NUMERICAL CALCULATION OF ELASTOHYDRODYNAMIC LUBRICATION NUMERICAL CALCULATION OF ELASTOHYDRODYNAMIC LUBRICATION METHODS AND PROGRAMS Ping Huang School of Mechanical and Automotive Engineering South China Universityof Technology People’s Republic of China Thiseditionfirstpublished2015 ©2015byTsinghuaUniversityPress.Allrightsreserved PublishedbyJohnWiley&SonsSingaporePteLtd,1FusionopolisWalk,#07-01SolarisSouthTower,Singapore 138628,underexclusivelicensegrantedbyTsinghuaUniversityPressforallmediaandlanguagesexcludingChinese andthroughouttheworldexcludingMainlandChina,andwithnon-exclusivelicenseforelectronicversionsin MainlandChina. RegisteredOffice JohnWiley&SonsSingaporePteLtd,1FusionopolisWalk,#07-01SolarisSouthTower,Singapore138628 Fordetailsofourglobaleditorialoffices,forcustomerservicesandforinformationabouthowtoapplyfor permissiontoreusethecopyrightmaterialinthisbookpleaseseeourwebsiteatwww.wiley.com. 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Setin11/13ptTimesbySPiGlobal,Pondicherry,India 1 2015 Contents Preface xi Introduction xv 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 vi Contents 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 for 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 method and program of 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 Contents vii 4 Numerical calculation method and program for 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 Iteration 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 iteration 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 method 90 5.3.1 Coefficient treatment in nonlubricated region 90 5.3.2 Determination of ρ H 90 e e 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 for isothermal EHL in point contact 99 6.1 Basic equations for 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 viii Contents 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 for 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 for EHL in point contact with multigrid method 133 7.5.1 Specification of 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 for 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