Numerical Calculation of Lubrication

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