dynamics of a rigid shaft supported by angular contact ball bearings

299 Pages·2008·18.99 MB·English

Contents
CONTENTS
Abstract 2
Dedication 4
Acknowledgements 5
Contents 6
Notation 11
CHAPTER 1General Introduction
1.1 Introduction 18
1.2 Angular Contact Ball Bearings as a Source of Vibrations 18
1.3 Damping of Vibrations 21
1.4 Objectives of the Thesis 22
1.5 Layout of the Thesis 22
1.6 Definition of Some Factors Influencing Ball Bearing Vibrations 23
CHAPTER 2 Previous Work
2.1 Introduction 30
2.2 Early Work on Ball Bearing Vibration 32
2.3 A Brief Look at Research Post 1960 34
2.4 Experimental Work 35
2.5 Elastic Contact Behaviour of Ball Bearings 41
2.6 Important Frequencies Associated with Ball Bearings 44
2.6.1 Natural Frequency 44
2.6.2 The Ball Passage Frequency 46
2.6.3 Flexural Vibrations of the Outer Ring due to Ball Loads 54
2.6.4 Natural Frequencies of Outer Ring 55
2.6.5 Natural Frequencies of Ball Motion 55
2.7 Vibration Monitoring and Frequencies due to
the Imperfections of Ball Bearings 57
2.7.1 Imperfections of Ball Bearings 59
2.7.1.1 Waviness 60
2.7.1.2 Oft-sized Balls 64
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Contents
2.7.1.3 Misalignment 66
2.7.2 Defects 70
2.8 Shaft-Ball Bearings with External Dampers 74
2.9 Elastomers as External Dampers 77
2.10 Background of the Project 81
CHAPTER 3 Mechanics of Ball Bearings
3.1 Introduction 84
3.2 Elastic Contact Modelling 85
3.2.1 Contact Mechanics 85
3.2.2 Calculation of the Stiffness Factor 89
3.3 Bearing Geometry 90
3.3.1 Calculation of Preloaded Contact Angle 90
3.3.2 Calculation of Deflection 92
3.3.3 Calculation of Total Force on a Bearing 100
3.4 Dynamic Modelling of a Vibrating Shaft Supported by Ball Bearings 102
3.4.1 Assumptions 103
3.4.2 Equations of Motion 104
3.4.3 Dimensionless Groups 108
3.4.4 Computational Solution of the Equations of Motion 111
3.4.4.1 Initial Conditions 111
3.4.4.2 Computer Flow Chart and the Solution Procedure 112
CHAPTER 4 Vibration Frequencies of a Shaft-Bearing System
4.1 Introduction 114
4.2 The Inherent System Frequencies 115
4.2.1 System Natural Frequency 115
4.2.2 Cage Speed 117
4.2.3 Ball Rotational Speed 120
4.2.4 Relative Speeds 121
4.2.5 Ball Passage Frequency 121
4.3 Manufacturing Malfunctions 123
4.3.1 Waviness 124
4.3.1.1 Inner Ring Waviness 126
4.3.1.2 Outer Ring Waviness 127
4.3.1.3 Ball Waviness 128
4.3.2 Off-sized Balls 129
4.3.3 Misalignment 130
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Contents
4.3.3.1 Deflection due to Misalignment 131
4.3.3.2 Contact Angle Variation due to Misalignment 132
4.3.4 Out of Balance of Mass Centre 134
4.3.5 Defects on the Rolling Surfaces 135
4.3.5.1 Defect on the Outer Race Surface 136
4.3.5.2 Defect on the Inner Race Surface 137
4.3.5.3 Defect on the Ball Surface 137
CHAPTER 5 Elastomers as External Dampers
5.1 Introduction 139
5.2 Properties of Elastomeric Materials 140
5.2.1 Large Deformations 140
5.2.2 Dynamic Properties of Elastomers 141
5.2.3 Geometry Effect 141
5.2.4 Frequency Effect 142
5.3 Formulation of Elastomers 144
5.3.1 Rectangular Cross-Section Buttons 146
5.3.2 Cylindrical Buttons 147
5.3.3 Ring Cartridges 149
5.3.4 0-Rings 150
5.4 Experimental Results Used in This Thesis 153
5.4.1 Material Selection 156
5.4.2 Cylindrical Button Cartridge Test Samples 156
5.4.3 Ring Cartridge Test Samples 157
5.4.4 0-ring Test Sample 158
5.5 Representation of the Experimental Data 159
5.5.1 Generalised Viscoelastic Models 160
5.5.2 Generalised Force-Displacement Relationships 161
5.5.3 Direct Fit of Data to Mathematical Expressions 161
5.6 Brief Description of the Subroutine 163
5.7 Modelling of The System 166
5.8 Excitation Frequency 168
CHAPTER 6 Results and Discussion: Ball Bearings
6.1 Introduction 171
6.2 Shaft-Ball Bearing Specifications 171
6.3 System Natural Frequency 174
6.4 Ball Passage Frequency 179
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Contents
6.4.1 Effect of Varying the Number of Balls 186
6.6.2 Effect of Varying the Preload 189
6.5 Waviness 192
6.5.1 Outer ring Waviness 192
6.5.2 Inner ring Waviness 201
6.5.3 Ball Waviness 205
6.6 Off-sized Balls 208
6.7 Misalignment 217
6.7.1 Both Bearings are Misaligned 217
6.7.1.1 Outer ring Misalignment 217
6.7.1.2 Inner ring Misalignment 218
6.7.1 LHS Bearings is Misaligned 218
6.8 Defects on the Rolling Surfaces 219
6.8.1 Outer race Defect 220
6.8.2 Inner race Defect 221
6.8.3 Defect on the Ball Surface 223
6.9 Vibrations in 5 Degrees of Freedom 224
CHAPTER 7 Results and Discussion: Elastomers as External Dampers
7.1 Introduction 228
7.2 Dynamic Properties of Elastomers 229
7.2.1 Cylindrical Buttons 229
7.2.2 Ring Cartridges 233
7.2.3 0-rings 234
7.3 Simulation of Elastomer External Dampers 236
with Mechanical Voigt Model
7.3.1 Cylindrical Buttons 237
7.3.2 0-rings 239
7.4 Elastomer Dampers in Assembly 241
CHAPTER 8 Overall Conclusions and Suggestion for Future Work
8.1 Introduction 250
8.2 Conclusions for Ball Bearings 250
8.2.1 System Natural Frequency 250
8.2.2 The Ball Passage Frequency 251
8.2.3 Outer ring Waviness 251
8.2.4 Inner ring Waviness 252
8.2.5 Ball Waviness 252
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Contents
8.2.6 Off-sized Balls 253
8.2.7 Misalignment 253
8.2.8 Defective Running Surfaces 254
8.3 Conclusions for Elastomers as External Dampers 255
8.4 Recommendation for Further Research 257
REFERENCES 259
APPENDIX 1 Equivalent Voigt Model 272
APPENDIX 2 Phase-Plane Representation 273
APPENDIX 3 Large Deformations of Elastomers 277
APPENDIX 4 Other Methods of Obtaining the Stiffness Coefficients of 279
Ring Cartridges
APPENDIX 5 Previously Derived Prediction Methods for Elastomers 284
APPENDIX 6 Voigt Model 289
APPENDIX 7 Ball Passage Frequency 291
APPENDIX 8 Fast Fourier Transform 294
APPENDIX 9 Runge-Kutta Method 296
APPENDIX 10 The Effect of Preload and Number of Balls on 297
The Natural Frequency
10
Notation
NOTATION
symbol Description Units
a the distance between the misalignment centre and bearing centre m
Or ball waviness amplitude m
or cross-sectional area of button (Chapter 5) m2
or width of the rectangular cross-sectional elastomer button (Chapter 5) m
Or distance between the external forces and the LHS bearing (see Fig.(3.16))m
al position of the LHS bearing from the C.G. (see Fig.(3.16)) m
A amplitude of the lobes m
or cross-sectional area of button (Chapter 5) m2
Or a constant (Chapter 5)
or distance between raceway groove centres m
b breadth of the rectangular cross-sectional elastomer button (Chapter 5) m
bi . position of the RHS bearing from the C.G. (see Fig.(3.16)) m
B total curvature, A / db
or a constant (Chapter 5)
c viscous damping factor N s/m
ce equivalent viscous damping factor N slm
d diameter m
or diameter of elastomer buttons (Chapter 5) m
db ball diameter m
di inner raceway diameter m
do, pitch diameter m
do outer raceway diameter m
D diameter of the buttons m
or shaft diameter at groove base m
E Young's modulus NI/n/2
Or error function (Chapter 5)
Eeff effective Young's modulus (Chapter 5) N/m2
11
Notation
E. static Young's modulus (Chapter 5) N/m2
E* complex Young's modulus of elasticity N/m2
E' real part of the complex Young's modulus of elasticity N/m2
E" imaginary part of the complex Young's modulus of elasticity N/rn2
f frequency Hz
fbp ball passage frequency Hz
fc cage rotation frequency Hz
fn natural frequency Hz
fw wave passage frequency Hz
nf n th super harmonic of one frequency Hz
F( p) curvature difference 1/m
F external forces N
external tensile force (Chapter 5) N
Or
g acceleration due to gravity m/s2
G shear modulus of elasticity N/m2
G* complex shear modulus of elasticity N/m2
Nim2
G real part of the complex shear modulus of elasticity
GG " imaginary part of the complex shear modulus of elasticity N/rn2
h the height of the buttons (Chapter 5) m
or time increment s
H compliance rn/N
H* complex compliance rn/N
H1 real part of the complex compliance m/N
H2 imaginary part of the complex compliance m/N
m4
I second moment of area
k material constant, proportional to hardness
K contact stiffness factor N/m3/2
Runge-Kutta interval constant
Or
K* complex stiffness N/m
K1 real part of the complex stiffness N/m
K2 imaginary part of the complex stiffness N/m
K e equivalent stiffness N/m
K, inner raceway/ball contact stiffness factor N/m312
K0 outer raceway/ball contact stiffness factor N/m3r2
/ length of the specimen (Chapter 5) m
12
Notation
L a characteristic length m
or mean circumferential length of a cartridge ring (Chapter 5) m
number of balls
m
or number of Voigt elements (Chapter 5)
mor mass of outer ring kg
mass of the shaft kg
M
or moment N m
n mode order
or shaft speed cp.m.
ni inner race (i.e., shaft) speed Ep.m.
no outer race speed Ep.m.
N number of data points
or number of wave lobes (Chapter 5)
or number of buttons (Chapter 5)
P empirical constant for a particular geometry
q instantaneous centre of movement m
3 radius m
inner groove radius m
r •
gi
outer groove radius m
rgo
inner raceway radius m
r•I
pitch radius m
rm

r outer raceway radius m
0
the amplitude of forcing function N
Po
preload N
PR
R ball centre focus radius under zero load m
R, inner radius of the ring cartridge or 0-ring (Chapter 5) m
R0 outer radius of the ring cartridge or 0-ring (Chapter 5) m
R mean radius of ring cartridge or 0-ring (Chapter 5 —see Fig.(5.8)) m
S entropy
S#. shape factor equivalent to loaded area divided by force free area
I time s
or thickness of the ring cartridge (Chapter 5) m
A t time step s
T temperature K
3 velocity m/s
13
Notation
or volume (Chapter 5) m3
Vi inner raceway velocity nVs
V° outer raceway velocity m/s
V c cage velocity m/s
w work done by the representative chain (Chapter 5) Nm
w 1 weighting constant
wz weighting constant
W total contact force between a ball and its race N
or the total work done (Chapter 5) Nm
Woff total contact force due to off sized ball N
x, y, z local coordinates attached to the shaft (see Fig.(3.17))
or local coordinates attached to the reference ball (see Fig.(3.15))
X, Y, Z fixed coordinates (see Fig.(3.17))
X1, Yi, Zi local fixed coordinates at the C.G. (see Fig.(3.17))
X '1, Y11,
Z'i intermediate coordinates following Euler's angles (see Fig.(3.17))
a contact angle rad.
a. Instantaneous contact angle rad.
i
a unloaded contact angle rad.
0
a preloaded contact angle rad.
P
shape factor
P
(5 contact deflection m
or phase angle as in Equ.(5.5) rad.
Oc deflection due to misalignment m
total instantaneous contact deflection for the i th baU m
3i
AS diameter difference of the off sized ball m
& amplitude of the vibration (Chapter 5) m
6* dimensionless contact deflection
lc elliptical eccentricity parameter
A wave length m
or extension ratio (Chapter 5) rad.
V Poisson's ratio
0 the angle between the fixed and moving
reference axes (see Fig.(3.15)) rad.
Or phase difference
14
Notation
e convergence criterion m
or strain
• rock angle of the shaft about y fads rad.
3 angle between neighbouring two balls Fig.(3.15)) rad.
(see
or number of chains (Chapter 5)
integration angle rad.
g
damping ratio
C
N1/12
a stress
angular position of balls rad.
0
arbitrary angular position of balls rad.
0'
P curvature 1/m
or misalignment angle (Chapter 5) rad.
Zp curvature sum 1 /m
to angular velocity of the shaft radis
ball angular velocity in the same lp ane ast htee cage speedd rad./s
ball angular velocity about its centre rad./s
wb
a) cage angular velocity rad./s
c
angular velocity about the instantaneous axis of zero velocity rad./s
WCL
a). inner race angular velocity radfs
I
to outer race angular velocity rad./s
cr
a) ball spinning velocity rad./s
s
12 The stiffness divided by the damping factor (Klc ) (Chapter 5) 1/s
17 angular speed vector of the shaft
V rock angle of the shaft about x axis rad.
the amplitude of vibration in x direction m
X0
the amplitude of vibration in y direction m
Yo
n the amplitude of the waves at the contact angle m
initial wave amplitude m
FO
3 maximum amplitude of the wave m
P
X complete elliptic integral of the first kind
3 complete elliptic integral of the second kind
radius of the loci groove centres of curvature for inner ring m
radius of the loci groove centres of curvature for outer ring m
910
/ a characteristic length m
15