Air Bearings

Farid Al-Bender, Katholieke Universiteit Leuven, BelgiumDr. Ir. Farid Al-Bender is Hon. Professor in the Department of Mechanical Engineering at KU Leuven, where his main areas of research included air bearing design and fabrication, tribology, friction modelling and non-linear system dynamics. He is the Director of the consultancy bureau Air Bearing Precision Technology and founder of Leuven Air Bearings company (now LAB Motion systems) where he is a board member.

All chapters are written in an authoritative yet easy-to-read manner. The introduction of similarity parameters and scale effects in different chapters and a nice blend of experimental comparisons to theoretical analyses sprinkled throughout will appeal to graduate students and researchers. In summary, this comprehensive book on air bearings is a carefully written, methodical, insightful, and welcome contribution to the tribology literature. —Michael Khonsari, Journal of Tribology, November 2021.Air bearings are a technology originally developed by the computer industry and which over time has been adopted by precision machining and by very high speed rotating machines. The monographs dedicated to this subject can be counted on the fingers of one hand and the work of Farid Al Bender is an important and welcome contribution. This book gives at the same time solid theoretical bases, presents physical models, details their mathematical formulations and describes a large variety of technical solutions. The reader is delighted by the wealth of information grouped into 17 carefully chosen chapters. —Mihai Arghir, Tribology International, November 2021.

• List of contributorsList of TablesList of FiguresPrefaceNomenclature1. Introduction1.1 Gas lubrication in perspective1.1.1 Short history1.2 Capabilities and limitations of gas lubrication1.3 When is the use of air bearings pertinent1.4 Situation of the present work1.5 Classification of air bearings for analysis purposes1.6 Structure of the book 1References2 .General Formulation and Modelling2.1 Introduction2.1.1 Qualitative description of the flow2.2 Basic equations of the flow2.2.1 Continuity equation2.2.2 Navier-Stokes momentum equation2.2.3 The (thermodynamic) Energy equation2.2.4 Equation of State2.2.5 Auxiliary conditions2.2.6 Comment on the solution of the flow problem2.3 Simplification of the flow equations2.3.1 Fluid properties and body forces2.3.2 Truncation of the flow equations2.3.3 Film flow (or channel flow)2.4 Formulation of bearing flow and pressure models2.4.1 The quasi-static flow model for axisymmetric EP bearing2.4.2 The Reynolds plus restrictor model2.5 The basic bearing characteristics2.5.1 The load carrying capacity2.5.2 The axial stiffness2.5.3 The feed mass flow rate2.5.4 The mass flow rate in the viscous region2.5.5 The tangential resistive, ”friction” force2.6 Normalization and similitude2.6.1 The axisymmetric flow problem2.6.2 Geometry2.6.3 Dimensionless parameters and similitude2.6.4 The Reynolds equation2.6.5 The bearing characteristics2.6.6 Static similarity of two bearings2.7 Methods of solution2.7.1 Analytic methods2.7.2 Semi-analytic Methods2.7.3 Purely numerical methods2.8 SummaryReferences3. Flow into the bearing gap3.1 Introduction3.2 Entrance to a parallel channel (gap) with stationary, parallel walls3.2.1 Analysis of flow development3.3 Results and discussion3.3.1 Limiting cases3.3.2 Method of solution3.3.3 Determination of the entrance length into a plane channel3.4 The case of radial flow of a polytropically compressible fluid between nominally parallel plates3.4.1 Conclusions on pressure-fed entrance3.5 Narrow channel entrance by shear-induced flow3.5.1 Stability of viscous laminar flow at the entrance3.5.2 Development of the flow upstream of a slider bearing3.5.3 Development of the flow downstream of the gap entrance3.5.4 Method of solution3.5.5 Conclusions regarding shear-induced entrance flow3.6 SummaryReferences4. Reynolds Equation: Derivation, forms and interpretations4.1 Introduction4.2 The Reynolds equation4.3 The Reynolds Equation for various film/bearing arrangements and coordinate systems4.3.1 Cartesian coordinates (x; y)4.3.2 Plain polar coordinates (r; _)4.3.3 Cylinderical coordinates (z; _) with constant R4.3.4 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 flat4.4.1 Stationary inclined upper surface, sliding lower member4.4.2 Pure surface motion4.4.3 Inclined moving upper surface with features4.4.4 Moving periodic feature on one or both surfaces4.5 Neglected flow effects4.6 Wall smoothness effects4.6.1 Effect of surface roughness4.7 Slip at the walls4.8 Turbulence4.8.1 Formulation4.9 Approximate methods for incorporating the convective terms in integral flow formulations and the modified Reynolds Equation4.9.1 Introduction4.9.2 Analysis4.9.3 Limiting solution: the Reynolds equation4.9.4 Approximate solutions to steady channel entrance problems4.9.5 Approximation of convective terms by averaging: the modified Reynolds Equation4.9.6 Approximation of convective terms by averaging in turbulent flow4.9.7 summary4.10 ClosureReferences5. Modelling of Radial Flow in Externally Pressurised Bearings5.1 Introduction5.2 Radial flow in the gap and its modelling5.3 Lumped parameter models5.3.1 The orifice/nozzle formula5.3.2 Vohr’s correlation formula5.4 Short review of other methods5.4.1 Approximation of the inertia (or convective) terms5.4.2 The momentum integral method5.4.3 Series expansion5.4.4 Pure numerical solutions5.5 Application of the method of “separation of variables”5.5.1 Boundary conditions on I5.5.2 Flow from stagnation to gap entrance5.5.3 The density function in the gap5.5.4 Solution procedure5.6 Results and discussion5.6.1 Qualitative trends5.6.2 Comparison with experiments5.7 Other comparisons5.8 Formulation of a lumped-parameter inherent compensator model5.8.1 The entrance coefficient of discharge5.8.2 Calculation of Cd5.8.3 The normalized inlet flow rate5.8.4 Solution of the static axisymmetric bearing problem by the Reynolds/compensator model5.9 SummaryReferences6. Basic Characteristics of Circular Centrally Fed Aerostatic Bearings6.1 Introduction6.2 Axial characteristics: Load, stiffness and flow6.2.1 Determination of the pressure distribution6.2.2 Typical results6.2.3 Characteristics with given supply pressure6.2.4 Conclusions on axial characteristics6.3 Tilt and misalignment characteristics (Al-Bender 1992; Al-Bender andVan Brussel 1992)6.3.1 Analysis6.3.2 Theoretical results6.3.3 Experimental investigation6.3.4 Results, comparison and discussion6.3.5 Conclusions on tilt6.4 The influence of relative sliding velocity on aerostatic bearing characteristics(Al-Bender 1992)6.4.1 Formulation of the problem6.4.2 Qualitative considerations of the influence of relative velocity6.4.3 Solution method6.4.4 Results and discussion6.4.5 Conclusions on relative sliding6.5 SummaryReferences7. Dynamic Characteristics of Circular Centrally Fed Aerostatic Bearing Films, and the Problem of Pneumatic Stability7.1 Introduction7.1.1 Pneumatic instability7.1.2 Squeeze film7.1.3 Active compensation7.1.4 Objeetives and layout of this study7.2 Review of past treatments7.2.1 Models and theory7.2.2 System analysis tools and stability criteria7.2.3 Methods of stabilization7.2.4 Discussion and evaluation7.3 Formulation of the linearized model7.3.1 Basic assumptions7.3.2 Basic equations7.3.3 The perturbation procedure7.3.4 Range of validity of the proposed model7.3.5 Special and limiting cases7.4 Solution7.4.1 Integration of the linearized Reynolds Equation7.4.2 Bearing dynamic characteristics7.5 Results and discussion7.5.1 General characteristics and Similitude7.5.2 The supply pressure response Kp7.5.3 Comparison with experiment7.6 SummaryReferences8. Aerodynamic action: Self-acting bearing principles and configurations8.1 Introduction8.2 The aerodynamic action and the effect of compressibility8.3 Self-acting or EP Bearings?8.3.1 Energy efficiency of self-acting bearings8.3.2 The viscous motor8.4 Dimensionless formulation of the Reynolds equation8.5 Some basic aerodynamic bearing configurations8.5.1 Slider bearings8.6 Grooved-surface bearings8.6.1 Derivation of the Narrow-Groove Theory (NGT) equation forgrooved bearings8.6.2 Assumptions8.6.3 Flow in the x-direction8.6.4 Flow in the y-direction8.6.5 Squeeze volume8.6.6 Inclined-grooves Reynolds equation8.6.7 Globally compressible Reynolds equation8.6.8 The case when both surfaces are moving8.6.9 Discussion and properties of the solution8.6.10 The case of stationary grooves versus that of moving grooves8.6.11 Grooved bearing embodiments8.7 Rotary bearings8.7.1 Journal bearings8.8 Dynamic characteristics8.9 Similarity and scale effects8.10 Hybrid bearings8.11 summaryReferences9. Journal Bearings9.1 Introduction9.1.1 Geometry and Notation9.1.2 Basic Equation9.2 Basic JB characteristics9.3 Plain Self-acting9.3.1 Small-eccentricity perturbation static-pressure solution9.3.2 Dynamic characteristics9.4 Dynamic stability of a JB and the problem of half-speed whirl9.4.1 General numerical solution9.5 Herringbone Grooved Journal Bearings (HGJB)9.5.1 Static characteristics9.5.2 Dynamic characteristics9.6 EP Journal Bearings9.6.1 Single feed plane9.6.2 Other possible combinations9.7 Hybrid JB’s9.8 Comparison of the three types in regard to whirl critical mass9.9 SummaryReferences10. Dynamic Whirling Behaviour and the Rotordynamic Stability Problem10.1 Introduction10.2 The nature and classification of whirl motion10.2.1 Synchronous whirl10.2.2 Self-excited whirl10.3 Study of the self-excited whirling phenomenon10.3.1 Description and terminology10.3.2 Half-speed whirl in literature10.3.3 Sensitivity analysis to identify the relevant parameters10.4 Techniques for enhancing stability10.4.1 Literature overview on current techniques10.5 Optimum Design of Externally Pressurised Journal Bearings for High-SpeedApplications10.6 Reducing or eliminating the cross-coupling10.7 Introducing external damping10.8 SummaryReferences  11. Tilting Pad Air Bearings11.1 Introduction11.2 Plane slider bearing11.3 Pivoted pad slider bearing11.3.1 Equivalent bearing stiffness11.4 Tilting pad journal bearing11.4.1 Steady state bearing characteristics11.4.2 Dynamic stiffness of a tilting pad bearing11.5 Dynamic stability11.6 Construction and fabrication aspects11.7 SummaryReferences12. Foil Bearings12.1 Introduction12.2 Compliant material foil bearings: state-of-the-art12.2.1 Early foil bearing developments12.2.2 Recent advances in macro scale foil bearings12.2.3 Recent advances in mesoscopic foil bearings12.3 Self-acting tension foil bearing12.3.1 Effect of foil stiffness12.4 Externally-pressurised tension foil bearing12.4.1 Theoretical Analysis12.4.2 Practical Design of a Prototype12.4.3 Experimental Validation12.5 Bump foil bearing12.5.1 Modeling of a foil bearing with an idealised mechanical structure12.6 Numerical analysis methods for the (compliant) Reynolds equation12.7 Steady-state simulation with FDM and Newton-Raphson12.7.1 Different algorithms to implement the JFO boundary conditions infoil bearings12.7.2 Simulation procedure12.7.3 Steady-state simulation results & discussion12.8 Steady-state properties12.8.1 Load capacity and attitude angle12.8.2 Minimum gap height in middle bearing plane and maximum load capacity12.8.3 Thermal phenomena in foil bearings & cooling air12.8.4 Variable flexible element stiffness and bilinear springs12.8.5 Geometrical preloading12.9 Dynamic properties12.9.1 Dynamic properties calculation with the perturbation method12.9.2 Stiffness and damping coefficients12.9.3 Influence of compliant structure dynamics on bearing characteristics12.9.4 Structural damping in real foil bearings12.10Bearing stability12.10.1 Bearing stability equations12.10.2 Foil bearing stability maps12.10.3 Fabrication Technology12.11SummaryReferences13 .Porous Bearings13.1 Introduction13.2 Modelling of porous bearing13.2.1 Feed flow: Darcy’s law13.2.2 Film flow: modified Reynolds equation13.2.3 Boundary conditions for the general case13.2.4 Solution procedure13.3 Static bearing characteristics13.4 Dynamic bearing characteristics13.5 Dynamic film coefficients13.6 Normalisation13.6.1 Aerostatic porous journal bearing13.6.2 Aerostatic porous thrust bearing13.7 Validation of the numerical models13.8 SummaryReferences14 .Hanging Air Bearings and the Over-expansion Method14.1 Introduction14.2 Outline14.2.1 Problem statement14.2.2 Possible solutions14.2.3 Choice of a solution14.3 Problem formulation14.4 Theoretical analysis14.4.1 Basic assumptions14.4.2 Basic equations and definitions14.4.3 Derivation of the pressure equations14.4.4 Normalisation of the final equations14.4.5 Solution procedure14.4.6 Matching the solution with experiment: empirical parameter values14.5 Experimental verification14.5.1 Test apparatus14.5.2 Range of tests14.6 Bearing Characteristics and Optimization14.7 Design methodology14.8 Other details14.9 Brief comparison of the three hanging-bearing solutions14.10Aerodynamic hanging bearings14.10.1 Inclined and tilting pad case14.11SummaryReferences15. Actively Compensated Gas Bearings15.1 Introduction15.2 Essentials of active bearing film compensation15.3 An active bearing prototype with centrally clamped plate surface15.3.1 Simulation model of active air bearing system with conicity control15.3.2 Tests, results and discussion of the active air bearing system15.3.3 Conclusions15.4 Active milling electro-spindle15.4.1 Context sketch15.4.2 Specifications of the spindles15.4.3 Spindle with passive air bearings15.4.4 Active spindle15.4.5 Repetitive Controller design and results15.5 Active manipulation of substrates in the plane of the film15.6 Squeeze-film (SF) bearings15.6.1 Other configurations15.6.2 Assessment of possible inertia effects15.6.3 Ultrasonic levitation and acoustic bearings15.7 SummaryReferences16. Design of an active aerostatic slide16.1 Introduction16.2 A multiphysics active bearing model16.2.1 General formulation of the model16.2.2 Structural flexibility16.2.3 Fluid dynamics16.2.4 Dynamics of the moving elements16.2.5 Piezoelectric actuators16.2.6 Controller16.2.7 Coupled formulation of the model16.3 Bearing performance and model validation16.3.1 Test setup for active aerostatic bearings16.3.2 Active bearing performance and model validation16.3.3 Discussion on the validity of the model16.3.4 Analysis of the relevance of model coupling16.4 Active aerostatic slide16.4.1 Design of the active slide prototype16.4.2 Identification of active slide characteristics16.4.3 Active performance16.5 SummaryReferences17. On the Thermal Characteristics of the Film Flow17.1 Introduction17.2 Basic considerations17.2.1 Isothermal walls17.2.2 Adiabatic walls17.2.3 one adiabatic wall and one isothermal wall17.3 Adiabatic-wall Reynolds equation and the thermal wedge17.3.1 Results and discussion17.3.2 Effect of temperature on gas properties17.3.3 Conclusions on the aeordynamic case17.4 Flow through centrally fed bearing: formulation of the problem17.5 Method of solution17.5.1 Solutions17.6 Results and discussion17.7 SummaryReferencesIndex