Numerical Calculation of Elastohydrodynamic Lubrication: Methods and Programs

The book not only offers scientists and engineers a clear inter-disciplinary introduction and orientation to all major EHL problems and their solutions but, most importantly, it also provides numerical programs on specific application in engineering. • A one-stop reference providing equations and their solutions to all major elastohydrodynamic lubrication (EHL) problems, plus numerical programs on specific applications in engineering • offers engineers and scientists a clear inter-disciplinary introduction and a concise program for practical engineering applications to most important EHL problems and their solutions • brings together a number of case studies in one text, each being solved using solution methods which share common features and methods

Table of Contents:
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 ρeHe 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