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# Numerical Calculation of Lubrication: Methods and Programs

ISBN: 978-1-118-45119-9 October 2013 450 Pages

## Description

Focusing on basic lubrication problems this book offers specific engineering applications. The book introduces methods and programs for the most important lubrication problems and their solutions. It is divided into four parts. The first part is about the general solving methods of the Reynolds equation, including solutions of Reynolds equations with different conditions and their discrete forms, such as a steady-state incompressible slider, journal bearing, dynamic bearing, gas bearing and grease lubrication. The second part gives the ‘energy equation solution’. The third part introduces methods and programs for elasto-hydrodynamic lurbication, which links the Reynolds equation with the elastic deformation equation. The final part presents application lubrication programs used in engineering.

• Provides numerical solution methodologies including appropriate software for the hydrodynamic and elasto-hydrodynamic lubrication of bearings
• Offers a clear introduction and orientation to all major engineering lubrication problems and their solutions
• Presents numerical programs for specific applications in engineering, with special topics including grease-lubricated bearings and gas bearings
• Equips those working in tribology and those new to the topic with the fundamental tools of calculation

With an emphasis on clear explanations, the text offers a thorough understanding of the numerical calculation of lubrication for graduate students on tribology and engineering mechanics courses, with more detailed materials suitable for engineers. This is an accessible reference for senior undergraduate students of tribology and researchers in thin-film fluid mechanics.

## Related Resources

##### Student

View Student Companion Site

Preface xv

Part 1 NUMERICAL METHOD FOR REYNOLDS EQUATION 1

1 Reynolds Equation and its Discrete Form 3

1.1 General Reynolds Equation and Its Boundary Conditions 3

1.1.1 Reynolds Equation 3

1.1.2 Definite Condition 3

1.1.3 Computation of Lubrication Performances 5

1.2 Reynolds Equations for Some Special Working Conditions 6

1.2.1 Slider and Thrust Bearing 6

1.2.2 Journal Bearing 7

1.2.3 Hydrostatic Lubrication 8

1.2.4 Squeeze Bearing 9

1.2.5 Dynamic Bearing 9

1.2.6 Gas Bearing 10

1.3 Finite Difference Method of Reynolds Equation 10

1.3.1 Discretization of Equation 11

1.3.2 Difference Form of Reynolds Equation 12

1.3.3 Iteration of Differential Equation 13

1.3.4 Iteration Convergence Condition 13

2 Numerical Method and Program for Incompressible and Steady Lubrication of One-dimensional Slider 17

2.1 Basic Equations 17

2.1.1 Reynolds Equation 17

2.1.2 Boundary Conditions 18

2.1.3 Continuity Equation 18

2.2 Numerical Method for Incompressible and Steady Lubrication of One-dimensional Slider 18

2.2.1 Discrete Reynolds Equation 19

2.3 Calculation Program for Incompressible and Steady Lubrication of One-dimensional Slider 20

2.3.1 Introduction 20

2.3.2 Calculation Diagram 21

2.3.3 Calculation Program 21

2.3.4 Calculation Results 24

3 Numerical Method and Program for Incompressible and Steady Lubrication of Two-dimensional Slider 25

3.1 Basic Equations 25

3.2 Discrete Reynolds Equation 26

3.3 Calculation Program for Incompressible and Steady Lubrication of Two-dimensional Slider 27

3.3.1 Introduction 27

3.3.2 Calculation Diagram 27

3.3.3 Calculation Program 28

3.3.4 Calculation Results 31

4 Numerical Method and Program for Incompressible and Steady Lubrication of Journal Bearing 33

4.1 Basic Equations 33

4.1.1 Axis Position and Clearance Shape 33

4.1.2 Reynolds Equation 34

4.2 Numerical Method for Incompressible and Steady Lubrication of Journal Bearing 35

4.2.1 Dimensionless Reynolds Equation 35

4.2.2 Discrete Form of Reynolds Equation 36

4.3 Calculation Program for Incompressible and Steady Lubrication of Journal Bearing 37

4.3.1 Calculation Diagram 37

4.3.2 Calculation Program 38

4.3.3 Calculation Results 40

5 Numerical Method and Program for Incompressible Squeeze Lubrication 41

5.1 Basic Equation 41

5.2 Numerical Method and Program for Rectangular Plane Squeeze 42

5.2.1 Basic Equations 42

5.2.2 Numerical Method 42

5.2.3 Calculation Diagram 43

5.2.4 Calculation Program 44

5.2.5 Calculation Results 47

5.3 Numerical Method and Program for Disc Squeeze 47

5.3.1 Basic Equations 47

5.3.2 Numerical Method 48

5.3.3 Calculation Diagram 48

5.3.4 Calculation Program 49

5.3.5 Calculation Results 52

5.4 Numerical Method and Program for Journal Bearing Squeeze 52

5.4.1 Basic Equations 52

5.4.2 Numerical Method 54

5.4.3 Calculation Diagram 54

5.4.4 Calculation Program 55

5.4.5 Calculation Results 60

6 Numerical Method and Program for Dynamic Bearing 61

6.1 Basic Equations 61

6.2 Numerical Method for Trace of Journal Center 65

6.2.1 Introduction 65

6.2.2 Calculation Steps 66

6.3 Calculation Program for Dynamic Journal Bearing 67

6.3.1 Introduction 67

6.3.2 Calculation Diagram 67

6.3.3 Calculation Program 68

6.3.4 Calculation Results 82

7 Numerical Method and Program for Gas Lubrication 85

7.1 Basic Equations 85

7.1.1 General Reynolds Equation of Gas Lubrication 85

7.2 Numerical Method of Gas Lubrication 86

7.2.1 Basic Equations of Steady and Isothermal Gas Lubrication 86

7.2.2 Numerical Method 87

7.3 Calculation Program for Gas Lubrication 88

7.3.1 Calculation Program and Solutions of One-Dimensional Gas Lubrication 88

7.3.2 Numerical Program and Solutions of Two-Dimensional Gas Lubrication 91

7.3.3 Numerical Program and Solutions of Journal Bearing Gas Lubrication 94

8 Numerical Method and Program for Rarefied Gas Lubrication 97

8.1 Basic Equations 97

8.2 Numerical Method of Rarefied Gas Lubrication 99

8.2.1 Rarefied Gas Lubrication Model 99

8.2.2 Treatment of the Ultra-Thin Gas Film Lubrication Equation 100

8.3 Discretization and Iteration of Modified Reynolds Equation 101

8.3.1 Discrete Equation 101

8.3.2 Iteration Method 101

8.4 Calculation Program for Rarefied Gas Lubrication of Slider 102

8.4.1 Procedures Introduction 102

8.4.2 Calculation Diagram 102

8.4.3 Calculation Program 102

8.4.4 Calculation Results 106

9 Numerical Method and Program for One-dimensional Grease Lubrication 107

9.1 Basic Equations 107

9.1.1 Introduction 107

9.1.2 Constitutive Equations of Grease 108

9.1.3 Reynolds Equation 109

9.2 Numerical Method of One-Dimensional Grease Lubrication 109

9.3 Calculation Program of One-Dimensional Grease Lubrication 110

9.3.1 Calculation Diagram 110

9.3.2 Calculation Program 111

9.3.3 Calculation Results 113

Part 2 NUMERICAL METHOD FOR ENERGY EQUATION 115

10 Energy Equation and its Discrete Form 117

10.1 Basic Equations 117

10.1.1 Simplified Energy Equation 118

10.1.2 Boundary Conditions 118

10.1.3 Numerical Method 119

10.2 Influence of Temperature on Lubricant Performance 120

10.2.1 Viscosity–Temperature Equation 120

10.2.2 Density–Temperature Equation 120

10.3 Numerical Method for Thermal Hydrodynamic Lubrication 121

10.3.1 Methods and Program for One-dimensional Thermal Hydrodynamic Lubrication 121

10.3.2 Numerical Method and Program for Two-dimensional Thermal Hydrodynamic Lubrication 124

11 Numerical Method and Program for Incompressible and Steady Thermal Hydrodynamic Lubrication of Journal Bearing 131

11.1 Basic Equations 131

11.1.1 Reynolds Equation 131

11.1.2 Energy Equation 132

11.1.3 Viscosity–Temperature Equation 132

11.2 Numerical Method 132

11.2.1 Discrete Reynolds Equation 132

11.2.2 Discrete Energy Equation 133

11.2.3 Temperature–Viscosity Equation 133

11.3 Calculation Program 133

11.3.1 Calculation Diagram 133

11.3.2 Calculation Program 134

11.3.3 Calculation Results 138

Part 3 NUMERICAL METHOD FOR ELASTIC DEFORMATION AND THERMAL ELASTOHYDRODYNAMIC LUBRICATION 141

12 Numerical Method and Program for Elastic Deformation and Viscosity–Pressure Equation 143

12.1 Basic Equations of Elastic Deformation 143

12.1.1 Film Thickness Equation 143

12.1.2 Elastic Deformation Equation 143

12.2 Numerical Methods and Programs of Elastic Deformation 145

12.2.1 Numerical Method and Program of Elastic Deformation Equation in Line Contact 145

12.2.2 Numerical Method and Program of Elastic Deformation Equation in Point Contact 148

12.3 Viscosity–Pressure and Density–Pressure Equations 155

12.3.1 Viscosity–Pressure Relationship 155

12.3.2 Viscosity–Pressure–Temperature Relationship 156

12.3.3 Density–Pressure Relationship 156

13 Numerical Method and Program for EHL in Line Contact 159

13.1 Basic Equations 159

13.2 Numerical Method 160

13.2.1 Dimensionless Equations 160

13.2.2 Discrete Equations 161

13.2.3 Iterative Method 162

13.2.4 Selection of Iterative Methods 163

13.2.5 Relaxation Factors 164

13.3 Calculation Program 164

13.3.1 Calculation Diagram 164

13.3.2 Calculation Program 165

13.3.3 Calculation Results 171

14 Numerical Method and Program for EHL in Point Contact 173

14.1 Basic Equations 173

14.2 Numerical Method 174

14.2.1 Dimensionless Equations 174

14.2.2 Discrete Equations 175

14.3 Calculation Program 176

14.3.1 Calculation Diagram 176

14.3.2 Calculation Program 177

14.3.3 Calculation Results 186

15 Numerical Method and Program for Grease EHL in Line Contact 187

15.1 Basic Equations 187

15.1.1 Reynolds Equation 187

15.1.2 Film Thickness Equation 187

15.1.3 Viscosity–Pressure Equation 188

15.1.4 Density–Pressure Equation 188

15.2 Numerical Method 188

15.2.1 Dimensionless Equations 188

15.2.2 Discrete Equations 189

15.3 Calculation Program 189

15.3.1 Calculating Diagram 189

15.3.2 Calculation Program 190

15.3.3 Calculation Results 199

16 Numerical Method and Program for Grease EHL in Point Contact 201

16.1 Basic Equations 201

16.1.1 Reynolds Equation 201

16.1.2 Film Thickness Equation 201

16.1.3 Elastic Deformation Equation 202

16.1.4 Viscosity–Pressure Equation 202

16.1.5 Density Equation 202

16.2 Numerical Method 202

16.2.1 Dimensionless Equations 202

16.2.2 Discrete Equations 203

16.3 Calculation Program 204

16.3.1 Calculation Diagram 204

16.3.2 Calculation Program 205

16.3.3 Calculation Results 214

17 Numerical Method and Program for Thermal EHL in Line Contact 215

17.1 Basic Equations 215

17.1.1 Reynolds Equation 215

17.1.2 Energy Equation 215

17.1.3 Film Thickness Equation 216

17.1.4 Elastic Deformation Equation 216

17.1.5 Roelands Viscosity–Pressure–Temperature Equation 216

17.1.6 Density–Pressure–Temperature Equation 217

17.2 Numerical Method 217

17.2.1 Dimensionless Equations 217

17.2.2 Discrete Equations 218

17.3 Calculation Program 220

17.3.1 Calculation Diagram of Multigrid Method 220

17.3.2 Calculation Diagram of Temperature 221

17.3.3 Calculation Program 222

17.3.4 Calculation Results 236

18 Numerical Method and Program for Thermal EHL in Point Contact 237

18.1 Basic Equations 237

18.1.1 Reynolds Equation 237

18.1.2 Energy Equation 237

18.1.3 Film Thickness Equation 238

18.1.4 Elastic Deformation Equation 238

18.1.5 Roelands Viscosity–Pressure–Temperature Equation 239

18.1.6 Density–Pressure–Temperature Equation 239

18.2 Numerical Method 239

18.2.1 Dimensionless Equations 239

18.2.2 Discrete Equations 241

18.3 Calculation Program 242

18.3.1 Calculation Diagram 242

18.3.2 Calculation Program 242

18.3.3 Calculation Results 261

19 Numerical Method and Program for Thermal Grease EHL in Line Contact 263

19.1 Basic Equations 263

19.1.1 Reynolds Equation 263

19.1.2 Energy Equation 264

19.1.3 Film Thickness Equation 264

19.1.4 Elastic Deformation Equation 265

19.1.5 Viscosity–Pressure–Temperature Equation 265

19.1.6 Density–Pressure–Temperature Equation 265

19.2 Numerical Method 265

19.2.1 Dimensionless Equations 265

19.2.2 Discrete Equations 267

19.3 Calculation Program 268

19.3.1 Calculation Diagram 268

19.3.2 Calculation Program 268

19.3.3 Calculation Results 287

20 Numerical Method and Program for Thermal Grease EHL in Point Contact 289

20.1 Basic Equations 289

20.1.1 Reynolds Equation 289

20.1.2 Energy Equation 290

20.1.3 Film Thickness Equation 290

20.1.4 Elastic Deformation Equation 291

20.1.5 Roelands Viscosity–Pressure–Temperature Equation 291

20.1.6 Density–Pressure–Temperature Equation 291

20.2 Numerical Method 291

20.2.1 Dimensionless Equations 291

20.2.2 Discrete Equations 293

20.3 Calculation Program 294

20.3.1 Calculation Diagram 294

20.3.2 Calculation Program 295

20.3.3 Calculation Results 310

Part 4 CALCULATION PROGRAMS FOR LUBRICATION ANALYSIS IN ENGINEERING 311

21 Lubrication Calculation Program for Herringbone Grooved Journal Bearing of Micro Motor 313

21.1 Basic Theory of Lubrication Calculation of Herringbone Groove Bearing 313

21.1.1 Journal Center Position and Film Thickness 313

21.1.2 Reynolds Equation 314

21.1.3 Boundary Conditions 315

21.1.4 Flux Calculation 316

21.1.5 Temperature Calculation 316

21.2 Program for Performance Calculation 318

21.2.1 Lubrication Performances 318

21.2.2 Calculation Program 318

21.3 Calculation Results 326

21.4 Instruction for HBFA Software Package 332

21.4.1 Package Contents 332

21.4.2 Program Installation 332

21.4.3 Program Operation 333

22 Lubrication Optimization Program of Herringbone Grooved Journal Bearing of Micro Motor 337

22.1 Method of Optimization Calculation 337

22.1.1 Requirements of Parameter Optimization 337

22.1.2 Optimization Model 337

22.1.3 Optimization Methods and Steps 338

22.2 Program Layout of Optimization Calculation 338

22.2.1 Optimization Program Diagram 338

22.2.2 Calculation Program 339

22.2.3 Parameters in Program 352

22.3 Optimization Calculation Examples 352

22.3.1 Example 1: Optimization Calculation for Static Load 352

22.3.2 Example 2: Optimization Calculation for Static Flux (Eccentricity Ratio e is Constant) 354

22.3.3 Example 3: Optimization Calculation for Static Flux (Load W is Constant) 354

22.3.4 Example 4: Optimization Calculation for Dynamic Load 354

22.3.5 Example 5: Optimization Calculation for Dynamic Flux (Eccentricity e is Constant) 354

22.3.6 Example 6: Optimization Calculation for Dynamic Flux (Load W is Constant) 355

22.4 Instructions for HBOA Software Package 355

22.4.1 Program Package 355

22.4.2 Program Execution 356

23 Calculation Program for Gas Lubrication of Hard Disk/Head in Ultra Thin Film 361

23.1 Basic Equations of Gas Lubricating Film of Hard Disk/Head 361

23.1.1 Basic Equations 361

23.1.2 Gas Film Thickness 362

23.1.3 Poiseuille Flow Rate 362

23.2 Discrete Equation and Special Treatments 363

23.2.1 Iterative Scheme Considering High Bearing Numbers 363

23.2.2 Abrupt Changes between Steps on ABS 364

23.3 Calculation Program 364

23.3.1 Calculation Diagram 364

23.3.2 Calculation Program 366

23.3.3 Calculation Results 371

24 Calculation Program of Flight Attitude of Magnetic Head 373

24.1 Search Strategy for Flight Attitude 373

24.2 Calculation Program 375

24.2.1 Program Introduction 375

24.2.2 Calculation Diagram 376

24.2.3 Calculation Program 376

24.2.4 Calculation Results 386

References 389

Index 391

 Numerical Calculation of Lubrication Visit the companion website to access the programs discussed in the book.