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Air Bearings. Theory, Design and Applications. Edition No. 1. Tribology in Practice Series

  • ID: 4464972
  • Book
  • January 2021
  • 592 Pages
  • John Wiley and Sons Ltd

Comprehensive treatise on gas bearing theory, design and application

This book treats the fundamental aspects of gas bearings of different configurations (thrust, radial, circular, conical) and operating principles (externally pressurized, self-acting, hybrid, squeeze), guiding the reader throughout the design process from theoretical modelling, design parameters, numerical formulation, through experimental characterisation and practical design and fabrication.

The book devotes a substantial part to the dynamic stability issues (pneumatic hammering, sub-synchronous whirling, active dynamic compensation and control), treating them comprehensively from theoretical and experimental points of view.

Key features:

  • Systematic and thorough treatment of the topic.
  • Summarizes relevant previous knowledge with extensive references.
  • Includes numerical modelling and solutions useful for practical application.
  • Thorough treatment of the gas-film dynamics problem including active control.
  • Discusses high-speed bearings and applications.

Air Bearings: Theory, Design and Applications
is a useful reference for academics, researchers, instructors, and design engineers.  The contents will help readers to formulate a gas-bearing problem correctly, set up the basic equations, solve them establishing the static and dynamic characteristics, utilise these to examine the scope of the design space of a given problem, and evaluate practical issues, be they in design, construction or testing.

Note: Product cover images may vary from those shown

List of contributors

List of Tables

List of Figures



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


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


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


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


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


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


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


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


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


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


10.6 Reducing or eliminating the cross-coupling

10.7 Introducing external damping

10.8 Summary


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


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



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


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



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


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


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



Note: Product cover images may vary from those shown
Farid Al-Bender
Note: Product cover images may vary from those shown