Elastohydrodynamic lubrication for line and point contacts : asymptotic and numerical approaches

703 Pages·9.19 MB·English

by  Ilya I Kudish

Elastohydrodynamic
lubrication for linE
and Point contacts
Asymptotic and Numerical
Approaches
K19013_FM.indd 1 5/6/13 5:06 PM
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Elastohydrodynamic
lubrication for linE
and Point contacts
Asymptotic and Numerical
Approaches
ilya i. Kudish
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
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CRC Press
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Version Date: 20130523
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To Rima
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Contents
Preface xvii
About the author xxiii
I Basic Properties of Solids and Fluids Involved in
Lubricated Contacts 1
1 Basic Properties and Rheology of Lubricating Oils 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Rheology Relationships for Lubricating Oils . . . . . . . . . . 4
1.2.1 Definition of Viscosity . . . . . . . . . . . . . . . . . . 4
1.2.2 The Effect of Temperature on Viscosity . . . . . . . . 5
1.2.3 The Effect of Shear Rate on Viscosity . . . . . . . . . 8
1.2.4 The Effect of Pressure on Viscosity . . . . . . . . . . . 9
1.2.5 The Effect of Suspended Contaminants on Viscosity . 10
1.2.6 Density . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.7 Thermal Conductivity . . . . . . . . . . . . . . . . . . 14
1.3 Polymer Thickening and Shear Stability . . . . . . . . . . . . 15
1.3.1 Molecular Weight . . . . . . . . . . . . . . . . . . . . . 15
1.3.2 Dilute Solution Rheology . . . . . . . . . . . . . . . . 16
1.3.3 Shear Stability . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . 19
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Basic Properties of Elastic Solids 23
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Material Density . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Material Elastic Properties . . . . . . . . . . . . . . . . . . . 25
2.4 Material Thermal Diffusivity and Specific Heat Capacity . . . 26
2.5 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
vii
viii Contents
II Asymptotic Methods and Relationships Relevant
to Elastohydrodynamic Lubrication Theory 29
3 Basics of Asymptotic Expansions and Methods 33
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Ordering, Order Sequences, and Asymptotic Expansions . . . 33
3.3 Asymptotic Sequences and Expansions . . . . . . . . . . . . . 35
3.4 Asymptotic Methods . . . . . . . . . . . . . . . . . . . . . . . 36
3.5 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.6 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . 37
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 Basics of the Theory of Elastohydrodynamically Lubricated
(EHL) Contacts 41
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Simplified Navier-Stokes and Energy Equations . . . . . . . . 41
4.3 Some Classic Results for Smooth Elastic Solids . . . . . . . . 44
4.3.1 Formulas for Elastic Half-Space . . . . . . . . . . . . . 44
4.3.2 Formulas and Results for Elastic Half-Plane . . . . . . 45
4.4 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . 47
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
III EHL Problems for Lightly Loaded Line and Point
Contacts 49
5 Lightly Loaded Lubrication Regimes for Line and Point
Contacts 53
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Lightly Loaded Lubrication Regimes for Line Contacts . . . . 54
5.2.1 PerturbationSolution of the EHL Problem . . . . . . 57
5.2.2 Thermal EHL Problem. . . . . . . . . . . . . . . . . . 62
5.2.3 Numerical Method for Lightly Loaded EHL Contacts . 64
5.3 Asymptotic Approach to Lightly Loaded Point EHL Problems 70
5.4 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.5 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . 73
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
IV Isothermal EHL Problems for Heavily Loaded
Line Contacts with Newtonian Lubricant 75
6 Asymptotic Approaches to Heavily Loaded Lubricated Line
Contacts 79
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Contents ix
6.2 Numerical Solution for EHL Contacts . . . . . . . . . . . . . 81
6.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 82
6.2.2 Numerical Procedure . . . . . . . . . . . . . . . . . . . 83
6.2.3 Some Numerical Results . . . . . . . . . . . . . . . . . 88
6.3 Pre-Critical Lubrication Regimes . . . . . . . . . . . . . . . . 91
6.3.1 Equivalent Problem Formulations . . . . . . . . . . . . 94
6.3.2 Asymptotic Analysis of the Problem . . . . . . . . . . 95
6.3.3 Asymptotic Analysis of the System of Equations
(6.19)-(6.23). . . . . . . . . . . . . . . . . . . . . . . . 102
6.4 Practical Use of the Asymptotic Solutions . . . . . . . . . . . 104
6.5 Compressible Fluids in Heavily Loaded Contacts . . . . . . . 105
6.6 Over-CriticalLubrication Regimes . . . . . . . . . . . . . . . 107
6.6.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 107
6.6.2 Structure of the Solution. . . . . . . . . . . . . . . . . 108
6.6.3 Auxiliary Gap Function h (x) . . . . . . . . . . . . . 119
H
6.6.4 Solutions in the (cid:2) -Zones. . . . . . . . . . . . . . . . . 122
q
6.6.5 Asymptotic Analysis of the Inlet and Exit (cid:2)0-Zones . . 125
6.7 Choosing Pre- or Over-Critical Lubrication Regimes . . . . . 131
6.8 Analysis of the Ertel-Grubin Method . . . . . . . . . . . . . . 133
6.9 Numerical Solution of Asymptotic Equations . . . . . . . . . 137
6.9.1 Numerical Solution in the Inlet Zone . . . . . . . . . . 138
6.9.2 Numerical Solution in the Exit Zone . . . . . . . . . . 141
6.9.3 Some Numerical Results in the Inlet and Exit Zones . 145
6.10 Numerical Validation of the Asymptotic Analysis . . . . . . . 153
6.11 Numerical Precision, Grid Size, and Stability Considerations . 157
6.12 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.13 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . 162
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
V Isothermal and Thermal EHL Problems for Line
Contacts and Lubricants with Newtonian and Non-
Newtonian Rheologies 167
7 Thermal EHL Problems for Line Contacts 171
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.2 Formulation of TEHL Problem . . . . . . . . . . . . . . . . . 171
7.3 Analytical Approximations for Newtonian Fluids in TEHL
Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.4 Numerical Solutions of Asymptotic TEHL Problems for New-
tonian Lubricants . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.5 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.6 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . 183
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185