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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.
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.
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.
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.
Über den Autor
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.
Inhaltsverzeichnis
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...
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...
Details
| Erscheinungsjahr: | 2021 |
|---|---|
| Fachbereich: | Technik allgemein |
| Genre: | Technik |
| Rubrik: | Naturwissenschaften & Technik |
| Medium: | Buch |
| Seiten: | 592 |
| Inhalt: | 592 S. |
| ISBN-13: | 9781118511497 |
| ISBN-10: | 1118511492 |
| Sprache: | Englisch |
| Einband: | Gebunden |
| Autor: | Al-Bender, Farid |
| Hersteller: | Wiley |
| Maße: | 260 x 210 x 40 mm |
| Von/Mit: | Farid Al-Bender |
| Erscheinungsdatum: | 11.01.2021 |
| Gewicht: | 1,635 kg |
Über den Autor
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.
Inhaltsverzeichnis
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...
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...
Details
| Erscheinungsjahr: | 2021 |
|---|---|
| Fachbereich: | Technik allgemein |
| Genre: | Technik |
| Rubrik: | Naturwissenschaften & Technik |
| Medium: | Buch |
| Seiten: | 592 |
| Inhalt: | 592 S. |
| ISBN-13: | 9781118511497 |
| ISBN-10: | 1118511492 |
| Sprache: | Englisch |
| Einband: | Gebunden |
| Autor: | Al-Bender, Farid |
| Hersteller: | Wiley |
| Maße: | 260 x 210 x 40 mm |
| Von/Mit: | Farid Al-Bender |
| Erscheinungsdatum: | 11.01.2021 |
| Gewicht: | 1,635 kg |
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