Journal Bearing Lubrication Manoj Kumar Singh Thesis

“Influence of stochastic and deterministic surface
roughness on the performance of circular and non-
circular fluid film journal bearings”
A thesis
Submitted in fulfillment of the
Requirements for the award of the degree
Of
Master of Technology
In
Mechanical Engineering (CAD/CAM)
By
Manoj Singh
Roll No: 09M-334
Department of Mechanical Engineering
National Institute of Technology, Hamirpur (H.P)
June 2011

“Influence of stochastic and deterministic surface
roughness on the performance of circular and non-
circular fluid film journal bearings”
A thesis
Submitted in fulfillment of the
Requirements for the award of the degree
Of
Master of Technology
In
Mechanical Engineering(CAD/CAM)
By
Manoj Singh
Roll No: 09M-334
Under the supervision of
Prof. Rakesh Sehgal
Dr. R. K. Sharma
Department of Mechanical Engineering
National Institute of Technology, Hamirpur (H.P)
June 2011

CERTIFICATE
I hereby certify that the work which is presented in the dissertation entitled, “Influence of
stochastic and deterministic surface roughness on the performance of circular and
non-circular fluid film journal bearings” in partial fulfilment of the requirements for the
award of degree Master of Technology in CAD/CAM and being submitted in Mechanical
Engineering Department of National Institute of Technology, Hamirpur (H.P) is the
bonafide work done by me under the supervision of Prof. Rakesh Sehgal and Dr. Rajesh
Sharma.
The matter presented in this dissertation has not been submitted for the award of any other
degree of this or any other university.
Date:……………….. (Manoj Singh)
Roll No- 09M334
This is to certify that the above statement made by the candidate is correct and true to the
best of our knowledge.
Sign by
Supervisor Co-Supervisor
(Dr. R. Sehgal) (Dr. R.K. Sharma)
Professor Associate Professor
MED- NIT Hamirpur (HP) MED-NIT Hamirpur (HP
The M.Tech Viva-Voce examination of MANOJ KUMAR SINGH, has been held
on____________________
Supervisor Co-Supervisor Head of Department External Examiner
(Dr. Rakesh Sehgal) (Dr. R. K. Sharma)

ii
A C K N O W L E D G E M E N T
It gives me great pleasure to express my sincere gratitude to my major
professor and research advisor Dr. Rakesh Sehgal (Professor, NIT Hamirpur, HP) for
giving me this wonderful opportunity to work under him. His constant academic and
professional guidance has been the core to the success of this research. I am indebted
for his valuable time spent in guidance, teaching and patience shown during all stages
of this research.
I give my gratitude to my Co-supervisor Dr. R. K. Sharma (Associate Professor, NIT
Hamirpur H.P). Under whom I was first exposed to the field of surface roughness on
journal bearings and also for his encouragement and support to pursue this field
further.
I am very thankful to Dr. Amit Chauhan (Assistant professor, UIET, Punjab) and Mr.
Saurabh Kango (Phd. Scholar, NIT Hamirpur, HP) for their valuable help of
programming support. It has been a great experience working with them and has
helped me broaden my application of skills to various fields. Also thankful to all my
friends for being supportive and helpful at all times.
Manoj Singh

iii
A B S T R A C T
This work presents the comparative study of the performance of different
bearing designs/profiles for different types of bearing surface roughness. The journal
bearing profiles considered are circular, elliptical and off-set halves. The different
types of roughness and wave texture considered for inner surface of bearings are
positive half wave, positive full wave and sinusoidal wave texture.
By using the film thickness equations for smooth bearings and modified film
thickness equations for rough bearing (considering all types of roughness and surface
textures individually as well as in combination) profiles in Reynold’s equation, a
comparison of bearing performance parameters such as load carrying capacity,
friction force, coefficient of friction and side leakage has been carried out for smooth
and rough bearing surfaces in case of all the three bearing profiles. Further,
thermohydrodynamic analysis using PTPA method has been carried out for all the
three bearing profiles considering smooth bearing surfaces as well as longitudinal
stochastic roughness. It is concluded that the bearing performance in general is
positively influenced by different types of roughness, wave texture and combination
of roughness and texture. Combined stochastic roughness and wave texture influence
the bearing parameters considerably. Comparison of all types of roughness and wave
texture indicate that the combination of longitudinal stochastic roughness and
transverse full wave texture gives maximum enhancement of bearing performance
parameters.
Computed results indicate that the load carrying capacity and friction force increases
whereas the coefficient of friction and oil side leakage decreases.
Among different bearing configurations/profiles it is found that the non-circular
bearings (elliptical and off-set halves) provide better stability in comparison to
circular bearing profile.
On the basis of thermal analysis it is observed that temperature rise in rough bearing
surfaces is more in comparison to smooth bearing surfaces for all the three bearing
profiles. The order of temperature rise is maximum in case of circular followed by
elliptical and off set halves bearing profiles thus causing the off-set halves journal
bearing to run coolest.

iv
C O N T E N T S
CERTIFICATE i
ACKNOWLEDGEMENT ii
ABSTRACT iii
CONTENTS iv
NOMENCLATURE viii
LIST OF FIGURES xii
LIST OF TABLES xxxi
CHAPTERS
1 Introduction 1-19
1.1 Hydrodynamic Journal bearings 1
1.1.1
1.1.2
Circular Journal Bearings 3
Multilobe Journal Bearings /Non-Cylindrical Bearing
Bore
6
1.1.2.1 Two Lobe Bearings 7
1.1.2.2 Three-Lobe Bearings 8
1.1.2.3 Offset-Halves Bearings 9
1.1.2.4 Advantages of Multi Lobe Bearings 11
1.2 Roughness Theory 12
1.2.1 Stochastic roughness 12
1.2.2 Surface texturing / Deterministic asperities 17
1.3 Outline of Thesis 19
2 Review of Literature 20-42
2.1
2.2
2.3
Circular Smooth Journal Bearings 20
Non-Circular Bearings 23
Thermal Analysis 27
2.4 Rough Circular journal bearings 29
2.5 Rough Elliptical Bearings 40
2.6 Research gaps 40
2.7 Objective of Present Study 41
3 Mathematical modeling 43-64

v
3.1 Film Thickness Equations 43
3.1.1 Film thickness equations for circular
journal bearings
43
3.1.2 Film thickness equations for offset-halves
journal bearings
43
3.1.3 Film thickness equations for elliptical journal
bearings
43
3.1.4 Wave texture film thickness equation 44
3.2 Reynolds Equation 45
3.3 Rough film thickness function 46
3.3.1 Evaluation of Rough film Thickness 47
3.4 Pressure distribution 49
3.4.1 Pressure equation For Longitudinal roughness
with Finite Difference Method (FDM)
49
3.4.2 Pressure equation for wave texture (full wave
half wave and sinusoidal wave roughness) with
finite difference method (FDM)
51
3.4.3 Boundary conditions for Reynolds equation 52
3.4.4 Computational Procedure 52
3.4.4.1 Reynolds Equation 53
3.5 Thermal Analysis 53
3.5.1 Energy equation 53
3.5.2 Velocity equation 53
3.5.3 Temperature profile expression 53
3.5.4 Heat conduction equation 55
3.5.5 Boundary conditions 55
3.5.6 Viscosity Equation 56
3.6 Bearing parameters 56
3.6.1 Load Capacity 56
3.6.2 Friction Force 56
3.6.3 Oil flow 56
3.7 Numerical solution 57
3.7.1 Numerical solutions procedure for the effect of 57

vi
stochastic roughness
3.7.2 Numerical solutions procedure for the effect of
deterministic roughness
57
3.7.3 Numerical solution for combined effect of
stochastic and deterministic roughness
58
3.7.4 Numerical solution for Thermal Analysis 59
3.7.5 Flow chart for stochastic surface roughness 61
3.7.6 Flow chart for deterministic surface roughness 62
3.7.7 Flow chart for combined effect of stochastic and
deterministic surface roughness
63
3.7.5 Flow chart for Thermal effect 64
4 Results and Discussion
65-141
4.1 Bearing Geometry 65
4.1.1 Circular bearing profile
65
4.1.2 Off-set halves bearing profile
68
4.1.3 Elliptical bearing profile
69
4.2 Input / Operating Parameter
70
4.3 Model validation
71
4.4 Results and discussions
74
4.4.1 Film thickness variation / Combined film thickness
74
4.4.2 Pressure distribution
77
4.4.3 Comparison of load carrying capacity for different
types of bearing profiles
80
4.4.4 Comparison of friction force for different types of
bearing profiles
101
4.4.5 Comparison of coefficient of friction for different
types of bearing profiles
111

vii
4.4.6 Comparison of oil side flow for different types of
bearing profiles
121
4.5 Thermal Analysis 131
4.5.1 Model validation 131
4.5.2 Thermal Results 132
4.6 Variation of load carrying capacity, friction force and
coefficient of friction for different types of bearing profiles
134
5 Conclusion and scope for future work 142
5.1 Conclusions 142
5.2 Scope for future work 142
6 References 143-146

viii
N O M E N C L A T U R E
Am Asperity Amplitude (m)
Half total range of random film thickness variable (roughness parameter)
C Radial clearance, m
Cm Minimum clearance when journal centre is coincident with geometric
centre of the bearing, m
CP Specific heat of the lubricating oil, J/Kg Deg. C
Eccentricity, mm
Elliptical ratio
Expected value
Probability density function
F Normalized friction force (N)
Nominal film thickness (m)
H Film thickness, random variable [ ]
Koil Thermal conductivity of lubricating oil, W/m Deg. C
L Length of bearing (m)
N Shaft speed (rpm)
OL Center of lower lobe
OU Center of upper lobe
Dimensional pressure (M Pa)
P Load per unit Area ⁄
Q Side oil flow (m3
/s)

ix
Rb Bearing radius, mm
Rj Journal radius, mm
T Lubricating film temperature, Deg. C
Ta Ambient temperature, Deg. C
Tb Bush temperature, Deg. C
T0 Oil inlet temperature, Deg. C
u, w Velocity components in X- and Z-directions, m/s
U Shaft speed (m/sec)
Lu Velocity of lower bounding surface, m/s
Uu Velocity of upper bounding surface, m/s
WL Dimensional load (N)
W Asperity wavelength (m)
W1 Fixed load (N)
x, y, z Coordinates in circumferential, radial, and axial directions
Interval between two nodes in length side (Z-axis)
21, Eccentricity Ratio from 0-180 Deg. & 180-360 Deg. respectively
 Attitude angle
21, Attitude angles from 0-180 Deg. & 180-360 Deg. respectively
 Absolute viscosity, Pas
ref Absolute viscosity at oil inlet temperature, Pas
 Density of lubricating oil, Kg/m3
 Temperature viscosity coefficient of lubricant, (K-1
)

x
 Barus viscosity-pressure index, Pa-1
Degree Celsius
Angular direction (radians)
Interval between two nodes in circumference of bearing
Eccentricity ratio
Standard distribution
Shear stress(N/m2
)
Relative eccentricity of the journal(m)
Angular velocity (radian/sec)
Off-set factor
Texture film thickness (m)
Abbreviations:
STC Stochastic roughness
TRN Transverse
TRN WV Transverse wave
SIN WV Sinusoidal wave
HLF WV Half wave
FL WV Full wave
UN Uniform
LN Longitudinal
LHWT Longitudinal half wave texture
LFWT Longitudinal full wave texture
LSWT Longitudinal sinusoidal wave texture
THWT Transverse half wave texture
TFWT Transverse full wave texture

xi
TSWT Transverse sinusoidal wave texture
LS+LHWT Combine effect of longitudinal stochastic roughness and longitudinal
half wave texture
LS+LFWT Combine effect of longitudinal stochastic roughness and Longitudinal
full wave texture
LS+LSWT Combine effect of longitudinal stochastic roughness and longitudinal
sinusoidal wave texture
LS+THWT Combine effect of longitudinal stochastic roughness and transverse
half wave texture
LS+TFWT Combine effect of longitudinal stochastic roughness and transverse
full wave texture
LS+TSWT Combine effect of longitudinal stochastic roughness and transverse
sinusoidal wave texture

xii
L I S T OF F I G U R E S
Fig.
No.
Title P. No
1.1 Schematic of circular journal bearing 4
1.2 Schematic of operation of hydrodynamic lubrication in journal
bearing
4
1.3 Stribeck curve 6
1.4 Schematic of two lobe journal bearing 9
1.5 Schematic of three lobe journal bearing 9
1.7 Schematic of off-set halve journal bearing 10
1.9 Longitudinal roughness on bearing surfaces 16
1.10 Transverse roughness on bearing surfaces 17
1.11 Sinusoidal wave texture profile 18
1.12 Positive half wave texture profile 18
1.13 Positive full wave texture profile 19
3.1 Grid network for pressure distribution 49
4.1 (a) Circular bearing with sinusoidal wave textured area 66
(b) Circular bearing with positive half wave textured area 66
(c) Circular bearing with positive full wave textured area 66
(d) Circular bearing profiles with combined effect of stochastic
roughness and sinusoidal wave texture
67
(e) Circular bearing profile with combined effect of stochastic
roughness and positive half wave textured area
67
(f) Circular bearing profile with combined effect of stochastic
roughness and positive full wave textured area
68
4.2 (a) Off-set halves bearing profiles with sinusoidal wave texture 68
(b) Off-set halves bearing profiles with positive full wave texture 69
(c) Off-set halves bearing profiles with positive half wave texture 69
4.3 Elliptical bearing with positive full wave textured area 70
4.4 Elliptical bearing with positive sinusoidal wave textured area 70
4.5 Comparison of center plane pressure distribution at eccentricity 72

xiii
ratio=0.6, C=100 , N=100 rpm, D=0.1, L/D = 1
4.6 Comparison of load carrying capacity at C=100 , ,
N=100 rpm, D=0.1, and L/D = 1
72
N=300 rpm, D=0.1, c=20 % of C and L/D = 1 for longitudinal
stochastic roughness
73
N=300 rpm, D=0.1, c=20 % of C and L/D = 1 for uniform stochastic
roughness
73
4.9 Comparison of central plane pressure distribution at eccentricity
ratio=0.6, Cm=120 , C=200 , D=0.1, L/D = 1 and
speed=3000rpm for off-set halve bearing profile.
73
4.10 Comparison of oil film thickness at eccentricity ratio=0.6,
Cm=120 , C=200 , D=0.1, L/D = 1 and speed=3000rpm for
off-set halves, elliptical profile and circular bearings profile.
74
4.11 Comparison of film thickness at eccentricity ratio=0.6, C=200 ,
A=7.5 , W=0.009m and speed=3000rpm for circular bearing
profile with texture and without texture.
75
4.12 Comparison of film thickness at eccentricity ratio=0.6, Cm=120 ,
C=200 , A=7.5 , W=0.009m and speed=3000rpm for off-set
halves bearing profile with texture and without texture
75
4.13
Comparison of film thickness at eccentricity ratio=0.6, Cm=120 ,
C=200 , A=7.5 , W=0.009m and speed=3000rpm for elliptical
bearing profile with texture and without texture.
76
4.14 Film thickness for off-set halve journal bearing considering transverse
sinusoidal wave texture at eccentricity ratio=0.6, Cm=120 ,
C=200 , speed=3000rpm, A=7.5 and W=0.009m
76
4.15 Film thickness for off-set halves journal bearing considering
longitudinal sinusoidal wave texture at eccentricity ratio=0.6,
Cm=120 , C=200 , speed=3000rpm, A=7.5 and
W=0.009m
77
4.16 Comparison of pressure distribution for different geometry of bearing 78
4.17 Comparison of pressure distribution at eccentricity ratio=0.6, 78

xiv
C=200 , A=7.5 , W=0.009m and speed=3000rpm for circular
bearing profile with texturing (full wave and sinusoidal wave) and
without texturing.
4.18 Comparison of pressure distribution at eccentricity ratio=0.6,
Cm=120 , C=200 , A=7.5 , W=0.009m and
speed=3000rpm for off-set halves bearing profile with texturing (full
wave and sinusoidal wave) and without texturing.
78
4.19 Comparison of pressure distribution at eccentricity ratio=0.6,
Cm=120 , C=200 , A=7.5 , W=0.009m and speed=3000rpm
for off-set halves bearing profile with texturing (full wave and
sinusoidal wave) and without texturing
79
4.20 Pressure distribution at eccentricity ratio=0.6, Cm=120 ,
C=200 and speed=3000rpm for off-set halves smooth bearing
profile
79
4.21 Pressure distribution at eccentricity ratio=0.6, Cm=120 ,
C=200 , A=7.5 , W=0.009m and speed=3000rpm for off-set
halves full transverse texture bearing profile.
80
4.22 (a) Comparison of load carrying capacity for smooth bearings 81
4.22 (b)Comparison of load carrying capacity for longitudinal stochastic
roughness
81
4.23 Comparison of load carrying capacity for longitudinal stochastic
roughness & longitudinal sinusoidal wave texture
82
roughness & transverse sinusoidal wave texture
82
Roughness & longitudinal half wave texture
83
Roughness & transverse half wave texture
83
roughness & longitudinal full wave texture
83
roughness & transverse full wave texture
84
4.29 (b) Comparison of load carrying capacity for combined effects of 85

xv
longitudinal stochastic roughness & longitudinal Sinusoidal wave
texture
4.30 (b) Comparison of load carrying capacity for combined effects of
longitudinal stochastic roughness & transverse Sinusoidal wave
texture
85
4.31 (a) Comparison of load carrying capacity for combined effects of
longitudinal stochastic roughness & longitudinal half wave texture at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
86
(b) Comparison of load carrying capacity for combined effects of
longitudinal stochastic roughness & longitudinal half wave texture at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
eccentricity ratio
86
longitudinal stochastic roughness & transverse half wave texture at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
86
longitudinal stochastic roughness & transverse half wave texture at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
eccentricity ratio
87
longitudinal stochastic roughness & longitudinal full wave texture at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
87
(b) Comparison of load carrying capacity for combined effect of
longitudinal stochastic roughness & longitudinal full wave texture
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
eccentricity ratio
87
longitudinal stochastic roughness & transverse full wave texture at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
88

xvi
longitudinal stochastic roughness & transverse full wave texture at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
eccentricity ratio
88
uniform stochastic roughness & longitudinal Sinusoidal wave texture
eccentricity ratio=0.6, Cm=120 , C=200 , A=7.5 , c=20%C,
W=0.009m and speed=3000rpm
88
at Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
eccentricity ratio
89
uniform stochastic roughness & transverse Sinusoidal wave texture at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
89
uniform stochastic roughness & transverse Sinusoidal wave texture at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
eccentricity ratio
89
4.37 (a) ) Comparison of load carrying capacity for combined effects of
uniform stochastic roughness & longitudinal half wave texture
90
uniform stochastic roughness & longitudinal half wave texture at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
eccentricity ratio
90
uniform stochastic roughness & transverse half wave texture
90
(b) Comparison of load carrying capacity for combined effect of 91

xvii
uniform stochastic roughness & transverse half wave texture at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
eccentricity ratio
uniform stochastic roughness & longitudinal full wave texture
91
uniform stochastic roughness & longitudinal full wave texture at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
eccentricity ratio
91
uniform stochastic roughness & transverse full wave texture
92
(b) Comparison of load carrying capacity for Combine effect of
uniform stochastic roughness & transverse full wave texture at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
eccentricity ratio
92
4.42 (a) ) Comparison of load carrying capacity for combined effects of
texture and longitudinal sinusoidal wave texture at eccentricity
ratio=0.6, Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m
and speed=3000rpm
93
4.43 Comparison of load carrying capacity for combined effects of
texture and longitudinal sinusoidal wave texture at Cm=120 ,
C=200 , A=7.5 , c=20%C, W=0.009m and eccentricity ratio
93
4.44 (a) Comparison of load carrying capacity for combined effect of
texture and transverse sinusoidal wave texture at eccentricity
94

xviii
ratio=0.6, Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m
and speed=3000rpm
texture and transverse sinusoidal wave texture at Cm=120 ,
94
longitudinal stochastic roughness & longitudinal half wave texture
and longitudinal half wave texture at eccentricity ratio=0.6,
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
94
4.46 (b) Comparison of load carrying capacity for combined effect of
longitudinal stochastic roughness & longitudinal half wave texture
and longitudinal half wave texture at Cm=120 , C=200 ,
A=7.5 , c=20%C, W=0.009m and eccentricity ratio
95
longitudinal stochastic roughness & transverse half wave texture and
transverse half wave texture at eccentricity ratio=0.6, Cm=120 ,
C=200 , A=7.5 , c=20%C, W=0.009m and speed=3000rpm
95
(longitudinal stochastic roughness & transverse half wave texture) and
transverse half wave texture at Cm=120 , C=200 , A=7.5 ,
c=20%C, W=0.009m and eccentricity ratio
95
(longitudinal stochastic roughness & longitudinal full wave texture)
and longitudinal full wave texture at eccentricity ratio=0.6,
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and speed
=300rpm
96
and longitudinal full wave texture at Cm=120 , C=200 ,
96

xix
longitudinal stochastic roughness & transverse full wave texture
transverse full wave texture at eccentricity ratio=0.6, Cm=120 ,
96
longitudinal stochastic roughness & transverse full wave texture
transverse full wave texture at Cm=120 , C=200 , A=7.5 ,
97
(uniform stochastic roughness & longitudinal Sinusoidal wave
texture) and longitudinal sinusoidal wave texture at Cm=120 ,
98
(uniform stochastic roughness & transverse Sinusoidal wave texture)
and transverse sinusoidal wave texture at eccentricity ratio=0.6,
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
98
(uniform stochastic roughness & transverse sinusoidal wave texture)
and transverse sinusoidal wave texture at Cm=120 , C=200 ,
98
4.52 (a) Comparison of load carrying capacity for Combined effect of
(uniform stochastic roughness & longitudinal half wave texture) and
longitudinal half wave texture at eccentricity ratio=0.6, Cm=120 ,
99
(b) Comparison of load carrying capacity for Combined effect of
(uniform stochastic roughness & longitudinal half wave texture) and
longitudinal half wave texture at Cm=120 , C=200 ,
99
4.53 (a) Comparison of load carrying capacity for Combined effect of
uniform stochastic roughness & transverse half wave texture and
transverse half wave texture at eccentricity ratio=0.6, Cm=120 ,
99

xx
(uniform stochastic roughness & transverse half wave texture) and
transverse half wave texture at Cm=120 , C=200 , A=7.5 ,
100
(uniform stochastic roughness & longitudinal full wave texture) and
longitudinal full wave texture at eccentricity ratio=0.6, Cm=120 ,
100
(uniform stochastic roughness & longitudinal full wave texture) and
longitudinal full wave texture at Cm=120 , C=200 , A=7.5 ,
100
(uniform stochastic roughness & transverse full wave texture) and
transverse full wave texture at eccentricity ratio=0.6, Cm=120 ,
101
(uniform stochastic roughness & transverse full wave texture) and
transverse full wave texture at Cm=120 , C=200 , A=7.5 ,
101
4.56 Comparison of friction force variation with considering longitudinal
stochastic roughness for different bearing profile at Cm=120 ,
C=200 , c=20%C, W=0.009m and speed=3000rpm
102
4.57 Comparison of friction force for combined effects of longitudinal
stochastic roughness & longitudinal Sinusoidal wave texture for
different bearing profile at Cm=120 , C=200 , A=7.5 ,
c=20%C, W=0.009m and speed=3000rpm
102
stochastic roughness & transverse Sinusoidal wave texture for
different bearing profile at Cm=120 , C=200 , A=7.5 ,
103
4.59 Comparison of friction force for combined effect of longitudinal 103

xxi
stochastic roughness & longitudinal half wave texture for different
bearing profile at Cm=120 , C=200 , A=7.5 , c=20%C,
stochastic roughness & transverse half wave texture for different
bearing profile at Cm=120 , C=200 , A=7.5 , c=20%C,
103
stochastic roughness & longitudinal full wave texture for different
bearing profiles at Cm=120 , C=200 , A=7.5 , c=20%C,
104
stochastic roughness & transverse full wave texture for different
104
4.63 Comparison of friction force for combined effects of uniform
different bearing profiles at Cm=120 , C=200 , A=7.5 ,
104
stochastic roughness & transverse Sinusoidal wave texture for
105
105
105
106

xxii
106
4.69 Comparison of friction force for combined effects of (longitudinal
stochastic roughness & longitudinal Sinusoidal wave texture) and
longitudinal sinusoidal wave texture for different bearing profiles at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
107
stochastic roughness & transverse Sinusoidal wave texture) and
transverse sinusoidal wave texture for different bearing profiles at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
107
4.71 Comparison of friction force for combined effect of (longitudinal
stochastic roughness & longitudinal half wave texture) and
longitudinal half wave texture for different bearing profiles at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
108
stochastic roughness & transverse half wave texture) and transverse
half wave texture for different bearing profiles at Cm=120 ,
108
4.73 Comparison of friction force combined effects of for (longitudinal
stochastic roughness & longitudinal full wave texture) and
longitudinal full wave texture for different bearing profiles at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
118
stochastic roughness & transverse full wave texture) and transverse
full wave texture for different bearing profiles at Cm=120 ,
109

xxiii
4.75 Comparison of friction force for combined effects of (uniform
stochastic roughness & longitudinal sinusoidal wave texture) and
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
109
stochastic roughness & transverse sinusoidal wave texture) and
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
109
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
110
110
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
110
111
4.81 Comparison of coefficient of friction with considering longitudinal
stochastic roughness for different bearing profiles at Cm=120 ,
112
4.82 Comparisons of coefficient of friction for combined effects of 112

xxiv
texture for different bearing profiles at Cm=120 , C=200 ,
A=7.5 , c=20%C, W=0.009m and speed=3000rpm
4.83 Comparisons of coefficient of friction for combined effects of
texture for different bearing profiles at Cm=120 , C=200 ,
112
longitudinal stochastic roughness & longitudinal half wave texture for
113
longitudinal stochastic roughness & transverse half wave texture for
113
longitudinal stochastic roughness & longitudinal full wave texture for
113
longitudinal stochastic roughness & transverse full wave texture for
114
for different bearing profiles at Cm=120 , C=200 , A=7.5 ,
114
uniform stochastic roughness & transverse Sinusoidal wave texture
for different bearing profiles at Cm=120 , C=200 , A=7.5 ,
115
uniform stochastic roughness & longitudinal half wave texture for
115

xxv
uniform stochastic roughness & transverse half wave texture for
115
uniform stochastic roughness & longitudinal full wave texture for
116
uniform stochastic roughness & transverse full wave texture for
116
4.94 Comparison of coefficient of friction for combined effect of
(longitudinal stochastic roughness & longitudinal sinusoidal wave
texture) and longitudinal sinusoidal wave texture for different bearing
profiles at Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m
and speed=3000rpm
117
4.95 Comparison of coefficient of friction for combined effects of
(longitudinal stochastic roughness & transverse sinusoidal wave
texture) and transverse sinusoidal wave texture for different bearing
and speed=3000rpm
117
(longitudinal stochastic roughness & longitudinal half wave texture)
and longitudinal half wave texture for different bearing profiles at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
117
(longitudinal stochastic roughness & transverse half wave texture) and
transverse half wave texture for different bearing profile at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
118

xxvi
speed=3000rpm
4.98 Comparison of coefficient of friction for combined effect of
and longitudinal full wave texture for different bearing profiles at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
118
(longitudinal stochastic roughness & transverse full wave texture) and
transverse full wave texture for different bearing profiles at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
118
4.100 Comparison of coefficient of friction for combined effects of (uniform
stochastic roughness & longitudinal sinusoidal wave texture) and
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
119
stochastic roughness & transverse sinusoidal wave texture) and
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
119
4.102 Comparison of coefficient of friction for combined effect of )uniform
longitudinal half wave texture at Cm=120 , C=200 ,
119
120
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
120

xxvii
speed=3000rpm
4.105 Comparison of coefficient of friction for combined effect of (uniform
120
4.106 Comparison of oil side flow with considering longitudinal stochastic
roughness for different bearing profile at Cm=120 , C=200 ,
121
4.107 Comparisons of oil side flow for combined effects of longitudinal
different bearing profiles at Cm=120 , C=200 , c=20%C,
122
4.108 Comparison of oil side flow for combined effects of longitudinal
stochastic roughness & transverse sinusoidal wave texture for
122
122
123
123
123
4.113 Comparison of oil side flow for combined effects of uniform 124

xxviii
stochastic roughness & longitudinal sinusoidal wave texture for
4.114 Comparison of oil side flow for combined effects of uniform
stochastic roughness & transverse sinusoidal wave texture for
125
4.115 Comparison of oil side flow for combined effect of uniform stochastic
roughness & longitudinal half wave texture for different bearing
and speed=3000rpm
125
125
126
126
4.119 Comparison of oil side flow for combined effects of (longitudinal
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
126
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
127

xxix
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
127
127
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
128
128
4.125 Comparison of oil side flow for combined effects of (uniform
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
128
4.126 Comparison of oil side flow for Combined effects of (uniform
transverse sinusoidal wave texture for different bearing profile at
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
129
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
129

xxx
129
Cm=120 , C=200 , A=7.5 , c=20%C, W=0.009m and
speed=3000rpm
130
130
4.131 Variation of load carrying capacity for longitudinal stochastic
roughness with increasing Roughness parameter at Cm=120 ,
C=200 and speed=3000rpm
130
4.132 Validations of temperature distributions at 4000rpm, , K=0.13,
C=200 and Cm=120 for smooth elliptical journal bearing.
132
4.133 Comparison of center plane oil film temperature distribution for
circular journal bearing by considering longitudinal stochastic rough
and smooth surfaces.
133
4.134 Comparison of center plane oil film temperature distribution for
elliptical journal bearing by considering longitudinal stochastic rough
133
4.135 Comparison of center plane oil film temperature distribution for off-
set halve journal bearing by considering longitudinal stochastic rough
133

xxxi
L I S T OF T A B L E S
Table
No
Title Page No.
4.1 The input parameters for calculating the results of
different types of bearing profile
72
4.2 Percentage variation of load carrying capacity,
comparison of smooth bearing surface of circular journal
bearing with different roughness and wave texture
136
4.3 Percentage variation of friction force, compare to smooth
bearing surface of circular journal bearing with different
roughness and wave texture
137
4.4 Percentage variation of coefficient of friction, compare to
smooth bearing surface of circular journal bearing with
different roughness and wave texture
137
4.5 Percentage variation of load carrying capacity, compare
to smooth bearing surface of elliptical journal bearing
with different roughness and wave texture
138
bearing surface of elliptical journal bearing with different
roughness and wave texture
139
smooth bearing surface of elliptical journal bearing with
139
4.8 Percentage variation of load carrying capacity, compare
to smooth bearing surface of off-set halves journal
bearing with different roughness and wave texture
140
bearing surface of off-set halves journal bearing with
141
smooth bearing surface of off-set halves journal bearing
with different roughness and wave texture
141

xxxii
4.11 Percentages of load variation for off-set halves journal
bearing with increasing amplitudes of transverse full
wave texture.
142

1 | P a g e
CHAPTER-1
INTRODUCTION
This chapter provides historical and theoretical details on the hydrodynamic
journal bearings for circular, offset-halves and elliptical profiles with and without
using the surface roughness parameter. There is enormous information available for
the working principles and their uses.
1.1 HYDRODYNAMIC JOURNAL BEARING
A journal bearing, simply stated, is a cylinder which surrounds the shaft and is
filled with some form of fluid lubricant. In this bearing a fluid is the medium that
supports the shaft preventing metal to metal contact. The most common fluid used is
oil, whereas in special applications using water or a gas may also be used.
Hydrodynamic journal bearings are defined as the mechanical components that
support the external loads smoothly due to their geometry and relative motion of
mating surfaces in the presence of a thick film of lubricant. These bearings are
extensively used in high speed rotating machines because of their low friction, high
load capacity, and good damping characteristics. Hydrodynamic journal bearings have
many different designs to compensate for differing load requirements, machine
speeds, cost, or dynamic properties. The prime aim of lubrication is to separate the
surfaces completely by a fluid film. This eliminates wear and considerably reduces
the friction losses. Liquids are usually considered to be incompressible for most
bearings performance calculations. However, at higher pressure, the compressibility
of the lubricant become significant and affects fluid film stiffness and damping of
journal bearing system.
The mass of the fluid is generally neglected in the analysis of journal bearing systems.
However, with the use of low viscosity fluids and owing to the fact that the present
day machinery is operated at high speed, the effect of fluid film inertia may be quite
significant. Hydrodynamic principles, which are active as the shaft rotates, create an
oil wedge that supports the shaft and relocates it within the bearing clearances. In a
horizontally split bearing the oil wedge will lift and support the shaft, relocating the
centreline slightly up and to one side into a normal attitude position in a lower
quadrant of the bearing. The pressure in the lubricant film in hydrodynamic bearings

2 | P a g e
is generated by wedge-action where the relative movement of the surface drags the
lubricant into decreasing space. The resultant of the bearing film forces, which act
normally to the journal at each point around the bearing, will be equal and opposite of
the externally applied force on the shaft. For given eccentricity of the journal within
the bearing, the pressure force giving rise to the hydrodynamic load is primarily
dependent upon speed, viscosity and bearing projected area. The starting and stopping
is the chief cause of wear in the hydrodynamic bearing because there is no pressurized
fluid film present to avoid the contact of two bearing surfaces at zero operating speed.
The normal attitude angle will depend upon the shaft rotation direction with a
clockwise rotation having an attitude angle in the lower left quadrant. External
influences, such as hydraulic volute pressures in pumps or generator electrical load
can produce additional relocating forces on the shaft attitude angle and centreline
position. The development of fluid film lubrication mechanisms has been observed by
Petrov (1883) in Russia and Tower (1883) in England. In 1886, Reynolds presented
his classical analysis of bearing hydrodynamics, which forms the basis of present days
bearing study.
The hydrodynamic theory of lubrication of journal bearings is older than a century. In
his famous experiment Tower has shown first the pressure distribution in the
lubricating oil film in the clearance of journal bearings in 1883. Also in this year,
Petroff measured the friction torque of oil lubricated sliding bearings and created a
formula to calculate it. The modern period of lubrication began with the work of
Osborne Reynolds (1842-1912). Reynolds research was concerned with shafts
rotating in bearings and cases. He also noted that as the shaft gained velocity, the
liquid flowed between the two surfaces at a greater rate. This, because the lubricant is
viscous, produces a liquid pressure in the lubricant wedge that is sufficient to keep the
two surfaces separated. Under ideal conditions, Reynolds showed that this liquid
pressure was great enough to keep the two bodies from having any contact and that
the only friction in the system was the viscous resistance of the lubricant. The rotating
shaft drags a wedge of fluid beneath it that develops a pressure great enough to
support the journal and eliminate contact friction between the journal and the bearing
in ideal situations. In hydrodynamic lubrication the fluid is assumed not to slip at the
interface with the bearing surface. That means the fluid in contact with the bearing
surfaces moves at the same velocity as the surface. Over the thickness of the fluid
there is a velocity gradient depending on the relative movement of the bearing

3 | P a g e
surfaces. If the bearing surfaces are parallel (or concentric) the action motion of the
lubricant will not result in a pressure which could support any bearing load. However,
if the surfaces are at a slight angle the resulting lubrication fluid velocity gradients
will be such that a pressure results from the wedging action of the bearing surfaces.
Journal bearings are widely used in rotating machinery, especially when shafts are
submitted to both high speeds and heavy applied loads.
1.1.1 Circular Journal Bearing
Lubrication reduces friction between two surfaces (such as sliding surfaces of
a bearing and a shaft) in relative motion. It is typically categorized as boundary,
mixed and hydrodynamic lubrication, by authors Heywood (1988), Becker (2004) and
Gleghorn and Bonassar (2008). When a journal bearing operates under boundary
lubrication, the sliding surfaces of the bearing and shaft are practically in direct
contact and friction is at its highest level. Lower friction levels are achieved through
the use of mixed lubrication, where the sliding surfaces are partially separated by the
lubricant, and of hydrodynamic lubrication, where the sliding surfaces are completely
separated by the lubricant.
To illustrate how friction varies under different lubrication conditions, Stribeck curves
(or diagrams) have been used widely in different engineering sciences. In Stribeck
curves, the friction coefficient is presented as a function of a dimensionless parameter
calculated from the dynamic viscosity, angular speed and pressure. The above-
mentioned parameter is typically called the duty parameter or Hersey number. The
minimum of the friction coefficient is reached at the critical value of the duty
parameter, at the dividing line between the mixed and hydrodynamic lubrication
zones.
Heywood (1988) presented a Stribeck curve for a journal bearing. Methods for the
calculation of Stribeck curves were studied by Kraker et al. (2007). They calculated
the friction coefficient as a function of the journal frequency at different values of the
projected bearing pressure.
The basic configuration of the circular journal bearing consists of a journal which
rotates relative to the bearing which is also known as bush (Fig.1.1).
The operation of hydrodynamic lubrication in journal bearing has been illustrated in
Fig 1.2. Before the rotation commences at rest the shaft rests on the bearing surface.
When the journal starts to rotate, it will climb the bearing surface gradually as the

4 | P a g e
speed is further increased; it will then force the lubricant into the wedge-shaped
region.
Fig. 1.1 Circular journal bearing
Fig. 1.2 Operation of hydrodynamic lubrication in journal bearing
When more and more lubricant is forced into a wedge-shaped clearance space, the
shaft moves up the bore until an equilibrium condition is reached and now, the shaft is
supported on a wedge of lubricant. The moving surfaces are then held apart by the
Bearing
Journal
Lubricant

5 | P a g e
pressure generated within the fluid film. Journal bearings are designed such that at
normal operating conditions the continuously generated fluid pressure supports the
load with no contact between the bearing surfaces.
This operating condition is known as thick film or fluid film lubrication and results in
a very low operating friction. On the other hand if the lubricant film is insufficient
between the relatively moving parts, it may lead to surface contact and the
phenomenon is normally known as boundary lubrication. It is known that the
coefficient of friction of a journal bearing changes with operating conditions as shown
in Fig. 1.3. The vertical axis indicates the coefficient of friction f = F/P and the
horizontal axis represents the bearing number μU/P, where F = frictional force, P =
journal load, μ = coefficient of viscosity, and U = circumferential velocity of the
journal (the part of a shaft supported by a bearing). The curvehas a minimum point
where corresponds to small value of coefficient of friction usually of the order of
0.001. For larger values of bearing number the coefficient of friction increases along a
straight line through the rate of increase is small. With a decrease in the bearing
number from the point of minimum coefficient of friction, in contrast, the frictional
coefficient increases rapidly, but does not exceed a certain fixed value. Since the
diagram is based on the careful, extensive experiments during (1902) carried out by
Richard Stribeck (1861 – 1950) of Germany, it is called the Stribeck diagram. The
diagram exhibits clearly the features of the frictional coefficient of a journal bearing.
The reason why the curve in the Stribeck diagram takes such a form is as follows.
First, consider the region where the bearing number is sufficiently large (the region on
the right of the minimum point, or the region where, for example, the circumferential
speed is sufficiently high). In this region, the frictional coefficient increases at a very
low rate, its value being of the order of 0.001. The reason for this is that a sufficiently
thick oil film is formed between the two surfaces in relative motion, and the two
surfaces do not contact each other directly. The frictional force in this case is
attributable to the viscosity of oil and is proportional to the shear rate of the oil film.
The bearing load is supported by the pressure produced in the oil film. Since the two
surfaces do not contact directly, wear hardly takes place. This is an ideal state of
lubrication and is called hydrodynamic lubrication. This occurs at rotation start-up at
slow speed operation or if the load is too heavy. This regime results in bearing wear
and a relatively high friction value. If a bearing is to be operated under boundary
lubricating conditions, special lubricants must be used. Amongst hydrodynamic

6 | P a g e
bearings, circular journal bearing is the most familiar and widely used bearing. Simple
form of this bearing offers many advantages in its manufacturing as well as in its
performance. However, the circular journal bearings operating at high speed
encounter instability problems of whirl and whip. Instability may damage not only the
bearings but also the complete machine.
Fig. 1.3 Stribeck curve
Moreover, these bearings usually experience a considerable variation in temperature
due to viscous heat dissipation. This significantly affects the bearing performance as
lubricant viscosity is a strong function of temperature. Moreover, excessive rise in
temperature can cause oxidation of the lubricant and, consequently, lead to failure of
the bearing. Pressure also influences the viscosity of the lubricant to certain extent.
Usually viscosity increases exponentially as the pressure increases which in turn
increases the load capacity of the journal bearing. Researchers have studied the
behavior of circular journal bearing by adopting various numerical approaches to
simulate the performance in accordance with the real conditions.
1.1.2 Multilobe Journal Bearings /Non-Cylindrical Bearing Bore
Journal bearings are widely used for supporting rotating shafts in a wide
variety of applications. Due to the manufacturing tolerances, deflection of journal and
bearing support, asymmetric bearing load, etc, journal bearings may quite often
operate in the misaligned condition.
Distributed across the entire shaft diameter, there are as many individual
hydrodynamic carrying forces directed at the center of the shaft as there are lobes.

7 | P a g e
The strength of the individual hydrodynamic force is, among other things, dependent
on the width of the wedge gap. The vector total of all the individual carrying forces
represents the effective load capacity of the bearing towards the outside. This results
in a strong cantering effect being applied to the shaft which produces good
concentricity and generates a defined shaft position. By matching the lubricant
viscosity to the shaft´s peripheral speed and the wedge gap shape, the degree of the
hydrodynamic carrying force and the bearing friction can be varied to meet individual
requirements.
The characteristic of the multilobe bearing is the non-cylindrical bearing bore. This
deviates from conventional, cylindrical, hydrodynamic bearings by having two or
more lobes. The lobe radius (Rb) is larger than the shaft radius (Rj) by a specific
amount. These differences in the radius of the shaft and the lobe results in the
formation of a wedge gap in each arc. This gap begins at the oil inlet groove, an
axially positioned at the widest point of the respective arc. As a rule, the narrowest
point of the gap lies in the center of the lobe. When the shaft begins rotating, basic
theory dictates that the lubricant‟s adhesive effect on the shaft and lobes acts to pull
the lubricant into this gap, which narrows in the direction of rotation. Peak pressure
develops between the shaft and the bearing. Once this pressure reaches a certain level,
it lifts the shaft off the bearing. Thus, the shaft and the bearing are separated by the
lubricant gap. In other words, the shaft operates hydrodynamically with no metal to
metal contact.
1.1.2.1 Two Lobes Bearings
The geometry of the two lobe bearing is shown in Fig. 1.4. The bore is made
of two arcs of larger radius than for a circular bearing. It forms two pads with
opposing forces. In order to simplify the manufacturing process, the bearing bore is
machined after two shims are placed at a split between two halves of a round sleeve.
After round machining, the two shims are removed and the ellipse type shape is
achieved. In fact, the shape is not precisely elliptical, but the bearing has larger
clearances on the two horizontal sides and smaller clearance in the upper and bottom
sides. In this way, the bearing operates as a two-pad bearing, with action and reaction
forces in opposite directions.
The elliptical or „lemon-bore‟ bearing is a variation of the cylindrical axial groove
bearing with a reduced clearance in one direction. As a result, the elliptical bearing is

8 | P a g e
less susceptible to self-excited vibrations or instability at high speeds than the plain
cylindrical journal bearing. Furthermore, it is relatively of low cost and easily
manufactured. Therefore, it is a machine component commonly found in high speed
machinery such as turbines or turbo gear boxes. They are usually manufactured by
boring the circular bearing profile with shims inserted at the joint of the two bearing
halves. The shims are then removed and the bearing cap reassembled, resulting in a
reduced clearance in the direction perpendicular to the joint. However, the increased
viscous heat loss in the lubricating film will result in a larger increase of the
temperature in the lubricating film compared to that in the cylindrical bearing. For
horizontally split bearings, this design creates an increased vertical pre-load onto the
shaft.
1.1.2.2 Three-Lobes Bearings
Various designs have been developed to prevent the undesired effect of
bearing whirl. An example of a successful design is the three-lobe journal bearing
shown in Fig 1.5. It has three curved segments that are referred to as lobes. During
operation, the geometry of the three lobes introduces preload inside the bearing. This
design improves the stability because it increases the bearing stiffness and reduces the
magnitude of the cross-stiffness components. The preferred design for optimum
stability is achieved if the center of curvature of each lobe lies on the journal center
trajectory. This trajectory is the small circle generated by the journal center when the
journal is rolling in contact with the bearing surface around the bearing. According to
this design, the journal center is below the center of each of the three lobes, and the
load capacity of each lobe is directed to the bearing center.
The calculation of the load capacity of each lobe is based on a simplifying assumption
that the journal is running centrally in the bearing. This assumption is justified
because this type of bearing is commonly used at low loads and high speeds, where
the shaft eccentricity is very small.
An additional advantage of the three-lobe bearing is that it has oil grooves between
the lobes. The oil circulation is obviously better than for a regular journal bearing
(3600
). This bearing can carry higher loads when the journal center is over an oil
groove rather than over the center of a lobe [22].

9 | P a g e
Fig. 1.4 Two lobe bearing
Fig. 1.5 Three lobes bearing
1.1.2.3 Offset-Halves Bearings
Offset-Halves Bearings are frequently used in gearboxes connecting turbine
and generator for the power generation industry. Where primary direction of force,
constant direction of rotation is found, high bearing load capacity, long service life
and high stiffness and damping values are the main criteria. As a rule, if this
equipment is operated at full power, these requirements can be met by lemon bore
bearings. However, the equipment must often be operated at lower performance
levels, particularly in times of reduced current needs. It is precisely under these
conditions that lemon bore clearance bearings may produce unstable conditions,
resulting in equipment shut-down in order to avoid damage. Thus Offset-Halves

10 | P a g e
Bearings prove to be a technical alternative to conventional lemon bore shaped
bearings.
Fig. 1.6 Offset-halves bearing
Fig. 1.7 Offset-halves bearing
Journal
Bearing
Upper lobe
Lower lobe
Journal
LOAD

11 | P a g e
The goal is thus to find a hydrodynamic bearing which, on the one hand, has the
durability of a lemon bore bearing while, on the other hand, it shows the stiffness and
damping properties which permit light loads at high rotational speeds. The Offset-
Halves Bearings ideally meets these requirements, as numerous theoretical studies
and practical applications have proved (e.g. some lemon bore bearing installations
showed unstable operating behaviour. This was completely eliminated by retrofitting
with Offset-Halves Bearings.). The bore shape of the Offset-Halves Bearings (shown
in Fig. 1.6, 1.7)can be described by the horizontal offset of two cylindrical half-shells
with respect to the bearing radius Rb. “Horizontal” in this instance is generally
considered to be the direction of the bearing split line for split bearings. The shaft
radius, RJ, is smaller by the radial bearing clearance, C, than the remaining part of
one-half of the bore diameter.
As can be seen from the geometry of the bore, the bearing design lends itself to bi-
directional rotation. It also offers the advantage of a long, minimally convergent inlet
gap, resulting in high load-bearing capacity and simultaneously high coolant
throughout (hydrodynamic lateral flow). At the same time, the externally applied
force and the compression resulting from the horizontal displacement of the bearing
halves accurately holds the shaft in the lubricant film. This effect produces excellent
hydrodynamic characteristics, such as elastic rigidity and damping by the generated
oil film.
To judge conventional lemon bore bearings against Offset-Halves Bearings, it is
helpful to compare the most important hydrodynamic bearing characteristics. These
are, for example, friction, oil flow, as well as stiffness and damping over the
characteristic load bearing capacity range.
1.1.2.4 Advantages of Multi Lobes Bearing:
 Virtually no metal to metal contact between shaft and bearing while the
machine is operating.
 Dampened, low-oscillation, noise and wear-free shaft operation. If the oil
supply is operating properly, virtually unlimited bearing service life.
 Several supportive lubrication films distributed around the shaft circumference
guarantee that the shaft is generally centred, thus significantly improving
concentricity.
 Permits high continuous loading.

12 | P a g e
 Shock loads of several times the level of the continuous load are acceptable
 Low friction losses.
 Good lubricant flushing and cooling affects structural adaptability to every
existing machine Construction.
1.2 ROUGHNESS THEORY
Roughness means that most parts of a surface are not flat but form either a
peak or a valley. The typical amplitude between the peaks and valleys for engineering
surfaces is about one micrometre. The profile of a rough surface is almost always
random unless some regular features have been deliberately introduced. The random
components of the surface profiles look very much the same whatever their source,
irrespectively of the absolute scale of size involved.
1.2.1 Stochastic roughness
Early research primarily focused on stochastic surface roughness, which
occurs naturally during manufacture. The manufacturing processes are commonly
engineered to create stochastic features including preferential groove orientations
during machining, controlled porosity or optimum asperity statistical distribution
during ceramic forming, or an array of micro vanes that occur due to elastomer
deformation in rotary lip seals.
The study of the effects of surface roughness on the hydrodynamic lubrication of
various bearing systems has been a subject of growing interest. This is mainly because
of the reason that, in practice all bearing surfaces are rough. The study of the effect of
surface roughness has a greater importance in the study of porous bearings as the
surface roughness is inherent to the process used in their manufacture. In general, the
roughness asperity height is of the same order as the mean separation between the
lubricated contacts. In such situations, surface roughness affects the performance of
the bearing system.
Characterization of surface topography is important in applications involving friction,
lubrication, and wear (Thomas, 1999). In general, it has been found that friction
increases with average roughness. The effect of roughness on lubrication has also
been studied to determine its impact on issues regarding lubrication of sliding
surfaces, compliant surfaces, and roller bearing fatigue. Finally, some researchers

13 | P a g e
have found a correlation between initial roughness of sliding surfaces and their wear
rate. Such correlations have been used to predict failure time of contact surfaces
The stochastic study of Tzeng and Saibel (1967) has fascinated several investigators
in the field of tribology. Patir and Cheng (1978-1979) proposed an average flow
model for deriving the Reynolds type equation which is applicable to any general
surface roughness structure. Chritensen (1970) proposed a new stochastic averaging
approach for the study of roughness effects on the hydrodynamic lubrication of
bearings. Christensen and Tonder (1969) presented a comprehensive general analysis
for the two types of one dimensional surface roughness patterns Viz. transverse and
longitudinal, based on the general probability density function and this approach
formed the basis for the study of surface roughness effects by several researchers
(Gururajanand Prakash2003). In all these studies it is assumed that, the probability
density function for the random variable characterizing the surface roughness is
symmetric with zero mean. However, in general due to non-uniform rubbing of the
surfaces, especially in slider bearings the distribution of surface roughness may be
asymmetrical. In view of this, Andharia et al. (2001)studied the effect of surface
roughness on the performance characteristics of one-dimensional slider bearings with
an assumption of the probability density function for the random variable
characterizing the surface roughness is asymmetrical with a non-zero mean. All these
studies are limited to the study of surface roughness effects on bearing performance
with Newtonian lubricants. All previous developments were based on the highly
unrealistic assumption of perfectly smooth bearing surfaces. In reality, however,
engineering surfaces are covered with asperities. Even for a ground surface, asperities
might reach 1.25 in height and ten times this value in lateral spacing; the lateral
distance we equate with the in-plane characteristic length Lxz. The minimum film
thickness in a journal bearing, say, of diameter D = 25 mm operating at eccentricity
ratio = 0.5 is hmin = 12.5 ; this minimum film thickness is selected here to
represent the across-the-film characteristic length, Ly. Because the average asperity
height is one order of magnitude smaller than the minimum film thickness, one might
be tempted to ignore surface roughness altogether. However, the local characteristic
lengths are of the same order of magnitude, Ly = Lxz =12.5 , violating the thin film
assumption of lubrication analysis, and it becomes questionable whether the Reynolds
equation is at all valid. In cases when the lubrication approximation still holds even

14 | P a g e
though the surfaces are rough, one may be dealing with Reynolds roughness. When
there is significant pressure variation across the film due to surface roughness, to the
extent that the lubrication approximation is no longer valid, Stokes equation instead of
Reynolds equation must be employed; thus dealing with Stokes roughness (Elrod,
1973). Just where the demarcation between these two roughness regimes lies, is not
currently known. Compounding the difficulties is the fact that the asperity height
distribution for most machined surfaces is random, and statistical methods must be
applied when attempting to model lubrication between rough surfaces.
The classical theory of hydrodynamic lubrication does not consider the surface
roughness of the elements having relative motion. This theory is applicable in thick
film lubrication, when the load is very high and film thickness is very small, there is a
probability of asperity –asperity contact. Rough surface has been modelled as a
stochastic process by Chistensen in hydrodynamic bearing. Both one dimensional
(longitudinal and transverse) and two dimensional (Isotropic) models of roughness
were considered for roughness slope of about 10-12. It has been reported that surface
roughness has significant effect on steady state characteristics of hydrodynamic
bearing when roughness height is of the same order of magnitude as film thickness.
Majumdar and Hamrock (1981) have studied the effect of roughness on finite journal
oil bearing. The effect of surface roughness parameter, surface pattern, eccentricity
ratio and length to diameter ratio on hydrodynamic load and side leakage was
investigated. For an accurate prediction of journal bearing performance
characteristics, the consideration of the surface roughness in the analyses is
imperative. Good bearing properties in any part are obtained when the surface has
large number of hills and valleys, as the hills in an irregular surface reduce the metal-
to-metal contact and the valleys help to retain the film of lubricating oil. Due to oil
storage between the surfaces of journal and bearing it reduces the metal to metal
contact of these two surfaces. Christensen and Tonder (1973) have given three
different types of the roughness models namely one-dimensional longitudinal,
transverse roughness and two dimensional isotropic roughness models. In one
dimensional longitudinal roughness model, the roughness is assumed to have the form
of long narrow ridges and valleys running in the direction of the sliding, while in case
of one dimensional transverse roughness model, the roughness is assumed to have the
form of long narrow ridges and valleys running in the direction perpendicular to
sliding. In isotropic roughness model, the roughness is assumed to be uniformly

15 | P a g e
distributed over the bearing surface with no preferred direction. Patir and Cheng
(1978) proposed an average flow model for determining effects of three-dimensional
roughness for deriving Reynolds type equations applicable to any general roughness
structure. In their extended work Patir and Cheng (1979) included deriving the shear
flow factor for various roughness configurations. This model has been used by a
number of researchers. Khalil and EI-Shorbagy (1985) found that surface roughness
has pronounced effect on the operating characteristics of bearings, especially at lower
values of the lubricant film thickness and higher values of the wave number. The
surface roughness always increases frictional power and decreases lubricant flow rate.
Andres (1990b) studied the pocketed hybrid journal bearing by considering the effect
of surface roughness along with fluid inertia effect. The surface roughness in his
analysis has been modelled by an effective roughness depth varying from 10 to 30%
of radial clearance and showed an improvement of 20% in the dynamic performance
of the bearing. However, the work reported by San Andres (1993) did not take into
account the height distribution of the surface irregularities and the surface pattern,
which are the inherent properties of any finished surfaces.
Nagaraju et al. (2002) studied the effects of surface roughness on the performance of
capillary compensated hole-entry hybrid journal bearing of symmetric and
asymmetric configurations. It has been observed that the surface roughness heights
are typically of same order as that of fluid film thickness of the journal bearing. So the
performance of journal bearing system gets altered. For a hole-entry journal bearing
system operating in the hydrostatic mode of operation, the transversely oriented
roughness pattern provides a higher load carrying capacity as compared to a
corresponding similar bearing with a smooth surface, whereas the longitudinally
oriented roughness pattern provides a lesser load capacity. Inclusion of surface
roughness effects in the analysis affects the bearing dynamic coefficients. Further, the
maximum enhancement in the stability threshold speed margin is found to be of the
order of 41%, 85% and 131% for transverse, isotropic and longitudinal roughness
pattern, respectively for an asymmetric hole-entry journal-bearing configuration.
Sharma et al (2002b) described the static and dynamic performance of an orifice
compensated hole-entry hybrid journal bearing system considering the combined
influence of surface roughness and journal misalignment. The concepts of stochastic
process to the problem of surface roughness in hydrodynamic bearing, two different

16 | P a g e
models of hydrodynamic lubrication in conjunction with rough bearing surfaces are
developed.
The first of these models is associated with a one-dimensional, longitudinal
roughness. The second model applies to a one-dimensional, transverse roughness. The
surfaces roughness is said to be one dimensional if its roughness varies only in one
direction, such as in the x direction as shown in Fig 1.8. In this figure, the rough
surface is considered to be the stationary bearing surface which is sliding against
moving flat surfaces.
This condition is typical lubrication process with one sided rough. The roughness is
said to be longitudinal if sliding is along the ridges means, in these models the
roughness is assumed to have the form of long, narrow ridges and valleys running in
the direction of sliding(X-direction) (shown in Fig. 1.9) and transverse if it is normal
to the surfaces such as, in these models the roughness is assumed to have the form of
long, narrow ridges and valleys running in the direction of perpendicular to the
direction of sliding. (Shown in Fig 1.10)
Fig. 1.8 One directional roughness
Fig. 1.9 Bearing with longitudinal roughness
Longitudinal Roughness
Uh
V
V=0, Longitudinal
Uh=0, Transverse

17 | P a g e
Fig. 1.10 Bearing with transverse roughness
1.2.2 Surface texturing / Deterministic asperities
Deterministic asperities are patterned surface features with arbitrarily
specified geometries that are controllable and repeatable.
A texture surface is a surface whose form is composed of such repeated patterns.
Commonly, micro asperities can be created by photo etching, laser texturing or by a
ultra-violet photolithographic process [23].
Deterministic micro asperities show potential for enhancement of lubrication in
conformal contacts as found in many bearing and seal designs. Several manufacturing
methods have been proposed for deterministic micro asperities. Of these, laser
texturing has emerged as the most viable option. The resulting asperities can be
positive (protuberances) or negative (recesses) and can have heights (depths) from 1–
1000 microns and be patterned over surface areas up to about 150mm. Surface
metrology indicates submicron accuracy of form and 13 nm Ra roughness on the
asperity tops (land). Tribology testing in a non-pressurized oil bath indicates full film
conditions and shows a 14–22% reduction in friction coefficient for a thrust surface
covered with the micro asperities [23].
Micro asperities are the micro and nano-sized peaks and valleys on a surface that
constitute the surface roughness. Depending on the size, shape, and distribution of
these asperities, the hydrodynamic lubrication characteristics of the surface can vary
significantly.
On fluid bearings and seals, control of the lubrication properties using micro
asperities can alter load capacity, friction torque, dynamic stiffness, and damping
coefficients, among others. This, in turn, significantly affects energy consumption,
reliability, and vibration in rotating machines. An important distinction is that of
deterministic micro asperities versus deterministic macro asperities. Macro asperities
Transverse Rough ness

18 | P a g e
are typically large area surface features with extremely low height to diameter aspect
Ratios (0.0001).
Unlike micro asperities, macro asperities have found widespread application since
1970s. Examples include the use of sinusoidal waves, full wave and half wave (Figs.
1.11, 1.12, & 1.13) in the form of longitudinal and transverse type as discussed
earlier, on mechanical seal faces. Macro surface features are typically few in number
and therefore can be manufactured with comparative ease using processes including
grinding and chemical etching. By contrast, deterministic micro asperities are orders
of magnitude smaller in average diameter, significantly greater in number and have
larger aspect ratios 0.001–10. These properties make it extremely difficult to cost
effectively manufacture large fields of asperities with controllable and repeatable
geometry. Laser texturing has found application to end face mechanical seals, and
reciprocating automotive components. In laser texturing, negative asperities recesses
are cut into the surface using a focused laser.
Fig. 1.11 Sinusoidal wave textures
Fig. 1.12 Positive half wave textures
Fig. 1.13 Positive full wave textures
Amplitude
W
Amplitude
W
Amplitude
e
W
Surface

19 | P a g e
1.3 Outline of the Thesis
The work carried has been reported in five chapters of the thesis. Introduction
of the journal bearings for different bearing surfaces and outline of the thesis has been
presented in Chapter-1. Review of the relevant literature is introduced in Chapter-2.
The governing equations and numerical procedures for different types of journal
bearings with specific surface roughness have been described in Chapter-3.
Analytic results for different bearing parameter (load carrying capacity, friction force,
coefficient of friction & oil side flow) and temperature distribution for different
bearing profile at specific operating conditions have been presented in Chapter-4.
Conclusions of the present work with scope for future work are presented in Chapter-
5.

20 | P a g e
CHAPTER-2
LITERATURE REVIEW
This chapter provides details of research carried out on hydrodynamic
bearings for circular, offset-halves and elliptical journal bearings with and without
using the surface roughness parameter. There is enormous information available on
the theoretical and analytical work of the smooth and rough circular journal bearings.
However, such works pertaining to non-circular journal bearings especially elliptical
and offset-halves journal bearings while considering different types of roughness
parameter are limited and hence, are the main areas of focus in the present study.
2.1 CIRCULAR SMOOTH JOURNAL BEARING
Hydrodynamic journal bearings are widely used in industry because of their
simplicity, efficiency and low cost. They support rotating shafts over a number of
years and are often subjected to many stops and starts. The mechanism of pressure
development, temperature and load caring capacity in full and split journal bearing
have been analysed by many researchers.
Newkirk and Taylor (1925) experimental reported a new kind of self-excited rotor
dynamic instability in hydrodynamic type journal bearings. They observed that during
this instability the rotor orbits in its bearing at a frequency approximately half of the
rotor speed. They reported that at the onset of this self-excited instability, the rotor
behavior is unlike critical speed resonance where the amplitude of motion builds up as
the rotor reaches its critical speed and then decreases as it passes the critical speed. At
the inception of this non-synchronous whirling, the amplitude of the rotor motion
continuously builds up at the frequency of approximately half of that of the rotor
speed and never dies down. They concluded that these types of lateral vibrations of
rotor are due to the action of lubricating oil film and referred to this self-excited rotor
dynamic instability as oil whirl. They found that oil whirl can be prevented by
shutting off the oil supply to the bearings. From their experiments they also concluded
that these sub-synchronous vibrations may be prevented by misaligning the bearings
slightly, by the use of friction damped bearings or by avoiding the lightly loaded
shafts. Hagg (1984) provided some theoretical insights into the phenomenon of oil
whirl. He stated that during the stable motion or steady state condition of rotor in a
bearing, the hydrodynamic fluid forces developed by the oil film are equal to the

21 | P a g e
external load. But during the whirling motion of the shaft, the hydrodynamic fluid
forces overcome the external load and act as an “energy source” accelerating the shaft
in circular orbit. Pinkus in 1957 conducted an extensive experimental investigation on
oil whirl.
During his work he investigated the effects of loading, speed, viscosity, amount of oil,
unbalance, flexibility and external excitation on oil whirl. His work led to very
important conclusions such as the fact that the whirl is independent of balancing.
Pinkus (1958) also reported that a lower lubricant temperature tended to have
stabilizing effects on the rotor bearing system which is opposite to findings reported
by Newkirk and Lewis stating (1956) that a higher lubricant inlet temperature
promoted stability.
Pinkus (1958) derived the solution of Reynolds differential equation for finite journal
bearing having 1000& 750 arcs and their results were applied for partial bearing as
well as to three groove and four groove full journal bearing for L/D ranging from ½ to
4 and eccentricity ratio up to 0.95 for calculating minimum film thickness, power loss,
oil flow and load carrying capacity. Further author has applied the solution for 1500
arc to a full two groove bearing, in that case the forces and flow were identical for the
partial and full bearing, but here at least two often times three lobes develop pressure
affecting the load capacity and magnitude of flow and finally conclusion comes that
the value of attitude angle rises with a decreases in L/D ratio and with rising arc
angle, the relative improvement in load capacity decreases with the L/D ratio.
Holmes in 1960 presented that it is possible to predict the oil whirl threshold speed
by considering oil film as elastic and a viscous system. He developed a stability chart
for the prediction of onset of oil whirl by expressing the dynamic oil film forces
acting on the shaft / journal in terms of the linearized velocity dependent (damping)
and displacement dependent (stiffness) terms. He tabulated the stability chart by using
a linear perturbation stability analysis. It should be mentioned that the idea of
representing the dynamic response characteristics of a journal bearing by means of
stiffness and damping coefficients originates from Stodola and Hummel in 1926.
Their aim was to improve the calculation of critical speed of a rotor by including the
flexibility of bearing oil film.
Pinkus (1961) introduced a new parameter α/β ratio where represent the load angle
and represent the bearing arc angle, author mentioned α/β = 0.5 for constant
centrally loaded bearing and calculated the Sommerfeld number, side oil flow and

22 | P a g e
pressure distribution for partial arc 1500, 1000, 750 at eccentricity ratio 0.2-0.95 for
L/D ratio ¼ to 1. Author concluded that two general families of problems such as: (1)
Partial arc bearing with off-set, in this it is a simply partial journal bearing with the
load family at anywhere along the arc. From the data calculated by author it is
possible to calculate the performance of non- centrally loaded partial arc bearing. (2)
Grooved bearing: author described that, in practice full bearing contain axial or
circumferential groove and these convert such full bearing in to random partial arc
bearing. The value of α/β will be different for the various bearing arc.
Gethin and Medwell (1986) presented an isothermal condition for full films journal
bearing as well as incomplete film analysis extended to the cavitation zone that
existing bearing used in the bi-rotational system. This system is then considered to
operate in the starved mode and, since these conditions are met more commonly at
high speeds, the lubricant flow to be assumed as nonlaminar. Author demonstrates the
effect of including the cavitation region of the bearing in the calculation of the
dimensionless power, for the case where there is a complete film at the entry to the
converging section of bearing. Results show that if the cavitated zone is excluded,
then the power loss is underestimated by approximately 40% over a wide range of
eccentricity ratio. In the case of compliant journal bearings the study of hydrodynamic
behavior was reported by Heshmat et al. (1983) where the authors solved the
Reynolds equation using the Newton-Raphson method and reported the effect of
various structural, geometrical and operational variables on the bearing behavior.
Also, estimation of load capacity for foil gas journal bearing was made using a “Rule
of Thumb” by Dellacorte et al. (2000). This was based on the first principles and data
available in the literature and it relates bearing load capacity to the bearing size and
speed through an empirically-based, load-capacity coefficient. It reported that the
“first generation” compliant support elements have a relatively low load-carrying
capacity compared to the more advanced ones developed by Heshmat (1994) which
achieved a breakthrough load-carrying capacity of 670,000 Pa at 59,700 rpm. This
advanced design, which is referred to as the “third generation”, has unique compliant
support elements where the elastic structural properties are modified with the use of
multi-stage bumps and advanced solid lubricant coating. This design showed overall
improvement of bearing performance at higher speeds, including a better load-
carrying capacity.

23 | P a g e
Dammak (2005) has studied the Reynolds pressure boundary conditions, the influence
of the eccentricity ratio ε and the length-to-diameter ratio λ on the pressure
distribution of two-dimensional journal bearings.There are many new boundary
conditions taken into consideration with the influence of the bearings on the
hydrodynamic effect. In order to solve the Reynolds equation to obtain the film
pressure distribution of the journal bearing system, the Reynolds conditions are
used.The numerical solution of journal bearing lubrication has been presented using
the finite element method. It is shown that there is a considerable difference between
the pressure profiles using the Reynolds and half-Sommerfeld pressure boundary
condition. The results agree more and more when λ increases. The short bearing
assumption gives simpler solution than the infinite long bearing assumption.Author
concluded that the modern bearings tend to be shorter than those used a few decades
ago. Ratios of length to diameter λ (L/D) are commonly in the range of 0.25 to 0.75.
This results in flow in the z direction (and the end leakage) being a major portion of
the total lubricant flow. In lubrication mechanics, the Reynolds pressure boundary
condition is widely applied to the analysis of journal bearings.
Nuruzzaman and Khalil (2010) presenteda comparative study of pressure distribution
and load capacity of a cylindrical bore journal bearing. For calculating the pressure
distribution and load capacity of a journal bearing, isothermal analysis was carried
out. Using both analytical method and finite element method, pressure distribution in
the bearing was calculated. Moreover, the effects of variations in operating variables
such as eccentricity ratio and shaft speed on the load capacity of the bearing were
calculated. The analytical results and finite element results were compared and were
also validated with the available published results.
2.2 NON-CIRCULAR BEARING
Pinkus and Lynn (1956) derived the power losses for elliptical and three-lobe
bearings, both symmetrical and asymmetrical as functions of the bearing parameters
and bearing ellipticity. Further authors gave expressions for two cases. In the first one
by assuming a complete oil film and in the second one by taking into account the
incompleteness of the oil film in the diverging sections of the bearing. They also
presented the analysis of elliptical bearings based on the numerical solution of
Reynolds equation for finite bearings. The solution of the differential equation carried

24 | P a g e
out by authors was supplemented by additional work on the nature of the oil flow,
power loss, and eccentricity in elliptical bearings.
Wilcock (1961) worked towards the possibility of displacing the lobe centers of two-
lobe journal bearings orthogonally with respect to the mid-radius of the lobe. The
author showed that when the lobe displacement is in a direction opposite to the shaft
surface motion, and the bearing is centrally loaded, shaft stiffness orthogonal to the
load vector is substantially increased. At the same time, vertical stiffness essentially
remains unchanged and minimum film thickness is decreased; particularly at low
loads, while oil flow is increased. Author also carried out an analysis for a bearing
having in cross-section two arcs (each subtending an angle of 1500), L/D=1/2, and
with the arc centers each displaced from the geometric center by half the radial
clearance.
Black and Murray (1974) presented a theory which allows the characteristics of
bearings operating in the laminar or turbulent regimes to be evaluated by a similar
method, using less storage requirements than finite difference methods and bearings
of different geometries can be easily analysed using the program structure. When
multi-lobe bearing configurations are being considered, the load magnitudes and
directions are dependent on the bearing characteristics and cannot be directly
calculated. The authors constructed a databank to provide information on circular,
partial arc, offset halves, and lemon bore bearings operating in the laminar and
turbulent regimes, together with a fast interpolation sub program.
Flack et al. (1980) developed the pressures profile in four-lobe bearings, both
experimentally and analytically. Then they tested a four-lobed bearing 25.4mm in
diameter with the load vector „on pad‟ and „off pad‟. Static pressures were measured
on the Centre line of the bearing and the experimental data was compared with two
sets of theoretical results. The authors used Half-Sommerfeld and Reynolds boundary
conditions in the theoretical predictions. It is observed by the authors that the trends
of the pressure versus rotational speed for the experimental data and the theoretical
solution are the same for the Half-Sommerfeld condition but sometimes differ for the
Reynolds condition.
Singh and Gupta (1982) considered the stability limits of elliptical journal bearings
supporting flexible rotors. They solved Reynolds equation numerically for several
values of the eccentricity ratio (0.2-0.8), the L/D ratio, and the dimensionless velocity
of the journal centre. The authors observed that the operating load, ellipticity, L/D

Journal Bearing Lubrication Manoj Kumar Singh Thesis

Journal Bearing Lubrication Manoj Kumar Singh Thesis

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