Air Bearings E-Book

Farid Al-Bender

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Tribology in Practice Series (PEP)
• Sprache: Englisch

Seitenzahl: 1057

Veröffentlichungsjahr: 2021

Beschreibung

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.

Leseprobe

Air Bearings

Theory, Design and Applications

Farid Al‐Bender

KU Leuven, Department of Mechanical EngineeringLeuvenBelgium

This edition first published 2021

© 2021 John Wiley & Sons Ltd

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The right of Farid Al‐Bender to be identified as the author of this work has been asserted in accordance with law.

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Library of Congress Cataloging‐in‐Publication Data applied for

HB ISBN: 9781118511497

Cover image: Courtesy of Farid Al‐Bender

Cover design by Wiley

To the memory of my parents

To Katherine

To the children

List of Contributors

Name

Affiliation

Contribution

Tobias Waumans

Xeryon, Leuven, Belgium

Chapter 10

Peter Vleugels

ASML BV, Netherlands

Major part of

Chapter 12

Steven Cappa

CAPPA precision engineering, Belgium

Major part of

Chapter 13

Gorka Aguirre

IDEKO, Spain

Chapter 16

Marius Nabuurs

KU Leuven, Belgium

Chapter 11

List of Figures

Figure 1.1

Classification of air bearings according to pressure generation (dynamics) and morphology (kinematics). The bottom row depicts compliant‐surface bearings.

Figure 2.1

General bearing configuration and notation (

and

denote respectively the supply and atmospheric pressure). Source: Adapted from Al‐Bender F 1992.

Figure 2.2

Schematic flow configuration of inlet flow to EP bearing. (Not to scale.) Source: Al‐Bender F 1992.

Figure 2.3

(a) Feed flow and its transition to (b) film flow.

Figure 2.4

Comparison between potential flow in the entrance region and ideal sink flow. Source: Al‐Bender F 1992.

Figure 2.5

Entrance flow into a slider bearing: open shear flow transition to Couette–Poiseuille channel flow.

Figure 2.6

Film flow configuration for EP case. (Not to scale.) Source: Al‐Bender F 1992.

Figure 2.7

Film flow configuration for the slider‐bearing case. (Not to scale.)

Figure 2.8

Circular centrally fed bearing geometry and notation. Source: Al‐Bender F 1992

.

Figure 3.1

Schematic and terminology of the entrance problem. (Not to scale.)

Figure 3.2

A method of “separation of variables” for the solution of laminar boundary layer equations of narrow channel flows. Journal of Tribology 114, 630–636. 1992 by permission of ASME.

Figure 3.3

Velocity profile functions in forward and reverse flow. Source: Al-Bender and Van Brussel, 1992 by permission of ASME.

Figure 3.4

. Source: Al-Bender and Van Brussel, 1992 by permission of ASME.

Figure 3.5

Velocity and pressure development in the entrance of a plane channel. Source: Al-Bender and Van Brussel, 1992 by permission of ASME.

Figure 3.6

Radial channel flow: Notation. Source: Al-Bender and Van Brussel, 1992 by permission of ASME.

Figure 3.7

Qualitative streamline field for (a) moderate taper (large flow rate) and (b) large taper (small flow rate) sliders and associated entrance velocity profiles.

Figure 3.8

Development of the flow upstream of a slider bearing; Blasius BL (a) versus Skiadis BL (b). (Not to scale.)

Figure 3.9

Normalised velocity profile of the Sakiadis boundary layer.

Figure 3.10

Shear‐flow entrance into a plane uniform gap channel. Development of (a) velocity profile and (b) pressure (including head loss). (Not to scale)

Figure 3.11

Velocity profile and its two components (a) just downstream of entrance and (b) Fully developed.

Figure 3.12

Shear‐flow entrance parameters (a) head loss, (b) entrance length and (c) coefficient of discharge as a function of

.

Figure 3.13

Shear‐flow entrance parameters for small values of

(a) head loss, (b) entrance length and (c) coefficient of discharge as a function of

. Note that the pressure is expressed as

to facilitate evaluation and comparison.

Figure 3.14

Shear‐flow entrance into a plane uniform gap channel with reverse flow component. Development of (a) velocity profile and (b) pressure (including head gain). (Not to scale)

Figure 3.15

Shear‐flow entrance parameters from left to right: head gain, entrance length and

as a function of

. Note that the pressure is expressed as

to facilitate evaluation and comparison.

Figure 3.16

Global pressure development, upstream and downstream the gap entrance.

Figure 4.1

Control volume of the Reynolds Equation.

Figure 4.2

Polar and spherical coordinates.

Figure 4.3

Fixed inclined upper surface and moving flat lower one.

Figure 4.4

Pure surface motion with various surface velocities.

Figure 4.5

Inclined moving upper surface (with possible features).

Figure 4.6

Moving and stationary grooves.

Figure 4.7

Moving and stationary grooves with sliding surface.

Figure 4.8

Moving and stationary grooves with sliding surface: the two situations are not equivalent.

Figure 4.9

Probable flow pattern at the transition from land to groove. Source: Van der Stegen RHM 1997.

Figure 4.10

Schematic of the 2‐D flow configuration.

Figure 4.11

The boundary‐layer velocity profile.

Figure 4.12

Profile function

for different values of

.

Figure 4.13

Pressure distribution in an inclined slider with and without inertia effects; large

.

Figure 4.14

Schematic of squeeze film bearing configuration.

Figure 5.1

Various restrictor types.

Figure 5.2

Entrance region and notation. (Source: Al‐Bender and Van Brussel

1992

by permission of ASME.)

Figure 5.3

Qualitative development of the velocity profile integral function in radial channel flow. Source: Al‐Bender and Van Brussel 1992 by permission of ASME.

Figure 5.4

Comparison of pressure distributions for converging and diverging incompressible radial flow. Experimental data points adapted from McGinn (

1955

). Source: Al‐Bender F and Van Brussel H 1992 by permission of ASME.

Figure 5.5

Comparison of pressure distributions for a bearing with large radius ratio. Experimental data points adapted from Lowe (

1970

). Source: Al‐Bender F and Van Brussel H 1992 by permission of ASME.

Figure 5.6

Comparison of pressure distributions for a bearing with large radius ratio. Experimental data points adapted from Mori (

1969

). Source: Al‐Bender F and Van Brussel H 1992 by permission of ASME.

Figure 5.7

Comparison of pressure distributions for a bearing with small radius ratio. Experimental data points adapted from Mori (

1969

). Source: Al‐Bender F and Van Brussel H 1992 by permission of ASME.

Figure 5.8

Comparison of pressure distributions for a bearing with small radius ratio. Experimental data points adapted from Vohr (

1966

). Source: Al‐Bender F and Van Brussel H 1992 by permission of ASME.

Figure 5.9

Comparison of pressure distributions for a convergent gap air bearing. Source: Al‐Bender F and Van Brussel H 1992 by permission of ASME.

Figure 5.10

Comparison of pressure profile for an inherently restricted bearing (detail view provided in Figure 5.11).

,

for

µm;

,

for

µm. Experimental data from Belforte et al. (

2007

). Source: Waumans T, Al‐Bender F and Reynaerts D 2008 by permission of ASME.

Figure 5.11

Detail view of Figure 5.10. Source: Waumans T, Al‐Bender F and Reynaerts D 2008 by permission of ASME.

Figure 5.12

Comparison of pressure profile for an orifice restricted bearing with a shallow feeding pocket. Experimental data from Belforte et al. (

2007

). Source: Waumans T, Al‐Bender F and Reynaerts D 2008 by permission of ASME.

Figure 5.13

Comparison of pressure profile for an orifice restricted bearing with a feeding pocket. Experimental data from Belforte et al. (

2007

). Source: Waumans T, Al‐Bender F and Reynaerts D 2008 by permission of ASME.

Figure 5.14

Calculated values of the Coefficient of discharge

. Source: Al-Bender F 1992.

Figure 6.1

Geometry and notation. Source: Al‐Bender F 1992.

Figure 6.2

Normalised pressure distribution corresponding to incompressible fluid flow for

.

Figure 6.3

Pressure distributions for

and three values of

, for compressible flow.

Figure 6.4

Pressure distributions for

and, from bottom to top,

.

Figure 6.5

Clearer presentation: the effect of higher

. Curves from bottom to top are for

.

Figure 6.6

Pressure distribution for different conicities, from bottom to top,

.

Figure 6.7

The ratio of dimensionless load‐to‐flow as a function of the dimensionless conicity, when the minimum gap

is taken as the nominal gap for calculating the flow rate.

Figure 6.8

The ratio of dimensionless load‐to‐flow as a function of the dimensionless conicity, when the mean gap

is taken as the nominal gap for calculating the flow rate.

Figure 6.9

The dimensionless load and flow as a function of the dimensionless conicity.

Figure 6.10

Dimensional load as a function of the gap‐height and concity for bearing with

bar;

bar;

mm;

mm.

Figure 6.11

Dimensional flow rate as a function of the gap‐height and conicity for bearing with

bar;

bar;

mm;

mm.

Figure 6.12

The dimensionless bearing load as a function of the dimensionless feed number

(or

) and the dimensionless conicity

.

Figure 6.13

The dimensionless bearing flow as a function of the dimensionless feed number

(or

and the dimensionless conicity

.

Figure 6.14

The dimensionless bearing load‐to‐flow ratio as a function of the dimensionless feed number

(or

) and the dimensionless conicity

. The ratio increases with

and decreases with

, when the minimum gap

is taken as the nominal gap for calculating the flow rate.

Figure 6.15

The dimensionless bearing load‐to‐flow ratio as a function of the dimensionless feed number

(or

) and the dimensionless conicity

. The ratio increases with

and also increases with

, when the mean gap

is taken as the nominal gap for calculating the flow rate.

Figure 6.16

The dimensionless bearing stiffness as a function of the dimensionless feed number

(or

) and the dimensionless conicity

.

Figure 6.17

The load increases monotonically with both

and

.

Figure 6.18

The flow increases monotonically with both

and

.

Figure 6.19

The load‐to‐flow ratio decreases with

(especially for small

). It also decreases with

, when the minimum gap

is taken as the nominal gap for calculating the flow rate.

Figure 6.20

The load‐to‐flow ratio decreases with

(especially for small

). It increases with

, when the mean gap

is taken as the nominal gap for calculating the flow rate.

Figure 6.21

The dimensionless stiffness as a function of

and

. For any given

, there is a unique value of

that yields maximum stiffness. As

increases, the value of the optimum

increases.

Figure 6.22

Parametrised graphs corresponding to the previous figure to facilitate reading the values.

Figure 6.23

Tilt notation. Source: Republished with permission from Elsevier, from Al‐Bender F and Van Brussel H 1992; permission conveyed through Copyright Clearance Center, Inc.

Figure 6.24

Static tilt stiffness‐pressure distribution. Source: Republished with permission from Elsevier, from Al‐Bender F and Van Brussel H 1992; permission conveyed through Copyright Clearance Center, Inc.

Figure 6.25

Static tilt damping‐pressure distribution. Source: Republished with permission from Elsevier, from Al‐Bender F and Van Brussel H 1992; permission conveyed through Copyright Clearance Center, Inc.

Figure 6.26

Dynamic tilt stiffness‐pressure distribution. Source: Republished with permission from Elsevier, from Al‐Bender F and Van Brussel H 1992; permission conveyed through Copyright Clearance Center, Inc.

Figure 6.27

Dynamic tilt damping‐pressure distribution. Source: Republished with permission from Elsevier, from Al‐Bender F and Van Brussel H 1992; permission conveyed through Copyright Clearance Center, Inc.

Figure 6.28

Static tilt stiffness versus load capacity. Source: Republished with permission from Elsevier, from Al‐Bender F and Van Brussel H 1992; permission conveyed through Copyright Clearance Center, Inc.

Figure 6.29

Static tilt damping coefficient versus load capacity. Source: Republished with permission from Elsevier, from Al‐Bender F and Van Brussel H 1992; permission conveyed through Copyright Clearance Center, Inc.

Figure 6.30

Influence of the conicity on the static stiffness and damping. Source: Republished with permission from Elsevier, from Al‐Bender F and Van Brussel H 1992; permission conveyed through Copyright Clearance Center, Inc.

Figure 6.31

Dynamic tilt Nyquist diagram. Source: Republished with permission from Elsevier, from Al‐Bender F and Van Brussel H 1992; permission conveyed through Copyright Clearance Center, Inc.

Figure 6.32

Test apparatus. Source: Republished with permission from Elsevier, from Al‐Bender F and Van Brussel H 1992; permission conveyed through Copyright Clearance Center, Inc.

Figure 6.33

The different test types. Source: Republished with permission from Elsevier, from Al‐Bender F and Van Brussel H 1992; permission conveyed through Copyright Clearance Center, Inc.

Figure 6.34

Comparison of stiffness values (flat bearings.) Source: Republished with permission from Elsevier, from Al‐Bender F and Van Brussel H 1992; permission conveyed through Copyright Clearance Center, Inc.

Figure 6.35

Comparison of damping values (flat bearings.) Source: Republished with permission from Elsevier, from Al‐Bender F and Van Brussel H 1992; permission conveyed through Copyright Clearance Center, Inc.

Figure 6.36

Comparison of stiffness values (convergent bearings). Source: Republished with permission from Elsevier, from Al‐Bender F and Van Brussel H 1992; permission conveyed through Copyright Clearance Center, Inc.

Figure 6.37

Comparison of damping values (convergent bearings). Source: Republished with permission from Elsevier, from Al‐Bender F and Van Brussel H 1992; permission conveyed through Copyright Clearance Center, Inc.

Figure 6.38

Plane bearing with relative sliding velocity. Source: Al‐Bender F 1992.

Figure 6.39

The influence of sliding on the pressure distribution. Source: Al‐Bender F 1992.

Figure 6.40

The influence of sliding on the pressure distribution obtained from the F.D. solution.

.

Figure 6.41

The influence of sliding on the pressure distribution obtained from the F.D. solution.

.

Figure 6.42

The influence of sliding on the load carrying capacity. Source: Al‐Bender F 1992.

Figure 6.43

The influence of sliding on the mass flow rate. Source: Al‐Bender F 1992.

Figure 6.44

The influence of sliding on the centre of force. Source: Al‐Bender F 1992.

Figure 6.45

The friction force as a function of the conicity. Source: Al‐Bender F 1992.

Figure 6.46

The normalised coefficient of friction as a function of the sliding number. Source: Al‐Bender F 1992.

Figure 6.47

The influence of sliding on the load capacity and stiffness, for a flat bearing. Source: Al‐Bender F 1992.

Figure 6.48

The influence of sliding on the load capacity and stiffness, for a convergent gap bearing. Source: Al‐Bender F 1992.

Figure 6.49

Auto‐centering scaife, for diamond polishing, supported on three circular aerostatic pads. Source: Al‐Bender F, Vanherck P and Van Brussel H 1989.

Figure 7.1

Dynamic pressure distributions pertaining to stiffness

and damping

at small squeeze numbers.

. Source: Al‐Bender F 1992.

Figure 7.2

Dynamic pressure distributions pertaining to stiffness