Thorough
Trade Review
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.
Table of Contents
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
