Bearing Design In Machinery

Bearing Design In Machinery
Engineering Tribology And Lubrication
Avraham Harnoy
Table of Contents
Preface
Symbols
Chapter 1 Classification and Selection of Bearings
1.1 Introduction
1.2 Dry and Boundary Lubrication Bearings
1.3 Hydrodynamic Bearing
1.4 Hydrostatic Bearing
1.5 Magnetic Bearing
1.6 Rolling Element Bearings
1.7 Selection Criteria
1.8 Bearings for Precision Applications
1.9 Noncontact Bearings for Precision Application
1.10 Bearing Subjected to Frequent Starts and Stops
1.11 Example Problems
Chapter 2 Lubricant Viscosity
2.1 Introduction
2.2 Simple Shear Flow
.2.3 Boundary Conditions of Flow
2.4 Viscosity Units
2.5 Viscosity–Temperature Curves
2.6 Viscosity Index
2.7 Viscosity as a Function of Pressure
2.8 Viscosity as a Function of Shear Rate
2.9 Viscoelastic Lubricants
Chapter 3 Fundamental Properties of Lubricants
3.1 Introduction
3.2 Crude Oils
3.3 Base Oil Components
3.4 Synthetic Oils
3.5 Greases
3.6 Additives to Lubricants
Chapter 4 Principles of Hydrodynamic Lubrication
4.1 Introduction
4.2 Assumptions of Hydrodynamic Lubrication Theory
4.3 Hydrodynamic Long Bearing
4.4 Differential Equation of Fluid Motion
4.5 Flow in a Long Bearing
4.6 Pressure Wave
4.7 Plane-Slider Load Capacity
4.8 Viscous Friction Force in a Plane-Slider
4.9 Flow Between Two Parallel Plates
4.10 Fluid-Film Between a Cylinder and Flat Plate
4.11 Solution in Dimensionless Terms
Chapter 5 Basic Hydrodynamic Equations
5.1 Navier–Stokes Equations
5.2 Reynolds Hydrodynamic Lubrication Equation
5.3 Wide Plane-Slider
5.4 Fluid Film Between a Flat Plate and a Cylinder
5.5 Transition to Turbulence
5.6 Cylindrical Coordinates
5.7 Squeeze-Film Flow
.Chapter 6 Long Hydrodynamic Journal Bearing
6.1 Introduction
6.2 Reynolds Equation for a Journal Bearing
6.3 Journal Bearing with Rotating Sleeve
6.4 Combined Rolling and Sliding
6.5 Pressure Wave in a Long Journal Bearing
6.6 Sommerfeld Solution of the Pressure Wave
6.7 Journal Bearing Load Capacity
6.8 Load Capacity Based on Sommerfeld Conditions
6.9 Friction in a Long Journal Bearing
6.10 Power Loss on Viscous Friction
6.11 Sommerfeld Number
6.12 Practical Pressure Boundary Conditions
Chapter 7 Short Journal Bearings
7.1 Introduction
7.2 Short-Bearing Analysis
7.3 Flow in the Axial Direction
7.4 Sommerfeld Number of a Short Bearing
7.5 Viscous Friction
7.6 Journal Bearing Stiffness
Chapter 8 Design Charts for Finite-Length Journal Bearings
8.1 Introduction
8.2 Design Procedure
8.3 Minimum Film Thickness
8.4 Raimondi and Boyd Charts and Tables
8.5 Fluid Film Temperature
8.6 Peak Temperature in Large, Heavily Loaded Bearings
8.7 Design Based on Experimental Curves
Chapter 9 Practical Applications of Journal Bearings
9.1 Introduction
9.2 Hydrodynamic Bearing Whirl
9.3 Elliptical Bearings
9.4 Three-Lobe Bearings
.9.5 Pivoted-Pad Journal Bearing
9.6 Bearings Made of Compliant Materials
9.7 Foil Bearings
9.8 Analysis of a Foil Bearing
9.9 Foil Bearings in High-Speed Turbines
9.10 Design Example of a Compliant Bearing
Chapter 10 Hydrostatic Bearings
10.1 Introduction
10.2 Hydrostatic Circular Pads
10.3 Radial Pressure Distribution and Load Capacity
10.4 Power Losses in the Hydrostatic Pad
10.5 Optimization for Minimum Power Loss
10.6 Long Rectangular Hydrostatic Bearings
10.7 Multidirectional Hydrostatic Support
10.8 Hydrostatic Pad Stiffness for Constant Flow-Rate
10.9 Constant-Pressure-Supply Pads with Restrictors
10.10 Analysis of Stiffness for a Constant Pressure Supply
10.11 Journal Bearing Cross-Stiffness
10.12 Applications
10.13 Hydraulic Pumps
10.14 Gear Pump Characteristics
10.15 Flow Dividers
10.16 Case Study: Hydrostatic Shoe Pads in Large Rotary Mills
Chapter 11 Bearing Materials
11.1 Fundamental Principles of Tribology
11.2 Wear Mechanisms
11.3 Selection of Bearing Materials
11.4 Metal Bearings
11.5 Nonmetal Bearing Materials
Chapter 12 Rolling Element Bearings
12.1 Introduction
12.2 Classification of Rolling-Element Bearings
12.3 Hertz Contact Stresses in Rolling Bearings
12.4 Theoretical Line Contact
.12.5 Ellipsoidal Contact Area in Ball Bearings
12.6 Rolling-Element Speed
12.7 Elastohydrodynamic Lubrication in Rolling Bearings
12.8 Elastohydrodynamic Lubrication of a Line Contact
12.9 Elastohydrodynamic Lubrication of Ball Bearings
12.10 Force Components in an Angular Contact Bearing
Chapter 13 Selection and Design of Rolling Bearings
13.1 Introduction
13.2 Fatigue Life Calculations
13.3 Bearing Operating Temperature
13.4 Rolling Bearing Lubrication
13.5 Bearing Precision
13.6 Internal Clearance of Rolling Bearings
13.7 Vibrations and Noise in Rolling Bearings
13.8 Shaft and Housing Fits
13.9 Stress and Deformation Due to Tight Fits
13.10 Bearing Mounting Arrangements
13.11 Adjustable Bearing Arrangement
13.12 Examples of Bearing Arrangements in Machinery
13.13 Selection of Oil Versus Grease
13.14 Grease Lubrication
13.15 Grease Life
13.16 Liquid Lubrication Systems
13.17 High-Temperature Applications
13.18 Speed Limit of Standard Bearings
13.19 Materials for Rolling Bearings
13.20 Processes for Manufacturing High-Purity Steel
13.21 Ceramic Materials for Rolling Bearings
13.22 Rolling Bearing Cages
13.23 Bearing Seals
13.24 Mechanical Seals
Chapter 14 Testing of Friction and Wear
14.1 Introduction
14.2 Testing Machines for Dry and Boundary Lubrication
14.3 Friction Testing Under High-Frequency Oscillations
14.4 Measurement of Journal Bearing Friction
14.5 Testing of Dynamic Friction
14.6 Friction-Testing Machine with a Hydrostatic Pad
.14.7 Four-Bearings Measurement Apparatus
14.8 Apparatus for Measuring Friction in Linear Motion
Chapter 15 Hydrodynamic Bearings Under Dynamic Conditions
15.1 Introduction
15.2 Analysis of Short Bearings Under Dynamic Conditions
15.3 Journal Center Trajectory
15.4 Solution of Journal Motion by Finite-Difference Method
Chapter 16 Friction Characteristics
16.1 Introduction
16.2 Friction in Hydrodynamic and Mixed Lubrication
16.3 Friction of Plastic Against Metal
16.4 Dynamic Friction
Chapter 17 Modeling Dynamic Friction
17.1 Introduction
17.2 Dynamic Friction Model for Journal Bearings
17.3 Development of the Model
17.4 Modeling Friction at Steady Velocity
17.5 Modeling Dynamic Friction
17.6 Comparison of Model Simulations and Experiments
Chapter 18 Case Study: Composite Bearing—Rolling Element
and Fluid Film in Series
18.1 Introduction
18.2 Composite-Bearing Designs
18.3 Previous Research in Composite Bearings
18.4 Composite Bearing with Centrifugal Mechanism
18.5 Performance Under Dynamic Conditions
18.6 Thermal Effects
Chapter 19 Non-Newtonian Viscoelastic Effects
19.1 Introduction
19.2 Viscoelastic Fluid Models
.19.3 Analysis of Viscoelastic Fluid Flow
19.4 Pressure Wave in a Journal Bearing
19.5 Squeeze-Film Flow
Chapter 20 Orthopedic Joint Implants
20.1 Introduction
20.2 Artificial Hip Joint as a Bearing
20.3 History of the Hip Replacement Joint
20.4 Materials for Joint Implants
20.5 Dynamic Friction
Appendix A Units and Definitions of Material Properties
Appendix B Numerical Integration
Bibliography
.Symbols
NOMENCLATURE FOR HYDRODYNAMIC
BEARINGS
a~ ¼ acceleration vector
a ¼ tan a, slope of inclined plane slider
B ¼ length of plane slider (x direction) (Fig. 4-5)
C ¼ radial clearance
c ¼ specific heat
e ¼ eccentricity
F ¼ external load
Ff ¼ friction force
F(t) ¼ time dependent load; having components FxðtÞ, FyðtÞ
h ¼ variable film thickness
h
n ¼ hmin, minimum film thickness
h0 ¼ film thickness at a point of peak pressure
L ¼ length of the sleeve (z direction) (Fig. 7-1); width of a plane slider
(z direction) (Fig. 4-5)
m ¼ mass of journal
N ¼ bearing speed [RPM]
n ¼ bearing speed [rps]
O; O1 ¼ sleeve and journal centers, respectively (Fig. 6-1)
.p ¼ pressure wave in the fluid film
P ¼ average pressure
PV ¼ bearing limit (product of average pressure times sliding velocity)
q ¼ constant flow rate in the clearance (per unit of bearing length)
R ¼ journal radius
R1 ¼ bearing bore radius
t ¼ time

t¼ ot, dimensionless time
U ¼ journal surface velocity
V ¼ sliding velocity
VI ¼ viscosity index (Eq. 2-5)
W ¼ bearing load carrying capacity, Wx, Wy, components
a ¼ slope of inclined plane slider, or variable slope of converging
clearance
a ¼ viscosity-pressure exponent, Eq. 2-6
b ¼ h2=h1, ratio of maximum and minimum film thickness in plane
slider
e ¼ eccentricity ratio, e=C
f ¼ Attitude angle, Fig. 1-3
l ¼ relaxation time of the fluid
r ¼ density
y ¼ angular coordinates (Figs. 1-3 and 9-1)
t
xy; tyz; txz ¼ shear stresses
s
x:sy; sz ¼ tensile stresses
o ¼ angular velocity of the journal
m ¼ absolute viscosity
mo ¼ absolute viscosity at atmospheric pressure
n ¼ kinematic viscosity, m=r
NOMENCLATURE FOR HYDROSTATIC BEARINGS
A
e ¼ effective bearing area (Eq. 10-25)
B ¼ width of plate in unidirectional flow
di ¼ inside diameter of capillary tube
_Eh
¼ hydraulic power required to pump the fluid through the bearing and piping
system
_Ef
¼ mechanical power provided by the drive (electrical motor) to overcome the
friction torque (Eq. 10.15)
_ E
t ¼ total power of hydraulic power and mechanical power required to maintain
the operation of hydrostatic bearing (Eq. 10-18)
h
0 ¼ clearance between two parallel, concentric disks
H
p ¼ head of pump ¼ Hd
Hs
.Hd ¼ discharge head (Eq. 10-51)
H
s ¼ suction head (Eq. 10-52)
k ¼ bearing stiffness (Eq. 10-23)
K ¼ parameter used to calculate stiffness of bearing ¼ 3kAeQ
L ¼ length of rectangular pad
l
c ¼ length of capillary tube
pd ¼ pump discharge pressure
pr ¼ recess pressure
ps ¼ supply pressure (also pump suction pressure)
Dp ¼ pressure loss along the resistance
Q ¼ flow rate
R ¼ disk radius
R0 ¼ radius of a round recess
Rf ¼ flow resistance ¼ Dp=Q
Rin ¼ resistance of inlet flow restrictor
Tm
¼ mechanical torque of motor
V ¼ fluid velocity
W ¼ load capacity
Z ¼ height
Z1 ¼ efficiency of motor
Z2 ¼ efficiency of pump
k ¼ constant that depends on bearing geometry (Eq. 10-27)
b ¼ ratio of recess pressure to the supply pressure, pr=ps
m ¼ fluid viscosity
g ¼ specific weight of fluid
NOMENCLATURE FOR ROLLING ELEMENT
BEARINGS
a ¼ half width of rectangular contact area (Fig. 12-8)
a, b ¼ small and large radius, respectively, of an ellipsoidal contact area
d ¼ rolling element diameter
di; do ¼ inside and outside diameters of a ring
E
eq ¼ equivalent modulus of elasticity [N=m2]
E^ ¼ elliptical integral, defined by Eq. 12-28 and estimated by Eq. 12.19
F
c ¼ centrifugal force of a rolling element
h
c ¼ central film thickness
hmin; hn ¼ minimum film thickness
k ¼ ellipticity-parameter, b=a , estimated by Eq. 12.17
L ¼ An effective length of a line contact between two cylinders
m
r ¼ mass of a rolling element (ball or cylinder)
.n
r ¼ number of rolling elements around the bearing
p ¼ pressure distribution
pmax ¼ maximum Hertz pressure at the center of contact area (Eq. 12-15)
qa ¼ parameter to estimate, E, defined in Eq. 12-18 ^
r ¼ deep groove radius
R1; R2 ¼ radius of curvatures of two bodies in contact
R1x; R2x ¼ radius of curvatures, in plane y; z, of two bodies in contact
R
1y; R2y ¼ radius of curvatures, in plane x; z, of two bodies in contact
R
eq; ¼ equivalent radius of curvature
R
r ¼ race-conformity ratio, r=d
R
s ¼ equivalent surface roughness at the contact (Eq. 12-38)
Rs1 and Rs2 ¼ surface roughness of two individual surfaces in contact
R
x ¼ equivalent contact radius (Eqs. 12-5, 12-6)
Rd ¼ curvature difference defined by Eq. 12-27
t* ¼ parameter estimated by Eq. 12.25 for calculating tyz in Eq. 12-24
T^ ¼ elliptical integral, defined by Eq. 12.28 and estimated by Eq.
12-22
UC ¼ velocity of a rolling element center (Eq. 12-31)
Ur
¼ rolling velocity (Eq. 12-35)
W ¼ dimensionless bearing load carrying capacity
W ¼ load carrying capacity
Wi; Wo ¼ resultant normal contact forces of the inner and outer ring races in
angular contact bearing
W
max ¼ maximum load on a single rolling element
N ¼ bearing speed [RPM]
a ¼ viscosity-pressure exponent
a ¼ linear thermal-expansion coefficient
a
r ¼ radius ratio ¼ Ry=Rx
L ¼ a ratio of a film thickness and size of surface asperities, Rs (Eq.
12-39)
d
m ¼ maximum deformation of the roller in a normal direction to the
contact area (Eq. 12-7, 12-21)
x ¼ ratio of rolling to sliding velocity
t
xy; tyz; txz ¼ shear stresses
s
x; sy; sz ¼ tensile stresses
m0 ¼ absolute viscosity of the lubricant at atmospheric pressure
n ¼ Poisson’s ratio
o ¼ angular speed
oC ¼ angular speed of the center of a rolling element (or cage)
[rad=s]
r ¼ density
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