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Vice President, Kanazawa Institute of Technology 7-1 Ohgigaoka, Nonoichi, Ishikawa 921-8501, Japan Professor Emeritus, University of Tokyo /LEUDU\RI&RQJUHVV&RQWURO1XPEHU ,6%16SULQJHU9HUODJ7RN\R%HUOLQ+HLGHOEHUJ1HZ<RUN ,6%16SULQJHU9HUODJ7RN\R%HUOLQ+HLGHOEHUJ1HZ<RUN 3ULQWHGRQDFLGIUHHSDSHU 7KLV(QJOLVKWUDQVODWLRQLVEDVHGRQWKH-DSDQHVHRULJLQDO 5\ ØWDL=\XQNDWX+\GURG\QDPLF/XEULFDWLRQE\<XNLR+RUL 3XEOLVKHGE\<RNHQGR/WG <XNLR+RUL 6SULQJHU9HUODJ7RN\R 3ULQWHGLQ-DSDQ 7KLVZRUNLVVXEMHFWWRFRS\ULJKW$OOULJKWVDUHUHVHUYHGZKHWKHUWKHZKROHRUSDUWRIWKHPDWHULDO LVFRQFHUQHGVSHFLILFDOO\WKHULJKWVRIWUDQVODWLRQUHSULQWLQJUHXVHRILOOXVWUDWLRQVUHFLWDWLRQ EURDGFDVWLQJUHSURGXFWLRQRQPLFURILOPVRULQRWKHUZD\VDQGVWRUDJHLQGDWDEDQNV 7KHXVHRIUHJLVWHUHGQDPHVWUDGHPDUNVHWFLQWKLVSXEOLFDWLRQGRHVQRWLPSO\HYHQLQWKH DEVHQFHRIDVSHFLILFVWDWHPHQWWKDWVXFKQDPHVDUHH[HPSWIURPWKHUHOHYDQWSURWHFWLYHODZVDQG UHJXODWLRQVDQGWKHUHIRUHIUHHIRUJHQHUDOXVH 6SULQJHULVDSDUWRI6SULQJHU6FLHQFH%XVLQHVV0HGLD springeronline.com 3ULQWLQJDQGELQGLQJ+LFRP-DSDQ Prefaces To the original Japanese edition: Hydrodynamic lubrication occupies an important position in mechanical engi- neering; however, books on the subject are seldom seen in Japan. Not so many books have been published on the subject overseas either. This book consists of some historical and theoretical introductions (Chapters 1 – 4) and the results of research by myself, by myself and co-workers who were in my laboratory, and by a few of my very close colleagues (Chapters 5 – 9). Ref- erences are given at the end of each chapter. The material has been taken from the above-mentioned research partly because of the ease in getting permission for the use of figures in published papers. At the manuscript stage, I received kind advice from my colleagues, approxi- mately in the order of the contents of the book, Drs. Yoshitsugu Kimura, Masato Tanaka, Takahisa Kato, Shige-aki Kuroda, Akira Hasuike, Kyung Woong Kim, Jun  ichi Mitsui, and Satoru Kaneko. Especially, Drs. Masato Tanaka and Takahisa Kato kindly examined the whole manuscript and gave me valuable suggestions. I am very much obliged to all of them. Concerning publication of this book, I give sincere thanks to Mr. Kiyoshi Oikawa, the President, Mr. Nobuyuki Miura, the Director, and Mr. Kaoru Shimada, the Editor of Yokendo, Ltd October 2002, Tokyo Y. H . To the English edition: On publication of the English edition, I have corrected a few errata that were found during translation and added Fig. 5.16, which was drawn after the publica- tion of the original edition. I express my sincere thanks to the staff members of the publisher, Springer-Verlag, Tokyo. October 2005, Tokyo Y. H . Contents 1 Friction, Wear, and Lubrication 1 1.1 Friction,Wear,andLubrication—Tribology 1 1.2 VariousFormsofLubrication 2 1.2.1 SolidFriction 4 1.2.2 Hydrodynamic Lubrication . . . 6 1.3 Meanings of Tribology . 7 References 8 2 Foundations of Hydrodynamic Lubrication 9 2.1 Tower’s Experiment . . . . 9 2.2 Reynolds’ Theory of Hydrodynamic Lubrication . . 11 2.2.1 InterpretationofReynolds’Equation 18 References 22 3 Fundamentals of Journal Bearings 23 3.1 CircularJournalBearings 25 3.1.1 CrossSectionofaBearing 25 3.1.2 Shape of the Oil Film 26 3.1.3 Bearing Length (Bearing Width) 27 3.1.4 Boundary Conditions for the Oil Film 27 3.2 InfinitelyLongBearings 29 3.2.1 OilFilmPressure 29 3.2.2 Infinitely Long Bearing Under Sommerfeld’s Condition 31 3.2.3 Infinitely Long Bearing Under G ¨ umbel’s Condition 37 3.3 Short Bearings . . 41 3.3.1 OilFilmPressure 41 3.3.2 Characteristics of a Short Bearing Under G ¨ umbel’s Condition 42 3.4 Finite Length Bearings 43 References 46 VIII Contents 4 Fundamentals of Thrust Bearings 47 4.1 InfinitelyLongPlanePadBearings 48 4.1.1 BasicFormulae 49 4.1.2 Basic Characteristics . 49 4.2 Finite Length Plane Pad Bearings 54 4.3 Sector Pad Bearings 55 4.3.1 Reynolds’ Equation in Cylindrical Coordinates . 55 4.3.2 Numerical Solution of a Sector Pad 57 4.4 Additional Topics . . 58 4.4.1 Influence of Deformation of the Pad . . 58 4.4.2 Magnetic Disk Memory Storage . . . . . . 59 References 60 5 Stability of a Rotating Shaft — Oil Whip 63 5.1 OilWhip 64 5.2 OilWhipTheory 67 5.2.1 OilFilmPressure 68 5.2.2 OilFilmForce 71 5.2.3 Linearization of the Oil Film Force . . . 72 5.2.4 Equations of Motion . 75 5.2.5 Stability Limit 76 5.2.6 Occurrence of Oil Whip — Hysteresis . 84 5.2.7 Coordinate Axes 88 5.3 Stability of Multibearing Systems 89 5.4 Influence of Earthquakes on Oil Whip 92 5.4.1 Basic Equations 94 5.4.2 Examples of Simulation 95 5.5 Limit Cycle in an Unstable Domain 98 5.5.1 Approximate Nonlinear Analysis of Journal Bearing Characteristics 98 5.5.2 ResultsofAnalysis 101 5.6 FloatingBushBearings 102 5.7 ThreeCircularArcBearings 106 5.8 PorousBearings 109 5.8.1 Governing Equations 109 5.8.2 Stability of a Shaft System . . . 110 5.9 Chaos in Rotor–Bearing Systems 111 5.10 PreventionofOilWhip 113 References 114 6 Foil Bearings 119 6.1 Basic Equations 121 6.2 Finite Element Solution of the Basic Equations 122 6.2.1 Reynolds’ Equation 122 6.2.2 Equation of Balance for the Foil 125 Contents IX 6.2.3 SolutionProcedure 126 6.3 Characteristics of Foil Bearings 126 6.3.1 Single Cylinder Heads 127 6.3.2 Double Cylinder Heads . . 128 6.3.3 Comparison with Experiments 130 6.4 Additional Topics . . . . . . 130 6.4.1 Magnetic Tape Memory Storage . 130 6.4.2 FoilDisk 131 References 136 7 Squeeze Film 137 7.1 Basic Equations 138 7.2 Squeeze Between Rigid Surfaces 141 7.2.1 Squeeze Without Fluid Inertia . 141 7.2.2 Squeeze with Fluid Inertia 142 7.2.3 Sinusoidal Squeeze Motion . . . 144 7.3 Sinusoidal Squeeze by a Rigid Surface (Experiments) . . . . . 145 7.3.1 Mild Sinusoidal Squeeze . 145 7.3.2 Intense Sinusoidal Squeeze — Cavitation . . 146 7.4 Sinusoidal Squeeze with a Soft Surface . . . . . . 149 7.4.1 Low-Frequency Squeeze . 150 7.4.2 High-Frequency Squeeze . 153 7.4.3 Results of Experiment and Calculation 154 References 159 8 Heat Generation and Temperature Rise 161 8.1 Basic Equations for Thermohydrodynamic Lubrication 162 8.2 Generalized Reynolds’ Equation . 163 8.2.1 BalanceofForces 163 8.2.2 FlowVelocity 164 8.2.3 Continuity Equation . 164 8.2.4 Generalized Reynolds’ Equation 165 8.3 Energy Equation 166 8.3.1 General Energy Equation . 166 8.3.2 Energy Equation 168 8.3.3 Transformation of the Energy Equation 170 8.4 Temperature Distribution in Bearings 171 8.5 Temperature Analyses of Tilting Pad Thrust Bearings — Sector Pads 172 8.5.1 Basic Equations 173 8.5.2 Boundary Conditions 175 8.5.3 Numerical Analyses 175 8.5.4 Examples of Three-Dimensional Analyses of Temperature Distribution 177 8.5.5 Comparisons of Three-Dimensional, Two-Dimensional, andIsoviscousAnalyses 178 X Contents 8.5.6 Analysis Considering Inertia Forces 180 8.5.7 Comparison of Calculated Results and Experiments 184 8.6 Temperature Analyses of Circular Journal Bearings 185 8.6.1 Basic Equations 187 8.6.2 Boundary Conditions 187 8.6.3 Comparison of Calculated Results and Experiments 189 References 193 9 Turbulent Lubrication 197 9.1 Time-Average Equation of Motion and the Reynolds’ Stress . . . . . . . 198 9.2 TurbulentFlowModel 201 9.2.1 MixingLengthModel 201 9.2.2 k-ε Model . . 203 9.3 Turbulent Lubrication Theory Using the Mixing Length Model 204 9.3.1 ModifiedMixingLength 204 9.3.2 Turbulent Velocity Distribution Between Two Surfaces . . . . . 206 9.3.3 TurbulentReynolds’Equation 208 9.3.4 Turbulent Coefficients of Fluid Film Seals . 209 9.4 Comparison of Analyses Using the Mixing Length Model with Experiments 211 9.4.1 Turbulent Static Characteristics of Fluid Film Seals 211 9.4.2 Turbulent Dynamic Characteristics of Fluid Film Seals . 213 9.5 Turbulent Lubrication Theory Using the k-ε Model 214 9.5.1 Application of the k-ε Model to an Oil Film 215 9.5.2 TurbulentReynolds’Equation 216 9.6 Comparison of Analyses Using the k-ε Model with Experiments . . . 218 9.7 Reduction of Friction in a Turbulent Bearing by Toms’ Effect . . . . . . 222 9.8 TaylorVorticesinaJournalBearing 224 References 226 Index 229 The references at the end of each chapter are listed as a rule in chronological order. Symbols The meanings of symbols are as follows, unless otherwise stated. A :area b : bearing width (length) B : pad width B x , B z : nondimensional pressure gradient B p : bearing parameter c : radial clearance (= R b − R j ) c p : specific heat at constat pressure c  : specific heat at constat volume D : diameter e : eccentricity (= O b O j ) f : coefficient of friction (Section 1.2.1) f : frequency of squeeze motion (Chapter 7) F : force G x , G z : turbulent coefficient G c : term of centrifugal force h : film thickness h : enthalpy (Section 8.3.3) h c : coefficient of heat transfer H : indentation hardness (Section 1.2.1) H : mean stress function (Section 7.4.1) J : functional k : turbulent energy k o , k s : thermal conductivity of oil and solid parts k x , k z : reciprocal of turbulent coefficient l : mixing length l m : modified mixing length L : bearing length (width) L s : frictional loss m : inclination of pad (= h 1 /h 2 ) M : frictional moment N : rotational speed of shaft p : pressure, film pressure p m : bearing pressure (= P/(2DL)) P : oil film force P 0 : oil film force (resultant) P 1 : bearing load P : nondimensional load capacity q : flow rate of oil Q : heat flux Q s : generated heat r,θ : polar coordinates R : radius Re : bearing Reynolds’ number (= Uc/ν) R h : local Reynolds’ number (= Uh/ν) R t : turbulent Reynolds’ number (= k 2 /εν) r,θ,z : cylindrical coordinates XII Symbols S : Sommerfeld number (= (R/c) 2 µN/p m ) t : time t : thickness of pad (Section 8.5) T : tension (Chapter 6) T : temperature T 2 : temperature of pad (Section 8.5) u, , w : components of fluid velocity u ∗ : frictional velocity (=  τ w /ρ ) u + : = u/u ∗ U, V, W : surface velocity U : internal energy (Section 8.3) V : generalized velocity (Section 8.3) x, y, z : rectangular coordinate y + : = u ∗ y/ν X, Y, Z : rectangular coordinate axes α : coefficient of expansion α  : coefficient of cubic expansion β : wrap angle (Chapter 6) β : viscosity index (Chapter 8) δ ij : Kronecker’s delta  : strain  : small quantity (Sections 3.4, 6.2.3) ε : turbulent dissipation η : mixing coefficient θ : attitude angle κ : eccentricity ratio (= e/c) κ k : Karman’s constant λ : secondary coefficient of viscosity µ : coefficient of viscosity ν : coefficient of kinetic viscosity (= µ/ρ) ν : Poisson’s ratio (Section 7.4.1) Π : functional ρ : density σ : normal stress σ y : yield stress (Section 1.2.1) τ : shear stress τ w : surface shear stress τ + : = τ/τ w φ : angle Φ : permeability (Section 5.9) Φ : dissipation energy (Chapter 8) ϕ : angle (Sommerfeld transform) ψ : angle ω : angular velocity of rotation Ω : angular velocity of whirling Suffix i, j, k : 1,2,3 a : ambient b : bearing j : journal s : solid part t :turbulent ( ¯ ) : non-dimensional [...]... Tokyo, 19 82 2 Foundations of Hydrodynamic Lubrication The essence of hydrodynamic lubrication was first clarified experimentally by British railroad engineer Beauchamp Tower (18 45 – 19 04) in 18 83 [1] [2] Based on Tower’s experiments, Osborn Reynolds (18 42 – 19 12), the physicist, formulated a theory of lubrication in 18 86 [3] Since then, Reynolds’ theory has been the foundation of the theory of hydrodynamic. .. Holm [1] in relation to electric contacts 1. 2.2 Hydrodynamic Lubrication Hydrodynamic lubrication, which is the subject of this book, is an ideal state of lubrication in that friction and wear hardly occur Figure 1. 3 shows three typical examples in which hydrodynamic lubrication is important These are a journal bearing, an air-floating slider in a magnetic disk memory device, and an animal joint Fig 1. 3... loss of 10 billion yen All because of tribology ” References 1 R Holm, “Electric Contacts”, H Gebers F¨ rlag, Stockholm, 19 46 o 2 F.P Bowden and D Tabor, “The Friction and Lubrication of Solids - Part I”, Oxford U.P., Oxford, 19 50 3 Norimune Soda, “Friction and Lubrication (in Japanese), Iwanami Zensho Series, Iwanami Shoten, Tokyo, 19 54 4 D Dowson, “History of Tribology”, Longman, London, 19 79 5... Forms of Lubrication 5 In mathematical form, it can be written as follows: F = fP (1. 1) where f is the coefficient of friction It is said that this law was first discovered by Leonardo da Vinci (14 52 – 15 19) of Italy in the fifteenth century, and was later rediscovered by G Amonton (16 63 – 17 05) and C A Coulomb (17 36 – 18 06) of France in the eighteenth century independently Today, the law is called Amonton’s... physicist, formulated a theory of lubrication in 18 86 [3] Since then, Reynolds’ theory has been the foundation of the theory of hydrodynamic lubrication Recently developed theories of elastohydrodynamic lubrication, thermohydrodynamic lubrication, turbulent hydrodynamic lubrication, and others are regarded as extensions of Reynolds’ theory The pioneering works of Tower and Reynolds are reflections of Britain’s... 2 .1 Tower’s Experiment Figure 2 .1, which is a simplification of a drawing from Tower’s famous paper of 18 83 [1] , shows the main part of Tower’s friction test rig for a bearing used in rolling stock The bearing is a partial bearing, and bearing bush A covers the upper half of the journal A load (weight of the vehicle) acts on the journal from above through bearing cap B and bearing bush A The lower part. .. If the shear strength of the contact area through the film per unit area is denoted by sl , the frictional force F will be given as follows: F = {sm α + sl (1 − α)}Ar (1. 3) F = f P where f = {sm α + sl (1 − α)}/σy (1. 4) or 6 1 Friction, Wear, and Lubrication The frictional force is proportional to the load in this case also It is clear in the above argument that only the true contact area is related... two surfaces becomes severer and the frictional coefficient will increase further The situation at this stage is a mixture of 4 1 Friction, Wear, and Lubrication the hydrodynamic lubrication described above and boundary lubrication which will be explained later and is called mixed lubrication Since all surfaces in practice, however smooth they may be, have minute asperities, the above-described process... and lubrication efficiently, and to do this, the importance of the research had to be signaled to the public It was thought that the best way for this was to unify these classical subjects as one concept and to give it a new name — tribology 1. 2 Various Forms of Lubrication Although the main subject of this book is hydrodynamic lubrication, it is worthwhile to initially consider the various forms of lubrication. .. follows 1 Frictional resistance is nearly constant, regardless of the bearing load 2 The frictional coefficient is very small (usually of the order of 1/ 1000) 3 Frictional resistance increases with sliding speed 10 2 Foundations of Hydrodynamic Lubrication Fig 2 .1 Tower’s test rig A, bearing bush; B, bearing cap 4 Frictional resistance decreases with a rise in temperature Furthermore, he pointed out that . Rotor–Bearing Systems 11 1 5 .10 PreventionofOilWhip 11 3 References 11 4 6 Foil Bearings 11 9 6 .1 Basic Equations 12 1 6.2 Finite Element Solution of the Basic Equations 12 2 6.2 .1 Reynolds’ Equation 12 2 6.2.2. . Contents 1 Friction, Wear, and Lubrication 1 1 .1 Friction,Wear,andLubrication—Tribology 1 1.2 VariousFormsofLubrication 2 1. 2 .1 SolidFriction 4 1. 2.2 Hydrodynamic Lubrication . . . 6 1. 3 Meanings. Topics . . . . . . 13 0 6.4 .1 Magnetic Tape Memory Storage . 13 0 6.4.2 FoilDisk 13 1 References 13 6 7 Squeeze Film 13 7 7 .1 Basic Equations 13 8 7.2 Squeeze Between Rigid Surfaces 14 1 7.2 .1 Squeeze Without