DESIGN AND DEVELOPMENT OF MAGNETIC- HYDRODYNAMIC HYBRID JOURNAL BEARING
K. P. LIJESH
DEPARTMENT OF MECHANICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY DELHI
DECEMBER 2015
© Indian Institute of Technology Delhi (IITD), New Delhi, 2015
DESIGN AND DEVELOPMENT OF MAGNETIC-HYDRODYNAMIC HYBRID
JOURNAL BEARING
by
K. P. LIJESH
Department of Mechanical Engineering
Submitted
in fulfilment of the requirements of the degree of Doctor of Philosophy
to the
Indian Institute of Technology Delhi
December 2015
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Certificate
This is to certify that the Doctoral Thesis titled “Design and Development of Magnetic- Hydrodynamic Hybrid Journal Bearing” being submitted by Mr. K. P. Lijesh to Indian Institute of Technology Delhi, New Delhi, India, in fulfilment for the requirement of the degree of Doctor of Philosophy, is a bona fide record of research work carried out by him under my guidance and supervision.
To the best of my knowledge, the results embodied in this research work have not been submitted in the part or full to the other University or Institute for the award of any degree or diploma.
Dr. Harish Hirani Professor Department of Mechanical Engineering
Indian Institute of Technology Delhi India Date:
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Acknowledgements
I express my deep sense of gratitude to my Ph. D. supervisor, Prof. Harish Hirani (Mechanical Engineering Department, IIT Delhi) for his encouragement, motivation, guidance and constant support throughout the research work.
I am thankful to the Chairman and Members of the Student Research Committee, Prof. J.
K. Dutt, Prof. S. P. Singh and Prof. N. Tondon for their interest, guidance and valuable suggestions. I am thankful to Prof. S. K. Saha, Head of Mechanical Engineering Department, IIT Delhi for providing me the necessary facilities.
I would like to thank Mr. Rajinder Singh (Turbomachinery lab), Mr. K. N. Madana Sundaran (Vibration and Research lab), and Mr. S. Babu (Vibration and Instrumentation lab) for their helps and support to conduct experiments. I am very much thankful to Dr. S.M. Muzakkir, Mr. Venkat Vinoth Chandhran K for their constant support throughout my research. I am grateful to my friends and co-workers, Dr. Chiranjit Sarkar, Mr. Kamal, Mr. Pawan for their cooperation.
Finally, I express a debt of gratitude to my parents for loving inspirations.
K. P. Lijesh
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Abstract
Journal bearings are known for their low friction and negligible wear characteristics under normal operating conditions. A well designed journal bearing is able to tolerate some misalignment, some fluctuations in speed, load, contamination of lubricant by moisture, wear debris and unexpected dirt. However, their operation shifts to boundary lubrication regime when subjected to severe operating conditions like heavy load and slow speed. In boundary lubrication, high compressive stresses on the asperities are induced at the contact zones which results in wear of the bearing during operation. The wear of the bearing results in the performance degradation of the machines resulting in reduced life of machinery. Therefore, for satisfactory and longer service life of bearing, the wear of the bearing needs to be prevented or minimized.
A review of the literature has revealed that the existing theoretical models are unable to predict the wear of the bearing considering all significant influencing parameters including the elastic and plastic deformation of asperities. Therefore, in the present research work, a semi- theoretical model has been developed, that simulates the wear of the bearing operating under boundary lubrication regime. The proposed semi-theoretical model estimates the wear of the bearing by considering the actual bearing surface profile. The wear constant to estimate wear is determined experimentally by performing experiment on conformal block on disc that simulates the journal bearing operation. The validation of the semi-theoretical model is carried out by experimental investigations and its suitability in predicting wear under boundary lubrication conditions is established. It was established that wear is a localized phenomenon and the geometric deviations at the contact zone significantly affects the bearing performance. Therefore, to
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minimize the wear of bearing, the cylindricity at the contact zone must be low. However, manufacturing of such bearing is costly.
An effort is made in the present research work to present a novel design of a bearing, involving hybridization of different technologies, which is able to successfully prevent or minimize the wear of the bearing while operating in the boundary lubrication regime. Based on the detailed literature review of various hybridized technologies, the hybridization of hydrodynamic bearing with Passive Magnetic Bearing (PMB) was identified to be most cost effective. However, due to very low load carrying capacity it required many design challenges to be able to make this hybridization successful.
In the present research work, a simpler, less complex, faster and reasonably accurate method to estimate the load carrying capacity and stiffness of a full ring axially polarized PMB is proposed. To this end, simple analytical equations have been developed by considering all the design variables. The developed equations are validated against the results obtained using numerical methods and experiments. The bearing optimization, considering minimization of magnet volume as the objective function, has been carried out to demonstrate the accuracy and usefulness of the developed equations.
The experiments were performed on hybrid bearing to study its performance under severe operating conditions. It was experimentally determined that the failure of hybrid bearings occurs due to the brittleness of high strength neodymium magnet and insufficient support load by magnets. The isolation of magnets by using rubber was shown to reduce the failure of magnet due to its brittle nature. A novel structure of PMB consisting of small sized square magnets inserted into aluminum structure has been proposed to enhance the structural damping of high strength
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neodymium magnets. A theoretical model to simulate the cuboid magnets in form of sector magnets has been developed and validated experimentally. A Rotation Magnetized Direction (RMD) configuration is designed and developed for magnetic bearing to achieve enhanced load carrying capacity. A theoretical model is developed for RMD configuration by considering principle of superposition.
On the basis of theoretical and experimental studies, feasibility of each approach is critically examined and a novel design of a hybrid bearing is presented. Its performance is validated theoretically and experimentally.
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Contents
Certificate ... ii
Acknowledgements ... iii
Abstract ... iv
Contents ... vii
List of Tables ... xxi
Nomenclature ... xxiii
CHAPTER 1 INTRODUCTION ... 1
1.1 Background and motivation ... 1
1.2 Objectives of the work ... 7
1.3 Organization of the thesis ... 8
1.4 Brief summary of research contributions ... 10
CHAPTER 2 LITERATURE REVIEW ... 12
2.1 Boundary Lubrication ... 13
2.2 Boundary Lubrication Models ... 15
2.3 Wear Model ... 17
2.4 Geometrical parameters affecting bearing performance ... 22
2.5 Hybridization of Bearing Technology ... 25
2.5.1 Hydrodynamic with Hydrostatic Bearing ... 26
2.5.2 Hydrodynamic with Magnetic Bearings ... 27
2.6 Enhancement of Load and Stiffness of PMB ... 35
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2.7 Enhancement of Damping of PMB ... 38
2.8 Epilogue ... 39
CHAPTER 3 STUDY OF GEOMETRICAL PARAMETERS OF JOURNAL BEARING 41 3.1 Introduction ... 41
3.2 Development of the theoretical model ... 43
3.2.1 Acquire 3D profile of the bearing ... 45
Develop the acquired bearing profile with respect to journal surface ... 45
3.2.2 Calculation of asperities radii (Rasp) ... 46
3.2.3 Assumption of initial eccentricity, Number of cycles=N ... 46
3.2.4 Estimation of the film thickness at each node ... 47
3.2.5 Calculation of at every node ... 47
3.2.6 Estimation of load when the journal bearing operates in the hydrodynamic regime ... 48
3.2.7 Estimation of load when the journal bearing operates in EHL regime ... 49
3.2.8 Load taken by the Asperities ... 50
3.2.9 Estimation of wear coefficient (K) ... 51
3.2.10 Wear Model ... 53
3.2.11 Increment of eccentricity: ... 54
3.2.12 Completion of Loop ... 54
3.3 Results and Discussion ... 54
3.4 Epilogue ... 60
CHAPTER 4 DEVELOPMENT OF ANALYTICAL EXPRESSION FOR MAGNETIC BEARING ... 61
4.1 Introduction ... 61
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4.2 Identification of design variables ... 63
4.3 Results and discussion ... 66
4.4 Analytical expressions for force and stiffness ... 67
4.5 Determination of corrective coefficients ... 68
4.6 Validation of proposed analytical expressions ... 74
4.7 Optimization ... 76
4.8 Epilogue ... 81
CHAPTER 5 DESIGN AND DEVELOPMENT OF BEARING SETUP... 82
5.1 Design and Development of Experimental Setup for PMB ... 82
5.1.1 Motor and Drive ... 83
5.1.2 Bearing ... 84
5.1.3 Loading Arrangement ... 85
5.1.4 Sensors ... 87
5.1.4.1Eddy current Proximity Sensors ... 87
5.1.4.2 Accelerometer ... 88
5.1.5 Data Acquisition System (DAQ) ... 89
5.1.6 Developed Experimental Setup for PMB for Static Performance Evaluation ... 89
5.2 Experimental Setup for Hybrid Bearing: ... 94
5.3 Epilogue ... 96
CHAPTER 6 DEVELOPMENT OF HYBRID BEARING ... 97
6.1 Experiment on Hybrid Bearing ... 97
6.1.1 Wear ... 99
6.1.2 New Pole formation ... 100
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6.1.3 Magnetic Flux Density ... 100
6.1.4 Crack Formation ... 102
6.2 Magnet with Cracks ... 104
6.3 Isolation using Rubber ... 110
6.3.1 Selection Procedure for Rubber as Isolator ... 111
6.3.2 Pendulum Tests ... 111
6.3.3 Hysteresis test ... 114
6.3.4 Hammer tests ... 115
6.3.5 Compression Test ... 117
6.4 Experiment on PMB with Selected Rubber ... 118
6.5 Effect of Young’s modulus of butyl rubber on hybrid bearing ... 122
6.6 Modelling and Simulation of Radial and Axial Load Carrying Capacities ... 129
6.7 Experimentation on Bearing with Proposed Structure ... 132
6.8 Mathematical Formulation of Proposed Bearing ... 134
6.9 Magnets Stacking ... 145
6.10 Epilogue ... 146
CHAPTER 7 HYBRID BEARING USING RMD CONFIGURATION ... 148
7.1 Introduction ... 148
7.2 Mathematical Modelling of RMD Configured PMB ... 149
7.3 Development of RMD configuration ... 152
CHAPTER 8 CONCLUSIONS ... 163
SCOPE FOR FUTURE WORK ... 165
Publications ... 175
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Publications in International Journals: ... 175
Patent ... 176
Publications in Conferences ... 176
Under review ... 177
Brief Bio-Data of Author ... 178
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List of Figures
Figure1.1 Sugar mill bearing, (a) Bearing, (b) Journal ... 2
Figure 1.2 Failed sugar mill bearing ... 3
Figure 1.3 Surface profile in angular direction at different axial length, (a)Axial length=0.01m, (b) Axial length=0.015m, (c) Axial length=0.020m ... 4
Figure 1.4 Journal bearing with wear near oil hole... 5
Figure 1.5 Failed Hybrid bearing ... 6
Figure 1.6 Arrangement of magnets to increase the load carrying capacity [Yonnet et al (1991)]7 Figure 2.1 Different lubrication regime, (a) Hydrodynamic, (b) Mixed, (c) Boundary ... 12
Figure 2.2 Roughness with circular and non-circular profile, (a) Circular profile, (b) Non circular profile ... 16
Figure 2.3 Rnning-in and steady state wear, (a) Specific wear, (b) Cummulative wear ... 17
Figure 2.4 Stages in wear Process, (a) Journal separated from Bearing, (b) Journal in partial contact with Bearing, (c) Smoothening of bearing surface by Journal ... 18
Figure 2.5 Geometry of a Worn Journal Bearing ... 20
Figure 2.6 Worn out bearing [Dufrane et al (1983)], (a) Case 1, (b) Case 2,(c) Case 3 ... 21
Figure 2.7 Circularity of the bearing, (a) Front view, (b) Cross sectional Side view ... 23
Figure 2.8 Bearing profiles considered by Vijayaraghavan et al (1991), (a) Elliptical, (b) Semi elliptical, (c) Three lobed epicycloids, (d) circular profile ... 24
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Figure 2.9 Cylindricity of the bearing, (a) Front view, (b) Cross sectional Side view ... 25
Figure 2.10 Hybrid Hydrodynamic + Hydrostatic bearing (Santos and Russo (1998)) ... 26
Figure 2.11 Schematic representation of an AMB ... 28
Figure 2.12 Base motion, (a) Vertical motion, (b) Horizontal motion, (c) Tipping motion, (d) Non- uniform motion ... 31
Figure 2.13 Backup bearing ... 31
Figure 2.14 Different configuration of hybrid bearing, (a) Tan et al (2002), (b) Hirani and Samanta (2007), (c) Muzakkir et al (2013) ... 34
Figure 2.15 (a) Halbach arrangement of Magnets with direction of polarization, (b) Magnetic field generated by radial magnets, (c) Magnetic field generated by axial magnets (d) Resultant Magnetic ... 35
Figure 2.16 Arrangements of magnets to enhance the load carrying capacity [Yonnet et al (1991)], (a) Back to back arrangement, (b) RMD configuration ... 36
Figure 2.17 Bearing for increasing the Stiffness by stacking 40 magnets [Moser et al (2006)] . 37 Figure 2.18 Magnets Arrangement with Damping Imlach (2002)... 38
Figure 3.1 Talyrond Machine and zoomed view, (a)Talyrond machine, (b) Zoomed view ... 42
Figure 3.2 Surface profile of the bearing ... 42
Figure 3.3 Flow Chart of the mathematical model ... 70
Figure 3.4 Schematic diagram of bearing profile with respect to Journal Surface ... 45
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Figure 3.5 Asperities model, (a) Asperities, (b) Enlarged view of one asperity ... 46
Figure 3.6 Conformal block on disk setup, (a) Front View of test setup, (b) Conformal block and disk ... 52
Figure 3.7 Conformal block ... 52
Figure 3.8 Wear vs sliding distance plot of two conformal block on disk ... 53
Figure 3.9 Tansient wear ... 55
Figure 3.10 Worn-out bearing surface profile (after 4500 rotations of the journal) ... 56
Figure 3.11 300 Profiles of generated journal ... 84
Figure 4.1 Radial Magnetic bearing, (a) Front View, (b) Sectional side view ... 61
Figure 4.2 Percentage mean square values, (a) Force, (b) Stiffness ... 66
Figure 4.3 Comparison of numerical and analytical equations for ɛ ... 68
Figure 4.4 Comparison of numerical and analytical equations for β=C (1-0.2ɛ2) ... 70
Figure 4.5 Comparison of numerical and analytical methods for Rm, (a) Mean value, (b) Lowest value ... 70
Figure 4.6 Comparison of numerical and analytical equation for α, (a) Comparison for α, (b) Comparison for ... 72
Figure 4.7 Comparison of numerical and proposed equations for b, (a) Numerical approach, (b) Numerical and analytical approach ... 73
Figure 4.8 Force vs L for dimensions provided in Tan et al (2002) ... 79
1 00111 5.4
0.4 . T'
0.9T'
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Figure 4.9 Force vs L for dimensions provided in Muzakkir et al (2014) ... 80
Figure 4.10 Force vs L for bearing described in Samanta and Hirani (2007) ... 80
Figure 5.1 Experimental setup for PMB ... 82
Figure 5.2 Motor and drive, (a) 3 Phase induction motor (b) Frequency drive ... 83
Figure 5.3 Passive Magnetic Bearing ... 84
Figure 5.4 Stator magnet arrangement, (a) Full ring, (b) half ring, (c) sector magnets ... 85
Figure 5.5 Loading arrangements, (a) Static, (b) Disc ... 85
Figure 5.6 Base plate, (a) Isometric view of base plate, (b) Top View, (c) Side view ... 86
Figure 5.7 Proximity sensor and proximitor ... 88
Figure 5.8 Accelerometer ... 88
Figure 5.9 DAQ system, (a) cRio 9202 (b) Voltage Module, (c) DAQ for acelertometer ... 89
Figure 5.10 Static loading magnetic bearing setup, (a) View 1, (b) View 2... 90
Figure 5.11 Determination of deflection at bearing end ... 91
Figure 5.12 Magnetic bearing setup for dynamic loading ... 92
Figure 5.13 Vertical and Horizontal signals for full ring stator, (a)Vertical, (b) Horizontal, (c) Orbit plot ... 93
Figure 5.14 Vertical acceleration signal comparison for PMB ... 94
Figure 5.15 Hybrid bearing experimental test setup ... 94
Figure 5.16 Tank and lubricant ... 95
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Figure 5.17 Hybrid Bearing Test setup ... 95
Figure 5.18 Journal bearing ... 96
Figure 6.1 Repulsive force for half ring stator magnet ... 98
Figure 6.2 Acceleration Vs time plot in vertical direction, (a) Start of experiment, (b) End of experiment... 98
Figure 6.3 Wear of rotor and stator magnets, (a) Stator Magnet, (b) Rotor magnets ... 99
Figure 6.4 Pole formation in stator, (a) Front view, (b) Rotor magnets ... 100
Figure 6.5 Pole Finder Sheet used on a rotor magnet ... 100
Figure 6.6 Magnetic flux density (T) of rotor before and after test ... 101
Figure 6.7 Crack in stator magnet, (a) Front view, (b) Top view ... 102
Figure 6.8 Acceleration plot for various failure of hybrid bearing, (a) Good hybrid bearing, (b) Wear in stator, (c) Wear in rotor, (d) Demagnetized rotor, (e) New pole formation in stator, (f) Crack in stator ... 104
Figure 6.9 Magnet considered for Magnetic field analysis, (a) Virgin magnet, (b) Magnet with crack ... 105
Figure 6.10 Mesh of magnet, (a) Increase in number of elements with one pass, (b) Magnet with mesh ... 105
Figure 6.11 Mid Plane for determination of magnetic field ... 106
Figure 6.12 Magnetic field plot for a good magnet, (a) Magnitude of magnetic field, (b) Magnetic field vector plot ... 107
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Figure 6.13 Magnetic field plot for 1/4th thickness crack of the magnet, (a) Magnitude of magnetic
field, (b) Magnetic field vector plot, (c) Polarization of magnet ... 107
Figure 6.14 Magnetic field plot for 1/2 thickness crack of the magnet, (a) Magnitude of magnetic field, (b) Magnetic field vector plot ... 108
Figure 6.15 Magnetic field plot for 3/4th thickness crack of the magnet, (a) Magnitude of magnetic field, (b) Magnetic field vector plot ... 108
Figure 6.16 Magnetic field plot for through thickness crack in the magnet, (a) Magnitude of magnetic field, (b) Magnetic field vector plot ... 108
Figure 6.17 Variation of magnetic field for different width cracks, (a) Depth of crack, (b) Width of crack... 110
Figure 6.18 Experimental Setup for pendulum tests ... 111
Figure 6.19 Pendulum test for different rubbers, (a) Natural rubber, (b) Nitrile Butadine Rubber, (c) Hypalon rubber, (d) Butyl rubber ... 113
Figure 6.20 Experimental Setup for hysteresis tests ... 114
Figure 6.21 Hysteresis plot for different extension, (a) Butyl rubber, (b) Hypalon rubber ... 114
Figure 6.22 Damping test for rubber ... 116
Figure 6.23 Damping ratio for magnet and different thickness of the rubber, (a) Magnet, (b) Rubber of thickness 1mm, (c) Rubber of thickness 1.5mm, (d) Rubber of thickness 2 mm ... 117
Figure 6.24 Force vs. deflection for different thickness rubber ... 118
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Figure 6.25 Acceleration and displacement signals of magnet bearing, (a) Acceleration signal in horizontal direction, (b) Acceleration signal in vertical direction, (c) Displacement signal in
horizontal direction, (d) Displacement signal in vertical direction... 119
Figure 6.26 Orbit plot of magnetic bearing ... 120
Figure 6.27 Acceleration at different frequency for magnetic bearing ... 121
Figure 6.28 Hybrid bearing ... 121
Figure 6.29 Acceleration at different frequency for hybrid bearing, (a) Vertical Acceleration (b) Horizontal Acceleration ... 121
Figure 6.30 Acceleration at different frequency for hybrid bearing, (a) Vertical Acceleration (b) Horizontal Acceleration ... 123
Figure 6.31 Acceleration amplitude at different operating frequency ... 124
Figure 6.32 Full ring magnet divided into 12 sectors with Magnetic remanence value (T), (a)Configuration 1, (b) Configuration 2, (c) Configuration ... 125
Figure 6.33 Sector magnet ... 126
Figure 6.34 Radial magnetic Field value for different configuration in (kA/m), (a)Configuration, (b) Configuration 2, (c) Configuration... 127
Figure 6.35 Axial magnetic Field value for different configuration in (kA/m) ... 128
Figure 6.36 Vertical force for different configurations, (a) Vertical Force, (b) Explanation of force variation ... 130
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Figure 6.37 Axial Magnetic forces for different configurations, (a) Axial force, (b) Description of
force variation ... 131
Figure 6.38 Aluminum structure with slots for sector/square magnets, (a) Aluminum structure with sector slots, (b) Aluminum structure with sector slots ... 133
Figure 6.39 Sector magnets, (a) Isometric view, (b) Front view ... 135
Figure 6.40 Cuboidal magnets, (a) Isometric view, (b) Front view ... 135
Figure 6.41 Comparison of vertical forces between square and sector magnets ... 138
Figure 6.42 Proposed magnetic bearing, (a) Proposed Structure without square magnets, (b) Proposed structure with square magnets ... 139
Figure 6.43 Full ring and proposed bearing (a) Full ring magnet, (b) Proposed structure with square magnets ... 140
Figure 6.44 Vertical and Horizontal signals for full ring stator, (a) Bearing with full ring stator (b) Bearing with proposed stator ... 141
Figure 6.45 Vertical acceleration signal comparison for full ring and proposed magnetic bearing, Full Ring Magnetic Bearing, (b) Proposed Magnetic Bearing ... 143
Figure 6.46 Amplitude of acceleration of full ring ... 144
Figure 6.47 Full ring bearing with crack ... 144
Figure 6.48 Proposed RMD bearing ... 144
Figure 6.49 Back to Back stacking arrangement, (a) Schematic arrangement, (b) developed stator ... 145
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Figure 6.50 Magnetic flux density plot, (a) Gauss meter, (b) Magnetic field plot ... 146
Figure 7.1 RMD configuration... 148
Figure 7.2 Configuration 3: RMD configuration, (a) Front View, (b) Side view ... 149
Figure 7.3 The radial and perpendicular polarized magnetic bearings ... 150
Figure 7.4 Comparison of RMD configured and full ring PMB ... 152
Figure 7.5 Rotor and Stator for RMD configuration, (a) Radial polarized assembly, (b) Rotor, (c) Stator Figure 7.6 RMD configuration... 153
Figure 7.7 Magnetic flux density plot ... 154
Figure 7.8 Theoretical and experimental comparison of RMD and full ring PMB ... 154
Figure 7.9 Hybrid bearing stator ... 156
Figure 7.10 Proposed hybrid bearing ... 157
Figure 7.11 Different configurations of magnetic bearing considered for present case, (a) Configuration 1, (b) Configuration 2, (c) Configuration 3, (d) Configuration 4 ... 158
Figure 7.12 Rotor for RMD ... 158
Figure 7.13 Hybrid bearing, (a) Bearing and magnets, (b) Assembled hybrid bearing ... 159
Figure 7.14 Bearings, (a) Bearing 1, (b) Bearing 3, (c) Bearing 4, (d) Bearing 2, (e) Bearing 5, (f) Bearing 6 ... 160
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List of Tables
Table 3.1 Maximum pressure value for different convergence values ... 49
Table 3.2 Parameters used in mathematical model ... 55
Table 3.3 Result of theoretical model ... 59
Table 4.1 Range of variables ... 64
Table 4.2 Range, factors and levels of the variables ... 64
Table 4.3 Variables values considered for sensitivity ... 65
Table 5.1 Specification of Eddy current sensors ... 88
Table 5.2 Parameter of the proposed magnetic bearing ... 90
Table 5.3 Theoretical and experimental comparison of load for PMB ... 91
Table 6.1 Magnetic flux density of rotor surface before and after test ... 101
Table 6.2 Pendulum test results for different rubbers ... 113
Table 6.3 Area of hysteresis curve for different elongation for Hypalon and Butyl rubber ... 115
Table 6.4 Damping ratio for different thickness of rubber ... 117
Table 6.5 Modulus change for different elongation for butyl in different liquid ... 122
Table 6.6 Parameter of the rotor considered for present work ... 130
Table 6.7 Parameter of the proposed magnetic bearing ... 139
Table 6.8 Theoretical and experimental comparison of load for the full ring and proposed bearing ... 140
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Table 6.9 Comparison of acceleration signal for the full ring and proposed bearing ... 142 Table 7.1 Diameter of fabricated Bearings ... 160 Table 7.2 Diameter of fabricated Bearings ... 161
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Nomenclature
1
Br Magnetic Remanence, Tesla
2
Br Magnetic Remanence, Tesla
C Radial clearance, m
Db Bearing diameter, m
Dj Journal diameter, m
e Eccentricity Ratio
Fcalculated Calculated value of F-statistic Fcritical Tabulated value of F-statistic
Fehl Load shared by elastohydrodynamic lubrication, N Felas Load shared by asperities in elastic deformation, N Fh Load shared by hydrodynamic lubrication, N Fplas Load shared by asperities in plastic deformation, N
Force component in θ direction, N Force component in radial direction, N
E1, E2 Young’s Modulus of journal and bearing material respectively, GPa
h Film thickness, m
h0 Minimum film thickness, m
H Hardness of journal
k Wear coefficient, m2/N
L Bearing length, m
l Length of rotor magnet (m)
n, m Number of nodes in axial and angular direction
F
Fr
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N Journal speed, rpm
Non dimensional nodal pressure
PL Projected load, N
Rq,bearing RMS of the bearing surface roughness, μm Rq, journal RMS of the journal surface roughness, μm R1 Inner radius of rotor magnet, mm
R2 Outer radius of rotor magnet, mm R3 Inner radius of stator magnet, mm R4 Outer radius of stator magnet, mm
s, S Semi-perimeter of triangle, Sommerfeld Number
t Time, second
T Sliding time, second
TN Time segments
U Sliding velocity, m/s
V Wear volume, m3
w Depth of wear, m
Non-dimensional incremental wear
W Load, N
Ys Yield strength of bearing material, GPa
z Coordinate in Z-direction
Node length in axial direction, L/n
Angular variable of rotor, Radian
Angular variable of stator, Radian
j
pi,
w
z
xxv Node length in angular direction
Elastic deformation during point contact, m Non-dimensional elastic deformation Elastic deformation of asperities, m
δc Critical deformation, m
Dynamic viscosity, Pa.s Specific film thickness
0 Permeability of free space, H/m
σ Standard deviation of the surface roughness (m)
v1, v2 Poisson’s ratio of journal and bearing material respectively Number less than 1, for convergence in Newton Raphson method
e
