28Rolling Element

Bearings

28.1 Introduction28.2 Rolling Element Bearing Types

Ball Bearings • Roller Bearings • Special Bearings

28.3 Bearing Materials28.4 Contact Mechanics

Theory of Hertzian Contact • Bearing Internal Contact Geometry • Non-Hertzian Contact

28.5 Bearing Internal Load DistributionBearings under General Load Conditions • Bearings under Pure Eccentric Thrust Load

28.6 Bearing LubricationElastohydrodynamic Lubrication • Effect of Spin Motion • Effect of Lubricant Starvation • Lubrication Methods

28.7 Bearing KinematicsOuter Raceway Control • Inner Raceway Control • Minimum Differential Spin

28.8 Bearing Load Ratings and Life PredictionBasic Load Ratings • Bearing Life Prediction

28.9 Bearing Torque CalculationStarting Torque • Running Torque

28.10 Bearing Temperature AnalysisHeat Generation • Heat Transfer

28.11 Bearing Endurance TestingTesting Procedure and Data Analysis

28.12 Bearing Failure AnalysisContact Fatigue • Surface Depression and Fracture • Mechanical Wear • Corrosion • Electric Arc Damage • Discoloring and Overheating

28.1 Introduction

Rolling element bearings are typical tribological components. They utilize rolling contacts between therolling elements and raceways to support load while permitting constrained motion of one body relativeto another. The standard configuration of a rolling element bearing comprises inner and outer rings, aset of rolling elements arranged in a row between the inner and outer rings, and a retainer or cage tomaintain a proper annular spacing between the rolling elements (Figure 28.1). Some bearings also haveseals as integrated components. Due to their wide availability and versatility, rolling element bearingsare, perhaps, the most widely used bearing type. Rolling element bearings are characterized by little or

Xiaolan AiThe Timken Company

Charles A. MoyerThe Timken Company (retired)

no sliding motion. They usually generate less friction and have low starting torque compared to hydro-dynamic bearings. Unlike hydrodynamic bearings, the performance of rolling element bearings is lesssusceptible to changes in load, speed, and temperature. Most rolling element bearings are capable ofcarrying both radial and thrust loads. Because of rolling contact, the dependency on lubricant is not ascritical. This makes rolling element bearings easier to maintain. Well-designed and well-built rollingelement bearings can operate over wide ranges of load and speed. Such features often put rolling elementbearings on the top of the bearing selection list.

28.2 Rolling Element Bearing Types

Various means have been used to categorize rolling element bearings such as by the geometry of therolling elements, by the manner in which a bearing is used or by certain technical features that the bearingpossesses. However, the most common classification is by the rolling element geometry. There are twomajor categories: ball bearings and roller bearings. Within each category, bearings can be further dividedinto sub-categories. Each type of bearing has characteristics or properties that make it particularly suitablefor certain applications. The main factors to be considered when selecting the optimum bearing type are:

• Available space

• Load condition

• Speed

• Temperature

• Misalignment

• Mounting and dismounting procedures

• Dynamic stiffness

• Motion error

• Noise factor

This section outlines the most popular bearing types used in various automotive and industrialapplications. For comprehensive listings on bearing types and usage, the reader is referred to variousbearing catalogs provided by bearing manufacturers.

28.2.1 Ball Bearings

28.2.1.1 Deep-Groove Ball Bearings

A deep-groove ball bearing consists of a set of balls rolling between an inner and outer raceway(Figure 28.1). Both inner and outer raceways have high shoulders on each side. A deep-groove bearingcan carry significant radial load and, due to the high degree of conformity between the balls and raceways,moderate thrust loads. When the elastic deflection of bearing races is used to introduce the balls intobearing raceways, bearings can have uninterrupted raceway grooves and are capable of carrying substantial

FIGURE 28.1 A single-row, deep-groove, radial ball bearing. (Courtesy of SKF Bearing Industries Co.)

load in either direction, even at high speeds. This type of bearing is often referred to as Conrad type. Toincorporate the greatest possible number and size of balls, a deep-groove ball bearing may rely on a fillingslot to introduce balls into the bearing. This type of bearing is referred to as Max type or “filling slottype.” Max-type bearings are capable of carrying higher radial loads.

28.2.1.2 Angular-Contact Ball Bearings

Angular-contact ball bearings can be regarded as a variation of deep-groove ball bearings. Unlike deep-groove bearings, an angular-contact bearing has at least one race ring that has only one side shoulder.Angular-contact ball bearings are capable of carrying an appreciable thrust load in one direction withor without a radial load. Because an angular-contact bearing must have a thrust load acting on it, noendplay (lateral movement) exists within the bearing. The line that connects the nominal contact betweenthe inner raceway and a ball and the contact between the outer raceway and the ball lies at an angle withthe radial plane of the bearing. This angle is called the contact angle. It ranges from 15 to 40° forcommercially available angular-contact ball bearings.

Angular-contact ball bearings are commonly used in pairs to accommodate heavier radial loads, two-directional thrust loads, or various combinations of radial and thrust loads. Tandem mounting is usedfor heavy unidirectional thrust loads and duplex mounting of either face-to-face or back-to-back is usedfor two-directional thrust loads.

28.2.1.3 Double-Row Ball Bearings

Double-row ball bearings are similar in design to single-row ball bearings. They are designed with one-piece inner and outer rings. Most double-row ball bearings are made with opposite contact angles inrows like two face-to-face or back-to-back angular-contact ball bearings in duplex mounting. They arealso made as two deep-groove rows. Like two single-row ball bearings in paired mounting, a double-rowball bearing is often used to carry heavy radial loads. It is also capable of carrying thrust loads in eitherdirection. There are advantages to using a double-row ball bearing over two single-row ball bearings inpaired mounting. Double-row ball bearings take less axial space and the internal relationship betweentwo rows is predetermined and is not, or less, affected by mounting practice.

28.2.1.4 Thrust Ball Bearings

Thrust ball bearings generally have a 90° contact angle. However, ball bearings with contact angle greaterthan 45° are also classified as thrust ball bearings. A thrust ball bearing whose contact angle is 90° canonly be used to support thrust loads.

28.2.2 Roller Bearings

28.2.2.1 Cylindrical Roller Bearings

In cylindrical roller bearings, the rollers are axially guided between integral flanges on at least one of thebearing rings. The most popular type is the single-row design, which offers various flange arrangements.Figure 28.2 shows a typical single-row cylindrical bearing with two flanges on the outer ring. Cylindrical

FIGURE 28.2 A single-row, radial cylindrical roller bearing. (Courtesy of SKF Bearing Industries Co.)

roller bearings have exceptionally high radial load-carrying capacity. They are also capable of carrying asmall amount of thrust load when two flanges are positioned on opposing rings at opposite sides. Intheory, true rolling motion exists in cylindrical bearings under radial load. Therefore, cylindrical bearingshave lower starting torque and lower operating temperature than other roller bearings. This makes themsuitable for high-speed applications.

To achieve greater load capacity without increasing the tendency of roller skewing on the raceways,cylindrical bearings are frequently constructed in two or more rows rather than with one row of longerrollers.

28.2.2.2 Tapered Roller Bearings

Tapered roller bearings have tapered inner and outer raceways, with tapered rollers guided between themby a very accurately positioned flange known as the rib (Figure 28.3a). The inner and outer racewayshave different contact angles. The extensions of the inner and outer raceways and the rollers are designedto converge at a common apex point on the axis of rotation (Figure 28.3b). The on-apex design resultsin true rolling motion of the rollers on the raceways along the line of contact. At the contacts betweenthe rib and roller-ends, however, sliding and spinning exist.

The tapered raceways allow a tapered roller bearing to carry combined radial and thrust loads or thrustloads only. The ratio of thrust to radial load capacity is determined by the contact angle between the

a

FIGURE 28.3 Tapered roller bearings. (a) A single-row, tapered roller bearing. (Courtesy of the Timken Company).(b) “On-apex” design results in true rolling motion.

b

outer raceway and the axis of rotation. The long roller-raceway contact gives the tapered roller bearinga high load-carrying capacity.

Tapered roller bearings are used in pairs. One bearing is adjusted against the other to achieve thedesired endplay or pre-load. To acquire greater radial load-carrying capacity and eliminate problems ofaxial adjustment or thermal growth, tapered roller bearings are designed in two rows. Most double-row,tapered roller bearings have a single inner race and double outer races; or a single outer race and doubleinner races. Tapered roller bearings are also available in four-row arrangements.

28.2.2.3 Spherical Roller Bearings

Most spherical roller bearings have an outer raceway that is a portion of a sphere. The bearings areinternally self-aligning and permit angular displacement of the shaft relative to the housing. The mostpopular design has two rows of rollers that lie at an angle relative to the axis of the bearing. The rollersare in either symmetrical or asymmetrical barrel shape. The curvature of rollers in the direction transverseto the rotation conforms closely to the inner and outer raceways. The high degree of conformity betweenthe rollers and raceways makes spherical roller bearings suitable for heavy-duty applications. Because ofthe non-zero contact angle, spherical bearings are also capable of carrying a certain thrust load in eitherdirection along with the radial load.

Unlike cylindrical roller bearings or tapered roller bearings, true rolling motion at contact between therollers and raceways cannot be achieved with spherical bearings. Therefore, spherical roller bearings haveinherently higher frictional torque than cylindrical bearings and are not suitable for high-speed applications.

28.2.2.4 Roller Thrust Bearings

Cylindrical roller thrust bearings are the simplest of this type. A typical cylindrical roller thrust bearingis comprised of a pair of parallel thrust plates (washers): a row of cylindrical rollers sits between thethrust plates and a cage retaining the rollers. Cylindrical roller thrust bearings inherently experience alarge amount of sliding as a result of the spin motion between the rollers and raceways. For this reason,cylindrical roller thrust bearings are limited to slow-speed applications. To reduce the magnitude of rollerspinning, several rollers can be used in each cage pocket rather than a single long roller.

The spin motion of rollers on raceways can be eliminated by using tapered rollers with an “on-apex”design, as illustrated in Figure 28.3b. Bearings of such design are referred to as tapered roller thrustbearings. Because at least one bearing raceway is tapered, an outboard flange is required to confine rollersfrom being expelled in the radial direction. Sliding contact exists between the outboard flange and roller-ends. Friction forces generated at the sliding contact limit the bearings to relatively slow-speed applications.

28.2.3 Special Bearings

28.2.3.1 Clutch Bearings

A clutch bearing combines the functionality of a bearing and a clutch together. It transmits torque betweenthe inner and outer rings in one direction and allows free overrun in the opposite direction. Whentransmitting torque, either the inner ring or the outer ring can be used as the input member. Thetransition from the overrun to locked operation normally occurs with small backlash. Clutch bearingsare generally used in indexing, backstopping, or overrunning.

28.2.3.2 Smart Bearings

The functionality of a bearing can be extended by integrating electronic systems. The speed-sensingbearing used in automotive wheel application is a typical example of smart bearings. A typical speed-sensing bearing is a self-contained, double-row tapered roller bearing featuring an integral sensing systemdesigned to provide speed information for anti-lock brake systems (ABS) and traction control systems(TCS). The sensing system contains a target wheel within the bearing and a microchip speed-sensingelement that detects the rotational speed of the target wheel. The sensor’s speed information can also beused for the vehicle dynamics control system, navigation system, speed control, speedometer, odometer,trip computer, and other purposes.

Future smart bearings will feature sensing elements capable of detecting speed, temperature, load,vibration, and other operating parameters that can be used not only for system control, but also forbearing and bearing system health monitoring.

28.3 Bearing Materials

A bearing’s endurance life is largely determined by the strength of bearing material in relation to thecontact stresses the bearing experiences during operation. While a well-designed internal geometry andgood surface finish effectively reduce the actual contact stress and thus prolong the bearing’s service life,high-quality bearing material is essential. Bearing materials are selected on the basis of strength, fatigueresistance, wear resistance, elevated temperature resistance, corrosion resistance, toughness, hardenabilityand dimensional stability.

A great majority of rolling element bearings are fabricated from vacuum-refined, high-quality low-alloy or carbon steels. Bearing steels can be categorized as high-carbon steels, which contain 0.8% ormore carbon by weight; and low-carbon steels, which contain less than approximately 0.2% carbon.High-carbon steels, with chromium-alloy additions, are usually through-hardened to ensure a surfaceRockwell hardness of 58 to 64 HRc. For large bearings, particularly those with thick cross-sections,increased amounts of manganese, chromium, silicon, molybdenum, and/or nickel are introduced toenhance the hardenability. Through-hardened steels are used extensively in ball bearings while case-hardened steels are predominately used in roller bearings. Table 28.1 lists common grade and respectivechemical compositions for high-carbon steels.

Low-carbon steels are alloyed with nickel, chromium, molybdenum, and manganese to increase hard-enability. Carbon is diffused into the steel under controlled temperature and atmospheric compositionto increase carbon content at surface and subsurface layers to approximately 0.56 to 1.10%. The carburizedsteel is heat-treated to provide a hardened surface case for high contact load-carrying capability and atough, ductile core for heavy shock load absorption. The softer core also helps to retard surface crackpropagation. Low-carbon steels are historically used in tapered roller bearings, although they are well-suited for other types of bearings. Table 28.2 lists the most commonly used carburizing steels.

The hardness of bearing steels drops sharply when the tempering temperature is exceeded. Becausemost bearing steels are tempered between 160 and 280°C, bearings should not be used at temperatures

TABLE 28.1 Chemical Composition of High-Carbon Bearing Steels

Grade

Chemical Composition(%)

ApplicationC Si Mn Cr Mo P S

ASTM-A295(50100) 0.98–1.10 0.15–0.35 0.25–0.45 0.40–0.60 ≤0.10 ≤0.025 ≤0.025 Not commonly used ASTM-A295 (51100) 0.98–1.10 0.15–0.35 0.25–0.45 0.90–1.15 ≤0.10 ≤0.025 ≤0.025

JIS-G4805 (SUJ1) 0.95–1.10 0.15–0.35 ≤0.50 0.90–1.20 — ≤0.025 ≤0.025

ASTM-A295-94 (52100) 0.98–1.10 0.15–0.35 0.25–0.45 1.30–1.60 ≤0.10 ≤0.025 ≤0.025 Typical steels for small and medium-size bearings

ASTM-A535-85 (52100) 0.95–1.10 0.15–0.35 0.25–0.45 1.30–1.60 ≤0.10 ≤0.015 ≤0.015BS-535A99 0.95–1.10 0.10–0.35 0.40–0.70 1.20–1.60 — ≤0.025 ≤0.025DIN-100Cr6 0.90–1.05 0.15–0.35 0.25–0.40 1.40–1.65 0.30 ≤0.025 ≤0.025JIS-G4805 (SUJ2) 0.95–1.10 0.15–0.35 ≤0.50 1.30–1.60 — ≤0.025 ≤0.025NF-100C6 0.95–1.10 0.15–0.35 0.20–0.40 1.35–1.60 ≤0.08 ≤0.030 ≤0.025

ASTM-A485-89 grade 1 0.90–1.05 0.45–0.75 0.95–1.25 0.90–1.20 ≤0.10 ≤0.025 ≤0.025 Used for large-size bearingsASTM-A485-89 grade 2 0.85–1.00 0.50–0.80 1.40–1.70 1.40–1.80 ≤0.10 ≤0.025 ≤0.025

JIS-G4805 (SUJ3) 0.95–1.10 0.40–0.70 0.90–1.15 0.90–1.20 — ≤0.025 ≤0.025

ASTM-A485-89 grade 3 0.95–1.10 0.15–0.35 0.65–0.90 1.10–1.50 0.20–0.30 ≤0.025 ≤0.025 Used for ultra-large-size bearings

ASTM-A485-89 grade 4 0.95–1.10 0.15–0.35 1.05–1.35 1.10–1.50 0.45–0.60 ≤0.025 ≤0.025JIS-G4805 (SUJ5) 0.95–1.10 0.40–0.70 0.90–1.15 0.90–1.20 0.10–0.25 ≤0.025 ≤0.025

above 120°C, if tempered at 160°C; or above 200°C if tempered at 280°C. When bearings are used atelevated temperatures, steels with high-temperature hardness are required. Table 28.3 lists some of thewidely used high-temperature bearing steels along with their chemical compositions.

Corrosion-resistant alloys should be used for bearings in applications where they are exposed tocorrosive environments. Steels with high chromium content possess good corrosion resistance. Of thealloys listed in Table 28.3, 440C, BG-42, and ASM 5749 are good candidates for corrosion-resistantbearings. Non-ferrous bearing materials such as ceramic also provide excellent corrosion resistance andcan be considered.

TABLE 28.2 Chemical Composition of Carburizing Bearing Steels

Grade

Chemical Composition(%)

ApplicationC Si Mn Ni Cr Mo P S

ASTM-A534-90 (5120H)

0.17–0.23 0.15–0.35 0.60–1.00 — 0.60–1.00 — ≤0.035 ≤0.040 Used for small-size bearingsASTM-A534-90

(4118H)0.17–0.23 0.15–0.35 0.60–1.00 — 0.30–0.70 0.08–0.15 ≤0.035 ≤0.040

ASTM-A534-90 (8620H)

0.17–0.23 0.15–0.35 0.60–0.95 0.35–0.75 0.35–0.65 0.15–0.25 ≤0.035 ≤0.040

DIN 20NiCrMo2 0.23 0.35 0.90 0.70 0.60 0.25 ≤0.025 ≤0.025JIS-G4052/4103

(SCr420H)0.17–0.23 0.15–0.35 0.55–0.90 — 0.85–1.25 — ≤0.030 ≤0.030

JIS-G4052/4103 (SCM420H)

0.17–0.23 0.15–0.35 0.55–0.90 — 0.85–1.25 0.15–0.35 ≤0.030 ≤0.030

JIS-G4052/4103 (SNCM220H)

0.17–0.23 0.15–0.35 0.60–0.95 0.35–0.75 0.35–0.65 0.15–0.30 ≤0.030 ≤0.030

ASTM-A534-90 (4320H)

0.17–0.23 0.15–0.35 0.40–0.70 1.55–2.00 0.35–0.65 0.20–0.30 ≤0.035 ≤0.040 Used for medium-size bearingsJIS-G4052/4103

(SNCM420)0.17–0.23 0.15–0.35 0.40–0.70 1.55–2.00 0.35–0.65 0.15–0.30 ≤0.030 ≤0.030

ASTM-A534-90 (9310H)

0.07–0.13 0.15–0.35 0.40–0.70 2.95–3.55 1.00–1.45 0.08–0.15 ≤0.035 ≤0.040 Used for large-size bearingsJIS-G4052/4103

(SNCM815)0.12–0.18 0.15–0.35 0.30–0.60 4.00–4.50 0.70–1.00 0.15–0.30 ≤0.030 ≤0.030

TABLE 28.3 Chemical Composition of Special Bearing Steels

Grade

Chemical Composition(%)

Temperature Limits(°C) ApplicationsC Si Mn Ni Cr Mo W V Co

M-50 0.82 ≤0.25 0.25 ≤0.10 4.12 4.25 ≤0.25 1.00 ≤0.25 420 Used for high-temperature bearingsM-50-NiL 0.15 0.18 0.15 3.50 4.00 4.00 — 1.00 ≤0.25 315

CBS-1000 0.15 0.50 0.50 3.00 1.05 4.50 — 0.38 314–425CBS-600 0.19 1.10 0.55 — 1.45 1.00 — — — 230–315TBS-600 1.03 1.03 0.70 — 1.45 0.30 — — — 315VASCO X-2 0.22 0.90 0.30 — 5.00 1.40 1.35 0.45 — 380M-10 0.87 0.25 0.25 — 4.00 8.00 0.75 1.90 — 430M-1 0.83 0.30 0.30 — 3.75 8.50 1.75 1.15 — 480M-2 0.83 0.30 0.30 — 4.15 5.00 6.15 1.85 — 380SKH4 0.78 ≤0.40 ≤0.40 — 4.15 — 18.00 1.25 10.00 450

440-C 1.10 ≤1.00 ≤1.00 ≤0.60 17.00 ≤0.75 — — — 200 Used for high-temperature and corrosion-resistant bearings

BG-42 1.15 0.30 0.50 — 14.50 4.00 — 1.20 — 480ASM 5749 1.15 0.21 0.12 0.10 4.13 4.80 1.40 1.08 7.81 480

Ceramics are superior to ferrous alloys in corrosion, heat, and wear resistance, but are limited inapplication because they have lower toughness and fracture resistance and are not able to sustain shockloads. Furthermore, ceramic bearing materials are more expensive than ferrous alloys and very expensiveto machine. New engineering ceramics, to a certain degree, have improved some of the material propertylimitations and have increased use in bearing applications. Today, a fair number of bearings are madetotally, or partially, from a variety of ceramics, including alumina, silicon carbide, titanium carbide, andsilicon nitride. With the advances in material science and manufacturing technology, the demands forceramic bearings will continue to grow. Table 28.4 provides the properties of engineering ceramics forbearing applications.

28.4 Contact Mechanics

Rolling element bearings are typical mechanical components that operate under concentrated-contactconditions. Loads carried by rolling element bearings are transmitted through the discrete contactsbetween the rolling elements and the two raceways. Even under moderate bearing load, the stresses atthe contact are quite high, being on the order of 1 to 4 GPa. Well-designed bearings are, however, capableof carrying an appreciable amount of load. This is attributed to the fact that contact stress increasesslowly with the applied load (to one third power for point contact and one half power for line contact)and that the material is in general compression. In rolling element bearings, point contact refers to theconjunction of two surfaces such that under no load, the initial contact is a single point. As load isapplied, the contact develops into a finite area of a generally elliptical shape. Point contact exists betweenthe rolling elements and raceways of ball bearings, and of roller bearings with high-crown on rollers andraceways. It also exists in roller bearings between the roller-ends and flanges. Line contact is the con-junction of two surfaces such that the initial contact under no load is a straight line. When loaded, theline spreads to form an elongated rectangle. A concept of “modified line contact” is frequently used(Lundberg et al., 1947). The modified line contact is the major form of contact between the rollingelements and raceways for roller bearings. This chapter section outlines the classical Hertzian theory ofcontact that allows quick calculations of contact stress and deformation to be made from the appliedload, material properties, and the internal geometry. Issues of non-Hertzian contact are also discussed.

28.4.1 Theory of Hertzian Contact

28.4.1.1 Point Contact

When two elastic solids contact under load W, a contact area develops. For a point contact, the area, ingeneral, assumes an elliptical shape and has a semi-major, a, in one direction and a semi-minor, b, inthe perpendicular direction. For purposes of discussion, assume a lies in the x-direction; b lies in the

TABLE 28.4 Properties of Engineering Ceramics and Bearing Steels

MaterialDensity(g/cm3)

Hardness(HV)

Young’s Modulus

(GPa)

Flexural Strength(MPa)

Fracture Toughness(MPa-m1/2)

Linear Thermal

Expansion Coefficient×10–6/°C

Thermal Shock

Resistance(°C)

Thermal Conductivity

(W/m-K)

Electric Resistance

(Ω-cm)

Silicon nitride (Si3N4)

3.1–3.3 1500–2000 250–330 700–1000 5.2–7.0 2.5–3.3 800–1000 12–50 1013–1014

Silicon carbonate (SiC)

3.1–3.2 1800–2500 310–450 500–900 3.0–5.0 3.8–5.0 400–700 46–75 100–200

Alumina (AL2O3)

3.6–3.9 1900–2700 300–390 300–500 3.8–4.5 6.8–8.1 190–210 17–33 1014–1016

Partly stabilized zirconia (ZrO2)

5.8–6.1 1300–1500 150–210 900–1200 8.5–10.0 9.2–10.5 230–350 2–3 1010–1012

Bearing steel 7.8 700 208 — 14–18 12.5 — 50 10–5

y-direction; and Rx and Ry are principal relative radii (composite radii) of curvature at origin of thecontact. The ratio of the semi-minor axis b to the semi-major axis a is defined as ellipticity k = b/a (0 ≤k ≤ 1). The ellipticity is related to the ratio of surface curvature radii, kr = Ry/Rx, through a transcendentalequation, which can be approximated as follows (Johnson, 1985):

(28.1)

Denote Re = (RxRy)1/2 as the geometrical average radius of surface curvatures, the geometrical averagecontact size, denoted as c = (ab)1/2, can be obtained as:

(28.2)

where E ′ is referred to as the effective Young’s modulus and is determined from the Young’s moduli E1

and E2, and Poisson ratios ν1 and ν2 of the contacting bodies,

(28.3)

F(e) in Equation 28.2 is an auxiliary function, varying only with the eccentricity of the contact ellipse,e = (1 – k2)1/2. For e > 0 (kr < 1), F(e) can be approximated in terms of kr by the following equation:

(28.4)

For the circular contact where e = 0, F(e) is equal to unity, F(e) = F(0) = 1.Having defined the shape k and size c of the contact ellipse, the semi-major and semi-minor axes, a

and b, maximum Hertzian pressure pk, and maximum deformation at center of the contact δ can bedetermined.

(28.5)

(28.6)

(28.7)

(28.8)

where K(e) is the complete elliptical integral of the first kind. The value can be determined by the followingapproximation (Hamrock et al., 1983):

(28.9)

k kr≈ 2 3

cWR

EF e

e=( )

′

( )3

2

1 3

′ = − + −

−

EE E

21 11

2

1

22

2

1

ν ν

F e kk k

kkr

r r

r

r( ) ≈ − −−

< <0 726

1

10 11 6

4 3

1 3

.ln

a k c W= ∝−1 2 1 3

b k c W= ∝1 2 1 3

pW

cWh =

π∝3

2 2

1 3

δ =( )

( ) ′

∝

k e

F e

W

E RW

e

1 2 2

2

1 3

2 39

2

K

K e k kr r( ) ≈ π − π −

< ≤

2 21 0 1ln

The stresses in x- and y-directions at the center of the contact are:

(28.10)

(28.11)

The maximum shear stress occurs at a certain depth along the z-axis below the surface. The maximumshear stress has been used as the critical stress for contact fatigue analysis. Table 28.5 gives the maximumshear stress along with its depth as functions of ellipticity. Detailed information regarding stress calcu-lations at arbitrary points below the contact surface is given in Jones (1964) and Sackfield (1983a,b).

28.4.1.2 Line Contact

In line contact, one of the principal radii, Rx, approaches infinity (Rx → ∞). Accordingly, the ratio ofsurface curvature radii kr reduces to zero (kr = Ry /Rx = 0), and Re is redefined as Re = Ry. The contactarea becomes a long strip of width 2b. Assume Wl is the contact load per unit length. The half-contactwidth b and Hertzian pressure ph can be expressed as:

(28.12)

(28.13)

Assume the deformation is zero at a certain preset depth (z = Re) for both contacting surfaces. Themaximum composite deformation δ at the center of contact can be approximated by:

(28.14)

Alternative empirical equations for calculating surface deformation have been developed for modifiedline contact (Lundberg, 1939; Kunert, 1961). Assume two cylinders with parallel axes are pressed togetherunder load W. The effective contact length is leff . The total deformation of the contact surfaces can beapproximated by a power-law function

(28.15)

TABLE 28.5 Maximum Shear Stress and Its Depth

k 0 0.2 0.4 0.6 0.8 1.0

z/b 0.785 0.745 0.665 0.590 0.530 0.480τmax/ph 0.300 0.322 0.325 0.323 0.317 0.310

σ ν νx hpk

k= − + −( ) +

2 1 21

σ ν νy hpk

= − + −( ) +

2 1 21

1

bW R

El e=

π ′

8

1 2

pW

bWh

l=π

∝2 1 2

δν ν ν ν

=π ′

π ′

−

+( )−

+( )

W

E

E

WR

E El

l

e

2

2

1 11 1

1

2 2

2

ln

δ = ⋅′

CW

l

t

efft

It is suggested that for steel, C = 4.05 × 10-5, t = 0.925, and t ′ = 0.85 when W is in Newtons and leff

and δ are in millimeters. For simplicity concerns, t = 9/10 has been widely adopted.The stresses at a general point (y, z) within the contact solids are given by McEwen (1949)

(28.16)

(28.17)

(28.18)

where

(28.19)

(28.20)

Various critical stresses (orthogonal shear stress, maximum shear stress, von Mises stress, etc.) for contactfatigue life calculations can be derived from the stress components.

28.4.2 Bearing Internal Contact Geometry

The previous section provided useful equations for contact analysis. To use these equations, bearinginternal geometry must be defined. As can be seen, the shape and dimension of the contact ellipse orstrip, the maximum Hertzian pressure, and surface deformation all relate to two important parameters:(1) the principal effective curvature ratio, kr = Ry /Rx, of the contact surfaces and (2) the geometricalaverage radius, Re = (RxRy)1/2 (for line contact, kr = 0 and Re is defined as Re = Ry). As will be seen, thesetwo parameters also play important roles in lubricant film thickness and bearing torque calculations.This section defines the two geometry parameters for commonly used bearing types.

28.4.2.1 Ball Bearings

The internal geometry of a ball bearing is determined by the raceway-groove radii ri and ro in the bearing’saxial plane, ball diameter db, bearing pitch diameter Dp, and the inner- and outer-raceway contact anglesβi and βo. The radii of inner- and outer-raceway grooves and the pitch diameter are expressed in termsof ball diameter db as ri = gi(db/2), ro = go(db/2) and Dp = Gdb, respectively (Figure 28.4).

For the inner-raceway contact, it can be shown that:

(28.21)

σ yhp

bm

z n

m nz= − + +

+

−

1 22 2

2 2

σzhp

bm

z n

m n= − − +

+

1

2 2

2 2

τ yzhp

bn

m z

m n= −

+

2 2

2 2

m sign z b y z y z b y z= ( ) − +( ) +

+ − +( )

1

242 2 2

22 2

1 2

2 2 2

1 2

n sign y b y z y z b y z= ( ) − +( ) +

− − +( )

1

242 2 2

22 2

1 2

2 2 2

1 2

kG gr

i

i

= −

−

1 1

1cos β

(28.22)

Similar equations are obtained for the outer-raceway contact:

(28.23)

(28.24)

where

(28.25)

and di and Do are the inner- and outer-raceway diameters measured at the central radial plane. Thetheoretical radial clearance δ0 is negligibly small, and this is not included in the calculation.

In the absence of centrifugal forces, βi and βo are identical. For bearings under pure radial loads, βi =βo = 0.

FIGURE 28.4 Internal geometry of a ball bearing.

RG g

de

i

i

b= −

−

−

1 11

2

1 2 1 2

cosβ

kG gr

o

o

= +

−

1 1

1cos β

RG g

de

o

o

b= +

−

−

1 11

2

1 2 1 2

cos β

GD

d

d

do

b

i

b

= −

= +

1 1

28.4.2.2 Roller Bearings

A roller bearing’s internal geometry is defined by the roller central or mean diameter dr, effective pitchdiameter Dp*, inner-raceway contact angle βi, and outer-raceway contact angle βο (Figure 28.5). Theeffective pitch diameter Dp* is related to the inner-raceway mean diameter dm and outer-raceway meandiameter Dm through the following equation:

(28.26)

where Dp = (Dm + dm)/2 is the bearing’s pitch diameter.For spherical roller bearings, Dm is defined by the diameter of the outer-raceway sphere Do and the

contact angle of the outer raceway βo; for example, Dm = Docosβo.For most roller bearings, including spherical roller bearings, line contact exists between the raceways

and rollers. Therefore, kr = 0. At the inner-ring raceway and roller contact, the geometrical average radius is:

(28.27)

At the outer-ring raceway and roller contact, the geometric average radius is expressed as:

(28.28)

where G* is the ratio of effective pitch diameter to roller mean diameter, G* = Dp*/dr .

FIGURE 28.5 Internal geometry of a tapered roller bearing.

D D D dp p m mi o o i* tan tan= + −( ) +

−

1

2 2 2

β β β β

Rd

Ge

m

o i

=−( )2

1

2* cos β β

RD

Ge

m

o i

=−( )2

1

2* cos β β

For spherical roller bearings with symmetric rollers, βi = βo. For cylindrical roller bearings, βi = βo = 0.For high-crowned rollers and raceways, rollers may not fully engage with raceways along the roller

length when the applied load is low relative to the load under which the crowns were designed. In suchcases, elliptical contact exists. Assume the crown curvature radius of the rollers is Rw, and the inner andouter raceways have crown curvature of radii of ri = giRw and ro = goRw, respectively. The followingrelationships exist when βo – βi is small.

At the inner-ring raceway,

(28.29)

(28.30)

At the outer-ring raceway,

(28.31)

(28.32)

Equations 28.29 through 28.32 are applicable to spherical bearings in elliptical contact. In this case,bearing raceways are considered as being concavely crowned. Accordingly, negative values are used forgi and go.

28.4.2.3 Roller-End and Flange Contact

For tapered roller bearings, rollers are retained by flanges, also known as ribs, at both ends of the innerring. The rib located at the large end of the inner ring sees a setting force from each roller. The large endof a roller is a portion of a sphere with the radius Rs being a fraction of the apex length La, Rs = ζLa (0 <ζ < 1). Elliptical contact exists between the rib and each roller-end. The rib and roller-end geometry aredesigned such that the center of the contact is located at a height H, roughly half the rib height, and themajor axis of contact ellipse is oriented along the sliding and rolling direction of rollers. Assume the ribangle is θ. As seen from Figure 28.6, the following relationships exist:

(28.33)

(28.34)

(28.35)

(28.36)

kG g

d

Rri

i

r

w

= −

+

∗1 1

1

2

cos β

RG g

R de

i

i

w r= −

+

∗

−

1 11

2

1 2 1 2 1 2

cos β

kG g

d

Rro

o

r

w

= +

+

∗1 1

1

2

cos β

RG g

R de

o

o

w r= +

+

∗

−

1 11

2

1 2 1 2 1 2

cos β

H hRs= cosθ

hgr=

− −( )−( )

sin

cos

γ θ

γ θ

gD

Rrr

s

=2

kh g

r

i

i i

= −+( )

+( ) +1

sin

cos

β θ

β θ

(28.37)

(28.38)

γ = (βo – βi)/2 is the roller half-included angle. Ri is the base-radius of the inner raceway at the large end,and Dr is the base-diameter of rollers at the large end. The above equations also apply to cylindricalbearings where γ = βi = βo = 0.

28.4.3 Non-Hertzian Contact

Hertzian contact exists only in the idealized situation. For practical applications, Hertzian assumptionsmay not always be satisfied. While this, in general, does not undermine the usefulness of the Hertziantheory, it gives rise to the necessity of non-Hertzian contact. For finite line contact as seen in roller bearings,plane strain assumption is not applicable at the end portions of the contact where edge loading occurs.Moreover, contact surfaces are far from perfectly smooth. Surface imperfections or damage, whetherresulting from manufacturing processes or from mishandling during transportation and assembling orfrom operation, are practically inevitable. A surface imperfection serves as a stress-raiser. It can causeperceptible stress deviation from the Hertzian theory. Under thin-film lubrication conditions, surfaceroughness also plays an important role in altering contact stresses. Life reduction due to surface roughnessbecomes an important issue. In this section, issues regarding non-Hertzian contact are discussed.

28.4.3.1 Edge Stress and Roller Crowning

When a finite roller contacts a raceway of greater length, the material in the raceway is in greater tensionalong the contact strip at roller ends than at the central portion of the roller. Correspondingly, the rollerreceives a high compressive stress at its ends. This condition of edge-loading is shown in Figure 28.7where a spherical roller is in contact with the raceway under heavy load conditions. To prevent edge-loading, rollers and raceways are usually partially or fully crowned to relieve the edge stresses. Crowningthe contact members of a bearing also gives the bearing protection against edge stress resulting from

FIGURE 28.6 Roller-end and rib contact geometry.

R k Re r s= −1 2

gR

Rii

s

=

misalignment (Hartnett, 1984). The amount of crowning is determined by the maximum load the bearingwas designed for and by the degree of misalignment anticipated in the application. Substantial studieshave been conducted over the years to provide methods of designing profiles based on a uniform contactpressure or contact stresses along the length of contact. For more detailed discussion on roller profiling,the reader is referred to Lundberg (1939), Poon et al. (1978), Rahnejat et al. (1979), Hartnett (1984), andReusner (1987).

28.4.3.2 Effect of Surface Scratch and Indentation

Surface scratches or indentations are common surface defects. They result from mishandling during thetransportation and assembly processes. Lubricant contamination is another major source of surfacedamage. The presence of a surface scratch or indentation can result in a noticeable stress concentrationat the edges of the defect. It brings the maximum stress closer to the surface, causing surface-originatedspall initiation and subsequent propagation. Figure 28.8 shows the contact pressure and interior vonMises stress distributions from the reference (Ai et al., 1999) for a scratched roller in contact with itsraceway. The scratch has slightly raised shoulders on both edges. Pressure spikes and high stress concen-trations at the edges of the scratch are clearly demonstrated. Studies have indicated that the magnitudesof the pressure spike and stress concentration are largely dependent on the ratio of depth to half-widthof the scratch or indentation. The ratio represents the average slope of the indentation profile. A deepslope gives rise to high contact pressure and stresses.

FIGURE 28.7 Contact pressure in a spherical roller and raceway contact under heavy load conditions.

FIGURE 28.8 Contact pressure and interior von Mises stress for a roller in contact with a scratched surface. (FromAi, X. and Sawamiphakdi, K. (1999), Solving elastic contact between rough surfaces as an unconstrained strain energyminimization by using CGM and FFT techniques, ASME J. Tribol., 121, 639-647. With permission.)

1.00

0.50

0.00

Pre

ssur

e, P

= p

/Ph

Y coordinate (mm)

X co

ordi

nate

(mm

)

0.00

-10.

00-5.0

00.005.

0010.0

0

28.4.3.3 Effect of Roughness

With increasing demand on power density, bearings are often required to operate under thin-lubricant-film conditions. When the lubricant film is penetrated by the surface roughness, surface asperity causesperceptible stress concentration on the surface or in the near-surface layer. Consequently, surface distress-related contact failure, mostly in the form of surface pitting and spalling, becomes an important issue.The contact problem under this condition is known as rough contact. Unlike smooth Hertzian contact,the contact area is now a myriad of micro-contacts of highly irregular shapes within the macro-contact.The contact pressure at these micro-contacts can be high enough to cause localized property changesboth in contact bodies and in lubricant.

Because the contact surfaces in rolling element bearings are, in general, smoother than other machinecomponents such as gears, the mutual dependency between asperities is dramatic. Micro-contact areasare often connected and the ratio of actual to nominal contact areas is close to unity. Figure 28.9 depictsa typical surface roughness profile produced by grinding processes. The center-line-average roughness is0.11 µm and average asperity slope is about 1 degree. When the surface engages in contact, micro-contactareas develop. Accordingly, contact stresses of different magnitudes are generated. Figure 28.10 showscontact pressure and footprint in a boundary-lubricated contact. As demonstrated in Figure 28.10, surfaceroughness causes perceptible pressure fluctuation. While reducing surface roughness decreases pressurerippling, lowering the surface slope has a dominating impact on reducing pressure fluctuation and peakpressure. In life-sensitive applications, bearing raceways must be honed or properly machined to providegood surface qualities.

28.5 Bearing Internal Load Distribution

Bearing load is distributed among rolling elements. The load each rolling element receives is dependent,in general, on the azimuth position of the rolling element. The load distribution is important forpredicting bearing contact stress, torque, temperature, and fatigue life. A bearing’s internal load distri-bution can be obtained by analyzing contact deflections and bearing geometry constraints. As a load isapplied to the bearing, deformations of various magnitudes occur at different azimuth locations. Thedeformation must comply with certain geometrical constraints so that the bearing’s geometrical integrityis not violated. Assume the outer ring of the bearing is firmly seated in a housing and the inner ring iswell-fitted on a solid shaft or a thick wall hollow shaft. In the absence of centrifugal forces, the load eachroller receives can be determined from the deformation/load relationship outlined in Section 28.4.1. Inthis section, bearing load distribution is derived.

28.5.1 Bearings under General Load Conditions

When a bearing is subjected to a radial and an axial load Fr and Fa, respectively, the centers of the innerring and outer ring are displaced relative to each other in both radial and axial directions. Assume themovement is δr in the radial direction and δa in the axial direction. The deformations at any rollingelement position x is expressed by

(28.39)

where β is the contact angle and δe is the diametrical effective clearance.The equation can be rearranged in terms of δmax = δ(0):

(28.40)

δ ψ δ β ψ δ β δ β( ) = + −r aecos cos sin cos

2

δ ψ δε

ψ( ) = − −( )

max cos1

1

21

FIGUR

E 28.9 Surface roughness profile: ground surface.

FIGURE 28.10 Contact pressure distribution and footprint for contact with ground surfaces.

where ε is the load zone parameter and is defined as:

(28.41)

For pure thrust load, ε approaches infinity (ε = ∞).The deflection at azimuth position ψ is the sum of deformations between the rolling element and each

raceway:

(28.42)

From the load and deformation relationship defined in Section 28.4.1, the load can be expressed in termsof the total deformation as

(28.43)

where

(28.44)

kt(i) and kt(o) are defined by Equation 28.8 for point contact and by Equation 28.15 for line contact.Accordingly, t = 3/2 for point contact and t = 40/37 or 10/9 for line contact. Assume Wmax as the maximumroller load. It can be easily shown from Equation 28.43 that:

(28.45)

Thus, the load distribution can be written as:

(28.46)

To achieve the state of static equilibrium, the following conditions must be satisfied:

(28.47)

(28.48)

where Jr(ε) and Ja(ε) are Sjövall integrals (Sjövall 1933) and are defined as:

(28.49)

ε δδ

β δδ

= + −

1

21

2a

r

e

r

tan

δ δ δ= +( ) ( )o i

W ktt= δ

k k kt t o

t

t i

tt

= +

( )

−

( )−

−1 1

W

W

t

ψ δ ψ

δ( )

=( )

max max

W W

t

ψε

ψ( ) = − −( )

max cos1

1

21

F ZW Jr r= ( )max cosε β

F ZW Ja a= ( )max sinε β

J dr

t

x

x

l

l

εε

ψ ψ ψ( ) =π

− −( )

−∫1

21

1

21 cos cos

(28.50)

ψl is the half load-zone angle and is a function of the load-zone parameter ε.

(28.51)

The values of Sjövall integrals are listed in Table 28.6. For an angular-contact bearing, with a givencontact angle β, under the combined radial and thrust load condition, ε can be found from the followingequation with the aid of Table 28.6.

(28.52)

The maximum load of a rolling element is given by:

(28.53)

For a radial bearing, with zero contact angle β = 0, under pure radial load, ε can be found from thefollowing equation with the aid of Table 28.6.

(28.54)

TABLE 28.6 Jr(ε) and Ja(ε) for Single-Row Bearings

εPoint Contact (t = 3/2) Line Contact (t = 10/9)

Jr(ε) Ja(ε) JR(ε) Jr(ε) Ja(ε) JR(ε) Ja(0.31)(ε)

0 0/0a 0/0a 0 0/0a 0/0a 0 0/0a

0.1 0.1156 0.1196 0.0051 0.1268 0.1319 0.0126 0.17090.2 0.1590 0.1707 0.0306 0.1737 0.1885 0.0512 0.24540.3 0.1892 0.2110 0.1229 0.2055 0.2334 0.1493 0.30550.4 0.2117 0.2462 0.5988 0.2286 0.2728 0.4938 0.35910.5 0.2288 0.2782 ∞ 0.2453 0.3090 ∞ 0.40950.6 0.2416 0.3084 1.2554 0.2568 0.3433 0.8704 0.45670.7 0.2505 0.3374 0.5799 0.2636 0.3766 0.4909 0.50840.8 0.2559 0.3658 0.3934 0.2658 0.4098 0.3659 0.56070.9 0.2576 0.3945 0.3074 0.2628 0.4439 0.2995 0.61961.0 0.2546 0.4244 0.2546 0.2523 0.4817 0.2523 0.70721.25 0.2289 0.5044 0.1741 0.2078 0.5775 0.1697 0.81431.67 0.1871 0.6060 0.1128 0.1589 0.6790 0.1092 0.87502.5 0.1339 0.7240 0.0662 0.1075 0.7837 0.0638 0.92275 0.0711 0.8558 0.0294 0.0544 0.8909 0.0283 0.9637∞ 0 1 0 0 1 0 1

a 0/0 denotes a special number that any product or division by this number is zero.

J da

t

x

x

l

l

εε

ψ ψ( ) =π

− −( )

−∫1

21

1

21 cos

ψ εε ε

εl( ) =

−( ) ≤

π >

−cos 1 1 2 1

1

JJ

J

F

F

r

a

r

a

εε

εβ( ) =

( )( ) = tan

WZ

F

J

F

Jr

r

a

a

max =

+

1

2 2

F Zk Jr t R e

t

= ( )ε δ

When δe ≥ 0,

0 ≤ ε ≤ 1/2

and

(28.55a)

When δe < 0,

1/2 < ε < ∞

and

(28.55b)

The values of JR(ε) for various ε are listed in Table 28.6.For angular-contact ball bearings under pure thrust load, the contact angle β is usually greater than

the initial contact angle β00 that exists under zero load. The actual contact angle can be determined fromthe following equation (Eschmann, 1964).

(28.56)

where δ(o) is the deformation at the contact between the ball and outer race. It can be calculated fromEquation 28.8. ∆l is the distance between the centers of curvatures of the inner- and outer-racewaygrooves.

(28.57)

where db is the ball diameter, gi and go are the ratios of the inner and outer raceway groove radii to balldiameter.

28.5.2 Bearings under Pure Eccentric Thrust Load

When a pure thrust load Fa is applied on a thrust or angular-contact bearing, each rolling element sharesan equal amount of load; that is,

(28.58)

However, if the load is applied eccentrically, the radial planes of the two raceways are no longer paralleland they tilt at an angle φ to each other. The load is distributed unevenly among the rolling elements.The equations for δ(ψ), W(ψ), and Fa derived in the previous section are applicable provided that theload-zone parameter ε is redefined as:

(28.59)

J JR r

t

ε ε εε( ) = ( ) −

1 2

J JR r

t

ε ε εε( ) = ( ) −

2 1

sinsin

cos sin

ββ δ

β β δ

=+

+ +

( )

( )

∆

∆ ∆

l

l l

o

o

00

2 200 00

2

∆l g gd

i ob= + −( )2

2

WF

Zaψ

β( ) =sin

ε δφ

= +

1

21

2 a

pD

where Dp is the bearing pitch diameter.Because of eccentric loading, a tilting moment M develops, where

(28.60)

The eccentricity el is determined from the following equation:

(28.61)

28.6 Bearing Lubrication

28.6.1 Elastohydrodynamic Lubrication

Under healthy operating conditions, contact surfaces in a rolling element bearing are separated and,therefore, protected by a thin layer of self-pressurized lubricant film. The applied load is transmittedfrom one surface to the other surface through this pressurized lubricant film. The formation of thelubricant film is the subject of the Elastohydrodynamic Lubrication (EHL) theory and has been describedelsewhere (Dowson et al., 1977; Gohar, 1988). The lubricant film between the contact surfaces has analmost uniform thickness hc at the central region. At the outlet, a distinct constriction occurs and thefilm reaches its minimum thickness hmin. The thickness of the lubricant film in relation to surfaceroughness plays important roles in the determination of frictional torque, heat generation, wear, andfatigue failure. Equations derived from the numerical regression of EHL simulations or experimentalmeasurements are available for film-thickness calculations.

Based on numerical simulation results, Dowson and Higginson (1961) established an empirical for-mula to determine the minimum film thickness at the outlet of the EHL conjunction for line contactproblems under fully flooded, isothermal conditions. Dowson (1968) revised the equation to make itcompatible with the law of dimensionless analysis. In dimensional form, the equation reads:

(28.62)

where η and α are viscosity and pressure-viscosity coefficient of the lubricant, respectively; E ′ is theeffective Young’s modulus; Re is the geometrical average radius given in Section 28.4.2; L is the contactlength; ue is the entrainment speed and is given in the following section as a function of bearing geometryand rotational speed for various bearings; and w is the applied contact load.

The central film thickness is approximately 4/3 of the minimum film thickness for most of the EHLregime. However, the ratio of central film thickness to minimum film thickness reduces asymptoticallyto unity as load increases and speed decreases. The ratio also reduces under conditions of lubricantstarvation.

For point contact problems, Hamrock and Dowson (1977a) presented a minimum-film-thicknessequation. The equation was restricted to circular contact or transverse elliptical contact where the majoraxis of the ellipse lies perpendicular to the direction of entraining motion.

The restriction on the direction of entraining motion was lifted and generalized solutions for entrain-ment along either axis, or along any direction inclined to the principal axes, were presented by Chittendenet al. (1985a,b). A revised minimum film thickness equation was expressed in terms of the directionalsurface curvature ratio k*r (0 < k*r < ∞); that is:

M e FD

ZW Jl ap

r= = ( )2 max sinε β

eD J

Jl

p r

a

=( )( )2

ε

ε

h E R L u we emin. . . . . . ..= ′( ) ( ) ( )− −2 65 0 54 0 7 0 03 0 43 0 13 0 7 0 13α η

Lube & material properties Geometry Operation conditions

1 244 344 1 24 34 1 24 34

(28.63)

When the entraining motion is in the direction of the major axis of ellipse, k*r >1 and k*r = 1/kr. Whenthe entrainment is along the direction of the minor axis of ellipse, k*r <1 and k*r = kr.

The ratio of film thickness to the composite rms value of surface roughness, σ = (σ12 + σ2

2)1/2, is calledthe lambda (λ) ratio. It is the most used parameter among other quantities associated with EHL. Itindicates the condition of lubricant film formation and somewhat reflects the severity of asperity contactbetween contact surfaces.

Three lubrication regimes — from full-film lubrication to mixed lubrication and to boundary lubri-cation — can be distinguished based on the value of the lambda ratio. When the lubricant film thicknessexceeds three times the composite rms roughness, a full separation of the contact surfaces is achieved.The contact load is carried almost entirely by the lubricant film. In full film regime, contact stresses areless affected by surface roughness. The overall performance can be predicted by the classical EHL theoryconsidering smooth surfaces. When the lambda value becomes less than three but greater than one, aperceptible amount of asperity contact occurs. Local lubricant film can be interrupted at the tip of tallasperities. This regime is called partial EHL or mixed lubrication regime. The contact load is sharedbetween lubricant film and contacting asperities. The load sharing ratio and contact friction forces arestrongly dependent on the lambda value (Tallian, 1972). A majority of rolling element bearings operatein this regime. As the lambda ratio becomes less than one, the contact falls in the boundary lubricationregime where asperity contact predominates. Severe surface distress is expected. The overall performancecan be well-predicted by the dry contact analysis.

28.6.2 Effect of Spin Motion

In addition to rolling motion, spinning — a rotation of a rolling element about an axis normal to thecontacting surfaces — often exists at the contacts between the rolling elements and raceways. A studyon point EHL contact (Taniguchi et al., 1996) shows that spin motion reduces the minimum filmthickness. The effect on center film thickness is, however, insignificant. The reduction in the minimumfilm thickness depends on load and speed parameters. An adjustment factor for the minimum filmthickness was given by Taniguchi et al. (1996) as:

(28.64)

where Ω is the spin-to-roll ratio defined as:

(28.65)

and—W is the non-dimensional load defined by:

(28.66)

where ωs is spin angular velocity.

28.6.3 Effect of Lubricant Starvation

When lubricant supply to the inlet of the contact is insufficient, reduction in film thickness occurs. Thisphenomenon is referred to as lubricant starvation. Under such conditions, the thickness of the lubricant

h E u w R k ee e rkr

min. . . . . . * . ..

*= ′( ) ( ) −

− − − −3 68 10 49 0 68 0 117 0 68 0 073 0 466 0 233 0 67 2 3

α η

Lube & material properties Operating conditions Geometry configurations

1 2444 3444 1 244 344 1 244444 344444

ϕspin

h

hW= = −min

min

..0

0 471 3 47 Ω

Ω = 2ω s e

e r

R

u k

Wk w

E Rr

e

=′ 2

film is largely determined by the availability of lubricant supplied at the inlet to the contact. The usageof lubricant becomes more efficient. Lubricant at the inlet zone is entrained into the contact with littleor no side flows. The reduction in film thickness has been studied by many researchers, includingWolveridge et al. (1971), Wedeven et al. (1971), Kingsbury (1973), Chiu et al. (1974), and Elrod et al.(1974). A concept of inlet lubricant-air-meniscus distance to the center of contact was adopted to quantifythe severity of lubricant starvation. Empirical equations were subsequently developed to correlate thereduction in lubricant film thickness to inlet meniscus distance (Castle et al., 1972; Hamrock et al.,1977b). The primary limitation of these meniscus-distance-based equations is that they only apply tothe starvation regime where the meniscus is clearly observed outside the Hertzian contact. As the inletmeniscus distance approaches the half Hertzian contact width, the approach breaks down.

In a recent work, inspired by the work of Kingsbury (1973), Chevalier et al. (1998) used the availablelubricant film thickness at inlet to define lubricant supply conditions. An equation based on the inletavailable film thickness was developed to estimate the reduction in central film thickness for circularcontact problems:

(28.67)

where hc is the central film thickness; hcf is the central film thickness under fully flooded conditions; and–γ represents the ratio of available film thickness at the inlet to the modified fully flooded film thicknessat the contact center by considering lubricant compressibility.

(28.68)

–ρ = ρ/ρ0 is the non-dimensional lubricant density; ρ is the lubricant density; ρ0 is the density underambient pressure; hspl is the thickness of the supplied lubricant at the inlet; and q is an exponent and itincreases as load increases or the entrainment speed decreases. For most of EHL regime, q varies from2 to 5. As q → ∞, Equation 28.67 reduces to

(28.69)

Equation 28.69 underlines the physics. When lubricant supply at the inlet is abundant, the thickness oflubricant film is governed solely by load, speed, and contact geometry. Increasing lubricant supply at theinlet has no effect on film thickness. However, when the lubricant supply at the inlet is insufficient, theformation of lubricant film relies heavily on the availability of lubricant at the inlet. The film thicknessat the contact center cannot exceed what can be generated at the inlet from the limited lubricant supply.

As–γ decreases, the central film thickness hc decreases faster than the minimum film thickness hmin.

The ratio of hc to hmin approaches unity. When–γ < 0.5, Equation 28.67 can be used quite accurately for

calculating reductions in minimum film thickness.The concept of inlet lubricant availability also applies to elliptical contact as well as line contact. For

transverse elliptic contact (kr* < 1) and line contact problems, side flows at the inlet to the contactencounter greater resistance. Consequently, q assumes a greater value when Equation 28.67 is used for atransverse elliptic contact or line contact problem.

28.6.4 Lubrication Methods

In the proceeding equations and text, oil lubrication is considered and the EHL film between contactingsurfaces is implied. Thus, for fluid lubrication, the determination of films and operating expectations

ϕ γ

γstrv

c

cfqq

h

h= =

+1

γρ

=h

hspl

cf

ϕ γ

γγstrv =

+= ( )

∞∞ 11min ,

are straightforward. In selecting a lubricant and method of lubrication, it is recommended that the firstchoice be as simple a selection as possible. For a simple system, straight mineral oil usually is the firstchoice. With reasonable temperature generated (temperature ranging from –50°C to 300°C), this systemcould be a simple sump.

28.6.4.1 Oil Bath

The oil bath or closed sump is a popular means to lubricate bearings. A proper oil level should be aboutat the middle to almost the top of the bottom rolling element. For many systems, this provides for filmformation and removes sufficient heat from the bearings. If load, temperature, or system speed leads toproblems, then the lubricant viscosity or type of lubricant may need to be changed. When a wide rangeof temperature in the system is anticipated, a lubricant with higher viscosity index may be used. Forbearings used at elevated temperature, a higher viscosity oil, additives or, in special cases, a syntheticlubricant such as polyalphaolefin should be considered.

28.6.4.2 Circulating Oil Systems

If the above suggestions do not solve the problems, then a system may require circulating the lubricantthrough the bearings with an oil circulation system. The oil circulation system usually includes an externalreservoir and pump and, in some cases, a heat exchanger that will maintain lubricant and bearing at arequired temperature range. Very often, the appearance of too much wear debris or contamination mayalso indicate the need for filtration to be added to the system. Considering the environmental pressuresto maximize lubricant life and reduce fluid contamination, the addition of a filtration system includinga monitoring procedure may be required so that the quality of the oil going through the bearings andperhaps gears of the overall system can be continually assessed.

28.6.4.3 Oil Mist and Oil Jet

Oil mist lubrication can be used for large bearing systems or high speed. For this type of lubrication,small amounts of oil droplets are carried by air onto bearings with an atomizer. This provides enoughlubricant for fluid film formation and some cooling. It uses very little oil but can be limited in use,depending how well the mist can be contained within the system. An oil jet may be required for ultra-high speed bearings because at such speeds, the lubricant would be driven off the bearing components.Thus, a jet velocity on the order of 10 to 20 m/s allows the lubricant spray to reach the internal bearingsurfaces and provides the necessary fluid contact.

28.6.4.4 Greases

Grease is a lubricant made up of about 10% soap thickener and 80% base oil, with the addition of rustand oxidation inhibitors and specific additives to provide anticorrosion, viscosity index improvement,or high film strength, depending on the need for extended life under required temperature, load, speed,and specific dry to wet conditions. It is recognized that more rolling bearings are lubricated with greaserather than oil because grease lubrication is simpler, more economical, and can provide sealing to bearingsand thus keep out contaminants and resist water ingress more than oils. Conventional greases are usuallylimited to lower speeds ranging up to 3500 rpm (Boehringer, 1992). Lithium-based greases are commonlyused because they can be pumped, have resistance to small particle ingress, and can be stored. For arolling element bearing, grease that fills about 30 to 50% of the bearing inside cavity is recommended.Greater fill can cause higher temperatures or the grease may be forced out from the bearings.

As to its shortcomings, grease does not flush out wear particles generated within a bearing as well andmaintaining the proper amount of lubricant within a bearing is not as easily controlled as is possiblewith an oil. Finally, grease will not be the choice of lubricant when heat generated in a system must bedissipated quickly.

Considering the EHL film formation with grease, tests on six lithium greases (Dalmaz and Nantuo,1987) indicated that bearing fatigue life related to the viscosity of the base oil in the grease and thereforeto the EHL film thickness. It has been recommended (Chetta et al., 1992) that an EHL film reductionfactor of 0.5 to 0.7 be used with grease in a rolling element bearing. This was also suggested by Aihara

and Dowson (1978), especially under longer operation when base oil moves from the grease and resultsin bearing starvation.

28.6.4.5 Solid Lubrication

Other lubrication methods are solid lubrication and gas lubrication. Gas lubrication is usually notconsidered for rolling element bearings, but solid lubrication is used when severe conditions must beaddressed. When temperatures are extreme or specific contaminants must be avoided, solid lubricationcan be used although actual usage is much less than oil or grease, considering the costs and that mostsolid lubricants are not off-the-shelf products. Solid lubricant advantages include: there may be little orno movement of the lubricant, low volatility, and functionality at very low or high temperatures. Thedisadvantages are poor thermal conductivity, thermal expansion different than metals, and wear withoutmeans to replace the solid material worn away. Because solid lubricants are very special, their selectionis more complex than the selection of oils and greases.

The first solid lubricant was probably graphite, although molybdenum disulfide has been recognizedand used for quite some time. Above 350°C, graphite provides low friction and may adhere firmly tometal surfaces but the purity and specific form of graphite must be known to provide worthwhileproperties as a solid lubricant. Unfortunately, the same may be true for other solid lubricants, bothnatural and those developed in the laboratory. Because solid lubricants have different characteristics,relevant information must be reviewed when considering the use of solid lubricants. Information onsolid lubricants is available (see, e.g., Landsdown (1998) and Sliney (1992)).

Whatever the lubricant used in rolling element bearings, it should be selected based on the under-standing of the overall system the bearings operate in and the ranges of speed, load (two most important),and temperature, along with the materials of the bearings and other components such as cage materialor seals so that expected bearing life can be achieved.

28.7 Bearing Kinematics

Kinematic analysis is very important for evaluating a bearing’s performance. It determines the entrain-ment speed ue as well as the sliding and spinning velocities required for lubricant film thickness calculationand subsequent bearing torque and temperature analyses. In this section, rolling speed (entrainmentspeed), sliding speed, and spin angular velocity at both inner- and outer-raceway contacts are discussed.

Consider a reference coordinate system in a bearing’s axial plane as shown in Figure 28.11. To facilitatethe discussion, assume for the moment that the rotating axes of rolling elements are fixed in space andno orbital movement exists. Denote ωr, ωi, and ωo as the angular velocities of the rolling elements, theinner raceway, and the outer raceway, respectively, in relation to the reference coordinate system. Tosimplify the analysis, further assume that ωr lies in the same plane as ωi and ωo. The assumption is validfor roller bearings with minimum or no roller skewing. The assumption is also valid for ball bearingsunder low and moderate speed applications where the gyroscopic moment of balls is negligible. Inaddition, this assumption may hold true for ball bearings under heavy loads where the gyroscopic momentis resisted by the frictional moment at the inner- and outer-raceway contacts. In general cases, a rollingelement contacts the inner raceway at a contact angle βi and the outer raceway at a contact angle βo. Asthe rolling element revolves about the bearing’s axis (in general cases), the rolling element also rotatesabout its own axis that lies at an angle βr with respect to the bearing axis. At the inner-raceway contact,as shown in Figure 28.11, the angular velocity of the rolling element ωr resolves into two components,one parallel with the contact surface, ωrcos(βr – βi), one perpendicular to the contact surface, ωrsin(βr –βi). Similarly, the angular velocity ωi also resolves into two components, ωicosβi and ωisinβi in relationto the contact surface. The rolling speed, also known as the entrainment speed, at contact is:

(28.70)u Dei dr

i

i r d i p i= −( ) − +

1

41γ ω

ωβ β γ β ωcos cos

where γd = De /Dp. De is the diameter of rolling elements measured at the central radial plane of eachrolling element, and Dp is the bearing pitch diameter.

Should gross slip occur at the inner-raceway contact, the sliding speed is:

(28.71)

The spin angular velocity of a rolling element normal to the contacting surfaces at the inner-racewaycontact can be expressed as:

(28.72)

Similarly, at the outer-raceway contact, the rolling speed, sliding speed, and spin angular velocity can beobtained as:

(28.73)

(28.74)

(28.75)

FIGURE 28.11 Rolling element and raceway contact.

v Dir

i

d i r d i p i= −( ) + −

1

21

ωω

γ β β γ β ωcos cos

ω ωω

β β β ωsir

i

i r i i= −( ) +

sin sin

u Deor

o

d o r d o p o= −( ) + +

1

41

ωω

γ β β γ β ωcos cos

v Dor

o

d o r d o p o= −( ) − −

1

21

ωω

γ β β γ β ωcos cos

ω ωω

β β β ωsor

o

o r o o= −( ) −

sin sin

In applications, bearing rotational speeds are usually prescribed by the absolute angular velocities ofthe inner and outer rings,

–ωi and–ωo. The following relationships exist between the relative and absolute

angular velocities:

(28.76)

(28.77)

where ωc is the angular velocity of the cage or the rolling element set.Now one has eight equations (Equations 28.70 to 28.77) and eleven unknown variables. Eight of the

eleven unknown variables are placed on the left-hand side of the above equations. These unknownvariables are not attainable unless the three remainder unknowns, ωr, βr, and ωc, are solved. Obviously,the three unknowns are governed by a bearing’s internal geometry design and surface frictional conditionsat the inner- and outer-raceway contacts. Rather sophisticated numerical analysis of forces and momentsis required to determine these unknown quantities (Harris, 1966, 1971a,b; Gupta, 1975, 1979a,b,c,d).Several computer programs are publicly available. The reader is referred to COSMIC, University ofGeorgia (Athens), for detailed program listings.

As a practical matter, approximations can be made to simplify the analysis. For roller bearings, theorientation angle βr of the rotation axis for each roller is constrained by bearing geometry. Only twounknown variables ωr and ωc need to be determined. The two unknowns are usually obtained by assumingnon-gross slip conditions at both the inner- and outer-raceway contacts (vi = vo = 0). From the non-grossslip conditions, it can be shown that

(28.78)

(28.79)

where

(28.80)

The entrainment speed and spin angular velocity at the roller and inner raceway contact are determinedas:

(28.81)

(28.82)

ω ω ωi i c= −

ω ω ωo o c= −

ω γ βγ β β

ω ωr

d o

d o r

o i

cm= +

−( )−

−cos

cos

1

1

ω ω ωc

c o i

c

m

m= −

−1

mc

i r

o r

d o

d i

=−( )−( )

+−

cos

cos

cos

cos

β β

β βγ βγ β

1

1

um

m

Dei

c d i

c

o ip=

−( )−

−

1

1 2

γ βω ω

cos

ωγ

β β β β ω ωsi

d

i i r ic

c

o i

m

m= −

−( ) +

−

−

1

1cos tan sin

The entrainment speed and spin angular velocity at the roller and outer raceway contact are:

(28.83)

(28.84)

For ball bearings, the motion of balls is more complex. Each ball has an additional degree of freedom.The orientation angle βr of the rotation axis for each ball is unknown. Three spin control hypotheses,known as outer raceway control, inner raceway control, and minimum differential spin, are available.

28.7.1 Outer Raceway Control

Experiments have revealed that balls of an angular-contact ball bearing normally operate with minimalspinning motion at one raceway, with nearly all the spinning motion occurring at the other raceway(Shevchenko and Bolan, 1957; Jones, 1959). Where spinning motion prevails is mainly dictated by frictionand contact conditions and by contact conformability or between the ball and raceway.

Outer raceway control theory assumes, in addition to non-gross slip at both inner- and outer racewaycontacts, that no spin exists at the outer raceway contact, ωso = 0, and all spinning motion occurs at theinner raceway. The rotating axis of each ball intersects the extended tangent of the contact between theball and outer raceway at bearing axis. For relatively high-speed, lightly loaded and oil-film-lubricatedbearings, the outer raceway control theory is a rational approximation. The load and the spin-resistingfrictional moment are greater at the outer raceway contact than at the inner raceway contact due tocentrifugal effects. With the additional constraint ωso = 0, given by the outer raceway control theory, theorientation angle βr of the rotation axis for each ball can be determined.

(28.85)

The entrainment speed at both the inner- and outer-raceway contacts and the spin angular velocityat the inner-raceway contact are given in Equations 28.81, 28.83, and 28.82, respectively.

28.7.2 Inner Raceway Control

The inner raceway control theory assumes, in addition to the non-gross slip conditions at both innerand outer contacts, that there is no spin at the inner contact, ωsi = 0. The rotating axis of each ballintersects the contact tangent of ball and inner raceway at bearing axis. Under this constraint, it can beshown that:

(28.86)

Equations for uei and ueo have the same forms as those for outer raceway control. The spin angular velocityat the outer-raceway contact is given in Equation 28.84.

Inner raceway control probably occurs only in marginal lubricated bearings operated at low speeds.The formability of lubricant film is lower at the inner raceway than at the outer raceway. Consequently,the spin-resisting moment is higher at the inner raceway than at the outer raceway. Inner raceway control

um

Deo

d o

c

o ip= +

−−

1

1 2

γ β ω ωcos

ωγ

β β β β ω ωso

d

o o r o

c

o im= +

−( ) −

−

−

1 1

1cos tan sin

tansin

cosβ β

γ βro

d o

=+

tansin sin

cos cosβ β γ β

β γ βri d i

i d i

= −−

2

2

may prevail when the inner-raceway groove is designed to give a greater contact conformability withballs in the bearing’s axial plane than the outer-raceway groove. Thus, the inner-raceway contact hasgreater ellipticity and consequently a higher spin-resisting moment than the outer-raceway contact if thecentrifugal forces are negligible.

28.7.3 Minimum Differential Spin

For oil-lubricated bearings, in which elastohydrodynamic films exist at both inner- and outer-racewaycontacts, spin occurs at both contacts. It has been shown that even under less favorable lubricationconditions, spinning and rolling occur simultaneously at both inner- and outer-raceway contacts. Thespin moment at the inner raceway counteracts the spin moment at the outer raceway to achieve equi-librium. The minimum differential spin hypothesis assumes that no gross sliding exists at inner- andouter-raceway contacts, and that spin angular velocities at the inner and outer raceways have the samemagnitude. The arithmetical summation of spin angular speed is zero, ωsi + ωso = 0. The minimumdifferential spin theory tends to approximate the minimum friction energy principle. Should the spin-resisting moments at both the inner and outer raceways be the same, minimum differential spin theoryreflects exactly the minimization of frictional energy. The minimum differential spin theory results inthe following approximation:

(28.87)

Rolling speeds and spin angular velocities for the inner- and outer-raceway contacts can be obtainedfrom Equations 28.81 and 28.83 and from Equations 28.82 and 28.84, respectively.

Because of the “on-apex” design of tapered roller bearings, no spin exists at either the inner- or outer-raceway contacts if rollers are not skewed. However, the same cannot be said of spherical bearings.Perceptible spin occurs at both inner- and outer-raceway contacts.

28.8 Bearing Load Ratings and Life Prediction

28.8.1 Basic Load Ratings

Bearing load rating is an important basis for bearing selections. Bearing manufacturers provide two basicload ratings: the basic static load rating and the basic dynamic load rating. The concept of dynamic loadrating is introduced to reflect the fatigue nature of bearing failure.

Rolling contact fatigue is a unique form of material failure that occurs at the surface or sub-surfacelayer due to repeated stresses. Because of the micro-scale inhomogeneities of bearing material, the strengthor resistance to fatigue of bearing material varies from point-to-point. Even under identical stress condi-tions, there will be a wide dispersion in bearing lives. For this reason, bearing life is associated with afailure probability or a survival probability. The concept of percentage life is used. For example, L10 liferefers to the number of hours at a certain speed or the number of revolutions that 10% of a group ofsupposedly identical bearings are expected to fail, or 90% of the bearings are expected to survive under aspecific load. L10 life can also be interpreted as the number of hours at a certain speed or the number ofrevolutions that an individual bearing will survive with 90% reliability under the specified load conditions.

Research over the years has shown that for a given failure probability, bearing life decreases as loadincreases. ISO adopts L10 = 106 revolutions as the standard rating life to establish the basic dynamic loadrating. The basic dynamic load rating is defined as a constant load applied to a bearing with a stationaryouter ring that will endure the standard rating life of 1 million revolutions, L10 = 106. The basic dynamicload rating for radial bearings is a central radial load of constant direction, denoted as C1, while the basicdynamic load rating for thrust bearings is an axial load in the same direction as the central axial and is

tansin sin

cos cosβ β β

β βro i

o i

≈ ++

denoted as C1a. The basic dynamic load ratings for various bearings are calculated by the equations shownin Table 28.7.

The basic static load rating is used to determine the maximum permissible load that can be appliedto a non-rotating bearing. The static radial load rating C0, and the static thrust load rating C0a, are basedon a maximum contact stress between rolling element and bearing raceway at the center of the contactin a non-rotating bearing with a 180° and 360° load zone, respectively. The maximum stress varies slightlyfor different bearing types. Table 28.8 lists the maximum contact stresses.

The maximum stress level listed in Table 28.8 may cause visible light Brinell marks on bearing raceways.These marks will not have a measurable effect on fatigue life when the bearing is subsequently rotatingunder lower application loads.

28.8.2 Bearing Life Prediction

28.8.2.1 Bearing Life Theory

In studying the failure of brittle engineering materials, Weibull (1939) concluded that the strength of amaterial was a statistical variable, and assumed that the first initial crack led to the break of the entirestructure under stress. By applying the calculus of probability, Weibull stated that the survival probabilityS could be expressed by:

(28.88)

where Γ(σ) is a material characteristic and is a function of stress conditions σ.Lundberg et al. (1947) extended Weibull’s weakest link theory to ductile bearing materials by arguing

that a bearing’s life consists of crack initiation life and crack propagation life, and that initiation lifepredominates the bearing’s life. Based on extensive bearing tests, Lundberg and Palmgren proposed anempirical relationship:

TABLE 28.7 Basic Dynamic Load Rating

Ball Bearinga Roller Bearing

Radial bearing bmfc (i cos β)7/10 Z 2/3 Dw9/5 bmfc (iLwe cos β)7/9 Z3/4 Dwe

29/27

Angular contact bearing bmfc (cos β)7/10 tanβZ 2/3 Dw9/5 bmfc (Lwe cos β)7/9 tanβZ 3/4 Dwe

29/27

Thrust bearing β = 90° bmfc Z 2/3 Dw9/5 bmfc Lwe

7/9 Z 3/4 Dwe29/27

a When Dw > 25.4 mm, use 3.645Dw7/5 for Dw

9/5.Note: bm: Rating factor for contemporary bearing material and manufacturing quality, the

value of which varies from 1.0 to 1.3, depending on bearing type and design.fc: Factor that depends on the geometry of the bearing components, the accuracy towhich the bearing components are made, and the material. The values of fc for variousbearings are given in ISO 281-1990.i: Number of rows of rolling elements in a bearing.β: Normal contact angle of a bearing.Ζ: Number of elements per row.Dw, Dwe: Diameter at the central plane of a rolling element (mm).Lwe: Effective contact length of rollers (mm).

TABLE 28.8 Maximum Permissible Contact Stress

Bearing TypeMaximum Stress

(MPa)

Self-aligning ball bearings 4600Non-self-aligning ball bearings 4200Roller bearings 4000

ln1

Sdv

V

= ( )∫ Γ σ

(28.89)

where τ0 is the maximum orthogonal shear stress, z0 is the depth below the load-carrying surface at whichthe maximum shear stress occurs, V is the volume of stressed material, and N is the number of stresscycles before failure with survival probability of S. c, e, and h are exponents.

Assume Hertzian contact exists between the rolling elements and raceways. For an assembly of one ormore contacts, Equation 28.89 can be expressed in terms of bearing load P by substituting bearinggeometry and by applying Hertzian contact theory:

(28.90)

where n is an exponent (n = 3 for ball bearings and n = 10/3 for roller bearings). Lc10 is the referencerating life under which the dynamic load rating C is established. In ISO standards, Lc10 = 106 revolutionsand, accordingly, C = C1. As shown in Table 28.7, the influences of bearing geometry and material qualityare lumped into the dynamic load rating C1. P is the equivalent load and can be calculated from thefollowing equation.

(28.91)

Fr and Fa are bearing loads in radial and axial directions, respectively. Factors X and Y are determinedbased on the ratio of Fa/Fr . The values of X and Y are given in ISO 281-1990 and bearing catalogs forspecific bearings.

28.8.2.2 Bearing Life Adjustment Factors

Actual bearing operating conditions may vary perceptibly from the reference conditions under whichthe basic dynamic load ratings are established. To permit the quantitative evaluation of environmentdifferences, such as reliability requirement, bearing material quality and processing, operation conditions,and failure criteria, various life adjustment factors have been introduced to the basic life Equation 28.90by bearing manufacturers. ISO 281-1990 combines three factors into the following life equation to givethe adjusted life:

(28.92)

Lna is the adjusted bearing life, in revolutions, and a1, a2, and a3 are life adjustment factors. The definitionof life adjustment factors and the variables that have been incorporated in each factor are discussed asfollows.

Life adjustment factor for reliability (a1): For applications that require bearings to have reliability other than 90% (S = 0.9), the life adjustmentfactor can be obtained from:

(28.93)

where e is the Weibull slope; e = 1.5 is often recommended.

ln1 0

0S

dvN

zV

V

c e

h= ( ) ∝∫Γ σ τ

LC

PL

n

c10 10=

×

P XF YFr a= +

L a a a Lna = 1 2 3 10

aS

e

1

1

1

1 0 9=

ln

ln .

Life adjustment factor for special bearing properties (a2): This factor allows improvements in bearing material and process to be incorporated in the bearing lifeestimate. For bearings made from standard bearing quality alloy and carbon steels, a value of a2 = 1 isrecommended. Bearings are also made from higher grade steels with better controlled refining processes.These premium steels contain fewer and smaller inclusion impurities than standard steels and providethe benefit of extending bearing fatigue life where fatigue life is limited by nonmetallic inclusions. Whenbearings made from premium steels are used, a value of a2 greater than 1 can be adopted, provided thatthe maximum contact stress is less than 3 GPa and that favorable lubrication conditions are maintained.However, if a reduction in life is expected as the result of a special heat treatment, a reduced a2 valuemay be considered.

Values of a2 range from 0.6 to 6.0 for different chemistries, melting practices, clearness levels and metalworking, heat treatment, and surface modification practices. Information on a2 can be found in Zaretsky(1992).

Life adjustment factor for operating conditions (a3): Variables considered in the life adjustment factor for operating conditions are load distribution, temper-ature, and lubrication. The inclusion of load distribution and temperature requires extensive analyticaland experimental work. There is no general consent among bearing manufacturers regarding these effectson bearing life. ANSI/ABMA and ISO decided not to give general recommendations.

Adequacy of lubrication is central to a3. The ANSI/ABMA and ISO standards are based on nominallubrication conditions where the lambda ratio λ is equal to or slightly greater than 1. A value of a3 = 1is recommended for such conditions. a3 increases as λ increases and reduces as λ decreases.

Life adjustment factors were introduced to account for variances not included in the basic life model(Equation 28.90). However, choosing appropriate values of life adjustment factors is not as easy as itmight appear. Environmental and design factors are often mutually dependent. Changing one factor canaffect others. Some factors may not even be multiplicative. Not to restrict various means of combiningthe appropriate life factors, the ISO bearing life rating technical committee is currently working on aproposal in which a2 and a3 will be replaced by a single factor axyz. This factor is derived from a stress-based life calculation model that incorporates all possible factors affecting the generalized stress field.

There are still effects on bearing life that cannot be properly accounted for by life adjustment factors.To incorporate these effects, extensive analytical work and bearing life tests are required. The ISO andANSI/ABMA may further revise their life adjustment factors and/or life calculation method to reflect theadvance of modern bearing technology.

It is worth noticing that alternative bearing-life models are also available for bearing fatigue lifeprediction. These models can generally be classified into two categories: the engineering model and theresearch model. Tallian (1992) provided a comprehensive review of some of the published models.Discussions of various fatigue-life prediction models are beyond the scope of this handbook and thusare not presented here.

The ISO and various bearing manufacturer associations may have their own practice for the selectionof life calculation methods and life adjustment factors. For a specific application, seek a bearing manu-facturer for advice.

28.9 Bearing Torque Calculation

When efficiency and energy consumption of a mechanical system are of concern, a bearing’s frictionaltorque is often analyzed to determine its acceptance. Bearing torque calculations are also performed toassess bearing heat generation. When a bearing’s torque is obtained under normal running conditions,it is referred to as running torque. For angular-contact ball bearings and tapered roller bearings, torqueis often used to determine the appropriate amount of pre-load. Bearing torque obtained under slowspeed conditions or setup conditions is referred to as starting torque or setup torque. This section

addresses bearings’ starting torque as well as running torque and provides empirical and analyticalequations for torque predictions of angular-contact ball bearings and tapered roller bearings.

The major source of bearing torque derives from the rolling resistance and, if applicable, the slidingresistance at the contacts between the rolling elements and raceways. Bearing torque also results fromthe sliding resistance at the contacts between the rolling elements and cage, and between the rollingelements and flange, if a flange exists. In moderate and high-speed applications, lubricant churning isan additional source of frictional resistance. The rolling resistance is caused by internal friction in thelubricant and bearing material and by the micro-slip. Sliding resistance results primarily from surfaceasperity interactions introduced by the sliding and/or spinning motion. Under high-speed and light-loadconditions, gross sliding occurs at raceway contacts, which further complicates the torque analysis.

A complete analytical approach for bearing torque predictions is very difficult, if not impossible.Torque models used by bearing manufacturers are mostly established through experimentation. However,there are cases where bearing torque models can be derived analytically or semi-analytically.

28.9.1 Starting Torque

28.9.1.1 Starting Torque for Angular-Contact Ball Bearings

For angular-contact ball bearings, the starting torque results primarily from the slippage associated withspin motion between the balls and outer raceway. At slow speed, as suggested by inner-raceway controltheory, the spin-resisting moment at the inner raceway prevails. Thus, the starting torque M is soughtby only considering the spin moment at contacts between the balls and outer raceway.

(28.94)

where Msj is the torque caused by Coulomb friction forces for the contact at the j th azimuth location.Poritsky et al. (1947) integrated the friction force over a contact ellipse and showed that:

(28.95)

where Wj, aj, and βj are the element load, the half major axis of contact ellipse, and the contact angle atthe j th azimuth location, respectively (see Section 28.4). µa is sliding friction coefficient for the dry orboundary lubricated contact. E(ej) is the elliptic integral of the second kind, the value of which can beobtained from mathematical handbooks or from the following empirical equation (Hamrock et al.,1983):

(28.96)

where kr is defined in Section 28.4.If the bearing is under pure thrust load, as in the bearing setup situation, bearing torque can be

expressed in terms of the axial load Fa as:

(28.97)

where k is the ellipticity ratio at the outer-raceway contact, Re is the geometric average radius at the outerraceway contact, and F(e) can be calculated from equations given in Section 28.4.

M Msj j

j

Z

==

∑ sin β1

M W a esj a j j j= µ ( )3

8E

E e k kr r( ) ≈ + π −

≤ ≤1

21 0 1

ME

F e e

k

R

ZFa

Lubricationand Material

e

Internal Geometry

a

Axial Load

= µ′

( ) ( )

0 429

1 3

1 3

4 3.sin

123 1 24444 34444

E

β

28.9.1.2 Starting Torque of Tapered Roller Bearings

Although the “on-apex” design of tapered roller bearings enables rollers to achieve primarily true rollingmotions on the raceways at every point along the line of contact, sliding and spinning inherently occurat the rib and roller-end contact for every roller. Under slow-speed or starved lubrication conditions,and when the load on a tapered roller bearing exceeds a certain level, usually about the level of pre-load,the sliding friction at the rib and roller-end contact becomes a decisive factor. Bearing torque caused byrolling resistance at roller and raceway contacts is comparatively negligible. This is due, in large part, tothe inability of forming a full lubricant film at the rib and roller-end contact to separate the matingsurfaces under these conditions. Severe asperity-to-asperity contact occurs. The friction torque from therib and roller-end contact is perceptibly high. Therefore, the starting torque of a tapered roller bearingis sought by considering only the sliding and spinning motion at the rib and roller-end contact.

The frictional torque at the rib and roller-end contact Mrib consists of two components: a sliding torqueMsld and a spinning torque Mspn:

(28.98)

(28.99)

(28.100)

where γ is the half included angle of rollers; H is the rib contact height and is determined by the bearinginternal geometry from Equation 28.33; and χ is a factor depending primarily on rib and roller-endgeometry and load distribution. For most applications, χ ≈ 0.5 is a good approximation.

The starting torque can be calculated from the following approximation:

(28.101)

28.9.2 Running Torque

28.9.2.1 Running Torque of Ball Bearings

The prediction of running torque for ball bearings is largely dependent on empirical equations. Basedon experimental studies, Palmgren (1959) resolved the total torque into a load-related moment Ml anda lubricant viscous moment Mv. The load-related moment reflects elastic rolling resistance, includingmicro-slip and hysteresis loss, while the viscous-related moment accounts mostly for the losses of lubri-cant film shearing and bulk lubricant churning.

(28.102)

This general approach has been adopted by various bearing manufacturers. However, the empiricalequations established by various bearing manufacturers for Ml and Mv calculations may vary to reflectthe differences in internal geometry design. Catalogs of manufacturers must be consulted for appropriatetorque calculation equations. NSK uses the following equations for load- and viscous-related momentscalculations for ball bearings under high-speed applications:

(28.103)

(28.104)

M M Mrib sld spn= +

M HFsld a a= µ cosγ

M HFspn a a= µχ γcos

M HF HFa a a a= µ +( ) ≈ µ1 1 5χ γ γcos . cos

M M Ml v= +

M D F N mml p a= × ⋅( )−0 672 10 3 0 7 1 2. . .

M D n Q N mmv p ia b= × ⋅( )−3 47 10 10 3 1 4. . η

where Dp = Bearing pitch diameter (mm)Fa = Axial load (N)ni = Inner ring speed (rpm)η = Lubricant viscosity at outer ring temperature (MPa.s)Q = Lubricant flow rate (kg/min)

The exponents a and b are functions of bearing angular speed only and are given, respectively, by:

(28.105)

(28.106)

28.9.2.2 Running Torque of Tapered Roller Bearings

Running torque of tapered roller bearings results primarily from the hysteresis and viscous-related rollingresistance at the contacts between the rollers and raceways, and from the slide-and-spin-related slidingresistance at the contacts between the roller-ends and rib face. As the speed of the bearing increases, andwhen ample lubricant is available at the inlet to the contact, contact surfaces between the roller-ends andrib may be fully separated by a layer of lubricant film. The thickness of the lubricant film at the roller-end and rib contact is approximately twice the film thickness at the roller and raceway contact. Conse-quently, the friction loss from the roller-end and rib contact reduces substantially. Figure 28.12 showsschematically the variation of frictional torque as a function of bearing speed for tapered roller bearings.

For a tapered roller bearing, the running torque is composed of three components: the inner racewaytorque Mrace(i), the outer raceway torque Mrace(o), and the rib torque Mrib.

(28.107)

The raceway torque for a combined radial and axial load is obtained by advancing the work of Aihara(1987).

(28.108)

FIGURE 28.12 Bearing torque as a function of speed.

a ni= −24 0 37.

b ni= × +−4 10 0 039 1 6. .

M M M Mrace i race o rib= + +( ) ( )

MR R F

R l Erace i TH SP D DSe o a

e o( )

( )=

′

φ φ ρ ζ

αγβ

0 312

0 31

2.

.

cos

sin

(28.109)

where

(28.110)

is a thermal factor that is introduced to consider the inlet shear heating. φTH is dependent only on

(28.111)

where–β is the temperature coefficient of viscosity, ηo is lubricant viscosity, and kcd is the thermal

conductivity of lubricant.

(28.112)

is a speed factor, and ρD = is the aspect ratio of rollers.

(28.113)

is a load distribution factor. Ja(0.31) is called the extended Sjövall integral and is defined as:

(28.114)

The values of Ja(0.31) are listed in Table 28.6 for line contact (t = 10/9).

Other parameters used in the above equations are defined as follows:

α = Pressure-viscosity coefficient of lubricantβ0 = Contact angle at the outer raceway contactl = Roller lengthD = Roller mean diameterZ = Number of rollersRe = Effective contact radiusRi, Ro = Mean radii of the inner and outer raceways

The rib torque is expressed in a similar form as Equation 28.101:

(28.115)

MR R F

R l Erace o TH SP D DSe i a

e o( )

( )=

′

φ φ ρ ζ

α β0 31

20 31

.

.

sin

φTHL

= ×+

1 76 10

1 0 29

2

0 78

.

. .

Lu

ke

cd

= βη02

φ αηSP

e

e

u

R=

0

0 658.

1D----

ζDSa

a

ZJ

ZJ

0 310 31

0 31

..

.

( ) ( )=

( )

J da

t

x

x

l

l

0 31

0 31

1

21

1

21

.

.

cos( )−

=π

− −( )

∫ ε

ψ ψ

M HFrib a= +( )µ1 χ γcos

Under normal running conditions, contacts between the roller-ends and rib face are partially or fullyseparated by a lubricant film. Friction coefficient µ at these contacts is reduced by virtue of the lowshearing resistance of lubricant film. The reduction in µ is determined by the load sharing ratio ϑ, whichin turn is a function of the lambda ratio (λ):

(28.116)

(28.117)

µa and µf are friction coefficients for dry contact (asperity contact) and lubricated contact, respectively.The empirical approach as outlined for running torque calculations of ball bearings is also applicable

for various other bearing types, including tapered roller bearings, when a quick and simple torqueestimate under normal operating conditions is desired. The reader is referred to Brandlein et al. (1999)or Harris (1991) for details.

28.10 Bearing Temperature Analysis

The expected bearing operating temperature is important for designing the bearing arrangement, lubri-cation and sealing, and for determining the proper bearing setting and fitting practice. Excluding extra-neous heat, the operating temperature of a rolling element bearing under medium speed and loadconditions is relatively low. Table 28.9 provides reference values for the average operating temperaturein various applications (Brandlein et al., 1999). When exposed to extraneous heat, the temperature thata bearing assumes can be very high. Table 28.10 lists the typical operating temperature for bearingsexposed to extraneous heat (Brandlein et al., 1999).

The operating temperature of bearings in a mechanical system is determined by the amount of heatgenerated within the bearing and the amount of heat that is transferred to or away from the bearing.

TABLE 28.9 Operating Temperature of Bearings in Different Machines Operating at 20°C

Approximate Operating TemperatureBearing Application (°C)

Cutter shaft of planing machine 40Bench drill spindle 40Horizontal boring spindle 40Circular saw shaft 40Double-shaft circular saw 40Blooming and slabbing mill 45Lathe spindle 50Vertical turret lathe 50Wood cutter spindle 50Calender roll of a paper machine 55Backup rolls of hot strip mills 55Face grinding machine 55Jaw crusher 60Axle box bearing/locomotive or passenger coach 60Hammer mill 60Wire mill roll 65Vibratory motor 70Rope stranding machine 70Vibrating screen 80Impact mill 80Ship’s propeller thrust block 80Vibrating road roller 90

µ = µ + −( )µϑ ϑa f1

ϑ = −e 1 8 1 2. .λ

The actual system in which bearings are operated can be very complex. However, the principle oftemperature analysis remains the same. The principle is based on the law of energy conservation, knownas the first law of thermodynamics, and on the rate equations of heat transfer.

Consider a control volume, a region of space bounded by control surfaces through which energy andmass can pass. The control volume can be a finite region or a differential (infinitesimal) region. Forbearing temperature analysis, it is customary to use finite-control volumes. These control volumes arebounded by surfaces that define a bearing’s boundary dimensions. The law of energy conservation statesthat the net rate at which thermal and mechanical energy enters a control volume, plus the rate at whichthermal energy is generated within the control volume, must equal the rate of increase in energy storedwithin the control volume. Under steady-state conditions, the amount of heat generated in the bearingis carried away by the net dissipated heat flow.

(28.118)

To assess the temperature level at which a rolling element bearing operates, the heat generated andthe heat transferred through various modes must be determined. This section intends to provide thebasic elements being used by the bearing industry in temperature analysis.

28.10.1 Heat Generation

The power loss in a rolling element bearing primarily comes from the frictional loss during operationthat manifests itself mainly in a form of heat generation. Thus, the rate of heat generation in the bearingcan be estimated from the bearing’s frictional torque through the following equation:

(28.119)

where ω is the rotational speed and M is the bearing torque.For ball bearings and roller bearings, the frictional torque can be obtained by equations given in the

previous section. The frictional torque equations for ball bearings are based on empirical formulae. Theeffect of sliding between the balls and cage pockets is included and so is lubricant churning effect. Thus,

(28.120)

For roller bearings, however, the friction between the rollers and cage pockets is not considered. Inaddition, lubricant churning loss is not included in the torque calculation. Lubricant churning effects

TABLE 28.10 Operating Temperature of Bearings Exposed to Extraneous Heat

Bearing Application Extraneous Heat

Approximate Operating Temperature

(°C)

Electric traction motor Electric heat from armature, housing cooled by air 80–90Hot gas fans Heat transmission to the bearing via the shaft from the

impeller exposed to hot gas90

Water pump in a vehicle engine Heat from cooling water and engine 120Turbo-compressor Dissipation of compression heat through shaft 120Internal combustion engine crankshaft Dissipation of combustion heat through crankshaft;

cooled housing120

Dryer rolls of paper machine Heating steam of 140–150°C through bearing journal 120–130Calender for plastic substances Supply of heat carrier at 200–240°C through bearing

journal180

Wheel bearings of kiln truck Heat radiation and transmission from kiln space 200–300

˙ ˙ ˙E E Egen out in= −

H Mf = ω

E Hgen f=

must be accounted for when a bearing operates at a relatively high speed and churns a fair amount oflubricant. The rate of heat generation due to viscous drag can be approximated by:

(28.121)

where Z = Number of rollersξ = Density of oil-air mist; weight of lubricant in bearing cavity divided by the free volume

within the bearing boundary dimensionl = Effective roller lengthDrm = Roller mean diameterDp = Bearing pitch diameterωc = Orbital angular speed of rollersg = Gravitational accelerationcv = Drag coefficientφVC = Effective volume correction factor, 0 < φ VC < 1

Therefore, for roller bearings, the rate of heat generation is calculated as:

(28.122)

28.10.2 Heat Transfer

Heat generated in bearings is dissipated into the system via three basic modes: conduction, convection,and radiation. It is possible to quantify the heat transfer processes using rate equations. Because heatconduction within a solid and heat convection between the solid and fluid are the primary forms of heattransfer in a bearing system, the rate equations for these two modes are addressed below. The rate equationfor radiation, if needed, can be found elsewhere (Pinkus, 1990).

28.10.2.1 Conduction

Heat conduction is caused by the diffusion of energy from the more energetic to the less energeticmolecules or particles through random motion. The rate of heat transfer via conduction is given byFourier law:

(28.123)

where kcd is the thermal conductivity; A is the area normal to the heat flux; and ∂T/∂l is the temperaturegradient along the direction of heat flux.

28.10.2.2 Convection

Heat convection is the result of energy diffusion due to random molecular motion and energy transferdue to bulk motion. The actual physical events associated with heat conduction may be quite complexwhen latent heat exchange is involved. Regardless of the particular nature of the convection heat transferprocess, the appropriate rate equation is, in simple form, known as Newton’s cooling law:

(28.124)

where Ts is the surface temperature and Tr is the reference temperature of the fluid. For external flow,the free stream temperature Tr = T∝ is used; for internal flow, the reference temperature Tr refers to themean temperature Tr = Tm. The proportionality constant hcv is the heat convection coefficient or filmconductance. It encompasses all the parameters that influence convection heat transfer. In particular, it

H cZ lD D

gl VC v

rm p c=( )

φξ ω

2 95

32

.

E H Hgen f l= +

q k AT

lcd cd= ∂∂

q h A T Tcv cv s r= −( )

depends on conditions of the boundary layer, which is affected by surface geometry, the nature of fluidmotion, and the thermodynamic and transport properties of the fluid. Any study of convection ultimatelyreduces to a study of determining hcv . The equation provided by Eckert (1950) can be used as anapproximation for hcv when considering the convection heat transfer from bearing to oil that contactsthe bearing assuming a laminar flow field (Harris, 1991):

(28.125)

where kcd and ν are the thermal conductivity and kinematic viscosity of the oil; Pr is the Prandtl numberof the oil; and uc is the cage speed.

The equation is also suitable for calculating the convection heat transfer from the housing inside-surface to oil, taking uc equal to 1/3 of the cage speed and Dp equal to the housing diameter.

The temperature obtained through heat flow analysis using a finite-control volume outlined abovegives the average bearing bulk temperature. The actual temperature at the contact between the rollingelements and raceways, or between the roller-ends and flange, can be perceptibly higher than the bulktemperature. For flash temperature analysis at bearing contact, the reader is referred to Gao et al. (2000).

There are cases where comprehensive numerical programs must be used to accomplish bearing tem-perature analysis. When such need arises, seek a bearing manufacturer for advice.

28.11 Bearing Endurance Testing

Bearing endurance testing is performed to establish life ratings, conduct quality auditing, and evaluatematerial, heat treatment, internal geometry, and surface finish improvements. Great variation can beexpected despite the fact that bearings are run under “identical” conditions. Thus, statistical methods inplanning and interpretation of bearing endurance tests have become a necessity. Accordingly, testingprocedures have been developed to optimize the tests based on statistics.

28.11.1 Testing Procedure and Data AnalysisExtensive testing results over the years have indicated that bearing life is approximately a power lawfunction of load under a given failure probability. On a log-log plot, the life and load relation falls on astraight line. A family of such straight lines can be obtained, with each corresponding to a certain failureprobability. For a given load, the cumulative percentage of failure assumes the Weibull distribution. OnWeibull probability paper, the graphical presentation of the relationship between the failure probabilityand bearing life also assumes a straight line under constant load conditions.

Bearing endurance testing is a time-consuming and costly process. To reduce testing time and cost,bearings are usually tested under accelerated conditions. Increasing applied bearing load is perhaps themost commonly adopted means for test acceleration. Care must taken to ensure that the load increasedoes not alter the bearings’ failure mode. As a general guideline, the maximum applicable load shouldnot cause any plastic deformation. For this consideration, the applied load should not produce a maxi-mum Hertzian stress in excess of 3.3 GPa.

Even with accelerated tests, it is neither economical nor possible to test the entire bearing population.Instead, only a finite collection of bearings is tested. Statistical analysis is therefore used to derive theunderlying life distribution for the general population from the finite bearing samples. Several methodsexist to design testing procedures that reduce testing time. Sudden death testing, sequential testing, andparallel testing are exemplary test strategies. Among the various test methods, sudden death tests are themost recommended test procedures in the bearing industry.

In sudden death tests, bearing samples of size m are divided into l groups with n bearings in eachgroup (m = nl). When the first failure occurs in a group, the test is suspended for that group. After allgroups are tested, l failures are obtained. Each failure represents an estimate of Lq life that is associated

h ku

Dcv cdc

p

=

0 0332 1 3

1 2

. Pr ν

with a definite percentage q of the bearing population failing as if the entire population were tested. Thispercentage is called rank. Because the true rank is not known, estimations are made such that in a longrun, the positive and negative errors of the estimate cancel each other. A rank with such property is calledmedian rank and is denoted as q = r0.5. For a detailed discussion on the determination of median ranks,the reader is referred to Johnson (1951).

The l estimates of Lq are tabulated in ascending order. The median rank for the j th estimate in l canbe obtained approximately from the following equation (Johnson 1974):

(28.126)

Plotting Lq as abscissa in logarithm scale and 1/(1 – r (l)0.5) as ordinate in log-log scale, the Weibull

distribution of Lq is established. The best estimate of Lq is the median life corresponding to the 50% level.An α% confidence band can be constructed by seeking the (50 ± α/2)% rank values of the life estimateLq. The reader is referred to Johnson (1974) for detailed discussion.

To ensure meaningful results, test controls are required. Caution must be exercised to make sure thatthe test bearings are free from material and manufacturing defects, and that all parts conform toestablished dimensional and form tolerances. Moreover, test conditions must be adequate for the variablethat is under evaluation. The failure mode is of fatigue nature and no other variables or artifacts alterthe outcome of test results. Finally, each failed bearing should be carefully examined to exclude non-fatigue-related failures in the life estimate and to determine if the endurance test series was adequatelycontrolled before and during the test.

28.12 Bearing Failure Analysis

Present-day bearings seldom fail when properly installed and maintained. Premature failures resultprimarily from damage to bearings while handling before and during installation, and damage causedby improper installation, setting, and operation. When premature failure occurs, post-mortem investi-gations are often conducted to determine the causes of failure so that similar failures can be prevented.

In many cases, failures are easily identified visually on bearings, but the root cause of these failures isdifficult, if not impossible, to determine. This section intends to provide some possible cause-and-effectrelationships to help the reader determine the causes of the most common types of bearing failure sothat preventive actions can be taken.

A variety of means have been proposed to classify the types of bearing failure, some based on themechanisms or causes of failure, and others based on failure appearances or failure locations, or acombination of all of the above. In this section, bearing failures are categorized into a number of majortypes according to the mechanisms. Each type may further be distinguished by specific modes based onthe nature of the failure.

28.12.1 Contact Fatigue

Contact fatigue can have different appearances, depending on the initial causes. It ranges from surfacepitting, peeling, to spalling or flaking, and to section cracking in rare cases. Whatever the initial causes,cyclic stress is the common factor.

Pitting. Pitting is indicated by the development of small cavities in the contact path. It is a surface-originfatigue failure associated primarily with asperity contact and small debris dents that act as stress raisers.Pitting can also result from water corrosion. In such cases, pits usually have rough irregular bottoms(Figure 28.13).

Peeling. Peeling is characterized by the formation of cracks that originate at a very shallow angle to thesurface and propagate parallel to the surface at depths much smaller than the maximum nominal stress.

rj

l

l

0 5

0 3

0 4.

.

.( ) = −

+

Peeling may appear as a frosted area. Thus, it is sometimes called frosting. Peeling occurs when a relativelylarge percentage of the surface asperities are in direct contact with the mating surface. Asperity-scaleplastic deformation is inevitable. The fact that depth of peeling often correlates with surface roughnesssupports the roughness asperity theory. Progressive superficial pitting can also lead to peeling.

Spalling. Spalling is identified by the formation of large and deep cavities in the loaded area (Figure 28.14).In severe cases, a spall can extend along or across the bearing raceway, resulting in the removal of a largevolume of material from the surface. The progressive state of spalling is sometimes called flaking. Spallingis considered the most common form of fatigue failure. It can be either surface or subsurface initiated. Asurface-originated spall is usually associated with surface defects that act as stress raisers. Debris dents, nicks,and grinding grooves are common sites for surface-originated spalling. Surface-originated spalls often showan arrowhead growth pattern (Figure 28.14a). A multitude of small surface defects results in micro-spallingor pitting, which could potentially culminate in micro-peeling or frosting. Large surface defects tend to

FIGURE 28.13 Pitting (a) proceeded with spalling, and (b) caused by acid corrosion. (From Walp, H.O. (1971),Interpreting Service Damage in Rolling Type Bearings — A Manual on Ball & Roller Bearing Damage, ASLE, ParkRidge, IL. With permission.)

FIGURE 28.14 (a) Spall initiated at surface damage. (From Harris, T.A. (1991), Rolling Bearing Analysis, 3rd ed.,John Wiley & Sons, New York. With permission.) (b) Spall nucleated at subsurface on cylindrical bearing inner-ringraceway. (From Derner, W.J. and Pfaffenberger, E.E. (1984), CRC Handbook of Lubrication, Vol. II, Booser, E.R. (Ed.),CRC Press, Boca Raton, FL. With permission.)

(a) (b)

(a) (b)

ROLLING

cause deep and large spalls that penetrate to the depth of the maximum Hertzian shear stress. Subsurface-originated spalling is associated with stress concentrations caused by constituents in the matrix, particularlyoxide-type inclusions (Figure 28.14b). When the Hertzian stresses are dominant, cracks propagate trans-granularly to the maximum Hertzian stress depth, causing material to break off from the surface.

Flaking. Flaking is identified by material break-off with conchoidal and ripple patterns extending evenlyacross or along the loaded part of the race (Figure 28.15). Although there is no clear distinction betweenflaking and spalling, some consider flaking as a severe state of spalling. Flaking can be surface or subsurfaceoriginated. Surface-originated flaking is usually caused by a multitude of debris dents. Subsurface-originated flaking most likely results from an inclusion inside the material. Edge loading tends to resultin flaking more severe on one side of the raceway.

Transverse cracking. Transverse cracking seldom occurs for case-carburized bearing components. It mainlyaffects bearings made from through-hardened steels. Transverse cracking is the result of crack propagationunder cyclic tensile stress due to press fit or flexing of the section. This type of failure is usually precededby another mode of fatigue.

28.12.2 Surface Depression and Fracture

Unlike contact fatigue, surface depressions and fractures of bearing components are usually inflicted bystatic load or impact load. The load for inflicting such damage is greater than those that cause fatigue failure.Under some circumstances, surface depressions themselves may not cause significant concern. However,the consequences are usually detrimental. Whenever possible, this type of damage should be avoided.

Brinelling. Brinelling is characterized by the plastic deformation of bearing surfaces due to extreme staticor repeated shock loads. Brinelling is identified as dents or grooves on bearing raceways conforming to

FIGURE 28.15 (a) Flaking with conchoidal pattern extending across the loaded part of the race. (From Neale, M.J.(1973), Tribology Handbook, John Wiley & Sons, New York. With permission.) (b) A greatly advanced flaking oninner raceway of a cylindrical roller bearing. (From Walp, H.O. (1971), Interpreting Service Damage in Rolling TypeBearings — A Manual on Ball & Roller Bearing Damage, ASLE, Park Ridge, IL. With permission.)

the shape of rolling elements (Figure 28.16). Indentations by foreign objects do not fall into this category.Brinelling may not immediately terminate bearing operation if vibrations caused by brinelling marks donot arouse significant concern. However, the post over-rolling of brinelling marks will lead to crackinitiation and subsequent propagation. Bearings with brinelling marks are eventually failed by spalling.

Gouges or nicks. Improper handling and installation often inflict bearings with gouges or nicks. Grossgrinding operation can also leave nicks on bearing raceways. These surface defects serve as stress raisersand are more likely to cause surface-originated pitting or spalling under cyclic loads.

Debris bruises. Debris bruises are surface indentations caused by foreign particles that are entrained intothe contacts between the rolling elements and bearing raceways during operation. Unless the surfacesare severely indented, the proceeding failure modes, mostly pitting or spalling, are the ultimate modesof failure.

Fracture. On rare occasions, bearings suffer from component fracture, a sudden failure caused primarilyby high stresses that exceed the material’s ultimate strength limits. Fracture occurs in bearings withimproper tight fit and bearings with high internal tensile stresses that resulted from improper heattreatment.

Cage damage or breakage. Cage damage and cage breakage are occasionally seen in bearing applications.Cage damage is usually a result of mishandling or improper installation. Cage breakage is primarilyassociated with torsional vibration or excessive acceleration and deceleration. Cage breakage can alsoresult from excessive roller skewing due to misalignment or loss of internal geometry.

28.12.3 Mechanical Wear

Mechanical wear is a gradual surface deterioration by relative motion. Compared to sliding bearings,mechanical wear in rolling element bearings is far less severe. This is particularly true for cylindrical andtapered roller bearings whose surfaces are subjected primarily to pure rolling contact. There are funda-mentally three mechanical wear mechanisms existing in rolling element bearings: adhesive wear, abrasivewear, and fretting wear.

FIGURE 28.16 Brinelling marks on the inner raceway of a tapered roller bearing. (Courtesy of the Timken Company.)

Adhesive wear. Adhesive wear is characterized by material transfer from one surface to the mating surface.It is often found on roller-ends and the corresponding guide face of the flanges, as well as at the interfacesbetween shafts and inner rings and between housings and outer rings. Adhesive wear can also occur atcontacts between the rolling elements and raceways when substantial sliding exists. Adhesive wear appearsin various forms, depending on its severity.

1. Scoring. Scoring occurs when lubricant film thickness is inadequate to separate the contact surfaces.Asperity-to-asperity contact occurs. Under high normal and tangential stresses, heat generationat the asperity becomes excessive. It gives rise to strong adhesive or welded junctions at isolatedspots. With relative motion between the contacting asperities, junctions are torn apart, resultingin noticeable material transfer from surface to surface.

2. Scuffing. Scuffing is progressive scoring that occurs on a greater scale when the more weldedjunctions appear and the isolated scoring spots are joined together (Figure 28.17). Scuffing isidentified by severe surface roughening, and is accompanied by perceptibly high friction andtemperature.

3. Smearing. Smearing is sometimes called galling. It is considered scoring on a grand scale. It involvessevere plastic deformation and massive material transfer from surface to surface. Although thefinal smearing mode may represent the gross breakdown of various surface films and near surfaceregion, it may be triggered by the deterioration in surface topography as a result of progressivescoring or scuffing.

4. Seizing. Seizing is the final stage of adhesive wear. Welding between contacting asperities is sosevere that it seizes the bearing, resulting in catastrophic failure. Seizing is, however, rarely seenin rolling element bearings.

Abrasive wear. Abrasive wear occurs when asperity-scale plastic deformation leads to material removaland wear debris. It involves abrasive foreign particles harder than bearing materials, often nonmetallic.Abrasive wear is identified by microscopic furrows and the dulling of contact surfaces (Figure 28.18).Excessive wear results in a loss of bearing dimension or internal geometry, which in turn affects bearingperformance.

Fretting. Fretting results from micro-movement between mating surfaces. Fretting mostly occurs duringtransportation or machine idling. It is recognizable by the grooves worn into the raceway by vibrationalmovement between the rolling elements and raceways (Figure 28.19a). Fretting also occurs betweenbearing rings and the surfaces that they contact (Figure 28.19b). It usually indicates an inadequate fit.Fretting fatigue is now recognized as one of the outcomes of continuous fretting.

FIGURE 28.17 (a) Scuffing at a roller end; surface showing spiral scuffing marks; (b) scuffing streaks in racewayof ball thrust bearing caused by light load and high speed. (From Walp, H.O. (1971), Interpreting Service Damage inRolling Type Bearings — A Manual on Ball & Roller Bearing Damage, ASLE, Park Ridge, IL. With permission.)

FIGURE 28.18 Advanced stage of abrasive wear and corrosion in inner raceway and rollers of a spherical rollerbearing. (From Walp, H.O. (1971), Interpreting Service Damage in Rolling Type Bearings — A Manual on Ball & RollerBearing Damage, ASLE, Park Ridge, IL. With permission.)

FIGURE 28.19 (a) Fretting grooves worn into the raceways by axial movement of the rollers during transportation.(Courtesy of the Timken Company.) (b) Fretting on the outer diameter of an outer race ring caused by nonuniformseat in housing. (From Walp, H.O. (1971), Interpreting Service Damage in Rolling Type Bearings — A Manual on Ball& Roller Bearing Damage, ASLE, Park Ridge, IL. With permission.)

28.12.4 Corrosion

Corrosion is a chemical wear. The common forms of corrosive wear seen in bearing applications areetching and hydrolysis.

Acid etching. Acid etching occurs when a bearing operates in an acid environment. Acid substances reactwith and dissolve steel surfaces to form salts. As a result, numerous irregular and dark-bottomed pits arecreated on bearing surfaces, a condition that subsequently leads to surface-originated spalling.

Hydrolysis. Bearings exposed to mist conditions are susceptible to hydrolysis, a corrosion by oxidation.Hydrolysis is identified by localized irregular pits. The affected area may appear red to dark brown,depending on the state of oxidation.

28.12.5 Electric Arc Damage

Improperly grounded electrical equipment can cause bearings to suffer from electric arcing damage bylocalized material melting and metallurgical property alteration. Apparent visual effects range fromrandom isolated pits to fluted patterns composed of numerous pits (Figure 28.20).

28.12.6 Discoloring and Overheating

Discoloring is associated with heat generation at the contact where gross sliding occurs. The heat oxidizescontact surfaces, producing oxidation colors from dark brown to blue. Excessive heat can cause all partsof a bearing to show tempering colors. Overheating causes the lubricant to decompose, diminishing itsability to lubricate. The major impact is, however, the loss of material strength to contact fatigue.

Table 28.11 lists the most common failure modes and the possible causes.

FIGURE 28.20 Electric arc damage on bearing raceways showing fluted patterns. (From Harris, T.A. (1991), RollingBearing Analysis, 3rd ed., John Wiley & Sons, New York. With permission; Widner, R.L. and Littmann, W.E. (1976),Bearing Damage Analysis, NBS 423, National Bureau of Standards, Washington, D.C.)

TABLE 28.11 Failure Modes and the Possible Causes

Possible Causes

g Overheating

Dis

colo

rin

g

Material/Surface Condition Improper heat treatmentInclusionPoor surface finishInadequate surface profile

Mounting Improper storageImproper handlingIncorrect assembling

⊗ Incorrect setting⊗ Misalignment

DeflectionImproper fit

Environment Water in lubricantDuctile foreign particlesBrittle foreign particlesDefective sealsHousing cleannessChemical contaminationElectric current

Operating Conditions Impact or shock loadHigh static overloadHeavy loadVibration or oscillations

Lubrication ⊗ Quality of lubricant⊗ Quantity of lubricant

Failure Mode

Mechanism Contact Fatigue Surface Depression and Fracture Wear Corrosion Electric Arcin

Appearance Pit

tin

g

Peel

ing

Spal

ling

Flak

ing

Cra

ckin

g

Bri

nel

ling

Nic

ks

Deb

ris

bru

ises

Frac

ture

Cag

e br

eak/

dam

age

Adh

esio

n

Abr

asio

n

Fret

tin

g

Aci

d et

chin

g

Hyd

roly

sis

⊗ ⊗⊗ ⊗

⊗ ⊗⊗ ⊗

⊗ ⊗⊗ ⊗⊗ ⊗

⊗ ⊗ ⊗ ⊗⊗ ⊗⊗ ⊗ ⊗

⊗ ⊗ ⊗ ⊗

⊗ ⊗⊗ ⊗ ⊗ ⊗⊗ ⊗ ⊗

⊗ ⊗ ⊗⊗ ⊗

⊗⊗

⊗ ⊗ ⊗⊗

⊗ ⊗ ⊗⊗ ⊗

⊗ ⊗ ⊗ ⊗ ⊗ ⊗⊗ ⊗ ⊗ ⊗ ⊗ ⊗

References

Ai, Xiaolan and Sawamiphakdi, K. (1999), Solving elastic contact between rough surfaces as an unconstrainedstrain energy minimization by using CGM and FFT techniques, ASME J. Tribol., 121, 639-647.

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