Piet M. Lugt

Tribology Series

Grease Lubrication in Rolling Bearings

GREASE LUBRICATIONIN ROLLING BEARINGS

Tribology Series

Bhushan Introduction to Tribology, 2nd Edition March 2013Bhushan Principles and Applications to Tribology, 2nd

EditionMarch 2013

Lugt Grease Lubrication in Rolling Bearings January 2013Honary and Richter Biobased Lubricants and Greases: Technology

and ProductsApril 2011

Martin and Ohmae Nanolubricants April 2008Khonsari andBooser

Applied Tribology: Bearing Design andLubrication, 2nd Edition

April 2008

Stachowiak (ed) Wear: Materials, Mechanisms and Practice November 2005Lansdown Lubrication and Lubricant Selection: A

Practical Guide, 3rd EditionNovember 2003

Cartier Handbook of Surface Treatment and Coatings May 2003Sherrington, Roweand Wood (eds)

Total Tribology: Towards an IntegratedApproach

December 2002

Kragelsky and Tribology: Lubrication, Friction and Wear April 2001Stolarski and Tobe Rolling Contacts December 2000Neale and Gee Guide to Wear Problems and Testing for

IndustryOctober 2000

GREASE LUBRICATIONIN ROLLING BEARINGS

Piet M. LugtSKF, The Netherlands

A John Wiley & Sons, Ltd., Publication

This edition first published 2013C© 2013 John Wiley & Sons, Ltd

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Library of Congress Cataloging-in-Publication Data

Lugt, Piet M., author.Grease lubrication in rolling bearings / Piet M. Lugt.

pages cm. – (Tribology in practice series)Includes bibliographical references and index.

ISBN 978-1-118-35391-2 (hardback) – ISBN 978-1-118-48396-1 (obook) – ISBN 978-1-118-48397-8 (epub)1. Roller bearings–Lubrication. 2. Lubrication and lubricants. I. Title.

TJ1071.L78 2013621.8′9–dc23

2012031584

Cover photograph courtesy of SKF Maintenance Products

A catalogue record for this book is available from the Library of Congress.

ISBN: 978-1-118-35391-2

Typeset in 10/12pt Times by Aptara Inc., New Delhi, India

Dedicated to:Marjo, Michiel and Marijn

Contents

Preface xvii

Series Preface xix

List of Abbreviations xxi

1 Introduction 11.1 Why Lubricate Rolling Bearings? 11.2 History of Grease Lubrication 21.3 Grease Versus Oil Lubrication 3

2 Lubrication Mechanisms 52.1 Introduction 52.2 Definition of Grease 62.3 Operating Conditions 62.4 The Phases in Grease Lubrication 72.5 Film Thickness During the Bleeding Phase 8

2.5.1 Ball Bearings 82.5.2 Roller Bearings 10

2.6 Feed and Loss Mechanisms During the Bleeding Phase 102.7 Film Thickness and Starvation (Side Flow) 112.8 Track Replenishment 122.9 Grease Flow 13

2.9.1 Non-Newtonian Rheology 142.10 Wall-Slip 152.11 Oxidation 162.12 EP Additives 162.13 Dynamic Behaviour 172.14 Grease Life 17

2.14.1 Temperature 182.14.2 Speed 192.14.3 Load 192.14.4 Bearing Type 202.14.5 Grease Type 202.14.6 Environment 21

viii Contents

3 Grease Composition and Properties 233.1 Base Oil 24

3.1.1 Natural Triglyceride and Wax Ester Base Oils 263.1.2 Mineral Oils 263.1.3 Synthetic Oils 30

3.2 Base Oil Viscosity and Density 413.2.1 Viscosity–Temperature 443.2.2 Viscosity–Pressure–Temperature 453.2.3 Density, Compressibility 47

3.3 Thickener 493.3.1 Soap Greases, Simple Greases 503.3.2 Complex Greases 513.3.3 Non-soap Thickeners 523.3.4 Mixed Thickeners 523.3.5 Mechanical Structure 533.3.6 Oil Retention 563.3.7 Properties of Different Types of Grease Thickeners 56

3.4 Additives 613.4.1 Corrosion Inhibitors 623.4.2 Antioxidants 623.4.3 EP/AW Additives 63

3.5 Solid Fillers/Dry Lubricants 663.5.1 MoS2 and Graphite 663.5.2 Nanoparticles 663.5.3 ZnO 663.5.4 Teflon (polytetrafluoroethylene) 663.5.5 Polyethylene 66

3.6 Compatibility 673.7 Polymer Grease 67

4 Grease Life in Rolling Bearings 714.1 Introduction 714.2 Relubrication Intervals and Grease Life 714.3 The Traffic Light Concept 72

4.3.1 Low Temperatures 744.3.2 Extreme Low Temperature 754.3.3 Extreme High Temperature 75

4.4 Grease Life as a Function of Temperature in the Green Zone 754.5 SKF Relubrication and Grease Life 764.6 Comparison Grease Life/Relubrication Models 784.7 Very Low and High Speeds 82

4.7.1 Speed Ratings and Speed Factors 824.7.2 High Speed 824.7.3 Very Low Speeds 85

4.8 Large Rolling Bearings 854.9 Effect of Load 86

Contents ix

4.9.1 Varying Load 864.9.2 Direction of Load 894.9.3 Very Heavy Loads 89

4.10 Effect of Outer-Ring Rotation 904.11 Cage Material 904.12 Bearing Type 91

4.12.1 Roller Bearings 914.12.2 Hybrid Bearings 91

4.13 Temperature and Bearing Material 924.14 Grease Fill 944.15 Vertical Shaft 954.16 Vibrations and Shock Loads 964.17 Grease Shelf Life/Storage Life 97

5 Lubricating Grease Rheology 995.1 Visco-Elastic Behaviour 995.2 Viscometers 102

5.2.1 Parallel Plate and Cone-Plate Viscometers 1035.2.2 Errors in Rheometry Measurements 1035.2.3 Errors in Thin Film Parallel Plate Rheometry Measurements 105

5.3 Oscillatory Shear 1085.3.1 Theory 1085.3.2 Application to Grease 1105.3.3 Effect of Thickener Concentration 112

5.4 Shear Thinning and Yield 1125.4.1 Grease 1125.4.2 Lubricating Oil 116

5.5 Yield Stress 1185.5.1 The Concept 1185.5.2 Influence of Temperature 1195.5.3 Consistency 120

5.6 Wall-Slip Effects 1225.7 Translation Between Oscillatory Shear and Linear Shear Measurements 125

5.7.1 Viscosity 1255.7.2 Yield Stress 126

5.8 Normal stresses 1265.9 Time Dependent Viscosity and Thixotropy 1285.10 Tackiness 133

5.10.1 Introduction 1335.10.2 Tackifiers 1345.10.3 Pull-Off Test 1355.10.4 Other Tests 136

6 Grease and Base Oil Flow 1376.1 Grease Flow in Pipes 137

6.1.1 Approximation Using the Newtonian Pipe Flow Equations 137

x Contents

6.1.2 Non-Newtonian Fluid 1386.1.3 Bingham Rheology 1396.1.4 Sisko Rheology 1406.1.5 Power Law Rheology 1406.1.6 Herschel–Bulkley Rheology 1406.1.7 The Darcy Friction Factor 1426.1.8 Transient Effects 1446.1.9 Air in Grease 1446.1.10 Entrance Length 1456.1.11 Solid Particles in Grease Flow 1456.1.12 Wall-Slip/Slip Layer 1456.1.13 Impact of Roughness 1476.1.14 Grease Aging in Pipes 149

6.2 Grease Flow in Rolling Bearings 1496.2.1 Churning 1496.2.2 Flow Through Bearing Seals 1526.2.3 Relubrication 1526.2.4 Grease Flow Around Discontinuities 1536.2.5 Creep Flow 1536.2.6 Flow Induced by Vibrations 155

7 Grease Bleeding 1577.1 Introduction 1577.2 Ball Versus Roller Bearings 1587.3 Grease Bleeding Measurement Techniques 1587.4 Bleeding from the Covers and Under the Cage 1597.5 A Grease Bleeding Model for Pressurized Grease by Centrifugal Forces 161

7.5.1 Oil Bleeding Model 1627.5.2 Quality of the Model 166

8 Grease Aging 1718.1 Mechanical Aging 172

8.1.1 Softening of Grease in Rolling Bearings 1728.1.2 Hardening of Grease in Rolling Bearings 179

8.2 Grease Oxidation 1798.3 The Chemistry of Base Oil Film Oxidation 181

8.3.1 Chemical Reactions 1818.4 Oxidation of the Thickener 1838.5 A Simple Model for Base Oil Degradation 1858.6 Polymerization 1868.7 Evaporation 1868.8 Simple Models for the Life of Base Oil 187

8.8.1 Booser’s Oil Life Model 1878.8.2 Two Phase Model 188

Contents xi

9 Film Thickness Theory for Single Contacts 1919.1 Elasto-Hydrodynamic Lubrication 192

9.1.1 History 1929.1.2 The Navier–Stokes Equations 1939.1.3 The Reynolds and Thin Film Equation 1949.1.4 Cavitation 198

9.2 Contact Geometry and Deformation 1989.2.1 Rigid Bodies 1999.2.2 Elastic Deformation 200

9.3 EHL Film Thickness, Oil 2029.3.1 Example: 6204 Bearing 205

9.4 EHD Film Thickness, Grease 2059.4.1 Measurements 2059.4.2 Film Thickness Models for Grease Rheology 207

9.5 Starvation 2129.5.1 Starved Oil Lubricated Contacts 2129.5.2 Starved Lubrication EHL Models 2139.5.3 Base Oil Replenishment 2199.5.4 Starved Grease Lubricated Contacts 222

9.6 Spin 225

10 Film Thickness in Grease Lubricated Rolling Bearings 22710.1 Thin Layer Flow on Bearing Surfaces 228

10.1.1 Contact Replenishment in Bearings 22810.1.2 Thin Layer Flow Induced by Centrifugal Forces 23110.1.3 Combining the Thin Layer Flow on the All Bearing Components 233

10.2 Starved EHL for Rolling Bearings 23410.2.1 Central Film Thickness 23410.2.2 Combining Lightly Starved and Severely Starved 237

10.3 Cage Clearance and Film Thickness 23910.4 Full Bearing Film Thickness 241

11 Grease Dynamics 24511.1 Introduction 24511.2 Grease Reservoir Formation 24511.3 Temperature Behaviour 24611.4 Temperature and Film Breakdown 24911.5 Chaotic Behaviour 249

11.5.1 Reconstruction of the Temperature Dynamics Using Time DelayedEmbedding 249

11.5.2 Estimation of the Time Delay τ 25111.5.3 Calculation of the Dimensions d and m 25111.5.4 Calculation of the Lyapunov Exponents 252

11.6 Quantitative Analysis of Grease Tests 25311.7 Discussion 254

xii Contents

12 Reliability 25712.1 Failure Distribution 25812.2 Mean Life and Time Between Failures 26112.3 Percentile Life 26412.4 Point and Interval Estimates 265

12.4.1 Graphical Methods for Point Estimates 26512.4.2 Suspended Tests, Censored Data 26712.4.3 Weibull Parameters η and β: Maximum Likelihood Method 26912.4.4 Bias of Point Estimates 27212.4.5 Confidence Intervals for β 27312.4.6 Confidence Intervals and Unbiased Point Estimates

for Life Percentiles 27312.4.7 Estimate Precision 274

12.5 Sudden Death Testing 27512.5.1 Maximum Likelihood Method for a 3-Parameter Weibull

Distribution 28012.6 System Life Prediction 281

13 Grease Lubrication and Bearing Life 28313.1 Bearing Failure Modes 28313.2 Rated Fatigue Life of Grease Lubricated Rolling Bearings 285

13.2.1 Introduction 28513.2.2 The Lubrication Factor 28713.2.3 The Contamination Factor ηc 28813.2.4 The Stress-Life Modification Factor aslf 289

13.3 Background of the Fatigue Life Ratings of Grease Lubricated Bearings 28913.3.1 Fatigue Life and Endurance Testing in the Period 1940–1960 28913.3.2 Fatigue Life and Endurance Testing After 1960 29113.3.3 The Reliability of Grease Lubricated Bearings 292

13.4 Lubricant Chemistry and Bearing Life 29613.4.1 Anti-Wear Additives 29713.4.2 EP Additives 29713.4.3 The Influence of Lubricant Additives on Bearing Life 297

13.5 Water in Grease 30413.5.1 Introduction 30413.5.2 Film Thickness 30413.5.3 Water in Oil and Bearing Life 30413.5.4 Concentration of Water 30513.5.5 Water in Grease 306

13.6 Surface Finish Aspects Related to Grease Lubrication 306

14 Grease Lubrication Mechanisms in Bearing Seals 30914.1 Introduction 30914.2 Lubrication Mechanisms for Elastomer Contact Seals 30914.3 Sealing Action of Grease 312

14.3.1 Migration of Contaminant Particles in the Pocket 313

Contents xiii

14.3.2 Migration of Contaminant Particles in the Vicinityof the Sealing Contact 316

14.4 Softening and Leakage 31914.5 Compatibility 32014.6 A Film Thickness Model for Bearing Seals 320

14.6.1 Oil Feed 32114.6.2 Oil Loss 321

14.7 Some Examples Showing the Importance of Sealing and Grease 324

15 Condition Monitoring and Maintenance 32715.1 Condition Monitoring 32715.2 Vibrations and Acoustic Emission 32815.3 Lubcheck 33115.4 Consistency Measurement 33115.5 Oil Bleeding Properties 33215.6 Oil Content 33215.7 Particle Contamination 33215.8 Spectroscopy 333

15.8.1 Infrared (IR) Spectroscopy 33315.9 Linear Voltammetry 33415.10 Total Acid Number 33515.11 DSC – Differential Scanning Calorimetry 33515.12 Oxidation Bomb 33615.13 Water 336

16 Grease Qualification Testing 33916.1 Introduction 33916.2 Standard Test Methods 339

16.2.1 Penetration/Grease Consistency 33916.2.2 Worked Penetration 34116.2.3 Shell Roll Stability 34116.2.4 Dropping Point 34316.2.5 Emcor 34416.2.6 Oil Separation 34616.2.7 Water Resistance 34716.2.8 Low Temperature Torque 34816.2.9 Flow Pressure 34916.2.10 4-Ball Weld Load 34916.2.11 4-Ball Wear Scar 35016.2.12 High Speed Grease Life Testing, RHF1 35116.2.13 R0F 35316.2.14 R0F+ 35416.2.15 R2F, Using the Special Spherical Roller Bearing 35616.2.16 R2F, Using Standard Bearings 35716.2.17 V2F 35816.2.18 FE8 359

xiv Contents

16.2.19 FE9 36016.2.20 A-Frame Cycle Test 36016.2.21 Cold Chamber Test 36116.2.22 BeQuiet+ 36216.2.23 Fafnir Friction Oxidation Test 36416.2.24 Copper Corrosion Test 36516.2.25 EP Reaction Test 36616.2.26 Compatibility with Preservatives/Process Fluids 36716.2.27 Compatibility Tests for Polymeric Materials 36716.2.28 Remaining Oil Percentage, or Thickener/Oil Ratio 36816.2.29 ROF/ROF+ 36916.2.30 R2F and FE8 Comparison 37016.2.31 ASTM D 3527 Life Performance of Wheel Bearing Grease 37316.2.32 ASTM D 5483 Oxidation Induction Time of Lubricating Greases by

Pressure Differential Scanning Calometry 37316.2.33 Linear Sweep Voltammmetry 373

16.3 Some Qualification Criteria for Grease Selection 37316.3.1 Low Temperature Limit 37316.3.2 Low Temperature Performance Limit 37416.3.3 High Temperature Performance Limit 37416.3.4 High Temperature Limit 37416.3.5 Minimum Speed 37516.3.6 Maximum Speed 375

16.4 Pumpability 375

17 Lubrication Systems 37717.1 Single Point Lubrication Methods 37917.2 Centralized Grease Lubrication Systems 38017.3 Pumps 382

17.3.1 Shovel Pump for Pumping High Viscous Grease 38217.3.2 Method to Create a Positive Head Pressure by Using a

Follower Plate 38417.4 Valves 38417.5 Distributors 38617.6 Single-Line Centralized Lubrication Systems 386

17.6.1 Single-Line System and Venting 38717.6.2 Prelubrication Distributors 38717.6.3 Relubrication Distributors 39017.6.4 Strengths and Weaknesses of Single-Line Systems 392

17.7 Dual-Line Lubrication Systems 39317.7.1 Description 39317.7.2 Strengths and Weaknesses of the Dual-Line System 394

17.8 Progressive Lubrication Systems 39417.8.1 Description 39417.8.2 Strengths and Weaknesses of Progressive Systems 397

Contents xv

17.9 Multi-Line Lubrication System 39717.10 Cyclic Grease Flow 39717.11 Requirements of the Grease 398

17.11.1 Grease Pumpability 39817.11.2 Venting Pressure for Single-Line Systems 39917.11.3 Oil Separation/Bleeding 40017.11.4 Cleanliness 40017.11.5 Compressibility 40117.11.6 Homogeneity 40117.11.7 Additives 40117.11.8 Compatibility 40217.11.9 Delivery Resistance or Pressure Losses 402

17.12 Grease Pumpability Tests 40217.12.1 Flow Ability 40317.12.2 Delivery Test 408

A Characteristics of Paraffinic Hydrocarbons 413

References 415

Index 439

Preface

Technology development and bearing development have gone hand-in-hand. There are morethan 50 billion bearings operating in the world at any time. They are the most widespreadmachine element after nuts and bolts [412]. The continuous increase in performance is placingvery high demands on bearings in many applications. The load carrying capacity of bearingshas increased enormously over the years and energy losses have been reduced. In practice thismeans that for the same type and size of bearing, the service life has become much longerand the frictional torque has been reduced. Long service life and low friction in bearings canonly be obtained by proper lubrication, that is, by having a lubricating film separating therolling elements from the rings such that roughness interaction is prevented. In the case of oillubrication, the films can easily be calculated using classic Elasto-Hydrodynamic Lubrication(EHL) models. In the case of grease lubrication, this is much more difficult. Several aspectsplay a role here, such as oil bleeding, oil flow and starvation. But mechanical and thermalaging aspects of the grease or its components also have an influence on the ability to form alubricating film.

The challenge in grease research is primarily three-fold. The first challenge is to developgreases that will provide longer life and/or are able to operate under more severe conditions(extreme low and high temperature and speed). The second challenge is the developmentof predictive tools, such as numerical models or expert systems. The third challenge is todesign bearing-systems that will increase grease life by, for example, optimizing the greaseflow. All these aspects require a fundamental understanding of the lubrication mechanisms oflubricating greases.

The bearing industry has a particular interest in understanding grease lubrication. More than90% of all rolling element bearings are greased and sealed for life, effectively making greasea bearing component, similar to rolling elements and seals. In addition, the internal design ofthe bearing has an impact on the performance of the grease. This book gives an overview ofthe existing knowledge on the various aspects of grease lubrication and the state of the artmodels that exist in the public literature today.

In other words, this book reviews the physical and chemical aspects of grease lubrication,primarily directed towards lubrication of rolling bearings. It is intended for researchers andengineers in the petrochemical and bearing industries. It may also be of interest for teachingin postgraduate courses.

I have used material and information from various experts in the field of grease lubrication,rolling bearings, seals and lubrication systems. The following persons contributed to much ofthe material in the various chapters: Dave M. Pallister, Chapter 3, Grease composition and

xviii Preface

Chapter 8, Grease Aging; Pieter Baart, Chapter 7, Grease bleeding and Chapter 14, Sealing;Marco T. van Zoelen and Cornelis (Kees) H. Venner, Chapters 9 and 10, Film thickness; JohnH. Tripp and Slavco Velickov, Chapter 11, Grease dynamics; Antonio Gabelli, Chapter 13,Bearing Life; Raimund Stockhammer and Paul Conley, Chapter 17, Lubrication systems.

John H. Tripp is the main author of Chapter 12, Reliability. Much of the text from Chapter16 originates from documents from Ben Huiskamp.

I utilized various experts to review parts of this book: Bas v.d. Vorst (rheology), SebastienBlachere (reliability), Rihard Pasaribu (grease aging), John Tripp (grease flow), Pieter Baart(rheology), Brian Murray and Alan Thomson (Condition Monitoring and Maintenance) andDick Meijer (grease composition). Marylou Rood created many of the figures and WalterVerhaert edited the full document.

Many thanks to the people of the SKF reference group: Alejandro Sanz, Hakan Lindgren,Domenico Bosco, Frank Berens, Frank Fiddelaers, Victoria van Camp, Gerwin Preisinger, Fer-dinant Schweitzer, Filip Rosengren, Goran Lindsten, Cornelia Haag, Jurgen Kreutzkaemper,Risto Kuukkanen, Rihard Pasaribu and Steve Lane for their critical review of the documentand constructive comments.

I would like to express my sincere thanks to Alejadro Sanz for originating this project andfor his continuous support throughout the writing process.

I hereby acknowledge Alexander de Vries, Alan Begg, Edward Holweg and Eva Karlssonfor their permission to commence this work and Alexander de Vries for his approval of thefinal document.

Piet M. LugtSKF Engineering & Research Centre, The Netherlands

Series Preface

There are more than 20 billion grease lubricated rolling bearings working in various mechanicaldevices across the world. Experience shows that about 80% of premature bearing failures aredue to lubrication problems. This is a long-awaited book addressing the important topic ofrolling contact bearing lubrication by greases.

The book opens with a discussion on grease lubrication mechanisms and then followsby describing grease composition and properties, grease life in rolling bearings, rheologicalproperties, flow characteristics and grease ageing. The text then proceeds to calculations ofgrease film thickness in elastohydrodynamic contacts, beginning with the theory and endingwith the temperature effects on grease dynamics. The next section explaining the theory ofreliability is followed by a description of the effects of grease lubrication on bearing life. Greaselubricated seals are also discussed in a separate chapter. The book finishes with chapters oncondition monitoring, grease testing standards and grease lubrication systems. The interestedreader will be able to find all information relevant to greases and grease lubricated rollingbearings in this book.

The strength of this book is its comprehensiveness. The fundamentals of grease propertiesand the lubrication of rolling bearings are illustrated through practical applications, with anemphasis on bearing life and reliability. The topic has been thoroughly researched by theauthors and all the relevant areas are meticulously covered. The material is presented in aneasily accessible manner.

Based on the contents and the level of detail, this book can be recommended for advancedundergraduate and postgraduate courses in the subject areas of tribology, machine design,reliability and maintenance. Practicing engineers and designers will also find the book veryuseful as a reference. The book is a valuable addition to Wiley’s Tribology Book Series.

Gwidon StachowiakUniversity of Western Australia

List of Abbreviations

a = Acceleration [m·s−2]a = Constant in the Walther equation Chapter 3 [cSt]a = Radius of spherical particle Chapter 14 [kg m−3]ax ,ay = Half Hertzian contact width [m]a+,a− = Location of the boundary of the pressurized

regionChapter 9 [m]

asl f = Stress-life factor Chapter 13 [-]A = Speed factor A = b f × n × dm Chapter 4 [rev· mm·min−1]A = Surface area [m2]AW = Anti-Wear (additive) Chapter 13 [-]b f = Bearing factor Chapter 4 [-]bbrg = Bearing factor Chapter 13 [-]b = Soap fibre diameter Chapter 7 [m]b1,2 = Lubrication factor constants Chapter 13 [m]b = Lubricant film width in seal contact Chapter 14 [m]B = Bearing width [m]

Bin∗ = Bingham number Bin∗ = τy

K

(D

2uav

)nChapter 6 [-]

c = Mutual approach of two spherical bodies incontact

Chapter 9 [m]

c = Stress-life exponent of the rolling contact Chapter 13 [-]c = Constant in the Walther equation Chapter 3 [log10 log10

cSt/log10 K ]c1,2 = Contamination factor constants Chapter 13 [-]C = Correlation function Chapter 11 [-]C = Dynamic capacity of a rolling bearing Chapter 13 [N]C = Concentration Chapter 13 [%]Cs = Concentration at the surface Chapter 13 [%]CEY = Computerized Evaluation of Yield Chapter 5 [Pa]d = Bearing bore diameter [m]d = Dimension of an attractor Chapter 11 [-]dc = Correlation dimension Chapter 11 [-]dd = Drop diameter in a wetting test Chapter 13 [m]de = Elastic deformation Chapter 13 [m]dr = Roller diameter Chapter 10 [m]drr = Distance between two rollers Chapter 10 [m]dm = Pitch diameter Chapter 10 [m]D = Bearing outer diameter or pipe diameter Chapter 6 [m]

xxii List of Abbreviations

D = Diffusion coefficient Chapter 13 [m2s−1]D = Deborah number Chapter 5 [-]e1 = Yielding energy density Chapter 5 [Pa]E = Young’s modulus [Pa]Ec = Complete elliptic integral Chapter 13 [-]EHL = Elasto-Hydrodynamic Lubrication Chapter 9 [-]

E ′ = Reduced elastic modulus 2E ′ = 1−ν2

1E1

+ 1−ν22

E2Chapter 9 [Pa]

E = Activation energy Chapter 3 [J·mol−1]EP = Extreme Pressure (additive) Chapter 13 [-]f = Specific body force Chapter 9 [N·m−3]f = Fibre volume fraction Chapter 7 [-]f = Darcy friction factor 64/Re Chapter 6 [-]f = Probability density function Chapter 12 [s−1]f0 = Initial fibre-volume fraction Chapter 7 [-]fmax = Maximum fibre-volume fraction Chapter 7 [-]fm0 = Initial fibre volume fraction Chapter 7 [-]fs = fs = ρω2r ∂r

∂s Chapter 9 [N·m−3]F = Load Chapter 9 [N]F = Cumulative distribution function Chapter 12 [-]Fa = Axial load [N]Fr = Radial load [N]Fc,r = Force on particle in radial direction Chapter 14 [N]Fc = Elliptic integral Chapter 13 [-]Fd,r = Drag force on particle in radial direction Chapter 14 [N]Fbody = Body force Chapter 7 [N]Ff riction = Friction force Chapter 7 [N]Flip = Seal lip force Chapter 14 [N]g = Gravitational acceleration Chapter 7 [m·s−2]G = Shear modulus Chapter 5 [Pa]G ′ = Storage modulus Chapter 5 [Pa]G = Duty parameter Chapter 14 [-]G ′′ = Loss modulus Chapter 5 [Pa]G∗ = Complex modulus G∗ = G ′ + iG ′′ Chapter 5 [Pa]G = Bearing mass Chapter 4 [kg]G = Material parameter G = αE ′ Chapter 9 [-]Ga = Factor for pressure drop in pipe with Sisko model Chapter 6 [-]G p = Grease quantity for relubrication Chapter 4 [g]h = Film thickness Chapter 9 [m]h = Gap height Chapter 5 [m]h = Hazard function (=p) Chapter 12 [s−1]k = Permeability Chapter 7 [m2]hcs,0 = Initial starved film thickness Chapter 9 [m]hc = Central film thickness Chapter 9 [m]hm = Minimum film thickness Chapter 9 [m]h = Planck’s constant (6.63×10−34) Chapter 3 [J·s]hd = Drop height in a wetting test Chapter 4 [m]h00 = Parameter in film thickness equation Chapter 9 [m]h = Film thickness Chapter 9 [m]h = Free surface layer thickness Chapter 9 [m]

List of Abbreviations xxiii

h∞ = Average (over the length) layer thickness Chapter 9 [m]h0,∞ = Initial layer thickness at the centerline Chapter 10 [m]hc,0 = Initial central film thickness Chapter 10 [m]hi = Initial layer thickness Chapter 9 [m]hcff = Central fully flooded film thickness Chapter 9 [m]hcs = Central, starved film thickness Chapter 9 [m]hEHL = Hydrodynamic film thickness Chapter 9 [m]hD = Central starved film thickness according to

DamiensChapter 10 [m]

h R = Residual layer film thickness Chapter 9 [m]hT = Total film thickness Chapter 9 [m]hZ = Central starved film thickness according to

Van ZoelenChapter 10 [m]

H = Shannon entropy Chapter 11 [-]H = Cumulative Hazard function Chapter 12 [-]k = Boltzmann’s constant k = 1.38 × 10−23J/K or

k = 8.62 × 10−5eV/Kk = Permeability Chapter 7 [m2]k = Reaction rate coefficient. Unit depends on

reaction orderk f = Bearing grease life factor (GfT) Chapter 4 [-]K = Load correction factor in the presence of water Chapter 13 [-]K = Grease consistency index τ = τy + K γ n Chapter 5 [Pa·sn]K = Plastic viscosity τ = τy + K γ Chapter 5 [Pa·s]K0 = Grease consistency index at ambient pressure Chapter 9 [Pa·sn]K = Constant in the Walther equation Chapter 3 [log10 log10 cSt]K = Grease consistency index

τ = τy + K γ n + ηbγ

Chapter 5 [Pa·sn]

K ′ = Grease consistency indexτ = [

τ ny + (K ′γ )n]1/n

Chapter 5 [Pa·sn]

lt = Total length of the track Chapter 9 [m]L p = Percentile life for bearing life Chapter 12 [MRevs]L p = Percentile life for grease life Chapter 12 [hour]L pq = Percentile life p at confidence limit percentile

qChapter 12 [hour]

L = Mean life Chapter 12 [hour] or [MRev]L p = Reference life at p ◦C Chapter 4 [hour] or [MRev]L = Dimensionless material parameter

L = αE ′(

E ′ Rxη0us

)− 14

Chapter 9 [-]

L = Length used in various contexts [m]Lentr = Entrance length Chapter 6 [m]L p = Maximum likelihood estimate for L p Chapter 12 [hour] or [MRev]L ′

p = Mean unbiased estimate of L p Chapter 12 [hour] or [MRev]L ′′

p = Median unbiased estimate of L p Chapter 12 [hour] or [MRev]L p = Estimate of L p with sudden death testing Chapter 12 [hour] or [MRev]L = Likelihood Chapter 12 [s−1] or [-]m = Mass [kg]m = Phase space dimension Chapter11 [-]m = Shear thinning parameter m = 1/n Chapter 6 [-]

xxiv List of Abbreviations

m = Normal stress parameter Chapter 5 [-]M = Dimensionless load number

M = FE ′ R2

x

(E ′ Rxη0us

) 34

Chapter 9 [-]

M = Torque Chapter 5 [N·m]M = Molecular weight Chapter 3 [-]Moil = Mass of oil Chapter 8 [kg]N = Avogadro’s number (6.02×1023) Chapter 3 [mol−1]NN = Non-Newtonian Chapter 5 [-]N1 = Normal stress difference Chapter 5 [Pa]N2 = Normal stress difference Chapter 5 [Pa]n = Shear thinning parameter

τ = τy + K γ n + ηbγ

Chapter 5 [-]

n = Number of overrollings Chapter 13 [-]n = Rotational speed [rev·min−1]n = Total number of measurements point over a

length LChapter 13 [-]

n = Number of bearings Chapter 12 [-]n0 = Population of test bearings Chapter 12 [-]nc = Number of contacts in a bearings Chapter 10 [-]nmax = Limiting speed with grease lubrication Chapter 4 [rev·min−1]nopt = Speed where droplets detach from inner-ring

surfaceChapter 4 [rev·min−1]

n dm = Speed number n × dm Chapter 4 [mm·min−1]p = Exponent in the life equation Chapter 13 [-]p = Pressure [Pa]p = Instantaneous failure probability rate Chapter 13 [s−1]pbody = External body force per unit volume Chapter 7 [N·m−3]p f riction = Friction force per unit volume Chapter 7 [N·m−3]�p = Pressure difference Chapter 6 [Pa]ph = Maximum Hertzian pressure Chapter 9 [Pa]pr = Constant in Roelands equation

pr = 1.962 · 108Chapter 3 [Pa]

pN N = Pressure in a non-Newtonian fluid Chapter 6 [Pa]P = Equivalent load Chapter 13 [N]

Pe = Peclet number Pe = 6πηa3 γ

kT Chapter 5 [-]Pen = Penetration (ISO 2137 test) Chapter 5 [1/10 mm]Pu = Fatigue load limit Chapter 13 [N]qy = Mass flow in y-direction integrated over the

trackChapter 9 [kg·s−1]

q = Fluid velocity Chapter 7 [m·s−1]qx , qy = Volume flow per unit length Chapter 9 [m2·s−1]q = Specific mass flow (mass flow per unit of

length)Chapter 9 [kg·s−1·m−1]

q = Fluid velocity Chapter 7 [m s−1]q = Integrated mass flow flux to the side of the

trackChapter 9 [kg·s−1]

q = Pivotal function for the life percentileconf.interval estmn.

Chapter 12 [-]

Q = Flow rate Chapter 5 [m3· s−1]

List of Abbreviations xxv

r = Ratio combined layer and uncompressed fullyflooded film

Chapter 9 [-]

rev = Revolution [-]r = Radius [m]r = Fibre radius Chapter 7 [m]r = Number of failures Chapter 12 [-]R = Larger radius [m]R = Ideal gas constant (8.31) [J ·mol−1·K−1]R = Reliability Chapter 12 [-]R = Precision ratio for β: R = v0.95(r,n)

v0.05(r,n) Chapter 12 [-]Roller = Change of penetration (roll stability) Chapter 8 [1/10 mm]

Re = Reynolds number Re = ρu D

ηChapter 6 [-]

Reav = Reynolds number using ηw Chapter 6 [-]

Rq = Roughness parameter Rq =√

1

L

∫ L

0z2dx Chapter 13 [m]

Rsk = Roughness parameter Rsk = 1

n R3q

i=n∑i=1

z3i Chapter 13 [-]

Ro = Outer radius, see Figure 7.2 [m]Rc = Radial position of the seal contact Chapter 14 [m]s = Coordinate on axisymmetric surface Chapter 10 [m]S(t) = Probability that a bearing survives a time t Chapter 12 [-]t = Time [s]ttr = Transition time Chapter 10 [s]tc = Characteristic time tc = η/G Chapter 5 [s]T0 = Temperature at which η0 has been measured Chapter 3 [K]T = Temperature [K]Tc = Temperature at the centre of the EHL film Chapter 9 [K]T (◦C) = Temperature with unit Celsius Chapter 3 [◦C]Tg = Glass transition temperature Chapter 3 [K]u = Velocity [m·s−1]u = Pivotal function for the life percentile

conf.interval estmn.Chapter 12 [-]

um = Mean velocity (um = (u1 + u2)/2) Chapter 9 [m·s−1]u p = Velocity of particle Chapter 14 [m·s−1]us = Entrainment velocity (us = u1 + u2) Chapter 9 [m·s−1]us = Slip velocity Chapter 9 [m·s−1]us = Shaft velocity Chapter 14 [m·s−1]uav = Average velocity Chapter 5 [m·s−1]U = Dimensionless number U = η0us

2E ′ R Chapter 9 [-]v = Pivotal function (normalized shape parameter) Chapter 12 [-]V = Volume Chapter 7 [m3]Vp = Percentage free volume in a bearing filled with

greaseChapter 4 [%]

w = Exponent relating load to stress Chapter 13 [-]w = Load per unit width Chapter 9 [N·m−1]W = Dimensionless load number W = F

E ′ R2 Chapter 9 [-]W = Dimensionless wear parameter Chapter 13 [-]W = Width of grease reservoir Chapter 14 [m]

xxvi List of Abbreviations

Wi = Weissenberg number Chapter 14 [-]xcg = Effective thickness of reaction layer Chapter 13 [m]z = Viscosity–pressure coefficient Chapter 3 [-]z p = Half the thickness of the film where plug flow

occursChapter 9 [m]

z = Number of rolling elements Chapter 10 [-]Z = Load cycle number Chapter 8 [-]Z0 = Load cycle reference number Chapter 8 [-]x, y, z = Coordinates (running direction,across the

track, height)[m]

X, Y = Dimensionless co-ordinates X = xax

,Y = yay

Chapter 9 [-]

ys = Slip layer thickness Chapter 6 [m]yl = Transition from viscous flow to plug flow Chapter 6 [m]Yt = Reconstruction state vector Chapter 11 [-]Z = Load cycle number Z = 8L/(π D) Chapter 6 [-]ZDDP = Zinc Di-alkyl Di-thio Phosphate Chapter 13 [-]α = Viscosity–pressure coefficient Chapter 3 [Pa−1]α = Surface angle (sometimes also α′) [rad]

α = Relative radius of plug flow α = τy

τw

Chapter 6 [-]

β = Shape parameter in the Weibull distribution Chapter 12 [-]β = Maximum likelihood estimate for β Chapter 12 [-]βW = Value of β for a selected Weibull distribution Chapter 12 [-]β ′′ = Median unbiased estimate of β Chapter 12 [-]β ′ = Mean unbiased estimate of β Chapter 12 [-]↔β = Median value of β.

↔β= v0.50β Chapter 12 [-]

γ = Shear Chapter 5 [-]γm = Shear for Doraiswamy rule Chapter 5 [-]γR = Shear at the outer radius of the plate-plate

rheometerChapter 5 [-]

γ = Resistance to side flow parameter Chapter 9 [-]γ = Shear rate Chapter 5 [s−1]γc = Characteristic shear rate Chapter 5 [s−1]γw = Shear rate at the wall Chapter 6 [s−1]γw,N = Shear rate at the wall for a Newtonian fluid Chapter 6 [s−1]δ = Phase shift G′′

G′ = tan δ Chapter 5 [-]�q = Roughness parameter

�q =√

1L

∫ L0

(θ − θ

)2dx

Chapter 13 [rad]

ε = Roughness on pipe surface Chapter 6 [m]ζ = Lim. shear stress factor τL = τL0 + ζ p;

0.02 < ζ < 0.15Chapter 9 [-]

η = Dynamic viscosity Chapter 3 [Pa s]ηoil = Base oil viscosity Chapter 5 [Pa·s]ηb = Viscosity at γ → ∞ (usually it is assumed

ηb = ηoil )Chapter 5 [Pa s]

ηb = Lubrication penalty factor in the life equation Chapter 13 [-]ηc = Contamination penalty factor in the life

equationChapter 13 [-]

ηg = Viscosity at the glass transition temperature Chapter 3 [Pa·s]

List of Abbreviations xxvii

ηi = Viscosity at γ → 0 Chapter 5 [Pa·s]η0 = Dynamic base oil viscosity at ambient

pressure and T = T0

Chapter 3 [Pa s]

ηw = Viscosity at the wall Chapter 5 [Pa·s]η1 = Required viscosity for adequate lubrication Chapter 13 [Pa·s]η∗ = Complex viscosity Chapter 5 [Pa·s]η = Scale parameter Chapter 12 [s]η = Maximum likelihood estimate for η Chapter 12 [s]ηW = Value of η for a selected Weibull distribution Chapter 12 [s]θ = Fractional film content Chapter 9 [-]θ = Fiber tilting angle Chapter 7 [rad]θ = Slope in a roughness profile Chapter 13 [rad]κ = Ratio of contact size in running and transverse

directionChapter 9 [-]

κd = κd = 1.03(

Ry

Rx

)0.63Chapter 9 [-]

κ = Ratio of viscosity and required viscosity Chapter 13 [-]λ = Ratio radii of curvature λ = Rx/Ry Chapter 9 [-]λ = Ratio of film thickness and combined

roughnessChapter 13 [-]

λ1 = Normal stress parameter Chapter 5 [-]λ = Lyapunov exponent Chapter 11 [-]ν = Poisson’s ratio Chapter 12 [-]νcSt = Kinematic viscosity with unit cSt Chapter 3 [cSt]ρ = Density Chapter 3 [kg m−3]ρp = Density of particle Chapter 14 [kg m−3]ρg = Density of grease Chapter 14 [kg m−3]ρ = Ratio compressed and uncompressed density Chapter 9 [-]ρc = Dimensionless density ρc = ρ(ph)/ρ0 Chapter 10 [-]ρ0 = Density at ambient pressure Chapter 3 [kg m−3]σ = Stress [Pa]σ = Surface tension Chapter 9 [N·m−1]τ = Shear stress [Pa]τ = Time delay Chapter 11 [s]τ = Minimum life (L0) Chapter 12 [s][Revs]τy = Yield stress Chapter 5 [Pa]τy0 = Yield stress at T = T0 and ambient pressure Chapter 5 [Pa]τc = Characteristic time τc = η

ρω2 h2i

Chapter 10 [s]

τL = Limiting shear stress τL = τL0 + ζ p Chapter 5 [Pa]τL0 = Limiting shear stress at ambient pressure Chapter 5 [Pa]τw = Wall shear stress Chapter 5 [Pa]τy,∞ = Yield stress after severe aging Chapter 8 [Pa]τy,0 = Yield stress of fresh grease Chapter 8 [Pa]τy = Dimensionless yield stress τ = τy hc

2η0umChapter 9 [-]

υ = Correlation exponent Chapter 11 [m]ψ = Normal stress coefficient Chapter 5 [s−1]� = Bearing circumference coordinate (angle) Chapter 10 [rad]ω = Frequency Chapter 5 [s−1]ω = Parameter uav = ωumax Chapter 6 [-]

xxviii List of Abbreviations

ω = Angular speed Chapter 9 [rad · s−1]ωi = Angular speed inner raceway Chapter 9 [rad · s−1]ωo = Angular speed outer raceway Chapter 9 [rad · s−1]ωR = Angular speed rollers Chapter 9 [rad · s−1]

Subscriptsav = Average0 = Reference or startx, y, z = x,y,z directionsr = Radial directionθ = Circumferential directionN = NewtonianN N = Non-Newtonianw = Wall