Air Bearings

List of contributors List of Tables List of Figures Preface Nomenclature 1. Introduction 1.1 Gas lubrication in perspective 1.1.1 Short history 1.2 Capabilities and limitations of gas lubrication 1.3 When is the use of air bearings pertinent 1.4 Situation of the present work 1.5 Classification of air bearings for analysis purposes 1.6 Structure of the book 1 References 2 .General Formulation and Modelling 2.1 Introduction 2.1.1 Qualitative description of the flow 2.2 Basic equations of the flow 2.2.1 Continuity equation 2.2.2 Navier-Stokes momentum equation 2.2.3 The (thermodynamic) Energy equation 2.2.4 Equation of State 2.2.5 Auxiliary conditions 2.2.6 Comment on the solution of the flow problem 2.3 Simplification of the flow equations 2.3.1 Fluid properties and body forces 2.3.2 Truncation of the flow equations 2.3.3 Film flow (or channel flow) 2.4 Formulation of bearing flow and pressure models 2.4.1 The quasi-static flow model for axisymmetric EP bearing 2.4.2 The Reynolds plus restrictor model 2.5 The basic bearing characteristics 2.5.1 The load carrying capacity 2.5.2 The axial stiffness 2.5.3 The feed mass flow rate 2.5.4 The mass flow rate in the viscous region 2.5.5 The tangential resistive, ”friction” force 2.6 Normalization and similitude 2.6.1 The axisymmetric flow problem 2.6.2 Geometry 2.6.3 Dimensionless parameters and similitude 2.6.4 The Reynolds equation 2.6.5 The bearing characteristics 2.6.6 Static similarity of two bearings 2.7 Methods of solution 2.7.1 Analytic methods 2.7.2 Semi-analytic Methods 2.7.3 Purely numerical methods 2.8 Summary References 3. Flow into the bearing gap 3.1 Introduction 3.2 Entrance to a parallel channel (gap) with stationary, parallel walls 3.2.1 Analysis of flow development 3.3 Results and discussion 3.3.1 Limiting cases 3.3.2 Method of solution 3.3.3 Determination of the entrance length into a plane channel 3.4 The case of radial flow of a polytropically compressible fluid between nominally parallel plates 3.4.1 Conclusions on pressure-fed entrance 3.5 Narrow channel entrance by shear-induced flow 3.5.1 Stability of viscous laminar flow at the entrance 3.5.2 Development of the flow upstream of a slider bearing 3.5.3 Development of the flow downstream of the gap entrance 3.5.4 Method of solution 3.5.5 Conclusions regarding shear-induced entrance flow 3.6 Summary References 4. Reynolds Equation: Derivation, forms and interpretations 4.1 Introduction 4.2 The Reynolds equation 4.3 The Reynolds Equation for various film/bearing arrangements and coordinate systems 4.3.1 Cartesian coordinates (x; y) 4.3.2 Plain polar coordinates (r; _) 4.3.3 Cylinderical coordinates (z; _) with constant R 4.3.4 Conical coordinates (r; _) (_ = _ = constant) 4.3.5 Spherical coordinates (_; _) (r = R = constant) 4.4 Interpretation of the Reynolds Equation when both surfaces are moving and not flat 4.4.1 Stationary inclined upper surface, sliding lower member 4.4.2 Pure surface motion 4.4.3 Inclined moving upper surface with features 4.4.4 Moving periodic feature on one or both surfaces 4.5 Neglected flow effects 4.6 Wall smoothness effects 4.6.1 Effect of surface roughness 4.7 Slip at the walls 4.8 Turbulence 4.8.1 Formulation 4.9 Approximate methods for incorporating the convective terms in integral flow formulations and the modified Reynolds Equation 4.9.1 Introduction 4.9.2 Analysis 4.9.3 Limiting solution: the Reynolds equation 4.9.4 Approximate solutions to steady channel entrance problems 4.9.5 Approximation of convective terms by averaging: the modified Reynolds Equation 4.9.6 Approximation of convective terms by averaging in turbulent flow 4.9.7 summary 4.10 Closure References 5. Modelling of Radial Flow in Externally Pressurised Bearings 5.1 Introduction 5.2 Radial flow in the gap and its modelling 5.3 Lumped parameter models 5.3.1 The orifice/nozzle formula 5.3.2 Vohr’s correlation formula 5.4 Short review of other methods 5.4.1 Approximation of the inertia (or convective) terms 5.4.2 The momentum integral method 5.4.3 Series expansion 5.4.4 Pure numerical solutions 5.5 Application of the method of “separation of variables” 5.5.1 Boundary conditions on I 5.5.2 Flow from stagnation to gap entrance 5.5.3 The density function in the gap 5.5.4 Solution procedure 5.6 Results and discussion 5.6.1 Qualitative trends 5.6.2 Comparison with experiments 5.7 Other comparisons 5.8 Formulation of a lumped-parameter inherent compensator model 5.8.1 The entrance coefficient of discharge 5.8.2 Calculation of Cd 5.8.3 The normalized inlet flow rate 5.8.4 Solution of the static axisymmetric bearing problem by the Reynolds/compensator model 5.9 Summary References 6. Basic Characteristics of Circular Centrally Fed Aerostatic Bearings 6.1 Introduction 6.2 Axial characteristics: Load, stiffness and flow 6.2.1 Determination of the pressure distribution 6.2.2 Typical results 6.2.3 Characteristics with given supply pressure 6.2.4 Conclusions on axial characteristics 6.3 Tilt and misalignment characteristics (Al-Bender 1992; Al-Bender and Van Brussel 1992) 6.3.1 Analysis 6.3.2 Theoretical results 6.3.3 Experimental investigation 6.3.4 Results, comparison and discussion 6.3.5 Conclusions on tilt 6.4 The influence of relative sliding velocity on aerostatic bearing characteristics (Al-Bender 1992) 6.4.1 Formulation of the problem 6.4.2 Qualitative considerations of the influence of relative velocity 6.4.3 Solution method 6.4.4 Results and discussion 6.4.5 Conclusions on relative sliding 6.5 Summary References 7. Dynamic Characteristics of Circular Centrally Fed Aerostatic Bearing Films, and the Problem of Pneumatic Stability 7.1 Introduction 7.1.1 Pneumatic instability 7.1.2 Squeeze film 7.1.3 Active compensation 7.1.4 Objeetives and layout of this study 7.2 Review of past treatments 7.2.1 Models and theory 7.2.2 System analysis tools and stability criteria 7.2.3 Methods of stabilization 7.2.4 Discussion and evaluation 7.3 Formulation of the linearized model 7.3.1 Basic assumptions 7.3.2 Basic equations 7.3.3 The perturbation procedure 7.3.4 Range of validity of the proposed model 7.3.5 Special and limiting cases 7.4 Solution 7.4.1 Integration of the linearized Reynolds Equation 7.4.2 Bearing dynamic characteristics 7.5 Results and discussion 7.5.1 General characteristics and Similitude 7.5.2 The supply pressure response Kp 7.5.3 Comparison with experiment 7.6 Summary References 8. Aerodynamic action: Self-acting bearing principles and configurations 8.1 Introduction 8.2 The aerodynamic action and the effect of compressibility 8.3 Self-acting or EP Bearings? 8.3.1 Energy efficiency of self-acting bearings 8.3.2 The viscous motor 8.4 Dimensionless formulation of the Reynolds equation 8.5 Some basic aerodynamic bearing configurations 8.5.1 Slider bearings 8.6 Grooved-surface bearings 8.6.1 Derivation of the Narrow-Groove Theory (NGT) equation for grooved bearings 8.6.2 Assumptions 8.6.3 Flow in the x-direction 8.6.4 Flow in the y-direction 8.6.5 Squeeze volume 8.6.6 Inclined-grooves Reynolds equation 8.6.7 Globally compressible Reynolds equation 8.6.8 The case when both surfaces are moving 8.6.9 Discussion and properties of the solution 8.6.10 The case of stationary grooves versus that of moving grooves 8.6.11 Grooved bearing embodiments 8.7 Rotary bearings 8.7.1 Journal bearings 8.8 Dynamic characteristics 8.9 Similarity and scale effects 8.10 Hybrid bearings 8.11 summary References 9. Journal Bearings 9.1 Introduction 9.1.1 Geometry and Notation 9.1.2 Basic Equation 9.2 Basic JB characteristics 9.3 Plain Self-acting 9.3.1 Small-eccentricity perturbation static-pressure solution 9.3.2 Dynamic characteristics 9.4 Dynamic stability of a JB and the problem of half-speed whirl 9.4.1 General numerical solution 9.5 Herringbone Grooved Journal Bearings (HGJB) 9.5.1 Static characteristics 9.5.2 Dynamic characteristics 9.6 EP Journal Bearings 9.6.1 Single feed plane 9.6.2 Other possible combinations 9.7 Hybrid JB’s 9.8 Comparison of the three types in regard to whirl critical mass 9.9 Summary References 10. Dynamic Whirling Behaviour and the Rotordynamic Stability Problem 10.1 Introduction 10.2 The nature and classification of whirl motion 10.2.1 Synchronous whirl 10.2.2 Self-excited whirl 10.3 Study of the self-excited whirling phenomenon 10.3.1 Description and terminology 10.3.2 Half-speed whirl in literature 10.3.3 Sensitivity analysis to identify the relevant parameters 10.4 Techniques for enhancing stability 10.4.1 Literature overview on current techniques 10.5 Optimum Design of Externally Pressurised Journal Bearings for High-Speed Applications 10.6 Reducing or eliminating the cross-coupling 10.7 Introducing external damping 10.8 Summary References   11. Tilting Pad Air Bearings 11.1 Introduction 11.2 Plane slider bearing 11.3 Pivoted pad slider bearing 11.3.1 Equivalent bearing stiffness 11.4 Tilting pad journal bearing 11.4.1 Steady state bearing characteristics 11.4.2 Dynamic stiffness of a tilting pad bearing 11.5 Dynamic stability 11.6 Construction and fabrication aspects 11.7 Summary References 12. Foil Bearings 12.1 Introduction 12.2 Compliant material foil bearings: state-of-the-art 12.2.1 Early foil bearing developments 12.2.2 Recent advances in macro scale foil bearings 12.2.3 Recent advances in mesoscopic foil bearings 12.3 Self-acting tension foil bearing 12.3.1 Effect of foil stiffness 12.4 Externally-pressurised tension foil bearing 12.4.1 Theoretical Analysis 12.4.2 Practical Design of a Prototype 12.4.3 Experimental Validation 12.5 Bump foil bearing 12.5.1 Modeling of a foil bearing with an idealised mechanical structure 12.6 Numerical analysis methods for the (compliant) Reynolds equation 12.7 Steady-state simulation with FDM and Newton-Raphson 12.7.1 Different algorithms to implement the JFO boundary conditions in foil bearings 12.7.2 Simulation procedure 12.7.3 Steady-state simulation results & discussion 12.8 Steady-state properties 12.8.1 Load capacity and attitude angle 12.8.2 Minimum gap height in middle bearing plane and maximum load capacity 12.8.3 Thermal phenomena in foil bearings & cooling air 12.8.4 Variable flexible element stiffness and bilinear springs 12.8.5 Geometrical preloading 12.9 Dynamic properties 12.9.1 Dynamic properties calculation with the perturbation method 12.9.2 Stiffness and damping coefficients 12.9.3 Influence of compliant structure dynamics on bearing characteristics 12.9.4 Structural damping in real foil bearings 12.10Bearing stability 12.10.1 Bearing stability equations 12.10.2 Foil bearing stability maps 12.10.3 Fabrication Technology 12.11Summary References 13 .Porous Bearings 13.1 Introduction 13.2 Modelling of porous bearing 13.2.1 Feed flow: Darcy’s law 13.2.2 Film flow: modified Reynolds equation 13.2.3 Boundary conditions for the general case 13.2.4 Solution procedure 13.3 Static bearing characteristics 13.4 Dynamic bearing characteristics 13.5 Dynamic film coefficients 13.6 Normalisation 13.6.1 Aerostatic porous journal bearing 13.6.2 Aerostatic porous thrust bearing 13.7 Validation of the numerical models 13.8 Summary References 14 .Hanging Air Bearings and the Over-expansion Method 14.1 Introduction 14.2 Outline 14.2.1 Problem statement 14.2.2 Possible solutions 14.2.3 Choice of a solution 14.3 Problem formulation 14.4 Theoretical analysis 14.4.1 Basic assumptions 14.4.2 Basic equations and definitions 14.4.3 Derivation of the pressure equations 14.4.4 Normalisation of the final equations 14.4.5 Solution procedure 14.4.6 Matching the solution with experiment: empirical parameter values 14.5 Experimental verification 14.5.1 Test apparatus 14.5.2 Range of tests 14.6 Bearing Characteristics and Optimization 14.7 Design methodology 14.8 Other details 14.9 Brief comparison of the three hanging-bearing solutions 14.10Aerodynamic hanging bearings 14.10.1 Inclined and tilting pad case 14.11Summary References 15. Actively Compensated Gas Bearings 15.1 Introduction 15.2 Essentials of active bearing film compensation 15.3 An active bearing prototype with centrally clamped plate surface 15.3.1 Simulation model of active air bearing system with conicity control 15.3.2 Tests, results and discussion of the active air bearing system 15.3.3 Conclusions 15.4 Active milling electro-spindle 15.4.1 Context sketch 15.4.2 Specifications of the spindles 15.4.3 Spindle with passive air bearings 15.4.4 Active spindle 15.4.5 Repetitive Controller design and results 15.5 Active manipulation of substrates in the plane of the film 15.6 Squeeze-film (SF) bearings 15.6.1 Other configurations 15.6.2 Assessment of possible inertia effects 15.6.3 Ultrasonic levitation and acoustic bearings 15.7 Summary References 16. Design of an active aerostatic slide 16.1 Introduction 16.2 A multiphysics active bearing model 16.2.1 General formulation of the model 16.2.2 Structural flexibility 16.2.3 Fluid dynamics 16.2.4 Dynamics of the moving elements 16.2.5 Piezoelectric actuators 16.2.6 Controller 16.2.7 Coupled formulation of the model 16.3 Bearing performance and model validation 16.3.1 Test setup for active aerostatic bearings 16.3.2 Active bearing performance and model validation 16.3.3 Discussion on the validity of the model 16.3.4 Analysis of the relevance of model coupling 16.4 Active aerostatic slide 16.4.1 Design of the active slide prototype 16.4.2 Identification of active slide characteristics 16.4.3 Active performance 16.5 Summary References 17. On the Thermal Characteristics of the Film Flow 17.1 Introduction 17.2 Basic considerations 17.2.1 Isothermal walls 17.2.2 Adiabatic walls 17.2.3 one adiabatic wall and one isothermal wall 17.3 Adiabatic-wall Reynolds equation and the thermal wedge 17.3.1 Results and discussion 17.3.2 Effect of temperature on gas properties 17.3.3 Conclusions on the aeordynamic case 17.4 Flow through centrally fed bearing: formulation of the problem 17.5 Method of solution 17.5.1 Solutions 17.6 Results and discussion 17.7 Summary References Index