CALIFO~~IA
STATE UNIVERSITY, NORTHRIDGE
A VISCOUS FRICTION MODEL
A thesis submitted in partial satisfaction of the
requirements for the degree of Master of Science in
Engineering
by
Phyllis Lynn Dahl
May, 1982
The Thesis of Phyllis Lynn Dahl is approved:
California State University, Northridge
ii
ACKNOWLEDGEMENTS
I express a profound appreciation to my parents; Dorothy for her
incessant patience and support, and Philip for his invaluable counsel
and advisement during the course of this study.
A deep sense of
gratitude is also expressed to my husband Ron for the long hours he
spent typing and editing this document.
His endless encouragement,
diligence, and support have truly made a difference.
Finally, I
express thanks to The Aerospace Corporation for the use of their
laboratory facilities and personnel to perform the experimental portion
of this study.
iii
TABLE OF CONTENTS
page
LIST OF FIGURES •
vi
LIST OF PLATES
viii
NOMENCLATURE
ix
ABSTRACT
xi
Chapter
1.
INTRODUCTION
1
2.
BACKGROUND ON DYNAMICS OF FRICTION
3
3.
4.
5.
SOLID FRICTION
3
FLUID FRICTION
5
COMBINED FRICTION •
9
MATHEMATICAL MODEL DEVELOPMENT
12
THE SOLID FRICTION MODEL
12
THE VISCOUS HYPOTHESIS
15
A COMBINED FRICTION MODEL •
17
VISCOUS FRICTION MODEL VERIFICATION •
24
VFM SOLUTION
24
COUETTE SOLUTION
25
EXPERIMENT
34
EXPERIMENTAL RESULTS
44
CONCLUSION
47
NOTES •
•
BIBLIOGRAPHY
52
54
iv
APPENDIXES
A.
TEST PROCEDURE
57
v
LIST OF FIGURES
Figure
Page
1.
Solid Friction Models
2.
Fluid Friction Models
3.
Maxwell Model Element
4.
Dynamic Viscous Friction Models
5.
Bingham Model Element
6.
Combined Friction Models •
7.
Block Diagram of the Solid Friction Model
8.
SFM Predicted Hysteresis Loops
9.
Block Diagram of the Combined Friction Model
4
..
...
...
..
...
8
.......
11
..
...
6
9
11
...
.......
14
15
..
.....
19
10.
Block Diagram of the Viscous Friction Model
11.
Block Diagram of the Prandtl Model
21
12.
Block Diagram of
22
13.
Block Diagram of
14.
Block Diagram of the Newtonian Model
15.
Couette Flow Problem
16.
Steady-State Velocity Profile of Couette Flow
17.
Velocity-Time Profile Across the Gap
18.
Torque Profile at the Fixed Wall •
19.
Viscometer Design
20.
Physical Characteristics of Cylinders and Annuli •
38
21.
Experimental Set-Up
40
22.
Dry Pivot Friction Torque
.
the Coulomb Model . . . .
the Hooke Model . . . .
22
..
23
............
vi
...
• • • •
.........
25
27
30
32
•••••••••••••
..
20
...
........
35
42
...
45
23.
Experimental Friction Torque and Velocity
24.
Experimental Friction Torque-Time History
46
25.
Comparison of Experimental Results With Couette Solution •
SO
26.
Comparison of Experimental Results With VFM Solution • • •
51
vii
LIST OF PLATES
PLATE
page
I.
VISCOMETER:
CUP tVITH THREE DRUMS
II.
LABORATORY TEST EQUIPMENT
•••
III. EXPERIMENT DURING TEST PROCEDURE •
viii
..
........
........
.....
39
43
44
NOMENCLATURE
English Symbols
symbol
units
a
spatial frequency of sine term
rad/cm
A
coefficient of sine term
em/sec
th
n
A
area of viscometer wall, steady state
em
B
steady state viscous friction coefficient
dyne-cm-s/rad
c
coefficient of cosine term
em/sec
f
function
F
Force
dyne
Coulomb force level
dyne
G
shear modulus
dyne/em
h
viscometer gap
em
H
viscometer height
em
i
solid friction model parameter
k
Maxwell model spring stiffness
dyne-cm/rad
thermal conductivity of fluid
cal/gm- C
n
w
F
c
coefficient of sine term
em/sec
A
2
0
m
eiqenvalue of solution
m
n
n
R
radius of viscometer drum
em
t
time
sec
th
eigenvalue
ix
2
T
torque
dyne-em
T
c
Coulomb torque level
dyne-em
Temp
max
maximum centerline temperature between
two parallel plates
oc
temperature of both parallel plates
oC
velocity of fluid element
em/sec
velocity of moving wall
em/sec
u
spatially dependent velocity
em/sec
un
n
X
linear displacement
em
y
distance from fixed wall
em
z
distance from wall
Temp
u
u
w
w
th
velocity term
em/sec
Greek Symbols
y
elemental shear strain
rad
9
angle
rad
viscosity
dyne-s/em
\)
kinematic viscosity
2
em /sec
p
density
gm/cm
(J
friction model rest stiffness
dyne-cm/rad
friction model variable stiffness
dyne-cm/rad
shear stress
dyne/em
first time constant
sec
transportation delay time
sec
T
X
2
3
2
T
T
n
w
n
th
time constant
sec
shear stress at fixed wall
xi
dyne/em
2
ABSTRACT
A VISCOUS FRICTION MODEL
by
Phyllis Lynn Dahl
Master of Science in Engineering
A viscous friction model is developed which represents the dynamic
or transient behavior of viscous friction.
Stated as a general rule of
finite viscous friction, the viscous friction force lags the velocity
that produces it and in steady state it is proportional to that
velocity.
An experiment is designed and performed which validates the
analytical model.
A new combined friction model is also presented
which represents elastic-plastic or viscous behavior.
This combined
model exhibits the unifying characteristic of reducing to various
historical solid and viscous friction models in addition to the
proposed viscous friction model.
xii
Chapter 1
INTRODUCTION
The analyst in search of a mathematical model for viscous friction
has usually assumed friction force simply to be proportional to
velocity.
This model has been found adequate in nearly all control
systems analyses and has gone unquestioned to almost the same extent as
Newton's second law of motion.
In cases where the system dynamic model
appears to break down and fails to duplicate experimental data over the
spectrum of interest, elusive non-linear effects have been blamed and
an improvement in the fidelity of the model is often attempted by
adding dynamic fudge factors.
A brief exploration is made of the historical models that have been
used to mathematically describe the elastic, plastic and viscous flow
deformation of matter.
Recently, the solid friction model was
developed for use in simulations of dynamic systems which involved
mechanical elements that are subject to sliding and rolling friction.
1
An approach similar to the one taken with the solid friction model
is used to develop a viscous friction model which represents the
dynamic or transient behavior of viscous friction.
The improved
viscous friction model is linear; it does not include non-linear
effects.
Stated as a general rule of finite viscous friction, the
viscous friction force lags the velocity that produces it and in steady
state is proportional to that velocity.
1
2
An experiment is designed and performed which validates the
proposed analytical model.
Dynamical relations are derived for the
force on a stationary plate produced by laminar flow of a viscous fluid
between that plate and another moving parallel plate.
The response of
the plate drag force to a step change in velocity is derived and shows
the time lag behavior.
Velocity profiles across the gap are presented
to help explain the physical nature of the lag effect.
The force
velocity transfer function for the parallel plate or Couette flow
problem obtained from the step response analyses is found to be an
infinite series of first order lag terms.
A reasonable approximation
to this viscous friction model is obtained by using the first or
longest time constant lag term.
A new combined friction model is also put forth that realistically
represents the elastic-plastic or viscous characteristics of materials
and devices.
This combined model exhibits the unifying characteristic
of breaking down to various historical solid and viscous friction
models in addition to the solid friction model and the proposed viscous
friction model.
Chapter 2
BACKGROUND ON DYNAMICS OF FRICTION
The dynamics of friction in engineering systems and devices that
use bearings of all types, including rotating bearings, sliding
bearings, dashpots, and dampers have been represented in the past by
equations of the form
(1)
The first term on the right represents classical Coulomb or solid
friction, the second term represents classical Newtonian or viscous
friction, and the third term represents turbulent and other kinds of
non-viscous flow.
in this thesis.
The latter term and its concepts will not be treated
Due to the high performance requirements of modern
control systems it has been necessary to seek and refine high fidelity
simulation models which improve the representation of the classical
approach typified by Equation 1.
SOLID FRICTION
Perhaps the most rudimentary part of Equation 1 is the Coulomb
friction term.
2
This term has been refined and improved upon by the
solid friction model, referred to in this thesis as the SFM.
The
improvement was brought about through the observation that experimental
3
4
data showed rolling friction to be a function not only of velocity but
also of displacement.
This behavioral characteristic is represented in
Figure 1 which displays friction torque vs. angular displacement for
continuing positive velocity motion.
The torque-displacement
characteristic for positive velocity in this Figure is seen to be
generally elastic-plastic.
!
Coulomb ( 1784)
St. Venant
-
Prandtl (1904)
- - - ( Davidenkov (1938)
Solid Friction
Model ( 1968)
DISPLACEMENT , 8
Figure 1.
The model of Prandtl,
displacement,
3
Solid Friction Models
which is elastic up to a characteristic
ec , and subsequently plastic at a constant friction
level, T , was an improvement over the sudden step up to the constant
c
5
friction level, T , of the earlier Coulomb model.
c
model
The DavidenkoV
4 was a later improvement over the Prandtl model for
elastic-plastic materials that mathematically represent a continuously
changing function from elastic to plastic behavior.
Both the Prandtl
and Davidenkov models have been available for many years, but not
generally used in the study of dynamical systems.
The reason for this
is that there was no simple mathematical way of representing the
hysteretic behavior in the repeated or successive reversals in velocity
for other than simple Coulomb friction prior to the introduction of the
solid friction model.
The development of the higher fidelity solid
friction model coincided with the high speed and high capacity
capabilities that were being developed in modern computer technology.
FLUID FRICTION
Not as crude, generally speaking, as the early Coulomb model of
solid friction is the Newtonian model of viscous friction, 5 which is
the second term in the general friction Equation 1.
This simple
Newtonian model depicted in Figure 2, is linear, i.e.,
6
---Newton ( 1687)
I
Blasius (1908)
Nikuradse
(1932)
1UJ
:::J
0
0:::
0
1-
•
VELOCITY , 8
Figure 2.
Fluid Friction Models
Generally, it has been known that a non-linear dependence on velocity
is present in the mathematical representation of liquid shearing
friction.
The deviation from linear behavior is referred to as the
pseudoplastic or dilatant property of liquids and is a subject of
interest in the field of rheology.
In a steady state velocity sense, any fluid response not
representable by Equation 2 may be termed non-Newtonian.
In the past,
a number of relationships have been used in an effort to model this
non-linear behavior.
The simplest relationship that fits experimental
data being the power law model,
7
(3)
For most materials the exponent n is less than unity and the material
flows more easily the faster it is sheared and the viscosity thereby
decreases with shear velocity.
pseudoplastic.
This type of behavior has been termed
For a few materials, the exponent n is greater than
unity and the viscosity increases with shear velocity.
This type of
behavior has been called dilatant.
For most dynamic non-steady state modeling problems the linear
Newtonian model has been adequate and used almost exclusively in the
study of dynamics of systems with the usual engineering devices.
However, it is noted in the Newtonian model of Equation 2 that viscous
friction changes instantaneously with velocity.
It has been known, for
example in shear dampers and dashpots that operate in the laminar flow
regime, that dynamical effects which are not represented by the simple
linear Equation 2 are present.
This is taken to mean that no dynamical
time dependence is represented in Equation 2.
An improvement to the
Newtonian model can be made by employing the Maxwell model. 6
This
model is represented by the spring and dashpot series combination,
shown in Figure 3, and has the equation of motion
(4)
The steady state behavior of the Maxwell model is the same as the
Newtonian model, i.e., T
•
= Ba.
However, a dynamical term is present
as a result of the introduction of the spring term to the model which
brings in elastic behavior in addition to the viscous behavior.
8
Figure 3.
Maxwell Model Element
The dynamic response behavior of the Newton and Maxwell models is
illustrated for a step change in velocity in Figure 4, where the
Maxwell model shows a friction varying continuously with time, and the
Newton model shows friction changing instantaneously with the step
change in velocity.
Thus, the Maxwell model appears to appropriately
represent the observed dynamical behavior in shear dampers and
dashpots.
However, the solutions for the shear damper and dashpot
equations for non-steady Couette flow do not indicate a physical
7 8
mechanism that would explain the spring in the Maxwell model. '
In
fact the Couette solutions indicate that the dynamic term in Equation
4, (B/k)T, if it is an appropriate model, would be due to inertia
effects of the fluid mass in the shear damper gap rather than a spring
effect which is totally absent in a pure Newtonian fluid.
Nonetheless,
the Maxwell model would appear to appropriately introduce, as a first
approximation, a dynamic term which represents the dynamic effects of
9
. . Ir
~
WI
;::)
Ol
0::
~I
I
(1965)
I
I
I
TIME , t
T
=
J(9, 9,• t) Dynamic
9•
Figure 4.
Couette flow.
=
Function of Time
u(t)
Dynamic Viscous Friction Models
Therefore, the Maxwell model appears to be a good model
in representing the dynamics of viscous friction.
COMBINED FRICTION
When the solid and viscous friction effects are combined, it might
be expected that the Maxwell model would adequately represent linear
elastic and Newtonian viscous combined friction.
This turns out to be
the case in some situations but it should be pointed out that the
elastic term does not allow for the plastic deformation characteristics
of solid friction.
The elastic portion is due to elastic properties
10
within the fluid and cannot be considered as part of the solid friction
component.
Bingham9 evidently understood this shortcoming and added
a Coulomb friction element to the elements of the Maxwell model, as
shown in Figure 5.
The Bingham model has the equation of motion,
(5)
where,
T
= -k(e 1- e) 2
(6)
The responses of the Maxwell and Bingham models to a step input in
velocity are shown in Figure 6.
The Bingham model exhibits a linear
response with time due to the linear spring for the region of friction
torque below the Coulomb friction level, T , and a response identical
c
to the Maxwell response thereafter.
A SFM in conjunction with a
viscous component of friction could be expected to behave in a manner
similar to the Bingham model since the SFM has a steady state Coulomb
level as well as a static spring effect.
11
T
Figure 5.
Bingham Model Element
Bingham (1923)
Maxwell (1855)
TIME , t
T =
j (8, 8, t )
Dynamic Function of Time
•
8 = u (t)
Figure 6.
Combined Friction Models
Chapter 3
MATHEMATICAL MODEL DEVELOPMENT
The solid friction model was developed for use in simulations of
dynamic systems which involved mechanical elements that are subject to
sliding and rolling friction.
The foundation of this model was based
on the hypothesis that solid friction resulted from quasi-static
contact bonds which were continuously formed and subsequently broken.
This SFM has been proven to accurately simulate transient torque
behavior seen in tests in the laboratory on ball bearings and other
devices that exhibit solid friction.
It is desired to approach viscous friction from a similar viewpoint
to see if it is possible to develop a viscous friction model (VFM) that
more accurately represents the dynamic or transient behavior of viscous
friction.
THE SOLID FRICTION MODEL
Solid materials that undergo elastic-plastic deformation as a
function of time have a time rate of deformation given by;
dy
dt
1
dT
= G
dt
(7)
where we have considered the elemental shear strain, y, to be
produced by the shear stress, T.
The shear modulus, G, is taken to
12
13
be a non-linear function of
y
or
T
and so it includes elastic as
well as plastic shear modulus characteristics.
Now convert the
elemental or microscopic Equation 7 to macroscopic terms by
substitution of the torque T for
the angular deformation e for
T,
y, and the angular stiffness b for G.
Solving
•
for T we obtain
dT
Cit=
•
(8)
2:9
Equation 8 was the key relation upon which the non-linear solid
friction model was built.
The non-linear stiffness o is usually
considered to be a function of the deformation angle e but it can
alternatively be considered a function of the shear stress or torque.
A simple general stiffness relation was empirically developed that is
expressed as
10
T
T
•li
T < Tc
sgne
c
(9)
where o is the rest stiffness or the slope dT/de when T : 0,
T
c
is the saturation or running friction, and i is an exponent that
describes the plasticity or brittleness of the material or the
friction process.
The stiffness can conversely be thought of as the
slope of the torque vs. angular deflection characteristic,
E
dT
(10)
=de
and the shape of the characteristic can be found analytically by
substituting Equation 10 into Equation 9 and integrating.
example, for i
=1
•
and 9 > 0 we obtain for 9
=0
at T = 0,
As an
14
T
T
=
-(~)9
T (1 - e a
)
c
•
e > o
(11)
•
When the rate reverses, i.e. 9 < 0, a new solution must be obtained,
using as initial conditions those values of T and 9 existing at the
instant of rate reversal.
Successive rate reversals produce the
hysteresis behavior of solid friction.
The hysteresis behavior of solid friction and of solid materials is
nicely represented by the solution of Equations 9 and 10 as depicted in
the block diagram in Figure 7.
Typical torque vs. deflection
hysteresis loops obtained from the solution of Equations 9 and 10 are
illustrated in Figure 8 •
•
•
(J
T
1
T
s
•
.E(T, 9)
•
sgn 9
Figure 7.
Block
Dia~ram
of The Solid Friction Model
Examination of this solid friction or materials properties model
reveals that there is a time dependence inherent in the solution or
simulation of the process.
This is the result of using •9 as the
input independent variable and solving for the torque as a function of
time in the simulation.
Note that if torque is plotted vs. angle as in
Figure 8, the time dependence does not explicitly appear but the rate
15
dependence is evident from the motion reversals.
Note· also that the
hysteresis curves are independent of the rate magnitude from Equation
9.
These observations lead to the conclusion that the solid friction
model represents solid friction and solid materials elastic-plastic
behavior adequately in a manner which is independent of the rate
magnitude as is the case from the facts of experimental observations.
T
Limit , Tc
Figure 8.
SFM Predicted Hysteresis Loops
This representation is what was desired and expected of the solid
friction model.
Questions that naturally arise next are: 1) can
viscous friction be represented somehow by the solid friction model, or
2) can the solid friction model be modified generally to include
viscous friction?
The following paragraphs will shed light on these
questions.
THE VISCOUS HYPOTHESIS
In an heuristic approach to adapting the SFM to apply to viscous
friction, suppose we look at the solid friction model equation
16
•
T
• •
(12)
T = a(l - T sgn e)e
c
where it is assumed that i=l and that the saturation level of friction
Tc varies with velocity according to Equation 3.
Equation 12 then
•
becomes, for Newtonian fluids with n = 1 (ie. T = Be)
c
B•
-T
+ T
0'
= Be•
(13)
•
The sgn term has been dropped because the e which was introduced has
the sgn property and so obviates the need for it.
Equation 13 could be
construed as a dynamical equation for viscous friction torque.
It is
identical in form to the Maxwell rheological model Equation 4 and is
referred to in this thesis as the viscous friction model (VFM).
It has
been arrived at or deduced in this section independent of the Maxwell
rheological model using the solid friction model as a basis.
The VFM
indicates that the viscous friction torque T lags velocity •a with a
time constant of B/a seconds.
The parameter B/a for fluids was
thought to be related to the time required for the laminar velocity
profile to change, as in Couette flow, however, it has a form identical
to the Maxwell model with the friction stiffness a equivalent to the
spring stiffness k in the Maxwell model.
The steady state viscous
• as could have been anticipated.
friction torque is Be,
The validity
of this viscous friction model has not been established at this point
but the groundwork has been laid by the hypothesis that a combination
of viscous and solid friction is accomodated within the framework of
the basic solid friction model.
Because of the usefulness of the solid friction model in modeling
17
and simulating many friction processes imbedded in dynamic systems, it
was deemed desirable to include or add the viscous contribution to the
total friction of elements in a unified solid and viscous friction
model.
So before proceeding to the testing of the viscous friction
model as postulated in Equation 13, we shall treat the combined effects
of solid and viscous friction in a combined friction model.
A COMBINED FRICTION MODEL
The SFM was originally developed from Equation 7 which was applied
in this thesis as an infinitesimal element material property relation
derived from elastic-plastic laws.
Following this same approach we
depart from the macro world of friction producing devices and focus on
the microscopic deformations of materials at this time to include both
elastic-plastic and viscous properties.
The deformation processes
involve the fundamental properties of materials in a microscopic sense,
and so the combined friction model (CFM) to be derived can be
considered a combined property rheological model in the microscopic
sense.
Materials that undergo elastic-plastic deformation in a
quasi-static fashion and at the same time undergo viscous flow
deformation can be considered to have a total time rate of deformation
gi ven by
ll
(14)
where we have considered elemental shear strain
y
produced by the
18
shear stress T.
The net shear strain rate dy/dt is the sum of the
elastic-plastic strain rate (1/G)(dy/dt) and the viscous contribution
(1/~)T.
The shear modulus G is taken to be non-linear and so
includes the elastic as well as plastic shear stiffness
characteristics.
Thus, G is a function of strain G(y) or
alternatively can be modeled as a function of stress G(T).
This is
the approach taken in arriving at Equation 8 and is also taken here.
The viscosity
~
is assumed to be a constant or a function of
•
•
temperature but not T, y, G, T, nor y.
We now convert the basic governing Equation 14 from elemental or
microscopic terms to macroscopic terms by substituting T for T, e
for y, B for
~,
and E for G.
In this context, T stands for
torque, e for angle, B for drag or friction coefficient of angular
velocity, and E for angular stiffness.
Then, solving the resulting
•
equation for T we obtain
dT
•
E
dt=Ee-BT
(15)
where we assume the same form for the stiffness as in Equation 9.
•
I
E(T,e) = a 11-
T
., i
T
sgne
T < T
c
c
where again a is the "rest" stiffness and T
c
is the usual Coulomb
or saturation level of torque in the quasi-static elastic-plastic
portion of the deformation.
Now that viscous flow deformation is present, under the usual
stable condition that (dT/dt) approaches zero in steady state in
(16)
19
appropriate dynamic situations, Equation 15 can be solved to yield
•
T = Be
(17)
which is the classical Newtonian viscous flow drag or friction relation.
The block diagram of Equation 15 is presented below in Figure 9.
This block diagram represents a math model for a visco-elastic-plastic
material or a system component or element that exhibits such behavior •
•
•
T
8
1
T
s
1
B
Figure 9.
Block Diagram of The Combined Friction Model
If we assume from Equation 16 that I is constant, this implies
that T/T
c
is very small and can be considered negligible.
Then
I
can be replaced by a pure elastic characteristic a and the block
diagram can be configured as shown in Figure 10.
This block diagram
represents, in at least one case, the dynamic lag behavior of a
viscometer or dashpot or other element as described by the Couette flow
solution in Chapter 4.
model of Equation 17.
It is an improvement to the simple Newtonian
20
•
8
(J
•
T ..
+
\,
-
-s1
T
'
(J
B
Figure 10.
Block Diagram of The Viscous Friction Model
It should be pointed out that the combined friction model is valid
for a single physical macro or micro element and that it may not be
applicable to describe two separate elements.
As an example, if solid
friction and viscous friction occur simultaneously but are independent,
then a single model is not appropriate and two separate models should
be employed and connected in series or parallel or as the physics of
the system require.
The combined friction model shown in Figure 9 encompasses most of
the features of the various friction models which were described in the
background section.
This unifying characteristic is unusual and is
projected to have many applications.
Elastic-Plastic Special Cases
When the steady state viscous coefficient B is increased to
infinity the CFM reduces to the Solid Friction Model, represented in
Equation 18 and Figure 7.
21
•
I9 =
•
T
.I i .
all - - sgne e
I
T
I
c
ITI < T
c
T
(18)
ITI = T
c
0
When the steady state viscous coefficient B is increased to
infinity and the friction function exponent i is set to zero the CFM
reduces to the Prandtl model, as noted in Equation 19 and Figure 11 •
ae•
ITI < T
•
c
(19)
T =
ITI > T
c
0
-
•
•
T
8
1
T
s
Figure 11.
Block Diagram of The Prandtl Model
If the steady state viscous coefficient B is defined to be infinite
and the rest stiffness a becomes infinite, the CFM reduces to the
Coulomb model, represented by Equation 20 and Figure 12
•
T = T sgne
c
(20)
22
Figure 12. Block Diagram of The Coulomb Model
Finally, if B becomes infinite and the Coulomb friction force, or
saturation level of torque T , becomes infinite then the CFM reduces
c
to the linear elastic Hooke model, represented by Equation 21 and
Figure 13 below.
T
= oe
(21)
____8_·~-~~~---~--~----T______
Figure 13.
Block Diagram of The Hooke Model
Viscous Special Cases
When the Coulomb friction level, or saturation level of torque T
c
is raised to infinity, the CFM reduces to the viscous friction model
where the angular stiffness term
rest stiffness o.
~
is constant and replaced by the
The VFM is found from the CFM Equations 15 and 16
to reduce to Equation 13 and is represented by the block diagram in
Figure 10.
23
When the saturation level of torque T in the CFM equations iS
c
raised to infinity and the time constant of the model B/a is defined
to be zero, the CFM reduces to the classical Newtonian model,
represented by Equation 22 and Figure 14 •
T = Be•
6
Figure 14.
(22)
·1. .
_s_ _:-__T_,.,..._
Block Diagram of The Newtonian Model
Chapter 4
VISCOUS FRICTION MODEL VERIFICATION
This section presents both an analytic and an experimental
validation of the VFM.
An experiment was envisioned which would verify
both the viscous hypothesis and the viscous friction model.
The test
would utilize a viscometer where a viscous fluid is contained in a gap
between an inner cylindrical drum and an outer cup.
The torque
response to a sudden stop from a steady flow was expected to reveal the
characteristic lag phenomenon.
A solution to the Navier-Stokes
equation for flow between parallel plates would relate the physical
characteristics of the experiment to the solution predicted by the VFM.
THE VFM SOLUTION
The analytical solution of the VFM for the desired test conditions
is obtained by assuming the experimental angular velocity of the
viscometer drum to be a step change from a previously existing steady
state velocity,
stopped.
•a, to zero, i.e. the viscometer drum is suddenly
The resulting steady torque induced by the fluid flow prior
to time zero is, from Equation 13
T
= B\l•
(23)
The solution from the VFM Equation 13 for this viscous drag problem is,
24
25
(24)
COUETTE SOLUTION
The unsteady viscous fluid flow field resulting from time varying
boundary conditions is referred to as Couette flow.
To analytically
determine the appropriate experimental parameters for the linear
viscous friction model, it is desired to solve the one-dimensional
Navier-Stokes equation.
A solution for a non-steady Couette flow
condition is calculated for the case where the relative motion of the
parallel plates is suddenly stopped or reversed.
12 13 14
'
'
The viscous flow system is modeled as having two infinite parallel
plates, initially one moving relative to the other with velocity uw'
and with a viscous fluid of viscosity
~
or kinematic viscosity v
between them as shown in Figure 15.
Wall
y
FoRAG = 11 d u]
dy
Figure 15.
Aw
WALL
Couette Flow Problem
p '
26
The assumptions which are made for this derivation are:
1.
The properties of the fluid that we are dealing with may
be treated as continuous and homogeneous.
2.
The tangential stresses are neglected.
3.
The temperature is considered to be constant, thus the
loss of heat is ignored.
4.
The density of the fluid is considered constant.
S.
The fluid velocity at a surface is tangential to that
surface and equal to the surface velocity.
Selection of the x-axis along the wall in the direction of u
w
leads
us to obtain the one-dimensional Navier-Stokes Equation for viscous flow
au
at
(25)
-=
The boundary conditions for the problem being considered are:
1)
The fluid velocity at the stationary wall is zero.
u(O,t)
2)
=0
(26)
For all t < 0, the moving wall moves at velocity uw
prior to bringing it to a sudden stop at t = 0.
Therefore, at t
u(h,O)
3)
= 0,
= uw
(27)
The flow is assumed to be laminar.
The geometry of the
steady velocity profile of the fluid at time t < 0 is
linear as shown in Figure 16.
Therefore, for t = 0
u
u(y,O)
= hw
y
(28)
27
y
__._-L-------------y=O
Figure 16.
4)
Steady State Velocity Profile of Couette Flow
For t > 0 , u(h,t)
=0
(29)
Employ the method of separation of variables and try the solution
u(y,t) = emtU(y)
where U(y) is the gap dependent variable.
(30)
Substitution of Equation 30
into Equation 25 yields
(31)
or
(32)
If we let
a
2
Equation 25 now becomes
m
\1
(33)
28
(34)
A solution to Equation 34 is
U = A sin(ay) + C cos(ay)
(35)
Satisfying the boundary condition u(O,t)
c
=0
requires that
= 0
(36)
In order to satisfy the boundary condition u(h,t) = 0, we must have
A sin(ah) = 0
(37)
or
n1f
= 0,1,2,3,
n
a=-
h
••••
(38)
For each n we therefore obtain a solution to Equation 34
U
n
=A
n
sin(n1ry)
h
(39)
By substituting Equation 38 into Equation 33 we find the eigenvalues
2
m = n
(mr) v
h
(40)
Using Equations 39 and 40 in Equation 30
(41)
Summing over all n, we obtain the general solution
29
2
-(E.!) vt
...
u(y,t)
= f Ae h
sin(n~y)
(42)
n=l n
To satisfy the initial condition given by Equation 28
...
u(y,O)
= f
Ansin(n~y)
(43)
n=l
We must therefore expand u(y,O) in a half-range series of sines,
2 h uw
n~y
An = -h I -h y sin(-h) dy
(44)
0
and if we let
n~y/h
=
z
2u
A
n
A
n
=
n~
~
n 2~
I
[z sin(z)Jdz
(45)
0
= (-l)n+l
2u
w
(46)
n~
By substitution into Equation 42 we obtain the solution to our problem
2
..,
u(y,t) =
r
n=l
n+l 2u
(-1)
-2:.
n~
-(E.!)
e h
(47)
A plot of the velocity profiles for several times, t > 0 is shown in
Figure 17.
The velocity profile, as a function of time, tends
asymptotically with time from the initial steady state linear velocity
profile to the final zero velocity as seen in the figure.
30
U " UWALL
Figure 17.
Velocity-Time Profile Across the Gap
Finding the force at the fixed wall,
TAw
(48)
au(O,t)
ay
(49)
Fw (O,t)
=
where
T = l.l
is the shear stress at the wall, and Aw is the area of the fixed wall
Then
=
r
2
n1T
2u w -(--h) vt
au(y,t) =
(-l)n+l ~ cos(n1Ty) --e
ay
n=l
h
h
n1r
(SO)
31
au(O,t) =
ay
Q)
r (-l)n+l
(51)
n=l
Using equations 49 and 51 in 48
Fw (t)
=
2AwI.IUw
h
Q)
r
(-l)n+l
(52)
n=l
The torque on the outer drum (from the experimental configuration)
T (t)
w
=
(R + h)F (t)
w
(53)
where,
R _ radius of the viscometer inner drum
h _ gap width
Finally,
•
2A 9Rvp(R+h)
T (t) = _w_~-w
h
ao
L (-l)n+l
(54)
n=l
where
•e = u /R = viscometer drum angular velocity.
w
A plot of T (t)/T (0) is shown in Figure 18.
w
w
In the experiment to be described in the next section, the
viscometer drum was driven such that the steady angular velocity of the
drum was reversed cyclicly from +u to -u to +u , etc.
w
w
w
The velocity
profile solution given by Equation 47 is not valid for this input
condition.
The correct solution was found to be
32
Analytical Solution:
T.w (t) = ~
LJ 2 (-1
o
[
-a
~
~
...:
-·-·
·-·-.
.~
--
-.
Tw(O)
'· '·
'·,
0
)n+1 exp [- (n1T)2
-·- vt
n=1
'· '· '· ................
h
'·
-. -
.............. __
·-............... -·-
2.0
1.0
Figure 18.
l
Torque Profile at Fixed Wall
2
u(y,t)
uwy
4uw
m
(-l)n+1
h
"If
n=l
n
= --+-r
-(~)
vt
h
e
sin(n1ry)
h
(55)
The resulting torque on the viscometer cup was then found to be
identical to Equation 54 in form although a bias term is present and a
factor of two in magnitude is introduced.
If we equate the VFM solution, Equation 24, to the dominate first
term from the Couette flow solution in Equation 54 the following
approximating equation is obtained.
33
(56)
T
w
We then obtain expressions for the VFM parameters in terms of the
experimental configuration.
B
The viscous friction coefficient is
2A vpR(R + h)
= ___
w__~------
(
h
dyne-em-s
rad
)
(57)
and the time constant of the VFM is
(58)
and the apparent shear stiffness of the VFM is
C1
=
2(~) 2 ~ 2 AwR(R + h)
(
ph
dyne-em-s
rad
2
)
(59)
Equation 57 shows us that B has the proper units of a viscous friction
coefficient and thus corroborate the VFM solution.
Also, the B/a
coefficient has the units of time and thus behaves like a time
constant, as the VFM stipulates.
a
It is interesting to note that the
coefficient seen in Equation 59 exhibits the units of a stiffness,
but contains no terms which indicate elastic properties of the fluid.
This is expected because no elastic properties were stipulated in the
definition of the Couette flow problem.
However, the VFM can be
interpreted to contain what in fact is an inertia or mass property of
the fluid and not a stiffness property at all.
If one views the
dynamic lag characteristic in the VFM as the time required for the
velocity profile to change, this implies that viscosity and inertia
34
properties of the fluid are governing as Equation 58 suggests.
The
viscosity and inertia or density properties of the fluid were not
explicitly modeled in the VFM and yet the VFM response shows their
effect.
This leads to the conclusion that the a or stiffness term
in the VFM is related to the viscosity, density, and geometry of the
Couette boundary value problem as Equation 59 indicates and is not a
stiffness at all.
EXPERIMENT
Through experimental testing, it is desired to verify and validate
the postulated viscous friction model.
It is expected that the
experimental data will show the characteristic lag for a given viscous
fluid and test configuration.
This subtle effect must be measured
within the capability of test hardware and thus poses considerations
for the experiment hardware design and assembly.
We wish to observe the behavior of a liquid in a viscometer when
it is viscously sheared under isothermal conditions in a narrow
annulus between vertical coaxial cylinders, one cylinder rotating with
respect to the other.
The viscometer is composed of a fixed vertical outer cylinder, or
cup, which is mounted on the torque measuring device.
The
experimental set up to accomplish this is shown in Figure 19.
The
torque on the stationary cylinder or cup is measured as a function of
time after the cessation or reversal of motion of the inner rotating
cylinder or drum.
The torque is measured by a device which moves
minutely with the torque being applied to it by the cup.
The inner
35
Torque
Motor
Drive Shaft
Pivot
Bearing
spotface
:..)
20• drill angle I
m
\bpunch
·.
: :
.
go·
.
. ,:
Mounting Platform
Torque Measuring Device
Figure 19.
Viscometer design
drum is accurately aligned with the outer cup so that the annulus gap
is uniform.
bearing.
The alignment during rotation is maintained by the pivot
By fine vertical adjustment of the drum coupling to the
36
motor drive shaft which can support the weight of the drum, the pivot
bearing solid friction effects are minimized.
The pivot bearing was
selected in the design to allow a vertical adjustment that would
minimize vertical loading on the bearing and thus the pivot bearing
solid friction contribution to total friction.
The drum is coupled to
the drive shaft of a torque motor which is driven so that any
prescribed drum motion time function may be achieved.
In order to minimize the viscous drag from end effects not
accounted for in the Couette solution the following methodology was
employed.
A large space with tapering walls was machined on the
bottom of the inner cup to trap an air bubble.
The air has low
viscosity compared to the oil in the gap and so causes a minimal drag
contribution, which can be neglected, compared to the annular gap
viscous drag friction.
Several items of concern arise when the viscometer dimensions are
to be finalized.
First of all, the test must be operated in
compliance with the assumptions made for the Couette flow solution.
Second, additional geometric considerations must be recognized ie.,
the flow must be laminar, the temperature and end effects must be
negligible, and the lag effect must be substantial enough to be
measured.
To determine whether the laminar flow condition prevails, we check
the point at which turbulent flow may be anticipated in the gap due to
shearing the fluid at very high rates.
The critical velocity is given
bylS
u
w
(60)
37
where u
w
is the velocity at radius R of the inner cylinder, and v
is the kinematic viscosity of the fluid.
The laminar flow constraint
is to limit the drum rotation rates to values below this velocity.
Temperature control is important because the lag time constant is
proportional to viscosity which is temperature sensitive.
With the
complicated equipment used it was deemed impractical to control the
temperature of the entire apparatus and thus thermal influence was
unavoidable.
In evaluating whether the viscous sample will remain
nearly constant at room temperature leads us to the analysis of the
problem of heat generation for Couette flow.
The solution to this
problem of a Newtonian fluid in simple shear between two parallel
plates with both plates held at temperature Temp
u
Temp
- Temp =
max
w
where Temp
max
2
w
n p h
w
is given as 16
2
---==---8kt
(61)
is the maximum centerline temperature of the fluid in
the viscometer gap and kt is the thermal conductivity of the fluid.
If the maximum temperature differential in the fluid across the gap is
small then our assumption of an isothermal system is valid.
Otherwise,
a large temperature differential would cause an error between the
experimental and predicted time constants.
An analysis of Equation 61
relative to this concern suggests that we minimize the drum velocity and
the gap width for a given test sample.
In order to assure that the time constant was sufficiently large to
be measurable, a computer program was created to help provide the best
design criteria for the sizing of the viscometer.
The aforementioned
laminar flow and thermal criteria were applied to the Couette flow
38
solution Equation 54 with the added restriction that the lag time
constant be greater than 20 milliseconds.
It was anticipated that 20
milliseconds would be within the measurement capability of the
experiment equipment.
In order to achieve the desired measurement for a
large range of viscosities, three different drum diameters were
considered for fabrication.
The characteristics of the viscometer
cylinder and annulus determined by this program are given in Figure 20.
PLATE I shows the inner and outer cups used in the experiment.
Cylinder
Radius
(em)
Annulus
Length, H
(em)
Annulus
Width, h
(em)
-
-
Outer
2.5400
Inner I
2.3813
10.1600
0.1588
Inner II
2.2225
10.1600
0.3175
Inner III
1.9050
10.1600
0.6350
Figure 20.
Physical Characteristics of Cylinders and Annuli
The experimental layout which is designed for these validation
tests can be seen in Figure 21.
A periodic forcing function is
achieved by a square wave output from a function generator which is
used as a command input to an angular rate servo.
consists of a servo amplifier, a D.
c.
The rate servo
torque motor, and a tachometer.
The torque motor output shaft angular velocity is measured by the
tachometer, and fed back to the servo amplifier which determines the
error between the command signal and the tachometer signal.
It then
39
PLATE I.
VISCOMETER:
CUP WITH THREE DRUMS
amplifies and applies current correction to the torque motor to reduce
the error.
The torque motor output shaft drives the inner drum of the
viscometer through a coupling device.
The inner drum rotates about a
pivot bearing that was designed to minimize the dry friction torque
contribution to the measured torque.
The viscous fluid between the
inner and outer cylinders is placed under shear by the angular motion
of the inner cylinder.
The torque on the outer cylinder is expected to
give us the decay characteristic predicted by the VFM solution Equation
56 and / or the Couette solution Equation 55.
The torque measuring
device used at the base of the viscometer is a McFadden Electronics Co.
40
SERVO
AMP.
FUNCTION
GENERATOR
POWER
SUPPLY
VISCOMETER
HP
PLOTTER
FLOPPY
DISK
O·SCOPE
NORLAND
DIGITAL t--'-T--;
ANALYZER
PWR.
SUPPLY
I
TORQUE
DYNAMOMETER
I
I
Figure 21. Experimental Set Up
Servo Torque-Balance Reaction Dynamometer Model llOA.
The mounting
platform is part of the torque meter and is air bearing supported for
axial and radial loads.
When a torque is applied from the viscometer
test fixture to the platform, a small angular displacement of the
mounting platform occurs.
This displacement is sensed by a feedback
41
position transducer which in turn sends a signal to a control and power
electronics unit that supplies current to a torquer.
The torquer
output torque reacts against the input applied torque to reduce the
displacement error and thus provide a torque balance, and limits the
platform motion to very small angular displacements.
It is the output
of the torquer which is measured to be opposite but equal to the torque
produced by the viscometer outer cup wall.
The current to the torquer
is proportional to torque produced and is readily measured as a voltage
which can be recorded.
The continuous recording of the torque decay characteristic is
achieved through the use of a Norland 3001 Digital Processing
Analyzer.
The three signals of interest, rate command, viscometer drum
angular velocity, and viscometer cup torque measured at the stationary
viscometer wall, are continuous signal inputs to the Norland.
The
Norland digitizes and stores over 1000 data points over a selected
period of time and operates on these input signals.
Peripheral
equipment such as an oscilloscope, x-y plotter and floppy disc memory
aid in the manipulation and recording of the data.
A test procedure
was created in order to standardize the testing process.
An outline of
the test procedure can be found in Appendix A.
The dry pivot torque characteristics of the test hardware are
presented in Figure 22.
The viscometer drum position and rate
measurements and viscometer cup torque response are plotted as a
function of time.
The torque responds to a step change in rate rather
quickly, taking approximately 25 milliseconds to reach the maximum
torque overshoot.
Therefore, the 20 millisecond response assumption
used in the design criteria computer program was not far off.
Care had
42
Torque on Cup
0
I
0
0
0
Viscometer Drum Rate
N
I
0
20
40
60
80
Time, msec
Figure 22.
Dry Pivot Experiment, Friction Torque
to be exercised to assure that the predicted decay time constant of the
VFM be a magnitude or two greater than this.
The validation test was performed using an oil with a kinematic
viscosity rating of 164 centistokes, about the consistency of cooking
oil.
An inner cylinder with 0.6350 em annulus width was chosen for the
selected viscosity oil.
PLATE II displays the equipment assembled for
the test, and the viscometer undergoing dynamic testing is shown in
PLATE III.
43
PLATE II.
LABORATORY TEST EQUIPMENT
44
PLATE III.
EXPERIMENT DURING TEST PROCEDURE
EXPERIMENTAL RESULTS
The command rate was kept substantially under the predicted rate
for the onset of turbulance for the test geometry.
Figures 23 and 24
are presented as raw data obtained from the experiment.
Figure 23
shows the experimental friction torque and velocity curves as a
function of time.
The torque responds to a step change in rate quite a
bit slower than the dry friction torque response, as anticipated.
The
"viscous" torque response takes approximately 250 milliseconds to reach
45
the maximum torque overshoot, ahout two magnitudes greater than the
solid or dry friction effect, as was desired.
This same "viscous"
friction torque versus time plot is presented on a log-linear format in
Figure 24.
The delay experienced as a function of the viscous fluid is
clearly visible.
150~--------------------------------------------------~
-
Velocit
(,)
Q)
.5
~
~
Q)
- -....
"'C
N
>.
0
!:
Q)
:::l
C"
(,)
0
~
~
~
0
0
-:5
-150
80
120
160
Time (msec)
Figure 23.
Experimental Friction Torque and Velocity
46
"" ' ' '
.1
.~
~
"\
'~
N
0
~
s::
1
\
\
'\.
\
~
'\
'
.001
40
.
60
80
100
120
140
160
Time (msec)
Figure 24.
Experimental Friction Torque Time History
The comparison of this raw data and the results predicted from the
VFM are discussed in the following section.
Chapter 5
RESULTS AND CONCLUSIONS
The agreement between theory and experiment for the case of a
Newtonian fluid is illustrated in Figure 25 which shows the
experimentally obtained friction torque and velocity curves plotted as
a function of time.
A theoretical fit derived from the Couette
solution equation is compared to the experimental data.
The Couette
solution predicted a slightly higher torque measured at the wall and a
slightly smaller time constant.
3.5 and 8.0 percent respectively.
The differences were determined to be
This agreement appears to justify
the neglect of end and temperature effects, as well as to indicate the
accuracy achieved in this kind of experiment with the apparatus
described in Chapter 4.
Generally speaking the VFM models the viscous time lag to a
relatively high degree of accuracy using the predominate, or first
constant plus a time delay.
If the VFM without the empirical time
delay is used, a relatively poor approximation is obtained as seen in
Figure 26.
An increase in accuracy of the VFM without the time delay
would be obtained with the use of additional first order VFM lag
models in parallel that can be derived based on the accurate Couette
solution.
The delay effect was not expected nor predicted by the
simple VFM Equation 13.
Therefore, the VFM Equation 13 should be
modified to include the transportation delay effect.
The Laplace
transformed Equation 13 with the delay included takes the form
47
48
(62)
where the single time constant Tl is as before
2
T
1
= -(£)
(63)
'\1
1T
and the delay time constant that fits the test data and the Couette
flow solution is
TD = Tl ln 2
(64)
The validation of the viscous friction model presented above in
addition to the solid friction model validations from 1968 to the
present lead to the conclusion that these models are useful and that
the combined friction model if and when similarly validated will be
useful in the future.
The special cases of the combined friction
model were presented in section three.
It is interesting to note that
the mathematical reductions and assumptions made there have physical
meanings.
The common mathematical limiting conditions made for the
elastic-plastic special cases had the effect of increasing the steady
state viscous coefficient, B, to infinity.
On a physical basis, this
means the viscosity approaches infinity, or this viscous material
behaves as a solid.
Conversely for the viscous special cases, the
Coulomb saturation level of torque was raised to infinity.
This
implies that the solid material characteristic with a saturation
torque level no longer exists, and the solid material behaves like a
viscous one.
49
It must be reemphasized that the combined friction model is
valid for a single physical material element only.
If solid friction
and viscous friction occur simultaneously, but are independent, then
two separate models should be employed and connected in series or
parallel as the physics of the system require.
Because the combined friction model does not contain an inertia
term, the viscous lag effect seen in the Couette solution is not
actually present in its first order differential equation.
The VFM
does not contain an inertia term either, however, the stiffness term
a can be interpreted to be such if the solid material properties
producing friction are negligible.
The VFM then becomes useful in
representing the dynamic lag characteristic of viscous dampers and
other devices.
so
.10
--~
Experimental
Couette Fit
for T w = .166 &
-
T1
=.027
+150.
N
0
I
c:
Velocity
Q,)
::::l
C"
:...
~
c:
-
EXPT
DATA
COUETTE
FIT
%
DIFF.
.027
.025
8.0
.166
.172
3.5
0
u
Tl
:...
{SEC)
u..
-
Tw
{IN OZ)
u
Q,)
If)
0
'
C)
-Q,)
0
"C
>.
·u
0
~
-150.
40
60
80
Time (ms)
Figure 25.
Comparison of Experimental Results With Couette Solution
51
T(s)
BO(s)
h2
7j = rr2 "
e-ros
=
7jS+1
T0 =
r 1 Jn
2
~~~~~~~~~~----~~------~~----------------~---
1Q)
:::l
2"
~
c:
0
-
·-u
""'
LL.
6
VFM without Delay
T(s)
•
BO(s)
=
1
TS+1
8~---L----L---~----~--~----~--~----~--~----~---100
Time (msec)
Figure 26.
Comparison of Experimental Results With VFM Solution
52
NOTES
1 P.R. Dahl, "A Solid Friction Model," The Aerospace
Corporation, El Segundo, California, TOR-158(3107-18)-1, 1968, p. 1-24.
2 D.T. Greenwood, Principles of Dynamics (New Jersey:
Prentice-Hall, 1965) pp. 111-116.
3 L. Prandtl, "tlver Flussigkeitsbewegung bei schrkleuner Reibung
Verhandlungen,", IIIrd Inter. Math. Kongres Heidelburg 1904, (1905),
484-491.
4 N.N. Davidenkov, "Energy Dissipation in Vibration," J. Tech.
Phys., 8, No. 6, (1938) P• 483.
5 Sir Issac Newton, Mathematical Principles of Natural
Philosophy, trans. A. Motte and F. Cagori (Berkeley, Univ. of
California Press, 1966), p. 385.
6 N.H. Polakowski, and E.J. Ripling, Strength and Structure of
Engineering Materials (New York: Prentice Hall, 1965), 8, pp. 210-213.
York:
7 H. Schlichting, Boundary Layer Theory trans. J. Kesten (New
McGraw-Hill 1979), pp. 90-92.
8 J. Steinheuer, "Eine exakte Ltisung der interstation~ren
Couette-Stromung," Proc. Scientific Soc. of Braunschweig, (1965) pp.
154-164.
9 Polakowski, p. 215.
10 P.R. Dahl, "Solid Friction Damping of Mechanical Vibrations,"
AIAA Journal, 14, No. 12 (1976), 1675.
11 Polakowski, p. 215.
12 Schlichting, PP• 90-92.
13 G.G. Stokes, "On The Effect Of The Internal Friction Of
Fluids On The Motion Of Pendulums," Math and Phys. Papers, (1901),
1-141.
14 Steinheuer pp. 154-164.
53
15 D.C. Bogue, and J. White, "Engineering Analysis of
Non-Newtonian Fluids," AGARDograph, No. 144 (1970), p.ll.
16 Bogue p. 11.
54
BIBLIOGRAPHY
Bogue, D.c. and J.L. White. "Engineering Analysis of Non-Newtonian
Fluids." AGARDograph, No. 144 (1970), 1-34.
Dahl, P.R. "A Solid Friction Model." The Aerospace Corporation, El
Segundo, California, TOR-158(3107-18)-1, 1968.
Dahl, P.R. "Measurement of Solid Friction Parameters of Ball
Bearings." Incremental Motion Control Systems and Devices
Symposium Proceedings, (1977), pp. 49-60.
Dahl, P.R. "Solid Friction Damping of Mechanical Vibrations."
Journal, 14, No. 12 (1976), pp. 1675-1682.
Davidenkov, N.N. "Energy Dissipation in Vibrations."
8, No. 6 (1938), 483.
AIAA
J. Tech. Phys.,
Draper, c.s. and M. Finston. "High Accuracy Mechanical Integration by
Shear in Viscous Liquids." Proceedings of National Academy of
Sciences, No. 4 (1959), 528-49.
Eckert, E.R.G. Introduction to the Transfer of Heat and Mass.
York: McGraw-Hill, 1950.
Eirich, F.R., ed. Rheology, Theory and Applications.
Academic Press, 1958.
New
New York:
Goldstein, s. ed. Modern Developments in Fluid Dynamics.
Oxford: Claendon Press, 1952.
Greenwood, D.T.
1965.
Principles of Dynamics.
Lamb, Sir Horace.
Hydrodynamics.
New Jersey:
New York:
Prentice-Hall,
Dover, 1945.
Lazan, B.J. Damping of Materials and Members in Structural Mechanics.
London: Pergamon Press, 1968.
Marris, A.W. and J.T.s. Wang, eds. "Symposium on Rheology." ASME
Applied Mechanics and Fluid Engineering Conference Proceedings,
New York: American Society of Mechanical Engineers, 1965.
McKelvey, J.M.
Polymer Processing.
New York:
Wiley, 1962.
55
Newton, Sir Isaac. Mathematical Principles of Natural Philosophy•
Motte, A. and and F. Cagori, eds. Berkeley: Univ. of California
Press, 1966.
Pipes, L. A. Applied Mathematics for Engineers and Physicists.
York: McGraw-Hill, 1946.
New
Prandtl, L. Uver Fltissigkeitsbewegung be! schrkleuner Reibung.
Verhandlungen. IIIrdinter. Math. Kongres Heidelburg 1904,
(1905), 484-491.
Polakowski, N.H. and E.J. Ripling. Strength and Structure of
Engineering Materials. New York: Prentice Hall, 1965.
Schlicting, H. Boundary Layer Theory.
McGraw-Hill, 1979.
Sneddon, I.N.
Steinheuer, J.
StrBmung.
154-164.
Fourier Transforms.
Trans. J. Kestin.
New York:
New York:
McGraw-Hill, 1951.
"Eine exakte LBsung der instationaren Couette Proc. Scientific Soc. of Braunschweig, XVII, (1965),
Stokes, G.G. On the Effect of the Internal Friction of Fluids on the
Motion of Pendulums. Cambridge: Math. and Phys. Papers, (1901),
1-141.
Van Vazer. J.R., J.W. Lyons, K.Y. Kum, and R.E. Colwell. Viscosity
and Flow Measurement, A Laboratory Handbook of Rheology. New
York: Interscience Publishers, 1963.
56
APPENDIX A
TEST PROCEDURE
57
1.
Assemble test hardware
2.
Verify proper equipment set up
3.
Obtain scale factors for torque meter and tachometer
4.
Obtain time response of test hardware
5.
Obtain pivot dry friction torque characteristics
6.
Test specified oil using a specified viscometer gap
a)
Check equation for the predicted parameters
b)
Fill annulus with fluid
c)
Record data in log
d)
1)
date
2)
run number
3)
sample interval of analyzer
4)
torque range selection on analyzer
5)
veloctiy range selection on analyzer
6)
scale factor for torque data
7)
scale factor for velocity data
8)
dynamometer meter range
9)
command signal frequency
10)
wall velocity
11)
n-point filter for data; n = ?
Record data on floppy disc
1)
command
2)
rate
3)
torque
58
4)
e)
f)
time
Plot the following parameters
1)
torque vs time (linear scale)
2)
velocity vs time
3)
torque vs position
4)
torque vs time (log - linear scale)
Thoroughly cleanse viscometer