, fluidinertia forces are small compared to the viscous fluid forces. Hence a particle of fluid is subjected only to fluid pressure and shear stresses asshown in Fig. 12.2.
u=o
12.1. Cylinder rolling on a plane with lubricant between cylinder and plane.
420
LUBRICANT FILMS IN ROLLING E ~ ~ ~ T - ~ CCONTACTS E ~ A Y
12.2. Stresses on a fluid particle in a two-dimensionalflow field.
Noting the stresses of Fig. 12.2and recognizing that static equilibrium exists, the sum of the forces in any direction must equal zero. Therefore,
and (12.2) ~iffe~entiating equation (12.1) once with respect to x yields (12.3) ~ubstitutingequation (12.3) into (12.2) one obtains (12.4) Assuming forthe moment that splay is constant, equation (12.4) may be integrated twice with respect to z. This procedure gives the following expression for local fluid velocityu: (12.5) The velocity 17 may be ascribed to the fluid adjacent to the plane that
~
R
Q
421
D L ~~R I C ~ A TCI Q ~
translates relative to a roller. At a point on the opposing surface, it is proper to assume that u = 0, that is, at x = 0, u = T/ and at x = h, u = 0. ~ubstitutingthese boundary conditions into equation (12.5),it can be determined that
(12.6)
in which h is the film thickness. Considering the fluidvelocities surrounding the fluid particle as shown by Fig. 12.3, one can apply the law of continuity of flow in steady state, that is, mass influx equals mass efflux. Hence, since density is constant for an incompressible fluid
Therefore,
(12.8)
Differentiating equation (12.6) with respect to y and equating this to equation (12.8) yields
v
F
~ 12.3 GVelocities ~ associated ~ with a fluid particle in a two-dimensional flow field.
42
L
~
R
I FILMS C ~ IN ROLLING E L E ~ ~ - ~ C CONTACTS E ~ A ~
(12.9) Integrating equation (12.9) with respect to x,
(12.IO)
and (12.11) Equation (12.11) is commonly called the Reynolds equation in two dimensions.
To solve the Reynolds equation, it is only necessary to evaluate film thickness as a function ofy, that is, h = h(y).For a cylindrical rollernear a plane as shown by Fig. 12.4, it can be seen that
121.4. Film thickness hCy) in the contact between a roller and plane.
423
h=ho-t”’
(12.12)
m
in which hO is the minimum film thickness. Substituting equation (12.12) into (12.11) gives (12.13) Equation (12.13) varies only in y ; hence (12.14)
The integration of equation (12.14) to determine p ( y ) is rather intricate. This equation was integrated by Sterniicht et al. E12.31 as follows:
+-
4
32
(12.15)
and pa,,
=
0.76rlU(2R)1’2(h0)-3’2 (12.16)
In equation (12.15), tan /3 = y ~ ’ ( Z R h ~ ) and l ’ ~ K is a constant ~ e ~ e ~ d i n ~ on the boundary condition. If both surfaces are construed to be rotating cylinders, then
u = u1+ uz
(12.17)
in which subscripts 1 and 2 refer to the respective cylinders. Moreover, an equivalent cylinder radius % is defined as $8, = (R,1
+ R,1)-1
Note that for an outer raceway R i l is negative. To find the load which will be carried,
(12.18)
424
LUBRICANT FILMS IN ROLLING E L E ~ ~ - ~ C E CON'I'ACTS ~ A S
&'
=
P ( Y ) dY
(12.19)
in which &' is the supported load per unit length of the cylinder. Unfortunately, the film thicknesses indicated by subsequent calculations are far smaller than the composite surface roughness achievable on rolling bearings. Thus, it is apparent that hydrodynamic lubrication by an isoviscous fluid alonecannot explain the existence of a satisfactory fluid film in the contact zone between rolling element and raceway. The foregoing analysis represents a limiting solution for light loads.
IS scosit~~ ~ r i a twith i o ~Pressure According to the sample calculations of compressive contact stress in Chapter 6, the normal pressure between contacting rollingbodies is likely to be of the magnitude of 700 N/mm2 (100,000 psi) and higher. In normal fluid filmbearing applications pressures of this magnitude do not exist and, consequently, it is usually assumed that viscosity is unaffected by pressure. Figure 12.5 shows someexperimental data on viscosityvariation with pressure for different bearing lubricants. It is to be noted that fluid viscosity is an exponential function of pressure such that between the contacting surfaces in a loaded rolling bearing assembly, viscosity can be several orders of magnitude greater than its value at zero pressure. In 1893, Barus [12.4] established an empirical equation to describe the variation of viscosity with pressure for a given liquidat a given temperature; an isothermal relationship. Barus's equation is usually stated as follows: q = q0e@
(12.20)
In equation (12.20), the viscosity-pressure coefficient h is a constant at the given temperature. In 1953, an ASME [12.5] study published viscosity vs pressure curves for various fluid lubricants. Based on the ASME data, it is apparent that theBarus relationship is a crude approximation since the viscosity-pressure coefficient tends to decrease with both pressure and temperature for most fluid lubricants. It has been established, however, that thelubricant film thickness that obtains in a concentrated contact is a function of the mechanical properties of the lubricant entering the contact. Therefore, for the purpose of determining the thickness
Pressure, psi x 1000
~
I 12.6. ~Pressure ~ viscosity E of lubricants (ASME data 112.51).
of the lubricant film, the viscosity-pressure coefficient at atmospheric pressure has been utilized. Roelands [12.6] later established an equation defining the viscositypressure relationship for given fluids,but including the influence of temperature on viscosity as well.
fin (12.21), pressure is expressed in kgf/cm2 and temperature i ponents So and z are determined empirically foreach lubricant pressures, equation (12.21) indicates viscosities substantially lower than produced using the Barus equation (12.20). ab and VanArsdale fl2.71 developed an expression for viscosity vs pressure and temperaturethat can be applied to several of the l u ~ r i c ~ n t s employed in the ASME E12.151 study.
42
L
~
R
I FILMS C ~ IN ~
= Al(E -
ln
-
+ A,(”
-
L E LL E~ ~ ~E ~ - ~ C E
+ A 2 ( 2 - 1)
1)
TO
+ A3(:
O
1)p + 1)
Po
2
A4(2
(2
-
-
1)
(12.22)
1)
In (312.221, temperature is stated in OK. Reference [12.6] providesvalues of the coefficients 4 for the various lubricants tested in reference [12.153. As an example, the coefficients for the diester fluid viscosityvs pressure curve of Fig. 12.5 are:
A,
11.78
A3 -7.7 A,
*
14.31
This fluid may be considered representative of an aircraft power transmission fluid lubricant. Sorab and Vanhsdale [12.7] demonstrate that equation (12.22) is superior to the Roelands equation in approximating the ASME viscosity-pressure-temperature data. Nevertheless each of the approximations has only been demonstrated over the 0-1034 MPa (0-150 kpsi) pressure range and 25-218°C(77-425°F) temperature range of the ASME data. Contact pressures and temperatures in many ball and roller bearing applications exceed these ranges; therefore, it becomes necessary to extrapolate these data s~bstantiallybeyond the range of the experimentation. This is not critical for the determination of lubricant film thicknesses. In the estimation of bearing friction, however, lubricant viscosity at pressures higher than 1034 MPa and at temperatures greater than 218°C has a great influence on the magnitudes of friction forcescalculated and hence on the accuracy of the calculations. Harris E12.81 introduced the use of a sigmoid curveas defined by equation (12.23) to fit the ASME E12.51 data. (12.23) In (12.23) C,
. . . 6,
are constants determined from the curve-fitting
procedure for a given lubricant at a given temperature. Figure 12.6 illustrates the sigmoid curves for the ASME data for a Mi~-L-7808estertype lubricant at 37.8, 98.9, and 218.3"C (100, 210, and 425°F). The salient featureof the sigmoid viscosity vspressure curve is the virtually constant viscosity value at extremely high pressures. As noted by .9, 12.101, the fluid in a high pressure, concentrated contact undergoes transformation to a gZassy state; Le., the fluid essentially becomes a solid during its time in the contact. It therefore appears reasonable to assume that fluid viscosity becomesessentially constant with pressure during its time in the contact. To accurately predict bearing friction torque, this becomes an important consideration forthe use of a sigmoid curve to describe lubricant viscosity in the contact. Conversely, using a sigmoid curve to approximate lubricant viscosity at atmospheric and low pressures does not provide the accuracy of either the Roelands [12.61 or Sorab et al. E12.71 model. Either of these models may be used in the estimation of lubricant viscosity to calculate lubricant film thickness.
Because of the fluid pressures present between contacting rolling bodies causing the increases in viscosity noted on Fig. 12.5, the rolling surfaces
0
1000
2000
3000
4000
5000
Pressure - MPa 12.6. Viscosity vs pressure and temperature Mil-L-7808 ester-type lubricant (sigmoid curve fit to ASME data-extrapolation from 1000-4000 MPa).
428
L
~
R
I FILMS C ~ IN ROLLING E L E ~ ~ T - CONTACTS ~ ~ ~ ~ A Y
deform appreciablyin proportion to the thickness of a fluid film between the surfaces. The combinationof the deformable surface with the hydrodynamic lubricating action constitutes the “elastohydrodynamic” problem. The solution of this problem established the first feasible analytical means of estimatin~ the thickness of fluid films,the local pressures, and the tractive forces that occur in rolling bearings. Dowson and Higginson [12.11] for the model of Fig. 12.7 used the following formulation for film thickness at any point in the contact: Y2
Y2
m1
m 2
h=h*+-+-
+ ut1 + ut2
(12.24)
Solid displacements w are calculated for semiinfinite solid in a condition of‘ plane strain. Since the width of the loaded zone is extremely small compared to the dimensions of the contacting bodies, an approximation that w1 = u t 2 is valid. Hence for the equivalent cylinder radius
the film thickness is given by
.7.
Forces and velocities pertaining to the equivalent roller.
429
Y2
h=hO+-+w 2%
(12.25)
To solve the plane strain problem, the following stress function was assumed: (12.26) Using this stressfunction, the stressesdue to a narrow strip of pressure over the width ds in the y direction are determined as follows: = -
2y2zp ds d y 2 + x2)2
(12.27)
a, = -
2z3p ds d y 2 + x2)2
(12.28)
2yz2p ds d y 2 + z2)2
(12.29)
cry
Tyz
- -
cry and crz are normal stresses andT~~is the shear stress.By Hooke's law, the strains aregiven by
(12.30) (12.31) (12.32)
I in which G is the shearmodulus of elasticity and ( is Poisson's ratio. In plane strain
E
=
av
-,
aY
E, =
aw ax
-,
and
eYz =
av aw -+ ax ay
Using these relationships, and equations (12.27)-(12.32), it can be established that at the surface, that is, at z = 0:
43
L
~
I FILMS C ~ IN R
R
;1
m = - -2(1 -
TE
O E L E~ ~ ~ - ~ C E ~~ A Y
p In ( y - S ) d S
+ constant
C O ~ A C T
(12.33)
To solvefor UI, Dowson et al. [12.11] divided the pressure curve into segments and represented the pressure thereunder by p
= 61
+ J2S + 13S2
(12.34)
in which 11,J2, and l3are constants for that segment. Using p in this form, equation (12.33) can be integrated to obtain surface deformation. This procedure, of course, is used for an assumed pressure distribution. To obtain ho, the Reynolds equation is used in accordance with the pressure variation of viscosity. (12.35)
Performing the indicated diffe~entiatio~ and rearranging yields:
At the inlet and at the outlet of the contact: (12.37)
such that equation (12.36) becomes (12.38)
At the outlet end of the pressure curve dh/ dy = 0. This condition applies to the point of minimum film thickness. At the inlet, equation (12.38) is solved by (12.39)
Thus, if viscosity and speed are known, the value of h for the point at which equation (12.38) is satisfied in the inlet region can be evaluated for a given pressure curve. Solving (12.39) for h, (at inlet) gives
(12.40) Once h, has been determined, then the entire film shape can be estimated by using the integrated form of the Reynolds equation, that is, (12.41) ~ubstitutionof d p / d y from equation (12.39) for the point at which h = h, determines that he = 2hb/3, At other positions y , film thickness h may be determined from the following cubic equation developed from (12.41)
At the point of maximum pressure d p / d y = 0 and equation (12.36) becomes (12.43) In cases of most interest the pressure curve is predominantly Hertzian such that P =Po [I -
(s)l”~
(12.44)
in which p o is the maximum pressure and b is the semiwidth of the contact zone. Thus at y = 0, p = p o and equation (112.43) becomes -d = h
dy
p0h3 3q0eAP0U~2
(12.45)
~onsequently,if h is small (as it must be in arolling bearing under load) and viscosity is high (as itwill become becauseof high pressure), d h / d y is very small and the film is essentially of uniform thickness. This result owson et al. [12,11],and also by Grubin E12.121.
In a laterpresentation Dowson et al. E12.131 indicated that dimensionless film thickness H = h / %could be expressed as follows:
432
LUBRICANT FILMS IN ROLLING E
~
~
~
- C~ O~ T C ACTS E
where (12.47) (12.48) 8 = =
mt
(12.49)
E 1- 52
(12.50)
In the expression for H and in equations (12.47 ) and (12.48), the equivalent radiusin thedirection of rolling fora ball or roller bearing is given by:
13
9 l = - (1 T y )
2
(12.51)
In (12.51), the upper sign refers to the inner raceway contact and the lower sign to the outer raceway contact. The velocitieswith which fluid is swept into the rolling element-racewaycontacts is given by equations (12.52) and (12.53) forthe inner andouter raceway contacts respectively. (12.52)
Dowson et al. [12.13] presented the results shown by Figs. 12.8and 12.9 for 9 = 2500 and 5000 corresponding to bronze rollers and steel rollers = 0.00003correrespectively lubricated by a mineral oil. Theload sponds to approximately 483 N/mm2 (70,000 psi) and QZ= 0.0003 corresponds to 1380 n/mm2 (200,000 psi) approximately.Dimensionless corresponds to surface velocities on the order of 1524 speed = mmlsec (5 ftlsec) for an equivalent roller radius of 25.4 mm (1in.) operating in mineral oil. Note from Figs. 12.8 and 12.9 that the departure from the Hertzian pressure ~istributionis less significant as load increases. The second pressure peak at the outlet end of the contact corresponds to a local decrease in the film thickness at that point. Otherwise, the film is es-
~
~
IS0
33
-1
0
1
0
-1
1
Ylb
Ylb 14
12
2 10 G 8 6
4
2 -1
1 0
1
Ylb
- =(b)3 X8= 2500zi= IO+ 12.8. Pressure distribution and film thickness for high loadconditions (reprinted from [l2.13]by permission of the Institution of Mechanical Engineers).
i
sentially of uniform thickness. The latter condition was confirmed by tests conducted by Sibley and Orcutt E12.141. ~dditionally,Dovvson et al. E12.131 demonstrated the effect of the distorted pressure distribution on maximumsubsurface shear stress.Figure 12.10 shows contours of ~ ~ ~Note~that~the /shear p stress ~ increases ~ . in thevicinity of the second pressure peak and tends towards the surface. This condition was indicated in Chapter 6.
Grubin E12.121 developed a formula for minimum film th contact, that is, the thickness of the lubricant film be berance at thetrailing edge of the contact on the e~uivalent roller surface and the opposin~ surface of the relative flat. The Grubin formula is base
43
S IN ROLL^^ ELE
-2
-1
1
0 Ylb
l2
"2
r
0 1 Yib (a) (& = 5000 ;i5r = 3 X 10-5, i7= 10-11
-1
-2
-1
0
1
Yl b Zjz=
(b) 8 = 2500 3X i7=
10-11
12.9. Pressure distribution and film thickness for light load conditions (reprinted from 112.131 by permission of the Institution of Mechanical Engineers).
on the assumption that the rolling surfaces deform as if dry contact occurs and is given in dimensionless format. (12.54) where H o = ~ O / ~ ~ . Based uponanalytical studies and experimental results, Dowson et al. C12.151 established the following formula to calculate the minimum film thickness:
(12.55)
A s i ~ i f i c a nfeature t of both equations is the relatively large dependency of film thickness on speed and lubricant viscosity and the comparative insensitivity to load. Testing conducted by Sibley et al. C12.141 using radiation techniques seemed to conform the Grubin equation; however, the agreement between the Dowson and Grubin formulas is apparent. Today, owson equation is recommended as representative of line contact lub~cationconditions.
f 3 3 3
43
LUBRICAIYC FILMS IN ROLLING E L E ~ ~ - ~ C E CO ~ ~AA C STS
The foregoing equations describe the minimum lubricant film thickness. The film thickness at the center of the contact is approximated by
Archard and Kirk [12.16] described the minimum film thickness between two spheres as (12.5'1) It is interesting to note once again the relative insensitivityto load. Using a ball-disk test rig (see Chapter 19) with a clear sapphire disk and interferomet~, is it possible to obtain photoflaphs of the lubricant film thickness distribution in a moving ball-disk contact. Figure 12.11 shows the horseshoe pattern corresponding to the high pressure ridge associated with the minimum lubricant film thickness. The central or ~ Z a t e afilm ~ thickness is enclosed by the horseshoe.
.
Photograph of ~uid-lub~cated steel ball-sapphire disk contact.Interferometricfringesindicate variation of film thickness and hence pressure (see Wedeven 112.171).
A more generalized formula for minim~mlubricant film thickness in an elliptical area point contact wassubsequently developed by Hamrock and Dowson E12.181. (12.58)
where qzfor point contact is given by
-
Q
Qz = 5
(12.59)
Sometimes for elliptical point contact an equivalent line contact load is considered as follows: Qez =
3&
(12.60)
In equation (12.58), K is the ellipticity ratio alb. The central or plateau lubricant film thickness is given by (12.61)
Kotzalas E12.191 conducted a study of lubricant film formation using both Roelands equation (12.21) and a fitted sigmoid curve (12.23) to define lubricant viscosity vs pressure at a given temperature. He established that the calculated lubricant film thickness distributions are substantially identical irrespective of which of the two models for viscosity vs pressure is used.
1. The 209 cylindrical roller bearing of Example 9.2 is lubricated by a naphthenic oil having a kinematic viscosity of 100 SSU at operating temperature. Considering that the bearing supports a 4450 N (1000 lb) radial load at a shaft speed of 10,000 rpm, estimate the minimum lubricant film thickness. max
=
1335 N (300 lb)
Fig. 9.11
23 = 10 mm (0.3937 in.)
Ex. 2.7
1 = 9.6 mm (0.378 in.)
Ex. 2.7
Avallone and Marks [12.20] give the following formula for k i ~ e ~ a t i c viscosity in stokes (cm2/sec)for a mineral oil at a nominal temperature:
43
L
~
R
I FILMS C ~ IN ROLLING E
~
~
~
-
CONTACTS ~ C E
(12.62) vb = 0.207 stokes (cm2/sec)
(12.63)
E'
(-)
=
0.1122 0.207
=
0.01934 mm2/N (1.33 X
=
E 1-
(12.50)
52
- 206'g00 1 - (0.3)2 9 =
227,400 N/mm2(33 x
lo6 lblin.2)
M'
(12.49)
=
0.01934
X
=
0.207
0.86
X
1 cm 10 mm
x-------x-
In.
ia2/lb)
= 1.780 X
227,400 = 4398
g 1 X -
cm3
io3 g
l m IO3mm
N sec 0
mm2
2.59
X
lb sec
7 (12.51)
- lo(' - 0'1538! = 4.231 mm (0.167 in.) 2
nm = 0.4231ni
~
A
2T 60
mi - mm = (ni - nm) X =
ni(l
Ex. 9.2
2T 60
- 0.4231) X -
- 1.15471. "
60
IZi
(8.14)
- 6.34871. -
60
ni (12.52)
=
3.324 X
lo4 rnrnlsec (1308 in./sec)
voui u. = 2zr$J$
-
(12.48)
lo-'
3.324 X lo4 2 X 2.274 X lo5 X 4.231
1.780 -
X
X
=
3.075 X
-
1335 9.6 X 2.274 X lo5 X 4.231
=
1.44 X 10-4
(12.47)
=
1.704 X
X
4.231
=
0.000721mm
=
0.721 pm (28.4 X
in.)
The average or “plateau” film thickness is
=
0.000961 mm
=
0.961 pm (37.9 x
lov6in.)
2.2. Estimate the minimum lubricant film thickness assuming the bearing of Example 12.1 is lubricated by a mineral oil having a kinematic viscosity of 40 SSU at operating temperature. (12.62) =
2.26
X
X
32
-
1’95 - 0.01138 cm2/sec 32
( ~ ) 0.163
h = 0.1122
=
(12.63)
0.01206 mm2/N (0.8316 X
ho = 7.209
X
(
~100ssv h40SSU)o*54 0.01206
ia2/lb)
(
%l0OSSU vb-40SSU)o*7
0.54
0.01138
(
=
7.2090
=
(0.01934) 0.207 0.0000733mm = 0.0733 pm (2.89 X
X
0.7
)
in.)
As indicated previously, maximum Hertz pressures occurring in therolling element-raceway contacts typically fall in the range of 1000-2000
ET L
~
R
I ~ CI C ~ T I O ~ ING ~ EFFECT^
MPa (approximately 150-300 kpsi), however, in modern bearing applications, particularly endurance tests, it is not unusual for maximum Hertz pressure to reach 4000 MPa. To prevent damage to laboratory test equipment and the materials under test, experiments used to confirm the lubricant film thickness equations providedabove have typically been confined to pressures not exceeding 1500 MPa. Venner [12.21] conducted EHL analyses at high pressures and concluded that lubricant films predicted by the equations, both minimum and central lubricant film thicknesses, are somewhat thinner than calculated by these equations. Using a tungsten carbide ball on a sapphire disk and ultrathin film interferometry and digital techniques, Smeeth and Spikes L12.221 measured lubricant fill thicknesses at maximum Hertz pressures up to 3500 MPa. They confirmed Venner's conclusions, finding that, above contact loading of 2000 MPa, minimum lubricant film thickness varies inversely as dimensionless load to the 0.3 power as compared to the 0.073 power indicated by equation (12.58). The data shown by Smeeth et al. E12.221 can be represented by equations (12.64) and (12.65). Theseequations define the ratio of film thickness resulting from veryhigh pressure to that calculated using equations (12.58) and (12.61) for minimum and central film thicknesses respectively.
(12.64) hchp
-
0.8736 - 8543
"
lo-'
pim
(12.65)
C ',
At high bearing operating speeds, some of the frictional heat generated in each concentrated contact is dissipated in the lubricant momentarily residing in the inlet zone of the contact. This effect, examined first by Cheng [12.23], tends to increase the temperature of the lubricant in the contact. Vogels E12.241 gives the following expression for viscosity:
in which Tbis in "C and Al, A,, and ,E3 are parameters to be defined for each lubricant. Three temperature-viscosity data points are required to determine Al, A,, and ,E3 as follows:
4
L ~ R I C FILMS ~ T IN ROLLING ELE
A,
=
A3T1
-
T3
1-A3
(12.68) (12.69) (12.70)
If only two temperature-viscosity data points are known and A2 can be fixed to 273, equation (12.66) can be simplified to:
where 1" is now in "E(. and qrefis the absolute viscosity at reference temperature TrepSince Trefis generally room temperature and since T b is usually higher than room temperature, equation (12.71) generally takes the form:
showing that as temperature increases, lubricant viscosity decreases. It is clear that the lubricant film thickness will reduce as a result of temperature increase in the contact. Cheng E12.251 and subsequently urch and Wilson E12.261, Wilson [12.27], and Wilson and Sheu [12.28] developed thermal reduction factors for lubricant film thickness from numerical solutions of the thermal EHL problem for rolling-sliding contacts. Gupta et al. E12.291 recommended the film thickness reduction factor of equation (12.73). 1- 13.2~)L0.42 =
1 + 0.2131. + 2.23S0*83)L0.64
(12.73)
where p o is the Hertzian pressure and dimensionless parameters L and S are defined as follows: (12.74) (12.75) particularly for line contacts, Hsu and Lee[12.30]provided (12.76).
equation
F R l C T I O N ~EATI IN^ EFFECTS 1
. Consider that the lubricant used in the 209 cylinr bearing of Example 12.1 is SAE 10W achievi cosity of 100 SSU at 54.4"C (100°F). According to 112.201, has a viscosity of 45 SSU at 983°C (210°F).What is the average lubricant film thickness in thebearing of Example 12.1 considering thermal effects?
In equation (12.69), assume A, = 273.2'C. Since v = q / p and since in the temperature range considered oil density may be assumed constant, equation (12.69) becomes for T in OK,
-
372.1 * 327.6 (372.1 - 327.6)
qb = 1.780
9
lo-'
I\J sec/mm2
Ex. 12.1
From equation (12.71) q " - - pqb -d = dT T2
3466 0.01780 = -5.749 (327.6),
7 7 = u1 + u2 = 33.22 m/sec (1308 in./sec)
10-4
N sec m2 * "K
~
Ex. 12.1
According to MacAdams 112.311, the thermalconductivity of a mineral oil in the temperature of this example is 0.1385 W/m * "C (0.08 Btu/ h r ft OF) 0
44
(12.74) -
(33.22)' -5.749 4 0.1385
"
lom4= 1.145 Ex. 12.1
9 = 4398 -
Q~= 1.44 * 10-4
Ex. 12.1
For simple rolling u 1 = u2 and from equation (12.75) S
=.
0.
-
1 (12.76) 1 + 0.0766 * (4398)0.687 * (1.44 * 10"4)0.447 (1.145)0*527 * 1
=
0.6657
c _
hci = 0.961 pm (37.9 *
hbi =
&ktci
=
low6in.)
0.6657 * 0.961
=
Ex. 12.1 0.6397 pm (25.2
9
in.)
The basic formulas for calculation of lubricant film thickness assume an adequate supply of lubricant to the contact zones. The condition in which the volume of lubricant on the surfaces entering the contact is insufficient to develop a full lubricant film is called s t ~ r v ~ t Factors ~ o ~ . to determine the reduction of the apparentlubricant film thickness have been developed as functions of the distance of the lubricant meniscus in the inlet zone from the center of the contact. As yet, no definitive equations have been developed to accurately calculate the aforementioned distance; therefore, the meniscus distance has to be determined experimentally. Figure 12.12 illustrates the concept of meniscus distance. References E12.32-12.361 give further detail about this concept. In consideration of the meniscus distance problem, a condition of zero reverse flow is defined. Under thiscondition, the minimum velocity of the point situated at the meniscus distance from the contact center is, by definition, zero. If the meniscus distance is greater, the latter point will have a negative velocity, that is, reverse flow. The zero reverse flow condition is therefore a quasistable situation because no lubricant is lost to the contact owing to reverse flow. In the case of a minimum quantity of lubricant supplied, for example, oil mist or grease lubrication, the lubricant film thickness reduction factor owingto starvation effects, according
S T ~ V A T I O NOF L
~
~
I
C
44
~
(a)
(b)
12.12. Meniscus distance in (a)hydrodynamic and (b) elastohydrodynamic lu-
brication.
to references E12.321 and E12.351, lies between 0.71 (in pure rolling) and 0.46 (in pure sliding). Castle et al. [12.351 give the following equation for line contact: (12.77)
where Yb
”
1 (12.78)
It is clear that @ is zero if the meniscus distance should equal b and in that case 50, = 0. Accordingly, the accurate estimation of the meniscus distance is necessary to the effective employmentof a lubricant starvation factor. In theabsence of this value, the condition of zero reverse flow provides a practical limitation and a starvation factor of ps= 0.70. Thermal effectson lubricant filmformation under conditions approaching lubricant starvation are extremely significant owingto the absence of excess lubricant to help dissipate frictional heat generation in the contacts. Accordingly, the lubricant film reduction factors for thermal effects and starvation are not multiplicative and a combined factor is required. Goksem et al. [12.32] derived the following expressionfor elast o h y d r o ~ ~ a mline i c contact:
where L is given by equation (12.74) and
4
L
~
R
I FILMS C ~ IN ROLLING E
L
E
~
~
~
(12.80) For the zero reverse flow condition, the combined reduction factor forthe central lubricant film thickness is
For point contact, equations (12,79)-(12.81) can be used in conjunction with (12.60) for equivalent line contact loading. Considering the 209 cylindrical rollerbearing of Examples 12.1 and 12.3, what would be the average lubricant film thickness if oil mist lubrication wereused instead of fully flooded lubrication?
L
=
1.148
Ex. 12.3
(PT
=I:
0.704
Ex. 12.3
For thermal plus starvation effects, use
=
0.572
hbi = fishci =
0,572
X
0.961 = 0.550 pm (21.7 X
in.)
In the methods and equations used in the calculatioli of lubricant film thickness thus far only the macrogeometries of the rolling components have been considered; i.e., the surfaces of the components have been assumed to be smooth. In practice, each ball, roller or raceway surface has a roughness superimposed upon the principal geometry. This roughness, or more correctly surface topo~aphysimilar to the earth’s surface superimposed upon the spherical surface of the planet, is introduced by the surface finishing processes during component manufacture. In recent
history, substantial manufacturing development efforts have been expended to produce ultra-smooth rolling componentsurfaces, Figure 12.13 schematically illustrates a rough rolling component surface. For a given surface, the roughness is most commonly defined by the arithmetic average (MI) peak-to-valley distance. This is easily measurable using stylus devices such as the Talysurf machine. Using surfacemeasuring devices, more extensive properties of surface microgeometry can alsobe measured; see McCool [12.3’7]. Some ofthese will be discussed surface I roughnesses, R A , as fine as 0.05 pm in Chapter 13. To date, & (2 pin.) have been produced on ball bearing raceways approaching 600 mm (24 in.) diameter. Balls larger than 25 mm (I in.) diameter are routinely produced with R A values of 0.005 pm (0.2 pin.). It is, however, not certain that R A = 0 is an ideal microgeometry from a lubrication effectiveness or surface fatigue endurance standpoint. Depending on the thickness of the lubricant film relative to the roughnesses of the rolling contactsurfaces, the direction of the roughness pattern can aEectthe film-building capability of the lubricant. If the surface roughness has a pattern wherein the microgrooves are transverse to the direction of motion, this could resultin a beneficial lubricant filmbuilding effect. Conversely, if the lay of the roughness is parallel to the direction of motion, the effect can beto produce a thinner lubricant film. The most successful applications of rolling bearings are those in which fluid lubricant films over the rolling element-raceway contacts are sufficiently thick to completely separate those components. This is generally defined by the parameter A as follows: .S. RMS Surface Roughness 1 * 5 urn
I
12.13. Isometric viewof a typical honed and lapped surface showing “roughness” peaks.
448
LUBRICANT FILMS IN ROLLING E L E ~ E ~ - CONTACTS ~ C E ~ ~ ~
(12.82) In (12.82), ho is the minimum lubricant film thickness, s, is the rms roughness of the raceway surface, and, sRE is the rms roughness of the ball or roller surface. In general, the rms roughness value is taken as 1.25 RA. Patir and Cheng [12.38] first investigated the effect of the lay of surface topography on the lubricant film thickness generated. They developed a correction factor shown by Fig. 12.14 as a function of topography correlation factor I‘ and A. The topography correlation factor is: (12.83) where Ay and Ax are “correlation”lengths pertaining to distances between “hills”and “valleys”on a surface in thedirections transverse and parallel (longitudinal) to motion, respectively. Tgnder and Jakobsen [12.39], using a ball-on-disk test rig and optical interferometry, confirmed Patir and Cheng’s general conclusion that transverse lay tends to generate thicker films than does longitudinal lay. Kaneta et al. [12.40] in a similar experimental effort determined that, in the thin film region(A < l), film thickness for surfaces with transverse
a
\pst:
1
0
n 12.14. Lubricant film factor due to surface roughness orientation (from 112.383).
SURFACE T
O
P
44
O EFFECTS ~ ~ ~
lay tends to increase with slide/roll ratio due to deformation of asperities. When A > 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 three-dimensional plots of point contact pressure and film thickness distribution for transverse, longitudinal, and oblique topographical lays. Fig. 12.15-12.1’7 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 three-dimensional pressure fluctuations in which the maximum pressure may be greater than that produced by transverse roughness lay. It is to be noted that 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 x-direction (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 thex-direction (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 thex-direction (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 < 1.5-the mean lubricant film thicknesses will tend to be less than those predicted by the equations given for rolling contacts with smooth surfaces. The amount of the reduction may only be determined by testing; empirical relationships need to be developed.
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 E12.46-12.491 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 Herschel-Bulkley 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 Herschel-Bulkley 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 35-80°C (95-176°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 8-10 (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 element-raceway 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 cage-rolling 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
PHD
c,
EHD
Low contact pressure, no significant surface deformation No significant surface deformation, lubricant viscosity increases with pressure Surface deformation and lubricant viscosity increase with pressure
c,
2 s
22
1
-
u
~
H = (128
-2
(
2)1'2[
l
+
~
(2)+
(12.91) 0.131 tan-'
)
+ 2.6511 1.1631
For the PHD regime in line contacts, data from El2.551 have been used to establish the following espression for minimum film thickness:
where C2 = log,, (618U0.6617) (12.93)
C,
=
log"' (1.285U0.0025)
(12.94)
and C, is given by equation (12.89). In equation (12.92),C, is given by
c4= c2+ clc3(c:- 3) - o.094c1 (c:- O.??c,- 1)
(12.95)
Dalmaz [12.56] also developed numerical results for point contact film thicknesses in the PHD regime; an analytical relationship was not then established.
455
i
10-6
10-5 Dimensionless load
II
gs
1o4
12.18. Film thickness vs speed and load for a line contact (from (12.541).
456
LUBRICANT FILMS
IN ROLLING E L E ~ N T - ~ C E ~CONTACTS AS
In the foregoing discussionit has been demonstrated analytically that a lubricant film can serve to separate the rolling elements from the contacting raceways. Moreover, the fluid friction forces developed in thecontact zones between the rolling elements and raceways can significantly alter thebearing‘s mode of operation. It is desirable from the standpoint of preventing increased stresses caused by metal-to-metal contact that the minimum film thickness should be sufficient to completely separate the rolling surfaces. The effect of film thickness on bearing endurance is discussed in Chapter 23. A substantial amount of analytical and experimental research from the 1960s through the 1990s has contributed greatly to the understanding of the lubrication mechanics of concentrated contacts in rolling bearings. Perhaps the original work of Grubin [12.121 will prove to be as significant as that conducted by Reynolds during the 1880s. Besides acting to separate rolling surfaces, the lubricant is frequently used as a medium to dissipate the heat generated by bearing friction as well as remove heat that would otherwise be transferred to the bearing from surroundings at elevated temperatures. This topic is discussed in Chapter 15.
12.1. H. Martin, “Lubrication of Gear Teeth,”Engineering 102, 199 (1916). 12.2. J. Osterle, “On the Hydrodynamic Lubrication of Roller Bearings,” Wear 2, 195 (1959). 12.3. 13. Sternlicht, P. Lewis, and P. Flynn, “Theory of Lubrication and Failure of Rolling Contacts,” ASME Trans., J ; Basic Eng. 213-4226 (1961). 12.4. C. Barus, “Isothermals, Isopiestics, and Isometrics Relative to Viscosity,” Arner. J ; Science 45, 87-96 (1893). II,” 12.5. ASME Research Committee on Lubrication “Pressure-Viscosity Report-Vol. ASME (1953). 12.6. C. Roelands, “Correlation Aspects of Viscosity-Temperature-Pressure Relationship of Lubricating Oils” (Ph.D. thesis, Delft University of Technology, 1966). 12.7. J. Sorab and W. Vanhsdale, “A Correlation for the Pressure and TemperatureDependence of Viscosity,” Tribology Trans. 3 4 4 , 604-610 (1991). 12.8. T. Harris, “Establishmentof a New RollingBearing Life Calculation Method,” Final Report, U. S. Navy Contract N68335-93-C-0111 (January 15, 1994). 12.9. S. Bair and W. Winer, “Shear Strength Measurementsof Lubricants at High Pressure,” Trans. ASME, J ; Lubrication Technology, ser. F 101, 251-257 (1979). 12.10. S. Bair and W. Winer, “Some Observations in High Pressure Rheology of Lubricants,” Trans. ASME, J ; Lubrication Technology, Ser. F 104, 357-364 (1982). 12.11. D. Dowson and G. Higginson, “A Numerical Solution to the ElastoHydrodynamic Problem,” J ; Mech. Eng. Sei. 1(1),6 (1959).
12.12. A, Grubin, “Fundamentals of the Hydrodynamic Theory of Lubrication of Heavily Loaded Cylindrical Surfaces,” in Investigation of the Contact Machine Components, ed. Kh. F. Ketova (Translation of Russian Book No. 30, Chapter 21, Central Scientific Institute of Technology and Mechanical Engineering, Moscow (1949). 12.13. D. Dowson and G. Higginson, “The Effect of Material Properties on the Lubrication of Elastic Rollers,’’ J: Mech. Eng. Sei. 2(3) (1960). 12.14. L. Sibley and F. Orcutt, “Elasto-hydrodynamic Lubrication of Rolling Contact Surfaces,”ASLE Trans. 4, 234-249 (1961). 12.15. D. Dowson and G. Higginson, Proc. Inst. Meeh. Eng., Vol. 182, Part 3A, 151-167 (1968). 12.16. G. Archard and M. Kirk, “Lubrication at Point Contacts,” Proc. Royal SOC.Ser. A 261,532-550 (1961) 12.17. B. Hamrock and D. Dowson, “Isothermal Elasto-hydrod~amicLubrication of Point Contacts-Part III-Fully Flooded Results,” Truns. ASME, J: Lubrication Technology 99,264-276 (1977). 12.18. L. Wedeven, “OpticalMeasurements in Elasto-hydrod~amicRolling Contact Bearings,” Ph.D. Thesis, University of London (1971). 12.19. M. Kotzalas, “Power Transmission Component Failure and Rolling Contact Fatigue Progression,”Ph.D. Thesis, Pennsylvania State University (1999). 12.20. E. Avallone and T. Baumeister, Marks Standard Handbook for Mechanical Engineers, 9th ed., McGra~-Hill,New York (1987). 12.21. C. Venner, “Higher Order Multilevel Solvers for the EHL Line and Point Contact Problems,”ASME Trans., J: Tribology 116, 741-750 (1994). 12.22. M. Smeeth and H. Spikes, “Central and Minimum Elastohydrod~amicFilm Thickness at High Contact Pressure,” ASME Trans., J: Tribology 119,291-296 (1997). 12.23. H. Cheng, “A Numerical Solution to the ElastohydrodynamicFilm Thickness in an Elliptical Contact,” Trans. ME, J: Lubrication Technology 92, 155-162 (1970). der Viscositat von Flussigkeiten,” 12.24. H. Vogels, “DasTemperaturabh~~g~eitsgesetz Phys. 21. 22,645-646 (1921). 12.25. H, Cheng, “A Refined Solution to the Thermal-Elastohydrod~amicLubrication of Rolling and Sliding Cylinders,”ASLE Trans. 8(4), 397-410 (1965). 12.26.L. Murch and W. Wilson, “A Thermal Elastohydrodynamic Inlet Zone Analysis,” Trans. AS^^, J: Lubrication Technology 97(2), 212-216 (1975). 12.27. A. Wilson, “An Experimental Thermal Correction for Predicted Oil Film Thickness in Elastohydrodynamic Contacts,” in Thermal Effects in Tribology: Proceedings of the 6th Leeds-Ijton S y ~ p o s i u mon Tribology, 1979 (1980). 12.28. W. Wilson and S. Sheu, “Effect of Inlet Shear Heating Due to Sliding on Elastohydrodynamic Film Thickness,”Trans. ASME, J: Lubrication Technology 105(2),187188 (1983). 12.29. P. Gupta, H. Cheng, D. Zhu, N. Forster, and J. Schrand, “Viscoelastic Effectsin MilL-7808Type Lubricant, Part I: Analytical Formulation,” Tribology Trans. 3S(2), 269-274 (1992). 12.30. C. Hsu and R. Lee, “An Efficient Algorithm for Thermal Elastohydrodynamic Lubrication under R o l l ~ g / S l i d i ~Line g Contacts,” J: ~ibration,Acoustics and Reliability in Design ll6(4), 762-768 (1994). , ed., Mc~raw-Hill, NewYork (1954). 12.31. W. MacAdams, Heat ~ a n s m i s s i o n3d 12.32. P. Goksem and R. Hargreaves, “The Effect of Viscous Shear Heating in Both Film Thickness and Rolling Traction in an EHL Line Contact-Part 11: Starved CondiLion,” Trans. A ~ M E J: , Lubrication Technology 1
12.33. D. Dowson, “Inlet Boundary Conditions,” Leeds-Lyon Symposium (1974). 12.34. P. Wolveridge, K. Baglin, and J. Archard, “The Starved Lubrication of Cylinders in Line Contact,”Proc. Inst. Mech. Eng, 185, 1159-1169 (1970-71). 12.35. P. Castle and D.Dowson, “A Theoretical Analysis of the Starved Elastohydrodynamic Lubrication Problem,’,Proc. Inst. Mech. Eng. 131, 131-137 (1972). 12.36. B. Hamrock and D. Dowson, “Isothermal Elastohydrod~amicLubrication of Point Contact-Part IV StarvationResults,” Trans. ASME? J Lubrication ~echnology99, 15-23 (1977). 12.37. J. McCool, “Relating ProfileInstrument Measurements to the Functional Performance of Rough Surfaces,” Trans. ASME7 J Tribology 109,271-275 (April 1987). 12.38. N. Patir and H. Cheng, “ERect of Surface Roughness Orientation on the Central Film Thicknessin EHD Contacts,” in Elastohydrodynamics and Related Topics:Proceedings of the 5th Leeds-Lyon S y ~ p o s i u mon TriboZogy7 1978, 15-21 (1979). 12.39. E(. Tgnder and J. Jakobsen, “Interferometric Studies of EfYects of Striated Roughness on Lubricant Film Thickness Under Elastohydrodynamic Conditions,”Trans. ASME>J Tribology 114, 52-56 (January 1992). 12.40. M. Kaneta, T. Sakai, and H. Nishikawa, “ERects of Surface Roughness on Point Contact EHL,”Tribology Trans. 36,(4), 605-612 (1993). 12.41. L. Chang, M. Webster, and A. Jackson, “On the Pressure Rippling and Roughness Deformation in Elastohydrodynamic Lubrication of Rough Trans. ASME7 J; Tribology 115,439-444 (July 1993). of Surface Textureon EHL Point Contacts,”Trans. 12.42. X. Ai and H. Cheng, “The Effects ASME>J ; Tribology 118,59-66 (January 1996). 12.43. G. Guangteng and H. Spikes, “An Experimental Study of Film Thickness in the Mixed Lubrication Regime,”in EZastohydrodynamics ’996:~undamentalsand Applications in Lubrications and Traction: Proceedings of the 23rd Leeds-Lyon Symposium on Elastohydrodynamics, 1996, 159-166 (1997). 12.44. P. Cann, J. Hutchinson, and H. Spikes, “The Development of a Spacer Layer Imaging Method (SLIM) for Mapping Elastohydrod~amicContacts,” Di~oZogyTrans. 3 9 9 915-921 (1996). 12.45.G. Guangteng, P. Cann, A, Olver, and H. Spikes, “Lubricant Film Thickness in Rough Surface, Mixed Elastohydrod~amicContact,”ASME Paper 99-TRIB-40(October 1999). 12.46. A. Wilson, “The Relative Thickness of Grease and Oil Films in Rolling Bearings,” Proc. Inst. Mech. Eng. 193, 185-192 (1979). 12.47.H.Munnich and H.Glockner, “Elastohydrod~amic Lubrication of GreaseLubricated Rolling Bearings,”ASLE Trans. 23,45-52 (1980). 12.48. J. Palacios, A. Cameron, and L. Arizmendi, “Film Thickness of Grease in Rolling Contacts,”ASLE Dans. 24, 474-478 (1981). 12.49. J. Palacios, “Elastohydrod~amicFilms in Mixed Lubrication: An Experimental Investigation,” Wear 89, 303-312 (1983). 12.50. J. Kauzlarich and J. Greenwood, “Elastohydrod~amicLubrication with HerschelBulkley Model Greases,” ASLE Trans, 15, 269-277 (1972). 12.51. J. Palacios and M. Palacios, “Rheological Properties of Greases in EHD Contacts,” Tribology Int. 17, 167-171 (1984). 12.52. P. Cann, “Starvation andReflow in a Grease-Lubricated Elastohydrod~amicContact,” TriboZogy Dans. 39(3), 698-704 (1996). 12.53. P. Cann, “Starved Grease Lubrication of Rolling Contacts,” TriboZogy Trans. 42(4), 867-873 (1999). 12.54. D. Dowson and G. Higginson, “Theoryof Roller Bearing Lubrication and Deformation,” Proc. Inst. Mech. Eng. 117 (1963).
12.55. P. Markho and D. Clegg, “Reflections on Some Aspects of Lubrication of Concentrated Line Contacts,”Trans. ASME, J ~ubrication ~e~hnology 101,528-531 (1979). 12.56. G. Dalmaz, “Le film mince visquex dans les contacts hertziens en regimes hydrodynamique et elastohydrod~amique”(Docteur d’8tat BS Sciences thesis, I.N.S.A. Lyon, 1979).
This Page Intentionally Left Blank
Symbol a
d
Description Semimajor axis of contact ellipse True average contact area Apparent contact area Semiminor arris of contact ellipse Separation of mean plane of summits and smooth plane Summit density Elastic moduli of bodies I and 2
Reduced elastic modulus Tabular functions for the ~ r e e n ~ o o d - ~ i l l i a m smodel on Lubricant film thickness Central or plateau lubricant film thickness
Units mm (in.) mm2 (in.)2 mm2 (in.)2 mm (in.) mm (in.) mm-2 (in.)-2 MPa (psi) MPa (psi) mm (in.) mm (in.)
4
FRICTION IN ~ L U I ~ - L ~ R ROLLING I C A ~ ~E
Symbol
Description
L
E
~
~
- CONTACTS ~ C E ~
Units
Zeroth-order spectral moment, EiZ
q
s2
Second-order spectral moment Fourth-order spectral moment Contact density Plastic contact density Local contact pressure Maximum contact pressure Applied load Asperity-supported load Fluid-supported load Summit sphere radius Root mean square (rms)value of surface profile Temperature Sliding velocity Deflection of summit Variable governing asperity density Yield strength in simple tension Summit height relative to summit mean plane Distance between surface and summit mean plane Surface profile Bandwidth parameter Shear rate Absolute viscosity Lubricant film parameter, h / s Coulomb friction coefficient Poisson's ratio for bodies 1 and 2 Standard deviation of summit heights for bodies 1. and 2 Standard deviation for summit heights for composite surface Shear stress Shear stress dueto fluid Limiting shear stress influid Shear stress inNewtonian fluid lubrication
pm2 (pin.2) mm-2 (in.-2) mme2 (in.-2) mm-2 (in.-2) MPa (psi) MPa (psi) N (lb) N (lb) N (lb) mm (in.) pm (pin.) "C ( O F ) mmlsec (in./sec) pm (pin.) pm (pin.) MPa (psi) mm (in.) mm (in.) mm (in.) sec-l N"sec/m2 (lb-sec/in.2)
mm (in.) mm (in.) MPa (psi) MPa (psi) MPa (psi) MPa (psi)
~
~
GE
Symbol
Description Gaussian probability density function
Units mm-l (in.-l)
In its full complexity, a rolling element-raceway contact caanot be represented by a simple analytical expression. The combined action of an applied load and kinematic constraints produces some combination of rolling, sliding, and spinning motions. These motions act to draw lubricant into the contact where, its properties altered by the pressure and temperatures that vary throughout the contact region, it forms a film that serves to separate the contacting bodies to an extent depend in^ on both the microgeometry of the bodies, and theproperties of the lubricant. When the separating film is small relative to the composite surface roughness, a myriad of microcont~ctsof highly irregular shapes forms within the macrocontact, causing pressure, temperature, and film thickness perturbations on a microscale. Moreover, these microcontacts may deform plastically as well as elastically with the result that the microgeometry varies with Lime, Sliding and spinning motions on the macrocontact act to shear the separating lubricant film and, if separation is only partial, to drag the A tangential force is producedfrom microcontacts across each other. these combined effects. This tangential or traction force alters the stress distri~utionin the solids and is a critical factor in determining fatigue itude of the fluid contribution to the traction depends on roperties under the locally variable pressure and temperature and shear rates that prevail in themacrocontact. Thecontribution to the traction caused by the sliding microcontacts will depend on the local film conditions or the nature of the surface boundary films that result from oxidation and additives present in the lubricant. As discussed in Chapter 12, several researchers have attem~tedto model the eff'ect of surface roughness, i.e., microgeometry, on the thickness of the lubricant films in rolling/sliding concentrated contacts. efforts have fre~uentlyincluded the estimation of fluid friction a effect on the lubricant film thickness. In general, it hasbeen determine that thefriction and the resultant localized temperature rise in thecontact has little effect on lubricant film thickness; as indicated in Chapter 12, lubricant film thickness depends on events occurring at the inlet to the contact and not in the contact proper.In these analyses, the indicated solutions have been obtained by numerical analysis, most recently using finite element methods re~uiringmeshes of several thousand nodes and tes to hours of calculation evenwith hi
464
FRICTION IN F L ~ - L ~ R I C A T ROLLING E ~ E L E ~ ~ - ~ C C OE~ A~ C A T S~
culation. While this approach is useful for research purposes, it does not suffice for use in the determination of ball and roller bearing performance in practical engineering applications. This chapter describes an approach that synthesizes state-of-the art models forlubricant film thickness and asperity load sharing into a practicable, analytical description of a real, rolling element-raceway contact.
In calculating the lubricant film thickness in Chapter 12, it is assumed that thesurfaces are perfectly smooth. Theassumption is now made that when the surfaces are rough the lubricant film thickness, calculated as if the surfaces were smooth, separates the mean planes of the rough surfaces, as shown in Fig. 13.1. The surfaces fluctuate randomly about their mean planes in accordance with a probability distribution. The root-mean-square (rms) value of this distribution is denoted a, for the upper surface and a2 for the lower surface. When the combined surface fluctuations at a given position exceed the gap h due to the lubricant film, a microcontact occurs. At the microcontacts the surfaces deform elastically and possibly plastically. The aggregate of the microcontact areas is generally a small fraction (-45%) of the nominal area of contact. A microcontact model uses surface microgeometry data to predict, at a minimum, the density of microcontacts, the real area of contact, and the elastically supported mean load. One of the earliest and simplest microcontact models is that of Greenwood and Williamson (GW) E13.11. this model applicable to isotropic surfaces have been et al. [13.2] and by O’Gallaghan and Cameron E13.31. lso treated a strongly anisotropic surface. One of the most comprehensive models yet developed is A ~ P E E13.51, ~ ~ which I ~ requires a nine-parameter microgeometry description and accounts for anisotropic as well as isotropic surfaces. A comparison of various micro-
E 13.1. Asperity contacts through partial oil film.
5
C R O ~ E O ~ T RAND Y ~ C ~ O C O ~ A C T S
contact models conducted byMcCool L13.61 has shown that the G~ model, despite its simplicity, compares favorably with the other models. Because it is much easier to implement than the other models, the GW model is the microcontact model recommended here.
For the contact of real surfaces Greenwood and Williamson [13.1] developed one of the first models that specifically accounted for the random nature of interfacial phenomena. Themodel applies tothe contact of two flat plastic planes, one rough and the othersmooth. It is readily adapted to the case of two rough surfaces as discussed further below. In the GW model the rough surface is presumed to be covered with local high spots or asperities whose summits are spherical. The summits are presumed to have the same radius R, but randomly variable heights, and to be uniformly distributed over the rough surface witha known density DSUM of summitshnit area. The mean heightof summits liesabove the mean heightof the surface as a whole by the amount Z, indicated in Fig. 13.2. The summit heights x, are assumed to follow a Gaussian probability law with a standard deviation u-,.Figure 13.3 shows the assumedform for the summit height distribution or probability density function (pdf) f(z,). It is symmetrical about the mean summit height. The probability that a summit has a height, measured relative to the summit mean planein the interval(z,? z, + dz,) is expressed in termsof the pdf as f(z,) dz,. The probability that a randomly selected summit has a height in excess of some value d is the area under thepdf to the right of d. The equation of the pdf is (13'.1)
SOthe probability that a randomly selected summit has height in excess of d is S U ~ ~ HEIGHT I T ~IST~I~UTIO~ .
~
.
~
s 13.2. Surface and summit mean planes and distributions.
-
13.3. Distribution of summit heights.
This integration must be performed numerically. for tun at el^, however, the calculation can be related to tabulated areas under the standard normal curve for which the mean is 0 and the standarddeviation is 1.0. Using the standard normal density function #(x), the probability that a summit has a height greater than d above the summit mean plane is calculated.
(13.3)
where F&) is the area under the standard normal curve to the right of the value t. Values Fo(t)for t ranging from 1.0 to 4.0, are given in column 2 of Table 13.1. It is assumed that when large flat surfaces are pressed together,their mean planes remain parallel. Thus, if a rough surface and a smooth surface are pressed against each other until the summit mean plane of the rough surface and the mean plane of the smooth surface are separated by an amount d, the probability that a randomly selected summit will be a microcontact is PEsummit is a contact] = P[z, > dl
=
Fo (d/a$)
(13.4)
Since the number of summits per unit areais DSTIM, the average expected number of contacts in. any unit area is
E 13.1. Functions for the G r e e n ~ o o d - ~ i l l i a ~ sModel on 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.2 3.4 3.6 3.8 4.0
0.5000 0.4602 0.4207 0.3821 0.3446 0,3085 0.2743 0.2420 0.2119 0.1841 0.1587 0.1357 0.1151 0.9680 x 10-1 0.8076 X 10-1 0.6681 x 10-1 0.5480 x 10-1 0.4457 x 10-1 0.3583 x 10-1 0.2872 x 10-l 0.2275 x 10-1 0.01786 0.01390 0.01072 0.8198 x 0.6210 x 0.4661 X low2 0.3467 x 0.2555 x 0.1866 x 0.1350 x 6.871 x lo-* 3.369 x lo-* 1.591 X lo-* 7.235 X 3.167 x
0.3989 0.3509 0.3069 0.2668 0.2304 0.1978 0.1687 0.1429 0.1202 0.1004 0.8332 x 10-1 0.6862 x 10-1 0.5610 x 10-1 0.4553 x 10-1 0.3667 x 10-1 0.2930 x 10-1 0.2324 x 10-1 0.1829 x 10-1 0.1428 x 10-1 0.1105 x 10-1 0.8490 x 6.468 X 4.887 X 3.662 x 2.720 x 2.004 X 1.464 X 1.060 x 7,611 X lo-* 5.417 X lo-* 3.822 X 1.852 X lo-* 8.666 x 3.911 X 1.702 X 7,145 x
0.4299 0.3715 0.3191 0.2725 0.2313 0.1951 0.1636 0.1363 0.1127 0.9267 x 0.7567 x 0.6132 x 0.4935 x 0.3944 x 0.3129 x 0.2463 x 0.1925 x 0.1493 x 0.1149 x 0.8773 x 0.6646 x 0.4995 x 0.3724 x 0.2754 x 0.2020 x 0.1469 x 0.1060 x 0.7587 x 0.5380 X 0.3784 x 0.2639 X 0.1251 X 0.5724 X 0.2529 x 0.1079 X 0.4438 X
10-1 10-1 10-1 10-1 10-1 10-1 10-1 10-1 10-1 10-1
10-2
.
lo-* lo-* lo-*
Given that a summit is in contact because its height x, exceeds d , the summit must deflect by the amount u) = z, - d , as shown in Fig. 13.4. For notational simplicity the subscript on x, is henceforth deleted. For a sphere of radius 12 elastically deflecting by the amount u), the solution gives the contact area
A
=
TRW = TR(Z- d ) =
ma2
x
d
(13.6)
where a = contact radius. The correspondingasperity load is (13.7)
where E' = [(l- v?)lE, + (1 - ~i4)lE~I-l and Ei, vi (i = 1,2) are "Young's moduli and Poisson's ratios for the two bodies. The maximum Hertzian pressure in the microcontact is (13.8)
Both A and P are functions of the random variable z. The average or expected value of functions of random variables are obtained by integrating the function and the probability density of the random variable over the space of possible values of the random variable. The expected summit contact area is thus (13.9)
which transforms to (13.10)
where
(13.11)
F,(t) is also given in Table 13.1. The expected total contact area as a fraction of the apparent area is obtained as theproduct of the average asperity contact area contributed by a single randomly selectedsummit and the density of summits. Thus, the ratio of contact to apparent area, AJA,, is (13.12) By the same argument the total load per unit area supported by asperities is (13.13)
where (13.14)
F3/2(t)is also given in Table 13.1. s2ie
e2
A contacting summit will experience some degree of plastic flow when the maximum shear stressexceeds half the yield stress insimple tension. In thecontact of a sphere and aflat, the maximum shear stressis related to the maximum Hertzian stress p o by
egree of plastic deformation is present at a contact if T~~~
> 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=h-Z,
(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 fluid-supported 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 10-4 1.8 X X 32.65
1.77 X
pm-2 (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 10-3)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 asperity-supported load is Q, = Tab =
(~)
= T X
3
X
0.33
X
31.6
98.3 N (22.1 lb)
The fluid-supported 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 non-Newtonian lubricant behavior on the EHL model. Bell [13.ll] specifically studied the effects of a Ree-Eyring 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 non-Newtonian 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 non-Newtonian lubricant will significantly influence lubricant film thickness. Non-Newtonian lubrication does, however, significantly influence friction in the contact. Due to friction, lubricant temperature in the contact rises during rolling element-raceway 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 Ree-Eyring modelfor analysis of EHL traction. Equations (12.21)-(12.23), introduced in Chapter 12, can provide the viscosity-pressure-temperature 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 components-i.e., rolling elements and raceways-and the contact deformations of these components under load, both rollingand sliding motions occur in most rolling element-raceway 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 slide-roll 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 rolling-sliding contacts and found that equation (12.1) pertains only to a situation involving a relatively low slide-to-roll 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 7-1im 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,7-N 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 asperity-asperity 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 full-scale 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 performance-related 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,multi-thousand 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, 300-319 (1966). 13.2. A. Bush, R. Gibson, and T. Thomas, “TheElastic Contact of a Rough Surface,” Wear 35, 87-111 (1975). 13.3. M. O’Callaghan and M. Cameron, “Static Contact under Load Between Nominally Flat Surfaces,’’ Wear 36, 79-97 (1976). 13.4. A. Bush, R. Gibson, and G. Keogh, “Strongly Anisotropic Rough Surfaces,” ASME Paper 78-LUB-16 (1978). 13.5. J. McCool and S. Gassel, “The Contactof Two Surfaces Having Anisotropic Roughness Geometry,”ASLE Special ~ubzication(SP-71,29-38 (1981). 13.6. J, McCool, “Comparison of Models for the Contact of Rough Surfaces,” Wear 107, 37-60 (1986). 13.7. A. Bush, R. Gibson, and G. Keogh, “The Limitof Elastic Deformation in the Contact of Rough Surfaces,” Mech. Res. Cornrn. 3, 169-174 (1976). 13.8. P. Nayak, “Random Process Model of Rough Surfaces,” Trans. ASME, J Lub. Technology 93F, 398-407 (1971). 13.9. R. Sayles and T. Thomas, “ThermalConductances of a Rough Elastic Contact,”AppZ. Energy 2, 249-267 (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 Ree-Eyring Fluid,” ASLE Trans. 5, 160-171 (1963). 13.12. F. Smith, “Rolling Contact Lubrication-The Application of Elastohydrod~amic Theory,” ASME Paper 64-Lubs-2 (April 1964). 13.13 B. Gecim and W. Winer, “A Film Thickness Analysis for Line Contacts under Pure Rolling Conditions with a Non-Newtonian Rheological Model,” ASME Paper 80C2/ LUB 26 (August 8, 1980).
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),303-312 (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), 258-265 (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, ‘“Micro-EHL in Lubricated Concentrated Contacts,’’A5”E Trans., J. Tribology 112, 392-397 (1990). 13.19. E. Trachman and H. Cheng, “Thermal and Non-Newtonian EEects on Traction in Elastohydrodynamic Contacts,” Proc. Inst. Mech. Eng. 2nd Symposium on Elastohydrodynamic Lubrication, Leeds, 142-148 (1972). 13.20. J. Tevaamerk and IC. Johnson, “A Simple Non-Linear Constitutive Equation for EHD Oil Films,” ‘Wear 35, 345-356 (1975). 13.21. T. Harris and R. Barnsby, “Tribological Performance Prediction of Aircraft Turbine Mainshaft Ball Bearings,” Tribology Trans. 41(1), 60-68 (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 load-carrying ability. A notable exception to the foregoing generalization is, of course, the hydrostatic gas bearing; however, such a bearing is not self-sustaining, 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 element-raceway 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. Roller-raceway 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 roller-raceway 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 grease-lubricated 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 asperity-asperity 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 ball-raceway 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 dry-film 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 element-raceway 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. Ball-raceway contact ellipse showing “locked” region and microslip region-radial 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 gas-lubricant 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 element-cage 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 end-flange 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 End-Flange 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 angular-contact 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 ball-raceway 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 theball-raceway 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 angular-contact 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 thrust-loaded 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 ball-raceway 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 ball-raceway 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 thrust-loade~angular-contact 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
=
n-1b
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. Roller-raceway 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 element-raceway 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 oil-lubricated ball and roller bearings. Rolling element centrifugal forces in such applications tend to cause higher normal load at theouter raceway-rolling element contact as compared to the inner race,way-rolling 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 fluidfilm-lubricatedrollingelementraceway 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 raceway-rolling 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 mist-type 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 element-cage friction and cage-bearing ring friction as well as cage-lubricant friction also affect skidding.
One of the most important applications with regard to skidding is the mainshaft angular-contact 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 rolling-sliding components. For assumed contact loading, the frictional forces acting in thecontact areas are given by
FXln= anbn J-1
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<= 0
(14.75)
n=O
n=i
=
n=o
x
n=i
n=o
n=i
n=o
where
and J is the polar momentof inertia. FVin equation (14.72)is determined from equation (14.2). A ball-riding case with negligible friction in the ball pockets is assumed. Since onlya simple thrust load is assumed, cage speed is identical to ball orbital speed wm. The unknowns in equations (14.69)--(14.75) are inner and outer raceway-ball contact deformatio~s9 ball contactangles or position variables, bearing axial deflection, and ball wyr9wzf, and corn.Hence, there are nine unknowns and seven speeds, mXf, equations. The remaining two equations pertaining to ball position are obtained from Chapter 9. The solution of the equations requires the use of a computer. These equations were first solved by Harris [14.18] using the simplif~ngassumption of an isothermal Newtonian lubricant, adequately supplied to the ball-raceway contacts. Figures 14.23 and 14.24 showthe comparison of the analytical results with the experimental data of Shevchenko and Bolan [14.19] and Poplawski and Mauriello [14.20]. Notethe deviations from the outer raceway control appro~imation.
DING AND CAGE FORCES
o
---
data Test Raceway control theory Harrisanalysis 14.181
0.49 14 21
N Thrust/ball (475 ib)
0.48
.-+.r
0.47
2
-a
8
0.46
8 CI
u-
f 0.45 I
B 0.44 0.43 0.42
0.52
0
2000 4000 6000 8000 10000 Shaft speed ( rpm)
1000
2000
I
I
0.50
.-0
E
N
4000 3000
I
I
9000 rpm
0.48
-8 a. 0.46 (0
f
\
I
8 0.44
\
"
"
0
0.42
Oa4'
400200
600 1000 800
Thrust load per bail ( Ib)
. Experimental data of Shevchenko and Bolan[14.19]for contact ballbearing with three 28.58 mm (1.125 in.) balls.
an angular-
N
“ -
0
0*380
100
200
300
Raceway control theory Harris analysis i 14.181 data Test
400
Thrust load ( Ib)
N
500
I
0.46
35,000rpm
0.44
.-0
c,
E
p
0.42
P
0
c,
Y-
(CI
f I
0.40 0.38
0.36 0
100 200 300 Thrust load (lb)
400
14.24. Experimental data o f Poplawski and Mauriello 114.201 vs analyticaldata of Harris [14,18]for a 35 mm * 62 mrn angular-contact ball bearing.
Parker [14.21] established an empirical formula to estimate the percentage of the bearing “free space” occupied by fluid lubricant. Using Parker’s formula it is possible to calculate the effective fluiddensity 6 in equation (14.3) and hence Fu in equation (14.72). The effective density so dete~minedis given by equation (14.78).
3
(14.78) This equation was developed from ball bearing tests.
Skidding is a problem in cylindrical roller bearings used to support the mainshaft in aircraft gas turbines. These bearings, which are used principally for locationare very lightly loaded whileoperating at high speeds. Harris E14.221 indicates the method to predict skidding in this application. considering the roller-raceway contacts to be divided into laminae as in Chapter 7, the sliding velocity at a given “slice”is given by
(14.79) where I>,is the equivalent roller diameter at laminum A. It is assumed in equation (14.79)that one- ird of the elastic deformation occursin the roller and two-thirds in the raceway. Furthermore, owing to assumed zero clearance between the roller and cage pocket, roller orbital speed is constrained to equal cage speed, hence, woj = urn- 0, and wij = ai wxn*
Fluid entrainment velocities are given by equations (12.52) and (12.53) and the lubricant film thicknesses by equations (12.55) and (12.56).The contact frictional shear stresses rhnjcan be determined using equation (13.39). Fromequation (6.45), the pressure or normal stress at each laminum is given by (14.80) where t
=y
’ l b and q
=
QlZ.Contact friction is then given by
Fnj
= 2wn
bhnj h=l
I
1
h=k
0
‘hnj
dt
(14.81)
Figure 14.25 shows the forces and moment acting on a roller in a radially loaded cylindricalroller bearing with negligiblerollerend-
62
14.25. Forces acting on a roller.
flange friction. From Fig. 14.25, the equilibrium equations (14.82) and (14.83) obtain n=i
E
n=o
CnQnj - F c j =
0
n =o,i co = 1; ci = -1
(14.82)
where Fcj is given by equation (5.529, and n=i
where the viscous drag force is given by equation (14.3). Note that if there is clearance between the roller and the cage webbing, then the roller is free to orbit at other than cage speed and equation (14.83) is d ~fric. nonzero, being equal to the inertial load, rnd,m* ~ w ~ j /The tional moments aboutthe roller axis due to shear stresses are given by
+
and n=i n=o
DING AND CAGE FORCES
5
Finally, the radial equilibrium equation for the bearing is (14.86) and if the bearing operates at constant speed, the sum of the moments on the cage in the circumferential direction must equate to zero, or (14.87) where F C L is given by equation (14.4). As in Chapter 7, normal loads Qnj can be written in terms of contact deformations, and bearing radial deflection can be related to contact deformations and radial clearance. Accordingly, equations (14.82), (14.83), (14.85), (14.86),and (14.87), a set of 32 + 2 equations, can be solved for 8,. Sij, am,coj, and QCGj.Reference E14.111 gives the general solution for all types of roller bearings (and ball bearings); that is, for five degrees of freedom in applied bearing loading, freedom for each roller(and ball) to orbit at a speed other than cage speed(cow instead of corn),and any shape of raceway and/or roller. Harris [14.22] using a simpler form of the analysis, considering only isothermal lubrication conditions and neglecting viscousdrag on the rollers, nevertheless managed to demonstrate the adequacy of the analytical method. Figure 14.26, taken from reference [14.221, compares analytical data against experimental data on cage speedvs applied loadand speed. The analysis further indicated that skidding tends to decrease as applied load is increased and is relatively insensitive to the type of lubricant. Several aircraft engine manufacturers assemble their bearings in an “out-of-roun~ outer raceway to achieve the load distribution of Fig. 14.27 as a means of minimizing skidding.This artificial loading of the bearing increases the maximum roller loadand doubles the number of the rollers so loaded. Figure 14.28, taken from reference [14.22]illustrates theeffect on skidding of an out-of-round outer raceway. Another method to miniis to use a few, for example, three, equally spaced hollow rollers that provide an interference fit with the raceways under zero radial load and static conditions. Figure 14.29, from reference [14.23], illustrates such an assembly, while Fig. 14.30 indicates the effectiveness to minimize skidding. Figure 14.31, taken from the commentary to reference 114.231,confirms the adequacy of the analytical method by showing a high degree of skidding for 2-4 90% hollow rollers tested in a 207 cylindrical roller bearing.
52
FRICTION IN ROLL IN^ ~ E ~ I N ~ S N
3000
2000
4000
6000
10,000
8000
I
i
200c E
E? d a, Q P)
F
0
l0OC
a
500
1000
1500
2000
2500
Bearing load, Ib 14.26. Cage speed vs load and inner ring speed for cylindrical roller bearing, lubricant-diester type according to MIL-L-7808 Specification. Z = 36 rollers, I = 20 mm (0.787 in.), D = 19 mm (0.551 in.), dm= 183 mm (7.204 in.), Pd = 0.0635 mm (0.0025 in).
7. Distribution of load among the rollers of a bearing having an out-ofround outer ring and subjected to radial load Fr.
mm
0
0.1 I
I
~
0 35
0.010
6500 rpm
0. 15
Out-of-round, in.
14.28. Cage speed vs out-of-round and inner ring speed. Lubricant-diester type according to MII-L-7808 Specification. Z = 36, i = 1, I = 20 mm (0787 in.), D = 14 mm (0.551 in.), dm = 183 mm (7.204 in.), Pd = 0.0635 mm (0.0025 in.), Fr = 222.5 N (50 b ) .
roller bearing having three preloaded annular rollers.
CAGE ~ Q ~ I Q N
FORCES
E
rollers
Et
"0
4
8
12
16
Inner raceway speed X
20
24
io3 rpm
14.31. Cage speed vs inner raceway speed: 207 roller bearing, F, = 0, Pd = -0.061 mm (-0.0024 in.), 90% hollow rollers, lubricantMIL-L-6085Aat 0.85 kg//min.
s
s
With respect to rolling element bearing performance, cage design has become more important as bearing rotational speeds increase. In instrument ball bearings undesirable torque variations have been traced to cage dynamic instabilities. In the development of solid-lu~ricatedbearings for high-speed, high-temperature gas turbine engines, the cage is a major concern. A key t o successful cage design is a detailed analysis of the forces acting on the cage and the motions it undergoes. Both steady-state and dynamic formulations of varying complexity have been developed.
The primary forces acting on the cage are due to the interactions between the rolling element and cage pocket
53
FRI~TIONIPi ROL~IPiGB
A lubricant viscous drag force (ICDRAG) develops on the cage surfaces resisting motion of the cage. Centrifugal body forces (shown as FcF)due t o cage rotation make the cage expand uniformly outward radially and induce tensile hoop stresses in thecage rails. An unbalanced force (FuB), the magnitude of which dependson how accurately the cage is balanced, acts radially outward. H ~ d r o d ~ a mshort i c bearing theory can be used to model the cageland interaction as indicated in C14.241. The contact between the rolling or element and cage pocket can be hydrodynamic, elastoh~drod~amic, elastic in nature, depending on the proximity of the two bodies and the m a ~ i t u d eof the rolling element forces. In mostcases the rolling element-cage interaction forces are small enough that hydrodynamic lubrication considerations prevail.
In a previous sectionit was de~onstrated that anal~ical means exist t o predict skidding in ball and roller bearings in any ~ui~-lubricated application. All of the calculations, even for the least complex application, require the use of a computer, As a spin-off from the skidding analysis, rolling element-cage forces are determined. For an out-o€-round outer raceway cylindrical rollerbearing under radial load, Fig.14.33, from ref-
- 14 - 12 - 10
3
2
-1
N
-
*
Azimuth ( degrees, 1
14.33. Cage-to-roller load vs azimuth for a gas turbine main shaft cylindrical roller bearing. Thirty 12 mm X 12 mm rollers on a 152.4mm (6-in.) pitch diameter. Roller i.d./o.d. = 0.6, outer ring out-of-roundness = 0.254mm (0.01in.), radial load = 445 N (100 lb), shaft speed = 25,000rpm.
erence E14.251 illustrates cage web loading for steady-state, centric cage rotation. Whereas the analysis of 114.241 considered onlycentric rotation in the radial plane, Kleckner and Pirvics E14.261 used three degrees of freedom in the radial plane; that is, the cage rotational speed and two radial displacements locating the cage center in the plane of rotation. The corresponding cage equilibrium equations are
(14.90) where I V Y ? Wz = components of FcLin they and x directions Fcpj = cage pocket normal force for the jth rolling element fcpj = cage pocket friction force forthe jth rolling element The cage coordinate system is shown in Fig. 14.34.
F R I ~ T I OIN ~ ROLLING E
GS
roller
' 1 wz
TCL 14.34. Cage coordinatesystem.
Equations (14.88) and (14.89) represent equilibrium of cage forces in the radial plane of motion. The summation of the cage pocket normal forces and friction forces equilibrate the cage-land normal force. Equation (14.90) establishes torque equilibrium for the cage about its axis of rotation. The cage pocketnormal forces are assumed to react at the bearing pitch circle. The signof the cage-land frictiontorque TcLdepends on whether the cage is inner ring land-riding or outer ring land-riding. In the formulation of [14.26] each roller is allowed to have different rotational and orbital speeds.
ie element bearing cages are subjected to transient motions and forces due to accelerations caused by contact with rolling elements, rings, and eccentric rotation. In some applications, notably with very high speed or rapid acceleration, these transient cage effects may beof sufficient magnitude to warrant evaluation. The steady-state analytical approaches discusseddo not address the time-dependent behavior of rolling element bearing cage. Several researchers have developed analytical models for transient cage response 114.24, 14.27-14303. plexity of the calculations involved, such performance an extensive time on present-day c o ~ p ~ t e r s . neral, the cage is treated as a rigid body subjected to a complex of forces. These forces may includethe following: pact and frictional forces at the cage-rolling element interface mal and fri~tionalforces at the cage-land surface (if land-
533
CAGE M O T I O A.lW ~ ~ FORCES
Cage mass unbalance force
. Gravitational force
e
Cage inertial forces Others (i.e., lubricant drag on the cage and lubricant churning forces) Forces 1and 2 are intermittent. For example the cage might or might not be in contact with a given rolling element or guide flange at a given time, depending on the relative position of the bodies in question. Frictional forces can be modeled as hydrodynamic, EHL, or dry f~ction,depending on the nature of the lubricant, contact load, and geometry. Both elastic and inelastic impact models appear in the literature. General equations of motion for the cage may be written. The Euler equations describing cage rotation about its center of mass (in Cartesian coordinates) are as follows:
Ixhx- ( I y - I,)
oyo,= Mx
(14.91)
Iyoy - ( I , - I,) ozox= My
(14.92)
I,&,
(14.93)
-
(I, - I,)
oxwy =
M'
where I,, Iy, I' are the cage principal moments of inertia, and ox,my, o, are the angularvelocities of the cage about the inertial x, y , z axes. The total moment about each axis is denoted by M,, M y , and M', respectively. The equations of motion fortranslation of the cage center of mass in the inertial reference frame are
mrx = F,
(14.94)
Fy
(14.95)
nary
=
where m is cage mass, rx, ry? r, describe the position of the cage center of mass, and F,, Fy, F, are the netforce components acting on the cage. Oncecageforce and moment components are determined, accelerations can be computed. Numericalintegration of the equations of motion (with respect to discrete time increments) will yield cage translational velocity? rotational velocity, and dis~lacementvectors. In some approaches [14.24] fl4.281 the cage dynamics model is solved in conjunction with roller and ring equations of motion. Other researchers have devised less cumbersome approaches by limiting the cage to im-plame motion [14.271 or by considering simplified dynamic models forthe rolling elements 114.293.
FRICTION IN ~ O ~ I N B G
Meeks and Ng [14.29] developed a cage dynamics model for ballbearings, which treats both ball-and ringland-guided cages.This model considers six cage degrees of freedom and inelastic contact between balls and cage and between cage and rings. This model was used to perform a cage design optimization study for a solid-lubricated, gas turbine engine bearing [14.301. The results of the study indicated that ball-cage pocket forces and wear are significantly affectedby the combination of cage-land and ballpocket clearances. Using the analytical model to identify more suitable clearance values improved experimental cage performance.Figures 14.35 and 14.36 contain typical output data from the cage dynamics analysis. In Fig. 14.35 the cage center of mass motion is plotted versus time for X and Y (radial plane) directions. Thetime scale relates to approximately five shaft revolutions at a shaft speed of 40,000 rpm. Figure 14.36 shows plots of ball-cage pocket normal force fortwo representative pockets positioned approximately 90” apart. In addition to the work of Meeks [14.30], Mauriello et al. E14.311 succeeded in measuring ball-to-cage loading in a ball bearing subjected to combined radial and thrust loading. They observed impact loading between balls and cage to be a significant factoron high speed bearing cage design.
Thus far in this section, rollers have been assumed to run “true” in cylindrical, spherical, and tapered roller bearings. In fact, due to slightly imperfect geometrythere is an inevitable tendency for unbalance of frictional loading betweenthe roller-inner raceway and roller-outer raceway contacts, and thus a tendency for rollers to skew. Additionally, in a misaligned radial cylindrical roller bearing, as indicated sc~ematicall~ in Fig. 7.23, rollers are “squeezed” at one end and thereby forced against the “guide” flange. The latter causes a roller end-flange frictional force and hence a roller skewing momentthat must be substantially resisted by the cage. In tapered roller bearings, even without misalignment, the rollers are forced against the large end flange and skewing moments occur, The thrust load applied to radial cylindrical roller bearings, as discussed in Chapter 7, results in a roller skewing moment that is augmented by unbalance of raceway-roller friction forces, as indicated in Fig. 14.37. In most cases roller skewing is detrimental to roller bearing operation because it causes increased friction torque and frictional heat generation as well as necessitating a cage strong enough to resist the roller moment loadi~g.
0'10
I
0 08
0 06
-
Ball pocket clearance = 0.2 mm (0.008IN) Race land clearance = 0.4 mm (0.016IN)
2ot
0.04 e . I
I
0
-
"4
x
0 02
. . 6
-
-C
-0.oc
- 0.02 -0.04
-0.ot-
1 12
1.20
128
1.36
1.44
1 6105 2
Time (5) ( x
(a>
Tlme (sf ( x 10")
(b)
14.35. Calculated cage motion versus time. (a)Prediction of cage motion, X vs time. ( b ) Prediction of cage motion, Y vs time (from [14.301).
FRI~TIONIN'R O ~ I NI3' ~ 12 01
10 c l c
8 OC
9
60C
4 00
2 00
0 00
12 00 50
10.00
40
0
0
Time (sed ( X 10-2)
04 14.36. Calculated ball-pocket force vs time. ( a )Prediction of cage ball-pocket force vs time (pocket No. 1). (b)Prediction of cage ball-pocket force vs time (pocket No. 4) (from 114.301).
Thenotion that rollers skew until skewingmoment equilibrium is achieved has implications beyond those of roller end-flange load determination. In spherical roller bearings with symmetric rollerprofiles, proper management of roller skewing can reduce frictional losses and
Qij 14.37. Normal, axial, and frictional loadingof a roller at azimuth Jlj; in a radial cylindrical roller bearing subjected to radial and thrust applied loading.
corresponding frictiontorque. Early spherical roller bearing designs employing asymmetrical roller profiles, because of their close osculations and primary skewing guidance from cage and flange contacts, exhibit greater friction than current bearings with symmetrical roller designs. The temperature rise associated with friction limits performance in many applications. Designing the bearings so that skewing e~uilibrium is provided by raceway guidance alone lowers lossesand increases loadcarrying capacity. Kellstrom E14.32, 14.331 investigated skewing equilibrium in spherical roller bearings considering the complex changes in roller force and moment balance causedby roller tilting and skewing in the presence of friction. Any rolling element that contacts a raceway along a curved contact surface will undergosliding in the contact. For an unskewed rollerthere will be at most two points along each contactwhere the sliding velocity is zero. These zero sliding points form the generatrices of a theoretical "rolling" cone, which represents the contact surface on which pure kinematic rolling would occur for a given roller orientation. At all other points along the contact, sliding is present in the direction of rolling or opposite to it, depending on whether the roller radius is greater or less than the radius to the theoretical rolling cone. This situation is illustrated in Fig. 14.38. Friction forces or tractions due to sliding will be oriented to oppose the direction of sliding on the roller. In the absence of tangential roller forces from cageor flange contacts, the roller-raceway traction forces in
FRICTION IN ROLL IN^ I3 Points of rolling
Points of rolling
Sphericalrollerbearing, sprne and forcedirection: 0 out of page;page.
oller-tangentialfrictionforcedi-
each contact must sum to zero. Additionally, the sum of the inner and outer raceway contact skewing momentsmust equal zero. These two conditions will determine the position of the rolling points along the contacts and thus the theoretical rolling cone. These conditions are met at the equilib~umskewing angle. If the moments tend to restore the roller to the equilibriu~skewing angle when it is disturbed, the e~uilibrium skewing angle is said to be stable. As a roller skews relative to its contacting raceway a sliding component is generated in the roller axial direction and traction forces are developed that oppose axial sliding. These traction forces may be beneficial in that, if suitably oriented, they help to carry the axial bearing load, as indicated in Fig. 14.39. Those skewing angles that produce axial tractions opposing the applied axial load and reducing the roller contact load require applied axial load are termed positive (Fig. 14.39a). Conversely, those skewing angles producing axial tractions that add to the applied axial load are termed negative (Fig. 14.393). Fora positive skewing rollerthe normal contact loading is reduced, and an improvement in contact fatigue life achieved. The axial traction forces acting on the roller also produce a second effect. These forces, acting in different directions on the inner and on outer ring contacts, create a moment about the roller and cause it to tilt.
\ \
I I
AQ\,
1
', I \ I
\I
.
(b)
Y
Forces on outer raceway of axially loaded spherical roller bearing with positive and negative skewing. (a)Positive skewing angle. ( b ) Negative skewing angle.
The tilting motion respositions the inner and outer ring contact load distributions with respect to the theoretical points of rolling and distribution of sliding velocity. Detailed evaluations [14.32, 14.331 of this behavior have shown that skewing in excess of the equilibrium skewing angle generates a net skewing moment opposingthe increasing skewing motion. A. roller that skews less than theequilibrium skewing an generate a net skewing momenttending to increase the skew angle.This set of interactions explains the existence of stable equilibrium skewing angles. To apply this concept to the design of spherical roller bearings, specific design geometries over a wide range of operating conditions must be evaluated. There are tradeoffsinvolvedbetweenminimizingfriction losses and maximizing contact fatigue life. Some designs may exhibit unstable skewing controlin certain operating regimes or stable skewing equilibrium and require impractically large skewing angles. Computer programs that predict spherical roller bearing performance contribute to more accurate evaluations. See Fig. 14.40, whichshows the possible tradeoffs between frictional power loss and calculated fatigue life for a
5
FRICTION IN ROLLING B
0.08r
~
C
0.07
B
2
0.06
0
= 0.97
v
20
1812 16
22
14
24
26
L,, Fatigue life (100h)
14.40. Study of frictional power loss vs calculated fatigue lifeof spherical roller bearing with equilibrium skewing control. QIP = ratio of bearing power loss to applied load. 0, = outer raceway osculation. Oi = inner raceway osculation.
bearing design using skewing control.Results are shown forseveral Values of outer and inner raceway osculation.
e
ie
Exclusive of an analytical approach to determine bearing friction torque, Palmgren [14.4] empiricallyevaluated bearing friction torque due to all mechanical friction phenomena with the exception of friction owing to the quantity of lubricant contained within the bearing boundary dimensions; that is, within the bearing cavity. Data were compiled on each basic bearing type. Palmgren [14.4] gave the following equation to describe this torque: (14.97) in which fi is a factordependingupon bearing load. For ball bearings,
bearing design and relative (14.98)
~
ING F ~ I ~ ~ I TO^^^ O N
in which F, is static equivalent load and C, basic static load rating (these terms are explained in Chapter 21 covering plastic deformation and static capacity). Table 14.3 gives appropriate values of x and y. Values of 6, are generally given in m~ufacturers'catalogs alongwith data to enablecalculation of Fs. The internal designs of roller bearings have changed both from macrogeometrical and microgeometrical bases since the publicati~nby Palmgren 114.41. Therefore, Table 14.4 as updated according to data from 114.341 givesempirical values of f,for roller bearings. For moderndesign,double-row, radial spherical roller b e ~ i n ~ s , SKP 114.341 uses the formula:
M,
(14.99)
= f,F"db
in which constant f, and exponents a and b depend upon the specific bearing series. As the internaldesign of these bearings is specific t o S the catalog [14.34] should be consulted to obtain the required values of fi, a, and b.
14.3 Values of z and y ~~
Ball Bearing Type "
Nominal Contact Angle
z
Y
0.006-0.004~ 0.001 0.0008 0.0003
0.55 0.33 0.33 0.40
-
Radial deep groove Angular contact Thrust Double-row, self-aligning
0" 30-40"
90" 10"
aLower values pertainto light series bearings; higher values pertain to heavy series bearings.
LE 14.4, f i Roller Bearings Roller Bearing Type
fi
Radial cylindrical with cage Radial cylindrical, full complement Tapered Radial needle Thrust cylindrical Thrust needle Thrust spherical
0.000~-0.0004u 0.00055 0.0004 0.002 0.0015 0.0015 0.00023-0.0005u
pertainto light series bearings; higher values pertain to heavy series bearings.
a Lower values
F p in equation (14.97) depends on the magnitude and direction of the applied load. It may be expressed in equation form as follows for radial ball bearings:
F p = 0.9 Fa ctn a - O . l F r
(14.100)
F p = Fr
(14.101)
or
Of equations [14.100], the one yieldingthe larger value of F p is used. For deep groove ball bearings, with nominal contact angle O", the first equation can be a p ~ r o x i m a by ~e~
F p = 3Fa - 0.Wr
(14.101)
F p = 0.8Fa ctn a
(14.102)
For radial roller bearings,
F p = Fr ~ g ~ ithe n ,larger value of F p is used. For thrust bearings, either ball or roller, F p = Fa. These values of torque as calculated from equation (14.97) appear to be reasonably accurate for bearings operating under reasonable load and ~ E14.351 a used ~ these~ relatively slow speed co~ditions.( cessfully in the thermal evaluation of a submarine propeller shaft thrust bearing assembly.)
e Complex methods forcalculating viscous friction forces in lubricat~drolling b e a ~ n g swere indicated in Chapter 12, which dealt with elastohydrod~amiclubrication. In lieu of those methods to estimate friction torque, a simpler, empirical method was developed to cover standard bearing types. For bearings that operate at moderate speeds and under not-~xcessive load, P a l m ~ e nE14.41 determined empirically that viscous frictiontorque can be expressed as follows:
ENG ~ R I C ~ IOR^^ O ~ ~u
=
43
1 0 - ~ f , ( ~ ~ n ) ~v,n' ~ 2 d ~2000
Mu = 160 X 10-7f0dg v,n
5 2000
(14.103) (14.104)
in which v, is given in centistokes and n in revolutions per minute. In equations (14.103) and (14.104), f, is a factor depending upon type of bearing and method of lubrication. Table 14.5 as updated in [14.34] gives values off, for various types of bearings subjected to different conditions of lubrication. Equations (14.103) and (14.104) are valid for oils having a specific gravity of approximately 0.9. Palmgren [14.4] gavea more complete formula for oils of different densities. For grease-lubricated bearings, kinematic viscosity v, refers to the oil within the grease, and the equation is approximately valid shortly after the addition of lubricant. Radial cylindrical rollerbearings with flanges on both inner andouter rings can carry thrust load in addition to the normal radial load. In this case, the rollers are loaded against one flange on each ring. The bearing friction torque due to the roller end motions against properly designed and manufactured flanges is given by
.
Values of fo vs Bearing Type and Lubrication Type of Lubrication
Bearing Type
Grease
Oil Mist
Deep groove ball" Self-aligning ball" Thrust ball ~ ~ l a r - c o n t aball" ct Cylindrical roller with cage" full complement Spherical rollerc Tapered roller" Needle roller Thrust cylindrical roller Thrust spherical roller Thrust needle roller
0.7-2b 1.5-=Zb 5.5 2
1 0.7-lb 0.8 1.7
2 1.5-2b 1.5 3.3
4 3-4b 3 6.6
0.6-lb 5-10b 3* 5-7b 6 12 9
1.5-2.8b
2 .2-4b 5-10b 3.5-7b 6 12 3.5 2.5-5' 5
2.2-4b*d
-
14
-
1.7-3.5b 3 6
Oil Bath
Oil Bath (vertical shaft) or Oil Jet
-
7-14b 8--10b7d 24 8 5-10b 11
aUse 2 X f,value for paired bearings or double row bearings. bLower valuesare for light series bearings; higher valuesare for heavy series bearings. "Double row bearings only dFor oil bath lubrication and vertical shaft, use 2 X f,.
544
~ I ~ T I O INNROLL
(14.105) Values of ff are given in Table 14.6 when Fa/FrI 0.4 and the lubricant is sufficiently viscous. UE?
A reasonable estimate of the friction torque of a given rolling bearing under moderate load and speed conditionsis the sum of the load friction torque, viscous friction torque, and roller end-flange friction torque, if any, that is,
M
=
M,
+ Mv + Mf
(14.106)
Since M , and Mv are based on empirical formulas, the effect of rolling element-cage pocketsliding friction is included. For high speedball bearings for which frictiondue to spinning motions becomes important, the equations previously given should enable a calculation of friction torque. This torque should be added to that of equation (14.106). It must be remembered also that equation (14.106) does not account for friction torque due to seals, which in most instances far exceeds the friction torque of the bearing alone. 2. Estimate the total friction torque for a 209 cylindrical roller bearing rotating at 10,000 rpm and supporting a radial load of 4450 N (1000 lb). The bearing is lubricated by a mineral oil bath, the oil having a kinematic viscosity of 20 centistokes.
dm= 65 mm (2.559 in.)
2.7 Ex.
Ex. 2.7
L) = 10 mm (0.3937 in.)
Ex. 2.7
y = 0.1538
2
Z
LE 14.6, Values of
f f
=
14
=(0.378 mm 9.6
2.7 in.)
Ex. 2.7 Ex.
for Radial Cylindrical RollerBearings Type of Lubrication
Bearing Type With cage, optimum design With cage, other designs Full complement, singlerow Full complement, double row
Grease
Oil
0.003
0.002
0.009
0.006 0.003 0.009
0.006 0.015
B
E
~~~ I C GT I O TORQUE ~
From Table 14.4, assume f l having a cage.
Ml
=
0.0003 for a medium series bearing
= f1F;dm
(14.97)
=
0.0003
=
86.78 N mm (0.7677 in. * lb)
X
4450
X
65
von = 20 x 10,000 = 200,000 (14.103)
M,
=
10-7 f 0 ( v 0 n ) ~ 3 d ~
For oil bath lubrication from Table 14.5, assume f o series bearing,
M, M
=
10-7
=
281.8 N mm (2.493 in. * lb)
=
M1 + M"
=
86.8
X
3
X
=
3 for a medium
~ 2 0 0 , 0 0 0 ~X~(6513 3
+ M;.
(14.106)
+ 281.8 + 0 = 368.6 N
*
mm (3.261 in. * lb)
~ ~ . ~ .
le Estimate the rolling friction torque and viscous friction torque of the 219 angular-contact ball bearing operatinga t a shaft speed of 10,000 rpm and a thrust load of 22,250 N (5000 lb). The bearing is jet lubricated by a highly refined mineral oil having a kinematic viscosity of 5 centistokes a t operating temperature.
dm = 125.3 mm (4.932 in.)
D
=
22.23 mm (0.875 in.)
Ex. 2.6 Ex. 2.3
a = 40" (nominal)
Ex. 2.3
y = 0.1359
Ex. 2.6
f
Ex. 2.3
=
0.5232
2 = 16
6, = (p,iZD2cos a
Ex. 2.5 (21.8)
FRICTIO~IN RO
From Table 21.2 a t y
C,
0.1359, q, = 15.48
=
=
15.48
=
93,760 N (21,070 lb)
= XsFr
F,
From Table 21.2, X ,
f1
(22.2312 cos 40"
+ Y,Fa
(21.15)
+ 0.26 X
=
0.5
=
5785 N (1300 lb)
=
.dF,/C,)Y
X
0.001; y
=
X
0.5; Y,= 0.26 for a
=
F,
From Table 14.3, x
1 X 16
X
0
40°,
22,250
(14.98)
0.33 for a
=
=
=
40"
f l = 0.001 (~,785/93,760)0.33 =
0.0003988
Fb = 0.9Fa ctn a"
Ml
=
23,860 N (5363 lb)
X
(14.100)
22,250 ctn 40" - 0.1
0.9
X
0
= flfigd,
(14.97)
=
0.0003988
=
1192 N * mm (10.55 in. * lb)
=
6.6 for oil jet lubrication
von = 5 =
X
=
X
23,860
X
125.3
10,000 = 50,000
10-7~0(v0n)2'3d~
=
M
O.lFr
=
From Table 14.5, f o
M,
-
X
6.6
X
(14.103)
(50,000)~3X (125.313
1762 N * mm (15.59 in. * lb)
+ Mv + M f = 1192 + 1762 + 0
=
Ml
=
2954 N mm (26.13 in. * lb)
(14.106)
Rolling bearings are sometimes called~nti~iction bearings to emphasize the small amount of frictional power consumed during their operation. Notwithstanding, it has been shown in this chapter that therolling process does involve frictional power losses from various sources. Recent basic research had done muchto define the mechanics of rolling friction, and for certain ideal conditions of rolling, estimates of rolling friction torque .can be made. The operation of industrial rolling bearings that employ curved raceways, cages, and seals is, however, far from ideal in that sources of frictional power lossother than rolling are present in the bearings. Therefore, although it is important to understand themechanics of rolling friction, empirical data are usually required to define friction torque of rolling bearing assemblies.Theseempirical dataare presented in the previous section. Rolling bearing friction is manifested as temperature rises in therolling bearing structure and lubricant unless effective heat removal methods are employed or naturally occur. When excessive temperature level occurs, the rolling bearing steel suffers loss in its ability to resist rolling surface fatigue and the lubricant undergoes deterioration such that it is ineffective. ~ubsequently,rapid bearing failure may beanticipated. Bearing thermalanalysis and methods of heat removal are discussed fixrther in Chapter 15. Rolling bearing friction also tends to retard motion. In sensitive control systems such as those employing instrument bearings, torque due to bearing friction can significantly affectrotor speed.
14.1. R. Drutowski, “Energy Losses of Balls Rolling on Plates,” Friction and Wear, Elsevier, h s t e r d a m , 16-35 (1959). 14.2. R. Drutowski, “Linear Dependence of Rolling Friction on Stressed Volume,” Rolling Contact P h e n o ~ e n aElsevier, , h s t e r d a m , (1962). 14.3. A. Jones, “Motions in Loaded Rolling Element Bearings,” ASME Trans., J. Basic Eng., 1-12 (1959). 14.4. A. Palmgren, Ball and Roller Bearing Engineering, 3rded., Burbank, Philadelphia, 34-41 (1959). 14.5. 0. Reynolds, Philos. Trans. Royal Soc. London, 166, 155 (1875). 14.6. H. Poritsky, J. Appl. Mech., 72, 191 (1950). 14.7. B. Cain, J. Appl. Mech., 72, 465 (1950). 14.8. I(. Johnson, “Tangential Tractions and Micro-slip,”Rolling Contact Phenomena, Elsevier, h s t e r d a m , 6-28 (1962).
54
FRICTION IN R Q L L B~ ~
~
14.9. H. Heathcote, Proc. Inst. Automobile Eng. London, 15, 569, (1921). 14.10. J. Greenwood and D. Tabor, Proc. Phys. Soc. London, 71,989 (1958). 14.11. T. Harris, Rolling Element Bearing Dynamics,” Wear, 23, 311-337 (1973). 14.12. V. Streeter, Fluid Mechanics, McGraw-Hill, New York, 313-314 (1951). 14.13. E. Bisson and W. Anderson, Advanced Bearing Technology,NASA SP-38 (1964). 14.14. B. Rydell, “New Spherical Roller Thrust Bearings, the E Design,” Ball Bearing 6, SKF, 202,l-7 (1980). 14.15. P. Brown, L. Dobek, F. Hsing, and J. Miner, “Mainshaft High Speed Cylindrical Roller Bearings for Gas Turbine Engines,” U. S. Navy Contract N00140-76”C-0383, Interin Report FR-8615 (April 1977). 14.16. T. Harris, “Ball Motion in Thrust-Loaded, Angular-Contact Ball Bearings with Coulomb Friction,” ASME Dans., 6 Lub-Tech., 93, 32-38 (1971). 14.17. T. Tallian, G. Baile, H. Dalal, and 0.Gustafson, Rolling BearingDamage Atlas, SKF Industries, Inc., King of Prussia, PA, 119-143 (1971). 14.18, T. Harris, “An Analytical Method to Predict Skidding in Thrust-Loaded, AngularContact Ball Bearings,”ASME Duns., 6 Lub.Tech., 93, 17-24 (1971). 14.19, R. Shevchenko and P. Bolan, ‘Tisual Study of Ball Motion in a High Speed Thrust Bearing,” SAE Paper No. 37 (January 14-18, 1957). 14.20 J. Poplawski and J . Mauriello, “Skidding in Lightly Loaded, High Speed, Ball Thrust Bearings,” ASME Paper 69-LUBS-20 (1969). 14.21 R. Parker, “Comparison of Predicted and Experimental Thermal Performance of Angular-Contact Ball Bearings,” NASA Tech. Paper 2275 (1984). 14.22 T. Harris, “An Analytical Method to Predict Skidding in High Speed ‘Roller Bearings,” ASLE Trans., 9, 229-241 (1966). 14.23 T. Harris andS. Aaronson, “An Analytical Investigation of Skidding in a High Speed, Cylindrical Roller Bearing Having Circurnferentially Spaced, Preloaded Annular Rollers,”Lub. Eng., 30-34 (January 1968). 14.24 C. Walters, “The Dynamics of Ball Bearings,’’ASME Trans., 6 Lub. Tech., Vol. 93, (11, 1-10 (January 1971). 14.25. F. Wellons and T. Harris, “Bearing Design Considerations,” Interdisciplina~ Approach to theLubrication of Concentrated Contacts, NASA SP-237, 529-549 (1970). 14.26. R. Kleckner and J. Pirvics, “High Speed Cylindrical Roller Bearing A n a l y s i s - S ~ Computer Program CYBEAN, Vol. I: Analysis,” SKF Report AL78P022, NASA Contract NAS3-20068 (July 1978). 14.27. J, Kannel and S. Bupara, “A Simplified Model of Cage Motion in Angular-Contact Bearings Operating in the EHD Lubrication Regime,”ASME Dans., J. Lub. Tech., 100, 395-403 (July 1978). 14.28. P. Gupta, “Dynamics of Rolling Element Bearings-Part I-IS7 Cylindrical Roller Bearing Analysis,”A S M ~Trans., 6 Lub. Tech., 101, 293-326 (1979). 14.29. C. Meeks and K. Ng, “The Dynamics of Ball Separators in Ball Bearings-Part I: Analysis,” ASLE Paper No. 84-ANI-6C-2 (May 1984). 14.30. C. Meeks, “The Dynamics of Ball Separators in Ball Bearings-Part 11: Results of Optimization Study,” ASLE Paper No. 84-ANI-66-3 (May 1984). 14.31. J. Mauriello, N. Lagasse, A. Jones, and W. Murray, “Rolling Element Bearing Retainer Analysis,” U. s. Army ANIRDL Technical Report 72-45 (November 1973). 14.32. M. Kellstrom, and E. Blomquist, “Roller Bearings Comprising Rollers with Positive Skew Angle,”U. S. Patent 3,990,753 (1979).
~
REFERENCES
54
14.33. M. Kellstrom, “Rolling Contact Guidance of Rollers in Spherical Roller Bearings,” ASME Paper 79-LUB-23 (1979). 14.34. SKI?, General Catalog 4000 US, 2nd ed. (1997). 14.35. T. Harris, “Prediction of Temperature in a Rolling ContactBearing Assembly,”Lub. Eng., 145-150 (April 1964).
This Page Intentionally Left Blank
Symbol
Units
Description Specific heat Rolling element diameter Diameter Thermal emissivity Temperature coefficient Acceleration due to gravity Grashof number Film coefficient of heat transfer
w
Heat flow Conversion factor, IO3 N * mm = 1W sec Thermal conductivity Length of heat conduction path Friction torque Rotational speed
W (Btu/hr)
sec/g "C (Btullb * mm (in.) m (ft) 0
0
m/sec2 (in./sec2) W/m2 * "C (Btu/hr * ft2
"Iio
OF)
ROLLING ~
Symbol
Pr 4
Re %
s T us u W
Yo X E
rl V LL)
R a C
f j 0
r S V
1 2
Description Prandtl number Error function Reynolds number Radius Area normal to heat flow Temperature Fluid velocity Velocity (ftlsec) Weight flow glsecrate Width Distance in x-direction Error Absolute viscosity Fluid kinematic viscosity Rotational velocity Rotational velocity
~
I TN E G~ E ~ T ~ E S
Units
m2 (ft2) "C ( O F ) (ftlsec)mlsec mlsec (lblsec) m (Et;) (ft)m cp (lb seclin.2) m2/sec (ft21sec) radlsec radlsec
SUBSCRIPTS Refers to air or ambient condition Refers to heat conduction Refers to friction Refers to rolling element position Refers to oil Refers to heat radiation Refers to spinning motion Refers to heat convection Refers to temperature node Refers to temperature node, and so on
The temperature level at which a rolling bearing operates is a f~nction of many variables amongwhich the following are predominant: bearing load bearing speed bearing friction torque ~ubricanttype and viscosity bearing mounting andlor housing design environment of operation
In the steady-state operation of a rolling bearing, as for any other machine element, whatever heat is generated internally is dissipated. Therefore, the steady-state temperature level of one bearing system compared to another system using identical sizes and number of bearings is a measure of the relative ability of that system’s efficiency of heat dissipation. Of course, if the rate of heat dissipation is less than the heatgeneration rate, then an unsteady state exists and the system temperature will rise until lubricant distress occurs, with ultimate bearing failure. The temperature level at which this occurs is determined largely by the type of lubricant and the bearing material. This dissertation is limited to the steady-state thermal operation of rolling bearings since this is a common concern of bearing users regarding satisfactory operation. Most rolling bearing applications perform at temperature levels that are relatively cool and therefore do not require any special consideration regarding thermal adequacy. This is due to either one of the following conditions:
. The bearing heat generation rate is low because of light load and/ or relatively slow speed. . The ability to remove heat from the bearing is sufficient becauseof
location of the bearing assembly in a moving air stream or because or adequate heat conduction through adjacent metal.
Some applications occur in certain adverse environmental conditions such that it is certain that external cooling is required. A rapid determination of the bearing cooling requirements may then suffice to establish the cooling capability that must be applied to the lubricating fluid. In other applications it is not obvious whether external cooling is required, and it may be economically advantageous to establish analytically the thermal conditions of bearing operation.
though rolling bearings have been called ~ ~ t i f r i c t ibearings, o~ nevertheless, they exhibit a small amount of friction during rotation. This should be evident since if friction were notpresent, the rolling elements would slip on the rotating ring rather than roll. ~ c t i o nin a rolling bearing as in most other mechanis~srepresents a wasteful power dissipation manifested in the form of heat generation. his frictional power must be effectively removed or an unsati~factory temperature condition will obtain in the bearing.
Having determined the bearing friction torque by methods of Chapter 14, one can obtain the bearing frictional power loss in watts from the following equation:
Hf= 1.047 X
nM
(15.1)
For a high speed ball bearing, the local heat generation rate due to sliding at each contact is given by (15.2)
where J is a constant converting N mlsec to watts, anduk is the sliding velocity in the direction of the differential friction force dFk. The rate of heat generation at a raceway contact due to spinning alone is given by (15.3)
Equations (15.2) and (15.3) must be evaluated separately for inner and outer raceway contacts at each rolling element location. Having established values for the ratesof heat generation due to friction, it is then possible to determine the bearing temperature structure .. by heattransfer analysis.
1, For the 218 angular-contact ball bearing of Examples 9.1 and 14.1, estimate the rate of heat generation due to spinning motion at the inner raceway contacts for 22,250 N (5000 lb) thrust load at 10,000 rpm shaft speed. Ex. 6.5
2 = 16
48.8"
Fig. 9.4
cy, = 33.3"
Fig. 9.4
D
22.23 mm (0.875 in.)
Ex. 2.3
dm = 125.3 mm (4.932 in.)
Ex. 2.3
cyi
=
=
Y ' = - d=m-
22*23 125.3 = 0.1774
Assuming simple rollingat the outer raceway contact
tan ,6
sin ab cos a, + y'
=
sin(33.3") = 0.5419 cos(33.3") + 0.1774
-
(8.59)
/3 = 28.5"
1 - y' COS ai - 1 - 0.1774 * cos(48.8") = 0.4498 1 + cos(cr, - ao) 1 + COS (48.8" - 33.3")
~rn-%=
"
n
(L,
(8.63)
n,
=
0.4498 * 10,000 = 4498 rpm (8.57)
=
(1- 0.1774 * cos(48.8'))tan(48.~ - 28.5")
+ 0.1774 wo = --urnfor =
@o -
-
"
Y'
sin(48.8") = 0.4602
inner raceway rotation
wrn -
(8.50)
Y'
or
n, = 2.594
X
4498 = 11,670 rpxn
MSi= 43.19 N mm (0.382 in. * lb) HG = 1.047 X
Since 2
=
16,
X
Msij
=
1.05 X
=
52.92 VV (180.6 ( B t d h r )
X
111670 X 43.19
Ex. 14.1 (15.1)
ROLLING B
~
~
G
Hf = 16 X HG =
16 X 52.92
,+ A4, = 2860 N
*
=
846.7 W (2889 Btu/hr)
mm (25.27 in. * lb)
Ex. 14.3
The total heat generation rate is calculated as follows: ,+ = ,
ot
1.047 X
nM,,,
(15.1) 3003 W (10248 Btu/hr)
=
1.05 X
X
10,000 X 2860
=
3003 + 847
=
3850 W (13140 Btulhr)
=
Figure 15.1 shows heat generation rates for this bearing for a load range of 44,500 N (0-10,000 lb) and a shaft speed of 10,000 rpm.
There exist three fundamental modes for the transfer of heat between masses having different temperature levels. Theseare conduction of heat within solid structures, convection of heat from solid structures to fluids in motion (or apparently at rest), and radiation of heat between masses separated by space. Although other modes exist, such as radiation to gases and conduction within fluids, their effects are minor for mostbearing applications and may usually be neglected.
etio eat conduction? which is the simplest form of heat transfer, may be described for the purpose of this discussion as a linear function of the difference in temperature level within a solid structure, that is, (15.4) The quantity S in equation (15.4) is the area normal to the flow of heat between two points and 6 is the distance between the same two points. The thermal conductivity lz is a function of the material an is generally minor for ever, the latter varia heat conduction in a r 11 be neglected here.
HEAT T ~ ~ F E R
557 N
0
18,000
10,000
20,000
30,000
40,000
I
I
50,0(
1 5000
4000
3000
II) 4 . d
c
i2 2000
1000
0
Bearing thrust load, Ib
F ~ ~ U R15.E1. Friction heat generation vs load; 218 angular-contact ball bearing,10,000 rpm, 5 centistokes oil,jet lubrication.
within a cylindrical structure such as a bearing inner or outer ring, the following equation is useful:
He =
2 ~ k ? 3 ( T-~To) ln (ti't,/&)
(15.5)
In equation (15.~9, TO is the width of the annular structureand $&and $j$ are the inner andouter radii defining the limits of the structure through
which heat flow occurs. If % = 0, an arithmetic mean area is used and the equation assumes the form of equation (15.4).
Heat convection is the most difficult form of heat transfer to estimate quantitatively. It occurs within the bearing housing as heat is transferred t o the lubricant from the bearing and from the lubricant to other structures within the housing as well as to the inside walls of the housing. It also occurs between the outside of the housing and the environmental fluid-generally air, but possibly oil, water, another gas, or a working fluid medium. eat convection froma surface may generally be described as follows:
in which h,, the film coefficient of heat transfer, is a function of surface and fluid temperatures, fluid thermal conductivitx fluid velocity adjacent to the sudace, surface dimensions and attitude, fluid viscosity, and density. It can be seen that many of the foregoing properties are temperature dependent. Therefore, heat convection is not a linear function of temperature unless fluid properties can be considered reasonably stable over a finite tem~eraturerange. Heat convection within the housing is most difficult to describe, and a rough a~prosimationwill be used forthe heat transferfilm coefficient. Since oil is used as a lubricant and viscosity is high, laminar flow is assumed. Eckert [15.1] states for a plate in a laminar flow field:
h,
=
0.0332kPr1’3 (
~ ) 1 ’ 2
(15.7)
The use of equation (15.7) taking u, equal to bearing cage surface velocity and x equal to bearing pitch diameter seems to yield workablevalues for h,, conside~ngheat transferfrom the bearing to the oil that contacts the bearing. For heat transfer from the housing inside surface to the oil, taking u, equal to one-third cage velocityand x equal to housing diameter yields adequate results. In equation (15.7), vo represents kinematic viscosity and Pr the Prandtl number of the oil. If cooling coils are submerged in the oil sump, it is best that they be aligned parallel to the shaft so that a laminar cross flow obtains. In this case Eckert El5.11 shows for a cylinder in cross flow, the outside heat transfer film coefficient may be approsimated by
(15.8) in which 9 i s the outside diameter of the tube and h, is the thermal conductivity of the oil. It is recommended that u, be taken as approximately one-fou~hof the bearing inner ring surface velocity. Theforegoing approximations forfilmcoefficient are necessarily crude. If greater accuracy is required, Reference [15.1] indicates more refined methods for obtaining the film coefficient. In lieu of a more elegant analysis, the values yielded by equations (15.7) and (15.8), and (15.9) and (15.10) that follow, should suffice forgeneral e n ~ n e e r i purn~ poses. In ~uiescentair, heat transferby convection fromthe housing external surface may be approximated by using an outside film coefficient in accordance with equation (15.9) (see Jakob and Hawkins [15.2]):
h, = 2.3
X
10-5 (T-
1~~10.25
(15.9)
For forced flow of air of velocity u, over the housing, Reference [15.l] yields:
hv
=;
0.03 -
(15.10)
in which $h is the approxi~atehousing diameter. Palmgren [15.3] gives the following formulato approximate the external area of a bearing housing or pillow block: (15.11) in which g h is the m ~ i m u mdiameter of the pillow block and 96, is the width. The calculationsof lubricant film thickness as specified in Chapter 12 depend upon the viscosit~of the lubrieant entering the rolling/s~i contact, whilethe calculations of traction over the contact as specified in Chapter 13 depend upon the viscosity of the lubricant in the contact. detailed performince lubricant viscosity is a function of tem~erature~ nce analysis of ball and roller bearings entails the estimation of terneratures of lubricant both entering, and residing in, the individual o this requires the e s t i ~ ~ t i of o nheat dissipation rates from the rotating components and rings. The coefficien transfer for a rota tin^ sphere (ball) is provided by lows:
h D k
V" - 0.33 Reg5
(15.12)
where Re,, the Reynolds number for a rotating ball, is given by (15.13) In equation (15.13), 13 is the diameter of the ball, o is ball speed about its own axis, and v is the lubricant kinematic viscosity. Equation (15.12) is valid for 0.7 < Pr < 217 and Gr, < 0.1 Re;. The Grashof number is given by
where 3 is the thermal coefficient of fluid volume expansion, g is acceleration due to gravity, T8is the temperature at theball surface, and Tw is the fluid stream temperature. The Prandtl number is given by
Pr
=
rlgc -
k
(15.15)
where c is specific heat of the fluid. For a rotating cylindrical ring or roller, (15.16) In equation (15.16),D is the outside diameter of the ringor roller. Equation (15.16) is valid for Re, < 4 io5.
The remaining mode of heat transfer t o be considered is radiation from the housing external surface to surrounding structures. For a small struct~re in a large enclosure reference El5.21 gives (15.17) in which tem~eratureis in degrees Kelvin (absolute). Equation (15.17) being nonlinear in tem~eraturesis sometimes written in the following form:
H,
=
h,S(T
-
Ta)
(15.18)
in which
h, = 5.73
X
IO-' &(T+ T J T 2
+ Tz)
(15.19)
~quations(15.18) and (15.19) are useful for hand calculation in which problem T and Ta are not significantly different. Upon assuming a temperature T for the surface, the pseudofilm coefficient of radiation h, may be calculated. Of course, if the final calculated value of T is significantly di~erentfrom that assumed, then the entire calculation must be repeated. Actually, the same consideration is true for calculation of h, for 1 , are dependent upon te~perature, theasthe oil film. Since FL, and z sumed temperature must be reasonably close to the final calculated temperature. How close is dictated by the actualvariation of those properties with oil temperature.
Because of the discontinuities of the structures that comprise a rolling bearing assembly, classical methods of heat transfer analysis cannot be applied to obtain a solution describingthe system temperatures. sical methods is meant the description of the system in terms of dif5erentia1 equations and the analytical solution of these equations. Instead, methods of finite difference as demonstrated by Dusinberre E15.51 must be applied to obtain a mathematical solution. For finite difference methods appliedto steady-state heattransfer, various points or nodes are selected throughout the system to be analyzed. At each of these points, temperature is determined. In steady-state heat transfer, heat influx t o any point equals heat efflux; therefore, the sum of all heat-flowing toward a temperature node is equal to zero. Figure 15.2 is a heat flow diagram at a temperature node; demon st rat in^ that the nodal temperature is affected by the temperatures of each of the four indicated surrounding nodes. ( ~ t h o u the ~ h system depicted by Fi . 15.2 shows only four s~rroundingnodes, this is purely by choice of the number of nodes may be greater or smaller.) Since the sum of the heat flows is zero, therefore
it is assumed that heatflow oc is nonsymmetrical, making all rent. Furthermo~e, thematerial i
5
so that thermal conductivity is different for all flow paths. ~ubstitution of equation (15.4) into (15.20) therefore yields
h4S4
+ k3S3 -(T3 - To)+ -(T4 - To)= 0 %3
(15.21)
g4
By rearranging terms, one obtains
or
FITl + F2T2+ F3T3+ F4T4-
i=4
(15.23) i= 1
Dividing by 2Fi yields
~ ~ Y $ OF I $
FLOW
(15.24)
More concisely, equation (15.19) may be written
&Ti = 0
(15.25)
in which the +i are influence coefficientsof temperature equal to Fi/XFi. If the material were isotropic and a symmetrical grid was chosen, then +o = 1 and the other = 0.25. In the foregoingexample,only heat conductionwas illustrated. If, however, heat flow between points 4 and 0 was by convection, then according to equation (15.5), F4 = hv4S4.For a multinodal system, a series of equations similar to (15.24) may be written. If the equations are linear in temperature T,they may be solved by classical methods for the solution of si~ultaneouslinear equations or by numerical methods (see eference i15.61). The system may include heat generation and be further complicated, however, by nonlinear terms caused by heat radiation and free convection. Considerthe example schematicallyillustrated by Fig. 15.3. In that illustration, heat is generated at point 0, dissipated by free convection and radiation between points 1 and 0 and dissipated by conduction between points 2 and 0. Thus
+i
(HfO+ Hl-O,, + Hl-o,, + H2-0
=
0
(15.26)
The use of equations (15.41, (15.61, (15.91, and (15.17) gives
Hfo+ 2.3
X
10-5S1(T1- T0)1+25 + 5.73 X
+-K2S2 (T2- To)= 0
ESl(T;' - T;) (15.27)
E '2
or Q Heat generated
ROLLI~GB
+ Flv(Tl - T0)1.25+ FJT:
E
~ TE ~
G
- Tg) + F2(T2- To) == 0
(15.28)
A system of nonlinear equations similar to equation (15.28) is difficult to solve bydirect numerical methods of iteration or relaxation. Therefore, the Newton-Raphson method [15.6] is recommended for solution. The Newton-Raphson method states that for a series of nonlinear functions q iof variables Tj:
(15.29) Equation (15.29) represents a system of simultaneous linear equations which may be solved for (error on q). Then, the new estimate of Tj is Ti
==
Tj(0) + ej
(15.30)
and new values qi may bedetermined. The processis continued until the functions qi are virtually zero. Witha system of nonlinear equations similar to equation (15.28), such equations must be linearized according to equation (15.29). Thus, let equation (15.28) be rewritten as follows:
NOW
(15.32) ~ u ~ s t i t u tofi oequations ~ (15.31) and (15.32)into (15.29) yields one equation in variables eo, el, and e2. he syst~mof nonlinear equations is solved for To, rms error is sufficiently small, for e~ample,less than 0.1".
23072 double-row radial spherical roller bearing has a 444.5 mm (17.5 in.) pitch diameter and is mounted in thepillow block shown by Fig. 15.4. The bearing is operated at a shaft speed of 350 rpm while it supports a 489,500 N (110,000 lb) radial load. The bearing is lubricated by 100 SSU (20 centistokes) oil in a bath at operating temperature. The pillow block that houses the bearing is situated inan atmosphere of quiescent airat temperature 48.9"C (120°F). Estimate the bearing and sump oil temperatures.
MI From t15.71 use f i
=
0.001, a
=
= f1F"dL
1.5, and b
(14.99) =
-0.3
M I = 0.001. * (~~9500)1.5(444.5)-0.3 =
5.499
lo4 N
mm (486.5 in. lb) 0
From Table 14.5, use f o = 7 for bath lubrication.
uon = 20 350
Mv = =
lo4 N
2.250
=
0
*
7 (7000)2/3(444.5)3
(14.103)
mm (199.1 in * lb)
5.499 104 + 2.250 = 104 e
lo4 N
7.749
Hf = 1.047 =
7000
fo(uon)2/3d~ =
+ M, =
=
*
mm (685.6 in lb)
nM = 1.047
9
350 7.749
lo4
(15.1)
2840 W (9687 Btu/hr)
Since this problem is for illustrative purposes, it hasbeen designedto be as simple as possible such that all equations and methods of solution maybe demonstrated. Therefore, the followingconditionswillbe assumed: Nine temperat~renodes are sufficient to describe the system shown by Fig. 15.4. The inside of the housing is coated with oil and may be described by a single temperature. The inner ring raceway may be described by a single te~perature. The outer ring raceway may be describedby a single t e ~ ~ e r a ~ u r e .
15.4,
Temperature node selection.
The housing is symmetrical about the shaft centerline and vertical . Thus, heat transfer in the circumferential direction need not be considered. The sump oil may be considered at a single temperature. The shaft ends at the extremities of the housing are at ambient temperature. onsidering the temperature nodes indicated in Fig. 15.4, the heat t ~ a n ~ fsystem er is that indicated by Table 15.1. Table15.1 also indicates which e~uations are used to determine heat flow, film coefficient of heat transfer, and rate of heat generation. The heat flow areas andlengths of
I
u0
l
l
I
u0
/
I
I
I
I
R ~ L L ~ G ~~~~
flow path are obtained from the dimensions of Fig. 15.4, consideringthe location of each temperature node. ased on Table 15.1 and Fig. 15.4, a set of nine simultaneous, nonlinthis ear equations with unknown variables Tl-T9 can be developed. Since system is nonlinear in temperature because of free convection and raon of diation from the housing to ambient, the ~ e w t o n - ~ a ~ h smethod equations (15.29) will be used to obtain a solution. The final values of temperature are shown in the proper location in Fig. 15.5. The system chosenfor evaluation was necessarily simple. A more realistic system would consider variation of bearing temperature in a circumferential directionalso.For this case,viscous torque maybe considered constant with respect to angular position;however,load torque varies as the individual rolling element load on the stationary ring but may be consideredinvariant with respect to angular position on the rotatingring. A three-dimensional analysis such as thatindicated by load torque variation on the stationary ringshould, however, showlittle (369"F)
15.5. Temperature distribution, natural convection of air.
variation intemperature around the bearing rings so that a twodimensional system should suEce for most engineering applications. Of course, if temperatures of structures surrounding or abutting the housing are significantly different, then a three-dimensional study is required. A three-dimensional study will require a computer. It is not intended that the results of the foregoing methodof analysis will beof extreme accuracy, but only that accuracy will be sufficient to determine the approximate thermal level of operation such that corrective measures may betaken in theevent excessive steady-state operating temperatures are indicated. Moreover, in the event that cooling of the assembly is required, the same methods may be used to evaluate the adequacy of the cooling system. Generally, the more temperature nodes selected or the finer the grid, the more accurate will be the analysis.
Having established the operating temperatures in a rolling bearing assembly while using a conventional mineral oil lubricant and lubrication system, and having estimated that thebearing and/or lubricant temperatures are excessive, it then becomes necessary to redesign the system to either reduce the operating temperatures or make the assembly compatible with the temperature level. Of the two alternatives, the former is safest when considering prolongedduration of operation of the assembly; however, when shorter finite lubricant life and/or bearing life are acceptable, it may be expeditious and even economical to simply accommodate the increased temperature level by using special lubricants and/ or bearing steels. The later approach is effective when spaceand weight limitations preclude the use of external cooling systems and is further necessitated in many applications in which the bearing is not the prime source of heat, such as in aircraft gas turbine engines.
For situations in which the bearing is the prime source of heat and in which the ambient conditions surrounding the housing do not permit an adequate rate of heat removal, simply placing the housing in a moving air stream may be sufficient to reduce operating temperatures. For instance, placing the housing of Fig. 15.4 in a fanned air stream of 15.2 m/sec (50 ft/sec) velocity will create a heat transfer coefficient of about 2.386 x W/mm2 * "C (4.2 Btu/hr * ft2 * O F ) on the outside surface of the housing, which is approximately 4 times that for the natural con-
vection system and gives a maximum bearing temperature of 168°C (334°F)as opposed to 221°C (429°F)obtained for free convection. Figure 15.6 shows the remaining system temperatures. Thus, in this case, the system temperatures can be significantly reduced if a fan is used to circulate air over the housing. If a fan is used, increased heat transfer from the housing to the air stream may be effected by placing fins on the housing. This increases the effective area for heat transfer from the housing. Consider that the external area of the housing of Fig. 15.4 is doubled by use of fins. Using W/mm2 "C (4.2 Btu/hr * ft2 * O F ) for the filmcoefficient of 2.386 X the moving air stream and 2 times the external housing area yields a maximum bearing temperature of 149°C (300°F).Figure 15.7 showsother system temperatures. M e n the bearing is not the prime source of heat, cooling of the housing will generally not suffice to maintain the bearing and lubricant cool.
L
A
15.6. Temperature distribution, outside air velocity 15.2 mlsec (50 ftlsec).
IGH TIE
E CONS
IONS
1
15.7. Temperature distribution, air velocity 15.2 rn/sec (50 ftlsec) finned housing.
For example, consider a shaft temperature of 260°C (500°F) instead of th the aforementioned movingair system in operation, the m ~ i m u mbearing temperature for the reference system is 196°C (385°F).(Figure 15.8 shows other system temperatures.) Thus, it is necessary to cool the lubricant and permit the lubricant to cool the bearing. The most effective way of accomplishing this is to pass the oil through an external heat exchanger and direct jets of the cooled oil on the bearing. To save space when a supply of moving coolant is readily available, it may be possible to place heat exchanger coilsdirectly in thesump. The cooled lubricant is then circulated by bearing rotation. The latter method is not quite as efficient thermally as jetcooling although bearing friction torque and heatgeneration may beless by notresorting to jet lubrication and the attendantchurning of excess oil. The adequacyof either system of cooling may bedemonstrated approximately by assuming that oil tem-
57
b
A
. Temperature distribution, outside air velocity 15.2 mlsec (50 Wsec). perature is maintained at anaverage of 60°C (140°F)with shaft temperature of 260°C (500°F)as above and ambient temperature of 49°C (120°F) in quiescent air. Maximum bearing temperature is thereby suppressed to 93°C (2OO"F),which would appear to be a satisfactory operating level. (see Fig. 15.9 for other temperatures.) It was intended to demonstrate in the foregoing discussion that it is possible to estimate with a reasonable degree of accuracy the temperatures occurring in an oil-lubricated rolling bearing assembly more, if the bearing and oil temperatures so calculated are excessive, it is possible to determine the type and degree of cooling capability required to maintain a satisfactory temperature level. Several researchers have applied the foregoing methods to effectively predict temperatures in rolling bearing applications. Initially, Harris i15.8, 15.91 applied the method to relatively slow speed spherical roller
15.9. Temperature distribution, oil is cooled to 66°C (150°F) average temperature.
bearings. ~ubsequently7 these methods have been successfully appliedto both high speed ball and roller bearings [15.10-15.lZ]. Good agreement with experimentally measured temperatureshas been reported [15.14] using the steady-state temperaturecalculation op- ' tion of ~ ~ ~ E Ra computer T H 7 programto analyze the thermomechanical performance of shaft-rolling bearing systems. Figure 15.10 shows a nodal network model and the associated heat flow paths for a 35-mmbore ball bearing. Figure 15.11 shows the agreement achieved between calculated and experimentally measured temperatures. It must be pointed out, however, that construction of a thermal model that accurately models a bearing often requires a considerable amount of effort and heat transferexpertise.
4 Oil sump (known temperature)
b
support bearing
" "
Inner ring
(a)
15.10. Bearing system nodal network and heat flow paths for steady-state thersis. ( a ) Metal, air, and lubricant temperature nodes: metal or air node; o lubricant node; * lubricant flow path. ( b ) Conduction and convection heat flow paths (from [15.14]).
-
As indicated previously, accurate calculation of lubricant film thickness and traction in a rolling contact depends on the determination of lubricant viscosity at theappropriate temperatures. For lubricant film thickness, this means calculation of the lubricant temperature entering the contact. For traction, this means calculation of the lubricant temperature for its duration in the contact. In Reference [15.14], the heat transfer system illustrated by Fig. 15.12 was used. Designating subscript k to represent the raceway and j the rolling element location, the following heat flow equations describe the system.
440
~
Predicted Experi- Shaft speed, rpm mental
j,,
7
0”.
g w =I
c
E
g.
I
360
-
a3
2
&
320-
Y
0 3
280 aj
t
I
I
I
1
-
I
480
440 U-
-
c
3
7 440
‘5
320
is 420
280
0
5 0 0 1
I 1m
J
m
Total iubricant flow rate, cm3/min
1 .1 2 .3 .4 .5
0
Total lubricant flow rate, gpm
w 4 w
Total lubricant flow rate, c m 3 h i n
2 0
.1 .2 .3 .4 Total lubricant flow rate. gpm
15.11. Comparison of predicted and experimental temperatures using S (a>Inner raceway temperature. (b)Outer raceway t e m p e r ~ t ~(c> e .Oil-out temperature. (d)Bear-
ing heat generation (from [15.141).
.5
57
ROLLING BEARING T
E
~
E
~
1-rolling element 2-rolling element surface r
6-lubricant out
5-ring 15.12. Rolling element-lubricant-raceway-~ng temperature node system.
Since the lubricant is essentially a solid slug during its time in the contact, heat transfer from the film to the rolling body surfaces is by conduction. Then, assuming the minute slug exists atan average temperature T3,
The lubricant slug is transported through the contact; it enters at temand exits at 7';. Therefore, for heat transfer totally within perature the slug,
7':
Finally, the lubricant acts as a heat sink carrying heat away from the contact.
In high speed bearing frictional performance analyses such as those indicated in Chapter 14, the rolling-sliding contact heat transfer analyses are performed thousands of times to achieve consistent solutions. The analyses are begun by assuming a set of system temperatures. Lubricant viscosities are then determined at these temperatures, and frictional heat generation rates arecalculated. These are subse~uently used to recalculate temperaturesand te~perature-depen~ent parameters.
T
~
~
FERE~CES
The process is repeated until the calculated temperatures substantially match the assumed temperatures. This method, while producing more accurate calculations for bearing heat generations and friction torques, requires rather sophisticated computer programs for its execution. For slow speed bearing applications in which the bearing rings are rigidly supported, the simpler calculations for bearing heat generations illustrated by Examples 14.2 and 14.3 will usually sufice.
The temperature level at which a rolling bearing operates dictates the type and amount of lubricant required as well as the materials from which the bearing components may be fabricated. In some applications the environment in which the bearing operates establishes the temperature level whereas in other applications the bearing is the prime source of heat. In either case, depending on the bearing materials and the endurance required of the bearing, it may be necessary to cool the bearing using the lubricant as a coolant. General rules cannot be formulated to determine the temperatur~ level for a given bearing operating under a given load at a given speed. The environment in which the bearing operates is generally different for each specialized application.Using the friction torque formulas of Chapter 14 to establish the rate of bearing heat generation in conjunction with the heat transfer methods presented in this chapter, however, it is possible to estimate the bearing system temperatures with an adequate degree of accuracy.
15.1. E. Eckert, Introduction to the Transfer ofHeat and Mass, McGraw-Hill, New York (1950). 15.2. M. Jakob and G. Hawkins, Elements ofHeat Transferand Insulation,2nd ed., Wiley, New York (1950). 15.3. A. Palmgren, Ball and Roller Bearing Engineering, 3rd ed., Burbank, Philadelphia (1959). 15.4. I?. Kreith, “Convection Heat Transfer in Rotating Systems,’’Adv. in Heat Transfeq 6 , 129-251 (1968). 15.5. G. Dusinberre, Numerical Methods in Heat Transfeq McGraw-Hill,New York(1949). 15.6. 6. Korn and T. Korn, Mathematical Handbook forScientists and Engineers, McGraw-Hill, New York (1961). 15.7. SKY, General Catalog 4000 US, 2nd ed., 49 (1997). 15.8. T. Harris, “Prediction of Temperature in a Rolling Bearing Assembly,” Lubr. Eng., 145-150 (April 1964).
15.9. T. Harris, “HOWto Predict Temperature Increases in Rolling Bearings,” Prod. Eng., 89-98 (Dec. 9, 1963). 15.10. J. Pirvics and R. Kleckner, “Prediction of Ball and Roller Bearing Thermal and n Kinematic Performance by Computer Analysis,” in Adu. Power ~ a n s m i s s ~ oTech., NASA Conference Publication 2210, 185-201 (1982). 15.11. H. Coe, “Predicted and Experimental Performance of Large-Bore High Speed Ball and Roller Bearings,” in Adu. Power D a n s ~ i s s i o nTech., NASA Conference Publication 2210, 203-220 (1982). 15.12. R. Kleckner and G. Dyba, “High Speed Spherical Roller Bearing Analysis and Comparison with Experimental Performance,” in Adu. Power Transmission Tech., NASA Conference Publication 2210, 239-252 (1982). 15.13. VV. Crecelius, “User7s Manual for SKF Computer Program S W E R T H , Steady State and Transient Thermal Analysis of a Shaft Bearing System Including Ball, Cylindrical, and Tapered Roller Bearings,” SKF’Report AL77P015, submitted toU.S. Army Ballistic Research Laboratory (February 1978). 15.14. R. Parker, “Comparison of Predicted and Experi~entalThermal Performance of Angular-Contact Ball Bearings,”NASA Tech. Paper 2275 (February 1984). 15.15. T. Harris and R. Barnsby, “Tribological Performance Prediction of Aircraft Gas Turbine Mainshaft Ball Bearings,” DiboZogy Dans 41(1), 60-68 (1998).
The functional performance and endurance of a “dimensionally perfect” bearing with ideal internal geometries and surfaces, correct mounting, and preferential operating conditions is significantly influenced by the characteristics of its materials. Major criteria to be considered for satisfactory bearing performance include material selection and processing with resultant physical properties. This chapter contains brief descriptions of various bearing steel analyses, melting practices, manufacturing process variables, and the influence of these factors on the physical and metallurgical and properties with respect to bearing performance. It also contains discussion concerning metallicand nonmetallic materials used for cages, seals, and shields.
Rolling bearing steels, from their inception, were selected on the basis of hardenability, fatigue strength, wear resistance, and toughness. h e r -
ican Iron & Steel Institute (AISI) 521000 steel, an alloy machinable in its annealed condition and exhibiting high hardness in the heat-treated state, was introduced around 1900 and is still the most-used steel for ball bearing plus most roller bearing applications. For large bearing sizes, particularly with respect to cross-sectional thickness, modifications to this basic analysis incorporating silicon, manganese, and molybdenum were introduced. Carburizing steel came into being when the tapered roller bearing was introduced. Over the years more demanding product requirements promoted the introduction of high speed steels and stainless steels for high temperature operating conditions and corrosion res.istance.
The largest tonnage of bearing steels currently produced is the category of through-hardening steels, Table 16.1 lists common grade designations and respective chemical compositions for this family of alloys. Through-hardening steels are classified as h ~ e r e u t e c t o i d - t steels ~e when containing greater than 0.8% carbon by weight and essentially containing less than 5% by weight of the total alloying elements. Assuming satisfactory material availability, the bearing producer selects the appropriate grade of steel based on bearing size, geometry, dimensional characteristics, specific product pedormance requirements, and manufacturing methods and associated costs.
Table 16.2 outlines the grade designations and corresponding chemical compositions of the common carburizing steels. These steels are classified as h~oeutectoidsteels; their carbon contents are generally below 0.80%. Carburizing steels are alloyed with nickel, chromium, molybdenum,and manganese to increase hardenability. The higher hardenability grades are used in applications requiring ring components of heavier cross section. Carbonis diRused into the surface layer of the machined componentsto appro~imately0.65-1.10% carbon during the heat treatment operation to achieve surface hardnesses comparable to those attained with through-hardening grades of steel.
Demands for good bearing performance under hostile operating conditions have consistently increased. The aerospace industry, in particular, requires products capable of operating at increasingly higher speeds, heavier loads, and with high reliability. Other challenging applications include performingin corrosive atmospheres, at cryogenic temperatures,
1
G STEELS
.1. Chemical Composition of Thro~gh-HardeningBearing Steels ~-
Composition (%) Grade"Mn ASTM~-A295(52100) ISO" grade l,683/WII ASTM-A295 (51100) DINd 105 Cr4 ASTM-A295 (50100) DIN 105 Cr2 A~TM-A295(5195) ASTi"A295 (K19526) ASTM-A295 (1570) ASTM-A295 (5160) ASTM-A485 grade 1 IS0 Grade 2,6831XVII ASTM-A485 grade 2 ASTM-A485 grade 3 ASTM-A485 grade 4 DIN 100 Cr Mo 6 ISO-grade 4,683/XVII
C 0.98 1.10 0.98 1.10 0.98 1.10 0.90 1.03 0.89 1.01 0.65 0.75 0.56 0.64 0.95 1.05 0.85 1.00 0.95 1.10 0.95 1.10 0.92 1.02
Cr
MO
Si
0.25 0.45 0.25 0.45 0.25 0.45 0.75 1.00 0.50 0.80 0.80 1.10 0.75 1.00 0.95 1.25 1.40 1.70 0.65 0.90 1.05 1.35 0.25 0.40
0.15 0.35 0.15 0.35 0.15 0.35 0.15 0.35 0.15 0.35 0.15 0.35 0.15 0.35 0.45 0.75 0.50 0.80 0.15 0.35 0.15 0.35 0.25 0.40
1.30 1.60 0.90 1.15 0.40 0.60 0.70 0.90 0.40 0.60
0.70 0.90 0.90 1.20 1.40 1.80 1.10 1.50 1.10 1.50 1.65 1.95
-
0.10
-
0.10 -
0.10 ___.
0.10
-
0.10
-
0.10
-
0.10
-
0.10
-
0.10 0.20 0.30 0.45 0.60 0.30 0.40
"Phosphorus and sulfur limitation for each alloy0.025% is masimum (each element). bAmerican Society for Testing Materials[16.1]. International Organization for Standards.
and in hard vacuum. To achieve satisfactory bearing operation in the critical applications associated with these conditions, it is necessary to minimize adverse effect on fatigue life due to undesirable nonmetallic inclusions. This required substantial changes in steel-melting practice. Vacuum induction-melted, vacuum arc-rernelted (VIWAB) alloy steels such as M50 and BG42 were developed to provide the required high reliability in resistance to fatigue. M50NiL l16.3, 16.41 was developed as a high temperature operation, carburizing steel to provide throughcracking resistance for very high speed bearing inner rings. More recently, Cronidur 30E16.5,16.61, a steel rich in nitrogen and chromium, has been introduced to satisfy requirements for high temperature operation with both corrosion resistance and high reliability fatigue resistance. Furthermore, in many aircraft power transmissions, to minimize weight and space, bearing inner raceways are made integral with a one-
.
Chemical Composition of Carburizing Bearing Steels Composition. (%)
Gradea
S a b 4118
SAE 8620, IS0 12 DIN 20 NiCrMo2 SAl3 5120 AFNOR" 1863 SAE 4720, IS0 13 SAE 4620 SAE 4320, IS0 14 SAE E9310
SAE E3310
muPP
C 0.18 0.23 0.18 0.23 0.17 0.22 0.17 0.22 0.17 0.22 0.17 0.22 0.08 0.13 0.08 0.13 0.10 0.15
Mn 0.70 0.90 0.70 0.90 0.70 0.90 0.50 0.70 0.45 0.65 0.45 0.65 0.45 0.65 0.45 0.60 0.45 0.65
Si
Ni
Cr
Mo
0.15 0.35 0.15 0.35 0.15 0.35 0.15 0.35 0.15 0.35 0.15 0.35 0.15 0.35 0.15 0.35 0.15 0.35
-
0.40 0.60 0.40 0.60 0.70 0.90 0.35 0.55
0.08 0.15 0.15 0.25
0.40 0.70
-
0.90 1.20 1.65 2.00 1.65 2.00 3.00 3.50 3.25 3.75 3.75 4.25
-
0.40 0.60 1.00 1.40 1.40 1.75 1.35 1.75
0.15 0.25 0.20 0.30 0.20 0.30 0.08 0.15
aGrades are listed inASTMA534 [16.2]; phosphorus and sulfur limitation for each alloy is 0.025% maximum (each element). Society of Automotive Engineers. cFrench Standard.
piece gear-shaft component. Thus, the steel used must satisfy the fati~e-resistancerequirements for the gear as well as the shaft and supporting bearings. Pyrowear 675 carburizing steel is frequently employed in such applications. Table 16.3 lists the chemical co~positions of several special bearing steels.
During the past 30 years high quality bearing steels have been melted by the electric arc furnace melting process. During the oxidizing period in the furnace cycle, impurities such as phosphorus and some sulfur are removed from the steel. Further refining removes dissolved oxides and other impurities that might negatively affectthe steel's performance. Unfortunately, this furnace practice by itself does not remove undesirable
.37
.3. Chemical Composition of Special Bearing Steels Typical Composition (%) Grade
C
Mn
Si
Cr
Ni
V
Mo
W
N
”~
M50 €36;-42 440-C CBS-60 CBS-1000 VASCO X-2 M50-NiL Pyrowear 675 0.10 EX-53 Cronidur 30
*
0.80 1.15 1.10 0.20 0.15 0.22 0.15 0.07
0.25 0.50 1.00 0.60 3.00 0.30 0.15 0.65
0.31
-
0.25 0.30 1.00 1.00 0.50 0.90 0.18 0.40 0.98 0.55
4.00 0.10 1.00 14.50 1.20 17.00 1.45 1.05 4.50 0.35 3.00 0.45 5.00 4.00 3.50 1.00 13.00 2.60 0.60 - 2.13 0.94 0.12 1.05 15.2
4.25 4.00 0.75 1.00
-
-
1.40 4.00 1.80
1.35
1.02
-
-
0.38
Note: M50 NiL and Pyrowear 675 are surface-hardening steels.
gases absorbed by the molten steel during melting and entrapped during solidification. Although vacuum ladle degassing was patented in 1943 [16.7]?it was not until the 1960s through the 1970s that cost-effective vacuum degassing production facilities were used to principally remove oxygen and hydrogen and further improve material quality. These processes provide bearing steels with the good machinability, hardenability? and homogeneity required for manufacturing economy and good product performance. The advent of high-performance?aircraft gas turbine engines and the corresponding demand for premium-quality bearing steels led to the development of sophisticated induction and consumable electrode vacuum melting techniques. Electroslag remelting capability was developed in concert with these two vacuum processes. These special melting techniques have significantly influencedthe development of higher-alloy tool steels for elevated temperature and corrosion-resistant bearing applications.
The demand upon the steel industry to provide higher-quality alloy steels promoted the development of the cold charge in the basic lined, electric arc furnace. These furnaces let high-alloy scrap and lower-grade alloy scrap be mixed with plain carbon scrap; economical operation and product quality are achieved through proper selection and weight control of these materials. owing the exact chemistry of the scrap charge reduces the consumption of more costly alloy additions to the melt and minimizes introduction of‘ undesirable “tramp” alloying elements. An
abundance of certain trace metallic elements, occurring unintentionally in an MSI 52100 steel, has shown negative correlation on fatigue life in ball bearings E16.81. The furnace melter selects scrap that, when melted, contains a lower percentage of the specified alloying elements for the specific grade analysis and permits final adjustment with selected alloying and carbon additions. The furnace charge must not contain elements that are not to be part of the final heat chemistry, Raw material selection for a furnace charge is also judged by physical size and weight. Light scrap takes up furnace volume. Heavyscrap reduces pro~ctionto furnace walls and roof during meltdown. A proper balance and distribution of scrap must be maintained to enable proper thermal and electrical dist~butionduring the initial stage of meltdown. Variation from these parameters can produce “off-analysis” heat. Careful attention must also be given to slagmaking materials, such as lime, silica, and fluorspar, to assure consistent quality and slag performance.
The basic electric arc furnace (Fig. 16.1)[16.9] is circulaz, lined with heat-resistant brick, and contains three electrodes in a removable roof. The charge is blended to provide efficient melting, the electrodes are lowered, and arcing begins. A layer ofcomplex: slag is produced that covers the molten surface layer and absorbs impurities from the steel. During this oxidizing period a carbon boil occurs, which producesgases
Water-cooled roof ring
16.1. Basic electric arc furnace.
~TEEL
CT
from the molten bath. This “complex” slag is then replaced with a “reducing” slag to decrease oxygen levels. Because the molten metal in the refining cycle is less active than in the oxidizing cycle, furnaces may be equipped with inductive stirrers. These stirrers generate a magnetic field that imparts a circulatory motion to the molten bath, enhancing both temperature control and homogeneity of chemicalcomposition throughout the melt. m e n the chemical compositionof the molten steel has been adjusted to the desired range and the proper pouring temperature has been reached, the “heat” is “tapped” or removed from the furnace. Material not to be vacuum treated is then ready for “teeming” into ingot molds.
Material quality produced in basic electric arc furnace process can be improved through vacuum degassing by various methods, including the ladle, stream, D-H (Dortmund-Horder), and R-H (Ruhrstahl-Heraeus) processes. In conjunction with these refining practices, inert gas shrouding may be incorporated with both bottom pouring and uphill teeming. These methods economically reduce undesirable gases and remove nonmetallic inclusions from the molten steel.
L a d e ~ e g u s s i n The ~ . ladle degassing process requires a ladle of molten metal to be placed in a sealed, evacuated chamber. Gases resulting from pressure differentials cause turbulence and a moderate stirring action in the molten bath. Ladle degassing units may be equipped with induction stirrers andlor injection devices for bubbling inert gases such as argon or helium through the molten metal to further agitatethe bath. See Fig. 16.2 El6.101. With this technique the quantity of metal exposed to the vacuum is greatly increased, which allows small alloy additions to be made to the melt. egassing. Themolten metal in stream degassing is poured termediate or “pony” ladle, as illustrated in Fig. 16.3 E16.111, into the vacuum chamber containing an empty ladle. As the molten metal enters the vacuum chamber, the stream explodes into fine spray or droplets that fall into a second, or teeming, ladle. When this ladle is full, the vacuum seal is broken and the molten metal is ingot molds. der proates por a refractory-lined vessel, whichmoves s into the molten into the v a c ~ ~ m
58 ALLOY ADDITION
CHAMBER COVER
VACUUM CHAMBER INDUCTION TIRRING COIL
!
/METAL FLOW
STAINLESSSTEEL LADLE (NONMAGNETIC)
16.2. Schematic arrangement of equipment for ladle degassing. (a)Gas stirring. ( b ) Induction stirring.
EX~AUST GASES TO
16.3. Schematic illustration of stream degassing method.
chamber, and is agitated vigorously as degassing begins. As the degassed steel flows back into the ladle, it is mixed with the remaining steel and the cycle is repeated. A total cycle requires approximately 20 sec and is repeated 40 to 60 times during the entire degassing operation. losses occurringduring the cycle are compensated by a graphite, electrically resistant, heating element positioned in the upper part of the vacuum chamber. Vacuum-sealed hoppers allow alloys to be added during the operation. When the vacuum degassing cycle is completed, the vacuum vessel is purged with an inert gas so that the accumulated flammable gas in the vessel will not be ignited before the nozzle is raised above the surface level of the liquid steel and the ladle is removed.
-H ( ~ ~ ~ r s ~ ~ ~ Z Process. - H e r ~The e ~R-H s ) (Ruhrstahl-Heraeus) vacuum chamber (Fig. 16.4) [16.12] straddles the ladle containing the mol-
16.4. Principle of operation of the R-H (Ruhrstahl-Heraeus) degassing process.
ten metal. T w o vertical legs extending downward from the base of the vacuum chamberare submerged just below the surface level of the metal in the holding ladle. The vacuum chamber is evacuated, and a pressure imbalance is created by flowing inert gas into one leg extension. The resulting pumping action circulates the molten metal between the holding ladle and the vacuum chamber. Outgassing of the molten metal occurs as it enters the vacuum chamber. The circulation process continues until the specified gas content is obtained.
Ladle metallurgy of molten steel is conducted in a ladle furnace outside the normal constraints of the initial electric arc melting furnace. The ladle furnace is equipped with independent electrodes for both temperature control and electromagnetic stirrers for bath circulation. Therefore there is no need to superheat steel in theelectric arc furnace to compensate for subsequent temperature drops experiencedin standardladle degassing practices as previously described. Lance injection permits powdered alloys to be inserted deep into the ladle. Argon is used as a carrier gas for these powders; the resulting bubbling actionhelps disperse particles uniformly throughout the molten bath. Lanceinjectioncombined with wirefeedingprovidesinclusion shape control, reduction of sulfur content, and improvement of fluidity, chemical homogeneity, and overall microcleanliness. Ladle furnace technology permits very rapid meltdown of scrap in the electric arc furnace and improved refining capability in a subsequent ladle furnace operation. This system generates improved productquality with a correspondingly improvedeconomy in steel melting.
ss ~onsistentdemands for cleaner bearing steels have resulted in themelting and refining process, termed the M-R process. This method employs a twin-shell, electric arc melting furnace in parallel with a EA ladle-refiningunit.* The twin-shell furnace has two vesselswith oxy-fuel burners andtwo roofs-that is, one contains graphite electrodes, and the other has no electrodes. Melting occurs in one furnace while charging and preheating of scrap can be carried out in the other shell. In the melting furnace carbon and phosphorus contents are adjusted to values below the final m ~ i m u mlimits. The fu then tapped, and the ladle of molten metal is transferred to the ladle-refining furnace an independent electrode roof. ipment provides many metallur~caloptions, including vacuum degassing, desulfuriz~ng,deox*This method was developed by SKI? Steel AB in conjunction with ASEA in Sweden.
idizing, and adjusting the chemical compositionof the molten steel. The additional ability to induction stir under conditions of close temperature control permits precipitation deoxidation with aluminum, resulting in steels with very low contents of oxygen and nonmetallic inclusions. The sequence of steps in the melting and refining process is illustrated in Fig. 16.5 (16.131.
t ~ ~ h i ~ h - ~ uSteel rity Vacuum I n ~ u ~ t i o n ~ e Z tYet i n more ~. sophisticated steel melting processes were introduced during the 1960s for producing ~ltrahigh-p~rity steels, also called “clean”bearing steels, which are essentially free from deleterious nonmetallic-t~einclusions. In vacuum induction melting, selected scrap material containing few impurities and comparable in chemical composition to the alloy grade being melted is charged into a small electrical induction furnace. The
SKF twin-shell
11
16.5. M-R steel process.
furnace (Fig, 16.6) 116.141 is encapsulated in a large vacuum chamber containing sealed hoppers strategically located for adding required alloys. out gas sin^ of the melt occurs early in a very rapid meltdown and refining period. Mter the melting cycle, furnace tilting and pouring the molten metal into ingot molds take place. The molds are automatically manipulated into and out of the pouring position while still within the vacuum-sealed chamber.This vacuum induction melting furnace process was among the first vacuum processing methods employedin manufacturing premium aircraft quality bearing steels. One of its primary functions today is to provide electrodes used to produce ultrahigh-purity, vacuum arc remelted steels.
Vacuum Arc Remelting. Vacuum technology for bearing steel alloy production, as described in the foregoing sections on vacuum degassing an induction vacuum melting, provided a way to reduce the gas content and nonmetallic inclusionsin steel. Steel electrodes, meltedin furnaces using vacuum technology can be remelted by still more sophisticated techniques, such as the consumable electrode vacuum melt practice, to provide material for bearings requiring the utmost reliability. This process, illustrated in Fig. 16.7, involves inserting an electrode of the desired chemical compositioninto a water-cooled, copper moldin which a vacuum is created. Charging
Vacuum chargedischarge chamber
16.6. Schematic view of induction vacuum melting furnace.
electrode
16.7. Vacuum are remelting furnace.
A n electrical arc is struck between the bottom face of the electrode and a base plate of the same alloy co~position.As the electrode is conit is automatically lowsumed under extremely high vacuum conditions, ered and the voltage is controlled to maintain constant melting ecause the soli~ification pattern is controlled, the remelted uct is essentially free from center porosity and ingot product has improvedmechanical properties, particu ransverse direction. Aircraft bearing material specifications for critical applications specify the VIM-VA steel melting practice. e ~ The ~eleetroslag ~ ~ e~f i n i process n~~ (E . ilar to that of the consumable electrode vacuum melting that a l i ~ slag ~ bath i ~ positione~ at the base of the elec the ~lectrical ~esistancer e ~ u i r efor ~ melting. The sla the ~ u r ~ a chamber ce in the molten sta
tudinal directions.
Lee
s
~ r o ~Forms. ~ c t Rolling bearings cover a broad spectrum of sizes with ring components varying in both cross-sectional configurationsand material grades. Balls or rollers vary in size and shape to accommodate their mating ring components. Generally, these components are manufactured from forgings,tubing, bar stock, or wire. The bearing manufacturer orders from the steel producer the form and condition of the raw material best suited for the selected method of processing and that will meet bearing performance plus customer requirements. Regardless of the melting practice employed, the resulting ingots are stripped from the molds, homogenized in soaking pits, rolled into blooms or billets, and subsequently conditioned to remove surface defects. Billets are reheated and hot-worked into bars, tubes, forgings, or rolled rings. Additional cold-workingoperations convert the hot-rolled tubes and bars into cold-reduced tubes, bars, and cold-drawn wire. AS an ingot passes back and forth between the rolls on a blooming mill, its cast structure is broken up and refined. Continued hot-working operations elongate and break up nonmetallic inclusions and alloy segregation. Hot mechanical working operations also permit plastic deformation of the material into desired shape or form. The subsequent cold-working of material results in induced stresses and improved machinability. Cold-working also produces changesin themechanical properties, improved surface finish, and closer tolerances. Dimensional tolerances and eccentricities of coldworked tubes are far superior to those of hot-rolled tubes. ost mill product formsrequire thermal treatmentat the mill before or after final finishing so that theforms are ready for machining or forming. The thermaltreatment mayinclude annealing, normalizing, or stress relieving. The product is then straightened,if necessary,inspected, and readied for transport. r o ~ ~ c t ~ ~ sThe ~ esteel ~ t producer i o ~ . performs two major functions to satisfy customer product quality requirements. First, an inspection plan emented to ensure that during the various man duct meets the specified quality limits. Secon n its final form to ensure that it conforms to edetermined inernal standards. Nondestructive testing methods, includtching, magnetic particle, eddy current, ultrasonic, and ssure tests, in conjunction with standard dimensional testing of the ~roductare successfully usedin statistical process control nd final ins~ectionprograms. rous in~ustrial,military, aerospace, or other indepen~entcust o ~ e s~ecifications r u ally form part of the purchaser’s requirement for bearing quality steels ese standards necessitate testing to satisfy heat characteristics, i ~ c l u ~ i nchemical g analyses, hardenability, macrotech
and microinclusion ratings. Special productrequirements inclu defects, microstructure?hardness, and dimensional tests.
uality ~ e ~ u i ~ e ~ e Cleanliness, nts. segregation, and mic~ostructure are steel product characteristics that influence bearing manufacture and subsequent product performance. The cleanliness of steel pertains to nonmetallic inclusions entrapped during ingot solidificationthat cannot be subsequently removed. Segregation pertains to an undesirable nonuniform distribution of alloying elements. Microstructure of the mill product relates directly to its suitability for machining andlor forming by the bearing manufacturer. The producer therefore incorporates tests early in the manufacture of the product and before shipment to be sure that pertinent mill and customer quality requirements are met. ~ZeanZiness. Steel quality with respect to nonmetallicinclusionsdepends on the initial raw material charge, selection of the melting furnace typelpraetice, and control of the entire process, including teeming and conditioning of the ingot molds. Exogenous inclusions result from erosion or breakdown of furnace refractory material or from other dirt particles outside the melt that are entrapped during tapping and teeming. Indigenous inclusions are products of deoxidation occurring within the melt. Inclusions less than 0.5 mm (0.020 in.) long are considered microinclusions; larger ones are considered macroinclusions. ~ o n ~ e t a l lincluic sions are classified by their composition and morphology as sulfides, aluminates, silicates, and oxides. Occasionally, nitrides are included in rating steel cleanliness. t is well establis~esthrough testing that nonmetallic inclusions are detri~entalto rolling contact fatigue life. The data further indicate that -type inclusions, based on their size, shape, and trimental than soft deformable-type particles apsulating harder, nonmeta coon or cushion around them so that the concentration under cyclic loading, sulfi sions can also enhance machinability
.
Oxygen content improvement of through-hardening AIS1 52100 (SKI? M-R) steel from 1965 to 1982 (from 116.151).
.
Reduction in amount of macroinelusions (mm/m2) in through-hardene~ AIS1 52100 steel (SKI?NI-R) from 1965 to 1982 (from [16.161).
E
for AISI 52100 steel to have high oxygen content; for example, 35 parts per million (ppm) and substantial amounts of macro the 1~70s, the acid open hearth furnace was introduc anliness; that is, oxygen content in the form of oxides ppm, and ~acroinclusionsover a period of time were process as shown by Fig. 16.8, and subsequently similar vacuum degassing processes, loweredthe oxy to 10 ppm while maintaining a low level of macroi .lo, the decrease of oxygen content from 3 tenfold improvementin bearing fatigue life. any methods have been devised for detecting or qua fin in^ nonmetallic inclusions. One of the most popular i scopic examination at l O O X magnification of polished specimens of a predete~minedsize from specific ingot locations and comparing the worst fieldobserved against ndard photographs weighted on a numbered erial specifications containing this meth rating system [16.121. evaluation stipulate th ting frequency and the acceptance or reje limits. Industry standards a ceptance tests such as fracmagnaflm ture, stepdown 2301), and visual ratingcounting indications on m ces to determine material respect to nonmetallic inclusion content. -quality aircraft bearing steel with respect to detection of inclusions is automatically checked on a 100% ba basis by employing both eddycurrent and ultrasonic test proc ~ e g r e g ~ ~onu ~ ouniform ~. solidification rate of molten metal within an ingot mold might lead to segregation of alloying constituents. the rapid freezing rate occurring in the mold-ingot interface, this out-
.IO. Fatiguelife of AISI all bearings vs oxygen content (from 116.151).
ermost portion or shell will solidify first and form columnar crystals. At the centermost portion of the ingot, which cools at a much slower rate, the grains are equiaxed.Eecausesolidification is not spontaneous throughout the ingot, the molten metal that freezes last will also become richer in alloying elements such as manganese, phosphorus, and sulfur, because the eleme s have inherently lower melting points. ~ e ~ e g a t i is o nonly slightly improved by thermal treatment andhotworking. Precautions must be taken in the melting and pouring practices, such as incorporating special molds designed to control rates of freezing and thereby prevent or minimize segregation during solidification. acroetching of properly prepared billet or bar slices, employing dilute, hot h ~ d r o c h l o ~acid c solution, which preferentially attacks these numerous alloying constituents, is used to reveal material se 0th macroscopic and microscopic methodsof inspection are te steel structure. Discs cut from the ends of bars or billets for macroscopic examination are prepared according to industry standards, acid etched, and examined with the unaided eye or under magnification generally notexceeding lox, Although numerous etching reagents are available, the generally recommended and accepted solution is hot dilute, hydrochloric (muriatic) acid. In addition to the detection of alloy segregates, material may be microstructurally evaluated for other objectionable characteristics such as “pipe,” porosity,blow holes, decarburization, excessive inclusions,cracks, and banding. The aerospace industry uses hot acid etching and f o r ~ n g sure conformance of grain flow patterns to a previously ination of steels involves a more detailed study of ifications generally between 100X and l000X . Numerous reagents are available to help identify and rate specific microconstituents. The steel producers’ manufacturing processes incorpo~ating thermal c cles influencethe resultingmicrostruct~reof the finished mill
arbides should be uniform in size and well arbides increase.
T R ~ T
OF STEEL
low-carbon grades, such as AIS1 8620, is appropriate for cold-forming operations but is considered gummy and unsatisfactory in machining operations. Each material grade in its finished mill product form must exhibit the proper microstructure and hardness so that it can be economically converted to its designated configuration.
Many of the mechanical properties of finished bearing components are developed by manufacturing methods that dictate the form and condition for raw material. Generally, raw material is produced by either hot- or cold-reduction processes and furnished as tubing, bars, wire, and forgings. Cold-reduction for producing bars, tubing, balls, and rollers of AIS1 52100 will lower both the austenite transformation temperature during heating for hardening and the martensitestart temperature upon cooling [16.17]. Theresulting fracture grain size of a ring produced from a coldreduced tube will be finerthan thatof an identical ring from a hot-rolled annealed tube. Although the volume change for the hot-rolled and coldreduced components is the same, the ring from the cold-reduced tube will have a smaller diameter after heat treatment. s and tubes are elongated during manufacture and display direcproperties; that is, the mechanical properties are different in the longitudinal direction compared to the transverse direction. Forgin the bar into ring components provides a more homogeneous product. rovide beneficial grain flow conforming to the rollin aring endurance tests demonstrate that end grain is detrimental to rolling contact fatigue life [16.181. Raw material intended for machining operations before heat treatment should be received in a readily machinable condition. Thematerial should have sufficient stock to render an “as-machined”component free from carburization, decarburization, and/or other surface defects.
eat treatment of bearing steel components necessitates heating an cooling under controlled atmospheric conditions to impart terial characteristics and ropert ties such as hardness, a di carbon surface layer, high fracture toughness or d ~ c t i l i t ~ ~ proper grain size, or re stren h, i m ~ r o v emachina~ility, pecific thermal cycles that produce these materialc
are called annealing, normalizing, hardening, carburizing, tempering, and stress relieving. Selective thermal cycles provide distinctive microstructures such as bainite, martensite, austenite, ferrite, and pearlite. Iron and carbon are thebasic constituents in bearing steels along with specific amounts of manganese, silicon, or other alloying elements such as chromium,nickel,molybdenum, vanadium, or tungsten. Bearing steels have a distribution in carbon content from 0.08% minimum (AIS1 3310) to 1.10% maximum (AIS1 52100). Beginning with ingot solidification, bearing steels take on a crystalline structure. These crystals are composed of atoms placed at fixed locations within a unit cell. Spacings remain constant at fixed temperature. Although there are 14 different space lattice types, the bearing metallurgist deals primarily with only three: body-centeredcubic(bcc),face-centeredcubic(fcc), and bodycentered tetragonal (bct). See Fig. 16.11. These types of three-di~ensionalcells have different physical and mechanical properties because of differences in atomic spacings; they also have a different solubility for atoms of other alloying elements. An atom of one or more such alloying elements residing in thehigh-carbon bearing steel may be substituted for an iron atom. Elements with very small atomic radius, such as carbon, whichis about one eighth the size of iron, can be placed in the interstitial spaces in the lattice. Pure iron has a bcc structure at room temperature andan fcc structure within a specificelevated temperature range. The temperature upon
Body-centered cubic
Face-centered cubic
~ody-centeredtetragonal
.11. Crystal structures of steel.
AT T
N" OF S T E E ~
heating or cooling, at which the atoms shift from one unit-cell type to another, is called a transformation temperature. These alterations can be observed in the time-temperature cooling curve foriron, as shown by Fig. 16.12. Pure iron is bcc below 912°C (1673°F)and fcc above. added to iron, the transformation temperature is lower over a broader temperature range. This information is displayed in the -carbon phase diagram in Fig. 16.13. ecause bearing steels rarely exceed 1.1% carbonabd their heat treatment does not exceed a metal temperature of 1302°C (237 (C-0.70,Cr-4.00,-1.00,V--lS.OO)],only a section of the phase diagram, in g. 16.14, will be required for further discussion. is dissolvable in water. Carbon is dissolvable in molten iron just as salt t is this action occurringin olid solution that enables alteration of mechanical properties of steel. igh-carbon-chromiumbearing steels, as receivedfrom the steel producer, are generally in a soft, spher annealed condition suitable for machining. The microstructure c of spheroidal carbide particles in a ferritic matrix. This mixture of ferrite and carbide that exists at room temperature transforms to austeni approximately ~27°C(1340°F). Theaustenite is capable of dissolvin larger ~uantitiesof carbon than that contained within the ferrit altering the cooling rate from the austeniti~ing temperature, the distri-
1500 ~-
" " " " " " " "
Time
16.12. Time-temperature cooling curve for pure iron.
.13. Iron-carbonphasediagram.
&&on Content % 16.14. Section of iron-carbon phase diagram.
EAT
OF STEEL
bution of the resulting ferrite and carbide can be modified, thus giving a wide variation in resultant material properties. Based on carbon content, steel can be put into three categories: eutectoid,hypoeutectoid, and hypereutectoid. Eutectoid steels are those 0% carbon, which upon heating above 727°C (1340°F) bestenite. This composition upon cooling fromthe austenitic range to approximately 727°C (1340°F) simultaneously forms ferrite and cementite. This product is termed pearlite, and it will revert to austenite is reheated to slightly above 727°C (1340°F). ypoeutectoid steels are those containing less than 0.80% carbon. The iron-carbon diagram indicates that for a 0.40%carbon steel approximately 843°C (1550°F)is required to dissolve all the carbon into the austenite. Under conditions of slowcooling, ferriteseparates from the austenite until the mixture reaches 727°C (1340°F). At this point the remaining austenite, containing 0.80% carbon, transforms into pearlite. The resulting microstructure is a mixture of ferrite and pearlite. The pearlite will dissolve into solid solution when it is reheated to approximately 727°C (1340°F).At temperatures above 727°C (1340°F)the ferrite will dissolve into austenite. The iron-carbondiagram indicates the existing phases when veryslow heating and cooling rates are enacted.
e The time-temperature transformation diagram is an isothermal transformation diagram. Steel will transform when cooled rapidly from the austenitizing temperature to ower temperature than the minimum at which the austenite is stable. agrams for various grades of steel have been developed at specific austenitizing temperatures to depict the time required for the austenite to begin to transform and to be completely transformed at any constant temperature studied. Figure 16.15 [16.19] shows an isothermal time-temperature transformation (TTT) for a typical high-carbon steel (AISI 52100). Theshape and the of the curves change with increased alloy content, grain size of the austenite, and austenitizing temperature. Figure 16.16 [16. TTT diagram for a typical alloy steel (AISI 4337).
es A eutectoid steel, upon slow cooling, will transform to pearlite at approximately 727°C (1340°F). Ifthis same steel specimen is quenched into a liquid medium controlled at a temperature just below 727°C (1340" a coarse pearlite structure will result. As the temperatureof the holding medium is lowered, however, the diffusion of carbon atoms is decreased and the lattice spacing of ferrite and cementite is reduced, thus produc-
Temperature OF "C 1472 800
Austenltlzed al860"C ( 1580°F )r 10 mlnutes
1292 700
250
2
g
u)
345
9
1112 600
400 510
360
932 500
385 445
752 400
515 572 300
615 695
392 200
212 100
0
0
2 1 I
1
10 I
1
Seconds
.
4 I
l
Mlnutes
l
10 I
30 1
1
2
4
10
1
1
I
I
Hours
Time-temperature t r ~ s f o r m a t i o ndiagram for AIS1 52100 steel (from
[16.191).
earlitic microstructu~e.These microstructures indicate that the on of pearlite is a nucleation and growth process. At still lower temperatures carbon atoms move more slowly, and the resulting transformation productis bainite, which consistsof ferrite needles containing a fine dispersion of cementite. Under still further cooling, the transformation product martensite is formed, which consists of a very fine, needlelike structure. Martensite forms athermally involving a shear mechanism in the microstructu~e;is itnot a product of isothermal transformation. The ~ u e n c h i nmust ~ be done veryrapidly into a m e d i u ~ such as molten salt or oil at a controlled temperature to prevent the austenite ing to a soft transformation product such as pearlite. diagrams reveal the microstructures that form at a single constant temperature; however, steel heat treatment uses rapid cooling,
AT
"F 1472
"C 800
Austenitized temperature 850°C
(1562°F)
1292 1112
200 235
600
v . nln
932
500
752
400
e 572 Martensite
300
p)
1;'
Bainite 50%
I
3
c1
e
8
c L-
420
I
~
~
50%
10
1 2 4
Seconds Hours
Minutes
1 0 3 0 1 2 4 10
Time-temperature transformation diagram
for AIS1 4337 steel (from
116.201).
and transformation occurs over a range of temperatures. Continuous cooling transformation (CCT) curves have been developed to explain the resulting transformations. Figure 16.17 is a cooling transformation dia-
An AIS1 52100 steel Jominy test bar 25.4 mm (1in.) in diameter by n.) long may be used to explain the value of e is austenitized at 843°C (1550°F)and, while is sprayed with a stream of water on the lower end face. e then varies from the quenched surface to the extreme opposite end, which cools much slower.~icrostructurescan then be correlated with various cooling rates occurring alongthe length of the bar.
Y ardness should not be confused with hardenability. Hardness is resistance to penetration and is normally measured by an indenter of fixed geometry applied under static load in a direction perpendicular to the material surface being tested. ardenability pertains to the depth of hardness achievable in an alloy. The alloying elements in the steel, as witnessed by the movement of the isothermal transformation curves to the right on the TTT diagram, permit additional cooling time from the austenitizing temperature to the point of martensite transformation.
O F
“C
1652 900 1
I
1
1
I I
I
I
I
1
1
I
I I
I
I
I
I
5
10
30
1472 1292 1112
E 4 3
E
932 752
c 572 392 212
1
I 5
60 1040 20 Seconds
1
2
Minutes
16.17. Continuous cooling transformation diagram for AIS1 52100 steel.
This positive effect of alloy additions to steel readily explains the need for the numerous modifications of the basic AIS1 52100 for varying section thicknesses of the bearing components. ardenability is also influenced by the effect of grain size and the ee of hot-working. The hardenability of a coarse-grained steel is much greater than that of a fine-grained steel. Hot-working of material into pro~essivelysmaller bar sizes correspondingl~reduces the hardenability spread found in ingots and blooms by reducing the segregation of carbon and other alloying elements normally experienced during ingot solidification. Higher austenitizing temperaturesand longer soaking times at te~peratures that promote grain coarsening also enhance hardenability by permitting more carbon to go into solid solution. ecause hardenability is a measure of the depth of hardness achieved under perfect heat treatment parameters, it is possible to quench bar products varying in diameter and to measure the resulting crosssectional hardnesspatterns to determine hardenability ~rossman E16.211 defined the ideal critical size of a bar processed in this manner to be one in which the core hardens in an “ideal quench” to 50% martensite and fully to 100% martensite at the surface. Ideal quench is one in which the surface of the heated test specimen insta~taneouslyreaches
the quenchmedium temperature. uenching identical specimens into media of less severity reduces the extent of hardening. Under these conditions ~rossmandefined the smaller bar diameter that ha martensite in the core, as the “actual critical diameter.” the development of severity quenchcurves alues) relating to he ideal critical diameter and the actual cri e Jominy end-quench hardenability test is standardized E16.2 with respect to specimen geometry, apparatus, water temperature, and flow rate such that all results can be rated on a comparison basis. The hardness ratings at 1.6-mm (0.062~-in.)intervals when plotted against specimen length provide a curve indicative of the hardenability of an alloy. End-q~enchhardenability data normally incorporate both the maximum and minimum hardenability limits anticipated under specific heat treatment parameters E16.231.
tices used for beari com~onents are either surface-hardening. eat treatments for the‘ to ing ma~ensiticgrades are substantially c in that they necessitate heating (to an ng temperature), quenching, washing, and tempering. Time-temperature parameters, primarily based on weight and cross-sectional thickness of the part being processed, have been established for the various throughhardening bearing alloys. ing components, particularly of large diameter and thin section thickness, require elaborate means of handling to minimize physical damage. In furnace construction, precautions are taken to avoid mass loading and excessive weight, which could adversely influence the geometry of the parts during heating or quenching. Furnace manufacturers generally use natural gas or electricity as theirheat sourcefor equipment. Arrangements forprotective atmospheres are normally provided to minimize carburization or decarburization of the high-carbon-chromium steel parts during processing. Furnaces of comparableconstruction and processing capability are also selected for the heat treatmentof carburizing grades of steel except that the atmosphere is controlled to provide the carbon potential necessary for carbon to diffuse into the steel. Precise, uniformfurnace temperatures are maintained and controlled, providing exact reproducibilityof processing cycles. Adequate quenching facilities are provided for salt, oils, water, or synthetic-type quenchants. Temperature control, agitation, and fixtures are used independently or in combination to reduce distortion in the heat-treated components. Induction heating, often using synthetic-type quenchants, can be used for automated heat treatment of special bearing components. This can
be a selective-type heat treatment inwhich on1 face is hardened. e parts are washed to remove all quenchant resiTempering furnaces are generally electric or gas ither batch loaded or automatically transported ces are being used t o process bearing co~ponents: earths, rotary drum, rotary hearth, s yor belt/cast link, and pusher t tomated salt lines using programmable hoists are in req~iringaustenitizing temperatures of ~ 0 2 - ~ ~ 0 2 * ~
~ e ~ t ~ The e ~high t . hardness
etic oil controlled between bainite-hardened components, 57-62. ~lthoughbainite-ha~denedcomponents do not require subsequent thermal t~eatment,martensitehardened components are tempered.
te. The martensite start ( ) temperature is lowered asthe zing temperatureand the e at tem~erature are increased, permitting more carbon to go into solid solution. ~orrespondingly, the tendency exists for more austenite to be retained uring the martensite transformation. Themorphology of the resultant ~ a ~ e n s i also t e depends on the dissolved carbon content: high amounts of dissolved carbon are associated with plate martensite formation, and low a a tendency to form lath martensite. lgh austenitizing te~peraturesalso have the tendency to coarsen the material grain size. This condition is evby both visual and lowpower m a ~ i ~ c a t i oofn fracture surfaces. rly heat-treated, highcarbon-chromium grades of steel show a fine, “silkylike”appearance on fracture faces. ~ t e quenching, r components are washed and temp ere^ t o relieve stressesand improve toughness. Tempering at temperatures at or
NT OF STEEL
slightly above the MS point will also transform retained austenite to bainite. The penalty for tempering at higher temperatures is loss of hardness, which can adversely affectload-carryingcapacity and endurance of the bearing component. Components of lower hardness are also more prone to handling and functional surface damage than their hardercounterparts are. ching into a 1o~-temperaturemedium[49-82”C uce thermal shock and no nun if or^ phase transformation stresses. Components with nonuniform cross sections and/or sharp corners can warp or fracture. Transformation stresses may be reduced by quenching the part into a hot oil or hot salt medium controlled at a te~peraturebetween 177 and 218°C (350-425”F), in the uppermost portion of the martensite transformation range. Temperature equalization throughout the crosssection of the component permits uniform phase transformation to progress during subsequent air coolin to room tem~erature.Although the as-quenched hardness is normally tempering cyclesfor martempered parts are similar to those used in straight ma~ensitehardening operations. ainite hardening is an “austempering”-typeheat treatment m theaustenitizing temperature temperature, which is the lower bainite transformation zone. It baths between 220 and 230°C (425-450”F) are normally type of heat treatme can be added to the quench bath to achieve the critical quench avoiding the formation of undesirable soft constituents. hardening grades of steel are again selected on the basis of component cross-sectional area. e higher the hardenability is, the greater the permissible cross-sectio area or thickness of a given component. As alloy content increases, the “nose” and “knee” of the transformation curve are pushed further to the right, which lengthens the time for bainite transmally require 4 hours or more for complete transforare achieved oncornmation to bainite. Hardness values of 57-63 nents processed in this manner. Subsequent pering is not required. uenching into molten salts and holding at these temperatures sipificantly reduce stresses induced due to thermal shock and phase transforainite hardening produces components with small compressive surface stresses, in contrast to martensite hardening, which produces small tensile stresses in the as-quenched surface layers. A bainite microstructure is coarser, with a more “feathery” needle than that produced in straight martensite hardening.
e ~ e ~ ~ Surface o ~ s hardening . is done by altering the chemical composi-
tion of the base material-for example, by carburizing or carbonitriding-orby selectively heattreatingthe surface layer of a given high-carbon bearing steel component. Induction and ~ame-hardening practices are used to fabricate production bearings. Laser beam and electron beam processes are also possible, depending onthe hardness depth required. Surface hardening of bearing steels produces well-defined depths of high surface hardness and wear characteristics. igh residual compressive stresses, present in the surface layer, enhance rolling and bending fatigue resistance. The surface layer is supported by a softer and tougher core which tends to retard crack propagation. ~ ~ r ~ ~ rThe i carburizing ~ i ~ ~ source . ormedium (gas, liquid, or solid) supplies carbon for absorption and diffusion into the steel. The same precautions followed for through-harden in^ furnace operations are followed in carburizing to minimize handling damage, reduce part distortion, and provide process economy. The normal carburizing temperature with the carbon diffu range is $99-9~~'C(~6~0-1$OO0F) creasing with temperature. Therefore, it is easier to control narrow case depth ranges at the lower carburizing temperatures. Based on the alloy steel being processed, time, temperature, and atmospheric composition determine the resulting carbon gradient. The resulting carbon content affects thehardness, amount of retained austenite, and microstructure of the carburized case. The hardness profile and compressive stress field depend on the carbon profile. though the practice of quenching directly from the carburizing furnace is used to heat treat bearing components, it is general practice to reharden carburized components to develop both caseand core properties and, at thesame time, to employ fixture-quenchingdevices to reduce part distortion. Based on the grade of steel being carburized, the carbon potential of the furnace atmosphere must be adjusted so that large carbides and/or a carbide network are not formed. Those alloyingelements such as chromium which lower the eutectoid carbon content are most likely to form globular carbides. Carbon canbe further precipitated to the grain boundaries if the steel is then slowcooledbefore quenching.These grain boundary carbides and/or the carbide network can reduce mechanical properties. Choosing the bearing material not only involves considering proper surface hardness and microstructure, but it also must incorporate core properties to prevent case crushing. Resistance to case crushing is generally provided by increasing subsurface strength. Therefore, a material with a section thickness and hardenability that will providea core hard-
TREA~NT OF STEEL
ness R, 30 to 45 is selected. Carburizing grades should be fine grained to minimize sensitivity to grain growth at high carburizing temperatures. Direct quenching from the carburizing furnace has theadvantage that one can obtain a case microstructure free of soft constituents, such as bainite, while using a leaner alloy steel. This heat treatment practice offers less part distortion than is experienced in reheating and quenching, particularly if the temperature is lowered to 8164343°C(15001550°F)beforequenching.Adversely, this practicecanproduce parts with too much retained austenite and possible microcracking. The excess austenite in thecase couldpermit plastic deformation of components under heavy load; microcracking could provide initiation points for fatigue. ~ i c r o c r a c ~ ncan g be minimized by keeping the carbon content in the as-carburized component lower than eutectoid level. Reheating at the lower austenitizing temperature and quenching tend to reduce microcracking. Gas carburizing is common to the roller bearing industry because gas flow rates and carbon potential of the atmosphere may be accurately controlled. Gasespresent in furnace atmospheres include carbon dioxide, carbon monoxide, water vapor, methane, nitrogen, and hydrogen. Over a period of time at a predetermined temperature, the specified case depth is established. This effective case depth (ECD) is generally defined as the perpendicular distance from the surface to the farthest point where the hardnessdrops to R, 50. Normal ECDs forbearing component range between 0.5 and 5 mm (0.020-0.200 in.) with a surface carbon content between 0.75 and 1.00%. Carburized components are tempered after quenching to increase their toughness. Cold-treating might be introduced to transform retained austenite to martensite. Additional tempering is then required, ~ a r ~ o n i ~ r i ~ iarbo n ~ on . it riding is a modified gas-carburizing process. ecause of the health hazards and ecological problems in disposing of cyanide salts, the preferred method is to use a gaseous atmosphere. an elevated temperature an atmosphere is generated having a given c bon potential to which ammonia is added. Nitrogen and carbon are diffused into the steel forming the hard,wear-resistant case. hard carbonit~dedcases are generally shallow in nature, ranging from approximately 0.07 to 0.75 mm(0.003-0.03 in.), produced at furnace temperatures ranging from 788 to 843°C (1450--1550"F), the case-core interface is easily differentiated. These same beneficialshallowcase characteristics can also be achieved in components requiring excessively heavy case depths. In thisinstance the parts aregenerally carburized to the heavy casedepths and thenreheated in a carbonitriding atmosphere. Ammonia added to the carburizing atmosphere dissociates to form nascent nitrogen at the work surface. The combinationof the carbon and nitrogen being adsorbed into the surface layer of the steel lowers the
critical cooling rate of the steel; that is, the hardenability of the steel is significantly increased by the nitrogen. This characteristics permits lower-cost materials, such as AIS1 1010 and 1020, to be processed to the esired high hardness by oil quenching and thus minimizes distortion uring heat treatment. All parameters being constant, the carbonitrided component will evice a more uniform case depth than that produced by carburizing. ause nitrogen lowers the transformation temperature, carbonitrided components will have more retained austenite than carburize nents of the same carbon content. These high levels of austeni reduced by increasing the carbonitriding temperature, contr surface conc~ntrationsto approximately 0.70-0.85% carbon, keepingthe ammonia content at a minimum during processing, and introducing a ion cycle before quenching. e presence of nitrogen in the carbonitrided case also enhances resistance to tempering. Carbonitrided components are tempered in the O F ) range to increase toughness and maintain a min190-205°C (375imum hardness ating. Induction heating is a means for rapidly bringing the surface layer of a high-carbon-low-alloy bearing steel component into the austenitic temperature range, from whichit can be quenched directly to martensite. Induction heating is accomplished by passing an alternating current through a work coil or inductor. A concentrated magnetic field is then induced within the coil. This magnetic field will in turn induce an electrical potential in a part placed within the coil. Since a part represents a closed circuit, the induced potential establishes an electrical current within the part. Heating of the part is then the result of the material's resistance to the flow of induced current. generating equipment is selected according to frequency rets. Motor generators have historically beenused to provide medium-frequencyranges from 1-10 k z and to provide deep, hardened surface layers. These units are currently being replaced by solid-state inverters using silicon-controlled-rectifier (SCR) switching devices. Radio frequency generators provide frequencies ranging from 100 to 500 k for very shallow case depth requirements. The chief factors influencing the success of the induction-heating operation are frequency selection, power density, heating time, and coupling distance: ~re~~en sezection-the cy size of the part and the depth of heating desired dictate the frequency requirement. Power density-the watts available per square mm of inductor surface influence the depth to which a part can be surface hardened.
ating time-the heating time required to bring the part to temperature is a critical factor with respect to overheating and resulting case depth. ~ o ~ ~ Zdistance-the ing coupling distance is defined as the distance between the coil and the part surface. uenching of induction-hardened components is generally accomplished by either a spray or immersion method. Spray quenching involves a pressure deposition of the quenchant onto the component by a series of holes machined into the inductor or by a separate quench ring. The immersion method necessitates dropping the partout of the inductor into an agitated quench bath. The required physical and metallurgical properties in high-carbon-chromiumbearing steel can be achieved by using a synthetic quench in lieu of water and/or oil. Concentrations may be adjusted to provide maximum quenchability while minimizing the tendency for cracking. All su~ace-hardenedcomponents require tempering after a quenching. Although the case depth may be similar to those achieved by carburizing, a steeper hardness gradient exists in the case-core transition zone. Properly induction-hardened AIS1 52 100 steel bearing components will generally achieve hardness values of R, 65-67 as quenched. If the part before heat treatment is in the annealed condition, the microstructure of the as-hardened surface zone will consist of fine spheroidal carbide and a matrix of untempered martensite. When it is examined for fractures, a fine grain size can be seen. is used primarily in the heat treat~ Z a m e - ~ ~ r d e n i nFlame-hardening g. ment of very large rings of high-carbon-low-alloy steel components more than 1 m (approximately 3 ft) in diameter. A combustible gas is mixed with oxygen to fire a cluster of burners directed to selectively heat the ring component as it rotates at a fixed rate through the impinging flame. The depth of the heat-affected zone is a function of the dwell time of the part at the heat source. The rotating part, upon reaching the proper austenitic temperature, is water quenched. The core material, being in the unaffected heat zone, remains in the annealed condition. Subsequent tempering is mandatory to relieve stresses and increase the ductility of the as-quenched component. Flame-hardening is not a capital intensive process froman equipment standpoint. It is very versatile for selectivehardening and rapidly adaptable for changing ring sizes with varying cross-sectional confi~rations and thicknesses. The progressive zone heating method means an overlap will occur after 360" are completed. The resulting overtempering effect in the heat
sink zone will result in a spot of lower hardness. Precautions must be taken to minimize the thermal and transformation stresses at this overlap point to prevent cracking.
er Knowledge of size and shape changes of rolling bearing components occurring in heat treatmentis critical to subsequent manufacturing operations and to the component's functional suitability. Basic high-carbonlow-alloy steel with a bcc structure expands rapidly (Fig. 16.18) as it is heated to approximately 727°C (1340°F)due to the coefficient of thermal expansion. At this critical temperaturethe material, as previously stated, undergoes a phase transformation to an fcc structure (i.e., austenite), resulting in component shrinkage. The specific volumeof austenite is less than that of ferrite. If the material is heated to still higher temperatures within the austenite range, it continues to increase in volume due to the coefficient of thermal expansion. Conversely, when rapidly cooled, thematerialshrinks to themartensite transformation te~perature.The martensite, formed as the part continues to contract while cooling to room temperature, is a bct structure. The resulting increase in volume, due to this transformation occurring at such low temperatures, stresses the material. Because it is virtually impossible in production heat treatment to achieve complete transformation from the austenitic (fcc) structure to the untempered martensite (bct), varying
16.18. Volume changes during heating and quenching (hardening) of high-carbon bearing steel.
Tcritical
Temperature
("C)
amounts of austenite, depending on the severity of the quench, are retained in the as-quench microstructure. Components must be thermally treated to reduce the residual stresses andto provide required structural stability. ional changes occurring in bearing steels essentially depend on tation of finecarbidefrom martensiteand decomposition or transformation of retained austenite. Because changes can also be induring bearing operation, due to the temperature or stress envint, the manufacturer must select the appropriate heat treatment to provide required stability. Temperingof high-carbon-chromium steels generally occurs in the range 66-260°C (150-500°F). At these temperatures fine carbide is precipitated, and the tempered martensite remains essentially bct with some shrinkage. Tempering in the range 20~-2$$”C (400-550°F) results in a time-temperature-dependent decomposition of retained austenite to bainite and volume increase. Loss of hardness at high temper~turesis prevented by tempering below ~ 6 0 ° C(500°F). The annealed microstructures of high-speed steels, providing maximum machinability, contain numerous hard metallic carbides, such as tungsten, molybdenum, vanadium, or chromium, imbeddedin a soft ferritic matrix. Unlike the high-carbon-chromium steels, temperatures far above the critical temperature must be attained to dissolve the desired amount of these hard carbide particles. Carbide precipitation is avoided by rapidly cooling the steel from the austenitizing temperature into the martensite transformation temperature range. After further cooling to room temperature, the structure normally contains 20-3096 retained austenite. Heating to temperatures required for tempering high-carbonchromium steel produces only slight tempering of the martensite. Between 427 and 593°C ($OO-llO”F), “secondary hardening” occurs; that is, the austenite is conditioned and subsequently transforms to martensite uponcoolingback through the MS temperature transformation range. ~ u l t i p l tempering e at these high temperatures is required to complete the transformation of austenite to martensite and to precipitate very fine alloy carbides, which are responsible for the secondary hardening phenomenon and provide for the high-temperature hardness retention characteristic of high-speed steels. Subzero treatments are often used after the initial quench or intermittently between tempering cycles to complete austenite-to-martensite transformation upon cooling. However, because cold-treatment sets up high internal stresses in the as-quenched components, it is generally recommended that cold-treating be practiced only after the first tempering cycle. Corrosion-resistant steels, for example, AISI 440C and 5749), are generally heat-treated incorporating deep freezing immediately after rapid cooling from the austenitizing temperature. AIS1 440C may be subsequently multiple tempered at approximately 149°C (300°F)
or 316°C (600°F), dependingon the product hardness requirements. Because of its alloy composition 13642is heat-treated according to standard practicesforhigh-speed steels-that is, multiple tempering at 524°C (975°F)incorporating ref~gerationcycles. Retained austenite, present inthe case microstructure of casecarburized steels, is a relatively soft constituent providing some tolerance for stress concentrations arising from inclusions, handling damage, and surface roughness. Case properties are preserved by generally tempering bearing components from 135-196’6 (285-385°F). The coreis stable at normal bearing operating temperatures.
~lasticity. The elastic properties of rolling bearing steel are not significantly affected by heat treatment. Hence the modulus of elasticity at normal temperatures is 202 M/mm2 (29.3 X lo6 psi) for both throughhardened and case-hardened steels. The limit of elastic behavior-that is, the stressunder maximum uniasial loading givinginsignificant plastic deformation or permanent setis described for rolling bearing steels by a 0.2% offset yield strength% remaining plastic strain. Figure 16.19 illustrates that strength properties tend to decline as transformation temperature increases for a given rolling bearing steel composition. ~ l t i ~ a t e ~ t r e n gUltimate th. strength is the stressat which the sample breaks in the uniaxial test described before;it is significantly affectedby heat treatment. For through-hardened AIS1 52100 ultimate strength for martensitic steel generally lies between 2 ~ 0 and 0 3500 N/mm2 (420-510 ksi). For the best case-hardening bearing steels, for example, ultimate strength approximates 2600 N/mm2 (380 ksi). ~ a t i g ~ e ~ ~ r Fatigue e n ~ t h strength . is determined in a cyclic push-pull or reversed bending test as the maximum stress that can be endured with no failure before accumulation of 10 million cycles. These data depend strongly on heat treatment, surface finish and treatment, testconditions, and so on.Accordingly, it is difficult to generalize and no numerical values are given herein. It is best to test the individual steel. ~ o ~ ~ ~ n Two e s stest . methods are used to determine toughness of bearing steels: the fracture toughness test and the impact test. In the first, a plain stress value KIc is measured in N/mm2-m1’2;this is the stress related to the defect size that can be tolerated without incipient structural failure. For martensitic AIS1 52100, ICIc falls between 15 and 22,
1600
1400 n
"E 1200
E
\
t
.-12 1000
5a
g 92
.s
800
ti 600 400 200 350
400
450 550 500 600 750 700 650 Transformation temperature("C)
16.19. Properties of a 0.8% C steel vs transformation temperature.
depending on heat treatment. A slight increase in ICIc occurs as temperature increases. Case-hardening steel tends to have greater fracture toughness than through-hardening steel. A ICIc value of 60 is not uncommon for surface-hardened steel. The second test-the impact of a hammer blow of defined energy on a sample-measures the energy absorbed in breaking the sample. For martensite-hardened AIS1 52100 this is only 4.5 J (3.3 ft-lb) compared to 172 J (127 ft-lb) for the soft annealed material.
dness. The manner in which carbon is distributed in steel dictates resulting hardness and mechanical properties. Althoughcarbon makes by far the greatest contribution to hardness, increasing the alloy also increases hardness. dness, a material's resistance to penetration and, hence, wear,can sured by static or dynamic methods. Static testin through a penetrator of defined geometry, rdness tester employed, either the depth of the size of the indentation becomes the measurement of the material ha Fig. See 16.20. ic testing involves bouncing a diamond-tipped hammer from a specific height onto the surface of the test specimen. The resultant re-
Depth
I (4
d (b)
(e)
16.20. Hardness tests. (a)Rockwell R,; indentation body: diamond cone; load 150 kgf (including 10-kgf preload), indentation depth for Rc 63: 74 ,urn, hardness testing range: R, 20-67. ( b )Vickers; indentation body: diamond pyramid 136", indentation depth: td-for V 782 = R,: 22 ,urn, hardness testing range: up to V 2000. ( e ) Brinell; Indentation body: hardened steel ball (Dl, hard metal ball (Dl, Indentation depth: -Qd, Hardness testing range: up to 400 B (-42.5 R,), up to 600 B (-57 Rc).
bound height is a measure of material hardness. The scleroscope is the only piece of equipment based on the dynamic test principle. Hardness attainable is a function of carbon content, as shown by Fig. 16.21. In general, as hardness is increased, toughness decreases for a given alloy steel. ~ e s Stress. ~ ~ Stresses ~ ~ Zinduced in a component through fabrication or thermal treatments aretotally eliminated upon uniform heating and soaking in the austenite temperaturerange. uenching of the component can once again generate tremendous int a1 stressesinthepart. Through-hardening of the martensitic high-carbon steels may produce surface tensile stresses that can producepart distortion or even cracking. ~urface-hardening heattreatments, including carburizing, carbonitriding, induction or ~ame-hardening,generally produce parts face compressive stresses. Regardless of the heating rnetho austenitizing, subsequent thermal cycles with or without subzero treatment canappreciably alterthe established stressstateintheasquenched parts. The stresses induced ina through-hardened component during quenching are principally the result of temperature variances and nonuniform phase transformations. Bearing rings, essentially being thin hoops of varying cross-sectional thickness, are prone to both size and
NT OF STEEL
AT TR
o Plain carbon steel Alloyed steel
0
0
0.4
0.2
0.6
0.8
1 .o
c (wt. %) 16.121. Maximum hardness vs carbon content.
shape changes, Fixture quenching, employed to retain the co~ponents9 as-machined dimensional characteristics, may hamper quench rnedi~m flow and induce additional nonuniform stress distribution in the part because of mechanical restraints that do not adapt to size or shape changes, The machined undercuts, grooves, filling slots, oil holes, and flanges have sharp corners and recesses provide additional focal points for stress concentration. The high-carbon-chromi~rnbearing steels under recommended austenitizing temperatures have an S temperature range of appro~imately 2 0 4 - 2 ~ 2 ' ~(400-4~0'
loying elements in this perature. ~ o a r s e -
ing bath, wash the parts, and subject them to a tempering cycle. low-tempering temperature for a long time is equivalent to a high temperature for a short time from the standpoint of reducing the residual component stresses. This sequence of operation may be inte~uptedby a subzero treatment following the washing operation to ermit the completion of the austenite-to-martensite transformatio~. in this manner are very prone to cracking because of the resulting high residual stress state. These parts must be tempered as soon as they warm to room temperature. ~urface-hardeningheat treatments,by a diffusion processaltering the composition of the material or by the rapid heating of a selected surface area of a homogeneous steel, are developed and controlled to provide surface compressive stresses with normal counterbalancing tensile stresses in the core. Induction surface hardening of an appropriate material to the proper case depth results in the maximum Compressive stress being located at the case-core transition zone. The m a ~ i t u d eof this co~pressivestress ina surface-hardened high-carbon steel alloy will normally be less than that produced in a carburized part at its point of maximumcompressive stress, which is at the approximate midpoint of the total case depth. This point corresponds to the carbon content of approximately 0.50%. Tempering of as-quenched, surface-hardened components with or without support of subzero treatment will generally reduce the retained austenite level and modestly alter the level of compressive stresses.
Contact fatigue failure in rolling bearings occurs when local material stresses exceed the local fatigue limit; cracks are initiated and thenpropagated. Even if the stresses induced by cyclic loading between rolling elements and raceways are generally below the fatigue limit, additional stresses can be caused locally by material inhomogeneities and defects ac ress concentrators. These stressesare superimposed on those ar to normal bearing operation. Inhomogeneities occurring from the steelmaking process are distributed throughout the entire material: for example, 'slag inclusions and pores. They can also be the ~ a n u f a c t u r i nprocess, ~ where they are mainly limit zones: for example, ring marks, scratches, and grinding burns. The different types of material inhomogeneities and defects are the reason that bearings fatigue in two observable modes. In one mode, failure is caused by a stress-raising inhomogeneity in the subsurface region where the normal stresses induced by cyclic loading are max~mum, mode, failure is initiated by an inhomogeneity or defect in the surface,
thus increasing the surface stresses. Tallian E16.241 provides a compendium of results concerning rollingbearing failure modes and causes.
terial inhomogeneities that can lead to subsurfacemacro- and microinclusions, pores, and bandings. ses surrounding these inhomogeneities depend on the nature, size, distribution, shape, and interface between the inhomogeneity and the base material. acroinclusions are impurities ori~natingfrom the ecause of their large size [above 0.5 mm (0.02 in.)], high brittleness, very low cold deformability, and irregular shape. always cause early failure when situated in the surface or subsurface zones.
c r o i ~ c Z ~ s ~ oAll ~ oxide-type s. slag microinclusions are brittleand either in stringers or in globpractically impossibleto deform. They occur ove 40 pm (0,0016 in.) diameter for the g1 their negative influence onlife is great; seeFig.16.10. stresses can occur in the matrix surrounding the inclusions difference in thermal contraction between matrix and particle during . They are called tessellated stresses. A relationship exists besize and size distribution of oxides, on the one hand, and the n content of the steel, on the other hand; the higher the oxygen content, the bigger the maximum size and the total number of oxides. ~~~Z c Z ~ s~ i oSulfide ~~s . inclusions e are relatively soft and easily colddeformed. Therefore, high stress peaks surrounding sulfide inclusions can be partially re uced by plastic deformation. In some cases sulfides surro~ndthe oxides and, by plastic deformation, close the voids so that the oxides become less dangerous. The allowable sizeand distribution of eater than those of oxides.
~
res are generated by gas bubbles enclosed during steel solidoften occur as clusters. If their surfaces are not oxidized, ed during warm- and cold-processing of the steel. In the oxidized condition of the surface they have a negative influence on fatigue life and, regardless size and shape, can be treated similarly to globular-type slag inclusions. igh local differences in steel chemical composition (se gations) caused by improper solidification conditions can lead to difl'erent
component microstructures after hardening. The high stresses between microstructural zones, originating from the different MS points, and the reduced fatigue limits, can lead to early fatigue failures.
foregoing defects, other erial defects in the sur-tempered or rehardened ing burns, decarburized layers, and marks can lead to surface-initiated failure.
rind in^ Burns. Grinding burns occur under improper grinding conditions of hardened components. The material is then locally tempered, yielding insufficienthardness, or it is rehardened. The heat-affected zone varies between a few microns and a few hundred microns. In the latter case tensile stresses up to 1200 N/mm2 (175 ksi) have been measured. ~ e c a r ~ u r i ~ a t iDecarburization o~. is the result of heat treatment in an oxygen rich atmosphere. The carbon content in the surface is thereby depleted. The different MS points in the base material and the surface e rise to residual tensile stresses. ~ a r and ~ s~ n d e n t ~ t i o n s Marks . and indentations occur because of incorrect handling of components during manufacturer or mounting. During bearing operation the normallyinduced stresses can add to the ual tensile stress caused around oxides ecarburized zones, bandor areas of grinding burns. The resulti stresses can get so high that fatigue can be initiated. Thevoids arou indentations, and . * defects are discontinuities in the surfacearks, material from which can also begin.
advent of the aircraft
rifice of fatigue endurance when compared to bearings fabricated from ecause of light applied loading, however,fatigue endurance is not a major consideration in such applications. The chemical compositions of some of the foregoing steels are given in Table 16.3. Theexploration of space andthe continuing development of the aircraft gas turbine engine provided the demand for yet increased development of exotic materials. Examples are sapphire for balls, precipitation-hardening stainless steels, and nickel-based superalloys. Additionally, the nuclear power industry created the need for cobalt alloys such as L-605, Stellite-3, and Stellite-6. Powdermetal-forming techniques have now provided the means to create steeZs of differential properties; for example, extremely hard, corrosion-resistant surfaces combined with tough, high strength substrates. The requirement for aircraft gas turbine engine mainshaft ball and roller bearings to operate at ever-increasing speeds initiated the search for a relatively high temperature capability, fracture-tough steel. Because of the bearing ring hoop stresses caused by ring centrifugal stresses and rolling element centrifugal forces at high speeds, fatigue spalls under such conditions canlead to fracture of rings fabricated from through-hardening steels such as M50. Thus, the operating speeds of aircraft gas turbine engines were limited to approximately 2.4 million dN (bearing bore in mm x shaft speed in rpm). With the development of M50-~il,a case-hardening derivative of M50 whose chemical composition is shown in Table 16.3, this limitation has been overcome. Figure 16.22 from Spitzer [16.29] shows the effect of the higher fracture toughness of M50-~ilon speed capability. The need for bearings to operate at ultrahigh temperatures has triggered the development of cemented carbides and ceramics as rolling bearing materials. Materials such as titaniumcarbide, tungsten carbide, silicon carbide, sialon, and particularly silicon nitride are being investigated. At elevated temperatures, these materials retain hardness, have corrosion resistance, and provide someunique properties, some of which ous, such as low specific gravity for silicon nitride. Conversely, other properties of these materials, such as extremely high elastic modulus and low thermal coefficient of expansion for silicon nitride as compared to steel, create significant bearing design problems that must beovex+comeif these materials are to succeed for use in rolling bearing structure c o ~ p o n e n ~particularly s, rings. Ceramic materials such as silicon nitride, as illustrated by Fig. 16.23, commence life as powders that after a series of processes, principal among which is hot isostatic ~ressing, are transformed into highly engineered bear in^ components. Table 16.4, excerpted fromPallini [16.30], gives significant mechanical prope~tiesand allowable operating te~peraturesfor several of the materials describedabove. ons side ring the low density and high elastic
BEARING STRUCT
Hoop stress, N/mm2
, Comparison ofM50
vs M5O-Nil steel-ring hoop stress vs bearing diV
(from [16.291).
3. Silicon nitride begins life as a powder before a series of processes transforms it into a highly engineeredbearing component (SKI?photograph).
s Properties of Special Bearing Structural Materials Elastic Modulus Rockwell N/mm2 of Thermal Hardness C Useful Max X 103 Expansion (room Temperature Specific (psi X Poisson's 10-6/"C Material temp) "C ( O F ) Gravity lo6) (10-6/"F) Ratio Coefficient
440C stainless steel M50 tool steel M2 tool steel T5 tool steel T15 tool steel Titanium carbide cermet Tungsten carbide Silicon nitride Silicon carbide Sialon 201
62
260(500)
64
320(600)
66
480(900)
65
560(1050)
8.8
190(28)
0.28
67
590(1100)
8.2
190(28)
0.28
67
800(1470)
6.3
390(57)
0.23
78
815( 1500) 14.0
533(77.3)
0.24
7.6
190(28)
190(28) 7.6
0.28
10.1(10O0C) (5.61)
0.28
12.3(3OO0C) (6.83) 12.3(300"6) (6.83) 11.3 (6.28) 11.9 (6.61) 10.7 (5.94)
0.28
78
1200(2200)
310(48) 3.2
0.26
90
1200(2200)
410(59) 3.2
0.25
78
1300(2372)
3.3
288(42)
0.23
5.9 (3.28) 2.9 (1.61) 5.0 (2.78) 3.0 (1.67)
modulus of hot isostatically pressed (HIP) silicon nitride (Si,N4) as compared to steel, Figs. 9.7-9.9 compared performanceparameters of a 218 angular-contact ball bearing having HIP silicon nitride balls with those of the same bearing having steel balls. It can be seen that, athigh speed, eoff betweenreduced ball load and increased co there i stress. 16.24 from [16.30]indicates the frictional prope silicon nitride when usedwith various types of lubricants. It 1s apparen that for sustained low friction operation, beari tride balls require oil lubrication. It is usual fo to have raceway c ~ ~ a t u radii r e rm = 0. bscript m refers to the rac stresses. f for tun n; therefore, ball b
es increases in contact itride balls may have
1O .O 0.90
0.80 0.70
0.60 0.50 0.40 0.30 0.20 0.10
0.00
0.50
1.00
1.50 2.00 Slip 76
2.50
3.00
3.50
16.24. Traction coefficient vs percentage slip in the contact between two bodies of hot pressed silicon nitride. Contact stress is 2068 N/mm2 (300,000 psi); nominal speed is 3800 mm/sec (150 in./sec) (from L16.301).
- Operating conditions (1) 25°C (77"F)-dry
contact (2) 370°C (698"Fb-graphite lubrication (3) 538°C (lOOO"F)-graphite lubrication (4)25°C (77"F)--oil lubrication
their designs optimized for Hertz stress (and hence fatigue life) or friction. ons side ring the basically straight contour raceways, this option is not available for cylindrical or tapered roller bearings. ure 163.24 further shows that irrespective of fluid or ~ ~ y lubri- ~ Z , the friction coefficient of HIP silicon nitride is depende~ton the temperature andenvironment. It has been de~onstrated,howwhen lubricant flow is interrupted, steel bearings with silicon nitride balls will sustain operation longer without seizure as comto the same bearings with steel balls. ile the compressive strength of con nitride is excellent, the strength is only about 30%th 0 steel. The fracture toughness is alsoonly a small percentage of that of 50 steel, let alone u~hermore,although in rolling contactunder heavy load ds to fail by surface fatigue and even tends to have tigue life than steel, any disruption of the surface can lead to mbling of the surface under continued operation. tion is therefore an important consideration in
~
The generally stated function of the rolling bearing cage is to maintain the rolling elements at properly spaced intervals for assembly purposes. t is sometimes inferred that, in normal bearing operation, the cage is not necessary;rather it“goes along forthe ride”; that is, it is not a highly stressed com~onentrequiring the strength of the accompanying ring e are more exceptionsto this statementthan examples. nshaft and accessory airc 4340 steel (AMS 6414 or the hardness range of I t ,28-35. These cag S 2412) to provide corrosion resistance and added lubricity. aring applications not only do the rolling elements contact the cage pockets,but the cages themselves are either inner ringor outer are manufactured from many types of material, including aluminum, S-Monel, graphite, nylon, and cast iron, the major bearing product lines use brass or steel. In ball bearings principally, but also in some roller bearings, plastics are replacing these metals.
Plain low-carbon strip steel, suitable for cold-forming (0.1-0. used in thebulk fabrication of pressed, two-piece, or ~nger-retention-t~e Two-piece cages are joined by mechanical loc e material has a tensile strength of 300-400 I 4340 machined steel cages previously ment ons have appro~mately0.4% G for increased tionally, low-carbon steel tubes and forgings are used bearing ap~licationsth need unique features for lubrication or any steel cages are surface hardened or proved wear characteristics.
making ball and roller bearings. Other nonferrous brass alloys may be ally cast, but they are hot-worked by upsetting or ring rolling to meet specific productre~uirements.Cage blanks may also be produced by e x t r ~ ~ i the n g centrifugally cast billet.
ilicon-iron-bronze(Cu: 91.5%, Zn: 3.5%, Si: 3 , 0%) is an alloy r e c o ~ ~ e n d efor d ball and roll temperatures up to 316°C (600" extensively hot-worked and extr promote optimum
ma-
articular1 n lon
mental factors.
Some examples of polymeric cage roperties of cage polymeric materials are Low coefficient of thermal expansion Good physical property retention, especially strength and fle~ibility, throughout the temperature rangeof operation ~ o m p a t i b i ~ iwith t y lubricants and environmental factors evelopment of a suitable cage design to minimize ~riction and roper lubrication
Polymeric cage designs. ( a ) Snap cage for ball bearing (nylon 6,6). ( b ) Cage for cylindrical roller bearing (nylon 6,6). (e> Cage for hig~-an~lar-contact 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 ~ Fabric-reinforced phenolic resin cages have been used for many years in high-speed 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 glass-fiber reinforcement are required for bearing series employing such cage material. Figure 16.26from[16.32] illustrates the endurance capability of 25% glass-fiber-filled 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~h-temperatureresins with and without glass-fiber 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% glassfiber-fill (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 high-temperature ~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 lubricant-temperature 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 ester-based 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 "snap-in" 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 low-t~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: Elastomer-basic polymer that determines the ranges of final product properties Curing agents, activators, accezerators-determine degree and rate of elastomeric vulcanization (cross-linking) PZasticizers-improve flexibility characteristics and serve as processing aids ~ntio~i~ants-improveantifatigue 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
r-4
SE
CH2=CH-CH=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 low-tern requirements and thermal resistance required. rubber seal is used in many standard beari a1 cost is low compared to other elastomers oldable, which allows one-step processing of complex lip shapes. ~ubricantcompatibility with petroleum-based lubric~ntsis good for high acrylonitrile versions. This elastomer is suitable for applications to 100°C(212°F) and is therefore not indicated for high-temperature 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 high-temperature and food-contacting bearing applications, Silicon rubbers have a backbone structure made up of silicon-oxygen linkages, which give excellent thermal resistance. A typical polymer is R
I
R
I -0-Si-O-Si-O-Si-O-Sii I R
R
R
I
I
R
R
I
I
R
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 high-temperature performance to 180°C (356" ) and good low-temperature 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 low-temperature 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 high-temperature and lubricant-compatibilitychartypical 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 run-in by preventing metal-to-metal contacts and providing a lubricant reservoir Prepared surfaces for other surface coatings-that 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 run-in 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 wear-resistant coatings for cages. In response to aircraft bearing requirements, silver plating over a nickel- or copper-struck 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 process-coating 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 850-1050°C (1562-1~22"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~e-platingtechniques 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, lithium-based 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 8450-Nil bearing rings provide long-term 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 high-pressure 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 MIL-Lextreme 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 vacuum-melted AISI 440C stainless steel. Target duration for bearing operation is only a few hours.
16.1. American Society for Testing and Materials, Std A.295-84, High Carbon Ball and Roller Bearing Steels; Std A485-79,“High Hardenability Bearing Steels.” 16.2. American Society for Testing and Materials, Std A534-79, ““CarburizingSteels for Anti-Friction Bearings.” 16.3. J. Braza, P. Pearson, and C. Hannigan, “The Performanceof 52100, M-50, 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, “Degassing-Then and Now,” Ironand S ~ e e Z ~ a k e26-32 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., 5,347-364 (1962). 16.9. United States Steel Corp., Making, Shaping, and Treating of Steel, 9th ed., 551 (1971). 16.10. United States Steel Corp., Making, Shaping, and Treating of Steel, 9th ed., 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 Steels-Properties and Processes,” Ball Bearing J 217,32-44 (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 Rolling-Contact,”NASA TN 3933 (1975). 16.19. SKF Steel, The Black Book, 194 (1984). 16.20. SKF Steel, The Black Book, 151 (1984). 16.21. M. Grossman, Principles of Heat Treatment, American Society for Metals (1962). 16.22. American Society for Testing and Materials, Std A255-67,“End-Quench Test for Hardenability of Steel” (1979). 16.23.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, McGraw-Hill, New York (1985-1986). 16.27. Metal Finishing Guidebook and Directory 85, Metals and Plastics Publications, Hackensack, N.J. (1985). 16.28. A. Graham, Electroplating Engineering, 3rd ed., Van Nostrand Reinhold, New York (1971). 16.29. R. Spitzer, “New Case-Hardening Steel Provides Greater Fracture Toughness,’’Ball Bearing J, SKF, 234, 6-11 (July 1989). , “Turbine Engine Bearings for Ultra-High Temperatures,” Ball Bearing 34, 12-15 (July 1989). 16.31. A. Olschewski,“High Temperature Cage Plastics,” Ball Bearing J, SKF, 228, 13-16 (November 1986). 16.32. H. Lankamp, “Materials for Plastic Cages in Rolling Bearings,” Ball Bearing J, SKF, 227,14-18 (August 1986). 16.33. R. Maurer and L. Wedeven, “Material Selection for SpaceShuttle Fuel Pumps,’’Ball Bearing J, SKF, 226,2-9 (April 1986). 16.34. Delta Rubber Company, Elastomer Selection Guide.
This Page Intentionally Left Blank
escription a C Ca F H Li O P
R, R' , R" S VI
w
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 petroleum-based 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, petroleum-based oils exhibit good performance characteristics at relatively inexpensive costs. Synthetic hydrocarbo~fluids are manufactured from petroleum-based 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 non-petroleum sources. Other “synthetic” fluids have unique properties and can be manufactured from non-petroleum-based 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 low-to-moderate 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. ~ick-feedand oil-ring 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 debris-induced 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 high-performance
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 start-upuntil 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
'c Oil
discharge
\ \
.
Under-raceway 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 fluorocarbon-based 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 moderate-to-high 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 soap-based 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~m-bas~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. Resin-bonded 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 heat-cured or air-cured material. Heatcured materials are generally superior to the air-dried 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 solid-lubricated 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 high-current 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 lubricant-saturated 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 liquid-lubricated 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 steady-state 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 index-that 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 start-up. 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 con-87% carbon, 11-14% 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 viscosity-temperature characteristics for lubrication. Usually such crudes are low in asphalt and trace materials. The earliest commercial crude, nnsylvania, was of this type.
C
c
cI
/ / \
c
\
/
cI1
C Arorhatic Cbenzene ring)
/
C
c I c\
\
c I c /
C Naphthenic (saturated ring)
i -C-
I I I I --c-c--6°C" I
i
I
I -C-
I Paraffinic (saturated chain)
.3. Chemical structures of mineral oils.
Naphthenic-based 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 wear-minim~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 light-tomoderate 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 metal-to-metal 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 sulfur-chlorine-phosphorus 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 85-95. 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 D-567.
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 petroleum-based 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
3
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ci m m m
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coeoeow
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oc-u3* ri
ri
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1000
500
200
"d
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 3-11°C (5-20°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 lubricant-base 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 asload-carrying ability and high-speed operation. Also, many very high-temperature 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 low-viscosity index. Their outstanding characteristic is fire resistance, and assuch these fluids are often used as hydraulic fluids in high-temperature 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 viscosity-temperature relationships, good low-temperature 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 -RO-CH~-CH-O-R”-
I
R’
FI
.
Chemical structure of a polyglycol.
0 II
RO-P-OR‘ 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 viscosity-temperature 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 high-temperature 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 high-temperature hydraulic fluids and lo~-volatilitygreases. ~luorinatedpolyethers as a class represent the highest-temperature 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 radiation-induced 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 oxygen-handling 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~e-enhancingadditives. 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 D-942. 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 D-566. ~ 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 99-149’6 (210-300”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 D-566. L o ~ - T e ~ ~ e r a t uTorque. re Low-temperature torque is the extent to which a low-temperature grease retards therotation of a slow-speed ball bearing when subjected to subzero temperature. The method of testing is described by ASTM specification D-1478. 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 D-1742;the sample is determined by supporting on a 74-pm 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 D-97. The pour point together with measured lowtemperature viscosities givesan indication of the low-temperature 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) 445-475 400-300 355-385 3 10-340 265-295 220-250 175-205 130-160 85-115
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 oil-bleed in^ with respectto temperature capabilit~, water-resistance, 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 greases-Lithium 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 complex-based 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. High-quality lithium soap greases of both types have excellent service histories in rolling bearings and have been usedextensively in prelubricated; that is, sealed and grease~-~or-Zi~e applications. Lithium-based 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 ~ ~ greases-The 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 ncalcium-thick~ned greases is the calcium n the soap is modified by adding an complex-based 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 greases-Sodium ~ ~ 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). Sodium-base greases have been superceded by more water-resistant products in applications such as electric motors and front wheel bearings. Sodium complex-base greases have subsequently beendeveloped having upper and lower operating temperature limits of 1140°C (284°F) and -20°F (-4"F), respectively. ~luminum comple~ greases-Aluminum stearate used in rolling bearings, but aluminum compl being used more often. Greases formed from the complex soap perform favorably on water-resistance tests; however, the upper operating temperaturelimit is somewhat low at Ii0"C (230°F) compared to other types of high-quality greases. The lower operating temperature limit is satisfactory at -30°C (-22°F). These greases find use in rolling mills and food-processing plants. on-soap-~ase greases-Organic thickeners, including ureas, amides, and dyes, are used to provide higher temperature capability than
~
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 high-temperature ball bearing greases for the electric motor industry. The recommended upper operating temperature limit for polyurea-base 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 mixed-that is, soap and nonsoap or differing soap types-dramatic 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 oils-that is, petroleum and siliconeoils-canproduce 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 oil-flow
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,tire-cordwinding, 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
F
~ 17.9.~ Polymer-lubricated ~ 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 corrosion-control 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“shear-stren~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 low-shear-strength 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. Initially9two-cycle 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 micro-organisms 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 run-outs, 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 labyrinth-type 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 once-through 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 soot-type 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. Deep-groove 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 grease-laden 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 single-stage 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 well-designed seal, is generally greater than the fric-
17.12. Bearing pillow block with felt seals.
67
I
I
1
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 fluid-tight 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 start-up 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 outof-roundness. 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 by-product 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 double-li~ 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. Double-lip seal. Inside lip is for fluid retention; outside lip is for dust exclusion.
17.18. Garter seal X-section. 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 two-stage 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 (1997-01).
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 load-life 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 mm-l (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 load-carrying 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 fatigue-spalledarea. 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 Lundberg-Palmgren 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 load-carrying 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 single-row bearings superimposed on the dispersion curve
Number of bearings failed, I%;
18.6. Fatigue life comparison of a single-row bearing to a two-row bearing. A group of single-row bearings programmed for fatigue testing, was numbered by random selection,no. 1-30, 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 two-row bearingsand the shorter life of the two plotted as the life of a two-row 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 double-row 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 double-row bearing subjected to the same specified loadingas a single-row of identical design is less than the life of a single-row 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 load-carrying 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 load-carrying 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 stress-depth 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 element-raceway 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 Lundberg-Palmgren 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 through-hardened AIS1 52100 steel. Palmgren [l8.9] states that for commonly used bearing steels, e is the range 1.1-1.5. For modern, ultraclean, vacuum-remelted 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 = 0.40-a 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 $b
since 8, = 0.9Sa, 1 1 In- = 1nsb 0.9
+ ln-1 sa
Therefore, 1 In -
sa
+ 0.1053 ==
(~~
By subtraction
or (L& + The additional Llo life is given by
LiO
=
- La
X
0.1053
(18.21)
In equation (6.71) it was established that at point contact (6.71)
More simply, ro =
Tvmax
(18.22)
in which T is a function of the contact surface dimensions, that is, bla (see Fig. 6.14). From equation (6.47) the maximum compressi~e stress within the contact ellipse is (6.47)
Furthermore, from equations (6.38) and (6.40) a and b are (6.38) (6.40)
in which (18.23)
By equation (6.58)
in which [ is a function of bla per equation (6.72) and (6.14). ~ubstitutingequations (6.47) and (6.40) into (18.15)yields (18.24)
Letting d equal the raceway diameter, then 2 =
and
TG?
(18.25) Rearranging equation (18.25)
From equations (6.38) and (6.40),
Q
EO~P 3a*(b*)2
-
"
ab2
(18.27)
Create the identity
and substituting equations (18.27) and (18.28) into (18.26) yields In
S
Ch-l
[
Ed>=P
]
3a*(b*)2
(c+h-1)/2
):(
(c-h-1)/2
( ~ )
(c-h+1)/2
dD2-hueLe (18.29)
~ubstitutingequation (6.38) for the semimajor axis a in point contact into equation (18.29):
(18.30) Rearranging equation (18.30) gives
Equation (18.31) can be further rearranged. Recognizing that the probability of survival s is a constant for any given bearing application,
~
~ CAPACITY ~ (e-h+2)/3
( ~ ) Letting T
IzI
=
\
ILIFE OF CA ROLL^^ CONTACT
[
T'dD 2-hue
(18.32)
gh-l('*)c-l(b*)c+h-l
Tl and J
= J1
when bla
=
1, then
I
- [(~~ ($)
]
h -(1L ) x p ) ( 2 ~ + h - 2 ) / 3 (,*)c-l('*)c+h-lE
(18.33) f)" ( 3 - h )
Further rearrangement yields (18.34)
in which A, is a material constant and
[(g)'($)
(L)xp)(2e+h-2)/3 -3/(c-h+2)
h-1
@
=
x
(,*)c-l(b*)c+h-l
(18.35)
For a given probability of survival the basic dynamic capacity of a rolling element-raceway 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+h-5)/(c-h+2)
(18.36)
P'or a bearing of given dimension,by equating equation (18.34)to (18.36), one obtains QL(3e)/(c-h+2)
=
Qc
(18.37)
or
( ~ )
(e-h+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 load-life 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
c--h=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 roller-raceway 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
$2,
Figure 18.9 from reference [18.1]establishes the validity of this assumption. ~ubstituting(18.52) and (18.47) into (18.51) gives
di>
= 0.0706
[~1 ~] 222
X
r
( ~ ) 0.3
0.41
(1
7)1.39
(18.53)
~4-l’~
The number of stress cycles u per revolution is the number of rolling elements which pass a given point (under load) on the raceway of one ring while the other ring has turned through one complete revolution. Hence fromChapter 8 the number of rolling elements passing a point on the inner ringper unit time is
=
0.5Z(l
+ y)
(18.54)
For the outer ring, U, =
0,5Z(l - 7)
(18.55)
or
in which the upper sign refers to the innerring and the lower signrefers to the outer ring.
F I G ~ E
L18.11).
Substitution of equation (18.56) into (18.53) gives the following expression for @: 0.41
@ =
0*089
(1 T (1
($x 5)
741.39 0.3 ,,,)1/3
( ~ )
2-lf3 (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 D1.82-1/3 (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 through-hardened to Rockwell C = 61.7-64.5. 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 through-hardened 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 Lundberg-Palmgren 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
+
through-hardened 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 self-aligning 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
single-row ball bearings and to 88.2 (6700)for double-row, deep-grooveball 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 groove-type bearings. This increaseis accommodated by a factor b,, that augments the above-indicated 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 subsurface-initiated fatigue failure of the raceways. Ball failure was not consideredapparently because it was not frequently observed in the Lundberg-Palmgren 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 Lundberg-Palmgreln assumption. It is perhaps more correct to assume that Lundberg and Palmgren did not observe significant numbers of ball-fatigue 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 ball-fatigue 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 ball-fatigue 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 Lundberg-Palmgren 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 Lundberg-Palmgren 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 Single-Row 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
Double-Row 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 two-row bearing does not have twice the basic dynamic capacity of a single-row 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 (=)
3-6
(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 single-rowball bearings and 35.9 (2730) for double-row deep-grooveball 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
r-l
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)O-33
tan &0-67D1.8
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 thrust-loaded angular-contact 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 angular-contact 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*8Z-V3(18.6'7) =
93.2
=
17,040 N (3830 lb)
Oe4'
(1- 0. 1169)1.39 (1+ 0.1169)V3
*Strictly speaking, an angular-contact 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 22,250-N (5000-lb)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
(
1.8z-l/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)-1-11+ (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 roller-raceway 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 0.41-0.56 0.6-0.8
Outer Raceway 0.38-0.6 0.6-0.8
7~~
F A T I G ~LIFE OF A ROLLEYG BEARING 4
L P
=
( ~ )
(18.145)
As with point-contact 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 point-contact 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)
(L1-9/8+ 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 point-contactbearings, 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 point-contact 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
0.4-0.5 0.5-0.6 0.6-0.8
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 self-aligning 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 Single-row bearing Double-row 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. Roller-RacewayContact 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 <1
737
BEARING RQLLING FATIGm A L I m OF
If both inner and outer raceway contacts are line contacts and A = 0.45 to accountforedgeloading and nonuniform stress dist~bution, curve 1of Fig. 18.13 showsthe variation of load with life by using equation (18.160) and the fourth power slope. Assuming u = 1.36 and using equation (18.167), curve 2 illustrates the approximation to curve 1. The shaded area shows the error which occurs when using the approximation. Points A on Fig. 18.13 represents 5% error. If outer and inner raceway contacts are point contacts for loads arbitrarily less than 0.21C ( L , = 100 million revolutions), then for h = 0.65 curve 1 of Fig. 18.14 shows the load-life variation of the bearing. Note the inverse slope of the curve decreases from 4 to 3 at L = 100 million revolutions. Curve 2 of Fig. 18.14 shows the fit obtained while using equation (18.167) and v = 1.20. Lastly, if one raceway contact is line contact and the other is point variation for h = 0.54. contact,curve 1 of Fig.18.15showsload-life Transformation from pointto line contact is arbitrarily assumed to occur at I; = 100 million revolutions. Curve 2 of Fig. 18.15 shows the fit obtained while using equation (18.167) and v = 1.26. In Fig. 18.13, using equation (18.167), fatigue lives between 150 and 1500 million revolutions have a calculational error less than 5%. Similarly, in Fig. 18.14 lives between 15 and 2000 million revolutions have less than 5% calculational error, and in Fig. 18.15 lives between 40 and 10,000 million revolutionshave less than 5% calculation error. Sincethe foregoing ranges represent probable regions of roller bearing operation, Lundberg et a1 [18.8] considered that equation (18.167) was a satisfactory approximation by which to estimate fatigue life of roller bearings,
0.2
1
~
10
100 500 lo00 Llo Fatigue life, revolutions x lo6
1( Ooo
G 18.13. ~ AI,, vs E FIC for line contact at both raceways.
4 3 2
0.5 0.4
0.3
0.2 Llo Fatigue life, revolutions x IO6
F I G ~ E 18.14. L,, vs FIC for point or line contact at both raceways.
18.15. L,, vs FIC for combination of line and point contact (reprinted
from
118.81).
Accordingly, the data in Table 18.10 were developed. Equation (18.156) becomes
c = 207hv
{ + [ ( );
y2/9(1
X
(1 rt
143/108] 9/2}-2/9
1.04: -
1
y)29/27
X y)1/4
(iZ cos a)7’gZ3’4D 29/27
(18.169)
EFFECT OF STEEL C Q ~ Q S I T O NANI) P R O C E S S ~ GON FATIGUE
.
73
LIFE
u and h for Roller Bearings
Condition Modified line contact in cylindrical roller bearings 0.83 1.36 Modified line contact in spherical and tapered roller bearings Combination line and point contact (inner and outer raceways) Line
h
U
hU
0.61 0.57 0.78
0.54 0.45
1.36 0.68
1.26 1.36
0.61
int and Line C~ntact For thrust roller bearings operating with a combination point and line contact at the raceways, equations (18.162) and (18.163) become
C,
=
552hqvj2"
(1 -t. y)29/27 ( I cos a)7/9 tan &314D29/27a (1 rf: y)V4
z 90" (18.170)
CE All of the equations pertaining to basic dynamic capacity of a raceway contact QC,or bearing raceway Ci or C,, or of an entire bearing C developed thus far are based on bearings fabricated from AISI 52100 steel hardened at least to 58 Rockwell C. This is air-melted, air-cast steel whose chemical compositionis shown in Table 16.1. Moreover,the equations are based uponsteel processing methodsand manufacturing methods that existed at the time the original load rating standards were created. Equation (18.107) gives a recommended reduction in basic dynamic capacityif the steel is not as hard as58 Rockwell C . Additionally, if the bearing material hardness is greater than 58 Rockwell C as occasioned by a special heat treatment, the bearing fatigue life can tend to be greater than that predicted by the standard life rating formulas. As shown in Fig. 18.16 from reference [18.14], this effect is diminished if the bearing operates at elevated temperature. Developments in theprocessing of rolling bearing steel have been continuous since 1960, the date corresponding to the introduction of carbon vacuum-degassed ( 0 ) steel in the United States. Improvements in melting practices have yielded bearing steels of significantly increased fatigue endurance capability. For instance, AISI 52100 steel melted in a
740
FATIGUE LIF'E:
L ~ B E R G - ~ THEORY ~ G ~ AN'l3 N RATING S
S
2.2
2.0
1.8 400 1.6
300
"C
1.4 200 1.2 100
1 .o
58 62
60
66
64
Rockwell C hardness at room temperature
~ I 18.16.~ Nomograph ~ for E determining relative life at bearing operating temperature as a function of room temperature hardness.
vacuum has fewer impurities and therefore tends t o yield increased rolling bearing fatigue life. Walp et al. [18.15] determined that thepresence of traces of metallic impurities such as aluminum, copper, and vanadium is detrimental to fatigue life. Remelting of vacuum-melted steel while using a consumable electrode furnace tends to produce a more homogeneous steel, which inherently has a better rsistance to fatigue. Some rolling bearing steels afford increased corrosion resistance in certain applications however at the expense of potentially decreased fatigue endurance. In Chapter 16, the various bearing steels and steel manufact~ringprocesses are discussed in detail. W e n rolling bearings are fabricated of special steels, the expected increased or decreased fatigue lives maybe estimated using life adjustment factors discussed later in Chapter 23. Koistinen [Is.161 demonstrated a two-stage heat-treating process that forms a surface layer in compression conducive to increased fatigue endurance. The rationale for the increased endurance is discussed in Chapter 23. Additionally, for AIS1 52100 and Mi50 bearing steels, ring and rolling element forming processesthat tend to eliminate end grain in the bearing rolling contact surfaces have been used to manufacture rolling bearings of increased endurance capability. The endurance characteristics of case-hardened bearing steels must also be considered. These steels, covered in detail in Chapter 16, have
the advantage of a tough, fracture-resistant core as well as a fatigueresistant, hard surface layer (case). Since the information in Chapter 6 on subsurface shear stressindicates the critical stress occurs closeto the raceway or rolling element surface, case-hardened steels are appropriate for rolling bearings. In fact the fracture-resistant core is essential in many applications. The Lundber~-Palm~enmethod has historically been applied to case-hardened steel bearings also; however, the effect of the compressive stresses resulting from heat treatment is not properly accommodated. In Chapter 23 life calculation for case-hardened steel bearings is discussed. One may question the value of the data developed in this chapter concerning Llo fatigue life and basic dynamic capacity since these data are based only upon air-melted AIS1 52100 steel of a type no longer acceptable in modern industrial practice. The answer is threefold: (1)it is important to understand the origin of the load rating standards in current worldwide use in order to use them ef5ectively and accurately, (2) a comparison may be conducted between the endurance characteristics of similar and dissimilar bearings of different manufacturers on the basis of geometry alone, and (3) the equations may be used to optimi~e rolling bearing design for any given application.
To accommodate the improvements in bearing geometrical accuracies afforded by modern manufacturing methods and the improvements in the modern basic rolling bearing steel chemistries and metallur~es,ANSI and IS0 standards have included in the formulas for basic load rating* “a ratingfactor for contemporary, normally usedmaterial and manufacturing quality.” IS0 [18.13] uses the b,, factor directly in the equations for basic dynamic radial load rating* and basic dynamic axial load rating;* for example, equation (18.103) for radial ball bearings becomes:
IS0 E18.131 gives values of the factor b,, which may be applied to the formulas for basic dynamic capacity for each of the various executions of *The terms basic loading rating, basic dynamic radial load rating (or basic dynamic axial load rating), and basic dynamic capacity may be used interchangeably. The last term was that created by Lundberg and Palmgren. ?ANSI Il8.101 recommends using D raised to the 1.4 power in lieu of 1.8 for bearings having balls of diameter greater than 25.4 mm (1 in.). In this case, for metric units calcualtion of basic load rating, f,, values must be multiplied by 3.647; that is, f,, = 3.647 X f,, (tabular value).
742
F A T I G LIFE: ~ L ~ B E R G - P ~ ~ GTHEORY R E ~ AND RATIN'G S
T
~
ball and roller bearings as deemed appropriate. Table 18.11 summarizes these bm values. The ANSI load rating standards f18.10 and 18.121 have incorporated the b, factors into f,, factors which are simply fCm = b, x f,. Tables 18.12 and 18.13 give the f C m values to be used in the standardbasic load rating equation (18.173) forradial ball bearings.
C
= fcm(iCOS
a)0*7Z2f3D1.8*
(18.173)
Similarly, for thrust ball bearings, Tables 18.14 and 18.15 give the fcm values to be used in the standard basic load rating equations (18.174) and (i8.175),
C, C,
= fcm(cosa)0.7tan f
a,Z2f3D1.8* a Z 90"
(18.174)
a = 90"
(18.175)
22f3D1.8"
cm
For radial roller bearings, Table 18.16and Table 18.17 provide f,, values for use in the standardbasic load rating equation (18.176).
C
= fcm(iZCOS
a)7f9Z3'4D 29f27
(18.176)
And forthrust roller bearings of various executions, Tables18.18through 18.11. Rating Factor for Contemporary Bearing Factor Bearing Type
Radial ball (except filling slot and self-aligning types) Angular-contact Filling slot ball Thrust Radial Radial Radial machined rings with roller needle Radial roller Drawn cup needle Thrust Thrust roller Thrust cylindrical Thrust needle
Factor b,, Rating
1.3 1.3 1.1 1.1 1.15 1.1 1.1 1.1 1.0 1.1 1.15 1.o 1.0
*ANSI [18.10] recommends using L) raised to the 1.4 power in lieu of 1.8 for bearings having balls of diameter greater than 25.4 mm (1 in.). In this case, for metric units calculation of basic load rating, f,,, values must be multiplied by 3.647; that is, f,,, = 3.647 X f,. (tabular value).
~
S
4
#& Q,
drn
Single-Row Radial-Contact Groove Ball Bearings; Single and Double-Row Angular-Contact Groove Ball Bearings; Insert Bearingsb
0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40
5908 5888 5859 5829 5790 5750 5701 5641 5592 5528 5464 5394 5325 5256 5177 5108 5029 4940 4861 4782
D cos a
Filling Slot Ball Bearings
Double-Row Radial Contact Groove Ball Bearings
Single-Row and Double-Row Self Aligning Ball Bearings
Single-Row Radial-Contact Separable Ball Bearings
4999 4983 4957 4932 4899 4866 4824 4774 4732 4678 4623 4565 4506 4448 4381 4322 4255 4180 4113 4046
5592 5582 5553 5523 5510 5444 5394 5345 5296 5236 5177 5118 5049 4980 4910 4831 4762 4683 4604 4525
3399 3478 3567 3636 3705 3774 3833 3893 3942 3982 4011 4041 4061 4071 4080 4080 407 1 4051 402 1 3992
3092 3171 3251 3330 3399 3478 3547 3616 3675 3735 3794 3843 3893 3932 3962 3992 402 1 4031 4041 4041
=Useto obtain C in pounds when L) and d , are given in inches. Values o f f , for intermediate values of I> cos ald, are obtained by linear interpolation. Insert bearings are not in accordance with L18.131.
LE 18.14, Metric Values for fcm for Thrust Ball Bearings"
Db -
fcm
D cos ab
dm 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35
a = 90"
dm 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30
47.71 58.76 66.43 72.41 77.35 81.77 85.54 89.05 92.30 95.29 98.02 100.62 103.09 105.43 107.51 109.72 111.67 113.62 115.44 117.26 118.95 120.64 122.33 123.89 125.45 126.88 128.31 129.74 131.04 132.47 133.77 135.07 136.24 137.54 138.71
-
fcm
a = 45"
54.73 67.21 75.66 82.29 87.49 91.91 95.55 98.67 10 1.40 103.61 105.43 106.99 108.29 109.33 110.11 110.63 111.02 111.15 111.15 111.02 110.76 110.37 109.85 109.20 108.42 107.64 106.60 105.69 104.52 103.48
-
-
a = 60"
a = 75"
50.96 62.53 70.46 76.57 81.38 85.54 88.92 91.78 94.38 96.46 98.15 99.58 100.75 101.79 102.44 102.96 103.35 103.48 103.48 103.35
-
aUse to obtain C in Newtons when D and dm are given in mm. &Valuesof fcm for D / d m or D cos aid, and/or angles other than those shown are obtained by linear interpolation. cFor thrust bearings a! > 45". Valuesfor a! = 45" permit interpolation of values for a between 45" and 60".
748
FATIGUE LIFE L
~
~
E
R
G
- TBXORY P ~ GAND ~ RATING ~ $
T
~
LE 18.15. Inch Values for fern for Thrust Ball Bearings" Db -
fcm
D cos ab
dm
a = 90"
dm
a = 45""
a = 60"
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30
4159 5108 5750 6254 6649 6985 7262 7499 7706 7874 8013 8131 8230 8309 8368 8408 8438 8447 8447 8438 8418 8388 8349 8299 8240 8181 8102 8034 7944 7864
3873 4752 5355 5819 6185 6501 6758 6975 7173 7331 7459 7568 7657 7736 7785 7825 7855 7864 7864 7855
-
-
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35
3626 4466 5049 5503 5879 6215 6501 6768 7015 7242 7450 7647 7835 8013 8171 8339 8487 8635 8773 8912 9040 9169 9297 9416 9534 9643 9752 9860 9959 10068 10167 10265 10354 10453 10542
Lm
.__
-
a = 75"
-
__.
__.
-
aTJse to obtainC in pounds whenD and d, are given in inches. bValues off,, for Dld, or D cos ald, andlor angles other than those shown are obtained by linear interpolation. cFor thrust bearings a > 45".Valuesfor a = 45" permit inte~olationof values for a between 45" and 60".
~
$
749
LOAD RATING ElTANDARDS
T ~ L 18.16. E Metric Values for fcm for Radial Roller Bearingsa
D cos ab dm
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37
Cylindrical Roller Bearings, Tapered Roller Bearings, and Needle Drawn Cup Roller Bearings with Rings Machined 57.310 66.880 73.150 77.770 81.510 84.590 87.120 89.210 91.080 92.620 93.830 95.040 95.810 96.470 97.020 97.350 97.570 97.680 97.680 97.570 97.350 97.020 96.580 96.250 95.590 95.040 94.380 93.720 92.840 92.070 91.300 90.420 89.430 88.440 87.450 66.460 85.360
Needle Roller Bearings 52.100 60.800 66.500 70.700 74.100 76.900 79.200 81.100 82.800 84.200 85.300 86.400 87.100 87.700 88.200 88.500 88.700 88.800 88.800 88.700 88.500 88.200 87.800 87.500 86.900 86.400 85.800 85.200 84.400 83.700 83.OOO 82.200 81.300 80.400 79.500 78.600 77.600
Spherical Roller Bearings 59.915 69.920 76.475 81.305 85.215 88.435 9 1.080 93.265 95,220 96.830 98.095 99.360 100.165 100.855 101.430 101.775 102,005 102.120 102.120 102.005 101.775 101.430 100.970 100.625 99.935 99.360 98.670 97.980 97.060 96.255 95.450 94.530 93.495 92.460 91.425 90.390 89.240
750
L ~ B E R G - P ~ ~ G RTHEORY EN AND RATING STANDARDS
FATIGUE LIF'E
D cos cxb dm 0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50
Cylindrical Roller Bearings, TaperedRoller Bearings, and Needle Roller Bearings with Machined Rings 84.370 83.270 82.060 80.960 79.750 78.540 77.330 76.120 74.910 78.315 73.700 77.050 72.380 75.670 71.060 74.290 69.850 73.025
Drawn Cup Needle Roller Bearings 76.700 75.700 74.600 73.600 72.500 71.400 70.300 69.200 68.100 67.000 65.800 64.600 63.500
Spherical Roller Bearings 88.205 87.055 85.790 84.640 83.375 82.110 80.845 79.580
a t h e to obtain C in Newtons whenD and f, are given in mm. bValuesof z, for intermediate valuesof D cos ald, are obtainedby linear interpolation.
18.23 give fcm values to be used in the standardbasic load rating equations (18.177) and (18.178).
C, = fcm(Z cos a)7f9 tan 6,
=f
L7/923/4D 29/27
cm
&3/4D29f27
90"
(18.177)
a = 90"
(18.178)
a
rjt
With regard to the use of the standardload rating equations and tabular data on fcm, X and Y factors to estimate rolling bearing fatigue life endurance, certain limitations should be observed:
1. Load ratings pertain only to bearings fabricated fromproperly hardened, good quality steel. Rating life calculations assume the bearing inner and outer rings are rigidly supported. 3. Rating life calculations assume the bearing inner and outer ring axes are properly aligned. Rating life calculations assume the bearing has only a nominal internal clearance during operation. With regard to ball bearings, the raceway groove radii must fall within 0.52 5 f / D 5 0.53. e
e
s
LO
Inch Values for f,,, for Radial Roller Bearings" "
-
D cos 2 dm
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35
Cylindrical Roller Bearings, TaperedRoller Bearings, and Needle Bearings oilerwith Machined Rings 5 149 6009 6573 6987 7324 7600 7828 8016 8184 8322 8431 8539 8609 8668 8718 8747 8767 8778 8778 8767 8747 8718 8678 8648 8589 8539 8480 8421 8342 8273 8204 8125 8036 7946 7857
Drawn Cup Needle Roller Bearings 4681 5463 5975 6352 6658 6909 7116 7287 7440 7565 7665 7763 7826 7880 7925 7952 7970 7979 7979 7970 7952 7925 7889 7862 7808 7763 7709 7655 7584 7521 7458 7386 7305 7224 7143
~pherical Roller Bearings 5383 6282 6871 7305 7657 7945 8183 8380 8556 8700 8814 8927 9000 9062 9114 9145 9166 9176 9176 9166 9 145 9114 9073 9041 8980 8927 8865 8803 8721 8649 8577 8494 8401 8308 8214
F ~ T I LIFE: G ~ L ~ B E R G - P ~ ~ G RTHEORS E N AND
.
(Continued)
dm
Cylindrical Roller Bearings, Tapered Roller Bearings, and Needle Roller Bearings with Rings Machined
0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50
7768 7669 7580 7482 7373 7274 7 165 7057 6948 6840 6731 6622 6503 6384 6276
D cos
ab
Drawn Cup Needle Roller Bearings 7062 6972 6891 6802 6703 6613 6514 6415 6316 6218 6119 6020 5912 5804 5705
Spherical Roller Bearings 8121 8018 7925 7822 7708 7605 7491 7377 7263 7151 7037 6923 6799 6675 6561
aZrse to obtain C in pounds when I) and d, are given in inches. 6Values off,, for intermediate values of I) cos cxld, are obtained by linear interpolation.
With regard to roller bearings, the load ratings pertainonly to bearings manufactured to achieve optimized contact.This involves good roller guidance by flanges or cage as well as optimum roller and/ or raceway profile. For both ball and roller bearings, no stress concentrations may occur due to loading conditions. In a ball bearing this condition can be caused if the applied thrust load is sufficient to cause the balls to override the land edges.
*
Estimate the L,, life of the 209 radial ball bearing according to the ANSI method. Use a shaft speed of 1800 rpm as in Example 18.4. Z=9
D 2.5
=
2.1
Ex.
12.7 mm (0.5 in.)
Ex. y = 0.1954
8.12, for y
=
0.1954, fcm
*Example 18.7 continues on page 759.
=
77.93
2.1 Ex.
U
0
tn
8 3
TABLE 18.19. Inch Values for f,, for Tapered Roller Thrust Bearings"
D cos a!'
D ' dm
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30
90" 10400 12127 13271 14149 14870 15481 16024 16498 16942 17336 17711 18057 18382 18688 18974 19251 19507 19764 20000 20227 20444 20661 20869 21066 21254 21441 21619 21796 21974 22132
a
=
dm 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26
-
= 50"" 10824 12610 13764 14633 15314 15876 16340 16725 17050 17317 17544 17731 17869 17988 18066 18126 18155 18165 18155 18126 18076 18017 17938 17849 17741 17632
a!
-
aUse to obtain C, in pounds when D and d, are given in inches. bValues off, for intermediate values of Dld, or D cos ald, are obtained by linear interpolation. "Applicablefor 45" < a < 60". "Applicable for 60" 5 a < 75". "Applicablefor 75" 5 a < 90".
a!
= 650d 10568 12304 13439 14278 14949 15491 15945 16330 16646 16912 17129 17307 17445 17553 17642 17692 17721 17731 17721 17692
= 80"" 10420 12136 13251 14090 14741 15284 15728 16103 16419 16675 16892 17070 17208 17317 17396
a!
TABLE 18.20. Metric Values for Am for Cylindrical Roller Thrust Bearings and Needle Roller Thrust Bearings"
D cos ab 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30
!
*
4 cn cn
a = 90"
dm
a = 5OoC
a = 650d
a = 80""
105.4 122.9 134.5 143.4 150.7 156.9 162.4 167.2 171.7 175.7 179.5 183.0 186.3 189.4 192.3 195.1 197.7 200.3 202.7 205.0 207.2 209.4 211.5 213.5 215.4 217.3 219.1 220.9 222.7 224.3
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26
109.7 127.8 139.5 148.3 155.2 160.9 165.6 169.5 172.8 175.5 177.8 179.7 181.1 182.3 183.1 183.7 184.0 184.1 184.0 183.7 183.2 182.6 181.6 180.9 179.8 178.7
107.1 124.7 136.2 144.7 151.5 157.0 161.6 165.5 168.7 171.4 173.6 175.4 176.8 177.9 178.8 179.3 179.6 179.7 179.6 179.3
105.6 123.0 134.3 142.8 149.4 154.9 159.4 163.2 166.4 169.0 171.2 173.0 174.4 175.5 176.3
-
-
aUse to obtain C, in Newtons when D and d, are given in mm. bValues off, for intermediate values of D l d , or D cos a l d , are obtained by linear interpolation. "Applicablefor 45" < a < 60". dApplicablefor 60" 5 a < 75". "Applicable for 75" 5 a < 90".
-
I I I I I
I I I I I
TABLE 18.22. Metric Values for f,, for Spherical Roller Thrust Bearings" D b dm 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 !
L
4 CJl 4
D cos ab CY
=
90"
121.210 141.335 154.675 164.910 173.305 180.435 186.760 192.280 197.455 202.055 206.425 210.450 214.245 217.810 221.145 224.365 227.355 230.345 233.105 235.750 238.280 240.810 243.225 245.525 247.710 249.895 251.965 254.035 256.105 257.945
a
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 -
=
50""
126.155 146.970 160.425 170.545 178.480 185.035 190.440 194.925 198.720 201.825 204.470 206.655 208.265 209.645 210.565 2 11.255 2 11.600 211.715 2 11.600 2 11.255 210.680 209.990 209.070 208.035 206.770 205.505 -
"Use to obtain C, in Newtons when D and d, are given in mm. bValues of f,, for intermediate values of D l d , or D cos ald, are obtained by linear interpolation. "Applicable for 45" < a < 60". dApplicable for 60" 5 a < 75". "Applicable for 75" 5 a < 90".
650d 123.165 143.405 156.630 166.405 174.225 180.550 185.840 190.325 194.005 197.110 199.640 201.710 203.320 204.585 205.620 206.195 206.540 206.655 206.540 206.195
CY
=
80"" 121.440 141.450 154.445 164.220 171.810 178.135 183.310 187.680 191.360 194.350 196.880 198.950 200.560 201.825 202.745
(Y
=
-
-
_.
c = f,,(i =
cos a)O. 77.93 X (9)2’3 (12.7)1*8= 32,710 N (735
(18.173) #
Fe = Fr = 8900 N (2000 lb)
=
(32,710/8900)3= 49.64 million revolutions
=
49.64
106/(60 X 1800) = 459.6 hr
X
To compare this result with that of Example multiply the resultof that example by b;. Fro ~ccordingly, thecomparable I;, 426.2 hr. It is evident that the as that calculated in Example 18.4. In this particular case, however, the difference is not s i ~ i f i c a n t , Estimate the Ll0 fatigue life of the xample 7.3 according to ings is 4450 N (1000 lb).
Z
E
=
14
x. 2.7
=
10 mm (0.3987 in.)
x. 2.7
=
9.6 mm (0.37$ in.)
x. 2.7
y = 0.1538
Using Table 18.16, f,,
=
II,
97.15
=
X
97.15
(1 X 9.6)7’9(14)3/4( 10)29/27
= (c/Fe)10/3
=
(48,430/4450)10’3= 2856 million revolutions
To compare the result of this SI method calculation with that of Example 18.6 it is necessary to introduce b, factor into the latter set lations. From Table 18.11, b,, is 1.1for cylindrical roller bear~ X IO6 or 1442 nce, the bearing I;,,life would be ( ~ 1X )985 x lo6 revolutions. It is apparent that the ANSI method which does
not account for the precise load distribution gives onlyan a~proximate life estimate and in this case tends to overstate the bearing fatigue e n ~ u r a ~ capability. ce The 22317 two-row spherical roller bearing of rotation of 900 rpm, a
=
12”
x. 2.7
z= 14 Z
ble 18.16,
=
20.76 mm (0.8
=
25 mm (0.9843 in.)
fcm =
x. 2.7
97.68
c = fcm(iZ cos aI7 X
1.5 tan a
(2 x 20.71 x cos 12°)7’Q(14)3’4(25)2g’27
=
39g9300N (89,720 lb)
=
22,250 N (5000 lb)
x. 7.
=
89,000 N (209000lb)
x. 7.
=
22,250/89,000 = 0.25
=
1.5 tan 12” = 0.3189
Since F a / F ~< 1.5 tan a, from Table 18.7use X
Y
=
=
1and Y
I; =
0.45 ctn a
0.45 etn 12” = 2.117
Fo = . A T r 1- YFa =
=
(18.129)
136,100 N (30,950 lb) (C/Fa)10’3
=
(399~300/136~000)10’3
=
36.15 million revolutions
(18.160)
or 1; = 36.15 X 106/(60 X 900) = 669.5 hr
I
The rolling bearing industry was among the first to use fatigue life as a design criterion. As a result, the space-age term reZi~~iZity, which is synu i u ~ Z to , rolling bearing manonymous with ~ r o b ~ b i Z i t y o ~ ~ ~isrfamiliar ufacturers and users. The conceptsof L,, or rating life and L5, life are used as yardsticks of bearin performance. By means o load rating standards, the rolling bearing industry established relatively uncomplicated methodsto evaluate the ratinglife. Thesestandards enjoy worldwide acceptance, and they can be applied to compare the adequacy of diverse bearing types from different manufacturers for use in most en~neeringapplications. In certain applications, however, the simple use of mulas to rate rolling bearing performance can lead to inaccurate estimates of fatigue life. These applications include those involving high speed, flexible bearing ring support structures, extremely slow speed, and unusual loading conditions. For these situations, methods to estimate fatigue life will bepresented in Chapter 23. These methods relyon the basic Lundberg-Palm~en theory; however, they typically require more complex analyses of bearing contact angles, internal load distribution, internal speeds, lubrication, and so on. Suchanalyses require the use of a computer. nd IS0 standard methods terminate with the calculation ing life based on a bearing fabricated from a good quality, ring steel; however, they indicate how life may be influenced by such parameters as alternate rolling component mate~als,lubrication, reliability, and combinations of these. Modern imp~ovements in rolling bearing steels and methods of bearing manufacture have succeeded in yielding bearings capable of substantially increased endurance. In many applications, modern rollingbearings will not experience rolling e failure. In other words, they may be said to have infinite cepts are also presented in detail in Chapter 23.
18.1. G,Lundberg and A. Palmgren, “Dynamic Capacity of Rolling Bearings,”Acta Polytech. Me&. Eng. Ser. 1, R.S.A.E.E., No. 3, 7 (1947). 18.2. T. Martin, S. Borgese, and A. Eberhardt, “Microstructural Alterations of Rolling Bearing Steel Undergoing Cyclic Stressing,” ~~E Preprint 65-WAI CY-4, Winter Annual Meeting, Chicago (November 1965). 18.3. P. Hoel, Introduction to ~athernaticaz Statistics,2nd ed., Wiles, New York (1954). 18.4. W. Weibull, “A Statistical Representationof Fatigue Failure in Solids,”Acta PoZytech. Mech. Eng. Ser. 1, R.S.A.E.E., No. 9, 49 (1949). 18.5. W. Weibull, “A Statistical Theory of the Strength of Materials,” Proc, R. Swedish Inst. Eng. Res., No. 151, Stockholm (1939).
18.6. T. Tallian, “Weibull Distribution of Rolling Contact Fatigue Life and Deviations Therefrom,’’ASLE Trans. 5(1)(April 1962). 18.7. J. Lieblen, “A New Method of Analyzing Extreme-ValueData,”Technical Note 3053, NACA (January 1954). 18.8. 6;. Lundberg and A. Palmgren, “Dynamic Capacity of Roller Bearings,”Acta Polytech. Mech. Eng. Ser. 2, R.S.A.E.E., No. 4, 96 (1952). 18.9. A. Palmgren, Ball and Roller Bearing Engineering, 3rd ed., Burbank, Philadelphia (1959). 18.10.AmericanNational Standards Institute, American ~ a t i o n a lStandard (ANSI I A..FBMA)Std 9-1990, “Load Ratings and Fatigue Life for Ball Bearings.’’ 18.11. J. Lieblein and M. Zelen, “Statistical Investigation of Fatigue Life of Ball Bearings,” National Bureau of Standards, Report No. 3996 (March 28, 1955). 18.12.AmericanNational Standards Institute, American ~ a t i o n a lStandard (ANSI:/ ~ B M Std A 11-1990, ~ “Load Ratings and Fatigue Life for Roller Bearings.” 18.13. International Organization for Standards, International Standard I S 0 281 I 1, “RollingBearings-DynamicLoad Ratings and Rating Life-Part I; Calculation Methods’’ (1999). 18.14. E. Zaretsky, E. Bamberger, T. Harris, W. Kacmarslcy, C. Moyer, R. Parker, and J. Sherlock,Life Adjustment Factors for Ball and Roller Bearings, ASME Engineering Design Guide (1971). 18.15. H. Walp, T. Morrison, T. Tallian, and G. Baile, “The Effectof Material Variables on the Fatigue Life of AIS1 52100 Steel Ball Bearings,”ASLETrans. 5(2) (1962). 18.16. D. Koistinen, “Heat Treated Steel Article,” U.S. Patent No. 3117041 (January 7, 1964).
Description
nits
capacity asic dynamic (W load quivalent applied (1b) h Lubricant film thickness pm (pin.) R Lubricant film parameter K Ratio of actual lubricant viscosity to viscosity required for adequate separation of rolling contact surfaces CT Rms surface roughness pm (pin.)
In service evena rolling contactbearing that is well lubricated, properly adequately protected from the effects of abrasives or moisfrom rolling contact fatigue if the loading is suf~cientlyhigh. It is i ~ p o r t a n for t the user to be able to predict reliably the len
service that can be achieved from a bearing in a specific application. As indicated in Fig. 1.11, the ability to make these types of predictions is hampered since rolling contactfatigue is probabilistic, similar to human life and the service lifeof light bulbs. Therefore,the life of any one bearing operating in a specific environment can differ significantly from that of an apparently identical unit. This distribution of rolling bearing fatigue data is illustrated in Fig.. 18.5. Over the years, statistical procedures have been established for the analysis of bearing fatigue data. A “rating” life system has been defined that uses two specific points on the failure distribution curve to compare the fatigue endurance of bearing designs. These points are the Llo life, or the life that 90% of the bearings can be expected to survive, and the life, which 50% of the bearings can be expected to exceed. Concurrently, testing procedures have been developed to measure bearing fatigue life (endurance). As shown in Chapter 18, these experimental techniques have been utilized to collect a quantity of data which allowed the derivation of theoretical life prediction formulas based on a calculated bearing basic,dynamic capacity, the equivalent radial load applied to the bearing, and a number of environmental and manufacturingrelated factors. This basic concept is routinely used to select bearings that will yield the desired performances in specific applications. earing endurance tests are used to evaluate bearing materials, new heat treatment processes, and improvedforming or surface finishing techniques. The spread in experimental fatigue data and thelimitations of the statistical analysis techniques require that many bearings be tested for a long time to obtain valid estimates of bearing life. Testing 1y related to the manufacturing costs of the testspecimens of the test process; therefore, conducting a life test series on full bearing assemblies is expensive. Effort has thus been expended to develop life testing techniques using simple partial bearings or single rolling contacts. his chapter discusses the concepts and p~ilosophies of conducting endurance tests on complete rollingbearing assemblies and on elemental rolling contact co~figurations.
The ability to use life test data collected on a particular bearing type and size under a specific set of operating conditions to predict general bearing performance requires a systematic relationship between applied load and life. The basic form of this relationship was definedby Lundberg almgren E19.11 as equation (19.1).
s indicated in Chapter 23, this equation has undergone several modifications since it was first proposed; however, for the purposes discussion, the basic format is used. In equation (19.I), exponen for ball bearings and 10/3 for roller bearings. The form of the 1 relationship for ball bearings was graphically illustrated a straightline on a log-log chart. This formula provides th experimental life data collected under one set of test conditions to estabbearing performance under a wide range of co initiation of a fatigue-originated spalling fail ing contact in atypical application is about 10 to 15 years. It is therefore obvious that any practical laboratory test sequence must be ted ted under acceleratedconditions if the necessary data are to be ac within a reasonable time. Several potential methods exist for test accelolling contact fatigue modes exist, however, that compete in bearings t o produce the final failure [19.Z]. Care must be the taken to ensure that themethod of test acceleration does not change failure mode. In an endurance test-that is, a test serie establish the experimental life of a lot of bearings for co theoretical lives andlor other test data-this means that the primary failure mode must remain fatigue related. Generally, there have been two methods of test acceleration usedin most endurance testing: increasing the applied load levelsandlor increasing the operating speed. The experimental results obtained under increased load levels canbe rather easily extrapolated to other conditions by using the basic load-life formula. Thus, this is probably the most widely used method of life test acceleration. It is important to note that consistency must be maintained with the basic assumptions used in the derivation of the life formulas. The stresses generated at and below the bearing raceways shouldremain within the elastic regime. Exceeding elastic stress levels will produce modifications to the load-life relationship as reported in E19.31 and illustrated in Fig. 19.1. Intuitively; it is surprising that such extreme loads can produce apparent life increases of significant magnitude over theoretical predictions. This phenomenon is, however, readily explained by the plastic deformations produced in the raceways, creating a conforming contact pattern thatthen significantly reducesthe existing stress levels. Testing in thisregime producesresults that areinconsistent with operating practice and cannot be reliably extrapolated. The practical load limit for testing acceleration is usually considered to be that load that produces a maximum Hertz stress of approximately 3300 N/mm2 (475 ksi). Some cases require special consideration in general life testing application. One involves testing self-aligning ball bearings. The spherical
1000
r
600 400
E
200
100
.
1
5 10 20 30 Life,L , , (millions of rev)
50
100
Life relationship deviation due to overloading. Theoretical (“-1.
~qm-
imental (----1.
geometry of the outer ring raceway needed to provi e the self-ali~nin~ oduces circular oint contactsbetween balls and raceway s loads are increased, th resses more rapidly than is considered
ndard internal geometries mations at lower than and before initiating an
are often insufficient forthe heavy loads
of the ratinglife. Also, extra care must be used whencalculating the life. Experimental indications are that mo~ifyingeffects of the lubrican films thicker than those generally roller bearings tend to ~enerate
calculated. These thicker films will then produce greater than expected life enhancements. To insure the proper interpretation of life test data, these effects must be established before extrapolating the data to sets of operating conditions containing different speeds or lubricants. The second usual method of accelerating life testing is by increasing operating speed. perating a rolling bearing assembly faster produces a lation of stress cycles, but not necessarily a shorter st time. The increased operating speedalsoproduces thicker ms, which then enhance bearing endurance. The life enhanceme ct may overshadow the increased stressing rate, and testtimes will be increased, As the operating speeds are increased to even higher speed levels, other life-confoundin eff'ects can come into play. Under these operating conditions ball or roller centrifugal loads are increased, which causes increased outer raceway loading, increased clearance, and more concentrated innerraceway loading that reduce assembly life,as shown by 23.4'
The magnitude of this effect incr ses rapidly, since centrifugal force varies with the square of the speed. aring size also has a major eEect, cing variations in both rollingelement mass and radius effects are particularly important in angular-contact ba the centrifugal forces alter theouter ring contact angle even more significant life alteration effects. ter often used to express the severity of bearing speed conditio bearing bore diameter expressed in millimeters of the bearing in revolutions per minute. It is earings as those with d N values of 1mi n under high-speedconditions,soph niques are required to reliably calculate beari with collected life data. sing speed to accelerate bearing test programs has its limitations. d limitations exist standard in bearin because designs, e inademetal or molded plastic cage quate for high-speed operation. Excessive heat generation rates may occur at the raceway contacts, which have been designed primarily for maximum load-carryingcapabilities at lower speeds,and component precision may bealtered due to the dynamic loadingof high-speed operation. effects can also produce significant life effects on highome effects are insufficient cooling or the inadequate cooling medium, creating thermal gradients in the bearings that affect internal clearances an ing speeds will also produce higher bearing o The lubricants used then must be capable sure to these high temperatures without suffering degradation. The con-
7
ING E
~
~ T ECS T ~ E G
duct of high-speedlife tests requires extra care to ensure that the failures obtained are fatigue related and not precipitated by some speedoriginated performance malfunction. In Chapter 12 it was shown that rolling contacts in ball and roller bearings operate in the EHL regime and that the thickness of the lubricant film generated byEHL is a strong function of bearing internal speeds, increasing as these speeds are increased. Furthermore, in Chapter 23 it is shown that the fatigue life of a rolling bearing is a function of the thicknesses of the lubricant films it generates compared to the r o u g ~ ~ e s s of e sthe rolling contact surfaces, Bearing fatigue life tends to increase as the surfaces in rolling contact are effectively separated by lubricant films. The adequacy of lubrication is currently expressed as A, the ratio of the lubricant film thickness to the composite rms roughness of the eo~taetingsurfaces. Values of A in excess of 3 tend to yield extended bearing fatigue life. To ensure an adequate lubricant film, a sufficient lubricant supply must be available to preclude lubricant s ~ a r ~ a t (see i o ~Chapter 12). Lubricant starvation is a particular consideration when performing endurance testing with grease lubrication at high speeds. n Chapter 23, the means to quantify the lubrication-associatedeffect peed on bearing endurance is demonstrated. Particularly in the reme of marginal lubrication, the effect is complex owing to the interactions of rolling component surface finishes and chemistry, lubricant chemical and mechanical properties, lubrication adequacy, contaminant types, and contamination levels. Considering adequate lubricant supply and minimal contamination, increasing testing speed to the point that complete separation of rolling contact surfaces is achieved has tended to extend test duration appreciably, thus thwarting thedesired acceleration of endurance testing. Testing at speeds slow enough to cause operation in the marginal lubrication regime can achieve shortened testing time; however, the above-indicated side effects must be considered in theevaluation of test results.
An individual bearing may fail for several reasons; however, the results of an endurance test series are only meaningful when the test bearings fail by fati~e-relatedmechanisms. The experimenter must control the test process to ensure that this occurs. Some of the other failure modes that can be experienced are discussed in detail by Tallian [19.2]. The following paragraphs deal with a few specificfailure types that can affect the conduct of a life test sequence. In Chapter 23, the influence of lubrication on contact fatigue life is discussed from the standpoint of EHL film generation. There are also
other lubrication-related effects that can affect the outcome of the test series. The first is particulate contaminants in the lubricant. Depending on bearing size, operating speed, and lubricant rheology, the overall thickness of the lubricant film developedat the rolling element-raceway contacts may fall between 0.05 and 0.5 pm (2-20 pin.), Solid particles larger than the film can be mechanically trapped in the contact regions and damage the raceway and rolling element surfaces, leading to substantially shortened endurances. This has been amply demonstrated by acPherson [19.6] and others. Therefore, filtration of the lubricant to the desired level is necessary to ensure meaningful test results.The desired level is determined by the application which the testing purports to approximate. If this de filtration is not provided, effects of contamination must be considered when evaluating test results. Chapter 23 discusses the effect of various degrees of particulate contamination, and hence filtration, on bearing fatigue life. The moisture content in the lubricant is another important consideration. It has long been apparent that quantities of free water in the oil cause corrosion of the rolling contactsurfaces and thus have a detrimental effect on b e a ~ n glife. It has been further shown by Fitch [19.7] and others, however, that water levels as low as 50-100 parts per million (ppm) may also have a detrimental effect, even with no evidence of corrosion. This is due t o h~drogenembrittlement of the rolling element and raceway material. See also Chapter 23. Moisture control in test lubrication systems is thus a major concern, and the effect of moisture needs to be considered during the evaluation of life test results. A m ~ i m u m of 4.0 ppm is considered necessary to minimize life reduction effects. The chemical composition of the test lubricant also requires consideration. Most commercial lubricants contain a number of pro~rietaryadditives developed for specific purposes; for example,to provide antiwear properties, to achieve extreme pressure andlor thermal stability, and to provide boundary lubrication in case of marginal lubricant films. These additives can also affect the endurance of rolling bearings, either immediately or after esperiencing time-related degradation. Care must be taken to ensure that the additives included in the testlubricant will not suffer excessive deterioration as a result of accelerated life test conditions. Also for consistency of results and comparing lifetest groups, it is good practice to utilize one standard testlubricant from a particular producer for the conduct of all general life tests. The statistical nature of rolling contact fatigue requires many test samples to obtain a reasonable estimate of life. A bearing life test sequence thus needs a long time. A major job of the experimentalist is to ensure the consistency of the applied test conditions throughout the entire test period. This process is not simple because subtle changes can occur during the test period. Such changes might be overlooked until
their effects become major.At that time it is often too late to salvage the collected data, and the test mustbe redone under better controls. For example, the stability of the additive packages in a test lubricant can be a source of changing test conditions. Some lubricants have been known to suffer additive depletion after an extended period of operation. The degradation of the additive package can alter theE the rolling contacts,altering bearing life. Generally,the tests used to evaluate lubricants do not determine the conditions of the additive content. Therefore if a lubricant is used for endurance testing over a long time, a sample of the fluid shouldbe returned to the producer at regular intervals, say annually, for a detailed evaluation of its condition. Adequate temperature controls must also be employedduring the test. The thickness of the EHL film is sensitive to the contact temperature. ost test machines are located in standard industrial environments where rather wide fluctuations in ambient temperature are experienced over a period of a year. In addition, the heat generation rates of individual bearings can vary as a result of the combined effects of normal manufacturing tolerances. Both of these conditions produce variations in operating temperature levels in a lot of bearings and affect the validity of the life data. A means must be provided to monitor and control the operating temperature level of each bearing to achieve a degree of consistency. A tolerance level of ~fr3 6 is normally considered adequate for the endurance test process. The deterioration of the condition of the mounting hardware used with the bearings is another area requiring constant monitoring. The heavy loads used for life testing require heavy interference fits between the bearing inner rings and shafts. Repeated mounting and dismounting of bearings can produce damage to the shaft surface, which in turn can alter the geometry of a mounted ring. The shaft surface and the bore of the housing are also subjectto deterioration from fretting corrosion. Fretting corrosion results from the oxidation of the fine wear particles generated by the vibratory abrasion of the surfaces, which is accelerated by the heavy endurance test loading. This mechanism can also produce significant variations in the geometry of the mounting surfaces, which can alter the internal bearing geometry. Such changes can have a major effect in reducing bearing test life. The detection of bearing failure is also a major consideration in a life test series. The fatigue theory considers failure as the initiation of the first crack in the bulk material. Obviously there is no way to detect this occurrence in practice. To be detectable the crack must propagate to the surface and produce a spa11 of sufficient magnitude to produce a marked effect on an operating parameter of the bearing: for example, noise, vibration, and/or temperature. Techniques exist for detecting failures in application systems. The ability of these systems to detect early signs of
failure varies with the complexity of the testsystem, the type of bearing under evaluation, and other test conditions. Currently no single system exists that can consistently provide the failure discrimination necessary for all types of bearing life tests. It is then necessary to select a system that will repeatedly terminate machine operation with a consistent minimal degree of damage. The rate of failure propagation is therefore important. If the degree of damage at test termination is consistent among test elements, the only variation between the experimental and theoretical lives is the lag in failure detection. In standard through-hardened bearing steels the failure propagation rate is quite rapid under endurance test conditions, and this is not a major factor, consideringthe typical dispersion of endurance test data and the degree of confidence obtained fromstatistical analysis. This may not, however, be the case with other experimental materials or with surface-hardened steels or steels produced by experimental techniques. Care must be used when evaluating these latter results and particularly when comparing the experimental lives with those obtained from standard steel lots. The ultimate means of ensuring that an endurance test series was adequately controlled is the conduct of a post-test analysis. This detailed examination of all the tested bearings uses high-ma~ificationoptical inspection, higher-magnificationscanning electron microscopy, metallurgical and dimensional examinations, and chemical evaluations as required. The characteristics of the failures are examined to establish their origins and the residual surface conditions are evaluated for indications of extraneous effects that may have influenced the bearing life. This technique allows the experimenter to ensure that the data areindeed valid. The “DamageAtlas” compiled by Tallian et al. [19.8] containing numerous black and white photographs of the various bearing failure modes can provide guidance for these types of determinations. This work was subsequently updated by Tallian E19.91, now including color photographs as well. The post-test analysis is, by definition, after the fact. To provide control throughout the test series and to eliminate all questionable areas, the experimenter should conduct a preliminary study whenever a bearing is removed fromthe testmachine. In thisportion of the investigation each bearing is examined optically at magnifications up to 30X for indications of improper or out-of-control test parameters. Examples of the types of indications that can be observed are given in Figs. 19.2-19.6. Figure 19.2 illustrates the appearance of a typical fatigue-originated spa11 on a ball bearing raceway. Figure 19.3 contains a spalling failure on the raceway of a roller bearing that resulted from bearing misalignment, and Fig. 19.4 contains a spalling failure on the outer ring of a ball bearing produced by fretting corrosion onthe outer diameter. Figure 19.5 illustrates a more subtle form of test alteration, where the spalling fail-
19.2. Typical fatigue spalling failure (from ref. [19.11]).
ure ori~natedfrom the presence of a debris dent on the surface. Figure es an example of a totally different failure mode produced bythe loss of internal bearing clearance due to thermal unbalance of the system. The last four failures are not validfatigue spalls and indicate the need to correct the test methods. Furthermore, these data points would need to be eliminated from the failure data to obtain a valid estimate of the experimental bearing life.
Specific requirements have been established for a testsample to be used in an endurance test sequence. The statistical techniques used to evaluate the failure data require that the bearings be statistically similar assemblies. Therefore the individual components must be manufa~tured in the same processing lot from one heat of material. Generally, it is considered prudent to manufacture the total bearing assembly in this manner; however, when highly experimental materials or processes are considered, this is often not cost effective or even possible.In those cases the inner ring, the most critical element in a bearing assembly from a fatigue point of view, can be used as the test element, with the other components being manufactured from standard material. The ef5ects of failures occurring on the other parts can be eliminated during analysis of the test data.There is some risk in thisapproach because it is possible that too many failures could occur on these nontest parts, rendering it
1~,3. Bearing failure precipitated by misalignment (ref. [19.111).
impossible to calculate an accurate life estimate for the material under evaluation. In the cases cited, however, this risk is small because an initial resultindicating the superior performance of an experimental process is usually sufficient to justify continued developmental effort even without a firm numerical estimate. In any case, additional life tests would be required to establish the magnitude of the expected lot-to-lot variation before adopting a new material or implementing a new manufacturing process. The number of' bearings to be tested and the test strategy to be employed must also be carefully considered. Statistical analysis provides a numerical estimate of the value of the experimental life enclosed by upper- and lower-bound estimates at specified confidence levels. The precision of the experimental life estimate can be defined by the ratio of these upper and lower confidence limits, and the experimental aim is to minimize this spread. The magnitude of the confidence interval de-
.4. Spall precipitated by fretting corrosion crack (ref. [19.111).
19.5. Spall initiating at surface damage.
19.6, Failure resulting from thermal imbalance (ref. 119.111).
creases as thesize of the test lot is increased; the cost of conducting the test series also increases with test lot size. Therefore,the degree of precision required in the testresult should be established during the planning stages to define the size of the testlot to achieve the required result. The test strategyemployed also affectstesting precision. The classical method of performing endurance tests is to use one large group running each individual bearing to failure. This process is time consuming, but it provides the best experimental estimates of both the L,, and L5, lives. Primary interest is, however, in the magnitude of the experimental Llo, so considerable time savings can be achieved by curtailing the test runs after a finite operating period equal t o at least three times the achieved experimental L,, life. Recentlyit has been shownthat additional savings in test time accompanied by increases in test precision can be obtained by using a sudden death test strategy119.lo]. In this test approach the original test lot is subdivided into smaller groups of equal sizes. Each subgroup is then run as a unit until one bearing fails, at which time the testing of that subgroup is terminated. Figure 19.7 illustrates the effect of both lot size and test strategy on the precision of life test estimates obtainable from an endurance testing series. To provide an accurate life estimate for the variable under evaluation, the experimenter must be sure that the testbearings are free from material and manufacturing defects and that all parts conform t o established dimensional and form tolerances. Although this is an obvious requirement, it is not always easy to attain. Experimental materials might respond quite differently to standard manufacturing processes, or they could require unique processing steps that are not yet totally de-
E L E ~ N TTEST^^
Conventional test 14 12
10 9 Rspo
7 6
5 4
2 1 120
Sudden death test
I 1 2 3 4 5 6 7 8 9 10
12
14
16
18
20
Rl! N = size of test series (total number of bearings) M = number of bearing failures G = number of groups NG = group size (all groups are run to one failure) p = true Weibuli slope
.7. Efliect of lot size on confidence of life test results (from ref. [19.101).
.
Typical Metallurgical Audit Parameters
100% ond destructive Tests: Ring Raceways Only
Magnaflux for near-surface materials defects Etch inspection for surface processing defects Sa~ple
~ ~ sTests: t r ~ All ~ t Components ~ve
Microhardness to 0.1 mm (0.004 in.) depthbelow raceway surfaces Microstructures to 0.3 mm (0.012 in.) depthbelow raceway surfaces Retained austenite levels Fracture grain size Inclusions ratings
fined. Experimental manufacturing processes require additional verification, or their use might produceunexpected variations in related metallurgical or dimensional parameters. Therefore, adequate test control is achieved by detailed pretest auditing of the test parts to supplement the standard in-process evaluations. Tables 19.1 and 19.2 contain lists of those metallurgical and dimensional parameters considered mandatory in a typical pretest audit, as well as anindication of the number of samples that need to be checked in each case. These lists are not complete. Other parameters could be evaluated beneficially if time and money permit.
N ific characteristics are desired in an endurance test system to e control requirements of a life test series. An individual test run uses a long time, so the test machine must be capable of running unattended without experiencing variation in the applied test parameters, such as load(s), speed, lubrication conditions, and operating temperature. Thebasic test system components that could also be subjectto fatigue, such as load support bearings, shafts, and load linkages, should be many times stronger than the test bearings so that test runscan be com~letedwith the fewest interruptions from extraneous causes. The assembly of the test machine should have a minor in~uenceon the experimental test ~onditions to minimize variations between i n d i v i ~ ~ a l test runs. For example, the alignment of the test bearing relative to the shaft should be automatically ensured by the ass ly of the test housing. If not, a simple direct means of monitoring a~justingthis parameter must be provided. Again, since a test s setups, easy assembly and disassembly of the test system is nd time and manpower requirements for on, the test system must be easy
.2.
Typical Dimensional Audit
Parameters 100%Assembled Bearings
Radial looseness Average and peak vibration levels Statistical S a ~ ~ufl Ring e Grooves
Diameter and waviness Radius and form Cross-groove surface texture Statistical Sample of Balls
Diameter and out-of-round Set size variation Waviness Surface texture
should be capable of operating reliably and efficiently foryears to ensure the long-term compatibilityof test results. Basically, design simplicityis a key ingredient in meeting all of these demands. A more comprehensive discussion of the design philosophy of life test rigs has been published in [19.11]. Figure 19.8 illustrates some of the typical endurance test rig confi~rationsdiscussed there. The application of these design concepts to actual endurance test systems will be briefly addressed. Figure 19.9 is the schematic of an S rig for testing 35-50-mm-bore ball and roller bearings under radial, ial, or combined loads. Figure 19.10 is an actual photo~aph. Operating speed may be varied within limits to achieve a given test condition and bearing lubrication can be provided by grease, sump oil, circulating oil, or air-oil mist. Practical life test rig designs will vary? depending on the type of bearing to be tested and its normal operating mode. For example, Fig. 19.11 illustrates a four-bearing test rig concept used in the life testing of tapered roller bearings [19.12]. In thisinstance, while testing is conducted under an externally applied radial load, the bearing also sees an internally induced axial load. The size of this latter load is a function of the itude of the applied radial load, the fixed axial locations of the bearing cups and cones in the test housing, and the basic internal design of the test bearings. Figure 19.112 shows a rig using this design concept. Tests are often conducted to define the life of bearings as used in specific applications. Theyare frequently called life or endurance tests, but, more correctly, they are extended duration performance tests. The same
2P B
A
B
D
T
P
2P
P
P
I
P
(a)
D
P
P (C)
(dl
(e)
F
19.8. Typical test rig configuration. A = test bearing. 13 = load bearing, T = thrust load, P = radial load, D = drive (from ref. [19.11]).
basic test practices and rig considerations are required for these tests, but some modificationsof philosophy are required to simulate the major operating parameters of the application while achieving realistic test developed acceleration. An example of this type of tester is the “A-frame” tester shown in Fig. 19.13, which is used for evaluating automotive wheel hub assemblies. This tester simulates an automotive wheel bearing envi~onmentby using actual car mounting hardware, combined radial and axial loads applied at the tire periphery to produce moment loads on the bearing assembly, grease lubrication, and forced-air cooling. Dynamic wheel loading cycles equivalent to those produced by vehicle lateral loading are applied cyclically to simulate a critical driving sequence. Testing is conducted in the sudden death mode so that hub unit life in this simulated environment can be calculated firom standard life test statistics.This test provides a way to compare the relative performance of automotive wheel s using life data generated under conditions similar to those of actual ap~lications.
ecause numerous test samples are required to obtain a useful experimental life estimate, conducting an endurance test series on full-scale bearings is expensive. Theid~ntificationof simpler, less costly lifetesting
T
G
b
19.9. Schematic layout of endurance test system. A = load bearing, B = test bearing, P = radial load, IT = thrust load, D = drive, E = strain gage, F = hydraulic cylinder, G = pressure gage (from ref. [19.12]).
methods has therefore been a longstanding objective. Theuse of elemental test confi~rations offers a potential solution to this need. In this approach, a test specimen having a simplified geometry (e.g.,flat washer, rod, or ball) is used and rolling contact is developed at multiple test locations. The ambition is to extrapolate the life data generated in an element test to a real bearing application, thus saving calendar time and cost as compared to life data generated using full-scale bearing tests. This objective has historically not been achieved, generally because all of the operating parameters influencing fatigue life of rolling-sliding r contacts were not reducedto stresses; rather, as is shown in ~ h a p t e23, they were evaluated as life factors. The only stress directly evaluated in both element and full-scale bearing endurance testing has been the ertz or normal stress acting on the contact. Lubrication,contamination, sudace topo~aphy,and material effects have been evaluated as life factors. To be able to extrapolate the life data derived fromelement testing to full-scale bearing life data, it is necessary to evaluate both data sets from the stan~pointof applie~and induced stresses as compared to material s t r e n ~ hThe . methods to accomplish this for full bearings are de-
F ~ ~ U R19.10. E R2 tester.
veloped in Chapter 23; Harris [19.13] developed a similar method for balls endurance tested in V-ring test rigs. Even without direct correlation of life test databetween elements and full-scale bearings, element testing has proven useful in the ability to rank performance of various materials in initialscreening sequences or in adverse environments, such as extremely low or high temperature, oxidizing atmospheres, and vacuum. Therefore, discussion of element life testing techniques is warranted, evenwhen the test data evaluation techniques are not such as to permit direct correlation with full-scale bearing life test data. Caution must always be used, however, because the precision of the ranking process is open to question. Performance reversals have sometimes been experienced when comparing the screening element test results when the materials have been retested in actual
Radial toad 19.11. Schematic layout of tapered roller bearing test c o n ~ ~ r a t i o n .
bearings. Such reversals can be avoided if boththe element test data and actual bearing test data evaluations are based on the total stressconsideration. The oldest and perhaps most widely used element test configuration is the rolling four-ball machine developed in theearly 1950s [19.141, This system uses four 12.7-mm (0.5-in.) diameter balls to simulate an angularcontact ball bearing operating with a vertical axis under a pure thrust load. One ball is the primary test element serving as the inner ring of the bearing assembly. It is supported in pyramid fashion on the remaining three balls, which rotate freely in a conforming cup at a predeterminedcontact angle. A modification of this test method, the rolling five-ball tester, was subsequently developed at NASA Lewis Research Laboratories [19.15], and it uses four balls in the intermediate position. This latter system, illustrated in Fig. 19.14 has been used to generate an extensive amount of life test dataon standard andexperimental bearing materials. Another widely used element test system is the RC (rolling contact) tester developed at General Electric E19.161 (Fig. 19.15).The test element in this configuration is a 4.76-mm (0.11875-in.) rod rotating under load between two 95.25-mm (3.75-in.) diameter discs. The rod can be axially repositioned to achieve a number of rolling contacttracks on a single test
ELE
.12. Tapered roller bearing tester.
specimen. ~nfortunately, thisconfiguration is not as cost effective as it pears. Stress concentrations will occur at the edges of the rod unless the discs are profiled in the asial direction. This significantly increases the cost of manufacturing the discs. uring operation, fatigue failures on the rod also tend to damage the disc surfaces, requiring that these be refinished at regular intervals. An interesting modification of this element test concept was developed to eliminate the need for the espensive test discs [19.17].In this version, illustrated in Fig. 19.16, three s t a n d a r ~ 'balls supported in standard tapered roller bearing outer rings serve the function of the discs and increase the number of test contacts. The test specimen is a cylindrical rod which can be used for several tests. Another element test configuration is the single-ball tester developed for evaluating ~ n i t e dTechnolo~es ~orporation as turbine engine bearings [l9.18].This system s balls fromapprosimately 19-65 mm (0.75-~.50 in.) in two V-ring raceways ith lubrication to simulate the application, It is this system for which arris [19.13] developed a stress-based ball life predictio~method. The fatigue limit stress values (see ~ h a p t e 23) r
GE
~
~ T ECS T ~ G E
E
L
E T E~ S T~~ G
19.13. A-frame hub unit tester.
determined from this analysis for ~W~ M50 steel balls compared favorably with values determined from endurance testing of V I W A M50 steel bearings [19.19].
As indicated at the beginning of the previous section, to be able to correlate the results of full-scale bearing endurance testing with those of element endu~ancetesting it is necessary to evaluate and combine all of the stressesacting on each contact. Perhaps the most s i ~ i f i c a nstresses t in determi~ing the extent of bearing life are the shear stresses occurring in the rolling element-race~ay contacts. All of the element testing devices described thus far are based on rolling motion. Sliding motions, as
O L L I N ~ $ L ~ I N ~
C FRICTIO~ O ~ ~ C TESTING T
.14, NASA rolling five-ball test system.
.
GE/polymet rolling contact disc machine.
ed and defined in Chapter 8, do nevertheless occur in thecontacts. surface frictional shear stresses accompan~ng thesliding motions to be included in the evaluation of fatigue endurance. To experimentally d e t e r ~ i n ethe m a ~ i t u d eof the frictional stresses occurrin contacts, rolli~g-sliding disc es have been developed. The illustrated by Fig. 19.18. The e developed by N6lias et al. [1 cs are contoured to produce elli ntact areas as illustrated by drostatic cylindrical bearings to p tion force ~easurement.The frict plied force W ratio is called the traction coeffi~ient.Us methods of Chapter 13, the effective local(x,y) friction coefficients can be~stimatedfrom the test results. In Chapter 23, it is shown how the test device of Fig. 19. has been used to determine the characteristics of the effect of friction fatigue of the rolling-sliding contacts in ball and roller bearings. ping the test rig with the contaminated lubrication system of Fig. ,Ville and N6lias [19.21] investigat cts of particulate conation on rolling-sliding contact fa will be discussed further in Chapter 23.
1. Specimen 2. Ball 3. Tapered bearing cup 4. retainer Ball 5. Compression spring 6. Upper cup housing
-7.Spring retainer plate housing cup 8.Lower 9. Shock mount 10. Load application bolt 11. Spring calibration bolt
. Ball-rod rolling contact fatigue tester.
19.17. Pratt and Whitney Aircraft single-ball tester.
.18. Schematic drawingof rolling-sliding disc testing device [19.20].
ball-disc test rig, initially developed by Wedeven [19.211 and shown . 19.21, was designed to determine the natureof lubricant films in olling velocity may be varied by varying the ball drive e angle and the radius at which the ball contacts the disc. sing element of' a clear material such as sapphire or ass and optical
T FRICTION T E ~ T I N ~
I 19.19. Illustration of elliptical contactarea generated by the rolling-sliding disc test device 119.201.
TF""
0. Schematic diagram of lubrication contamination system used junction with rolling-sliding test rig 119.211.
in con-
.21. Ball-disc traction test rig 119.221.
determine contact traction force versus slide-roll ratio; the graphical display of Fig.19.22 was output by the test rig. A mathematical model of L circular point contact may also be developed,and by the matching of analytical and experimental data, thelocalized friction components which comprise the traction may be determined. The rig can be equipped with an environment chamber to allow evaluation of traction coefficient under conditions of high and low temperaI
J..l
f:
.$j 0.02 -
E 8
0.00
""-"
ts
0 ._/
35 -0.02-
f?
I-. I I
I I I I I I I I I
I
I
I
-10
0
10
% Slip
. ball-disc test rig.
Curve of traction coefficient vs percent sliding obtained from Wedeven
3. Interferograms taken using the ball-disc test and showing a contaminant particle ( a ) entering an EHL point contact; (b) in the center of the contact; and (e) exiting the contact.
ture, and high vacuum. s shown by Fig. 19.23, it further permits optical examination of the circular point contact under the effects of lu~ricant particulate CQntamination.
, it was demonstrated that although ball and roller bearing fatigue life rating and endurance formulas are founded in theory, they are semiempirical relationships requiring the establishment of various constants to enable their use. These constants, which depend upon the bearing raceway and ing element materials, can be establishe~only by appropriate testin cause of the stochastic nature of rolling bearing procedures necessarily require bearing and/ fatigue endurance, t or material populations of sufficient sizeto render the test results meaningful, Sample sizing effects are discussed in detail in Chapter Historically, to establish sufficiently accurate rating form stants, it has been necessary to test complete bearings. opment of stress-based life factors as shown in Chapter 23, however, it is now possible to use element-testing methods to determine many of these constants. For example, endurance testing of balls in V-ring test s rigs may be used to determine the basic material fatigue s t r e n ~ h of various materials. On the other hand, some of the stressesthat influence bearing life depend on the raceway forming and surface finishing methods. To duplicate these effects, the exact component may need to be endurance tested.
19.1. G. Lundberg and A. Palmgren, “Dynamic Capacityof Rolling Bearings,”Acta Pdytech. Mech. Eng. Ser. 1, R S U E , No. 3, 7 (1947). 19.2. T. Tallian, “On Competing FailureModes in Rolling Contact,”ASLE Trans. 10,418439 (1967).
19.3. R. Valori, T. Tallian, and L. Sibley, “Elastohydrodynamic Film Effects on the Load Life Behavior of Rolling Contacts,” ASME Paper 65-LUBS-11(1965). 19.4. 6. Johnston, T. Andersson, E. Van ~ e r o n g e n and , A. Voskamp, “Experience of Element and Full Bearing Testing over Several Years,” in Rolling Contact Fatigue Testing of Bearing Steels, ASTM STP 771, ed. J. J. C. Hoo, American Society for Testing and Materials, Philadelphia (1982). 19.5. G. Lundberg and A. Palmgren, “Dynamic Capacity of Roller Bearings,” Acta Polytech. Mech. Eng. Ser. 2, RSAEE, No. 4, 96 (1952). 19.6. R. Sayles and P. MacPherson, “Influence of Wear Debris on RollingContact Fatigue,’’ in Rolling Contact Fatigue Testing of Bearing Steels, ASTM STP 771, ed. J. Hoo, 255-274 (1982). 19.7. E. Fitch, An Encyclopedia of Fluid Contamination Control, Fluid Power Research Center, Oklahoma State University (1980). 19.8. T. Tallian, G. Baile, H. Dalal, and 0. Gustafsson, Rolling Bearing Damage: A Mor~hological Atlas,SKF Industries, Philadelphia (1974). 19.9. T. Tallian, Failure Atlas for Hertz Contact Machine Elements, ASME Press, New York (1992). 19.10. T. Andersson, “Endurance Testing in Theory,”Ball Bearing J. 19.11. G. Sebok and U. Rimrott, “Design of Rolling Element Endurance Testers,” ASME Paper 69-DE-24 (1964). 19.12. R. Hacker, “Trials and Tribulations of Fatigue Testing of Bearings,” SAE Technical Paper 831372 (1983). 19.13. T. Harris, “Prediction of Ball Fatigue Life in a BalllV-Ring Test Rig,”ASME Trans., J. Tribology 119, 365-374 (July 1997). 19.14. F. Barwell and D. Scott, Engineering 1 19.15. E. Zaretsky, R. Parker, and W. Anderson, “NASA Five-Ball Tester-over 20 Years of Research,” in Rolling Contact Fatigue Testing of Bearing Steels, ASTM STP 771, ed. J. J. C. Hoo, American Society for Testing and Materials, Philadelphia (1982). 19.16. E. Bamberger and J. Clark, “Development and Application of the Rolling Contact Fatigue Test Rig,” in Rolling Contact Fatigue Testing of Bearing Steels, ASTM STP 771, ed. J. Hoo (1982). 19.17. D. Glover, “A Ball-Rod Rolling Contact Fatigue Tester,” in Rolling Contact Fatigue Testing of Bearing Steels, ASTM STP 771, ed. J. J. C . Hoo, American Society for Testing and Materials, Philadelphia (1982). 19.18. P. Brown, G. Bogardus, R. Dayton, and D. Schulze, “Evaluation of Powder-Processed Metals for Turbine Engine Ball Bearings,’’ in Rolling Contact Fatigue Testing of Bearing Steels, ASTM STP 771, ed. J. J. C. Hoo, American Society for Testing and Materials, Philadelphia (1982). 19.19. T. Harris and J. McCool, “On the Accuracy of Rolling Bearing Fatigue Life Prediction,” ASME Trans., J. Tribology 118, 297-310 (April 1996). 19.20. D. NBlias, M.-L. Dumont, F. Couhier, G. Dudragne, and L. Flamand, “Experimental and Theoretical Investigation of Rolling Contact Fatigue of 52100 and M50 Steels under EHL or Micro-EHL Conditions,” ASME Trans. J. Tribology 1’20, 184-190 (April 1998). 19.21. F. Ville and D. Nhlias, “Early Fatigue FailureDue to Dents in EHL Contacts,” Presented at the STLE Annual Meeting, Detroit (May 17-21, 1998). 19.22, L. Wedeven, “Optical Measurements in Elastohydrod~amic Rolling Contact Bearings” (Ph.D. dissertation, University of London, 1971). 19.23. Wedeven Associates, Inc., “Bridging Technology and Application through Testing,” Brochure (1997).
Symbol
L
Fz
Description Probability density function Cumulative distribution function Hazard rate Cumulative hazard rate Failure order number Number of samples -1n (1 - p ) Number of subgroups in a sudden death test Sample size of a sudden death subgroup Sample size A probability value A probability value A pivotal function used forsudden death test analysis Number of failures in a censored sample Ratio of upper to lower confidence limits for ,6 Median ratio of upper to lower confidence limits for xo.lo
ODS TO ~
Symbol
X xP
P rl
~
Y EZ
E
CE
Description Pivotal function fortesting for diEerences among k estimates of xo.lo Pivotal function for setting confidence limits on xp Pivotal function for setting confidence limits on xp using k data samples Pivotal function for setting confidence limits on P Pivotal function for setting confidence limits on P using k data samples Pivotal function for testing whether k Weibull populations have a common P A random variable The pth percentile of the distribution of the random variable x The Weibull shape parameter ibull scale parameter
statistical distributions have been used to describe the random ility of the life of manufactured products. Such choices can be variously justified. For example, if a product has a reservoir of a substance that is used up at a uniform rate through the product’s life, and if the initial supply of the substance varies from item to item according to a normal (~aussian)distribution, then the product life will be normally distributed. ~orrespondingly,if the initial amount of the substance fola distribution, item life will be gamma distributed. 11 distribution is a popular product life model generally justified by its property of describing, under fairly general circumstances, the way that the smallest values in large samples vary among sets of large samples. Thus if item life is determined by the smallest life among many potential failure sites, it is reasonable to expect that life will vary from item to item according to a Weibull distribution. Another property that makes the Weibull distribution a reasonable choice for some products is that it can account for a steadily increasing failure rate characteristic of wear-out failures, a steadily decreasing failure rate characteristic of a product that benefits from “burn-in,” or a constant failure rate typical of products that fail due to the occurrence of a random shock. The two-parameter Weibull distribution was adoptedby Lundberg and almgren E20.11 to describe bearing fatigue life on the strength of the excellence of the empirical fit to bearing fatigue life data. As will be described in Chapter 23, because of improvements in both the material and the methods of finishing bearings, it has been found recently that
there is, under moderate load, a finite probability that a bearing will endure for an inde~nitely long period. The Weibull model cannot describe this aspect of fatigue life. Nonetheless, under the relatively high loads common in fatigue testing practice, the Weibull model will closely approximate the observed fatigue life behavior of rolling bearings. What follows is an account of the principal features of the Weibull distribution that areof interest to bearing engineers and an explanatio~ of the methods of inferring the Weibull parameter values from life test data acquired under various experimental schemes.
en it is said that a random variable, for example,bearing life, follows it is implied that theprobability the two-parameter ~ e i b u ldistribution, l that anobserved value of that variable is less than some arbitrary value x can be expressed as rob(1ife < x) = F ( x ) = 1 -
x, q, p > 0
(20.1)
Thefunction F ( x ) is known as the cumulative distribution function (CDF). The constants q and p are the scale parameter and the shape parameter, respectively. The functionF ( x )may be thought of as the area under a curve f ( x ) between 0 and the arbitrary value x. This curve is known as the probability density function (pdf) and has the form
Figure 20.1 is a plot of the Weibull pdf for various values of p that a wide diversity of distribution forms are encompassed by th bull family, depending on the value of p. For /3 = 1.0 the Weibull bution reduces to the exponential distribution. For G , in the range 3.03.5 the Weibull distribution is nearly symmetrical and approximates the normal pdf. The ability to assume such a range of shapes accounts for the extraordinary applicability of the Weibull distribution to many types of data.
e
es
The average or expected value of a random variable is a useful measure of its “central tendency”; it is a single numerical value that can be considered to typify the random variable. It is defined as
7
'
ST~TISTIC ~ ~T ~ O TO D ~S
~ END Y
~
X
20.1. The two-parameter Weibull distribution for various values of the shape parameter p.
(20.3) The value of the integral in(20.3) is (20.4) I?( ) is the widely tabulated gamma function. Table 20.1 givesvalues of
r(l/p + 1) for p ranging from 1.0 to 5.0.
In reliability theory E(x)is known as themean time between failures and is commonly referred to as MTBF. It represents the average time between the failures of two consecutivelyrun bearings-that is, the time between the failure of a bearing and the failure of its replacement. It does not represent the mean time between consecutivefailures in a group of si~ultaneouslyrunning bearings. For this latter situation, provided /3 j t 1.0, MTBF will vary with the failure order number. For example, the mean time between the first and second failures in samples of size 20 is different from the mean time between the 19th and 20th failures. The scatter of a random variable is often characterized by a quantity known as variance, defined as the average or expected value of the squared deviation of the variable from its expected value; that is,
The value of this integral is
E
. Values of r ( l / p + 1) and r(21p + 1) - r 2 ( l / p + 1) 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.5 3.0 3.5 4.0 5,O
1.0000 0.9649 0.8407 0.9236 0.9114 0.9027 0.8966 0.8922 0.8~93 0.8874 0.8862 0.$873 0.8930 0.8997 0.9064 0.9182
1.0000 0.7714 0.6197 0,5133 0.4351 0.37~7 0.3292 0,2919 0.2614 0.2360 0.2146 0.1441 0.1053 0.0811 0,0647 0.0442
Values of the quantity [l?(2/p + 1)- r2(1/p+ 1)are listed in for p ranging from 1.0-5.0. The units of cr2 are the square of the units inwhich life is ~ e a s u ~ e d , (hr)2. The square root of cr2 is often preferred as a measure of scatter because it is expressed in the same units as the random variable itself. It is called the standarddeviation. Neither the variance nor the standard deviation are cited much forthe Weibull distribution; it is more usual to convey the magnitude of scatter by citing the values of a low percentile and a high percentile.
Equation (20.1) gives the probability that theobserved value of a random variable is less than an arbitrary value. The inverse problem is to find the value of x,say xp,for whichthe probability is a specified value p such that the life will not exceed it. The term xp is defined im~licitly as
The solution of equation (20.7) is
important special case in rolling bearing since it is historically customary xo.lois referred of their tenthpercentile life. In the literature r consistency with the statistical literature on the distribution, xo.lois used in this partof the chapter. (20.9)
Also of some interest is the median life, xo,50:
quation ( 2 0 . ~shows ) that the ratio of two percentiles, say xp and xq,is (20.11)
hus
1,therefore, ~0 . 5 0= 5.54. This supports the rule, often quo ing bearing industry, that is roughly five times
(x*.1*).
From equation (20.1) the probability that a bearing survives a life denoted S(x),is given by
X,
Taking natural logarithms twice of both sides of equation (20.12), leads to
TE
s a linear function of ln (x).On special graph paper, called ility paper, for whichthe ordinate is ruled proportionately and the abscissa is logarithmically scaled, the values of versus the associated values of x plot as a straight line. If in thedesign of the paper the same cycle lengths are used for the logarithmic scale on both coordinate axes,the slope of the straightline representation will be ibull shape parameter or numerically equal to p. In any case, the ibull slope will be related to the slope of the straight line represenon, and in some designs of probability paper an auxiliary scale is ided to relate the shap rameter and the slope. bullprobability paper onwhich the disigure 20.2 is a piece o = 15.0 is represented. It was constructed Lion with ,6 = 1.0 and by passing a 45' line t ugh the point correspondingto the failure prob0.9) and the life value xo.lo= 15.0.From this ability value F = 0.1 plotmaybe read the percentile asthe abscissa value at which a horizontal line at the ordinate value F = 0.2 intersects the straightline. hin graphical accuracy, xo,20= 32.0. Inversely, the pr prior to the life x = 52.0 is read to be roughly 30%. ibull population on probability paper thus offers a graphical alter-
10
32.0 52.0 100
1000
X
~ ~Graphical . ~representation . of the Weibull population for which p = 1.0,
xo.lo= 15.0.
native to the use of equations (20.1) and (20.8) for the computation of probabilities and percentiles. The graphical approach is sufficiently accurate for most purposes. The primary use of probability paper is not, however, forrepresenting known Weibulldistributions but for estimating the Weibull parameters from life test results.
Up to this point it has been assumed that the Weibull parameters are known, and additionally required quantities such as probabilities, percentiles, expected values, variances, and standard deviations have been calculated in terms of these known parameters. This is a common situation in bearing application engineering, in which, given a catalog calculation of xosl0(Llo) and the standard Weibull slope of p = 1.1, it is required to compute the median life or the TBF, and so on. In developmental work involving newvariables such as materials, lubricants, or finishing processes, the focus is on determining the effect of these factors on the Weibull parameters. Accordingly, a sample of bearings modified from standard in some way is subjected to testing under standardized conditions of load and speed until some or all fail. When all fail, the sample is said to be uncensored. In a censored sample some bearings are removed from test prior to failure. Given the lives to failure or to test suspension for the unfailed bearings, the aim is to deduce the underlying Weibull parameters. This process is called estimation, because it is recognized that, since lifeis a random variable, identical samples will result in different test lives. The Weibull parameter values estimated in any single sample must themselves therefore be regarded as observed values of random variables that will vary from sample to sample according to a probability distribution known as the sampling distribution of the estimate. The scatter in thesampling distribution will decrease with sample size. Sample size thus affects the degree of precision with which the parameters are determined by a life test. The precision is expressed by an uncertainty or confidence interval within which the parameter value is likely to lie. An. estimation procedure that results in the computation of a confidence interval is called interval estimation. A procedure that results in a single numerical value for the parameter is called point estimation. Point estimates in themselves are virtually useless, since without some qualification there is no way of judging how precise they are. Accordingly?an analytical technique is given in the sequel for computing interval estimates of Weibull parameters. It is recommended that this technique be supplemented, however, with a point estimate obtained graphically. The graphical approach to estimation gives a synoptic view
801
EST~TION IN SINGLE SAMPLES
of the entire dist~butionand offers the opportunity to detect anomalies in the data thatcould easily be overlooked if reliance is placed entirely on an analytical technique.
Assume that a sample of n bearings is tested until all fail. The ordered times to failure are denoted x1 < x2 < * x,. Consider the CDF of the Weibull population from which the sample was drawn. If this function were known, it would follow that the lives xi and the values F(xi), i = 1, . . . , n, would plot as a straight line on Weibull probability paper. It has been shown that even though the function F(x) is not known, nonetheless F(xi)will vary in repeated samples according to a known pdf. The mean or expected value of F(x,) has been shown to equal il(n + 1).The median value of F(xi),also known as the median rank, has been shown by Johnson f20.21to be approximately (i - 0.3)l(n + 0.4). The procedure then is to plot the mean or median value of F(xi)against xifor i = 1, 2, . . . , n. The tradition in thebearing industry is to use the median rather than the mean as a plotting position choice, but the difference is small compared to the sampling variability. Table 20.2 lists the ordered lives at failure for a sample of size n = 10, alongwith the actual and approximated values of the median ranks. is adequate within the limits of graphical apHence the appro~mation proximation. The median ranks are shown plotted against the lives in Fig. 20.3. The straight line fitted to the plotted points represents the graphical estimate of the entire F(x) curve. Estimates of the percentiles of interest are thenread from the fitted straight line. For example, to within graphical accuracy the xo,lovalue is estimated as 15.3. The Weibull shape pa9
.
9
Random Uncensored Sample Size of n
= 10
Failure Order No. (i)
Life
Median Rank
i - 0.3 n + 0.4
1 2 3 4 5 6 7 8 9 10
14.01 15.38 20.94 29.44 31.15 36.72 40.32 48.61 56.42 56.97
0.06697 0.16226 0.25857 0.35510 0.45169 0.54831 0.64490 0.74142 0.83774 0.93303
0.06731 0.16346 0.25962 0.35577 0.45192 0.54808 0.64423 0.74038 0.83654 0.93269
~ T ~ T I ~ T I C THO^^ TO
ZE E
CE
95 90
80 70 60
50 40
g
30
v
3 20 4
10 8
6 4 2
.3. Probability plot for uncensored random sample for size n
=
10.
rameter, estimated simply as the measured slope of the straight line, is roughly 2.2. The same graphical approach applies for right-censored data in which the censored observations achieve a longer running time than the failures do. The full sample size n is used to compute the plotting positions, but only ithe failures are plotted. When there is mixed censoring-that is, there are suspended tests among the failures-the plotting positions are no longer computable by the method given since the suspensions cause ambiguity in determining the order numbers of the failures. Several alternative approaches are available for this situation, with generally negligible differences among them. The method known as hazard plotting due to Nelson l20.31 is recommended because it is easy to use. Column 1 of Table 20.3 gives the lives of failure or test suspension in a sample of size n = 10. Of the 10 bearings, r = 4 have failed, and the lives at failure are marked with an “ F in Table 20.3. Similarly the lives at test suspension are indicated by “”s” The lives in column 1 are in ascending order of time on test irrespective of whether the bearing failed. Column 2, termed the reverse rank by Nelson [20.3], assigns the value n to the lowest time on test, the value n - 1 to the next lowest, and so on. Column 3, called the hazard, is the reciprocal of the reverse rank, but is computed only forthe failed bearings. Column 4 is the cumulative hazard and contains for each failure the sum of the hazard values in column 3 for that failure and each failure that occurred at an earlier running time. Thus for the second failure the cumulative hazard is
.
Co~putationof Plotting Positions for Hazard Plot ~~
Hazard Reverse Rank
Life
10 9 8 7 6 0.4540 5 4 3 2 1
0.569 S 8.910 F 21.410 S 0.1429 F 21.960 32.620 S 0.3649 39.290 F 42.990 S 50,400 F 53.270 S 102.600 S
(h)
Cumulative Hazard ( H )
-
-
-
0.1111
0.1111
0‘1052
-
=
1 - e-H
-
-
0.2540
0.2243
-
-
F
0.2000
-
-
-
-
0.3333
0.7873
0.5449
-
-
-
-
-
0.2540 = 0.1111 + 0.i429. The cumulative hazard can then be plotted directly against life on probabilitypaper that hasbeen designedwith an extra “hazard” scale. If paper of this type is unavailable, it is only necessary to compute an estimate of the plotting position applicable to ordinary probability paper by transforming the cumulative hazard H to F = 1 - exp (-H). This computation is shown in column 5 of Table 20.3. Figure 20.4 shows the resultant plot. Note that as in the right censored
2 x Probability plot formixed censoring. Plotting positions are calculated based on cumulative hazard. 20.4.
804
S T ~ T I~ STTHIOC D~ S
TO ANALYZl3 ENXI
case only the failures are plotted. The suspended tests have played a role, however, in determining the plotting positions for the failures.
The method of maximum likelihood is a general approach to the estimation of the parameters of probability distributions. The central idea is to estimate the parameters as the values for which the observed test sample would most likely have occurred. Consider an uncensored sample of size n. The “likelihood”is the product of the probability density functions f ( x ) = xP-’/qP exp [-(x/$ P3 evaluated at each observed life value. The maximum likelihoodestimates of q and p are thevalues that maximize this product. For censored samples with r < n failures the likelihood functioncontains, in lieu of the density function f(x), the term 1 - F ( x ) = exp [-(x/$ PI evaluated at each suspended life value. It can be shown that the maximum likelihood (NIL) estimate of p, denoted by a caret (“), is the solution of the following nonlinear equation: i=r
x x$ x x$
i=n
In xi
i=S
i=n
=o
(20.14)
i= S
This equation has only a single positive solution according to [20.4].That solution is readily found by the Newton-Raphson method, although in highly censored cases the initial guess used to start the method might need to be modified to avoid convergence to a negative value for p. Having determined p from equation (20.14), the ML estimate of q is obtained: (20.15) The NIL estimate of a general percentile is (20.16) where l i p is defined as
kp = -1n (1 - p )
(20.17)
Confidence limits can be set if the censoring mode corresponds to the
suspension of testing when the rth earliest failure occurs. This type of censoring is customarily referred to as type 11 censoring, as contrasted to type I censoring in which testing is suspended at a predetermined running time. In type I censoring the number of failures r is a random variable. In type II censoring the number of failures is predetermined by the experimenter. The basis for confidence intervals for /3 is that the random function u(r, n) = p/p follows a sampling distribution that depends on the sample size n and censoring number r, not on the underlying values of p or q. unctions with this property are known as pivotal functions. The sampling distribution of u(r, n ) cannot be found analytically, b termined empirically to whatever precision is needed by sampling. In the onteCarlomethod repeated samples computer simulation from a Weibull distribution having arbitrary parameter values, say p = 1.0 and q = 1.0. The ML estimate is formed for each sample and d ed by the underlying population value of ,El to yield a value of u(r, n). typically l0,OOO such values the percentiles may be computed from the sorted set and their average equated to the expected value of the dist~bution. Denoting the 5th and 95th percentiles of u(r, n) as uo,05(r,n ) and u0.&, n ) leads to the following 90% confidenceinterval for p:
p
The raw maximum likelihood estimates of the Weibull parameters are biased; that is, both the average and the median of the ,El estimates in an indefinitely large number of samples will differ somewhat from the true p value for the population from which the samples were drawn. It is possible to correct the raw NIL estimate so that either its average or its median will coincide with the underlying population value of ,El. cause the distribution of u(r, n ) is not symmetrical, it is necessary to choose whether the adjusted estimator should be median or mean unbiased. Medianunbiasedness is recommended becausethen the ML point estimate will have the reasonable property that it is just as likely to be larger than the underlying true value as to be smaller. It is shown in [20.4] that themedian unbiased estimate of p, denoted by writing above the symbol, is above the s p b o l , is expressible as
n), and uo,95(r, n ) for n from Table 20.4 gives values of uo,05(r,n), vOdO(r, 5-30 and various values of r.
CE
.
5th, 50th, and 95th Percentiles of u(r, n ) and u(r, n, 0.10)
u(r, n, 0.10)
~ ( rn,) r 3 5 3 5 10 5 15 10 15 5 10 15 20 20 5 10 15 20 30
n 0.500.05 5 5 10 10 10 15
15 20 20 20 30 30 30 30 30
0.6351 0.6795 0.6208 0.6482 0.7561 0.6430 0.7130 0.7715 0.6432 0.7047 0.7459 0.7949 0.6430 0.6996 0.7410 0.7662 0.8259
0.95 0.500.05 1.6510 1,2346 1,7223 1.3117 1.1031 1.3321 1.1269 1.0679 1.3353 1.1328 1.0754 1.0476 1.3475 1.1309 1.0819 1.0569 1.0290
6.7596 2.8146 7.6478 3,2791 1.8363 3.3937 1.9428 1.5634 3.5078 1.9913 1.6327 1.4454 3.4437 2.0236 1.6771 1.5182 1.3353
- 1.2672 - 1.1422
0.8483 0.4465 - 1,4304 0.4313 -0.9571 0.3737 -0.8794 0.2125 -0.9223 0.2435 -0.6184 0.1933 -0.7648 0.1393 -0.9601 0.1482 -0.7274 0.1604 -0.7055 0.1215 -0.6740 0.0958 - 1,1306 0.0176 -0.6348 0.0931 -0.6038 0.0928 -0.5955 0.0790 -0.5672 0.0536
0.95
R
9.9607 10.6 8.98 4.4453 4.14 92.4 7.0208 12.3 135 3.7698 5.06 36.7 2.49 2.1304 15.3 18.2 5:28 2.9446 1.9477 2.72 9.75 1,5091 2.03 8.41 13.8 5.45 2.5445 8.89 1.7473 2.83 1.4431 2.19 7.37 1.2262 1.82 6.13 8.12 1.6920 5.36 1.3891 2.89 5.99 5.39 1.2152 2.26 5.04 1.1130 1.98 4.22 1.62 0.9147
Correcting the bias of the estimate and settingconfidence limits for a general percentile xp depends on the pivotality of the random function u(r, n, p ) = ,h ln (gplxp).Given percentiles of u(r, n, 0.10) determined by Monte Carlo sampling, a 90% confidence interval on xo.locan be set up:
A median unbiased estimate of xo.locan be computed as (20.21) Values of the 5th,5Oth, and 95th percentiles of u(r,n, 0.10) are also given in Table 20.4. Using the NIL method, calculate median unbiased point estimates and 90% confidence limits for /3 and xo,lofor the uncensored sample of size n = 10 listed in Table 20.2. Using a computer program that solves equation (20.14) for /3 and equation (20.16) for xo.lo,it is found that the raw ML estimates are ,h = 2.58 and xo.lo= 16.55. The following applicablevalues are taken from Table 20.4.
0, 10) = 1.103,
2.58
uo.50
(10, 10, 0.10)
=
0.
2.58
From equation (20.19) the median unbiased estimate of ,El is (20.19) From equation (20.20), 90% confidencelimits for xo.loare computed as
7.25 < xo,lo< 23.3.
(20.20)
The median unbiased estimate of xo,lois
=
{ [- ( ~ ) ] }
16.6 exp
=
15.3
The median unbiased estimates of xo,loand p agree closely with the values estimated graphically.
A popular test strategy in the bearing industry is the "sudden death" test. In sudden death testing a test sample of size n is divided into I
80
ST~TISTIC~ ~ T H O TO ~ S
CE
subgroups each of size rn (pz = Zm). When the first failure occurs in each subgroup, testing is suspended on that subgroup. When the test is over, there are Z failures, the first failures in each of the Z subgroups. To estimate p, substitute thesefirst failures directly in equation (20.14). Confidence limits for p are then computed from equation (20.18) with r = n = 1. That is, the first failures are treated as members of an uncensored sample whose size is equal to the number of subgroups. Table 20.5 gives percentiles of u(Z, Z) for Z = 2-6. The value of 90.10,determined by using the sample of' first failures and equation (20.16), is denoted 90,10,.The ML estimate applicable to the complete sample is then computed as in [20.5]:
90% confidence limits for xo.lomay be computed as
A median unbiased estimate of xo.lois computed from
Table20.5gives values of the percentiles of the random function q(1, m, p ) required for these calculations. ,
A sudden death test conducted with Z = 3 subgroups, each of size rn = 10, yielded the following values for the subgroup first failures: 4.72,6.64, and 14.17. Using these lifetimes as inputto a computer program that solves equation (20.14) yieldsthe shape parameter estimate = 2.27. The estimated value of xo.lo based on these three failure lives is 90.10s= 3.59. From equation (20.22) the estimate of xo,lo for the parent population is
The followingvalues are taken from Table 20.5:
The median unbiased estimate of p is
CE
ST~TISTIC
(20.19) 90% confidence limits for p are (20.18) .27
2.27
The median unbiased estimator of xo.lois -40.50
-0.133
=
9.34 (20.24)
90% confidence limits for xo.loare
e value of the estiest sample. As the interval approach each other; that is, the ratio of the upper to lower ends of the confidence a ~ ~ r o a c 1.0. ~ e sFor finite sample sizes this ratio was suggested as a useful measure of the precision of esti~ation.From equa18) the confidence limit ratio R for p estimation is (20.25) for various n and r are given in Table 20.4 for conventional tests and Table 20.5 Note for that for a given sample ) the as number of failures r size n, the recision im increases.
r xo.lothe ratio of the upper to lower confidence limits contains the m variable The approach taken in E20.61 in this case is to use as a precision measure the ~ e d value ~ ~of nthis ratio,denoted expression fop this median ratio contains the unknown value o shape parameter p. For planning purposes one may use an historical or, alternatively, the valu 50 are given in Table 20.4 den test testing.
*
from a quantitative factor,
bearing fatigue life will inclu from each other with respect qualitative factor is dismperature or load, which of qualitative factors incl
It was shown in [20.7] that more preciseestimates can be made if the samples making up the complete investigation are analyzed is is possible if it can be assumed that thesamples are drawn 11 populations, which, although they might differ in their scale ~arametervalues, nonetheless have a common value of p. ~pplicabletabular values for carrying out the analyses presuppose that the sample size n and the number of failures r are the same for each sample in the set, so henceforth this is assumed to be the case. It is thus assumed that k groups of size n have been tested until the rth first step is to deter failure occurred in each is ~lausible that the gro common value of p. oup individually to determine the Val mallest of the k p estimates are then ~etermined,and values did differ amongthe groups, this ratio le 20.6 gives the value of the 90th percentile of the ratio w = for various r, n, a 11 populations that sampling from k did have a co f p. Thus, values atio of largest to smallest shapeparameter estima exceeding those in Table ups do have a common v occur only 10% of the time if the These values maybeused as th tical values in deciding whether a common ,El assumption is justified. aving determined that thecommon p assumption is reasonable, this common p value can be estimated using the data ineach group,by solving the nonlinear equation
81
CriticalValues n, k ) forTesting homo gene it^ of k Weibull Shape Parameter Estimates e
n
r
5 5 10 10 10 15 15 15 20 20 20 20 30 30 30 30 30
3 5 3 5 10 5 PO 15 5 10 15 20 5 10 15 20 30
k k=k=4=3 2 5.45 2.77 6.04 3.21 1.87 3.20 2.02 1.65 3.31 2.11 1.72 1.52 3.28 2.11 1.78 1.61 1.41
8.73 3.59 9.93 4.35 2.23 4.48 2.44 1.87 4.54 2.50 1.96 1.70 4.47 2.54 2.05 1.82 1.53
11.0 4.23 12.5 5.16 2.47 5.28 2.69 2.05 5.28 2.80 2.15 1.80 5.38 2.90 2.27 1.95 1.61
k=5
k = 10
13.4 4.69 15.4 5.69 2.61 5.90 2.90 2.16 6.24 3.00 2.30 1.90 6.11 3.10 2.40 2.06 1.67
22.2 6.56 26.8 7.98 3.16 8.39 3.56 2.45 9.05 3.69 2.70 2.14 8.95 3.88 2.82 2.40 1.84
Dl
where denotes the ML estimate of the common ,E? value and x i ( j )denotes t h e ~ ordered t ~ failure time within the ith group. Confidence limits for p may be set analogously to equation (20.117) as follows:
(20.27) where ul(r,n, k ) = puted from
&lp. A median unbiased estimate of /3 may be com-
0.7 gives percentiles of ul(r, n, k ) needed for setting ~0~ confidence limits and for bias co ction for various values of n, r, and k . The scale ~ a r a m e t e for r the ith up may be reestimated with p1 as follows:
13
ION IN SETS OF T W
20.7. Percentiles of Functions Needed for Analyzing Sets of Weibull
Data 0.95 n
0.50 r
k 0.05
0.05 0.95
0.50
5 5 5 5 5
3 3 3 3 3
2 3 4 5 10
0.7618 0.8431 0.8915 0.9331 1.038
1.505 1.471 1.462 1.446 1.430
3.833 3.022 2.673 2.485 2.030
-1.175 -1.135 -1.131 -1.112 -1.084
0.6861 0.6111 0.6124 0.6216 0.5889
5 5 5 5 5
5 5 5
2 3 4 5 10
0.7773 0.8265 0.8609 0.8888 0.9545
1.195 1.184 1.178 1.177 1.167
2.056 1.816 1.703 1.621 1.453
-0.9715 -0.8715 -0.8321 -0.7907 -0.7314
10 10 10 10 10 10 10 10 10 10 15 15 15 15
5 5 5 5 5
0.7677 0.8281 0.8679 0.8904 0.9669
1.256 1.242 1.234 1.230 1.219
10 10 10 10 10
2 3 4 5 10 2 3 4 5 10
0.8130 0.8508 0.8794 0.8939 0.9405
1.067 1.082 1.077 1.076 1.073
2 3 4 5 2 3 4 5
0.768 0.822 0.857 0.891
15 15 15 15
5 5 5 5 10 10 10 10
1.269 1.256 1.246 1.239 1.106 1.100 1.097 1.094
2.326 2.039 1.877 1.768 1.575 1.516 1.414 1.355 1.319 1.235 2.414 2.080 1.927 1.815
15 15 15 15 20 20 20 20
15 15 15 15 5 5 5 5
0.836 0.871 0.890 0.902
20 20 20 20 20 20 20 20
10 10 10 10 15 15 15 15
20 20 20 20
20 20 20 20
2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5
25 25 25 25
5 5 5 5 10 10 10 10
2 3 4 5 2 3 4 5
25 25 25 25
5 5
tl
u1
*1
0.90
2.833 3.294 3.605 3.733 4.158
3.752 4.192 4.342 4.512 4.707
0.3875 0.3473 0.3410 0.3470 0.3336
5.325 3.864 3.417 3.073 2.439 2.750 2.255 1.980 1.801 1.471
1.441 1.742 1.950 2.082 2.409
-0.8691 -0.8526 -0.8303 -0.7987 -0.7994 -0.7161 -0.6590 -0.6117 -0.5883 -0.5766
0.2856 0.2727 0.2761 0.2794 0.2733 0.1587 0.1460 0.1448 0.1519 0.1450
2.424 1.959 1.729 1.587 1.361 1.444 1.204 1.066 0.9899 0.8384
1.576 1.900 2.084 2.209 2.549
1.802 2.111 2.285 2.392 2.702 1.999 2.296 2.426 2.547 2.823
0.8509 1.062 1.171 1.256 1.468
1.040 1.231 1.330 1.406 1.598
-0.879 -0.876 -0.848 -0.864
0.203 0.190 0.193 0.189
1.976 1.645 1.474 1.377
1.569 1.899 2.110 2.206
2.02 2.308 2.475 2.548
1.611 1.489 1.412 1.375
-0.668 -0.613 -0.685 -0.573
0.157 0.145 0.154 0.138
1.352 1.152 1.033 0.987
0.873 1.080 1.200 1.289
1.080 1.254 1.381 1.457
1.369 1.302 1.258 1.230
0.108 0.098 0.097 0.095
...
2.428 2.078 1.916 1.836
-0.615 -0.559 -0.515 -0.497 -0.937 -0.912 -0.918 -0.937
0.671 0.827 0.920 0.987 1.575 1.922 2.121 2.233
0.796 0.836 0.866 0.885
1.056 1.052 1.049 1.048 1.278 1.267 1.251 1.246 1.109 1.104 1.100 1.099
1.650 1.507 1.446 1.402
-0.621 -0.587 -0.573 -0.550
0.125 0.125 0.115 0.117
0.826 0.857 0.882 0.896
1.064 1.062 1.058 1.059
1.425 1.337 1.289 1.263
0.855 0.881 0.898 0.911 0.764 0.822 0.866 0.894
1.040 1.037 1.037 1.036 1.287 1.264 1.256 1.251
1.300 1.242 1.208 1.187 2.427 2.102 1.952 1.846
0.791 0.832 0.864 0.885
1.117 1.107 1.103 1.100
1.652 1.527 1.456 1.401
-0.569 -0.531 -0.497 -0.472 -0.531 -0.486 -0.461 -0.433 -1.002 -0.996 -0.977 -0.986 -0.605 -0.586 -0.575 -0.557
0.0982 0.0988 0.0914 0.0955 0.0723 0.0746 0.0715 0.0714 0.0776 0.0786 0.0740 0.0719 0.107 0.0957 0.0987 0.0946
0.817 0.949 1.036 1.101 2.020 2.348 2.486 2.580 1.072 1.282 1.384 1.475 0.809 0.964 1.057 1.120
0.799 0.845 0.866 0.888
0.761 0.818 0.864 0.892
0.119 0.127 0.118 0.119
0.904 0.805 0.772 1.660 1.403 1.299 1.226 1.231 1.035 0.953 0.915 1.013 0.874 0.800 0.743 0.874 0.7422 0.665 0.623 1.407 1.246 1.144 1.098 1.099 0.979 0.887 0.838
0.871 1.098 1.213 1.304 0.672 0.830 0.922 0.999 0.566 0.702 0.785 0.836
0.678 0.803 0.883 0.941
1.560 1.910 2.114 2.231
1.200 2.308 2.469 2.562
0.871 1,091 1.219 1.289
1.061 1.278 1.387 1.466
U1
V1
n 25 25 25 25 20 20 20 20
25 25 25 25
25
25 25 25 25
5
30 30 30 30
5
10 10
20 20 20 20 25 25 25 30 30 30
30 30 30 30
r 0.817 15 15 15 15 0.874 0.892 0.905 0.864 0.890 25 0.905 25 25 0.916 0.761 0.827 5
5 10
10
30 30 30 30
0.813 15 15 15 15
30 30 30 30 30 30 30 30
0.833
30 30 30 30
0.873 0.895 0.911 30
0.889 0.903 0.853 0.882 0.899 25
k 2 3 4 5
tl
0.05
0.50
0.95
0.05
0.50
0.95
0.90
0.95
0.855 0.878 0.893
1.069 1.063 1.061 1.061
1.447 1.358 1.308 1.277
-0.551 -0.501 -0.477 -0.458
0.0969 0.0830 0.0847 0.0843
0.974 0.849 0.753 0.710
0.673 0.835 0.932 0.996
0.827 0.972 1.072 1.118
1.045 1.043 1.041 1.041
1.336 1.268 1.236 1.208
-0.519 -0.469 -0.435 -0.426
0.0762 0.0710 0.0692 0.0681
0.857 0.745 0.661 0.618
0.567 0.701 0.785 0.836
0.682 0.811 0.882 0.931
1.255 1.204 1.178 1.051 2.436 2.120 1.963 1.838
-0.494 -0.445 -0.409 -0.387 - 1.066 - 1.070 - 1.065 - 1.062
0.0629 0.0607 0.0541 0.0570 0.0177 0.0342 0.0331 0.0275
0.752 0.655 0.581 0.539 1.289 1.151 1.049 1.019
0.601 0.710 0.779 0.814
0.0874 0.0804 0.0819 0.0795 0.0823 0.0767 0.0734 0.0768
1.020 0.896 0.828 0.787 0.886 0.776 0.721 0.692
0.497 0.615 0.686 0.736 1.591 1.938 2.148 2.256 0.880 1.099 1.228 1.313 0.669 0.832 0.940 1.004
0.813 0.965 1.067 1.136
2 3 4 5 2 3 4 5 2 3 4 5
0.841
0.871 0.895
1.032 1.030 1.030 1.028 1.285 1.267 1.256 1.256
2 3 4 5 2 3 4
0.789 0.831 0.861 0.883
1.117 1.109 1.108 1.104
1.674 1.528 1.454 1.410
1.071 1.068 1.065 1.066 1.049 1.045 1.045 1.045
1.458 1.365 1.319 1.288
-0.587 -0.576 -0.570 -0.570 -0.513 -0.583 -0.470 -0.460
1.349 1.280 1.243 1.219
-0.490 -0.442 -0.430 -0.410
0.0737 0.0680 0.0664 0.0701
1.083 0.704 0.629 0.621
0.560 0.698 0.781 0.842
0.679 0.796 0.884 0.942
1.037 1.035 1.033 1.033 1.026 1.025 1.024 1.024
1.280 1.226 1.195 1.175 1.224 1.184 1.156 1.142
-0.481 -0.425 -0.396 -0.380 -0.458 -0.406 -0.387 -0.368
0.0572 0.0593 0.0575 0.0599
0.882 0.627 0.583 0.549
0.595 0.699 0.771 0.816
0.0530 0.0513 0.0467 0.0493
0.668 0.574 0.529 0.496
0.500 0.609 0.683 0.736 0.448 0.552 0.618 0.665
5 2 3 4 5 2 3 4 5 2 3 4 5
0.815 0.875 0.893 0.871
0.913
0.921
2.060 2.367 2.515 2.607 1.095 1.303 1.407 1.492
0.534 0.635 0.698 0.740
The value of xpi may be estimated from
for xo.lo may be computed as follows: ~ o n ~ d e n limits ce (20.31)
where u1 = In ($o~lo/xo~lo) is the h sample generalization o f u ( r ,n, O.lO>, he median unbiased estimate of xo.lomay be computed as
Now that xo.lohas been estimated for each group using the mate ,bl of the common shape parameter, the next question of interest is whether these xo.lovalues differ significantly. That is, are the apparent differences amongthe xo.loestimates real, or could they be due to chance? To test whether the underlying true xo.lovalues could all be equal, the magnitude of variation that cou ccur in the estimated values due to chance alone must be assessed. s can bedoneby using the random function tl(r,n, k ) defined by
where (32:0.10)max and (32:o.lo)~in are the largest and smallest values of 32:o.10 calculated among the k samples. The 90th or 95th percentiles of tl(r, n, k ) may be used to assess the observed difference in thexoel0values. Any two samples, say sample i and sample j , for which values the quantity In ~(32:0~10)i/(32:0~10)j1 exceeds (tl)o.9a,may be declared to differ from each other at the 10% level of si~nificance. ~orrespondi~gly, using the 95th percentile of tl(r,n, k ) results in a5% significance leveltest for the equality of the xo,lovalues.
Dl
Ten bearings are fabricated from each of performed. The r timates of p and xo.lofor each group are given below:
. ~ncensoredlife testsare
Group No.
1 2.59 5.06
B
~0.10
2 2.32 2.60
3 3.13 4.72
4 1.94 3.49
rS-
5 3.65 8.83
To test whether a common assumption is plausible, the ratio of the largest to the smallest shape parameter estimates is com~uted:
of a constant
.
I0 and k = 5, it is fou ta are consistent with the p~ausibility of a com ) is calculated and ,bl = es of 32: ai by e ~ u a t i o ~
the median unbiase
(~0.~~)
81
$ T A T I $ T I C ~METHODS TO ~
Group No. Raw ML estimate &o Median unbiased estimate i&, Lower confidence limit Upper confidence limit
1 4.84 4.55 3.25 6.13
2 2.81 2.65 1.89 3.56
3 3.80 3.57 2.55 4.82
~ E
4 4.87 4.58 3.27 6.18
~~
~ Z C
5 6.34 5.97 4.25 8.04
From Table 20.7 the 90th percentile of tl(lO, 10, 5 ) is 1.26. The smallest ratio of raw fo.lo values that will be significant, denotedSSR (shortest significant ratio) is defined implicitlyby 1.26 =
p1 In (SSR) = 2.480 In (SSR).
Solving for SSR gives
SSR
=
(i::)
exp -
=
1.66.
Thus groups for which the ratio of raw ML estimates of xo.lo exceeds 1.66 may be declared different. This would include Groups 5 and 3, because their xo,lo estimates are in the ratio 6.341330 = 1.67 > 1.66. Note that the confidence limits on xo.lo overlap for these two groups. In general, if the confidence limits do not overlap, the groups differ. However the groups may differas in thepresent case eventhough the confidence limits do overlap.
Chapter 19 explored the reasons for, and concepts and methods of, endurance testing of ball and roller bearings and components. In thischapter the means to relate such test results to applications of standard and special bearing products were covered. The applicability of the Weibull distribution to such test datawas demonstrated. It is further shown that, having specified a Weibull population, for example, by a catalog calculation of theoretical life, it is possible to calculate other characteristics that may occasionally beof interest, such as the mean time between failures. he methodology for forminga graphical estimate of a Weibull popun using a censored or an uncensored data sample was describedand
s were given forusing the method of maximum li~elihoodfor d interval estimation of the Weibull parameters from either red or sudden death test samples. ally, the procedure was given for analyzing ibull samples in sets. eters because it exes more precise estimated p tracts in~ormationfrom the entire set of data.
EE
20.1. G. Lundberg and A. Palmgren, “Dynamic Capacity of Rolling Bearings,” Acta Polytech. Mech. Eng. Ser. 1, R.S.A.E.E., No. 3, 7 (1947). 20.2. L. Johnson, Theory and Technique of Variation Research, Elsevier, New York (1970). 20.3. W. Nelson, “Theory and Application Hazard Plotting for Censored Failure Data,” ameter for Maximum Likelihood Estimates,” ~ E E E D a n 20.5. J. McCool, “Analysis of Sudden Death Tests of Bearing E n d u r a n c e , ” A S Dans. ~~ 1 8-13 (1974). 20.6. J. McCool, “Censored Sample Size Selection for LifeTests,”Proceedings 1973 ~ n n u a l ~ e l i a b i ~ iand t y ~ a ~ n t a i n a b i l i t y S y m ~ o s IEEE i u m , Cat. No. 73CHO714-64 (1973). 20.7. J. McCool, “Analysis of Sets of Two-Parameter Weibull Data Arising in Rolling Contact Endurance Testing,” in Rolling Contact Fatigue Testing of Bearing Steels,ASTM STP 771, ed. J. J. C. Hoo, American Society forTesting and Materials, ~hiladelphia, 293-319 (1982).
This Page Intentionally Left Blank
Symbol
~escription
Units
asic static load rating all or roller diameter
rlD Factor of safety 'ckers hardness umber of rows
oove curvature radius ller contour radius adial load factor of rolling elements per row
mm (in.) mm (in.) I\J (lb) mm (in.) mm (in.)
Symbol a Y 68 rl P 0
v;$
a i iP 0
r S
Units Contact angle cos ald, ermanent deformation ardness reduction factor Curvature Yield or limit stress Load rating factor
0
mm (in.) mm-l (in.-l) /mm2 (psi)
efers to inner raceway efers to incipient plastic flow of‘ material efers to outer raceway efers to radial direction efers to static loading
any structural materials exhibit a strain limit under loadbeyond which full recovery of the original elemental dimensions is not possible when the load is removed. Bearing steel loaded in compression behaves in a similar manner. Thus when a loaded ball is pressed on a bearin raceway, an indentation may remain in the raceway and the ball may exhibit a 66flat?7 spot after load is removed. These ~ermanentdeformations, if they are sufficiently large, can cause excessive vibration and possibly stress concentrations of considerable magnitude.
In practice, permanent deformations of small magnitude occur even under light loads. Figure 21.1 taken from reference [21.4] shows a very large ma~ificationof the contacting rolling element surfaces in a typical ball bearing both in thedirection of rolling motionand tr~nsverseto that ~irection. Figure 21.2, also from reference [21.4], shows an isometric view of a ground surface having spatial properties similar to hone raceway surfaces. Noting the occurrence of “peaks and valleys” evenwith a finely finished surface, it is apparent that prior to distributing a load between rolling element and raceway over the entire contact area thus giving an average compressive stress o- = the load i ibu , giving a lar only over the smaller area of contacting
C ~ C ~ T I O OFNPE
N'I' D E F O R ~ T I O N
"E----2b-
Raceway
21.1. Ball and raceway contacting surfaces (greatly magnified).
stress than cr. Thus, it is probable that the compressive yieldstrength is exceeded locally and both surfaces are somewhat flattened and polished in operation. According to Palmgren [21.1] this flattening has little effect on bearing operation because of the extremely small magnitude of deformation. It may be detected by a slight change in reflection of light from the surface. It was shownin Chapter 6 that therelative approach of two solidsteel bodies loaded elastically in point contact is given by 6 = 2.79 X 10-4 &*
PSf3
(6.43)
in which 6" is a constant depending on the shapes of the contacting surfaces. As the load between the surfaces is increased, deformation gradually departs from that depicted by equation (6.43) and becomes larger for any given load (see Fig. 21.3). The point of departure is the bulk compressive yield strength. Based on empirical data for bearing uality steel hardened between63.5 and 65.5Rockwell 6, 1.11 developed the following formula to describe permanent deformation in point contact:
in which p i s is the curvature of body I in plane 1, and so on. raceway contact, equation (21.1) is
vs load in point contact.
in which the upper signs refer to the innerraceway contacta signs refer to the outer raceway contact. r roller-raceway the following e~uationobtains:
is the roller contour radius and r is the groove ra rmulas are valid for pe anent deformation in the vicinity of the compressive elastic limit (yie oink) of the steel, ial ball bearing of mate the maximum permanent deformation at the inner r~ceway. omp pare this value to the maximum elastic deformation. =
12.7 mm (0.5 in,)
fi = 0.52 4536 N (1019 lb)
x. 7.1
amax= 0.0604 mm (0.00238 in.)
x. 7.1
rnax
=
P, = 0.0150 mm (0.0006 in.) 2.5
x. y
=
0.1954
6,
=
5.25 X 10-7 D3 Q2 [l
+
~]
(1 -
$)
) ('
1001954 0.1954 =
2.521 X
mm (9.93 X
(21.2)
-
in.)
On the other hand the elastic deformation ai, at t,b
=
2 X
=
0" is
0.0150 0.0604 - -= 0.0454 mm (0.00179 in.) 2
Thus isi0 >> 6,. For line contact between roller and raceway, the following formula may be used to ascertain permanent deformation with the same restrictions as earlier: 6, =
(21.4)
According to Lundberg et al. E21.61, the deformation predicted by equation (21.4) occursat the end(s) of a line contact whenthe raceway length tends to exceed the roller effective length. The corresponding deformation in the center of the contact is 6J6.2 accordingto reference 121.61.Palmgren [21,1] stated that of the total permanent deformation, ap~roximatelytwo-thirds occur in the ring and one-third in therolling element. Palmgren's data were based on indentation tests carried out in the 194Os, and the data were dependent on the measurement devices available then. Later, some of these testswere repeated using modern measurement devices. The following conclusions were reached: The amount of total permanent indentation occurrin~due to an between a ball and a raceway appears to be less than that given by equation (21.1). The amount of permanent deformation that occurs in the ball surface is virtually equal to that occurring in the raceway, whenballs have not been work hardened.
~TATICL
~ccordingly,it can be stated that permanent deformations calculated according to equations (21.1-2 1.4) will tend to be greater than will actually occur in modern ball and roller bearings of good quality steel and with relatively smooth surf'ace finishes.
As indicated earlier, some degree of permanent deformation is unavoidable in loaded rolling bearings. Moreover, experience has demonstrated that rolling bearings do not generally fracture under normal operating loads. Experience further has shown that permanent deformations have little effect on the operation of the bearing if the magnitude at any given contact point is limited to a maximum of 0 . 0 0 0 ~If. the de~ormations become much larger, the cavities formedin the raceways cause the bearing to vibrate and become noisier although bearing friction does not appear to increase significantly. earing operation is usually not impaired in any other manner; howeve indentations together with conditions of marginal lubrication can lead to surface-initiated fatigue. The basic static load rating of a rolling bearing is defined as thatload applied to a nonrotating bearing that will result in permanent deformation of 0.0001D at theweaker of the inner or outer raceway contacts occurring at the position of the maximum loaded rolli other words in equations (21.2)-(21.4), S,/D = 0,0001 at concept of an allowable amount of permanent deform with smooth minimal vibration and noise operat' ~ t i n u e to s be the basis of the IS0 standard [21. 1.2, 21.31. In the latest revision of the IS0 sta that contact stresses at the center of contact at the maximum loaded rolling elements as shown in Table 21.1 yield permanent deformations of 0.OOOlD for the bearing types indicated. The ANSI standards [21.2, use the same criteria. r most radial ball bearing and roller bearing applications the maximum loaded rolling element load according to Chapter 7 may be approximated by
.
Contact Stress That Causes 0 . 0 0 0 Permanent ~ ~eformation Contact Stress
pe
Bearing ~~~
Self-aligning ball bearing Other ball bearings Roller bearings
4600 4200
4000
(66~7000) (609,000) (5807000)
8
P
E
~
~ DEFO N T
ION ."I)B~~~
STATIC G
(7.24) in which i is the number of rows of rolling elements. Setting Fr = 6, = yields
C,
=
0.2iZQm,, cos
a
(21.5)
Considering the stress criterion, equations (6.25), (6.34),and (6.36) may be used to determine Qmaxcorresponding to 4200 N/mm2 (609,000 psi) for standard radial ball bearings. Substituting for Qmaxin equation (21.5) yields the equation cos a c, = 23.8iZD2(a~b~)~
(21.6)
if the maximum stress occurs at the inner raceway and
6, =
23.8iZD2(az bzI3 cos a
(21.7)
if the maximum stress occurs at the outer raceway. Reference E21.51 reduces these equations to
C,
=
p,iZD2 cos a
(21.8)
where values of p, are given in Table 21.2 for standard ball bearings. The corresponding formula for radial roller bearings as taken from reference E2 1.51 is
C,
=
44(1 - y)iZZD cos a
(21.9)
For thrust bearings, (7.26) Setting F, = C,, yields
C,,
=
iZQm,, sin a
~orrespondingly, the standard stress criterion formula is
(21.10)
1.2. Values of
for Ball Bearings"
Radial and Radial Angular-Contact Groove Type
Self-Aligning
Thrust
Metricb 14.7
Inch" 2 120
Metricb
Inch"
Metricb
Inch"
0.00
1.9
284
61.6
8950
0.01 0.02 0.03 0.04 0.05
14.9 15.1 15.3 15.5 15.7
2180 2220 2270 2300 2350
2.0 2.0 2.1 2.1 2.1
290 297 301 307 313
60.8 59.9 59.1 58.3 57.5
8820 8680 8540 8430 8320
0.06 0.07 0.08 0.09 0.10
15.9 16.0 16.2 16.4 16.4
2400 2430 2480 2440 2410
2.2 2.2 2.3 2.3 2.4
319 325 332 338 344
56.7 55.9 55.1 54.3 53.5
8210 8100 7990 7870 7790
0.11 0.12 0.13 0.14 0.15
16.1 15.9 15.6 15.4 15.2
2370 2340 2290 2260 2220
2.4 2.4 2.5 2.5 2.6
351 357 363 370 376
52.7 51.9 51.2 50.4 49.0
7710 7630 7500 7390 7270
0.16 0.17 0.18 0.19 0.20
14.9 14.7 14.4 14.2 14.0
2190 2140 2110 2070 2040
2.6 2.7 2.7 2.8 2.8
382 389 397 403 409
48.8 48.0 47.3 46.5 45.7
7150 7030 6910 6780 6670
0.21 0.22 0.23 0.24 0.25
13.7 13.5 13.2 13.0 12.8
2000 1960 1920 1890 1850
2.8 2.9 2.9 3.0 3.0
417 423 430 438 446
44.9 44.2 43.5 42.7 41.9
6540 6420 6300 6200 6110
0.26 0.27 0.28 0.29 0.30
12.5 12.3 12.1 11.8 11.6
1820 1780 1750 1730 1690
3.1 3.1 3.2 3.2 3.3
452 459 467 473 481
41.2 40.5 39.7 39.0 38.2
6010 5880 5760 5660 5570
0.31 0.32 0.33
11.4 11.2 10.9
1670 1630 1600
3.3 3.4 3.4
488 496 503
37.5 36.8 36.0
5490 5370 5244
Y
. ~Continued~ Radial and Radial
~~lar-À on tact
Self-Aligning Type Y
10.7
10.3 10.0 8 6 4
Groove
Thrust
Inchc Metricb Metricb Inchc Metricb Inchc
0.34 0.35 0.36 0.37 0.38 0.39 0.40
3.5 10.5 3.5 3.6 3.6 3.7 3.8 3.8
1560 15305040 1490 1460 1440 1400 1370
34.6
511 519 526 534 541 549 558
35.3
5120
aBased on modulus of elasticity = 2.07 X lo5 N/mm2 (30 X lo6 psi), Poisson’s ratio = 0.3. bUse to obtain C, in Newtons when I) is given in millimeters, ‘Use to obtain C, in. pounds when I) is given in inches.
C,,
= p,ZD2 sin a
(21.11)
where 9,is given by Table 21.2. For thrust roller bearings with line contact,
C,,
=
220(1 - y)ZZD sin a
(21.12)
en hardness of the surfaces is less than specified lower limit of validity, a correction factor may be applied directly to the basic static capacity such that (21.13) in which (21.14)
is the Vickers hardness. A graph of Vickers hardness versus 1 C hardness is shown in Fig. 21.4. Equation (21.14) was developed e ~ p i r i c a ~by lyS The values of q1 depend on type of contact and are given by Table 21.3. ys has a maximum value of unity.
To compare the load on a nonrotating bearing with the basic static capacity, it is necessary to determine the staticequivalent load, that is, the
Rockwell C hardness 21.4. Vickers hardnessvs Rockwell C hardness.
1.3. Values of q1 rll
1 1.5 2 2.5
Type of Contact Ball on plane (self-aligning ball bearings) Ball groove on Roller on roller (radial roller bearings) Roller on plane
pure radial or pure thrust load-whichever is appropriate-that would cause the same total permanent deformation at the most heavily loaded contact as the applied combined load. A theoretical calculation of this load may be made in accordance with methods of Chapter 7. In lieu of the more rigorous approach, for bearin~ssubjected to combined radial and thrust loads the static equivalent load may be calculated as follows:
If Fr is greater than F8 as calculated in equation (21.15), use F, equal to Fr. Table 21.4 taken from reference E21.23 gives values of X8 and Y, for radial ball bearings.
PIE
.
Values o f X s an
Radial contact groove ball bearing ~ ~ l a ~ - c o n tgroove act ball bearings a = 15" a = 25" a = 30" a = 35" a = 40" Self-align in^ ball bearings
0.6
0.5
0.5 0.5 0.5 0.5 0.5 0.5 0.5
0.47 0.42 0.38 0.33 0.29 0.26 0.22 ctn a
0.6
0.5
0.94 0.84 0.76 0.66 0.58 0.52 0.44 ctn a
"Pois always 2 Fr. &Valuesof Yofor intermediate contact angles are obtained by linear interpolation. "Permissible maximum value of F,ICo depends on the bearing design (groove depth and internal clearance).
~ e r t a i nto bearings havin
or thrust bearings the static e~uivalentload is given by
and Ysfor Radial Roller gle-Row aringsb
rr, a elf-ali~ingtapered and roller bearings a .f 0"
0.22 0.5
YS ctn a
1
0.44 ctn a
"The ability of radial roller bearings with a = 0" to support axial loads varies considerably with bearing design and execution. The bearing user should therefore consult the bearing manufacturer for. recommendations regarding the evaluation of equivalent load in cases where bearings with a = 0" are subjected to axial load. &Fais always 2 F,.The elastic equivalent radial load for radial roller bearings with a = 0", and subjected to radial load only is Fs = F,.
(21.16)
ctn a, the accuracy of equation ( iminishes and the theoretical approach according to Chapter 7 is warranted.
that the load which will fracture bearing rolling elements or raceways is greater than 8 6 , (see reference [
ting bearing may d this load acts c revolutions of bearing rotation, In thismanner the ons that occur are uniformly istributed over the ents, and the bearing ret ins satisfactory oper hand the load is of short duration, unevenly elop even whenthe bearing is rotating his situation, it is necessary to use a ating exceeds the m ~ x i ~ uam of longer duration, the basic static load t impairing operation of the bearing, of bearing service a factor of safety ad rating. Therefore the allowable 1
(21.17)
able 21.6 gives satisfactory values of FS for various types of service.
.
Factor of Safety for Static Loading ~
~~~
Factor Service of Safety
FS FS FS
2 2 2
0.5 1 2
Smooth shock-free operation Ordinary service Sudden shocks and high requirements for smooth running
1.2. The 218 angular-contact ball bearing is subjected to simultaneously applied radial and thrust loads of 17,800 N (4000 lb) each. Using the IS0 standard, estimate the factor of safety for the basic static load rating of the bearing. f = 0.52
Ex. 2.3
Z
Ex. 7.5
=
16
1) = 22.23 mm (0.875 in.)
a
==
Ex. 2.3 Ex. 2.3
40"
dm= 125.3 mm (4.932 in.) 1) cos y="---
Ex. 2.6
a!
Ex. 2.6
dm
From Table 21.2 at y = 0.1358,
C,
=
15.48
X
= 93,760
(PB =
15.48
1 X 16 X (22.23)2 COS (40")
N (21,070 lb)
From Table 2 1.4,
X,
=
0.5,
Fa = X,Fr
Y
=
0.26
+ YsFa
= 0.5 X 17,800
(21.15)
+ 0.26
X
17,800 = 13,530
N (3040 lb)
Therefore, use F8 = 17,800 N (4000 lb)
F,
CS -
FS
Smoothness of operation is an important consideration for modern ball and roller bearings. I n t e ~ p t i o n in s the rolling path such as caused by
permanent deformations result in increased friction, noise, and vibration. Chapter 25 discusses the noise and vibration phenomenon in substantial detail. In this chapter, the discussion centered on bearing static load ratings, which, if not exceeded whilethe bearing was not rotating, would preclude permanent deformations of significant magnitude. The ratings were based on a maximum allowable permanent deformation of 0.0001~.~ubsequently,it was determined that for various types of ball and roller bearings, this deformation could berelated to a value of rolling element-raceway contact stress. In accordance with this stress, basic static load ratings are developed for each rolling bearing type and size. Generally, a load of magnitude equal to the basic static load rating cannot be continuously applied to the bearing with the expectation of obtaining satisfactory endurance characteristics. Rather, the basic static load rating is based on a sudden overload or, at most, one of short duration compared to the normal loading during continuous operation. Exceptions to this rule are bearings that undergo infrequent operation of short duration, for example, bearings on doors of missile silos or dam gate bearings. For these and simpler applications, bearing design may be based on basic static load rating rather thanon endurance of fatigue. In Chapter 22, the concept of a shakedown limit stress is discussed, which pertains to raceway subsurface microstructural alterations that occur during rotation while the current static load ratings are based upon damage ~ u r i n g nonrotation. Because of relatively slow speeds of rotation and infrequent operation, neither vibration nor surface fatigue may be as significant in such applications as excessive plastic flow of subsurface material. The bearings could thus be sized to eliminate or mi~imizesuch plastic flow and ultimately bearing failure.
21.1. A. Palmgren, Ball and Roller Bearing Engineering, 3rd ed., Burbank, Philadelphia (1959). 21.2. American National Standard, ANSI I ~ B Std~ 9-1990, A “Load Ratings and Fatigue Life for Ball Bearings.” ~ 11-1990, A “Load Ratings and Fa21.3. American National Standard, ANSI I ~ B Std tigue Life for Roller Bearings.” 21.4. R. S. Sayles and S, Y. Poon, “Surface Topography and Rolling Element Vibration,” Precis. Eng. 137-144 (1981). 21.5. International Standard IS0 76, “RollingBearings-Static Load Ratings” (1989). 21.6. G. Lundberg, A. Palrngren, and E. Bratt, “Statiska Barform&ganhos Kullager och Rullager,” ~ u ~ l a g e r t i d ~ i n3g e(1943). n
This Page Intentionally Left Blank
Symbol 0
ICI d h
Units Angle of incidence of x-rays to diffracting planes Angle between specimen normal and normal to diffracting planes Spacing of crystallo~aphicplanes -ray wavelength
O
0
A A
Much work has been devoted to characterizing alterations of the microstructure, hardness, and residual stress distribution occurring in rolling bearing components during bearing operation. The results of this work provide a technological foundation that serves as a basis for continuing analytical and experimental investigations into the fundamental mechanisms involved in bearing failure. 835
~aterials-orientedinvestigations have been primarily phenomenological in nature; that is, attempts have been made to carefully controltest conditions and to correlate material response characteristics with test conditions and bearing performance. ~ u b s e ~ u eanalyses nt of the metallurgical mechanisms involved are prerequisites for establishing reliable analytical models for pre~ictingperformance. This chapter reviews the microstructural alterations resulting from rolling contact and the attendant changes in residual stress state. Observed effects of residual and applied bulk stresses, combined with rolling contact stress, on bearing performance are discussed in terms of fatigue life, failure mode, and dimensional stability.
The microstructure of martensitically hardened and tempered 52100 steel is shown in Fig. 22.1. The microstructure consists primarily of plate martensite E22.11, with 5-8 vol.% of (Fe,Cr)& type carbides E22.21, and up to 20 vol.%retained austenite, depending on austenitizing and tempering conditions. Temperedhardness is generally Rc 58-64. The lower values of retained austenite content and hardness are characteristically associated with higher tempering temperatures.
22.1. Microstructure of hardened and tempered AIS1 52100 steel.
C R O ~ ~ R ~~T C E ~TT ~I O N D Sm TO ROLLING CONTACT
37
The hardened and tempered microstructures of the near-surface (case) and subsurface (core)regions of carburized bearing steel (e.g., AIS1 8620), are shown in Fig. 22.2. The case microstructure is predominantly ma~ensitic(plate type). The volume fractions of carbide and retained austenite vary widely, depending on the carburizing conditions, steel analysis, and t e m ~ e r i ~ procedures g used. The microstructure of the core region-that is, the subsurface material unaffected by carburizing-is also primarily martensitic, but, due to the low carbon content (gradually 0.2 wt.% or less), is of the lath type rather than plate martensite. ~haracterizationof these two ma~ensite morphologies is detailed in i22.11 and i22.31. With the relatively rare exceptions of single-stress-cycle bearing failure (e.g., fracture from heavy impact loading and wear), all materialrelated bearing failures involveaccumulation of irreversible plastic deformation during cyclic stressing. Theilatter is the classical definition of metallurgical fatigue. The variety of bearing failure manifestations arises from the conditions that promote fatigue initiations and the manner in which the operating conditions sustain propagation of the fatigue mechanisms. Fatigue failure may be initiated by an exogenous inclusion in the bearing steel, by mechanical disruption of the integrity of the rolling contact surface (e.g., debris dents or scratches), or by a corrosion pit, to name just a few. The subsequent bearing failure, which by definition precludes continued functionality, results from progression of fatigue damage from the initiating cause. ~pecificationsfor microstructural characteristics, hardness, depth of hardening, amount of retained austenite, and so forth, to a large extent have been based upon bearing performance experiences in the field and the results of bearing endurance tests conducted under controlled operating conditions. Several decades of collective experience provide today’s phenomenologically based compendiumof materials knowledge concerning re~uirementsfor and responses to rolling contact stressing. Using this experience to provide bearing steel quality heat treatment, manufacturing procedures, and operating conditions that minimize the likelihoodof experienced early-failure modes has permitted the study of material response to long-term rolling contact stressing. The objectives of such studies have been oriented toward gaining insight into the fundamental mechanisms involved in rolling contact fatigue and to assess fatigue life potential under “ideal” operating conditions. Alterations of the microstructure that have been observed under these conditions are described in the next section.
Marked alteration of the near-surface microstructure of endurancetested bearing inner rings has been reported in the literaturesince 1946
22.2. Microstructure of carburized, hardened, and tempered AISI 8620 steel, (a)Case. ( b ) Core.
OSTRUCT
[22.4-22.131. The alterations are principally characterized by differences in the etching response of the microstructure in the region just beneath the raceway surface (see Fig. 22.3), and are most heavily concentrated at a depth corresponding to the maximum shear stress associated with the Hertzian stress field in the contact E22.5, 22.63. Three aspects of microstructural alterations have beendescribed ronologically characterized 1122.81: the dark etching + 30" bands, and DER + 30" bands + 80" bands. The structural changes obtained as a function of stress level and number of inner ring revolutions are diagrammed in Fig. 22.4. Opticalmicro~aphs of the structural alterations, in parallel sections.,are shown in Fig. 22.5. The first alteration is the formation of the DER. Transmission electron microscopy identified the DER as consisting of a ferritic phase., containing an inhomo~eneously distributed excess carboncontent (equivalent to that of the initial martensite)mixed with residual parent martensite. A stress-induced process of martensitic decay is indicated E22.81. The second manifestation of altered microstructure 1122.8, 22.91 is the formation of white etching, disc-shape regionsof ferrite, about 0.1-0.5 pm (40-200 pin.) thick, and inclined at anangle of approximately 30" to the raceway circumferen~ial tangent. These regions are sandwiched between carbide-
.3. Orientation of viewing sections and location of region of ~icrostructural alterations in a 309 deep-groove ball bearinginner ring (from L22.81).
NO S l f f ~ ~ U ~ A L C ~ ~ A N ~ E
0.E.R . 4 30' D.E.R.
BANDS
D.E.R.+3O0+8d BANDS
~Evolullo~s 22.4. Observed microstructural alterations as afunction of stress level and number of inner ring revolutions (&om E22.81).
rich layers. The third feature, initially reported in 122.71, is a second set of white etching bands, considerably larger than the 30" bands and inclined at 80" to the raceway tangent in parallel sections. These discshaped regions are abut 10 pm (0.0004 in.) thick and consist of severely plastically deformed ferrite E22.81. As shown in Fig. 22.5, both the 30" and 80" bands incline toward the surface in the direction of rolling element motion. Reversingthe direction of rotation reverses the orientation of the bands. The characteristic angular orientations of these white etching bands have not been satisfactorily explained. ~ r y s t a l l o ~ a p htexturing ic in thenear-surface region of ball bearing inner rings after high-stress operation has, however, been observed by Voskamp 122.61 Angular characteristics of the texturing are reported to be consistent with white etching band orientations, suggesting that the bands have a crystallo~aphicallydetermined nature. Moreover, the crystal alignment appears to accentuate the plane of' least resistance to fracture; that is, the cube planes of body-centeredcubic (bcc) Fe become aligned parallel to the raceway surface with the so-called [I101 direction parallel to the direction of rolling. Hardness has been reported to increase slightly in the early stages of testing and thento decrease markedly in the region associatedwith microstructural alteration 122.8,22.131. The number of stress cycles required to produce the sequence of events leading to microstructural alteration may be signi~cantlyreduced by increasing the temperatureof the test specimen [22.13]. Localized changes in retained austenite con-
UE TO R O L L ~ GC O ~ ~ C T
2.5. Optical micrographsof structural changes in 309 deep-grooveball bearing inner rings (parallel section). (a) DER in early stage. ( b ) Fully developed DER and 30" bands. (c) DER, 30" bands, and 80" bands. AIS1 52100 steel (fiom i22.81).
tent and residual stress level are also associated with these microstructural alterations. See next section. Another manifestation of microstructural alteration found in bearing rolling contact components that have experienced substantially heavy ~e loading is commonly calleda "butterfly," becauseof w i n ~ - l i emanations from a "body" composed of a non~etallicinclusion. An example of a butterfly is shown by ig. 22.6. ~haracteristica~ly, as shown in 122.14and illustrated by Fig. 22.7, after natal acid etching the butterfly wings appear white in contrast to the surrounding matrix of martensite and are oriented at an angle of 40-45" to the raceway track in a direction determined by the direction of the friction force acting on the surface. They can occur at depths well below the depth of the m ~ i m u mshear stress and are characte~isticallyassociated with microcrac~srunning alon the smooth edges of the wings. s form around oxide, silicate, and tita~iumnitride pa njun~tionwith manganese sulfide or carbide particles
~
T
E RESPONSE R ~ TO R
O
L C ~O ~ ~~ C T
.
Optical micrographof white etching regions associated with a subsurface, nonmetallic inclusion.
FIG
.7. Subsurface"butterfly" in M50 steel ring, shown at 45" orientation to the rolling contact surface. The surface friction force is directed to the left. From Nklias[22.171.
22.16,22.18]. Wing developmentdepends on stress level an of stress cycles [22.16].~ u t t e ~ igenerally es form only under very heavy loading. A comprehensive characterization of the microstructural features of butterfly wings [22.19] concluded that they consist of a dispersion of ultra-fine-~ainedferrite and carbide, very similar in nature and for~ationmode to the 30" and 80" white etching bands described earlier. Further, the wings are probably initiated by preexisting cracks associated with non~etallicinclusion bodies.[2~.14-22.16,22.191. ~ubsequent crack and wing growth proceed together. ite etching bands and butterflies are striking manifestatio~sof high stress, high cycle, rolling contact.M i l e it has been difficult to positively identify them as failure-initiating characteristics, Nhlias [22.17] has indicated that fatigue failure does not seemto occur in their absence.
3
Therefore, the stress below which they do not occur after a substantial number of cycles might be identifiedas thelower value of a fatigue limit stress. See Chapter 23.
esi.
tresses
Residual stress is that stress which remains in a material when all externally applied forces are removed. Residual stresses arise inan object from any process that produces a nonunifo~mchange in shapeor volume. These stresses may be induced mechanically, thermally, chemically, or by combinations of these processes [22.20]. If a relatively thin sheet of malleable material, such as copper, is repeatedly struck with a hammer, the thickness of the sheet is reduced, and the length and width are corresponding increased, preserving constancy of volume. Ifthe same number of equally intensive hammer blows were uniformly deliveredto the surface of a copper block several inches thick, the depth of penetration of plastic deformation would berelatively shallow with respect to the block thickness. The deformed surface layer would be restrained from lateral expansion by the bulk of subsurface material, which experienced less defor~ation,Consequently, the heavily deformedsurface material would be like an elastically compressed spring, prevented from expanding to its unloaded dimensionsby its association with elastically extended subsurface material. The resulting residual stress profile is one in which the surface region is in residual compression and the subsurface region is in a balancing residual tension. This example is a literal description of the shot-peening process, wherein a surface is bombarded with pellets of steel or glass. A highly desirable residual-stress pattern is established for componentsthat experience high?cyclic tensile stresses at thesurface during service. The magnitude of tensile stress experienced by the component during service is functionally reduced by the amount of residual compressive stress, thereby providingsignificantly increased fatigue liyes for parts such as springs and shafts. The shot-peening example illustrates the essential characteristics of a surface in which residual stress has been induced:
. Nonuniformity of plastic deformation-that .
is, being near-surface only-encourages the surface material to expand laterally. Subsurface material, which experiencedless plastic deformation, is elastically strained (in tension, in this example) as it restrains expansion of the surface material, thereby inducing residual compressive stress in thesurface region.
0
resulting state of residual stress is a reflection of the elastic ponents of strain in thesurface and subsurface r are in equilibrium, providing a balanced tensile-co tem.
t treatment, such as is used for hardening rolling bearing compoexert very s i ~ i f i c a n tinfluence over the s ending on the steel analysis, austeniti~i quen~hingseverity, component geometry, sectionthic heat treatme~tcan provide either residual compress tensile stress in the surface of the hardened eomp Temperature gradients are established from the surface to the center of a part during quenching in a hardening treatment. fferential thermal contractionassociated with these gradients provi s fornonuniform voluplastic deformation, giving rise to residual stresses. ~ddit~onally, metric changes associated with the phase transfor~ationstaking place during heat treatmentof steel occur at different times during quenching at the partsurface and interior due to the thermalgradients established. These sequential volumetric changes, combined with differential thermal contractions, are responsible for the residual-stress state in a hardened steel component. The sequenceand relative m a ~ i t u d e of s these contributing factors determine the stressm a ~ i t u d and e whether the surface is in residual tension or compression. Grinding of a hardened steel component to finished dimensions also residual surface stress. Generally, neglecting the effects of nding practices that generate excessive heat and produce mie alterations, it is found that the residual-stress effects associated with grinding are confined to material within the first 50 microns (0.002 in.) of the surface. Good grinding practice, as applied to bearing rings, produces circumferential residual compressive stress ina shallow surface layer. Grinding also involves someplastic deformation of the surface, producing residual compression as described earlier. The residual-stress state ina finished bearing ring is therefore a function of heat treatment and grinding. If properly ground, the residual stress in a throu~h-hardenedbearing ring will be 0 to slightly compressive. Thesubsurface residual stress conditions willbe determined by the prior heat treatment.
*
are welldeveloped
determine residual stress in ay diffractionequipment and and described in theliterature [22.19, e solids,consist of atoms arra rms of interplanar distances. In
~
S STRESSES ~ IN ~ OUL L I N GBEARING ~ ~
0
~
0
~
~
4s s
crystal of a metal the orientation of these planes of atoms is consistent everywhere within the crystal. Most metallic objects of interest here are not single crystals but polycrystalline; that is, they consist of many crystals. Each crystal or grain in the microstructure of a polycrystalline metal is delineated from its neighbors by mismatch in theorientation of the crystallographic planes. The region of mismatch or disorder between neighboring grains is called the grain boundary. In the unstressed condition the distances between crystallographic planes assume equilibrium values. If elastically stressed in tension (Le., a tensile stress component perpendicular to the planes), the interplanar distance is increased. In compression the distance is decreased. Consequently, if the equilibrium interplanar distance, the stressed interplanar distance, and theorientation of the planes with respect to the stressaxis are known, the elastic strain conditions are defined. ~ u l t i p l ~ nthe g strain by the elastic modulus forthe material being studied provides the value of residual stress. X-ray diffraction is used to measure interplanar distances. The technique is therefore used to measure elastic strain, from which the associated residual stress is calculated. The relationship stating the conditions that must be met for x-ray diffraction to occur was first formulated by Bragg [22.24] in 1912 and is known as Bragg’s law: A = 2d sin 0
(22.1)
where h is the wavelength of the x-rays used, d is the interplanar spacing, and 0 is the angle of incidence of the x-ray beam to the diffracting planes. What Bragg’s law states is that for a given x-ray wavelength h and interplanar spacing d, there is an angle of incidence 0 such that the x-rays penetrating the specimen surface will experience constructive interference and emerge from the surface at an angle 0 to the planes of spacing d. With appropriate detection equipment the emergent (diffracted) x-rays can be detected, and the precise diffraction angle can be determined. The orientations of a polycrystalline specimenand a particular family of crystallo~aphicsplanes (in randomly oriented grains) to the x-ray beam in residual-stress determination are shown in Fig. 22.8. Initially, the specimen is oriented such that the normal to the specimen surface (A) and the normal to the diffracting planes ( B )are coincident (Fig. 22.8a); that is, the diffracting planes are parallel to the specimen surface. The diffraction angle,0 is determine^ for these planes. The specimen is then rotated to an orientation shown in Fig. 22.86. In this orientation the normal to the diffracting planes makes an angle with the normal to the specimen surface. These planes, being at a nonzero angle to residual stress acting in the direction parallel to the specimen surface, are elastically separated from their equilibrium spacesidual compressive stress gives a smaller value of d. According
+
* =
Orientations of specimen surface, diffracting planes, and x-ray beam for residual-stress determination via x-ray diffraction. (.
SSES IN ROLLZNG BEARING C O M P O ~ ~ T S
847
to equation (22.I), for fixed h a decrease in d requires an increase in the value of sin 0-that is, a larger diffraction angle 0. Conversely, residual tensile stress gives a larger value of d , corresponding to a smaller 0. Therefore, comparing the 0 values obtained with the two specimen orientations-that is, Fig. 22.8a,b-will show an increase or decrease, indicating residual compression or residual tension, respectively.The magnitude of the change in 0 is related to the magnitude of residual stress by a calculated stress factor [22.20, 22.22, 22.231. Values of residual stress as functions of depth below the surface (i.e., residual stress profiles) are obtained by successive material removal and x-ray residual-stress determinations. Material removal is most appropriately performed by electrochemicalmeans. Mechanical removalof material would introduce alterations to the residual stress profile being measured, and therefore may not be employed. Fixed installation x-ray equipment is most often used to obtain the residual stress measurements. The measurements may be obtained using the Ruud-~arrett position-sensitive scintillation detector [22.251. This detector operates by converting incident x-rays into light by means of a cadmium-zinc sulfide scintillation coating on one end of a pair of coherent, flexible fiber optic bundles. These transport thelight to an electronics package, which amplifies the intensity of the light and converts it into an electrical signal. This is shown sche~aticallyby Fig. 22.9. The device is interfaced with a computer, whichstores and processes the large amount of data generated. The software package incorporated in the computer uses a number of algorithms to correct electronicand mechan-
Liquid Cooled Heat Sink
2.9. Schematic illustration of the position-sensiti~escintillation detectorelectronics package.
848
MATERIAL RESPONSECONTACT TO ROLLING
ical hardware inconsistencies as well as x-ray focusing errors. Specific data-fitting algorithms are applied for x-ray background and peak location. Accurate residual stress measurements using the device can be obtained to a depth of approximately 0.0016 mm (0.0006 in.) below the surface of most metals. Therefore, successive electrochemical removalof surface layers is required to obtain residual stress profiles versus depth.
Iling Contact Associated with the microstructural alterations resulting from rolling contact stressing, significant changes in residual stress andretained austenite contact have been reported [22.6,22.10,22.11,22.13,22.26,22.27].
n
N I
Depth (mm)
22.10, Residual stress and percent retained austenite decomposition vs depth below raceway surface for various numbers of inner ring revolutions. Bearing: 6309 deepgroove ball bearing. Material: AIS1 52100, heat-treated hardness of R , 64. ( a )Maximum contact stress: 3280 N/mm2; depth of maximum orthogonal shear stress: 0.19 rnrn; depth e ~ next page). of maximum unidirectional shear stress: 0.30 mm ( e o n t i n ~on
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Depth (mm)
(b)
22.10. ( b )Maximum contact stress: 3720 N/mm2; depth of m a ~ m u m orthogonal shear stress: 0.21 mm; depth of maximum unidirectio~al shear stress: 0.33 mm (from 122.263).
The forms of the changes in tangentialresidual stress andretained austenite content profiles are illustrated in Fig. 22.10. The low and high loadvalues correspond to the two levelsof ma~irnum contact stress indicated in Fig. 22.4,depicting the number of revolutions of which the various mic~ostructural alterationsoccur. omp par is on of Figs. 22.4 and 22.10 indicates that si~ificant changes in residual-stress profile and retained austenite content precede any observable alterations in microstructure, The residual-stress data of Fig. 22.10 show peak Values at increasing depths co~espondingto increasing numbers of stress €or d ~ e ~ o r n ~ o s of i t~ i o~~t a i austenne~ cycles. A similar form is i ~ d i c a t ~ ite, with peak effect depths being slightly less than for residual stress. The data in Fig. 22.10 for the high rn~imurn-contact stress indicate more rapid rates of change for bothresidual stress andretained austenite content. ~ u b s e ~ u e testing nt conducted by the author on VI 50 steel balls loaded and run at 27~0-4000 ~ / ~ r m n a 2~ i m stress demonstrated the same tendency. This testing further indicated,
ERIAL R E S ~ O ~ STO ER
O CONTACT ~
~
however, that at thelower loading,the residual stress tends to dissipate after a significant amount of running time. This implies that for bearings operated under common industrial applied loading situations; forexample, less than 2000 N/mm2 maximum Hertz stress, residual stress due to mechanical working during manufacture does not tend to affect endurance. The slight differences in thedepths at which the peak values occur in residual stress and retained austenite decomposition imply correlation with the maximum shear stress andmaximum orthogonal shear stress, respectively. Earlier work [22.13] supports the correlation of peak residual stress values with the maximum shear stress. There appears to be no direct relationship between retained austenite decomposition and the generation of residual compressive stress, nor any indication of which, if either, of these processes triggers microstructural alterations [22.261.
Experimental studies on the fatigue life of rolling bearing components have indicated a positive effect of residual compressive stress over a “zero” stressstate E22.27-22.30, 22.34, 22.351. Another investigation E22.311 indicated that in addition to the positive effect of compressive stress superimposed on Hertzian contact stresses, there was a definite negative effect of superimposed tensile stress. In E22.221 the negative effect of tensile stress was demonstrated, but it was also concludedthat there was no advantage of high residual compressive stress over a zero stress state superimposed on the Hertzian stress field. Another study revealed that bearings tested with inner ring raceways at two levels of residual compressive stress showednosignificantdifl‘erence in life E22.331. The variety of methods used to induce the residual stresses in components for rolling contact fatigue testing included prestressing of the inner rings by running them in bearings at a load higher than the subsequent test load (thereby inducing subsurface residual compression in described previously)E22.27, 22.29, 22.321, bulk loading of test elements by shrink-fitting rings on a shaft or press-fitting into a housing E22.311, and altering the chemistry of the surface during heat treatment to provide residual compression in thequenched and tempered surface E22.33, 22.341, As discussed earlier, the subsurface residual compressive stress generated in a bearing ring during high-contact-stress operation is accompanied by changes in hardness, microstructure, and c ~ s t a l l o ~ a ~ h i c texture. Therefore, prestressing by high-stress operation before testing could introduce significant factors to rolling contact fatigue life and to residual compressive stress.
SSES IN R O L L ~ G~
~
I ~ N OG ~
O
~
~
T
S
Bulk loading of rolling contact test specimens by heavy inte~erencefitting on shafts or in housings provides stress profiles across the specimensection thatare quite different fromself-contained, balanced, residual-stress profiles within a freestanding component. Although such a test scheme might accurately indicate performance trends for bearing applications in which bulk ring loading is similarly experienced, it is not clear that such interact with a Hertzian stress field in thesame manner as a true residual state of stress. Alteration of surface chemistry by infusion of nitrogen or carbon to provide residual surface compressive stress also changesthe microstructural characteristics, the mechanical properties of the surface region, and, perhaps, physical properties such as frictioncoefficient.Consequently, resolution of the separate influence of residual stress is obscured. As indicated previously, Voskamp E22.61 determined that realigning crystals, caused by “over-rolling,”causes a plane of “weakness” aligned parallel to the raceway to form below the raceway. Moreover, he postulated that residual tensile stresses created by such over-rolling tend to cause fatigue failure to occur with propagation beneath and parallel to the raceway. Voskamp E22.61 further shows the photographic evidenceof such failures in highly loaded bearings; for example, 5200 maximum contact stress. Voskamp and ~ittemeijerE22.361, in a further effort to establish the effect of residual stresses on bearing fatigue life, identified three stages of material response:
. A material strengthening stage ( .
s ~ ~ ~ during e ~ which o ~ ~a de) crease occurs for the plastically deformed material volume An effectively stationary (stable) stage during which no change occurs in the plastically deformed volume A final stage during which material softening accompanied by pronounced plastically deformed material volume increase occurs, leading to fatigue failure.
They suggest that carbondiffusioninduced by local temperature increases occur~ngduring rolling contactis the key mechanism leading to e damage. They further suggest that the probability for crack initiation and growth at nonmetallic inclusions is enhanced. Finally, they imply that a higher initial load applied to bearings during the shakedown stage tends to prolong endurance by modiffing the 2nd and 3rd stage responses to rolling contact;this being the resultof increased work hardening. Regardless of direct or indirect association with rolling contactfatigue life, there is general agreement that a residual or applied bulk compres-
~
T
E ~ SRP O ~N S E TO ROLL^^ C O ~ A C T
sive residual stress is a more desirable situation in a rolling bearing component than a residual or applied bulk tensile. It further appears that excessive residual compressive stress can also lead to early bearing component fatigue failure.
Shakedown is described as a self-stabili~ationof material under a cyclically applied load of such magnitude that the material yield stress has been exceeded. Thus, a permanent change has occurred in the material below the rolling contact surfaces. Plastic flow of material has occurred in the structure, generally within a limited region. As a result of the plastic deformations produced during one load cycle, residual stresses occur after the load is removed, keeping the material in equilibrium. During the next load cycle, the residual stresses act together with the stresses caused by the externally applied load. If the load is not too heavy, the amount of plastic flow is less than during the previous cycle. If the load causes stresses in excess of the shakedown limit, however, the plastic flow continues and, in fact, spreads until failure occurs. In Chapter 21, static capacities and loading were basedon permanent deformations occurring in nonrotating bearings. Subsequently during bearing rotation, the indentations or rolling contactsurface deformations were consideredto impair bearing endurance and/or cause undue vibration. The permanent deformations occurringduring the shakedown process are the resultof rolling contactduring normal bearing rotation, and they do not eventually impair the rolling contact surface unless the shakedown limit has been exceeded. It is conceivable therefore that a bearing “static capacity” criterion could bebased on the shakedown limit. It is possible to apply the distortion energy yield criterion; that is, the von Mises yield criterion (see reference E22.371) to bearing steel. Experimental investigations have indicated a variation of‘ the von Mises yield limit with heat treatmentparameters. Yhland E22.381 states thatfor normal heat-treated through-hardened carbon chromiumsteel; that is, AIS1 ~ 2 ~ 0 as 0 ,measured in tension tests, the von Mises yield limit stress is in the region of 1800-2000 N/mm2 (260,000[22.39] developed a method to calculate the shakedown limit considering the fore~oingyield criterion. For a well-lub~catedbearing operating in line contact, the shakedown limit is ap~roximately2.31 times the yield stress insimple tension. For a point contactbearing, the shakedown limit is appro~imatel~ ~ . 7 7 ~ ~It @ is further ~ d . ~ossiblet o evaluate a sit~ation in which no plastic flow occurs; that is, loading is relatively light and subsurface stresses are therefore low. In this case, for a bearing operating in line contact the “incipient plasti arings the limit is 1 . 5 6 ~ ~ ~ ~ ~ . @Id; for point contact shows a interesting co fromreference [22.
EFF'ECTS OF BULK STRESSES ON ~
6000
I
t
I
T
I
E RESPONSE R ~ TO ROLLING CONTACT
1
I
I
I
~~~
I
5500
5000
-
N
E
E z 4500
b
4000
3500
3000
22.11. Comparison of stress limits-shakedown, permanent deformation vs. b l a (from [22.38]).
incipient yield, and 0.0001D
stresses obtaining for shakedown limit, incipient plastic flow limit, and a permanent deformation of O.OOOU.2. From Fig. 22.11, it is seen that the shakedown limit, that is, stress, permanent deformation stress. eater than the0 . 0 0 0 1 ~
8
There is relatively little information in theopen literature on the effects of bulk bearing ring stresses, imposed by mounting and c e n t r i f ~ ~efal
85
~
T
E RESPQ~SE R ~ TO R
O
~ CQWACT ~ G
fects, on bearing operation, and failure manifestations. However, significant consequences are associated with them, involving both dimensional instability and catastrophic fracture of bearing rings. For many decades a peculiar failure manifestation has been observed in bearings installed with heavy interference fit between the inner ring bore andshaft. Manycommonlyexperienced with heavilyloaded through-hardened roller bearings, the failure is characterized by through-section fracture of the inner ring on an axial plane. A typical fracture surface is shown in Fig. 22.12. The fracture surface is predominantly planar and characteristically exhibits a semicircular region of stable fatigue crack propagation originating at a raceway surface. The remaining portion of the fracture surface is characteristic of unstable crackpropagation-that is, rapid fracture. Spalling might or might not be associated with the axially oriented crack on the raceway surface. Such failures are almost exclusively experienced with through-hardened bearing rings. The traditional remedies for this failure experience have been to reduce the interference fit (if this can be donewithout permitting relative motion betweenthe inner ring bore and shaft surface during operation) or to use a case-hardened inner ring. This type of failure assumed greater significance whenit was observed in aircraft gas turbine engine bearings. This observation prompted the first published analysis of the mechanisms involved [22.35, 22.40, and 22.411. The describedcharacteristics of the failures are identical to those indicated here. The circumferential tensile stresses associated with inner ring failures wereinduced ~ e n t r i f ~ ~ a(due l l y to high-speed rotation)
.
Fracture surface of a spherical roller bearing inner ring that failed due to excessive shaft fit (about halfactual size).
EF’F’ECTS OF BULK STRESSES ON ~~R~
RESPONSE TO R O L L ~ G
CO~ACT
rather than from heavy shaft fit, but the magnitude of 172-207 N/mm2 (25-30 ksi) agrees closely with hoop stress values calculated for fitinduced ring fractures. The failure scenario outlined in [22.41] describes the role of circumferential tensile stress of relatively modest magnitude in producing bearing ring fracture. In classical subsurface-initiated fatigue failure, a crack initiates below the surface at a stress raiser such as a nonmetallic inclusion or carbide cluster. The crackpropagates radially outward toward the surface. It also propagates radially inward but, in theabsence of circumferential tensile stress, does not reach significant depth. During continued bearing operation, this crack participates in the formation of a spall. In the presence of circumferential tensile stress of sufficient magnitude [172 N/mm2 (25 ksi) or greater], radially inward stable crack propagation continues to the point at which the critical crack sizeis reached. The critical crack size is defined by the magnitude of the Circumferential tensile stress and the plane strain fracturetoughness of the bearing steel. When the critical crack sizeis reached, a rapid through-section fracture occurs. Rapidfracture occurs on a plane perpendicular to the circumferential tensile stress. Carburized materials provide both residual compressive stress in the high-hardness surface region and fracturetoughness that increases with decreasing hardness from the surface to the core region.Centrifugally or fit-induced circumferential tensile stress should be offsetto some degree by residual circumferential compression. The increased fracture toughness accom~odateslarger crack size. The individual cont~butionsfrom these sources to the successful use of carburized inner rings in heavy interference fit applications have not been described in the open literature. In E22.311, rolling contact fatigue experiments were performed with through-hardened inner ring specimens containing 1000 N/mm2 (145 ksi) circumferential tensile stress. From the foregoing discussionit is not surprising that the inner ring failed by axial fracture with only minute indications of fatigue crack propagation (the critical crack size at this level of circu~ferentialtension is estimated to be about 0.127 mm (0.005 in.). Perhaps of greater significance is the comparison of the running times to fracture for the high-tensile-stressed rings to running times accumulated by the “zero-stress”baseline. Ringfractures were experienced in 20 to 30 hr in the stressed ring tests, whereas baseline rings ran for 440 to 960 hr with no failures. This indicates that significant life reduction could be associated with stress conditions that promote ring fracture (e.g., bulk tensile loading). The results of bearing tests performed with circumferentiall~tensilestressed inner rings [345 N/mm2 (50 ksi)] indicated both very early failure and radial crack propagationin through-hardened AISI M50 bearing steel C22.421. Similarly tested bearings, with inner rings made of a carburizing version of AISI M50.(i.e., M50 Nil), completed substantially
856
MATERIAL CONTACT ~ROLLING SPONSE TO
Dimensional instability of bearing components in service, and particularly growth of bearing inner rings, is a known problem. For manyyears it was common knowledge that dimensional instability of hardened and tempered steel was due to retained austenite being transformed to martensite or bainite, depending on the thermal exposure conditions.Volume changes associated with the transformation of retained austenite, however, have not always been of sufficient m a ~ i t u d eto account for observed dimensional changes. Additionally, substantial dimensional changes of hardened and tempered bearing steel components have been reported with no reduction in retained austenite content [22.43, 22.441. Overhaul statistics from U.S. railroads indicate that inner ringgrowth of carburized railway axle bearings is the major cause of bearing rejection at overhaul [22.45]. Metallurgical phase transformations [22.45] and accumulation of microplasticdeformation [22.461 have beencited as causes. A detailed investigation of retained austenite andresidual-stress profiles in railway journal bearings that exhibited innerring bore growth, however, showed no alteration of either retained austenite content or residual-stress profile [22.431; instead, ring growth correlated with retained austenite content and bearing operating temperature. Rings with higher retained austenite content ehibited morebore growth. Increased operating temperature produced increased growth. The absence of any indications of microstructural alteration or residualstress buildup lead to the conclusion that a creep mechanismis involved in producing the observed change in inner ring dimensions, as opposed to metallurgical phase change or microplastic deformation. Data in E22.441 are consistent with the findings in L22.431. A cleverly designed fixture was usedto determine the dimensional stability of hardened AIS1 52100 steel containing 0-15% retained austenite. Testing was performed from -34 to 74OC (-30-165"F) with constant applied tensile stress levels of 0, 69, and 138 N/mm2 (10and 20 ksi). Test times up to 1700 hr were used. The results indicated that dimensional change (length increase) increased with increasing amounts of retained austenite. More dramatically, for a given austenite content and test temperature (which was well belowthe specimen tempering temperature), large increases in specimen length were associatedwith the application of relatively modest tensile stress. See Fig. 22.13. Also, no reduction in retained austenite content was detected, even aftertestsresultingin s i ~ i f i c a n length t extensions. A creep mechanism similar to that proposed in [22.421 could be indicated. Additionally, these data imply that assessment of growth potential via unstressed thermal exposure tests
ksi) ksi)
-20
I
I I 0 12 2 10 48
I 6
I
I I 1418 16 Time - Hrs x 100 i
I
I
I 20
22.13. Permanent increase in length (U)of AIS1 52100 tensile specimens at various stress levels and temperatures. Hardness (23, 64) and retained austenite content (15%)remained unchanged in specimens tested at 2900 N/mmz and 74°C (165°F) (from E22.441).
may provide an underestimate for an application such as a heavy interference-fitted bearing ring. Further, increasing interference fit to compensate for such an underestimate may result in more rapid loss of fit. See Fig. 22.13. Since this can be done without decomposition of transformation of the initial retained austenite content, the potential for continued growth is preserved. Clearly this would not bethe case if growth was experiencedat theexpense of retained austenite content. This could be a significant consideration when making judgments pertaining to such matters as bearing refurbishment. Retained austenite content remains a primary consideration in terms of bearing ring dimensional stability, with definite correlations between growth potential and initial austenite content. The mechanismsby which dimensional change is effected, however, are not understood. Published work E22.471 indicates desirable contributions of retained austenite to rolling bearing performance. Consequently,it may be technologically imprudent to totally ignore possible performance advantages in pursuit of austenite-free dimensional stability.
c
E
Because of the extremely high rolling element-raceway contactstresses that occur during operation of many ball and roller bearings, the microstructure of the bearing material, that is, steel, undergoes significant
~
T
E ~ $RP O ~N $ ETO R O L L ~ GC O ~ ~ C T
change. This phenomenon has been investigated for many years, and yet it has not thus far been possible to quantitativel~relate such microstructural changes to the imminence of rollingcontact fatigue. It is known that not only do the applied stresses cause such microstructural alterations, but also bearing operating temperatures wellbelow steel tempering temperatures significantly affect the rate and amount of microst~cturalchange. One typeof microstructural change, that is, shakedown, appears to be related to the yield strength of the steel. A shakedown limit might be established as one bearing operating criterion. Investigation in this important area of material science is continuing.
22.1. J. van de Sanden, “Martensite Morphology of Low Alloy Commercial Steels,’’ Practical metal log^ 17, 238-248 (1980). 22.2. J. Rescalvo, “Fracture and Fatigue Crack Growthin 52100, M50 and 18-4-1 Bearing Steels,”Ph.D. thesis, Department of Materials Science and Engineering, Massachusetts Institute of Technology (June 1979). 22.3. G. Gauss, Principles of Heat Treatment of Steel, American Society for Metals, 6175 (1980). 22.4. A. Jones, “Metallurgical Observationsof Ball Bearing Fatigue Phenomena,” Proc. ASTM 46, 1(1946). 22.5. T. Tallian, “On CompetingFailure Modes in Rolling Contact,”ASLE Trans. 10,418439 (1967). 22.6. A. Voskamp, “MaterialResponse to Rolling Contact Loading,”ASME Trans., J Tribology 1079359-366 (1985). 22.7. T. Lund, Jern~ontoretsAnn. 153, 337 (1969). 22.8. H. Swahn, P. Becker, and 0. Vingsbo, “Martensite Decay During Rolling Contact Fatigue in Ball Bearings,”Metallurgical Trans. A 7A, 1099-1110 (Aug. 1976). 22.9. J. Martin, S. Borgese, and D. Eberhardt, “Microstructural Alterations of Rolling Bearing Steel Undergoing CyclicStressing,”Trans. ASME 59,555-557 (Sept. 1966). 22.10. A Gentile, E. Jordan, and A. Martin, “Phase Transformations in High-Carbon High Hardness Steels Under Contact Loads,”Trans. AIME 233, 1085-1093 (June 1965). 22.11. J. Bush, W. Grube, and G. Robinson, ‘‘Microstructuraland Residual Stress Changes in Hardened Steel Due to Rolling Contact,” Trans. ASM 54, 390-412 (1961). 22.12. M. Kuroda, Trans. Jpn. Soc. Mech. Eng. 26, 1256-1270 (1960). 22.13. H. Muro, and N. Tsushima, “Microstructural, Microhardness and ResidualStress Changes Due to Rolling Contact,”Wear 15,309-330 (1970). 22.14. H. Styri, Proc. ASTM 51,682-700 (1951). 22.15. R. Tricot, J. Monnot, and L. Luansi, Metals Eng. Quart. 12, 39-42 (1972). 22.16. W. Littmann and R. Widner,“Propagationof Contact Fatiguefrom Surface and Subsurface Origins,”J Basic Eng. 88, 624-636 (1966). 22.17. D. NBlias, “Contribution a l’etude des roulements,” (Dossier habilitation a Diriger des Recherches, Laboratoire de MBcanique des Contacts, ~R-CNRS-I~S deALyon No. 5514, December 16, 1999). 22.18. L. Uhrus, “Clean Steel,” Iron and Steel Institute, London, 104-109 (1963).
22.19. P. Becker, ‘‘Microstructural Changes Around Non-Metallic Inclusions Caused by Rolling-Contact Fatigue of Ball-Bearing Steels,’’ Metals Technology, 234-243 (June 1981). 22.20. “Residual Stress Measurements by X-Ray Diffraction,” SAE J784a, 2nd ed. Society for Automotive Engineers, New York (1971). 22.21. D. Koistinen, “TheDistribution of Residual Stresses in Carburized Cases and Their Origins,” Trans. ASM 50,227-238 (1958). Addison-Wesley, Reading, Mass. (1959). 22.22, B. Cullity, Elements of X-ray Dif~action, 22.23. C. Gazzara, “The Measurement of Residual Stress with X-ray Diffraction,” Rept. AD-A130 614, Army Material & Mechanics Res. Center (May 1983). 22.24. W. Bragg, “The Diffraction of Short Electromagnetic Wavesby a Crystal,” Proc. Carnb. Phil. Soc. 17,43 (1912). 22.25. C. Ruud and D. Carpenter, Operators Manual for theD-1OOOAStress analyze^ Denver X-Ray Instrument, Inc., Colorado (1985). 22.26. A. Voskamp, R. Osterlund, P. Becker, and 0. Vingsbo, “Gradual Changes in Residual Stress and Microstructure During Contact Fatigue in Ball Bearings,” Metals TechnoZogy, 14-21 (Jan. 1980). 22.27. E. Zaretsky, R. Parker, and W. Anderson, “A Study of Residual Stress Induced During Rolling,” J ; Lub. Tech. 91,314-319 (1969). 22.28. R. Scott, R. Kepple, and M. Miller, “The Effectof Processing-Induced Near-Surface Residual Stress on Ball Bearing Fatigue,” in Rolling Contact Phenomena, ed. J. B. Bidwell, Elsevier, 301-316 (1962). 22.29. E. Zaretsky, R. Parker, W. Anderson, and S. Miller, “Effectof ComponentDifferential Hardness on Residual Stress and Rolling-ContactFatigue,” NASA TND-2664(1965). 22.30. C. Foord, C. Hingley, and A. Cameron, “Pitting of Steel Under Varying Speedsand Combined Stresses,” J ; Lub. Tech. 91, 282-290 (1969). 22.31. R. Kepple and R. Mattson, “Rolling Element Fatigue and Macroresidual Stress,” J ; Lub. Tech. 92,76-82 (1970). 22.32, W. Littmann, Discussion to Reference 31, J ; Lub. Tech. 92,81 (1970). 22.33 C. Stickels and A. Janotik, “Controlling Residual Stresses in 52100 Bearing Steel by Heat Treatment,” Metal Progress, 34-40 (Sept. 1981). 22.34 D. Koistinen, “TheGeneration of Residual Compressive Stresses in theSurface Layers of Through-Hardening Steel Components by Heat Treatment,” Trans. ASM 57, 581-588 (1964). t M M , Journal Aircraft 22.35 J. Clark, “Fracture Failure Modes in L i g h ~ e i g hBearings,” 12,No. 4 (1975). 22.36. A. Voskamp and E. Mittemeijer, “The Effectof the Changing Microstructure on the Fatigue Behaviour During CyclicRolling Contact Loading,” in Microstructural Changes During RollingContact Fatigue, Metal Fatigue in the Subsurface Region of Deep Groove Ball Bearing Inner Rings,A. Voskamp, Proefschrift ter verkrijging van der graad doctor aan de Technische Universiteit Delft (January 8, 1997). 22.37. M. Spotts and T. Shoup, Design of Machine Elements, 7th ed., 123-127, Prentice Hall, Englewood Cliffs, N.J. (1998).
22.39. G. Rydholm, “OnInequalities and Shakedown in Contact Problems, Linkoping Studies in Science and Technology,” Dissertation No. 61, Linkoping, Sweden (1981). 22.40, E. Bamberger, E. Zaretsky, and H. Signer, “Endurance and Failure Characteristics of Main-Shaft Jet Engine Bearings at 3 X lo6 DN,”ASME Trans. J ; Lub. Tech. 95(4) (1976).
86
iMATERW ~ S P O N S E TO ROLLING COWACT
22.41. E. Bamberger, “Materials for RollingElement Bearings,”presented at ASME-ASLE International Lubrication Conference, San Francisco, Calif. (August 1980). 22.42. J. Clark, “Fracture Tough Bearings for High Stress Applications,” presented at ~ ~ / S ~ / A S M E 21st / ~ Joint E E Propulsion Conference, Monterey, Calif. (July 8-10, 1985). 22.43. A. Voskamp and B. Schalk, “Ring Growthin Case Hardened Railway Journal Roller Bearings,” presented at the 2nd International Heavy Haul Railway Conference, Pueblo, Color. (September 1982). 22.44. E. Mikus, T. Hughel, J, Gerty, and A. Knudsen, “The Dimensional Stability of a Precision Ball Bearing Material,” Trans. ASM 52, 307-315 (1960). 22.45. J. McGrew,A.ECrawler, and G. Moyar, “Reliability of Railroad Roller Bearings,” ASME Trans. J Lub. Tech. 99, 30-40 (1977). 22.46. R. Steel, Discussion to Reference 45, ASME Truns. J Lub. Tech. 99, 39 (1977). 22.47. J. Seehan and M. Howes, “The Effectof Case Carbon Content and Heat Treatment on the Pitting Fatigue of 8620 Steel,” SAE Conf. Congress, No. 720268 (January 1972).
Symbol
Description Semimajor axis of contact ellipse Fatigue life reduction factor for clearance Fatigue life factor for oscillatory motion Fatigue life factor for steel ~eliabi~ity-life factor Material-life factor Lubrication-life factor ~ o n t a m ~ n a t i o ~ - factor life Stress-life factor Semiminor axis of contact ellipse bearing pitch diameter raceway diameter Simpson’s rule coefficients Bearing basic dynamic capacity Particulate contamination parameter Rolling element diameter
Units mm (in.)
mm (in.) mm (in.) mm (in.)
N Ob) mm (in.) 861
86
Symbol
~PLICATIO LOAD ~ AND LIFE FACTORS
Description Applied axial load Applied radial load Equivalent applied load Fatigue limit load Filter rating ~ i n i m u mlubricant film thickness Stress concentration factor due to particulate contamination Fatigue life Rating life Median life Speed Number of stress cycles Load on a roller-raceway contact laminum Basic dynamic capacity for a roller-raceway contact laminum Rolling element load Basic dynamic capacity for a ball-raceway contact xlb Oil bath and grease contamination parameter Probability of survival Composite rms surface roughness of mating surfaces Temperature Stress cycles per revolution Stressed volume Width of a roller-raceway contact laminum Depth at which ro occurs Number of rolling elements per bearing row Contact angle Filter eEectiveness ratio for particles of size x Pm Bearing axial deflection Bearing radial deflection D / d m cos a hO/SF kinematic viscosity kinematic viscosity for adequate lubrication von Mises stress
Units
mm3 (in.3) mm (in.) mm (in.) O,
rad
mm (in.) mm (in.)
mm2/sec (in3sec) mm21sec (in.21sec) MPa (psi)
Symbol
Units Maximum orthogonal shear stress Oscillation angle Rolling element azimuth angle
TO
*
CD)
b i j k n r RE
MPa (psi) rad rad O,
O,
SUBSCRIPTS Refers to ball Refers to inner ring or raceway Refers to rolling element azimuth location Refers to roller-raceway contact laminum location Refers to nonrotatin~raceway Refers to rotating raceway Refers to equivalent rotating bearing or rolling element
The Lundberg-Pal~~en theory and the standard load and fatigue life calculations which resulted [23.1-23.51 are only the first step toward determining the bearing fatigue lives in applications.Use of the standard methods should be limited to those applications in which the internal geometries and rolling component materials of the bearings employed conform to the standard specifications, and the operating conditions are bounded as follows: The bearing outer ring is mounted and properly supported in a rigid housing. The bearing inner ring is properly mounted on a nonflexible shaft. The bearing is operated at a steady speed under invariant loading. Operational speed is sufficiently slow such that rolling element centrifugal and gyroscopic loading are insignificant. Bearing loading can be adequately defined by a single radial load, a single axial load, or a combination of these. Bearing loading does not cause significant permanent deformations or material transformations. For bearings under radial loading, mounted internal clearance is essentially nil. For angular-contact ball bearings, nominal contact angle is constant. For roller bearings, uniformloading is maintained at each rollerraceway contact. The bearing is adequately lubricated.
864
LOAD AF’PLICATION
AND LIFE FACTORS
Many applications can be considered to be included within these conditions. In many applications, these simple conditions are exceeded. For errample, many applications do not operate at steady speed or load, rather, under a load-speed cycle. Furthermore, the bearing may support, as indicated in Chapter 7, combined radial, axial?and moment loadingunder which distribution of internal loading is significantly different from the standard limitations. Bearings may operate at speeds which cause substantial rolling element inertial loading and variation in contact angles between inner and outerraceway contacts. These conditions maybe addressed by applying the Lundberg-Palm~entheory in detail using computer programs to perform the complex calculations. Since the development of the Lundberg-Palm~entheory, the ability of lubricant to separate rolling elements from raceways, as discussed in Chapter 12, was established. This condition has beenshown to have probably the most profound effect onextending bearing fatigue life compared to any other. Improvements in modern bearing steel manufacturing methods have provided steels of very high cleanliness and homogeneity?as compared to the basic air-melt AIS1 52100 steel used in the development of the Lundberg-Palmgren theory and standards.With the advent of substantially extended life, increased reliability bearing life prediction canbe considered. Finally, as theimprovements in bearing manufacture and lub~cation were applied, it became apparent that, similar to other steel structural members subjected to cyclic loading, bearing raceways and rolling elements also exhibit an endurance limit in fatigue. This means that in a given application, a ball or roller bearing does not have to fail in fatigue providing applied loading and conditions of operation are such that the bearing material fatigue Limit stress is not errceeded. All of the foregoing conditions willbe addressed in this chapter.
G INTE
ON
When the distribution of load among the balls is different from that resulting from the applied loading conditions specified in the load rating standards, it is necessary to revert to the Lundberg-Palm~enload-life relationships for individual ball-raceway contacts. For example, for a contact on a rotating raceway: = Lrj
( ~ ) 3
(23.1)
where Qcr.is the basic dynamic capacity of the contact of ball j on the
EFFECT OF
BMING
LOAD ~
~~~
I
S
~
~ ON ~FATIGUE I O N LIFE
6&
rotating raceway, and Qd is the load acting on the contact. It is to be noted that the capacity may be different from point to point around the raceway because contact angle may vary with azimuth angle. For a nonrotating raceway contact, as determined from equation (18.71), (23.2)
It is also to be noted that the ball-raceway load may diEer between raceways due to ball inertial loading. From equations (23.1) and (23.2), it may be determined that the life of a bearing having a complement of Z balls is given by j=Z
x
j=Z
j=1
L,."
+
x
L;!?
(23.3)
j=1
where exponent e is the slope of the Weibull distribution. It is further to be noted that life calculated according to equation (23.3) does not include ball lives. As indicated in Chapter 18, there is ample evidence that balls, as well as raceways, can succumb to fatigue failure. Assuming that in rolling bearings subjected to reasonable levels of loading, the balls contact the raceways over defined tracks, startingwith equations (18.42)and (18.53), an equation for basic dynamic capacity of the ball portion of a ballraceway contact can be developed. In (18.53), recognizing there are two stress cycles per ball revolution, it can be determined that (23.4) Using equation (23.~4)~ equation (23.3) forbearing life becomes
In using equation (23.5),it must be recognizedthat bearing life is defined in revolutions of the rotatingring. For example, for simple rolling motion, the number of ball revolutions per inner ring revolution as determined from equation (8.14) is (23.6) Therefore, the ball lives indicated in (23.5) must first be divided by the ratio of equation (23.6). In cases where skidding occurs in the bearing
8
~ P L I ~ ~ T LOAD ION
and ball speeds are calculated according to methods in Chapter 14, the ratio of equation (23.6) may be replaced bythe calculated speed ratio. Also, in using equation (23.51, it must be recognized that the Weibull slope for ball failures may be somewhat different from that for raceway failures. For example, in a fatigue failure investigation of VIWAR M50 steel balls, the data of Harris E23.61 indicated an average Weibull slope of 3.33. In such a case, an average value of e may be used in equation (23.5).
In Chapter 7, it was shown that to determine the distribution of load among the rollers for nonstandard applied loading, the roller-raceway contacts may be divided into a number of laminae. Hence for a rollerraceway contactof length I , if the contact is divided into m laminae, each of width w ,I = m w and k=m
(23.7)
Therefore, referring to equations (23.1) and (23.2) for ball bearings and considering, as indicated in Chapter 18, a 4th power load-life relationship for line contact, the following equations may be written for the fatigue lives roller-raceway contact laminae: (23.8)
(23.9)
Accordingly, roller bearing fatigue life may be calculated using equation (23.9). j=Z k=m k = lj = lk = lj = 1
j=Z k=m
)'"
(23.10)
Roller bearing fatigue life including the lives of the rollers may be calculated using equation (23.11).
(23.11)
For rollers, the basic dynamic capacityis given by
EFFECT OF B
G ~E~~
LOAD ~ I S T R ~ ~ ION O FATIGTJEl N LIFE
(23.12)
As for balls, roller life must be reduced by the speed ratio for use in equation (23.11).
The fatigue life of a rolling bearing is strongly dependent on the maximum rollingelement load Qmm;if Qmaxis significantly increased, fatigue life is significantly decreased.h y parameter that affects Qma, therefore, affects bearing fatigue life. One such parameter is diametral clearance. In Chapter 7, the effect of clearance on load distribution in radial bearings was examined. The effect of clearance on bearing fatigue life may be expressed in terms of the standard rating life; for example, LlOc= aCL,,. Figure 23.1 from reference [23.7], which gives the L,, life reduction factor t t c as a function of E, the extent of rolling element loading, was developed by using the load distribution data of Chapter 7 in accordance with the contact life and bearing life equations (23.1)-(23.3) and (23.8)(23.10). An increase in Qmaxfor rigidly supported bearings is accompanied by a decrease in the numbers of rolling elements loaded. This decrease in load zone, however,has less effect on the mean effective rolling element load than does the increase in Qmax.Figure 7.3 illustrates the variation of load distribution among the rolling elements for different amounts of diametral clearance.
€
F
I
~ 23~.1. EFatiguelifereductionfactorbased
on diametral clearance.
~ P L I C A T I OLOATJ ~ AND LIFE FACTORS
le 23.11. The 209 radial ball bearing of Example 7.1 has 0.0150 mm (0.0006 in.) diametral clearance that gives E = 0.434 for a radial load of 8900 N (2000 lb).The ANSI loadrating method is based on E = 0.5, that is, zero clearance. What reduction in Llolife may be expected, consideringthe standard ratinglife? From Fig. 23.1, at E = 0.434, ($ = 0.93
at
E =
0.5, C$
=
1
Consequently, a 7% reduction in rating life is required.
If one or both rings of a rolling bearing bend under the applied loads such as ina planet gear application f23.8, 23.91 or other aircraft bearing applications in which ring and housing cross sections are optimized for aircraft weight reduction, then load distribution may be considerably different from that of a rigid ring bearing. Depending on the flexibility of the ring and bearing clearance, it may be possible for a flexible ring to yield superior endurance characteristics when compared to a rigid ring bearing. Figure 23.2 from reference [23.8], shows the variation of bearing fatigue life with outer ring section and clearance for a planet gear bearing as shown by Fig. 7.35. The load distribution obtained is illustrated by Fig. 7.44. When the bearing rings are flexibly supported, it may be possible to alter bearing design and obtain increased fatigue life. Harris and Broschard 123.93 applied clearance selectively at theplanet gear bearing maximum load positions by making the bearing inner ring elliptical. Figure 23.3 demonstrates the variation of fatigue life with diametral clearance and o u t - o ~ r o u Out-of-round ~~. is the difference between the major and minor axes of the elliptical ring. A further reference f23.101 also demonstrates that rolling bearing ring dimensions can be optimized to maximize fatigue life.
Operation at high speeds, as shown in Chapter 9, affects the bearing load distribution due to the increased magnitude of rolling element centrifugal forces and gyroscopic moments. Thestandard methods of fatigue life calculation E23.3-23.51do not account for these inertial forces and moments and subsequent effects such as changes in ball bearing contact
E ~ C OF TBEARING
~E~~
0
0.02
869
LOAT) R I S ~ ~ ~ ON I OFATIGUE N LIFE
mm
0.04
0.06
0.08
0.10
Diametral clearance; in. 23.2. Planet gear bearing section moment of inertia.
life vs diametralclearanceandouterringcross-
angles. Hence, the deviation in fatigue life fromthat calculated according to the standard method can be considerable.In Chapter 9, methods were developed to calculate load distribution in high speed ball and roller bearings. Methods for using these load distributions in theestimation of fatigue life have been given in this chapter. Figure 23.4 demonstrates the variation of life with load and speed for the 218 angular-contact ball bearing of Example 9.1. Note that the data shown in Fig. 23.4 do not consider the effect of skidding, which results ina reduction in ball orbital speed, and hence reduced ball centrifugal and gyroscopic loading. This, in turn,tends to result in an increase in fatigue life; however, depending on the thicknesses of the lubricant films separating the balls from the raceways, sliding in the ball-raceway contacts, with its potential deleterious effect on fatigue endurance, may more than eliminate the beneficial effect of reduced inertial loading. Figure 23.5 compares the fatigue life of the 218 angular-contact ball bearing operating at high speed with lightweight silicon nitride balls to that of the bearing having steel balls. Whereas the silicon nitride balls operate with reduced inertial loading, the elastic modulus of HIPped silicon nitride (see Chapter 16) is approximately 50%greater than that of
PLICATION LOAD AND LIFE FACTORS
23.3. Bearing life vs diametral clearance and out-of-round.
steel. This results in reduced contact area between the steel raceways and ceramic balls; therefore, Hertz stresses are increased, causing reduction in fatigue life. Thus, the beneficial effect of lightweight balls is counteracted. By decreasing the radii of the raceway grooves somewhat, the Hertz stresses may be decreased. This, however, causes an increase in frictional stresses andhigher operating temperatures which mayhave 'to be accommodated by cooling the lubricant or bearing. Optimum bearing design may be achieved for a given application by parametric study using a bearing performance analysis computer program. It can be seen from Fig. 23.5 that there is little difference.in the fatigue life performance of the bearing under relatively heavy loading. Figure 23.6 shows life vs speed for the 209 cylindrical roller bearing of Example 9.2. Skidding eRects are not included in these illustrations.
Misalignment in nonaligning rolling bearings distorts the internal load distribution, and thus alters fatigue life. In Chapter 7, methods were described to determine the misalignment angle in ball and roller bearings as a function of the applied moment. In ball bearings, the load distribution from ball to ball is altered by misalignment; in roller bearings, however, the distribution of roller load per unit length becomes nonuniform as shown by Fig. 7.26. The variable load per unit length is given by equation ('7.112).
23.4. L,, life* vs thrust load and speed; 218 angular-contact ball bearing,a0 = 40". *Note that the b, factor was not included in the L,, life calculations; the graphical data should be used for comparison purposes only.
The analysis of roller bearing lives indicated in Chapter 18 pertained only to bearings having uniform distribution of load per unit length along the roller length at each roller-raceway contact. As indicated in Chapter 7, roller-raceway loading varies not only from contact to contact, but also from laminum to laminum along a contact. The methods definedin Chapter 7 allow the determination of the load per unit length qnjkat each
872,
APPLICATION LOAD AND LIFE FACTORS 7e+9 6e+9
{ -steel balls
...... silicon nitride balls
Oe+O
1 t
I
I
0
10000
20000
I
I
30000
40000
50000
-
Applied Thrust Load N F
I ~ 23~.5. ELife vs thrust load for a 218 angular-contact ball bearing operating at approximately 1.50 million dn.* "Bearing bore in millimeters timesshaft speed in rpm.
roller-raceway laminum contact; where n = 1. (outer raceway) or 2 (inner raeeway),j = 1 . . Z , and Jz = 1to rn. It should be apparent that misalignment can quicklylead to edge Zoading in theroller-raceway contacts; edge loadingof even small magnitude can rapidly diminish fatigue life. In Chapter 6, references were cited indicating that the magnitude of edge (23.13)
stressing can be calculated for any roller-raceway contact profile. The calculations require a computer and substantial time even for a single An approximation can,however,be roller-racewaycontactcondition. used in the laminated contact calculations of Chapter 7. Accordingly, stress concentration factors ke may be applied to the end laminae in the life calculations. For example, For rn
=
20 laminae, ke = 5.4 appears to provide reasonable results.
~~~~
LOAD D I S T R ~ ~ I OON FATIGUE N
LIFE
73
~ ~ . ~ ,
Life vs speed; 209 cylindrical roller bearing with zero mounted clearance supporting 44,500 N (10,000 lb) radial load.
As a u ~ e n t e dby equation (23.13), equations (23.8)-(23.10~ may be used to estimate the effect of misalignment on roller bearing life. Figure 23.7 from reference [23.11] shows the effect of misali~menton the life of a 309 cylindrical roller bearing as a function of roller crowning and applied load. Table 23.1 from reference E23.121 indicates, based on exerience data inmanufacturers catalogs, maximum acceptable misalignments for the various rolling bearing types.
~ ~ L I C A T LO IO~ 160 r
LIFE FACTORS
Radial load = 15,800 N (C14) (3530Ib)
Radial load=31,600 N (C/2) (7100 Ib)' 100
90
f
80
e d
70
2
"0
y.
= 7.7 mm (0.303 in.) = 4.8 mm (0.188 in.)
in.) = 4.8 mm (0.188 in.)
60 50
40
4 $ii30 2
0" 20 10
'0
5
10
15 25
020
20
~ i ~ l i g n m e (min) nt
15 5
10
25
isa alignment ( m i d
23.7. Life vs misalignment for a 309 cylindrical rollerbearing as a function of crowning and applied load.
LE 23.1. Limitations on Rolling Bearing Misali~ment Misalignment Angle Bearing Type
(min)
Cylindrical roller Tapered roller Spherical roller Deep goove ball
3-4 3-4 30 12-16
(rad) 0.001 0.001 0.0087
0.0035-0.0047
Many applications do not involve invariant applied loading;rather, they undergo a defined load vs time cycle. To analyze these applications, use is made of the P a l m ~ e n - ~ i n erule, r which is defined by equation (23.14).
EFFECT OF
LE LO^^^ ON F
~
T LIFE I ~
~
1
(23.14)
This equation refers to a set of bearing operating conditions TL,each individual condition being designated i, each condition having a potential fatigue life Lirevolutions, and the bearing operating only for Ni revolutions where Ni< Li. To determine the life of a rolling bearing subjected to a time-variant load, it is necessary to determine a mean effective bearing load such that
L
=
(~~
(23.15)
in which p = 3 for point contact bearings, and p = 4 for line contact bearings. Actually, there is little difference in Fmcalculated for p = 3 as compared to Fm calculated for p = 4; therefore, a cubic mean effective load is frequently used for rollerbearings with little error. Consider a bearing subjected to a load F , for N,revolutions and F2for N2revolutions. The calculatedfatigue lives for these loads are (23.16) (23.17)
~ubstituting(23.16) and (23.17) into (23.14) yields (23.18)
Dividing by L the total bearing fatigue life, and substituting for L according to equation (23.15) yields
(23.19)
ence,
87
~ ~ L I C A T I OLOAD N AiTD LIE% FACTORS
Fm= ~
N
F~N2)1’F l
(23.20)
It is readily apparent that L = Nl + N2. Thus, in general for k loads each operating for N k revolutions, the mean effective load is given by (23.21) in which N is the total number of revolutions in one load cycle. In integral format, equation (23.21) becomes (23.22) For a cyclic load of period r, (23.23) in which Ft is a defined function of time.
3.2. The 22317two-row spherical roller bearing of Example 18.9 was shown to have a basic load rating of 399,300 N ($9,720 lb). Assuming the bearing experiences the following repetitive loading cycle while operating at 900 rpm shaft speed, estimate the L,, fatigue life according to standard (ANSI, ISO) methods.
.
1 2 3
20 30 10
89,000 44,500 22,250
(20,000) (10,000) (5000)
22,250 0 22,250
Condition 1
Fel = 136,100 N (30,950 lb)
Condition 2
Fe2= 44,500 N (10,000 lb)
Condition 3
FalF~ = 22,250122,250 = 1 1.5 tan a
=
0.3189
FalFr> 1.5 tan a FromTable 18.7, X
=
0.67
Y = 0.67 ctn a
~
(5000)
(~000)
Ex.18.9
x. 18.9
877
EFFECT OF VARIABLE LOADING ON FATIGUE LIFE
Ex. 2.9
a = 12"
Y
=
0.67 ctn (12") = 3.152
Fe3 = XF, - YFa =
0.67
X
22,250
(18.166)
+ 3.152 X 22,250
= 85,040 N (19,100 lb)
Fm = (X F ~ 0 ' 3 ~ ~ ) 3 ' 1 0
(23.21)
20( 136,100)10/3 + 30(44,500)10'3+ 10(85,040)10'3 20 + 30 + 10 =
101,700 N (22,860 lb)
L = (C/Fm)10'3
(23.14)
= (39~~300/101,700)10'3 = 95.48
million revolutions
L = 95.48 =
X
106/(60 X 900)
1768 hr
Some special cases of fluctuating load may now be defined. Palmgren [28.13] states thatfor a bearing load that varies nearly linearly between Fdn and FmBxas shown in Fig. 23.8, the following approximation is valid:
Fm = QFmin+ QFmax
(23.24)
If the load variation is truly linear, then (23.25) and
T
" " " "
P
T O 73
Emin
4
evolutions
N
i"
23.8. Load vs time; nearly linear.
~ P L I ~ A T LOAD I O ~ AND LIFE FACTORS
{I,’ [
1+
Fm= Fmin
~~
-
1)
..)’”
(23.26)
in which z is a dummy variable. A bearing load may be composed of a steady load F , upon which a sinusoidally varying load of amplitude F3 in phase with F , is superimposed as shown in Fig. 23.9. In this case,
Ft in which
w
=
F,
+ F3 cos w t
(23.27)
is the circular frequency in radians per second. Thus,
Fm= F ,
IT
r o (1
+ F , cos ot
(23.28)
.A more general case of loading is that of a steady load F,, a rotating load F2, and a sinusoidally varying load of amplitude F3 (in phase with F,) simultaneously applied to a rolling bearing. Figure 23.10demonstrates this form of loading. Maximum bearing load occurs whenF,, F2, and F3 assume the same line of action. Steady loads F, are caused by the weight of machine elements on a shaft and also by their imposed loading such as gear or belt loads. Rotating loads F2 are caused by unbalance in spinning mechanisms, either intentional or not. Sinusoidal this loads F3 are caused by inertial forces of reciprocating machinery. For general loading, one may simplystate that Fm=
+ F2 + F3)
in which
Time
IG
23.9. Sinusoidal load F3 superimposedon a steady load
(23.29)
FATIGWC LIFE OF O S C I L ~ T ~BG E~INGS
E: 23.10. Loadvector diagram for generalizing bearing loading consistingof steady load F,, rotating load F2, and sinusoidal load F3 in phase with
F, .'
[$I,"
( [ F , + (F,
+ F3) cos +I2
-t (F,
L
+m
=
sin
+)2)P'2
dQ l l h -1
F,
f
F2 + F3
(23.30)
SKI? has developed a series of curves depicting this relationship in terms of the relative magnitudes of F,, F,, and F3. Figure 23.11a applies to point contact, that is, p = 3, and Fig. 23.1lb applies to line contact, p = 4. Note that when F, = 0, the loading of Fig. 23.12 occurs. Thus, the lowest curves of Figs. 23.11 refer to that loading situation. If F , and F2 are individually absent, that is, for the sinusoidal bearing loading demonstrated by Fig. 23.12; then according to Fig. 23.11, +m = 0.75 for point contact and +m = 0.79 for line contact. Figure 23.13 demonstrates a bearing loading in which F3, the sinusoidal load, acts 90" out-of-phase to the steady load Fl. Figures 23.14a and 23.14b yield values of #m for this type of loading for point and line contact, respectively. W e n steady load F , is absent and maximum F3 occurs 90" out-of-phase with F2, then Fig. 23.15 illustrates the bearing loading with time, and Fig. 23.16 gives values of +m for point and line contact.
scillating bearings do not turn through complete revolutions. If one refers t o the f r e ~ u e ~ co fy oscillation as n cycles per minute, then an
APPLICATION LOAD AND LIFE FACTORS
23.10.
" " " " " "
xI: 23.12. Sinusoidal bearing load.
23.13. Load vector diagram for generalized bearing loading consistingof steady load F,, rotating load F, and sinusoidal loadF3, 90" out-of-phasewith F,.
oscillating bearing operating at frequency n will have a longer fatigue life than the same bearing rotating at n rpm under the same load. This is due to a lesser mean equivalent load per rolling element. One oscillation (cycle) refers t o the motion of the bearing from one extreme position to the other, and return.The time to traverse the angular am~litude $ of oscillation s t a ~ i n g from 0" position is one-fourth of a cycle, or d4, in which T is the period of the oscillation. To determine the fatigue life of an oscillatingbearing, the applied load may be converted to an equivalent load for a rotating bearing, thus acco~ntingfor the decreased number of stress cycles. For a given bearing a given fatigue life, equation (18.31) yields the following relationship for point contact:
r stress cycls per revol~~ion, in which $ is a constant, u is the n ~ m b e of
PLICATION LO
3.14.
+m
vs F,I(F,
LIFE F ~ C T O R ~
+ F2 + F3)for the generalized loading of Fig. 23.13.
3
23.16. Load vector diagram for bearing loading consistingof a rotating load Fz and a sinusoidal load F3. Maximum F3 occurs 90"out-of-phase withFz.
+
~ 2 J ~ E ~j ) Z
23.16 da, vs F 2 / ( F 2f Fa)for the loading of Fig. 23.15.
and e is the Weibull slope. It was shown in equations (18.37) and (18.38) that C-hS-2 3e
herefore,
=
3 "'p
(~3.32)
~PLICATIO LO ~
u=(g
LIFE FACTOR^
(23.33)
Using the subscript RE for the equivalent rotating bearing, osc for an oscillating bearing, and R for a rotating bearing, from equation (23.33), (23.34)
The length of arc stressed during one complete revolution of a rotating bearing is 2nr, in which r is the raceway radius. The length of arc stressed during one complete oscillation is 44" in which bj is the amplitude of oscillation in radians. It is apparent that (23.35)
Thus, (23.36)
In this example is the load of a contact yielding a fatigue life of L million revolutions. Therefore,for an oscillating bearing
L
=
(~~
(23.37)
in which
for oscillation angle in radians and degrees respectively. for bearings subjected oupert E23.141 refined the foregoi ), the load zone being mbined radial and thrust load, defined by (7.61)
85
FATIGm LIFE OF O S ~ I L ~ T mBIURmGS G
the number of stress cycles experienced by each sector. Accordingly, an oscillation-life factorQoSc was developed such that (23.39)
Figure 23.17 gives Qosc vs oscillation angles and extent of load zone. Defining an oscillation factor as d24, the relative effects of the life calculated using equations (23.37) and (23.38) or (23.39) are shown by Fig. 23.18. It can be seen from Fig. 23.18, which comparesbearing oscillatory life calculated using the Houpert method E23.141 with that calculated using the simple method of equations (23.37) and (23.381, that for relatively large extent of load zone, the simple method can suffice. For relatively larger angles of oscillation and narrow extent of the load zone, however, the Houpert method is preferred. W e n 4/90 < 1/Z, in which is is the number of rolling elements per row, then a strong possibility exists that indentation of the raceways will occur. In this situation, surface fatigue may not be a valid criterion of failure, in view of the vibration which urthermore, it is possible that lubricant film thicknesses adequate to separate the rolling contact componentsurfaces may not be
10
a
6
4
2
0 0
1
2
3
&
E ~ 3 . 1 ~CtbBbSc . vs load zone extent and oscillation angle.
4
88
~ P L I ~ A T LOAD I O ~ AND LIFE FACTORS
0
1
2
3
4
E
FI
23.18. Oscillation-life accuracyratio vs extent of load zone; from [23.141.
generated, resulting in surface wear and excessive friction heat generation.
Based upon an analysis of the endurance data pertaining to more than 2500 test bearings, Tallian [23.15] confirmed that a Weibull distribution fits the test data in the most used cumulative failure probability region, that is, between 5 = 0.07 and 5 = 0.60. Everywhere outside of this region, fatigue life is greater than that predicted by a Weibull distribution. Generally, bearing users are not concerned about fatigue lives in extherefore, the upper failure region, cess of the median life, that is, LEO; which deviates from the Weibull distribution, is not of interest here. However, rolling bearing users are extremely interested in the failure region between 5: = 0 and 3: = 0.10. Aerospace applications demand rolling bearings having better than 90% reliability (3= 0.1, s = 0.9). In fact, reliability of better than 99% is not an uncommon requirement. Automobile ~anufacturers are on record with multi-year unconditional warranties on parts including bearings and hence would like bearings of greater reliability. Considering the improved bearing steels which are available, it is possible to manufacture bearings having increased fatigue
~
L
~
I AND L FATIGUE I ~ LIFE
life. f t may therefore be necessary to specify the fatigue of a given rolling bearing application in terms of Ll, L,, and so on, instead of L,,. Figure 23.19, taken from Tallian’s paper [23.15], shows the deviation from the Weibull distribution for probability of failure 3: less than 10%. Note that below approximately a standardized bearing life of 0.004, that is, below =
(&)”
1n 2
=
0.004
there is apparently no decrease in fatigue life. Figure 23.20, which is a similar plotshownby Harris 123.161 on semilogarithmic coordinates, demonstrates more dramatically the significance of Tallian’s research. It is apparent that “no-failure”fatigue life may be predictedwith an attendant reliability of 100% (S = 1.0, 3: = 0).According to Tallian [23.15], the no-failure fatigue life may be approximated by
LNF 0.05 L,,
(23.40)
Tallian [23.15] reasoned that bearing fatigue life may be separated into two discrete phases:
Standardized bearing lives
F
I
~ 23~.19.E Life distribution in the early failure region.
8 8 ~
APPLICATION LOAD ANI) LIFE FACTORS
u0
FIGURE 23.20. Fraction L,,life vs probability of failure.
1, The time between commencement of rotation and initiation of the crack below the surface, that is, La. The time necessary for the crack to propagate to the is, Lb. e
Hence, the fatigue life measured represents the sum of the two durations:
i!&may be called an excess life since La is the fatigue life predicted by Weibull theory. Ifa crack is initiated when rotation has justcommenced,
Lb is very much greater than La and L is not a valid Weibull life measurement. If a crack is initiated a considerable time after rotation has started, then Lb is insi~ificantcompared to La and L is a reasonably accurate Weibull lifeestimator. Thus, deviation of early failures from the Weibull distribution is explained. In equation format, one maydetermine the fatigue life at reliabilities other than s = 0.9 as follows: In -
+ "Ye
Ve
(23.42) in which %e is the standardi~ed excesslife. Tallian [23.15]gives the schedule for "Ye presented in Table 23.2.
I!CS AND F A T I G ~ LIFE
8
23.2, Excess Standardized Life Cyc Probability of Survival (%)
Standardized Theoretical Life
s 99.9 99.9 22 5 2 95 95 > S > 40
cy 2s 0.001 0.001 < < 0.05 0.05 Cy < 0.6
Standardized Life Cy
+ Cye = 0.004
In(%+ cye)
=
0.690 ln(0.328 Cy)
cye
=
0.013
Increase in reliability may be assigned to a given bearing by reduction of its basic dpamic capacity. Harris [23.16] developed Fig. 23.21, which determines the reduction in basic dynamic capacity required to achieve
3.3. In example18.6, the Llo fatigue life of the 218 angular-contact ball bearing that supports a 22,250 N (5000 lb) thrust load while the shaft rotates at 10,000 rpm was estimated at 525 hr according to the basic Lundberg-Palm~en formulas. Consider that the bearing is fabricated from contemporary 52100 steel and according to contemporary manufacturing methods; estimate the fatigue life that may be attained with 99% reliability. 5 = 100 - 99 = 1%
From Table 18.11, b,, = 1.3, therefore
(18.70, 18.75) Therefore,
or
Ll0 = 2.20L10 From Fig. 23.20, LIL;,, hr.
=
0.23; therefore, I;,= 0.23 2.20
9
525 = 265
The 209 radial ball bearing of Example 18.7 achieved life of 4620 hr. Assuming contemporary steel and manufacturing processes, the bearing has a basic load rating C = 32,710 N (7350 lb). To achieve the same life with 95%reliability, what e basic load rating of the required ball bearing? romFig.23.20, C’/C = 0.845; therefore, C’ = 32,71010.845 = 38,710 N ( 8 ~ 9 9lb). In the ANSI standards [23.3-23.4], the following table of reliabilitylife factors Ili;vs reliability is provided.
In Chapter 12, it was indicated that if a rolling bearing is adequately designed and lubricated, the rolling surfaces can be separated by a lubricant film. Endurance testing of rolling bearings as shown by Tallian et al. [23.17] and Skurka [23.18] has demonstrated the considerable effect of lubricant film thickness on fatigue life. In Chapter 12, methods
LE 23.3. Reliabi1it~-LifeFactors % Reliability (%)
0.33
(4 I
90 95
0.62
96
0.53
97 98 99
0.44 0.21
~ I C A ~ I OON N F A ~ I LIFE G ~
for estimating this lubricant film thickness were given. It was also demonstrated that lubricant film thickness is sensitive to bearing speed of operation and lubricant viscous properties and, moreover, the film thickness is virtually insensitive to load. The test results reported in references [23.17]and [23.18] showedthat at high speeds of operation, a considerable improvement in fatigue life occurs. Moreover, a similar effect can be achievedby using a sufficiently viscous lubricant at slower speeds. The effectiveness of the lubricant film thickness generated depends upon its magnitude relative to the surface topo~raphiesof the contacting elements, that is, rolling elements and raceways. In other words, a bearing with very smooth raceway and rolling element surfaces requires less of a lubricant film than does a bearing with relatively rough surfaces (see Fig. 23.22). The relationship of lubricant film thickness to surface roughness has been signifiedin rolling bearing literature by A, which utilizes the simple rms value of the rou~hnessesof the surfaces of the contacting bodies. Tallian [23.19] among many other researchers introduced the use of asperity slopes as well as height of asperity peaks. Chapter 13, covering microcontact phenomena provides additional, morecomplex means to evaluate the efEectof a “rough” surface on contact and hence bearing lubrication and performance. UsingA, Harris [23.20] indicated the effect of lubrication on bearing fatigue life in Fig. 23.23. Accordingto reference
Smooth surface bearing
Rough surface bearing
Low speed
High speed
Low speed
23.22. Illustration of the effect of surface roughnesson the lubricant film thickness requiredto prevent metal-to-metal contact.
~PLICATIO LOAD ~ AM) LIFE FACTORS
A =5 function of film thickness and surface roughness ~~~~~
23.23. Percent film vs A.
E23.201, if A is numerically equal to or greater than 4, fatigue life can be expected to exceed standard Llo estimates by at least 100%.On the other hand, if A is less than unity, the bearing will probably not attain the calculated Llo estimates because of surface distress such as smearing or pulling which can lead to rapid fatigue failure of the rolling surfaces. Fig. 23.23 shows the various operating regions just described. In Fig. 23.23, the ordinate, that is, percent film, is a measure of the time percentage during which the “contacting”surfaces are fully separated by an oil or lubricant film. Tallian E23.191 showed a more definitive estimate of rolling bearing fatigue life vs A as did Skurka [23.18]. Bamberger et al. [23.121 show the co~binationof the foregoing in Fig. 23.24, recommending the use of the mean curve. Experimental data indicate that for A > 4, the L/Llo ratios given by Fig. 23.24 are substantially greater for accurately manufactured, bearings lubricated by minimally contaminated oil. Using a microtransducer to measure the pressure distribution in an oil-lubricated line contact in the direction of rolling, it was shown in [23.21] that the edge stress in a line contact is substantially reduced if i ~ g bodies.Thus, an adequate lubricant film separates thec ~ ~ t ~ c trolling in thissituation, the lubricant film tends to permit an increase in fatigue life by reducing the magnitude of normal stress at the end(s) of heavily loaded contact. In the 209 cylindrical rollerbearing of Example 12.1, the rollers have surface finishes of 0.102 pm (4-pin.) rms and the
EF~CT OF L ~ R I C A T I O NON FATIGUE LIFE
3*5
r-
reference From
123.331
3.0
2.5
2.0 L LlO
1.5
1.O
0.5
0
0.6 0.8 1
2
4
6
810
Film parameter ( A )
F I ~ 23.24. ~ E Lubrication-life factor vs A.
raceways have surface finishes of 0.203 p m (8-pin.) rms. Determine the effect of lubrication on fatigue life.
ho = 0.731 pm (28.4 pin.)(naphthenic oil)Ex.
SF
12.1
+ SF?)"" = [(0.102)2+ (0.203)2]1'2 =
(SF;
=
0.227 p m (8.94 pin.)
0.726 -- 3.2 -
0.227
According to Fig. 23.24, the expected fatigue life would be between 1.7 and 3.3 L,, or a mean value of 2.5 Llo.
3.6. Estimate the effect of lubrication on the life of the 209 cylindrical roller bearing of Example 12.2.
~ P L I C A T I OLO ~
LIFE F A ~ T O R ~
0.0734 pm (2.89 pin.)
ho -=”-
SF
0.0734 - 0.32 0.227
From Fig. 23.24 it can be seen that a reduction in Llo fatigue life will most likelybe experienced. If any degree of gross sliding occurs in the roller-raceway contacts, the fatigue life reduction maybe very severe.
*
The foregoing example showsthe deficiency in using only h to define the effect of lubrication on fatigue life. It must be recognized that A is only a qualitative measure of lubrication effectiveness. In Chapter 18, it can be determined that fatigue life is a strong function of stresses acting on the contacts between mating rolling surfaces. For example, it can be shown using Lundberg-Palm~entheory that for point contacts L; oc
7;;9*3
Furthermore, according to Chapter 13, lubricant film thickness in conjunction with the Hertz stresses andsliding velocities acting on the contact areasdetermines the contact surface shear stresses. It is these shear stresses in conjunction with the Hertz stresses which determine the effect of lubrication on fatigue life. How to include the surface shear stresses in the prediction of bearing fatigue life will be discussed later in this chapter.
In Chapter 16, the various materials used for the fabrication of bearing rings, balls, and rollers were discussed in detail concerning their chemical compositionsand structuralproperties, and in Chapter 18, the influence of steel composition and heat treatment on fatigue endurance was further discussed. It was seen in Chapter 18 that the effect on fatigue endurance of the basic steels used in modern bearing manufacture may be assumed to be included in the bm or f,,, factors in the calculation of basic loadrating C.Currently, the s t a ~ d a r dsteel is assumed to be carbon vacuum degassed (CVI)) 52100, through-hardened at least to Rockwell C 58. Many roller bearings, particularly tapered roller bearings, are, however, manufactured from carburized (case-hardened)steel. Since the load and life rating methods for such bearings are assumed to be included in the standardsE23.3-23.51, it must be further assumed that the endurance performances of the CVI) 52100 through-hardened steel and the basic carburizing steels are equivalent.
EFFECT OF ~
T
E ANI) R ~~ T
E PROCESSING R ~ ON FATIGUE L I m
To attain high temperature, long-life performance,WWAR M50 tool steel was developed for aircraft gas turbine mainshaft bearing applications. This vacuum-induction-melted, vacuum-arc-remelted steel provides excellent fatigue endurance characteristics for bearing rings and rolling elements. Due to the necessity to operate modern gas turbine mainshaft bearings at ultra-high speed-for example, 3 million dn-a carburizing version of this steel, VIWAR M50NiL, was developed. See Chapter 16 for M5ONiL chemistry and properties. In this case, it is intended that the “softer” core will arrest any fatigue cracks which emanate in thehardened case and thus prevent through-cracking of bearing rings. A number of specialty steels have been developed to provide superior corrosion resistance while not sacrificing fatigue endurance properties; for example, Cronidur 30. Additionally, ceramic materials, for example, hot isostatically pressed silicon nitride, are now being used in the manufacture of balls and rollers. STLE 123.221 has attempted to codify the effect of some these materials on rolling bearing fatigue life. Moreover, STLE [23.22]has also separated the effects of heat treatment and metalworking. A material-life factor Wsteel has been recommended such that (23.43) = ~ ~ ~ e n s C ~ e a t t r e a t Q j p r o c e s s . The datain Tables23.4-23.6wereobwhere tained from [23.22]. From the foregoing tabular data, it can be deter-
LE 23.4.
23.5.
Treatment
@heattreat
CtChem
vs Steel Type
Steel Type
%hem
MSI 52100 M50 M5ONiL
3 2 4
vs Hsat Treatment
Heat
Air-melt Carbon vacuum degassed (CVD) Vacuum arc remelted (VAR) Double V U Vacuum induction melted vacuum arc remelted (VIWAR)
ckeattreat
1 1.5 3 4.5 6
896
LOAD APPLICATION
LE 23.6, Process
Qprocess vs
AND LIFE FACTORS
Metalworking Process
Metalworking
@process
Deep-groove ball bearing raceways ball Angular-contact bearing raceways Angular-contact ball bearing raceways-forged rings Cylindrical roller bearings
1.2 1 1.2 1
mined that an angular-contact bearing with forged rings manufactured steel would be given an (!&teel = 28.8. from W W m ~ 5 0 N i L No value has been universally established to date for hot isostatically pressed silicon nitride. Endurance testing of single balls in balllv-ring endurance test rigs (see Chapter 20) has, however, yielded highmultiples of the endurance for steel balls tested under the same loading conditions. To date, owing to relative weakness in tensile strength inbending tests and extremely low coefficient of thermal expansion (see Chapter 161, silicon nitride has been principally used for ball and rollers in high precision, high speed applications;for example, machine tool spindle bearings.
INATION ON F
A
~ LIFE I ~
~
Excessive contamination in the lubricant will severely shorten bearing fatigue life. The standards E23.3-23.51 and manufacturers' catalogs contain warning statements about this. Contaminants may be either particulate or liquid, usually water. Evensmall amounts of contaminants have significant limiting effects on bearing fatigue life. Particulate contaminants such as gear wear metal particles, alumina, silica, and so on will cause dents in the raceway and rolling element surfaces, which disrupt the lubricant films which tend to separate the rolling body surfaces. This tends to locally increase the frictional shear stresses produced inthe rolling-sliding contacts. Furthermore, the raised material on the shoulder of the dent tendsto cause stress concentrations. Ville and N6lias E23.231 using a two-disk, rolling-sliding test rig, demonstrated the stress concentration phenomenon. They further showed that combined rolling-sliding motion is a more severe condition with regard to generation of surface distress and fatigue than rolling alone. Both the film disruption and dent shoulder stress-increasing effects accelerate the onset of rolling contact fatigue and component failure. Figure 23.25 from a study of the effects of surface topography on fatigue failure by Webster et al. E23.241 indicates the relative risk of failure effected by the shoulders of dents. Hamer et al. i23.25-23.261 pointed out that even relatively soft particles can generate significant denting assuming bearing speeds and loads are sufficiently high. They
EFFECT OF C O ~ ~ ~ A T ON I OF N A
~ LIFE I ~
~
RISK PEAKSASSOCIATED DENTSHOULDERS
PEAKS ASS~CIA~ED SUB-SURFACE
1 0.5 0.0
-.5
0
Note: Z/B = 0 re^^ surfaw
F I ~ 23.26, ~ E Plot showing relative risk of fatigue failure throughout raceway subsurface including effectof dent shoulders (from i23.26-23.291).
further indicate that particle diameter to lubricant film thickness ratio appears to be a critical parameter with regard to denting. In Figs. 23.2623.29, Ville [23.271 characterized the types of dents generated by hard and soft particles. Using the same rolling-sliding disk endurance test rig of [23.23], Ville 123.271 showed that fatigue microspallingcommenceson the surface ahead of the dent in the friction direction. See Fig.23.29. Ville i23.271 also demonstrated the dent location using transient EHL analysis. See Fig. 23.31. Xu et al. C23.281 in an analytical and experimental study presented similar results to those of Ville [23.271.They also showed that thelocation of spa11 initiation depends onthe EHL and dent condition, and that spalls can initiate at either the leading or trailing edge of the dent, depending on the direction of surface traction.
F I G ~ E 23.26. Dent generated by a ductile metallic particle; for example, M50 steel (from 123.261).
23.2~. Dent generated by hard brittle material;for example, Arizona road dust (from 123.271).
.
Coarse dent generated by ceramic material a t slow speed; for example boron carbide or silicon carbide at 2.51 m/sec (98.8 in./sec) (from [23.271).
23.2~~.Fine dents generated by ceramic material at high speed; for example, boron carbide or silicon carbide at 20 m/sec (787.4 in./sec) (from 123.271).
(a) faster surface
(b) slower surface
23.29. Surface distress(in dottedellipses)associated with dent in rollingsliding motion, endurance tested 52100 steel components. Solid arrows signify rolling direction; dashed arrows signify friction direction (from 123.271).
Theexperimental data of Sayles and MacPherson [23.291 demonstrated theeffect of different levels of particulate contamination on bearing fatigue life by endurance testing cylindrical roller bearings with varying degrees of absolute lubricant filtration;for example, from 40 pm (0.0016 in.) down to 1 pm (0.00004 in.). Particulate matter was deemed
NATION ON F
~
T LIFE I ~
~
23.30. For the slower surface in Fig. 23.29, formation of microspalls ahead of the dent in the sliding direction on the surface of a 52100 steel component after 60 * lo6 stress cycles at 3500 MPa (5.08 lo6 psi). Rolling speed is 40 m/sec (1575 in./sec); slideroll ratio = +0.015. (from 123.271). Solid arrow signifies rolling direction; dashed arrow signifies friction direction.
-
typical of that generated in gearboxes. Figure 23.32 is a photograph of dents incurred under the ~ a y l e s - ~ a c ~ h e r sC23.291 on operating conditions with 40 pm (0.0016 in.) filtration. The dents areapproximately 1030 pm (0.0004-0.0012 in.) long and about 2 pm (0.00008 in.) deep. Comparing this depth with the thickness of a good lubricant film (A > l..5), it can be determined that the film can easily collapse in the dent. Evaluation of the ~ayles-~acPherson[23.29] operating conditions according to the methods discussed in Chapters 7, 9, and 12-15, indicates h values from approximately 0.45 at 40 pm (0.0016 in.) filtration to nearly 1using magnetic filtration. filter rating. According to Figure 23.33 from [23.29], showslifevs Fig. 23.33, significant improvement in life is achieved with finer lubricant filtration level; however, little improvement in life is achieved for filtration level less than 3 pm. Thus, there appears to be a limit to fine data tended to be confilter effectiveness. ~ a y l e s - ~ a c ~ h e r sE23.291 on firmed by Tanaka et al. E23.301, who, by using sealed ball bearings in an automotive gearbox, managed to increase fatigue life severalfold, compared to that of open (no seals or shields) bearings in the same application. Consideringthe lubricant film conditionsof the testprogram, the data of Fig. 23.33 have been curve-fitted to contamination-life factor equation (23.44). (23.44)
where FR is the filter rating. Based on test results using 3 pm and 49 pm filtration, Needelman and Zaretsky E23.311 recommend the following equation for the reduction of fatigue life due to particulate contamination:
~PLICATIO LOAD ~ AND LIFE FACTORS
2
0,15
1'5 $
a
0,05
0,5 0
25
3 d
0,l
1
E
0,05
0
E
0
,
2.51
*
"
-0.2
t
2
O'
-0'2
-0'1
0
0.1
0.2
I
0.3
-0.1
0.1
0
01
02
0.4
0.5
23.31. Comparison of results of numerical simulations and tests for two opposite slide-roll ratios. The upper row shows pressure distribution and film thickness over the line contact, the middle rows show zoom views of the film thickness around the dent and lines of constant maximum shear stress in the metal, and the lower row shows dent area micro~aphs(from [23.271).
==
1.8(.Z?R)-o*2s
(23.45)
It is apparent that equations for fatigue life reduction due to particulate contamination must be applied with care since they depend on the type of particles as well as thesize and on the bearing lubrication conditions. Presence of water in the lubricant is thought to effect hydrogen emb r i t t l e ~ eof ~ the t surface steel, creating stress concentrat~ons and shortening fatigue life. Figure 23.34, from i'23.321 illustrates thelife reduction effect
EFFECT OF C O ~ ~ A FATIGUE T I O ONN
~
LIFE
901
I 23.32.GDenting~caused by ~ particulate contamination (from L23.291).
13 12 11 10
-$ 0
a
g
9 7
6
X
a , 5 z E 4 3 3 2 1 0 Filter rating (pm) 23.33. Bearing fatigue life
vs degree of lubricant filtration (from [23.291).
02
23-34. Effect of water contamination on rolling bearing life (from [23.32]),
Table 23.7, from E23.321 for IS0 220 circulating oils indicates that the effect of water in the lubricant also varies with the composition of the lubricant. It appears that adding 0.5%water to lubricant A caused a life reduction by a factor of 3, which is consistent with the dataof Fig. 23.34. The results for the remaining lubricant variants, however, demonstrate a wide variation in bearing life indicating a significant endurance deLE 23.7. Bearing Fatigue Life for 0.5% Water Concentration in Various Lubricants Lubricant
LIO
L50
A (no water) A B C D E F G H I
59.2 20.8 66.7 33.4 54.5 20.8 23.9 32.1 66.8 47.4
171.4 61.2 195.7
77 195 61.2 168 143 410 122
C
O
03
~ FA TI^^ I ~LIFE~FACTORS ~
pendency on the lubricant composition as well as on the amount of contained moisture. Because of this, life reduction equations need to based on the combination of lubricant type, specific composition, and amount of contained moisture. Reference [23.221 provides further discussion on the effect of moisture in the lubricant.
It may be observed that nonstandard loading conditions can be accommodated in the estimation of bearing fatigue life by determining the bearing internal load distribution and applying the contact lifeequations presented at the beginning of this chapter. User-friendly computer programs to perform the calculations using the equations and methods presented in Chapters 7-9 are readily available for operation on inexpensive personal computers. To apply the effects of increased reliability, nonstandard materials, lubrication, and contamination, the simple approach of cascading the life factors has been most frequently taken, and is recommended in reference E23.221 and various bearing manu~acturerscatalogs. This approach uses the following equation: (23.46)
In (23.46):
% is the reliability-life factor as determined from Table 23.3. & is the material-life factor as determined from Tables 23.4-23.5 or similar empirical data.
% is the lubrication-life factor determined using Fig. 23.29 or similar empirical data. a4 is the contamination-life factor using equation (23.43), (23.45), similar empirically derived data. Lnais the adjusted fatigue life at reliability n.
or
This simple calculational approach has been used since the 196Os, when the first improvements in bearing steels and understanding of the riile of lubricant films in bearing fatigue endurance occurred. It does not however recognizethe interde~endency of the various life factors. fore, it must be used judiciously. For example,the ANSI standards E23.3, 23.41 state, “It may not be assumed that the use of a special material, process, or design will overcome a deficiency in lubrication. Values of greater than 1. should therefore normally notbe applied if Q3 is less than 1. because of such deficiency.” The contamination-life factor is strongly
904
LOAD APPLICATION
AND LIFE FACTORS
dependent on the thickness of the lubricant film compared to the size of foreign particulate matter; inlarge bearings it is far less significant than in small bearings.
Lundberg and Palmgren formulas represented a significant development in rolling bearing technology; however, it was not possible to correlate the fatigue of bearing surfaces in rolling contact so calculated to structural fatigue. Nor was it possible to correlate rolling contact fatigue in bearings to fatigue of elemental surfaces in rolling contact. Moreover, materials tested for structural fatigue have typically exhibited a fatigue limit as shown by curves similar to Fig. 18.1. According to Fig. 18.1, for cyclic loading less than the fatigue limit, fatigue, for all practical purposes, does notoccur. Onthe contrary7 rolling bearing applications according to the standard methods of calculations were characterized by a finite fatigue life in any application. Innumerable modern rolling bearing applications, however, have defied this limitation. Data for bearings of standard design, accurately manufactured from high-quality steel-that is, having minimal impurities and homogeneouschemical and metallurgical structures L23.321, have demonstrated that infinite fatigue life is a practical consideration in some rolling bearing applications. Since the Lundberg and Palmgren formulas did not address the concept of a possible infinite fatigue life and did not relate to structural fatigue, an improvement in these formulas beyond application of some empirical life adjustment factors was required. The Lundberg and Palmgren theory considers that a fatigue crack begins at a point below the surface in rolling contact, at which point a large-magnitude orthogonal shear stress coincides with a weak point in the material. Such weak points are assumed to be randomly distributed throughout the material. As demonstrated in Chapter 6, the orthogonal shear stress resultsfrom a concentrated load applied normal to the surfaces in contact, giving a Hertzian surface stress distribution similar to that of Fig. 6.6 for point contact.Figure 6.13 shows the orthogonal shear stress distribution in the subsurface material. Knowledge of pressure distributions in EHLcontacts has demonstrated that such pressure distributions can be substantially different from the pure Hertzian distribution indicated in Fig. 6.6. different an EHL pressure distribution can be compared distribution. Moreover, ifthe surfaces are nonideal-that is, not smoothbut having perturbations on the smooth surface-then concepts of micro-EHL, as discussed in Chapter 13, obtain. Additionally7in their analysis Lundberg and Palmgren did not accommodate surface shear stresses, which can
substantially alter the subsurface shear stresses, as indicated byFig. 6.18. In Fig. 6.i8, the subsurface stress determined is that from the distortion energy theory of von Mises; a similar situation would occur consideringsubsurfaceorthogonal shear stresses. There is a difference of opinion concerning which subsurface stress effectively causes rolling contact fatigue. The depths below the surface at which maximum orthogonal shear stress and maximurn von Misesstress occur are slightly different, the latter occurring at a depth approximately 50% deeper than the former. Whichever stress is considered most detrimental, the effect of surface shear stress is to bring the maximum subsurfaceshear stress toward the surface. The m ~ i m u mof the subsurface shear stress is estimated to occur on the surface when the applied surface shear stress is approximately 30% of the applied normal stress. In general, shear stresses of this magnitude do not occur over the entire concentrated contact area in an effectiveEHLcontact.Such stresses couldoccur in micro-EHL contacts existing within the overall contact area. m e n the maximum subsurface shear stress occurs at the surface, the possibility of surfaceinitiated fatigue, as compared to subsurface-initiated fatigue, occurs. Tallian [23.33] considers competing modes of fatigue failure-that is, surface initiated and subsurface initiated. Rigorous mathematical solution requires the consideration of failure at any point in the material from the surface intothe subsurface, consistent with the applied stresses both normal and tangential to the surface. The Lundberg and Palmgren theory did not cover this generalized stress situation. The basic equation stated by Lundberg and Palmgrenis
In this equation T~ is the masimum orthogonal shear stress, zo is the depth at which it occurs, V is the stressed volume, N is the number of stress cycles, and 8 is the probability of survival of the stressed volume. rom Fig. 23.35, the stressed volume of Lundberg and Palmgren is proportional to the product of the major axis of the contact ellipse,the depth to m ~ m u m shear stress, and the circumference of the r a c e ~ a ycontact. This proportionalityis valid for geometrically perfect, contacting surfaces between which only normal stresses occur. If significant surface shear e tendency toward surface-initiated fatigueis ignored. and Palmgren theory also does not account thefor bearing operatingtemperature and its effect on material properties, let alone account forthe eEect of temperature on lubrication and henceon surface shear stresses. Furthermore, the theory does not consider the rate at is absorbed by the surfaces in rolling contact.
AND LIFE FACTO^^
" L " " " "
23.3~. Volume at risk to fatigue in rolling contact according to Lundberg and Palmgren.
speeds are used simply to convert predicted fatigue lives in millions of revolutions to time values. Nor are hoop stresses induced by ring fitting on shafts or in housings or by centrifugal loading accommodated. Finally, the development of microstructural alterations and residual stresses below the raceways, induced by rolling contact, as indicated by Voskamp E23.501 must be considered.
Considering the foregoing Lundberg-Palm~entheory limitations, Ioannides and Harris L23.341 developed the basic equation (23.48)
In this formula, a fatigue crack is presumed incapable of being initiated until stress criterion Tiexceeds a threshold value of the criterion Tlimit at a given elemental volume AVi. It is evident that the crack threshold criterion Tlimitcorresponds to an endurance limit. To be consistent with the ~undberg-Palm~en theory, the stresscriterion would bethe orthogonal shear stress range 2r0; however, another criterion, such as the von Mises or m mum shear stressmay be used.In equation (23.48), in lieu of the stress volume used by Lundberg and Palm~en-that is, 2 ~ ~ ~ 0 d , in which d is the raceway diameter-only the incremental volume over which Ti> 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 theLundberg-Palm~en relationship (23.47), is obtained. (23.49)
in which A is a constant pertaining to the overall material and z ' is a stress-weighted 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 Ioannides-Harris theory using octahedral shear stress as the fatigue-initiating stress to 62 different applications involvingdeep-groove and angular-contact 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 Lundberg-Palmgren 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 Ioannides-Harris 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~e-initiatingstress c~terion ter~inationand ap~licationof all residual, applied, and induced on the material of the rolling element-race~aycontacts elopment and ap~licationof a stress-life 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 TH-BBAN* and -RBAN.* Moreover, it should be apparent thatthe concept of a stress-life factor fulfills the requirement for the interdependency of the various fatigue life-influencing 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 stress-life factor, which comprises all life-influencing stresses acting on rolling element-raceway 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 reliability-life factor q,which is not stress-related. 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 stress-life 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 Mises-Hencky 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 failure-initiating 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 non-standard loading, is distributed over the rolling elements. The rolling element loads which are applied perpendicular to the contact areas result inpressure-type (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~lar-contactball 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 element-raceway 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 rolling-sliding 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 1500-3500 MPa (2.18-5.08 * lo5 psi) under lubricant film parameter A. ranging approsimately from 0.6-1.3. 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~lar-contactball 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), 5-10 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 life-r 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 oil-IS0
13/10--p,
= 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. 23.42-23.45, 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 oil-IS0
17/14-&5
2
75 (from
-45. CLvs
K
and dm for filtered circulating oil-IS0
19/16--p,,
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 bath-IS0
‘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 bath-IS0
15/12-22
= 4-5 (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 forwell-machinedground gears; R, = 3 to 5 when bearin~sand gears are lubricate^ with the same oil. The C,values obtained using Figs. 23.42-23.50 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 STRESS-LIFE 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 bath-IS0 17/14-22
=
CL
6-7 (from 123.421).
/
0.5 0.4
0.3 0.2 0.1
23.50. C, vs
K
and d,for oil bath-IS0 21/18-22
=
11-15 (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 micro-cracks 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 failure-initiating 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 failure-initiating
. 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
0.1-0.5 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
slide-to-roll 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 press-fitted 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 press-fitted 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 stresses-for 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 near-surface 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 (arrow-tip lines ininserts) (from N6lias 123.451).
ises stress as thefatigue failure-initiating criterion, the stress tenalculated for everysubsurface point (x,y, x). The basicequaarris theory-that 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 cross-sectional 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 stress-life 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 a-a dgo,where zo is the depth to the m a ~ m u m shear stress roeIn the analysis of the stress-life 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 Lundberg-Palmgren 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 loading-shaded area indicates effective life-influencing stresses.
As compared to the Lundberg-Pal~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 element-raceway 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 failure-initiating criterion, Harris andMcCool E23.351 indicated some values for various bearing steels. In v-ring 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/ v-ring 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 curve-fitting 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 stress-life factor is presented as “axye)).
7
THE STRESS-LIFE FACTOR
.
Fatigue Limit Stress (Von Mises Criterion) for Bearing Steel Steel
C W 52100 Carburized steel (common types) V I W m M50 M5ONiL Induction-hardened 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 stress-life 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 <
item
M m
K
<0.41 0.41
0.7786 0.1909
0.8783 0.0576
1 < ~ < 4 0.7789 0.07174
23.12 for the various bearing types. Ioannides et al. 123.421 present sets of curves of QL vs K and bearing pitch diameter for the different basic bearing types; Fig. 23.55illustrates such curves forradial ball bearings. To assist in calculating K = v/vl, equation (23.67) may be used to define kinematic viscosity ml. (23.67)
where n is speed in rpm. When n
2
v1 =
1000,
4500 -
a
(23.68)
Ze 23.6. The common design, 209 deep-groove ball bearing of Example 18.7 was determined to have an Llo life of 49.64 lo6 revolutions. Since the bearing operates at 1800 rpm shaft speed, Llo = 459.6 hr. Assume that the bearing is just adequately lubricated by circulating oil, that is K = 1, and that it is properly protected from ingress of Contaminants by a lubricant filter of nominal rating 20 pm (there is only negligible debris in the bearing and lubricant). Determine the 99% reliability life of the bearing according to equation (23.46). For 99% reliability, Table 23.3 gives (& = 0.21
23.12.
item hrg
W
9.2
c/e
lc/brg,
w, and c/e for Bearing Types (from E23.401)
Radial Ball Bearings
Radial Roller Bearings
Thrust Ball Bearings
Thrust Roller Bearings
0.5 0.333 9.3
0.15 0.40
0.16 0.333 9.3
0.06 0.4 9.2
THE S T R E ~ S - L ~FACTOR E
23.55.
(&,L
vs CLFlim/Fe for radial ball bearings (from [23.421).
(23.46) From Example 18.7, the basic load rating for the bearing C = 32,710 N (7351 lb). This utilizes the b, factor; from Table 18.11 for deepgroove ball bearings, properly manufactured from good quality Cvp> 52100 steel, b, = 1.3.Tables23.3-23.5yield a material-life factor Q2 = 5.4 for this material. To utilize this value in (23.46), however,the basic load rating C must be reduced by b,; that is, C ' = Clb,,. Therefore, C' = 25,160 N (5654 lb). As indicated above, A = KO.' (approximately).Therefore, A = 1;contact surfaces are reasonably well separated by lubricant films, and % = 1. According to [23.31] the contamination-life factor may be estimated using equation (23.45); that is,
~ P L I C ~ T LOAD I O ~
=
21.78 IO6revolutions
Estimate the 99% reliability life of the 209 bearing of Example 23.6 using the IS0 Standard [23.5] integrated system method. To perform this calculation, it is necessary to apply a fatigue limit stress. As indicated by Fig. 23.55, this will be accomplished using a Several bearing manufacturers now supply this fatigue limit load Flim. parameter for each bearing in their general catalog. Accordingly, for their 6209 bearing, SW [23.531 give a value .Flimit = 915 N (206 lb). For normal cleanliness, Table 23.9 gives C,
FromFig.23.53, at
K
(~~
=
=
0.6; therefore,
1.3.
1,ttsL
P
L,
=
qcySL
=
0.21 1.1
=
13.55 * IO6revolutions (23.51)
Estimate the 99% reliability life of the 209 bearing of Example 23.6 using the computer program T H - B B ~ This . program calculates life accordingto the more detailed methods presented in thistext. The ball loading wasthat established in Example 7.1. The raceway surface roughnesses were input as 0.075 pm (3 pin.) M. The bearing was assumed to be lubricated with a light turbine oil filtered to 20 pm. Table 23.13 illustrates the operating parameters for the individual ball-raceway contacts. In the calculation of fatigue life, the fatigue limit stress was selected from Table 23.10 for CVI) 52100 steel;
3.113. Results of TH-BBAN Calculations Contact NO. 1 2&9 3&8 4&7 5&6
Q MPa N Ob) 4536 (1019) 2846 (639) 61 (14) 0 0
gmax,i
gmaX.0
(kpsi)
MPa (kpsi)
Ai
A0
3066 (445) 2624 (381) 723.4 (105) 0 0
2563 (372) 2193 (318) 609.6 (88.4) 0 0
0.866 1.045 1.718 0 0
1.145 1.338 2.052 0 0
aVmlimit = 750 MPa (109,000 psi). Based on the above values, the calculated Llo = 304.1 lo6 revolutions. Thus, L, = 0.21 * 304.1 ,. lo6 = 63.87 * IO6revolutions. The calculations using TH-BBAN tend to bemoreprecise than those of Examples 2.6 and 2.7. They use the total stress patternacting on the ball-raceway contacts, not just the Hertz stresses. The calculational results do not include fatigue of the ball surfaces.
The Lundberg- almgren theory to predict fatigue life was a significant advancement in thestate-of-the-art of ball and roller bearing technology, affecting the internal design and external dimensions for 40 years. The EHL theory, introduced by Grubin and further advanced by scores of researchers, initially affected bearing microgeometry, but later, because of the possibility of increased endurance together with improved materials resulted in "downsizing" of ball and roller bearings. The IoannidesHarris theory, in its ability to apply the total stress pattern to predict life in any bearing application, and inits use of a fatigue stress limit for rolling bearing materials, carries the development to the next plateau by substantially increasing understanding of the significance of material quality and concentrated contact surface integrity. It is now apparent that a bearing, manufactured from material that is clean and homogeneous, whichoperates with its rolling/sliding contacts free from contaminants, andwhich is not overloaded, maysurvive without fatigue. In fact, Palmgren E23.541 initially considered the existence of a fatigue limit stress; however, the rolling bearings sets which were tested in the development of the Lundberg-Palmgren theory failed rather completely under the test loading, and he abandoned the concept. During the early 198Os, when the Ioannides-Harris theory was under development, as indicated by [23,551 and other modern bearing fatigue investigations, fatigue testing consumed substantial calendar time, often requiring year and more with no bearing failures after more than 500 million revolutions. This chapter converts the Ioannides-Harris theory into practice. The life theory is stress-based, as opposed to the factor-based, modifiedLundberg-Palmgren theory (Standard methods [23.3-23.51) exemplified by equation (23.46). Rather, the Ioannides-Harris theory utilizes the base Lundberg-Palmgrenlife equations (23.15),(23.1)-(23.2), or (23.8)integratesthe effecton (23.10) together with a single factorwhich fatigue life of all stresses acting on the bearing contact material. An accurate life prediction for any bearing application only depends on the successful evaluation of the appropriate stresses. With the appli~ation of
*
932
PLICATION LOAS) FACTORS AND LIFE
modern computers and computational methods, these stresses arebeing subjected to increasingly greater scrutiny. With the current availability of powerful, inexpensive desktop and laptopcomputers, engineers worldwide have the capability to use rolling bearing performance analysis computer programs which can effectively employ the methods described in this textfor such analysis.
23.1. G, Lundberg and A. Palmgren, “Dynamic Capacityof Rolling Bearings,”Acta Polytech. Mech. Eng. Ser, 1, Royal Swedish Acad. Eng., No. 3, 7, (1947). 23.2. G. Lundberg and A. Palmgren, “Dynamic Capacity of Roller Bearings,”Acta Polytech, Mech. Eng. Ser. 2, Royal Swedish Acad, Eng., No. 9, 49, (1952). (ANSI I 23.3. AmericanNational Standards Institute, American National Standard M M A ) Std. 9-1990 “Load Ratings and Fatigue Life for Ball Bearings” (July 17, 1990). 23.4. AmericanNational Standards Institute, American National Standard (ANSI I MA) Std. 11-1990“Load Ratings and Fatigue Life for Roller Bearings” (July 17, 1990). 23.5. International Organization for Standards, International Standard I S 0 281 f 1, “Rolling Bearings-Dynamic Load Ratingsand Rating Life (2000). 23.6. T. Harris, “Predictionof Ball Fatigue Lifein a BallfV-Ring Test Rig,”ASMETrans., J: Tribology, Vol. 119, 365-374 (July 1997). 23.7. T. Harris, “How to Compute the Effects of Preloaded Bearings,’’Prod. Eng., 84-93 (July 19, 1965). 23.8. A. Jones and T. Harris, “Analysis of Rolling Element Idler Gear Bearing Havinga Deformable Outer Race Structure,” M M E Trans., J. Basic Eng., 273-277 (June 1963). 23.9. T. Harris and J. Broschard, “Analysisof an Improved Planetary Gear Transmission Bearing,”ASME Trans., J: Basic Eng., 457-462 (September 1964). Lu23.10. T. Harris, “Optimizingthe Fatigue Life of Flexibly Mounted, Rolling Bearings,” brication Eng., 420-428 (October 1965). 23.11. T. Harris, “The Effect of isa alignment on the Fatigue Life of Cylindrical Roller Bearings Having Crowned Rolling Members,” ASME Trans,, J: Lubr. Tech., 294300 (April 1969). 23.12. E. Bamberger, 11‘.Harris, W. Kacmarsky, C. Moyer, R. Parker, J. Sherlock, and E. t for Ball and Roller Bearings, ASME Engineering Zaretsky, Life ~ d j u s t m e nFactors Design Guide (1971). 23.13. A. Palmgren, Ball and Roller Bearing Engineering, 3rd Ed., Burbank, Philadelphia (1959). 23.14. L. Houpert, “BearingLife Calculation in Oscillatory Applications,’’Tribology Tkans., Vol, 42, 1, 136-143 (1999). 23.15. T. Tallian, ‘Weibull Distribution of Rolling Contact Fatigue Life and Deviations Therefrom,’’ASLE Trans., 6 (1) (April 1962). 23.16. T. Harris, “Predicting Bearing Reliability,”Mach. Des., 129-132 (January 3, 1963). 23.17. T. Tallian, L. Sibley, and R. Valori, “Elastohydrod~amicFilm Effects on the LoadLife Behavior of Rolling Contacts,” ASMEPaper 65-LUBS-11, ASME Spring Lubrication Symposium, NY (June 8, 1965).
~ F E ~ ~ C E S
933
23.18. J. Skurka, ‘‘Elastohydrod~amicLubrication of Roller Bearings,” ASME Paper 69LUB-18 (1969). 23.19. T. Tallian, “Theory of Partial Elastohydrod~amicContacts,’’ Wear, 21, 49-101 (1972). 23.20. T. Harris, “The Endurance of Modern RollingBearings,”AGMA Paper 269.01, American Gear Manufacturers’ Ass’n Roller Bearing Symposium, Chicago (October 26, 1964). 23.21. M. Schouten, kbensduur vanOverbrengingen, TH Eindhoven (November10,1976). 23.22. STLE, Life Factors for Rolling Bearings, E. Zaretsky, Ed. (1992). 23.23, F. Ville and D. Nklias, “Early Fatigue Failure Due to Dents in EHL Contacts,” Presented at theSTLE Annual Meeting, Detroit (May 17-21, 1998). 23.24. M. Webster, E. Ioannides, and R. Sayles, “The Effect of Topo~aphicalDefects on the Contact Stress and Fatigue Life in Rolling Element Bearings,” Proc. 12th Leeds&yon Symposium on Tribology, 207-226 (1986). 23.25. J. Hamer, R. Sayles, and E. Ioannides, “Particle Deformation and Counterface Damage When RelativelySoft Particles Are Squashed between Hard Anvils,” Tribology Trans., Vol. 32, 3, 281-288 (1989). 23.26. R. Sayles, J. Hamer, and E. Ioannides, “The Effects of Particulate Contamination in Rolling Bearings-A State of the Art Review,” Proc. Inst. Mech. Eng., Vol. 204, 29-36 (1990). 23.27. F. Ville, Contamination of Oil by Solid Particles, I n ~ n t a t i o nProcess and Surface Fatigue, Ph.D. thesis, Laboratoire de MBcanique des Contacts, UMR CNRS-INSA No. 5514, Lyon, France (November 16, 1998). 23.28, G. Xu, F.Sadeghi, and M. Hoeprich, “Dent Initiated Spa11Formation in EHL Rolling/ Sliding Contact,”ASME Trans, J Dibology, Vol. 120, 453-462 (July 1998). 23.29. R. Sayles and P. MacPherson, “Influenceof Wear Debris on Rolling Contact Fatigue,” ASTM Special TechnicalPublication 771, J. Hoo, Ed., 255-274 (1982). 23.30, A. Tanaka, IC. Furumura, and T. Ohkuna, “Highly Extended Life of Transmission Bearings of ‘Sealed-Clean’ Concept,”SAE Technical Paper, 830570 (1983). 23.31, VV. Needelman and E. Zaretsky, “New Equations Show Oil Filtration Effect onBearing Life,” Power Transmission Design, Vol. 33, 8, 65-68 (1991). 23.32. R. Barnsby, T. Harris, S. Ioannides, W. Littmann, T. Losche, Y. Murakami, W. Needelman, H. Nixon, and M. Webster, “Life Ratings for Modern Rolling Bearings,” ASME Paper 98-TRIB-57, Presented at theASME/STLE Tribology Conference,Toronto (October 26, 1998). 23.33. T. Tallian, “On Competing Failure Modes in Rolling Contact,” ASLE Trans, 10, 418439 (1967). 23.34. E. Ioannides and T. Harris, “A New Fatigue Life Model for RollingBearings,”ASME Trans., J. Tribology, Vol. 107, 367-378 (1985). 23.35. T.Harris and J. McCool, “On the Accuracy of Rolling Bearing Fatigue Life Prediction,” ASME Trans., J. Tribology, Vol 118,297-310 (April 1996). 23.36. T. Harris, “Predictionof Ball Fatigue Life in a Ball/V-Ring Test Rig,” ASME Trans., J Tribology, Vol. 119, 365-374 (July 1997). 23.37. R. Juvinall andIC.Marshek, Fundamentals ofMachineComponent Design, 2nd Ed., Wiley, New York (1991). 23.38. H. Thomas and V. Hoersch, “Stresses Due the Pressure of One Elastic Solid upon Another,” Univ. Illinois, Bull., 212 (July 15, 1930). 23.39. D. NBlias, M.-L. Dumont, F. Couhier, G. Dudragne, and L. Flamand, “Experimental and Theoretical Investigation of Rolling Contact Fatigue of 52100 and M50 Steels Under EHL or Micro-EHL Conditions,” ASME Trans. J Tribology, Vol. 120, 184190 (April 1998).
934
~PLICATIO LOAD ~ ANI) LIFE F ~ C ~ O R ~
23.40. 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 Trans., J Tribology, Vol 109, 627-629 (1987). 23.41. X. Ai and H. Cheng, “The Influence of Moving Dent on Point EHL Contacts,” Tribology Trans., Vol. 37, 2,323-335 (1994). 23.42. E. Ioannides, G. Bergling, and A. Gabelli, “An Analytical Formulation for the Life of Rolling Bearings,” Acta Polytechnica Scandinavica, Mech. Eng. Series No. 137, Finnish Institute of Technology (1999). 23.43. International Organization for Standards, International Standard I S 0 4406, “Hyof Contamination by Solid draulic Fluid Power-Fluids-Method for Coding Level Particles” (1987). 23.44. International Organization for Standards, International Standard I S 0 4372, “Hydraulic Fluid Power- Filters-Multi-Pass Method for Evaluating Filtration Performance” (1981). 23.45. D.NQlias,Contribution a L’etude des Roulements, Dossier #Habilitation a Diriger des Recherches, Laboratoire de MQcanique des Contacts, ~MR-CNRS- INS^ de Lyon No. 5514 (December 16, 1999). Standards Institute (19721, American National Standard 23.46. AmericanNational (ABMA/ANSI) Std 7-1972, “Shaft, and Housing Fits for Metric Radial Ball and Roller Bearings” (Except Tapered Roller Bearings)? (1972). Standards Institute (19871, American National Standard 23.47 AmericanNational (ABMA /ANSI) Std 19.1-1987,“Tapered Roller Bearings-Radial, Metric Design” (October 19, 1987). Standards Institute (19751, American National Standard 23.48 AmericanNational A MA /AiVSI)Std 19.2-1994, “Tapered Roller Bearings-Radial, Inch Design” (May 12, 1994).
23.50. A. Voskamp, “Material Response to Rolling Contact Loading,” ASME Paper 84TRIB-2 (1984). of Fail23.51. T. Harris and W.-K. Su, “Lundberg-Palmgren Fatigue Theory: Considerations ure Stress and Stressed Volume,”ASME Trans., J Tribology, Vol. 121,85-89 (1999). 23.52. H.-J. Bohmer, T. Losche, F.-J. Ebert, and E. Streit, “The Influence of Heat Generation in the Contact Zone on Bearing Fatigue Behavior,”ASME Trans., J Tribology, Vol. 121, 462-467 (July 1999). 23.53. SKF, General Catalog 4000 US, 2nd Ed. (1997). 23.54. A. Palmgren, “The Service Lifeof Ball Bearings,”Zeitschrift des Vereines Deutscher Ingenieure, 68(14) 339-341 (1924). 23.55. T. Andersson, “EnduranceTesting in Theory,” Ball Bearing J , 217,14-23 (1983).
Symbol
F h
iforce v R T Tb Tf u
a-
A 13
Units Shear force Mean plane separation, lubrication film thickness Normal Radius of surface Total temperature between contacting surfaces ulk temperature of a component Flash temperature generated during a tribological encounter Surface velocity ~ombinedsurface rou~hness Film thichesslsurface roughness ratio SU~SCRIPTS efer to individual surfaces
N (1b) p m (pin.)
N (1b) mm (in.)
"C ( O F ) "C (OF) "C (OF) mm/sec (in./sec) p m (pin.)
936
The forces transmitted in a bearing give rise to stresses of varying magnitudes between surfaces in both rolling and sliding motion, As a result of repeated loads in concentrated contacts, changes occur in the contact surfaces and in theregions below the surfaces. These changes cause surface deterioration or wear. Wear is the loss or displacement of material from a surface. Material loss may be loosedebris. Material displacement may occur by local plastic deformation or the transfer of material from one location to another by adhesion. Wlnen wear has progressed to the degree that it threatens theessential function of the bearing, the bearing is considered to hhve failed. Through experience and detailed failure analysis, the bearing engineer recognizes distinct classes of failure. They are listed in Table 24.1. These failures are defined without presupposing the exact mechanism by which they occur. They are defined in engineering terms based on a description of observations. Theobservations and theirclassi~cations reflect the remaining evidence of a complicated sequence of events involving many physical and chemical processes that preceded it, including those in the manufacturing of the original surfaces. Associated with the physical and chemical interactions on the surfaces are several mechanistic wear processes. They are listed in Table 24.2. Wearprevention is accomplished by forming lubricating films by hydrodynamic lubrication, elastohydrodynamic lubrication (E boundary lubrication. During the surface life of a bearing, the lubrication
E 24.1. BearingFailure Classification Due to Wear Mild mechanical wear Adhesive wear Smearing Corrosive (tribochemical) wear Plastic flow Surface indentation Abrasive wear Surface distress Pitting Fatigue spalling
.
MechanisticWearProcesses Plastic flow Adhesion Chemical reaction Fatigue
STRUC
37
S OF A L ~ R I C A T E CONTACT ~
and wear processes are interactive and competitive. The topic of wear cannot be divorced from the topic of lubrication, and it is essential that each contactarea within a bearing be considered as a tribological system. The tribological interactions of a system are described schematically in Fig. 24.1. Numerous technical options exist for improving wear performance through materials, lubricant base stocks, additives, finishing processes, and surface modification technologies.An appropriate bearing design for a particular application is derived fromthe synergistic assembly of many tribological contributions.
Load-carrying capacityis derived from the integral strengthof four general regions of a lubricated contact; these are indicated in Fig. 24.2. r o is- created ~ ~ ~by the generation of an EHL The E ~ ~ / ~ i cregion film, which ona global scale is derived fromthe hydrodynamic pressure generated in the inlet region of the contact; on a local scale it is derived from the micro-EH~action associated with the local topo~aphyof the surfaces. The EHL/micro-EHL region is typically less than 1 pm (40 pin.) thick.
E H ~ i c r o - ~ ~ Surface Film Formation Adhesion Abrasion Plastic Flow TribochemicalReaction
~ i b o l o ~ c aInduced ~ly Property Changes of System.Components
Tribological interactions of a lubricated concentrated contact system (from 124.111.
Core
4.2.
Structural elements of a lubricated contact.
The sur~ace~1~ region contains the outer layers of the surface, which consist of surface oxides, adsorbed films,and chemical reaction films derived from the lubricant and its additives. It is usually less than l pm (40 pin.) thick. The near-sur~aceregion contains the inner layers of the surface, including a finely structured and highlyworked by layer as well as other deformed layers. Thesedeformed layers, ch have a different microstructure than the material below them, may arise from surface pr~parationtechniques, such as grind g and honing, or they may be induced during operation (e.g., run-in). ardness andresidual stress can and they could be substantially different from the manear-surface region may extend to 50 pm (0.002 in.)
r concentrated contacts a s u ~ s u r ~ a cregion e can be defined, which m (0.002-0.04 in.) below the surface. This region is cted by the mechanical processes that produce the surface or the asperity-in~ucedchanges that occur durin microstructure and hardness may still be different from t stresses might still be present. rial below it, and s i ~ i f i c a n residual These stresses andmicrostr~ctures,however, are the resultof macroprosuch as heat treating, surface hardening, and forging. For typical an contact pressures the ~ ~ i m shear u m stress is located within surface region. In other words, the detrimental global contact stresses are co~municatedto the subsurface region where fatigue be. This fatigue is called subsurface-initiated. etween the near-surface region and the subsurface region is a “quiescent zone” that resides below the surface where the local asperity and
surface defect stresses are not significant and the stress field from the macroscopic Hertzian contact stress is not yet appreciable. This zone is quiescent fromthe point of view of stress and the accumulation of plastic flow and fatigue damage. The existence of the quiescent zone is impord to rolling contact fatigue. It inhibits the propagation of the stress field in the near-surface region and the stress field in the subsurface region. th regard to rolling contact fatigue, major material improvements have been made that reduce the risk of subsurface-initiated fatigue. The risk of su~ace-initiatedfatigue is now a more dominant factor.
es The lubricated contact system can be characterized by global processes associated with the lubricated contact as a whole, and by local rocesses associated with local features of the system-that is, those de the t o p o ~ a p ~ ofythe surface, the ~icrostructure of the unde terials, or the presence of wear debris. An. inherent practica with the control or prediction of wear is that the failure p is initiated generally on a local level, but it is influenced interaction of both global and local processes. Some local processes are definable on reasonably good scienti ever, their interactions in a real system seem to be the missing. This section discusses some of the important rocesses in connection with wear.
.
The ~o~mation of an ng the local stresses
film contributes to wear reduction by en the surfaces and by creating a lu-
The formationof an sure generated in the ounds, which allows the viscous properties of' and the operating co tool for predicting tions; however, it is not sufficient to predict wear. This is partly because L is primarily an inlet phen enon; that is, its major action occurs m a region displaced from the rtzian region where the more local
,/-PRESSURE I I
events involved in wear initiation take place. The severity of these local events can be significantly influenced by the EHL process by the thickness of the EHL film. It determines what may be called the “degree of asperity interaction.” Thus, the EHL process is viewed as a quantitative €oundation upon which a predictive capability of wear can beestablished by incorporating several less quantifiable processes. ~ ~ r T ~e ~~ ~c e e~ Surface ~ ~ ~ temperature r e . is not a lubrication process but a key link between lubrication and wear because it significantly influences the viscous properties of the lubricant that control the thickness film, and it is a major driving force in theformation of chemfilms. ~dditionally,it determines the rate of lubricant degradation and influences the strength of surface films as well as the flow properties of the material inthe near-surface region. ~onsequently,it is not s~rprising that the total temperature level is a frequently used criterion for failure. From a simplistic point of view the total temperature (27) is the sum of a bulk temperature (Tb)of the bearing component and the flash temperature (Tf)associated with the instantaneous temperaturerise derived from the friction within the lubricated contact. Flash temperature m arise from the traction of the lubricant film as well as from the ener dissipated from the adhesion, plastic flowof surface films, and deformation of the material within the near-su~aceregion. The global magnitude of Tf can be predicted if simplif~ngassumptions about the coefficient of friction and convection heat transfer are made.
,A
= hla;, a mostuseful engineer in^ 1m thickness h to the t a; of the interacting surfaces. It is a gree of asperity interaction. Thus, whe er thanfor lowerA, beca~selocal asp
~ I ~ O L O PROCESSES ~ I C ~ ASSOCIATED WITH WEAR
941
been significantly reduced. Its connection with surface-initiated fatigue seems to be more obvious than failure modes associated with wear. The latter failure modes generally appear at low A (e.g., A < I) where, unfo~unately,it loses much of its meaning. When cr is on the same order of magnitude as h, the surface topography becomes intimately involved in the lubrication process itself in the form of micro-EHL. This comes about from a global standpoint where the orientation of the topographical features can infhence the average film thickness. It also comes about from a local standpoint through the generation of micro-EHL pressures associated with topographical features, as shown in Fig. 24.4. Measurements [24.2] imply that theenergy dissipation due to friction becomes concentrated at specific local topographical sites, giving rise to local stresses and temperatures even without physical contact.
~ o ~ n d Lubric~tion. ~ r y It is well known that surface films are important to boundary lubrication because they prevent adhesion and provide a film that is easy to shear. These films may be in the form of oxides, adsorbed films fromsurfactants, and chemical reaction films from other additives. These surface films are schematically shown in Fig. 24.5. Their reactions and interactions are complex. Most studies on the subject have focused on the chemical identificationor phenomenolo~cal effect of surface films, but little is known about the mechanism of protection, the means of removal, or the rate of reformation. At high tem-
24.4.
Micro-EHL lubrication.
2
1
ADSORBED
FILM OXIDE
L
I
METAL
24.5.
METAL
I
Surface films.
peratures the oxidation of the base fluid can contribute to surface film formation. There have been manystudies on the catalytic effect of metals on the bulk oxidation of fluids. Similar oxidative processes can occur under the thermal stress environment in the contact region where intermediate oxidationspeciescan react with the surface or organometallic material. These reactions can influenceboundary lubrication in several ways, such as by corrosive wear, by competition with other additives, or by forming polymeric material-that is, a friction polymer. The contribution of surface films in preventing wear is complex, The time and spatial distribution of the various surface films within the contact seems to be important, particularly with regard to the accumulation of material (including debris of all sorts) in depressions and the formation of films at asperity sites. In view of the complexity of surface films, one wonders what the real lub~cating“juice”is in a real system.
~ ~ t e ~ ~ c Perhaps t ~ o ~ the s . most important quantity in connection with wear is the deformation attributes of the near-surface region. It is unfortunate that there is little understan~ingof near-surface mechanical properties or the attributes needed to complement the various lubricating mechanisms to improve wear resistance. To maintain surface integrity, the near-surface region must prevent micro~ractureand maintain a viable surface finish even in the presence of plastic flow. he interac~ionof tribological processes in an asperity encounter is shown sc~ematicallyin Fig. 24.6.
~ O L O ~ P~OCESSES I C ~ ASSOC~TE WITH ~
”2
\
.
Surfaceplastic flow.
The severity of interaction is reflected in thenormal load influenced by the thickness of the EHL film h. shear force F is inL films,along with fluenced by the various surface films and micr exact mechanism the flow properties inthe near-surface region whereby shear stressis applied to the near-surface region is not known. This could come about through metal-to-metal adhesion but it is also possible to have sufficient compressiveand shear stressesapplied locally through a thin lubricant film. In any case, the severity of intbraction is important to the initiation and propa~ationof wear. It will determine whether the result is (I) a benign elastic encounter, (2) a further accumulation of plastic fracture sites that can lead to the generation of wear particles (e.g., microspalling, mild mechanical wear,and delamination), (3) oxidative or corrosive wear, or (4) the advancing of * transfer, whichcan lead to smearing. ith these events are discussed in conhe wear processes ass nection with a descri~ti only recognizedwear modes. ear can be defined in terms of four “mechanistic wear processes”: adhesion, plastic defor~ation9 fatigue, and chemical reactions i24.31.The or “pitized wear modes in bearing technology, such as ‘6smearin~” ting,” are not singly connectedto these wear processes but are associated e interaction of the processes both si~ultaneouslyand sequenhe importance of connecting the wear process with the commonly accepted failure mode is associated with engineer in^ decisions required to overcome wear problems through lubrication, material selection, design, or allowable operating conditions.
944
rn
A ~ ~ e ~ i oUnder n , high normal and tangential stresses, boundary films trapped between contacting asperities can be stretched until they rupture, which allows the formation of metal-to-metal contact on an atomic scale and gives rise to strong adhesive or welded junctions. With relative motionbetween the contacting asperities, junction growth occurs by plastic deformation. Fracture ultimately takes place, and it can occur at a location other than theoriginal interface, resulting in material transfer from surface to surface. The formation and rupture of adhesive junctions is accompanied by very high local temperatures that can formreaction films on the newly formedsurface and change the mechanical properties of the underlying material. “Adhesivewear” occurs when the adhesive transfer of material is the important or controlling mechanism. Smearing. Smearing is adhesive wear on a large scale, which occurs between rolling element and raceway when sliding is substantial. The severe plastic deformation that accompanies smearing is shown in Fig. 24.7. Smearing is sometimes called scuffingor galling. The precise mechanism of smearing is not well understood but it does involve the gross failure of the surface and is accompanied by an increase in friction and contact temperature. A. current view of smearing E24.41 is that under conditions yet to be defined it is a gradual breakdown in the lubrication of interacting asperities, the nature of which may be boundary, microEHL, or a mi~tureof the two. Although the final smearing mode may represent the gross breakdown of various lubricating films and thenearsurface region, it may be triggered by the deterioration in surface topography as a result of adhesive wear or local plastic flow. eaction. The mechanistic wear processes associated with fatigue and plastic flow are the result of material deforination caused by t of wear and its control involves chemical restress. A s i ~ i f i c a n part action processes with the environment. The environment is defined as that portion of the contact system that is not an intrinsic part of the surfaces. The environment includes the surroun~ing atmosphere as well lubricating films. re chemical reactions should be distin~ishedfrom “tribochemical” reactions, which are a conse~uenceof the tribological interactions between the contacting surfaces. “Corrosion” results from reaction of the surface with the ambient environment under the prevailing ambient conditions; triboc~emicalbehavior is activated by mechanical interaction of the contacting surfaces. Corrosion often occurs on bearing components ~ecauseof improper handling or storage resulting from the absence or val of a ~rotectivefilm. A n example of corrosion is shown in Fig.
~ O L O ~ PROC~SSES I C ~ A S S O C ~ ~WITH D WEAR
4.7. Smearing. (a) Smeared material crossing preexisting finishing marks of a honed roller. (b) Replica electron micrograph showing adhesive wear [AI, original machining marks [B], and microcracks [Cl. (c) Etched metallogra~hiccross section through smeared area showing white etching bands at surface attributed to rehardening as a result of overheating (from [24.131).
Preventing adhesive wear is done by forming tribochemical films. These films may be formed from oxygen in the atmosphere or from antiwear or extremepressureadditives in the lubricant. “Tribochemical wear” generally involves a continuous process of surface film formation and removal. The formation process involves chemical reaction or adsorption of chemical species on the surface. The removal process results from mechanic all^ induced crack formation and abrasion of the reaction products in thecontact. The process introduces “clean,” that is, activated, local areas where new tribochemical films can be formed and subsequently removed. The tribochemical process introduces thermal chanicalactivation of the near-su~ace region, which can c chemical reactivity as a result of increased asperi changes in the microstructure and mechanical p
~
.
Corrosion resulting from reaction of the surf‘ace with the environment. Roller subsequently run in a bearing leaving a multitude of dark-bottomedpits, a condition that subsequently creates surface-originatedspalling (from f24.131).
near-surface layer due to high local temperatures and mechanical working. Under favorable operating conditions tribochemical reactions may be associated with “mild wear.” ild wear is associated with low wear rates and smooth surfaces frequently characterized by oxidation faces and subsequently removed-that is, oxidative wear [ Unfavorable operating conditions can produce “severe wear,’ surfaces are extensively disturbed and may be ch~racterizedby extensive adhesion and plastic flow rather than oxidative wear. Severe wear can r e v e ~ t e dby increasing the rate of chemical re~ctionsto form protective surface films at the same rate as clean activated local areas are enerated. Inthis way a balance canbe obtained between adhesive wear and “chemical wear.” ““Corrosive wear” is a term used when chemical wear dominates the adhesive wear mode by a wide margin. The rate of mica1 wear is controlled by additive composition and concentration. optimum additive formulation is achieved when there is a balance between adhesive and chemical wear for a given degree of contact se-
verity E24.71. This balance between adhesiveand chemical wear is shown schematically in Fig. 24.9.
lastic ~ e f o r ~ a t i o n . epending on geometry, relative hardness, and load, the shape of a contacting surface can be permanently deformed, on both a macroscopic and a ~icroscopicscale, as a result of mation. On a macroscopic scale the overload of rolling elements under static conditions can cause “Brinell marks” or distort the entire rolling track under operating conditions. A much more d~structivemacroscopic plastic deformation process occurs when the thermal balance between the rolling element bearing components becomesunstable because more heat is generated than is removed. A thermal run~waycan cause the bearing materials to soften and flow plastically until the entire bearing geometry has been destroyed. Almost all wear processes involve plastic flow on a microscopic scale. he plastic de~ormation thatoccurs from overru~ningof ha such as contaminants and wear debris, is “denting.,’ Figure example. “Plowing” occurs when there is displacement of mat presence of sliding or combined rolli “Abrasive wear” occurs whenthe plastic deformation leads to material removal and wear debris. The interaction of hard rolling el~ments softer separator materials often lead to abrasive wear, as shown in 24.12.
General plastic deformation of asperities and ridges on rolling contact surfaces is generally referred to as “surface distress,” or at least the initial stagesof surface distress. The final stages of surface distress involve
I 24.9.
1
I
A Lubricant reactivity
4
Adhesivelcorrosive wear balance (from [24.7]).
.IO. Debris dent showing local plastic deformation (surface distress) at a dent shoulder [AI as a result of substantial rolling over (from E24.131).
the loss of material through microfracture and pitting. Figures 24.13 and are examples of surface distress. ~ ~ ~ The i final ~ ~ mechanistic e . wear process is fatigue. by cyclically repeated stresses on the contact surface, permanent damage within the material. Damage begins as a er repeated stress cycles, cracks can propagate and eventually s of surface material. Fatigue may initiate andpropagate from the macrostresses induced in the subsurfa resulting in “spalling” characterized by relatively large craters. e can also be initiatedinthenear surface region as a icrostresses from perities or surface defects, such as dents the combined micro- and macrostress fi e ~uiescentzone and into the subsur~a
~ O L O G I VIEW C ~ OF
4.11, Plowing (from [24.131).
time spalling can occur. “Pitting” and delamination,’ occur when crack propa~ationis confined to the near-surface region.eseprocesses are associated with the final stages of surface distress discussed above. The microstructural material changes and theory for spalling fatigue are discussed further in Chapters 18, 22, and 23.
he previous section reviewedthe tri~ologicalprocesses associatedwith ubrication and wear. In a realsystem these processes interact and compete with one another in a complicated way so that the contrib the individual processes to the overall picture is not very clear. stan~ingthe parts of the tribological system is important for s bearing materials, l~brication,bearing selection, life prediction
.13. Inception of surface distress. (a)Surface distress begins with the plastic burnishing of asperity ridges [B]. ( b ) Higher magnification of (a).The small pit (probably preexisting) is surrounded by an area that has partially been burnished away (the slanting white mark is an artifact) (from [24.13]).
24.14. Final stages of surface distress of a ball bearing inner ring shown at three different magnifications. (a) Frosted appearance. (b) Multiple spalling of burnished surface. ( e ) Smooth appearance of plastically flowed material [A] (from 124.131).
decisions, and failure analysis. From an engineering standpoint it is essential to have a phenomenolo~calview of wear in addition to an understanding of the constitutive parts of the tribological processes.In this way the complicated tribological processes can be reduced to a description of simpler observed behavior as a result of esternal operating conditions. For example, the wear rate for a given system can be observed as a function of time, loads, velocities, temperatures, and lubricant film thicknesses. The phenomenological approach is useful if the behavior is orderly with respect to the controlling variables. Figure 24.15 is a schematic representation for unlubricated, boundary lubricated, and fluid film lubricated systems 124.81.
Type of lubricant 1. Un~ubricated 2. Boundary lubricated 3. Fluid film lubricated
Time (4
Fixed conditions Material Lubricant Viscosity Geometry Finish Lubricant quantity Ambient temperature Ambient atmosphe~
3
e L
m
2” Load
Temperature (4
Q)
+”
E. L
m
2” Velocity
Film thickness
(c)
(e>
24.15. Phenomenological view of wear (from [24.8]).
Several interesting phenomenological investigations have been conducted using simple sliding contact test rigs that demonstrate the usefulness of this approach. Begelinger et al. [24.9],using a simple sliding ball-on-ring test assembly, demonstrated the usefulness of the phenomenological approach for establishing a failure map as shown in Fig. 24.16. The map defines various transitional regions as functions of load, speed, and temperature. These transitions are identified by changes in friction and wear characteristics, and the regions separated by the transitions are characterized by various regimes of lubrication. Region I is associated with EHL and micro-EHL;region I1 is characterized by
3
VS
Y-
F I ~ U 24.16. ~ E Load-velocity failure map (from i24.91).
boundary lubrication; and region I11 reflects unlubricated or smearing (scuffing) behavior. One of the most useful parameters for characterizing the phenomenological viewof lubricated wear is A. Chapters 14 and 18 described how this parameter can be used along with a detailed characteri~ationof surface topography to predict asperity contact severity as a function of the mean plane separation of the surfaces. Detailed studies were conducted in [24.10] to show the wear behavior and fatigue behavior as functions of A. The result is shown in Fig. 24.17.
The ultimate failure of surfaces in a rolling contactbearing is the result of a complex sequenceof events involving tribological processes of lubrication and wear. Fre~uently, anevent that initiates a wear process can be successfully rescued by a lubrication process. On other occasions a minor wear process proceedingat one contact location can initiate a more devastating wear process at a more critical location. Examples of these will be given. Figure 24.18c,d shows debris dents typical of those found in bearings contaminated with hard particles [24.11]. The stress concentrations at the shoulders of defects, similar to debris dents, frequently lead t o initiation of spalling fatigue as shown in Fig. 24.18a,b. The role of EHL in reducing the stresses at the defect site was studied in i24.121. If the
- 0.1 - 0.05
0
.$1 El
cr
0.01
0.001 0.0005
jj
2
.-0
3
t 0.0002
1 \
area
0.000 1
\ 40.00005 0.4
0.6 0.8 1
1.5 2
3
4
5
A F 24.17. Radiotracer-measuredrollingfour-ballwear rate andfractionalasperity contact area vs A. For a, = 0.13 mm2, A = Mineral oil, B = Synthetic ester, C = Sodium grease.
defect dimensions are small compared with the inlet dimensions where EHL pressure is generated, the stress concent~ationat the edge of the defect canessentially be eliminated. It was further found that with EHL, the trailing edge of the dent should have higher local stress than the leading edge; however,the micro-EHL behaviorat the leading edge gives a much lower film thickness, as shown in Fig. 24.19. This corresponds to frequent observations of fatigue spalls initiating at the trailing edge of defect sites, as shown in Fig. 24.18c,d, and that surface distress appears more frequently at the leading edge due to the lower film thickness. This can also be seen in the same figure. When the defects are large, as shown by the groove in Fig. 24.20a, the local EHL is not effective in providing surface separation, which results in surface interaction and the initiation of boundary film formation. The identification of the boundary films is shown in Fig. 24.20b,c. An example of interactive wear modes is shown in Fig. 24.21. In Fig. 24.21a, hard particles are shown imbedded in the cage material of a
CO
ATIONS FOR
P~OTECTION
I
Direction of Ball Travel
(c)
F ~ G ~ R24. E18. Spallsand prespall cracking (from [24.11]).
spherical roller bearing. The hard particles initiated an abrasive wear process on the roller in Fig. 24.123. The abrasive wear has altered the geometry of the roller, as shown in Fig. 24.21~. Stress concentrations induced by the altered profile of the roller produced the final failure of the bearing by way of a fatigue spa11 on the inner ring, as shown in Fig. 24.21d.
eco~men~ations for wear protection can be derived from a knowledge of the tribological processes associated with lubrication and wear along with a good phenomenological view of wear. Table 24.3 contains a summary of the wear processes and theirtribological implications.A detailed overview of the control of rolling contact failure through lubrication is given in L24.101.
4
956
co
0
0
d-
0
0
l
d-
0
I
0
I
1")
0 nJ
0
I
2
R E C O ~ ~ ~ T I O FOR N SWEAR, PROTEXTION
Auger Maps iron phosphate
chrome oxide
FIGURE 24.20. Boundary films at defect sites.
Concave end”+-
. ( a ) Hard particles embedded in separator. ( b ) SEM of embedded particle. (e)Abrasively worn rollerand typical surface profile. ( d )Fatigue spa11 on inner ring due to stress concentration of worn roller and/or debris denting.
The topicof wear in rolling element bearings is complex, and an attempt to present the subject in simple design criteria and formulas has been avoided. Wear is not an intrinsic property of materials but a complex sequence of events of an entire system. The difficultiesof wear prediction and prevention are associated with several factors. First, the stressfield that drives the process of wear is not well defined and occurs on both macro- and microscales. Second, the properties of materials that resist wear are usually those associated with the near-surface region generally characterized by insufficiently defined microstructures and mechanical
REF~RE~~ES
961
and chemical properties significantly different from the bulk material. Third, chemical reaction processes occur in parallel with stress-induced mechanical deformation and ultimately are interdependent, Fourth, the description of wear modes is generally confused by the proliferation of terminology and a lack of definitive connection between wear mode and wear process. Consequently, the subject of wear has generally been presented not in quantitative terms with spegific materials of construction but in terms of tribological processes associatedwith specific structural elements of a lubricated contact. Nevertheless, in [24.14] a method to predict wear life in ball and roller bearings was presented. Since it is possible to eliminate significant wear in rolling bearings by effective lubrication, sealing, a n d or shielding in most applications, the need to be able to predict wear rate in a bearing application is far less important than the need to prevent wear. Consequently, effortsby most major rollingbearing manufacturers have been aimed at prevention, and the approach in E24.141 has not been generally used in industry.
24.1. H. Czichos, “Importance of Properties of Solids to Friction and Wear Behavior,” TriboZogy in the 8@s, NASA Conference Publication 2300, Vol. I, Sessions 1 to 4, Proceedings of an International Conference held a t NASA-Lewis Research Center, Cleveland, Ohio (April 18-21, 1983). 24.2. L. Wedeven and C. Cusano, “Elastohydrod~amicContacts-Effects of Dents and Grooves on Traction and Local Film Thickness,” NASA TP 2175 (June, 1983). 24.3. A. Dorinson and K. Ludema, Mechanics and Chemistry in Lubrication, Tribology Series, 9, Elsevier (1985). 24.4. A. Dyson and L, Wedeven, “Assessment of Lubricated Contacts-Mechanisms of Scuffing and Scoring,” NASA TM-83074 (February, 1983). between Mild and 24.5. J. Lancaster, “The Formationof Surface Films at the Transitions Severe Metallic Wear,’’ Proc. Roy. Soc. A. 273,466-483 (1963). 24.6. T. Quinn, “NASA Interdisciplinary Collaboration in Tribology-A Review of Oxidation Wear,” NASA Contractor Report 3686. 24.7. C. Rowe, “Lubricated Wear,” in Wear Control Handbook, ASME, 143-160 (1980). 24.8. M. Peterson, “Design Considerations for Effective Wear Control,’’ in Wear Control Handbook, ASME, 413-473, (1980). 24.9. A. Begelinger, A. deGee, and G. Solomon, “Failure of Thin Film LubricationFunctionOriented Characterization of Additives and Steels,” ASLE 23, 23-34 (1980). 24.10. T. Tallian, “Rolling Contact Failure Control Through Lubrication,”Proc. Inst. Mech. Engrs. 182,205-236 (1967-68). 24.11. R. Parker, “Correlation of Magnetic Perturbation Inspection Data with Rolling Element Bearing Fatigue Results,’’ASME Trans. J Lub. Tech. 97, Ser. F, No. 2, 151158 (Apr. 1975).
24.12. L. Wedeven, “Influence of DebrisDent onEHD Lubrication,” M L E (1978). 24.13. T. Tallian, G, Baile, H. Dalal, and 0.Gustafsson, Rolling Bearing Damage Atlas, SKF Industries, Inc., Revere Press, Philadelphia (1974). 24.14. J. Brandlein, P. Eschmann, L. Hasbargen, and R. Weigand, Ball and Roller Beari n g s - ~ h e oDesign, ~, and Application, 3rd ed., 205-212, Wiley, New York (1995).
Symbol
Description Peak displacement amplitude Rolling element diameter Decibels relative logarithmic amplitude Pitch diameter Reaction force by bearing Frequency Acceleration due to gravity Mass Speed Radius Radial deviation Time Waves per circumference Vertical displacement
Units mm (in.) mm (in.) mm (in.)
N Ob) rps, Hz mm/sec2 (in./sec2) kg (lb-sec2/ in.) rPm? rps mm (in.) mm (in.) sec mm (in.) 963
964
VIBRATION, NOISE,
ANI3 C O ~ I T I O N~
Symbol
z 4
h 0 C
i 0
O
~
~
O
Units Number of balls or rollers Radial deflection Wavelength Angular measure
mm (in.) mm (in.) rad
SUBSCRIPTS Refers to cage Refers to inner ring, shaft of raceway Refers to outer ring or raceway
This chapter provides a practical overview of bearing vibration. Where relevant, reference is also made to noise, sometimes resulting from excessive bearing vibration. Common bearing applications in which noise andlor vibration are important are described. Machine vibration or noise levels, whether excessive or not, are affected by bearings in threeways: as a structural element defining in part a machine’s stiffness; as a generator of vibration by virtue of the way load distribution within the bearing varies cyclically; as a vibration generator because of geometrical imperfections frommanufacturing, installation or wear and damage after continued use. Illustrations of manufacturing and installation problems are shown, in some cases with the use of vibration measurement taken from machines after bearing installation. Descriptions are also given of methods used in rolling bearing factories to evaluate bearing component quality, control manufacturing processes, and minimize bearing vibration. Detection of progressive bearing deterioration in operating machinery by vibration ~easurements has become more economicaland reliable in recent years. Some aspects of such machinery monitoring are considered.
In many cases objectionable airborne noise from a machine results from measurable vibration of machine components. Correlation between bearing noise and machine vibration measurements has been reported E25.1, 25.21. Therefore, with respect to rolling bearings, the terms “noise” and “vibration”usually denote similar and related phenomena. Their relative importance to the bearing user may differ, depending onwhere and how the machine is used.
R
~
ON .AND NOISE-SEN SIT^ APPLICATIONS
965
Regardless of which seems to be more important in a particular application, noise and vibration may both be indicators in new machines of quality problems with bearings, machine components, or assembly methods. Such problems canlimit the functional capabilities of the machine, and they can reduce the potential useful life of the bearings or the machine itself. In cases where a machine has successfully pedormed its function and is approaching the normal time for repair or replacement, the first indication may come fromincreased levels of noise or vibration.
~Qise-~en~iti~e
~~plicatiQns
The application that has been the major driving force for reduced noise is that of small and medium electric motors, primarily utilizing deepgroove ball bearings. Figure 25.1 shows such an application. The outer ring of the bearing at the left end of the motor is free to move axially under controlled thrust load of a spring to remove axial clearance from within the bearing. This allows for thermal expansion of the shaft and motor assemblywithout loss of preload whilesimultaneously preventing excessive bearing loads or distortion of motor components. Quiet running characteristics of the electric motor are required in office equipment and household appliances where noise may bean irritant. Noise is also a problem in building heating and airconditioning systems, where motor or fan support bearing noise can be transmitted and amplified through duct work or air columns. Also included in this category are drive systems of elevators, using larger electric motors with deep-
F I G n E 25.1. Electric motor.
groove ball bearings and cylindrical roller bearings and spherical roller bearings in pillow blocks to support cable sheaves. Aside fromir~tation, excessive noise in the latter application might make passengers concerned. Automotive applications also requiring quiet running performance include alternators (deep-groove ball bearings and needle roller bearings), transmissions, differ~ntials(tapered roller bearings), and fans. ~bjectionalnoise might becharacterized by volume or sound level and pitch or frequency. Sound from a machine may be more irritating if a particular frequency is dominant. Possibly even more objectionable are intermittent or transient sounds that vary with time in either pitch or volume at regular or irregular intervals. Such effects might be more easily heard than measured, since common measuring methods may be acquiring data over time periods that are long compared to short-duration transient sounds or vibrations. In addition, transient sound or vibration is sometimes most significant whena machine is coming up to operating speed or coasting down. Even in vacuum cleaners or dishwashers this effect is sometimes heard. ualitative audible evaluation of airborne sound from machinessuch ectric motors is performed as a routine inspection in many cases. Audible evaluation is also performed by processing the output of a vibration measurement transducer through a loudspeaker. This is useful for detecting transients and also in some cases foridentiffing the cause of excessive measured vibration. The vibration parameter monitored could be velocity or acceleration rather than displacement, which overemphasizes low-frequency Vibration. The United States Navy has made extensive demands on bearing manufacturers with respect to bearing vibration and noise reduction as well as boundary dimensionand runningaccuracy tolerances E25.31. This stems in part from the requirement to make submarines more difficult to detect by monitoring sound transmitted through water. time, improvements in reliability and reduced maintenance costs are achieved. Extensive research efforts on bearing vibration have been sponsored by the U.S. Navy E25.41.
Applications where bearing and machine vibration are more important than noise fall into two categories. In some cases the machine must be capable of high running and positioning accuracies to function properly. In other cases the major concerns are safety, if vibration causes catastrophic failure, and the economic impact of reduced machineutilization and increased repair cost if vibration foreshortens the life of components. Not only is noise intrinsically less important than vibration in these categories, but it may also be incapable of indicating a significant problem. This would occur if the predominant frequency of high-amplitude vibration falls outside the audible range; for example,rotating imbalance
ION AND NOISE-SEN SIT^ ~ P L I C ~ T I O N S
7
in a machine running at 1800 rpm (30 Hz). In addition, abnormal noise might be undetectable because of ambient noise where the machine operates or because of normal noise fromthe process the machine performs. Bearing applications where machine accuracy might be affected by vibration include machine tools.Grinding spindles often must be capable of producing components with size and two- or three-point roundness in.). Figure 25.2 shows a grinding wheel spinwithin a micron (4 X dle using precision double-row, cylindrical roller bearings and a doubledirection, angular-contact ball thrust bearing to achieve high radial and axial stiffnesses. The cylindrical roller bearings have tapered bores for accurately controlling preload. Precisionangular-contact ball bearings in matched sets are also widely applied in spindles. In addition to si,ze controland roundness, precision spindles must be capable of producing even finer levels of geometrical accuracy such as relatively low levels of surface roughness and circumferential waviness in.). Vibration can amplitudes of much less than a micron (4 X contribute to excessive roughness or waviness and can also producechatter, a more severe form of waviness that can cause permanent metallurgical damage to hardened steel parts. Other machines in which vibration might prevent the required accuracy from being achieved include rolling mills for sheet steel, paper, and chemical films. Computer disc drives are a further example, requiring nonrepeatable bearing runout accuracy of no more than one quarter to one half of a micron (1-2 X in.) for the spindle and head combined. Similarly, gyroscopebearings require good dynamic running accuracy as well as very low torque levels. Cases where running accuracy is not as important as safety and machine reliability often involve machines that are producing or transmitting highhorsepower, have massive rotating components, and are
Double row cylindrical
5.2.
Machine tool spindle.
running at high speeds relative to the size of the equipment. Eccentric mass produces large and potentially destructive forces in these applications. As discussed in Chapter 26, such equipment may operate at speeds above resonant frequencies, so large amplification of vibration could occur as equipment is run up to speed. Examples include compressors, pumps, and turbines. Applications in this section are examples where machine noise or vibration is important. More demanding applications continually arise, requiring greater accuracy, higher speeds and loads, and improved reliability. Therefore, bearing manufacturers have continuously emphasized improvementof bearing quality with respect to noise and vibration through ongoing developmentof machines and methods for manufacturing and inspection.
Rolling bearings have three effects with respect to machine vibration. The first effect is as a structural element that acts as a spring and also adds mass to a system. As such, bearings define, in part, the vibration response of the system to external time-varying forces. The second and third eEects occur because bearings act as excitation sources, producing time-var~ngforces that cause system vibration. In one case this excidesign of rolling bearings and cannot be avoided. tation is inherent in the In the other case these forces result from imperfections, which usually are avoidable.
With sufficient load, the bearing is a stiff structural member of a machine. It is a spring whose deflection varies nonlinearly with force, in contrast to the usual linear spring characteristics assumed in dynamic models, such as the single-de~ee-of-freedom spring-mass-da~permodel discussed in Chapter 26. As a first appro~mation,it may be adequate to estimate machine vibration response by considering the bearing as a linear spring. In this case a bearing spring constant is determined by taking theslope of the force-deflection curveof the bearing at thenormal operating load. Theapproximation may be insufficientin cases requiring precise knowledgeof transient vibration response, particularly near machine resonant frequencies. In these cases extensive mathematical modeling and experimental modal analysis are performed, bothof which are beyond the objectives of this chapter. If it is sufficient to consider bearing stiffness as a constant, under a specific set of operating conditions, then this approximation can be derived from equations in Chapter 10.
RATION
earing stiffness increases with increasing load, a characteristic referred to as a “hardening” spring. Larger nominal operating loads or built-in preload would result in smaller variations in dynamic bearing deflection when subjected to a particular dynamic load variation. Similarly, increased bearing stiffness raises the value of a resonant frequency associated with this spring, since a resonant frequency is inversely proportional to the square root of stiffness. Moreover, radial stiEness decreases with increasing contact angle, whereas the reverse is true for axial stiEness. Therefore, responseto dynamic loadvariation will depend strongly on the direction of such loads relative to that of the nominal load that governs contact angle. Since the bearing “spring” is nonlinear, it is evident that sinusoidal deviations from the nominal load will not cause sinusoidal bearing deen the load is greatest, the increase to nominal be less than the decrease from nominal bearin when the load is at its lowest value. If large dynamic fluctuations in load le for are experienced, say in a radially loaded bearing, then it is the load zone to alternate from the bottom to the top of the ou eway. If the bearing has radial internal clearance, there is the ~ossibilityof tially no loading at all on the outer raceway for brief instances. conditions could arise because of external loading or conditions within the bearing.
second effect of bearings on machine vibration occurs becausebearcarry load with discrete elements whose angular position, with respect to the line of action of the load, continually changes with time. This mere change of position causes the inner andouter raceways to undergo ‘odic relative motionevenif the bearing is geometrically perfect. alysis of this motion is described in f25.41. The following example illustrates how variable elastic compliance vibration occurs. Consider a 204 radial ball bearing with eight 7.938mm (0.3125-in.)balls. The bearing supports a 4450-N (1000-lb)radial load. Figure 25.3 shows the bearing at two different times. In Fig. 25.3a ball 1 is located directly under the load, and balls 1, 2, and 8 carry the load. In Fig. 2 5 . 3 ~balls 1 and 8 straddle the load s ~ m e t rically; and balls 1, 2, 7, and 8 carry the load. Obviously, the radial ~eflectionis different in each situation. With methods from Chapter 6 the radial deflection in thefirst case is estimated to be 0.04323 mm (0.001702 in.). In the second case the deflection is approximately 0.04353 mm (0.0017~4in.). The bearing eflection is less in the former arrangement. The position of the ball ig. 2 5 . 3 ~ gives a stiffer bearing at that instant. ceway have a~proachedcloser to the outer race
(4
.3. (a)Angular position of ball set, time #l/Z (cage rotation frequency)].
(b) = 0.
( b ) Angular position, time =
it takes for one half of the ball spacing to pass a point on the outer raceway to reach the position shown in Fig. 25.33. The shaft will return to its original position as ball 1 comes under the load line. This frequenc~of vibration is therefore equal to the cage rotational frequency multipliedby the number of balls; that is, the frequency of this vibration occurs at thefrequency of balls passing the outer raceway. This example illustrates vertical elastic compliance vibration. zontal motion also occurs, at thesame vibrational frequency, as th set assumes angular positions that areasymmetrical with respect to the 0th vertical and horizontal vibration amplitudes are nonsinusoidal as a result of the nonlinear deflection characteristics. The existence of this type of vibration, which occurs evenwith a geometrically perfect bearing, is one reason why bearing damage detectionis best performed by monitoring frequencies other than the fundamental bearing frequencies.
The first effect that bearings have on machine vibration arises from geometrical imperfections.Theseimperfectionsalways present to varyingdegrees in; manufactured componentsayles and Poon E25.61 discuss three mechanisms by which imperfections in bearings cause vibration: waviness (Fig.25.4) and other form errors causing radial or axial motion of raceways; microslip together with asperity collisions and entrained debris that break through the lubricant film and shocks due to local elastic deformations caused by summits that do not break the lubricant film.
CH
ION
. Vibration from raceway waviness. he local elastic contacts are of approximately the same size or areas. At any instant there may be only tzian deformation zone, dependingon the type of bearing ishing processes employed; example, for honing and lapping. Elastic deformations of the type discussed occurrapidly, and the time separating one such contact from the next is brief. A major contribution to bearing vibration in the higher fre~uencies,for s thought to be the result of such deformanature, however, they are capable of exciting lower-fre~uencyresonances. Controlling component wavinessand other types of errors from manufacturing, distortion, or damage occurring while the bearing is assembled to the machine, is a high priority. The effectsof such form errors on machine vibration or noise can be si~ificant.
igure 25.4 represents a bearing with waviness on the outer raceway. It is assumed that the bearing supports a mass and that the outer ring is rigidly supported by a housing. If no waviness is present on the surface of the bearing raceways, a force balance in the vertical direction is (25.1) If waviness is present, then for an approximation it will be assumed that the mass will move up and down as a rigid body, with reaction force produced in the bearing as a result of the acceleration of the mass. this case the force balance is
For waviness that can be approximated as sinusoidal the e y = A sin ( 2 ~ f ~ )
(25.3)
and
The frequency f is the rate at which balls pass over a complete wave cycle. The assumption of only vertical motion of the mass implies two conditions: (1)that the wave peaks are always in phase with balls, and (2) that variation of the ball set angular position within the load zone has no influence on the direction of motion. For illustrative purposes, these simplifying assumptions will sufficeto demonstrate the importance of relatively small form errors. ~ombiningequations (25.2) and (25.4) and rearranging,
For sufficient waviness amplitude and passage frequency, the right side of the e~uationcan vanish, in which case the bearing force (left side of the equation) vanishes, or it can become negative, in which case the bearing produces a negative force to restrain the motion of the mass. this case the load zone wouldalternate from the bottom to the top of the outer raceway. If the bearing has clearance, it could become unloaded in either direction at some instant. The following example gives an estimate of waviness amplitude that would cause this condition. For a bearing with 50 waves on the circumference of way, estimate the amplitude of the waviness that could cause sufficient acceleration to momentarily unload the speed is 1800 rpm (30 Hz), and the cage speed is 11.rps. e rate at which any ball passes over a wave cycle is the product of the cage speed and the number of waves per circumference of the outer raceway-in this case, 550 wave cyclesper second. The condition for bearing unloading would occur when
or
E ~ I N G IN S MACHINE VIBRATION
A = - - -g
-
(271-f)2 (271= 8.208 X
980 X 550)2
IOv4mm (3.231 X
in.)
Wave amplitude is usually expressed in terms of peak-to-valley amplitude with units of microns (pm) (1 pm = I O + m). In these terms, the waviness peak-to-valley amplitude is 1.64 pm. Bearing raceway wavinessof this amplitude and frequency are in excess of acceptable levels. Although wavy componentsof this type rarely occur, they can occurdue to improper manufacturing procedures or manufacturing machine malfunctions. Examples of excessively wavy components are presented later on.
Three examples are discussed to reveal sources of noise and vibration problems in small electric motorswith newly installed bearings. The first example shows measurements of bearing distortion that, occurring as a result of assembly to faulty machine components, produced waviness on the inner raceway. The second example showsthe effect of improper assembly, and the third illustrates the effect of a defective bearing on motor vibration.
3 Assembly of bearings in housings or on shafts with poorly controlled geometry can distort the bearing components and produce wavy running surfaces that aEect the machine vibration or noise. Johansson 125.61 discusses effects of inner raceway distortion on electric motorvibration. Figure 2 5 . 5 ~shows ~ a circumferential trace of a motor shaft-bearing journal where a 6 mm bore deep-groove ball bearing was mounted. The motor was rejected after assembly for audible noise and for vibration as determined by hand-turning the armature while it was supported by the bearing. The shaft speed in the power tool application was 23,000 rpm. Theshaft exhibits a three-point out-of-roundness condition of approximately 24 pm (0.001 in.). Shaft diameter tolerance for the particular application would normally be held within a total spread, from one shaft to another, of 8 pm (0.0003). Figure 25.53 shows traces of the bore and ball groove after disassemblyfrom theshaft, with both surfaces being less than 1 pm (0.0000~in.) o~t-of-round. Figure 25.5~shows a trace of the ballgroove as mounted on theshaft. This indicates that the raceway in the mounted condition exhibited 16 p m (0.0006 in.) of three-point out-of-
(a> 5 . 5 ~ . Motor shaft circumference (each radial division is 2 pm) (from [25.7]).
round. Note the two local imperfections onthe raceway. Thelargest is approxi~ately2, ,urn (0.0000~in.) deep and is located on one of the lobes. The origin of this defect was not determined, although it probably occurred during press-fit assembly or from damage during running. he bearing has six balls, so the three high points on the distorted r raceway could be in contact with three balls, while the other three balls carry no load. This effect would lowerthe bearing st either axial or radial, which depends on the number of balls. other hand, larger individual ball loads are expected to be generated during parts of the rotational cycle of the shaft, tending to raise stiffness, but simultaneously causing large axial vibration. The cage operating speed can be calculated by using equatio~sof z. The cage speed relative to the Chapter 8 to be appro at which a wave cycle passes a ny of the high points passes from one ball to the next is equal to the product of the cage speed relative to the inner raceway and the number of balls (1470), harmonically
ROLE OF
.
(b) Inner raceway after disassembly (each radial division. is 1 pm) (from
[25.71).
related to the wave passage frequency, becausethe number of balls is a multiple of the number of waves. Accordingly,there is the potential for large-~mplitudevibrations with two fundamental frequencies (735 z) well into the audible range. In addition, numerous high harmonics of each w o ~ l dbe expected,with the potential for excitation of various structural resonances in the motor. This example illustrates a loose assembly that contributes to noise an ~ibration. In this case airborne sound measurement in the form of frequency spectrum analysis is used to identify the sourcc?: ofthe problem. 1 frequency spectrum analyzer is a com~uter-basedint transforms time-sampled d into the frequency domain through Fourier series analysis. owledge of dominant fre~uenciesin vibration signals can often reveal sources of a specific problem, An additional benefit of the technique is that storage and
TIO ON, NOISE,
~ O ~ I T I OMNO ~ T O R ~ ~
(c> 2 5 . 5 ~ . Inner raceway mounted on shaft (from [25.71).
documentation of data arefacilitated. Figure 25.6a shows a simplified time signal, which might be obtained as a voltage signal from a transducer such as an accelerometer or eddy current displacement sensor. This signal might be displayed on an oscilloscope. The same signal can be sampled and Fourier transformed into the frequency domain, as shown in Fig. 25.6b, to show the amplitude as a function of frequency. In this case the time signal consists of only a few frequencies, which in the frequency spectrum show up distinctly. Figure 25.7a shows the frequency spectrum of a vacuum cleaner motor rejected for noise after assembly. The spectrum was obtained from airborne noise measurement utilizing a microphone. The amplitude scale is uncalibrated and logarithmic; each vertical division represents an amplitude increase of a factor of 10, with lower frequencies being attenuated, as is common in sound measurement. The normal operating speed of the motor is 20,000 rpm. Neither the sound spectrum nor qualitative audible evaluation revealed anything abnormal en themotor was turned off and was coasting down, however, a distinct rumble was heard at a speed subsequently esti-
Time
Amplitude
FI
.
(b) Simplified time signal. (a)!Pime domain. (b) Frequency domain,
Frequency
”++
(a)
Frequency
“+
t b) .7, Electric motor sound spectrum. (a)Loose bearing mount. ( b )Normal mo-
tor.
mated to be around 10,000 rpm. The spectrum of Fig 25.7~1 was then obtained with the motor running at that speed. It seemed likely that some system resonance was occurringas themotor coasteddown and passed a critical speed. The spectrum shows distinct frequency peaks, determined to be 165.5, 325, 487.5, 650, 975,and 1137.5 Hz.A similar good motor (e.g., .7b) shows only a peak at 975. Rotor endplays o 094, mm (0.0037 in.) for the motor of Fig. 25.7a a (0.002 in.) for the other motor. The peak at 975 Hz for the good motor was expected to be the blade pass frequency on the motor fan, which has six blades. This means that the actual running speed was ~ 7 5 0 rpm. In thiscase the measured spectral peaks on the noisy motor were all harmonically related to the running speed. Harmonics of the running speed usually occur when mechanical looseness, which could be caused by improper bearing mounting [25.8], exists. Looseness would result in low stifkess, lower resonant frequency, and “play” in the system. The imbalance force, rotating at the shaft
’
speed, could then produce significant vibration amplitude at that frequency, The harmonics probably result from either of two effects: (1) directional stiffness variation or (2) shocks occurring if the load zone shifts from one side of the bearingto the other in an unstable manner. The noisy motor was disassembled. A spring clip, which retains the bearingouter raceway in a plastic housing, had been improperly seated during assembly, resulting in a loose bearing mount. and reassembly reduced the noise to an acceptable level. this problem could have been made with vibration transducersrather than sound measurement. The audible evaluation of the transie~t vibration during coast-down also provided a clue to the source of the problem. Noise emitted from a small electric motor with a deis discussed. Figure 25.8 shows vibration spectraof two small electric motors (3600 rpm) on the same plot. These data were obtained by screwing a small accelerometer into a nut glue end cover. The frequency range investigated was to 10,000 ering the most important part of the audible frequency rate at usual machine operating speeds. The rms (root mean square) vibration amplitudeat each frequency is plotted in volts dB, where the dB value is equal to20 log,, (voltage/ reference voltage). In thiscase the reference voltage is 1.0. Each major
FREO. SPAN+ 0 Hz
AV~~AGEI 0
10 KHz /
8% 60.0 Hz
FIG
5.8. Motor vibration spectra.
division on the plot is equal to 10 dB,with the amplitude scale ranging V). Theaccelfrom -40 dB(0.01 V) full scale down to -120 dl3 erometer output is 0.010 V/g of acceleration. One motor is seen to have vibration amplitudes from 5 to 25 dl3 higher than the other motor over much of the frequency range. The motor with the higher vibration also gave torque readings more than a factor of 2 higher than theother motor. Frequencyspectra at various points on the motors were taken with the same results. Spectra in other frequency ranges were taken with no conclusiveresults regarding theorigin of the vibration. Qualitative evaluation of audible noise indicated a low-amplitude clicking sound that repeated at a fairly high rate. Themotorswere disassembled, theshaftsand housingswere checked for geometryand found to be normal, and the bearing torques were measured. The bearings from the motor with high vibration had rubbing seals, whereas the other set hadnoncontacting seals. The four bearings were vibration tested, with seals removed,on a standard bearing test apparatusdiscussed in thenext section. It was foundthat one ofthe bearings from the motor with high vibration also gavehigh readings on the inspection tester. Spectrum analysis indicated harmonics of the frequency at which balls pass a point on the outer raceway. ~xaminationof the outer racewayrevealed a defect that appeared to be related to manufacturing and hadescaped detectionin final inspection vibration testing. The discussionand examples of this section have viewed several ways in which bearings can affect or cause machine vibration. The irregularities on bearing surfaces that exist from manufacturing, assembly into the machine, or from deterioration after long-term use provide a source of forced excitation to rotating machine members or structural components that can increase stress, accelerate wear, increase frictional losses, and possibly cause catastrophic machine failure. Other forms of bearing distortion or imperfections occurringas a result of assembly include misalignment (housing centers out of line from each other or not parallel), rinell damage, contamination by debris, and bell-mounted housings preventing axial movement of the outer ring (e.g., in small electric motors with spring preload).
enerally, a part is said to be round in a specific cross section if there exists a point within that x-section from which all other points on the
periphery are equidistant. The first-mentioned point is of course the tenter of the circle, and the x-section is a perfect circle as in Fig. 25.9a. If the x-section is not a perfect circle as in Fig. 25.93, it is said to be outof-round with the ‘ ‘ ~ ~ t - ~ f - r ~specified ~ n d n eas~the ~ ~difference ~ in distance of points on the periphery from the center. Thus, out-of-roundness in Fig. 25.93 is rl - r2. In addition to the basic profile of Fig. 25.93, an irregular profile similar to Fig. 25.9~is usually present in manufactured machine elements, and this includes rollingbearing raceways and rolling elements. The irregular surface of Fig. 25.9~is of substantial importance to bearing frictional performance and endurance; this was discussed in Chapters 13 and 18. The lobed surface of Fig. 25.93 is also s as it is a causative factor of bearing vibration. The important e s is, s ,the number of lobes per circumference. called ~ ~ u ~ ~that ~ a v i n e s scan occur in the machining process. A round bar or ringtype element is compressed at thepoints of contact in a chuck, three jaw or five jaw, causing stresses in the part. The part is then “turned” or ground perfectly circular; however, when it is released from the chuck, the stresses are released, and the part becomes lobed. Centerless grinding also causes waviness where the original bar stock is irregular; perhaps due to the previous machining operation. Waviness is measured by equipment such as the Taylor-Hobson Talyrond; Talyrond traces are shown in Fig. 25.10. They were also shown in Fig. 25.5(a)-(c). For a tapered roller bearing mounted in an SKI! WCL tester, shown schematically by Fig. 25.11, Waland E25.91 examined the correspondence between waviness and the resulting vibration spectrum. For a bearing rolling elements and if p and q are integers equal to or greater than 1 and 0, respectively, then for vibrations in the radial direction measured at a point on the 0.d. of the outer ring, the vibration circular fre~uencies as functions of inner ring, outer ring, and roller waviness are given in Table 25.1.
.
Illustration of a round surface.
TION ON, NOISE, AND C
ITION ~
O
~
T
O
25.10. Talyrond traces of circular parts indicating Waviness.
Electrodynamic pickup . .
25.11. Schematic view of VKL vibration tester.
.
Vibration Frequency vs Waviness
ComponentWaviness of OrdersCauseVibrationwithCircularFrequencies Inner ring Outer ring Roller
k=qZt-p k=qZt-p k (even)
In Table 25.1, mi, wc, and mr are theinner ring, cage, and roller angular velocities, respectively. Rigid body vibrations are indicated when p = 1; that is, the bearing outer ring moves as a rigid body. Forp > 1,vibrations are of the flexural type with p equal to the number of lobes outer circumference of the outer ring deflection curve. For a waviness spectrum obtained at aninner ring speed of 900 rpm, for a bearing with an accentuated inner ring waviness, Yhland 125.91 obtained the vibration spectrum at 1800 rpm shown by Fig. 25.12. Also shown by Fig. 25.12 is the
R
~
~
10
2 34 5 768 9 1 0 15 2 Waves per circumference
Peaks caused by waviness of order:
2
50
8
16
32 24
40
100 150 200 300 400 500 Speed: 900-1800 rev/min.
1000 1500
~ 5 . ~ 2Waviness . and vibration spectra from inner ring with accentuated wavi-
ness L25.91.
Talyrond trace of the inner ring; the tested tapered roller bearing contained very smooth rollers and outer ring. As compared to the inner ringdistortion shown in Fig. 25.5, waviness is a more uniform type of form error. Figure 25.13 shows circumferential traces of two spherical roller bearing inner raceways. One raceway has a peak-to-valley amplitude of approximately 4 pm (0.00016 in.), the other approximately 9 pm (0.000~6in.); each has nine wave cycles per circumference. From Fig. 25.13 the radial geometrical deviation of the surface from a true circle (shown onthe traces as thedashed line) can be approximated as a function of angular position. The shape is approximately sinusoidal. The radial amplitude variation is then
where A R
= =
peak amplitude circumferential distance measured from a starting point 0 = 0.
84
F~~~~
TION ON, NOISE, AND C O ~ I T I O ~
~ O ~ T O ~ I N
25.13. Spherical roller bearinginner ring waviness, machine setup error. Each radial division equals 1 pm,
h = wavelength of one cycle or A =
circumference 2vR =number of waves W
(25.7)
Therefore, from equations (25.6) and (25.7)
r = A sin ~0
(25.8)
and r = sin 98, approximating a sinusoidal wave. This type of discrete frequency wavinessis an excitation source for vibration as well as generation of dynamic forcevariations on bearing components. Defects of the magnitude shown in Fig. 25.13 are rare and are easily detected when they occur. Waviness of relatively low number (usually odd) of waves per circumference occurs because of inaccuracies in grinding machine tooling or setup involving sliding contact shoesupports for the workpiece as itis ground. Contactof two high points simultaneously on two shoes causes more material to be removed by the wheel at a position opposite the shoes; conversely,low points on the shoes result in less material being removed, producing high points on either side of the wheelwork contact zone. This condition is detectable with conventional in-process gaging usedto control diameter, provided that gages are setup for three-point diameter measurement rather than for two point. Other types of imperfections canresult from machine malfunction. See Fig. 25.14, which showsa circumferential trace of a spherical roller. This component was ground “on-centers”and does not show a waviness pattern of the type seen before. A machine controlsystem malfunction, however, allowed the roller to be released from the grinding station before the wheel was retracted, with the result that the roller contacted the wheel and produced a localized flat spot on the roller surface over approximately 5% of the circumference to a maximum depth of 18 pm (0.00072 in.). In contrast to waviness from incorrect setup, three-point diameter measurement is not likely to reveal this imperfection unless measurements are made around the entire circumference. More subtle defects can also occur. They may becharacterized by much smaller deviations from true geometrical formand require more detailed component inspection, such as waviness testing or vibration testing of assembled bearings. Examples of such defects are shown in Figs. 25.15 and 25.16. These traces show, respectively, a spherical roller and a cylindrical roller with lower-amplitude and higher-frequency waviness than the examples of
25.14. Roller fiat spot, machine malfunction (2 pm per radial division).
Fig. 25.13. The roller of Fig. 25.15 has 36 waves with an average peakto-valley amplitude of approximately 1 pm (0.00004 in.). The roller of Fig. 25.16 has over 100 waves with a peak-to-valley amplitude of less than 0.5 pm (0.00002 in.). Both rollers were identified as causes of noise and vibration in assembled bearings. The cylindrical roller bearing corresponding to Fig. 25.16 had been installed in a large electric motor. The motor emitted a periodic audible noise at slow running speed. With a stopwatch, the repetition rate of the noise could be associatedwith each revolution of the cage. Subsequent investigation on a test rig traced the noise to this particular roller.
Component inspection for wavinessahs been performed for many years and is described in the literature[25.9,25.10].This inspection is used to assess the degree of radial deviations from a true circle on the circumference of a component. This is accomplished by rotating the component on a hydrodynamic spindle and applying a contacting transducer perpendicular to the surface of the component. The transducer is a stylus that follows the radial deviations and produces a voltage output propor-
25.15. Roller waviness.
tional to the instantaneous radial rate of change of the displacement of the stylus; that is, the signal from the transducer is propo~ionalto velocity. This propo~ionalityexists over a wide frequency range, such as 10,000 Hz, which allows reasonably high test speeds to be used. The voltage signal from the transducer is amplified and bandpass fib filter bands tered into three or more bands typically 2.5 octaves wide. The combined with the selected testing speed encompass a broad range of wavinessfrequencies.Frequenciesshown in the previouscomponent traces extend to approximately 100 waves per circumference. Waviness testing equipment and procedures in use cover a range well beyond this. The rms value of the signal in each filter band is obtained and compared to specifications to determine acceptance or rejection of the lot of components being inspectedand to provide information for corrective action on the manufacturing processes. Frequency spectrum analysis is also becoming widely appliedas instruments become less expensive and more suitable for use in the factory. The following discussion illustrates some reasons why velocity measurement hasseveral advantages over displacement for evaluating bearing component quality. The cylindrical roller of Fig. 25.16 was rotated on
~ B ~ T I O NOISE, N , AND ~ O ~ I T I OMN O ~ T O R ~ ~
F
~ 25.16.~ Low-amplitude, ~ E high-frequencywaviness.
a waviness testing machine, and the amplified output of the velocity transducer was analyzed with a frequency spectrum analyzer. Results are shown in the plot of Fig. 25.17. The abscissa covers a frequency range of 0-2000 Hz, and the roller was tested at 720 rpm (12 Hz), so this frequency range would detect a dominant wave pattern up to 166 waves per circumference (2000/12). The ordinate gives rms voltage in dB referenced to 1 V, ranging from -10 dB full scale (0.3162 V) to -90 dB (3.162 X lo-' V). Nominal calibration for the velocity transducer and amplifier is 3.0 pV/pin.-sec. A cursor mark is indicated at the peak, with corresponding coordinates printed below the frequency axis, The peak occurs at 1250 Hz, with rms amplitude at -23.43 dB. Since roller test speed was12 Hz, the frequency at which. peak occurs corresponds to approximately 104 waves per circumference. 5.7. For the test arrangement previously indicated, determine the rms radial velocity of the predominant waviness and the number of waves per circumference. Also estimate its average peakto-valley amplitude in microns.
$8 EFFECT ITS RANGE:
-25 dBV
SlATUS:
PAUSED
-10 dBV
10 dB /DIV
-90 START' t 0 HZ X: 1250 HZ
l3wx
Y: -23. 4 3 dE?V
F I G ~ 25.17. E Roller waviness velocity spectrum.
From Example 25.4, rms voltage is determined from -23.43 dl3 = 20 log,, V. Therefore, V = 0.067375. Rms velocity is determined by dividing the voltage by the transducer-amplifier conversion of 3 pV/pin.-l/sec-l. Therefore,
1 ~ 1 ~ ~ ~ =
(22,458 pin./sec) 570 pm/sec
In equation (25.8) the radial deviation r was given as a function of angular position 0 on the component and the numberof waves w. In the waviness test apparatus the part is rotated, so the radial deviation r is a function of time. Any angular locationon.the part also becomes a function of time with respect to the fixed location of the transducer being used to measure the radialdeviations: 0 = 2TNt
(25.9)
where N is the rotational testspeed (rps). Therefore,
(25.10) and
990
VIBRATION, NOISE, AND CONDITION MONITORING
r
=A
sin (2vNtW)
(25.11)
The radial velocity, measured by the transducer, is the change of the radial deviation with respect to time:
t
=
2 v M A cos (2nNtW)
(25.12)
and rrms= 1.414 v M A
(25.13)
From the measured rms velocity the peak amplitude A can be estimated: 22,458 = 1.414 71.(720/60) 104A A
=
0.1029 pm (4.05 x
in.)
The peak-to-valley amplitude is twice that value and is therefore estimated to be 0.206 pm (8.1 X lop6in.) Table 25.2 summarizes estimates of average peak-to-valley displacement and rms velocity values from equation (25.13) of the components of Figs. 25.5c, 25.13, 25.15, and 25.16. Peak-to-valley displacement of these components, one of which was determined as mounted on a shaft with excessive three-point out-ofroundness, varies over a range of approximately 80:1, whereas the number of waves per circumferences varies over a range of approximately 35:l. The velocity values, however, are within a range of only 4:l for TABLE 25.2. Waviness Displacement Velocity Amplitudes Peak-t o-Valley Amplitude
rmsa Velocity
Component Trace Figure
(pin.)
Pm
Waves per Circumference
(pin./ sec)
Pm/ sec
25.5~ 25.9a 25.9b 25.11 25.12
(629.9) (157.5) (354.3) (9.37) (8.11)
16b 4 9 1 0.206"
3 9 9 36 104
(50,368) (37,776) (84,997) (37,776) (22,458)
1,279 960 2,159 960 570
"Displacement assumed sinusoidal; therefore drldtl,, the rpm of the waviness test spindle. bValue estimated as mounted on out-of-round shaft. "Calculated from measured velocity in Example 25.6.
=
(0.707)(2.rrNw/60), where N is
NONROUNDNESS EFFECT AND ITS MEASUREMENT
991
components of diverse size and configuration. Additionally, transducers and instrumentation for velocity measurement systems do not require as great a dynamic range as displacement measurement. Numerical specifications of similar magnitude are also more easily developed and applied. Some types of defects that can arise on components are very local in nature. Detection of these defects may not be feasible with waviness testing, which may only acquire data from one or two circumferences on components being tested. Balls have numerous potential axes of rotation. Therefore, visual component inspection and vibration testing of assembled bearings provides more definitive assurance of final bearing quality.
Vibration Testing
.?
Aside from defects that are not discovered in waviness testing, vibration testing of the assembly allows the detection of damage occurring in assembly, such as binding or excessively loose case, Brine11 damage to raceways or scuffing of balls, and distortion of raceways from incorrect insertion of seals or shields with bearings tested after grease insertion. Certain types of geometrical problems may also be detectable in vibration testing. These can include, for example, oversized rolling elements, improper cross-groove form on raceways, or groove runout to side faces of the raceways. In addition, testing can reveal contamination by dirt or inferior grease quality. Figure 25.18 shows a manually operated vibration testing apparatus for relatively small bearings, for example, up to 100 mm 0.d. Similar equipment is in use for larger diameter bearings, and automatic versions are implemented on production lines. The main elements of the system are the test station and the vibration signal analysis instrument. The test station consists of a hydrodynamic spindle, an air cylinder for applying load to the bearing being tested, and an adjustable slide for positioning the velocity transducer. The spindle is belt driven by the motor mounted beneath the stand. A schematic representation of the system is shown in Fig. 25.19. The inner raceway of the bearing mounts on a precision arbor fastened to the spindle, which rotates at 1800 rpm. A specified thrust load is applied to the side face of the nonrotating outer ring. The tip of the velocity transducer is lightly spring-loaded on the outer diameter of the outer ring. The loading tool (not shown) consists of a thin-walled steel ring molded into a neoprene annulus. The ring contacts the side face of the outer ring. The tool and load combinations are sufficiently compliant to allow radial motion of the outer ring to occur as balls roll over wavy surfaces or defects in the ball grooves. The voltage signal from the transducer is input to the analysis instrument, which amplifies the signal,
Tran
F I G ~ 25.19. E Vibration test schematic.
The calculation of the fundamental pass frequencies of rolling contact bearings is used to establish component waviness testing speeds and filter bands that coincide with vibration measurement bands. In addition, knowledge of these frequencies is useful, though not alwaysessential, in machinery condition monitoring. Derivation of these equations is presented in Chapter 8. Results are given here for the case of a stationary outer ring and rotating innerring, as is used in the vibration test system described before and is most often the case in typical bearing applications. Assuming no skidding of rolling elements, the frequencies of in-
terest are related to the rotational speed of the inner ring N , the pitch the number of diameter of bearing dm, the rolling element diameter balls or rollers Z, and the contact angle a. The rotational speed of the cage fc is f = -
c
2
(
1"
;.) cos
(25.14)
The rotational speed of the inner ringrelative to the cage is the rate at which a fixed point onthe inner ringpasses by a fixed point on the cage. This relative speed is
The rate at which balls pass a point in the groove of the outer raceway (also called the ball-pass-outer-racewayfrequency or outer raceway defect frequency)is (25.16) is also that frequency at which variable elastic compliance vibration occurs. The rate at which a point in the inner ring groove passes balls (also called the ball-pass-inner-racewayfrequency or inner raceway defectfrequency) is
fbpor
(25.17)
The rate of rotation of a rolling element about its own asis is (25.18)
A single defect on a ball or roller would contact both racewaysin one ball or roller revolution so that the defect frequency is 2fR. In addition, the defect could contact one or both sides of the cage pocket; however, this usually will have little influence on vibration measured external to the bearing. viness Within reach of the vibration measurement frequency bands, the number of waves on a component that influence a particular band can be calculated. For outer raceway waviness any ball rolls over all the waves in
the outer raceway groove in one cage revolution. Therefore, the ball passage frequency over an individual wave cycle on the outer raceway is fc X number of waves per circumference. Consequently, dividing the filter frequencies by f, determines the number of waves per circumference of the outer raceway producing vibration within a particular band. For balls, the filter bandfrequencies are divided by f R to determine the number of waves per circumference of a ball producing vibration within a band. Similarly, for the inner raceway the band frequencies are divided by fci. The lobes of low orders of inner raceway waviness, such as two-and three-point out-of-roundness, however, can causeflexure of the outer ring af(two- or three-point lobing) and vibration at two or three times N,, fecting readings in the low-frequency band.
.
Compute the cage rotational frequencies, ball rotay, and estimate thewaviness orders for each component that fall within vibration testing bands for a 203 ball bearing withan assumed contact angle of 12" and test speed of 1800 rpm (30 a 203 ball bearing, ball diameter = 6.747 mm (0.2656 in.) and pitch diameter = 28.5 mm (1.122 in.).
fc
=
1 - - cos 1"
2 =
d", .)
2 "i
(25.14)
6.747 cos 12" 28.5
11.53 Hz (25.15)
2 = fR
6.747 cos 12" 28.5
18.47 Hz
= Nidm [l -
(
1)~cos) a!
2
]
(25.18)
[
-
30 X 28.5 (6.747 cos 12°)2] 12 X 6.747 28.5
=
59.97 Hz
Therefore, calculations indicate that waviness orders of the components falls within the vibration test bands approximately as follows:
50-300 HZ
Outer raceway 27-156 Inner raceway 17-97 6-30 Balls
300-1800 HZ
4-26 2"-16 2-5
1800-10,000 HZ 157-868 97-541 31-167
*Including two- and three-point out-of-roundness.
Similar values are obtained for the range of ball bearings sizes tested in this manner: for example, deep-groove ball bearings in the 2 and 3 series. Waviness testing procedures are established to correlate with average waviness ranges over a wide range of bearing sizes. In addition, the range of waviness orders tested in either vibration or waviness measurement corresponds approximately to wavelengths of the size of the small axis of the Hertzian contact ellipse in typical applications such as electric motors L25.91.
Defects other than waviness can contribute to bearing vibration or noise. Some of them can be difficultto detect with the conventional three-band inspection method. Such defects include local defects on raceways or balls, dirt, grease with improper constituents or properties, and cages with incorrect clearance or geometry. Some of these defect types may produce brief disturbances spaced widely apart in time, which, as a consequence, have only a small effect on the average measured vibration in the inspection bands. For example, a single localized defect onthe inner raceway or on an individual ball will be remote from the transducer location during most of the time that it takes for the inner raceway (or cage in the case of a defective ball) to make one revolution.Such defects impacting the outer ring will momentarily excite its various natural frequencies. The lowest of these natural frequencies is a rigid body mode (individual balls act as springs) and higher natural frequencies being modes of outer ring flextural vibration. These modesresult from bending of the outer ring into shapes that have an integral number of lobes, as analyzed in [25.41. Resonant vibration of the outer ring amplifies the effects of these local defects, and the resonant vibration can be used to detect their presence. Thereforevibration measurement is supplemented with the use of a peak detector instrument whose functions are shown schematically in Fig. 25.20. Although the figure showspeaks of relatively constant amplitude, which might bethe case foran outer raceway defect, maximum peak values are obtained and evaluated because they can vary with time in the case of ball and inner raceway defects.
”~\”+”“--””
, I
1
PEAKDETECTOR ”*”
I NSTRW4ENT
_ “ ” ”
F
I 25.20.~ Peak detection. ~
Vibration analysis is one of the most common methods usedto evaluate the condition of bearings in an operating machine. As previously shown, such measurements may be used for machines with bearings in new condition as well as for machines whose bearings are deteriorating and approaching the end of their useful lives. If a machine’s vibration response to known excitation forces has been determined through techniques such as finite element analysis and modal analysis, then vibration measurements during its in-service operation can define the dynamic characteristics of the forces acting on the machine. Vibration data can also be used to infer forcing characteristics and condition of machine components, including bearings. General methods for evaluating data include one or more of the following: Comparison of data with guidelines developed empiricallyon similar types of equipment E25.8, 25.11, 25.121 2, Comparison of data from similar or identical machines in service within the same factory 3. Trending of data from one machine overtime 31,
98
~ B ~ T I O NOISE, N , ABiD C O ~ I T I O N~ e
O
~
Evaluation of data in an absolute sense with no prior history. For example, by evaluating time signals or frequency spectra to associate vibration with specific machine compon~nts
Many machine problems can be traced to faults other than damaged bearings. If a moderate1 etailed vibration analysis capability is not .available,however,beqri are oftenreplacedunnecessarily. The b e ~ n n i n gof pro~essivebearing damage, which can be called incipient failure, is often characterized by a sizeable local defecton one of the components. Whenthis occurs, subsequent rolling overof the damage will produce repetitive shocks or short-duration impulses. It can be surmised that such impulses might appear, if they could be measured, ~ b. as those in Fig. 2 5 . 2 1 ~and Figure 2 5 . 2 1 ~could ~ represent, for example, the effect of successive rolling elements passing over a damaged area on the outer raceway. Similarly, Fig. 25.2lb might represent the effect of inner raceway damage interacting with several rolling elements in the load zone of a radially loaded bearing without preload. In this case the damage enters the load zone onceper revolution of the shaft.The locationof the rolling elements with respect to the load zone will vary somewhat from one shaft revo-
25.21. Impulse train. ( a )Outer raceway damage. ( b )Inner raceway damage. (c) Resonant vibration.
T
O
FAILING B E ~ I N G SIN ~
C
~
S
9
lution to the next. If a sensor were placed on the bearing housing to measure the resulting vibration from the series of impacts, it may show a response as in Fig. 25.21~.This vibration corresponds to lightly damped oscillation of some system natural frequency greater than therepetition frequency of the train of impacts. It could, for example, be a resonance of the bearing outer ring or of the housing or sensor itself. Such resonant response is excited by harmonics that exist in the periodic nonsinusoidal forcing function. Figure 25.22 shows the time history of an electrical signal representing a pulse train with a fundamental frequency of 160 Hz; Fig. 25.23 is the frequency spectrum of that signal. It contains all harmonics of the fundamental. Impulsive occurrencesin bearings, therefore, can cause system vibration at many frequencies that can be harmonically related. Forcing harmonics that arenear system resonant frequencies cancause significantly amplified vibration response compared to the vibration at nonresonant frequencies. In the early states of failure the impulse might have little effect on the amplitude of vibration at the fundamental bearing pass frequencies. In addition, significant normal machine vibration could occur at these lower frequencies, so small changes in vibration amplitude initially may be difficultto detect. Higher-order harmonics, with spacing related to specific component frequencies, however, might be detectable at higher frequencies if the sensor and mounting method provide sufficient response at the higher frequencies. Small accelerometers studmounted to electrically isolated nuts and glued to a surface on the machine work satisfactorily. Magneticmounting is faster but it requires a better surface and frequency is lower. The following example illustrates the effect of local bearing damage on vibration response.
.
e A test rig was used to run two 205 ball bearings, each mounted on a pillow block at opposite ends of a shaft. The shaft
2~.22. Periodic impulse-time
domain.
~ B ~ T I O NOISE, N , AND C O ~ I T I O N
~ONITOR~G
26.23. Frequencyspectrum.
was belt driven at 1690 rpm by a pulley mountedon the shaft between the bearings. The outer raceway of one of the bearings contained localized damage, approximately circular in shape of 1.6 rnm (0.0625 in.) diameter, located in thebottom of the groove and in theload zone. The radial bearings each contained nine balls of 7.938 mm (0.3125in.) diameter on a pitch diameter of 39.04 mm (1.537 in.). The calculated bearing frequencies are
Ni = -= 28.2 Hz 60 L) cos a!
= fbpor
=
2 11.23 HZ
Zfc = 9
X
7.938 cos 0" 39.04
11.23 = 101.1 HZ
13 cos =
(
a!
28.2 2 7.938 x39.04 1) 16.97 Hz +
(25.14)
(25.16) (25.15)
fbPk =
Zfii = 9
X
16.97 = 152.7 Hz (25.16)
Figure 25.24 shows spectra of the two bearings on the same plot. The frequency span is 0-10,000 Hz and’full-scale amplitude is -20 dB. Data were obtained with a stud-mounted accelerometer, 0.010 V/g. The vibration amplitudes of the damaged bearing are 20 dBgreater than those of the undamaged bearing at most frequencies above3000 Hz. The spectrum of the damaged bearing also shows peaks, about 10 of them in each 1000 segrnent, whose spacing corresponds to Zfi. Better resolution of peak spacing would require using a narrower frewith finer quency span in regions of the spectrum or an i~strument resolution. Nevertheless, the figure illustrates themajor effectof local bearing damage on vibration in the higher-frequency regions. Figure 25.25, taken on the damaged bearing over 2500 Hz, clearly shows harmonics of the ball passage frequency overthe outer raceway from 500 to 1250 Hz. Amplitudes of harmonics from 700 to 1200 Hz were approximately 10 dB greater than vibration amplitudes of the undamaged bearings in this range. Below 700 Hz, amplitudes of the two bearings were the same. Depending on the presence of other sources of machine vibration and the magnitude of bearing damage that exists, it might also be possible to successfully identify a problem at low frequencies.
CH 4t
- ZOdeV
FS
, Vibration of damaged and undamaged bearings.
1
2.5 KHr 1
FfER SPAN=\ 0 Hz
A V E R A ~ ~ $3
0% 15.0 HZ
2 ~ . Z ~Low-frequency , vibration of damaged bearing.
Figure 25.26 shows part of a digitized time sample taken from the damaged bearing, indicating some perturbation at a spacing corresponding to l/Zfcsec. The previousillustration considers onlya single local damagethat can be associated with a specific bearing component with frequency spectrum analysis. Numerous casesof such failure detection are reported in the literature [25.13, 25.143. ost forms of damage preceding bearing failure will result in progressive wear and roughening of component surfaces and irregular running geometry. Suchirregularities may producevibration that can clearly be identified with specific com~onents,or they could produce vibration whose amplitude varies randomly in time and frequency content. In either case machine vibration measurement in one form or another can be used to periodically assess bearing condition. particularl~damaging form of failure is caused by electrical current passing through bearings in large motors. Arcing erodes the bearing and creates numerous large damaged areas. Sample data show detection of such damage in a 12OO-rpm, 1000-hpvertical pump motor using axial vibration on the pump motor base. The top bearing is a 170-mm (6.69-in.) bore, 40" angular-contact ball bearing. In contrast to an 800-hp pump with rotating imbalance, discussed earlier, vibraspectra indicated harmonics, as see in Fig. 2 5 . 2 7 ~ ~ ccurate evaluation of harmonics was performed with cepstrum analysis [25.15], which determines periodicity within a spectrum, al-
sec.
.o25 sec.
+
Time .0025 sec/div
2 ~ . 2 ~Damaged . bearing-time
domain.
lowingidentification to analysis of frequency bands within the spectrum. Harmonics with a spacing of 101.7 Hz were identified, and they are corresponded to the outer raceway ball pass frequency. The bearing was removed and replaced after less than nine months of operation. Data from the pump after rebuild are shown in Fi and a photograph (Fig.25.28) illustrates the nature of damage on the outer raceway. Analysis and means of prevention of this type of damage are presented in [25.16-25.18].
Aside from evaluation of ~ibrationspectra to identify specific machine frequencies, data can be obtained or analyzed by other means to trend the onset of failure. athew and Alfredson [25.19] present a comprehensive evaluation of vibration parameters over the life of bearings run to advanced stages of damageprogression or failure. Conditions under which bearings were tested include bearing components with initial damage, contained lubrication, overload condition leading to cage collapse, and sudden loss of lubrication. Parameters that might be obtained with relatively low cost instrumentation include peak acceler eration over a broad frequency band, and shock puls evaluates vibration at a frequency corresponding to the resonant frequency of the accelerometer (32 Hz). Other parameters were calculated by performing arithmetic operations ontwo frequency spectra, one of
0 PJ
0
m
0 fi
n v
00
~ ~ . Pump ~ 8 .motor outer raceway failure by electrical arcing.
which was usually the initial spectrum obtained when tests were begun. The calculated parameters were then trended. Statistical functions, including probability density,skewness, and kurtosis were also evaluated. The results indicated that several parameters evaluated from frequency spectra were successfultrend indicators, generally providing a 30-dB increase or more bythe time a test runwas completed. Onesuch parameter is simply obtained by subtracting the initial spectrum from each new spectrum and computing the rms of the resulting spectrum. This value is then trended over the duration of tests. Aside from computationsof trend parametersfrom completevibration spectra, the shock pulse method was reported t o provide successful detection for all tests except the case of total lubrication loss. For the successful tests the shock pulse values are estimated to have increased 40 e test with l u ~ r i ~ a t i oloss, n seizure occurred in 2 hr. This sug-
gests t ~ ~ a o~ ~
e ~v i= ~ ~r ~~r a~~bey~ a~obetter ~~ einitial ~ i~ ~ ~~ € a ~ o r - f r e ~ u e nvibration c~ unless the components have time t o unient gradual distress to be detectable in the high-frequency
e,
In the foregoing sections,it has been ~ e ~ o n s t r a t that e d monitoring bearing vibrations and c o m ~ a r i n the ~ vibration signals against a baseline for
1006
VIBRATION, NOISE, AND CONDITION MONITORING
satisfactory bearing operation may be used as a means t o detect impending bearing failure. According to definitions of bearing failure discussed thus far-for example, initial spalling or pitting of rolling contact surfaces-the occurrence of abnormal signals may indicate that bearing failure has already occurred. On the other hand, the bearing, although running rough and with increased friction, generally will continue t o rotate after initial surface damage, permitting effective machinery use. Eventually, the rolling contact surfaces will be completely destroyed and the machinery will cease to function due t o bearing seizure or excessive vibratory loading and component fracture. These last conditions represent potential catastrophe from the minimum standpoint of unscheduled machinery downtime and excessive costs, or worse, from the standpoint of loss of human life in life-critical applications. The latter would include, for example, air transport applications and applications which handle hazardous fluids. From the time at which excessive vibration signal is experienced to the time at which the machinery no longer functions represents a duration in which action may be taken to prevent catastrophic events. Historically, many applications have relied on preventive maintenance to minimize unscheduled downtime due to bearing failure. Based on calculations of bearing endurance, either from fatigue of rolling contact surfaces or other wear phenomena, or based on past experience of bearing failures, periodical stoppages of machinery are scheduled, during which bearings are inspected and replaced. Frequently, inspection does not occur and rolling bearings are simply replaced. The problem with this procedure, in addition to the cost of taking equipment out of service and losing production and revenue, is that the bearings which had been in operation were mostly likely not prone to failure; however, they might be replaced with bearings which could fail. Once a rolling bearing has experienced sustained operation, it has passed the period in which birth defects cause early failures, and under proper mounting, applied load, speed, and lubrication conditions, it will continue to operate without failure. Thus, presuming proper operation, it is usually best to allow the bearing to run without interruption once an initial operating period has been successfully achieved. Maintenance is considered the largest controllable cost in modern industry. Based on bearing condition monitoring, which provides operational information on impending failure, and prognostic knowledge of the duration of effective bearing performance, taking failure-prevention action after the first signals of impending bearing failure have been received, but prior to occurrence of catastrophic events is a more cost-effective procedure than preventive maintenance; unnecessary machinery downtime is avoided. Of course, condition-monitoring sensors and techniques must be proven reliable, and life prognostication methods must be proven sufficiently accurate. This procedure is called conditionbased maintenance (CBM).
1007
CONDITION-BASEDMAINTENANCE
CBM is a relatively new concept, and additional information is required for its effective implementation. The initial CBM consideration is that, more often than not, the actual load-speed-temperature operating conditions experienced by the bearing are significantly different from the design. Bearing types and sizes are selected based on a design duty cycle of the machinery. Therefore, at any instant, prediction of remaining bearing life should be based on actual accumulated conditions of operation. In describing the Health Usage and Monitoring System (HUMS) technology under development for helicopter maintenance, Cronkhite [25.20] illustrated the potential for increased endurance of helicopter mechanical components; see Fig. 25.29. Design conditions tend t o be conservative to assure reliable operation, and the lower shaded area on Fig. 25.29 indicates that actual life will tend to exceed design life. Load and speed-sensing equipment are readily available to enable the acquisition of such data. Miniature microcomputers are also available at reasonable cost to enable detailed evaluation of the acquired data. With regard to prediction of bearing life t o the initial spall, the actual load-speed-temperature operation of the bearing must be accommodated in the analysis. Instead of considering failure at the occurrence of the initial spall, the ability to detect incipient spalling becomes important. This knowledge would most likely indicate additional time in which t o take failureprevention action. With the continued development of micro-sized pressure, temperature, and ultrasound sensors which can be embedded in close proximity to, or directly in, the bearing, it appears probable that effective means to sense incipient fatigue failure will eventually be available. Figure 23.30 shows a stress pin, a miniature pressure sensor, embedded in the outer ring of a tapered roller bearing. This sensor does not
+
Retirement extension
Current service life
1 //A\ I
Service limit without monitoring
Time
FIGURE 25.29. Component life consumption in actual service compared to design.
~
.
_I; ~
x__c
--
Stress ins
(b) 5.30. Stress pins inserted in the cup of a tapered roller bearing (a)locations to determine axialstress distribution; (b) circumferential locations to determinedistribufor tion of load among the rollers-showing wireless connectionof analog/digital converter transmission of signal (courtesyof Oceana Sensor Technologies, Virginia Beach, Virginia).
impair bearing function, and may be usedto determine whether bearing loading conforms to design. If bearing loading is substantially in excess of design, this is an indication of failure occurrence or incipient failure. ~ d d i t i o n a l lsince ~ bearing operation will continue a&er the occurrence of the initial spall, al~orithmsto establish the time available for effective operation aRer this event can be developed. Kotzalas L25.211, using a ~ a l l / v - ~ i nrig, g progressed ball spalls past the initial surface f l a ~ i nuntil ~ the entire track was eventually destroyed. Figure 25.31 sion. Figure 25.32 illustrates the voltage signal from loading r e ~ a i n srather lowfor more t ~ a n 1.5 . h ~ ~ reven. s , UII heavy loading, high speed o~erationof the test c~ndition.
(b)
(a)
(e)
FI
~ ~ . ~AIS1 l .52100 steel, 22.22mm (0.875in.) diameter ball. (a)Showinginitial spa11 after endurance testing at 45,000 rprn, 3170MPa (460,000 psi) maximum Hertz stress, and 71.1"C (160"F), lubricant supply temperature in a ballbring rig; ( b ) showing progression to 33% spalling over track; ( c ) showing progressionto 100%spalling over track (from t25.211).
A ~ ~ r o m e t Signal e r vs. Time for Ball 34A
3
,
0.5
0
I
I
20
40
I
5
I
80 80 Time (m~nutes)
I
l
100
120
140
. Accelerometer voltagesignal for ball operating conditions of Fig. 25.31. disk test rig to measure traction coefficient of the faiZed balls, Kotzalas [25.21] determined that an effective lubricant film was generated even in the presence of gross spalling, and it is presumed this film was instrumental in assuring thecontinued operation of the test ball. The increase in vibration after 100 minutes points to the breakdown of the lubricant film, metal-to-metal contact, component temperature rise, and eventual seizure or fracture, depending on heat dissipation paths. Figure 25.33 shows the effect of increased load on the time from initial spalling
1010
VIBRATION, NOISE, AND ~
O
~
~
I
O~ NO
~
AccelemmeterSignal M. Time for B.ll21A
3.5
5
I
3t
D
25.33. Accelerometer voltage signal for an AIS1 52100 steel, 22.22 mm (0.875 in.) diameter ball, endurance tested at 45,000 rpm, and 37.8"C (100°F) lubricant supply temperature ina ball/v-ring rig at (a) 3170 MPa (460,000psi) maximum Hertz stress; and ( b ) 3860 MPa (560,000 psi) maximum Hertz stress.
to component failure. It can be seen that Hertz stress magnitude, and hence load magnitude, has a profound effect on the duration of spall progression. Moreover, comparing Fig. 25.33~~ (373°C (100°F))with Fig. 25.32 (71.1'6 ( 16OoF)),it may bedetermined that lubricant film thickness and heat dissipation rate have significant effect onthe rateof spall progression; the lower the temperature, the longer is the duration of spall progression to component failure. Kotzalas E25.211 also correlated the ball traction coefficient with the degree of spall pro~ession,and degree of spall progression with the accelerometer signal. Thus, it was possible to correlate friction with the accelerometer signal. Therefore,accelerometer signal, so correlated, might be used to indicate friction, which may be used in an on-board computer program to more accurately estimate remaining bearing life.
This chapter provides an indication of how bearings can affect the vibration of machines, as a result of either inherent design characteristics or im~erfectionsand deviations from ideal running geometry within the bearing. Some examplesillustrated that such imperfections and geometric deviations can occur during bearing component manufacture, during assembly of a bearing into a machine, or from bearing deterioration during operation. Each can have a ~ronouncedeffect on machine vibration, either by altering stiffness properties or by acting as a source of forces to directly generate vibration.
T
O
R
~ F E ~ N C E S
1011
It was also shown that detection of vibration frequencies and amplit ~ in machintudes is used as a means to determine the ~ e ~ ofZbearings ery. Using condition-monitoring, it is possible to detect bearing fatigue failure in machinery and to determine the location of the failed bearing. Recognizing that bearing function does not ceasewith the initial rolling component surface spa11 and using the prognostic methods associated with condition-based maintenance techniques, it is possible to estimate how long the machine may be expected to continue to function reasonably.
25.1. T. Tallian and 0. Gustafsson, “Progress in Rolling Bearing Vibration Research and Control,”ASLE Paper 64C-27 (October 1964). 25.2. R. Scanlan, “Noise in Rolling-Element Bearings,”ASME Paper 65-WAIMD-6 (November 1965). 25.3. Military Specification Mil-B-1793lD (Ships), “Bearings, Ball, Annular, for Quiet Operation” (April 15, 1975). 25.4. 0. Gustafsson, T. Tallian et al., “FinalReport on the Study of Vibration Characteristics of Bearings,” US. Navy Contract Nobs-78552, U.S. Navy Index No. NE 071 200 (December 6, 1963). 25.5. R. Sayles and S. Poon, “Surface Topography and Rolling Element Vibration,’’ in Precision Engineering, IPC Business Press(1981). 25.6. L. Johansson, “BearingNoise in Electric Motors,”Ball Bearing J. 200 (1979). 25.7. J. Hyer and D. Sileo, “Some Practical Considerations in the Selection and Use of Ball Bearings in Small Electric Motors,” Small Motor Manufacturers Association (March 1985). 25.8. J. Mitchell, machine^ Analysis and Monitoring, PennWell, Tulsa, Chap. 9 and 10 91981). 25.9. E. M. Yhland, (‘Waviness Measurement-An Instrument for Quality Control in Rolling Bearing Industrx” Proc. Inst. Mech. Eng., 182,Pt. 3K, 438-445 (1967-68). 25.10. 0. Gustafssonand U. Rimrott,“Measurement of SurfaceWaviness of RollingElement Bearing Parts,”SAl3 Paper 195C (June 1960). 25.11. International Organization for Standardization, “Acoustics, Vibration and Shock,” IS0 Standards Handbook 4 (1980). Tools for Use in 25.12. S. Norris, “Suggested Guidelines for Forced Vibration in Machine Protective Maintenance and Analysis Applications,” in Vibration ~ n a l y s i sto Improve Reliability and Reduce Failure, ASME H00331 (September, 1985). 25.13. J. Taylor, “Identificationof Bearing Defects by Spectral Analysis?”A ~ M E J. Mech. Design 102 (April 1980). by Spectral 25.14. J. Taylor, “AnUpdate of Determination of Antifriction Bearing Condition Analysis,’’ VibrationInstitute (April 1981). 25.15. R. Randall, “Cepstrum Analysis and Gearbox Fault Diagnosis,” Bruel& Kjaer Application Note 233-80. 25.16. E. Wallin, “Preventionof Bearing Damage Caused by the Passage of Electric Current,” Ball Bearing J. 163 (1968).
25.17. S. Andreason, “Passageof Electric Current Through RollingBearings,”Ball Bearing J. 163 (1968). 25.18. A. Boto, “Passage of Electric Current Through a Rolling Contact,” Ball Bearing J: 163 (1968). 25.19. J. Mathew, and R. Alfredson, ‘‘The Condition Monitoring of Rolling Element Bearings Using Vibration Analysis,” ASME Paper 83-WAINCA-1 (November 1983). 25.20. J. Cronkhite, “Practical Application of Health and Usage Monitoring (HUMS) to Helicopter Rotor, Engine, and Drive Systems”, Paper presented at the American Helicopter Society 49th Annual Forum,St. Louis, Mo. (May 19-21,1993). 25.21. M. Kotzalas, “Power Transmission Component Failure and Rolling Contact Fatigue Progression” (Ph.D.thesis in Mech. Eng., Pennsylvania State Univ., August 1999).
Symbol
I3
B1
c D
e F Fo
I JG
K N/mm"
KI M
Description
Units
Bearing totalcurvature = (f, + fi - 1) Damping matrix Damping coefficient N-sec/mm (lb-seclin.) Ball diameter mm (in.) Eccentricity mm (in.) Applied force N (1b) Total amplitude of harmonic forcing function N (lb) Identity matrix moment Mass of inertia N-mm-sec2(in.-1b.-sec2) Bearing deflection constant (x = 1.5 for ball bearing, 1.1for roller bearings) (lblin.") Stiffness matrix Large body mass (rotating unbalance case) kg (lb-sec2/in.) 1013
1014
Symbol
ROTOR D ~ ~ I AND C SCRITICAL SPEEDS
Description Mass Bearing internal diametral clearance Ball normal load Radius of gyration Stiffness Time Displacement (lateral) isp placement a ~ p l i t u d e Number of rolling elements Contact angle (loaded bearing) Contact angle (unloaded bearing) Elastic deflection Angular displacement or inclination Angular displacement amplitude Phase angle Eigenvalue Damping factor Frequency Natural or critical frequency Frequency of vibration
Units kg (lb-sec2/in.) mm (in.) N (1b) mm (in.) Nlmm (lb/in.) see mm (in.) mm (in.) rad rad mm (in.)
O, O,
rad rad O, rad
O,
O,
radlsec rad/sec
SUBSCRIPTS Refers to first or second natural frequency Refers to axial direction Refers t o critical Refers to ith frequency or ith value Refers to inner raceway Refers t o amplitude value Refers to outer raceway Refers to radial direction Refers to coordinate direction Refers to coordinate direction Refers to coordinate direction
GE In 1948, DenHartog [26.1] published the first text to address the mathematical principles of vibrating motion in mechanical systems. These principles have been built upon and extended over the years to provide vital tools to machine designers. With the advent of digital computer
D
~
E FORCED D ~
~
T
~
Q
N
1015
S
techniques, sophisticated design toolshave been created that provide the ability to predict critical speeds and rotor behavior in high-speed, shaftbearing systems. The specific topic of rotor-bearing dynamics rose in sophistication and impo~anceto such a degree that in 1965 the US. Air Force sponsoredthe construction of a 10-part design guide on the subject 126.21. The guide was later updated in 1978 [26.3]. The majorityof this book deals with the technologies of rolling bearing design ranging through materials, performance prediction, lubrication, and fatigue life. This chapter deals with the one characteristic of rolling bearings that directly in~uences shaft-bearing dynamic behaviorstiffness. The subject of bearing-rotor systems interaction will be addressed in a threefold manner. First, the basics of mechanical vibration to form a foundation for understand in^ the analytics involved in rotor dynamics are presented. Second, the concept of bearing stiffness and the nature of its behavior in the rotor system environment are considered. Finally, a brief overview of rotor dynamic analysis is given.
The basic principlesof flexible rotor dynamics stem from the mathematical representation of dampedforced vibrations [26.4]. Figure 2 6 . 1 ~ ~ shows the system as a mass with viscous damping being forced by a
C
k
cwx
kx
cx
FIGURE 26.1, (a)Free-body diagram of a viscously damped system with a harmonic forcing function. (b) Vector relationship of (a).
101
harmonic exciting forcePo. The equation of motion that follows from the free-body diagram is
+ ex + kx = Fo sin wt
mx
(26.1)
The solution of this differential equation is characterized by two parts: the complementary function and the particular integral. The particular solution is of the form
x
=X
sin (wt
(26.2)
- ct,)
where X is the amplitude of oscillation and cf, is the phase of the displacement with respect to the exciting force. Vectorily, this is shown in Fig. 26.16. From the vector diagram it is seen that
6 x = [(k - wtw2)2 +(~w)~]l/~
(26.3)
and ct,
=
tan-1
(k
-
)
mu2
(26.4)
Dividing by the stiffness k allows the expressions to be made dimensionless. Thus
and tan ct, =
cdk 1- mw2/k
The following quantities are now defined: wn = (k/m)2 = natural frequency
C,
=
2m0,
=
critical damping parameter
[ = CIC, = damping factor
cdk
=
( C / C c ) ( C c ~ /=k )(250/0,)
Thus, equations (26.5) and (26.6) can be rewritten as
(26.6)
17
Xk -= F,
1 {[l- (W/WJ2I2 + (2JW/Wn)211'2
(26.7)
and tan (f)
=
2~#/Wn
1-
(26.8)
(W/WJ2
Equations (26.7) and (26.8) are plotted in Fig. 26.2 to show the relationship between frequency( d u n )and damping ( J ) on amplitude and phase. The curves show the impact of damping factor on amplitude and phase, particularly in the region of resonance. Constructing vector diagrams of the system such as Fig. 26.3 yieldsa better understanding of the effects of frequency. For values of d u nthat are much less than 1 (Fig. 26.3a), both inertial and damping forces are small, resulting in a small phase angle (f) and an impressed force nearly equal to the spring force. Figure 26.3131 shows the situation for @/con= 1. Here the phase angle is go", and the inertial and spring forces balance each other, allowing the impressed force to overcome the damping force. At large values of wIwn (Fig. 26.3~)the phase angle is approaching 180", and theimpressed force is stretched out to compensate fora large inertial force.
.O
2.0 Fr
I
I
I .O
ratio
I
I
Frequency rotio
&
26.2. Plot of vibration amplitude for the system of Fig. 26.1.
ROTOR D ~ ~ I m C SC R I T I SPEEDS ~ ~
101
In summary, the differential equation of motion and its complete solution are written (26.9)
x(t) =
The most common sourceof harmonic excitation in rotor-bearing systems is rotating unbalance. A simple rotating unbalance system with a spring and damper support can be represented as shown in Fig. 26.4. The unbalance can be represented by a mass rotating at some radius represented by an eccentricity e. If the mass is rotating with angular velocity co and x is the displacement of the nonrotating mass A4 - rn, then the displacement of rn is
26.4. Schematic of a simple rotating unbalance system.
ED ~ O R ~ E ~D
~
~
1
x,
0
=
x
~
s
+ e sin wt
(26.11)
The differ~ntialequation of motion is thus
(M - m) x
d2 +m(x + e sin at) = -Kx dt
- e2
(26.12)
which rearranges to ~x
+ cx + kx = mea2 sin wt
(26.13)
Equation (26.13) is the same as eqaation (26.1), where F, is represented by mew2. It follows that thenondimensional form of the solution is (26.14) and
The solution is presented graphically in Fig. 26.5. Assume a system as shown schematicallyin Fig. 26.4.
A combination of rotating unbalanced causes a resonant amplitude of 1.2 mm (0.0472 in.) Upon considerablyincreasing the speed of rotating of the unbalanced masses, the resonant amplitude seemed to level out at 0.16 mm (0.0063 in.) Calculate the apparent damping factor. From equation (26.14) the resonant amplitude is
1.2
=
hen the ratio of w to mn is very large, the equation becomes
ROTOR ~
~ AND CRITICAL ~ SP CE ~ D S
1.0
0
0
I .o
3.0
Frequency ratto
2.0
3.0
4.0
4.0
5.0
5.0
&
26.5. Plot of vibrration amplitude for a system with a rotating unbalance.
0.16
me -
M
Solving both equations simultaneously yields
1.2
=
0.16 oy 4
y = 0.0666
~onsidera free-body diagram that can be constructed to represent a rigid beam on elastic supports [26.5]. It would be represented by a large unsymmetrical body supported by unequal springs. Such a system is shown in Fig. 26.6. The mass of the body is m, and its mass center is located To define the motion of the body, a coordinate x will be used to fy linear position and a coordinate 0 to specify angular position. will be measured from some reference position or neutral plane.
S
k, (xc + b, e)
. Free-body diagram of a coupled system (rigid beam), ecause of this coordinate mix, the system is said to be coupled, The nature of th6 coupling can only be determined by analysis and depends on the coor~inatesselected in defining the equations of motion. For instance, if x G were measured as the linear displacement of the mass from its mass center 6, then the equations of motion become nzji.G
JGS
+ (k1 + k2) X G + (k2b2 - klb1) 6 = 0
+ (kzb, - klb1)
XG
+ (klby + k2bi) 6
0
(26.16)
These equations are said to be “statically coupled” through the coupling coefficient k2b2 - klbl. This system model closely resembles the rotorbearing system situation. Assuming motionsto be harmonic, sin (wt
+ (2;)
6=
(26.17)
Solving for amplitude gives
+ k2)& + (k2b2 - klbl) (k ,b2 - klbl) X0 + (klby + k 2 b 3 (kl
(26.18)
The followiig simplifications are now made: - klbl)/m = b* (klby + k 2 b i ) / m = C*
where JG = mr2,r being the radius of gyration with respect to 6. tions (26.18) now become
etting the deter~inantequation to zero and solving for the principal frequencies yields
1022
ROTOR DYNAMICS AND CRITICAL SPEEDS
CO:,~=
21 ( c*7 + a*
[(f
k
(y)] } 2
- a*)2
+
1/2
(26.20)
Example 26.2. Consider a rigid beam system, as depicted by Fig. 26.7a, with the following data: b , = 400 mm (15.75 in.), k , = 100 N / mm (5708 lb/in.), b 2 = 200 mm (7.874 in.), k 2 = 70 N/mm (399.6 lb/ in.). Assume that the beam mass is 5.1813 kg (0.0296 lb-sec2/in.)and that the radius of gyration is 300 mm (11.81 in.) Determine the natural frequencies.
k 2 b 2 - klbl m
=
70
X
200 - 100 X 400 5.181
-5018 mm/sec2 (- 197.6 in./sec2)
k,by
+ k 2 b $ = 100 X m
3.628
f
X
(" +
4002 + 70 5.181
[(s
a*
-
4{40.3 + 32.8
+_
a*)2
7.33 rad/sec and
[56.48 + 1119]1/2)
o2 =
FIGURE 26.7. Free-body diagram of system for Example 26.2.
]}
+ (T) 2b*
53.7, 19.4 rad2/sec2 m1 =
2002
lo6 mm2/sec2(1.429 X lo5 in.2/sec2)
2 r2
Thus,
X
4.4 rad/sec.
1023
COUPLED VIBRATORY MOTION (RIGID SHAFT)
In matrix notation the amplitude equations (26.18) are k + k2 k 2 b 2 - klbl [ k i b , - klbl klb: + k 2 b i ]
[z]-.[ [z]
(26'21)
=
This presents the coupling term more visibly as the off-diagonal elements of the spring matrix. Further, if the spring moments are equal, then k 2 b 2 - klbl = 0 and the solutions become -I=(-->
k l + k2
1127
)
klb: JG + k2bi
- 2 = (
(26.22)
These solutions represent the translation (x)and rocking ( 6 ) natural frequencies. If k 2 b 2 - klbl # 0 , then the frequencies are mixed. That is, the first frequency is translation with some rocking, and the other frequency is predominantly rocking with some translation. Associated with each eigenvector solution or natural frequency is a natural mode shape. The mode shape is defined mathematically as the ratio of the displacement and rotation amplitudes at the mass center. For the uncoupled case ( k 2 b 2- klbl = 0) they are
x, = w fb* a*' e, = a$r2b
-
c*
(26.23)
-
Thus, when the system is decoupled, the behavior is governed by two distinct single-degree-of-freedomsystems. Conversely, it can be shown that, at the extreme of the coupled case, the system behaves as a single-degree-of-freedomsystem. This occurs when (26.24) This means (26.25) This condition can only be satisfied when
(k)
3= k2 and
2
(26.26)
This gives
A series of n masses connected by sprin is shown in Fig, am. This re~resentsthe nerd case of a ~ u l t i - d e ~ e e of-~eedomsystem. The diff~rentiale ~ u a t i o of ~ $motion are thuswritten as
and in matrix form they
m1 0 0
0 m 2
0
0 0 m3
0
... ...
+
k +k2 -k2 0
* * * * * * *
*
e
-k2 ,122 k3
+
0 -k3
-k3
i
* 9
e
*
e
*
... * * *
0 0 0
kn+l + kn,
X
This is a particular case of the general equation
where the square matrices [ml and [k]are the mass matrix and stiffness matrix, respectively, and are given by mll m21 m22 m23
and
m 1m21 3
* *
rn In
* * '
m2n
[A
M I(26.35) {X}= 0
-
where 2 = -Xwith h = w2. h is known as the direct frequency factor. The determinant of equation (26.35), W-Mj=O
(26.36)
is the characteristic equation of the system. The roots of this characteristic equation (eigenvalues)represent the naturalfrequencies of the system:
The eigenvectorscan now be determined by substituting hi into equation (26.35) and solving for the corresponding mode shapes Xi.For an nde~ee-of-freedomsystem there are n eigenvalues and n eigenvectors. It is also possible to solve this system by methods using the adjoint matrix (see any reference on matrix algebra). Let
=A
-
hT
Then by definition the inverse is (26.39)
lBlI
=
(26.40)
B adj 13
or, resubstituting for -
I/I
=
[A - MI adj [A
h71
If the eigenvalue is introduced into the equation (I termi~antis zero and
[O]
=
[A
-
Ail'] adj [A - hill
(26.41) =
hi),then the de(26.42)
From equation (26.35), for some ith mode,
[A - h,Il [Xi] =0 It must follow that the adjoint matrix adj
(26.43)
- hill must be a column matrix of the eigenvectors Xi(multiplied by some arbitrary constant).
.
Considering the generalized system of Fig. 2 6 . ~per, form a complete eigenvalueleigenvector solution[26.4]. The equations of motion in matrix form are
[- ']I("'> + 0
2m
[ 2-kk
2,
-2k 'l(xl} x2
~ssuming harmonic motion, substituting h the inverse of the mass matrix yield
=
{~}
= w2, and
(26.44)
multiplying by
2klm - h -klm -% k l m k l m - h
(26.45)
The characteristic equation is thus h2
-
3
(k)2
0
A, = 2.366
):(
fk) h +; \ml
=
(26.46)
from which the eigenvalues follow as
(k)
v2
A,
=
0.634
v2
(26.47)
Returning to equation (26.45) and substituting in the eigenvalues, the adjoint matrix: is obtained: adj [A ~ubstitutingA,
=
-
Ail]
=
I
[ ~ ~ A,~ m k l m2 k l m - hi
(26.48)
0.634 ( k l m ) 1 / 2in (26.48) gives
~} [ =
0 366 1.000 k 0:500 1.3661
1/2
(m)
Normalizing each column to unity results in the first eigenvector
26.9.
System for Example 26.3.
1
R O ~ O RI)
~}
0 732 0.732 k = [1:000 1.000](m)
v2
or
k
0.732
=
{l*ooo}(~)v2
Likewise, it can be shown that
x2=
-2.73 {l.OO
k }(~),,
The two normal modes are shown schematically in Fig. 26.10.
A cornmon design parameter for oil film bearings is stiffness, which is defined as follows:
s = d25’ -
I
(26.49)
dl3
n other words, stiffness 5 is the inverse of the rate of change of bearing deflection with applied load. To find the stiffness of a particular rolling bearing under a given type of loading, one must first determine the load distri~utionamong the rollers and then relate the m ~ i m ~rolling rn elradial ement load to the applied load. This is simply done for single-row ball bearings with nominal clearance for which (7.24)
~ u ~ s t i t u t i for ng
max
in equation ( 1 0 . ~yields ) 1.o
w2 = (2.366 k/rn)”*
. Mode results of Example 11.3.
ss (26.50) From equation (26.50) according to (26.49), the stiffness of the bearing is S = 329002D1'2
(26.51)
From equation (26.51)it is apparent that thestiffness of a ball bearing is a nonlinear relationship because it is dependent on the square root of radial deflection. In this respect a ball bearing is unlike a simple spring for which deflection is linear with respect to load. Roller bearin~sreact in similar fashion except that load varies with deflection to the 1.11 power and stiffness varies directly with deflection to the 0.11 Another relatively simple case is that of an an~lar-contact d to simple axial load. The loadis distributed equally among
where a is the contact angle in the loaded condition; ao is the contact angle in the unloaded condition, givenby cos ao = 1 - pd
(2.9)
is the mounted internal diametral clearance. seen that a thrust load Fa applied to the inner ring results
and since
equation ( ~ . was ~ ~ obtai~ed: ) 1.5
- 1
103
ROTOR D Fa
=
ZK,,(BD)1.5
(z
~
sin a --
~ ANB C CRITICAL S SPEEDS
(7.30)
1)1*5
and Sa =
BD sin ( a - a")
(7,36)
cos a
Computing bearing stiffness involves solving equation (7.30) for bearing equilibrium by iteration. In Chapter 7, it was shown that K,, is also a function of a and therefore must be determined by iteration schemes as well. In Chapter 7, however, it was further shown that for a thrustloaded, angular-contact ball bearing K,, can be replaced by KD0-5/B1*5, is determined by Fig. 7.5. Hence (7.30) became 1.5
(7.33)
Since stiffness
%="Wa da
I
S =
d6,
da d6, ll2
s=
[cos3a (cos a" - cos a ) + 1.5 sin2 a]
.
A 218 angular-contact ball bearing is loaded axially (4000 lb). Determine the axial stiffness. gular-contact ball bearing, Z = 16, D = 22.23 mm 0.0464, a" = 40". FromFig. 7.5, = 896.7 N/mm2
(130,000 psi). The new contact angle can be determined from equation (7.30) by iteration:
312
17,800 (1~)(22.23)2(896.7) remains unchanged, iteration yields a
=
41.36".
S = -
%~~
B cos a'
(E -1)
v2
[cos3a (cos a'
16 X 22.23 X 896.7 cos 40' 0.0464cos40' (cos 41.36' X
=
cos a'
[(cos3 41.36')(cos 40"
8.522
X
-
-
cos a) + 1.5 sin2 a]
I>
cos 41.36')
v2
(26.52)
+ 1.5 sin2 41.36'1
lo5 N/mm (4.864 X lo6 lb/in.)
Bearing stiffness behavior can be extrapolated to the whole bearing. Thus, the bearing locations in a rotor-bearing system can be modeled as spring supports. Remembering rigid beam theory, it is seen that there is a need to analytically represent a bearing location with both lateral and angular stiffnesses. In the analytical model the reaction force and the reaction moment of each bearing are felt by the rotor through a single location on the rotor axis [26.61. Schematically this is illustrated in Fig. 26.11. A typical linear spring restraining the lateral displacement and a torsional spring opposing the rotation are attached to the same point on the rotor axis. A complete description of the characteristics of the support bearings, however, involves much more than the specification of the two spring constants [26.6] for the following reasons: The lateral motion of the rotor axis is concerned with two displacement components and two inclination components. The restraining characteristics may include cross-coupling among various displacement/inclination coordinates. The restraining force/moment may not be temporally in phase with the ~isplacement/inclination. The restraining characteristics of the bearing may dependon the rotor speed or the vibration frequency or both.
Tss?s 26.11. Basic model concept for bearing stiffness,
ROTOR D
~
~ AND C CRITIC^ S SPEEDS
Bearing pedestal compliance might not be negligible. To accommodate the above concerns, the support bearing characteristics are described analytically as a 4-de~ee-of-freedomimpedance matrix:
where [WN] is a columnvector containing elements that are the two lateral displacements (a, 8,) and the two lateral inclinations (Or, 6J of the rotor axis at some bearing location N . With a right-hand Cartesian coordinate system as shown in Fig. 26.12,directions can be defined. From the figure the x axis is coincident with the spin vector of the rotor or the rotor axis. This results in thex axis and y axis being perpendicular and in the direction of radially applied external static loads. Intuitively, Sx, Sy and 0,, are, respectively, the displacements and rotations at the appropriate axis. Returning to equation (26.53), it is noted that [ZN]is a complex 4 x 4 matrix that represents the stiffness and damping coefficients at the bearing support. According to common notation this is
where [K]' is the stiffness matrix and [BN]is the damping matrix, v is the frequency of vibration. Most commonly, lateral, linear, and angular displacements do not interact with each other, so the nonvanishing portions of [KJ and [BN]are separate 2 X 2 matrices. Thus
(26.55)
(26.56)
Here the four 2 X 2 matrices characterize the support and are constructed as follows:
B Y
F I G ~ E26.12. Bearing-shaft coordinate system.
load
(26.57) (26.58) (26.59) (26.60) Figure 26.13 depicts an angular-contact ball bearing arranged with an Y
FIG^^ 26.13. Basic ball bearing configuration and coordinate system.
orthogonal xyz coordinate system. The outer ring is assumed fised in space while the inner ring is permitted to displace with respect to the coordinate system. For complex:loading such as combined radial, asial, andmoment loading, the location of the inner ring is defined by the three lineardisplacements s,, s,, 6, and the two angular displacements $,, $., The components of the 2 X 2 stiffness matrices of equations (26.57) and (26.59) then become
(26.61)
I as,as,
I (26.62)
The axial stiffness components are not presented, given that theanalysis is commonly lateral. The relationship between the individual rolling element as well as total bearing stiffness and external applied loadagrees with Hertzian contact theory and is nonlinear. Figure 26.14 illustrates a one-dimensional loaddisplacement curve. Since the theories upon which rotor dynamic analyses are built are linear, the bearing stiffnesses are linearized, as shown in Fig. 26.14, by taking a tangent to the load-deflection function near the point of static equilibrium.
cts
As already developed, the deflection of a rolling element bearing under load is due to the contact compliances at both the inner and outer raceways. At higher operating speeds centrifugal force acting on a rolling element increases the load at the outer raceway contact, which causes, for a thrust-loaded, angular-contact ball bearing, a load decrease at the inner raceway contact. See Fig. 26.15. Owing to the nonlinear nature of the load-deflection characteristic and the load-speed behavior described before, the stiffness of a rolling element bearing is speed-dependent. As speed increases, there is an additional effect that occurs simultaneously with the contact load effects
C
6%
26.14. Linearization method for calculating bearing stiffness.
P P
P-
.
Angular-contact ball bearing contact angle change to due load and speed. (a)Free contact angle (no load). ( b )Contact angleunder load. ( c ) Contact angles under load and high speed.
already described; the outer raceway contact angle decreases and outer raceway contact stiffness increases while the inner sees an opposite effect. The radial stiffness components can be formulated in terms of the individual contact stiffnesses as follows:
k , = k , cos a,
(26.63)
A?,
(26.64)
=
ki cos ai
Combining as springs in series yields
ROTOR ~
100
~
~ AND I CEITICAL C S SPEEDS
300 Normal load (N)
F I G ~ 26.16. E Effectof load and speed on bearing radial stiffness.
k,
=
1+
ki cos ai (kilko)cos ailcos ao'
(26.65)
As a result of the kinematic changes,the increase inaicauses a decrease in the numerator ki cos ai.Since this is the dominant term, k , undergoes a decrease with increasing speed. Figure 26.17 illustrates the combined effect of load and speed on bearingradial stiffness. Plotted against nor-
I
0.6 0
1
1
I
10
20
30
Rotational speed ( lo3rpm)
F ~ G 26.17. ~ E Effect of contact angle and speed on radial stiffness.
TERI~TIC$ C
OF B~~
$TI
mal load is normalized radial stiffness value that is the ratio of radial stiffness at load and speed to the radial. stiffnessat static conditions.
From the analytical develop~entof Section 26.6,the interdependence of load and stiffness with contact angle and deflection was established. it can be shown that From these~de~elopments (26.66) If speed effects are neglected, increasin~the contact angle will decrease the bearing radial stiffness; see Fig. 26.17, where, for any given speed, stiffness decreases with increased contact angle. F u ~ ~ ethe r , effect is more dramatic at higher speed.
d effect in radial bearings is the effect upon load disa1 clearance. Of particular interest to the stifkess ch~racteris~ics of bearings is the condition of preload or negative internal 6.18 illustrates the preloading effect on load distribu~ionin a' radiall~loaded bearing. As preload is increased (clearance reduced or more negative), the number of elements sharing a given radial load Fr is increased. This lowersthe individual contact load and thereby decreases stiffness. Eventually, a preload level will be reached that will create a 100% load zone, thereby creating an increase in stiffness with. further preloading.
(a)
(b)
2633. Load dist~butionin a radially loaded bearing. ( a ) Zero clearance. (b) Negative clearance (preload).
1038
ROTOR DYNAMICS AND CRITICAL SPEEDS
The combined load-preload effect on bearing radial stiffness warrants further explanation [26.61. In general, the radial stiffness versus radial load curve for an angular-contact bearing is composed of three distinct behavior regions. Figure 26.19 illustrates this for a typical angularcontact ball bearing with heavy preload. In the region of light radial load the stiffness is nearly constant. The reason is that in this region, the preload is dominant and the radial load is not sufficient to effect a change. In the middle region or region of medium radial load, there is a slight drop in stiffness to some minimum value. Here the radial load is nearly equal to, and then surpasses, the preload. This causes greater load sharing among the elements, therefore reducing total stiffness. The third region shows the effect of the radial load dominating the system, resulting in a linearly increasing stiffness.
Shaft Bending Effects The interactive nature of the shaft-bearing system is terms of its mechanical behavior dictates that shaft deflections influence bearing deflection and, hence, stiffness. One such shaft deflection is angular deflection or misalignment, which results from the bearing resistance to applied moments. Figure 26.20 illustrates a cylindrical roller bearing experiencing shaft misalignment. It is obvious that shaft bending (misalignment) will influence the load distribution. The general trend is that as shaft stiffness decreases, bending increases, resulting in increased bearing stiffness.
1
0.2 lo2
i
5
I
I
lo3 Radial load
5
I
lo4
(N)
FIGURE 26.19. Effect of preload on bearing radial stiffness (angular-contact ball bearing-heavy preload).
ROTOR DYNAMICS ANALYSIS
1039
FIGURE 26.20. Misalignment of bearing rings.
ROTOR DYNAMICS ANALYSIS Critical Speed The major objective of rotor dynamics analysis is t o allow development of rotating machinery that will be free from vibrational problems detrimental to its performance. This is generally a two-step process consisting of a critical speed analysis and a synchronous response analysis. Critical speed analysis is the process of determining the natural frequencies of a rotor-bearing system and identifying the mode shapes associated with them. This must be done t o insure that the machine operating speed is located at a frequency safely spaced from any undamped natural frequencies. If allowed to operate at or near a natural frequency, a machine may enter an unstable and destructive vibration situation.
Synchronous Response Synchronous response analysis allows the further examination of the rotor-bearing system behavior as a function of operating speed. It relates rotor displacement and bearing loads to operating frequency. Both of the foregoing analytical techniques are accomplished by computer programs constructed using the mathematical principles reviewed in this chapter. In general, the programs mathematically model the shaft-bearing system as a series of rigidly connected, multispan beams supported on springs. Division of the rotor into segments allows for the proper modeling of variations, such as cross-sectional thickness, material, location of gears or discs, and location of bearing supports.
2040
ROTOR DYNA,REICS"I3 C
Figure 26.21 is an example of a sha~-bearingsystem model. Section mass canbe either distributed across the element or lumped at the ends of the element. The problem is then solved by a transfer matrix technique. Thetransfer matrix is derived directly fromthe differential equation describing the dynamic behaviorof each beam s e ~ e n t .
Associated with each identified natural frequency (critical speed) is a normalized mode shape. An estimation of the relative displacement of each shaft station is computed.Figure26.22 illustrates two common bending modes associated with systems involving stiff bearings. These Station
26.21. Model representation of a s h a ~ - b e a ~systeq, ng
F I ~ 26~.22,E Typical mode shape results for a flexible rotor-bearing system.(a)First bending mode, ocl. (b) Second bending mode,ut,,.
are the shapes with which the system would vibrate if excited at the corresponding critical speed. Rotor-bearing systems can be escited into vibration modes by any number of periodic forces. The most common cause of periodic forces in rotating machinery is mass unbalance. Mass unbalance may be a result of inadequate balancing techniques, shaft bending due to gravity, debris deposits, or unstable rotor structures. Analyzing the effects of unbalance forces on shaft vibration and bearing loads is done by synchronous response analysis. By solving the differential equation of motion for each shaft segment, with the harmonic driving forces being represented by the unbalance mass, vibration amplitude can be computed. Then by sweeping through a range of frequencies (operating speeds) synchronous response plots for each shaft station can be constructed. Figure 26.23 illustrates the typical results of a synchronous response. They identify the location of the critical operating speeds and allow the determination of safe operating frequencies based and ) bearing load (Fig. on severity of vibration amplitude (Fig. 2 6 . 2 3 ~ ~ 26.2223). sponse peak identifies Q)
3 +-._. a.
6
0 ._. +E
-0
> Shaft rotational speed
(a)
I
Response mak identifies
FIGURE 26.23. Typical synchronous response plots for a flexible rotor-bearing system. (a)Shaft station vibration amplitude.( b ) Bearing station load magnitude.
1
ROTOR D
~
~
C CRITIC^ S S~~EDS
Rotating shaft-bearing systems have a natural tendency to form a first bending mode at certain speeds. Theshaft motion under these conditions is complex and is referred to as whirling. Whirling is defined as the rotation of the plane made by the bent shaft and the line of centers of the bearings. Whirling is most prevalent in systems with little or no bearing damping, such as systems involving rolling bearings. The shaft whirl may take place in the same direction (natural precession) as that of the rotating shaft or in the opposite direction (reverse precession). In addition, the whirling speed may be equal to (synchronous) or different from (asynchronous) the shaft rotational speed. In general, whirl is a self-excited force. Whirlingin lightly damped systems may be the result of internal friction. Studies into thisphenomenon [26.71 have pointed out the importance of the use of flexible foundations for reducing the whirl threshold forhigh-speed systems with rolling bearing supports and thereby achieving rotor stability.
~nderstanding the mechanical interactions between the various components (shaft, bearing, seals, impellers, etc.) of rotating machinery is critical to their successful operations. Without incorporating this understanding in thedesign stages of new machinery, catastrophic failures of shafting and/or bearings could occur. Since digital computers have made complex: and repetitive calculations quite simple, engineers and designers have been applying the foundations of mechanical vibrations into useful designtools. Rotor dynamics and theanalysis of rotor-bearing systems is an outgrowth of their efforts.
26.1. J. DenHartog, ~ e c ~ a n i c Vibrations, al 4th ed., McGraw-Hill, New York (1956). 26.2. Air Force Aero Propulsion Laboratory Report AFAPL-TR-65-45, Parts I-S, “Rotor Bearing Dynamics Design Technology” (June 1965). 26.3. Air Force Aero Propulsion Laboratory Report AFAPL-TR-78-6, Parts I-IV, “Rotor Bearing Dynamics Design Technology” (February 1978). ‘26.4. W. Thomson, Theory of Vibration with ~ ~ ~ Z i c a t i o2nd n s , ed., Prentice-Hall, Englewood Cliffs, N.J. (1972). 26.5. R. Vierck, Vibration Analysis, Harper & Row, New York (1979). 26.6. A Jones and J. McGrew, “Rotor Bearing Dynamics Design Guide, Part 11, Ball Bearings,,, AFFAPL-TR-78-6, Part I1 (February 1978). 26.7. E. Gunter, “Dynamic Stability of Rotor-Bearing Systems,” NASA SI?-113, U.S. Government Printing Office, Washington, DC (1966).
Although rolling bearings are extremely reliable, failure can occur by improper operation or manufacture. With early failure-that is, within the anticipated service liferequirement-postmo~em investigations are frequently conducted to prevent another failure after the component is replaced. To determine rolling bearing failure, one must first gather data about the alloys from whichthe bearing was made and theirsuitability for the application, as well as operating conditions relative to loading, lubrication system, and external environment. Data must then be evaluated with respect to the manner in which the bearing failed. A bearing is deemed to fail when it does not perform as intended; this is frequently much sooner than when it ceases to function. Figure 27.1 is owc chart of the data-gathering process.
When possible, preliminary examination of the hardware should include observations of general features on related components before, and dur1043
104
ANALYSIS OF BE^^^ F ~ L ~ E S Application
Residual Applied
F
I
Lubricant
Temperature
~ 27~.1. EFlowchart of failure data-gatheringprocess.
ing, removal from equipment. Observations should be noted for future reference when more detailed examinations are conducted on the bearing. Similarly,general features should then be noted forthe bearing components during disassembly.
1SASS
GS
Disassembly varies with the types of bearings encountered, so it is beyond the scope of this chapter to deal with specifics for eachtype. W a t ever method is employed, care should be exercised to avoid inadvertent damage, which could bemisleading later in theinvestigation. If destructive methods are required to separate bearing components, the bearing should first be esamined to avoid destroying areas that contain regions of interest. Components should be identified with regard to corresponding sides of rings and rolling elements. Samples of grease or residual lubricant should be collected forfuture reference in theevent the failure is found to be lubricant related. If destructive methods have been used to separate components, extraneous debris should be removed, without rotating the rolling elements. Water and other fluids that come into contact with the bearing components during disassembly should be removed immediately to avoid corrosion. Degreasing should be followed by immersion into a preservative.
SMS
Precision bearing components are produced to fine surface finishes on the order of fractions of microns (microinches).W e n rolling contact sur-
faces are marred, stress conditions can be imposed that introduce a potential to reduce bearing life significantly. Abusive handling can induce nicks and dents that are harmful, particularly when located in regions tracked by rolling elements. Displaced metal particles generated by nicks and scuffing-type damageintroduce secondary effects when they dislodge and indent the raceway. Permanent indentation created by rolling element overload is called brinelling, a type of damage destined to result infailure. Brinelling may occur by an overload mechanism, such as dropping the bearing or improper mounting techniques. Initial s i p s of brinelling are signaled by noisy bearing operation, Raceways may be damaged whenthey are subjected to vibratory motion while rolling elements are not rotating. This type of damage, called false brinelling, can occur beforeand aftermounting on equipment. False brinelling damage has been observed in bearings that were subjected to vibration during transit aswell as on equipment that lay idle fora period of time.
Wear generally results in gradual deterioration of bearing components, which in turn leads to loss of dimensions and other associated problems. Failure by wear does not mean that bearings will be removed solely because of change in fit or clearances. Secondary conditions arising from wear can become the predominant failure mechanism. Lubricants may be affected or become contaminated to the degree where lubrication is severely diminished. Stress raiserscould begenerated that may serve as sites for crack initiation. Adhesive wear is described in Chapter 24 and is involved in removal of material and possible transfer to mating components. Under properly lubricated conditions, mating components’ microscopic asperities could yield and be flattened by cold work. Under these conditions the bearing might function adequately for its projected life. When lubricating conditions become inadequate, b how ever, increased friction results in metalto-metal contact,giving rise to localizeddeformation and friction welding. Operating forces cause increased plastic deformation by tearing the locally friction-welded regions from the matrix. One component is now pitted, and the other contains the transferred metal. This condition couldbeprogressive,depending on operating conditions.Generally, lighter adhesive damage is described as scuffing and scoring, whereas more gross damage is described as seizing and galling. Figure 27.2 shows a cylindrical rollerbearing raceway whichhas worn due to sliding motions between rollers and raceways. Figure 27.3 shows smearing damage leading to increased friction, high metal temperatures, material softening, and plastic movement of the raceway metal.
104
~ S T I G ~ T I O AND N ANALSSIS OF ~
~
I FN
G ~
27.2. Smearing damage on a cylindrical roller bearing inner raceway.
27.3. Smearing damage and sliding in a ball bearing raceway has led to increased frictionalheating and gross plastic deformation of the softened steel.
Abrasive wear occurs when hard particles become entrained between bearing components,moving relative to contacting surfaces. Coarser hard particles can induce microscopic furrows, whereas fine particles may produce a highly polished surface. Abrasive particles may originate from adjacent components in the mechanism's system and be transported by the lubricant. Oxidation products and carbides from ferrous components serve as abrasive media. Fretting, a wear mechanism described under corrosion, is a prime example of oxidation byproducts.
Lubrication problems may be associated with an inadequate lubricant, an inefficient lubricating system, or a combination of these conditions. Ideally, rolling elements are separated from the raceways by an EHL film, thereby minimizing friction between,and wear of, the bearing eom-
~
ponents [27.1]. Improperly lubricated bearings produce varying condisurface deterioration and tions, which lead to progressivecontact reduced life. Initial stages of wear involve the plastic deformation of grinding furrow asperities, which, in subsequent cold-working, fracture to produce extremely fine platelets of steel. This stage may also be accompaniedby adhesion of microscopic asperities that delaminate and pass on into the lubricant. These cold-worked,hard particles, which alsocontain carbides, serve as abrasive media. After a time, original grinding furrows in the rolling element tracks are worn smooth t o produce a glazed condition. Continued operation will generally lead to a deterioration stage that manifests itself as a frosted condition and, sequentially, to scuffing. Microscopic pits and crevices created by this adhesion mechanism serve as stress raisersfor the initiation of microspalls. Figure 27.4 illustrates the frosted surface condition and micropitting. In a lubrication system where the quantity of lubricant to vital areas is too low, bearing component temperatures increase. This in turn increases lubricant bulk temperature, decreasing viscosity and effecting increased friction that makes the situation progressively worse, surface degradation will be accelerated, and the surfaces will di the process progresses. Figure 27.5 shows a cylindrical which has undergone gross sliding in the absence of E: ciently thick to adequately separate the rollers and innerraceway in the contacts. Figure 27.6 shows a needle roller bearing which has experienced roller skewing, thermal excursion with attendant overheating and discoloration, cage fracture, and seizure.
Cracking of bearing components mayoriginate as a function of operating stress conditions via overload or cyclic loading (fatigue). ~dditionally, manufacturing-related cracks may derive from the steelmaking process andlor working processes. Withthe exception of cracks arising from inclusions and hydrogen in steels, cracks associated with Steelmaking and primary working processes seldom survive through secondary working processes. Cracks that arise from bearing manufacture secondary working processes frequently are associated with heat treatment and grindi~g.The nature of the processes are such that sequential operations tend to promote rapid crack propagationin bearing steels, if any are present d ~ r i n g manufacture. An obvious exception is cracking that began at the final ind ding stages. In view of the foregoing, cracking problemsencountered are usually operationally related as compared to manufacturing-related, occurring via cyclic loading. Stress states may be complex, arising from combined effects of component residual stress state, static stresses imposed by mounting, and stresses superimposed by applied loads.
1048
~ S T I G A T I ANI) O ~ ANALYSIS OF B
27.4.
~
~
Surface frostingand micropitting in a ball bearing raceway.
Cylindrical rollerbearing inner raceway whichhas experienced gross sliding and high friction, showing heat discoloration. Guide flange damage indicates roller skewing may have causedthe gross sliding. e
degree of contribution may be deduced by the direction of crack gation becausepropagation tends to proceed normal to the acting stresses.
~orrosion-initiatedfailures are often difficult to re Applications involving moisture-laden environ~ent face o~idationand produce rust particles and pits. potentia~media for prod~cingrapid wear via abrasion, and the pits can
G
E
ATION SPECIFIC ION OF
CO~ITIO~S
....
~"
(a)
(b)
27.6. Needle roller bearing: (a)With damage to ends of cage pockets due to
roller skewing; ( b ) subsequent cage failure due to overheating, Discolorationof outer ring is apparent.
function as stress raisers providing sites for crack initiation. The series of events that occurs during subsequent operation could alter the conditions initiating thefailure by the time a bearing fails and is examined. earings that operate in an environment where water is absorbed in the lubricant may be subjected to pitting corrosion by hydrolysis. Sulfur and chlorine contaminants in lubricants could dissociate from their respective compounds and react with the water to attack the steel on a microscopic scale[26.2]. Pits would then provide sites for spa11 initiation. Should spalling occur in this environment, it would be assisted by the corroding speciesand accelerate the rateof crack propagation. dissociated from water is also reported to assist and accelerate crack propagation [26.3]. In the event the contaminating species were to become available in higher concentrations, the lubricant wouldbecome acidified, and a more general corrosion could be observed. Fretting corrosion is an oxidation wear mechanism generated by relative motion between ferrous components that contact bearing surfaces. oducts are various stable an ableoxides of iron ( , which are colored red, an 27.7 shows fretting corros einnerring bore of a ball was caused by an insufficient interference fit between the re and the shaft outside diameter.
ents revolve in rin
E 27.7. Fretting corrosion. on the inner ring bore of a ball bearing.
teration of the normal rolling element path might be caused by misalignment associated with assembly or machine operating characteristics. Assembly deficienciesare self-explanatory, Excessive deflection and axial looseness are examples of machinery deficiencies. Since the stress distribution will be adversely altered, fatigue life will be reduced and propensity for spalling will be increased. The relationship between tracking pattern and failure mode must be evaluated with respect to cause and effect. Characteristic loading patterns in ball bearings are illustrated in Fig. 27.8. For roller bearings, misalignment and heavy loading give in-
IG
.
within a bearing.
(a) Load distribution
CIFIC C O ~ I ~ I O N S
I
I *
7.8. ( c o ~ ~ (b) ~ Normal ~ ~ eload ~ zone ) inner ring relativeto load. (c) Normal load zone outer ring rotating relativeto load or load rotating in phase with inner ring.(d) Normal load zone. Axial load. (e) Load zone when thrust loads are excessive. 0") Normal load zone combinedthrust and radial load. (g) Load zone frominternally preloaded bearing supporting radial load. (h)Load zones produced by out-of-round housing pinching bearing outer ring.(i) Load zone produced when outer ringis misaligned relative to shaft, ( j )Load zones when inner ring is misaligned relative to housing.
10
ES
dications through the location of initial spalling. For example, Fig. 27.9 shows a tapered roller bearing with spalling occurring on the trackedge near the small end flange. This points to misalignment as a probable cause. Figure 27.10 shows a cylindrical rollerbearing inner raceway with at theextremities of the track. This is an indicator of edge loading and extremely heavy load (assuming the rollers andlor raceways were properly crowned).
g r i ~ e Z Z is i ~a~term describing depressions in raceways. Roller element indentations may be induced by inadvertent overload while mounting
(a>
(b)
27.9. Misaligned tapered roller bearing. (a) Discoloration shows load zone on cup raceway; ( b ) spalling at raceway edge near small end flange of cone (courtesy of the Timken Company).
Cylindrical roller bearing with spalling damage at extremities of roller tra~~-indicatedvery heavy loadingor insufficient crowning (courtesyof EKE'). I
~ ~ U A T I O OFN~ ~ E C ~ I C
C O ~ I T I O N ~
1. Brinell marks on tapered roller bearing raceway (courtesy of the Timken Company).
1
STIGATION AND ANALYSIS OF B
E
~ F ~ ~G ~
indicative of the roller. An example of this condition is shown in Fig. 27.12. s
False brinelling is a condition that resembles true brinelling, but it is generated by a different mechanism. True brinelling is generated by plastic deformation of the steel, whereas false brinelling is generated by a corrosion-wear mechanism. Nonrotating bearings subjected to vibration wear by fretting corrosion between rolling elements and raceways. Co~rosionproducts accumulate, which then proceed to accelerate wear by abrasion. Surface irregularities created by wear serve as initiation sites for spalls during subsequent operation. The EHL film may also be affected locally, effecting marginal lubricating conditions. Figure 27.13 depicts false brinelling.
re
s
Careless handling is the primary cause for scores and digs that mar bearing surfaces. Depressions of this type plastically deform the steel, usually displacing metal that is subsequently cold-worked by the rolling
.12. Spalled area of raceway containing a brinelled region in which fatigue crack propagation is normal to the roller track (ma~ification2X).
E
S
ION
UATION OF ~ P E C I ~ ICO C
.I% False brinelling of a tapered roller bearing cup raceway; the insert shows the corrosion products in the “brinell” mark (courtesy of the Timken Company).
ventually, the cold-worked slivers fracture, producing frage system that may be coined in the roller path; crowns are generated around the indentations, which are subject to the effects discussed under brinelling. ~ o t c h e dregions created by the scores function as stress raisers. sharpness of the notches and their locations with respect to operating forces are important factors affecting the propensity for crack initiation. Stress intensity is greater for sharply notched surface discontinuities.
dhesion damage manifests itself as a ~ u i l d u of~metal on a component, res~ltingfrom metal transfer from the interact in^ component. In turn, this interacting component, if examined during early stages of pr ntain corresponding pitted regions where metal had been 7.14 displays the characteristic features associated with metal transfer.
suspended in the lubrica
been severely dented crown features discus
'7.14. Adhesive wear. (a)Topography of scuffed region displayi~gmetal transfer (ma~ification1.5X). ( b ) Cross-sectional view exhibiting deformation associated with scuffing ( ~ a ~ i f i c a t i o n lox).
. Ball bearing inner raceway severely dented by hard:]?articles, s~rfaceareas are subject to surface distress and s less l u ~ ~ i c a n t s y s tcontain e~s d a t e matter and ~ e n e ~ aa sel t~
E
ION OF ~PECIFICC
.
Particle dent encircled by distressed crowned region having dark gray border. As discussed in Chapter 23, the spa11 initiated adjacent to the distress region according to the friction direction( m a ~ i ~ c a t i 90x1. on
.
Abrasive wear. (a)Surface area polished by fine coal particle conveyed by the lubricant system ( ~ a ~ i ~ c a t1.5X). i o n ( b ) SEM p h o t o ~ a p hshowing coal particle indentations (ma~ification200 X ).
way.
. Static corrosion caused by moisture between rolling element sand racepear red or black, de ending on the state o re sites for spall initiation upon subse~uent
earings that operate at relatively high temperat calized pitting, as depicted in een associated with sulfur and ehlo their respective ions in a lubricant containing moisture can lead to hydrolysis, whereby localized anodesare established that provide sites for ~ i t t i n gcorrosion. In turn, pits provide sites for spall initiation when located in the roller path. enerally, when bearings operate under conditions nt remains unaffected with respect to its composition, the steel remains relatively bright. common problems associa permanent discoloration of ng steel surf~cesareheat r example, frictional heat generated by abrading metal components canoxidize th ces,producingoxidationcolors r a n ~ n gfrom straw color to blue. between 177°C (350 cal and associated elationship, howe position of the lubricant, which in turn p en a lubricant decomposes, lubricatin~ability is the conditions disc~ssed in the preceding paras may be s~bjectedto corrosion effects.
.
Pitting corrosion. Sulfur concentrations detectedby EDS. Insert displays characteristic dissolution features ( ~ a ~ i ~ c a t500X i o n1.
tting ~orrosion. movement between a bear face it contacts prear on a microscopicscale. ring surfaces interact with internal housing and shafts pic interaction produces oxidationwear product rfaces appear smooth to the unaided eye, but reveal microscopic crevices indicative of the direction etting features on the respective bearing surfac an i n a d e ~ u afit. t ~ This condi n can occur when mating surface areas are not intimately supported. ngs are mounted to resist movement by applied loadsvia ~ r i c t i owhi ~, s developed betweencontacting surfaces is isusually accom~lishedby press-fitting or by therexpand or contract rings to fit shafts andhousings, ed wear by fretting results indimensional changes en wear is excessive, rings will rotate and overheat. also been observedto initiate from surface areas sub-
$TI~ATIO
.
SI$ OF I3
Fretting corrosion wearpattern shows directionof motion ( m a ~ i ~ c a t i o n
40x1,
jected to frettin crevices.
. Cracks initiate by fatigue at col
cone raceway with
tin^ due to electric
arcin
Electric arcing also causes metallurgical properties to be altered to effects intense, highly localized temperatures at is rapidly dissipated as it is conducted into the mass. Temperature transitions, as evidenced metallurgically, hibit remelting, au~tenitization,and retempering. Surface an surface regions are markedly affected. Initial surface zones m of brittle untempered martensite followed by high temperature, retemtemperatures dissip~tewith depth, the effects are less en arc events are random, the effects are more easily identified in themicrostructure. A single arc generates a he~ispherically shaped, affected region consisting of the zones discussed above. Succeeding arc events superimpose heat-treatment effects on the previously affected region. An example of a surface and microstructure in a fluted region is shown by Fig. 27.22. Arcingalso alters the previously existing stress field. zones eshibit high residual compressive stresses, whereas adjacent retempered zones counteract these stresses with residual tensile stresses. Under cyclic loading this tensile stress zone is vulnerable to fatigue initiation and propagation.
~ ~ ~ Z sometimes Z i ~ ~ called , ~
~ is a stable, ~ stress-related i ~ crack ~ mech, anism caused by subcritical cyclic loading. Spalls may surface-initiated or subsurface-initiat~d.Theprocesses are called ~ ~ t ~ ~ ~ initiating spalling usually occurs by progressive deterioration contact surfaces whenfilms are insufficiently thick to ade~uately separate the sur ofa1 the previously described mechanisms also exhibit co could potentially culminate in surface" become microscopicallypitted.
ed with stress concentrations
e
2. Electric arc damage. ( a )Topography in fiuted regionof surface consisting of pits (ma~ification250X); ( b ) cross-sectional view showing effect onthe microstructure (magnification 500X).
.
Surface-initiatedspalling showingtypicalarrowheadgrowth (ma~ification15X 1.
pattern
direction of crack propagation is dictated by the effects of the applied stresses. The microcracks propagate trans~anularlyresulting in a relatively deep spa11 as illustrated by Fig. 27.24. Inner rings usually have tensile stresses resulting from interference fitting of the inner rings on the shafts. so, high speed rotation of the bearin results in centrifugally induced ensile hoop stresses. Ifexcess can influence the dir~ction of crack pro~a~ation. to the surface, cracks may propagate radially inpropa~ationcan also be abetted by the lubri ts high hydrostatic pressure. 73'1, also results in crack br oisture also abets fatigue
27.24.
Subsurface-initi~tedspa11 in a ball bearing raceway,
crack propagationby virtually eliminating the endurance limit. Cracking can then occur at any applied stress if the number of stress cycles is satisfied. This condition is called corrosion fatigue.
ee The foregoing discussions involved macroscopicfeatures identified with the various failure mechanisms. Macroscopic features, however, do not always represent the failure-initiating mechanism. It is important that the failure mode be identified at the origin area, preferably the initiation site itself. This is not always possible with conventional optical microscope techniques due to the inherent loss of focusing ability at higher magnifications. A SEM provides greater depth of focus and theability to study irregular surfaces. Identifying modes of initiation at crack origins will often provide data leading to the cause of failure, despite damage incurred in other regions. These data will also provide an insight to the types and relative magnitudes of the applied loads. Four basic modesof fracture will be presented with regard to how they appear in bearing steels. Rolling bearings may be manufactured from various grades of steel, but only those that aretermed “bearing steel^'^ are considered here. These steels consist of high-carbon and low-alloy compositions.
Under abnormal loading conditions in bearings, microvoid coalescence, commonly called dimples,represent the ductile mode of cracking due to
overloading. In this mode cracks propagate transgranularly under loads that transmit tensile force components. This mode can be generated by pure tensile, shear, or torsional loading. The last is not common in bearing failure mechanisms. Variations in the microscopic features identify the type of load experienced. Due to the high hardnesses and corresponding low ductilities to which precision bearings are man~factured,diff'erences in surface texture are ns less distin~ishable at lower m a ~ i ~ c a t i o nAs s . SEM m a ~ i ~ c a t i o are increased, dimples will be resolved as hemispherical voids, They differ from ductile metals because the tear edges do not elongate significantly. See Fig. 27.25, Microvoids initiate in thesteel matrix where carbides, inclusions, and matrix i m p e ~ e c t i o ~ reside. s In bearing steels dimples are found to be relatively uniform in size, suggesting that the indigenous spherical carbides are the predominant initiation sites. Carbides are sometimes obsewed within the hemispherical voids. Dimples indicate the direction of the applied force by their shapes. When dimple formations are equiaxed, the acting force is normal to the fracture. Dimples exhibiting increasingly more oval shapes indicate that the forces exerted increasingly greater shearcomponents. ~ i m p l modes e
FI
27.25. ~icrovoidcoalescence.Hemisphe~cal voids are depicted, which are charn acteristic of bearing steels( m a ~ i ~ c a t i o500x1.
CTO
of failure are sometimes observedin bearing flange fracture when thrust loads are excessive.
Cleavage is a rapid overloading mode of failure resulting in bearing fractures. A fracture may initiate by cleavage, or it may initiate by a different mode and propagate by cleavage. In other words, a crack may initiate and propagate by one or more modes until it reaches a critical size. At this point the crack will expand rapidly; that is, fracture occurs. This unstable crack propagation occurrence is related to the steel's fracture toughness property, which is influenced by composition, microstructure, temperature, and loading rate. If a bearing steel is impacted with sufficient force, cleavage will be observed across the section thickness, including the initiation site. Cleavage is a low energy fracture that propagates transgranularly alongspecific crystallographic planes. It appears as flat planes that change orientation from grain to grain. Fan-shaped features are evident on the facets. These features arise from second-orderplanes, giving the appearance of steps, and are called river patterns. These patterns are typically forked,indicating the direction of crack propagation towardthe converging feature within a grain. Matrix carbides interfere with the normal cleavage progression, and bearing steels contain numerous carbides that are precipitated within the prior austenite grains during the temper treatment. Therefore, bearing steels do not exhibit the normal cleavage features. The crackappears to propagate around carbides and develop smaller cleavage facets within grains. This condition is called quasicleavage, whichis displayed in Fig. 27.26. Hence,cleavage is moredifficult to identify in bearing steels. Quasicleavage is indicative of unstable crack propagation in bearing steels occurring due to sudden overload. er
Intergranular fracture is a low energy mode of cracking that starts at grain boundaries. This condition is an embrittlement mechanism, which reduces more tightly bonded grain boundary energy areas. Embrittlement of bearing steels has been caused by improper heat treatment, whereby a brittle phase is precipitated at the grain boundaries. This has ecipitation o f phosphorous at the prior beenshown to occurby th austenite grain boundaries. nchcracks exhibit this behavior. Hydrogen gas can alsoembrittle the steel and cause cracks to progress intergranularly in the affected region. Hydrogen may be dissolved during the melting process and diffuse to form gas pockets during solidification.
Quasicleavage. ( a )Characteristic smallercleavage facets within crystals (magnification 1500X). ( b ) Fan-shaped features on a cleaved facet (magnification 2500X).
1 processes that provide hydrogen gas en~ironments are po sources for embrittlement by a b s o ~ t i o nof hydrogen at thesurfac of cracking is readily identified by its g. ~ 7 . ~Secondary 7. cracks are often o es similar to those of other steels at lower ification may reveal small sphericalpockets where carbides reside.
e features are the most d i f ~ c ~tol tidentify in ~ e a r i ~ ~ s consist of spherical carbides dispersedin a fine mar-
.
Intergranular failure. ( a ) General view of quench crack (magnification
250X). ( b )Higher magnification showingintergranular features (ma@ification 1OOOX >.
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)~ubsurface-initiatedfatigue 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 low-energy 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 failure-containing 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 all-inclusive; 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. 282,205-236 (1967-68). 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 Rolling-Contact Fatigue,” ASLE Trans. 23,23-39 (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 65-WA/CF-4 (1965). 27.6. R. Osterlund, 0. Vingsbo, L. Vincent,and P. Guiraldeng, “Butterflies in Fati~ed Ball Bea~ngs-For~ationMechanisms and Structures,”Seand. J: ~etaZZ.11 (1982). 27.7. S. Way, “Pitting Due to Rolling Contact,”J: AppZ. Mech. 2, A49-A58 (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 mm-N "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 10-11 2.26
76
3.39 10-l1
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 10-4 3.35 10-3 3.84 10-5 2.15 105 7.86 104 3.84 10-5 1.24 10-5 1.24 10-5 3.84 10-5 3.84 10-5 1.24 10-5 1.24 10-5 1.92 10-5 3.84 10-5 3.84 10-5 1.92 10-5 1.24 10-5 2.24 4.36 10-4 6.98 10-4 1.81 10-4 7.68 10-5 4.36 10-4 6.98 10-4 5.24 10-4 1.81 10-4 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 10-6 jJ.17 4.18 10-7 7.83 10-7 0.0045 0.0045 1.01 10-5 2.78 10-4 4.36 10-7 3.12 107 1.14 107 4.36 10-7 8.71 lo-' 6-71 lo-' 4.36 10-7 4.36 * 10-7 8.71 lo-' 8.71 10"' 2.18 lo-' 4.36 10-7 4.36 10-7 2.18 lo-' 8.71 lo-' 2.09 10-5 8.71 lo-' 2.53 10-5 4.33 * ~m 10-7 8.71 lo-' 2-53 10-5 1.90 10-5 4.33 8.71 0.0090 0.0090 0.0090 &.24 10-5 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 10-7 1.60 10-9 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 10-7 5.25 10-7 2.52 10-7 6.03 23.8 23.8 44.0 220 1.274 * 32900
8624 1.45 2.32 10-3 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 10-l1 1.98 10-17 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 Angular-contactball bearings, 19 angle, 20 automotive wheel application, 41 back-to-back arrangement, 21, 369 ball friction forces due to gyroscopic moment, 176 double-row, 19, 22 diametral play, 55 duplex set, 369 face-to-facearrangement, 21, 369 groove curvature radii, 19 limiting thrust load, 379 self-aligning,22 single-row, 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 Asperity-asperity 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 angular-contact 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 Back-to-backbearing arrangement, 21, 368 Back-up 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 double-row, 15, 18 filling-slot, 15, 18 free angle of misalignment, 58 free contact angle, 55 free endplay, 55 friction in ball-raceway 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 single-row, deep-groove,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 five-ball endurance tester, 782 Pratt & Whitney v-ring/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 bearing-shaft systems, 410 permissible static load, 831 shaft supported by three bearings: non-rigid 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 deep-groove 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 Cam-follower 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 Case-hardening: 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 deep-groove ball bearing, 12 double-row bearings, 262 single-row bearings, 256 Combining fatigue life factors, 903 Composite shear stress on contact surface, 475 Condition-based maintenance, 1005 Cone, 28,77 Conrad assembly, 12, 13 Consumable electrode vacuum melting of steel, 583 Contact: asperity and fluid-supported load, 472 deformation (elastic), 195, 234, 340 roller-raceway skewing effect, 286 dynamic capacity, 699 elastohydrodynamiclubrication friction, 476 ellipse, 193 fatigue life, 699 flange-roller end, 330 heat transfer (in), 574 lubricated, structural elements of, 936 near-surface 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 angular-contact 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: asperity-asperity 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 double-row, 26 endplay, 73, 76 friction torque due to roller end-flange 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 multi-row, 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 Deep-groove ball bearing, 11 spherical outside surface, 14 Deflections: radial ball bearings, 366 roller bearings, 277, 366 self-aligningball bearings, 366 thrust bearings, 367 Deformation: bands in subsurface, 212 brinelling, 1052 contact, 234 rolling, 487 skewing effect in roller-raceway 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 Double-row ball bearings, 15 Dowson, 1, 428, 434, 453 Dry-film lubrication, 418 silicon nitride components, 624 Duplex ball bearings, 370 back-to-back arrangement, 368 face-to-facearrangement, 368 Dynamic loading: crank-reciprocatingload, 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
Micro-EHL, 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 contamination-lifetest 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 Face-to-facebearing 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 Fatigue-initiating 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 double-row 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 fatigue-i~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
material-life 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 stress-life factor, 910 testing, 764 ball-rod rolling contact fatigue tester, 787 design considerations, 777 elements, 779 General Electric Polymet rolling contact endurance tester, 786 INSA rolling-slidingdisc endurance tester, 788 NASA five-ball endurance tester, 782 Pratt & Whitney v-ring/ball endurance tester, 783 sample size selection, 810 SKF A-frame 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 Filling-slot ball bearings, 15 Filtration of lubricant, 649 fatigue life effect, 898 Finite element method of analysis, 221, 302 Fit: classification, 86 line-to-line, 78 standards for practice, 84 tolerance classes, 85 Five-ball endurance test rig, 782
EX
Five degrees of freedom in loading, 357 Flaking, 1061 Flame-hardening 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: ball-disc 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 ball-raceway contacts, 496 forces and moments in roller-raceway 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 ball-raceway contacts, 497 seal, 494 shear stresses in ball-raceway 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 element-raceway 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 contamination-lifetest system, 789 Instrument ball bearings, 15 Interference fit: effect on clearance, 55, 119 surface finish effect, 126 Ioannides-Harris 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 Line-to-line fit, 78 Liquid lubricants, 654 mineral oils, 655 Loading: bearing, 135 classification, 85 combined radial and thrust, 256 double-row 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 non-Newtonian, 476 oil sump, 648 once-through,651 polymeric, 653 regimes, 453 solid, 653 wear, 939 Lundberg-Palm~enfatigue life theory, 208, 219,688,692,694, 794,931 case-hardening steel bearings (application to), 740 limitations of the theory, 863, 904 stress-life relationship, 894 Macrogeometry, 48 Marquenc~ng,607 Martensite, 606, 612 Material effect on fatigue life, 894 Material-life 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 Melting-Refining (M-R) 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 bearing-shaft systems, 410 NASA five-ball endurance test rig, 782 Near-surface region of contact, 938 Needle roller bearing, 26 cam follower application, 44 thrust bearing, 39 Newtonian fluid, 419, 451 Newton-Raphson 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 Once-through 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 Out-of-round raceway, 525 Outside diameter (o.d.1, 49 Overriding ring land, 379 Oxygen in steel, 583 Palmgren-Miner 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 v-ring/ball endurance test rig, 783 Preloading, 368 deflection, 372 isoelasticity, 378 radial, 375 Press-fitting, 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 Roller-raceway: contact laminum: load, 274, 280 deformations due to skewing, 286 heat transfer, 574 Rollers: axial loading, 177 centrifugal force, 355 corner-flange contact, 228 crowning, 26 deformation components, 272 eccentricity of loading, 276, 284 end-ring 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 Rolling-mill 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 deep-groove 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 oil-lubricated bearings (for), 673 shields (with), 17, 675 solid-lubricatedbearings (for), 673 torque, 679 Self-aligning: 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 flange-rollerend contact and, 330 roller axial loading, 179 roller-raceway d e f o ~ a t i o n s286 , SKI? A-frame automotive wheel hub bearing endurance test rig, 779 Skidding motion, 315, 347, 360, 515 ball bearings (in), 518 cylindrical roller bearings (in), 523 out-of-round raceway(to control), 525 Slewing bearings, 5 , 7
Sliding friction: cage, 493 distribution of forces in ball-raceway contacts, 502 gyroscopic motion(due to), 488 rolling motion (in), 488 skewing, effect of, 179 spherical-roller bearings, 35 tapered roller bearings, 28 Sliding motion, 313 deformation (cause), 316 Sliding velocity: ball bearing inner raceway, 321 ball bearing outer raceway, 320 ball-raceway contacts (in), 498 distribution in ball-raceway contacts, 501 gyroscopic motion, 314 roller end-flange, 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 barrel-shaped rollers, 32 clearance, 68 contact angle, 66 curvature diflerence, 71 curvature sum, 71 free endplay, 66 high speed, 357 hourglass-shaped rollers, 32 load classification, 85 osculation, 70 pitch diameter, 66 planet gear bearing, 44 roller skewing, 34, 537 single-row, 35, 37 sliding friction, 35 steel-making (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 Spin-roll 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 two-parameter, 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 case-hardening, 4, 580, 608 fatigue life effects, 741, 894 cleanliness, 593 macroinclusions, 594 oxygen content, 594 cobalt alloys: L-605, 621 Stellite 3, 621 Stellite 6, 621 composition (effect onfatigue life), 739 dimensional instability of components, 856 fatigue failure modes, 618 subsurface"initiated, 619 surface-initiated, 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 time-temperature-transformation (TTT) curve, 601 high temperature, 569 induction heating, 610 inhomogeneities, 619 low carbon for cages, 625 machinability, 596 marq~enching,607
Martensite, 606, 612 material-life 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: Ioannides-Harris theory, 907 Lundberg-Palmgren theory, 694 Stresses: fatigue-initiating, 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 Stress-life 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 flame-hardening, 611 induction heating, 610 residual stress, 616 Surface-initiated fatigue, 620 Surface shear stresses, 215 ball-raceway 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 double-direction, 35 double-row, 29 endplay, 78 endurance test rig, 782 four-row, 35 high speed, 357 misalignment, 272, 1052 pitch diameter, 77 roller: angle, 77 end-flange 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: ball-rod rolling contact fatigue tester, 787 design considerations, 777 General Electric P o l p e t rolling contact endurance tester, 786 INSA rolling-slidingdisc endurance tester, 788 NASA five-ball endurance tester, 782 Pratt & Whitney v-ring/ball endurance tester, 783 SKI? A-frame 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 bearing-shaft systems: non-rigid shaft, 404 rigid shaft, 400 Through-hardening 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 angular-contact ball bearings, 375 Truck wheel application, 43 Tungsten carbide, 621 Two bearing-shaft 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 micro-sensors, 1007 shock pulse method, 1005 frequencies in bearing operations, 993 multi-degree-of-freedomsystem, 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 end-flange, 330 smearing, 944, 1046 tribological processes, 939 Wedeven ball-disc test rig, 790 Weibull distribution, 692, 794 application, 800 maximum likelihoodmethod, 804 slope, 695, 706, 799 sudden death test analysis, 807 two-parameter, 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 X-Ray diffraction, 844 X and Til factors: point contact radial bearings, 720 point contact thrust bearings, 725 radial roller bearings, 734 static loading, 830
