HYBRID AIR BEARINGS FOR HIGH SPEED

TURBO MACHINERY

By

Guang Pu

A thesis submitted to
The University of Birmingham
for the degree of

DOCTOR OF PHILOSOPHY

School of Mechanical Engineering

The University of Birmingham
December 2015
 University of Birmingham Research Archive
e-theses repository

This unpublished thesis/dissertation is copyright of the author and/or third

parties. The intellectual property rights of the author or third parties in respect
of this work are as defined by The Copyright Designs and Patents Act 1988 or
as modified by any successor legislation.

Any use made of information contained in this thesis/dissertation must be in

accordance with that legislation and must be properly acknowledged. Further
distribution or reproduction in any format is prohibited without the permission
of the copyright holder.
 ABSTRACT
This PhD project is set out to develop a type of hybrid journal air bearings with reduced reliance

on the supply of compressed air for mobile turbomachinery applications. The research work

covers hydrostatic and hybrid journal air bearings with non-compliant clearance boundaries.

The approach adopted combined numerical analysis based on CFD and experimental

verification of the designs.

The research can be divided into three sections. In the first section, numerical approaches to

model hydrostatic and hybrid journal air bearings with a fixed clearance boundary were

developed based on finite difference method (FDM) and finite volume method (FVM)

respectively. The computational time of the iterative process was reduced using the Newton’s

method and successive relaxation. The rotor used for dynamic analysis was modelled using

finite element method based on the Timoshenko beam theory with gyroscopic effect. The

governing equation of motion for a general rotor-bearing system with a constant rotational

speed was provided. This section also includes a discussion on the analytical methods to predict

the performance of a rotor bearing system with both linear perturbation analysis and non-linear

transient analysis.

In the second section, theoretical and experimental studies were performed on hydrostatic

journal air bearings. Performance of the bearings was investigated in non-rotational and

rotational conditions. The analysis on stability and natural frequencies of rotor bearing system

was performed using the linear bearing model in two cases. In the first case (CASE I), the

system consists of a rotor and two journal bearings without viscoelastic support outside the

bearing sleeve, while in the second case (CASE II), the same rotor-bearing system has

I
viscoelastic support. A test rig was designed and manufactured based on the system

configuration in CASE II. The unbalance responses of the rotor in the test rig were predicted

using non-linear transient analysis and measured experimentally at two sensor positions from

50k rpm to 100k rpm in rotor speed. The experimental and prediction results were compared

and discussed.

In the third section, theoretical and experimental study were performed on hybrid journal air

bearings. The proposed hybrid journal air bearings combine hydrostatic air bearings with orifice

restrictors and herringbone grooved hydrodynamic air bearings. The design allows the rotor to

be lifted by compressed air during stop and low speed sessions. When the speed of the rotor is

sufficiently high, the rotor can be fully self-suspended. A novel herringbone groove design has

been proposed to improve bearing reaction forces at given static equilibrium positions. The

influence of some design parameters on the dynamic performance of hybrid air bearings are

then investigated. The analysis on natural frequencies and stability of the rotor supported by the

hybrid air bearings were performed in the same cases as in previous section. The unbalance

responses of the rotor bearing system in CASE II were predicted using non-linear transient

analysis and measured experimentally in two working conditions from 50k rpm to 120k rpm

rotor speed at the two sensor positions. In the first condition, the compressed air supply pressure

(𝑃𝑠 ) is maintained at bar. The responses of the rotor showed a synchronous component

(unbalance excitation). In the second working condition, the bearings are fully self-acting with

no external compressed air supply. The responses of the rotor showed a synchronous component

(unbalance excitation) and a clear sub-synchronous component around 820Hz (self-excited

whirl).

II
Through the theoretical and experimental investigations of the hybrid journal air bearings, the

objectives of the project have been implemented and the aims have been met. The hybrid air

bearings have been fitted to a rotor from a turbocharger for a 2Litres diesel engine. The feature

of the hybrid air bearing has been proven working in experiments by cutting off compressed air

when the rotational speed is sufficiently high. A novel type of herringbone grooves has been

developed for hybrid journal air bearings for improved performance.

III
Dedicated to my family

IV
 ACKNOWLEDGEMENTS

Above all, I would like to express great appreciation to my research supervisor, Professor Kyle

Jiang for his guidance, idea and support on the research work during this PhD project. The way

in which he approaches research and work inspires me. I am very grateful for his efforts in

improving my communication and presentation skills.

I would like to thank Professor Damien Walmsley at School of Dentistry in the University of

Birmingham for kindly providing high resolution laser vibrometer. I owe my sincere thanks to

Vahid Nasrikkahi, Pavel Pencheve and Tahseen Jwad in laser machining process. Without them,

the innovative idea cannot be put in practice. I should also thank the members of the

Nanotechnology Group, for being kind and supportive and my collogues Muhao, Jianyi, Yang,

Changzhao, Haichun, Tianshi, Sagar, Jason, Susan, Guanxiong for sharing their knowledge and

expertise throughout the completion of this work. I would like to thank my parents for their

support throughout my graduate study journey. They are always being there as a sounding board.

Finally, I would also like to thank my wife, Xiaoxu Zou, for her greatest tenderness and

unswerving support during my difficult times.

V
 LIST OF PUBLICATIONS AND

ACHIEVEMENTS

List of publications:
• Yunluo Yu, Guang Pu, and Kyle Jiang, Modelling and analysis of the static
characteristics and dynamic responses of herringbone-grooved thrust bearings, to be
presented in The 3rd International Conference on Mechanical Engineering and
Automation Science (ICMEAS 201 ), O ctober, 201

• Yunluo Yu, Guang Pu, and Kyle Jiang, Numerical modelling and analysis of
hydrostatic thrust air bearings for high loading capacities and low air consumption, to
be presented in The 3rd International Conference on Mechanical Engineering and
Automation Science (ICMEAS 201 ) , O ctober, 201

Patent:
Kyle Jiang and Guang Pu, Improvements in or Relating to Gas Bearings, International patent
WO201 0606 9A 1, 13 April, 201

Grant Awarded:
July 2015, Innovate UK Project 10 05 (P roof of Concept), ‘Hiperturbo’, £100,000, for the
development of a high performance turbocharger with the proposed hybrid air bearings.

Competition Awards:
The air bearings developed in the project were installed into a turbocharger for a reduced
turbo lag and increased speed. We were invited to take part in the following competitions and
received prizes.

VI
• “A High performance Turbocharger”, The Second prize, the International New Energy
and Intelligent Vehicle Competition of CIEC (Suzhou region), Suzhou, China, October
2016

• “A High performance Turbocharger”, The third prize, The final of the International New
Energy and Intelligent Vehicle Competition of CIEC, Beijing Diaoyutai State Guest
Houses, Beijing, January 201

VII
 TABLE OF CONTENTS
ABSTRACT .......................................................................................................I

ACKNOWLEDGEMENTS ........................................................................... V

LIST OF PUBLICATIONS AND ACHIEVEMENTS ............................... VI

TABLE OF CONTENTS ............................................................................VIII

LIST OF TABLES........................................................................................ XII

LIST OF FIGURES ....................................................................................XIII

NOMENCLATURE .................................................................................... XIX

CHAPTER 1: INTRODUCTION .................................................................. 1

1.1 Background ................................................................................................................... 1
1.2 Aim and objectives ....................................................................................................... 2
1.3 Contributions ................................................................................................................ 4
1.4 The outline of the Thesis .............................................................................................. 5

CHAPTER 2: LITERATURE REVIEW....................................................... 7

2.1 Introduction ..................................................................................................................
2.2 Hydrostatic air bearings ................................................................................................ 8
2.2.1 Modelling of orifice restrictors of hydrostatic bearings ........................................ 9
2.2.2 Modelling of porous media restrictors of hydrostatic bearings ........................... 13
2.3 Hydrodynamic air bearings ........................................................................................ 16
2.3.1 Herringbone grooved hydrodynamic air bearings ............................................... 1
2.3.2 Foil air bearings ................................................................................................... 23
2.3.3 Tilt pad air bearings ............................................................................................. 26
2.4 Hybrid air bearings ..................................................................................................... 2
2.5 Numerical techniques in modelling of air bearings .................................................... 33
2.6 Modelling of rotor bearing system and stability analysis ........................................... 34
2. S ummary..................................................................................................................... 3

CHAPTER 3: NUMERICAL ANALYSIS OF AIR BEARINGS .............. 39

VIII
 3.1 Introduction ................................................................................................................ 39
3.2 Modelling of gas-lubricated journal bearings with non-compliant boundaries .......... 40
3.2.1 Discrete scheme of Reynolds Equation ............................................................... 40
3.2.2 Assumptions and challenges in modelling air bearings ....................................... 44
3.2.3 Mesh and boundary conditions ............................................................................ 46
3.2.4 FDM, FVM, iterative strategies and numerical techniques ................................. 51
3.2.5 Validation of the bearing model in the static equilibrium analysis ...................... 63
3.2.6 Analysis of truncation errors................................................................................ 65
3.3 Modelling of rotor ...................................................................................................... 6
3.3.1 Finite element rotor model and impact tests ........................................................ 68
3.3.2 Governing Equations of rotor dynamic system ................................................... 3
3.4 Linear perturbation analysis ....................................................................................... 4
3.4.1 Linearization of bearing forces with perturbation method .................................. 5
3.4.2 Static equilibrium stability analysis ..................................................................... 80
3.4.3 Limitations of the proposed linear perturbation analysis .................................... 82
3.5 Non-linear transient analysis ...................................................................................... 83
3.5.1 Bearing models in non-linear transient analysis .................................................. 83
3.5.2 Non-linear transient stability analysis ................................................................. 86
3.5.3 The conditions for the proposed non-linear transient analysis ............................ 8
3.6 Summary..................................................................................................................... 88

CHAPTER 4: HYDROSTATIC JOURNAL AIR BEARINGS ................. 90

4.1 Introduction ................................................................................................................ 90
4.2 Modelling of hydrostatic journal air bearings ............................................................ 92
4.2.1 Orifice flow model in hydrostatic journal air bearings........................................ 93
4.2.2 Modified orifice flow model................................................................................ 98
4.3 Non-rotational performance of hydrostatic journal air bearings .............................. 100
4.3.1 Effect of eccentricity on the flow rate ............................................................... 101
4.3.2 Optimization of radial clearance, orifice diameter and supply pressure ........... 103
4.4 Theoretical studies on rotational performance of hydrostatic journal air bearings .. 109
4.4.1 Static equilibrium analysis of hydrostatic journal air bearings at rotational
condition ..................................................................................................................... 109
4.4.2 Stiffness and damping coefficients of hydrostatic journal air bearings ............. 114

IX
 4.4.3 Analysis of stability and natural frequencies of a rotor bearing system using
linear bearing model ................................................................................................... 122
4.5 Non-linear transient analysis and experimental verification .................................... 132
4.5.1 Non-linear transient analysis of hydrostatic journal air bearings ...................... 133
4.5.2 Hydrostatic journal air bearing test rig and experiment configuration .............. 13
4.5.3 Experiments on unbalance responses ................................................................ 141
4.5.4 Limitations of experiments ................................................................................ 14
4.6 Summary................................................................................................................... 148

CHAPTER 5: HYBRID JOURNAL AIR BEARINGS ............................ 150

5.1. Introduction ............................................................................................................. 150
5.2 Modelling of hybrid journal air bearings .................................................................. 153
5.2.1 Finite volume model of hybrid journal air bearings .......................................... 153
5.2.2 A novel herringbone groove geometry .............................................................. 15
5.3 Theoretical studies on rotational performance of hybrid journal air bearings ......... 161
5.3.1 Analysis of bearing reaction forces at given equilibrium positions .................. 161
5.3.2 Stiffness and damping coefficients of hybrid journal air bearings .................... 166
5.3.3 Analysis on stability and natural frequencies of rotor bearing system using linear
bearing model ............................................................................................................. 1 8
5.4 Non-linear transient analysis and experimental verification of hybrid air bearings. 186
5.4.1 Manufacturing of the novel herringbone groove ............................................... 18
5.4.2 Unbalance responses of hybrid journal air bearings with compressed air supply
.................................................................................................................................... 189
5.4.3 Unbalance responses of hybrid journal air bearings without supply of
compressed air ............................................................................................................ 19
5.5 Summary................................................................................................................... 205

CHAPTER 6: CONCLUSIONS AND FUTURE WORK ........................ 206

6.1 Summary................................................................................................................... 206
6.2 Conclusions .............................................................................................................. 20
6.3 Suggestions for future work ..................................................................................... 210

APPENDIX A ............................................................................................... 212

APPENDIX B ............................................................................................... 215

X
APPENDIX C ............................................................................................... 219

APPENDIX D ............................................................................................... 227

APPENDIX E ............................................................................................... 239

APPENDIX F ............................................................................................... 249

REFERENCES............................................................................................. 259

XI
 LIST OF TABLES
Table 3.1 Bearing parameters of the hydrostatic journal air bearings to be analysed .............. 64
Table 3. 2 Finite element model of rotor R-1 ........................................................................... 69
Table 3. 3 A comparison of Eigen-frequencies from the rotor model and impact hammer tests
.................................................................................................................................................. 1
Table 4. 1 Design parameters of the hydrostatic journal air bearing ........................................ 91
Table 4. 2 Comparison of the dimensionless bearing reaction forces on different grid sizes and
the experiments ......................................................................................................................... 9
Table 4. 3 Design parameters of hydrostatic journal air bearings used to investigate the effect
of eccentricity on flow rate ..................................................................................................... 101
Table 4. 4 Optimal radial clearance at different orifice diameters ......................................... 10
Table 4. 5 Rotational speed to achieve same Λ for different radial clearances and percentage
differences of bearing forces .................................................................................................. 111
Table 4. 6 Dimensions of the hydrostatic journal air bearings used in analysis ..................... 123
Table 4. D imensions of hydrostatic journal bearings used in the test rig ............................ 138
Table 4. 8 Unbalance information of R-1 ............................................................................... 140
Table 5. 1 Dimensions of hybrid journal air bearings studied ................................................ 152
Table 5. 2 Dimensions of hybrid journal air bearings used in simulation .............................. 159
Table 5. 3 Design parameters and restrictor setup of hybrid air bearings to be studied ......... 162
Table 5. 4 Design parameters and restrictor setup of hybrid air bearings to be studied ......... 16
Table 5. 5 Design parameters and restrictor setup of hybrid air bearings to be studied ......... 1 8
Table 5. 6 Design parameters of the novel herringbone groove ............................................. 189
Table 5. Unbalance information of R-1 ............................................................................... 190

XII
 LIST OF FIGURES
Figure 2.1 a) Orifice restrictors b) Orifice restrictors with pocket c) Porous material restrictors

.................................................................................................................................................... 8

Figure 2.2 Pink’s model of flow through orifice restrictors [ ]................................................ 10

Figure 2.3 a) The pressure in the thrust bearings was measured with an off-set, r, from the

location of an orifice; and b) The test bench[ , 8] .................................................................... 11

Figure 2.4 Comparison of the pressure distribution around an orifice between CFD and

experimental results, G. Belforte and T. Raparelli [9, 10] ........................................................ 12

Figure 2.5 Formation of air film in hydrodynamic journal air bearings .................................. 1

Figure 2.6 Hydrodynamic air bearing with herringbone grooves ............................................ 20

Figure 2. a) Position of the interpolation points for the enhanced groove geometry b)

Traditional groove shape and the enhanced groove shape [40] ................................................ 21

Figure 2.8 Two designs of foil air bearings [44] ...................................................................... 24

Figure 2.9 Pivot at the back of a pad in a tilt pad air bearing ................................................... 2

Figure 2.10 Flexure pivot hybrid gas bearings designed by Zhu and San Andrés [ 0 ]............ 29

Figure 2.11 Grooved hybrid air bearings designed by P. Stanev, F. P. Wardle, J. Corbett[ 2] . 31

Figure 2.12 The elimination of self-excited whirl by introducing mesh metal wire damper in

parallel with a hybrid compliant air bearing [ 4] ..................................................................... 33

Figure 2.13 A rotor to be supported by the proposed hybrid air bearings ................................ 35

Figure 2.14 A 4-DOF model of shaft element and global coordinates used in Timoshenko

beam theory .............................................................................................................................. 35

Figure 3.1 Five-point discrete scheme ...................................................................................... 43

Figure 3.2 Mesh and boundary conditions for static air journal bearings ................................ 4

Figure 3.3 A flow diagram of the FDM approach .................................................................... 52

XIII
Figure 3.4 a) A conventional herringbone grooved journal from [93] b) A cross-section view

in axial direction of grooves in the circled area from ............................................................... 5

Figure 3.5 A flow diagram of the FDM approach .................................................................... 58

Figure 3.6 a) Controlled volume surrounding a node in FVM approach b) Projected view of

the controlled volume on the meshed surface .......................................................................... 59

Figure 3. S chematic drawings of hydrostatic journal air bearings with equally distributing

pocketed orifice restrictors [ ] .................................................................................................. 63

Figure 3.8 Mesh for the static air journal bearing .................................................................... 64

Figure 3.9 Pressure distribution from the bottom boundary to the axial symmetry plan,

eccentricity ratio, 0.4 ................................................................................................................ 64

Figure 3.10 Dimensionless bearing force VS. Eccentricity ratio, ............................................ 65

Figure 3.11 The rotors used in this project and their finite element models. ........................... 68

Figure 3.12. The first four Free-free undamped modes of R-1 at zero speed .......................... 0

Figure 3.13 Bode plots of impact hammer tests on free-free rotor R-1 at zero speed .............. 1

Figure 3.14 Campbell diagram and mode shape of rotor R-1 with undamped isotropic

support.. .................................................................................................................................... 2

Figure 3.15 Static equilibrium configuration and coordinate system of linear perturbation

analysis. .................................................................................................................................... 4

Figure 3.16 The algorithm of time dependent finite difference scheme. .................................. 85

Figure 4.1 Mesh of bearing surface in FDM ............................................................................ 93

Figure 4.2 Modified orifice flow model ................................................................................. 100

Figure 4.3 Effect of eccentricity on mass flow rate of static air journal bearings for different

supply pressure and radial clearance combinations ................................................................ 102

XIV
Figure 4.4 The effect of radial clearance on bearing reaction forces for hydrostatic journal air

bearings ................................................................................................................................... 104

Figure 4.5 The effect of radial clearance on flow rate at different supply pressures for

hydrostatic journal air bearings .............................................................................................. 104

Figure 4.6 Bearing reaction forces in relation with mass flow rate of hydrostatic journal air

bearings ................................................................................................................................... 105

Figure 4. E ffect of supply pressure on bearing reaction forces versus mass flow rate. ....... 106

Figure 4.8 Increasing rate of bearing reaction forces versus mass flow rate.......................... 106

Figure 4.9 Effect of orifice diameters on bearing reaction forces to static load..................... 108

Figure 4.10 Optimal bearing reaction forces to static load and associated flow rate for

different orifice diameters ...................................................................................................... 108

Figure 4.11 Pressure distribution for journal bearings ........................................................... 110

Figure 4.12 Pressure profile at symmetry plane ..................................................................... 110

Figure 4.13 Relation of bearing reaction forces and eccentricity at different compressibility

numbers .................................................................................................................................. 112

Figure 4.14 Attitude angles for different compressibility numbers. ....................................... 113

Figure 4.15 Attitude angle for different compressibility numbers ......................................... 114

Figure 4.16 Stiffness and damping coefficients at concentric journal position ...................... 115

Figure 4.1 The effect of eccentricity ratio on the bearing stiffness and damping coefficients

................................................................................................................................................ 11

Figure 4.18 The effect of compressibility number on bearing stiffness and damping

coefficients.............................................................................................................................. 118

Figure 4.19 The effect of orifice diameter on bearing stiffness and damping

coefficients…..........................................................................................................................119

XV
Figure 4.20 The effect of supply pressure on bearing stiffness and damping coefficients..... 120

Figure 4.21 The effect of bearing length to diameter ratio on bearing stiffness and damping

coefficients.............................................................................................................................. 121

Figure 4.22 Schematic views of R-1 with linearized journal bearing .................................... 124

Figure 4.23 Schematic views of the R-1 with linearized journal bearing and viscoelastic

support. ................................................................................................................................... 126

Figure 4.24 The Campbell diagrams for Case I & II and the bearing sleeve with linear

dampers. .................................................................................................................................. 128

Figure 4.25 Stability maps based on SESA ........................................................................... 129

Figure 4.26 Frequency ratio of self-excited whirl to rotation speed for CASE I ................... 130

Figure 4. 2 Stability map of CASE II, 𝛾 = 0.153.................................................................. 131

Figure 4.28 Stability map and predictions of whirling frequency ratio of the rotor bearing

system in CASE I ................................................................................................................... 132

Figure 4.29 Trajectory of the journal centre at 120k rotor speed of rotor R-1 in CASE I and II

................................................................................................................................................ 136

Figure 4.30 Cross-section view of prototype hydrostatic bearing test rig .............................. 13

Figure 4.31 High speed air bearing test bench ....................................................................... 139

Figure 4.32 Top views of waterfall plots in a run-up test at sensor position B to 100k rpm. . 142

Figure 4.33 Unbalance responses of sensor position A & B obtained at 100k rpm in speed . 145

Figure 4.34 Peak vibration velocities at sensor positions....................................................... 146

Figure 4.35 Unintended change on the shaft journal during a balancing process .................. 146

Figure 5.1 A proposed hybrid journal air bearing.. ................................................................. 150

Figure 5.2 The Model of a hybrid journal air bearing.. .......................................................... 154

XVI
Figure 5.3 Dimensionless bearing reaction forces versus eccentricity ratio at various

compressibility numbers. ........................................................................................................ 156

Figure 5.4 Pressure distribution of hybrid journal air bearings at a given static equilibrium

configuration. .......................................................................................................................... 15

Figure 5.5 Air flow over the groove ....................................................................................... 158

Figure 5.6 Comparisons between herringbone grooves and their effects. .............................. 160

Figure 5. Bearing reaction forces at the given SEP of multiple groove number 𝐺𝑛𝑢𝑚 and

maximum groove depth ratio (𝐻𝑔 ) combinations. .................................................................. 164

Figure 5.8 The influence of groove number to bearing reaction forces at the given SEP.. ... 164

Figure 5.9 Influence of maximum groove depth ratio on bearing reaction forces. ................ 165

Figure 5.10 Influence of groove angle to bearing reaction force at given SEP ...................... 166

Figure 5.11 The effect of whirling frequency ratio on the stiffness and damping coefficients of

hybrid journal air bearings at concentric journal position. ..................................................... 168

Figure 5.12 The effect of compressibility number on the synchronous stiffness and damping

coefficients of hybrid journal air bearings. ............................................................................. 169

Figure 5.13 The effect of supply pressure on the synchronous stiffness and damping

coefficients of hybrid journal air bearings. ............................................................................. 1 0

Figure 5.14 The effect of maximum groove depth on the stiffness and damping coefficients of

hybrid journal air bearings. ..................................................................................................... 1 2

Figure 5.15 The effect of groove angle on the stiffness and damping coefficients of hybrid

journal air bearings. ................................................................................................................ 1 3

Figure 5.16 The effect of groove number on the stiffness and damping coefficients of hybrid

journal air bearings. ................................................................................................................ 1 4

XVII
Figure 5.1 The effect of groove width ratio on the stiffness and damping coefficients of

hybrid journal air bearings. ..................................................................................................... 1 5

Figure 5.18 The effect of grooved area fraction on the stiffness and damping coefficients of

hybrid journal air bearings. ..................................................................................................... 1 6

Figure 5.19 Campbell diagrams for Cases I & II. .................................................................. 182

Figure 5.20 Stability maps and whirl frequency ratio based on SESA.. ................................ 184

Figure 5.21 Stability map based on SESA of CASE II, 𝑃𝑠 = 1, 𝛾 = 0.25 ............................. 185

Figure 5.22 The rotor used in the hybrid air bearings and the novel herringbone grooves.. .. 188

Figure 5.23 Top views of waterfall plots in run-up test to 120k rpm. Measurements were made

at sensor position A in horizontal direction. ........................................................................... 191

Figure 5.24 Unbalance responses of sensor positions A & B obtained at 120.9k rpm in speed.

Supply pressure maintained at ba r. ....................................................................................... 193

Figure 5.25 Peak synchronous vibration velocities of rotor at sensor position A and B of

various rotor speeds ................................................................................................................ 194

Figure 5.26 The theoretical shaft response of R-1 in CASE II with hybrid air bearings in

response to out of phase unbalance. ....................................................................................... 196

Figure 5.2 Top views of waterfall plots generated from experimental data.. ....................... 198

Figure 5.28 Frequency ratio of sub-synchronous vibration to rotational speed at various

speeds. .................................................................................................................................... 198

Figure 5.29 Unbalance responses of sensor positions A & B obtained at 98.4k rpm in speed

................................................................................................................................................ 200

Figure 5.30 Comparisons of prediction results with experimental results.. ........................... 203

XVIII
 NOMENCLATURE

𝛼 Surface correcting coefficient

𝛼𝑔 Ratio of groove width to total width (groove plus ridge)

𝛽𝑔 Angle of herringbone grooves, degree

𝛾 Deformation ratio of the O-ring

𝛾𝑔 Fraction of bearing length occupied by grooves

𝜀 Eccentricity ratio, e/c

𝜁 Bearing length to diameter ratio

𝜂 Viscosity of fluid, 𝑝𝑎 ∙ 𝑠

𝜃 Circumferential coordinate

∆𝜃 Distance between two points in 𝜃 direction

𝜃𝑥𝑖 , 𝜃𝑦𝑖 Rotational displacement of 𝑖 𝑡ℎ Node under right hand rule

𝜆 A complex root of the equation

𝜆𝑚 Mean free molecular path of air and atmospheric pressure

𝜆𝑔 Groove width

ℎ0
𝑣0 Characteristic speeds in 𝑧 direction, 𝑣0 = 𝑢0 𝑙0

𝜉 Coordinate along the circumferential direction

𝜌 Density of the fluid, 𝑘𝑔/𝑚3

𝜌0 Local density in the fluid film, 𝑘𝑔/𝑚3

ωt
τ Dimensionless time, 2

𝜔 Rotational speed, rad/s

XIX
 𝜔𝑠𝑟 Relaxation factor

Angular frequency of the journal orbit (whirling frequency),

𝜔𝑤
rad/s

𝛤 Projected boundaries of the controlled volume

𝛬 Compressibility number or bearing number

𝛺 Whirling frequency ratio or perturbation frequency ratio

∇ Differential operator

𝑐 Radial clearance, 𝜇𝑚

𝐴𝑟 Restricted area, 𝑚2

𝑏0 Characteristic lengths in 𝑧 direction

𝑏𝑠 Damping coefficients of the support

𝐶𝑑 Discharge coefficient of orifices

𝐶𝑐𝑜𝑟𝑓 Correction coefficient

𝑑0 Orifice diameter, mm

𝑑𝑙 Unit length of each cell boundary at a controlled volume in FVM

𝑑𝑖,𝑗 Damping coefficients, Ns/m

𝑑𝑝𝑜𝑐 Pocket diameter, mm

Equivalent damping coefficients at a static equilibrium position,

𝑑𝑥𝑥 , 𝑑𝑥𝑦 , 𝑑𝑦𝑥 , 𝑑𝑦𝑦
Nm/s

𝐷 Diameter, mm

𝐷𝑘𝑛 Inverse Knudsen number

e Eccentricity, 𝜇𝑚

𝑓𝑝 Excitation frequency

XX
 Bearing forces in the ∆𝑥𝑗 And ∆𝑦𝑗 Directions at the static
𝐹𝑋0𝑏𝑟𝑔 , 𝐹𝑌0𝑏𝑟𝑔
equilibrium position

𝒈 Body accelerations (per unit mass) acting on the continuum

𝐺𝑛𝑢𝑚 Number of herringbone grooves

ℎ Air film thickness, 𝜇𝑚

ℎ0 Characteristic film thickness, 𝜇𝑚

Local clearance at the downstream of an orifice at the entrance to

𝐻𝑑
air film

ℎ𝑔 Maximum depth of herringbone grooves, 𝜇𝑚

𝐻𝑢 Local film clearance upstream at the volume under the orifice

𝐻 Dimensionless film thickness, ℎ/𝑐

𝐻𝑔 Maximum groove depth ratio, ℎ𝑔 /𝑐

Interpolation values of P at the boundary of two adjacent

𝑖 − 1/2
controlled volume in FVM

ID Inner diameter, mm

𝐼𝑚𝑔() Imaginary part of a complex number

I_D Lumped diametral moment of inertia

I_P Polar moment of inertia

Interpolation values of H at the boundary of two adjacent

𝑗 − 1/2
controlled volume in FVM

𝑘𝑖,𝑗 Stiffness coefficients, N/m

𝐾𝑑 Entrance loss coefficient

𝐾𝑛 Knudsen number

XXI
 Equivalent stiffness coefficients at a static equilibrium position,
𝑘𝑥𝑥 , 𝑘𝑥𝑦 , 𝑘𝑦𝑥 , 𝑘𝑦𝑦
N/m

𝐾𝑥𝑥 , 𝐾𝑥𝑦 , 𝐾𝑦𝑥 , 𝐾𝑦𝑦 , Dimensionless form of stiffness and damping coefficients at a

𝐷𝑥𝑥 , 𝐷𝑥𝑦 , 𝐷𝑦𝑥 , 𝐷𝑦𝑦 static equilibrium position

𝑙 Length of the bearing, mm

𝑙0 Characteristic lengths in 𝜉 direction

Outward unit vector normal to the boundaries of each cell at a

𝑛⃗
controlled volume in FVM

𝑵𝒐𝒓𝒊 Number of orifices

OD Outer diameter, mm

𝑝 Pressure in the air film, 𝑏𝑎𝑟

𝑝̅ Flow pressure, bar

𝑝0 Flow pressure at the downstream of an orifice, bar

𝑝𝑎 Ambient pressure，1bar

𝑃𝑎𝑣𝑔 Average pressure, bar

𝑝𝑐 Local static pressure at the pocket edge, bar

𝑃𝑑 Flow pressure downstream at the entrance, bar

𝑝𝑑𝑦𝑛 Dynamic pressure, bar

𝑝𝑠 Flow pressure at the upstream of an orifice, bar

𝑃𝑢 Flow pressure upstream at the volume under the restrictor

𝑝
𝑃 Dimensionless pressure, ⁄𝑝𝑎

𝑃𝑠 Supply pressure, bar

q Displacement vector in governing equation of motion

XXII
 𝑄𝑝 Slip flow corrector for Poiseuille flow

𝑄𝑐 Slip flow corrector for Couette flow

ℛ(∆𝜃 2 ) Lagrange remainder

𝑅𝑒() Real part of a complex number

𝑅𝑠𝑝𝑒𝑐 Gas constant

𝑟0 Bearing radius, mm

t Time, s

𝑇 Temperature in Kelvin, K

𝒖 Flow velocity, m/s

𝑢 Averaged velocity, m/s

Characteristic speeds in 𝜉 direction, journal surface speed in

𝑢0
journal bearings

W Lumped mass

𝑥𝑖 , 𝑦𝑖 Translational displacement of 𝑖 𝑡ℎ Node

𝑥𝐽0 , 𝑦𝐽0 Static equilibrium position of the journal centre

∆𝑥𝐽 , ∆𝑦𝐽 Displacement of the journal centre in the x and y directions

∆𝑥̇ 𝐽 , ∆𝑦̇ 𝐽 Velocity of the journal centre in the x and y directions

Dimensionless amplitude of the displacement in the ∆𝑥𝑗 And ∆𝑦𝑗

𝑋𝐽1 , 𝑌𝐽1
Directions

z Coordinate along the axial direction

Z Dimensionless axial coordinate

𝑚̇𝑜𝑟𝑖 Flow rate through an orifice, 𝑘𝑔/𝑠

𝑚̇𝑜𝑟𝑖
Flux term or flow rate per unit area
∂ξ𝜕𝑧

XXIII
 𝜕𝑃 𝜕𝑃
Changes of dimensionless pressure introduced by the
, 𝜕𝑌
𝜕𝑋𝐽 𝐽
displacement of the journal centre

𝜕𝑃 𝜕𝑃
Changes of dimensionless pressure introduced by the velocity of
, 𝜕𝑌 ̇
𝜕𝑋𝐽̇ 𝐽
the journal centre

[𝟎] Null matrix

[𝐈] Unit matrix

C Damping

[𝑪𝒔 ] Structural damping matrix

[𝑪𝒃 ] Bearing’s damping matrix

[𝑪𝒗 ] Structure Damping matrix of the bearing sleeve

[F] Force vector

[𝑭𝒃𝒓𝒈 ] The bearing force vector

[𝑭𝒈 ] Static gravitational force vector

[𝑭𝒖𝒃 ] Synchronous unbalance excitation force vector

[𝑮] Gyroscopic matrix

[𝑪𝑮𝒔𝒚𝒔_𝟏 ] System damping and gyroscopic matrix

𝑱 System characteristic matrix

K Stiffness

[𝑲𝒔 ] Structural stiffness matrix derived from strain energy

[𝑲𝒃 ] Bearings’ stiffness matrix

[𝑲𝒗 ] Stiffness matrix of the bearing sleeve

[𝑲𝒔𝒚𝒔_𝟏 ] System stiffness matrix

M System mass and inertia

XXIV
[𝑴] Mass and inertia matrix

[𝑴𝒔 ] Shaft structure mass and inertia matrix

[𝑴𝒗 ] Bearing sleeve mass and inertia matrix

XXV
CHAPTER 1 INTRODUCTION

CHAPTER 1: INTRODUCTION

This thesis presents a study on the development of hybrid air bearings with a fixed clearance

boundary for micro turbomachinery. The research was driven by the needs of high speed and

mobile turbomachinery and stationary compressed air source cannot be used, such as

turbochargers and micro gas turbine engines.

This chapter introduce the research conducted in this project. A brief background of air bearings

is presented. The aim, objectives and contributions to the knowledge of the field are then

introduced, followed by a summary of the thesis organisation.

1.1 Background

Air bearings are bearings that use a thin air film to provide exceedingly low friction between

surfaces. They can provide oil-free and frictionless precision motions. They are widely applied

in precision and high-speed machines, including high speed spindles, turbo expander/generators

etc. Air bearings can be classified as either ‘hydrostatic’, ‘hydrodynamic’ or ‘hybrid’.

Hydrostatic air bearings rely on external compressed air source to form the air films and lift the

rotor of the machine during their operation. This makes them not suitable for portable devices.

Hydrodynamic air bearings require high relative speeds between surfaces to lift loading, but

have a dry friction issue at low speed. Hybrid air bearing are externally pressurized but with

enhanced features used in self-acting bearings.

1
CHAPTER 1 INTRODUCTION

In this study, a hybrid journal air bearing is proposed. It is designed as a combination of a

hydrostatic bearing with orifice restrictors and a hydrodynamic bearing with herringbone

grooved journal. The hybrid bearings work as hydrostatic air bearings to lift the rotor by

compressed air at low speeds, which occur at the start and the end of a rotation session. When

the rotational speed goes over a threshold, the bearings will be self-acting to form air films and

lift the rotor. The supply of compressed air can be then switched off to reduce air consumption.

As either the start or stop session usually takes a few seconds, the proposed hybrid air bearings

only need a very small compressed air source to support them. In applications such as

turbochargers and micro gas turbine engines, an air reservoir could be sufficient to supply air

for the bearings to start. The reservoir can be refilled with the air from the compressor of these

machines. Therefore, the proposed hybrid air bearings have the potential to reduce the reliance

of compressed air and be applied in turbomachinery.

1.2 Aim and objectives

The aim of this PhD project is to develop a type of hybrid journal air bearings with reduced

reliance on the supply of compressed air for mobile turbo machinery applications, typically

including turbochargers and micro gas turbine engines used as range extenders for electric

vehicles. This research will study the proposed hybrid air bearing theoretically and

experimentally.

Three challenges are posed for this research. The first is the modelling of hybrid journal air

bearings and the rotor structure they support, to enable the performance of the rotor bearing

system to be predicted. The second is to apply the proposed hybrid air bearings to a rotor of a

2
CHAPTER 1 INTRODUCTION

micro turbomachine and prove the bearings work in both hydrostatic and hydrodynamic modes.

The third is to verify the accuracy of the rotor bearing model with experiments.

The objectives of the research to achieve the research aim mentioned above are set out below:

1. Review the state of the art of the research relating to air bearings and their applications in

high-speed turbomachinery.

2. Develop numerical approaches to model journal air bearings and verify the validity of the

methodology used by comparing some predicted performance with references.

3. Use appropriate methodology to model the rotor bearing system. The rotor is modelled as

linear using finite element method based on Timoshenko beam theory with gyroscopic

effect.

4. Perform theoretical studies using the developed model on rotor bearing system with

hydrostatic journal air bearings. Verify numerical analytical methodology by comparing

predicted performance with experimental ones.

5. Perform theoretical studies using the developed model on the proposed hybrid journal air

bearings. Design hybrid air bearings for the developed test rig. Verify the predicted

performance of the system with experiments.

3
CHAPTER 1 INTRODUCTION

1.3 Contributions

By accomplishing the aim and objectives, this research has three novel contributions to air

bearing knowledge and applications:

1. A type of hybrid journal air bearing has been developed for applications in micro

turbomachinery. The hybrid air bearings have been fitted to a rotor from a turbocharger for

2Litres diesel engines. The feature of the hybrid air bearing has been proven to work in

experiments by cutting off compressed air when the rotational speed is sufficiently high.

2. A novel type of herringbone grooves has been developed for hybrid air bearings in the

project for improved performance. The grooves are also applicable to hydrodynamic air

bearings.

3. The proposed hybrid journal air bearing design (combinations of hydrostatic air bearings

with orifice restrictors and herringbone grooved self-acting bearings) is modelled using

finite volume method. The bearing model is used in the non-linear transient rotor dynamic

analysis and can give adequate predictions of the unbalance response of a non-symmetric

rotor bearing system. The literature survey prior to this project shows the research work on

the same type of hybrid journal air bearings is limited to using narrow groove theory and

linear perturbation analysis.

4
CHAPTER 1 INTRODUCTION

1.4 The outline of the Thesis

This thesis consists of six chapters. Chapter 1 is the introduction of the research topics covered

by this thesis. The aim of the project, objectives and thesis outline are included.

Chapter 2 gives a comprehensive literature review summarising the previous published work in

the core areas relevant to this thesis. This chapter starts with research works on hydrostatic air

bearings with focus on the modelling of the bearing, mainly the restrictor system. This is then

followed by reviews of hydrodynamic air bearings. Literatures reviewed in this area is focused

on herringbone grooved self-act bearings. Literatures of foil air bearings and tilt pad air bearings

are included but not reviewed in depth. Previous research works on hybrid air bearings are

reviewed and followed by a list of areas in which the present thesis proposes to improve on. At

the end, the appropriate numerical techniques in modelling air bearings and rotor bearing

system are introduced.

Chapter 3 begins by introducing the theories of fluid dynamics relevant to modelling of air

bearings. A numerical model based on finite difference method is developed for hydrostatic

journal air bearings. A numerical mode based on finite volume method is developed for hybrid

journal air bearings. Several numerical techniques are applied to provide a reliable and stable

iteration process. Next, the methodology used to model rotor bearing system is introduced. The

rotor used in this project is then modelled with some verifications on its dynamic properties

using impact hammer tests. At the end, the analysis methods used to predict the performance of

rotor bearing system are discussed under two approaches: linear perturbation analysis and non-

linear transient analysis.

5
CHAPTER 1 INTRODUCTION

Chapter 4 presents the theoretical and experimental studies on hydrostatic journal air bearings

with orifice restrictors. It begins with a comparison of different flow models for orifice

restrictors and how they are improved in this thesis. Following on, the effects of orifice

diameters, radial clearances and supply pressures on bearing’s performance are investigated

under non-rotational and rotational conditions. The analysis on stability and natural frequencies

of rotor bearing system are performed using linear bearing models on two cases, namely CASE

I & II respectively. Then a test rig is designed according to the system configuration in CASE

II. The unbalance responses of the rotor in the test rig are predicted using non-linear transient

analysis and measured experimentally.

Chapter 5 is focused on the investigation of hybrid journal air bearings with orifice restrictors

and herringbone grooved journal. It starts with setting the numerical model for the hybrid

journal air bearing. A modified groove profile with cosine spline is then proposed as an

enhancement. The influences of some design parameters on performance of hybrid journal air

bearings are then investigated under rotational condition. The analysis on stability and natural

frequencies of rotor bearing system supported by hybrid journal air bearings are performed on

the two cases used in Chapter 4. Finally, experimental investigations on unbalance responses

are carried out on hybrid journal air bearings using the developed test rig under two working

conditions: with compressed air supply and without. The unbalance responses under these two

conditions are also predicted using non-linear transient analysis. The experimental and

prediction results are compared and discussed.

Chapter 6 concludes the research work in this PhD project. The major research conclusions are

drawn. Future research work on hybrid air bearings and their applications is suggested.

6
CHAPTER 2 LITERATURE REVIEW

CHAPTER 2: LITERATURE REVIEW

2.1 Introduction

This chapter gives a comprehensive review on the techniques relevant to the air bearings

research project. Firstly, literatures of hydrostatic, hydrodynamic and hybrid journal air

bearings are reviewed. Secondly, numerical techniques used in modelling of air bearings are

introduced. Finally, literatures on modelling and stability analysis of rotor bearing system are

discussed.

Air bearings use a thin film of pressurized air to support load and provide exceedingly low

friction. Properties of this thin air film are governed by the compressible Reynolds Equation.

Air bearings have been developed for over 50 years. Early studies on air bearings were

completely based on experiments and engineering practice. The results were used to design air

bearings in an empirical method [1-3]. More recent literatures on air bearings tends to use

numerical approaches to solve the compressible Reynolds Equation and predict the dynamic

performance of the air bearings. Experiments are generally included as verifications to the

numerical approaches. The review of the air bearing technology in this section will be focused

on hydrostatic air bearings, hydrodynamic air bearings and hybrid air bearings. Foil air bearings

are briefly reviewed within the hydrodynamic air bearings to cover an important category of air

bearings. However, as this thesis is focused on developing a type of hybrid air bearings with a

fixed clearance boundary, foil bearings have compliant clearance boundaries and is outside the

scope of this project. They will not be reviewed in depth, so as the tilt pad air bearings.
CHAPTER 2 LITERATURE REVIEW

2.2 Hydrostatic air bearings

In hydrostatic air bearings, a thin air film is maintained between two opposing surfaces using

externally pressurized air. These bearings can lift an object at zero rotational speed. The

pressurized air continuously flows into bearing clearance through restrictors and escapes into

the atmosphere at the two ends of the bearing. Three types of restrictors can be used in

hydrostatic air bearings: capillary compensators, orifices, and porous media. The latter two are

commonly used because of their compact structures and advantages in installation. Figure 2.1

shows some typical restrictor designs used in hydrostatic air bearings.

Figure 2.1 a) Orifice restrictors b) Orifice restrictors with pocket c) Porous material restrictors

The mechanism of hydrostatic air bearings makes their performance highly depend on the

restrictor configurations. A major research area of hydrostatic air bearings is the bearings’

performance with different restrictor configurations. It is studied both theoretically and

experimentally by many researchers. An important part of these research work is the modelling

of flow properties through restrictors used in the bearings. Numerical studies on hydrostatic air

bearings with orifice restrictors normally using a separation of variable method [4] to calculate

the overall flow rate through an orifice. One approach is to set up physical equations which

8
CHAPTER 2 LITERATURE REVIEW

accurately describe the edge loss effect by introducing correction factors [1]. The other

approach is based on the CFD to simulate the flow velocity field. Numerical studies on

hydrostatic air bearings with porous restrictors adopted Darcy’s law [5] to model the porous

media. The 3D Cauchy’s law is commonly applied with finite volume method to calculate

pressure drop and the across flow rate through the media. Other related methods require

investigation of micro structures in porous materials [6]. Experimental work on hydrostatic air

bearings involves two parts: (1) Measurements of steady state pressure distribution in air film

using thrust bearings. This allows the model of restrictors and bearings to be verified; (2)

Measurements of stability on the rotor dynamics.

In this section, literatures of hydrostatic air bearings with orifice restrictors are reviewed first

with focus on the modelling of the restrictors. Researches on hydrostatic air bearings with

porous media are discussed afterwards. The mathematical modes of them provided by different

researchers are shown with comments on their limitations.

2.2.1 Modelling of orifice restrictors of hydrostatic bearings

An early study on hydrostatic air bearings with orifice restrictors from E. G. Pink [ ] reported

an effect of orifices in air bearings known as ‘entrance loss’. When the flow enters the bearing

clearance, the flow velocity is increased and results in a significant conversion of static pressure

into kinetic energy within air film. E. G. Pink provided an accurate mathematic model to take

which took this effect into account. He also reported that there would be a recovery on static

pressure downstream of the restrictors. The entrance loss effect and pressure recovery in [ ] are

shown in Figure 2.2. Bearing reaction forces to static load of hydrostatic air bearings predicted

using Pink’s model in [ ] showed a good agreement with experimental data. However, a main

9
CHAPTER 2 LITERATURE REVIEW

limitation of his work is that the measurements and predictions were made only at non-

rotational condition.

Figure 2.2 Pink’s model of flow through orifice restrictors [ ]

Investigations on hydrostatic bearings with orifice restrictors were extended with the help of

CFD. Flow properties through orifice restrictors were investigated and modelled with the

bearing numerically. Masaaki Miyatake [8] studied the effect of micron-level (30um to 50um

diameter) orifices in hydrostatic air thrust bearings and showed an increase in the load capacity

under the same mass flow rate. G. Belforte and T. Raparelli [9] improved the studies on the

entrance loss effect by means of performing a series of CFD analysis on bearings with plain

orifice restrictors. They used an experimental method to identify the discharge coefficient

experimentally [10]. The results were compared with theoretical predictions. Figure 2.3 shows

the measurement method and test bench used in [10]. Figure 2.4 shows a comparison of the

10
CHAPTER 2 LITERATURE REVIEW

results between their numerical and experimental work. G. Belforte and T. Raparelli also

concluded that the radius around an orifice restrictor in the thrust bearings where pressure

recovery occurred could be approximated by the following relation:

𝑑0 Equation 2.1
𝑟𝑖 = + 40 ∗ h
2

where 𝑟𝑖 is the radius around an orifice and h the average air film thickness.

a) b)

Figure 2.3 a) The pressure in the thrust bearings was measured with an off-set, r, from the
location of an orifice; and b) The test bench[ , 8]

11
CHAPTER 2 LITERATURE REVIEW

Figure 2.4 Comparison of the pressure distribution around an orifice between CFD and
experimental results, G. Belforte and T. Raparelli [9, 10]

The above studies were mainly performed on hydrostatic thrust air bearings. The results proved

that CFD could be employed as a reliable tool to investigate the performance of orifice

restrictors. There were also plenty of works on flow properties of orifice restrictors in

hydrostatic journal air bearings based on CFD, i.e. the influence of the discharge coefficient

[11], and taking dynamic flow effects into account [12]. A few of them involved a similar

measurement as presented in [10]. It should also be noted that the manufacturing errors of

orifice restrictors had a significant influence on bearing reaction forces in both journal and

thrust bearings. This influence was summarized in K. J. Stout’s work [13].

The work presented in this thesis adopted the plain orifice as restrictors. The mathematical

model used is based on the results in [ , 9] and [10]. Other correction factors in [11-13] were

12
CHAPTER 2 LITERATURE REVIEW

also considered to further improve the accuracy of the model in predicting the performance of

the bearing. The improved model and Equations will be explained in Chapter 3.

2.2.2 Modelling of porous media restrictors of hydrostatic bearings

Porous materials have been applied to hydrostatic air bearings for a long time. Among all

different types of restrictors, porous material restrictors provide the most uniform pressure

distribution, which results in high bearing reaction forces and stiffness. Porous hydrostatic air

bearings are convenient to manufacture and can be machined into complex geometries, as the

pores inside the material serve as built-in restrictors.

The literature from the 19 0s laid down the fundamentals for analyzing externally pressurized

porous bearings with assumptions that flow in porous materials was one dimensional and

incompressible. These early works were reviewed by Sneck in 1968 [14] and then updated by

Majumda[15] in 19 6. Later, other corrections, such as compressible and slip flow, were

included to improve the accuracy of predictions. Mori [16] extended the theory by introducing

the method of equivalent clearance, which allowed for the two-dimensional flow to be

accounted. One of the commonly adopted 3D models to predict the flow in the porous material

for hydrostatic air bearings was developed by Majumdar [1 , 18]. Based on this theory, Rao

[19-21] studied the rotational performance of porous hydrostatic journal air bearings. The

equivalent stiffness and damping coefficients were calculated and the results were compared

with previous solutions, showing better accuracy. He also suggested that if the thickness of the

porous material was small compared with the radius of the bearing, the flow direction could be

considered as radial only.

13
CHAPTER 2 LITERATURE REVIEW

All above literatures have not addressed tangential velocity slip effect at the porous-fluid film

interface. K. C. Singh [22] and his co-workers developed a method to predict the steady-state

performance of an annular thrust bearing based on the Beavers-Joseph slip velocity conditions

which took the tilt and anisotropy of the porous materials into account. In Singh’s report, there

were three important conclusions: the slip effect was less predominant if the flow film was

uniform; it was more significant in materials with lower permeability and slip coefficient; and

it became more dominant at higher radius to film thickness ratio.

Kwan [23] summarized most of the previous work and carried out a series of experiments and

measurements based on hot isostatically pressed porous alumina. In his work, several

specimens with different porosities ranging from 0.1 to 0.4 were tested. The results showed that

the slip coefficient was widely spread according to open porosity, but had a good agreement

with the Beaver-Joseph model when porosity was less than 0.15. In a later publication by Kwan

[24] and his co-worker, a simplified method for the correction of the velocity slip effect is

proposed. They compared the flow rate of air passed through two solid surfaces and that through

one porous surface in the experiments. As the results implied the flow passage in the presence

of a porous surface was higher, an equivalent clearance could be applied to flow rate Equations.

They also explained that this equivalent clearance differed from measurements of either porous

surface roughness or pore size. The exact relationship between equivalent clearance and micro

structure of porous materials was not given and could only be acquired through experiments.

There were also some reports to compare the performance of different types of restrictors.

Mohamed Fourka [25] compared plain orifice, orifice with pocket and porous medium in

hydrostatic thrust air bearings. The results showed that the highest bearing reaction force to

14
CHAPTER 2 LITERATURE REVIEW

static load was achieved by bearings with porous media. Pocket bearings had higher stiffness

but it decreased sharply as film thickness increased. In comparison, porous air bearings had a

wide range of settlement and better stability. Another advantage of using porous inserts was that

there would not be any local pressure loss at the entrance to the bearing surface [26]. Plain

orifice bearings could also achieve high bearing reaction forces and stiffness but at the cost of

high air consumption. The author suggested the design of the restrictor system should be a

compromise among load capacity, stiffness and air consumption based on the application of the

bearings.

To improve the stability of porous hydrostatic journal air bearings and prevent ‘pneumatic

hammer’ caused by the internal cavities in the porous material, porous materials with surface-

restricted layers caught the interests of researchers in recent years. Kwan also studied porous

hydrostatic thrust air bearings with a two-layered structure [23]. Each layer of the porous bulk

had a different permeability. The porous bearings with a similar structure were studied by a

group of researchers in Japan [2 -30]. They discovered that there existed a surface-restrict layer

after grinding of porous metals. This surface-restrict layer had lower permeability than the

porous material itself. The report from Togo [2 ] showed one of the benefits of having this layer:

the threshold of instability of journal bearings was improved significantly at high speeds in the

experiments. The prediction from numerical studies carried out by other researchers [28-30]

showed good agreement with experimental data. Additionally, these studies also implied that

surface-restrict layers not only increased load capacity but also reduced friction. However, it

was impossible to measure the permeability of surface-restricted layers. Yuta Otsu [31]

introduced a surface restriction ratio to relate it to the permeability of porous material. By

comparing flow rate passed through a porous medium with and without a surface-restrict layer,

15
CHAPTER 2 LITERATURE REVIEW

Yuta also showed that the thickness of the surface-restricted layer had little influence on the

characteristics of bearings.

For the purpose of reducing the size of porous air bearings, the shape of compressed air supply

area (the channel from which compressed air are fed to porous material) was investigated by

Yoshimoto [30] to avoid deformation of thin porous media. He compared the performance of

hydrostatic thrust air bearings with annular groove to supply air with those using a full circile

to supply air and concluded that static stiffness was mainly influenced by the outer diameter of

the supply area and a surface-restricted layer could reduce the effect of the supply area on

bearing’s characteristics.

2.3 Hydrodynamic air bearings

Hydrodynamic air bearings are also known as ‘self-acting air bearings’. Unlike hydrostatic air

bearings, a thin pressurized air film in these bearings is generated from the relative motion

between two opposing surfaces instead of relying on external pressurized gas source. In

conventional hydrodynamic air bearings, the boundary of air film is non-compliant and the

pressure in the air film is built up when air is dragged into physical wedges between the two

bearing surfaces. Some sub-structures can be applied on bearing surfaces to enhance the

performance, for example, spiral grooved journal and multiple lope bearing sleeve. Figure 2.5

indicates the mechanism of pressure build-up in a plain journal bearing.

Another well-known type of hydrodynamic air bearings are foil air bearings. The core of a foil

bearing is a compliant, spring like foil lining which supports the rotor at zero speed. Once spin

16
CHAPTER 2 LITERATURE REVIEW

speed of the rotor is fast enough, the foil and the rotor are separated by a thin air film generated

from the rotation via viscosity effect. This flexible clearance boundary makes foil air bearings

clearly distinguished from their hydrodynamic relevants.

Figure 2.5 Formation of air film in hydrodynamic journal air bearings, ω is rotating direction
of journal

Literature found so far on conventional hydrodynamic air bearings is focused on spiral grooved

or herringbone grooved air bearings, which related most to the research work in this project.

Literature on foil air bearings is also included in this section to cover the knowledge.

2.3.1 Herringbone grooved hydrodynamic air bearings

Conventional hydrodynamic air bearings usually use some sort of surface structures to facilitate

the pressure built-up. These surface structures are machined permanently either onto the

stationary bearing sleeves or the rotating components. Some typical designs are known as multi-

1
CHAPTER 2 LITERATURE REVIEW

lobe bearings and spiral grooved bearings. The latter are also known as herringbone grooved

bearings, which are widely used because of their excellent stability and load capacity compared

with others. The review of conventional hydrodynamic air bearings is focused on herringbone

grooved bearings in this section.

The working principle of herringbone bearings is developed from step sliders. It makes use of

the pressure surge caused by sudden reduction in fluid film thickness. A herringbone grooved

air bearing has a number of groove-ridge pairs, which makes the overall pressure profile into a

saw teeth shape. Grooves can either be engraved fully or partially on the surface with a certain

angle inclined to the axial direction. These bearings are normally light loaded and need to

operate at high speeds (several ten thousand to several hundred thousand rpm) with extremely

low radial clearance ranging from 2 to 10μm. In practice, such small clearance demands a high

precision manufacturing process. On the other hand, there is no need for an external gas source,

which simplifies bearing systems and makes the machine portable. Hydrodynamic air bearings

are normally applied in oil-free turbo-machinery and instrumentation, such as optical scanner.

The development of hydrodynamic herringbone bearings started around 1965. Early work was

focused on the prediction of the bearings’ characteristics using the narrow groove theory (NGT)

[32, 33]. In such analysis, several assumptions are made. First, it is assumed that the

herringbone bearings have an infinite number of grooves with infinitesimal width. In this case,

the pressure profile in the air film becomes smooth. Secondly, deflection used to analyse

rotational performance should stay within small amplitude, for example, within 0.2 eccentricity

ratio. These assumptions limit the use of NGT to herringbone air bearings with large groove

number. Later in 1969, Cunningham[34] compared the empirical formulae for herringbone

18
CHAPTER 2 LITERATURE REVIEW

journal bearings in experiments for six different rotors and concluded that load capacity was

over predicted at low groove number and the altitude angle was smaller. After that time,

research work on the herringbone bearings was transferred into numerical analysis based on the

finite difference method (FDM), finite volume method (FVM) and finite element method

(FEM). Bonneau [35] used a finite element method with the Petrov-Galerkin weighted discrete

scheme to analyze herringbone journal bearings with 4 to 16 grooves. The method produced a

relatively accurate prediction on the load-deflection relation. Although the finite element

method had advantages in handling complex geometries, the process required a long

computational time and was complex in coding. The finite difference method and finite volume

method were used by other researchers in attempting to find a fast and accurate solution to a

problem. Kobayashi [36] carried out a numerical analysis of herringbone journal bearings based

on a finite difference scheme in a skewered coordinate system. The prediction in load capacity

had a good agreement with Cunningham’s experiment when the rotor speed was 38k rpm.

However, the predicted loading capacity was lower than reality for a rotor speed at 60k rpm and

higher for a rotor speed less than 20k rpm. It should be noted that in [36], Kobayashi considered

the differences between having a grooved journal and having a grooved sleeve in numerical

analysis. Kobayashi also presented a non-linear bearing model using semi-implicit Crank-

Nicoson algorism [3 ] , which had advantages in generating convergent numerical solutions

over others.

Another important research stream in the study of herringbone bearings is the optimization in

the design of bearings’ geometry. In [36], Cunningham noted that groove depth had a significant

influence on the load capacity of the bearings. Gad [38] and his co-workers investigated the

influence of the groove profile by introducing the bevelled-step groove design. Oil lubricated

19
CHAPTER 2 LITERATURE REVIEW

herringbone bearings with this groove profile showed an increase in load capacity and lower

friction torque compared to a rectangular groove profile. Although, the groove geometry in

Gad’s publication was based on oil-lubricated herringbone bearings, the results still serve as a

reference for gas-lubricated herringbone bearings and a further improvement of the

performance can be achieved by modifying the groove profile.

There are other attempts to optimize the gas-lubricated herringbone bearings. In Ikeda [39] and

his co-workers’ research, a hydrodynamic air bearing with non-uniform herringbone grooves

was proposed for high speed spindles. The grooves were designed to grow narrower towards

the centre along axial direction with curved pitch as shown in Figure 2.6. The numerical model

was based on the finite volume method under the boundary-fitted coordinate mesh. Their results

showed a 23% increase in the critical speed and had good agreement with experimental data.

Schiffmann [40] suggested a groove design with variable width, depth and local pitch at four

different locations in the grooved region along the axial direction, as shown in Figure 2. . The

enhanced groove design increased the clearance to diameter ratio up to 80% while maintaining

the same stability margin. A table of the optimum groove parameters for different bearing length

to diameter ratios and compressibility numbers was summarized as guidelines for designs.

Schiffmann’s bearing model was based on the narrow groove theory.

Figure 2.6 Hydrodynamic air bearing with herringbone grooves a) uniform distributed groove
pattern, and b) non-uniform distributed groove pattern [39]

20
CHAPTER 2 LITERATURE REVIEW

Figure 2. a) Position of the interpolation points for the enhanced groove geometry b)
Traditional groove shape and the enhanced groove shape [40]

Although the optimization strategies in [39, 40] showed improvement on the performance of

herringbone air bearings, neither of them considered the effect of the optimization on the groove

profile. As suggested in [39], the original proposed groove geometry should also be shallower

towards the centre along axial direction, but it cannot be achieved because of the limitation of

fabrication. Additionally, the grooves were manufactured on stationary parts in their

experiments, where the effects of rotating groove were not involved. In Schiffmann’s work [40],

the bearing model based on the narrow groove theory restricted their results to herringbone air

bearings with large groove numbers. In project, the conventional rectangular groove profile was

optimized into a curved profile of cosine waves and machined using an advanced laser system.

The detail of the proposed novel groove design was discussed in Chapter 4.

The effect of using viscoelastic support on the stability of herringbone air bearings was also

discussed by some researchers [41-43]. In these works, the supporting structure were illustrated

as a spring and damper element in parallel with their equivalent stiffness and damping

coefficient defined as Equations 2.2 and 2.3. Tomioka and Miyanaga [41] showed that the

21
CHAPTER 2 LITERATURE REVIEW

stability of the system could be effectively improved when the stiffness of the support was close

to that of herringbone air bearings. In their case, the ideal damping coefficient of the support

was between 0.2 and 0.6. They also found that the viscoelastic support system could have worse

stability than the rigid support system if the damping coefficient of the support was less than

0.1 [42]. Experiments were also provided to verify the predictions. The empirical equations to

describe the dynamic properties of O-rings were concluded as shown in Equations 2.2 and 2.3

[43].

𝑘𝑠 = 𝑘 ′ exp(𝑘 ′′ 𝑓𝑝 ) Equation 2.2

1
𝑏𝑠 = 𝑏 ′ + exp(𝑏′′𝑓 Equation 2.3
𝑝)

where 𝑘𝑠 and 𝑏𝑠 are the stiffness and damping coefficients of the support, 𝑘 ′ , 𝑘 ′′ , 𝑏 ′ , 𝑏 ′′ the

dynamic properties of the O-rings determined by the least square method from experimental

data, and 𝑓𝑝 the excitation frequency.

The analysis of stability in Tomioka and Miyanaga’s work was performed using both linear

perturbation method and non-linear transient method. The two methods had good agreement

with each other. However, their rotor model was based on a symmetric rigid rotor supported by

two identical herringbone bearings without considering gyroscopic effects. The bearing model

was still based on NGT. The research presented in this has extended their study to a non-

symmetric rotor system with herringbone grooves fabricated on the rotor. The gyroscopic

22
CHAPTER 2 LITERATURE REVIEW

effects were also considered in the rotor model used. The model of rotor used in this project is

introduced in Chapter 3. Because the viscoelastic support in present work was formed by nitrile

rubber O-rings, Equations 2.2 and 2.3 are used as references.

2.3.2 Foil air bearings

A foil bearing is made of elastic metal with one end foils fixed on a stationary sleeve. A foil air

bearing is made of elastic metal foils fixed on a stationary sleeve. The journal of the shaft is

supported by these spring-loaded foil linings. Once the rotating speed of the journal is

sufficiently high, the foils deform and are pushed away from the shaft by the increasing pressure

within the air film. To avoid wear, the shaft must accelerate to a ‘lift-off’ speed quickly and

work at high speeds to maintain the air pressure and keep the foils away from the shaft.

The development of foil air bearings started from the 1960s and they were first put into

commercial applications on air cycle machines for airliners [44]. The application of foil

bearings then expanded into different kinds of oil-free turbo machinery, such as turbo blowers

and turbo generators. The advantages of foil air bearings over other types of hydrodynamic air

bearings are their compliant clearance boundary, which provides excellent resistance to thermal

expansion, high coulomb damping characteristic and shock load stability [44]. Figure 2.8

illustrates two types of foil air bearings: a multi-leaf foil air bearing and a bump foil air bearing.

23
CHAPTER 2 LITERATURE REVIEW

a) b)

Figure 2.8 Two designs of foil air bearings [44] a) Schematic of a multi-leaf foil journal
bearings, and b) Schematic of bump foil journal bearings

Unlike other air bearings, the numerical model of a foil air bearing requires interactive between

air film and the compliance foil lining. An early numerical model was introduced by Heshmat

[45] in which the top foil was assumed to be ideal without sag between the bumps but with

membrane stress. The pressure in the air film was still governed by the Reynolds Equation while

the pressure dependent clearance was considered. In later work, it was a general practice to

neglect the deflection of the foil structure in axial direction [46]. Foil air bearings were

investigated using both theoretical analysis and experiments in [4 -49].

Besides the foil air bearing showed in Figure 2.8, there were also studies on other types of foil

bearings. San Andrés [50] compared foil air bearings using metal mesh wire with bump type

foil air bearings and showed that the former offered larger damping to dissipate mechanical

energy. Feng [51] analysed the performance of a novel hybrid bump-metal mesh foil bearing.

Studies of foil air bearing applied in micro-turbo machinery also draw the attention of the author.

Most rotors used in turbo machinery are non-symmetric, performance of foil air bearings in

24
CHAPTER 2 LITERATURE REVIEW

these applications was investigated together with finite element rotor dynamic model [52]. Lee,

Park and Sim carried a series of researches into foil bearings for turbocharger applications [53-

55]. The rotor dynamic performance of lobed foil air bearings was studied and tested to evaluate

the effects of mechanical preload and bearing clearance [55]. The results reveal that a decrease

in radial clearance or an introduction of mechanical preload could improve the performance of

foil air bearings. They also carried out a feasibility study of a foil air bearing supported

turbocharger for a two-litre diesel engine [53]. In their latest work [54], the rotor dynamic

performance of the turbocharger with foil air bearings was compared to that supported by

floating ring bearings. The tests showed that the turbocharger equipped with foil air bearings

provided 20% higher rotational speed and showed improvements in rotor dynamic performance.

Although the finite element based rotor model was used in [53-55], the foil air bearings were

modelled as a linear system using perturbation method, in which bearing forces were

represented by equivalent stiffness and damping coefficients.

In the case that foil air bearings are modelled as non-linear as reported in [56-58], a non-

simultaneous routine was generally adopted to solve the bearing model and the system

governing equations of motion. This is also the case in non-linear transient analysis of

conventional air bearings. However, this routine does not reflect the simultaneously interaction

among rotor, air film and the foil structure. It is also inherently time consuming in computations

as solutions require sufficiently small time steps to achieve required accuracy. To overcome the

issue, Bonello and Pham [59, 60] proposed a novel algorithm which enables to achieve the

solution of the whole system simultaneously. The proposed method was cross-verified between

different transformations (finite difference transformation and mesh free Galerkin Reduction)

of the Reynolds Equation. They also proved that the novel Galerkin Reduction significantly

25
CHAPTER 2 LITERATURE REVIEW

reduced the computational time. This simultaneous routine was then applied on stability

analysis of a turbocharger with foil air bearings in [60]. The simultaneous routine can also be

helpful in the non-linear rotor dynamic analysis on other types of air bearings.

2.3.3 Tilt pad air bearings

Pivot pad air bearings, also known as tilt pad bearings, are originally designed as oil-lubricated

hydrodynamic bearings for high-speed turbo machinery. The first hydrodynamic pivot pad

bearing was invented by Kingsbury [61]. In the 1960s, with the increasing demand for oil-free

turbo machinery, researchers started to investigate the possibility of using gas-lubricated pivot

pad bearings and investigated their performance characteristics. Pivot pad air bearings are

normally made of four or five individual pads which are supported by a pivot at the back, as

depicted in Figure 2.9. These pads can wobble about the pivot. This feature allows a better

stability compared with the conventional hydrodynamic air bearings under varying working

conditions. The difficulty in predicting the flow properties for pivot pad bearings was the

complexity of geometry. For example, a single pad will provide three degrees of freedom about

pitch, roll and yaw axes. Timothy [62] and his co-workers summarized the detailed theoretical

development of pivot pad bearings, up until very recent developments, in their review article.

Unlike other types of hydrodynamic air bearings, the recent research works on the pivot pad

bearings involved identification of the pad transfer function. The new approach helped develop

more accurate models in predicting the rotational performance. Early work was reported by

Wilkes and Childs [63]. However, the complex structure of pivot pad bearings and the assembly

technique made it difficult to fit them in micro turbo machinery.

26
CHAPTER 2 LITERATURE REVIEW

Pivot

Figure 2.9 Pivot at the back of a pad in a tilt pad air bearing

2.4 Hybrid air bearings

The term ‘gas-lubricated hybrid journal bearing’ was first reported by Lund [64]. However, in

his definition, it was an externally pressurized gas journal bearing including hydrodynamic

effect caused by the rotating journal. The hybrid bearing concept has been extended to describe

an air bearing which works in either hydrostatic or hydrodynamic depending on the rotational

speed of the rotor. This kind of hybrid bearings can lift at low rotational speeds with externally

compressed air and fully self-act without external air supply when rotational speed is

sufficiently high.

Researches on hybrid air bearings can be found in both conventional and compliant air bearings,

for example hybrid foil air bearings. Because the research in this thesis is only concerned with

non-compliant air bearings, the work on hybrid foil air bearings, such as investigated in [65,

66], is not discussed here.

2
CHAPTER 2 LITERATURE REVIEW

Ives and Rowe [6 ] reported hybrid journal bearings combined with hydrostatic and

hydrodynamic bearings. Their effort showed that this combination did allow the bearing to

support load at a wide range of speeds, including zero speed. Other researchers also reported

works based on different combinations of the gas-lubricated bearings. Osbourne and San

Andrés [68] reported a design of gas-lubricated journal bearings based on the combination of

three lobe bearings and hydrostatic air bearings with orifices. The experimental results showed

an increase in critical speed and threshold whirl frequency ratio (whirling frequency over

journal rotating frequency), which allowed for a higher stable operation, compared with that of

purely hydrostatic/hydrodynamic air bearing designs. The results also indicated that by

increasing supply pressure, better stability could be achieved. In another report published by

Osbourne and San Andrés later [69], predictions of rotor dynamic performance on a test rotor

were discussed. The predicted stable operation speed was somehow much lower than that

observed from the test. They suggested that the current physical models of lubrication and

orifice flow need to be modified for hybrid bearings to fit the experimental observations. Zhu

and San Andrés [ 0] also developed flexure pivot hydrostatic gas bearings which were a type

of hybrid bearing in actually. The design was modified from four-pad pivot pad bearings by

adding an inclined inject nozzle at the spare space between each pad, Figure 2.10. The results

demonstrated an enhanced performance compared with the original four pads’ design. In a later

publication by San Andrés [ 1] , the effect of supply pressure in hybrid bearings in [ 0] on the

vibration while crossing system critical speeds was investigated. It was found that the use of

external pressurization stiffened the bearings and increased the rotor-bearing system critical

speeds. However, the damping ratio of the system decreased accordingly, which made the

system more sensible to imbalance when crossing critical speeds. Additionally, the experiments

in [ 1 ] showed no external pressurization would be necessary for operation beyond the critical

28
CHAPTER 2 LITERATURE REVIEW

speeds of the rotor-bearing system. In particular, the critical speeds during coast down could be

fully eliminated over an extended operating speed range by manually control the external

supply pressure.

Figure 2.10 Flexure pivot hybrid gas bearings designed by Zhu and San Andrés [ 0]

The advantages of hybrid air bearings demonstrated in [68- 1 ] inspired the author to further

investigate into this area and provide hybrid air bearings with compact structures for micro

turbo machinery. The hybrid air bearings proposed in the thesis is a combination of hydrostatic

air bearings with plain orifice restrictors and hydrodynamic air bearings with herringbone

grooves fabricated on the surface of the rotor. The herringbone grooves are also enhanced by a

novel groove profile investigated in this research. Figure 2.11 gives a demonstration of the

proposed hybrid air bearings. The most similar researches to the present work was made by

29
CHAPTER 2 LITERATURE REVIEW

Stanev [ 2] and Zhang [ 3] . They both used the same combination proposed here, referring to

Figure 2.11. According to their investigations, this type of hybrid air bearings showed

improvement on non-rotational and rotational performance. Zhang’s work [ 3] was more

focused on the application in power MEMS with the gas rarefaction effects considered.

Although the influence of restrictor location and fabrication defects were discussed, it was for

the thrust bearings only and less interested in for the applications discussed in this thesis. In

Stanev’s work [ 2] , the performance of the hybrid journal air bearing was investigated. It was

noticed that the stability of this hybrid bearing was increased significantly for a compressibility

number over 10 compared with that of the hydrostatic air bearings and was lower for a

compressibility number between 3 and . Stanev concluded that improvement on stiffness and

stability of the bearings could be substantial for hybrid journal bearings in this configuration

operating at a compressibility number over 30. However, there were some limitations on the

work in [ 2] . Firstly, Stanev’s hybrid air bearing model was based on the narrow groove theory

which limits the model to bearings with large groove number. There was a need for precision

models of the proposed hybrid air bearings using other numerical techniques, for example, finite

difference/volume/element method. Secondly, the parametric studies on the performance,

especially the stability analysis of rotor bearing system was based on the linear perturbation

method, in which bearing forces were linearized and represented by equivalent stiffness and

damping coefficients. In the approach in [ 2] , the analysis was only applied to a symmetric and

rigid rotor supported on two identical bearings. No modelling of rotor was involved. Also, the

steady-state amplitude of the self-excited whirl of the bearings could not be predicted using the

linear rotor bearing model. Thus, the rotor dynamic behaviour of a non-symmetric rotor

supported by such hybrid air bearings was not fully explained and it would be of great potential

for turbo machinery applications.

30
CHAPTER 2 LITERATURE REVIEW

Figure 2.11 Grooved hybrid air bearings designed by P. Stanev, F. P. Wardle, J. Corbett[ 2]

With reference to the limitations in Stanev’s work and the optimization on herringbone grooved

air bearings in [38-40], the work presented in this thesis extends the studies on the proposed

hybrid air bearings in following aspects:

1) A numerical model based on finite volume method is provided for the hybrid air

bearings which is suitable for any groove number and took the effects of rotating

grooves into account.

2) A novel profile of herringbone groove pattern in air bearings is fabricated to enhance

the hybrid air bearings. Its positive effects on the performance are investigated and

confirmed.

31
CHAPTER 2 LITERATURE REVIEW

3) Both linear perturbation method and non-linear transient method are performed to study

the rotational performance of hybrid air bearings and the rotor dynamic configuration

they support. The former is used in theoretical studies of bearings to understand the

influence of different design parameters and give predictions on the stability and natural

frequencies of rotor bearing system. The latter is applied with finite element rotor

dynamic model of a non-symmetric rotor to predict unbalance responses and compare

with experimental measurements.

In an attempt to apply the proposed hybrid air bearings to turbomachinery, external dampers

are used outside stationary bearing sleeves. The benefits of this arrangement were not only

investigated by the authors in [41-43] but also by Ertas [ 4] and Delgado [ 5] . The latter two

used metal wire mesh as external dampers in parallel with compliant hybrid air bearings. One

improvement by using this arrangement was the elimination of self-excited whirl as the results

shown in Figure 2.12 [ 4] . The use of external dampers could also help reduce the sensitivity

of the rotor-bearing to imbalance while crossing the critical speeds of the system as addressed

in [ 1 ] for hybrid air bearings.

32
CHAPTER 2 LITERATURE REVIEW

Figure 2.12 The elimination of self-excited whirl by introducing mesh metal wire damper in
parallel with a hybrid compliant air bearing [ 4]

2.5 Numerical techniques in modelling of air bearings

tasks in this project. The nature of the work is to solve the Reynolds Equation numerically. As

a second order partial differential equation, there are three discrete schemes which can be used:

the finite difference method (FDM), the finite element method (FEM) and the finite volume

method (FVM). In this area, NASA has provided several numerical approaches which are

commonly used nowadays. Florin suggested that by applying Newton’s method, it could reduce

the computational time significantly. Newton’s method was also proven to have better

numerical stability in getting converged solutions [ 6] . Wang [ - 9] carried out a series of

33
CHAPTER 2 LITERATURE REVIEW

studies on numerical methods which can be applied to solving a compressible Reynolds

Equation. These works included comparisons of different methods with respect to

computational time and numerical stability, the applications of Newton’s method, successive

relaxation and the G-S method. He also suggested different stopping criterion in the iterative

process and analysed the truncation error of numerical approaches. The above literature was all

based on FDM and they have good agreement with experimental results for hydrostatic air

bearings and plain journal bearings. In the analysis of herringbone bearings, FDM is limited by

the complex surface patterns. In this case, FEM is considered as an alternative approach.

Challenges in the FEM approach were the interpolation of shape functions at each node in the

discrete scheme. Faria [80] suggested a higher order shape function based on index function

which showed relatively good accuracy. FVM was also suggested by some researchers. The

FVM scheme for air bearing modelling can be derived by integrating the Reynolds Equation

using the Green’s theorem in a controlled volume surrounding a point in the air film. This

approach showed advantages in dealing with film discontinuities and some pocket restrictor

configurations [81].

The modelling of the proposed hybrid air bearings in this thesis is based on the FVM scheme.

Some numerical techniques used in the FDM scheme is also adopted to improve the

computational speed and numerical stability.

2.6 Modelling of rotor bearing system and stability analysis

In this project, the rotor supported by hybrid air bearings is non-symmetric and has built-on

components such as a turbine and a compressor, as shown in Figure 2.13. A well accepted

34
CHAPTER 2 LITERATURE REVIEW

method to model such a rotor is using finite element method based on the Timoshenko beam

theory with gyroscopic effect [82]. In this method, the rotor is modelled as linear. The shaft

elements are formed of two nodes with four degrees of freedom (4-DOF) on each. The built-on

components were modelled as lumped masses and moment of inertia. Figure 2.14 shows a

typical shaft element used in the model and its coordinate system. This linear rotor model can

be combined with either linear or non-linear bearing models to give predictions on the

performance of the system.

Figure 2.13 A rotor to be supported by the proposed hybrid air bearings

𝑦𝑗
𝜃𝑦𝑗
𝑦𝑖 𝑥
𝑗
𝜃𝑦𝑖 𝜃𝑥𝑗
𝑥𝑖

𝜃𝑥𝑖

Figure 2.14 A 4-DOF model of shaft element and global coordinates used in Timoshenko
beam theory

35
CHAPTER 2 LITERATURE REVIEW

The general governing equations of motion for a modelled rotor bearing system with constant

rotational speed, 𝜔 , are shown in Equation 2.4 [83]. The techniques of adding non-linear

bearing model and external damper were described in [84] for several situations. They were

used as references for modelling rotor bearings system in this project.

𝑴𝑞̈ + 𝑲𝑞 + (𝑪 + 𝜔𝑮)𝑞̇ = 𝐹 Equation 2.4

𝑇
𝑞 = [… 𝑥𝑖 𝑦𝑖 𝜃𝑥𝑖 𝜃𝑦𝑖 … ] Equation 2.5

where q is the system displacement vector to be solved, 𝑥𝑖 , 𝑦𝑖 the translational displacement of

𝑖 𝑡ℎ node, 𝜃𝑥𝑖 , 𝜃𝑦𝑖 the rotational displacement of 𝑖 𝑡ℎ node under right hand rule, M, K, C, G the

system mass and inertia, stiffness, damping and gyroscopic matrices, 𝜔 the rotational speed and

F the force vector that contains all forces/moments, including unbalance excitations, shaft bow

forces, gravitational and/or static loads and all nonlinear interconnection forces.

One important topic in the studies of air bearings is the stability of the rotor bearing system,

which refers to the stability of static equilibrium position (SEP) of a journal to small

perturbations. Two approaches can be used to analyse the stability of a rotor bearing system:

the linear perturbation analysis and the non-linear transient analysis. In the linear perturbation

analysis, the centre of journal is assumed to whirlr around the SEP under a given frequency

with small amplitude. Bearing forces are linearized and represented by equivalent stiffness and

damping coefficients. The stability of the linearized system can then be evaluated by omitting

36
CHAPTER 2 LITERATURE REVIEW

the force vector in Equation 2.4 and examining the eigenvalues [60, 85] . However, the linear

model cannot predict the steady-state amplitude of the self-excited whirl. In the non-linear

transient analysis, the bearing is modelled as non-linear. The bearing forces are no longer

approximated by the equivalent stiffness and damping coefficients but forces applied to the

rotor. This approach allows the steady-state amplitude of the self-excited whirl to be predicted

[86].

In this thesis, the both approaches are employed. The linear perturbation analysis is used to

study the influence of design parameters on rotational performance of hybrid air bearings and

give predictions on the stability and natural frequencies of rotor bearing system theoretically.

The non-linear transient analysis is used to predict unbalance responses of rotor bearing system

and compare with those observed from experiments.

2.7 Summary

This chapter outlines the aspects relevant to the proposed hybrid air bearings and techniques

used in modelling and analysis of rotor bearing system. The literature survey on hybrid air

bearings has highlighted the necessity and novelty of the research presented within this thesis.

Literature reviewed on hydrostatic air bearings was focused on works that involve modelling

of the bearing, especially the restrictor system and its interactions with air film. The orifice

restrictors are usually modelled using empirical equations with correction on the entrance loss

effect. Modelling of porous media restrictors is commonly based on 3D Cauchy’s law.

3
CHAPTER 2 LITERATURE REVIEW

In conventional hydrodynamic air bearings, spiral or herringbone grooved bearings are often

used because of their excellent stability compared to others. Numerical models of herringbone

grooved bearings are usually based on one of the following approaches: NGT, FDM, FEM and

FVM. Research works also show that the performance of this type of air bearing can be

improved using optimized groove geometry and viscoelastic supports. Hydrodynamic air

bearing with compliant bearing boundaries, such as foil bearings and tilt pad bearings are

reviewed to cover the relevant knowledge.

Hybrid air bearings combine the features of hydrostatic and hydrodynamic air bearings.

Different combinations are possible in this category, for example, hybrid herringbone grooved

bearings and hybrid pivot pad bearings. The relevant literatures of hybrid air bearings show that

progress has been made in improving the load capacity and stability. By means of adjusting the

supply air pressure, hybrid air bearings can also change the system’s critical speeds.

Modelling of air bearings requires the Reynolds Equation to be solved numerically. Newton’s

method and successive relaxation techniques are commonly applied with several mesh based

algorisms to provide the solution.

Flexible rotors used in rotor dynamic analysis are modelled using finite element method based

on the Timoshenko beam theory with gyroscopic effect. The governing equation of motion for

a rotor bearing system can either be used in a linear analysis approach or a non-linear approach

to give predictions on the stability information of the system.

38
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

CHAPTER 3: NUMERICAL ANALYSIS OF

AIR BEARINGS

3.1 Introduction

This chapter introduces the theories used for analysing the performance of air bearings and the

development of numerical air bearing models to be used in the study as presented in the

following chapters. The bearing models are derived for gas-lubricated journal bearings with

non-compliant boundaries. Truncation errors are provided for analysing the accuracy of the

bearing models. Both linear perturbation analysis and non-linear transient analysis are

explained in order to investigate the stability and unbalance responses of gas-lubricated journal

bearings. The rotor used in this project is modelled using finite element method based on

Timoshenko beam theory.

This chapter starts with an introduction to the theories of fluid dynamics relevant to gas-

lubricated bearings, including the effects of slip flow. Next, air bearing models are developed

with assumptions and numerical techniques. The proposed numerical approach is then used to

analyse hydrostatic journal air bearings with pocketed orifice restrictors for verification. The

validity and accuracy of the rotor model is investigated using an impact hammer on a free-free

rotor condition. Then, the linear perturbation analysis and its limitations are discussed.

Afterwards, non-linear transient analysis is carried out based on a non-simultaneous routine.

Finally, a summary is provided at the end of the chapter.

39
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

3.2 Modelling of gas-lubricated journal bearings with non-compliant

boundaries

In General, gas-lubricated bearings can be modelled using the compressible Reynolds Equation.

In the study, the Reynolds Equation is solved numerically by means of finite difference methods

(FDM) and finite volume method (FVM). Other flow effects can be considered by adding

boundary conditions before solving the Reynold Equation.

3.2.1 Discrete scheme of Reynolds Equation

The performance of a journal air bearing largely depends on five important parameters: the

pressure (𝑝), the rotational speed (𝜔), the air film thickness (ℎ), the bearing radius (𝑟0 ) and the

length of the bearing (𝑙). Their relationships are described in the Reynolds Equation, as shown

in Equation 3.3. The Reynolds Equation is derived from the Navier-Stokes Equations, Equation

3.1, and mass continuity Equation, Equation 3.2. It represents the momentum conservation as

a simplified form of the Navier-Stokes Equations and mass conservation by satisfying the mass

continuity of the gas.

𝜕𝒖 1 1
+ 𝒖 ∙ ∇𝒖 = − ∇𝑝̅ + 𝜂∇2 𝒖 + 𝜂∇(∇ ∙ 𝒖) + 𝒈
𝜕𝑡 𝜌 3 Equation 3.1

40
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

𝜕𝜌
+ ∇ ∙ (𝜌𝒖) = 0 Equation 3.2
𝜕𝑡

where 𝜌 is the density of the fluid, 𝒖 the flow velocity, ∇ the differential operator, 𝑝̅ the flow

pressure, 𝜂 the viscosity of fluid, and 𝒈 the body accelerations (per unit mass) acting on the

continuum.

The Reynolds Equation for journal air bearings is given as Equation 3.3. It is convenient to

unwrap journal bearings from a radial symmetry plane by means of spreading the circular

surface of the journal flat.

∂ 𝑝ℎ3 𝜕𝑝 ∂ 𝑝ℎ3 𝜕𝑝 𝜕(𝑝ℎ) 𝜕(𝑝ℎ)

( )+ ( )=𝑢 +
∂ξ 12𝜂 𝜕𝜉 ∂z 12𝜂 𝜕𝑧 𝜕𝜉 𝜕𝑡 Equation 3.3

where 𝜉 is the coordinate along the circumferential direction, z the coordinate along the axial

direction, p the pressure in the air film, h the local thickness of the air film. 𝜂 the dynamic

viscosity of air, 𝑢 an averaged velocity of the rotating journal and stationary sleeve, and t the

time.

The Reynolds Equation is a second order partial differential equation (PDE) which does not

have an analytical solution, but can be solved by using numerical approaches. In practice, it is

convenient to normalize the parameters and rewrite Equation 3.3 into its dimensionless form,

as shown in Equation 3.5.

41
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

Normalized parameters are:

𝜉 𝑧 𝑝 ℎ 𝜔𝑟0 ωt
𝜃= ,Z = ,𝑃 = ,𝐻 = ,𝑢 = ,τ =
𝑟0 𝑟0 𝑝𝑎 𝐶 2 2 Equation 3.4

where 𝐶 is for the radial clearance, 𝑝𝑎 the ambient pressure and 𝑟0 the radius of the journal

bearing.

The dimensionless form of Equation 3.3 is:

𝜕 𝜕𝑃 𝜕 𝜕𝑃 𝜕(𝑃𝐻) 𝜕(𝑃𝐻)
(𝑃𝐻 3 ) + (𝑃𝐻 3 ) = 𝛬 + 𝛬 ,
𝜕𝜃 𝜕𝜃 𝜕𝑍 𝜕𝑍 𝜕𝜃 𝜕𝜏

6𝜂𝜔𝑟0 2 Equation 3.5

𝛬=
𝑝𝑎 ℎ0 2

where 𝛬 is known as the compressibility number or bearing number. Most of the performance

characteristics of air bearings can be determined with respect to 𝛬.

In this study, the performance of air bearings was investigated using the FDM for hydrostatic

bearings and the FVM for hydrodynamic and hybrid bearings. A discretizing scheme is needed

in both numerical methods. In this research, the five-point central difference scheme is adopted.

42
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

The scheme is derived from the Taylor series expansion with second order accuracy. If there

are five discrete points in the air film, as shown in Figure 3.1, the pressure at point (𝑖, 𝑗) can be

expressed using the Taylor expansion.

Figure 3.1 Five-point discrete scheme

𝜕𝑃 ∆𝜃 2 𝜕 2 𝑃 ∆𝜃 3 𝜕 3 𝑃
𝑃(𝑖, 𝑗 + 1) = 𝑃(𝑖, 𝑗) + ∆𝜃 + + … Equation 3.6
𝜕𝜃 2! 𝜕𝜃 2 3! 𝜕𝜃 3

𝜕𝑃 ∆𝜃 2 𝜕 2 𝑃 ∆𝜃 3 𝜕 3 𝑃
𝑃(𝑖, 𝑗 − 1) = 𝑃(𝑖, 𝑗) − ∆𝜃 + − … Equation 3.7
𝜕𝜃 2! 𝜕𝜃 2 3! 𝜕𝜃 3

Subtracting Equation 3. f rom Equation 3.6 and dividing the result by ∆𝜃 lead to Equation

3.8:

𝜕𝑃 𝑃(𝑖, 𝑗 + 1) − 𝑃(𝑖, 𝑗 − 1)
= + ℛ(∆𝜃 2 ) Equation 3.8
𝜕𝜃 2∆𝜃

43
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

where ∆𝜃 is the distance between two points in 𝜃 direction and ℛ(∆𝜃 2 ) the Lagrange

remainder. It determines the accuracy of the discrete scheme and is a function of ∆𝜃. The same

discrete scheme can be applied to other partial difference terms.

𝜕𝑃 𝑃(𝑖 + 1, 𝑗) − 𝑃(𝑖 − 1, 𝑗)
= + ℛ(∆Ζ2 ) Equation 3.9
𝜕Ζ 2∆Ζ

𝜕 2𝑃 𝑃(𝑖, 𝑗 + 1) + 𝑃(𝑖, 𝑗 − 1) − 𝑃(𝑖, 𝑗)

2
= + ℛ(∆𝜃 2 ) Equation 3.10
𝜕𝜃 ∆𝜃 2

𝜕 2𝑃 𝑃(𝑖 + 1, 𝑗) + 𝑃(𝑖 − 1, 𝑗) − 𝑃(𝑖, 𝑗)

2
= 2
+ ℛ(∆Ζ2 ) Equation 3.11
𝜕Ζ ∆Ζ

The five-point central difference scheme is adopted because of its accuracy and numerical

stability in iterative computations [8 ].

3.2.2 Assumptions and challenges in modelling air bearings

Before starting the development of bearing models, some general assumptions need to be made

to simplify the problem. They are as follows:

 In an air film, the inertia effect is neglected [88].

44
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

 No turbulence flow is considered in the proposed bearing models.

 The air properties are assumed to follow the ideal gas law.

 The viscosity of air is considered as a constant.

 The pressure within the air film does not vary along the depth of the air film.

The above assumptions are applied to all bearing models proposed in this thesis.

From [88], the Reynolds Numbers are defined in the field of fluid film lubrication as:

𝑖𝑛𝑡𝑒𝑟𝑡𝑖𝑎 𝜌0 𝑢0 ℎ0 2
𝑅𝜉 = = Equation 3.12
𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝜂0 𝑙0

𝑖𝑛𝑡𝑒𝑟𝑡𝑖𝑎 𝜌0 𝑣0 ℎ0 2
𝑅𝑧 = = Equation 3.13
𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝜂0 𝑏0

where 𝜌0 is the local density in the fluid film, ℎ0 the characteristic film thickness. 𝑙0 and 𝑏0 are

the characteristic lengths in 𝜉 and 𝑧 direction, 𝑢0 and 𝑣0 are the characteristic speeds in 𝜉 , 𝑧

ℎ0
direction. 𝑢0 normally refers to the journal surface speed in journal bearings and 𝑣0 = 𝑢0 .
𝑙0

𝜂0 is the absolute viscosity. All parameters are in SI units.

One challenge in the development of simulation models for air bearings is that the numerical

solution to the Reynolds Equation may become unstable and diverge during iterative

computations if many bearing design parameters are involved. For example, when three design

parameters need to be considered for a plain journal bearing, which are the radius 𝑟0 , the length

45
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

𝑙 and the radial clearance 𝑐, the discrete Reynolds Equation will yield a converged solution

under all reasonable conditions. However, for a hydrostatic journal air bearing with orifice

restrictors, if three additional design parameters are involved, i.e. supply pressure 𝑃𝑠 , orifice

diameter 𝑑0 and number of orifices, the discrete Reynolds Equation may not yield a converged

solution at some conditions, when the supply pressure is high. This can be resolved using other

numerical techniques, such as the Newton’s method introduced in Section 3.2.4

The other sources of divergence are the high compressibility number and eccentricities. High

values of these two indicate the pressure distribution within the air film will change rapidly at

the physical wedge and cause divergence. This can be overcome by using fine mesh locally to

smoothen the pressure profile.

In addition, the Green’s theorem, Newton’s method and successive relaxation are applied to the

modelling and computations of the air bearings. With all the analysis and measures taken,

repeated simulation practice shows that the proposed model has good numerical stability for a

wide range of design and working parameters, while maintaining a reasonable computational

time.

3.2.3 Mesh and boundary conditions

The mesh grid used for the bearing models is based on the five-point central difference scheme.

As shown in Figure 3.2, the surface of the journal bearing is unwrapped flat from a radial

symmetric plane. The unwrapped surface is meshed with a uniform grid under the following

principles:

46
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

- The locations of the restrictors coincide with the grid nodes for hydrostatic journal air

bearings;

- The locations of the restrictors and the apexes of herringbone grooves both coincide

with the grid nodes for hybrid/hydrodynamic journal air bearings;

- The coordinate system is placed on the stationary component;

- There are M nodes in Z direction and N nodes in 𝜃 direction as indicated in Figure 3.2.

The boundary conditions applied to all journal bearing models are periodic boundary and

ambient pressure boundary. Other boundary conditions, such as orifice boundary and slip flow

boundary, are added by modifying the Reynolds Equation at specific nodes in the mesh. The

flow continuity boundaries at the edges of herringbone grooves in hydrodynamic/hybrid journal

air bearing are self-satisfied by using the FVM approach.

θ
ω
O1 O2

Periodic
Boundary, Orifice
pn-1 = pn+1 Attitude angle
Boundar
Z Ambient Boundary, p = pa
Periodic
Boundary,
pn+1 = pn-1

Figure 3.2 Mesh and boundary conditions for static air journal bearings

4
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

The orifice boundary is added to the nodes aligned with restrictors. The Reynolds Equation

𝑚̇
(Equation 3.3) at these nodes is modified as Equation 3.14. The term, ∂ξ𝜕𝑧
𝑜𝑟𝑖
represents the flux

which is caused by the orifice restrictors.

∂ 𝑝ℎ3 𝜕𝑝 ∂ 𝑝ℎ3 𝜕𝑝 𝑚̇𝑜𝑟𝑖 𝜕(𝑝ℎ) 𝜕(𝑝ℎ)

( )+ ( )+ =𝑢 +
∂ξ 12𝜂 𝜕𝜉 ∂z 12𝜂 𝜕𝑧 ∂ξ𝜕𝑧 𝜕𝜉 𝜕𝑡 Equation 3.14

where 𝑚̇𝑜𝑟𝑖 is the flow rate through an orifice. It can be calculated by solving Equations 3.15

𝑚̇
and 3.16. 𝑚̇𝑜𝑟𝑖 refers to the flow rate in a ‘choking’ condition. ∂ξ𝜕𝑧
𝑜𝑟𝑖
is the flux term or flow rate

per unit area. The both can be expressed as follows.

1
2 𝛾+1 2 𝛾
2𝛾 𝑝0 𝛾 𝑝0 𝛾 𝑝0 2 𝛾−1
𝑚̇𝑜𝑟𝑖 = 𝐶𝑑 ∗ 𝐴𝑟 ∗ 𝑝𝑠∗ { 𝑅 𝑇 [( ) − ( ) ]} , > [ ]
𝛾 − 1 𝑠𝑝𝑒𝑐 𝑝𝑠 𝑝𝑠 𝑝𝑠 𝛾+1

Equation 3.15

1 1 𝛾
2𝛾 2 2 𝛾−1 𝑝0 2 𝛾−1
𝑚̇𝑜𝑟𝑖 = 𝐶𝑑 ∗ 𝐴𝑟 ∗ 𝑝𝑠 ∗ [ ] [ ] , ≤[ ] Equation 3.16
𝛾+1 𝛾+1 𝑝𝑠 𝛾+1

where 𝑅𝑠𝑝𝑒𝑐 is the gas constant, 𝑇 the temperature in Kelvin, 𝑝0 the flow pressure at the

downstream of an orifice, 𝑝𝑠 the flow pressure at the upstream of an orifice and will be given at

fixed value depends on the actual pressure used in the bearing, and 𝛾 the specific heat ratio of

48
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

air. In most conditions, 𝛾 equals to 1.4. 𝐶𝑑 is the coefficient of discharge. It is a factor relating

the actual mass flow rate to the theoretical mass flow rate for an orifice. 𝐴𝑟 is the restricted area.

For application to air bearings, 𝐴𝑟 can be calculated as 𝜋𝑑0 2 or 𝜋𝑑0 ℎ whichever is smaller. ℎ

is the local air film thickness.

In fluid dynamics, the fluid velocity at a solid boundary can be treated either as non-slip

condition or slip condition. The non-slip condition assumes that the fluid velocity at all fluid-

solid boundaries equals to that of the solid boundaries. On the other hand, the slip flow condition

assumes that the fluid has a non-zero velocity, relative to the solid boundaries. In the air bearing

models, these two conditions refer to the air velocity at the journal surface and stationary sleeve

surface. In most of the cases, the no-slip boundary will be employed at the fluid-solid

boundaries. The Reynolds Equation (Equation 3.3) can then be used straightforwardly. If the

slip boundary is applied, the Reynolds Equation needs to be modified accordingly.

Here the slip flow condition is determined by the Knudsen number, Equation 3.1 , which is

defined as the ratio of the mean free molecular path to characteristic dimension. The

characteristic dimension for air bearing applications is the radial clearance. When the Knudsen

number, 𝐾𝑛 , is greater than 0.01, the dimensionless first order slip flow correctors, 𝐷𝑘𝑛 and 𝑄𝑝 ,

are added to the Reynolds Equation (Equation 3.5) for the Poiseuille and Couette flow terms to

satisfy the slip flow boundary.

𝜆𝑚 𝑝𝑎
𝐾𝑛 =
𝑝ℎ Equation 3.17

49
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

√𝜋 √𝜋𝑝ℎ
𝐷𝑘𝑛 = = Equation 3.18
2𝐾𝑛 2𝜆𝑚 𝑝𝑎

where 𝐷𝑘𝑛 is the inverse Knudsen number, and 𝜆𝑚 = 0.064𝜇𝑚 the mean free molecular path

of air at 21 degrees and atmospheric pressure. If 𝑄𝑝 is the slip flow corrector for Poiseuille flow

and 𝑄𝑐 is the slip flow corrector for Couette flow, they can be expressed as:

𝛼 √𝜋 𝐷
𝑄𝑝 = + Equation 3.19
2 6

𝐷
𝑄𝑐 = Equation 3.20
6

where 𝛼 = (2 − 𝜎)/𝜎 is the surface correcting coefficient and 𝜎 = 0.8 for practical surfaces

[89].

By substituting Equation 3.19 and Equation 3.20 into Equation 3.5, the dimensionless Reynolds

Equation with slip flow boundary can be generated as Equation 3.21. This equation will serve

as the governing equation in FDM and FVM when slip flow boundary condition is considered.

∂ 𝜕𝑃 ∂ 𝜕𝑃 𝜕(𝑄𝑐 𝑃𝐻) 𝜕(𝑃𝐻)

(𝑄𝑝 𝑃𝐻 3 ) + (𝑄𝑝 𝑃𝐻 3 ) = Λ + 2Λ
∂θ 𝜕𝜃 ∂Z 𝜕𝑍 𝜕𝜃 𝜕𝜏

Equation 3.21

50
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

3.2.4 FDM, FVM, iterative strategies and numerical techniques

In simulation of hydrostatic journal air bearings, the FDM is adopted in iterative computations.

In simulation of hydrodynamic and hybrid journal air bearings, the FVM is adopted in iterative

computations.

In the FDM approach, the Reynolds Equation with orifice boundary conditions (Equation 3.14)

is discretized straightforwardly using Equations 3.8 to 3.11. The discrete equation has both first

and second order partial differentiation of P. It is then solved numerically using the algorithm

shown in Figure 3.3.

51
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

Figure 3.3 A flow diagram of the FDM approach

52
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

In order to improve the numerical stability and convergence rate of the iterative process,

Newton’s method is adopted [90]. It is a general algorithm for linearizing second order partial

differential equations, and is often combined with a successive relaxation method for either

good numerical stability, or fast converging rate. The linearization procedure is demonstrated

below.

In Newton’s method, 𝑓 is defined as a function of 𝑃:

∂ 𝜕𝑃 ∂ 𝜕𝑃 𝜕(𝑃𝐻)
𝑓(𝑃) = (𝑃𝐻 3 ) + (𝑃𝐻 3 ) − Λ Equation 3.22
∂θ 𝜕𝜃 ∂Ζ 𝜕Ζ 𝜕𝜃

In reference with Newton’s method, the following expression stands:

𝑓(𝑃𝑛 ) + 𝑓 ′ (𝑃𝑛 )(𝑃𝑛+1 − 𝑃𝑛 ) = 0

Equation 3.23

where the superscript 𝑛 and 𝑛 + 1 denote the 𝑛𝑡ℎ and (𝑛 + 1)𝑡ℎ iterative step.

By means of introducing parameter, 𝛿 𝑛 = 𝑃𝑛+1 − 𝑃𝑛 , and apply Taylor expansion to 𝑓(𝑃𝑛 )

within the interval [𝑃𝑛 , 𝑃𝑛 + 𝛿 𝑛 ], there exists some point in this interval, denoted by 𝑃𝑛 + 𝛽𝛿 𝑛

for some 𝛽 ∈ [0,1], such that:

53
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

𝑓(𝑃𝑛 + 𝛽𝛿 𝑛 ) = 𝑓(𝑃𝑛 ) + 𝛿 𝑛 𝛽𝑓 ′ (𝑃𝑛 ) + 𝑅((𝛿𝛽)2 )

Equation 3.24

And:

𝑑𝑓(𝑃𝑛 + 𝛽𝛿 𝑛 )
|𝛽=0 = 𝛿 𝑛 𝑓 ′ (𝑃𝑛 ) = −𝑓(𝑃𝑛 )
𝑑𝛽 Equation 3.25

Expressing the term 𝑓(𝑃𝑛 + 𝛽𝛿 𝑛 ) with Equation 3.22 and making differential with respect to

𝛽, one can get the following equation:

𝑑𝑓(𝑃𝑛 + 𝛽𝛿 𝑛 )
|𝛽=0
𝑑𝛽
𝜕 3 𝑛
𝜕𝑃𝑛 3 𝑛
𝜕𝛿 𝑛
= 2[ (𝐻 𝛿 +𝐻 𝑃 )
𝜕𝜃 𝜕𝜃 𝜕𝜃
𝜕 3 𝑛
𝜕𝑃𝑛 3 𝑛
𝜕𝛿 𝑛 𝜕(𝛿 𝑛 𝐻)
+ (𝐻 𝛿 +𝐻 𝑃 )] − Λ
𝜕Ζ 𝜕Ζ 𝜕Ζ 𝜕𝜃

Equation 3.26

𝑑𝑓(𝑃 𝑛 + 𝛽𝛿 𝑛 )
Substituting expressions of |𝛽=0 and 𝑓(𝑃𝑛 ) into Equation 3.25, the equation to be
𝑑𝛽

solved at 𝑛𝑡ℎ iterative step is transformed into:

54
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

𝜕 3 𝑛
𝜕𝑃𝑛 3 𝑛
𝜕𝛿 𝑛 𝜕 3 𝑛
𝜕𝑃𝑛 3 𝑛
𝜕𝛿 𝑛 𝜕(𝛿 𝑛 𝐻)
2[ (𝐻 𝛿 +𝐻 𝑃 )+ (𝐻 𝛿 +𝐻 𝑃 )] − Λ
𝜕𝜃 𝜕𝜃 𝜕𝜃 𝜕Ζ 𝜕Ζ 𝜕Ζ 𝜕𝜃
𝜕 𝑛 3
𝜕𝑃𝑛 𝜕 𝑛 3
𝜕𝑃𝑛 𝜕(𝑃𝑛 𝐻)
=− (𝑃 𝐻 )− (𝑃 𝐻 )+Λ
𝜕𝜃 𝜕𝜃 𝜕Ζ 𝜕Ζ 𝜕𝜃

Equation 3.27

Applying central difference equations for 𝛿 , Equation 3.2 can be expressed as a linear

algebraic equation with respect to 𝛿:

𝑎𝑖,𝑗 𝛿𝑖,𝑗 + 𝑏𝑖,𝑗 𝛿𝑖,𝑗−1 + 𝑐𝑖,𝑗 𝛿𝑖,𝑗+1 + 𝑑𝑖,𝑗 𝛿𝑖−1,𝑗 + 𝑒𝑖,𝑗 𝛿𝑖+1,𝑗 + 𝐶𝑜𝑛𝑖,𝑗 = 0

Equation 3.28

where (i,j) denotes the point at ith row and jth column in the mesh. 𝑎𝑖,𝑗 , 𝑏𝑖,𝑗 , 𝑐𝑖,𝑗 , 𝑑, 𝑒𝑖,𝑗 , 𝐶𝑜𝑛𝑖,𝑗 are

the terms related with H, P, 𝜃, Ζ only and they are listed in Appendix A.

The introduced parameter 𝛿𝑖,𝑗 can be found with a successive relaxation method.

𝑏𝑖,𝑗 𝛿𝑖,𝑗−1 + 𝑐𝑖,𝑗 𝛿𝑖,𝑗+1 + 𝑑𝑖,𝑗 𝛿𝑖−1,𝑗 + 𝑒𝑖,𝑗 𝛿𝑖+1,𝑗 + 𝐶𝑜𝑛𝑖,𝑗

𝛿𝑖,𝑗 𝑛+1 = 𝛿𝑖,𝑗 𝑛 + 𝜔𝑠𝑟 [ ]
𝑎𝑖,𝑗

Equation 3.29

where 𝜔𝑠𝑟 is the relaxation factor ranging from 0 to 2. For 1 < 𝜔𝑠𝑟 < 2, the method is known

as successive over-relaxation, and can increase the convergence rate. On the other hand, the

55
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

method becomes successive under relaxation and can make a non-convergence case converged.

The optimized relaxation factor can be pre-determined with respect to the grid size in the

discrete scheme [91].

Although a linear system, represented by Equation 3.28 can be solved quickly in most numerical

tools, there are some limitations of this procedure. Firstly, in Equation 3.24 and Equation 3.25,

𝛽 is set as 0. This may not be true when small clearances are involved in the simulation. In

Cheng’s report [92], Newton’s method failed to converge at 6𝜇𝑚 bearing radial clearance with

0.1 eccentricity ratio. Additional numerical treatments, such as the rate cutting method or pre-

conditioned conjugate gradient method (PCG), are required to make the computations converge.

Secondly, the flux term in Equation 3.14 for nodes with orifice restrictors disturbs the stability

of convergence. The pressure at the nodes of an orifice needs to be calculated in a separate

process first to balance the flow rate into and out of the boundary around the point. This pressure

is then used as an initial boundary condition to calculate the pressure distribution at other nodes.

Thirdly, the above procedure is not suitable for fluid problems with fluid film discontinuity

which will be the case of hydrodynamic and hybrid bearings investigated in this project.

The aforementioned limitations with the FDM approach can be resolved by adopting the

numerical approach based on finite volume method (FVM). The hydrodynamic and hybrid

journal air bearings in this research adopt the herringbone groove surface patterns, as shown in

Figure 3.4 a). At the edges of a groove, there is a sudden change in the air film thickness and

results fluid film discontinuity issues presented in Figure 3.4 b). If FDM is used, additional

boundary conditions need to be applied to all groove edges to ensure the continuity of mass

flow, which makes the computations complex. In this case, the FVM can serve as a useful tool

56
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

to overcome the problem with FDM and simplify the process. The flow chart of the FVM

approach is shown in Figure 3.5.

Figure 3.4 a) A conventional herringbone grooved journal from [93] b) A cross-section view
in axial direction of grooves in the circled area from a). The air film discontinuity is marked at
the interface of one groove-ridge pair.

5
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

Figure 3.5 A flow diagram of the FDM approach

The FVM approach is similar to the FDM except that it calculates flow properties in a controlled

volume surrounding each node. The divergence terms in the Reynolds Equation need to be

converted to surface integrals using divergence theorem. The controlled volume surrounding a

node of the mesh can be illustrated as Figure 3. 6 a). The Reynolds Equation (Equation 3.5) can

be integrated using Green’s Theorem along the boundaries of the controlled volume and lead to

58
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

Equation 3.31. The projected area of the controlled volume on the meshed surface can be

divided into four cells, as in Figure 3. 6 b), for the ease of computations.

Figure 3. 6 a) Controlled volume surrounding a node in FVM approach b) Projected view of

the controlled volume on the meshed surface

Equation 3.30 is the transformation of the Reynolds Equation used in the FVM approach. The

physical meaning of Equation 3.30 is that the net inlet flow rate through all boundaries equal to

the accumulating rate of the mass in the controlled volume. It is assumed that the pressure and

clearance vary linearly in the controlled volume. The integrals of the divergence terms in

Equation 3.30 can then be expressed numerically using the discrete form for each cell boundary.

Equation 3.31 and Equation 3.32 give an example of these integrals along the left and bottom

boundaries of Cell 1. Similar expression can be derived for other cells in the same manner, and

they are listed in Appendix B. Equation 3.30 is rewritten as Equation 3.33 for the convenience

of computations.

59
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

∂ 𝜕𝑃 ∂ 𝜕𝑃 𝜕(𝑃𝐻)
∮ [ (𝑃𝐻 3 − Λ𝑃𝐻) + (𝑃𝐻 3 )]𝑛⃗ ∗ 𝑑𝑙 = ∬ Λ ∂θ𝜕𝑍
Γ ∂θ 𝜕𝜃 ∂Z 𝜕𝑍 𝜕𝜏

Equation 3.30

where 𝛤 denotes the projected boundaries of the controlled volume used in FVM. 𝑛⃗ is the

outward unit vector normal to the boundaries of each cell. 𝑑𝑙 is the unit length of each cell

boundary.

𝑃𝑖,𝑗 − 𝑃𝑖,𝑗−1 ∆𝑍
𝑄𝜃1 = [− 𝑃𝑖,𝑗−1/2 𝐻𝑖−,𝑗−1/2 3 + Λ𝑃𝑖,𝑗−1 𝐻𝑖−,𝑗−1 ]
Δ𝜃 2 2 2

Equation 3.31

𝑃𝑖,𝑗 − 𝑃𝑖−1,𝑗 ∆𝜃
𝑄𝑍1 = − 𝑃𝑖−1/2,𝑗 𝐻𝑖−,𝑗−1/2 3 Equation 3.32
Δ𝑍 2

4
𝜕(𝑃𝐻)
∑(𝑄𝜃𝑘 + 𝑄𝑍𝑘 ) = ∬ Λ ∂θ𝜕𝑍
𝜕𝜏 Equation 3.33
𝑘=1

where the subscripts 𝑖 − and 𝑗 − 1/2 denote the interpolation values of P and H. Subscripts 𝜃1

and 𝑍1 denote the integrals for the left boundary and bottom boundary of Cell 1 respectively.

60
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

The discrete formula of Equation 3.33 can be reordered into a polynomial form with respect

to terms including 𝑃𝑖,𝑗 , 𝑃𝑖−1,𝑗 , 𝑃𝑖+1,𝑗 , 𝑃𝑖,𝑗−1 , 𝑃𝑖,𝑗+1 :

𝐴𝑖,𝑗 𝑃𝑖,𝑗 2 + 𝐵𝑖,𝑗 𝑃𝑖,𝑗 + 𝐶1𝑖,𝑗 𝑃𝑖,𝑗−1 2 + 𝐶2𝑖,𝑗 𝑃𝑖,𝑗+1 2 + 𝐷1𝑖,𝑗 𝑃𝑖−1,𝑗 2 + 𝐷2𝑖,𝑗 𝑃𝑖,𝑗+1 2
𝜕(𝑃𝐻)
+ 𝐸1𝑖,𝑗 𝑃𝑖,𝑗−1 + 𝐸2𝑖,𝑗 𝑃𝑖,𝑗+1 = ∬ Λ ∂θ𝜕𝑍
𝜕𝜏

Equation 3.34

where 𝐴𝑖,𝑗 , 𝐵𝑖,𝑗 , 𝐶1𝑖,𝑗 , 𝐶2𝑖,𝑗 , 𝐷1𝑖,𝑗 , 𝐷2𝑖,𝑗 , 𝐸1𝑖,𝑗 , 𝐸2𝑖,𝑗 are constant coefficients that do not contain

P in each iteration step. The mathematic expressions of these coefficients are listed in Appendix

B.

Newton’s method and successive relaxation method can still be applied with minor

modifications on the discrete formula. It is assumed that 𝑃𝑖,𝑗 is the only variable in one iterative

step. The following function with respect to 𝑃𝑖,𝑗 can be formed:

61
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

𝑓(𝑃𝑖,𝑗 + 𝛿𝑖,𝑗 ) = 𝐴𝑖,𝑗 (𝑃𝑖,𝑗 + 𝛿𝑖,𝑗 )2 + 𝐵𝑖,𝑗 (𝑃𝑖,𝑗 + 𝛿𝑖,𝑗 ) + 𝐶1𝑖,𝑗 (𝑃𝑖,𝑗−1 + 𝛿𝑖,𝑗−1 )2
+ 𝐶2𝑖,𝑗 (𝑃𝑖,𝑗+1 + 𝛿𝑖,𝑗+1 )2 + 𝐷1𝑖,𝑗 (𝑃𝑖−1,𝑗 + 𝛿𝑖−1,𝑗 )2
+ 𝐷2𝑖,𝑗 (𝑃𝑖+1,𝑗 + 𝛿𝑖+1,𝑗 )2 + 𝐸1𝑖,𝑗 (𝑃𝑖,𝑗−1 + 𝛿𝑖,𝑗−1 ) + 𝐸2𝑖,𝑗 (𝑃𝑖,𝑗+1
+ 𝛿𝑖,𝑗+1 )

Equation 3.35

𝑑𝑓(𝑃𝑖,𝑗 + 𝛿𝑖,𝑗 )
𝑓 ′ (𝑃𝑖,𝑗 + 𝛿𝑖,𝑗 ) = = 2𝐴𝑖,𝑗 (𝑃𝑖,𝑗 + 𝛿𝑖,𝑗 ) + 𝐵𝑖,𝑗
𝑑(𝑃𝑖,𝑗 + 𝛿𝑖,𝑗 )

Equation 3.36

Apply Newton’s method in Equation 3.23.

𝑓(𝑃𝑖,𝑗 + 𝛿𝑖,𝑗 )
𝑑𝛿 =
𝑓 ′ (𝑃𝑖,𝑗 + 𝛿𝑖,𝑗 ) Equation 3.37

𝛿𝑖,𝑗 𝑛+1 = 𝛿𝑖,𝑗 𝑛 − 𝜔𝑠𝑟 𝑑𝛿 𝑛

Equation 3.38

In each iterative step, 𝛿𝑖,𝑗 is updated by Equation 3.3 until 𝑑𝛿 reaches a preset difference range;

(𝑃𝑖,𝑗 + 𝛿𝑖,𝑗 ) is then output as the pressure distribution within the air film.

62
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

3.2.5 Validation of the bearing model in the static equilibrium analysis

The proposed numerical approach was first verified by comparing the static equilibrium

position with the experiment results reported by Pink [ ].

The bearing used in [ ] is a hydrostatic journal air bearing shown in Figure 3. . The

compensation system used is orifice restrictors with pockets. There are two rows of restrictors

located symmetrically, and each row had eight restrictors. The dimensions of the bearing are

listed in Table 3.1. The CFD simulation was carried out on a 29 × 97 grid, Figure 3.8. The

pressure distribution and bearing force at different eccentricities are shown in Figure 3.9 and

Figure 3.10 The predicted dimensionless bearing force agrees well with the experiment results.

The maximum error is less than 5%.

Figure 3. Schematic drawings of hydrostatic journal air bearings with equally distributing
pocketed orifice restrictors [ ]

63
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

Table 3.1 Bearing parameters of the hydrostatic journal air bearings to be analysed
Diameter, Length, Orifice Pocket Supply
Clearance Diameter,
𝑫 𝑳 Diameter, Pressure,
,𝒄 𝒅𝒑𝒐𝒄
(mm) (mm) 𝒅𝟎 𝑷𝒔
85.2 85.2 18μm 0.2mm 2.6mm .8bar

Number of Row of
Location of the restrictors to the edge of the bearing, 𝑎
restrictors restrictors
8 per row 2 𝑙1 = 𝑙2 = 0.25𝐿

Figure 3.8 Mesh for the static air journal bearing

Figure 3.9 Pressure distribution from the bottom boundary to the axial symmetry plan,
eccentricity ratio, 0.4

64
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

Figure 3.10 Dimensionless bearing force VS. Eccentricity ratio,

𝐹
𝐹̅ = (𝑝 − 𝑝𝑏𝑟𝑔)2𝑟 𝑙, 𝐹𝑏𝑟𝑔 is the bearing force in newton.
𝑠 𝑎 0

3.2.6 Analysis of truncation errors

In the above numerical analysis, there are two types of errors. One is round-off error, which

terminates the iterative process once it is satisfied. The round-off error does not accumulate and

is normally pre-set in the order of 1e-6.

The other error is the truncation error, which comes from the Lagrange remainder derived from

the Taylor’s expansion used in the finite difference scheme for the Reynolds Equation. It is

related with the grid size. The difference between a true solution and a converged numerical

solution mostly depends on the truncation error. The following procedure has been adopted to

analysis the truncation error [ 9] .

65
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

In the proposed simulation model, the average pressure (𝑃𝑎𝑣𝑔 ) is defined by integrating the

pressure of the air film and dividing by the bearing area. It is used to analyse the truncation

error. The true solution can be expressed as:

𝑃𝑎𝑣𝑔 𝑇𝑆 = 𝑃𝑎𝑣𝑔 𝑁𝑆 + 𝐸𝑟𝑟𝑇𝐸

Equation 3.39

ErrTE is the truncation error term and can be rewritten as:

∆𝑥
𝑚𝑟 (𝑃𝑎𝑣𝑔 (∆𝑥) − 𝑃𝑎𝑣𝑔 ( 𝑚 ))
𝐸𝑟𝑟𝑇𝐸 = Equation 3.40
∆𝑥 𝑟 (𝑚𝑟 − 1)

where 𝑟 is the order of the iterative method. It can be determined by carrying out three iterative

∆𝑥 ∆𝑥
processes on different grid sizes, for example applying ∆𝑥, and 𝑚2 to Equation 3.40.
𝑚

∆𝑥
(𝑃𝑎𝑣𝑔 𝑁𝑆(∆𝑥) − 𝑃𝑎𝑣𝑔 𝑁𝑆 ( ))
𝑟 = 𝑙𝑜𝑔𝑚 𝑚
∆𝑥 ∆𝑥 Equation 3.41
(𝑃𝑎𝑣𝑔 𝑁𝑆 ( 𝑚 ) − 𝑃𝑎𝑣𝑔 𝑁𝑆 ( 2 ))
𝑚

66
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

The truncation error analysis was applied to simulations in section 3.2.5. The results yielded an

error level of 8%. The same method can be applied to all other simulation models used in this

thesis. The peak pressure can also be used to evaluate the truncation error.

3.3 Modelling of rotor

In this project, air bearings will be designed to support the rotor used in a turbocharger for a 2-

liter diesel engine. The shaft diameter is increased to 20mm over a length of 20mm at two

journal bearing locations to accommodate the size of air bearings used in this project, as shown

in Figure 3.11 a). The rotor is referred as R-1. The rotor contains add-on elements, such as shaft

extensions, compressor and turbine. The mechanical structure of these rotors is modelled using

a finite element method (FEM) based on the Timoshenko Beam Theory to describe their

dynamic properties [82, 84]. The rotor model will be coupled with bearing models to give

predictions on the vibrational information of the system in Chapters 4 and 5, which is then

compared with the experimental data for verification.

a) Rotor R-1

6
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

b) Model of R-1

Figure 3.11 The rotors used in this project and their finite element models. a) Rotor R-1 - a
rotor for turbochargers and b) Finite element model of rotor R-1 and the global coordinate
system.

3.3.1 Finite element rotor model and impact tests

The finite element model of rotor R-1 composes of a set of finite rotor segments with 8-DOF,

as introduced in Section 2.3.4. The schematic drawing of the model is illustrated as Figure 3.11

b). Rotor R-1 has a length of 118.3mm and a total mass of 230.9 grams. Its model is defined by

20 elements and 21 nodes. The Inconel turbine of R-1 is represented as a disc with the same

mass and moment of inertia of the turbine wheel. The aluminium compressor is modelled as

one disc on the steel shaft retaining the same mass and the moment of inertia. Table 3.2

describes detailed finite element model information of the rotor, including element length (L),

outer diameter (OD), inner diameter (ID), lumped mass (W), lumped diametral moment of

inertia (I_D) and polar moment of inertia (I_P). Figure 3.12 shows the first four free-free

undamped modes at zero speed of the rotors, predicated by the finite element models.

68
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

Table 3.2 Finite element model of rotor R-1

Node L(mm) OD(mm) ID(mm) W(g) I_D(g*mm2) I_P(g*mm2)

1 6.67 2.50 0 0 0 0

2 10.00 2.50 0 2.84 31.80 48.33

3 10.00 3.70 0 0 0 0

4 7.91 3.70 0 26.04 3637.91 5563.02

5 4.34 5.00 0 0 0 0

6 6.46 5.00 0 15.08 953.79 1884.96

7 3.50 10.00 0 0 0 0

8 4.20 10.00 1.94 0 0 0

9 2.80 10.00 6.06 0 0 0

10 10.40 10.00 6.06 0 0 0

11 1.85 9.00 6.06 0 0 0

12 1.85 9.00 6.06 0 0 0

13 10.30 10.00 6.06 0 0 0

14 2.70 10.00 6.06 0 0 0

15 7.60 10.00 0 0 0 0

16 1.40 7.65 0 0 0 0

17 9.20 4.50 0 0 0 0

18 9.20 4.50 0 56.00 3514.14 7669.08

19 4.60 4.50 0 0 0 0

20 4.60 4.50 0 0 0 0

21 0 4.50 0 0 0 0

69
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

To verify the validity of the rotor model, impact hammer tests have been performed to find the

undamped natural frequencies of the flexible rotor. Figure 3.13 illustrates the bode plots

generated from the impact hammer tests on free-free rotor. The associated natural frequencies

of the first two flexible modes are listed in Table 3.3 and compared with the predictions from

rotor models. It can be seen that the predictions on the natural frequencies have good

agreements with experimental observations. The proposed rotor model is equivalent to the rotor

used in experiments in dynamics.

Figure 3.12. The first four Free-free undamped modes of R-1 at zero speed

0
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

Figure 3.13 Bode plots of impact hammer tests on free-free rotor R-1 at zero speed

Table 3.3 A comparison of Eigen-frequencies from the rotor model and impact hammer tests
Eigen-frequencies of flexible modes
Prediction from FEM
Rotor Impact hammer tests Error
model
3111Hz 31 0Hz 1.8%
R-1
6050Hz 5950Hz 1.6%

The gyroscopic effect will result in the natural frequencies splitting along with rotational speeds

into forward and backward whirl. To demonstrate this effect, R-1 is assumed to be supported

on undamped isotropic supports of stiffness 𝑘𝑥𝑥 = 𝑘𝑦𝑦 = 1.7𝑒 7 N/m. Results are presented in

the Campbell diagram shown in Figure 3.14.

1
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

a) Campbell diagram of R-1

b) Mode shape plotted at 1.6 Hz

Figure 3.14 Campbell diagram and mode shape of rotor R-1 with undamped isotropic support.
The dash line in Campbell diagram is the synchronous line, ‘F’ refers to forward mode and
‘B’ refers to backward mode.

2
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

3.3.2 Governing Equations of rotor dynamic system

The governing equation of motion for a general rotor-bearing system, with rotor modelled as in

Section 3.3.1 and a constant rotational speed, 𝜔, is [84]:

̈ + [𝑲𝒔 ]{𝑞} + ([𝑪𝒔 ] + ω[𝑮]){𝑞}

[𝑴]{𝑞} ̇ = [𝑭𝒃𝒓𝒈 ] + [𝑭𝒈 ] + [𝑭𝒖 ]

Equation 3.42

𝑇
𝑞 = [… 𝑥𝑖 𝑦𝑖 𝜃𝑥𝑖 𝜃𝑦𝑖 … ] Equation 3.43

where 𝑞 is the system displacement vector to be solved, 𝑥𝑖 , 𝑦𝑖 the translational displacement of

𝑖 𝑡ℎ node and 𝜃𝑥𝑖 , 𝜃𝑦𝑖 the rotational displacement of 𝑖 𝑡ℎ node under right hand rule. [𝑴] is the

mass and inertia matrix, [𝑲𝒔 ] the structural stiffness matrix derived from strain energy, [𝑪𝒔 ] the

structural damping matrix, [𝑮] the gyroscopic matrix, [𝑭𝒃𝒓𝒈 ] the bearing force vector, [𝑭𝒈 ] the

static gravitational force vector, and [𝑭𝒖𝒃 ] the synchronous unbalance excitation force vector.

The expressions of the element matrices are given in Appendix C. The global matrices

[𝑴], [𝑲𝒔 ], [𝑪𝒔 ] and [𝑮] are assembled using the element matrices with the method described in

[84]. This governing equation will be applied to both linear and non-linear rotor dynamic

analysis in this project.

3
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

3.4 Linear perturbation analysis

The concept of linear perturbation analysis was first suggested by Lund [94], who applied the

procedure on both gas and oil bearings. In this analysis, bearing forces were linearized using

Taylor’s expansion and expressed using stiffness and damping coefficients with respect to a

static equilibrium configuration. Figure 3.15 shows a static equilibrium configuration of journal

bearings and the coordinate system used for performing perturbation analysis. In the figure, the

stationary bearing sleeve is represented by the outer circle with its centre at 𝑂1. The shaded

circle represents the rotating journal whose centre is 𝑂2. Bearing forces are indicated by blue

arrows in horizontal and vertical directions. Static load is indicated by red arrow. In this case,

𝑂2 is placed coinciding with the static equilibrium position. The dash circle is the trajectory of

𝑂2 with small perturbations. ∆𝑥𝑗 and ∆𝑦𝑗 are the displacement coordinates of 𝑂2 relative to the

static equilibrium position.

Figure 3.15 Static equilibrium configuration and coordinate system of linear perturbation
analysis.

4
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

3.4.1 Linearization of bearing forces with perturbation method

In the linear perturbation analysis, it is assumed that the geometrical centre of the journal orbits

near a static equilibrium position with infinity small amplitude as shown in Figure 3.16, and the

motion of the journal is in a harmonic form.

With the assumptions, variation of air film thickness and pressure in the Reynolds Equation

(Equation 3.5) is expressed by Equation 3.44 and3.45.

𝐻(𝜃, 𝑍, 𝑋𝐽 , 𝑌𝐽 ) = 𝐻0 (𝜃, 𝑍, 𝑋𝐽0 , 𝑌𝐽0 ) + ∆𝑋𝐽 𝑠𝑖𝑛𝜃 − ∆𝑌𝐽 𝑐𝑜𝑠𝜃

Equation 3.44

𝜕𝑃 𝜕𝑃 𝜕𝑃
𝑃(𝜃, 𝑍, 𝑋𝐽 , 𝑌𝐽 ) = 𝑃0 (𝜃, 𝑍, 𝑋𝐽0 , 𝑌𝐽0 ) + ∆𝑋𝐽 + ∆𝑋̇ 𝐽 + + ∆𝑌𝐽
𝜕∆𝑋𝐽 𝜕∆𝑋̇ 𝐽 𝜕∆𝑌𝐽
𝜕𝑃
+ ∆𝑌̇ 𝐽
𝜕∆𝑌̇ 𝐽

Equation 3.45

The normalized parameters in the above equations are:

𝑥𝐽0 𝑦𝐽0 ∆𝑥𝐽 ∆𝑦𝐽 ∆𝑥̇ 𝐽 ∆𝑦̇ 𝐽

𝑋𝐽0 = , 𝑌𝐽 = , ∆𝑋𝐽 = , ∆𝑌𝐽 = , ∆𝑋̇ 𝐽 = , ∆𝑌̇ 𝐽 =
𝑐 𝑐 𝑐 𝑐 𝜔𝑐 𝜔𝑐

Equation 3.46

5
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

where 𝐻0 and 𝑃0 are the dimensionless air film thickness and pressure at the static equilibrium

position respectively, 𝑥𝐽0 and 𝑦𝐽0 the static equilibrium position of the journal centre, ∆𝑥𝐽 and

∆𝑦𝐽 the displacement of the journal centre in the x and y directions, ∆𝑥̇ 𝐽 and ∆𝑦̇ 𝐽 the velocity of

the journal centre in the x and y directions, 𝑐 the radial clearance of the bearing, 𝜔 the rotation

𝜕𝑃 𝜕𝑃
speed of the rotor, and the changes of dimensionless pressure introduced by the
𝜕𝑋𝐽 𝜕𝑌𝐽

𝜕𝑃 𝜕𝑃
displacement of the journal centre and and 𝜕𝑌 ̇ the changes of dimensionless pressure
𝜕𝑋𝐽̇ 𝐽

introduced by the velocity of the journal centre.

The displacement of the journal centre is assumed as follows:

∆𝑋𝐽 = 𝑋𝐽1 𝑒 𝑠𝑡 , ∆𝑋̇𝐽 = 𝑠∆𝑋𝐽 , ∆𝑋𝐽̈ = 𝑠 2 ∆𝑋𝐽

Equation 3.47

∆𝑌𝐽 = 𝑌𝐽1 𝑒 𝑠𝑡 , ∆𝑌𝐽̇ = 𝑠∆𝑌𝐽 , ∆𝑌𝐽̈ = 𝑠 2 ∆𝑌𝐽

Equation 3.48

where 𝑋𝐽1 and 𝑌𝐽1 are the dimensionless amplitude of the displacement in the ∆𝑥𝑗 and ∆𝑦𝑗

directions. 𝜔𝑤 is the angular frequency of the journal orbit. 𝑠 = 𝜆 + 𝑖𝜔𝑤 and 𝑖 = √−1.

Equation 3.45 can be simplified by using Equations 3.4 a nd 3.48.

𝑃(𝜃, 𝑍, 𝑋𝐽 , 𝑌𝐽 ) = 𝑃0 (𝜃, 𝑍, 𝑋𝐽0 , 𝑌𝐽0 ) + ∆𝑋𝐽 𝑃1 + ∆𝑌𝐽 𝑃2

Equation 3.49

6
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

𝜕𝑃 𝜕𝑃 𝜕𝑃 𝜕𝑃
𝑃1 = +𝑠 , 𝑃2 = +𝑠
𝜕∆𝑋𝐽 𝜕∆𝑋̇ 𝐽 𝜕∆𝑌𝐽 𝜕∆𝑌̇ 𝐽 Equation 3.50

Substituting Equation 3.44 and Equation 3.49 to the Reynolds Equation (Equation 3.5), and

neglecting terms at the order of ∆X J and ∆YJ , one can get the zeroth-order and first-order

lubrication equations, Equations 3.51 and 3.52, respectively.

The zeroth-order lubrication equation can be written as follows, which stands for pressure at a

static equilibrium position:

∂ 𝜕𝑃0 ∂ 𝜕𝑃0 𝜕(𝑃0 𝐻0 )

(𝑃0 𝐻0 3 )+ (𝑃0 𝐻0 3 )=Λ Equation 3.51
∂θ 𝜕𝜃 ∂Z 𝜕𝑍 𝜕𝜃

The first-order lubrication equation can be written as follows, which describes the pressure

changes because of the whirl of the journal:

∂ 𝜕𝑃𝑗 𝜕𝑃0 𝜕𝑃0

(𝑃0 𝐻0 3 + 3𝑃0 𝐻0 2 𝐻𝑗 + 𝐻0 3 𝑃𝑗 )
∂θ 𝜕𝜃 𝜕𝜃 𝜕𝜃

∂ 𝜕𝑃𝑗 𝜕𝑃0 𝜕𝑃0

+ (𝑃0 𝐻0 3 + 3𝑃0 𝐻0 2 𝐻𝑗 + 𝐻0 3 𝑃𝑗 )
∂Z 𝜕𝑍 𝜕𝑍 𝜕𝑍
Equation 3.52
𝜕 𝑠
=Λ (𝑃0 𝐻𝑖 + 𝑃𝑖 𝐻0 ) + ∙ 2Λ(𝑃0 𝐻𝑗 + 𝑃𝑗 𝐻0 )
𝜕𝜃 𝜔

𝑗 = 1, 2

where 𝐻1 = 𝑠𝑖𝑛𝜃 and 𝐻2 = −𝑐𝑜𝑠𝜃. 𝜔 is the rotation speed of the rotor.

CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

By solving Equations 3.51 and 3.52, the bearing forces can be represented using equivalent

stiffness and damping coefficients. These coefficients can be obtained by means of calculating

the pressure changes 𝑃1 and 𝑃2 from Equation 3.52. The stiffness and damping coefficients can

then be expressed as Equation 3.53 to Equation 3.60.

2𝜋 𝐿
𝑘𝑥𝑥 𝑐
𝐾𝑥𝑥 = = − ∫ ∫ 𝑅𝑒(𝑃1)sin(𝜃) 𝑑𝑍𝑑𝜃
𝑝𝑎 𝑙𝑑0 0 0
Equation 3.53

𝑘𝑥𝑦 𝑐 2𝜋 𝐿
𝐾𝑥𝑦 = = − ∫ ∫ 𝑅𝑒(𝑃1)cos(𝜃) 𝑑𝑍𝑑𝜃
𝑝𝑎 𝑙𝑑0 0 0
Equation 3.54

𝑘𝑦𝑥 𝑐 2𝜋 𝐿
𝐾𝑦𝑥 = = − ∫ ∫ 𝑅𝑒(𝑃2)sin(𝜃) 𝑑𝑍𝑑𝜃
𝑝𝑎 𝑙𝑑0 0 0
Equation 3.55

𝑘𝑦𝑦 𝑐 2𝜋 𝐿
𝐾𝑦𝑦 = = − ∫ ∫ 𝑅𝑒(𝑃2)cos(𝜃) 𝑑𝑍𝑑𝜃
𝑝𝑎 𝑙𝑑0 0 0
Equation 3.56

2𝜋 𝐿
𝑑𝑥𝑥 𝑐𝜔𝑤
𝐷𝑥𝑥 = = − ∫ ∫ 𝐼𝑚𝑔(𝑃1)sin(𝜃) 𝑑𝑍𝑑𝜃
𝑝𝑎 𝑙𝑑0 0 0
Equation 3.57

𝑑𝑥𝑦 𝑐𝜔𝑤 2𝜋 𝐿
𝐷𝑥𝑦 = = − ∫ ∫ 𝐼𝑚𝑔(𝑃1)cos(𝜃) 𝑑𝑍𝑑𝜃
𝑝𝑎 𝑙𝑑0 0 0
Equation 3.58

𝑑𝑦𝑥 𝑐𝜔𝑤 2𝜋 𝐿
𝐷𝑦𝑥 = = − ∫ ∫ 𝐼𝑚𝑔(𝑃2)sin(𝜃) 𝑑𝑍𝑑𝜃
𝑝𝑎 𝑙𝑑0 0 0
Equation 3.59

8
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

𝑑𝑦𝑦 𝑐𝜔𝑤 2𝜋 𝐿
𝐷𝑦𝑦 = = − ∫ ∫ 𝐼𝑚𝑔(𝑃2)cos(𝜃) 𝑑𝑍𝑑𝜃
𝑝𝑎 𝑙𝑑0 0 0
Equation 3.60

where 𝑘𝑥𝑥 , 𝑘𝑥𝑦 , 𝑘𝑦𝑥 , 𝑘𝑦𝑦 are the stiffness coefficients of the air film, 𝑑𝑥𝑥 , 𝑑𝑥𝑦 , 𝑑𝑦𝑥 , 𝑑𝑦𝑦 the

damping coefficients of the air film, 𝐾𝑥𝑥 , 𝐾𝑥𝑦 , 𝐾𝑦𝑥 , 𝐾𝑦𝑦 , 𝐷𝑥𝑥 , 𝐷𝑥𝑦 , 𝐷𝑦𝑥 , 𝐷𝑦𝑦 their dimensionless

form, 𝑐 the radial clearance of the bearing, 𝑙 the length of the bearing, 𝑑0 the diameter of the

bearing, 𝑝𝑎 the ambient pressure, 𝜔𝑤 the angular frequency of whirling, and 𝐿 = 2𝑙/𝑑0 , the

dimensionless bearing length. 𝑅𝑒() represents the real part of a complex number. 𝐼𝑚𝑔()

represents the imaginary part of a complex number.

In the coordinate system shown in Figure 3.15, the bearing forces in the x and y directions can

be represented in a matrix regarding a static equilibrium position as follows:

𝐹𝑋𝑏𝑟𝑔 𝐹𝑋0𝑏𝑟𝑔 𝑘𝑥𝑥 𝑘𝑥𝑦 ∆𝑥𝐽 𝑑𝑥𝑥 𝑑𝑥𝑦 ∆𝑥̇ 𝐽

[ ]=[ ]− [ ][ ] − [ ][ ]
𝐹𝑌𝑏𝑟𝑔 𝐹𝑌0𝑏𝑟𝑔 𝑘𝑦𝑥 𝑘𝑦𝑦 ∆𝑦𝐽 𝑑𝑦𝑥 𝑑𝑦𝑦 ∆𝑦̇ 𝐽 Equation 3.61

where 𝐹𝑋0𝑏𝑟𝑔 and 𝐹𝑌0𝑏𝑟𝑔 are the bearing forces in the ∆𝑥𝑗 and ∆𝑦𝑗 directions at the static

equilibrium position.

9
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

3.4.2 Static equilibrium stability analysis

In a static equilibrium stability analysis (SESA), the static gravitational force vector is

counteracted by the static forces (𝐹𝑋0𝑏𝑟𝑔 , 𝐹𝑌0𝑏𝑟𝑔 ) given by the bearing force linearization [85].

The unbalance force vector is also omitted from the equation. The governing equation (Equation

3.42) can be reformed as:

̈ + ([𝑲𝒔 ] + [𝑲𝒃 ]){𝑞} + ([𝑪𝒔 ] + [𝑪𝒃 ] + 𝛚[𝑮]){𝑞}

[𝑴]{𝑞} ̇ =0

Equation 3.62

where [𝑲𝒃 ] and [𝑪𝒃 ] is the linearized bearing stiffness and damping matrices. They contain the

stiffness and damping coefficients of bearing extracted from linear perturbation analysis.

With the assumptions that the perturbation is in harmonic form, the displacement vector {𝑞}

can be found as follows:

{𝑞} = {𝑞0 }𝑒 𝜆𝑡
Equation 3.63

where 𝜆 is a complex root of the equation.

80
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

For general use, it is convenient to reform the second order equation, Equation 3.62 using state

space method into first order form. Substituting Equation 3.63 and:

𝑞
{𝑢} = {𝑞̇ }
Equation 3.64

into Equation 3.62 with after some manipulation leads to the Equation 3.65,

[0] [I]
[ −1 −𝟏 ] {𝑢0 } = 𝜆[𝐼]{𝑢0 }
−[𝑴𝒔 ] [𝑲𝒔𝒚𝒔 ] −[𝑴𝒔 ] [𝑪𝑮𝒔𝒚𝒔 ] Equation 3.65

where [0] and [I] are null and unit matrices. [𝑲𝒔𝒚𝒔 ] = [𝑲𝒔 ] + [𝑲𝒃 ] and [𝑪𝑮𝒔𝒚𝒔 ] = [𝑪𝒔 ] +

[0] [I]
[𝑪𝒃 ] + 𝛚[𝑮]. 𝑱 = [ −1 −𝟏 ] is the system characteristic matrix.
−[𝑴𝒔 ] [𝑲𝒔𝒚𝒔 ] −[𝑴𝒔 ] [𝑪𝑮𝒔𝒚𝒔 ]

Equation 3.65 is in the form of a standard eigenvalue problem. The eigenvalues of the

augmented system appear in complex conjugate pairs, as do their respective eigenvectors. On

solution of Equation 3.65, the eigenvalues are found in the form of:

𝜆𝑖 = 𝑠𝑖 + √−1 ∗ 𝜔𝑖 Equation 3.66

81
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

where 𝑖 denotes the 𝑖 𝑡ℎ eigenvalue. The imaginary part, 𝜔𝑖 , is the whirl speed. For assurance of

stability, all the real parts, 𝑠𝑖 , must be negative.

The stability of the system can be investigated by means of examining the leading eigenvalue

[60], 𝜆𝐿 (i.e. the one whose real part is nearest to +∞), of the system characteristic matrix J. If

the real part of 𝜆𝐿 is negative, the system is stable. Otherwise, the system is unstable and 𝜔𝐿

gives corresponding whirl speed.

In the project, SESA were performed on rotor bearing systems based on R-1. Because there is

no additional static load, the gravitational force of the rotor will determine the static equilibrium

positions of the both journal bearings at a given rotational speed. According to the axial

locations of rotor centre of gravity and the two journal bearings, the equivalent gravitational

force applied is 1.04N at bearing location 1 and 1.2N at bearing location 2 with reference to R-

1 model. At each bearing location, the static equilibrium position is calculated by letting

𝐹𝑋0𝑏𝑟𝑔 = 0 and 𝐹𝑦0𝑏𝑟𝑔 equals to the equivalent gravitational force.

3.4.3 Limitations of the proposed linear perturbation analysis

The limitations of the linear perturbation analysis used in this project can be summarized as

below:

1. The orbit of the journal centre is assumed to follow a harmonic motion.

82
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

2. As the linear bearing model is applied, the computations cannot find the steady state

amplitude of the self-excited whirl, or the so-called limited cycle.

These limitations can be overcome using non-linear transient analysis, which is introduced next

3.5 Non-linear transient analysis

Non-linear transient analysis can analyse the orbit of the journal without assumptions of small

harmonic vibrations to the journal behaviours. Bearing forces in the analysis are no longer

approximated by stiffness and damping coefficients. This analysis requires the governing

equations of the rotor dynamic system, coupled with non-linear bearing models, to be solved as

functions of time. Non-linear transient analysis can be used to study the stability and unbalance

responses of the rotor bearing system. In this section, the non-linear transient analysis used in

the study is introduced.

3.5.1 Bearing models in non-linear transient analysis

The bearing models of both the FDM and the FVM approaches consider time as an independent

𝜕(𝑃𝐻)
variable. The time dependent differential term, Λ , in the Reynolds Equation (Equation 3.5)
𝜕𝜏

needs to be treated numerically in the non-linear transient analysis, so that it can be coupled

with the equations of motion of the rotor-bearing system. Here, it is expressed using two finite

difference schemes with respect to time.

83
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

For bearing models based on the FDM approach, an implicit difference scheme is applied to

generate a discrete form of the time dependent term in the Reynolds Equation as Equation 3.6 ,

and the algorithm is shown in Figure 3.16 a).

𝑛
∂ 3
𝜕𝑃 ∂ 3
𝜕𝑃 (𝑃𝑖,𝑗 𝐻𝑖,𝑗 )𝑛 − (𝑃𝑖,𝑗 𝐻𝑖,𝑗 )𝑛−1
[ (𝑃𝐻 − Λ𝑃𝐻) + (𝑃𝐻 )] =Λ
∂θ 𝜕𝜃 ∂Z 𝜕Z 𝑖,𝑗 Δ𝜏

Equation 3.67

where n denotes the values at current time step 𝑡𝑛 and (n - 1) the values at previous time step

𝑡𝑛−1

For bearing models based on the FVM approach, the Crank-Nicolson semi-implicit scheme is

applied to transform Equation 3.32 into Equation 3.68. The algorithm is shown in Figure 3.16

b). This scheme is always numerically stable and suitable for unsteady gas-bearing problems

[3 ] .

4
ΛΔ𝜃ΔΖ 𝑃𝑖,𝑗 (𝑛) ∑4𝑘 𝐻𝑖,𝑗,𝑘 (𝑛) − 𝑃𝑖,𝑗 (𝑛−1) ∑4𝑘 𝐻𝑖,𝑗,𝑘 (𝑛−1)
̅̅̅̅̅
∑(𝑄 ̅̅̅̅̅
𝜃𝑘 + 𝑄𝑍𝑘 ) =
4 Δ𝜏
𝑘=1

Equation 3.68

84
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

where the subscript k corresponds to each of the cell numbers shown in Figure 3. used in the

FVM approach. The definitions of ̅̅̅̅̅ ̅̅̅̅̅

𝑄𝜃𝑘 and 𝑄 𝑍𝑘 are:

1 1
̅̅̅̅̅
𝑄𝜃𝑘 = (𝑄𝜃𝑘 (𝑛) + 𝑄𝜃𝑘 (𝑛−1) ) 𝑎𝑛𝑑 𝑄
̅̅̅̅̅
𝑍𝑘 = (𝑄 (𝑛) + 𝑄𝑍𝑘 (𝑛−1) )
2 2 𝑍𝑘

Equation 3.69

Figure 3.16 The algorithm of time dependent finite difference scheme. a) Algorithm of the
implicit scheme in the FDM approach. b) Algorithm of the Crank-Nicolson semi-implicit
scheme in the FVM approach

85
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

3.5.2 Non-linear transient stability analysis

In non-linear transient stability analysis, the governing equation (Equation 3.42) can be used

directly. It needs to be solved with the non-linear bearing model as function of time. The

solution can be found by adopting a non-simultaneous iterative process to couple the bearing

models with governing equation of motion for the rotor bearing system [42, 56]. This process

involves the following steps:

1) To initiate the iterative process, the displacement of R-1 model at time step 𝑡𝑛−1 will

be given with zero velocities. The pressure in the air film is pre-set as ambient.

2) By using the displacement and velocity of bearing locations at the previous time step

𝑡𝑛−1 , the Reynolds Equation in bearing models is solved to yield the bearing forces for

time step 𝑡𝑛 , 𝐹𝑋𝑏𝑟𝑔 (𝑛) and 𝐹𝑌𝑏𝑟𝑔 (𝑛) .

3) The bearing forces are then treated as algebraic and applied to the governing equation

of motion. The latter is solved as state equations using Matlab ODE (e.g. ode23/ode23s)

solvers. The solution gives the displacement and velocity states of all nodes in R-1

model at time step 𝑡𝑛 , including those of the bearing locations.

4) Repeat step 2 to 3 until a pre-set maximum simulation time is reached.

The above non-simultaneous routine will be used in Chapter 4 and 5 to give predictions on the

rotor bearing system responses to unbalance excitation at various constant rotor speeds. The

86
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

rotor responses at each speed will be calculated with appropriated initial conditions set by

MATLAB ODE (ode23/ode23s) solvers for the first 00 shaft revolutions. The steady state of

the simulation is assumed to be achieved in the last 100 revolutions, from which the response

data are collected and analysed.

The stability of the system can be identified by means of examining the time history of journal

trajectory: In the absence of unbalance excitation ([𝑭𝒖𝒃 ] is null), if the static equilibrium

position (SEP) of the bearing is stable, free perturbation decays and a converged trajectory can

be observed. Otherwise, the growth of free perturbation is contained within a limit cycle and

the trajectory gives the steady state amplitude of self-excited whirl. The whirl speed of the limit

cycle can be analysed by performing Fast Fourier Transform (FFT) of the time history. In the

presence of unbalanced forces (unbalance excitation), a stable SEP shows an orbit with

frequency that is equal to the rotational speed. On the other hand, an unstable SEP with

unbalanced excitation will lead to an orbit with a non-synchronous component of self-excited

whirl and a synchronous component driven by unbalanced forces.

3.5.3 The conditions for the proposed non-linear transient analysis

There are some restrictions on the conducted transient analysis using the rotor model in junction

with non-linear bearing models. They are listed as follows:

1) The non-linear transient analysis is based on the constant rotational speed;

8
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

2) Non-linear bearing models are coupled with the rotor model by applying the bearing

force vector into the governing equations of motion for the system under a non-

simultaneous routine. In reality, they react to each other simultaneously. Sufficiently

small-time steps need to be maintained in the integration of the rotor ODE;

3) The mass, stiffness, damping and gyroscopic matrices represent the dynamic properties

of the rotor. Non-linear bearings and viscoelastic-support are modelled individually. The

gyroscopic effects are considered as a function of rotational speed;

4) The rotor is assumed to be axial symmetric and modelled with a Timoshenko beam

element. Only the lateral motions are considered.

3.6 Summary

This chapter gives theories and numerical techniques adopted in the study of air bearings and

the rotor dynamic configuration. In this area, hydrostatic journal air bearings are modelled using

the finite difference method. Hydrodynamic and hybrid journal air bearings are modelled using

the finite volume method. Various boundary conditions applied to the bearing models are

explained. A static equilibrium analysis was performed using the proposed bearing model of

hydrostatic air bearings. The results were compared with experimental data from [16] and

showed good agreement. Rotor R-1 used in the project were modelled using finite element

method based on Timoshenko beam theory. The rotor model was verified by means of

performing impact tests on free-free rotor. The techniques to identify the stability of rotor

bearing system are provided as linear perturbation analysis and non-linear transient analysis.

88
CHAPTER 3 NUMERICAL ANALYSIS OF AIR BEARINGS

The both analytical approaches will be applied in Chapters 4 and 5 to estimate vibrational

performance of the rotor bearing system. The conditions of these analysis are stated.

89
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

CHAPTER 4: HYDROSTATIC JOURNAL AIR

BEARINGS

4.1 Introduction

In this chapter, non-rotational and rotational performance of hydrostatic journal air bearings are

investigated. The performance is first analysed using the numerical approach presented in

Chapter 3. The predicted rotational performance is then compared with experimental ones for

verification of the proposed numerical approaches.

In section 4.2, modelling of hydrostatic journal air bearings is introduced with a focus on

refining restrictor boundary conditions to improve the accuracy of the model.

In section 4.3, the model of hydrostatic journal air bearings is used to study the non-rotational

performance. The influence of some design parameters on bearing reaction forces to static load

are studied for the optimal design.

Section 4.4 presents theoretical studies on the rotational performance of hydrostatic journal air

bearings using linear perturbation analysis. The influence of bearing design parameters on

equivalent bearing stiffness and damping coefficients is investigated. The analysed coefficients

are then used to build up a linear bearing model and combined with the rotor model (R-1) to

give predictions on the stability and natural frequencies of the system.

90
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Section 4.5 describes a non-linear transient analysis (NTA) to predict unbalance responses of

practical rotor-bearing systems, which is then verified by experiments. The model and

techniques used in the NTA for hydrostatic journal air bearings are explained at the first place.

Experiments on rotational performance is performed with R-1 rotor in a speed range from 50k

to 100k rpm. Experimental data are compared with prediction from NTA at the end.

For the convenience of reading, the nomenclatures of the design parameters of hydrostatic

journal air bearings are presented below. Table 4.1 lists the range of these parameters used in

this chapter.

– 𝑟0 , radius of the bearing, mm

– 𝑙, length of the bearing, mm

– 𝑐, radial clearance, μm

– 𝑃𝑠 , supply pressure (absolute), bar

– 𝑑0 , orifice diameter, mm

– 𝑁𝑜𝑟𝑖 , number of orifices

Table 4.1 Design parameters of the hydrostatic journal air bearing

Length to
Radius of the Radial Orifice Supply
diameter Number of
bearings, 𝒓𝟎 clearance, 𝒄 diameter, 𝒅𝟎 Pressure,
ratio, 𝒍/𝟐𝒓𝟎 orifices, 𝑵𝒐𝒓𝒊
(mm) μm (μm) 𝒑𝒔 (bar)

4 & 10 0.5 to 2 5 to 10 50 to 300 3 to 7.5 4 to 8

91
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

4.2 Modelling of hydrostatic journal air bearings

Hydrostatic journal air bearings are modelled based on the methods presented in Chapter 3. The

Reynold’s Equation is solved numerically using finite difference method. The bearings are

considered symmetric about the middle lines of the axial length and the both ends are open to

atmosphere. Finite difference (FD) grid only needs to cover half of the bearing length. Orifices

are used as restrictors and the centre of each orifice coincides with a node in the FD grid. Figure

4.1 a) shows a typical grid and the boundary conditions. The red circles represent the orifice

restrictors which are coincided with grid nodes.

Since orifice flow model is applied as a boundary condition, accuracy of the bearing model will

largely rely on it in addition to the grid size. It is necessary to investigate flow properties through

the orifice restrictors in air bearings and develop accurate flow models. This section will focus

on the development of a refined orifice flow model to be used as boundary conditions.

92
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

a)

b)

Figure 4.1 Mesh of bearing surface in FDM a) Grid and boundary conditions for single-row
restrictor system; b) Grid and boundary conditions for double-row restrictor system

4.2.1 Orifice flow model in hydrostatic journal air bearings

A commonly used orifice flow model is a set of empirical equations which stand for normal

and choked flow conditions such as Equations 3.15 and 3.16 introduced in Chapter 3. However,

they are not suitable to be directly applied to modelling of hydrostatic journal air bearings. This

is because of an effect known as entrance loss. Such effect is a result of a rapid increase in

93
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

dynamic pressure locally around an orifice when air flow is pressurized into bearing clearance

[3].

A well accepted orifice flow model with consideration of the entrance loss effect was provided

by Pink [3]. In his model, a correction coefficient C𝑐𝑜𝑟𝑓 was used to adjust overall flow rate

through an orifice restrictor in hydrostatic journal air bearings. The corrected flow rate can be

formulated for normal and choking conditions respectively as Equations 4.1 and 4.2.

1
2 2 𝛾+1 2
𝑑𝑚 𝜋𝑑0 2𝛾 𝑝0 𝛾 𝑝0 𝛾
= 𝐶𝑜𝑟𝑓 ∗ 𝐶𝑑 ∗ 1 ∗ 𝑝𝑠∗ { 𝑅𝑇 [( ) − ( ) ]} ,
𝑑𝑡 2 2 𝛾−1 𝑝𝑠 𝑝𝑠
4(1 + 𝛿𝑙 )
𝛾
𝑝0 2 𝛾−1
>[ ] Equation 4.1
𝑝𝑠 𝛾+1

1 1 𝛾
2𝛾 2 2 𝛾−1 𝑝0 2 𝛾−1
𝑚̇𝑜𝑟𝑖 = 𝐶𝑜𝑟𝑓 ∗ 𝐶𝑑 ∗ 𝐴𝑟 ∗ 𝑝𝑠 ∗ [ ] [ ] , ≤[ ]
𝛾+1 𝛾+1 𝑝𝑠 𝛾+1

Equation 4.2

1
2 2
1 + 𝛿𝑙
𝐶𝑜𝑟𝑓 = [ ] Equation 4.3
1 + 𝐾𝑑 𝛿𝑙 2 𝐶𝑑 2

𝜋𝑑0 2
𝛿𝑙 = Equation 4.4
4𝜋𝑑𝑟 ℎ

94
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

where 𝐾𝑑 is the entrance loss coefficient introduced by Vohr [95]. It relates static pressure in

the pockets, static pressure at the edges of pockets, and dynamic pressure:

𝑝𝑝 − 𝑝𝑐 = 𝐾𝑑 ∗ 𝑝𝑑𝑦𝑛
Equation 4.5

where 𝑝𝑐 is the local static pressure at the pocket edge and 𝑝𝑑𝑦𝑛 the dynamic pressure. From

Vohr’s experimental work [95], 𝐾𝑑 is a function of the Reynolds number defined for air

bearings [88], 𝑅𝑒:

𝐾𝑑 = 0.15 + 0.000225𝑅𝑒 , 𝑖𝑓 0 < 𝑅𝑒 < 2000

Equation 4.6

Or

𝐾𝑑 = 0.6, 𝑖𝑓 2000 < 𝑅𝑒

Equation 4.7

Re is given by:

𝑑𝑚 2
𝑅𝑒 = ∗
𝑑𝑡 𝜋𝜇𝑑𝑟 Equation 4.8

For un-choked orifice flow, 𝐶𝑑 should be considered as a variable with respect to orifice

diameter and the pressure drops.

95
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

By taking the effect of the orifice diameter into account, Cd can be written as:

𝐶𝑑 ∗ = 0.52 + 2.1 ∗ 𝑑0 − 3 ∗ 𝑑0 2
Equation 4.9

By taking the effect of the pressure drop into account, Cd can be written as:

𝛾
𝐶𝑑 ∗ 𝑝0 2 𝛾−1
𝐶𝑑 = 𝑝 , 𝑖𝑓 >[ ]
1.174 − 0.327 ∗ 𝑝0 𝑝𝑠 𝛾+1 Equation 4.10
𝑠

where 𝑑0 is the orifice diameter. Under ‘choking’ conditions, 𝐶𝑑 is regarded as a constant equal

to 0.8.

In this project, Pink’s orifice flow model was initially applied directly to the bearing model.

However, numerical study showed that this orifice flow model tended to underestimate bearing

forces in static equilibrium analysis when the grid size was smaller than the pocket diameter,

or when it was applied to a plain orifice in comparison with experimental data from other

research work. Table 4.2 shows the results obtained by using Pink’s flow model on different

grid sizes and comparisons between predicted bearing forces at zero rotation speed for a

hydrostatic journal air bearing with parameters from Table 4.1. It can be seen bearing reaction

force to static load dropped 6% to 8.8% when a fine grid size is applied.

96
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Table 4.2 Comparison of the dimensionless bearing reaction forces on different grid sizes and
the experiments, the eccentricity ratio is 0.4 and 0.8

Dimensionless pocket diameter, dpoc/r0 = 0.061

Coarse Grid, 29x97, Fine Grid, 49x177
Δ𝜃 = 0.065, Δ𝑌 = 0.0714 Δ𝜃 = 0.0357, Δ𝑌 = 0.0417
Modified orifice
𝜺 Pink’s flow model Pink’s flow model Experiment
flow model
W Difference W Difference W Difference W
0.4 0.2499 1.2% 0.2322 6.0% 0.2448 0.8% 0.247
0.8 0.3350 1.8% 0.3109 8.8% 0.3582 5.0% 0.341

One possible cause for this issue is that the local pressure drop and pressure recovery only occur

at a radius around the pocket or plain orifice. G. Belforte and T. Raparelli [9] summarized the

radius where pressure recovery occurs with CFD and experiments for hydrostatic thrust air

bearings. The radius can be expressed as Equation 4.11.

𝑑0
𝑟𝑖 = + 40 ∗ ℎ Equation 4.11
2

where 𝑟𝑖 is the radius around an orifice.

When the grid size is smaller than the radius of the pocket or the orifice, Pink’s flow model

underestimated the pressure at the orifice nodes. An alternative flow model was then developed

and introduced in section 4.2.2 for fine grid size. The discrete effect of mass flow is considered.

9
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

4.2.2 Modified orifice flow model

The modified orifice flow model is based on the mass continuity equation. It is considered as

the flow rate through an orifice into the volume under a restrictor equals to that into the bearing

clearance, Figure 4.2 a). Equations 4.1 and 4.2 are adopted to describe the flow rate through an

orifice while 𝐶𝑑 is considered as a variable calculated by Equation 4.10. 𝑃𝑢 is the average

pressure in the volume under an orifice restrictor. The flow rate at the entrance to bearing

clearance is hereby given as:

1
2 𝛾+1 2 𝛾
2𝛾 𝑃𝑑 𝛾 𝑃𝑑 𝛾 𝑃𝑑 2 𝛾−1
𝑄(𝑃𝑑 , 𝑃𝑢 , 𝐻𝑑 , 𝐻𝑢 ) = 𝛤𝑠 ∗ 𝑃𝑢 ∗ { 𝑅𝑇 [( ) − ( ) ]} , > [ ]
𝛾−1 𝑃𝑢 𝑃𝑢 𝑃𝑢 𝛾+1

Equation 4.12

1 1 𝛾
2𝛾 2 2 𝛾−1 𝑝𝑑 2 𝛾−1
𝑄(𝑃𝑑 , 𝑃𝑢 , 𝐻𝑑 , 𝐻𝑢 ) = 𝛤𝑠 ∗ 𝑃𝑢 ∗ [ ] [ ] , ≤[ ] Equation 4.13
𝛾+1 𝛾+1 𝑝𝑢 𝛾+1

12𝜂 √𝑅𝑠𝑝𝑒𝑐 𝑇𝐶𝑑 𝜋𝑑𝑝𝑜𝑐 (𝐻𝑑 + 𝐻𝑢 )

𝛤𝑠 = ∗ Equation 4.14
𝑝𝑎 𝑐 2 2

𝑃𝑑
𝐶𝑑 = 0.9093 − 0.0751
𝑃𝑢 Equation 4.15

where 𝑃𝑑 and 𝐻𝑑 are flow pressure and local clearance downstream at the entrance and 𝑃𝑢 and

𝐻𝑢 the flow pressure and local film clearance upstream at the volume under the restrictor.

98
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

The empirical formula Equation 4.15, provided by Neves [11], can be used to calculate 𝐶𝑑 for

un-choked flow to account for the entrance loss effect. For a choking condition, 𝐶𝑑 is regarded

as a constant and equal to 0.8.

For a grid configuration as shown in Figure 4.2 b), the flow properties at the restrictor nodes

are described by the algebraic formula as in Equation 4.16, instead of the Reynolds Equation.

𝑃𝑖,𝑗 is the pressure at the restrictor nodes. 𝑃𝑖−1,𝑗 , 𝑃𝑖+1,𝑗 , 𝑃𝑖,𝑗−1 and 𝑃𝑖,𝑗+1 are the pressure at the

four adjacent points. 𝑄𝑘 (k = 1, 2, 3, 4) is the flow rate into the bearing clearance through a

quarter of the orifice edge and can be calculated using Equations 4.12 and 4.13. The algebraic

formula can be written as:

𝑄𝑜𝑟𝑖 (𝑃𝑖,𝑗 , 𝑃𝑠 , 𝐻𝑖,𝑗 ) = ∑ 𝑄𝑘 (𝑃𝑑 , 𝑃𝑢 , 𝐻𝑑 , 𝐻𝑢 )

𝑘=1

𝑄1 (𝑃𝑑 , 𝑃𝑢 , 𝐻𝑑 , 𝐻𝑢 ) = 𝑄1 (𝑃𝑖−1,𝑗 , 𝑃𝑖,𝑗 , 𝐻𝑖−1,𝑗 , 𝐻𝑖,𝑗 )

𝑄2 (𝑃𝑑 , 𝑃𝑢 , 𝐻𝑑 , 𝐻𝑢 ) = 𝑄2 (𝑃𝑖+1,𝑗 , 𝑃𝑖,𝑗 , 𝐻𝑖+1,𝑗 , 𝐻𝑖,𝑗 ) Equation 4.16

𝑄3 (𝑃𝑑 , 𝑃𝑢 , 𝐻𝑑 , 𝐻𝑢 ) = 𝑄3 (𝑃𝑖,𝑗−1 , 𝑃𝑖,𝑗 , 𝐻𝑖,𝑗−1 , 𝐻𝑖,𝑗 )

𝑄4 (𝑃𝑑 , 𝑃𝑢 , 𝐻𝑑 , 𝐻𝑢 ) = 𝑄4 (𝑃𝑖,𝑗+1 , 𝑃𝑖,𝑗 , 𝐻𝑖,𝑗+1 , 𝐻𝑖,𝑗 )

99
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

a) b)

Figure 4.2 Modified orifice flow model. a) Pressure and flow direction under an orifice
restrictor b) Flow into the bearing clearance through the edge of an orifice restrictor for a grid
size where Δ𝜃 = Δ𝑌 = 𝑑𝑝𝑜𝑐 /2𝑟0

4.3 Non-rotational performance of hydrostatic journal air bearings

In this section, model of hydrostatic journal air bearings is used to study the non-rotational

performance. The influence of design parameters on bearing reaction forces to static load and

compressed air consumptions at given static equilibrium configurations is investigated. The

optimization of design parameters is for achieving maximum bearing reaction force while

reducing compressed air consumptions.

100
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

4.3.1 Effect of eccentricity on the flow rate

For all types of hydrostatic journal air bearings, bearing reaction forces increase directly with

eccentricity as shown in Figure 3.10. At non-rotational configuration, the attitude angle, which

is defined as angle between line of journal centre and housing centre to bearing reaction force,

is zero for symmetrical orifice arrangements such as in Figure 4.1. Before investigating the

effect of other design parameters, it is necessary to understand how flow rate varies with

eccentricity. Local film thickness under each orifice restrictor is different from others due to

the eccentricity and will change accordingly. The flow rate of hydrostatic journal air bearings

with the geometry listed in Table 4.3 is analysed to investigate the effect of eccentricity.

Table 4.3 Design parameters of hydrostatic journal air bearings used to investigate the effect
of eccentricity on flow rate
Radius of Length to
Orifice Number of Radial Supply
the diameter
diameter, orifices per clearance, Pressure, 𝑷𝒔
bearings, 𝒓𝟎 ratio, 𝒍/𝟐𝒓𝟎
𝒅𝟎 (μm) row, 𝑵𝒐𝒓𝒊 𝒄 (μm) (bar)
(mm)
4 1 300 6 5, 10, 15 4, 6

101
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Figure 4.3 Effect of eccentricity on mass flow rate of static air journal bearings for different
supply pressure and radial clearance combinations

102
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Figure 4.3 shows the variation of mass flow rate with eccentricity at different supply pressures

and radial clearances. In Figure 4.3 a) to c), the radial clearance increases from 5 𝜇𝑚 to 15 𝜇𝑚

at two supply pressures. The percentage of the mass flow rate variation reduces from 74% down

to around 10%. With a larger radial clearance, the effect of eccentricity on mass flow rate

becomes less significant. In Figure 4.3 d), it can be found that mass flow rate increases with the

supply pressure and radial clearance.

4.3.2 Optimization of radial clearance, orifice diameter and supply pressure

Hydrostatic journal air bearings with low air consumption while maintaining reasonable bearing

reaction forces to static load reduce the demand on air supplies and enable the applications of

hydrostatic air bearings to be expanded. The influence of radial clearance, the diameter of

orifice restrictors and the effect of supply pressure on the bearing reaction forces and mass flow

rate are investigated in this section to find optimized combinations.

Figure 4.4 shows that the bearing reaction force has a parabolic relationship with the radial

clearance. The optimized radial clearance at which maximum bearing reaction forces to static

load varies with the supply pressure. The optimized radial clearance becomes smaller when

higher supply pressure is used. On the other hand, both supply pressure and radial clearance

have a linear effect on mass flow rate, Figure 4.5. This is because the restricted area is the

annular region at the edge of orifice restrictors.

103
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Figure 4.4 The effect of radial clearance on bearing reaction forces for hydrostatic journal air
bearings with 8mm diameter, a length to radius ratio of 1, and a single row of orifice
restrictors at a 0.4 eccentricity ratio

Figure 4.5 The effect of radial clearance on flow rate at different supply pressures for
hydrostatic journal air bearings with 8mm diameter, a length to radius ratio of 1, and a single
row of orifice restrictors at a 0.4 eccentricity ratio

It should be noticed that when the mass flow rate of the supplied air is limited at a certain level,

as the dashed line indicated in Figure 4.5. The radial clearance at this condition for each supply

104
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

pressure in Figure 4.5 can be used to calculate bearing reaction forces using Figure 4.4. It can

be found that bearing reaction forces are improved by using high supply pressure and low radial

clearance. For example, bearings with 6 bar supply pressure and 12μm clearance have up to 40%

higher bearing forces compared to bearings with 4.5bar supply pressure and 15μm clearance,

Figure 4.4. However, the air consumption of the former is much lower as shown in Figure 4.6.

In general, the bearing reaction forces always increase with supply pressure at the cost of air

consumption, as shown in Figure 4. and Figure 4.8. The increasing rate of bearing reaction

forces drops rapidly while the increasing rate of mass flow does not vary a lot. The efficiency

of improving bearing reaction forces to static load by using higher supply pressure is less

significant.

Figure 4.6 Bearing reaction forces in relation with mass flow rate of hydrostatic journal air
bearings

105
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Figure 4. Effect of supply pressure on bearing reaction forces versus mass flow rate. Journal
bearings with 15 𝜇𝑚 radial clearance, 8 mm diameter, length to radius ratio as 1, single row
orifice restrictors

Figure 4.8 Increasing rate of bearing reaction forces versus mass flow rate

The effect of orifice diameters on bearing reaction forces and mass flow rate was investigated.

The study was carried out on hydrostatic journal air bearings with 3 bar supply pressure and 0.4

eccentricity ratio. The dimensions of the bearings were the same as listed in Table 4.3. The

results are plotted in Figure 4.9 and Figure 4.10. The optimal radial clearance is defined as the

106
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

radial clearance where maximum bearing reaction forces are achieved. The optimal radial

clearances for some given orifice diameters have been found from tests and are listed in Table

4.4. It is observed from tests that the optimal radial clearance decreases with the reduction of

orifice diameters. Figure 4.9 shows that curves of bearing reaction forces become sharp as a

result of reduced orifice diameter. For example, for an orifice diameter of 100 𝜇𝑚, the bearing

reaction forces can decrease by 3% from the maximum in zone A, then drop rapidly once the

bearing radial clearance moves away from it. However, for an orifice diameter of 300 𝜇𝑚, the

reduction of bearing reaction forces is maintained at less than 3% in a much wider range B. In

other words, the latter has a higher tolerance of manufacturing errors. In Figure 4.10, the

relationships between the maximum bearing reaction forces and the associated mass flow rate

are plotted. It can be found that the variation of the maximum bearing reaction forces is less

than 1%, but the mass flow rate increases seven times when orifice diameter increases from

100 𝜇𝑚 to 300 𝜇𝑚.

Table 4.4 Optimal radial clearance at different orifice diameters

Orifice diameter (μm)
300 250 200 150 100
Optimal radial
22 20 17 14 11
clearance (μm)

10
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Figure 4.9 Effect of orifice diameters on bearing reaction forces to static load

Figure 4.10 Optimal bearing reaction forces to static load and associated flow rate for
different orifice diameters

108
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

4.4 Theoretical studies on rotational performance of hydrostatic journal air

bearings

This section focuses on theoretical studies on the rotational performance of hydrostatic journal

air bearings from three aspects using the linear perturbation analysis approach proposed in

Chapter 3. Firstly, the static equilibrium analysis is performed at varies configurations.

Secondly, bearing forces are represented by stiffness and damping coefficients with respect to

the static equilibrium configurations. Thirdly, the stability and natural frequencies are analysed

for rotor-bearing system using linear bearing models.

4.4.1 Static equilibrium analysis of hydrostatic journal air bearings at rotational condition

At a given static load and rotational speed, the static equilibrium configuration of a hydrostatic

journal air bearing largely depends on the compressibility number Λ and the restrictor setup.

The hydrodynamic effect increases with the compressibility number. From its definition in

Equation 3.3, it can be increased by either increasing the radius of the journal or reducing radial

clearance. For a rotor with a given dimension, a small radial clearance is preferred to achieve a

high compressibility number and therefore a high dynamic flow effect. Figure 4.11 shows the

pressure distribution of a hydrostatic journal air bearing when Λ is 0 and 1.86. There is a

significant pressure build up at the convergent zone of the air film because of the journal

rotation. Figure 4.12 gives the pressure profiles at the symmetry plane of the two cases

respectively.

109
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Figure 4.11 Pressure distribution for journal bearings, 𝑟0 = 4 mm; 𝑙 = 8 mm; 𝑑0 = 150 μm;
𝑐 = 14 μm; a) ω = 0, Λ = 0, b) ω = 200000 rpm, Λ = 1.86

Figure 4.12 Pressure profile at symmetry plane, 𝑟0 = 4 mm; 𝑙 = 8 mm; 𝑑0 = 150 μm; 𝑐 =
14 μm; a) ω = 0, Λ = 0, b) ω = 200000 rpm, Λ = 1.86

The relationship between bearing reaction forces and eccentricity at various compressibility

numbers is presented in Figure 4.13. The results are calculated for bearings with 8mm diameter

and 8 mm length. The solid line represents 11μm radial clearance, while the dashed line

110
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

represents 8μm radial clearance. In the both cases, the rotational speeds are adjusted to achieve

the same values for 𝛬. There are slight differences for the two cases with the same 𝛬 and the

difference becomes smaller with a higher 𝛬.

Using low radial clearance is a preferable method to enhance the rotational performance of

hydrostatic journal air bearings. Table 4.5 lists the maximum percentage difference on bearing

forces with respect to compressibility number. Although bearings with the shown geometry

achieve maximum bearing forces with 11μm radial clearance at non-rotational condition in

Figure 4.9, the rotation speed must be much higher to maintain the same bearing reaction forces

compared to bearings with an 8 μm radial clearance at rotational condition.

Table 4.5 Rotational speed to achieve same Λ for different radial clearances and percentage
differences of bearing forces
Bearing Geometry: r: 4mm l: 8mm, 𝒅𝟎 :100𝛍𝐦 𝑷𝒔 : 3bar
Λ 0.75 1.51 2.56 4.84 7.12
Rotation C = 8μm 26450 52895 89920 170000 250190
speed, rpm C = 11μm 50000 100000 170000 321500 473000

Percentage difference 9.80% 4.60% 1.10% 1.60% 3.70%

r - radius, l - length, C - radius clearance, 𝒅𝟎 – orifice diameter

111
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Figure 4.13 Relation of bearing reaction forces and eccentricity at different compressibility
numbers, solid line for c = 11 μm and dashed line for c = 8 μm

Because of the rotation of the journal, the pressure in air film is no longer distributed

symmetrically on the surface of the bearing. The angle between the line of centres of the journal

and bearing sleeve to bearing reaction force at a static equilibrium position is called the attitude

angle. The static equilibrium position varies with compressibility numbers and restrictor setup.

The track of these equilibrium positions at different eccentricity ratios forms a static equilibrium

locus curve. When the static load applied to an air bearing changes, the static equilibrium

position moves along this curve. It can be used as an indicator of attitude angles. Also, the

rotational performance of air bearings is often studied according to the static equilibrium

positions. In Figure 4.14, the static equilibrium locus curves are plotted for the cases shown in

Table 4.5.

112
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Figure 4.14 Attitude angles for different compressibility numbers. a) Attitude angle for
bearing with 8 μm radial clearance at various compressibility numbers. b) Attitude angle for
bearing with 11 μm radial clearance at various compressibility numbers

113
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Figure 4.15 Attitude angle for different compressibility numbers. Eccentricity ratio is 0.4

In Figure 4.14 a), the attitude angle varies between 20° and 40° for an 8 μm radial clearance.

While in Figure 4.14 b), the range is between 15° and 30°. This suggests that the attitude angle

is greatly influenced by rotational speed. Figure 4.15 shows the prediction on the variation of

attitude angle with Λ. As listed in Table 4.5, smaller radial clearance requires lower rotation

speed to achieve the same values of Λ. It can be concluded that a combination of low radial

clearance and rotation speed operation will lead to relatively high attitude angles.

4.4.2 Stiffness and damping coefficients of hydrostatic journal air bearings

Bearing forces can be represented by Equation 3.61 when a static equilibrium position and a

rotational speed are given. The associate stiffness (𝑘𝑖,𝑗 ) and damping (𝑑𝑖,𝑗 ) coefficients can be

calculated by means of assigning a whirling frequency (𝜔𝑤 ) of the journal centre and using

Equations from 3.53 to 3.60. It is convenient to represent the whirling frequency as a ratio to

114
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

rotational speed. This ratio is noted as the whirling frequency ratio ( 𝛺 ), or perturbation

frequency ratio.

Figure 4.16 gives the variation of 𝑘𝑖,𝑗 and 𝑑𝑖,𝑗 of a hydrostatic journal air bearing with whirling

frequency ratio at the concentric journal position. Note that the principal stiffness coefficients

increase rapidly with whirling frequency showing a typical gas bearing hardening effect [ 0] .

On the other hand, the direct damping coefficients decrease dramatically as the whirling

frequency rises. Similar behaviour can be found on the cross-coupled stiffness coefficients. It

indicates that hydrostatic journal air bearings become quite stiff with little or zero viscous

damping coefficients at high perturbation frequencies.

Figure 4.16 Stiffness and damping coefficients at concentric journal position, r = 4mm; l = 8
mm, when c = 6 μm; 𝜔 = 100,000 rpm; 𝛬 = 5.06; 𝑑0 = 100 μm; 𝑃𝑆 = 3 bar

115
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

In this section, the variation of 𝑘𝑖,𝑗 and 𝑑𝑖,𝑗 at synchronic whirling frequency (𝛺 = 1) will be

investigated in several cases to reveal the effect of different design parameters and working

conditions. The principal coefficients are plotted in solid lines and the cross coupled coefficients

are plotted in dash lines.

Figure 4.1 shows the variation of these coefficients versus the eccentricity ratio. In this case,

the compressibility number (𝛬) is 7.9 with 4bar supply pressure at restrictors. The difference in

the amplitude of two principal stiffness coefficients increases with the eccentricity ratios while

the cross coupled stiffness coefficients are still very similar. The amplitude of all stiffness

coefficients increases with eccentricity ratios. The damping coefficients have no significant

change in this case.

116
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Figure 4.1 The effect of eccentricity ratio on the bearing stiffness and damping coefficients,
𝑙
where 𝑑0 = 0.25 𝑚𝑚, 𝑟 = 2 = 10 𝑚𝑚, 𝑐 = 10 𝜇𝑚, 𝛬 = 7.9, 𝑃𝑠 = 4 𝑏𝑎𝑟

The variation of bearing radius (r), radial clearance (C) and rotational speed (𝜔) will change

the compressibility number (𝛬). Their effects on 𝑘𝑖,𝑗 and 𝑑𝑖,𝑗 are similar and studied as the

effect of 𝛬 . Figure 4.18 gives the values of 𝑘𝑖,𝑗 and 𝑑𝑖,𝑗 at various 𝛬 . The amplitude of the

stiffness coefficients increases significantly with 𝛬 while the damping coefficients show

slightly variation. The figure shows hardening effect of hydrostatic journal air bearings

introduced by high compressibility number.

11
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Figure 4.18 The effect of compressibility number on bearing stiffness and damping
coefficients, where 𝑑0 = 0.25𝑚𝑚, 𝑟 = 𝑙/2 = 10𝑚𝑚, 𝑐 = 10𝜇𝑚, 𝑃𝑠 = 4𝑏𝑎𝑟

The influence of design parameters investigated in this section is analysed based on the static

equilibrium configuration at 0.1 eccentricity ratio from Figure 4.1 . The design parameters

being studied are the orifice restrictor diameters (do), supply pressure (Ps) and length to diameter

ratio (ζ = l/2𝑟0 ).

118
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Figure 4.19 presents the effect of the orifice restrictor diameters on 𝑘𝑖,𝑗 and 𝑑𝑖,𝑗 . The stiffness

coefficients increase gradually with orifice diameter (do) from 0.15mm to 0.35mm. When do is

larger than 0.35mm, there is no further increase on the stiffness coefficients. If the orifice

diameter is larger than 0.85mm for this bearing configuration, the orifices lose their function as

restrictors and introduces sharp decreasing of stiffness. The damping coefficients show a similar

trend. In general, the variation of orifice diameters has limited effects on 𝑘𝑖,𝑗 and 𝑑𝑖,𝑗 .

Figure 4.19 Effect of orifice diameter on bearing stiffness and damping coefficients, where
𝑟 = 𝑙/2 = 10𝑚𝑚, 𝑐 = 10𝜇𝑚, 𝑃𝑠 = 4𝑏𝑎𝑟

119
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Figure 4.20 shows the variation of stiffness and damping coefficients at different supply

pressures (Ps). The eccentricity ratio is 0.1 and the orifice diameter is 250μm. The principal

stiffness coefficients increase linearly with the supply pressure. On the other hand, the cross

coupled stiffness coefficients reach the maximum amplitude when Ps about 3bar. It is found

that the supply pressure has no effect on the damping coefficients of a hydrostatic journal

bearings.

Figure 4.20 The effect of supply pressure on bearing stiffness and damping coefficients,
where 𝑑0 = 0.25𝑚𝑚, 𝑟 = 𝑙/2 = 10𝑚𝑚, 𝑐 = 10𝜇𝑚

120
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Figure 4.21 shows the variation of stiffness and damping coefficients with the length to bearing

diameter ratios ζ. The stiffness coefficients increase significantly with 𝜁. This is because the

bearing surface area has been increased with large 𝜁 . Although the principal damping

coefficients increase with 𝜁, the amplitude of this improvement is very limited. In this analysis,

the maximum value of 𝜁 is within 2.

Figure 4.21 The effect of bearing length to diameter ratio on bearing stiffness and damping
coefficients, where 𝑑0 = 0.25𝑚𝑚, 𝑟 = 10𝑚𝑚, 𝑐 = 10𝜇𝑚, 𝛬 = 7.9, 𝑃𝑠 = 4𝑏𝑎𝑟

121
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

From the above analysis on the stiffness and damping coefficients, it can be found that the

compressibility number (𝛬), eccentricity ratio (𝜀), supply pressure (Ps) and bearing length to

diameter ratio (𝜁) have direct influences on the stiffness coefficients. By means of adopting a

large orifice diameter (do), the stiffness coefficients can be improved in a limited margin at the

cost of high compressed air consumption. The damping coefficients of a hydrostatic journal air

bearings are mainly influenced by the journal whirling frequency ( 𝜔𝑤 ) or the excitation

frequency. They cannot be improved by means of adjusting bearing design parameters.

The analysis on the stiffness and damping coefficients of hydrostatic journal air bearings can

assist the bearing design and provide useful information for stability and natural frequencies

predictions of a rotor bearing system.

4.4.3 Analysis of stability and natural frequencies of a rotor bearing system using linear

bearing model

In this section, the stability and natural frequencies of a rotor bearing system supported by

hydrostatic journal air bearings are investigated. Rotor R-1 is supported by two identical journal

bearings and two thrust bearings to constrain the axial movement. The major dimensions of the

journal bearings are listed in Table 4.6.

122
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Table 4.6 Dimensions of the hydrostatic journal air bearings used in analysis
Length to
Radius of the Radial Orifice Supply
diameter Number of
bearings, 𝒓𝟎 clearance, 𝒄 diameter, 𝒅𝟎 Pressure,
ratio, 𝒍/𝟐𝒓𝟎 orifices, 𝑵𝒐𝒓𝒊
(mm) μm (μm) 𝑷𝒔 (bar)

10 1 13 250 7 6

The hydrostatic journal air bearings under study here are represented as linear bearing units

with stiffness and damping coefficients extracted from the perturbation analysis in Section 4.4.2.

The bearing models are added into the rotor dynamic model to predict stability and critical

speeds of the rotor bearing system.

The finite element rotor dynamic model of R-1 is adopted with the hydrostatic bearing model

to form a linear time invariant (LTI) system. The stability of this system can be analysed using

the static equilibrium stability analysis (SESA) proposed in Chapter 3.

Two cases are studied in this section:

Case I: R-1 with linear journal bearings only. The system is illustrated in Figure 4.22. The

bearings support is taken to be rigid. The characteristic matrix of the system equations of motion

is expressed in Equation 4.17. The stiffness and damping matrix of the journal bearings are

added into the global matrices following the process from [84] as shown in Appendix C.

123
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

a) R-1 supported by linearized journal bearings

b) Linearized journal bearing – view in Z axis

Figure 4.22 Schematic views of R-1 with linearized journal bearing

where [𝑴𝒔 ] is the shaft structure mass and inertia matrix, [𝑲𝒔𝒚𝒔_𝟏 ] the system stiffness matrix,

[𝑲𝒔 ] the shaft structure stiffness matrix, [𝑲𝒃 ] the bearings’ stiffness matrix, [𝑪𝑮𝒔𝒚𝒔_𝟏 ] the

124
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

system damping and gyroscopic matrix, [𝑪𝒔 ] the shaft structure damping coefficients, [𝑪𝒃 ] the

bearing’s damping matrix, and [𝑮] the shaft structure gyroscopic matrix.

Case II: R-1 is supported by linear journal bearings with linear viscoelasticity. The system is

illustrated in Figure 4.23. The bearing sleeve is supported by a viscoelastic support made from

4 O-rings in parallel. The whole system is connected to a rigid body. The dynamic properties

of the O-rings can be calculated using the empirical Equations 4.19 [43]. The characteristic

matrix of the system equation of motion is expressed as Equation 4.18. In addition to the

conditions as given in Case I, the mass and inertia matrix, stiffness and damping matrix of the

viscoelastic support are also assembled into the global matrices following the process

introduced in [84]. Since the bearing sleeve supported by the O-rings are not rotating, there is

no need to use the gyroscopic matrix of it.

a) R-1 supported by linearized journal bearings with bearing sleeve supported by O-rings

125
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

b) Axial view of the rotor bearing system

Figure 4.23 Schematic views of the R-1 with linearized journal bearing and viscoelastic
support. a) The R-1 supported by linearized journal bearings with bearing sleeve supported by
O-rings; b) A linearized journal bearing with bearing sleeve supported by O-rings – view in Z
axis

Equation 4.18

where [𝑴𝒔 ] is the shaft structure mass and inertia matrix, [𝑲𝒔𝒚𝒔_𝟐 ] the system stiffness matrix,

[𝑲𝒔 ] the shaft structure stiffness matrix, [𝑲𝒃 ] the bearings’ stiffness matrix, [𝑪𝑮𝒔𝒚𝒔_𝟐 ] the

system damping and gyroscopic matrix, [𝑪𝒔 ] the shaft structure damping coefficients, [𝑪𝒃 ] the

bearing’s damping matrix, [𝑮] the shaft structure gyroscopic matrix, [𝑴𝒗 ] the bearing sleeve

126
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

mass and inertia matrix, and [𝑲𝒗 ] and [𝑪𝒗 ] the stiffness and damping matrix of the bearing

sleeve respectively.

𝑘𝑜 = (5.57𝛾 − 0.636) ∙ exp((0.754𝛾 − 0.0111) ∙ 𝑓𝑝 )

Equation 4.19
𝑐𝑜 = (0.483𝛾 − 0.074) + 1/exp((−33.3𝛾 + 11.8) ∙ 𝑓𝑝 )

where 𝛾 is the deformation ratio of the O-ring and 𝑓𝑝 the excitation frequency in kHz.

The natural frequencies of the two cases identified from theoretical analysis are presented as

Campbell diagrams in Figure 4.24 a) and b), with the speed up to 150k rpm. The first two

modes are rigid modes and the third one is bending mode. It can be found that the natural

frequencies of all three modes drops significantly because of viscoelastic support. For example,

the natural frequency of mode 1 drops from 1680Hz to 383Hz. The differences in forward and

backward whirl is also reduced for mode 2 and 3. This implies the system in Case II is a much

‘soft’ system in comparison with Case I. Figure 4.24 c) is a Campbell diagram of the bearing

sleeve supported by O-rings. It is part of the system in Case II and has its own natural

frequencies. Because it is not rotating and there is no gyroscopic effect, the natural frequencies

of all three modes stay as constants throughout the rotor speed range. It also worth noting that,

for this system, natural frequencies of the first two rigid modes are quite close to each other,

470Hz and 657Hz respectively, while the bending mode occurs at 27900Hz.

12
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

(a) (b)

(c)
Figure 4.24 The Campbell diagrams for Case I & II and the bearing sleeve with linear
dampers. The dash line is the synchronous line. ‘F’ denotes forward whirl and ‘B’ denotes
backward whirl. a) Case I: linear rotor with linear bearings. b) Case II: linear rotor with
linear bearings and linear dampers. c) Bearing sleeve with linear dampers.

128
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

The results of the two cases in stability analysis are shown in Figure 4.25 a) and b). In the

stability maps, the real part of the leading eigenvalue of the system characteristic matrix is

plotted against rotational speeds from 100k to 200k rpm. The system in CASE I become

unstable when rotation speed slightly goes over 110k rpm and remains for the rest of the speed

range. By means of introducing the viscoelastic support in CASE II, the stability of the system

is greatly improved throughout the speed range. Figure 4.26 shows the predicted dominant

whirling frequency ratio for CASE I after the system became unstable.

a) Stability map of CASE I

b) Stability map of CASE II, 𝛾 = 0.2

Figure 4.25 Stability maps based on SESA a) stability map of CASE I. b) stability map of
CASE II

129
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Figure 4.26 Frequency ratio of self-excited whirl to rotation speed for CASE I

The above analysis shows that viscoelastic support has the capability of improving the system

stability. However, this will rely on careful selections of its dynamic properties as reported in

[43]. From Equation 4.19, the dynamic properties of O-rings used in this project depend on both

excitation frequencies and the O-ring deformation ratio, 𝛾. The improper deformation of O-

rings will introduce instability rather than stabilize the system. For example, Figure 4. 2 shows

the stability map of CASE II when 𝛾 is 0.153. In comparison with Figure 4.25 a), the system

becomes unstable at just over 50k rpm.

130
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Figure 4. 2 Stability map of CASE II, 𝛾 = 0.153

It is interesting to note that there is a kink in the plot of leading eigenvalues in Figure 4. 2 ,

pointed by a green arrow. This effect has been reported in other research works, e.g. [60, 96].

It happens when one of the 𝑛𝑠𝑡𝑎𝑡𝑒 eigenvalues supersedes another to become the leading

eigenvalue [60]. There will also be a shift in the whirling frequency whenever the change in

leading eigenvalue occurs. To extend the study on this effect, an analysis was performed on

CASE I with a lower supply pressure at 5bar and speed up to 250k rpm. The results are

presented in Figure 4.28. From the stability map, the change of leading eigenvalue happens

around 170k rpm. An abrupt whirling frequency shift can also be observed at the same speed.

131
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

a)

b)

Figure 4.28 Stability map and predictions of whirling frequency ratio of the rotor bearing
system in CASE I, 𝑃𝑠 = 5𝑏𝑎𝑟 a) stability map b) predicted whirling frequency ratio

4.5 Non-linear transient analysis and experimental verification

The performed analysis using linear bearing models in previous section provided useful

information on the system’s natural frequencies and stability in frequency domain. It can serve

132
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

as a useful tool to assist the design of rotor bearing system supported by air bearings. A

limitation of this approach is that it cannot give predictions on the steady state amplitude of

self-excited whirl, which can only be calculated using a nonlinear model. In [94], it is also

reported that the use of linear bearing models in predicting synchronous unbalance responses

has limitations. To overcome the limitations, a non-linear transient analysis approach is applied

here and then verified with experiments in the unbalance responses.

Experiments in this section were performed with a prototype manufactured based on the rotor

bearing configuration shown in Figure 4.23, CASE II. The viscoelastic support can not only

improve stability but also reduce the amplitude when the rotor speed passes the system natural

frequencies. The rotor was balanced to achieve G1 standard for a service speed at 100k rpm.

The unbalance responses of this device were measured up to the service speed and compared

with predictions using non-linear bearing model.

4.5.1 Non-linear transient analysis of hydrostatic journal air bearings

The non-linear transient analysis follows the non-simultaneous routine described in Chapter 3.

The air-film ODEs were uncoupled from the system ODE and treated as algebraic rather than

state equations [60]. In the case that the bearing sleeve is supported by O-rings (CASE II), its

equations of motion needs to be solved with that of the rotor at the same time.

In the analysis of CASE II, the mechanical structure of the bearing sleeve is modelled using

Timoshenko beam theory with no gyroscopic effect. The O-rings to support it are still modelled

133
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

as linear with equivalent stiffness and damping coefficient from [97] to avoid the reliance on

the excitation frequency in Equation 4.19.

The governing equations of motion for the bearing sleeve is:

[𝑀𝑣 ]{𝑞𝑣̈ } + [𝐾𝑣 ]{𝑞𝑣 } + [𝐶𝑣 ]{𝑞}

̇ = [−𝐹𝑏𝑟𝑔 ]
Equation 4.20

where {𝑞𝑣 } is the states vector of bearing sleeve. The number of states is 4(𝑁𝑣 + 1) and 𝑁𝑣 is

the number of nodes used to model the structure. [𝑴𝒗 ] and [𝑲𝒗 ] are the structure mass/inertia

matrix and stiffness matrix respectively. [𝑲𝒗 ] also has the stiffness coefficients of O-rings

added to their locations. [𝑪𝒗 ] is the damping matrix. [𝑪𝒗 ] only contains the damping

coefficients of O-rings. [−𝑭𝒃𝒓𝒈 ] is the bearing force vector. The negative sign indicates that the

direction of bearing forces applied on the bearing sleeve and its viscoelastic support is opposite

to that on the rotor.

The governing equation of motion, Equation 3.42, is related to Equation 4.20 by the bearing

forces vector and they are solved together using the ODE solvers provided in Matlab. In

addition, the left-hand side of both equations is in the standard form required by the state space

method. They can be assembled together and form global state equations that contains all the

states from the rotor and the bearing sleeve.

134
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

The nonlinear transient analysis was first applied on CASE I and CASE II with perfectly

balanced rotors. Figure 4.29 presents the responses of bearing journal at turbine side to free

perturbation at 120k rpm. The red arrows in the figures indicate the position of journal centre

at time zero in simulations. From SESA, the system in CASE I is unstable at this speed and can

be stabilized by using viscoelastic supports as in CASE II. Figure 4.29 a) shows the trajectory

of the journal centre at turbine side in CASE I. It forms a limit cycle which has a whirling

frequency at 769Hz. It implies the system is unstable. Figure 4.29 b) shows the journal orbit in

CASE I but with unbalance excitations. It can be found that the added unbalance has no effect

on the self-excited orbit (as far as the fundamental frequency and steady state amplitude of the

limit cycle is concerned). It also can be found the whirl ratio predicted by non-linear transient

analysis is 0.38, which is quite close to the prediction from SESA (0.42).

At the meantime, trajectory of the same point in CASE II only shows a decaying free vibration

and eventually converged to the static equilibrium position of the bearing at this speed, Figure

4.29 c). That is to say the system is stable.

a) Limit cycle of journal bearing at turbine side of perfect balanced rotor R-1 in CASE I at
120k rpm rotor speed.

135
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

b) Limit cycle of the bearing journal at turbine side of rotor R-1 in CASE I at 120k rpm rotor
speed with unbalance excitation.

c) Converging orbit of journal bearing at turbine side of perfect balanced rotor R-1 in CASE
II

Figure 4.29 Trajectory of the journal centre at 120k rotor speed of rotor R-1 in CASE I and
II

The unbalance responses predicted by the non-linear transient analysis are demonstrated with

the experimental results in Section 4.5.3. The bearing forces are calculated using the actual

bearing geometry measured from manufactured components. The unbalance on the rotor is

calculated using the information provided by the balancing service provider.

136
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

4.5.2 Hydrostatic journal air bearing test rig and experiment configuration

A test rig was designed for experiments on the hydrostatic journal air bearings. Figure 4.30

shows a cross section view of the test rig. The flow channel of compressed air in the hydrostatic

journal air bearings is indicated by blue arrows. The measurements on dimensions of the

bearing components are listed in Table 4.7 and compared with their design values.

Figure 4.30 Cross-section view of prototype hydrostatic bearing test rig

13
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Table 4. Dimensions of hydrostatic journal bearings used in the test rig

Bearing Design Manufactured
dimensions value value

Radius, 𝒓𝟎
10.00 10.03
(mm)

Length, 𝒍
20.00 20.15
(mm)
Radial
clearance, 𝒄
10.00 13.00
(μm)

Orifice
diameter, 𝒅𝟎
0.25 0.27
(mm)

Supply
pressure 7.00 7.00+/-0.5
(Gauge, bar)

The test rig is mounted on a VSR balancing system supplied by Turbotechnic, which is used as

a test bench as shown in Figure 4.31. The bench is connected to a compressed air reservoir and

can supply massive air flow in a short time to drive the turbine wheel. The speed of the rotor

can be controlled using a speed control valve manually, which adjusts the air flow during the

turbine wheel.

138
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Figure 4.31 High speed air bearing test bench

The unbalance responses of the rotor were measured at two sensor positions marked in Figure

4.30 and both match with a node in the shaft rotor dynamic model (See Figure 3.12 ). Sensor

position A matches with Node 2 and sensor position B matches with Node 12. The vibration of

the rotor was measured at constant speed in horizontal and vertical direction (X & Y) using a

laser vibrometer. The direct measurements of the vibrometer are the velocities of the vibration.

139
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

They are compared with predictions on the velocity states of the nodes from non-linear transient

analysis.

In the measurements, the laser vibrometer is placed on the floor with the test bench. The

measured vibration velocities are therefore relative to ground. However, in the simulation, the

vibration velocities are relative to the casing of the device. A pre-measurement on the casing

was first performed. It was found the amplitude of vibration from the casing is no more than

10% percent of the measurement from the rotor. In this case, the casing is considered as fixed

and its vibration is neglected. The measurements on the rotor are used directly to compare with

simulation results.

Rotor R-1 used in the hydrostatic bearing experiments was balanced to G1 standard for a service

speed at 100k rpm. The rotor was balanced using two-plane dynamic balancing approach. The

equivalent unbalance residuals are concentrated on two planes and listed in Table 4.8.

Table 4.8 Unbalance information of R-1

Left hand correction Right hand correction
plane plane

Unbalance residual
0.020 0.022
(g*mm)

Unbalance phase angle 45° 60°

Positions on Rotor Compressor disc Turbine disc

Position on Rotor
Node 4 Node 18
model

140
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

4.5.3 Experiments on unbalance responses

The experimental study on unbalance responses of the rotor is presented in this section and

compared with simulation results. Because of the limitation of using a manual speed control

valve, it is almost impossible for the rotor to be tested consistently and repeatedly at a target

speed below 50k rpm. As the unbalance responses at a high-speed region are the main interest

of this study, the responses of the rotor were measured at speeds of 50k rpm, 60k rpm, 0k rpm,

80k rpm, 90k rpm and 100k rpm respectively. A 5% difference in the provide power of drive

was allowed in each test.

From the analysis shown by Figure 4.24 b) of Section 4.4.3, the system has no natural frequency

within the test speed range from 50k to 100k rpm. In experiments, the bearing sleeve is a non-

rotating component which was excited by bearing forces. Therefore, a resonance should be

observed at 660Hz. Figure 4.32 are the event time waterfall plots [98] in a run-up test to 100k

rpm. Plot (a) was measured from the rotor directly, while Plot (b) was measured from the

bearing sleeve. It shows relatively good agreement with the predictions on the natural

frequencies from the linear analysis approach. Both plots present a synchronous component (1x)

only. This indicates the system is stable within the speed range as predicted from the SESA.

Figure 4.32 also shows the acceleration using the compressed air driven mechanism is not

uniform and might be different in each individual test.

141
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

a)

b)

Figure 4.32 Top views of waterfall plots in a run-up test at sensor position B to 100k rpm. a)
vibration velocity measured from rotor and b) vibration velocity measured from bearing
sleeve

142
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

The experimental unbalance responses and related simulation results are shown Appendix D.

In each group of the figures, the dominating vibration frequency and the peak vibration

amplitude are marked on the FFT plot. The simulations were produced using the actual rotation

speed (1x component in frequency spectrum) acquired from experiments. At each speed, the

rotor responses were calculated for the first 0 0 shaft revolutions and a steady state was

assumed to be achieved at the last 100 revolutions. The response data were collected for the last

100 revolutions and analysed using FFT. Figure 4.33 shows a case of these measurements and

predictions at 100k rpm rotor speed for both sensor positions.

a) Measurements at sensor position A

143
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

b) Predictions at sensor position A

c) Measurements at sensor position B

144
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

d) Predictions at sensor position B

Figure 4.33 Unbalance responses of sensor position A & B obtained at 100k rpm in speed

The presented results, also the results in Appendix D, have a relatively good agreement between

experimental and prediction results. They show that the vibrations of the system are dominated

by synchronous component and no self-excited whirl has been spotted. The amplitudes of the

unbalance responses from experiments and simulations are very similar, as shown in Figure

4.34. However, the predicted amplitudes of vibrations at the both sensor positions are smaller

than those observed. This can be caused by two reasons: first, the rotor was slightly altered at

both journal positions during balancing as marked in Figure 4.35. Second, the rotor was

balanced alone and then reassembled back into the device. Extra imbalance can be expected in

the process without further field balancing. There was a significant increasing on vibration level

at sensor position A when the speed was close to 100k rpm (1.6 k Hz). This is likely to imply

that the system is running towards the 1st bending natural frequency as shown in Figure 4.24 b).

145
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

a) Sensor position A

b) Sensor position B

Figure 4.34 Peak vibration velocities at sensor positions

Figure 4.35 Unintended change on the shaft journal during a balancing process

146
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

There are also marginal differences between the simulation and experimental results. Some of

the measurements also have a small portion of low frequency vibration around 100Hz to 120Hz,

referring to Figure D-3 c) in Appendix D for example. This was introduced by excitations from

the mounting bench, which has a motor running at the same frequency as long as the bench is

switched on. The X & Y vibration velocities from a simulation for hydrostatic bearings are

identical. However, the difference between them is slightly larger in reality, Figure 4.34. This

is because the bearing components, the assembly and the mounting are not isotropic in the two

directions which are not considered in the simulation.

4.5.4 Limitations of experiments

The experiments in this section are designed to verify the non-linear transient analysis of

hydrostatic journal air bearings. The test rig was designed to simulate a system in CASE II.

However, the rig only provided limited access for measurements of vibrations on the rotor. Most

of the shaft sections, including the turbine wheel were sealed or covered due to the mounting

and driving mechanism (like a turbocharger) to achieve high speed operations. This limited

gaining sufficient information to identify the mode shape of the rotor, e.g. cylindrical, conical,

bending or mixed. This can only be done by gathering vibration information, including the

phase angle on multiple points of the rotor. Accessing such information would involve redesign

of the test rig, manufacture, balancing, assembly, and repeated tests, which would require much

more time than currently allowed for this PhD study. However, this work is likely to be arranged

in the follow up investigation.

14
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

Another limitation is the use of laser vibrometer to measure vibrations. The single point laser

vibrometer only provided accurate measurements of absolute vibrational velocities. It cannot

give relative positions of the shaft centre at the two sensor positions in rotating condition. That

is to say the trajectory of the shaft centre relative to ground (or casing) cannot be detected. As

the tests show that the vibration amplitude of the stand (excitation from mounting) is negligible,

the vibration velocities of the rotor measured are reasonable to be used to compare with the

simulation results. The latter does not simulate excitations from the mounting and consider the

system is mounted on a fixed rigid body.

4.6 Summary

This chapter provides a comprehensive study on hydrostatic journal air bearings and the rotor

dynamic structure they support. A novel orifice model was adopted to improve the accuracy of

bearing model for predictions of bearing reaction forces at a static equilibrium configuration.

The non-rotational performance was then investigated with a focus on the optimization of

bearings to achieve the maximum bearing forces while reducing compressed air consumption

rate. The rotational performance was studied using the linear perturbation analysis. Bearing

forces were represented by equivalent stiffness and damping coefficients with given static

equilibrium configurations. The effects of different design parameters on these dynamic

coefficients were investigated. The research shows a ‘hardening’ effect at high rotation speeds.

It results in a lack of damping which cannot be improved effectively by adjusting bearing design

parameters, but can be done by introducing dampers, such as O-rings as shown in CASE II. The

bearing stiffness and damping coefficients were also used in the rotor dynamic models for two

cases (CASE I & II) to predict the stability and natural frequencies of the rotor dynamic system

148
CHAPTER 4 HYDROSTATIC JOURNAL AIR BEARINGS

supported by hydrostatic journal air bearings. In the end, non-linear transient analysis was

performed together with experimental verifications on unbalance responses. The analysis and

experiments were focused on the rotor bearing system in CASE II and they have good

agreement on the synchronous unbalance responses. No self-excited whirl was observed at this

stage for the tested speed range from 50k to 100k rpm. The limitations of the experiments were

also discussed.

149
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

CHAPTER 5: HYBRID JOURNAL AIR

BEARINGS

5.1. Introduction

In this chapter, rotational performance of hybrid journal air bearings is investigated using the

numerical approach developed in Chapter 3. The proposed hybrid journal air bearing consists

of orifice restrictors located on a stationary bearing sleeve and herringbone grooved journal, as

shown in Figure 5.1. This combination allows the bearing to lift at zero speed with external

compressed air supply and self-suspending without the supply of compressed air when the

rotational speed is sufficiently high.

Figure 5.1 A proposed hybrid journal air bearing. The stationary bearing sleeve is presented in
cross section view to show orifice restrictors, which are used to supply compressed air when
rotational speed is zero or low.

150
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

The research in this area is presented in five sections. After the introduction above, Section 5.2

presents the modelling of hybrid journal air bearings. The FVM approach proposed in Chapter

3 has advantages in simulating air bearings with herringbone grooves and is adopted here. The

validity of this approach was first tested on hydrodynamic air bearings with the same

configuration. The predicted bearing reaction forces at a given static equilibrium position are

compared with those in published references. A novel herringbone groove geometry is also

introduced in this section. The design can help increase bearing reaction forces to a static load

and is applied to the hybrid bearings in this project.

Section 5.3 presents theoretical study on the rotational performance of hybrid journal air

bearings using linear perturbation analysis. The influence of bearing design parameters on

equivalent bearing stiffness and damping coefficients is investigated. The analysed coefficients

are then used to build up linear bearing model and combined with the rotor model (R-1) to give

predictions on the stability and natural frequencies of the system.

In section 5.4, the non-linear transient analysis (NTA) is used to study rotational performance

of rotor-bearing systems described in Section 5.3 with a focus on stability and unbalance

responses. NTA is adopted to give predictions on unbalance responses of the rotor in the test

rig developed in Chapter 4. The journal bearings in the test rig are replaced by hybrid air

bearings. The predicted unbalance responses are then presented and compared with

experimental results.

For the convenience of writing, nomenclature of design parameters of hybrid journal air

bearings are presented below. Table 5. 1 lists the range of these parameters used in this chapter.

151
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

– 𝑟0 , radius of the bearing, mm

– 𝑙, length of the bearing, mm

– ζ, bearing length to diameter ratio

– 𝑐, radial clearance, μm

– 𝑃𝑠 , supply pressure (absolute), bar

– 𝑑0 , orifice diameter, mm

– 𝑁𝑜𝑟𝑖 , number of orifices

– 𝐺𝑛𝑢𝑚 , number of herringbone grooves

– 𝛽𝑔 , angle of herringbone grooves, degree

– ℎ𝑔 , maximum depth of herringbone grooves, μm

– 𝐻𝑔 , maximum groove depth ratio, ℎ𝑔 /𝑐

– 𝛼𝑔 , ratio of groove width to total width (groove plus ridge)

– 𝛾𝑔 , fraction of bearing length occupied by grooves

Table 5. 1 Dimensions of hybrid journal air bearings studied

a) Dimensions of hybrid journal air bearings

Length to
Radius of the Radial Orifice Supply
diameter Number of
bearings, 𝒓𝟎 clearance, 𝒄 diameter, 𝒅𝟎 Pressure,
ratio, 𝜻 orifices, 𝑵𝒐𝒓𝒊
(mm) μm (mm) 𝑷𝒔 (bar)

10 0.5 to 2 8 to 13 0.25 1 to 7.5 6

152
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

b) Dimensions of hybrid journal air bearings - continued

Angle of Maximum depth Ratio of groove Fraction of bearing
Number of width to total length occupied by
grooves, 𝜷𝒈 of groove, 𝒉𝒈
grooves, 𝑮𝒏𝒖𝒎 width, 𝜶𝒈 grooves, 𝜸𝒈
(degree) μm

10 0.5 to 2 10 to 13 0.25 0 to 1

5.2 Modelling of hybrid journal air bearings

In this project, models of hydrostatic journal air bearings are based on the method proposed in

Chapter 3. The Reynold’s Equation is solved numerically using finite volume method (FVM).

Similar to the hydrostatic journal air bearing model, the hybrid bearing is symmetric and open

to atmosphere at both ends. Calculation only need to be performed on half of the bearing axial

length. Orifices are applied as boundary conditions with flow model proposed in Chapter 4. The

centre of each orifice coincides with a node on the axial symmetry plane of FVM grid.

5.2.1 Finite volume model of hybrid journal air bearings

In Figure 5.2 a), the FVM grid of a hybrid journal air bearing is presented with the circular

surface of bearing sleeve spread flat. The total volume to perform numerical calculation is

enlarged (not in scale) and presented as empty volume between the surfaces of the journal and

the bearing sleeve. Herringbone grooves are modelled in a way that all groove edges pass

through a node on FVM grid. To simplify computations, the groove edge can also be aligned to

the diagonal line of the controlled volume around a node, as shown in Figure 5.2 b) and c). If

153
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

Δ𝑍
this is adopted, the groove angle, 𝛽𝑔 , determines the grid aspect ratio, Δ𝜃, and they follow the

Δ𝑍
relation: 𝑡𝑎𝑛𝛽𝑔 = .
Δ𝜃

a)

b) c)

Figure 5.2 The Model of a hybrid journal air bearing. a) FVM grid and boundary conditions,
b) The controlled volume around a node coinciding with a groove edge, and c) A top view of
the controlled volume when the groove edge is aligned to the diagonal line.

154
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

In modelling air bearings with herringbone grooves, particularly hydrodynamic bearings, a

conventional approach is based on the narrow groove theory [99], which assumes the bearing

has infinite grooves with infinitesimal width. This approach is valid for bearing with large

groove numbers, for example 20 grooves [42]. Other numerical approaches involve the use of

neutral coordinate system for the surface of the bearing, in which one of the axes coincides with

the groove edges [36] and finite element method [100].

Before moving on to investigating the performance of hybrid air bearings, the model of

hydrodynamic air bearings with herringbone grooves was tested by means of removing the

orifice boundaries from the model. Predictions on bearing reaction forces at a given static

equilibrium configuration were compared with the method proposed in [36], in which the

Reynold’s Equation was solved on a neutral coordinate system. The reference also provided

experimental data from [93]. Figure 5.3 shows the comparisons. The method in [36]

overestimated the bearing force when compressibility number is below 8.6 and changes to

underestimating when compressibility is higher. The proposed FVM model agrees well with

experiments with less than 5% difference in the compressibility number (Λ) range. Only one

exception occurs at Λ = 26.8 and eccentricity ratio is greater than 0.3. However, this is not the

condition for the air bearings in this project to operate at. The FVM model can serve as an

adequate approach to model air bearings with herringbone grooves.

155
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

a) Predictions of bearing reaction forces based on the numerical model in [36]

b) Predictions of bearing reaction forces using the proposed numerical approach

Figure 5.3 Dimensionless bearing reaction forces versus eccentricity ratio at various
compressibility numbers: the dashed line is experimental data and the solid line is the
predictions from the proposed model. a) Results from [36] b) Predictions using the proposed
bearing model.

156
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

5.2.2 A novel herringbone groove geometry

In Figure 5.4, pressure distribution of a hybrid air bearing is predicted at a given static

equilibrium position for two cases: one with compressed air supply at 2.5bar, and the other

without compressed air supply. It can be found that when compressed air is cut-off, the orifice

restrictors will introduce leakage in the air film and disturb the pressure build-up around them

and result in reduction of bearing reaction forces. To compromise this issue, a novel groove

profile is proposed.

Figure 5.4 Pressure distribution of hybrid journal air bearings at a given static equilibrium
configuration. a) Compressed air supply at 2.5bar. b) No external compressed air supply

Unlike conventional rectangular profile, the novel herringbone groove is constructed on a

curved profile formed by cosine spline. This unique design can enhance bearing reaction forces

at a static equilibrium position. Figure 5.5 shows a comparison between the conventional design

and proposed design. The edges of the grooves in herringbone bearings can be divided into the

leading edges and the trailing edges. At the leading edge, the air film forms a convergent zone,

15
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

while at the trailing edge, the air film is divergent. With reference to [4 ] , the flow passage at

the trailing edge of a rectangular groove is suddenly enlarged which results in vortex and

pressure loss, as shown in Figure 5.5 a). In theory, this can be avoided by means of modifying

the trailing edge with a spline as shown in Figure 5.5 b). The gradually enlarged air film can

avoid pressure loss and improve the bearing reaction forces. The spline used in this thesis is

cosine spline and can be described using Equation 5.1.

Figure 5.5 Air flow over the groove. a) conventional rectangular groove profile. b) Spline
groove profile

1 2𝜋
ℎ𝑔 = ̅̅̅
ℎ 𝑔 ∗ (1 + cos ( ∗ 𝜅𝑔 ))
2 𝛤𝑔 Equation 5.1

̅̅̅
where ℎ𝑔 is the groove depth, ℎ 𝑔 the maximum groove depth, 𝛤𝑔 the period of cosine spline,

and 𝜅𝑔 the distance of any point on spline relative to the start point of a tailing edge.

158
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

To verify the improvement of using the curved groove profile, numerical simulation was

performed on two hybrid air bearings with dimensions shown in Table 5.2 and no supply of

compressed air. One of the bearing has adopted curved grooves and the other one uses

conventional rectangular ones. The groove profiles of the two journals are spread flat and

presented in Figure 5.6 a). Pressure distributions at a quarter bearing length are compared in

Figure 5.6 b). The bearing with curved groove profile shows the pressure recovery at the trailing

edge as predicted. Figure 5.6 c) compares bearing reaction forces at a given static equilibrium

position with various compressibility numbers for the two bearings. The comparison shows that

the proposed groove profile has improved the bearing reaction forces 16%.

The proposed groove profile will be applied to the hybrid air bearings and all related analysis

on hybrid air bearing from here onwards in this thesis is based on this novel herringbone design.

Table 5.2 Dimensions of hybrid journal air bearings used in simulation

Dimensions
Radius (mm) 10
Length (mm) 20
Radial clearance (𝝁𝒎) 10
Groove depth (𝝁𝒎) 10
Groove angle (°) 30
Groove number 8
Groove to ridge width ratio 1
Groove length ratio 0.5

159
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

a)

b)

c)

Figure 5.6 Comparisons between herringbone grooves and their effects. a) Curved groove
profile, 𝐻 is dimensionless film thickness; 𝜃 is circumferential coordinate; b) Comparison of
pressure distribution at the same location of conventional and curved groove. c) Bearing
reaction forces at the same static equilibrium position with various compressibility number.

160
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

5.3 Theoretical studies on rotational performance of hybrid journal air

bearings

This section focuses on theoretical studies on the rotational performance of hybrid journal air

bearings. The bearings investigated in this section adopt the novel groove profile proposed in

Section 5.2.2. First, bearing reaction forces at given static equilibrium configurations are

investigated. Second, bearing forces are represented by stiffness and damping coefficients with

respect to the static equilibrium configurations using linear perturbation analysis. Third, the

stability and natural frequencies are analysed for rotor-bearing system using linear bearing

models.

5.3.1 Analysis of bearing reaction forces at given equilibrium positions

In this section, numerical calculations were performed on hybrid journal air bearings to

investigate the influence of design parameters on bearing reaction forces at given static

equilibrium positions. The effects of compressibility number and restrictor setup are very

similar to that of hydrostatic bearings and will not be discussed here again. Instead, the

influences of groove number (𝐺𝑛𝑢𝑚 ), maximum groove depth ratio (𝐻𝑔 = ℎ𝑔 /𝑐) and groove

angle (𝛽𝑔 ) are studied. The dimensions of the hybrid air bearings being studied in this section

are given in Table 5.3. The bearing reaction forces of hydrostatic journal air bearings with the

same dimensions are also analysed for comparisons. The static equilibrium position (SEP) is

chosen at eccentricity ratio of 0.4. The attitude angle is adjusted in each case to ensure the

bearing reaction force is in vertical direction only.

161
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

Table 5.3 Design parameters and restrictor setup of hybrid air bearings to be studied

Design parameters Value

Radius (mm), 𝒓𝟎 4

Length (mm), 𝒍 8

Radial clearance (𝛍𝐦), 𝒄 6

Orifice diameter (𝛍𝐦), 𝒅𝟎 300

Supply pressure (bar), 𝑷𝒔 3

ratio of groove width to total width (groove plus
0.6
ridge), 𝜶𝒈
fraction of bearing length occupied by grooves, 𝜸𝒈 0.6

Figure 5. shows bearing reaction forces at the chosen SEP for multiple groove number (𝐺𝑛𝑢𝑚 )

and maximum groove depth ratio (𝐻𝑔 ) of the hybrid bearing. The red dash line in the plots is

the result calculated from hydrostatic bearings at the same condition. It is found that bearing

reaction forces change with compressibility number (𝛬) more rapidly in hybrid bearings than

in hydrostatic bearings. When 𝛬 goes over a threshold (pointed by a black arrow in plots),

hybrid bearings can provide higher reaction forces than hydrostatic bearings. This threshold of

𝛬 can be reduced by means of increasing 𝐻𝑔 or 𝐺𝑛𝑢𝑚 . Figure 5.8 presents the change on

threshold of 𝛬 when 𝐺𝑛𝑢𝑚 is increased. One interesting fact observed from the figures is that

there is a point at which bearing reaction forces calculated at different 𝐻𝑔 merged together. The

value of 𝛬 at this point is regarded as a ‘stagnation value’. At this compressibility number, if

the same static load is applied, hybrid air bearings with different 𝐻𝑔 will operate at the same

SEP. However, their dynamic performance can be quite different. For example, in the case that

the bearing forces are represented by stiffness and damping coefficients, the coefficients will

change with 𝐻𝑔 . This effect has been further investigated and explained in Section 5.3.2.

162
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

a)

b)

163
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

c)

Figure 5. Bearing reaction forces at the given SEP of multiple groove number 𝐺𝑛𝑢𝑚 and
maximum groove depth ratio (𝐻𝑔 ) combinations. Groove angle (𝛽𝑔 ) is 30 degrees. 𝐻𝑔
increases from 0.5 to 2 in all plots. a) 𝐺𝑛𝑢𝑚 = 4; b) 𝐺𝑛𝑢𝑚 = 6; and c) 𝐺𝑛𝑢𝑚 = 8.

Figure 5.8 The influence of groove number to bearing reaction forces at the given SEP.
Threshold 𝛬 of each case is marked in the figure.

164
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

Figure 5.9 is an enlarged view of the cases in Figure 5. c) when 𝐻𝑔 equals to 1 and 2, and 𝛬 is

between 30 and 50. An additional case that 𝐻𝑔 = 3 is plotted together. It shows bearing reaction

forces do not always increase with 𝐻𝑔 . In this particular case, maximum bearing reaction forces

are achieved when 𝐻𝑔 = 2.

Figure 5.9 Influence of maximum groove depth ratio on bearing reaction forces.

The influence of groove angle to bearing reaction forces is presented in Figure 5.10. The angle

of herringbone grooves (𝛽𝑔 ) is changed from 20° to 30° for one of the cases in Figure 5. c)

(𝐺𝑛𝑢𝑚 = 8, 𝐻𝑔 = 2, 𝛬 = 31). The maximum bearing reaction force is achieved when 𝛽𝑔 = 24°.

165
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

Figure 5.10 Influence of groove angle to bearing reaction force at given SEP

5.3.2 Stiffness and damping coefficients of hybrid journal air bearings

In this section, the influence of herringbone groove configurations on equivalent stiffness (𝑘𝑖,𝑗 )

and damping (𝑑𝑖,𝑗 ) coefficients is investigated. Bearing forces of the proposed hybrid journal

air bearings are represented using Equation 3.61 regarding a static equilibrium position at a

given rotational speed. The equivalent stiffness (𝑘𝑖,𝑗 ) and damping coefficients (𝑑𝑖,𝑗 ) are

calculated by means of assigning a whirling frequency (𝜔𝑤 ) of the journal centre and using

Equations from 3.53 to 3.60. In the analysis, the whirling frequency ratio (𝛺 = 𝜔𝑤 /𝜔) was

defined as the ratio of whirl frequency (𝜔𝑤 ) to rotational speed (𝜔). Dimensions and restrictor

setup of the hybrid journal air bearings investigated in this section are listed in Table 5.4.

166
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

Table 5.4 Design parameters and restrictor setup of hybrid journal air bearings to be studied

Design parameters Value

Radius (mm), 𝒓𝟎 10

Length (mm), 𝒍 20

Radial clearance (𝛍𝐦), 𝒄 10

Orifice diameter (𝛍𝐦), 𝒅𝟎 250

Supply pressure, 𝑷𝒔 1 to 6.5

For the convenience of writing, the principal stiffness and damping coefficients are noted as

𝑘𝑖,𝑖 and 𝑑𝑖,𝑖 respectively. The cross-coupled stiffness and damping coefficients are noted as

𝑘𝑖,𝑗 and 𝑑𝑖,𝑗 respectively.

Similar to the analysis performed on hydrostatic journal air bearings, the variation of stiffness

and damping coefficients of the hybrid journal air bearings with whirling frequency ratio (𝛺) at

the concentric journal position are studied and the results are presented in Figure 5. 11. The

change of all coefficients with 𝛺 is the same as those of hydrostatic bearings. However, the

amplitude of the principal stiffness coefficients is smaller. For example, under synchronous

excitation, 𝑘𝑦𝑦 is 8.90𝑒 6 N/m for the hybrid bearings and is 1.15𝑒 7 N/m for hydrostatic

bearings with the same dimension and rotational speed. On the other hand, the hybrid air

bearing shows improvement with principal damping coefficients. For example, at synchronous

excitation, 𝑑𝑦𝑦 is 8.50Ns/m for the hybrid bearing while it is only 1.74 Ns/m for a hydrostatic

bearing at the same configuration.

16
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

Figure 5. 11 The effect of whirling frequency ratio on the stiffness and damping coefficients
of hybrid journal air bearings at concentric journal position. The coefficients are calculated at
𝜔 = 100𝑘 𝑟𝑝𝑚, 𝑃𝑠 = 3.5𝑏𝑎𝑟, 𝑐 = 10𝜇𝑚, ℎ𝑔 = 13𝜇𝑚, 𝛽𝑔 = 30°, 𝐺𝑛𝑢𝑚 = 18, 𝛼𝑔 = 𝛾𝑔 = 0.6

Figure 5.12 presents the variation of synchronous stiffness and damping coefficients with

compressibility number (𝛬) at the given static equilibrium position (eccentricity ratio, 𝜀 = 0.1).

The plots indicate a combination effect of bearing radius (r), radial clearance (c) and rotational

speed (𝜔). The figure shows the hardening effect as what was observed in hydrostatic journal

air bearings. It is interesting to note that the cross-coupled stiffness of hybrid bearings will reach

their maximum value when 𝛬 is around 10. In comparison, the cross-coupled stiffness of

hydrostatic bearings increases with 𝛬.

168
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

Figure 5.12 The effect of compressibility number on the synchronous stiffness and damping
coefficients of hybrid journal air bearings. The coefficients are calculated at 𝜀 = 0.1, 𝜔 =
150𝑘 𝑟𝑝𝑚, 𝑐 = 10𝜇𝑚, ℎ𝑔 = 13𝜇𝑚, 𝛽𝑔 = 30°, 𝐺𝑛𝑢𝑚 = 18, 𝛼𝑔 = 𝛾𝑔 = 0.6

The effect of restrictor setup is very like what have been investigated on hydrostatic bearings

except supply pressure (𝑃𝑠 ), as shown in Figure 5.13. In hybrid air bearings, herringbone

grooves enable the principal damping coefficients to be increased with 𝑃𝑠 . In hydrostatic air

bearings, the damping coefficients are not influenced by 𝑃𝑠 . The cross-coupled damping

coefficients are very small (almost zero). When 𝑃𝑠 is 1 bar, i.e. under ambient pressure, the

results represent the coefficients of hybrid air bearing operating with no external compressed

air supply.

169
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

Figure 5.13 The effect of supply pressure on the synchronous stiffness and damping
coefficients of hybrid journal air bearings. The coefficients are calculated at 𝜀 = 0.1, 𝜔 =
150𝑘 𝑟𝑝𝑚, 𝑐 = 10𝜇𝑚, ℎ𝑔 = 13𝜇𝑚, 𝛽𝑔 = 30°, 𝐺𝑛𝑢𝑚 = 18, 𝛼𝑔 = 𝛾𝑔 = 0.6

The influence of herringbone groove configurations on the stiffness and damping coefficients

are investigated using the static equilibrium position at 0.1 eccentricity ratio. The herringbone

groove structure parameters being studied are maximum groove depth (ℎ𝑔 ), groove angle (𝛽𝑔 ),

groove number (𝐺𝑛𝑢𝑚 ), groove width ratio (𝛼𝑔 ) and the fraction of grooved area to bearing

length (𝛾𝑔 ). The simulation results are summarized and plotted from Figure 5.14 to Figure 5.18.

The coefficients in the figures are calculated at the synchronous whirl frequency.

10
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

The effect of maximum groove depth (ℎ𝑔 ) on principal stiffness coefficients and damping

coefficients is presented in Figure 5.14. In the plot, ℎ𝑔 = 0μm implies there is no herringbone

grooves (hydrostatic bearings). The principal stiffness coefficients have a maximum value when

ℎ𝑔 = 10μm. Increasing ℎ𝑔 over 15μm considerably reduces the principal stiffness (reduced

over 12% when ℎ𝑔 = 30μm ). It is interesting to note that the cross-coupled stiffness

coefficients also reach their minimum at a groove depth ℎ𝑔 close to 8μm . Meanwhile, the

principal damping coefficients increase significantly with ℎ𝑔 and reaches the maximum at ℎ𝑔 =

20μm . In comparison with hydrostatic bearings ( ℎ𝑔 = 0μm ), the principal damping

coefficients are increased from 0.9Ns/m to 5.9Ns/m. By means of comparing the effect of ℎ𝑔

on the principal stiffness and damping coefficients, it can be found that the increased damping

property is at the cost of reduced bearing stiffness. It also worth noting, when ℎ𝑔 = 10μm,

principal stiffness coefficients of hybrid air bearings are the same as hydrostatic bearings (ℎ𝑔 =

10μm). However, the cross coupled stiffness coefficients are reduced 20% and the damping

coefficients are increased to 4Ns/m from 0.9Ns/m

11
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

Figure 5.14 The effect of maximum groove depth on the stiffness and damping coefficients of
hybrid journal air bearings. The coefficients are calculated at 𝜀 = 0.1, 𝜔 = 150𝑘 𝑟𝑝𝑚, 𝑐 =
8𝜇𝑚, 𝑃𝑠 = 2.5𝑏𝑎𝑟, 𝛽𝑔 = 30°, 𝐺𝑛𝑢𝑚 = 18, 𝛼𝑔 = 𝛾𝑔 = 0.6

Figure 5.15 shows the influence of groove angle (𝛽𝑔 ). When 𝛽𝑔 is between 16° and 30°, the

change on principal stiffness coefficients is not significant (4% variation), although maximum

stiffness is achieved 𝛽𝑔 = 25° . Further increasing in 𝛽𝑔 will cause the principal stiffness

coefficients drop considerably. On the other hand, the cross-coupled stiffness coefficient (𝑘𝑥𝑦 )

present an ascending trend with 𝛽𝑔 (descending for 𝑘𝑦𝑥 ). For example, 𝑘𝑥𝑥 is reduced 12% at

𝛽𝑔 = 46° in comparison with 𝑘𝑥𝑥 at 𝛽𝑔 = 26°. In the same process, 𝑘𝑥𝑦 is increased 38%. The

damping coefficients decrease with 𝛽𝑔 . However, the principal damping coefficients are only

12
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

reduced from 5Ns/m to 3.5Ns/m when 𝛽𝑔 increases from 16° to 48°. The groove angle only

has marginal influence on the damping coefficients.

Figure 5.15 The effect of groove angle on the stiffness and damping coefficients of hybrid
journal air bearings. The coefficients are calculated at 𝜀 = 0.1, 𝜔 = 150𝑘 𝑟𝑝𝑚, 𝑐 =
8𝜇𝑚, 𝑃𝑠 = 2.5𝑏𝑎𝑟, ℎ𝑔 = 10𝜇𝑚, 𝐺𝑛𝑢𝑚 = 18, 𝛼𝑔 = 𝛾𝑔 = 0.6

Figure 5.16 shows the effect of groove number (𝐺𝑛𝑢𝑚 ). It is found that 𝐺𝑛𝑢𝑚 has very limited

effect on the principal stiffness coefficients. The overall variation is less than 5% when 𝐺𝑛𝑢𝑚

changes from 6 to 24. However, the amplitude of cross-coupled stiffness coefficients is reduced

13
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

about 28%. On the other hand, the principal damping coefficients increase linearly with 𝐺𝑛𝑢𝑚 .

When 𝐺𝑛𝑢𝑚 increases from 6 to 24, the principal damping coefficients are increased from

2Ns/m to 4.9Ns/m.

Figure 5.16 The effect of groove number on the stiffness and damping coefficients of hybrid
journal air bearings. The coefficients are calculated at 𝜀 = 0.1, 𝜔 = 150𝑘 𝑟𝑝𝑚, 𝑐 =
8𝜇𝑚, 𝑃𝑠 = 2.5𝑏𝑎𝑟, ℎ𝑔 = 10𝜇𝑚, 𝛽𝑔 = 30°, 𝛼𝑔 = 𝛾𝑔 = 0.6

In Figure 5.1 , the effect of groove width ratio (𝛼𝑔 ) is presented. It is found that all principal

and cross-coupled stiffness coefficients increase with 𝛼𝑔 . At the same time, the principal

14
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

damping coefficients only decrease with 𝛼𝑔 slightly (drop from 3Ns/m to 2.5Ns/m). This

indicated the influence of 𝛼𝑔 is mainly on the stiffness coefficients.

Figure 5.1 The effect of groove width ratio on the stiffness and damping coefficients of
hybrid journal air bearings. The coefficients are calculated at 𝜀 = 0.1, 𝜔 = 150𝑘 𝑟𝑝𝑚, 𝑐 =
8𝜇𝑚, 𝑃𝑠 = 2.5𝑏𝑎𝑟, ℎ𝑔 = 10𝜇𝑚, 𝛽𝑔 = 30°, 𝐺𝑛𝑢𝑚 = 18, 𝛾𝑔 = 0.6

The effect of grooved area fraction (𝛾𝑔 ) is shown in Figure 5.18. 𝛾𝑔 = 1 implies the journal

surface is fully grooved. It is found that the principal stiffness coefficients drop quickly with

𝛾𝑔 when it is greater than 0.6. However, the amplitude of cross-coupled stiffness coefficients

15
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

is reduced with the increase of 𝛾𝑔 . The plot also indicates 𝛾𝑔 can improve principal damping

coefficients significantly.

Figure 5.18 The effect of grooved area fraction on the stiffness and damping coefficients of
hybrid journal air bearings. The coefficients are calculated at 𝜀 = 0.1, 𝜔 = 150𝑘 𝑟𝑝𝑚, 𝑐 =
8𝜇𝑚, 𝑃𝑠 = 2.5𝑏𝑎𝑟, ℎ𝑔 = 10𝜇𝑚, 𝛽𝑔 = 30°, 𝐺𝑛𝑢𝑚 = 18, 𝛼𝑔 = 0.6

Throughout the analysis from Figure 5.14 to Figure 5.18, the effect of herringbone groove

configurations on the stiffness and damping coefficients of hybrid journal air bearing can be

summarized as follows:

16
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

(a) The influence of herringbone groove parameters groove depth (ℎ𝑔 ), groove width ratio

(𝛼𝑔 ) and grooved area fraction ( 𝛾𝑔 ) will influence the volume of air film. Other

parameters will not.

(b) The principal damping coefficients can be improved by increasing ℎ𝑔 and 𝛾𝑔 at the cost

of reduced principal bearing stiffness coefficients. By changing ℎ𝑔 , one can get the

maximum principal and minimum cross-coupled stiffness coefficients with same value

of ℎ𝑔 . However, 𝛼𝑔 mainly has effect on the stiffness coefficients. The principal

stiffness coefficients increase with 𝛼𝑔 , which is different from ℎ𝑔 and 𝛾𝑔 .

(c) 𝛽𝑔 mainly has effect on changing the stiffness coefficients, while 𝐺𝑛𝑢𝑚 has more effect

on changing the principal damping coefficients.

(d) With reference to [85, 94] and study on hydrostatic journal air bearing in Chapter 4, the

cross-coupled stiffness in air bearings are the source to cause instability. In hybrid air

bearings, the stability of the bearing can be improved by manipulating the groove

configurations to reduce the cross-coupled stiffness coefficients and improve the

damping coefficients.

One can also manipulate the groove configurations to achieve different bearing stiffness

and damping coefficients to meet the requirement of rotor dynamic configuration

supported by the bearings. For example, one can improve the bearing’s damping

coefficients using larger ℎ𝑔 , 𝛾𝑔 and 𝐺𝑛𝑢𝑚 , at the same time, selecting proper values of

𝛼𝑔 and 𝛽𝑔 to avoid too much compromise on reducing bearing stiffness.

1
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

5.3.3 Analysis on stability and natural frequencies of rotor bearing system using linear

bearing model

In this section, the stability and natural frequencies of a rotor bearing system with hybrid air

bearings are investigated. The configurations of the rotor bearing system under study here are

the same as CASES I & II. The hybrid air bearings are represented as linear bearing units with

stiffness and damping coefficients extracted from the linear perturbation analysis in Section

5.3.2. The dimensions and groove configurations of the bearings are listed in Table 5.5.

Because there is no change in the mechanical configurations in both cases, only the

characteristic matrices of the system governing equations of motion are presented as Equation

5.2 and Equation 5.3. The analytical method is the same as that performed on hydrostatic air

bearings in Chapter 4.

Table 5.5 Design parameters and restrictor setup of hybrid air bearings to be studied

Design parameters Value

Radius (mm), 𝒓𝟎 10

Length (mm), 𝒍 20

Radial clearance (𝛍𝐦), 𝒄 12.5

Orifice diameter (𝛍𝐦), 𝒅𝟎 250

Supply pressure (bar), 𝑷𝒔 1 and 7

Maximum groove depth (𝛍𝐦) 13

Groove angle (degree), 𝜷𝒈 28.5

Number of grooves, 𝑮𝒏𝒖𝒎 18

ratio of groove width to total width (groove plus
0.6
ridge), 𝜶𝒈
fraction of bearing length occupied by grooves, 𝜸𝒈 0.6

18
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

Characteristic matrix of the governing equation of motion for the rotor bearing system presented

in Chapter 4, CASE I:

[0] [1]
𝐽1 = [ −1 −𝟏 ]
−[𝑴𝒔 ] [𝑲𝒔𝒚𝒔_𝟏 ] −[𝑴𝒔 ] [𝑪𝑮𝒔𝒚𝒔_𝟏 ]

[𝑲𝒔𝒚𝒔_𝟏 ] = [[𝑲𝒔 ] + [𝑲𝒃 ]] Equation 5.2

[𝑪𝑮𝒔𝒚𝒔_𝟏 ] = [[𝑪𝒔 ] + [𝑪𝒃 ] + [𝑮]]

where [𝑴𝒔 ] is the shaft structure mass and inertia matrix, [𝑲𝒔𝒚𝒔_𝟏 ] the system stiffness matrix,

[𝑲𝒔 ] the shaft structure stiffness matrix, [𝑲𝒃 ] the bearings’ stiffness matrix, [𝑪𝑮𝒔𝒚𝒔_𝟏 ] the

system damping and gyroscopic matrix, [𝑪𝒔 ] the shaft structure damping coefficients, [𝑪𝒃 ] the

bearing’s damping matrix, and [𝑮] the shaft structure gyroscopic matrix.

Characteristic matrix of the governing equation of motion for rotor bearing system presented in

Chapter 4, CASE II:

[0] [1]
𝐽2 = [ −1 −𝟏 ]
−[𝑴𝟐 ] [𝑲𝒔𝒚𝒔_𝟐 ] −[𝑴𝟐 ] [𝑪𝑮𝒔𝒚𝒔_𝟐 ]

[𝑴𝒔 ] [𝟎]
[𝑴𝟐 ] = [ ]
[𝟎] [𝑴𝒗 ]
Equation 5.3
[𝑲 ] + [𝑲𝒃 ] −[𝑲𝒃 ′ ]
[𝑲𝒔𝒚𝒔_𝟐 ] = [ 𝒔 ]
−[𝑲𝒃 ′ ]𝑇 [𝑲𝒗 ]

[𝑪𝒔 ] + [𝑪𝒃 ] + [𝑮] −[𝑪𝒃 ]

[𝑪𝑮𝒔𝒚𝒔_𝟐 ] = [ ]
−[𝑪𝒃 ] [𝑪𝒗 ]

19
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

where [𝑴𝒔 ] is the shaft structure mass and inertia matrix, [𝑲𝒔𝒚𝒔_𝟐 ] the system stiffness matrix,

[𝑲𝒔 ] the shaft structure stiffness matrix, [𝑲𝒃 ] the bearings’ stiffness matrix, [𝑪𝑮𝒔𝒚𝒔_𝟐 ] the

system damping and gyroscopic matrix, [𝑪𝒔 ] the shaft structure damping coefficients, [𝑪𝒃 ] the

bearing’s damping matrix, [𝑮] the shaft structure gyroscopic matrix, [𝑴𝒗 ] the bearing sleeve

mass and inertia matrix, and [𝑲𝒗 ] and [𝑪𝒗 ] the stiffness and damping matrix of the bearing

sleeve respectively.

The analysis on natural frequencies is performed at the working condition that external

compressed air supply pressure (𝑃𝑠 ) is maintained at bar. Figure 5.19 shows the natural

frequencies of the two cases identified from theoretical analysis as Campbell diagrams, with

the rotor speed up to 150k rpm. Mode shapes of the rotor are obtained at 1330Hz. In the plots,

the first two modes are rigid modes and the third and fourth one (if it is shown) are bending

modes. It can be seen that natural frequencies of all modes are reduced in comparison with

systems running on hydrostatic air bearings, referring to Figure 4.24 in Chapter 4. For example,

the natural frequency of mode 2 in CASE I is 1315 Hz, while it is 1680 Hz for hydrostatic air

bearings; the natural frequency of mode 3 in CASE II is 1470 Hz, while it is 1790 Hz for

hydrostatic air bearings. A forth mode also appears in the speed range of CASE II. They are

the result of hybrid bearings with given geometry having lower stiffness than that of the

hydrostatic bearings with similar geometry investigated in Chapter 4.

180
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

a) b)

c)

181
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

d)

Figure 5.19 Campbell diagrams for Cases I & II. The dash line is the synchronous line. ‘F’
denotes forward whirl and ‘B’ denotes backward whirl. a) Case I: linear rotor with linear
bearings. b) Case II: linear rotor with linear bearings and linear dampers. c) Mode shape
obtained at 1330Hz of the rotor in CASE I. d) Mode shape obtained at 1330Hz of the rotor in
CASE II.

From the Campbell diagram, it is seen the frequency of the backward whirl of mode 2 in CASE

I, mode 2 and 3 in CASE II gradually ascends with increasing rotor speed, which leads to a

very small difference with the forward whirl of these modes. Similar effect was observed for

the two cases with hydrostatic air bearings. This has also been reported in [85], in which the

analysis was made on a turbocharger rotor supported by floating ring bearings.

The stability analysis is performed in two working conditions for both cases. In the first working

condition, the compressed air supply pressure (𝑃𝑠 ) is maintained at bar. In the second working

condition, the bearings are fully self-acting with no external pressurized air (the supply pressure

is ambient, 𝑃𝑠 = 1bar).

182
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

The results of stability analysis are shown in Figure 5.20. In the stability maps, the real part of

the leading eigenvalue of the system characteristic matrix is plotted against rotational speeds

from 10k to 250k rpm. Plots a and b show the both cases are stable within the speed range when

compressed air supply is maintained at 7bar. In comparison, the system of CASE I supported

by hydrostatic air bearings becomes unstable at 110k rpm (See Figure 4.25 a). This proves the

use of hybrid air bearings can greatly improve the stability of a rotor bearing system. Plots c

and d are the stability maps when the bearings are working as hydrodynamic ones. At this

condition, the bearings can lift the rotor only when its rotational speed is over 20k rpm. The

system in CASE I is unstable within the speed range while the system in CASE II becomes

stable once the rotational speed goes over 200k rpm. It indicates the viscoelastic support can be

used to improve the stability. The whirl frequency ratio corresponding to stability map c and d

are presented in Figure 5.20 e) and f) respectively. As discussed in Section 4.4.3, the abrupt

whirling frequency shift is observed again in both cases when one leading eigenvalue

supersedes another. The shift of whirling frequency and leading eigenvalue are marked with

green arrows.

183
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

a) b)

c) d)

e) f)

Figure 5.20 Stability maps and whirl frequency ratio based on SESA. a) stability map of
CASE I, 𝑃𝑠 = 7, b) stability map of CASE II, 𝑃𝑠 = 7, c) stability map of CASE I, 𝑃𝑠 = 1, d)
stability map of CASE II, 𝑃𝑠 = 1, 𝛾 = 0.2, e) whirl frequency ratio of CASE I, 𝑃𝑠 = 1, f)
whirl frequency ratio of CASE II, 𝑃𝑠 = 1, 𝛾 = 0.2.

184
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

The above analysis shows that the viscoelastic support has the capability of improving the

system stability when hybrid air bearing is operating with no supply of externally compressed

air.

The deformation ratio (𝛾) of the O-rings under the current study is selected as 0.2. This is based

on the actual O-ring deformation evaluated from the testing device used in experiments.

Although the viscoelastic support with this configuration can improve the stability, it may not

be ideal. For example, the rotor bearing system in CASE II is still unstable under 190k rpm.

This is a speed range where the rotor in the size of R-1 typically working at. A further study on

stability was performed by increasing 𝛾 to 0.25. The stability map is shown in Figure 5.21. It is

found that the viscoelastic support with increased O-ring deformation ratio can stabilize the

system when hybrid air bearings working with no compressed air supply.

Figure 5.21 Stability map based on SESA of CASE II, 𝑃𝑠 = 1, 𝛾 = 0.25

185
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

5.4 Non-linear transient analysis and experimental verification of hybrid air

bearings

In this section, the non-linear transient analysis performed in Chapter 4 is applied to hybrid air

bearings and the results are then verified by experiments in unbalance responses. The bearing

test rig developed is also used here. The rotor is remanufactured with the novel herringbone

grooves fabricated on two journals of the rotor to form the hybrid air bearings. In order to

operating at a higher rotational speed, this rotor was balanced to achieve G1 standard for a

service speed at 150k rpm. The unbalance responses of the rotor were measured up to 120k and

compared with predictions from the non-linear transient analysis.

This section begins with an introduction to the manufacturing process of the novel herringbone

groove with its geometry measured. The predicted unbalance responses of the test rig are then

discussed under two occasions. In the first one, the rotor is running with hybrid air bearing

using compressed air supply pressure (𝑃𝑠 ) at bar and tested up to 120k rpm. The discussion is

focused on the synchronous responses. In the second one, the rotor is initially lifted by hybrid

air bearings with 𝑃𝑠 = 7bar. Once the rotor accelerates to a speed between 50k and 60k rpm,

compressed air supply to the hybrid journal bearings is cut off and the bearings are self-acting.

The theoretically studies and experiments were performed from 0k rpm up to 120k rpm. The

discussion is focused on the unbalance responses with sub-synchronous components (self-

excited whirl).

186
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

5.4.1 Manufacturing of the novel herringbone groove

In the proposed hybrid air bearings, the novel herringbone grooves are the key feature and

fabricated using the latest laser machining technology. Figure 5.22 a) shows the manufactured

rotor. Some design parameters and the groove profile measuring place are indicated on an

enlarged view of the grooved journal in Figure 5.22 b). The measured values of these parameters

are given in Table 5.6. The groove profile is measured using Alicona microscope and presented

in Figure 5.22 c). It is designed as a cosine weave spline with 13μm maximum depth and

1 40 μm span (half cycle). The measured dimensions of the grooves were used in the non-linear

bearing model.

a) R-1 with herringbone grooved journal

18
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

b) Enlarged view of the herringbone grooved journal

c) Measured groove profile

Figure 5.22 The rotor used in the hybrid air bearings and the novel herringbone grooves. a)
The manufactured rotor, b) Enlarged view of the herringbone grooved journal, and c)
Measured groove profile: the red dash line is the design profile; the blue curve is the
measured profile.

188
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

Table 5.6 Design parameters of the novel herringbone groove

Design parameters Value

Number of grooves, 𝑮𝒏𝒖𝒎 18

Maximum groove depth (𝛍𝐦) 13

Groove angle (degree), 𝜷𝒈 28.5

Groove width (mm), 𝝀𝒈 1.74

Groove length (mm), 𝜸𝒈 6.3

5.4.2 Unbalance responses of hybrid journal air bearings with compressed air supply

In this section, the experimental studies on unbalance responses of the rotor were measured

when compressed air was supplied to the hybrid air bearings at b ar. The test rig introduced in

Chapter 4 was used. The responses of the rotor were measured at the same sensor positions and

compared with simulation results. The bearing components had no significant changes in

dimensions with that used in hydrostatic bearing tests. The responses of the rotor were measured

at constant speeds 0k, 80k, 90k, 100k, 110k and 120k rpm respectively. A 5% difference in the

power of the provide drive was allowed in each test.

Rotor R-1 used in hybrid air bearing tests has been presented in Figure 5.22 a). It was balanced

to G1 standard for a service speed at 150k rpm. The rotor was balanced using two-plane

dynamic balancing approach. The equivalent unbalance residuals are concentrated on two

planes and listed in Table 5. .

189
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

Table 5. Unbalance information of R-1

Left hand correction Right hand correction
plane plane

Unbalance residual
0.008 0.005
(g*mm)

Unbalance phase angle 220° 78°

Positions on Rotor Compressor disc Turbine disc

Position on Rotor
Node 4 Node 18
model

From the analysis shown by Figure 5.19 b) of Section 5.3.3, the system has a natural frequency

at 14 0 Hz within the test speed range. Figure 5.23 is the event time waterfall plots in a run-up

test to 120k rpm. It shows the vibrations of the rotor are dominated by the synchronous

components (1x) and have a critical speed at 1400 Hz. The results have good agreement with

the predictions on the natural frequencies. The rotor bearing system is stable at this working

condition up 120k rpm rotor speed (no sub-synchronous vibration) as expected.

190
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

Figure 5.23 Top views of waterfall plots in a run-up test to 120k rpm. Measurements were
made at sensor position A in horizontal direction.

The experimental unbalance responses and related simulation results from non-linear transient

analysis at the given constant rotor speeds are shown in Appendix E for hybrid air bearings with

ba r compressed air supply. In each group of the figures, the dominating vibration frequency

and the peak vibration velocity amplitude are marked on the FFT plot. The simulations were

produced using the actual rotor speed (1x component in frequency spectrum) acquired from

experiments. At each speed, the rotor responses were calculated for the first 00 shaft

revolutions and a steady state was assumed to be achieved at the last 100 revolutions. The

response data were collected for the last 100 revolutions and analysed using FFT. Figure 5.24

shows a case of these measurements and predictions at 120.9k rpm rotor speed for both sensor

positions. By comparing Figures 5.24 a & b, it can be found that the vibration frequencies and

magnitudes of the measured and simulation results are very similar, and both peak at 2015Hz

(unbalance excitations), but the peak amplitudes of the vibration velocities are 2.91mm/s

measured from experiments, and 2.80mm/s acquired from simulation. The difference between

them is 3%, which can be neglected. Figures 5.24 c & d show that the vibration frequencies and

191
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

magnitudes of the measured and simulation results are very similar again, but the peak

amplitudes of the speeds are too small to give meaningful readings.

a) Measurements at sensor position A

b) Predictions at sensor position A

192
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

c) Measurements at sensor position B

d) Predictions at sensor position B

Figure 5.24 Unbalance responses of sensor positions A & B obtained at 120.9k rpm in speed.
Supply pressure maintained at ba r.

193
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

Figure 5.25 Peak synchronous vibration velocities of rotor at sensor position A and B of
various rotor speeds

Figure 5.25 compares the sychronous vibration velocities of the rotor measured in experiments

(solid lines) and predicted from non-linear transient analysis (dash lines) at the two sensor

positions for the test speeds. The presented results, also the results in Appendix E, have a

reasonable agreement between experiments and predictions. Both of them show that the

vibrations of the rotor are dominated by synchronous component and no self-excited whirl has

been spotted. The results also show that there is an increasing on the amplitude of vibration

when the rotor speed is close to the critical speed of the system (1400Hz). This could also be

found from the watfall plot in Figure 5.23. It is found that the unbalance responses at sensor

position B have exceedingly small amplitudes in comparison with that from sensor position A.

This can be explained using the shaft responses to out of phase unbalance from non-linear

194
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

transient analysis shown in Figure 5.26. Figure 5.26 a) illustrates the shaft responses at 0k rpm.

The sensor position B is placed at the waist of the conical orbits. When the system work at the

rotor speed higher than 1400Hz, for example at 120k rpm, the shaft is likely to working at a

bending mode as illustrated in Figure 5.26 b). The maximum displacement of the shaft centre

happens at the position close to sensor position A. The shaft responses at sensor position B are

still very small. The disparancy between experimental and prediction results is also significant

than that of sensor position A. This is because the vibration velocities at this position are low

and close to reaching the lower sensiticity limit of the vibrometer. The vibration signals are

merged with sensor noise and difficult to be filtered out. From the above analysis, it can be said

that, for the tested speeds, the numerical model developed are dynamically equivalent to the

rotor bearings system at both sensor positions when compressed air are supplied to the bearings

at bar.

195
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

a)

b)

Figure 5.26 The theoretical shaft response of R-1 in CASE II with hybrid air bearings in
response to out of phase unbalance. The orbits of each node are obtained using the last 50
shaft revolutions in simulation. The dash line indicates positions of the shaft centre at each
node at the last simulation time step. Shaft responses at two sensor positions are marked in
red. a) rotor speed at 0k rpm. b) rotor speed at 120k rpm

196
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

5.4.3 Unbalance responses of hybrid air bearings without supply of compressed air

This section presents the experimental and theoretical studies on unbalance responses of the

rotor with hybrid journal air bearings operating without the supply of compressed air. The

compressed air supply was manually switched off during the experiments when rotor speed

went over 50k rpm. The responses of the rotor were measured at constant speeds of 0k, 80k,

90k, 100k, 110k and 120k rpm respectively.

Figure 5.2 shows the waterfall plot to indicates the responses of the rotor before and after the

compressed air supply to hybrid air bearings was switched off. Besides the synchronous

component, it is found that a sub-synchronous component quickly builds up when there is no

compressed air supply. The frequency of this sub-synchronous component is locked around

820Hz. However, its ratio to the synchronous frequency descends with the rotational speed as

shown in Figure 5.28. The appearance of sub-synchronous vibrations indicates the system is

unstable as the analysis shown by Figure 5.20 d). The descending whirl frequency ratio was

also predicted in Figure 5.20 f).

19
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

Figure 5.2 Top views of waterfall plots generated from experimental data. Measurements
were made at sensor position A in horizontal direction.

Figure 5.28 Frequency ratio of sub-synchronous vibration to rotational speed at various

speeds using the experimental data from Figure 5.2 .

The unbalance responses measured from experiments and ralated simulation results from non-

linear transient analysis at the given constant rotor speeds are provided in Appendix F for hybrid

air bearings with no compressed air supply. In each group of the figures, the sub- and

synchronous vibration frequencies and their peak vibration velocity amplitude are marked on

198
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

the FFT plot. The simulations were conducted using the actual rotor speed (1x component in

frequency spectrum) acquired from experiments. At each speed, the rotor responses were

calculated for the first 0 0 shaft revolutions and a steady state was assumed to be achieved at

the last 100 revolutions. The response data were collected for the last 100 revolutions and

analysed using FFT.

Figure 5.29 shows a case of these measurements and predictions at 98.4k rpm rotor speed for

both sensor positions. The results illustrate the instability corresponds to a case of ‘half-

frequency whirl’ (as far as the fundamental frequency to speed ratio is concerned) in both

simulation and experiment.

a) Measurements at sensor position A

199
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

b) Predictions at sensor position A

c) Measurements at sensor position B

d) Predictions at sensor position B

Figure 5.29 Unbalance responses of sensor positions A & B obtained at 98.4k rpm in speed

200
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

In Figure 5.30, the peak amplitude of vibration velocities (both sub- and synchronous

components) and sub-sychronous whirl frequency ratio are summarized and compared at the

constant rotor speeds between experimental and prediction results at the two sensor positions.

Figure 5.30 a) shows the peak amplitude of vibration velocities at sensor position A in horizontal

directions. The experimental results are marked as solid lines at various rotor speeds. The blue

line is the amplitude of sub-synchronous vibration velocites while the orange line represents

the amplitude of syhchronous vibration velocites. The prediction results are marked as discrete

points at the rotor speeds at which the simulations were performed (triangle for sub-sychronous

and diamond for synchronous). The responses at sensor position B are ploted in Figure 5.30 b)

following the same rule. The whirl frequency ratio of sub-synchronous vibrations to rotor

speeds are presented in Figure 5.30 c) with prediciton from SESA added. It can be seen that

there are good agreement between the simulations and experiments at most test speeds.

At sensor position A, the prediction results clearly reflect the change on amplitude of the

vibration velocities, especially the steady state amplitude of sub-synchronous vibration (self-

excited whirl). The differences between experimental and predicted synchronous vibration are

mainly the result of the extra imbalance introduced by reassembly of a balanced rotor to the test

rig. There are only marginal discrepancies at sensor position B. However, this is the result of

the vibration velocity at this position being close to the lower sensiticity limit of the vibrometer.

The experimental measurements may not be as accurate as that at sensor positon A.

201
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

a) Comparisions on the amplitude of peak vibrations velocities at sensor position A.

Blue dots are the amplitude of sub-sychronous vibration velocity measured in experiments.
Orange squares are the amplitude of sychronous vibration velocity measured in experiments
Grey triangles are the predicted amplitude of sub-sychronous vibration velocity.
Yellow diamonds are the predicted amplitude of sub-sychronous vibration velocity.

b) Comparisions on the amplitude of peak vibrations velocities at sensor position B.

Blue dots are the amplitude of sub-sychronous vibration velocity measured in experiments.
Orange squares are the amplitude of sychronous vibration velocity measured in experiments
Grey triangles are the predicted amplitude of sub-sychronous vibration velocity.
Yellow diamonds are the predicted amplitude of sub-sychronous vibration velocity.

202
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

c) The blue dots are experimental observations. Prediction results from non-linear transient
analysis are represented by diamonds. Prediction results from SESA are represented by
squares

Figure 5.30 Comparisons of prediction results with experimental results. The amplitude of
peak vibration velocities of the sub- and sychrounous vibration components at the two sensor
positions are compared in a) and b) respecitively. Plot c) compares the frequency ratio of sub-
sychronous vibration components to rotatinal speed.

Meanwhile, predicitons on the whirling frequency ratios of self-excited whirl from non-linear

transient analysis have a good agreement with experimental observations, as shown in Figure

5.30 c). The differences between experimental and prediction results for rotor speed from 8k

rpm to 120k rpm is less than 9%. The main discrepancy on the whirl frequency ratio occurs at

a rotor speed of 6 .2k rpm. At this speed, the sub-sychronous frequency is predicted to be

696Hz while it is observed at 810Hz in experiments, i.e. 14. % in difference. In experiments,

the frequency of sub-synchronous vibration is quickly locked at 820Hz. While in the

simulations, this frequency ascends with the rotor speed slightly from 696Hz and is locked at

855Hz when the rotor speed goes over 100k rpm. On the other hand, it is found that the whirl

frequency ratio varies with speed in the same way between the SESA and non-linear approach,

203
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

but the whirl ratio predicted by SESA is lower. Althought the whirl frequency ratio observed in

experiments is higher than predictions and descends slightly more rapidly, predictions from

non-linear transient and SESA have an adequate agrement with experimental results.

The simulations and experiments performed in this section both prove that the proposed hybrid

air bearings work well without the supply of compressed air from 60k rpm up to 120k rpm. The

non-linear transient analysis performed successfully predicted the frequency and steady state

amplitude of self-excited whirl at several test speeds. Predictions on frequency ratio of self-

excited whirl from SESA are also acceptable in comparisons with experimental results. It can

be said that the numerical analysis proposed (both linear and non-linear analysis) are

dynamicaly equivalent to the actural rotor bearing system at the two sensor positions within the

given test rotor speed range. Although instability of the rotor bearing system appears when

compressed air is switched off, both simulation and experimental results show that is contained

within a limited cycle by the bearings and no contacts between rotor and bearings were observed

in experiments.

Based on the discussions of unbalnace responses in Section 5.4.2 and 5.4.3, it can be stated that

the numerical analysis performed shows the portential to predict rotational performance of a

rotor bearing system supported by the proposed hybrid journal air bearings. In general, the

linear approach will give predictions in frequency domain, including natural frequenices and

frequency ratio of self-excited whirl. The non-linear transient analyis on the other hand can

predict the shaft orbits in time domian and is capable of predicting the steady state amplitude

of self-excited whirl. Further improvement can be made on the non-linear transient analysis as

part of the follow up investigations by considering mounting excitations and non-constant

rotation speed (run-up or run down simulations) as discussed in [85].

204
CHAPTER 5 HYBRID JOURNAL AIR BEARINGS

5.5 Summary

This chapter provides a comprehensive study on hybrid journal air bearings and the rotor

structure they support. The finite volume model of hybrid air bearings is presented. The validity

of using FVM to model air bearings with herringbone grooves was examined and compared

with references. A novel herringbone groove design was adopted to increase the bearing

reaction forces of hybrid air bearings at a given equilibrium configuration. The rotational

performance was studied using the linear perturbation analysis. The effects of herringbone

groove configurations on the equivalent stiffness and damping coefficients of bearing forces at

given static equilibrium configurations were investigated. The theoretical studies showed

hybrid air bearings had better damping properties in comparison with hydrostatic air bearings.

The stability and natural frequencies were analysed for the system in CASE I & II with a rotor

supported by hybrid air bearings. Non-linear transient analysis was performed with

experimental verifications on unbalance responses at two working conditions of hybrid air

bearings. In the first working condition, compressed air was supplied to the hybrid air bearings

at ba r. Both experimental and prediction results showed that the system was stable and there

was no self-excited whirl. In the second working condition, compressed air was supplied

initially to lift the rotor at ba r and was switched off once the rotor speed went over 50k rpm.

Self-excited whirl was observed in both simulations and experiments when the hybrid air

bearings were running without supply of compressed air. The experimental and prediction

results were compared and discussed.

205
APPENDIX A

CHAPTER 6: CONCLUSIONS AND FUTURE

WORK

This chapter summarises the research work of this thesis, highlights important findings, and

draw conclusions of the research. It also recommends future work to extend this study.

6.1 Summary

This PhD project was set up to provide a deeper insight into hybrid journal air bearings and

explore their possible applications in high speed and mobile micro turbo machinery, such as

turbochargers and micro-gas turbine engines. Two main challenges need to be resolved in order

to achieve the project aims. One is to develop a valid model to predict the dynamics

performance of the hybrid air bearings and the rotor dynamic structure and help the design of

hybrid air bearings. The other is to apply the proposed hybrid journal air bearings on a micro

turbomachine and verify both the proposed hybrid air bearings and the modelling method

through experiments. This PhD thesis presents the research to reach the project’s targets.

The research scope covers hydrostatic and hybrid journal air bearings with non-compliant

boundaries. The approach adopted combines numerical analysis based on CFD and

methodologies in rotor dynamic analysis with experimental verifications of the designs. The

stability and natural frequencies of the system are predicted using a linear approach. The

unbalance responses are predicted using non-linear transient analysis. Repeated experiments

were carried out on a test rig. A rotor taken from a turbocharger was modified to accommodate

206
APPENDIX A

the required size of hybrid air bearings. The experiments were performed in ambient

temperature and no thermal effects were involved at this stage. The experimental and simulation

results have an adequate agreement in a rotor speed range from 50k rpm to 120k rpm in the

design specifications. The idea to eliminate the reliance on compressed air supply to the

bearings at high speeds has been proven working by both simulation and experiments. Air

supply to the bearings was switched off at speed of 50k to 60k rpm and the bearings can fully

self-suspended and support the rotor to maintain the speed and accelerate up to 120k rpm.

6.2 Conclusions

Through the theoretical and experimental investigation of the hybrid journal air bearings, the

objectives of the project have been implemented and the aims have been met. The following

conclusions can be drawn from the research:

1. In the theoretical studies on hydrostatic journal air bearings, it is found the bearing reaction

forces to static load is determined by the supply pressure. The orifices’ diameter and radial

clearance have a combined effect on the optimal design. The optimal orifice diameter

decreases with the radial clearances. When orifice restrictors with a 100 μm diameter are

used, the bearing reaction forces to static load varies sharply with the radial clearance.

Orifice restrictors with larger diameters provide a smooth change instead.

2. The linear perturbation analysis on the rotational performance of hydrostatic journal air

bearings shows a ‘hardening effect’ at high rotor speed results in very weak damping. The

damping property (indicated by the equivalent damping coefficients) of hydrostatic journal

20
APPENDIX A

air bearings with plain orifices cannot be improved effectively by adjusting bearing design

parameters such as radial clearance (c), supply pressure (𝑃𝑠 ) or orifice diameter (𝑑0 ), except

increasing the bearing length to diameter ratio (𝜁).

3. The numerical modelling approach proposed in this research is valid in predicting natural

frequencies and stability of a rotor bearing system with hydrostatic journal air bearings and

viscoelastic damp. The unbalance responses of the same system are predicted using non-

linear transient analysis from 50k rpm to 100k rpm rotor speed. The experimental and

prediction results have a good agreement at the sensor positions where the system responses

were measured.

4. In the theoretical studies on hybrid journal air bearings, a novel herringbone groove

configuration is proposed and proven effective in improving bearing reaction forces at a

given static equilibrium position investigated. The herringbone groove configuration is

novel in a way that a cosine spline is used to form the profile of it. The theoretical study

shows a maximum 16% increasing on bearing reaction forces in comparison with

conventional herringbone groove design. The study also shows the hybrid journal air

bearings with different herringbone groove configurations can achieve same bearing

reaction forces to static load at the same static equilibrium position and rotational speed.

However, their dynamic properties are significantly different with each other.

5. The linear perturbation analysis on the rotational performance of hybrid journal air bearings

also shows a ‘hardening effect’ at high rotor speed. However, the damping property

(indicated by the equivalent damping coefficients) of hybrid journal air bearings can be

208
APPENDIX A

improved by adjusting multiple herringbone groove configurations with compromise on the

stiffness. Increasing supply pressure (𝑃𝑠 ) of hybrid journal air bearings can also improve the

damping property, which is quite different from hydrostatic journal air bearings. In summary,

the damping properties of hybrid journal air bearing can be improved by increasing the

maximum groove depth (ℎ𝑔 ), grooved area fraction (𝛾𝑔 ) and groove number (𝐺𝑛𝑢𝑚 ). At the

same time, one can select proper values of groove width ratio (𝛼𝑔 ) and groove angle (𝛽𝑔 )

to increase the stiffness of the bearing.

6. In the case that no compressed air is supplied to hybrid air bearings, the stability analysis

using the linear approach shows the system will be unstable from 20k rpm up to 200k rpm

and would be stable afterwards. In experiments, the compressed air supply is switched off

when rotor speed goes over 50k rpm. The responses of the system measured at the sensor

positions from 0k rpm up to 120k rpm show a synchronous component (from unbalance

excitation) and a clear sub-synchronous component around 820Hz (from self-excited whirl).

This indicated the system is unstable. The stability predictions from the linear approach

within this speed range is valid. Tests at higher speeds are unavailable at this stage, as 120k

rpm (or a slight higher speed, e.g.125k rpm) is the limit by means of driving the turbo rotor

with compressed air using available facilities. Predictions of stability from 120k rpm

onwards need further investigation or proof.

7. Based on the theoretical and experimental work presented in this thesis, the analytical

methods (both linear and non-linear) adopted can be used as adequate tools to analyse the

rotational performance of a rotor bearing system supported by the hybrid air bearings. The

experiments also prove the reliance on compressed air supply of the proposed hybrid journal

209
APPENDIX A

air bearings can be reduced at high speeds. With the specifications of the hybrid journal air

bearings and the rotor dynamic structure designed in this project, the suggested speeds when

compressed air supply can be switched off should be no less than 50k rpm.

6.3 Suggestions for future work

This thesis presents a research effort to explore a type of hybrid air bearings for high speed and

portable turbo machinery applications. The results obtained in the research can be regarded as

a solid foundation for future work. Future effort is required to extend the related research and

to complete the works initiated in this PhD thesis which still need further investigation.

The following is a list of proposed further research topics:

1. Non-linear transient analysis with non-constant rotor speed.

The current non-linear transient analysis is only valid with constant rotor speed. The

responses of the rotor bearing system during run-up and run-down is not fully clear. This

can be done by improving the rotor bearing system model and perform further theoretical

studies. Similar studies were found in [85] which could be a starting point.

2. Investigations on the performance of the rotor bearing system with thermal effect being

considered.

The current theoretical and experimental study are all based on constant temperature, more

210
APPENDIX A

specifically the lab temperature (20°𝐶 to 25°𝐶). This is an important limitation on apply the

proposed hybrid air bearings on realistic micro turbomachinery, which normally work in

environment that involves high temperature. This could be further investigated theoretically

by considering the shaft expansion in the numerical model. Experiments are also possible,

for example, tests could be performed by means of driving the turbo rotor using heated air.

3. Investigations on the stability of the rotor bearing system supported by hybrid air bearings

at higher speeds (from 130k rpm and upwards).

In the static equilibrium stability analysis of the rotor bearing system, it is found there would

be shift in the self-excited whirl frequency. This should also be observed in experiments if

proper condition applies. The stability analysis on the rotor bearing system with hybrid air

bearings in a case study presented in the thesis (CASE II) also shows the system can be

stabilized even with compressed air switched off if the rotor speed goes over 200k rpm.

This prediction can be validated if tests at higher speeds are available. In current

experiments, the maximum rotor speed is limited by the size of compressed air reservoir

(350 Litres), which stores compressed air to drive the turbo rotor in the test rig. Increasing

the size of the reservoir to 900 Litres could remove this limit.

4. Investigations on the mode of rotor bearing system supported by hybrid air bearings.

The mode shape of a rotor bearing system could be identified by measuring the rotor

responses at multiple shaft locations. This could be beneficial to further verify/improve

the current numerical model used in the non-linear transient analysis.

211
APPENDIX A

APPENDIX A

In this project, numerical model of hydrostatic journal air bearings is based on the finite

difference method introduced in Chapter 3. The finite difference transformation of the Reynolds

Equation is linearized using Newton’s method to improve the numerical stability and converge

rate. Equation 3.28 is the resulting system after applying Newton’s method. Its coefficients are

given here:

−2𝑃𝑖,𝑗 𝐻𝑖,𝑗 3 2
3 𝜕 𝑃 2 𝜕𝐻 𝜕𝑃 2𝑃𝑖,𝑗 𝐻𝑖,𝑗 2 A-1
𝑎𝑖,𝑗 = + 𝐻𝑖,𝑗 ( 2 )𝑖,𝑗 + 3𝐻𝑖,𝑗 ( ) ( )𝑖,𝑗 −
∆𝜃 2 𝜕𝜃 𝜕𝜃 𝑖,𝑗 𝜕𝜃 ∆𝑍 2

3 𝜕 2𝑃 𝜕𝐻
+ 𝐻𝑖,𝑗 ( 2) − 6 ( )
𝜕𝑍 𝑖,𝑗 𝜕𝜃 𝑖,𝑗

3 𝜕𝑃 A-2
𝑃𝑖,𝑗 𝐻𝑖,𝑗 3 𝐻𝑖,𝑗 (𝜕𝜃 )𝑖,𝑗
𝑏𝑖,𝑗 = −
∆𝑍 2 ∆𝑍

3 𝜕𝑃 A-3
𝑃𝑖,𝑗 𝐻𝑖,𝑗 3 𝐻𝑖,𝑗 (𝜕𝜃 )𝑖,𝑗
𝑐𝑖,𝑗 = +
∆𝑍 2 ∆𝑍

3 𝜕𝑃 2 𝜕𝐻 A-4
𝑃𝑖,𝑗 𝐻𝑖,𝑗 3 𝐻𝑖,𝑗 (𝜕𝜃 )𝑖,𝑗 3𝐻𝑖,𝑗 𝑃𝑖,𝑗 ( 𝜕𝜃 )𝑖,𝑗 6𝐻𝑖,𝑗
𝑑𝑖,𝑗 = − − +
∆𝜃 2 ∆𝜃 2∆𝜃 2∆𝜃

212
APPENDIX A

𝜕𝑃 𝜕𝐻 A-5
𝑃𝑖,𝑗 𝐻𝑖,𝑗 3 𝐻𝑖,𝑗 3 ( ) 3𝐻𝑖,𝑗 2 𝑃𝑖,𝑗 ( )
𝜕𝜃 𝑖,𝑗 𝜕𝜃 𝑖,𝑗 6𝐻𝑖,𝑗
𝑑𝑖,𝑗 = − − +
∆𝜃 2 ∆𝜃 2∆𝜃 2∆𝜃

3 𝜕𝑃 2 𝜕𝐻 A-6
𝑃𝑖,𝑗 𝐻𝑖,𝑗 3 𝐻𝑖,𝑗 (𝜕𝜃 )𝑖,𝑗 3𝐻𝑖,𝑗 𝑃𝑖,𝑗 ( 𝜕𝜃 )𝑖,𝑗 6𝐻𝑖,𝑗
𝑒𝑖,𝑗 = + + −
∆𝜃 2 ∆𝜃 2∆𝜃 2∆𝜃

𝜕 2𝑃 𝜕 2𝑃 𝜕𝑃 2 𝜕𝑃 2 A-7
3 3
𝐶𝑜𝑛𝑖,𝑗 = 𝑃𝑖,𝑗 𝐻𝑖,𝑗 [( 2 ) + ( 2 ) ] + 𝐻𝑖,𝑗 [( ) + ( ) ]
𝜕𝜃 𝑖,𝑗 𝜕𝑍 𝑖,𝑗 𝜕𝜃 𝑖,𝑗 𝜕𝑍 𝑖,𝑗

𝜕𝐻 𝜕𝑃 𝜕𝑃 𝜕𝐻
+ 3𝐻𝑖,𝑗 2 𝑃𝑖,𝑗 ( ) ( ) − 6𝐻𝑖,𝑗 ( ) − 6𝑃𝑖,𝑗 ( )
𝜕𝜃 𝑖,𝑗 𝜕𝜃 𝑖,𝑗 𝜕𝜃 𝑖,𝑗 𝜕𝜃 𝑖,𝑗

where 𝑖 and 𝑗 denotes the point at 𝑖 𝑡ℎ row and 𝑗 𝑡ℎ column in a finite difference mesh.

The partial differential terms in A - 1 to A - a re expressed below using the second order

central finite difference formulae.

𝜕𝐻 𝐻𝑖,𝑗+1 − 𝐻𝑖,𝑗−1 A-8

( ) =
𝜕𝜃 𝑖,𝑗 2∆𝜃

𝜕𝑃 𝑃𝑖,𝑗+1 − 𝑃𝑖,𝑗−1 A-9

( ) =
𝜕𝜃 𝑖,𝑗 2∆𝜃

213
APPENDIX A

𝜕𝑃 𝑃𝑖+1,𝑗 − 𝑃𝑖−1,𝑗 A-10

( ) =
𝜕𝑍 𝑖,𝑗 2∆𝑍

𝜕 2𝑃 𝑃𝑖,𝑗+1 + 𝑃𝑖,𝑗−1 − 2𝑃𝑖,𝑗 A-11

( 2
) =
𝜕𝜃 𝑖,𝑗 ∆𝜃 2

𝜕 2𝑃 𝑃𝑖+1,𝑗 + 𝑃𝑖−1,𝑗 − 2𝑃𝑖,𝑗 A-12

( 2) =
𝜕𝑍 𝑖,𝑗 ∆𝑍 2

214
APPENDIX B

APPENDIX B

In this project, numerical model of hybrid journal air bearings is based on the finite volume

method (FVM) introduced in Chapter 3. To perform the finite volume transformation, the

Reynolds Equation was integrated using Green’s theorem along the boundaries of the controlled

volume surrounding a node on the meshed surface.

Figure B.1 a) Controlled volume surrounding a node in FVM approach b) Projected view of

the controlled volume on a meshed surface

The finite volume was divided into four cells, as shown in Figure B.1 b). 𝑄𝜃𝑖 and 𝑄𝑍𝑖 (𝑖 =

1,2,3,4) are integrals of the Reynold equation along the boundaries of the controlled volume in

𝜃 and 𝑍 direction.

Expressions of 𝑄𝜃𝑖 and 𝑄𝑍𝑖 (I = 1,2,3,4) are given here:

215
APPENDIX B

𝑃𝑖,𝑗 − 𝑃𝑖,𝑗−1 ∆𝑍 B-1

𝑄𝜃1 = [− 𝑃𝑖,𝑗−1/2 𝐻𝑖−,𝑗−1/2 3 + Λ𝑃𝑖,𝑗−1 𝐻𝑖−,𝑗−1 ]
Δ𝜃 2 2 2

𝑃𝑖,𝑗 − 𝑃𝑖−1,𝑗 ∆𝜃 B-2

𝑄𝑍1 = − 𝑃𝑖−1/2,𝑗 𝐻𝑖−,𝑗−1/2 3
Δ𝑍 2

𝑃𝑖,𝑗+1 − 𝑃𝑖,𝑗 ∆𝑍 B-3

𝑄𝜃2 = [− 𝑃𝑖,𝑗+1/2 𝐻𝑖−,𝑗+1/2 3 + Λ𝑃𝑖,𝑗+1 𝐻𝑖−,𝑗+1 ]
Δ𝜃 2 2 2

𝑃𝑖,𝑗+1 − 𝑃𝑖,𝑗 ∆𝜃 B-4

𝑄𝑍2 = − 𝑃𝑖−1/2,𝑗 𝐻𝑖−,𝑗+1/2 3
Δ𝑍 2

𝑃𝑖,𝑗 − 𝑃𝑖,𝑗−1 ∆𝑍 B-5

𝑄𝜃3 = [− 𝑃𝑖,𝑗−1/2 𝐻𝑖−,𝑗−1/2 3 + Λ𝑃𝑖,𝑗−1 𝐻𝑖+,𝑗−1 ]
Δ𝜃 2 2 2

𝑃𝑖+1,𝑗 − 𝑃𝑖,𝑗 ∆𝜃 B-6

𝑄𝑍3 = − 𝑃𝑖+1/2,𝑗 𝐻𝑖+,𝑗−1/2 3
Δ𝑍 2

𝑃𝑖,𝑗+1 − 𝑃𝑖,𝑗 ∆𝑍 B-7

𝑄𝜃4 = [− 𝑃𝑖,𝑗+1/2 𝐻𝑖−,𝑗+1/2 3 + Λ𝑃𝑖,𝑗+1 𝐻𝑖+,𝑗+1 ]
Δ𝜃 2 2 2

𝑃𝑖+1,𝑗 − 𝑃𝑖,𝑗 ∆𝜃 B-8

𝑄𝑍4 = − 𝑃𝑖+1/2,𝑗 𝐻𝑖+,𝑗+1/2 3
Δ𝑍 2

Equation 3.34 is the finite volume transformation of the Reynolds Equation. Its coefficients

are expressed below:

216
APPENDIX B

𝐻𝑖−,𝑗−1/2 3 𝐻𝑖−,𝑗+1/2 3 𝐻𝑖+,𝑗−1/2 3 𝐻𝑖+,𝑗+1/2 3

𝐴𝑖,𝑗 = ∆𝑍 + ∆𝑍 + ∆𝑍 + ∆𝑍
4∆𝜃 4∆𝜃 4∆𝜃 4∆𝜃

𝐻𝑖−,𝑗−1/2 3 𝐻𝑖−,𝑗+1/2 3 𝐻𝑖+,𝑗−1/2 3

+ ∆𝜃 + ∆𝜃 + ∆𝜃 B-9
4∆𝑍 4∆𝑍 4∆𝑍

𝐻𝑖+,𝑗+1/2 3
+ ∆𝜃
4∆𝑍

𝐻𝑖−,𝑗−1 3 𝐻𝑖+,𝑗−1 3 𝐻𝑖+,𝑗−1 3

2 2 2 𝐻𝑖+,𝑗+1/2 3 B-10
𝐵𝑖,𝑗 = 𝛬(− ∆𝑍 − ∆𝑍 + ∆𝑍 + ∆𝑍)
4 4 4 4

𝐻𝑖−,𝑗−1 3 𝐻𝑖−,𝑗+1 3
𝐶1𝑖,𝑗 = −( 2
∆𝑍 + 2
∆𝑍) B-11
4∆𝜃 4∆𝜃

𝐻𝑖+,𝑗−1 3 𝐻𝑖+,𝑗+1 3
𝐶2𝑖,𝑗 = −( 2
∆𝑍 + 2
∆𝑍) B-12
4∆𝜃 4∆𝜃

𝐻𝑖−,𝑗−1 3
2 𝐻𝑖−,𝑗+1/2 3 B-13
𝐷1𝑖,𝑗 = −( ∆𝜃 + ∆𝜃)
4∆𝑍 4∆𝑍

𝐻𝑖+,𝑗−1 3
2 𝐻𝑖+,𝑗+1/2 3 B-14
𝐷2𝑖,𝑗 = −( ∆𝜃 + ∆𝜃)
4∆𝑍 4∆𝑍

𝐻𝑖−,𝑗−1/2 3 𝐻𝑖+,𝑗−1/2 3
𝐸1𝑖,𝑗 = 𝛬(− ∆𝑍 − ∆𝑍) B-15
4 4

21
APPENDIX B

𝐻𝑖+,𝑗−1 3
2 𝐻𝑖+,𝑗+1/2 3 B-16
𝐸2𝑖,𝑗 = 𝛬( ∆𝑍 + ∆𝑍)
4 4

218
APPENDIX C

APPENDIX C

Timenshenko Beam Element in FEM

The Timoshenko finite element matrices for homogeneous beam elements are listed here. By
assuming with thin element, the diametral moment of inertia 𝐼𝑑 = 𝜌𝐴𝑏 (𝐷𝑜 2 + 𝐷𝑖 2 )⁄16 and
polar moment of inertia 𝐼𝑝 = 2𝐼𝑑

Shape factor:

where ν is the Poission ratio, 𝐷𝑖 is element inner diameter and 𝐷𝑜 is element outer diameter.

Transverse shear factor:

12𝐸𝐼
∅=
𝜅𝐴𝑏 𝐺𝐿𝑏 2

where 𝐸 is Young’s modulus, 𝐺 the shear modulus, 𝐼 the polar moment of inertia, A𝑏 the
cross-section area and 𝐿𝑏 the element length.

219
APPENDIX C

Element translational mass matrix, 𝑴𝒆 𝒕

𝑴𝒕𝒆𝟎 =

156 0 0 22𝐿𝑏 54 0 0 −13𝐿𝑏

0 156 −22𝐿𝑏 0 0 54 13𝐿𝑏 0
0 −22𝐿𝑏 4𝐿2𝑏 0 0 −13𝐿𝑏 −3𝐿2𝑏 0
𝜌𝐴𝑏 𝐿𝑏 22𝐿𝑏 0 0 4𝐿2𝑏 13𝐿𝑏 0 0 −3𝐿2𝑏
420(1 + ∅)2 54 0 0 13𝐿𝑏 156 0 0 −22𝐿𝑏
0 54 −13𝐿𝑏 0 0 156 22𝐿𝑏 0
0 13𝐿𝑏 −3𝐿2𝑏 0 0 22𝐿𝑏 4𝐿2𝑏 0
[−13𝐿𝑏 0 0 −3𝐿2𝑏 −22𝐿𝑏 0 0 4𝐿2𝑏 ]

𝜌𝐴𝑏 𝐿𝑏
𝑴𝒕𝒆𝟏 = ∗
420(1 + ∅)2

394 0 0 38.5𝐿𝑏 126 0 0 −31.5𝐿𝑏

0 294 −38.5𝐿𝑏 0 0 126 31.5𝐿𝑏 0
0 −38.5𝐿𝑏 7𝐿2𝑏 0 0 −31.5𝐿𝑏 −7𝐿2𝑏 0
38.5𝐿𝑏 0 0 7𝐿2𝑏 31.5𝐿𝑏 0 0 −7𝐿2𝑏
126 0 0 31.5𝐿𝑏 294 0 0 −38.5𝐿𝑏
0 126 −31.5𝐿𝑏 0 0 294 38.5𝐿𝑏 0
0 31.5𝐿𝑏 −7𝐿2𝑏 0 0 38.5𝐿𝑏 7𝐿2𝑏 0
[−31.5𝐿𝑏 0 0 −7𝐿2𝑏 −38.5𝐿𝑏 0 0 7𝐿2𝑏 ]

𝜌𝐴𝑏 𝐿𝑏
𝑴𝒕𝒆𝟐 = ∗
420(1 + ∅)2

140 0 0 17.5𝐿𝑏 70 0 0 −17.5𝐿𝑏

0 140 −17.5𝐿𝑏 0 0 70 17.5𝐿𝑏 0
0 −17.5𝐿𝑏 3.5𝐿2𝑏 0 0 −17.5𝐿𝑏 −3.5𝐿2𝑏 0
17.5𝐿𝑏 0 0 3.5𝐿2𝑏 17.5𝐿𝑏 0 0 −3.5𝐿2𝑏
70 0 0 17.5𝐿𝑏 140 0 0 −17.5𝐿𝑏
0 70 −17.5𝐿𝑏 0 0 140 17.5𝐿𝑏 0
0 17.5𝐿𝑏 −3.5𝐿2𝑏 0 0 17.5𝐿𝑏 3.5𝐿2𝑏 0
[−17.5𝐿𝑏 0 0 −3.5𝐿2𝑏 −17.5𝐿𝑏 0 0 3.5𝐿2𝑏 ]

𝑴𝒆 𝒕 = 𝑴𝒕𝒆𝟎 + ∅𝑴𝒕𝒆𝟏 + ∅2 𝑴𝒕𝒆𝟐

220
APPENDIX C

Element rotational mass matrix, 𝑴𝒆 𝒓

𝑴𝒓𝒆𝟎 =

36 0 0 3𝐿𝑏 −36 0 0 3𝐿𝑏

0 36 −3𝐿𝑏 0 0 −36 −3𝐿𝑏 0
0 −3𝐿𝑏 4𝐿2𝑏 0 0 3𝐿𝑏 −𝐿2𝑏 0
𝐼𝑑 3𝐿𝑏 0 0 4𝐿2𝑏 −3𝐿𝑏 0 0 −𝐿2𝑏
30𝐿𝑏 (1 + ∅) −36
2 0 0 −3𝐿𝑏 36 0 0 −3𝐿𝑏
0 −36 3𝐿𝑏 0 0 36 3𝐿𝑏 0
0 −3𝐿𝑏 −𝐿2𝑏 0 0 3𝐿𝑏 4𝐿2𝑏 0
[ 3𝐿𝑏 0 0 −𝐿2𝑏 −3𝐿𝑏 0 0 4𝐿2𝑏 ]

𝑴𝒓𝒆𝟏 =

0 0 0 −15𝐿𝑏 0 0 0 −15𝐿𝑏
0 0 15𝐿𝑏 0 0 0 15𝐿𝑏 0
0 15𝐿𝑏 5𝐿2𝑏 0 0 −15𝐿𝑏 −5𝐿2𝑏 0
𝐼𝑑 −15𝐿𝑏 0 0 5𝐿2𝑏 15𝐿𝑏 0 0 −5𝐿2𝑏
30𝐿𝑏 (1 + ∅) 2 0 0 0 15𝐿𝑏 0 0 0 −15𝐿𝑏
0 0 −15𝐿𝑏 0 0 0 −15𝐿𝑏 0
0 15𝐿𝑏 −5𝐿2𝑏 0 0 −15𝐿𝑏 5𝐿2𝑏 0
[−15𝐿𝑏 0 0 −5𝐿2𝑏 −15𝐿𝑏 0 0 5𝐿2𝑏 ]

𝑴𝒓𝒆𝟐 =

0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 10𝐿2𝑏 0 0 0 5𝐿2𝑏 0
𝐼𝑑 0 0 0 10𝐿2𝑏 0 0 0 5𝐿2𝑏
30𝐿𝑏 (1 + ∅) 0
2 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 5𝐿2𝑏 0 0 0 10𝐿2𝑏 0
[0 0 0 5𝐿2𝑏 0 0 0 10𝐿2𝑏 ]

𝑴𝒆 𝒓 = 𝑴𝒓𝒆𝟎 + ∅𝑴𝒓𝒆𝟏 + ∅2 𝑴𝒓𝒆𝟐

221
APPENDIX C

Element gyroscopic matrix, 𝑮𝒆

𝑮𝒆𝟎 =

0 −36 3𝐿𝑏 0 0 36 3𝐿𝑏 0

36 0 0 3𝐿𝑏 −36 0 0 3𝐿𝑏
−3𝐿𝑏 0 0 −4𝐿2𝑏 3𝐿𝑏 0 0 𝐿2𝑏
𝐼𝑝 0 −3𝐿𝑏 4𝐿2𝑏 0 0 3𝐿𝑏 −𝐿2𝑏 0
30𝐿𝑏 0 36 −3𝐿𝑏 0 0 −36 −3𝐿𝑏 0
−36 0 0 −3𝐿𝑏 36 0 0 3𝐿𝑏
−3𝐿𝑏 0 0 𝐿2𝑏 3𝐿𝑏 0 0 −4𝐿2𝑏
[ 0 −3𝐿𝑏 −𝐿2𝑏 0 0 3𝐿𝑏 4𝐿2𝑏 0 ]

𝑮𝒆𝟏 =

0 0 −15𝐿𝑏 0 0 0 −15𝐿𝑏 0
0 0 0 −15𝐿𝑏 0 0 0 −15𝐿𝑏
15𝐿𝑏 0 0 −5𝐿2𝑏 −15𝐿𝑏 0 0 5𝐿2𝑏
𝐼𝑝 0 15𝐿𝑏 5𝐿2𝑏 0 0 −15𝐿𝑏 −5𝐿2𝑏 0
30𝐿𝑏 0 0 15𝐿𝑏 0 0 0 15𝐿𝑏 0
0 0 0 15𝐿𝑏 0 0 0 15𝐿𝑏
15𝐿𝑏 0 0 5𝐿2𝑏 −15𝐿𝑏 0 0 −5𝐿2𝑏
[ 0 15𝐿𝑏 −5𝐿2𝑏 0 0 −15𝐿𝑏 5𝐿2𝑏 0 ]

𝑮𝒆𝟐 =

0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 10𝐿2𝑏 0 0 0 −5𝐿2𝑏
𝐼𝑝 0 0 10𝐿2𝑏 0 0 5𝐿𝑏 5𝐿2𝑏 0
30𝐿𝑏 0 0 0 0 0 0 0 0
0 0 0 −5𝐿𝑏 0 0 0 0
0 0 0 −5𝐿2𝑏 0 0 0 −10𝐿2𝑏
[0 0 5𝐿2𝑏 0 0 0 10𝐿2𝑏 0 ]

𝑮𝒆 = 𝑮𝒆𝟎 + ∅𝑮𝒆𝟏 + ∅2 𝑮𝒆𝟐

222
APPENDIX C

Element stiffness matrix, 𝑲𝒆

𝑲𝒆𝟎 =

12 0 0 6𝐿𝑏 −12 0 0 6𝐿𝑏

0 12 −6𝐿𝑏 0 0 −12 −6𝐿𝑏 0
0 −6𝐿𝑏 4𝐿2𝑏 0 0 6𝐿𝑏 2𝐿2𝑏 0
𝐸𝐼 6𝐿𝑏 0 0 4𝐿2𝑏 −6𝐿𝑏 0 0 2𝐿2𝑏
𝐿𝑏 (1 + ∅) −12
3 0 0 −6𝐿𝑏 12 0 0 −6𝐿𝑏
0 −12 6𝐿𝑏 0 0 12 6𝐿𝑏 0
0 −6𝐿𝑏 2𝐿2𝑏 0 0 6𝐿𝑏 4𝐿2𝑏 0
[ 6𝐿𝑏 0 0 2𝐿2𝑏 −6𝐿𝑏 0 0 4𝐿2𝑏 ]

𝑲𝒆𝟏 =

0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 𝐿2𝑏 0 0 0 −𝐿2𝑏 0
𝐸𝐼 0 0 0 𝐿2𝑏 0 0 0 −𝐿2𝑏
𝐿𝑏 (1 + ∅) 0
3 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 −𝐿2𝑏 0 0 0 𝐿2𝑏 0
[0 0 0 −𝐿2𝑏 0 0 0 𝐿2𝑏 ]

𝑲𝒆 = 𝑲𝒆𝟎 + ∅𝑲𝒆𝟏

Assembly of the element matrices into global matrix

The element mass, stiffness and damping matrices are assembled into the global matrix
following the routine introduced in [84]. The Matlab toolbox Rotor Software V1 are used as
reference. Figure C-1 gives the schematic illustration of the assembling process on global
stiffness matrix using a model with 4 elements (5 nodes) as an example. [𝐾𝑒𝑖 ] (i = 1,2,3,4) is
the element stiffness matrix as shown above. [𝐾𝑠 ] is the global stiffness matrix of the shaft
model. The process is the same for other element matrices.

223
APPENDIX C

Figure C-1 Assembling the element stiffness matrix to the global stiffness matrix of the shaft
model [84].

Assembling of the rotor bearing system:

CASE I: Linear rotor model with linear bearing model

The process of adding equivalent stiffness coefficients into the rotor bearing system of CASE
I is demonstrated in Figure C-2 and C-3 using the aforementioned 4 elements rotor model.
The linear bearing is assumed to be add to the location of Node 2 and Node 4 The process is
the same to the system damping matrix [𝐶𝑠𝑦𝑠_1 ].

224
APPENDIX C

Figure C-2 Bearing stiffness matrix used in the assembling of the rotor bearing system in
CASE I when linear bearing model is used

Figure C-3 The rotor bearing system stiffness matrix of CASE I after assembly the bearing
stiffness matrix

CASE II: Linear rotor model with linear bearing and viscoelastic support

225
APPENDIX C

The assembly process of CASE II was introduced in [84]. The overall process is similar to
CASE I. The bearing stiffness matrix (as well as the damping matrix) are first assembled into
the two sub systems: Sub-system 1 is rotor with linear bearing model; Sub-system 2 is linear
bearing model with model of bearing sleeve and linear O-ring model. The two sub-systems
are then assembled together as shown in Section 4.4.3.

226
APPENDIX D

APPENDIX D

a) Measurements at sensor position A

b) Predictions at sensor position A

22
APPENDIX D

c) Measurements at sensor position B

d) Prediction at sensor position B

Figure D-1 Unbalance responses of sensor positions A & B obtained at 50. k rp m in speed

228
APPENDIX D

a) Measurements at sensor position A

b) Predictions at sensor position A

229
APPENDIX D

c) Measurements at sensor position B

d) Predictions at sensor position B

Figure D-2 Unbalance responses of sensor positions A & B obtained at 60.2k rpm in speed

230
APPENDIX D

a) Measurements at sensor position A

b) Predictions at sensor position A

231
APPENDIX D

c) Measurements at sensor position B

d) Predictions at sensor position B

Figure D-3 Unbalance responses of sensor positions A & B obtained at 69.2k rpm in speed

232
APPENDIX D

a) Measurements at sensor position A

b) Predictions at sensor position A

233
APPENDIX D

c) Measurements at sensor position B

d) Predictions at sensor position B

Figure D-4 Unbalance response of sensor positions A & B obtained at 83.0k rpm in speed

234
APPENDIX D

a) Measurements at sensor position A

b) Predictions at sensor position A

235
APPENDIX D

c) Measurements at sensor position B

d) Predictions at sensor position B

Figure D-5 Unbalance responses of sensor positions A & B obtained at 89.8k rpm in speed

236
APPENDIX D

a) Measurements at sensor position A

b) Predictions at sensor position A

23
APPENDIX D

c) Measurements at sensor position B

d) Predictions at sensor position B

Figure D-6 Unbalance responses of sensor position A & B obtained at 100k rpm in speed

238
APPENDIX E

APPENDIX E

a) Measurements at sensor position A

b) Predictions at sensor position A

239
APPENDIX E

c) Measurement at sensor position B

d) Predictions at sensor position B

Figure E-1 Unbalance responses of sensor positions A & B obtained at 6 . 5k rpm in speed.
Supply pressure maintained at ba r.

240
APPENDIX E

a) Measurements at sensor position A

b) Predictions at sensor position A

c) Measurements at sensor position B

241
APPENDIX E

d) Predictions at sensor position B

Figure E-2 Unbalance responses of sensor positions A & B obtained at 8. 4k rpm in speed.
Supply pressure maintained at ba r.

a) Measurements at sensor position A

242
APPENDIX E

b) Predictions at sensor position A

c) Measurements at sensor position B

d) Predictions at sensor position B

243
APPENDIX E

Figure E-3 Unbalance responses of sensor positions A & B obtained at 90.0k rpm in speed.
Supply pressure maintained at ba r.

a) Measurements at sensor position A

b) Predictions at sensor position A

244
APPENDIX E

c) Measurements at sensor position B

d) Predictions at sensor position B

Figure E-4 Unbalance responses of sensor positions A & B obtained at 100.8k rpm in speed.
Supply pressure maintained at ba r.

a) Measurements at sensor position A

245
APPENDIX E

b) Predictions at sensor position A

c) Measurements at sensor position B

d) Predictions at sensor position B

Figure E-5 Unbalance responses of sensor positions A & B obtained at 111k rpm in speed.
Supply pressure maintained at ba r.

246
APPENDIX E

a) Measurements at sensor position A

b) Predictions at sensor position A

24
APPENDIX E

c) Measurements at sensor position B

d) Predictions at sensor position B

Figure E-6 Unbalance responses of sensor positions A & B obtained at 120.9k rpm in speed.
Supply pressure maintained at ba r.

248
APPENDIX F

APPENDIX F

a) Measurements at sensor position A

b) Predictions at sensor position A

249
APPENDIX F

c) Measurements at sensor position B

Figure F-1 Unbalance responses of sensor positions A & B obtained at 6 .2 k rpm in speed

a) Measurements at sensor position A

250
APPENDIX F

b) Predictions at sensor position A

c) Measurements at sensor position B

d) Predictions at sensor position B

Figure F-2 Unbalance responses of sensor positions A & B obtained at 80.4k rpm in speed

251
APPENDIX F

a) Measurements at sensor position A

b) Predictions at sensor position A

252
APPENDIX F

c) Measurements at sensor position B

d) Predictions at sensor position B

Figure F-3 Unbalance responses of sensor positions A & B obtained at 90k rpm in speed

253
APPENDIX F

a) Measurements at sensor position A

b) Predictions at sensor position A

c) Measurements at sensor position B

254
APPENDIX F

d) Predictions at sensor position B

Figure F-4 Unbalance responses of sensor positions A & B obtained at 98.4k rpm in speed

a) Measurements at sensor position A

255
APPENDIX F

b) Predictions at sensor position A

c) Measurements at sensor position B

d) Predictions at sensor position B

Figure F-5 Unbalance responses of sensor positions A & B obtained at 109.2k rpm in speed

256
APPENDIX F

a) Measurements at sensor position A

b) Predictions at sensor position A

25
APPENDIX F

c) Measurements at sensor position B

d) Predictions at sensor position B

Figure F-6 Unbalance responses of sensor positions A & B obtained at 120k rpm in speed

258
REFERENCES_______________________________________________________________

REFERENCES
[1] Pink E, Stout K. Design procedures for orifice compensated gas journal bearings based on
experimental data. Tribology International. 19 8; 11(1):63- 5.
[2] Wilcock DF, Carbino J. Design of gas bearings. LA: Mechanical Technology Incorporated;
19 2 .
[3] Pink E, An experimental investigation of externally pressurised gas journal bearings and
comparison with design method predictions. Proc. Gas Bear. Symp; 19 6.
[4] Ronze J, Weisstein EW. Separation of Variables: MathWorld--A Wolfram Web Resource.
Available from: http://mathworld.wolfram.com/SeparationofVariables.html.
[5] Whitaker S. Flow in porous media I: A theoretical derivation of Darcy's law. Transport in
porous media. 1986;1(1):3-25.
[6] Zhang X, Deeks LK, Bengough AG, Crawford JW, Young IM. Determination of soil
hydraulic conductivity with the lattice Boltzmann method and soil thin-section technique.
Journal of Hydrology. 2005;306(1):59- 0.
[ ] Pink E, Stout K. Orifice restrictor losses in journal bearings. Proceedings of the Institution
of Mechanical Engineers. 19 9;193(1) :4 -52.
[8] Miyatake M, Yoshimoto S. Numerical investigation of static and dynamic characteristics of
aerostatic thrust bearings with small feed holes. Tribology International. 2010;43(8):1353-9.
[9] Belforte G, Raparelli T, Trivella A, Viktorov V, Visconte C. CFD Analysis of a Simple
Orifice-Type Feeding System for Aerostatic Bearings. Tribology Letters. 2015;58(2):1-8.
[10] Belforte G, Trivella A, Raparelli T, Viktorov V. Identification of discharge coefficients of
orifice-type restrictors for aerostatic bearings and application examples: INTECH Open Access
Publisher; 2011.
[11] Neves M, Schwarz V, Menon G. Discharge coefficient influence on the performance of
aerostatic journal bearings. Tribology International. 2010;43(4): 46 -51.
[12] Wu D, Wang B, Luo X. Research on load characteristics of aerostatic journal bearing with
pocketed orifice-type restrictor. Automation and Computing (ICAC), 2014 20th International
Conference on; 2014: IEEE.
[13] Stout K, Pink E. Orifice compensated EP gas bearings: the significance of errors of
manufacture. Tribology International. 1980;13(3):105-11.

259
REFERENCES_______________________________________________________________

[14] Sneck HJ. A survey of gas-lubricated porous bearings. Journal of tribology.

1968;90(4):804-9.
[15] Majumdar B. Gas-lubricated porous bearings: a bibliography. Wear. 19 6;36(3) :269- 3.
[16] Mori H, Yabe H, Ono T. Theory of externally pressurized circular thrust porous gas bearing.
Journal of Fluids Engineering. 1965;8 (3) :613-20.
[1 ] Majumdar B. Analysis of externally pressurized porous gas journal bearings—I. Wear.
19 5;33(1) :25-35.
[18] Majumdar B. Analysis of externally pressurized porous gas journal bearings—II. Wear.
19 5;33(1) :3 -43.
[19] Rao N, Majumdar B. Dynamic characteristics of gas-lubricated externally pressurized
porous bearings with journal rotation II. Wear. 19 8;50(2) :201-10.
[20] Rao N, Majumdar B. Dynamic characteristics of gas-lubricated externally pressurized
porous bearings with journal rotation: I. Wear. 19 8;50(1) :59- 0.
[21] Rao N. Tilt stiffness and damping of externally pressurized porous gas journal bearings.
Wear. 19 8;4 (1) :31-44.
[22] Singh K, Rao N, Majumdar B. Effects of velocity slip, anisotropy and tilt on the steady
state performance of aerostatic porous annular thrust bearings. Wear. 1984;9 (1) :51-63.
[23] Kwan Y. Processing and fluid flow characteristics of hot isostatically pressed porous
alumina for aerostatic bearing applications. 1996.
[24] Kwan Y, Corbett J. A simplified method for the correction of velocity slip and inertia effects
in porous aerostatic thrust bearings. Tribology international. 1998;31(12): 9 -86.
[25] Fourka M, Bonis M. Comparison between externally pressurized gas thrust bearings with
different orifice and porous feeding systems. Wear. 199 ;210(1) :311- .
[26] Belforte G, Raparelli T, Viktorov V, Trivella A. Feeding system of aerostatic bearings with
porous media. ASME 8th Biennial Conference on Engineering Systems Design and Analysis;
2006: American Society of Mechanical Engineers.
[2 ] Togo S, Akiyama Y, Okano M. Experimental Study on Externally Pressurized Porous Gas
Bearings. Journal of Japan Society of Lubrication Engineers. 19 2;1 ( 6):360-&.
[28] Yoshimoto S, Tozuka H, Dambara S. Static characteristics of aerostatic porous journal
bearings with a surface-restricted layer. Proceedings of the Institution of Mechanical Engineers,
Part J: Journal of Engineering Tribology. 2003;21 (2) :125-32.

260
REFERENCES_______________________________________________________________

[29] Miyatake M, Yoshimoto S, Sato J. Whirling instability of a rotor supported by aerostatic

porous journal bearings with a surface-restricted layer. Proceedings of the Institution of
Mechanical Engineers, Part J: Journal of Engineering Tribology. 2006;220(2):95-103.
[30] Yoshimoto S, Kohno K. Static and dynamic characteristics of aerostatic circular porous
thrust bearings (effect of the shape of the air supply area). Journal of tribology.
2001;123(3):501-8.
[31] Otsu Y, Miyatake M, Yoshimoto S. Dynamic characteristics of aerostatic porous journal
bearings with a surface-restricted layer. Journal of tribology. 2011;133(1):011 01.
[32] Vohr JH, Chow CY. Characteristics of herringbone-grooved, gas-lubricated journal
bearings. Journal of Fluids Engineering. 1965;8 ( 3):568- 6.
[33] Malanoski S, Pan C. The static and dynamic characteristics of the spiral-grooved thrust
bearing. Journal of Fluids Engineering. 1965;8 (3 ):54 -55.
[34] Cunningham RE, Fleming DP, Anderson WJ. Experiments on the steady state
characteristics of herringbone-grooved air-lubricated journal bearings: National Aeronautics
and Space Administration; 1969.
[35] Bonneau D, Absi J. Analysis of aerodynamic journal bearings with small number of
herringbone grooves by finite element method. Journal of tribology. 1994;116(4):698- 04.
[36] Kobayashi T. Numerical analysis of herringbone-grooved gas-lubricated journal bearings
using a multigrid technique. Journal of tribology. 1999;121(1):148-56.
[3 ] Srivastava V, Tamsir M. Crank-Nicolson semi implicit approach for numerical solutions
of two-dimensional coupled nonlinear Burgers' equations. International Journal of Applied
Mechanics andEngineering. 2012;1 (2) :5 1.
[38] Gad A, Nemat-Alla M, Khalil A, Nasr A. On the optimum groove geometry for herringbone
grooved journal bearings. Journal of tribology. 2006;128(3):585-93.
[39] Ikeda S, Arakawa Y, Hishida N, Hirayama T, Matsuoka T, Yabe H. Herringbone‐grooved
bearing with non‐uniform grooves for high‐speed spindle. Lubrication Science.
2010;22(9):3 -92.
[40] Schiffmann J. Enhanced Groove Geometry for Herringbone Grooved Journal Bearings.
Journal of Engineering for Gas Turbines and Power. 2013;135(10):102501.
[41] Tomioka J, Miyanaga N. Stability Threshold of Herringbone Grooved Aerodynamic
Journal Bearings with External Stiffness and Damping Elements. Journal of Advanced
Mechanical Design, Systems, and Manufacturing. 2013; (6) :8 6 -8 .

261
REFERENCES_______________________________________________________________

[42] Miyanaga N, Tomioka J. Effect of support stiffness and damping on stability characteristics
of herringbone-grooved aerodynamic journal bearings mounted on viscoelastic supports.
Tribology International. 2016;100:195-203.
[43] Miyanaga N, Tomioka J. Effect of Dynamic Properties of Support O-Rings on Stability of
Herringbone-Grooved Aerodynamic Journal Bearings. Tribology Online. 2016;11(2):2 2 -80.
[44] Agrawal GL, Foil air/gas bearing technology—an overview. ASME 199 International Gas
Turbine and Aeroengine Congress and Exhibition; 199 : American Society of Mechanical
Engineers.
[45] Heshmat CA, Xu DS, Heshmat H. Analysis of gas lubricated foil thrust bearings using
coupled finite element and finite difference methods. Journal of tribology. 2000;122(1):199-
204.
[46] Peng Z, Khonsari M. Hydrodynamic analysis of compliant foil bearings with compressible
air flow. Transactions-American Society of Mechanical Engineers Journal of Tribology.
2004;126(3):542-6.
[4 ] San Andrés L, Rubio D, Kim TH. Rotordynamic performance of a rotor supported on bump
type foil gas bearings: experiments and predictions. Journal of Engineering for Gas Turbines
and Power. 200 ;129 (3):850- .
[48] San Andrés L, Kim TH. Thermohydrodynamic analysis of bump type gas foil bearings: a
model anchored to test data. Journal of Engineering for Gas Turbines and Power.
2010;132(4):042504.
[49] Wilkes JC, Wade J, Rimpel A, Moore J, Swanson E, Grieco J, et al. Impact of Bearing
Clearance on Measured Stiffness and Damping Coefficients and Thermal Performance of a
High-Stiffness Generation 3 Foil Journal Bearing. ASME Turbo Expo 2016: Turbomachinery
Technical Conference and Exposition; 2016: American Society of Mechanical Engineers.
[50] San Andrés L, Chirathadam TA. A metal mesh foil bearing and a bump-type foil bearing:
Comparison of performance for two similar size gas bearings. Journal of Engineering for Gas
Turbines and Power. 2012;134(10):102501.
[51] Feng K, Zhao X, Huo C, Zhang Z. Analysis of novel hybrid bump-metal mesh foil bearings.
Tribology International. 2016;103:529-39.
[52] Choe BS, Kim TH, Kim CH, Lee YB. Rotordynamic behavior of 225 kw (300 hp) class
pms motor–generator system supported by gas foil bearings. Journal of Engineering for Gas
Turbines and Power. 2015;13 (9) :092505.

262
REFERENCES_______________________________________________________________

[53] Lee Y-B, Kwon SB, Kim TH, Sim K. Feasibility study of an oil-free turbocharger
supported on gas foil bearings via on-road tests of a two-liter class diesel vehicle. Journal of
Engineering for Gas Turbines and Power. 2013;135(5):052 01.
[54] Lee Y-B, Park D-J, Sim K. Rotordynamic performance measurements of an oil-free
turbocharger supported on gas foil bearings and their comparisons to floating ring bearings.
International Journal of Fluid Machinery and Systems. 2015;8(1):23-35.
[55] Sim K, Lee Y-B, Kim TH. Effects of mechanical preload and bearing clearance on
rotordynamic performance of lobed gas foil bearings for oil-free turbochargers. Tribology
Transactions. 2013;56(2):224-35.
[56] Lee D-H, Kim Y-C, Kim K-W. The dynamic performance analysis of foil journal bearings
considering coulomb friction: rotating unbalance response. Tribology Transactions.
2009;52(2):146-56.
[5 ] Bhore SP, Darpe AK. Nonlinear dynamics of flexible rotor supported on the gas foil journal
bearings. Journal of Sound and Vibration. 2013;332(20):5135-50.
[58] Song J-h, Kim D. Foil gas bearing with compression springs: analyses and experiments.
Journal of tribology. 200 ;129(3) :628-39.
[59] Bonello P, Pham H. The efficient computation of the nonlinear dynamic response of a foil–
air bearing rotor system. Journal of Sound and Vibration. 2014;333(15):3459- 8.
[60] Bonello P, Pham H. Nonlinear dynamic analysis of high speed oil-free turbomachinery
with focus on stability and self-excited vibration. Journal of Tribology. 2014;136(4):041 05.
[61] Simmons J, Advani S. Paper II (iii) Michell and the development of tilting pad bearings.
Tribology Series. 198 ; 11:49-56.
[62] Dimond T, Younan A, Allaire P. A review of tilting pad bearing theory. International Journal
of Rotating Machinery. 2011;2011.
[63] Wilkes JC. Measured and Predicted Transfer Functions Between Rotor Motion and Pad
Motion for a Rocker-Back Tilting-Pad Bearing in LOP Configuration. ASME 2011 Turbo Expo:
Turbine Technical Conference and Exposition; 2011: American Society of Mechanical
Engineers.
[64] Lund J. On the Hybrid Gas Lubricated Journal Bearing. DTIC Document, 1962.
[65] Kim D, Park S. Hybrid air foil bearings with external pressurization. ASME 2006
International Mechanical Engineering Congress and Exposition; 2006: American Society of
Mechanical Engineers.

263
REFERENCES_______________________________________________________________

[66] Wang YP, Kim D. Experimental Identification of Force Coefficients of Large Hybrid Air
Foil Bearings. Journal of Engineering for Gas Turbines and Power. 2014;136(3):032503.
[6 ] Ives D, Rowe W. The performance of hybrid journal bearings in the superlaminar flow
regimes. Tribology Transactions. 1992;35(4):62 -34.
[68] Osborne DA, San Andrés L. Experimental Response of Simple Gas Hybrid Bearings for
Oil-Free Turbomachinery. Journal of Engineering for Gas Turbines and Power.
2006;128(3):626-33.
[69] Wilde DA, San Andrés L. Comparison of Rotordynamic Analysis Predictions with the test
response of simple gas hybrid Bearings for oil free Turbomachinery. ASME Turbo Expo 2003,
collocated with the 2003 International Joint Power Generation Conference; 2003: American
Society of Mechanical Engineers.
[ 0 ] Zhu X, San Andrés L. Rotordynamic performance of flexure pivot hydrostatic gas bearings
for oil-free turbomachinery. Journal of Engineering for Gas Turbines and Power.
200 ;129(4) :1020- .
[ 1 ] San Andrés L, Ryu K. Hybrid gas bearings with controlled supply pressure to eliminate
rotor vibrations while crossing system critical speeds. Journal of Engineering for Gas Turbines
and Power. 2008;130(6):062505.
[ 2 ] Stanev P, Wardle F, Corbett J. Investigation of grooved hybrid air bearing performance.
Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body
Dynamics. 2004;218(2):95-106.
[ 3 ] Zhang X-Q, Wang X-L, Liu R, Zhang Y-Y. Modeling and Analysis of Micro Hybrid Gas
Spiral-Grooved Thrust Bearing for Microengine. Journal of Engineering for Gas Turbines and
Power. 2013;135(12):122508.
[ 4 ] Ertas BH. Compliant hybrid journal bearings using integral wire mesh dampers. Journal
of Engineering for Gas Turbines and Power. 2009;131(2):022503.
[ 5 ] Delgado A. Experimental identification of dynamic force coefficients for a 110 MM
compliantly damped hybrid gas bearing. Journal of Engineering for Gas Turbines and Power.
2015;13 ( ) :0 2502.
[ 6 ] Waumans T, Peirs J, Reynaerts D, Al-Bender F. On the dynamic stability of high-speed gas
bearings: stability study and experimental validation. Sustainable Construction and Design.
2011;2(2):342.

264
REFERENCES_______________________________________________________________

[ ] Wang N, Chang C. An application of Newton's method to the lubrication analysis of air-

lubricated bearings. Tribology Transactions. 1999;42(2):419-24.
[ 8 ] Wang N, Chang S-H, Huang H-C. Comparison of Iterative Methods for the Solution of
Compressible-Fluid Reynolds Equation. Journal of tribology. 2011;133(2):021 02.
[ 9 ] Wang N, Chang S-H, Huang H-C. Stopping Criterion in Iterative Solution Methods for
Reynolds Equations. Tribology Transactions. 2010;53(5): 39 -4 .
[80] Faria MTC. Performance Characteristics Curves for Gas Lubricated Herringbone Groove
Journal Bearings. Proceedings of the COBEM 2005: 18 th International Congress of
Mechanical Engineering; 2005.
[81] Arghir M, Alsayed A, Nicolas D. The finite volume solution of the Reynolds equation of
lubrication with film discontinuities. International journal of mechanical sciences.
2002;44(10):2119-32.
[82] Bazoune A, Khulief Y, Stephen N. Shape functions of three-dimensional Timoshenko beam
element. Journal of Sound and Vibration. 2003;259(2):4 3 -80.
[83] Chen W, Gunter E. Instruction to Dynamics of Rotor–Bearing System. Victoria, BA,
Canada, Trafford Publishing. 2005.
[84] Cao J. Transient Analysis of Flexible Rotors with Nonlinear Bearings, Dampers and
External Forces [Doctoral dissertation]: University of Virginia; 2012.
[85] Tian L, Wang W. Nonlinear dynamics of rotor bearing systems in automotive turbochargers:
LAP LAMBERT Academic Publishing; 2013.
[86] Miyanaga N, Tomioka J. Stability Analysis of Herringbone-Grooved Aerodynamic Journal
Bearings for Ultra High-Speed Rotations. International Journal of Materials, Mechanics and
Manufacturing. 2016;4(3):156-61.
[8 ] Quarteroni A, Sacco R, Saleri F. Numerical mathematics: Springer Science & Business
Media; 2010 Nov 30.
[88] Hamrock BJ, Schmid SR, Jacobson BO. Fundamentals of fluid film lubrication: CRC press;
2004.
[89] Lee Y-B, Kwak H-D, Kim C-H, Lee N-S. Numerical prediction of slip flow effect on gas-
lubricated journal bearings for MEMS/MST-based micro-rotating machinery. Tribology
international. 2005;38(2):89-96.
[90] Kincaid DR, Cheney EW. Numerical analysis: mathematics of scientific computing. 2 ed:
American Mathematical Soc.; 2002.

265
REFERENCES_______________________________________________________________

[91] Simoncini V, Szyld DB. Recent computational developments in Krylov subspace methods
for linear systems. Numerical Linear Algebra with Applications. 200 ; 14(1):1-59.
[92] Lo C-Y, Wang C-C, Lee Y-H. Performance analysis of high-speed spindle aerostatic
bearings. Tribology International. 2005;38(1):5-14.
[93] Cunningham RE, Fleming DP, Anderson WJ. Experimental Load Capacity and Power Loss
of Herringbone Grooved Gas Lubricated Journal Bearings. Journal of tribology.
19 1;93(3) :415-22.
[94] Lund J. Review of the concept of dynamic coefficients for fluid film journal bearings.
ASME J Tribol. 198 ; 109(1):3 -41.
[95] Vohr J. A study of inherent restrictor characteristics for hydrostatic gas bearings.
Proceedings of Fourth Gas Bearing Symposium: University of Southampton; 1969.
[96] Holmes R, Brennan M, Gottrand B. Vibration of an automotive turbocharger–a case study.
Proceedings 8th International Conference on Vibrations in Rotating Machinery; 2004.
[9 ] Belforte G, Raparelli T, Viktorov V, Colombo F. A Theoretical Study of a High Speed Rotor
Supported by Air Bearings Mounted on O-Rings. Fifth International Conference on Tribology,
AITC-AIT 2006; 2006.
[98] Swanson E, Powell CD, Weissman S. A practical review of rotating machinery critical
speeds and modes. Sound and vibration. 2005;39(5):16- .
[99] Chu L, Li W-L, Shen R, Tsai T. Dynamic characteristics of grooved air bearings in
microsystems. Proceedings of the Institution of Mechanical Engineers, Part J: Journal of
Engineering Tribology. 2009;223(6):895-908.
[100] Faria MTC. Some performance characteristics of high speed gas lubricated herringbone
groove journal bearings. JSME International Journal Series C Mechanical Systems, Machine
Elements and Manufacturing. 2001;44(3): 5 -81.

266