Numerical Calculation of Elastohydrodynamic Lubrication: Methods and Programs

This book is a one–stop reference providing equations and solutions to all major elastohydrodynamic lubrication (EHL) problems, in addition to numerical programs on specific applications in engineering. It offers scientists and engineers a clear interdisciplinary introduction and orientation to all major EHL problems and their solutions by first taking them through the basic equations, numerical methods, and calculation programs for Reynolds equation, elastic deformation and energy equation. Readers then progress to problems of isothermal or thermal EHL in line contact, point contact, and ellipse contact. Finally, readers are introduced to the more challenging EHL problems of grease, electric double layer effect, unsteadiness, roughness, and micro–polar fluid.

 Systematic introduction of EHL problems, starting from basic concepts to challenging problems and applications

 Features numerical simulations of the Reynolds equation, elasticity equation of the surfaces, and energy equations

 Offers engineers and scientists a clear interdisciplinary introduction and concise programs for practical engineering applications

 Enhances learning through numerous case studies, with common features and methods in solutions highlighted.

Numerical Calculation of Elastohydrodynamic Lubrication: Methods and Programs is a handy and practical numerical method guide for engineers and technicians. Advanced students in tribology, mechanical engineering and materials science courses will also find it a useful text.

 

Introduction xi

Preface xiii

Nomenclature xvii

1 Basic equations of elastohydrodynamic lubrication 1

1.1 Basic equations 1

1.1.1 One–dimensional Reynolds equation of elastohydrodynamic lubrication 1

1.1.2 Two–dimensional Reynolds equation of EHL 3

1.1.3 EHL Reynolds equation with two direction velocities 3

1.1.4 Time–dependent EHL Reynolds equation 4

1.2 Film thickness equation without elastic deformation 4

1.2.1 Film thickness equation in line contact 4

1.2.2 Film thickness equation in point contact 5

1.2.3 Film thickness equation in ellipse contact 5

1.3 Surface elastic deformation 6

1.3.1 One–dimensional elastic deformation equation 6

1.3.2 Two–dimensional elastic deformation equation 6

1.4 Viscosity and density equations varying with pressure and temperature 7

1.4.1 Viscosity equations 7

1.4.2 Density equation 9

1.5 Load balancing condition 10

1.5.1 Load balancing equation 10

1.5.2 Numerical calculation of load balancing 11

1.6 Finite difference method of Reynolds equation 15

1.6.1 Discretization of equation 15

1.6.2 Different forms of Reynolds equation 17

1.6.3 Iteration of differential equation 18

1.6.4 Iteration convergence condition 18

2 Numerical calculation method and program of elastic deformation 21

2.1 Numerical method and program of elastic deformation in line contact 21

2.1.1 Equations of elastic deformation 21

2.1.2 Numerical method of elastic deformation 22

2.1.3 Calculation diagram and program 23

2.1.4 Example 25

2.2 Numerical method and program of elastic deformation in point contact 27

2.2.1 Equation of elastic deformation 27

2.2.2 Numerical method for elastic deformation in point contact 28

2.2.3 Calculation diagram 29

2.2.4 Program 30

2.2.5 Example 31

2.3 Numerical calculation method and program for elastic deformation in ellipse contact 33

2.3.1 Contact geometry 33

2.3.2 Contact pressure and contact zone 36

2.3.3 Calculation program 38

2.3.4 Calculation results 44

2.4 Calculation of elastic deformation with multigrid integration technique 44

2.4.1 Principle of multigrid integration 44

2.4.2 Calculation programs and examples 49

3 Numerical calculation method and program for energy equation 61

3.1 Energy equation 61

3.1.1 One–dimensional energy equation 62

3.1.2 Two–dimensional energy equation 62

3.2 Numerical method and program for thermal hydrodynamic lubrication 64

3.2.1 One–dimensional thermal hydrodynamic lubrication 64

3.2.2 Two–dimensional thermal hydrodynamic lubrication 68

4 Numerical calculation method and program of isothermal EHL in line contact 75

4.1 Basic equations and Nondimensional equations 75

4.1.1 Basic equations 75

4.1.2 Nondimensional equations 76

4.1.3 Discrete equations 77

4.2 Numerical calculation method and program 78

4.2.1 Iterative method 78

4.2.2 Program and example 79

5 Newton Raphson method and programs to solve EHL problems in line contact 87

5.1 Basic equations 87

5.2 Newton Raphson iterative method 88

5.2.1 Coefficient matrix 88

5.2.2 Calculation of variables in equations 89

5.3 Numerical method and program of Newton Raphson 90

5.3.1 Coefficient treatment in nonlubricated region 90

5.3.2 Determination of eHe 90

5.3.3 Determination and correction of initial rigid film thickness 91

5.3.4 Calculation Program 91

5.3.5 Example 96

6 Numerical calculation method and program of isothermal EHL in point contact 99

6.1 Basic equations of isothermal EHL in point contact 99

6.1.1 Basic equations 99

6.1.2 Nondimensional equations 100

6.2 Numerical calculation method and program 101

6.2.1 Differential equations 101

6.2.2 Iteration method 102

6.2.3 Calculation diagram 102

6.2.4 Calculation program 106

6.2.5 Example 111

7 Numerical calculation method and programs of multigrid method for isothermal EHL 113

7.1 Basic principles of multigrid method 113

7.1.1 Grid structure 113

7.1.2 Equation discrete 114

7.1.3 Restriction and extension 114

7.2 Nonlinear full approximation scheme of multigrid method 116

7.2.1 Parameter transformation downwards 116

7.2.2 Correction of lower grid parameters 116

7.2.3 Parameter transformation upwards 118

7.2.4 V and W loops 118

7.3 Key factors to solve EHL problem with multigrid method 119

7.3.1 Iteration methods 119

7.3.2 Relaxation factors selection 121

7.4 Program of EHL in line contact with multigrid method 122

7.4.1 Specification of program 122

7.4.2 Calculation program 123

7.4.3 Example 133

7.5 Program of EHL in point contact with multigrid method 133

7.5.1 Specification of the program 133

7.5.2 Calculation program 134

7.5.3 Example 143

8 Numerical calculation method and program for isothermal EHL in ellipse contact 145

8.1 Basic equation 145

8.1.1 Reynolds equation 145

8.1.2 Film thickness equation 146

8.1.3 Viscosity pressure equation 146

8.1.4 Density pressure equation 147

8.1.5 Load balancing equation 147

8.2 Calculation program 148

8.2.1 Calculation diagram 148

8.2.2 Calculation program 149

8.2.3 Example 157

9 Numerical calculation method and program of isothermal EHL in elliptical contact with two–dimensional velocities 159

9.1 Basic equations 159

9.2 Velocity treatment 160

9.3 Numerical calculation method and program 161

9.3.1 Flowchart 161

9.3.2 Program 161

9.3.3 Example 171

10 Numerical calculation method and program for thermal EHL 173

10.1 Basic equations for thermal EHL 173

10.1.1 Thermal EHL in line contact 173

10.1.2 TEHL in point contact 176

10.2 Viscosity and temperature across film thickness 179

10.2.1 Calculation of fluid velocity field 179

10.2.2 Continuity equation 181

10.2.3 Energy equation 182

10.2.4 Temperature boundary conditions 185

10.2.5 Calculation of linear equation set 187

10.2.6 Program to calculate temperature 187

10.3 Numerical calculation method and program for TEHL in line contact 189

10.3.1 Flowchart 189

10.3.2 Program 190

10.3.3 Example 201

10.4 Numerical calculation method and program of TEHL in point contact 202

10.4.1 Flowchart 202

10.4.2 Calculation program 204

10.4.3 Example 216

11 Numerical calculation method and program of grease EHL 217

11.1 Basic equations of grease EHL 217

11.2 Numerical calculation and program of isothermal grease EHL 220

11.2.1 Line contact problem 220

11.2.2 Point contact problem 231

11.3 Numerical calculation method and program of thermal GEHL 243

11.3.1 Line contact problem 243

11.3.2 Point contact problem 262

12 Numerical calculation method and program of EHL considering effect of electric double layer 279

12.1 Structure of electric double layer 279

12.2 Reynolds equation considering EDL effect 279

12.2.1 Modified Reynolds equation 279

12.2.2 Expression of electroviscosity 284

12.3 Calculation program and example 285

12.3.1 Calculation program 285

12.3.2 Example 293

13 Numerical calculation method and program of time–dependent EHL in line contact 295

13.1 Time–dependent EHL Reynolds equation 295

13.1.1 Nondimensional Reynolds equation 295

13.1.2 Discrete Reynolds equation 296

13.2 Numerical calculation method and program 296

13.2.1 Iteration method 296

13.2.2 Calculation diagram 297

13.2.3 Calculation program 297

13.2.4 Example 304

14 Numerical calculation method and program of isothermal EHL with rough surface 305

14.1 Film thickness equation with surface roughness 305

14.2 Calculation of rough surface EHL problem with Newton Raphson method 306

14.2.1 Sine roughness 306

14.2.2 Single depression 311

14.3 EHL solution with random roughness in line contact 317

14.3.1 Description of the program 317

14.3.2 Calculation program 317

14.3.3 Example 321

14.4 EHL solution with random roughness in point contact 322

14.4.1 Description of the program 322

14.4.2 Calculation program 322

14.4.3 Example 326

14.5 Random roughness generation program 328

15 Numerical calculation method and program of micropolar fluid EHL 329

15.1 Reynolds equation of micropolar fluid EHL 329

15.1.1 Basic equations 329

15.1.2 Reynolds equation 330

15.2 Calculation program of EHL of micropolar fluid in line contact 334

15.2.1 Description of the program 334

15.2.2 Calculation program 334

15.2.3 Example 338

15.3 Calculation program of EHL with micropolar fluid in point contact 339

15.3.1 Calculation program 339

15.3.2 Example 343

References 345

Index 000

Ping Huang South China University of Technology, China