Farid Al-Bender, Katholieke Universiteit Leuven, Belgium
Dr. 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.

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...