Elastohydrodynamics

by Gohar, Ramsey

• ISBN13: 9781860941702
• ISBN10: 1860941702
• Edition: 2nd
• Format: Hardcover
• Copyright: 2002-02-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

Elastohydrodynamic lubrication (EHL) is a difficult topic, embracing several disciplines, which can cause many problems for engineers and scientists. This up-to-date volume explains the subject both theoretically and experimentally. Moreover, with a refreshing approach and using several novel techniques of application, it provides lucid coverage of new and important findings. Here, in one volume, are the results of much research over the last forty years. The author's clear explanation of the theory of EHL is authoritatively applied to a wide range of related topics, with physical explanations wherever possible. Many of the experimental techniques described were carried out at the Imperial College Lubrication Laboratory, where the application of interferometry (a means of measuring the EHL film thickness) was pioneered.

Table of Contents

Preface	v
Preface to the Second Edition	vii
Introduction	1	(6)
References	4	(3)
Material Properties	7	(22)
Introduction	7	(1)
The Lubricant	7	(12)
Viscosity	7	(2)
Kinematic Viscosity (v)	9	(1)
Effect of Temperature	9	(1)
Viscosity Index	10	(1)
Effect of Pressure on Viscosity	11	(4)
Density	15	(1)
Refractive Index	16	(1)
Thermal Conductivity	16	(1)
Non-Newtonian Behaviour	17	(2)
Lubricant Solidification at High Pressures	19	(1)
The Bounding Surfaces	19	(7)
The Elastic Behaviour Within the Bodies	26	(1)
Closure	26	(3)
Notation	26	(1)
References	27	(2)
Elastostatic Contacts	29	(28)
Introduction	29	(1)
Non-Conforming Contact Elastostatics	30	(1)
Hertzian Contact Theory	30	(3)
Assumptions	30	(1)
Geometry Equation	31	(1)
Elastic Displacement	32	(1)
Axisymmetric Pressure Distribution	33	(4)
Contact Between Spherical Surfaces	37	(1)
Line Contact	38	(4)
Line Contact Between Cylindrical Surfaces	42	(2)
General Case of Touching Point Contact	44	(6)
Footprints Having Only One Axis of Symmetry	50	(1)
Contact Between Regular Wavy Surfaces	50	(3)
Closure	53	(4)
Notation	54	(1)
Dimensionless variables	55	(1)
References	55	(2)
The Governing Equations of Elastohydrodynamic Lubrication	57	(26)
Introduction	57	(1)
The Reynolds Equation	57	(7)
Assumptions made in the Derivation of the Reynolds Equation	58	(1)
Equilibrium of Forces on a Lubricant Element	59	(1)
Velocity Distribution	60	(1)
Mass Continuity	61	(3)
Special Forms of the Reynolds Equation	64	(3)
The Reynolds Equation in Polar Coordinates	67	(2)
The Energy Equation	69	(3)
The Lubricant Film Geometry Equation	72	(1)
The Elastic Deflection Equation	73	(2)
The Lubricant Equations of State	75	(1)
The Load Equation	75	(1)
The Governing Dimensionless Groups	75	(8)
Notation	79	(1)
Dimensionaless variables and groups	80	(1)
References	80	(3)
Simple Concepts of Non-Conforming Contact Lubrication	83	(56)
Introduction	83	(1)
Lubrication of Rigid Line Contacts	83	(15)
Pressure Distribution	83	(5)
Load	88	(1)
Inlet Boundary Conditions	89	(1)
The Partially Flooded Inlet Boundary	89	(4)
The Starved Inlet Boundary	93	(1)
Outlet Boundary Conditions	94	(1)
The Swift--Steiber (SS) Boundary Condition	94	(2)
The Separation Exit Boundary Condition	96	(1)
Rigid Piezoviscous Line Contacts	96	(2)
Lubrication of Rigid Point Contacts	98	(6)
Pressure Distribution	99	(2)
Load	101	(1)
Rigid Piezoviscous Point Contacts	102	(1)
Load at which Surface Deformation becomes Important	103	(1)
Experimental Evidence	104	(3)
An Approximate Theory of Elastohydrodynamic Lubrication	107	(14)
Introduction	107	(1)
Grubin's Solution for Ehd Line Contacts	108	(4)
The Load-Bearing and Exit Zone in Ehd Line Contacts	112	(5)
Point Contact Approximate Solutions	117	(4)
An Approximate Theory for Rough Surfaces Elastohydrodynamic Lubrication	121	(10)
Introduction	121	(1)
Steady State Analysis	121	(5)
Transient Analysis	126	(5)
Closure	131	(8)
Notation	132	(2)
Dimensionaless variables	134	(1)
Suffixes	135	(1)
References	135	(4)
Numerical Methods for Determining Oil Film Thickness and Pressure Distribution in Elastohydrodynamic Lubrication	139	(30)
Introduction	139	(1)
Assumptions	139	(1)
Governing Equations Used for the Numerical Solutions	140	(14)
Elastic Deflection	140	(1)
Flat-Roofed Pressure Elements	141	(4)
Quadratic Surface Roofed Pressure Elements	145	(2)
Film Shape	147	(2)
Equations of State of the Lubricant	149	(1)
Density	149	(1)
Viscosity	149	(1)
Reynolds Equation	149	(3)
Finite Difference Form of the Reynolds Equation	152	(1)
Load	153	(1)
Solution Methods	154	(10)
Introduction	154	(1)
The Forward Iterative Method	155	(1)
Boundary Conditions	155	(1)
Initial Condition	155	(1)
Computation Algorithm	155	(2)
Convergence Criteria and Number of Elements Used	157	(1)
Relaxation Factors	157	(1)
Computing Time and Units Used	158	(1)
Inverse Solution Method	158	(4)
Other Solution Methods	162	(1)
The Newton--Raphson Method (NR)	162	(1)
Introduction to the Multigrid Method	163	(1)
Closure	164	(5)
Notation	164	(1)
Dimensionaless variables	165	(1)
Suffixes	166	(1)
References	166	(3)
Film Thickness and Pressure Distribution in Elastohydrodynamic Lubrication	169	(56)
Introduction	169	(1)
Results for the Lubricant Film Thickness in Ehd Line Contacts	169	(12)
Film Geometry (Infinite Line Contacts)	169	(2)
Film Geometry (Finite Line Contacts)	171	(3)
Ehd Line Contact Film Thickness Formulae	174	(1)
Formulae for Oil Film Thickness (Line Contacts)	174	(4)
Comparison of Theoretical Predictions with Experimental Results	178	(3)
Results for the Lubricant Film Thickness in Ehd Point Contacts	181	(13)
Film Geometry	181	(5)
Ehd Point Contact Film Thickness Formulae	186	(4)
Comparison of Experimental Results with Theoretical Predictions	190	(2)
Results at High Maximum Pressures	192	(1)
Some Results Using Multigrid and Multi-Level Multi-Integration Methods	193	(1)
Regimes of Lubrication in Point and Line Contacts	194	(4)
Point Contacts	194	(3)
Line Contacts	197	(1)
Lubricant Thermal and Starvation Effects in Ehl	198	(10)
Thermal Effects (Flooded Contacts)	198	(5)
Starvation Effects (Isothermal)	203	(4)
Combined Thermal and Starvation Effects	207	(1)
Ehl of Rough Surfaces	208	(10)
Partial Ehl	208	(8)
Early Research on Micro Ehl	216	(1)
Waviness Features on a Stationary Surface	216	(2)
Closure	218	(7)
Notation	219	(1)
Dimensionless variables and groups	220	(2)
Suffixes	222	(1)
Abbreviations	222	(1)
References	222	(3)
Friction and Non-Linear Lubricant Behaviour in Elastohydrodynamic Contacts	225	(48)
Introduction	225	(1)
Newtonian Treatment of Friction	226	(16)
Rigid-Isoviscous Conditions	226	(4)
Rolling Friction in Ehd Isothermal Contact	230	(5)
Friction in Ehd Sliding Contacts	235	(1)
Effect of Emperature	236	(6)
A Rheological Model for the Lubricant	242	(12)
Introduction	242	(1)
Viscoelastic Model	242	(4)
The Limiting Shear Stress	246	(1)
Disk Machine Experiments	247	(7)
Application to Practical Ehd Contacts	254	(5)
Thermal Effects	254	(4)
Regimes of Friction Maps	258	(1)
The Reynolds--Eyring Equation	259	(7)
Closure	266	(7)
Notation	267	(2)
Subscripts for the rheological model	269	(1)
Dimensionless groups	269	(2)
References	271	(2)
Stress and Fatigue in Elastohydrodynamic Contacts	273	(40)
Introduction	273	(3)
Stresses Under Ehd Contacts	276	(8)
Orthogonal Stresses	276	(2)
Failure Stress Hypotheses	278	(1)
Maximum Shear Stress Hypothesis	278	(4)
Alternating Orthogonal Shear Stress Hypothesis	282	(1)
Deformation Energy Hypothesis	283	(1)
Life Prediction	284	(19)
Lundberg and Palmgren's Model	284	(6)
Effect of Ehl on Fatigue Life	290	(1)
Complete Ehd Film	290	(2)
Partial Ehd Film	292	(3)
Localized Stress Field Below Asperities	295	(2)
Unified Life Models	297	(1)
Descriptions of the Modifying Factors	298	(2)
Effect of Ehl on Life Model	300	(3)
Effect of Debris Denting on the Fatigue Life of Rolling Element Bearings	303	(1)
Life of Gears	303	(2)
Closure	305	(1)
Appendix	305	(8)
Computation of Orthogonal Shear Stresses	305	(3)
Notation	308	(1)
Dimensionless variables	309	(1)
References	310	(3)
Experimental Methods in Elastohydrodynamics	313	(28)
Introduction	313	(1)
Measurement of Friction Force	314	(4)
Vacuum-Deposited Transducers	318	(6)
Measurement of Pressure distribution	318	(2)
Temperature Measurement	320	(1)
Measurement of Film Thickness	320	(2)
Methods of Deposition	322	(1)
Vacuum Deposition	322	(1)
Sputtering	322	(1)
The Mask	322	(1)
Direct Laser Milling of the Transducer	322	(1)
Calibration	323	(1)
The Electric Circuit	323	(1)
Measurement of the Ehd Film by Interferometry	324	(13)
Introduction	324	(1)
Two-Beam Interferometry	325	(2)
Multiple-Beam Interferometry	327	(1)
Measuring the Film Thickness	328	(2)
Further Interferometry Applications	330	(1)
Coloured Fringes	330	(1)
Light of Two Discrete Wavelengths (Duochromatic System)	330	(2)
Ultra-Thin Films	332	(3)
Ultra-Thin Ehl Films Between rough Surfaces	335	(2)
Thermography	337	(1)
Methods of Detecting the Presence of Ehd Films	338	(1)
Electrical Resistance	338	(1)
`Before and after' inspection	339	(1)
Closure	339	(2)
References	339	(2)
Some Advanced Numerical Methods for Determining the Oil Film Thickness in Elastohydrodynamic Lubrication	341	(48)
Multigrid (Multi-Level) and Multi-Level Multi-Integration Methods	341	(14)
Introduction	341	(2)
The Multigrid Method	343	(1)
The Multigrid Algorithm	344	(2)
The Non-Linear Full Approximation scheme	346	(1)
Integrid Operators	347	(2)
Cycling Strategies	349	(2)
Application of Multigrid Methods to Time-Dependent Problems	351	(1)
The Multi-Level Multi-Integration Method (MLMI)	352	(3)
Numerical Methods to Solve Ehl Problems	355	(10)
The Governing Equations	355	(1)
Discretization of the Governing Equations	356	(1)
Reynolds Equation	356	(1)
The Film Thickness Equation	357	(1)
Equations of State	358	(1)
The Load Balance Equation	358	(1)
Relaxation Schemes	358	(5)
The Effective Influence Newton Method (EIN)	363	(1)
Time-Dependent Problems	364	(1)
Application of Multigrid and Multi-Level Multi-Integration to Ehl Problems	365	(1)
Results	366	(17)
Introduction	366	(1)
A Single Depression on a Moving Surface	366	(4)
Wavy Features on a Moving Surface	370	(3)
Roughness Features on a Stationary Surface	373	(3)
Ehl Transient Problems for Smooth Surfaces	376	(7)
Closure	383	(6)
Notation	384	(1)
Multigrid and MLMI terms	385	(1)
Dimensionless variables	385	(1)
Abbreviations	386	(1)
References	386	(3)
Various Topics Related to Elastohydrodynamics	389	(50)
Introduction	389	(1)
Application of Ehl to Gear Tooth Contacts	389	(12)
Introduction	389	(3)
Geometry	392	(1)
Kinematics	393	(1)
Load	394	(1)
Criterion for Determining Validity of the Quasi-Steady Solution	394	(1)
Example	395	(3)
Temperature and Friction	398	(3)
Roller Bearings with Spherical-Ended Elements	401	(9)
Accomodation of Axial Loads	401	(3)
Estimation of Minimum Film Thickness	404	(1)
Contact Geometry	404	(2)
Ehd Film Thickness	406	(3)
Applications	409	(1)
Rebounding Lubricant Layers	410	(8)
Variable Speed Transmissions	418	(2)
Ehd Films as Vibration Dampers	420	(7)
Vibrations of Rolling Element Bearings	420	(1)
Governing Equations	421	(1)
The Integrated Pressure Under a Ball	421	(2)
The Second Governing Equation	423	(2)
The Elastic Deflection Equation	425	(1)
Vibration Response	425	(2)
Ehd Lubrication of Impacting Non-Conforming Surfaces	427	(7)
Closure	434	(1)
Appendix	434	(5)
Navier--Stokes Equations	434	(2)
Reynolds Equation	436	(1)
References	436	(3)
Future Trends in Ehl	439	(4)
Rough Surface and Partial Ehl	439	(1)
Gear Lubrication	439	(1)
Ehl Treated as Part of a System	440	(1)
The Effect of Wear Debris on the Fatigue Life of Rolling Bearings	440	(1)
Transient Ehl	440	(1)
Experimental Techniques	441	(1)
Abnormal Phenomena	441	(1)
Closure	441	(2)
Index	443