Bearing Design in Machinery Episode 3 Part 12 doc

Unlike the metal-on-metal curve, the dynamic f –U curve with UHMWPE has
considerable hysteresis for dry friction. This effect suggests that the friction of
polymers involves viscous friction.
Several cycles are measured, and the curve shows a good repeatability of
the cycles—except for the first cycle (dotted line), which has a higher stiction
force. Unlike what we see with metal-on-metal testing, the friction coefficient
increases with the velocity, reaching a maximum of f ¼ 0:26. In this case, the
breakaway friction at each cycle approaches zero. However, at the first cycle of
the experiment (dotted line), there is a higher stiction force, and the breakaway
friction coefficient is near 0.2.
Figure 20-4 is for identical conditions, but lubrication is provided with a
very light (low viscosity) oil, m ¼ 0:001 N-s=m
2
. This curve simulates the friction
in an actual joint implant. The curve indicates that even for a low viscosity and
speed, the bearing operates in the boundary and mixed lubrication regime, and the
friction decreases versus sliding velocity. This curve also shows a considerable
hysteresis. For lubricated surfaces, the first cycle (dotted line in Fig. 20-4) also
demonstrates a higher stiction force of f ¼ 0:25 while the following cycles have a
reduced maximum breakaway coefficient of f ¼ 0:2.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.
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Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.
Appendix A
Units a nd De¢nitions of Material
Properties
A.1 UNIT SYSTEMS
The traditional unit system in the United States has been the Imperial system,
often referred to as the British system, although in United Kingdom the Imperial
system was replaced by the SI International System (Syste
`
me Internationale,

French). In the United States, the engineering societies are in favor of adopting
SI, and most engineering publications and textbooks currently use SI units. Many
engineering companies are in transition from Imperial to SI units, so engineers
must be familiar with the two systems. For this reason, this text uses both
systems, although most of the example problems are presented in SI units.
The SI is based on three units: mass, length, and time. The unit of mass is
the kilogram (kg), that of length is the meter (m), and that for time is the second
(s). The unit of force is the Newton (N), which is defined by Newton’s second law
as the force required to accelerate 1 kg of mass at the rate of 1 m
2
=s.
Gravitational acceleration is g ¼ 9:81 m
2
=s, so the weight (force exerted by
gravity at the earth’s surface) of 1 kg mass is
F ¼ mg ¼ 1 Â9:81 ¼ 9:81 N ðA-1Þ
The unit of energy (or work) is the Joule (J), which is equivalent to N-m. The unit
of power, which is energy per unit of time, is the watt (W). The watt is equivalent
to J=s, or, in basic SI units, N-m=s.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.
Pressure or stress is force per unit area. The SI unit is the pascal (Pa), which
is equivalent to N=m
2
. This is a small unit, and prefixes such as kPa (10
3
Pa) and
MPa (10
6
Pa) are often used.
In SI units, very large or very small numbers are often needed in practical

problems, and the following prefixes serve to indicate multiplication of units by
various powers of 10:
m ðmicro-Þ¼10
À6
k ðkilo-Þ¼10
3
m ðmilli-Þ¼10
À3
M ðmega-Þ¼10
6
c ðcenti-Þ¼10
À2
G ðgiga-Þ¼10
9
For example, the well-known Imperial unit of pressure is psi (lb
f
=in:
2
).
1 psi is ¼ 6895 N=m
2
ðPaÞ¼6:895 kPa:
A second example is the modulus of elasticity of the steel:
E ¼ 2:05 Â10
11
Pa ðN=m
2
Þ¼2:05 Â10
5
MPa; ¼ 2:05 Â10

2
GPa:
A.2 DEFINITIONS OF MATERIAL PROPERTIES
A.2.1 Density, r
Material density r is mass per unit volume. The SI unit of density is kg=m
3
.In
Imperial units, the density is lb
m
=ft
3
,orlb
m
=in
3
. For example, the density of
water at 4

C is 1000 kg=m
3
, and in imperial units it is 62:43 lb
m
=in:
3
.
The conversion is
1kg=m
3
¼ 0:06243 lb
m

=ft
3
:
A.2.2 Speci¢c Weight, g
Specific weight, g, is the gravity force (weight) per unit volume of the material
g ¼ rg ðA-2Þ
The SI unit of density is N=m
3
. For example, the specific weight g of water at 4

C
is 9810 N=m
3
, obtained by the equation
g
water
¼ rg ¼ 1000 Â 9:81 ¼ 9810 N=m
3
:
The Imperial unit of specific density is lb
f
=ft
3
,orlb
f
=in
3
. For example, the
specific weight g of water at 4


Cis62:4lb
f
=ft
3
. The conversion is
1lb
f
=ft
3
¼ 157:1N=m
3
1N=m
3
¼ 0: 00636 lb
f
=ft
3
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.
A.2.3 Speci¢c Gravity, S
Specific gravity, S, of a material is the ratio of its specific weight to the specific
weight of water at 4

C. It is also the ratio of its density to the density of water at
4

C. For example, if the density of a steel is 7800 kg=m
3
, its specific density is
7800=1000 ¼ 7.8 (specific gravity is a dimensionless ratio).
A.2.4 Speci¢c Heat, c

Specific heat, c, is the amount of heat that must be transferred to a unit of mass of
a material to raise its temperature by one degree. For gas, the specific heat
depends if the unit of mass has a constant pressure, c
p
, or if the unit of mass has a
constant volume c
v
. The specific heat of a material is a function of its
temperature. The SI unit of specific heat is J=Kg-

C (a widely used unit is
KJ=Kg-

C), and the Imperial unit is BTU=lb
m

F.
The conversion ratio is
1 BTU=lb
m

F ¼ 2326 J=Kg-

C
1 BTU=lb
m

F ¼ 2:326 KJ=Kg-

C

A.2.5 Thermal Conductivity, k
The thermal conductivity is a measure of the rate of heat transfer through a
material. It is the coefficient k in the Fourier Law of heat conduction
q ¼ÀkA
@T
@x
ðA-3Þ
where q is the rate of heat transfer, A is the area normal to the temperature
gradient @T=@x.
The SI unit of thermal conductivity is Watt per meter per Celsius degree,
W=m-C. The Imperial unit of thermal conductivity is BTU=h-ft-

F. The conver-
sion ratio is
1W=m-C ¼ 0:57782 BTU=h-ft-

F:
A.2.6 Absolute Viscosity, m
The absolute viscosity, m, is a measure of the fluid resistance to flow. The
viscosity and its units are presented in Chap. 2. The SI unit of absolute viscosity
is N-s=m
2
(or Pa-s). An additional widely used unit is the poise, (P) (after
Poiseuille), which is dyne-s=cm
2
, and a smaller traditional unit is centipoise (cP).
1 centipoise; ðcPÞ¼10
À2
 poise
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.

An Imperial unit for the viscosity is the reyn (after Osborne Reynolds), which is
lbf -s=in:
2
.
Conversions
1. 1 centipoise is equal to 1:45 Â 10
À7
reyn
2. 1 centipoise is equal to 0.001 N-s=m
2
3. 1 centipoise is equal to 0.01 poise
4. 1 reyn is equal to 6:895 Â10
3
N-s=m
2
5. 1 reyn is equal to 6:895 Â10
6
centipoise
6. 1 N-s=m
2
is equal to 10
3
centipoise
7. 1 N-s=m
2
is equal to 1:45 Â 10
À4
reyn
A.2.7 Kinematic Viscosity, n
The kinematic viscosity, n, is the ratio of the absolute viscosity and density

n ¼
m
r
ðA-4Þ
The SI unit of kinematic viscosity is m
2
=s. Additional widely used traditional unit
is the stokes (St) (after Stokes), which is cm
2
=s, and a smaller unit is the
centistokes (cSt), which is mm
2
=s.
The common Imperial unit is in:
2
=s.
Conversions
1 centistokes, cSt ¼ 10
6
m
2
=s
1 stokes, St ¼ 10
4
m
2
=s
1m
2
=s ¼ 6:452 Â10

À4
in:
2
=s
1m
2
=s ¼ 10
À4
stokes
1in:
2
=s ¼ 0:00155 cSt
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.
Appendix B
Numerical Integration
The pressure wave along the bearing is solved by integration of Eq. 4-13.
Although some integrals can be solved analytically, complex functions can be
solved by numerical integration. This appendix is a survey of the various methods
for numerical integration, and examples are presented. A simple numerical
integration is demonstrated by means of a spreadsheet computer program,
which is favored by engineers and students for its simplicity, and because the
spreadsheet program can be used for graphic presentation of the pressure wave.
The methods of approximate numerical integration are based on a summa-
tion of small areas of width Dx below the curve, which are approximated by
various methods that include the midpoint rule, rectangle rule, trapezoidal rule,
and Simpson rule.
B.1 MIDPOINT RULE
Integration by midpoint rule is an approximation. The area below the curve is
approximated by the sum of the rectangular areas, as shown in Fig. B-1.
The integral is approximated by the following equation:

ð
b
a
f ðxÞdx %
b À a
n
P
n
i¼1
f ðx
j
Þ
x
j
¼
x
iÀ1
þ x
i
2
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.
B-3 TRAPEZOI DAL RULE (FIG. B-3)
The integral is approximated by the following equation:
ð
b
a
f ðxÞdx % T
T ¼
Dx
2

½f ðx
0
Þþ2f ðx
1
Þþ2f ðx
2
Þþ 2f ðx
iÀ1
Þþf ðx
i
Þ
Dx ¼
b À a
n
The endpoints, at points a and b, are counted only once, while all the other points
have the coefficient 2.
B-4 SIMPSON RULE (FIG. B-4)
The Simpson rule is based on approximating the graph by parabolas rather than
straight lines. The parabola is determined each time by the three consecutive
points through which it passes.
f ðx
iÀ1
Þ; f ðx
i
Þ and f ðx
iþ1
Þ
The area from ðx
iÀ1
)to(x

iþ1
Þ is
A
i
¼
Dx
3
ðf ðx
iÀ1
Þþ4f ðx
i
Þþf ðx
iþ1
Þ
FIG. B-3 Integration by the trapezoidal rule.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.
finally summing the rectangular areas to obtain the total area under the curve (see
Fig. B-5).
The numerical integration is according to the equation
ð
b
a
f ðxÞdx %
P
n
i¼1
f ðx
i
ÞDx
i

Dx
i
¼
b À a
n
In this problem, the function is
f ðxÞ¼3x
2
0 x 2
ð
2
0
3x
2
dx ¼
P
n
i¼1
f ðx
i
ÞDx
i
x
i
¼ x
iÀ1
þ Dx
The summation is performed with the aid of a spreadsheet program (Table B-1).
The first and second rows are added for explanation. The number of rectangles is
selected (n ¼ 200), resulting in uniform Dx

i
¼ 0: 01. The third column shows the
values x
i
, and the fourth column shows the respective values of the function f ðx
i
Þ.
The fifth column lists the areas of the rectangles obtained by the product f ðx
i
ÞDx
i
.
The sixth (last) column lists the sum of the rectangles to the last x
i
. The solution
of this numerical integration is at the bottom of this column. The exact solution of
this integration is 8, and the errors of the various methods are compared in Table
B-2. The best precision is obtained using the Simpson method.
FIG. B-5 Approximate integration by summation of rectangles.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.